decision-analysis(ppt文档).ppt

DecisionAnalysisDecisionAnalysisTherearemanykindsofdecisionmakinginthefaceofgreatuncertaintythatdecisionanalysisisdesignedtoaddress.Decisionanalysisprovidesaframeworkandmethodologyforrationaldecisionmakingwhentheoutcomesareuncertain.DecisionmakingcanbedividedintoDecisionmakingwithoutexperimentationDecisionmakerisnotawareoftheprobabilityofpossiblestatesofnature.DecisionmakingwithexperimentationDecisionmakerhassomeinformationontheprobabilityofpossiblestatesofnature.DecisionMakingunderUncertainty(withoutExperimentation)Whatthedecisionmakerknows:FeasiblealternativesunderconsiderationforhowtoproceedwiththeproblemofconcernPossiblestatesofnatureResultingpayoffforeachcombinationofanactionandastateofnatureItisuncertainwhichstateofnaturetakesplaceWhatthedecisionmakerdoesntknow:TheprobabilitythateachstateofnaturetakesplaceDecisionMakingunderUncertainty(withoutExperimentation)Example:FFCompanyisgoingtodecidehowmuchproductionforanewtypeofproductshouldbeimplemented.Thecompanyhasnoinformationaboutthemarketdemandfortheproduct,butthepayoffforeachpossiblestateofmarketisknown.N1(largedemand)N2(smalldemand)S1(large-batchproduction)30-6S2(middle-batchproduction)20-2S3(small-batchproduction)105Denoteby(Si,Nj)thepayoffforthesituationwhereFFchoosesSiandthenatureofmarketdemandisNj.TheMaximinPayoffCriterion(pessimisticcriterion)Considertheproblemfromtheviewoftheworstsituation:Maximinpayoffcriterion:Foreachpossibleaction,findtheminimumpayoffoverallpossiblestatesofnature.Next,findthemaximumoftheseminimumpayoffs.Choosetheactionwhoseminimumpayoffgivesthismaximum.TheMaximaxPayoffCriterion(optimisticcriterion)Considertheproblemfromtheviewofthebestsituation:Maximaxpayoffcriterion:Foreachpossibleaction,findthemaximumpayoffoverallpossiblestatesofnature.Next,findthemaximumofthesemaximumpayoffs.Choosetheactionwhosemaximumpayoffgivesthismaximum.EquallyLikelyCriterionDecisionmakerassumesthatthechancethateverystateofnaturetakesplaceisthesame:Thechancethateverypossiblestateofnaturetakesplaceis1/”thenumberofstatesofnature”.Thenthedecisionmakermaximizestheexpectedpayoff.HurwiczDecisionCriterionAcompromisebetweenthepessimisticcriterionandtheoptimisticcriterion:Determineaparameter(01),andcomputeCVi=max(Si,Nj)+(1-)min(Si,Nj)PicktheactionassociatedwiththelargestvalueofCVi.set=0.7TheSavageCriterion(RegretCriterion)DecisionmakerwantstominimizeregretTakethemaximumpayoffundereverypossiblestateofnatureasanidealobjective,theregretvalueofanactionisdefinedasthedifferencebetweenthepayoffobtainedbytakingtheactionandtheidealobjective.Foreachpossibleaction,findthemaximumregretoverallpossiblestatesofnature.Next,findtheminimumofthesemaximumregretvalues.Choosetheactionwhosemaximumregretgivesthismaximum.DecisionMakingunderUncertainty(withExperimentation)Whatthedecisionmakerknows:FeasiblealternativesunderconsiderationforhowtoproceedwiththeproblemofconcernPossiblestatesofnatureResultingpayoffforeachcombinationofanactionandastateofnatureItisuncertainwhichstateofnaturetakesplaceThepriorprobabilitythateachstateofnaturetakesplaceTheMaximumLikelihoodCriterionIdentifythemostlikelystateofnature(theonewiththelargestpriorprobability).Forthisstateofnature,findtheactionwiththemaximumpayoff.Choosethisaction.AssumethatinthepreviousexampletheprobabilitythatN1takesplaceis0.3.ExpectedValueCriterionAccordingtothepriorprobabilitiesofthestatesofnature,computetheexpectedpayoffsforallpossibleactions,choosetheactiongivingthemaximumexpectedpayoff.E(Si)=P(Nj)(Si,Nj)DecisionTreesForsomecomplicateddecision-makingproblems,e.g.,sequentialdecision-makingproblems,wesometimescouldemploydecisiontrees.Thenodesofadecisiontreearereferredtoasforks,andthearcsarecalledbranches.Adecisionfork,representedbyasquare,indicatesthatadecisionneedstobemadeatthatpointintheprocess.Achancefork,representedbyacircle,indicatesthatarandomeventoccursatthatpoint.Anendnode,representedbyatriangle,indicatesthataresultisreachedatthatpoint.DecisionTreesDecision forkS1S2S3large-batch productionmiddle-batch productionSmall-batch productionN1(large demand);P(N1)=0.330-6N2(small demand);P(N2)=0.72010-254.84.66.56.5Decision forkChance forksEnd nodesN1(large demand);P(N1)=0.3N2(small demand);P(N2)=0.7N2(small demand);P(N2)=0.7N1(large demand);P(N1)=0.3DecisionTreesMakeadecisionbyconstructingadecisiontree:ConstructadecisiontreefromlefttorightDrawadecisionforkandseveralchanceforks.Eachchanceforkcorrespondstooneaction,andthearcconnectingthedecisionforkandachanceforkrepresentsthecorrespondingaction.Drawendnodesandconnectthemtothechanceforks.Eacharcconnectingachanceforkandanendnoderepresentsonepossibleresult.Computetheexpectedpayoffsforalltheactions,fromrighttoleft.Marktheexpectedpayoffsbesidesthecorrespondingactions.Picktheactionassociatedwiththemaximumexpectedpayoffastheoptimalone,andmarkontheotheractions.SensitivityAnalysisInreallife,theestimationoftheparameters,e.g.,probabilitiesofnaturestates,payoffsforpossibleactions,isusuallyimprecise.Byemployingsensitivityanalysis,wecanfindtherangeoftheparameterssuchthattheoptimaldecisionstaysiftheparametersareintherange.Inpreviousexample,letP(N1)=p,P(N2)=1-pE(S1)=p30+(1-p)(-6)=36p6E(S2)=p20+(1-p)(-2)=22p2E(S3)=p10+(1-p)(+5)=5p+5SensitivityAnalysisE(S1)E(S2)E(S3)010.35ptake S3take S1ExpectedValueofPerfectInformation(EVPI)Asweallknow,wecanmakebetterdecisionsifwegetmoreinformationabouttheproblemconcerning.However,wehavetopaymoreongettingmoreinformation.Therefore,weshouldevaluatethecostsandthegainsofobtainingperfectinformation,andthendeterminewhethertogetperfectinformation.Inthepreviousexample,withoutperfectinformation,weadopttheexpectedvaluecriterionandtakeS3astheoptimaldecision.Themaximumexpectedpayoffis0.3 10+0.7 5=6.5,whichwedenotebyEVWOPI.ExpectedValueofPerfectInformation(EVPI)Ifwehaveperfectinformation,thenweareawareoftheunderlyingstateofnaturebeforewemakedecisions:IfweknowthatthestateofnatureisN1,thewesurelytakeactionS1andgetpayoffof30.Theprobabilityoftheaboveeventis0.3.IfweknowthatthestateofnatureisN2,thewesurelytakeactionS3andgetpayoffof5.Theprobabilityoftheaboveeventis0.7.Therefore,afterweobtaintheperfectinformation,theexpectedpayoffwegetiscalledEVWPI,whichis0.3 30+0.7 