數(shù)據(jù)、模型與決策(運籌學)課后習題和案例答案004

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Chapter4

linearprogramming:formulationandapplications

ReviewQuestions

4.1-1 Determinewhichlevelsshouldbechosenofdifferentadvertisingmediatoobtainthemosteffectiveadvertisingmixforthenewcereal.

4.1-2 Theexpectednumberofexposures.

4.1-3 TVcommercialsarenotbeingusedandthatistheprimarymethodofreachingyoungchildren.

4.1-4 Theyneedtochecktheassumptionthatfractionalsolutionsareallowedandtheassumptionofproportionality.

4.2-1 Eachfunctionalconstraintinthelinearprogrammingmodelisaresourceconstraint.

4.2-2 Amountofresourceused≤Amountofresourceavailable.

4.2-3 1) Theamountavailableofeachlimitedresource.

2) Theamountofeachresourceneededbyeachactivity.Specifically,foreachcombinationofresourceandactivity,theamountofresourceusedperunitofactivitymustbeestimated.

3) Thecontributionperunitofeachactivitytotheoverallmeasureofperformance.

4.2-4 Thethreeactivitiesintheexamplesaredeterminingthemostprofitablemixofproductionratesfortwonewproducts,capitalbudgeting,andchoosingthemixofadvertisingmedia.

4.2-5 Theresourcesintheexamplesareavailableproductioncapacitiesofdifferentplants,cumulativeinvestmentcapitalavailablebycertaintimes,financialallocationsforadvertisingandforplanningpurposes,andTVcommercialspotsavailableforpurchase.

4.3-1 Forresource-allocationproblems,limitsaresetontheuseofvariousresources,andthentheobjectiveistomakethemosteffectiveuseofthesegivenresources.Forcost-benefit-tradeoffproblems,managementtakesamoreaggressivestance,prescribingwhatbenefitsmustbeachievedbytheactivitiesunderconsideration,andthentheobjectiveistoachieveallthesebenefitswithminimumcost.

4.3-2 Theidentifyingfeatureforacost-benefit-tradeoffproblemisthateachfunctionalconstraintisabenefitconstraint.

4.3-3 Levelachieved≥Minimumacceptablelevel.

4.3-4 1) Theminimumacceptablelevelforeachbenefit(amanagerialpolicydecision).

2) Foreachbenefit,thecontributionofeachactivitytothatbenefit(perunitoftheactivity).

3) Thecostperunitofeachactivity.

4.3-5 Theactivitiesfortheexamplesarechoosingthemixofadvertisingmedia,personnelscheduling,andcontrollingairpollution.

4.3-6 Thebenefitsfortheexamplesareincreasedmarketshare,minimizingtotalpersonnelcostswhilemeetingservicerequirements,andreductionsintheemissionofpollutants.

4.4-1 Distribution-networkproblemsdealwiththedistributionofgoodsthroughadistributionnetworkatminimumcost.

4.4-2 Anidentifyingfeatureforadistribution-networkproblemisthateachfunctionalconstraintisafixed-requirementconstraint.

4.4-3 Incontrasttothe≤formforresourceconstraintsandthe≥formforbenefitconstraints,fixed-requirementconstraintshavean=form.

4.4-4 Factory1mustship12lathes,Factory2mustship15lathes,Customer1mustreceive10lathes,Customer2mustreceive8lathes,andCustomer3mustreceive9lathes.

4.5-1 Twonewgoalsneedtobeincorporatedintothemodel.Thefirstisthattheadvertisingshouldbeseenbyatleast5millionyoungchildren.Thesecondisthattheadvertisingshouldbeseenbyatleast5millionparentsofyoungchildren.

4.5-2 Twobenefitconstraintsandafixed-requirementconstraintareincludedinthenewlinearprogrammingmodel.

4.5-3 Managementdecidedtoadoptthenewplanbecauseitdoesamuchbetterjobofmeetingallofmanagement’sgoalsforthecampaign.

