Socio-&on. Plan. Sci. Vol. 19, No. 2, pp. 101-107, Printed in the U.S.A.
0038.0121/85 $3.00 + .I0 0 1985 Pergamon Press Ltd.
1985
AN ALTERNATIVE SOLUTION METHOD FOR GOAL PROGRAMMING PROBLEMS: THE LEXICOGRAPHIC GOAL PROGRAMMING CASE N. K. KWAK Saint Louis University, St. Louis, MO 63108, U.S.A.
and MARC J. SCHNIEDERJANS Department
of Management,
University
of Nebraska-Lincoln,
Lincoln,
NE 68588-0491,
U.S.A.
(Received 18 April 1983; revised 2 March 1984) Abstract-This article provides an example of the use of an alternative goal programming solution procedure. The solution procedure is applied to solve a lexicographic goal programming problem. In the process of illustrating the problem, a number of unique technical problems and their solution procedures are discussed. This study provides a flowchart as a procedural guide to aid in its use. This article also provides a comparative analysis of GP algorithms. The results demonstrate the tableau element reduction capabilities of the proposed alternative goal programming solution procedure.
INTRODUCHON
The recent advances in GP algorithms and increased computer technology/capabilities have permitted most real-world, large-scale GP problems to be solved. Despite these advances, the need to further reduce the number of tableau elements required in a GP algorithm continues for several reasons:
(GP) is one of many techniques for dealing with the modeling, solution, and analysis of multiple and conflicting objectives in constrained linear optimization problems. The types of multiobjective linear problems requiring a goal programming solution have been expanded and defined considerably since Chames and Cooper introduced the concept of “goal programming” [l]. As goal programming models were improved to more accurately reflect the decision environment they were designed to model, complications inevitably arose. One complication concerned the weighting of goals in the objective functions [2]. The problem that arose was one of finding a valid means by which to calculate representative weightings. One approach to avoid this difficulty is to eliminate the mathematical weighting from the model. With this approach, a goal programming problem becomes a lexicographic goal programming problem. The goals in the lexicographic problem are not differentiated by any weighting system, but instead are ordinally ranked in order of preference. To solve the lexicographic ,programming problem, decision makers have a choice of two approaches: Linear
goal
programming
(1) the multiphase simplex method, or (2) the sequential linear goal programming
(1) A further reduction in tableau elements will permit larger-scale GP problems to be solved on an existing computer system, thus increasing the problem-solving capabilities of a given computer system. (This reduction is particularly important with the increased utilization of microcomputers.) (2) A reduction in tableau elements permits solutions for those existing GP problems and future GP problems that cannot be solved with existing computer facilities because of their size. (3) A reduction in the tableau elements necessary to obtain a solution reduces computational tedium, which can enhance an educator’s presentation of GP concepts to students. In retrospect, there is a need for an algorithm that can solve not only weighted goal programming problems but also GP problems that use the lexicographic approach. Ideally, this new algorithm would also substantially reduce the required elements in the computational tableaus so as to minimize computer space usage and thus allow larger-scale GP problems to be solved. Such an algorithm has been developed by Schniederjans and Kwak [lo] and descriptively presented and illustrated in their study by solving a weighted GP problem. The results of their study demonstrated the ability of their solution technique to solve the weighted GP problem and do so with a reduction in the required computational elements for each tableau. The purpose of this study is to illustrate the same alternative GP solution procedures ability to solve the lexicographic GP problem, thus demonstrating its broader base of solution application. In addition,
method.
