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Thanksareduetomysupervisor,Dr.ElifAkcal,forherinsights,perspectives,guidanceandsupport.IthankDr.SlaCetinkaya,Dr.JosephP.GeunesandDr.SelcukErengucforservingasmydissertationcommitteemembersaswellfortheircontributionsandvaluablesuggestions.IalsothankDr.HalitUsterforhisvaluablecontributions,insights,andsuggestions.IextendmysinceregratitudetoDr._IsmailS.BakalandZekiC.Tasknfortheirwarmfriendship.Itwouldhavebeenmuchmoredicultformetogetthroughthisprocesswithouttheirendlesssupportandunderstanding.Ithankmyparents,HaticeandBakiKarakayal;mybelovedsisters,FusunandRaziye;andmylittlebrother,Burak,fortheirunconditionalloveandsupport.Finally,Ithankmywife,andbestfriendandcondantinlife,Gunseli,forhersupportandencouragementtocompletemydegree.Mywifeandmydaughter,Sla,werethebiggestmotivationforthiswork.Iknewthatwewouldbetogetherforeveruponthecompletionofmydegree.IalsowouldliketoacknowledgethenancialsupportprovidedbytheNationalScienceFoundationunderGrantNo.CMMI-0522960. 4
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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1BackgroundandMotivation .......................... 12 1.2Overview .................................... 13 1.3Impact ...................................... 14 2LITERATUREREVIEW .............................. 15 2.1Introduction ................................... 15 2.2DeterministicDiscreteTimeModelsforInventoryandProductionPlanningforProductRecovery .............................. 15 2.3StochasticModelsforInventoryandProductionPlanningforProductRecovery 19 2.4Multi-ItemCapacitatedLotSizing ...................... 21 3REPLENISHMENTANDDISPOSALPLANNINGFORMULTIPLEPRODUCTSWITHREUSABLERETURNS .......................... 25 3.1Introduction ................................... 25 3.2SharedReplenishmentandProduct-SpecicDisposalCapacities ...... 26 3.3SolutionApproach ............................... 30 3.3.1ObtainingLowerBounds ........................ 32 3.3.2ObtainingUpperBounds ........................ 36 3.3.3SubgradientOptimization ....................... 39 3.4ComputationalExperiments .......................... 39 3.4.1RandomTestInstanceGeneration ................... 40 3.4.2DiscussionofComputationalResults ................. 42 3.5SharedReplenishmentandDisposalCapacities ................ 45 3.6ConcludingRemarks .............................. 48 4MANUFACTURING,REMANUFACTURING,DISPOSALPLANNINGFORMULTIPLEPRODUCTSWITHREMANUFACTURABLERETURNS .... 56 4.1Introduction ................................... 56 4.2ModellingAssumptionsandMathematicalFormulationofMRDPP .... 58 4.3SolutionApproaches .............................. 63 4.3.1ALagrangianDecompositionApproachforMRDPP ......... 65 5
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................... 67 4.3.1.2Obtainingupperbounds ................... 72 4.3.1.3Subgradientoptimization .................. 80 4.3.2ALagrangianRelaxationApproachforMRDPP ........... 81 4.3.2.1Obtaininglowerbounds ................... 82 4.3.2.2Obtainingupperbounds ................... 83 4.3.2.3Subgradientoptimization .................. 84 4.4ComputationalExperiments .......................... 85 4.4.1RandomTestInstanceGeneration ................... 85 4.4.2DiscussionofComputationalResults ................. 86 4.5ConcludingRemarks .............................. 89 5PROCUREMENTPLANNINGFORMULTIPLEPRODUCTSWITHREUSABLERANDOMRETURNS ................................ 96 5.1Introduction ................................... 96 5.2ModellingAssumptions ............................ 97 5.3PPP1:SinglePeriodandSingleProduct ................... 98 5.4PPP2:SinglePeriodandMultipleProducts ................. 102 5.5PPP3:MultiplePeriodandSingleProduct .................. 107 5.6PPP4:MultiplePeriodandMultipleProduct ................ 114 5.7ConcludingRemarks .............................. 119 6CONCLUSION .................................... 120 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 129 6
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Table page 3-1Parametervaluesusedincomputationalexperiments. ............... 51 3-2PerformanceoftheLRApproach:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). ............................ 51 3-3EectofTimeBetweenReplenishment(TBR)andTimeBetweenDisposal(TBD)SetupsontheAverageandMaximumOptimalityGaps(%)oftheLRApproach. 52 3-4EectofReplenishmentandDisposalCapacityTightnessontheAverageandMaximumOptimalityGaps(%)oftheLRApproach. ............... 52 3-5EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)oftheLRApproach. ..................... 52 3-6PerformanceofCPLEX(withaTimeLimitofthe30SecondsandaStoppingGapof0.1Percent):AverageandMaximumOptimalityGaps(%). ....... 52 3-7EectofTimeBetweenReplenishment(TBR)andTimeBetweenDisposal(TBD)SetupsontheAverageandMaximumOptimalityGaps(%)ofCPLEX. ..... 53 3-8EectofReplenishmentandDisposalCapacityTightnessontheAverageandMaximumOptimalityGaps(%)ofCPLEX. .................... 53 3-9EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)ofCPLEX. .......................... 53 3-10InuenceofIncreasingtheTimeLimitandtheStoppingGaponthePerformanceofCPLEX:NumberofUnsolvedInstances,AverageandMaximumOptimalityGaps(%). ....................................... 53 3-11PerformanceoftheLRApproachfortheRDPPwithSharedReplenishmentandDisposalCapacities:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). .................................. 54 3-12PerformanceofCPLEXfortheRDPPwithSharedReplenishmentandDisposalCapacities(withaTimeLimitofthe30secondsandaStoppingGapof0.1Percent):AverageandMaximumOptimalityGaps(%). ................... 54 3-13InuenceofIncreasingtheTimeLimitandtheStoppingGaponthePerformanceofCPLEXfortheRDPPwithSharedReplenishmentandDisposalCapacities:NumberofUnsolvedInstances,AverageandMaximumOptimalityGaps(%). 55 4-1ParameterValuesUsedinComputationalExperimentsofMRDPP ....... 93 4-2PerformanceoftheLDApproach:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). ............................ 93 7
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............ 93 4-4EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. ..................... 94 4-5PerformanceoftheLRApproach:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). ............................ 94 4-6EectofTimeBetweenActivitySetupsandCapacityTightnessontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. ............ 94 4-7EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. ..................... 94 4-8PerformanceofCPLEX(withaTimeLimitof400Seconds):AverageandMaximumOptimalityGaps(%). ................................ 95 8
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Figure page 3-1Problemsetting. ................................... 50 3-2Pseudo-codeoftheIdenticationofSetIpt. .................... 50 3-3Pseudo-codeoftheIdenticationofSetI .................... 50 4-1ProblemSettingofMRDPP ............................. 90 4-2Pseudo-codeoftheLDApproachfortheMRDPP. ................. 91 4-3Pseudo-codeoftheGAApproachfortheLD. ................... 92 4-4Pseudo-codeoftheLRApproachfortheMRDPP. ................. 92 9
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Weconsiderthecharacterizationoftheoptimalinventoryandproductionplanningpoliciesformulti-productclosed-loopsupplychainsystemsfordirectreuseandvalue-addedrecoverybyconsideringaclassofdeterministicdiscretetimedynamicdemandandreturnmodels.Inparticular,weconsiderthreeproblems.First,motivatedbytheanoriginalequipmentmanufacturer(OEM)inthepowergenerationequipmentindustrythatprovidessparepartkitsforpowerturbinemaintenanceservices,westudyareplenishmentanddisposalplanningproblemthatarisesinsettingswherecustomerreturnsareinas-good-as-newcondition.Thesereturnscanbeplacedintoinventorytosatisfyfuturedemandorcanbedisposedof,incasetheyleadtoexcessinventory.WeproposeaLagrangianRelaxationapproachthatreliesontherelaxationofthecapacityconstraintsanddevelopasmoothingheuristicthatusesthesolutionoftheLagrangianproblemtoobtainnear-optimalsolutions.Ourcomputationalresultsdemonstratethattheproposedapproachisveryeectiveinobtaininghigh-qualitysolutionswithareasonablecomputationaleort.Next,totaketheuncertaintyassociatedwithdirectlyreusablecustomerreturnsintoaccountexplicitly,weconsideraprocurementplanningproblemencounteredbyacatalogueretailerthatalsoreceivescustomerreturns.Tothisend,wedevelopadiscretetimedynamicdeterministicdemandandstochasticreturnmodelforthecharacterizationofinventoryandproductionplanningpolicy.Wedevelopsolutionalgorithmsusingcontinuousoptimizationtechniques,andidentifytheoptimalsolution 10
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PohlenandM.T.Farris 1992 ; RogersandTibben-Lembke 1999 ).Toreducetheeconomicalandenvironmentalburdenofsolidwaste,productrecoverypracticesarebecomingincreasinglypopular. Productrecoverymanagement(PRM)encompassestheplanningandmanagementofabroadsetofactivitiestoreclaimtheeconomicalandenvironmentalvalueresidinginend-of-useorend-of-lifeproducts.Theeconomicalvaluethatcanbereclaimedfromtheend-of-useandend-of-lifeproductsisrelatedtothecostsavingsrealizedbyrecoveringproducts,components,andmaterialsforreuse.Similarly,theenvironmentalvalueinvolvestheenvironmentalimpactreductionrealizedbydiminishingtherelianceonlandllingandincinerationforsolidwastedisposal. Historically,boththetheoryandpracticeofsupplychainmanagementhasplacedemphasisontheforwardchannel,whichconsistsofsuppliers,manufacturers,distributors,retailers,andcustomers.Recentinterestinproductrecovery,however,hasextendedthescopeoftraditionalsupplychainmanagementbydrawingattentiontothereversechannel,whichconsistsofnal-users,retailers,collectors,andremanufacturers.Sinceproductrecoveryprocessesreturnedandusedproductsinthereversechannel,theeciencywithwhichtheseproductsareprocessedhasadirectimpactontheprotabilityofproductrecoverypractices.Moreover,asreturnsareprocessedorremanufacturedtosatisfythecustomerdemand,thisimpactstheforwardchannelowsandintroducesastrong 12
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Acloserexaminationofexistingproductrecoverypracticesrevealthatitispossibletodistinguishbetweentwoformsofproductrecovery: 13
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Guide 2000 ),thegeneraltopicofinventoryandproductionplanningforproductrecoveryhasreceivedconsiderableattentioninrecentyears.Foracomprehensivereviewoftheliterature,wereferthereaderto( DekkerandvanderLaan 2003 ; Inderfurthetal. 2004 ; InderfurthandTeunter 2003 ; vanderLaanetal. 2004 ; RichterandDobos 2004 ).Acarefulexaminationoftheinventoryandproductionplanningforproductrecovery,however,revealsthatthislineofworkmainlyconcentratesonvalueaddedrecovery( Golanyetal. 2001 2005 ; Lietal. 2006b a ; Richter 15
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, 2003 ; RichterandSombrutzki 2000 ; RichterandWeber 2001 ; Yangetal. 2005 )withoneexceptionthatfocusesondirectreuse( BeltranandKrass 2002 ). Motivatedbypracticalapplicationsinhandlingreturnsincatalogueretailingandmanagingaircraftengineservicecomponentkits,BeltranandKrass( BeltranandKrass 2002 )considerasingleitemuncapacitatedlotsizingforasystemwherereturnsreceivedfromcustomersareplacedinnishedgoodsinventorywithoutanyprocessing.Theirmodelisaimedtodeterminetheoptimalquantityofitemstoprocure,quantityofitemstodispose,andinventorylevelforitemsthroughouttheplanninghorizonwiththeobjectiveofminimizingthesumofconcaveandpiecewisedierentiableprocurement,disposal,andinventoryholdingcosts.Toobtainthesolutiontothismodel,theydevelopaDPalgorithmthatrunsinpolynomialtime. RichterandSombrutzki ( 2000 )considerthesingleitemuncapacitatedlotsizingproblemforapureremanufacturingsystemwithxed-chargeremanufacturingandlinearinventoryholdingcosts.Theirmodelseekstheoptimalquantityofitemstoremanufactureineachperiodandtheinventorylevelsforremanufacturableandremanufactureditemsthroughouttheplanninghorizonwiththeobjectiveofminimizingthesumofremanufacturingsetupandinventoryholdingcosts.Theyshowthatamodicationoftheclassicallotsizingalgorithmcanbeusedtoobtainthesolution( WagnerandWhitin 1958 ).Theyalsoconsidertheprobleminthecontextofahybridremanufacturingandmanufacturingsystem,consideringxed-chargemanufacturingcostswherethemodelseeksalsotheoptimalquantityofitemsmanufacturedineachperiod.However,theydonotprovideasolutionapproachforthisgeneralproblem,butconsideraspecialcaseoftheproblemwherethequantityofremanufacturableitemsavailableatthebeginningoftheplanninghorizonissucientlylargetocovertheentiredemandthroughouttheplanninghorizon.Theydevelopadynamicprogramming(DP)approachtoobtainthesolution. RichterandWeber ( 2001 )extendthemodelformulationsgivenby RichterandSombrutzki ( 2000 )byincludinglinearremanufacturingandmanufacturingcostsand 16
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RichterandGobsch ( 2003 )concentrateonthesingleitemuncapacitatedlotsizingproblemforahybridmanufacturingandremanufacturingsystemconsideringxed-chargemanufacturingandremanufacturingaswellaslinearnewmaterialpurchase,remanufacturableitemacquisition,andinventoryholdingcosts.Theirmodelseekstheoptimalquantityofremanufactureditemsandmaterialtopurchase,quantityofnewitemstomanufacture,quantityofremanufacturableitemstoremanufacture,andinventorylevelsfornewmaterials,remanufacturableitems,andremanufactured/newitemsthroughouttheplanninghorizonwiththeobjectiveofminimizingthetotalcosts.TheyprovideanestedDPapproachtosolvethisgeneralproblem.Theyalsoanalyzetwospecialcasesoftheproblemwhere(i)nomaterialorremanufacturableiteminventoryisheldand(ii)noremanufacturedornewiteminventoryisheld. Golanyetal. ( 2001 2005 )considerasingleitemuncapacitatedlotsizingproblemforahybridmanufacturingandremanufacturingsystemwithmanufacturing,remanufacturing,disposalandinventoryholdingcosts.Theyshowthattheproblem(i)isNP-hardforgeneralconcavecostsand(ii)canbetransformedintoaminimumcostnetworkowproblemifallcostsarelinear. Yangetal. ( 2005 )showthattheproblemremainsNP-hardforstationaryconcavecostsanddevelopaDPapproachfortheproblem.Byexploringthecharacteristicsoftheextremepointsolutions,theydevelopaDPbasedheuristicapproachthatrunsinpolynomialtime. Lietal. ( 2006b )consideramulti-itemuncapacitatedlotsizingproblemforahybridmanufacturingandremanufacturingsystem,wheredownwardsubstitutionofitemsisallowed,i.e.,ahighergradeitemcanbesubstitutedforalowergradeitemtosatisfythedemand.Wenotethatthegradedoesnotrefertowhethertheitemisneworremanufactured,buttothecharacteristicsoftheitem,e.g.,afasterprocessorcanbesubstitutedforaprocessorwithalowerspeed,oramemorycardwithmorestoragespace 17
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Lietal. ( 2006a )developatwo-itemmodelinacapacitatedsettingwhereemergencyprocurementsareallowedincaseofapotentialshortage.Theirmodelseekstheoptimalquantitytomanufacture,remanufacture,andprocureforeachitem,remanufacturableandnew/remanufacturedinventorylevelsforeachitemaswellasthequantityofalowergradeitemusedtosatisfythedemandforahighergradeitemwiththeobjectiveofminimizingthesumofxed-chargemanufacturingandremanufacturingaswellaslinearmanufacturing,remanufacturing,emergencyprocurement,andinventoryholdingcosts.Althoughdownwardsubstitutionisallowedtosatisfythedemand,thecostofsubstitutionisassumedtobezero.TheauthorsdevelopahybridDPandgeneticalgorithm(GA)approachtoobtainasolutionfortheproblem,whereGAapproachisusedtospecifythetimingofthemanufacturingandremanufacturingsetups,whereastheDPalgorithmisusedtodeterminethemanufacturing,remanufacturing,substitution,andemergencyprocurementquantities. 18
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Theexistingcontinuousreviewmodelsattempttocomputeinventoryandproductioncontrolpolicyparametersmainlyforvalue-addedrecoverysystems.Assumingthatallthereturnsareremanufacturedoneatatimeuponreceipt,alineofworkfocusesonsystemswithasinglestockpoint Bayndretal. ( 2003 2005 ); Fleischmannetal. ( 2002 ); Heyman ( 1977 ); vanderLaan ( 2003 ); Toktayetal. ( 2000 ); YuanandCheung ( 1998 ).Anotherlineofworkfocusesonsettingswherereturnsareremanufacturedinbatches Arasetal. ( 2004 ); MuckstadtandIsaac ( 1981 ); Teunteretal. ( 2000 ); Teunter ( 2002 ); Teunteretal. ( 2004 ); vanderLaan ( 2003 ); vanderLaanetal. ( 1996a b ); vanderLaanandSalomon ( 1997 ); vanderLaanetal. ( 1999a b ); vanderLaanandTeunter ( 2006 ).Theemphasisismainlyontheanalysisofsingle-itemsettings Arasetal. ( 2004 ); Bayndretal. ( 2003 2005 ); Fleischmannetal. ( 2002 ); Heyman ( 1977 ); MuckstadtandIsaac ( 1981 ); Teunteretal. ( 2000 ); Teunter ( 2002 ); Teunteretal. ( 2004 ); Toktayetal. ( 2000 ); vanderLaan ( 2003 ); vanderLaanetal. ( 1996a b ); vanderLaanandSalomon ( 1997 ); vanderLaanetal. ( 1999a b ); vanderLaanandTeunter ( 2006 ); YuanandCheung ( 1998 ),consideringbothnite Teunteretal. ( 2000 ); Teunter ( 2002 ); Teunteretal. ( 2004 )andinnite Arasetal. ( 2004 ); Bayndretal. ( 2003 2005 ); Fleischmannetal. ( 2002 ); Heyman ( 1977 ); MuckstadtandIsaac ( 1981 ); Toktayetal. ( 2000 ); vanderLaan ( 2003 ); vanderLaanetal. ( 1996a b ); vanderLaanandSalomon ( 1997 ); vanderLaanetal. ( 1999a b ); vanderLaanandTeunter ( 2006 ); YuanandCheung ( 1998 )planninghorizons.Onlyalimitedamountof 19
