<%BANNER%>

Efficient Approaches to Integrated Requirements Planning Problems in Supply Chain Optimization


PAGE 4

Ioweadebtofgratitudetomysupervisor,EdwinRomeijn,forhispatienceduringmydissertationandallthethingsIhavelearnedbybeinghisstudent.Iamverygratefultomymum,FeyzaAlisan,tomyfather,MutAlisan,tomybrotherOnurAlisan,andUmut'sfamilyfortheircontinuoussupport.Myfatherneverstoppedsendingmeemailsandmakingmesmile.MaybeIhaveknownhimbetterduringthesefouryearsthanthroughoutmyentirelife.AlthoughNietzschesawhopeasthelatestevilleftPandora'sbox,hopeisstillthegreatestthingfortheworldandme.Umut,Icouldnotdoitwithoutyou. iv

PAGE 5

page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ ix ABSTRACT .................................... xi CHAPTER 1INTRODUCTION .............................. 1 2CAPACITATEDREQUIREMENTSPLANNINGWITHPRICINGFLEXIBILITYANDGENERALCOSTANDREVENUEFUNCTIONS 9 2.1Introduction ............................... 9 2.2BackgroundInformation ........................ 10 2.2.1DynamicLot-SizingProblem .................. 10 2.2.2FullyPolynomialTimeApproximationSchemesforCLSP .. 14 2.3ModelFormulation ........................... 18 2.3.1RequirementsPlanningwithPricingFlexibility ........ 18 2.3.2CapacitatedEconomicLot-SizingFormulation ........ 21 2.4PropertiesoftheNewProcurementCostFunction .......... 24 2.5Results .................................. 28 2.5.1DynamicProgramming ..................... 28 2.5.2FullyPolynomialTimeApproximationScheme ........ 30 2.6Conclusions ............................... 34 3LOT-SIZINGWITHNON-STATIONARYCUMULATIVECAPACITIES 36 3.1Introduction ............................... 36 3.2ModelFormulation ........................... 37 3.2.1Two-LevelLot-SizingwithNon-stationaryProductionCapacities ........................... 37 3.2.2Single-LevelLot-SizingwithCumulativeCapacities ..... 40 3.3GeneralCostFunctions ......................... 41 3.3.1ProofofNP-Hardness ...................... 42 3.3.2AFullyPolynomialTimeApproximationScheme ...... 43 3.4ConcaveCosts .............................. 47 3.4.1Introduction ........................... 47 v

PAGE 6

........................ 49 3.4.3ComputingtheSubplanCosts ................. 50 3.5AllowingforBacklogging ........................ 54 3.6SummaryandDirectionsforFutureResearch ............ 55 4CAPACITATEDPRODUCTIONANDSUBCONTRACTINGINASERIALSUPPLYCHAIN ......................... 57 4.1Introduction ............................... 57 4.2ProblemFormulationandSolutionApproach ............. 63 4.2.1ProblemFormulation ...................... 63 4.2.2SolutionApproach ........................ 66 4.2.3OnlyBackloggingattheManufacturerLevel ......... 69 4.3ModelswithoutSubcontractingOpportunities ............ 71 4.3.1SubplanProperties ....................... 71 4.3.2InventoryHoldingandBackloggingattheManufacturerLevel 72 4.3.3ModelwithonlyBackloggingattheManufacturerLevel ... 77 4.3.4Evaluation ............................ 78 4.3.5ManagerialInsights ....................... 79 4.4ModelswithSubcontractingOpportunities .............. 80 4.4.1GeneralPropertiesoftheSubplans .............. 80 4.4.2ConcaveCosts .......................... 81 4.4.3SubcontractingversusOvertimeProductionOption ..... 84 4.5TheUncapacitatedCase ........................ 87 4.5.1BothInventoryHoldingandBackloggingattheManufacturer 88 4.5.2OnlyBackloggingattheManufacturer ............ 90 4.6SummaryandConcludingRemarks .................. 91 5MULTI-WAREHOUSEMULTI-RETAILERNEWSVENDORPROBLEM 93 5.1IntroductionandRelatedLiterature .................. 93 5.2Multi-SourcingTransportationProblem ................ 98 5.2.1ModelFormulation ....................... 98 5.2.2PropertiesoftheModelandtheOptimalSolution ...... 100 5.2.3LagrangeanRelaxationApproach ............... 102 5.2.4MinimumConvexCostFlowRepresentation ......... 105 5.2.5Fixed-ChargeTransportationCosts .............. 107 5.2.6GeneralDemandDistributions ................. 109 5.3ABranch-and-PriceAlgorithmforSingle-SourcingTransportationProblem ................................ 111 5.3.1ModelFormulation ....................... 111 5.3.2ASet-PartitioningFormulation ................ 114 5.3.3ThePricingProblemforLPSP ................. 118 5.3.4AGeneralClassofProblems .................. 119 5.3.5SolvingPricingProblemtoOptimality ............ 125 5.3.6BranchingRuleforBranch-and-Price ............. 126 vi

