数据模型与决策运筹学课后习题和案例答案

ReviewQuestions

12.1-1 Thedecisionalternativesaretodrillforoilortoselltheland.

12.1-2 Theconsultinggeologistbelievesthatthereis1chancein4ofoilonthetractofland.

12.1-3 Maxdoesnotputmuchfaithintheassessment.

12.1-4 Adetailedseismicsurveyofthelandcouldbedonetoobtainmoreinformation.

12.1-5 Thepossiblestatesofnaturearethepossibleoutcomesoftherandomfactorsthataffectthepayoffthatwouldbeobtainedfromadecisionalternative.

12.1-6 Priorprobabilitiesaretheestimatedprobabilitiesofthestatesofnaturepriortoobtainingadditionalinformationthroughatestorsurvey.

12.1-7 Thepayoffsarequantitativemeasuresoftheoutcomesfromadecisionalternativeandastateofnature.Payoffsaregenerallyexpressedinmonetaryterms.

12.2-1 Themaximaxcriterionidentifiesthemaximumpayoffforeachdecisionalternativeandchoosesthedecisionalternativewiththemaximumofthesemaximumpayoffs.Themaximaxcriterionisfortheeternaloptimist.

12.2-2 Themaximaxciterioncompletelyignoresthepriorprobabilitiesandignoresallpayoffsexceptforthelargestone.

12.2-3 Themaximincriterionidentifiestheminimumpayoffforeachdecisionalternativeandchoosesthedecisionalternativewiththemaximumoftheseminimumpayoffs.Themaximincriterionisforthetotalpessimist.

12.2-4 Themaximincriterionignoresthepriorprobabilitiesandignoresallpayoffsexceptthemaximinpayoff.

12.2-5 Themaximumlikelihoodcriterionfocusesonthemostlikelystateofnature,theonewiththelargestpriorprobability.

12.2-6 Criticismsofthemaximumlikelihoodcriterioninclude:1)thiscriterionchoosesanalternativewithoutconsideringitspayoffsforstatesofnatureotherthanthemostlikelyone,2)foralternativesthatarenotchosen,thiscriterionignorestheirpayoffsforstatesofnatureotherthanthemostlikelyone,3)ifthedifferencesinthepayoffsforthemostlikelystateofnaturearemuchlessthanforanothersomewhatlikelystateofnature,thenitmightmakesensetofocusonthislatterstateofnatureinstead,and4)iftherearemanystatesofnatureandtheyarenearlyequallylikely,thentheprobabilitythatthemostlikelystateofnaturewillbethetrueoneisfairlylow.

12.2-7 Bayes’decisionrulesaystochoosethealternativewiththelargestexpectedpayoff.

12.2-8 Theexpectedpayoffiscalculatedbymultiplyingeachpayoffbythepriorprobabilityofthecorrespondingstateofnatureandthensummingtheseproducts.

12.2-9 CriticismsofBayes’decisionruleinclude:1)thereusuallyisconsiderableuncertaintyinvolvedinassigningvaluestopriorprobabilities,2)priorprobabilitiesinherentlyareatleastlargelysubjectiveinnature,whereassounddecisionmakingshouldbebasedonobjectivedataandprocedures,and3)byfocusingonaverageoutcomes,expectedpayoffsignoretheeffectthattheamountofvariabilityinthepossibleoutcomesshouldhaveonthedecisionmaking.

12.3-1 Adecisiontreeisagraphicaldisplayoftheprogressionofdecisionsandrandomeventstobeconsidered.

12.3-2 Adecisionnodeindicatesthatadecisionneedstobemadeatthatpointintheprocess.Aneventnodeindicatesthatarandomeventoccursatthatpoint.

12.3-3 Decisionnodesarerepresentedbysquareswhilecirclesrepresenteventnodes.

12.4-1 Sensitivityanalysismightbehelpfultostudytheeffectifsomeofthenumbersincludedinthemodelarenotcorrect.

