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Stochastic Inventory Control in Dynamic Environments


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Iwishtoexpressmysinceregratitudetothemembersofmysupervisorycom-mittee,Dr.J.P.Geunes,Dr.A.Paul,Dr.H.E.Romeijn,Dr.Z.J.ShenandDr.S.Urasev,fortheirassistanceandguidance.IwouldalsoliketothankDr.AydinAlptekinoglu,fortakingtimetoreviewmydissertation.Especially,Iamgratefultomychair,Dr.H.EdwinRomeijn,forhissupport,encouragementandpatiencethroughoutthestudy.AsIwillpursuemyowncareer,nothingismorebenecialthanhavinghisneexampleofajustperson,anindustriousandthoroughscholar,andaresponsibleadvisor,tolookupto.Myparentsandwifehavegivenmetremendoussupportformystudyabroad.Theirtrustandlovearetheinvaluablewealthinmylife,butnoacknowledgementcouldpossiblystateallthatIowetothem. iv

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Page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ ix ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 1.1GeneralDescription .......................... 1 1.2LiteratureReview ........................... 2 1.3OutlineofDissertation ........................ 11 2INVENTORYCONTROLINASEMI-MARKOVMODULATEDDE-MANDENVIRONMENT .......................... 14 2.1Introduction .............................. 14 2.2ModelFormulation .......................... 14 2.2.1TheDemandProcess ..................... 14 2.2.2GammaDistributedStateTransitionTimeswithObservableStageTransitions ....................... 15 2.2.3DenitionsandNotation ................... 16 2.2.4ProblemFormulation ..................... 19 2.3ModelAnalysis ............................ 21 2.3.1OptimalPolicy ......................... 21 2.3.2OptimalPolicywithSemi-MarkovModulatedPoissonDe-mands ............................. 23 2.3.3DeterminationoftheOptimalInventoryPosition ...... 29 2.3.4TotalPolicyCosts ....................... 31 2.4MonotonicityResults ......................... 32 2.4.1MonotonicityofOptimalInventoryPositionswithinaGivenState .............................. 33 2.4.2MonotonicityofOptimalInventoryPositionsbetweenStates 38 2.4.3ImplicationsoftheMonotonicityResults .......... 41 2.5AnAlgorithmtoComputetheOptimalInventoryPolicy ..... 43 2.5.1ContinuousPhase-TypeDistriutedWorldTransitionTimeandLeadTime ........................ 43 v

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............. 54 2.6AnExtension:DemandArrivesFollowingaGeneralRenewalProcess 59 2.6.1GeneralizationoftheDemandProcessModel ........ 59 2.6.2TheOptimalInventoryPolicy ................ 61 2.7Summary ............................... 63 3MODELSWITHPARTIALLYOBSERVABLEWORLDSTATES .... 65 3.1Introduction .............................. 65 3.2ASimpleModelwithTwoWorldStates .............. 65 3.2.1EectsoftheUnobservableWorld .............. 66 3.2.2ExponentialTransitionTimeDistribution .......... 70 3.2.3ComputationoftheOptimalInventoryPosition ...... 73 3.2.4AnExtension ......................... 78 3.3MultipleWorldStatesModels .................... 79 3.3.1ModelswithMultipleWorldStatesWhichareVisitedinaFixedSequence ........................ 79 3.3.2ARecursiveFormula ..................... 85 3.3.3OptimalInventoryPosition .................. 88 3.3.4MoreGeneralMultipleStatesWorldModels ........ 90 3.4Summary ............................... 92 4JOINTPRICINGANDINVENTORYCONTROLINDYNAMICEN-VIRONMENT ................................ 93 4.1Introduction .............................. 93 4.2JointPricingandInventoryControlinPriceSensitivePoissonDemandEnvironment ......................... 94 4.2.1TheModelthatPriceCanOnlyBeSetOnce ........ 94 4.2.2AlgorithmtoComputetheOptimalPriceandInventoryPosition ............................ 102 4.2.3TheModelthatPriceCanBeSetContinuously ...... 105 4.3Semi-MarkovModulatedPrice-SensitivePoissonDemand ..... 106 4.3.1TheModel ........................... 107 4.3.2ThePriceCanOnlyBeSetOnceforEachState ...... 108 4.3.3ApproximateModelsforSemi-MarkovModulatedPoissonDemand ............................ 111 4.3.4ApproximateModelthattheWorldProcessisMarkovianandOnePriceforEachState ................. 113 4.4Summary ............................... 113 5ASTOCHASTICMULTI-ITEMINVENTORYMODELWITHUN-EQUALREPLENISHMENTINTERVALSANDLIMITEDWARE-HOUSECAPACITY ............................. 115 5.1Introduction .............................. 115 vi

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................ 115 5.3SolutionApproaches ......................... 119 5.3.1EqualReplenishmentIntervals ................ 119 5.3.2HeuristicsforaTwo-itemCase ................ 124 5.3.3HeuristicsfortheGeneralMulti-itemCase ......... 131 5.3.4ProofofOptimalityofHeuristicsforSimultaneousReplen-ishmentCase .......................... 135 5.4NumericalResults ........................... 135 5.5Summary ............................... 139 REFERENCES ................................... 142 BIOGRAPHICALSKETCH ............................ 147 vii

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Table Page 5{1Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionofthetightnessofthestoragecapacity(2items). ............................. 138 5{2Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionofdemandvariabilitybetweenitems(2items). .................................. 138 5{3Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionoftheunderagepenaltycostsoftheitems(2items). .............................. 138 5{4Relativecostassociatedwithcoordinatingreplenishmentsfordierentitemstooccuratthesametime,asafunctionofthetightnessofthestoragecapacity. .............................. 138 5{5Relativecostassociatedwithcoordinatingreplenishmentsfordierentitemstooccuratthesametime,asafunctionofdemandvariabilitybetweenitems(2items). ......................... 140 5{6Relativecostassociatedwithcoordinatingreplenishmentsfordierentitemstooccuratthesametime,asafunctionoftheunderagepenaltycostsoftheitems. ............................. 140 5{7Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionofthetightnessofthestoragecapacity(3items). ............................. 140 5{8Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionoftheunderagepenaltycostsoftheitems(3items). .............................. 140 viii

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Figure Page 2{1Optimalinventorypositionaty 53 2{2Optimalinventorypositionaty+1 .................. 53 2{3Optimalinventorypositionaty1 .................. 54 3{11=1,2=1:2,=0:4,andnodemandbeforet 74 3{21=1,2=2,=0:4,andnodemandbeforet 74 3{31=1,2=3,=1:5,and1demandbeforet,s1=1 ......... 75 3{41=1,2=2,=1,andnodemandbeforet 75 4{1Optimalinventorypositiony() .................... 103 4{2Protfunction ............................... 103 5{1Amulti-iteminventorysystem. ..................... 117 5{2Aninventorysystemwithunequalreplenishmentintervals. ...... 124 5{3Aninventorypolicyforthenonintrusiveheuristic. ........... 125 5{4Aninventorypolicyforthegreedyheuristic. .............. 127 5{5Aninventorypolicyforthesharingheuristic. .............. 130 ix

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Thisdissertationstudiessomeissuesinstochasticinventorycontrol. Therstfocusofthedissertationisonstochasticcontinuous-timeinventorycontrolproblemsforasingleitemindynamicenvironments.Thedemandprocessismodeledasasemi-MarkovchainmodulatedPoissonprocess.Itisshownthatamyopicpolicyisoptimaliftheproductscanbepurchasedorbought-backatasingleprice.Conditionsonthesemi-Markovchainunderwhichproductswillneverbereturnedisderived.Analgorithmtodynamicallycomputetheoptimalpolicyforaspecialcaseofthemodelisalsoprovided.Thisdemandmodelisnextextendedtoasemi-Markovmodulatedrenewalprocess,andseveralresultsaregeneralizedtothismorerealisticmodel. ThenextfocusofthedissertationisaclassofMarkovmodulatedPoissonde-mandprocessesinwhichthetransitionsbetweenthedierentstatesoftheworldisunobservable.Abasicmodelwithtwodemandstatesisrststudied,andtheoptimalinventorypolicyisderived.Analgorithmtocomputethispolicyisalsoprovided.Nextthebasicmodelisextendedtomultiplestates,andarecursiveformulaisgivenwhichcanbeusedtocomputetheoptimalpolicy. x

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Theotherfocusofthedissertationisonstochasticinventorymodelsformultipleitemswithbothequalandunequalreplenishmentintervalsunderlimitedwarehousecapacity.Theoptimalityconditionforequalreplenishmentintervalscaseisgiven,threeheuristicsareimplemented,anditisprovedthattheseheuristicsprovidetheoptimalsolutionsinthecaseofequalreplenishmentintervals.Extensivenumericaltestsareconducted,andtheheuristicsyieldhighqualitysolutionsinverylimitedtime. xi

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Distributionsystemsoftencontainasetofregionalwarehouses,eachofwhichstoresavarietyofitemssuppliedbymultiplemanufacturersinordertoserveare-gionalpopulationofcustomers.Eectivelymanagingtheinventoryofmultipleitemsunderlimitedwarehousestoragecapacityiscriticalforensuringgoodcustomerser-vicewithoutincurringexcessiveinventoryholdingcosts.Eachregionalwarehousemanagerthusfacesthechallengeofcoordinatingtheinventorylevelsanddeliveriesofmultipleitemsinordertomeetdesiredservicelevelswhileobeyingwarehousecapacitylimits.Supplierstosuchregionalwarehousesmustecientlymanagethetradeostheyfacebetweeninventoryandtransportationcosts,whichoftenleadsdierentsupplierstopreferdierentwarehousereplenishmentfrequencies.These 1

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dierentreplenishmentfrequencypreferences,combinedwithvaryingdegreesofde-manduncertainty,furthercompoundthechallengesthewarehousemanagerfacesineectivelyutilizinglimitedwarehousecapacity. 6 ],HadleyandWhitin[ 29 ]).Undersomespecialdemandmodels,e.g.,Poissondemandprocess,orincaseoflinearorderingcosts,an(s;S)policycanbesimpliedtoabasestockpolicy(s;Q)wheresisagainthereorderpoint,andQistheorderquantity.EarlierworksinthisareaarewellsummarizedbyLeeandNahmias[ 41 ].Recentdevelopmentsinthisareahavemainlybeenfocusedonthedeterminationoftheoptimalparameters,orthedesignofgoodheuristicsthatresultinnearoptimalsolutions,e.g.,FedergruenandZheng[ 20 ]andGallego[ 25 ]. Thesecondclassofproblemsdealswithtime-dependenceandadaptivedeci-sionmakinginadynamicdemandenvironment.Mostoftheresearchworkdoneinthisareadealswithdiscrete-timemodels,wherethedynamicnatureofthede-mandprocessisreadilyrepresentedviaadynamicprogrammingapproach.TherstmathematicalformulationofproblemsofthistypewasintroducedinArrowetal.[ 3 ],andlaterenrichedbyBellmanetal.[ 7 ].Karlin[ 36 ]extendstheirresultsby

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studyinginventorymodelswheredemandsareindependent,butnotnecessarilyiden-ticallydistributed,overtime.Heshowedthatastate-dependentbasestockpolicyisoptimal.Moreover,inanysetofconsecutiveperiodsforwhichthesequenceofdemanddistributionsdecreasesstochastically,theoptimalbasestocklevelalsode-creases.Intheseearlierworks,thedemandsindierentperiodsareassumedtobeindependent.Veinott[ 56 ]extendstheseresultstoaninnitehorizon,discrete-time,multi-productdynamicnonstationaryinventoryproblem.Thedemandsindierentperiodsarenotnecessarilyindependent.Underlinearorderingcostsandtheassump-tionthatdisposalofexcessinventoryisallowedatthesamepriceasreplenishmentofinventory,Veinottderivesconditionsunderwhichamyopicbasestockorderingpolicyisoptimal.Lovejoy[ 44 ]considersaperiodicreview,dynamic,single-productinventorymodelwithlinearorderingcosts.Heconsidersbothdisposalandnondis-posalmodels,andderivesboundsontherelativelosscomparedtotheoptimalcostthatisincurredbyrestrictingconsiderationtotheclassofmyopicinventorypolicies. Recentstudiesmodelthedependenceofdemandindisjointtimeintervalsasresultingfromtheeectofsomeunderlyingevents.Theseunderlyingeventsoccurastimepasses,andtheymayaectthepropertiesofthecurrentandfuturedemandprocess.SongandZipkin[ 52 ]andZipkin[ 63 ]providesomeexamplesofeectsthatmaycharacterizethestateoftheworld,suchasweather,economy,technology,cus-tomerstatus,etc.TheyusuallymodeltheunderlyingeventsasaMarkovprocess,eitherincontinuousordiscretetime.Inparticular,SongandZipkin[ 52 ]assumeademandprocessthatisgovernedbyanunderlyingcoreprocess,calledtheworld,whichisacontinuous-timeMarkovchainwithdiscretestatespace.ThedemandprocessisthenaPoissonprocesswhoseratedependsonthecurrentstateoftheworld.Theyshowthatiftheorderingcostsarelinearinthequantityordered,thenastate-dependentbasestockpolicyisoptimal.Ifaxedorderingcostisincurred,thenastate-dependent(s;S)policyisoptimal.Theyalsoshowthatifthedemand

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processsatisesacertainmonotonicityproperty,theoptimalpolicywillinheritthismonotonicity.Theyalsoconstructaniterativealgorithmtoapproximatetheoptimalpolicies,andanexactalgorithmforthelinearcostmodel.Later,SongandZipkin[ 53 ]utilizethesimilarmodeltoshowhowtomanageinventoryunderadeterioratingdemandenvironment. ThememorylesspropertyinherentinthemodelofSongandZipkin[ 52 ]meansthatorderingdecisionsareonlymadeinresponsetoanevent,i.e.,achangeofthestateoftheworldortheoccurrenceofademand.Thispropertyallowsthetransfor-mationofthecontinuous-timemodelintoanequivalentdiscrete-timemodel,andtheythenemployadiscrete-timedynamicprogrammingapproach.However,theimplicitassumptioninthesemodelsthatthetimebetweenstatetransitionsismemorylessisnotalwaysreasonable.Forexample,ifthepropertiesofthedemandprocessareweather-related,e.g.inthecaseofseasonaldemands,thisassumptionisclearlynotsatised.Knowledgeoftheamountoftimethatwehavespentinagivenseasongenerallyprovidesuswithsomeinformationonhowsoonthisseasonwillendandwhatthenextseasonwillbe. ThemodelweproposeiscloselyrelatedtothemodelofSongandZipkin[ 52 ].Inparticular,weextendtheirworldmodelbyrelaxingtheassumptionthatthetimebetweenstatetransitionsismemoryless.Thismeansthatthechangesinthede-mandprocessaredescribedbyasemi-MarkovprocessinsteadofanordinaryMarkovprocess.Themaineectofthisrelaxationisthatateverypointintimeinagivenstate,theprocessoffuturedemandsisdierent.Thus,anoptimalpolicymayrequiremakingorderingdecisionscontinuouslyintime.Thismeansthattheelapseoftimeitselfprovidesuswithinformationaboutthefuture.Forexample,theabsenceofdemandsforacertainamountoftimemaycauseustoadjusttheinventoryposition.HeymanandSobel[ 31 ]alsostudiedsemi-Markovdecisionprocesses.Butnotlike

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us,theyrestrictedthatthedecisionscanbemadeonlyattheepochofstatetran-sitionsandconcludedthattheoptimizationofinnitehorizondiscountedmodelisessentiallythesameastheoptimizationofdiscrete-timeMarkovdecisionprocesses.Obviously,allowingdecisionmakingsatanytime,aswedohere,makestheproblemmorecomplicated,andthebehavioroftheoptimalpolicyalsodiersfromthatforMDPs,aswewillshow.Tobeabletodealwiththisadditionalcomplexity,wemainlyfocusonamodelinwhichdisposalofexcessinventoryisallowedatthesamepriceasreplenishmentofinventory(analogoustothediscrete-timemodelofVeinott[ 56 ]).Forthiscase,wederiveconditionsunderwhichamyopicbasestockorderingpolicyisoptimal. Mostreallifeinventorycontrolproblemsfaceonlypartiallyobservabledemand.Thetrueunderlyingdistributionofthedemandisnotdirectlyobserved,andonlydemandoccurrencesareobserved.Scarf[ 50 51 ]studiedaninventoryprobleminwhichtheparameterofthedemandisunknown,butaprioriBayesiandistributionischosenfortheparameter.HeusedBayesianmethodstosolvetheinventorycontrolproblemsandcharactertheoptimalorderingpolicy.Azoury[ 4 ]extendedtheresultofScarf[ 51 ]bystudyingdynamicinventorymodelsundervariousfamiliesofdemanddistributionswithunknownparameters.HederivedtheoptimalBayesianpolicy,andshoweditscomputationisnomoredicultthanthecorrespondingcomputationwhenthedemanddistributionisknown.Lovejoy[ 43 ]studiedinventorymodelswithuncer-taindemanddistributionswhereestimatesoftheunknownparameterareupdatedinastatisticalfashionasdemandisobservedthroughtime.Heshowedthatasimpleinventorypolicybaseduponacriticalfractilecanbeoptimalornear-optimal.Later,Lovejoy[ 44 45 ]extendedthestudybyshowingtherobustnessofboundsonthevaluelossrelativetooptimalcostofmyopicpolicieswhichmaybestoppedearlier. KurawarwalaandMatsuo[ 38 ]gaveacombinedforecastingandinventorymodelaccordingtothecharacteristicsofshort-lifecycleproducts.Theyproposedaseasonal

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trendgrowthmodelandusedoptimalcontroltheorytogettheoptimalinventorypol-icy.TreharneandSox[ 55 ]studiedapartiallyobservableMarkovmodulateddemandmodelinwhichtheprobabilitydistributionsforthedemandineachperiodisde-terminedbythestateofanunderlyingdiscrete-timeMarkovchain,andpartiallyobserved.Theyshowedthatthatsomesuboptimalcontrolpolicies,open-loopfeed-backcontrolandlimitedlook-aheadcontrol,whichaccountformoreoftheinherentuncertaintyinthedemandprocesses,almostalwaysachievemuchbetterperformancethanthetypicallyusedCEC(certaintyequivalentcontrol)policy. Mosttraditionalinventoryproblemsconcernthedeterminationofoptimalre-plenishmentpoliciesindierenttypesofenvironments,wherethedemandprocessisoftenassumedtobegiven.Intheseproblems,productsellingpricesarenotadecisionvariable,butgivenasknownparameters,althoughtheymaychangefromperiodtoperiod.Therefore,theaimistominimizetheexpectedoperatingcosts,becausetheexpectedrevenuesarenotcontrollable. Morerecentdevelopmentsinindustrialpracticecombiningpricingandinventorymanagementhaveshowngreatsuccess,andhavestimulatedtheneedforresearchintocombiningpricingandinventorycontrolpolicies.Whitin[ 61 ]proposedincludingthepricingintoinventoryplanningdecisions.Hestudiedthesingleperiodnewsvendormodelwithpricedependentdemandandconsideredtheproblemofsimultaneousdeterminationofasinglepriceandorderingquantity.Thomas[ 54 ]consideredasingleitem,periodicreview,nitehorizonmodelwithaxedorderingcostandpricesensitivedemandprocess.Heconjecturedthata(s;S;p)policywasoptimal:theinventoryreplenishmentisgovernedbyadynamic(s;S)policy,wheretheoptimalpricedependsontheinventorylevelatthebeginningofareviewperiod.Healsoconstructedacounterexamplewhichdemonstratesthatiftheavailablepricechoiceisrestrictedtoadiscreteset,thispolicymaynotbeoptimal.

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PetruzziandDada[ 47 ]providedanexcellentreviewonpricingdecisionsinthenewsvendorproblem,andinadditionextendedthesingleperiodmodeltoamulti-periodone.Theyconcludedthatinmostpapersonpricingtherandomnessindemandisassumedindependentoftheitempriceandcanbemodelledeitherinanadditiveoramultiplicativeway.Theypointedoutthatadicultyinmulti-periodmodelsresultsfromtheassumptionthatinventoryleftoverscannotbedisposedof.Theyshowhowrevisingthisassumptionandallowingforthepossibilityofsalvagingleftoversissucienttoyieldastationarymyopicpolicyforthemultipleperiodproblem.Bycuttingothelinksbetweenperiods,allresultsandmanagerialinsightavailableforthesingleperiodmodelapplydirectlytothemultipleperiodmodel. FedergruenandHeching[ 19 ]analyzedasingleitemperiodicreviewmodelwheredemandsdependontheitem'sprice,orderingcostsarelinearintheorderedamount,andallstockoutsarebacklogged.Theystudiedbothniteandinnitehorizonmodels,usingbothexpecteddiscountedandtimeaveragedprotcriteria.Theyderivedthestructureofanoptimalcombinedpricingandinventorystrategyforalltheirmodelsanddevelopedanecientvalueiterationmethodtocomputetheoptimalstrategies.Theyshowedthatabase-stocklistpricepolicyisoptimalfortheirmodel:ineachperiodtheoptimalpolicyischaracterizedbyanorder-up-tolevelandapricewhichdependsonthestartinginventorylevelbeforeorderingatthebeginningofeachperiod.Ifthestartinginventorylevelbeforeorderingisbelowtheorder-up-tolevel,anorderisplacedtoraisetheinventoryleveltothelevel.Otherwise,noorderisplacedandadiscountpriceisoered.Thediscountpriceisanon-increasingfunctionofthestartinginventorylevel. Recently,ChenandSimchi-Levi[ 12 13 ]generalizedtheabovemodelbyincor-poratingaxedcostcomponent.Theyshowthatthe(s;S;p)policyproposedbyThomas[ 54 ]isindeedoptimalforadditivedemandfunctionswhentheplanninghori-zonisnite;whentheplanninghorizonisinnite,thispolicyisoptimalforboth

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additiveandgeneraldemandprocessesunderbothdiscountedandaverageprotcri-teria.Theyalsointroducetheconceptofsymmetrick-convexfunctionsandusethistoprovideacharacterizationoftheoptimalpolicy. Thoughperiodicalreviewmodelshavebeenstudiedquiteextensively,continuous-reviewjointpricingandinventorycontrolproblemshavereceivedfarlessattentionintheliterature.Li[ 42 ]consideredacontinuoustimeintegratedpricingandinventoryplanningstrategiesmodelwheredemandandproductionarebothPoissonprocesses.Theintensityofthedemandprocessdependsontheitem'schosenprice.Heshowedthatiforderingandholdingcostsarebothlinearabarrierpolicyisoptimal.Healsogaveanimplicitcharacterizationoftheoptimalpricingpolicywhendynamicpricingisallowed.FengandChen[ 23 ]studiedacontinuousreviewmodelthatisrelatedtoourswherethedemandismodelledasprice-sensitivePoissonprocess.Theyre-stricttheavailablepricestoagivenniteset(specically,onlytwocandidateprices),andassumezeroleadtimes.Theyshowthata(s;S;p)policyisoptimalwhenxedorderingcostsarepresent. Inourproblem,wemodelthedemandasaMarkovmodulatedPoissonprocess(seealsoSongandZipkin[ 52 ]).Inparticular,thedemandprocessisaPoissonprocesswhoserateisgovernedbyanunderlyingMarkovchainthatrepresentsthestateoftheworld.WeintroducepricingexibilityintothismodelbyallowingtherateofthePoissonprocessineachstatetodependonthepriceoftheproduct.Recently,ChenandSimchi-Levi[ 12 13 ]generalizedtheabovemodelbyincorporatingaxedcostcomponent.Theyshowthatthe(s;S;p)policyproposedbyThomas[ 54 ]isindeedoptimalforadditivedemandfunctionswhentheplanninghorizonisnite;whentheplanninghorizonisinnite,thispolicyisoptimalforbothadditiveandgeneraldemandprocessesunderbothdiscountedandaverageprotcriteria.Theyalsointroducetheconceptofsymmetrick-convexfunctionsandusethistoprovideacharacterizationoftheoptimalpolicy.

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Distributionsystemsoftencontainasetofregionalwarehouses,eachofwhichstoresavarietyofitemssuppliedbymultiplemanufacturers.Eectivelymanagingtheinventoryofmultipleitemsunderlimitedwarehousestoragecapacityiscriticalforensuringgoodcustomerservicewithoutincurringexcessiveinventoryholdingcosts.Supplierstosuchregionalwarehousesmustecientlymanagethetradeostheyfacebetweeninventoryandtransportationcosts,whichoftenleadsdierentsupplierstopreferdierentwarehousereplenishmentfrequencies.Forexample,manufacturerswhosupplyitemswithahighvalue-to-weightratiotypicallynditmoreeconomi-caltosendrelativelyfrequentshipmentsinsmallquantities,whilethosewhosupplyitemswithalowvalue-to-weightratiooftenprefertodeliverylargequantitieslessfrequently(seeBallou[ 5 ]).Thesedierentreplenishmentfrequencypreferences,com-binedwithvaryingdegreesofdemanduncertainty,furthercompoundthechallengesthewarehousemanagerfacesineectivelyutilizinglimitedwarehousecapacity. Stochasticinventorymodelsinvolving(production)capacityconstrainedperio-dic-reviewpolicieshaveattractedtheattentionofmanyresearchers.Evans[ 18 ]rstconsidersthisissuebymodelingperiodic-reviewproductionandinventorysystemswithmultipleproducts,randomdemandsandaniteplanninghorizon.Hedevelopstheformoftheoptimalpolicyformulti-productcontrolforsuchasystem.Sincethen,muchoftheliteraturehasstudiedperiodic-review,single-productsystemswithproductioncapacityconstraints.FlorianandKlein[ 24 ]andDeKoketal.[ 37 ]charac-terizethestructureoftheoptimalsolutiontoamulti-period,single-itemproductionmodelwithacapacityconstraint.FedergruenandZipkin[ 21 22 ]showthatamodi-edbase-stockpolicyisoptimalunderbothdiscountedandaveragecostcriteriaandaninniteplanninghorizon.Themodiedbase-stockpolicyrequiresthat,whenini-tialstockisbelowacertaincriticalnumber,weproduceenoughtobringtotalstockuptothatnumber,orasclosetoitaspossible,giventhelimitedcapacity;otherwise,wedonotproduce.Theyalsocharacterizetheoptimalpolicybyderivingexpressions

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fortheexpectedcostsofmodiedbase-stockpolicies.KapuscinskiandTayur[ 35 ]provideasimplerproofofoptimalitythanFedergruenandZipkin[ 21 ]fortheinnite-horizondiscountedcostcase,basedonresultsfromBertsekas[ 8 ].Ciaralloetal.[ 15 ]andWangandGerchak[ 58 ]analyzeaproductionmodelwithvariablecapacityinasimilarenvironmentasFedergruenandZipkin[ 21 ].WangandGerchak[ 57 ]alsoincorporatevariablecapacityexplicitlyintocontinuousreviewmodels. DeCroixandArreola-Risa[ 16 ]studyaninnite-horizonversionofthecapacitatedmulti-productcase.Theyestablishtheoptimalpolicyforthecaseofhomogeneousproducts,andproposeaheuristicpolicyforheterogeneousproductsbygeneralizingtheoptimalpolicyforthehomogeneousproductcase.Productsarecalledhomoge-neousiftheyhaveidenticalcostparametersandtheirdemandsareidenticallydis-tributed.Glasserman[ 27 ]addressesasimilarproblemtoDeCroixandArreola-Risa[ 16 ]inacontinuous-reviewsystem.Hepresentsaprocedureforchoosingbase-stocklevelsandcapacityallocationthatisasymptoticallyoptimal,butassumesthataxedproportionoftotalcapacityisdedicatedexclusivelytoeachproduct.TheuseofasymptoticanalysisissimilarinspirittoAnantharam[ 1 ].HisstaticallocationproblemcontrastswiththedynamicschedulingproblemaddressedinWein[ 60 ]andZhengandZipkin[ 62 ]andthepriorityschemeinCarretal.[ 11 ].LauandLau[ 39 40 ]presentformulationsandsolutionproceduresforhandlingamulti-productnewsboyproblemundermultipleresourceconstraints.NahmiasandSchmidt[ 46 ]alsoinvestigateseveralheuristicsforasingle-period,multi-iteminventoryproblemwitharesourceconstraint. Anily[ 2 ]andGallegoetal.[ 26 ]studyamulti-itemreplenishmentproblemwithdeterministicdemand.Anily[ 2 ]investigatestheworst-casebehaviorofaheuristicforthemulti-itemreplenishmentandstorageproblemandderivesalowerboundontheoptimalaveragecostoverallpoliciesthatfollowstationarydemandandcostparameters.Gallegoetal.[ 26 ]considertwoeconomicorderquantitymodelswhere

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multipleitemsuseacommonresource:thetacticalandstrategicmodels.Theyderivealowerboundonthepeakresourceusagethatisvalidforanyfeasiblepolicy,usethistoderivelowerboundsontheoptimalaveragecostforbothmodels,andshowthatsimpleheuristicsforeithermodelhaveboundedworst-caseperformanceratios.Additionalliterature,e.g.,RosenblattandRothblum[ 48 ],Goyal[ 28 ],HartleyandThomas[ 30 ],JonesandInman[ 34 ]andDobson[ 17 ],dealswithdeterministicinventorymodelswithwarehousingconstraints. Althoughmuchoftheliteratureisdevotedtomulti-item,periodic-reviewsystemswithaproductioncapacityconstraint,littlehasbeendoneforstochasticinventorymodelswithawarehouse-capacityconstraint.Veinott[ 56 ]rstconsidersamulti-productdynamic,nonstationaryinventoryproblemwithlimitedwarehousecapacity.Heprovidesconditionsthatensurethatthebasestockorderingpolicyisoptimalinaperiodic-reviewinventorysystemwithanitehorizon.IgnallandVeinott[ 33 ]showthat,inthestationarydemandcase,amyopicorderingpolicyisoptimalforasequenceofperiodsunderallinitialinventorylevels.Recently,Beyeretal.[ 9 10 ]useadynamicprogrammingapproachtoderivetheoptimalorderingpolicyfortheaveragecostproblemandshowtheconvexityofthecostfunction,aswedidinthisstudycoincidentally.Theyalsoshowtheoptimalityofthemodiedbase-stockpolicyinthediscountedcostversionoftheproblem.Inthispaper,weextendtheirresultsbyexplicitlycharacterizingtheoptimalmyopicpolicyunderrelaxedassumptionsonthedemanddistributions. 2 ,weproposeaninventorymodelunderasemi-MarkovmodulatedPoissondemandenvironment.Wegivethedescriptionofthemodel,andsomeprop-ertiesofthemodel.Thenwegivetheoptimalinventorypolicy.Weshowthatifthedemandprocessobservessomemonotoneproperty,thentheoptimalinventoryposi-tionwillalsoshowasimilarpattern.Foraspecialphasetypeleadtimedistribution,

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wegiveonealgorithmtoactuallycomputetheoptimalinventorypositions.WealsoextendthemodeltoamoregeneralcasewhererenewalprocessisinplaceofPoissonprocessforthedemandprocess. InChapter 3 ,werestudytheinventorymodelinChapter 2 .Butthistimethestateoftheunderlyingworldisnolongerobservable,andwecanonlyobservetheactualdemandarrivals.Werststudyatwoworldstatesmodel,andgivetheformoftheoptimalinventorypolicyforthismodel,andproposeonealgorithmtosolveit.Wethenextendthestudytoamultipleworldstatesmodelandgivearecursiveformulatodeterminetheprobabilityoftheunderlyingworldineachstate,andhelpdeterminetheoptimalinventorypolicy. InChapter 4 ,weincludethepricingdecisionsatthesametimethatthein-ventorystrategyisdetermined.Thepricewillaectthedemandprocess,andwearemaximizingthetotalexpecteddiscountedprotsinaninnitehorizon.Werststudythejointpricingandinventorymodelunderaprice-sensitivePoissondemandenvironmentwithoutMarkovmodulation.Inthecasewherethepricecanonlybesetonceatthebeginning,wegivesomepropertiesthatcanbeusedtodeterminetheoptimalsolution,andderiveanalgorithmtocomputetheoptimalsolution.Wethenstudythemodelwherepricecanbecontinuouslyset.Nextweextendthestudytothesemi-MarkovmodulatedPoissondemandenvironment,andshowthatwithcertainapproximation,themodelcanbesolvedinthesimilarwayasinaPoissondemandenvironment. InChapter 5 ,westudydiscretetimestochasticinventorymodelsformultipleitemswithbothequalandunequalreplenishmentintervalsunderlimitedwarehousecapacity.Weproposethreeecientandintuitivelyattractiveheuristics.Weshowthattheseheuristicsprovidetheoptimalreplenishmentquantitiesinthecaseofequal

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replenishmentintervals.Forthegeneralmodel,anumericalcomparisonoftheheuris-ticsolutionstotheoptimalsolutionsshowsthattheheuristicsyieldhighqualitysolutionsinverylimitedtime.

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2.2 ,weformulateourmodelandstudyaspecialcaseofourdemandprocessthatreducestoaMarkovmodulateddemandenvironment.Then,inSection 2.3 ,weshowthatforourgeneraldemandprocessamyopicpolicyisoptimal,andcharacterizetheoptimalpolicyparameters.InSection 2.4 ,wederivesucientconditionsonthedemandsthatimplythatthedisposaloptionwillneverbeusedandthemyopicpolicyisthusoptimaleveninthecasewheredisposalisnotallowed.InSection 2.5 ,weproposeanalgorithmtocomputetheoptimalinventorypolicyforaspecialcaseofourinventorymodel.InSection 2.6 ,weproposeanextensionofthemodel,wherethedemandprocessinagivenstateoftheworldisageneralrenewalprocessinsteadofaPoissonprocess.WeendthechapterinSection 2.7 withsomeconcludingremarks. 2.2.1TheDemandProcess 14

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andthetransitionprobabilitymatrixisP=(pij)i;j2I.Givenacurrentstateiandanextstatej,thedistributionfunctionofthetransitiontimefromitojisdenotedbyGij.Weassumethetransitiontimesareindependentofeachother.Whenthecoreprocessisinstatei,theactualdemandprocessfollowsaPoissonprocesswithratei,whereweassumethat=supi2Ifigisnite.Wecallthisdemandprocessasemi-MarkovmodulatedPoissondemandprocess.Thisdemandprocessisexogenousandisnotaectedbyanyorderingdecisions.Clearly,ifthedistributionsGijareexponentialdistributionswhoserateonlydependsoni,thedemandprocessreducestoaMarkovmodulatedPoissondemandprocess,asintroducedbySongandZipkin[ 52 ].Beforewedescribeandanalyzeourmodel,werstbrieydiscussacaseinforwhichthetransitiontimesarenotexponentiallydistributed,butneverthelessresultsinaMarkovmodulatedPoissonprocess.

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probabilitiesgivenbyp(i;k);(i;k+1)=1k=1;:::;ri1;i2Ip(i;ri);(j;1)=piji;j2I: 52 ].However,ifwecannotdirectlyobservethestagechangesofthecoreprocessintheabovemodel,thentheirresultscannotbeapplied.

