Optimization under Uncertainty

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Optimization under Uncertainty Sensitivity Analysis and Regret
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Tan, Chin Hon
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Industrial and Systems Engineering
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Hartman, Joseph C
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Richard, Jean-Philippe
Geunes, Joseph P
Koehler, Gary J

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optimization -- regret -- risk -- sensitivity
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Solving for the optimal solution of any problem under uncertainty is generally challenging. This dissertation explores optimization under uncertainty from the perspective of sensitivity analysis and regret. Sequential decision problems can often be modeled as Markov decision processes. Classical solution approaches assume that the parameters of the model are known. However, model parameters are usually estimated and uncertain in practice. As a result, managers are often interested in determining how estimation errors affect the optimal solution. We illustrate how sensitivity analysis can be performed directly for a Markov decision process with uncertain reward parameters using the Bellman equations. In particular, we consider problems involving (i) a single stationary parameter, (ii) multiple stationary parameters and (iii) multiple non-stationary parameters. We illustrate the applicability of this work on a capacitated stochastic lot-sizing problem. In sensitivity analysis, we study the stability or robustness of an optimal solution with respect to uncertainties in the model. If the optimal solution is constant across all possible scenarios, the uncertainties in the model parameters can be ignored. However, these uncertainties need to be addressed if the optimal solution differs under different possible scenarios. Research in psychology and behavioral decision theory suggests that regret plays an important role in shaping preferences in decisions under uncertainty. Despite their relevance, feelings of regret are often ignored in the optimization literature and the representation of regret, when present, is limited. Regret theory describes choice preferences based on the rewards received and opportunities missed. We show that regret-theoretic choice preferences are described by multivariate stochastic dominance, present regret-based risk measures and illustrate how they can be adopted within the mean-risk framework. Research also suggest that people are willing to exchange direct material gain for regret reduction. We consider an equipment replacement problem under horizon uncertainty. We present stochastic dynamic programming formulations and explore solutions which minimize either expected costs or maximum regret. We identify the critical time period where optimal decisions diverge for different horizon realizations and design a lease option contract such that owners can lower the regret that may result from a given horizon realization, while opening a possible source of revenue for a leasor. Finally, we also study the optimal number of products to offer under various conditions in a heterogeneous market. In the absence of regret, consumers are happier when presented with more choices and a company that wishes to capture a broad market share needs to provide a rich product line. However, the relationship between the optimal number of products and the number of market segments are reversed hen consumers are regret averse. In general, the product line should be narrow when outcomes are uncertain and consumers experience regret. In addition, it is almost surely optimal for the firm to offer a single product when the outcomes of choices are highly uncertain and/or consumers are highly regret averse. We also show that the optimal number of products to offer is non-increasing when the cost of introducing variety into the product line increases uniformly and obtain a tight upper bound on the expected optimal number of products to offer when fixed costs are uniform. However, interestingly, the optimal number of products to offer can increase when regret aversion increases.
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by Chin Hon Tan.
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Thesis (Ph.D.)--University of Florida, 2012.
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OPTIMIZATIONUNDERUNCERTAINTY:SENSITIVITYANALYSISANDREGRETByCHINHONTANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012ChinHonTan 2

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Idedicatethisdissertationtomyparents. 3

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ACKNOWLEDGMENTS IwouldliketogratefullyandsincerelythankDr.JosephC.Hartmanforhispatience,encouragementandsupportinmypursuitofproblemsthatexciteme.IwouldalsoliketothankmycommitteemembersDr.JosephP.Geunes,Dr.JeanPhilippeRichardandDr.GaryKoehlerfortheirvaluablecommentsthatimprovedthecontentandpresentationofthiswork.Last,butnotleast,Iwouldliketothankmyparentsfortheirsupportandunderstandingthroughoutmygraduatestudies.ThisworkissupportedinpartbytheNationalScienceFoundationundergrantCMMI-0813671. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2SENSITIVITYANALYSIS .............................. 14 2.1TypesofSensitivityAnalysis .......................... 14 2.2DynamicProgrammingandMarkovDecisionProcesses .......... 17 2.3LinearProgramsandMarkovDecisionProcesses ............. 21 3SENSITIVITYANALYSISINMARKOVDECISIONPROCESSESWITHUNCERTAINREWARDPARAMETERS .............................. 24 3.1StationaryRewards .............................. 26 3.1.1SingleParameterSensitivityAnalysis ................. 30 3.1.2ToleranceApproach .......................... 33 3.2Non-StationaryRewards ............................ 34 3.3ToleranceGap ................................. 37 4REGRET ....................................... 43 4.1MultivariateStochasticDominance ...................... 45 4.2RegretinNormativeDecisionAnalysis .................... 48 5EQUIPMENTREPLACEMENTANALYSISWITHANUNCERTAINFINITEHORIZON ....................................... 53 5.1MinimizingExpectedCost ........................... 56 5.2MinimizingMaximumRegret ......................... 59 5.3LeasingOptionstoFurtherMitigateRisk ................... 63 6OPTIMALPRODUCTLINEFORCONSUMERSTHATEXPERIENCEREGRET 78 6.1ModelDescription ............................... 82 6.1.1ConsumerChoiceModel ........................ 83 6.1.2ProblemFormulation .......................... 84 6.2OptimalNumberofProductstoOffer ..................... 86 6.2.1LimitingCases ............................. 88 6.2.2ExperimentalResults .......................... 91 5

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6.2.3UpperBoundonE[jj] ........................ 93 7SUMMARYANDFUTURERESEARCH ...................... 97 7.1SensitivityAnalysis ............................... 97 7.2Regret ...................................... 99 REFERENCES ....................................... 103 BIOGRAPHICALSKETCH ................................ 113 6

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LISTOFTABLES Table page 3-1Singleparametersensitivityanalysisand. .................... 42 3-2Computing. ..................................... 42 5-1Conditionalprobabilitiesofthevariousscenarios ................. 70 5-2Boundaryconditionsatt=8 ............................ 70 5-3Boundaryconditionsatt=6 ............................ 70 5-4Discountedcoststhroughperiods6,7and8periodswithanassetofagei .. 70 5-5x,yandzvaluesforpolicy(i) .......................... 70 5-6x,yandzvaluesforpolicy(ii) .......................... 70 5-7Optimalpolicies:P(T=6)=0.6,P(T=7)=0.2,P(T=8)=0.2andn=4 71 5-8Optimalpolicies:P(T=6)=0.2,P(T=7)=0.2,P(T=8)=0.6andn=4 72 5-9Optimalpolicies:P(T=6)=0.6,P(T=7)=0.2,P(T=8)=0.2andn=8 73 5-10Optimalpolicies:P(T=6)=0.2,P(T=7)=0.2,P(T=8)=0.6andn=8 74 6-1Respectiveuijsvalues ................................ 95 6-2Estimatedexpectedoptimalnumberofproductstooffer,^E[jj ......... 96 7

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LISTOFFIGURES Figure page 5-1Visualizationofdynamicprogrammingformulation ................ 75 5-2Discountrateandoptionprice ........................... 76 5-3Numberofleaseperiodsandoptionprice ..................... 77 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATIONUNDERUNCERTAINTY:SENSITIVITYANALYSISANDREGRETByChinHonTanMay2012Chair:JosephC.HartmanMajor:IndustrialandSystemsEngineering Solvingfortheoptimalsolutionofanyproblemunderuncertaintyisgenerallychallenging.Thisdissertationexploresoptimizationunderuncertaintyfromtheperspectiveofsensitivityanalysisandregret. SequentialdecisionproblemscanoftenbemodeledasMarkovdecisionprocesses.Classicalsolutionapproachesassumethattheparametersofthemodelareknown.However,modelparametersareusuallyestimatedanduncertaininpractice.Asaresult,managersareofteninterestedindetermininghowestimationerrorsaffecttheoptimalsolution.WeillustratehowsensitivityanalysiscanbeperformeddirectlyforaMarkovdecisionprocesswithuncertainrewardparametersusingtheBellmanequations.Inparticular,weconsiderproblemsinvolving(i)asinglestationaryparameter,(ii)multiplestationaryparametersand(iii)multiplenon-stationaryparameters.Weillustratetheapplicabilityofthisworkonacapacitatedstochasticlot-sizingproblem. Insensitivityanalysis,westudythestabilityorrobustnessofanoptimalsolutionwithrespecttouncertaintiesinthemodel.Iftheoptimalsolutionisconstantacrossallpossiblescenarios,theuncertaintiesinthemodelparameterscanbeignored.However,theseuncertaintiesneedtobeaddressediftheoptimalsolutiondiffersunderdifferentpossiblescenarios. Researchinpsychologyandbehavioraldecisiontheorysuggeststhatregretplaysanimportantroleinshapingpreferencesindecisionsunderuncertainty.Despitetheir 9

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relevance,feelingsofregretareoftenignoredintheoptimizationliteratureandtherepresentationofregret,whenpresent,islimited.Regrettheorydescribeschoicepreferencesbasedontherewardsreceivedandopportunitiesmissed.Weshowthatregret-theoreticchoicepreferencesaredescribedbymultivariatestochasticdominance,presentregret-basedriskmeasuresandillustratehowtheycanbeadoptedwithinthemean-riskframework. Researchalsosuggestthatpeoplearewillingtoexchangedirectmaterialgainforregretreduction.Weconsideranequipmentreplacementproblemunderhorizonuncertainty.Wepresentstochasticdynamicprogrammingformulationsandexploresolutionswhichminimizeeitherexpectedcostsormaximumregret.Weidentifythecriticaltimeperiodwhereoptimaldecisionsdivergefordifferenthorizonrealizationsanddesignaleaseoptioncontractsuchthatownerscanlowertheregretthatmayresultfromagivenhorizonrealization,whileopeningapossiblesourceofrevenueforaleasor. Finally,wealsostudytheoptimalnumberofproductstoofferundervariousconditionsinaheterogeneousmarket.Intheabsenceofregret,consumersarehappierwhenpresentedwithmorechoicesandacompanythatwishestocaptureabroadmarketshareneedstoprovidearichproductline.However,therelationshipbetweentheoptimalnumberofproductsandthenumberofmarketsegmentsarereversedwhenconsumersareregretaverse.Ingeneral,theproductlineshouldbenarrowwhenoutcomesareuncertainandconsumersexperienceregret.Inaddition,itisalmostsurelyoptimalforthermtoofferasingleproductwhentheoutcomesofchoicesarehighlyuncertainand/orconsumersarehighlyregretaverse.Wealsoshowthattheoptimalnumberofproductstoofferisnon-increasingwhenthecostofintroducingvarietyintotheproductlineincreasesuniformlyandobtainatightupperboundontheexpectedoptimalnumberofproductstoofferwhenxedcostsareuniform.However,interestingly,theoptimalnumberofproductstooffercanincreasewhenregretaversionincreases. 10

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CHAPTER1INTRODUCTION Formorethanhalfacentury,operationsresearchershaveemployedvariousoptimizationtools,includingmathematicalprogramminganddynamicprogramming,tosolveavarietyofproblems.Advancesinoptimizationtechniquesandcomputingpowerhaveallowedoperationsresearcherstotackleproblemsatascalepreviouslyunimaginable.Despitetheseadvances,solvingfortheoptimalsolutionunderuncertaintyisgenerallystillverychallenging.Thisdissertationexploresoptimizationunderuncertaintyfromtheperspectiveofsensitivityanalysisandregret. Insensitivityanalysis,westudythestabilityorrobustnessofanoptimalsolutionwithrespecttouncertaintiesinthemodel.Iftheoptimalsolutionisconstantacrossallpossiblescenarios,theuncertaintiesinthemodelparameterscanbeignored.However,theseuncertaintiesneedtobeaddressediftheoptimalsolutiondiffersunderdifferentpossiblescenarios. Whentheoptimalsolutiondiffersacrossdifferentscenarios,thedecisionmakermayseeksolutionsthatmaximizeexpectedrewardsreceived(orminimizeexpectedcostsincurred).Asolutionthatmaximizesexpectedrewardsoutperformstheothersolutionsinthelongrun.However,suchsolutionsmaynotbeappropriateforsingle-eventproblems.Inaddition,suchsolutionsdonotaccountfortheriskpreferenceofthedecisionmaker.Krokhmaletal.(2011)providesanextensivereviewondifferentapproachestooptimizationunderriskinmathematicalprogrammingmodels.Inaddition,researchershavehighlightedthattheprobabilitiesofpossibleoutcomesmaynotbeavailable.Underthiscondition,researchersproposendingrobustsolutionsthatareinsensitivetouncertaintiesinthemodel.TheinterestedreaderisreferredtoBeyerandSendhoff(2007)andBertsimasetal.(2011)forasurveyonrobustoptimization.Inthisdissertation,wefocusontheuseofMarkovdecisionprocesses(MDPs)tosolvesequentialdecisionproblemsunderrisk. 11

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Researchinpsychologyandeconomicssuggeststhatsatisfactionisinuencedbyfeelingsofregret(ConnollyandZeelenberg2002).Furthermore,anticipatedregretcanaffectthedecisionmakingprocess(seeZeelenberg(1999)andEngelbrecht-WiggansandKatok(2009)).Forexample,managerswhoareconcernedabouttheopinionsofshareholdersandsuperiorsmaybemoreinterestedindecisionsthatminimizeregretresultingfromlostopportunities,ratherthanoptionsthatmaximizeprots.Despitetheirrelevance,feelingsofregretareoftenignoredintheoptimizationliterature.Inthisdissertation,weidentifythegaps,illustratetheimportanceofregretconsiderationbyshowinghowregretcandrasticallychangetheoptimalsolutionandhighlightpromisingareasoffutureresearch. InChapter2,wediscussdifferentconceptsofsensitivityanalysisandreviewtheliteratureonsensitivityanalysisindynamicprogramsandMDPs.Inaddition,wehighlighttherelationshipbetweenlinearprogramsandMDPsandhighlighttheneedforsensitivityanalysisapproachesstrictlyforMDPs.InChapter3,weillustratehowsensitivityanalysiscanbeconductedforaMDPwithuncertainrewardparameters.Asingleparameterapproachandatoleranceapproachareproposedforproblemsinvolvingsingleandmultipleuncertainrewardparameters,respectively.Inaddition,weextendthetoleranceapproachtoproblemsinvolvingnon-stationaryrewardsandinvestigatethesensitivityofoptimalsolutionsfordifferentproblems. InChapter4,wegiveabriefoverviewofregrettheory.Weobtainnecessaryandsufcientconditionsforregret-theoreticchoicepreferencesandillustratehowtheycanbeusedtoaidthedecisionmakingprocessbyeliminatingnecessarilyinferiorsolutions.Inaddition,weillustratehowregretcanbeviewedasameasureofriskandbeadoptedwithinthemean-riskframework.WeconsideranequipmentreplacementproblemwherethehorizonisuncertaininChapter5.Wehighlightthatthepolicythatminimizesexpectedcostandthepolicythatminimizesmaximumregretcandiffer.VandeVenandZeelenberg(2011)recentlyshowedthatpeoplearewillingtoexchangedirectmaterial 12

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gainforregretreduction.Weillustratehowanoptioncontract,whichallowsthedecisionmakertoleasetheequipmentacrosstheuncertainhorizonatafavorablerate,canreduceregretforthedecisionmakerandgeneraterevenuefortheleasor.InChapter6,weconsidertheeffectsofregretonconsumerbehavior.Wepresentachoicemodelwhereconsumersatisfactionisaffectedbyfeelingsofregretbutchoicepreferencesareindependentofanticipatedregret.Weobtaintheoreticalandexperimentalresultsfortheoptimalnumberofproductstoofferinaheterogeneousmarketunderourproposedchoicemodel.Inparticular,weshowthattherelationshipbetweentheoptimalnumberofproductsandthenumberofmarketsegmentsarereversedwhenregretintensityishighandoutcomesarevariables. WeconcludewithasummaryandlistfutureresearchdirectionsinChapter7. 13

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CHAPTER2SENSITIVITYANALYSIS Thevalidityofanysolutiondependsontheaccuracyofthemodelandthedataforagiveninstance.However,theparametersofthemodelareoftenuncertainandestimatedinpractice.Hence,therobustnessorstabilityofthesolutionwithrespecttochangesinthemodelparametersisofinterest.Oneapproachistosolvetheproblemfordifferentparametersandtrytoinfertherelationshipbetweenthemodelparametersandtheoptimalsolutionfromtheresultingsolutions(see,forexample,Puumalainen(1998)andTilahunandRaes(2002)).However,thisapproachhastwodrawbacks.First,whenthesizeoftheproblemorthenumberofscenariostoanalyzeislarge,thisbruteforceapproachcanbeverytime-consumingandmaynotbepracticalinpractice.Second,unlessthepropertiesregardingtherelationshipbetweenthemodelparametersandtheoptimalsolutionsareknown(forexample,continuous,monotonic,etc.),thesolutionforasetofparametervaluesmaynotprovideinsightsonthesolutionforanothersetofparametervalues.Forexample,supposethatissomemodelparameter.Onecannotguaranteethattheoptimalobjectivevaluefor=2mustbebetweentheoptimalobjectivevaluesfor=1and=3.Fromatheoreticalandpracticalperspective,moresophisticatedapproachesforconductingsensitivityanalysisaredesired. 2.1TypesofSensitivityAnalysis Inthissection,wepresentvariousconceptsofsensitivityanalysisintheoptimizationliteraturethroughamulti-stageproblem.LetSanddenotethesetofstatesandthesetofpolicies,respectively.LetA(s)denotethesetofactionsavailablewiths2S.Weconsiderproblemsofthefollowingform: V=TXt=0tr(st),(2) whereVisthevaluefunctionofthesystemunderpolicy2throughthehorizonT.Tcanbeeitherniteorinnite.st2Sisthestateofthesystemattimetunder 14

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and(s)2A(s)istheactionthatistakenatstatesunder.Notethattheactionthatisselecteddependssolelyonthestateandisindependentoft.Thisstationarypolicyassumptionisnotrestrictivesinceadditionalstatescanalwaysbedeclaredtoaddressthedependencyofont.raistherewardassociatedwithactionaandistheper-perioddiscountfactor.Ifthestatetransitionsarestochastic,stisrandomandthevaluefunctionisexpressedas: V=E"TXt=0tr(st)#.(2) LetdenoteanoptimalpolicyandVdenotetheoptimalvaluefunction.Theobjectiveistondapolicythatmaximizesthevaluefunction, V=V=max2V.(2) WeareinterestedinhowVandchangewhenmodelparametersvary.Werstconsiderthecasewherethevariationsinthemodelparameterscanbeexpressedbyasingleparameter(i.e.,univariate).Forexample,supposethatisallowedtovarybetween0.85and0.95.Wemodelthisbysetting=0.9+,where2[)]TJ /F4 11.955 Tf 9.3 0 Td[(0.05,0.05].Thismodelingapproachallowsfordependenciesbetweenthemodelparametersandisparticularlyusefulwhenmodelingsimplevariationsintransitionprobabilities,wherethesumofprobabilitiesmustaddtoone. Supposethattheproblemissolvedfor=(0)andtheoptimalvaluefunctionandoptimalpolicy,V(0)and(0),areobtained.Sensitivityanalysisaimstoansweroneormoreofthefollowingquestions: 1. WhatisdV(0) dj=(0)? 2. Whatistherangeofvaluesforwhich(0)isoptimal? 3. WhatisVif=(0)+,forsome2R? 4. Whatisif=(0)+,forsome2R? 15

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Questions 1 and 2 arespecicto(0)andcanonlybeansweredafter(0)hasbeenidentied.Hence,thesequestionsareaddressedinaposterioranalysis(see,Fernandez-BacaandVenkatachalam(2007))orpostoptimalanalysis(see,Sotskovet.al.(1995)andWallace(2000)).Theanswerstoquestions 3 and 4 donotrequireadvancedknowledgeofV(0)and(0).HoweverV(0)and(0)canoftenbeusedtoobtainorapproximateVand(seeSection 2.2 )andknowledgeofV(0)and(0)mayprovideinsightstotheanswersofquestions 3 and 4 Besidesposterioranalysis,researchershavealsoconsideredprioranalysiswhereknowledgeofV(0)and(0)isnotrequired.Forexample,inverseparametricoptimizationaimstondthevaluewhereapolicyisoptimalorgood(see,Eppstein(2003),Sunetal.(2004)andKimandKececioglu(2007)).PrioranalysiswillnotbediscussedbuttheinterestedreaderisreferredtoFernandez-BacaandVenkatachalam(2007)foradiscussiononthistopic.Inadditiontouncertaintiesinthemodelparameters,researchershavealsopointedoutthatthemodelisgenerallyanapproximationoftheactualproblemandhencethereisdoubtonthestructureofthemodelitself.LiteraturethataddressmodeluncertaintiesincludeBriggsetal.(1994),Chateld(1995),Draper(1995)andWalkerandFox-Rushby(2001). Theanalysiscanalsobeclassiedasbeingeitherlocalorglobal(McKay,1979).Question 1 takesalocalperspectiveandisconcernedwithchangesinthe"-neighborhoodof(0).Wagner(1995)considersquestion 2 tobelocalaswellasitisconcernedwiththepropertiesof(0).Incontrast,questions 3 and 4 explorevariationsacrossawiderrangeofvaluesandpossiblydifferent. Intheunivariateproblem,thestabilityofV(0)and(0)areaddressedbyquestions 1 through 4 .However,thenotionofstabilityismorecomplicatedinthemultivariateproblem(i.e.,multipleparameters).Thetypicalapproachtosensitivityanalysisforamultivariateproblemistovaryoneparameteratatimewhileholdingtheotherparametersconstant.Thisissometimesreferredtoasordinarysensitivityanalysis 16