12.4-2 Itassuresthateachpieceofdataisinonlyoneplaceanditmakesiteasyforanyonetointerpretthemodel,eveniftheydon’tunderstandTreePlanordecisiontrees.

12.4-3 Adatatabledisplaysresultsofselectedoutputcellsforvarioustrialvaluesofadatacell.

12.4-4 Ifthereislessthana23.75%chanceofoil,theyshouldsell.Ifit’smore,theyshoulddrill.

12.5-1 Perfectinformationmeansknowingforsurewhichstateofnatureisthetruestateofnature.

12.5-2 Theexpectedpayoffwithperfectinformationiscalculatedbymultiplyingthemaximumpayoffforeachalternativebythepriorprobabilityofthecorrespondingstateofnature.

12.5-3 Thedecisiontreeshouldbestartedwithachancenodewhosebranchesarethevariousstatesofnature.

12.5-4 EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

12.5-5 Ifthecostofobtainingmoreinformationismorethantheexpectedvalueofperfectinformationthenitisnotworthwhiletoobtainmoreinformation.

12.5-6 Ifthecostofobtainingmoreinformationislessthantheexpectedvalueofperfectinformationthenitmightbeworthwhiletoobtainmoreinformation.

12.5-7 IntheGoferbrokeproblemtheEVPI>Csoitmightbeworthwhiletodotheseismicsurvey.

12.6-1 Posteriorprobabilitiesarerevisedprobabilitiesofthestatesofnatureafterdoingatestorsurveytoimprovethepriorprobabilities.

12.6-2 Thepossiblefindingsarefavorablewithoilbeingfairlylikely,orunfavorablewithoilbeingquiteunlikely.

12.6-3 Conditionalprobabilitiesneedtobeestimated.

12.6-4 Thefivekindsofprobabilitiesconsideredareprior,conditional,joint,unconditional,andposterior.

12.6-5 P(stateandfinding)=P(state)*P(finding|state).

12.6-6 P(finding)=sumofP(stateandfinding)foreachstate.

12.6-7 P(state|finding)=P(stateandfinding)/P(finding).

12.6-8 Bayes’theoremisusedtocalculateposteriorprobabilities.

12.7-1 Adecisiontreeprovidesagraphicaldisplayoftheprogressionofdecisionsandrandomeventsforaproblem.

12.7-2 Adecisionneedstobemadeatadecisionnode.

12.7-3 Arandomeventwilloccurataeventnode.

12.7-4 Theprobabilitiesofrandomeventsandthepayoffsneedtobeinsertedbeforebeginninganalysis.

12.7-5 Whenperformingtheanalysis,startattherightsideofthedecisiontreeandmoveleftonecolumnatatime.

12.7-6 Foreacheventnode,calculateitsexpectedpayoffbymultiplyingthepayoffofeachbranchbytheprobabilityofthatbranchandthensummingtheseproducts.

12.7-7 Foreachdecisionnode,comparetheexpectedpayoffsofitsbranchesandchoosethealternativewhosebranchhasthelargestexpectedpayoff.

12.8-1 Consolidatethedataandresultsintoonesectionofthespreadsheet.

12.8-2 Performingsensitivityanalysisonapieceofdatashouldrequirechangingavalueinonlyoneplaceonthespreadsheet.

12.8-3 Adatatablecanconsiderchangesinonlyoneortwodatacells.

12.8-4 One.

12.8-5 Yes.Thespidergraphcanconsiderchangesinmanydatacellsatatime.

12.8-6 SensIt’sspidergraphassumesthateachdatavaluevariesbythesameamount.Sensit’stornadodiagramovercomesthislimitation.

12.9-1 Utilitiesareintendedtoreflectthetruevalueofanoutcometothedecision-maker.