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thecurrentstateA(t)sincethelaststatetransition,sayS(t),andweletS(t)=fS(u):0utg: Weassumethattheinventorylevelisreviewedcontinuously,andweneedtodecideonhowtoadjusttheinventorypositionateachpointintime.Weassumethatbothinventoryordering(i.e.,andupwardsadjustmentoftheinventoryposition)andinventorydisposal(i.e.,adownwardsadjustmentoftheinventoryposition)arepossible,bothatthesameprice(seealsoVeinott[ 56 ]).Thiswillbeareasonableassumptioninthecontextofconsignmentsales.Thisisanincreasinglypopularbusinessarrangementwhereforexampletheretailerdoesnotpayitssupplieruntiltheitemsaresold.Ownershipofthegoodsisthereforeretainedbythesupplier.Suchanarrangementiswidelyusedindistributionchannelsforartsandcrafts,aswellasindustrialandconsumergoodssuchaselectronics,furniture,food,books,journalsandnewspapers,etc.(seealsoWangetal.[ 59 ]).Itmayalsobeapplicabletosettingswheresupplierspromisetobuybackunsoldgoodsasaservicetotheircustomers,inordertobuildabetterrelationshipandimprovetheeciencyoftheentiresupplychain.Finally,thisstrategymaybeattractiveforproducersofcopyrightedproducts,i.e.,books,software,musicCDs,etc.Thevalueoftheseproductsliesintheircontentorknowledge,whilethecostsofproducingthemediaarerelativelylow.Theriskofpromisingbuybackisnothigh,whilethechancesofsellingcanbegreatlyincreasedbyattractingmoreretailerstodistributethegood.Intheremainderofthispaper,wewillsimplytalkabouttheplacementoforders,withnegativevaluescorrespondingtodisposal.Althoughweallowdisposal,weassumethatitisnotpossibletocancelorchangeanorderthatisalreadyplacedbutwhichhasnotyetbeendelivered.

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Inventoriesincurholdingcosts,whereasunsatiseddemandisbackloggedandincursapenaltycost.Inourmodel,weassumethatholdingandpenaltycostsarelinearwithstationaryrateshandp,respectively.Wecancombinethesetwokindsofcoststogether,refertothemsimplyasinventorycosts,andrepresentthembythefollowingcostfunction:^C(x)=theinventorycostratewhentheinventorylevelisx=8><>:hxifx0pxifx<0: 52 ],weassumethattheordering(purchasing)costs,saycarepaidwhentheorderisreceived(i.e.,aftertheleadtime).Allcostsarediscountedatarateof.Atthetimewhenanorderingdecisionismade,theobservedunitorderingcostisthusadiscountedone,whichwedenotebyc= Wewilloftenbeusingthetotaldemandoccurringduringtheleadtime.Notethatthistotaldemanddependsontheobserveddemandhistory,butnotthehistoryoftheinventoryposition,sinceweassumedthatthedemandprocessisexogenous.Also,asremarkedinSongandZipkin[ 52 ],ifwerequirethattheleadtimesdonotcrossintime,thatis,ordersthatareplacedearlierthanotheronescanneverarrivelaterthanthese,thentheyarenotindependent.FollowingSongandZipkin[ 52 ],weignoretheimpactofleadtimehistoryinmakingorderingdecisionsbecausewelacktheabilitytocollectandprocesssuchinformation,andtreatthembystandardapproachproposedbyHadleyandWhitin[ 29 ].So,wedeneDt;h(t)Ltobetherandomvariablerepresentingthetotaldemandoccurringduringthetimeinterval(t;t+L],givendemandhistoryinformationh(t)2H(t).Forxed`,denotethedistributionfunctionofDt;h(t)`byFDt;h(t)`(z),anddene~FDt;h(t)L(z)=Z10e`FDt;h(t)`(z)dFL(`):

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Nowtheconditionalexpecteddiscountedholdingandshortagecostrate,attheendofaleadtimestartingfromthecurrenttimet,andviewedattimet,giventhatthedemandhistoryish(t),andthecurrentinventoryposition(afterorderingdecision)isy,canbewrittenas

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Iftheinitialinventorypositionisx,theinitialdemandhistoryh(0)isobserved,andsomeorderingpolicyyisfollowed,thetotalcostsmaybeexpressedas =cx+Ecy(0;h(0))+Z10etcd(y(t;H(t)))+EZ10etcd(D(t))+EZ10etC(y(t;H(t));t;H(t))dt: Inequation( 2.1 ),theexpectationistakenovertheentiredemandprocessfromtime0totheinnitehorizon.Insidetheexpectation,thersttermrepresentstheorderingcostattime0;thesecondtermrepresentstheorderingcostsforreplenishinginven-toryduringtheentireinnitehorizonsincetherateatwhichweorderattimetisd(y(t;h(t))+D(t));andthethirdtermrepresentsthetotalinventoryholdingandshortagecosts.Weconsideronlythosepoliciesthatmakethetotalcostsnite.Wewillshowlaterthatsuchpoliciesdoindeedexist. Inequation( 2.2 ),notethatthetotalexpectedreplenishingcostsofallthede-mands,EZ10etcd(D(t))

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Nowconsiderarealizationoftheexpressioninsidetherstexpectationoftheabovecostfunction,i.e.,weconsideraxedhistorypathh(t)startingfromh(0).Thisexpressioncanthenbesimpliedasfollows: SoweobtainthatEcy(0;h(0))+Z10etcd(y(t;H(t)))=EZ10etcy(t;H(t))dt 2.3.1OptimalPolicy

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positionisfoundbysolvingasingleone-dimensionaloptimizationproblem.Ourrstlemmaconstructssuchamyopicpolicyforeachtimet. bythedenitionofy(t;).Therefore,bydenotingthedistributionofthehistoryuptotimetbyFH(t),wehaveEC(y(t;H(t));t;H(t))+cy(t;H(t))=ZC(y(t;h(t));t;h(t))+cy(t;h(t))dFH(t)(h(t))ZC(y0(t;h(t));t;h(t))+cy0(t;h(t))dFH(t)(h(t))=EC(y0(t;H(t));t;H(t))+cy0(t;H(t))

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2.3.1 .Ifitexistsforallt0,thenthepolicyy=fy(t;):t0gisanoptimalorderingpolicythatminimizesW(x;h(0)jy)amongallpoliciesy. 2.3.1 2.2.1 .Wewillshowthatinthatcasetheoptimalpolicyonlydependsonthecurrentstateofthecoreprocessandtheamountoftimethathasbeenspentinthisstatesincethelaststatetransition.

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costratefunctiontoC(y;i;s)=EheL^C(yDi;sL)i 2.4 )byC(y(t;H(t));A(t);S(t)).Thetotalexpectedcostfunctionforoursemi-MarkovmodulatedPoissondemandmodelthusreducestoW(x;h(0)jy)=Z10etEC(y(t;H(t));A(t);S(t))+cy(t;H(t))dt:

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Notethatsincetheleadtimedemandcanonlyassumeintegralvalues,allpointsatwhichthefunctionsC(;i;s)andfi;s()arenondierentiableareintegral.Inaddition,yi(s)isintegralaswell.Wewillnowgivesomepropertiesofthecostfunctionsandoptimalpolicy;thesepropertiesaresimilartotheonesobtainedbySongandZipkin[ 52 ]fortheMarkovmodulateddemandmodel. (b) Ifc
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of`timeunits.Thisimpliesthat Sofory<0,C0+(y;i;s)=pE[eL] and(f0i;s)+(y)=C0+(y;i;s)+c=(cp)E[eL]: 2.5 )wehavelimy!+1C0+(y;i;s)=hE[eL]0 andlimy!+1(f0i;s)+(y)=hE[eL]+c0: (c) Ifcp0,then(f0i;s)+(y)=(h+p)Z10e`FDi;s`(y)dFL(`)+(cp)E[eL]0 forally,i,ands,soyi(s)=.

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BytheresultofLemma 2.3.3 ,weconcludethatweneedtoassumethatc


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(2.7) (2.8) whereinequality( 2.7 )followsfromLemma 2.3.4 ,andinequality( 2.8 )holdssinceeL1.Then,bythedenitionofyi(s)andLemma 2.3.3 ,fi;s(yi(s))fi;s(0)<1andyi(s)<1.Thus,foreveryxedtandeveryh(t)2H(t)forwhichA(t)=i;S(t)=s,theoptimalpolicyfortimetstatedinLemma 2.3.1 doesexist,andy(t;h(t))=yi(s): 2.3.2 ,thepolicyfy(t;H(t))=yA(t)(S(t)):t0gisanoptimalpolicy.

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Furthermore,theoptimaltotalexpectedcostsatisesW(x;h(0)jy)=Z10etEC(yA(t)(S(t));A(t);S(t))+cyA(t)(S(t))dt=Z10etEfA(t);S(t)(yA(t)(S(t)))dtZ10etpE[L]dt=1 andisthusnite. 2.5 ),wehaveyi(s)=argminy:(f0i;s)+(y)0=argminy:C0+(y;i;s)+c0=argminy:Z10e`(h+p)FDi;s`(y)dFL(`)pE[eL]c=argminy:Z10e`FDi;s`(y)dFL(`)pE[eL]c h+p: sothat h+p:(2.9) ThismeansthattheoptimalpolicydependsonthecostparametersonlythroughtheratiopE[eL]c h+p=(pc)E[eL]

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Itiseasytoseethatthisratioisalwaysbetween0and1.Incasetheleadtimeisdeterministic,theexpressionfortheoptimalpolicycanbesimpliedto h+p=argminy:FDi;sL(y)pc h+p: Wenextderiveamoreexplicitexpressionofthemyopicinventorycostratefunction: FDi;s`(y)dFL(`)=Z10e`(h+p)yFDi;s`(y)pyhE[Di;sL]+(h+p)Z1yzdFDi;s`(z)dFL(`)=(h+p)yZ10e`FDi;s`(y)dFL(`)pyE[eL]hE[Di;sL]+(h+p)Z10e`Z1yzdFDi;s`(z)dFL(`): Notethatiftheinequalityinequation( 2.9 )or( 2.10 )isinfactanequality,thelastterminequation( 2.11 )reducestozero,andtheoptimalcostratereducestofi;s(yi(s))=(h+p)Z10e`Z1yi(s)zdFDi;s`(z)dFL(`)hE[Di;sL]:

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However,thisgenerallycanonlyhappeniftheleadtimedemanddistributioniscontinuous,whichisnotthecaseinourmodel. Finally,wewouldliketostressthesimilarityoftheexpressionsinequations( 2.9 )and( 2.11 )withtheoptimalpolicyandcostinthestandardnewsvendorproblem.Itturnsoutthatwecanndtheoptimalpolicyforourmodelbysolvingonenewsvendorproblemforeachiands. Weassumethataninventorypolicycharacterizedbyyi(s)isadopted.ThendeneVi(x)tobetheexpectedtotalcostsofthispolicyfromthetimewhenthecoreprocessjustentersstateiwhentheinitialinventorypositionisx,anddiscountedtothetimeoftransition.Thetotalcostscanbedividedintotwocomponents:thetotalcostsduringourcurrentstayinstatei,andthetotalcostsaftertransitioningawayfromstatei.Therstcomponentcanbedeterminedbyconditioningonthetimeuntilthenexttransition,whosedistributionfunctionisequaltoGi(t)Xj2IpijGij(t):

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Sincetheinitialinventorypositionxisnotaectedbytheorderingdecisionsmadeforstateiperiod,letVi=Vi(x)+cx: 2.3 )tosimplifytheexpressionforthecostswhileinstatei.Nowobservethatwecaninprinciplecompute,foralli,thetotalcostsfromthetimeoftransitiontostateibysolvingasystemoflinearequationsifwecancomputethetotalcostswhileinstateiforalli.Computingthetotalcostswhileinstateiisclearlystillnontrivial,butmucheasierthandirectlytryingtocomputetheinnitehorizoncosts.

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WesaythatarandomvariableXhasaConditionalPoissondistributionwithrandomparameter,whereisanonnegativerandomvariable,iftheconditionalrandomvariableXj=hasaPoissondistributionwithparameter.Thefollowinglemmawillthenproveusefullaterinthissection. 49 ]).Therefore,theassump-tioninthetheoremsaysthatE[x(1)]E[x(2)].NowdenotethedistributionofnbyHn(n=1;2).Then,forn=1;2,Pr(Xnx)=Z10Pr(Xnxjn=)dHn()=E[Pr(Xnxjn]=E[x(n)]:

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Returningtothefocusofthissection,denotethestateofthecoreprocessafterttimeunitsiftheprocesshascurrentlybeeninstateiforstimeunitsby[A(t+s)jA(s)=i].Wewillshowthatthefollowingconditionimpliesthatthefunctionyiisincreasingins: 2.4.2 ensuresthattheleadtimedemandsarestochasticallyincreasingwhileinagivenstate. 2.4.2 forsomei2I,thentheleadtimedemandDi;sLisstochasticallyincreasingins,i.e.,Di;sLstDi;s0Lforall0s
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Sincethisinequalityholdsforallxedleadtimes`,thedesiredresultfollowsforthestochasticleadtimecaseaswell. 2.4.2 2.4.2 forsomei2I,thentheoptimalinventorypositionyi(s)isincreasingins. 2.4.2 issatised.Observethatinequality( 2.12 )intheproofofLemma 2.4.3 saysthat,forxedleadtime`,forall0s
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Therefore,itimmediatelyfollowsthatCondition 2.4.2 issatisedforalli2I,andthereforetheoptimalinventorypositionyi(s)isincreasingins.Infact,wecanuseappropriatemodicationsofLemma 2.4.3 andTheorem 2.4.4 toshowthattheinventorypositionyi(s)isconstantins,whichcorrespondswiththeresultofSongandZipkin[ 52 ]. 2. SupposethattheinterarrivaldistributionsGijdependonionlyand,moreover,areincreasingfailurerate(IFR).Inaddition,supposethattransitionscanonlybemadetostateswithahigherdemandrate,thatis,pij>0impliesthati
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=PrZ`z0[A(t)jA(0)=Ji]idtxi` 2.13 )followsfromthefactthatthedistributionoftheremainingtransitiontimeZsiisindependentofthenextstatevisited.Ifz`,wehaves(z)=PrZ`0idtxjZsi=z=Pr(i`x)=1fi`xg Sincewechosethevaluesof`0andxarbitrarily,weconcludethatCondition 2.4.2 issatisedforstatei. 2.4.2 forsomei2Iand,inaddition,thetransitiontimedistributionsGijfromthatstatehavecon-tinuousdensities.Thentheoptimalinventorypositionfunctionyiisastepfunctionthatcanonlyhavestepsize1intheinventorypositionspace.

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2.4.2 issatised,andsupposethatwehavebeeninthisstateforstimeunits.Thenrecallthat,foraxedleadtime`,wehaveDi;s`ConditionalPoissonZ`0[A(t+s)jA(s)=i]dt: 2.4.4 Sincetheleadtimedemandisadiscreterandomvariablethathasstrictlypositiveprobabilityateverynonnegativeintegerd,weconcludethat,forxeds,~FDi;sL(d)isastrictlyincreasingfunctionofdford=0;1;2;:::.Thismeansthatallfunctionsinthefamilyf~FDi;sL(y):y0g,viewedasfunctionsofy,arestepfunctionsthatstrictlyincreaseateachintegralvalueofy. Weconcludethatyi(s)isastepfunctionthat,ateachstep,increasesbyexactly1. 2.4.5 ,aslongasthecoreprocessremainsinagivenstate,eachorderis,withprobability1,forasingleunitonly.

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thefollowingconditionforapairofstatesi;j2I,thentheinventorypositionneverdecreasesifatransitionismadefromstateitostatej. 2.4.6 impliesamonotonicityrela-tionshipbetweentheleadtimedemandsindierentstates. 2.4.6 forstatesi;j2I,thenDi;sLstDj;0Lforalls0: 2.4.6 andLemma 2.4.1 thenimplythat Sincethisinequalityholdsforallxedleadtimes`,thedesiredresultfollowsforthestochasticleadtimecaseaswell. 2.4.6 forthattransition. 2.4.6 forstatesi;j2Ithenyi(s)yj(0)foralls0:

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2.14 )intheproofofLemma 2.4.7 saysthat,forxedleadtime`,andforally0FDi;s`(y)FDj;0`(y): 2.9 ). Iftheunderlyingcoreprocessisacontinuous-timeMarkovprocessand,inaddition,[A(t)jA(0)=i][A(t)jA(0)=j]w.p.1,forallt0 foralli;j2Isuchthatii.

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NownotethatPrZ`0[A(t+s)jA(s)=i]dtxjZsi=0=PrZ`0[A(t)jA(0)=i+1]dtx: 2.4.1 wehaves(0)=PrZ`0[A(t+s)jA(s)=i]dtxjZsi=0E[s(Zsi)]=PrZ`0[A(t+s)jA(s)=i]dtxE[s(Zsi)]s(0) wherewehaveusedthefactthatJii+1,andthelastinequalityfollowsfromthefactthatthefunctionsisdecreasing.Sincewechosethevaluesof`0andxarbitrarily,thisimpliesthatCondition 2.4.6 issatisedfori;i+12I. 2.4.2 foralli2Iand,inaddition,Condition 2.4.6 issatisedforalli;j2Isuchthatpij>0.Thentheoptimalpolicyresultsinasequenceofinventorypositionsthatisnondecreasing. 2.4.4 saysthattheoptimalinventorypositionneverdecreasesaslongasweareinagivenstate.SinceCondition 2.4.6 issatisedwheneveritispossibletotransitionfromstateitostatej,andTheorem 2.4.8 saysthattheoptimalinventorypositionneverdecreaseswhenwemovetoanewstate.

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clearlynotoptimal.However,notethatundertheconditionsofTheorem 2.4.9 ,themyopicinventorypositionwillalwaysincrease.Thefollowingtheoremnowprovidesasucientconditionunderwhichdisposalisneverdesirable,sothatthemyopicpolicyremainsoptimalevenifdisposalisnotallowed. 2.4.2 foralli2Iand,inaddition,Condition 2.4.6 issatisedforalli;j2Isuchthatpij>0.If,initially(attime0),thecoreprocesshasbeeninstatei2Iforstimeunits,andtheinitialinventoryxisnolargerthanyi(s),thenthemyopicpolicyistheoptimalpolicy. 2.4.8 impliesthatthemyopicpolicywillneverprescribeareductionininventoryposition,itremainsoptimalevenwhendisposalisnotallowed. 2.4.10 ontheinitialinventorylevelisviolated.ThistheoremisacontinuousanalogofTheorem6.2inVeinott[ 56 ]. 2.4.2 foralli2Iand,inaddition,Condition 2.4.6 issatisedforalli;j2Isuchthatpij>0.Thenthepolicy,say~y,thatdoesnotorderuntiltheinventorypositiondropsbelowthelevelthatisprescribedbythemyopicpolicy,andfollowsthemyopicpolicyafterthat,istheoptimalpolicy.

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notplaceanyorders.Sincethecostratefunctionateachpointintimeisconvex,andthemyopicpolicyisthesmallestminimizerofthecostratefunction,policy~ywillminimizethetotalcostovertheinterval0tTamongallfeasiblepolicies.AftertimeT,Corollary 2.4.10 implieshatthemyopicpolicywillbeoptimal.Sothepolicydescribedinthistheoremisanoptimalpolicy. h+p:( 2:9 ) Notethattheoptimalinventorypolicyisthusacollectionoffunctions,oneforeachstateoftheworld.Wethereforecannotexpecttobeabletocomputeinnitetime(orrepresentusingnitestoragespace)theentireoptimalinventorypolicyforourmodel.Inaddition,sisacontinuousvariable,whichfurthercomplicatestheaprioricomputationoftheoptimalinventorypolicy.Instead,wewillinthissectiondevelopanalgorithmthatconstructspartsoftheoptimalpolicyasneededforaspecialcaseofourmodel.Inaddition,wedenoteandtorepresenttheKroneckerproductandKroneckersum,respectively. 2.9 )weseethatthekeyistheleadtimedemanddistributionfunctions~FDi;sL(y).However,ingeneralitisverydiculttocomputetheseleadtimedemanddistributionsdirectly(seeZipkin[ 63 ]).SongandZipkin[ 52 ]designedanalgorithmtocomputethemyopicpolicyforaspecialcaseoftheirMarkovmodu-latedPoissondemandmodel.Specically,theydevisedawaytocomputethecostratefunctionbyassumingthattheleadtimehasacontinuousphase-type(CPH)

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distribution,whichcanbemodeledbythetimeuntilabsorptionofacontinuous-timeMarkovchain.ThentheystudythebehaviorofajointprocessconsistingoffourMarkovprocesses:theworldprocess,demandprocess,leadtimeprocess,andtheprocessusedtorepresentcontinuous-timediscounting.Aftersomenontrivialtransformations,theycancomputetheleadtimedemanddistributionsandcostratefunction. Inthissection,weapplytheideaofthisalgorithmtoourmorecomplicateddemandmodelwhentheleadtimeLhasacontinuousphasetypedistribution.Inaddition,weassumethatthetransitiontimeforleavingeachworldstateisalsocon-tinuousphasetypedistributed.Assumethatwecannotobservethephasechangeswithinthistransitionperiod.(RecallfromSection 2.2.2 that,ifthephasechangesoftheErlangtransitiondistributions,whicharespecialCPHdistributions,areob-servable,thenwecantransformthecoreprocessintoaMarkovprocessbyusinganextendedstaterepresentation(A(t);r(t))andatransformedtransitionprobabilitymatrix.)WerstdenotetheprobabilitymassfunctionofDi;s`forxed`,giveni;s,bybi;s(dj`)=Pr(Di;s`=d) thenforrandomleadtimeL,wedeneqi;s(d)=EL[eLbi;s(djL)]=Z10e`bi;s(dj`)dFL(`): forintegervaluesofy.Inaddition,wecanwrite~FDi;sL(y)=Z10e`FDi;s`(y)dFL(`)=Z10e`yXd=0bi;s(dj`)!dFL(`)=yXd=0qi;s(d)

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forintegervaluesofy.Itiseasytoseethat1Xd=0qi;s(d)=EL[eL]: 52 ]todevelopoural-gorithm.Deneetobeacolumnvectorwhoseelementsareall1,whileeiisaunitcolumnvectorwheretheithelementis1andallotherelementsare0.Forcomplete-nesssake,wenextbrieyreviewsomeresultsfromSongandZipkin[ 52 ]thatweneedtofurtherdevelopouralgorithm.WeassumethattheleadtimeLhasacontinuousphase-typedistribution(;M),whereisrowvectorwithnonnegativecomponentswhosesumisnolongerthan1,andMisannonsingularmatrixwhoseo-diagonalentriesareallnonnegativeandwhosediagonalentriesaswellasrowsumsareallnonpositive.LetUbeacontinuous-timeMarkovchainwith+1states,wherethelaststateisanabsorbingstate,initialprobabilitydistribution[;1e],andgenerator264MMe00375:

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Inaddition,weassumethattheworldtransitiontimefromstateiisacontinuousphase-typedistribution(&i;Hi),whichcanrepresentedasthetimeuntilabsorptionofacontinuous-timeMarkovchainVwithritransientstatesandoneabsorbingstates.Weassumethatforalli,&ie=1.IfwecanobservethephasechangesofthisMarkovchain,wecantranslatetheworldprocessAintoaMarkovchainwithstate(i;r),whereristhephaseoftheV.LetQdenotethegeneratorofthetransformedworldprocessofdimensionPmi=1ri.Forexample,fori6=j,therateq(i;r0);(j;r00)=(Hie)Ter0pij(&j)Ter00,where(Hie)TrepresentsthetransposeofmatrixHie. Foracontinuousphasetypedistribution(&i;Hi),theprobabilitythatitisineachstateafterstimeunits,denotedbyi;s,isthesolutiontothefollowingdierentialequation(s): withboundaryconditioni;0=&i.Itiseasytosolvethati;s=&ieHis.Thusgiventhatthelasttransitionwasintoworldstatei,theconditionalprobabilitythatthephaseoftheCPHdistributionafterspendingstimeunitsinthecurrentstate,denotedbyr(s),isequaltor,denotedbyRi;s(r),canbecomputedasRi;s(r)=i;ser SinceaCPHdistributionisinterpretedintermofaMarkovchain,giventhecurrentstateoftheMarkovchain(i.e.,thephaseoftheCPH),thetimethathas

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elapsedsincetheCPHdistributionstarted,s,becomesirrelevantduetothememo-rylesspropertyofacontinuous-timeMarkovchain.Sowecandene Pr(Di;s`=djr(s)=r)=Pr(Di;0`=djr(0)=r)bi(dj`;r) (2.16) aswellas ByconditioningonthephaseoftheCPHworldtransitiontime,wehavebi;s(dj`)=riXr=1Ri;s(r)Pr(Di;s`=djr(s)=r)=riXr=1Ri;s(r)bi(dj`;r) wherewehaveusedequation( 2.16 )andqi;s(d)=Z10e`riXr=1Ri;s(r)Pr(Di;s`=djr(s)=r)dFL(`)=riXr=1Ri;s(r)Z10e`Pr(Di;s`=djr(s)=r)dFL(`)=riXr=1Ri;s(r)qi(djr) wherewehaveusedequation( 2.17 ).NowwecanuseSongandZipkin'sapproachtocomputethefunctionqi(djr)foreveryiandr.Thedierencehereisthatweuseacompositestate(i;r)toreplacetheworldstateiinSongandZipkin'smodel,andchangeourworldprocessintoaMarkovprocess,aswedidinSection 2.2.2 WeassumethattheworldprocessA(notewehaveconvertedthestateofthisprocessinto(i;r))andthedemandprocessDstopchangingandremainxedwhenU(t)=+1.ThusforanyrealizationoftheprocessA;UandD,thenalvalueof

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therstofDispreciselyarealizationoftheleadtimedemand.Toincorporatethediscountfactor,weconstructanauxiliarycontinuous-timeMarkovchainJ,indepen-dentofA;UandD,withtwostates,aninitialstate0anda\killing"state1.WestatewithJ(0)=0,andstate1isabsorbing.WhileU(t),thetransitionratefromstate0to1isthediscountfactor;whenU(t)=+1,theprocessJstopschangingandremainsxed.ThustheprobabilitythattheprocessJisnotkilledbytheendoftheleadtimeisPr(X>L)=Z10Pr(X>`jL=`)dFL(`)=Z10e`dFL(`)=E[eL]: 52 ],wecangetqi(djr)=Pr(D(L)=d;J(L)=0jA(0)=i;r(0)=r):

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Nowwecanexpressthediscountedcostratefunctionas (2.18) where~C(y;i;r)=hyXd=0(yd)qi(djr)+p1Xd=y(dy)qi(djr)=(h+p)yXd=0(yd)qi(djr)+p1Xd=0(dy)qi(djr)=(h+p)yXd=0(yd)qi(djr)+p1Xd=0dqi(djr)pyEL[eL]=p(eir)(IH)2(IHK1)(ee)p(y1)(IM)1Me+(p+h)(eir)"yXd=0(yd)Hd#(IHK1)(ee): 2.3 ,theoptimalinventorypositionywillminimizefi;s(y)ifaty

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therightderivativeisgreaterthanorequalto0,whiletheleftderivativeissmallerthan0. Weneedaresultregardingthechangesofoptimalinventorypositionwheneachoftheworldtransitiontimedistributioniscontinuous.TheproofisverysimilartothatofTheorem 2.4.5 ,thusomittedhere. 2.5.1 ,aslongastheworldstateremainsunchanged,itwillchangebyone,eitherincreasingordecreasing.ThissituationisillustratedinFigure 2{1 throughFigure 2{3 .Sotodetermineafterhowlongtheoptimalinven-torypositionwillchangetoadierentvalue,weneedtocomputetheleftandrightderivativesof~C(y;i;r)atyforallr,andmakinguseofequation( 2.18 ).Inotherwords,weneedtosolveeachofthefollowingtwoequations, (2.19) Onlythesolutionsforstotheabovetwoequationsarecandidatetimesatwhichtheoptimalinventorypositionwillchange.Lets+1s+2ands1s2bethesolutionstoequations( 2.19 )and( 2.20 )thatarestrictlygreaterthans0,respectively.Onlythesesolutionsarecandidatetimesatwhichtheoptimalinventorypositionwillchange.Notethatitispossiblethateitheroftheseequationsdoesnothavesuchasolution.Ifneitherequationhassuchasolution,weknowthatthecurrent

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optimalinventorypositionwillcontinuetobeoptimalaslongastheworldstatedoesnotchange.Forequation( 2.19 ),wecheckitscandidatesolutionsasdescribedaboveinincreasingorder,tondoutthesmallestoneatwhichtheright-hand-sideoftheequationhasanegativederivative,anddenoteitbys0.Ifnosuchsolutionexist,welets0=1.Wefollowasimilarprocedureforequation( 2.20 ),exceptthatwenowchoosethesolutionatwhichthederivativeisgreaterthan0.Denotethatsolutionbys00.Ifs0s00,thenafters00timeunitstheoptimalinventorypositionwilldecreasebyone.Ifboths0ands00areinnite,thentheoptimalinventorypositionwillremainunchangedunlesstheworldstatechanges.Notethatitisnotpossiblethats0=s00<1,whichwouldmeanthatattimes0=s00thefunctionC(y;i;s)+cyhasapositiveleftderivativeandanegativerightderivative,whichcontradictsthefactthatitisconvex. Eachtimetheworldjustenteranewstatei,wethenknowprobabilitythattheworldisineachstate,whichisderiveddirectlyfromtheinitialdistributionoftheworldtransitiontime,i.e.,Ri;0(r)=&er,andwecomputetheoptimalinventorypositionforthistimepoint.Wecanthenrepeattheproceduresdescribedabovetocomputewhentheoptimalinventorypositionswillchange. Nowturntothecalculationoftheleftandrightderivativeof~C(y;i;r)withrespecttoyforsomexedi,r.Therightderivativeatintegervaluey(recallthatwehaveprovedtheoptimalinventorypositionscanonlybeintegers)is~C(y;i;r)0+=pE[eL]+(p+h)yXd=0qi(djr) andtheleftderivativeis~C(y;i;r)0=pE[eL]+(p+h)y1Xd=0qi(djr):

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Ify<0,thentherightderivativeatyispE[eL]=p(IM)1Me; ify0,therightderivativecanbewrittenaspE[eL]+(p+h)(eir)[yXd=0Hd](IHK1)(ee): 52 ]asdescribedabovetocompute~C(y;i;r)andtheoptimalvalue,anddenoteitbyy0.Setk=0,anddenotes0=0. 2.21 )and( 2.22 ) (2.21) Lets+1s+2:::bethesolutionstoequation( 2.21 )thatarestrictlygreaterthansk.Checktheminincreasingorder,andlets0bethesmallestsolutionat

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Figure2{1: Optimalinventorypositionaty Optimalinventorypositionaty+1 whichtheleft-hand-sideofequation( 2.21 )hasanegativederivative,lets0=1ifnosuchsolutionexists. Similarly,lets1s2:::bethesolutionstoequation( 2.22 )thatarestrictlygreaterthansk.Lets00bethesmallestsolutionatwhichtheleft-hand-sideofequation( 2.22 )hasapositivederivative,lets00=1ifnosuchsolutionexists.

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Figure2{3: Optimalinventorypositionaty1 ItisobviousthatthecomputationinvolvestheevaluationoftheprobabilityRi;s(r).AndforgeneralCPHworldtransitiontime,itisdiculttohandleRi;s(r).Inthenextsection,wewillgiveanimplementablealgorithmbyconsideringtheworldtransitiontimeasErlangdistributed.

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thefollowingformhjr=8>>>><>>>>:iifr=jiifr=j+10o/w: (lr+j+1)!(rj)!=l+1i(1)l+1r+j(l+1)! (l+1r+j)!(rj)!=l+1i(1)l+1r+j0B@l+1rj1CA:

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Tosummarize,wehavehl+1jr=8>>>><>>>>:l+1i(1)l+1r+j0B@l+1rj1CAifrjandrjl+10o/w: 2.5.2 ,wehavei;ser=&ieHiser=&i1Xl=0(His)l (r1)!(lr+1)!=(is)r1

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Fromtheprevioustwolemmas,forErlang(i;ri;i)distribution,wecangetthat =(is)r1eis ByreplacingRi;s(r)bythevaluesinequations( 2.19 )and( 2.20 ) 0=riXr=1(is)r1 0=riXr=1(is)r1 Atthetimewhentheworldjustenteranewstatei,i.e.,s=0,weknowthattheworldmustbeintherststageoftheErlangdistribution,soRi;0(1)=1,andRi;0(r)=0forallr=2;:::;ri.Thus,fors=0,wehaveC(y;i;0)=~C(y;i;1).Wecanthenrepeattheproceduresdescribedabovetocomputewhentheoptimalinventorypositionswillchange. Wehavethefollowingalgorithmbymakingtheaccordingchanges. 52 ]asdescribedabove,computetheoptimalvalueanddenoteitbyy0.Setk=0,anddenotes0=0.

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2.27 )and( 2.28 ) (2.27) Lets+1s+2:::bethesolutionstoequation( 2.27 )thatarestrictlygreaterthansk.Checktheminincreasingorder,andlets0bethesmallestsolutionatwhichtheleft-hand-sideofequation( 2.27 )hasanegativederivative,i.e.,riXr=2(i)r1~C(yk;i;r)0++c Similarly,lets1s2:::bethesolutionstoequation( 2.28 )thatarestrictlygreaterthansk.Lets00bethesmallestsolutionatwhichtheleft-hand-sideofequation( 2.28 )hasapositivederivative,i.e.,riXr=2(i)r1~C(yk;i;r)0+c

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2.6.1GeneralizationoftheDemandProcessModel Bykeepingalltheothermodelassumptionsasbefore,theresultsdevelopeduptoSection 2.3.1 continuetoholdwithoutanychangessincethePoissonnatureofthedemandprocessisnotused,i.e.,thecostfunctionandthegeneralformoftheoptimalpolicy(asafunctionoftheentirehistory)remainthesame.However,nowthehistorycannolongerbesummarizedbythecurrentworldstateiandthetimespentinthisstatesonly.Thepropertiesofthesemi-MarkovmodulatedrenewaldemandprocessimplythatDt;h(t)Ldependsonthehistorythroughnotonlythestatethattheprocessiscurrentlyin(i)andhowlongithasbeeninthatstate(s),butalsohowlongithasbeensincethelastdemandoccurred,whichwewilldenotebyB(t).Toreectthisfact,werewritetheleadtimedemandasDi;s;bLwhenthecore

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processhasbeeninstateiforstimeunits,andbtimeunitshaveelapsedsincethelastdemand. Oneimportantissuethatweneedtopayattentiontoisthatbeforetherstoccurrenceofademandwithinaworldstate,theamounttimesincethelastdemandbgenerallyisnotequaltotheamountoftimesspentinthecurrentstate,sincethelastdemandwilllikelyhaveoccurredwhileintheprecedingstate(orevenearlier,ifnodemandoccurredwhileintheprecedingstate).Thus,bnotonlyincludestimespentinthecurrentstate,butalsosomeamountoftimespentinpreviousstate(s).However,sincethedistributionofthetimebetweendemandschangesbetweenstatesoftheworld,neithersnorbseemstobeanaccuratemeasureofthetimesincethelastdemandforthecurrentinterarrivaltimedistribution. Tohandlethissituation,weshouldinfactletthedemandprocessinagivenstatebeadelayedrenewalprocess,wherethedistributionoftherstinterarrivaltimedependsonthetimesincethelastdemandaswellasthepreviousstatevisited.Inparticular,supposewearecurrentlytransitioningfromstatejintostateiwithgenericinterarrivaltimeXiKi,andletthetimesincethelastdemandbeb.WethenlettherstinterarrivaltimebedistributedasXij;i(b)jXi>j;i(b) wherej;iissomefunctionthattransformstheamountoftimethathaselapsedsincethelastdemand.Forconvenience,wewillinfactsimplyredenebtobeequaltoj;i(b)atthemomentoftransitionfromstatejtostatei.Notethatthepreviousdemandmightactuallyhaveoccurredinastatethatwasvisitedbeforestatej.Inthatcase,byrecursivelyupdatingthetimesincethelastdemandusingtheappropriatetransformationfunctions;willappropriatelydenetherstinterarrivaltimedistributionineachstate.