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(seeWendall(2004)andFilippi(2005)).However,ordinarysensitivityanalysisdoesnotallowforsimultaneousvariationsofdifferentparameters.Bradleyetal.(1977)providedrestrictionsonthevariationsintheright-hand-side(RHS)termsandobjective-functioncoefcientstoguaranteetheoptimalityofthecurrentoptimalbasis(i.e.,optimaldecisionvariables)inalinearprogram.Theytermtheserestrictionsasthe100percentrule.Wendell(1985)wasinterestedintherangethateachparameterwasallowedtovarywithoutviolatingoptimality.Hisapproach,whichhetermedthetoleranceapproach,isdifferentfromordinarysensitivityanalysisinthatsimultaneousvariationsofdifferentparametersareconsidered.Thetoleranceapproachhasbeenappliedtodifferentproblemsbyvariousresearchers(see,forexample,Wendell(1985),Hansenetal.(1989),RaviandWendell(1989),Wondolowski(1991)andFilippi(2005)). 2.2DynamicProgrammingandMarkovDecisionProcesses Dynamicprogramming(DP)hasbeensuccessfullyappliedtoawiderangeofproblemsinavarietyofelds(see,forexample,HeldandKarp(1962),PruzanandJackson(1967),EltonandGruber(1971)andYakowitz(1982))sinceitwasproposedbyRichardBellmaninthemiddleofthelastcentury(Bellman1952).OneofthemainadvantagesofDPliesinitsabilitytoaccountforthedynamicsofasystemanditisoftenusedbyoperationsresearcherstosolveproblemsinvolvingsequentialdecisions.Inaddition,unlikemostoptimizationtechniques,DPisablehandleawiderangeofcostand/orrewardfunctions.Givenitsversatility,itisnotsurprisingthatDPisusedacrossvariousdisciplines,includingarticialintelligence,control,economicsandoperationsresearch.However,DPdoeshaveitsdrawbacksaswell.Thesizeofadynamicprogramgrowsrapidlywiththedimensionoftheproblem.ThisiscommonlyreferredtoasthecurseofdimensionalityandhasbeenthemainchallengeinapplyingDPapproachestosolvingrealworldproblems.Inlightofthecomputationaldifcultiesthatarisefromthecurseofdimensionality,researchershaveproposedavarietyofapproximationapproaches,includingstateaggregationandvaluefunctionapproximation,tosolve 17

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largedynamicprograms.TheinterestedreaderisreferredtoPowell(2007)foradetailedpresentationofapproximateDP. AdynamicprogramcomprisesasetofstatesS,withasetofdecisionsA(s),availableateachs2S.Eacha2A(s)transitsthesystemfromastatestoanotherstates0andresultsinarewardofra.Inthedeterministicproblem(Equation( 2 )),thestatetransitionsareknownwithcertainty.Inthestochasticproblem,thestatetransitionsareuncertain(Equation( 2 ))andtheprobabilityoftransitingfromstatestostates0undera2A(s)isdenotedbyPa(s0).ThestatetransitionsareMarkovian(i.e.,onlydependentonthestates0andactiona)andstochasticdynamicprogramsareoftenreferredtoasMarkovdecisionprocesses,orMDPs.Wefocusonthestochasticproblem.However,wenotethatthediscussionandresultsalsoapplytodeterministicDPsincethedeterministicproblemismerelyaspecialcaseofthestochasticproblem. MosttextbooksdivideMDPsintotwobroadcategories:i)nitehorizonandii)innitehorizon(see,White(1993),Puterman(1994)andPowell(2007)).Finitehorizonproblemsareusuallymodeledasacyclicgraphswhereeachnodecorrespondstoaparticularsataparticulart.ThecommonsolutionapproachfornitehorizonproblemsistodenetheboundaryconditionsVT(s)foreachstateattimeTandrecursivelysolveforVinabackwardmanner: Vt)]TJ /F12 7.97 Tf 6.59 0 Td[(1(s)=maxa2A(s)(ra+Xs02SPa(s0)Vt(s0))t=0,1,...,T)]TJ /F4 11.955 Tf 11.96 0 Td[(1,8s2S,(2) whereVt(s)isthevaluefunctionofstatesattimet.Boundaryconditionsdonotexistforinnitehorizonproblems.Innitehorizonproblemsareexpressedascyclicgraphsandvalueand/orpolicyiterationapproachescanbeusedtoobtaintheoptimalsolution.TheseapproachesarebasedontheoptimalityconditionsthatarecommonreferredtoastheBellmanequations(see,Puterman(1994)): V(s)=maxa2A(s)(ra+Xs02SPa(s0)V(s0))8s2S.(2) 18

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ItisclearfromEquations( 2 )to( 2 )thatViscontinuousin.AltmanandShwartz(1991)showedthatVremainscontinuousininaconstrainedMDP.Inaddition,italsofollowsfromEquations( 2 )to( 2 )thatViscontinuousandmonotonewithrespecttoeachindividualreward.However,therewardsmaynotbeindependentfromeachother.Forexample,thecostofproducing10itemsmaybetwicethecostofproducing5items.Inthesecases,weareinterestedinthechangesintheoptimalsolutionwithrespecttotheunitcostofproduction,ratherthanthecostofproducing10itemsspecically.Intheseproblems,therewardsareexpressedas:ra=ca(), whereisavectorthatrepresentstheparametersthatareofinterestandca:P!R,wherePisthesetcontaining.WhiteandEl-Deib(1986)showedthatifraandVT(s)areafneinforalla2Aands2S,theoptimalvaluefunctionofeachstate,includingV,ispiecewiseafneandconcavein. Modelingtheuncertaintiesinthetransitionprobabilitiesiscomplicatedbytheadditionalrequirementthattransitionprobabilitiesmustaddtoone.Researchershavemodeledtheuncertaintiesinthetransitionprobabilitiesinavarietyofways(see,forexample,SatiaandLave(1973),WhiteandEldeib(1994),Givanetal.(2000),Kalyanasundarametal.(2002)andIyengar(2005)).Muller(1997)showedthattheoptimalvaluefunctionismonotoneandcontinuouswhenappropriatestochasticordersandprobabilitymetricsareused. Clearly,VisdependentonT.Inparticular,ifalltherewardsarenon-negative(non-positive),Vwillbemonotonicallynon-decreasing(non-increasing)inT.However,theformerisnotanecessaryconditionanditispossibletoprovethemonotonicityofVforcertainproblems,eveniftherewardsdifferinsign.Forexample,TanandHartman(2010)showedthatthecostofowningandoperatingequipmentisnon-decreasinginT 19

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ifthediscountfactorandmaintenancecostarenon-negativeandthesalvagevalueisnon-increasingwithage. AlthoughthevalueofVissensitivetochangesinthemodelparameters,theirimpactonisnotasapparent.Infact,theoptimalpolicyisoftenrobusttominordeviationsinthemodelparameters.WhiteandEl-Deib(1986)consideredaMDPwithimpreciserewardsandwereinterestedinndingthesetofpoliciesthatareoptimalforsomerealizationoftheimpreciseparameters.Theyprovidedasuccessiveapproximationprocedureforanacyclicnitehorizonproblemandapolicyiterationprocedurefortheinnitehorizonproblem.Harmanec(2002)computedthesetofnon-dominatedpoliciesforanitehorizonproblemwithimprecisetransitionprobabilitiesbygeneralizingtheclassicalbackwardrecursionapproach(Equation( 2 )). Researchershavehighlightedthatthecurrentdecisionistheonlydecisionthatthedecision-makerhastocommittoinmanycasesandhenceheorshemightbesolelyinterestedintheoptimalityofthecurrentdecision(i.e.,decisionattimezero).Hopp(1988)wasinterestedinthetoleranceonthevaluefunctionsofthelaterstages.Inparticular,hederivedlowerboundsonthemaximumallowableperturbationsinthestatevalueofthefuturestagessuchthatthetimezerooptimaldecisionremainunchanged.Hisnumericalresultsindicatethatthetoleranceofthevaluefunctions(i.e.,allowableperturbation)increasesgeometricallywithtime.Inaddition,researchershavealsoproposedndingaforecasthorizonsuchthattheoptimaltimezerodecisionisnotaffectedbydatabeyondthathorizon.ResearcherswhohavestudiedthisproblemincludeBeanandSmith(1984),Hopp(1989),Beanetal.(1992)andCheevaprawatdomrongetal.(2003). Inmostcases,itisnotnecessarytore-solvetheproblemwhenthemodelparameterschange.Veryoften,thenewandVcanbeefcientlycomputedorapproximatedfromthecurrentsolution.Forexample,theoptimalsolutionstoshortestpathproblemscanbeeasilyrevisedwhenthearclengthschange(Ahujaetal.,1993). 20

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TopalogluandPowell(2007)providedanefcientapproachtoapproximatethechangeintheoptimalvaluefunctionforadynamiceetmanagementmodel.Theirapproachwasbasedonthespecialstructurethatresultedfromtheirvaluefunctionapproximation.ForMDPproblemswithnospecialstructure,theBellmanequation(Equation( 2 ))canbeusedtoverifytheoptimalityofthecurrentsolution.Iftheoptimalityconditionsareviolated,thepolicyiterationapproachcanbeusedtodeterminethenewsolution. 2.3LinearProgramsandMarkovDecisionProcesses Thereisarichliteratureonsensitivityanalysisforlinearprograms(seeGalandGreenberg(1997)andBazaraaetal.(2005)).Inaddition,ithasbeenlongrecognizedthataMDPcanbeformulatedandsolvedasalinearprogram.TheLPmethodforMDPswasrstproposedbyManne(1960)andelaborateduponbyDerman(1962)andPuterman(1994).ItcanbeshownthataMDPcanalwaysbeformulatedbythefollowinglinearprogram(Powell,2007):minXs2Svss.t.vs)]TJ /F5 11.955 Tf 11.96 0 Td[(Xs02SPa(s0)vs0ra8s,avs2Rs=1,2,...,jSj. TheoptimalvaluefunctionVcorrespondstovs0,wheres0isthecurrentstate(i.e.stateattime0)andisdenedbytheconstraintsthatarebindingatoptimality.Manne'sobservationhasanimportanttheoreticalimplication.Ithighlightsthatanoptimalsolutionofadynamicprogramwillhavethesamepropertiesasanoptimalsolutiontoitsassociatedlinearprogram.Thisisparticularlyinsightfulassensitivityanalysisiswell-establishedinLP. Foreverylinearprogram,whichwewillrefertoastheprimalproblem,thereisanotherassociatedlinearprogram,whichiscommonlyreferredtoasthedualproblem.Therelationshipbetweenthetwoproblemsprovidesinsightonthestabilityofthesolutionintheprimalproblem.Foreachconstraintintheprimalproblem,thereisan 21

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associatedvariableinthedualproblem.Thelatteriscommonlyreferredtoasthedualvariable.Thecomplementaryslacknesstheorem,whichwasrstproposedbyGoldmanandTucker(1956)andextendedbyWilliams(1970),highlightstherelationshipbetweenthevariablesandconstraintsoftheprimalanddualproblemsatoptimality.Inparticular,ifavariableinoneproblemispositive,thecorrespondingconstraintintheotherproblemmustbebinding.Iftheconstraintisnotbinding,thecorrespondingvariableintheotherproblemmustbezero.Thevaluesoftheoptimaldualvariables,alsoknownasshadowprices,correspondtothemarginalchangeintheobjectivevaluewhentheRHSoftheassociatedconstraintisperturbed.ThisresultcanbeapplieddirectlytoDPproblemswheretherewardsofindividualactionsareindependent.Ifasmallperturbationtoarewardchangessomestatevaluefunctionvs,itfollowsfromthecomplementaryslacknesstheoremthattheassociatedactionmustbeoptimal.Iftheactionisnotoptimal,itsassociatedrewardcanbeperturbedbyasmallamountwithoutchangingthevalueofanyvs. Thecomplementaryslacknesstheoremandeconomicinterpretationoftheshadowpricesthatarepresentedareonlyvalidwhentherewardsareindependentofeachother.Whentherewardsaredependent,theindividualrewardsareexpressedasfunctionsofindependentparameters(Equation( 2 ))andparametricanalysisapproachescanbeusedtoanalyzethesensitivityofthesolutionwithrespecttotheseindependentparameters(seeWardandWendell(1990),GalandGreenberg(1997)andBazaraaetal.(2005)). AlthoughLP-basedsensitivityanalysisapproachescanbeappliedtoMDPs,thereareacoupleofreasonsforaseparatestudyofsensitivityanalysisinMDPs.First,thenumberofconstraintsintheLPformulationofaMDPislarge(Powell,2007)andhenceMDPsarerarelyformulatedandsolvedaslinearprogramsinpractice.Second,thetypeofsensitivityanalysisthatisrequiredforthetwoproblemclassescandiffer.For 22

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example,theanalystmaybeinterestedintheoptimalityofthecurrentdecision(i.e.,decisionattimezero)ratherthantheoptimalityofabasisinthelinearprogram. Inthenextchapter,weillustratehowthemarginalchangeintheobjectiveandthetolerancesoftheindividualuncertainparameterscanbeobtaineddirectlyfromthewell-knownBellmanequations.Ourproposedapproachisgeneralandcanbeappliedtoproblemsinvolvingmultiplenon-stationaryrewardparameters. 23

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CHAPTER3SENSITIVITYANALYSISINMARKOVDECISIONPROCESSESWITHUNCERTAINREWARDPARAMETERS TheMarkovdecisionprocess(MDP)frameworkhasbeenusedbyresearcherstomodelavarietyofsequentialdecisionproblemsbecauseitcanaccountforthedynamicsofacomplexsystemandhandleawiderangeofrewardfunctions.AMDPisdenedbyasetofstates,withasetofpotentialactionsassociatedwitheachstate.Classicalsolutionapproachesassumethattheparametersofthemodel,includingrewards,transitionprobabilitiesandthediscountfactor,areknown(seePowell(2007),Puterman(1994)andWhite(1993)).However,theseareoftenestimatedanduncertaininpractice.Forexample,itisdifculttoquantifythecostofnothavinganiteminthestoreuponthearrivalofacustomer(stock-outcost). WhiteandEl-Deib(1986)identiedoptimalpoliciesforsomerealizationoftheimpreciseparameters,termednon-dominatedpolicies,foraMDPwithimpreciserewards.Harmanec(2002)studiedasimilarproblemwheretheimprecisionwasdenedinthetransitionprobabilities,ratherthantherewards.However,adecisionmakercanonlyimplementasinglepolicyinpractice.Oneapproachistoassumethattheimprecisionisresolvedinthemostpessimisticscenario(seeIyengar(2005)andNilimandElGhaoui(2005)).Thisisoftenreferredtoasthemax-minpolicy.However,ithasbeenhighlightedthatmax-minpoliciescanbeoverlyconservativeandmaynotbepracticalinreality(Wallace,2000). Managersareofteninterestedinhowanoptimalsolutionchangeswithdeviationsinthemodelparameters.Thetypicalapproachtoansweringthisquestionistosolvetheproblemfordifferentvaluesoftheuncertainparameter,butthiscanbeverytime-consumingwhentheproblemislarge.Forexample,TopalogluandPowell(2007)wereinterestedinthebenetsofaddinganextravehicleorloadinadynamiceetmanagementmodel.SandvikandThorlund-Petersen(2010)wereinterestedintheconditionswherethereisatmostonecriticalrisktolerancevalue,suchthatthe 24

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knowledgeofwhethertheindividual'srisktoleranceisaboveorbelowthatvalueissufcientforidentifyingthepreferreddecision. Inthischapter,weconsideraMDPwhererewardsareexpressedasafnefunctionsofuncertainparameters.ProblemsofthisformaboundintheMDPliteratureincluding,thelot-sizingproblem(MuckstadtandSapra,2010),theequipmentreplacementproblem(TanandHartman,2010),thesequentialsearchproblem(Limetal.,2006)andvariousresourceallocationproblems(see,forexample,Erkinetal.(2010),CharalambousandGittins(2008)andGlazebrooketal.(2004)).BoundsontheperturbationsinthestatevaluesforagivenpolicyarecomputedinMitrophanovetal.(2005).Weareinterestedinthemaximumrangeparametersareallowedtovarysuchthatapolicyremainsoptimal.Hopp(1988)derivedboundsontheminimumperturbationsinthefuturestatevaluesrequiredtochangethecurrentoptimaldecision(i.e.,attime0)andextendedtheresultstoperturbationsintherewardsateachstate.Ourmodelallowsfordependenciesbetweentheuncertaintiesintherewardsassociatedwithdifferentactionsandstates.Anotherimportantdifferenceisthatwearenotderivingbounds,butcomputingtheactualrangeofvaluesourparametersareallowedtovary. Asingleparameteranalysisprovidesinsightonthestabilityofthesolutionwithrespecttoaparticularparameter.However,estimationerrorscanexistformultipleparameters.Wendell(1985)proposedndingatolerancelevelwhichindicatesthemaximumpercentageparametersareallowedtovaryfromtheirbasevaluesuchthattheoptimalbasisofalinearprogramremainsoptimal.WeillustratehowthemaximumtolerancecanbeobtainedforourMDPwhenmultipleuncertainparametersareallowedtovarysimultaneously.Inaddition,weallowtheseparameterstobenon-stationary. First,weobtaintherangeinwhichasingleparameterandmultipleparametersareallowedtovarywhilemaintainingtheoptimalityofthecurrentsolution(Propositions 3.1 and 3.2 ).Second,weillustratehowthemaximumallowabletolerancecanbecomputedwhenuncertainparametersarenon-stationary(Proposition 3.3 )andshowthatitcannot 25

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begreaterthantheallowabletoleranceofthestationaryproblem(Theorem 3.3 ).Third,wederivetheconditionswherethetolerancesofthestationaryandnon-stationaryrewardsproblemarethesame(Corollary 2 )andtheconditionswheretheydiffer(Theorem 3.4 ).Inparticular,weshowthat,undermildassumptions,thetolerancesoflot-sizingproblemswithuncertainorderingcostsandbacklogpenaltiesdifferwhenthemaximumallowabletoleranceisassociatedwithanactionthatchangesthereorderpoint(Theorem 3.5 ). Inthenextsection,wedescribeourstationaryrewardsmodelandillustratehowsingleparametersensitivityanalysiscanbeperformedforthisproblem.Inaddition,wedemonstratehowthemaximumallowabletolerancecanbecomputedwhentheuncertainparametersareallowedtovarysimultaneously.Next,weillustratehowthemaximumallowabletolerancecanbecomputedforthenon-stationaryrewardsproblem.InSection 3.3 ,westudyanddiscussthedifferenceinthemaximumallowabletoleranceofthetwoproblems. 3.1StationaryRewards Consideranitestate,niteaction,innitehorizonMDP.LetSandA(s)denotethesetofstatesandthesetofactionsavailablewiths2S,respectively.Eachas2A(s)transitionsthesystemfromstatestostates0withprobabilityPas(s0).Let~rasdenotetherewardassociatedwithas,expressedasanafnefunctionofNuncertainparameters:~ras=as0+as~x, whereas0issomeknownconstant,~x=(~x1,~x2,...,~xN)0representstheuncertainparametersandas=(as1,as2,...,asN)therespectiveknowncoefcients.Weassumethatanestimationof~xiisavailablefori=1,2,...,N.Letx=(x1,x2,...,xN)0denotethevectorofestimatedparametervalues.Inaddition,letrasdenotetheestimatedrewardassociatedwithas:ras=as0+asx. 26

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Let=(1,2,...,N)0denotethecorrespondingestimationerror,denedasfollows:i=~xi)]TJ /F7 11.955 Tf 11.96 0 Td[(xi xi, andas=(as1,as2,...,asN)thecorrespondingcoefcient:asi=asixi, suchthat~rascanbere-expressedasfollows:~ras=ras+as. Asinsensitivityanalysis,weareinterestedinthestabilityofthesolutionobtainedusingtheestimatedparametersx.Inthissection,weobtaintherelationshipbetweentheestimationerrorsandthetotalrewardreceived.Inaddition,wecomputetherangeoferrorvalueswherethecurrentsolutionremainsoptimal.Here,weconsideraproblemwherethevalueofisuncertainbutstationary.Sincetherewardsarestationary(i.e.,donotvarywithtime),theremustexistastationaryoptimalpolicywheretheactionisdeterminedsolelybythestateoftheprocess(Puterman1994).Letanddenotetheperiodicdiscountfactorandthesetofallpossiblestationarypolicies,respectively.Weassumethatrasisboundedand<1toensurethatthevaluefunctionisnite.Let(s)2A(s)denotetheactionthatistakenatstatesunder2andletVs()denotethevaluefunctionofstatesunderpolicyforagiven.Thevaluefunctionofastatecanbeexpressedbythefollowingrecursiveequation: Vs()=r(s)+(s)+Xs02SP(s)(s0)Vs0()8s,.(3) NotethatVs()dependsonthevaluefunctionsoftheotherstates.Hence,itisconvenienttoexpressEquation( 3 )inmatrixform. LetV()andr=)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(r(1),r(2),,r(jSj)0denoteavectorofstatevaluesandrewards,respectively.Inaddition,letandPdenoteamatrixofuncertainreward 27

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parametersandtransitionprobabilities,respectively:V()=)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(V1(),V2(),,VjSj()0,r=)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(r(1),r(2),,r(jSj)0,=0BBBBBBB@(1)1(1)2(1)N(2)1(2)2(2)N............(jSj)1(jSj)2(jSj)N1CCCCCCCA andP=0BBBBBBB@P(1)(1)P(1)(2)P(1)(jSj)P(2)(1)P(2)(2)P(2)(jSj)............P(jSj)(1)P(jSj)(2)P(jSj)(jSj)1CCCCCCCA.V()canbeexpressedas: V()=r++PV()(I)]TJ /F5 11.955 Tf 11.95 0 Td[(P)V()=r+V()=(I)]TJ /F5 11.955 Tf 11.95 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1(r+)V()=(I)]TJ /F5 11.955 Tf 11.95 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1r+(I)]TJ /F5 11.955 Tf 11.95 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1V()=V(0)+(I)]TJ /F5 11.955 Tf 11.96 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1. (3) LetV()=maxV().Foragiven(including=0),thepolicythatmaximizesV()canbeobtainedthroughvalueand/orpolicyiterationapproaches(Puterman1994).LetedenoteapolicythatmaximizesV(0).ItfollowsfromEquation( 3 )that,withintheregionwhereeistheoptimalpolicy,themarginalchangeinV()is)]TJ /F19 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F5 11.955 Tf 11.95 0 Td[(Pe)]TJ /F12 7.97 Tf 6.59 0 Td[(1e. Inlinearprograms,sensitivityanalysisisperformedbyderivingasetofnecessaryandsufcientconditionsforoptimalitybasedonthereducedcostofeachvariableand 28