4.6-1 Mixedproblemsmaycontainallthreetypesoffunctionalconstraints:resourceconstraints,benefitconstraints,andfixed-requirementconstraints.

4.6-2 TheSave-ItCblemisanexampleofablendingproblemwheretheobjectiveistofindthebestblendofingredientsintofinalproductstomeetcertainspecifications.

4.7-1 Alinearprogrammingmodelmustaccuratelyreflectthemanagerialviewoftheproblem.

4.7-2 Largelinearprogrammingmodelsgenerallyareformulatedbymanagementscienceteams.

4.7-3 Thelineofcommunicationbetweenthemanagementscienceteamandthemanagerisvital.

4.7-4 Modelvalidationisatestingprocessusedonaninitialversionofamodeltoidentifytheerrorsandomissionsthatinevitablyoccurwhenconstructinglargemodels.

4.7-5 Theprocessofmodelenrichmentinvolvesbeginningwitharelativelysimpleversionofthemodelandthenusingtheexperiencegainedwiththismodeltoevolvetowardmoreelaboratemodelsthatmorenearlyreflectthecomplexityoftherealproblem.

4.7-6 What-ifanalysisisanimportantpartofalinearprogrammingstudybecauseanoptimalsolutioncanonlybesolvedforwithrespecttoonespecificversionofthemodelatatime.Managementmayhave“what-if”questionsabouthowthesolutionwillchangegivenchangesinthemodelformulation.

4.8-1 ThePonderosaproblemfallsintothemixedproblemcategory.TheUnitedAirlinesproblemisbasicallyacost-benefit-tradeoffproblem.TheCitgoproblemisadistribution-networkproblem.

4.8-2 ThePonderosaproblemhas90decisionvariables,theUnitedAirlinesproblemhasover20,000decisionvariables,andtheCitgoproblemhasabout15,000decisionvariables.

4.8-3 TwofactorshelpedmakethePonderosaapplicationsuccessful.Oneisthattheyimplementedafinancialplanningsystemwithanatural-languageuserinterface,withtheoptimizationcodesoperatinginthebackground.Theothersuccessfactorwasthattheoptimizationsystemusedwasinteractive.

4.8-4 ThemostimportantsuccessfactorintheUnitedAirlinesapplicationwasthesupportofoperationalmanagersandtheirstaffs.

4.8-5 ThefactorsthathelpedmaketheCitgoapplicationsuccessfulweredevelopingoutputreportsinthelanguageofmanagerstomeettheirneeds,using“what-if”analysis,thesupportofoperationalmanagers,and,mostimportantly,theunlimitedsupportprovidedthemanagementsciencetaskforcebytopmanagement.

Problems

4.1 a)

Datacells: B2:E2,B6:E7,H6:H7,B13,andD13

Changingcells: B11:E11

Targetcell: H11

b) Thisisalinearprogrammingmodelbecausethedecisionsarerepresentedbychangingcellsthatcanhaveanyvaluethatsatisfytheconstraints.Eachconstrainthasanoutputcellontheleft,amathematicalsigninthemiddle,andadatacellontheright.Theoveralllevelofperformanceisrepresentedbythetargetcellandtheobjectiveistomaximizethatcell.Also,theExcelequationforeachoutputcellisexpressedasaSUMPRODUCTfunctionwhereeachterminthesumistheproductofadatacellandachangingcell.

c) Let T=numberofcommercialsonTV

M=numberofadvertisementsinmagazines

R=numberofcommercialsonradio

S=numberofadvertisementsinSundaysupplements.

MaximizeExposures(thousands)=140T+60M+90R+50S

subjectto 300T+150M+200R+100S≤4,000($thousands)

90T+30M+50R+40S≤1,000($thousands)

T≤5spots

R≤10spots

and T≥0,M≥0,R≥0,S≥0.

4.2 a&c)

b)

(x1,x2)

Feasible?

