llr
Université du Qu ébec
École de technologie supérieure
DÉPARTEMENT DE GÉNIE DE LA PRODUCTION AUTOMATISÉE
GES-802
Analyse de faisabilité
NOTES DE COURS
Par Ali Gharbi Professeur
Révisé : Août 2009
Table des matières
CHAPITRE 1:
ESTIMATION DES MARCHÉS, REVENUS ET COÛTS
CHAPITRE 2:
INTRODUCTION À LA NOTION DE RISQUE ET D'INCERTITUDE
CHAPITRE 3:
ANALYSE ÉCONOMIQUE TRADITIONNELLE ET AVANCÉE SOUS RISQUE ET INCERTITUDE
CHAPITRE 4:
TECHNIQUES DE DÉCISION STATISTIQUE AVEC INFORMATIONS ADDITIONNELLES
CHAPITRE 5:
L'ARBRE DE DÉCISION
CHAPITRE 6:
ÉLÉMENTS DE BASE DE LA SIMULATION DES SYSTÈMES
CHAPITRE 7:
ÉVALUATION DES NOUVELLES TECHNOLOGIES : UNE APPROCHE MULTICRITÈRE
CHAPITRE 8:
DÉCIDER FACE À LA COMPLEXITÉ: ANALYSE HIÉRARCHIQUE DES PROCÉDÉS (AHP)
CHAPITRE 9:
ÉLÉMENTS DE CONCEPTION D'UN PROJET
ANNEXE 1:
MANUEL D'UTILISATION DU LOGICIEL EXPERT CHOICE
estimating markets, revenues, and costs
chapter
1
5.1 INTRODUCTION In any planning analysis for strategie manufacturing investments, there is great need for reasonably ace urate estimates of market conditions (e.g., market size and share) : pricing structure, and cost patterns . Severa! techniques for forecasting the size of the market for new or m_odified products are presented in .. SecÜoiï- 5~2. Quantitative forecasting l)let_hod s covered are regression analysi~. an~_t:_){p9J1.en~iaL~ll}.QQ!J.ürrg; ·· ~~ye . techn_i_qyes inelude the D_~d.techn.o~alforecasting. Once an estimate of market demand for a new or modified product has been preparea, the priçing policy and revenues associated with that product become a matter of conce~;; to...~an~gëëiïëni~Yhis ..toplc iii the..subject of SectiQJJ $.3. Numerous techoiq~-~s for ~stimating capital, material, I~.J:>or , an·d.. overhead costs are dis,c.ussed in Section 5.4. Finally , the impact of inflation on. the cost estimatin&g~oces~ - -i~_ add_r:essed in Section 5.5.
-
Estimating Markets, Revenues, and Costs
Chap. 5
5.2 FORECASTING THE MARKET
A forecast is useful if it reduces the uncertainty surrounding an event, and by doing so results in a decision that has increased value in excess of the cost required to produce the forecast. Forecasts are also useful to givc management an image of the firm it guides. Companies are often what they want themselves to be. An open-minded, optimistic vision of the future can lead to growth and profit; a narrow. pessimistic view can lead to stagnation or worse . 5.2.1 Forecasting Procedure
A conscientious forecaster will follow a logical step-by-step procedure in developing and revising forecasts . The first step is to decide what response, or quantity, to forecast. Th en pertinent numerical data are gathered and summarized in graphical form, whenever possible. Often the data represent sorne response as a function of time, such as sales per quarter. Such data are referred to as comprising a rime series. The time series should be examined for underlying patterns, and the forecaster should attempt to explain these patterns . For example, a large sudden increase in sales m~y ~e explained by increased government spending. A long-lerm steady nse m sales may be a result of graduai public acceptance of a new product. Typical forecasts based on lime-series data merely ex tend historical patterns into the future and do not result in predicting abrupt changes (tuming points) in activity levels . ln this respect, the forecaster should search for causal relationships between the response variable (e.g., sales) and other variables . For example, a rise in interest rates is usually tollowed by a decline in new construction. This is a causal relationship thal is useful in forecasting major upswings and downturns in the construction industry. The next step of the forecasting process is to apply expert judgment to forecasts obtained by utilizing causal relationships and/or time-series analysis . The result of this evaluation, whether it takes the form of a Del phi study or simply a group's or an individual's opinion, should be documented and included in the forecast. Thus forecasting involves a coordinated approach to collecting data, identifying and mathematically testing causal relationships , and/or selecting a procedure for projecting time-series data, and choosing a structured approach for eliciting exper1 opinion concerning the problem under consideration. One of the most effective forecasting strategies is first to use mathematical techniqu es based on pas t da ta and then t introduce judgment in • tt mpting 1 d c id h w th futu r pr bobly wi ll di ff r fr m th p st. A pr irn • udva nr n of this opproa h is that 1t r lu th lî\111 11 r f f11 tM~ 1 whi h h11111 tn 1 ri n • 11111 1 1 ' 1ppl 11 Il wl ly 'll nll 11 11 1111lh 11111
Sec. 5.2
, Forecasting the Market
83
cal techniques and informedjudgment, both methods can serve as checks on each other and tend to minimize gross errors . To evaluate any quantitative forecasting technique, the procedure first should be defined and theo applied to representative data. For example, if one's goal is to forecast sales one quarter in advance, one could start by applying the proposed forecasting technique to, say. the last three years of quarterly data. Begin with the first quarter and apply the procedure to take each new quarter's actual sales into account when developing the next quarter's forecast. Theo for the historical data one can measure the difference between the forecast and actual sales. The difference is the forecasting error, and severa! techniques should be evaluated in an effort to minimize this error.
!' 1
J 1
i.
1 1
5.2.2 Forecastlng New Products
Because CIM usually requires rethinking the way a firm does business, modified existing products and new products form the basis of forecasting and eventually justifying the enabling technology. When a company wishes to forecast with reference to a particular product, it must consider the stage of the product's li fe cycle for which it is making the forecast. The availability of data and the possibility of establishing relationships between the factors depend directly on the maturity of a product, and hence the !ife-cycle stage is a prime determinant of the forecasting method to be used (e.g .. see Ref. [1]). At each phase in the li fe of a product, from conception to maturity, the decisions that management must make are characteristically quite different, and they require different kinds of information as a base. ln the earl y stages · · of product development, the manager wants answers to questions such as these: • • • • • •
What are the alternative growth opportunities of pursuing product X? How have established products similar to product X fared? Should we enter this business, and if so, in what segments? How should we allocate R&D efforts and funds? How successful will different product concepts be ? How will product X fit into the markets 5 or W years from now?
'1
:~
Es tima tlnJ.! Ma rkets, R euenu es , and
osts
Chap . 5
product that is still "only a gleam in the eye," information about its likely performance can be gathered in a number of ways, provided that the market in which it is to be sold is a known entity. For example, one can compare a proposed product with competitors' present and planned products, ranking ·J it on quantitative scales for different factors. If this approach is to be successful, it is essential th at the persons (hopefully. experts) who pro vide the basic data come from different disciplines-marketing, R&D, manufacturing, legal, and so on-and that their opinions be unbiased . Frequently, however, the market for a new product is weakly defined, the product concept is still fluid, and history seems irrelevant. This has been the typical case for gas turbines, modular housing, pollution measurement deviees, and computer terminais. Before a product can enter its (hopefully) rapid penetration stage, the market potential must be t~sted and the .product must be introduced. Then market testing may be advtsable . At thts stage, management needs answers to such questions as: • What shall our marketing plan be-which markets should we enter, and with what production quantities and priees? • How much manufacturing capacity will the early production stages require? • As demand grows, where should we build this capacity? • How shall we allocate our R&D resources over time? Significant profits depend on finding the right answers, and it is therefore economically feasible to expend relatively large amounts of effort and money on obtaining good forecasts (short, medium, and long range) . A sales forecast at this stage should provide three points of information: the date when rapid sales will begin, the rate of market penetration du ring the growth phase, and the ultimate levei of penetration, or sales rate, during the maturity phase. 5.2.3 Quantitative Forecasting Techniques
Two relatively simple, yet extremely useful, techniques for obtaining initial time series forecasts of demand for a new or modified product are desc ribed in this section: ( 1) regression analysis and (2) exponential smoothing . Results of quantitative forecasting are coupled with subjective forecasts to develop the final projection of market conditions.
Sec. 5.2
Forecasting tite Market
a person using regression might forecast sales for a particular product for the next month to be, say, $425,000 and also can state that she is, for example, 95% confident that actual sales will be between $390,000 and $460,000. This assumes that basi.c market conditions and trends will remain unchanged . Confidence mtervals are useful in identifying a change in the trend of a set of data. If two or three consecutive observations fall outside these timits, the a~aly~t can be fairly sure that the basic relationship has changed since the htstoncal dat~ were gathered. This information may prove financially valuable by enabhng one to re act quickly to a change in trend. If the trend changes, however, the forecaster is advised to discard or substantiafly discount data regarding the previous trend . In linear regression involving one independent and one dependent variable, the relationship that is used to fit n data points ( 1 :s; ; :s; n) is y= a+ bx
(5-1) A mathematical statement of expressions used to estimate a and b in the simple linear regression equation 5-1 is as follows:
i i
b =
X;Yai -
i= l
a=
Xi xi
Yai
i= l
XJ-
;,. ,
(5-2) X;
i= l
Ya - bi
(5-3) Here x and are averages of the independent variable and dependent variable, respecttvely, for the n data points .
J?
Example 5-l A durable goods manufacturer has found persona! disposable income in its market region in a given quarter to be strongly related to sales in the following quarter. !JI~se data are listed and summarized in Table 5-1. Because a plot of these data md1cates an approximately linear relationship between the dependent variable (on . the Y axis) and the independent variable (on the x axis), linear regression is used to fit an equation to the data. The data and calculations summarized in Table 5- J are utilized below to determine the linear regression equation : n
b =
n
L X;Yai - XL Yai i• l
i:l
•
•
1=1
1= 1
2: xl - X·2: x;
5.2.3.1 Linear Regression Analysis Regression is a statistical method of fitting a Ii ne through data to minimize squared error. It is exact; however, graphing might be used to .provide a satisfactory approximation. With linear regression, statistics can be used to obtain a forecast and a confidence interval for the forecast . For example,
85
_ 2,626,817 - 250.6(9,788) - 1,416,926 - 250.6(5,01 2)
173,944.2 160,918.8 = 1. 081 a =
Ya -
bi= 489.4 - 1.081(250.6) = 218 .5
y=
218 .5 + 1.081x
· ~
'
~
Sec. 5.2
Table 5-l CALCULATIONS FOR S IMPLE LINEAR REGRESS ION
Forecasting the Marker
Figure 5- l shows the plotted data a nd the calculated regression tine (as weil as confidence li mits, to be discussed later) . As an ex ample of how the regression equation is used , suppose that disposable income for the previous quarter is $3 10 x 10 6 . Then our forecast or estima te of sales for the current quarter, y, is
J
Data Point, i (period)
X;
X;Y ai
121 118 271 190 75
43,560 30,680 119,240 76,000 27 ,000
14,641 13,924 73,441 36, 100 5,625
480
263 334 368 305 210
131,500 193,720 206,080 154,025 100,800
69, 169 111,556 135,424 93,025 44 , 100
tl 12 13 14 15
602 540 415 590 492
387 270 2 18 342 173
232,974 145,800 90,470 20 1,780 85, 11 6
149,769 72,900 47 ,524 116,964 29,929
16 17 18 19 20
660 360 410 680 594
370 170 205 339 283
244,200 61,200 84 ,050 230,520 168 , 102
136,900 28,900 42,025 114,921 80,089
9,788
5,012
2,626,81 7
1,416,926
Ya i 360 260 440 400 360 500 580 560
2 3 4
5 6 7 8 9 10
505
Totals
x!
87
= 218 .5
+ 1. 081(310)
= 553 .6
$553 .6 x JOl
or
To calcula te a con fidence interval for our forecast, we must obtai n the standard deviation of a single esti mated value, s(y). Th is quan tity is a measure of the tightness of the individual data points about the regression tine ,
2: (y.;_ y,)2]
[
s( y ) =
" ,,. 1
n _
l/2
[
1 + .!_ + n
·
2
- 2 ] 112 (xo - x)
L"
(x;
- iV
(5-4)
i• l
Here x 0 is a value of the indepe nde nt variable being used to obtai n the quarter! y sales forecast, y, and it lies in the ra nge of observed x 1 (i .e., Xmin :5 x 0 :5 xm. , ) :
y =a +
bxo
-~
.....~
700 1-
..~""'/.
~'~
" LX; "'
2:" y., = 9,788
5,0 12
i ::. l
i •l
"
<=
.
LX;
" Lx!=
!.:..!__ = 5,012 = 250 6 "
20
.
2: y., i = l,
=
9
·;~ 8
= 489 .4
~ 0 -8c;
i• l
2:" x,y.
.!!
s
1,416 ,926
"
y.=
~
&~
0
:: >
= 2,626,817
i•l
•,
>-
i
..
x1 •
1)
for period i disposnble lnco me in precedl ng period (S IO')
.
,/. '••
,.,_, oP vo
/
t:
"
y., - actual qu arterly sales ($ 10
~"'
~,tJi
500
.~f :x 1
.,...
,,
0 K~y :
~,l'
600
200
llii!H•••IJI•I•• ''"'•ln (lf'lfoUI"u •lllo rl•• · '< lu il ll httiO II I rtoll•rtl
1•1•'"" ~. . . !li
l 'h tl l llllllliltl l -
lull l t~NI ~~-IIlrl ll1111 f111 1' \"' "1'111 \ 1,
1 1
xx
EstimatinK Markets, Revenues, and Costs
Chap . 5
Sec. 5 .2
Foreca.ftÎIII( th e Market
The 100( 1 - a) percent confidence interval for the forecast of actual quarterly sales is defined to be between
Y+
(upper limit)
(la/l,n - l)s(y)
(5-5)
and
Y-
(lower limit)
(tan,n - l)s( y)
Table 5·2 CALCULATIONS TO DETERMINE CONFIDENCE INTER VAL Data Point , i (period)
(5-6)
2 3 4
The term tan ,n-l is an appropriate value of the t statistic . A table of t values and instructions for using this table are given in Appendix C of Ref. [14] as weil as in most statistics texts. For a 95% confidence interval (a = 0.05, a/2 = 0.025) and for n data points, there are n - 2 degrees of freedom. This mea ns that with repeated fore cas ts involving n data points, approximately 95% of the confidence intervals calcu lated with the equation above will con tain the true value of the response variable if the underlying causal rel ationship does not change . The information necessary to determine the co nfidence interval for this forecast is summarized in Table 5-2 and is used with Eq . 5-4 to perform the following calculations . 1 - [52,482.8] l/l [ --18· 1 + 20
s(y)-
+
5 6 7 8 9 10
(31 0 - 250.6) 2] 112 160,919.6 .
For a 95% confidence interval we have
y,
(y.,- y,)'
121 118 271 190
349.3 346. 1 511.4 423 .9 299.6 502.8 580.0 616 .3 541! .2
114 .5 7,413 ,2 5,098.0 571 .2 3,648.2
75
480 602 540 4 15 590 492
263 334 368 305 2 10 387 270 218 342 173
636 .8 510.4 454.2 588 .2 405.5
7.8 0.0 3,169.7 1,866.2 1, 190.2 1,211.0 1!76.2 1,536.6 3.2 7.482.2
660 360 410 680 594
370 170 205 339 283
618 .5 402 .3 440.1 585.0 524.4
1,722.2 1,789.3 906.0 9,025.0 4,844.2 52,482.8
(upper limit)
Uan,n-2)s(y) = 553 .6- 2. 1009(55 .88) = 436
(lower limit)
To summarize, our forecast of next quarter's sales is $553,600, and we are 95% confident that the true value (outcome) of quarterly sales will occur between $436,000 and $671,000.
The correlation coefficient is a measure of the strength of the relationship between two variables only if the variables are linearly related. In Example 5-1 the correlation coefficient, r, can be determined as follows :
1.0-
-
i) 1
16,796.2 17,582.8 416 .2 3,672.4 30,835.4 153 .8 6,955.6 13,782.8 2,959.4 1,648.4 18 ,605.0 376.4 1,1162.8 8,354.0 6,021.8 14,256.4 6,496.4 2,079.4 7,814.6 1,049.8 160,9 19.6
(y., -
.Y.J'
16,744 52,624 2,440 7,992 16,744 11 2 8,208 4,984 243 88 12,679 2,560 5,535 10, 120 7 29,104 16,744 6,304 36,328 10,941 240,501
y1 = estimated quarterly sales ($10 1 ) for period i x 1 = disposable income in preceding period ($10•) From calculations associa led with Table 5-1 : y = 218.5 + 1.081x; and
x = 250.6 and .Y. = 489 .4
value of r indicates that one variable decreases as the other increases, while the variables both increase (or decrease) at the same time when r is positive. The closer ris to -1 or +1, the more "perfect" is the correla tion . Using Eq, 5-7, the correlation coefficient for data in Table 5-2 is n
n
r =
(x1
y. 1 = actual quarter! y sales ($ 10 1 ) for period i
= 671
2:
445 .5
Key:
and by utilizing Eqs. 5-5 and 5-6, these limits are obtained:
+ 2.1009(55 .88)
x,
360 260 440 400 360 500 580
Totals
to.o2S.I8 = 2. 1009
Uan ,n- 2)s(y) = 553 .6
y.,
560 505
Il 12 13 14 15 16 17 18 19 20
= 55.88
Y+ .v -
1!9
(Yai - Y;)
:_i::_l- - -
2: (Yai -
2:
2
r=
1.0 -
2: (Yai -
Ya) 2
Ya) 2
j
52,483 1.0 - 240,501
i=l
i-l
where - 1 < r < 1. If there is no relationship at ali (a shotgun etfect) between the dependent and independent variables , r will be zero , or nearly zero . A negative
(Ya;- Y;Jl
.:-. ':..:.. 1- - - -
= V0.78I8
=0.88
This value of r indicates a fairly good relationship between the independent and dependent variables .
Vi
-\.
Estimating Markets, Revenues, and Costs
90
Chap . 5
Table 5-3 EXPON ENTI A L SM OOTHING EXAMPLE
5.2.3.2 Exponential Smoothing
Period Number,
A major ad van tage of the exponential smoothing method compared to simple linear regression is that it permits the forecaster to place relatively more weight on current data rather than treating ali data points with equal importance. Forecasting equations can quickly be revised with a relatively small number of calculations as each new data point is collected . Also, exponential smoothing does not assume linearity. The main disadvantage of exponential smoothing concerns the basic assomption that trends and patterns of the past will continue into the future. However, it is more sensitive to changes than is linear regression. Because time-series analysis cannot predict turning points in the future, expert judgment and/or analysis of suspected causal factors should be used in interpreting results of forecasting methods discussed in this chapter. The basic exponential smoothing model that we shall discuss and illustrate is shown as follows: S, = ot'x, + (1 - a')S1-
5
Il 12 13 14 15 16 17 18 19 20
new = a' ( new ) + (1 _ a') (previous) estimate datum estimate This term, a', the smoothing constant, merely provides a relative weighting for the new datum point compared to previous estimates. In general, a' should lie between 0.01 and 0.30, but the analyst should not hesitate to use a value outside this range if it gives better results with representative historical data . Single exponential smoothing is Hlustrated with data listed in Table 5-3 that are graphed in Fig. 5-2. Based on Eq. 5-8, the following are example calculations forS,, which is termed a "smoothed statistic" for period t. S, = a'x, + (1 - a')S,_1 s 1 = 0.3(50) + o. 7(50) = 50
s2 = 0.3(52) +
.1: '1
. 1 1
o. 7(50)
= 50.6
s]
=
o.3(47) + o.7(50;6)
s4
=
o.J(5I) + 0.7(49.52)
49.52
= =
80
50.00 50.00 50.60 49 .52 49 .96 49 .67 49 . 17 49 .72 46 .80 47 . 16 48 .61
50 52 47 51 49 48 51 40 48 52 51 59 57
6 7 8 9 10
or
S, (a' = 0.30)
-
0 l 2 3 4
(5-8)
1
Demand. x, (1,000 units)
1
49.33 52.23 53 .66 56.76 60 . 13 62. 19
64 68 67 69 76 75 80
64.23 67 .76 69 .93 72.95
1
,...,.....
-:J 70 'ë
0"
~ ill 5 2 -g
,.,
60
,.,lt!
~
.
/
j '
49.96
ln determining S 1 above, a value of S0 mu st be estimated. Ifthere is no trend in the initi al data, an estima te based on the average of the flrst few da ta roi nts is adequo tc . He re S0 wos cstimutcd lO be 50. T o bcttcr undet·stnnd th e mc:n nlog of ~·Xf'I1111Cnilu l NllloO th in!(, the fol o u ~ h owN how d!111 1111 lll lh'tll \ 1 , 1 , 1 , 11 1, '" 11 lnd111l cd
·-
~-•/
.1''
...../ - - . Demand. x,
l'/
-- - JJ.
Exponont la lly lffiOOthod
esti mete, $, 1 14
1 1
1 10
-
1tl
l'ori;tl l lllllltli•t
1 1 1~11 1•1 ,1 A,
1
~ l lllllllllhol 'J tl lllll li lttW ~~ tiltlltl
'1
Estimating Markets , Reve nues, and Costs
92
Clwp . 5
S, = a'x, + (1 - a ')S,_ 1
(5-9)
+
(5-1 0)
where Sr - I =
a'Xr-l
(1 - a')S,-2
and then S,
=
a'x, + a'(l - a')x, _ 1 + (1 - a') 2Sr- 2
lt is possible to continue substituting smoothed values in the same fashion until we get the following: S, = a'x, + a '(l - a')x, _ 1 + a'(l -
a') 2X 1- z
+ a '(l - a') 3x,_3 + · · · +
(5-12) (1 - a')'So
As one can see, every previou s value of x is included in S, . The i values are weighted so that the values most distant in time have the smallest weighting factors . A large a' will place very little weight on remote data. The following calculations illustrate the weighting of the different data points included in S 4 above as described by Eq. 5-12.
s.
= 0.3(51) + 0.3(0.7)47 + 0.3(0.7) 252 + 0.3(0.7) 350 + (0 .7) 450 = 0.30(51) + 0.21(47) + 0. 147(52) + 0. 1029(50) + 0.2401(50) = 49.96
This agrees with our calculation of S4 earlier. Table 5-3 shows a continuation of the example above for hypothetical demand data over 20 periods. It should be noted that the higher the value of a '. the cl oser the new estimate will be to the most recent datum point. 5.2.4 Subjective Forecasting Techniques
The quantitative forecasting methods discussed above are underpinned by the premise that the future is an extension of the past. However, the future will contain events that today are poorly understood or completely unanticipated . Hence the aim of this section is to explain and illustrate two techniques for developing subjective (i.e., qualitative) information for purposes of making a forecast : (1) the Delphi method and (2) technology forecasting. 5.2.4.1 The Delphi Method
Most decision makers draw on the ad vice of experts as they form their judgments . Often the decision situation is highly complex and poorly understood, so that no single person can be expected to make an informed decision . The traditional approach to decision making in such cases is to obtain
Sec. 5.2
Forecasring the Market
93
expert opinion through open discussions and to attempt to determine a consensus among the experts. However, results of panel discussions are sornetimes unsatisfactory because group opinion is highly influenced by dominant individuals and/or because a majority opinion may be used to create the "band wagon effect. •' The Delphi method attempts to overcome these difficulties by forcing persons involved in the forecasting exercise to voice their opinions anonymously and through an intermediary. The intermediary acts as a control center in analyzing responses to each round of opinion gathering and in feeding back opinion to participants in subsequent rounds . By follo wing such a procedure, it is hoped that the responses will converge on a consensus forecast that turns out to be a good estimator of the true outcome . Two premises underlie the Delphi method . The first is that persons who are highly knowledgeable in a particular field make the most plausible forecasts . Second, it is believed that the combined knowledge of severa! persons is at least as good as that of one person . Typically, the technique is initiated by writing an unambiguous description of the forecasting problem and sending this, along with relevant background information, to each participant in the study. Often the participants are invited to list major areas of concern in their particular specialty as they may relate to the problem being addressed by the study. The first questionnaire sent out might request the opinion of each expert regarding likely dates for the occurrence of an event identified in the problem statement. Because responses to this type of question will normally reveal a spread of opinions, interquartile ranges are customarily computed and presented to the experts at the beginning of the second round . Interquartile ranges identify upper and Iower quartile values in the continuum of responses such that 50% of the responses fall within that range . In the second round of the Del phi technique, the participants are asked to review their response in the first round relative to interquartile ranges from that round . They then have the opportunity to revise their estimates in tight of the group response . At this point, participants can request that additional information relevant to the forecasting problem be gathered and sent to them . If an estima te departs appreciably from the group median , the res pondent who furnished it is asked to give reasons for his or her position . Frequent) y, ali panelists are urged to conceive statements that challenge or support estimates falling outside the interquartile or sorne other range of responses . These reasons, along with routine second-round estimates for the entire group, are again analyzed and statistically summarized (usually as interquartile ranges, although other measures capable of showing group convergence or divergence could be used) . In those cases where a third-round questionnaire is fe ft necessary, participants receive a su rn mary of second-round responses plus a request to
1~
l"j ;·
1. 1
,,1
1Ji.:
;: .. :ii 1
.1
:·
i:
~
. \~
:L· ~
!:· .:
' !•
: ij
·;i,. . !~
1
"ii
94
Estimating Markets, Revenues, and Costs
Chap . 5
reconsider and/or explain their estimate in view of group responses in the second round . They are again asked to reassess their earlier responses and possibly to explain why their estimates do not conform to the majority of group opinion . An example of quantitative results of the Delphi technique is summarized in Fig. 5-3. The problem of concern to a manufacturer of large earthmoving equipment was to develop a forecast of total company sales during fi scal year 19XX. Six marketing and sales experts (A through F) were asked to con sider historical company and industry data and anonymously prepare a forecast of bookings for a particular product. First-round questionnaire results are shown in the top part of Fig. 5-3. The median response in round 1 was 229 and the interquartile range of the responses was 85. Results of round 1 were fed back to each participant along with additional information pertaining to the forecast that each person requested . The second and third rounds were completed in a similar manner. Notice how forecasts in the three iterations of questioning tend to converge, with a final median forecast value of 260 unit s and an interquartile range of 47 .
..
~ N,._
"'
1':::3 Il
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Il
Trend extrapolation is often used to make technological forecasts. This technique is based on a historie time series for a selected technological parame ter. It often is assumed that the factors influencing historical data are likely to re main constant rather than to change in the future . Usually, a single-fun ction paramete r suc h as speed, horsepower, or weight is extrapolated . A good trend extrapo lation depends on selection and predictio n of key para rneters of performance. T he trend undcr stud y shoul d be cnpnble of qun nti fi cn tion in ordcr th at it cnn be portroycd nurn cricnll y, 011d nn ndcqun t (lut o hnsc ~ h o u l d cxist o n which to hn sc 11 roli11blo trend lloo. An oxurnpl n 1 ~ f)ltliiO ilt cd ln I•'IH. ~ il . Nntko tlit1t tll11 y n x l r~ 1
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Technological forecasting is a name given to a myriad of specialized forecasting techniques . It provides procedures for data collection and anal· ysis to predict future technological developments and the impacts such developments will have on the environment and lifestyles of people. These techniques seek to make potential technological developments explicit, but more important , they force decision makers to try to anticipate future developments . Technology forecasting is a method that can be used to estimate the growth and direction of a technology. A typical question that technology forecasting attempts to answer is: What will be the machining tolerances of numerically controlled machine tools in the future? In dealing with operating pa rameters, the forecast is limited to specifie technical units and dimen·
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foreseen technology interactions such as unprecedented changes or inexplicable discoveries. The substitution curue is based on the belief that a product or technology that exhibits a relative increase in performance over an older (i.e., established or conventional) product or technology will eventually substitute for the one having lesser performance. The relative increase in performance is the important factor in the substitution of one technology for another. A basic assumption with this method is that once the substitution of one technology for another has begun it will irreversibly continue to completion. Listed below are sorne common examples of the substitution effect:
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An example of linear trend extrapolation.
An advantage of trend extrapolation is that historical data are often readily obtainable. A straight-line or fitted-curve projection of the future is easily understood and used . A drawback to extrapolation stems from the assumption that factors that shaped the past will continue to hold basically unchanged in the future . Trend extrapolation techniques cannot predict un-
The forecast starts with the observation that a new technology is starting to displace an older technology. A measurement term that best defines the fraction of total usage of each technology must be selected, and limeseries data are gathered for both technologies . These data are used to establish the initial takeover rate and to predict the year in which takeover will reach 50%. A typical substitution effect for two technologies is shown in Fig. 5-5 . Forecasting by analysis of precursor events uses the correlation of performance trends between two innovative technologies . Because technological advance usually follows a pattern of continuous increase, situations frequently occur in which an indicator of technical progress lags another by a given period of time. lt is thus possible to utilize the leading technology to predict the status of the lagging technology over a ti me period equal to the lag ti me. The frequent! y cited example of precursor events shown in Fig. 5-6 concems the historical relationship between maximum speed of military aircraft to the maximum speed of commercial aircraft . In this example, it was found that the speed of commercial aircraft followed the speed of military aircraft by six years in the 1920s and eleven years in the 1950s [9): As a result, it was predicted that commercial aircraft with speeds of Mach 2 may be expected no later than 1970, or if such aircraft were not introduced at this time, aircraft with speeds of Mach 3 will be introduced near 1976. The forecast in this situation says, in effect, that there is a logical time for the introduction of Mach 2 aircraft but if other forces
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5.3 PRICES AS A RETURN ON INVESTMENT* Attention in Section 5.2 was directed at various techniques to forecast market demand (sales dollars and/or units of production) for existing and new manufactured products . Once the annual sales in dollars, S, or production volume in units, q, have been estimated for a particular product, it is normall y ncccssary to develop revenue cstimates so that bencfit s of product chnngcs, inclu di ng new or mod iflcd mnnufacturi ng proccsscs/mu tcrials, cnn
be assessed . In this section the objective is to demonstrate how priees can be set through policy adopted by a firm and then used to estimate annual revenues for selected product lines. Es timation of costs is considered in Section 5.4. h
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the circumstances. Maximization of profit, return on investment, market share, and/or total sales are typical objectives. For example, market share can be the dominant pricing objective for standard products, while custommade products may focus on return on investment. Cost is a significant factor in ali pricing decisions-il is needed to calculate the minimum or "floor priee'' that must be met or exceeded. A significant portion of total product cost is shifting to equipment-related costs as facilities become more automated. Clearly, capacity utilization needs to be recognized in setting priees, and in the long term priees will need to recover full costs plus provide an adequate return on the capital invested. In procedures of this type, the costs of production should be based on the long-run average to allow for abnormal operations which will occur from time to time. These considerations can usually be estimated from an economie break-even chart (discussed in Chapter 2) at sorne reasonable percentage of plant capacity. The acceptable rate of return on investment varies with ali the economie factors applicable to a given situation, product, company, or industry . Historical values are commonly-used guides in setting acceptable earning rates . They average from a low of perhaps 5% for the manufacture of durable goods up to perhaps 50% for a new pharmaceutical product. A general average may be 12 to 15%; this range can serve as a criterion in setting priees. To visualize the mathematical calculations, the following relations are useful, where the fiscal period is 1 year, 1 is the effective income tax rate , 1 is the investment, and S is the sales dollars : return on investment, i*
= net profit/i nvestment =
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Any of these equations may be solved for any one of the variables when ali the others are known or specified. The values for priees and costs may be for total output or per unit of output. It should be observed further that the markup (S - Cr) can be based on any component or combinations of components desired by management, but diligence should be applied that no costs are omitted. Furthermore, the basis should be a logical one, utilizing modern costing principles, so that erroneous priee levels are avoided, which may impair the number of sales (if set too high) or which may produce !osses in the long run (if priced too low). To demonstrate certain pricing procedures, a typical example is presented.
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Example S-2 In manufacturing products M and Nin amounts of 20,000 and 16,000 units annually, a company analysis shows the following data. It also hasan effective income tax rate of 52%.
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Chap.5
Priees as a Return on Investment
103
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Sec. 5.3
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For product M : p = _S_ = (1 - 0.52)220,000 20,000 20,000(0.48 - 0.20) = $16.80 For product N : p = _ S_ = 16,000
Solution (a)
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Estimatin g Mark ets , Reve nues, otid Cos ts
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Chap . 5
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Sec. 5.4
Cost Estimating: Perspective and Approaches
10.5
3. Integration with other company plans 4. Evaluation against company objectives for market position, sales volume, profit, and investment 5. Provision of operating controls, assuming that the project is adopted
A summary of the unit priees is:
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Product M
Product N
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$13.60 13 .00 17.15 12.00 14.38
Since there are numerous variations in the pricing procedures illustrated , any policy requires clear statement to avoid misunderstanding of exactly how a final priee was established. The results in parts (a) and (b) are perhaps the most meaningful.
5.4 COST ESTIMATING: PERSPECTIVE AND APPROACHES
In Chapter 2 we described various cost concepts that are important to economie analyses. A key point was that the costs which are important to economie comparisons are the marginal (variable, or incrementai) costs for the future, and that these costs may weil include opportunities foregone as weil as actual cash flows . The basic difficulty in estimating costs for purposes of making economie analyses is that most prospective projects are unique; that is , substantially similar projects have not been undertaken in the past under conditions that are the same as expected for the future . Hence, data that can be used in estimating costs directly and without modification often do not exist. lt may be possible, however, to gather data on certain past outcomes th at are related to the outcomes being estimated and then to make forecasts regarding anticipated future conditions . Whenever an economie analysis is for a major new product or process, estimating for that analysis should be an integral part of a comprehensive planning procedure. Such comprehensive planning would require the active participation of at )east the marketing, design engineering, manufacturing, finance, and top-management functions. It would generally include the foiIowing ingredients :
1. A realistic master plan for product development, testing, phase into production, and operation 2. Provision for working capital and facilities requirements
Obviously, such comprehensive planning is costly in time and effort , but wh en a new product or process has major implications for the future of a finn, it is generally a sound rule to devote a greater rather than a fesser amou nt of effort to complete planning, including estimates for the economie analyses that are a part of the planning. The application of this rule, of course, is bounded by constraints of limited time and talent. Following the rule will tend to minimize the chance of poor decisions or Jack of preparedness to implement projects once the decision to invest has been made. As technology continues to accelerate, quick and reliable methods are needed for determining the costs associated with adding new and/or improving existing manufacturing processes. The methods discussed below are excellent for estimating costs oflabor, equipment, materials , operations, and engineering. 5.4.1 Sources of Data
The variety of sources from which information can be obtained is too great for complete enumeration . The following four major sources, which are ordered roughly according to decreasing importance, are described in subsequent sections: (1) accounting records, (2) other sources within the finn, (3) sources outside the firm, and (4) research and development. 5.4.1.1 Accounting Records It should be emphasized that although data available from the records of the accounting function are a prime source of information for economie analyses, these data are very often not sui table for direct, unadjusted use . In its most basic sense, accounting consists of a series of procedures for keep. ing a detailed record of monetary transactions between established categories of accounts, each of which has an accepted interpretation useful for its own purposes . The data generated by the accounting function are often inherently misleading .for economie analyses, not only because they are based on past results, but also because of the following limitations. First, the accounting system is rigidly categorized. These categories for a given firm may be perfectly appropriate for operating decisions and financial summaries, but rarely are they fully appropriate to the needs of economie analyses for longer-term decisions . Another limitation of accounting data for obtaining estimates is the misstatements embedded by convention into accounting practice . These are
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106
Estimating Markets , Revenues, and Costs
Chap . 5
based philosophically on the idea that management should avoid overstating the value of its assets and should therefore assess them very conservatively . This leads to such practices as (1) not changing the stated value of one's resources as they appreciate due to rising market priees, and (2) depreciating assets over a much shorter fife than actually expected. As a result of such accounting customs, the analyst should always be careful about treating such resources as cheaply (or, sometimes, as expensively!) as they might be represented. The final limitations of accounting data are its illusory precision and implied authoritativeness. Even though it is usual to present data to the nearest dollar or cent, the records are not nearly that accurate in general. Further, the results are often affected by arbitrary allocations and optional charges and/or credits. In summary, accounting records are a good source of historical data, but they have severe limitations when used in making estimates for economie analyses. Further, accounting records seldom directly contain the marginal costs, especially marginal opportunity costs, appropriate for economie analyses.
5.4.1.2 Other Sources within the Finn The usual firm has a large number of people and records which may be excellent sources of estimates or information from which estimates can be made . Colleagues, supervisors, and workers can pro vide insights or suggest sources that can readily be obtained. Examples of records that exist in most firms are sales, production, inventory, quality, purchasing, industrial engineering , and personnel. Table 5-4 provides a list of the types of data that might be needed for cost estimating purposes, together with typical sources (mostly intrafirm) for the data .
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5.4.1.3 Sources Outside the Finn There are innumerable sources outside the firm that can provide information helpful for estimating. The main problem is to determine which sources potentially are most fruitful for particular needs. Published information such as technical directories, trade joumals, U .S. government publications, and comprehensive reference books offer a wealth of informati o n to the knowledgeable or persistent searcher. T'c r·so nal contacts are excelle nt potential sources . Vendors, sn lespcoplc, profcss iona l ncq uain tnnccs , custo mcrs, bankll, govcn, mc nt fi KC ncics , · hamhc r·:~ of co mm ci'Cc, nnct vvcn co mpctil ors nrc ofl lln wlll lnA to ftr rn l~ h dt1d lui
Sec. 5.4
Cost Estimating : Perspective and Approaches
107 Table 5-4
TYPES AND SOURCES OF COST ESTIMA TING DATA
Description of Data General design specifications Quantity and rate of production Assembly or Jayout drawings General tooling plans and list of proposed subassemblies of product Detail drawings and bill of material Test and inspection procedures and equipment Machine tool and equipment requirements Packaging and /or transportation requirements Manufacturing routings and operation sheets Detail tool, gage, machine, and equipment requirements Operation analysis and workplace studies Standard time data Material release data
Subcontractor cost and de li very data Area and building requirements Historical records of previous cost es timates (for comparison purposes, etc .) Current costs of items presently in production
Sources Product engineering and/or sales department Request for estimate or sales department Product engineering or sales department or customer' s contact man Product engineering or manufaciuring engineering Product engineering or sales department Quality control or product engineering or sales department Manufacturing engineering or vendors of materials Sales department or shipping department or product engineering (government specifications) Manufacturing engineering or methods engineering Manufacturing engineering or material vendors Methods engineering Special charts , tables , time studies, and technical books and magazines Manufacturing engineering and/or purchasing department or materials vendors Manufacturing engineering and /or purchasing department or customer Manufacturing engineering or plant layout or plant engineer Manufacturing engineering or cost department or sales department Cost department or treasurer or comptroller
Source: Reproduced by permission of ASTME, Manufacturing Planning and Esrimaring Handbook, McGraw-Hill Book Company, New York , 1%3, pp. 3-20.
5.4.1.4 Research and Development
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Es timatÏIII( Markets , Reue11ues, and Costs
10!1
Chap. 5
are very important estimates to be made and when the sources mentioned above are known to be inadequate. 5.4.2 Estimating Methods
Severa! methods of estimating exist. However, we limit our discussion to the (1) conference, (2) comparison, (3) unit, and (4) detailed anal y sis methods . The first three methods are preliminary techniques for projects that have not been designed yet or that are in the design phase. The detailed analysis method is used for preparing final estimates .
5.4.2.1 Conference Method In the conference method, representatives from different functions in a firm work together to estimate the various manufacturing costs (and revenues) that are needed. Primary advantages of this method are (1) its speed and (2) the fact that it causes people with a diversity of backgrounds to focus on the estimate(s). It is also useful when a company does not have-a formai estimating depart ment. The main disadvantage of the conference method is th at it may lack accuracy.
t
•,1
ost Es timating : Perspective and Approaches
109
• Operating cost per mile • Maintenance cost per day of use These factors may be multiplied by the appropriate unit to provide the total estimate. The following examples may be used for breaking quantities to be estimated into units that can be estimated readily: 1. In different units (Example: dollars per week, to convert to dollars per year) 2. A proportion instead of a number ( Example : percent defective , to convert to number of defects) 3. A number instead of a proportion (Examp/e : number defective and number produced, to convert to percent defective) 4. A rate instead of a number (Example. : miles per gallon, to convert to gallons consumed) S. A number instead of a rate (Example : miles and hours traveled, to convert to average speed) 6. Using an adjustment factor to increase or decrease a known or estimated number (Example: defectives reported, to convert to total defectives)
5.4.2.2 Comparison Method ·i
-!
·· "i·
In the comparison method an accumulation of historical data and experience is utilized. According to a publication by the Society of Manufacturing Engineers (SME): "The estimator applies up-to-date costs derived from si milar parts to the project, adjusting these costs to suit material, labor, and processing variations. Care should be taken when using data from larger or smaller manufacturing quantities. For instance , smaller production lot sizes will usually increase setup costs per item in the lot. Converse! y, Iabor costs normally decline with increased efficiency developed over longer production runs" [15] . The comparison method , like the conference method, is used for quick estimates.
5.4.2.3 Unit Method The most popular of the preliminary estimating methods is the unit · method, which iovolves using an assumed or estimated per unit factor. Sorne examples are: • • • •
Capital cost of plant per kilowatt of capacity Fuel cost per kilowatt-hour generated Capital cost per installed phone Revenue per long-distance cali
Although the unit method is useful for preliminary estimating, the values can be very misleading because there is no consideration of the principle of economies of sc ale or economies of scope .
5.4.2.4 Factor Technique The factor technique is an extension of the unit method in which one sums the product of one or more quantities or components involving unit factors and adds these to any components estimated directly . That is ,
c
=
2: cd + 2: Ji
x V;
(5-19)
whère C = value (cost, priee, etc.) being estimated Cd = cost of selected components estimated directly fi = cost per unit of component i U; = number of units of component i
Suppose that we need a slightly refined estimate of the cost of a hou se consisting of 1,500 ft 2, two porches , and a garage. Using unit factors of$40/ft 2, $2,000/porch, and $3,000/garage, we can calculate the estimate as $40 x 1,500 + $2,000 x 2 + $3,000
=
$67,000
1
1
1
' V'
110
Estimating Markets, Revenues, and Costs
Chap.5
5.4.2.5 Detailed Analysis Method The d.etailed analysis method is the most reliable form of estimating and is normally performed by one estimator (with input from other knowledgeable people as needed) . The following steps should be followed in preparing a de tai led analysis estimate [ 15]:
Cos/ Estimating : Perspective and Approaches
Sec. 5.4
'\
Ill
·::-
,,...
where TS1 = standard ti me for work element j, j = 1, 2, . . . , rn NT; = normal time for work element rn = number of elements
~~.
:·.
The standard times for ali elements are then summed to provide the total standard time, TS, per unit. lfTS is in minutes, picccs pcr hour may be calculated as 60/TS.
1. Calculate raw material usage, including scrap allowances and salvage 2. 3. 4. 5. 6.
material (direct material). Process each individual component (write the operation sheet). Compute the production time (direct labor) for each operation . Determine the equipment required (new, rework, or on hand) . Determine the required tools, gages, and special fixtures . Determine any additional equipment needed for inspection and tes ting.
'
5.4.3.2 Labor-Hour Reports
:r.
.,.f
Labor-hour reports may be used as a basis to determine tabor hours for nonrepetitive work or when time studies are not available . These are merely summaries of hours worked laken from job tickets , with whatever adjustments are appropriate for estimating the times desired.
~~:
.'
k
·),~; i\!l
5.4.3.3 Work Sampling Even though the conference and comparison methods are quicker, most manufacturers use a detailed analysis for a preliminary estimate. It provides the most accurate results and may be used again as the final estimate aftei- revisions are made. 5.4.3 Labor Analysis
Labor is one of the most important expenses to be considered in cost estimating . To provide an estimate of tabor costs, the amount of lime involved must first be measured . Four methods of measuring labor time inelude standard time studies, Iabor-hour reports, work sampling, and predetermined time systems.
5.4.3.1 Standard Time Studies ' 'A standard ti me may be defined as the time required by an average worker to complete a given task while working at a normal pace" [10) . When a job is considered to be repetitive, the stopwatch procedure may be used to obtain this standard time . The following seven steps are involved: (1) divide the task into elements; (2) observe and record time to complete each element each time it is performed; (3) determine the pace (rating) at which the work was performed ; (4) determine allowances for personal needs, fatigue, and unavoidable delay s; (5) deJete observed times that are not performed under normal working situations or conditions; (6) determine the normal ti me; and (7) ca lcul ate the standard time by add ing all owa nces to th e no rnu1 1 tim c . Thu s
Tl'\
NT, 1 (N 'l'
' doc l11 ml)
(5-20)
Work sampling is another method of obtaining tabor hours for nonrepetitive work. This method does not require a stopwatch or historical reports and is convenient and inexpensivc . According to Ostwald :
: j
l'
A work sampling study consists of a number of observations taken pertaining to the specifie activities of the person(s) or machine(s) at random intervals. These observations are classitied into predetined categories directly related to the work situation. Du ring the course of the work sampling study, tally marks are made by the technician, such as "work," "idle," or "absent." The key to the accuracy is the number of observations, which may vary according to the requirements . [Il, p. 42]
1;
11
L '
Work sampling is a statistical technique ; therefore, probability theory must be utilized to plan for or describe accuracy of results . Work sampling statistics generally are based on the binomial distribution in which the mean is equal to Np; and the variance is equal to Np;(l - p;), where N = number of observations and p; = probability thal event i will occur. Since N is usually large, this bino:nial distribution can be approximated by a normal distribution. To begin the process, N ' observations must be made so that p; may be estimated :
Ni
Pi = N'
where
11
r 1 1
(5-21)
"j
pi
= obscrved proportion of occurrence of an even t i expressed as a dec ima l b.sorvution R of e ve nt 1 N; • in itinl 11111111 l llfldlllil llh NCI VIl { iO n ~ N ' • luitl nl 11111111
1 -
11 2
Estimating Markets, Revenues, and Costs .
Chap . 5
Once p; has be en calculated, the estimator must decide on the maximum desirable interval, /, as a measure of the accuracy of the estimate of p 1• Suppose that p' = 50%; if a maximum interval of 2% is desired, the estimate of p ranges from p'- 1/2 top'+ 112 (49 to 51%). The estimator must also decide on the confidence leve!. A confidence levet of 90% is common for work-sampling studies. From a normal distribution table (see Appendix D), for a confidence leve! of 90%, Z = 1.645. Other values of Z (number of standard deviations for a standard normal distribution) for different confidence levels are listed in Table 5-5 . The number of observations of event i necessary to achieve the desired confidence lev el and interval may be found from the following equation [ 11]: N = 4Z2p;(J .,...- p;) 1
(5-22)
[2
Once these observations have been taken , the labor hours may be calculated as H = (N;IN)HR(l + PF&D) ·' Np
Cost Estimating: Perspective and Approaches
113
initial observations the operator has been busy . From these values we canuse Eqs . 5-21 and 5-22 to find the number of observations necessary to achieve the desired confidence levet and interval :
p;
=
:;~
N
=
4(1.645)2(0 .83333)(0 . 1667)
= 0.8333
(0 .04)2
1
=
940
It turned out th at a total nu rn ber of observations, N = 1,262 were needed to observe 940 in which the ope ra tor was busy . The opera tor worked at a rate of about 115%; therefore, R = 1.15. The PF&D allowance factor is 0. 15 . The study was done over a period of H = 6 hr , in which 20 parts were machined. From Eq . 5-23, the standard time in hours per part is
H = (940/ 1,262)(6 hr)( 1.15)( 1 + 0. 15) s 20 = 0.295 hr per part
(5-23)
5.4.3.4 Predetennined Time Systems
where H, = standard labor hours per job element i N 1 = total number of event i observations N = total number of observations H = total tabor hours worked during study R = rating factor PF&D = allowance decimal NP = work units accomplished during period of observing this event
A predetermined time system is defined to be "an organized body of information, procedures, and techniques employed in the study and evaluation of work elements . . . " [10, p. 34] . Such systems are often preferred to time studies because they are considered more accurate and because they are generally more acceptable to workers . Numerous systems exist, but MTM (methods time measurement) appears to be the most widely used . Malstrom describes MTM as follows:
Example 5-4 Suppose that it is desired to know H, for machining a part using work sampling. A confidence of 90% and an interval of 4% have been chosen, and 125 times out of 150 Table 5-5 Z
It assigns times for the performance of elemental motions that a re completed by the hands, arms, and other parts of the human body . Times required to complete different types of motions are assigned specifie values. Ti me value s are expressed intime measurement units (tmu). One tmu is assigned the value of 10 - ~ hr. or 0.036 sec. [10, pp. 34-35]
VALU ES FROM STANDARD N ORMAL DISTRIBUTION TABLE
Area Between Limits (Confidence Level) (%)
68 90
95 99
Number of Standard Deviations -z to +Z :!:1.000 :!: 1.645 :!: 1.960 :!:2.576
lt should be noted that extensive training is required to use MTM with competence. Area Outside Limits (%) 32 10 5
Source : Phillip F. Ostwald, Cos/ Estimating, 2nd ed.,
Prentice-Hall, lnc ., Englewood Cliffs, N.J., 1984, pp. 4344. Reproduced by permission of the publisher.
5.4.3.5 Leaming Curves For manufacturing in which a large amount of direct tabor is involved, the average tabor time required to produce a unit is typically found to decrease over time . As a general rule , each time the cumulative production doubles, the total time required per unit is reduced by 4%. In this case when unit time is plotted against cumulative production, the plot is known as a 96% learning curve . An 80% learning curve would show a 20% reduction
-
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114
Estimating Markets, Revenues, and Costs
:i,.,
Chap . 5
::~
(i.e .. rate of improvement) in cumulative average labor ti me wh en production doubles. This simple mode! is an exponential function [8]:
i
Y;= y,;-h
where Y1
.. !
= cumulative
average direct labor hours or cost through the ith production unit (thus Y10 would represent the cumulative average ti me required for the first ten units, while Y1 would represent that for the first unit produced) i = cumulative production count b = the learning curve exponent
To solve for b, suppose that as cumulative production doubles from 20 to 40, there is a 20% reduction in cumulative average Jabor hours per unit . Then
-a < .,..,' f<
."~ _g...
~ j ~ ;,.;.-
...."' 0ô
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:0 z
:>
E ..;, ::l
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~ '·
-
0 · c.~ .J::J ..c;ro::l 0
Thus. -h log 2 = log(0.8), or -h = log(0 .8)/log 2. This can be solved to find th at h = 0.3219. Fin ally, the equation Y; = Y 1;- 0 · 3219 represents an 80% learning curve. In general, the learning curve exponent, b, can be easily computed: b = log of percent learning/log 2. For example, b = 0. 1520 for 90% learning and b = 0.2345 for 85% learning. The following example shows the use of the learning curve .
5
> 0
....
0.80 = Y4o = Y,(40) - h = 2-h Y2o Y,(20) - h
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Example 5-5
.j
1
One thousand units of a new product are to be produced. Direct labor has been cstimated to average 8 hours per unit. Six months after production has begun, management has requested the following information:
. "~
c ,g '(; ü
]~
=~::t~~~ i
j ,
u~
i'1 ·
1. Average labor hours per unit, to date
2. Labor hou rs per unit, for th e latest month' s units 3. The % learning curve thal production has followed . The cumulative labor hours a nd units produced are listed in Table 5-6. Dividing the cumulative labor hours by the cumulative production shows that the average labor hours per unit to date is equal to 15.75 labor hours . For the latest month's production, an average of 410/42 = 9.76 labor hours per unit was reached .
1
In examining the data , it can be see n that the cumulative production at the end of mon th 6 ( 132 units) is double the cumulative production at the end of mon th 4 (66 units) . The le arning curve ca n be approximated by dividing the cumul a tive average labor hours pcr unit during mo nth 6 by the cu mulati ve ave rage labor hours pcr unit during mo nth 4 . The lcarn ing curvc in effect is approxi motely 75%, as shown lo w .
1'..
1t 7 ,~ hli)\)1' lit HII ~/11•1 1 1 ü •ll h,hn, 1111111 ~A u1T1
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c
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=sc: ·r;
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, 1 ~__,....., ,._.................
11 6
Es tim atin11 Marke ts , R evenues, and C:os ts
Clwp . 5
To determine the equation for this leaming curve, the parameter b must be calculated. This can be accomplished by applying equation 5-25 and solving for b, as shown below .
.>tC.
5.4
Cost Estimati1111 : Perspective a11 d Approaclr es
c. = cb
1
2- b = 0.75 -b log 2 = log 0.75 b = 0.415
Th us the equation is Y;
=
Y, ; -o.•l5
5.4.4 Capital Cost Analysis
Frequently, the larges! cost of a project is attributable to new equipment, buildings, and in sorne cases, entire production plants. Severa! methods for estimating capital costs are discussed below .
5.4.4.1 Analogy Costing Normally, the information necessary for a detailed estima te of capital costs is not readily available du ring the initial stages of a project. Therefore, the first estimate is usually fairly crude and uses analogy costing. Analogy costing is based on similarity among production plants with known capital costs. Equipment for which a cost estimate is desired is located at an existing installation, and by analogy an estimate is prepared for the new equipment. ln this situation differences in cost from the existing to the new equipment would typically arise because of cost inflation and/or productivity improvements . However, these two influences on cost have a tendency to counteract each other in such a way that two projects, whose installations are separated by only a few years, may have estirrÎated costs that are approximately the same. given that the existing plant was not grossly over- or underdesigned . Other factors must be considered when the same plant is to be built in a different location, such as different delivery charges, different site preparations, and different costs of supplementary equipment and facilities to support operations [4] .
5.4.4.2 Exponential Costing Exponential cos ting may be used when the proposed plant has a different production capacity than the existing plant. According to de la Mare, " the principle of exponential cos ting states that for many real-world production systems proportionate increases in production capacity can be achieved by Jess than proportionate increases in capital cost. This principle is a spe-
~ ~
t
cial manifestation of the law of increasing returns to scale, and is known as the law of increasing technical returns to scale" [4, p. 151]. The following general equation represents most types of equipment:
Y,( 132)- h Y(66) - b = 0.75
Reducing and solving, we find :
~~.
11 ,
(Q·)Il Qb
~
4$
(5 -26)
where Ca = capital cost of the proposed facility Cb = known capital cost of an existing facility Q. = production capacity of the proposed facility Qb = production capacity of the existing plant f3 = cost exponent factor, which can range from 0.4 to greater than 1.00, but is usually in ·the range 0.5 to 0.8 Table 5-7 provides typical cost exponent factors for selected types of industrial equipment. Equation 5-27 permits one to obtain an estimate of the capital cost of a proposed project by including factors to adjust for the effects of inflation and productivity differences as follows :
Qu)(J
Ca = Cb ( Qb 1,1,.
(5-27)
where 1, = index of cost inflation lp = index of productivity improvement The accuracy of the exponential costing method depends largely on the similarity between the two projects, and the accuracy of the cost exponent factor {3. Generally, error ranges from ± 10 to ± 30% of the actual final cost . A certain steam-generating boiler in the utility plant of a manufacturing complex produces 50,000 lb /hr of saturated steam. This boiler was purchased for $250,000 eight years ago . If the priee index for this type of boil er has increased at an average rate of 12% per year for the past 8 years, and productivity improvements have averaged 4% per year du ring this lime, how much would a 150,000-lb/hr boiler cost now? The cost exponent factor for this boiler is 0.50.
Utilizing Eq. 5-27,
c. = with / 1 = (1 + 0.12)8 and lp estimated as follows:
c. == =
= (1
(Q.)Il 1,1p
cb Qb
- 0.04) 8 , the cost of the proposed boiter can be
$250,000 [ ( ~;~,;:) 0. ( 1.12)8(0.96)8] 5
$250,000[(3.00) 0·1(2.476)(0.7214)]
= $773,441
~.....!:.:)
'J , .
Sec. 5.4
Cost Estimating : Perspective and Approaches
119
:;
i
5.4.5 Materials Analysis
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Depending on the product, materials can be a very large, and therefore important, part of a cost estimate. ln many industries it is common for materials costs to comprise 50% or more of total product cost. Ma teri al cost analysis involves making and using a bill of materials, deciding whether to make or purchase parts, measuring the amou nt of raw material that goes into each unit, and finding the costs of parts and raw materials.
~
~
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Q
u
5.4.5.1 Bill of Materials
ci
1-
The bill of mate rials is a complete listing of ali parts (mate rials , su bas· semblies, purchased items, etc.) going into the final prodpct. lt also lists the part number, the quantity required to produce one unit, the material specifications , whether the part is purchased or made, and perhaps other products in which the parts are used as weil as a cross-reference to similar parts. This information is vital to the cost estimator so that an accurate estimate may be made.
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. 5.4.5.2 Decision to Purchase or Make·
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In most instances the decision to purchase or make parts is determined by considering the incrementai costs of each option . That is, parts will be purchased rather than made when the purchase priee is Jess than the incrementalcost to manufacture . There are situations, though , where it is desirable to select the more expensive option . Suppose that a part can be purchased for Jess than the cost of in-house manufacture, but the vendor cannot guarantee delivery when the parts are needed. ln this case the part probably should be made in-house to avoid the possibility of missing the due date. Furthermore, sometimes a part may not be readily available for purchase without risking the Joss of competitive information .
.
Q.
5.4.5.3 Estlmates of Raw Materials and Purchased Parts Costs Once the estimator has determined which parts are to be purchased and which are to be manufactured, the next step is to obtain cost estimates for both raw materials and purchased parts . According to Malstrom : In estimating material costs, severa! factors about the part to be purchased should interest the cost esti mator. Examples of fac tors to be considered inelude the qua ntit y to be purchased, whether or not the part has bee n purchascd beforc, and the lcns th of tin1C requircd to obt11ln 11n occumtc priee qu Oll•t lon. (10, p. SS I l lll) (l fC IH ll y 11111 111( hi N i lld~·JI) II,U,' 'IIIL• • ollon k t• pltlll ll l OIIIJ'IIIt' l ill(l litiiiiiii' Y
11 n
IIH
1·
120
Estimatinf( Markets. Revenues, and Costs Table 5-8 HISTORICAL DATA
Dale
Order Quantity
Unit Cost
3/84 9/85 8/86 12/87
250 1.000 160 730
1.65
1.25 1.!!7 1.72
Sec. 5.4
12 1
Cost Estimatinf( : Perspective and Approaclzes
for numerous products), there could be a significant difference in the per unit cost.
5.4.6 Other Costa The estimator should recognize that costs for working capital, setups, run cycles, tooling, and factory burden need to be considered when relevant.
!(
5.4.6.1 Working Capital are easily accessible. Table 5-8 provides a list of historical data. First, the estimator picks two order quantities that bracket the order quantity desi red . Suppose that the desired order quantity is 424 ; he or she must then choose the quantities of 250 and 730. If today 's date is 6/88 and the average inflation for this material has been 1% per mon th du ring the last 4 years, the unit cost today of an order size of 250 would be*
V25o = $1.65(1 + o.ol)· = $2.74 11
For a lot size of 730, the unit cost today would be
u730
= $1.72(1
+ o.Ol)6
730 - 424) = $1.83 + ( ($2.74- $1.83) = $2.41 _ 730 250
There are severa! problems with this method . For example, inflation rates do not usually remain constant. As a result, this method is recommended only for parts with a low unit cost. When the unit cost of a part is high, it is normal to obtain at least one priee quotation even if the part has been purchased before. A priee quote from a manufacturer's catalog is acceptable if it is up to date. "Substitution is acceptable as long as the priee of the part may be obtained from a catalog in a period of time equal to or Jess than that required to obtain a quote" [10, p. 56]. If the dollar value of the order is Jess than the cost to ob tain the estimate, the estimator may make an educated guess, use a manufacturer's catalog, or use historical data to obtain a reasonable estimate. In instances where a priee is not readily available, a part number match may be used as a basis for estimating. The estimator should also be aware that priee breaks may exist in the purchase of parts. If the company is willing to purchase sufficient quantity of a part at one ti me (perhaps by combining requirements • See Section 5.5, Eq. 5-37: (1 + 0.01)" = (F/ P, 1%, 51), which merely adjusts "RS" into "A$" by compounding inflation at 1% per mon th for the 51 months from 3/84 to 6/88 .
-.,.. !!f!S, #'~~-?.91(0;A .'-·: '...,
·' · · ··~ -- ~- ~..------ - - ---
- - - -- -· __ _. _
1; 1
(5-28)
·''
Table 5-9 shows an example of current as sets and liabilities for a company. Using equation 5-28, working capital= $4.35 x 10 6 - $1 .2 x 10 6 = $3 . 15 x W. Although working capital is constantly being liquidated and regenerated, it is not available for other uses, so it must be regarded as an investment.
:f
working capital = current assets - current liabilities
,1
t
\·
1
,.
= $1.83
Interpolation can then be used to obtain a unit cost for an order quantity of 424 units. If we assume thal the relationship is linear, it can be solved as u424
Working capital is the moncy required to transform a new capital project into an operating process, excluding fixed assets (such as equipment, machinery, or buildings). A common accounting definition is:
;
5.4.6.2 Setup Costs
! '.
Setup is the work required to prepare the machine , process, or bench for operation . Normally, the setup ti me can be fou nd by using standard data, but a study of the elements necessary to perform the setup may be required . These elements may include operator punching in or out, obtainTable 5-9 A COMPANY's A SSETS AND LIABILITIES
Fixed assets Current assets Cas h ln ven tory Accounts receivable
$ 8,000,000
$
50 ,000 3,300,000 1,000,000
$4,350 ,000 Current liabilities Accounts payable Short-lerm debt
$ 700 ,000 500,000
$1,200,000 Working capi tal
$ 3,150,000
Total assets employed
$11 ' 150,000
Source: R. F . de la Mare , Manufacturing Sy.rtem.r Economie.<, Cassell Education al LTD.. London . 1982, p. 167. Reproduced by permission of the publisher.
1' 1 ~
i
,\ 122
Estimating Markets, Revenues, and Costs
123
Cost Estimating: Perspective and Approaches
Sec. 5.4
ing tools, doing paperwork, making machine adjustments, and positioning necessary materials . Also included in the setup are tasks performed after a run cycle is completed-such as teardown, returning tooling, and cleaning workstation. Normally, the setup cost is handled as an overhead charge for mass production [Il].
·' 5.4.6.3 Cycle Time Costs Cycle time is the time required to produce 1 unit of a product after setup has been completed. Depending on the process, there are many different methods of calculating the cycle time of a machine . As an example, the steps in a metal cutting operation are ( 1) load work, (2) ad vance tool, (3) retract tool, and (4) unload work. The time to load and unload the workpiece is called the handling lime and may include the time to ad vance and retract the tool from the piece. The handling cos/ may be found by
·c:
..
o.
handling cost = Coth
where tm = machining time, min L = length of eut for metal cutting, in f = feed rate, in/rev N
d . = rotary cuttmg spee = = diameter, in
12 v
The unit cost decreases as cutting velocity increases, as shown in Fig. 5-7b . Note that the diameter, D , may be for either the tool or the workpiece, de pending on the type of machine . The diameter of the tool is used for milling and drilling; in lathe turning, the diameter of the workpiece is used.
Cutting speed
Cutting speed
le)
(d)
ing to Ostwald [Il], the aue rage /ife of a cutting tool may be fou nd by the
1 .
machining cost = Colm
il u
Figure 5·7. Graphie costs for four parts of machine turning economies . Source : Ostwald, Phillip F. , Cosr Esrimaring, Seco nd Edition . Eng)ewood Cliffs. NJ : Prentice-Hall , lnc ., 1984, p. 271. Reproduced by permission of the publisher.
rrD , rev mm
D V = cutting speed, ft/min
8.
Tool ·changing
~
where C0 = direct labor cost, dollars/min th = time for handling, min See Fig. 5-7a as an example of handling cost plotted against cutting speed. To find the time to machine 1 unit, the following equation is used [10, p. 270) : L LrrD tm = JN = 12 Vf
ï:::1
::1
T
=
K""
v
where T = average tool !ife, minutes per cutting edge n, K = empirical constants resulting from regression analysis and field studies 0 s n ::5 1, K ;:::: 0 When V = 200ft/min and the tool-life equation is VT 0· 16 = 400 , then 76 min of machining may be done before the tool must be indexed to a new The tool-changing cost per operation may be found by tool-changing cost = Co te
5.4.6.4
To olin~
(5 -33)
Costs
'utti l1fl, t ou l ~ 11111y h ilC (lll l è uull, 1\nd th c r c f () lt' 11\IINI h ll ~ h t11 (1i' I W
(5-32)
. Il
,.1 ~-
l. 124
Esrimaring Mark ets , Revenues , and Cosrs
Chap. 5
Sec. 5.4
Cosr Esrim a rin g : Perspective and Apprnaches
Figure 5-7c shows the relationship of tool-changing cost to cutting speed. The cast of cutting tools must also be calculated: tool cost per operation = C1
(t;)
(5-34)
where C, denotes the tool cost in dollars and T is the average tool !ife in minutes . Ostwald observes that "for insert tooling, tool cost is a function of the insert priee, and the number of cutting edges per insert. For regrindable tooling, the tool cost is a function of original priee, total number of cutting edges in the li fe of the tool , and the cost to grind per edge. " As shown in Figure 5-7d, the tool cos! increases as cutting speed increases [Il , p. 272].
Design engineering estimate for
No.t
Customer
Ali costs that cannot, conveniently and economically, be allocated to a particular product are called factory burden or ouerhead. Burden is typically divided into two categories: fixed and variable . Fixed burden consists of costs that do not change with production volume, such as salaries , mortgage payments, rent, and insurance. Variable burden increases or decreases as production volume rises or falls . Examples include electricity to operate equipment. indirect materials and labor , and certain tooling . Often, companies allocate burden to different cost centers and/or products in proportion to sorne activit y basis such as direct labor hours or prime (tabor + materials) cost.
h~ ow~
Based on quantity of
1
lnquiry or quota no.
480 u>t.U&
Date
Ouri ng period of
Type of labor
f ~g..:.~ee.lt,
producn
Ve.ûg~ 6olt te 6.t ha~d w eed ~t educ ~.lt
Description
Hours
Rate per hour
Extended la bor
~te~e.all c h
Suei'I.Û.I>.t ,
5.4.6.5 Factory Burden
Speed 1teduce1t
125
6 e ~olt
de.ûg n
Engwee.lt , de.ûg~ Tech~c.i.a~ ,
40
$33 . 00
$1. 32 0
800
$30 . 00
$ 24,000
160
$IL 75
$3. 000
80
$17 . 00
$1,36 0
e.lec..tM.ca.t
Vu.i.g~e.lt / e~glt.
a.i.M
Tech. Wlt.i..telt
1U~.tlta.tolt 01t46 ûma ~ P1tov~.i.o~ng 6peeia.t~.t
Mode.l 6h op
5.4.7 Engineering Costa
When the product to be produced is similar to an existing product, engineering costs can be estimated by using historical records . The estimate must include costs of basic engineering services such as product design, specifications, and bill of materials. as weil as maintenance engineering for the li fe of the product. The costs are made up of wages , housing, and utilities. In addition, costs for projects that did not materialize into final products must be considered in the wage costs [Il) . Spreadsheets such as th at shown in Fig. 5-8 may be used in figuring the cost of tabor for engineers. When repre senting engineering rate per unit , C,, the following equation may be used : C
=
•
total engineering cost projected production quantity
(5 -35)
where the " production quantity" can be in any relevant units , which are often " hours worked ." Sometimes· factors are used to multiply by expected salaries to provide an estimate . They can range from 1.8 to 3.0 and depend on the complexity,
To.ta.t du .i.g~ e ~g .i.~ ee.lt.i.~g la bolt -
Figure S-8.
- ---
1080
$29 ,680
- --- -
Estimating fo r engineering cos ts. Source : Ostwald , Ph ill ip F., Cmt
Esrimating , Seco nd Edition. Engle wood Cliffs. NJ : Prentice-H all , lnc . , 1984, p. 318. Reproduced by permission of the publi sher .
novelty, and sec recy of the work . Other factors , su ch as drawing size, are a!so being used to estimate engineering design costs . 5.4.8 Comprehensive Cost Estimating Example
The XYZ company has decided to btlild a new plant that is very similar to an existing plant. The plant will be used to produce 5,000 units of the part shown in Fig. 5-9. The loss due to scrap is expected to be 2.5% . The company that sells the raw materials to the XYZ company will buy back any scrap or waste for $0.95 per cubic inch . The recovery cos! is $0.20 per cubic inch. Approximately 15% of the waste is not recovered for resale . Histori-
1
0:
1
· l'
126
Estimating Markets, Revenues, and Costs 0.25-in. dia .
0.66-in. dia. - +-.,--------,------'------.
0.25-in.
T
" I
dia.~
Sec. 5.4
Solution
The actual volume of material required per part is calculated to be 1 1.5375 inches- . The total material necessary to produce 5,000 units is therefore (1.5375 inches-1)(5,000) = 7,688 inches 3 . The amount of m'a terial available to sell back consists of both scrap and waste. The amount of scrap per parti s ( 1.5375 inche s 3 )(0.025) = 0.0384 inches 3• The amount ofwaste (from drill ing holes) is (1 -
o. t s>f( 0 ·~ 6Y<2>
+ (2)( 0 -~ 5 Y}n 2 C0.25 in thick) = o. l663 in l
.127
From the historical data we can find the cost of the raw material if it is purchased on February 1, 1990. * 23
U3,000
= $1.89
(1
0.12) + 12"" = $2.38
Us,ooo = $1.45 ( 1 +
0; 12) 12""
37
= $2.10
By interpolation, the cost of a unit of raw material is $2. 12 per cu bic inch . Therefore, the total cost of materials is (
($2 . 12/in 3 )(7 ,688 in 3 )
-
($0.1535)(5,000) = $15,531 ~
The capital cost of the plant may be approximated by using Eq . 5-27 as :
Figure 5-9. Sample part .
cal data show that on January 1, 1987, the cost of this material was $1.45 per cu bic inch, in a quantity of 8,000 inches 3 • On March 1, 1988, the unit cost of 3 3,000 inches was $1.89 per cubic inch . For the past 4 years we will assume that inflation rates have remained stable at an annual rate of 12% and that the date wh en the material is to be purchased is February 1, 1990. The approximate capital cost of the existing facility which was purchased recently was $12,000, and its production capacity is 20 units per day. It is desirable for the future facility to have a production capacity of 30 units per day. Capital cost can be estimated using Eq. 5-27 with /, = 1.12, lp = 1. 10, and (3 = 2/3. Studies were do ne at the existing facility showing thal approximately 0.35 hr is necessary to produce each unit. This time was achieved over a production of 3,500 units and production has followed an 85% Iearning curve. Labor costs $15 per hour, including benefits. The parts are produced in lot quantities of 25 and the machine setup time is 1 hr. Part of the setup involves changing the cutting tool. The cost of the tool is $20. Factory burden is 40% of direct material. Estimate the manufacturing cost from the information given .
Cost Estimating : Perspective and Approaches
,, . 1
30)0.6667 CA= $12,000 ( 20 (1.12)(1.10) = $19,373 The average labor hours over the production of ali 5,000 units may be found by use of the Iearning curve and Equation 5-24:
b
= log(0.85)/log 2 = 0.2345
So
Y; = y 1; - o.2345 For Y1 = 2.03, we find Y;= (2 .03)(5,000)- 0 2345
=
0.275
The labor cost of producing 5,000 units is (0.275 hr/unit)(5,000 units)($15/hr)
= $20,625
The setup cost may be found by (5 ,000/25)(1 hr)($15/hr) = $3,000 The cutting tool cost is (5 ,000/25)($20) = $4,000 Factory burden is 40% of direct material , and thus is (0.40)($2.12)(7 ,688) = $6,519 From the above costs, the total manufacturing cost, Cr, may be calculated
as: Mat criai $1
Producti on la bor 1
•
$20,
tl ~ lnw .~rollto !l \ ,, , tlq
Setup
Cutting tools
$1,000
$4,000 + $6,5 19 - $69.048
Factory bu rdc n = '-T
i \/ tn t •U IIillllll olllloiM 111 11 ~11•111111 t '1• tl ~ t 1111111111
.
\ i;
~
128
Estimating Markets, Revenues, and Costs
Chap. 5
Sec. 5.5
Cmt Estimating Relationships and Inflat ion
l'
.
.~
129
1
5.5 COST ESTIMATING RELATIONSHIPS AND INFLATION
Except for Sections 5.4.5.3 and 5.4.8 we have assumed that prices.up to this point will remain relatively unchanged over si.Jbstantial periods of time, or that the effect of such changes is the same on any alternatives considered. U nfortunatèly. the se are not realistic assumptions in general. Inflation can, indeed, affect the economie comparison of alternatives. Hence the nature of inflation and methods of taking inflation into account will be considered below. 5.5. 1 Actual Dollars versus Real Dollars
Inflation describes the situation in which priees of fixed amounts of goods and services are increasing. As priees ri se, the value of mo ney, that is, its purchasing power (in real dollars , as defined below), decreases correspondingly . Let us defi ne two distinct kinds of dollars (or other monetary units such as pesos or ru bles) with which we can work in economie analyses, if done properly: 1. Ac tuai dollars: the actual nu rn ber of dollars as of the point in ti me they occur and the usual kind of dollar terms in which people think. Sornetimes called then-current dollars , or current dollars , or even infiated dollars, they will be denoted as "A$" whenever a distinction needs to be made in this book. 2. Real dollars : dollars of purchasing power as of sorne base point in time, regardless of the point in time the actual dollars occur. Sornetimes called constant worth dollars, or constant dollars, or even uninfiated dollars, they will be denoted as "R$" whenever a distinction needs to be made in this book . If the base point in time, k, needs to be specified (it is usually the time of the study or the initial investment), that can be shown with a superscript [i .e ., R$Ck>].
Actual dollars at any time, n, can be converted into real dollars at time n, of purchasing power as of any base time k , by R$~k> = A$n ( l
)"-k= + f, 1
'
· data for estima ting inflation rates (either general, such as cost of living, or for
specifie commodities and services) are cost indexes. Cost indexes are dimensionless· nu rn bers th at provide a means for converting past costs to costs at any later time, or vice versa . There are many cost indexes and they cover almost every area of interest. Sorne are based on national averages; others are very specialized by type of item and perhaps locality . Indicative values for se veral frequent! y used indexes are shown in Table 5-10. Probably the most commonly-used measure of the general inflation rate is the Consumer Priee Index of the U .S. Department of Labor . Note that "1-year change" (i.e. , percent increase or decrease for each year) in Table 5-10 is also shown for two of the indexes . Additionally, the compounded an nuai rate of inflation for the 8-year period is shown at the bottom of the column for each index . If we want to fi nd the cost of something at ti me n and know its cost as of sorne (base) time k, then the relation ship that can be used is Cn
ck
i
iJ
; r
·v ' ··
,, i
i 1~
ln
=
liq
"h
where Cn and Ck are costs and ln and h are the appropriate index values at times n and k, respective! y. Solving gives
c.
=
ck
(i)
~
~ ~
'' ! '
: i
(5-38)
;
Recognizing that Cn could be A$n, and Ck cou id be R$~kl , we can substitute into Eq . 5-38 to find A$n =
(R$~kl)
i
1 ! i
(5-39)
i
Combining Eqs . 5-37 a nd 5-39, we find that a;,
(F/P,f7o, n - k) =
hln
(5-40)
il
1;
/;:
1
Example 5-7
A$n(PIF,f%, n - k)"'
(5-36)
Use the Consumer Priee Index in Table 5-10 to e stimate the 1983 cost of products that cost $1 million in 1980. Then find the annual compounded rate of inflation .
R$~k 1 (FIP,J%,
(5-37)
Solution
Similarly, A$n
= R$fkl( J + f)n -k =
n - k)
where fis the average inflation rate per period over the n periods. By ''inflation rate" is meant the average rate of inflation for the particular items over the period in question . Sorne of the most valuable sources of • lnterest factors and symbols are explained in chapter 6. Fac tors are tabled in Appendix A.
Using Eq. 5-38 yields
/83)
CsJ = Cso (lw
. . (298.4) . . = ( $ 1 mtlhon) . = $ 1. 209 mtlhon 246 8
\ '(--..>
~
~
... . .
·
'
131
Cost Estimating Relationships and Inflation 0"'
- ' )( w
"
c
~
.,., w
0
.-
"' e~
1-f-
8'"
u
"
>;-
..!!
.Cl
0
z
0::
"'0
w ..J
~
~
:~
...
< )(
~-
~
.5"
MNC\~'o.DN\0000
O..:v-\v\r-o:oOO..:O..:..Or-\
Using Eq. 5-39, we obtain
"c
(FIP,f%, 83 - 80) =
:::J
&.
E E
ë..
8
0..
-
Il ~ N C
0
a-
z
"'
,_-
OOr"'-"tf"'o:t""':f"\O~f""''
v\NOOMoôv\c---:v\ Nf'""''~V"'\Ot'--C\0-
-
-
-
-
-
· -
-
e
(J.
0\
~
....; 0 ....;
C'l N
0
5.5.2 Real lnterest Rate, Combined lnterest Rate, and Inflation Rate
c:i
00
~ c
~
"'
.c~
><
.::"" ~~ "' " >:'"
0.0
-ooV"lr-f"f""Vl
a\v-l...or..:....:t'f"\o..or"")
~ c(
<.)
0..
s
~
~ c
8
u
Il
-
0
><
~
~-=
NV""'V)"'=tVOO"'=t-"':f'
....:c:::)_:v-)r--:...0No\OÔ -.cr-ooQ\-""':fr-ooa-..
r-
oci
----NNNNN
.;
·-
ÇQ~ "' O
~25:2D
0
<
:~
0..
"'
~'- E u ~ ro:O-r-c "'"'--r-ou r.n · -Cl\ > 0 0-
. ·E
(J.
.
gg;;::ggr;:~:2~~ ---NNNN
0
r.l'l
UJ ~
o>.o"" ='
«1
o-8ooNOI""'V"'\O Oo\OMr--:..OO..:N~
oooos=~~:!:~~
o
o\
~
•
>o
.....
:l 0\
0
.Cl
cu ..J
"'
~~g
::l
«1
L.o
0
~ -~
OIJ
,_ "
c
[JJ
•
'-
0
0
·-
"" ..J
c
.c>
E 0
'-'-'-0 0 0 0 <.)
z
-c
~
êg ti
~
5 >-
Vl\0('000\0-~r"'
~~~~~~~~~
-c
-c
[JJ '-
1. Real interest rate: Increase in real purchasing power expressed as a percent per period, or the interest rate at which R$ outflow is equivalent to R$ inflow . lt is sometimes known as real monetary rate or uninfiated rate and denoted as i, when it needs to be distinguished from ic (below) . 2. Combined interest rate : Increase in dollar amount to cover real interest and inflation expressed as a percent per period, and is the interest rate at which A$ outflow is equivalent to A$ inflow. It is sometimes known as actual rate or infiated rate and is denoted as ic whenever it needs to be distinguished from i, (above). 3. Inflation rate : As defined previously, the increase in priee of given goods or services as a percent per period. It is denoted asf.
oo.o
Because the real interest rate and the inflation rate have a multiplicative or compounding efTect,
.... " ';j'g.
ic = (1 + i,)(l + f) - 1
(5-41)
ic = i, + J + (i, X f)
(5-42)
~_..::
....
~
~.<:
.,-,u
~I'O ·= "'..,
00 .Cl
a-
't:
--:.~
.
E E E ::: ~~ la la la g c(< 4.)
- -~~
0.0
....
~
:::J [JJ :::J ~ ÇQ_ÇQ<
~ ~~ - 00 ~~
..........,..
g .-:. 8 ~
j.Bj5~111
.. -:::J r- "c V)- r - -
>o 0
~
·~
'-..5'-CO~t-..,
::J
·~ ~ 8 a..--"' ....
E
"'0
~
0
~ ·~~~c2~ .8§.8~~~
~<""§Il ~ ..... 0
(,)
~
~5!9~.o-'-·
u
-6
Vi
·;: < ·;:
"' c .. ï: -~ ~ v; ~
]
..· 5
~
·~
0
.9 ë
00\0"'=t'V'lr--V""'
"'
:::1
~
0
u
Let us define severa! types of rates and show how they are used :
ôè
·c
" E
298.4 _ = 1.209 246 8
From compound iriterest tables, (FIP, 8% , 3) = 1.2.597 and (FIP, 6%, 3) = 1.1910 . lnterpolating, the inflation rate for that 3-year periO
0
<.)
.§
[JJ
"' 1:!
00
~
'.:J-
"'Q~f"'\ "'
y
~ . !3~
"''?>J: ?.
u
u
a. o. o. 0
0
u u 0 Cl
0
::J
0
u
-Li: .,.~c
1 '
ic- f
= 1 +f
(5-43)
Where fis not large relative to the accuracy desired , th en
0
,. K. vivi vivi V'\ <+
::i ::i ::i ::i $<
ic ô!! i, + f
nd
(5-44)
,, 1
!·, ·1
·~
Il\
132
Estimating Markets , Revenues, and Costs
Co.ft Es timating R elationship.f and Inflation
5.5.3 What lnterest Rate to Use in Economy Studies
133
real rate, i, = 4% inflation rate, f
In general, the interest rate that is appropriate for time-value calculalions in economy studies depends on the type of cash flow estimates as follows :
= 6%
combined rate i, = (1 + 0.04)(1 + 0.06) = 0.1024 = 10.24%
Method
If Cash Flows Are Estimated in Terms of:
Then the lnterest Rate to Use ls:
Actual $, AS Real$ , RS
Combined interest rate , i, Real interest rate , i,
A B
"" JO%
A$96 =
The above is made intuitively consistent if one thinks in terms of method A as working with inflated (actual) dollars and interest , and method 8 being applicable to uninflated (real) dollars and interest. Method A is the most natural to use because we usually think in terms of A$. Since interest paid or earned is based on A$, it is a combined interest rate , ic. method 8 is sometimes easier to use.
=
A~,(FI P,
$10,000(FIP , 10.24%, 5) = $16,282
(Note : If 10% approximation were used, answer would be $16,105 .)
R$~ 1 1 = R$~ 11 (F/P , i,% , 96 - 91)
= $10,000(FIP, 5.5.4 Summary of Formulas for Relating Single Sum A$ and R$ over Time R$~kl (= A$•)
k
A$n N S . . R$~·~- ote: uper~cnpt needed only to clarify base lime .
Time scale
Method A
A$ (inflated $)
B
R$ (uninfla ted $)
From R$ to A$, o r from AS to R$ , at a given time
4%, 5)
= $12,167
In 1996 dollars, beginning with 1996 equivalent in 1991 dollars : A$96 = R$~n(FIP, f%, 96- 91)
,·
j
= $12,167(F/P, 6%, 5) = $16,282
i
5.5.5 Manipulating Series Which Are Uniform in R$
n
Base time for ex pressing R$ Type of Dollars or Conversion
1,
ic% , 96 - 91)
Moving Forward in Time
Moving Backward in Time
A$. = AS,(F/P , i" n - k)
A$,= A$.(PIF, i" n - k)
RS~" = RS\"(F/P , i,, n - k)
RSI" = R$~ 1 (PIF, i., n - k)
If cash flows expressed in R$ are uniform each year (and th us the A$ ' s increase each year at the average rate of inflation, J), they can be conveniently converted to equivalent worth(s) at other point(s) in time using uniform series formulas at the firm's i, = real MARR . Thus, if there is a uniform series in which each end-of-period pa y ment, A, for n periods, is expressed in R$
lnflating (at given time)
Detlating (at given time)
Phk> = (A
(5-46)
F~"> = (A<•>)(FIA, i" n)
(5-47)
AS. = R$~"(F/ P .f, n - k)
' '
'
\! 1
i 1
R S~" = AS.(PIF.J, n - k)
Example 5-8 A certain expense at the end of 1991 is estimated to be $10,000. The end of 1991 is the base point for considering inflation. (Thus $10,000 = A~ 1 = R$~~ 0 . ) Find its equivalent worth in 1996 for the following circumstances (paralleling the formulas above) . and us ing the following rates .
The superscripts make clear the base point in time at which the dollars (like present or future equivalents) are expressed . Normal! y k = 0, but estimates can be converted to any base point in ti me at the inflation rate, f, using Eq . 5-36 or 5-37. An equivalent formula for finding the present worth of a uniform series that is escalating at the inflation rate f for n years is
1
1 ·' 1
(
~
~
Estimating Markets, Revenues, and Costs
134
Chap . 5
Sec . 5.6
Summary and Prologue
d '. '
~
135
.~
0
Note that
ic -
-
A<0>(FIP,f,
~
eral ways to convert a series of cash ftows subject to whatever rate of inflation (which might vary from year to year) into either R$ or A$, and then to manipulate them (in this case, into present worths) at the general inflation rate .
p _ A(FIP,f, 1)[1 -(PIF, ic%, n)(FIP,f%, n)]
J
1) = A(l). Thus
~
lî
p _ Am[l -(PIF, ic%, n)(FIP,f%, n)] 0
ic -
-
For the special case in which ic 5-48 and 5-49 become
J
=f, so thal the real monetary rate = 0, Eqs.
P~ 1 = A
Given the same individual salary situation as in Example 5-9, the only difference being that the salary will inflate (escalate) at 8%1year, which differs from the general inflation rate of 10% . To repeat, the end-of-tirst-year salary A$ 1 = R$\n = $21,600, i, = 5%./ = 10%, and ic = 15.5%. (a) Show her salary for 3 years expressed in R$ and in A$.
and
P& > = 0
(b) Show the present worth (as of beginning of the tirst year) of both ways
Am(PIF,f, l)(n)
of expressing this salary.
