Linear Programming Programming Formulation Formulation Exercises from from Textbook Textbook ISM 4400, Fall 2006 Page 1 !"4
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 7
7-14 7-14
The Electr Electrocom ocomp p Corporat Corporation ion manufa manufactu ctures res two two electric electrical al product products: s: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner taes ! hours of wiring and " hours of drilling. Each fan must go through " hours of wiring and 1 hour of drilling. #uring the ne$t production period% "4& hours ho urs of wiring time are a'ailable and up to 14& hours of drilling time maybe used. Each air conditioner sold yields a profit of ("). Each fan assembled may be sold for a (1) profit. *ormulate and sol'e this +, production mi$ situation to find the best combination of air conditioners and fans that that yields the highest profit. se the corner point graphical approach. ap proach. +et 1 / the number of air conditioners scheduled to be produced " / the number of fans scheduled to be produced
0a$imie 5ub6ect to:
") ! "
2 2 2
1) " %
"4 14 8 &
3ma$imie profit 3wiring capacity 3drilling capacity 3non-negati'ity
9ptimal 5olution: 1 / 4& " / & ,rofit / (1%;&&
7-1)
Electrocomp Electrocomp<ecause Electrocomp incurred an o'ersupply of fans in the preceding period% management also insists that no more than ?& fans be produced during this production period. @esol'e this product mi$ problem to find the new optimal solution. +et 1 / the number of air conditioners scheduled to be produced " / the number of fans scheduled to be produced
0a$imie 5ub6ect to:
") ! "
2 2 2
1) "
%
"4 14 8 " ? 8 &
3ma$imie profit 3wiring capacity constraint 3drilling capacity constraint 3aAc contract constraint 3ma$imum B of fans 3non-negati'ity constraints
9ptimal 5olution: 1 / 4& " / & ,rofit / (1%;&&
Linear Programming Formulation Exercises from Textbook ISM 4400, Fall 2006 Page " !"4
7-1
candidate for mayor in a small town has allocated (4&%&&& for last-minute ad'ertising in the days preceding the election. Two types of ads will be used: radio and tele'ision. Each radio ad costs ("&& and reaches an estimated !%&&& people. Each tele'ision ad costs ()&& and reaches an estimated 7%&&& people. =n planning the ad'ertising campaign% the campaign manager would lie to reach as many people as possible% but she has stipulated that at least 1& ads of each type must be used. lso% the number of radio ads must be at least as great as the number of tele'ision ads. Dow many ads of each type should be used Dow many people will this reach +et 1 / the number of radio ads purchased " / the number of tele'ision ads purchased
0a$imie 5ub6ect to:
!%&&& "&&
2 2
7%&&& )&&
4&%&& 8 1 8 1 8 8 &
%
3ma$imie e$posure 3budget constraint 3at least 1& radio ads purchased 3at least 1& tele'ision ads 3B of radio ads 8 B of tele'ision 3non-negati'ity constraints
*or solution purposes% the fourth constraint would be rewritten as: 1 F " 8 & 9ptimal 5olution: 1 / 17)
7-17
" / 1&
E$posure / );)%&&& people
The 9utdoor *urniture Corporation manufactures two products% benches and picnic tables% for use in yards and pars. The firm has two main resources: its carpenters 3labor force and a supply of redwood for use in the furniture. #uring the ne$t production cycle% 1%"&& hours of labor are a'ailable under a union agreement. The firm also has a stoc of !)&& feet of good-quality redwood. Each bench that 9utdoor *urniture produces requires 4 labor hours and 1& feet of redwoodG each picnic table taes labor hours and !) feet of redwood. Completed benches will yield a profit of (; each% and tables will result in a profit of ("& each. Dow many benches and tables should 9utdoor *urniture produce to obtain the largest possible profit se the graphical +, approach. +et 1 / the number of benches produced " / the number of tables produced
0a$imie 5ub6ect to:
; 4 1&
2 2 2
"& !) %
1%"& !%)& 8 &
9ptimal 5olution: 1 / "".)
" / ")
3ma$imie profit 3labor hours constraint 3redwood capacity 3non-negati'ity constraints ,rofit / ("%?".)&
Linear Programming Formulation Exercises from Textbook ISM 4400, Fall 2006 Page ! !"4
7-1?