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Intuitively,itseemsclearthatweshouldchoosethefunctions;tobenonde-creasing.Twointerestingextremecasesare;(b)=0,wherewesimply\forget"thetimesincethelastdemandatthemomentoftransitionbetweenworldstates,and;(b)=b,whereweignorethefactthattheinterarrivaltimebetweendemandsisdierentindierentstates.IfwedenethegeneralizedinverseofdistributionfunctionKbyK(p)=minfy:K(y)pg 2.3.2 toaccommodatethechangesinthedemandmodel.Forgiveny,t,andh(t),wecansimplifytheinventorycostratefunctiontoC(y;i;s;b)=EheL^C(yDi;s;bL)i

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Next,wegeneralizesomeofthekeyresultsobtainedforthesemi-Markovmod-ulatedPoissondemandprocessthatcontinuetoholdforournewdemandmodel.DenotetherightderivativesofC(;i;s;b)andfi;s;b()byC0+(y;i;s;b)=lim"#0C(y+";i;s;b)C(y;i;s;b) (b) Ifc
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2.3.5 Finally,wewillgiveamoreexplicitcharacterizationoftheoptimalinventoryposition,whichcaninprinciplebeusedtocomputetheoptimalpolicy,aswellasthecostoftheoptimalpolicy.Dene~FDi;s;bL(y)=Z10e`FDi;s;b`(y)dFL(`): h+p: h+p=argminy:FDi;s;bL(y)pc h+p:

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thatcase,themyopicpolicyremainsoptimalevenifnegativeordersarenotallowed.Foraspecialcasewhereboththeworldtransitiontimedistributionsandleadtimedistributionarecontinuousphasetypedistributed,wegiveanalgorithmtocomputetheoptimalinventorypositions.Finally,weextendthemodelbyrelaxingthedemandprocessfromasemi-MarkovmodulatedPoissonprocesstoageneralsemi-Markovmodulatedrenewalprocess,andseethatthisrelaxationreallydoesnotaecttheformoftheoptimalpolicy.

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2 toamorecomplexcaseinwhichthedemandprocessisastate-dependentPoissonprocess,buttheunderlyingcoreprocess(world)isnotdirectlyobservable.Whatwecanobserveisonlythearrivalofthecustomerdemands,andwecanusethatinformationtoobtaininferenceonthestateoftheworld.Thisscenarioisverycommoninrealsituations,andthusofsignicantpracticalinterest. Thechapterisorganizedasfollows.InSection 3.2 westudyamodelwithonlytwoworldstates.Wedescribethedierencebetweenthismodelandourpreviousmodelsandshowhowthefactthattheworldstateisunobservableaectstheopti-malpolicy.Wealsogivetheformofthisoptimalinventorypolicyandprovideanalgorithmtoincrementallydeterminetheoptimalpolicy.Thenweextendthebasictwo-statemodeltoamultiple-stateoneinSection 3.3 andderivearecursiveformulatohelpdeterminetheoptimalinventorypolicy.InSection 3.3.4 wegeneralizethisresulttoamoregeneralmultiple-statemodel.Finally,wesummarizethechapterinSection 3.4 andprovidesomefutureresearchdirections. 65

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state1tostate2.Asbefore,weassumethatthedemandprocessisindependentofthereplenishmentdecisions.Implicitly,wetreatthesystemasifwestartourobservationswhentheworldjustentersstate1.Putdierently,eveniftheinventorysystemstartedatsomepointinthepastbeforeobservationsstart,weassumethatweknowthedistributionGofthetimethatthesystemwillremaininstate2.Notethatifthereisapositiveprobabilitythatthetransitiontostate2hasalreadyhappenedatthetimeobservationsstartthiscanbeincorporatedbydeningGtohaveapositiveprobabilitymassat0. RecallthatthecumulativedemandbytimetisdenotedbyD(t).Then,attimetwewillhaveobservedD(t)=fD(u):0utg: 2.3.1 canbeusedwithoutchanges,exceptforthecontentofthehistoryH(t). H(t)=fN(t);X1;X2;:::;XN(t)g=fN(t);S1;S2;:::;SN(t)g

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Wedenotethestateoftheworldbythestochasticprocessf(t);t0g.Inparticular,(t)isarandomvariablethatisequaltoiiftheworldisinstatei(i=1;2)attimet.Then((t);H(t))isajointmixedrandomvariablewithjointprobabilitydensityfunctionf(i;h(t)).DenotetheconditionalprobabilitythattheworldisinstateiattimetforeverytgiventhatthehistoryinformationuptotimetisH(t)=h(t),byp(i;t;h(t))=fjH(ijh(t)).ByconditioningontheworldchangingstateattimeS=s,anddenotingtheconditionaldensityfunctionofhistorybyfHjS(h(t)js),weobtain Themeaningofeachdensity(conditionaldensity)functionfshouldbeclearwithinitscontext. NowletuslookattheconditionaldensityfunctionofhistoryH(t)givenS=smoreclosely.Forh(t)=fN(t)=n;S1=s1;:::;Sn=sng,letfHjS(n;s1;:::;snjs)denotetheconditiondensityofN(t)=n;S1=s1;:::;Sn=sngiventhatS=s,andletfHjN;S(s1;:::;snjn;s)denotetheconditionaldensityofS1=s1;:::;Sn=sngiventhatN(t)=nandS=s. Ifs>t,theworldisstillinstate1bytimet,and

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Ifs
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ItiseasytoseethatthisisacontinuousfunctionoftifthetransitiontimedistributionfunctionGiscontinuous. Nowwecomputetheleadtimedemanddistributiongivenhistoryh(t).Letg(s)bethedensityfunctionofthetransitiontimedistribution,andg(sjh)betheconditionaldensityfunctionofthetransitiontimedistributiongivenhistoryh(t).Conditioningonlyonwhichstatetheworldisinnowisnotenough,sincewealsoneedtoknowhowlongtheworldhasbeeninthecurrentstatetodeterminethetheremaininglifetimedistribution.Sowhatweneedistoconditiononthetimeofthestatetransition.Forxedleadtime`, 3.1 )that (3.3) =FD2;0`(z)1p(1;t;h(t))+FD1;t`(z)p(1;t;h(t)):

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Whentheleadtimeisstochastic,weobtain ~FDt;h(t)L(z)=p(1;t;h(t))Z10e`FD1;t`(z)dFL(`)+(1p(1;t;h(t)))Z10e`FD2;0`(z)dFL(`): Thenwecanwrite Itisobviousthattheconditionalprobabilityfunctionp(1;t;h(t))aswellasthecostratefunctionsC(y;1;t)andC(y;2;0)playkeyrolesinthepartiallyunobservablemodel.Tomakethemodeltractable,weconsiderthecasewherethestatetransitiontimedistributionGisexponentialwithrate,i.e.,theworldprocessisaMarkovprocess,inmoredetailintheremainderofthissection. (3.6)

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wherewedeneDiLtobetherandomvariablerepresentingthetotaldemandduringaleadtime,startedfromnow,giventheworldiscurrentlyinstatei.Thelastequalityin( 3.6 )followsfromthefactthattheworldprocessfollowsaMarkovprocessanddemandisaPoissonprocess,andifweknowwhichstatetheworldisinnow,thetimethathaselapsedinthecurrentstatebecomesirrelevant.Weseethatweobtainthesameresultasbyconditioningonthetransitiontime.Andaccordingly, (3.7) whereC(y;i)isdenedastheconditionalexpecteddiscountedholdingandshortagecostrate,attheendofaleadtimeandviewedfrom,giventhattheworldiscurrentlyinstatei,andthecurrentinventoryposition(afterorderingdecision)isy.(SeealsoSongandZipkin[ 52 ].) Whencomparingequations( 3.5 )and( 3.7 )weseethatincaseGistheexpo-nentialdistributiontheinstantaneouscostratefunctionsimpliesconsiderably,andthedependenceontimeandhistoryisthenrestrictedtotheconditionalprobabilityfunctionp(1;t;h(t)).Intheremainderofthissectionwefocusonthecomputationandanalysisofthisfunction.Theseresultswillthenbeusedinthenextsectiontocomputetheoptimalinventorypolicy. UsingthefactthatG(t)=1etanddening=12+weobtain 1+e(12)t= G(t)Pnk=0(2=1)nkRsk+1ske(12)sdG(s)=1 1+e(12+)tPnk=0(2=1)nkRsk+1ske(12+)sds=1 1+etPnk=0(2=1)nkRsk+1skesds:

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Case1: If6=0then 1+(=)etPnk=0(2=1)nk(eskesk+1)=1 1+(=)etPn1k=0(2=1)nk(eskesk+1)+esnet=1 1=+(=)etPn1k=0(2=1)nk(eskesk+1)+esn =1 1=+(=)e(tsn)1+esnPn1k=0(2=1)nk(eskesk+1): limt!1p(1;t;h(t))=8><>:0if>01 1=if<0: ispositive,anditincreasesmonotonelyintifitisnegative.Moreintuitively,itfollowsthatthefunctionp(1;t;h(t))decreasesmonotonelyintiftheprobabilitythatweareinstate1atthetimeofthenthdemandexceedsthelimitingprobability1=(1=)andincreasesmonotonelyintotherwise. Case2: If=0then 1+etPnk=0(2=1)nkRsk+1skesds=1 1+Pnk=0(2=1)nkRsk+1skds=1 1+Pnk=0(2=1)nk(sk+1sk)=1 1+t+Pn1k=0(2=1)nk(sk+1sk)sn:

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Itiseasytoseethat,inthiscase,thefunctionp(1;t;h(t))decreasesmonotonelyand limt!1p(1;t;h(t))=0: 3{1 through 3{4 illustratethedierentbehaviorsoftheprobabilityfunc-tionp(1;t;h(t))withdierentparameters. Weclosethissectionbyprovidingasummaryregardingtheprobabilityfunctionp(1;t;h(t)).Thisfunctionisalwaysmonotone,butthenatureofmonotonicityde-pendsonthesignof=12+aswellastheobservedhistory.Inparticular,if0theprobabilityalwaysdecreasesmonotonelyto0,whileif<0theprobabilitywillconvergemonotonelytoapositivelimit. 2.3.2 forthepartiallyunobservabledemandmodel.First,thefollowingtheoremshowsanimportantpropertyoftheoptimalpolicy. 3.2 ),weseethatp(1;t;h(t))isacontinuousfunctionoftsincethetransitiontimedistributionisexponentialandthuscontinuous.Fromequations( 3.6 )and( 3.4 ),itisobviousthat~FDt;h(t)L(z)iscontinuousintalso.Finally,theleadtimedemandisaadiscreterandomvariablehavingstrictlypositiveprobabilitymassateverynonnegativeintegervalueduetothenatureofthePoissonprocess.SofromLemma 2.3.1 ,eachtimetheoptimalinventorypositionchanges,itwillchangetoaneighboringinteger,i.e.,eitherincreaseby1ordecreaseby1. 2.5 andinSongandZipkin[ 52 ]tocomputetheoptimalinventorypolicyforthe

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Figure3{1:

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Figure3{3: Figure3{4:

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partiallyunobservabledemandmodel.FromTheorem 2.3.2 ,theoptimalpolicymin-imizesthefunctionC(y;t;h(t))+cyforeveryt.Wewanttondy(t;h(t)),andwecanusethemethodinSection 2.5 tocomputetheoptimalinventorylevels.Notethaty(t;h(t))isjustthepointatwhichtherightderivativeofC(y;t;h(t))+cyisnosmallerthan0,whiletheleftderivativeissmallerthan0.Usingequation( 3.7 ),itiseasytoseethatC(y;t;h(t))0+=p(1;t;h(t))(C(y;1))0++(1p(1;t;h(t)))(C(y;2))0+C(y;t;h(t))0=p(1;t;h(t))(C(y;1))0+(1p(1;t;h(t)))(C(y;2))0 52 ]. Att=0oratapointintimewhereademandoccurs,sayt,wecancomputetheprobabilityp(1;t;h(t)aswellastheoptimalinventorypositiony(t;h(t)atthatpointintime(forconveniencedenotedsimplybyyinthefollowingiftheargumentsareclearfromthecontext).Aswewillshowbelow,weareabletodeterminethetimeatwhichtheoptimalinventorypositionwillchangeifnonewdemandoccursbythattime.UsingananalogousapproachasinSection 2.5 ,weneedtosolvethefollowingtwoequationstodeterminethetimetatwhichtheoptimalinventorypositionwillchangeifnonewdemandoccursuptotimet: or,equivalently,

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Denotethesolutiontoequation( 3.10 )byt0andthesolutiontoequation( 3.11 )byt00.If6=0wecanuseequation( 3.8 )tondthesesolutionsexplicitly: Similarly,if=0wecanuseequation( 3.9 )toobtain (3.14) Onlysolutionsfort0andt00whicharelargerthansnwillbeconsidered.Ifeitherorbothofthesesolutionsarelessthanorequaltosn,itsimplymeansthattheoptimalinventorypositionwillnotincrease(ordecrease,orneither)fromtimesnonwardsifnonewdemandoccurs.Itwillprovetobeconvenienttoreplaceavalueoft0ort00thatdoesnotexceedsnby1.Moreover,afterobtainingt0andt00,wealsoneedtocheckthederivativeoftherightderivativeatt0andthederivativeofleftderivativeatt00.Ifthederivativeoftherightderivativeofthecostratefunctionatt0isnonnegativethentheoptimalinventorypositionwillnotchangetoy+1andwesett0=1.Similarly,Ifthederivativeoftheleftderivativeofthecostratefunctionatt00isnonpositivethentheoptimalinventorypositionwillnotchangetoy1andsowesett00=1. Now,ift0t00thenattimepointt00theoptimalinventorypositionwilldecreaseby1unit.Ift0andt00arebothinnitythentheoptimalinventorypositionwillremainunchangeduntilanewdemandoccurs.Notethatitis

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notpossiblethatt0=t00<1sincethiswouldmeanthatrightaftertimet0theleftderivativeatyispositiveandtherightderivativeatyisnegative,whichviolatestheconvexityofthecostratefunction. Tosummarize,wehavethefollowingalgorithm 52 ],computetheoptimalvalue,anddenoteitbyy0.Setn=0andsn=0.Also,setm=0andtm=0.(nrecordsthenumberofdemandsthathaveoccurredsofar,whilemrecordsthenumberoftimesthattheoptimalinventorypositionhaschangedsofar.) 3.12 )and( 3.13 )orequations( 3.14 )and( 3.15 )(withyreplacedbyym).Ift0
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startingtimeisp1(sothattheprobabilitythattheworldisinstate2is1p1)andletGdenotetheconditionaldistributionofthetimeremaininginstate1giventhatwearecurrentlyinthatstate.Thenbyconditioningonwhetherweareinstate1attime0weobtainthatequation 3.2 becomes 1+e(12)t= G(t)Pnk=0(2=1)nkRsk+1ske(12)sdG(s): Letusstartwithm=3,thatis,thereare3worldstates,andtheywillbeencounteredintheorder1,2,3.Oncetheworldentersstate3,itwillstaythereforever.Thetransitiontimeinstatei,i=1;2,isexponentiallydistributed,withratei.Instatei,thedemandprocesspossessesratei. Asintheprevioussection,weuse(t)equaltoitoindicatewhethertheworldisinstateiattimet,i=1;2;3.Denotetheconditionalprobabilitythattheworld

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isinstateiattimetforeverytgiventhatthehistoryinformationuptotimetisH(t)=h(t),byp(i;t;h(t))=fjH(ijh(t)).FollowingfromBayes'rule,weobtain Tocomputethisconditionalprobability,weneedtoconditiononbothpossibleworldstatetransitiontimesinstates1and2,namelyT1andT1+T2,whereT1G1andT2G2.ThescenariosneedtobeconsideredareT1>t,T1+T2>t>T1,andt>T1+T2.LetusconsidercomputingthehistorydensityfunctionfH(h(t))rst.ByconditioningonT1=1(andT2=2),anddenotetheconditionaldensityfunctionofhistoryinformationbyfHjT1(h(t)j1)(andfHjT1;T2(h(t)j1;2)),weobtain sincethedierentworldstatetransitiontimeareindependent.

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Itisobviousthatthedensityfunctionofthehistoryh(t)canbedecomposedasfH(h(t))=f(1;h(t))+f(2;h(t))+f(3;h(t)) andf(1;h(t))=Z1tfHjT1(h(t)j1)dG1(1)f(2;h(t))=Zt0Z1t1fHjT1;T2(h(t)j1;2)dG2(2)dG1(1)f(3;h(t))=Zt0Zt10fHjT1;T2(h(t)j1;2)dG2(2)dG1(1) ThefollowingcomputationandthenotationsusedaresimilarasthoseinSection 3.2.1 .Forexample,forhistoryh(t)=fN(t)=n;X1=x1;:::;Xn=xng=fN(t)=n;S1=s1;:::;Sn=sngwhereXiistheinterarrivaltimeoftheithdemandandSiisthearrivaltimeoftheithdemand,letfHjT1(n;s1;:::;snj1)denotetheconditionaldensityofN(t)=n;S1=s1;:::;Sn=sngiventhatT1=1,andletfHjN;T1(s1;:::;snjn;1)denotetheconditionaldensityofS1=s1;:::;Sn=sngiventhatN(t)=nandT1=1.ForadditionalconditionsonT2,T3,thenotationsarestraightforward. If1>t,thentheworldisinstate1attimet,and If1
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respectively.Attimet,forgiven1and2,thesenumbersareknown,denotedbyn1(1)andnn1(1)respectively,andtheoccurrencesofdemandsindierentworldstatesareindependent!Thenweobtain (3.19) wherewelets0=0andsn+1=t. Similarly,ift>1+2,thentheworldisinstate3attimet.WedenotethenumbersofdemandsoccurredineachstatebyN1,N2andN3respectively.Forgiven1,2and3,thesenumbersareknown,denotedbyn1(1),n2(2)andnn1(1)n2(2)respectively.Thenbyfollowingthesimilarproceduresasabove,weobtain

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Ifsk<1
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and Bytimetallthedemandoccurrencetimessk;k=0;nareknown,andwecancompute( 3.21 ),( 3.22 )and( 3.23 ).Pluggingtheresultsintoequation( 3.16 ),wecancomputetheconditionalprobabilitythattheworldisinstatei(i=1;2;3)attimet

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givenh(t), Denotefk(h(t))=fk(Dk(u):0ut)torepresentthedensityfunctionthatpartofthehistorywhichstartsatthestartingtimeofworldstatektotaketheinstancefDk(u):0utg.Wealsodenotefk(h(t);j)torepresentthe(joint)densityfunctionofhistoryinformationuptotimet,h(t),andtheworldstateattimet,j,giventhatthehistoryobservationstartsatthestartingtimeofstatek. NowbyconditioningontherststatetransitiontimeTm=m,wegettwopossibilities:iftm,weconsideritaproblemwithm1states,startingattimem.DenotetheconditionaldensityfunctionofhistorywhichstartsinworldstatemgivenTm=mbyfm(h(t)jm),we

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obtain whereform>t,fm(h(t)jm)=emtnmGm(t),whichisderivedfromtheresultsoftheprevioussection. Forthecasem
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Sothegeneralrecursiveformulacanbewrittenas

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andtherecursiveformulagivesthesameresultsasthosegivenbydirectmethodintheprevioussection. Weneedtopointoutthatthisrecursiveformulaonlyeasestherepresentationoftheconditionalprobabilityfunctionsgivenhistoryh(t),p(i;t;h(t))=fm(h(t);i) 3.3.1 ,andthecomputationremainsthesamecomplicated. ThelastequationfollowsfromthefactthattheworldisaMarkovprocessanddemandprocessisaPoissonprocess.Forstochasticleadtime, ~FDt;h(t)L(z)=mXi=1p(i;t;h(t))Z10e`FDi`(z)dFL(`): ThenwecanwriteC(y;t;h(t))=EheL^C(yDt;h(t)L)i=ZZe`^C(yz)dFDt;h(t)`(z)dFL(`)=mXi=1p(k;t;h(t))ZZe`^C(yz)dFDi`(z)dFL(`)=mXi=1p(i;t;h(t))C(y;i)

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whereC(y;i)hasthesamedenitionsasinSongandZipkin[ 52 ]. InprincipalwecanndtheoptimalinventorypositionsbyusingthesimilarlyalgorithmasinSection 3.2.3 ,butnowtheformoftheprobabilityfunctionp(i;t;h(t))becomesmuchmorecomplicated,andtocomputethetsatwhichtheleftorrightderivativesofC(y;t;h(t))are0isnotaneasytasknow. Tocomputetheoptimalinventoryposition,weagainneedtoassumethattheleadtimedistributioniscontinuousphase-type.Theoptimalinventorylevely(t;h(t))isjustthepointatwhichtherightderivativeofC(y;t;h(t))+cyisnosmallerthan0,whiletheleftderivativeissmallerthan0.ItfollowsC(y;t;h(t))0+=mXi=1p(i;t;h(t))(C(y;i))0+C(y;t;h(t))0=mXi=1p(i;t;h(t))(C(y;i))0: 3.2.3 ,andthedetailsareomittedhere.Themajordicultyhereistogetthesolutiont0andt00. Everytimewhenanewdemandoccurs,weshouldupdatealltheprobabilitiesp(i;t;h(t))atthetimet,andgetthenewoptimalinventorypositionatthattime

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accordingly.Thenwerepeattheproceduretocomputethenexttimepointthattheoptimalinventorypositionchangesifnodemandoccurs. Westudyamultiple-statemodelwithmstates.Asintheprevioussection,were-indextheworldstatesandnumberthemfrommthrough1.Also,weletfDm(u):0utgrepresenttheobservationofthecumulativedemandcurvebetweentime0andtwherethestatestartsinstatem.Denotefk(h(t))=fk(Dk(u):0ut)torepresentthedensitythatpartofthehistorywhichstartsatthestartingtimeofworldstatektotaketheinstancefDk(u):0utg. Ifthehistorystartsinworldstatem,conditioningontherststatetransitiontimeTm=m,andthenextworldstatevisited,i,wegettwopossibilities:iftm,thehistoryinformationinformationbytcanbedividedintotwoparts:theobservationsbeforem,i.e.,h(m),whichfollowedastationaryPoissonprocesswithparametermwithnm(m)demandoccurrences,andtheobservationsafterm.SincetheworldprocessisMarkovian,thesetwopartsofobservationsareindependent.Forthehistoryaftermandbeforetwheretheworldprocessentersstateiattimem,itisexactlythesameastheobservationsbetweentime0andtimetmforai-stateworldmodel.Denotingthattheconditionaldensityfunctionofhistoryh(t)giventherststatetransitiontimembyfm(h(t)jm),whiletheconditionaldensityfunctiongiventhersttransitiontimemandthenextstateibyfm(h(t)ji;m),weobtain

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Usingthenotationdenedabove,weobtainform
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andfk(fDk(u):0utg;j)=Zt0eknk()kk1Xi=jpkifi(fDi(u):0utg;j)dGk()for1kmand1j
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Thechapterisorganizedasfollows.InSection 4.2 ,westudythejointpricingandinventorymodelunderaprice-sensitivePoissondemandenvironmentwithoutMarkovmodulation.Werststudythemodelwherepricecanonlybesetonceatthebeginning,andgivesomepropertiesthatcanbeusedtodeterminetheoptimalsolution.Wealsogiveanalgorithmtocomputetheoptimalsolution.Nextwestudythemodelwherepricecanbecontinuouslyset.InSection 4.3 ,weextendthestudytothesemi-MarkovmodulatedPoissondemandenvironment,andshowthatwithcertainapproximation,themodelcanbesolvedinthesimilarwayasinaPoissondemandenvironment.Inallthemodels,weassumethattheorderingcostislinear,andthereisapositiveorderleadtime. 93

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Theinventorylevel(position)isreviewedcontinuously,andanordercanbeplacedanytime.Ordersplacedwillarriveafteraxedpositiveleadtime`,sothediscountedorderingcostbecomesc= Thedemandissensitivetotheprices:ifthepriceis,thenthedemandratewillbe().Wewilloftenbeusingthetotaldemandoccurringduringtheleadtime.Notethatthistotaldemanddependsonthepricingdecisionsduringtheleadtime.Ifthepricecannotbechangedaftertheinitialchoice,thenthisleadtimedemandisPoissonrandomvariablewithmean`.Ifthepricecanbechangedcontinuously,thentheleadtimedemanddependsontheentirepricefunctionduringtheleadtime. Weconsidertherevenuefromsellingtheproductandthreetypesofcosts:order-ing,holding,andshortagecosts,andourobjectiveistomaximizethetotalexpecteddiscountedprotsovertheinnitehorizon.

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Nowtheconditionalexpecteddiscountedholdingandshortagecostrate,attheendofaleadtime,andviewednow,giventhatthecurrentinventoryposition(afterorderingdecision)isy,canbewrittenas +()cy1 ()=maxy1

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Theoptimalpriceshouldbechosentomaximize(),andthemaximumprotovertheinnitehorizonis =max()=maxmaxy1 Foraxedprice,itiseasytoshow(ref.SongandZipkin[ 52 ])thatthemyopicinventorycontrolpolicywhichminimizecy+C(y)isoptimal,i.e.,theoptimalinventorypositioncanbedeterminedas h+p: Beforeweproceedfurther,wemakesomecommonassumptionsaboutthede-mandratewithregardtotheprice: 4.2.1 ,denotedby. Foraxedprice,theleadtimedemandisaPoissondistributedrandomvariablewithrate()`.Withoutlossofgenerality,wecanscaletheleadtimetobe1,anddonotconsiderthe`explicitlyinthefollowing.Andfrom 4.2.2 ,weassumethat()isstrictlydecreasing,thusandhaveone-to-onecorrespondence.Sowewilluseasthecontrolvariableinsteadofthroughoutthechapter.WeuseDinsteadofD`;torepresenttheleadtimedemand,andFinsteadofFD`;torepresentitscumulativedistributionfunction(cdf).Wealsousey()insteadofy()torepresenttheoptimalinventorypositioncorrespondingtoprice(orequivalently,demandrate).Wemakethefollowingassumptionsregarding.

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Itiseasytoverifythat(()c)isconcavefortwomostfrequentlyusedcasesofdemandratefunctions():linearfunction,()=(1b);andexponentialfunction,()=eb. Withalltheseassumptions,theleadtimedemandisaPoisson()randomvari-able,andthecdfisF(y)=yXx=0Pr(D=x)=yXx=0ex Sothecdfofleadtimedemandisconvexfor>y,andconcavefor
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4.3 ),andthefactthatF(y)isacontinuousanddecreasingfunctionof. 4.2.4 ,thefunctionC(y();)isdiscontinuousin.Denotethediscontinuitypointsofy()by1<2<.Foreachsuchrangeof,y()isconstant,andC(y(m+1);)iscontinuousin.Forexample,for2(m;m+1],y()=y(m+1),wherey(m+1)satises h+p: Andfor2(m;m+1),F(y(m+1))>pc h+p: 4{1

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ThoughC(y();)isdiscontinuousin,wecanshowthatthefollowingpropo-sitionholds. wherethesecondtolastequationfollowsequation( 4.4 ).SoC(y();)+cy()isindeedcontinuouseverywhereinforitsallowablerange. wherethelastinequalityfollowsagainthedenitionofyi(m+1)inequation( 4.4 ).Wesummarizethisresultinthefollowingproposition:

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Thesecondorderderivativewithrespecttofor2(m;m+1]form=1;2;:::equalsd2[C(y(m+1);)+cy(m+1)] 4.3 ))tohelpdeterminetheoptimaldemandrate,andthustheoptimalprice.Recallthatweassumethat()isstrictlydecreasing,thereversefunctionexists,anddenotedby().LetR()=(()c).NowwewanttomaximizeMP()=()=R()C(y();)cy()overallfeasible.Byassumption 4.2.3 ,MP()isapiecewiseconcavefunctionforeachrange(m;m+1].Andforsomem,and2(m;m+1),MP()=R()C(y(m+1);)cy(m+1)=R()e24(h+p)y(m+1)1Xj=0F(j)+ppy(m+1)35cy(m+1):

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whiletheleftderivativeattherightendpointm+1,denotedbyMP0(m+1),is (4.6) whereweusethefactthatFm+1(y(m+1)1)=Fm+1(y(m+1))Pr(Dm+1=y(m+1))=pc h+pPr(Dm+1=y(m+1)): Wecanobtainthefollowingproposition 4.6 ).AlsonotethatMP0+(m)isdecreasinginm,i.e.,MP0+(m)>MP0+(m+1).OnceMP0+(m)dropstolessthanorequalto0,thevalueofMP()willkeepdecreasing,thuscannotbeoptimalafter.AndifMP0(m+1)>0,thevalueofMP()willkeepincreasingwithinthisrange,thuscannotbeoptimal.Sooursearchfortheoptimalpriceisrestrictedtotheintervalsof(m;m+1]whereMP0+(m)>0MP0(m+1)0:

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4{2 illustratesthepossibleshapeoftheprotfunction.Weseethattheoptimizationproblemisaglobaloptimizationproblem,andasearchoveralltheintervalssatisfyingProposition 4.2.8 maybenecessary.Wenextintroducesuchanalgorithmtondtheoptimalprice. ; ]where

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Figure4{1: Optimalinventorypositiony() Figure4{2: Protfunction

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weknowthatthestationarypointin(m;m+1]isnogreaterthan~.Sowechoosethesmalleroneof~andtobeournewsearchupperboundfortheoptimal,denotedby^. Nextstepistodetermineallthediscontinuouspointsofy()for2[ ;^],whichsatisfyF(y())=pc h+p: h+pg,thenlety1=y01.WethenwanttodeterminethevalueofsuchthatF(y1)=pc h+p,denotedby1.This1isuniquesinceFisastrictlydecreasingfunction.Tondoutthe1,weapplytheChernoBoundsforPoissonrandomvariables(ref.Ross[ 49 ])tonda,foraxedt>0,suchthatF(ym)1e(et1)tym>pc h+p: h+pwithintheinterval(;0),sinceFisdecreasingin.Nextwelety2=y11,andndy2whichsatisesF(y2)=pc h+pinthesameway.Wecontinuethisprocedureuntilwereachthelowerbound Forinterval[m;m+1],werstcheckwhetherthebothendpointssatisfytheoptimalitynecessaryconditionMP0+(m)>0MP0(m+1)0:

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upthebestone.Sotheoptimalpriceshouldbesettothevaluecorrespondstothisoptimaldemandrate,andsolveacorrespondingoptimalorder-up-tolevel.Wesummarizetheseproceduresinthefollowingalgorithm. Step0.SolveR0(~)c=0for~,andlet^=minf~;g. Step1.Letk=0,k=^,andset=^andMP=MP().Solveyk=argminyfFDk(y)pc h+pg. Step2.Ifk< h+p.Letk=k1. Step3.UsebinarysearchmethodtolocatethekthatsatisesFDk(yk)=pc h+pwithintheinterval(;k+1). Step4.Checkwhetherthefollowingtwoinequalitieshold:MP0+(k)>0MP0(k+1)0: Step5.UsebinarysearchmethodtodeterminetheuniquerootforMP0()=0(stationarypoint)within[k;k+1],denoteitby_k.LetMP=maxfMP;MP(_k)g,andlet=argmaxfMPg.GobacktoStep2. Step6.Outputastheoptimaldemandrate,and()astheoptimalprice,andy()astheoptimalorder-up-tolevel.Theoptimalprotis1

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Onekeydierencebetweenthismodelandthepreviousmodelwherepricecanonlybesetonceisthat,duringtheleadtime,theremaybemultiplepricestakingeect,andthusthedemandratewillchange.Buttoourknowledge,noonehastriedthismodelformulationduetoitscomplexity.FedergruenandHeching[ 19 ]discussedaheuristictreatmentwhenaleadtimeisconsidered.Theyassumedthatthepriceselectedismaintainedoverthenextorderleadtime.Weadoptthisassumption,andthusourmodelisreallyanapproximationoftheprecisemodel. SimilarlyasinSection 4.2.3 ,bydiscretizingtimeattheeventepochofdemandoccurrence,wegetthefollowingrecursionv(x)=max2maxyc(yx)1 +()cy1

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introducethedemandmodel,anddenesomenewnotations.Thenweanalyzethemodel,andshowundercertainassumptions,thesemi-MarkovmodulatedPoissondemandmodelcanbedecomposedintodierentstate-dependentPoissondemandmodels,andthesimilarresultsfromtheprevioussectionfollow. 2 ,whichisasemi-Markovmodulatedprice-sensitivePoissondemandprocess.Asbefore,theunderlyingcoreprocesswhichrepresentsthestateoftheworldisacontinuous-timesemi-MarkovprocesswithstatespaceIf0;1;2;:::gandthetransitionprobabilitymatrixisP=(pij)i;j2I(wherepii=0foreachi2I).Weassumethatthetransitionprobabilitiesarenotaectedbyprice.Whenthecoreprocessisinstateiandthepricefortheproductis,theactualdemandprocessfollowsaPoissonprocesswithratei().Asaspecialcase,thecoreprocessisacontinuous-timeMarkovprocess,withrateiforstatei.Givencurrentstateiandnextstatej,thetransitiontimedistributionofthesemi-MarkovprocessisGij,whoseexpectedvalueis1=ij.Notethatthetimetheworldstaysinstateibeforeleav-ingithasdistributionGi,whereGi(x)=Pj2IpijGij(x),anditsexpectedvalueisPj2Ipij=ij. Wewanttodetermineajointpricingandinventorycontrolpolicythatmaximizesthetotalexpecteddiscountedprotsoveraninnitehorizon.Theinventorylevel(position)isreviewedcontinuously,andanordercanbeplacedanytime.WekeepalltheotherassumptionsaboutthedemandmodelsunchangedasthePoissondemandmodel,exceptthatnowtheyareallstate-dependent.Werststudythemodelwherepricecanbesetonlyonceatthebeginningofeachworldstate,andthenbrieygeneralizetheresulttothecasewherepricecanbecontinuouslyset.