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Fleischmannetal. ( 2002 ); Heyman ( 1977 ),andthemainemphasisofthesemodelsisonanalyzingsystemperformanceunderaparticularinventorycontrolpolicy.TheunderlyingsystemistypicallymodeledasacontinuoustimeMarkovchain Arasetal. ( 2004 ); Bayndretal. ( 2005 ); MuckstadtandIsaac ( 1981 ); vanderLaan ( 2003 ); vanderLaanetal. ( 1996a b ); vanderLaanandSalomon ( 1997 ); vanderLaanetal. ( 1999a b ); vanderLaanandTeunter ( 2006 ); YuanandCheung ( 1998 )oraqueueingnetwork Bayndretal. ( 2003 ); Toktayetal. ( 2000 ).Tocomputetheinventorycontrolpolicyparameters,numericaloptimizationtechniques Arasetal. ( 2004 ); Bayndretal. ( 2003 2005 ); MuckstadtandIsaac ( 1981 ); vanderLaan ( 2003 ); vanderLaanetal. ( 1996a b ); vanderLaanandSalomon ( 1997 ); vanderLaanetal. ( 1999a b ); vanderLaanandTeunter ( 2006 ); YuanandCheung ( 1998 )oranalyticalapproximations Toktayetal. ( 2000 ); vanderLaan ( 2003 ); vanderLaanandTeunter ( 2006 )areused.Amorerecentlineofworkalsousessimulationmodelingandanalysistocomputeeectivevaluesforpolicyparametersortostudysystemperformance Teunteretal. ( 2000 ); Teunter ( 2002 ); Teunteretal. ( 2004 ); Toktayetal. ( 2000 ). Themajorityofexistingperiodicreviewmodelsaimtocomputeinventoryandproductioncontrolpolicyparametersforvalue-addedrecoverysystems.Whilesomemodelsaredevelopedconsideringasinglestockpoint BuchananandAbad ( 1998 ); Cohenetal. ( 1980 ); FleischmannandKuik ( 2003 ); Inderfurth ( 1997 ); KelleandSilver ( 1989 ); KiesmullerandScherer ( 2003 ); KiesmullerandvanderLaan ( 2001 ); MostardandTeunter ( 2006 ); Nakashimaetal. ( 2002 2004 ); VlachosandDekker ( 2003 ); Whisler ( 1967 ),othersfocusonsettingswithtwostockpoints Bayndretal. ( 2007 ); Inderfurth ( 1997 2004 ); Kiesmuller ( 2003 ); KiesmullerandMinner ( 2003 ); KiesmullerandScherer ( 2003 ); Mahadevanetal. ( 1999 ); Simpson ( 1978 ); TeunterandVlachos ( 2002 ),andthegeneralemphasisoftheexistingmodelshasbeenonsingle-itemsettingsconsideringsingleperiod Bayndretal. ( 2007 ); Inderfurth ( 2004 ); MostardandTeunter ( 2006 ); Vlachosand 20
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( 2003 ),nitehorizon BuchananandAbad ( 1998 ); Cohenetal. ( 1980 ); HeisigandFleischmann ( 2001 ); Inderfurth ( 1997 ); KelleandSilver ( 1989 ); KiesmullerandMinner ( 2003 ); KiesmullerandScherer ( 2003 ); KiesmullerandvanderLaan ( 2001 ); Simpson ( 1978 ); TeunterandVlachos ( 2002 ); Whisler ( 1967 ),andinnitehorizon FleischmannandKuik ( 2003 ); HeisigandFleischmann ( 2001 ); Kiesmuller ( 2003 ); Mahadevanetal. ( 1999 ); Nakashimaetal. ( 2002 2004 ); Whisler ( 1967 )models. Alimitedstreamofresearchonperiodicreviewmodelscharacterizestheoptimalpolicyusingdierentialcalculustechniquesinsingleperiodsettings Bayndretal. ( 2007 ); Inderfurth ( 2004 ); MostardandTeunter ( 2006 ); VlachosandDekker ( 2003 )anddynamic-programming-basedoptimizationtechniquesinnitehorizonsettings Cohenetal. ( 1980 ); FleischmannandKuik ( 2003 ); Inderfurth ( 1997 ); Simpson ( 1978 ); Whisler ( 1967 ).Also,Markovdecisionprocessmodelsaredevelopedtocomputetheoptimalinventorycontrolparameters Nakashimaetal. ( 2002 2004 ).Otherexistingworkintheareafocusesonanalyzingsystemperformanceunderaparticularinventorycontrolpolicyandusesdynamicprogramming BuchananandAbad ( 1998 ); Cohenetal. ( 1980 ); KelleandSilver ( 1989 ); KiesmullerandScherer ( 2003 )orsimulationbased Kiesmuller ( 2003 ); KiesmullerandMinner ( 2003 ); KiesmullerandvanderLaan ( 2001 ); Mahadevanetal. ( 1999 ); TeunterandVlachos ( 2002 )approachestocomputethecontrolparameters. Brahimietal. 2006 ).Itispossibletoclassifytheexistingworkonlotsizingproblemsaccordingtothe(i)numberofproducts(i.e.,single-vs.multi-item)considered,(ii)numberoflevelsinthebill-of-materials(i.e.,single-vs.multi-level)considered,and(iii)restrictiononthecapacityavailabilityofproductionresources(i.e.,uncapacitatedvs.capacitated)takenintoaccount( Brahimietal. 2006 ; Karimietal. 2003 ).Inthiscontext,theclassofproblemsweconsiderintheproposed 21
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Therearecomprehensivereviewsofsingle-levelsingle-item( Brahimietal. 2006 )andmulti-item( Karimietal. 2003 )capacitatedlotsizingmodelsintheliterature.Morerecently,JansandDegraeve( JansandDegraeve 2007 )reviewthemeta-heuristicapproachesdevelopedfordicultlotsizingproblems.Thegeneralsingle-itemcapacitatedlotsizingproblem( Florianetal. 1980 )aswellassomespecialcases( BitranandYanasse 1982 )areNP-hard.Thegeneralmulti-itemcapacitatedlotsizingproblemisstronglyNP-hard( ChenandThizy 1990 ).Inwhatfollows,wereviewsomeoftheexistingworkthatdevelopsexactandheuristicsolutionapproachesforthemulti-itemcapacitatedlotsizingproblem. Alineofexistingworkfocusesonthedevelopmentofexactsolutionapproachesforthesingle-levelmulti-itemcapacitatedlotsizingproblem( Baranyetal. 1984 ; EppenandMartin 1987 ; Leungetal. 1989 ).While Baranyetal. ( 1984 )proposecutgenerationtechniquewhicharethefacetsofthesingle-itemuncapacitatedlotsizingproblem, Leungetal. ( 1989 )exploretheuseofthefacetsofthesingle-itemcapacitatedlotsizingproblem. EppenandMartin ( 1987 )proposealternativeformulationsthatyieldmuchtighterlinearrelaxationmodelsforboththesingle-andmulti-itemcapacitatedlotsizingproblem. BelvauxandWolsey ( 2000 2001 )developaframeworkforthesolutionofthemulti-level,multi-resource,multi-itemcapacitatedlotsizingproblemwithbackloggingthatperformssomepreprocessing,generatescuttingplanes,andutilizesaprimalheuristicforthelotsizingproblem. Thereareanumberofpapersthatdevelopsspecializedconstructiveand/orimprovementheuristicsforthemulti-itemcapacitatedlotsizingproblem( Dogramacietal. 1981 ; DixonandSilver 1981 ; Eisenhut 1975 ; Gunther 1987 ; KarniandRoll 1982 ; KrcaandKokten 1994 ; LambrechtandVanderveken 1979 ; MaesandWassenhove 1986 ; SelenandHeuts 1987 ; Trigeiro 1989 ). Eisenhut ( 1975 )proposesasingleforward-pass 22
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LambrechtandVanderveken ( 1979 )and DixonandSilver ( 1981 )proposeprocedurestoremovecapacityinfeasibilities. Dogramacietal. ( 1981 )and KarniandRoll ( 1982 )proposemulti-phaseheuristicapproachesthatobtainimprovedsolutionsfrominitialfeasiblesolutions. MaesandWassenhove ( 1986 )proposeaexibleapproachthatcanusedierentrulesfortheaprioriorderingofitems,criteriausedtodecidewhetherornottoincludethedemandforaniteminthecurrentperiod,andsearchstrategy.Theirapproachincludesprocedurestoensurefeasibilityandtoimproveagivenfeasiblesolution.Alltheseapproachesbuildaplanusingarollinghorizonapproach. KrcaandKokten ( 1994 )focusonitemsratherthanperiods,andoneachiterationoftheapproach,afeasibleproductionplanforanitemisdeveloped.Thatis,oncetheplanforanitemisgenerated,theavailablecapacityineachperiodisupdated.Then,aDPbasedprocedureisusedtogeneratetheplanforthenextitem. Gunther ( 1987 )proposetostartfromalot-for-lotsolutionandbuildacapacityfeasiblesolutionusingapriorityindextocombinelots,alook-aheadproceduretoensurecumulativecapacityfeasibility,andax-upproceduretoremoveperiodcapacityinfeasibilities. SelenandHeuts ( 1987 )proposetheuseofamodiedpriorityindexforGunther'sheuristic. Trigeiro ( 1989 )proposesamulti-passapproachfortheversionoftheproblemwithsetuptimes.Intherstforwardpassaninitialplanisbuiltusingamodiedversionofthewell-knownSilver-Mealpriorityindex.Inthesecondbackwardpass,capacityinfeasibilitiesareremovedbyshiftingexcessquantities.Thenalthirdpassensuresthattheresultingsolutionsatisesthepropertiesofanextremepointsolution. Anotherlineofworkfocusesonthedevelopmentofrelaxation-basedheuristicsforthemulti-itemcapacitatedlotsizingproblem( ChenandThizy 1990 ; Diabyetal. 1992b a ; MillarandYang 1994 ; Thizy 1991 ; ThizyandWassenhove 1985 ; Trigeiro 1987 ; Trigeiro 23
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, 1989 ). ThizyandWassenhove ( 1985 )and Trigeiro ( 1987 )proposeLagrangianrelaxationapproachesthatfocusontherelaxationofthecapacityconstraints.While ThizyandWassenhove ( 1985 )developatransportationproblembasedupperboundingheuristictoconstructafeasibleschedulefromthedualsolutiontotheoriginalproblem, Trigeiro ( 1987 )implementsasmoothingheuristicforthesamepurpose. ChenandThizy ( 1990 )provideacomprehensiveoverviewofrelaxationapproachesandshowthattheLagrangianrelaxationofthecapacityconstraintsyieldthetightestlowerboundontheoptimalsolution. Thizy ( 1991 )investigatesthequalityofboundsgeneratedbyalternativeLagrangiandecompositionapproaches. Trigeiroetal. ( 1989 )developaLagrangianrelaxationbasedsolutionapproachfortheversionoftheproblemwithsetuptimes.Inaddition, Diabyetal. ( 1992a )and Diabyetal. ( 1992b )considersingle-andmulti-resource,respectively,versionoftheproblemanddevelopLagrangianrelaxationbasedsolutionapproaches. MillarandYang ( 1994 )alsoconcentrateontheversionoftheproblemwithsetuptimesandstudyLagrangianrelaxationanddecompositiontechniques. 24
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Morespecically,theproblemisconcernedwithdeterminingwhen,andhowmuch,toreplenishordisposeofeachtypeofproduct.Weconsidertheprobleminadeterministicdemandandreturnmodelingframeworkoveraniteplanninghorizonunderthepresenceofprocessingcapacitiesforreplenishmentanddisposalactivities.Wedevelopanintegerprogrammingformulationtoseekanoptimalsolutionthatcharacterizesthe 25
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3.2 and 3.5 ,respectively. Theremainderofthispaperisorganizedasfollows.WeproceedwithconsideringtherstvariantoftheRDPP.Section 3.2 presentsamathematicalformulationforthisproblem,andthesolutionapproachisdescribedinSection 3.3 .ResultsfromcomputationalexperimentsarepresentedandanalyzedinSection 3.4 .Section 3.5 detailshowtheproposedmathematicalformulationandthesolutionapproachcanbemodiedforaddressingthesecondvariantofRDPPwherebothreplenishmentanddisposalcapacitiesaresharedamongmultipleproducttypes.Finally,conclusionsandfutureresearchdirectionsaresummarizedinSection 3.6 26
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Thecustomersmayorderseveralkitsbutusefewerkitsthanordered.ThekitsthatarenotusedbythecustomercanbereturnedtotheOEM.Inourwork,weconsiderfullkitsonly.Thatis,partialkitsarenotacceptedbytheOEM.Hence,theOEMhastwosourcesforreplenishingtheinventoryofassembledkits:returnsreceivedfromthecustomersandnewkitsassembledbyusingthepartsinthepartinventories.Iftoomuchinventoryofaparticulartypeofassembledkitaccumulates,thentheOEMcan`dispose'oftheexcessinventorybydisassemblingthekitsandrestockingpartsinpartinventories.Thedisassemblyofkitscanbeperformedonworkbencheswhichdonotneedtobecongured;however,thelaborrequiredimposesadisassemblycapacityrestrictionthatlimitsthequantityofkitsthatcanbedisassembled.Therefore,thereisadisassemblycapacityrestrictionthatlimitsthequantityofaparticulartypeofkitthatcanbedisassembled,butthedisassemblycapacityisnotsharedamongdierenttypesofkits. Toensureuninterruptedsparepartkitavailability,theOEMhastocoordinatereplenishment(i.e.,assembly)anddisposal(i.e.,disassembly)decisionsforthekits,whichyieldsthemulti-itemcapacitatedlotsizingproblemofinterestinthischapter.TheproblemsettingisshowninFigure 3-1 .Specically,giventheforecasteddemandsandreturnsformultipletypesofproducts(kits)aswellasthesharedreplenishmentandproduct-specicdisposalcapacityrestrictionsineachtimeperiod,ourobjectiveis 27
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Wenotethatwithoutlossofgeneralitytheleadtimesassociatedwithreplenishmentanddisposalactivitiescanassumedtobezero.Moreover,therearenospeculativemotivesthatencouragereplenishmentwiththepurposeofdisposalinthefuture.Theproblemparametersareasfollows: 28
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Wenotethatwithoutlossofgeneralitytheleadtimesassociatedwithreplenishmentanddisposalactivitiescanassumedtobezero.Moreover,therearenospeculativemotivesthatencouragereplenishmentwiththepurposeofdisposalinthefuture.Thedecisionvariablesareasfollows: TheRDPPwithsharedreplenishmentandproduct-specicdisposalcapacitiesisformulatedasminPXp=1TXt=1(Fmptympt+ptxmpt)+PXp=1TXt=1(Fdptydpt+ptxdpt)+PXp=1TXt=1"ptipt (3{2)xdptCdptydptp=1;:::;P;t=1;:::;T; (3{3)ip;t1+xmpt=D0pt+xdpt+iptp=1;:::;P;t=1;:::;T; (3{4)xmpt (3{5)xdpt (3{6) 29
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(3{7)xmpt;xdpt;ipt;0andintegerp=1;:::;P;t=1;:::;T; (3{8)ympt;ydpt2f0;1gp=1;:::;P;t=1;:::;T: Objectivefunction( 3{1 )minimizesthesumofthexedcostofperformingreplenishmentanddisposalsetupsaswellasthelinearreplenishment,disposal,andinventoryholdingcoststhroughouttheplanninghorizon.Constraintset( 3{2 )ensuresthatthetotalreplenishmentcapacityrequiredforalltheproductsdoesnotexceedtheavailablereplenishmentcapacityinperiodt.Constraintset( 3{3 )ensuresthatthedisposalcapacityrequiredforeachproductpdoesnotexceedtheavailabledisposalcapacityforproductpinperiodt.Constraintset( 3{4 )istheinventorybalanceequationforeachproductpineachperiodt,whichensuresthatforeachproductp,thesumofthequantityininventorycarriedintoperiodtandthequantityproducedinperiodtisequaltothesumofthenetdemandinperiodt,thequantitydisposedinperiodt,andthequantityininventorycarriedtoperiodt+1.Constraintset( 3{5 )ensuresthatareplenishmentsetupisperformedforproductpinperiodtifthequantityofproductpproducedinperiodtispositive;thisquantityshouldnotexceedthecumulativedemanduntiltheendoftheplanninghorizon.Similarly,constraintset( 3{6 )ensuresthatadisposalsetupisperformedforproductpinperiodtifthequantityofproductpdisposedinperiodtispositive;thisquantityshouldnotexceedthecumulativereturnsreceivedsincethebeginningoftheplanninghorizon.Constraintset( 3{7 )initializesthebeginningandendinginventorylevelsforeachproductp.Constraintsets( 3{8 )and( 3{9 )ensuretheintegralityofthedecisionvariables. 3{2 )and( 3{3 ).Lett0betheLagrangianmultiplierassociatedwiththereplenishmentcapacityconstraintinperiodtandbetheTdimensionalLagrangianvectorassociated 30
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3{2 ).Similarly,letpt0betheLagrangianmultiplierassociatedwiththedisposalcapacityconstraintforproductpinperiodtandbethePTdimensionalLagrangianmatrixassociatedwithconstraintset( 3{3 ).Givenand,theLagrangianproblemobtainedbyrelaxingconstraintsets( 3{2 )and( 3{3 )canbestatedasfollows:minPXp=1TXt=1(Fmptympt+(pt+pt)xmpt)+PXp=1TXt=1((FdptCdptpt)ydpt+(pt+pt)xdpt)+PXp=1TXt=1"ptiptTXt=1tCmt subjectto( 3{4 ){( 3{9 ). Foragivenand,theLagrangianproblem( 3{10 )canbesolvedecientlytoobtainalowerbound.Moreover,usingthesolutiontotheLagrangianproblem,asmoothingheuristiccanbeutilizedtoidentifyasolutionthatdoesnotviolatereplenishmentanddisposalcapacityrestrictionswhenndinganupperbound.Usingtheselowerandupperbounds,thevaluesofthemultipliersandcanbeupdatedaccordinglyusingsubgradientoptimization. Letkbetheiterationcounter.Supposethattheterminationcriteriaaredenedby,thetolerancegapbetweenthebestlowerandupperbound,andkmax,themaximumnumberofiterations.Also,UBkandLBkdenotetheupperandlowerbounds,respectively,ontheobjectivefunctionvalueoftheLagrangianproblematiterationk.Similarly,UBandLBdenotetheincumbentupperandlowerbounds,respectively,ontheobjectivefunctionvalueoftheLagrangianproblem.Then,thegeneralLRprocedureweusecanbestatedasfollows: 1. Initialize,kmaxandtheLagrangianmultipliers.Setk=1;UB=UBk=1;andLB=LBk=. 2. While((LB(1)UB)and(kkmax)) (a) FormulateandsolvetheLagrangiandualproblemtoobtainLBk. 31
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If(LBk>LB)thensetLB=LBk. (c) ApplythesmoothingheuristictondafeasiblesolutiontoobtainUBk. (d) If(UBk
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3{11 )canbenegativesincept0.Weobservethatintheoptimalsolutiontothisproblem,if(FdptCdptpt)<0,thenydpt=1.Then,wecanmodifytheobjectivefunction( 3{11 )by(i)replacingFdptwithF0dptwhereF0dpt=maxf(FdptCdptpt);0gfort=1;:::;T;and(ii)addingaconstanttermPTt=1minfFdptCdptpt;0g.Consequently,foragivenand,themodiedsingle-productLagrangiansubproblemisgivenby:minTXt=1(Fmptympt+(pt+pt)xmpt)+TXt=1(F0dptydpt+(pt+pt)xdpt)+TXt=1"ptipt+TXt=1minfFdptCdptpt;0g subjectto( 3{12 )-( 3{17 ). Thismodiedsingle-productLagrangiansubproblemresemblesthesingle-itemuncapacitatedlot-sizingproblemwithreturns(SULSPwR)(in BeltranandKrass ( 2002 ))withonedierence:thereisanitedisposalcostassociatedwithanyexcessinventoryattheendoftheplanninghorizoninthesubproblemabove,whileanyexcessinventoryattheendoftheplanninghorizonhasnovaluein BeltranandKrass ( 2002 ).Thisdierencedoesnotinuencethestructuralpropertiesoftheoptimalsolutionpresentedin BeltranandKrass ( 2002 )butitdoesrequireamodiedDPalgorithm.Next,weproceedwithareviewofthepropertiesoftheoptimalsolutiontotheSULSPwRpresentedin BeltranandKrass ( 2002 ),andthenweextendtheDPalgorithmin BeltranandKrass ( 2002 )toobtainanoptimalsolutiontoourmodiedsingle-productLagrangiansubproblem.Asin BeltranandKrass ( 2002 ),intheremainderofthispaper,aperiodintheplanninghorizoniscalledadecisionperiod,ifareplenishmentordisposalsetuphastobeperformed.