The dean of the Hestern College of >usiness must plan the school
0inimie 5ub6ect to:
"%)&&
2
!%&&&
2 % 9ptimal 5olution: 1 / 4&
7-1;
8 8 8 8 & " / "&
! "
3minimie faculty salaries 3schedule at least !& undergrad 3schedule at least "& grad courses 3schedule at least & total courses 3non-negati'ity constraints
Cost / (1&%&&&
05 Computer Corporation manufactures two models of minicomputers% the lpha 4 and the >eta ). The firm employs fi'e technicians% woring 1& hours each per month% on its assembly line. 0anagement insists that full employment 3i.e.% all 1& hours of time be maintained for each worer during ne$t montheta ) model. 05 wants to see at least 1& lpha 4s and at least 1) >eta )s produced during the production period. lpha 4s generate (1%"&& profit per unit% and >eta )s yield (1%?&& each. #etermine the most profitable number of each model of minicomputer to produce during the coming month. +et 1 / the number of lpha 4 computers scheduled for production ne$t month " / the number of >eta ) computers scheduled for production ne$t month
0a$imie 5ub6ect to:
1%"&& "&
2 2
1%?&& ")
% 9ptimal 5olution: 1 / 1&
/ ?& 8 1 8 1 8 & " / "4
3ma$imie profit 3full employment% ) worers $ 1& 3mae at least 1& lpha 4 computers 3mae at least 1) >eta ) computers 3non-negati'ity constraints
,rofit / ())%"&&
Linear Programming Formulation Exercises from Textbook ISM 4400, Fall 2006 Page 4 !"4
7-"&
winner of the Te$as +otto has decided to in'est ()&%&&& per year in the stoc maret. nder consideration are stocs for a petrochemical firm and a public utility. lthough a long-range goal is to get the highest possible return% some consideration is gi'en to the ris in'ol'ed with the stocs. ris inde$ on a scale of 1I1& 3with 1& being the most risy is assigned to each of the two stocs. The total ris of the portfolio is found by multiplying the ris of each stoc by the dollars in'ested in that stoc. The following table pro'ides a summary of the return and ris:
5toc ,etrochemical
Estimated @eturn 1" J
tility
@is =nde$ ; 4
The in'estor would lie to ma$imie the return on the in'estment% but the a'erage ris inde$ of the in'estment should not be higher than . Dow much should be in'ested in each stoc Hhat is the a'erage ris for this in'estment Hhat is the estimated return for this in'estment +et 1 / the number of dollars in'ested in petrochemical stocs " / the number of dollars in'ested in utility stocs 0a$imie 5ub6ect to:
.
2 2 F
!
. " %
)&%&& & 8 &
9ptimal 5olution: 1 / ("&%&&&
3ma$imie return on 3limit on total in'estment 3a'erage ris cannot e$ceed 3non-negati'ity constraints
" / (!&%&&&
@eturn / (4%"&&
The total ris is !&&%&&& 3; $ ("&%&&& 2 4 $ (!&%&&&% which yields an a'erage ris of 3!&&%&&&A)&%&&& / .
7-"1
@eferring to the Te$as +otto situation in ,roblem 7-"&% suppose the in'estor has changed his attitude about the in'estment and wishes to gi'e greater emphasis to the ris of the in'estment. Kow the in'estor wishes to minimie the ris of the in'estment as long as a return of at least ?J is generated. *ormulate this as an +, problem and find the optimal solution. Dow much should be in'ested in each stoc Hhat is the a'erage ris for this in'estment Hhat is the estimated return for this in'estment +et 1 / the number of dollars in'ested in petrochemical stocs " / the number of dollars in'ested in utility stocs
0inimie 5ub6ect to:
; .
2 2 F
4 . %
)&%&& 8 & 8 &
9ptimal 5olution: 1 / (1%.7
3minimie total ris 3limit on total in'estment 3a'erage return must be at least 3non-negati'ity constraints
" / (!!%!!!.!!
Total ris / "?!%!!!.!! 3which
equates to an a'erage ris of "?!%!!!.!!A)&%&&& / ).7. The total return would be (4&&& 3.1" $ 1%.7 2 .&; $ !!%!!!.!!% which 6ust happens to be a return of e$actly ?J 3(4&&&A()&%&&&.
Linear Programming Formulation Exercises from Textbook ISM 4400, Fall 2006 Page ) !"4
7-"4
The stoc broerage firm of >lan% +eibowit% and Heinberger has analyed and recommended two stocs to an in'estors< club of college professors. The professors were interested in factors such as short term growth% intermediate growth% and di'idend rates. These data on each stoc are as follows: 5toc +ouisiana Las and Trime$ =nsulation *actor 5hort term growth potential% per dollar in'ested =ntermediate growth potential 3o'er ne$t three years% per dollar in'ested #i'idend rate
,ower
.!