Example 5·9 A person saJary to inflation. 0. 155 , or
who is earning $21,600 salary for {assumed end of) year 1 expects that inflate (escalate) at 10%/year, which is the same as the general rate of If her real monetary rate , i,. is 5%, then i, = 0.05 + 0.10 + 0.05{0. 10)"' 15 .5%. Find the present worth, P&01 for 3 years of salary.
ln A$. : ( 1)
Year n
(a) Use the approach of Eq. 5-46 to find P&11 and then convert to Pb >. 0
(b) Use Eq . 5-49.
3
(a) P&11 = Alll(PIA, i,. 3) = $21,600(PIA, 5%, 3) = $58,821
Pb01 = Fb1J(PIF,J, 1)
= $58,821 (Pl F,
10% , 1)
= $53,474
(b) p
$21,600[1 -(PIF, 15.5% , 3)(FIP, 10% , 3)] 0. 155- 0.10 = $53,496
[same as for part {a) except for round-off erro r)
5.5.6 Manipulating Series That lnflate (Escalate) at Rate Different from General Inflation
Whcn a cas h flow series ex pressed in A$ inflates or escalates at a rate differen t from genera l infla tio n. it will no t be a unifo rm series wh en cxnrcsllcd in R$. T he use of whot might be cnll cd "d ifferentiai oscolatio n ITI ICII" Cll n he Il hnmly C(l!)lP III OI Î() tllli CO II VCfl ic n CC in llti Ch Il Cll llC, h ui WC will nol ll ln Ntt ltll' llt t nl h
11 -
Il
S21,600(FIP, 8%, 0) = 21,600 21,600(FI P , 8%, 1) = 23,328 21,600(FIP, 8%, 2) = 25,194
2
Solution
(2)
A$. = A$ 1(FI P, 8% ,
(3)
= (1)
x
(PI F, 15 .5%, n )
Pw~··
0.8658 0.7495 0.6489
$18,701 17,485 16,348
(2)
$52,534
ln R$~0>: Year
1
2 3
RS~01
11
= AS.(PI F,
10%,
11)
S21,600(PIF, 10%, IJ = 19,637 23 ,328(PIF, 10%, 2) = 19,278 25 ,194(PIF, 10%, 3) = 18,928
(PIF, 5%,
0.9524 0.9070 0 .8638
11)
Pw~··
$18,702 17,485 16,350 $52 ,537
Note in the lower column {1) that even though her A$ salary is going up (at 8% per year) , the R$ (purchasing power) of that salary is going down {approximately 10%- 8% = 2% per year). The present worths of both ways of ex pressing the sa lary are the same, except for minor round-oiT error.
l·
5.6 SUMMARY AND PROLOGUE ln 1hls chnpt cr w dMIIJWCCiil'lllr
y f'or
usscd ~c v c r ul toni es • elul eu lo ocvelo nmcnt of lh c h c ru; JII ~ lllld C(lS IS \l fii (IV!IO{' I: d OHIIHtfhl' lU
1 ~
136
Estimating Markets , Revenues, and Costs
Chap . 5
ing systems . Specifically, we addressed forecasting the market for new/ modified products; estimating revenues and costs, including consideration of learning curves; and dealing with inflation in the estimating process . Once monetary benefits and costs of investing in new and innovative technology are quantified , they need to be combined into economie measure(s) of merit. This is the important subject of Chapter 6 (before-tax analyses) and Chapter 7 (after-tax analyses, including replacement studies) .
REFERENCES 1. Chambers. J . C., S . K. Mullick, and D. D. Smith, "How to Choose the Right
2. 3. 4. 5.
6. 7. 8. 9. 10. Il. 12. 13. 14. 15.
Forecasting Technique," Harvard Business Review, July-August 1971, pp. 45-74. Corr, A. V., "The Role of Cost in Pricing," Management A ccounting , November 1974. DeGarmo, E. P., W. G. Sullivan, and J. R. Canada, Engineering Economy, 7 ed., Macmillan Publishing Company, New York, 1984. de la Mare, R. F ., Manufacturing Systems Economies, Holt, Rinehart and Winston, New York, 1982. Draper. N . R .. and H. Smith, App/ied Regression Analysis, John Wiley & Sons, !ne .. New York, 1978. Gilchrist, W., Statistical Forecasting, John Wiley & Sons, lnc., New York, 1976. Jones, B. W. , Inflation in Engineering Economie Analysis, John Wiley & Sons, !ne. , New York, 1982. Kleinfeld, l. H ., Engineering and Managerial Economies , Holt, Rinehart and Winston, New York, 1986. R. C. Lenz, Jr., Technologica/ Forecasting, ASD-TDR-63-414, Aeronautical Systems Division, Air Force Systems Command, June 1962. Malstrom, E. M., What Every Engineer Shou/d Know about Manufacturing Cos/ Estimating, Marcel Dekker, !ne., New York , 1981. Ostwald, P. F., Cost Estimating , 2nd ed ., Prentice-Hall, Inc., Englewood Cliffs, N .J., 1984. Porter, M. E., Competitive Strategy, Techniques for Analyzing Industries and Competitors, The Free Press, New York, 1980. Schweyer, H. E., Analytical Models for Managerial and Engineering Economies , Reinhold Publishing Corporation, New York, 1964. Sullivan , W. G .• and W. W. Claycombe, Fundamentals of Forecasting, Reston Publishing Co., !ne., Reston, Va., 1977. Vernon, 1. R. (ed .), Realistic Cost Estimating for Manufacturing; Society of Manufacturing Engineers. Dearborn, Mich., 1968.
Chap. 5
Exercises
137
16. Wheelwright, S. C., and S. Makridakis, Forecastinf( Methodsfor Management , 2nd ed ., John Wiley & Sons , lnc .. New York, 1977.
EXERCISES 5-1. List as many considerations as you can th at might affect the choice of forecasting techniques by a local company familiar to you . 5-2. If a company manufactures a "faddish" li ne of sporting goods equipment, how would its forecasting strategy differ from that of another company producing aluminum beverage containers ? 5-3. Explain how .the different stages of a product's !ife cycle will influence the choice of a forecasting strategy. S-4. (Section 5.2.3) Suppose that you own a small company that manufactures metal castings for severa! large automotive companies. Over the past severa! years you have found that quarter! y new-car sales tend to lag the prime interest rate by 3 months . You would like to make a forecast ofnext quarter' s car sales so that the size of your work force can be anticipated. There is a direct relationship between car sales and demand for castings that your company produces. The following data are gathered : lntere st Rate Year
2
3
Quarter
(% )
Next Quarter Sales ($M)
1 2 3 4
8.00 8.25 8.50 8.25
1 2 3 4 1 2 3 4
ln te rest Rate Year
Quarter
(%)
Ne xl Quarter Sales ($M)
$23 17 18 20
4
1 2 3 4
7.00 7.50 7.50 8.25
$25 26 17 20
7.75 7.25 7.70 7.25
21 25 24 29
5
1
8.75 8.50 7.50 7.00
18 22 23
7.50 7.75 7.25 7.00
24 23 26 30
6
2 3 4
:·.:
\'
L
1
;.j :tl :t
j
1\Jj ft
11
15
7.50
(a) Calculate a linear regression equation for these data, assuming that the interest rate is the independent variable . (b) Calculate the correlation coefficient. (c) Make a forecast of sales for the next quarter based on this quarter's prime interest rate of 7.50%. 5-S. (Section 5. 1.3) Consider the following ti me-series data of demand for a certain company's product.
\ ~ ·~
' '"
i1 138
Estimating Markets, Revenues, and Costs Mon th
4
5 6 7
8 9 10
Chap. 5
Demand (Booked Orders)
Mon th
Demand (Booked Orders)
3,1109 2,641 2.934 3.239 3,490 2,569 3,205 2,56 1 3,047 2,607
Il
3,387 3,138 2,908 3,512 3,291 2,804 3,096 3,106 3,195 3,605
12 13 14 15 16 17 18 19 20
(a) Plot these data on a piece of graph paper. (b) Apply single exponential smoothing to the data when a'= 0.20 and make a forecast for T = 1 month into the future . (c) Repeat part (b) when a' = 0.05 and T = 3 months.
5-6. (Section 5.2.4) Discuss the principal advantages and disadvantages of the Delphi method of forecasting . 5-7. (Section 5 .2.4) In your class, attempt torun a Delphi study to determine the priee of a compact disk player 3 years from now. Was group consensus af. fected by conducting two or three rounds of the procedure? 5-8. (Section 5.2.4) Try to think of sorne products that a re presently in the carly stages of a substitution curve effect. List them and try to estimate when the newer product will take more than half the market. 5-9. (Section 5.2.4) How could trend extrapolation be used to forecast future innovations in the aerospace industry? What performance characteristics do you believe are important here? 5-10. (Section 5.3) (a) Determine the priees for two products A and B, basing priee on a return on investment of 15%. (b) At the same priee as in part (a) , what is the markup based on (1) total costs. (2) incrementai costs, and (3) value added in production? Available data for a year's operation. with an effective income tax rate of 52% and 10,000 annual units of each product , are as follows: Product
Initi al investment . 1
A
B
s 90,000
$120,000
20,000 30.000 50 .000
40,000
1()(),00()
10 ,000
Cos t ~
Mnterin ls, ~ m ircct, Cn ther ~ . tl xed 1\!HII,
I.S.OOO 15.000
Chap . 5
''
Exercises
139
5-11. (Section 5.4.2) Use the factor technique to estimate the cost of installing a local area network in a factory environment having the following characteristics . One large building on a single leve! will require a total of 3,000 ft of coaxial (broadband) cable to network its six departments . Six network interface units (NIUs) will be required and a total of 50 taps will have to be made to connect ali the anticipated workstations and programmable deviees . Two modems are needed in addition to one network manager/analyzer that costs $30,000. The information necessary to make the estimate may be obtained from the worksheet shown . How accurate do you think such an estimate would be? Component 1. 2. 3. 4.
Interbuilding connections lntrabuilding connections Cable installation Equipment a. Broadband CA TV amplifier Taps Splitters NI Us Modems b. Baseband NI Us Repeaters Taps/transceivers c. Network manager Network analyzer
'(
Cost Estimating Relationship $100- 150 per foot $20-50 per foot $20 per foot
x x x
$500- 1500 $17- 20 each $5- 15 $500-$1 .000 per port $1,000 each
x x x __ = __ x __ = _ _ x __ = __
$600 per port SI ,200-1 ,500 each $200- 300 each $10,000-30,000 $30,000
x x x __ = __
1
'
5-U. (Section 5.4.2) A residential builder just finished constructing a 3,000-ft 2 home for $96,000. This cost did not include the lot or utility access fees . A detailed breakout of costs for this job is as follows :
1 j: ,. \.
Item
Fraction of Finished Cost
Lumber and carpentry Electrical wiring Plumbing Concrete and masonry Wall board Flooring Foundation preparation A cce~so ri es and npplin nces Heuti ng and air contlilionlng
0.20 0 . 10 0. 14 0.09 0.04 0.06 0.02 0 .0
RoonnR l'ul nt ln M ~ ~~ nllllriOUIIN
O.llY () , 1()
1) ,()/
tl 01
1 IMI
\
L\.
\,t
140
Estimating Markets, R evenues, and Costs (a) What is the unit cost for the just-finished home? (b) If a 4,000-ft 2 home is to be built, es ti mate the total cost from the answer to
part (a) and compare it with the total of estimated item costs based on the breakout given above . 5-13. (Section 5.4.3) Develop the standard time to perform a manual assembly task having the following elements and compute the standard number of assemblies that can be completed each hour.
Element 1 2 3
Normal Time (min)
Allowance
4.2
15 15 15
1.7
2.9
Exercises
1(
(b) If a person had take-home pa y of $20,000 in 1978, how much should it have been in 1981 in order for that person to be living comparably to 1978? (c) If, during 1978-1981 a person was earning 8% through investments (such as moncy-market accounts}, what would have been his real rate of return? S-19. (Section 5.4) (a) It is desired to estimate the 1990 construction cost of a new plant for which 1985 component costs, and applicable rates of inflation, are as follows :
(%)
5-14. (Section 5.4.3) For a particular activity that is being work sampled, determine the number of observations necessary to ens ure a confidence interval of 95% and a maximum interval of 5% . The observed proportion of occurrence of this activity is 0.4 (i.e . , 40% of the random observations have identified this activity in progress) . Next, repeat this exercise when the confidence interval is lowered to 90%. 5-15. (Section 5.4 .3) In a production run of R-130jet engines, the cumulative time to produce units 1 through 100 is 80,000 hr. (a) Calculate the unit time for the fiftieth engine with a 71 % leaming curve in effect. (b) Calculate the cumulative time to produce units 1 through 40 with a 71% leaming curve. 5-16. (Section 5.4.4) A 100-kW diesel generator cost $140,000 seven years ago when a certain equipment cost index was arbitrarily set at 100. A similarly designed generator rated at 150 kW is now being proposed and the cost index is 140. The cos! exponent factor. {3. is O.7 for this type of equipment. (a) Determine the estimated cost of the proposed generator by using the approptiate cost estimating relationship. (b) Repeat part (a) when {3 = 0.4 . 5-17. (Section 5.4.4) The Neptune Manufactuting Company is consideting abandoning their old plant , built 23 years ago, and constructing a new plant that has 50% more square footage . The original cost of the old plant was $300,000 and its capacity, in terms of standardized production units, was 250,000 units per year. Capacity of the proposed plant is to be 500,000 units per year. During the past 23 years, costs of plant construction have risen by an average of 5% per year. If the cost exponent factor is 0.8, what is the estimated cost of the new plant ? 5-18. (Section 5 .5) Answer the following questions based on the Consumer Priee Index in Table 5-10 . (a) What is the annual compounded rate of inflation in living costs from 1978 through 1981 ?
l
141
La bor Building materials Equipment Total
19!!5 Cost (Millions)
Projected Annual Inflation Rate ( 1985 through 1990) (%)
$1
10
5 3
15
,, ~
f j
1 ~. !
0 i
$9
(b) Using the answer to part (a) and using a (combined) MARR of 25% , find the equivalent worth of the 1990 construction cost as of 1985 . S-20. (Section 5.5) Estimate the salary you hope to be making 5 years from now, stated in today's purchasing power (i.e. , RS\0l) . If the average inflation rate for goods and services you expect to bu y is 10% compounded annually, wh at will your hoped-for salary need to be in actual dollars (i .e., A$ 5 )? S-21. (Section 5.5) (a) You hope to prepare for someone's college education costs by depositing enough in a bank savings account today to provide, as of 15 years from now, an amou nt equivalent to $20,000 of today's purchasing power. If the bank will pa y 8% and the estimated inflation rate is 5% (both compounded annually), what lump sum should you deposit today ? (b) Repeat part (a) except that the estimated inflation rate is 10% per year . S-22. (Section 5.5) Labor costs for alternatives X and Y are estimated on different bases as follows: End of Year n
Ait. X (AS.)
Ait. B (RS~"J
1 2
S200M 210M 220M 230M
S190M 190M 190M 190M
3
4
If the combined (real and inflation) MARR = 25% and the real MARR 15%/yr, show which alternative has the lower equivalent : (a) Present worth of costs at year O. (b) Future worth of costs at year 4.
l
-
vv
Chapitre 2
INTRODUCTION À LA NOTION DE RISQUE ET D'INCERTITUDE
2.1
INTRODUCTION: Dans le monde de l'industrie et du commerce les décideurs sont conscients qu'ils n'auront pas toujours raison quand il s'agit d'évaluer un projet ou de choisir entre plusieurs possibilités. En effet, le profil d'un «Cash flow» est influencé par les événements futurs (international, national , local .. .) qui sont incertains et qui ne peuvent être contrôlés par le décideur. Il n'est pas possible de prévoir à l'avance l'arrivée de tels événements et la rentabilité d'un certain projet ne peut être évaluée avec certitude, qu'à la fin de celui-ci. Donc, la rentabilité d'un projet donné dépend d'un facteur risque qui peut entraîner qu'elle sera différente de celle prévue au moment de la prise de décision. Nous avons considéré, jusqu'à maintenant dans toutes les études économiques effectuées, connaître tous les éléments avec certitude; nous pouvions donc définir la valeur d'un paramètre économique par un seul nombre. Mais en réalité dans une situation concrète ces paramètres: vie utile, valeur de récupération, revenus et coûts ne peuvent être déterminés par une valeur unique. Pour plusieurs études économiques, il devient nécessaire ou préférable de considérer l'aspect relié aux risques et à l'incertitude introduit par la variation des différents paramètres. Dans les chapitres 3,4 et 5, nous allons voir différentes techniques pour tenir compte du risque et de l'incertitude dans une analyse économique.
2.2
QU'EST-CE-QUE LE RISQUE ET L'INCERTITUDE Avant de passer aux différentes techniques d'analyse économique avec risque et incertitude, nous tenterons d'éclaircir cette notion dans les sections suivantes: ·
2. 2.1 Terminologie Les situations de décision pouvent être classées en trois catégories: certitude, risque et incertitude. Toutes les études économiques qui ont été conduites dans les chapitres précédents ont été réalisées dans des conditions de certitude, c'est-à-dire que la probabilité de succès a été considérée égale à 1.0 pour chaque projet évalué (fig. 2.1 a) . Cependant en réalité, les exemples où les paramètres sont estimés avec certitude sont rares. On parle alors de risque ou d'incertitude.
2-2 Les analystes présentent une distinction classique entre le risque et l'incertitude pour un paramètre ou une analyse économique; par le risque, l'analyste connaît les probabilités que les états de la nature, les revenus et les coüts se produisent (figure. 2.1 b) alors qu'en présence d'incertitude il ne possède pas la connaissance de la fréquence de leur distribution (figure. 2. l.c). La figure 2.1 montre graphiquement ces différences .
ŒJrnl\.Œ
ROCl.E
inœtitu:E
j_ m a=3
\Ae\Jile 7 friS (a)
\Ae Uile
?
~; \Ae\Jile
V" 7 friS
(b)
(c)
figure 2.1
D'autres auteurs différencient le risque de l'incertitude en affirmant que pour le risque ils peuvent utiliser la distribution de la probabilité d'un paramètre alors que pour l'incertitude ils n'ont .pas confiance que l'estimé de cette probabilité soit correcte. 2.2.2. Causes du risque et de l'incertitude Nous allons maintenant considérer quelques unes des principales causes du risquee d l'incertitude:
1.
Un nombre insuffisant d'investissement similaire Chaque firme a un faible nombre d'investissement d'un type particulier. L 'e des revenus défavorables peut difficilement être statistiquement compensé par les revenus favorables; ce phénomène devient critique si une compagnie in es ·. un montant considérable par rapport à ses ressources totales.
2-3
2.
Erreurs subjectives dans les données L'analyse économique dépend de l'optimisme ou du pessimisme de l'analyse, ou des facteurs intangibles qui ne devraient pas influencer l'analyse économique.
3.
Changement dans l'environnement économique invalidant l'expérience précédente Il arrive souvent que l'analyste fait ses prédictions futures à partir des données accumulées précédemment mais sans ajustement pour les conditions futures prévues. Ceci implique qu'il assume implicitement que le futur se déroulera comme le passé; parfois des circonstances peuvent invalider cette hypothèse, par exemple la crise d'énergie.
4.
Mauvaise interprétation des données Différents facteurs influençant les données peuvent conduire à une mauvaise interprétation.
5.
Erreurs d'analyse L'analyse technique et économique peut comporter des erreurs.
6.
Disponibilité d'administrateurs compétents La limitation des ressources humaines et physiques à l'intérieur d'une
organisation peut handicaper certains projets relativement à d'autres. 7.
Récupération de l'investissement Une industrie qui va se procurer de l'équipement très spécialisé doit se rendre compte qu'il pourra être reveridu avec beaucoup plus de risque que de l'équipement d'application générale.
8.
Désuet De nouveaux procédés peuvent rendre l'équipement actuel désuet; ce dernier facteur constitue aussi une cause plus ou moins grande de risque.
2
De Garmo et Canada C) dans leur livre, regroupent en quatre parties les cau risque et de l'incertitude: a)
L'imprécision possible des valeurs utilisées dans l'étude économ1 l'estimation des revenus entraîne généralement le plus d'erreur.
b)
Le genre d'industrie considéré; par exemple une entreprise minière co généralement plus de risque qu'une épicerie.
c)
Le type d'usine et d'équipement; de l'équipement spécialisé se re end difficilement que de l'équipement régulier.
d)
Le temps qui doit s'écouler avant que toutes les conditions de l'étude .so·=-: réalisées; un temps plus long diminue la probabilité que tous les a< demeurent tels qu'estimés, donc le risque augmente puisqu'on ne prophétiser exactement ce que l'avenir apportera. La méthode de la péri recouvrement fournit une indication de ce niveau de risque.
2.2.3 Faiblesse du traitement probabiliste Les probabilités utilisées dans une analyse du risque constituent généralem éléments non vérifiables et très subjectifs. De plus, les différentes probabilités u · · ' peuvent dans une même étude économique différer en qualité et en quantité entree es. L'anal yste se doit de réaliser que l'utilisation de probabilités ne fait pas disparaî risque et l'incertitude mais fournit des informations supplémentaires qui aidero · à prise de décision. L'utilisation des limites de confiance des probabilités aidera · gestionnaire à prendre une décision. Pour les problèmes industriels les ex recommandent d'utiliser une limite de confiance de 95%; c'est-à-dire d 'ajoute valeur moyenne plus ou moins deux fois la déviation standard (2a). 2.2.4 Moyens de changer le degré d'incertitude Une firme peut prendre différents moyens pour changer son degré d' ince · mentionnons: 1)
Obtenir des informations techniques et économiques supplémentaires.
2)
Augmenter le nombre d'activités pour avoir suffisamment de pro d'investissement pour que les valeurs moyennes utilisées soient statistique valables.
3)
Diversifier les opérations; choisir des produits que les cycles économiq influencent différemment.
2-5 2.2.5 Choix entre différents niveaux de risques et retours sur l'investissement Les économistes acceptent généralement qu'un projet plus risqué doit rapporter un retour sur l'investissement plus élevé. La connaissance d'une relation entre le risque et le retour permettrait de comparer tous les projets sur une même base. Malheureusement cette relation ne peut être déterminée d'une façon générale et précise; la politique d'une firme par rapport au risque doit donc être connue avant qu'une décision d'investir puisse être prise. Cette politique dépend grandement des préférences du responsable décisionnel et du risque total auquel la firme doit déjà faire face. 2.2.6 Guide pour considérer le risque et l'incertitude Le risque et l'incertitude existent sous une variété de forme et de degré dans toutes les analyses économiques. L•importance de considérer le risque et l'incertitude pour un investissement augmente, avec l'augmentation de la complexité du projet et de son importance économique relativement à la valeur totale de la firme. Une analyse économique supplémentaire tenant compte du risque et de l'incertitude se justifie si elle coûte moins que les économies potentielles qui en résulteraient; en pratique leurs estimations deviennent difficiles. En général, lorsqu'une firme désire effectuer une étude économique, soit pour un investissement ou un remplacement, elle considérera les différentes étapes telles que le montre la figure 2.2. Ces étapes s'appliquent aussi bien à des projets mutuellement exclusifs ou non. Elle fait voir l'existence de quatre points où les projets peuvent être acceptés ou rejetés. Le point 1: comprend l'ensemble de l'argent que la compagnie devra payer si elle accepte le projet. Elle effectuera un choix immédiatement si les montants pour chaque projet demeurent faibles comparés à ceux pour une étude supplémentaire. Ce critère subjectif dépend de l'état financier de l'entreprise, de la grosseur des projets généralement considérés et de la disponibilité des ressources pour une analyse plus poussée. Le point 2: considère le rapprochement des choix; si l'un se détache nettement la compagnie arrêtera sa décision. Le point 3: exan1ine les analyses déjà effectuées en fonction de l'importance relative du projet pour la compagnie, elle peut alors décider de faire une étude plus précise de certains éléments. Le point 4; la compagnie répète les considérations du point 3.
2 -6
2.3
DÉCISION SOUS «INCERTITUDE COMPLÈTE» Les différentes techniques d'analyse sont utilisées selon qu'on soit dans le cas de risque ou celui d'incertitude. Dans le présent chapitre nous allons développer les techniques d'anal se économique dans le cas d'un investissement sous incertitude COMPLÈTE c'est-à-dire dans une situation ou l'analyste ne peut pas assumer des probabilités aux différents états de la nature. Supposons qu'une décision doit être prise entre plusieurs possibilités A1 , A2 , ••• ~;le décideu désire choisir la meilleure. n y a rn conditions futures, sous appelées «état de la nature» SI s1 ... Sm. Ces états peuvent survenir indépendamment du contrôle de celui qui doit prendre la décision. Prenons P;i = Valeur pour celui qui prend la décision s'il choisit la possibilité A; et qu 'en même temps, l'état de la nature Si se produit. Supposons que les valeurs positives indiquent des situations favorables alors que celles négatives en montrent des non-favorables. Ces valeurs peuvent être des revenus, des valeurs présentes, des utilités ou tout autre indice caractérisant un investissement. Les différents choix et états de la nature peuvent être présentés sous la forme d'une matrice, tableau 2.1 illustre une matrice de décision.
Tableau 2.1: matrice de décisions ÉTATS DE LA NATURE 'aucun contrôle du décideur Possibilités (contrôlé par le décideur)
sl
s?
---
SI
---
SM
A
p
Pn
---
p.
---
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A?
p?l
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---
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---
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c
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---
1
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1
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1
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2-7
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déçerœs reca m OOées p:u u-e amy1yoo éiectrori~
2-9
Maintenant considérons l'exemple présenté dans le tableau 2.2
Tableau 2.2 Exemple d'une matrice de décision ÉTATS DE LA NATURE A\S. 1 J Possibilités (actions)
récession
prospérité normale
grande prospérité
Sl
S2
S3
Aucune construction
Al
+25
-15
-50
Construire une petite usine
A2
-50
+40
+50
Construire une grosse usine
A3
-150
+10
+200
Une analyse économique sous certitude comporte un seul état de la nature c'est-à-dire une seule colonne. La meilleure possibilité consiste à prendre la valeur la plus élevée dans la colonne unique (ex: en cas de récession (S 1) aucune construction (A 1)). Mais, maintenant dans une analyse sous incertitude avec plusieurs états de la nature donc plusieurs colonnes, quelles possibilités choisir? En premier, éliminons toutes les possibilités qui mdépendamment du principe de choix que le décideur utilisera montrent des infériorités évidëntes, et ensuite, on procédera au choix entre les autres possibilités en tenant compte de l'incertitude. Exemple 2.1 Considérons la matrice du profit du tableau 2.3 qui représente la valeur présente des profits de projets (A 1 , A2 , A3 , A4 et A5) selon quatre états de la nature (S 1 , S2 , S3 et S4 ).
2- 10
Tableau 2.3: matrice de profits ÉTATS DE LA NATURE Possibilités
s
s?
S,
s~~
A
4
4
0
2
0
4
A?
2
2
2
2
2
2
A,
0
8
0
0
0
8
A4
2
6
0
0
0
6
As
0
1
1
0
0
1
minimum · maximum
L'analyste se rend immédiatement compte par inspection de la matrice qu'indépendam men e l'état de la nature qui surviendra, il aura toujours avantage à choisir la possibilité A2 au li de As. Il peut donc éliminer immédiatement la possibilité As de toutes considérations futures. On se réfère à cette situation en mentionnant que la possibilité A2 domine la possibilité A5 • D'une façon générale s'il y a deux possibilités Ag et Ah tel P gj ~phi pour toutes valeurs de · alors la possibilité Ah domine la possibilité Ag qui peut-être éliminé de cette façon. ll restera. plusieurs possibilités et l'analyste aura besoin de procédures supplémentaires pour faire choix. 2.3.1 Étude de quelques principes sous «incertitude>> Nous allons maintenant considérer différents principes de choix que les experts no suggèrent. Nous discuterons aussi des points forts et des faiblesses qu 'on principes. 2.3.1.1
Principes de l'opportunité égale Les analystes se réfèrent souvent à ce principe sous les noms de <
2-11
La possibilité A1 a donc une valeur espérée égale à: E(A 1)= IAx4+ 1Ax4+ IAxO+ 1Ax2=2.5
De la même façon les autres possibilités ont comme E(A0=E(A 3)=E(A4)=2.0
v~eurs
espérées:
Le choix se porte donc sur la possibilité A1 qui permet d'avoir un profit espéré maximal de 2.5.
2.3.1.2
Principe maximin ou minimax Ce principe consiste à choisir la possibilité qui maximise la valeur minimale qu'on pourrait recevoir ou minimiser la valeur maximale qu'on pourrait débourser. En d'autres mots, l'analyste choisit la possibilité Ai de la façon suivante dans le cas d'une matrice de profits. maximum
minimum Pij J
Ce principe reflète une attitude très pessimisme et conservatrice; en effet, il suppose qu'indépendamment de la possibilité choisie, l'état de la nature le plus défavorable va se produire. Dans le cas d'une matrice contenant des coûts au lieu des profits, le principe consistera à choisir l'action qui minimisera le coût maximal. Le principe s'appellera alors minimax: minimum
maximum p IJ.. j
Dans l'exemple 2.1, le principe maximin choisit la possibilité A2 laquelle correspond le maximum (2) des minimums de (0,2,0 et 0) respectivement pour les possibilités (A 1 , A2 , A3 et A4 ), (tableau 2.3). 2.3.1.3
Principe maximax ou minimin Ce principe consiste à choisir la possibilité Ai, qui maximise la valeur maximum des profits, c'est-à-dire l'analyste choisit la possibilité Ai associée avec le maximum Pij dans le cas d'une matrice de profits. Dans le cas d'une matrice de coûts, ce principe permet de sélectionner la possibilité qui minimise le coût minimum. minimum minimum Pij i j Cette règle optimisme et aventureuse suppose que, indépendamment de la possibilité choisie, le meilleur état de la nature possible associé à cette possibilité va arriver.
2- 12 Pour l'exemple 2.1, le principe maximax choisit la possibilité A3 à laquelle correspond le maximum (8) des maximums de (2,8,6,et 1) respectivement pour les possibilités (A 1 , A2 , A3 et A4) , (Tableau 2.3). 2.3.1.4
Principe d'Hurwicz Hurwicz a généralisé les deux principes ci-dessus pour tous les niveaux d 'optimisme entre le pessimisme extrême de MAXIMIN et l 'optimisme exagéré de MAXIMAX . L'analyste choisit une indice d'optimisme o< a< 1. Alors pour chaque possibilité il calcule la relation:
Hi = a max Pii+(1-a) min Pii 1 1 Dans le cas de matrice de profits, on choisit la possibilité qui maximise Hi et dans le cas de matrice de coûts, on choisit la possibilité qui minimise Hi, cette relation se définie comme le principe d 'Hurwicz. Aux deux limites lorsque le degré d'op ti mis o égale zéro et un , on retrouve respectivement les principes maximin et maximax .
a=O=>Hi= min Pii j = max min P ..IJ =>maximin i j
a=1 =>Hi= max Pii j Max Hi => max max Pi(>maximax i j La figure 2.3 nous fait voir le principe d'Hurwicz appliqué à l'exemple 2 . 1.
2-13
Valeur du critére d'Hurwicz
86
~ A3A4
4
A1
~ ~
A2
0.25 Indice d'optimiste: a
Supposons un degré d'optimisme a=0.25 dans l'exemple 2.1 H 1 pour H 2 pour H 3 pour H4 pour
A 1 = 0.25 x 4 A2 = 0.25 x 2 A3 = 0.25 x 8 A4 = 0.25 x 6
+ O. 75 x 0 + 0.75 x 2 + O. 75 x 0 + O. 75 x 0
= = = =
1 2 = > choisir A2 ou A3 2 1.5
Dans l'exemple 2.1, le choix se portera sur la possibilité A2 pour des degrés d'optimisme (a) inférieur à 0.25. Alors qu'il choisira A3 pour des degrés supérieurs à 0.25. Pour un degré d'optimisme égale à la valeur 0.25, le principe n'indique aucune préférence entre les possibilités A2 et A3 • Dans la relation d'Hurwicz, le degré d'optimisme a influencera les profits maximaux et les coûts minimaux à moins de l'utilisation d'une autre convention.
2.3.1.5
Principe du regret minimax (Savage) Considérons de nouveau l'exemple 2.1, si après qu'une possibilité ait été choisie et les mesures prises pour son application, le·décideur constate que l'état de la nature S2 se produit, il aimerait certainement avoir choisi la possibilité A3 • S'il avait dans les faits choisi une autre possibilité, disons A2 il regretterait son choix actuel. Pour un état de la nature donné la mesure du regret qu'il peut avoir; s'appelle «REGRET». Pour une matrice content ~ possibilité et Sm états de la nature les regrets pourront être calculés . par la re1atwn
max {Pii}-Pi1=Ri1.. '
L'application de la définition du regret à une possibilité A2 et l'état de la nature S2 de l'exemple 2.1, donne un regret de R22 =8-2=6. Considérant d'une façon similaire les
2-14
15 autres combinaisons possibles (Ai, Sj) de la matrice de cet exemple l'analyste pourra construire la «matrice du regret», (tableau 2.3). Tableau 2.4: matrice de regrets ÉTAT DE LA NATURE Possibilités
sl
s2
s3
s4
Regret maximum pour pour la possibilités A
A1
0
4
2
0
4
A?
2
6
0
0
6
A3
4
0
2
2
4
A4
2
2
2
2
2
Dans cet exemple, le choix de la possibilité A4 minimisera le regret maximal .
. 2.3.2 Problème de choix de principe En résumé les différents principes considérés jusqu'à maintenant choisissent: L'opportunité égale - > A 1 Hurwicz pour a< 0.25- > A2 Hurwicz pour a> 0.25- > A3 Regret minimax - > A4 Nous avons donc converti le problème de choisir une possibilité en celui de choisir principe. Pour faciliter ce choix, considérons maintenant quelques unes des propriétés de ces principes.
2.3.2.1
Problèmes dans l'utilisation du principe d'opportunité égale Si l'on utilise le principe d'opportunité égale, le choix d'une possibilité est sensibl 2 la façon dont les différents états futurs sont présentés. Considérons l'exemple suivant: Exemple 2.2 On désire choisir entre deux pièces d'équipement; l'une étant d'application générale, l'autre ayant un emploi bien spécifique. Or, on peut obtenir trois co possibles pour utiliser la machine. Il est aussi possible que l'on ait aucun contra
2-15 ce cas la machine d'application générale pourra être utilisée pour une autre application, mais la machine d'application spécialisée demeurera inutilisée. Donc la liste des états futurs possibles: S1 S2
aucun contrat un contrat
A1 A2
on achète la machine spéciale on achète la machine générale
La matrice, présentée dans le tableau 2.5, montre les profits (pertes) correspondant à chaque possibilité et état de la nature; avec le principe d'opportunité égale p 1 =p2 =lf2
Tableau 2.5: matrice de profits (2 états de la nature) ETAT DE LA NATURE Possibilités
s
s?
A
-1
6
A?
1
5
Le profit espéré résultant du choix de chacune des actions égales: (p 1 =p2 = 1/z)
E{A 1) = 1/z(-1)+ 1/z{6) =2.5 E(A2)= 1/z ( 1)+ 1/z(5) =3.0 Ce principe permet de sélectionner la possibilité A2 , car elle a un profit espéré (3.0), supérieur
à celui de la possibilité A1 (2.5) . Maintenant, supposons que 1'analyste veut solutionner le même problème en présentant les états de la nature (obtention d'un contrat) de la façon suivante: S1 S2 S3 s4
aucun contrat contrat A contrat B contrat c
La matrice des profits devient celle présentée au tableau 2.6
2-16
Tableau 2.6: matrice de profits (4 états de la nature) ÉTAT DELA NATURE Possibilités
SL
S,
s3_
sd.
A1
-1
6
6
6
A,
1
5
5
5
Le principe d'opportunité égale nous donne donc maintenant:
Les profits espérés des deux possibilités deviennent: E(AI) = 1A(-1)
+
1
A (6)
+
E(A2) = lA( 1)
+
1
A (5)
+ 1A (5) +
A (6)
1
+
A (6) = 4.25
1
1,4 (5)
= 4.00
Maintenant avec le principe d'opportunité égale l'analyte choisi la possibilité A1, car elle génère un profit espéré de 4.25 supérieur à celui de la possibilité A2 • Pour éviter cet inconvénient, l'analyste devrait établir la liste des «états de la na de façon que chacun ait la même probabilité d'arriver. Ceci implique qu'il conruu d'avance les probabilités, alors que sous «incertitude complète>> il assume ne pas les connaître. 2.3 .2.2
Problème dans l'utilisation du principe d'Hurwicz Considérons maintenant des exemples pour illustrer deux problèmes que le principe d'Hurwicz engendre le tableau 2. 7 illustre une matrice des profits. Exemple 2.3 Soit la matrice des profits (tableau 2. 7).
2-17
Tableau 2.7: matrice de profits ÉTATS DELA NATURE Possibilités
SI
s2
max Pu
.i
min p ..IJ j
A1
1
4
4
1
A?
3
3
3
3
H 1 = a(4) +(l-a) 1= 3 a+l H 2 = a(3) + (l-a) 3= 3 Le choix de la possibilité A1 donne selon le critère d 'Hurwicz les profits de 3a + 1, alors que pour celui de la possibilité A2 , il donne un profit constant de 3, qui ne dépend pas du degré d'optimisme a.
Égalant ces deux équations, 3a+ 1 =3l'analyste trouve une valeur de 2/3 pour le degré d'optimisme a au point d'intersection. Le choix se porte donc sur la possibilité A2 pour un degré d'optimisme inférieur à 2/3 , car ces profits égalent à 3, alors que ceux de la possibilité A1 demeurent inférieurs, car il égalent 3a + 1. Maintenant, ajoutons une constante 5 à chaque élément de la première colonne de la matrice du tableau 2.7; le tableau 2.8 fait voir la nouvelle matrice.
Tableau 2.8: matrice de _Qrofits
1
1
ÉTATS DE LA NATURE Possibilltés
SI
s2
max p ..IJ
min Pu j
.i A
6
4
6
4
A?
8
3
8
3
Le critère d 'Hurwicz permet de calculer les profits pour
la possibilité A1 : H 1 de a(6) + (l-a) 4= 2 a+4 la possibilité A2 : H2 de a(8) + (l-a) 3= 5 a+3
2-18
Le point d'intersection devient:
5a+3 = 2a+4 La solution de cette équation donne une degré d'optimisme a, égale à 113 , donc po toute valeur de a inférieur à 113, le choix se déplace de la possibilité A2 à la possibilité A 1 par rapport à la solution précédente, car 2a+4 demeure supérieur à 5a+ 3. Les experts réfèrent à ce problème comme à un manque de «linéarité de la colonne- . Si le futur S 1 se produit, l'avénement de l'état de la nature S 1 procurera cinq unités supplémentaires de profit indépendamment de la possibilité choisie. Le changemen de la possibilité A2 à A 1 dépend donc, seulement de l'état de la nature et non pas de la possibilité choisie; idéalement la décision prise ne devrait pas être influencée. Le principe du regret minimum et de 1'opportunité égale possèdent la linéarité d la colonne. Exemple 2.4 Considérons la matrice des profits présentée dans le tableau 2.9, pour illustrer deuxième problème associé au principe d'Hurwicz: Tableau 2.9: matrice de profits ÉTATS DE LA NATURE Possibilités
sl
s?
s~
s
s~
S,.,
s7
SR
SQ
slO
A1
1
0
0
0
0.
0
0
0
0
0
A?
0
1
1
1
1
1
1
1
1
1
L'analyste qui utilise le critère d'Hurwicz considérera les possibilités A 1 et A2 co équivalentes, car pour:
H1 =a(1) +(1-a) O=a H2 =a(1) +(1-a) O=a
e
et pour
Mais la plupart des gens auront une préférence intuitive pour la possibilité _ impliquant une notion préconçue au sujet des probabilités, ce qui ne peut être le cas lorsqu'on traite d'un problème sous incertitude complète. Pour cet exemple l'anal qui utilise le principe d'opportunité égale choisira la possibilité A2 •
2-19 2.3.2.3
Problèmes associés au principe du regret minimax Savage a établi ce principe pour contourner le très grand conservatisme du principe MAXIMIN (ou MINIMAX). Le tableau 2.10, illustre une matrice de profit où l'utilisation du principe de Savage a un avantage sur celui du principe maximin.
Tableau 2.10: matrice de profits . ÉTATS DE LA NATURE Possibilités
SI
s2
min p IJ.. j
A,
0
1000000
0
A,
1
1
1
L'analyste qui utilise le principe maximin choisit la possibilité A2 , mais intuitivement la possibilité A1 semble plus désirable, mais comme on peut le voir à partir de la matrice du regret montrée au tableau 2.11, le principe du regret minimax choisit la possibilité A1 •
Tableau 2.11: matrice de regrets ÉTATS DE LA NATURE Possibilités
s_j
s2
Regret maximum
A,
1
0
1
A"
0
999999
999999
Par ailleurs, le principe du regret minimax a aussi des problèmes propres;
Exemple 2.6 Les tableaux 2.12 et 2.13 illustrent respectivement une matrice des profits etla matrice de regrets correspondants.
2-20
Tableau 2.12: matrice de profits ÉTATS DE LA NATURE Possibilités
s,
s2
s3
A,
1
6
4
A?
5
3
6
Tableau 2.13: matrice de regrets ÉTATS DE LA NATURE Possibilités
s,
s2
s3
regret maximum
A,
4
0
2
4
A?
0
3
0
3
L'utilisation du principe du regret minimax permet de choisir la possibilité Maintenant, supposons qu'une 3e possibilité A3 devient possible. La matrice des pro est présentée dans le tableau 2.14 et celle des regrets par le tableau 2.15.
Tableau 2.14: matrice de profits ÉTATS DELA NATURE Possibilités
S,
s?
S,
A.,
1
6
4
A?
5
3
6
A,
4
8
1
~.
:.s
2-21 Tableau 2.15: matrice de regrets ÉTATS DE LA NATURE Possibilités
SI
s2
s3
regrets maximum
A,
4
2
2
4
A?
0
5
0
5
A_3
1
0
5
5
L'ajout d'une troisième possibilité A3 , entraîne que l'analyste choisit maintenant en utilisant le principe de regret minimax la possibilité A 1 • Ce résultat désappointe, car le fait de considérer une nouvelle possibilité ne devrait pas changer le choix fait parmi les possibilités précédemment considérées. L'exemple suivant illustre ce point. Une personne désire acheter une nouvelle auto après avoir considéré les modèles Ford et Chevrolet, elle choisit un modèle Ford. Avant d'avoir complété la transaction, elle apprend qu'un nouveau détaillant des modèles Dodge vient de s'installer près de chez elle. Elle va donc voir ces autres modèles pour déterminer s'ils présentent des avantages par rapport au modèle Ford choisi. Mais voilà qu 'après avoir vu les modèles Dodge, cette personne se ravise et décide maintenant de choisir une auto de modèle Chevrolet. L'utilisation du principe du regret minimax dépend donc de l'addition d'une nouvelle possibilité qui peut-être ou non pertinente. Les principes d'opportunité égale et d 'Hurwicz ne possèdent pas ce problème.
2.3.3 Choix d'un principe Plusieurs auteurs ont listé les propriétés désirables qu'un principe devrait avoir. L'une des plus connue a été faite par Milnor (2). Il liste les dix conditions auxquelles on aimerait qu'une bonne règle de décision respecte: 1.
Classification: Classifier les diverses actions.
2.
Symétrie: La classification devrait être indépendante de l'ordre des colonnes et des rangées de la matrice.
2-22 3.
Forte domination: L'analyste préfère la possibilité A; à Ak, s'il préfère la valeur P;i à Pki pour tous lesj.
4.
Continuité: Dans une matrice convergente, la classification ne doit pas être inversée à la limite.
5.
Linéarité: La classification ne devrait pas changer si l'on remplace les valeurs de la matrice (P;i) par a Pij + ~ ou a> 0
6.
Addition d'une rangée: L'addition de nouvelles possibilités, i.e. de nou e es rangées à la matrice, ne devrait pas changer la classification des possibilités.
7.
Linéarité de la colonne: L'addition d'une constante à tous les éléments d' colonne ne devrait pas changer la classification des possibilités.
8.
La duplication d'une colonne: L'addition à la matrice d'une nouvelle colonn
identique à une existante ne devrait changer la classification des possibilités. 9.
Convexité: Si les possibilités A; et Ak s'équivaillent aucune n'est préférée .
2.
Addition d'une rangée: L"addition d'une nouvelle rangée où aucun élémen de cette rangée n'est préféré aux éléments correspondants de toutes rangées d ,. existantes ne devrait pas changer la classification des possibilités.
Milnor (2) a prouvé qu'aucun des principes discutés précédemment ne satisfait les dix conditions. On peut voir facilement que la règle de Laplace viole la condition Minimax la condition 7, Hurwicz la condition 7 et 9 et Savage la condition 6. Milnor a aussi prouvé mathématiquement que le principe: de Laplace satisfait les conditions 1, 2, 3, 6 et 7 Minimax satisfait les conditions 1, 2, 3, 4, 6, 8 et 9 d'Hurwicz satisfait les conditions 1, 2, 3, 4, 5, 6 et 8 de Savage satisfait les conditions 1, 2, 3, 4, 7, 8, 9 et 10 Pour choisir un principe, l'analyste doit donc d~ider des propriétés les plus importan pour un cas bien particulier et choisir celui qui en satisfait le plus. La liste d propriétés ne peut être très longue.
2-23
Problème no 2.1 Établir un principe personnel qui vous permet de prendre une décision sous «incertitude complète». Monter un exemple comment ce principe peut-être utilisé. Quels sont les avantages et inconvénients de votre principe?
Problème no 2.2 Appliquer les différents principes de choix que nous avons vu aux cours à la matrice de coûts suivante: ÉTATS DE LA NATURE Possibilités
S,
s?
s1
s"
A1
18
18
10
14
A?
14
14
14
14
A1
5
26
10
10
A"
14
22
10
10
A_5_
10
12
12
10
Problème no 2.3 Former une nouvelle matrice en prenant chaque coût de la matrice du problème 2.2 et en lui additionnant 2 et en multipliant le résultat par 3. Utiliser les mêmes principes de choix qu'au problème 2.2. Qu'est-ce-que ces résultats vous suggèrent sur les propriétés de différents principes?
Problème no 2.4 Dans la matrice de profits ci-dessous, quelle possibilité choisiriez-vous en utilisant le principe du regret minimal?
2-2 ÉTAT DE LA NATURE Possibilités
s,
s2
s3
A
8
7
4
A?
10
0
4
A,
1
9
5
At
5
6
7
1 . De Garmo et Canada, Engineering Economy
Chapitre 3 ANALYSE ÉCONOMIQUE TRADITIONNELLE ET AVANCÉE SOUS RISQUE ET INCERTITUDE
3.1
INTRODUCTION: Dans ce chapitre nous allons développer d'autres techniques d'analyse économique avec risque et incertitude qu'on classera en deux grandes catégories soit, celles d'analyse traditionnelle et celles d'analyse avancée. L'accent sera mis sur l'analyse économique avec risque, c'est-à-dire le cas où l'analyste connaît les probabilités des divers paramètres économiques, notamment, que les états de la nature, les revenus et les coûts se produisent.
3.2
BREF RAPPEL DES NOTIONS DE BASE DE PROBABILITÉS La probabilité peut-être définie comme le rapport du nombre de cas favorables à l'événement
sur le nombre de cas possibles. Si un événement A peut survenir «S>> fois à partir d'un nombre total de «n>> cas possibles et également identiques, la probabilité que l'événement survienne sera de:
P(A)
=
s n
où P(A) = probabilité que l'événement survienne s = nombre de cas où l'événement peut survenir n = nombre total de cas possibles. Par exemple, en jouant à pile ou face, la probabilité d'obtenir pile est de 1/2 et la probabilité d'obtenir face est de 1h. Quand un dé est lancé sur la table, la probabilité d'obtenir 1 est de 1/6, la probabilité d'obtenir 2 est de 1/6, ... , la probabilité d'obtenir un 6 est de 116. Aux cartes, la probabilité d'obtenir du carreau est de 13/52. ll en est de même pour le pique, le coeur et le trèfle. La définition précédente est utile dans les cas de situations finies où «S>> et «n>> sont connus.
Pour une situation infinie, où n = "", la définition conduira toujours à une probabilité de O. Ainsi, dans un cas de situation infinie, la probabilité qu'un événement survienne est proportionnelle à la distribution de l'univers.
3.2.1 Théorèmes de probabilité Théorème 1. La probabilité est exprimée par un nombre variant entre 1.00 et 0, où la valeur 1.00 représente la certitude que l'événement arrivera et 0, la certitude que l'événement n'arrivera pas.
3-2 soit un événement E; 0:5; P(E;) :5; 1
(équation 3.1)
Théorème 2. Si P(A) est la probabilité que l'événement A survienne, la probabilité que A n'arrive pas est donnée par P(A)=l.OO- P(A) P(A)
+ P(A) = 1
(équation 3.2)
Exemple: Si la probabilité d'obtenir une pièce d'équipement électrique défectueuse es 0.04, la probabilité d'obtenir une bonne pièce d'équipement est: P(B) = 1.00 - P(A) = 1.00- 0.04 = 0.96 Ainsi la probabilité d'obtenir une pièce d'équipement acceptable est de 0.96. Théorème 3. Si A et B sont deux événements mutuellement exclusifs, alor la probabilité que soit l'événement A ou soit l'événement B survienne est la somme de leurs probabilités respectives. P(A ou B)
= P(A) + P
(équation .3)
(B)
L'expression mutuellement exclusive signifie que le fait qu'un événement se produise rend l'autre événement impossible. Ainsi, si en jetant un dé, on obtient un (événement A) , il n'est pas possible d'obtenir un 5. Théorème 4. Si les événements A et B sont indépendants, la probabilité q e l'événement A ou l'événement B ou que A et B se produisent est donnée par P(A ou B ou les deux) = P(A)
+ P(B) - P(A et B)
(équation 3. )
Un événement indépendant est celui dont la réalisation ne présente pas d'influences la probabilité de réalisation de l'autre ou des autres événements. Exemple: Déterminer la probabilité d'obtenir un valet ou un carreau en tiran t u carte d'un paquet. P(A ou B ou les deux) = P(A)
+ P(B) - P(A et B)
4 13 1 P(valet ou carreau) =- + - - 52 52 52 16 =-=0.308 52
3-3 Théorème 5. La somme des probabilités des événements relatifs à une situation est égale à 1. P(A)
+ P(B) + ...
P(N) = 1.00
(équation 3.5)
Exemple: Un inspecteur examine 3 produits d'un sous-groupe pour déterminer s'ils sont défectueux. L'expérience nous dit que la probabilité de ne trouver aucun produit défectueux d'un échantillon de 3 est 0.89, que la probabilité de trouver un produit défectueux d'un échantillon de 3 est 0.06, et la probabilité de trouver deux produits défectueux d'un échantillon de 3 est 0.03. Quelle est la probabilité de trouver 3 produits défectueux d'un échantillon de 3? Cette situation ne comprends que 4 événements: 0, 1, 2 et 3 produits défectueux. P(O) 0.89
+ P(l) + P(2) + P(3) = + 0.06 + 0.03 + P(3) =
1.00 1.00
P(3) = 0.02 Théorème 6. Si A et B sont deux événements indépendants, la probabilité que les deux événements arrivent est le produit de leurs probabilités respectives. P(A et B) = P(A) x P(B)
(équation 3.6)
La probabilité que le projet A soit rentable est de 0.30 et la probabilité Exemple: que le projet B soit rentable est de 0.10. Quelle est la probabilité que les deux projets soient rentables.
P(A et B) = P(A) x P(B) = (0. 30) (0.10) = 0.03 Théorème 7. Si A et B sont deux événements dépendants, la probabilité que les deux événements se produisent est le produit de la probabilité que A arrive et que si A arrive, B arrivera aussi. P(A et B) = P(A) P(B/ A)
P(B/A)
=
P(A et B) si P(A) *O P(A)
(équation 3. 7)
(équation 3.8)
Le symbole P(B/ A) est défini comme étant la probabilité que 1'événement B se produise, si 1'événement A est arrivé. Un événement dépendant est celui dont la réalisation influence la probabilité de l'autre ou des autres événements.
3 Exemple: Une boîte contient 50 engrenages et 3 d'entre eux sont défectueux . i un échantillon de 2 est enlevé, quelle est la probabilité que les deux soient défectueux? La probabilité que le premier soit défectueux (événement A) est P(A) = 3/50. La probabilité que le second soit défectueux est P(B/A) = 2/49, étant donné que la
première pièce défectueuse a déjà été enlevée. P(A et B) = P(A) x P(B/ A) P(A et B) = (3/50) x (2/49) = 0.002 3.2.2 Valeur espérée et variance Soit X 1, X2 .. . X, des événements mutuels exclusifs qui représentent des re en possibles et P 1, P 2 , ... Pn, les probabilités de ces événements tel que:
La valeur espérée des revenus (moyenne) est de: m
E(x) = P.. =
L
(équation . 10)
xi Pi
i~ I
La mesure de dispersion de la distribution de probabilité appelé variante V(x) ou ci es
donnée que d'expression. m
o2
=V(n) =L
((xi- E(x)) 2 Pi
(équation . 10)
i~ I
o 2 = V(x) = E(x 2) - [E(x)f
(équation . 11
Exemple 3.1: Une compagnie considère l'investissement dans un projet don 1 revenus probabilistes sont donnés au tableau 3 .1.
Tableau 3.1: donnée pour l'exemple 3.1 Revenu X.$
Probabilité des revenus P(xi)
-llO 000
0.8
530 000
0.2
3-5 Déterminer les quantités suivantes: a) b)
La valeur espérée du projet La variante du projet
Solution: a)
b)
3.3
de l'équation 3.9 E(X) = X1 P(Xl) + X2 P(X2) = -110 000 x (0.8) + 530 000 x (0.2)
= 18 000$
de l'équation 3.10 a 2 = V(X) = (X 1 - E(X 1))2 x P(X 1) + (X2 - E(X2)) 2 x P(X2) = (-110 000- 18 000)2 x (0.8) + (530 000-18 000) 2 x (0.2) 2 a = V(X) = 6,553.10 10$
ANALYSES TRADITIONNELLES Nous allons en premier voir quelques unes des analyses traditionnelles le plus importantes utilisées pour considérer le risque.
1. 2. 3. 4. 5. 6.
Jugement intuitif Ajustement conservateur Optimiste-pessimiste Sensibilité Point mort Escompte du risque
3.3.1 Jugement intuitif Cette méthode consiste à effectuer l'analyse économique en supposant que l'analyste connaît tous les paramètres économiques avec certitude et applique un jugement subjectif sur le risque de chaque projet; il peut renverser la décision basée sur les résultats quantitatifs s'il considère le risque suffisamment grand.
3.3.2 Ajustement conservateur Cette procédure consiste à changer les estimés d'un ou plusieurs paramètres d'une analyse dans une direction conservatrice, de façon à diminuer le risque qu'une situation survienne qui ne soit pas aussi favorable que celle prédite par l'analyse économique. L'analyste peut faire l'étude à différents degrés de conservatisme. Le choix de valeurs conservatrices pour les paramètres économiques augmente la certitude d'un résultat favorable.
3-6 Exemple 3. 2: Considérons l'achat d'une nouvelle machine pour imprimer les circuits électriques e remplacer celle existante, ceci devrait permettre à une compagnie de réaliser une économie. L 'achat et l'installation coûtera 10 000$ , avec certitude. Elle estime produire 5 000 circuits de plus par année durant les six prochaines années et obtenir une économie annuelle de 5 200$. La compagnie estime que la nouvelle machine aura une vie utile de six ans , possèdera alors une valeur de récupération de zéro et coûtera 000$ par an de plus pour 1'opération et la maintenance. La banque financera cet achat à un taux annuel effectif de 5%. Considérant la valeur des paramètres estimés comme exacte, on peut calculer le profi annuel réalisé durant les six prochaines années. Récupération du capital avec un retour 10 000 (A/P, 5 , 6) Coût d'opération et de maintenance supplémentaire Coût total annuel Revenu total annuel Profit annuel a)
(3 9 52
)
12
Pour tenir compte de la variation possible des valeurs des estimés, la compagnie appliquera un jugement conservateur en utilisant un taux d'intérêt de 25 %. Les coûts et les revenus annuels deviennent:
Récupération du capital avec une retour 10 000 (AfP, 25 , 6) Coût d'opération et de maintenance supplémentaire
(3 390 ) (2 )
Coût total annuel Revenu total annuel
(5 39 52
Profit (perte) annuel
-(190 )
Un taux d'intérêt de 25% entraînera une perte de 190$ par an et la compagnie pourra décider de ne pas effectuer le remplacement. b)
D'une façon similaire la compagnie pourra conserver le taux d' intérêt de 5 o appliquer un jugement conservateur en réduisant la période de récupération de 6 à 3 ans. Les coûts et les revenus annuels deviennent:
3-7
Récupération du capital avec une retour 10 000 (AfP, 5, 3) Coût d'opération et de maintenance supplémentaire
(3 670$) (2 000$)
Coût total annuel Revenu total annuel
(5 670$) 5 200$
Profit (perte) annuel
-(470$)
De nouveau le jugement conservateur rend le remplacement non favorable. 3. 3. 3 Optimiste-pessimiste Cette procédure consiste à changer la valeur estimée pour un ou plusieurs paramètres d'une façon favorable (optimiste) et non-favorable (pessimiste), pour déterminer l'effet de ces variations sur les résultats de l'analyse économique. L'utilisation de méthodes statistiques permet de mesurer le degré d'optimisme. La valeur pessimiste correspond à l'estimé des résultats lorsque l'analyste attribue à chaque paramètre économique la valeur la moins favorable qui a une chance raisonnable de se produire. Cette valeur ne correspond pas à la plus mauvaise valeur qui peut se produire. L'estimé doit être fait pour chaque paramètre individuel. La valeur optimiste correspond à l'estimé des résultats lorsque l'analyste attribue à chaque paramètre la valeur la plus favorable qui a une chance raisonnable de se produire. Canada dans son livre défmit un estimé optimiste comme la valeur d'un paramètre dont l'analyste prévoit qu'elle sera atteinte pas plus que 5% du temps, alors qu'un estimé pessimiste correspond à la valeur d'un paramètre dont il prévoit qu'elle sera plus favorable que la valeur finale pas plus que 5% du temps.
La figure 3.1 illustre une représentation possible des valeurs pessimistes et optimistes des revenus.
valeur prévue
estimé pessimist;»
~ ..~~~~~~---.x~~~~~~•..,.
5%
2
a
2
a
~estimé optim iste
5%
distribution des revenus
Fig. 3.1
o: déviation standard Cette méthode optimisme-pessimisme apporte des informations additionnelles dans une étude économique; elle montre les conséquences résultantes des déviations de la valeur prévue. Les valeurs pessimistes utilisées dans cette analyse peuvent correspondre aux valeurs conservatrices de l'anal yse précédente. Exemple 3.3 Appliquons cette analyse à la situation décrite dans le tableau 3. 1 Tableau 3.1: donnée de l'exemple 3. 3 Optimiste
Prévue (estimé)
10 000
10 000
10 000
7
5
4
Valeur de récupération $
2 000
2 000
2 000
Profit annuel, $
6 000
5 000
4 500
Coût annuel $
2 200
2 200
2 400
8
8
8
Investissement $ Vie utile, ans
Taux d'intérêts %
Pessimiste
3-9
Calculons les profits (coûts) équivalents annuels en considérant successivement les valeurs pessimiste, prévue et optimiste des différents paramètres. AE (pessimiste) = -[(10 000-2 000) (A/P, 8%, 4) +2 000 (.08)] - 2 400 + 4 500 = -(470$) AE (prévue) = -[(10 000- 2 000) (A/P, 8% , 5) +2 000 (.08)] - 2 200 + 5 000 = 630$ AE (optimiste) = -[(10 000- 2 000) (A/P, 8%, 7) +2 000 (.08)] - 2 200 + 6 000 = 2 100$ Les informations additionnelles dans des conditions d'optimiste et de pessimiste aideront l'investisseur à prendre une décision en considérant les profîts et les pertes possibles dans ces conditions extrêmes. Note: Dans cet exemple, l'analyste a considéré que tous les éléments pessimistes ou optimistes se produisent en même temps, mais il pourrait considérer différentes combinaisons de ces valeurs. 3. 3. 4 Analyse de sensibilité Une analyse de sensibilité apporte des informations supplémentaires à une simple analyse sous certitude. Elle permet de déterminer le degré de sensibilité des résultats d'une analyse économique aux variations des différents paramètres utilisés. Les experts, parfois, définiront le concept de sensibilité comme le changement relatif de grandeur d'un ou plusieurs paramètres, dans une analyse économique, qui va renverser le choix des actions possibles: cette analyse s'effectue d'une façon similaire à celle optimiste-pessimiste. L'une des façons les plus intéressantes pour étudier la sensibilité consiste à mettre en graphique les résultats de tous les paramètres indépendants considérés; l'analyste montre la variation de chacun sur une abscisse commune, comme pourcentage de la valeur estimée; l'ordonnée représentant le critère économique utilisé pour évaluer le projet. La figure 3.2 montre un exemple de ce type:
-10
Q)
::::J
laux d'int,
erêt
c: c:
m
-100 -50 0 50 100 Déviation de la valeur estimée, % Fig.3.2 Analyse de sensibilité Dans ce cas particulier la variation des coûts annuels et de la vie utile ont une influence beaucoup pl importante sur la variation des profits annuels, que celle du taux d'intérêt; l'analyste devrait leu accorder une attention particulière. Exemple 3.4 Une société 0 désirant construire un nouvel atelier de fabrication de pâte doi déterminer le rendement en pâte le plus économique. Une analyse préliminaire a permis d'identifier les coûts du bois et du raffinage comme les paramètres les plus importants sur le coût de la fabrication de la pâte. La société a conduit une étude d sensibilité sur ces deux paramètres. La figure 3.3 montre l'influence des variations de ± 60% respectivement dans les coûts du bois et de l'énergie sur le coût de fab rication de la pâte, pour trois niveaux de rendement. À titre d'illustration cette analyse permet de constater que si les coûts d'éne ·e augmentaient par 50%, il n'y aurait pratiquement pas d'avantage économique d produire une pâte à 90% de rendement au lieu de 70%. De même, si une compagni pouvait réduire ses coüts en bois d'environ 30%, elle pourrait utiliser aussi écono · quement des pâtes à bas (50%) et haut rendements (90%). On peut facilem comprendre que ce genre d'information a grandement aider la compagnie à prendre décision concernant son investissement.
3-11
Rendement en pâte
50
coût dubois
7
du coût d'én
8
6 -60°/o
-40
-20
0
+40
+600!o'
Variation du coût individuel Fig.3.3: coût de production de pâte à différents niveaux de rendement
3- 1 3.3.4.1
Autre type d'analyse de sensibilité Les experts suggèrent aussi une autre façon d'étudier la sensibilité; elle consiste à mettre en graphique deux des paramètres importants, par exemple le taux d 'intérêt et la vie utile, et de tracer une ou des iso-courbes, pour le critère économique choisi, tels les profits annuels. Une ligne d'indifférence limitera deux zones ; l'une où l'analyste suggère l'investissement et l'autre où ille déconseille. La figure 3 . illustre cette méthode.
8
Investir
6 4
pas investir
2
0 0 2
4
6
8 10 12 14 16 18 20
Taux d'intérêt, % Fig . 3.4 : sensibilité d'un projet à deux paramètres importants Exemple 3.5 Appliquons cette analyse de sensibilité aux valeurs prévues de l'exemple considéré pour illustrer l'analyse optimiste-pessimiste. L'analyste désire connaî l'effet d'une diminution de la vie utile de 40% et d'une augmentation du t2. d'intérêt de 50%, car il considère ces deux éléments comme critiques. La mé od des coûts annuels équivalents donne: CA= + 5 000-2 200 -[(10 000-2 000) (A/P, 12%, 3) + 2 000 (. 12)] =- 773$ Précédemment pour un taux d'intérêt de 8% et une durée de vie utile de 5 ans. cette méthode indiquait un profit annuel de 630$. Dans le même exemp l'analyste désire trouver pour quelle valeur du taux d'intérêt la décision d 'in changera. Il suffit donc de trouver la valeur du taux <;l'intérêt pour que les co· annuels égalent zéro. CA=
5 000-2 200 -[(10 000-2 000) (A/P, i% , 5) + 2 000 (i %)] =0 pour i = 12% i = 20%
CA= +330 CA= -300
3- 13
Donc par interpolation, ou graphiquement l'on trouve que:
. ( 1=12%
330. 330+300
l
(20%-12%)=16.2%
3.3.5 Analyse du point mort L 'analyste utilise cette analyse dans des études économiques où il existe une incertitude particulière, concernant un des paramètres. Il définit le point mort comme la valeur du paramètre auquel la rentabilité du projet devient marginale. Dans le cas de plusieurs choix, il définit le point mort comme la valeur du paramètre auquel les choix deviennent également désirables. Exemple 3.6 Supposons que dans l'exemple 3.3 , l'estimation de la vie utile soit très incertaine, l'analyste désire déterminer la vie utile minimal permettant de justifier le projet. Égalant les coûts annuels équivalents à zéro l'anal yste détermine la durée de vie utile (N) minimal. 5 000-2 200-((10 000-2 000) (A/P,8%,N)-2 000(.08)]=0 2640 =0.330 (AIP, 8%,N)=Dans les tables on trouve un 8000 temps de 3. 7 ans qui correspond à la valeur de 0.330. Exemple 3.7 Une compagnie considère la fabrication d'un nouveau produit, elle dispose des données suivantes: Prix de vente:$/unité Coût d'équipement:$ Frais fixe par année:$ Coût d'opération et de maintenance:$/heure Temps de production /1000 unité, heure Durée de vie du projet, ans TRAM,%
12,50 200 000 50 000 25 100 5 15
La compagnie assume une valeur de récupération des équipements nulle après 5 ans et prend X= volume de vente /année. La valeur annuelle de l'investissement est: VA = -200 000 (A/P, 15, 5)- 50 000- 0.1 * (25) X + 12.5.X VA = -109 660 + 10.X
Au point mort VA
= 0 =>
X
= 10 966 unités/ année
3-1 La figure 3.5 montre une représentation graphique de ce problème.
1
1
1
1
1
1
1
1 1 1
0~
~-..)'b
·~\~
~e~e
~o"
~e\
0~~
J 1
Coûts annuel ~e'b
1
~e
l 1
20 00
1
0
5000
1
1
1
1
1
1
11000
1
1
1
150 000
X (Ventes annuelles) Fig. 3.5 : graphique du point mort
3.3.6 L'escompte du risque Cette analyse utilise un taux d'intérêt proportionnel au degré de risque; lorsque ce dernier croît le retour sur l'investissement doit être plus élevé. L'analyste trou e généralement difficile et subjectif d'établir un taux d'intérêt associé à un degré de risque. Il doit connaître les revenus et les coûts pour chaque proje . L'interprétation de cette analyse doit être changée lorsqu'un projet comporte u revenu net négatif, car un taux d'intérêt plus élevé pour un risque supéri entraînera une valeur présente du coût plus faible pour le projet. Cette analyse ne s'applique pas aux projets ayant une durée de vie utile très faible, car dans ce cas l'analyste n'escompte pas les revenus et les coûts. Pour éviter d'avoir à attribuer un taux d'intérêt différent à chaque projet selon le risque qu'il comporte, l'analyste peut calculer les valeurs présentes pour cha e projet correspondant à divers taux d'intérêt. Mettant en graphique la val présente en fonction du taux d'intérêt, pour les projets les plus risqués il détermin les taux d'intérêt ou de retour correspondant à une même valeur présente.
3-15
Taux d'intérêt, 0/o Fig. 3.5 : exemple d'analyse avec escompte de risque Les projets A et B ont une même valeur présente de 10 lorsque l'analyste escompte les revenus du projet B par un taux de 5% et celui du projet A par 20%. On peut voir aussi, que s'il escompte les revenus du projet A à un taux suffisamment élevé, si le projet comporte un haut niveau de risque, la valeur présente du projet B peut devenir supérieure à celle du projet A. Cette analyse fournira un élément de plus pour la prise de décision. 3.4
ANALYSE AVANCÉE Après avoir considéré quelques analyses traditionnelles, nous allons maintenant voir quatre (4) méthodes d'analyse économique plus avancées: 1. 2. 3. 4.
Monétaire probabilistique Estimé - variance Escompte des variables Simulation de Monte-Carlo
3.4.1 Analyse monétaire probabilistisque L'anal yse monétaire probabilistique comprend l'estimation de certains paramètres dont l'analyste considère la variation comme importante en terme de probabilité. Ces valeurs estimées peuvent être modifiées pour obtenir les caractéristiques
3-16 désirées de la mesure du mérite (ex. la valeur présente). Les résultats d'une telle étude peuvent se présenter sous l'une des formes suivantes: a)
L'estimé des bénéfices: i) ü) üi)
b)
3.4.1.1
valeur prévue état futur le plus probable niveau d'aspiration
L'estimé des bénéfices et de leurs variations possibles.
L'estimé des bénéfices Considérons en premier l'analyse reliée à l'estimé des bénéfices. Posons: Pi = probabilité que l'état futur possible Si se produise Pi ~ 0 pour tous les j et j
= rn
:E
j = 1
P. J
1
Dans une situation sous risque l'analyste connaît ces probabilités ou peut les estimer alors que dans une situations sous incertitude complète, il n'a pas confiance que l'estimé des probabilités soit correct.
3.4.1.1.1
Vale urs prévues La règle de la valeur prévue sert souvent pour faire un choix dans une situatio comportant des risques. L'analyste choisit la possibilité qui maximise le gain pré ou minimise les pertes prévues.
Pour la possibilité A;, on calcule le gain ou la perte prévue à partir, de la sommation du produit de chacun des gains ou pertes par la probabilité qu'il se produise: j = rn
L
j = 1
P IJ.. P.J
La détermination des probabilités constitue l'un des principaux problèmes lors l'analyste désire utiliser la méthode de la valeur prévue. Elles peuvent être basées sur des données historiques ou sur un traitement statistique rigoureux , mais dans la plupart des cas l'analyste doit se servir de son intuition et de son j ugemen . ne possède aucune assurance que le futur se déroulera d'une façon identique a passé même si les probabilités peuvent avoir pour base des données historiques.
3-17 En revanche, l'obligation de penser et d'estimer la valeur des différents facteurs dans le futur procurera des résultats supérieurs à ceux provenant d'aucune réflexion . Certains analystes trouvent utile d'utiliser des distributions statistiques (ex: Poisson , Normale) pour déterminer les valeurs prévues. Parmi les distributions de probabilité théoriques , qui sont le plus utilisées dans l'analyse économique avec risque, on cite les distributions Normale et Beta. La figure 3.6 montre un exemple de distribution Normale. Les probabilités qu'une variable aléatoire X, normalement distribuée ait une valeur dans un intervalle de a, 2a ou 3a de chaque côté de la moyenne sont respectivement 68.26, 95.45 et 99.73% .
99.73%
Fig. 3.6 : exemple d'une distribution normale
Dans certains cas, on ne peut représenter la distribution de probabilité par une distribution théorique connue. Une approche qui peut-être utilisée pour estimer une distribution de probabilité subjective consiste à déterminer un estimé pessimiste, optimiste et la valeur la plus probable d'une variable aléatoire. La règle de la valeur prévue constitue une méthode très répandue; pourquoi est-elle bonne? Considérons une séquence de décisions ayant des résultats indépendants: j = rn
E
Pj ou j
= 1,2 ,3, .. . m
j = 1
z'r.. égale la somme des valeurs des résultats des N premières décisions lorsque l'analyste applique le principe de la valeur prévue. Ici , par hypothèse pour chaque décision l'analyste choisit la possibilité qui maximise VP ; z·Nsymbolise cette somme partielle. Alors que: ~ égale la somme des valeurs des résultats des N premières décisions lorsque l'analyste applique tout autre principe de choix. Lorsque le nombre de
3-1 décision N croit et tend vers l'infini la probabilité que z·Nsoit plus grand que ~ tend vers un. Ceci constitue la «Loi du succès à long terme» . Plusieurs volu mes de statistiques présentent la preuve de cette loi. 3.4.1.1.2
L'état futur le plus probable Cette règle consiste à trouver l'état de la nature possible, Sj, qui possède la probabilité maximal de se produire; pour cet état de la nature choisir la possibilité ~ qui maximisera Pij· Cette règle considère que l'état de la nature le plus probable va se produire avec certitude. En pratique, beaucoup d'investisseurs prennent des décisions de cette façon .
3.4.1.1.3
Niveau d'aspiration Pour un iüveau d'aspiration K, sélectionnez la possibilité Ai qui maximisera la probabilité que la valeur du résultat soit plus grande ou égale à K; K étant une mesure d'un niveau de rentabilité que le propriétaire d' une firme désire obten· pour un projet. Cette règle reflète le fait que la plupart des humains essaient de découvrir et de sélectionner des possibilités satisfaisantes et non pas nécessairement optimales. Exemple 3.7 Pour mieux comprendre l'application de ces règles, considérons la matrice de profits (tableau 3.2); elle montre aussi les probabilités qu'ont chaque état de la nature de se produire. Tableau 3.2: matrice de profits ÉTATS DE LA NATURE Possibilités actions
SI p_l
= .2
S2
S3
p2 = .7
P1 = .1
Al
50
120
2 000
A,
130
130
130
AJ
500
110
100
Calculons en premier la valeur prévue pour chaque action: VA(A 1) = 50* .2 + 120 * .7 + 2 000 * .1 = 294 VA(A 2) = 130 * .2 + 130 * .7 + 130 * .1 = 130 VA(A 3) = 500 * .2 + 110 * .7 + 100 * .1 = 187
3-19 Donc ici la règle de la «valeur prévue>> va permettre de choisir la possibilité A 1, car 294> 130 et 187. L'état de la nature S2 possède la probabilité la plus élevée de se produire donc la règle de l'état futur le plus probable va sélectionner pour cet état de la nature l'action qui rapportera le profit maximal; ici, la possibilité A2 qui donnera un profit de 130 comparé à 120 et 110 respectivement pour A1 et A3 • Maintenant, supposons que l'investisseur désire un profit supérieur ou égal à 300 (K = 300), il considérera pour chaque possibilité la ou les probabilités d'obtenir un tel profit. Pour la possibilité A1 seul l'état de la nature S3 qui a une probabilité de 0.1 de se produire pourra lui procurer un profit de 2 000. Pour la possibilité A2 aucun état de la nature ne pourra lui procurer un tel profit. Pour la possibilité A3 , l'état de nature S1 qui a une probabilité de 0.2 de se produire lui procurera un profit de 500. Donc, la règle de niveau d'aspiration, pour un niveau de profit de 300, sélectionnera la possibilité A3 qui a une probabilité de 0.2 de lui procurer ce profit alors que les possibilité A1 et A2 ont respectivement des probabilités de 0.1 et 0.0. · Plusieurs autres principes de choix peuvent être présentés, mais lequel devons-nous utiliser? Beaucoup d'études utilisent le principe de la «valeur prévue» principalement à cause de la «loi du succès à long terme». En revanche, cette loi s'applique à une longue séquence de décisions. Or, la plupart des investisseurs ont à effectuer un seul investissement ou un nombre restreint ce qui n'assure pas la validité de la loi du succès à long terme. Exemple 3.8 Considérons un exemple familier à chacun d'entre nous, soit celui d'acheter ou non une assurance automobile. Les statistiques publiées montrent qu'un individu a une chance sur mille d'avoir un accident (P 1 = 0.001); donc, la probabilité de ne pas avoir d'accident égale 0.999. Si une police d'assurance coût 600$ et que le coût moyen d'un accident égale 40 000$, l'individu doit-il ou non acheter une assurance? La matrice du tableau 3.3 montre les différents coûts: Les règles de la valeur prévue (VP(A 1) =-600 et E(A 2) =-40) de l'état futur le plus probable (S~ et du niveau d'aspiration pour K supérieur à -600 sélectionnent toutes la possibilité A2 c'est-à-dire de ne pas acheter d'assurance. Mais beaucoup de personnes dans ce cas vont préférer payer 600$ pour éliminer même une chance très faible de perdre une somme considérable de 40 000$.
3-20 Tableau 3.3: donnée de l'exemQle 3.8
1
1
ÉTATS DELA NATURE
Possibilités
Avoir un accident SI
ne pas avoir d'accident s2
3.4.1.2
P=O.OOl
P=0.999
A1 : acheter de l'assurance
-600
-600
A 2 : ne pas acheter d'assurance
-40 000
0
Estimé des bénéfices et de leurs variations possibles La règle de la valeur prévue peut-être considérablement améliorée lorsque 1' analyste ajoute aux valeurs prévues une mesure de la variation possible des bénéfices et si possible une description complète de variation de la probabilité des bénéfices.
Pour mieux comprendre cette notion, considérons l'exemple suivant: Exemple 3.9 Un projet A a un bénéfice annuel prévu de 10 000$ avec une déviation standard de 25 000$. Un projet Ba un bénéfice annuel prévu de 8 000$ avec une déviatio standard de 4 000$. Un investisseur qui désire prendre le minimum de risq ue pourra choisir le projet B qui a un bénéfice annuel prévu inférieur au projet A. Supposons que les bénéfices prévus possèdent une distribution normale; la forme de la courbe de distribution des bénéfices procurera à l'investisseur des informations supplémentaires et lui permettra de mieux jug~r des avantages respectifs des différents choix. La figure 3.7 montre les bénéfices de l'exemple distribués normalement.
3-21
projet B probabilité de perte pour le projet A
~
=8 000
=4 000
0
projet A
~
~
-20
-10
0
10
20
30
=10 000
= 25 000
40
Fig. 3.7 : distribution de bénéfices
On peut voir, que le projet A a une probabilité plus élevée d'encourir une perte que le projet B; le projet B pourrait être alors choisi. Une distribution statistique des bénéfices autre que normale pourrait changer la décision. Les séries de Taylor peuvent servir pour approximer les variations simples des différents paramètres économiques, mais 1'analyste doit recourir aux techniques de simulation pour l'étude des variations plus complexes.
3.4.1.3
Exemple sur l'analyse avancée
3.4.1.3.1
Exemple sur la méthode monétaire probabilistique Le département de génie industriel a fourni aux cadres d'une usine les informations suivantes concernant un projet d'investissement. 10 000
Investissement,$ durée de vie, ans 3
5 7 Valeur de récupération, $ Revenu annuel, $ Coût annuel , $ Financement (taux d'intérêt annuel effectif), %
Probabilité
0.3 0.4 0.3 2 000 5000 2 200 8
3-22 Les cadres peuvent donc calculer à l'aide de la méthode des revenus (coûts) annuels équivalents les bénéfices prévus et leur déviation standard. Pour une vie de 3 ans VA=5 000-2 200 -[(10 000-2 000) (A/P, 8%, 3) + 2 000 (8%)] =- 460$ Pour une vie de 5 ans VA=5 000 - 2 200 -[(10 000-2 000) (A/P, 8%, 5) + 2 000 (8%)] =
630$
Pour une vie de 7 ans VA=5 000-2 200 -[(10 000-2 000) (A/P, 8%, 7) + 2 000 (8%)] = 1 110 L'investissement aura donc un revenu annuel prévu de: VAprévu= -460$ (0.3) + 630$ (0.4) +1110 (0.3) = 446$ Les bénéfices auront comme variance: 2
Variance(a =L [(AE)2 AE
= ( -460? = 401000
* probabilité(AE)] - (AEprévu) 2 * 0.3 + (630? * 0.4 + (1110) 2 * 0.3- (446) 2
a= déviation standard= V401000 = 631$ Pour une limite de confiance de 95% les bénéfices pourront varier entre - 81 6S e 1 708$ car;
AE = JL± 2a AE = 446$ + 2 * 631$ AE = - 816$ à 1 708$ Les cadres savent maintenant que l'investissement a un bénéfice prévu de 44 mais qu'il pourra encourir aussi une perte de 816$ et un bénéfice maximum de 1 708$ pour les limites de confiance fixées.
3.4.1.3.2
Exemple sur la valeur et l'utilité prévues Un investisseur qui a la courbe utilitaire ci-dessous désire étudier la possibili é d'investir dans un projet qui a des revenus prévus en fonction de différen probabilités.
3-23
30
20
10
1/ /1~--------------------------- -2 000
i
5 000
1
10 000
15 000
20 000
1 1
Dollars
Revenu
Probabilité
$
20 000 000 10 000 000 0 -2 000 000
0.05 0.15 0.30 0.50
L'investisseur désire effectuer l'étude en utilisant les méthodes de l'utilité et de la valeur prévues. Le tableau 3.4 ci-dessous montre ces différents calculs. Tableau 3.4 Revenu (1 000 000$)
Probabilité du revenu
A *B (1 000 000$)
Utilité du revenu
B*D
(A)
(B)
(C)
(D)
20
0.05
1.0
26.3
1.31
10
0.15
1.5
20.0
3.00
0
0.30
0
1.0
0.30
-2
0.50
-1.0
-10.0
-5.00
Revenu prévu 1. 5
utilité_Qrévue
-0.39
3-24 La méthode de l'utilité prévue indique que le projet n'est pas favorable car il a une utilité prévue négative alors que l'utilité d'un revenu de 0 égale un. Par contre, . la méthode de la valeur prévue indique un projet favorable, car elle montre un revenu prévu supérieur à zéro.
3.4.2 Estimé- variance: Cette méthode d'analyse économique, parfois appelée «méthode équivalente à la certitude», consiste à réduire les considérations économiques en une mesure simple des bénéfices prévus ainsi que de leur variation. Mathématique, elles 'exprime par la relation:
où
V= f.1.- Aa V= estimé - variance f.1. = bénéfices prévus
a= déviation standard des bénéfices A= coefficient d'aversion du risque
Les experts suggèrent de prendre
u "f.l. A=___ 2
ou u "
correspond
à la dérivée seconde de la fonction utilité versus bénéfices. Cette valeur de A s'applique lorsque l'utilité marginale de l'argent diminue; ce cas se rencontre souvent en pratique. 3.4.3 L'escompte des variables Une façon de considérer une augmentation du risque avec des temps de plus en plus éloignés consiste à escompter les différents revenus possibles par des taux différents; des taux plus élevés accorderont une pondération plus faible aux bénéfices les plus éloignés. Les experts suggèrent des modèles mathématiques pour établir les différents taux d'intérêt, mais ici nous nous limiterons au principe. 3.4.4 Simulation de Monte-Carlo Méthode développée durant les années 40 par Von Neuman, Vlan et Fermi pour résoudre certains problèmes de design d'écrans anti-rayonnement. Elle es coûteuse expérimentalement et difficile à résoudre analytiquement. Elle consiste à représenter un problème déterministe par un processus stochastique dont les distributions de probabilité satisfont les relations mathématiques du problème déterministe complexe. La première étape consiste à construire un modèle analytique qui représente l'investissement considéré (exemple une équation pour la valeur présente) .
3-25 Dans la deuxième étape, l'analyste développe une distribution de probabilité pour chaque facteur qui est sujet à incertitude dans le modèle. Il peut utiliser des données historiques ou une approche subjective. À partir des distributions de probabilités pour chaque facteur dans le modèle, Il génère au hasard une réponse tentative. La répétition de ce mode d'échantillonnage un nombre important de fois, permet de générer une distribution de fréquence pour la réponse (ex. Valeur Présente). On peut utiliser n'importe laquelle des méthodes de mesure de rentabilité. La distribution de fréquence résultant peut être utilisée pour obtenir des données probabilistiques sur le problème original. La simulation Monte-Carlo est une technique efficace pour l'analyse économique sous risque . Elle permet de varier les facteurs simultanément, en tenant compte, aussi bien de leur tendance centrale que de leur dispersion, et cela, quelque soit la distribution de probabilité qui les représente. Le chapitre 6 permet d ' expliquer plus en détail cette technique.
3-26
Problème no 3.1 Des amateurs de tennis désirent construire deux cours intérieurs au coût de 125 000$. Ce tennis intérieur aura une durée de vie utile de 10 ans et une valeur de récupération de 20 000$. Les coûts d'entretien sont estimés à 23 000$ par année. S'ils peuvent louer au taux de 7$ par heure par cours, combien d'heures devront-ils louer chaque année pour opérer au point mort? Le taux d'intérêt bancaire est actuellement de 10% :
Problème no 3.2 Une ligne de production sera requise encore pour trois ans . Elle a actuellement des coûts annuels de 310 000$; l'on prévoit qu'ils vont demeurer constants. Un ingénieur industriel a soumis un plan pour une nouvelle conception de la ligne de production. Cette nouvelle conception coûterait 150 000$ à installer et il y a 50% des chances qu'elle permettrait de réduire les coûts annuels de fonctionnement à 210 000$. Cependant, il y a une probabilité de 25% que les coûts annuels de la nouvelle ligne augmentent de 20 000$ dans chacune des deux dernières années. Il y a aussi 25% que l'augmentation soit de 75 000$ dans chacune des deux dernières années. Doit-on conserver la ligne actuelle ou installer la nouvelle?
