商务与经济统计第21章课件

SlidesbyJOHNLOUCKSSt.Edward’sUniversityStatisticsforBusinessandEconomics,11EAnderson/Sweeney/WilliamsSlidesbyChapter21

DecisionAnalysisProblemFormulationDecisionMakingwithProbabilitiesDecisionAnalysis withSampleInformationComputingBranchProbabilities UsingBayes’TheoremChapter21

DecisionAnalysisPrProblemFormulationThefirststepinthedecisionanalysisprocessisproblemformulation.Webeginwithaverbalstatementoftheproblem.Thenweidentify:thedecisionalternativesthestatesofnature(uncertainfutureevents)thepayoffs(consequences)associatedwitheachspecificcombinationof:decisionalternativestateofnatureProblemFormulationThefirstsProblemFormulationAdecisionproblemischaracterizedbydecisionalternatives,statesofnature,andresultingpayoffs.Thedecisionalternativesarethedifferentpossiblestrategiesthedecisionmakercanemploy.Thestatesofnaturerefertofutureevents,notunderthecontrolofthedecisionmaker,whichmayoccur.Statesofnatureshouldbedefinedsothattheyaremutuallyexclusiveandcollectivelyexhaustive.ProblemFormulationAdecisionTheconsequenceresultingfromaspecificcombinationofadecisionalternativeandastateofnatureisapayoff.PayoffTablesAtableshowingpayoffsforallcombinationsofdecisionalternativesandstatesofnatureisapayofftable.Payoffscanbeexpressedintermsofprofit,cost,time,distanceoranyotherappropriatemeasure.TheconsequenceresultingfromDecisionTreesAdecisiontreeprovidesagraphicalrepresentationshowingthesequentialnatureofthedecision-makingprocess.Eachdecisiontreehastwotypesofnodes:roundnodescorrespondtothestatesofnaturesquarenodescorrespondtothedecisionalternativesDecisionTreesAdecisiontreeDecisionTreesThebranchesleavingeachroundnoderepresentthedifferentstatesofnaturewhilethebranchesleavingeachsquarenoderepresentthedifferentdecisionalternatives.Attheendofeachlimbofatreearethepayoffsattainedfromtheseriesofbranchesmakingupthatlimb.DecisionTreesThebranchesleaDecisionMakingwithProbabilitiesOncewehavedefinedthedecisionalternativesandstatesofnatureforthechanceevents,wefocusondeterminingprobabilitiesforthestatesofnature.Theclassicalmethod,relativefrequencymethod,orsubjectivemethodofassigningprobabilitiesmaybeused.BecauseoneandonlyoneoftheNstatesofnaturecanoccur,theprobabilitiesmustsatisfytwoconditions:P(sj)>0forallstatesofnatureDecisionMakingwithProbabiliDecisionMakingwithProbabilitiesThenweusetheexpectedvalueapproachtoidentifythebestorrecommendeddecisionalternative.Theexpectedvalueofeachdecisionalternativeiscalculated(explainedonthenextslide).Thedecisionalternativeyieldingthebestexpectedvalueischosen.DecisionMakingwithProbabiliExpectedValueApproach where:N=thenumberofstatesofnature

