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LINEAR PROGRAMMING FORMULATION PROBLEMS AND SOLUTIONS

7-14 The Electrocomp Corporation manufactures two electrical products: air conditioners and largefans. The assembly process for each is similar in that both require a certain amount of wiring anddrilling. 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$t production period% "4& hours ofwiring time are a'ailable and up to 14& hours of drilling time maybe used. Each air conditionersold yields a profit of ("). Each fan assembled may be sold for a (1) profit. *ormulate and sol'ethis +, production mi$ situation to find the best combination of air conditioners and fans thatyields the highest profit. se the corner point graphical approach.

+et 1/ the number of air conditioners scheduled to be produced"/ the number of fans scheduled to be produced

0a$imie ")1 2 1)" 3ma$imie profit5ub6ect to: !1 2 "" "4& 3wiring capacity constraint

"1 2 " 14& 3drilling capacity constraint1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ 4& "/ & ,rofit / (1%;&&

7-1) Electrocomp

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7-1 candidate for mayor in a small town has allocated (4&%&&& for last-minute ad'ertising in thedays preceding the election. Two types of ads will be used: radio and tele'ision. Each radio adcosts ("&& and reaches an estimated !%&&& people. Each tele'ision ad costs ()&& and reaches anestimated 7%&&& people. =n planning the ad'ertising campaign% the campaign manager would lieto 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$imie !%&&&1 2 7%&&&" 3ma$imie e$posure5ub6ect to: "&&1 2 )&&" 4&%&&& 3budget constraint

1 8 1& 3at least 1& radio ads purchased" 8 1& 3at least 1& tele'ision ads purchased1 8 " 3B of radio ads 8 B of tele'ision ads

1% " 8 & 3non-negati'ity constraints

*or solution purposes% the fourth constraint would be rewritten as:1 F " 8 &

9ptimal 5olution: 1/ 17) "/ 1& E$posure / );)%&&& people

7-17 The 9utdoor *urniture Corporation manufactures two products% benches and picnic tables% foruse in yards and pars. The firm has two main resources: its carpenters 3labor force and a supplyof redwood for use in the furniture. #uring the ne$t production cycle% 1%"&& hours of labor area'ailable under a union agreement. The firm also has a stoc of !)&& feet of good-qualityredwood. Each bench that 9utdoor *urniture produces requires 4 labor hours and 1& feet ofredwoodG each picnic table taes labor hours and !) feet of redwood. Completed benches willyield a profit of (; each% and tables will result in a profit of ("& each. Dow many benches andtables 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$imie ;1 2 "&" 3ma$imie profit5ub6ect to: 41 2 " 1%"&& 3labor hours constraint

1&1 2 !)" !%)&& 3redwood capacity constraint1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ "".) "/ ") ,rofit / ("%?".)&

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7-1? The dean of the Hestern College of >usiness must plan the schooleta )s produced during the production period. lpha 4s generate (1%"&& profitper unit% and >eta )s yield (1%?&& each. #etermine the most profitable number of each model ofminicomputer 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$imie 1%"&&1 2 1%?&&" 3ma$imie profit5ub6ect to: "&1 2 ")" / ?&& 3full employment% ) worers $ 1& hours

1 8 1& 3mae at least 1& lpha 4 computers" 8 1) 3mae at least 1) >eta ) computers

1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ 1& "/ "4 ,rofit / ())%"&&

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7-"& winner of the Te$as +otto has decided to in'est ()&%&&& per year in the stoc maret. nderconsideration are stocs for a petrochemical firm and a public utility. lthough a long-range goalis to get the highest possible return% some consideration is gi'en to the ris in'ol'ed with thestocs. ris inde$ on a scale of 1I1& 3with 1& being the most risy is assigned to each of thetwo 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 Estimated @eturn @is =nde$

,etrochemical 1"J ;tility J 4

The in'estor would lie to ma$imie the return on the in'estment% but the a'erage ris inde$ ofthe in'estment should not be higher than . Dow much should be in'ested in each stoc Hhat isthe 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$imie .1"1 2 .&" 3ma$imie return on in'estment5ub6ect to: 1 2 " )&%&&& 3limit on total in'estment

!1 F "" & 3a'erage ris cannot e$ceed 1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ ("&%&&& "/ (!&%&&& @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 hisattitude 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 ?Jis generated. *ormulate this as an +, problem and find the optimal solution. Dow much shouldbe in'ested in each stoc Hhat is the a'erage ris for this in'estment Hhat is the estimatedreturn 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 ;1

2 4"

3minimie total ris5ub6ect to: 1 2 " )&%&&& 3limit on total in'estment

.&41 F .&"" 8 & 3a'erage return must be at least ?J1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ (1%.7 "/ (!!%!!!.!! Total ris / "?!%!!!.!! 3whichequates to an a'erage ris of "?!%!!!.!!A)&%&&& / ).7.The total return would be (4&&& 3.1" $ 1%.7 2 .&; $ !!%!!!.!!% which 6ust happens tobe a return of e$actly ?J 3(4&&&A()&%&&&.