The multiphase simplex method is basically a modified version of Dantzig’s simplex method for solving linear programming minimization problems [3]. The two most common versions of the multiphase simplex method are by Lee [4] and Ignizio [5, 61. The sequential linear goal programming method’s major feature is that it allows GP problems to be run on conventional linear programming (LP) computer programs. Kornbluth [7] originally described the sequential linear GP algorithm while Dauer and Krueger [S], and Arthur and Ravindran [9] improved its efficacy. 101
N. K. KWAK and MARC J. SCHNIEDEWANS
102
Table l(a). Initial tableau
E Priority
Z
OPO
0PO OPl 09
I
+
d4 + d5 + dl + d2
I
OP3
d4
RHS
-
-
x1
x2
dl
d*
d3
d4
d5
0
0
0
1
1
1
0
0
-34
2
-72
06
-12
1
-7
0
0
-10
1
0
this article provides a summarization of the descriptive solution procedure discussed in Ref. [lo]. The summarization is provided in a flowchart format which enhances its educational utilization. This study also presents a comparison of the Schniederjans and Kwak algorithm with two other commonly used GP algorithms. The purpose of the comparison is to clearly show the reduction in computation possible when using the proposed GP algorithm over other existing algorithms. AN ILLUSTRATIVE
0 0
constraint in the objective function. problem is the resulting lexicographic minimize
Z = OP, d; + OP, d; + P1 d; + P2 d? + PX d; subject to 2x, + 3x, + dq - d: = 34 6x, + 4x2 + d; - d; = 72
EXAMPLE
Problem statement Assume we have the following situation:
The following formulation:
x,+x,+d;-d:= linear GP problem
12
x2 + d; - d: = 7 x,+d;-d;=
10
minimize Z = PI d,-i- P2 d;+
P3 d;
subject to 2x, + 3x, < 34 6x, + 4x, < 72 x,+xz+d;-d:=
12
x2 + d; - df = 7 x, + d; - d: = 10 xj, d;, d; > 0
(i = 1, 2, 3;j = 1, 2)
and Pk = preemptive
priority levels (where P, %- P2 + P3)
xj = decision
and Xi, dy , dt 3 0. Since the linear systems constraints are strict requirements in the problem formulation, they are assigned a priority level, PO, which indicates they should be forced out of the solution basis before all others.? The negative deviational variables, d; and d;, are placed in the objective function to be consistent with the direction of the original inequality in the linear systems constraint (i.e. <, thus allowing negative deviation from the right-hand-side value). Since d; and d; were not originally part of the objective function, a zero is attached to their contribution to Z in the model.$ Initial tableau setup Given the lexicographic problem, the next step to finding its solution would be to structure the problem
variables
d;, df = deviational
variables
Z =total minimized
deviation.
This problem can be converted into a lexicographic GP problem by (1) converting the remaining linear systems constraints to goal constraints; (2) eliminating any weighting of the goals; and (3) ranking each goal
t The treatment of linear systems constraints in a GP problem was not addressed in the earlier study [ 101. The approach demonstrated here can be used on any type of GP problem formulation. $ This zero value should not be misinterpreted as a mathematical weight. Similar treatment to the decision variables in this model and other simplex models are common practice.
10.
11.
Other eleslents are obtained by: new element = old elemant - (the product of two comer elwents/pivot).
12.
Design a new tableau. Interchange the pivot row and C01lKtn Others variables. remain the same.
ing new element.
Divide the old pivot row elements by the pivot and change the sign to obtain the
I
Total absolute deviation element is found by ~itiplyiog the urioritv
13
I
4.
3
I
2.a.
+
..-*I
Are there any negative elearents in the first row under column variables?
Select another negative eleaent in the first column.
I
l
elements
Exiunine .b& Cir-r
4.a.
5.a.
NO I+l
t
I
Fig. 1. Flowchart for computation procedure.
Exaaine the ratios, The pivot element comes front the smallest ratio and is circled.
them into the top row corresponding elements.
Are there positive elements in the pivot row?
priority. Pivot elerent comes from this row.
Are there aoY negative elements in the first Column?
I
L
Yes
Select largest positive elesent with the highest priority in the first C5lUMIl. Pivot element comes from this row.