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ndingtherangeofvaluesforwhichtheseconditionsholdBazaraaetal.(2005).Intheory,MDPscanbeformulatedaslinearprograms(Manne,1960)andtheallowablevaluescanbeobtainedbyapplyingresultsfromparametriclinearprogramming(seeGalandGreenberg(1997)orWardandWendell(1990))onthedualoftheassociatedlinearprogram(TanandHartman,2011).However,thesetofnecessaryandsufcientconditions(i.e.,Bellmanequations)isreadilyavailableforMDPs(Bellman1957).WestatetheBellmanequationsforthisproblem,re-expresstheminacompactformanduseittoobtainthemaximumallowableerrorforthesingleparameterandmultipleparameterprobleminSections 3.1.1 and 3.1.2 ,respectively. LetPas=(Pas(1),Pas(2),...,Pas(jSj)).NotethatP(s)isthesthrowofP.TheBellmanequationsforthestationaryrewardsproblemare:Vs()=maxas2A(s)fras+as+PasV()g8s2S, andeisoptimalifandonlyif: re(s)+e(s)+Pe(s)Ve()ras+as+PasVe()8s2S,as2A(s).(3) Dene:ce,as=re(s))]TJ /F7 11.955 Tf 11.95 0 Td[(ras+Pe(s))]TJ /F19 11.955 Tf 11.96 0 Td[(Pas)]TJ /F19 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F5 11.955 Tf 11.96 0 Td[(Pe)]TJ /F12 7.97 Tf 6.58 0 Td[(1re, and:be,as=e(s))]TJ /F19 11.955 Tf 11.95 0 Td[(as+Pe(s))]TJ /F19 11.955 Tf 11.95 0 Td[(Pas)]TJ /F19 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F5 11.955 Tf 11.95 0 Td[(Pe)]TJ /F12 7.97 Tf 6.59 0 Td[(1e. Notethatce,asisthemarginaldecreaseintheestimatedrewardthatresultsfromasingleperturbationoftheactionats,whilebe,asisthemarginalchangeintheestimationerrorthatresultsfromthatactionperturbation.Usingourdenitionsofce,asandbe,as,thenecessaryandsufcientoptimalconditionsexpressedinEquation( 3 )canberewrittenas: ce,as+be,as08s2S,as2A(s).(3) 29

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LetHdenotetheregionwhereeisoptimal:H=f:Ve()V(),82g. Theorem3.1. GivenEquation( 3 ),Hisclosedandconvex. Proof. ItfollowsfromEquation( 3 )thatHistheintersectionofclosedhalf-spaces.Hence,Hisclosedandconvex. 3.1.1SingleParameterSensitivityAnalysis Insingleparametersensitivityanalysis,weareinterestedinthesetofivalueswhereeremainsoptimalwhenj6=i=0.ItfollowsfromTheorem 3.1 thatthereexistconstants(l)i,(u)i2Rsuchthateremainsoptimalwhenj6=i=0andi2[(l)i,(u)i]. Proposition3.1. GivenEquation( 3 ),(l)i=8><>:1be,asi08s2S,as2A(s)maxbe,asi>0,8s,as)]TJ /F8 7.97 Tf 10.49 4.71 Td[(ce,as be,asiotherwise, and:(u)i=8><>:be,asi08s2S,as2A(s)minbe,asi<0,8s,as)]TJ /F8 7.97 Tf 10.49 4.71 Td[(ce,as be,asiotherwise. Proof. Letbe,asidenotetheithentryofbe,as.Settingj6=i=0andusingourdenitionsofce,asandbe,asi,weobtainthefollowingnecessaryandsufcientoptimalityconditionsfromEquation( 3 ): i)]TJ /F7 11.955 Tf 23.11 8.09 Td[(ce,as be,asiwhenbe,asi>0,(3) and: i)]TJ /F7 11.955 Tf 23.11 8.09 Td[(ce,as be,asiwhenbe,asi<0.(3) Giventhattheseholdforallvaluesofiatoptimality,theextremevaluesareofinterestandthepropositionfollowsfromEquations( 3 )and( 3 ). 30

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Next,weillustratehowsingleparametersensitivityanalysiscanbeconductedforacapacitatedstochasticlot-sizingproblemwithuncertainorderingcostandbacklogpenalty.TheinterestedreaderisreferredtoMuckstadtandSapra(2010)foranintroductiontolot-sizingproblems. Example1:Capacitatedstochasticlot-sizingproblem .Consideralot-sizingproblemwheretheprobabilitydistributionofdemandineachperiodisstationaryandgivenbyP(D=0)=P(D=1)=P(D=2)=1=3.Eachitemsellsfor$150.Theinventorycapacityofthesystemis3andbackloggingisallowed.Thereareatotalof5states,S=f)]TJ /F4 11.955 Tf 15.28 0 Td[(1,0,1,2,3g.Theindexofeachstaterepresentstheamountofinventorythatisavailableatthebeginningoftheperiod.Ordersareplacedatthestartoftheperiodandanordermustbeplacedifthereisnoinventory.Hence,thesetoffeasibleactions(i.e.,orderquantity)foreachstateisA()]TJ /F4 11.955 Tf 9.3 0 Td[(1)=f2,3,4g,A(0)=f1,2,3g,A(1)=f0,1,2g,A(2)=f0,1gandA(3)=f0g.Weassumethatthestockarrivesattheendoftheperiodinwhichitisorderedandthedemandisalsorealizedattheendoftheperiod.Theproductioncostofeachitemis$20andtheholdingcostofeachitemis$5perperiod.Thevalueoftheorderingcostandbacklogpenaltyareunclear,butbelievedtobe$40and$100,respectively.WeanalyzeeachuncertainparameterindependentlysuchthatN=1foreachcase.Theproblemextendsacrossaninnitehorizonwithaper-perioddiscountfactorof0.9. Let1and2denotetheestimationerrorintheordercostandbacklogpenalty,respectively.Itfollowsthattheexpectedrewardassociatedwith(s)is:r(s)=8><>:(0.9)(150)(1=3+2=3))]TJ /F4 11.955 Tf 11.96 0 Td[(20(s))]TJ /F4 11.955 Tf 11.96 0 Td[(40)]TJ /F4 11.955 Tf 11.95 0 Td[(100ifs=)]TJ /F4 11.955 Tf 9.3 0 Td[(1(0.9)(150)(1=3+2=3))]TJ /F4 11.955 Tf 11.95 0 Td[(5s)]TJ /F4 11.955 Tf 11.95 0 Td[(20(s))]TJ /F4 11.955 Tf 11.95 0 Td[(40I(s)otherwise. Inaddition,wedenethefollowingindicatorvector:I(s)=8><>:1if(s)10otherwise 31

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and=0BBBBBBBBBB@)]TJ /F7 11.955 Tf 9.3 0 Td[(I()]TJ /F12 7.97 Tf 6.59 0 Td[(1))]TJ /F4 11.955 Tf 9.3 0 Td[(100)]TJ /F7 11.955 Tf 9.3 0 Td[(I(0)0)]TJ /F7 11.955 Tf 9.3 0 Td[(I(1)0)]TJ /F7 11.955 Tf 9.3 0 Td[(I(2)0)]TJ /F7 11.955 Tf 9.3 0 Td[(I(3)01CCCCCCCCCCA. Thetransitionprobabilitiesaredenedby:Pas(s0)=8><>:1=3if)]TJ /F4 11.955 Tf 11.95 0 Td[(2s0)]TJ /F4 11.955 Tf 11.96 -.16 Td[((s+as)00otherwise. SolvingtheMDPdescribedabovewiththepolicyiterationapproach,weobtaine=(4,3,2,0,0)andVe(0)=($723.5,$843.5,$858.5,$908,$928.5)0.ItfollowsfromEquation( 3 )thatthemarginalchangeinVe()is()]TJ /F4 11.955 Tf 9.29 0 Td[(196,)]TJ /F4 11.955 Tf 9.3 0 Td[(196,)]TJ /F4 11.955 Tf 9.3 0 Td[(196,)]TJ /F4 11.955 Tf 9.3 0 Td[(168,)]TJ /F4 11.955 Tf 9.3 0 Td[(156)01+()]TJ /F4 11.955 Tf 9.3 0 Td[(100,0,0,0,0)02. Therearetwouncertainparametersinthisproblem.Weanalyzetheoptimalregionofewithrespectto1bysetting2=0.Thecorrespondingce,asandbe,asivaluesofeachactionarelistedinTable 3-1 .Atstate-1,theoptimaldecisionistoorder4unitswhen=0.However,iftheorderingcostdecreasesbymorethan46%(with2remaining0),ordering3unitswillresultinhigherexpectedprotsthanordering4units.Inaddition,thedecisiontoorder4unitsatstate-1isbetterthanthatofordering2unitssolongastheorderingcostdoesnotdeclineby200%ofitsestimatedvalue(i.e.,$40(1)]TJ /F4 11.955 Tf 12.64 0 Td[(2.00)=)]TJ /F4 11.955 Tf 9.3 0 Td[($40).Assumingthattheorderingcostmustbenonnegative,itfollowsthata)]TJ /F12 7.97 Tf 6.59 0 Td[(1=2issuboptimalwhen2=0.ItfollowsfromTable 3-1 that(l)=maxf)]TJ /F4 11.955 Tf 15.27 0 Td[(2.00,)]TJ /F4 11.955 Tf 9.3 0 Td[(0.46,)]TJ /F4 11.955 Tf 9.3 0 Td[(1.23,g=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.46and(u)=minf0.04,1g=0.04.Hence,eremainsoptimalforall12[)]TJ /F4 11.955 Tf 9.3 0 Td[(0.46,0.04]giventhat2=0. Inasimilarfashion,weobtainthateremainsoptimalforall22[)]TJ /F4 11.955 Tf 9.3 0 Td[(0.03,1],giventhat1=0,fromtheresultsinTable 3-1 .2 32

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3.1.2ToleranceApproach Whentherearemultipleuncertainparameters,Wendell(1985)proposedndingatolerancelevel,whereisthemaximumratiouncertainparametersareallowedtovaryfromtheirbasevaluesuchthattheoptimalbasisofalinearprogramremainsoptimal.Notethatis,bydenition,non-negative.Geometricallyspeaking,thisentailsndingthelargesthypercubethatiscontainedinthecriticalregion(i.e.,H).Wendellshowedthatthemaximumallowabletolerance,whichwedenoteby,canbeobtainedbyndingthemaximumtolerancewithrespecttoeachconstraintindependently.Followingasimilarapproach,weillustratehowthetolerancelevelcanbecomputedforaMDPwithuncertainrewards.Lets,asdenotethemaximumtoleranceallowablebyEquation( 3 )forstatesandactionas:s,as=maxfy:ce,as+be,as0andjijyfori=1,2,...,Ng. Proposition3.2. GivenEquation( 3 ),=mins,asce,as PNi=1jbe,asij. Proof. TondthemaximumallowabletoleranceforeachconstraintexpressedbyEquation( 3 ),weconsidertheworstcasescenariowhere: i=8><>:s,asifbe,asi0)]TJ /F5 11.955 Tf 9.3 0 Td[(s,asotherwise.(3) Sincece,asis,bydenition,non-negative,weobtainfromEquation( 3 )thefollowingexpressionfors,as:s,as=ce,as PNi=1jbe,asij. Sincecannotbelargerthananyoftheindividualtolerancess,as:=mins,ass,as=mins,asce,as PNi=1jbe,asij. 33

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WereconsiderExample 1 ,allowingforsimultaneousperturbationsintheorderingcostandbacklogpenalty,asfollows. Example2:Toleranceapproach .ConsiderExample 1 again.Whentheorderingcostandbacklogpenaltyareallowedtoperturbsimultaneously,itfollowsfromTable 3-1 andProposition 3.2 that=0.02andisassociatedwiththeactiona1=0.Thisimpliesthatewillremainoptimalsolongastheorderingcostandbacklogpenaltydonotdeviatefromtheircurrentestimatesbymorethan2%.Inparticular,itissuboptimaltoorderwhens=1ifweunderestimatetheorderingcostandoverestimatethepenaltycostbymorethan2%each.2 3.2Non-StationaryRewards Inthissection,weconsiderthenon-stationaryrewardsproblemwhereuncertainparametersareallowedtovaryateachperiod.Letdenotethetoleranceforthenon-stationaryrewardsproblem.Inaddition,let!representtheestimationerrorinthenon-stationaryrewardproblem,where!s,i,tdenotesthevalueofiatstatesatperiodt.Wesaythat!isstationaryif!s,i,t1=!s,i,t2forallt1,t2,sandi.If!isstationary,idependsonlyonthestatethattheprocessisinandwedenotethevalueofiatstatesby!s,i.LetNSandSTdenotethesetscontainingallnon-stationary!andstationary!foragiven,respectively.LetPs,t(i)denotetheprobabilityofbeinginstateiattimetunderpolicygiveninitialstates.Thevaluefunctionofstatesgivenand!is: Vs(!)=1Xt=0Xi2StPs,t(i)"r(i)+NXj=1(i)j!i,j,t#.(3) Wesaythateis-optimalif: Ves(!)Vs(!)8s2S,2,!2NS.(3) Theconditionmustholdforallpossiblesetsofscenariosacrosstheinnitehorizon. 34

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Since,thereareinnitelymanyelementsinNS,itisimpossibletoevaluatethe-optimalityofapolicywithConditions( 3 ).Theorem 3.2 highlightsthatwecanlimitouranalysistostationary!. Theorem3.2. GivenEquation( 3 ),eis-optimalifandonlyif: Ves(!)Vs(!)8s2S,2,!2ST.(3) Proof. First,weprovethateis-optimalifConditions( 3 )hold.Weprovethisbycontradiction.AssumethatCondition( 3 )holdsandeisnot-optimal.Thisimpliesthattheremustexistsome!02NSnST,s2Sand02suchthatVes(!0))]TJ /F7 11.955 Tf 11.95 0 Td[(V0s(!0)<0.ItfollowsfromEquation( 3 )thatVes(!0))]TJ /F7 11.955 Tf 11.95 0 Td[(V0s(!0)=1Xt=0Xi2St"Pes,t(i)re(i))]TJ /F7 11.955 Tf 11.96 0 Td[(P0s,t(i)r0(i)+NXj=1Pes,t(i)e(i)j)]TJ /F7 11.955 Tf 11.96 0 Td[(P0s,t(i)0(i)j!i,j,t#. Weconstructastationary!00bysetting:!00i,j=8><>:ifPes,t(i)e(i)j)]TJ /F7 11.955 Tf 11.95 0 Td[(P0s,t(i)0(i)j<0)]TJ /F5 11.955 Tf 9.29 0 Td[(otherwise. Notethat!002ST.Inaddition,Ves(!00))]TJ /F7 11.955 Tf 13.19 0 Td[(V0s(!00)Ves(!0))]TJ /F7 11.955 Tf 13.19 0 Td[(V0s(!0)<0,contradictingConditions( 3 ).Therefore,Conditions( 3 )implythateis-optimal.TheproofforthereversedirectionisstraightforwardandfollowsfromtheobservationthatSTNS. Theorem 3.2 providesasetofconditionsthatcanbeusedtoevaluatethe-optimalityofapolicy.However,thenumberofpoliciesincangrowrapidlywiththesizeoftheproblem.Corollary 1 providesamorecompactsetofconditions.First,wemakethefollowingdenitions:V(!)=)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(V1(!),V2(!),,VjSj(!)0and!s=(!s,1,!s,2,,!s,N)0. 35

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Corollary1. GivenEquation( 3 ),eis-optimalifandonlyifthefollowingBellmanequationsaresatised:re(s))]TJ /F7 11.955 Tf 10.41 0 Td[(ras+e(s))]TJ /F19 11.955 Tf 11.95 0 Td[(as!s+Pe(s))]TJ /F19 11.955 Tf 11.95 0 Td[(PasVe(!)08s2S,as2A(s),!2ST. Proof. Foragiven!,theBellmanequationsarenecessaryandsufcient:Ves(!)Vs(!)8s2S,2,re(s))]TJ /F7 11.955 Tf 11.95 0 Td[(ras+e(s))]TJ /F19 11.955 Tf 11.96 0 Td[(as!s+Pe(s))]TJ /F19 11.955 Tf 11.96 0 Td[(PasVe(!)08s2S,as2A(s). Thisimpliesthat:Ves(!)Vs(!)8s2S,2,!2ST,re(s))]TJ /F7 11.955 Tf 11.95 0 Td[(ras+e(s))]TJ /F19 11.955 Tf 11.96 0 Td[(as!s+Pe(s))]TJ /F19 11.955 Tf 11.95 0 Td[(PasVe(!)08s2S,as2A(s),!2ST andthecorollaryfollowsfromTheorem 3.2 Next,weillustratehow,themaximumallowabletoleranceforthenon-stationaryproblem,canbeobtainedfromCorollary 1 .Let(I)]TJ /F5 11.955 Tf 11.96 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1s,idenotetheentryinthesthrowandithcolumnofthematrix(I)]TJ /F5 11.955 Tf 11.96 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1.Forstationary!,thevaluefunctionofastatecanbeexpressedas: Vs(!)=Xi2S(I)]TJ /F5 11.955 Tf 11.95 0 Td[(P))]TJ /F12 7.97 Tf 6.59 0 Td[(1s,i r(i)+NXj=1(i)j!i,j!.(3) Dene:fe,asi,j=8><>:(1+Ge,ass)e(s)j)]TJ /F4 11.955 Tf 11.96 0 Td[(asjifi=sGe,asie(i)jotherwise, whereGe,asidenotetheithentryofPe(s))]TJ /F19 11.955 Tf 11.96 0 Td[(Pas)]TJ /F19 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F5 11.955 Tf 11.95 0 Td[(Pe)]TJ /F12 7.97 Tf 6.59 0 Td[(1.SubstitutingEquation( 3 )intoCorollary 1 andusingourdenitionsofce,asandfe,asi,j: ce,as+Xi2SNXj=1fe,asi,j!i,j08s2S,as2A(s),!2NS.(3) 36

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Proposition3.3. GivenEquation( 3 ),=mins,asce,as PNj=1de,asj. Proof. Lets,asdenotethemaximumtoleranceallowablebyEquation( 3 )forstatesandactionas:s,as=maxfy:ce,as+Xi2SNXj=1fe,asi,j!i,j0andj!i,jjyfori2S,j=1,2,...,Ng. Similartothestationaryrewardcase,weconsidertheworstcasescenarioandobtainthefollowingexpressionfors,asfromEquations( 3 ):s,as=ce,as PNj=1de,asj, wherede,asj=Pi2Sjfe,asi,jj.Hence,themaximumallowabletoleranceforthenon-stationaryrewardsproblemis:=mins,ass,as=mins,asce,as PNj=1de,asj. Were-examineourcapacitatedstochasticlot-sizingprobleminExample 3 byallowingtheorderingcostandbacklogpenaltytovaryovertime. Example3:Non-stationaryrewards .ConsiderExample 1 again.Whentheorderingcostandbacklogpenaltyareallowedtoperturbsimultaneouslyateachperiod,itfollowsfromTable 3-2 andProposition 3.3 that=0.01andisassociatedwiththeactiona1=0.Thisimpliesthateremainsoptimalsolongastheorderingcostandbacklogpenaltydonotdeviatefromtheircurrentestimatesbymorethan1%acrossallperiods.2 3.3ToleranceGap Atthestartofthischapter,weclaimedthatthemaximumallowabletoleranceobtainedundertheassumptionthattheparametersarestationarymaybeoverly 37

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optimisticifthevaluesoftheuncertainparametersvaryacrossthehorizon.Inthissection,weprovideaformalproofforthisstatementandhighlighttheconditionswherethetolerancesofthestationaryandnon-stationaryrewardsproblemarethesameandtheconditionswheretheydiffer. Ifthecurrentdecisione(s)isreplacedbyas,jbe,asjjandde,asjrepresentthemarginalchangesintheestimationerrorofthestationaryandnon-stationaryrewardsproblem,respectively.Recallthats,asands,asdenotethemaximumallowabletoleranceforstatesandactionasinthestationaryandnon-stationaryproblem,respectively.anddenotethemaximumallowabletoleranceforthestationaryandnon-stationaryproblem,respectively.ItfollowsfromPropositions 3.2 and 3.3 that=ifjbe,asjj=de,asjforallasandj.Ifthatisnottrue,theallowabletolerancesofthestationaryandnon-stationaryrewardsproblemmaydiffer.Inparticular,>ifjbe,asjj
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Lemma2. jbe,asjjs,asifandonlyifthereexistssomeparameterjandstatess1,s22Swherefe,ass1,jispositiveandfe,ass2,jisnegative. Proof. Ifthereexistssomeparameterjandstatess1,s22Swherefe,ass1,jispositiveandfe,ass2,jisnegative,itfollowsfromLemma 2 thatjbe,asjjs,as. Ifs,as>s,as,itimpliesthatthereexistssomeparameterjwherejbe,asjj
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Corollary 2 providesasufcientconditionforjbe,asjj=de,asj.Inparticular,whenthecontributionofitothevaluefunctionisrestrictedtojustasinglestate,theimpactofiontheallowabletoleranceisthesame,regardlessofwhetheriisstationaryornot.ThevalidityofTheorems 3.3 and 3.4 andCorollary 2 areillustratedinthefollowingexample. Example4:Tolerancegap .InExamples 2 and 3 ,weobtain=0.02and=0.01,respectively.ThisresultisconsistentwithTheorem 3.3 whichstatesthat. Example 2 highlightsthat1,0=.willbestrictlygreaterthanifthereisadifferenceintheallowabletoleranceassociatedwithnotorderingwheninventoryisone.Undere,theoptimalactionistoordertwounits.Ifnoorderisplaced,anorderingcostisavoidedandtheprobabilityofenteringstate1remainsthesame.Hence,fe,01,1isnegative.However,theprobabilityofenteringstate)]TJ /F4 11.955 Tf 9.3 0 Td[(1isincreased(i.e.,Ge,0)]TJ /F12 7.97 Tf 6.59 0 Td[(1isnegative).Sincee()]TJ /F12 7.97 Tf 6.59 0 Td[(1)1isalsonegative,fe,0)]TJ /F12 7.97 Tf 6.59 0 Td[(1,1ispositive.Hence,itfollowsfromTheorem 3.4 that>. Since(s)2=0foralls0,itfollowsfromCorollary 2 thatjbe,as2j=de,as2forallas.Thisisvalidatedbycomparingthevaluesofjbe,as2jandde,as2inTables 3-1 and 3-2 ,respectively.Thisimpliesthattheimpactof2ontheallowabletolerancesforthestationaryandnon-stationaryrewardsproblemsarethesameandthatanyreductionintheallowabletoleranceofthenon-stationaryproblemisduetotherelaxationofthestationaryassumptionon1.2 InExample 1 ,wefoundthattheoptimalpolicyistobringtheinventoryleveluptothreewhenevertheinventorydropsbelowtwo.Werefertothisasanorder-up-topolicy.Next,wewillshowthat>iftheactionassociatedwithchangesthereorderpointforgenerallot-sizingproblemsundermildassumptions. Consideralot-sizingproblemwherepidenotestheprobabilitythatdemandisi.Weassumethatpiisstationary(i.e.,remainsthesameacrossthehorizon).Thereisconstantleadtime,adiscountfactor0<<1,linearproductioncostandaconvex 40