TotalContribution

(2,2)

Yes

$100

(3,3)

Yes

$150

(2,4)

Yes

$160

Best

(4,2)

Yes

$140

(3,4)

No

(4,3)

No

d) Let x1=levelofactivity1

x2=levelofactivity2

MaximizeContribution=$20x1+$30x2

subjectto 2x1+x2≤10

3x1+3x2≤20

2x1+4x2≤

20

and x1≥0,x2≥0.

e) OptimalSolution:(x1,x2)=(3.333,3.333)andTotalContribution=$166.67.

4.3 a)

b) Let x1=levelofactivity1

x2=levelofactivity2

x3=levelofactivity3

MaximizeContribution=$50x1+$40x2+$70x3

subjectto 30x1+20x2≤

500

10x2+40x3≤600

20x1+20x2+30x3≤

1,000

and x1≥

0,x2≥

0,x3≥

0.

4.4 a&c)

b) Belowarefivepossibleguesses(manyanswersarepossible).

(x1,x2,x3,x4)

Feasible?

P

(30,30,30,30)

Yes

$1110

(40,40,40,40)

No

(35,39,30,40)

Yes

$1336

(35,39,34,40)

Yes

$1368

(37,39,35,40)

Yes

$1398

Best

4.5 a) Theactivitiesaretheproductionratesofproducts1,2,and3.Thelimitedresourcesarehoursavailableperweekonthemillingmachine,lathe,andgrinder.

b) Thedecisionstobemadearehowmanyofeachproductshouldbeproducedperweek.Theconstraintsonthesedecisionsarethenumberofhoursavailableperweekonthemillingmachine,lathe,andgrinderaswellasthesalespotentialofproduct3.Theoverallmeasureofperformanceistotalprofit,whichistobemaximized.

c) millingmachine: 9(#unitsof1)+3(#unitsof2)+5(#unitsof3)≤500

lathe: 5(#unitsof1)+4(#unitsof2)≤350

grinder: 3(#unitsof1)+2(#unitsof3)≤150

sales: (#unitsof3)≤20

Nonnegativity: (#unitsof1)≥0,(#unitsof2)≥0,(#unitsof3)≥0

Profit=$50(#unitsof1)+$20(#unitsof2)+$25(#unitsof3)

d)

Datacells: B2:D2,B5:D7,G5:G7,andD12

Changingcells: B10:D10

Targetcell: G10

Outputcells: E5:E7

e) Let x1=unitsofproduct1producedperweek

x2=unitsofproduct2producedperweek

x3=unitsofproduct3producedperweek

MaximizeProfit=$50x1+$20x2+$25x3

subjectto 9x13x2+5x3≤500hours

5x1+4x2≤350hours

3x1+2x3≤150hours

x3≤20

and x1≥

0,x2≥

0,x3≥

0.

4.6 a) TheactivitiesaretheproductionquantitiesofpartsA,B,andC.Thelimitedresourcesarethehoursavailableonmachine1andmachine2.

b&d)

c) Belowarethreepossibleguesses(manyanswersarepossible).

(x1,x2,x3)

Feasible?

P

(500,500,300)

No

(350,1000,0)

Yes

$57,500

(400,1000,0)

Yes

$60,000

Best

e) Let A=numberofpartAproduced

B=numberofpartBproduced

C=numberofpartCproduced

MaximizeProfit=$50A+$40B+$30C

subjectto 0.02A+0.03B+0.05C≤

40hours

0.05A+0.02B+0.04C≤

40hours

and A≥0,B≥

0,C≥

0.

4.7

4.8 a&c)

b)

(x1,x2)

Feasible?

C

(7,7)

No

(7,8)

No

(8,7)

No

(8,8)

Yes

$880

Best

(8,9)

Yes

$930

(9,8)

Yes

$940

d) Let x1=levelofactivity1

x2=levelofactivity2

MinimizeCost=$60x1+$50x2

subjectto 5x1+3x2≥60

2x1+2x2≥30

7x1+9x2≥126

and x1≥0,x2≥0.

e) OptimalSolution:(x1,x2)=(6.75,8.75)andTotalCost=$842.50.