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Asaresult,theplanninghorizoncanbedividedintoregenerationintervalseachofwhichhas Moreover,underamodiedzero-inventorypolicy,itcanbeshownthateachreplenishmentordisposalsetupshouldsatisfythenetdemandofanintegernumberofperiods,i.e.,amodiedzero-inventorypolicysatisestheexactrequirementsproperty. 3-2 andFigure 3-3 ,respectively.Letisaparameter. 34
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Letptdenotethemaximumnumberofperiodsbywhichadecisionperiod(eitherwithareplenishmentordisposalsetup)forproductpcanbepostponedbeyondperiodt.Thatis,wehavept=maxfw:twTandSt0pt0;8tt0wg.Letgp(t)denotetheoptimalvaluefunctionforproductpinperiodt,whichisdenedastheminimumcostofsatisfyingdemandandprocessingreturnfromperiodtuntiltheendoftheplanninghorizonwithzerostartinginventoryforproductp.TherecurrencerelationoftheDPalgorithmcanbestatedasgp(t)=min8>>>>>>>><>>>>>>>>:minv2ft;:::;ptgv1P`=t("pl(S`pt))+(Ipv)minv2ft;:::;ptg8><>:v1P`=t("pl(S`pt))+8><>:0pt=TandSTpt=0(I
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LetLBkpdenotetheoptimalsolutiontothesingle-productLagrangiansubproblemforproductpatiterationkobtainedusingtheDPalgorithmdescribedabove.Then,thelowerboundfortheLagrangianproblematiterationk,LBk,isgivenbyLBk=PXp=1LBkpTXt=1tCmt 3{10 )canbeidentiedecientlywiththeDPapproachdescribedinSection 3.3.1 .Asthisoptimalsolutionmayviolatethereplenishmentanddisposalcapacityrestrictions,wedevelopasmoothingheuristictoremovesuchinfeasibilities.Thisapproachissimilarto,yet,extends,theonepresentedin Trigeiroetal. ( 1989 ).Oursmoothingheuristichasthreephasesandisguaranteedtoremoveallreplenishmentanddisposalcapacityviolationsforafeasibleinstanceoftheproblem.Therstphaseoftheapproachisaforwardpass(i.e.,startsfromthebeginningoftheplanninghorizonandproceedstowardsthelastperiod)thatadjustsdisposalquantitiestoeliminatedisposalcapacityviolations.Thesecondphaseadjustsreplenishmentquantitiestoeliminatereplenishmentcapacityviolationsusingabackwardpass(i.e.,startsfromtheendoftheplanninghorizonandproceedstowardstherstperiod)andanadditionalforwardpassifneeded.Thethird,andlast,phaseeliminatesanyunnecessarysetupsand/orinventoryfortheproductsinthreesteps. 36
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1. 2. (a) Eliminatereplenishmentcapacityviolationsforperiodst>1:Webeginfromthelatestperiodtwherethereplenishmentcapacityisviolatedandconsideralltheproductsthatareproducedinthisperiod.Ifthereplenishmentquantityassociatedwithaproductissmallerthantheamountofcapacityviolation,thenweconsidertwooptionstoshifttheentirereplenishmentquantityto(i)themostrecentperiodwithareplenishmentsetupor(ii)thepreviousperiod.Ifthereplenishmentquantityassociatedwithaproductislargerthantheamountofcapacityviolation,thenweconsiderthreeoptionstoshift(i)theminimumamountpossibletoreduce/eliminatethecapacityviolationtothepreviousperiod;(ii)theentirereplenishmentquantitytothepreviousperiodor(iii)theentirereplenishmentquantitytothemostrecentperiodwithareplenishmentsetup.Foreachoption,wedeterminethemarginalcostofeliminatingthereplenishmentcapacityviolation,whichisobtainedbydividingthetotalcostassociatedwithimplementingtheoptionbythetotalamountthatcanbeshiftedbytheoption.Theseoptionsareexecuteduntilthereisnomorereplenishmentcapacityviolationinthecurrentperiodt.Then,theprocedure 37
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(b) Eliminatereplenishmentcapacityviolationinperiodt=1:Weconsideralltheproductsthatareproducedintherstperiod.Foralltheproducts,weconsidertheoptionofshiftingtheminimumamountpossibletoeliminatetheproductcapacityviolationortheentirereplenishmentquantitytothenextperiod.Wedeterminethemarginalcostofeliminatingthereplenishmentcapacityviolationforeachproduct.Theviolationiseliminatedbyshiftingasmanyproductsasnecessarytothenextperiod(t+1),startingwiththeproductthathasthelowestmarginalcost.Then,theprocedureisrepeatedforperiod(t+1).Attheendofthisstepwhenthecurrentperiodhasnoreplenishmentcapacityviolation,allreplenishmentcapacityviolationsareguaranteedtobeeliminated. 3. (a) Eliminateanyunnecessarysetups:Givenaproduct,westartfromperiodt=T.Ifbothreplenishmentanddisposalsetupsareperformedfortheproductinthecurrentperiod,thenthereplenishmentanddisposalquantitiesfortheproductcanbereducedbytheminimumofthesetwoquantities,i.e.,eectivelyeliminatingthethesetupthatisassociatedwiththesmallerbatchsize.Attheendofthisstepwhenthecurrentperiodist=1,unnecessarysetupshavebeeneliminatedfortheproduct.Theprocedureisrepeatedforalltheproducts. (b) Adjustreplenishmentquantitiestoeliminateanyunnecessaryinventory:Givenaproduct,westartfromperiodt=T.Ifthereisanyunusedreplenishmentcapacity,thentheproductswithpositiveincominginventoryareconsidered.Foreachsuchproduct,thecostreductionassociatedwithincreasingthereplenishmentquantityintanddecreasingthereplenishmentquantityint0(wheret0isthemostrecentreplenishmentsetupperiod,andendinginventoryispositiveforallperiodsfromt0uptoandincluding(t1)fortheproduct)isidentied.Ifthecostreductionispositive,thentheshiftisperformed,i.e.,themaximumamountofreplenishmentforproductpthatcanbemovedfromperiodt0toperiodtismoved.Theprocedurecontinuesuntileithertherearenootherproductswithpositivesavingsorallthereplenishmentcapacityisusedinperiodt.Then,theprocedureisrepeatedforperiod(t1).Attheendofthisstepwhenthecurrentperiodist=1,unnecessaryinventoryduetoreplenishmenthasbeeneliminatedfortheproduct.Thisstepisrepeatedforalltheproducts. (c) Adjustdisposalquantitiestoeliminateanyunnecessaryinventory:Foragivenproduct,westartfromperiodt=1.Ifthereisanyunuseddisposalcapacityandtheinventoryispositiveinallperiodsuptothenextdisposalsetupperiod,t0,thenthecostreductionassociatedwithdecreasingthedisposal 38
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(3{19) whereakisthestepsizeatiterationk.Weupdatethestepsizebyhalvingitsvaluewhenthereisnoimprovementinthelowerboundinaxednumberofiterations. 39
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1. (a) GenerateintegerDptforp=1;:::;Pandt=1;:::;Tuniformlyfrom[(1'D)D;(1+'D)D]. (b) Obtaintherealizedvalueofexpecteddemandforp=1;:::;Pusing^Dp=PTt=1Dpt=T. 2. (a) Settheexpectedvalueofthereturnsperperiodforp=1;:::;PusingRp=RD^Dp. (b) GenerateintegerRptforp=1;:::;Pandt=1;:::;Tuniformlyfrom[(1'R)Rp;(1+'R)Rp]. (c) LetD0+ptandD0ptdenotethepositiveandnegativenetdemandrespectively,i.e.,D0+pt=maxf0;D0ptg,andD0pt=maxf0;D0ptg,forp=1;:::;Pandt=1;:::;T.Similarly,letT+(T)denotethenumberofperiodswherenetdemandis 40
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3. (a) Setthereplenishmentcapacityfort=1;:::;TusingCmt= (b) Setthedisposalcapacityforp=1;:::;Pandt=1;:::;TusingCdpt=bD0minfRpt;D0ptgc. 4. (a) Settheexpectedvalueofreplenishmentsetupcostforp=1;:::;PusingFmp="pt(minf^D0+p;^Dpg)TBR2=2. (b) GenerateFmptforp=1;:::;Pandt=1;:::;Tuniformlyfrom[(1'Fm)Fmp;(1+'Fm)Fmp]. 5. (a) Settheexpectedvalueofdisposalsetupcostforp=1;:::;PusingFdp="pt(minf^D0p;^Rpg)TBD2=2. (b) GenerateFdptforp=1;:::;Pandt=1;:::;Tuniformlyfrom[(1'Fd)Fdp;(1+'Fd)Fdp]. Todevelopasetofrandominstancesofrealisticsize,wevarythenumberofproductsPandthelengthoftheplanninghorizon,T.Specically,weconsiderfourlevelsforP(20,30,40,and50)andthreelevelsforT(12,24,and36).Consequently,wehave12classesoftestinstances.Moreover,weconsidervelevelsforthecorrelationbetweentotaldemand 41
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3-1 .Foreach(P;T)weconsider80dierentproblemsettings,andforeachsettingwegenerate10randomtestinstances.Therefore,wegenerateandsolve9,600instancesintotal. MillarandYang ( 1994 )).Inourexperiments,theLRapproachterminateswhenthegapbetweenthelowerandupperboundislessthan0.5%oramaximumof500iterationsareexecuted.Thestepsizeissetinitiallyto2andhalvedifthelowerboundvalueisnotimprovedinveconsecutiveiterations. WesummarizetheoptimalitygapsobtainedbytheLRapproachanditsCPUrequirementsinTable 3-2 .Aswesolveatotalof9,600testinstances,eachcellinthistablecorrespondstotheaverageormaximumvaluerepresenting800instancesofthesamesize.Wenotethattheobservedaverageoptimalitygapoverallinstancesis0.25percent,whereasthemaximumisslightlyabove2percent.Wealsoobservethatforagivennumberofproducts,theaverageandmaximumvaluesoftheoptimalitygapsbecomesmallerasthelengthoftheplanninghorizonincreases.Foragivenplanninghorizonlength,theaverageandmaximumvaluesoftheoptimalitygapsarenotsignicantlyinuencedbyanincreaseinthenumberofproducts.Whenthenumberofproductsislargeand/ortheplanninghorizonisrelativelylong,(i)theinuenceofa`misplaced'setupontheoverallobjectivefunctionvaluedecreasesand(ii)thesmoothingheuristic 42
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Moreover,weobservethattheLRapproachiscomputationallyecientastheobservedaverageCPUrequirementoverallinstancesislessthanonesecond,whereasthemaximumislessthan22seconds. InTables 4-3 4-4 ,and 3-5 ,weprovideadetailedanalysisoftheinuenceofthedatacharacteristicsontheperformanceoftheLRapproach. Notsurprisingly,weobservethatthequalityoftheoptimalitygapsdeterioratesas(i)theTBRandTBDvaluesincreaseand/or(ii)thereplenishmentand/ordisposalcapacitiesbecomemorerestraining.ItisworthwhiletonotethattheinuenceofanincreaseinTBR(and/oradecreaseinreplenishmentcapacity)appearstobemoreinuentialthananincreaseinTBD(and/oradecreaseindisposalcapacity),whichcanbeattributedtoproblemcharacteristics.Thatis,asthereplenishmentcapacityissharedamongmultipleproducttypes,anincreaseinTBR(oradecreaseinreplenishmentcapacity)inuencesreplenishmentdecisionsformultipleproductssimultaneously,anda`mislocated'replenishmentsetupdecisioninuencestheobjectivefunctionvaluemorethana`mislocated'disposalsetupdecision.Inaddition,weobservethatthecorrelationbetweenreturnsanddemandhasaslightinuenceonthequalityofthegapsobtainedbytheLRapproach. AlthoughtheresultsinTable 3-2 provideempiricalevidencefortheeectivenessoftheproposedLRapproachtoaddresstheRDPP,aninterestingquestionthatremainsunansweredistheeectivenessoftheB&Capproachforobtainingsolutionstotheproblem.Tothisend,weconductedapreliminaryexperimentwherewesolvedaninstanceusingtheLRapproachandrecordedtheCPUtimerequired.Then,settingthetimelimittothisvalue,weattemptedtosolvethesameinstancewithCPLEX.Fornumeroustestinstances,weobservedtheoptimalitygapstobequitelarge.Therefore,weconductedan 43
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3-2 )andthestoppinggapto0.1percent.WesummarizetheresultsofthisexperimentinTable 3-6 Again,eachcellinthistablerepresentstheaverageormaximumvalueof800observationsfortestinstancesofthesamesize.Wenotethattheaverageoptimalitygapoverallinstancesisaround8percentwhereasthemaximumisaround34percent.InTables 3-7 3-8 ,and 3-9 ,weprovideadetailedanalysisoftheinuenceofdatacharacteristicsontheperformanceofCPLEX.Again,weobservethatthequalityofoptimalitygapsdeterioratesas(i)theTBRandTBDvaluesincreaseand/or(ii)thereplenishmentand/ordisposalcapacitiesbecomemorerestraining.WealsoobservethatthequalityoftheoptimalitygapsismoresensitivetochangesintheTBRandTBDvaluesthantothoseinthetightnessofthereplenishmentand/ordisposalcapacities. TotestwhethertheperformanceofCPLEXwouldimproveifthetimelimitwereincreasedandtheoptimalitygapwerereduced,weconductedafollow-upexperimenttoconsiderinstanceswithT=12only.Inparticular,weincreasedthetimelimitto300seconds(i.e.,aten-foldincrease)andincreasedthestoppinggapto0.5percent(i.e.,ave-foldincrease).InTable 3-10 below,wesummarizetheresultsfromthisexperiment;wereportthenumberoftestinstancesthatcannotbesolvedbyCPLEX(i.e.,theoptimalitygapcannotbereducedto0.5percentin300seconds)aswellastheaverageandmaximumoptimalitygapsuponterminationfortheunsolvedinstances.Forexample,weobservethatonly183(outof800)testinstanceswithsize(20,12)weresolvedbyCPLEXwithin30secondswithastoppinggapof0.1percent.Increasingthetimelimitto300secondsandreducingtheoptimalitygapto0.5percentallowedCPLEXtosolve9additionalinstances(i.e.,aonepercentincrease).Keepingthenumberofproductsconstantandincreasingthenumberoftimeperiods,weobservedthatincreasingthetimelimitandthestoppinggapdidnothelpCPLEXobtainsolutionsforadditionaltestinstances. 44
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ThemathematicalformulationinSection 3.2 canbemodiedforthisvariantoftheRDPPbyreplacingconstraintset( 3{3 )withPXp=1pxdptCdtt=1;:::;T whichensuresthatthetotaldisposalcapacityrequiredforallproductsdoesnotexceedtheavailabledisposalcapacityinperiodt. TheproposedLRapproachcaneasilybemodiedaswell.Inparticular,theapproachisnowbasedontherelaxationofconstraintsets( 3{2 )and( 3{21 ).Let=ftgdenotethevectorofLagrangemultiplierst0;t=1;:::;T,eachofwhichisassociatedwiththedisposalcapacityconstraintinperiodt=1;:::;T.TheLagrangianproblemobtainedbyrelaxingconstraintsets( 3{2 )and( 3{21 )forgivenandcanbestatedasminPXp=1TXt=1(Fmptympt+(pt+pt)xmpt)+PXp=1TXt=1(Fdptydpt+(pt+pt)xdpt)+PXp=1TXt=1"ptiptTXt=1(tCmt+tCdt) (3{22)subjectto( 3{4 ){( 3{9 ). 45
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3{22 )isconstant,andtheproblem( 3{22 )isseparableintoPsubproblems,eachofwhichcorrespondstoaproductp.Thesingle-productLagrangiansubproblem(forproductp)nowcanbestatedasminTXt=1(Fmptympt+(pt+pt)xmpt)+TXt=1(Fdptydpt+(pt+pt)xdpt)+TXt=1"ptipt 3{12 ){( 3{17 ). EachofthesesubproblemscanalsobesolvedecientlywiththeDPalgorithmpresentedinSection 3.3.1 ,andalowerboundcanbeobtainedeasily.Toobtaintheupperbounds,wemodifyPhase1ofthesmoothingheuristicpresentedinSection 3.3.2 asdiscussednext.Eliminatedisposalcapacityviolations: (a) Weclassifytheproductsthataredisposedofinthisperiodintotwosets.Theproductsforwhichthedisposalquantityissmallerthanorequaltotheamountofdisposalcapacityviolationareincludedintherstset.Thesecondsetcomprisesoftheproductsforwhichthedisposalquantityislargerthattheamountofdisposalcapacityviolation.Fortheproductsintherstset,weconsidertoshifttheentiredisposalquantityfortheproduct(i)tothenextfutureperiodwithanexistingdisposalsetup,or(ii)tothenextperiod(t+1).Inevaluatingthetotalcostchangeassociatedwiththeseoptions,weaccountforchangesinthetotalunitdisposalandinventoryholdingcosts.Moreover,forthesecondoption,wealsotakeintoaccountthexedcostiftheshiftrequirestheschedulingofanadditionaldisposalsetupinperiod(t+1).Similarly,fortheproductsinthesecondset,weconsidertoshift(i)theamountofcapacityviolationtothenextperiod(t+1),or(ii)theentiredisposalquantitytothenextperiod(t+1),or(iii)theentiredisposalquantitytothenextfutureperiodwithanexistingdisposalsetup.Again,inevaluatingtheseoptions,weaccountforthechangesinthetotalunitdisposalandinventoryholdingcosts.Moreover,fortherstandsecondoptions,wetakeintoaccountthexeddisposalsetupcostiftheshiftrequirestheschedulingofanadditionaldisposalsetupinperiod(t+1). Foreachproduct,wepicktheoptionthatyieldstheminimummarginalcost.Thenwepicktheproduct(and,hence,theoption)leadingtothebestmarginalcostandshifttheamountimpliedbytheoptionfortheproduct.Werepeattheprocedure 46