Problème no 3.3 Considérant la matrice de profits annuels en milliers de dollars ci-dessous: ÉTATS DE LA NATURE SI
s2
s3
s4
s5
s6
P=.l
.1
.4
.2
.1
.1
Al
12
5
-8
-3
6
9
A?
7
0
1
5
20
7
Al
3
3
7
9
-5
5
A4
0
12
15
2
8
-200
A~
-10
22
9
0
4
12
Actions
a)
Quelle(s) action(s) devriez-vous choisir pour minimiser votre probabilité de faire des pertes?
3-27 b)
Quelle(s) action(s) choisiriez-vous si vous aviez un niveau d'aspiration d 'au moins 9 000$?
Problème no 3.4 Deux (2) projets A et B ont les profits prévus selon les probabilités montrées dans la matrice ci-dessous. Lequel choisiriez-vous et pourquoi? A Probabilité B Probabilité Profits
0.2 0 -11
0.2 0.3 5
0.2 0.4 -3
0.2 0.2 13
0.2 -0.1 21
Problème 3.5 Il existe plusieurs méthodes pour détecter des soudures imparfaites. Une compagnie considère actuellement deux méthodes possibles. La méthode 1 coûte 0.50$ par inspection et détecte les défauts 80% par temps. La méthode 2 coûte 2.00$ par test et détecte toujours une mauvaise soudure. Lorsqu'une soudure défectueuse est non détectée, la compagnie estime à 30.00$ les pertes qu'elle doit encourir pour le remplacement et la mauvaise publicité qui en résulte. La probabilité d'avoir une soudure défectueuse est de 0.05. Utilisant la méthode de la "valeur prévue", détermine quelle méthode devrait être utilisée ou est-il préférable pour la compagnie de n'en utiliser aucune.
Problème 3. 6 Appliquer les différents principes de choix («valeur prévue», «l'état futur le plus favorable>>, «niveau d'aspiration») à la matrice suivante; les valeurs de cette matrice sont des profits. ÉTATS DE LA NATURE P 1 =0.1
P 2=0.7
P 3=0.2
Actions
SI
s2
s3
At
200
10
-60
A2
150
30
0
A3
35
35
35
Chapitre 4 TECHNIQUES DE DÉCISION STATISTIQUE AVEC INFORMA TIONS ADDITIONNELLES 4.1
INTRODUCTION: Plusieurs techniques statistiques ont été développées pour aider l'investisseur à prendre une décision. Même en présence d'incertitude des informations additionnelles peuvent être obtenues par expérience. Dans ce chapitre on verra l'importance d'informations additionnelles sur la prise de décision dans le cas de risque.
4.2
Statistique Bayesian Pour tenir compte des informations additionnelles dans le cas de calcul des probabilités, on fait appel aux statistiques Bayesian. Les statistiques Bayesian sont caractérisées par l'ajustement des probabilités connues à «priori» pour un paramètre inconnu a des probabilités «posteriori» plus certaines. Ces probabilités peuvent être obtenues de simples évidences ou d'études supplémentaires.
4.2.1 Théorème de Bayes La probabilité que 2 événements se produisent est donnée par la théorie des probabilités. P(A et B) = P(A/B)*P(B) ou
P(A et B)
= probabilité que deux événements A et B se produisent ensemble
= probabilité que A se produise lorsque B s'est déjà produit
P(A/B) P(B)
= probabilité que B se produise
On a aussi que: Donc
[équation 4.1]
P(A et B)
= P(B et A)
P(B et A) = P(B/A)*P(A) et il résulte que: P(A/B)*P(B)
= P(B/A)*P(A)
Le théorème de Bayes peut être déduit si l'on assume que P(B) et P(A)
P(AIB)
=
P(BIA) P(A) P(B)
[équation 4.2]
*0 [équation 4.3]
P(BIA)
= P (Al B) P(B)
P(A)
[équation . ]
Exemple 4.1 La probabilité d'obtenir l'as de pique d'un jeu de 52 cartes = 1/52 ou peut être calculé:
P(as et pique)= P(as/pique) P(pique) = (1 as/13 piques) (13 piques/52 cartes) = (1/13) (13/52) = 1152
P(as/pique)
= P(pique/as P(piqu
= (1 pique/4 as) (4 as/52 cartes)
(13 piques/52 cartes)
= (114 (1/13)
= 1/13
1/4 Donc, en général s'il yan états de la nature S 1 , S2 , S3 , ••• Sn et le résultat d 'u ne étud additionnelle tel que échantillonner, est X tel que X soit discrète et P(X) y; 0 e 1 probabilités à priori P(S;) onfété établies, alors le théorème de Bayes peut s'écrire:
P(XIS;) P(S) P(SJX) = P(X)
[équation .
La probabilité postérieure P(S/X) est la probabilité d'avoir Si si le résultat d 'une é d additionnelle donne X. La probabilité de X et Si de se produire P(X/S;) P(S;) est la probabilité conjointe de X et Si. La somme de toutes les probabilités conjointes égale à la probabilité de X. Donc l'équation [4 .5] peut s'écrire: P(S/X) = P(X/S~;} P(X/S;) P(S;)
L
[équation .6]
4-3 Le tableau 4.1 représente un modèle pour 1'application du théorème de Bayes .
Tableau 4.1: application du théorème de Bayes 1
2
3
4
5
État de la nature
Probabilité à priori
Probabilité d'obtenir 1' échantillon X
Probabilité conjointe
Probabilité postérieur P(S./X)
SI
P(S 1)
P(X/S 1)
P(X/S 1) P(S 1)
P(X/S 1lX(S. 1} P(X)
s2
P(S 2)
P(X/S2)
P(X/S 2) P(S 2)
P(X/S~J
P(X) S;
P(S;)
P(X/S;)
P(X/S;) P(S;)
P(X/SJ__ffiïl P(X)
sn
P(SJ
P(X/SJ
P(X/Sj P(SJ
~JflS.J
P_{X) n
n
P(S) =1. 1
L
n
P(XIS)P(S)=P(X
i =l
L
P(SJX)=
i =l
Colonne: 1 S; : les états de la nature possibles 2 P(S;): la probabilité à priori S; estimée (la somme égale 1) 3 P(X/S;): la probabilité conditionnelle d'obtenir un échantillon ou des résultats d'une étude supplémentaire X connaissant l'état de natureS; 4 P(X/S;) P(S;): la probabilité commune d'obtenir S; et X; la somme de cette colonne est P(X), la probabilité que 1'échantillon ou les résultats d'une étude supplémentaire soit X. 5 P(S/X): la robabilité postérieure deS;, étant donné qu'une étude additionnelle a produit le résultat X; la iième valeur égale à la iième entrée dans la colonne 4 divisée par la somme de la colonne
4.
Exemple 4.2 Une production requiert que le procédé soit arrangé pour fabriquer 200 unités. Si les ajustements sont bien faits, des défauts dans les unités produites se produiront avec une probabilité de o.o-. Un mauvais ajustement avec une probabilité de 0.2 alors le taux des défauts dans les unités produites est de 0.25. Posant S1 pour un bon ajustement, S2 pour un mauvais et X l'événemen qu'un échantillon contienne un défaut. On désire calculer la probabilité postérieure que les ajustements soit bien fait ou mal fait étant donné que X s'est produit. Solution: Dans ce cas P(S 1)
= O. 8 et P(S 2) = 0.2
Le tableau 4.2 résume les calculs:
s.
1
P(S)
P(XIS)
P(X/S)P(S)
P(S/X)
SI
0.80
0.05
0.04
4/9 = 0.44
s?
0.20
0.25
0.05
5/9 = 0.56
Li
P(Si) = 1.0
L
=P(X)= 0.09
i
L
P(S/X) = 1.00
i
Tableau 4.2: résultat de l'exemple 4.8 Donc la probabilité a priori de 0.8 que l'ajustement soit bien fait est révisée, par le fait qu 'on a trouvé une pièce défectueuse, en une probabilité postérieure de 0.44. Exemple 4.3 Considérons un investissement ayant un retour (valeur présente nette) de 6 000$ si l'événemen 1 se produit et 4 000$ si c'est l'événement S2 qui survient. Les probabilités à priori sont de 0.4 pour S 1 et 0.6 pour S2 • Donc le retour prévu E(R): E(R) =0.4 (6 000$) + 0.6 (- 4 000$) = 0 Supposant que la possibilité de ne pas investir à un retour prévu nul; donc il y aura alors ni perte ni gain.
4-5 Un étude additionnelle permettra d'obtenir soit X1 montrant un retour de 6 000$ ou X2 pour une perte de 4 000$. Si S1 se produit, X1 sera indiqué avec une probabilité de 0.8 ; de même pour S2 avec une probabilité 0.6. Le problème se résume ainsi:
E(S 1)
=
6 000$
E(S 2)
= - 4 000$
P(X/S ~
= 1.0 - P(X 2 /S~ = 0.4
Les probabilités postérieures résultantes de l'étude additionnelle peuvent être calculés et la décision d'investir ou non peut maintenant être déterminée en fonction du résultat X de l'étude additionnelle. Probabilités postérieures étant données que X 1 se produit.
s.
P(SJ
P(X/ SJ
P(X/SJP(SJ
SI
0.4
0.8
0.32
0.32/0.56=0.57 _,
s2
0.6 \
0.4
0.24
0.24/0.56=0.43
1
I:= 1.0
L
=P(X) = 0.56
P(S/X 1)
I: 1.00
Quand xl se produit la probabilité d'avoir SI change de 0.4 (priori) à 0.57 (posteriori). Le retour prévu devient alors:
E(R/X 1)=0.57 (6,000$)
+ 0.43 (-4,000$) = 1 714$
Donc si X1 se produit, le projet devra être entrepris pour obtenir un retour de 1 714$ . Probabilité postérieures étant donné que X2 se produit:
si
P(SJ
P(XiSJ
P(X 21SJP(SJ
P(S/X2)
s,
0.4
0.2
0.08
0.18
s2
0.6
0.6
0.36
0.82
L: = o.44
L: Loo
L: = Lo
Il résulte une probabilité de 0.18 d'obtenir S1 du fait que X 2 se produit; le retour prévu devien alors; E(R/X2)=0.18 (6 000$) + 0.82 (-4 000$) = -2 182$
Comme E(R/X2) est négatif on préférera ne pas investir si X2 se produit préférant un retour de zéro. Donc on aura: 0 si X=X 2 E(R)= [ 1 714$ si X =X 1 Comme P(X 1) =0.56 et P(X 2) = 0.44l'étude additionnelle donnera le retour suivant. E(R) = E(R/X 1) P(X 1) + E(R/X2) P(X2) = (1 714$) (0.56) + (0) 0.44 = 960$
Donc en général le .retour prévu suite à une étude additionnelle (ou échantillonnage) Xj est: E ( R 1 s 1)
=~
max [E (actions) 1xi] P(Xj)
J
ou E(R/SI) est le retour prévu étant donné les informations de l'échantillon. Le changement de la valeur prévu de 0 à 960$ s'appelle souvent la valeur prévue provenant de l'échantillon (VPPE) qui se représente: VPPE = E(R/SI) - E(R)
4-7 Exemple 4.4 Un lot de cinq articles doit être accepté (probabilité a 1) ou rejeté (probabilité~). L'état de la nature est caractérisée par le nombre de probabilités. M d'articles défectueux parmi les 5; donc M=O, 1, 2, 3, 4 ou 5. Pour simplifier, supposons que les pertes égalent deux fois le nombre d'articles défectueux (2M) dans un lot qui est vendu; et égal le nombre de bons articles dans un lot qui est rejeté (5-M). Déterminez la possibilité Bayes si les probabilités à priori sont P(O) = 0.6, P(1) = 0.2, P(2) = P(3) = 0.1, P(4) = P(5) =O. Solution
Le tableau 4.3, ci-dessous, résume les données
Tableau 4.3 Données pour l'exemple 4.4 Nombre d'article défectueux S; = Action: a 1 (accepté)= Action: ~ (rejeté)= P(S) (à priori)=
1 2 4 .2
0 0 5 .6
2 4 3 .1
3 6 2 .1
4
5
8
10
1 0
0 0
Calculons les pertes si l'on accepte et rejete respectivement les lots. B(a 1) = 0 * 0.6 =1.4 B(a2)
5 * 0.6 = 4.3 -
+ 2 * 0.2+
0.1
* 4 + 0.1 * 6 + 0 * 8
+ 4 * 0.2 + 3 * 0.1
+ 2
* 0.1 +
1
*0 +0 *0
Donc, accepter les lots entraînera une perte moindre Maintenant, à partir des données précédentes et des probabilités conditionnelles pour les données X étant données et l'état de la nature M ci-dessous M P(X/M) =
5
si X = 1 (mauvais)
1 - M
5
si X = 0 (bon)
Calculons successivement les probabilités à posteriori et la nouvelle action Bayes.
4-8 Sachant que P(X et M) = P(X/M) P(M)
0
X\M
1 (.6) = .6 0 (.6) = 0
(bon) 0 !(mauvais) 1
5
=0 =0
=0 =0
(bon) 0 !(mauvais) 1
1/5 (0) l4-!5 (0)
Donc, P(X = 0) P(X = 1)
= 0.6 + 0.16 + 0.04 + 0 = 0.86 = 0.04 + 0.04 + 0.06 + 0 = 0.14
0 (0) 1 (0)
Calculant les probabilités à posteriori: P(M/X)
0
1
0
0.6 30 .86 43 0 -=0 0.14
0.16 8 - - =0.86 43 0.04 2 ---0.14 7
1
P(X et M) P(X)
3
4
5
0.04 2 0.86 43 0.06 3 - --0.14 7
0 -=0 0.86 0 -=0 0.14
0 -=0 0.86 0 -=0 0.14
2
X\M
---
3
4/5 (.2) = .16 3/5 (.1) = .06 2/5 (.1) = .04 1/5 (.2) = .04 2/5 (.1) = .04 3/5 (.1) = .06
4
X\M
2
1
0.06 3 0.86 43 0.04 2 --- 0.14 7
----
----
On désire avoir le minimum de perte; les pertes peuvent être calculées pour X = 0 c'est-à-dire lorsque l'échantillon testé est bon. Donc, les pertes totales égalent la somme des pertes pour les différents états de la nature.
30 . 8 3 2 40 =- (0)+- (2)+- (4)+- (6)+ 0 (8)=43 43 43 43 43 Donc, si on prend la possibilité a2 c'est-à-dire si on rejette le lot lorsque l'échantillon testé est bo (X =0) les pertes totales seront: =5
* -30 43
+
4
8 *43
+
3
* -3
43
+
2
2 43
195 43
*- --
40 195 pour X = 0, la possibilité Bayes est a 1• En d'autres mots, lors ue Donc comme - < 43 43 l'échantillon testé est bon, on accepte le lot. De la même façon si l'échantillon testé est mau ais. on peut calculer les pertes pour les 2 possibilités a 1 et a2 • Pertes pour action a 1:
4-9 =0
*
0 + 2
2
*-
4
+
7
2
*
+
7
*
6
3
7
30
--
7
Pertes pour action a2 : =5
*
(0) + 4
2
*-
+
7
20 Donc, comme -
3
*
2
7
30 <-
7
+
2
*
3
7
20
--
7
az
, la possibilité Bayes est
7
pour X =
1, on rejette le lot si
l'échantillon testé est mauvais de façon à minimiser les pertes.
4.2.2 Méthode de Bayes appliquée à un échantillonnage Nous avons vu précédemment qu'une compagnie pouvait réaliser une économie, en testant un échantillon dans un lot de cinq. Maintenant le management désire savoir s'il peut réaliser une économie supplémentaire en testant un deuxième échantillon. Dans ce cas les probabilités à posteriori trouvées précédemment après avoir testé un échantillon deviendront les probabilités à priori avant de tester le deuxième échantillon. Le tableau ci-dessous montre ces probabilités.
X\M 0
0 30
-
43
1
1
2
43 2
8
3 43 2
7
7
-
0
-
-
3
4
5
2
0
0
0
0
43 3 -
7
Considérant maintenant qu'un échantillon a déjà été testé et qu'il reste seulement quatre échantillons dans le lot. Les relations ci-dessous caractériseront 1'état de la nature lorsque le testeur aura trouvé un premier échantillon bon. P(X/M)=
M
si
4
M P(X/M)= 1 - 4
x2
= 1 (mauvais)
si X 2 = 0 (bon)
Par contre lorsqu'il aura trouvé un premier échantillon mauvais les relations suivantes caractériseront 1' état de la nature.
-10 P(X/M)=
M- 1 4
P(X/M)= 1
si X2 = 1 (mauvais)
M-1 4
si X2 = 0 (bon)
---
M a été définit comme le nombre de mauvais échantillon dans le lot initial.
À partir de ces relations calculons en premier les probabilités conjointes en utilisant la relation P (X et M) = P(X/M) P (M) lorsque le testeur a trouvé un premier échantillon bon. X\M 0 (bon)
1 (mauvais)
1
2
8 3 * 4 43 0.14
1 3 - *2 43 0.035
8 1 - *4 43 0.046
1 3 - *2 43 0.035
0 30 1 *43 0.70 30 0 *43 0
-
3
4
1 2 *4 43 0.011
0 *0
-
0
2 3 - *4 43 0.035
0 * 1 0
Étant donné que le testeur a trouvé un premier échantillon bon: La probabilité que le deuxième échantillon soit bon égale: P(X2 =0) = 0.70 + 0.14 + 0.035 + 0.011 = 0.885 De même, la probabilité que le deuxième échantillon soit mauvais égale: P(X2 = 1) = 0.046 + 0.035 + 0.035 = 0.115 , hantl11on mauv ais. ons1 erons mamtenant e cas ou' 1e testeur a trouve un premier ec
X\M 0 (bon)
0
1
2
3
4
0 *0 0
2 1 *7 0.286
3 2 - *4 7 0.214
1 3 - * 2 7 0.214
- *0
1 4
0 1 (mauvais)
0 *0 0
2 0* 7 0
2 7 4 0.071
-1 *-
1 3 *2 7 0.214
-
3 *0 4
-
0
4-11 Étant donné que le testeur a trouvé un premier échantillon mauvais: La probabilité que le deuxième échantillon soit bon égale: P(X2 =0) = 0.286 + 0.214 + 0.214 = 0.714
De même, la probabilité que le deuxième échantillon soit mauvais égale: P(X2 = 1) = 0.071 + 0.214 = 0.286 On peut maintenant calculer les probabilités à posteriori que le deuxième échantillon soit bon ou mauvais étant donné que le testeur a trouvé un premier échantillon bon . X\M 0 (bon)
1 (mauvais)
0 .70
1 . 14
2
3
4
.011
0
--
- . -
--
.035
--
.885 0.79
.885 0.158
.885 0.0395
.885 0.0124
0
.046 -
.035 -
.035 -
0
0
.115 0.40
.115 0.30
. 115 0.30
0
Calculons de la même façon pour le cas où le testeur a trouvé un premier échantillon mauvais . X\M
0
1
2
3
4
0 (bon)
0
.286 .714 0.400
.214 .714 0.300
--
.214 .714 0.300
0
1 (mauvais)
0
0
.214
0
0
0
.071 .286 0.249
--
--
-
-
-
.286 0.748
Considérons maintenant la logique des résultats. Lorsque le testeur a testé un seul échantillon et qu'ill'a trouvé bon, il résultait une probabilité à posteriori de 3/43 (0.0698) d'avoir deux échantillons de mauvais dans le lot. Si maintenant il teste un deuxième échantillon et qu'ille trouve encore bon, la probabilité à posteriori deviendra (.0395). Il est donc normal que la probabilité de trouver deux échantillons mauvais dans le lot diminue du fait qu'on en a trouvé un deuxième de bon. On pourrait démontrer un raisonnement similaire pour les autres probabilités.
-1
Connaissant les probabilités à posteriori, calculons les pertes lorsque le testeur trou e u premier échantillon bon et un deuxième: a)
bon accepter le lot: 0.79 * 0 + 0.158 * 2 + 0.0395 * 4 + 0.0124 : 0 + 0.316 + 0.158 + 0.0744 = 0.548 rejeter le lot: 0.79 * 5 + 0.158 * 4 + 0.0395 * 3 + 0.0124 : 3.95 + 0.632 + 0.118 + 0.024 = 4.72
*6
*2
+0
Le management minimise ses pertes en acceptant le lot.
b)
mauvais accepter le lot: 0.40 * 2 + 0.30 * 4 + 0.30 * 6 : 0.80 + 1.20 + 1.80 = 3.80 rejeter le lot : 0.40 * 4 + 0.30 * 3 + 0.30 * 2 : 1.60 + 0.90 + 0.60 = 3.10 Le management minimise ses pertes en rejetant le lot.
Maintenant calculons les pertes lorsque le testeur trouve un premier échantillon mau ais et un deuxième: a)
bon accepter le lot: 0.40 * 2 + 0.30 * 4 + 0.30 * 6 : 0.80 + 1.20 + 1.80 = 3.80 rejeter le lot : 0.40 * 4 + 0.30 * 3 + 0.30 * 2 : 1.60 + 0.90 + 0.60 = 3.10 Le management minimise ses pertes en rejetant le lot.
b)
mauvais On peut immédiatement prévoir que le management rejettera le lot puisqu 'ille fai lorsque le testeur trouve un deuxième échantillon bon. accepter le lot: 0.249 * 4 + O. 748 * 6 : 0.996 + 4.48 = 5.48 rejeter le lot : 0.249
*3
+ O. 748
~
2
4-13 : 0.747
+
1.49 = 2.24
Le management minimisera ses pertes en rejetant le lot. Donc, pour minimiser l'ensemble de ses pertes, le management adoptera comme stratégie de contrôle: Rejeter le lot lorsque le testeur trouve un premier échantillon mauvais et ne pas en tester un deuxième. S'il trouve un premier échantillon bon il en teste un deuxième; un mauvais échantillon entraînera, le rejet du lot alors qu'un bon échantillon son acceptation. Précédemment l'acceptation du lot sans aucun test entraînait des pertes de 1.4. Le test d'un échantillon permettait de réduire les pertes à 1.2. Le test d'un deuxième échantillon et l'application de la stratégie de contrôle ci-dessus entraînera des pertes de: 0.86 (0.885 * 0.548 + 0.115 * 3.10) + 0.14 * 2.85 0.86 (0.485 + 0.359) + 0.40 = 1.13 Donc le test d'un deuxième échantillon permettra de réduire les pertes de 1.20- 1.13 = 0.07 Dépendamment des coûts pour effectuer ces tests le management décidera d'effectuer, aucun test un ou deux échantillons par lot.
4.2.3 L'utilisation de données dans la prise de décision Par données, on entend les résultats d'une ou plusieurs observations que l'on croît intimement reliés à l'état de la nature. La disponibilité de telles données va fournir une certaine illumination sur l'état de la nature facilitant ainsi le choix d'une action. Une utilisation intelligente de ces données va permettre d'obtenir des pertes inférieures à celles qu'on auraient eues sans leur utilisation. Pour obtenir les données, on doit effectuer une expérience (étude) dont les résultats doivent être différents selon les divers états de la nature. Exemple 4.5 Supposons qu'une personne entrant dans un édifice peut voir les lumières sur les planchers inférieurs et supérieurs d'un ascenseur. Lorsque quelqu'un demande l'ascenseur la lumière, s'allume et s'éteint dès que la porte de l'ascenseur s'ouvre. Les données peuvent être exprimées ainsi: x = nombre de lumières allumées.
x= 0 : x= 1: x= 2:
personne n'a demandé l'ascenseur. l'ascenseur a été demandé sur un des planchers et peut ou non être e mouvement. l'ascenseur a été demandé sur les deux étages (sous-sol et premier étage)· peut alors penser que l'ascenseur ne fonctionne pas puisqu'elle a été demandé depuis un certain temps.
Supposons que l'on connaisse les probabilités pour x = 0, 1, 2 pour chaque état de la na et le tableau des pertes. Dans ce cas particulier, on pourrait assimiler les pertes à l 'éner ·e nécessaire pour monter un escalier. Plus on devra monter de niveau plus on perdra de l'énergie.
Tableau des pertes États de la nature
Actions
Fonctionne
Ne fonctionne pas
A 1 (descendre au sous-sol)
0
6
A2 (monter au premier)
1
5
Probabilité pour chaque état de la nature SI x
s2
Fonctionne Ne fonctionne pas
0
.6
.1
1
.3
.4
2
.1
.5
On peut se demander d'où peuvent provenir de telles informations? Il y a deux sources possibles: . a) b)
par observation sur une longue période de temps . de considérations théoriques; exemple; un modèle mathématique qui tiendrait compte du débit de personnes entrant dans l'édifice et celles utilisant l'ascenseur. On note qu'on doit aussi faire des déterminations pour établir les paramètres du modèle.
4-15 Les probabilités postérieures utilisant la relation de Bayes, sachant par 1'expérience passée que l'ascenseur fonctionne 7 fois sur 10, c'est-à-dire P(S 1) = 0.7, P(S:z) = 0.3 P(SJX)= P(X/S) P(Si) P(X)
État de la nature x=O
x = 1
x=2
(.3)(.7) 7 -(.3)(.7) + (.4)(.3) 11
7 (.1) (. 7) -(.1)(.7) + (.5)(.3) 22
(.4)(.3) 4 -(.3)(.7) + (.4)(.3) 11
(.5) (.3) 15 (.1)(.7) + (.5)(.3) 22
fonctionne S 1 : (.6)(.7) 14 -(.6)(. 7) + (.1)(.3) 15 ne fonctionne pas S2 : (;1)(.3) 1 -(.6)(.7) + (.1)(.3) 15
Il nous faut maintenant une règle pour prendre une décision. Si la personne entrant dans 1' édifice trouve deux lumières allumées X=2, il sait alors que P(S/2) = 7/22 et P(S 2/2) = 15/22. Se servant de ces valeurs les pertes postérieurs peuvent alors être calculées: La possibilité a 1: descendre =
La
possibilité~:
monter =
15 90 7 0*-+6*-=22 22 22 15 82 7 1*-+5*-=22 22 22
Donc, pour X = 2 les pertes sont inférieures en prenant la possibilité a2 c'est-à-dire de monter. De même pour X = 1 La possibilité a 1: descendre =
7
4
0*-+6*11 11
24 11
1
La possibilité a2 : monter =
7
4
1*-+5* 11 11
27 11
Donc, pour minimiser les retards on choisit a 1 , descendre Pour X= 0
ai : 0 * 14 15
+
6
1
6
15
15
*-
14 15
~:1*- + 5*
19
15
15
Donc la règle sera de descendre au sous-sol s'il n'y a pas de lumière allumée ou une seule lumière et de monter au premier étage si les deux lumières sont allumées. Donc en utilisant les probabilités à priori que 1' ascenseur fonction c'est-à-dire O. 7, on aura comme perte prévue: Si on descend: E = 0.7 * 0 + 0.3 * 6 = 1.8 Si on monte: E = 0.7 * 1 + 0.3 * 5 = 2.2 Donc, si on descend toujours, on aura une perte de 1. 8 qui est une possibilité Bayes qui minimise les pertes. Maintenant, si on utilise les résultats de 1'étude, quelle sera alors la perte? Comme P(X) =
E P(XIS) P(S) i
P(X = 0) = 0.6 * 0.7 + 0.1 * 0.3 = 0.45 P(X = 1) = 0.3 * 0.7 + 0.4 * 0.3 = 0.33 P(X = 2) = 0.1 * 0.7 + 0.5 * 0.3 = 0.22 Total = 1.0
- (.45) +
82 24 (.33) + (.22) = 1 11 22
Donc, dans ce cas, le prix que rapporte l'échantillonnage est de: 1.80- 1. 72 = 0.08
4-17
On peut noter que ce problème pourrait être solutionner en considérant l'ensemble des règles possibles. Dans ce problème, il y a deux possibilités: a 1 : descendre et a2 monter. De plus, il y a trois valeurs possibles observées: X = 0, 1, 2; donc, il y a 23 c'est-à-dire 8 combinaisons ou règles possibles (d) de décisions.
x 0 1 2 En calculant les pertes pour chaque règle possible et en choisissant celle qui donne la plus faible perte, on obtiendrait le même résultat que précédemment. Mais, on peut immédiatement voir que la règle d3 signifie: que si on observe 0 ou 2 lumières allumées on choisit la possibilité a2 , c'est-à-dire monter et la possibilité a 1 descendre si une seule lumière est allumée. Les règles d 1 et d 8 ignorent les données. On remarque aussi que les règles d4 et d 5 sont opposées.
4-17
On peut noter que ce problème pourrait être solutionner en considérant 1' ensemble des règles possibles. Dans ce problème, il y a deux possibilités: a1 : descendre et a2 monter. De plus, il y a trois valeurs possibles observées: X = 0, 1, 2; donc, il y a 23 c'est-à-dire 8 combinaisons ou règles possibles (d) de décisions.
x 0 1
2
En calculant les pertes pour chaque règle possible et en choisissant celle qui donne la plus faible perte, on obtiendrait le même résultat que précédemment. Mais, on peut immédiatement voir que la règle d3 signifie: que si on observe 0 ou 2 lumières allumées on choisit la possibilité a2 , c'est-à-dire monter et la possibilité a 1 descendre si une seule lumière est allumée. Les règles d 1 et d 8 ignorent les données. On remarque aussi que les règles d4 et d5 sont opposées.
4-18 Problème 4.1 La compagnie Papier désire augmenter sa production; elle considère actuellement les trois choix suivants: cl: faire un grand agrandissement C2 : faire un petit agrandissement C3 : louer les facilités additionnelles nécessaires
La compagnie demeure incertaine sur son choix, car la demande pour ses produits pourra être: élevée (S 1), bonne (SJ, normale (S 3) ou faible (S 4). Son département de marketing évalue les revenus et probabilités reliés à chaque niveau de demande de la façon suivante:
Revenus et probabilités de la demande (Revenus en million de dollars) Choix
élevée(S 1) bonne(SJ normale (S 3) P(Sl) 0.2 P(S 1 ) 0.45 P(S 1 ) 0.25
faible (S 4) P(SJ_ 0.1
Grand agrandissement (C 1)
18
12
6
-12
Petit agrandissement (C 2)
12
12
9
-6
Location (C1)
13
10
6
-1
Considérant sa difficulté à arrêter son choix, la compagnie décide de faire appel à un consultant. Ce dernier s'engage à prédire trois niveaux .de la demande avec certaines probabilités; le tableau suivant montre ses prédictions.
Les probabilités de la prédiction du consultant Demande future
élevée-bonne
bonne-normale
normale-faible
xl
x2
x3
0.6
0.3
0.1
s2 normale s3
0.3
0.5
0.2
0.1
0.3
0.6
faible S4
0.1
0.1
0.8
~levée
S1
bonne
À titre d'exemple si le consultant prédit une demande «élevée-bonne» dans 60% des cas cette demande se révélera dans les faits «élevée» P (X 1 1 S 1) = 0.6
a)
Déterminer les revenus prévus pour chacun des choix avant de faire appel au consultant?
4-19
b)
Déterminer les probabilités postérieures à l'étude du consultant?
c)
Déterminer les revenus prévus après l'étude du consultant?
d)
Quel est le montant maximal que la compagnie Papier voudra payer le consultant?
Problème no 4.2 Un étudiant possède trois (3) dés; un vert, un rouge et un blanc. Le dé vert a cinq (5) faces marquées H et une face marquée T, de même, le dé rouge a deux (2) faces marquées H et quatre (4) faces marquées Tet le blanc trois (3) faces marquées H et trois (3) faces marquées T. L'étudiant tire les trois (3) dés sans que vous puissiez les voir. Il en cache deux (2) et vous informe que celui non caché montre la face H. Démontrez à l'aide du théorème de Bayes le choix de la couleur du dé que vous feriez pour maximiser vos chances.
Problème no 4.3 Lorsqu'une machine est ajustée correctement 9 sur 10, des articles proçluits sont acceptables. Lors d'un mauvais ajustement, il y a 0.4 chance de produire des articles acceptables. La probabilité que la machine soit bien ajustée est de 0.95. a)
Si le premier article testé après un ajustement est non acceptable, quelle est la probabilité que la machine soit bien ajustée?
b)
Si les deux premiers articles testés sont acceptables, quelle est la probabilité que la machine soit bien ajustée? Problème 4.4
Nous avons vu, aux cours, l'application de la méthode de Bayes à l'échantillonnage (voir notes). Avant de prendre une décision finale sur le nombre d'échantillons qu'il faudra tester, votre patron désire connaître quelques informations supplémentaires; veuillez lui fournir ces informations. Étant donné que le testeur a trouvé un premier échantillon bon et un deuxième mauvais: a)
Quelles seront les relations qui caractériseront 1'état de la nature?
b)
Quelles seront les probabilités a priori avant de tester le troisième échantillon?
c)
Quelle est la probabilité que le troisième échantillon soit bon?
4-20
d)
Déterminer les probabilités a posteriori que le troisième échantillon soit bon.
e)
Sans rien calculer, si les coûts pour tester ce troisième échantillon sont nuls, quelle est l'économie maximale qu'on pourra réaliser en le testant? (maximum 5 lignes)
Problème 4.5 La société électronique doit décider si elle va produire un nouveau capteur; elle devra alors investir 5 $ millions. La demande pour ce capteur n'est pas connue. Si la demande était élevée, la société ferait un profit de 2 $ millions par année durant cinq (5) ans. Si la demande est modérée, le profit serait de 1,6$ million par année durant quatre (4) ans. Pour une demande faible, le profit ne serait que de 0,8 $ million par année durant quatre (4) ans. On estime à 10% les chances d'une faible demande et à 50%, celles pour une demande élevée. Le taux d'intérêt est actuellement de 10%. a)
Utilisant la méthode de la valeur présente, la société doit-elle faire l'investissement?
b)
L'un des cadres désire que l'on fasse une étude de marché; quel est le montant maximal que l'on pourrait être prêt à dépenser pour cette étude?
c)
Un consultant propose une étude au coût de 75 000$ qui a les caractéristiques suivantes?
Si la demande est
L'étude indiquera favorable ou non-favorable avec les probabilités favorable '&.
non-favorable
élevée
0.9
0.1
modérée
0.5
0.5
faible
0
1.0
Est-ce que la société doit demander l'étude? d)
Si l'étude indique non-favorable, la société doit-elle produire le capteur? Quel est le profit prévu dans ce cas?
4-2 1 Problème 4.6 Un éditeur d'une maison d'édition doit déterminer s'il va accepter ou non un manuscrit pour publication. Il a déjà dépensé 1 000$ pour le développement de ce manuscrit. Il a le choix entre: - A 1: rejeter le manuscrit et perdre 1 000$ -A 2: accepter le manuscrit sans consulter un· expert extérieur - A 3: obtenir l'avis d'un expert extérieur au coût de 800$ Les données sont résumées dans le tableau ci-dessous: Décision et résultats siÉvaluation de l'éditeur 2ubliés {10 cas~
Évaluation de l'expert ~20 cas~
Retour si _Qublié
Mauvais Acceptable Très bon
$
MO : pas publier
4
9
0
0
0
Ml : demande faible
3
3
2
0
-10 000
M2 : bonne vente
2
1
1
1
50 000
M3 : grand succès
1
0
2
1
200 000
Il y a trois niveaux de succès possibles si le livre est publié: Ml, M2 et M3. La dernière colonne donne la valeur présente des revenus moins les coûts de publication pour chaque condition de marché. Le tableau indique aussi les décisions antérieures prises après l'avis de l'expert. L 'éditeur a accepté 6 des 10 derniers manuscrits soumis; un a eu un très grand succès. L'expert en a classé 20 comme mauvais, acceptable et très bon et charge 800$ par étude. a)
Quel est le profit prévu si l'éditeur décide sans consulter l'expert?
b)
Quel est le profit prévu après consultation de l'expert?
c)
Quel est le montant maximal que l'éditeur sera prêt à payer pour l'expert?
Problème 4.7 Un inventeur a développé un instrument pour mesurer le taux de bûchettes dans le papier. Une compagnie considère actuellement la possibilité d'acheter les brevets reliés à cet instrument pour un montant de 40 000$. Elle pense qu'il y actuellement une chance sur cinq que l'instrument puisse être commercialisé avec succès; dans un tel cas, elle obtiendrait des revenus nets de 150 000$ par année durant les cinq prochaines années. Dans les autres cas l'instrument ne rapportera aucun revenu. Cette compagnie utilise un taux d'intérêt annuel effectif de 15%.
4-22
a)
La compagnie doit-elle l'acheter?
b)
Si un département de marketing peut fournir une information avec certitude du marché futur, combien la compagnie pourrait être prête à payer pour ce service?
c)
Ce même département de marketing peut aussi faire une étude de marché avec une probabilité de 0.7 que si le résultat de l'étude s'avère positif la commercialisation sera un succès et une probabilité de 0.9 que si l'étude montre un résultat non-favorable l'instrument ne pourra être commercialisé. Déterminer combien la compagnie voudra payer pour cette dernière étude?
Chapitre 5
V ARBRE DE DÉCISION Dans les problèmes décisionnels l'analyste doit reconnaître que des décisions prises immédiatement influenceront directement celles qui devront être prises dans le futur. Très souvent les industriels prennent des décisions sans considérer les effets à long terme; une décision qui semble bonne actuellement pourrait placer la compagnie dans une position défavorable et parfois non compétitive par rapport à d'autres à prendre dans l'avenir. L'utilisation des arbres de décision s'avère un moyen très efficace pour représenter l'interaction entre une séquence de décision. Ce diagramme des événements futurs qui pourraient influer sur la décision présente permet d'éclairer le décideur sur les choix à faire en présence de risques pour atteindre les objectifs fixés. L'arbre de décision explicite graphiquement la séquence des décisions à prendre et les divers événements qui peuvent arriver. Il montre les mêmes informations qu'un tableau de coûts mais d'une façon beaucoup plus clairement, particulièrement dans le cas de situations complexes. Pour Belzile et al (1) la procédure consiste à : bien se situer dans le temps; définir les événements possibles qui peuvent arriver (tenir compte de l'échelle de temps); déterminer les actions qui peuvent être entreprises; déterminer la valeur en dollars ou en utilité de chaque action combinée avec l'événement; associer à chaque événement une probabilité d'occurrence; trouver la valeur espérée de chaque solution possible; choisir la possibilité qui offre le résultat espéré le plus élevé (en dollars ou en utilité). Graphiquement l'arbre se compose d'une série de noeuds et de branches. Chaque branche représente une action possible. Le noeud localisé à l'extrémité gauche de chaque branche représente une chance que l'événement ou l'état de la nature se produise. Toute autre branche à la droite du noeud représente d'autres actions possibles associées aussi à un état de la nature représenté par le noeud. Un coût, un profit ou une utilité correspond à chacune des actions représentées; il est indiqué à l'extrémité droite de chacune des brariches terminales. Dans le présent texte les carrés représentent les points de décision et les cercles les points de chance. Un arbre de décision comprendra donc toujours une combinaison d'actions possible entièrement sous le contrôle du décideur reliées par des points de chance, lesquels sont soumis aux des lois des probabilités dont la décision ne possède aucun contrôle sur leur occurrence. L'arbre de décision contiendra seulement les actions et les états de la nature qui ont une importance pour l'investisseur et dont les conséquences désirent être comparées. De plus, cette approche assume comme hypothèse qu'aucun changement important n'influencera les différentes situations dans le temps.
5-2
Un arbre de décision ne donne pas au décideur la réponse à un problème décisionnel mais l 'aide plutô à déterminer quelle action en un point particulier du temps va produire l'utilité ou le profit le plus élevé. Il aide ainsi à visualiser l'information pour prendre une décision. Le décideur devra toujours considérer les gains prévus en fonction des risques à encourir. La nature de ces risques , dépendant de la façon qu'illes percevra, influencera non seulement les hypothèses qu'il a faites mais aussi la stratégie à suivre pour en tenir compte. La méthode de construction d'un arbre de décision peut
ê~e
résumée ainsi :
1.
identifier les points de décision et les actions présentes à chacun des points;
2.
identifier les points de risque et la valeur des revenus ou utilités qu'engendreront les actions à chaque point;
3.
estimer les valeurs nécessaires pour faire l'analyse; plus spécifiquement déterminer les probabilités des différents états de la nature, les coûts, les gains ou les utilités associés à chacune des actions;
4.
analyser les valeurs des choix pour prendre une décision.
Solution d'un arbre de décision En pratique, l'analyste utilise une procédure à contre courant pour analyser le problème représenté par un arbre de décision. Il effectue l'analyse en partant aux bouts des branches et en se déplaçan t ers le tronc en appliquant les deux règles suivantes : 1.
2.
s'il rencontre un point de chance, c'est-à-dire associé à un état de nature, il calcule la valeur prévue du coût, du gain ou de l'utilité en se basant sur les valeurs placées à la droite du point; s'il rencontre un point de décision, c'est-à-dire une action à prendre, il choisit le profit ou l'utilité maximale ou les coûts minimaux en considérant les branches de droite adjacentes au point.
En procédant de cette façon, il peut éliminer certaines actions de considération future; ceci est particulièrement important dans le cas de situation complexe qui exige la construction d'arbre très grand et l'utilisation d'ordinateurs pour leur solution.
Référence 1.
Belzile R., Mercier G., Rassif F., <
5-3 Arbre dé décision Exemple La société PAPIER VERT désire fabriquer un nouveau type de papier sanitaire, à la fois plus économique et plus écologique, à partir d'un nouveau procédé de fabrication de pâte mécanique. Malgré les études techniques et l'analyse de marché effectuées, la société n'est pas certaine si la qualité de son produit permettra un volume de vente important. La société considère actuellement deux possibilités; la première consiste à construire un atelier de fabrication de pâte mécanique de 500 t/d et de tenter de prendre une part importante du marché. La deuxième possibilité serait de construire une usine de seulement 250 t/d et de faire une nouvelle étude après deux ans. Si la demande est suffisamment élevée, la société construirait une autre unité de 250 t/d pour minimiser l'influence de ses coûts indirects de fabrication et maximiser ses profits. Par contre, si la demande pour ce nouveau produit était faible, la société se retirerait du marché et réorienterait la fabrication de ses 250 t/d vers la fabrication de papiers ayant une valeur ajoutée inférieure. Les résultats de l'analyse de marché font voir qu'il y a 60 % de chance que la demande initiale soit élevée et demeure élevée durant vingt ans et 10 % de chance qu'elle soit initialement élevée et qu'elle devienne faible après deux ans; donc, il y a 70 %de chance que la demande initiale soit élevée pour ce type de papier et 30 % qu ' elle soit faible dans deux ans. Il y a 86 % (60/70) des chances qu'une demande élevée resterait élevée et seulement 14% ( 10/70) qu'elle devienne faible. Si, avec toute la publicité qui entourerait le lancement de ce nouveau produit, la demande initiale était faible, elle demeurerait certainement faible. Pour sa part, le département de génie industriel de la société estime que les coûts de construction seraient respectivement de trente et soixante millions de dollars pour un atelier de 500 et 250 t/d. L'addition de 250 t/d, après deux ans, coûterait vingt millions de dollars. Le tableau 1 fait voir l'estimé des profits selon que l'on aurait une capacité de production de 250 et 500 t/D et que la demande soit élevée ou faible. La figure 1 illustre, sous la forme d'un arbre de décision , l'ensemble des informations . Cette représentation n'apporte par de solution particulière au problème de la société PAPIER VERT, mais elle aide ses dirigeants à visualiser la suite des décisions à prendre . Cette société exige un taux de retour minimum de 10 5/6 sur cet investissement qu i permettrait de consolider sa position sur les marchés internationaux.
5-4 L'analyse utilise la procédure à contre-courant pour solutionner cet arbre, en commençant au point de décision 2 et en attribuant une valeur monétaire qui lui permettra de prendre une décision au point 1. Le tableau 2 illustre l'analyse de la décision au point 2 et fait voir les valeurs présentes prévues. Donc, si la société décide maintenant, c'est-à-dire au point 1, de construire l'atelier de 250 t/d, elle tiendra compte que l'addition de 250 t/d après deux ans lui procurerait un profit net de quarante-quatre millions de dollars. La figure 2 illustre l'arbre de décision simplifié et ne contenant plus que le point de décision 1. Multipliant les valeurs présentes des revenues prévues par les probabilités. L'analyste obtient les profits pour la construction d'une usine de 500 t/d et d'une usine de 250 t/d et cela, après que la décision au point 2 a été prise de procéder à une addition de 250 t/d dans les deux ans. Le tableau 3 illustre les calculs des valeurs présentes prévues au point de décision 1. La construction immédiate d'un atelier de 500 t/d entraînerait: 0.6 x 85.1
+
0.10 x (17.3
+
6.78)
+
0.30 x 8.51-30 =
26.1
Celle d'un atelier de 250 t/d:
o. 7
x {7.81 + 36.4) + 0.3 x 29.8- 16
=
23.9
Donc, la maximisation du revenu proviendrait de la construction d'un atelier de 500 t/d immédiatement. Avec ces informations et d'autres pertinentes, le management de cette société pourra prendre un décision finale.
5-5
Tableau 5.1 Estimation des profits annuels en fonction de la grosseur de l'atelier et du volume de la demande
Atelier capacité, t/d
État de la nature de la demande
Revenu 1 an (millions $)
500
élevée faible
10 1
250
élevée faible ·
4.5 3.5
élevée faible
0.5
250
+ 250
9
Si la demande reste faible, la société réorientera la production et le profit annuel ne sera que de 3.5 millions de dollars.
5-6
Tableau 5.2 Les valeurs présentes prévues au point de décision 2
Choix
État de la nature de la demande
Probabilité
Revenu an (millions $) durant 18 ans
ajouter 250 Ud
élevée
0.86
9
73.8
63.5
ajouter 250 Ud
faible
0.14
0.5
4.10
0.57
Total - l'investis.
64.1
Valeur présente (millions $)
Valeur prévue escomptée (millions$}
NETTE
.2.L 44.1
pas de changement
élevée
0.86
4.5
36.9
31.7
pas de changement
faible
0.14
3.5
28.7
4.02
Total - l'investis.
35.7 0
NETTE
35.7
Tableau 5.3 Les valeurs présentes prévues au point de décision 1
État de la nature de la demande
Choix
Probabilité
Revenu an (millions $)
Nombre d'années
Revenu escompté (millions $)
Valeur prévue escomptée (millions $)
élevée
0.60
10
20
85.1
51 .1
initiale élevée et après faible
0.10
10
2
17.3
1
18
6.78
faible
0.30
1
20
8.51
2.55
Total - l'investis. NETTE
56.1 ~ 26.1
2.41
'
élevée
0.70
4.5 et valeur de la décision au point 2
2
31 .0
44.1 à la fin de la 29 année faible
- - - - -- - -
-
--
--
0.30
3.5
7.81
36.4
20
29.8
8.94
Total - l'investis. NETTE
39.9 -16 23.9
5-8
Demande
~~'0 ~
~
~
.~0
~
0"'
10
Élevée Faible
0.1
Faible
0.3
10 1 1
0.86
9
0.14
0.5
~
0.86
4.5
0.14
3.5
18
0.3
3.5
20
t':l"''llée
"
r...ri>~x~
0~
il'~.
il'~
Nombre d'années
0.6
~0
~ <:!
Revenu/an
Élevée
~ ~0 ~ ~
c?
Probabilité
'-.__.. / ~
20 2 18 )
20 18
)
18
~
19
~~
~~
~
~
~
~,
~~-
i:\e'llée
18
'7.
6'
'-
• i:I/Ofe
/
/:'êl/6
v&o.a "-
0
Faible
2
Figure 5.1 L'arbre de décision pour la construction d'un atelier de fabrication de pâte mécanique
20
ans
Demande
Probabilité
Revenu/an
Nombre d'années
0.6
10
20
0.1
10
2
1
18
1
20
~e-lee /
~0
~cP
·-:-.<..0 ~ ~v ~0 0
~?
cP
~o
· ~? ~~
,~~0
)
Élevée initiale Faible après ):-~.
"6;$
~
0.3
Provenant de l'addition de
~
.t.:
~d' ~~ .
'?k
(\)d'
250 tld après 2 ans
~
~d>
(9~
\$'(9.
.,
~(9. ....
~
~
0.7
+4.5
2
0.3
3.5
20
'7..
6'
Figure 5.2 L'arbre de décision simplifié: après que la décision a été prise de procéder à l'addition de 250 t/d après 2 ans
5-10
Problème 5 .1 Un jeune débrouillard a décidé de tirer avantage de ses connaissances en statistiques pour faire quelques dollars durant ses vacances. Il s'est acheté deux chapeaux identiques; dans le fond intérieur il a écrit X dans l'un et Y dans l'autre. Dans le chapeau marqué d'un X, il dépose 5 balles de golf; 3 blanches et 2 jaunes, alors que dans celui identifié par un Y, il y dépose aussi 5 balles, mais cette fois, 4 jaunes et 1 blanche. Il propose à des amis les paris suivants: Payer 10 $ et identifier le chapeau marqué d'un X. S'il l'identifie correctement, il gagnera 8 $,s'il se trompe, il perdra son 10 $; Payer 2 $ et tirer une balle de l'un des chapeaux, après qu'il ait vu si elle est blanche ou jaune il peut ou non payer 10 $ et tenter d'identifier le chapeau marqué d'un X. Il pourrait décider de tirer 2 ou 3 balles et payer 2 $ pour chacune d'elles avant de payer 10 $ pour identifier le chapeau . L'ami a toujours la possibilité de refuser le pari.
a)
Tracer un arbre de décision pour montrer toutes les possibilités du jeux.
b)
Solutionner l'arbre et déterminer la possibilité la plus intéressante pour l'ami.
Problème 5.2
La compagnie Boisbriand désire investir 100 millions de dollars dans la construction d'un nouveau complexe résidentiel. Elle prévoit des revenus de la location de ces résidences qui varieront avec la situation économique de la province. Le tableau 1 décrit ces prédictions. Tableau 1 Variation des revenus nets État de la nature de l'économie
Valeur actuelle des revenus nets - Millions $
Probabilité d'occurrence
A
140
0.2
Normal . .... . ... .....
B
120
0.5
Récession
c
50
0.3
Prospère
•
•
0
••
•••
••
0.
••••••••••
0
5- 11 Comme la compagnie n'a pas encore pris la décision finale d'investir, elle cons idère la possibilité d'engager un consultant . Ce dernier exige un montant de 100 000 $ et s'engage à prédire l'état de la nature avec certaines précisions. Ces dernières proviennent des résultats de ses études précédentes. Le tableau 2 résum e ces prédictions. Tableau 2 Les probabilités que le consultant a déjà obtenues d'études similaires
Prédictions du consultant
Les probabilités d'obt en ir les états de la nature
A
8
c
An
0.8
0.1
0.1
Bn
0 .1
0.9
0.2
en
0 .1
0.0
0.7
a)
Démontrer, à l'aide d'un arbre de décision, si îa compagnie l'investissement en se basant sur ses propres prédictions .
doit fa ire
b)
Est-il avantageux pour la compagnie d'engager le consultant?
c)
Quel est le montant maximal que la compagni'e pourrait payer à un consultant?
d)
Si une information parfaite pouvait être obtenue, combien la compagnie pourraitelle payer?
··- -·-···- -- - -
-----
13.3 SEQUENTIAL DECISION MAKING: USE OF DECISION TREES
Decision trees, also commonly called decision flow networks and decision diagrams, are powerful means for depicting and facilitating the analysis of important problems, especially those that involve sequential decisions and variable outcomes over time. Decision trees have great usefulness in practice because they make it possible to look at a large complicated problem in terms of a series of smaller simple problems and they enable objective anal ysis and decision ma king which includes explicit consideration of the risk and effect of the future. The name decision tree is descriptive of the appearance of a graphical portrayal, for it shows branches for each possible alternative for a given decision and branches for each possible outcome (event) which can result from each alternative. Such networks reduce abstract thinking to a logical visual pattern of cause and effect. When costs and benefits are associated with each branch and probabilities are estimated for each possible outcome , then analysis of the How network can clarify choices and risks. 13.3.1 Deterministic Exemple
The most basic form of decision tree occurs when each alternative can be assumed to result in a single outcome-that is, when certainty is assumed. The replacement problem in Fig. 13-4 illustrates this. The problem as shown retlects that the decision on whether to replace the old machine with the new machine is not just a one-lime decision , but rather one which recurs periodically. That is, if the decision is made to keep the old machine · at decision point 1, then la ter, at decision point 2, a choice again has to be made. Similarly, if the old machine is chosen at decision point 2, then a choice again has to be made at decision point 3. For each alternative , the cash inflow and duration of the project is shown above the arrow and the cash investment opportunity cost is shown below the arrow . For this problem, one is concerned initially with which alternative to choose at decision point 1. But an intelligent choice at decision point 1 should take into account the later alternatives and decisions which stem from it. Hence, the correct procedure in analyzing this type of problem is to start at the most distant decision point, determine the best alternative and quantitative result of that alternative, and then " roll back" to each sucees-
Vl N
342
Risk and Uncertainty
Chap . 13
1<
-$2M ,.,.. x,.~&:
~
~
~ x ,.6'
'.r
1-.....
".s: '.r~'\. {s-~~ .9J-,..
Figure 13-4.
.s:
~
J--:
': 6'
x
J-,..
Deterministic replacement example .
Table 13-5 M ONETA RY ÜUTCOMES AND D EC ISIONS AT EAOI POI NT- DETERMINISTIC REPLACEMENT EXAMPLE OF FI G.
3 2
Alternative
{ { {
Monetary Out come
Old New
S3M(3) - $2M S6.5M(3)- SISM
Old New
$7M + $3 .5M(3) - SIM $6,5M(6) - Sl7M
Old New
= S 7.0M
13-4'
For decision tree analyses, which involve working from the most distant decision point to the nearest decision point, the easiest way to take into account the timing of moncy is to use the present worth approach and th us discount ali monetary outcomes to the decision points in question. To demonstrate, Table 13-6 shows computations for the same replacement problem of Fig . 13-4 using an interest rate of 25% per year. Note from Table 13-6 that when taking into account the effect of timing by calculating present worths at each decision point, the indicated choice is not only to keep the "old" at decision point 1, but also to keep the " old '' at decision points 2 and 3 as weil. This result is not surprising sincc the high interest rate tends to favor the alternatives with lower initial investments, and it also tends to place Jess weight on long-term returns . 3.3.2 Typical Example with Random Outcomes
The following is a brief description of a typical automation problem for which the analytical capabilities of decision trees are very useful . A decision maker must decide whether to automate or not to automate a given process. He is uncertain whether the results will turn out to be poor. fair, or excellent-depending on the technological success of the automation project. The net payoffs for possible outcomes (expressed in net present worths) are $-90M, $40M , and $300M, respectively . The initially estimatcd probabilities thal each outcome will occur are 0.5 , 0.3. and 0.2. respectively. Figure 13-5 is a decision tree depicting this simple situation . Table 13-7 shows calculations to determine that the best choice for the firm is to
Choice Old
Poor S - 90M (0.5)
New
Exc. SJOOM (0.2)
=S
1.5M = Sl6.5M
= S22.0M S22.0M + S4M(3) - SO.SM = $33 .2M S5M(9)- $15M = SJO.OM
• lnterest = 0%; that is . ignoring timing.
343
13.3.1.1 Dctenninistic Example Considcring Timing
sive decision point, repeating the procedure until finally the choice at the initial or present decision point is determined . By this procedure, one can make a present decision which directly takes into account the alternatives and expected decisions of the future. For simplicity in this example, timing of the monetary outcomes will first be neglected . which means that a dollar has the same value regardless of the year in which it occurs. Table 13-5 shows the necessary computations and decisions . Note that the monetary outcome of the best alternative at decision point 3 ($7 .0M for the "old") becomes part of the outcome for the "old" alternative at decision point 2. Similarly, the best alternative at decision point 2 ($22 .0M for the "new") becomes part of the outcome for the "old " alternative at decision point 1.
Decision Point
Sequential Decision Making : Use of Decision Trees
By following the computations in Table 13-5 , one can see that the answer is to keep the "old" now and plan to replace it with the "new" at the end of 3 years (at decision point 2). But this does not mean that the old machine should necessarily be kept for a full 3 years and then a new machine bought without question at the end of 3 years. Conditions may change at a ny time, thus necessitating a fresh analysis-probably a decision tree ana lysis-based on estimates which are reasonable in light of conditions at that later time .
Old + $3M/yr, 3 yr 3yr
%...
Sec. 13.3
Fair S40M 10.3)
Old
Figure 1:3·5. Automation problem diagram before consideration of techno(ogy study . ($ are in net PW)
Not automate
10
Y'-...... ·
.......
Chap. 13
Risk and Uncertainty
Sec . 13.3
25%
PER YEAR FOR
DETERMINISTIC REPLACEMENT EXAMPLE OF fiG.
Decision Point
3
Alternative
{ {
{
Old New Old New Old New
PW of Monetary Outcome
13-4
Fair $40M 10.3)
Fair $40M 10.22)
Exc. $300M~
Exc. $300M 10.05)
Choice
$3M(P/A,3) - S2M · S3M( 1. 95) - $2M = S3.85M $6.5M(PIA.3)- $18M $6.5M(I.95)- SI8M = -$5 .33M $3.85(PIF.3) + S3.5M(P/A.3)- SIM $3.85(0.512) + $3.5M(I.9~)- SIM = $7.79M $6.5M(PIA,6)- $17M S6.5M(2.95) - SI7M = $2.18M S7.79M(P/F.3) + $4M(P/A.3)- $0.8M $7.79M(0.512) + $4M(I.95) - $0.8M • SI0.99M $~.0M(PIA.9) - $15M $5.0M(3 .46) - SI5M = S2.30M
--
Old
$0
~~ .l'o-f, "o/. ")-
Not automate
o,
"o:'J.
"l
~
"),
""'
'J. ' _ ,
10
Poor $-90M (0.43)
.
"\
Q, ~;.
Old
Poor 1-00M (0.73)
Poor $-90M (0.6)
Table 13-6 DECISIONS I'IT EI'ICH POINT WITH INTEREST =
345
Sequential Decision Making: Use of Decision Trees
Fair $40M (0.34)
q"'*
.J\tt'-e
---
Exc . $300M (0.23)
Not automate
10
Poor $ - 90M 10.21)
Old
Fair $40M (0.37) Exc. $300M (0.42) 10
Table 13-7 EXPECTED NET
PW
CALCULATION FOR THE AUTOMATION
PROBLEM BEFORE CONSIDERATION OF TECHNOLOGY
Figure 13-6. Automation example with technology study laken in consideration . ($are in net PW be fore consideration of any cost for "Technology Study").
FEASIBILITY STUDY
Automate: S-90M(0.5) + $40M(0.3) + $300M(0.2) = $27M Not automate : = $0
automate based on an expected* net PW of $27M versus $0 if it does not automate. Nevertheless. this may not be a clear-cut decision because of the risk of a $90M Joss and because the decision maker might reduce the risk by · obtaining further information. Suppose that it is possible for the decision maker to make a further technology study at the PW cost of $10M. The study will disclose that the enabling technology is either "shaky," "promising," or "solid." lnstead of using Bayesian statistics for revision of probabilities at this point (this topic will be covered in Section 13 .3.5 and Appendix 13-A), let us assume that the probabilities of the various possible automation outcomes, given the technology study outcomes, are as shown in Fig. 13-6, which is a decision tree diagram for the entire problem. • Expected (long-run average. or mean) monetary value will be used as a choice criterion for problems involving probabilistic monetary outcomes al most throughout this book. [Exception : Where certain monetary equiva lents are used, as in Section 13.4].
The tree diagram shows expected future events (outcomes), along with their respective cash flows and probabilities of occurrence . The rectangular blocks represent (decision) points intime at which the decision maker must elect to take one and only one of the paths (alternatives) considered. These decisions are normally based on a quantifiable measure, su ch as money, which has been determined to be the predominant "cost" or "reward" for comparing alternatives . The general approach is to find the action or alternative that will maximize the expected net PW equivalent of future cash flows at each decision point, starting with the furthermost decision point(s) and then "rolling back" until the initial decision pointis reached . The circular (chance) nodes represent points at which uncertain events (outcomes) occur. At a chance node the expected value of ali paths leading (from the right) into the node can be calculated as the sum of the anticipated value of each path multiplied by its respective probability. (The probabilities of ali paths leading into a node must sum to 1.) As the project progresses through time, the chance nodes are automatically reduced to a single outcome based on the "state of nature" which occurs at that time. The solution to the problem in Fig. 13-6 is shown in Table 13-8. It can be noted that the alternative "technology study" is shown to be best with an expected net PW of $34.62M.
346
Risk and Uncertainty
Chap. 13 Table 13-8
ExrF.\THl NFT
PW
CALCUl.ATIONs FOR THE AuTOMATION PROBLEM WITH CoNSIDERATION oF TECHNOLOGY STUDY
Deci~ion
Poinl 2A
Alternai ive {Automate
Expecled Net PW $-90M(0.73) + $40M(0.22) $ + JOOM(0.05)
Not automate 28
{Automate
{Automate
$ - 90M(0.21) + $40M(0 .37) $ + JOOM(0.42)
Not automate Technology study
Not automate
= $43 .9M
Automate
= $12l.9M
Automate
so
Not automate
{'"'".'"
= $-4 1.9M $0
S- 90M(0.43) + S40M(0.34) $+ 300M(0.2Jl
Not automate
2C
Choice
$0
From labie 1J-7 From table 13-7 $0(0.41) + $43. 9M(0.35) $+ 121.9M(0.24) - $10M
$27M $0
= $34.62M
Technology study
13.3.3 Decision Tree Steps
Now thal decision trees (diagrams) have been introduced and the mechanics of using the diagrams to arrive at an initial decision have been illustrated, the steps involved can be summarized as follows : 1. Identify the points of decision and alternatives available at each point. 2. Identify the points of uncertainty and the type or range of possible outcomes at each point (layout of decision flow network) . 3. Estimate the values needed to make the analysis, especially the probabilities of different outcomes and the costs/returns for various outcornes and alternative actions. 4. Analyze the alternatives, starting with the most distant decision point(s) and working back, to choose the best initial decision .
The example above used the expected net PW as the decision criterion. However. if outcomes can be expressed in terms of utility units, the decision ma ker can use the expected utility as a decision criterion. Alternative) y, the decision maker may be willing to express his or her certain monetary equivalent for each chance outcome node and use thal as his or her decision criterion as explained in Sections 13.4 and 13 .5. Because a decision diagram can quickly become discouragingly , if not unmanageably, large, it is generally best to start out by structuring a problem
Sec. 13.3
Sequential Decision Making: Use of De_cision
Tre~s
347
simply by considering only major alternatives and outcomes in order to gel an initial understanding or "feel" for the problem . Then one can develop more information on alternatives and outcomes which seem sufficiently important to affect the final decision until one is satisfied thal the study is sufficiently complete in view of the nature and importance of the problem and the lime and stndy resources available . 13.3.4 General Principles of Diagramming
The proper diagramming of a decision problem is, in itself, general! y very useful to the understanding of the problem , and it is essential to correct subsequent analysis. The placement of decision points and chance outcome nodes from the initial decision point to the· base of any later decision point should give a correct representation of the information thal will and will not be available when the decision maker actually has to make the choice represented by the decision point in question. The decision tree diagram should show the following:
1. Ail initial or immediate alternatives among which the decision maker wishes to choose 2. Ali uncertain outcomes and future alternatives thal the decision maker wishes to consider because they may directly affect the consequences of initial alternatives 3. Ali uncertain outcomes thal the decision maker wishes to consider because they may provide information thal can affect his or her future choices among alternatives and hence indirectly affect the consequences of initial alternatives It should also be noted !hat the alternatives at any decision point and the outcomes at any outcome node must be:
1. Mutually excl usive ; thal is, no more than one can possibly be chosen . 2. Collectively exhaustive; that is, sorne one must be chosen or something must occur if the decision point or outcome node is reached . Figure 13-6 retlects these points . For example, decision points 2A , 28 , and 2C are each reached only after one of the mutually exclusive results of the technology study are known; and each decision point retlects ali alternatives to be considered at thal point. Further, ali possible outcomes to be · considered are shown as evidenced by the fact thal the probabilities sum to 1.0 for each chance node .
'-h 1
..._
'-\
Risk and Uncertainty
-•48
Chap . 13
13.3.5 Use of Statistics to Evaluate the Worth of Further lnvestigative Study
As was included in the automation example above, one alternative that frequcntly exists in an investment decision problem is further research or investigation before deciding on the investment. This means making an intensive objective study. hopefully by a fresh group of people . It may involve su ch aspects as undertaking additional research and development study, making a new analysis of market demand, or possibly studying anew future operating costs for particular alternatives . The concepts of Bayesian statistics pro vide a means for utilizing subsequent information to modify estimates of probabilities and also a means for cstimating the economie value of further investigation study . To illustrate, consider the one-stage decision situation shown in Fig. 13-7 in which each alternative has two possible chance outcomes: "high" or "low" demand . lt is estimated that each outcome is equally likely to occur, and the monetary result expressed as PW is shown above the arrow for each outcome. Again. the amount of investment for each alternative is shown below the respective !ines. Based on these amounts, the calculation of the expected monetary values (in net PW) is shown in Table 13-9, which indicates that the "new FMS" should be selected. To demonstrate the use of Bayesian statistics, suppose that one is considering the advisability of undertaking a fresh intensive investigation before deciding upon the "old system" versus the "new FMS." Suppose also that this further study would cost $2.0M. To use the Bayesian approach. it is necessary to assess the conditional probabilities that the investigation or "technology study" will yield certain results. These probabilities reflect explicit measures of management's confidence in .the ability of the ·investigation to predict the outcome. Sample assessments are shown in Table 13-1 O. As an explanation, P(ltill) means the probability that the predicted demand is "high" (Ir), given that the actual demand will turn out to be "high" (Il) .
Sequential Decision Making: Use of Decision Tree.t
Sec. /3.3
Appendix 13-A contains a formai statement of Bayes' theorem as weil as a tabular format for ease of calculations in the discrete outcome· case . Tables 13-1 1 and 13-12 use this format for revision of probabilities based on the data in Table 13-10 and the prior probabilitics of0 .5 that the dcmand will be high and 0.5 that the demand will be low. Table 13·9 EXPECTF.D NF.T PW FOR PRORLHI IN FIG. 13·7 Old system: S45M(0.5) + $27.5M(0.5) - SJOM New FMS: S80M(0.5) + $48M(0.5) - $35M
~
S21i.25M S29.0M
P(hj HJ ~ O. 70
P(/1jL)
= 0.20
P(~H) =
P(~L)
0.30
= 0.80
• lnvestigation-predictcd dcmand : h , high; /, low. Actual demand : Il, high ; L, Jow. Table 13·11 CoMPUTATION OF PoSTERIOR PRORABIUTIF.S GIV~N THAT (NVESTIGATION-PREDICTED DEMAND ls HIGII Ch) (1)
(2)
(4)
(5)
= (2)(3)
= (4)Œ(4)
Joint Probability
Posterior Probability , P (Statejlr)
(3)
State (Actual Demand)
Prior Probability, P (State)
Confidence Assessment. P (lrjState)
H L
0.5 0.5
0.70 0.20
:E
=
0.35 0. 10 0.45
0.7R 0.22
Table 13-12 CoMPUTATION oF PosTERIOR PROBABILITIES GIVEN THAT INVESTIGATION-PREDICTED DEMAND (S LOW (/)
$45M
(0.5)
$27 .5M
(0.5)
( 1)
(2)
0) ~
H
~
Table 13-10 MANAGEMF.NT's AssESSMENT OF CoNFIDENCE IN INVESTIGATION RESULTS'
PW of returns H
349
$SOM
(0.5)
$48M
(0.5)
Figure 13-7. One-stage FMS replace· ment problem.
State (Actua l Demand)
Prior Probability, P(State)
H L
0.5 /
0.5'~
(4) (2)(3)
(5) = (4)/L(4)
Confidence Assessment, P (ljState)
Joint Probability
Posterior Probability, P (Statej/)
0.30 0.80
0.15 0.40
0.27 0.73
:E = 0.55
V\ \
~
Risk and Uncertainty
350
Chap . /3
The probabilities calculated in Tables 13-11 and 13-12 can now be used to assess the "technology study" alternative . Figure 13-8 shows a decision tree diagram for this alternative as weil as the two original alternatives . Note the demand probabilities are entered on the branches according to whether the investigation indicates "high" or "low" demand.
PW of benefits $45M
H
$27 .5M
(0.5) (0.5)
Sec. 13.3
Sequential Decision Making : Use of Decision
The expected outcome for the "technology study" alternative can now be calculated. This is done by the standard decision tree "rollback" principle. This is shown in Table 13-13. His worthy of note thal the 0.45 and 0.55 probabilities that investigation-predicted demand will be "high .. and "low:· respectively, are obtained from the totals in column (4) of the Bayesian revision calculations shown in Tables 13-11 and 13-12. Thus, from Table 13-13, it can be seen that the "new FMS .. alternative with an expected net PW of $29.0M is the best initial course of action by a slight margin. Wh ile the figures used here do not reftect any ad van tage to this technology study, the ad v an tage potentially can be great.
A 3.3.6 H
$45M
(0.78)
L
$27.5M
(0.22)
H
$80M
(0.78)
L
$48M
(0.22)
H
$45M
(0.27)
$27.6M
(0.73)
Teehnology study ~ - 2M
Expected Value of Perfect Information
The expected value of perfect information (EVPI) is the maximum expected loss due to imperfect information or foreknowledge as to wh at will be the outcome(s) in a situation involving probabilities associated with the possible returns or costs . It is described more complete( y in Appendix 13-B. Calculations of the EVPI for the cxample replacement problem of Figure 13-7 are shown in Table 13-14. It can be seen in the table thal the expected net PW with perfect foreknowledge, $31.25M, is greater than the expected net PW of the preferred alternative before the further investigation. $29 .0M (from Table 13-9), by $2.25M, which is measure of the maximum expected value of further investigation . For the same example problem except with the different probabilities resulting from "technology study" as given in Fig. 13-8 and Table 13-13. the technology study results in a decrease in the expected net PW of $1.70M. (i.e., $29.0M for the "new FMS" minus $27 .3M for "technology study .. ). If the "technology study" had cost $0 instead of $2 .0M assumed in the prob-
a
EXPECTED
Decision Point H
$80M
(0.27)
$48M
(0.73)
2A
{
2B {
H
figure 13-8.
$80M
(0.5)
$48M
(0.5)
351
Tree .~
1
{
NET
Table 13-13 13-8
PW FOR REPLACEMENT PRORLEM OF FIG .
Choice
Expected Net PW
Alternative Old system New FMS
$45M(0.78) + $27 .5M(0.22)- SIOM = S31.15M $80M(0.78) + $48M(0.22) - $35M = S37 .%M
New FMS
Old system
$45M(0.27) + $27.5M(0.73)- SIOM = $22.23M
Old system
New FMS
$80M(0.27)
Technology study Old system New FMS
+ S48M(0.73) - $35M
= S21.64M
$37.96M(0.45) + $22 .23M(0.55)- S2M = S27.31M (from Table 13-9) (from Table 13-9)
S26 .25M S29.0M
New FMS
Replacement problem with alternative of further investigation.
0
-'\-)
3.54
.,
Risk and Uncertainty
Chap . 13
Sec. 13.5
355
Alternative Method of Decision Tree Analysi.t
13.5 ALTERNATIVE METHOD OF DECISION TREE ANALYSIS; NOTATION/ANALYSIS CONVENTIONS H
Sorne analysts and decision makers prefer to show the criterion values at the end of each possible pa th through the tree. (Note: If the monies along the path are at significantly different points in time, the criterion values should be expressed in terms of equivalent worths, such as PW or FW.) Then the "rollback" technique can be used to determine the optimal choice at each decision point and obtain the same initial decision as when using the previous method. · Suppose, for the replacement problem in Fig. 13-8, that it is desired to evaluate the alternatives using the certainty equivalent (CE) criterion while formally considering th at the decision maker's initial asset position is, say, $10M . Figure 13-10 depicts this as weil as a hypothetical solution to the problem. Note !hat the initial asse! position is shown in a dashed oval to the left of the initial decision point. and that the solid ovals at the extreme right indicate the outcomc's criterion values, (including the initial asset position) for each path of branches. The conventions which produce the results in Fig. 13-10 are as follows: 1. At each chance node (circle), the CE is shown in a rectangular box and represents the value of being at that node, and th us takes into account everything subsequent to (i.e., to the right of) the node. 2. At each decision point node (square) the CE of the best alternative is shown in a rectangular box. Further, ali alternatives except the best at each decision node are marked with double slashes across their respective paths. To complete the solution, the decision maker must specify his CE for ali chance outcome nodes. Note that the uppermost node is the same as in Fig. 13-9a. As in thal example, let's assume thal the decision maker decides that a certain $35.5M is just as desirable as 0.5 chance at $45M and 0.5 chance at $27.5M . This is shown in the small block above the node. Subjectively determined CEs are shown at ali other nodes. The "rollback" analysis procedure is applied but this time taking into account only the probabilities . and not the cash flow s. along each path (sin ce the effect of the cash flows along each path has already been laken into account by the equivalent worth outcomes (in ovals) at the end of each path). The final "rollback" to decision point 1 shows thal the alternative of "technology study" (with a CE of $35.7M) is better than either "old system·· (with a CE of $35.5M) .or "new FMS" (with a CE of $34M). This differs from the expected monetary (net PW) results in Table 13-13 as typi· cally would be anticipated.
145M(0. 5)~ 127.5M 10.51 ~
(~r Initial asset position
H $SOM
(0. 271~
$48M
(0. 73)~
~
180M(0.5)~
'""""! Outcome1
in cl udlng
ini tia18'SS!et po~ition
•
Figure 13·10. Replacement problem with alternative of further investigation (showing outcome criterion values) at the end of each branch and subsequent analysis using CE criterion and including initial asse! position . • Example for lowest branch path: SIOM - SJ5M + S48M = SZJ M.
V\
'
~
j. 3)~
Risk and Uncerrainty
Chap. 13
Table 13-14 EXPECTED VALUE OF PF.RFECT INFORMATION FOR ÜNE·STAGE REPLACEMENT PROBLEM OF FIG. 13-7 (ALL $ARE IN PW) If demand
Oulcomes for alternatives would he :
i~:
High
Low
Old: New:
$4 .~M ~OM
Then the And the The prior Thus the preferred monetary probability expected alternative is: oulcome is: estimate is: value is:
- $10M = $J5M - 35M = 45M
New FMS
$45M
0.5
Old: 27.5M - IOM = 17.5M Old system 17.5M 0.5 New: 48M - 35M = 13~ Expected net PW with perfect foreknowledge: Expected net PW (from Table 13-9) = S29.0M :. EVPI = SJ1.25M - S29.0M = $2.25M (in net PW)
S22.50M 8.75M
r = S3J.25M
lem . it would have resulted in an increase in the expected net PW of $-1.7M + $2.0M = $+0.3M . Thal $0.3M represents the "expected value of sample information·:: i.e ., how much out of the possible maximum EVPI of $2.25M can be achieved by this particular "technology study ."
13.4 USE OF CERTAIN MONETARY EQUIVALENTS Any problem involving variable outcomes, such as the replacement problem in Fig . 13-8. could just as weil be analyzed using the so-called certain monetary equivalent (CME) or certainty equivalent (CE) criterion rather than an expccted monetary value criterion . A certainty equivalent (CE) for any situation involving variable outcornes is a monetary amou nt (lump sum) which the decision maker considers to be equally as valuable (to pay or to receive) as the variable possible outcomes for thal situation. So-called "risk-averse" (risk-avoiding) decision . makers typically have (or will choose) a CE which is Jess than the expected monetary value (EMV) for a given situation. The amount by which the EMV is greater than the CE, called a "risk premium" (RP), is a measure of how much the decision maker is willing to give up, on the average , to eliminate the variability (riskiness) in a decision problem . Thus CE
= EMV
- RP
and
RP
=
EMV - CE
While RP is positive for a risk-averse decision maker (i .e., he or she would o pt to "bu y out" of a ris ky situation), it is negative in the case of a so-called "risk-seeking" (risk-taking) decision maker, and it is 0 in the case of a "riskneutral" decision maker.
Sec. 13.4
53
Use of Certain Monetary Equivalents
Example 13-l(a) Suppose thal it is desired for a decision maker to specify his CE for the situation in the left part of Fig. 13-9. The question to be answered can be phrased as: Wh at amount for certain (to be received or paid) would be just as good Cor bad) as a 50% chance of $45M and a 50% chance of $27 .5M? Another way of phrasing this is : "How much would you be wiUing to pay (or get paid) in place of [sorne probabilistic payoff situation]?" The decision maker might consider th at situation not very ris ky and answer, say $35 .5M. The EMV for the situation is $36.25M . which mcans that RP = $36.25M - $35 .5M = $0.75M. Example 13-l(b) Suppose thal the outcomes in Example 13-l(a) are made more risky (highly variable) by increasing the $45M by $40M to $85M and decreasing the $27.5M by $40M to $-12.5M . The situation is shown in the righi part of Fig . 13-9. The EMV is stiJl $36.25M, but the CE answer by a given decision maker might vary considerably according to his perception of the impact of the new possible outcomes . If the decision maker is very slightly risk averse for the amounts involved, he might stiJl have a CE close to the $35 .5M in Example 13-l(a) . However, if he is significantly concerned about the impact of a possible low outcome su ch as the - $12 .5M . his CE might be significantly lower , such as , say, $28 .0M. If so. his RP = $3t>.25M $28.0M = $8.25M.
A decision maker's certainty equivalent for a set of uncertain cash flow outcomes should depend on the complete context in which the cash flow occurs . Thal is, the decision maker should be cognizant of the firm's financial position when deciding on a CE . One way this could be taken into account is to specify the decision maker' s financial position before the initial decision is to be made. For example, perhaps net liquid assets (such as net assets thal can be converted into cash within a month or within a year) might be laken as the most appropriate mcasurc of the firm ' s initial asse! position.
(0.5) (0.51
Choice 2 CE
a
S27 .5M
10.5)
S85M
10.5)
S- 12.5M
7 +- (to equal choie• 11 (a)
Figure 13-9. determined .
S45.0M
(b)
Example of outcomes for which certainty equivalents are lo be
Risk and Uncertainty
3Sb
Chap. 13
Sec. 13.6
j 57
Stochastic Decision Trees
13.6 STOCHASTIC DECISION TREES
,,o~
lt is generally true that the number of possible outcomes that could result from the choice of an alternative is greater than the few outcomes typically assumed in decision tree problems so that the number of branches is kept to manageable size. Hence, information (and sometimes, objectivity) is sacrificed by the assumed reduction of branches. It is corn mon that the possible out cornes emana ting from an alternative could be described most adequately as a continuous distribution . These are sometimes called stochastic decision
'/
Not autom .
$0
trees .
As an example of a problem for which sorne kind of stochastic decision tree analysis is probably applicable, consider the automation problem in Fig. 13-6. Only thrce possible outcomes were considered in case the process were automated. However. the problem would have been more adequately mode led if, say, ten or more possible monetary outcomes were considered if the process were automated, taking into account the variable costs and benefits of automation. lndeed, the problem probably can be represented best by continuous probability distributions . Monte Carlo simulation can be used to analyze such decision situations efficiently to arrive at probability distributions of the decision criterion for ali initial alternatives. Section 13.7 will explain Monte Carlo simulation in general. Often, variable outcomes are thought to have a continuous distribution . but the decision maker or analyst feels that it is not.feasible or reasonable to specify the distribution (i.e., the decision problem fits the classical definition of uncertainty). A useful way to diagram an uncertain outcome·for a decision tree problem is shown in Fig. 13-11 . Note that this outcome "fan" can show maximum and minimum values if these can be estimated. As a further example, suppose the above automation problem involves corltinuous outcomes for which probabilities cannot be estimated but for which the minimum and maximum extremes for each outcome can be estimated. In this case, a diagram representing the problem (with assumed max-
«'•~'ve
~,)
Figure 13-11. Representation of a continuous outcome for which probabilities are not known.
Figure 13·1Z. Automation problem assuming continuou s outcomes for which probabilities may not be known , but extremes can be estimated .
imums and minimums) could be as shown in Fig. 13-12. Note that even the outcomes of the technology study are represented to have continuous , but unknown, probability distributions. White information in this form does not le nd itself to mathematical calculation of decision criteria such as EMV, the mere diagramming of the problem can be very helpful in reaching a decision. Indeed, a decision maker can be asked to determine his CE at any outcome node even though probabilities of the various outcomes are not known. Any such CE would be a gross guesstimate (certainly, it would be more subjective than if probabilities of various possible outcomes were known), but at !east it represents the decision maker's intuitive judgment under the circumstances and can be used for comparing alternatives .
~ ~
r. ,
J74
Risk and Uncertainty
Chap. 13
/)rcüinn Trers (Srctions 13.3 tllrou!!ll 13.6) :
B-9. Given : There is a 60% chance !hat Dynamics General Corporation will require a conversion from haselmnd Ethernet cabling in a certain manufacturing plant to hroadhand Ethernet cabling at a rate of 1000 feet per year (there is a 40% chance that the conversion will occur at a rate of 500 feet per year). The recabling will be required if a new government contract is negotiated with them now. which is 10 years prior to expiration of the present con tract. The VP of Manufacturingfeels that if he wait.r for the present contract to expire in 10 years. there is a 25% chance that the new contract will require 5000 feet per year of conversion from baseband to broadband Ethernet cable and a 75 % chance of conversion at the rate of 1000 feet per year. Each 1000 feet of Ethernet c<~ble conversion (baseband to broadband) costs $100,000. Pwpert y ta x for either type of cable is the sa me and should not be C
IJ-10. A purchasing manager is faced with deciding whether or not to stock a large supply of metal. The uncertain variable is the future priee of the metal. The following are present worths of consequences and prior probabilities for the various perceived outcomes : Future Priee
P(Furure Priee)
PW If Stock
PW If Don't Stock
High Medium Low
O.J
$ 100,000 - 10,000 -100,000
$0 $0 $0
0.5 0.2
For $6.000 it is possible to hire a consulting firm that would be able to make a fairly accurate forecast in terms of whether the priee will go up or down as follows :
Then the probabilities rhe consultant will predict the priee will go up or down are as follows:
If the future is going to be:
Up
Down
High Medium Low
0.9
0.4 0.8
0.1 0.6 0.2
Chap . 13
Exercise.r
375
(a) Diagram the problem in the form of a decision trec . (b) Determine what would be the best alternative using the EMY criterion . (c) Determine the maximum expected value of the consulting firm ' s ~ervices if the firm could perfectly predict the future. (d) Determine the best alternative if you were the decision ma ker using your own unique CE values. 13·11. The Norva Company has already spent $80,000 developing a new electronic gage and is now considering whether or not to market it. Tooling for production wou id cost $50,000. If the gage is produced and marketed, the company estimates that there is only one chance in four thal the gage would be success· fui. If successful, the net cash infiows would be $100.000 per year for 8 years . If not successful, the net cash outnows wou id be $30,000 per year for 2 years, after which lime the venture would be terminated. The minimum at · tractive rate of return on money is 20% per year. (a) Draw a decision tree and determine the best alternative using the expected net PW criterion . (b) If there is a market research group that can provide perfect information about the success of this product, what would be the most the company should be willing to pay for the group's service? (c) Suppose the market research group can make a market survey thal with probability 0 .8 will predict a successif the gage will actually tu rn out to be a success and with probability 0.9 will predict failure if the gage will tu rn out to be unsuccessful. Should the survey be undertaken first? What is the expected value of the survey to the company?
13-U. Given the following two-stage decision situation shown in Fig. E 13-12. determine which is the best initial decision . Notice thal sorne annual costs are negatively signed, which makes them positive cash flows. Use the expected PW of costs method and a MARR of 12%. To give the problem a physical context, the following letter symbols have been employed for each alternative: BSW: Build small warehouse. RLW: Rent large warehouse. BA: Build addition. NC: No change.
13-13. A firm must decide between purchasing an automatic machine which costs $50,000 and will last 10 years and have 0 salvage value, or purchasing a manual machine which costs $20,000 and will last 5 years and have 0 salvage value. If the manual machine is purchased initially, a ft er 5 years a decision will have to be between a manual machine having the same characteristics affecting cost as the first manual machine and a semiautomatic machine cost· ing $40,000 which would have a $20,000 salvage value after 5 years of life. The annual operating tosts for each of the machines are as follows : automatie, $10,000; manual, $14,000; semiautomatic, $11.000. (a) Graphically construct a decision tree to represent this situation .
~
-
N
Risk and Uncertainty
376 First cost and resale
Annual cost (or savings, if negative)
Chap. /3
Chap. 13
Exercises
Jll
$20,000/ yr, 25 yr
Probabillties
value of alternatives
1 15,000/yr, 15 yr Keep 11 0,000/ yr, 10 yr, 10 salvoqe
$-2,000/yr (10 yr)
0.15
11 ,000/yr (3 yr) and 110,000/yr (7 yr)
0.15
Abandon $0/yr, 10 yr Salvage • 120,000 $0/yr, 15 vr. salvage • $25,000
1- 8,000/yr (7 yr)
0.2
12.000/ yr (7 yr)
0.8 $20,000/yr, 15 yr Keep $15,000/yr, 15 yr, 10 salvag<~
$1 ,000/yr (7 yr)
Abandon 10/yr, 15 yr
0. 2
Salvage • $30,000 118,000/yr (7 yr)
-<-.