4.9 a&c)

b) Belowarefivepossibleguesses(manyanswersarepossible).

(x1,x2,x3,x4)

Feasible?

C

(32,4,0,6)

No

(33,4,0,6)

Yes

$17,400

Best

(33,5,0,6)

No

(33,4,1,6)

Yes

$17,900

(33,4,1,7)

Yes

$18,200

4.10 a&d)

b) (x1,x2,x3)=(1,2,2)isafeasiblesolutionwithadailycostof$3.48.Thisdietwillprovide210kgofcarbohydrates,310kgofprotein,and170kgofvitaminsdaily.

c) Answerswillvary.

e) Let C=kgofcorntofeedeachpig

T=kgoftankagetofeedeachpig

A=kgofalfalfatofeedeachpig

MinimizeCost=$0.84C+$0.72T+$0.60A

subjectto 90C+20T+40A≥200

30C+80T+40A≥

180

10C+20T+60A≥150

and C≥0,T≥0,A≥0.

4.11 a&d)

c) (x1,x2,x3)=(100,100,200)isafeasiblesolution.Thiswouldgenerate$400millionin5years,$300millionin10years,and$550millionin20years.Thetotalinvestedwillbe$400million.

d) Answerswillvary.

f) Let x1=unitsofAsset1purchased

x2=unitsofAsset2purchased

x3=unitsofAsset3purchased

MinimizeCost=x1+x2+x3($millions)

subjectto 2x1+x2+0.5x3≥400($millions)

0.5x1+0.5x2+x3≥100($millions)

1.5x2+2x3≥300($millions)

and x1≥

0,x2≥0,x3≥0.

4.12 a) Theactivitiesareleasingspaceineachmonthforanumberofmonths.Thebenefitismeetingthespacerequirementsforeachmonth.

b) Thedecisionstobemadearehowmuchspacetoleaseandforhowmanymonths.Theconstraintsonthesedecisionsaretheminimumrequiredspace.Theoverallmeasureofperformanceiscostwhichistobeminimized.

c) Month1:(M11molease)+(M12molease)+(M13molease)+(M14molease)+(M15molease)≥30,000squarefeet.

Month2:(M12molease)+M13molease)+(M14molease)+(M15molease)+(M21molease)+(M22molease)+(M23molease)+(M24molease)≥20,000squarefeet.

Month3:(M13molease)+(M14molease)+(M15molease)+(M22molease)+(M23molease)+(M24molease)+(M31molease)+(M32molease)+(M33molease)≥40,000squarefeet.

Month4:(M14molease)+(M15molease)+(M23molease)+(M24molease)+(M32molease)+(M33molease)+(M41molease)+(M42molease)≥10,000squarefeet.

Month5:(M15molease)+(M24molease)+(M33molease)+(M42molease)+(M51molease)≥50,000squarefeet.

Nonnegativity:(M11molease)≥0,(M12molease)≥0,(M13molease)≥

0,(M14molease)≥0,(M15molease)≥0,(M21molease)≥0,(M22molease)≥0,(M23molease)≥0,(M24molease)≥0,(M31molease)≥0,(M32molease)≥0,(M33molease)≥0,(M41molease)≥0,(M42molease)≥0,(M51molease)≥

0.

Cost=($650)[(M11molease)+(M21molease)+(M31molease)+(M41molease)+(M51molease)]+($1,000)[(M12molease)+(M22molease)+(M32molease)+(M42molease)]+($1,350)[(M13molease)+(M23molease)+(M33molease)]+($1,600)[(M14molease)+(M24molease)]+($1,900)[M15molease]

d)

Datacells: B4:P8,B10:P10,andS4:S8

Changingcells: B13:P13

Targetcell: S13

Outputcells: Q4:Q8

e) Letxij=squarefeetofspaceleasedinmonthiforaperiodofjmonths.

fori=1,...,5andj=1,...,6-i.