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Foraxedpricevector^,wedeneDi;s;^`tobetherandomvariablerepresentingtheleadtimedemandgiventhattheworldhasbeeninstateiforstimeunits,andthepricevectoris^.Andtheconditionalexpecteddiscountedinventoryandbackloggingcostrateattheendofaleadtime,asviewedfromnow,giventhatthecurrentworldstateisiandinventoryposition(afterordering)isy,andpriceisi,canbewrittenasC(y;i;s;^). Foraxedpricevector,theexpectedrevenueswillbexed.Theoptimalinven-torypolicyminimizesthetotaldiscountedexpectedcosts.Itdependsonthecurrentstateoftheworldandhowlongithasbeeninthecurrentstate,andtheentirepricevectorwhichdeterminethedemandratewithineachstate,butnothingelsesincetheworldisasemi-Markovprocess.Wecanrepresentthisinventorypolicyasyi(s;).Wehavesolvedforthispolicybeforeundertheassumptionifthereturnisallowed. Denei(x;)tobetheoptimalexpectedtotalprotsbyfollowingtheoptimalinventorypolicystartingfromthetimewhenthecoreprocessjustentersstateiwhentheinitialinventorypositionisx,anddiscountedtothetimeoftheoccurrenceofstatetransition,foraxedpricevector.Thetotalprotscanbedividedintotwocomponents:thetotalprotsduringtheworld'scurrentstayinstatei,andthetotalprotsaftertransittingintoadierentworldstate.Therstcomponentcanbedeterminedbyconditioningonthetimeuntilthenexttransition,whosedistributionfunctionisequaltoGi(t)Xj2IpijGij(t):

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Notethatthisdistribution,andthereforetherstprotcomponent,doesnotdependonthenextstatevisited.However,forthesecondprotcomponentweneedtoconditiononboththetimeofthetransitionaswellasthenextstateitself. Wecancomputetheexpectedrevenueearnedinstateidiscountedtothestartingtimeofthestateinthisway:supposetherearendemandsoccurredinstatei,andtheyoccursattimes1;s2;:::;snrespectively,wherethetimeisrelativetothestartingtimeofthisstate.Thenthetotalrevenueinstateiisi(es1+es2+:::+esn): whereSisuniformallydistributedon[0;].Itistruesincegivenndemandsoccurred,thedistributionofarrivaltimeofPoissonprocessisjustliketheorderstatisticsofnindependentuniformallydistributedrandomvariables.Andtheorderofdemandsbecomesirrelavantwhenallofthemaresummedup. Thentheexpectedrevenueinstateicanbecomputedas

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where~Gi()isthelaplacetransformofGi,~Gi()=Z10edGi(): i(x;)=(ic)i(i)1 i()=(ic)i(i)1 whereTijGijdenotesthetimespentinstateiwhenthenextstateisj,andwehavealsousedasimilarderivationasbeforetosimplifytheexpressionforthecostswhileinstatei.

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Andtheoptimalprotsovertheinnitehorizoncanbedeterminedasi=maxi(): Theoptimalexpectedtotalprotsofthebestjointpricingandinventorypolicyfromthetimewhenthecoreprocessjustentersstateiwhentheinitialinventorypositionisx,anddiscountedtothetimeoftheoccurrenceofstatetransition,i(x),

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canberepresentedasthefollowingrecursiveway: i(x)=maxi((ic)i(i)1 i=maxi(ic)i(i)1

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Forthecasewherethepriceisallowedtochangecontinuously,ifwemaketheapproximationthatthepriceselectedatthebeginningofaleadtimeismaintainedovertheleadtime,thentheoptimalpricingisagaintosetthepriceonlyonceatthebeginningofeachworldstate,thesameasforthePoissondemandcase,andthesolutionproceduresremainunchanged. i=maxi1 Weseethatinequation( 4.7 ),thetotalprotsfromthenextworldstateonwardsdonotdependonyi;i.Sodeterminingtheoptimalpriceforstateiisequivalenttosolve maxi(ic)i(i)C(yi(i);i;i)cyi(i):(4.8) ThisisexactlythesameprotfunctionweareseekingtomaximizeforthestationaryPoissondemandmodel.Thuswithourassumptionoftheapproximateleadtime,andthepossibilitytoreturn,wecandecomposetheMarkovmodulatedPoissondemandmodelintoseparateStationaryPoissondemandmodels,oneforeachworldstate.

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extendthestudytothesemi-MarkovmodulatedPoissondemandenvironment,andshowthatwithcertainapproximation,themodelcanbesolvedinthesimilarwayasinaPoissondemandenvironment. Futureresearchwillfocusonthedesignofmoreeectivealgorithmsforndingtheoptimalpriceineachstatebasedonthedeeperanalysisoftheprotfunction.Wewillalsostudytheprecisemodelforsemi-MarkovmodulatedPoissondemandprocessmodelinsteadoftheapproximatemodelwestudiedinthischapter.Finally,wewillextendourmodelstothesituationswherereturningexcessinventoryisnotallowed.

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14 ].Werstdiscussastochasticmulti-iteminventorymodelunderbothequalandunequalreplenishmentintervalswithlimitedwarehousecapacityinSection 5.2 .InSection 5.3 ,weprovidetheoptimalpolicyforthecaseofequalreplenishmentintervalsunderlimitedwarehousecapacity,extendingresultsbyChoi[ 14 ]andBeyer[ 10 ]fromcontinuoustogeneraldemanddistributions.WethenreneandgeneralizethethreeheuristicsthatwereproposedbyChoi[ 14 ]toapproximatetheoptimalreplenishmentquantities.Wealsoprovethatintheequalreplenishmentintervalscase,alltheheuristicsprovidetheoptimalsolutiontotheproblem.Section 5.4 presentsnumericalresultsillustratingandcomparingtheperformanceoftheheuristics.InSection 5.5 ,wediscussconclusionsfromourstudyandbrieysummarizepotentialfutureresearchdirections. 115

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replenishmentcycles.Sincereplenishmentcyclesarexedandadeliveryoccursineachreplenishmentcycleforeverysupplier,weassumeanyxeddeliverycostsareconstantandthereforeoutsidethecontrolofthewarehousedecisionmaker.Ourmodeldoesnotthereforeconsiderxedorderingcosts.Inaddition,sincealldemandswillbesatisedwewillalsonotconsideranyvariableorderingcosts.Weconsideraniteplanninghorizon.Werststateourmodelassumptionsandthenformulatetheexpectedcostfunctiontodetermineanoptimalreplenishmentpolicy.Weusethefollowingnotationtodescribethiscapacityconstrainedmulti-iteminventorymodelwithunequalreplenishmentintervals:

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DenethevectorofdecisionvariablesQj=(Qi;j;i2Rj)consistingofthereplenishmentquantitiesoftheitemsinRjthatarereplenishedattimej.Itiseasytoseethattheoptimalreplenishmentquantityofanitemdependsnotonlyontheitem'sowninventorylevel,butalsoonthecurrentinventorylevelofallotheritemsaswellasthewarehousecapacity.Figure 5{1 illustratesthetrackingofinventorylevelsofallitems,aswellasthereplenishmenttimeinstants. Figure5{1: Amulti-iteminventorysystem.

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Inordertoformulateourproblemasastochasticdynamicprogrammingproblem,wedeneIj=(I1;j;:::;I`;j)tobethestateofthesysteminstagej(i.e.,attimej).ForagivenstateIj1,thecurrentinventorylevelofindividualitemsinstagejcanbeexpressedrecursivelyasfollows:Ii;j=8><>:Ii;j1+Qi;j1Pjt=j1+1di;tifi2Rj1Ii;j1Pjt=j1+1di;totherwise. Wethendenegj(Ij;Qj)tobetheexpectedholdingandpenaltycostsforallitemsthatarereplenishedinstagejovertheirrespectivereplenishmentintervals,giveninitialinventoriesIj=(I1;j;:::;I`;j)andreplenishmentsizesQj=(Qi;j;i2Rj):gj(Ij;Qj)=Xi2RjEhi(Ii;j+Qi;jDi)++pi(DiIi;jQi;j)+: WedenetheterminalcostsGm+1()=0tobe0forallpossiblenalinventorylevelsremainingattheendofhorizon.Thedynamicprogrammingrecursion( 5.1 )includesthefullcostsofallreplenishmentcyclesthathavenotbeencompletedatthetimehorizon,T.Ifthenitehorizonrepresentsatruncationofanunderlyinginnitehorizonproblem,thismitigatesend-of-studyeects.However,inthecaseofatrulynitehorizonproblem,wemaytruncateallreplenishmentcyclesatthehorizonbydeningaterminalreplenishmentinstantm+1=T+1andcorrespondingRm+1=f1;:::;`g.Notethatthiswouldusuallyimplythatthenalreplenishmentintervalfor,say,itemihasalengthdierentfromTi,andtheendinginventoryshouldpossiblyalsobevaluedusingdierentcoststhanpiandhi.Thisisinprincipleeasyto

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dobyappropriatelymodifyingthedistributionofDiinthenalreplenishmentcycle.Infact,inasimilarwaywecouldalsohandlereplenishmentintervallengthsthatarevaryingovertheplanninghorizon.However,foreaseofnotationandexpositionwehaveomittedthisgeneralization.Finally,theoptimalcostovertheentireplanninghorizonisthenequaltoG1(I1),whereI1representstheinitialinventorylevels. 5.3.1EqualReplenishmentIntervals 5.1 )forthegeneralcaseofunequalreplenishmentintervals,wewillrststudythecaseofequalreplenishmentintervalsinmoredetail.Inparticular,wewilldeterminetheoptimalpolicyforthiscaseunderasomewhatmildersetofassumptionsonthedemanddistributionsthanhasbeenconsideredintheliteraturetodate.Inthiscase,Tiisidenticalforallitems,andj=(j1)T1+1andRj=f1;:::;`gforj=1;:::;m.Thedynamicprogrammingrecursion( 5.1 )thensimpliestoGj(Ij)=minQj:P`i=1(Ii;j+Qi;j)+Vfgj(Ij;Qj)+E[Gj+1(Ij+1)]g:

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inventoryleveljustafterreplenishmentisalwaysnonnegative:~Qi;j=8><>:Qi;jifIi;j+Qi;j0Ii;jotherwise. Clearly,~Qi;jQi;jforalli.Nevertheless,itiseasytoseethattheseorderquantitiessatisfythecapacityconstraint.IfIi;j+Qi;j<0,wehavethatE[(Ii;j+Qi;jDi)+]=0andE[(Ii;j+~Qi;jDi)+]=E[(0Di)+]=0,aswellasE[(DiIi;jQi;j)+]>E[Di].Thisimmediatelyimpliesthatgj(Ij;~Qj)gj(Ij;Qj): and0~Ii;j+1>Ii;j+1otherwise foranyrealizationofdemands.ThisthenimpliesthatE[Gj+1(~Ii;j+1)]E[Gj+1(Ii;j+1)] sincethetotalfuturecostsarenonincreasinginIi;j+1whenIi;j+1<0.Thisprovesthedesiredresult. 33 ]andBeyeretal.[ 10 ]showthat,inastochastic,multi-iteminventorymodelwithlimitedwarehousecapacityandwhereallitemsarereplenishedsimultaneously,amyopicpolicyisoptimaliftheproductdemandsarestationaryandindependent,thecostfunctionsareseparable,andtheinventoryafteranyreplen-ishmentisalwaysnonnegative.Choi[ 14 ]usedthisresulttoderiveanexplicitformofthismyopicpolicyforourinventorysysteminthecasewherethedemanddistri-butionsareabsolutelycontinuous.Thefollowingtheoremcharacterizestheoptimal

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replenishmentpolicyforgeneraldemanddistributions.Beforewestatethetheo-rem,werstintroducesomeadditionalnotationanddenitionsthatareusedinthecharacterizationoftheoptimalpolicy.NotethatifFisthecumulativedistributionfunctionofarandomvariableitiscontinuousfromtheright.WewilldenotearelatedfunctionthatiscontinuousfromtheleftthroughF(y)=limh#0F(yh): hi+pifori=1;:::;`Fi(Si)pi hi+pifori=1;:::;`0: 33 ]andBeyeretal.[ 10 ]showedthattheoptimalpolicyisamyopicpolicy.Therefore,wecandeterminetheoptimal

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inventorypolicyforeachreplenishmentintervalseparatelybysolvingthefollowingoptimizationproblem:minimize`Xi=1E[hi(Ii;j+Qi;jDi)++pi(DiIi;jQi;j)+] subjectto`Xi=1(Ii;j+Qi;j)VIi;j+Qi;j0i=1;:::;`: subjectto`Xi=1SiVSi0i=1;:::;`: Deningi(Si)=E[hi(SiDi)++pi(DiSi)+]

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andintroducinganonnegativeLagrangemultiplierwiththecapacityconstraint,theKKTconditionsfortheoptimizationproblemaregivenby: (5.2) (5.3) (SeeHiriart-UrrutyandLemarechal[ 32 ]).Nownotethat@i(Si)=(hi+pi)Fi(Si)pi;(hi+pi)Fi(Si)pi 5.2 )canbewrittenas(hi+pi)Fi(Si)pi+0(hi+pi)Fi(Si)pi+0 Nownotingthatcondition( 5.3 )saysthatthecapacityconstraintcanonlybenon-bindingif=0,weobtainthedesiredresult.

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hi+piV#: 5.3.2 iscontinuous.Theresultthenfollowsimmediatelyfromthefactthatcondition( 5.2 )canbewrittenas(hi+pi)Fi(Si)pi+=0: 14 ],butonlyworkwithdiscretedemandmodels.Wegeneralizetheresultsandmakethemworkforbothdiscreteandcontinuousdemandmodels. 5{2 ). Figure5{2: Aninventorysystemwithunequalreplenishmentintervals.

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5.3.2 whichconsideredtheproblemunderequalreplenishmentintervals.Inparticular,inthisheuristicwedetermineapolicybyignoringthefactthatthetwoitemsarenotreplenishedsimultaneously,andsimplycomputeorder-up-tolevelsasinthecasewithequalreplenishmentintervals.Wewilldenotetheseorder-up-tolevelsbySN1andSN2.Notethattheselevelscanbeinterpretedasindividualitemcapacities,i.e.,thispolicyactsasifthewarehouseispartitionedintodedicatedsectionsforthetwoitems.Clearly,ifneitheritemviolatesitsindividualcapacity,thejointcapacityconstraintisalsosatised,regardlessofthetimingoftheitemreplenishments.Wethereforecallthispolicyanonintrusiveone,sincethevalueofthedemand(inparticular,anexceptionallysmalldemand)foraparticulargivenitemintheperiodfollowingitsreplenishmentwillnotimposeanyadditionalconstraintonthereplenishmentamountfortheotheritem.Notethat(intheabsenceofpositivelowerboundsonthedemands),anyorder-up-topolicywithlevelsS01;S02suchthatS01+S02>Vdoesnotenjoythisproperty!Theactualpolicygivenbythisheuristicbecomes:QNi;j=VNiIi;j 5{3 foranillustrationofthispolicy). Figure5{3: Aninventorypolicyforthenonintrusiveheuristic.

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Notethat,if=0andthusthecapacityconstraintisnotbinding,SN1andSN2aretheunconstrainedoptimalorder-up-toquantitiesforeachitemindividually.Therefore,itiseasytoseethatthenonintrusiveheuristicenjoysthedesirableprop-ertythatitndstheoptimal(i.e.,unconstrained)solutionwheneverthecapacityconstraintisnotbinding. Ontheotherhand,theobviousdrawbacktothisheuristicisthat,whenthecapacityconstraintisbinding,atthetimeofreplenishmentof,say,item1therewillusuallybeavailablecapacityinthewarehousethatwillremainunuseduntilthereplenishmentofitem2.Thisintuitivelyappearstobeawasteofresourcessinceatleastsomeofthatavailablecapacitycouldbeusedforitem1,sincedemandforitem1willlikelyfreeupthisspacebeforeitisneededbyitem2.AGreedyHeuristic 5{4 ,itemi(i=1;2)isreplenisheduptoitsunconstrainedoptimalinventorylevelorthetotalavailablecapacityatthewarehouse,whicheverissmaller.Thatis,theorder-up-tolevelisSGi;j=minFipi

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Figure5{4: Aninventorypolicyforthegreedyheuristic. Ontheotherhand,theobviousdrawbacktothisheuristicisitsgreedynature.Atthetimeofreplenishmentofsayitem1,potentiallyallavailablecapacityisusedforreplenishingthatitem,possiblyleavingtoolittlespaceforitem2atitsreplenishmenttimetobeabletoachieveanadequatecost(andcustomerservice)level.ASharingHeuristic Thesharingheuristicweproposeissimilarinspirittotherstheuristicinthatwedetermine,apriori,individualpseudo-capacitiesforeachofthetwoitems,whichwewilldenotebyV01andV02.Thesumoftheseindividualcapacitieshoweverwill,ingeneral,exceedthetotalwarehousecapacityV,reectingthefactthateitheritemcantemporarilyusesomeofthestoragespaceintendedfortheother,therebycountingonthisspacetobefreedupbydemandbeforetheotheritemactuallyneedsit.Asnoted

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above,thiscannotbeguaranteed.Wedealwiththissituationinasimilarmannerasinthegreedyheuristic,bylimitingtheorder-up-toquantitytotheactualavailablespaceinthewarehouseatthetimeofreplenishment.Thepolicythenbecomes:QSi;j=minV0iIi;j;QGi;j: whereSNiistheoptimalcapacityusedinthenonintrusiveheuristic.Itiseasytoseethatshouldbenonnegative,andthatwithrespecttoeachindividualitemwewouldliketomaximizethevalueof.Intheremainder,wewillderivesomesuitablecandidatevaluesfor.Assumingthatcapacityisverytight,andwethereforeusuallyorderuptoV0i,theinventorylevelofitem2whenitem1isreplenishedwillonaveragebeequaltoV02E[d2]=SN2(1)E[d2].Tomakesurethattheactualavailablecapacityatthewarehouseisnotconstrainingonaverage,weimposetheconstraintV01+(V02E[d2])+VwhichisequivalenttoSN1+E[d1]+(SN2(1)E[d2])+VorSN1+E[d1]+(SN2E[d2]+E[d2])+V:

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bychoosing=1min(1;2): 2(1+2): 5{5 foranillustrationofthisheuristic.SummaryofHeuristics

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Figure5{5: Aninventorypolicyforthesharingheuristic. hi+pii=1;2Fi(Si)pi hi+pii=1;20 andlet(;SN1;SN2)beanoptimalsolutiontothisproblem.

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(i) (ii) (iii) 5.3.2 arelimitedtothetwo-itemcase,inwhichproductsarereplenishedatdierentpointsintime.Inrealwarehousesystems,however,itismorelikelythatmorethantwoitemsarereplenished,andthatreplenishmentsofsomeitemsmaytakeplaceatthesametime.Basedontheresultsfromthetwo-itemcase,wecanextendtheproposedheuristicstomulti-itemcases.Let`2bethenumberofitemsinthesystemandletbbetheleastcommonmultipleoftheindividualintervallengths(orthelengthofonecommoncycle).IfTirepresentsthereplenishmentintervallengthofitemiasdenedearlierinthissection,bi=b=Tirepresentsthenumberofreplenishmentsofitemiinonecommoncycle.Inthefollowingthreesections,wewillgeneralizethethreeheuristicstothisgeneralcase.

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bi(hi+pi)i=1;:::;`Fi(Si)bipi bi(hi+pi)i=1;:::;`0 andletting(;SN1;:::;SN`)denoteanoptimalsolution,wherethevaluesSNiaretheorder-up-tolevels.Notethatforthisheuristic,theprocesstodetermineindividualcapacitiesinthemulti-itemcaseisthesameregardlessofwhethercertainitemsarereplenishedsimultaneously.Thisfollowsdirectlyfromthefactthat,inthenonintru-siveheuristic,wedeterminetheindividualcapacityofeachitemundertheassumptionthatallitemsarereplenishedatthesametime.Theindividualcapacitiesarethereforenotaectedbywhetheritemsarereplenishedsimultaneouslyintheactualreplenish-mentscheduleornot.Andweseethatthisheuristicistime-independent,andnotaectedbytheactualinventorylevelsatthetimeofreplenishments. Notethat,if=0andthusthecapacityconstraintisnotbinding,SN1andSN2aretheunconstrainedoptimalorder-up-toquantitiesforeachitemindividually.Therefore,itiseasytoseethatthenonintrusiveheuristicenjoysthedesirableprop-ertythatitndstheoptimal(i.e.,unconstrained)solutionwheneverthecapacityconstraintisnotbinding.

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bi(hi+pi)i2RjFi(Si)bipi bi(hi+pi)i2Rj0:

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5{2 ,wehavee1;2=1,e2;3=1,e1;4=1,andsoon.RecallthatthesetofitemsthatarereplenishedinthejthreplenishmentperiodisgivenbyRj.Then,similarlytothetwo-itemcase,wedenethepseudo-capacityforitemsi2RjbyV0i=SNi+TiE[di]i2Rj

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Asinthetwo-itemcase,wewillrestricttheorder-up-tolevelinthesharingheuristictothegreedylevel:SSi=minV0i;SGi: 5.3.2 .Furthermore,ifallitemsarereplenishedatthesametime,thengreedyheuristicwillbeidenticaltothenonintrusiveheuristicsinceallitemsibelongtoRjatanreplenishmenttime.Finally,inthesharingheuristicrecallthatV0iSNibyconstruction,regardlessofthevalueof.SinceintheequalreplenishmentcasewehaveSGi=SNi,weobtainthatSSi=minV0i;SGi=minV0i;SNi=SNi 5.2 .Wehave

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testedbothtwo-itemandthree-itemmodels,withplanninghorizonsofT=60andT=12,respectively(inordertoallowustocomputetheoptimalpolicies). Forthetwo-itemmodels,wehavegeneratedprobleminstanceswithdierentrelativelevelsoftheunitholdingandpenaltycostparameters:hi=2andpi2f3;10g.Thedemandperperiodforeachitemisassumedtofollowadiscreteuniformdistributionwithalowerboundof0andanupperboundU12f5;10;15;20gandU2=40U12f20;25;30;35g,inordertostudytheeectof(non-)homogeneitiesinthedemanddistributionsoftheitems.Theinitialinventorylevelsforallitemsaresetto0,whichshouldnothaveamajorimpactonthereplenishmentpolicyinthelongrun.Foreachinstance,werstdeterminedthestoragecapacityusedintheunconstrainedoptimalsolution.Wethenconsideredprobleminstancesinwhichthecapacityisequalto90%,80%,60%,and40%ofthisvalue.Finally,wevariedthereplenishmentcyclelengths(2or3)aswellasthetimingsofthereplenishments. Wesolvedallprobleminstancesusingthestochasticdynamicprogrammingre-cursionderivedinthischapter.Wethensimulatedthesystemforallproposedheuris-ticsandcomparedtheexpectedcostofeachheuristicpolicytotheexpectedcostoftheoptimalpolicy.Tables 5{1 5{3 showmeasuresoftherelativeerroroftheheuris-ticsolutionsascomparedtotheoptimalsolutions.Inparticular,eachentryofthetablesgivestheaverageerrorobtainedfromallinstanceswiththespeciedparam-eters.Thetablesalsoshowtheaveragetimerequiredforsolvingtheproblemtooptimality.Thetimerequiredfortheheuristicsisnegligible. Thetablesclearlyshowthat,ingeneral,theperformanceofallheuristicstendstoimprovewithmorehomogeneityinthedemanddistributionsorwithhighercapacitylevels.Thelatterobservationshouldnotbesurprisingsinceweshowedthatallheuristicsprovidetheoptimalsolutionwhenthecapacityconstraintisnotbinding.Furthermore,inalmostallcasesvariant2ofthesharingheuristicseemstooutperformtheotherheuristics,withvariant3ofthatheuristicareasonablealternative.However,

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duetothefactthatvariant2issomewhateasiertoimplementinpracticethereseemstobelittlereasontoapplythatvariant.Asomewhatsurprisingresultisthatthegreedyheuristicseemstooutperformvariant1ofthesharingheuristicformostofthetestedinstances. Tables 5{4 5{6 focusonlyoninstanceswherethelengthofthereplenishmentintervalsisequalto2forbothitemsandthereplenishmentsarestaggered.Intheseinstances,notethatthenonintrusiveheuristicinfactprovidestheoptimalsolutioncorrespondingtothecasewheretheitemsarereplenishedatthesametime.Theerrorassociatedwiththenonintrusiveheuristicthereforealsomeasuresthecostassociatedwithcoordinatingthereplenishmentoftheitemstotakeplaceinthesameperiods.Theseresultsshowthatthebenetofstaggeringreplenishmentsislargestwhencapacityisverylimitedanddemandsarehomogeneous.AnunrelatedbutneverthelessinterestingobservationthatcanbemadebycomparingtheresultsinTables 5{4 5{6 totheresultsinTables 5{1 5{3 isthatthetimerequiredbythestochasticdynamicprogrammingmethodisquitereasonablewhenonlyoneitemisreplenishedineachreplenishmentperiod,butincreasesdramaticallywhenmultipleitemsarereplenishedsimultaneouslyinsomereplenishmentperiods.Thisfollowsimmediatelyfromthefactthatthestatespaceofthestochasticdynamicprogramincreasesrapidlyinthenumberofitemsthatcanbereplenishedinanyperiod. Forthethree-itemmodels,wehaveagainconsidereddierentrelativelevelsoftheunitholdingandpenaltycostparameters:hi=2andpi2f3;10g,dierentcapacitylevels,anddierentreplenishmentschedules.However,intheinterestoftimewehavefocusedonprobleminstanceswithonlyasingledemandpattern,i.e.,U1=U2=10andU3=15.Tables 5{7 and 5{8 showtheresultsofthesetests.Theperformanceoftheheuristicsforthethree-itemcasesseemstofollowthesamepatternasforthetwo-itemcases.Althoughingeneralthesolutionerrorsarelargerforthree-iteminstances,thesolutionsfoundbyvariant2ofthesharingheuristicare

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Table5{1: Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionofthetightnessofthestoragecapacity(2items). Capacity Time(opt.) Nonintrusive Greedy Sharing (%ofunconstrained) (h:mm:ss) 1 2 3 90 3:45:24 2.17 0.26 1.40 0.24 0.26 80 2:50:51 6.57 1.56 4.17 0.86 1.22 60 1:28:43 12.15 5.36 7.97 1.35 2.51 40 0:36:08 10.91 6.29 7.66 2.08 2.82 Table5{2: Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionofdemandvariabilitybetweenitems(2items). Meandemands Time(opt.) Nonintrusive Greedy Sharing (h:mm:ss) 1 2 3 20,20 2:44:31 10.23 3.80 6.56 1.54 1.87 15,25 2:42:12 9.53 4.35 6.16 1.27 2.00 10,30 2:05:13 7.64 3.40 5.30 1.16 1.84 5,35 1:09:09 4.40 1.92 3.18 0.56 1.11 Table5{3: Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionoftheunderagepenaltycostsoftheitems(2items). Penaltycosts Time(opt.) Nonintrusive Greedy Sharing (h:mm:ss) 1 2 3 3,3 1:37:01 7.51 1.87 4.69 0.83 1.48 10,10 2:44:04 10.98 4.15 6.95 1.24 2.46 3,10 2:22:09 5.71 5.01 4.26 1.26 1.09 10,3 1:57:50 7.59 2.44 5.30 1.20 1.80 Table5{4: Relativecostassociatedwithcoordinatingreplenishmentsfordierentitemstooccuratthesametime,asafunctionofthetightnessofthestoragecapacity. Capacity Time(opt.) Nonintrusive (%ofunconstrained) (h:mm:ss) 90 0:09:23 2.57 80 0:07:51 8.19 60 0:04:56 15.76 40 0:02:32 14.77

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stillquiteacceptable,especiallyconsideringthetimerequiredtondtheoptimalsolution. Thenonintrusiveheuristic,whichusesaseparatecapacityforeachproductforreplenishment,isveryeasytoimplement,butisnotabletodealaseectivelywiththescarceresourceastheothertwoheuristicsformostproblems.Thisbehaviorisnotunexpectedbecausethisheuristicismorelikelytoretainunusedcapacityforsomeproductswhileothersmaysuerfromthelackoftheresource.Thegreedyheuristicusesareplenishmentpolicythatreplenishesproductsuptothetotalavailableca-pacityofthesystematthetimeofreplenishment.Thisheuristicoutperformsthenonintrusiveheuristicinmostcases,butsometimesstillleadstopoorperformanceduetothefactthatitistooaggressiveinreplenishingproductsinagivenperiodsothattheotheritemssuerfromthelackofresourceinthefollowingreplenishmentperiods,causinghighwarehouseshortagepenaltycosts.Thesharingheuristic,how-ever,attemptstocombinethepositivequalitiesofbothotherheuristicsbydeningindividualcapacitiesforeachitem,butallowsthesumoftheindividualcapacitiestoexceedthetotalwarehousecapacity{toreectthepossibilityofsharingsomeofthewarehousespaceamongitemsduetothedierentreplenishmentschedules.Inpar-ticular,thesecondvariantofthisheuristicseemstoenjoyaverygoodperformanceoverallinstancesstudied.

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Table5{5: Relativecostassociatedwithcoordinatingreplenishmentsfordierentitemstooccuratthesametime,asafunctionofdemandvariabilitybetweenitems(2items). Meandemands Time(opt.) Nonintrusive (h:mm:ss) 20,20 0:08:00 13.47 15,25 2:42:12 12.49 10,30 2:05:13 9.81 5,35 1:09:09 5.52 Table5{6: Relativecostassociatedwithcoordinatingreplenishmentsfordierentitemstooccuratthesametime,asafunctionoftheunderagepenaltycostsoftheitems. Penaltycosts Time(opt.) Nonintrusive (h:mm:ss) 3,3 0:04:40 10.01 10,10 0:07:44 14.64 3,10 0:06:47 7.47 10,3 0:05:32 9.18 Table5{7: Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionofthetightnessofthestoragecapacity(3items). Capacity Time(opt.) Nonintrusive Greedy Sharing (%ofunconstrained) (h:mm:ss) 1 2 3 90 35:02:18 16.27 3.11 8.91 2.34 4.92 80 17:30:36 20.88 4.22 14.91 3.77 7.08 65 9:45:32 23.60 4.93 18.60 4.95 9.27 50 5:04:38 22.40 4.73 17.66 6.07 8.71 Table5{8: Error(in%)ofthesolutionobtainedbytheheuristicsascomparedtotheoptimalsolution,asafunctionoftheunderagepenaltycostsoftheitems(3items). Penaltycosts Time(opt.) Nonintrusive Greedy Sharing (h:mm:ss) 1 2 3 3,3,3 10:47:29 17.02 2.73 10.11 1.97 5.85 10,10,10 22:51:43 23.46 4.81 18.91 3.91 7.81 3,3,10 15:22:36 17.45 3.25 15.17 6.22 6.87

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Weconcludethischapterbypointingoutpossibleextensionsofthestochasticmodelsconsideredinthisstudy.Thecurrentoptimizationmodelisestablishedforaxedwarehousecapacityovertime.However,theapproachusedinthischaptercanbeusedtodeterminehowmuchoneshouldbewillingtopayforextrawarehousecapacitybysimulatingdierentvaluesofthiscapacity.Oneinterestingextensionworthpursuingistoactuallyincorporateavariablewarehousecapacityintotheproblem,forsituationswhereextrawarehousecapacitycanbeleasedtoaccommodatehigherinventorylevels.Anotherfutureresearcheortmayconsiderarelaxationoftheassumptionofzerodeliveryleadtimesbetweenmanufacturersandthewarehouse.Finally,onemayconsidertheoptimizationofthereplenishmentschedule(whichwehaveassumedgiven),andevenextendconsiderationtocaseswithnonzeroxedorderingcosts. Themajorcontributionofthisstudyisthatweextendtheaforementionedmulti-item,periodic-reviewinventorymodelbyrelaxingtheassumptionofidenticalreplen-ishmentschedulesforthedierentitems.Toourknowledge,noonehasconsideredthecaseinwhichproductsfromdierentmanufacturershavedistinctreplenishmentschedulesorunequalreplenishmentintervallengths(wecallthisthecaseofunequalreplenishmentintervals).Unfortunately,exceptforrelativelysmallproblems,itisdiculttodetermineanoptimalreplenishmentpolicyinthiscase.Inthischapter,wethereforedevelopthreeecientandeectiveheuristicstodeterminereplenish-mentquantitiesunderunequalreplenishmentintervals.Weshowthateachoftheseheuristicsprovidestheoptimalinventoryorderingquantitiesforthecasewherethereplenishmentintervalsforthedierentitemscoincide.Extensivenumericaltestsareemployedthatcomparetheperformanceoftheheuristicstotheoptimalpolicies.Theseresultsnotonlyshowthathigh-qualitysolutionscanbeobtainedinverylim-itedtime,butalsosuggestsguidelinesonwhichheuristictouseforvariousclassesofinstances.