12.9-2

12.9-3 Undertheassumptionsofutilitytheory,thedecision-maker’sutilityfunctionformoneyhasthepropertythatthedecision-makerisindifferentbetweentwoalternativecoursesofactionifthetwoalternativeshavethesameexpectedutility.

12.9-4 Thedecision-makerisofferedtwohypotheticalalternativesandaskedtoidentifythepointofindifferencebetweenthetwo.

12.9-5 Thepointofindifferenceisthevalueofpwherethedecision-makerisindifferentbetweenthetwohypotheticalalternatives.

12.9-6 Thevalueobtainedtoevaluateeachnodeofthetreeistheexpectedutility.

12.9-7 Maxdecidedtodotheseismicsurveyandtoselliftheresultisunfavorableordrilliftheresultisfavorable.

12.10-1 TheGoferbrokeproblemcontainedthesameelementsastypicalapplicationsofdecisionanalysisbutisoversimplified.

12.10-2 Aninfluencediagramcomplementsthedecisiontreeforrepresentingandanalyzingdecisionanalysisproblems.

12.10-3 Typicalparticipantsincludemanagement,ananalyst,andagroupfacilitator.

12.10-4 Amanagercangotoamanagementconsultingfirmthatspecializesindecisionanalysis.

12.10-5 Decisionanalysisiswidelyusedaroundtheworld.

Problems

12.1 a) Max(A1)=6,Max(A2)=4,Max(A3)=8.Maximax=8withalternativeA3.

b) Min(A1)=2,Min(A2)=3,Min(A3)=1.Maximin=3withalternativeA2.

12.2 a) Max(A1)=30,Max(A2)=31,Max(A3)=22,Max(A4)=29.Maximax=31withA2.

b) Min(A1)=20,Min(A2)=14,Min(A3)=22,Min(A4)=21.Maximin=22withA3.

12.3 a)

StateofNature

Alternative

Sell10cases

Sell11cases

Sell12cases

Sell13cases

Buy10cases

$50

$50

$50

$50

Buy11cases

$47

$55

$55

$55

Buy12cases

$44

$52

$60

$60

Buy13cases

$41

$49

$57

$65

PriorProbability

b) Max(Buy10)=$50,Max(Buy11)=$55,Max(Buy12)=$60,Max(Buy13)=$65.

Maximax=$65withbuying13cases.

c) Min(Buy10)=$50,Min(Buy11)=$47,Min(Buy12)=$44,Min(Buy13)=$41.

Maximin=$50withbuying10cases.

d) Themostlikelystateofnatureistosell11cases.Underthisstate,sheshouldbuy11caseswithapayoffof$55.

e)

Jeanshouldbuy12cases.Themaximumexpectedpayoffis$53.60.

f)

Jeanshouldpurchase12cases.Themaximumexpectedpayoffis$55.20.

Jeanshouldpurchase12cases.Themaximumexpectedpayoffis$54.40.

Jeanshouldpurchase11cases.Themaximumexpectedpayoffis$53.40.

12.4 a) Max(Conservative)=$30million

Max(Speculative)=$40million

Max(Countercyclical)=$15million

Maximax=$40millionwiththespeculativeinvestment

b) Min(Conservative)=–$10million

Min(Speculative)=–$30million

Min(Countercyclical)=–$10million

Maximin=–$10millionwitheithertheconservativeofcountercyclicalinvestment.

c) Thestableeconomyisthemostlikelystateofnature.

Thespeculativeinvestmenthasthemaximumpayoffforthisstate($10million).

d) Thecountercyclicalinvestmenthasthemaximumexpectedpayoffof$5million.

12.5 a) Thecountercyclicalinvestmenthasthemaximumexpectedpayoffof$8million.

b) Thespeculativeinvestmenthasthemaximumexpectedpayoffof$5million.

c&d)

e)

f) Parta)Partb)

g)

h)

Counter-cyclicalandconservativecrossatapproximatelyp=0.62.