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holdingcost.Inaddition,eachorderincursanuncertainorderingcostandeachbacklogitemincursanuncertainpenaltycost.AsinExample 1 ,wemodeltheuncertaintiesintheorderingcostandbacklogpenaltyby1and2,respectively.Theobjectiveistondthepolicythatminimizesthelongrunexpectedcosts. ThisproblemcanbeformulatedasaMDPwherethestatesrepresenttheamountofinventoryavailableandtheactionsrepresenttheamountofinventorytoorder.Itiswell-knownthatanoptimalorder-up-topolicyexistsforthisproblemforagiven1and2VeinottandWagner(1965). Theorem3.5. Forthelot-sizingproblemdescribedabove,>ifthereexistss,as=whereaschangesthereorderpointandpi<1 2foralli. Proof. First,weconsiderthecasewheree(s)>0andas=0.Ifnoorderisplacedats,theremustexistsomestates0. Whene(s)=0andas>0,theremustexistsomestates0. Theorem 3.5 highlightsthatatolerancegapexistsforlot-sizingproblemswheretheactionassociatedwithchangesthereorderpointwhenpiareboundedfromaboveby1 2.Since<1,theupperboundisgreaterthan0.5forallproblemsandis,inouropinion,areasonableassumptionformostpracticalproblems. 41

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Table3-1. Singleparametersensitivityanalysisand. sasce,asbe,as1be,as2)]TJ /F7 11.955 Tf 9.3 0 Td[(ce,as=be,as1)]TJ /F7 11.955 Tf 9.29 0 Td[(ce,as=be,as2s,as -1240.8520.4030.00-2.00-1.360.81-135.5012.000.00-0.46-0.46-140.000.000.00---0140.8520.4030.00-2.00-1.360.81025.5012.000.00-0.46-0.46030.000.000.00---100.85-19.6030.000.04-0.030.02115.5012.000.00-0.46-0.46120.000.000.00---200.000.000.00---2134.5028.000.00-1.23-1.23300.000.000.00--Table3-2. Computing. sasce,asde,as1de,as2s,as -1240.8520.4030.000.81-135.5012.000.000.46-140.000.000.00-0140.8520.4030.000.81025.5012.000.000.46030.000.000.00-100.8560.4030.000.01115.5012.000.000.46120.000.000.00-200.000.000.00-2134.5052.000.000.66300.000.000.0042

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CHAPTER4REGRET Ithaslongbeenrecognizedthatadecisionbasedsolelyontherewardreceivedisnotadequatefordescribinghowindividualsmakedecisionsunderuncertaintyinpractice.In1738,DanielBernoullihighlightedtheneedtoconsidertheperceivedutilityoftheoutcome.Twocenturieslater,VonNeumannandMorgenstern(1944)proposedasetofaxiomsthatguaranteetheexistenceofautilityfunctionwheretheexpectedutilityofapreferredchoiceisgreaterthantheutilityofalesspreferredalternative.However,researchershaveidentiedinstanceswherepredictionsfromexpectedutilitytheorydeviatefromobservationsinpracticeandhaveproposedalternativetheoriestoexplaintheseobservations(Leland2010). Researchinpsychologyandbehavioraldecisiontheorysuggeststhatregretplaysanimportantroleinshapingpreferencesindecisionsunderuncertainty(seeZeelenberg(1999),ConnollyandZeelenberg(2002)andEngelbrecht-WiggansandKatok(2009)).Bell(1982)andLoomesandSugden(1982)highlightedthatgapsbetweenthepredictionsofexpectedutilitytheoryandobservationsinpracticecanbeaddressedwhenfeelingsofregretandrejoicingaretakenintoaccount.Theyproposedusingamodiedutilityfunctionwhichdependsonbothrewardandregrettomodelthesatisfactionassociatedwithadecision.Theirmodel,commonlyreferredtoasregrettheory,wasinitiallydevelopedforpairwisedecisions.Subsequently,itwasgeneralizedformultiplefeasiblealternativesinLoomesandSugden(1987)andanaxiomaticfoundationforthetheorywaspresentedinSugden(1993).Quiggin(1990)derivedthenecessaryandsufcientconditionsforapairwisechoicetobepreferredoveranotherforallregret-theoreticdecisionmakersandprovidedasetofsufcientconditionsforproblemsinvolvingmultiplechoicesinQuiggin(1994).Recently,Bleichrodtetal.(2010)illustratedhowregrettheorycanbemeasuredandtheyfoundthatindividualsareaversetoregretanddisproportionatelyaversetolargeregret. 43

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LetSandCdenotethesetofpossiblescenariosandchoices,respectively.Theoutcomeofchoicecunderscenariosisdenotedbyx(s)c.Letx(s)crepresentavectorofoutcomesassociatedwithchoicecunderscenarios:x(s)c=x(s)c,)]TJ /F7 11.955 Tf 9.3 0 Td[(x(s)1,)]TJ /F7 11.955 Tf 9.3 0 Td[(x(s)2,...,x(s)c)]TJ /F12 7.97 Tf 6.58 0 Td[(1,)]TJ /F7 11.955 Tf 9.29 0 Td[(x(s)c+1,...,)]TJ /F7 11.955 Tf 9.3 0 Td[(x(s)jCj. Letux(s)cdenotetheutilityofobtainingx(s)candmissingoutontheotheropportunities(i.e.,incurring)]TJ /F7 11.955 Tf 9.3 0 Td[(x(s)1,)]TJ /F7 11.955 Tf 9.3 0 Td[(x(s)2,etc.).Sincetheattractivenessofachoiceisgenerallynon-decreasingintherewardreceivedandnon-increasingintheopportunitiesmissed,itisreasonabletoassumethatuisnon-decreasinginitsarguments.UnlikeVonNeumannandMorgenstern's(1944)utilityfunction,theregret-theoreticutilityfunctionuisdependentontheoutcomeofpossiblechoices,ratherthantheselectedchoicealone.Letpsdenotetheprobabilityofscenariosoccurring.Inregrettheory,choicepreferencesaredescribedasfollows:c1c2,Xsps)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(u)]TJ /F19 11.955 Tf 5.48 -9.68 Td[(x(s)c1)]TJ /F7 11.955 Tf 11.95 0 Td[(u)]TJ /F19 11.955 Tf 5.48 -9.68 Td[(x(s)c20, wherec1c2denotesthatc1isatleastaspreferredasc2. Expectedutilitytheorypredictsthatc1c2iftheoutcomesofc1stochasticallydominatestheoutcomesofc2(Levy2006).However,violationsofstochasticdominance(i.e.,c2c1whentheoutcomesofc1stochasticallydominatestheoutcomesofc2)havebeenobservedinpractice.Inparticular,BirnbaumandNavarrete(1998)presentsasetofproblemswheresuchbehaviorissystematicallyobserved.Regrettheorypredictsstochasticdominanceviolations(LoomesandSugden,1987).Quiggin(1990)showedthatforapairwiseproblem(i.e.,jCj=2)whereSisniteandscenariosareequallyprobable,c1c2ifandonlyifthereexists,abijectionofSontoitself,suchthat(i))]TJ /F12 7.97 Tf 6.58 0 Td[(1=and(ii)x(s)1x((s))2foralls.NotethatCondition(ii)impliesthatoutcomesofchoice1stochasticallydominatestheoutcomesofchoice2.Inaddition,healsohighlightedthatConditions(i)and(ii)aresufcientforproblemsinvolvingmultiple 44

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choices(i.e.,jCj>2)underadditionalconditionsontheoutcomesoftheoptimalchoiceunderthebijection(Quiggin1994). UsingQuiggin'sresults,ananalystisabletoconstructaregret-theoreticefcientsetbyeliminatingchoicesthatareclearlyinferior(i.e.,suboptimalforallregret-theoreticutilityfunctions).However,thereareseverallimitations.First,thenumberofscenariosareassumedtobeniteinQuiggin(1990,1994).However,therearemanyproblemswherejSjisinnite(forexample,whenoutcomesarecontinuouslydistributed).Second,theconditionsareonlynecessaryandsufcientforpairwiseproblems.WhenjCj>2,theconditionsaresufcient,butnotnecessary(i.e.,theefcientsetmaynotbeminimalinsize).Third,thenumberofpossiblebijectionsgrowsexponentiallywiththenumberofscenariosanddeterminingifthereexistsathatsatisesconditions(i)and(ii)isgenerallychallenging.Inthenextsection,wepresentasetofconditionsthataddressestheselimitations. 4.1MultivariateStochasticDominance Intheprevioussection,wepresentedregrettheoryasamodelwheretheoutcomeofeachchoiceundereachscenarioisexplicitlymodeled.Here,wepresentanimplicitformofthemodelwhereSisnotexplicitlydened.Rathertheoutcomeofeachchoiceismodeledasarandomvariablethatcanbedependentontheoutcomeofotherchoices. LetXcdenotetherewardofchoicecwhereXcisarandomvariabledescribedbysomeknownprobabilitydistributionfunctionfXc.LetXcdenoteamultivariaterandomvariabledenedas:Xc=(Xc,)]TJ /F7 11.955 Tf 9.3 0 Td[(X1,)]TJ /F7 11.955 Tf 9.3 0 Td[(X2,...,)]TJ /F7 11.955 Tf 9.3 0 Td[(Xc)]TJ /F12 7.97 Tf 6.59 0 Td[(1,)]TJ /F7 11.955 Tf 9.3 0 Td[(Xc+1,...,)]TJ /F7 11.955 Tf 9.3 0 Td[(XjCj). Regret-theoreticchoicepreferencesfortheimplicitformarestatedasfollows:c1c2,E[u(Xc1))]TJ /F7 11.955 Tf 11.96 0 Td[(u(Xc2)]0.Lisalowersetif(x1,x2,...,xjCj)2Limplies(y1,y2,...,yjCj)2Lwhenyixiforalli. 45

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Considertwomultivariaterandomvariables,X1andX2.WesaythatX1stochasticallydominatesX2ifandonlyif:P(X12L)P(X22L)foralllowersetsLRjCj. WedenotethisrelationshipbyX1sX2.ItiswellknownthatX1sX2ifandonlyifE[u(X1)]E[u(X2)]forallnon-decreasingu(see,forexampleShakedandShanthikumar(2007)).Therefore,itfollowsfromourdenitionofXcthatmultivariatestochasticdominanceisnecessaryandsufcientforchoicepreferenceoverallnon-increasingregret-theoreticutilityfunctions.Statedformally: Xc1sXc2,E[u(Xc1))]TJ /F7 11.955 Tf 11.96 0 Td[(u(Xc2)]0forallu2U,(4) whereUdenotesthesetofnon-decreasingregret-theoreticutilityfunctions.UnliketheconditionsinQuiggin(1994),Condition( 4 )isbothnecessaryandsufcient.Inaddition,itiseasiertocheckandappliestoproblemswherejSjisinnite. Next,wepresentconditionswherechoicepreferencesinregrettheoryareconsistentwiththatofexpectedutilitytheory(i.e.,c1c2ifXc1sXc2).Webeginwiththefollowingtheorem.LetXi(k)denotethek-thelementofXi. Theorem4.1. Xc1sXc2,Xc1(k)sXc2(k)forallk. Proof. First,weshowthat: Xc1sXc2)Xc1(k)sXc2(k)forallk.(4) Weprovethisresultbyshowingthat( 4 )holdsforeachk.Withoutlossofgenerality,letchoicesc1=1andc2=2.SinceXc1(1)=Xc1andXc2(1)=Xc2,( 4 )holdsfork=1.SinceXc1(2)=)]TJ /F7 11.955 Tf 9.3 0 Td[(X2andXc2(2)=)]TJ /F7 11.955 Tf 9.3 0 Td[(X1and)]TJ /F7 11.955 Tf 9.3 0 Td[(X2s)]TJ /F7 11.955 Tf 12.19 0 Td[(X1,( 4 )holdsfork=2.SinceXc1(k)=Xc2(k)=)]TJ /F7 11.955 Tf 9.3 0 Td[(Xk,( 4 )holdsfork>2aswell.Theproofinthereversedirectionisstraightforward. 46

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Theorem 4.1 highlightsthattheoutcomesofc1stochasticallydominatestheoutcomesofc2ifandonlyifeachmarginaldistributionofXc1stochasticallydominatesthatofXc2. Corollary3. WhenXiareindependent,Xc1sXc2ifandonlyifXc1sXc2. Proof. WhenXiareindependent,stochasticdominanceofeachmarginaldistributionisnecessaryandsufcient.Therefore,itfollowsfromTheorem 4.1 that:Xc1sXc2,Xc1(k)sXc2(k)forallk,Xc1sXc2. Corollary 3 highlightsthatwhenrewardsareindependent,choiceswithoutcomesthatstochasticallydominatearepreferredbyallregret-theoreticdecisionmakers.Next,weshowthatchoicepreferencesinregrettheoryarealsoconsistentwiththatofexpectedutilitytheorywhenwelimitourselvestoadditiveregret-theoreticutilityfunctions.LetU1Udenotethesetofallnon-decreasingadditiveregret-theoreticutilityfunctions. Corollary4. E[u(Xc1))]TJ /F7 11.955 Tf 11.96 0 Td[(u(Xc2)]0forallu2U1ifandonlyifXc1sXc2. Proof. StochasticdominanceofeachmarginaldistributionisnecessaryandsufcientforE[u(Xc1))]TJ /F7 11.955 Tf 12.26 0 Td[(u(Xc2)]0forallu2U1(LevyandParoush1974).Therefore,itfollowsfromTheorem 4.1 that:E[u(Xc1))]TJ /F7 11.955 Tf 11.96 0 Td[(u(Xc2)]0forallu2U1,Xc1(k)sXc2(k)forallk,Xc1sXc2. Corollary 4 statesthatchoiceswithoutcomesthatstochasticallydominatearepreferredbyallregret-theoreticdecisionmakerswithadditiveutilityfunctions,evenifrewardsaredependent. 47

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4.2RegretinNormativeDecisionAnalysis Theconceptofregrethasbeenadoptedbyvariousresearchersinnormativedecisionanalysis.In1951,Savageproposedndingasolutionwithminimalmaximumregret.Ithasbeenarguedthatthesesolutionstendtobelessconservativethansolutionsthatmaximizeminimumrewardbecausetheformerconsidersmissedopportunitiesoverallpossiblescenariosratherthantherewardintheworstcasescenarioalone.Themin-maxregretcriteriahasbeenusedinavarietyofproblems,includinglinearprogramswithimpreciseobjectivecoefcients(InuiguchiandSakawa,1995)andnetworkoptimizationwithintervaldata(AverbakhandLebedev,2004).TheinterestedreaderisreferredtoKouvelisandYu(1997)andAissiaetal.(2009)forasurveyofmin-maxregretproblems. Despiteahistoryofmorethanhalfacentury,therepresentationofregretintheoptimizationliteraturehasbeenrestrictedtothedeviationand,toalesserextent,theratiooftheoptimalrewardtothereceivedreward.Inpractice,regretcandependonasubsetofmissedopportunities,ratherthantheoutcomeoftheoptimalalternativealone. LetYcdenotetheregretassociatedwithchoicec,whereYcissomefunctionoftherewardreceivedandopportunitiesmissed.Let'denotetheregretfunction:Yc='()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc). Sinceregretisnon-increasingintherewardreceivedandnon-decreasingintheopportunitiesmissed,weassumethat'isnon-decreasingin)]TJ /F19 11.955 Tf 9.29 0 Td[(Xc. Intheoptimizationliterature,regretiscommonlyexpressedasthedifferencebetweentheoptimalrewardandtherewardreceived:Yc='a()]TJ /F19 11.955 Tf 9.29 0 Td[(Xc)=maxi2SXi)]TJ /F7 11.955 Tf 11.95 0 Td[(Xc. Werefertothisasabsoluteregretand'aastheabsoluteregretfunction.Inaddition,researchershavealsoproposedexpressingregretasaratiooftheoptimalandreceived 48

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reward:Yc='r()]TJ /F19 11.955 Tf 9.29 0 Td[(Xc)=maxi2SXi)]TJ /F7 11.955 Tf 11.96 0 Td[(Xc Xc. Werefertothisasrelativeregretand'rastherelativeregretfunction.TheinterestedreaderisreferredtoKovelisandYu(1997)andAissiaetal.(2009)forasurveyofproblemsthatminimizeabsoluteorrelativeregret. Next,wegeneralizeabsoluteregretbyintroducingthefollowingtwonotionsofregret.First,weexpressregretasthedifferencebetweenthe-percentilerewardandthereceivedreward:Yc='p()]TJ /F19 11.955 Tf 9.29 0 Td[(Xc)=X)]TJ /F7 11.955 Tf 11.95 0 Td[(Xc, whereXistherewardatthe-percentile.Werefertothisaspercentileregretand'pasthepercentileregretfunction.Notethatpercentileregretreducestoabsoluteregretwhen=1. Next,weexpressregretasthedifferencebetweentheaverageoftheklargestrewardsandthereceivedreward:Yc='k()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc)=1 kkXi=1X(i))]TJ /F7 11.955 Tf 11.96 0 Td[(Xc, whereX(i)denotestheithlargestrewardinS.Werefertothisask-averageregretand'kasthek-averageregretfunction.Notethatk-averageregretreducestoabsoluteregretwhenk=1. Letdenotethesetofnon-decreasing'.Observethat'a,'r,'p,'k2.Thefollowingtheorem,Theorem 4.2 ,highlightsthatstochasticdominancerelationsarereversedunderreward-to-regret-transformation(i.e.,ifoutcomesofc1stochasticallydominatesoutcomesofc2.thenregretofc2stochasticallydominatesregretofc1)whentheoutcomeofchoicesareindependent.Thisresultisgeneralandholdsforallformsofregretthatarenon-decreasingin)]TJ /F19 11.955 Tf 9.29 0 Td[(Xc,includingthefournotionsofregretpresentedinthissection. 49

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Theorem4.2. Supposethat'2andXiareindependent.IfXc1sXc2,thenYc2sYc1. Proof. ItfollowsfromCorollary 3 thatwhenXiareindependent:Xc1sXc2)Xc1sXc2)E[u(Xc1))]TJ /F7 11.955 Tf 11.96 0 Td[(u(Xc2)]0,forallnon-decreasingu)E[u()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc1))]TJ /F7 11.955 Tf 11.96 0 Td[(u()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc2)]0,forallnon-decreasingu)E[('()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc1)))]TJ /F5 11.955 Tf 11.95 0 Td[(('()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc2))]0,forallnon-decreasing)E[(Yc1))]TJ /F5 11.955 Tf 11.96 0 Td[((Yc2)]0,forallnon-decreasing)Yc2sYc1. Since'rinvolvesamaxoperatorandthedivisionofadependentrandomvariable,provingTheorem 4.2 intheabsenceofCorollary 3 isnon-trivial,evenforthespecialcaseofrelativeregret.Next,weillustratehowregretcanbeviewedasameasureoftheriskandbeadoptedwithinthemean-riskframework.Letrcdenotetheriskassociatedwithchoicec.Themean-riskframeworkseeksachoicecthatmaximizesthefollowingobjective:g(c)=E[Xc])]TJ /F5 11.955 Tf 11.95 0 Td[(rc, whereissomeconstant.Conventionalriskmeasuresarebasedontheoutcomesassociatedwithachoiceandthevalueofisdenedaszero,positiveandnegativeforariskneutral,riskaverseandriskseekingdecisionmaker,respectively(Krokhmaletal.,2011). Wedenearegret-basedriskmeasureasfollows:rc=(Yc), whereissomenon-decreasingfunction.Sincethesatisfactionofadecisionisnon-increasinginregret,thevalueofisnon-negativeinmean-regretmodels. 50

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Wesaythat'isseparableifthereexistssomerandomvariableeXsuchthatYc='()]TJ /F19 11.955 Tf 9.3 0 Td[(Xc)=eX)]TJ /F7 11.955 Tf 12.21 0 Td[(Xc.NotethatthereisnoindependencerestrictionbetweeneXandXc.Let1denotethesetofnon-decreasingseparable'.Observethat'a,'p,'k21.Next,weshowthatbeingstochasticdominanceinregretisnecessaryandsufcientformean-regretchoicepreferenceif'21. Theorem4.3. If'21,isnon-decreasingand0,Yc2sYc1,E[Xc1])]TJ /F5 11.955 Tf 11.96 0 Td[((Yc1)E[Xc2])]TJ /F5 11.955 Tf 11.95 0 Td[((Yc2). Proof. First,weshowthatE[Xc1]E[Xc2]if'21andYc2sYc1.WhenYc2sYc1,E[Yc1]E[Yc2].Thisimpliesthat:E[Yc1]E[Yc2]E[eX)]TJ /F7 11.955 Tf 11.96 0 Td[(Xc1]E[eX)]TJ /F7 11.955 Tf 11.95 0 Td[(Xc2]E[eX])]TJ /F3 11.955 Tf 11.96 0 Td[(E[Xc1]E[eX])]TJ /F3 11.955 Tf 11.95 0 Td[(E[Xc2])]TJ /F3 11.955 Tf 9.3 0 Td[(E[Xc1])]TJ /F3 11.955 Tf 28.56 0 Td[(E[Xc2]E[Xc1]E[Xc2]. Thesecondinequalityfollowsfromthefactthat'21.Inaddition,itfollowsfromstandardresultsinstochasticdominancethat(Yc1)(Yc2)forallnon-decreasing.ThereforeYc2sYc1)E[Xc1])]TJ /F5 11.955 Tf 12.03 0 Td[((Yc1)E[Xc2])]TJ /F5 11.955 Tf 12.03 0 Td[((Yc2),forallnon-decreasingand0. Next,weprovidetheproofforthereversedirection.SinceE[Xc1])]TJ /F5 11.955 Tf 12.94 0 Td[((Yc1)E[Xc2])]TJ /F5 11.955 Tf 12.51 0 Td[((Yc2)forall0,(Yc1)(Yc2)forallnon-decreasing,whichimpliesYc2sYc1. Theorem 4.3 highlightsthatregretdominanceisnecessaryandsufcientformean-regretpreferencesforallnon-decreasingwhenregretisseparable.However, 51