Take t e1emcnr> LtllU first colunw :.._ . . __^..&_
c -I
1
104
N. K. KWAKand MARCJ. SCHNIEDERJANS Table l(b). Second tableau RHS
ciority
0PO
I
f H
*2
d;
d;
d;
d;
d;
0
0
1
1
1
0
0
d-i -10
1/3
05/3
0
0
0
1
-l/3
12
116
-2f3
0
0
0
0
-l/6
0
2
X1 +
OPI
dI
0
116
I/3
1
0
0
0
-l/b
09
dZ
-7
0
1
0
I
0
0
0
or3
d:
0
0
1
0
-l/6
*
2
l/6
-213
into the initial simplex tableau. This is accomplished by the follo~ng steps:? (1) Express goal and systems constraints in terms of a positive deviational variable, d’. (2) Place decision variables followed by negative deviational variables (i.e. nonbasic variables) in column headings. (3) Express the existence of the objective function variables in the objective function “2” row. (4) List priorities (and weights)* in the “Priority” column. The initial tableau for the lexico~aphic problem is presented in Table l(a). Examining the Priority column, it can be seen that all of the weights attached to priority levels are zero. This is due to the fact that there are no positive deviational variables in the objective function. The values under the nonbasic variables in the objective function row (the Z row) are similarly weighted. From this it is apparent that this solution technique can easily accommodate either the weighted or lexicographic GP problem with no additional considerations as to the nature of the problem. Solution procedure The solution procedure, first introduced in Ref. [lo], can be summarized into the step-by-step flowchart presented in Fig. 1. Since the problem has been formulated into its initial tableau (step l), the next step is to examine the “RHS”’ column elements for negative values because the basic variables cannot be negative (step 2). As can be seen in Table l(a), all RHS coiumn elements are negative. Next, we select the greatest negative value at the highest priority level (i.e. -72 at PO) as our pivotal row (step 3). This row determines the variable to leave the solution basis (i.e. d:). To determine the variable to enter the solution basis (step 4), we select only the positive values in the pivot row (step 5) and divide them into t It is suggested that readers review Ref. 1101for a more complete explanation. $ The le~co~ph~c problem weighting of priori&s w&l consist of either a one (d- is in the objective fun~on) or zero (d- is not in the objective function).
the elements in the Z row (step 6). The smallest ratio determines the pivot column and the variable to enter the solution basis (step 7). In this problem the numerator of each ratio is zero, resulting in three ratios of zero (O/6, O/4 and O/l). If the numerator were any other value than zero, the smallest ratios will occur where the denominator is the largest. As such, the pivot column can be selected based on the largest positive value in this type of zero-tie situation, Having found the pivot row and pivot column, the next step in the solution procedure is to design the next tableau (step 8). In designing the new tableau, the exiting and entering variables to the solution basis must be exchanged. Note in Table l(b), the priority column has no priority value attached to the entering variable .x1, This is because x1 is not in the objective function and thus has no priority. Note also that d: in the RHS row has a slash (/) placed over it. This is to remind problem solvers that d; will not reenter the solution basis because d: is an artificial variable and does not exist in the original problem formulation. Consistent with the Dantzig simplex method, artificial variables leaving the solution basis are never allowed to return. Indeed, the columns headed by artificial variables may actually be dropped from the tableau to save needless computation and complication. The remaining elements in Table l(b) can be obtained by following the solution steps in Fig. 1. Those procedures are repeated until all basic variables become nonnegative. The optimal solution is presented in Table l(e). The solution basis reveals that x1 = 13/2, x2 = 7, d: = 3/2, d; = 5, and d; = 712, with a Z = 7/2. To determine the underachievement of the goals at each priority level, we would have to su~titute the basic deviational variables into the lexicographic problem’s objective function. Since d: is not in the objective function and as stated, d; makes a zero contribution to Z, then d, at P3 makes the only contribution to Z. In this case only Px has a positive value of 712, which is reflected in the solution basis for d; and Z. COMPARATIVE ANALYSIS To demonstrate the superiority of the ~h~~e~ans and Kwak algorithm in reducing the tableau element
105
An alternative solution method for goal programming problems Table l(c). Third tableau
Priority
Z
0
x2
6
Xl
8
OPl
+ dl
2
OP2
d;
-1
OP3
+ d3
-2
-l/5 3/10 l/l0 -l/5 3110
l/5
0
0
0
-315
-215
0
0
0
0
-3110
l/5
1
0
0
0
-l/l0
315
0
1
0
0
0 l/5
0
0
1
215
-3/10
315
-215
Table l(d). Fourth tableau 1
/Priority i zI O 0
IYZ(710 1
I
OPl
0
1
0
0
1
1
0
-1
0
-315
1
0
/
x1
j 13/2
1
0
l/2
0
3/2
0
0
-312
(
d;
]
)
0
l/2
1
l/2
0
0
-l/2
0
0
3/2
OP2
d5
5
1
-3
0
-5
OP3
df
-712
0
I/?