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[1] V.Anantharam.Theoptimalbuerallocationproblem.IEEETransactionsonInformationTheory,35:721{725,1989. [2] S.Anily.Multi-itemreplenishmentandstorageproblem:heuristicsandbounds.OperationsResearch,39:233{243,1991. [3] K.J.Arrow,T.Harris,andJ.Marschak.Optimalinventorypolicy.Econometrica,19:250{272,1951. [4] K.S.Azoury.Bayessolutiontodynamicinventorymodelsunderunknownde-manddistribution.ManagementScience,31:1150{1160,1985. [5] R.H.Ballou.BusinessLogisticsManagement.PrenticeHall,UpperSaddleRiver,NewJersey,4thedition,1999. [6] M.Beckmann.Aninventorymodelforarbitraryintervalandquantitydistribu-tionsofdemands.ManagementScience,8:35{57,1961. [7] R.Bellman,I.Glicksberg,andO.Gross.Ontheoptimalinventoryequation.ManagementScience,2:83{104,1955. [8] D.P.Bertsekas.DynamicProgramming:DeterministicandStochasticModels.PrenticeHall,EnglewoodClis,NewJersey,1988. [9] D.Beyer,S.P.Sethi,andR.Sridhar.DecisionandControlinManagementSciencesinhonorofProfessorAlainHaurie,chapterAverage-costoptimalityofabase-stockpolicyforamulti-productinventorymodelwithlimitedstorage,pages241{260.KluwerAcademicPublishers,Dordrecht,TheNetherlands,2001. [10] D.Beyer,S.P.Sethi,andR.Sridhar.Stochasticmulti-productinventorymodelswithlimitedstorage.JournalofOptimizationTheoryandApplications,111:553{588,2001. [11] S.A.Carr,A.R.Gullu,P.L.Jackson,andJ.Muckstadt.Exactanalysisofthenob/cstockpolicy.TechnicalReport,CornellUniversity,Ithaca,NewYork,1993. [12] X.ChenandD.Simchi-Levi.Coordinatinginventorycontrolandpricingstrate-gieswithrandomdemandandxedorderingcost:thenitehorizoncase.Op-erationsResearch,52:887{896,2004. 142

PAGE 154

[13] X.ChenandD.Simchi-Levi.Coordinatinginventorycontrolandpricingstrate-gieswithrandomdemandandxedorderingcost:theinnitehorizoncase.MathematicsofOperationsResearch,29:698{723,2004. [14] J.Choi.StochasticProductionandInventoryModelswithLimitedResources.PhDdissertation,UniversityofFlorida,Gainesville,Florida,2001. [15] F.W.Ciarallo,R.Akella,andT.E.Morton.Aperiodicreviewproductionplan-ningmodelwithuncertaincapacityanduncertaindemand.ManagementScience,40:320{332,1994. [16] G.A.DeCroixandA.Arreola-Risa.Optimalproductionandinventorypolicyformultipleproductsunderresourceconstraints.ManagementScience,44:950{961,1998. [17] G.Dobson.Theeconomiclotschedulingproblem:aresolutiontofeasibilityusingtimevaryinglotsizes.OperationsResearch,35:764{771,1987. [18] R.V.Evans.Inventorycontrolofamultiproductsystemwithalimitedproduc-tionresource.NavalResearchLogisticsQuarterly,14:173{184,1967. [19] A.FedergruenandA.Heching.Combinedpricingandinventorycontrolunderuncertainty.OperationsResearch,47:454{474,1999. [20] A.FedergruenandY.Zheng.Anecientalgorithmforcomputinganopti-mal(r;q)policyincontinuousreviewstochasticinventorysystems.OperationsResearch,40:808{813,1992. [21] A.FedergruenandP.Zipkin.Aninventorymodelwithlimitedproductionca-pacityanduncertaindemandsi:theaverage-costcriterion.MathematicsofOperationsResearch,11:193{207,1986. [22] A.FedergruenandP.Zipkin.Aninventorymodelwithlimitedproductionca-pacityanduncertaindemandsii:thediscounted-costcriterion.MathematicsofOperationsResearch,11:208{215,1986. [23] Y.FengandF.Y.Chen.Jointpricingandinventorycontrolwithsetupcostsanddemanduncertainty.Workingpaper,ChineseUniversityofHongKong,HongKong,China,2003. [24] M.FlorianandM.Klein.Deterministicproductionplanningwithconcavecostsandcapacityconstraints.ManagementScience,18:12{20,1971. [25] G.Gallego.Newboundsandheuristicsfor(q;r)policies.ManagementScience,44:219{233,1998. [26] G.Gallego,M.Queyranne,andD.Simchi-Levi.Singleresourcemulti-itemin-ventorysystems.OperationsResearch,44:580{595,1996.

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[27] P.Glasserman.Allocatingproductioncapacityamongmultipleproducts.Oper-ationsResearch,44:724{734,1996. [28] S.K.Goyal.Anoteon\multi-productinventorysituationwithonerestriction".JournaloftheOperationalResearchSociety,29:269{271,1978. [29] G.HadleyandT.M.Whitin.AnalysisofInventorySystems.PrenticeHall,EnglewoodClis,NewJersey,1963. [30] R.HartleyandL.C.Thomas.Thedeterministic,two-product,inventorysystemwithcapacityconstraint.JournaloftheOperationalResearchSociety,33:1013{1020,1982. [31] D.P.HeymanandM.J.Sobel.StochasticModelsinOperationsResearch,VolumeII:StochasticOptimization.McGraw-Hill,1984. [32] J.-B.Hiriart-UrrutyandC.Lemarechal.ConvexAnalysisandMinimizationAlgorithmsI:Fundamentals.Springer-Verlag,Berlin,Germany,1996. [33] E.IgnallandA.F.Veinott.Optimalityofmyopicinventorypoliciesforseveralsubstituteproducts.ManagementScience,15:284{304,1969. [34] P.C.JonesandR.R.Inman.Whenistheeconomiclotschedulingproblemeasy?IIETransactions,21:11{20,1989. [35] R.KapuscinskiandS.Tayur.Acapacitatedproduction-inventorymodelwithperiodicdemand.OperationsResearch,46:899{911,1998. [36] S.Karlin.Dynamicinventorypolicywithvaryingstochasticdemands.Manage-mentScience,6:231{258,1960. [37] A.DeKok,H.Tijms,andF.VanderDuynSchouten.Approximationsforthesingle-productproduction-inventoryproblemwithcompoundpoissondemandandservicelevelconstraints.AdvancesinAppliedProbability,16:378{402,1984. [38] A.A.KurawarwalaandH.Matsuo.Forecastingandinventorymanagementofshortlife-cycleproducts.OperationsResearch,44:131{150,1996. [39] H.LauandA.H.Lau.Thenewsstandproblemunderprice-dependentdemanddistribution.IIETransactions,20:168{175,1988. [40] H.LauandA.H.Lau.Thenewsstandproblem:acapacitatedmultiple-productsingle-periodinventoryproblem.EuropeanJournalofOperationalResearch,94:29{42,1996. [41] H.L.LeeandS.Nahmias.HandbooksinOperationsResearchandManagementScience:LogisticsofProductionandInventory,chapterSingleproduct,single-locationmodels,pages1{55.ElsevierScience,Amsterdam,TheNetherlands,1993.

PAGE 156

[42] L.Li.Astochastictheoryoftherm.MathematicsofOperationsResearch,13:447{466,1988. [43] W.S.Lovejoy.Myopicpoliciesforsomeinventorymodelswithuncertaindemanddistributions.ManagementScience,36:724{738,1990. [44] W.S.Lovejoy.Stoppedmyopicpoliciesinsomeinventorymodelswithgeneralizeddemandprocesses.ManagementScience,38:688{707,1992. [45] W.S.Lovejoy.Suboptimalpolicies,withbounds,forparameteradaptivedecisionprocesses.OperationsResearch,41:583{599,1993. [46] S.NahmiasandC.Schmidt.Anecientheuristicforthemulti-itemnewsboyproblemwithasingleconstraint.NavalResearchLogisticsQuarterly,31:463{474,1984. [47] N.C.PetruzziandM.Dada.Pricingandthenewsvendorproblem:Areviewwithextensions.OperationsResearch,47:183{194,1999. [48] M.RosenblattandU.Rothblum.Onthesingleresourcecapacityproblemformulti-iteminventorysystems.OperationsResearch,38:686{693,1990. [49] S.M.Ross.StochasticProcesses.Wiley,NewYork,NewYork,2ndedition,1996. [50] H.E.Scarf.Bayessolutionofthestatisticalinventoryproblem.AnnalsofMath-ematicalStatistics,30:490{508,1959. [51] H.E.Scarf.Someremarksonbayessolutiontotheinventoryproblem.NavalResearchLogisticsQuarterly,7:591{596,1960. [52] J.SongandP.Zipkin.Inventorycontrolinauctuatingdemandenvironment.OperationsResearch,41:351{370,1993. [53] J.SongandP.Zipkin.Managinginventorywiththeprospectofobsolescence.OperationsResearch,44:215{222,1996. [54] L.J.Thomas.Priceandproductiondecisionswithrandomdemand.OperationsResearch,22:513{518,1974. [55] J.T.TreharneandC.R.Sox.Adaptiveinventorycontrolfornonstationaryde-mandandpartialinformation.ManagementScience,48:607{624,2002. [56] A.F.Veinott.Optimalpolicyforamulti-product,dynamic,nonstationaryin-ventoryproblem.ManagementScience,12:206{222,1965. [57] Y.WangandY.Gerchak.Continuousreviewinventorycontrolwhencapacityisvariable.InternationalJournalofProductionEconomics,45:381{388,1996.

PAGE 157

[58] Y.WangandY.Gerchak.Periodicreviewproductionmodelswithvariablecapacity,randomyieldanduncertaindemand.ManagementScience,42:130{137,1996. [59] Y.Z.Wang,L.Jiang,andZ.-J.Shen.Channelperformanceunderconsignmentcontractwithrevenuesharing.ManagementScience,50:34{47,2004. [60] L.M.Wein.Capacityallocationingeneralizedjacksonnetworks.OperationsResearchLetters,8:143{146,1989. [61] T.M.Whitin.Inventorycontrolandpricetheory.ManagementScience,2:61{68,1955. [62] Y.ZhengandP.Zipkin.Aqueueingmodeltoanalyzethevalueofcentralizedinventoryinformation.OperationsResearch,38:296{307,1990. [63] P.Zipkin.FoundationsofInventoryManagement.McGraw-Hill,NewYork,NewYork,2000.

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JieCaowasborninYidu,HubeiProvince,thePeople'sRepublicofChina.Heearnedhisbachelor'sdegreefromDepartmentofControlScienceandEngineering,HuazhongUniversityofScienceandTechnology,majoringinautomaticcontrol,andmaster'sdegreefromtheInstituteofSystemsEngineering,HuazhongUniversityofScienceandTechnology,majoringindecisionsupportsystems.In2001,hejoinedtheIndustrialandSystemsEngineeringdepartmentattheUniversityofFlorida.HeexpectstogethisdoctoratedegreefromtheIndustrialandSystemsEngineeringdepartment,theUniversityofFlorida,inAugust2005. 147


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Permanent Link: http://ufdc.ufl.edu/UFE0011469/00001

Material Information

Title: Stochastic Inventory Control in Dynamic Environments
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0011469:00001

Permanent Link: http://ufdc.ufl.edu/UFE0011469/00001

Material Information

Title: Stochastic Inventory Control in Dynamic Environments
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0011469:00001


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STOCHASTIC INVENTORY CONTROL IN DYNAMIC ENVIRONMENTS


By

JIE CAO















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005


































Copyright 2005

by

Jie Cao

















To my parents and my wife, for their love and support















ACKNOWLEDGMENTS

I wish to express my sincere gratitude to the members of my supervisory com-

mittee, Dr. J.P. Geunes, Dr. A. Paul, Dr. H.E. Romeijn, Dr. Z.J. Shen and Dr.

S. Urasev, for their assistance and guidance. I would also like to thank Dr. Aydin

Alptekinoglu, for taking time to review my dissertation.

Especially, I am grateful to my chair, Dr. H. Edwin Romeijn, for his support,

encouragement and patience throughout the study. As I will pursue my own career,

nothing is more beneficial than having his fine example of a just person, an industrious

and thorough scholar, and a responsible advisor, to look up to.

My parents and wife have given me tremendous support for my study abroad.

Their trust and love are the invaluable wealth in my life, but no acknowledgement

could possibly state all that I owe to them.















TABLE OF CONTENTS
Page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ................... .......... viii

LIST OF FIGURES ..................... ......... ix

ABSTRACT ... .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. x

CHAPTER

1 INTRODUCTION ........................... 1

1.1 General Description ........... .............. 1
1.2 Literature Review ........................... 2
1.3 Outline of Dissertation ......... .............. 11

2 INVENTORY CONTROL IN A SEMI-MARKOV MODULATED DE-
MAND ENVIRONMENT ................... ...... 14

2.1 Introduction ......... .......... ........ 14
2.2 Model Formulation ........ ........... .... 14
2.2.1 The Demand Process ........ ............ 14
2.2.2 Gamma Distributed State Transition Times with Observable
Stage Transitions ..... .................. 15
2.2.3 Definitions and Notation .............. .. 16
2.2.4 Problem Formulation ..... ........... .... 19
2.3 Model Analysis ............... ....... .. 21
2.3.1 Optimal Policy .......... ..... ... .... 21
2.3.2 Optimal Policy with Semi-Markov Modulated Poisson De-
mands . . ......... .......... 23
2.3.3 Determination of the Optimal Inventory Position . 29
2.3.4 Total Policy Costs ...... ......... .... 31
2.4 Monotonicity Results ....... . . ...... 32
2.4.1 Monotonicity of Optimal Inventory Positions within a Given
State ............. .... .......... 33
2.4.2 Monotonicity of Optimal Inventory Positions between States 38
2.4.3 Implications of the Monotonicity Results . ... 41
2.5 An Algorithm to Compute the Optimal Inventory Policy . 43
2.5.1 Continuous Phase-Type Distriuted World Transition Time
and Lead Time .................. .... .. 43









2.5.2 FiE,/lai Distributed World Transition Time and Continuous
Phase-Type Distributed Lead Time . . 54
2.6 An Extension: Demand Arrives Following a General Renewal Process 59
2.6.1 Generalization of the Demand Process Model . ... 59
2.6.2 The Optimal Inventory Policy ............... .. 61
2.7 Sum m ary .. ... .. .. .. .. ... .. ... ... 63

3 MODELS WITH PARTIALLY OBSERVABLE WORLD STATES .... 65

3.1 Introduction ............. . . ...... 65
3.2 A Simple Model with Two World States . . ...... 65
3.2.1 Effects of the Unobservable World . . ..... 66
3.2.2 Exponential Transition Time Distribution . ... 70
3.2.3 Computation of the Optimal Inventory Position . 73
3.2.4 An Extension .................. ..... .. 78
3.3 Multiple World States Models ..... . . .. 79
3.3.1 Models with Multiple World States Which are Visited in a
Fixed Sequence .................. .. .. .. 79
3.3.2 A Recursive Formula .................. .. 85
3.3.3 Optimal Inventory Position . . . ..... 88
3.3.4 More General Multiple States World Models . ... 90
3.4 Summary .................. ............ .. 92

4 JOINT PRICING AND INVENTORY CONTROL IN DYNAMIC EN-
VIRONM ENT .................. ............ 93

4.1 Introduction ....... . . ............ 93
4.2 Joint Pricing and Inventory Control in Price Sensitive Poisson
Demand Environment ......... . . .... 94
4.2.1 The Model that Price Can Only Be Set Once . ... 94
4.2.2 Algorithm to Compute the Optimal Price and Inventory
Position ..... . ... ..... 102
4.2.3 The Model that Price Can Be Set Continuously ...... 105
4.3 Semi-Markov Modulated Price-Sensitive Poisson Demand ..... 106
4.3.1 The Model .......... ..... ...... 107
4.3.2 The Price Can Only Be Set Once for Each State ...... 108
4.3.3 Approximate Models for Semi-Markov Modulated Poisson
Demand ........ ..... ......... 111
4.3.4 Approximate Model that the World Process is Markovian
and One Price for Each State ..... . . 113
4.4 Summary ............... ......... .. 113

5 A STOCHASTIC MULTI-ITEM INVENTORY MODEL WITH UN-
EQUAL REPLENISHMENT INTERVALS AND LIMITED WARE-
HOUSE CAPACITY .............. . . .. 115

5.1 Introduction ............... .......... .. 115









5.2 Model Assumptions and Formulation . . 115
5.3 Solution Approaches .................. ..... .. 119
5.3.1 Equal Replenishment Intervals . . 119
5.3.2 Heuristics for a Two-item Case . . ..... 124
5.3.3 Heuristics for the General Multi-item Case . ... 131
5.3.4 Proof of Optimality of Heuristics for Simultaneous Replen-
ishment Case .................. .. ..... 135
5.4 Numerical Results .................. ...... .. 135
5.5 Summary .................. ............ 139

REFERENCES .................. ............. .. 142

BIOGRAPHICAL SKETCH .................. ......... 147















LIST OF TABLES
Table Page

5-1 Error (in .) of the solution obtained by the heuristics as compared
to the optimal solution, as a function of the tightness of the storage
capacity (2 item s). .................. ..... ..... 138

5-2 Error (in .) of the solution obtained by the heuristics as compared to
the optimal solution, as a function of demand variability between items
(2 item s). . . . . . .. . .... 138

5-3 Error (in .) of the solution obtained by the heuristics as compared to
the optimal solution, as a function of the underage penalty costs of the
item s (2 item s). .................. ...... ... .. 138

5-4 Relative cost associated with coordinating replenishments for different
items to occur at the same time, as a function of the tightness of the
storage capacity. .................. .. ...... 138

5-5 Relative cost associated with coordinating replenishments for different
items to occur at the same time, as a function of demand variability
between items (2 items). .................. .... 140

5-6 Relative cost associated with coordinating replenishments for different
items to occur at the same time, as a function of the underage penalty
costs of the items. .................. ..... ..... 140

5-7 Error (in .) of the solution obtained by the heuristics as compared
to the optimal solution, as a function of the tightness of the storage
capacity (3 item s). .................. ..... ..... 140

5-8 Error (in .) of the solution obtained by the heuristics as compared to
the optimal solution, as a function of the underage penalty costs of the
item s (3 item s). .................. ...... ... .. 140
















LIST OF FIGURES


Optimal inventory position at y

Optimal inventory position at y

Optimal inventory position at y

A1 1, A2 = 1.2,p = 0.4, and n

A1 1-, A2 = 2,/p 0.4, and no

A1 1A, A2 = 3,/t 1.5, and 1 d

A1 1A, A2 = 2,/p 1, and no de

Optimal inventory position y*()

Profit function . ...

A multi-item inventory system.


7*

7*

I*

3

d(

ei

In


inventory system with unequ

inventory policy for the nonir

inventory policy for the greed

inventory policy for the shari:


Figure

21

22

23

31

32

33

34

41

42

51


Page

... . . 5 3

+ 1 ................. .53

1 ................. .54

demand before t . . 74

demand before t . . 74

nand before t, s = 1 . ... 75

land before t . . 75

... . . 0 3

. . .. . 0 3

... . . 17

al replenishment intervals . ... 124

itrusive heuristic. . . ... 125

y heuristic. .. . . 127

ng heuristic. . . 130















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STOCHASTIC INVENTORY CONTROL IN DYNAMIC ENVIRONMENTS

By

Jie Cao

August 2005

C('! i: H. Edwin Romeijn
Major Department: Industrial and Systems Engineering

This dissertation studies some issues in stochastic inventory control.

The first focus of the dissertation is on stochastic continuous-time inventory

control problems for a single item in dynamic environments. The demand process

is modeled as a semi-Markov chain modulated Poisson process. It is shown that

a myopic policy is optimal if the products can be purchased or bought-back at a

single price. Conditions on the semi-Markov chain under which products will never

be returned is derived. An algorithm to dynamically compute the optimal policy for

a special case of the model is also provided. This demand model is next extended to

a semi-Markov modulated renewal process, and several results are generalized to this

more realistic model.

The next focus of the dissertation is a class of Markov modulated Poisson de-

mand processes in which the transitions between the different states of the world is

unobservable. A basic model with two demand states is first studied, and the optimal

inventory policy is derived. An algorithm to compute this policy is also provided.

Next the basic model is extended to multiple states, and a recursive formula is given

which can be used to compute the optimal policy.









The inventory models with simultaneous ordering and pricing decisions are stud-

ied next. The demand process is dependent on the price. The joint pricing and

inventory model under a price-sensitive Poisson demand environment is studied, and

an algorithm to compute the optimal solution is given. Next the study is extended to

the semi-Markov modulated Poisson demand environment, and it is shown that with

certain approximation, the model can be solved in the similar way as in a Poisson

demand environment.

The other focus of the dissertation is on stochastic inventory models for multiple

items with both equal and unequal replenishment intervals under limited warehouse

capacity. The optimality condition for equal replenishment intervals case is given,

three heuristics are implemented, and it is proved that these heuristics provide the

optimal solutions in the case of equal replenishment intervals. Extensive numerical

tests are conducted, and the heuristics yield high quality solutions in very limited

time.















CHAPTER 1
INTRODUCTION

1.1 General Description

Stochastic inventory control has long been one of the central issues in supply

chain management. This is in part because efficient inventory management can both

maintain a high customer service level and reduce unnecessary over-stock expenses

which may take up a significant part of an organization's total costs. Even after over

50 years' study and thousands of papers published in this area, inventory problems

still continue to provide many new and challenging fields for researchers to explore.

One modern direction is to study the optimal inventory behavior under more com-

plicated but more realistic demand environments. In addition, more recent research

starts studying the effects of including pricing decisions into the traditional inventory

problems.

Distribution systems often contain a set of regional warehouses, each of which

stores a v ,ii I v of items supplied by multiple manufacturers in order to serve a re-

gional population of customers. Effectively managing the inventory of multiple items

under limited warehouse storage capacity is critical for ensuring good customer ser-

vice without incurring excessive inventory holding costs. Each regional warehouse

manager thus faces the challenge of coordinating the inventory levels and deliveries

of multiple items in order to meet desired service levels while obeying warehouse

capacity limits. Suppliers to such regional warehouses must efficiently manage the

tradeoffs they face between inventory and transportation costs, which often leads

different suppliers to prefer different warehouse replenishment frequencies. These









different replenishment frequency preferences, combined with varying degrees of de-

mand uncertainty, further compound the challenges the warehouse manager faces in

effectively utilizing limited warehouse capacity.

1.2 Literature Review

Stochastic inventory control models can be roughly divided into two in i i" classes

based on the nature of the demand process. The first class deals with the station-

ary behavior of an inventory system and its corresponding control policies. In the

literature, the control policies considered for such systems are often continuous time

review policies, and the long-run average cost is often used as the performance mea-

sure. The demand process is either assumed to be stationary, or at least assumed to

have a limiting distribution. These types of problems have been worked on exten-

sively since the 1950s, and it has been well-known for decades that an (s, S) policy,

with reorder point s and order-up-to level S, is optimal under mild conditions on the

cost structure (Beckmann [6], Hadley and Whitin [29]). Under some special demand

models, e.g., Poisson demand process, or in case of linear ordering costs, an (s, S)

policy can be simplified to a base stock policy (s, Q) where s is again the reorder

point, and Q is the order quantity. Earlier works in this area are well summarized by

Lee and N liii [41]. Recent developments in this area have mainly been focused

on the determination of the optimal parameters, or the design of good heuristics that

result in near optimal solutions, e.g., Federgruen and Zheng [20] and Gallego [25].

The second class of problems deals with time-dependence and adaptive deci-

sion making in a dynamic demand environment. Most of the research work done

in this area deals with discrete-time models, where the dynamic nature of the de-

mand process is readily represented via a dynamic programming approach. The first

mathematical formulation of problems of this type was introduced in Arrow et al.

[3], and later enriched by Bellman et al. [7]. Karlin [36] extends their results by









studying inventory models where demands are independent, but not necessarily iden-

tically distributed, over time. He showed that a state-dependent base stock policy

is optimal. Moreover, in any set of consecutive periods for which the sequence of

demand distributions decreases stochastically, the optimal base stock level also de-

creases. In these earlier works, the demands in different periods are assumed to be

independent. Veinott [56] extends these results to an infinite horizon, discrete-time,

multi-product dynamic nonstationary inventory problem. The demands in different

periods are not necessarily independent. Under linear ordering costs and the assump-

tion that disposal of excess inventory is allowed at the same price as replenishment

of inventory, Veinott derives conditions under which a myopic base stock ordering

policy is optimal. Lovejoy [44] considers a periodic review, dynamic, single-product

inventory model with linear ordering costs. He considers both disposal and nondis-

posal models, and derives bounds on the relative loss compared to the optimal cost

that is incurred by restricting consideration to the class of myopic inventory policies.

Recent studies model the dependence of demand in dl-i iinl time intervals as

resulting from the effect of some underlying events. These underlying events occur

as time passes, and they may affect the properties of the current and future demand

process. Song and Zipkin [52] and Zipkin [63] provide some examples of effects that

may characterize the state of the world, such as weather, economy, technology, cus-

tomer status, etc. They usually model the underlying events as a Markov process,

either in continuous or discrete time. In particular, Song and Zipkin [52] assume

a demand process that is governed by an underlying core process, called the world,

which is a continuous-time Markov chain with discrete state space. The demand

process is then a Poisson process whose rate depends on the current state of the

world. They show that if the ordering costs are linear in the quantity ordered, then

a state-dependent base stock policy is optimal. If a fixed ordering cost is incurred,

then a state-dependent (s, S) policy is optimal. They also show that if the demand









process satisfies a certain monotonicity property, the optimal policy will inherit this

monotonicity. They also construct an iterative algorithm to approximate the optimal

policies, and an exact algorithm for the linear cost model. Later, Song and Zipkin

[53] utilize the similar model to show how to manage inventory under a deteriorating

demand environment.

The memoryless property inherent in the model of Song and Zipkin [52] means

that ordering decisions are only made in response to an event, i.e., a change of the

state of the world or the occurrence of a demand. This property allows the transfor-

mation of the continuous-time model into an equivalent discrete-time model, and they

then employ a discrete-time dynamic programming approach. However, the implicit

assumption in these models that the time between state transitions is memoryless

is not ahb--iv- reasonable. For example, if the properties of the demand process are

weather-related, e.g. in the case of seasonal demands, this assumption is clearly not

satisfied. Knowledge of the amount of time that we have spent in a given season

generally provides us with some information on how soon this season will end and

what the next season will be.

The model we propose is closely related to the model of Song and Zipkin [52].

In particular, we extend their world model by relaxing the assumption that the time

between state transitions is memoryless. This means that the changes in the de-

mand process are described by a semi-Markov process instead of an ordinary Markov

process. The main effect of this relaxation is that at every point in time in a given

state, the process of future demands is different. Thus, an optimal policy may require

making ordering decisions continuously in time. This means that the elapse of time

itself provides us with information about the future. For example, the absence of

demands for a certain amount of time may cause us to adjust the inventory position.

Heyman and Sobel [31] also studied semi-Markov decision processes. But not like









us, they restricted that the decisions can be made only at the epoch of state tran-

sitions and concluded that the optimization of infinite horizon discounted model is

essentially the same as the optimization of discrete-time Markov decision processes.

Obviously, allowing decision makings at any time, as we do here, makes the problem

more complicated, and the behavior of the optimal policy also differs from that for

MDPs, as we will show. To be able to deal with this additional complexity, we mainly

focus on a model in which disposal of excess inventory is allowed at the same price

as replenishment of inventory (analogous to the discrete-time model of Veinott [56]).

For this case, we derive conditions under which a myopic base stock ordering policy

is optimal.

Most real life inventory control problems face only partially observable demand.

The true underlying distribution of the demand is not directly observed, and only

demand occurrences are observed. Scarf [50, 51] studied an inventory problem in

which the parameter of the demand is unknown, but a priori B ,i, i, distribution is

chosen for the parameter. He used B li- i i methods to solve the inventory control

problems and character the optimal ordering policy. Azoury [4] extended the result

of Scarf [51] by studying dynamic inventory models under various families of demand

distributions with unknown parameters. He derived the optimal B li-, -i i, policy, and

showed its computation is no more difficult than the corresponding computation when

the demand distribution is known. Lovejoy [43] studied inventory models with uncer-

tain demand distributions where estimates of the unknown parameter are updated in

a statistical fashion as demand is observed through time. He showed that a simple

inventory policy based upon a critical fractile can be optimal or near-optimal. Later,

Lovejoy [44, 45] extended the study by showing the robustness of bounds on the value

loss relative to optimal cost of myopic policies which may be stopped earlier.

Kurawarwala and Matsuo [38] gave a combined forecasting and inventory model

according to the characteristics of short-life cycle products. They proposed a seasonal









trend growth model and used optimal control theory to get the optimal inventory pol-

icy. Treharne and Sox [55] studied a partially observable Markov modulated demand

model in which the probability distributions for the demand in each period is de-

termined by the state of an underlying discrete-time Markov chain, and partially

observed. They showed that that some suboptimal control policies, open-loop feed-

back control and limited look-ahead control, which account for more of the inherent

uncertainty in the demand processes, almost ahb--iv achieve much better performance

than the typically used CEC (certainty equivalent control) policy.

Most traditional inventory problems concern the determination of optimal re-

plenishment policies in different types of environments, where the demand process is

often assumed to be given. In these problems, product selling prices are not a decision

variable, but given as known parameters, although they may change from period to

period. Therefore, the aim is to minimize the expected operating costs, because the

expected revenues are not controllable.

More recent developments in industrial practice combining pricing and inventory

management have shown great success, and have stimulated the need for research into

combining pricing and inventory control policies. Whitin [61] proposed including the

pricing into inventory planning decisions. He studied the single period newsvendor

model with price dependent demand and considered the problem of simultaneous

determination of a single price and ordering quantity. Thomas [54] considered a

single item, periodic review, finite horizon model with a fixed ordering cost and price

sensitive demand process. He conjectured that a (s, S,p) policy was optimal: the

inventory replenishment is governed by a dynamic (s, S) policy, where the optimal

price depends on the inventory level at the beginning of a review period. He also

constructed a counterexample which demonstrates that if the available price choice

is restricted to a discrete set, this policy may not be optimal.









Petruzzi and Dada [47] provided an excellent review on pricing decisions in the

newsvendor problem, and in addition extended the single period model to a multi-

period one. They concluded that in most papers on pricing the randomness in demand

is assumed independent of the item price and can be modelled either in an additive

or a multiplicative way. They pointed out that a difficulty in multi-period models

results from the assumption that inventory leftovers cannot be disposed of. They show

how revising this assumption and allowing for the possibility of salvaging leftovers

is sufficient to yield a stationary myopic policy for the multiple period problem. By

cutting off the links between periods, all results and managerial insight available for

the single period model apply directly to the multiple period model.

Federgruen and Heching [19] analyzed a single item periodic review model where

demands depend on the item's price, ordering costs are linear in the ordered amount,

and all stockouts are backl..--.-.,. They studied both finite and infinite horizon models,

using both expected discounted and time averaged profit criteria. They derived the

structure of an optimal combined pricing and inventory strategy for all their models

and developed an efficient value iteration method to compute the optimal strategies.

They showed that a base-stock list price policy is optimal for their model: in each

period the optimal policy is characterized by an order-up-to level and a price which

depends on the starting inventory level before ordering at the beginning of each

period. If the starting inventory level before ordering is below the order-up-to level,

an order is placed to raise the inventory level to the level. Otherwise, no order is

placed and a discount price is offered. The discount price is a non-increasing function

of the starting inventory level.

Recently, C'!. i and Simchi-Levi [12, 13] generalized the above model by incor-

porating a fixed cost component. They show that the (s, S,p) policy proposed by

Thomas [54] is indeed optimal for additive demand functions when the planning hori-

zon is finite; when the planning horizon is infinite, this policy is optimal for both









additive and general demand processes under both discounted and average profit cri-

teria. They also introduce the concept of symmetric k-convex functions and use this

to provide a characterization of the optimal policy.

Though periodical review models have been studied quite extensively, continuous-

review joint pricing and inventory control problems have received far less attention in

the literature. Li [42] considered a continuous time integrated pricing and inventory

planning strategies model where demand and production are both Poisson processes.

The intensity of the demand process depends on the item's chosen price. He showed

that if ordering and holding costs are both linear a barrier policy is optimal. He also

gave an implicit characterization of the optimal pricing policy when dynamic pricing

is allowed. Feng and ('C!, i [23] studied a continuous review model that is related

to ours where the demand is modelled as price-sensitive Poisson process. They re-

strict the available prices to a given finite set (specifically, only two candidate prices),

and assume zero lead times. They show that a (s, S,p) policy is optimal when fixed

ordering costs are present.

In our problem, we model the demand as a Markov modulated Poisson process

(see also Song and Zipkin [52]). In particular, the demand process is a Poisson process

whose rate is governed by an underlying Markov chain that represents the state of

the world. We introduce pricing flexibility into this model by allowing the rate of

the Poisson process in each state to depend on the price of the product. Recently,

C'!, i, and Simchi-Levi [12, 13] generalized the above model by incorporating a fixed

cost component. They show that the (s,S,p) policy proposed by Thomas [54] is

indeed optimal for additive demand functions when the planning horizon is finite;

when the planning horizon is infinite, this policy is optimal for both additive and

general demand processes under both discounted and average profit criteria. They

also introduce the concept of symmetric k-convex functions and use this to provide

a characterization of the optimal policy.









Distribution systems often contain a set of regional warehouses, each of which

stores a variety of items supplied by multiple manufacturers. Effectively managing

the inventory of multiple items under limited warehouse storage capacity is critical for

ensuring good customer service without incurring excessive inventory holding costs.

Suppliers to such regional warehouses must efficiently manage the tradeoffs they face

between inventory and transportation costs, which often leads different suppliers to

prefer different warehouse replenishment frequencies. For example, manufacturers

who supply items with a high value-to-weight ratio typically find it more economi-

cal to send relatively frequent shipments in small quantities, while those who supply

items with a low value-to-weight ratio often prefer to delivery large quantities less

frequently (see Ballou [5]). These different replenishment frequency preferences, com-

bined with varying degrees of demand uncertainty, further compound the challenges

the warehouse manager faces in effectively utilizing limited warehouse capacity.

Stochastic inventory models involving (production) capacity constrained perio-

dic-review policies have attracted the attention of many researchers. Evans [18] first

considers this issue by modeling periodic-review production and inventory systems

with multiple products, random demands and a finite planning horizon. He develops

the form of the optimal policy for multi-product control for such a system. Since

then, much of the literature has studied periodic-review, single-product systems with

production capacity constraints. Florian and Klein [24] and De Kok et al. [37] charac-

terize the structure of the optimal solution to a multi-period, single-item production

model with a capacity constraint. Federgruen and Zipkin [21, 22] show that a modi-

fied base-stock policy is optimal under both discounted and average cost criteria and

an infinite planning horizon. The modified base-stock policy requires that, when ini-

tial stock is below a certain critical number, we produce enough to bring total stock

up to that number, or as close to it as possible, given the limited capacity; otherwise,

we do not produce. They also characterize the optimal policy by deriving expressions









for the expected costs of modified base-stock policies. Kapuscinski and Tayur [35]

provide a simpler proof of optimality than Federgruen and Zipkin [21] for the infinite-

horizon discounted cost case, based on results from Bertsekas [8]. Ciarallo et al. [15]

and Wang and Gerchak [58] analyze a production model with variable capacity in

a similar environment as Federgruen and Zipkin [21]. Wang and Gerchak [57] also

incorporate variable capacity explicitly into continuous review models.

DeCroix and Arreola-Risa [16] study an infinite-horizon version of the capacitated

multi-product case. They establish the optimal policy for the case of homogeneous

products, and propose a heuristic policy for heterogeneous products by generalizing

the optimal policy for the homogeneous product case. Products are called homoge-

neous if they have identical cost parameters and their demands are identically dis-

tributed. Glasserman [27] addresses a similar problem to DeCroix and Arreola-Risa

[16] in a continuous-review system. He presents a procedure for choosing base-stock

levels and capacity allocation that is .,-imptotically optimal, but assumes that a

fixed proportion of total capacity is dedicated exclusively to each product. The use

of .i,,ii!illl c analysis is similar in spirit to Anantharam [1]. His static allocation

problem contrasts with the dynamic scheduling problem addressed in Wein [60] and

Zheng and Zipkin [62] and the priority scheme in Carr et al. [11]. Lau and Lau

[39, 40] present formulations and solution procedures for handling a multi-product

newsboy problem under multiple resource constraints. N i, ii i- and Schmidt [46]

also investigate several heuristics for a single-period, multi-item inventory problem

with a resource constraint.