P(sj)=theprobabilityofstateofnaturesj

Vij=thepayoffcorrespondingtodecision alternativediandstateofnaturesjTheexpectedvalue(EV)ofdecisionalternativediisdefinedasTheexpectedvalueofadecisionalternativeisthesumofweightedpayoffsforthedecisionalternative.ExpectedValueApproach where:ExpectedValueApproach BurgerPrinceRestaurantisconsidering openinganewrestaurantonMainStreet. Ithasthreedifferentmodels,each withadifferentseatingcapacity. BurgerPrinceestimatesthattheaverage numberofcustomersperhourwillbe 80,100,or120.Thepayofftableforthe threemodelsisonthenextslide.Example:BurgerPrinceExpectedValueApproach BurgeExpectedValueApproachPayoffTable$6,000$16,000$21,000$8,000$18,000$12,000$10,000$15,000$14,000AverageNumberofCustomersPerHours1=80s2=100s3=120ModelAModelBModelCExpectedValueApproachPayoffExpectedValueApproachCalculatetheexpectedvalueforeachdecision.Thedecisiontreeonthenextslidecanassistinthiscalculation.Hered1,d2,d3representthedecisionalternativesofmodelsA,B,andC.Ands1,s2,s3representthestatesofnatureof80,100,and120customersperhour.ExpectedValueApproachCalcula..2.4d1d2d3s1s1s1s2s3s2s2s3s3Payoffs10,00015,00014,0008,00018,00012,0006,00016,00021,000234ExpectedValueApproachDecisionTree..2.4d1d2d3s1s1sExpectedValueApproach3d1d2d3EMV=.4(10,000)+.2(15,000)+.4(14,000)=$12,600EMV=.4(8,000)+.2(18,000)+.4(12,000)=$11,600EMV=.4(6,000)+.2(16,000) +.4(21,000)=$14,000ModelAModelBModelC214ChoosethemodelwithlargestEV,ModelCExpectedValueApproach3d1d2d3ExpectedValueofPerfectInformationFrequently,informationisavailablethatcanimprovetheprobabilityestimatesforthestatesofnature.Theexpectedvalueofperfectinformation(EVPI)istheincreaseintheexpectedprofitthatwouldresultifoneknewwithcertaintywhichstateofnaturewouldoccur.TheEVPIprovidesanupperboundontheexpectedvalueofanysampleorsurveyinformation.ExpectedValueofPerfectInfoTheexpectedvalueofperfectinformationisdefinedasExpectedValueofPerfectInformationEVPI=|EVwPI–EVwoPI|EVPI=expectedvalueofperfectinformationEVwPI=expectedvaluewithperfectinformationaboutthestatesofnatureEVwoPI=expectedvaluewithoutperfectinformationaboutthestatesofnaturewhere:TheexpectedvalueofperfectStep1: Determinetheoptimalreturncorrespondingtoeachstateofnature.Step3: SubtracttheEVoftheoptimaldecisionfromtheamountdeterminedinstep(2).Step2: Computetheexpectedvalueoftheseoptimalreturns.ExpectedValueofPerfectInformationEVPICalculationStep1:Step3:Step2:ExpectedEVPI=.4(10,000)+.2(18,000)+.4(21,000)-14,000=$2,000ExpectedValueofPerfectInformationCalculatetheexpectedvaluefortheoptimumpayoffforeachstateofnatureandsubtracttheEVoftheoptimaldecision.EVPI=.4(10,000)+.2(18,000)DecisionAnalysisWithSampleInformationKnowledgeofsample(survey)informationcanbeusedtorevisetheprobabilityestimatesforthestatesofnature.Priortoobtainingthisinformation,theprobabilityestimatesforthestatesofnaturearecalledpriorprobabilities.Withknowledgeofconditionalprobabilitiesfortheoutcomesorindicatorsofthesampleorsurveyinformation,thesepriorprobabilitiescanberevisedbyemployingBayes'Theorem.Theoutcomesofthisanalysisarecalledposteriorprobabilitiesorbranchprobabilitiesfordecisiontrees.DecisionAnalysisWithSampleAdecisionstrategyisasequenceofdecisionsandchanceoutcomes.Thedecisionschosendependontheyettobedeterminedoutcomesofchanceevents.Theapproachusedtodeterminetheoptimaldecisionstrategyisbasedonabackwardpassthroughthedecisiontree.DecisionAnalysisWithSampleInformationDecisionStrategyAdecisionstrategyisasequeAtChanceNodes: Computetheexpectedvaluebymultiplyingthepayoffattheendofeachbranchbythecorrespondingbranchprobability.AtDecisionNodes: Selectthedecisionbranchthatleadstothebestexpectedvalue.Thisexpectedvaluebecomestheexpectedvalueatthedecisionnode.DecisionAnalysisWithSampleInformationBackwardPassThroughtheDecisionTreeAtChanceNodes:AtDecisionNo

BurgerPrincemustdecidewhethertopurchasea marketingsurveyfromStantonMarketingfor$1,000. Theresultsofthesurveyare"favorable"or "unfavorable".Theconditional probabilitiesare:DecisionAnalysisWithSampleInformationExample:BurgerPrinceP(favorable|