0
312
computations, a comparative anaysis was undertaken. The Ignizio [6] multiphase simplex method and the Arthur and Ravindran [9] sequential linear goal programming methods were selected for comparative purposes. A sample of 25 GP problems of different sizes was randomly selected in part from Lee [ 1 l]
215
-312
and Schniedejans [ 121. The 25 problems are defined on Table 2 by their number of decision variables, number of goal constraints, and number of priorities. FORTRAN computer programs of the Ignizio [6], Arthur and Ravindran 191, and Schniederjans and Kwak [lo] algorithms were developed, verified, and
c Table l(e). Fifth
01
5
tableau and optimal
solution
RHS
Z
OPl OPO lP3
7/2
0
-l/2
1
-312
1
-2/5
3/2
x2
7
0
0
0
-1
0
-3f5
1
Xl + dl
13/2
0
l/2
0
312
0
0
-312
312
0
l/2
1
l/2
0
0
-l/2
5
1
-3
0
-5
0
0
5
712
0
-l/2
0
-3/2
1
-2J5
3/2
d5 d3
106
N.
K. KWAK
Table
Prob. NO.
No. of Decision Variables
No. of Goal Constraints
and MARC J. SCHNIEDERJANS
2. Comparison
No. of Priorities
of GP problem compu~tion
Ignizi0 No. of Tableaus
Total Tableau Elements
Algorithm Arthur & Ravindran No. of Total Tableaus Tableau Elements
Schniederjans No. of Tableaus
& Kwak Tot.%1 Tableau Elements
3’ 4
4 3
108 80
3
2 3 3
4 3 4
5 5 5
4 5 6
168 252 280 480
4 3 4 4 3 4
60 52 60 144 85 It30
2 2 4 b
48 60 112 240
7 8
4 4
4 5
6 6
6 R
594 I380
5 7
210 612
4 7
180 420
9 10 I1 12 13 14
5 5 15 20 20 20
5 10 20 20 20 24
6 6 10 10 10 10
8 14 2.0 27 24 33
12 14 * 27 24 33
680 2004 * 24772 21666 40118
8 10 15 21 24 34
528 1760 11345 18081 20664 38250
15 16 17 IS 19 20 21 22 23 24
25 28 28 35 30 45 40 50 65 80
25 30 30 30 32 60 60 60 80 100
10 10 10 15 15 15 15 15 15 15
41 58 63 74 61 95 103 111 142 214
73185 136880 148680 203130 174887 719625 780225 924075 1902090 4454410
* 78 63 * 60 117 * 104 * *
25
90
100
15
246
5403390
*
:
22
3
2
4 5 6
*so1utton
not
2 3
4
968 3584 15600 33210 29520 50490
* 197820 120020 * 128778 747525 * 524373 * * *
2 3
45 48
37 43 63 70 54 84 95 101 127 181
49062 78647 115227 132370 108702 517524 585295 68387 1450467 3308861
216
4166856
I
optimal
used to solve the 2.5 sample GP problems. Except as noted, each algorithm generated the same optimal solution. The number of tableaus and total tableau elements (i.e. in all tableaus) necessary to obtain an optimal solution were recorded by algorithm on Table 2. In examining the algorithm columns in Table 2, several comparative observations can be made. First, the Schniedejans and Kwak algorithm required a fewer number of GP tableaus to reach an optimal solution in 21 out of 25 problems when compared to the Ignizio algorithm and 13 out of 18 problems (i.e. 7 of the 25 had to be excluded because the algorithm did not generate an optimal solution) when compared to the Arthur and Ravindran algorithm. Second, the Schniederjans and Kwak algorithm required a fewer number of total GP tableau elements to reach an optimal solution in 25 out of 25 problems when compared to the Ignizio algorithm, and 16 out of 18 problems when compared to the Arthur and Ravindran algorithm. Similar results to these two observations were obtained by Olson [4, 131. Third, from its failure to obtain solutions in 7 of the 25 problems, it appears the Arthur and Ravindran algorithm is more limited in the type of problem it can solve than the Ignizio and the Schniedetjans and Kwak algorithms. Similar results were obtained by Olson [ 141 and corrective measures in their algorithm are currently under study by Arthur and Ravindran. Finally, it should be observed that the Ignizio algorithm required less tableaus (i.e. note Problem No. 14) to generate a solution, and the Arthur and Ravindran algorithm required less tableaus and less total tableau elements (i.e. note Problem Nos. 5, 6, and 14), than the Schniedesjans and Kwak algorithms.