Anily [2] and Gallego et al. [26] study a multi-item replenishment problem with

deterministic demand. Anily [2] investigates the worst-case behavior of a heuristic

for the multi-item replenishment and storage problem and derives a lower bound on

the optimal average cost over all policies that follow stationary demand and cost

parameters. Gallego et al. [26] consider two economic order quantity models where









multiple items use a common resource: the tactical and strategic models. They

derive a lower bound on the peak resource usage that is valid for any feasible policy,

use this to derive lower bounds on the optimal average cost for both models, and

show that simple heuristics for either model have bounded worst-case performance

ratios. Additional literature, e.g., Rosenblatt and Rothblum [48], Goyal [28], Hartley

and Thomas [30], Jones and Inman [34] and Dobson [17], deals with deterministic

inventory models with warehousing constraints.

Although much of the literature is devoted to multi-item, periodic-review systems

with a production capacity constraint, little has been done for stochastic inventory

models with a warehouse-capacity constraint. Veinott [56] first considers a multi-

product dynamic, nonstationary inventory problem with limited warehouse capacity.

He provides conditions that ensure that the base stock ordering policy is optimal

in a periodic-review inventory system with a finite horizon. Ignall and Veinott [33]

show that, in the stationary demand case, a myopic ordering policy is optimal for a

sequence of periods under all initial inventory levels. Recently, B. Vr et al. [9, 10]

use a dynamic programming approach to derive the optimal ordering policy for the

average cost problem and show the convexity of the cost function, as we did in this

study coincidentally. They also show the optimality of the modified base-stock policy

in the discounted cost version of the problem. In this paper, we extend their results

by explicitly characterizing the optimal myopic policy under relaxed assumptions on

the demand distributions.

1.3 Outline of Dissertation

In C'! lpter 2, we propose an inventory model under a semi-Markov modulated

Poisson demand environment. We give the description of the model, and some prop-

erties of the model. Then we give the optimal inventory policy. We show that if the

demand process observes some monotone property, then the optimal inventory posi-

tion will also show a similar pattern. For a special phase type lead time distribution,









we give one algorithm to actually compute the optimal inventory positions. We also

extend the model to a more general case where renewal process is in place of Poisson

process for the demand process.

In ('!i lpter 3, we restudy the inventory model in C'!i lpter 2. But this time the

state of the underlying world is no longer observable, and we can only observe the

actual demand arrivals. We first study a two world states model, and give the form

of the optimal inventory policy for this model, and propose one algorithm to solve

it. We then extend the study to a multiple world states model and give a recursive

formula to determine the probability of the underlying world in each state, and help

determine the optimal inventory policy.

In C'!i lpter 4, we include the pricing decisions at the same time that the in-

ventory strategy is determined. The price will affect the demand process, and we

are maximizing the total expected discounted profits in an infinite horizon. We first

study the joint pricing and inventory model under a price-sensitive Poisson demand

environment without Markov modulation. In the case where the price can only be

set once at the beginning, we give some properties that can be used to determine

the optimal solution, and derive an algorithm to compute the optimal solution. We

then study the model where price can be continuously set. Next we extend the study

to the semi-Markov modulated Poisson demand environment, and show that with

certain approximation, the model can be solved in the similar way as in a Poisson

demand environment.

In ('!i lpter 5, we study discrete time stochastic inventory models for multiple

items with both equal and unequal replenishment intervals under limited warehouse

capacity. We propose three efficient and intuitively attractive heuristics. We show

that these heuristics provide the optimal replenishment quantities in the case of equal







13

replenishment intervals. For the general model, a numerical comparison of the heuris-

tic solutions to the optimal solutions shows that the heuristics yield high quality

solutions in very limited time.















CHAPTER 2
INVENTORY CONTROL IN A SEMI-MARKOV MODULATED DEMAND
ENVIRONMENT

2.1 Introduction

This chapter is organized as follows. In Section 2.2, we formulate our model

and study a special case of our demand process that reduces to a Markov modulated

demand environment. Then, in Section 2.3, we show that for our general demand

process a myopic policy is optimal, and characterize the optimal policy parameters.

In Section 2.4, we derive sufficient conditions on the demands that imply that the

disposal option will never be used and the myopic policy is thus optimal even in

the case where disposal is not allowed. In Section 2.5, we propose an algorithm to

compute the optimal inventory policy for a special case of our inventory model. In

Section 2.6, we propose an extension of the model, where the demand process in a

given state of the world is a general renewal process instead of a Poisson process. We

end the chapter in Section 2.7 with some concluding remarks.

2.2 Model Formulation

2.2.1 The Demand Process

We consider inventory systems for managing the demand for a single product.

Denote the stochastic process representing the cumulative demand for this product

at each point in time by {D(t),t > 0}. We also assume that there is some under-

lying core stochastic process that models the state of the world, which is denoted

by {A(t), t > 0}. Although we provide our basic analysis of the inventory model

for general demand processes governed by such an underlying core process, the main

results of this chapter assume that the core process is a continuous-time semi-Markov

process. The embedded Markov chain's state space is denoted by I C {0, 1, 2,...},









and the transition probability matrix is P = (pij)ijEl. Given a current state i and a

next state j, the distribution function of the transition time from i to j is denoted by

Gij. We assume the transition times are independent of each other. When the core

process is in state i, the actual demand process follows a Poisson process with rate

Ai, where we assume that A = sup,,j{Aj} is finite. We call this demand process a

semi-Markov modulated Poisson demand process. This demand process is exogenous

and is not affected by any ordering decisions. Clearly, if the distributions Gij are

exponential distributions whose rate only depends on i, the demand process reduces

to a Markov modulated Poisson demand process, as introduced by Song and Zipkin

[52]. Before we describe and analyze our model, we first briefly discuss a case in for

which the transition times are not exponentially distributed, but nevertheless results

in a Markov modulated Poisson process.

2.2.2 Gamma Distributed State Transition Times with Observable Stage
Transitions

A special case of our model is obtained if the time until a transition from state i

takes place has a Gamma distribution with parameters (ri, i) (where ri is a positive

integer), independent of which state is visited next. That is, Gi is the Gamma

distribution with parameters (ri, Vi) for all j e I. Now recall that we can write a

Gamma(ri, i) random variable as the sum of ri independent random variables which

are exponentially distributed with parameter vi. So, if we have the ability to observe

not only the state transitions, but also the -i i,'." changes between the ri successive

stages of the Gamma distributed transition times directly, then we can transform

the underlying semi-Markov process to a continuous-time Markov process using the

following method. We redefine the state space of of the underlying core process to

be I x {1,..., ri}. If the core process is in state (i, k), we are currently in world

state i, and have completed k 1 stages of exponentially distributed duration with

parameter vi in this state. The new embedded Markov chain has one-step transition









probabilities given by


P(i,k),(i,k+) = 1 k= 1,...,ri 1;i E I

P(i,r),(j,1) Pij ij IC

All other transition probabilities are zero. It is clear that we now have a Markov

modulated Poisson demand process that falls within the framework of Song and

Zipkin [52]. However, if we can not directly observe the stage changes of the core

process in the above model, then their results cannot be applied.

2.2.3 Definitions and Notation

For convenience, we merge the two stochastic processes describing the demands

and the state of the world into a single demand /i,.:i/. ;/ process {H(t), t > 0}, where


H(t) (A(t),D(t)).

The entire history up to time t, for all stochastic processes, is denoted by


A(t) {= A(u): 0 < u< t}

S core state history up to and including time t

D(t) = D(u) :0 < u< t}

S cumulative demand history up to and including time t

H(t) H(u) : 0< u < t}

(A(t), D(t))

history up to and including time t.


The observations of the history process are important because they provide informa-

tion on the future of the demand process, and thus may affect the ordering decisions.

Note that we can extract from the demand history process another stochastic process

that represents, at each point in time t, the amount of time that has been spent in









the current state A(t) since the last state transition, -,i S(t), and we let


S t)= {Su) :0

Finally, we let (H(t) denote the sample space of the demand history up to time t,

H(t).

We assume that the inventory level is reviewed continuously, and we need to

decide on how to adjust the inventory position at each point in time. We assume

that both inventory ordering (i.e., and upwards adjustment of the inventory position)

and inventory disposal (i.e., a downwards adjustment of the inventory position) are

possible, both at the same price (see also Veinott [56]). This will be a reasonable

assumption in the context of consignment sales. This is an increasingly popular

business arrangement where for example the retailer does not p li its supplier until

the items are sold. Ownership of the goods is therefore retained by the supplier. Such

an arrangement is widely used in distribution channels for arts and crafts, as well as

industrial and consumer goods such as electronics, furniture, food, books, journals

and newspapers, etc. (see also Wang et al. [59]). It may also be applicable to settings

where suppliers promise to buy back unsold goods as a service to their customers, in

order to build a better relationship and improve the efficiency of the entire supply

chain. Finally, this strategy may be attractive for producers of copyrighted products,

i.e., books, software, music CDs, etc. The value of these products lies in their content

or knowledge, while the costs of producing the media are relatively low. The risk of

promising buyback is not high, while the chances of selling can be greatly increased

by attracting more retailers to distribute the good. In the remainder of this paper,

we will simply talk about the placement of orders, with negative values corresponding

to disposal. Although we allow disposal, we assume that it is not possible to cancel

or change an order that is already placed but which has not yet been delivered.









Inventories incur holding costs, whereas unsatisfied demand is backl-. 1.--- .1 and

incurs a penalty cost. In our model, we assume that holding and penalty costs are

linear with stationary rates h and p, respectively. We can combine these two kinds

of costs together, refer to them simply as inventory costs, and represent them by the

following cost function:


C(x) =the inventory cost rate when the inventory level is x

hx if x > 0

-px if x < 0.

Orders placed will arrive after a potentially stochastic lead time L with distribution

FL. As in Song and Zipkin [52], we assume that the ordering (purchasing) costs, -,i c

are paid when the order is received (i.e., after the lead time). All costs are discounted

at a rate of a. At the time when an ordering decision is made, the observed unit

ordering cost is thus a discounted one, which we denote by c = cE[e-L].

We will often be using the total demand occurring during the lead time. Note

that this total demand depends on the observed demand history, but not the history

of the inventory position, since we assumed that the demand process is exogenous.

Also, as remarked in Song and Zipkin [52], if we require that the lead times do not

cross in time, that is, orders that are placed earlier than other ones can never arrive

later than these, then they are not independent. Following Song and Zipkin [52],

we ignore the impact of lead time history in making ordering decisions because we

lack the ability to collect and process such information, and treat them by standard

approach proposed by Hadley and Whitin [29]. So, we define Djh(t) to be the random

variable representing the total demand occurring during the time interval (t, t + L],

given demand history information h(t) E Ti(t). For fixed denote the distribution

function of Dh(t) by FD,h(t) (z), and define


FL ') j((z)dFL (









Now the conditional expected discounted holding and shortage cost rate, at the

end of a lead time starting from the current time t, and viewed at time t, given

that the demand history is h(t), and the current inventory position (after ordering

decision) is y, can be written as


C(y, t, h(t)) = eaLC(y Dt))t

= Oe- (^y z)dFDt,-^) ')dFL )

I C(y-z)dFDt,h(t)().


2.2.4 Problem Formulation

In our problem, we consider three types of costs: ordering, ]. '.1iii.: and shortage

costs, and our objective is to minimize the total expected discounted costs over the

infinite horizon. With a slight abuse of notation, we define an ordering J ''.. ;/ to

be a family of functions that prescribes, for each time t and each potential history

observed up to that time, the desired inventory position at that time. Note that we

have assumed that the demand process does not depend on any ordering decisions

that we make. This, together with the fact that we may place negative and positive

orders at the same price, implies that the ordering policy does not depend on the

inventory position immediately preceding an ordering decision. More formally, we

define an ordering policy y to be a family of functions, {y(t, -) : H-(t) -+ R; t > 0},

where y(t, -) prescribes that, if we have observed history h(t) up to time t, we place

a (possibly negative) order that brings the inventory position to y(t, h(t)), for every

t > 0. It will be convenient to refer to the function y(t, -) as the 1 '.. :; for time t.
We -i- that a policy y* is optimal if this policy minimizes the expected total future

discounted costs over all policies.








If the initial inventory position is x, the initial demand history h(O) is observed,
and some ordering policy y is followed, the total costs may be expressed as

W(x, h(O) y)

E c(y(0, h(O)) x) + j e-tcd(y(t, H(t)) + D(t))+

Se-C(y(t, H(t)), t, H(t))dt (2.1)

-cx + E c((o0, h()) + e-cd(y(t,H(t))) +
0
E e-r cd(D(t)) + E e- tC(y(t, H(t)), t, H(t))dt (2.2)

In equation (2.1), the expectation is taken over the entire demand process from time
0 to the infinite horizon. Inside the expectation, the first term represents the ordering
cost at time 0; the second term represents the ordering costs for replenishing inven-
tory during the entire infinite horizon since the rate at which we order at time t is

d(y(t, h(t)) + D(t)); and the third term represents the total inventory holding and
shortage costs. We consider only those policies that make the total costs finite. We
will show later that such policies do indeed exist.
In equation (2.2), note that the total expected replenishing costs of all the de-
mands,
E I e- t(D(t))

as well as the value of the initial inventory position (i.e., the term ex) are not influ-
enced by the choice of policy y. Therefore, we will omit these terms in our analysis
and redefine the cost function as

W(x, h(0) y)

E cy(o h(0)) + j e-acd(y(t,H(t))) +E o -tC(y(, H(t)), t, H(t))dt
JO J0









Now consider a realization of the expression inside the first expectation of the above

cost function, i.e., we consider a fixed history path h(t) starting from h(O). This

expression can then be simplified as follows:

0y


e-acd(y(t, h(t))) + cy(0, A

Sc ae-aTdrd(y(t,
0C
= c ae- T d( (t, h(t
0o
roo
= c e-o(U(,-, h(,-)) _


= c ea (T, h(r-))dT
0o
= c ae- (T, h(T))dT
0C


= ace-aty(t, h(t))dt.


So we obtain that


h(t))) + y(O, h(O))

)))dr + y(0, h(0))


y(O, h(O))dT + y(O, h(O))

- ae-a7(0, h(O))dT + y(O, h(O))
0
- y(, h()) J ae- adT + y(0, h())
Jo


(2.3)


E cy(0, h(O)) + e atcd(y(t, H(t))) E e-at(,. ,(t, H(t))dt
Jo J J


and the cost function reduces to


W(x, h(O) y)

S -at,, ,(t,H(t,))dt + L [-atC(y(t,H(tf)),t,H(t))dt]

S e-atE [C(y(t, H(t)), t, H(t)) + (t, H(t))] dt.
0J


(2.4)


2.3 Model Analysis

2.3.1 Optimal Policy

In this section, we will show that under our model assumptions, the optimal

ordering policy is a i',,. '/'.: one, in which, at each point in time, the optimal inventory


(0))









position is found by solving a single one-dimensional optimization problem. Our first
lemma constructs such a myopic policy for each time t.
Lemma 2.3.1 For every fixed t, and every possible hI:/.. h(t) E H(t), let y*(t, h(t))
denote an optimal solution to the problem

min C(y, h(t)) + (. iI
y

if it exists. Then y*(t, -) is a p,'.1. ;/ for time t that minimizes


E [C(y(t, H(t)),t, H(t)) + o. ,,(t, H(t))].

Proof: For any fixed t and every possible h(t) e H(-{t), following any other policy for
time t, v y'(t, -) will result in the inventory position y'(t, h(t)), and

C(y*(t, h(t)), t, h(t)) + o, i*(t, h(t)) < C(y'(t, h(t)), t, h(t)) + (, i'(t, h(t))

by the definition of y*(t, .). Therefore, by denoting the distribution of the history up
to time t by FH(t), we have


E [C(y*(t, H(t)), t, H(t)) + o. ,i*(t, H(t))]


< f [C(y'(t,h(t)),t,h(t)) + (. ,'(t,h(t))] dFH(t)(h(t))


SE[C(y'(t, H(t)) t, H(t)) + r, ,,'(t, H(t))]

which shows the desired result. D

We call y*(t, -) a myopic optimal policy for time t in the sense that it only seeks
to minimize the "current" discounted cost rate at time t rather than the expected
total future discounted costs. The following theorem shows that the optimal policy
coincides with the myopic policy under our model assumptions.









Theorem 2.3.2 Let y*(t, -) be /. I,, .l as in Lemma 2.3.1. If it exists for all t > 0,

then the .. 1. ,I y* = {y*(t, ) : t > 0} is an optimal ordering J..p. ,i that minimizes

W(x, h(0) y) among all policies y.

Proof: Since the decision variables, i.e., the inventory positions at any time t, are

unrestricted and independent of each other by the assumption that negative orders

are allowed, minimizing W(x, h(0)|y) over all policies can be decomposed into mini-

mization problems


min E [C(y(t, H(t)), t, H(t)) + o, ,I(t, H(t))]
Y(t,.)

for all fixed t > 0. Thus the conclusion follows from Lemma 2.3.1. O


Note that so far we have not used any specific details of the demand process.

So our expression of the cost function in fact holds for all continuous-review inven-

tory models with linear ordering cost in which the demand process is independent

of ordering decisions, as long as negative orders are allowed. In the next section,

we return to the semi-Markov modulated Poisson demand process as introduced in

Section 2.2.1. We will show that in that case the optimal policy only depends on the

current state of the core process and the amount of time that has been spent in this

state since the last state transition.

2.3.2 Optimal Policy with Semi-Markov Modulated Poisson Demands

The properties of the semi-Markov modulated Poisson demand process immedi-

ately imply that the lead time demand Dth(t) depends on the history only through

the state that the process is currently in and how long it has been in that state. To

reflect this fact, we rewrite the lead time demand as D" when the core process has

been in state i for s time units. For given y, t, and h(t), since the lead time demand

can be simplified, we can (with a slight abuse of notation) also simplify the inventory








cost rate function to

C(y, i, s) E e-aL(y D)]

which is equivalent to C(y, t, h(t)) if the history h(t) -,i- that A(t) =i and S(t) =
s. We can then replace C(y(t, H(t)), t, H(t)) in the total expected discounted cost
formula (2.4) by C(y(t, H(t)), A(t), S(t)). The total expected cost function for our
semi-Markov modulated Poisson demand model thus reduces to


W(x, h(O) y)


j e-atE [C(y(t, H(t)), A(t), S(t)) + o u(t, H(t))] dt.
Jo/


Let us now define the function

f,s(y)= C(y, i, s) + I,

for every fixed i and s, which can be viewed as the cost rate function if the inventory
position is y at the time when the core process has been in state i for s time units.
With this definition, the objective function can be written as


W(x, h(0) y)


Se-fA(t),s(t)(y(t, H(t),t))dt.
Jo/


Due to the discrete nature of the demand process, the functions C(., i, s) and
fi,(.) will not be everywhere differentiable. Therefore, it will be convenient to define
for every i and s the right derivatives of C(-, i, s) and fi,s(-):

f, i {C(y + E, i, ) C(y, i, )
C (y, i, s) lim
+ ( ,s(Y + E) fi,s(Y)
(fi,s)+(y) i= hm )

C7'(y, i, s) + ac.


Also, let


y*(s) = inf{y : (ff,l)+(y) > 0}.








Note that since the leadtime demand can only assume integral values, all points at
which the functions C(-, i, s) and fi,s(-) are nondifferentiable are integral. In addition,
iI (s) is integral as well. We will now give some properties of the cost functions and
optimal policy; these properties are similar to the ones obtained by Song and Zipkin
[52] for the Markov modulated demand model.
Lemma 2.3.3

(a) C(y, i, s) and fi,s(y) are both convex in y for all i and s, so that ,I (s) minimizes


(b) If ac < p, then /I (s) is finite and nonnegative for all i and s. In addition,
fi,s(y) is nonnegative for all i and s.
(c) If ac > p, then /I (s) = -oo for all i and s.
Proof:

(a) C(y, i, s) is a convex function in y because C(x) is a convex function in y, and
C(y, i, s) is a positive weighted average of convex functions. The convexity of
fi,s (y) and optimality of y (s) then follow immediately.
(b) Note that

C(y, i, s)
= [eC-L(y D'")]

e~ C(y z) dFDi L )dFL%)
0 Jo
S e-a j h(y- z)dFDi,(z)+ pz-y)dFDi,(z) dFL()
JO I0 y

where FDi,, (z) is the distribution function of the lead time demand when the
demand process has been in state i for s time units, conditional on a lead time









of f time units. This implies that


C' (y, i,s)


S00 (
10 a- I(h+p)F (y)


(h +p) C e-aFD (y)dFL) f
Jo f


So for y < 0,


-pE[e-L]


and


C'(y, i, s) + ac

(ac p)E[e-aL.


If ac p < 0 then (f,,)+(y) < 0 for all y < 0 and all i and s. So I/ (s) > 0 for
all i and s. Thus, for all y we have


fi,s(y) > f>,s(y;(s))


C(y (s), i, s) + i(s)> 0.


Furthermore, by equation (2.5) we have


lim C' (y,i,s)
y-+o00



lim (Lf,)+(y)
yv+Oc


ShE[e-aL] > 0



hE [e-aL] + ac > 0.


So y*(s) < +oo for all s and i.
(c) If cr p > 0, then


(fi,) (y) (h + p) J

for all y, i, and s, so ,/ (s) -


e-aFDiF,, ()dFL(f) + (ac


p)E[e-aL] > 0


p}dFL ()


pE[e-L].


(2.5)


and









By the result of Lemma 2.3.3, we conclude that we need to assume that


ac < p


to obtain a reasonable model. Intuitively, if this condition is not met, then we would

alhiv-, prefer to postpone ordering and p iv the shortage penalty, and thus never

place any orders. In the remainder of the paper, we will therefore assume that this

condition is satisfied.

Before we continue, we derive a property of lead time demand.

Lemma 2.3.4 For ,.;, i c I and s > 0,


LD" -st LD

where D represents the lead time demand when the demand process is a stat:., ~;

Poisson process with rate A.

Proof: For fixed lead time the conditional distribution of D ", given a realization

of the core process, is Poisson distributed (see, e.g., Ross [49]). Since the demand

rates of all the states are bounded from above by A, the mean of this Poisson random

variable is also bounded from above by XA, which is true for all the possible realizations

of the core process. This implies that


D'" 0 and i e I. (2.6)


Since inequality (2.6) holds for all fixed the desired result follows for the case of a

stochastic lead time L. O


We are now ready to derive the form of the optimal policy for our system, which

is a continuous-time analog of Theorem 6.1 in Veinott (1965).









Theorem 2.3.5 Under the semi-Markov modulated Poisson demand model, the my-

opic j]. /,. ,; y* /. f,' l by

y*(t, H(t)) yA(t)(S(t)) for all t > 0

exists and its total 1./1.:. ;, costs are finite. Thus, the optimal inventory position at
time t only depends on the state at time t and the amount of time that has elapsed
since the core process last entered that state.
Proof: First, note that for every fixed i and s we have


f,(0) = C(0, i, s)
= E[e-L(^(-D )]

E[e-,LpD "]

< E[e-aLpDA (2.7)

< pAE[L] (2.8)

where inequality (2.7) follows from Lemma 2.3.4, and inequality (2.8) holds since
e-aL < 1. Then, by the definition of (s) and Lemma 2.3.3, fi,s(,i (s)) < fi,s(0) < oo

and y*(s) < oo. Thus, for every fixed t and every h(t) E 7t(t) for which A(t) =
i, S(t) = s, the optimal policy for time t stated in Lemma 2.3.1 does exist, and


y*(t, h(t)) = y* (s).

Thus, by Theorem 2.3.2, the policy {y*(t,H(t)) y*A)(S(t)) : t > 0} is an optimal
policy.









Furthermore, the optimal total expected cost satisfies


0
W(x, h(O)ly*) = e-tE [C(y(t)(S(t)), A(t), S(t)) + ,, (t(S(t))] dt

=e-atE [\fA(t),s(t) (y(t) (S(t)))] dt

< e-atpXE[L]dt

-pAE[L]
a

and is thus finite. D

In the next section, we will give a more explicit characterization of the optimal

inventory position, which can in principle be used to compute the optimal policy, as

well as the cost of the optimal policy.

2.3.3 Determination of the Optimal Inventory Position

Using equation (2.5), we have


y(s) = argmin {y : (f:)+(y) > 0}

= argmin (y : Cf'(y, i, s) + ac > 0}

SargminY: e-a(h + p)F) ()dF(t) pE[ -] > -ac

argmin Y: C e-aFD (y) dFL(f) > pE .

For notational convenience, we may define


F y (Y)= e-'FD (y)dFL(f)

so that

y:(s) = argmin : FDs(y) > PE[ ac (2.9)
11 h+p
This means that the optimal policy depends on the cost parameters only through the

ratio
pE[e-, ] ac (p ac)E[e-]
h+p h+p









It is easy to see that this ratio is alv--, between 0 and 1. In case the lead time is
deterministic, the expression for the optimal policy can be simplified to

y7(s) argm in :e-LFD(y)> aL

argmin Y:F FDs (y) > (2.10)
S L nh+p

We next derive a more explicit expression of the myopic inventory cost rate
function:


C(y, i, s)
Se- h {y- z)dFDz () +p z y)dFi,,z) dFL()
JO 0 f I Jy f
Se- hyFD i, () hzdFxs (z) + pzdFDs(z) pYFDy) dFL(f)

e- (h +p)yFD y) py hE[D}] + (h + p) j zdFDs ()} dFL()
(h + p) y ( L-

(h + p)y e- FD, (Y)dFL() pyE[e -L] hE[D"l] +

(h + p) e- ~ zd ,()d(FL(D).
O y

This expression may be used to determine the optimal cost rate by substituting the
optimal policy in this expression:


fi,s (Y(S))

= C(y(s), i, s) + (s)

(h+ p) e- zdFD,,(z)dFL (-) -hE[D"~] +
Jo 1i',) f

i (s)(+p) e-hp aFD, (y (s))dFL() (pE ac) (2.11)

Note that if the inequality in equation (2.9) or (2.10) is in fact an equality, the last
term in equation (2.11) reduces to zero, and the optimal cost rate reduces to

f.(y (s)) (h + p) e- \ zdFDFS(z)dFL() hE[Dl].
Ji~s\Y \a(l)









However, this generally can only happen if the lead time demand distribution is

continuous, which is not the case in our model.

Finally, we would like to stress the similarity of the expressions in equations (2.9)

and (2.11) with the optimal policy and cost in the standard newsvendor problem. It

turns out that we can find the optimal policy for our model by solving one newsvendor

problem for each i and s.

2.3.4 Total Policy Costs

In the previous sections, we have obtained the form of the optimal policy. How-

ever, the corresponding optimal expected total cost is very hard to evaluate. In this

section we will employ the underlying semi-Markov structure of our demand model

to derive an easier way to determine the optimal costs.

We assume that an inventory policy characterized by yi(s) is adopted. Then

define Vi(x) to be the expected total costs of this policy from the time when the core

process just enters state i when the initial inventory position is x, and discounted to

the time of transition. The total costs can be divided into two components: the total

costs during our current stay in state i, and the total costs after transitioning away

from state i. The first component can be determined by conditioning on the time

until the next transition, whose distribution function is equal to


Git)_ Ypij Gij~t).
jEI

Note that this distribution, and therefore the first cost component, does not depend

on the next state visited. However, for the second component we need to condition

on both the time of the transition as well as the next state itself. We can then express

Vi(x) in terms of the other values of this function as follows:


V'(x) c(y() -x) + e- SC(y(s), s)ds + j e- cdyis)} dGi(T) +


Sj o
Je SI








Since the initial inventory position x is not affected by the ordering decisions made
for state i period, let
i = Vi(x) + ex.

Using this definition for all states and all inventory positions, we obtain

V = cy(0) + /e- c" (ys),i,s)ds + e-cdy (s) }dG(r) +
0 0 Jo0

jEI jE I O
ao C ^i (
= (0) + j { e- SC(y(s), ,s)ds+

Se-Scdyi(s) ce-oyi(r) dGi(r) +


Pij -a7 e VdG(r)
jel 0
= j e-as[C(yi(S), s) + o. (s)]ds dG () + SpjE[Ce-aTij]V

= G(s)e- [C(y), i, s) + o (s)]ds + pjE[Ce- aT1V

where Tij Gi denotes the time spent in state i when the next state is j, and we
have also used a similar derivation as in equation (2.3) to simplify the expression for
the costs while in state i. Now observe that we can in principle compute, for all i,
the total costs from the time of transition to state i by solving a system of linear
equations if we can compute the total costs while in state i for all i. Computing the
total costs while in state i is clearly still nontrivial, but much easier than directly
trying to compute the infinite horizon costs.
2.4 Monotonicity Results
In this section we will show that, if the demand process possesses certain mono-
tonicity properties, the optimal inventory positions over time inherit these properties.









2.4.1 Monotonicity of Optimal Inventory Positions within a Given State

In order to be able to analyze the behavior of the optimal inventory positions

II (s) while in a given state i E I, we will first derive a general stochastic dominance

result.

We -w that a random variable X has a Conditional Poisson distribution with

random parameter A, where A is a nonnegative random variable, if the conditional

random variable X A = A has a Poisson distribution with parameter A. The following

lemma will then prove useful later in this section.

Lemma 2.4.1 Let


X1 ~ Conditional Poisson(A1)

X2 ~ Conditional Poisson(A2).


If A1
Proof: Fix some x > 0, and define


x(A) = Pr(Xi > x |Ai A) = Pr(X2 > xA2 = ).

This function is increasing in A by the fact that a Poisson random variable is stochasti-

cally increasing in its mean (see Example 9.2(b) in Ross [49]). Therefore, the assump-

tion in the theorem z-,v that E [ox(Al)] < E [oz(A2)]. Now denote the distribution

of An by H, (n 1,2). Then, for n = 1,2,


Pr(X > x) = Pr(X, > xA,, A) dH,(A)
JO
SE [Pr(X, > xlA,]

SE [x (A,)].


This yields the desired result.








Returning to the focus of this section, denote the state of the core process after
t time units if the process has currently been in state i for s time units by [A(t +
s)lA(s) = i]. We will show that the following condition implies that the function ,
is increasing in s:
Condition 2.4.2 For all > 0 and all 0 < s < s',

o A[A(t+s)lA(s)=i]dt
The following lemma shows that Condition 2.4.2 ensures that the lead time demands
are stochastically increasing while in a given state.
Lemma 2.4.3 If the demand process -.,/:-i. Condition 2.4.2 for some i E I, then
the lead time demand DL" is .l/.. i,.I-/. Illj increasing in s, i.e.,

D"
Proof: Let i E I be such that Condition 2.4.2 is satisfied. Fix 0 < s < s' and
consider a fixed lead time By the theory of nonhomogeneous Poisson processes
(see, e.g., Ross (1996)), the lead time demand is a Poisson random variable when
conditioned on the core process A. This means that

D'8 ~ Conditional Poisson i A[A(t+s)A(s) i]dt)

De ~ Conditional Poisson ( A[A(t+s')A(s') i]dt

Condition 2.4.2 now implies that


O A[A(t+s) A(s)=i]dt
By Lemma 2.4.1, we then have


D'"

(2.12)









Since this inequality holds for all fixed lead times the desired result follows for the

stochastic lead time case as well. O


We are now able to show that the optimal inventory positions in a given state are

increasing over time if the demand process in that state is stochastically increasing

over time in the sense of Condition 2.4.2.

Theorem 2.4.4 If the demand process -.i-. Condition 2.4.2 for some i E I, then

the optimal inventory position yj (s) is increasing in s.

Proof: Let i E I be such that Condition 2.4.2 is satisfied. Observe that inequality

(2.12) in the proof of Lemma 2.4.3 -,v that, for fixed lead time f, for all 0 < s < s',

and for all y > 0

F s(y) > F^'sy).

By integration, we then obtain that for all 0 < s < s' and for all y > 0,


FDs(y) > FDi"y(.
L L

The result now follows immediately from the expression for the optimal inventory

position ti (s) in equation (2.9). O


We will next discuss two examples to which this result applies.


Examples

1. If the underlying core process is a continuous-time Markov process, the memo-

ryless property implies that the random variables


SA[A(t+s)lA(s)=idt and A[A(t+s')lA(s')=i]dt

both have the same probability distribution as


/ A[A(t)A(0)=i]dt.
Jo









Therefore, it immediately follows that Condition 2.4.2 is satisfied for all i e I,

and therefore the optimal inventory position y (s) is increasing in s. In fact, we

can use appropriate modifications of Lemma 2.4.3 and Theorem 2.4.4 to show

that the inventory position iI (s) is constant in s, which corresponds with the

result of Song and Zipkin [52].

2. Suppose that the interarrival distributions Gij depend on i only and, moreover,

are increasing failure rate (IFR). In addition, suppose that transitions can only

be made to states with a higher demand rate, that is, pij > 0 implies that Ai <

Aj. Then Condition 2.4.2 is satisfied for all i c I, and we obtain monotonicity

of the inventory positions.

Proof: Choose some state i E I, and let Zf be the amount of time remaining

in state i given that the core process has been in state i for s time units, and

denote its distribution by G'. To show that Condition 2.4.2 holds, we need to

show that for all > 0 and all x,

Pr ( A[A(t+s)A(s)=i]dt > x

is increasing in s. Now fix arbitrary values of f > 0 and x, and define


s(z) = Pr ( A[A(t+s)A(s)i]dt > xZ, = z .

Since

E[',,(Z$)] Pr ( A[A(t+s)|A(s)i]dt> x

we in fact need to show that E[,' (Zf)] is increasing in s. Since Gi is IFR, we

know that the random variables Z, are stochastically decreasing in s. In the

remainder of the proof, we will show that the function Qs(z) is decreasing in z

and independent of s, which then implies the desired result (see Ross, [49].

For any i c I, let Ji be a random variable that represents the next state reached

from state i by the core process. If 0 < z < we can then rewrite the function









is, as follows:




Pr Aiz + j A[A(t)IA(o) ]dt > x (2.13)

Pr (j (A[A(t)IA(o)=Ji] A) dt> x- Ai)
o/

where equality (2.13) follows from the fact that the distribution of the remaining

transition time Zf is independent of the next state visited. If z > we have

~(z) = Pr Aidt > xZ = z )= Pr (Ai > x) = l{ j>x}

where 1{} denotes the indicator function. Summarizing, we have


(z Pr (jz (A(t)A(o)=J] ) dt>x i) if z <
1{xij>x} if z >

which is clearly independent of s. Moreover, if x < Ai, the function Q, is

identically equal to 1 and therefore decreasing. If x > Ai, the assumption in

this example -,i- that A[A(t)IA(o)=JA] Ai > 0, so that in that case the function

8s(z) is decreasing as well.
Since we chose the values of > 0 and x arbitrarily, we conclude that Condition

2.4.2 is satisfied for state i. O

The next theorem shows that, under an additional mild regularity condition, the

optimal inventory position in a given state is a step function of time that increases

only by one unit in each step.