Conservativeandspeculativecrossatapproximatelyp=0.68.

i) Letp=priorprobabilityofstableeconomy

Fortheconservativeoption:

EP =(0.1)(30)+p(5)+(1–0.1–p)(–10)

=3+5p–9+10p

=15p–6

Forthespeculativeoption:

EP =(0.1)(40)+p(10)+(1–0.1–p)(–30)

=4+10p–27+30p

=40p–23

Forthecounter-cyclicaloption:

EP =(0.1)(–10)+p(0)+(1–0.1–p)(15)

=–1+0+13.5–15p

=–15p+12.5

Counter-cyclicalandconservativecrosswhen

–15p+12.5=15p–6or30p=18.5orp=0.617

Conservativeandspeculativecrosswhen

15p–6=40p–23or25p=17orp=0.68

Theyshouldchoosethecounter-cyclicaloptionwhenp<0.617,theconservativeoptionwhen0.617≤p<0.68,andthespeculativeoptionwhenp≥0.68.

12.6 a) Max(A1)=80,Max(A2)=50,Max(A3)=60.

Maximax=$80thousandwhenchoosingalternativeA1.

b) Min(A1)=25,Min(A2)=30,Min(A3)=40.

Maximin=$40thousandwhenchoosingalternativeA3.

c) S2isthemostlikelyoutcome.Forthisstate,themaximumpayoffof$50thousandoccurswithalternativeA2.

d) AlternativeA3hasthehighestexpectedpayoffof$48thousand.

e)

f) WhenthepriorprobabilityofS1is0.2,alternativeA2shouldbechosen,withanexpectedpayoffof$46thousand.

WhenthepriorprobabilityofS1is0.6,alternativeA1shouldbechosen,withanexpectedpayoffof$58thousand.

g)

12.7 a) Max(A1)=$220thousand,Max(A2)=$200thousand.

Maximax=$220thousandwhenchoosingalternativeA1.

b) Min(A1)=$110thousand,Min(A2)=$150thousand.

Maximin=$150thousandwhenchoosingalternativeA2.

c) S1isthemostlikelyoutcome.Forthisstate,themaximumpayoffof$220thousandoccurswithalternativeA1.

d) AlternativeA1hasthehighestexpectedpayoffof$194thousand.

e&f)

g)

Letp=priorprobabilityofS1.

ForA1:

EP =p(220)+(1–0.1–p)(170)+(0.1)(110)

=220p+153–170p+11

=50p+164

ForA2:

EP =p(200)+(1–0.1–p)(180)+(0.1)(150)

=200p+162–180p+15 =20p+177

A1andA2crosswhen50p+164=20p+177or30p=13orp=0.433.

TheyshouldchooseA2whenp≤0.433,A1whenp>0.433.

h)

Letp=priorprobabilityofS1.

ForA1:

EP =p(220)+(0.3)(170)+(1–0.3–p)(110)

=220p+51+77–110p

=110p+128

ForA2:

EP =p(200)+(0.3)(180)+(1–0.3–p)(150)

=200p+54+105–150p

=50p+159

A1andA2crosswhen110p+128=50p+159or60p=31orp=0.517.

TheyshouldchooseA2whenp≤0.517,A1whenp>0.517.

i)

Letp=priorprobabilityofS2.

ForA1:

EP =(0.6)(220)+p(170)+(1–0.6–p)(110)

=132+170p+44–110p

=60p+176

ForA2:

EP =(0.6)(200)+p(180)+(1–0.6–p)(150)

=120+180p+60–150p

=30p+180

A1andA2crosswhen60p+176=30p+180or30p=4orp=0.133.

TheyshouldchooseA2whenp≤0.133,A1whenp>0.133.

j) AlternativeA1shouldbechosen.