Company ."4
1.7
1.)
4J
?J
potential
Each member of the club has an in'estment goal of 31 an appreciation of no less than (7"& in the short term% 3" an appreciation of at least ()%&&& in the ne$t three years% and 3! a di'idend income of at least ("&& per year. Hhat is the smallest in'estment that a professor can mae to meet these three goals +et 1 / the number of dollars in'ested in +ouisiana Las and ,ower " / the number of dollars in'ested in Trime$ =nsulation Co. 0inimie 5ub6ect to:
. 1.7 .
2 2 2 2
. 1.)& . %
8 7" 8 )%&& 8 "& 8 &
9ptimal 5olution: 1 / (1%!);
3minimie total in'estment 3appreciation in the short term 3appreciation in ne$t three 3di'idend income per year 3non-negati'ity constraints
" / (1%?1?.1? Total in'estment / (!%177.1?
7-")
Hoofer ,et *oods produces a low-calorie dog food for o'erweight dogs. This product is made from beef products and grain. Each pound of beef costs (&.;&% and each pound of grain costs (&.&. pound of the dog food must contain at least ; units of Mitamin 1 and 1& units of Mitamin ". pound of beef contains 1& units of Mitamin 1 and 1" units of Mitamin ". pound of grain contains units of Mitamin 1 and ; units of Mitamin ". *ormulate this as an +, problem to minimie the cost of the dog food. Dow many pounds of beef and grain should be included in each pound of dog food Hhat is the cost and 'itamin content of the final product +et 1 / the number of pounds of beef in each pound of dog food " / the number of pounds of grain in each pound of dog food
0inimie 5ub6ect to:
. 1& 1"
2 2 2 2
. ; %
/ 8 8 8 &
3minimie cost per pound of dog food 1 3total weight should be one pound ; 3at least ; units of 'itamin 1 in a 1 3at least 1& units of 'itamin " in a 3non-negati'ity constraints
9ptimal 5olution: 1 / .7) " / .") Cost / (.?")
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 8
?-1
3,roduction problem Hinler *urniture manufactures two different types of china cabinets: a *rench ,ro'incial model and a #anish 0odern model. Each cabinet produced must go through three departments: carpentry% painting% and finishing. The table below contains all rele'ant information concerning production times per cabinet produced and production capacities for each operation per day% along with net re'enue per unit produced. The firm has a contract with an =ndiana distributor to produce a minimum of !&& of each cabinet per wee 3or & cabinets per day. 9wner >ob Hinler would lie to determine a product mi$ to ma$imie his daily re'enue. 3a *ormulate as an +, problem. 3b 5ol'e using an +, software program or spreadsheet. Carpentr y ! " !
Cabinet 5tyle *rench ,ro'incial #anish 0odern #ept. capacity 3hrs
,aintin g 1 1 "
*inishin g . . 1
Ket @e'enue per " "
+et 1 / the number of *rench ,ro'incial cabinets produced each day " / the number of #anish 0odern cabinets produced each day 0a$imie 5ub6ect to:
"? ! 1.) .
2 2 2 2
") " .
% 9ptimal 5olution: 1 / &
! "& 1" 8 8 8 & " / ;&
3ma$imie re'enue 3carpentry hours a'ailable 3painting hours a'ailable 3finishing hours a'ailable 3contract requirement on *.,. cabinets 3contract requirement on #.0. 3non-negati'ity constraints @e'enue / (!%;!&
?-"
3=n'estment decision problem The Deinlein and Nrarnpf >roerage firm has 6ust been instructed by one of its clients to in'est (")&%&&& for her money obtained recently through the sale of land holdings in 9hio. The client has a good deal of trust in the in'estment house% but she also has her own ideas about the distribution of the funds being in'ested. =n particular% she requests that the firm select whate'er stocs and bonds they belie'e are well rated% but within the following guidelines: 3a 0unicipal bonds should constitute at least "&J of the in'estment. 3b t least 4&J of the funds should be placed in a combination of electronic firms% aerospace firms% and drug manufacturers. 3c Ko more than )&J of the amount in'ested in municipal bonds should be placed in a highris% high-yield nursing home stoc. 5ub6ect to these restraints% the client
=n'estment +os ngeles municipal bonds Thompson Electronics% =nc. nited erospace Corp. ,almer #rugs Dappy #ays Kursing Domes
3a *ormulate this portfolio selection problem using +,. 3b 5ol'e this problem. +et 1 / dollars in'ested in +os ngeles municipal bonds " / dollars in'ested in Thompson Electronics ! / dollars in'ested in nited erospace 4 / dollars in'ested in ,almer #rugs ) / dollars in'ested in Dappy #ays Kursing Domes 0a$imie
.