PAGE 7

............... 127 5.4.1Fixed-ChargeTransportationCosts .............. 127 5.4.2MoreGeneralProcurementCostFunctions .......... 128 5.4.3CapacityExpansion ....................... 131 5.5Discussion ................................ 135 5.6ComputationalResults ......................... 137 5.6.1ExperimentalDesign ...................... 137 5.6.2Branch-and-PriceStrategies .................. 139 5.6.3AHeuristicforSSTP ...................... 141 5.6.4ComputationalResults ..................... 141 5.6.5DiscussionoftheResults .................... 156 5.7ConclusionsandFutureResearchDirections ............. 158 6CONCLUSIONS ............................... 160 APPENDIX ASUMMARYOFCOMPUTATIONALRESULTSFORHIGHCAPACITY 162 BSUMMARYOFCOMPUTATIONALRESULTSFORLOWCAPACITY 169 REFERENCES ................................... 176 BIOGRAPHICALSKETCH ............................ 181 vii

PAGE 8

Table page 4-1Runningtimesformodelswithoutsubcontractingopportunities. .... 91 4-2Runningtimesformodelswithsubcontractingopportunities. ....... 92 5-1Zeroxedcostoftransportation(highcapacity). ............. 143 5-2Zeroxedcostoftransportation(lowcapacity). .............. 144 5-3Lowxedcostoftransportation(highcapacity). .............. 144 5-4Lowxedcostoftransportation(lowcapacity). .............. 144 5-5Highxedcostoftransportation(highcapacity). ............. 145 5-6Highxedcostoftransportation(lowcapacity). .............. 145 5-7Thepercentageofthepricingproblemsthatbranch-and-boundisused. 145 5-8Branchingrulecomparison. ......................... 152 5-9Datasets. ................................... 156 A-1Zeroxed-costcase. ............................. 163 A-2Highxed-costcase. ............................. 165 A-3Lowxed-costcase. .............................. 167 B-1Zeroxed-costcase. ............................. 170 B-2Highxed-costcase. ............................. 172 B-3Lowxed-costcase. .............................. 174 viii

PAGE 9

Figure page 2-1NetworkowrepresentationofDLSP. ................... 13 2-2Demand(a)asadecisionvariable;(b)deterministic. ........... 22 3-1Productioncostfunctioninperiodt. .................... 43 3-2NetworkowrepresentationofLSP-CC. .................. 48 3-3Extremepointsolutionwith2subplansandtheassociatedarcows. .. 52 4-1Networkrepresentationwithinventoryholdingandbacklogging. .... 66 4-2Networkrepresentationwithbackloggingonlyatthemanufacturer. .. 70 4-3Networkrepresentationwithovertimeproduction. ............ 85 5-1Warehouse-retailernetwork,with3warehousesand4retailers. ..... 98 5-2Convexcostnetworkowrepresentationforn=m=3. ......... 106 5-3Transportationnetworkinthecaseofatspots. ............. 111 5-4j(Qj):(a)zjcannotbedetermined(b)zj=1. ............. 120 5-5 ................................ 121 5-6Pni=1Qjversus:(a)fractionalsolution(b)integersolution. ...... 124 5-7iversuseifortwocases. .......................... 133 5-8Branch-and-pricetreeofaproblem. .................... 142 5-9SSTPCPUtimes(highcapacity). ..................... 146 5-10SSTPCPUtimes(lowcapacity). ...................... 146 5-11SSTPsolutionvalues(highcapacity). ................... 147 5-12SSTPsolutionvalues(zeroxedcost). ................... 148 5-13SSTPsolutionvalues(lowxedcost). .................... 148 5-14SSTPsolutionvalues(highxedcost). ................... 149 ix