0.8
"J. O~"~
$20,000/ yr, 25 yr 118,000/ yr l10yr)
~
118,000/yr (3yr)and $21,000/yr (7 yr)
0.15
$23,000/yr (3 yr) and $21 ,000/ yr (7 yr)
0.56
123,000/yr (10 yr)
0.14
110,000/yr, 15 yr Keep $5,000/ yr, 10 yr
Abandon $0/yr , 10 yr
Figure E13-12. Two-stage decision situation for Exercise 13-12.
(b) Determine which decision would be made at each point using the PW of costs method and an interest rate of 10%. (c) At what interest rate would the decision between the manual and semiautomatic machine be reversed? 13-14. Suppose one is faced with the same alternatives and dollar outcome consequences as in the replacement problem depicted in Fig. 13-8. However, the initi al estimates of probability of demand are : high, 0.6; low, 0.4. Furthermore . management's assessment of confidence in further investigation results, using the notation in Table 13- 10 are
P(h !Hl P(hiL) P(liH) P(I!Ll
0.80 0.40 = 0.20 = 0.60
= =
Calculate the choice at each decision poini to determine the best initial dcci- . sion. How close is the initial decision with these revised probabilities to the initial decision for the original problem depicted in Fig. 13-8? 13-15. Figure E13 - 15 is a decision trec portrayal of a building lease versus buy problem with input data supplied. lnvestment requirements are shown as negative numbers : probabilities associated with each outcome are shown in
Abandon 10/yr, 15 yr
Figure E13-15.
Decision lree for Exercise 13- 15.
parentheses. The annual cash savings and duration of those savings are shown together at each relevant outcome. Salvage values in the cases of abandonments are assumed to occur at the end of the 25-year study period . Determine the best decision using the expected net PW method with a minimum required RR of 0%. 13-16. Set up a decision tree to reHect the persona! automobile alternatives which you expect over the next severa! years. Carry the trec far enough in ti me to show severa! decision points at which a decision must be made between keeping an old car and buying a new car (from a possible choice of several). Show your roughly estimated assumed certain investment costs and salvage values and an nuai operating costs for each alternative and determine the best initial decision using the PW method and a 10% minimum persona! opportunity cost of moncy. 13-17. Suppose, for the automation problem for which prior probabilities are shown in Fig. 13-5, the probabilities of the possible outcomes for the added study (technology study) are as follows :
V\ \ ~ ('0
Ri.
371!
Givcn lhnl lhc mllomnlion rc~uli~ will IUrn OUI 10 be : Pnor Fair Exce llcnl
Clzap. 13
Thcn lhc prohahilily lhal lhc lcchnnlogy ~lully will indicnle - - - - is: Shaky
Promi~ing
Sol id
0 .6
0.3 0.4 0.4
0. 1 0.3 0.5
O.J 0. 1
lise Rayes" thcorcm to cnlculn lc the posterior prohnhililies regarding whal the a utomation results will lurn oui ln he. givcn whal lhe lcchnology sludy in
J\ 1 ~
~
6-1
CHAPITRE6 ÉLÉMENTS DE BASE DE LA SIMULATION DES SYSTÈMES 6.1
Technique de Monte Carlo :
Méthode développée durant les années 40 par Von Neuman, Vlan et Fermi pour résoudre certains problèmes de design d'écrans anti-rayonnement. Elle est coûteuse expérimentalement et difficile à résoudre analytiquement. Elle consiste à représenter un problème déterministe par un processus stochastique dont les distributions de probabilité satisfont les relations mathématiques du problème déterministe complexe. La première étape consiste à construire un modèle analytique qui représente l'investissement considéré (exemple: une équation pour la valeur présente). Dans la deuxième étape, l'analyste développe une distribution de probabilité pour chaque facteur qui est sujet à l'incertitude dans le modèle. Il peut utiliser des données historiques ou une approche subjective.
À partir des distributions de probabilités pour chaque facteur dans le modèle. Il génère au hasard une réponse tentative. La répétition de ce mode d'échantillonnage un nombre important de fois, permet de générer une distribution de fréquence pour la réponse. On peut utiliser n'importe laquelle des méthodes de mesure de rentabilité. La distribution de fréquence résultant peut être utilisée pour obtenir des données probabilistes sur le problème original.
Exemple 6.1 : Calcul de
1t
Si on met aléatoirement un certain nombre de points (n) dans le carré. Un certain nombre de points appartiendront au cercle et d'autres non
6-2
- Quand n est grand, la surface du carré sera couverte. n =>
~=>
oo
surface cercle surface du carré
#cercle
# total
(2a)
nb. pt. _ds c ercle nb. pt. ds carré
n = 4·
2
n-4
nb. ds cercle nb. ds. carré
x ------
a
=1
M
= [: : ]
r 1 r2 des nombres aléatoires entre -1 et 1.
On a généré 30 pts, 24 e cercle et 6 e cercles. n = 4
* ~= 30
3 .2
Sin-oo, 1t-3 .14 Exemple 6.2 : Le pari
Supposons que l'on vous propose de jouer au jeu suivant: pour une mise de 20 dollars, on vous permet de jouer à pile ou face avec une pièce bien équilibrée et de gagner 2\ x étant le nombre de lancés nécessaires pour avoir pile. Accepterez-vous de faire la mise pour jouer? Quel est le montant maximum (supérieur ou inférieur à 20 $)que vous accepterez de risquer pour jouer? Votre réponse dépendra de votre goût pour ce type de risque. Je n'ai encore jamais rencontré quelqu'un qui dit accepter de miser jusqu'à 100 $. Pourtant. .. Une approche analytique de ce problème de pari suggérera le recours à l'espérance mathématique. Dans ce cas, on arrivera à la conclusion qu'il faut toujours jouer à ce jeu quel que soit le montant de la mise demandée, car l'espérance mathématique du gain est infinie! C'est ce qui ressort du calcul ci-après.
6-3
E ( ga i n)
E (gain )
= "
LJ
=L =
( 2n 2n - 1
2
.
22 )
1
!><)
Une approche de simulation est facile à concevoir et à mettre en oeuvre: essayez sans frais, avec une pièce de monnaie, de jouer à pile ou face et déterminez ce qu'auraient été vos revenus si vous aviez réellement joué. Vous pourriez obtenir les résultats suivants :
1 2
3 4 5
ftfp fp ffifp p p
Chaque essai est bien terminé avec une sortie de pile et les montants gagnés sont de 2 à une puissance égale au nombre de fois qu'il a fallu lancer la pièce pour avoir pile.
À la vue de ces résultats on va être frappé par le gros gain comme 32 ou le petit 2 selon qu'on a le tempérament de joueur ou non. En faisant la moyenne des gains on trouve 56/5, soit environ 11 . Même si ces résultats ne sont pas très significatifs (échantillon trop petit et probablement biaisé par la manipulation), ils livrent néanmoins un message intéressant: au delà de 30 la mise est trop risquée. La simulation ne nous donne pas LA solution; elle nous donne des indications nous permettant de prendre une décision selon notre tempérament. Cette situation est confortable pour le décideur car on ne lui impose pas une solution fut-elle optimale. L'exemple du pari va nous permettre d'illustrer aussi un aspect essentiel de la simulation Monte Carlo : la simulation de la réalisation d'un événement à partir de la probabilité attachée à la réalisation de cet événement. Une pièce de monnaie permet normalement de bien simuler le comportement d'une autre pièce de monnaie équivalente. Cependant les résultats obtenus par le jet d'une pièce de monnaie ne sont pas tout à fait aléatoires: les façons de jeter la pièce, de la saisir sont propres à l'opérateur et non au hasard. Une autre limitation de la technique de simulation par le jet de la pièce de monnaie apparaît lorsque l'on envisage de simuler la réalisation non pas de deux événements équiprobables (pile -face) mais de plusieurs (pile 1 fois ou 2 fo is ou 3 fois ou 4 fois en 4 lancés).
6-4
La simulation Monte Carlo permet d'arriver au même résultat que la simulation manuelle du jet de la pièce de monnaie. En faisant la correspondance suivante : pile= {0, 1, 2, 3, 4} face= {5, 6, 7, 8, 9} et en supposant que lors d'un tirage au sort parmi ces nombres équiprobables la sortie d'un chiffre traduit le fait que la pièce de monnaie a montré le côté (pile ou face ) correspondant. Par exemple, si le nombre tiré est 2, c'est pile; 8 c'est face, etc. Il ne restera plus alors qu'à trouver un système de génération de nombres aléatoires pour simuler le jet de la pièce de monnaie. Ce procédé a l'avantage d'être plus neutre. En plus, il permet de faire face au cas où il y a plus de deux événements. Si nous avons trois événements par exemple, il suffit de connaître la probabilité attachée à chacun de ces événements et de faire la correspondance des chiffres conséquemment.
Exemple 6.3 : Estimé des ventes annuelles On a estimé que les ventes annuelles d'un produit d'entretien ménager ont 60% de chance d'atteindre 200,000 dollars, 30% de chance d'atteindre 300, 000 dollars et 10% de chance d'atteindre 400,000 dollars. Pour simuler le comportement des ventes annuelles, on fera les correspondances suivantes sur la base de 10 :
200 000$ 300 000$ 400 000$
60% {0, 1, 2, 3, 4, 5} 30% {6, 7, 8} 10% {9}
Il ne restera plus qu'à tirer les nombres aléatoires pour simuler le comportement des ventes annuelles. Si les pourcentages n'étaient pas arrondis, il aurait fallu faire la correspondance sur la base de 100 nombres. Essayez pour l'exemple précédent : 61%, 29% et 10% respectivement. Essayez ensuite 61.5%, 28 .5% et 10%. Exemple 6.4 : Comparaison de résultats analytiques et de simulation
[!]
Une clinique médicale dispose de 5 centres.
[!]
Une ( 1) infirmière est affectée à chaque centre.
[!]
Il y a 3 infirmières réservistes qui peuvent être appelées pour remplacer celles qui s'absenteraient.
Les infirmières peuvent s' absenter en cas de maladie. La distribution de probabilité des infirmi ères
6-5
malades est représentée dans le tableau suivant :
Nb. de malades
0
1
.2
3
4
5
Probabilité
.2
.25
.2
.15
.1
.1
On désire simuler 25 jours de fonctionnement de la clinique à l'aide de la technique MONTE CARLO pour obtenir des estimés de :
[!] [!]
le taux d'utilisation des infirmières réservistes; la probabilité qu'au moins un centre médical soit fermé par manque d'infirmières.
Feuilles de statistiques pour la simulation du cas des infirmières.
NIG NRU NJCF
NIG NRU NJCF
Analytiquement Nombre moyen d'infirmières malades= Nombre moyen d'infirmières réservistes utilisées= La probabilité qu'au moins un centre médical soit fermé par manque d'infirmières =
6-6
Expérience de simulation Nombre moyen d'infirmières malades = Nombre moyen d'infirmières réservistes utilisées = La probabilité qu'au moins un centre médical soit fermé par manque d'infirmières =
Disque d'échantillonnage pour le cas des infirmières :
NIG U)
D
=0
NIG U) = 1
1
s Q
u E
NIG (j) =nombre d'infinnières malades au jour j
6-7
Logigramme du modèle de simulation
INITIALISAnoN NRU NJFC 0 NIG=O
=
=
NJCF = NJCF + 1 NRU= NRU+ 3 NIG = NIG + NIG UJ
UIR = NRU/75 PCF = NJCF /25
_j
1 Ill lill 1 1 1 1 Il 1 1 1 1 Il 1 Il 1 1 1 1 1 1 1 1 1 1 1 1 Il 1 1 1 1 1 1 L GLOSSAIRE NJCF = #jour oa au moins 1 centre fermé NRU = # réserviste -jours utilisés NIG (j) = 11 infinmière en grève au jour j UIR = utilisation des infinmières de réserve PCF = probabilrté qu'au moins un centre sort fenmé
1 11111111 Il l Ill 111 11 11 Ill 1 I l Ill lill 1 I l Il l I l 1 1
6-8
6.2
Considérations dans la sim.ulation de Monte-Carlo
Une question importante à connaître dans la simulation de Monte-Carlo est le nombre d'essais (ou d'exécution) nécessaire pour avoir une réponse satisfaisante. La réponse est que le nombre d'essais doit être assez élevé pour atteindre le régime permanent. On peut définir les conditions de régime permanent comme une situation dans laquelle le résultat d'essais consécutifs ne varie pas significativement. Dans un simulateur, on utilise habituellement une période de réchauffement avant de collecter les statistiques, l'utilisation des simulations de Monte-Carlo présente des avantages pour: Les situations dans lesquelles des méthodes de résolution analytique n'ont pas encore été trouvées; Des cas où les modèles analytiques sont très complexes à résoudre; Ne pas perturber le fonctionnement de la compagnie; Des situations pour lesquelles il est très difficile ou dangereux de créer les mêmes conditions d'opération pour chaque expérience; Faciliter l'expérimentation de plusieurs possibilités qui ne peuvent être réalisée en vraie vie; Des situations qui peuvent coûter chères et/ou prennent beaucoup de temps pour générer le résultat voulu. Les modèles analytiques et la simulation de Monte-Carlo ont les mêmes problèmes, à savoir, la validité du modèle. Cependant, dans le cas de la simulation de Monte-Carlo, la taille de l'échantillon doit être assez grande pour décroître la variation d'échantillonnage à un niveau acceptable. Les probabilités d'es événements doivent être basées sur les jugements de personnes impliquées dans le projet, par la suite, l'analyse doit être dosée sur ces estimés bien qu'ils soient subjectifs étant donné que chaque modèle est aussi bien que les estimés de ses paramètres d'entrée, la simulation de Monte-Carlo doit être utilisée avec le jugement du décideur.
6.3
GÉNÉRATION DE NOMBRES ALÉATOIRES Les systèmes habituellement simulés comportent des processus stochastiques; ex. :
demande pour un produit; temps d'opération sur une machine; arrivé de clients à une station de service; temps de réalisation d'un projet. etc.
Pour simuler adéquatement la réalisation des événements dont on connaît les probabilités de réalisation, il ne suffit pas de faire les correspondances que nous venons de voir. Il faut avoir aussi un système de génération de nombres équiprobables (ou aléatoires). Malgré l'abondance des séries de chiffre dans notre entourage: bottin de téléphone, numéro de rue, etc., on ne compte que deux sources fiables de nombres aléatoires: les tables de nombres aléatoires publiées (on en trouve dans les annexes de la plupart des manuels de statistiques et de simulation) et les générateurs de nombres aléatoires des ordinateurs.
6-9
Les tables se prêtent à l'utilisation manuelle. Les nombres contenus dans ces tables ont été soumis à divers tests statistiques pour s'assurer de leur caractère aléatoire dans toutes les directions: horizontale, verticale et oblique. Tous les chiffres sont équiprobables; il en est de même des nombres de deux chiffres, de trois chiffres. Lorsque l'on veut utiliser ces tables, il est conseillé de fixer la direction de lecture de nombres et la position de départ et de ne pas revenir sur ses pas. Voir la table de nombres aléatoires. Les ordinateurs possèdent des générateurs de nombres aléatoires. Il suffit généralement de les évoquer. On peut aussi programmer la génération des nombres aléatoires ou les calculer soi-même.
On a donc besoin de générer des variables aléatoires (dépendantes ou indépendantes) ayant des distributions spécifiées. C'est-à-dire un échantillonnage d'une certaine population. L'idée est : générer des variables aléatoires uniformément distribuées; -t générer des variables d'autres distributions. 6.3.1
Distributions uniformes Ses caractéristiques sont :
1) Chacun des nombres de la population a la même probabilité d'être dans la série : 2) Chacun des nombre est complètement indépendant de tous les autres déjà générés.
.... 6.3.1.1
Variables aléatoires indépendantes uniformément distribuées .
Méthode manuelle Pile ou face, dés, cartes, roulette, etc. Avantages : Très simples, honnêtes. Désavantages : Lentes, peu pratiques, impossible à reproduire.
6.3.1.2
Tables Pris dans n'importe quelle direction prédéterminée, à partir de n'importe quel point. A vanta ge : reproductibilité Désavantages : Lenteur de l'emploi, utilisation continuelle des mêmes séquences (tables limitées).
6-10
Table de nombres aléatoires
Table A.t.
RANOOM DIGITS
94737 87259 63856 66612 30712
08225 85982 14016 54714 58582
35614 13296 18527 46783 05704
24826 89326 11634 61934 23172
88319 74863 96908 30258 86689
05595 99986 52146 61674 94834
58701 68558 53496 07471 99057
57365 06391 51730 67566 55832
i4759 50248 03500 31635 21012
69607 37792 01488 66248 51453
24145 27282 56680 97697 03462
43886 94107 73847 38244 61157
86477 41967 64930 50918 65366
05317 21425 11108 55441 61130
30445 04743 44834 51217 26204
33456 42822 45390 54786 15016
34029 28111 86043 04940 85665
09603 09757 23973 50807 97714
92168 36463 47097 80400 94554
82530 07331 78780 45972 13863
19271 54590 04210 88239
86999 00546 87084 99708 91624
96499 03337 44484 45935 00022
12765 41583 75377 03694. 40471
20926 46439 57753 81421 78462
25282 40173 41415 60170 96265
39119 46455 09890 58457 55360
31567 07821 09056 19922 29923
53597 24759 10709 37025 02570
08490 47266 69314 80731 80164
73544 21747 11449 26179 36108
72573 72496 40531 16039 73689
30961 77755 02917 01518 26342
12282 50391 95878 82697 35712
97033 59554 74587 73227 49137
13676 31177 60906 13160 13482
29602 94135 87926 85039 66070
29464 94661 34092 19212 38480
99219 87ï24 34334 59160 74636
20308 88187 55064 83537 45095
82109 62191 43152 54414 86576
03898 70607 01610 19856 79337
82072 63099 03126 90527 39578
85199 40494 47312 21756 40851
13103 49069 59578 64783 53503
78166 94672 56406 67726 07516
82521 07912 70023 57805 45979
ï9261 26153 27734 94264 16i35
12570 10531 22254 77009 46509
10930 12715 27685 08682 17696
47564 63142 67518 18784 67177
77869 88937 63966 47554 92600
16480 94466 33203 59869 55512
43972 31388 70803 66320 17245
43070 36917 03919 46724 16108
22671 60370 82922 22558 61281
00152 80812 02312 64303 86823
81326 87225 31106 78804 20286
89428 02850 05762 14025
16368 47118 05573 70650 24909
Sï659 23790 17470 56117 38391
79424 55043 25900 06707 12183
57604 75117 91080 90035 89393
74541 82919 31388 17190 00466
75808 31285 26809 75522 88068
89669 01850
87680 72550 99360 07161 98745
72758 42986 92362 99745 97810
60851 57518 21979 48767 35886
55292 01159 41319 03121
95663 01786 75739 20046 90230
883:!6 98145
+till
77'158
15687 68631
44~35
1~97
9808~
28013 69264
6-11
6.3.1.3
Utilisation de formules récursives
Utilisation d'équations mathématiques de récursion (ex. : à partir d'un nombre, un autre nombre de la série est généré). Puis qu'on peut reproduire ces nombres, ils sont appelés pseudo-aléatoire. Cette reproductivité est leur avantage majeur.
6.3.1.3.1 Méthode du centre du carré Chaque nombre aléatoire composé den chiffres est formé des n chiffres du centre du carré du nombre précédent. Exemple : nombre aléatoire de 2 chiffres (0 ;!; Nombre
Source= 12
Xo= 12
Xo2 = 144
X 1 =44 X 2 =93 X 3 =64
Xj
;!; 99).
x = 1936 X/= 8649 2 1
(NB. On pourrait aussi alterner)
Désavantage : Dégénérescence rapide (cycle court) notons qu'aussitôt qu'un nombre apparaît pour la 2ième fois, il y a création d'un cycle.
6.3.1.3.2 Méthode du centre du produit Chaque nombre aléatoire Xj composé de n chiffre est formé des n chiffres du centre de de ~- 1 où K est une constante : Exemple :
Nombre de 2 chiffres (0;!;~;!;99) . Nombre source X 1=8; X 2 = 86. X 1 =8 X 2 = 82 X 3 =56 X 4 =59 X 1X 2 = 656 X 2X 3 = 4592
~- 1 *~
ou
6-12
6.3.1.3.3 Méthodes congruentielles Additives : "n+I =(a "n + C) (modulo rn) Multiplicatives : "n+I
=(a~
(modulo rn)
{I(modulo rn)= reste de la division deI par rn}
Exemple:
13 (mod 5) = 3. rn, c, a, sont des entiers positifs.
Exemple :
a= 2; rn= 11;
X0
= 4; c =O.
x 1 = (2*4) mod 11 = 8 x2 = (2*8) mod 11 = 5 x3 = (2*5) mod 11 = 10 ... Notons que les nombres générés sont entre 0 et 10. 0
:S
"n
:S
m-1.
TOUTES CES MÉTHODES POSSÈDENT 1 OU PLUSJEURS DES CARACTÉRISTIQUES: les nombres peuvent être reproduits; possibilité de cyclage; possibilité de dégénérescence (toujours même nombre); les n.a. sont± uniformes. Il existe des tests pour vérifier l'uniformité et l'auto-corrélation des séries (pas dans le cadre du cours) . La méthode Congruentielle est la plus utilisée en pratique : Poser rn = 2b (b entier positif) a = 8t ± 3 (t entier positif) a= 2 bn ko = entier impair. Tous les nombres seront :S 2b- 1. Il y aura au maximum 2b-2 récursions avant de former un cycle.
Exemple :
Si b = 21 ; 2 19 . Pour U(O, 1), diviser les nombres par 2b - 1.
6-13
Exemple 6.5 : Prévision de la demande La compagnie SOW Inc. cherche à se faire une idée plus précise de ce que sera la demande pour son nouveau modèle d'ordinateur. Ce type de produit informatique étant vite désuet, la compagnie ne veut pas se retrouver avec des invendus à la fin de la période de planification de trois mois. Utilisons des données historiques pour simuler la demande trimestrielle.
2 3 4
.09 .27 .37 .18 .09
3 4 2 1
Total= 11
Correspondances :
0 = {00, 01, 02, 03, 04, 05, 06, 1 = {09, 10, 11, 12, 13, 14, ... 2 = {36, 37, 38, 39, 40, 41, ... 3 = {73, 74, 75, 76, 77, 78, 4 = {91 , 92, 93, 94, 95 , 96, 97, 000
07, 08} '35} ' 72} 90} 98, 99} '
(9) (27) (37) (18) (9)
6-14
Tirage de nombres aléatoires: Dans la table des nombres aléatoires, prenons la colonne formée par les chiffres à l'extrême droite et associons chaque nombre trouvé avec sa demande correspondante. Simulons 20 semaines.
-
-=--,. . -. -~
"' ...._~P.: 18_qm_br.e~
-Je_atgire,..,,.....
::::-~ ~
f•
-
_. qemant;t~~sJ.!IlP.~.a _-
95 31 13 86 02 79 85 87 87 64 54 59 39 03 52 53 22 34 74 38
4 1 1 3 0 3 3 3 3 2 2 2 2 0 2 2 1 1 2 2
Total=>
39
Ce qui donne une demande de 1.95 en moyenne, arrondie à 2. On pourrait penser utiliser l'espérance mathématique pour faire l'estimé de la demande à partir des probabilités attachées à la réalisation de chaque volume de demande hebdomadaire. Il faut remarquer cependant que l'espérance mathématique est figée. Elle indique ce vers quoi tend la moyenne des demandes sur le long terme sans permettre à dame nature d'être capricieuse. La simulation permet ce caprice de la nature: par exemple, même si la demande de 1 unité ne survient normalement que 9 fois sur 100, la simulation peut générer plus ou moins que cette proportion, donnant à ce volume de demande un poids plus grand ou plus petit dans la moyenne calculée. L'espérance mathématique figera ce poids à 9 pour cent.
6-15
Exemple 6.6 : Gestion des stocks Modèle analytique de base Le problème de la gestion des stocks est bien classique: minimiser le coût total de cette gestion en faisant le trade-off entre le coût occasionné par les lancements de commandes d'approvisionnement et le coût de conserver des 1Ùveaux élevés d'inventaire. Un modèle très simple a été mis au point par l'approche analytique sur la base de quelques hypothèses simplificatrices. Soient : D A H L R Q
la demande annuelle le coût de lancement d'une commande le coût unitaire de maintient des inventaires le délais de livraison le niveau de stocks à partir duquel on doit commander la quantité à commander à chaque fois qu'il faut le faire
La quantité optimale à commander notée Q* pour minimiser le coût total de la gestion des stocks est donnée par la formule suivante :
Q* = /(2AD/H) et
R =dL (d étant la demande durant le délais de livraison R) Exemple : D = 1 000 unités par an; A = 20 $ par commande; H = 4 $ par unité; L
=
2 semaines; alors :
Q* = v'(2*20* 1000/4) R
= 100 unités = (1 000 unités/50 semaines) * 2 semaines =
40 unités
On le voit : la formule de la quantité optimale à commander est élégante et facile à utiliser. Cependant on comprend aussi qu'elle n'est appropriée que pour des cas d'une simplicité rarement rencontrée dans la vie réelle: demande connue, fixe et régulière; délais de livraison connus et constants, etc. La simulation permet de faire face aux situations plus réalistes.
Modèle de simulation Ici on va supposer que la demande varie à la semaine, que les délais de livraison sont susceptibles de changer; que des ruptures de stocks peuvent survenir et que cela comporte un coût estimé à 30 dollars par unité manquante. Le lancement d'une commande coûte comme précédemment 20 $; on suppose que l'on commence avec 40 unités en inventaire; le coût de maintient des inventaires est de 4 $ par unité par semaine. Par ailleurs on a obtenu les données suivantes sur les demandes hebdomadaires et les délais de livraison observés par le passé.
6-16
10 20 30 40
10 20 10 10
1 2 3
4 8 4
La simulation nous permettra de comparer plusieurs politiques alternatives dans la gestion des stocks, chacune des politiques consistant dans une réponse aux 2 questions : Combien commander (Q Quand (R= ?)
=
?)
Dans cet exemple manuel, on se contentera de faire la simulation pour la politique qui consiste à prendre Q = 40, R = 1 O. Les semaines successives traduisant le déroulement du temps; les nombres aléatoires pour simuler la demande de chaque semaine; les nombres aléatoires pour simuler les délais de livraison lorsqu'une commande est lancée; les demandes simulées; les délais simulés; les commandes lancées; les commandes reçues; les unités en stock; les coûts de maintien; les coûts de commandes et les coûts de rupture.
6-17
;;:. ~i$f'
! ,~· ri~ 1\..
t
··••.Do'.·· .:.~, ·-~~. D
d
)... '
~
;~
7)
1~
D
.
.~
..~
• · ~·~, t;.<,;,';Y~ v.··~vu
' "'(k;''
.&~~m ~
·-~'" - ~ '
c
'
ar
0 1
69
45
2
48
52
3 4
84 46
11
5
66
6 7
27
8 9
10
49 69 68 33
12
20 09
13
05
14 15
72
11
16 17 18 19 ?0
19 _76 38 41 74 95
46 52 77
30 20 40 20 30 20
20 30 30 20 20 10
JO
24 08 50
10 52 98 59 27 42
08
31
10 20 10
f)Q
_50
30_
80
30 40
40 40 40
40 80
40 40 40
40 _40 40
40 40 40
20
<.J._
~,
<
t 1-~
"'
•
' "'~!:.~
~ ·"<"t':.""-
~·nL!'
, •. 1>
. $)Q1i!~
B
H
A
40 10 (10)
40
20 20
60 30 10 (1 0)
240 120 40
10 20 40 20
40 80 160
10 40 40
10 40
40
60 50
(10)
80 40 40 160 240 200 120
20 20 20
)'IJ.fi.
Tot
300
60 320 20 240 120 60
300
320 60 80 160 80
20 20 20
60 20 60 160 240 200 120 80
80 0
·~.
.,~,
ru
20
30 20 40
··--.-
?0
100
1?0
6- 18
Exemples 6.8: Durée de vie d'un équipement
Considérons la distribution des probabilités de la durée de vie utile d'une pièce d'équipement : An
Probabilité .20 .40 .25 .15
3 5 7 10
LP= l.O
Pour simuler la vie utile on assigne un nombre pris au hasard à chaque valeur de façon à ce qu'il soit proportionnel aux probabilités respectives. On prend le même nombre de chiffres significatifs que pour les probabilités.
A
Nombres 00-19 20-59 60-84 85-99
3 5 7 10
On simule une durée de vie utile en prenant un nombre au hasard dans une table à cet effet. Si on obtient un nombre entre 00 et 19 la vie utile serait de 3 ans.
Exemple 6.9: Paramètre suivant une loi normale Considérons qu'un cash flow net annuel a une valeur moyenne de 50 000$, une déviation standard de 10 000$ et il suit une distribution normale. la valeur simulée ou cash flow pour une période de 5 ans peut être calculée AN
Déviation standard prise au hasard (DSH)
Cash Flow net [50 000 + DSH (10 000$)]
1
0.090
50 900
2
0.240
52 400
3
-0.448
45 520
4
0.295
52 950
5
-0.292
47 080
:E = 248 850 1 5 = 49 770$ Cette valeur moyenne se compare à la valeur connue de 50 000$; erreur de 0.46%
6-19
Exemple 6.10: Paramètre suivant une loi uniforme La distribution de probabilité qui décrit un événement au hasard est uniforme et continue avec une valeur minimale A et une valeur maximale B. La réponse simulée s'obtient: , Reponse = A+
OuNH: (NH)M:
N H
(B-A)
(N H) M
Nombre au choisi hasard Nombre au hasard maximal possible. (9 pour un chiffre significatif, 99 pour deux ... )
Par exemple, la valeur de récupération à l'année d'une pièce d'équipement suit une distribution uniforme et continue entre 8 000$ et 12 000$. La réponse simulée pour un nombre au hasard de 74 se calcule: Réponse = 8 000+_2i (12 000 - 8 000) = 10 990$ 99
L'utilisation adéquate de ces procédures avec un modèle précis permet d'obtenir des approximations valables. Un nombre suffisant d'essais doit être fait pour que la valeur moyenne cumulative demeure relativement constante.
Exemple 6.11: Simulation d'un cash flow complet
Considérons un investissement normalement distribué ayant une moyenne de - 50 000$ et une déviation standard de 1 000$. Une vie utile uniformément distribuée minimale de 10 ans et maximale de 14 ans. Les revenus annuels prévus $ 35 000 40 000 45 000
probabilités 0.4 0.5 0.1
Les dépenses annuelles sont normalement distribuées avec une moyenne de - 30 000$ et une déviation standard de 2 000$. Le gestionnaire désire connaître la probabilité que l'investissement soit profitable, il utilise un taux d'intérêt de 10% et la méthode de la valeur présente. Le tableau 6.4 illustre l'utilisation de la simulation de Monte Carlo pour 5 essais. Pour obtenir des résultats plus précis, il faudrait utiliser un ordinateur et des centaines d'essais. On doit se rappeler que la précision de la réponse dépend directement de la précision du modèle et des probabilités utilisées. La valeur présente moyenne des 5 essais égale 19 004/5
= 3 801$.
La solution de ce même
6-20
problème en utilisant 3 160 essais sur un ordinateur a permis d'obtenir une moyenne de 7 759$. La figure illustre les résultats provenant de l'ordinateur. La valeur médiane égale 6 700$ et 59.5% de toutes les simulations donne une valeur présente de 0 et plus. Donc, le management aura environ 4 chances sur 10 de ne pas atteindre le 10% de retour sur son investissement.
... ....
l:l• 10• 71• 71•
7•. 72•
.......... •• ... ... 70•
10•
lW•
................... ............... ..
.... .... .... JO•
............... •• .•.' .... . ......................... ......................... ......................... ......................... ...........................................
•2. oiO•
31• 31• :M• 32• 30. 21•
• •••••••••••••••••••••••••••••••••••••••••• • ••••••••••••••••••••••••
•• • •••• •• •• • •• •• • • ••••••••••••••••••••• •• ••• •••• ••• ••••• ••• •
2t• 24'
22. 10. tt'
,.,.
···················~····· • ••••••••••••••••••••••••••
• • • • • • • • • • • • • • • • • • • • • • •••••••••••••••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • • • •••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••
....................................................... . ................................................................ .
'
14'
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
' 10' 1.
..
............................. ····i······· .... .
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •••••••••••••••••••••••••••••••••••••••••••
4•
..................................
••••
......
• ·: • •• 1 •. : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : :
:: : :
: : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : : ; : : : • 1 •. . . ;
C:....
8
I'G
15
20
2$
30
35
oiO
41
JO
56
••
•
JO
•••••••••••••••••••••••••••••
Ill
70
71
JO
85
90
115
100
106
110
•.t211;:
l'-< E,.,.._
6-21
Tableau 6.4
Simulation de Monte Carlo
Déviation Standard Hasard (DSH)
Investissement ( I) 50 000$+DSH(1 000)
Nombre au hasard (NH)
1
-1.003
48 997
807
13.23
13
2
-0.358
49 642
657
12.63
13
3
1. 294
51 294
488
11.95
12
4
-0.019
49 981
282
11.13
11
5
0.147
50 147
504
12.02
12
Essai
Tableau 6.4
Vie utile (N) 10 +
( NH
) ( 14 - 1 0 ) 999
Durée de vie utile (N) arrondie
(suite)
Essai
Nombre au hasard
Revenus annuel (R) 35 000$ pour 0-3 40 000$ pour 4-8 45 000$ pour 9
Déviation standard au hasard DSH
Dépense annuelle 30 000 +DSH (2 000)
VP=-I+ (R-E) (P/A, 10 %, N)
1
2
35 000
- 0.036
29 928
-12 970
2
0
35
obo
0.605
31 210
-22 724
3
4
40 000
1. 4 70
32 940
- 3 189
4
9
45 000
1.864
33 728
+23 233
5
8
40 000
-1.223
27 554
+34 654 +19 004
6-22
1 r)
Problème à faire:
0'
J"
Une compagnie considère l'installation d'un nouvel équipement qui réduira les coûts de fonctionnement annuels de 5 000$. La valeur de récupération de 1' équipement est zéro à la fm de sa vie utile. Les coûts d'installation d~ équipements sont normalement distribué avec une moyenne de 25 000$ et une écart type de 2 500$. La vie utile de l'équipement a une distribution de fréquence montrée au tableau suivant:
durée de vie (année) Probabilité
8
10
12
14
16
0.1
0.25
0.35
0.2
0.1
La génération de 20 nombres aléatoires est montrée au tableau suivant: Exp. 1 2 3 4 5 6 7 8 9 10
Nombre aléatoire 48 86 25 89 40 16 20 15 34 33
Déviation nombre aléatoire 0.951 1.065 0.742 0.579 0.844 2.323 -0.800 0.485 0.396 1.925
Utiliser la simulation de Monte-Carlo pour l'échantillonnage pour déterminer les quantités suivantes: 1) 2) 3)
Le taux interne de rendement de chaque expérience; La distribution de fréquence des taux internes de rendement et les probabilités associés ; La moyenne et la variance des taux internes de rendement.
6-23
6.3.2
Autres distributions de probabilité Dans un modèle de simulation on doit utiliser la bonne distribution de probabilité. Dans l'annexe 1, on montre l' importance d' un tel principe. Une foi , la distribution de probabilité choisie, le simulateur est capable de la reproduire un nombre infinie de fois . La méthode utilisée le plus fréquemment est la méthode de la transformation inverse. Supposons qu'une variable aléatoire X a une distribution définie par : x
P[ X ~ x )
= Fx =
Jf
x
(x) d x
La fonction cumulative de distribution est définie de 0 à 1 inclusivement. Puisqu'on veut obtenir des valeurs successives de x qui correspondent à fx(x), il suffit de générer un nombre aléatoire uniforme entre 0 et 1, U(O, 1) et de l'égaler à Fx(x). La transformation inverse nous donnera x selon la distribution fx(x).
Mathématiquement:
générer r - U(0,1) fixer r =Fx(x) ainsi x =Fx"1(r)
La Figure 6.2 présente une illustration graphique du procédé.
fx{x)
aires égales
à noter ci-dessous: (re-r b)=(r!Jr 8 )
1
X'b
Fx, (x) 1 1
1 1
rb
=f=x
x
1 1
1 1
1 1
re =Fx (Xc) ·--t ·-- ·- - ·- - · - - ·-t - ·- - ·j- - ·1
Xc 1
1
1
• 1 1
(x 0 --:-- -- · -- · - - - -~. - - - - ~-
1 . 1
.
1 1
où
1-
1
1
- -.-- --- --- --- --!.. -1
x
Figureé.2 : Génération des n.a. x. et xb à partir der. et rb "'N(O, 1)
x=((rJ
6-24
6.3.2.1
Génération d'une distribution exponentielle X suit une expon.: Exp (li.) insi
= ll.e - ,\x
f x (x)
0 < x < "' ; 0
J
autrement
X
et
Fx (x)
=
f x ( t) dt = 1 - e
- ,\x
0
r
=1
1 - r
=e
fixons
ln (1 - r) - l n ( l - r) ,\
donc
- e -,\x
ou
r - U ( 0 , 1)
- ,\x
= ll.x =x
x =
- ln(l-r) ,\
= F- 1 X
(r)
Ainsi pour obtenir un échantillon d'une distribution exponentielle ayant une moyenne égale à 1/À., il suffit de générer R~U(O , 1) et d'effectuer la transformation inverse.
6.3.2.2
Génération d'une distribution normale x -N(J.t,
La génération de nombres aléatoires normalement distribués n'est pas aussi simple que pour la distribution exponentielle, car il est plus difficile d'obtenir la fonction inverse de Fx(x). Pour cette raison, on utilise la propriété suivante : «Si r1 et r2 sont 2 nombres
aléatoires ~
U(O, 1) indépendants, on a que
xl= (-2ln rlY'" cos 2nr2 x2 = (-2ln rl?' sin 2nr2 sont deux variables aléatoires normalement distribuées avec ~ =0 et cr2= 1» Pour générer des. nombres d'une distribution N()l,cr2) on utilisera l'une ou l'autre des identités pour obtenir un X ~ N(O, 1) et on le transformera comme suit : X= (-2 ln r 1Y'" cos 21tf2 Y= ~+(cr*X) Pour obtenir Y~N()l,cr2 ). Notons que cette transformation est basée sur l'identité Y-u cr
=Z~N(O, 1),
expression connue pour la loi normale.
6-25
6.3.2.3
Génération de variables stochastiques de distributions empiriques.
Il est habituellement utile de pouvoir générer des nombres entiers selon une distribution empirique (données expérimentales par exemple) non-exprimée sous forme analytique. Les données peuvent être regroupées sous la forme suivante : i
Valeur b
P(x
= b) = Pi
fx (b)
=
L
pj
j =1
Ou sous forme graphique: Fx (bi)
1- Pn
--- -- -------- -- --- - -.- - - - l
P1 + P2 + P3 P1 + P2 P1 bn
6.4
bi
RÉGIME TRANSITOIRE ET PERMANENT
On s'intéresse au système, via le modèle de simulation dans son régime permanent : (c'est-à-dire dans ses conditions normales d'opération) . Mais, les 1•r «moments» d'opération ne représentent que très peu les conditions normales d'opération car : 1- Les conditions initiales du modèle sont généralement vides (inoccupé, zéro) . Ces conditions peuvent être assez différentes des conditions normales de fonctionnement de système réel: - atelier rarement sans ouvrage; - file d'attente rarement vide (dépend du taux d'arrivée) - hôpital rarement sans malade. 2- Simuler consiste à générer des N.A. et les faire interagir : ce processus nécessite un certain nombre de valeur avant d'obtenir des états significatifs. Dans un simulateur on utilise habituellement une période de réchauffement avant de collecter les statistiques. Notons qu'il y a des systèmes qui n'entrent jamais en régime permanent ex.: économie nationale et industrielle, un régime permanent peut consister en une oscillation.
6-26
Exemple : Arrivée de clients à la caisse. Intervalle de 10 min. avec une probabilité de .5 Intervalle de 4 min. avec une probabilité de .5 Simule à l'aide d'une pièce de monnaie. Si pile prochaine arrivée dans 10 min. Si face prochaine arrivée dans 4 min.
T abi eau d' une expenence
#
1
1 2 3 4 5
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 ,,..
10 4 10 4 10 4 4 10 10 4 10 10 10 4 4 4 10 4 10 10 4 4 10 10
10 10 10 4 10 10 4 4 4 10 10 4 4 4 4
10 14 24 28 38 42 46 56 66 70 80 90 100 104 108 110 122 126 136 146 150 154 164 174 184 194 204 208 218 228 232 236 240 250 260 264 268 272 276 "'Of\
#
10. 7.0 8.0 7.0 7.6 7.0 6.6 7.0
41 42 43 44 45 46 47 48 49 50
7.7
51 52 53 54 55
56 57 58 59 60
7.1
7.6
7.4
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78
79
,..,..
1
10 4 4 4 4 4 10 10 4 4 4 4 4 10 4 4 10 10 4 4 4 10 10 10
10 10 4 10 10 4 10 4 4 10
4 10 10 4 4
290 294 298 302 306 310 320 330 334 338 342 346 350 360 364 368 378 388 392 396 400
410 420 430 440 450 454 464 474 478 488 492 496 506 510 520 530 534 538 ~·~
Il
6.9
#
81 82 83 84 85 86 87 88 89 90 91 92
6.6
6.6
6.8
6.8
-
-
-
Il
93 94 95 96 97 98 99 100 101 102 103 104 105 106 4 108 109 110 111 112 113 114 115 116 117 118 119 ,~ ,..
1
4 10 10 4 10 4 10 10 4 4 10 4 4 10 4 4 10 4 4 4 10 4 4 10 4 4 10 4 10 4 10 4 4 4 4 10 4 10
546 556 566 570 580 584 594 604 608 612 622 626 630 640 644 648 658 662 672 682 692 696 706 716 720 724 728 738 742 752 756 766 770 774 778 782 792 796 806 n•n
Il
#
6.7
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159
6.9
6.8
6.8
6.8
6.8
,,,..
Il
4 4 10 10 4 10 10 10 4 4 4 4 4 4 10 10 4 4 10 10 4 4 10 10 4 10 10
10 4 10 4 10 10 4 4 4 10 10 10
,,..
1
814 818 828 888 842 ·852 862 872 876 880 884 888 892 896 906 916 920 924 934 944 948 952 962 972 976 986 996 1006 1010 1020 1024 1034 1044 1048 1052 1056 1066 1076 1086 11'\C\L
Il
6.8
6.7
6.7
6.8
6.8
6-27
(min)
i 1
7.5 --.--1 1
1
7
-.--1
6.5 -.--6
Permanent
Transitoire
5.5 200
400
600
800
1000
Stabilité croissante à partir de 400 à 500 min x "" 6.8 min. On sait qu'à la longue ça tend vers 7 min. = (0.5 x 4 + 0.5 x 10) Le régime transitoire n'est pas très représentatif de la réalité - minimiser l'effet de l'état transitoire. 1. Utiliser des expériences très longues; pour négliger l'effet des résultats transitoires sur le total. Ceci peut être très coûteux en temps d'ordinateur. 2. Rejeter les résultats des premiers moments, ex. : ne pas compiler les premiers 50 jours, on compte lorsque 1 000 commande sont traitées .. . 3. Utiliser des conditions initiales très proches de celles attendues en régime permenent. Ca peut biaiser si le régime permanent n'est pas atteint.
6-28
A quel moment le régime transitoire se termine-t-il? Il n'y a pas de règle satisfaisante pour déterminer avec certitude et précision l'entrée en régime permanent. 1. Effectuer des expériences pilotes 2. Observer le comportement du système 3. Décider quoi rejeter Le nombre source est d'une grande influence sur la période d'instabilité.
Les méthodes suivantes ont été utilisées avec un certain succès : 1. Rejeter une suite de mesure. La première de la suite n' est ni maximum ni minimum des données suivantes (Conway) . 2. Observer une séquence de résultats : si les nombres d'obs où le résultat est > à J..l est nombre où le résultat est
=
le
3. Calculer la période K la plus longue (durant une expérience pilote) pour laquelle les résultats ont une auto-corrélation significative; rejeter une période initiale de même longueur pour les expériences suivantes (Fishman). 4. Le plus long cycle du modèle doit avoir été exécuté au moins 3 à 4 fois avant que les effets transitoires soient disparus (Tacher) . 5. Calculer en moyenne mobile des résultats et supposer que le régime permanent est atteint lorsque cette moyenne ne change pas de façon significative avec le temps. 6. Utiliser différents nombres sources pour constituer un échantillon de mesure à des durées de simulation * et effectuer des tests sur des moyennes.
Chap . 13
obtain estimates of the mean and variance of the net A W . Assume a mini· mum attractive rate of return of 10% . 13-20. A dicseltractor is expectcd to require an investmenl of $100.000 and lo have a !ife which ean be best describcd by a uniform distribution with a minimum of S ycars and a maximum of 15 years . The salvage value is cxpectcd to he $40,000 if the lifc is less than 8 ycars. $20,000 if the lifc is R to 12 ycars. and $15,000 if the life is 13 or more years . Show how lo build a distribution of the capi tal recovery cos! for this project by gcnerating tivc outcomcs using the Monte Carlo technique . Use an intcrest rate of 10% a nd round the proj e ct li ves to the nearest wholc ycar. From your rcsults cstimale the e xpcct ed value and variance of th e distribution of capital recovery co st.
13-24. Use lhe Monte Carlo simulation technique to obtain five pairs of present worth values for the alternatives described in Exercise 13-23. What percentage of the present worth combinat ions fa vors alternative A 1 ,\(rl'fft'
13-21. Projecl X is expected to require an inveslmenl of $40,()()() and to ha ve a li fe which is normally distribulcd with a mean of 5 years and a sland a nJ dev ia tio n of 1 year (rounded to the nearcsl intcger). The salvagc value is e xpe c ted 10 vnry nccording lo the relationship $!!,000 - $1 ,000 x Clife in yearq . Project Y is cxpccted to requirc an invc slmcnt o f .~ .5 0 . 000 a nd to h ~v e " lifc which is uniformly distrihuted hctween 5 and 15 yc:1rs fr o unded tn the nearest in leger). The salvage value is expectcd to be nil regMdl ess o f li fe . Show how to build a distribution of the difTerence in capit:ol re co vcry cos ts fo r the two projects (X minus Y) by gcncrating three out comcs usin~ th e Mon te Carlo technique . Interest is 15%. The project lives vary independently .
Cnrl,•
1.\-1!\ . The
~alva!!e v~htc
the lifc
,,f the
of a pro~pe~tive ns~et is n random variable depemlcnt upon ncn>rding ln the f'>llowing tnhlc :
as~et
Prnhnhility
s_,_ooo
l .ifc
,,r .Snlvn~c
VRille
SIO,OOO
Sl5,000
S20,000
$25,000
0.20 O.JO 0.50
0.20 0.50 O.JO 0.20
0.50 0.20 0. 10
0.30 0. 10
1
-
4 6
O.JO O.JO
R
Il is thought thal each of the asse! lives is equally likely to occur. If the in vcstmcnt in the asse! is $50.000 and the interest rate is 15%, show how one cnn ohtain a distribution of the cap.ital recovery cost for the asse! by sctting up a table and generating fi vc trial outcomcs . The n calculatc cstimates of the meon and variance.
1.'-19. The c stimatcd clement \>Utcomcs for a certain flexible manufacturing system arc a s follows : ln vestmenl
Normally distributed with a mean of $1,000.000 and a variance of 16,000,000
Li fe
.'i years with probability 0 .2
Net
;~nnunl c ;~sh
S;~Jv;~gc
vnl uc
flow
7 years with prohability 0.7 9 years with probability 0 . 1 Uniformly distrihutcd bctwcen $120,000 nnd $340,000 per year
0
Ali clement 0lllcomcs arc indcpendent of cach other. Demonstrate how to 0bt a in n distribution of the net A W by generating fivc outcomes . From this
)79
Exercise.1
13-22. Records for a certain invcntory item type indicate thal the numbcr of d a ys lead lime required for replenishment can be expected to have the fo llowing probability distribution :
Lead Time (days)
Probabilil y
1
0.25 0.50 0.25
2
3
Demand du ring any day of le ad ti me is a normally distributed random variable independent of the number of da ys lead ti mc and with a mean of 4 units and a standard deviation of 1 unit of the item . lnventory holding cosl.s average $100 per unit per yr and the opportunity cost of a stock-out (failing to have a unit when demanded) is estimated at $70 pcr unit. Show how to use the Monte Carlo technique to determine (eventu :o ll y l) the most econo mical si7.e of safety stock for the item , wherc safety stock is detined as the difTeren ce between the number of units in inventory whcn the item is reordered and the expec ted demand during the Jead lime period . lllustrate by simulating tive re o rder periods with safety stocks of 0 units a nd thcn 4 units . 13-23. Two invcstment alternatives, A and 13 , arc under consideration : one must be seleeted . Alternative A rcquircs an initi al invcstmcnt nf $15.000 in equir · ment; annual operating and maintenance costs arc anticip:oted to be no rmally distributed, with a mean of $6,000 and a stand:ord deviation of $1i00 : the terminal _salvage value at the end of the 10-year study pcriod is anti c ipated to be normally distributed with a mean of $2,000 and a standard deviation of $500. Alternative B req!Jires an investment of $5,000 and end-of-year annual expenditures which are normally distributed with a mean of $9,000 and a standard deviation of $900. Use a MARR of 10%. Generating five trials, estimate the orobability !hat alternative A is the oreferred alternative.
(}'\
tl
40
Economie Concepts and Accounting: Financial Considerations
Chap . 2
Determine which method is most economical if actual overhead costs for the two methods are thought to (a) Vary according to the allocation formula (b) Be the same for each method (c) Be $5,000 more for the new method than for the old method 2-14. (Section 2.4) Describe why the return on net worth calculation in Fig. 2-6 (for example) is a short-term and relatively narrow assessment of the benefits associated with an advanced manufacturing system investment.
computer-integrated manufaduring: the new technologies
chapter7 !·:
3.1 INTRODUCTION AND HISTORICAL CONTEXT
The gross national product (GNP) of the U.S. economy is a barometer of the nation ' s real wealth generation. As of the tnid-1980s it was comprised of service industries (63%), manufacturing industries (24%), and the extractive and construction industries (13%) . [7) Extractive industries include agriculture and mining, while service industries encompass banking, insurance, health care, entertainment, and so on . Because activities of a nation 's service industries by themselves can be said not to increase the financial , longlerm well-being of the populace, one can surmise from the percentages above that manufacturing industries are responsible for about two-thirds [24/ (24 + 13)] of this country' s ability to crea te real wealth. Any tech nol ogy that can significantly improve productivity in the manufacturing sector of the U.S. economy will have a multiplied effect on our GNP. The purpose of this chapter is to explore the various components of computer-integrated manufacturing (CIM) and to better understand why the justification of CIM systems offers a special challenge to engineering and management personnel. Manufacturing is the conversion of raw materials '
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Computer-lntegrated Manufacturing: the New Technologies
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into end products. It includes ali activities from the perception of a need for a product through the conception, design, and development of the product; to the production, marketing, and ultimately supporting the product in use. Ali these activities are closely related and interactive; therefore, they are best treated as parts of a single system. [4] Modem manufacturing began in 1798 when Eli Whitney accepted a govemment contract to deliver 10,000 muskets in two years. Up to that time arms had been produced by individual craftsmen who made custom-ordered pieces in small quantities . This method lacked both consistency and quality, and Whitney decided to di vide the production process into small jobs which required that ali component parts of the muskets be interchangeable. Whitney proceeded to design hand tools. to produce these parts accurately, uniformly, and quickly, and he trained his workers to use them. Eli Whitney's technique was the first démonstration of interchangeable parts and mass production. ln 1857, both priee and quality determined who would win government contracts in the competitive arms market when Samuel Colt began making his legendary revolver. Colt realized the need for accurate and efficient production and began concentrating on methods that would eliminate manua! fashioning of parts. His strategy was to assemble a staff of machinists who could build machine tools for the rapid production of precise forgings, jigs, and stamped parts. The success ofthese new machines led to new firms producing tools for newly emerging manufacturers of sewing machines and typewriters . In 1870, Isaac Singer combined new manufacturing techniques with aggressive marketing to dominate the emerging market for sewing machines . Singer realized that mass market sales required acceptance by both lower- and middle-class households, and could be attained only by producing a variety of models consistent in quality and reliability. To accomplish these objectives, he constructed a factory that ushered in a new era of manufacturing . With the employment of gauging, forging, and milling machines, Singer soon produced interchangeable parts accurate to the tenthousandth of an inch, and then used these parts as standard components placed inside a wide range of cabinets . In 1908, Henry Ford revolutionized the manufacturing assembly process. Assembly was organized according to the necessary flow ofwork rather than the nature of the operation involved. This technology rapidly advanced to the point where it reduced production time from 13 hours to 1 hour per automobile chassis . In 1946, ENIAC (the first digital computer) was produced. It weighed 30 tons, and used 18,000 vacuum tubes, 70,000 resistors, and 10,000 capacitors . Two decades later, much smaller and more powerful computers became the foundation for pioneering work in computer-aided design (CAD) as researchers emp)oyed interactive computer graphies to display and manipu-
Sec. 3.2
Automation of Manufacturing Operations
43
late !ines and shapes instead of numbers. This was expanded to computeraided manufacturing-including robotics, flexible manufacturing systems, and other automated systems. Effective implementation of advanced manufacturing technologies through a CIM system is the cornerstone of factory modernization. As a company goes through a modemization effort, it typically identifies numerous individual improvement projects. However, CIM must be viewed as a total system which provides an automatic link between product design, manufacturing engineering, and the factory floor . The methodology selected for screening and prioritizing the se individual projects, with consideration of the synergistic effect, is critical-it must provide a framework that ensures consistency with the company's strategie objectives.
3.2 AUTOMATION OF MANUFACTURING OPERATIONS Automation has been defined as "the technology concerned with the application of complex mechanical, electronic, and computer-based systems in the operation and control of production" [3]. This includes automatic machine tools, automated material handling systems, automatic assembly machines, robotics, computer-aided design, computer-aided manufacturing systems, and other computerized systems. Tying ali these systems together with an integrated data base architecture of computers is the basic goal of a CIM system. Automated systems are usually one of two types: fixed or programmable. Fixed systems feature high production rates for high volume demand, but they are inflexible, making changes in the process costly and difficult. Fixed systems often are used in automatic assembly lines, transfer !ines, oil refineries, and chemical processes. Programmable automation allows for flexibility by making the systems suitable for low-quantity production of different products. Figure 3-1 illustrates the integration of manufacturing functions through a common data base (in the center), as is perceived by the General Electric Company for the "factory of the future ." Benefits achieved through automation are numerous and expanding. A National Research Council report cited in Table 3-1 provides the findings of interviews in five major companies having automation projects that ali resulted in reduced work in process, excellence in design, lead-time reduction, and improved productivity and quality. The values shown are representative of intermediate benefits of those companies' 10- to 20-year efforts. Further benefits are expected as full integration is approached . Examples of automation that typify the associated benefits of new technology within the Generai.Electric Company are described in Fig. 3-2 . Descriptions of various types of manufacturing and the components of CIM systems are provided in the remainder of this chapter.
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Sec . 3.2
45
Automation of Manufacturing Operations
Examples of Automation within GE GE S
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ln C"xcn• of 2 , ~oo ltlt . ThcrC" :arc 1~0 •urf:ac n 10 lw m:achlnC"d, more th:an ~5 .,.lth tla,htt o ln:ancc, . Ni ne medium :and hu.,. y m:ac hlnc 10011 .,. o rklnlliO!lt:lhn, cach cqulppcd .,.hh Gcncnl EIC'Ctrlc OC adjuuahk·Jpccd splndlc drlvn co ncrolltn~t moton r:ant~lnR from o40 - 100 horKJ)().,.cr, :arc :an tmpor· une put o f ttw PMS . The m:achlnc tool' conu ln common drl.,.e h:ard ... :ue chroua,hout , ue cqulpptd ~lth torque prOfi:r:tm· mina :and pot~ Illo n orient fnr cool • c h:analna . :and SIOrc o vcr 500
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Th(' General f:kctrk Ct1mp:any 't Alrcrafc F.natnc- M:anuf:actu r.' lng OIYblon ln I!.Yrnd:ak, O H, boeR:~n :autom:atlna lu ftct ory hy :applytnaa f>NC ncc~orlc to 111 n:l•tlnll numC"rlnlly c nntrulled machine cool buc . Toda y the nc-t~ o rlr. h:u 101 NC :and CNC m11chl~•- pcrhap1 t~ IURCII diltrlhutcd numcrlc:al co ntrol nct~ o rk ln che ~ o rld , T'hnoe lOt NC mJchlnct run on p:art provam• do~nlo:adrd from an lntcgntc:d syucm ofmlnkomputcn . The: b:ulc dhtrlb· uted numC"tic.l control tystem , wlth che :additio n of :a nuchl nc IOOitt•tut m o nltOrlnJ IJICem (~hlch te ill :li 1 sJ:anCC ~hlch nuchlnes :arC" ln K"rvlcc- , a~2ltln1 maintenance, o r boelnR
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Computer-Integrated Mamifacruring : the New Technologies
Chap . 3 Table 3-1
BENEFITS ACHIEVED WITH DATA-DRIVEN AUTOMATION (BASED ON INTERVI EWS OF fiVE MAJOR COMPANIES)
Reduction in engineering design cos! Reduction in overall lead time Increased product quality as measured by yield of acceptable product Increased capability of engineers as measured by extent and depth of analysis ·in same or Jess time !han before lncreased productivity of production operation (complete assemblies) lncreased productivity (operating time) Reduction of work in process Reduction of personnel costs
15-30% 30-60% 2-5 times previous levels 35 times 40-70%
2-3 times 30-60% 5-20%
Source: National Research Cou neil, Computer Integration of Enginuring Design and Production, National Academy Press, Washington, D.C., 1984, p. 17. Reproduced with permission of the publisher.
3.3 BASIC TYPES OF MANUFACTURING FIRMS AND LEVELS OF AUTOMATION
Sec. 3.3
Basic Types of Manu/ac turing Firms
47
Mass production is a continuous specialized manufacture of identical products and is characterized by very high production rates, special-purpose equipment, and very high demand rates for the product. Typically, variable operating costs per unit decrease and total fixed costs increase as the degree (leve!) of automation increases. As batch size increases, production flow becomes more continuous and total costs are spread over more units. When batch sizes are quite large, more specialized fixed or " hard" automation typically is chosen because it increases capacity and minimizes total costs. Figure 3-3 shows typical total costs per unit for various production quantities and levels of automation/technology. Note by the dashed portions of the curves how a given leve! might be more costly than others for a particular production quantity. Current and future production volumes and processes must be understood to determine the type of automated system desired. Figure 3-4 reflects the "classic" conflict between capacity and flexibility that can be resolved by the application of a mid-volume, mid-variety manufacturing systems concept. The productivity versus flexibility arrows denote the divergent paths to greater achievement of those respective criteria. Flexible manufacturing systems (in the middle) make possible an optimum combination of these criteria for a wide range of production volumes and varieties.
There are three basic types of manufacturing firms as classified by their associated production volumes: Manuel teehnology, generel ·purpose machines
1. Job shop production 2. Batch production 3. Mass production
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Job shop production is characterized by low-volume production with smalllot size (sometimes one of a kind) and it is often used to meet specifie customer orders . This normal! y requires that the plant perform a great variety of work; therefore, the production equipment must be general purpose. Aircraft industry contracts, as weil as special machines and deviees required by the electronics industry, are in this category. Batch production involves the manufacture of medium-sized lots of the same item or product. The lots may be produced only once, or at regular intervals. The purpose of batch production is often to satisfy periodic eustomer demand for an item. The manufacturing equipment is usually general purpose, but designed for higher rates of production. Machine tools are often combined with special-madejigs and fixtures to accommodate diversified processes and increase production .
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Computer-lntegrated Manufacturing: the New Technologies
Chap.J
Sec. 3.4
Computer-lntegrated Mamifacturing
49
3.4.1.1 Computer-Aided Design
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Computer-aided design (CAD) involves the use of computerized tools to improve productivity in the areas of design, drafting, and testing. CADis actually a means of accumulating, manipulating, and displaying related graphical data electronically. The se graphical data can then be recalled for duplication or modification. A typical CAD system includes:
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3.4 COMPONENTS/ELEMENTS OF COMPUTERINTEGRATED MANUFACTURING* A computer-integrated manufacturing (CIM) system provides an automated link between product design, manufacturing engineering, and the factory floor. The major components of a CIM system include computer-aided engineering, management, and control systems; manufacturing processes; and information integration. Each of these components is described in greater detail below. 3.4. 1 Computer-Aided Engineering
Computer-aided engineering (CAE) includes those hardware and software tools, such as interactive graphies systems, that can be used in the product design and manufacturing engineering functions . The functional systems associated with CAE require integration of both design and manufacturing data bases. A description of these systems and the required integration follows. • This section was written substantially by Earle Steinberg and James A. Brimson and excerpted with minor changes from A Framework for Computer Jntegrated Manufacturing, Touche Ro ss, lnc .. Houston . Tex .. 1985 . Reprinted with permission of the authors.
• CRT workstations where engineers can create or modify a part by "drawing it" on the screen using a number of different deviees, such as: • A standard computer terminal keyboard • An electronic stylus that the engineer uses like a pen to draw on the screen • An electronic sensitive menu tablet of commands that the engineer can select • A digitizer that can con vert an existing print to X-Y coordinate points which can be stored in the CAD system and retrieved on the terminal • Automatic drafting machines that produce high-quality drawings from data stored in the system A CAD system can provide engineers, designers, and drafting personnel with a number of significant benefits: • Time and effort required to develop designs is reduced because the computer software can be utilized to draw figures, calculate dimensions, rotate or section views, check interferences, and revise existing designs. • The quality of designs, and their associated products, are improved as the increased productivity of design personnel allows more designs to be modeled, tested, and evaluated before engineers select the final, production version . • The costs associated with the design of tooling and fixtures can be reduced by designing them on the CAD system, in conjunction with the design of the part. • Communication with engineering and production personnel is improved by taking advantage of CAD's ability to display any view of an object quickly and correctly and provide high-quality drawings.
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Computer-lntegrated Manufacturing : the New Technologies
Chap. 3
rapid development of computer technology, in both the hardware and software areas . This development will probably continue, perhaps at an increasing rate, into the foreseeable future. The effect of this on companies using CAD will be: • Significantly increased display speed and pictures/drawings of higher quality • More complex designs, tests, and simulations which will be made possible through more powerful computers with more real memory • Improved priee/performance ratios which will facilitate the support of more workstations, extending the power of CAD to more areas of the company, without significantly increasing costs ·\'
3.4.1.2 Design Analysis System
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In most CAD systems, a wide variety of anal y ti cal software is available for mathematical tes ting and modeling of part designs. Computer-aided testing (CAT) uses a CAD-generated mode! of a product design to simulate the operating conditions and performance of different materials. The design can be interactively improved on the basis of the results of the simulations. Another important element of a CAD system is the capability to analyze the impact of product design on manufacturing cost. It is extremely important to conduct cost trade-offs earlyin the design process since it is at this point that most of the production and logistic support costs are locked in. and the economie leverage to reduce cost is the greatest. The manufacturing cost design subsystem should provide the following capabilities: • Cost comparisons with differing manufacturing processes • Performance/manufacturing cost trade-off information • Cost trade-offs of materials chosen for the product 3.4.1.3 Group Technology
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Group technology is a method of classifying parts into families according to similar shapes, or common manufacturing process operations. This is accomplished through a coding mechanism that assigns specifie codes to each significant part characteristic. These codes are then combined to form a unique identifier for each part, describing it from an engineering and manufacturing perspective. Thus group technology facilitates a systems approach to the redesign and teorganization of the factory into a flexible or cellular manufacturing system. Such a system consists of a number ofwork cells, which can be viewed as a cluster or collection of machines designed and arranged to produce a specifie group of component parts. Group technology plays a key role in
Sec. 3.4
Computer-Jntegrated Manufacturing
51
setting up a cellular manufacturing system because it provides a computeroriented tool that the manufacturing engineer can use to identify and design the work cell, and the product engineer canuse to determine if similar parts are in existence, thereby reducing the number of parts and process plans. For standard families of parts, standard routings can be prepared that need only slight modification for a newly designed part. The result is typically faster, more correct routings at Jess cost. Such standard routings can then be used to group production machines in logical cells dedicated to the manufacture of one or more families of parts. In addition, similar parts or families of parts requiring approximately the same tooling and machinery can be grouped so that production setup time is reduced. Benefits of group technology and related work cells include: • Reduced machine setup times because work cells are designed to perform the "common" operations identified by group technology • Less work-in-process inventory because lot sizes can usually be reduced, smoothing the production flow • Less material handling because the flexible nature of the work cell allows more operations to be performed within the confines of the cell • Easier analysis of production costs of parts in a given family because outliers can be readily reviewed to determine whether the cost is wrong or unusually high • Rapid development of cost estimates for new parts based on the costs associated with other parts in the family • Easier tracing of the effects of priee increases on components and raw materials back to the part families, and identification of the associ ated impact on product cost The end re suit of the work ce li approach to manufacturing is increased throughput, lower costs, and better cost control. 3.4.1.4 Computer·Aided Process Planning Computer-aided process planning (CAPP) refers to the method of preparing routings or process plans using computer assistance. Recently, two approaches have evolved for this tas k. The first and most popular of the se is the variant approach where process plans are generated for families of parts that have been classified under the group technology concept. In this technique, the process planner calls up existing routings based on similarly coded families of parts. Then, using a "same as-except for" technique, the family's process plan is quickly amended to cover the individual part in question. To create a plan from scratch in the variant approach, the process planner can select from a computerized menu of operations based on part characteristics. Here the process plan is built, line by line, using company-
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Computer-lntegrated Manufacturing: the New Technologies
Chap . 3
oriented standard text. The procedure is somewhat similar to a manual process. but the standard text, of course, is stored on the company's data base ready for retrieval, modification, or hard-copy duplication. The second, far more sophisticated computer-aided process planning technique is the generative method. Here, the computer analyzes the part under consideration and, based on part geometry, material, and so on, generales a process pl~n automatically. Not only is the sequence of operations generated, but the computer also selects the company's best suited machine tools and calculates the probable machine time for each operation. From this base it is a simple matter for ·process plan ners to review and edit the plan, and revise time allowances after initial sample parts are run. CAPP typically increases the product planner's productivity and improves the accuracy of the resulting plans. Both the variant and generative methods use a "building block" approach to developing plans based on common attributes within the part's family, reducing the planning effort and thereby improving productivity. Accuracy is increased because the new plan is "assembled" from pieces of already existing plans, which have been tested in actual production. The generative method is just beginning to be implemented in U.S. manufacturing plants . The key to its more universal adoption will be the development of better three-dimensional solid geometry modeling software . In addition, much work remains to be done on the logic of process planning based on this part geometry. However, the generative method is an appropriate area for further investigation in modemization programs. 3.4.1.5 Machine Program Generators
Once the part's geometry is defined, it then becomes the basis for future part machining operations . In the past, a design engineer wou Id finish a blueprint, and the NC Programmer would have to take that blueprint and redefine the geometry before he/she could start programming. With an integrated system, the programmer starts with something in digital format. Machine program generators pi-ovide a bridge between CAD and CAM, where designs can be turned into the instructions necessary for computer controlled machines. The programmer does not have to create new instructions from scratch, but as with CAPP, he or she can build new instructions from the instructions of older family members. This is called family-of-parts programming and increases the programmer's flexibility and productivity .
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Sec. 3.4
Computer-lntegrated Mamifacturing
53
production planning for product families, master production scheduling for · customer order promising and shipping schedule development, master requirements planning for component scheduling and purchasing plans, production activity control for shop operations scheduling, capacity requirements planning for shop loading, priority planning for daily dispatch, and inventory planning for material control. These systems address the production questions of wh en and how many, and facilitate the solution of production problems, including machine utilization , control techniques, planning capabilities, quality control, excess work in process, and poor use of skilled tabor. To be effective in this area, certain requirements must be met in the areas of planning, facility loading, and dispatching. 3.4.2.1 Production Planning
Production planning translates the dollar goals expressed in the bu si ness plan into Ievels of production for product families. This aggregate-level plan is tested against available plant resources through the process of resource requirement planning and represents top management's key "control knob" on the business. 3.4.2.2 Master Production Scheduling
A master production schedule represents a disaggregation of product family production plans into individual end items (or major components when "planning bills" are used). Mas ter production scheduling must also simulate the resources , including machine types, and personnel and inventory levels, required to accommodate the proposed production schedule or product mix. The simulation capability uses forecast work load profiles of each product and projected resource loads for the existing master schedule. Net changes in each type of resource are identified , as are underloads and overloads of existing resource capacities . This process is referred to as rough-cut capacity planning . 3.4.2.3 Material Requirements Planning Material requirements planning (MRP) involves computerized techniques for scheduling the replenishment of material, based on the master production schedule and the components needed to build the product. MRP provides the ability to update priorities and schedules frequently white providing suggested start and completion dates for ali open and planned work and purchase orders.
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Factory management and control systems provide for the flow of production information through the integration of data from both engineering and manufacturing functions . The planning and control function includes
3.4.2.4 Production Activity Control
Operations scheduling establishes projected start and completion dates for al! activities on open or planned work orders and is required to automati-
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Computer-lntegrated Manufacturing: the New Technologies
Chap. 3
cali y verify availability of tools and materials for generating shortage lists. It calls for delivery of required tools and materials when a selected Joad is released, generates initial shop fioor Joad, and identifies bottleneck conditions . This function provides for interactive intervention to undo bottlenecks and a simulation capability for use in interactive intervention. Production activity control also includes the dispatch function which releases work to specifie machines and operators. While the dispatch function can be either automatic or manual, a simulation and optimization capability to assist in assigning jobs and operators to machines is required for either approach. Furthermore, if a manual system is being used, it must have the capability to make detailed recommendations upon request. 3.4.2.5 Capacity Requirements Planning
The information produced in capacity requirements planning is used by manufacturing management, quality assurance management, shop fioor supervisors, purchasing, accounting, and personnel departments. At a minimum, capacity planning must: access inventory files and process planning files, convert production requirements for personnel, identify overload conditions, and provide the capability to evaluate alternatives . 3.4.2.6 lnventory Control
The activities and techniques of maintaining the stock of items at desired levels, whether raw materials, work in process, or finished products is generally referred to as inventory control. These activities involve stocking levels, safety stocks, and lot sizes; and focus on managing the company's investment in inventory assets. 3.4.2. 7 Purchasing Systems
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Purchasing systems serve a dual role . First, they provide information to buyers to aid them in negotiating with vendors and placing/expediting orders. Second, the systems interface with MRP systems to provide feedback on actual order delivery schedules, order quantities, and lead times. This information must be both timely and accurate, if MRP is to schedule material replenishments correctly . 3A.2.8 Maintenance Planning and Control Systems
Scheduling preventive maintenance and collecting maintenance statistics are now computerized . These systems allow the company to monitor such maintenance activitie s as:
Sec. 3.4
Computer-lntegrated Manufacturing
55
• Mechanics' productivity • Work order processing for requested jobs • Requirements for parts and special tools These systems also allow the capacity planning systems to include scheduled downtime in their calculations and to adjust work center loads accordingly . 3.4.2.9 Engineering Data Base and Change Control Systems
The engineering data base is also frequent! y referred to as the engineering bill of materials. It allows engineers to maintain their product engineering information on the computer in terms of parts lists, specifications, and process requirements. The power of this tool is achieved when it is integrated with the manufacturing bill of materials to facilitate engineering changes, and reconciliation of product information . The engineering change control capability enhances efficiency by maintaining the history of product changes as they are released from engineering. However, its greatest values may lie in its capability to facilitate the incorporation of the changes into the manufacturing bill of materials and its ability to effectively manage inventory investment by controlling effectivity dates . 3.4.2.10 Tool Control Systems
The relationship between planning and controlling production and the need to coordinate tooling requirements is being addressed by many of today's more advanced manufacturing and control systems. Time-phased requirements for specifie tools are generated, based on production schedules. This information is supplied to the tooling design and fabrication department to identüy the tool requirement dates necessary to support the production schedules. 3.4.2.11 Energy Management
The increasing cost of energy in factories has led to the development of computer software to manage energy resources . These systems can monitor energy consumption, turn lights on and off, adjust heating or air conditioning and provide management with data to include energy costs in their business decisions. 3.4.2.U Warehouse Systems
Software packages designed to help warehouse managers max•m•ze space utilization, make more efficient use of labor, and improve customer
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Sec. 3.4
Computer-lntegrated Manufacturing
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service are now available for a wide range of warehouses. These systems provide a variety of functions, including: • Order processing • Forecasting • Finished goods inventory control Systems offering these capabilities are available for companies from one warehouse location to a worldwide warehousing network. Systems for the latter generally are capable of playing a coordinating role in the management of inventories and purchasing activities. Properly implemented, the integrated planning and control system eliminates the problem of missing resources. It ensures that ali resources needed to perform a given job are identified, coordinated, and available when thatjob arrives at its workstation. This reduces the amount ofwork in process, normally held as butTer stock in input queues. Control of material movement reduces the size of the output queues . These capabilities support reductions in the amount of material held as in-process inventory. Flow times for material moving through the production process are reduced accordingly. Utilization rates for machines and workers increase, since the workers are doing productive work rather than waiting for tools, parts, and instructions. One of the m<\ior challenges in implementing factory management and control systems is the difficulty in linking together different hardware and software packages. There are currently many vendors and little standardization of technical specifications . Additionally, a company that is modernizing must decide on the value of off-the-shelf versus custom-tailored systems . The modernization process should assess projected costs involved in changing existing organizational procedures to fit the software systems being offered rather than modifying the software to reflect organizational needs.
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The automated shop floor includes computerized production machines and equipment that facilitate the production process. CIM requires highly automated and intelligent production facilities which are controlled by sensors, minicomputers, and microcomputers. A CIM system has many benefits from the machinists' point of view. Among these are: • Increased accuracy • Increased repeatability (1,000 parts, for example, can ali be created alike)
• Decreases in the time it takes to go from a part print to the finished product • Decreases in scrap due to better utilization of raw materials • Shorter production time frames • Increased machine flexibility 3.4.3.1 Computer-Aided Manufacturing
There are many definitions of CAM in use toda y. For purposes of this discussion, we define CAM as NC, CNC, and DNC machine tool hierarchies, robotics, AS/RS, FMS, PCs and related technologies.
1. NC. Numerical control (NC) refers to a category of progràmmable machine tools. In NC, a machine is programmed with a set of instructions designed for a particular work part or job. Based on the instructions, the machine will move its cutter tool through a series of X, Y, and Z coordinates, performing the desired machining operations. One of the advantages ofNC is that, when thejqb changes. the program can be changed. This feature gives NC its flexibility . Instead of making major changes to production equipment, relatively simple changes are made to a program. The first NC machines were programmed by a punched paper tape that contained the instructions for the speci fi e job. Many of these machines are in use toda y and they are weil sulted for a number of applications. Advances in computer powe r hov e led 10 an extension of the NC concept to computer numericnl control (CN and direct numerical control (DNC). 2. Standardized CNC controls . Computer numerical co ntrol (CNC:) places a dedicated computer in or alongside the NC machin Paper tapes are no longer necessary and machine tool in structions co n be created and stored electronically in the computer's memory or on tape cassettes or diskettes . In modern CNC machines , programs can bè created, edited. or changed ifnecessary on the machine . CNC ellminates problems of .physical tape storage or deterioration . Thu s CN machines perform the same function as NC machines, although their method of receiving operating instructions is different . 3. Direct NC. Direct numerical control (DNC) offers real-time computer control of more than one NC machine at a time . Many NC programs are stored in a central computer's memory , on tape or disk . Not onl y can the computer control multiple NC machines simultaneously, but it '-J can gather feedback from each machine asto part production rates and 1..0 machine status. With DNC and CNC capability at each machine , programs can be downloaded from the DNC to the CNC machine for running. This saves the amount of memory needed for each machinr
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Computer-lntegrated Manufacturing: the New Technologies
Chap. 3
and means that the program only has to be created and loaded in one machine (the host DNC)-not every machine that can run the job. Over the past decade, NC machining has been proven to allow a productivity gain of approximately 3 to 1 in most applications. Other benefits of NC machining are, of course, reduced setup time, better part quality, Jess scrap and rework, and reduced operator attention. 3.4.3.2 Robotics Robots are computer-controlled deviees that automatically perform a programmed sequence of operations. Most industrial robots in use today were designed for specifie work such as painting, welding, plastic molding, and assembly, and were, for the most part, laborsaving deviees utilized sole! y for the performance of noncomplex production work . In addition to such specific-purpose robots, many others perform material handling operations such as workpiece loading/unloading and workpiece transfer in conjunction with other automated machinery. The benefits of robots include: • Cost. Robots are becoming quite economical relative to the wages of workers they displace. • Precision. Robots have a higher degree of precision than humans, and with proper preventive maintenance can be highly reliable. Precision re fers both to the concept of accuracy and repeatability. • Availability. Robots are tireless and can work three shifts with only ti me off for preventive maintenance; th us robots are excellent for overloaded or complex structured work. • Safety and environment. Robots have reduced the exposure of hu man workers to undesirable environments such as noise, heat, dirt, and machine hazards. The trend in robot technology is toward universality and versatility to produce a standard unit that can handle diverse manufacturing problems. The real future usefulness of robots will be determined by the power of the computers that operate them, the sensors (touch and sight) by which they receive real-time data, and the software that con trois them. In the future, as pattern recognition software, artificial intelligence, sensory equipment, and adaptive feedback mechanisms become available at even lower costs, robots should be weil qualified to perform many assembly and parts selection tasks now commonly performed by human beings. 3.4.3.3 Programmable Controllers and Microprocessors Programmable control/ers (PCs) are microprocessor-based deviees that integrate automated machines and equipment to provide a factory with
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Computer-Integrated Manufacturing
59
real-time control. Unlike · old fashioned, hard-wired, relay-based controllers, PCs are relatively inexpensive, solid-state deviees that can be reprogrammed quickly to perform new tasks. Better still, different task sequences or programs can be stored in the PC's memory or on tape so that process control changes can be effected with a change of tape cassette or with a typed-in command. The ultimate advantage ofPCs and microprocessors is that any number of them can be controlled by central computers. These deviees can then be used to report back on factory floor or process status and can be reprogrammed quickly for different jobs when necessary. 3.4.3.4 Flexible Material Handling Automated material handling systems permit an improved integration of material flow to and within manufacturing functions . Sorne of the available technologies include wire guided vehicles, smart carts, and dedicated transfer tines. The automated material handling system must be flexible to permit changes in lot sizes, design, and product mix . 3.4.3.5 Automatic Storage-Retrieval System Automatic storage and retrieval systems (ASIRS) are computer-operated part pickers and stockers. Generally, the Iarger pallet-sized systems for · warehouses are called AS-RS while smaller stockroom bin sized systems are labeled automatic part pickers-stockers. Working either from direct computer-generated and Iinked pick or stock instructions, or from manually generated pick or stock instructions, these machines either will deliver a batch of parts to an open or random location, while recording the location for future reference, will stock a given part in a preprogrammed location, or will go to a selected location to pick a part. 3.4.3.6 Computer-Aided Inspection Computer-aided inspection (CAl) is another new use of computerized . engineering design data in manufacturing, but this time in the quality control ·• area. In CAl, coordinate measuring machines (CMMs) controlled by software that draws data from the company's engineering design data base automatically measure parts to determine if they have been manufactured to design tolerances specified on the part's drawing. These machines have a probe that automatically moves to a programmed point, takes a measurement, and displays or records the result. Sorne of the more advanced machines, becoming available or under development, replace the probe with a laser, significantly increasing the machine's accuracy. The computer can also print out both the required and actual measurement or dimension and/or the deviation, if significant, for the part under consideration . More important, the CMMs also enable these measurements to be stored automatically
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Chap. 3
in a data base, to be used as a basis for further statistical analysis on the part, or perhaps as an earl y indicator of machine tool wear. These data also serve to record permanent) y for the company the data necessary to meet traceability requirements and demonstrate that a part was manufactured correctly. 3.4.3.7 Flexible Manufacturing Systems
Flexible manufacturing systems (FMS) permit the continuous manufacture of different items within a family of parts in small batches within a dedicated machining facility. Flexible manufacturing systems, as shown in Fig. 3-5, use the concept of integrated raw material storage, part picking, part transportation, and DNC machining that are linked together in such a way that the parts being worked on can travel from raw material storage to finished goods storage in different sequences under the control of computers. The central FMS computer schedules and tracks ail production and material movement in the FMS center. Based on a family of similar parts, an FMS can be reprogrammed quickly through downloaded instructions from a central computer to individual machines, conveyors, and part pickers to perform a new set of tasks. The major benefit of an FMS. is fiexibility of manufacturing resource assignment along with computer-controlled operation . 3.4.3.8 Bar Coding The expanding role of the computer as a tool for management and production is directly tied to the company's ability to provide it with timely, Vortlc.ol drill
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accurate data. Many businesses are discovering that bar code scanning is the fastest, most accurate, !east labor intensive, and lowest-cost method for delivering that data. Companies are using bar codes to track work in process, aid in physical inventories, control stockroom inventories and lots, and track shipments to customers. Wherever data entry is required by production, shipping, receiving, or inventory personnel, a potential application for bar coding exists.
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3.4.3.9 Laser Machining Where extremely precise and/or delicate machining is required , lasers are receiving much attention . Laser machining is capable of maintaining very tight tolerances, in part because they are unaffected by the tool wear inherent in more conventional types of machining . Recent advances in laser machining systems, especially in the areas of laser positioning and process control, have significantly enhanced the capabilities of laser machining . Lasers can be used to machine complicated or unusual shapes, and are excellent solutions to machining delicate components which dis tort easily, because there is no machine tool contact with the workpiece. lt is important to note one central feature of ali CAM applications that we have touched on so far . With CAM, the skill requirement for an operation is transferred from the operator to the programmer who creates ali the.. machine instructions. No longer must a person running an NC lathe be a skilled machinist. lnstead, the person can be a Jess expensive and more readily-available machine operator. The same trend is evident in ali applications of computers in manufacturing processes. On the other hand, CIM is promoting the integration of design and manufacturing. With CIM these two previously separate groups work from the same data base almost concurrently in the design process to design the part, the tools and fixtures needed to manufacture the part, the bill of materials for the part, and the process plan for the manufacturing of the part . In fact, sorne companies have created cells of people who work as a team through one to three CRTs to carry out the design and manufacturing engineering process. Each cell is responsible for a product family or product group. People in each team gain energy from their continuous interaction and not incidentally come to have considerable pride in their group efforts . Such a team approach to manufacturing design and operations is the way sorne people view manufacturing in the future.
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successful system incorporates software links capable of allowing various systems to communicate directly with each other. This is crucial to the integration of a new system with other in-place systems. The task of processing and controlling information in the factory of the future requires a reconceptualization of the role of data processing and the use of new techniques .
Sec. 3.4
Summary
63
Traditionally, data processing systems were designed to expedite and/ or automate transaction processing, record keeping, and business reporting. Decision support systems are designed to aid in making and implementing decisions. These systems facilitate development of computer models which retlect "real-world" situations, simulations, and answering "what if" questions. In this regard, they allow the manager to assess the impact of a decision before implementing it within the company .
3.4.4.1 Data Base Approach As companies migrate toward the factory of the future, data bases that support the production processes are becoming more electronic and less paper oriented . Common shared data is a critical element in this process . Much of the potential advantage of factory automation lies in integrating engineering with manufacturing functions. The key is minimizing the need for manual intervention between functions and processes. 1. Data base management systems (DBMS) . File-oriented information systems have been the mainstay of computer applications since their original ion . These traditional approaches are inadequate for the factory of the future since they hinder data sharing. The concept of data base management systems (DBMS) is to overcome this limitation by making data independent of application programs. 2. Information network. While a centralized data base is technologically possible, the factory of the future will probably remain a distributed data base environment due to the nature of the automation process. The advanced manufacturing technology explosion has occurred concurrently with the development of minicomputer and microprocessor technology . This technology, along with local area networking which permits efficient transfer of files between data bases, has provided greater ftexibility and responsiveness at the machine and process levels where the transformation process occurs.
The adoption of a decentralized data base concept will not by itself ensure the benefits of this approach. ln order to maximize the value of a DBMS, the design of data structures must be organized by logical function rather than by narrow application. The value of a functional orientation is that data can be shared by everyone in the organization, as required. 3.4.4.2 Decision Support Systems Although very few management fonctions have been automated, advances in information retrieval, processing, and display technologies have led to significant computer applications that help people perform management functions. Si nee the purpose of these systems is to support managers responsible for making and implementing decisions rather than to replace them, these applications are often called decision support systems.
3.4.4.3 Local Area Network Approach A local area network is a collection of computer hardware and software designed for the specifie purpose of facilitating communications between computer deviees within an area su ch as a factory . ln a distributed processing environment, the local area network links various deviees together, vastly increasing the total available computer capabilities. 1. Distributed data processing . One of the key challenges of the factory of the future will be the design of a networking system. As the trend toward distributed automation continues with systems such as CAD and programmable controllers, it will be the role of the local area network to tie the se potential "islands of automation" together. Sophisticated users must design the network with enough flexibility to facilitate future factory modemization efforts. The local are a network will allow the creation of a more complete data base. This will be a distributed data base built up through the various nodes on the network, not imposed through a centralized computer system. 2. Data communications . One of the major problems with designing a local area network system today is that most commercially availabl e systems are closed systems where the equipment that provides the data communications function uses vendor specifie protocols. This closed system restricts a company's freedom to choose the best possible equipment to meet specifie needs. Open systems architecture appears to be emerging in importance . The Manufacturing Automation Protocol (MAP) is a popular evolving industry standard for communications among shop floor deviees .
