数据、模型与决策(运筹学)课后习题和案例答案

CHAPTER10

NONLINEARPROGRAMMING

ReviewQuestions

10.1-1Inbothcases,decisionsneedtobemaderegardingthelevelsofanumberofactivities,

wheretheseactivitylevelscanhaveanyvaluethatsatisfiesanumberofconstraints.The

decisionsregardingactivitylevelsaretobebasedonanoverallmeasureofperformance.

10.1-2Theonlywaytodistinguishanonlinearprogrammingmodelfromalinearprogramming

modelistoexaminetheformulasenteredintotheoutputcells.Itisaanonlinear

programmingmodelifoneormoreoftheseformulasisnonlinearinsteadoflinear.

10.1-3(1)Nonlinearprogrammingisusedtomodelnonproportionalrelationshipsbetween

activitylevelsandtheoverallmeasureofperformance,whereaslinearprogramming

assumesaproportionalrelationship.(2)Constructingthenonlinearfbrmula(s)neededfora

nonlinearprogramisconsiderablymoredifficultthandevelopingthelinearformulasused

inlinearprogramming.(3)Solvinganonlinearprogrammingmodelisoftenmuchmore

difficultthansolvingalinearprogrammingmodel.

10.1-4Thecontributionofeachactivitytothevalueoftheobjectivefunctionisproportionalto

theleveloftheactivityinalinearprogram.Whenanactivityhasanonproportional

relationship,thisassumptionisviolated.

10.1-5Theslopeofthegraphneverincreasesbutsometimesdecreasesastheleveloftheactivity

increases.

10.1-6Thiskindofgraphmightoccurbecauseovertimeneedstobeusedtoincreasethelevelof

theactivitybeyondthefirstkink.

10.1-7Itiscommontoassumeaquadraticformoralogarithmicform.

10.1-8Typeswheretheactivitieshavedecreasingmarginalreturns.

10.1-9TheSolverbeginsclimbingthemountainuntilitreachesthepeak.

10.1-10ApplyingtheExcelSolverrepeatedlywithavarietyofstartingsolutionsandthen

adoptingthebestofthefinalsolutionsgivesabetterchanceofobtainingtheoptimal

solution.

10.2-1Asimplenonlinearprogrammingproblemhasthesameconstraintsasalinear

programmingmodel,anonlinearfunctionfortheobjectivefunction,anddecreasing

marginalreturnsforeachactivitythatviolatestheproportionalityassumptionoflinear

programming.

10.2-2Insteadofhavingobjectivefunctionlines,thereareobjectivefunctioncurves.

10.2-3Theobjectivefunctionmustbeaquadratic.

10.2-4Inportfolioselection,atrade-offisbeingsoughtbetweenexpectedreturnandrisk.

10-1

10.3-1Theprofitgraphmusthaveakinkwheretheslopechanges(decreases)inordertoapply

separableprogramming.

10.3-2Alinearprogrammingmodeliseventuallyformulated.

10.3-3Convertingtheproblemintoalinearprogrammingproblemtendstomakeitquickerto

solve,makesavailabletheSensitivityReport,andonlyrequiresestimatingtheprofitata

fewpoints.

10.3-4TheExcelSolvercanoftenreadilysolvesuchproblemsandnoapproximationisneeded.

10.4-1TheStandardSolveroftenhasdifficultysolvingnonlinearprogrammingmodelsifthe

constraintsarenonlinear,oriftheprofitgraphsarenotsmooth,ortheprofitgraphshave

increasingmarginalreturns.

10.4-2OneapproachistoruntheSolvermanytimes,eachtimestartingwithadifferentinitial

solutionenteredintothechangingcells.SolverTablecanbeusedtostreamlinethis

approach.

10.5-1ThephilosophyofEvolutionarySolverisbasedongenetics,evolution,andthesurvivalof

thefittest.

10.5-2Theleveloffitnessisdeterminedbyevaluatingtheobjectivefunction.

10.5-3Mutationcanhelpthealgorithmgetunstuckifitisgettingtrappednearalocaloptimum.

10.5-4PremiumSolverincludesEvolutionarySolver,butthestandardSolverdoesnot.