PAGE 10

..... 149 5-16Heuristicandbranch-and-pricesolutiontimes(lowcapacity). ....... 150 5-17Heuristicandbranch-and-pricesolutionvalues. .............. 150 5-18Timecomparisonforpricingstrategies(highcapacity). .......... 151 5-19Timecomparisonforpricingstrategies(lowcapacity). ........... 151 5-20Pricingstrategies(lowcapacity). ...................... 152 5-21Columnsgeneratedfortwopricingrules(lowcapacity). .......... 153 5-22Pricingstrategies(highcapacity). ...................... 153 5-23Columnsgeneratedfortwopricingrules(highcapacity). ......... 154 5-24ComparisonofMSTPandSSTPsolutionvaluesasmchanges. ...... 154 5-25Rasmchanges. ............................... 155 5-26Rforallm;npairs. .............................. 155 5-27Locationsofthewarehousesandtheretailersindataset6. ........ 156 5-28EectsofparametersonR. ......................... 157 5-29Rforalldatasets. .............................. 157 x

PAGE 11

Instandarddynamiceconomiclot-sizingmodels,asequenceofdemandsforasinglegoodoveraniteanddiscreteplanninghorizonmustbesatisedatminimumproductionandinventoryholdingcost,wherethecostparametersanddemandsareassumedtobeknown.Inthisdissertation,westudyvariationsofdynamiceconomiclot-sizingmodelsbyintegratingmoredecisionsandaddingmoreconstraints.Foreachmodel,eitherweidentifypolynomiallysolvablecasesandpresentsolutionalgorithmsorpresentapproximationschemes. Inourrstmodel,weintegratepricingdecisionsandbackloggingundernonstationaryproductioncapacities.Wepresentafullypolynomialtimeapproximationschemeforthismodel. Inoursecondmodel,weconsiderasettingwhereanyremainingcapacityistransferredtothenextproductionperiod.Thisisincontrastwithtraditionalcapacitatedlot-sizingmodels,wherethequantityproducedineachperiodislimitedbysomecapacity,butanycapacityremainingattheendofaperiodisessentiallylost.WeprovethatthisproblemisNP-hardforgeneralcostfunctionsandprovideafullypolynomialtimeapproximationschemeforthecasewhenallcostfunctions xi

PAGE 12

Inourthirdmodel,weconsideratwo-echelonsupplychainconsistingofasupplierandamanufacturer,whereasequenceofdeterministicbutnonstationarydemandsofthemanufacturerforasinglegoodneedstobesatisedoveranitehorizon.Weassumestationaryproductioncapacities.Inparticular,weconsideranintegratedmodelthatminimizestotalsystemcosts,consistingofproduction,inventory,transportation,backlogging,andsubcontractingcosts. Inourlastmodel,weconsideramulti-warehousemulti-retailermodelinastochasticdemandsetting.Inthismodel,weconsiderasingleperiod,wherewedonotknowtheexactdemand,butitsdistribution.Moreover,warehouseshavecapacitiesontheamounttheycansupply.Foreachretailer,wehavenewsvendortypecosts,andwealsohavethecostsforassigningeachretailertoeachwarehouse.Wesolveanassignmentandcapacityallocationproblem. xii