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3.5 SUMMARY
In this chapter we have described the nature and major components of CIM within the context of developing an understanding of how technology can
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Computer-/nregrated Manufacturing: the New Technologies
Chap.J
improve critical performance, which contributes to the competitive posture of a manufacturing firm . Simply acquiring technology for the sake of sophistication without establishing performance measures and targets which support strategie objectives is both ex pensive and foolhardy. A clear understanding of how advanced manufacturing technology contributes to better performance in ter ms of cost, quality, delivery , ftexibility, reliability, and other factors should precede, not follow, consideration of new technology .
REFERENCES 1. Ayres, Robert U., and Steven Miller, "Robotics, CAM and Industrial Productivity," National Productivity Review, Winter 1981-1982, pp. 42-60. 2. Curtin, Frank T ., "Planning and Justifying Factory Automation," Production Engineering, May 1984, pp. 46-51. 3. Groover, Mikell P., Automation, Production Systems and Computer-Aided Manufacturing, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. 4. Harrington. Joseph, Understanding the Manufacturing Process, Marcel Dekker, Inc., New York, 1984. 5. Hegland , Donald E .• "Flexible Manufacturing- Y our Balance between Productivity and Adaptability," Production Engineering, May 1981, pp. 169-221. 6. National Research Council, Committee on the CAD/CAM Interface, Manufacturing Studies Board. Complller Integration of Engineering Design and Production, National Academy Press , Washington, D.C., 1984. 7. Steinberg, Earle, and James A. Brimson, A Frameworkfor Computer lntegrated Manufaclllring, Touche Ross, Inc ., Houston, Texas, 1985. 8. Teicholz, Eric, "Computer Integrated Manufacturing," Datamation, March 1984. pp . 169-174.
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3-1. Briefty identify reasons why the benefits presented in Table 3-1 might be achieved through information integration within a firm. 3-2. Explain sorne of the difficulties one might expect to encounter in using a computer to achieve computer-integrated manufacturing in the context of Figure 3-1. 3·3. Outline sorne important characteristics that distinguish job shop production from mass production. 3·4. What are sorne of the key benefits of CAD, and how might one go about quantifying them? 3-S. Why is group technology particularly useful in the design and management of flexible manufacturing systems?
Chap . 3
Exercises
65
3-6. Describe briefty the generative method of computer-aided process planning (CAPP). 3·7. Draw a block diagram to tie together the various components of factory management and control systems . Identify the types of information that pass from one block to another.
3-8. List sorne of the technologies that comprise computer-aided manufacturing. Summarize sorne of the costs and benefits of each. 3-9. What is a decision support system , and how does it relate to the generic subject of "information integration"? 3-10. Data communications systems are an integral part of the "factory of the future ." Research the topic "manufacturing automation protocol" (originally proposed by General Motors) and describe how this communication protocol is likely to reduce the costs of data-driven automation in manufacturing and service industries .
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ÉVALUATION
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TECHNOLOGIES:
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APPROCHE MULTICRITÈRE
1- Introduction:
Dans la vie de l'entreprise, il est rare que les critères d'évaluation des solutions alternatives soient uniques. L'objectif sera souvent multiple: par exemple, les gestionnaires cherchent, par le biais d'une implantation d'une nouvelle technologie, à réduire les coûts, améliorer la qualité, réduire les retards de livraison et maintenir la stabilité d'emploi. Fréquemment, ces objectifs sont contradictoires. En effet, les alternatives qui réduisent les coûts peuvent, au contraire, affecter la qualité ou le temps de livraison. C'est pourquoi des méthodes d'aide à la prise de décision face à des critères multiples ont été développées, particulièrement durant les vingts dernières années. Ces méthodes visent à fournir au décideur une solution satisfaisante en prenant en compte tous ces facteurs simultanément. Il importe de rappeler que dans un environnement mulitcritère, il n'est pas possible de parler réellement d'optimisation. En effet une solution indiscutablement optimale selon un ensemble de critères devrait être la meilleure selon chacun d'entre eux. Un tel cas ne peut être envisagé quand on sait, étant donné que les différents critères sont souvent conflictuels.
Par contre, lorsque 1' on veut comparer plusieurs options entre elles en se basant uniquement sur des facteurs tangibles (fmanciers), on peut appliquer les méthodes d'évaluation économiques (V AN, RSI, TRI, etc.) puis choisir la moins coûteuse. La principale difficulté est d'obtenir de
bons estimés car il faut prévoir les coûts à venir et non pas seulement ceux qui sont en vigueur au moment de l'étude. Mais l'introduction des facteurs intangibles, difficiles à quantifier, vient compliquer grandement le choix de la meilleure option. L'utilisation d'un modèle d'aide à la décision susceptible de considérer simultanément les facteurs tangibles et intangibles peut remédier à cette lacune. A cette fin, plusieurs modèles ont été expérimentés jusqu'à présent.
7.15 TI-Difficultés rencontrées par les méthodes traditionnelles:
Dans le processus d'évaluation, ces méthodes ne prennent en considération que les aspects financiers ce qui rend l'étude incomplète. D'une part, le coût de l'investissement risque d'être sous-estimé, d'autre part, les économies sont parfois difficiles à estimer. En effet, outre les gains directs de la main-d'oeuvre (gains de fonctionnement), on retrouve presque toujours un effet d'entraînement de plusieurs avantages indirects, tous provoqués par l'implantation d'une technologie appropriée. En plus de collaborer à la diminution des temps de cycle de production, la flexibilité du système de production augmente, les en-cours sont réduits, le temps de réponse est diminué. Ces avantages sont particulièrement intéressants quand on perçoit le processus de fabrication d'une façon dynamique et non pas comme un simple processus de fabrication statique. A cela, il faudrait ajouter la multiplicité des facteurs, souvent contradictoires, dont on doit tenir compte simultanément. Les méthodes d'évaluations traditionnelles ne tiennent pas compte de tous ces éléments et plus particulière~ent
des aspects stratégiques du projet.
Dès lors qu'il s'agit d'une décision stratégique, comme c'est souvent le cas des nouvelles technologies et des systèmes intégrés, le recours à d'autres méthodes devient nécessaire TI faudrait développer d'autres critères pour justifier ces technologies. Huber [1985] souligne qu'en 1984, une enquête a révélé que 91% des compagnies utilisaient la méthode du délai de récupération comme outil principal pour justifier les efforts d'automatisation de leurs entreprises. Selon lui, le modèle de justification des systèmes manufacturiers avancés doivent toujours inclure les bénéfices anticipés qui ne pouvaient pas être directement déterminés tel que l'impact de l'amélioration de la qualité. Ignorer ces bénéfices parce qu'ils sont difficiles
à évaluer serait, selon l'auteur, l'exclusion de toute tentative d'adoption de ces nouvelles technologies. Meredith et Suresh [1986], de leur côté, constatent que beaucoup de projets méritant d'être retenus ont été abandonnés parce que les avantages qualitatifs ne pouvaient pas être inclus dans la procédure de justification tandis que les économies réalisées sur les coûts directs étaient insuffisantes pour répondre aux obstacles financiers établis par l'entreprise. -ces technologies sont plus flexibles que les autres équipements ordinaires. Les avantages de cette flexibilité ne sont pas facilement capturés par les procédures traditionnelles;
7.16 -leurs synergies c'est-à-dire leurs effets conjugués.
En d'autres termes, elle suppose que le procédé actuel continuera à fonctionner dans les mêmes conditions que dans le passé, et sans aucun risque. Certains auteurs sont même allés jusqu'à qualifier la justification économique d'obstacle N°l pour l'implantation de ces nouvelles technologies. En fait, les entreprises ont tendance d'incorporer deux fois le risque d'un projet comportant des nouvelles technologies: -par l'utilisation d'un taux épreuve élevé; - et l'exclusion de certains bénéfices, difficiles à mesurer, dans les méthodes des estimés conservateurs. Remédier à cette double pénalité constitue un défi à toute approche de justification de ces nouvelles technologies.
7.17
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ll.l- Déf"mition: Les différentes méthodes développées sont souvent classées sous l'appellation "aide à la décision multicritère" (Vincke 1989). Celle-ci désigne généralement un ensemble de méthodes permettant d'agréger plusieurs critères afin de sélectionner une ou plusieurs "actions". Elle vise à fournir au décideur des outils qui lui permetteront de progresser dans la résolution d'un problème de décision où plusieurs objectifs, souvent contradictoires, doivent être pris en compte.
La première constatation , qui doit être faite, lorsqu'on aborde un tel problème est qu'il n'existe pas en général une décision qui soit la meilleure simultanément pour tous les points de vue. Le
7.18 mot "optimisation" n'a donc plus de sens dans un tel contexte Contrairement aux techniques de la recherche opérationnelle, les méthodes multicritères ne fournissent pas de solutions "objectivement les meilleures". La divergence des objectifs nécessite la recherche d'une solution de meilleurs compromis possible.
Pour appliquer ces méthodes, on doit nécessairement suivre les étapes suivantes: - dresser la liste des critères à prendre en considération; - dresser la liste des actions potentielles, ou envisageables; - juger chacune des solutions aux yeux de chacun des critères; - enfin, agréger ces jugements pour choisir la solution la plus satisfaisante. A l'exception de la dernière étape, les trois premières sont communes à toutes les méthodes multicritères.
Le lecteur pourra trouver dans les livres de Schârlig [1985] et de Vincke [1989] un panorama des différentes approches proposées jusqu'ici en analyse multicritère, englobant aussi bien les méthodes d'inspiration américaine que celles de "l'école française". Pour ce qui est des modes d'emplois détaillés et les justifications mathématiques des différentes méthodes décrites, les ouvrages de Roy [1985], Zeleny [1982] et Saaty [1980] répondent parfaitement à ce besoin .
Un boil système d'aide à la décision doit être: - simple à comprendre. Seuls les phénomènes importants doivent être inclus dans le modèle. -Robuste: - Facile à contrôler - Adaptatif: Le modèle peut être mis à jours en cas de présence de nouvelles données. - Exhaustif quant aux points importants - Aisé pour ce qui est de la communication. - pouvoir traiter différentes solutions alternatives; - pouvoir considérer de nombreux critères et objectifs souvent contradictoires; -permettre au décideur de structurer (de modéliser) le problème lui-même;
7.19 - permettre au décideur d'incorporer des données objectives mais aussi des données subjectives basées sur son expérience.
m.2- Les méthodes de pondération (weiwtin2 methods): La méthode la plus simple, et sans doute la plus utilisée, est la moyenne pondérée qui peut prendre la forme suivante:
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Cette méthode consiste en les étapes suivantes: 1- dresser la liste des critères ou caractéristiques des options; 2- affecter à chacun un poids qui mesure l'importance que lui attache le décideur; 3- attribuer un ordre numérique qui indique à quel degré l'équipement présente la caractéristique recherchée; 4- déterminer la cote poids-rang de chaque équipement et choisir le mieux coté.
C'est la forme la plus ancienne, et donc qui a vu beaucoup d'applications dans divers domaines, en particulier le domaine manufacturier. Par exemple, Nelson [1986] suggère un "modèle scoring", une variante importante de cette catégorie de modèles, pour déterminer les sections d'une usine qui devraient être modernisées. Les modèles "scoring" ont été également appliqués par Sullivan [1984] pour effectuer un choix entre un système manuel et un Robot en incluant les critères non-monétaires et par Udoka et al. [1990] pour évaluer les CIMS. Huang et Ghandfouroush [1984] procèdent en cinq étapes dans leur application du modèle "scoring" pour l'évaluation et la sélection de robots [p.44-47]: 1- développer une liste de robots disponibles sur le marché; 2- déterminer les facteurs critiques éventuels susceptibles d'écarter certains robots; 3- définir les mesures des facteurs objectifs: OFMi;
7.20 4- défmir les mesures des facteurs subjectifs: SFM;; On assigne à chacun des facteurs subjectifs un score sur une échelle de 0 à 10. 5- Déterminer)a mesure de performance en considérant les 2 mesures objectives et subjectives
ainsi calculées aux étapes 3 et 4: PM1 = (a).OFM1 + (1-a).SFM;.
Le coefficient (a) dépend de la préférence du manager accordée aux mesures subjectives et objectives. Nous nous limiterons à cette description sommaire, pour plus de détails, nous recommandons de consulter l'article en question. (voir l'exemple traîté à la fm de ce document).
Il importe de rappeler qu'à ce niveau de l'analyse, on suppose qu'on a déjà effectué un premier trie et qu'on est en présence d'actions efficaces, dans le sens où elles ne sont dominées par aucune autre action. Par définition, une action domine une autre si elle fait un score au moins aussi bon que la seconde sur tous les critères, et meilleur sur un critère au moins.
TIL 3- Pondération des critères: Dans la plupart des méthodes multicritères, l'importance relative des critères est représentée par des poids numériques. Ceci soulève le problème de détermination de ces poids et la traduction en termes quantitatifs la notion qualitative d'importance des critères. Ces paramètres ne possèdent pas de signification claire et ont généralement une grande influence sur les résultats fournis par les méthodes multicritères.qui les utilisent. De façon générale, les poids permettent de traduire l'importance qu'accorde le décideur aux différents critères. L'importance de ces poids varie selon la méthode utilisée.
IV- La méthode AHP (Analytic Hierarcby Process): AHP se distingue des autres méthodes d'aide à la décision multicritère dans sa façon de déterminer des poids. En effet, elle procède par comparaisons binaires de chaque niveau de la hiérarchie par
7.21
rapport aux éléments du niveau supérieur. Les comparaisons se font sur une échelle définie ( 1 à 9). Par ailleurs, elle est capable d'identifier et de prendre en considération les incohérences des décideurs. Un logiciel "expert choice" a été conçu à cet effet. Nous y reviendrons plus en détails un peu plus loin.
AHP, développée en 1971 par Saaty [1980], est l'une des méthodes qui a reçu une attention particulière. Elle a été appliquée dans plusieurs domaines en particulier dans le domaine manufacturier pour la justification et le choix d'une nouvelle technologie avancée (Arbel et Seidman 1984, Varney et al. 1985, Sullivan 1986, Wabalickis 1988, Kleindorfer et Partovi 1990, Frazelle 1985). Les deux premiers articles développent des hiérarchies séparées pour les bénéfices et les coûts, déterminent les vecteurs de priorité pour chaque hiérarchie avant de les combiner ensemble, et enfin calculent le ratio bénéfice-coût. Le classement se fera par ordre décroissant du ratio obtenu. Kleindorfer & Partovi [1990] ont suggéré l'utilisation de la méthode AHP pour choisir une technologie [p.215], alors que Cambron et al. [1991] s'en servent pour choisir le meilleur aménagement possible au niveau d'une usine. Frazelle [1985], pour sa part, propose deux méthodes multicritères (technique de pondération, et méthode AHP) pour choisirun système de manutention sur la base de cinq critères globaux retenus: . RSI (Retour Sur Investissement) . flexibilité . sécurité . compatibilité . maintenabilité
Ces deux méthodes d'évaluation ont été jugées très satisfaisantes par l'auteur, en particulier AHP qui prend en considération les incohérences dans les jugements à tous les niveaux de la hiérarchie. Nous présenterons l'exemple d'application utilisé par Frazelle un peu plus loin.
7.22 Le succès remarquable réalisé par cette approche est dû au fait qu'elle prend en considération aussi bien les aspects quantitatifs que qualitatifs des critères tout en assurant une certaine cohérence dans lesjugements. En effet, AHP représente une façon pratique de traiter quantitativement les aspects qualitatifs des critères de décision tels que la flexibilité, la qualité, la fiabilité, la sécurité, etc. De plus, AHP ne force pas le décideur à être parfaitement cohérent, mais elle offre l'avantage déterminant de pouvoir fournir une mesure de cohérence de ses jugements. Dans AHP, la quantité de données requise par l'analyste est relativement limitée. L'input est principalement basé sur l'expérience et le jugement de celui qui a conçu le système.
Les articles
de Zahedi (1986), Shim (1989) et Vargas (1990) présentent une excellente revue de la littérature à ce sujet. La figure suivante montre un exemple de hiérarchie utilisée dans AHP .
7.23
1
1
1
1
1
~~~~g
~~f~~
-
';11\
......
Décomposition hiérarchique (objectif, macro-critères, critères, solutions).
7.24
Exemple d'application:
Systèrre de rrm.iertiCll sc:tisfaisa1t
1
AGJS usine Entière
Il
Olaicts élévatars
~
Ill
Olaicts
élévatars usine altière
~B&C.
IV l'vtn:rail ~A
AGJS
~B&C.
Hiérarchie utilisée par Frazelle 1985 p.44.
Analyse de sensibilité: Quelle sera la réaction des ratios à des changements dans jugements du décideur.
les
7.25
Analyse de sensibilité: 0.38
0.377
0.32
0.30 1 1
0.297
i,-0.402 1
0
0.2
0.4
0.6
Valeur dea
0.8
1.0
Point rrort pour les opt 1et Ill.
Discussion: Option I > Option II pour tout a; Option I > Option III pour a > 0.4; Le point more entre opt.I et opt.III comme suit:
a (0.377) + (l-a)(0.320) = a (0.297) + (l-a)(0.378) = = = > a = 0.42
7.26
Bibliographie:
-
Arbel A.
and Seidmann A.
[ 1984),
11
Performance evaluation of
flexible manufacturing systems", IEEE 1984, pp.118-129. - Cambron, K.E. and Evans, G.W. [1991], "Layout Design using Analytic Hierarchy Process," Computers ind. engng, Vol.20, N° . 2, pp.211-229. -Canada J.R. and Sullivan W.G. [1989J,"Economic and multiattribute evaluation of advanced manufacturing systems", Prentice Hall Inc. 1989. - Frazelle Ed. [1985], "Suggested techniques enable multi-criteria evaluation of material handling alternatives", Industrial Engineering Feb. 1985, pp.42-48. - Huang P. Y. and Ghandfouroush P. [ 1984), "Procedures given for evaluating, selecting Robots", Industrial Engineering, April 1984, pp.44-48. Huber R.F. "Justification-Barrier to competitive [1985], manufacturing", Production, Sep. 1985, pp.46-51. Kleindorfer, P.R. and Partovi, F.Y. [1990],"Integrating manufacturing strategy and technology choice", European Journal of Operational Research, 47, pp.214-224. -
Madu,
C.R.
&
Georantzas,
N.C.
[1991),
"Strategie
thrustnof
manufacturing decisions: a conceptual framwork", IIE Transactions, June 1991, pp. 138-147. -Meredith, J.R. and Suresh N. for
advanced
[1986], "Justification techniques
manufacturing technologies",
Int.
J.
Prod.
Res.,
vol.24, N°5, pp.1043-1057 (1986). -Nelson C.A. systems
[1986], "A scoring model for flexible manufacturing
project
selection",
European
Journal
of
Operations
Research, vol.24/N°3 1986 ,pp.346-359. - Partovi, F. Y. and Burton, J. [ 1992], "An Anlytical Hierarchiy Approach to Facility Layout 11 , Computers Ind. Engng Vol. n 0 • 4, pp.447-457,1992. -Roy, B. [1985], "Méthodologie multicritère d'aide à la décision", Economica, Paris. - Saaty T.L. [1980], "The Analytic Hierarchy Process" McGraw-Hill,
7.27 New-York, 1980. -
Saaty T.L.
[1982],
"Decision Making for Leaders", Wadsworth,
Inc., 1982. - SchSrlig, A.
(1985], "décider sur plusieurs critères", Presses
Polytechniques Romandes. -
Shim,
Jung
P.
( 1989],
Hierarchy Process
"Bibliographical Research
(AHP) ",
Socio-Econ.
Plann.
on Analytic
Sei.
vol. 23,
N°3,
pp.161-167, 1989. - Tversky A. [1972], "Elimination by aspects: a theory of choice". Psychological review, 79, 4 (1972), pp.281-199 . - Udoka S.J., Nazermetz J.W. (1990], "Development of a methodology for
evaluating
Computer
Integrated
Manufacturing
(CIM)
implementation performance", Computer Industrial Engineering N°5 1-4, pp.145-149 (1990). -
Vargas,
L.G.
(1990],
"An overview of the Analytic Hierarchy
Process and its applications", E.J.O.R. 48, pp.2-8,
(1990)
Varney M. s. , Sul li van W. G, and Cochran J. K. [1985], "Justification of flexible manufacturing systems with the Analytical
Hierarchy
Process",
Proceedings
of
1985
IIE
Spring
Conference, IEE, Norcross,GA, 1985, p.181-190. - Vincke, Ph. [1989], "L'aide multicritère à la décision", éditions de l'U.L.B., Ellipses, 179 pages. - Wabalickis R.N.
[1988], "Justification of FMS with the Analytic
Hierarchy Process", Journal of Manufacturing Systems, vol.7, N°3, 1988, pp.175-182. - Zahedi F. [ 1986], "The Analytical Hierarchy Process - A survey of the method and its applications", Interfaces, 1986, 16(4), pp.96108. - Zeleny, M.
[1982], "Multiple criteria decision making", McGraw-
hill, New York.
210
Economie Analysis Methods: After Taxes
Chap. 7
Table 7-A-1 continued Line Item ( t 9) Depreciation
(20) Net. before-tax savings 121) Net after-tax savings (22l Net after-tax cash savings (23) Net after-tax cash flow ~~~~ Discount factor at 20% 125) PW at JO% (26) Discount factor at 20% 127) PW at 20% (2RI Discount factor at 40% (291 PW at 40% ln upper riJ!hl cnr11er : Net PW
Payback period
Description Yearly depreciation calculated using ACRS, straight line, or whatever method; line (4) multiplied by the yearly percentage [Note: If applicable, should reduce line (4) by part or ali of line (5) .] Total savings minus depreciation [line (18) minus line (19)] Line (20) multiplied by 1 - tax rate Line (21) plus line (19) Line (22) minus line (Il) plus line (10) For calculation of present worth Line (23) multiplied by line (24) For calculation of present worth Line (23) multiplied by line (26) For calculation of present worth Line (23) multiplied by line (28) Total net PW laken from the "total" column on right-hand side for the appropriate interest percentage [Note : If want net PW for sorne interest rate other than 10%, 20%(..Dt 40%, can till in appropriate discount factors (PIF), in line like (24), then calculate totals for line like (25).) Use li ne (23) to fi nd number of years b.efore cumulative savings will just equal the investment; may want to interpolate to gel answer to nearest 0. 1 year
multiattribute decision analysis: introduction and basic techniques
chapter
8.1 INTRODUCTION The form normally can also be used to facilitate calculations of the approximate discounted cash flow rate of retum (also "internai rate of return" ) for the project. Ali that needs to be done is to interpolate between the two closes! rates which have positive and negative total net PWs, respectively. In this case the reis no negative net PW for the three rates shown, bu.t the net PW would become 0 at about 60% . The form can also be used to calculate the payback period by adding the cumulative net cash ftows (li ne 23) over lime. The payback period is the number of years until the cumulative net cash How switches from negative to positive .
Much has been written regarding the inadequacies of traditional economie analyses in evaluating the prospective results (benefit s and. sometimes. th problems and costs) of advanced manufacturing syste ms. Multia!! ribu t (multiobjective, multicriterion, multifactor, etc.)* decision techniqu es sec m to provide an easily understood, yet comprehen sive , set of qu antit ative/ qualitative approaches to justify advanced manufacturing systems. Classic advice on the subject of decisions involving multiple fa cto rs was rendered in a letter from Benjamin Franklin to a friend, Jose ph Pri estl y. in 1772. • The word "attribute" is used to describe what is important in a dec ision problc rn a nd is often used interchangeably with "objective" and "criterion . " A liner distinction can he made as follows: an "objective" represents direction of improvement or preference for o ne or more attributes, while "criterion" is a standard or rule that guides decision mak ing . 211
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Multiallribute Decision Analysis
Chap. 8
Dear Sir, ln the affair of so mu ch importance to you, wherein you ask my ad vice, 1 cannot. for want of sufficient premises, advise you what to determine, but if you please 1 will tell you how. Wh en tho se difficult cases occur, they are diflicult. chiefly bec a use wh ile we have them und er consideration, ali the reasons pro and con are not present to the mind at the same time; but sometimes one set present themselves, and at other times another, the first being out of sight. Hence the various purposes or information that alternatively prevail, and the uncertainty !hat perplexes us. To get over this, my way is to divide half a sheet of paper by aline into two columns; writing over the one Pro, and over the other Con . Then, during three or four days consideration, 1 put down und er the different heads short hints of the different motives, thal at -different times occur to me. for or against the measure. When 1 have thus got them ali togcthcr in one view. 1 endeavor to cstimate their respective weights; and where 1 find two, one on each side, thal seem equal, 1 strike them both out. lfl find a reason pro equal to sorne two reasons con, 1 strike out the three. If 1 judge sorne two reasons con, equal to three reasons pro, 1 strike out the five; and thus proceeding 1 find at length where the balance lies; and if, after a day or Iwo of further consideration, nothing new thal is of importance occurs on either side, 1 come to a determination accordingly. And, though the weight of the reasons cannot be taken with the precision of algebraic quàntities, yet when each is thus considered, separately and comparatively, and the whole lies before me, 1 think 1 canjudge better, and am Jess liable to make a rash step, and in fact 1 have found great advantage from this ki nd of equation, and what might be called moral or prudential a]gebra. Wishing sincerely that you may determine for the best, 1 am ever, my dear friend. yours most affectionately. B. Franklin Although the term "moral or prudential algebra" does not seem to have tlourished, the spirit of Franklin's "rational" or "systematic" approach continues. lndeed, in the last two decades severa! very powerful techniques or methodologies have emerged. The methodologies to be described herein ditfer considerably regarding their assumptions and limitations . Awareness of su ch limitations will make the user more effective in the selection and proper use of these methodologies as decision guides.
8.2 SELECTION OF ATTRIBUTES
~t
v
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The selection of what attributes should be considered in the comparison of alternatives is critically important to the eventual outcome (alternative selected). lndeed, it is sometimes observed that the mere articulation of what attributes are important in the context of particular alternatives under consideration can lead one to a rational (presumably at !east somewhat close to
Sec. 8.2
Selection of Allributes
213
the best) choice without the forma! application of sorne quantitative or semiquantitative methodology . The most important usual theoretical restriction in the na ming of auributes is that they be independent of one another. In brief, but somewhat oversimplified words, any two attributes are said to be independent if th e preference order and the trade-offs for different levels of those attributes do not depend on the levels at which ali other attributes occur. Independence of attributes is difficult to achieve (or to know that one has achieved) literally . Rather, normally one specifies important attributes to be as independent as practicable (intuitively), and theo stands ready to discount the results of the analysis somewhat if he or she feels thal Jack of independence affects the results significantly . In selecting attributes, one might weil identify scores of attributes that may be important. As a practical matter, one normally would want to keep the number of attributes finally selected within manageable · limits by drop ping or not otherwise considering those having little or no chance of significantly affecting the choice among alternatives. There is no ideal number of attributes to consider. But it should be recognized thal too few means that sorne attribute(s) which are important have been ignored, while too many means that one may become bogged down with futile details . This results in Joss of val ua ble ti me and energy, if not possible opportunities forgone. due to diversion of attention away from the discovery of better alternatives or designs for the problem at hand. One way to reduce the number of attributes is to identify any attribute fo[.which each alternative in a particular decision-making situation is equally good (or bad) with respect to that attribute. Since only differences among alternatives are important in their comparison, such attrihutes contrihute nothing to the analysis and, in general, can be eliminatcd . Example 8-1 A firm may have decided upon, say, 15 significant attributes for capital inves1men1 comparisons . Further, it may have narrowed thal down to, say, 10 significanl attri · butes for replacement comparisons ("keep old" versus "bu y new" ) in general. For a particular situation, involving possible replacement of a manufacturing system, il may ·be determined quickly that, say, 4 out of the 10 have the sa me outcomes for ali alternatives. Th us the 4 can be eliminated , leaving only 6 attributes which need to be considered for th at particufar situation . Rather than starting with a large number of attributes and theo narrowing them down (as discussed above), the opposite might be more workablc in particular situations. Thal is, one might start with one or a few attributes (e.g. , only economie etfects and strategie impact), do an initial analysis for familiarization and exploration purposes, and theo add more attributes and analyze further as believed to be justified by the nature of the decision
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Multiattribute Decision Analysis
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Chap . 8
situation. Indeed, attributes can be added and further studies undertaken in successive increments until one is satisfied thal the lime and expense required for further study are not sufficiently justified by the potential added information which might affect the choice(s) among alternatives.
8.3 MISCELLANEOUS SEOUENTIAL ELIMINATION METHODS Sequential elimination methods were first categorized by MacCrimmon [1] and are decision rules (sometimes arbitrary) whereby an individual might be able to eliminate one or more alternatives to narrow the choice and perhaps even be led to a decision. These methods are applicable when one can specify values: (outcomes) for ali attributes and alternatives. Those values should be scalar (measurc able) or at !east ordinal (rank orderable). The methods do not consider weighting, if any, of attributes. A Iso, they are "noncompensatory," which means that they do not recognize possible trade-offs among attributes across alternatives . Thus the method in Ben Franklin's letter (above) violates the mechanics of sequential elimination methods . Table 8-1 port rays an example showing. outcomes (values) for four attributes and four alternatives. Note the two right-hand columns state ·t he "ideal" and the "standard (minimum acceptable)" values for each attribute .
This method takes two forms as described below :
1. Disjunctive constraint. An alternative will be retained (not eliminated) if it meets the standard for at /east one attribute. Table 8- 1 EXAMPLE MULTIAITRIBUTE PROBLEM FOR ILLUSTRATION OF "SEQUENTIAL ELIMI NATION METHODS"
Eslimaled Oulcome• for Allernative: Allribule
' t
l
A . Quality B. Ftexibility C. Serviceabili 1y D. Cost savings
75 Very good
50 8
2
3
4
Ideal
90 Good 38
80 Poor 35
60 Excellent 30
too Excellent
5
6
6
50 10
Misce/laneous Sequential Elimination Methods
215
2. Conjunctive constraint. An alternative will be retained only if it meets the standard for al/ attributes . Example 8·2 For the problem in Table 8-1 .(and using the standards in the right-hand column), set up a table thal indicates which alternatives do not meet the standard for which attributes. Then condude which alternatives mee! the following constraints.
(a) Disjunctive constraint. (b) Conjunctive constraint. Solution X for Allernalives Not Meeting Standard: 4
Attribute A B
c D
x
x
x
x
x
(a) Ali alternatives meet the standard for at !east one attribute. (b) Only alternative 1 meets the standard for ali attributes.
8.3.2 Alternative versus Alternative: Comparison across Attributes
8.3.1 Alternative versus Standard: Comparison across Attributes
li
Sec. 8.3
Standard (Minimum Acceptable) 70 Fair 30 7
The method is normally called a "dominance" check. If one alternative is better tl:lan or equal to sorne other alternative with respect to ali . attributes , and better for at !east one attribute, the other alternative is said to be dominated and .can be eliminated . Example 8·3 For the problem in Table 8-1, comparison of ali pairs of alternatives across ali attributes will reveal thal alternative 2 dominates alternative 3. That is, alternati ve 2 :>alternative 3 because, in the given attribute order, 90 > 80; good > poor ; 38 > 35; and 6 > 5. Further inspection by the interested reader will lind no other alternative that is dominated .
8.3.3 Alternative versus Alternative: Comparison across Alternatives
w 0
This method is of two types:
1. Lexi.c.agraphy. This involves first ranking attributes by importance . • Higher numbers are belier.
'-J
For the most important attrib ute, choose the alternative, if any, which
Multinltribute Decision Ana/ysis
216
.;. ; t.
Chap . 8
is best. If there is a tie between two or more alternatives, one should go to the second most important attribute and choose which of those " remaining" alternatives, if any, is best. This process continues in this way until a single alternative emerges , or until ali attributes have been examined. 2 . Elimination .by Aspects. This is like lexicographybecause it examines one attribute at a time, making comparisons among alternatives. However. it eliminates alternatives thal do not satisfy the standard (minimum acceptable for thal attribute) and continues un til ali 'alte rnatives except one have been eliminated, or until ali attributes have been exa mined . Example 8-4 For the problem in Table 8-1 , assume thal attribute importance rankings are A > B > C > D* and determine what choice or alternatives remain after applying (a) lexicography and (b) elimination by aspects, using the "standards" given on the right-hand side of the table.
Sec. 8.4
Graphical Techniques
Ll1
8.4 GRAPHICAL TECHNIQUES Graphical techniques are very powerful because they allow one to describe the nature of a multiattributed decision problem so thal decision makers can readily understand and absorb large amounts of information . They often do not include specifie "weighting" of attributes. which may be advantageou s because attribute weights are often nebulous and/or differ greatly among various decision makers.
8.4. 1 Alternatives-Attributes Score Ca rd A scorecard is a matrix of alternatives versus attributes together with numbers and/or other symbols to represent the outcome expected for each alternative with respect to each attribute . Table 8-2 is an example in which qualitative and quantitative results are shown. Ease of interpretation of the scorecard results to facilitate decision making can be obtained by such devices as symbols and/or colors for "best" and "worst" alternatives for each attribute.
Solution (a) For attribute A, alternative 2 is best and would be the choice. (Note : Ir, for instance . alternative 2 and, say, alternative 3 had been tied for best with respect to attribute A, then the choice would have been based on attribute B. Alternative 2 is better than alternative 3 with respect to attribute B and thus would have heen the choice .) (b) Refer to Table 8-1 (or the table in the solution for Example 8-2). Allribute
Alternatives Eliminated
Alternatives Left
A B
4 3 None 2
1' 2, 3 l, 2 1, 2 1
c D
Thus alternative 1 is the sole survivor. Any of the methods above can be tempered/altered with good judgment rcgarding the decision-making circumstances. In most cases it make·s se nse to allow what might be called "bands of imperfect discrimination" so thal one alternative is not judged better than another just because it has a slightly higher value for one attribute wh en using "lexicography," or the other alternative just barely feil below one standard when using "elimination by aspects ."
• A > B mea ns thal A is preferred to B, etc.
8.4.2 Shaded Circles to Portray Scorecard-Type Results Figure 8-1 shows the use of shaded circles to portray visually the relative evaluations of alternatives with respect to the five attributes as given in Table 8-2. A s shown in the key, the evaluations for each attribute except net present worth were categorized five ways from "exceptional" (full shading) down to "poor" (no shading) . Net present worth is shown to be "exceptional" (with full shading) for $500M, with proportionately less shading down to $0M. Wh ile the use of shaded ci rel es does cause loss of the specifie language and quantitative information ~iven in the original scoreboard in Fig . 8-1, it is ea sy to scan for relative comparisons. In examining the scorecard information in Table 8-2 and/or Fig. 8-1, one might be led to conclude thal alternative P-4 (and perhaps P-3) are definitely not as desirable as alternatives P-1 and P-2 . Perhaps a final deci sion can be made u..ing this method, but it should be recognized thal any relative weights/imi5ortances of the attributes have not been assigned formally. We will consider only alternatives P-1 and P-2 in demonstrating sorne ' other multiattribute methods .
8.4.3 Graphs of Evaluation Ratings for Attribute Outcomes Figure 8-2 shows sample graphs for depicting what are commonly called "evaluation ratings" or "seo ring functions" (arbitrarily on a scale of 0
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219
Graphical Techniques
Sec. 8.4
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Figure 8·1. Graphical portrayal (using shaded circles) for 4 alternatives and 5 auributes . j
to 10) correspond1tg to possible outcomes (from worst to best, respectively) for each of four attributes. The evaluation ratings could be "utility" values (usually on a scale of 0 to 1), as shawn in Chapt er 9. Although such graphs are not recommended as sufficient by themselves for comparing alternatives, they do help by portraying how ratings or scoring values vary with outcome changes for whatever attributes to facilitate the use of other decision-making techniques. 8.4.4 Polar Graphs
Figure 8-3 is an example polar graph showing evaluation ratings for alternatives P-1 and P-2 . Note that four of the rays drawn from the center 218
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w N
220
Multiallribute Decision Ana/ysis CIMS tactical ai ms
Chap. 8
Sec. 8.4
CIMS tactical ai ms
Riskiness, lack of
{how weil met)
221
Graphical Tech11iques
" Ideal" or "be$t" for
each attribute
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correspond to evaluation ratings on a scale of 0 to 10 (such as might be obtained from Fig. 8-2) and the rightmost ray corresponds to the attribute "net PW" (on a scale of $0 to $500M). Any attributes thal can be measured or given an evaluation rating can be included if the maximum and minimum for each a re defined. Note that after the appropriate evaluations or other measures are plotled (using linear scales) on each ray, those points are connected to form polyhèdrons for each alternative. T his aids one in judging differences between alternatives . However, one should not be misled into thinking that the overall desirability of each alternative is in proportion to the areas of their respective polyhedrons . This technique as weil as the other graphical techniques up to this point have not considered the relative weights or importances of the attributes, which is the subject of the next section .
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Multiallribute Decision Analysis
222
Sec. 8.6
Chap. 8
Table 8-3
8.5 RANKING OF ATIRIBUTES (OR ALTERNATIVES)
I LL USTRATION OF PREFERENCE CoMPARISONS'
8efore attribut es are weighted (or alternatives are evaluated) it is often desired to rank them in order of decreasing preference . This might be done by presenting the decision maker with a list of factors (usually attributé~,l but they could be alternatives) and asking him or her to rank them in order of preference . There is, however, a procedure that may make this task easier and provide a check on the internai consistency of the value judgments obtained . This is ca lied the method of paired comparisons and is illustrated by example below. Consider the five factors (attributes) in Table 8-2, which we now want to rank order. They are designated as follows: A. B. C. D. E.
223
Weighting of Attributes
A A B
c D E
-p
8
-
c
D
E
p p
p p p
p p
;
-
p
Number or Times Prererred 3
4
;
Il
-
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0
• "P" irrow ractor is prererred to column ractor or"; .. ir they are equally prererred .
CIMS tactical aims Net present worth Serviceability Manage ment/engineering effort Riskiness, lack of
The foregoing scheme of deducing rankings assumes rransiriuiry of preferences. Th at is, if X> Y and Y > Z, th en X must be> Z. If the number of times a gi ven factor is preferred to another is equal fort wo or more factors (except in the case of ties), there is evidence of lack of consistency, which suggests the need for questioning preference judgments for the factors affected.
The method of paired comparisons suggests that the factors be submitted to the decision maker two at a ti me for preference judgments. In general. ifthere areN factors. N(N- 1)/2 pairs must bejudged. 'Assume that the results of this process are indicated by the following list of preference statements (as normal, the symbol > means "is preferred to" and = means is "equally preferred to." etc.). 1. 2. 3. 4. 5.
A A A A 8
< > > > >
8 C D E
c
6. 7. 8. 9. 10.
8 B C C D
8.6 WEIGHTING OF ATIRIBUTES*
Weighting of attributes involves quantifying the relative importance of each. Thefollowing methoù ofweighting attributes (again called " factors") is based on two assumptions:
> D > E > D = E < E
A good way to depict the pairwise comparisons above and then determine rankings is shown by the matrix in Table 8-3. In that matrix, "P" is shown for each pair in which the row factor is preferred to the column factor. Note thal the diagonal of the matrix is empty (since a given factor cannot be preferred to itself). A good way to make sure thal ali pairs of factors have been considered is to recognize thal there should be a P either above or below the diagonal for ali pairs, or in case of equal preferences, an " = " is shown both above and below the diagonal. Note thal on..the righthand side of the mat rix is shown the number of times the factor in each row is preferred . Thus, for this example, it is found that the rank order is 8 > A > C = E > D.
1. It must be possible for the.Jecision maker to consider and judge the relative weight of any combination of factors. That is, it must be possible to cons id er not only the weight of F 1 , but also the weight of both F 1 and F2. 2. W_çights are assumed to be additive. That is, given the weight of F 1 and the weight of F 2 , the weight of both F 1 and F 1 is the sum of their individual weights. ~
The decision maker could proceed in weighting the fac tors as follows :
1. Rank the factors according to decreasing preference or importance by any method, such as ordinal scaling illustrated above. Once this is
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L24
Multiauribute Decision Analysis
Chap . 8
done, ir mighr be desirable to assign or reassign subscripts to indicate the rankings . Thus F, is ranked first, F 2 is ranked second, and so on. . 1 2. Let the weight of F 1 equal lOO [i.e., W(F 1) = 100] and weight the(other factors so asto retlectjudgment oftheir weight relative to the weight of F 1 • Th us the weights of F 2 • • • F N will range from a possible .high value of lOO to a possible low value of O. Ail further steps (3 through 6) serve ro retine these initial weightingjudgments and to ensure they are internally consistent. 3. Compare F 1 with the combination of F 2 + F 3 (the plus sign here means " and"). If F 1 is considered to have Jess weight than F + F , the 2 3 weighrs assigned in step 2 must satisfy the relation W(F ) < W(F ) + 1 2 W( FJ). If !he assigned weights do not satisfy this relation, W(F ) 1 should be adjusred until the relation above corresponds to thejudgment on relative weights. If, on the other hand, F 1 is considered to have greater weight than F 2 + F 3 , the inequality sign of the relation above would be reversed, and, ifnecessary, the weight of F 1 may be adjusted. 4. Compare F, with the combination F 2 + F 3 + F4 , and repeat the process of adjusting the value of W(F 1) if necessary.
5. The process of comparison ilnd adjustment is continued according to rhe following pattern (for a given problem, it is to be expected that many of these comparisons will not be needed, thus shortening the process): Compare F, with F 2 + F 3 + F4 + F5
2,:
W(F;)
i= l
Example 8-5 Suppose that management desires to establish weights for the five factors or attri· butes identified above. The following are the identifications and initial weighting assignments for the factors in rank order.
\
Factor (Attribute) Name
Identificatio n
Net present worth CIMS tactical aims Serviceability Riskiness, lack of Management/engineering effort
B A C E D
Initial Weighting As•ignment 1{)()
65 40 40
15
It is desired to illustrate the determination of weights in the spirit of, but not exactly following, the consistency checking procedure in steps 3 through 5 (above). The first consistency check will be to compare B with A + C. Suppose that the decision maker feels that B is slightly less important than the combination of A and C. This judgment agrees with the initial assignment of weights. That is, W(B) < W(A) + W(C), where W(B) is weight of factor B, etc . 100 < 65 + 40; 100 < 105 ; .'. OK
The next comparison could be bJtween A and C + E. Suppose the decision maker feels that A is more important fuan the combination of C and E . A check of this compared to the assigned weights shows Judgment: Weights:
. + FN ~
Compare FN- z wirhFN- 1
100 N
Judgment: Weights:
Compare F 1 with F 2 + F 3 + F4 + F 5 + · · · + FN Compare F 2 with F 3 + F4 Compare F 2 with F 3 + F4 + Fs
Compare F 2 with F 3 + F4 + F5 +
225
Weighting of Allributes
Sec. 8.6
+ FN
6. Once !he weights [ W(F1) + W(F 2 ) + · · · + W(FN)] have been adjusted and checked for consistency to the satisfaction of the decision ! maker, it is common to "normalize" them to sum to 100 points by multiplying each individual weight by
W(A) > W(C) + W(E) 65 < 40 + 40; 65 < 80; .' . NOT OK
Thus there is an inconsistency to be resolved . This might be done by increasing W(A) alone by mo~ than 15 points (i.e., 80 - 65), by decreasing W(C) and/or W(È) by a sum of more than 15 points, or by sorne combination ofthese adjustments for consistency . Suppose it is judged that W(A) should be increased by 5 points and that W(C) a nd W(E) each should be decreased by 10 points . With these changes, the weights are fou nd to be consistent with the judgment. Judgment: Weights :
W(A) > W(C) + W(E) 70 > 30
+ 30; 70 > 60; ... OK
Ifadjustments in point assignments are significant, it may be nece ssary to recto prior consistency checks. This process may be continued until one is satisfied with tht
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226
Mu/tialtribute Decision Ana/ysis
Chap . 8
Tabt~ ·8-4 C... LCULATION OF NORMALIZED FACTOR WEIGHTS
~~
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Nonnalized Factor Weight =
Weight Factor (Attribute) B. A. C. D. E.
W(·)
i
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Net present worth CIMS tactical aims Serviceability Management/engineering eiTort Riskiness , Jack of
100 70 30 20 30 LW(· ) = 250
100
LW(·)
[
1= W(-)J
x
L
> en )o
~
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\ w_eights . Table 8-4 shows the normalization (to sum to 100) of the factor weights, which presumably have been checked sufficiently for consistency.
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x 100%
Il
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Many numerical formula methods for assigning weights exist that are easier but generally much less defensible than the procedure advocated above (even if checking for consistency is not done completely). Severa! are described below, and example calculations are shown in Table 8-5 for the same five factors (altributes) as shown in Table 8-4. As in Table 8-4 , the weighls are expressed as percentages.
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11:::0 1 - 0 0 0 0 N
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8.6. 1 Formula Methods for Weighting
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1': (
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2. Rank sum weights. If R 1 is the rank position of attribute i (with 1 the highest rank, etc .) and there are N attributes, rank sum weights for each altribute, W;, may be calculated as
3:
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NNNMN-
Il
Il H
~-
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N - R; + 1 N
L (N -
x 100%
.
t:
R 1 + 1)
3. Rank reciproca/ weights. Rank reciprocal weights using the same notation as above may be calculated as
.
t'1
·t
W; ==
N
IIR 1
L (t iR;)
x 100%
(8-3)
"c:
00
(8-2)
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Multiatlribute Decision Analysis
Chap. 8
0 recipmcal
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Comparing the methods in Table 8-5, note that the rank method gives the highest weight for the first-ranked attribute . One might arbitrarily choose among the weighting methods above according to which provides the closest approximation to the independently judged weight for the highest ranked attribute. If we take that independently judged weight to be the 40% in Table 8-4, it is seen to be closest to the 44% for the rank reciprocal method.
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8.7 WEIGHTEO EVALUATION OF ALTERNATIVES
Suppose thal we are comparing two alternatives on the basis of how.well-they satisfy the live attributes with the weights developed in Table 8-4. The attributes, together with the subjective evaluation of how weil a particular alternative meets each on the ba sis of a sc ale of 0 to 10, are shown in Table 8-6. [Note: The se ·evaluation ratings roughly correspond to the " scorecard " (Table 8-2), and sample graphs in Fig. 8-2 . Additionally , the " net present worth " amounts have been converted to a scale of 0 to
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P-2
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7
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Factor
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Allernative
t-
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Table 8-6 ExAMPLE EVAL UATION RATtNG OF
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Once the evaluations have been made, the results can be calculated as in Table 8-7 to arrive at weighted evaluations of attributes for each alternative. Thus the summed weighted evaluation is 70.0 for alternative P-1 and 74.2.for. alternative..P-2, (using Equation 8.4 at the bottom of Table 8-7), which indicates that alternative P-2 is the better even though it happened to have the lower evaluation rating for three out of the live attribut es.
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Once weights have been assigned to attributes, the next step is to assign numerical values regarding the degree to which each alternative · satisfies each attribute. This is generally a difficult judgment task using an arbitrary scale of, say, between 0 and 10 or between 0 and 1,000 to refl,ect relative evaluations for eac!h alternative and each attribute. Example 8-6
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231
Weighted Evaluation of Alternatives
Sec. 8.7
P- 1
-··\ Project P- 1
10
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70
80
--
1 t:l 1;:; 2 1 1~'V 1
lE!!' 1 t& ·;: 1 1~:: 1 ti ·~,
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100
Normalized weight , cumulative
Figure 8-S.
Graphical portrayal or weighted evaluations ror two allemalives .
4_ 8.7.1 Tabular and Graphical Ways to Show Weighted Evaluation of Alternatives
~
figure 8-4.
Sa mple rormat• to racilltate calcul ation or weighted evaluation.
Figure 8-2 showed sorne typical graphs relating attribute outcomes to evaluaijpn ratings . Now we illustrate in Fig. 8-4 a tabular means which -shows the same information and in addition provides a format* for easily calculating weighted evaluations. The illustration is for project P-1 only, and we see once again that it has a weighted evaluation of 700 (out of 1,000) , which is the same as 70 out of 100. The same form can be used for comparing alternatives , together if desired . Figure 8-5 is a graphical way of depicting the weighted evaluation of alternatives P-l and P-2 as calculated in Table 8-7. The attribute weights are shown to scale on the horizontal axis , and the evaluation ratings are shown to · scale on the vertical axis . Hence the areas for each alternative are in proportion to their respective weighted evaluations. ln this case the difference does not stand out , but project P-2 does have a somewhat greater area. Figure 8-6 shows the differences in weighted evaluations for the two àlfernatives, which may facilitate graphical interpretation better than Fig. 8-5.
• Format adapted substantially rrom "objectives matrix" ror measuring productivity performance by Oregon Productivity Center, Corvalli s, OR; adapted with permission or ils highl y productive directo r, the tate Dr. James E. Riggs .
--.. 230
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Multiattribute Decision Ana/ysis
BL
P-2 > P-1
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Chap. 8
Normalized weight,
J3
Variation of Weight ed Evaluation Method
Objective Measure The objective measure for each alterative k, OMk, can be calculated by one of the following:
cumulative
0 -1
Sec. 8 .8
(a) If measure is returns or profits for each alternative k, cali it OFPt (such as net PW), then
40
-2
P-2
\
< P-1
Figure 8-6. Graphical portrayal of difference in weighted evaluations for Iwo alternatives.
_ OM k -
where K = total number of alternatives
OFPk K
L OFPk
(8-6)
k~ l
(b) Jf measure is costs, for each alternative k, cali it OFCk (such as PW-C or AC), then ·
8.8 VARIATION OF WEIGHTED EVALUATION METHOD THAT CONSIDERS OBJECTIVE (ECONOMIC) MEASURES AND SUBJECTIVE MEASURES*
OMk = OFC1 x S
(8-7 )
where
-~,
s=±L.
The following variation of the weighted evaluation mode! involves first transforming any objective (economie) measure of merit for each alternative into a score between 0 and 1 so that the sum of the (qQi~çH,ye) scores totals 1. For subjective attributes, such as lack of riskiness, nominal ratings are th en transformed into nu me rica! scores, and the relative importances .of the attribi.Jtes are weighted so thal the scores of each attribute over ali alternatives and the weighted (subjective) scores over ali attributes each sum to 1. Next, weights are assigned to the objective and the subjective score.s such that those two weights sum to 1. The combined weighted scores (which also sum to 1) show the relative desirability for each alternative. Expressed in formula form, the combined measure (weighted evaluation) for each alternative k, WEt. is WEk
=
(a)(OMk) + (1 - a)(SMk)
k=t OFCk
Example 8-7 Alternative CIM systems are expected to have the following total equivalent a nnu al costs.
OFC,
Il Ill
$130,000 150,000 165,000
Solution
where OMk = objective measure for alternative k SMk = subjective measure for alternative k a = relative importance weighting for OMk
• Method tirst developed in 1972 as the Brown-Gibson mode!. One signiticant version with generalized modifications was b.y ..E' •.-Y. Huang-a nd P. Ghandfo.{oush , " Procedu_r~s_Given for Ev alua ting , Sel-;;~ting. Robots ," Indus trial Engineering, Aprjl.!2~,_Qp. ·M-48 . Adapted with permis sion of publisher. . . ...... .. . .
Alternative k
Determine the OM for each alternative.
(8-5)
and where OMk, SMk, and a are each ~ 0 and :s 1. We will demon strate determination of OMk, SMk, and WEk by exampies below . The examples also include the use of certain formulas or approaches for determining weights and evaluation ratings which are merely different than the more general methods for determining weighted evaluations shown previously.
....
1
s = 130,000 +
1 1 150,000 + 165,000
= 0 .000020418 ~
and /
1 0 377 OMt = 130,000 x 0 .000020418 = ·
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w
Similarly, OMu= OMw =
= 0 .326
= 0 .297 L = 1.00
1.!)
235
Variation of Weighted Evaluation Method
Sec. 8.8
Subjective Measure The subjective measure for each alternative k, SM*, can be calculated for N subjective attributes , as we demonstrated earlier for weighted evalua. tions in Section 8. 7\ ~~
00
~
.§
z
N
~"' .. 00
..
i=
3 .5
< ~a:: .0
~
> ..
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0
ï= <
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<
<
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il:
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gg
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excellent = 4
~~ >>
=2
poor = 0
fair= 1 ..J_
Example 8-8 For the same alternative CIM systems for which objective measures were shown above, Table 8-8 gi'les mw subjective weights and evaluation ratings for each . Table 8-9 shows those numbers each divided by their respective totals, so they will add to 1. These we might cali "unitized" weights and ratings . Calculate the SM for each.
~8
Soo ... >.
8
good
very good = 3
5
~
(8-8)
0000
<... .8 "'
x (subjective evaluation rating;t)
For the example below, let us assume thal subjective attribute weights are assigned with a 20 being the highest, and that the subjective evaluation ratings are assigned as follows:
êê
~
t1l
ü
1)
1
>
z
2: (subjective attribute weight 1= 1
~
tl.JO:: H
z
0
SMk =
~§'
t;
o..>
::s
Solution
>
c
t1l
~
~"8~
NON
~
oo~
:!.
20 (2) 15 (3) 10(2) 10 (1) 5 (10 4 ) ~ •. = 60 7 + 60 8 + 60 6 + 60 4 + 60
ë
~"
8 ~8 ~] U.tll
Similarly (as shown on bottom of Table 8-9),
O>O
= 0 .302
SMn =
.:;"'
=
SM111=
" ·s 'E_ ,;j<
"'
.il:-§,·-
a::~
>.
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Table 8·9
.-:: 0
Cl
u
0 V\0 N--
0.378
L = 1.000
" " e o o " >-§ c....; .ÈiJ§:o
> ~ ·::: ::s u.o
= 0.320
UNITIZED SUBJECTIVE WEIGIITS AND EVALUATION RATINGS'
<"-l
U nitized Evaluation Ratings for Alternatives
S"'l~
Unitized Weight
~
20160 \ ' 15/60 \ 10/60 1 10/60 /
H ·-
5160/
L
a
J.OQ
/
Subjective Attribute
1
Il
Ill
2n 3/8
417
117 4/8
2/6
2/6
2/6
114 4/10
0 3/10
3/4 3/10
SM 1 0.320
0.302
0.378
Quality of results Ease of use Competitive Adaptability Expandability
1/8
l: Evaluation Ratings 1.0 1.0 1.0 1.0 1.0
L SM, =
'-.! ~
0
1.0
• Final SMs are shown at bottom for each alternative .
234
.....
Multiattribute Decision Analysis
l36
Chap . 8
Chap. 8
237
References
a = 0.3 helps 'èonfirm the results above for the three alternatives. The graph shows, among other things, that alternative 1 is preferred to alternative li for ali values of a , and that alternative 1 becomes preferred to alternative Ill for ali a slightly greater than 0.4 . lndeed, the " break-even" value of a can be calculated by equating the WEs for alternatives 1 and Ill as follows:
Weighted Evaluation
The final weighted evaluation results for each alternative k (WE*) for any a ( 2!: 0 and :5 1) can now be calculated as WE* = (a)(OM*) + ( 1 - a)( SM*) . Example 8-9
{a)(0.377)
If a = 0.3, calculate the WEs.
+
(l - a)(0.320) = (a)(0.297)
+ (1
- a)(0.378)
a = 0.42
Solution
t·
WE 1: (0.3)(0.377) + (1 - 0.3)(0.320) = 0.337
8.9 SUMMARY AND PROLOGUE
WE 11 : (0.3)(0.326) + (1 - 0.3)(0.302) = 0.309 WE 111 : (0.3)(0.297) + (1 - 0.3)(0.378) = 0.354 L
=
1.000
Thus . for a = 0.3, alternative III is slightly better than alternative 1, which is somewhat better than alternative IL Very often it is useful to depict the effects of differing weights on the weighted evaluation as illustrated in Fig. 8-7. The vertical dashed line at
0.38
0.377
0.37 0.36 ·c
0.35
·~ ~
0 .34
Il .c 0>
0.33
'0
~
0.326 0 .32
~
0.31 0.42 = " breakeven" for alts. 1 and Ill
0.30
"'1 0.297
REFERENCES
0.29 0
0.4
0 .2
0.6
0.8
1.0
a '"' weight of objective measure
1.0
Figurr 8-7. ple 8-9.
This chapter has concentrated on basiq{raphical, tabular, and additive weighting techniques for aiding in the analysis and resolution of multipl e attribu._te deeisiôn problems . In the following four chapters we explain ot he r more detailed, quantitative methodologies; namely , use of the multiattribute utility method, analytic hierarchy process, goal programming, and expert system technology. It should be recognized th at presentation of most of the methodologies in this and other chapters is based on "assumed certainty ." Thal is, we normally assume that required input estimates (such as weights, ratings, utili~functions, trade-offs, etc.) are known quantities . Of course, most real decision-making situations involve uncertainties . The main recommended approach to dealing with uncertainties is to perform sensitivity studies to determine the relative effect on the indicated decision(s) due to changes in important input estimates for whatever methodology and to display the results in easy-to-understand ways . The determination of what methodology(ies) to use depends on the nature of the decision problem and the preferences of the decision maker(s). Although this determination is a very inexact science, it can be shown J)lat the use of widely differing models will often have Jess effect on the probable quality of the solution than does the unintended omission of possible outcomes, alternatives, or important criteria. Whatever methodology is chosen should make the decision maker more confident and ultimately make his or her job easier.
0.8 (1 -
a)
a
0.6 0.4 0.2 weight of subjective measure
1. MacCrimmon, Ken R., Decision Making among Multiple Attribute Alternatives : 0
Weighted evaluation for ali possible weights ofOM and SM in Exam-
A Survey and Consolidated Approach, Memo RM-4823 -ARPA, Rand Corporation, December 1968. 2. Canada, John R. and Robert L. Edwards, Should We Alllomate Now? Evaluation of Computer lntegrated Manufacturing Systems, Industrial Extensiotl Service, North Carolina State University, 1986.
'-.1
-t:>. f-'-
238
Multiallribwe Decision Analysis
Chap . 8
EXERCISES 8-1. !Section 1!.4) Select a class or type of significant decision problem in your work or personallife. Name the three or more significant auri butes to be considered for thal class of problem. Assume that the weighting of these attributes is nebulous, so you think il desirable to develop only a matrix of alternatives versus attributes, with entries to show how weil each alternative meets each attribute in whatever measures are appropriate, such as in dollars, lime, rank , and so on. Th en use co lors or other symbol codings to facilitate ease of understanding of the differences by the persons making the final selection among the alternatives . 8-2. (Sections 8.5 through 8.7) Select a class or type of decision problem in your work or personallife involving multiple objectives (not necessarily same as for Problem 8- 1). Na me three to live most important objectives for that class of problem. as you see them, and do the following : (a) Using the method of paired comparisons, rank the objectives in order of decreasing importance . (b) Weight them and check for consistency and then normalize the weight to sum to 100. (c) ldentify two to four alternatives for the decision problem and evaluate how weil each alternative meets each objective on a scale of 0 to 10. Then multiply the evaluation ratings by the objective weights and sum these for each objective to compute a weighted evaluation for each alternative . 8-3. (Sections 8.5 through 8.7) (a) Weight the relative importance of the three or four most important aUributes you would consider in selecting a job. Assume that these are the only attributes you will quantitatively consider. Show how you make comparisons for internai consistency and normalize the factor weights to sum to 100. (b) Using the attribute weights developed above, obtain a weighted evaluation of two alternative jobs in which you might be interested by evaluating how weil each job satisfies each attribute using a scale from 0 to 10. 8-4. (Sections 8.5 through 8.7) (a) Weight the relative importance of the three to six most important auributes you would consider in selecting a persona! car. Assume that these are the only attributes you will quantitatively consider. Show how yo6. make comparisons for internai consistency and normalize the attribute weights to su m to 1,000. (b) Using the attribute weights developed above, obtain a weighted evaluation of three alternative makes of cars you might consider for purchase by eva luating how weil each make of car satisfies each attribute. Use a scale from 0 to 20.
:1 .1 ,,
1 1i
8-5. (Sections 8.4 through 8.7) Describe a significant multiattribute decision proble m of existing or potential meaningful interest to you. Examples are : housing, car, job, spouse, !ravel, and equipment. ldentify at !east four most .relevant attributes which are as independent of each other as possible and at !east two
~
Chap . 8
Exercises
,
~.,
i0 ,L
239
.. .--
'
or three mutually exclusive alternatives. (Note : ln the process of further analysis, you might weil add or delete alternatives and/or attributes .) (a) Describe the problem as meaningfully as possible using two or more graphical techniques . What conclusion (choices), if any, can you make from these?1 (b) Rank order the attributes. Can you combine this with one or more of the graphical techniques in part (a) to make choices? If so. describe . (c) Weight the aUri butes by the method or formula(s) of your choice and show at !east two checks and/or revisions of weights for consistency between initial judgments and preferences . (d) For each attribute show a graph of attribute out come (x axis) versus evaluation rating (y axis) . (e) Using parts (c) and (d), show calculations of_tl~ weighted evaluations for each alternative and display the results graph~lly for ease of comparison . 8-6. (Section 8. 1) For' ~-significant multiattribute decision problem of existing or potential meaningful interest to you (preferably an entirely different problem from Exercises 8-1 through 8-5), apply "Ben Franklin's method" of listing pros and cons and canceling out combinations thereof. His description as sumed just one alternative or course of action is being considered . If two or more altematives are considered, his process can be used for each alternative by il self, and further cancellations can be do ne if one can fi nd one or more pros which ar-e-equal for ali alternatives, or one or more cons which are equal for ali alternatives . 8-7. (Sections ~.4 through 8.7) De scribe a significant multiattribute decision problem of exasting meaningful interest to a "client" (friend , business person, or relative). Describe the problem as clearly as possible using words, tables, graphs, and so on, in sufficient detail to facilitate analysis by two or more methods described in this chapter. [Note : Problem should consist of at !east three (independent as possible) attributes and at !east two alternatives . ln the process of analysis you might weil add or del ete alternatives and/or attributes , • \ but do not go below the minimums above.) 1 \ 8-8,1 (Section !r.'3) Given the following matrix of outcomes for alternatives and ,· attributes (with higher numbers being better), show what you can conclude using the following sequential elimination method s: (a) Conjunctive constraint. (b) Dominance . (c) Lexicography, with rank order of attributes D > C > B > A.
Alternative Attribute
t
2
3
"Ideal"
Standard (Minimum Acceptable)
A
60 7
75 7 Excellent 8
90 8 Fair 8
100 JO Excellent 10
70 6 Good 6
B
c
Poor
D
7
'.J ~
N
240
Multiallribute Decision Analysis
Chap. 8
Chap. 8
"
Exercises
8-9. (Sections 8.3, 8.6, and 8. 7) Management has asked you to look into moving the "cutting" operation from where it is now to a new location. Management has given you three alternative areas, which are mutually exclusive, to consider. After inspecting the three areas and considering which factors reHect significant differences among the alternatives, you have decided on five independent ( attributes by which to evaluate the alternatives, listed in descending order of
A Band Ç
r- ·• '
(Note : The cost for operating the cutter once the operation has been moved is independent of the area chosen and hence is the same for each area.) The data of these factors for the three alternatives are as follows:
D
El -
Area 1
A
5000 ft
ll
Good Excellent
c D
$7500
E
60,000 ft'
Area Il 3000 ft Very good Very good $3000 85,000 ftl
Ideal
Area Ill
Standard
750ft Good Good $8500
Oft
Excellent Excellent
3000 ft Good Good
25,000 ft'
10,000 ftl
$5000 25,000 ft'
so
0 s x s ~.000 ft 3,000 s x s 10,000 ft Excellent Very good Good Fair Poor $0 s x s $10,000
-x/600 + 10 -xli ,400 + 5017
15,000 s x s 25,000 ftl 25,000 s x s 100,000 ~
35/2 - x/2.000 20/3 - x/15 ,000
7.5 5.0
2.5 0 -x/1,000 + 10
/
Item
Alignment A
Alignment B
Land Bridges Pavement Grade and drainage Erosion control Clearing and grading
s 4,044,662
s 4,390,000
10,134,000 4,112,500 7,050,000 470,000 188,000
8,701,000 -4,-«;2,.500 7,650,000 510,000 204,000 $25,917,500
Total
$25,999,162
Nonmonetary attributes are : Alignment A
( Determine which alternative is to be chosen using: (a) Alternative versus aliernative : comparison across alternatives . (l) Lexicography . (2) Elimination by aspects . (b) Weighted evaluation of alternatives using the rank reciprocal method for weighting attributes . (c) Weighted evaluation of alternatives using uniform or equal weights for attributes . For parts (b) and (c) the following are evaluation ra ting functions on a scale of 0 to 10.
10.0
\_./ ing plant. Based on the data and information given below, comparison c a n be made of the two alignments. The better one must be chosen as the proposed highway alignment that will be the connector between an interstate highway and tho proposed site. The route length for Alignment A is 4.7 miles and for Alignment B, 5.1 miles. The monetary costs are as follows:
Worst 10,000 ft Poor Poor $10,000 150,000 ftl
Evaluation Rating
:r ; 8-1~) Two highway alignments have been proposed for access to a new manufactur-
Alternative Anribute
Range for Attribute Outcome
A.uribute
importance:
A . Distance traveled from cutting operation to next operation [more distance is worse] B. Stability of foundation [strong (excellent) to weak (poor)] C . Access to loading and unloading [close (excellent) to far (poor)] D. Cost of the moving operation E. Storage capacity
241
Maintenance Noise pollution Cost savings (on gas) Accessibility to another m~or roadway Impact on wildlife Relocation of residences Road condition
Alignment B
Very good Excellent U.S . Highway 41
High Good Poor None
Little 2 Flat
Little 3 Hill y
Moct~rate
""'-J ~
Which highway alignment would you recommend? Show ali work.
w
242
Multiattribute Decision Analysis
Chap . 8
Chap. 8
Alternative
Attribute
Domestic 1 $8,400
$10,000
$9,300
25 mpg
30mpg
35 mpg
Gasoline Excellent
Diesel Excellent
5 o/11 of 10
7 OU I of JO 6
9
4 Excellent Fair Po or Excellent Very good
OUI
Very good Very good Good Very good Excellent
Po or
8-12. Utilize any three multiattribute techniques described in this chapter to analyze the following important si tuation; Y ou need to determine which job to take, given acceptable offers from firms A, B, and C. You first determine a set of attributes thal are most important in evaluating and then construct a decision matrix with alternatives as columns and attributes as rows, and fil! in. (Results are given in the matrix below .) Analyze the alternatives with the different techniques according to your perceptions (be complete-il look years to gel to this point!) .
All ri bute
A
B
c
Starting salary Opportunity for advancement Management attitude Location Type of work Opportunity for continuing education
$30,000 Excellent Very good Good Excellent Very good
$28,500 Excellent Excellent Fair Poor Good
$33,000 Fair Good Poor Very good Fair
A B
15 8 14 13
12 20 14 12
would be best by; (a) Lexicography. -(' (b) Elimination by aspects . (c) Dominance cheekt (assuming attribute B is eliminated from consideration). 8-14. (Sections 8.6 and 8.7) Given the same attributes and alternatives with evaluation ratings as in Exescise 8-13, show which alternative would be best using the weighted evaluation methodology and each of the following attribute weighting formulas . (Note : Normalize the summed weighted evaluations to equal 100.)
~
(a) Uniform. (b) Rank sum, with A > B > C > D. (c) Rank reciprocal, with A > B > C > D. 8-15. (Section 8.4) Construct a polar chart depicting the evaluation ratings for the .two alternatives in E~rcise 8-13. 8-16. (Section 8.7) Construct a graphical portrayal of Exercise 8- 14(b) with cumulative weights on the x axis and evaluation ratings on the y axis . 8-17. (Section 8.8) Volunteer alumni A and B have contributed $100M and $ 150M , respectively (higher is better!). However, evaluations of their leadership success (on a 1 to 5 maximum scale) are 4 and 3, respectively . Contributions (objective) are considered to be half as important as leadership success (subjective). Using the weighted evaluation method combining these measures, which should be the" Alumnus of the Year"? What part of 1.0 total "weighted 8-18.
Alternative (Finn)
2
If the attributes are ranked A > B > C > D in order of importance, the minimum standard evaluation rating is 10 for each. Show which alternative
of JO
4 Good Very good Excellent Poor
1
ID
Foreign
Gasoline Very good
Attribute
,c
1-
Alternative
Priee Gas mileage Type of fuel Corn fort Aesthetic appeal Passengers Ease of servicing Performance on road Stereo system Ease of cleaning upholstery Storage space
243
"
8-13. (Section 8.3) The follow)ng are evaluation ratings (on a scale of 0 to 20) of how weil each of two alternatives satisfies each of four attributes.
8·11. (a) Use the weighted evaluation mode! to make a selection of one of the three automobiles for which sorne data are given below. State your assumptions regarding miles driven each year; !ife of the automobile (how long you would keep it); market (resale) value at end of life; interest cost; priee of fuel; cost of annual maintenance; and attribute weights and other subjectively based determinations. (b) Use the data developed in part (a) and the weighted evaluation method in which objective and subjective measures are combined (OMk and SMk, as in Section 8.8) to make a. selection. Do your answers in parts (a) and (b) agree? Explain why they should (or should not) agree.
Domestic 2
Exercises
brown ies" does he score? (Section 8.8) For Exercise 8-17, assume the relative weight, a, for the objective measure (contributions) is not known . Plot the weighted evaluations for each of the alumni for a ranging from 0 to 1.0 on the x axis .
,
'-.1 ~ ~
CHAPITRE 8
Décider face à la complexité.
In traduction Our present complex environment calls for a new logic-a new way to cope with the myriad factors that affect the achievement of goals and the consistency .of the judgments we use to draw valid conclusions. This approach should be justifiable and appeal to our wisdom and good sense . lt should not be so complex that only the educated can use it, but should serve as a unifying tool for thought in general. That we are ali very much creatures of the moment needs little debate . Try to reconstruct what was said two minutes before in a conversation and you will soon discover that, even for a short duration, your recollection is fuzzy. Even when you remember, the precision of your recollection is usually less than exacting. Our understanding of the world not only needs repetition to improve our recoll.ection and precision, but it also depends very much on the in/ensi/y of our participation . lt is an exaggeration to conclude that humans are logical creatures. It is more accurate to say that our judgment depends on the totality of our impressions, even if they cannot ,be logically and rigorously justified . For better understanding we need to deal with experience as an ongoing process . We need concentra tion, repetition, diversity, debate, and, when necessary, consensus. Severa! good !essons have been learned in recent years from political leadership. Having a clearly defined goal has again been demonstrated to be the core of successful statecraft. Being consistent is another vital ingredient . Persuasion and support for one's view are still considered two of the main attributes of a great leader. What ensues is largely designed to help leaders get their points across . More and more people are finding it hard to put ali the ir trust in the unspoken, unjustified, and intuitive thinking of their leaders' decisions on ·complex matters. Whatever internai mechanism leaders have need s to be 1
00 1-'-
2
INTRODUCTION
articulated and understood . Just as language itself and the rules of thought had to be organized in a formai manner long ago, so must we now organize our thought processes so they lead us to good decisions. We_should be able to say th at, given the information, we agree on the methq_g._o(maklng the decision (but not necessarily on the quality o~_ that dec!~ior:t)_. ThÊ! matter will then become a common concern rather than a mystical phenomenon. The following pages offer the reader a method for organizing the information an.
1 Making Decisions in a Complex World This chapter deals with the following questions : How can we understand com.plex problems involvi_ng a great many factors? What are the basic processes of thought and behavior involved in making decisions? What role should ethics play in the decision-making process? Why do we need a new way of thinking about complex problems?
...
• How might a group of people holding varying opinions debate an issue in search of an acceptable compromise? What can they do about their differences?
3
MAKING D EC ISIO N S IN A CO MI'I.EX WlliU .D
COPING WITH COMPLEXITY To the best of our understanding, the world is a complex system of interacting elements. The economy, for example, depends on energy and other resources; the availability of energy depends on geography and politics; poli tics depends on military strength; military strength depends on technology; technology depends on ideas and resources; ideas depend on politics for their acceptance and support; and so on . In such an intricate network of factors, first causes and final effects cannot be identified easily . Our minds have not yet evolved to the point where we can clearly see these ultimate relationships and readily resolve important issues like nuclear energy, world trade, and environmental regulations . ln our complex world system, we are forced to cope with more problems than we have the resources to handle. To deal with unstructured social, economie, and political issues, we need to order our priorities, to agree that one objective outweighs another in the short term, and to make tradeoffs to serve the greatest common interest. But it is often difficult to agree on which objective outweighs another-'-particularly in complex issues where a wide margin of error is possible in making tradeoffs . Leaders may be confused by the diverse information provided by their assistants; they may need help in identifying differences of opinion and seeing positions where compromise can be reached . They may need to know which important issues must be researched in depth to obtain better information and how sensitive the out come is to slight or drastic changes in opinion and judgments. Intuitive thought processes that serve us weil in the familiar routine of daily )ife can . mislead us on complicated matters where sources of information and opinions are varied . lncreasingly we need to articulate and map out the issues to see whether what we think and what we feellead us to the same kind of answers . Most of us be lieve !ife is so complicated thal in order to solve problems we need more complicated ways of thinking . Yet thinking even in simple ways can be taxing. If we struggle to examine collections of only a few ideas at a lime, how can we understand complex problems involving a great many factors? Simple thinking about such problems leads to combi nations of ideas whose structure is not unlike a dish of spaghetti in which ali strands are separate- but tangled.
RATIONALE FOR A NEW FRAMEWORK What we need is not a more complicated way of thinking, since it is d iffic ult eno ugh to do simple thinking . Rather, we need to view our problems in an organized but complex framework thal allows for interaction and int erdependence among factors and still enables us to think about
ORGANIZIN G KNOWLEDGE FOR DF.CISIONS
5
them in a simple way . This new way of thinking should be accessible to ali without straining our innate capabilities. The analytic hierarchy process (AHP) described in this book provides such a framework . lt enables us to make effective decisions on complex issues by simplifying and expediting our natural decision-making processes. Basically the AHP is a method of breaking down a complex, un structured situation into its component parts; arranging these pa rt s, o r variables, into a hierarchie order; assigning numerical values to subj ect ive judgments on the relative importance of each variable ; and synthes izing the judgments to determine which variables have the highest priority and should be acted upon to influence the outcome of the situat ion . The AHP also provides an effective structure for group decisio n m aking by imposing a discipline on the group's thought processes. The necessity of assigning a numerical value to each variable of the problem helps decision makers to maintain cohesive thought patterns and to reach a conclusion. ln addition , the consensual nature of group decision making improves the consistency of the judgments and enhances the reliability of the AHP as a decision-making tool. In this chapter we consider the human behavioral and thought pro cesses involved in decision making . These natural processes form the basis for the analytic hierarchy process, which is described in Chapter 2 and explained in greater detail in Chapters 3, 4, and 5 . The rest of the book shows how the AHP can be applied to a variety of decision-making situations and problems .
ORGANIZING KNOWLEDGE FOR DECISIONS The AHP can best be appreciated by first looking into how the human mind organizes knowledge for decisions . The Iwo fundamental ap proaches humans have developed so far for analysis are the deductive approach and the inductive or systems approach .
The Deductive Approach and the Systems Approach We can analyze a system logically by representing it as a network and structuring it into chains and cycles. ln analyzing natural systems, for example, biologists break down networks into food chains, hydrologie cycles, and so on . After structuring the network, we look for explanations for the functioning of ils parts . Then, by an act of imagination , as no rules · of logic exist for combining these piecemeal explanations, we synthesize an explanation for the whole netw ork . But this sci entific, d eductive approach ignores the feedback mechanisms among the parts and betwee n !he parts and the environment thal affect the whole system . 00 UJ
-·-·-- -· ~--~------------.----------~--~-· 7
MAKING DECI SIONS IN A COMPLE X WOR LD
ORGANIZING KN O WL EDGE FOR DECISIONS
Systems theorists have pointed out that we can better understand an en tire system by examining it from a general, holistic perspective thal does not give as much attention to the function of the parts. For example, a caris better understood by observing how il functions in the environment than by studying the operation of its mechanical parts . In this way we see it as a whole; we can simultaneously perceive how the car runs and how it interacis with other cars, road conditions, traffic signais, and so on . Clearly both the deductive and the systems approach contribute to our understanding of complex systems. We can benefit by combining the two within an integrated, logical framework : the anaJytic hierarchy process. The AHP enables us to structure a system and its environment into mutu• ally interacting parts and then to synthesize them by measuring and ranking the impact of these parts on the entire system. We will see how this process works with systems as diverse as a transport network in the Sudan and the steel industry in the United States. By providing a new logic for synthesis, this structured approach to decision making eliminates much of the guesswork and confusion of our ordinary method of synthesizing an overall explanation for a system from piecemeal explanations arrived at through deduction .
son's ability to persuade others to accept his or her ideas, like a politician selling himself in a campaign . People, then, not only have different feelings about the same situa tion, but their feelings change or can be changed by discussion, new evidence, and interaction with other experienced people . Usually the out· come is a compromise of many viewpoints involving substantial change in individual attitudes . The fact is that when we make decisions, person a! preference and persuasion usually prevail over clear and straight logic . Our actual d.e cision-making processes have been illuminated by recent studies conducted by behaviorists, other psychologists, and brain researchers. Let us examine their findings .
The Role of Logic, Intuition, and Experience The everyday way we proceed to understand and 5olve problems is to use logical deduction to argue through familiar matters . For example , it is easy to reason that to increase capital one must obtain a good retum on it by investing it or by obtaining a good interest rate . In this case we know that in order for capital to increase from size A to size B, money has to be added to A somehow. That much we can say with bold certainty. But we tend to treat larger, imprecise perceptions of a problem by relying on feeling, experience, emotion, appeal to other people's understanding, and sometimes even force . Many political problemsin advanced as weil as Jess developed countries are handled in such a fashion . In unstructured situations people often act on the ir" gut feelings " rather th an strict! y on rational grounds . Logic plays a role mostly in arranging words and ideas after the conclusions have been reached . People in the public and private sectors tend to cooperate in defining and structuring their problems broadly and richly so that ali their ideas can be included . But when they need to explain which factors have the greatest impact on the outcome of a decision, not even experts with the clearest logic can hold fast to their positions in the face of objections. As a result they are willing to compromise. Thus decisions are based not so much on the clarity of ideas or amount of information exchanged as on the persistence of sorne participant in the decision-making process and on that per-
Behaviorist Theories Those who study and explain ways, reasons, and consequences of human and animal behavior are finding it difficult to uphold the idea that humans are rational animais . Their theories are helping to create an atmosphere in which people are accepted as they really are rather than as they were idealistically portrayed during the Renaissance and the Age of Rationalism . Human behavior is enormously complex; the many theories explaining human action are deep and multilayered , and probably ali con tribute to our understanding of human behavior . Instinct-Drive Theory. Sorne theorists consider rational thinking to be but a thin veneer over human behavior. Much of our action is driven by instinct-patterns woven into the mind , bone , and muscle . Just as wasps have an instinct for nest building and birds have their characteristic songs, humans also follow certain unleamed patterns of behavior, such as seeking food, mating, avoiding pa in, caring for young , and so on . Although instinct-drive theory describes such patterns, it does not explain them . lt is inadequate to account for most adult behavior, including sentiment, value, ambition, attitude, taste, and inclination . Reason-Impulse Theory . We tend to regard ourselves as rational animais capable of making choices based on objective, or real , criteria . We fee! thal most of our decisions flow from logical necessity , not from whim and caprice . Although we may acknowledge that needs and persona! motives are the driving forces behind human behavior, we contend that we use reason to attain our goals efficiently and without harm or in jury . Through reason we get what we want within the limits of available resources . And many of us ultimately do learn to apply rational techn iq u es to d eci sion making, regardless of what our persona! wishes m ay dict ate . But critics of this view say that our so-called rea so n is an abyss o f (X)
_p.
8
ORGANIZING KNOWLEDGE FOR DECISIONS
MAKING DECISIONS IN A COMPLEX WORLD
unconscious or barely conscious urges and habits that overwhelm the intellect . They argue that human relationships are essentially governed by irrational, emotional forces; rational appeal in most cases plays only a minor role . These reason-impulse theorists hold that our actions arè on imitation, habit, suggestion, or other subrational forms of thinking and are rarely due to pure logic. Planned actions are the result of analysis based on preferences as to which objectives are served best-and preferences are strongly influenced by habit and training rather than by rational thinking.
1,.
bas.ed
Dynamic Field Theory. Other behaviorists point to the influence of environmental factors on human behavior. We act in response to a "dynamic field" of stresses and tensions when we perceive the environment to deny or fulfill the satisfaction of our wants and needs. The hierarchy of human needs that motivate behavior has been examined by Abraham H . Maslow and others; these needs range from the most basic physiological and safety and security needs to sophisticated self-actualization and esthetic needs.
previous experience. lt is an iterative, or repeated, process of adding knowledge that···e laborates on or expands existing knowledge . Leaming can" be conscious and intentional, as in memorizing facts, or it can be unconscious and unintentional, as in discovering physically from experience that eating green apples results in a stomachache . We generally agree that people who experience a phenomenon firsthand are the ones who can best shed light on our understanding of it; indeed, knowledge derived from experience is basic to ali understanding .
Brain Research Findings
,. ~
Leaming Theories Most people tend to assume that the way we think and th~Jo.gi.ç. ~e use to develop our thinking are innately human _and that the basics of human knowledge have come down tous as a packa.ge from heaven : But recent learning theories argue that we learn mainly by triaLmd error and through feeling rather than through logic. Stimulus-response theory, ..for example, maintains that we gravitate somewhat randomly in certain directions to satisfy our needs and desires; acts that bring satisfaction are reinforced . Gestalt theory tells us that even what we consider to be " insight" -perceiving the necessary relationships in a situation-results from feeling . Generally we have vague feelings and inklings about what we think we have experienced, but we are not attentive enough to register ideas and feelings sharply. We have no systematic way of reconstructing from memory that which we did not learn, understand, or memorize consciously. Most of our daily experience goes before our senses and passes through our feelings like a hazy cloud that slightly moistens the environment but makes little difference to the growth of our understanding. Our understanding does grow when a particular experience connects weil with our earlier experiences, not mere! y with our knowledgè;·-or when it shocks us, capturing our attention involuntarily as it intrudes into our being (sometimes unconsciously) either pleasantly or forcefully. L.~~_r:_ning can be defined as the ability to recognize a specifie act in the light _()f
9
1r
1 •:•(:
1
The importance of intuition, feeling, and experience in human behav ior and decision making is further underscored by the findings of brain researchers . In particular, they have discovered a distinction in the func tiorïs of the two halves of the neocortex of the brain-the left is the logical, rational, and calciiiating member; the right is the intuitive, creative, and v~rbally inarÙculate half. The verbal half's job is to interpret for ourselves and the world the décisions of its mute brother. Note that the decisions are actually made by the intuitive, not the logical, half. The left hemisphere mere! y arranges and puts into words the insights of the right. Studies of human perception show that our senses both condition and limit whatever enters our consciousness . Older people who have experienced !ife longer often recognize that illusion is primary experience . (Perhaps this awareness results from the aging of the senses, but more likely it is due to the wisdom of age.) Our senses shape our world; thus we can never interpret the universe with absolute accuracy . The qualities we study are simply those we can perceive, and the laws we develop are concoctions of our sense-limited brains. Another !ife form could have many more senses than we have; for example, it could have a magnetic eye, could perceive the colors of the spectrum in white light, or could see through objects. As a result, its consciousness would be different from ours. Of course, our senses have expanded through human inventions such as the microscope, telescope, and X ray. We can perceive many qualities not directly accessible to our senses, and these are only a small portion of the potential total. If we had more senses acting at once, we would probably have difficulty sorting out our perceptions and understanding their relations , given the present stage of the evolution of our brains. Although sense data can be organized in chronological sequence, brain events are not rigidly bound by time . Ideas can occur before or after other ideas . Because the way we think is fluid, we are freer to arrange our ideas in the manner we desirefor good or evil purposes.
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MAKING DECISIONS IN A COMPLEX WORLD
ETHICAL CONSIDERATIONS
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When complexity makes normal !ife appear difficult and even hopeless to many, a strong individual may emerge to take over leadership charismatically by declaring that ali the pain and confusion are simply due to a single problem that must be solved; Hitler is an infamous example. Such leaders draw attention to one problem and persuade or force others to believe in them. By oversimplifying the situation, they banish ali other problems to narrow the perspective and make themselves appear logical in their explanations; they linearize the workings of a system in a deductive fashion . We accept new ideas as truths on the basis of the manner in which they cohere with knowledge we already possess. A system of beliefs could be perfectly consistent, and yet each belief could be false . Severa! examples come to mind . The Gregorian calendar (1582) was based on a set of consistent assumptions that the earth is the center of the universe. Yet it was so accurate that the accumulated error was one day every 3323 years . Our calendar is in error every 20,000 years . This is an example of a consistent prevailing view that was in error according to modern astronomy but produced good results . An example of a consistent theory of our ti me that produced bad results until recently involves the humble golf bali. People used to believe that it is the perfectly spherical golf bali thal travels farthest when hit-until it was discovered that the more a bali is used, the more dents and dimples it develops, which serve as wings countering drag and sustaining it longer in the air. Now golf balls are made according to precisely dimpled patterns. Consistent thinking with no real validity is frequently espoused by lunatics and other mental cases (sometimes of a seemingly respectable genre) . Beliefs can be deduced from one another in perfect consistency depending solely on the observance of formai, linear relationships between them . Thus one may develop a consistent deductive system with no real validity in this world . Because it is possible in complex, unstructured situations to present convincing arguments that have little correspondence to reality and may harm society, one must apply certain ethical standards to the decision· making process. The philosopher Alasdair Maclntyre of Boston University has identified four qualities thal should characterize a decision maker's approach to dealing with social issues :
Truthfulness by not oversimplifying complexity . Our political and legislative processes demonstrate thal it is easier to consider issues such as environmental protection or health care in a narrow, piecemeal way than to look at ali the critical variables, fit them together, and determine their priorities and implications. ln the short run a simplistic approach
PERSPECTIVE
11
may satisfy local contending parties, but it is no way to get at the answers to complex problems.