120customersperhour)=.9P(favorable|

100customersperhour)=.5P(favorable|

80customersperhour)=.2 BurgerPrincemustdecidDecisionTree(tophalf)s1(.148)s1(.148)s1(.148)s2(.185)s2(.185)s2(.185)s3(.667)s3(.667)s3(.667)$10,000$15,000$14,000$8,000$18,000$12,000$6,000$16,000$21,000I1(.54)d1d2d324561DecisionAnalysisWithSampleInformationDecisionTree(tophalf)s1(.1DecisionTree(bottomhalf)s1(.696)s1(.696)s1(.696)s2(.217)s2(.217)s2(.217)s3(.087)s3(.087)s3(.087)$10,000$15,000$14,000$8,000$18,000$12,000$6,000$16,000$21,000d1d2d337891

I2(.46)DecisionAnalysisWithSampleInformationDecisionTree(bottomhalf)s1

I2(.46)d1d2d3EMV=.696(10,000)+.217(15,000)+.087(14,000)=$11,433EMV=.696(8,000)+.217(18,000)+.087(12,000)=$10,554EMV=.696(6,000)+.217(16,000)+.087(21,000)=$9,475

I1(.54)d1d2d3EMV=.148(10,000)+.185(15,000)+.667(14,000)=$13,593EMV=.148(8,000)+.185(18,000)+.667(12,000)=$12,518EMV=.148(6,000)+.185(16,000)+.667(21,000)=$17,855456789231$17,855$11,433DecisionAnalysisWithSampleInformationI2d1d2d3EMV=.696(10,000)+ExpectedValueofSampleInformationTheexpectedvalueofsampleinformation(EVSI)istheadditionalexpectedprofitpossiblethroughknowledgeofthesampleorsurveyinformation.EVSI=|EVwSI–EVwoSI|EVSI=expectedvalueofsampleinformationEVwSI=expectedvaluewithsampleinformationaboutthestatesofnatureEVwoSI=expectedvaluewithoutsampleinformationaboutthestatesofnaturewhere:ExpectedValueofSampleInforStep1: Determinetheoptimaldecisionanditsexpectedreturnforthepossibleoutcomesofthesampleusingtheposteriorprobabilitiesforthestatesofnature.Step2:

Computetheexpectedvalueoftheseoptimalreturns.ExpectedValueofSampleInformationEVwSICalculationStep1:Step2:ExpectedValue

I2(.46)d1d2d3$11,433$10,554$9,475

I1(.54)d1d2d3$13,593$12,518$17,855456789231$17,855$11,433DecisionAnalysisWithSampleInformationEVwSI=.54(17,855) +.46(11,433) =$14,900.88I2d1d2d3$11,433$10,554$9,4ExpectedValueofSampleInformationIftheoutcomeofthesurveyis"favorable”, chooseModelC.Iftheoutcomeofthesurveyis“unfavorable”,chooseModelA.EVwSI=.54($17,855)+.46($11,433)=$14,900.88ExpectedValueofSampleInforExpectedValueofSampleInformationSubtracttheEVwoSI(thevalueoftheoptimaldecisionobtainedwithoutusingthesampleinformation)fromtheEVwSI.EVSICalculationEVSI=.54($17,855)+.46($11,433)-$14,000=$900.88Conclusion BecausetheEVSIislessthanthecostofthesurvey,thesurveyshouldnotbepurchased.ExpectedValueofSampleInforComputingBranchProbabilities

UsingBayes’TheoremBayes’Theoremcanbeusedtocomputebranchprobabilitiesfordecisiontrees.Forthecomputationsweneedtoknow:theinitial(prior)probabilitiesforthestatesofnature,theconditionalprobabilitiesfortheoutcomesorindicatorsofthesampleinformationgiveneachstateofnature.Atabularapproachisaconvenientmethodforcarryingoutthecomputations.ComputingBranchProbabilitiesStep1ComputingBranchProbabilities