MinimizeC=$650(x11+x21+x31+x41+x51)+$1,000(x12+x22+x32+x42)

+$1,350(x13+x23+x33)+$1,600(x14+x24)+$1,900x15

subjectto x11+x12+x13+x14+x15≥30,000squarefeet

x12+x13+x14+x15+x21+x22+x23+x24≥20,000squarefeet

x13+x14+x15+x22+x23+x24+x31+x32+x33≥40,000sq.feet

x14+x15+x23+x24+x32+x33+x41+x42≥10,000squarefeet

x15+x24+x33+x42+x51≥50,000squarefeet

and xij≥

0,fori=1,...,5andj=1,...,6-i.

4.13

4.14 a) Thisisacost-benefit-tradeoffproblembecauseitasksyoutomeetminimumrequiredbenefitlevels(numberofconsultantsworkingeachtimeperiod)atminimumcost.

b)

c) Let f1=numberoffull-timeconsultantsworkingthemorningshift(8a.m.-4p.m.),

f2=numberoffull-timeconsultantsworkingtheafternoonshift(12p.m.-8p.m.),

f3=numberoffull-timeconsultantsworkingtheeveningshift(4p.m.-midnight),

p1=numberofpart-timeconsultantsworkingthefirstshift(8a.m.-12p.m.),

p2=numberofpart-timeconsultantsworkingthesecondshift(12p.m.-4p.m.),

p3=numberofpart-timeconsultantsworkingthethirdshift(4p.m.-8p.m.),

p4=numberofpart-timeconsultantsworkingthefourthshift(8p.m.-midnight).

MinimizeC=($14/hour)(8hours)(f1+f2+f3)+($5/hour)(4hours)(p1+p2+p3+p4)

subjectto f1+p1≥

6

f1+f2+p2≥8

f2+f3+p3≥

12

f3+p4≥6

f1≥2p1

f1+f2≥2p2

f2+f3≥2p3

f3≥

2p4

and f1≥

0,f2≥

0,f3≥

0,p1≥0,p2≥

0,p3≥

0,p4≥

0.

4.15 a) Thisisadistribution-networkproblembecauseitdealswiththedistributionofgoodsthroughadistributionnetworkatminimumcost.

b)

c) Let xij=numberofunitstoshipfromFactoryitoCustomerj(i=1,2;j=1,2,3)

MinimizeCost=$600x11+$800x12+$700x13+$400x21+$900x22+$600x23

subjectto x11+x12+x13=400

x21+x22+x23=500

x11+x21=300

x12+x22=200

x13+x23=400

and x11≥

0,x12≥0,x13≥0,x21≥

0,x22≥0,x23≥

0.

4.16 a) Requirement1:ThetotalamountshippedfromMine1mustbe40tons.

Requirement2:ThetotalamountshippedfromMine2mustbe60tons.

Requirement3:ThetotalamountshippedtothePlantmustbe100tons.

Requirement4:ForStorage1,theamountshippedout=theamountin.

Requirement5:ForStorage2,theamountshippedout=theamountin.

b)

c) Let xM1S1=numberofunitsshippedfromMine1toStorage1

xM1S2=numberofunitsshippedfromMine1toStorage2

xM2S1=numberofunitsshippedfromMine2toStorage1

xM2S2=numberofunitsshippedfromMine2toStorage2

xS1P=numberofunitsshippedfromStorage1tothePlant

xS2P=numberofunitsshippedfromStorage2tothePlant

MinimizeCost=$2,000xM1S1+$1,700xM1S2+$1,600xM2S1+$1,100xM2S2

+$400xS1P+$800xS2P

subjectto xM1S1+xM1S2=40

xM2S1+xM2S2=60

xM1S1+xM2S1=xS1P

xM1S2+xM2S2=xS2P

xS1P+xS2P=100

xM1S1≤

30,xM1S2≤

30,xM2S1≤

50,xM2S2≤50,xS1P≤

70,xS2P≤

70

and xM1S1≥

0,xM1S2≥

0,xM2S1≥

0,xM2S2≥

0,xS1P≥

0,xS2P≥

0.