12.8 a)

StateofNature(Weather)

Alternative

Dry

Moderate

Damp

Crop1

20

35

40

Crop2

30

45

Crop3

30

25

25

Crop4

20

20

20

PriorProbability

b)

c) Crop1hasthehighestexpectedpayoffof$31,500.

d) Whenthepriorprobabilityofmoderateweatheris0.2,Crop2hasthehighestexpectedpayoffof$35,250.

Whenthepriorprobabilityofmoderateweatheris0.3,Crop2hasthehighestexpectedpayoffof$33,750.

Whenthepriorprobabilityofmoderateweatheris0.4,Crop2hasthehighestexpectedpayoffof$32,250.

Whenthepriorprobabilityofmoderateweatheris0.6,Crop1hasthehighestexpectedpayoffof$31,000.

12.9 Whenx=50,alternativeA3hasthehighestexpectedpayoffof$5,600.

Whenx=75,alternativeA1hasthehighestexpectedpayoffof$7,400.

BarbaraMillershouldpayamaximumof$1,800toincreasexto75.

12.10 a) AlternativeA2hasthehighestexpectedpayoffof$1,000.

b) Withperfectinformation,chooseA1forwhenthestateisS1,A2whenthestateisS2,andA3whenthestateisS3.

EP(withperfectinformation)=(0.2)(4)+(0.5)(2)+(0.3)(1)=$2,100

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=$2,100–$1,000=$1,100.

c)

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=$2,100–$1,000=$1,100.

d) Sincetheinformationwillcost$1,000andthevalueisnomorethan$1,100,itmightbeworthwhiletospendthemoney.

12.11 a) AlternativeA1hasthehighestexpectedpayoffof$35.

b) Withperfectinformation,chooseA1forwhenthestateisS1,A1whenthestateisS2,andA2whenthestateisS3.

EP(withperfectinformation)=(0.5)($50)+(0.3)($100)+(0.2)(–$10)=$53

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=$53–$35=$18

c)

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=$53–$35=$18

d) Betsyshouldconsiderspendingupto$18toobtainmoreinformation.

12.12 a) AlternativeA3hasthehighestexpectedpayoffof$35,000.

b) IfS1occursforcertainthenchoosealternativeA3(payoffis$10,000).

IfS1doesnotoccurforcertainthenthechanceofS2occurringis3/8andthechanceofS3occurringis5/8.SochooseA1(expectedpayoffis$66,250).

A1: (3/8)(10)+(5/8)(100)=66.25

A2: (3/8)(20)+(5/8)(50)=38.75

A3: (3/8)(10)+(5/8)(60)=41.25

EP(withinformation)=(0.2)(10)+(0.8)(66.25)=55

EVI=EP(withinformation)–EP(withoutmoreinformation)

=55–35=$20,000

Themaximumamountyoushouldpayfortheinformationis$20,000.

ThedecisionwiththisinformationwouldbetochooseA3ifS1willoccur.OtherwisechooseA1.Theexpectedpayoffis$55,000(excludingthepaymentforinformation).

c) IfS2occursforcertainthenchoosealternativeA2(payoffis$20,000).

IfS2doesnotoccurforcertainthenthechanceofS1occurringis2/7andthechanceofS3occurringis5/7.SochooseA3(expectedpayoffis$45,714).

A1: (2/7)(–100)+(5/7)(100)=42.857

A2: (2/7)(–10)+(5/7)(50)=32.857

A3: (2/7)(10)+(5/7)(60)=45.714

EP(withinformation)=(0.3)(20)+(0.7)(42.857)=38

EVI=EP(withinformation)–EP(withoutmoreinformation)

=38–35=$3,000

Themaximumamountyoushouldpayfortheinformationis$3,000.

ThedecisionwiththisinformationwouldbetochooseA2ifS2willoccur.OtherwisechooseA3.Theexpectedpayoffis$38,000(excludingthepaymentforinformation).

d) IfS3occursforcertainthenchoosealternativeA1(payoffis$100,000).

IfS3doesnotoccurforcertainthenthechanceofS1occurringis2/5andthechanceofS2occurringis3/5.SochooseA3(expectedpayoffis$10,000).