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theresultsofTheorem 4.3 maynotholdfornon-separableregret.Example 5 highlightsthatregretdominanceisnotsufcientformean-regretpreferencesunderrelativeregret. Example5:Relativeregretinmean-regretmodels .Considertwochoices,c1andc2,andtwomutuallyexclusiveeventsE1andE2suchthatP(E1)=P(Xc1=1,Xc2=2)=0.5andP(E2)=P(Xc1=5,Xc2=3)=0.5.Computingtherelativeregretofc1andc2(i.e.,Yc=maxi2SXi Xc)]TJ /F4 11.955 Tf 11.36 0 Td[(1),wegetP(Yc1=0)=P(Yc1=1)=P(Yc2=0)=P(Yc2=0.67)=0.5,whichimpliesthatYc1sYc2.However,E[Xc1]=3>E[Xc2]=2.5.Sincec1>c2,amean-regretdecisionmakerwithasufcientlysmallwillpreferc1,eventhoughitregretdominatesc2.2 52

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CHAPTER5EQUIPMENTREPLACEMENTANALYSISWITHANUNCERTAINFINITEHORIZON Equipmentreplacementanalysisisconcernedwithidentifyingthelengthoftimetoretainanassetanditsreplacementsoversomedesignatedhorizonofanalysissuchthatcapitalandoperatingcostsareminimized.Thereasonsforreplacinganassetaregenerallyeconomic,includingdeteriorationoftheassetitself,leadingtoincreasedoperatingandmaintenance(O&M)costsanddecreasedsalvagevalues,andtechnologicaladvancesinpotentialreplacementassetsonthemarketthatmayprovidesimilarserviceatlowercost. Whilemuchofthevastliteratureonequipmentreplacementanalysisisconcernedwiththemodelingofdeteriorationandtechnologicalchange(see,forexample,Oakfordetal.(1984),Beanetal.(1994)andRegnieretal.(2004)),itshouldbeclearthattheselectedhorizoncanhaveadrasticeffectonthesolutionapproachandresultingsolutions(deSousaandGuimaraes1997).Inaninnitehorizonproblemwithstationarycosts,theoptimalpolicyistorepeatedlyreplacetheassetatitseconomiclife,theagewhichminimizesequivalentannualcosts.Ifthehorizonisnite,thenthepolicyofreplacinganassetatitseconomiclifeisonlyoptimalifthehorizonisamultipleoftheeconomiclife.Otherwise,asequenceofassetreplacementsmustbefound,generallythroughtheuseofdynamicprogramming(seeBellman(1955),Wagner(1975),orHartmanandMurphy(2006)). Inthecaseofnonstationarycosts,constantagereplacementpoliciesarerarelyoptimal(seeFraserandPosey(1989))forinnitehorizonproblems.Thus,mostsolutionsfocusonndingtheoptimaltimezerodecision.Generally,aforwarddynamicprogrammingalgorithmissolvedsuchthattheinitialreplacementdecision(keeporreplacetheassetattimezero)isoptimalforanyhorizongreaterthanadesignatedsolutionorforecast(nite)horizonT(seeBeanetal.(1985)andChandandSethi(1982)). 53

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Weareconcernedwithanequipmentreplacementproblemthathasanitehorizon,butthehorizonlengthisunknownattimezero,andmaynotbecomeknownuntilreached.Wetermthisproblemtheequipmentreplacementproblemunderanuncertainnitehorizon.Wehaveencounteredanumberofinstancesinpracticeinwhichthistrulycapturesreality,including:(1)Aproductionlineistobeclosed,butcontinuestooperatewhilecustomersplaceorders;(2)Theexpectedlifeofamineisextendedthroughthediscoveryofneworedeposits;(3)Makeshiftsolutionsareprovidedwhileinfrastructureisbuilt,suchasacompanyprovidingbusserviceforitsworkerswhileatramsystemisconstructed;and(4)Amilitarybaseisslatedforclosing,butadatehasnotbeensetandmostlikelywillbedebatedheavilybygovernmentofcials. Intheaboveexamples,assetsareutilizedinproductionorservicetomeetcapacityneedsandtherequiredlengthofserviceisnite,butuncertain.Itshouldbenotedthattheexamplesillustratecaseswheretherequiredservicecanextendbeyondorfallshortoftheexpectedlengthofservice.Furthermore,thelengthoftimeinwhichanassetisneededintheseinstancescanbeshort(lessthanayear)orlong(nearlyadecade).Incaseswheretheexpectedservicerequirementsareshort,itisconceivablethatacompanywouldmerelyutilizecurrentassetsovertheremainingperiodsofservice.However,whentherequiredlengthofserviceisextensiveorhighlyuncertain,thenitshouldbeclearthatequipmentreplacementdecisionsshouldbeconsideredovertimeintheinterestofsavingmoney.Inaddition,thecompanymayexploreoptionsoutsideoftraditionalownership,suchasleasing. Theuseofdynamicprogramminghasbecomeprevalentinsolvingequipmentreplacementproblemsbecauseiteasilyaccountsforthedynamicnatureoftheproblem(keeporreplacedecisionismadeperiodically)anditovercomesthetraditionalassumptionthatassetsarerepeatedlyretainedforthesamelengthoftime.Whilepreviouslydevelopeddynamicprogrammingformulationsdiffer,theycanallberepresentedbyacyclicgraphsinwhichthestatesofthesystemarerepresentedby 54

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nodesandthearcsrepresentdecisions.Aseachdecisioncarriesacost,thegoalistondtheshortest(minimumcost)paththroughthenetwork,beginningwithanassetofknownageattimezeroandmakingoptimaldecisionsthroughthehorizontimeT.Inourproblem,thevalueofTisuncertain.(Notethatthisisnotanoptimalstoppingproblemasthedecision-makercannotchoosethenitehorizonT.Rather,itisaproductoftheenvironment.) Wespecicallydenetheequipmentreplacementproblemwithanuncertainnitehorizonasfollows:AnassetisrequiredtobeinserviceforTperiodswhereTisarandomvariablesuchthatTsTTl.ThevaluesofTsandTlareknown,alongwiththeprobabilitydistributionofT.Itisassumedthattheassetissoldattheendoftherealizedhorizon.Attimezero,anassetofagenisownedandit,anditsreplacements,mayberetainedthroughageN,ifnotreplacedsooner.Itisassumedthattheequipmentcanperforminitsintendedcapacityineachperiodoverthehorizon. Ashighlightedpreviously,theoptimalreplacementdecisionisdependentonthehorizonandtheremaybeaperiodbeforeTswheretheoptimaldecisiondivergesfordifferenthorizonrealizations.Forexample,itmaybeoptimaltoretaintheequipmentattheendofperiod2ifT=7butifT=8,theoptimaldecisionmaybetoreplaceitattheendofperiod2.Wetermperiod2,tc,thecriticaldecisionperiod.Iftheuncertaintyofthehorizonisnotresolvedbytc,theriskoflosscannotbeeliminated. Weassumethatthecostsforpurchasing,operatinganddisposingoftheassetanditspotentialreplacementsovertimeareknown.Thus,wearenotfacedwithanetworkofstochasticarclengths,whichisfairlycommonintheliterature.Rather,wearefacedwithanetworkwithanuncertaindestination(state),whichisnotascommonintheliterature. Whilewebelieveourproblemisunique,thereexistsproblemsintheliteraturewithsimilarcharacteristics.Asignicantnumberofstudiesconsidertheuncertainlifetimeofequipment(seeSivazlian(1977),forexample)thatisreplaceduponfailure,suchasan 55

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appliance.Thatis,economicreplacementdecisionsarenotmadeperiodicallyovertimeasincapitalequipmentreplacementanalysis. Usingamathematicalapproachsimilartofailureanalysis,PliskaandYe(2007)determinedoptimalinsurancepurchasesandincomeconsumptionforawageearner.Clearly,thelifetimeofthepersonisuncertainandnite.However,thesedecisionsareonlymadethroughtheperson'sretirementagewhileweexaminedecisionsthroughtheuncertainhorizon. Trietsch(1985)studiedanapplicationinmilitarytargetplanning.Thedecisionsincludethetrajectoryandspeedoftheaircraftgivenpotentialtargetsupondepartureuntiltheultimatedestinationismadeknownatsomepointontheroute.Astheproblemiscontinuous,ittakesonthecharacteristicsofadynamicfacilitylocationproblem,asonemustnotheadtowardsapossibledestination,asinourdiscreteproblem. Inthischapter,westudytheequipmentreplacementproblemunderanuncertainhorizonindetail.Firstweexamineatraditionalproblemwhichminimizescost.Next,weturntondingarobustsolutionwhichminimizesmaximumregret.Inaddition,weproposeanddesignanoptioncontracttoleaseanassettohedgeagainsttherisksthatresultfromthehorizonuncertainty.Whileleasingoptionshavebeennotedintheequipmentreplacementliterature(seeHartmanandLohmann(1997)),toourknowledge,theyhavenotbeenproposedtohedgerisk. Inthenexttwosections,weillustratehowtoobtaintheoptimalreplacementpolicythatminimizesexpectedcostandalsotheoptimalreplacementpolicythatminimizesthemaximumregret.Afterwhich,weillustratehowanoptimaloptioncontract,thatgeneratesrevenuefortheleasorandreducestheriskfortheleasee,canbedesigned. 5.1MinimizingExpectedCost Deneft(i)astheminimumexpectedcostofmakingoptimalreplacementdecisionsforani-periodoldassetattimetthroughthehorizonT2[Ts,Tl].Weassumethatattheendofperiodt2[Ts,Tl)]TJ /F4 11.955 Tf 11.96 0 Td[(1],itismadeknownwhethertheproblemcontinues 56

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totimet+1.Thedecisioniswhethertokeep(K)orreplace(R)theassetaftereachperiod.Theassetissoldattheendoftherealizedhorizon. O&Mcostsforani)]TJ /F1 11.955 Tf 9.3 0 Td[(periodoldassetattimetaredenedasmt(i)whilesalvagevalues(assumedtoberevenues)aredenedasst(i).Thepurchasecostforanewassetinperiodtisdenedasct.Thereisonlyoneavailablechallenger(replacementasset)ineachperiod.AssetpurchasesandsalesareassumedtooccuratthebeginningoftheperiodwhileO&Mcostsoccurattheendoftheperiod.Allcostsarediscountedaccordingtotheperiodicdiscountfactor. ThedynamicprogrammingrecursionisdenedasstochasticinperiodsTsthroughTlanddeterministicinperiods0t<>:K:[mt+1(i+1)+ft+1(i+1)],R:ct)]TJ /F7 11.955 Tf 11.96 0 Td[(st(i)+[mt+1(1)+ft+1(1)]1CA+P(T=tjTt)()]TJ /F7 11.955 Tf 9.29 0 Td[(st(i)),Tst<>:K:[mt+1(i+1)+ft+1(i+1)],R:ct)]TJ /F7 11.955 Tf 11.95 0 Td[(st(i)+[mt+1(1)+ft+1(1)],0t
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6,7and8,arcsconnecttonodeT,representingtheterminationoftheproblem,and,forperiods6and7,anintermediatenode,signalingthattheproblemmaycontinueforatleastanotherperiod.Twoarcs,representingkeepandreplacedecisions,emanatefromeachintermediatenode.Notethatthearcsemanatingfromthestatenodesatperiods6and7areprobabilistic. Example6:MinimizingExpectedCost .Consideraeight-yearoldassetownedattimezero.Theassetmustbeinserviceforatleast6years(Ts),buttheservicemaylastforamaximumof8years(Tl).Anewassetcanbepurchasedinanyperiodfor$150,000.O&Mcostsforanewassetare$15,000intherstyear,increasingatacompoundingrateof10%eachyearwhilethesalvagevaluesdecline20%peryear.Anassetcanberetainedfor20years,atwhichtimeitmustbereplaced,suchthatitdoesnothavetobereplacedinthespeciedtimeframe.Theannualinterestrateis10%andtheprobabilitydistributionofthehorizonisP(T=6)=P(T=7)=0.2andP(T=8)=0.6.Notethattheseprobabilitiesarestationaryandareassumedtonotchangewithtime,deningtheconditionalprobabilitiesofP(T=tjTt)=0.2,0.25and1,fort=6,7and8,respectively. TheassociatednetworkforthisproblemisillustratedinFigure 5-1 .Notethatinperiod8,thereareninepossibleendconditions(ages1through8and16).Theseagesdenetheboundaryconditions( 5 )listedinTable 5-2 Workingthroughrecursions( 5 )and( 5 )denestheoptimalsolutionofreplacingtheassetattheendofthesecondperiod.Thus,forthevariouspossiblehorizonsofT=6,7and8,theretiredassetisage4,5,or6,respectively.Toprovidesomeframeofreference,notethattheeconomiclifeofthisassetis11years. Theexpecteddiscountedcostofthispolicyis$206,987.IfitisknownthatT=6,theoptimalreplacementpolicyistoretiretheinitialassetattheendofthesixthperiod.Thediscountedcostofthispolicyis$171,661.WhenT=7,itisoptimaltoreplacetheinitialassetattheendofthethirdperiodandretireitattheendoftheseventhperiod. 58

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Thediscountedcostofthispolicyis$200,161.WhenT=8,replacingtheinitialassetattheendofthesecondperiodandretiringitattheendoftheeighthperiodresultsinaminimumdiscountedcostof$218,392.Thus,thepolicywhichminimizesexpectedcostsfollowsthedeterministicsolutionsforT=8,butdivergeswhenT=6orT=7.2 Thisleadsustoinvestigatewhethermorerobustpoliciesarepossibleandwhethertheriskoflossfromdifferenthorizonrealizationscanbemitigated.Alsonotethatthisexampledenesperiod2asthecriticaldecisionperiod,becausetheoptimalpathsofdecisionsdivergeatthispointfordifferentrealizationsofthehorizon. 5.2MinimizingMaximumRegret Whileminimizingexpectedcostsisanappropriateobjectivefunctioninmanysituations,itmaynotbeappropriateforarisk-aversedecision-maker.Thatis,thepolicydeterminedwhenminimizingcostsmayexposethepossibilityofgreatlossesifcertainrealizationsoccur.Here,weexploremorerobustsolutionsbyminimizingmaximumregret.Theregretforeachhorizonrealizationisthedifferencebetweentheoptimalsolution,giventhatrealization,andthecostobtainedfromthechosenreplacementpolicy. Obtainingtheoptimalsolutionforeachpossiblehorizonisstraightforwardasadeterministicdynamicprogramcanbesolvedforwardsfromaninitialcondition,proceedingwithkeepandreplacedecisionsthrougheachpossiblehorizonlength.Thisisthemethodemployedinmanyequipmentreplacementproblemswhichguaranteeanoptimaltimezerosolutionfortheinnitehorizonproblem(seeBeanetal.(1985)). Theforwarddynamicprogrammingrecursionthatobtainsthepolicywhichminimizesmaximumregretacrosstheuncertainhorizonisdenedwiththefollowinginitialconditionsattime0: g0(i)=8><>:)]TJ /F7 11.955 Tf 9.3 0 Td[(s0(i),i=n1,otherwise(5) 59

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andtherecursionisdenedasfollowsfortheremainingperiodst,where1tT: gt(i)=8><>:K:t)]TJ /F12 7.97 Tf 6.59 0 Td[(1[st)]TJ /F12 7.97 Tf 6.59 0 Td[(1(i)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+(mt(i))]TJ /F7 11.955 Tf 11.96 0 Td[(st(i))]+gt)]TJ /F12 7.97 Tf 6.59 0 Td[(1(i)]TJ /F4 11.955 Tf 11.96 0 Td[(1),2iNR:t)]TJ /F12 7.97 Tf 6.59 0 Td[(1[ct+(mt(1))]TJ /F7 11.955 Tf 11.95 0 Td[(st(1))]+minj=1,2,...,Nfgt)]TJ /F12 7.97 Tf 6.59 0 Td[(1(j)g,i=1.(5) Therecursionfollows,asdescribedinParkandSharp(1990),inthatthecostingassumestheassetissoldattheendofeachperiod.Modeledthisway,theproblemcanbeterminatedafteranyperiod.Iftheproblemcontinues,keepinganassetrequirespurchasingtheassetatitssalvagevalue,thusnegatingtheprevioussaleforanetcashowtransactionofzero.Iftheassetisreplaced,thenanewassetispurchasedwiththesalvagevaluefortheoldassetalreadyreceived. Notethatthenetwork(Figure 5-1 )forthebackwardrecursionisthesamehere.However,therecursion( 5 )differsinthattherstexpressionrelatestoanassetthatiskept(movesfromagei)]TJ /F4 11.955 Tf 12.18 0 Td[(1toi)whilethesecondexpressionndstheminimumpathresultinginanewasset(i=1)attimet. TheadvantageofsolvingtheforwarddeterministicdynamicprograminoursettingisthatitdenestheoptimalcoststhroughanyperiodT2fTs,Tlgwithanassetofagei,whichisrequiredforthecomputationofregret.Tondtheminimalmaximumregretpath,onecouldenumerateallpossiblepathsfromperiod0toperiodTl.However,wecanreducethenumberofpathsbynotingthattheminimumcostpathistakenfromperiod0toTs,astheproblemisdeterministicintheseperiods.Therefore,itisonlynecessarytoenumeratethepathsfromTstoTl.Inaddition,itmaynotbenecessarytoexplicitlyenumerateallpossiblepaths.Whentwosub-pathsendatthestate,thedominatedsub-pathmaybeeliminatedfromfurtherconsideration.Adominatedsub-pathhasahigherexpectedcostandahighermaximumregret.AnalgorithmforndingtheminimalmaximumregretispresentedinTanandHartman(2010).Notethatintheworstcase,itispossible,althoughunlikely,thatthereisno 60

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dominatedsub-path,deningthemaximumnumberofpathstobeexplicitlyevaluatedas2Tl)]TJ /F8 7.97 Tf 6.59 0 Td[(TsN. Example7:MinimizingMaximumRegret .ConsiderExample 6 again.SolvingEquation( 5 )withthisdatadenesthediscountedcostsolutionsthroughperiods6,7and8asshowninTable 5-4 .Theminimumcostsolutionineachperiodisinbold. Supposethati=1attheendofperiod6.Keepingthatassetforthenexttwoperiodswouldresultinanassetofages2and3attheendofperiods7and8,respectively.ExaminingthedatainTable 5-4 ,theregretexperiencedwouldbe$3,241,$1,681and$5,353forperiods6,7and8,respectively.Thesevaluesaremerelythedifferencesbetweenthefunctionalvaluesg6(1),g7(2)andg8(3)andtheoptimalsolutionsofg6(14),g7(4)andg8(6),respectively. Supposethatwewanttoreachstate1ofperiod7(i.e.,replacementdecisionismade)bypassingthroughstate1ofperiod6.Thecostofthispathis$202,450+($174,902-$171,661)=$205,691.Thetermintheparenthesisistheadditionalcostthathasbeenincurredasaresultofreachingstate1ofperiod7viaanon-minimumcostpath.Themaximumregretofthispathismaxf$174,902)]TJ /F4 11.955 Tf 12.33 0 Td[($171,661,$205,691)]TJ /F4 11.955 Tf -445.5 -23.91 Td[($200,161g=$5,530.Observethatthepathendingatstate1ofperiod7whichpassesthroughstate14ofperiod6hasacostof$202,450andamaximumregretof$2,289.Thispathdominatesthepathwhichpassesthroughstate1ofperiod6andhence,thedominatedpathcanbediscarded. UsingthealgorithmpresentedinTanandHartman(2010),thepolicythatminimizesmaximumregretistoretaintheinitialassetforfourperiodsandkeepthereplacementthroughthehorizon.Thus,theassetageattheendofhorizonsoflength6,7and8,is2,3,and4,respectively.Thisresultsinaminimalmaximumregretof$5,112fromperiod6. Notethatthepolicywhichminimizesmaximumregretcarriesanexpectedcostof$208,084,whichisgreaterthantheminimumpossibleexpectedcost.However,theoptimalpolicyforminimizingexpectedcostsexposestheownertoapotentiallossof 61

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$7,855atT=6.Thus,thepolicywhichminimizesexpectedcostsmaynotalwaysbethemostattractivetoarisk-aversedecision-maker.Notethatnoreplacementpolicycaneliminateallpotentiallossesinthisexample.2 Itisclearthatreplacementdecisionsrequireextensiveforecastingofdata,includingfutureexpectedcosts.Were-examinethepreviousexampleswithsensitivityanalysistoillustratehowthesolutionschangewithdifferentdataforecasts. Example8:SensitivityAnalysis .Togainabetterunderstandingoftheproblem,thereplacementpoliciesthatminimizeexpectedcostandmaximumregretarerecomputedfordifferentratesofincreasingO&Mcostsandratesofdecreasingsalvagevalues.Inaddition,weconsidertwodifferentinitialassetagesandtwodifferenthorizonrealizationdistributions. TheresultsareillustratedinTables 5-7 through 5-10 ,wheretheoptimalreplacementpolicyisdescribedbylistingthe(endof)periodsinwhichreplacementsoccur.A-representsnoreplacementsoverthehorizon.ColumnsE[Cost]andMRlisttheexpectedcostandmaximumregretofthepolicy,respectively.Tables 5-7 and 5-8 consideraninitialassetofage4whileTables 5-9 and 5-10 consideraninitialassetofage8. Ofthe36scenariosexamined,therewere14scenariosinwhichtheregretcouldnotbeeliminated.ThesetendtooccurwhentheratesoftheO&Mcostincreasesandsalvagevaluedecreasesarehigh.Thisisreasonableasretaininganassetbeyonditseconomiclifecanbequitecostlyinthesesituations.Notethatregretisalsomorelikelywhenstartingwithanolderasset,evenwhentheratesarenotashigh. Thedifferentobjectivesofminimizingexpectedcostsandminimizingmaximumregretledtoeightdifferentreplacementpoliciesinthe36scenariosanalyzed.Thedifferencesinmaximumregretsinthoseeightcasesvariedfromafromamere5.6%toover140%.Clearly,minimizingexpectedcostalonemaynotbeappropriateforarisk-aversedecision-maker. 62