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7-"4 The stoc broerage firm of >lan% +eibowit% and Heinberger has analyed and recommendedtwo stocs to an in'estors< club of college professors. The professors were interested in factorssuch as short term growth% intermediate growth% and di'idend rates. These data on each stoc areas follows:

5toc

*actor+ouisiana Las and,ower

Trime$ =nsulationCompany

5hort term growthpotential% per dollarin'ested

.! ."4

=ntermediategrowth potential3o'er ne$t threeyears% per dollarin'ested

1.7 1.)

#i'idend ratepotential 4J ?J

Each member of the club has an in'estment goal of 31 an appreciation of no less than (7"& inthe short term% 3" an appreciation of at least ()%&&& in the ne$t three years% and 3! a di'idendincome of at least ("&& per year. Hhat is the smallest in'estment that a professor can mae tomeet 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 1 2 " 3minimie total in'estment5ub6ect to: .!1 2 ."4" 8 7"& 3appreciation in the short term1.71 2 1.)&" 8 )%&&& 3appreciation in ne$t three years.&41 2 .&?" 8 "&& 3di'idend income per year

1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ (1%!); "/ (1%?1?.1? Total in'estment / (!%177.1?

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7-") Hoofer ,et *oods produces a low-calorie dog food for o'erweight dogs. This product is madefrom 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 ofMitamin ". pound of beef contains 1& units of Mitamin 1 and 1" units of Mitamin ". pound ofgrain 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 ineach 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 .;&1 2 .&" 3minimie cost per pound of dog food5ub6ect to: 1 2 " / 1 3total weight should be one pound

1&1 2 " 8 ; 3at least ; units of 'itamin 1 in a pound1"1 2 ;" 8 1& 3at least 1& units of 'itamin " in a pound

1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ .7) "/ .") Cost / (.?")

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59+T=9K5 T9 5E+ECT ,@9>+E05 *@90 CD,TE@ ?

?-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 throughthree departments: carpentry% painting% and finishing. The table below contains all rele'ant

information concerning production times per cabinet produced and production capacities foreach operation per day% along with net re'enue per unit produced. The firm has a contract withan =ndiana distributor to produce a minimum of !&& of each cabinet per wee 3or & cabinets perday. 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.

Cabinet 5tyleCarpentry

3DoursACabinet,ainting

3DoursACabinet*inishing

3DoursACabinetKet @e'enue

per Cabinet 3(

*rench ,ro'incial ! 1.) .7) "?#anish 0odern " 1 .7) ")

#ept. capacity 3hrs !& "&& 1")

+et 1/ the number of *rench ,ro'incial cabinets produced each day"/ the number of #anish 0odern cabinets produced each day

0a$imie "?1 2 ")" 3ma$imie re'enue5ub6ect to: !1 2 "" !& 3carpentry hours a'ailable

1.)1 2 " "&& 3painting hours a'ailable.7)1 2 .7)" 1") 3finishing hours a'ailable

1 8 & 3contract requirement on *.,. cabinets

" 8 & 3contract requirement on #.0. cabinets1% " 8 & 3non-negati'ity constraints

9ptimal 5olution: 1/ & "/ ;& @e'enue / (!%;!&

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?-" 3=n'estment decision problem The Deinlein and Nrarnpf >roerage firm has 6ust been instructedby one of its clients to in'est (")&%&&& for her money obtained recently through the sale of landholdings in 9hio. The client has a good deal of trust in the in'estment house% but she also has herown ideas about the distribution of the funds being in'ested. =n particular% she requests that thefirm 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 high-

ris% high-yield nursing home stoc.5ub6ect to these restraints% the client

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?-! 3@estaurant wor scheduling problem. The famous O. 5. Chang @estaurant is open "4 hours aday. 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 worersneeded during the si$ periods into which the day is di'ided. Chang

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?-4 3nimal feed mi$ problem The >attery ,ar 5table feeds and houses the horses used to pulltourist-filled carriages through the streets of Charleston

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?- 30edia selection problem The ad'ertising director for #i'ersey ,aint and 5upply% a chain offour retail stores on Chicago

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?-11 3College meal selection problem Nathy @oniger% campus dietician for a small =daho college% isresponsible for formulating a nutritious meal plan for students. *or an e'ening meal% she feelsthat 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 " gramsof proteinG and 3) no more than )& grams of carbohydrates. 9n a particular day% @onigereans 1"? !." &.? 7 "? &.)?5pinach 11? 14.1 1.4 14 1; 1.17,otatoes "7; "." &.) ? ! &.!!