PAGE 13

AccordingtoZipkin(2000),everyinventoryliesbetweentwoactivitiesorprocesseswhichwecallsupplyanddemand.Supplyprocessesareproduction,transportationoranyactivitythataddstothecurrentinventory,whiledemandprocessesaretheactivitiesthatsubtractmaterialfromthecurrentinventory. Inaninventorymodel,supplyanddemandprocessescanbemodelledinvariousways.Moreover,thetimecanbemodelledasdiscreteorcontinuous.Indiscretetimemodels,wemodeltimeasanitesequenceofdiscretetimepointsandaperiodisdenedastheintervalbetweentwoconsecutivetimepoints.Inthedynamiclot-sizingproblem(DLSP)(introducedbyWagnerandWhitin1958,demandsofasingleitemareobservedandorderingdecisionsaremadeatdiscretetimepointsatasinglelocation.Inaddition,thedemandsineachperiodandthecostparametersareknownandvaryovertime.Wecanorderasmuchaswewant,andwereceiveourorderimmediately.Therefore,DLSPhasanuncapacitatedsupplyprocessandatime-varyingdemandprocess.InDLSP,wetrytominimizetotalordering(production)andinventorycostsduringaniteplanninghorizon. ThesinglelocationvariantoftheDLSPhasreceivedalotofattentionintheliteratureanditissolvableinpolynomialtimeinthelengthofthetimehorizon(seeWagnerandWhitin1958).AggarwalandPark(1990),FedergruenandTzur(1991),andWagelmansetal.(1992)proposemoreecientalgorithmsforthespecialcasesofDLSP.Zangwill(1969)proposesanO(T3)dynamicprogrammingalgorithmwhenabackloggingopportunityispresent. Whenproductioncapacitiesarepresent,weobtainthecapacitatedlot-sizingproblem(CLSP).IncontrasttotheDLSP,thisproblemisknowntobeNP-hard 1

PAGE 14

formanyspecialcases(seeFlorianetal.1980andBitranandYanasse1972).Aninterestingandimportantspecialcasethatallowsapolynomialtimealgorithmariseswhenproductioncapacitiesarestationary.ThepapersthatpresentpolynomialtimealgorithmsinthecaseofstationaryproductioncapacitiesareFlorianandKlein(1972),Florianetal.(1980),andvanHoeselandWagelmans(1996).ChungandLin(1988)andvandenHeuvelandWagelmans(2006)presentpolynomialtimealgorithmsforanotherspecialcaseoftheCLSPwhenthexed-costsofproductionarenon-increasing,theholdingcostsarelinear,thelinearpartoftheproductioncostfunctionsarenon-increasing,andthecapacitiesarenon-decreasingovertime. AmoregeneralsettingofDLSPconsidersthedeterminationofoptimalpricelevels,andtherebydemands,forthegoodineachperiodjointlywiththeproductionandinventorydecisionsthattogethermaximizeprot.ThisproblemisstudiedbyThomas(1970)intheabsenceofcapacitiesandwhenthepriceoftheproductineachperiodisadecisionvariable.KunreutherandSchrage(1971)andGilbert(1999)considertheuncapacitatedproblemwhenasinglepricemustbeusedovertheentirehorizon.KunreutherandSchrage(1971)providedboundsontheoptimalsolutionvalueundertimevaryingproductioncostassumptions,whileGilbert(1999)assumedatime-invariantproductionsetupandholdingcosts,andprovidesanexactpolynomialtimealgorithm.ArecentpaperbyvandenHeuvelandWagelmanspresentsanoptimalpolynomialtimealgorithmforKunreutherandSchrage'sproblem.Geunesetal.(2005)presentspolynomialtimealgorithmsfortheuncapacitatedandstationaryproductioncapacitiescaseswhentherevenuefunctionineachperiodisapiecewiselinearconcavefunctionoftheprice.Billeretal.(2005)developedagreedyalgorithmforthevariantoftheprobleminwhichproductioncapacitiesarenonstationarybutallcostfunctionsarelinear.Gilbert(2000)alsoconsidersasituationinwhichpricingandproduction