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Itisalsointerestingtonotethevarianceintheageoftheassetduringtheuncertainperiodsforeachofthescenarios.Forthescenariosinwhicharegretmustbeincurred,thetendencyistoreplacetheassetatsomepointbeforeTs,suchthattheassetisrelativelyyoungwhenenteringtheuncertainperiods.Thesearealsothecasesinwhichtheequivalentannualcostcurvedeningtheeconomiclifeoftheassetissteep,deninggreaterriskofregret.Fortheattercurvesandlongereconomiclives,thetendencyistoretaintheassetslonger,thusenteringtheuncertainperiodswitholderassets.Notethattheinitialageoftheassetdoesplayastrongroleinthisoutcome.2 5.3LeasingOptionstoFurtherMitigateRisk Theprevioussectionhighlightsacriticalissue:riskcanbereducedbyminimizingmaximumregret,butunlesstheassociatedpolicyalignswiththeoptimalpolicyforeachhorizonrealization,theriskoflosscannotbeeliminated.However,itmaybepossibletolowertheriskifavendoriswillingtoprovidethedecision-makerwiththeoptiontoleaseanassetatafavorablerateiftherequiredhorizonextendsbeyondTs.Thevendorwill,ofcourse,chargeapremiumfortheexibilitythatitprovidestothedecision-maker. Here,wedesignaleasingoptioncontractthatcanfurthermitigatetheowner'sriskexposure.Specically,(i)wedesignaleasingcontractaccordingtolengthandprice;(ii)priceanoptiontomaketheleaseavailable;and(iii)determinewhetherthiscontractisbenecialtoboththevendor(leasor)anddecision-maker(leasee).Notethatwedesigntheoptioncontractsuchthatadecisionmustbemadeatthecriticaltimeperiod(tc),therstperiodwhereoptimalpoliciesfordifferenthorizonrealizationsdiverge. ConsiderthenetworkrepresentationofthedynamicprogrammingformulationinFigure 5-1 .Toincludealeaseoption,thedecisionsatthecriticaltimeperiodbecometwofold:(i)shouldtheoptioncontractbepurchasedand(ii)shouldtheassetbekeptorreplaced.Iftheoptionisnotpurchased,thenthenetworkisdenedasbefore.However,iftheleaseoptionispurchased,thenanother(parallel)networkistraversedinwhichleasingbecomesaviableoptionafterperiodTs.Notethatthepurchaseoftheoption 63

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contractgivestheownertherighttoexerciseitatalaterdateandleaseanasset;however,theownerisnotrequiredtodoso. Supposethatthedecision-makerpurchasesandexercisesanoptiontoleaseattheendofperiodTsthroughthehorizon.Letxdenotethepriceoftheoption(discountedtoperiod0)andytdenotetheleasingrateforperiodt.Ourgoalistominimizethemaximumregretzforthedecision-maker.Assumingthattheleasor(vendor)requiressomeminimumexpectedprot(netpresentvalue)!,wecansolvefortheleasingcontractvariablesasfollows:Minimizez (5)Subjecttoz minigTs(i)+x+tXi=Ts+1iyt!)]TJ /F4 11.955 Tf 11.95 0 Td[(minjgt(j)t=Ts,...,Tl (5)!x+TlXt=Ts+1P(Tt)t(yt)]TJ /F5 11.955 Tf 11.95 0 Td[() (5)x,yt0t=Ts+1,...,Tl (5)z2R (5) Theobjectivefunction( 5 )minimizesthemaximumregret,whichisdenedinConstraints( 5 ).Thetermintheparenthesesdenesthetotalcostincurredthroughleasingforeachrealizedperiodofthelease,Ts+1throughthehorizon,whileminjgt(j)denestheminimumcostincurredwhenowningtheassetthroughthehorizon.Constraint( 5 )denestheprotforthevendor,accordingtothedecisionvariablesxandyt,asbeinggreaterthan!.Thevalueofcapturestheperiodiccostsofthevendor. Thiscontractdesigncanlowerthedecision-maker'srisk.Thefollowingtheoremquantiesthedecision-maker'sexposure. Theorem5.1. If: gt(j)isnon-decreasingint,(5) 64

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and !TlXt=Ts+1P(Tt)t()]TJ /F7 11.955 Tf 11.96 0 Td[(yt),(5) thereexistsanoptimalleasingcontract(minimizesmaximumregret)wherez=x. Proof. Considertheproblemwherewerelaxthenon-negativityconstraintsonxandyt.ObservethattherelaxedproblemisalinearprogramwithTl)]TJ /F7 11.955 Tf 12.88 0 Td[(Ts+2decisionvariableswiththesamenumberofindependentconstraints.Thisimpliesthatattheoptimalsolution,allconstraintsarebinding.Solvingthesystemoflinearequationsyieldsthefollowingsolution: yt=)]TJ /F8 7.97 Tf 6.58 0 Td[(tminigt(i))]TJ /F4 11.955 Tf 11.96 0 Td[(minigt)]TJ /F12 7.97 Tf 6.59 0 Td[(1(i)t=Ts+1,...,Tl (5)z=x=!)]TJ /F8 7.97 Tf 23.18 15.29 Td[(TlXt=Ts+1P(Tt)t(yt)]TJ /F5 11.955 Tf 11.96 0 Td[() (5) IfCondition( 5 )holds,thecostoftheoptimalreplacementpolicywithouttheleasingoptionisnon-decreasinginthehorizonandhenceallytarenonnegative(fromEquation( 5 )).IfCondition( 5 )holds,itfollowsfromConstraint( 5 )thatxisnonnegative.ThereforeunderConditions( 5 )and( 5 ),thex,ytandz,asdenedbyEquations( 5 )and( 5 )areoptimal(sincetheyalsosatisfytherelaxedconstraints)andz=x. ThesetofconditionsdescribedinTheorem 5.1 arereasonable.TherstconditionholdsiftheO&Mcostsarenonnegativeandthesalvagevalueoftheassetisnon-increasingwithage(TanandHartman,2010),whichiscommonlyassumedinreplacementanalysis.Thesecondconditionsimplyimpliesthattheexpectedprotofthevendorcannotbetoolow. TheimplicationofTheorem 5.1 isthataleasingcontractcanbeconstructedsuchthatallpossibleregretiseliminatedfromtheproblem,withtheexceptionofthecostofthecontract,whichmustbeincurred.However,itisunlikelythattheleaserates,yt,will 65

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varyeachperiodunlesssystematically.Inaddition,thedecision-makermayconsiderinitiatingaleaseatsomeperiodlaterthanTs.Hereweassumethattheleaseratesarexedoncealeaseisinitiated,buttheratecandifferforcontractsofdifferentlengths.Letydenotetheconstantleaserateandkdenotetheperiodinwhichtheassetisrstleased(attheendoftheperiod).Itisassumedthatoncetheassetisleased,itisleasedthroughtheendofthehorizon. Asthevendor'sprotisdenedbythedifferenceinrevenuesreceivedminuscostspaid,observethatthefollowingrelationshipbetweenxandymusthold: x|{z}optionprice=)]TJ /F11 11.955 Tf 11.29 20.44 Td[( TlXt=k+1P(Tt)t!y| {z }expectedleaserevenue+ TlXt=k+1P(Tt)t!| {z }expectedvendorcosts+!|{z}expectedvendorprot.(5) Theoptionpricecanbeviewedasthecombinationofthreeseparatecomponents:expectedleaserevenue;expectedvendorcosts;andexpectedvendorprot.Equation( 5 )impliesthatifand!areconstant,aunitincreaseinxdecreasesybyexactlyPTlt=k+1P(Tt)t. Supposethatthedecision-makerfollowsareplacementpolicyandrj,jk,istheregret(priceoftheoptionnotincluded)incurredifT=j.Recallthatyiistheoptimalvariableleaserateforperiodiandtheleasingoptionisnotexerciseduntiltheendofperiodk.Letadenotetheperiodwiththehighestregret,ra,(minustheoptionprice)andbtheperiodwiththelowestvariableleaserate,yb=mintytwhereytisdenedinTheorem 5.1 .Notethatakandb>k.and!areassumedtobenon-negative.Thefollowingtheorem,Theorem 5.2 ,providestheoptimalcontractparametersundercertainconditions,aspresentedinthetheoremstatementbelow: Theorem5.2. ForareplacementpolicythroughperiodkTs,if: 0
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and !> TlXt=k+1P(Tt)t!(yb+ra)]TJ /F5 11.955 Tf 11.96 0 Td[(),(5) thereexistsanoptimalleasingcontractwherex=!+()]TJ /F7 11.955 Tf 11.96 0 Td[(y) TlXt=k+1P(Tt)t!,y=yb+raandz=x+ra. Proof. NotethatCondition( 5 ),withEquation( 5 ),guaranteesthatx>0.Bydenition,raisthemaximumregretexperiencedinallperiodsthroughk.Asybistheminimumleaserate,itguaranteesthatnoregretisexperiencedinperiodsgreaterthank.(Theoptionpriceisignoredinthesetwostatements.)Therefore,theminimum,maximumregretzthatisachievableisx+ra.Notethatincreasingytoyb+radecreasesxwhilenotincreasingtheregretexperienced,asraisalreadyexperiencedthroughperiodk.Furtherincreasingydecreasesx,butataslowerratethanincreasingtheregretinperiodsafterk,fromCondition( 5 ).Therefore,y=yb+raandz=x+ra.ThedenitionofxfollowsfromEquation( 5 ). IftheconditionsstatedinTheorem 5.2 donothold,theoptionpriceisdriventozero,asinthefollowingcorollary. Corollary5. ForareplacementpolicythroughperiodkTs,if: TlXt=k+1P(Tt)t>1,(5) or TlXt=k+1P(Tt)t!(yb+ra)]TJ /F5 11.955 Tf 11.96 0 Td[(),(5) theoptimaloptionpricex=0. Proof. Whenyyb+ra,themaximumregretoccursatT=a.Whenyyb+ra,themaximumregretoccursatT=b.Henceitfollowsthatthemaximumregretwillalwaysoccurateitherperiodaorperiodb. 67

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GivenCondition( 5 ),theregretatperiodaandbisincreasinginx.Therefore,itisoptimaltosetxatitslowestpossiblevalue,i.e.,x=0. GivenCondition( 5 ),ymustbelessthanorequaltoyb+raforx0tohold.ThisimpliesthatthemaximumregretoccursatT=a.SincethemaximumregretoccursatT=a,themaximumregretisminimizedbysettingxatthelowestpossiblevalue,i.e.,x=0. Underthestatedconditions,Theorem 5.2 providestheoptimaloptionpriceandthecorrespondingleaserateforagivenk-periodreplacementpolicy.Inaddition,italsofollowsfromthetheoremthatthereplacementdecisionsforperiodslessthankshouldbethosethatminimizethemaximumregretacrossperiodsofownership. Figures 5-2 and 5-3 examinethetradeoffinoptionpriceandotherparametersmoreclosely.Whenishigh,thereislittlediscountingonthecostofleasinginthefutureandhencethevendorwillneedtochargeahighoptionprice.ThisisillustratedinFigure 5-2 .Similarly,ifkissmall,thevendorwillneedtochargeahighoptionpricetoaccountforthepossibilityofalonglease.ThisisillustratedinFigure 5-3 .Thereforeinthesesituations,thehighoptionpricethatisrequiredmaynotbejustiedandthedecision-makermaybebetteroffeitherleasingwithoutanoptionortonotleaseatall. Corollary 5 illustratesthattheoptimaloptionpriceiszerowhenthestatedconditionsdonothold.Inthissituation,itmaystillbebenecialforthedecision-makertoleasetheequipment,butthedecision-makerisnotabletofurtherreducehisorherriskbypayinganupfrontpremiumtoreducetheleaserate. Twoissuesremain.First,wedonotknowtheoptimalk.However,wecaneasilyndtheoptimalsolutionforeachkvaluewithTheorem 5.2 andidentifythebestanswerfromthisset.Second,weneedtodetermineappropriate!andvalues.Weassumethattheoptioncontractisattractivetothevendorifitsexpectedprotisnon-negative.Underthisassumption,weset!=0,althoughinpractisesomemarginmayberequired.Thevalueofisusuallyunknowntothedecision-maker,howeverthedecision-makershould 68

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beabletoreasonablyestimatewithitsowncostdata.Inthefollowingexample,weexaminetheoptioncontractdesign. Example9:Optioncontract .AgainconsiderExample 6 .Therearetwopossibleoptionleasecontracts,(i)leasingforperiods7and8(k=6)and(ii)leasingforperiod8only(k=7).Forthisexample,!isassumedtobe0. For(i),thepolicywhichminimizesthemaximumregretfora6-periodproblemistosalvagetheinitialassetattheendofperiod6suchthatr6=$0.Theoptimalvariableleaseratesarey7=$55,540andy8=$39,080.SincePTlt=k+1P(Tt)t=(0.8)(1.1))]TJ /F12 7.97 Tf 6.58 0 Td[(7+(0.6)(1.1))]TJ /F12 7.97 Tf 6.59 0 Td[(8=0.690<1,Condition( 5 )holds.If>minfy7,y8g+r6=$39,080+$0=$39,080,Condition( 5 )holdsandtheoptimaloptionpriceandleaserateare0.690)]TJ /F4 11.955 Tf 12.13 0 Td[($26,982and$39,080respectively.Sincer6=$0,themaximumregretisalso0.690)]TJ /F4 11.955 Tf 12.14 0 Td[($26,982.If$39,080,Condition( 5 )holdsandtheoptimaloptionprice,leaserateandmaximumregretare$0,and$0respectively.ThesearesummarizedinTable 5-5 For(ii),thepolicywhichminimizesthemaximumregretfora7-periodproblemistosalvagetheinitialassetattheendofthehorizon(period7)suchthatr6=$0andr7=$1,746.Theoptimalvariableleaserateforperiod8is$39,080.SincePTlt=k+1P(Tt)t=(0.6)(1.1))]TJ /F12 7.97 Tf 6.58 0 Td[(8=.280<1,Condition( 5 )holds.If>y8+maxfr6,r7g=$39,080+$1,746=$40,825,Condition( 5 )holdsandtheoptimaloptionpriceandleaserateare0.280)]TJ /F4 11.955 Tf 12.76 0 Td[($11,427and$40,825respectively.Sincemaxfr6,r7g=$1,746,themaximumregretisalso0.280)]TJ /F4 11.955 Tf 11.27 0 Td[($9,681.If$40,825,Condition( 5 )holdsandtheoptimaloptionprice,leaserateandmaximumregretare$0,and$1,746respectively.ThesearesummarizedinTable 5-6 When>$42,142,policy(ii)resultsinasmallermaximumregretthanpolicy(i).Notethattheeconomiclifeoftheassetwas11yearswhichwasdenedfromanannualequivalentcostof$45,494.Henceitislikelythatisgoingtobegreaterthan$42,142andpolicy(ii)willbetheoptimalpolicy.2 69

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Table5-1. Conditionalprobabilitiesofthevariousscenarios. tP(T=tjTt) 60.4070.6781.00 Table5-2. Boundaryconditionsatt=8. if8(i)if8(i)if8(i) 1-$120,0004-$61,4407-$31,4572-$96,0005-$49,1528-$25,1663-$76,8006-$39,32216-$4,222 Table5-3. Boundaryconditionsatt=6. if6(i)if6(i)if6(i) 1-$40,4044$1037$18,0982-$24,6625$8,7548$21,1013-$11,4466$14,09410$25,043 Table5-4. Discountedcoststhroughperiods6,7and8periodswithanassetofagei. ig6(i)g7(i)g8(i) 1$174,902$202,450$228,1522$176,773$201,843$226,9423$178,093$200,866$223,7454$179,515$200,161$220,9285$181,530$200,243$219,0056$184,456$201,494$218,3927$204,146$219,3948$222,18514$171,66115$201,90716$231,876 Table5-5. x,yandzvaluesforpolicy(i). xyz 39,08000>39,0800.690)]TJ /F4 11.955 Tf 11.95 0 Td[(26,98239,0800.690)]TJ /F4 11.955 Tf 11.96 0 Td[(26,982 Table5-6. x,yandzvaluesforpolicy(ii). xyz 40,82501,746>40,8250.280)]TJ /F4 11.955 Tf 11.95 0 Td[(11,42740,8250.280)]TJ /F4 11.955 Tf 11.96 0 Td[(9,681 70

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Table5-7. Optimalpolicies:P(T=6)=0.6,P(T=7)=0.2,P(T=8)=0.2andn=4. RatesEcon.MinE[Cost]PolicyMinMaxRegretPolicyO&MSalvageLifePolicyE[Cost]MRPolicyE[Cost]MR 0.30.35(1,7)$247,668$2,889(1,7)$247,668$2,8890.30.24(0,3,7)$202,757$1,994(0,4)$202,825$8210.30.11(1,2,...,7)$111,007$0(1,2,...,7)$111,007$00.20.37(2)$213,548$0(2)$213,548$00.20.26(1)$182,504$2,425(2)$182,545$9820.20.11(1,2,...,7)$111,007$0(1,2,...,7)$111,007$00.10.312(-)$129,836$0(-)$129,836$00.10.211(-)$124,030$0(-)$124,030$00.10.17(2)$99,997$0(2)$99,997$0 71

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Table5-8. Optimalpolicies:P(T=6)=0.2,P(T=7)=0.2,P(T=8)=0.6andn=4. RatesEcon.MinE[Cost]PolicyMinMaxRegretPolicyO&MSalvageLifePolicyE[Cost]MRPolicyE[Cost]MR 0.30.35(2,7)$274,872$5,791(1,7)$275,165$2,8890.30.24(0,4)$224,389$821(0,4)$224,389$8210.30.11(1,2,...,7)$128,641$0(1,2,...,7)$128,641$00.20.37(2)$231,736$0(2)$231,736$00.20.26(2)$202,429$982(2)$202,429$9820.20.11(1,2,...,7)$128,641$0(1,2,...,7)$128,641$00.10.312(-)$146,377$0(-)$146,377$00.10.211(-)$141,715$0(-)$141,715$00.10.17(2)$116,358$0(2)$116,358$0 72

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Table5-9. Optimalpolicies:P(T=6)=0.6,P(T=7)=0.2,P(T=8)=0.2andn=8. RatesEcon.MinE[Cost]PolicyMinMaxRegretPolicyO&MSalvageLifePolicyE[Cost]MRPolicyE[Cost]MR 0.30.35(0,6)$281,870$5,499(0,5)$283,674$5,2050.30.24(0,3,7)$239,031$1,994(0,4)$239,099$8210.30.11(1,2,...,7)$144,852$0(1,2,...,7)$144,852$00.20.37(0)$250,336$0(0)$250,336$00.20.26(0,7)$222,678$1,330(0,7)$222,678$1,3300.20.11(1,2,...,7)$144,852$0(1,2,...,7)$144,852$00.10.312(-)$192,459$0(-)$192,459$00.10.211(6)$188,875$8,550(4)$190,422$5,1120.10.17(0)$135,044$0(0)$135,044$0 73

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Table5-10. Optimalpolicies:P(T=6)=0.2,P(T=7)=0.2,P(T=8)=0.6andn=8. RatesEcon.MinE[Cost]PolicyMinMaxRegretPolicyO&MSalvageLifePolicyE[Cost]MRPolicyE[Cost]MR 0.30.35(0,5)$306,515$5,205(0,5)$306,515$5,2050.30.24(0,4)$260,663$821(0,4)$260,663$8210.30.11(1,2,...,7)$162,486$0(1,2,...,7)$162,486$00.20.37(0)$271,930$0(0)$271,930$00.20.26(0,6)$245,211$1,752(0,7)$245,489$1,3300.20.11(1,2,...,7)$162,486$0(1,2,...,7)$162,486$00.10.312(-)$215,980$0(-)$215,980$00.10.211(2)$206,987$7,855(4)$208,084$5,1120.10.17(0)$153,841$0(0)$153,841$0 74

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Figure5-1. Visualizationofdynamicprogrammingformulationwithn=8,Ts=6andTl=8. 75

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Figure5-2. Effectofthediscountrateontheoptionprice. 76

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Figure5-3. Effectofthenumberofleaseperiodsontheoptionprice. 77

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CHAPTER6OPTIMALPRODUCTLINEFORCONSUMERSTHATEXPERIENCEREGRET Individualshavedifferentpreferencesandmanywillagreethatitisbettertohavemorechoicesthanless.Forexample,Nikeallowsitscustomerstodesigntheirownshoes,DellsellscustombuiltlaptopsandSubwayletsitscustomersdecidewhatgoesintoasandwich.Empiricalstudiesontherelationshipbetweenproductvarietyandrmprotssuggestthatgreaterproductvarietycanleadtohigherprots(see,forexampleKotlerandKeller(2006)andNestessineandTaylor(2007)),especiallywhenproduction,supplychainandmarketingstrategiesarealigned(seeKekreandSrinivasan(1990),Sazadehetal.(1996),BerryandCooper(1999)andRandallandUlrich(2001)).For-protcompaniesarenottheonlyorganizationsthatrecognizetheneedtooffervariety.TheUnitedStatesgovernmentallowsemployerstoautomaticallyenrollemployeesin401(k)pensionplans,butemployeescanselectdifferentfunds,orevenoptoutoftheplancompletely.TheSingaporegovernmentrequireseverycitizenandpermanentresidenttobeenrolledinasavingsplan,butallowsindividualstodeterminehowtheirsavingsareinvested. Theproblemofdetermininganoptimalproductlinehasbeenstudiedextensivelyintheoptimizationliterature.Anoptimalproductlineisobtainedbysolvinganoptimizationproblemwherethebenetsandcostsofintroducingvarietyaremodeledanalytically.Thebenetsareoftenmeasuredintermsofwelfareorrevenue,whichisbasedontheutilityandpriceoftheproduct,respectively.Itisgenerallyassumedthattheutilityofaproductisknownorfollowssomeknownprobabilitydistribution.Thepriceofaproductiseitherxed(see,forexample,VanRyzinandMahajan(1999))orobtainedendogenously(see,forexample,AydinandPorteus(2008)).Productionandinventoryholdingcostsaretwocostcomponentsthatarecommonlyconsideredinthesemodels.DobsonandKalish(1993)consideredamodelwithxedandvariableproductioncosts.Desaietal.(2001)studiedthetradeoffsbetweenlowermanufacturingcostsandlower 78