5=12.5.EVWPI-EVWOPI=6,thentheexpectedvalueoftheperfectinformation(EVPI)is6.Ifthecostsofgettingtheperfectinformationislessthan6,itisnecessaryforthedecisionmakertopayfortheperfectinformation.BayesDecisionRulePerfectinformationisusuallynotavailableinreallife.However,wecangetpartialinformationfromexperimentation.Priorprobability:probabilitygainedfromexperienceorknowledgeofexperts.Posteriorprobability:probabilityobtainedfrommodifyingthepriorprobabilitybytakingintoaccountthefindingsinexperimentation.BayesTheoremGiventhatthefindingfromexperimentationisIk,theprobabilitythattheunderlyingstateofnatureisNjiswhereBayesDecisionRuleFFCompanyisgoingtoinviteanothercompanytoperformmarketinvestigation(sampling).TheresultsofsamplingincludeI1meaningthatthemarketdemandislarge,andI2meaningthatthedemandissmall.Wehavethefollowingconditionalprobabilities:Howtomakedecisionsbyusingthesampleinformation?Ifittakes3toperformthesampling,isitnecessary?N1N2I1P(I1/N1)=0.8P(I1/N2)=0.1I2P(I2/N1)=0.2P(I2/N2)=0.9BayesDecisionRuleWeconstructadecisiontreetosolvethisproblem.WeneedtoknowTheprobabilityoftheresultsofsamplingP(I)Giventheresultofsampling,theprobabilityofthemarketdemandP(N|I)WecancalculateP(I1)=0.31P(I2)=0.69Computetheexpectedpayoffofeachnode.P(N/I)N1N2I10.77420.2258I20.08700.9130BayesDecisionRuleConclusion:Iftheresultofsamplingislarge-demand,FFcompanyshouldselectlarge-batchproduction;Iftheresultofsamplingissmall-demand,FFcompanyshouldselectsmall-batchproduction.BayesDecisionRuleFromthedecisiontree,wefindthattheexpectedpayoffis10.5302,higherthan6.5,theexpectedpayoffwithoutsampling.Thedifferenceiscalledtheexpectedvalueofsampleinformation(EVSI).EVSI=10.5302-6.5=4.0302IfthecostforsamplingislowerthanEVSI,theninvestigationofmarketisnecessary.BayesDecisionRuleEfficiencyofsampleinformation:Theefficiencyofsampleinformation=EVSI/EVPI 100%Inthepreviousexample,theefficiencyofsampleinformationis4.0302/6 100%=67.17%.RecallhowtocomputeEVSIandEVPI,weknowthattheefficiencyofsampleinformationisrelatedwithDifferencebetweenpayoffsofdifferentoutcomes;Accuracyofsampleinformation.UtilityTheorySupposethatanindividualisofferedthechoiceofacceptinga50:50chanceofwinning$100,000ornothingorreceiving$40,000withcertainty.Manypeoplewouldpreferthe$40,000eventhoughtheexpectedpayoffonthe50:50chanceofwinning$100,000is$50,000.UtilityTheorySomecaseswheretheexpectedpayoffisnotagoodcriterionAcompanymaybeunwillingtoinvestalargesumofmoneyinanewproductevenwhentheexpectedprofitissubstantialifthereisariskoflosingitsinvestmentandtherebybecomingbankrupt.Peoplebuyinsuranceeventhoughitisapoorinvestmentfromtheviewpointoftheexpectedpayoff.UtilityTheoryThereisawayoftransformingmonetaryvalues toanappropriatescalethatreflectsthedecisionmakerspreferences.Thisscaleiscalledtheutilityfunctionformoney.Thefollowingfigureshowsatypicalutilityfunctionu(M)formoneyM.Itindicatesthatanindividualhavingthisutilityfunctionwouldvalueobtaining$30,000twiceasmuchas$10,000andwouldvalueobtaining$100,000twiceasmuchas$30,000.UtilityTheory01234$10,000$30,000$60,000$100,000Mu(M)UtilityTheoryHavingthisdecreasingslopeofthefunctionastheamountofmoneyincreasesisreferredtoashavingadecreasingmarginalutilityformoney.Suchanindividualisreferredtoasbeingrisk-averse.However,notallindividualshaveadecreasingmarginalutilityformoney.Somepeoplearenotrisk-aversebutriskseekers,andtheygothroughlifelookingforthe“bigscore.”Theslopeoftheirutilityfunctionincreasesastheamountofmoneyincreases,sotheyhaveanincreasingmarginalutilityformoney.UtilityTheoryTheintermediatecaseisthatofarisk-neutralindividual,whoprizesmoneyatitsfacevalue.Suchanindividualsutilityformoneyissimplyproportionaltotheamountofmoneyinvolved.Althoughsomepeopleappeartoberisk-neutralwhenonlysmallamountsofmoneyareinvolved,itisunusualtobetrulyrisk-neutralwithverylargeamounts