UsingBayes’TheoremForeachstateofnature,multiplythepriorprobabilitybyitsconditionalprobabilityfortheindicator.Thisgivesthejointprobabilitiesforthestatesandindicator.Step2Sumthesejointprobabilitiesoverallstates.Thisgivesthemarginalprobabilityfortheindicator.Step3Foreachstate,divideitsjointprobabilitybythemarginalprobabilityfortheindicator.Thisgivestheposteriorprobabilitydistribution.Step1ComputingBranchProbabi

RecallthatBurgerPrinceisconsideringpurchasinga marketingsurveyfromStantonMarketing.Theresults ofthesurveyare"favorable“or"unfavorable".The conditionalprobabilitiesare:DecisionAnalysisWithSampleInformationExample:BurgerPrinceP(favorable|

120customersperhour)=.9P(favorable|

100customersperhour)=.5P(favorable|

80customersperhour)=.2Computethebranch(posterior)probabilitydistribution. RecallthatBurgerPrincPosteriorProbabilitiesP(favorable)=.54Total.54State

Prior

Conditional

Joint

PosteriorFavorable.048.185.6671.000.08/.5480100.2.5.9PosteriorProbabilitiesP(favorPosteriorProbabilitiesP(unfavorable)=.46Total.46State

Prior

Conditional

Joint

PosteriorUnfavorable.32.10.04.696.217.0871.000.32/.4680100.8.5.1PosteriorProbabilitiesP(unfavEndofChapter21EndofChapter21SlidesbyJOHNLOUCKSSt.Edward’sUniversityStatisticsforBusinessandEconomics,11EAnderson/Sweeney/WilliamsSlidesbyChapter21

DecisionAnalysisProblemFormulationDecisionMakingwithProbabilitiesDecisionAnalysis withSampleInformationComputingBranchProbabilities UsingBayes’TheoremChapter21

DecisionAnalysisPrProblemFormulationThefirststepinthedecisionanalysisprocessisproblemformulation.Webeginwithaverbalstatementoftheproblem.Thenweidentify:thedecisionalternativesthestatesofnature(uncertainfutureevents)thepayoffs(consequences)associatedwitheachspecificcombinationof:decisionalternativestateofnatureProblemFormulationThefirstsProblemFormulationAdecisionproblemischaracterizedbydecisionalternatives,statesofnature,andresultingpayoffs.Thedecisionalternativesarethedifferentpossiblestrategiesthedecisionmakercanemploy.Thestatesofnaturerefertofutureevents,notunderthecontrolofthedecisionmaker,whichmayoccur.Statesofnatureshouldbedefinedsothattheyaremutuallyexclusiveandcollectivelyexhaustive.ProblemFormulationAdecisionTheconsequenceresultingfromaspecificcombinationofadecisionalternativeandastateofnatureisapayoff.PayoffTablesAtableshowingpayoffsforallcombinationsofdecisionalternativesandstatesofnatureisapayofftable.Payoffscanbeexpressedintermsofprofit,cost,time,distanceoranyotherappropriatemeasure.TheconsequenceresultingfromDecisionTreesAdecisiontreeprovidesagraphicalrepresentationshowingthesequentialnatureofthedecision-makingprocess.Eachdecisiontreehastwotypesofnodes:roundnodescorrespondtothestatesofnaturesquarenodescorrespondtothedecisionalternativesDecisionTreesAdecisiontreeDecisionTreesThebranchesleavingeachroundnoderepresentthedifferentstatesofnaturewhilethebranchesleavingeachsquarenoderepresentthedifferentdecisionalternatives.Attheendofeachlimbofatreearethepayoffsattainedfromtheseriesofbranchesmakingupthatlimb.DecisionTreesThebranchesleaDecisionMakingwithProbabilitiesOncewehavedefinedthedecisionalternativesandstatesofnatureforthechanceevents,wefocusondeterminingprobabilitiesforthestatesofnature.Theclassicalmethod,relativefrequencymethod,orsubjectivemethodofassigningprobabilitiesmaybeused.BecauseoneandonlyoneoftheNstatesofnaturecanoccur,theprobabilitiesmustsatisfytwoconditions:P(sj)>0forallstatesofnatureDecisionMakingwithProbabiliDecisionMakingwithProbabilitiesThenweusetheexpectedvalueapproachtoidentifythebestorrecommendeddecisionalternative.Theexpectedvalueofeachdecisionalternativeiscalculated(explainedonthenextslide).Thedecisionalternativeyieldingthebestexpectedvalueischosen.DecisionMakingwithProbabiliExpectedValueApproach where:N=thenumberofstatesofnature