4.17 a) A1+B1+R1=$60,000

A2+B2+C2+R2=R1

A3+B3+R3=R2+1.40A1

A4+R4=R3+1.40A2+1.70B1

A5+D5+R5=R4+1.40A3+1.70B2

b) Let At=amountinvestedininvestmentAatthebeginningofyeart.

Bt=amountinvestedininvestmentBatthebeginningofyeart.

Ct=amountinvestedininvestmentCatthebeginningofyeart.

Dt=amountinvestedininvestmentDatthebeginningofyeart.

Rt=amountnotinvestedatthebeginninfofyeart.

MaximizeReturn=1.40A4+1.70B3+1.90C2+1.30D5+R5

subjectto A1+B1+R1=$60,000

A2+B2+C2–R1+R2=0

–1.40A1+A3+B3–R2+R3=0

–1.40A2+A4–1.70B1–R3+R4=0

–1.40A3–1.70B2+D5–R4+R5=0

and At≥

0,Bt≥0,Ct≥

0,Dt≥

0,Rt≥0.

c)

4.18 a) Letxi=percentageofalloyiinthenewalloy(i=1,2,3,4,5).

(60%)x1+(25%)x2+(45%)x3+(20%)x4+(50%)x5=40%

(10%)x1+(15%)x2+(45%)x3+(50%)x4+(40%)x5=35%

(30%)x1+(60%)x2+(10%)x3+(30%)x4+(10%)x5=25%

x1+x2+x3+x4+x5=100%

b)

c) Letxi=percentageofalloyiinthenewalloy(i=1,2,3,4,5).

MinimizeCost=$22x1+$20x2+$25x3+$24x4+$27x5

subjectto (60%)x1+(25%)x2+(45%)x3+(20%)x4+(50%)x5=40%

(10%)x1+(15%)x2+(45%)x3+(50%)x4+(40%)x5=35%

(30%)x1+(60%)x2+(10%)x3+(30%)x4+(10%)x5=25%

and x1≥

0,x2≥0,x3≥0,x4≥

0,x5≥

0.

4.19 a)

b) Letxij=numberofunitsproducedatplantiofproductj(i=1,2,3;j=L,M,S).

MaximizeProfit=$420(x1L+x2L+x3L)+$360(x1M+x2M+x3M)+$300(x1S+x2S+x3S)

subjectto x1L+x1M+x1S≤750

x2L+x2M+x2S≤

900

x3L+x3M+x3S≤

450

20x1L+15x1M+12x1S≤

13,000squarefeet

20x2L+15x2M+12x2S≤

12,000squarefeet

20x3L+15x3M+12x3S≤

5,000squarefeet

x1L+x2L+x3L≤

900

x1M+x2M+x3M≤

1,200

x1S+x2S+x3S≤750

(x1L+x1M+x1S)/750=(x2L+x2M+x2S)/900

(x1L+x1M+x1S)/750=(x3L+x3M+x3S)/450

andx1L≥

0,x1M≥

0,x1S≥

0,x2L≥0,x2M≥

0,x2S≥0,x3L≥

0,x3M≥

0,x3S≥

0.

4.20 a)

b) Letxij=tonsofcargoistowedincompartmentj(i=1,2,3,4;j=F,C,B)

MaximizeProfit=$320(x1F+x1C+x1B)+$400(x2F+x2C+x2B)