• Justice by evaluating costs and benefits and assig11ing costs to those who get the benefits. Everyone involved in a decision-making situation- family members deciding whether to purchase a home computer or corporate executives deciding which companies to invest in-should have a chance to weigh costs and benefits. Those who receive the benefits should be the ones who pay the costs, and vice versa. Justice demands not only thal everyone have a voice and a vote but also thal those who will bear the risks and dangers have more of a voice and vote than others. (People should also be informed about where and how they are paying for benefits, particularly for commodities whose priees are controlled or subsidized .) • Ability to plan for the unknown by calculating changes, determi11il18 where they are likely to occur, and deciding which przorities should dictate actio11 . Leaders must be able tb plan and deal both with projected futures , such as higher energy priees in 1985, and with desired and less predictable futures, such as energy independence by the year 2000. Flexibility in adapting to cha11ge by plallni11g , implementi11g , a11d, i11 respOIISe to new conditions , repla1111ing and reimplementing . This iterative approach is essentially a learning process; it tempers our tenden cy to let immediate needs dictate short -lerm solutions . For example, fle xibility is necessary in planning strategies for the use of alternative resources in dealing with the energy situation . As we will see, the analytic hierarchy process encourages socially re sponsible decision making by helping leaders to avoid oversimplification, to identify and evaluate costs and benefits, to plan for the future, and to adapt to change.
PERSPECTIVE Most of us have trouble coping with ordinary problems of society that cannot be understood bya deductive , linear, cause-and -effect explanation . Our respect for the scientific method, which relies on deduction , has led us to try to solve ali our problems through logical debate. As a result of our scientific education and because science usually deals with thing-s we can observe through our physical senses, we are made to fee! that there is precis_ion in what we do . Our senses ·are trained to be consistent in focus ing on their abjects; thus our minds have a sense of consistency in synthesizing and interpreting sense data . But when we deal direct! y with ideas rather than with sense perceptions, things tose their precision . The reason (X)
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-- ···- -·- - KEY CONCEPTS
MAKING DECISIONS IN A COMPLEX WO HLD
is that we use words whose meanings are imprecise. Philosophers have long recognized that primary language does not express thoughts or ideas but feelings and affections. Moreover, we cannot be exact in describing abstract relationships, and our understanding is conditioned by our states. of mind, feeling, and imagination at the moment we are thinking . Thought without language is impossible . Abelard says that "language is generated by intellect and generales intellect." According to the Chandogya Upanishad, "The essence of man is speech." We are very much creatures of the moment. At any given time our attention is captured by whatever our senses perceive. We cannot remember the past de arly, not even what the tomorrow of our persona! lives will be like, despite the fact that sorne of us may have lived more than ten thousand days on this earth, repeating the same pattern that many times . With ali this experience we are still unable to see the immediate future with adequate clarity. But sorne would try to venture predictions on serious issues of politics or economies in which they have had little experience . To understand and deal with what is going on in the world, we need to improve our recollection of events and the precision of our knowledge by reviewing the facts and organizing them in a logical framework. If we are to make decisions thal are rational and effective, we must participate intensely in the act of understanding the world around us. It is an exaggeration to say thal humans are logical creatures. More accurately, our understanding is filtered through our senses, and our judgment relies on often hazy impressions of reality. With experience and through the perceptions and opinions of other people, our views of reality may change and become more precise . For a better understanding of the world, we need to persevere in thinking matters through carefully and to debate with others who hold different views. But the complexity of social systems cannot await a full, logical analysis of situations on which our health, safety, and even survival depend. We need to rethink the traditional use of logic to derive knowledge. We also need to expand our analytic procedures to improve our understanding of situations in which not only time and space but also human behavior pla ys a fundamental role in determining the outcome . The analytic hierarchy process enables decision makers to represent the simultaneous interaction of many factors in complex, unstructured situations . lt helps them to identify and set priorities on the basis of their objectives and their knowledge and experience of each problem . As we have seen, our feelings and intuitive judgments are probably more representative of our thinking and behavior than are our verbalizations of them . The new framework organizes feelings and intuitive judgments as weil as logic so that we can map out complex situations as we perceive them . lt reflects the simple, intuitive way we actually deal with problems,
13
but it improves and streamlines the process by providing a structured approach to decision making .
KEY CONCEPTS
0
In our complex world system, we are forced to cope with more prob· lems than we have the resources to handle .
0
What we need is not a more complicated way of thinking but a frame · work thal will enable us to think of complex problems in a simple way . There are two fundamental approaches to solving problems : the de · ductive approach and the systems approach . Basically, the deductive approach focuses on the parts whereas the systems approach co ncen trates on the workings of the whole. The analytic hierarchy process, the approach proposed in this book, combines these two approaches into one integrated, logical framework .
0
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Humans are not often logical creatures . Most of the time we base our judgments on hazy impressions of reality and then use logic to defend our conclusions.
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The analytic hierarchy process organizes feelings and intuition and logic in a structured approach to decision making .
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15
WHAT IT IS AND HOW IT WORKS
WHAT IT IS AND HOW 1T WORKS
2 The Analytic Hierarchy Process
To introduce the analytic hierarchy process, consider the following example of a decision problem. The Brandywine River Region in Pennsylvania faces possible urbanization and its environmental effects. What actions should the people of the region take to maintain environmental quality? Should they allow development and invest money to prevent environmental deterioration or should they limit development? Planners who used the AHP to study this problem first defined the situation carefully, including as many relevant details as possible . Then they structured it into a hierarchy of levels of deüil (Figure 2-1) . The highest leve! was the overall objective of protecting environmental quality . The lowest included the final actions, or alternative plans, thal would con tribute positively or negatively to the main objective through their impact on the intermediate criteria . The alternatives were (A) to leave the area nonurbanized, (B) to allow partial urbanization, and (C) to allow total urbanization . The intermediate levels of the hierarchy comprised the Iwo basic criteria for evaluating environmental quality : (1) esthetic criteria, which were further structured into properties of vividness, intactness, and
Environmental Quality This chapter deals with the following questions: What are the three basic principles of logical analysis? How do these three principles relate to a new approach to decision making-the analytic . hierarchy process? What can we do when the usual scales of measurement-dollars, time, tons, and so forth-fail to measure intangible qualities?
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Why is the analytic hierarchy process such a powerful method for tackling complex political and socioeconomic problems?
! What can you expect to gain by using the analytic hierarchy process?
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Hydrology
Hierarchy for Brandywine River Regiof1
THE ANAL YTIC lliERARC HY PROCESS
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no noise or disturbances ; and (2) hydrologie criteria, subdivided into no flooding, water quality, and channel naturalness . This hierarchy graphically depicts the interdependence of elements in the problem; it both isolaies the relevant factors and displays them in the larger context of their relationship to each other and to the system as a whole. After developing the hierarchy , the planners judged the relative importance of ali the elements. They quantified these judgments by assigning them numbers from 1 to 9-and sometimes they disagreed . On many major issues where an impasse in the judgment of different people occurs, careful assessment of the differences in the intensity with which these people defend their preferences and opinions is necessary. Often words alone or logical argument cannot express the subtleties of deeply felt differences. But these differences can be measured by numbers, as we will see later on. After debate and compromise, the planners determined priorities for the elements of the hierarchy. Through a sequential process the judgments were synthesized and the desirability of each of the three alternative plans was estimated mathematically. The plan with the highest numerical value, and therefore priority (in this case, plan B), was the obvious best choice. Judgments on the relative importance of each element in the hierarchy were made by people who were knowledgeable about the Brandywine River Region and about problems of urbanization and environmental quality . Yet even experts can make mistakes in setting up a hierarchy or discriminating between pairs of elements to judge priorities. The AHP also tests the con sistency o f judgments; too gre at a departure from the perfectly consistent value indica tes a need to improve the judgments or to restructure the hi erarchy. Suppose we take a doser look at the question of consistency . The consistency is perfect if ali the judgments relate to each other in a perfect way . lf y ou say thal you pre fer spring three times more to summer and thal you prefer summer twice more to winter, then when you give the judgment comparing your preference of spring to winter it should be 6 and no t anythi n g else. The greater your deviation from 6, the greater your inconsistency. This observation applies to relations among ali the judgments g iven . We would have perfect consistency , then, if ali the relations checked out c~ rrectly . As we will see , there is a rather simple way of verifyi ng inconsistency and how much it deviates from perfect consistenc y . There is also a good w ay for interpreting what inconsistency means in prac tical terms . When we are revising judgments, this method is useful an d necessa ry . This a pproach to the Brandywine River problem illustrates the basic p rinciples o f the a n aly tic hierarchy process . Now let us take a doser look at these principle s .
PRINCIPLES OF ANAL YTIC THIN KING
17
PRINCIPLES OF ANALYTIC THINKING In solving problems by explicit logical analysis, three principles can be distinguished : the principle of constructing hierarchies, the principle of establishing priorities, and the principle of logical consistency. As suggested in the Brandywine River example, these natural principles of analytic thought underlie the AHP.
Structuring Hierarchies Humans have the ability to perceive things and ideas, to identify them , and to communicate what they observe. For detailed knowledge our minds structure complex reality into its constituent parts, and these in tum into their parts, and so on hierarchically. The number of parts usually ranges between five and nine . In the Brandywine River Region study, the idea of environmental quality was structured into six elements : vividness, intactness, no noise or disturbance, no flooding, water quality , and channel naturalness. By breaking down reality into homogeneous dusters and subdividing these clusters into smaller ones, we can integrale large amounts of information into the structure of a problem and form a m ore complete picture of the whole system . (This process is explored further in the next chapter.)
Setting Priorities Humans also have the ability to perceive relationships among the things they observe, to compare pairs of similar things against certain criteria, and to discriminate between both members of a pair by judging the intensity of their preference for one over the other. Then they synthesize their judgments- through imagination or, with the AHP, through a new logical process- and gain a better understanding of the whole system . In the Brandywine River Region study, the planners established relationships between the elements of each leve! of the hierarchy by comparing the elements in pairs. These relationships represent the relative impact of the elements of a given leve! on each element of the next higher leve!. ln this context the latter element serves as a criterion and is called a pro pert y . The result of this discrimination process is a uector of priority , or of relative importance, of the elements with respect to each property . This pairwise comparisoh is repeated for ail the elements in each leve!. The final step is to come down the hierarchy by weighing each vector by the priority of its property. Th is synthesis results in a set of net priority weights for the bottom leve!. The element with the h ighest we ight-plan B (part ial ur-
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19
THE ANA LYTI C HI ERARCHY PROCESS
MEASUREM ENT
banization) in our example- is the one that merits the most serious consideration for action , although the others are not ruled out entirely . This principle and the next are fully explained in Chapter S.
numbers are simply artifacts that gi ve us the illusion of greater precision than we are capable of feeling, or whether we are missing a great deal by not realizing that numbers are a creation of our minds to reflect feelings and distinctions. Perhaps we have not yet recognized and appreciated their value in solving complex, unstructured problems. In a moment we will briefly consider how numbers have come to be used in our lives to measure our perceptions of physical stimul i. Later we will see that numbers can also be used to reflect accurately our subj ective judgments and their intensity; they can be used to distinguish among intangible as well as physical stimuli. Chapter 5 describes a simple way to use numbers to synthesize outcomes that faithfully represent our intuitive feeling and understanding of what we perceive the outcomes to be . The advantage is that finer shades of differences in judgment can be identified for their effect on the outcome and that we can accommoda te different opinions in the decision -making framework .
Logical Consistency
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The third principle of analytic thought is logical consistency . Humans have the ability to establish relat ionships among objects or ideas in such a way that they are coherent-thal is, they relate well to each other and their relations exhibit consistency. Consistency means two things . The first is that similar ideas or objects are grouped according to homogeneity and relevance . For example, a grape and amarble can be grouped into a homogeneous set if roundness is the relevant criterion but not if flavor _is the criterion. The second meaning of consistency is that the intensities of relations among ideas or objects based on a particular criterion justify each other in sorne logical way . Thus if sweetness is the criterion and honey is judged to be five times sweeter than sugar, and sugar twice as sweet as molasses, then honey should be taken to be ten times sweeter than molas ses. If honey is judged to be only four times sweeter than molasses, then the judgments are inconsistent and the process may have to be repeated if more accurate judgments could be obtained . ln utilizing these principles, the analy tic hierarchy process incorporates both the qualitative and the quantitative aspects of human thought : the qualitative to define the problem and its hierarchy and the quantitative to express judgments and preferences concisely . The process itself is designed to integrale these dual properties. lt clearly shows thal for better decision making the quantitative is basic to making sou:-~d decisions in complex situations where it is necessary to determine priorities and make tradeoffs. To calculate priorities, we need a practical method of generating scales for measurement. MEASUREMENT People are generally wary, if not distrustful, when numbers are introduced into the traditional process of decision making . But appropriately chosen numbers can represent variat ions in feelings more faithfully than can words or rhetoric. In the face of complexity, we run out of words to express adequately our full awareness of what we sense to be taking place. Words limit the perspectives of our feelings. · Numbers are used in many different ways in our civilization to measure ali kind s of physi cal experience . We find this application acceptable . The question is whe ther we can extend and justify the use of numbers in so rne reasonable , easily understood way to reflect our feelings on various social, econo mie , and political matters . We need to look into whether
Evolution of Scales Our highly organized civili zation depends on scales to measure such qualities as time, length. temperature , and money . Such measures were not handed down through the burning bush but evolved historically . Time. Time is a fundamental quality of nature; ils measurement is at the foundation of science. The Sumerians were the first to divide the year and the day into units . Their year contained twelve months; and each montlt, thirty days. Egyptian priests divided the year into 365 days . The full sun light period of a day was divided into ten hours, corresponding to ten fi,ngers; dawn and dusk were each allotted one hour, bringing the total to twelve . The night was also allotted twelve hours, making twenty -four hours in a day . The daylight hours were marked by shadow docks, precursors of sundials, and the night hours were marked by the appearance of stars. Thus neither daylight nor night hours were uniform in length ; they depended on the seasonal transit of the sun and the rising of the stars. The real beginning of time measurement was the depsydra , a water dock that measured time by the emptying and filling of a vessel. As the depsydra became more corn mon , so too did the notion of lime as a thing in itself, a flowing reality measured independently of the heavens . ln the tenth century Arab scientists developed an improved sundial thal marked off the hours accu ratel y year round and was the first use of fixed lime units. In the thirteenth century mechanical docks m easured lime by uniform 00 periodic motions . Europeans developed precision in meas u ring lim e dur- ;...... ing the sev{mteenth and eighteenth centuries . In October 1960 the 0 Eleventh General Conference on Weights and Measures, meetin ~ in Paris,:
THE ANAL YTI C HIERARC IIY PROCESS
refined chronological precision by defining the second as the duration of 9,192,631,770 cycles of the radiation associa led wi th a specified transi lion, or change in energy leve!, of the cesium atom. Length. The Babylonians, Egyptians, Greeks, and Chinese ali had their own un ils and subunits of length and other physical qualities such as area, weight, and liquid volume. In the thirteenth century the English in~ro duced a standard yard and divided it into three feet of twelve inches each . They also defined a rodas five and a half yards and a furlong as one-eighth of a mile. In 1878 the yard was redefined as "the straight line or distance betwe:en the centers of two gold plugs or pins in the bronze bar . . . measured when the bar is at the temperature of sixty-two degrees of Fahrenheit's thermometer, and when it is supported by bronze rollers placed und er it in such a manner as best to a void flexure of the bar." The French Revolution contributed the metric system. In 1792 Louis XVI issued a proclamation directing two engineers to determine the length of the meler. They set out to measure the distance on the meridian from Barcelona, Spain, to Dunkirk, France, but civil war intervened and the task took years to complete. A provisional meler was established in 1795 and was adopted by the French Assembly four years later: The meler was "one ten-millionth part of a meridional quadrant of the earth ." Again in Oclober 1960 the meler was redefined as 1,650,753 .73 wavelengths in a vacuum of the orange-red line of the spectrum of krypton-86. Temperature. The mercury therm o meter was invented by the German physicist Gabriel Daniel Fahrenheit in the eighteenth century. He considered body temperature to be 100° on his scale (later found to be 98.6°, as he erred in determining his fixed points) and the temperature of the coldest thing he could produce in his laboratory- a mixture of salt and ice--to be 0°. The freezing and boiling points of pure water at sea leve! had to be 32° and 212°, respectively . Although the thermometer has been improved, Fahrenheit's scale survives unchanged . It is sometimes preferred to the Celsius scale, devised by the Swedish astronomer Anders Celsius in 1742, because it has smaller subdivisions . For highly accurate temperature readings, Lord Kelvin (William Thompson) first suggested the use of the gas thermometer in the nineteenth century. AbS()lute zero is equal to -273° Celsius, the temperature at which ali atomic activity stops . Money. Mo netary standards differ from the other units of measure in th al they are neither uniform nor consistent. The monetary value we attribu te to goods and services fluctu a tes according to supply and demand and perce iv ed desira bility or utility. The evolution of money is closely interwoven w ith tha l of ci vilization . In sorne cases, as in ancient Egypt, the p reva ilin g m o ne tary sys te m evolved from the political, social, and eco -
MEASUREMENT
21
nomic systems, whereas in ancien! Lydia these institutions followed the progress of monetary evolution. Although goods and services can be exchanged directly, the adoption of a monetary system simplifies trade and promotes higher productivity. The Need for a New Scaie Just as we can distinguish and measure physical relationshipsmeters for length, for example, and seconds for time--we are capabl e of doing the same with abstract relationships . We have the capacity for a range of feeling and discrimination thal permits us to develop rela tio nships among the elements of a problem and to determine which elem e nt s have the greatest impact on the desired solution . In dealing with concret e matters, such as repairing a car, we perceive the varying intensity of impact through our senses, by hearing a faulty motor or seeing a leak, or through their refinement by scientific instruments such as a voltmeter or pressure gauge. We carry out this process of measuring the priorities of impacts in order to solve problems . Soto determine the intensity of impact of the various components of a system, we must perform sorne type of measurement on a scale with units such as pounds, seconds, miles, . and dollars . But these sc ales li mit the nature of ideas we can deal with. Social, political , and other qualitative factors can in no reasonable way be assessed in terms of physical or economie measurement . What then can we do? We can devise a scale thal enables us to measure intangible qualities, just as scales evolved for measuring physical qualities . Chapter 5 presents such a scale to measure priority impacts in unstructured systems . This new way of assessing intangible qualities should hold up in areas where we already know the unit of measurement, which can then be used to validate the method . And in fact examples show that this approach to measuring priorities can be used to generale results conforming to classic ratio scale measurement in physics, economies, and other fields where standard mea sures already exist. To measure priorities, we compare one element with another . The old adage thal one cannot compare apples and oranges is false . Apples and oranges have many properties in common : size , shape, !aste, aroma, color, seediness, juiciness, and so on . We may prefer an orange for sorne properties and an apple for others; moreover, the strength of our preference may vary. We may be indifferent to size and color, but have a strong preference for !aste,. which again may change with the time of day . It is my thesis thal this sort of complicated comparison occurs in real !ife over and over a gain , and sorne kind of mathematical approach is required to help us determine priorities and make tradeoffs. This approach is the analytic hierarchy process:
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THE ANi\l.YTIC IIIFRARCIIY PROCESS
AHP : A FLEXIBLE MODEL FOR DECISION MAKING
Unlty: The AHP provides a single. easily understood, flexible mode! for a wide range of unstructured problems
AHP: A FLEXIBLE MODEL FOR DECISION MAKING
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These basic observations on human nature, analytic thinking, and measurement have led to the development of a useful mode! for solving problems quantitatively. The analytic hierarchy process is a flexible mode! that allows individuals or groups to shape ideas and define problems by making their own assumptions and deriving the desired solution from them . lt also enables people to test the sensitivity of the solution, or outcome, to changes in information . Designed to accommodate our human nature rather than force us into a mode of thinking that may violate our better judgment, the AHP is a powerful process for tackling complex poli tical and socioeconomic problems. The AHP incorpora tes judgments and persona! values in a logical way. lt depends on imagination, experience, and knowledge to structure the hierarchy of a problem and on logic, intuition., and experience to provide judgments . Once accepted and followed, the AHP shows us how to connect elements of one part of the problem with those of another to obtain the combined outcome. lt is a process for identifying, understanding, and assessing the interactions of a system as a whole . To define a complex problem and to develop sound judgments, the AHP must be progressively repeated, or iterated, over time; one can hardly expect instant solutions to complicated problems with which one has wrestled for a long time. The AHP is flexible enough to allow revisiondecision makers can both expand the elements of a problem hierarchy and change their judgments. lt also permits them to investigate the sensitivity of the outcome to whatever kinds of change may be anticipated. Each iteration of the AHP is like hypothesis making and testing; the progressive refinement of hypotheses leads to a better understanding of the system. The many practical applications of the AHP have generated sample hierarchies, sorne of which are presented in Chapter 4. With mi nor modification, sorne of thèse paradigms can be used to structure new problems. Another feature of the AHP is that it provides a framework for group participation in decision making or problem solving. We have seen that ideas and judgments can be questioned and strengthened or weakened by evidence that other people present. The way to shape unstructured reality is through participation, bargaining, and compromise. Indeed, the conceptualization of any problem by the analytic hierarchy process requires one to consider ideas, judgments, and facts accepted by others as essential aspects of the problem. Group participation can contribute to the overall validity of the outcome, although perhaps not tù the ease of implementation if the views diverge widely . Th us one could include in the process any information derived scientifically or intuitively. The process can be applied to real problems and is particularly useful for - " ""Ca ting resources, planning, analyzing the impact of policy, and
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Process Repetition: The AHP enables people to re fine their definition of a problem and to improve th~ir judgment and understanding through repetition
Complexlty : The AHP integrales deductive and systems approaches in solving complex problems
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Judgment and Consensus: \ The AHP does not insist on consensus but synthesizes a representative outcome from d1verse ......______ judgments
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Interdependance: The AHP can deal w1th the 1nterdependence of elements 1n a system and does not tnSISt on hnear th1nk1ng
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Tradeoffs: The AHP takes into consideration the relative priorities of factors in a system and enables people to select the best alternative based on the ir goals
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Measurement: The AHP provides a scale for measuring intangibles and a method lor establishing priorities
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Advantages of the Analytic Hierarchy Process
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1ΠANALYTIC lllERARCHY PROCESS
KEY CONCEPTS
resolving conflicts. Social and physical scientists, engineers, policymakers, and e yen laypersons can use the method without intervention by so-called experts; those who have a problem are ordinarily best informed about that particular problem. Currently the AHP is being widely used in corporate planning, portfolio selection, and benefit/cost analysis by government agencies for resource allocation purposes . And it is being used more widely on an international scale for planning infrastructure in developing countries and for evalüating natural resources for investment. Figure 2-2 summarizes the advantages of using the AHP as a new approach to problem solving and decision making .
The AHP can be used to stimulate ideas for creative courses of action and to evaluate their effectiveness. lt helps leaders determine ·what information is worth acquiring to evaluate the impact of relevant factors in complex situations. And it tracks inconsistencies in the participants' judgments and preferences, thereby enabling leaders to assess the quality of their assistants' knowledge and the stability of the solution . The next three chapters elaborai~ on this new approach to decision making . Here is what one can expect to gain by using it:
PERSPECTIVE
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The analytic hierarchy process reflects the way we naturally behave and think . But it improves upon nature by accelerating our thought processes and broadening our consciousness to include more factors than we would ordinarily consider. The AHP is a process of "systemic rationality" : lt enables us to consider a problem as a whole and to study the simultaneous interaction of its components within a hierarchy. This process is rather sophisticated. If we observed an individual, such as a child, who does not have to deal with complex issues or do severa! tasks at the same time, we most likely would not note a talent for hierarchie organization . Over long periods of time, however, the mind learns sho rtcuts and groups activities into clusters. This mental process gradually develops into a style for looking at the ·world and organizing it in su ch a way that we can deal with it efficiently. Sorne people have gone so far as to point out that nature itself organizes matter and !ife hierarchically . But it is difficult to separate this observation from the fact that to understand the complexity of nature, it is we who have to sort and arrange what we perceive and see hierarchically . Moreover, the interactio ns among elements of a hierarchy do not always have an inherent structure apart from what we can observe. We must identify and synthesize these interactions based on our objectives aild the knowledge and experience we have of each problem . The AHP addresses complex problems on their own terms of interaction . It a llows people to lay out a problem as they see it in ils complexity and to refine ils definition and structure through iteration . To identify criti cal problems, to define their structure, and to locale and resolve confli cts, the AHP calls for informati o n and judgments from severa! partici pan ts in th e process. Through a mathematical sequence it synthesizes their judgmen ts in to an overall est imate of the relative priorit ies of alternative co urses o f a ctio n . The pri orities yielded by the AHP are the basic units used in ali ty pes of an alysis; for example, they can serve as guidelines for alloca ti n g re so urces or as probabilities in making predictions.
1. A practical way to deal quantitatively with different kinds of functi o nal relations in a complex network. 2. A powerful tool for integrating forward (projected) and backward (desired) planning in an interactive manner that reflects the judgment s o f ali relevant managerial personnel. The output of this process is explicit rules for allocating resources among current and new st rategy offerings--or to satisfy a specifie set of corporate objectives-or under alternative environmental scenarios . 3. A new way to : Integrale hard data with subjective judgments about intangible fac tors. lncorporate judgments of severa! people and resolve conflicts among them . Perform sensitivity analysis and revision at low cost . Use marginal as weil as average pri orities to guide allocation . Enhance the capacity of management to make tradeoffs explicitly. 4. A technique complementing other on es (benefitlcost, priority, risk minimization) for selecting projects or activities . 5. A single replacement for a variety of schemes for projecting the future and protecting against risk and uncertainty . 6. A vehicle for monitoring and guiding organizational performance ta ward a dynamic set of goals .
KEY CONCEPTS
0
There are three basic prin ci pies of the analytic hierarchy process: 1. Hierarchie representation and decomposition , which we cali h ierarchie structuring-that is, breaking clown the problem into separa te elements. 2. Priority discrimination and synthesis, which we cali pri orit y setting-that is, ranking the element s by relative importance .
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3. Logical consistency-that is, ensuring that elements are grouped logically and ranked consistently according to a logical criterion. We cannot measure without a scale, but traditional scales such as ti me and money limit the nature of ideas we can deal with. Thus we need a new scale for measuring intangible qualities. The analytic hierarchy process is a flexible mode! that allows us to make decisions by combining judgment and persona! values in a logica! way.
3 Analyzing and Structuring Hierarchies This chapter deals with the following questions : •
Why are hierarchies fundamental to human thinking?
•
How can they be used to understand complex systems?
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What is the difference between a structural hierarchy and a functional hierarchy? How do we go about constructing a hierarchy?
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~ ANALYZING AND STRUCfURING IIIERARCHIES
HIERARCHIES: A TOOL OF THE MIND Complex systems can best be understood by breaking them down into the ir constituent elements, structuring the elements hierarchically, and then composing, or synthesizing, judgments on the relative importance of the elements at each leve! of the hierarchy into a set of overall priorities. This chapter explains how to structure problems hierarchically . Hierarchies are a fundamental tool of the human mind . They involve identifying the elements of a problem , grouping the elements into homogeneous sets, and arranging these sets in different levels . The simples! hierarchies are linear, rising or descending from one levet to another, such as physics hierarchies that rise from atoms to molecules to compounds and so on; the most complex are networks with interacting elements, such as systems representing the learning process of a child .
CLASSIFYING HIERARCHIES Hierarchies can be divided into two kinds : structural and functional. ln structural hierarchies, complex systems are structured into their constituent parts in descending order according to structural properties such as size, shape, color, or age. A structural hierarchy of"the universe would descend fro m galaxies to constellations to solar systems to planets, and so on , down to atoms, nucleii, protons, and neutrons . Structural hierarchies relate closely to the way our brains analyze complexity by breaking down the abject s perceived by our senses into clu sters, s ubclu sters, and still smaller clusters. ln contras!, functional hierarchies decompose complex systems into their con stituen t parts according to their essential relationships . A conf!ict over school busing to achieve integration can be structured into a cluster of major st akeholders (majority and minority communities, city officiais, board of education , federal government) ; a cluster of stakeholders' objective s (education for children, retention of power, and the like) ; and alternative outcomes (complete, partial, or no busing) . Such functional hierarchies help people to steer a system toward a desired goal- like conflict re so lution , efficient performance , or overall happiness . For the purposes of this book , functional hierarchies are the only ki nd thal need be considered . Each set of elements in a functional hierarchy occupies a leve! of the hierarch y. The top leve! , called the foCIIs , consists of only one element: the broad , ov erall objective . Subsequent levels may each have severa! elem ents, although thei r number is usually small- between five and nine. Beca use the elem ent s in one leve! are to be compared with one another against a criter ion in the nex t higher leve!. the elements in each leve! must be of the same order o f magnitude . If the d isparity between them is great, they shou ld be lo ng to different levels . For example , we cannat make a
CONSTRUCflNG HIERARCHIES
29
precise comparison between two jobs whose performances· differ in difficulty by a factor of 100 because our judgment would be subject to significan t error. lnstead, we first group simple jobs into a cluster and compare the cluster with a job one arder of magnitude more difficult to perform than a simple job. We then compare the jobs in the cluster among themselves according to difficulty of performance. When we compare the results of the two comparison processes, we obtain a net comparison of a simple job with the more difficult one . To avoid making large errors, we must carry out this process of clustering. By forming hierarchically arranged cl usters of like elements-simple jobs in this case--we can efficiently compl e te the process of comparing the simple with the very complex. Similarl y, to com pare small stones with boulders or atoms with stars, we mu st int etv e ne between them severa! levels of abjects of slightly differe(1t magnitud e to make the transition and comparison possible . Because a hierarchy represents a mode! of how the brain analyzes complexity, the hierarchy must be flexible enough to deal with that com plexity. The levels of a hierarchy interconnect like layers of cell tissue to form an organic whole that setves a certain function . A spiraling effect is noticeable when we move from the focus expanding the hierarchy to the leve! of simple elements. The expansion may be continued to the leve! of elements of minutest concern . The next chapter illustrates the flexibility of hierarchies with a variety of examples. Ali are functional hierarchies , but sorne are complete-that is, ali the elements in one leve! share every prop erty in the next higher level- and SOrne are i11 C0 111p /ete in thal Sorne ele ments in a leve! do not share properties . CONSTRUCTING HIERARCHIES No inviolable rule exists for constructing hi erarchies . The sample hierarchies offered throughout the book are presented no t to prescribe certain frameworks but to stimulate thinking about what ty pes of hierarchicallevels to choose and what kinds of elements to include in the lev els . The numbers of levels and elements may be more or less than those in the examples. The great variety of examples thal !end themselves to hierarchies s ug gests thal the Sl.\,bjects thal can be approached with the an .,Jytic hierarchy process are infinlt&. ln ali these areas we are limited only by o ur experi ences and feelings as represented by words from the di cti o na ry . Lan guages that are limited in vocabulary may pose problems o f amb iguit y or may no.t represent human experience adequately . Recogni zing these lim itations, we may be encouraged to innovate by creating the need ed vocab ulary .a nd other symbols, such as computer languages, to re pre se nt feelings and ideas that we m ay become aware of in the proces s of id entificatio n and structuring.
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One' s approach to constructing a hierarchy depends on the kind of decision to be made. If it is a matter of choosing among alternatives, we could start from the bottom leve! by listing the alternatives . The next leve! would consist of the criteria for judging the alternatives . And the top leve! would be a single element, the focus or overall purpose, in terms of which the criteria can be compared according to the importance of their contribution. Suppose we wanted to decide whether to buy one of five sports cars (Figure 3-1) . These alternatives form the bottom leve! of the hierarchy. The. criteria in terms of which the alternatives will be judged form another leve! and might include adequacy of salary, prestige, basic necessities, comfort, satisfaction of other needs, large savings account, and freedom from worry. The priorities of these criteria will be judged in terms of their contribution to the focus of the hierarchy: our overall happiness . Note that once we construct this hierarchy, it is not necessarily cast in bronze. We can always alter parts of it later to accommodate new criteria that we may think of or that we did not consider to be important when we first designed it. The computer programs that assis! us in the task are constructed with this flexibility in mind . After we rank the criteria and arrive at overall priorities for the alternatives, we might still have sorne doubts about the final decision . In that case we would simply go through the process again and perhaps change sorne of our judgments on the relative importance of the criteria. If the same alternative is still significantly ahead of the others in overall priority , we know that it is the right choice for us . So m e times the criteria the mselves must be examined in detail, so a leve! of subcriteria should be inserted between those of the criteria and the alternatives . To select a school from three possible choices, for example, severa! criteria might be used-such as educational , cultural, and social
Selecting a Best School
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31
CONSTRUCTING HIERARCHIES
ANALYZING AN D ST RUCTURING HI ERARC HIES
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(Figure 3-2) . The educational criterion might be broken down into subcri teria of (1) quality of teachers, (2) general standard of students, (3) disci pline, (4) preparation for college, and (5) learning environment; the other criteria could similarly be broken down as weil. In this case the subcri teria would be .compared in terms of the criterion to which they belonged and not to the other criteria. Such a hierarchy would then be called incom plete because the subcriteria are not ali compared in terms of ali the criteria of the next higher leve!. There is no limit to the number of levels in a hierarchy. If on e; ., un able
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ANAL YZING AND STRUCTURING HIERARCHIES
KEY CONCEPTS
to compare the elements of a leve! in terms of the elements of the next higher leve!, one must ask in what terms they can be compared and then seek an intermediate leve! that should amount to a breakdown of the elements of the next higher leve!. Th us a new leve! has been introduced to facilitate the analysis for comparisons and to increase the precision of the judgments. Now we can answer the main question: How much more does one element contribute than another to satisfying a criterion in the next higher leve! of the hierarchy? Hierarchies may be more complicated than the ones described above . Those related to projected planning and repeated later, for example, inelude the following levels:
KEY CONCEPTS
Uncontrollable environmental constraints Risk scenarios Controllable systemic constraints Overall objectives of the system Stakeholders Stakeholders' objectives (separate ones for each stakeholder) Stakeholders' policies (separate ones for each stakeholder) Exploratory scenarios (outcomes) Composite or logical scenario (outcome) But decision makers do not have to pursue a problem to the full leve) of detail indicated here . The depth of detail depends on how much knowledge one has about the problem and how much can be gained by using that knowledge without unnecessarily tiring the mind . PERSPECTIVE
Although hierarchies have been known for a very long ti me, the anal ytic hierarchy process makes it possible to generale new levels and arrange them in a logical fashion so they relate to each other naturally. By making paired comparisons of the elements in a level in terms of the elements of the next higher level, it is possible to decide on an appropriate choice of thal upper level. Moreover, when the elements of a level cannot be compared except in terms of finer criteria than identified so far, a new leve! must be created for this purpose . Thus the analytic aspects of the AHP serve as a stimulus to create new dimensions for the hierarchy . It is a process for inducing cognitive awareness . A logically constructed hierarchy is a by - product of the entire AHP approach. lndeed, experience indicates that there are a few patterns that ali decision hierarchies seem to follow . ln the next chapter we will see how this concept of hierarchies can be appli ed to a broad range of real situations.
33
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In a functional hierarchy, complex systems are broken down into their constituent parts according to their essential relationships .
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The top level of the hierarchy-the focus--consists of only one element: the overall objective. The other levels contain severa! elements (usually between five and nine) .
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There is no limit to the number of levels in a hierarchy .
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When the elements of a level cannot be compared readily , a ne w level with finer distinctions must be created .
0
Hierarchies are flexible. We can always alter them to accommoda te new criteria.
HOW TO STRUCTURE A HIERARCHY The basic principle to follow in structuring a hierarchy is to see if one can answer the question: "Can you compare the elements in a lower leve! in terms of sorne or all the elements in the next higher level?" A useful way to proceed is to come down from the goal as far as one can and then go up from the alternatives until the levels of the two processes are linked in a way to make comparison possible. Here are sorne suggestions. 1. Identify overall goal. What are you trying to accomplish? What is the main question? 2. Identify subgoals of overall goal. If relevant, iden~ify time horizons that affect the decision. 3. Identify criteria that must be satisfied to fulfill subgoals of the overall goal. 4. Identify subcriteria under each criterion. Note t hat subcriteria may be intervals of numerical values; or intensities such as high, medium, low; or excellent, very good, good, average, poor, and very poor. 5 . Identify in descending levels, as needed, actors, actor objectives, and actor policies in this order. 6. Identify alternatives or outcomes. 7. For yes-no decisions include for example doing and not doing the alternative. 8. It is often useful to construct two hierarchies, one for benefits and one for costs to decide on the best alternative, particularly in the case of yes-no decisions. Ratios of benefits to costs, or better, marginal benefits to costs, are formed and the alternative with the largest ratio is chosen. Answer the question: Which alternative yields the greatest ~ benefit (for benefits) or costs the most (for costs)7 ~ 'J
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How can hierarchies be applied to planning economie policies-for example, planning economie strategy for an underdeveloped country?
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How can hierarchies be used for estimating and predicting-predicting the presidential election, estimating the popularity of a rock group, and so on?
• How can hierarchies be used for measuring influences- for example, parental influence on a child's psychological we!l-being?
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ln later chapters we will discuss in detail specifie applications of the AHP to planning, conflict resolution, benefit/cost analysis , and reso urce allocation . This chapter offers examples of different kinds of hierarchies to suggest ways of approaching problems and structuring them into their constituent parts with enough detail to make reasonable decisions . Most prob1ems arise because we do not know the internai dynamics of a system in enough detail to identify cause-and-effect relationships . If we were able to do so, the problem could be reduced to OJ1e of soci al engineer ing, as we would know at what points in the system interventio n is n ecessary to bring about the desired objective. The crucial contribution o f the analytic hierarchy process is that it enables us to make practical decisions based on a "precausal" understanding-namely , on our feelings and judgments about the relative impact of one variable on a nother. ln sum , when constructing hierarchies one must includ e e no ugh relevant detail to dep ict the problem as thoroughly as possible . Co nsider thE environment surrounding the problem . Jdentify the issues <' •rib ut e!
How can hierarchies be applied to business decisions--choosing equipment, deciding whether to buy or lease, making financial decisions, and so on?
How can hierarchies be applied to persona! and domestic decisions--choosing a car, choosing a career, purchasing a house, and so on?
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that you feel contribute to the solution . ldentify the participants associated with the problem . Arranging the goals, attributes, issues, and stakeholders in a hierarchy serves two purposes: lt provides an overall view of the complex relationships inherent in the situation, and it permits the decision maker to assess whether he or she is comparing issues of the same arder of magnitude in weight or impact on the solution . The elements should be clustered into homogeneous groups of five to nine so they can be meaningfully compared to elements in the next higher levet. The only restriction on the hierarchie arrangement of elements is that any element in one level must be capable of being related to sorne elements in the next higher leve!, which serves as a criterion for assessing the relative impact of elements in the level below. The hierarchy does not need to be complete; thal is, an element in a given level does not have to function as a criterion for al/ the elements in the leve! below. Thus a hierarchy can be divided into subhierarchies sharing only a corn mon topmost element. Further, a decision maker can insert or eliminate levels and elements as necessary to clarify the task of setting priorities or to sharpen the focus on one or more parts of the system . Elements that are of Jess immediate interest can he represented in general terms at the higher levels of the hierarchy and elements critical to the problem at hand can be developed in grea ter depth and specificity . ln addition to identifying within a hierarchie structure the major factors that influence the outcome of a decision, we need a way to decide whether these factors have equal effects on the outcome or whether sorne of them are dominant and others so insignificant they can be igno·r ed . This .is accomplished through the process of priority setting. The task of setting priorities requires that the criteria, the subcriteria, the properties or features of the alternatives being compared, and the alternatives themselves are gradually layered in the hierarchy so. thal the elements in each level are comparable among themselves in relation to the elements of the next higher leve!. Now the priori lies are set for the · elements in each level severa! times--
BUSINESS DECISIONS
37
Hierarchy for Choosing Urethane Equipment To decide on the purchase of urethane manufacturing equipment (Figure 4-1), three principal considerations are priee, technical features, and service. "Technical features" is really a cluster of severa! specifie desired qualities to which it is further decomposed. The overall priorities obtained through successive priority setting represent the relative importance of the considerations bearing upon the purchase decision . The three products under consideration are then ranked with respect to each criterion or subcriterion . The overall priority gives the relative superiority of the brands . :"!·
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benefit/cost ratios are obtained . The qualities desired of a word processin g machine form the second leve! in the benefit hierarchy, and thei r p ri or it ic s represent the relative weights assigned by the user. The eq ui pmen t chara Level1 :
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The decision regarding company ownership or leasing of a piece of capital equipment (Figure 4-3) depends on its contribution to the company's profitability . This profitability has two dimensions : economie and intangible . The benefits depend on a number of factors that, in turn, depend on certain characteristics of the company . Buying or leasing would promote the~e characteristics to a varying extent . By setting priorities for the factors at a certain leve! with respect to the relevant factors at the previous leve! and finding the composite priorities, we can find to what extent, relatively speaking, the factors in the same leve! contribute to the firm's overall profitability . Extending this logic to the question of company ownership or leasing, we can say, in the judgment of the decision maker, which alternative is preferable. In this example we take the intangible benefits explicitly into co nsideration for a decision , so the subjective judgments of the decision maker
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teristics that contribute to these qualities fall in the third level. They are ranked with respect to each de si red quality, and the overall priority represents the relative importance ofeach characteristic. The overall priorities of the brands under consideration represent the relative superiority regard ing benefits expected from each. In the cost hierarchy , likewise, the overall priorities represent the relative weights of the costs . The benefit/cost ratio for each indicates the rela tive superiority of one machine over the others.
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41
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evaluation and then rank them according to their relative importance on the final outcome . Next we judge the alternative containers with respect to each criterion per unit of beverage delivered. This prioritization shows the desirability from the point of view of each criterion, and the composite priorities show the overall superiority of the containers in relative terms .
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To decide how to stagger industry work hours, we firs t examin e the relevant consequences of staggered work hours and judge ho w importan t they are with respect to one another. This is clon e b y ranking the re levan t criteria with respect to the focus (Figure 4-6). Next the various shift patterns under consideration are ra nke d with respect to each of the criteria above to see how much they wou id affect it in relative terms . The composite priorit ies indicate the overall desirability of the considerations in relative terms. The highest-priority shift is the most desirable decision .
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are also considered . This is unlike a conventional exercise where only the hard economie data are considered and then managerial judgment is used in a qualifying manner at the end .
Hierarchy for Choosing a Candidate for Management To select the proper person to fill a management position, we first identify the four areas of evaluation and then the specifie traits to which these areas contribute (Figure 4-4). Successive stages of priority setting give the overall priorities for the relevant traits in leve) 3, which represent the relative weights believed to be associated with them for a person to function effectively in that position. Once ali the candidates have been reviewed and the choice has been narrowed clown to a few seemingly equal candidates, they may be ranked with respect to each trait . The composite priority for each candidate then represents his or her relative superiority on the basis of overall judgment and is useful in comparing candidates and ranking them in order of preference .
Hierarchy for Choosing a Beverage Container To evaluate the desirability of different containers to be used by the so ft drink b e vera ge industry (Figure 4-5), we first consider the criteria for
Hierarchy for Choosing a Site for Combustion Turbines The problem of site selection for an electric utility company' was narrowed to four alternatives after preliminary screening (Figure 4-7). The company identified nine relevant factors they would have to consider in the selection process . Since many of the considerations were in conflict, they were first ranked to find their relative importance bearing upon the Level1 :
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This decision regarding resource allocation is done as a benefit/cost exercise involving the benefits expected to accrue from the projects and the costs expected to be increased thereby (Figure 4-8) . In the benefits hierarchy, the benefits are ranked according to their impact on the bank' s performance. The projects are ranked according to how far they can generale that benefit. The composite priorities represent their overall benefit contributions on a ratio scale. In the costs hierarchy, Iikewise, the various costs are ranked by their severity and the projects are ranked with respect to their contribution to that cos!. The resulting composite priorities represent their overall costs. The benefit/cost ratios measure the superiority of benefit for cost incurred and also each R&D project's expected attractiveness . Ratios comparing the greatest marginal benefits to costs are often more useful than simple benefitlcost ratios . Sometimes discounting of benefits and costs is more realistically done before forming such ~ - •ios .
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tance of these factors . The projects are ranked according to· their contribution to each factor. The composite priorities give relative measures of the benefits accruing from them . ln the costs hierarchy, in a similar way, we find the relative importance of a number of factors the company would like to avoid or minimize. The overall priorities of the projects in this hierarchy, th en, give the relative measure of the negative contribution of these projects. The benefit/cost ratios give the superiority of the benefits over costs on a ratio scale. The project with the highest marginal benefitfcost ratio is the best selection .
risk depends on the scenarios envisioned and will be achieved in varying degrees by following different alternatives of product and market . The decision mode! is thus represented in the form of a complete hierarchy (Figure 4-10) . By prioritizing the factors in one leve! with respect to each factor in the preceding leve! and finding the overall priorities, we can find the relative influence, feasibility, importance, or contribution, as the case may be, of the factors in a leve! with respect to the focus : the company's well -being. The priority of each course of action is therefore a relative measure of how far that product/market posture would achieve the desired well-being .
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There are severa! dimensions of the performance of a division in a corporation. The principal dimensions to be considered in this evaluation are government dealings, management, imports, and customers (Figure 4-11). There are severa! factors for each dimension . Leve! 3 of the hierarchy shows those pertaining to management alone; other factors can be simi larly included for the other dimensions . The overall priorities of the factors at level3 are the relative weights by which the evaluators would view performance in that area . Composite priorities of the various divisions with respect to ali the factors at this leve! show the relative performance rating of the division on an overall ba sis.
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The decision regarding a company's product/market posture depends on a number of externat factors thal determine how far the company can strive to maintain the status quo or expect an optimistic or a pessimistic environment. The company' s objective regarding economie growth and
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There are both benefits and costs in having a home comput er. Three choices are prioritized in two separate hierarch ies for benefits a nd costs (Figure 4-13) . In each hierarchy , the priorities are determin ed ·· ~oug h
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In choosing a career, a person desires satisfaction in severa! dimensions : intellectual, financial, social, and persona!. However, one's sources of satisfaction are from expectations of learning, growth, leisure, friends, and prestige. Each source of satisfaction may derive fulfillment from severa! dimensions, which therefore occupy a superior leve! (Figure 4-14) . The priority of each career with respect to any criterion of satisfaction reflects its desirability with respect to that criterion only . The overall priority of the career shows the overall preference for that career; the highestpriority career is the one preferred . (This is an example of an incomplete hierarchy, as factors in leve! 3 do not relate to ali the factors in leve! 2.)
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alternative as viewed by the in vestor. For a large investor, the priorities .a lso indicate the proportions in which the total investment could be distributed among the various alternatives available-that is, a representation of the investor's portfolio.
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To decide which mode to use for crossing a river would be beneficiai to the community as a whole . We consider the nature of benefits envisaged and enlist under each the details (Figure 4-18) . Setting priorities for the benefits gives an idea which ones the community regards as important. We can also establish priorities for the costs .
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Considerations that apply to purchasing a house are the criteria, which are prioritized to find their relative importance (Figure 4-17). ln the next leve!, these criteria are decomposed into further subcriteria, which are similarly prioritized . ln the next stage, the alternative houses under consideration are prioritized with respect to each criterion or subcriterion, and their overall priorities indicate the buyer's preferences for the houses in question . This is a comprehensive model in which the criteria pertaining to the neighborhood and those pertaining to the qualities of the house are dealt with simultaneously .
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Deciding on means to improve harbor capacity in a small country with three harbors is viewed as a joint political process. The three principal stakeholders are the evaluation group of specialists, the transportation committee of Jegislators, and the harbor bureau (Figure 4-20). Their priorities represent their influence in the matter. The objectives and considerations to be followed by each stakeholder are identified and prioritized . At the next stage, the three existing harbors are prioritized with respect to each objective to find out how much they contribute to the objectives. The overall priority at this levet shows the extent to which resources and attention should be devoted to each harbor development . The specifie programs to increase harbor capacity are next identified and prioritized with respect to each harbor . In the context of each harbor, these priorities show the effectiveness of each action in achieving the goal. The overall priorities, obtained by weighting the harbor priorities, reflect the effectiveness for the whole country. The actual policy that is adopted will consist of several individual programs pursued with varying degrees of emphasis as represented by these priorities.
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areas of energy resources to determine which one holds the most promise. With respect to each energy source, several criteria are considered that require attention; these criteria are prioritized according to the importance they command for the energy resource. For each criterion, several technical aspects are identified and then prioritized for relative importance. The overall priority at this levet shows the share of effort and resources that should be devoted to the technical field concerned. As a continuation of the process, each technical field is further subdivided into research areas and subareas that need attentio n. The overall priorities at each levet again represent the extent to wh ich resources should be devoted to those areas according to the best overall judgment of the decision makers .
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satisfy the criteria . The highest-priority storage system is the one preferred. ·
Hierarchy for Allocating Resources in Juvenile Correction · Programs A group of public officiais were interested in juvenile law enforcement and wanted to allocate resources in five programs the staff had suggested. To ~ ·~rt with, they considered three principal areas of correction and
61
PUBLIC PO UC Y DECISIONS
.CTICAL EXAMI'LES OF HIERARC HIES
prioritized them to find out how much attention they should receive (Fig· ure 4-22). Each principal area could be associated with the programs identified, so the programs were prioritized regarding their effectiveness with respect to each area . The overall priorities obtained after weighting by the area priorities show how much relative importance each program commands for the optimum juvenile correction system .
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Hierarchy for Analyzing School Busing Conflict The introduction of busing due to the 1954 Supreme Court ruling has been a source of friction in a certain school district. The minority community wants to introduce complete busing for racial integration at school; the majority community wants segregation to protee! the privileged position they now enjoy. To analyze the situation and judge the potential outcome, we rank the stakeholders according to their relative influence on the political scene (Figure 4-23). Then we prioritize the objectives of each stakeholder to see which objectives weigh more and should thus be pur· sued in preference to others. The overall priorities give a picture of the relative strengths of the forces at work on the scene . The outcomes under consideration here are three scenarios spanning the complete spectrum of possibilities. They are prioritized with respect to each stakeholder objective to find out which outcome is favored by thal objective. The overall priorities indicate the relative likelihood thal each possible outcome will o ccur. This exercise shows the interaction of various Level1 : Focus
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Education for Children
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factors and forces at work, so thal if one stakeholder wanted to influence the outcome, he or she could decide accordingly on a course of action such as coalition or persuading others to change their objectives . Level2 : Correction Are a
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Hierarchy for Analyzing Health Administration Conflict This hierarchy for resolving conflict in health administration finds the likelihood of various health plans being adopted as national po licy (Figure 4-24) . First we prioritize the principal actors regarding their relative influ· ence on the issue . Next their objectives are identified and prioritized to indicate the relative extent to which the actors are motivated by various considerations. At the next stage , we establish which policies of the actors would satisfy the objectives . The policies thal relate to the same objective are
PRACTI CAL EXAMPLES OF HIERARCHIES
PLANNING ECONOMIC POLICIES
prioritized to find the importance of the policies to serve the objective in question. The composite priorities show the overall influence of the polides in national health administration. There are three main health plans being debated nationally. By prioritizing the plans with respect to each policy, we find to what extent a plan satisfies the po licy in q uestion oTh us the overall priorities of the plans show the relative extent of support for the plans from various interested quarterso The overall priorities are also the likelihoods of the plans being adopted nation ally o The elements of that hierarchy are defined as follows :
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PLANNING ECONOMIC POLICIES
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Hierarchy for Estimating Popularity of Rock Groups
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reflection ability, RA; retention and assimilation, R; character and personality, CHP) 1. The school (S) 2. The family (F)
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Network of a Volleyball Team Volleyball is agame of teamwork and simple skills, but the play.e rs are required to change positions and possess al/ the skills . Since few players have ali the skills to the same extent, the coach has to use players with the proper mix of skills. Th us the coach has to know the relative importance of skills and play the proper combination of players. Since the various skills depend upon one another, we use systems analysis and nol hierarchical representation (Figure 4-32) . By prioritizing the skills for a good game of volleyball and prioritizing the players with respect to one another's skills, we get the relative standings of the players and skills in the ultimate analysis . This information helps the coach to select players with basic skills . PERSPECTIVE
And those for the home (H) are : 1. Ph ysical characteristics (PCH)
2. The child's toys (TS) 3 . The communication media (radio and television) (CM) This mode! has been evaluated for a certain community, at an average socioecono~ ic leve!, in a democratie regime, and for children eight years o ld with normal psychological characteristics.
The hierarchies just presented are only a few of those thal have actually been used to make decisions with the AHP. But they suggest the wide range of problems to which the AHP can be applied-from choosing a car to crossing a river. Specifically, the AHP can be used for the following kinds of decision problems: o
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The sample hierarchies in this chapter should give sorne insight into the need for various levels in different kinds of hierarchies . Different people may have their own id ea about how to deal with a problem, and that is how they should carry it out. Only when a group must act together, as in a corporation, do people need to agree on the structure of their problem. In forming a hierarchy, one should include as much detail as seems to be needed to understand the problem; the prioritization process will eliminate elements that are unimportant . A new leve! should be added to the hierarchy if it facilitates the comparison and evaluation of the elements in the leve! immediately below and contributes to improving precision in the judgments. One contribution of a good hierarchy is thal it enables people to make better guesses about the effects of the unknown by laying out its components and studying each separately instead of Jumping ev· erything together and making one big guess at the consequences of deci· sions made in the face of that unknown . The hierarchy can provide an effective buffer between reason and worry. Clearly the design of an analytic hierarchy-like the structuring of a problem by any other method-is more art than science. There is no pre· cise formula for identification or stratification of elements. But structur· ing a hierarchy does require substantial knowledge about the system or problem in question . A strong aspect of the AHP is that the experienced decision makers who specify the hierarchy also supply judgments on the relative importance of the elements-which brings us to the next topic: establishing priorities .
This cha pter deals with the following question s : How do we establish priorities in a decision problem? Why is the matrix useful in setting priorities? How do we synthesize our judgments to geta set of overall priorities? How can we check the consis tency of our judgments? And how important is consistency? What do we do when the elements we are rank ing overlap?
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THE NEED FOR PRIORITIES ln this chapter we complete the analytic hierarchy process by establishing priorities among the elements of the hierarchy, syn.thesizing our judgments to yield a set of overall priorities, checking the consistency of these judgments, and coming to a final decision based on the results of this process. Systems theorists point out thal complex relationships can always be analyzed by taking pairs of elements and relating them through their attributes . The abject is to find from many things those that have a necessary connection. This causal approach to understanding complexity is complemented by the systems approach, whose abject is to find the subsystems or dimensions in which the parts are connected . We have seen thal the analytic hierarchy process deals with both approaches simultaneously . Systems thinking is addressed by structuring ideas hierarchically, and causal thinking, or explanation, is developed through paired comparison of the elements in the hierarchy and through synthesis. The judgments we apply in making paired comparisons combine logical thinking with feeling developed from informed experience. The mathematical sequence described in this chapter is a more efficient method of arriving at a solution than the intuitive means we usually employ, but the end result is not necessarily more accurate. If the solution reached through the AHP d o es not fee) right to an experienced, well-informed decision maker, then he or she would do weil to repeat the process and restructure the hierarchy or improve the judgments . On the other hand, the AHP prov ides its own check on the consistency of judgments, and experience has shawn that the results of the AHP closely approximate decisions reached more laboriously in the business world. lt is important to note thal the calculations described here can be ca rried o ut by computer. My aim is not to dwell on the mathematics of the process-a mathematical supplement is available for those interested-but to ex plain how subjective judgments can be quantified and converted into a set of priorities on which decisions can be based .
SETTING PRIORITIES Tlw first step in esl<~blishing the priorities of elements in a decision problem is to make pairwise comparisons-that is, to compare the ele ments in pairs against a given criterion. For pairwise comparisons, a matri x is the preferred form . The matrix is a simple, well-established tool that o ffer s a fr ame work for tes ting consis.t ency, obtaining additional informati o n thr o ugh making ali possible comparisons, and analyzing the sensiti vi ty o f overall priorities to changes in judgment. The matrix approach uniqu ely re flects the dual aspects of priorities : dominating and dominated.
77
SEHING PRIORITIES
To begin the pairwise comparison process, start at the top of the hierarchy to select the criterion C, or property, that will be used for making the first comparison. Then, from the level immediately below, tak e th elements to be compared: A v A 2, A 3, and so on. Let us say there are se ven elements. Arrange these elements in a matrix as in Figure 5-1 .
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In this matrix compare the element A, in the column on the left with the elements A 1 , A 2 , A 3 , and so on in the row on top with respect to property C in the upper left-hand corner. Then repeat with colurrin ele ment A 2 and so on. To compare elements, ask: How much more strongl y does this element (or activity) possess- or contribute to, dominate, influence, satisfy, or benefit- the property than does the element with which it is being compared? The phrasing of the question is important. Il must reflect the proper relationship between the elements in one leve! with the property in the next higher leve!. If time or another probabilistic criterion is used, then ask : How much more probable or likely is one element than another? If the elements are dominated by the property rather than vice versa, ask how much more strongly the element is possessed, dominated, affected by, and so on, this property. In projecting an outcome, ask which element is more likely to be decisive or to result in the outcome. To fill in the matrix of pairwise comparisons, we use numbers to represent the relative importance of one element over another with respect to the property . Table 5-1 contains the scale for pairwise comparisons . lt defines and explains the values 1 through 9 assigned to judgments in comparing pairs of like elements in each leve! of a hierarchy against a criterion in the next higher leve!. Experience has confirmed that a scale of nine units is reasonable and reflects the degree to which we can discriminate the inten sity of relationships between elements. When using the scale in a social , psychological, or political context, express the verbal judgments first and then translate them to numerical values . The numerically translated judg-
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cording to their impact on a criterion? If the problem concerns simple ·ranking, and the degree to which the elements being ranked reflect the criterion is obvious, then one can simply assign numbers. To discriminate the relative strength with which each element possesses or contributes to the criterion (property), numbers can be used directly by starting with the smallest element and perhaps using it as a unit . This procedure may be useful in organizing one's thinking, but the logic is not clear and, moreover, feeling is not integrated into the process. For fine distinctions, the pairwise comparison matrix and scale provide a more satisfactory framework. When tradeoffs must be made among severa! criteria, the problem of ranking becomes complex. lt is no longer sufficient simply to assign arbitrary numbers. We must select with care the numbers used to express the strength with which each element possesses or contributes to the property in question . Such care ensures that in the end we obtain the correct overall priorities for the elements by considering ali tradeoffs . (These priorities can also then be used to allocate resources .)
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SYNTHESIS
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SYNTHESIS To obtain the set of overall priorities for a decision problem , w e have to pull together or synthesize the judgments made in the pairwise comparisons-that is, we have to do sorne weighting and adding to give us a single number to indicate the priority of each element. The following example explains how to synthesize. Suppose we want to decide which of three new cars- a Chevrolet, a Thunderbird, and a Lincoln-ta buyon the basis of comfort . We draw a matrix with the criterion "comfort" listed in the upper left-hand corner and the cars listed in the column on the left and in a row on top (Figure 5-2) . We
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ments are approximations; their validity can be evaluated by a test of consistency, which will be described later, and by real-life applications for which the answers are already known. When comparing one element in a matrix with itself-for example, A 1 with A 1 in Figure 5-1-the comparison must give unity (1), so fill in the diagonal of the matrix with 1s. Always compare the first element of a pair (the element in the left -hand column of the matrix) with the second (the element in the row on top) and estimate the numerical value from the scale in Table 5-1. The reciprocal value is then used for the comparison of the second element with the f1rst. For example, if the two elements are stones and the first is five times heavier than the second , then the second is one-fifth as heavy as the f1.rst . Why not simply use arbitrary numbers for ranking the elements ac-
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matrix deals with, say, seven elements, the number of judgments needed to fiJI the entries is 7 x 7 - 7 + 2 = 21. We subtract the seven unit entries down the diagonal and divide by 2 because half the judgments are reciprocals th at are entered automatically. We then ask: How much more comfortable is an average new Chev· rolet than an average new Thunderbird and an average new Lincoln? Based on our experience and persona! preference, our judgment is that a Chevrolet is one-half as comfortable as a Thunderbird and one-fourth as comfortable as a Lincoln . To state these judgments in terms of the quantifiers in the scale (Table 5-1), a Thunderbird is slightly more comfortable than a Chevrolet, and a Lincoln is between slightly and strongly more comfortable than a Chevrolet. Thus we enter the values 2 for the Thunderbird over the Chevrolet and 4 for the Lincoln over the Chevrolet. These num bers are the reciprocals of the two judgments comparing the Chevrolet with the other cars. Remember that the element that appears in the left-hand column is always compared with the element appearing in the top row, and the value is given to the element in the column as it is compared with the element in the row . If it is regarded less favorably, the judgment is a fraction . The reciprocal value is entered in the position where the second element, when it appears in the column, is compared with the first element when it ap· pears in the row . In this example, because the Chevrolet is regarded less favorably when compared with the other two cars, we enter 112 and 1/4 in the second and third positions of the first row and enter 2 and 4 in what are known as the transp ose positions in the first column . We then compare the Thunderbird with the Lincoln and enter a value of 1/2 in the second row, third column position , and its reciprocal 2 in the second column, third row position. We now have the three judgments needed to complete the pàirwise comparison matrix (Figure 5-2). Next we want to synthesize our judgments to get an overall es ti mate of the relative priorities of these cars with respect to comfort. To do so, we first add the values in each column (Figure 5-3) . Then we di vide each entry
Comfort
Comfort
C
T
L
4
2
1
Co lumrr total
7
3 .5
1.75
417
\·
Normaliz:ed Matrix
nally, we average over the rows by adding the values in each row of the normalized matrix and dividing the rows by the number of en tries in each :
t
1/7 + 117 + 1/7 = 117 = 0. 14
Il lj
!.
+ 417 + 417
li
= 4/7 = 0.57
i:
ij
ii "
This synthesis yields the percentages of overall relative priorities, or preferences, for the Chevrolet, the Thunderbird, and the Lincoln: 14, 29, and 57 percent, respectively . As far as comfort is concemed, the Thunderbird and the Lincoln are thus about twice and four times more preferable than the Chevrolet. The answer in this case was very simple, because ali the columns in the normalized matrix were the same . They turned out to be the same because the pairwise comparison matrix (Figure 5-2) was consistent . That is, from the relationship of the Chevrolet to the Thunderbird in the first row of the matrix,
~
112
417
l·.
.
1!4
L
4/7
.
1
217
3
•
1/2
117
2/7
2/7 + 217 + 217 = 217 = 0.29
·1··
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1/7 217 417
117
T
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l
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c
Figure 5-3
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Figure 5-4
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Synthesizing the Judgments
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-----------our minds, we would be afraid to accept new ideas. Ali knowledge has to be admitted into our narrow corridor between tolerable inconsistency and perfect consistency. Of course, a certain degree of consistency in setting priorities for elements or activities with respect to sorne criterion is necessary to get valid results in the real world . The AHP measures the overall consistency of judgments by means of a consistency ratio. The value of the consistency ratio should be 10 percent or less. If it is more than 10 percent, the judgments may be somewhat random and should perhaps be revised . Let us continue with the example of the cars and see how the AHP measures consiste ney . Suppose that we keep the first row of our pairwise comparison matrix in Figure 5-2 but do not pay much attention to consistency with our previous judgments. In comparing the Thunderbird with the Lincoln, we enter the value 1/4 in the second row, third column, and enter its reciprocal 4 in the third r.ow, second column (Figure 5-5) . Following the steps described
and that T = (1/2)L which is precisely what we have in the second row, third column entry. In other words, if the Chevrolet is preferred half as much as the Thunderbird and one-fourth as much as the Lincoln, then the Thunderbird must be preferred half as much as the Lincoln. The information in the first row is used to force judgmental consistency.
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83
CONSISTENCY
ESTABL.ISIIING PRIORITIES
CONSISTENCY ln decision · making problems it may be important to know how good our consistency is, because we may not want the decision to be based on judgments that have such low consistency that they appear to be random. On the other hand, perfect consistency is hard to live up to. Our judgments on the relative comfort of the three cars were consistent, but in real !ife specifie circumstances often influence preferences, and circumstances change. If apples are preferred to oranges, for example, and oranges are pre· ferred to bananas, then in a perfectly consistent relationship apples must be preferred to bananas. But the same individual may sometimes like bananas better than apples, depending on the time of day, the season, and other circumstances. In the example of the cars, we identified a couple of relationships that showed the strength of our preference for a Thunderbird over a Chevrolet and for a Lincoln over a Chevrolet and fNced these rela· tionships on the comparison between a Thunderbird and a Lincoln-the Thunderbird was preferred half as much as the Lincoln . But often such a relationship does not hold true . Violating it, which we do ali the time , leads to inconsistency. How damaging is inconsistency? Usually we cannot be so certain of our judgments that we would insist on forcing consistency in the pairwise comparison matrix . Rather, we guess our feelings or judgments in ali the positions except the diagonal ones (which are always 1), force the reciprocals in the transpose positions, and look for an answer. We may not be perfectly consistent, but that is the way we tend to work . (lt is also the way we grow . When we integrale new experiences into our consciousness, previous relationships may change and sorne consistency is !ost. As long as there is enough consistency to maintain coherence among the objects of our experience, the consistency need not be perfect.) lt is useful to remember that most new ideas that affect our lives tend to cause us to rearrange sorne of our preferences, thus making us inconsistent with our previous commitments. If we were to program ourselves never to change
c
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c
1
1/2
114
T L Co/um11
2
1
114
4
4
7
5 .5
Comfort
1
L
-
lolr1/
1
Figure 5-5
1.5
lnconsistent Matrix
earlier, we obtain the normalized matrix, its row sums, and the percent· ages of relative overall priorities (Figure 5-6) . The percentages, 13, 21 , and 66 percent, constitute the priority vector of the three cars with respect to comfort. The value of the priority vector is approximate . (We can find the
Comfort
T
L
Row Sums
Avrragr Row Sum
c
1/7
1/11
1/6
0.40
0.4013 = 0. 13
T
2/7
2/11
116
0 .63
0 .6313 = 0 .21
L
4/7
8/11
4/6
1.97
1.9713 = 0 .66
Figure 5-6
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Normalized Matrix, Row Sums, and Overall Prir- ' •ies
(X)
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EXTENDING THE PROCESS
ESTABLISIIING PRIORITIES
exact value, but the solution is complicated. Besicles, when the judgments are perfectly consistent, the two values are identical; nearly consislent, the values are close.) Although the standing of the Chevrolet has not been changed by much, the other two have changed by our reducing the value for the Thunderbird and raising it for the Lincoln . With inconsistency ali the values are changed . The question is: How significant is this change? Presumably we want to compare our inconsistency with the value it would have if the judgments were random . To do this, multiply the first column of the inconsistent matrix (Figure 5-5), changed to decimal form, by the relative priority of the Chevrolet (0 .13), the second column by that of the Thunderbird (0 .21), and the third column by that of the Lincoln (0.66). Then total the entries in the rows (Figure 5-7) .
Comfort
c
(0 . 13)
T (0 .21)
L (0 .66)
Comfort
J
c
T
L
Row Total 0.41
c
1
0.5
0.25
- c
0.17
2
1
0.25
1'
0.13 0.26
0.11
T
0.21
0.17
0.64
L
4
4
1
L
0.52
0.84
0.66
2.02
Figure 5-7
Totaling the Entries
Now take the column of row totals and divide each of its entries by the corresponding entry from the priority vector (figure 5-8). We can now find the average of the three entries in the last column of Figure 5-8: 3.15 + 3 .~5 + 3.06 = 9 .; 6 "" 3.09
[
oS
0.41] . .15] 0.64 ..,. 0.13]0.21 - [33.05 [ 2.02 0 .66 3.06
Figure 5-8
Determining
Àmax
A second approximation procedure is to compute the geometrie mean of the elements in each row-that is, to multiply the elements and then take the ir n th root. This step is followed by normalizing the resulting vector so that its components add to unity . ln general, the geometrie mean is a good approximation, particularly when the consistency is high . The calculation of À max can proceed as before. The geometrie mean for the inconsistent matrix of cars with respect to comfort yields 0. 16, 0.20, and 0.64. The exact solution by computer is 0.13, 0.21, 0.66, and Àm ax = 3.05, which nearly coïncides with the results of the column normalization pro cess described earlier. Note that row averaging followed by a normali zation of the resulting vector yields 0.13, 0.23, and 0.64 . There are many exa~ples for which this last process yields unsatisfactory results when the matrix is inconsistent. One way to improve consistency when it turns out to be unsatisfactory is to rank the activities by a simple order based on the weights obtained in the first run of the problem . A second pairwise comparison matrix is then developed with this knowledge of ranking in mind . The consistency should gen erally be better.
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To see how the process just described can be extended to an entire hierarchy, consider the problem of a woman who has recently earned her Ph.D . and is being interviewed for three jobs . Which one should she choose? Figure 5-9 shows how she structured the elements of the problem and arranged them in a hierarchy . Leve) 1, the focus, is overall job satisfaction; leve! 2 comprises the criteria that contribute to job satisfaction; and leve! 3 consists of the three job possibilities . The hierarchy is a complete one: Each element in a leve) is evaluated in terms of ali the elements in the next higher leve!. The woman compared the leve) 2 criteria in pairs with respect to job satisfaction and judged the relative importance of each criterion . She felt that research, for example, would be equally as important as loca tion in contributing to job satisfaction, but it would be slightly to strongly more important than colleagues. Figure 5-10 shows the pairwise comparison matrix of the criteria with respect to the focus . The last column gives the
j
j
By convention, the symbol for this number is consistency index (Cl) is 3.09 - 3
=
À max
(lambda max) . The
0 .~9 = 0.045
The random value of the CI for n = 3 is 0.58 .• The consistency ratio (CR) is 0.045/0.58 = 0.08, which indicates good consistency .
• If numerical judgments were taken at random from the scale 119, 1/8, 117, . . , 112, . . . , 1, 2, . . . 9, then using a reciprocal matrix we would have the following average consistencies for di fferent -o rder rando m matrices : Size of ma trix Rand<'m consis tency
1 0 .00
2 0.00
3 0.58
4
o.90
5 1.12
6 1.24
7 1.32
8 1.41
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INTERDEPENDENCE
ESTABLISHING PRIORITIES
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priorities: Reputation tums out to be the most important criterion, followed by opportunities for growth and benefits . Next she developed six matrices for comparing the three jobs with respect to each criterion (Figure 5-11). Ali three entries in the vector of priorÙies obtained in each of the six matrices and listed in the las! column of each are multiplied (weighted) by the priority of the corresponding criterion . These values are shown in Figure 5-12 . The results of this operation are then added to yield the overall priori lies for the jobs: A= 0.40, B = 0.34, and C = 0.26. The differences made apparent by this synthesis were sufficiently large for the woman to accept the offer for job A. Although we will not go into the measurement of consistency here, it is important to note thal in a hierarchy, the highest-level elements usually have the highest priorities. Inconsistency arising from comparison with respect to these elements is very damaging because of their high priority. The consistency index of a hierarchy is obtained by multiplying the consistency index of each matrix. by the priority of the criterion used for the comparison and adding ali such quanti lies . To evaluate the consistency of a hierarchy, compare the consistency index of the hierarchy with ils counterpart when the consistency indices of the matrices are replaced by average random judgment consistency indices for matrices of the same size.
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INTERDEPENDENCE So far we have considered how to establish the priority of elements in a hierarchy and how to obtain the set of overall priorities when the elements of each leve! are independent . But often the elements are interdependent. How do we account for overlapping areas, or commonalities, among such elements? There are t_wo principal kinds of interdependence
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A
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112
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c
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Research
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Vector of Priori ti es
8enefits
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Vector of Priorities
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Vector of Priorities
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Vector of Priorities
Reputalion
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Vector of Priori ti es
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5
0.28
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A
1
7
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1
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7
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1
5
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c
115
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c
117
117
1
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c
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115
1
0.05
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Figure 5-11 Six Matrices for Comparing Three Jobs
Research
Growth
8enelits
(0 . 16 )
Col/eagues
(0 .19)
Location
Reputation
(0.19)
(0 .05)
(0 .12)
(0 .30)
Vector of Ovn-all Priori ti es
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A
0.14 {0.16)
8
0.63 (0.16)
c
0.24 {0.16)
+ + +
0.10 {0. 19) 0.33 (0.19) 0.57 (0.19)
+ + +
0 .32 (0.19) 0.22 (0.19 ) 0.46 {0.19)
+ + +
0.28 (0.05) 0.65 (0.05) 0 .07 {0 .05)
+ + +
0
0.47 {0.12) 0.47 (0.12) 0.07 (0 .12)
+ + +
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=
0.40
0.17 {0.30)
=
0.34
0.05 {0 .30)
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m "'0 m
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Figure 5-12 Determining the Overall Priorities
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ESTAII! .ISIIINC I'RI O RITI ES
PERSPECTIVE
91
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among elements of a hierarchy leve!: additive interdependence and synergistic interdependence. ·
Additive lnterdependence
ln additive interdependence, each element contributes a share that is uniquely its own and also contributes indirectly by overlapping or interacting with other elements . The total impact can be estimated by examining the impacts of the independent and the overlapping shares and then combining the impacts. The effects of this simpler type of interdependence can be computed precisely since we can usually tell how much of an element's contribution is due toits independent properties and how much to ils effect on other elements . 6oth mechanization and farm size con tri bute to agricultural productivity, for example, but mechanization also influences farm size by enabling farmers to work larger areas . ln practice, most people prefer to ignore the rather complex mathematical adjustment for additive interdependence and simply rely on their own judgment. ln the example just given, mechanization would be assigned a hi~;her priority than farm size . The precise value would be determined subjectively from the descriptions in the pairwise comparison scale . Such judgments can also replace the more technical adjustment for synergistic interdependence.
lion additional criteria thal reveal the nature of the interaction . The overlapping element should be separated from its constituent parts. Ils impact is added to theirs at the end to obtain their overall impact. Synergy of interaction is also captured at the upper levels when clusters are compared according to their importance. Interactions within and between clusters are better seen and judged higher up in the hierarchy . Much of the problem of synergistic interdependence arises from the fuzziness of words and even the underlying ideas they represent . The full potential of interaction is never fully understood until il has taken place in practice . The qualities thal emerge cannot be captured by a mathematical process. Thus a set theoretical approach using Venn diagrams would not be useful, because we do not have a simple geometrie overlap of regions . What we have instead is the overlap of elements with other elements to produce an element with new properties thal are not discernible in ils parent parts. Note that if we increase the elements being compared by one more element and attempt to preserve the consistency of the.ir earlier ranking, we must be careful how we make the comparisons with the new element. For example, if we have been comparing apples, oranges, and bananas with respect to taste and then add melons to the problem , this new addition may change the priorities . Once we compare one of the previous elements with a n·ew one, ail other relationships are automatically set ; otherwise there would be inconsistency and the rank order might change.
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Synergistic lnterdependence
ln synergistic interdependence, the impact of the interaction of the elements is greater than the sum of the impacts of the elements, with due consideration given to their overlap. This type of interdependence occurs more frequently in practice than additive interdependence and amounts to . crea ting a new entity for each interaction . Power coalitions and the biology of marriage are examples of synergy. The analytic hierarchy process provides a simple and direct means for measuring interdependence in a hierarchy . The basic idea is thal wherever there is interdependence, each criterion becomes an objective and ali the criteria are compared according to their· contributions to thal criterion. This generales a set of dependence priorities indicating the relative derend e nce 0f each criterion on ali the criteria. These priorities are then weighted by the independence priority of each related criterion obtained from the hierarchy and the results are summed over each row, thus yielding the interdependence weights . By way of validation we find that this approach is compatible with, for example , what econometricians do in calculating input-output matrices . These ideas are illustrated in the mathemahcal supplement. With synergistic interdependence, one needs to introduce for evalua-
:: ·;
PERSPECTIVE The process of setting priorities captures the feelings and judgments of informed individuals by asking them to make pairwise comparisons among like objects as criteria. The judgments, which represent strength of preference, are simultaneously converted to numerical values to represent their intensity and are laid out in a matrix . The priorities are then derived from ali the judgments and the consistency of the judgments is calculated through the deviation of a single number from the order of a matrix . ln a hierarchy we synthesize the priorities in a leve! by weighting the priorities derived from each matrix by the weight of the criterion of comparison . To obtain overall priorities, we add the results for each element . Note thal prioritization from the top of the hierarchy downward includes less and less synergy as we move from the larger more interactive clusters to the small and more independent ones. lndependence can be treated in two ways . Either the hierarchy is structured in a way that identifies independent elements or depen d ence is allowed for by evaluating in separa te matrices the impact of ali the ele ments on each of them with respect to the criterion being consid ered . As before , a weighting process is then used to determine their prioriti es .
;·
(X)
_p. 0"
1
ESTAIJLISHING PRIORITIES
·~
If we already have an idea of the ranking of elements and wish to derive their priorities, the judgments we give must indicate this domi· nance in rank. Otherwise we must assume that the individual does not fully understand (or is inconsistent in) his or her subjective ranking.
KEY CONCEPTS
0
0
0
0
To establish the priorities of elements, we have to compare them in · pairs according to a criterion . A matrix is the best framework for this corn parison. To obtain the set of overall priorities for a decision, we have to synthe· size the results of the pairwise comparisons. That is, we have to com· bine our judgments to get an overall estimate of the relative ank of priori ti es. ln a hierarchy, the highest-level elements usually have the highest . priorities. They are the clusters thal give rise to smaller elements at the lower levels. Their priorities are divided by the weighting process among their descendants . The consistency of a hierarchy can be measured by multiplying the · consistency of each matrix by the priority of its criterion and adding. This result is then compared with a similar number obtained for random matrices of the same size . The ratio should be 10 percent or less. Grea ter inconsistency ind ica tes lack of information or lack of understanding.
1(
6 Step-by-Step Examples of the Process This chapter deals with the following questions : What are the basic steps of the analytic hierarchy process? How can the process be used to analyze high-level decisions like the one to rescue the host ages in Iran? How can the process be used to determine consumer preference? How can the process be used to estimate the economy's impact on sales? How can the process be used to select a stock portfolio? How valid is the analytic hierarchy process?
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STEP-BY-STEP EXAMPLES O F THE I'RO CESS
ANALYZING THE HOST AGE RESCU E OPERATION
AN OUTLINE OF THE STEPS
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We began our study of the analytic hierarchy process by laying out the elements of a problem as a hierarchy. We then made paired comparisons among the elements of a leve! as required by the criteria of the next higher leve!. These comparisons gave rise to priorities and finally, through synthesis, to overall priorilies. We measured consistency and deal! with inter· dependence. These basic steps of the process can ali be condensed into a brief outline . In broad terms, the process is stable , although certain steps may be given special emphasis in particular problems and, as noted below , repetition is generally necessary .
These are the basic steps we will follow in working out the examples of this chapter. In each case we will examine the problem, set up the hierarchy, carry out the pairwise comparisons, determine the prioritie s, synthesize the overali priorities, and examine the consistency. We begin with a recent application of the method to the Iran rescue operation . An intang i· .ble factor played an important role in President Carter's thinking but no t necessarily the thinking of his advisors- hence their different conclusion s regarding the rescue operation . The second example challenges us to select the best produ ct to man u facture from three alternatives by using six criteria . The produ ct that is chosen costs mor~ but is more desirable ali around . The third e xampl e shows how the AHP may be used to estimate the impact of energ y, recession, and inflation on a company's sales. Here percentage ranges a re prioritized according to likelihood of occurrence. The fourth example ill ustraies how the AHP can be used to set priorities for stocks accord ing to severa! criteria. By way of validation , the high-priority stocks did in fa ct apprecia te in value.
2. Structure the hierarchy from the overall managerial viewpoint (from the top levels to the leve! at which intervention to solve the problem is possible).
~~
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3 . Construct a pairwise comparison matrix of the relevant contribution or impact of each element on each goveming criterion in the next higher leve!. In this matrix, pairs of elements are compared with respect to a criterion in the superior leve!. In comparing Iwo elements most people prefer to. give a judgment thal indicates the dominance as a whole number . The matrix has one position to enter thal number and another to enter ils reciprocal. Thus if one element does not contribu te more than another, the other must contribute more than it. This number is entered in the appropriate position in the matrix and ils reciprocal is entered in the other position . An element on the left is by convention examined regarding ils dominance over an element at the top of the matrix .
4. Obt21in ali judgments required to develop the set of matrices in step 3. If there are many people participating, the task for each person can be made simple by appropriate allocation of effort , which we describe in a later chapter. Multiple judgments can be synthesized by using their geometrie mean . '
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8. Evaluate consistency for the entire hierarchy by multiplying each con sistency index by the priority of the corresponding criterion and adding the products. The result is divided by the same type of expression using the random consistency index corresponding to the· dimensions of each matrix weighted by the priorities as before . The consistency ratio of the hierarchy should be 10 percent or less. If it is not, the quality of informa tion should be improved-perhaps by revising the manner in which questions are posed to make the pairwise comparisons. If this measure fails to improve consistency, it is likely thal the problem ha s not been accurately structured-that is, similar elements have not been grouped under a meaningful criterion . A return to step 2 is then required, al though only the problematic parts of the hierarchy may need revision .
1. Define the problem and specify the solution desired.
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ANALYZING THE HOSTAGE RESCUE OPERATION
Hav ing collected ali the pairwise comparison data and entered the re· ciprocals together with unit entries down the main diagonal, the priorilies are obtained and consistency is tested .
It is customary for high-level managers and political leaders to make their most important decisions by depending on expert recommendations governed by their own persona! judgment and understanding . President Carter once said thal when he had to make a really crucial d ecisi o n b etween two alternatives , his experts were usually evenly spli t. ln essence , then , the hard decisions were still up to him .. If expert op ini o ns w ere accurate and thorough, leaders would become superfluous. In s p lt e o f splitting on their judgments the experts do raise questi o ns th al s timul ate decision makers and draw their attention to iss ues they m ay h ave n e glected to consider.
6. Perform steps 3, 4, and 5 for ali levels and clusters in the hierarchy . 7 . Use hierarchical composition (synthesis) to weight the vectors of priorilies b y the weights of the criteria, and lake the sum over ali weighted priority en tries corresponding to those in the next lower levet and so on . The re sult is an overali priorit y vector for the lowest leve! of the hier· a rchy . If there are sev era! o utcomes, their arithmetic average may be ta kP"' .
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Background of the Decision A great shroud of secrecy surrounds the decision to rescue the hostages in Iran . On 28 April 1980 a decision was made to send an American air rescue team to Iran to bring out the fifty-three American hostages from Teheran where they had been held since early November 1979. The mission was a complicated plan involving troops, airplanes, helicopters, a long flight, .a landing in the desert, a journey to Teheran, taking out the hostages, and returning to safety. It appears that not even the upper-leve! national security staff was informed about how the decision was made . It is said thal President Carter had the Joint Chiefs of Staff draw up a couple of plans with different options. He then discussed these plans with his dosest advisors . According to Time (5 May 1980) these advisors were Brown, Brzezinski, Christopher, Jordan, Mondale, Turner, and Vance. Carter asked for clarification of sorne 1 of the ideas, but he made the decision on his own. There is no indication that the president resorted to sorne kind of voting in the discussion on whether the operation should or should not be undertaken . It has been said thal this was a typical Carter-style decision in which he asked his advisors for details and then made the decision himself. There is no question that the likelihood of success of a rescue operation is a compelling factor in deciding whether to go or not to go. Of course, it is crucial how one defines success for such a mission . High success means no deaths among the hostages or military personnel; medium success means a few military and no hostage deaths; low success means a few military and a few hostage deaths. Greater !osses could mean anything from failure to disaster. Sorne experts in the Department of Defense who are familiar with that mission have indicated that it would be generous to assign a medium chance of success to the military operation. They also said thal this was known prior to 28 April and is not a question of hindsight. Time writes: "Carter himself conceded that 'the operation was certain to be difficult and it was certain to be dangerous .' He insisted th at the operation had ' an excellent chance of success. "' As we will see, the president's decision was consistent with his perception of the situation as it affected him ; but it was not necessarily a good decision for the nation . Suppose we do a sensitivity analysis of the subjective factors and the emphasis the president placed on them .