10.5-5(1)ThecomplexityoftheobjectivefunctiondoesnotimpactEvolutionarySolver.(2)By

evaluatingwholepopulationsofcandidatessolutionsthataren'tnecessarilyinthesame

neighborhoodasthecurrentbestsolution,EvolutionarySolverkeepsfromgettingtrapped

atalocaloptimum.

10.5-6(1)ItcantakemuchlongerthanthestandardSolvertofindafinalsolution.(2)Itdoesnot

performwellonmodelsthathavemanyconstraints.(3)Thebestsolutionfoundtypicallyis

notoptimal.

10-2

Problems

10.1a)

$40,000

$35,000

$30,000

$25,000

Profit$20,000

$15,000

$10,000

$5,000

$0

02004006008001,000

ProductionRate

b)Theproportionalityassumptionseemstobesatisfiedreasonablywellforthisproduct.

c)Thisproductappearstohavedecreasingmarginalreturns.

d)

ProductionRate

10-3

10.2a)Case1

Case2

Case3

10-4

b)Case1:decreasingmarginalreturns

Case2:increasingmarginalreturns

Case3:neitherincreasingnordecreasingmarginalreturns

c)Case1:increasingmarginalreturns

Case2:decreasingmarginalreturns

Case3:neitherincreasingnordecreasingmarginalreturns

d)Case1

Case2

10-5

Case3

LevelofActivity

Thequadraticformdoesnotseemtobeaverygoodfit.

10.3a)

70

60

50

Profit40

($millions)30

20

10

0

0100200300400500600700800

Sales(thousands)

b)Themicrochiphasneitherincreasingnordecreasingmarginalreturns.

10-6

c)

d)

e)TheExceloptioninpartd(polynomicaloforder3)doesabetterjoboffittingtheprofit

graphtothedata.

10.4a)

EstimatedProfit/Day

ProductionRateActualProfit/Day(PM，］。。??』收)Error

0$0$0$0

1$95$95$0

2$184$18()$4

3$255$255$0

4$320$320$0

b)

10-7

EstimatedProfit/Day

ProductionRateActualProfit/Day(P=$104R-$6R2)Error

0$0$0$0

1$95$98$3

2$184$184$0

3$255$258$3

4$320$320$0

c)Thequadraticfunction$100R-5R2providesaslightlybetterfittoallthedata.

10.5a)

BCDEFGH

3Stock1Stock2Stock3

4ExpectedReturn21%30%8%

5

6Risk(Stand.Dev.)25%45%5%

7

8JointRisk(Covar.)Stock1Stock2Stock3

9Stock10.040-0.005

10Stock2-0.010

11Stock3

12

13Stock1Stock2Stock3Total

14Portfolio69.7%10.3%20.0%100%=100%

15<=

1620%

17

18Minimum

19Expected

20PortfolioReturn

21ExpectedReturn19.3%>=18.0%

22

23Risk(Variance)0.0365

24

25Risk(Stand.Dev.)19.1%

Theexpectedreturnincreasesby1.3%andtherisk(standarddeviation)increasesby

about3.7%.

10-8

b)

BCDEFGH

3Stock1Stock2Stock3

4ExpectedReturn21%30%8%

5

6Risk(Stand.Dev.)25%45%5%

7

8JointRisk(Covar.)Stock1Stock2Stock3

9Stock10.040-0.005

10Stock2-0.010

11Stock3

12

13Stock1Stock2Stock3Total

14Portfolio87.8%12.2%0.0%100%=100%

15<=

160%

17

18Minimum

19Expected

20PortfolioReturn

21ExpectedReturn22.1%>=18.0%

22

23Risk(Variance)0.0598

24

25Risk(Stand.Dev.)24.4%

Theexpectedreturnincreasesby4.1%andtherisk(standarddeviation)increasesby

about9%.