+$360(x3F+x3C+x3B)+$290(x4F+x4C+x4B)

subjectto x1F+x2F+x3F+x4F≤12tons

x1C+x2C+x3C+x4C≤18tons

x1B+x2B+x3B+x4B≤

10tons

x1F+x1C+x1B≤20tons

x2F+x2C+x2B≤

16tons

x3F+x3C+x3B≤25tons

x4F+x4C+x4B≤13tons

500x1F+700x2F+600x3F+400x4F≤7,000cubicfeet

500x1C+700x2C+600x3C+400x4C≤9,000cubicfeet

500x1B+700x2B+600x3B+400x4B≤

5,000cubicfeet

(x1F+x2F+x3F+x4F)/12=(x1C+x2C+x3C+x4C)/18

(x1F+x2F+x3F+x4F)/12=(x1B+x2B+x3B+x4B)/10

and x1F≥

0,x1C≥

0,x1B≥

0,x2F≥

0,x2C≥

0,x2B≥

0,

x3F≥

0,x3C≥0,x3B≥0,x4F≥

0,x4C≥

0,x4B≥

0.

4.21 a)

b) Let M=numberofmen’sglovestoproduceperweek,

W=numberofwomen’sglovestoproduceperweek,

C=numberofchildren’sglovestoproduceperweek,

F=numberoffull-timeworkerstoemploy,

PT=numberofpart-timeworkerstoemploy.

MaximizeProfit=$8M+$10W+$6C–$13(40)F–$10(20)PT

subjectto 2M+1.5W+C≤5,000squarefeet

30M+45W+40C≤40(60)F+20(60)PThours

F≥

20

F≥2PT

and M≥

0,W≥

0,C≥

0,F≥0,PT≥

0.

4.22

4.23 a) ResourceConstraints:

Caloriesmustbenomorethan420.

Nomorethan20%oftotalcaloriesfromfat.

BenefitConstraints:

Caloriesmustbeatleast380

Theremustbeatleast50mgofvitamincontent.

Theremustbeatleast2timesasmuchstrawberryflavoringassweetener.

Fixed-RequirementConstraints:

Theremustbe15mgofthickeners.

b)

c) Let S=Tablespoonsofstrawberryflavoring,

CR=Tablespoonsofcream,

V=Tablespoonsofvitaminsupplement,

A=Tablespoonsofartificialsweetener,

T=Tablespoonsofthickeningagent,

MinimizeC=$0.10S+$0.08CR+$0.25V+$0.15A+$0.06T

subjectto 50S+100CR+120A+80T≥380calories

50S+100CR+120A+80T≤420calories

S+75CR+30T≤0.2(50S+100C+120A+80T)

20S+50V+2T≥50mgVitamins

S≥2A

3S+8CR+V+2A+25T=15mgThickeners

and S≥

0,CR≥

0,V≥

0,A≥0,T≥0.

4.24 a) ResourceConstraints:

Caloriesmustbenomorethan600.

Nomorethan30%oftotalcaloriesfromfat.

BenefitConstraints:

Caloriesmustbeatleast400

Theremustbeatleast60mgofvitaminC.

Theremustbeatleast12gofprotein.

Theremustbeatleast2timesasmuchpeanutbutterasjelly.

Theremustbeatleast1cupofliquid

Fixed-RequirementConstraints:

Theremustbe2slicesofbread.

b)

c) Let B=slicesofbread,

P=Tablespoonsofpeanutbutter,

S=Tablespoonsofstrawberryjelly,

G=grahamcrackers,

M=cupsofmilk,

J=cupsofjuice.

MinimizeC=$0.05B+$0.04P+$0.07S+$0.08G+$0.15M+$0.35J

subjectto 70B+100P+50S+60G+150M+100J≥400calories

70B+100P+50S+60G+150M+100J≤600calories

10B+75P+20G+70M

≤0.3(70B+100P+50S+60G+150M+100J)

3S+2M+120J≥60mgVitaminC

3B+4P+G+8M+J≥12mgProtein

B=2slices

P≥2S

M+J≥1cup

and B≥0,P≥

0,S≥0,G≥0,M≥

0,J≥0.