Analysis of the Decision The decision problem can be divided into two parts. The first partis to ide ntify the best military option among those available and evaluate its likelihood of success . A military option would also have to be examined by exp erts on foreign relations and intelligence . The second partis the process
ANALYZING THE HOST AGE RESCUE OPERATION
97
of making the go/no-go decision based on the body of knowledge provided by the experts . We now examine these two parts in detail. The likelihood of success was determined by military experts. They considered such factors as: Transferability: to the desert, to Teheran, then to the embassy • Rounding up : getting inside the compound, creating diversion, locat ing the hostages Rescue: subduing the captors, transfer to aircraft, departure (avoiding Iranian forces) It is not my intention here to analyze the possible details of their deliberations . In this instance the importance of the military factors and the likelihood of the mission's success were probably much more important than the reaction of our allies or the Russians . These factors do appear in the hierarchy for the problem, but one could have construed the medium likelihood of success from other sources. In fact, an analysis of the first part did produce a r11edium likelihood of success, but a Pentagon expert, when asked, strongly supported these findings. Thus we may assume that a medium likelihood of success was presented to the president, who then made his go/no-go decision . How he did this is what 1 wish to dwell upon . The hierarchie structure of both parts of the pro cess is shown in Figure 6-1. We find that the main factors that could have played an important role in President's Carter's mind are : Hostages' lives: the president's concem for the safe return of ali fifty three hostages • Carter's politicallife : the president's concern about the influence of the decision and his chances for reelection Military costs : The president's concem about the loss of soldiers' lives in the operation United States prestige: the president's concern about the influence of the decision on relations with foreign states and their subsequent image of the United States These factors differ in their impact on the president's decision . Moreover, their relative importance changes as the likelihood of success is changed . Now let us assess their priorities based on a medium likelihood of success of the rescue operation. We can carry out a pairwise comparison process by using the pairwise comparison scale (Table S-1) . Figure 6-2 shows, for example, that "hostages' lives" has strong dominance over "military costs" in the president's mind . Thus we assign the value 5 in the first row and third column position and its reciprocal value in the first column and third row position . We always compare the row factors o n the left over the column factors on the top .
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As can be seen from the column of priorities in Figure 6-2, the two main factors are "Carter's political !ife" and "United States prestige." " United States prestige" is important because the United States had to do something to assert its power despite the fact that there was a medium chance for success. "Carter's political !ife," a subjective factor, is the highest-priority factor. Under each factor we consider this question : Which alternative (go or no-go) is more favorable considering just that factor under a medium likelihood of success? The results are shown in Figures 6-3 to 6-6 . Under "hostages' lives" the decision was evenly split . The rationale for this judgment is that there was no immediate danger to the lives of the hostages . With respect to "Carter's political life," we note that the president was strongly in favor of performing the operation at that time; a few months before the election the polis were not predicting a good chance for his reelection . A successful operation would have resulted in a strong influence on Carter's public image. Failure would be painful to him, but it
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ANALYZING THE HOSTAGE RESCUE OPERATION
STEP-BY-STEP EXAMPLES OF THE PROCESS
Carter's Pol itical Life
Priority of Hostages' Lives
G
N
Priorities
1
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Figure 6-4
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Priority of Carter's Political Life
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Military Costs
101
ANALYZING THE HOST AGE RESCUE OPERATION
Table 6-1
Relative Priorities Given a Low Likelihood of Success Priori/y Given Low Like/ihood of Success
Priori/y Given Medium Likelihood of Success
Factor
Priority of Military Costs
Figure 6-5
U.S. Prestige
Priorities
1
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1
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0.8
1
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Hostages' lives Carter's political life Military costs U.S. prestige
0 .15
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0.54
0.39
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Priority of U.S. Prestige
would not hurt him as much as a success would benefit him . The influences of "military costs" and "United States prestige" on the decision are apparent . Composing the factors' weights with those of the go/no-go alternatives under each factor gives these results : Go:
69
No-go :
31
lt is very easy to appear smart after the events. Yet 1 am convinced that if the AHP had been applied before the decision was made, we would have obtained similar results. Seven people established the priorities presented here . The variance in their judgments was small. lt is possible to argue that if those seven people had set the priorities individually, they would .have arrived at the same decision . · lt is clear that "Carter's political !ife" dominated that decision . Sensitivity analysis shows that if the 75 percent in favor of go under that factor is changed to 38 percent, the outcome for go/no-go would have been even. I believe that 75 percent is a modes! estimate. Carter may not have gone through with the decision to go if his desire to be reelected approximately matched his concern about the hostages. · To pursue the analysis further, my colleagues and I examined the outcome of the decision under a "low likelihood of success" recommendation by the experts. (Presumably this would have been interpreted less optimistically by the president . Recall that he interpreted medium success as " excellent.") Our results are shown in Table 6-1.
In this case the "hostages' lives" become a more important issue and "Carter's politicallife" less important. The influence of these factors on the go/no-go decision is shown in Table 6-2 . The outcome of the decision under a "low likelihood of success" recommendation by the experts leads to the following results: Go:
0.41
No-go :
0 .59
The change from the previous result is due to a greater emphasis on the hostages' lives. Only if the hostages' lives were clearly in jeopardy in Iran would one have decided in favor of go. Moreover, "Carter's political life" would have had to be less important because a military operation with a low chance of success would have had a low chance of helping Carter. Now let us pull together the observations we have drawn from this analysis of the Iran rescue operation. For President Carter, the subjective factor-namely, his concern with his career-accounted for 54 percent of the total. Perhaps the dominance of this subjective factor was perceived by Table 6-2
Factor
Hostages' lives Carter's political life Military costs U.S. prestige
Influence of Factors on the Go/No-Go Decision Ml'dium Likelihood Go No-Go
Law Likelihood Go No-Go
0.50
0 .50
0.20
0.80
0 .75
0.25
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0.125
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Carter's aides; perhaps it accounts for Secretary of State Vance's resignation . Certainly "Carter's political !ife" would not have figured as importantly in Vance's analysis of the situation. He would have had to decide against the operation . Of course he may have had other political reasons of his own. This application of the AHP illustrates its effectiveness in analyzing high-level decisions. There is a value to analysis even after a decision has been made because the method is an efficient tool for deriving a lesson from previous mistakes. ln this example the results were consistent with reality. ln sorne situations, as we have seen, media information may be . sufficient to indicate what went into a decision. If used by decision makers, the process can sharpen thinking and reveal subjective factors. lt can show, for example, that persona! interest is weighted much higher than what is good for the business. lt can also show that there is not a clear commitment to the important objectives.
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The AHP is basically a simple, efficient technique for problem solving. ·The following step-by-step example demonstrates this simplicity; il can also serve as a mode! for using the process to solve other problems. A firm wants to determine consumer preferences for three different kinds of paper towel. The attributes considered most relevant from the consumer's perspective are (1) softness, (2) absorptiveness, (3) priee, (4) size, (5) design, and (6) integrity. The three kinds of paper towels, X, Y, and Z, possess ali these attributes, but at different levels of intensity: high (H), medium (M), and low (L). Given the consumer' s "bounded rationality"that is, the fact that consumers do not act on perfect or complete information and are satisfied with less than the economically most rational choice-we can best distinguish among the attributes by dividing them into this small number of intensity categories. The resulting hierarchy is shown in Figure 6-7. The problem of selecting the product with the greatest overall consumer preference is solved in the following manner:
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Attributes Product Desirability
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118
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Then add this row and divide each entry by the total to gel the normalized vector of desired attribute intensities:
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L-Price
H-Size
M-Design
Ff-lntegrity
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0.3924
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Priorities of the Attributes
Step 3: Group the priorities of the intensities (H, M, L) for each of the six attributes in columns and enter the priorities of the attributes, laken from Figure 6-8, above the columns (Table 6-3). Then multiply each column by the priority of the corresponding attribute to obtain the weighted vectors of priority for the intensities (Table 6-4) . • Step 4: Now select from each column the element with the highest priority to obtain the vector of desired attribute intensities:
>...,,, = 6.66; Cl= 0.12.
Figure 6-8
105
DETERMINING CONSUMER PREFERENCE
STEP-BY-STEP EXAMI'LES OF THE PROCESS
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Soft ness
Absorp .
Priee
Size
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0.1278
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0.0734
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0.0094
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H-Siu
M-Design
H-lntegrity
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0.5659
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0 .2185
0.3727
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lies above the columns (Table 6-5) . Then multiply each column by the normalized priority o f the corresponding attribute intensity to obtain the weighted vectors of priority for the desired attri bute intensities for each paper towel (Table 6-6). St ep 7: Add each of the three rows to obtain the overall priorities of the three paper towels. This sy nthesis yields the following priorit ies:
x=
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y= 0.2519
z=
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From these results we would select product X as most desirable fro m the customer's perspecti ve . Even though low priee was the desired attribute intensity with the highest priority, product X, whose priority was very low with respect to low priee, was the final choice . The reason for this choice is clear: X domi nated Y and Z on ali other desired attribute intensities . Thus the firm decided to market a superior but high-prieed product , a decisio n that is not inconsistent with real-world situations.
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117
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0 .0666
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Energy crisis Recession Ill fla lion
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0- 5
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10 - 15
15 - 20
Pri ority
0.2264
0.2007
1
0.2697
To get the expected value of sales affected by the energy crisis, recession , and inflation, multiply the mid point of each interval by the priority weight of that interval. For example, the mid point of 0-5 percent is 2.5 and so on . Thus
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0.0518
5
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1/4
0.1451
10 - 15
7
3
1
1/3
0.2904
1.~ -20
5
4
3
1
0.5127
As this example shows, the analytic hierarchy process can be used to estimate numbers- in this case percentages.
5-10
10-15
Priority
SELECTING A PORTFOLIO
(2.5 x 0.3033) + (7.5 x 0 .2264) + (12 .5 x 0.2007) + (17 .5 x 0.2697)
= 9.685 %
= 4.337; Cl = 0 .11.
Recessio11
1
0-5
15 -20 ;'
-
0-5
l
2
5
7
0.5232
S- 10
112
1
3
5
0.2976
10 - 15
i /5
1/3
1
3
0.1222
I S- 20
117
1/5
1/3
1
0 .0570
5-10
10- 15
15-20
Priority
.\ """ = 4.069 ; lnflatio•1 0 -5
Cl
= 0 .02 .
0- 5
1
2
5
7
0 .5232 0.2976 0 .1222
5- !Cl
1/2
1
3
5
10 - 15
115
1/3
1
IS-2 0
117
1/5
1/3
3 1
0.0570
.\ .,,." = 4.069; Cl = 0 .02.
Figu re 6-12
':
the process is extended to estimate the percentage of a company's sales affected by the energy crisis, recession, and inflation . First the sales of the company-a manufacturer of heavy equipment (oil drills and construction machines)-are divided into intervals : 0-5 percent , 5-10 percent, 10-15 percent, and 15-20 percent. Then a matrix is developed to compare the criteria (energy crisis, recession, inflation) in pairs with respect to future sales (Figure 6-11). Next matrices are developed to compare sales with respect to the criteria (Figure 6-12) . The overall priorities of the sales percentage intervals are
.'i - 10
Àma '
:'!
= 0.00.
Matrix for Comparing Criteria with Respect to Fu-
Enl'rgy Cr isis
1
Cl
109
SELECTING A PORTFOLIO
STEP-BY-STEP EXAMPLES OF THE PROCESS
Matrix for Comparing Sales with Respect to Criteria
This application is more complex than the preceding one. Here we use the AHP to select a portfolio. Our hierarchical mode! consists of three separate hierarchies : one based on extrinsic factors, one based on intrinsic factors, and a third based on the investor's objectives. The firms being considered are ranked (weighted) relative to the criteria in each hierarchy . The weights are then combined to get an overall preference list of firms . Figure 6-13 gives an overall view of the mode! we will use . The various factors and objectives that influence the selection of firm s for the portfolio include :
1
Extrinsic Fact ors (A) : These are the outside factors or environmental
J
characteristics thal affect an industry' s (or firm ' s) performance . The firm, however, has no direct influence on them . These factors are eco nomie, political, social , and technical. By incorporating the analysis o f the extrinsic variables we can determine the sensitivity of a firm to changes in these factors . Intri11sic Fact ors (8 ): These are the internai factors or operati o n al charac. teristics of the firm . They may be considered as a measure of the way the firm is making its decisions or, in general, a measure of the firm 's capac-
l l
j
co (J1 (J1
Risk Level
Risk Lev el
Risk Class
!
leve! we have the primary extrinsic factors that affect a firm's behavior. At the second leve! we have the criteria that influence each of the primary factors:
B
Extdnsic Factors
lntrinsic Factors
lnvestor's Objectives Political
B Criteria
Social
1 Firms
..
Firms
Firms
1 Portfolio
Figure 6-13
Employment conditions Elasticity of demand Elasticity of supply International economy Interest rates Government regulations International exposure Employment conditions Family disintegration Age distribution Educational achievement Employment conditions State of technology Government involvement
Economie
c
1 A Criteria
Ail: Criteria
AI: Primary Extrinsic Factors
1 A
111
SELECfiNG A PORTFOLIO
STEP-BY-STEP EXAMPLES OF THE PROCESS
Hierarchical Model for Selecting a Portfolio
ity to compete successfully . These factors are profitability, size , technology, and philosophy . • Investor's Objectives (C): These are the values that define the actions the · investor undertakes in the business world . We are aware, of course, of the great variety of objectives an investor may have, but to simplify the mode! we consider four mutually exclusive objectives : profitability, securi ty, excitement, and control. Since we are dealing with a mode! that is based on future conditions, we must consider risk-the uncertainty of future events. The mode! incorporates the uncertainty of the general business environment, · the firm's behavior (high, medium, low risk), and the risk class of the investor (high, medium , low) . As our frrst step we look at the hierarchy of extrinsic factors . At the ftrst
Technological
q
'
The extrinsic. factors are compared pairwise for a high-risk environment, a medium-risk environment, and a low-risk environment (Figure 6-14). ln each case we base the comparisons on the questions: Which factor has more impact on a firm 's behavior and by how much7 We can see that in a high-risk environment the group constructing the matrix believed the technology would have a generally strong influence on a firm's behavior compared to the other factors. We then compute the priorities for each of the factors for each risk leve! (Table 6-7). These priorities point out that in a high-risk environment, future technological factors have the greatest impact on a firm's behavior. In a medium-risk environment both the economie and technological factors are most influential; in a low-risk environment the social factors are
Firm
E
High Risk p s
T
1
E
Medium Risk p s T 1
E
Low Risk p s
T
Economie
1
3
4
1/3
1
5
7
1
1
5
1/3
4
Po/itical
113
1
3
115
1/5
1
3
1/5
1/5
1
1/5
2
Social Te ch nological
1/4
1/3
1
118
117
1/3
1
1/6
3
5
1
5
3
5
8
1
1
5
6
1
1/4
1/2
1/2
1
Figure 6-14
Pairwise Comparison of Extrinsic Factors
(X)
lJ1 0'1
STEP-BY-STEP EXAMPLES OF THE PROCESS
Table 6-7 Risle
Priorities for Each Risk Level
Economie
Po/itica/
Social
Technologicn/
High
0.25
0.12
0.06
0.57
Medium
0.43
0.11
0.05
0.41
Low
0.30
0.10
0.54
0.07
Political (0.10) :
State of technology Got•ernment irrvolvement
Figure 6-15
., ;j'
:· Î,
1.
(.
·1
l'
r ()
1
4
1/4
1
Comparison Matrix for Technology
Economie
Political
Social
Technological
0.32
0.11
0.22
0.36
Elasticity of Demand
Elasticity of Supply
0.16
0.45
0.07
International Economy
lnterest Rates
0.05
0.27
Employment Conditions
0.24
0.06
0.70
0.30
0.11
Educational Achievement
Employment Conditions
0.11
0.48
State of Technology
Government lnvolvement
0.80
0.20
To get the final weights for a criterion , we must multiply the criterion weights just found by the weight of the factor associated with that criterion:
Employment Conditions
Elasticity of Demand
Elasticity of Supply
0.05
0.14
0.02
International Economy
Interest Rates
0.02
0.09
Government Regulations
International Exposure
Employment Conditions
0.02
0 .01
0.07
Fa mil y Disintegration
Age Distribution
0.07
0.02
Educational Achievement
Employmenl Conditions
0.02
0.11
Economie:
Now we compare the criteria for each factor to see their order of importance . In each pairwise comparison we ask : Which criterion has more impact on the factor and by how much? ln the case of technology and its two criteria, we have the comparison matrix shown in Figure 6-15, which indicates that the state of technology has somewhat more impact on technology in general than does the government's involvement in it . This matrix and three others yield the following sets of weights for the various criteria:
Employment Conditions
International Exposure
Age Distribution
Technological (0 .36):
most important . Economie factors have appreciable impact at ali three levels whereas the impact of political actions is rather low: We could carry through our analysis un der each of the three risk levels, but for illustrative purposes we will use the average weight for each factor. A weighted average could be used if we wished to favor one type of future over the other. A veraging, we obtain these weights:
Economie (0 .32): ~
~
G
Government Regulations
Family Disintegration Social (0.22) :
5
d3
SELECTING A PORTFOLIO
Political:
·a : ~l',
Social:
CXJ lJl '-]
-
.. ··-· . ·-·-- _.. ·-
.. ...
.... --- --- - - ~---····-· ......
--------
...
-··----.. -·-
.. -·~ --- - ... -·· -··-··--.. ~ .. --·-····--·· ......
Technological:
State of Technology
Government Jnvolvement
0.29
0.07
115
SELECTING A PORTFOLIO
STEP-BY-STEP EXAMPLES OF THE PROCESS
The weights printed in boldface are the largest ones. We will shorten our list to include only these . In order to have the weights total1, we divide each weight by the total of ali the weights in the shortened list. We have our list of extrinsic criteria:
Table 6-8 summarizes the weights that the eight firms received for each criterion. A glanee at the table shows that in a technological environment Data General, Rockwell, and I.C. Industries are heavily favored , as one might expect. Tappan and Rockwell are beneficiaries if employment conditions are good-Tappan's consumer market is in appliances (80 percent of sales) and Rockwell's is in home and auto products (35 percent of sales) . To get an overall prioritized list of firms vis-à-vis the extrinsic factors, we multiply the weights of the firms for a given criterion by the weight of that criterion (shown in parentheses in Table 6-8). Then we add up these new weights for each firm :
State of Tecltnology
Employment Cortditions
Elasticity of Demand
Tappan
lng . Rand
I. C. Industries
A/lied Ch .
0.33
0.26
0. 16
0.16
0 .07
0.13
0 .03
lnterest Rates
Fa mil y Disintegration
Government Involvement in Technology
0.10
0.08
0.08
R
DG
8
c
4
115
117
6
5
4
1/4
116
7
6
5
1/2
114
6
5
115
1
1/6
117
4
3
2
6
1
1/3
7
6
T
IR
/C
Tappan
1
1/3
115
Ing . Rand
3
1
1/4
/. C. Industries
5
4
1
A/lied Ch .
1/4
1/4
Rockwe/1
5
4
AC
Data General Butler
7
6
4
7
3
1
9
7
1/6
117
116
1/4
117
1/9
1
113
Cltemetron
115
116
115
113
116
117
3
~ure6-16
Data General
Butler
Chemetron
0 .20
0 .21
0 .07
0.13
For purposes of illustration we will use only the four highest ones with their weights adjusted as we did when shortening the list of criteria . Here is the prioritized list of firms relative to the extrinsic criteria :
The firms being considered (eight in this illustration) are now prioritized relative to each of the six extrinsic criteria. This process is the same pairwise comparison and weighting procedure we have been applying throughout . When comparing the firms pairwise relative to the state of technology, for example, we ask : Which firm will respond more favorably to the technological environment of the future? The matrix for this criterion is sHown in Figure 6-16.
Technology
Rockwe/1
Matrix for Pairwise Comparison of Firms Relative to State of Technology
Rockwe/1
Tappan
Data General
I. C. Indu st ries
0.29
0.23
0.30
0.19
Now we turn to the intrinsic factors and go through the same process : Construct a two-level hierarchy of factors and criteria ; obtain weights for the factors, then for the criteria relative to the corresponding factor, and then an overalllist of weighted criteria; and, finally, prepare a prioritized list of firms . The intrinsic factors hierarchy consists of:
BI : Prr'ma ry lntrinsic Factors Profitability
Size
BI/: Criteria Management Quality Market share Earnings Innovativeness Diversity Payout ratio Sales Labor force Assets Market structure
00 U1
00
-- - ' ....
-·- ..
----------------------.
--- .. ---~- ·~-·------......
STEP-BY- STEP EXAMI'LES OF Tl lE PROCESS
BI: Primary lntrinsic Factors ~
ë~
gË
ct..e
~.,o
()
BII: Criteria
Technological control
8
0
~ ~
0 0
8
~
g :::!
0 0 0
R&D quality Age distribution of product Energy dependence Pollution effects Social responsibility Participatory decision making
~
0 0
;:>
(..)~
Business philosophy ·.:::"'
117
SELECTING A PORTFOLIO
()
~
>. ..
~] 0
~
·e"' ,, ;
l'i
-
0
-~
o-..: ~..5
]
~"'"' ...
"
'; t:; Q
~ ~ ~
0
0 0
s0
~ ~ ~ ~
0 0
0 0
8 !::> l8 :::! ~ ~ ~ ~
0
0 0
0 0
0
0
"' ,i; till
~
co1 ~
>.r c:s
."!:;:
';
-~
E
0
-
...
~:gCl
("'"')
t....
' ll')
0
0
0
0 0 0
0
(""')
0
N
("'"')
f"-..
...-4
...-4
N
t""""'
("'"')
0
0
0 0
~
o-""·.
~
()
"'-"' E: ()
~ :::! ~ ::: ci ci ci ci ci ci ci ci ~
Size
Technological Control
Business Philosophy
:1 ,.
0.51
0.17
0.26
0.06
:!
t
Gl
·~
Innovativeness
Management Quality
R&D Quality
Sales
0.46
0.21
0.17
0.16
\!
il
~j
·1 1· !'
,,
1i,,
li tl
~
The four firms in the extrinsic factors list are now compared pairwise relative to each of the intrinsic criteria . The results are summarized in Table 6-9.
>.
.. -"'
-'0"-~ ...., ()
8.~..1: V)
...
!::> ~ ~ 8 r::i ~ :::! <5 0 0 0 0 0 0 0 0
Table 6-9
~
i·
"'
~
·c:':iiv
....
"'t
"::s..<::::~c:
c; 0
fiico:c:.,~o
.. ev
lE
.~
-
.
fij-oUvV!::
o..
.
-v..>o:.,~
O..l.lll(j:Sg-:;;'51!
~ ..5 ...;
< co:
0
1'0
u
1
Weights of Firms for Intrinsic Criteria
(0.46)
Firm
J
jl
Thus profitability and technological control exert about 76 percent of the total influence on a firm's behavior when we consider only the intrinsic factors . Business philosophy does not seem to have much impact . After we get the weights of the criteria for each of the factors and multiply them, we obtain a list of sixteen factors. Using the four whose combined weight is about 60 percent of the total, we obtain an abbreviated list of weighted intrinsic criteria :
l8 ;::!: 8
::0 ~
Profitability
~~~
~ S, ·~
•1
0
'o-.,. \.Q
After comparing the four primary factors pairwise in each of the three levels of risk and taking the average of the resulting three weights, wc obtain the weighted list of factors :
:J
~
,Ë
-~
!•
-
:~ Cl
1: u
..... ""0 1
......
to
lnnovativeruss
(0.21) Managl'ment Quality
(0 . 17) R&D Qualily
Rock weil
0.34
0.32
0.12
0.25
Tappan Data General I.C. Industries
0.27
0.03
0.04
0.05
0.30
0.55
0.74
0.52
0.09
0.10
0.10
0.18
• Sales/assets ratios were used to compare the firms .
i
(O . Jti)
~
Sales•
1
?'1 Ul \!)
119
SELECTING A PORTFOLIO
STEI'-BY-STEP EXAMPLES OF Tl lE PROCESS
..... .....
The final list of weighted firms relative to the intrinsic criteria (after multiplying and adding) is · Rockwell
Tappa11
Data General
I.C. Industries
0.29
0.13
0.47
0.11
The investor's objectives with weights computed under an average risk class are
Profit
Control
Security
Excitement
0.34
0.28
0.25
0.13
.....
"'
N
Ir) 1"""'f
Ir) ""'"
""' ,....4
0 0 0 0
~ ~ ~ ~ c:i c:i c:i
0
...... ""
N
""
""""'
ci 0
Weights of the firms for the investor's objectives a~é given in Table 6-10. The final list of weighh~d firms relative to the' investor's objectives is Rockwell
Tappa11
Data General
I. C. Ind11stries
0.17
0.05
0.58
0.20
t-..
~
t-..
""""'
c:i ci
-~
1.4
......
~ ~ ~ ~
0
-
ci ci ci ci
.c:"'
bO
~
To obl'ain the final priority of the firms (the portfolio), we weight each criterion (extrinsic, intrinsic, objectives) and perform the multiplication and addition. Table 6-11 shows the final weights for three weighting schemes for the criteria. (The scheme 2: 1: 1 means weight of 2/4, 1/4, 1/4, and so forth.) We note that Data General and Rockwell International rank fust and second respectively in ali the weighting schemes. Moreover, I.C. Industries and Tappan maintain the same relative standing unless the investor's objectives are emphasized . A certain amount of stability is shown here. What if we want to use these weights for a guideline for allocating funds among the stocks? If we use the criteria weighting scheme (2: 1 : 1) where the extrinsie criteria are deemed twice as important as each of the
-;; c '-i:
.
...... ......
·"'
~ 0 ·.:::
Cil
:c ~
0
-
Cf\
v
N
"' '!Ji t:
·.:::: ·c=
Weights of Firms for Investor's Objectives (0.28)
(0 .25 )
Control
Securily
(0.13) Exci/emenl
Rock weil
0.27
0.06
0.04
0.38
Tappan
0.03
0.04
0.13
0.03
Data General
0.63
0.75
0.29
0.55
l.C. Industries
0.07
0.15
0.54
r-...
~
...-4
. ......
.-
v; .::
~~
Profil
...-4
"'-"' ·s
~ ~ ~ .~
·ê u .!.';<>-
(0 .34)
(f"')
c:i c:i ci ci
;>
tl)
Firm
!:::; ~ ~ ~ ci c) ci ci
v;,
~
Table 6-10
'
' ... v~ " t: 0 -Vi~~
Ë
~
0.04
aat-.. Ir) N N N ..... c:i ci c:i ci
= Cil
"'Cil f '.E ;
~0 g: ~ ~
:~i
c:Cil
"' ;:J
!!
~u
~ -a
"' _. Cl
(X)
())
0
121
STEP-BY-STEI' EXAMPLES OF THE PROCESS
PERSPECTIVE
other two, we will invest about 40 percent in Data General, about 26 percent in Rockwell International, about 19 percent in Tappan, and 15 percent in 1. C. Industries.
shows the ,usefulness of the AHP to make tradeoffs. With a little higher priee, one can make a more desirable product. The third example-estimating the economy's impact on salesshows how the AHP can be used to estimate actual numbers (in this case expréssed as percentages). Subjective priorities are converted to directly understood numbers . The fourth example-selecting a stock portfolio-deals with prediction as a result of understanding a problem, its important criteria, their relations, and their priorities. If the understanding is good, prediction can range from good to excellent. Such an approach was used to predict the outcome of the World Chess Championship match of 1978; predictions o f the number of games to be played and the number of games won by both players were excellent. The fourth application is being generalized to the entire stock market, but considerable knowledge, understanding, and testing are needed before one can expect to have a representative structure and workable priorities.
HOW VAUD IS THE PROCESS? The validity of the AHP as a decision-making tool has been confirmed· by comparing the priorities derived from the process with those achieved independently by decision makers. For example, the matrix of pairwise comparisons in Figure 6-17 was developed during a discussion with top planners of a large business corporation . They were asked . how the chairman of the board viewed the various sectors of the corporation's activity. The relative importance of the sectors in terms of the corporation's alloca· tion of effort was judged and priorities were computed . At the end of this short ex~rcise, two corporate planners left the room and returned with a book containing data on the amount of capital actually invested in each activity. This amount is shown in the last column of Figure 6-17 . Clearly the results obtained through the AHP closely approximated those achieved by traditional methods .
f,
q '
·. ·'
·4. JI
i~
.,·~ t~
ii! ;
.i
1
f
l1
!.
t
A
B
c
D
E
Priority
Actual lnvestment
1
7
6
4
2
0 .45
0 .45
B
117
1
1/2
113
1/5
0 .05
0.04
Busirress C flusirrrss D
116
2
1
113
114
0 .07
0 . 105
114
3
3
1
1/3
0 .14
0 .145
Bu sirrrss E
112
5
4
3
1
0.28
0 .25
Business A Bu sin~ss
Figure 6-17
.:·.';'
.
1 l
..J i" '
~i !{
Matrix of Pairwise Comparisons
,, it
PERSPECTIVE The first example in this chapter-the decision to rescue the hostages in Iran-shows how the AHP can be used to put forth a certain point of viel\· and compare the outcome with what is already known. The purpose m ay be simply to make hypotheses . This application also shows that if a decision maker's priorities are significantly different from those expected, he o r she may have important criteria in mind not considered by others. T he second example--determining consumer preference-gives a neat layout dealing with the marketability of a product . Moreover, it clearly
J i! ji
!J
'i
cn l 0'>
.........
8.62
EXEMPLES D'APPLICATION DANS LE DOMAINE MANUFACTURIER
Sélection d'un système de manutention
Systèrre de rraUeriiCJ'l
saisfam
1
AG\/S usine entière
Il
Oaicts élévatars
~
Ill
Oaicts
élévéiars usine entière
déf:t B&C.
Hiérarchie utilisée par Frazelle 1985 p.44.
rvb'D'ail dép A AG\/S déf:i B&C.
8.63 comparaisons binaires des critères par rapport à l'objectif global:
ROI Comp. Flex. Main. Sée.
ROI
Comp.
Flex.
Main.
Sée.
veet. de priorit.
1. 00 0.14 0.20 0.25 0.20
7.00 1. 00 4.00 7.00 5.00
5.00 0.25 1. 00 0.33 0.25
4.00 0.14 3.00 1.00 0.20
5.00 0.20 4.00 5.00 1.00
0.48 0.04 0.21 0.19 0.09
1. 79
24.00
6.83
8.34
15.20
1. 00
= 5 • 84 C.I.= 0.21 C.R.= 0.19
Àma x
Alternatives par rapport à alt.I
alt.II
COMPATIBILITÉ: alt.III
alt.IV
vect. de priorit.
alt.I alt.II alt.III alt.IV
1. 00 0.33 0.17 0.20
3.00 1. 00 0.33 0.25
6.00 3.00 1. 00 2.00
5.00 4.00 0.50 1.00
0.55 0.26 0.08 0.11
SOMME
1. 70
4.58
12.00
10.50
1. 00
Àmax
C.I.= C.R.=
4 . 14 0.05 0.05
8.64
Alternatives par rapport à ROI:
vect. de priorit.
alt.r alt.rr alt.rr alt.rv alt.r alt.II alt.III alt.rv
1. 00 0.33 0.14 0.13
3.00 1.00 0.25 0.14
7.00 4.00 1. 00 0.33
8.00 7.00 3.00 1.00
0.57 0.28 0.10 0.05
SOMME
1. 60
4.39
12.33
19.00
1. 00
= c.r.= C.R.=
À max
4.16 0.05 0.06
Alternatives par rapport à la FLEXIBILITÉ:
alt.II
alt.rrr
alt.r alt.rr alt.III alt.IV
1.00 2.00 3.00 0.50
0.50 1.00 3.00 0.50
0.33 0.33 1. 00 0.25
2.00 2 . 00 4.00 1. 00
0.16 0.23 0.51 0.10
SOMME
6.50
5.00
1. 92
9.00
1. 00
= c.r.= c.R.=
À max
alt.rv
vect. de priorit.
alt.r
4.08 0.03 0.03
•
8.65
Alternatives par rapport à la MAINTENABILITÉ:
alt.I
alt.II
alt.III
alt.IV
vect. de priorit.
alt.I alt.II alt.III alt.IV
1.00 1. 00 2.00 1. 00
1.00 1. 00 2.00 1. 00
0.50 0.50 1. 00 0.50
1.00 1.00 2.00 1.00
0.20 0.20 0.40 0.20
SOMME
5.00
5.00
2.50
5.00
1. 00
= 4.00 C.I.= o.oo C.R.: o.oo
)\max
Alternatives par rapport à la SÉCURITÉ:
vect. de priorit.
alt.I
alt.II
alt.III
alt.IV
alt.I alt.II alt.III alt.IV
1. 00 0.50 0.25 2.00
2.00 1. 00 0.50 3.00
4.00 2.00 1.00 9.00
· o.5o 0.33 0.11 1.00
0.27 0.15 0.07 0.52
SOMME
3.75
6.50
16.00
1. 94
1. 00
= 4.02 C.I.= 0.01 c.R.= 0.01
)\max
8 .66
ÉVALUATION GLOBALE: CRITÈRES
ROI
comp.
flex. maint. sée.
0.48
0.04
0.21
0.57 0.28 0.10 0.05
0 . 55 0.26 0.08 0.11
0.16 0.23 0.51 0.10
0.20 0.20 0.40 0.20
0.27 0.15 0.07 0.52
0.39 0 . 24 0.24 0.13
1.00
1.00
1. 00
1.00
1.00
1. 00
0.19
0.09
Poids des alt.
OPTIONS
alt.I alt.II alt.III alt.IV
258
9-6.
9· 7.
9-8.
9-9.
Multiattribute Decision Analysis: Utility Models
Chap. 9
(b) Long-lerm income/year (say, what you will make ultimately, expressed in "today's dollars" of purchasing power). (c) Job satisfaction (on a subjective scale of 0 for worst to JO for best). (d) Quality of nonwork life (however that is meaningful to you, on subjective qualitative scale such as very poor, poor, fair, etc.). (Section 9.5) For the attributes in Exercise 9-5, decide which is most important to you and ask yourself Question A to gel the corresponding weighting factor. Then use Question B to determine weighting factors for the other attributes. Defi ne two or more career!location alternatives for yourself and compare them using the MAUM and the results of Exercises 9-5 and 9-6. Are the results conclusive for you? What judgments and assumptions included in your use of the model would seem to be most questionable or uncertain and thus be worthy of further (sensitivity-type) exploration? Repeat Exercise 8-7 except analyze using the multiattribute utility method. For at )east one attribute, show the probabilistic trade-off (lottery-type) questions you needed to ask together with answers to obtain two or more utility values to plot lletween the "!lest" and "worst" outcomes. For other attributes you may make shortcut approximations by determining if each is concave or convex, upward or downward. and then sketching graphs for each which are satisfactory representations to your client. Then ask questions to determine scaling factors and determine which alternative is best. [Note: If you follow the recommended procedure for scaling factors, most likely ~ k; f. 1.0, and you should use the multiplicative mode!. After comparing alternatives by that model, "normalize" or "unitize" the scaling factors so ~k; = 1.0, and then compare the alternatives using the additive model. (lt is not theoretically correct to normalize the k1's to ena ble use of the additive model.) How much difference does use of the "correct" mode) make?] Describe a multiattribute decision problem involving at )east two significantly different alternative Jevels (degrees) for automation involving CAD, CAM, and/ or a compt,Jter-aided decision support system . A real problem of a client would be most preferable. but failing thal, fabricate it to be as realistic as possible. Consider at )east four important attributes and show analysis by the multiattri( bute utility method using guidelines given in Exercise 9-8.
multiattributè decision analysis: the analytic hierarchy process { ~
ch apte 10.1 INTRODUCTION
,,
,,
The analytic hierarchy process (AHP) was developed and documented pri· marily by Thomas Saaty [ 1, 2]. Applications of this methodology have been reported in numerous fields, such as transportation planning, portfolio selection, corporate planning, marketing, and others. The strength of the AHP method lies in its ability to structure a corn· plex, multiperson, multiattribute, and multiperiod problem hierarchically . Pairwise comparisons of the elements (usually, alternatives and attributes) can be established using a scale indicating the strength with which one element dominates another with respect to a higher-level element. This scaling process can then be translated into priority weights (scores) for comparison of alternatives.
259
260
Multiattribute Decision Analysis 1
Chap . JO
10.2 CONSTRUCTION OF HIERARCHY AND SOLUTION PROCESS OVERVIEW
We will concentrate on what Saaty calls "functional" hierarchies as applieh to multiattribute decision problems. He uses the lerm "element" to apply to the overall objective, attributes, subattributes, sub-subattributes, and so on; and alternatives of a problem as follows:
.... =§~~B~
~. ~~~E
.Eë~;c~ ~ 0 . ~
The top leve!, called the focus, consists of only one element-the broad , overall objective. Subsequent levels may each have severa! elements, although their number is very small-between 5 and 9. Because the elements in one leve! are to be compared with one another against a criterion in the next higher leve!. the elements in each leve! must be of the same order of magnitude
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5 ~ g ~:::;:
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u.,-., Zo~~~~
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Figure 10-1 shows a typical four-levet hierarchy applied to a career choice problem. Note thal, as always, the focus is at the top leve! and 1 the alternatives are at the lowest leve!. If any of the subattributes were ' further divided into sub-subattributes, those sub-subattributes wouldJ have constituted a new leve!. Or if one felt that the subattributes are not necessary, thal leve! could be eliminated, thereby making it a three-leveJ problem. The general approach of thé AHP is to decompose the problem and to make pairwise comparisons of ali elements (attributes, alternatives, etc .) on a given leve! with respect to the related elements in the leve! just above. The degree of preferenèe or intensity of the decision maker in the choice for each pairwise comparison is quantified on a scale of 1 to 9, and these quantities are placed in a matrix of comparisons. The suggested numbers to express degrees of preference between the two elements x and y are as follows:
'//ü
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.., :x
then the preference number to assign is:
equally important/preferred weakly more important/preferred strongly more important/preferred very strongly more important/preferred absolutely more important/preferred
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261
Multiatlribute Decision Analysis
Chap. JO
Sec . 10.3
The solution process consists of three stages, with an optional concurrent fourth stage as follows :
Automation Alternatives Example
263 Table 10-1
MATRIX oF PAIRED CoMPARISONS ( INCLUDING DECIMAL EQUIV-ALENTS) FOR ATTRIBUTES
,.
1. Determination of the relative importance of the attributes and subattri( butes, if any
2. Determination of the relative standing (weight) of each alternative with respect to each subattribute, if applicable, and then successively with respect to each attribute 3. Determination of the overall priority weight (score) of each alternative 4. Determination of indicator(s) of consistency in making pairwise comparisons
Decimal Equivalents 1
A. B. C. D. E.
CIMS tktical aims 1 Net present worth Serviceability Management/engineering effort Riskiness, lack of
'"'
c
A
B
1 3
i
5 6 5
1
! i !
i
6 1
+ i
A
D E
1 3 0.20 0. 17 0.20
7 6 3 1 i 1 1 1 4 1
r iÏ
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l1
1
10.3 AUTOMATION ALTERNATIVES EXAMPLE
Let us demons! rate the AHP method by comparison of automation alternatives P-1, P-2, and P-3 (originally given in Chapter 8). Figure 10-2 shows the alternatives and at tri butes structured in a hierarchy . We will show the attributes in the order (A, B, etc.) given in Table 8-3. This constitutes a threelevel hierarchy . 1 0.3_1 Priority Weights for Attributes
Table 10-1 shows a mat rix of preference numbers expressed by a decision maker for ali combinations of the five main attributes in Fig. 10-2. For example, Table 10-1 shows that, with respect to the overall goal, serviceabil-
1
l.
Lt'Vtl 1: Overall focu1 fobjectiwd
r
Levtl Il : Attr ibute 1
1
l' ! i 1
i j.: 1/1 r~
1,·
Lr.otl Hl :
Alternatl~s
Figure 10-2 .
Decision hierarchy for au tomation alternatives exa mple.
~
B 0.33 1 0. 17 0. 14 0. 17
c 5 6 1 0.33 1
D 6 7 3 1 4
E
5 6 1 0.25 1
4.57 1.81 13.33 21.0 13.25
!
ity (C) is weakl y more important than management/engineering effort (0) and is equally as important as;iskiness, Jack of (E), and so on. The onl y combinations that really requtre thoughtful judgment are those above or below the diagonal , because the "mirror-image" counterpart of each number is the reciprocal of !hat number and the diagonal entries are ali 1 (since any given element must surely be equally preferred with that element). Note th at the right-hand si de of Table 10-1 shows the sa me paired comparison results in decimal form and summed by columns to facilitate later calculalions. Figure IP=3 shows part of a questionnaire form which might facilitate the answering of pairwise comparison questions . The check marks shown resulted in the preference numbers in Table 10-1. We will not show furth er use of a form like Fig. 10-3 , but will show ali preference- comparisons in matrixform . After obtaining the pairwisejudgments as in Table 10-1, the next step is the computation of a vector of priorities or weighting of elements in the matrix . In terms of ma tri x algebra, this consists of calculating the "principal vector" (eigenvector) of the matrix, and then normalizing it to sum to 1.0 or 100%. Standard programs are available for computing the principal vector of a matrix . The following is an approximation method which is thought to provide sufficiently close results for most applications: Divide the elements of each column by the sum of thal column (i .e ., normal ize the column) and then add the elements in each resulting row and di vide this sum by the number of elements in the row . Table 10-2 shows the normalized matrix by dividing each element in Table 10-1 by the sum of its respective column. Finally, row entries in the las! two columns of Table 10-2 are comprised of the sum of the five element s in the row and the average of those row elements (principal vector) , respectively .
(X)
0'1 \.D
264
Multiattribute Decision Analysis
Chap . JO
Importance {or preference) of one attribute or alternative over another 0>
0>
~ ::
~
::
~
"'
0>
With respect to :
:;
Best overall
ë
~
Automated system: Attribute:
il
<
~
.
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>
><
'ii
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11.
9 8 7 6
0>
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c
e
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2 3 4 5
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e
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~
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'ii
11.
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:;
6 7 8 9
x
CIMS tactical ai ms A
O.
Ma~agement/
engineering effort
x
A
B
E. Riskiness, lacft'oof
x
Net present worth B
C. Service-ability
x
O. Management/ engineering effort
B
x
E. Risklness
x
c
Serviceability
c Management/ engineering effort
x
E. Aiskiness
x
0
E. Aiskiness
( Table 10-2 NORMALIZED MATRIX OF PAIRED COMPARISONS AND CALCULATION OF PRIORITY WEIGHTS (APPROX IMATE ATTRIBUTE WEIGHTS)
c
D E I
The results (principal vector) are thal the attributes have the following approximate priority weights: 0.288 A. CIMS "tactical aims 0.489 B. Net pres,ent worth 0.086 C. Serviceability 0.041 D. Management/engineering effort 0.096 E. Riski~ss, lack of
}: = 1.000 The consistency ratio (C.R.) for the comparison above is calculated in Appendix 10-A to be 0.07, which is within a 0.10 empirical upper limit sugg_ested by Saaty, so we will "accept" the attribute priority weightings above. If the calculated C.R. is greater than 0.10, this empirically indicates excessive intransitivities of preferences (either real jptransitivities or, more likely, inconsistencies in stated degrees ofpreferenéès). Normally, the C.R. can be reduced by reestimat~g preferences. ln general. the lower the C.R .. the more accurate is this approximate method of calculating priority weights.
O. Management/ engineering effort
Figure 10-3. Questionnaire form which could be used to facilitate preference comparisons (Numerical results were shown in Table 10-1).
A B
265
Automation Altematives Example
C. Service-ability
x
A
Attribute
B. Net present worth
x
A
(
'§ ~
Sec. 10.3
A
B
c
D
0.219 0.656 0.044 0.037 0.044
0. 184 0.551 0.1)94 0.077 0.094
0.375 0.450 0,075 0.025 0,075
0.286 0.333 0. 143 0.048 0. 190
0.377 0.454 0.075 0.019 0,075
= 1.000
1.000
1.000
1.000
1.000
t
E
-
Row
l: 1.441 2.444 0.431 0.206 0.478
1
Average
i _., = l:/5 0.288 0.489 0.086 0.041 0.096 1.000
1 0.3.2 Prlority Weights for Alternatives with Respect to Attributes
The next step is to make pairwise comparisons of each of the alternatives with respect to each of the attributes to which they relate in the next higher levet in Fig. 10-2 . .We illustra te this only with respect to the first attribute, "CIMS tactical aims." Table \0-3 shows illustrative pairwise comparisons, and Table 10-4 shows subsequent calculations paralleling those in Table 10-2, resulting in the following approximate principal vectors (priority weights which are often descriptively called "evaluation ratings") with respect to CIMS tactical aims. The results are:
.P-1 P-2
0.21 0.55 P-3 0.24 }: = 1.00
The consistency ratio for these pairwise comparisons is calculated in the end of Appendix 10-A to be 0.02, which is quite good. Figure 10-4 summarizes the results of evaluating the alternatives with respect to each of the five attributes . Note that the uppermost results are those for which calculations were discussed above and shown in Tables 10-3 and 10-4. The remainder are presented without showing the computational details. (X)
o,J
0
\
Table 10-3
Sec. 10.3
Automation Alternatives Examp/e
267
MATRIX OF PAIRED COM PARISON RESULTS FOR ALTERNATIVES (WITH RESPECT TO CIMS
Table 10-4
TACTICAL AIMS)
P-1 P-2 P-3
P-1
P-2
P-3
P-1
P-2
P-3
1 3 1
!
1 2 1
1 3 1
0.33 1 0.50
1 2 1
:L=5
1.83
4
1
!
~~0')
"i:
o·-
With Respect to : A. CIMS tactical aims
P-1
;t~
P-3
P-1 P-2
B. Net present worth
P-2
NORMALIZED MATRIJt AND PRIORITY WEIGHTS FOR ,ALTERNATIVES (WITH RESPECT TO CIMS TACTI CAI. AIMS)
~-1
2
0.55
P-3
0.24 :L = 1.00
P-1
0.12
P-2
3
P-3
5
3
0.55
Consistency Ratio (CR)
0.02
0.26
0.33
}; = 1.00 C. Serviceability
P-1
1
2
2
:L
l:/3
1 0.20 "-0.60 0.20
0.18 0.55 0.27
Q.25 0.50 0.25
0.63 1.65 0.72
0.21 0.55 0.24
:L = 1.00
1.00
1.00
1.00
1 0.3.3 Priority Weights for Alternatives
Table 10-5 summarizes ali priority weights, in
0.50
P-2
0.25
P-3
P-3
P-1 P-2 P-3
0.21 3
P-2
1
0.25
Table 10-5
0.00
SUMMARY OF PRIORITY WEIGHTS LABELED AS ATTRIBUTE WEIGHTS, EVALUATION RATINGS,' AND WEIGHTED EVALUATIONS
(
}; = 1.00 D. Management/engineering effort
P-1
1
3
P-2
7
0.63
6
0.30
P-3
P-1
0.04
0.07
:L E. Riskiness, lack of
Attribute
1
3
= 1.00
4
0.62
P-2
2
0.24
P-3
1
0.14 1.00
:L
=
0.02
Attribute weights Alternative P-1 P-2 P-3
B:
D: E: Management/ Riskiness, Service- Engineering Lack ability of Effort
A: CIMS Tactical Ai ms
Net Present Worth
C:
0.288
0.489
0.086
0.041
0.096
0.21 0.55 0.24
0.12 0.55 0.33
0.50 0.25 0.25
0.63 0.30 0.07
0.62 0.24 0. 14
Alternative Weighted Evaluation = :L Attribute Weight x Evaluation Rating
:L Figure 10-4. Su rn mary of ali paired comparisons and resulting priority weights for alternatives with respect to each attribute.
• Evaluation rntings in body of matrix .
~
0.248 0.484 0.268 1.000
co '-J l-'
266
268
Multiattribute Decision Analysis
Sec . 10.4
Clrap . JO
The weighted evaluation for each alternative can be obtained by multiplying the mat rix of evaluation ratings by the vector of attribute weights and summing over ali attributes . Expressed in conventional mathematical notation, we have
Best overall
system
weighted evaluation for alternative k
2:
A
attribute weight; x evaluation rating;k
Attribute
aJJ i auributes
Th us, for alternative 1, 0.288(0.21)
+ 0.489(0 . 12) + 0.086(0.50) + 0.041(0.63) + 0.096(0.62)
= 0.248 Alternatiw
which is shown in the right-hand column of Table 10-5. Thus project P-2 (with a priority weight or weighted evaluation of 0.484) is indicated to be considerably more desirable than either project P-l or P-3 (with priority weights of 0.248 and 0.268, respectively) . 10.3.4 (Optlonal) Added Explanatlon Regarding Example
Another way to show the structure of the automation alternatives ex- . ample problem and the results of ali priority weights (from Table 10-2 and Fig. 10-4) is given in Fig . 10-5. Using the results as displayed in Fig. l0-5 , one can calculate the priority weight (weighted evaluation) for any alternative mere) y by summing the products of weights for ali branches including thal alternative. Thus, for alternative P-1 , the weighted evaluation would be 0.2 1(0.288) + 0. 12(0.489) + . + 0.62(0.0%) = 0.248. 0
0
10.4 FMS JUSTIFICATION EXAMf'LE* The following example further illustrates AHP methodology by including subattributes (thus making a four-levet hierarchy) and also by using concurrent hierarchies to compare alternative benefits as weil as costs. Ma nufacturing company XYZ, producer of aircraft systems' components, has decided to enter into a new market which is highly cost competitive and functionally different from current product lines. Company management is faced with deciding what manufacturing strategy is best for this new product tine given the highly competitive environment and technological requirements as' Example is adapted from Mark S. Varney, W. G. Sullivan, and J. K. Cochran, "Justification of Flexible Manufacturing Systems with the Analytic Hierarchy Process," Proceedings of the /985 Annual International lndustria/ Engineering Conference, Institute of Industrial Engineers, with permission of the publisher.
269
FMS Justification Example
·: ,
CIMS tact/cal ai ms 0.288
B Net present worth
0.489
c Serviceability
0.086
D
Management/
Riskines.s,
enginéering
l.ck of
effort 0.041
0.096
;1\ 11\;1\ 11\;1\
P- 1 P-2 P-3
P-1
P-2 P-3
P-1 P-2 P-3
P- 1 P- 2
P-3
P- 1 P-2 P-3
0.21 o.55 o.24
0.12 0.55 0 .33
0.50 0.25 0 .25
0.63 0 .30 0.07
0.62 0.24 0.14
- - - - - - - - - - - - - -- - - - - - - - - Resul ts:
Alternative
Priority_}Yeight
P-1
:::h 0~48
P- 2
0.484
P-3
0.268
Figure 10-S. Decision hierarchy and priority weight results for automation allernatives example.
sociated with the new market. Th
CXJ "-.J N
l ... l Glob.ll twneliu
Multiallrihute Decision Analysis
211)
Chap. JO
costs and benefits in the traditional discounted casldlow calculations.produced recomm~ndations that were not favorable to further consideration of a FMS. To perform a complete analysis of the two alternatives: the IË-dêpart~ent constructcd a list of attributes which they believed impoùant in selecting a manufacturing strategy for the new product tine. After considerable discussion, they settled on the following evaluation attributes: Product quality Present worth of costs Manufacturing flexibility Market response Shop floor information requirements Operating risk Required technical/management support Delivery schedule performance Physical inventory Manufacturing excellence The next step in the evaluation process was to determine how the two alternatives could be evaluated against each other using the previously determined attributes and AHP methodology. Utili:z:ing the evaluation criteria (attributes) above, hierarchical structures for both benefits and costs were constructed as shown in Fig. 10-6. piven a hierarchical structure, it was theo possible to establish priorities among the elements in the hierarchy by using the AHP method of paired comparisons and matrix calculations.
Il
lU
IV
Glohill cosu
u .
Ill
1 0.4. 1 Benefits
Figure 10-6.
Figure 10-7 shows ali paired comparisons and the resulting priority weights for the benefits. Next the alternatives may be compared by combining the se priority weights, working from the lowest leve! (of subattributes) to the highest levet (focused objective). Figure 10-8 shows this in a format analogous to Table 10-5. The resulting weights for_benefits of the two alternatives are:* Traditional : FMS:
0 .216 O.784
-{
(better)
• Priority weights for FMS example calculated using a computer program which sornetimes results in answers slightly different from that if the approximation formulas ittustrated earlier were used .
Benefits and costs heirarchies for FMS example.
1 0.4.2. Costs Figure 10-9 shows ali paired comparisons and the resulting priority weights for the costs (for which the hierarchy was shown in the lower part of Fig. 10-6) . When making paired comparisons involving costs, one can take a higher cost to be "more important" and thus to have a preference number greaterthaï1-T . ·Frnally, Table 10-6 shows the combination of priority weigï-iTs--rorëomparison of alternative costs. The resulting weights for costs of the two alternatives are: Traditional:
0.277
FMS:
0.723 .
(better)
co 1
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w /
271
N
;::j
Leve/ With Respect to :
Il.
Strategie
Global Benefits Strategie Operational
Ill.
Operational
0.25 0.75
1
3
Physical lnventory
Quality Traditional FMS
IV. IV. IV. IV. IV. Key:
Quality
Delivery Schedule Performance · Flexibility
i
1 5 5
1
5 3
Traditional
FMS
1 5
1
Traditional FMS IV.
0.167 0.833
1
Physical lnventory
Operation al
Priority Weight
5
Physical inventory Quality Delivery schedule performance Flexibility Shop floor information
IV.
Mfg. Excellence
Market Response
Strategie Market res panse Manufacturing excellence
Ill.
Priority Weight
i.
i
Priority Weight
1
t
3 1
4 3
4 5
0.045 0.438 0.279
! !
1 i
1
f
4 1
0.162 0.076
Priority Weight
0.167 0.833
Traditional
FMS
Priority Weight
1 3
1
0.25 0.75
1
i
Shop Floor Information
Delivery schedule performance (sa me as quality matrix) Flexibility (same as physical inventory matrix) Shop floor information (same as quality matrix) Market response (sa me as physical inventory matrix) Manufacturing excellence (sa me as physical inventory matrix) in upper left-hand corner of the table~ each matrix is shawn at the leve! at which paired comparisons are being made, and with respect ta what element at the next higher level. For example. "li-global benefits" means that the comparisons are (between attributes) at leve! li with respect ta global benefits (at !eveil).
Figure 10-7. Summary of paired comparison and resulting priority weights for benefitsFMS example.
~
.... ~
"' tL. ' 8
_74
Multiattribute Decision Analysis
Strategie Subattribute Marketing Response
Manufacturing Excellence
Weight
0.167
0.833
Alternative Traditional FMS
0.167 0.833
0.167 0.833
Chap. JO
Sec. 10.4
Leve/ With Respect to:
Alternative Priority Weight Il.
Operational Subattribute
Weight
Ill.
Physical lnventory
Ouality
0.045
0.438
0.279
0.162
0.076
0.167 0.833
0.25 0.75
0.25 0.75
0.167 0.833
0.25 0.75
Flexibility
Shop F/oor Information
Alternative Priority Weight Ill.
Strategie Weight
Ope rational
0.25
0.75
Traditional FMS
0. 167 0.833
0.233 0.767
,
l
1
FMS
Priority Weighl
1
0.33 0.67
2
Traditional
0.618 0.297 0.086
FMS
Priority Weight
1
0.20 0.80
4
Required Technical Support Traditional · FMS
\
Traditional
FMS
Priority Weight
1
5
0.167 0.833
Figure 10-9. Summary of paired comparisons and resulting priorily weighls for costs-FMS example .
Table 10-6 Sut.H•IARY COMBINATION OF PRtORITY WEIGHTS FOR ATTRIBUTES AND ALTERNATIVES TO DETERMINE PRIORITY WEIGHTS FOR COSTS: FMS EXAMPLE Attribute
Traditiona/ FMS
5 5
l(ey: Same as for Fig. 10-7.
0.2 16 0.784
~ïgure 10-8. Summary combination of priority weights for sub-attributes, attribut es, and a lternatives to determine priority weights for benefits-FMS example.
Weight Alternative
1
Priority Weight
_../
Alternative Priori!y Weight
Alternative
Operating Risk
Required Technical Support
Operating Risk
3
Traditional
Present Worth of Costs
Traditions/ FMS
0.233 0.767 Ill.
Attribute
Global Costs
Traditional FMS
Alternative Traditional FMS
Present Worth of Costs
Present Worth of costs Operating risk Required technical support
0.1 67 0.833
Delivery Schedule Performance
275
FMS Justification Example
Present Worth of Costs
Operating Risk
Required Technical Support
0.618
0.297
0.086
0.33 0.67
0.20 0.80
0.167 0.833
Alternative Priority Weight
After plotting priority weights for benefits versus costs for bath alternatives, the decision was made to adopt the FMS in this particular manufacturing company. 1 0.4.3 Added Explanation of FMS Justification Example
Another way to show the structure of the FMS justification example (originally given in Fig. 10-6) and the results of ali priority weightings is shown in Fig. 10-10. 1 0.4.4 · Sensitivity Analysis for FMS Example
0.277 0.723
Given the results for the above FMS example, one might a,sk how these ratios would change given a change in judgment between attributes. This is an important question because various individuals addressing the same deci-
co '-.1 Ul
Sec . 10.4
277
FMS Justificalion Examp/e Table 10-7 SENSJTIVITY OF GLOBAL 8EN EFITS PRIORITY WEIGHTS (TRADITION AL, FMS) TO CHANGES
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sion problem will have different feelings regarding the importance of one attribute over another. lt is realistic to expect that the management and analysts will have differing or at !east uncertain perceptions and estimates. Therefore, an analyst should be prepared to answer how changes in judgments would affect the decision outcome. From Fig. 10-7 the observation can be made that the "quality" attribute accounts for approximately 33% of the global benefits weighting (i.e ., 0.438 of operationa!.)< 0.75 of global = 0,13). As an example of sensitivity analysis using AI-fi> ·methodology, we will vary the judgment of quality versus physical inventory (the !east welghted attribute) to see what impact that change would have on the global be.!Ù:.fits priority weight. Given the base j4dgment of quality as having a strong preference (scale factor 5) over physical in ven tory, we will vary .this judgment from a weak preference (scale factor 3) through a dominant preference (scale factor 9). Utilizing an interactive computer program,* it is easy to modify a judgment or multiple judgments and recompute the priority weights. In our example, the resultant priority weights for base case , weak , and-dominant preference, respective! y, are shown in Fig. 10-11. lt can be observed from Fig. 10-11 that the priori! y weights do change but not dramatically. This result would appear reasonable given that we did not reorder the attribute preferences but only modified the strength of preference judgment. Table 10-7 displays the global benefits priority weights for each of the three cases. This table demonstrates that the global priorities do not nece ssarily change.given a priority change in the next lower levet. Given the large judgmental variation we would expect the measure of judgmental inconsistency to change. Figure 10-12 demonstrates how judgmental inconsistency (as measured by Saaty's consistency ratio) increases with the decrease in strength of preference of quality over physical inventory . This result is expected, given that quality was consistently preferred over the other auributes in the paired comparisons. Given a 10% upper bound, Fig. 10-12 would indicate that the preference of quality over physical inventory ~hould be greater than approximately 6 to be consistent with the other paired corn• One such program is Expert Choice , avai1able from Decision Support Software , lnc., 1300 Vincent Place, McLean, VA 22101.
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280
Multiattribute Decision Analysis
Clzap . JO
REFERENCES
Clzap. JO
281
E.xercises
X : Upgrade existing machine Y: New semiautomatic machine
1. Saaty, Thomas L., Tlze Ana/ytic Hierarclzy Process , McGraw-Hill Book Company. New York, 1980. 2. Saaty, Thomas L., Decision Making for Leaders, Wadsworth Publishing Company, !ne., Belmont, Calif., 1982.
Z : Fu lly automatic machine The attributes important to the decision are: N: Net present worth R: Risk F: Future capability
EXERCISES
M : Management requirements
10-1. Select a class or type of decision problem in your working or persona! !ife involving four or more important attributes (ignore subattributes) and three or more mutually exclusive alternatives. Describe as weil as you need to facilitate analysis by AHP method . · (a) Calculate priority weights for the alternatives. (b) Calculate the consistency ratio for each preference matrix. How conclusive are the results? 10·2. Repeat Exercise 10-1 except analyze decision problem of a "client," who may be a manager, engineer, fellow student, or friend. Ask him or her enough to structure the problem as an hierarchy , including, of course, ali relevant pairwise comparisons. 10-3. An engineer expresses his preferences between major attributes with respect to his focus objective of career satisfaction as follows :
Money (Ml
(a) Determine priority weights for each matrix and for the alternatives. (b) Determine consistency ratios for all_gatrices. What are your conclu· sions? 1 (c) Show your results of part (a) in bar graph (histogram) form . lncorporate the results of part (b) in w,hatever way seems useful. Using the classical 1 to 9 scale, the following express degrees of preference, with respect to overall go~l :
N
(F)
(W)
(M)
The comparison matrices are given below:
-
A B
c
4 1
i
c
(W)
7 2
A B
c
1
3 1
6
7
5
6 3
R F 1 M
1
R: risk
N : net present worth
His evaluation of each of job types A, B, and C are expressed as follows with respect toM, W, and F : B
F
With respect to:
Work (W)
A
R
2
Family (F)
(M)
M
N
A
2
B
c
(F)
A B
c
A
B
2
c 5 2
x
x
y
z
1
1
; !
y
-
z
1
1
1
x y
z
x
y
z
1
l
1 6
1
1
With respect to: M: management requirements
F: future capability (a) Fi nd the priority weights for ali matrices, and for the job types . (b) Find consistency ratios for ali matrices. What are your conclusions from this and part (a)? 10-4. It is desired to determine which of three alternatives is best for a high-volume coil winding operation using the AHP mode!. The alternatives are:
1
x y
z
x
y
z
1
1
; !
1 1
1
x y
z
1
z
x
y
1
l
1
1
5 (X)
"-J (X)
• 2R2
Multiallrihute Decision Ana/ysis
Chap . 10
10-5. You have just been transferred! Your company has oiTered two alternatives for a new position with them, and you decide to stay with your present employer. Use AHP to select your new job.
Alternative
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Attribute
Dry Gulch
Rapid Creek
Salary Proximity lo relatives Promotion potenlial Commuting time (per day)
$40,000 2000 miles Fair 40 minutes
$35,000 500 miles Good 90 minutes
10-6. Select a persona! decision prob1em to which you can relate fairly readily (such as alternative jobs. cars, housing, spouse, etc.) and for which you have not previously calculated attribute weights using the AHP methodology. Identify two or three alternatives that might be compared (for reference). Then identify four most important attributes which are, in your mind, reasonably distinct and independenl. Rank order the attributes, placing the most important first, and so on. (a) Weight the attributes and normalize total to 100 points as exemplified by the traditional weighted evaluation method. Do a few checks for consistency until you are satisfied that those weights reasonably reflect your approximate weights. Do not alter based on any later thinking or results. (b) Weighl the attributes by use of the AHP method of pairwise comparisons based on Saaty's ratio preference/importance factors ofl to 9. Calculate your C.R. (consistency ratio). If it is below 0.10, great-otherwise, go back and re-ask yourself sorne or ali pairwise comparison questions and recalculate your weights until the C.R. is sO. IO. (c) Repeat part (b) except use Canada's ratio preference/importance factors of 1 to 3. The following reflects scales for parts (b) and (c) Saaty's Factors If x(row) is . . as y(column) equally important/ preferred weakly more importa nt/ preferred strongly more important/preferred very strongly more importantlpreferred absolutely more important/preferred
Appendix 10-A
Chap. 10
appendix 10-A: consistency ratio (C.R.)
The consistency ratio (C .R .) is an approximate mathematical indicator , or guide, of the consistency cif pairwise comparisons. lt is a function of what is called the "maximum eigenvalue" and size of the matrix (called a " consistency index") which is then compared against similar values if the pairwise comparisons had been mere! y random (called a "random index") . If the ratio of the consistency index to the random index (called a "consistency ratio") is no greater than 0.1, Saaty suggests the consistency is generally quite acceptable for pragmatic purposes . First multiply the mat rix of pairwise comparisons (in Table 10-1 ), cali it matrix [A] by the principal vector or priority weights (right-hand column of Table 10-2) [B] to gel a new vector [C]. [Cl [A]
[B~
liNo.
No.
1 3 5 7
1 A = 0.33 ! = 0.20 + = 0. 14
9
' = 0. 11
1 1.5 2 2.5 3
1
0.67 0.50 0.40 0.33
(d) What can you conclude regarding weighting results using Saaty's versus · Canada's factors regarding closeness to "traditional" (in part (a))?
1.605
0.489
2.732
0.33
5
6
3
1
6
7 6
0.20
0.17
1
3
0.17
0.14
0.33
1 0.25
0.041
0.212
0.20
0. 17
1
4
1
0.096
0.487,
x 0.086 = 0.446
1
Next, di vide each element in vector [ C] by its corresponding element D]
= [1.605 ~
[
0.288
= [5 .57
0.489 5.58
0.446 0.212 0.086 0.041
5.19 5.17
0.487] 0.096
5.07]
Now, average the numbers in vector [D] . This is an approximation of what is called the "maximum eigenvalue," denoted Àm,. : - 5.57 + 5.58 + 5.19 + 5.17 + 5.07- 5
Àmax-
liNo .
,. = t= .... = A=
0.28
in vector [B] to find a new vector [D] .
Canada's Factors
No.
5 1
5
-
.32
The consistency index (Cl) for a matrix of size N is given by the formula CI =
N _ 5.32 - 5 N - 1 - 5 _ 1 = 0.08
Àmax -
Random indexes (RI) for various matrix sizes, N , have been approximated by Saaty (based on large numbers of simulation runs) as
CIJ '-1 I.D
,1 2!14
Multiattribute Decision Ana/ysis 4
5
6
7
8
9
10
0.90
1.12
1.24
1.32
1.41
1.45
1.49
N
RI 1 0.00
0.00
0.58
Chap. JO 11 ·
1.51
For the example above, the RI = 1.12. The consistency ratio (C.R.) can now be calculated using the relationship C.R. = CI _ 0.08 RI- 1.12
= 0.07
Based on Saaty's empirical suggestion thal a C.R. = 0. 10 is acceptable, we would conclude thal the foregoing pairwise comparisons to obtain attrihute wcights arc reasonably consistent. Calculation for Alternative Comparisons in Table 10·3 The calculation of a consistency ratio for alternatives is directly parailei to thal for attributes as given above. For the pairwise comparisons in Table 10-3, the following calculations are shown without explanation. [B]
[A]
1 0.33 1
'] 2
1 0.50
1
3
[
[D
]
Àmu
=
0.24
1.66 0.73] 0.55 0.24
=
= 3.00 +
3.02 + 3.04 3
= 3•02
Cl=
;;·
~
[0.63] 1.66 0.73
0.63 [ 0.21
=
p,
J.' '
[Cl
[0.21]
x 0.55
1
[3 00 3 02 · ·
3 · 04]
1 ·1, ·sr.
•:';-
~~:
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3- 1
CI C.R. = RI
0.01 ~ 0.02
= 0.58
co co
0
9-1
CHAPITRE9 ELEMENTS DE LA CONCEPTION D'UN PROJET La conception, la construction et l'opération des ateliers de fabrication constituent l'une des principales responsabilités d'un ingénieur. Pour remplir cette responsabilité, il doit estimer les événements futurs et les conséquences des actions prises immédiatement. Par exemple, il devra estimer les revenus annuels, les coûts initiaux et récurrents, la durée de vie du projet et les valeurs de récupération des bâtiments et équipements. La théorie des probabilités et les modèles mathématiques présentés dans la documentation, précisent les procédures à suivre pour l'estimation ainsi que le niveau de précision des valeurs obtenues. Lorsque l'analyste utilise des informations plus precises et qu'il se sert de modèles mathématiques plus raffinés, il devrait obtenir des estimations plus précises. Mais l'obtention d'intrants plus précis entraîne des coûts supplémentaires pour l'analyse. Ostwald [1] a mis en graphique, ce concept; la figure 9-1 montre la relation entre le coût de l'analyse et le niveau de détail des intrants.
Coût$
Coût de
Coût de
l'estimation
l'erreur Niveau de précision
Figure 9-1 Relation entre le coût d'estimation et le niveau de précision atteint.
9-2
Le coût total d'une estimation devient la somme du coût de l'estimation et de celui qui sera encouru par l'erreur dans l'estimation. Le coût total atteint une valeur minimale pour un certain niveau de précision obtenu. Ce concept est tout à fait réaliste mais il devient en pratique difficile à utiliser. CLASSES D'ESTIMATION
L'American Association of Cost Engineers propose cinq (5) classes d'estimation et précise leur niveau de précision. 1.
Estimation ordre de grandeur: basée sur des coûts similaires encourus précédemment et sur l'expérience et le jugement de l'analyste. Elle coûte généralement moins de 0,2% du coût du projet et sert principalement à faire le tamisage de certains projets ou variantes d'un projet. Une telle estimation a une précision d'environ " 30% à " 50%.
2.
Étude estimative: basée sur l'évaluation du coût des principales pièces d'équipement; elle a une précision probable de " 30%.
3.
Estimation préliminaire: basée sur des méthodes approximatives et des coûts grossiers, certains éléments de coûts sont estimés individuellement et des spécifications de génie considérés. L'analyste inclut peu de détails et garde ses calculs au minimum. Cette estimation a comme objectif principal de déterminer si l'un doit effectuer un travail supplémentaire dans l'étude du projet proposé. Une telle estimation coûte environ 1,5% du coût du projet et a généralement une précision de" 20%.
4.
Estimation détaillée: basée sur la détermination détaillée des coûts et sur une analyse du potentiel de profit. En revanche, l'analyste ne donne pas les spécifications exactes des équipements et minimise le nombre de plans. Elle a une précision probable de ± 10% du coût du projet.
5.
Estimation du contracteur: basée sur les coûts attribués à chaque composante du projet. Les plans et spécifications doivent être établis et étudiés soigneusement, les quantités requises, les prix de vente ou d'achat, les coûts d'installation, la disponibilité et le coût de la main d'œuvre, les soumissions des fournisseurs, les coûts de la matière première, etc.. . doivent être vérifiés soigneusement. L'analyste fait parfois l'erreur d'omettre certains éléments de coût et sous évalue des quantités, des pertes et de la main d' œuvre: ce type d'estimation a une précision de l'ordre de " 5% et peut coûter environ 6% du coût du projet.
La section précédente fait voir que le ruveau de détail et de précision peut varier
9-3
considérablement d'un type d'estimation à l'autre. Selon De Garmo et al. [2] et White et al. [3], il varie avec les facteurs suivants: •
degré de facilité d'estimer les différents paramètres;
•
méthodes et techniques utilisées;
•
expérience et compétence des analystes;
•
temps et moyen fmancier disponibles par rapport à l'importance· du projet;
•
sensibilité des résultats de l'étude en rapport avec une estimation particulière;
L'analyste doit réaliser qu'une estimation plus détaillée nécessite des coûts plus élevés tels qu'illustrés dans la figure 9-1. Les cinq (5) types d'évaluation vont engendrer plus ou moins d'erreurs; les plus sophistiqués devraient minimiser les erreurs; d'estimation et de plus fournir une meilleure information sur le niveau d'erreur anticipé.
SOURCES DE DONNÉES
Pour effectuer les estimations, l'analyste doit faire appel aux données disponibles dans son organisation et dans la documentation. La section suivante présente les principales sources de références possiblement disponibles.
a)
Les dossiers et documents comptables Un système comptable a comme un des objectifs principaux de déterminer les coûts unitaires des matières premières, de la main d'œuvres directe et des frais généraux découlant de la fabrication d'un produit ou de l'offre d'un service. Les documents et dossiers comptables contiennent donc un grand nombre de données pouvant servir à l'estimation, mais elles doivent cependant être utilisées avec beaucoup de circonspection. En effet, le système comptable doit répondre à des conventions ainsi qu'à d'autres objectifs de l'entreprise et il arrive fréquemment que ces données ne reflètent pas la vraie situation économique devant être prise en considération ou encore qu'elles ne soient pas regroupées ou présentées de façon à en permettre l'extraction des renseignements recherchés. Par exemple, certains actifs pourront être sous-évalués ou surévalués; certains actifs poùrront être amortis sur des périodes plus courtes que la durée de leurs vies réelles et les frais généraux pourront être imputés de façon arbitraire à certains produits ou services sans lien direct avec l'origine réelle de ces frais. De plus, les documents comptables ne font généralement pas état des coûts incrémentiels ou différentiels, ni des coûts d'opportunité.
9-4
b)
Autres sources internes dans l'entreprise Plusieurs unités ou services de l'entreprise préparent des documents ou dossiers pouvant contenir des renseignements utiles; notamment, le département des ventes ou du marketing, ainsi que les services du personnel, de planification et de contrôle de la production, du contrôle de la qualité, des achats, du contrôle des stocks et de l'expédition, d'ingénierie d'usine et du génie industriel. De plus, les personnes impliquées peuvent souvent faire part de sources additionnelles de renseignements.
c)
Sources externes à l'entreprise Il existe un très grand nombre de sources de renseignements et même des banques d'information qui peuvent s'avérer utiles. Ces sources peuvent être classées comme suit: i)
Publications diverses Les ouvrages de références, répertoires techniques, revues spécialisées, volumes, publications du gouvernement ou des associations de manufacturiers donnent accès à une foule de renseignements. Les publications du Bureau fédéral de la statistique, du Bureau de la statistique du Québec, dont l'Annuaire du Québec et de différents ministères comme les ministères d'industrie et du commerce provincial et fédéral, Approvisionnement et Services de plusieurs organismes gouvernementaux comme le Centre de Recherche Industriel du Québec, le Service d'informations Techniques du Conseil de Recherche en Sciences Naturelles et en Génie du Canada peuvent être très utiles. Ces deux derniers organismes ont aussi des banques d'information à la disposition des entreprises. De plus, d'autres sources comme «News Front » et « Dunn and Bradstreet » aux États-Unis compilent et publient les données provenant des états financiers des principales entreprises américaines. Beaucoup de ces renseignements sont compilés pour un secteur industriel donné, ce qui en rend l'utilisation plus facile.
ii)
Contacts personnels Les représentants techniques, vendeurs, banques, commissaires industriels, services gouvernementaux, chambres de commerce et même, dans certains cas, les compétiteurs peuvent fournir des renseignements utiles.
9-5
iii)
Indices de coûts et de production
iv)
Ces indices permettent de convertir des coûts historiques en coûts actuels. Le chapitre 16 présente plus en détails ces indices. Consultants Si les renseignements désirés ne sont pas disponibles ou qu'ils ne sont pas suffisamment précis, l'analyste peut faire appel à des bureaux ou des agences spécialisées pour obtenir une analyse du marché ou une analyse technique permettant de générer les estimations requises. Dans certains cas, l'obtention des renseignements peut nécessiter la construction d'un prototype, ou d'une usine-pilote ou le lancement-pilote d'un nouveau produit ou service.
9-6
ESTIMATION DES COÛTS
De Garmo et al. [2] présentent trois (3) approches pour estimer les coûts:
•
une conférence de personnes qui devraient avoir de bonnes informations ou une base nécessaire pour estimer les coûts en question. La technique Delphi fait partie de cette catégorie;
• Une comparaison avec une situation similaire déjà connue qui permet de faire une extrapolation pour la situation sous étude;
• Des techniques quantitatives telles les indices de coûts, les facteurs de capacité, le chapitre 16 explicite certaines méthodes. Avant d'aborder en détails l'estimation préliminaire, les auteurs décrivent la technique Delphi. La technique Delphi
Gordon et Helmer [4] de la Société Rand ont été les premiers à présenter la technique Delphi. Cette technique considère l'opinion d'un groupe de personnes sur un sujet donné par l'utilisation de plusieurs questionnaires successifs contenant approximativement les mêmes questions. Les personnes impliquées doivent avoir une bonne connaissance du sujet et leurs réponses et commentaires demeurent anonymes durant l'ensemble de l'évaluation. L'objectif de l'utilisation de cette technique est d'arriver à une prédiction ou une estimation en influençant les points de vue de chacun par celui des autres. Elle évite les confrontations face à face et fournit des arguments anonymes pour la défense des opinions. Dans un premier questionnaire, les participants doivent exprimer leurs opinions ainsi que les justifications. Les résultats sont compilés, la moyenne et les quantiles inférieurs et supérieurs c.alculés. L'analyste communique ces informations à chaque participant et leur demande de reconsidérer leur opinion. Ceux dont l'estimation se situe à l'extérieur de la plage inter quantile doivent expliquer brièvement les raisons justificatrices. L'analyste compile de nouveau les résultats de ce deuxième questionnaire. Il envoie la nouvelle moyenne et les quantiles aux participants; normalement, l'étendue des estimés diminue à chaque évaluation. Ce procédé est répété autant de fois que jugé nécessaire pour obtenir l'estimation recherchée. Cette technique Delphi a comme avantages : •
fournir la précision d'un texte écrit;
9-7
•
permettre aux participants de reconsidérer leurs opinions suite à la connaissance de celles des autres;
•
éliminer l'influence parfois dominante de certains participants;
•
permettre à des individus qui ne peuvent se rencontrer en un temps et un lieu donnés de participer à une même évaluation;
•
éliminer les barrières psychologiques de la communication dans plusieurs situations réelles.
En revanche, cette technique est intuitive, non quantitative et l'analyste intermédiaire joue un rôle très important.
EXEMPLE9-1 L'Université du Québec est entrain de préparer son plan quinquennal. Pour équilibrer son budget, elle doit obtenir des estimations de clientèle dans ses divers programmes. Comme beaucoup de divergence existe sur l'estimation de sa clientèle en ingénierie pour 1999, elle décide d'utiliser la technique Delphi. Le secrétaire de l'Université envoie un questionnaire aux dix cadres et administrateurs qui sont directement impliqués avec la gestion des quatre (4) programmes de premier cycle en génie. La question s'intitule « Estimer au meilleur de votre connaissance le nombre d'étudiantes et d'étudiants équivalents temps complet que l'Université aura dans ses programmes de premier cycle en génie en 1999; justifier brièvement votre estimation ». Les réponses· doivent être retournées dans une enveloppe scellée au secrétaire qui garantit l'anonymat des réponses de chaque répondant. La figure 9-2a illustre les réponses compilées par le secrétaire; il a calculé la moyenne et les quantiles inférieure et supérieur. Par exemple, le répondant no 10 justifie une clientèle de 600 par une plus grande attraction d'étudiantes et de personnes provenant de la région due à une augmentation importante prévue des frais de scolarité. Pour sa part, le répondant no 1 justifie une faible clientèle par l'ouverture de nouveaux programmes dans d'autres universités. Le secrétaire transmet les résultats tels que montrés sur la figure 9-2a aux dix participants avec les raisons invoquées par chacun d'eux pour justifier son estimation. Il prend bien soin de toujours conserver l'anonymat des réponses des répondants. Maintenant, il demande aux répondants de faire une autre estimation à la lumière des nouvelles informations disponibles. Plus particulièrement, il demande à ceux qui ont des estimations à l'extérieur des quantiles de bien justifier leur choix. De nouveau, le secrétaire reçoit les réponses de ce deuxième questionnaire et en fait la compilation. La figure 9-2b montre les résultats.
9-8
Le secrétaire décide de répéter l'opération une troisième fois; la figure 9-2c illustre les résultats. Il considère cette estimation moyenne de suffisamment représentative et l'utilisera dans la préparation du plan quinquennal de l'Université. a)
Résultat du premier questionnaire
3
•
•
200
300
.;
6
5
ï
s
roo
9
10
500
600
•
•
Moyenne
397 Nombre d"étudiants
b)
Résultat du deuxième questionnaire
•
•
200
300
1
2
435
6789
•
·.~~
600
Moyenne 403
c)
Résultat du troisième questionnaire
1
•
•
200
300
2
4 3
6 5 ï
8
9
·~
•
10
500
Moyenne 401
Figure 9-2 Illustration de la méthode Delphi.
• 600
9-9
ESTIMATION PRÉLIMINAIRE L'estimation préliminaire est celle qui est le plus souvent utilisée par les ingénieurs qui doivent vérifier le potentiel d'un nouvel équipement ou d'un nouveau procédé avant d'en confier la réalisation à un bureau interne ou externe à la compagnie. La section suivante présente en détails, la méthodologie souvent employée pour faire une estimation préliminaire. Finalement quelques informations sur l'estimation détaillée complète ce chapitre. L'estimation préliminaire a comme principal objectif d'éliminer les projets indésirables avant d'encourir des coûts élevés et des pertes de temps importantes. La figure 9-3 illustre la méthodologie d'une estimation préliminaire.
9- 10
Idée suggestion
1
-
OUI
l
'
Établir la base de projet
Faisabilité économique
c
~Infonnations/
1
-
\7
::::1
NON
j 1Recommander un hrograrnme1 de recherc e
1
Diagramme f4 F1 d'écoulement simplifié
B
~ '
Faisabilité technique
E
But du projet
•
OUI
F21~d'écoulement Diagramme i~ simplifi
'
Résultats
'
~
J
'
/
'•
1
'
Comparaison de différents procédés ...... et élimination des procédés non compétitifs G
.....
NON
Recommander un programme D de recherche révisé
1 Diagramme F3 d'écoulement simplifié .
•
1
"\fonnatio~ suffisantes
,,
D
'
Résultats
H Calcul plus détaillé des procédés compétitifs j 1
I Spécification de l'équipement -
, 14J K
L
Établir les services nécessaires
' '
Rentabilité de chaque procédé
1
...
Donner de la documentation et de l'expertise Usine pilote J f CoOt d'investissement
'
Coût de fonctionnement
J
Écrire un rap~ort et les recommandations 1 pour la p ase de développement
Figure 9.3 : Méthodologie d'une estimation préliminaire
1
1
9-11
A)
IDÉE OU SUGGESTION Le projet peut provenir d'une idée pour réaliser un appareil ou un produit devant servir à une application spécifique. Cette idée origine parfois du département des ventes qui pour répondre à une demande d'un client désirerait avoir un produit de spécification donnée. Elle émane aussi souvent du département de recherche et de développement où les travaux en cours ont permis de mettre au point un équipement ou un produit qui pourrait satisfaire les besoins de la société. Le management peut aussi vouloir développer un nouveau produit ou modifier un produit pour élargir le marché de la compagme.
B)
BUT DU PROJET Indépendamment de l'origine de l'idée si elle est retenue, l'analyste doit préciser les objectifs pour la fabrication du produit et entreprendre une étude de faisabilité.
C)
ÉTUDE DE FAISABILITÉ L'analyste doit alors déterminer si techniquement et économiquement le projet peut être réalisée. Ce type d'étude fournit une indication sur les probabilités de succès du projet et des renseignements sur les informations nécessaires pour compléter l'évaluation. Peters et Timrnerhaus [5] proposent de considérer les quinze éléments suivants pour effectuer l'analyse de faisabilité. •
Matières premières (disponibilité, quantité, qualité et coût).
•
La cinétique et thermodynamique des réactions (équilibre, rendement, taux).
•
Services et équipements disponibles.
•
Services et équipements devant être achetés.
•
Estimation des coûts de production et de l'investissement en capital total.
•
Potentiels de profit (retour sur l'investissement).
•
Matériaux de construction.
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D)
•
Considérations sécuritaires.
•
Étude de marché (présent et futur, plage des prix, nombre de clients).
•
Étude sur les compétiteurs (statistiques de production, procédés utilisés, spécification du produit).
•
Spécifications et propriétés des produits.
•
Vente et service (modes de distribution, publicité, service technique requis).
•
Transport et livraison des produits.
•
Localisation de l'usine.
•
Brevet et restrictions légales.
INFORMATIONS ET RECHERCHE SUPPLÉMENTAIRE Si les informations pour établir la base du projet sont insuffisantes, un programme de recherche doit être initié pour les obtenir, soit en laboratoire, soit en usine pilote dépendamment des infom1ations recherchées et des coûts requis pour les obtenir. Lorsque l'analyste a obtenu suffisan1ment d'information, il peut s'attaquer à la conception préliminaire de chacun des procédés pouvant fournir le produit selon les normes désirées.
E)
BASE DU PROJET DE CONCEPTION L'analyste doit fixer en premier la base pour le projet de conception. Il doit établir les spécifications pour le produit, les facteurs reliés au fonctionnement du procédé tels le nombre de jours par année, les caractéristiques des services requis tels la pression de la vapeur nécessaire, la température de l'eau de refroidissement et le nombre d'unités à produire.
F)
DIAGRAMME D'ÉCOULEMENT DES PROCÉDÉS L'analyste doit déterminer les opérations unitaires nécessaires et préparer un diagramme d'écoulement simplifié pour chacun des procédés susceptibles de fournir le produit ou le service désiré. li doit déterminer à l'aide d'un bilan de matière et d'énergie, les débits, les compositions, les plages de températures et de pressions sur chacune des conduites de chacun des procédés. Il doit aussi préciser les autres éléments pertinents tels les rendements, les taux de réaction, les cycles de production, les enthalpies des vapeurs.
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Le choix des pièces d'équipements permet d'estimer les différents services requis ainsi que la main d'œuvre nécessaire à leur fonctionnement. Le choix des températures, des pressions et de la composition des divers débits permet de sélectionner les matériaux appropriés. Cette étape permet généralement d'éliminer quelques procédés qm souffrent de lacunes importantes. Les procédés qui ne montrent pas de lacunes évidentes doivent faire l'objet d'une étude comparative plus détaillée.
G)
COMPARAISON DES DIFFÉRENTS PROCÉDÉS Le choix du procédé le plus intéressant pour fournir le produit désiré peut être fait à partir d'une estimation détaillée mais souvent une simple comparaison de plusieurs facteurs permet d'en éliminer un ou deux à un coût bien inférieur.