PAGE 15

decisionsmustbemadeformultipleconstant-pricedproductsthatsharethesameproductioncapacity.DengandYano(2006)considerbothset-upcostsandnonstationaryproductioncapacitiesintheirproblemandshowthattakingpricesasdecisionvariablesdoesnotchangethestructureoftheoptimalproductiondecisionscharacterizedbyFlorianandKlein(1971).Geunesetal.(2005)presentsasolutionapproachforeconomiesofscaleinproduction(xed-coststructure),price-sensitivedemand(generalconcaverevenuefunctions),andtime-invariantproductioncapacities.Theyprovidepolynomial-timesolutionmethodsforthisproblemwhenwemaysetdierentpricesforthegoodineachperiodandwesetaconstantpriceforthegoodovertheentireplanninghorizon. Subcontractingistheprocurementofanitemorservicewhichisnormallycapableofeconomicproductionintheprimecontractor'sownfacilitiesandwhichrequirestheprimecontractortomakespecicationsavailabletothesupplier(Day1956).However,overtimeproductionoccursattheendoftheperiodandincursrelativelyhighproductioncosts.InanothersettingofDLSP,regularproductioncapacitylevelsarealsodecisionvariables.AtamturkandHochbaum(2001)investigatesthetrade-obetweenacquiringcapacity,subcontracting,production,andholdinginventoryundervariouscoststructures,wheretheamountofsubcontractingisuncapacitated.Merzifonluogluetal.(2006)determinestheoptimalprice,production,inventory,subcontracting,overtime,andregularproductioncapacitylevels. AnotherresearchareaarisingfromDLSPisamulti-leveldynamiclot-sizingproblem.The50thanniversaryissueofthejournalManagementScienceisdevotedtothemostinuentialpapersduringtheseftyyears.OneofthesepapersisDynamicVersionoftheEconomicLotSizeModelbyHarveyM.WagnerandThomasM.Whitin.HarveyM.Wagnerwritesacommentintheirpaperandpredictsthatacandidatepaperforthenext50thanniversaryissueofManagement

PAGE 16

Scienceisapaperthatpresentsanecientsolutionmethodtoadynamiclot-sizingmodelthatencompassesamulti-echelonenvironment.Zangwill(1969)studiedtheuncapacitatedversionofthemulti-levellot-sizingproblem,anddevelopedadynamicprogrammingalgorithmthatispolynomialinboththeplanninghorizonandthenumberoflevels.Leeetal.(2003)consideratwo-levelmodelwherethetransportationcostsarenonconcavefunctions.KaminskyandSimchi-Levi(2003)proposeathree-levelmodelinwhichtherstandthirdlevelsareproductionstages,andthesecondlevelisatransportationstage.Bothproductionstagesarecapacitated,whilethetransportationstageisuncapacitated.Theyconsiderlinearinventoryholdingcoststhatincreasewiththelevelofthesupplychain,andlinearproductioncostsatbothlevels1and3thatsatisfyatraditionalnonspeculativemotivescondition.Thetransportationcostsatthesecondlevelareofthexed-chargeorgeneralconcaveformandareassumedtosatisfyarestrictiveandnontraditionalnonspeculativemotivescondition.Theyeliminatethethird-levelproductiondecisionsandreducetheproblemtoatwo-levelmodel.Forthexed-chargetransportationcosts,theyprovideapolynomialtimealgorithm.Forconcavetransportationcosts,theyprovideapolynomialtimealgorithminthepresenceofstationaryproductioncapacities.VanHoeseletal.(2005)proposesapolynomialtimealgorithmwhichiscomputationallylessexpensiveevenwhenallcostfunctionsareconcave.Theyalsoconsidermulti-levelmodelsandproposeanalgorithmthatrunsinpolynomialtimeinthenumberofperiodsandexponentialinthenumberoflevelswhenallcostfunctionsareconcave.However,wheninventoryholdingcostsarelinearandtransportationcostseitherarelinearorhavexed-chargestructureandsatisfyaformofnonspeculativemotives,runningtimesarepolynomialinthenumberofperiodsandrelativelyinsensitivetothenumberoflevels.