P(sj)=theprobabilityofstateofnaturesj

Vij=thepayoffcorrespondingtodecision alternativediandstateofnaturesjTheexpectedvalue(EV)ofdecisionalternativediisdefinedasTheexpectedvalueofadecisionalternativeisthesumofweightedpayoffsforthedecisionalternative.ExpectedValueApproach where:ExpectedValueApproach BurgerPrinceRestaurantisconsidering openinganewrestaurantonMainStreet. Ithasthreedifferentmodels,each withadifferentseatingcapacity. BurgerPrinceestimatesthattheaverage numberofcustomersperhourwillbe 80,100,or120.Thepayofftableforthe threemodelsisonthenextslide.Example:BurgerPrinceExpectedValueApproach BurgeExpectedValueApproachPayoffTable$6,000$16,000$21,000$8,000$18,000$12,000$10,000$15,000$14,000AverageNumberofCustomersPerHours1=80s2=100s3=120ModelAModelBModelCExpectedValueApproachPayoffExpectedValueApproachCalculatetheexpectedvalueforeachdecision.Thedecisiontreeonthenextslidecanassistinthiscalculation.Hered1,d2,d3representthedecisionalternativesofmodelsA,B,andC.Ands1,s2,s3representthestatesofnatureof80,100,and120customersperhour.ExpectedValueApproachCalcula..2.4d1d2d3s1s1s1s2s3s2s2s3s3Payoffs10,00015,00014,0008,00018,00012,0006,00016,00021,000234ExpectedValueApproachDecisionTree..2.4d1d2d3s1s1sExpectedValueApproach3d1d2d3EMV=.4(10,000)+.2(15,000)+.4(14,000)=$12,600EMV=.4(8,000)+.2(18,000)+.4(12,000)=$11,600EMV=.4(6,000)+.2(16,000) +.4(21,000)=$14,000ModelAModelBModelC214ChoosethemodelwithlargestEV,ModelCExpectedValueApproach3d1d2d3ExpectedValueofPerfectInformationFrequently,informationisavailablethatcanimprovetheprobabilityestimatesforthestatesofnature.Theexpectedvalueofperfectinformation(EVPI)istheincreaseintheexpectedprofitthatwouldresultifoneknewwithcertaintywhichstateofnaturewouldoccur.TheEVPIprovidesanupperboundontheexpectedvalueofanysampleorsurveyinformation.ExpectedValueofPerfectInfoTheexpectedvalueofperfectinformationisdefinedasExpectedValueofPerfectInformationEVPI=|EVwPI–EVwoPI|EVPI=expectedvalueofperfectinformationEVwPI=expectedvaluewithperfectinformationaboutthestatesofnatureEVwoPI=expectedvaluewithoutperfectinformationaboutthestatesofnaturewhere:TheexpectedvalueofperfectStep1: Determinetheoptimalreturncorrespondingtoeachstateofnature.Step3: SubtracttheEVoftheoptimaldecisionfromtheamountdeterminedinstep(2).Step2: Computetheexpectedvalueoftheseoptimalreturns.ExpectedValueofPerfectInformationEVPICalculationStep1:Step3:Step2:ExpectedEVPI=.4(10,000)+.2(18,000)+.4(21,000)-14,000=$2,000ExpectedValueofPerfectInformationCalculatetheexpectedvaluefortheoptimumpayoffforeachstateofnatureandsubtracttheEVoftheoptimaldecision.EVPI=.4(10,000)+.2(18,000)DecisionAnalysisWithSampleInformationKnowledgeofsample(survey)informationcanbeusedtorevisetheprobabilityestimatesforthestatesofnature.Priortoobtainingthisinformation,theprobabilityestimatesforthestatesofnaturearecalledpriorprobabilities.Withknowledgeofconditionalprobabilitiesfortheoutcomesorindicatorsofthesampleorsurveyinformation,thesepriorprobabilitiescanberevisedbyemployingBayes'Theorem.Theoutcomesofthisanalysisarecalledposteriorprobabilitiesorbranchprobabilitiesfordecisiontrees.DecisionAnalysisWithSampleAdecisionstrategyisasequenceofdecisionsandchanceoutcomes.Thedecisionschosendependontheyettobedeterminedoutcomesofchanceevents.Theapproachusedtodeterminetheoptimaldecisionstrategyisbasedonabackwardpassthroughthedecisiontree.DecisionAnalysisWithSampleInformationDecisionStrategyAdecisionstrategyisasequeAtChanceNodes: Computetheexpectedvaluebymultiplyingthepayoffattheendofeachbranchbythecorrespondingbranchprobability.AtDecisionNodes: Selectthedecisionbranchthatleadstothebestexpectedvalue.Thisexpectedvaluebecomestheexpectedvalueatthedecisionnode.DecisionAnalysisWithSampleInformationBackwardPassThroughtheDecisionTreeAtChanceNodes:AtDecisionNo