Peters et Timmerhaus [5] proposent d'effectuer la comparaison à partir des huit éléments suivants: •
Facteurs techniques ·flexibilité du procédé, opération continue ou discontinue, régulations spéciales, rendement commercial, difficultés techniques prévisibles, consommation d'énergie, service auxiliaire requis, possibilité de développements futurs, problèmes potentiels de santé et sécurité,
•
Matières premières disponibilité présente et future, traitement nécessaire, entreposage requis, problèmes de manipulation,
•
Perte de produit et de sous-produit quantité des pertes, valeur des pertes, potentiel de vente et d'utilisation, méthodes d'élimination,
9-14
considérations environnementales, •
Equipement disponibilité, matériaux de construction, coûts initiaux, coûts d'installation et d'entretien, remplacement requis, équipement de conception spéciale,
•
Localisation de l'usine surface de terrain requise, facilité de transport, proximité des marchés et des sources de matières premières, disponibilité des services et de l'énergie, disponibilité de la main d'oeuvre, climat, restrictions légales et taxes,
•
Coûts matières premières, énergie, dépréciation, autres charges fixes, fonctionnement et administration, main d'œuvre spécialisée, estimation de la durée réelle de la vie, droits des licences et royautés, régulation environnementale,
H)
•
Facteur de temps date de la finition du projet, développement requis, opportunité des marchés, valeur de l'argent,
•
Considérations des procédés technologie disponible, matières premières communes à d'autres procédés, ligne de produits de la compagnie, objectif général de la compagnie.
CALCUL PLUS DÉTAILLÉ DES PROCÉDÉS COMPÉTITIFS
9-15
L'estimation préliminaire complète des procédés potentiellement compétitifs requiert la considération de plusieurs sujets. En plus des considérations fondamentales en génie et en économie, Peters et Timmerhaus [5] suggèrent de considérer les dix éléments suivants: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 1.
localisation de l'usine; disposition d'écoulement de l'usine; fonctionnement et régulation; services d'utilité; fondations et structures; entreposage; manutention des matériaux; disposition des déchets; santé et sécurité; brevets.
LOCALISATION DE L'USINE La localisation d'une usine peut influer considérablement sur les coûts de production et sur l'efficacité du département des ventes donc sur son niveau de rentabilité. L'objectif de l'analyse pour la localisation d'une usine est de trouver l'endroit qui permettra à la firme de maximiser ses profits ou en d'autres mots, de minimiser ses coûts de fonctionnement et de distribution. Garrett et Sllver [6] suggèrent de faire deux analyses différentes; l'une pour les facteurs qui ont un impact mesurable en terme de coût, l'autre pour les facteurs qualitatifs. Plusieurs auteurs notamment Aries [7] présentent une liste exhaustive des facteurs à considérer pour déterminer la meilleure localisation de l'usine. Souvent, l'analyste trouvera que plusieurs sites offrent des avantages et inconvénients équivalents. Donc, l'étude se limite parfois à trouver une bonne localisation. Peters et Timmerhaus [5] proposent de considérer les avantages et inconvénients des douze facteurs suivants en se limitant aux quatre premiers pour une analyse préliminaire. Dans ce dernier cas, l'analyste pourra identifier une ou deux régions, par exemple, au nord et au sud de Montréal. a)
Matière première: une source et idéalement plusieurs sources de matière première est l'un des principaux facteurs qui influencent la sélection d'un site particulièrement pour les types d'industrie qui en utilisent beaucoup. L'analyste doit notamment tenir compte de son prix, de la distance · et de la sûreté d'approvisionnement, de sa pureté et de son transport et entreposage.
b)
Marchés: la proximité des marchés pour la vente des produits finis et semifinis permet de minimiser les coûts et le temps de livraison.
9-16
c)
Énergie: la disponibilité et le prix des principales sources d'énergie requises pour le fonctionnement de l'usine, est un facteur important pour le choix d'un site. Pour des entreprises très énergivores, ce facteur doit recevoir un traitement particulier.
d)
Climat: les conditions climatiques d'une région ont une influence à la fois sur les coûts de construction et sur les coûts de fonctionnement. Le niveau d'élévation au dessus du niveau de la mer, l'humidité, le taux de précipitation, les températures maximales et minimales et les probabilités de catastrophes naturelles sont autant de facteurs à prendre en considération.
e)
Facilités de transport: la disponibilité de divers services de transport ainsi que leurs coûts doivent être pris en considération pour l'acheminement des matières premières, des produits finis ainsi que pour le personnel. Idéalement, des services devraient être disponibles par eau, par air, par route et par rails; au moins deux services devraient être disponibles.
f)
Eau disponible: beaucoup de procédés industriels utilisent encore des quantités importantes d'eau. Les industriels tentent d'en diminuer l'utilisation pour réduire les coûts de traitement requis avant leur rejet dans les cours d'eau. L'analyste doit vérifier les sources d'eau disponibles, leurs températures, leurs duretés et leurs contenus en matières bactériologiques.
g)
Disposition des déchets: la plupart des pays ont des lois pour contrôler le niveau et le type de déversement des usines à la fois dans l'air, dans l'eau et dans le sol. L'industriel doit s'assurer que le site choisi possède des capacités adéquates pour ·l'élimination des déchets; il doit prendre en considération la possibilité que les lois actuelles deviennent plus contraignantes pour leur élimination.
h)
Main d'œuvre: la disponibilité et le coût de la main d'œuvre dans une région donnée doivent être pris en considération. L'analyste doit notamment considérer pour la région choisie, les règlements sur le nombre d'heures de travail par semaine, la situation des entreprises compétitives, les taux d'absentéisme, les lois sur le travail, l'efficacité des travailleurs, le roulement de personnel et l'historique syndical.
i)
Taxes et restrictions légales: l'analyste doit considérer pour chaque localité étudiée, les taux de taxation sur la propriété, les mesures sociales telles l'assurance chômage et les compensations pour invalidité ainsi que les restrictions sur le zonage et les codes pour les bâtiments. Certaines localités offrent des réductions de taxes pour attirer les industries.
9-17 .
j)
Caractéristiques du site: les caractéristiques du sol et la topographie du site doivent être considéré car ils influent directement sur les coûts de construction. L'analyste doit aussi tenir compte du coût d'achat du terrain, des coûts locaux de construction ainsi que des conditions de vie dans la région. L'entreprise doit prévoir des surfaces de terrain supplémentaires en cas d'expansion.
k)
Protection contre les sinistres: l'analyste doit étudier l'historique de la région concernant notamment les tremblements de terre, les ouragans et les inondations. Il doit s'assurer des facilités régionales adéquates de protection contre les incendies.
1)
Facilités communautaires: la région doit être en mesure de fournir les facilités sportives et c ulturelles pour le personnel. Si elles ne sont pas adéquates, l'entreprise doit déjà prendre en considération qu'elle devra investir dans leur développement. Cet élément est très important pour assurer la satisfaction et la stabilité du personnel. Négliger l'importance de ces facilités peut entraîner un roulement excessif du personnel et une diminution de l'efficacité de fonctionnement de l'entreprise.
En résumé, la décision prise pour la localisation de l'usine est vitale pour la conception d'un système de production. Elle influe sur les coûts de transport de la matière première et des produits finis, sur les coûts de fonctionnement incluant la main d'œuvre, les taxes, la construction, le terrain, les services, l'énergie et plusieurs autres facteurs. Cette décision influe aussi sur les expansions futures et l'efficacité de la mise en marché. Elle peut faire la différence entre une entreprise prospère et une faillite, donc le choix doit être fait dans une planification à long terme. 2.
DISPOSITION D'ÉCOULEMENT DE L'USINE L'un des plus importants aspects de la conception d'un système de production après que le diagramme général a été fixé, est l'arrangement des machines, des matériaux, des entrepôts, du personnel et des services, de façon à ce que le système opère efficacement. Les managers trouvent de plus en plus important de développer la disposition d'écoulement avant la construction de l'usine et non l'inverse. L'objectif de la disposition d'écoulement est d'optimiser l'arrangement des machines, du personnel, des matériaux et des services de support à la production de façon à maximiser l'efficacité du système de production et de satisfaire les besoins des travailleurs, du management et des autres personnes associées avec la production. Pour établir une disposition appropriée, Peters et Timmerhaus [5] proposent de considérer les douze facteurs suivants:
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• • • • • • • • • • •
nouvel emplacement ou addition à un atelier déjà existant; type et quantité des produits manufacturés; type de procédé et de contrôle du produit; accessibilité et facilité des opérations; distribution économique des services à la production; type de bâtiments; considération de santé et de sécurité; problème de la disposition des déchets; équipements auxiliaires; espaces disponibles et requis; routes et voies ferrées; expansion future .
Pour sa part, Hopeman [8] suggère de tenir compte immédiatement dans l'étude des principaux changements que les entreprises doivent faire face: changement dans la demande pour le produit, l'ajout de nouveaux produits, les changements dans la conception du produit, la désuétude des machines et des procédés, problèmes de personnel des accidents industriels potentiels et la nécessité de réduire les coûts.
3.
FONCTIONNEMENT ET RÉGULATION Les méthodes que l'ingénieur choisit pour le fonctionnement et la régulation d'un procédé·influent directement sur la conception de l'usine et aident à fixer plusieurs des paramètres; par exemple, un procédé très automatisé nécessite moins de main d'œuvre. L'ingénieur doit donc planifier immédiatement lors de la conception de l'usine et l'achat d'équipement, les activités d'entretien car beaucoup de problèmes reliés à l'entretien proviennent d'une mauvaise conception originale. L'entretien peut être défini comme toutes les activités requises pour garder un équipement ou un procédé en condition adéquate de fonctionnement: ses coûts concernent aussi bien la main d'œuvre que les matériaux.
4.
SERVICES D'UTILITÉ Les industriels considèrent généralement comme services d'utilité, la vapeur, l'eau de refroidissement, l'eau dionée, l'énergie électrique, l'huile, le charbon, l'énergie nucléaire et éolienne, la réfrigération, l'air comprimé et le traitement des effluents. L'ingénieur doit détem1iner lors de la conception d'une usine si elle achètera son énergie de sources extérieures ou elle la produira elle-même. Il doit aussi tenir compte de sous-produits énergétiques que peut produire l'usine et prévoir deux sources indépendantes pour pouvoir poursuivre les opérations en cas de problèmes avec l'une d'elles. L'eau peut être obtenue d'un système municipal ou d'une source propre à l'usine. Elle
9-19
doit permettre d'assurer le refroidissement de certains appareils, d'alimenter le procédé et d'assurer différents besoins tels les besoins domestiques et sanitaires. Le renforcement des mesures antipollution qui oblige maintenant les usines à traiter leurs effluents avant leur déversement font en sorte qu'elles jugent souvent économique de minimiser l'utilisation d'eau et d'accroître son recyclage. La disponibilité des divers services d'utilité constitue un élément fondamental dans le choix d'un emplacement pour une usine. On a qu'à penser aux usines de pâtes et papiers qui consomment beaucoup d'eau et sont construites à proximité d'un cours d'eau et aux usines d'aluminium qui sont situées près des sources importantes en énergie électrique.
5.
FONDATIONS ET STRUCTURES Les fondations ont pour but d'assurer la distribution du poids pour éviter les contraintes excessives provenant des équipements lourds et de la vibration, lesquelles pourraient endommager la bâtisse. Le type de fondation dépend du type d'équipement et du procédé utilisés ainsi que des caractéristiques du sol. Par exemple, le rock peut supporter une pression de 3 * 105 kg/rn2 alors que sur la glaise, le pression doit être maintenue à moins de 1 * 104 kg/rn2. Dans les regwns froides, l'entrepreneur doit s'assurer que les fondations sont suffisamment profondes pour ne pas être influencées par la gelée. Les planchers doivent pouvoir résister à la chaleur et aux produits chimiques et ne pas être glissants sous les conditions de fonctionnement de l'usine. L'acier et le ciment demeurent des matériaux de construction très populaires; mais le bois, l'aluminium, la brique et d'autres matériaux sont aussi utilisés. Le choix doit tenir compte des coûts ainsi que des conditions particulières de l'usine.
6.
ENTREPOSAGE La conception d'un atelier de fabrication doit prévoir des facilités d'entreposage pour la matière première pour permettre le fonctionnement même lorsque survietment des problèmes d'approvisionnement. Elle doit aussi prévoir l'entreposage des produits semi-finis pour faire face aux bris de l'équipement et des produits finis pour approvisionner les clients même durant les arrêts de production. Les solides peuvent être entreposés en vrac à l'extérieur ou à l'intérieur dans des silos, des trémies, des magasins d'alimentation, des sacs et des barils.
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Pour l'entreposage des liquides, l'ingénieur choisit le type de contenant en considérant les conditions ambiantes de température, de pression de vapeur du liquide, de la toxicité et du pouvoir corrosif du liquide. Il traite différemment l'entreposage à pression atmosphérique, sous faible pression et à pression élevée. Il existe des régulations strictes pour l'entreposage à haute pression et parfois aussi pour celui à faible pression. L'entreposage des gaz à très faible pression se fait principalement dans des cylindres à gaz dont l'étanchéité peut être assurée à l'état humide par un liquide ou à l'état sec par un rideau de caoutchouc ou ·de plastique. Des réservoirs verticaux ou horizontaux ou spécifiques, servent à l'entreposage des gaz sous pression.
Des lois régissent l'entreposage des produits classés comme dangereux soit qu'ils sont poisons, inflammables, oxydants, corrosifs, explosifs ou sous haute pression.
7.
MANUTENTION DES MATÉRIAUX
Les problèmes reliés à la manutention des matériaux varient considérablement d'un type d'industrie à l'autre. Hopeman [8] suggère quelques principes qui s'appliquent à la plupart des situations: •
les matériaux doivent être déplacés sur les plus courtes distances possibles;
• les temps d'arrêt dans les déplacements doivent être maintenus le plus court possible;
• les unités de transport devraient transporter des matériaux dans les deux directions; •
le remplissage partiel des unités de transport devrait être évité;
•
la manutention manuelle doit être évitée lorsque des moyens mécaniques existent;
•
la gravité est la source d'énergie la moins chère pour le transport;
•
les déplacements en ligne droite doivent être favorisés;
•
les petites unités doivent être regroupées en contenants larges et de grosseur constante;
•
les différents matériaux et produits doivent être clairement identifiés.
9-21
Les unités de manutention peuvent être divisées en cinq classes: les convoyeurs, les grues, les oléoducs, les camions et les unités diverses. Les convoyeurs de différents modèles sont généralement fixés pour déplacer les matériaux d'un point à un autre. Ils ont comme caractéristiques principales: indépendants des travailleurs, assurent un débit régulier, suivent des chemins définis donc ont une flexibilité limitée. Peuvent servir pour l'entreposage temporaire, peuvent maintenir un produit en position et minimisent les pertes de produits. Les grues et les vérins permettent de monter et de descendre les matériaux; ils peuvent parfois aussi se déplacer horizontalement. Une grande variété existe, opérée par l'énergie mécanique, électrique et pneumatique. Ils ont comme avantages principaux de ne pas nécessiter de surface de plancher, d'être plus flexible que les convoyeurs, de permettre de placer des équipements ou des outils lourds et ont un pouvoir très important pour lever les matériaux. Les oléoducs consistent en des tubes fermés reliant deux ou trois points. Ils peuvent être faits de différents matériaux. Ils ont l'avantage de permettre le transport des liquides et des gaz à haute vitesse, à bas coût et à haut degré d'efficacité sans perte de matériaux. Des systèmes de vannes peuvent permettre d'accroître leur flexibilité . L'industrie utilise une très grande variété de camions qu'on divise en trois catégories. Les petits camions ou accompagnés de remorques qui circulent sur les routes et autoroutes. Les camions motorisés qui servent seulement à l'intérieur de l'usine tels les chariots élévateurs, les véhicules autoguidés, les tracteurs, les camions à bascule et les chariots ajustables. Ce sont des modèles compacts et versatiles; certains possèdent deux fourches ou des attaches particulières. Dans la dernière catégorie, on retrouve les véhicules manuels qui se déplacent généralement sur roues; ils coûtent peu et servent pour les travaux intermittents. Finalement, dans la catégorie des unités diverses, on retrouve les monte-charges, la table tournante, la plate-forme hydraulique, les machines pour transférer ou positionner automatiquement des produits, de l'équipement ou des outils. Le choix des unités pour la manutention des matériaux dépend principalement du type de production: continue ou intermittente. Il dépend aussi d'une combinaison de la nature chimique et physique des matériaux à transporter, du type de bâtiments, des distances et des quantités à transporter et du coût des unités de manutention disponibles.
8.
DISPOSITION DES DÉCHETS Les provinces canadiennes assument la responsabilité directe pour le contrôle de la pollution en provenance de sources stationnaires. Cependant, le gouvernement fédéral
9-22
poursuit l'objectif de promouvoir une approche uniforme pour le contrôle de la pollution au pays, et est autorisé à développer et publier des lignes directrices nationales. Les promoteurs d'un projet de construction d'usine doivent fournir au gouvernement provincial pour obtenir un certificat d'autorisation de construction d'une usine, une étude de l'impact que produira sur l'environnement, la réalisation de ce projet. On définit scientifiquement la pollution comme le résidu inutilisable d'un procédé quelconque ou d'opérations manufacturières destinées à la fabrication d'un bien de consommation. Cette pollution peut se retrouver sous les trois états de la matière, soit les états solide, liquide ou gazeux. Les déchets solides d'une usine doivent être éliminés par un système de gestion des déchets approuvé par le ministère de l'environnement. L'élimination peut se faire par recyclage, par pyrolyse ou par brûlage dans un appareil de combustion ou un incinérateur ou par enfouissement dans le sol~ des normes très précises régissent cette dernière possibilité. Une usine ne doit pas rejeter dans l'environnement un effluent contenant des matières en suspension et des matières ayant une demande biochimique en oxygène cinq jours au-delà d'aucune norme établie pour chaque élément de transformation utilisé. Des normes s'appliquant aussi sur le déversement de tout élément polluant tel les produits toxiques, les minéraux, les solutions acidiques ou alcalines, les écumes et la couleur. Le ministère de l'environnement exige aussi les plans et devis de tout dispositif de traitement biologique destiné à purger les eaux de procédé d'une usine existante qui doit faire l'objet d'un certificat d'autorisation s'il est susceptible d'en résulter une émission, un dépôt, un dégagement ou un rejet de contaminant dans l'environnement. La demande de l'industriel doit contenir une évaluation, en kilogrammes par tonne, des matières en suspension, des matières ayant une demande biochimique en oxygène cinq jours, de la couleur et des substances nutritives qu'il prévoit rejeter dans l'environnement après la mise en opération du dispositif de traitement biologique. Les lois fixent aussi des normes précises sur l'émission dans l'atmosphère de matières particulières. Le promoteur d'un projet doit donc considérer très attentivement les différentes sources de pollution de chacun des procédés considérés. Des délais importants dans l'obtention du certificat d'autorisation peuvent survenir si le promoteur n'a pas justifié à la satisfaction du ministère de l'environnement, le respect des normes gouvernementales.
9-23
9.
SANTÉ ET SÉCURITÉ Dans la plupart des pays industrialisés, il existe des lois régissant la sécurité et la santé des travailleurs. Ces lois ont pour but la création de milieux de travail convenables et l'élimination des pratiques et procédures dangereuses. Au Québec, la loi numéro 17 sanctionnée en 1979, régit l'ensemble des activités concernant la santé et sécurité du travail. Elle contient des règlements qui régissent notamment l'état de l'usine, les évacuations et les incendies, l'environnement, le danger des machines, la manutention et le transport du matériel, les espaces pour l'entretien et la réparation, le contrôle des substances dangereuses, l'hygiène et le bien-être des travailleurs, les équipements et la protection individuelle, le transport des travailleurs, les niveaux de radiation, de vibration et de bruit, les poussières, les fumées, les gaz, les contraintes thermiques, les réservoirs sous pression, les produits chimiques et les produits toxiques. L'ingénieur doit s'assurer lors de la conception de l'usine que· chacune des opérations rencontre les normes pour la protection des travailleurs .
JO.
BREVETS Un brevet est défini comme un monopole accordé à son propriétaire par lequel il a le droit exclusif d'exploiter, (de fabriquer, d'importer, d'offrir en vente, de vendre ou d'utiliser) une invention durant tout le terme du brevet. Le Bureau Canadien des Brevets [9] est l'organisme fédéral chargé de la délivrance des brevets au Canada et ce, pour une période maximale de 20 ans suivant la date de dépôt de la demande de brevet. La demande de brevet est rendue publique 18 mois après la date de dépôt. pour être brevetable, l'invention doit remplir trois conditions de base, être: • • •
Nouvelle, c'est-à-dire la première au monde; Utile, c'est-à-dire fonctionnelle et exploitable; Un apport inventif.
Il peut s'agir d'un produit, d'une composition, d'un appareil, d'un procédé ou d'une amélioration d'un de ces éléments. Le guide des brevets [9] résume les étapes à franchir en vue d'obtenir un brevet au Canada. • • • • • •
Trouver un agent de brevet. Effectuer une recherche préliminaire; si l'invention a déjà été brevetée, ne pas poursuivre plus avant. Aider l'agent à rédiger la demande. Déposer la demande. Demander qu'on examine la demande. Rechercher les antériorités et en étudier les revendications.
• • •
Répondre aux objections et aux demandes de l'examinateur. Étudier à nouveau la demande telle que modifiée et l'examinateur l'accepte ou exige d'autres modification. Aller en appel d'un rapport final dans lequel la demande est rejetée.
Un brevet obtenu au Canada ne protège pas une invention dans un autre pays. Pour obtenir une telle protection, il faut déposer une demande dans chacun de ces pays. D'autre part, si votre entreprise désire utiliser un équipement ou un procédé breveté, elle doit conclure une entente écrite avec les propriétaires du brevet. Elle pourrait alors avoir à payer soit un montant global, un versement périodique, un pourcentage des profits, un montant fixe pour chaque unité produite ou tout autre mode accepté par les parties impliquées. En revanche, on peut consulter les brevets canadiens sur place à l'endroit suivant: Place · du portage, rue Victoria, Hull, Québec. Une revue plus détaillée peut être effectuée à partir des nombreuses banques de données; par exemple, à partir du serveur DIALOG, on retrouve la banque INP ADOC qui contient les brevets (modèles déposés) pour toutes les technologies relatives à 56 pays; sur le serveur QUESTEL, on retrouve les banques de données relatives à l'information se rapportant principalement à des brevets européens, et particulièrement des brevets français.
I)
SPÉCIFICATION DE L'ÉQUIPEMENT Pour obtenir un atelier de fabrication qui fonctionne efficacement, l'ingénieur doit être capable d'agencer plusieurs pièces d'équipements. Il doit donner préférence aux pièces standard qui généralement, coûtent moins chers et ont une meilleure garantie. Pour la sélection de l'équipement, l'ingénieur doit en premier, faire une revue de la documentation qui lui procurera de nombreuses informations. Il peut aussi faire appel à l'expertise de d'autres, notamment des fournisseurs d'équipement. Il doit être capable de leur fournir les principales données de conception telles : • • • • • • •
l'identification de l'équipement; la principale utilisation; le mode de fonctionnement; les matériaux à traiter; les régulations et l'isolation nécessaires; les limites de tolérance; les informations particulières, telles les matériaux de construction, le temps de livraison requis, le nombre d'ouverture, le type de support.
En d'autres mots, pour l'estimation préliminaire, l'analyste doit être capable de préciser
la grosseur de l'équipement en termes de volume, débit et surface. Certains types d'appareils nécessitent très souvent des essais en usine pilote, notamment les réacteurs, les cristalliseurs, les épurateurs centrifuges et les filtres à plaques et rotatifs. Il en est de même pour les équipements spéciaux qui peuvent être requis. L'ingénieur doit s'assurer que l'étude en usine pilote respecte les facteurs d'échelle pour les principaux paramètres caractérisant les grandeurs, les débits et les capacités. Peters et Timmerhaus [5] présentent ces facteurs pour certaines pièces d'équipement. L'ingénieur doit aussi s'assurer que les pièces choisies ont un facteur de sécurité . suffisant pour tenir compte des changements avec le temps, des conditions de fonctionnement pouvant provenir, notamment de l'usure et des dépôts de saleté. La documentation suggère des facteurs de sécurité variant généralement entre 10 et 20% pour les équipements standard; le facteur dépend notamment: • • • • • J)
des considérations économiques; de la précision des données et des calculs; du potentiel de changement des opérations; des informations préalables; de l'utilisation d'équipement standard ou non.
SERVICES NÉCESSAIRES Le choix des équipements permet de fixer les divers services requis tels que l'eau, l'électricité et la vapeur. Aries [7] propose les quantités d'électricité, de vapeur et d'eau nécessaire par unité de produit pour plusieurs types de procédés.
K)
RENTABILITÉ DE CHAQUE PROCÉDÉ Le chapitre 3 décrit les principales méthodes d'analyse économique. Le chapitre 4 montre comment faire des choix entre plusieurs projets et fmalement, les chapitres 16 et 17 traitent en détail des diverses méthodes pour estimer le potentiel de rentabilité d'un procédé à partir des coûts d'investissement et des coûts de fonctionnement.
L)
RAPPORT ET RECOMMANDATION Suite à cette estimation préliminaire, l'analyste doit être en mesure de recommander ou non, d'effectuer une estimation détaillée pour chacun des projets qui possèdent un potentiel de profit suffisant pour satisfaire les exigences de la compagnie. La section suivante fait voir différentes infoimations sur les dix éléments proposés par
Peters et Timmerhaus [5] qui pourraient être intégrées aux éléments plus classiques de la conception d'un atelier ou d'une unité de production. ESTIMATION DÉTAILLÉE L'estimation préliminaire devrait avoir permis d'éliminer plusieurs des procédés considérés. Ceux qui ont montré les meilleurs potentiels de profit devront faire l'objet d'une estimation détaillée avant qu'une décision finale soit prise d'investir ou non dans le projet. Cette procédure peut varier d'un analyste à l'autre mais les principes de base sont toujours les mêmes. Grossièrement, on peut dire que cette analyse reprend les diverses étapes de l'estimation préliminaire mais en y ajoutant beaucoup d'information et de détails pertinents. A cette étape, l'analyste doit faire appel à une multitude d'experts notamment des avocats, des architectes et des ingénieurs spécialisés dans différentes disciplines. Ces experts lui permettront de finaliser les dessins et de préciser les coûts d'investissement et de fonctionnement et donc d'obtenir un potentiel de profit plus précis. Suite à cette étude détaillée, si l'analyste recommande d'investir dans un projet, des plans détaillés de construction devront être préparés. Des techniques telles le PERT et le CPM devront être utilisées pour planifier le déroulement des diverses activités reliées à la construction.
PRÉCISIONS DES ESTIMATIONS Nous avons vu précédemment que l'estimation préliminaire et détaillée de la rentabilité d'un projet nécessite l'utilisation de plusieurs valeurs estimatives. Ces dernières doivent souvent être additionnées, soustraites, multipliés et divisées. L'analyste doit donc considérer l'impact de ces opérations sur la précision de l'estimation globale.
RÉFÉRENCES
1.
Ostwald, Phillip F. , "Cost estimating", 2è édit., Prentice-Hall, 1984.
2.
De Garmo, Paul, E. , Sullivan, William, G., Bontadelli, James, A., "Engineering economy", 8è édit., MacMillan Publishing Corn., New York, 1989.
3.
White, John, A., Agee, Marvin, H., Case, Kenneth, E., ?Principes of engineering economie analysis?, 3è édit., John Wiley & Sons, Toronto, 1989.
4.
Gordon, T.J. et Helmer, 0., "Report on a long range forecasting study", Rand Corporation Report, p. 2982, 1964.
5.
Peters, Max S., Timmerhaus Klaus, D., "Plant design and economies for chemical engineers",4è édit., McGraw Hill, Toronto, 1991.
6.
Garrett, Leonard, J. , Silver, Milton, "Production management analysis", Harcourt, Brace & World Inc., New York, 1966.
7.
Aries, Robert, S., Newton, Robert, D., "Chemical engineering cost estimation", McGraw Hill, Toronto, 1955.
8.
Hopeman, Richard, J. "Production, concepts, analysis, controls", Charles E. Merrill Publishing Co., Columbus, Ohio, 1965.
9.
Consommation et corporation Canada, « Le guide des brevets», Ministère des approvisionnements et services Canada, Ottawa, 1991.
Table of Contents
ANNEXEI Manuel d'utilisation du logiciel Expert Choice
Introduction
9
Conventions
9
Getting Started 1 Building a Model in the TreeView
10
ModeiView Overview...... ..... .... ..... ..... ...... .... .. ... ... .... .. ... .. .... .... ......... ......... .. 10 Creating a File and Goal Description ......... ... .. ....... .... .. ................. .. ..... .... ... 10 To crea te a new file: .. .. .. ... ...... .... ........... .. .... .......................... ... .. ... 10 Adding the Objectives and Sub-objectives to ModeiView's TreeView ....... ... 11 To add an objective: ... ..... .... .. ..... ... .... ... ....... ......... .......... .. ............. 11 To add sub-objectives below an Objective: ... ... ....... ... ...... ...... .... .... 11 To add an objective to an existing model: .... :................... .. ............ 11 Renaming Nodes in the ModeiView's TreeView ... ....... ....... .. ..... ........ .... ... ... 11 To rename a node: ... .... .. ............... ... ... ... ............ .. .... .... .. .. .. .......... . 11 Deleting Nod es in the TreeView ...... .... ..... .. ... ..... ...... ............... ... ... ..... ....... . 12 To delete a node and ali of its descendants in the TreeView: ....... .. 12 TrashCan ... .... .. .. ... .... .... ... ... .. ............... ................. ......... ........ .......... ...... .... 12 To Open the TrashCan: ...... ..... ....... ......... ..... .... ............... ....... ....... 12 To move nodes back into the TreeView (hierarchy): ...... .......... ...... . 12 Copy Pl ex and Drag ...... ........... ..... ..... ........ .... ........ ..... ... ..... ..... ......... ..... .... 12 To move a node: .... ...... .... ........... .... ... ... .. .. .. ... ... ..... ... ................ .... 12 Adding Alternatives to the ModeiView..... .. ......... ... .... .. ............ ... ...... ........ ... 13 Renam ing Alternatives in the ModeiView ... ... : ...... ... ... ........... ....... .. .... ....... .. 13 lnactivating and Reactivating Alternatives ...... .... ...... .. ......... .. ..... .. ......... ...... 13 To inactivate alternatives from the Alternatives pane: ..... ... .. ... ........ 13 To inactivate alternatives from the Data Grid:..... .. ...... ... .... ...... ..... .. 13 To reactivate alternatives- only available from the Data Grid: ........ 14 Adding Information Documents .... ......... .. .... ...... ... ...................... ... ....... .... .. 14 Dragging to lmport Files to Information Documents: .... ..... ........ .... .. 15 Adding Notes ......... ........... ........ ...... ... .. ...... ... ....... ...... ... ...... ...... ..... ... ......... 15 Displaying Priorities in Mode1View ....... ... ... .. ... ....... ............ .... .. ........... .. .. .. .. 16 Displaying the Current Nod es Children in the Alternatives Pane .... ... .. .... .... 16 Deleting an Information Document or Note ........ ..... ............. .... ............. .. .... 16 To Undo the Last Pairwise Comparison: .... ........... ... ... .... ... ..... .. ... ......... ... ... 17 To Undo the Last judgment or Data Value Entered in the Data Grid:.. .. ..... .. 17 To Undo- Information Documents or Notes: .. ... .... ............. .. ... ... .. ..... ...... .. .. 17 Revert ...... .. ............... ..... ..... ..... .... .... ...... .... ...... .. ... ...... ..... .. ..... ... ..... ...... .... . 17 Opening a Model ... ...... .... ... ............... .... .................. ... ...... .. .. ....... .. ...... .... ... 17 To open another model: .. ... .... .. ... ... .... ......... ..... .. .. ......... ..... .. ...... .... ... ... .. .... 17 Converting to an lncomplete Hierarchy .. .... .... ...... ...... .. ......... ........ ........... .. . 18 Rollup .... ........ ...... ..... .. .... ... ....... ........ ..... ... .. .. .. ... .... .... .... ...... ... .. .. .... .... ... .. .. 18 Converting from Version 9.5 .. ...... .......... .. ...... ............ .. ... .. ... .... ..... ........ .... .. 19 File Structure ...... .... .. ....... ... ...... ...... .... ... .. .... ........ ... .. ........... ........ .... ..... .. ... 19
Structuring - Another Way to Build A Model
19
Top Down or Bottom Up .... ..... .................. .......... ...... .... .... ........ .. .... ........ ... . 19 First Create a File (Model) and Goal Description ..... .. ..... ...... .... ...... .. .. .... ... . 19 To create a new file: ..... ............ ... .................. .... .. .... .. ................... 19 Ena ble Structuring ........ .. ...... ... ........ ... ....... ... ....... ............ ... ..... ...... .. ......... . 20 For a Newly Created Model with Only a Goal Node: .. ..... ............ .... 20 For Existing Models: .... ................ ................. ...... ............ ....... ..... ... 20
1
Top Down Structuring .. ..... ..... ............. ....... .............. .............. .............. ...... 20 Adding Objectives and Sub-objectives to ClusterView ..... ... .... .......... .... 20 T o add objective (or sub-objective) to the ClusterView: .. .... ... ... .. ........ ......... 20 To group sub-objectives within the ClusterView: ........ ........... ... .... ... ........ .... 21 Moving Nodes in the ClusterView .................. ... .. .. .... .. ............. ..... ... ... .. 21 Deleting Nod es in the ModeiView's ClusterView ...... ... .. ... ...... .. ... ... ....... 21 T o del ete a node and ali of its descendants in the ClusterView: ........... .. .. ... 21 Adding Alternatives in Top Down Structuring .. ....... ............... ... ... ........ .. 21 Finish Top Down Structuring .... ....... .. .... .......... .......... ... ... .. ............. .. .... 21 Bottom Up Structuring ......... .. .......... .. .. ...... ......... .. .... .. .. .... ..... .. ... .. ....... ... ... . 22 Adding Alternatives in Bottom Up Structuring ......... .... ....... ........... ...... .. 22 Adding Pros and Cons for Each Alternative ... ........ ... ... ... .. ........ .. .... ... .. . 22 View the List of Ali Pros and Cons .................. ... .. .. .... ............. ... ....... ... 22 Building the Hierarchy ............. ..... ... ... .. ...... ..................... .. ... ....... ....... .. 23 Finish Button Up Structuring ....... ..... .. ... ..... ....... ... ......... .. ......... ...... ............. 23
Pairwise Comparison Process - Making Judgments
23
Ma king Paired Comparisons ....... ...... ..... ..... ... ...... ........... .. ............. .......... .. 23 Ma king Verbal Judgments ........ .... ........... ... ... ..... .. ............... ..... .... . 24 Making Numerical Judgments ........... .... ... ........... ...... ........... ..... ... .. 24 Making Graphical Judgments ................. ...... ..... .. ...... ... .... .. ........ .. .. 24 Structural Ad just .......... ........ .. ...................... ... ..... ... ..... ...... .... ..... ... 24 Examining and lmproving lnconsistency ... .......... ... ... ... ....... ........ ... 24 Making Verbal Judgments ........... .... ... ........ ...... .... ..... ............ ... ..... ....... ... ... 24 Judgments can be made any of the following ways: .......... ....... ..... . 25 To advance to the next judgment: ........ ... ... .... ..... ...... ....... ........ .. .... 25 To invert a judgment (to select the other element in the comparison)25 To enter judgments directly in the comparison matrix: ... ... .... ..... .. ... 26 Ma king Numerical Judgments ..... .......... ...... .. ....... .. ............. .. ... ... ... .......... .. 26 Judgments can be made any of the following ways: ... ... ........ ...... ... 26 T o advance to the next judgment: .................. .... ....................... ..... 26 To invert a judgment (to select the other element in the comparison)27 To enter judgments directly in the comparison matrix: ........... .... ..... 27 Making Graphical Judgments ... ..... .. ... ...... .. ........................ ................... ... .. 27 Judgments can be made any of the following ways: ..... .... .. ............ 27 To advance to the next judgment: ... ..... ..... ... .................................. 28 To invert a judgment (to select the other bar in the comparison) ... .'. 28 To enter judgments directly in the comparison matrix: ....... ............. 28 Direct Entry of Priorities ....... .... ...... ..... .... ......... ......... ..... .......... ... ......... ...... . 28 To directly assign weights: .... ....... ... .... .. .... ... ... ............................... 28 Ma king Diagonal Pairwise Judgments ......... ............... .......... ... .. ...... ....... .... 29 Making a Factor Dormant from the Pairwise Comparison Matrix ................. 29 To reactivate a factor: ................. ........ .... ..... ............. .... ........ .... ..... 29 Structural Adjust. ..... ....... .. .... ....... .... ... .... .. .. ................ ............ ...... ...... ..... ... 30 Examining and lmproving lnconsistency ......... ..... ..... .... ... .... ... .. .... .. ..... ..... .. 30 To view the most inconsistent judgment: .... .... .. .. ... ... ....... .. ... ... ..... .. 30 To lower the lnconsistency Ratio for a set of judgments you can either: Understanding lnconsistency ................ ..... ....... ...... ..... ........ .................... .. 31 lmporting & Exporting to/from any Pairwise Window .. ... .. .......... ... ..... .. ........ 32 lmporting Only the Upper Portion of the Matrix ............... .. ....... .. ....... .. ... ..... 33 lmporting a Row Vector ......... .. .. .... ..... ... .......... .. ... .. .. .............. .. ........... .. ..... 33 lmporting a Column Vector ...... .. ............. .. ......... ......... ...................... ..... ..... 33
Synthesis
33
Synthesis Overview..... .... .. ... ....... ....... ........... ... .... ............. ..... ............... ..... 33 Synthesizing Group Judgments .... .. ........ .... .... ... ............. ... ..... ..... .. 34
2
30
Synthesizing from the Data Grid .................. .... ... .......... .. .............. . 34 How to Synthesize .... ... ... ..... ......... ....... ...... .... .... .................... ..... .... ........ ... 34 Selecting Synthesis Type ........... ........... ............. ... ...... .. ........ ... ........... ... ... . 34
Senstivity-Graphs
34
Sensitivity Analysis .... ... .... .... ......... .... .... ....... .... ........ ... ......... ................. ..... 34 PerformancePerformance_Sensitivity ................... ... .... ........ .. .... .... 35 DynamicDynamic_Sensitivity ........... ... ..... ..... .. ... ... .. ... ...... ... .. ......... 35 GradientGradient_Sensitivity ............... ... .. .. ... .......... .. ......... ........... 35 Head to HeadHead_to_Head_Sensitivity ....... .. .. .... ... .. .. ..... ........ .. .. 35 Two DimensionaiTwo_Dimensionai_Sensitivity .. .. ...... .... ......... ..... .. 35 Performance Sensitivity .... ..... .. ...... ...... .. ......... .. .. .... ... ...... ...... ..... ... ...... ... .. .. 35 Dynamic Sensitivity ........... .. ... ........ .. ........ ................ ..... ..... ........ ................ 36 Gradient Sensitivity .......... .. ... .. ...... .. .. ... .... .... ........ ... ..... ... .. .... .... .. .. ..... .... .... 38 Head to Head Sensitivity ............... ....... ...... .. ....... .. ...... ............................... 39 Two Dimensional Sensitivity .. .... ... ... .... ........ ... .... .......... ... ... .................... .. .. 40
Introduction Use this manual to learn about Expert Choice. The manual is organized in chapters and subchapters (shawn as book icons). Before reading this manual, we recommend that you read the Quick Start Guid t: and/or The Tut orials.
The Getting Started 1 Creating a Mode! chapter in this help document provides steps to build your own mode! but doesn't provide as much instruction as either the Quick Start Guide or the Tutorials.
The second chapter explains Structuring and should be used to complement Lesson 2 of the Tutorials.
The Painvise Comparison Process chapter describes how to make judgments (paired comparisons). The Synthesis chapter describes how to view the results ofyour paired comparisons while the Sensitivity Analyses chapter describes how to view and interpret your results in graphies format. Much ofthis information is also described in Lesson 1 of the Tutorials.
The Data Grid Functions chapter explains how to use the grid and describes Ratings, Step Functions, and Utility Curves. This chapter complements Lessons 3 and 4 of the Tutorials. The Printing and Reports chapter describes how to print information while using standard industry conventions. Expert Choice's Menus are outlined in the next chapter.
The next two chapters describe Expert Choice's group and web (Internet) capabilities found in the Team and Enterprise versions. These chapters parallel Lessons 5 & 6 of the Tutorials.
A comprehensive glossary is found in the last chapter.
Conventions Menu commands are preceded by the word "select". For example: Select .Eile, Qpen. Menu commands can also be accessed by typing or clicking with the mouse. Commands that can only be invoked with the mouse are preceded with the word "click". For example: Click on fust row of the Data grid. "Press" is for a single letter or a combination ofletters. For example: Press Ctri-J. "Type" is used when you are required to enter data. Items that you need to enter are in bold. For example, Type Carl, or Type your na me.
9
Getting Started 1 Building a Model in the TreeView ModeiView Overview When you start Expert Choice the fust window that appears is a blank Mode!View. The Mode!View is divided into three major sections or panes.
?
The TreeView (the left pane) displays the hierarchy.
By default, nod es with children (objectives) are displayed next to yellow circles, and no des with no children ( covering objectives) are displayed next to black circles. If a node has children that have not been assessed, a red dot will appear in the center of the circle. When ali objectives in the TreeView have been assessed, the circles are replaced by squares that graphically display the priority of each factor.
?
The Alternatives pane (top-right) shows the acti ve alterna ti ves.
?
The lower-left pane displays the Information Document for the current (selected) node.
Note: The appearance of the Mode!View can be altered; to do this use the Vicw menu commaml s.
Creating a File and Goal Description To create a new file: ?
Select _Eile, t!ew.
?
Type a file na me for your model. Select a drive/pa th designation, if necessary. Then press Enter. ·
?
Type the goal description and press Enter.
?
An alternative way to create a new file is to click on the new file icon.
D1
Note: Expert Choice creates a mode! with only one node (the goal) and displays it in the ModelView's TreeView.
To leam how to build an actual mode!, go to the QUICK ST ART GUIDE . Also see Lessons 1 and 2 of the Tutorials.
10
Adding the Objectives and Sub-objectives to ModeiView's TreeView To add an objective: ?
Select the Goal node .
?
Select !:;,dit, Insert ,Çhild of Current Node.
?
Type a descriptive objective and press Enter.
?
When the new node appears, type the next objective and press Enter; or press Esc to stop inserting.
An alternative way to add objective: right-click on the Goal node and continue as described above .
jTip: Try to keep the number of nades under each parent under ni ne.
Note: When you add objectives below the goal, the circle next to the goal will change from black to yellow indicating that- other elements are below it. If a red dot appears in either colored circle this means judgments must be made. See: Pairwise Co mpariso ns.
To add sub-objectives below an Objective: ?
Select an objective that will have sub-objectives entered beneath it.
?
Select !:;,dit, Insert ,Çhild of Current Node and continue as described above.
To add an objective to an existing model: ?
Select a node (an objective).
?
To enter a node on the same leve!, select !:;,dit, Insert §.ibling of the Current Node, or
?
To enter ·a node beneath the selected node, !:;,dit, lnsert ,Çhild of Current Node .
Renaming Nodes in the ModeiView's TreeView To rename a node: ?
Select an objective in the TreeView to be renamed.
?
Select !:;,dit, !:;,dit Node.
?
Type the new name and press Enter.
11
Alternatively, you can right-click on a node you wish to rename and then select E_dit Node.
Deleting Nodes in the TreeView T o del ete a node and ali of its descendants in the TreeView: ?
Select the node in the TreeView to be deleted .
?
Select E_dit, .Qelete Node or press the DeJete key.
TrashCan Trash receives nades that have been deleted fro m the TreeView (objectives hierarchy). Once items are in the trash, they can be dragged and dropped back into the hierarchy.
To Open the TrashCan: ?
Select Yiew, TrashCa!!_, or click the TrashCan button.
\.!i!J 1
Note: This button will only be visible when nades are in !rash.
To move nodes back into the TreeView (hierarchy): Drag and drop the node from the trash onto its new parent node. It will become the fust child under the receiving node.
?
Copy Plex and Drag To copy the current node and ail of its descendants to the Trash Can:
?
Select E_dit, Copy Pie! to TrashCan.
?
Then, if desired, drag and drop the copied Pl ex from Trash to the receiving node.
Moving Nodes in the TreeView Y ou can move an objective and ils descendants from one part of the hierarchy to another.
To move a node: ?
Click and drag the node to be moved onto the destination node. The node will become the fust child under the destination node. Tip: If you want the nodes to appear in alphabetical order, select the parent of the cluster and then select Edit, Sort Cluster. ITip: To rearrange nodes within an objective, drag each node within the cluster in
12
reverse arder of the way you want them to appear in the cluster and drop them, one at a time, on the parent node of the cluster.
Adding Alternatives to the ModeiView Alternatives can be added to the mode! from the ModeiView using the Alternatives pane. ?
Select !;_dit, Alternative, and then select !nsert; alternatively right-click in the Alternative pane and then select !nsert or just sirnply click the Alternative burton.
?
~~
Type an alternative.
Note: Alternatives added from the ModelView are known as active alternatives and are simultaneously added to the Data Grid.
Alternatives can also added from the Data Grid.
Renaming Alternatives in the ModeiView ?
Select the alternative in the Alternatives window.
? Select .5_dit, ~lternative , and then select Edit Alternative Name. Alternately right-click on an alternative and then se.lect Edit Alternative Name. ?
Type the new alternative name and press Enter.
lnactivating and Reactivating Alternatives One or more of the active alternatives (those appearing in the Alternatives pane of the ModelView) can be made inactive. The alternatives will still be available on the Data Grid.
To inactivate alternatives from the Alternatives pane: ?
Right-click on an alternative in the Alternative pane then a drop down list will be displayed. Alternatively you can select &dit, Alternative.
?
Select either lnac!ivate or Inactivate A!l. Notice that the alternative will be removed from the Alternatives' pane (Y ou will, however, still be able to see the inactive alternative(s) in the Data grid).
To inactivate alternatives from the Data Grid: ? Right-click on an active alternative to remove the check mark. (Active alternatives are check-marked). ?
From the menu select &dit; then select &xtract Selected to Hierarchy. This will place a new set of active alternatives in the Alternatives pane ofthe ModelView.
13
To reactivate alternatives- only available from the Data Grid: ?
Right-click on an inactive alternative, then it will become check-marked and repeat as necessary.
?
Select E_dit; then select E_xtract Selected to Hierarchy. This will place a new set of active alternatives in the Alternatives pane of the Mode!View.
Adding Information Documents lnform;3tion Documents are used to document the decision-making process and can be created for each objective, alternative or paired comparison in the hierarchy. When used in a group setting they are primarily used as a way to communicate with participants and for presentation purposes. Typically information documents contain text ( entered by you or a facilitator using either the Team or Enterprise group enabled versions), that may describe the goal, give additional information asto why particular objectives or sub-objectives were selected, and tell us how paired comparisons were made. Information Documents are rich text objects and can include Microsoft Office Files (Word, Powerpoint, Excel, Access), as weil as other files that contain audio, pictures and video. Information documents can be created or viewed from the current node. When information exists the information icon on the toolbar is a red open book. gray (a closed book).
fa!/ If it do es not exist then the icon is
~
ln the ModeiView, information documents for either the objectives or alternatives are displayed in the lower right-hand pane.
?
To edit a document, select the node.
?
Then click the Information icon. Alternatively select !;dit, !nformation .
?
Type the information and when done, click the Information icon to close.
When at any pairwise comparison window, information documents are displayed in separate windows.
? To view or create a document, click the book icon. Alternatively, select !;dit, information. And continue as described above. Note: Ali paired comparisons are made with respect to a parent node. For example, if you are making paired comparisons for an objective that has three sub-objectives; the information document displayed would be for the objective.
When at the Data Grid, information documents are displayed in separate windows .
? Select a covering objective column or select a node in the TreeView, or select an alternative.
? Click the Information icon to either create or display the document for that node. Alternatively, select !;dit, Information .
14
?
Type the information and when done, click the Information icon to close.
Dragging to lm port Files to Information Documents: When an infonnation document is open you can drag other documents to it and/or create shortcuts to other files or programs. See Object Lin k ing & Embedcl ing.
Adding Notes Notes can be entered and viewed for elements in the hierarchy, alternatives, individual paired comparisons, and cells in the Data Grid as weil as covering objectives. When working with group models, each participant can enter their own notes to express their views, rationale, concerns and the like about the different parts of the decision process. A Note can only be viewed by the person who made it as weil as the facilitator. A participant note could be merged in Information document.
If a note exists for the current node or cel! in the Data Grid, the bottom portion of the Note leon will be yellow instead of completely black. 1"1 When a Note window is open, you can drag other documents to it and/or create shortcuts to other files or programs.
ln the ModeiView, a note is displayed for either an objective or alternative as a separate window.
?
To create or edit a note, select the node.
?
Then click the Note icon. Altematively select gdit, Note.
?
Type your note and when done, click the Note icon to close.
When at any pairwise comparison window, a note for the current comparison is displayed as a separate window.
?
Select the paired comparison .
?
To view or create a note for the paired comparison , click the icon . Alternatively select gdit, Note.
?
Type your note and when done, cl ick the Note icon to close.
Note: Ali paired comparisons are made with respect to a parent node. For example, if you are making paired comparisons for an objective that has three sub-objectives; the Note displayed would be for the objective.
When at the Data Grid , note is displayed for the current cell as a separate window.
?
Select a cell in the Data Grid .
?
Cli ck the Note icon to either create or display a note. Alternatively, se lect gdit,
Note.
15
?
Type your note and when done, click the Note icon to close.
To link or embed ether documents see object lin king and embedding.
Displaying Priorities in ModeiView In the ModelView's TreeView and Alternatives panes, the default priorities will be displayed after paired comparisons are made; or after priorities are entered directly or after alternatives are extracted from the Data Grid. The default is to display Local Priority.
To change the priority display white working with the current model:
?
Select Yiew; then select I:riorities
?
Choose one: !!lobai, 1ocal, J!oth, or ~one.
See Too ls, Opti ons, View to disable the display ofpriorities in ali your models.
Displayingthe Current Nodes Children in the Alternatives Pane At times you may desire to display the current node's children and their derived priorities in the alternatives pane instead of alternatives. To do this:
?
In the TreeView, select a node that bas children.
?
Select Yiew, View !:;_hildren of the Current Node. Alternatively, right-click on the alternatives' pane and then select View !:;_hildren of the Current Node. Note: If the children are currently displayed in the Alternatives pane then selecting Yiew, View will display both the alternatives and their priorities.
~lternatives
See Too ls, Options, Vicw to change the default display from alternatives to children of the current node in ali your models.
Deleting an Information Document or Note ?
First select the node or cell whose Information Document or Note is to be deleted .
?
Click either the Information Document or Note icon.
?
From the information or note window, select !ile, ~ew and then close the window . This will erase the contents of the Information Document or Note.
?
Ifyou want to abandon ali changes to an Information Document or Note during the current session, select !ile, ~bandon Changes.
Un do 16
Undo is used to undo a judgment when making paired comparisons or when entering data on the data grid . lt can also undo changes when editing an Information Document or a Note.
To Undo the Last Pairwise Comparison : ? From any pairwise comparison window when making judgments, select ~dit, then select
~ndo .
To Undo the Last judgment or Data Value Entered in the Data Grid: ?
When making judgments in the Data Grid, select ~dit , then select ~ndo.
To Undo- Information Documents or Notes: ?
When editing either select Edit, then select Qndo.
Tip: Use the standard windows keys to highlight keystrokes then press the Delete key to remove them. Tip: If you want to abandon ali changes to an Information Document or Note du ring the current session, select File, Abandon Changes.
Revert Revert is used to go back to a previous version of the database. Use Revert when you decide not to continue working with the changes just made to the mo del (during the current session). Note: Tllls feature is not available to participants in a group model. The facilitator can't use Revert while the model is on a network.
To revert from either the ModelView, Data or Formula Grids: Re~ert.
?
Select Edit,
?
A list ofshowing versions ofthe current model, ifavailable, will be displayed .
?
Double-click on the version of the model y ou want to re vert to.
Waming: Because revert goes back to a previous version of the database, that contains EVERYTHING, for ALL people, the facilitator MUST modif)r the mode! only when participants are not accessing the model.
Opening a Model To open another madel: ?
From the Mode!View, Data or Formula Grids select ,Eile, Open and select a model from the Open Dialogue box.
17
Or alternatively, select Eile and then select one of the last four models previously opened from the bottom portion of the File drop-down list. ?
If a mode! is currently open and has not been saved since changes were made : o
Select .Xes to save your changes and open another mode!.
o
Select Cancel to abandon the save and open request.
o
Select ~o not to save your changes. Then the next time you open this mode] the last saved version will be displayed.
Converting to an lncomplete Hierarchy This command changes the structure ofthe mode] placing the active alternatives from the Alternatives Pane under the covering objectives in the TreeView. See Complete Hierarchy.
?
Select _Eile
?
Select Couvert to !ncomplete Hierarchy
Warning: Since you will be changing the structure ofyour mode], it is highly recommended that you make a copy ofyour mode! using the File, SaveA_s command before converting.
Rollup lt is possible to 'rollup' sub-nodes between a selected node and the alternatives using the ModelView menu command E_dit Bollup. This command removes ali rolled up nodes from the mode! and places them in the Trash Can where they can be dragged/dropped back into the mode!. Note: To roll up you must have at a minimum two levels of objectives and alternatives, or three levels ofnodes in the TreeView.
The Rollup command has two subcommands: ?
Çurrent Node only
?
Current Node and Ali feers . Rolls up ali the peers (siblings) of the current node as weil as the current node itself.
Prior to executing the rollup, you will be prompted to save the model if changes have been made since the last save. If the mode! is saved prior to the rollup then the rollup can be undone by using the E_dit B,eve rt.
Tip: Rolling up from a node above the lowest level nodes or alternatives has no effect.
18
Converting from Version 9.5 EC2000 will convert your 9.5 models to EC2000 Access data base models. To do this:
?
From Expert Choice 2000, select Eile, Qpen ; then navigate to and select the 9.5
model. Note: Expert Choice 2000 will combine ali the 9.5 files into one database. Converting a Ratings Model If you convert a Ratings madel (that has a wk1 file as one its components) then the Ratings intensities in the lowest leve! of the hierarchy will be converted to the scales used in the Data Grid . (For more information Expert Choice 2000 and Ratings see the Data Grid and Ratings chapters.)
File Structure Expert Choice 2000 files have the extension of .AHP. There is only one file per mode! and it structure based Microsoft's Access. Expert Choice's Professional and Team versions use Access for the database structure, while Enterprise version uses Sequel.
Structuring - Another Way to Build A Model Top Down or Bottom Up Use this information to determine which process to use. Top Down Str uctul"ing is best used when you know more about the objectives than the alternatives.
Bottom np Struct ul"in g is best suited for situations where the alternatives are better understood than the objectives. The pros and cons of the alternatives are used to help identifY the objectives that can then be clustered into groups.
First Create a File (Model) and Goal Description To create a new file: ~ew .
?
Select file,
?
Type a file name for your model. Select a drive/path designation, ifnecessary. Then press Enter.
?
Type the goal description and press Enter.
?
An alternative way to create a new file is to click on the new file icon .
D
19
Note: Expert Choice creates a mode! with only one node (the goal) and displays it in the Mode!View's TreeView.
Enable Structuring Structuring is another approach to building a model. Y ou can either build your model using the either the Top-down or Bottom -up approach .
For a Newly Created Model with Only a Goal Node: ?
Select Iools, Qptions; then select the G eneral tab.
?
From the Structuring box, select the E nable button; then select Çlose.
?
From the Mode!View menu, select Yiew, then select the ÇlusterView pane; or click the ClusterView button .
l[j]J
For Existing Models: ?
If the ClusterView icon is no t displayed next to the TreeView icon in the ModeiView, fo llow the steps described above.
Now build your mode! using either Top-down app roac h or Bottom-up app roach.
Top Down Structuring Adding Objectives and Sub-objectives to ClusterView To add objective (or sub-objective) to the ClusterView: ?
Click the Qbjective 1 C riterion 1 Group button.
?
Type a descriptive obj ective and press E nter .
Alternat ively, click and drag a box in the C luster View pane and when the objective pop-up box is displayed type the description.
Note : When adding objectives or sub-objectives we do not distinguish between them until a subobjective is dropped onto its parent.
20
To group sub-objectives within the ClusterView: ?
Click, drag and drop one objective into another. Then the dragged and dropped node will become a sub-objective of the destination node:
Moving Nades in the ClusterView Click, drag and drop one objective or a cluster onto another. Then the dragged and dropped node or cluster will become a sub-objective or cluster of the destination node.
Deleting Nades in the ModeiView's ClusterView To delete a node and ali of its descendants in the ClusterView: ?
Select the node in the ClusterView to be deleted.
?
Press the DeJete key. Note: The node and ali of its descendants will be removed from the hierarchy and placed in the Trash Can. Deleted nodes can be dragged and dropped back into the hierarchy. See: Moving 0/odcs in the TreeView.
Adding Alternatives in Top Dawn Structuring Alternatives can be added to the model from the ModeiView using the Alternatives pane.
?
Select
~dit,
Alternative, and then select Jnsert; alternatively right-click in the Alternative pane
and th en select Jnsert or just simply click the Alternative button.
?
~1
Type a description for the alternative.
Note: Alternatives added from the ModelView are known as ac tive alt ernatives and are simultaneously added to the Data Grid.
Alternatives can also added from the Data Grid.
Finish Top Dawn Structuring Once objectives and sub-objectives as weil as the alternatives have been added to your model view the hierarchy .
1' J
1.
Select the TreeView button .
2.
Review the hierarchy to see ifanything is missing. If something is missing return to ClusterView pane and enter objectives, or enter objectives and alternatives directly in the ModelView by using the ~dit command.
Now you are ready to make pa ircd compa ri suns.
21
Bottom Up Structuring Adding Alternatives in Bottom Up Structuring Alternatives can be added to the mode! from the ModeiView using the Alternatives pane.
?
Select E_dit, Alternative, and then select !nsert; alternatively right-click in the Alternative pane and then select !nsert or just sirnply click the Alternative button.
?
~~
Type a description for the alternative.
Note: Alternatives added from the ModelView are known as ac tive alternatives and are simultaneously added to the Data Grid .
Alternatives can also added from the Data Grid .
Adding Pros and Cons for Each Alternative ?
Click the Pro/Con button. I±.J Alternatively, select View; then select Alternative Pro/Con pane. The Pro/Con pane with three buttons will appear in the ModelView window. Notice the highlighted alternative in the Alternatives pane also appears as the title bar.
?
Click either the Add fro or Add Çon button to enter a pro or con for the highlighted alternative. Note: The Pros are in blue while the Cons are magenta.
?
Ifyou would like to add pros and/or cons for a different alternative sirnply click on another alternative to make it the current alternative.
?
Repeat this process for ail alternatives.
?
When done view a list o f al i pros and cons.
Tip: Some alternatives will have the same pros or cons; it is not necessary to add them to each alternative's pro/con window.
View the List of Ali Pros and Cons ?
Use the pro/con list to convert pros and cons to objectives using our patented process.
?
From the menu, select ,Yiew; then select the Ail Pros/Cons pane. Another way to see this is to click the Ail Pros/Cons button.1:J Note: The Pros are in blue while the Cons are magenta.
In Bottom-up Structuring start build ing a hicra rchy by dragging and dropping (as weil as redefining) a pro or a con (from the Ali Pros/Cons pane) to the TreeView . .
22
Building the Hierarchy ?
Start at the top of the list of ali pros and cons. Drag either a pro or a con into the Tree View pane.
?
A dialogue box will appear with the pro or con definition. Now you have opportunity to redefine the wording of either the pro or con as an objective. For example a pro for a car might be large trunk. After re-definition the objective would be Trunk - large carrying capacity.
Tip: Sin ce alternatives are evaluated based on their preference with respect to objectives, the wording for cons must al most always be changed to state the objective (or objectives) that the con 'points to'. For example, the con, expensive, 'points' to low cost as an objective. ?
After either the pro or con is dragged to the TreeView notice that it is now grayed out in the Ali Pros/Cons List to indicate that is was already converted to an objective. Note: Gray can be removed from i pro or con by clicking.
Tip: Looking at the list of pros and cons, you can see that sorne pros and cons cou Id be repeated more than once because they are associated with more than one alternative; if this is the case, you wou Id not need to convert repeats. Those pros and cons not used in the conversion process will remain white. On the other hand, a single pro or con may 'point' to severa! objectives and can be dragged and dropped nu merous times. For example, a pro of 'size' for a large car may point to the following objectives: Comfort; Carrying capacity; Safety; Fuel Economy; Ease of Parking. ?
Repeat the above steps until the hierarchy of objectives is complete.
Finish Button Up Structuring ?
Select the Alts/Children!InfoDocs button. {bmc alt_button.bmp} Then the alternatives will replace the list of ali pros and cons.
?
Review the hierarchy to see ifanything is missing. If items are missing, you can retum to structuring. Altematively, enter objectives and alternatives directly into the TreeView using the _Edit comrnand.
Now you are ready to make paircd co mpariso ns.
Pairwise Comparison Process - Making Judgments Making Paired Comparisons Pai rwisç co lll pu ri sons are made from the Mode!View in one of the following ways :
?
From the Mode!View, select Assessment, rairwise. One of the three pairwise comparison windows will be displayed (Verbal is the default).
23
?
To select another window, click one ofthe tabs: Numerical l 3; 1 ·1, Verbal! AE\_ Graphical. 1
=
1,
or
1
Note: Ifsome comparisons have been previously made then the Assessment tabs will be displayed in the ModelView.
Making Verbal Judgments
Making Numerical Judgments
Making Graphical Judgments
Structural Adjust
Examining and lmproving lnconsistency
Also see the Assess me nt M enu Co mma nds.
Making Verbal Judgments The Verbal Comparisons window is divided into two sections. Verbal judgments are made in the top pane. Two elements are compared with respect to their parent. What makes Verbal comparison unique is that words are used to represent the magnitude of the scale. The slider bar on the right side of the pane is used to indicate which element is preferred and the strength ofthat preference is represented by a corresponding word . The two opposing sides of the scale represent each element being compared.
24
1
i' -~
- Extreme -Very Strong
1
1• 1
· Strong - Moderate -Equal - Moderate - Strong
1 -
1 1
~
- Very Strong - Extreme
1
The comparison matrix is displayed in the lower pane. The num er i c<~ l representations of the verbal judgments are displayed here as numbers from 1 to 9. If the row element (on the left) is preferred, then the judgment is displayed in black. If the column element is preferred, then the judgment is "inverted" and displayed in red . When enough judgments have been made to calculate priorities, they will also be displayed as bar graphs that overlay the row elements.
Judgments can be made any of the following ways: ?
Drag the slider bar with the mouse.
?
Click on a statement (i.e. Moderate, Strong) next to the bar (or between two statements).
?
Right-click on a statement to automatically advance to the next comparison. This is the fastest way!
To advance to the next judgment: Expert Choice is automatically configured to ad vance to the next judgment. Y ou can change this with the Too ls, Options menu; see the Calculation tab.
?
If Autoadvance is off, to advance to the next judgment click on a cell in the comparison matrix_
To invert a judgment (to select the other element in the comparison)
..,. ?
Click the Invert icon "-·· to select the other side of the comparison scale.
25
To enter judgments directly in the comparison matrix: ?
Click on the cell representing the comparison you want to judge and type a number from 1 to 9; see the numerical representations ofverbal_iudgments. When using this option it is highly recomrnended that you use the Numerical mode since you are probably saying that you prefer, for example, Apples to Oranges 3 times more with respect to Craving.
Also see: the Asscssmcnt Menu Co mman ds.
Making Numerical Judgments The Numerical Comparison Window is divided into two sections. Numerical judgments are made in the top pane. Two elements are compared with respect to their parent using a numerical sc ale. The slider bar is used to indicate which judgment is preferred and the strength ofthat preference. The two opposing sides of the scale represent each element being compared.
The comparison matrix is displayed in the lower pane. The numerical equivalents of the judgments are displayed here as numbers from 1 to 9. If the row element (on the left) is preferred, then the judgment is displayed in black. If the column element is preferred, then the judgment is "inverted" and displayed in red . Wh en enough judgments have been made to calcula te priorities, they will also be displayed as bar graphs that overlay the row elements.
Judgments can be made any of the following ways: ?
Drag the slider bar with the mou se.
?
Click on a number above the bar.
?
Right-click on a number to automatically advance to the next comparison. This is the fastest way!
T o advance to the next judgment: Expert Choice is automatically configured to ad vance to the next judgment. Y ou can change this with the Tools. Options mt:n u; see the Calculation tab.
?
If Autoadvance is off, to advance to the next judgment click on a cell in the comparison matrix.
26
Notice that as you make judgments, the numerical equivalents ofyour judgments will appear in the comparison matrix.
To invert a judgment (to select the other element in the comparison)
..,. ?
Click the lnvert icon 'W to select the other side of the comparison scale.
To enter judgments directly in the comparison matrix: ?
Click on the cell representing the comparison you want to judge and type a number from 1 to 9.
Also see the Asscssmem Menu Commands .
Making Graphical Judgments The Graphical Comparison view is divided into two sections: Graphical judgments are made in the top pane. Two elements are compared with respect to their parent with bar graphs. The lengths of the bars indicate the relative dominance of the elements. If they are equallength, th en the elements are equally important. If one bar is twice as long as the other, then it is twice as important. Relative dominance is also represented with a pie chart on the right side of the pane.
Campzm: 1hr:-
rdativ~ importanc~
with
r~sp~ctto:
Ma:ximin puck-stcpping abili1y
The comparison matrix is displayed in the lower pane. The numerical representations of the element (on the left) is preferred, graphicaljudgments are displayed here as numbers. Ifthe then the judgment is displayed in black. If the column element is preferred, then the judgment is "inverted" and displayed in red. When enough judgments have been made to calculate priorities, they will also be displayed as bar graphs that overlay the row elements.
row
Judgments can be made any of the following ways: ?
Drag either the blue or red bar with the mouse.
27
?
Right-drag one of the bars to automatically advance to the next comparison.
To advance to the next judgment: Expert Choice is automatically configured to ad vance to the next judgment. Y ou can change this with the Tools, Options menu; see the Calculation tab.
?
If Autoadvance is off, to advance to the next judgment click on a cell in the comparison mat rix.
Notice that as you make judgments, the numerical representation ofyour graphical judgments will appear in the comparison matrix.
To invert a judgment (to select the other bar in the comparison) ?
Click the Invert icon to select the other bar.
~
1
To enter judgments directly in the comparison matrix: ?
Click on the cell in the matrix representing the comparison you want to judge and type a number from 1 to 99. When using this option it is highly recommended that you use the Numerical mode since you are probably saying that you prefer, for example, Apples to Oranges 3 times more with respect to Carving.
Also see the Assessment Menu Co mmands.
Direct Entry of Priorities Y ou can directly assign priorities without having to make paired comparisons. This method is not recommended because it is not as accurate or justifiable.
To directly assign weights: ?
Click on any of the three pairwise comparison tabs.
?
Select Assessment Qirect.
?
Enter a value between zero and one for each objective, or drag a bar using the column to the right of"Value'.
?
When done press Esc.
?
When asked to Record Judgments, select Y.es.
28
Tip: Assume the length of the bars representa factor's importance relative to the others. For example, if the bar for Priee is twice as long as the bar for Quality, then Priee is considered twice as important. If the lengths of the bars are equal, th en the factors are of equal importance.
Making Diagonal Pairwise Judgments The default order when making pairwise comparisons is sequential order. This can be changed to either random or diagonal. Random randomly determines the next judgment while Diagonal takes you down the diagonal of the matrix. Use Diagonal Pairwi se to reduce the number of comparisons and length of time when making paired comparisons.
?
From the menu select Iools, Qptions, Calc!!lation.
?
In the Judgrnent Order box select .Qiagonal.
?
Then select Close.
?
Now make your paired comparisons.
?
Afier making the fust row of diagonal judgments click the Calculate button. Do this, when the icon turns from red
li 1to yellow.
mm
See Also: Diagonal Pairwise
Making a Factor Dormant from the Pairwise Comparison Matrix When a factor (objective, sub-objective or alternative) is made dormant it is excluded from the pairwise comparison process.
From any pairwise comparison window: ?
Press Ctrl and click a factor name (objective or alternative located in the rows of the matrix). The factor row selected will be blocked. In addition, any comparison made (orto be made) relating to that factor in the matrix will also be blocked. By blocked we mean the paired comparison(s) to be made or previously made will not be included in the prioritization process.
T o reactivate a factor: ?
Press Ctrl and click a factor name.
29
Structural Adjust Structural Adjust , available from any of the paired comparison modes, is used to adjust a set of priorities based on the number of the current node's grandchildren. Turning on Structural Adjust for a node weights the priorities of its children by their respective proportions to ali the grandchildren . For example, if a node has two children A and B, and A has 4 children and B has 2 children, then the node has 6 grandchi ldren in ali and structural adjusting multiplies A's priority by 4/6 and B's by 2/6. Thus the global priorities of A's children are not diluted simply because they belong to a large family. Think of a grand parent wanting to leave mo ney equally to grandchildren. Four sixths of the money must go to A's children and two sixths to B's children. This feature should be used when you wish to prevent the dilution of the global weight of a grandchild simply because it has many siblings. lt is rarely necessary to use Structural Adjust because even though an element, for example, is divided into many sub-elements, its full weight is distributed among them. When the alternatives are then compared under each of tho se subelements, the full weight of the element is distributed on down to the alternatives. A case where you would want to use structural adjust would be if the sets of alternatives in the bottom leve! do not have the same alternatives in each group. In this case we recommend that you convert your mode) to an lncomplctc 1-l icrarchy
To tum Structural Adjust on or off, click the §tructural adjust button . i Structural adjust . When on, the bar graphs of the objectives in the matrix will turn aqua.
?
Examining and lmproving lnconsistency The lnconsistency menu, available from any of the pairwise assessment windows, provides a convenient way to locate any inconsistencies among a set of pairwise judgments. The Inconsistency Ratio is located in the lowest left-hand cell ofthe frrst column of the matrix. A ratio ofO.IO or Jess is considered acceptable. To learn how inconsistency is computed see u nd erstancling rnconsistcncy
To view the most inconsistent judgment: ?
From any comparison mode, select the I!!consistency
?
From the pull-down list, select 1'
1
Selecting l" moves the cursor to the most inconsistent judgment in the set ofjudgments being compared; selecting 2"d moves to the second most inconsistent judgment and so on.
To lower the lnconsistency Ratio for a set of judgments you can either: ?
From the 1" most inconsistent judgment (and so on) yo u can change the judgment by making a new paired comparison, or
30
?
Select I!!consistency and then select !!_est Fit. Altemative ly, right-click on any cell in the matrix to see th at ce li' s best fit. Best Fit (displayed above the fust factor row in the matrix) shows the judgment for the current ce li that would best improve your consistency. If the Best fit is displayed in red then the judgment suggested when entered must be inverted, element.
~ 1making the column element preferred to the row
Note: If the comparison window is either Verbal or Graphical then the Best Fit shown is a numerical representation of either the Verbal or Graphical judgment.
Tip: Do not enter the suggested Best Fitjudgment unless you believe it to be true. Enter only a jud ment that represents your best understanding and knowledge. ?
After changing a judgment the new Inconsistency Ratio will be displayed . There is one exception, ifyou deferred automatic calculation of the priorities with using the Tools, Options, Calculations command, you must click the Calculate leon or the new inconsistency will not be displayed.
Understanding lnconsistency Using the Assessment Pairwise command, Best Fit, you can request suggestions for reducing inconsistency. However, it is important to note that the methodology does not preclude inconsistencies in judgments. On the contrary, many decisions must be made wh ile recognizing inconsistencies that exist in the real world . The conscious mind constantly attempts to understand what is sensed and perceived by relating it ali together in sorne coherent way. What do we mean by coherent? Let us illustrate with an example. Ifyou were to say that A>B, B>C and C>A, you have been inconsistent. Consistency of judgment fo llows this simple transitive property. But we are very seldom perfectly consistent in making comparative judgments, particularly when we deal with intangibles that have no sca!es of measurement. And, we shou!d not expect to be totally consistent. The real world often !acks consistency, and we must be able to reflect that in our models. For example, Team A can beat Team B, and Team B can beat Team C, yet Team C might then beat Team A. Expert Choice provides a measure ofyour logical rationality, called the Inconsistency Ratio, but does not force you to be consistent. The Inconsistency Ratio is calculated for each set ofjudgments. lt is important to emphasize that the objective is to make "good" decisions, not to minimize the Inconsistency Ratio. Good decisions are most often based on consistent judgments, but the reverse is not necessarily true. lt is easy to make perfectly consistent judgments that are nonsensical and result in terrible decisions. When the Inconsistency Ratio is zero we have complete consistency; when it is grea ter than zero there is sorne inconsistency. The larger the value of the Inconsistency Ration the more inconsistent the judgments. If it is 0. 10 or less the inconsistency is generally considered tolerable. If the Inconsistency Ratio is considerably more than 0. 10 (and certainly if it is as high as 0.20), then a re-examination of our judgments is probably in order. The degree of inconsistency that indicates a "significant" problem depends, of course, on the specifie situation where the mode! is applied. The number 0. 10 is given as a general guideline. Ifwe demanded perfect consistency we would find it difficult to grow and/or to leam new things. When we integrate new experiences into our consciousness, previous relationships may change and sorne consistency is !ost. As long as there is enough consistency to maintain coherence among the objects of our experience, the consistency need not be perfect. lt is useful to remember that most new ideas that affect our lives tend to cause us to rearrange sorne of our preferences, thus making us inconsistent with our previous commitments. lfwe were to program ourselves never to change our minds, we would be afraid to accept new ideas.
31
We are able to make judgments, which serve us better if they are admissible in sorne subjective corridors between tolerable inconsistency and perfect consistency.Thus, while consistency is a concem, without sorne inconsistency we would not grow by taking on additional information and readjusting our viewpoints. We may say that the intensity of our concem with consistency and inconsistency differ by an order of magnitude. This means that ifwe were to divide a unit of priorities among the two , consistency would be about 0.90 and inconsistency would be about 0.10. The ratio of the two is nearly an order of magnitude. To measure the inconsistency of ali the judgments made in the decision hierarchy, we take the inconsistency value of each set of comparisons and multiply it by the priority of the element with respect to which these comparisons are made, and add for ali the elements. This gives a single overali weighted number. To decide how acceptable this number is, we forma ratio with a similar number obtained by multiplying the corresponding random inconsistency value for an equal number of comparisons by the priority of the elements, and again add over each attribute. The resulting ratio should be 0.1 0 or Jess.
To lean how to examine and improve inconsistency see Examining and lmprov ing lnconsistency.
lmporting & Exporting to/from any Pairwise Win dow Judgments/data can be exported from Expert Choice to Excel where the data can be modified or entered and then re-imported to any Assessment window. The Preferred Method: ~opy
1.
From any Expert Choice Assessment rairwise window select _Edit
to Clipboard.
2.
From Excel use the Edit Paste command to import the extracted data.
3.
From Excel select Insert, Name, Define and then type a range name such as X. This marks the range to be pasted back into Expert Choice.
4.
ModifY spreadsheet data or enter judgrnents.
5.
From Excel select Edit, Go To and select the range defined in step 3.
6.
From Excel select Edit Paste (to the clipboard).
7.
Retum to the Expert Choice Assessment window and select _Edit ~opy from the Clipboard . Note: You must enter val id data in your spreadsheet. Y ou may enter who le or fractional numbers. Negative numbers indicate that the judgment or datais inverted (colurnn element is preferred to the row element).
Tip: Backup your database by selecting the file Save6s before cutting and pasting. You can also use the following methods to import data into an Assessment window.
32
lmporting Only the Upper Portion of the Matrix In the above method, we export and import the factors (objectives or alternatives) used in the paired comparisons. In this method we only import the paired comparisons. You must know what the matrix looks like. 1.
In Excel enter values for the upper portion of the matrix.
2.
Select the range and then select Edit Copy (to the clipboard).
3.
Go to the desired Assessment Pairwise window and select ~dit , faste to Clipboard .
lmporting a Row Vector This method only imports the paired comparisons. 1.
Type a data to be imported in consecutive cells in a row.
2.
Select the row range; then select Edit Copy (to the clipboard).
3.
Go to the desired Assessment Pairwise window and select ~dit, faste to Clipboard.
lmporting a Column Vector This method only imports the paired comparisons. 1.
Type a data to be imported in consecutive cells in a column.
2.
Select the column range ; then select Edit Copy (to the clipboard).
3.
Go to the desired Assessment Pairwise window and select
~dit,
faste to Clipboard.
Synthesis Synthesis Overview Synthesis is the process of weighting and combining priorities throughout the model after judgments are made to yield the final result. Global priorities are obtained for nodes throughout the model by applying each node's local priority and its parent's global priority. The global priorities for each alternative are then summed to yield overall or synthesized priorities. The most preferred alternative is the one with the highest priority. Y our answer is presented in the form of a bar graph of the overall priorities of the alternatives. Your best choice is the alternative with the longest bar. See: Synthesi ze Menu Commands. Y ou can select either the Ideal or the Distributive mode to synthesize your mode!. This will not change the mode! in any way, and you can switch back and forth between the two nodes. In general, the priorities from either mode are within a few percent age points of each other. After the synthesis is performed and you have your answer you may wish to do sensitivity analyses to determine how sensitive the results are to changes in the priorities of the objectives.
33
Synthesizing Group Judgments When using our group-enabled systems (Team or Enterprise), ali active participants' judgments are combined, using ModeiView's Edi t command Combine Participants Judgments/Data. Then the facilitator can synthesize the combined grouping .
Synthesizing from the Data Grid Synthesis is not used for Data Grid models as the overall weight for an alternative is determined from the Total co lumn of the grid . Since these weights may be interpreted as performance measures the weights of the alternatives are considered the results. An easy way to read the results is to sort on the Total Score column; then you will not only see the weights but the alternat ive's rankings.
Optionally, you can fine-tune the results of the Data Grid by making paired comparisons on sets of selected alternatives that ranked very close to each other. To do th is, mark and extract selected alternatives in groups of nine. Then you could make paired comparisons, synthesize to get the results and view sensitivity graphs. See: Data Grid Edit Menu Commands
How to Synthesize ?
From the ModelView's TreeView, select either the Goal Node or an obj ective.
?
From the menu select §ynthesize. The default mode of the synthesis is Ideal to change this se lect Distributive The default display is Sumrnary, to change this select the Details tab.
Selecting Synthe·sis Type The default mode is Ideal. To change to the synthesis type to distributive: 216
From the ModeiView, select §.ynthesize, With Respect to Goal
216
Select Qistributive. (If Distributive is selected select !deal).
Idea l Synthesis Di stributi ve Syn thesis
Senstivity-Graphs Sensitivity Analysis The purpose of our sensitivity analyses is to graphically see how the alternatives change with respect to the importance of the objectives or sub-objectives. There are five types of analyses. Each sensitivity analysis can be performed from the Goal or fro m a selected objective or subobjective. In ali cases, there must be at !east two levels below the selected node. These levels can be comprised of at !east one leve! of objectives and alternatives or two levels of only objectives.
?
From the ModeiView's TreeView, click on either the Goal node or an objective.
?
Select Sensitivity-Graphs.
34
?
Select one of the options.
If you are at the Goal, you will see how sensitivity the alternatives are to change with respect to the first leve! objectives. If you are not at the Goal node, you will be asked : "Sensitivity with respect to Current Node or Goal?"
? If you select .Yes, then the priorities of the alternatives are with respect the current node. lt is as if the other portions of the mode! did not exist. ? If you select ~o. then the priorities of the alternatives are with respect to the goal (the entire model). With this option, you see the variability of the alternatives with respect to the current node . Note: The results of the sensitivity analysis graphs will differ depending on the type ofsynthesis selected: Ideal (the default) or Distributive. The current type is shown in the status bar at the bottom of each graph. See: Se lect ing Synthes is Type
PerformancePerforman ce_Sensitivity DynamicDynamic_Sensitivity GradientGradient_Sensitivity Head to HeadHead_to_Head_Sensitivity Two DimensionaiTwo_Dimension al_Sensitivity
Note: Graphs can also be accessed from other graphs by clicking the appropriate icon on the button bar.
Performance Sensitivity The Performance graph displays how the alternatives perform with respect to ali objectives as weil as overall. It can be accessed from the ModelView's menu by selecting Sensitivity-Graphs and then selecting ferformance .
35
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Use the "left y-axis" to read each objective's priority. Use the "right y-axis" to read the alternative priorities with respect to each objective. The Performance graph is also dynamic, so you can temporarily alter the relationship between the alternatives and their objectives by dragging the objective bars up or down. Note: The !ines connecting the alternatives from one objective to another have no meaning; they are included to help your find where a particular alternative lies as you move from one objective to another.
See Commands Co mm on lo Ali Sensiti vity Graphs
There are two additional Options commands that are specifie to the Performance Sensitivity.
?
_ratterns - changes how the alternative !ines are displayed - from so lid to dashed and vice versa.
?
Qverlap - Overlap is a switch that permits or disables the overlapping of alternatives' labels.
?
§ort/Unsort- sorts in order of priority.
Dynamic Sensitivity Dynamic Sensitivity analysis is used to dynamically change the priorities of the objectives to determine how these changes affect the priorities of the alternative choices.
36
.ri.
It can be accessed from the ModelView's menu by selecting Sensitivity-Graphs and then selecting ;Qynamic.
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By dragging the objective's priorities back and forth in the Ieft column, the priorities of the alternatives will change in the right column. Ifyou think an objective might be more or Jess important than originally indicated, drag that objective's bar to the right or left to increase or decrease the objective's priority and see the impact on alternatives . For example, as the priority of one objective increases (by dragging the bar to the right), the priorities of the remaining objectives decrease in proportion to their original priorities, and the priorities of the alternatives are recalculated . See Command s Com mon to Ail Sensitivily Grap hs
There is one additional Options command that is specifie to the Dynamic Sensitivity.
?
!;;_omponents- is a switch that changes the presentation of the objectives and alternatives bars. The default setting displays mono-colored solid bars. IfÇomponents is turned on then the graph shows the breakdown of each of the objective's contribution to the priorities of each of the alternatives. In other words, Çomponents shows each alternative's share of the different objectives.
37
Gradient Sensitivity This graph shows the alternatives priorities with respect to one objective at a time. It can be accessed from the ModelView's menu by selecting Sensitivity-Grapbs and then selecting Qradient.
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The vertical red line represents the priority of the selected objective and is read from the X-Axis intersection. The priorities for the alternatives are read from the Y-Axis; it is determined by the intersection of the alternative's line with the objective's (vertical) priority line. To change an objective's priority, drag the red bar to either the left or right; then a blue bar showing the new objective's priority will be displayed. The Gradient Sensitivity shows "key tradeoffs" when two or more alternatives intersect each other. This is even more important if the intersection is close to the objectives priority.
Tip: When ali the alternatives never intersect each other, increasing the objective's priority will have no effect. See Cn mmand s Commonto /\I l Scns itivity (jraph s
38
There are two additional Options commands that are specifie to the Gradient Sensitivity.
?
ratterns - changes how the alternative !ines are displayed - from solid to dashed and vice versa.
?
Qverlap - Overlap is a switch that perrnits or disa bles the overlapping of alternatives' labels. There is also an X-Axis command that is used to select another objective.
Head to Head Sensitivity It shows how two alternatives compare to one another against the objectives in a decision. It can be accessed from the Mode!View's menu by selecting Sensitivity-Graphs and then selecting _!!ead to Head.
One alternative is listed on the left side of the graph and the other is listed on the right. The alternative on the left is fixed, while selecting a different tab on the graph can vary the alternative on the right. Down the middle of the graph are listed the objectives in the decision. If the lefthand alternative is preferred to the right-hand alternative with respect to an objective, a horizontal bar is displayed towards the left. If the right-hand alternative is better, the horizontal bar will be on the right. If the two choices are equal, no bar is displayed. The overall result is displayed at the bottom of the graph and shows the overall percentage that one alternative is better than the other; this is the composite difference. The overall priority can either be shown based on the objective weights (typical) or un-weighted.
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39
See Co mmands Co mmun to Ail S\:ns itivity Grap hs
There is one additional Options Command 216 Weighted - displays a composite result that takes the objective priorities into account. The un-weighted result displays the composite resultas if the objectives have equal priorities. The caption at the bottom of the graph tells you whether you are viewing a weighted or un-weighted result.
The Head to Head command is used to change the fixed alternative on the left side of the graph.
Two Dimensional Sensitivity This sensitivity graph shows how weil the alternatives perform with respect to any two objectives. lt can be accessed from the ModelView's menu by selecting Sensitivity-Graphs and then selecting 2-D.
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There is one additional Options command.
40
?
rrojection - is used to turn on and off the projection of alternatives. When on. projection not only shows how preferable the alternatives are with respect to the two selected objectives but it shows a composite projection line indicating the preference of each alternative taking into account ali the objectives' priorities. The farther to the right on the line, the better the alternative. Note: Y ou can drag a composite projection line down to see what would change if the X-Axis objective became more important. Conversely, dragging the projection lù1e up will increase the importance ofthe Y-Axis.
There are two menu comrnands to select objectives for the ~ Axis and Y Axis.