PAGE 17

Inthisthesis,werstaddresssomeextensionsoftheDLSP.InChapter2,werstgivebackgroundinformationaboutDLSPandfullypolynomialtimeapproximationschemes(FPTAS).Wethenintroducetherequirementsplanningwithpricingexibilityproblem(RPP).Thisproblemconsidersthepricing(orhowmuchdemandtosatisfy)andtheproductiondecisionsinasinglelocation.Intheliterature,polynomiallysolvablecasesofRPParewellstudied(seeThomas1970,Geunesetal.2005,andBilleretal.2003,DengandYano2006,vandenHeuvelandWagelmans2006,Gilbert1999,Geunesetal.2005).However,weconsiderRPPwithgeneralcostandrevenuefunctions,andalsonon-stationaryproductioncapacities.WepresenttherstFPTASforRPP. InChapter3,weagainconsiderasinglelocationmodel.However,wehaveadierentcapacityconstraint,whichwecallcumulativecapacityconstraint,whichstatesthattheremainingcapacitiesinthepreviousperiodscanbeusedinthecomingperiods.Weassumenon-stationaryproductioncapacitiesforeachperiod.WeshowthatthisproblemisNP-hardwithgeneralproductioncostfunctionsandzeroinventorycosts.Asopposedtothetraditionaldynamiclot-sizingproblemwithnonstationaryproductioncapacities,weshowthatthisproblemissolvableinpolynomialtimewhenallcostfunctionsareconcave. InChapter4,weanalyzeatwo-echelonsupplychainconsistingofasupplierandamanufacturer,whereasequenceofdeterministicbutnonstationarydemandsofthemanufacturerforasingleitemneedstobesatisedoveranitehorizon.OurresultsgeneralizevanHoeseletal.(2005)andAtatumturkandHochbaum(2001).Inparticular,weconsideranintegratedmodelthatminimizestotalsystemcosts,consistingofproduction,inventory(holdingandbacklogging),transportation,andsubcontractingcosts.Thisisappropriateincaseswhere,forexample,thesupplierisacaptivesubsidiaryofthemanufacturer.Ineachperiod,theproductionquantityislimitedbyanitebutstationarycapacity.Alternatively,unitsmay

PAGE 18

beobtainedbysubcontractingfromanoutsidesupplier,wherethecapacityoftheoutsidesuppliermayormaynotbelimited.Notethatthissubcontractingoptionmaybeusedregardlessofwhethertheproductioncapacityinagivenperiodisexhausted.However,wewillshowhowourapproachcanbemodiedtoallowforanalternativesourcingoptionthatcanonlybeusedifproductionisatfullcapacity,andthereforecanbeinterpretedasanovertimeproductionoption.Tothebestofourknowledge,thisstudyistherstmulti-leveldynamiclot-sizingmodelthatencompassessubcontractingandovertimedecisions.Wepresentpolynomialtimedynamicprogrammingalgorithmsforthevariationsofthemodelwheresubcontracting,production,andovertimedecisionsarecapacitatedoruncapacitated,andwhenbackloggingisallowedornot.Weconsiderthecasewhenallcostfunctionsareconcave.Moreover,wepresentthealgorithmswithimprovedrunningtimeswhenwehavelinearinventorycostsandeitherxed-chargetransportationcostswithspeculativemotivesorlineartransportationcosts.Wealsoconsiderthecasewhereholdingattheretailerisnotallowed;inotherwords,theretailerpreferstobackloginsteadofcarryinginventory.Forthiscase,weshowthattherunningtimeofthedynamicprogrammingalgorithmimprovesforcapacitatedproductioncases. Thesecondthemethisthesisaddressesisbasedonthenewsvendorproblem.Newsvendor-basedmodelsareusedwhenthedemandoftheitemisarandomvariableandtheitemhasashortusefullife.Inthesemodels,weconsiderasingleperiod,inwhichweorderatthebeginningoftheperiodandthendemandoccursattheendoftheperiod.Firstofall,wepayaunitpurchasecostforeachitemweorder.Moreover,accordingtotheobserveddemand,weincursomecosts.Anyinventoryremainingattheendoftheperiodhassomesalvagevalue,whichislessthanunitpurchasecost.However,thereisapenalty(forinstance,duetolossofgoodwill)associatedwithanyunitofdemandthatcannotbesatised.Since