.Intheabovecase,itisindifferentbetweentomakedecisionsdependingontheexpectedpayoffsandtomakedecisionsdependingontheexpectedutility.UtilityTheoryItalsoispossibletoexhibitamixtureofthesekindsofbehavior.Forexample,anindividualmightbeessentiallyrisk-neutralwithsmallamountsofmoney,thenbecomeariskseekerwithmoderateamounts,andthenturnrisk-aversewithlargeamounts.Inaddition,onesattitudetowardriskcanshiftovertimedependinguponcircumstances.UtilityTheoryThefactthatdifferentpeoplehavedifferentutilityfunctionsformoneyhasanimportantimplicationfordecisionmakinginthefaceofuncertainty:Whenautilityfunctionformoneyisincorporatedintoadecisionanalysisapproachtoaproblem,thisutilityfunctionmustbeconstructedtofitthepreferencesandvaluesofthedecisionmakerinvolved.(Thedecisionmakercanbeeitherasingleindividualoragroupofpeople.)UtilityTheoryThekeytoconstructingtheutilityfunctionformoneytofitthedecisionmakeristhefollowingfundamentalpropertyofutilityfunctions.FundamentalProperty:Undertheassumptionsofutilitytheory,thedecisionmakersutilityfunctionformoneyhasthepropertythatthedecisionmakerisindifferentbetweentwoalternativecoursesofactionifthetwoalternativeshavethesameexpectedutility.UtilityTheoryToobtaineither$100,000(utility=4)withprobabilitypornothing(utility=0)withprobability1-phasanexpectedutilityE(utility)=4p.Therefore,thedecisionmakerisindifferentbetweeneachofthefollowingthreepairsofalternatives:theofferwithp=0.25E(utility)=1ordefinitelyobtaining$10,000(utility=1)theofferwithp=0.5E(utility)=2ordefinitelyobtaining$30,000(utility=2)theofferwithp=0.75E(utility)=3ordefinitelyobtaining$60,000(utility=3)UtilityTheoryThescaleoftheutilityfunction(e.g.,utility=1for$10,000)isirrelevant.Itisonlytherelativevaluesoftheutilitiesthatmatter.Alltheutilitiescanbemultipliedbyanypositiveconstantwithoutaffectingwhichalternativecourseofactionwillhavethelargestexpectedutility.UtilityTheoryAcompanyisgoingtochooseamongtwoprojectsAandB.Becausethebudgetofthecompanyislimited,itcanonlytakeoneproject.Theprobabilitiesofthethreepossibilitiesofmarketdemandandthepayoffsofdifferentdecisionsaregiveninthefollowingtable.Becauseitsbadfinancialsituation,thecompanywillbebankruptifitsuffersfromalostover50milliondollars.UtilityTheoryFirst,wehaveE(S1)=0.360+0.540+0.2(-100)=18E(S2)=0.3100+0.5(-40)+0.2(-60)=-2E(S3)=0.30+0.50+0.20=0AlthoughE(S1)isthelargest,themanagementofthecompanywillstilltakeS3toavoidriskofbecomingbankrupt.Weconstructtheutilityfunctionforthecompany:Settheutilityofthemaximumpossiblepayoffgainedbytakingtheprojects100as10,thatisU(100)=10.Settheutilityoftheminimumpossiblepayoffgainedbytakingtheprojects-100as0,thatisU(-100)=0.Thevalueoftheutilityfunctionshouldincreaseasthepayoffincreases.UtilityTheoryHowtodeterminetheutilityofpayoff60?LetU(60)=p U(100)+(1-p)U(-100),thusnowthetaskistodeterminep.Asshownintheaboveformula,thefollowingalternativesshouldbeequivalentintheopinionofthemanagementofthecompany:Gain100milliondollarswithprobabilityp,andlose100milliondollarswithprobability1-p.Get60milliondollarsforcertainty.Asthemanagementofthecompanyisrisk-averse,thevalueofpwouldbesetasalargevaluethatiscloseto1,say,0.95.UtilityTheorySimilarly,wecansettheutilityofpayoffs40,0,-40,-60.Forexample,thevalueofpcorrespondingto40,0,-40,-60are0.90,0.75,0.55,0.40,respectively.WehaveU(40)=9.0,U(0)=7.5,U(-40)=5.5andU(-60)=4.0.