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sellingpricesthatresultfromcommonalitieswithinaproductline.NestessineandTaylor(2007)foundthatmanufacturerswithinefcientproductiontechnologiescanhavelowerproductioncoststhanthosewithmoreefcientproductionsystemsasitisoptimalfortheformertoofferlowerqualityproducts.AlptekinogluandCorbett(2008)analyzedthecompetitionbetweenamasscustomizerandamassproducerwithinagame-theoreticcontextandfoundthatamassproducercan,undercertainscenarios,coexistwithamasscustomizerthathasamoreefcientproductionsystem.Koketal.(2008)providesanexcellentreviewoftheliteraturethatconsiderstheinventoryaspectoftheproblem. Overthelasttwodecades,researchershaveproposedvariousconsumerchoicemodels,includingthemultinomiallogit(MLN)model,exogenousdemandmodelandlocationalchoicemodel.IntheMLNmodel,theutilityofaproducttoaspecicconsumerisdecomposedintodeterministicandstochasticcomponents.VanRyzinandMahajan(1999)showedthattheoptimalassortmentconsistsofthemostpopularproductsunderstaticchoiceassumptions(i.e.,theconsumerwillonlypurchasethemostattractiveproductfromtheassortmentifitisavailable)andconsideredamodelwheretheconsumerpurchasesthebestavailablealternativeinMahajanandvanRyzin(2001).Cachonetal.(2005)consideredamodelwheretheconsumermightchoosetoexploreotherstores,evenwhenanacceptableproductisavailable.Whensimilarproductsareavailableindifferentretailstores,thevalueofexploringdecreases,resultinginwiderproductassortmentswhenconsumersearchconsiderationsareincluded.SmithandAgrawal(2000)consideredanexogenousdemandmodelwheretheconsumerpurchasesanalternativeproductaccordingtosomesubstitutionprobabilitydistributioniftheirbestchoiceproductisnotavailable.KokandFisher(2007)illustratedhowtheprobabilitiesthataconsumerwillpurchaseasubstituteproductcanbeestimatedfromsalesdata.Inthelocationalchoicemodel,consumersselectproductsthatareclosesttohisorherpreference,asdenedonanattributespace(e.g.,percentageoffatcontentinmilk).Researchersthathaveconsideredlocational 79

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choicemodelsincludedeGroote(1994),Chenetal.(1998),GaurandHonhon(2006),AlptekinogluandCorbett(2008)andKuksovandVillas-Boas(2010).ThesemodelsarediscussedingreaterdetailinKoketal.(2008).Recently,Honhonetal.(2010)proposedachoicemodelwhichgeneralizestheMLNandlocationalchoicemodels. However,productionandinventorycostsarenottheonlyincentivesforlimitingthenumberofproductsoffered.Researchershavehighlightedthatindividualsmaybehappierwhenchoicesarelimited.Forexample,TverskyandShar(1992)foundthathavingalargechoicesetcausesindividualstodefertheirdecisionsorchoosedefaultoptions.Intwoseparateexperiments,IyengarandLepper(2000)foundthatpeopleweremorelikelytopurchasegourmetjamsandchocolateswhenalimitedarrayofchoicesispresented.Inathirdexperiment,theyfoundthatstudentsweremorelikelytoundertakeoptionalclassessayassignmentswhentheyweregiven6,ratherthan30,potentialessaytopicsfromwhichtochoose.Furthermore,individualswhowerepresentedwithalimitedarrayofchoicesreportedgreatersatisfactionwiththeirchoices.Ithasalsobeenobservedthatpeopleareunhappy,indecisiveormayevennotchooseatallwhenfacedwithachoiceamongseveralalternatives(seeTversky(1972),Sharetal.(1993),RedelmeierandShar(1995)andBrenneretal.(1999)).Forfurtherresearchonproductvarietyandconsumersatisfaction,theinterestedreaderisreferredtoChernev(2003),Schwartz(2004)andScheibehenne(2008). Researchinpsychologyandeconomicssuggeststhatthesatisfactionassociatedwithachoiceisinuencedbyfeelingsofregret(ConnollyandZeelenberg,2002).Inaddition,choicesmayalsobeinuencedbytheregretthatoneanticipates(seeZeelenberg(1999)andEngelbrecht-WiggansandKatok(2009)).Theeffectofregretonconsumerdecisionshasbeenstudiedinavarietyofsettings.BraunandMuermann(2004)examinedtheoptimalinsurancepurchasesofregret-averseconsumers.IronsandHepburn(2007)wereinterestedintheoptimalchoicesetforaspecicindividualwithregretconsiderationsunderthreesearchscenarios:(i)apredeterminednumberof 80

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choicesareexamined;(ii)thedecision-makerdynamicallydecideswhethertocontinuethesearchorselectfromthesetofexaminedchoices;and(iii)afterachoicehasbeenexamined,thedecision-makereitherselectsthatchoiceandstopsorrejectsitandproceedstoexaminethenextchoice(i.e.,norecall).Syametal.(2008)consideredaproblemwhereaconsumerhastodecidebetweenpurchasingastandardorcustomproduct.Engelbrecht-WiggansandKatok(2008)lookedathowfeedbackinformationaffectsbiddingbehavior.Recently,NasiryandPopescu(2011)studiedtheeffectsofanticipatedregretwithinanadvancesellingcontext. Wesolvefortheoptimalproductlineusingamodelthataccountsfortheregretthatisanticipatedandexperienced.Unlikepreviousresearch,thesatisfactionassociatedwithachoiceisdependentontheentireproductline,ratherthantheselectedproductalone.Inourmodel,consumersareuncertainaboutthetrueutilityofaproduct.Villas-Boas(2009)consideredasimilarproblemwheretheuncertaintiescanberesolvedbyperformingasearchorevaluationandfoundthattheoptimalnumberofproductstoofferisdecreasinginevaluationcosts.KuksovandVillas-Boas(2010)highlightedthatsearchandevaluationcostscancauseconsumerstorefrainfromsearchingorevaluatingwhenthenumberofalternativesarehighorlow.Weconsideraproblemwheretheuncertaintiesintheutilityoftheproductsareunresolvedatthetimeofpurchase.Thisistrueformanyproducts,rangingfromconsumergoodslikecarsandlaptopstoserviceslikemedicaltreatmentandnancialplanning.Tothebestofourknowledge,theproblemofndingtheoptimalproductlineunderregretconsiderationshasneverbeenexplored.Inpractice,theanticipatedregretandtheregretexperiencedmaydiffer.Forexample,anoptimisticinvestormayunderestimatetheregretassociatedwithabadinvestment.Forsimplicity,weassumethatthedifferencebetweenthetwoisnegligible.Thisassumptionisreasonablesinceindividualsareoftenabletogivereasonablyaccuratepredictionsoftheirfeelings(LoewnsteinandSchkade,1999). 81

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Thischapterproceedsasfollows.First,wepresentourmodel.Next,westudytheoptimalnumberofproductstoofferundervariouslimitingconditions.Inaddition,werunvariousnumericalexperimentsandderiveatightupperboundontheoptimalnumberofproductstooffer. 6.1ModelDescription ConsideramarketwithNsegments,whereeachsegmentisdenedbyauniquecustomertype.Letjdenotethemarketshareofsegmentjsuchthatj=1 Nforj=1,2,...,Nifallmarketsegmentsareequallylarge.Weconsideraproblemwheretheutilityofaproductisuncertainandonlyresolvedafterpurchaseaccordingtosomescenarios.LetSdenotethesetofpossiblescenarios,psdenotetheprobabilityofscenariosoccurringanduijsdenotetheutilityofchoiceiforsegmentjunderscenarios.Withoutlossofgenerality,weassumethatuijsispositive.Inaddition,weassumethatuijsisnite. LetxandXdenoteaproductandthesetofproductsthatthermcanoffer,respectively.Weassumethatanon-negativenitexedcostciisincurredifproductiismadeavailable.Thermdecidesonasetofproducts,X,tobemadeavailable(i.e.,productline)anddenotesthesetofpossibleproductlines.Letrijsdenotetheregretexperiencedbysegmentjunderscenarioswhenchoicei2ischosen: rijs=maxx2uxjs)]TJ /F7 11.955 Tf 11.96 0 Td[(uijs,(6) Notethatrijsis,bydenition,non-negative.Denenetsatisfactionastheweightedcombinationoftheutilityreceivedandtheregretexperienced: vijs=uijs)]TJ /F5 11.955 Tf 11.96 0 Td[(jrijs,(6) wherejissomenon-negativeconstantthatrepresentstheregretaversionofsegmentjandvijsisthenetsatisfactionofsegmentjwithchoiceiunderscenariosandproductline.Theweightedcombinationoftheutilityreceivedandtheregretexperienced 82

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hasbeenreferredtoastheagent'sutilityandnetutilityinIronsandHepburn(2007)andSyametal.(2008),respectively.Webelievethatthetermnetsatisfactionismoreappropriateinourcontext. 6.1.1ConsumerChoiceModel Clearly,thechoicesofconsumersaredependentontheproductsthatareavailableintheproductline.Let(j)denotetheproductthatisselectedbysegmentjwhenproductlineisoffered.Weassumethatconsumersarerationalandselectproductsthatmaximizetheirindividualexpectednetsatisfaction: (j)=argmaxi2Xspsvijs.(6) Syametal.(2008)proposedasimilarchoicemodelinstudyingconsumerpreferencesforstandardizedandcustomizedproducts.AsimilarchoicemodelwasalsoadoptedbyNasiryandPopescu(2011)instudyingthepurchasingbehaviorofrationalconsumersthatexperienceregretunderadvanceselling.Finding(j)withEquation( 6 )ischallengingbecauseitinvolvestheregretthatisexperiencedbyconsumers. Theorem6.1. GivenEquation( 6 ),(j)=argmaxi2Pspsuijs. Proof. (j)=argmaxi2Xspsvijs=argmaxi2Xspsuijs)]TJ /F5 11.955 Tf 11.95 0 Td[(jrijs=argmaxi2Xshpsuijs)]TJ /F5 11.955 Tf 11.96 0 Td[(jmaxx2uxjs)]TJ /F7 11.955 Tf 11.95 0 Td[(uijsi=argmaxi2Xshps(1+j)uijs)]TJ /F5 11.955 Tf 11.95 0 Td[(jmaxx2uxjsi=argmaxi2Xspsuijs Thelastequalityfollowsfromthefactthatargmaxi2Psps(1+j)uijs=argmaxi2Pspsuijsandjmaxx2uxjsisindependentofi. 83

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Theorem 6.1 statesthatrationalconsumersthatmaximizeexpectednetsatisfactionselectproductsthatresultinthehighestexpectedutilityfromtheproductline.Althoughanticipatedregretcancomplicatethedecisionprocess(i.e.,decisionmakeraccountingfortheregretthatisanticipated),itdoesnotaffectthenaldecisionofarationalconsumerthatisdescribedbyourchoicemodel(i.e.,Equation( 6 )). Inourmodel,weassumethatallconsumerswillpurchaseaproduct.Thisistrueifallproductsresultinpositiveexpectedutilityforeachconsumer.Ifthatisnottrue,aconsumermaychoosetonotmakeanypurchasewhennoneoftheproductsaresatisfactory(i.e.,allproductsresultinnegativeexpectedutility).Toallowforthis,weincludeadummyproductwithzeroxedcostsandzeroutilityforallconsumersacrossallscenariosinourproductsetXwhennecessary.Notethattheproblemcanalwaysbescaledbyapositivefactor.Hence,itdoesnotviolateourassumptionthatuijsispositive. 6.1.2ProblemFormulation Weareinterestedinndingaproductlinethatmaximizestheexpectednetsatisfactionofthemarket,whileaccountingforthexedcostsofintroducingproducts:=argmax2f(), wheref()istheexpectedperformanceofproductlinedenedas:f()=XjXsjps)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(u(j)js)]TJ /F5 11.955 Tf 11.96 0 Td[(jr(j)js)]TJ /F11 11.955 Tf 11.95 11.36 Td[(Xi2cixi. Whenj=0forallj,theproblemreducestoanuncapacitatedfacilitylocationproblem,whichisNP-hard(Cornuejolsetal.1990).Hence,thegeneralproblemisalsoNP-hard. Wesolvethisproblembyformulatingitasabinarylinearprogram.First,wedeneacoupleofnotations.Letwjsdenotethemaximumutilityofsegmentjunderscenariosandproductline:wjs=maxi2uijs. 84

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Thenetsatisfactionofsegmentjunderscenariosandproductline(i.e.,Equation( 6 ))canbere-expressedasfollows:vijs=(1+)uijs)]TJ /F5 11.955 Tf 11.95 0 Td[(jwjs. Notethatwjsisindependentoftheconsumer'schoice.Theoptimalproductlinecanbeobtainedbysolvingthefollowingbinarylinearprogrammingformulationfortheproductlinedesignproblemwithregretconsiderations,PLD-R:maxXjXsjps"(1+j) Xiuijsyij!)]TJ /F5 11.955 Tf 11.95 0 Td[(jwjs#)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xicixi (6)s.t.wjsuijsxi8i,j,s (6)yijxi8i,j (6)Xiyij18j (6)yij08i,j (6)xi2f0,1g8i (6)wjs2R8j,s (6) wherethedecisionvariablesxiandyijdenoteifchoiceiisavailableandifchoiceiisselectedbysegmentj,respectively.Theunrestricteddecisionvariableswjs,denedbyConstraint( 6 ),denotethemaximumutilityofsegmentjunderscenarios.Constraint( 6 )ensuresthatonlyavailablechoicesareselected.Constraint( 6 )ensuresthatxiarebinary.Sinceyijarevariablesthatindicateifachoiceisselectedbyasegment,theyshouldalsobebinary.Theorem 6.2 highlightsthatConstraints( 6 )and( 6 )collectivelyensurethatthereisanoptimalsolutionwhereeachconsumermakesexactlyoneselection. Theorem6.2. ThereexistsanoptimalsolutiontoPLD-Rwhereyij2f0,1gforalliandj. 85

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Proof. Weprovethisbycontradiction.AssumetheredoesnotexistanoptimalsolutiontoPLD-Rwhereyij2f0,1gforalliandj.First,wenotethatitfollowsfromConstraints( 6 )and( 6 )that0yij1.Itfollowsfromourassumptionthatthereexistsi0andj0suchthat00.Settingyi0j0=1andyi0j=0forallj6=j0maintainsoptimality(sinceallpositiveyvariableshavethesamecoefcient)andfeasibility(i.e.,doesnotviolateanyconstraintsinPLD-R).Thisargumentappliestoalli,contradictingtheassumptionthattheredoesnotexistanoptimalsolutionwhereyij2f0,1gforalliandj. Theorem 6.2 highlightsthatthereexistsanoptimalsolutionwhereyijisbinaryforalliandj.Inaddition,itfollowsfromtheproofofTheorem 6.2 thatanoptimalsolutionwhereyijisbinaryforalliandjcanbeeasilyobtainedbyarbitrarilysettinganon-zeroyvariabletotakethevalue1andtheothernon-zeroyvariablesassociatedwiththesamechoicetotakethevalue0. 6.2OptimalNumberofProductstoOffer Inthissection,westudytherelationshipsbetweenvariousproblemparametersandtheoptimalnumberofproductstooffer.Letjjdenotethenumberofproductsinanoptimalproductline.Whenmultipleoptimalsolutionsexist,letjjdenotethenumberofproductsinthesmallestoptimalproductline.Sinceanon-negativexedcostisincurredwhenaproductisintroducedintotheproductline,jjislikelytobesmallerwhenxedcostsarehigh.Webeginthissectionbyshowingthatjjisnon-increasingwhenthexedcostofintroducingproductsincreasesuniformly. Theorem6.3. jjisnon-increasingin,whereisauniformincreaseinthexedcostofintroducingeachproduct. Proof. Let>jjdenotethesetofproductlinesthatarestrictlylargerthanjj(i.e.,>jj=f:jj>jjg).Weprovethistheorembyshowingthatf()>f()forall 86

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2>jjwheneachciisincreasedby.Sinceisoptimalwhenxedcostsareci:XjXsjpsv(j)js)]TJ /F11 11.955 Tf 12.21 11.36 Td[(Xi2ciXjXsjpsv(j)js)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xi2ci,82XjXsjpsv(j)js)]TJ /F11 11.955 Tf 12.21 11.35 Td[(Xi2ciXjXsjpsv(j)js)]TJ /F11 11.955 Tf 11.96 11.35 Td[(Xi2ci,82>jjXjXsjpsv(j)js)]TJ /F11 11.955 Tf 12.21 11.36 Td[(Xi2ci)-222(jj>XjXsjpsv(j)js)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xi2ci)-222(jj,82>jjf()>f(),82>jj, Thethirdinequalityfollowsfromthefactthatjjjj. Theorem 6.3 statesthatjjisnon-increasingwhenthexedcostofintroducingaproductincreasesuniformly,whichsuggeststhatjjislikelytobesmallwhenxedcostsarehigh.WenotethattheresultofTheorem 6.3 isgeneral,holdingforallprobleminstances.Intuitively,onewouldexpectasimilarrelationshipbetweenjjandtheregretaversionofconsumers(i.e.,smallerjjwhenjarehigh).However,thatisnottrue,evenwhenallconsumersaredescribedbythesameregretaversionparameter.ThisisillustratedinExample 10 Example10:jjincreaseswith .ConsideraproblemwhereN=3,jXj=5,jSj=3,c1=c2=c3=c4=c5=0,p1=p2=p3=1 3and1=2=3=1 3.TherespectiveuijsvaluesarelistedinTable 6-1 .Giventhisinformation,wesolvePLD-R. When=0,productlinef4,5gisoptimalandf(f4,5g)=4.73.When=0.2,productlinef1,2,3gisoptimalandf(f1,2,3g)=4.6.2 Example 10 highlightsthatjjcanincreasewhenconsumersbecomemoreregretaverse.Thisoccurswhenasmallsetofriskyproducts(i.e.,highlyvariableoutcomes)isappealingtothemarket.However,theseproductsresultinhighregretandalargersetoflessriskyproducts(forexample,astandardproductwithminormodicationsfordifferentcustomertypes)canresultingreatermarketsatisfactionwhenregretaversionincreases.ThisisdiscussedingreaterdetailinSection 7.2 87

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Althoughjjcanincreasewhenregretaversionincreases,onewouldexpectjjtobenon-increasinginformostproblems.Next,welookathowtheexpectedoptimalnumberofproductstooffer,E[jj],varieswith,N,SandXbygeneratingaseriesofdifferentproblemsinthefollowingmanner: EachuijsfollowsaprobabilitydistributionUwithnitemeananduppersupportUmax.Inaddition,eachuijsisindependentandidenticallydistributed. ps=qs Psqs,whereeachqsispositiveandfollowsaprobabilitydistributionQwithnitemean.Inaddition,eachqsisindependentandidenticallydistributed. j=aj Psaj,whereeachajispositiveandfollowsaprobabilitydistributionAwithnitemean.Inaddition,eachajisindependentandidenticallydistributed. uijs,qsandjareindependentlydistributedforalli,jands. First,weshowthatofferingasingleproductcanbeoptimalundervariouslimitingcases.Next,wepresentourexperimentalresultsforvarioussetsofproblemparameters.Third,wecomputeatightupperboundonE[jj]. 6.2.1LimitingCases Inthissection,westudyvariouslimitingcases.Inparticular,weshowthatofferingasingleproductislikelytobeoptimalwhen,jSjorjXjapproachesinnity.Furthermore,wehighlightthattheexpectedoptimalnumberofproductstoofferundertheseconditionsis1. Theorem6.4. lim!1jj=1. Proof. Thisisaproofbycontradiction.Assumethatlim!1jj>1.Thisimpliesthatlim!1f()>lim!1f()forall2j1j,wherej1j=f:jj=1g.Notethatlim!1f()isniteforall2j1j.Thisfollowsfromthefactthatf()isindependentofforall2j1janduijs,jandciarenite.Sincelim!1f()>lim!1f()forall2j1jandlim!1f()isniteforall2j1j,lim!1f()isboundedfrombelowbyaniteconstant.Thisimpliesthatrijs=0foralli,jandsbecauselim!1f()=ifsomerijs>0.ItfollowsfromEquation( 6 )thatrijs=0foralli,jandsifandonly 88

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ifuijs=ukjsforallj2N,s2Sandi,k2.Sinceallchoicesinresultinthesameutilityacrossallscenariosforallsegmentsandciisnon-negative,thereexistsa12j1jsuchthatf(1)f(2)forall22r=0=f:rijs=0,8i,j,sg,contradictingtheassumptionthatjj>1. Theorem 6.4 statesthatitisoptimaltoofferasingleproductwhenapproachesinnity.Thisimpliesthatthermshouldofferasingleproductwhenconsumersareheavilyinuencedbyfeelingsofregret.Thisisbecausethelikelihoodofexperiencingregretincreaseswiththenumberofproductsoffered.ItfollowsfromTheorem 6.4 thatE[jj]=1asapproaches1. Corollary6. lim!1E[jj]=1. Proof. FollowsimmediatelyfromTheorem 6.4 Next,weshowthattherealmostsurelyexistsanoptimalproductlinethatconsistsofasingleproductwhenjSjapproaches1. Theorem6.5. limjSj!1P(jj=1)=1. Proof. TheexpectedutilityofsegmentjwhenchoiceiischosenisPspsuijs:limjSj!1Xspsuijs=limjSj!1Psqsuijs Psqs=limjSj!1Psqsuijs limjSj!1Psqsp=limjSj!1E[QU] jSj limjSj!1E[Q] jSj(bythelawoflargenumbers)=E[QU] E[Q]=E[Q]E[U] E[Q](bytheindependenceofQandU)=E[U], where=pdenotesequalitywithprobability1. 89

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Sinceeachpsandeachuijsareindependentandidenticallydistributed,allchoicesresultinthesameexpectedutility(i.e.,E[U])foreachsegmentandaproductlinethatonlyconsistsoftheproductwiththelowestxedcostisoptimalwithprobability1whenjSj!1. Theorem 6.5 statesthattherealmostsurelyexistsanoptimalproductlinethatconsistsofasingleproductwhenjSj!1.WhenjSjishigh,thereishighuncertaintyinthetrueutilityofeachproduct.Underthiscondition,decisionmakersarelikelytobeindifferentbetweenthechoicesandhence,itisoptimaltoofferasingleproduct.ItfollowsfromTheorem 6.5 thatE[jj]=1asjSjapproaches1. Corollary7. limjSj!1E[jj]=1. Proof. FollowsimmediatelyfromTheorem 6.5 Theorem 6.6 statesthattherealsoalmostsurelyexistsanoptimalproductlinethatconsistsofasingleproductwhenjXj!1. Theorem6.6. IfP(U=Umax)>0,limjXj!1P(jj=1)=1. Proof. Considerachoicei0whereui0js=Umaxforalljands.SinceP(U=Umax)>0andeachuijsisindependentandidenticallydistributed,thereisapositiveprobabilitythati02X.Hence,limjXj!1P(i02X)=1.Sincefi0gisoptimalandlimjXj!1P(i02X)=1,limjXj!1P(jj=1)=1. Aproductlineconsistingofasingleproductthatachievesmaximumutilityforallconsumersacrossallscenariosisclearlyoptimal.Whenthenumberofproductsthatarmcanofferisinnitelylarge,therealmostsurelyexistssuchaproductwithinthesetofproductsthatthermcanoffer.Therefore,therealmostsurelyexistsanoptimalproductlinethatconsistsofasingleproductwhenjXj!1.TheresultofTheorem 6.6 isbasedontheassumptionthateachproducthasapositiveprobabilityofachievingmaximum 90