BCDEFG

27RiskExpected

28MaxStock3Stock1Stock2Stock3(St.Dev.)Return

29

300%87.8%12.2%0.0%24.4%22.1%

315%83.3%11.7%5.0%23.1%21.4%

3210%78.8%11.2%10.0%21.8%20.7%

3315%74.3%10.7%15.0%20.4%20.0%

3420%69.7%10.3%20.0%19.1%19.3%

3525%65.2%9.8%25.0%17.8%18.6%

3630%60.0%10.0%30.0%16.5%18.0%

3735%47.8%17.2%35.0%15.6%18.0%

3840%40.2%21.7%38.1%15.4%18.0%

3945%40.2%21.7%38.1%15.4%18.0%

4050%40.2%21.7%38.1%15.4%18.0%

10.6a)Let=percentageofportfoliotoinvestinstock1.

S2=percentageofportfoliotoinvestinstock2.

22

MinimizeRisk=($5,000Si)+($30,000S2)

subjectto$12,50051+$20,00052NMinimumExpectedProfit

51+52<100%

andS\>0,S2>0.

10-9

b&c)Minimumacceptableexpectedprofil=$13,000

ABcDEF

1Stock1Stock2

2ExpectedProfit(on$50,000)$12,500$20,000

3

4Risk(Stand.Dev.for$50,000)$5,000$30,000

5

6Stock1Stock2Total

7Portfolio(%of$50,000)93.3%6.7%100%<=100%

8$46,667$3,333

9Minimum

10Expected

11PortfolioProfit

12ExpectedProfit$13,000>=$13,000

13

14Risk(Variance)$25,777,778

15

16Risk(Stand.Dev.)$5,077

Minimumacceptableexpectedprofit=$15,0000

ABCDEF

1Stock1Stock2

2ExpectedProfit(on$50,000)$12,500$20,000

3

4Risk(Stand.Dev.for$50,000)$5,000$30,000

5

6Stock1Stock2Total

7Portfolio(%of$50,000)66.7%33.3%100%<=100%

8$33,333$16,667

9Minimum

10Expected

11PortfolioProfit

12ExpectedProfit$15,000>=$15,000

13

14Risk(Variance)$111,111,111

15

16Risk(Stand.Dev.)$10,541

10-10

Minimumacceptableexpectedprofit=$17,000

ABcDEF

1Stock1Stock2

2ExpectedProfit(on$50,000)$12,500$20,000

3

4Risk(Stand.Dev.for$50,000)$5,000$30,000

5

6Stock1Stock2Total

7Portfolio(%of$50,000)40.0%60.0%100%<=100%

8$20,000$30,000

9Minimum

10Expected

11PortfolioProfit

12ExpectedProfit$17,000>=$17,000

13

14Risk(Variance)$328,000,000

15

16Risk(Stand.Dev.)$18,111

Minimumacceptableexpectedprofit=$19,000

ABcDEF

1Stock1Stock2

2ExpectedProfit(on$50,000)$12,500$20,000

3

4Risk(Stand.Dev.for$50,000)$5,000$30,000

5

6Stock1Stock2Total

7Portfolio(%of$50,000)13.3%86.7%100%<=100%

8$6,667$43,333

9Minimum

10Expected

11PortfolioProfit

12ExpectedProfit$19,000>=$19,000

13

14Risk(Variance)$676,444,444

15

16Risk(Stand.Dev.)$26,009

d)

日o|i-a|i-3c

$13,000$5,077$7,923-$2,231

$15,000$10,541$4,459-$16,623

$17,000$18,111-$1,111-$37,333

$19,000$26,009-$7,009-$59,027

Theportfoliowithaminimumexpectedprofitof$15,00()givesthehighest日amonth

thosethatalsogive|j-o>0.

10-11

10.7a)LetS\=percentageofportfoliotoinvestinStock1

S?=percentageofportfoliotoinvestinStock2

S3=percentageofportfoliotoinvestinStock3

S4=percentageofportfoliotoinvestinStock4

222

MinimizeRisk=(0.25S1)+(0.45S2)+(0.05S3)+(0.18)S4+2(0.04)SIS2+2(-

0.005)55+2(-0.01)S2S3+2(-0.015)SI54+2(-0.025)5254+2(0.003)5354

subjectto0.21S]+0.3052+O.O8S3+0.17S4>0.18min.expectedreturn

S1+S2+S3+S4=100%

andSi>0,52>0,S3>0,54>0.