Cases

4.1 a) ThefixeddesignandfashioncostsaresunkcostsandthereforeshouldnotbeconsideredwhensettingtheproductionnowinJuly.Sincethevelvetshirtshaveapositivecontributiontocoveringthesunkcosts,theyshouldbeproducedoratleastconsideredforproductionaccordingtothelinearprogrammingmodel.HadTedraisedtheseconcernsbeforeanyfixedcostsweremade,thenhewouldhavebeencorrecttoadviseagainstdesigningandproducingtheshirts.Withacontributionof$22andademandof6000units,maximumexpectedprofitwillbeonly$132,000.Thisamountwillnotbeenoughtocoverthe$500,000infixedcostsdirectlyattributabletothisproduct.

b) Thelinearprogrammingspreadsheetmodelforthisproblemisshownbelow.

TrendLineshouldproduce4,200WoolSlacks,4,000CashmereSweaters,7,000SilkBlouses,15,000SilkCamisoles,8,067TailoredSkirts,5,000WoolBlazers,40,000CottonMinis,6,000VelvetShirts,and9,244Button-DownBlouses.Thetotalnetcontributionofallclothingitemsis$6,862,933.However,withthetotalfixedcostof$860,000+3($2,700,000)or$8,960,000,TrendLinesactuallyloses$2,097,067.

c) Ifvelvetcannotbesentbacktothetextilewholesaler,thenthewholequantitywillbeconsideredasasunkcostandthereforeaddedtothefixedcosts.Theobjectivefunctioncoefficientsofitemsusingvelvetwillnolongerincludethematerialcost.Thenetcontributionofthevelvetpantsandshirtsarenow$175and$40,respectively.Therevisedspreadsheetmodelisasfollows.

Theproductionplanchangesconsiderably.TrendLinesshouldproduce3,178tailoredskirts(downfrom8,067),3,667velvetpants(upfrom0),60,000cottonminis(upfrom40,000),and15,763button-downblouses(upfrom9,244).Theproductiondecisionsforallotheritemsareunaffectedbythechange.Thetotalnetcontributionofallclothingitemsequals$840,000+$1,226,00+$2,025,000+$2,983,822.22=$7,085,822.Thesunkcostsnowincludethematerialcostforvelvetandtotals$9,200,000.Thelossnowequals$2,114,178.

d) WhenTrendLinescannotreturnthevelvettothewholesaler,thecostsforvelvetcannotberecovered.Thesecostarenolongervariablecostbutnowaresunkcost.Asaconsequencetheincreasednetcontributionofthevelvetitemsmakesthemmoreattractivetoproduce.Thiswaytherevenuesfromsellingtheseitemscancontributetotherecoveryofatleastsomeofthefixedcosts.InsteadofzeroTrendLinesnowproduces3,667velvetpants.Thesepantsalsorequiresomeacetateandthustheirproductionaffectstheproductionplanforallotheritems.Sinceitisnotoptimaltomakefulluseoftheorderedvelvetinpart(b)itcomesasnosurprisethatthelossinpart(c)isevenbiggerthaninpart(b).

e) Theunitcontributionofawoolblazerchangesto$75.25.

TrendLinesshouldproduce10,067skirts(upfrom8,067),theminimumof3,000woolblazers(downfrom5,000),and6,578button-downblouses(downfrom9,244).Theproductiondecisionsforallotheritemsareunaffectedbythechange.Thetotalnetcontributionofallclothingitemsis$6,527,933.33.Thetotallossis$2,432,067.

f) Theavailableacetatechangesfrom28,000to38,000squareyards.Theresultingspreadsheetsolutionisshownbelow.

TrendLinesshouldproduce14,733skirts(upfrom8,067)and356button-downblouses(downfrom9,244).Theproductiondecisionsforallotheritemsareunaffectedbythechange.Thetotalnetcontributionofallclothingitemsis$7,581,267.Thelossis$1,378,733.

g) WeneedtoincludenewdecisionvariablesrepresentingthenumberofclothingitemsthataresoldduringtheNovembersale.Thenewspreadsheetmodelisshownbelow.

Itonlypaystoproduce2,000moreCashmeresweaters.Theproductionplanforallotheritemsisthe