2
5ub6ect to:
.
2
2
.
2
2
.
2
2
.
3ma$imie return on in'estment
2
")&%&&
3total funds a'ailable
.
-
.
-
.
-
.
-
.
8
&
3municipal bond restriction
-.4
2
.
2
.
2
.
-
.
8
&
3electronics% aerospace% drugs
&
3nursing home as a percent of
-.)
2 1%
"% !% 4% )
8
& 3non-negati'ity constraints
9ptimal 5olution: 1 / ()&%&&& " / (& ! / (& 4 / (17)%&&& ) / (")%&&& @9= / ("&%!&&
?-!
3@estaurant wor scheduling problem. The famous O. 5. Chang @estaurant is open "4 hours a day. Haiters and busboys report for duty at !0.% 7 0.% 11 0.% ! ,.0.% 7 ,.0.% or 11 ,.0.% and each wors an ?-hour shift. The following table shows the minimum number of worers needed during the si$ periods into which the day is di'ided. Chang
Time ! .0I7 .0. 7 .0I11 .0. 11 .0I! ,.0. ! ,.0I7 ,.0. 7 ,.0I11 ,.0. 11 ,.0I! .0.
>usboys @equired ! 1" 1 ; 11 4
+et i / the number worers beginning wor at the start of time period i 3i/1%"%!%4%)% 3min. staff 8
!
3period 1
8
1
3period "
8
1
3period !
8
;
3period 4
8
1
3period )
8
4
3period
8
&
3non0inimie
5ub6ect to:
1
2
"
2
!
2
4
2
)
1
1
2
2
:
2
:
2
:
"
"
2
!
!
2
4
4
2
)
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% % % % % 1 " ! 4 ) :
?-4
3nimal feed mi$ problem The >attery ,ar 5table feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston
#iet @equirement 3=ngredients > C # E CostAlb +et
*eed 0i$ Enrich ed Lrain ! 1 ) 1.) .) (&.14
9at ,roduc t " . ! 1 . (&.&;
0inera l ,roduc 1 . " 1. (&.17
1 / the number pounds of oat product per horse each day " / the number pounds of enriched grain per horse each day ! / the number pounds of mineral product per horse each day
0inimie s.t.
. " . ! .
2 2 2 2 2 2 2
. ! ) 1.) .
.
2 2 2 2 2 2 2 %
! . " 1.) !
%
8 8 8 8 8
8
" ; ? ) &
3minimie cost 3ingredient 3ingredient > 3ingredient C 3ingredient # 3ingredient E 3ma$imum feed per day 3non-negati'ity
0inimum #aily @equirement " ; ? )
?-
30edia selection problem The ad'ertising director for #i'ersey ,aint and 5upply% a chain of four retail stores on Chicago
1 / the number of newspaper ads placed " / the number of TM spots purchased
0inimie 5ub6ect to:
;") . .
2 2 2
"%&&& . . %
8 8 8 &
. .
3minimie cost 3city e$posure 3suburb e$posure 3non-negati'ity
?-11
3College meal selection problem Nathy @oniger% campus dietician for a small =daho college% is responsible for formulating a nutritious meal plan for students. *or an e'ening meal% she feels that the following fi'e meal-content requirements should be met: 31 between ;&& and 1%)&& caloriesG 3" at least 4 milligrams of ironG 3! no more than )& grams of fatG 34 at least " grams of proteinG and 3) no more than )& grams of carbohydrates. 9n a particular day% @oniger
=ron 3mgAlb &." &." 4.! !." !." 14.1 "."
Calorie sA ";) 1"1 !;4 !)? 1"? 11? "7;
*ood =tem 0il Lround 0eat Chicen *ish >eans 5pinach ,otatoes
Table of *ood Malues and Costs *at ,rotein Carbs 3gmAlb 3gmAlb . 1 1 "" ; ?1 & ; 74 & &.) ?! & &.? 7 "? 1.4 14 1; &.) ? !