PAGE 19

demandisarandomvariable,theretaileraimstominimizehis/herexpectedcostinasingleperiod. ForsingleproductandsinglelocationnewsvendorproblemsLau(1997)presentsclosed-formexpressionsforcomputingtheexpectedtotalcostsforvariousdemanddistributions.Intheliterature,manystudiesextendthesingleproductsinglefacilitynewsvendorproblembyconsideringaxed-costassociatedwiththereplenishment(seeMoonandSilver2000),quantitydiscounts(seeJuckerandRosenblatt1985,PantumsinchaiandKnowles1991),pricedependentdemand(seeLauandLau1988),andtheuseofadierentobjectivesuchasmaximizationoftheprobabilitythatatleastacertainprotlevelisachieved(seeSankarasubramanianandKumanraswamy1983). Eppen(1979),ChenandLin(1989),andChangandLin(1991)considermulti-locationnewsvendorproblems.LauandLau(1996)considersamultiple-productmodelwithsingleandmultiplecapacityconstraints.ThesolutionofthemodelwithsinglecapacityconstraintsextendsHadleyandWhitin(1963)(pp306-307)byallowingallkindsofcontinuousdemanddistributions.Tosolvethemodelwithmultipleconstraints,theyemployactivesetmethods,whichisproposedinLuenberger(1973).Theyalsoconsiderthecasewherethedemanddistributionhasanonzerolowerbound.Theotheranalysisofmulti-commoditytypesystems,withorwithoutcapacityconstraints,canbefoundinSivazlianandStanfel(1975),MoonandSilver(2000),andLauandLau(1995).AnexcellentreviewonthenewsvendorproblemhasbeenrecentlyprovidedbyKhouja(1999). InChapter5,weextendtheclassicalnewsvendorproblemtoacapacitatedmulti-warehousemulti-retailersetting.Thisproblemintegratesinventoryandtransportationdecisionsinthepresenceofcapacitiesthatconstrainthequantityofproductthatcanbesuppliedfromeachwarehouse.Asintheclassicalnewsvendorproblem,weconsiderthecaseofasingleproduct.Weconsidertwoversionsof

PAGE 20

theproblem:withorwithoutsingle-sourcingrestriction,whichenforcesthateachretailercanbeshippedbyatmostonewarehouse.Wecomparethesolutionsofthesingleandmulti-sourcingversionsandreportthecaseswheretheiroptimalobjectivevalueshavesignicantdierence.Wesolvesingle-sourcingproblemusingbranch-and-pricealgorithmandourpricingproblemisaninterestingprobleminitsownright.Itisanonlinearintegerknapsackproblemwhoseitemsizescanbeinuencedandwepresenttwosolutionmethods,oneofwhichgivesanapproximatesolutionandtheotheranoptimalsolution.

PAGE 21

Sinceitiswellknownthateventhestandardeconomiclot-sizingproblemwithtime-varyingproductioncapacitiesisNP-hard,evenwhenallcostfunctionsareconcave(seeFlorianetal.1980),wefocusondevelopingapseudo-polynomial-timedynamicprogramming(DP)algorithmaswellasafullypolynomialtimeapproximationscheme(FPTAS)fortheRPP(seeSargutandRomeijn2006a). 9