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(b) Weconsideralltheproductsthataredisposedofinthecurrentperiodandclassifythemintotwosetsasbefore.Foreachproductintherstset(whosedisposalquantityissmallerthanorequaltotheamountofdisposalcapacityviolation),weconsidertoshifttheentiredisposalquantityoftheproducttothepreviousperiod.Foreachproductinthesecondset(whosedisposalquantityislargerthantheamountofdisposalcapacityviolation)weconsidertoshift(i)theamountofcapacityviolationtothepreviousperiod,or(ii)theentirereplenishmentquantitytothepreviousperiod.Inevaluatingtheseoptions,weaccountforthechangesinthetotalunitdisposalandinventoryholdingcostsaswellastheadditionaldisposalsetupcost,ifneeded.Moreover,inidentifyingtheamountthatcanbeshiftedbytheseoptions,webearinmindthatreturnsthatarenotreceivedcannotbedisposedof.Themarginalcostofeachoptionisobtainedbydividingthetotalcostchangeassociatedwithimplementingtheoptionbytheamountthatcanbeshiftedbytheoption. Foreachproduct,wepicktheoptionthatyieldstheminimummarginalcostwhichisobtainedbydividingthetotalcostchangeassociatedwithimplementingtheoptionbytheamountthatcanbeshiftedbytheoption.Then,wepicktheproduct(and,hence,theassociatedoption)leadingtothebestmarginalcostandshifttheamountimpliedbytheoptionfortheproduct.Werepeattheprocedure(startingwithclassifyingtheproductsintotwosets)untilthereisnodisposalcapacityviolationinthecurrentperiodt. Ifthepreviousperiodhasnodisposalcapacityviolation,alldisposalcapacityviolationsareguaranteedtobeeliminated.Otherwise,theroutineisrepeatedforthepreviousperiod. Consideringthelowerandupperboundvaluesateachiterationk,thenewsetofmultiplierscanbeupdatedusing TotestthecomputationaleciencyofourproposedapproachforthisvariantoftheRDPP,weobtainedsetoftestinstancesusingtherandominstancegenerationschemewedescribeinSection 4.4.1 bymodifyingthedisposalcapacityforeachperiod. 47
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3-11 Again,wesolveatotalof9,600instances.AsinTable 3-2 ,eachcellinthistablerepresentstheaverageormaximumvalueof800observationsfortestinstancesofthesamesize.Weobservethattheaverageoptimalitygapoverallinstancesis0.22percent,whereasthemaximumislessthan2percent.Therefore,withtheproposedmodications,theLRapproachcaneasilyberevisedtoobtainhigh-qualitysolutionsforthisvariantoftheRDPPaswell.Uponadetailedinvestigationofthecomputationalresults,itispossibletomakethesameobservationsdiscussedinSection 4.4.2 inregardstotheinuenceofthedatacharacteristicsontheperformanceoftheLRapproachforthisvariantoftheRDPP. TotesttheeectivenessoftheB&Capproachinobtainingsolutionsforthisproblemvariant,wesolvedthesetestinstancesusingCPLEXwithatimelimitof30secondsandastoppinggapof0.1percentasbefore.WesummarizetheresultsofthisexperimentinTable 3-12 .Wenotethattheaverageoptimalitygapoverallinstancesisaround8percentwhereasthemaximumisaround37percent.InTable 3-13 below,wesummarizetheresultsfromanexperimentwhereweincreasedthetimelimitto300secondsandincreasedthestoppinggapto0.5percent. WeobservethatCPLEXisabletoobtaintheoptimalsolutionformoretestinstanceswithsize(20,12)forthisvariantoftheRDPP.However,thenumberofinstancesforwhichCPLEXcandetermineandverifytheoptimalsolutiondecreasesasthenumberofproductsincreases.Therefore,weconcludethatCPLEXisnoteectiveforaddressingthisvariantoftheRDPPeither. 48
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AnimmediateextensionofourworkthatwouldbeofpracticalrelevanceistoconsiderageneralizationoftheRDPPwhere(i)thereplenishmentand/ordisposalsetupcostsdependonthespecicproductsincludedinthesetupand/or(ii)setuptimesareexplicitlymodeled.Moreover,ourmodelscanbeextendedbyconsideringdemandbacklogging.Also,theLRapproachpresentedinthispapercanbeextendedtosettingswherecertaincapacitatedresourcesaresharedbythereplenishmentanddisposalactivities.Anotherpossibleextensionistodeveloprolling-horizonheuristics(possiblywithprovablebounds)fortheRDPP. 49
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Problemsetting. Pseudo-codeoftheIdenticationofSetIpt. Pseudo-codeoftheIdenticationofSetI 50
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Parametervaluesusedincomputationalexperiments. Level(s) Value(s) Constant 100 Variabilityofdemand('D) Constant 0.5 Variabilityofreturns('R) Constant 0.5 Variabilityofxedreplenishmentsetupcost('Fm) Constant 0.2 Variabilityofxeddisposalsetupcost('Fd) Constant 0.2 Unitinventoryholdingcost("pt) Constant 1 Unitreplenishmentcost(pt) Constant 3 Unitdisposalcost(pt) Cost 1 Unitcapacityconsumption(p) Constant 1 Timebetweenreplenishmentsetups(TBR) Low [1,3] High [3,5] Timebetweendisposalsetups(TBD) Low [1,3] High [3,5] Replenishmentcapacitytightness(D0+) Low 1.5 High 2.5 Disposalcapacitytightness(D0) Low 1.5 High 2.5 Return/Demandratio(RD) Low 0.15 High 0.75 Table3-2. PerformanceoftheLRApproach:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). CPURequirement InstanceSize Avr.Gap Max.Gap Avr.Time Max.Time (P;T) (%) (%) (sec.) (sec.) (20,12) 0.30 1.74 0.08 0.55 (30,12) 0.28 2.34 0.07 0.78 (40,12) 0.28 1.41 0.07 1.05 (50,12) 0.28 2.02 0.06 1.25 (20,24) 0.25 1.43 0.15 2.94 (30,24) 0.24 0.91 0.11 4.30 (40,24) 0.24 0.81 0.16 5.83 (50,24) 0.24 0.83 0.17 7.36 (20,36) 0.22 0.68 0.22 8.73 (30,36) 0.22 0.91 0.28 13.19 (40,36) 0.22 0.76 0.27 17.50 (50,36) 0.21 0.62 0.19 21.80 Overall 0.25 2.34 0.15 21.80
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EectofTimeBetweenReplenishment(TBR)andTimeBetweenDisposal(TBD)SetupsontheAverageandMaximumOptimalityGaps(%)oftheLRApproach. Max. (TBR,TBD) Gap(%) Gap(%) (U[1,3],U[1,3]) 0.08 0.86 (U[1,3],U[3,5]) 0.13 2.34 (U[3,5],U[1,3]) 0.37 1.34 (U[3,5],U[3,5]) 0.40 2.02 EectofReplenishmentandDisposalCapacityTightnessontheAverageandMaximumOptimalityGaps(%)oftheLRApproach. Max. ('R;'D) Gap(%) Gap(%) (1.5,1.5) 0.33 2.34 (1.5,2.5) 0.30 1.74 (2.5,1.5) 0.19 2.02 (2.5,2.5) 0.17 1.38 EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)oftheLRApproach. Max. Gap(%) 0.15 0.28 1.64 0.30 0.28 1.74 0.45 0.26 1.51 0.60 0.20 0.94 0.75 0.22 2.34 PerformanceofCPLEX(withaTimeLimitofthe30SecondsandaStoppingGapof0.1Percent):AverageandMaximumOptimalityGaps(%). Avr.Gap Max.Gap (P;T) (%) (%) (20,12) 3.85 14.88 (30,12) 4.74 15.85 (40,12) 5.67 18.50 (50,12) 6.20 18.20 (20,24) 7.47 23.47 (30,24) 8.33 27.06 (40,24) 9.13 28.44 (50,24) 10.03 34.80 (20,36) 9.00 28.29 (30,36) 10.61 34.13 (40,36) 11.00 32.01 (50,36) 11.48 32.87 Overall 8.13 34.80
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EectofTimeBetweenReplenishment(TBR)andTimeBetweenDisposal(TBD)SetupsontheAverageandMaximumOptimalityGaps(%)ofCPLEX. Max. (TBR,TBD) Gap(%) Gap(%) (U[1,3],U[1,3]) 4.53 21.48 (U[1,3],U[3,5]) 4.62 22.56 (U[3,5],U[1,3]) 11.68 34.13 (U[3,5],U[3,5]) 11.69 34.80 EectofReplenishmentandDisposalCapacityTightnessontheAverageandMaximumOptimalityGaps(%)ofCPLEX. Max. (R;D) Gap(%) Gap(%) (1.5,1.5) 8.31 32.87 (1.5,2.5) 8.30 34.13 (2.5,1.5) 7.96 34.13 (2.5,2.5) 7.93 34.80 EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)ofCPLEX. Max. Gap(%) 0.15 3.04 12.16 0.30 5.62 34.13 0.45 8.45 23.74 0.60 10.95 34.80 0.75 12.57 32.87 InuenceofIncreasingtheTimeLimitandtheStoppingGaponthePerformanceofCPLEX:NumberofUnsolvedInstances,AverageandMaximumOptimalityGaps(%). 300Secondsand0.5%StoppingGap No.ofInstances GapUponTermination No.ofInstances GapUponTermination Unsolved Avr.Gap Max.Gap Unsolved Avr.Gap Max.Gap (P;T) (outof800) (%) (%) (outof800) (%) (%) (20,12) 617 4.33 14.88 608 3.85 13.16 (20,24) 774 7.57 23.47 738 7.22 21.34 (20,36) 800 9.00 28.29 800 8.40 26.06
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PerformanceoftheLRApproachfortheRDPPwithSharedReplenishmentandDisposalCapacities:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). CPURequirement InstanceSize Avr.Gap Max.Gap Avr.Time Max.Time (P;T) (%) (%) (sec.) (sec.) (20,12) 0.26 1.74 0.07 0.70 (30,12) 0.24 1.31 0.07 0.84 (40,12) 0.24 0.91 0.07 1.08 (50,12) 0.24 0.80 0.08 1.31 (20,24) 0.23 1.21 0.21 3.17 (30,24) 0.22 0.66 0.20 4.97 (40,24) 0.22 0.60 0.24 6.50 (50,24) 0.22 0.50 0.22 4.34 (20,36) 0.28 0.81 0.40 9.48 (30,36) 0.20 0.64 0.35 14.77 (40,36) 0.20 0.50 0.38 10.83 (50,36) 0.19 0.50 0.34 9.84 Overall 0.22 1.74 0.22 14.77 PerformanceofCPLEXfortheRDPPwithSharedReplenishmentandDisposalCapacities(withaTimeLimitofthe30secondsandaStoppingGapof0.1Percent):AverageandMaximumOptimalityGaps(%). Avr.Gap Max.Gap (P;T) (%) (%) (20,12) 3.89 14.62 (30,12) 4.78 16.37 (40,12) 5.72 18.39 (50,12) 6.25 18.84 (20,24) 7.51 23.68 (30,24) 8.38 26.92 (40,24) 9.20 28.01 (50,24) 10.13 36.97 (20,36) 8.99 28.45 (30,36) 10.55 32.54 (40,36) 11.13 32.20 (50,36) 11.50 28.91 Overall 8.17 36.97
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InuenceofIncreasingtheTimeLimitandtheStoppingGaponthePerformanceofCPLEXfortheRDPPwithSharedReplenishmentandDisposalCapacities:NumberofUnsolvedInstances,AverageandMaximumOptimalityGaps(%). 300Secondsand0.5%StoppingGap No.ofInstances GapUponTermination No.ofInstances GapUponTermination Unsolved Avr.Gap Max.Gap Unsolved Avr.Gap Max.Gap (P;T) (outof800) (%) (%) (outof800) (%) (%) (20,12) 616 4.38 14.62 579 4.04 13.76 (20,24) 774 7.61 23.68 738 7.26 21.56 (20,36) 800 8.99 28.45 800 8.42 26.34
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Sinceautomotivepartsremanufacturinghasbeenapopularpracticesincetheinceptionoftheautomotiveindustry,remanufacturableautomotivepartsarepurchasedandsoldascommodities.Therefore,insomecases,theretailersmaysellsomeofthepartstoindependentremanufacturersandreturnonlyafractionofthepartstotheOEM,whichlimitsthesupplyofremanufacturableparts.Insomeothercases,thequantityofremanufacturablepartsreceivedmaybemorethanthedemandforremanufacturedparts,inwhichcasetheOEMdisposesoftheremanufacturablepartsbysellingthemtobulkmetalrecyclersformaterialrecovery.Inordertoensureuninterruptedreplacement 56
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Uponacloserexaminationofautomotivepartsremanufacturingpractices,weobservethatthereareseveralfactorsthatinuencethecomplexityoftheinventoryandproductionplanningproblemsencounteredinthiscontext.Afactorthatinuencesthecomplexityoftheunderlyingproblempertainstothemanufacturingandremanufacturingcapacities.Thereareusuallyprocessingcapacityrestrictionsthatlimitthequantityofnewandremanufactureditemsthatcanbeacquiredfromnewpartandremanufacturingsuppliers.Someautomotiveparts(e.g.,enginesandtransmissions)aremanufacturedbytheOEManddierentpartssharethesamemanufacturingcapacity.Someotherparts(e.g.,alternators,distributors)arepurchasedfrompartsuppliersandhencetheydonotsharethesamemanufacturingcapacity.Remanufacturingofusedpartsareusuallyoutsourcedtoremanufacturingsuppliers.Typically,remanufacturingofsometypesofparts(e.g.,transmissionsortorqueconverters)areoutsourcedtoasingleremanufacturingsuppliertotakeadvantageoflearning-by-doingeectsandtobeabletonegotiatealowerunitcostduetoconsolidatedvolume.Therefore,somepartssharetheavailableremanufacturingcapacity.Forotherparts(e.g.,electroniccontrolunits,distributors),however,remanufacturingcanbeoutsourcedtodierentremanufacturingsuppliers,eachofwhichhashisownprocessingcapacity.Anadditionalrelatedfactorconcernstheavailabledisposalcapacity.Theremaybetransportationand/orrecyclingcapacityconsiderationsthatlimitthequantityofusedpartsthatcanbedisposedof. Inthischapter,weconcentrateontheManufacturing,RemanufacturingandDisposalPlanningProblem(MRDPP)withmultipleproducttypestoaddresstheinventoryandproductionplanningproblemforremanufacturableautomotiveparts.Morespecically,MRDPPisconcernedwithdeterminingwhenandhowmuchtomanufacture,remanufacture,anddisposeofeachtypeofproduct.Weconsidertheproblemina 57
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whileconsideringthemanufacturing,remanufacturing,anddisposalcapacitiesexplicitly. Theremainderofthischapterisorganizedasfollows.Section 4.2 presentsthemodellingassumptionsandamathematicalformulationfortheMRDPP.SolutionapproachesaredescribedinSection 4.3 .ResultsfromcomputationalexperimentsarepresentedandanalyzedinSection 4.4 .Finally,conclusionsandfutureresearchdirectionsaresummarizedinSection 4.5 4-1 .Therearetwostockpointsforeachproducttype,onefortheusediteminventory(UII),andonefortheserviceable(manufacturedorremanufactured)itemsinventory(MRII).UIIaccumulatesasreturnsofproductsarereceived.ExcessreturnsmaybedisposedoffromtheUII.ManufacturingandremanufacturingactivitiesreplenishtheMRII,andthedemandfortheproducttypeissatisedfromtheMRII.Aswenotedearlier,wefocusondiscretetimedynamicdemandandreturnmodelstoaddresstheinventoryandproductionplanningproblemencounteredinthissetting.Ourobjectiveistominimizethesumofthe 58
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Inourwork,wemakethefollowingmodelingassumptions: 1. Demandforallproducttypesthroughouttheplanninghorizonisknown.Demandforeachproducttypecanbesatisedbyusingremanufacturinguseditemsorbymanufacturingnewitems. 2. Returnsforallproducttypesthroughouttheplanninghorizonisalsoknown.ReturnsforeachproducttypecanberemanufacturedtoreplenishtheMRII(i.e.,satisfythedemand)orbedisposedof. 3. Manufacturingactivityincursaxedsetupandlinearvariablecosts.Theavailablemanufacturingcapacityissharedamongmultipleproducttypes. 4. Remanufacturingactivityincursaxedsetupandlinearvariablecosts.Thereisaprocessingcapacityassociatedwiththeremanufacturingactivityforeachproducttype. 5. Disposalactivityincursaxedsetupandlinearvariablecosts.Thereisaprocessingcapacityassociatedwiththedisposalactivityforeachproducttype. 6. ItemsinMRII-pofproducttypepincuralinearinventoryholdingcost. 7. ItemsinUII-pofproducttypepincuralinearinventoryholdingcost. 8. Backloggingofdemandisnotallowed. Theproblemparametersaredenedasfollows: 59