BurgerPrincemustdecidewhethertopurchasea marketingsurveyfromStantonMarketingfor$1,000. Theresultsofthesurveyare"favorable"or "unfavorable".Theconditional probabilitiesare:DecisionAnalysisWithSampleInformationExample:BurgerPrinceP(favorable|

120customersperhour)=.9P(favorable|

100customersperhour)=.5P(favorable|

80customersperhour)=.2 BurgerPrincemustdecidDecisionTree(tophalf)s1(.148)s1(.148)s1(.148)s2(.185)s2(.185)s2(.185)s3(.667)s3(.667)s3(.667)$10,000$15,000$14,000$8,000$18,000$12,000$6,000$16,000$21,000I1(.54)d1d2d324561DecisionAnalysisWithSampleInformationDecisionTree(tophalf)s1(.1DecisionTree(bottomhalf)s1(.696)s1(.696)s1(.696)s2(.217)s2(.217)s2(.217)s3(.087)s3(.087)s3(.087)$10,000$15,000$14,000$8,000$18,000$12,000$6,000$16,000$21,000d1d2d337891

I2(.46)DecisionAnalysisWithSampleInformationDecisionTree(bottomhalf)s1

I2(.46)d1d2d3EMV=.696(10,000)+.217(15,000)+.087(14,000)=$11,433EMV=.696(8,000)+.217(18,000)+.087(12,000)=$10,554EMV=.696(6,000)+.217(16,000)+.087(21,000)=$9,475

I1(.54)d1d2d3EMV=.148(10,000)+.185(15,000)+.667(14,000)=$13,593EMV=.148(8,000)+.185(18,000)+.667(12,000)=$12,518EMV=.148(6,000)+.185(16,000)+.667(21,000)=$17,855456789231$17,855$11,433DecisionAnalysisWithSampleInformationI2d1d2d3EMV=.696(10,000)+ExpectedValueofSampleInformationTheexpectedvalueofsampleinformation(EVSI)istheadditionalexpectedprofitpossiblethroughknowledgeofthesampleorsurveyinformation.EVSI=|EVwSI–EVwoSI|EVSI=expectedvalueofsampleinformationEVwSI=expectedvaluewithsampleinformationaboutthestatesofnatureEVwoSI=expectedvaluewithoutsampleinformationaboutthestatesofnaturewhere:ExpectedValueofSampleInforStep1: Determinetheoptimaldecisionanditsexpectedreturnforthepossibleoutcomesofthesampleusingtheposteriorprobabilitiesforthestatesofnature.Step2:

Computetheexpectedvalueoftheseoptimalreturns.ExpectedValueofSampleInformationEVwSICalculationStep1:Step2:ExpectedValue

I2(.46)d1d2d3$11,433$10,554$9,475

I1(.54)d1d2d3$13,593$12,518$17,855456789231$17,855$11,433DecisionAnalysisWithSampleInformationEVwSI=.54(17,855) +.46(11,433) =$14,900.88I2d1d2d3$11,4