Hhat combination and amounts of food items will pro'ide the nutrition @oniger requires at theleast 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 mealEtc.% down to 7/ the number of pounds of potatoes per student in the e'ening meal

0inimie .1 2 ".!)" 2 1.1)! 2 ".")4 2 .)?) 2 1.17 2 .!!7

5.T. 3Cal. ";)1 2 1"1" 2 !;4! 2 !)?4 2 1"?) 2 11? 2 "7;7 8 ;&&

3Cal. ";)1 2 1"1" 2 !;4! 2 !)?4 2 1"?) 2 11? 2 "7;7 1)&&

3=ron ."1 2 ."" 2 4.!! 2 !."4 2 !.") 2 14.1 2 "."7 8 4

3*at 11 2 ;" 2 ;! 2 .)4 2 .?) 2 1.4 2 .)7 )&

3,rotein 11 2 ?1" 2 74! 2 ?!4 2 7) 2 14 2 ?7 8 "

3Carbs. ""1 2 "?) 2 1; 2 !7 )&

1% "% !% 4% )% % 7 8 &

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?-1" 3Digh tech production problem Puitmeyer Electronics =ncorporated manufactures the followingsi$ microcomputer peripheral de'ices: internal modems% e$ternal modems% graphics circuitboards% C# dri'es% hard dis dri'es% and memory e$pansion boards. Each of these technicalproducts requires time% in minutes% on three types of electronic testing equipment% as shown inthe table the following table:

=nternal0odem

E$ternal0odem

Circuit>oard

C##ri'e

Dard#ri'e

0emory>oard

Test de'ice 1 7 ! 1" 1? 17Test de'ice " " ) ! " 1) 17Test de'ice ! ) 1 ! " ; "

The first two test de'ices are a'ailable 1"& hours per wee. The third 3de'ice ! requires morepre'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 unitsof each product as it can manufacture. The table that follows summaries the re'enues and

material costs for each product:

#e'ice@e'enue ,ernit 5old 3(

0aterial Cost,er nit 3(

=nternal modem "&& !)E$ternal modem 1"& ")Lraphics circuit board 1?& 4&C# dri'e 1!& 4)Dard dis dri'e 4!& 17&0emory e$pansion board "& &

=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 whichequipment

+et 1/ the number of internal modems scheduled for manufacture each wee"/ the number of e$ternal modems scheduled for manufacture each weeEtc.% down to / the number of mem. e$pansion boards scheduled for mfg. each wee

0a$imie 11.!)1 2 ;".;)" 2 1!).)&! 2 ?".)&4 2 "4;.?&) 2 1;1.7)

5.T. 71 2 !" 2 1"! 2 4 2 1?) 2 17 7"&&

"1 2 )" 2 !! 2 "4 2 1)) 2 17 7"&&

)1 2 1" 2 !! 2 "4 2 ;) 2 " &&&

1% "% !% 4% )% 8 &

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?-1) 30aterial blending problem malgamated ,roducts has 6ust recei'ed a contract to constructsteel body frames for automobiles that are to be produced at the new Qapanese factory inTennessee. The Qapanese auto manufacturer has strict quality control standards for all of itscomponent subcontractors and has informed malgamated that each frame must ha'e thefollowing steel content:

0aterial 0inimum ,ercent 0a$imum ,ercent

0anganese ".1 ".!5ilicon 4.! 4.Carbon ).&) ).!)

malgamated mi$es batches of eight different a'ailable materials to produce one ton of steelused 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-tonload of steel so that malgamated meets its requirements while minimiing cost.

0aterial'ailable 0anganese3J 5ilicon3J Carbon3J ,ounds'ailable Cost ,er,ound 3(

lloy 1 7&.& 1).& !.& Ko limit &.1"lloy " )).& !&.& 1.& !&& &.1!lloy ! 1".& ".& & Ko limit &.1)=ron 1 1.& 1&.& !.& Ko limit &.&;=ron " ).& ".) & Ko limit &.&7Carbide 1 & "4.& 1?.& )& &.1&Carbide " & ").& "&.& "&& &.1"Carbide ! & "!.&& ").& 1&& &.&;

+et 1/ the number of pounds of alloy 1 in one ton of steel"/ the number of pounds of alloy " in one ton of steelEtc.% down to ?/ the number of pounds of carbide ! in one ton of steel

0inimie .1"1 2 .1!" 2 .1)! 2 .&;4 2 .&7) 2 .1& 2 .1"7 2 .&;?

5.T. 30n- .71 2 .))" 2 .1"! 2 .&14 2 .&)) 8 4"

30n-ma$ .71 2 .))" 2 .1"! 2 .&14 2 .&)) 4

35i-min .1)1 2 .!&" 2 ."! 2 .1&4 2 .&")) 2 ."4 2 .")7 2 ."!? 8 ?

35i-ma$ .1)1 2 .!&" 2 ."! 2 .1&4 2 .&")) 2 ."4 2 .")7 2 ."!? ;"

3C-min .&!1 2 .&1" 2 .&!4 2 .1? 2 ."&7 2 .")? 8 1&

3C-ma$ .&!1 2 .&1" 2 .&!4 2 .1? 2 ."&7 2 .")? 1&7

lloy " lim. " !&&Carbide 1 lim. )&

Carbide " lim. 7 "&&

Carbide ! lim. ? 1&&

Heighs 1 ton 1 2 " 2 ! 2 4 2 ) 2 2 7 2 ? / "&&&

1% "% !% 4% )% % 7% ? 8 &