b)

BCDEFGHI

3Stock1Stock2Stock3Stock4

4ExpectedReturn21%30%8%17%

Ir

5

6Risk(Stand.Dev.)25%45%5%18%

7

8JointRisk(Covar.)Stock1Stock2Stock3Stock4

9Stock10.040-0.005-0.015

10Stock2-0.010-0.025

11Stock30.003

12Stock4

13

14Stock1Stock2Stock3Stock4Total

15Portfolio24.5%12.4%17.6%45.5%100%=100%

16

17Minimum

18Expected

19PortfolioReturn

20ExpectedReturn18.0%>=18.0%

21

22Risk(Variance)0.0095

23

24Risk(Stand.Dev.)9.8%

c)

BCDEFIGH

26MinimumRiskExpected

27ExpectedReturnStock1Stock2Stock3Stock4(St.Dev.)Return

28

298%7.5%3.9%85.6%3.1%3.8%10.1%

3010%7.5%3.9%85.6%3.1%3.8%10.1%

3112%11.6%5.9%69.2%13.3%4.4%12.0%

3214%15.9%8.1%52.0%24.0%5.9%14.0%

3316%20.2%10.2%34.8%34.7%7.7%16.0%

3418%24.5%12.4%17.6%45.5%9.8%18.0%

3520%28.8%14.5%0.5%56.2%11.9%20.0%

3622%24.5%30.9%0.0%44.6%16.0%22.0%

3724%19.9%47.7%0.0%32.4%22.3%24.0%

3826%15.2%64.5%0.0%20.2%29.6%26.0%

3928%10.6%81.3%0.0%8.0%37.2%28.0%

4030%0.0%100.0%0.0%0.0%45.0%30.0%

10-12

10.8a)LetR\=theproductionrateofproduct1perhour

/?2=theproductionrateofproduct2perhour

22

MaximizeProfit=$200/？1-$10()/?j+$3O()/?2-$100/?2

subjecttoR]+/?2W2(maximumtotalproductionrate)

andR\>0,/?2>0.

b)

ABCDEF

1UnitProfit=a(Rate)+b(Rate)；wlere

2Product1Product2

3a$200$300

4b-$100-$100

5

6Product1Product2TotalTotalProfit

7ProductionRate0.751.252$313|

8<=

92

10-13

10.9a)Theprofitgraphforpowersawsisshownbelow:

Productionratetorpowersaws(000s)

Productionrateforpowerdrills(000s)

10-14

b)

AB0DEF

1UnitProfitPowerSawsPowerDrills

2RegularTime$150$100

3Overtime$50$75

4Total

5UsedperUnitProducedUsedAvailable

6PowerSupplies1110,000<=10,000

7GearAssemblies2113,000<=15,000

8

9UnitsProducedMaximum

10PowerSawsPowerDrillsPowerSawsPowerDrills

11RegularTime3,0005,000<=3,0005,000

12Overtime02,000<=2,0003,000

13Total3,0007,000

14TotalProfit

15$1,100,000

3,000powersawsand7,000powerdrillsshouldbeproducedinNovember.

10.10a)

ABcDEF

1UnitProfitDoorsWindows

2GrossProfit($hundred)46

3MarketingCost($hundred)(Doors)32(Windows)2

4TotalResource

5UsedPerUnitProducedUsedAvailable

6Resource1135.6547<=8

7Resource2528.7735<=14

8

9TotalProfit

10DoorsWindows($hundreds)

11ProductionRate1.1551.5007.58

b)Profitdatafordoorswhenmarketingcostsareconsidered:

ProductionGrossMarketingIncremental

RateProfitCostNetProfitNetProfit

0000—

1$400$10()$300$300

2$800$800$0-$300

3$1200$2700-$1900-$1900

D4D034£>-£>3

Profitdataforwindowswhenmarketingcostsareconsidered:

ProductionGrossMarketingIncremental

RateProfitCostNetProfitNetProfit

0000—

1$600$200$400$400

2$1200$800$400$0

3$1800$18000-$400

W6W21V26W-2W2