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utility,whichisreasonableforcomputer-basedsimulations.ItfollowsfromTheorem 6.6 thatE[jj]=1asjXjapproaches1. Corollary8. IfP(U=Umax)>0,limjXj!1E[jj]=1. Proof. FollowsimmediatelyfromTheorem 6.6 6.2.2ExperimentalResults Weperformourexperimentsbygeneratingeachuijsfromauniformdistributionwithminimumandmaximumvalues,5000and10000,respectively.Inordertoobtainalargerrangeofjjvalues,westudyproblemswherexedcostsarelow(sincejjislikelytobe1whenxedcostsarehigh).However,multiplesolutionsarelikelytoexistwhenci=0foralli.Inourexperiments,wesetci=0.01forallitoreducethenumberofprobleminstanceswithmultipleoptimalsolutions.QandAfollowadiscreteuniformdistributionwitharangeof[1,32767]and[10,20],respectively.Thelargerangeofvaluesthatqscantakeallowsforthemodelingofimprobablescenarioswhileajtakesasmallrangeofvaluessincetheeffectsofverysmallmarketsegmentscanusuallybeignored.Weconsider3levelsforeachparameterthatwestudy(i.e.,,jSj,NandjXj)andsolve20instanceswitheachproblemparameterconguration.TheproblemsweresolvedusingCPLEX12.1ona3.40GHzIntelPentiumCPUwith2GBRAM.ThesolutiontimewasthegreatestwhenN=jXj=jSj=30and=0.3,averaging51.6minutesperprobleminstance.Estimatesoftheexpectedoptimalnumberofproductstooffer,^E[jj],areobtainedbycomputingtheaveragejjof20probleminstancesandarelistedinTable 6-2 Theexperimentshighlightthatjjisgenerallynon-increasingwithandjSj.Asincreases,theeffectofregretisampliedandasmallerproductlineismoreattractive.AsjSjincreases,thereisgreateruncertaintyinthetrueutilityofaproduct,whichgenerallyresultsinanincreaseinthelikelihoodofexperiencingregret.ThereforeasmallproductlineisdesirablewhenjSjislarge. 91

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Theexperimentsalsoindicatethatjjisgenerallynon-decreasingwithNwhenandjSjaresmall.Forexample,^E[jj]=8.45,13.60and16.96undertheproblemsettings(=0.1,jSj=10,N=10,jXj=30),(=0.1,jSj=10,N=20,jXj=30)and(=0.1,jSj=10,N=30,jXj=30),respectively.Thisresultisintuitive.Whenthereislittleuncertaintyintheutilityofproductsandconsumersarenotinuencedbyregret,itisoptimaltoofferalargeproductlinetoadiversemarket.Thissupportstheconventionalbeliefthatconsumersarebetteroffwithmorechoices.However,therelationshipbetweenjjandNisreversedwhenandjSjarelarge.Forexample,^E[jj]=2.10,1.10and1.00undertheproblemsettings(=0.5,jSj=20,N=10,jXj=30),(=0.5,jSj=20,N=20,jXj=30)and(=0.5,jSj=20,N=30,jXj=30),respectively.WhenandjSjaresmall,theeffectsofregretaresmallandhencethebenets(i.e.,higherutility)ofhavingalargeproductlineoutweighthecosts(i.e.,higherregret).However,theeffectsofregretarepronouncedwhenandjSjarehigh.Undertheseconditions,havingalargeproductlineincreasesconsumerregretanditisbetterforthermtoofferasmallnumberofproductsbyfocusingonselectedmarketsegments. TherelationshipbetweenjjandjXjisdrivenbytwoconictingforces.WhenjXjishigh,thereisahigherchancethattherewillbeaproductthatiscustomizedforaparticularsegment.However,thereisalsoahigherchancethattherewillbeaproductthatisappealingtoawideportionofthemarket.Theformersuggeststhatjjisnon-decreasingwithjXj,whilethelattersuggeststhatjjisnon-increasingwithjXj.Ourexperimentalresultsindicatethatjjisgenerallynon-decreasingwithjXjforsmallproblems.However,Theorem 6.6 highlightsthatjjisalmostsurely1whenjXjisverylarge. 92

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6.2.3UpperBoundonE[jj] Inthissection,weprovideatightupperboundforE[jj].WebeginbyderivingarecursiveexpressionforE[jj]whentherearenoxedcosts(i.e.,ci=0foralli)andregretconsiderations(i.e.,j=0forallj). Theorem6.7. Ifci=0foralliandj=0forallj:E[jj]=minfjXj,NgXn=1nP(jj=n), where:P(jj=n)=8><>:1 jXjN)]TJ /F16 5.978 Tf 5.76 0 Td[(1n=1)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXjnn jXjN)]TJ /F11 11.955 Tf 11.95 8.96 Td[(Pn)]TJ /F12 7.97 Tf 6.58 0 Td[(1k=1)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXj)]TJ /F8 7.97 Tf 8.94 0 Td[(kn)]TJ /F8 7.97 Tf 6.59 0 Td[(kP(jj=k)n>1. Proof. ItfollowsfromTheorem 6.1 thatjjisequaltothenumberofchoicesthatyieldsthehighestexpectedutilityforatleastonesegment.Sinceeachuijsisindependentandidenticallydistributed,eachchoicehasanequalchanceofbeingmostpreferred(i.e.,yieldingthehighestexpectedutility)byasegment.Specically,theprobabilitythatchoiceiismostpreferredbysegmentjis1 jXj. First,weconsiderthen=1case(i.e.,offersingleproduct).Theprobabilitythatchoiceiismostpreferredbyallsegmentsiis1 jXjN.SincetherearejXjchoices:P(jj=1)=jXj1 jXjN=1 jXjN)]TJ /F12 7.97 Tf 6.58 0 Td[(1. Next,weconsiderthegeneralncase(i.e.,offernproducts).LetP()denotetheprobabilitythatcontainschoicesthataremostpreferredbyallsegments.Theprobabilitythataparticularproductlineofsizencontainsachoicethatismostpreferredbysegmentjisn jXj.Furthermore,P(jjj=n)=n jXjN.Inaddition,notethatthereare)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXjnproductlinesofsizen.Forexample,consideraproblemwherejXj=3.Thereare)]TJ /F12 7.97 Tf 5.48 -4.38 Td[(32=3productlinesofsize2(i.e.,f0,1g,f0,2gandf1,2g).However,P(=f0g)isaccountedforinbothP(f0,1g)andP(f0,2g).Inparticular, 93

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aspecicproductlineofsizekisasubsetof)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXj)]TJ /F8 7.97 Tf 8.94 0 Td[(kn)]TJ /F8 7.97 Tf 6.59 0 Td[(kproductlinesofsizen,n>k.ThisimpliesthatP(jj=k)isaccountedfor)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXj)]TJ /F8 7.97 Tf 8.94 0 Td[(kn)]TJ /F8 7.97 Tf 6.59 0 Td[(ktimesinP(jjj=n).Hence,weobtainthefollowingrecursiveequation:P(jj=n)=jXjnn jXjN)]TJ /F8 7.97 Tf 12.8 14.94 Td[(n)]TJ /F12 7.97 Tf 6.59 0 Td[(1Xk=1jXj)]TJ /F7 11.955 Tf 17.94 0 Td[(kn)]TJ /F7 11.955 Tf 11.95 0 Td[(kP(jj=k). Since1jjminfjXj,Ng:E[jj]=minfjXj,NgXn=1nP(jj=n). Theorem 6.7 canbeusedtocomputetheexpectedoptimalnumberofproductstoofferwhentherearenoxedcostsandregretconsiderations.ItfollowsfromTheorem 6.3 thatthevalueobtainedbyTheorem 6.7 isalsoanupperboundontheexpectedoptimalproductlinesizeforproblemswithuniformnon-negativexedcostsandnoregretconsiderations.ThisresultisexpressedinCorollary 9 Corollary9. Ifthereexistssomec0whereci=cforalliandj=0forallj:E[jj]minfjXj,NgXn=1nP(jj=n), where:P(jj=n)=8><>:1 jXjN)]TJ /F16 5.978 Tf 5.76 0 Td[(1n=1)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXjnn jXjN)]TJ /F11 11.955 Tf 11.95 8.96 Td[(Pn)]TJ /F12 7.97 Tf 6.58 0 Td[(1k=1)]TJ /F6 7.97 Tf 5.48 -4.38 Td[(jXj)]TJ /F8 7.97 Tf 8.94 0 Td[(kn)]TJ /F8 7.97 Tf 6.59 0 Td[(kP(jj=k)n>1. Proof. FollowsdirectlyfromTheorems 6.3 and 6.7 WenotethattheresultsofTheorem 6.7 andCorollary 9 aregeneralandapplicabletoallsetsofproblemsthatsatisfytheproblemgenerationassumptionslistedinSection 6.2 .Furthermore,experimentalresultsinSection 6.2.2 indicatethatE[]isunlikelytoincreaseifweretoincreaseuniformlyacrosssegments,suggesting 94

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thatCorollary 9 alsoappliestoproblemswithuniformnon-negativexedcostsand.However,intheabsenceofaformalproof,thevalidityofthisclaimremainsanopenquestion. Table6-1. Respectiveuijsvalues. Segment1Segment2Segment3ProductS=1S=2S=3S=1S=2S=3S=1S=2S=3 15.05.04.05.04.04.05.04.04.025.04.04.05.05.04.04.04.05.034.04.05.04.04.05.05.05.04.047.10.07.17.10.07.10.012.00.050.012.00.00.012.00.07.10.07.1 95

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Table6-2. Estimatedexpectedoptimalnumberofproductstooffer,^E[jj. N=10N=20N=30jSjjXj=10jXj=20jXj=30jXj=10jXj=20jXj=30jXj=10jXj=20jXj=30 0.1106.357.408.457.7011.6013.608.6514.6016.960.1205.706.807.657.7510.9013.308.3513.8015.880.1305.056.207.256.5510.8012.657.9512.3512.920.3104.506.007.255.609.6012.056.9012.0016.200.3202.604.405.953.256.359.104.109.2512.650.3301.652.952.651.802.053.951.252.553.650.5102.555.005.952.906.609.854.109.6513.350.5201.151.452.101.001.051.101.001.001.000.5301.051.001.001.001.001.001.001.001.00 UB6.518.038.638.7812.8314.779.5815.7119.15 96

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CHAPTER7SUMMARYANDFUTURERESEARCH 7.1SensitivityAnalysis Dynamicprogramming(DP)isaversatileoptimizationmethodologythatcanbeusedtosolveavarietyofproblems.Conventionalsolutionapproachesassumethatthemodelparametersareknown.However,theparametersareoftenuncertaininpracticeandhencethestabilityofthesolutionisofinteresttothedecisionmaker.Thetypicalapproachistosolvetheproblemfordifferentrealizationsofparametervalues.However,thereareboththeoreticalandpracticalreasonsfordevelopingmoreefcientsensitivityanalysisapproachesfordynamicprograms.Sensitivityanalysisisawell-establishedtopicinLPandcanbefoundinmostLPtextbooks(forexample,Bazaraaetal.(2005)).However,thistopicisrarelymentioned,muchlessdiscussed,inmostDPtextbooks(see,White(1993),Puterman(1994),Bather(2000)andPowell(2007)). InChapter2,wehighlighttherelationshipbetweenDPandLPanddemonstratehowapproachesandresultsfromtheLPliteraturecanbeappliedtodiscretedynamicprograms.However,dynamicprogramsarerarelysolvedaslinearprogramsinpracticeandthereisaneedtoconsidersensitivityanalysisapproachesthatexploitthestructureofDPproblems(forexample,Bellmanequations). InChapter3,weexaminehowsensitivityanalysiscanbeperformeddirectlyforaMDPwithuncertainrewards.Forthesingleparameterproblem,weillustratehowtheoptimalregionofapolicycanbeobtainedbyconsideringtheregioninwhichthecurrentpolicyisoptimalwithrespecttoeachpossibleaction(Proposition 3.1 ).Whentheuncertainparametersareallowedtovarysimultaneously,wecomputethemaximumallowableerrorintheestimatedvaluessuchthatthecurrentsolutionremainsoptimal(Proposition 3.2 )andillustratehowthemaximumallowabletolerancecanbecomputedwhentheuncertainparametersarenon-stationary(Proposition 3.3 )byshowingthatitissufcienttoconsiderasubsetofpossibleestimationerrors(Theorem 3.2 ).Inaddition, 97

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wehighlightthatthemaximumallowabletoleranceofthestationaryproblemisatleastasgreatasthatofthenon-stationaryproblem(Theorem 3.3 )andderivetheconditionswherethetolerancesofthestationaryandnon-stationaryproblemsarethesame(Corollary 2 )andtheconditionswheretheydiffer(Theorem 3.4 ). Thisworkismotivatedbythefactthatrewardsareoftenestimatedanduncertaininpractice.Weillustratetheapplicabilityofthisworkthroughacapacitatedstochasticlot-sizingproblemwheretheorderingcostsandbacklogpenaltiesareuncertain.Othersequentialproblemsthatinvolveuncertainrewardsincludeequipmentreplacement(forexample,uncertainsalvagevalue),medicaldecisionmaking(forexample,valueofahumanlife)anddynamicassignment(forexample,valueofatask). ThesensitivityanalysisapproachesproposedinChapter3assumethatrewardscanbeexpressedasafnefunctionsofuncertainparameters.Oneextensionistoconsiderrewardsinvolvingmoregeneralfunctions.Wealsohighlighttheconditionswherestationaryuncertainparameterassumptionsleadtooverlyoptimistictolerancelevelsforagenerallot-sizingproblemundermildassumptions(Theorem 3.5 ).Anotherareaoffurtherresearchistoidentifyconditionswherethisistrueforothersequentialdecisionproblems. ThemonotonicityandcontinuityofVwithrespecttothemodelparametersarediscussedinSection 2.2 .Inparticular,wehighlightthatVisnotnecessarilymonotoneinT,butitispossibletoderivesufcientconditionsthatguaranteemonotonicity.Apotentialareaofresearchistoexploreifmoreofthesesufcientconditionscanbederivedforspecicdynamicprogrammingproblems(i.e.,lot-sizingproblem,knapsackproblem,etc.).AnotherinterestingareaofresearchistoidentifyadditionalpropertiesofVandwithrespecttothemodelparameters.Forexample,WhiteandEl-Deib(1986)highlightedthatVispiecewiseafneandconcaveinifraandVT(s)areafnein. Anumberofproblemsthatareencounteredinpracticearelargeanditisnotpossibletocomputetheexactsolutionsforthesedynamicprograms.Theseproblems 98

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areoftensolvedbyapproximatingthevaluefunctionand/oraggregatingthestatespace.TopalogluandPowell(2007)wereabletoexploitthestructureoftheirvaluefunctionapproximationtoefcientlyrevisethesolutionwhenthemodelparameterschange.Itwouldbeinterestingtoexplorehowsensitivityanalysiscanbeperformedfordifferentapproximatedynamicprograms.Wealsonotethatresearchershavetypicallyexploredproblemswheretheparameteruncertaintiesarerestrictedtoonegroupofparameters,suchasuncertaintiessolelyintherewardsortransitionprobabilities.However,parameteruncertaintyisrarelyconnedtoasingleclassofparametersinpractice.Anotherpromisingareaofresearchwillbetoexploreproblemswheredifferenttypesofparametersareallowedtovarysimultaneously. 7.2Regret Researchinpsychologysuggeststhatthesatisfactionresultingfromadecisionunderuncertaintyisofteninuencedbyfeelingsofregret.However,thedenitionofregretinnormativedecisiontheoryislimited.InChapter4,wereviewregrettheoryandillustratehowregret-theoreticchoicepreferencesaredescribedbymultivariatestochasticdominance.Inaddition,wepresentregret-basedriskmeasuresandillustratehowtheycanbeadoptedwithinthemean-riskframework. InChapter5,weexamineanequipmentreplacementproblemunderanitehorizon,buttheactualhorizonlengthisuncertain.Inourexperience,thisoccursofteninpractice,fromtheuncertaintyofwhenaproductionlineistobeshutdowntotheclosingofmilitarybases.Stochasticdynamicprogrammingformulationsarepresentedinordertominimizeexpectedcostsandmaximumregret.Asthereplacementpoliciesfortheseobjectivesareoftendifferent,wedesignanoptioncontracttoleasetheassetafterperiodTs,theearliesthorizontime,inordertomitigatetheriskoflossduetotheuncertainhorizon.Thecontractismadeavailableforagivenpriceattc,thetimeperiodwhenoptimaldecisionsforthedifferentobjectives(minimizingexpectedcostsandminimizingmaximumregret)rstdiverge. 99

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Theoptimalcontractparameters(optionpriceandperiodicleaserates)aredependentonanumberoffactors,includingtheprobablelengthofthelease,discountrate,andtheleasor'sexpectedprotmargin.Theoptionprice,whichmustbeincurredbythedecisionmakerwhetherheorsheexercisestheleaseornotandrepresentssignicantincometotheleasor,increaseswiththediscountrateandleaselength.Weprovidetheoptimalparametersintermsofthecostsandexpectedprotoftheleasor(Theorem 5.2 andCorollary 5 ). Thisworkillustratestheneedtolookatequipmentreplacementproblemswithcontractualobligationsmoreclosely.Weexaminetheparametersforaleasingoptioncontracttobepursuedinlieuofownership.Therearenumeroussituationswithregardstoassetownershipthatrequiresophisticatedcontracts.Forexample,manyowners,suchasairlines,outsourcetheserviceoftheirassetstothirdpartiesortheoriginalequipmentmanufacturer.Additionally,equipmentsellersarenowundergreaterscrutinytogeneratestatedbenetsundercontractorfacepenalties.Thatis,equipmentmustmeettheexpectationsofadvertisedtechnologicalimprovements.Thesesituationsopentremendousresearchopportunitiesintocontractdesigninlightofequipmentreplacementdecisions. InChapter6,weinvestigatetherelationshipbetweenanticipatedregretandchoices.Individualsexperienceregretinavarietyofsettings.However,choicescanbeindependentofanticipatedregret,evenwhensatisfactionisinuencedbyfeelingsofregret.Inpractice,individualsmaymakecomputationalmistakeswhenidentifyingtheoptimalchoice.Hence,anobjectivethatiseasytocompute(i.e.,independentofregret)ispreferredoveronethatismorecomplicated(i.e.,inuencedbyregret).Theorem 6.1 statesthatchoicepreferenceisindependentofanticipatedregretwhenthedecisionmakermaximizesexpectednetsatisfactiondenedbyEquation( 6 ).Itisnothardtoseethatthisresultextendsforalldenitionsofnetsatisfactionthatarelinearinregret. 100

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Inaddition,weobtaintheoreticalandexperimentalresultsontheoptimalnumberofproductstoofferundervariousconditionsinaheterogeneousmarket.Whenconsumerpreferencesvarygreatly,acompanythatwishestocaptureabroadmarketshareneedstoprovidearichproductline.However,thisissuboptimalwhenthecostofallowingforawideselectionishigh(Theorem 6.3 ).Therefore,acompanyseekingabroadmarketshareshouldfocusonreducingthecostofintroducingvarietyintotheirproductline.Dellisanexcellentexampleofhowarmcansupportawideconsumerbasewhenitisabletoofferawidevarietyatcompetitiveprices.However,theproductlineshouldbenarrowwhenoutcomesareuncertainandconsumersareaffectedbyregret(Section 6.2.2 ).Thisisbecauseawidearrayofproductsincreasesthelikelihoodandmagnitudeofregretexperienced.Forexample,nancialadvisorsandinsuranceagentsgenerallydonotpresentamultitudeofproductstotheircustomers. Oneinterestingobservationisthattheoptimalnumberofproductstooffercanincreasewhenregretaversionincreases(Example 10 ).Thishappenswhenasmallsetofriskyproductsresultsinhighexpectedmarketsatisfactionwhenregretaversionislow.Whenregretaversionincreases,expectedmarketsatisfactiondecreasesasaresultofincreasedregretanditmaybebettertoofferalargersetoflessriskyproducts.Forexample,althoughpeopleoftenregretfoodchoices,manyfast-foodrestaurantsofferavarietyofproductsintheirmenus.Oneobservationisthatthechoicesofferedbyfast-foodrestaurantsareusuallyrelativelysafeinthattheregretofmakingawrongchoice(forexample,purchasingahamburgerwhenachickensandwichresultsinthegreatestsatisfaction)isgenerallylow. Ourresearchalsosuggeststhatconsumersarelikelytobeindifferentbetweenchoiceswhentheassociatedoutcomesareextremelyvariable(see,proofofTheorem 6.5 ).Whentheoutcomeofchoicesarehighlyuncertain,itisalmostsurelyoptimalforthermtoofferasingleproduct(Theorem 6.5 ).Thispredictionisconsistentwithstudiesthathighlightthatpatientsareoftenuneasyandundecidedwhengivenawideselection 101

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oftreatmentstochoosefrom,relyingstronglyonmedicalprofessionalstodecidethetreatmentplanforthemwhentheassociatedoutcomesarehighlyuncertain(Schneider1998). Inouroptimalproductlineanalysis,weuseachoicemodelthatisbasedonabsoluteregret.OneextensionistoconsiderothernotionsofregretdiscussedinSection 4.2 .Inaddition,recentstudiessuggestthatindividualscanbedisproportionatelyadversetolargeregret(Bleichrodtetal.2010).Anotherextensionoftheproblemistoconsiderconsumersatisfactionthatisnon-linearinregret.Third,consumersmaynotalwaysberationalandselectchoicesthatmaximizeexpectednetsatisfaction(Simonetal.1995).Anotherinterestingextensionistoconsiderproductlinedesignwithregretconsiderationsunderboundedrationality. 102

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BIOGRAPHICALSKETCH ChinHonTanisaPh.D.candidateintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.HereceivedhisB.Eng.(2005)andM.S.(2009)inIndustrialandSystemsEngineeringfromtheNationalUniversityofSingaporeandtheUniversityofFlorida,respectively.Priortohisgraduatestudies,heworkedasareliabilityengineerwithShellEasternPetroleumandalsotaughtappliedmathematicsatRepublicPolytechnicinSingapore.Hisresearchinterestsincludedecisionanalysis,stochasticmodelinganddynamicprogramming.HeisamemberofIIEandINFORMS. 113