A1: (2/5)(–100)+(3/5)(10)=–34

A2: (2/5)(–10)+(3/5)(20)=8

A3: (2/5)(10)+(3/5)(10)=10

EP(withinformation)=(0.5)(100)+(0.5)(10)=55

EVI=EP(withinformation)–EP(withoutmoreinformation)

=55–35=$20,000

Themaximumamountyoushouldpayfortheinformationis$20,000.

ThedecisionwiththisinformationwouldbetochooseA1ifS3willoccur.OtherwisechooseA3.Theexpectedpayoffis$55,000(excludingthepaymentforinformation).

e) Withperfectinformation,chooseA3forwhenthestateisS1,A2whenthestateisS2,andA1whenthestateisS3.

EP(withperfectinformation)=(0.2)(10)+(0.3)(20)+(0.5)(100)=$58,000

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=58–35=$23,000

Amaximumof$23,000shouldbepaidfortheinformation.Withperfectinformation,chooseA3forwhenthestateisS1,A2whenthestateisS2,andA1whenthestateisS3.Theresultingexpectedpayoffis$58,000.

f) Themaximumamountyoushouldeverpayfortestingis$23,000.

12.13 a)

b)

c&d) Theoptimalpolicyistodoaseismicsurveyandsellifitisunfavorableordrillifitisfavorable.

12.14 a)

b)

c)

12.15 a)

StateofNature

Alternative

PoorRisk

AverageRisk

GoodRisk

ExtendCredit

-$15,000

$10,000

$20,000

Don’tExtendCredit

$0

$0

$0

PriorProbabilities

b) Extendingcreditmaximizestheexpectedpayoff($8,000).

c) Withperfectinformation,youwouldextendcreditiftheircreditrecordisaverageorgood,anddon’textendcreditiftheircreditrecordispoor.

EP(withperfectinformation)=(0.2)(0)+(0.5)(10)+(0.3)(20)=$11,000

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=$11,000–$8,000=$3,000.

Thisindicatesthatthecredit-ratingorganizationshouldnotbeused.

d) PF=PoorFinding AF=AverageFinding GF=GoodFinding

PS=PoorState AS=AverageState GS=GoodState

e)

f&g) Vincentshouldnotgetthecreditratingandsimplyextendcredit.

12.16 a) AlternativeA1maximizestheexpectedpayoff($100).

b)

EVPI=EP(withperfectinfo)–EP(withoutmoreinfo)=$220–$100=$120

Thisindicatesthatitmightbeworthwhiletodotheresearch.

c) P(stateandfinding)=P(state)P(finding|state)

i) P(PredictS1andActualS1)=(0.4)(0.6)=0.24

ii) P(PredictS1andActualS2)=(0.4)(0.4)=0.16

iii) P(PredictS2andActualS1)=(0.6)(0.2)=0.12

iv) P(PredictS2andActualS2

d) P(PredictS1)=0.24+0.12=0.36

P(PredictS2

e) P(state|finding)=P(stateandfinding)/P(finding)

P(ActualS1|PredictS1)=0.24/0.36=0.667

P(ActualS1|PredictS2)=0.16/0.64=0.250

P(ActualS2|PredictS1)=0.12/0.36=0.333

P(ActualS2|PredictS2

f)

g) IfS1ispredicted,thenchoosingalternativeA1maximizestheexpectedpayoff($233.33).

h) IfS2ispredicted,thenchoosingalternativeA2maximizestheexpectedpayoff($75).

i) Expectedpayoffgivenresearchis(0.36)($233.33)+(0.64)($75)–$100=$32.

j) TheoptimalpolicyistodonoresearchandsimplychooseA1.

k)

12.17 athroughd)

e)

12.18 a)

StateofNature

Alternative

Successful

Unsuccessful

Developnewproduct

$1,500,000

–$1,800,000

Don’tdevelopnewproduct

0

0

PriorProbabilities

b) Choosingtodeveloptheproductmaximizestheexpectedpayoff($400,000).

c) Withperfectinformation,Telemoreshoulddeveloptheproductifitwouldbesuccessful,anddon’tifitwillbeunsuccessful.