CostA ,ound &.& ".!) 1.1) ".") &.)? 1.17 &.!!
Hhat combination and amounts of food items will pro'ide the nutrition @oniger requires at the least total food cost +et 1 / the number of pounds of mil per student in the e'ening meal " / the number of pounds of ground meat per student in the e'ening meal Etc.% down to 7 / the number of pounds of potatoes per student in the e'ening meal 8
;
1)
8
4
)
8
"
) 0inimie
.
5.T. 3Cal.
";)
1
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2
1"?
2
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1
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2
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.? 74
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2
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% % % % % %
1 " ! 4 ) 7
8
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?-1"
3Digh tech production problem Puitmeyer Electronics =ncorporated manufactures the following si$ microcomputer peripheral de'ices: internal modems% e$ternal modems% graphics circuit boards% C# dri'es% hard dis dri'es% and memory e$pansion boards. Each of these technical products requires time% in minutes% on three types of electronic testing equipment% as shown in the table the following table:
Test de'ice 1 Test de'ice " Te st de'ice !
=nternal 0odem 7 " )
E$ternal 0odem ! ) 1
Circuit >oard 1" ! !
C# #ri'e " "
Dard #ri'e 1? 1) ;
0emory >oard 1 1 "
The first two test de'ices are a'ailable 1"& hours per wee. The third 3de'ice ! requires more pre'enti'e maintenance and may be used only 1&& hours each wee. The maret for all si$ computer components is 'ast% and Puitmeyer Electronics belie'es that it can sell as many units of each product as it can manufacture. The table that follows summaries the re'enues and material costs for each product:
#e'ice =nternal modem E$ternal modem Lraphics circuit board C# dri'e Dard dis dri'e 0em ory e$pa nsion board
@e'enue ,er
0aterial Cost
nit 5old 3( "&& 1"& 1?& 1!& 4!& "&
,er nit 3( !) ") 4& 4) 17& &
=n addition% 'ariable labor costs are (1) per hour for test de'ice 1% (1" per hour for test de'ice ". and (1? per hour for test de'ice !. Puitmeyer Electronics wants to ma$imie its profits. 3a *ormulate this problem as an +, model. 3b 5ol'e the problem by computer. Hhat is the best product mi$ 3c Hhat is the 'alue of an additional minute of time per wee on test de'ice 1 Test de'ice " Test de'ice ! 5hould Puitmeyer Electronics add more test de'ice time =f so% on which equipment +et 1 / the number of internal modems scheduled for manufacture each wee " / the number of e$ternal modems scheduled for manufacture each wee Etc.% down to / the number of mem. e$pansion boards scheduled for mfg. each wee 0a$imie
11.!)
2
;".;)
2
1!).)&
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7
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?-1)
30aterial blending problem malgamated ,roducts has 6ust recei'ed a contract to construct steel body frames for automobiles that are to be produced at the new Qapanese factory in Tennessee. The Qapanese auto manufacturer has strict quality control standards for all of its component subcontractors and has informed malgamated that each frame must ha'e the following steel content:
0aterial 0anganese 5ilicon
0inimum ,ercent ".1 4.! ).&)
Carbon
0a$imum ,ercent ". 4. ).!)
malgamated mi$es batches of eight different a'ailable materials to produce one ton of steel used in the body frames. The table below details these materials. *ormulate and sol'e the +, model that will indicate how much of each of the eight materials should be blended into a 1-ton load of steel so that malgamated meets its requirements while minimiing cost. 0aterial 'ailable lloy 1 lloy " lloy ! =ron 1 =ron " Carbide 1 Carbide "
0angane se 7& )) 1" 1 ) & & &
Carbide !
5ilico n 1).& !&.& ".& 1&.& ". "4.& ").& "!.&&
Carbo n ! 1 & ! & 1?.& "&.& ").&
,oun ds Ko limit !& Ko limit Ko limit Ko limit ) "& 1&&
Cost ,er ,ound 3( &. &. &. &. &. &. &. &.&;
+et 1 / the number of pounds of alloy 1 in one ton of steel " / the number of pounds of alloy " in one ton of steel Etc.% down to ? / the number of pounds of carbide ! in one ton of steel 8
4
4
8
?
;
8
1
1
!
)
"
1
/
"& 0inimie
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5.T. 30n-
.7
30n-ma$
.7
1
35i-min
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lloy " lim.
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Carbide 1 lim.
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Carbide " lim.
7
Carbide ! lim. Heighs 1 ton
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