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Wehavethefollowingdecisionvariables: 60
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AmathematicalprogrammingformulationforMRDPPisasfollows:minPXp=1TXt=1(Fmptympt+cmptxmpt+Frptyrpt+crptxrpt+Fdptydpt+cdptxdpt+kptjpt+hptipt) (4{1) subjecttoip;t1+xmpt+xrptDpt=iptp=1;:::;P;t=1;:::;T; (4{2)jp;t1+Rptxrptxdpt=jptp=1;:::;P;t=1;:::;T; (4{3)PXp=1pxmptKmtt=1;:::;T; (4{4)xrptKrptyrptp=1;:::;P;t=1;:::;T; (4{5)xdptKdptydptp=1;:::;P;t=1;:::;T; (4{6)xmpt (4{7)xrpt (4{8)xdpt (4{9)jp0;jpT;ip0;ipT=0p=1;:::;P; (4{10)xmpt;xrpt;xdpt;jptipt0andintegerp=1;:::;P;t=1;:::;T; (4{11) 61
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Objectivefunction( 4{1 )minimizesthesumofxedcostofperformingmanufacturinganddisposalsetupsaswellaslinearmanufacturing,disposal,andinventoryholding(inUIIandMRII)costsforallproductsthroughouttheplanninghorizon.Constraintset( 4{2 )givestheinventorybalanceequationsforMRIIforproductpineachperiodt,whichensuresthatthesumofthequantityininventorycarriedfromperiodt1toperiodt1,thequantitymanufacturedinperiodt,andthequantityremanufacturedinperiodtisequaltothesumofthedemandinperiodtandthequantityininventorycarriedfromperiodttoperiodt+1.Constraintset( 4{3 )istheinventorybalanceequationforUIIforproductpineachperiodt,whichensuresthatthesumofthequantityininventorycarriedfromperiodt1toperiodtandthequantityofreturnsreceivedinperiodtisequaltothesumofthequantityremanufacturedinperiodt,thequantitydisposedinperiodtandthequantityininventorycarriedtofromperiodttoperiodt+1.Constraintset( 4{4 )ensuresthatthemanufacturingcapacityrequiredforallproductsinperiodtdoesnotexceedtheavailablemanufacturingcapacityinperiodt.Constraintset( 4{5 )ensuresthattheremanufacturingcapacityrequiredforproductpinperiodtdoesnotexceedtheavailableremanufacturingcapacityforproductpinperiodt.Constraintset( 4{6 )ensuresthatthedisposalcapacityrequiredforproductpinperiodtdoesnotexceedtheavailabledisposalcapacityforproductpinperiodt.Constraintset( 4{7 )ensuresthatamanufacturingsetupforproductpisperformedinperiodtifthequantitymanufacturedforproductpinperiodtispositiveandthisquantityshouldnotexceedthecumulativedemanduntiltheendoftheplanninghorizon.Constraintset( 4{8 )ensuresthataremanufacturingsetupisperformedforproductpinperiodtifthequantityremanufacturedforproductpinperiodtispositiveandthisquantityshouldnotexceedthecumulativedemanduntiltheendoftheplanninghorizon.Similarly,constraintset( 4{9 )ensuresthatadisposalsetupforproductpisperformedinperiodtifthequantityofproductpdisposedofin 62
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4{10 )initializesthebeginningandendinginventorylevelsforallproductsinUIIandMRII.Constraintsets( 4{11 )and( 4{12 )ensuretheintegralityofthedecisionvariables. Capacitatedmulti-itemlotsizingproblemisaspecialcaseofMRDPPproblem.Morespecically,whenthereturnquantityiszeroineveryperiodforeveryproduct,theproblemisequivalenttheclassicalcapacitatedmulti-itemlotsizingproblem.Sinceclassicalcapacitatedmulti-itemlotsizingproblemisNP-Hard,soisMRDPP.Therefore,wefocusonobtainingnear-optimalsolutionsforMRDPP.TothisendwedevelopaclassofsolutionalgorithmsbasedonLagrangianDecompositionandRelaxationtechniquesusedinconjunctionwithGeneticAlgorithms. IntheLagrangianDecomposition(LD)approach,weusevariableredenitionandrelaxationofsomeconstraintsetstodecomposetheproblemintopolynomiallysolvablesubproblems.WeobservethattheremanufacturingquantitydecisionvariablesconnecttheUIIandMRIIinventorybalanceconstraintsets.Hence,weredenetheremanufacturingquantityandsetupvariablesandreplacetheremanufacturingvariableintheUIIinventorybalanceconstraintset.WekeeptheoriginalremanufacturingvariablesintheMRIIinventorybalanceconstraintset,whereaswereplacetheremanufacturingvariablesintheUIIinventorybalanceconstraintsetwiththenewremanufacturingvariables.Weaddanequality(linkage)constraintsettodenetheequivalenceoftheexistingandredeneddecisionvariablestotheoriginalproblemformulation.Wethenrelaxthissetof 63
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IntheLagrangianRelaxationapproach,weonlyrelaxthemanufacturingcapacityconstraintset( 4{4 ).TheresultingLagrangiansubproblemsaresingleitemMRDPPproblems,whichareNP-Hardintheirownright.ForagivensetofLagrangianmultipliers,wesolvethesesubproblemsusingCPLEXtoobtainalowerboundsolutionoftheMRDPP.Weobtainupperboundsolutionsutilizingaconstructiveheuristic,whichisasmoothingheuristicthatmakesuseofthelowerboundsolutionsobtainedfromCPLEX.Again,weupdatetheLagrangianmultipliersbythesubgradientoptimizationapproachusinglowerandupperboundsolutions. Next,weanalyzethesetwoapproachesindetailbygivingthecorrespondingLagrangianproblemformulations,algorithmsdevelopedtoobtainlowerboundsandupperbounds,andthesubgradientoptimizationapproach. 64
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Wealsodenenewcostparametersforthenewdecisionvariables: Letcupt=Fupt=0forp=1;:::;P,andt=1;:::;T.Thenewformulationcanbestatedasfollows:minPXp=1TXt=1(Fmptympt+cmptxmpt+Frptyrpt+crptxrpt+Fdptydpt+cdptxdpt+kptjpt+hptipt+Fuptyupt+cuptxupt) (4{13) subjecttoip;t1+xmpt+xrptDpt=iptp=1;:::;P;t=1;:::;T; (4{14)jp;t1+Rptxuptxdpt=jptp=1;:::;P;t=1;:::;T; (4{15)PXp=1pxmptKmtt=1;:::;T; (4{16)xrptKrptyrptp=1;:::;P;t=1;:::;T; (4{17)xdptKdptydptp=1;:::;P;t=1;:::;T; (4{18)xmpt (4{19)xrpt (4{20)xdpt (4{21) 65
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(4{22)xrpt=xuptp=1;:::;P;t=1;:::;T; (4{23)yrpt=yuptp=1;:::;P;t=1;:::;T; (4{24)jp0;jpT;ip0;ipT=0p=1;:::;P; (4{25)xmpt;xrpt;xupt;xdpt;jptipt0andintegerp=1;:::;P;t=1;:::;T; (4{26)ympt;yrpt;yupt;ydpt2f0;1gp=1;:::;P;t=1;:::;T: NotethatweexchangethexrptdecisionvariableswithxuptinUIIinventorybalanceconstraintset.Constraintset( 4{23 )ensuresthatxrptandxupttakethesamevalues.Similarly,constraintset( 4{24 )ensuresthatyrptandyupttakethesamevalues.Wealsoaddauxiliaryconstraintset( 4{22 )todeneboundsforxuptandyupt.Since,theunitcostofremanufacturingforxupt,andthexedcostforyuptarezero,theformulation,i.e.,( 4{13 )-( 4{27 ),isequivalenttotheoriginalformulation,i.e,( 4{1 )-( 4{12 ). Wethenrelaxtheconstraintsetsassociatedwithmanufacturing,remanufacturinganddisposalcapacities,i.e.,( 4{16 )-( 4{18 )aswellasthelinkageconstraintsetsbetweenremanufacturingdecisionvariables,i.e.( 4{23 )-( 4{24 ).Whenconstraintsets( 4{16 ),( 4{17 )and( 4{18 )arerelaxed,theresultingrelaxedproblembecomesseparableforeachproducttypepforp=1;:::;P.Moreover,whenconstraintsets( 4{23 )and( 4{24 )arerelaxed,theresultingrelaxedproblembecomesseparableforforwardandreversechannelsubproblems.Let=ftgdenotethevectorofLagrangianmultiplierst0;t=1;:::;T,associatedwiththemanufacturingcapacityconstraintset( 4{16 ).Let=fptg,=fptgdenotethevectorofLagrangianmultiplierspt;pt0;p=1;:::;P;t=1;:::;T,associatedwiththeproduct-specicremanufacturinganddisposalcapacityconstraintsets( 4{17 )and( 4{18 ),respectively.Similarlylet=fptg;!=f!ptgdenotethevectorsofLagrangianmultiplierspt;!pt2(;1);p=1;:::;P;t=1;:::;T,associatedwiththeconstraintssets( 4{23 )and( 4{24 ).Given;;,and!,theLagrangianproblemobtainedupon 66
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4{16 )-( 4{18 ),and( 4{23 )-( 4{24 )canbestatedasfollows:minPXp=1TXt=1(Fmptympt+(cmpt+t)xmpt+(FrptKrptpt)yrt+(crpt+pt+pt)xrpt+(FdptKdptpt)ydpt+(cdpt+pt)xdpt(Fupt!pt)yupt+(cupt!pt)xupt+kptjpt+hptipt)TXt=1(tKmt) (4{28) subjectto( 4{14 )-( 4{15 ),( 4{19 )-( 4{22 ),( 4{25 )-( 4{27 ). ForagivensetofLagrangianmultipliers,thesolutionofLagrangianproblemyieldsalowerboundsolution. WeprovidethespecicoutlineofourLDimplementationinFigure 4-2 where`,anddenotetheiterationcounters.Theterminationcriteriaaredenedby,thetolerancegapbetweenthebestlowerandupperbounds,and`max,themaximumnumberofiterationsofLD.UponthecompletionofmaxLDiterations,newindividualsaddedtoGApopulation.UponthecompletionofmaxLDiterations,GAisexecuted.Moreover,UB`andLB`denotetheupperandlowerbounds,respectively,atiteration`.Similarly,UBandLBdenotetheincumbentupperandlowerbounds,respectively.WeproceedwithadetaileddiscussionofhowwesolvetheLagrangianproblemtoobtainalowerbound(Section 4.3.1.1 ),identifyanupperboundusingthesmoothingheuristic(Section 4.3.1.2 )andGA,andupdatetheLagrangianmultipliers(Section 4.3.1.3 ),oneachiterationoftheLDapproach. 4{28 )canbesolvedeciently.First,wenotethatthelasttermintheobjectivefunctionofthisproblemisconstant.Furthermore,theproblem( 4{28 )isseparableintoPsubproblems.Sincethelinkbetweenthereturnandthedemandisbrokenbyrelaxingtheconstraintsets( 4{23 )-( 4{24 ),eachsubproblemcorrespondingtoaproducttypepisseparableintotwosubproblemsthatcorrespondtotheforwardandreversechannelactivitiesfortheproduct.LetLR1SP1p
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Thesingle-productforward-channelLagrangiansubproblemforproducttypep(LR1SP1p)canbestatedasfollows:minTXt=1(Fmptympt+(cmpt+t)xmpt+(FrptKrptpt)yrpt+(crpt+pt+pt)xrpt+hptipt) (4{29) subjecttoip;t1+xmpt+xrptDpt=iptt=1;:::;T; (4{30)xmpt (4{31)xrpt (4{32)ip0;ipT=0 (4{33)xmpt;xrpt;ipt0andintegert=1;:::;T; (4{34)ympt;yrpt2f0;1gt=1;:::;T: Notethatthecoecient(FrptKrptpt)ofbinaryvariableyrptintheobjectivefunction( 4{29 )canbenegativesincept0.Weobservethatintheoptimalsolutiontothisproblem,if(FrptKrptpt)<0,thenyrpt=1.Asaresult,wecanmodifytheobjectivefunction( 4{29 )by(i)replacingFrptwithF0rptwhereF0rpt=maxf(FrptKrptpt);0gfort=1;:::;T;and(ii)addingaconstanttermPTt=1minfFrptKrptpt;0g.Consequently,foragiven,,,andthemodiedLR1SP1pforproductpisgivenby:minTXt=1(Fmptympt+(cmpt+t)xmpt+(F0rpt)yrpt+(crpt+ptpt)xrpt+hptipt) (4{36) subjectto( 4{30 )-( 4{35 ) ThemodiedLR1SP1pforproducttypepissameasthesingle-itemuncapacitatedlot-sizingwithtwosupplysources.Therefore,thezero-inventorypropertyissatisedforthisproblem.WedevelopanO(T2)DPalgorithmsimilartoWagner-Whitinalgorithmto 68
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Letfp(t)denotetheoptimalvaluefunctionforproductpinperiodt,whichisdenedastheminimumcostofsatisfyingdemandfromperiodtthroughtheendoftheplanninghorizonwithzerostartinginventoryforproductp.TherecurrencerelationoftheDPalgorithmcanbestatedasfollows:fp(t)=min8>>>>>><>>>>>>:minv=t;:::;TFmpt+(cmpt+pt)Svpt+v1Pl=thplSvpl+1+fp(v+1)minv=t;:::;TF0rpt+(crpt+Krptpt+!pt)Svpt+v1Pl=thplSvpl+1+fp(v+1) 69
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(4{37) subjecttojp;t1+Rptxuptxdpt=jptt=1;:::;T; (4{38)xdpt (4{39)xupt (4{40)jp0;jpT=0 (4{41)xdpt;xupt;jpt0andintegert=1;:::;T; (4{42)yupt;ydpt2f0;1gt=1;:::;T: Notethatthecoecient(FdptKdptpt)ofbinaryvariableydpt,and(Fupt!pt)ofbinaryvariableyuptintheobjectivefunction( 4{37 )canbenegativesincept0and!>.Weobservethatintheoptimalsolutiontothisproblem,if(FdptKdptpt)<0,thenydpt=1.Similarly,if(Fupt!pt)<0,thenyupt=1.Accordingly,wecanmodifytheobjectivefunction( 4{37 )by(i)replacingFdptwithF0dpt,andFuptwithF0uptwhereF0dpt=maxf(FdptKdptpt);0gandF0upt=maxf(Fupt!pt);0gfort=1;:::;T;and(ii)addingaconstanttermPTt=1min(fFdptKdptpt;0g+fFuptKupt!pt;0g).Consequently,forgiven,,,,!themodiedLR1SP2pisgivenby:minTXt=1(F0dptydpt+(cdpt+pt)xdpt+F0uptyupt+(cupt!pt)xupt+kptjpt) (4{44) subjectto( 4{38 )-( 4{43 ) NotethatmodiedLR1SP2pissameastheuncapacitatedlotsizingwithtwosupplysources,wherereturnsareprocessedbyremanufacturingordisposalactivities.Wedevelop 70
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Letfp(t)denotetheoptimalvaluefunctionforproductpinperiodt,whichisdenedastheminimumcostofprocessingreturnfromperiod1throughperiodtwithzeroendinginventoryforproductp.TherecurrencerelationoftheDPalgorithmcanbestatedasfollows:fp(t)=min8>>>>>><>>>>>>:minv=1;:::;tF0upt+(cupt!pt)Rtpv+t1Pl=vkplRlpv+fp(v1)minv=t;:::;TF0dpt+(cdpt+Kdptpt)Rtpt+t1Pl=v(kplRlpv)+fp(v1) 71
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4{13 )canbeidentiedecientlywiththeDPapproachesdescribedinSection 4.3.1.1 .However,thisoptimalsolutionmayviolatethemanufacturing,remanufacturing,anddisposalcapacityrestrictions,aswellaslinkageconstraintsetsforremanufacturingquantityandsetupvariables,i.e.,( 4{23 )and( 4{24 ).Moreover,thequantityofremanufactureditemsmayexceedtheavailablequantityofreturns.Weemployasmoothingheuristictoobtainupperboundoneachiteration.WeuseboththelowerboundsolutionsandsolutionsobtainedaftersmoothingheuristicateveryxnumberofiterationstogenerateaninitialpopulationfortheGA.Betweeneveryxednumberofiterations,weusethegeneticalgorithmtoimprovethecurrentbestupperbound.Moreover,attheendoftheLagrangianiterations,weuseGAagain.BelowweexplainthedetailsofthesmoothingheuristicandtheGA. 4.3.1.1 mayviolatethemanufacturing,remanufacturing,anddisposalcapacityrestrictions,aswellasthelinkageconstraintsfortheremanufacturingquantityandthesetupvariables.NotethattheremanufacturingquantitiesfromthesolutionofLR1SP1pmayviolatetheavailablereturnquantity,similarlytheremanufacturingquantitiesfromthesolutionofLR1SP2pmayviolatethedemandquantity.Hence,weapplythesmoothingheuristictothesetwosolutionsdierbyonlytheremanufacturingquantities,wheretherstone(yr;xr)isobtainedbythesolutionofLR1SP1pandthesecondone(yu;xu)isobtainedbythesolutionofLR1SP2p.Werstxtheremanufacturingquantityandthesetupvariablesforbothsolutions,thenapplythesameproceduretoobtaintwoupperboundstoMRDPPaswellastwoindividualsforthepopulationoftheGA. 72
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Theaboveprocedureisrepeatedforalltheproductswithremanufacturingcapacityand/oravailablereturnorrequireddemandviolations.2.EliminatingUIIandMRIIinventorybalanceviolations: 74