Theorem 2.4.5 Suppose that the demand process -.iI. I Condition 2.4.2 for some

i E I and, in addition, the transition time distributions Gi from that state have con-

tinuous densities. Then the optimal inventory position function /, is a step function

that can only have step size 1 in the inventory position space.









Proof: Let i e I be such that Condition 2.4.2 is satisfied, and suppose that we have

been in this state for s time units. Then recall that, for a fixed lead time we have


D'" ~ Conditional Poisson ( A[A(t+s)A(s) idt)


By the assumptions on the transition time distributions, we conclude that the random

variables

SA[A(t+s)lA(s)=idt

have densities that are continuous as a function of s. This implies that, for all

d 0,, ,2,..., the probability Pr(D'8 = d) is continuous as a function of s. Thus, for

a stochastic lead time L, FDis(Y) is continuous in s for all y > 0 as well. In addition,

FDi, (y) is a decreasing function of s by the proof of Theorem 2.4.4.

Since the lead time demand is a discrete random variable that has strictly positive

probability at every nonnegative integer d, we conclude that, for fixed s, FDi, (d) is

a strictly increasing function of d for d = 0, 1, 2,.... This means that all functions in

the family {FDi, (y) : y > 0}, viewed as functions of y, are step functions that strictly
L
increase at each integral value of y.

We conclude that I/ (s) is a step function that, at each step, increases by exactly

1. O


As a final remark, note that, with probability 1, each order either replenishes a

reduction in inventory position due to demand, or is due to an increase in the optimal

inventory position. Therefore, under the conditions of Theorem 2.4.5, as long as the

core process remains in a given state, each order is, with probability 1, for a single

unit only.

2.4.2 Monotonicity of Optimal Inventory Positions between States

We will next analyze the relationship between the optimal inventory positions

in different states. In particular, we will show that if the demand process satisfies









the following condition for a pair of states i, j I, then the inventory position never

decreases if a transition is made from state i to state j.

Condition 2.4.6 For all > 0 and all s > 0,


SA[A(t+s)A(s)=i]dt
The following lemma shows that Condition 2.4.6 implies a monotonicity rela-

tionship between the lead time demands in different states.

Lemma 2.4.7 If the demand process -.,l./;. Condition 2.4.6 for states i,j E I,

then
D~" 0.

Proof: Fix s > 0, and consider a fixed lead time We then have

D"' ~ Conditional Poisson ( A[A(t+s)lA(s)=idt

Dj'o ~ Conditional Poisson ( A[A(t)IA(o)=j]dt)

Condition 2.4.6 and Lemma 2.4.1 then imply that

D9's
Since this inequality holds for all fixed lead times the desired result follows for the

stochastic lead time case as well. O

We are now able to show that the optimal inventory positions are increase when

a state transition is made if the demand process satisfies Condition 2.4.6 for that

transition.

Theorem 2.4.8 If the demand process -.,/l. Condition 2.4.6 for states i,j E I

then


y>(s) < yj(0) for all s > 0.









Proof: Observe that inequality (2.14) in the proof of Lemma 2.4.7 --,v that, for

fixed lead time f, and for all y > 0

FDi,(y) > FDjo(y).

By integration, we then obtain that for all y > 0,

FDi,(y) > FDjo y).

The result now follows immediately from the definition of the optimal inventory

position / (s) in equation (2.9). D

We will next discuss two examples to which this result applies.

Examples

1. If the underlying core process is a continuous-time Markov process and, in

addition,


[A(t) A(0) i] < [A(t) A(0) j] w.p. 1, for all t > 0

for all i,j e I such that Ai < Aj it immediately follows that Condition 2.4.6 is

satisfied for such pairs of states. Theorem 2.4.8 then corresponds to Theorem

8 in Song and Zipkin [52].

2. If the transition time distributions are arbitrary, but states are alv--,v- visited

in a predetermined sequence, i.e., p,i+ = 1 for all i E I, and, in addition,

A0 < A1 < A2 < then Condition 2.4.6 is satisfied for all i,j E I such that

j> i.

Proof: C'! ....- some state i E I, and fix arbitrary values of f > 0 and x. We

need to show that

Pr ( A[A(t+s)IA(s) i]dt > x < Pr ( [A(t)A(0) i+]dt >
Vo / Vo/









Now note that

Pr A[A(t+s)lA(s)i]dt > x Z o= 0 Pr / A[A(t)IA(o)i+1]dt > x

Using the notation of the proof of Example 2 in Section 2.4.1 we have

s(0) Pr( A[A(t+s)A(s) ]dt > xZf = 0)

F[, (Z)] Pr ( A[A(t+s)\A(s)=i]dt >

[' .(Z)] < s(O)

where we have used the fact that Ji i +1, and the last inequality follows from
the fact that the function is decreasing. Since we chose the values of > 0
and x arbitrarily, this implies that Condition 2.4.6 is satisfied for i, i + 1 e I. O

2.4.3 Implications of the Monotonicity Results

The results of the previous two sections can now be used to conclude our main
monotonicity result:
Theorem 2.4.9 Assume that the demand process -.i/.1 Condition 2.4.2 for all i E

I and, in addition, Condition 2.4.6 is .l.:.'/7 ,1 for all i,j E I such thatpij > 0. Then
the optimal .1.:'. ;/ results in a sequence of inventory positions that is nondecreasing.
Proof: Theorem 2.4.4 -,i- that the optimal inventory position never decreases as
long as we are in a given state. Since Condition 2.4.6 is satisfied whenever it is possible
to transition from state i to state j, and Theorem 2.4.8 z-v- that the optimal inventory
position never decreases when we move to a new state. O

So far, we have dealt entirely with a situation where disposal of inventory is
allowed at the purchase price. However, this may not alv--,v- be a reasonable assump-
tion. If disposal is not possible, then at each point in time the inventory position

is bounded from below by the inventory position immediately preceding an ordering
decision. In that case, the myopic policy may no longer be feasible, and therefore









clearly not optimal. However, note that under the conditions of Theorem 2.4.9, the

myopic inventory position will alv--x increase. The following theorem now provides a

sufficient condition under which disposal is never desirable, so that the myopic policy

remains optimal even if disposal is not allowed.

Corollary 2.4.10 Consider the case where disposal of inventory is not allowed. As-

sume that the demand process .il.:.. Condition 2.4.2 for all i E I and, in addition,

Condition 2.4.6 is -,/.:-l ./, for all i,j E I such that pij > 0. If, .:,/.:l.:ll;i (at time 0),

the core process has been in state i E I for s time units, and the initial inventory x

is no 1'i, than iI (s), then the ,;,. I',: 1 q'/'. ;I is the optimal 1j../' ;,

Proof: Since x < y*(s), the initial myopic inventory position y (s) can be reached

even if disposal is not allowed. At each subsequent point in time, an ordering de-

cision should either replenish a demand or adjust the inventory position according

to an optimal policy. Since Theorem 2.4.8 implies that the myopic policy will never

prescribe a reduction in inventory position, it remains optimal even when disposal is

not allowed. O


Our next result derives the optimal policy if the condition in Corollary 2.4.10 on

the initial inventory level is violated. This theorem is a continuous analog of Theorem

6.2 in Veinott [56].

Corollary 2.4.11 Consider the case where disposal is not allowed. Assume that the

demand process -i/.:-/. Condition 2.4.2 for all i E I and, in addition, Condition

2.4.6 is -,/.:-l ./, for all i,j E I such that pij > 0. Then the .1.:. ;/ -.'.; y, that does

not order until the inventory position drops below the level that is prescribed by the

I'i'. ',:' 1./ : ;I and follows the i;,,, 1 ;.' ; after that, is the optimal j .1./. ;

Proof: Let T denote the earliest time at which the inventory position drops to or

below the level prescribed by the myopic policy when applying policy y (where we

let T = +oc if this event never occurs). Up to time T, the inventory positions

resulting from this policy will be the lowest among all feasible policies since y does









not place any orders. Since the cost rate function at each point in time is convex,

and the myopic policy is the smallest minimizer of the cost rate function, policy y

will minimize the total cost over the interval 0 < t < T among all feasible policies.

After time T, Corollary 2.4.10 implies hat the myopic policy will be optimal. So the

policy described in this theorem is an optimal policy. O


2.5 An Algorithm to Compute the Optimal Inventory Policy

Recall that the expression for the optimal inventory position after spending s

time units in state i is given by


yi(s) = argmin y : FDs (y) > pE[ ac (2.9)
L, h+p

Note that the optimal inventory policy is thus a collection of functions, one for each

state of the world. We therefore cannot expect to be able to compute in finite time

(or represent using finite storage space) the entire optimal inventory policy for our

model. In addition, s is a continuous variable, which further complicates the a priori

computation of the optimal inventory policy. Instead, we will in this section develop

an algorithm that constructs parts of the optimal policy as needed for a special case

of our model. In addition, we denote & and E to represent the Kronecker product

and Kronecker sum, respectively.

2.5.1 Continuous Phase-Type Distriuted World Transition Time and Lead
Time

From equation (2.9) we see that the key is the lead time demand distribution

functions FD i (y). However, in general it is very difficult to compute these lead time

demand distributions directly (see Zipkin [63]). Song and Zipkin [52] designed an

algorithm to compute the myopic policy for a special case of their Markov modu-

lated Poisson demand model. Specifically, they devised a way to compute the cost

rate function by assuming that the lead time has a continuous phase-type (CPH)









distribution, which can be modeled by the time until absorption of a continuous-

time Markov chain. Then they study the behavior of a joint process consisting of

four Markov processes: the world process, demand process, lead time process, and

the process used to represent continuous-time discounting. After some nontrivial

transformations, they can compute the lead time demand distributions and cost rate

function.

In this section, we apply the idea of this algorithm to our more complicated

demand model when the lead time L has a continuous phase type distribution. In

addition, we assume that the transition time for leaving each world state is also con-

tinuous phase type distributed. Assume that we cannot observe the phase changes

within this transition period. (Recall from Section 2.2.2 that, if the phase changes

of the Ela.,. transition distributions, which are special CPH distributions, are ob-

servable, then we can transform the core process into a Markov process by using an

extended state representation (A(t),r(t)) and a transformed transition probability

matrix.) We first denote the probability mass function of D'" for fixed given i, s,

by

bi,s(d l) Pr(D'"=d)

then for random lead time L, we define


q,s(d) = EL[e- b,,(dL)] = -b(d dFL().

With these two notations, we can express the lead time demand distribution function

as

FDis (y) = bi,8(d I)
d=0
for integer values of y. In addition, we can write


FDi (Y) e -a FDi, ( y)dFL (f) b,(d dFL (f) j q,(d)
d dO
Sd=0 d=0










for integer values of y. It is easy to see that
00
q,8(d) =EL[e- ]
d=0

Now we can see that the task of computing FD, (y) can be accomplished by computing

qi,s(d). In addition, we can express the discounted cost rate function in terms of qi,s(d)

as


C(y,, s) = E e-LoC(y DV)]

EL -aL h (y d L -a)LQd\d


h (y d)q,(d)+p (d y)q,().
d=0 d=y


d=O d=y

We adopt the same assumptions as in Song and Zipkin [52] to develop our al-

gorithm. Define e to be a column vector whose elements are all 1, while ei is a unit

column vector where the ith element is 1 and all other elements are 0. For complete-

ness sake, we next briefly review some results from Song and Zipkin [52] that we need

to further develop our algorithm. We assume that the leadtime L has a continuous

phase-type distribution (T, M), where r is row vector with p nonnegative components

whose sum is no longer than 1, and M is an p x p nonsingular matrix whose off-

diagonal entries are all nonnegative and whose diagonal entries as well as row sums

are all nonpositive. Let U be a continuous-time Markov chain with p + 1 states,

where the last state is an absorbing state, initial probability distribution [r, 1 re],

and generator

M -Me

0 0

Then the time until absorption of this chain is distributed as L. We assume re = 1,

so that L has no mass at zero, i.e., L / 0.









In addition, we assume that the world transition time from state i is a continuous

phase-type distribution (<;, Hi), which can represented as the time until absorption

of a continuous-time Markov chain V with ri transient states and one absorbing

states. We assume that for all i, ye = 1. If we can observe the phase changes

of this Markov chain, we can translate the world process A into a Markov chain

with state (i, r), where r is the phase of the V. Let Q denote the generator of the

transformed world process of dimension i'1 ri. For example, for i / j, the rate

q(i,r');(j,r") = -(Hie)TerPijj(Gj)Ter,,, where (Hie)T represents the transpose of matrix

Hie.

For a continuous phase type distribution (<;, Hi), the probability that it is in each

state after s time units, denoted by ri,,, is the solution to the following differential

equation(s):


7,s = 7,sH (2.15)

with boundary condition 7i,o = ;. It is easy to solve that 7ri, = H Thus given

that the last transition was into world state i, the conditional probability that the

phase of the CPH distribution after spending s time units in the current state, denoted

by r(s), is equal to r, denoted by R,,s(r), can be computed as

7Fi,ser
R j, (r) 1


It is easy to verify that Ri,o(r) = ;er, the rth element of the initial probability

distribution y for state i.

Since a CPH distribution is interpreted in term of a Markov chain, given the

current state of the Markov chain (i.e., the phase of the CPH), the time that has









elapsed since the CPH distribution started, s, becomes irrelevant due to the memo-

ryless property of a continuous-time Markov chain. So we can define


Pr(D'" djr(s) = r)




as well as


j e- Pr(D dlr(s) r)dFL(f)
0JoO


Pr(D'

bi(de, r)


d|r(O) =r)


(2.16)


\ e-a Pr(D'o =- d|r(O) = r)dFL(f)

qi(dr). (2.17)


By conditioning on the phase of the CPH world transition time, we have


ri
Ri,s(r) Pr(D"'
r=1


dlr(s) = r)


ri

r=1

where we have used equation (2.16) and


Sri
,o RiR(r)) Pr(Ds'






r=i


djr(s) = r)dFL()


djr(s) = r)dFL(f)


where we have used equation (2.17). Now we can use Song and Zipkin's approach to

compute the function qi(d|r) for every i and r. The difference here is that we use a

composite state (i, r) to replace the world state i in Song and Zipkin's model, and

change our world process into a Markov process, as we did in Section 2.2.2.

We assume that the world process A (note we have converted the state of this

process into (i, r)) and the demand process D stop changing and remain fixed when

U(t) = p + 1. Thus for any realization of the process A, U and D, the final value of


bj,s(dl)









the first of D is precisely a realization of the lead time demand. To incorporate the

discount factor, we construct an auxiliary continuous-time Markov chain J, indepen-

dent of A, U and D, with two states, an initial state 0 and a killingg state 1. We

state with J(0) = 0, and state 1 is absorbing. While U(t) < p, the transition rate

from state 0 to 1 is the discount factor a; when U(t) = p + 1, the process J stops

changing and remains fixed. Thus the probability that the process J is not killed by

the end of the leadtime is

Pr(X > L) Pr(X > L )dFL() j e-adF(f) = E[e-a].

For fixed i, s, if the world is in the phase r of its CPH distribution, then we

consider the joint chain {D, A, U, J}. Using the generator of the joint process, we can

write differential equations to represent the dynamics of the system, and solve them.
Denote I as the identity matrix of size 7"i r", and

A = diag(Ai)

Ka = -[Q M-A I-al l ]

Ha = Ka[A I].

Then similarly as in Song and Zipkin [52], we can get


q4(dlr) = Pr(D(L) d, J(L)= 0A(0) i,r(0) r).

After several steps of transformations, we can get


(er, T)H [I- Ha aKa-][e e].


qi(dlr)










Now we can express the discounted cost rate function as


C(y, i, s)


y 00oo

d=O d=y
y ri 00 ri
S- d) Y Ri,(r)qi(dlr) +pY -y)Y R,,sr) r
d=O r=l d=y r=l
ri y oo
SRi,,(r){h (y d) q(d r) +pZ(d y)(dlr)}
r=l d=O d=y


E Ris(r) p(eir 0 )(I HJ)-2( H aK-)(e 0 e)-
ri
p(y l)rT(al M)-lMe(p + h)(e, 0r)

y d)H (
d=-O

Y Ri,,(r)CO(, i,r) (2.18)
r=1


where


(y, i, r) = h (y d)qi(d r) + p (d y)qi(dlr)
d=O d=y
y oo
S(h+p)y(y d)q(dlr)+py(d -y)q(dlr)
d=O d=O
y oo
S(h +p) (y-d)q(dlr) +p dq(dqr) -pyEL[e-
d=O d=O
= p(Ceir r)(I H)-2(I H2 aK )(e ) -

p(y 1)T(al M)- Me + (p + h)(eir 0 ) .

Y(y-d)Hi (I-Ho-aK-)(
d=0

We see that C(y, i, s) is represented as a convex combination of the functions

C(y, i, r) for all r = 1,..., r, where the weights depend on the value of s, as its right

and left derivatives. Thus, the left and right derivatives of fi,s(y) = C(y, i, s) + (. I/

at integer point y are simply (C(y, i, s))' + ac and (C(y, i, s))' + ac, respectively. As

shown in Section 2.3, the optimal inventory position y* will minimize fi,s(y) if at y*









the right derivative is greater than or equal to 0, while the left derivative is smaller

than 0.

We need a result regarding the changes of optimal inventory position when each

of the world transition time distribution is continuous. The proof is very similar to

that of Theorem 2.4.5, thus omitted here.

Theorem 2.5.1 Suppose that the transition time distribution Gij from i,:., state i

to state j, i,j E I, have continuous 1. ,;.:i;, Then if the world does not. lIi.', state,

every time that the optimal inventory position function ,I li.ij'- it will either

increase by 1 or decrease by 1.

If now the world has been in state i for so time units, and the current inven-

tory position is optimal, then as time s increases, the optimal inventory position

may change. By Theorem 2.5.1, as long as the world state remains unchanged, it

will change by one, either increasing or decreasing. This situation is illustrated in

Figure 2-1 through Figure 2-3. So to determine after how long the optimal inven-

tory position will change to a different value, we need to compute the left and right

derivatives of C(y, i, r) at y* for all r, and making use of equation (2.18). In other

words, we need to solve each of the following two equations,


> RP,(r) (y, i, r)) + ac 0 (2.19)


Z +R, r) (Cy, r))+ ac = 0. (2.20)
r=1
Only the solutions for s to the above two equations are candidate times at which the

optimal inventory position will change. Let s+ < s+ < ... and sa- < s < .. be the

solutions to equations (2.19) and (2.20) that are strictly greater than so, respectively.

Only these solutions are candidate times at which the optimal inventory position

will change. Note that it is possible that either of these equations does not have

such a solution. If neither equation has such a solution, we know that the current









optimal inventory position will continue to be optimal as long as the world state

does not change. For equation (2.19), we check its candidate solutions as described

above in increasing order, to find out the smallest one at which the right-hand-side

of the equation has a negative derivative, and denote it by s'. If no such solution

exist, we let s' = o0. We follow a similar procedure for equation (2.20), except that

we now choose the solution at which the derivative is greater than 0. Denote that

solution by s". If s' < s", then after s' time units the optimal inventory position will

increase by one; if s' > s", then after s" time units the optimal inventory position will

decrease by one. If both s' and s" are infinite, then the optimal inventory position

will remain unchanged unless the world state changes. Note that it is not possible

that s' = s" < oo, which would mean that at time s' = s" the function C(y, i, s) +o. I/

has a positive left derivative and a negative right derivative, which contradicts the

fact that it is convex.

Each time the world just enter a new state i, we then know probability that

the world is in each state, which is derived directly from the initial distribution of

the world transition time, i.e., Rio(r) = er,, and we compute the optimal inventory

position for this time point. We can then repeat the procedures described above to

compute when the optimal inventory positions will change.

Now turn to the calculation of the left and right derivative of C(y, i, r) with

respect to y for some fixed i, r. The right derivative at integer value y (recall that

we have proved the optimal inventory positions can only be integers) is


C(y, i, r) -pE[e-L] + (p + h) q(d|r)
d=O
and the left derivative is

Sy-1
'(Cy, i, r)) = -pE[e-R ] + (p + h) > q(d|r).
d=O









If y < 0, then the right derivative at y is

-pE[e-L] = pr(al M)- Me;


if y > 0, the right derivative can be written as


-pE[e-] + (p + )(e, 0)[Y H](I H- aK,)( ).
d=O
To summarize, we can compute the optimal inventory levels and the time when

the optimal inventory levels change by applying the following algorithm.

Step 1. At the beginning time of a new world state i, i.e. s = 0. If the

world has been in state i before, retrieve the stored optimal inventory posi-

tion curve for state i. Otherwise, solve the minimization of C(y, i, 0) + o. =

Err ,CerC(y, i, r) + (. ,r Use the method of Song and Zipkin [52] as described
above to compute C(y, i, r) and the optimal value, and denote it by y*. Set

k = 0, and denote s* = 0.

Step 2. If s* = c, go to Step 4 directly. Otherwise, compute the left and right

derivative of C(y, i, r) at ,I, for all r = 1 to ri as


(C(y, i r)) -pE[e&-L] + (p + h) q (d r)
d=0
-1
(c i, r)) -pE[eR ] + (p + h) qj(d|r).
d=0
Solve the following equations (2.21) and (2.22)
ri
> R(r) ((Yk(1,r))+ac 0 (2.21)
r=l

>R(r) (C(yki, r)) + ac = 0. (2.22)
r=1

Let s+ < s+ < ... be the solutions to equation (2.21) that are strictly greater

than s*. C'! i 1: them in increasing order, and let s' be the smallest solution at







53












y -l y y*+l

Figure 2-1: Optimal inventory position at y*












y*-l y y*+l

Figure 2-2: Optimal inventory position at y* + 1

which the left-hand-side of equation (2.21) has a negative derivative, let s' = oo

if no such solution exists.

Similarly, let s- < s- < ... be the solutions to equation (2.22) that are strictly

greater than sk. Let s" be the smallest solution at which the left-hand-side of

equation (2.22) has a positive derivative, let s" = oo if no such solution exists.

* Step 3. If s' < s", let s* = s' and iI, +1 = + 1; if s' > s", let s*+ = s" and

'+ = '-i 1. If both s' = oc and s" = oc, let s*, = oc and iI, +1 = '' Store

that y*s) = n, for s* < s < s* Let k k + 1.

* Step 4. If the world does not change to a different state, go back to Step 2; if

a new state is encountered, go back to Step 1.




















y*-l y y*+l

Figure 2-3: Optimal inventory position at y* 1

It is obvious that the computation involves the evaluation of the probability

Ri,8(r). And for general CPH world transition time, it is difficult to handle Ri,s(r).

In the next section, we will give an implementable algorithm by considering the world

transition time as Ei, a1,,. distributed.

2.5.2 Erlang Distributed World Transition Time and Continuous Phase-
Type Distributed Lead Time

Suppose that the world transition for every state is a special case of CPH distri-

bution, F, El.n.i distribution. We first prove two useful lemmas regarding the property

of the Fila ,.' distribution.

Lemma 2.5.2 If Hi is the generator of Erlang(i, ri, vi) distribution, then the jrth

element of H!, 1 < j, r < ri and I > O0, is


hv (- 1)-r+ if r>j andr -j<


0 o/w

Proof: If a CPH distribution is an Eiln ,(i, ri, vi) distribution, then the initial state

distribution is y = [1,0,..., O]T, and the jrth element of Hi, hjr, 1 < j, r < ri, has









the following form


vi ifr = j
ifr j+l
if o/w
o/w


We prove the lemma by induction. It can be verified easily that the lemma holds for

1 1, 2. Now suppose it is true for 1, and we are considering the jrth element of Hj+1

for 1 < r,j < ri,


hl+1
Jr


ri
Shkhkr
kl1


If r j +11, it is easy to check that h'1 = 0; if r

that


j, it is straightforward


S1+1

0


if r > j and r < j + + 1, then


hl,rhlhr-,r + hlrhrr


S(-t)-r V + (-1)-r+
Sr- 1-j ) r

S1( 1) -r++, l!(- r +j + 1) + l!(r- j)
1 -- (1 r + j +)!(r j)!

v (+l( 1)+l-r+-j (1 + 1)!
(1 + r + j)!(r-j)!


+l(-1l)+l-r+j I + t


-j V
/ 3 1i (


hl+1
'jr


hjr =

0


h+1 = i(-1) (-t)


/+1(- 1)+1
Vi -









To summarize, we have


Vi (-1)+-j if r > j and r -j< 1+
hl1+1
j-

0 o/w

So the lemma holds for 1 + 1. By induction, the lemma holds for all 1. O


Lemma 2.5.3 If a CPH distribution is anErlang(i, ri, vi) distribution, then

,i,)r-1 -vis
7ri, r (r- 1)!


Proof: Following the result of lemma 2.5.2, we have


iTTser = CHi sr
(His

l=0


l 0




+-1 -11

( (-1)' (+(his) -(s)-'
1 --1)r! (r 1)!(1- r + 1)!


(r- 1)! r-1 (1 -r + 1)!
(r)r-1-
(vs) ) 1 is









From the previous two lemmas, for Ei, la,(i, ri, vi) distribution, we can get that


R (r) 7 r ,se (2.23)
Er= 1 7i,ser
(vis)"-le-"i"
(r-1)!
1 i (vis)k- le-vi

(VlS)r 1
k=1 (k- 1)!
(r -1)!
1 .kl ((2.24)

By replacing Ri,s(r) by the values in equations (2.19) and (2.20)


r r- ri (i)k-1
r=i (r 1)! (y*, i, r)) + ac (V-S)!

r (,i)r-1 (C(y*, i, r) + ac -
S- 1)! r (2.25)
r=1
Sri Vr-ri ( iS) k-1

r= ( 1 1)! ((y*, r)) +ac

-1 + r-1 (2.26)
r=1

At the time when the world just enter a new state i, i.e., s = 0, we know that

the world must be in the first stage of the ,ila,,'. distribution, so Rio(1) = 1, and

Ri,o(r) = 0 for all r = 2,...,ri. Thus, for s = 0, we have C(y,i,0) = C(y,i,1).

We can then repeat the procedures described above to compute when the optimal

inventory positions will change.

We have the following algorithm by making the according changes.

Step 1. At the beginning time of a new world state i, i.e. s = 0. If the world has

been in state i before, retrieve the stored optimal inventory position curve for

state i. Otherwise, solve the minimization of C(y, i, 0) + (. i = C(y, i, 1) + o. '

Using the method of Song and Zipkin [52] as described above, compute the

optimal value and denote it by y*. Set k = 0, and denote s* = 0.









* Step 2. If s* = o, go to Step 4 directly. Otherwise, compute the left and right
derivative of C(y, i, r) at f,, for all r = 1 to ri as


(cy(, i, pr) -p[e-"] + (p + h) q(d|r)
d=O
-1
(c i, r))' -pE[e-L] + (p + h) qj(d r).
d=O
Solve the following equations (2.27) and (2.28)


S (r i), + asr-1 0 (2.27)


^ (vr-)1 (C i, zr)) +
S(- ) i r- 0. (2.28)
r1(r

Let s+ < s+ < ... be the solutions to equation (2.27) that are strictly greater
than s*. C'! i 1: them in increasing order, and let s' be the smallest solution at

which the left-hand-side of equation (2.27) has a negative derivative, i.e.,

E ---[ +, ] O
ar (7u)r-'1 c(I,, i, r) + ac )r-2 <0.
r! (S') < 0.
r=2
let s' = oo if no such solution exists.
Similarly, let s- < s- < ... be the solutions to equation (2.28) that are strictly

greater than s*. Let s" be the smallest solution at which the left-hand-side of

equation (2.28) has a positive derivative, i.e.,

S 1 ( !, ,r) + ()r- > 0.

r=2


let s" = oo if no such solution exists.









Step 3. If s' < s", let s+, = s' and i, + = ,' + 1; if s' > s", let s*+ = s" and

', + = 'i,'- 1. If both s' = oc and s" = oo, let s+, = oc and /i, +1 'K,' Store

that Wi (s)= i, for sk < s < s Let k = k + 1.

Step 4. If the world does not change to a different state, go back to Step 2; if

a new state is encountered, go back to Step 1.

2.6 An Extension: Demand Arrives Following a General Renewal
Process

2.6.1 Generalization of the Demand Process Model

In this section, we will discuss an important extension to the the models we stud-

ied so far. Instead of assuming that the demand arrivals in each world state follows a

Poisson process, we now relax the demand process by allowing the interarrival time

between demands within a world state to have a general world-dependent distribu-

tion. In other words, when the world is in state i, the actual demand process follows

renewal process, with the distribution of the time between successive demands de-

noted by Ki. We call this demand process a semi-Markov modulated renewal demand

process. Again the demand process is exogenous and is not affected by any ordering

decisions.

By keeping all the other model assumptions as before, the results developed up

to Section 2.3.1 continue to hold without any changes since the Poisson nature of

the demand process is not used, i.e., the cost function and the general form of the

optimal policy (as a function of the entire history) remain the same. However, now

the history can no longer be summarized by the current world state i and the time

spent in this state s only. The properties of the semi-Markov modulated renewal

demand process imply that D h depends on the history through not only the state

that the process is currently in (i) and how long it has been in that state (s), but

also how long it has been since the last demand occurred, which we will denote by

B(t). To reflect this fact, we rewrite the lead time demand as D ,b when the core









process has been in state i for s time units, and b time units have elapsed since the

last demand.

One important issue that we need to p iv attention to is that before the first

occurrence of a demand within a world state, the amount time since the last demand

b generally is not equal to the amount of time s spent in the current state, since the

last demand will likely have occurred while in the preceding state (or even earlier, if

no demand occurred while in the preceding state). Thus, b not only includes time

spent in the current state, but also some amount of time spent in previous statess.

However, since the distribution of the time between demands changes between states

of the world, neither s nor b seems to be an accurate measure of the time since the

last demand for the current interarrival time distribution.

To handle this situation, we should in fact let the demand process in a given

state be a /. /,v;. renewal process, where the distribution of the first interarrival

time depends on the time since the last demand as well as the previous state visited.

In particular, suppose we are currently transitioning from state j into state i with

generic interarrival time Xi ~ Ki, and let the time since the last demand be b. We

then let the first interarrival time be distributed as


X, j,(b)|X, > ,(b)


where '. is some function that transforms the amount of time that has elapsed

since the last demand. For convenience, we will in fact simply redefine b to be

equal to '.. (b) at the moment of transition from state j to state i. Note that the

previous demand might actually have occurred in a state that was visited before

state j. In that case, by recursively updating the time since the last demand using the

appropriate transformation functions 0.,. will appropriately define the first interarrival

time distribution in each state.









Intuitively, it seems clear that we should choose the functions f.,. to be nonde-
creasing. Two interesting extreme cases are 0.,.(b) = 0, where we simply 1. ,. I '

the time since the last demand at the moment of transition between world states,

and O.,.(b) = b, where we ignore the fact that the interarrival time between demands
is different in different states. If we define the generalized inverse of distribution

function K by

K'(p) = min{y : K(y) > p}

then a more reasonable choice of the conversion function would be

,(b) = K (K,(b)) .

2.6.2 The Optimal Inventory Policy

Now we are ready to characterize the optimal inventory policy for the semi-
Markov modulated renewal demand process described above. We first introduce

some notation similar to Section 2.3.2 to accommodate the changes in the demand

model. For given y, t, and h(t), we can simplify the inventory cost rate function to

C(y, i, s, b) = Ee-L D b)]

which is equivalent to C(y, t, h(t)) if the history h(t) -,i- that A(t) = i, S(t) = s and

B(t) = b. The total expected cost function for our semi-Markov modulated renewal

demand model thus reduces to

W(x, h(0) y) e-atE [C(y(t, H(t)), A(t), S(t), B(t)) + ,. '(t, H(t))] dt.

In addition, we define

fi,s,b() = C(y, i, s, b) + oi

for every fixed i, s and b, which can be viewed as the cost rate function if the inventory

position is y at the time when the core process has been in state i for s time units,

and b time units have been passed since the last demand.









Next, we generalize some of the key results obtained for the semi-Markov mod-

ulated Poisson demand process that continue to hold for our new demand model.

Denote the right derivatives of C(., i, s, b) and fi,s,b(') by

C(y, is,b) lim C(y + ,i, b) C(y, i, s, b)

(f,,b ) l fi,s,b(Y + E) fi,s,b(Y)
(fi,s,b)+(y) M= i

= C' (y, i, s, b) + ac.

Also, let

y (s, b) = inf{y : (f,s,b)+(Y) > 0}.

Note that since the lead time demand can only assume integral values, all points

at which the functions C(., i, s, b) and fi,s,b(') are nondifferentiable are integral. In

addition, iI (s, b) is integral as well.

Lemma 2.6.1

(a) C(y,i,s,b) and fi,s,b(y) are both convex in y for all i, s and b, so that II (s,b)
minimizes fi,s,b(Y).
(b) If ac < p, then y (s, b) is finite and nonnegative for all i, s and b. In addition

fi,s,b(y) is nonnegative for all i, s and b.

(c) If ac > p, then ,I (s, b) = -oo for all i, s and b.
We omit the proof because the arguments are very similar to the ones in Lemma

2.3.3.

Theorem 2.6.2 Under the semi-Markov modulated Poisson demand model, the my-

opic ] ,/. 1 y* 1. 7,, ,1 by


y*(t, H(t)) = y(t)(S(t), B(t)) for all t > 0









exists and its total '.V'. I costs are finite. Thus, the optimal inventory position at

time t only depends on the state at time t and the amount of time that has elapsed

since the core process last entered that state.

We also omit the proof to this theorem because the arguments are very similar to the

ones in Theorem 2.3.5.

Finally, we will give a more explicit characterization of the optimal inventory

position, which can in principle be used to compute the optimal policy, as well as the

cost of the optimal policy. Define

FDb (y) e-FDi,,b (y)dFL ().
L JOO

Then



yT(s,b) = argmin?: (fyb)+(Y) > 0}
S~ pE[e-"L] ac
arg min y : FDib(y) > pE [-a-
L h+p p

In case the lead time is deterministic, the expression for the optimal policy can be

simplified to

S -ae-p, acrc -
y (s, b) arg min y C L

r p- ac
arg min y: FDs,b(y) > (2.29)


2.7 Summary

In this chapter, we have studied an inventory control model in a semi-Markov

modulated demand environment. Under linear ordering, ]1, 1lii: and shortage costs,

and assuming that both positive and negative orders are allowed, we have derived

the optimal inventory policy. In addition, we have formulated sufficient conditions on

the demand process for the myopic inventory positions to be increasing over time. In







64

that case, the myopic policy remains optimal even if negative orders are not allowed.

For a special case where both the world transition time distributions and lead time

distribution are continuous phase type distributed, we give an algorithm to compute

the optimal inventory positions. Finally, we extend the model by relaxing the demand

process from a semi-Markov modulated Poisson process to a general semi-Markov

modulated renewal process, and see that this relaxation really does not affect the

form of the optimal policy.