EP(perfectinformation)=(0.667)(1.5)+(0.333)(0)=$1million.

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=$1,000,000–$400,000=$600,000.

Thisindicatesthatconsiderationshouldbegiventoconductingthemarketsurvey.

d)

e) Theyshouldconductthesurvey,anddeveloptheproductifthesurveypredictstheproductwillbesuccessful.Theexpectedpayoffis$520,000.

f)

12.19 a)

StateofNature

Alternative

Screen

–$1,500

–$1,500

Don’tscreen

–$750

–$3,750

PriorProbabilities

b) Choosingnottoscreenmaximizestheexpectedpayoff.Theexpectedcostis$1,350.

c) Withperfectinformation,theywouldscreenifp=0.25,anddon’tscreenifp=0.05.

EP(withperfectinformation)=(0.8)(–$750)+(0.2)(–$1,500)=–$900

EVPI=EP(withperfectinformation)–EP(withoutmoreinformation)

=(–$900)–(–$1,350)=$450.

Thisindicatesthatconsiderationshouldbegiventoinspectingthesingleitem.

d)

e) Theoptimalpolicyisnottopre-screenorscreen.

12.20 a)

StateofNature

Alternative

Sell10,000

Sell100,000

BuildComputers

$0

$54million

SellRights

$15million

$15million

b)

c) Theyshouldbuildcomputers,withanexpectedpayoffof$27million.

d)

e)

f) Letp=priorprobabilityofselling10,000.

ForBuild:

EP =p(0)+(1–p)(54)

=–54p+54

ForSell:

EP =p(15)+(1–p)(15)

=15

BuildandSellcrosswhen–54p+54=15or54p=39orp=0.722

Theyshouldbuildwhenp≤0.722,andsellwhenp>0.722.

12.21 a) Withperfectinformation,theyshouldbuildcomputersiftheywillsell100,000ofthem,andselltherightsiftheycouldonlysell10,000computers.

EP(withperfectinformation)=(0.5)(54)+(0.5)(15)=$34.5million

EVPI=EP(withperfectinformation)–EPwithoutmoreinformation)

=34.5–27=$7.5million.

b)Sincethemarketresearchwillcost$1millionitmightbeworthwhiletoperformit.

c)

d)

12.22 a) Theoptimalpolicyistodonomarketresearchandbuildthecomputers.Theexpectedpayoffis$27million.

b) Iftherightscanbesoldfor$16.5or$13.5million,theoptimalpolicyisstilltobuildthecomputerswithanexpectedpayoffof$27million.

Ifthecostofsettinguptheassemblylineis$5.4millionor$6.6million,theoptimalpolicyisstilltobuildthecomputerswithanexpectedpayoffof$27.6or$26.4million,respectively.

Ifthedifferencebetweenthesellingpriceandvariablecostofeachcomputeris$540or$660,theoptimalpolicyisstilltobuildthecomputerswithanexpectedpayoffof$23.7or$33.3million,respectively.

Foreachcombinationoffinancialdata,theexpectedpayoffisasshownbelow.Inallcases,theoptimalpolicyistobuildthecomputers(withoutmarketresearch).