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(a) Weclassifytheproductsthataremanufacturedinthisperiodintotwosets.Theproductsforwhichthemanufacturingquantityissmallerthanorequaltotheamountofmanufacturingcapacityviolationareincludedintherstset.Thesecondsetcomprisesoftheproductsforwhichthemanufacturingquantityislargerthattheamountofmanufacturingcapacityviolation.Foreachproductintherstset,weconsidertoshifttheentiremanufacturingquantityoftheproduct(i)tothemostrecentperiodwithanexistingmanufacturingsetup,or(ii)tothepreviousperiod(t1).Inevaluatingthetotalcostchangeassociatedwiththeabovetwooptions,weaccountforchangesinthetotalunitmanufacturingandMRIIinventoryholdingcosts.Moreover,forthesecondoption,wealsotakeintoaccountthexedcostiftheshiftrequirestheschedulingofanadditionalmanufacturingsetupinperiod(t1).Similarly,foreachproductinthesecondset,weconsidertoshift(i)theamountofcapacityviolationtothepreviousperiod(t1),or(ii)theentiremanufacturingquantitytothepreviousperiod(t1),or(iii)theentiremanufacturingquantitytothemostrecentperiodwithanexistingmanufacturingsetup.Again,inevaluatingtheseoptions,weaccountforthechangesinthetotalunitmanufacturingandMRIIinventoryholdingcosts.Moreover,fortherstandsecondoptions,wetakeintoaccountthexedmanufacturingsetupcostiftheshiftrequirestheschedulingofanadditionalmanufacturinginperiod(t1). Foreachproduct,wepicktheoptionthatyieldstheminimummarginalcostwhichisobtainedbydividingthetotalcostchangeassociatedwithimplementingtheoptionbytheamountthatcanbeshiftedbytheoption.Then,wepicktheproduct(and,hence,theassociatedoption)leadingtothebestmarginalcostandshifttheamountimpliedbytheoption.Werepeattheprocedure(startingwithclassifyingtheproductsintotwosets)untilthereisnomanufacturingcapacityviolationinthecurrentperiodt. Then,theroutineisappliedforthepreviousperiodwhichhasacapacityviolation,andthisiscontinueduntiltherstperiodisreached.Whenthecurrentperiodist=1,allmanufacturingcapacityviolationsareguaranteedtobeeliminatedwith,possibly,theexceptionoftherstperiod. (b) Weconsideralltheproductsthataremanufacturedintheperiodandclassifythemintotwosetsasbefore.Foreachproductintherstset(whosemanufacturingquantityissmallerthanorequaltotheamountofmanufacturingcapacityviolation),weconsidertoshifttheentiremanufacturingquantityoftheproducttothenextperiod.Foreachproductinthesecondset(whosemanufacturingquantityislarger
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Foreachproduct,wepicktheoptionthatyieldstheminimummarginalcostwhichisobtainedbydividingthetotalcostchangeassociatedwithimplementingtheoptionbytheamountthatcanbeshiftedbytheoption.Then,wepicktheproduct(and,hence,theassociatedoption)leadingtothebestmarginalcostandshifttheamountimpliedbytheoptionfortheproduct.Werepeattheprocedure(startingwithclassifyingtheproductsintotwosets)untilthereisnomanufacturingcapacityviolationinthecurrentperiodt. Ifthenextperiodhasnomanufacturingcapacityviolation,allmanufacturingcapacityviolationsareguaranteedtobeeliminated.Otherwise,theroutineisrepeatedforthenextperiod.5.Eliminateunnecessaryinventory: (a) Westartfromthelatestperiodtwherethereisamanufacturingsetupandunusedmanufacturingcapacity.WeconsideralltheproductsthataremanufacturedinthisperiodandhavepositiveincomingMRIIinventorytotheperiod. Foreachsuchproduct,wecomputethepotentialcostreductionassociatedwithincreasingthecorrespondingmanufacturingquantityintanddecreasingitint0(wheret0istheproduct'smostrecentmanufacturingsetupperiod,andtheproduct'sendinginventoryispositiveforallperiodsfromt0uptoandincluding(t1)).Then,wepicktheproductwiththelargestcostreductionandshiftthemaximumamountpossibleforthisproductfromt0totbearinginmindthemanufacturingcapacitiesinbothperiodsaswellastheinventorybalanceconstraintsfromt0tot.Here,themaximumamountpossibleisdictatedbytheminimumof(i)theamountofincominginventoryoftheproducttoperiodt,(ii)themanufacturingquantityinperiodt0,and(iii)theavailablemanufacturingcapacityinperiodt. Theprocedureiscontinueduntileitherthereisnootherproductleadingtoacostreductionorallthemanufacturingcapacityisused.Then,theprocedureisrepeatedforthepreviousperiodwithamanufacturingsetupandunusedmanufacturingcapacity. Whenthecurrentperiodist=1,ortherearenootherperiodswithamanufacturingsetupandunusedmanufacturingcapacity,unnecessaryinventoryduetoearlymanufacturinghasbeeneliminatedforalltheproducts. 77
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Givenaproduct,westartfromthelatestperiodtwherethereisaremanufacturingsetupfortheproduct.IfthereisanyunusedcapacityinperiodtandtheincomingMRIIinventoryispositiveinallperiodsfromthepreviousremanufacturingsetupperiodt0fortheproduct,thenwecomputethepotentialcostreductionassociatedwithdecreasingtheremanufacturingquantityint0andincreasingtheremanufacturingquantityint.Ifthecostreductionispositive,thenweshiftthemaximumamountpossiblethatcanbemovedfortheproductfromt0totbearinginmindtheremanufacturingcapacitiesinbothperiodsaswellastheUIIandMRIIinventorybalanceconstraintsfromt0tot. Here,themaximumamountpossibleisdictatedbytheminimumof(i)theminimumamountofMRIIinventoryoftheproductfromperiodt0toperiodt,(ii)theremanufacturingquantityinperiodt0,and(iii)theavailableremanufacturingcapacityinperiodt.Theprocedureisrepeatedforthepreviousperiodwitharemanufacturingsetupandunusedremanufacturingcapacity.Whenthecurrentperiodist=1ortherearenootherperiodswitharemanufacturingsetupandunusedremanufacturingcapacityfortheproduct,unnecessaryMRIIinventoryduetoearlierremanufacturinghasbeeneliminatedfortheproduct. Aboveroutineisrepeatedforalltheproducts. (c) Givenaproduct,westartfromtheearliestperiodtwherethereisadisposalsetupfortheproduct.IfthereisanyunuseddisposalcapacityinperiodtandtheUIIinventoryispositiveinallperiodsuptothenextdisposalsetupperiodt0fortheproduct,thenwecomputethepotentialcostreductionassociatedwithdecreasingthedisposalquantityint0andincreasingthedisposalquantityint.Ifthecostreductionispositive,thenweshiftthemaximumamountpossiblethatcanbemovedfortheproductfromt0totbearinginmindthedisposalcapacitiesinbothperiodsaswellastheUIIinventorybalanceconstraintsfromttot0.Here,themaximumamountpossibleisdictatedbytheminimumof(i)theminimumamountofUIIinventoryoftheproductfromperiodttoperiodt0,(ii)thedisposalquantityinperiodt0,and(iii)theavailabledisposalcapacityinperiodt.Theprocedureisrepeatedforthenextperiodwithadisposalsetupandunuseddisposalcapacity.Whenthecurrentperiodist=Tortherearenootherperiodswithadisposalsetupandunuseddisposalcapacityfortheproduct,unnecessaryUIIinventoryduetodelayeddisposalhasbeeneliminatedfortheproduct. Aboveroutineisrepeatedforalltheproducts. Recallthatweemploytheaboveheuristicusingremanufacturingquantitiesxrorxu)separately.Therefore,weobtaintwoupperboundsolutions.Ifanyoftheseupper 78
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4{2 )and( 4{3 ),withsomepenaltyaddedtotheobjectivefunction. TogeneratethepopulationforaniterationofGA,weusethelowerboundsolutionandsmoothingheuristic.Weobtainfourchromosomesateveryxednumberofiterations.Therstchromosomeisobtainedusingthesetupvariables'valuescorrespondingtothemanufacturingandtheremanufacturingfromthesolutiontotherstsubproblem,andthesetupvariables'valuescorrespondingtothedisposalfromthesolutionofthesecondsubproblem,i.e.,[ym;yr;yd].Thesecondchromosomeisobtainedbyusingthesetupvariables'valuesofmanufacturingfromthesolutionoftherstsubproblem,andthesetupvariables'valuesofremanufacturinganddisposalfromthesolutionofthesecondsubproblem,i.e.,[ym;yu;yd].Thethirdandfourthchromosomesareobtainedfromthemodicationofthesolutionsobtainedfromlowerboudingusingthesmoothingheuristics. AfteraxednumberofLagrangianiterationsareexecuted,GAisemployedusingthegeneratedpopulationuptothatiteration.Weusetheobjectivefunctionvalueasatnessvalueforeachindividualinthepopulation.Fromthecurrentpopulationwerstchoose 79
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WeprovidethespecicoutlineofourGAimplementationinFigure 4-3 whereMdenotesthegivenpopulationsize,denotestheiterationcounter,andmaxdenotesthenumberofgenerations.LetP1andP2denotetwoparentchromosomes.Similarly,letC1andC2denotetwochildrenchromosomes.WeletL(j)denotetheobjectivefunction(tness)valueofindividualj. AfterLagrangianiterationsstopseitherwithpredenedgaphasbeenreachedorpredenednumberofiterationsexecuted.WeusetheGAbyusingthemostrecentpopulationtondabetterupperboundsolution.Wereportthesolutionwithbestobjectivefunctionvalue. 80
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`);`+1pt=max(0;`pt+a`(UBLB`)(xrptKrptyrpt) `);`+1pt=max(0;`pt+a`(UBLB`)(xdptKdptydpt) `);`+1pt=`pt+a`(UBLB`)(xrptxupt) `;!`+1pt=!`pt+a`(UBLB`)(yrptyupt) `: Weupdatethestepsizebyhalvingitsvaluewhenthereisnoimprovementinthelowerboundinaxednumberofiterations. 4{4 ).Whenconstraintset( 4{4 )isrelaxed,theresultingLagrangianproblembecomesseparableforeachproductp;p=1;:::;P.EachsubproblemisasingleproductMRDPP,anditisNP-Hard.Hence,weuseCPLEXtosolvetheLagrangiansubproblems.Insteadofsolvingthesubproblemstooptimality,a1%optimalitygapisusedtoobtainsolutionsfaster. Let=ftgdenotethevectorofLagrangianmultiplierst0;t=1;:::;T,associatedwiththemanufacturingcapacityconstraintsinset( 4{16 ).ThentheLagrangianproblemcanbestatedasfollows:minPXp=1TXt=1(Fmptympt+(cmpt+t)xmpt+Frptyrt+crptxrpt+Fdptydpt+cdptxdpt+kptjpt+hptipt)TXt=1(tKmt) (4{45) subjectto( 4{2 )-( 4{3 ),( 4{5 )-( 4{12 ). 81
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WeprovidethespecicoutlineofourLRimplementationinFigure 4-4 where`anddenotetheiterationcounters.Theterminationcriteriaaredenedby,thetolerancegapbetweenthebestlowerandupperbounds,and`max,themaximumnumberofiterationsofLR.WhentherearemaxLRiterations,thenGAisapplied.Moreover,UB`andLB`denotetheupperandlowerbounds,respectively,atiteration`.Similarly,UBandLBdenotetheincumbentupperandlowerbounds,respectively.WeproceedwithhowwesolvetheLagrangianproblemtoobtainalowerbound(Section 4.3.2.1 ),ndanupperboundusingthesmoothingheuristic(Section 4.3.2.2 )andGA,andupdatetheLagrangianmultipliers(Section 4.3.2.3 ),oneachiterationoftheLDapproach. 4{45 )isseparableintoPsubproblems.LetLR2SPpdenotesthesubproblemforproductpforp=1:::;P.WenotethatthelasttermintheobjectivefunctionofthisproblemisconstantanddoesnoteectthesolutionoftheLagrangianproblem.ThesingleproductLagrangiansubproblem(forproductp),i.e.,LR2SPpcanbestatedasfollows:minTXt=1(Fmptympt+(cmpt+t)xmpt+Frptyrpt+crptxrpt+Fdptydpt+cdptxdpt+kptjpt+hptipt) (4{46) subjecttoip;t1+xmpt+xrptDpt=iptt=1;:::;T; (4{47)jp;t1+Rptxrptxdpt=jptt=1;:::;T; (4{48)xrptKrptyrptt=1;:::;T; (4{49)xdptKdptydptt=1;:::;T; (4{50)xmpt (4{51) 82
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(4{52)xdpt (4{53)jp0;jpT;ip0;ipT=0 (4{54)xmpt;xrpt;xdpt;jptipt0andintegert=1;:::;T; (4{55)ympt;yrpt;ydpt2f0;1gt=1;:::;T: WesolveLR2SPpforproductpbyCPLEXwith1%optimalitygapforp=1;:::;P.LetLBpdenotebestobjectivefunctionvalueobtainedfromCPLEX,thenthelowerboundfortheMRDPP,LB,isgivenby:LB=PXp=1(LBp)TXt=1(tKmt) 4{45 )isobtainedbysolvingLR2SPpforp=1;:::;PbyCPLEX.Thisoptimalsolutionmayonlyviolatethesharedmanufacturingcapacityrestrictions.Weemploysmoothingheuristicstoobtainanupperboundsolution.Moreover,weuseGAtoimprovetheupperboundsolution.WeusethelowerboundsolutionandsmoothingheuristictogeneratethepopulationforGAforxednumberofLagrangianiterations.Weobtainfourchromosomesateachiteration.WethenuseGAtondabettersolutionfromthecurrentpopulation. Therstchromosomeisobtainedusingthesetupvariables'valuesofthemanufacturing,remanufacturinganddisposalactivitiesobtainedfromthelowerboundsolution.Thesecondchromosomeisobtainedbyapplyingmutationtotherstchromosome.Werandomlychooseaxedsetofproducts,andforeachchosenproduct,werandomlychoosexedsetofperiods.Foreachoftheseproductsandperiods,werandomlychoosemanufacturing,remanufacturing,anddisposalactivityandchangethecurrentassociatedsetupvariablevalue,from1to0or0to1.Thethirdchromosomeisobtainedbyapplying 83
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RecallthatthelowerboundsolutionisobtainedinSection 4.3.2.1 mayonlyviolatethemanufacturingcapacityrestrictions.WexthemanufacturingcapacityviolationsbytheroutineexplainedinSection 4.3.1.2 .Iftheobtainedupperboundisbetterthanthecurrentbestupperbound,thenthecurrentbestupperboundisupdated.ThisupperboundsolutionisusedasathirdchromosomeforthecurrentLagrangianiteration.Thefourthchromosomeisgeneratedbyapplyingmutationtothisupperupperbound. AfteraxednumberofLagrangianiterations,GAisemployedusingthegeneratedpopulationuptothatiteration.WeusetheGAgiveninSection 4.3.1.2 .Theonlydierencecomesfromthesizeandqualityofthepopulation. Lagrangianiterationsstopseitherwithpredenedgaphasbeenreachedorpredenednumberofiterationshit.Weconductthegeneticalgorithmoncemoreusingthemostrecentpopulationtondabetterupperboundsolution.Wereportthesolutionwithbestobjectivefunctionvalue. Weupdatethestepsizebyhalvingitsvaluewhenthereisnoimprovementinthelowerboundinaxednumberofiterations. 84
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Inourexperiments,webeginbygeneratingthedemands,returns,andmanufacturing,remanufacturinganddisposalcapacitiesbyinducingcorrelationsamongthemforeach(P;T)pair.Werstgeneratethedemands,thenbyusingthecorrectionbetweendemandandreturn,andrealizeddemandvalue,wegeneratereturns.Wethengeneratethereturnparametersinducingcorrelationbetweendemandsandreturns.Withthegivencapacitytightness,wegeneratethemanufacturingcapacitiesbyusingaveragerealizeddemandineachperiod.Similarly,wegeneratetheremanufacturinganddisposalcapacitiesforeachproductandeachperiodbyusingthecapacitytightnesslevelandtherealizedreturnanddemandvalues. Inourexperiments,weconsiderinstanceswithconstantunitmanufacturing,unitremanufacturing,andunitdisposal,andinventoryholdingcostsforUIIandMRII.Wegeneratethexedmanufacturing,remanufacturing,anddisposalsetupcostsbasedontheinventoryholdingcosts.Inparticular,webeginbygeneratingthetimebetween 85
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Similarly,wegeneratethexeddisposalsetupcostforeachproductineachperiodconsideringtheaveragereturn,TBDvalue,inventoryholdingcost,andvariabilityofxeddisposalsetupcosts. Todevelopasetofrandominstances,wevarythenumberofproductsPandthelengthoftheplanninghorizon,T.Specically,weconsidertwolevelsforP(10,20)andtwolevelsforT(12,24).Consequently,wehave4classesoftestinstances.Moreover,weconsidervelevelsforthecorrelationbetweentotaldemandtototalreturnsthroughouttheplanninghorizon(from0.15forlowto0.75forhighinincrementsof0.15),twolevelsforeachofTBM,TBR,andTBD(U[1,3]forloworU[3,5]forhigh),andtwolevelsforeachmanufacturing,remanufacturinganddisposalcapacitytightness(1.5forlowor2.5forhigh).WeconsidertheparametervaluessummarizedinTable 4-1 Foreach(P;T)weconsider20dierentproblemsettings,andforeachsettingwegenerate10randomtestinstances.Therefore,wegenerateandsolve800instancesintotal. 86