CHAPTER 3
MODELS WITH PARTIALLY OBSERVABLE WORLD STATES

3.1 Introduction

In this chapter, we extend our models studied in C'!i pter 2 to a more complex

case in which the demand process is a state-dependent Poisson process, but the

underlying core process (world) is not directly observable. What we can observe is

only the arrival of the customer demands, and we can use that information to obtain

inference on the state of the world. This scenario is very common in real situations,

and thus of significant practical interest.

The chapter is organized as follows. In Section 3.2 we study a model with only

two world states. We describe the difference between this model and our previous

models and show how the fact that the world state is unobservable affects the opti-

mal policy. We also give the form of this optimal inventory policy and provide an

algorithm to incrementally determine the optimal policy. Then we extend the basic

two-state model to a multiple-state one in Section 3.3 and derive a recursive formula

to help determine the optimal inventory policy. In Section 3.3.4 we generalize this

result to a more general multiple-state model. Finally, we summarize the chapter in

Section 3.4 and provide some future research directions.

3.2 A Simple Model with Two World States

We start with a simple case in which there are only two world states, state 1

and 2. A state transition can only happen from state 1 to state 2, and once in state

2 the world will stay in that state forever. The transition time from state 1 to state

2 is a continuous random variable with distribution G. While in world state 1 or 2,

the demand process is a Poisson process with rate A1 or A2, respectively. We assume

that we know the values of A1 and A2, but we do not observe the transition from









state 1 to state 2. As before, we assume that the demand process is independent

of the replenishment decisions. Implicitly, we treat the system as if we start our

observations when the world just enters state 1. Put differently, even if the inventory

system started at some point in the past before observations start, we assume that we

know the distribution G of the time that the system will remain in state 2. Note that

if there is a positive probability that the transition to state 2 has already happened at

the time observations start this can be incorporated by defining G to have a positive

probability mass at 0.

Recall that the cumulative demand by time t is denoted by D(t). Then, at time

t we will have observed

D(t)= D(u) :0
Note that in this case the history information H(t) and H(t) contains only the demand

information since the world state is now unobservable. As in the previous chapter, we

still assume that negative orders are allowed. Under this assumption, the notation

and model up to Section 2.3.1 can be used without changes, except for the content

of the history H(t).

3.2.1 Effects of the Unobservable World

In this section we address the effects of the fact that the world process is un-

observable. The history information by time t contains the past cumulative de-

mand at any time point before t and can be fully characterized by how many de-

mands have occurred, N(t), and the interarrival times between consecutive demands,

X,X,... XN(t). So another way to represent the history is

H(t) {N(t),X1,X2,..., XN()}

S {N(t), S, S2, ..., SN(t)}


where Sk = 1 Xi is the arrival time of the kth demand.








We denote the state of the world by the stochastic process {A(t), t > 0}. In
particular, A(t) is a random variable that is equal to Ai if the world is in state i
(i = 1, 2) at time t. Then (A(t), H(t)) is a joint mixed random variable with joint
probability density function f(Ai, h(t)). Denote the conditional probability that the
world is in state i at time t for every t given that the history information up to
time t is H(t) = h(t), by p(i,t, h(t)) = fAi (Alh(t)). By conditioning on the world
changing state at time S = s, and denoting the conditional density function of history

by fHss(h(t)|s), we obtain

p(, t, h(t)) /AH(A, Ih(t))
f(Ai, h(t))
fH (h(t))
fo f(A, h(t)|s)dG(s)
fo fH s(h(t) s)dG(s)
too fH s(h(t)|s)dG(s)
jf fs fH(h(st)(s)dG(ss)
Sff s(sh(t)t)ss)d)dG(ss)

fo' fH s(h(t) s)dG(s) + ft fHis(h(t) s)dG(s)
The meaning of each density (conditional density) function f should be clear within
its context.
Now let us look at the conditional density function of history H(t) given S = s
more closely. For h(t) = {N(t) = n, S1 = si,..., S,= s,}, let fHis(n, sl,... s~s)
denote the condition density of N(t) = n, S = sl,...S, S sT given that S = s,
and let fHIN,s(S1, .. s, ln, s) denote the conditional density of S = s ,..., Sn= s
given that N(t) = n and S = s.
If s > t, the world is still in state 1 by time t, and

fHIs(h(t) s) = Pr(N(t)= nIS )fHN,S(S1,... ,s~n, s)
e-Alt(it)" n!
n! t"
Se-xltA.









If s < t, then the state transition has already happened before t. We denote the

number of demands that occur while in state 1 and 2 by random variables N1 and N2

respectively. For a given s, these two numbers are known by t, and we denote them

by n(s) and n n(s) respectively, and the occurrence of demands in states 1 and 2

are independent! Then


fIf s(h(t) s)











If Sk < s < Sk+1, for k =


= fIsMn~s),X1,...,xn(s);
n(s)+l
n n(s), Xi S, Xn(s)+2, T nS)
i= 1
= fs(n(s), X,..., xn(s)) Is)
n(s)+1
fins(n- n(s), X r, -s, n(,)+2,... ,n s)
i 1
e-aAisn(s) e-A2(t-s) n-n(s)
... then ns) and
0, 1,...., n, then n(s) k, and


fn s(h(t) s)


Sf(h(t) S s)dG(s)


where we let so = 0 and sn,+ = t.

Now we have


n fSk+1
A f n-k
k-0 Isk


e-Ase-2 (t-S) dG(s)


Sk n-k fs e-I Ae--A2(t -)dG(s) + fJ7 e-AtA"dG(s)
k-1 0 2 fsk ft
e-AltAG(t)
o ; nk J- e -Alse-A2(t-s)dG(s) + e-AltAG(t)
e-AltAG(t)
0o AlA k ss I 6-(A l-A2)sdG(s) + e-AtG)
1
1 + (e(ACI-A)t/G(t)) EL o(A2/A1)n-k fksskI e1-(A 2)sdG(s)


e-AsAke-A2(t-s) An-k
1 2


p(1, t, h(t))








It is easy to see that this is a continuous function of t if the transition time distribution
function G is continuous.
Now we compute the lead time demand distribution given history h(t). Let

g(s) be the density function of the transition time distribution, and g(slh) be the
conditional density function of the transition time distribution given history h(t).
Conditioning only on which state the world is in now is not enough, since we also
need to know how long the world has been in the current state to determine the the
remaining life time distribution. So what we need is to condition on the time of the
state transition. For fixed lead time ,

FD ,h(t)(z) /Pr(Dh(t) < zS= s)g(sh(t))ds

j Pr(D-'() < z S s)g(s h(t))ds +

j Pr(Dt'h(t) < z S s= )g(slh(t))ds

JFD 2, W fHs(h(t)|sg(s) + F tW fHs(h(t)|s)g(s)}
o fH (h(t)) fH (h(t))
ft Hs(h(t) s) fwjs(h(t)|s)
SD 2,t-) (s)ds DI,tz) g(s)ds.
JO t fH (h(t)) t t f(h(t))

Recall that in our current model there are only two world states, and once the world
enters state 2, it will remain in state 2 forever. Therefore, FD2,t-s is equivalent to
FD2,o, and can move out of the integral. (This argument does not generalize to a
model with more than two states, as we will see later). It follows from equation (3.1)
that

Sh(t) F jfo fH s(h(t)Is)dG(s) ft fBs(h(t) s)dG(s)
FD+(t) = FD-,oZ) FD- ,t Z) w
e fH(h(t)) fH(h(t))
FD2,o(z) (1 p(1, t, h(t))) + FDI,t (z)p(1, h(t)).







70

When the lead time is stochastic, we obtain
roo
F D h(t) (z) p(, t, h(t)) ,)dL
DL J f

+ (t p(1, t, h(t))) e-a'FD,o( )dFL(f). (3.4)
JO O

Then we can write


C(y, t,h(t)) E e-aLc(y Dt h(t))

= e-O C(y z)dFD ,h(t(z)dFL

= p(l, t, h(t)) J e-aCO(y -z)dFDi,t(z)dFL() +

(t p(1, t, h(t))) I ye-o z)dFDo (z)dFL (

p(l, t, h(t))C(y, 1, t) + (1 p(l, t, h(t))) C(y, 2, 0). (3.5)

It is obvious that the conditional probability function p(l, t, h(t)) as well as the cost

rate functions C(y, t) and C(y, 2, 0) p1 v key roles in the partially unobservable

model. To make the model tractable, we consider the case where the state transition

time distribution G is exponential with rate p, i.e., the world process is a Markov

process, in more detail in the remainder of this section.

3.2.2 Exponential Transition Time Distribution

If the state transition time distribution G is exponential with rate p, i.e., the

world is a Markov process, then by conditioning on which state the world is currently

in, the distribution of the lead time demand can be expressed as


FDt,(t) (z) = Pr(D''h() < z)

Pr(D (t) < z A(t) A )p(1l,t,h(t))

+Fr(Dp,h(t) < FDA( z )( p(,t,h(t))) (









where we define D' to be the random variable representing the total demand during a

lead time, started from now, given the world is currently in state i. The last equality

in (3.6) follows from the fact that the world process follows a Markov process and

demand is a Poisson process, and if we know which state the world is in now, the

time that has elapsed in the current state becomes irrelevant. We see that we obtain

the same result as by conditioning on the transition time. And accordingly,


C(y, t, h(t)) = p(, t, h(t))C(y, l) + (1 p(, t, h(t))) C(y, 2) (3.7)

where C(y, i) is defined as the conditional expected discounted holding and shortage

cost rate, at the end of a lead time and viewed from, given that the world is currently

in state i, and the current inventory position (after ordering decision) is y. (See also

Song and Zipkin [52].)

When comparing equations (3.5) and (3.7)we see that in case G is the expo-

nential distribution the instantaneous cost rate function simplifies considerably, and

the dependence on time and history is then restricted to the conditional probability

function p(l, t, h(t)). In the remainder of this section we focus on the computation

and analysis of this function. These results will then be used in the next section to

compute the optimal inventory policy.

Using the fact that G(t) =1 e-lt and defining A = A1 A2 + we obtain


p(1, t, h(t)) -
1 + o(e(Ai- /G(A) EZ o 2/A(1) k s -(A-A)sdGs)
1
1 + pe(Ai-A2+/)t (A2/AS k f+ e-(A1-A2+)ds
S1
1 + peAt nA2/A1 ).-k 8k1 Asd

We now distinguish between the cases A :/ 0 and A 0.







72

Case 1: If A / 0 then


p(l,t,h(t))
1
1 + (p/A)ext Eo 0(2/A1)n-k(-Ask e-Ak+1)

1 + (p/A)et ( -1 AAl) n-k-As e-As +) + -A" -t)
1
1 p/A + (p/A)eXt ( -(A2/A k ( +i) + e-As)
1
1 p/A + (//A)eA(t-S) (1 + e As 0(A2/A\ n-k1) -A( -Ask+))"

It is now easy to show that

S0 if A> 0
lim p(, t, h(t)) if >
t->ooI 1 A fA
1-Io/X if A < 0.

Moreover, if A > 0 the function p(1, t, h(t)) decreases monotonely in t. If A < 0,

the function p(l, t, h(t)) decreases monotonely in t if the constant
n-1
1 + eAs" (A2/Al) n-k( -Ask -As-k+)
k-0
is positive, and it increases monotonely in t if it is negative. More intuitively, it

follows that the function p(l, t, h(t)) decreases monotonely in t if the probability

that we are in state 1 at the time of the tth demand exceeds the limiting

probability 1/(1 p/A) and increases monotonely in t otherwise.

Case 2: If A = 0 then

1
p(1, t,h(t)) =
SlpesAt o 0(A2/A l -k sk+ I-Asd
1
1+ p E 0(o2/A1)~ -k ds
1
1 +Z 0(A2/Ao ) -k(Sk+l- Sk)
1
+ p (t )k (3.9)
1p(t + "-0 (A2/ 1n-j Sk+1 Sk) ST')









It is easy to see that, in this case, the function p(l, t, h(t)) decreases monotonely

and


limp(1,t,h(t)) = 0.
t->oo

Figures 3-1 through 3-4 illustrate the different behaviors of the probability func-

tion p(1, t, h(t)) with different parameters.

We close this section by providing a summary regarding the probability function

p(l, t, h(t)). This function is ah--iv- monotone, but the nature of monotonicity de-

pends on the sign of A = A1 A2 + p as well as the observed history. In particular, if

A > 0 the probability ahv-- decreases monotonely to 0, while if A < 0 the probability

will converge monotonely to a positive limit.

3.2.3 Computation of the Optimal Inventory Position

We next study the optimal inventory policy that is given in Theorem 2.3.2 for

the partially unobservable demand model. First, the following theorem shows an

important property of the optimal policy.

Theorem 3.2.1 The optimal inventory 1,..l. ; is a step function with step sizes 1 and

-1 as long as no demands occur.

Proof: From equation (3.2), we see that p(l, t, h(t)) is a continuous function of t since

the transition time distribution is exponential and thus continuous. From equations

(3.6) and (3.4), it is obvious that Ft,h(t) (z) is continuous in t also. Finally, the lead
L
time demand is a a discrete random variable having strictly positive probability mass

at every nonnegative integer value due to the nature of the Poisson process. So from

Lemma 2.3.1, each time the optimal inventory position changes, it will change to a

neighboring integer, i.e., either increase by 1 or decrease by 1. O


If the lead time distribution is continuous phase-type, we can now use the proba-

bility function we computed in the previous section as well as the analyses in Section

2.5 and in Song and Zipkin [52] to compute the optimal inventory policy for the












1.2


1


0.8

0.6

0.4


0.2


0
0 5



Figure 3-1: A1


10 15 20 25 30 35

t

= 1, A2 = 1.2,p = 0.4, and no demand before t


0 5 10 15

t

Figure 3-2: A1 = A2 = 2,p = 0.4, and no demand before t





























Figure 3-3: A1


1, A2 3,p 1.5, and 1 demand before t, sl


0 10 20 30 40 50 60 70
t


- 1, A2 = 2,p= 1, and no demand before t


0.255

0.25
0.245

0.24
0.235
0.23
0.225
0.22

0.215


Figure 3-4: A1









partially unobservable demand model. From Theorem 2.3.2, the optimal policy min-

imizes the function C(y, t, h(t)) + o. / for every t. We want to find y*(t, h(t)), and

we can use the method in Section 2.5 to compute the optimal inventory levels. Note

that y*(t, h(t)) is just the point at which the right derivative of C(y, t, h(t)) + o. '/ is

no smaller than 0, while the left derivative is smaller than 0. Using equation (3.7), it

is easy to see that


(C(y, t, h(t)))' = p(, t, h(t))(C(y, 1)) + (1 p(l, t, h(t))) (C(y, 2))I

(C(y, t, h(t)))'_ = p(l, t, h(t)) (C(y, t))' + (1 p(l, t, h(t))) (C(y, 2))'

where the left and right derivatives of C(y, 1) and C(y, 2) can be determined by the

method discussed in Song and Zipkin [52].

At t = 0 or at a point in time where a demand occurs, v t, we can compute

the probability p(1,t, h(t) as well as the optimal inventory position y*(t, h(t) at that

point in time (for convenience denoted simply by y* in the following if the arguments

are clear from the context). As we will show below, we are able to determine the time

at which the optimal inventory position will change if no new demand occurs by that

time. Using an analogous approach as in Section 2.5, we need to solve the following

two equations to determine the time t at which the optimal inventory position will

change if no new demand occurs up to time t:


p(l, t, h(t)) (C(y*, i))' + (1 p(l, t, h(t))) (C(y*, 2))' + ac 0

p(l, t, h(t)) (C(y*, 1))' + (1 p(l, t, h(t))) (C(y*, 2))' + ac 0

or, equivalently,

ac + (C(y*, 2))'
p(l, t, h(t)) c + (3.10)
)) (C(y*, 2))- (C *, 1))

ac + (C(y*, 2))
p(, t, h(t)) ( ))(3.11)
(C(y*, 2))'_ (C(y*, 1))'









Denote the solution to equation (3.10) by t' and the solution to equation (3.11) by
t". If A / 0 we can use equation (3.8) to find these solutions explicitly:


+ n + t In1+( %,)) I(-p--/A )]) (3.12)


+ Aln ((/A) [L + 23s Y:'' (CA1))' A 1)] (3.13)
2 (C(y*,2)) -(C(y* 1))



1 ac+(C ((C +,2C2 )) ) A/A k
t = s (3.15)



Similarly, if A these 0 we can use equation (3.9) to obtain

1i n (C(yl nt 2))+o (orC(e, o ))+ time onwards
t = S + .., 1. -2 1 Sk+1 Sk)


















the derivative of the right derivative of the cost rate function at t is nonnegative then
P tc + (C(y*, 2)) n
(3.14)
S1 (C(y*, 2))'_ (C(y*, 1))'_ ) --k
t =s +- 1 (/2/1)n-/Sk+1- Sk).
P ac + (C(y*, 2))_ ko
(3.15)

Only solutions for t and t" which are larger than s will be considered. If either or
both of these solutions are less than or equal to ,, it simply means that the optimal
inventory position will not increase (or decrease, or neither) from time sp onwards if
no new demand occurs. It will prove to be convenient to replace a value of t' or t" that
does not exceed s, by oo. Moreover, after obtaining t' and t", we also need to check

the derivative of the right derivative at t' and the derivative of left derivative at t". If

the derivative of the right derivative of the cost rate function at t' is nonnegative then
the optimal inventory position will not change to y* + 1 and we set t' = o00. Similarly,

If the derivative of the left derivative of the cost rate function at t" is nonpositive
then the optimal inventory position will not change to y* 1 and so we set t" = o00.

Now, if t' < t" then we conclude that at time point t' the optimal inventory

position will increase by 1 unit; if t' > t" then at time point t" the optimal inventory
position will decrease by 1 unit. If t' and t" are both infinity then the optimal

inventory position will remain unchanged until a new demand occurs. Note that it is









not possible that t' = t" < o0 since this would mean that right after time t' the left

derivative at y* is positive and the right derivative at y* is negative, which violates

the convexity of the cost rate function.

To summarize, we have the following algorithm

Step 1. At the beginning time of world state 1, we know that p(1, 0, h(0)) = 1.

By using the same method as in Song and Zipkin [52], compute the optimal

value, and denote it by y*. Set n = 0 and s, = 0. Also, set m = 0 and t, = 0.

(n records the number of demands that have occurred so far, while m records

the number of times that the optimal inventory position has changed so far.)

Step 2. Compute t' and t" according to either equations (3.12) and (3.13) or

equations (3.14) and (3.15) (with y* replaced by yb). If t' < tm or


p'(1, t', h(t')) (C(y*, 1)) + (1 p'(1, t, h(t'))) (C(y, 2)) + ac > 0

set t' = oo; if t" < ti or


p'(l, t", h(t")) (C(y*, 1))' + (1 p'(l, t", h(t"))) (C(yT, 2))' + ac < 0

set t" = oo.

Step 3. If no new demand occurs before min{t', t"}, then the optimal inventory

position changes to ym+i = y* + 1 at if < t" and to y+ y 1 at t" if

t' > t". Set t,m+ = min{t', t"}, m = m + 1, and return to Step 2.

If a new demand occurs at time s,5+ < min{t', t"} set t,m+ = s-,+ and compute

the optimal inventory position y, +. Let n = n + m m + and go to step

2.

3.2.4 An Extension

We next consider a minor extension to the model discussed above. Suppose that

we start observing the model at some point in time that is past the actual start of

the system, but that we know that the probability that the world is in state 1 at the









starting time is pi (so that the probability that the world is in state 2 is 1 -pi) and

let G denote the conditional distribution of the time remaining in state 1 given that

we are currently in that state. Then by conditioning on whether we are in state 1 at

time 0 we obtain that equation 3.2 becomes

1
p(,th(t)) = pi
p(t, h(t)) p 1 + (e(A-)t/G(t)) E -oA2/A1) n-k s+1 e-(AI-A2)sdG(s)

Note that, in case the time that will be spent in state 1 has an exponential distribution

with parameter p, we have that G is that same distribution and pi = e-" where s is

the amount of time that has elapsed since the start of the system until the start of

the observations.

3.3 Multiple World States Models

In the previous section, we studied models with only two world states. However,

there are usually more than two world states to consider in real life. Therefore, in

this section we consider models with multiple world states, i.e., m world states.

3.3.1 Models with Multiple World States Which are Visited in a Fixed
Sequence

We assume that we know all the world states, and the sequence they will appear.

One example for this is the seasonal change. Another example for this type of world

is the life cycle of products. We also assume that the demand in each world state

follows a Poisson process, and the parameters are all known. As before, we cannot

observe directly when the transition of world occurs, but only the demand.

Let us start with m = 3, that is, there are 3 world states, and they will be

encountered in the order 1,2,3. Once the world enters state 3, it will stay there

forever. The transition time in state i, i = 1, 2, is expor..n ,u/.:.all distributed, with rate

Ii. In state i, the demand process possesses rate Ai.

As in the previous section, we use A(t) equal to Ai to indicate whether the world

is in state i at time t, i = 1, 2, 3. Denote the conditional probability that the world








is in state i at time t for every t given that the history information up to time t is
H(t) = h(t), by p(i,t, h(t)) = fA H ih(t)). Following from B ,, -' rule, we obtain

p(i, t,h(t)) .(3.16)
fH(h(t))
To compute this conditional probability, we need to condition on both possible world
state transition times in states 1 and 2, namely T1 and T1 + T2, where T1 ~ G1 and
T2 ~ G2. The scenarios need to be considered are T1 > t, Ti + T2 > t > T1, and
t > TI +T2. Let us consider computing the history density function fH(h(t)) first. By
conditioning on T1 = 7r (and T2 r2), and denote the conditional density function
of history information by fHiT,(h(t)lr ) (and fHT,,T2 (h(t) 17, 72)), we obtain




rt roo
fHITs(h(t) 7-l)dGl(7-l) + f C fHI T,(h(t~ -l)dGl (-l)

= fHT,T2 (h(t)Irl, T2)dG2(T2)dG,(T1) +



ft t fHlT1,T2(h(t) T, r2)dG2(r2)+


fH 1, (h(t) 71,r72)dG2(72) d(
OO-T1
f frT (h(t) T1)dGI(T1)

Sf t fHT\,T2 (h(t)| 1, T2)dG2(72)dGi


s h t-7t1
SfH IT (h(t)|Tl)dG I(r).

since the different world state transition time are independent.


,i(-1) +


(T1) +


'r) +


(3.17)









It is obvious that the density function of the history h(t) can be decomposed as


fH(h(t)) = f(A1,h(t)) + f(A2, h(t)) + f(A3, h(t))


and


ft -o fH ,(h(t) r,

ft fH\T (h(t) 1,d .
JoL JT-
I o Tlf BT {t\r


2)dG2 (2)dGi(Tl)

T2)dG2 (T2)dGi (T)


The following computation and the notations used are similar as those in Section 3.2.1.
For example, for history h(t) = {N(t) = n, X1 = xl,..., X, = x,} = {N(t) = n, S

si,..., Sn = Sn} where Xi is the interarrival time of the ith demand and Si is the
arrival time of the ith demand, let fHTi (n, i, ..., sn, r7) denote the conditional density
of N(t) = n, S = si,..., Sn sn given that Ti = r, and let fH N,T (i, ..., sln, Ti)
denote the conditional density of S1 = si,...,S, = sn given that N(t) = n and
TI = TI. For additional conditions on T2, T3, the notations are straightforward.
If Tr > t, then the world is in state 1 at time t, and


fHr1(h(t) Ti)


Pr(N(t) = n|TI = Ti)fHIN,T~(Sl ..., sn, -T1)
e-Axt(Alt)" n!
n! t"
e-Alt^n'


f (,, h(t))


CA (i(t)iTl)dG1 (7)
e-AXltAG1(t).


(3.18)


If Tr < t < Tr + T2, then the world is in state 2 at time t. As before, we use
random variables N, and N2 to represent the number of demands in state 1 and 2


f(A,, h(t))

f(A2, h(t))

f(A3, h(t))









respectively. At time t, for given Tr and T2, these numbers are known, denoted by

nl(r1) and n ni(rT) respectively, and the occurrences of demands in different world

states are independent! Then we obtain


f HiT,T2 (h(t) T1,T2)


fHI1I,T2 (ni(i) ..., Xnl* (7);
ni(TI)+1
-n (Tri), xi- T1, nI(T1)+2,... 'n, X lT 2)
ii 1
fHIT ,T2 (71(T1), Xl,..., Xn(71)I|1, 72)
n1(Ti)+1
fHliT,T2 (n 1(), x 'ri, IXn(TI)+2,... X, T 7,T2)
i 1
--AiT1 /1(T1) -_A2(t--T)/ n-nl(T1)


If Sk < T1 < Sk+l, then Ni(rT) = k and N2(t T1) = n k, and we obtain


fBHT,,T2 (h(t) 7l, 72)


-AT 11 k-A2 (t-r ) n-k"
1 2


ot oo
f(A2, h(t)) j J f, (h(,T ) 1, }2)dG2(T2)dGi (T1)

= G2(t -T1)fHT,Trh(t) 1,2)dGl (l)


= k -k G2( T 1)e-AITI-A2 (t-T)dGI(Tl) (3.19)
k-0 Isk
where we let so = 0 and sn+l = t.

Similarly, if t > 71 + T2, then the world is in state 3 at time t. We denote the

numbers of demands occurred in each state by N1, N2 and N3 respectively. For given

TI, T2 and T3, these numbers are known, denoted by nl(Tr), n2(T2) and n nl(T1) -

n2(Q-,) respectively. Then by following the similar procedures as above, we obtain


fHlT,T2 (h(t) I1,72)


-A-T1 nl (7Tl) -A22 2(T2) -A3(t-T1--2) T n-- 1(T1) -2 (22)







83

If Sk < 71 < Sk+1, Sk+j < 71 + 72 < Sk+j+l, where k,j > 0 and k + j < n, then

n1 (T1) = k, n2(T2) = j, and n3(t 71 -) = n j k. Then


f(A3, h(t))


ft ft--



k 0- k


k=0 Sk


k=0 0 s


f(h(t) T,


I t--T
JO


71,T2 T2)dG2(T2)dGi(T1)


f(h(t) T = T, T2 T2)dG2(T2)dG1(T)


SJS -2 -X3(t-Tl-2) An-k dG2 72)

n-k s +k+l-Tl
yji A "Ale-XT k e-A2\\A C-At(t-1 -T2
j=l JS3+k-T--

-3-k dG2 (2) dGi(Ti)
1 1

e-XAIT A e-.-2 e-x3(t-Tl-T2) Xn-kdG2 72)
[Jo '


n-k -7
Sk /S.j+kc+l-T
-1 Ak -2 j 3(t- 1-2).


(3.20)


Now substitute GI(G2) with the exponential distributions with rate pl(/12) in

qualities (3.18), (3.19) and (3.20), we can perform the following computations:


f(A1, h(t))


e-Altj e-lt


n Sfk+1
Sn-k G2(t
k=0 sh
n
A k 1n-k (A2+P2)t
1 2 '^ A!


- T1)e- A e- A2(t-l)dGi(1)


(A + 1) (2
(A1+111) (2 + 112)


[e-((Al+pl)-(A2+p2))Se -((Al+pl)-(A2+p2))Ske+


f(A2, h(t))


(3.21)


(3.22)










and


f(A3, h(t))
Jk0 s 1 fk+-TI

k 0 L0o


k=0 sk


n

f- I


e-11 e-2 e-A3(t-1-72) A-kdG2 (2)+


Jk~ i Tj+k+lT2








n-k
-AITI j A3(t -T1) A -- "k 2
SA2 2 A3
A -kin dG 2T-A3(t- ) d-jk
j= 1

[A -(A2+2-A3)(S T1) -(A2 +2-A3)(Sj 1-T1)] dG (1)

A3tn--k P2 1
C3 A2+ A- A3 AI + + i2 A3

(g-(A +( 11- A3)psc A -(1 1-(Ai+- 3)S c+IS _

-11
A + 11- A2 [-2

(e-(C 1+-(A+ A2-P2) e-(A1+P1-A2-P2)S+1) +


n-k
Y xkje- -xtAr-j-k 12 11
1 2 3 A2 + 2 3- A3 A1 + 1 A2 P2

-(A2+P2-A3)j -(A2+P2- A3)Sj+ k


S-(Ai+pl-2-P2)sk -(Ai+pl-A2-2)sk+l]
L ~J


(3.23)


By time t all the demand occurrence times Sk, k = 0, n are known, and we can

compute (3.21), (3.22) and (3.23). Plugging the results into equation (3.16), we can

compute the conditional probability that the world is in state i (i = 1, 2, 3) at time t









given h(t),


f(As, h(t))
p(i, t, h(t)) f h(t))
f(h(t))
f (A, h(t))
f(A1, h(t)) + f(A2, h(t)) + f(A3, h(t))

It is obvious that as the number of states increases, the computations in this

section will be more complicated. Even the results will be too long to represent. We

next seek a simple way to represent the (conditional) density functions of the history.

3.3.2 A Recursive Formula

To analyze the more general multiple-state model, i.e., m states in total, the

above way of multiple conditioning is not applicable. Another approach is to derive a

recursive formula. For notational convenience in the recursion, we re-index the world

states and number them from m through 1 in the order of their occurrence as in the

previous section. To use the history information into the recursive fashion, we let

H(t)= {Dm(u :0< u< t

where the subscript m represents the observation of the cumulative demand curve

starts in state m.

Denote fk(h(t)) fk(Dk(u) : 0 < u < t) to represent the density function that

part of the history which starts at the starting time of world state k to take the

instance {Dk(u) : 0 < u < t}. We also denote fk(h(t),j) to represent the (joint)

density function of history information up to time t, h(t), and the world state at time

t, j, given that the history observation starts at the starting time of state k.
Now by conditioning on the first state transition time Tm = Tm, we get two

possibilities: if t < Tm, the world is still in state m at time t; if t > T,, we consider

it a problem with m 1 states, starting at time T,. Denote the conditional density

function of history which starts in world state m given Tm = Tm by fm(h(t) Tm), we









obtain


fm(h(t)) = fm (h(t) Tm)dGm(Tm)

= (h(t)T)dGm(T) + fm(h(t) m)dGm(Tm)
0 t

ifm(h(t) Tm)dGm(Tm) + e-AmtG,(t)

where for Tm > t, fm(h(t)l\Tm) = e-'mtXmGm(t), which is derived from the results of

the previous section.

For the case Tm < t, the total past history information information by t can

be divided into two parts: the observations before Tm, i.e., h(rm), which followed

a stationary Poisson process with parameter Am and nm(Trm) demands occurred in

state m, and the observations after Tm. Since the world process is Markovian, these

two parts of observations are independent. And for the history after Tm and before

t, it is exactly the same as the observations between time 0 and time t Tm for a

(m 1)-state world model. By using the notations defined above, we get for Tm < t,


fm(h(t)\ Tm)

fm(h(Tmn), m) fm-1 (Dm(Tm + u) Dm(Tm) : 0 < u < t Tm)

A=e-A mrm m('m) fm-1 ({Dm-(U) : 0 < U < t Tm})


where we denote


{Dm-(u) : 0 < u < Tm} {Dm(Tm + ) Dm(Tm) : 0 < u < Tm}


to be the observation of history from the entering time of state m 1 until t Tm

time later, and nm(Tm) to be the number of demands occurred in Tm time units in

world state m.








So the general recursive formula can be written as

fk(Dk() : 0 < u < t)
tt
= Ce- AA fk-_I (Dk-1(u) : 0 < u < t t7k) dGk(7-k) +

fk({Dk(u) :0 < u< t},k).

We have

fk({Dk(u) :0 < < t},k) = e-AktTGk(t) for 1 < k < m



and

fk({Dk(u) :0 t= e- k( fk-1 ({Dk- (u) : 0 < u < t k}, k dGk(Tk)

for < k < nmand
We apply this recursive formula to the case m = 3

f3(h(t))

f3(D3(U) :0 < U< t)

0 t

f3({D3(u) :0< u < t},A(t) 3)
J t n3 (T3) t T3 X^ -2)
S e- 3T3 A3 3) J2 fl (D (u) :0 < u < t- 3 2) dG2(2)

A2+e-( T3) n3 (3) G2(t)} dG3(73) + e- 3ttG3(t
ft t-T3
-T3An3 (T3) T- A2T2 A 2(T 2)e A-(tT3 2) ALl3(T73)-2(T2)dG2 7(T)dG3 T73
i e At)3 f 3 T G 2d G ) + )\ 3 t
+ Ce-A373 A3(73) -A2 (t-T3)A-n3(T3)G2(t)dG3 (73)+e- 3tA G3
0









and the recursive formula gives the same results as those given by direct method in

the previous section.

We need to point out that this recursive formula only eases the representation

of the conditional probability functions given history h(t),

fm(h(t), i)
p(i, t, h(t))=- ,
fm(h(t))

but to compute the probability, we still need to express it extensively, as in Section

3.3.1, and the computation remains the same complicated.

3.3.3 Optimal Inventory Position

For fixed lead time by conditional on which state the world is currently in, the

distribution of the lead time demand can be expressed as


FD ,,(t) (z) =Pr(D'h(t) < z)

= Pr(D ,h() < zA(t)= -A)p(i,th(t))
i=i

= ~D(z)p(i, t, h(t)). (3.24)
i=1
The last equation follows from the fact that the world is a Markov process and demand

process is a Poisson process. For stochastic lead time,


i= 1
Then we can write


C(y,t,h(t)) = Ee-aLC(y D (t))

JJ C(y z)dFDt,h(t)()dz FL ()


= p(k, t t)) e-a (y z)dF(z)dFL(


i ,t, t))C y,
i= 1










where C(y, i) has the same definitions as in Song and Zipkin [52].

In principal we can find the optimal inventory positions by using the similarly

algorithm as in Section 3.2.3, but now the form of the probability function p(i, t, h(t))

becomes much more complicated, and to compute the ts at which the left or right

derivatives of C(y, t, h(t)) are 0 is not an easy task now.

To compute the optimal inventory position, we again need to assume that the lead

time distribution is continuous phase-type. The optimal inventory level y*(t, h(t)) is

just the point at which the right derivative of C(y, t, h(t)) + (. / is no smaller than

0, while the left derivative is smaller than 0. It follows



i=n



i=
At the starting time, we know the probability of p(i, t, h(t)) for all i 1,..., m,

and we can compute the optimal inventory position as y*. As time goes on, if no

demand occurs, we can determine the next time when the optimal inventory position

will change given that no new demand occurs by that time by solving the following

two equations separately,



i=i

p(i,t,h(t))(C(y, i))' + ac = 0.
i=1

Solve these two equations separately, and denote the solutions by t' and t" respectively.

Then we can follow the procedure in Section 3.2.3, and the details are omitted here.

The 1n i Pr difficulty here is to get the solution t' and t".

Every time when a new demand occurs, we should update all the probabilities

p(i, t, h(t)) at the time t, and get the new optimal inventory position at that time