SellRights

Costof

AssemblyLine

SellingPrice–

VariableCost

Expected

Payoff

$13.5million

$5.4million

$540

$24.3million

$13.5million

$5.4million

$660

$30.9million

$13.5million

$6.6million

$540

$23.1million

$13.5million

$6.6million

$660

$29.7million

$16.5million

$5.4million

$540

$24.3million

$16.5million

$5.4million

$660

$30.9million

$16.5million

$6.6million

$540

$23.1million

$16.5million

$6.6million

$660

$29.7million

c)

d)

12.23 aandb)

12.24

12.25 a)

StateofNature

Alternative

WinningSeason

LosingSeason

Holdcampaign

$3million

–$2million

Don’tholdcampaign

0

0

PriorProbabilities

b) Choosingtoholdthecampaignmaximizestheexpectedpayoff($1million).

c) Withperfectinformation,LelandUniversityshouldholdthecampaigniftheywillhaveawinningseasonanddon’tholdthecampaigniftheywillhavealosingseason.

EP(withperfectinformation)=(0.6)(3)+(0.4)(0)=$1.8million

EVPI =EP(withperfectinfo)–EP(withoutmoreinfo)

=$1.8million–$1million=$800,000.

d)

e)

f&g) LelandUniversityshouldhireWilliam.Ifhepredictsawinningseasonthentheyshouldholdthecampaign,ifhepredictsalosingseasonthentheyshouldnotholdthecampaign.

12.26 a&c) (Note:thisdecisiontreecontinuesonthenextpage.)

b) Thecomptrollershouldinvestinstocksthefirstyear.Ifthereisgrowthduringthefirstyearthensheshouldinvestinstocksagainthesecondyear.Ifthereisarecessionduringthefirstyearthensheshouldinvestinbondsforthesecondyear.Theexpectedpayoffis$122.94million.

12.27 a&b) TheoptimalpolicyistowaituntilWednesdaytobuyifthepriceis$9onTuesday.Ifthepriceis$10or$11onTuesdaythenbuyonTuesday.

12.28 Theoptimalpolicyistosamplethefruitandbuyifitisexcellentandrejectifitisunsatisfactory.

12.29 a)

StateofNature

Alternative

Successful

Unsuccessful

Introducenewproduct

$40million

–$15million

Don’tintroducenewproduct

0

0

PriorProbabilities

Choosetointroducethenewproduct(expectedpayoffis$12.5million).

b) Withperfectinformation,MortonWardshouldintroducetheproductifitwillbesuccessful,anddon’tintroducetheproductifitwon’t.

EP(withperfectinformation)=(0.5)(40)+(0.5)(0)=$20million.

EVPI=EP(withperfectinfo)–EP(withoutmoreinfo)=20–12.5=$7.5million.

c) Theoptimalpolicyisnottotestbuttointroducethenewproduct.Theexpectedpayoffis$12.5million.

d) Ifthenetprofitifsuccessfulisonly$30million,thentheoptimalpolicyistoconductthetestmarketandonlyintroducetheproductifthetestmarketapproves.Theexpectedpayoffis$8.125million.

Ifthenetprofitifsuccessfulis$50million,thentheoptimalpolicyistoskipthetestmarketandintroducetheproduct,withanexpectedpayoffof$17.5million.

Ifthenetlossifunsuccessfulisonly$11.25million,thentheoptimalpolicyistoskipthetestmarketandintroducetheproduct,withanexpectedpayoffof$14.375million.

Ifthenetlossifunsuccessfulis$18.75million,thentheoptimalpolicyistoconductthetestmarketandonlyintroducetheproductifthetestmarketapproves.Theexpectedpayoffis$11.656million.

Foreachcombinationoffinancialdata,theexpectedpayoffisasshownbelow.Inallcases,theoptimalpolicyistobuildthecomputers(withoutmarketresearch).

NetProfitif

Successful

NetLossif

Unsuccessful

Optimal

Policy

Expected

Payoff

$30million

$11.25million

SkipTest,IntroduceProduct

$9.375million

$30million

$18.75million

Test,IntroduceifApprove

$7.656million

$50million

$11.25million

SkipTest,IntroduceProduct

$19.375million

$50million

$18.75million

Test,IntroduceifApprove

$15.656million

e)

f)

Bothchartsindicatethattheexpectedprofitissens