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WesummarizetheoptimalitygapsobtainedbytheLDapproachanditsCPUrequirementsinTable 4-2 .Aswesolveatotalof800testinstances,eachcellinthistablecorrespondstotheaverageormaximumvaluerepresenting200instancesofthesamesize.Wenotethattheobservedaverageoptimalitygapoverallinstancesis6.86percent,whereasthemaximumis15.34percent.Wealsoobservethatforagivennumberofproducts,theaverageandmaximumvaluesoftheoptimalitygapsbecomesmallerasthelengthoftheplanninghorizonincreases.Foragivenplanninghorizonlength,theaverageandmaximumvaluesoftheoptimalitygapsbecomesmallerbyanincreaseinthenumberofproducts. Moreover,weobservethattheLDapproachiscomputationallyecientastheobservedaverageCPUrequirementoverallinstancesislessthan17seconds,whereasthemaximumislessthan37seconds. InTables 4-3 and 4-4 ,weprovideadetailedanalysisoftheinuenceofthedatacharacteristicsontheperformanceoftheLDapproach. Weobservethatthequalityoftheoptimalitygapsdeterioratesasthetimebetweenactivitysetupsincreaseand/or(ii)theactivitycapacitiesbecomemorerestraining.Itisworthwhiletonotethattheinuenceofanincreaseincapacitytightnessistobemoreinuentialthananincreaseintimebetweenactivitysetups.Inaddition,weobservethattheoptimalitygapsobtainedbytheLDapproachdeterioratesasthecorrelationbetweenreturnsanddemandincreasesinboththeaverageandmaximumoptimalitygap. 87
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WesummarizetheoptimalitygapsobtainedbytheLRapproachanditsCPUrequirementsinTable 4-5 .Wenotethattheobservedaverageoptimalitygapoverallinstancesis0.95percent,whereasthemaximumis3.69percent.WeobservethesameinuenceofnumberofproductsandlengthofplanninghorizonontheoptimalitygapsoftheLRapproachasintheLDapproach.Consequently,thequalityofthegapidentiedbytheLRapproachisbetterforinstanceswithalargernumberofproductsand/oralongerplanninghorizon.OurcomputationalresultsshowedthattheproposedLDapproachismoreecientthanLRapproach,butLRapproachismuchmoreeectivethanLDapproach. InTables 4-6 and 4-7 ,weprovideadetailedanalysisoftheinuenceofthedatacharacteristicsontheperformanceoftheLRapproach. Surprisingly,weobservethatthequalityoftheoptimalitygapsofLRapproachdoesnotdeteriorateasmuchasLDapproachwhenthetimebetweenactivitysetupsincreaseand/or(ii)theactivitycapacitiesbecomemorerestraining.Moreinterestingly,averageoptimalitygapsalmostremainsameindependentfromthecapacitytightnessand/ortimebetweenactivitysetups.Inaddition,thecorrelationbetweenreturnsanddemanddoesnothavemucheectonthequalityofLRapproach. AlthoughtheresultsinTable 4-2 provideempiricalevidencefortheeectivenessoftheproposedLRapproachtoaddresstheMRDPP,aninterestingquestionthatremainsunansweredistheeectivenessoftheB&Capproachforobtainingsolutionstothe 88
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4-8 Again,eachcellinthistablerepresentstheaverageormaximumvalueof800observationsfortestinstancesofthesamesize.Wenotethattheaverageoptimalitygapoverallinstancesisaround6.22percentwhereasthemaximumisaround32percent.ThisclearlyshowsthatourLRapproachoutperformstheCPLEX. 89
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WeconductedextensivecomputationalexperimentstoinvestigatethecomputationaleciencyoftheLDandLRapproaches.OurcomputationalresultsshowedthattheproposedLDapproachiscomputationallymoreecientthantheLRapproach,buttheLRapproachismoreeectivethantheLDapproach. AnimmediateextensionofourworkthatwouldbeofpracticalrelevanceistoconsiderageneralizationoftheMRDPPwhere(i)themanufacturingand/orremanufacturingand/ordisposalsetupcostsdependonthespecicproductsincludedinthesetupand/or(ii)setuptimesareexplicitlymodeled.Moreover,MRDPPwithsharedremanufacturingcapacityand/orlineardisposalcostscanbepossibleextensions.Oursolutionalgorithmscanbeappliedtotheseproblemswithslightmodications.Also,theLRapproachpresentedinthispapercanbeextendedtosettingswherecertaincapacitatedresourcesaresharedbythemanufacturingandremanufacturingactivities. Figure4-1. ProblemSettingofMRDPP 90
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Figure4-2. Pseudo-codeoftheLDApproachfortheMRDPP. 91
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Pseudo-codeoftheGAApproachfortheLD. Figure4-4. Pseudo-codeoftheLRApproachfortheMRDPP. 92
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ParameterValuesUsedinComputationalExperimentsofMRDPP Level(s) Value(s) Constant 100 Variabilityofdemand Constant 0.5 Variabilityofreturns Constant 0.5 Variabilityofxedmanufacturingsetupcost Constant 0.2 Variabilityofxedremanufacturingsetupcost Constant 0.2 Variabilityofxeddisposalsetupcost Constant 0.2 UnitMRIIinventoryholdingcost Constant 1 UnitUIIinventoryholdingcost Constant 0.5 Unitmanufacturingcost Constant 3 Unitremanufacturingcost Constant 2 Unitdisposalcost Constant 1 Unitcapacityconsumption Constant 1 Timebetweenmanufacturing,remanufacturing,anddisposalsetups Low U[1,3] (TBM;TBR;TBD) High U[3,5] Manufacturing,remanufacturinganddisposalcapacitytightness Low 1.5 ('M;'R;'D) High 2.5 Return/Demandratio Low 0.15 (RD) High 0.75 PerformanceoftheLDApproach:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). CPURequirement InstanceSize Avr.Gap Max.Gap Avr.Time Max.Time (P;T) (%) (%) (sec.) (sec.) (10,12) 7.68 15.34 11.65 19.25 (20,12) 6.43 12.63 16.72 28.56 (10,24) 7.21 14.68 15.21 27.14 (20,24) 6.12 11.24 22.43 36.79 Overall 6.86 15.34 16.50 36.79 EectofTimeBetweenActivitySetupsandCapacityTightnessontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. Max. (TBM,TBR,TBD) ('M;'R;'D) Gap(%) Gap(%) (U[1,3],U[1,3],U[1,3]) (1.5,1.5,1.5) 5.95 13.23 (U[3,5],U[3,5],U[3,5]) (1.5,1.5,1.5) 8.95 17.34 (U[1,3],U[1,3],U[1,3]) (2.5,2.5,2.5) 4.92 10.72 (U[3,5],U[3,5],U[3,5]) (2.5,2.5,2.5) 7.62 16.34
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EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. Max. Gap(%) 0.15 4.79 9.45 0.30 5.57 10.79 0.45 6.67 14.84 0.60 7.86 16.92 0.75 9.42 17.34 PerformanceoftheLRApproach:AverageandMaximumOptimalityGaps(%)andCPURequirement(sec.). CPURequirement InstanceSize Avr.Gap Max.Gap Avr.Time Max.Time (P;T) (%) (%) (sec.) (sec.) (10,12) 0.98 3.69 14.85 52.25 (20,12) 0.94 2.43 27.25 98.68 (10,24) 0.97 3.21 44.52 138.53 (20,24) 0.92 2.98 72.91 356.12 Overall 0.95 3.69 39.88 356.12 EectofTimeBetweenActivitySetupsandCapacityTightnessontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. Max. (TBM,TBR,TBD) ('M;'R;'D) Gap(%) Gap(%) (U[1,3],U[1,3],U[1,3]) (1.5,1.5,1.5) 0.94 1.59 (U[3,5],U[3,5],U[3,5]) (1.5,1.5,1.5) 1.03 3.69 (U[1,3],U[1,3],U[1,3]) (2.5,2.5,2.5) 0.91 1.31 (U[3,5],U[3,5],U[3,5]) (2.5,2.5,2.5) 0.92 1.44 EectofCorrelationBetweenReturnsandDemandontheAverageandMaximumOptimalityGaps(%)oftheLDApproach. Max. Gap(%) 0.15 0.89 3.69 0.30 0.94 2.72 0.45 0.93 2.43 0.60 0.98 1.91 0.75 1.01 2.34
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PerformanceofCPLEX(withaTimeLimitof400Seconds):AverageandMaximumOptimalityGaps(%). InstanceSize Avr.Gap Max.Gap (P;T) (%) (%) (10,12) 3.98 8.98 (20,12) 6.23 17.42 (10,24) 5.42 14.61 (20,24) 9.25 32.05 Overall 6.22 32.05
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MostardandTeunter 2006 ).Insomecases,thereturnratecanbeashighas75percent( MostardandTeunter 2006 ).Sincealargeproportionofthesereturnedproductsarereceivedinverygoodcondition,thesereturnscanbeplacedintoinventorytosatisfyfuturedemandwithlittleprocessing,suchasinspection,cleaning,testing,repackagingetc.Atypicalcatalogueretaileroersmultipleproductsthroughitscataloguesandinternetsitesandthe(nishedgoods)inventoryforaproductcanbereplenishedbyeitherprocuringproductsfromthesuppliersorprocessingandrestockingthereturnsreceivedfromconsumers.Inthissetting,foreectiveinventoryplanningandcontrol,thereturnsreceivedfromconsumersmustbetakenintoaccountwhilemakingprocurementdecisions.Animportantissuethatneedstobetakenintoaccountistheuncertaintyassociatedwiththereturnstream,whichcanbesignicantinthecontextofcatalogueretailing. Inthecontextofsparepartkittingforindustrialequipmentmaintenance,atypicalsupplierhasservicecontractswithastablecustomerbase.Thisallowsthesuppliertousehistoricaldatatoestimatethefuturedemandandreturnquantitiesaccuratelyfordierenttypesofproducts.Inthecontextofcatalogueretailing,however,retailer'sabilitytoforecastreturnsaccuratelyislimited,asthequantityand/ortimingofconsumerreturnsarehighlyvariable;inuencedbyseveralfactorssuchasbuyer'sremorseormismatchbetweencustomer'sexpectationsabouttheproduct.Typically,aretailerprocurestheproductsfromasupplieratthebeginningofaniteplanninghorizon.Reusablereturnsreceivedfromcustomersthroughouttheplanninghorizoncanbeplacedintoinventoryastheyarrivetosatisfythedemandinfutureperiods.Clearly,foroptimalinventoryplanningandcontrolforthissetting,returnsmustbetakenintoaccountexplicitlywhendecidingontheprocurementquantityatthebeginningoftheplanninghorizon.Ourfocus 96
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Morespecically,theproblemisconcernedwithdeterminingtheprocurementquantitiesformultipleproducttypesatthebeginningoftheplanninghorizon.Weconsidertheproblemindeterministicdemandandstochasticreturnframeworkoveraniteplanninghorizonunderniteprocurementcapacity.Inourwork,weconsiderfourvariantsofthePPP.Webeginbyconsideringtheprobleminasingle-periodsettingforasingleproducttype(PPP1)andextendthismodeltoincludemultipleproducttypes(PPP2).Then,westudytheprobleminmultiple-periodsettingforasingleproducttype(PPP3)aswellasformultipleproducttypes(PPP4). Theremainderofthischapterisorganizedasfollows.Weproceedwiththemodellingassumptionsin 5.2 WedevelopmathematicalmodelsandsolutionapproachesforasingleproducttypeinSections 5.3 and 5.4 forPPP1andPP2,respectively.Weanalyzethesemodelswhentheprocurementcostsarelinearandwhenthereisaxedcomponentinadditiontothelinearcomponent.Similarly,weinvestigatemodelsandsolutionapproachesformultipleproducttypesinSections 5.5 and 5.6 forPPP3andPPP4,respectively.Finally,conclusionsandfutureresearchdirectionsaresummarizedinSection 5.7 97
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Thesequenceofeventsareasfollows: 1. Manufactureditemsforproductpareprocuredpriortothebeginningoftheplanninghorizon,denotedbyxp,forp=1;:::;P. 2. Returnsforproducttypepinperiodtarereceivedforp=1;:::;Pandt=1;:::;T. 3. Demandforproducttypepinperiodtissatisedforp=1;:::;Pandt=1;:::;T. 4. Excessstockforproducttypepattheendofperiodtiscarriedtothenextperiodandinventoryholdingcostincurredforp=1;:::;Pandt=1;:::;T. 5. Excessdemandisbackorderedorshortagecostisincurredforproducttypepinperiodt,forp=1;:::;Pandt=1;:::;T. Theobjectiveistominimizethesumofexpectedprocurement,returnprocessing,backordering,andinventoryholdingcosts. 98
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Rearrangingtheterms,thetotalexpectedcostcanbewrittenasfollows: Then,wecanformulatethePPP1asfollows: minE[C(x)] (5{3) subjecttoxd: Firstorderconditionfor( 5{2 )isgivenby: Secondorderconditionfor( 5{2 )isgivenby:d2E[C(x)] (5{6) WehaveE00[C(x)]0.Hence,( 5{2 )isconvex,andtherstorderconditioncanbeusedtocharacterizetheoptimalprocurementquantity,x.Notethatwhenxdrstderivativeispositive,sotherstorderconditionscanonlybesatisedwhenx
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+hx=dF1cn+h +h Notethatwehavecn<(+h)F(d)h.Hence,xalwayssatises( 5{4 ). Intheclassicalnewsboy(newsvendor)problem,thecriticalfractileplaysanimportantroletocharacterizetheoptimalorderquantity.Recallthatthecriticalfractileistheratioofthecostofunderagetothesumofthecostsofunderageandoverage.InPPP1,thedemandisdeterministic,butreturnsarerandom.Wecaninterpret( 5{7 )asfollows:Thecostofunderageinourproblemis(cn+h).Also,thecostofoverageis(cn).Thatis,ifweunderestimatethequantityofreturnsbyoneunit,and,asaresult,weprocureonemoremanufactureditem,thenthecostofthisextraunitisitsprocurementcostplustheinventoryholdingcost,i.e.,(cn+h).Similarly,ifweoverestimatethereturnbyoneunit,and,asaresultprocureoneunitmanufactureditemless,thenthecostofnotprocuringthisunitisthepenaltycostofunsatiseddemandminustheprocurementcost,i.e.,(cn).Ifweusethecriticalfractileformulausingtheseunderageandoveragecosts,weobtain( 5{7 ).Therefore,ifweknowthecostofunderage,cu,andcostofoverage,co,thenwecandeterminetheoptimalprocurementquantityusing:x=dFcu 5{2 ),canberewrittenasfollows:E[C(x;z)]=Cnz+ 100
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LetMdenoteasucientlylargenumber.Then,theproblemformulation( 5{3 )-( 5{4 )becomes:minE[C(x;z)] (5{9) subjecttoxMz (5{12) Theoptimalsolutionto( 5{9 )-( 5{12 )canbecharacterizedasfollows.Letx0betheprocurementquantityidentiedbysolving( 5{3 )-( 5{4 ).Ifx0>0,thenthetotalexpectedcostisE[C(x0;1)].Moreover,E[C(0;0)]denotesthetotalexpectedcostwhentheprocurementquantityiszero.Wecancomparethesetwovalues,andiftheformerissmaller,thentheoptimalprocurementquantityisx0(i.e.,x=x0);otherwisetheoptimalprocurementquantityis0(i.e.,x=0). Recallthatwhencn<(+h)F(d)h,x0=dF1(cn+h +h),thenE[C(x0;1)]=Cn+(cn+h)x0+(cr+h)E[r]hd+(+h)Zdx00(dx0r)f(r)dr=Cn+cnd(cn+h)F1(cn+h +h))+(cr+h)E[r]+(+h)ZF1(cn+h +h)0(F1(cn+h +h)r)f(r)dr WealsohaveE[C(0;0)]=(cr+h)E[r]hd+(+h)Zd0(dr)f(r)dr 101
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+h);z=1.Morespecically,wecanndaconditiononthexedprocurementcostsuchthattheoptimalprocurementquantityiseitherx=dF1(cn+h +h)orzero.Letdenotethebreak-evenvalueofthexedcost,whichcanbeidentiedasfollows:=(cr+h)E[r]hd+(+h)Zd0(dr)f(r)dr(cnd(cn+h)F1(cn+h +h))+(cr+h)E[r]+(+h)ZF1(cn+h +h)0(F1(cn+h +h)r)f(r)dr)=(+h)Zd0(dr)f(r)drhdcnd(cn+h)F1(cn+h +h))+(+h)ZF1(cn+h +h)0(F1(cn+h +h)r)f(r)dr! +h).Otherwise,ifCn,thenitisoptimalnottoprocure,i.e.,x=0. 102