This occasional occurrence can be, in part, attributed to the type of GP problem and was observed by Olson [ 141 to be dependent on a number of factors, including the number of positive and negative deviational variables that are basic in the optimal solution. CONCIXJDING REMARKS This study has illustrated the use of an alternative goal programming solution method to solve nonweighted lexicographic GP problems. The alternative GP solution method used in this article has been converted into a Fortran computer program [15]. The computer program can be used to solve GP problems consistent with the solution procedure presented in Fig. 1. The comparative analysis presented has demonstrated superiority of the Schniedejans and Kwak GP algorithm in reducing the number of tableaus and tableau elements required to determine an optimal solution in most problem situations. The analysis clearly illustrated the computational efficacy difference between the GP algorithms and verifies the statements presented in Ref. [lo] and subsequent comments [16, 171. The randomly selected 25 problems, though, are not a definitive sample. As observed in the comparative analysis, some problem situations permit the alternative GP algorithms to match or exceed the tableau element reduction capabilities of the Schniederjans and Kwak algorithm. It is suggested that further research be conducted to determine first, if classes of GP problems can be identified and, second, if a particular GP algorithm can be used to more easily solve these problems.
An alternative
solution
method
REFERENCES 1. A. Charnes and W. W. Cooper, Management Models and the Industrial Application of Linear Programming, Vols. 1 and 2. Wiley, New York (1961). in goal pro2. J. P. Ignizio, Advances and applications gramming. Working Paper, The Pennsylvania State University ( 1980). 3. G. B. Dantzig, A. Orden and P. Wolfe, The generalized simplex method for minimizing a linear form under linear inequality restraints. Paczy .I Math. 5, 35-47 (1955). 4. D. L. Olson, Computational comparison of goal programming algorithms. Presented at the October Joint National ORSA/TIMS meeting, San Diego, California (1982). 5. J. P. Ignizio, Goal Programming and Extensions. Heath, Lexington, Massachusetts (1976). 6. J. P. Ignizio, Linear Programming in Single and Multiple Objective Systems. Prentice-Hall, Englewood Cliffs, New Jersey (1982). 7. J. S. H. Kombluth, A survey of goal programming. OMEGA 1, 193-205 (1973). 8. J. P. Dauer and R. J. Krueger, An iterative approach to goal programming. Opl. Res. Q. 28, 671-681 (1977).
for goal programming
problems
107
9. J. L. Arthur and A. Ravindran, An efficient goal programming algorithm using partitioning and variable elimination. Mgnzt. Sci. 24, 867-868 (1978). 10. M. J. Schniedejans and N. K. Kwak, An alternative solution method for goal programming problems: a tutorial. J. Opl. Res. Sot. 33, 247-25 1 (1982). 11. S. M. Lee, Goal Programming for Decision Analysis. Auerbach, Philadelphia, Pennsylvania (1972). 12. M. J. Schniedejans, Linear Goal Programming. Petrocelli, Princeton, New Jersey (1984). 13. D. L. Olson, A comparison of goal programming algorithms. J. Opl. Res. Sot. 35, 347-354 (1984). 14. D. L. Olson, Computational comparison of goal programming algorithms on a minicomputer. Working Paper, Texas A&M University, Texas (1983). 15. N. K. Kwak and M. J. Schniedejans, Introduction to Mathematical Programming. Kreiger, Melbourne, Flor-
. . , ^^ .
ida (lY84).
16. N. K. Kwak and M. J. Schniedejans, An alternative method for solving goal programming problems: a reply. J. Opl. Res. Sot. 33, 359-360 (1982). 17. M. J. Schniedejans and N. K. Kwak, Comments on a note on computational methods in lexicographic linear goal programming. J. Opl. Res. Sot. 34,663-664 (1983).