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(5{16)PXp=1pxpK Objectivefunction( 5{15 )isseparableforeachproductp,butconstraint( 5{17 ),sharedprocurementcapacityconstraint,linkstheproducts.Weconsiderrelaxingconstraint( 5{17 ),usingaLagrangianmultiplier,>0.When( 5{17 )isrelaxed,( 5{15 )-( 5{17 )becomesseparableforeachproducttypepforp=1;:::;P.ForagivenLagrangianmultiplierwehavethefollowingsubproblemforeachproducttypep:minE[C(xpj)]=Zdpxp0(cnp+p)xp+crprp+p((dpxp)rp)fp(rp)drp+Z1dpxp(cnp+p)xp+crprp+hp(rp(dpxp))fp(rp)drp) subjecttoxp0 (5{19) Aftersomealgebraicmanipulation,( 5{18 )canberewrittenasfollows:E[C(xpj)]=(cnp+p+hp)xp+(crp+hp)E[rp]hpdp+(p+hp)Zdpxp0(dpxprp)fp(rp)drp Firstorderconditionfor( 5{20 )is:E0[C(xpj)]=(cnp+p+hp)(p+hp)Zdpxp0fp(rp)drp Secondorderconditionfor( 5{20 )is:E00[C(xpj)]=8>><>>:(p+hp)fp(dpxp)xp
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5{22 )isnon-negative,( 5{20 )isconvex.Hence,theoptimalprocurementquantityforagivenLagrangianmultipliercanbeidentiedusingtherstorderconditionsfortheproducttypes.ForagivenLagrangianmultiplier,,theoptimalprocurementquantityxpisgivenby:xp=8>>><>>>:dpFpcnp+p+hp 5{17 ),thenitisoptimalto( 5{15 )-( 5{17 ).However,ifthissolutiondoesnotsatisfy( 5{17 ),thenwecansearchfortheoptimalvalueofthatyieldsthebestsolutionthatsatises( 5{17 ).First,weidentifythelargestpossiblevaluethatcantake,whichisgivenbythemaximum((p+hp)Fp(dp)hpcnp)=pvalueforp=1;:::;P. Wedevelopthefollowingalgorithmwhichseeksfortheoptimalsolutionto( 5{15 )-( 5{17 )byperformingasearchovertheLagrangianmultiplier.Let>0beanerrorboundonthecapacityconstraintsuchthatifasolutionthathasacapacityusagein[K;K]isidentied,thealgorithmstops. Initialization:Initialize.Setk0;k0;min0;maxmaxp=1;:::;Pn(p+hp)Fp(dp)hpcnp 104
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FindanInitialSolution:Forp=1;:::;Pndxkp:xkp=(dpFp(cnp+hp FindaSolution:Setk=min+max ReportSolution:Reportxp=xkpforp=1;:::;P,and=k. InAlgorithmAPPP2,werstinitializetheiterationcounter,,andminimumandmaximumvaluesthatcantake.Wethenndaninitialsolutionconsideringnocapacityconstraint,i.e.,when=0,inStep1.Ifthissolutionisfeasible,thenitisalsooptimalto( 5{15 )-( 5{17 ).Otherwise,wesearchfortheoptimalvalueiterativelybyhalvingtheintervalofateachiteration.Whenwendasolutionthatsatises( 5{17 )withanerrorboundof,westopthesearchandreportthecurrentsolutionasanoptimalsolution. Wealsoanalyzetheproblemiftheprocurementcosthasaxedcostcomponentinadditiontothelinearcomponent.Assumethatthereisaxedcost,Cnpforproductpforp=1;:::;P.Letzpbeadecisionvariableforproductpthattakesthevalueofonewhentheprocurementquantityispositive,i.e.,xp>0,andzerootherwise.( 5{15 )-( 5{17 )canberewrittenwhenthereisaxedcostasfollows:minE[C(!x;!z)]=PXp=1CnpzpZdpxp0cnpxp+crprp+p((dpxp)rp)fp(rp)drp+Z1dpxpcnpxp+crprp+hp(rp(dpxp))fp(rp)drp) 105
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(5{26) WecanstillusetheAlgorithmAPPP2.However,thesolutionofthesubproblemforproductpmaychangedependingonthexedcost.Morespecically,itmaybelesscostlynottoprocureproducttypepwhenthereisaxedcost.RecallthatinSection( 5.3 ),weidentifytwosolutions,i.e.,whenz=1andz=0.Then,weevaluatetheobjectivefunctionvaluesofthesesolutions.Theoptimalsolutionisthesolutionthatyieldsthelowerobjectivefunctionvalue.Subproblemforproducttypepcanbesolvedutilizingasimilarapproach.Wenextpresentthesubproblemformulationforproductpwhenthecapacityconstraint( 5{24 )isrelaxed:minE[C(xp;zpj)]=Cnpzp+(cnp+p+hp)xp+(crp+hp)E[rp]hpdp+(p+hp)Zdpxp0(dpxprp)fp(rp)drp subjecttoxpMzp (5{29) Foranygiven,zpcanbeeitherzeroorone.Hence,thesolutionsforbothcasescanbeevaluatedtoidentifytheoptimalsolution.Foragiven,whenzp=1theprocurementquantity,x0pisgivenby:x0p=8>>><>>>:dpFpcnp+p+hp
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ThesecondandthirdstepsofAlgorithmAPPP2canbemodiedtosolvethesubproblems.Thatis,foragivenLagrangianmultiplier,twosolutionsforeachproducttypephavetobecompared,i.e.,(x0p;1)and(0,0),andtheonethatyieldsthelowerobjectivefunctionvalueistheoptimalsolutionfortheproduct.Hence,wecanuseAlgorithmAPPP2,whenthereisaxedcostofprocurement. (5{30) subjecttox+r1(w)d1=I1(w) (5{31)It1(w)+rt(w)dt=It(w)t=2;:::;T (5{33) 107
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5{30 )minimizesthetotalexpectedcost.Thersttermisthetotalprocurementcost,secondtermistheexpectedprocessingcostofreturns,andthethirdandfourthtermsaretotalexpectedinventoryholdingandbackorderingcosts,respectively.Constraint( 5{31 )ensurestheinventorybalanceconstraintfortherstperiod.Constraintset( 5{32 )ensurestheinventorybalanceforperiods2throughendoftheplanninghorizon.Constraint( 5{33 )ensuresthenonnegativityofprocurementquantity. Byusing( 5{31 )and( 5{32 ),wecanndthenetendinginventoryinanyperiodtintermsofprocurementquantity,x,cumulativedemand,Dt,andcumulativereturn,Rt(w)asfollows: Notethatforanyperiodtifx
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Since( 5{38 )isnon-negative,Lt(x)isconvex. Wenowreformulatetheproblemwithoutinventoryvariablesbyusing( 5{34 )and( 5{36 )asfollows:minE[C(x)]=cnx+TXt=1(crtE[rt]+Lt(x)) (5{39) subjecttox0 (5{40) Recallthattheinventoryholdingandbackorderingcostsateachperioddependsontheprocurementquantity,x.Byusingdierentintervalsofx,theobjectivefunction,( 5{39 ),canberewrittenasfollows:
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TherstorderconditionofE[C(x)]isasfollows: Notethatthederivative,E0[C(x)],isnondecreasinginxfortworeasons,(i)Fj(Djx)isnonincreasinginx,and(ii)tincreasesinthesummationPTj=t(hj+j)Fj(Djx).WeformallypresentthispropertyinthefollowingProposition. Proof. 5{41 ),xcanbeinanyoftheT+1intervals.LetDT+1bedenedasDT+1=1.Intervalforperiodtcanbedenedas[Dt1;Dt)fort=1;:::;T+1.ItisenoughtoshowthatthederivativeofE[C(x)]is(i)nondecreasinginanyintervalt,i.e.,[Dt1;Dt),and(ii)nondecreasingthroughincreasingintervals,i.e.,t0andt00with1t0
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Thedierencebetweentherstderivativevalueswhenprocurementquantitiesarex2andx1is: SinceFj(Djx1)Fj(Djx2)forj=t;:::;T. (ii)Letx1andx2betwoprocurementquantitieswithx12[Dt01;Dt0)andx22[Dt001;Dt00),where1t0
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Proof. (i) (ii) (iii) Proof. (i) SinceE0[C(Dt)]<0,E0[C(DT)]>0andE0[C(x)]continuousandnondecreasing,thereexistssomeDt0,thenx=0,otherwise,0
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Initialization:Initializek1;xmin0;xmaxD1 Calculaterstderivativevaluesatxminandxmax:E0[C(x)]jxmin=cn+TXt=1htTXj=k(hj+j)Fj(Djxmin)E0[C(x)]jxmax=cn+TXt=1htTXj=k(hj+j)Fj(Djxmax) Checkconditions: { ifE0[C(x)]jxmax<0,thenkk+1;xminDk1;xmaxDk,andgotoStep1. { ifE0[C(x)]jxmax=0,thenx=xmax,andgotoStep4. { ifE0[C(x)]jxmax>0,thenkk,andgotoStep3. Findtheoptimalsolutionvialinesearch:Notethatx2(Dk;Dk+1).Solvethefollowingequation,andidentifyx. ReportSolution:Reportx. Wenotethatwhenk=T,therstderivativeatx=DTisalwayspositive,hence,thesolutioncannotbeunbounded.AlgorithmAPPP3identiesasolutionafteranitenumberofiterations,becausesameintervalwillnotberevisited.TheperformanceofthealgorithmalsodependsonhowecientlythelinesearchidentiestheoptimalsolutionatStep3. 113
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subjecttoxp+rp1(w)dp1=Ip1(w)p=1;:::;P Objectivefunction( 5{42 )minimizesthesumofexpectedprocurement,processing,inventoryholdingandbackorderingcoststhroughouttheplanninghorizon.Constraintset( 5{43 )capturestheinventorybalanceequationsforproductpinperiod1.Constraintset( 5{32 )representstheinventorybalanceequationsforproductpineachperiodtfort=2;:::;T,whichensuresthatforeachproductpthesumofquantityininventorycarriedintoperiodtandthequantityofreturnsprocessedisequaltothesumofthedemandinperiodtandthequantityofinventorycarriedtoperiodt+1.Constraint( 5{45 )ensuresthatthetotalprocurementcapacityrequiredforalltheproductsdoesnotexceedtheavailableprocurementcapacity.Constraintset( 5{46 )ensuresthenonnegativityofthedecisionvariables. Byusing( 5{43 )and( 5{44 )foreachproducttypep,wecanndthenetendinginventoryinperiodtintermsofprocurementquantityxpfortheproducttypep
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Notethatforanyproductpforanyperiodtifxp0denotetheassociatedLagrangianmultiplier.When( 5{50 )isrelaxed,( 5{49 )-( 5{51 )becomesseparableforeachproducttype.ThecorrespondingLagrangian 116
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ForagivenLagrangianmultiplierwehave:minE[C(!xj)]=PXp=1(cnp+p)xp+TXt=1crptE[rpt]+Lpt(xp)!K subjecttoxp0p=1;:::;P Notethatthelasttermof( 5{52 )isconstantandtheproblem( 5{52 )-( 5{53 )isseparableforeachproducttypep.Hence,foragivenLagrangianmultiplierwehavePsubproblems.Subproblemforproducttypepisasfollows:minE[C(xpj)]=(cnp+p)xp+TXt=1crptE[rpt]+Lpt(xp) subjecttoxp0 (5{55) SubproblemforproducttypepissameasPPP3problem.Therefore,AlgorithmAPPP3canbeusedtosolveeachsubproblem.WedevelopanalgorithmsimilartoAlgorithmAPPP2thatseeksfortheoptimal.WerstsolveeachsubproblemwhentheLagrangianmultiplierisequaltozero,i.e.,=0.Ifthesolutionobtainedsatisesthecapacityconstraint,thenthissolutionisalsooptimaltooriginalproblem,( 5{49 )-( 5{51 ).Otherwise, 117
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Initialization:Initializek0;k0;min0; FindanInitialSolution:Findxkpbysolvingsubproblem( 5{54 )-( 5{55 )forproductpbyusingAlgorithmASPMS,forp=1;:::;P FindaSolution:Setk=min+max 5{54 )-( 5{55 )forproductpbyusingAlgorithmASPMS,forp=1;:::;P ReportSolution:Reportxp=xkpforp=1;:::;P,and=k. InAlgorithmAPPP4werstinitializetheiterationcounter,andboundsof.Wethenndaninitialsolutionconsideringnocapacityconstraint,i.e.,when=0inStep1byusingtheAlgorithmAPPP3.Ifthissolutionisfeasiblethenitisalsooptimalto( 5{49 )-( 5{51 ).Otherwise,wesearchforoptimalvalueiteratively,byhalvingtheintervalofateachiteration.Whenwendasolutionthatsatises( 5{51 )withanerrorboundof,westopthesearchandreportthecurrentsolutionasanoptimalsolution. Assumethatthereisaxedcostcomponentforprocurementforeachproductp,Cnp,inadditiontolinearcostcomponent.WecanuseAlgorithmPPP4tosolvethisproblem. 118
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AnimmediateextensionofourworkthatwouldbeofpracticalrelevanceistoconsiderageneralizationofthePPPwhere(i)demandcanbemodeledasarandomvariabletotakedemanduncertaintyintoaccountand/or(ii)multipleprocurementopportunitiesareconsideredthroughouttheplanninghorizon. 119
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Inourstudy,wefocusedonthecharacterizationoftheoptimalinventoryandproductionplanningpoliciesformulti-productCLSCsystemsfordirectreuseandvalue-addedrecoverybyconsideringaclassofdeterministicdiscretetimedynamicdemandandreturnmodels.Wealsoconsideraprocurementplanningproblemfordirectreusesystemswithdeterministicdiscretetimedynamicdemandandstochasticreturns.Mostofthecurrentliteratureoninventoryandproductionplanningforproductrecoverymodelsforuncapacitatedsingle-productenvironments.Hence,ourcontributiontoliteratureisrelatedtoconsideringmulti-productenvironmentswithcapacityrestrictions.Moreover,wedevelopecientsolutionapproachesthatarebasedondynamicprogramming,relaxation,decomposition,andmeta-heuristicapproachestoaddressthisclassofmodels. Intherstpartofourstudy,wefocusedonreplenishmentanddisposalplanningproblemformulti-productdirectreusesystems.TheproblemofinterestwasmotivatedbyOEMinthepowergenerationequipmentindustrythatprovidessparepartkitsforpowerturbinemaintenanceservices.Toensureuninterruptedsparepartkitavailability,theOEMhastocoordinatereplenishmentanddisposaldecisionsforthekits.Wedeterminethetimingofthereplenishmentanddisposalsetups,alongwiththeassociatedreplenishmentanddisposalquantitiesformultipleproducttypesthroughoutaniteplanninghorizonsoastominimizevariablereplenishment,disposal,andinventoryholdingcostsaswellasxedreplenishmentanddisposalcosts.Inparticular,weconsideredtwovariantsoftheproblem.Intherstvariant,thereplenishmentcapacityissharedamongmultipleproducttypeswhilethedisposalcapacityisproductspecic.Inthesecondvariant,boththereplenishmentanddisposalcapacitiesaresharedamongmultipleproducttypes.Wedevelopedecientsolutionapproachforbothvariantsoftheproblems.Ourcomputationalresultsshowthattheproposedapproachesiseectiveinobtaininghigh-qualitysolutionsforrealistically-sizedinstancesoftheRDPPwithareasonable 120
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Inthesecondpartofourstudy,westudiedamanufacturing,remanufacturinganddisposalplanningproblemwithmultipleproducttypesforvalue-addedrecoverysystems.Problemofinterestismotivatedbytheoriginalequipmentmanufacturerintheautomotiveindustrythatprovidesvehiclemaintenanceandrepairservicesforvehiclesforwhichreplacementparts(products)areoftenneeded.Wedevelopedtwosolutionapproaches:LagrangianDecomposition(LD)andLagrangianRelaxation(LR).InLDapproach,wedecomposetheproblemintosubproblemsthataresolvedbypolynomialtimeDPapproaches.InLRapproach,however,weuseCPLEXtosolvetheresultingsingleproductproblems.OurcomputationalresultsshowedthatalthoughtheLDapproachiscomputationallymoreecient,theLRapproachiseectiveinndinghigh-qualitysolutions.ConsideringageneralizationoftheMRDPPwhere(i)themanufacturingand/orremanufacturingand/ordisposalsetupcostsdependonthespecicproductsincludedinthesetupand/or(ii)setuptimesareexplicitlymodeledwouldbepracticallyrelevantextensionofourwork.Moreover,MRDPPwithsharedremanufacturingcapacityand/orlineardisposalcostscanbepossibleextensions.Oursolutionalgorithmscanstillbeappliedtotheseproblemswithslightmodications.Also,theLRapproachpresentedinthispapercanbeextendedtosettingswherecertaincapacitatedresourcesaresharedbythemanufacturingandremanufacturingactivities. Inthelastpartofourstudy,wefocusedonprocurementplanningproblemformulti-productdirectreusesystems.Theproblemofinterestwasmotivatedbycatalogueretailingapplicationswherereturnsareinas-good-asnewconditionandcanbeusedtosatisfydemandwithlittleprocessing.Tothisend,weconsideredadiscretetimedynamicdeterministicdemandandstochasticreturnmodelforthecharacterizationofoptimal.Westudytheprocurementplanningproblemsincatalogueretailingapplications 121
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_IbrahimKarakayalearnedB.S.andM.S.degreesinindustrialengineeringfromtheMiddleEastTechnicalUniversityinAnkara,Turkey,in2001and2003,respectively.HebeganhisgraduatestudiesattheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridainAugust2003.HewasaninternintheEnterpriseOptimizationGroupatUnitedAirlinesinChicago,Illinois.Hismainareasofresearchareproductionplanningandcontrol,mathematicalprogramming,andclosed-loopsupplychains. 129
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