PAGE 22

Ourapproachisbasedonareformulationoftheintegratedpricingandlot-sizingproblemasapurelot-sizingproblemwithacomplexprocurementcostfunction(thatitselfinvolvessolvinganoptimizationproblem)inSection 2.3 .Wenextproceed,inSection 2.4 ,byanalyzingpropertiesofthisnewprocurementcostfunctionunderdierentcoststructuresintheoriginalproblem.WethendrawheavilyuponearlierpapersbyFlorianetal.(1980),inwhichaDPalgorithmfortheeconomiclot-sizingproblemisproposed,aswellasvanHoeselandWagelmans(2001),inwhichaFPTASfortheeconomiclot-sizingproblemisdeveloped.Inparticular,Section 2.5 isdevotedtotheanalysisoftherunningtimeofthesetwoapproachesasappliedtothelot-sizingformulationoftheRPP.WeendthechapterinSection 2.6 withsomeconcludingremarks. 2.2.1DynamicLot-SizingProblem WecanstatetheDLSPforasinglefacilitywithgeneralcosttermsas: minTXt=1(pt(xt)+ht(It))

PAGE 23

subjectto(DLSP) whereTisthelengthofthetheplanninghorizon.Foreachperiodt=1;:::;T,wedene: Ourdecisionvariablesare: IntheDLSPformulation,constraints 2{1 maintainbalanceofinowandoutowatthefacilityandconstraints 2{2 enforcethenonnegativityofthevariablesxtandIt. WagnerandWhitin(1958)developsoneofthesimplestandearliestdynamiclot-sizingmodelintheliterature.Inthismodel,orderingcosthasxed-coststructure,inventoryholdingcostsarelinear,backloggingisnotallowed,andproductionisuncapacitated.Letusdenotethexed-costoforderinginperiodtasst.ThesizeofthedecisionspaceofthisproblemisO(2T),sinceineachperiodwedecideeithertoorderornot.However,WagnerandWhitin(1958)proposesa

PAGE 24

forwarddynamicprogrammingrecursionthatworksinO(T2)time:F(t)=min0j
PAGE 25

Figure2-1: NetworkowrepresentationofDLSP. thesumofdemandsofsomeconsecutiveperiodsincludingperiodt.ThechoiceofproductionperiodinasubplanresultsinaO(T3)runningtime. Weobtaincapacitatedlot-sizingproblem(CLSP)byaddingthefollowingconstraints:xtCtfort=1;:::;T; Wecancharacterizeasubplanbyitsdemandperiods.Inparticular,asubplan(t1;t2)isdenedasaconsecutivesequenceofperiodst1+1;:::;t2,whereIt1=It2=0andIi>0fori=t1+1;:::;t21.

PAGE 26

Therefore,FlorianandKlein(1971)concludesthatwhenproductioncapacitiesareconstant,C,weproduceatfullcapacityforperiods,where=bdt1++dt2

PAGE 27

Woeginger(2001)statesthatanFPTASisthestrongestpossiblepolynomialtimeapproximationresultthatwecanderiveforanNP-hardproblem. VanHoeselandWagelmans(2001)providestherstFPTASforCLSPwithnon-stationaryproductioncapacities,whenwehave Theyassumethatalldemands,capacities,production,andinventorylevelsareinteger.Moreover,allcostfunctionscanbeevaluatedinpolynomialtimeatanyvalueintheirdomainandarescaledsuchthattheyareintegervalued.TheystatethatitisunlikelytogetanFPTASwiththetraditionaldynamicprogrammingformulations(forexampletheformulationinFlorianetal.1980)anddeneadierentpseudo-polynomialdynamicprogrammingformulation.TheydeneFt(b)asthemaximuminventoryamountthatwecanhaveattheendoftheperiodtwithbudgetb.Fortherstperiod: whereBisanyupperboundontheobjectivefunctionvalue. Forperiodst=2;:::;Tandb=0;:::;B,weconditionthebudgetwespendinperiods1;:::;t1(saya)andwespendtheremainingamount,bainperiodt.Theyshowthatthebestendinginventorylevelofperiodt1,whenwespendtotallyaduringperiods1;:::;t1isFt1(a).Therefore,Ft(b)iswrittenintermsofallFt1(a)suchthatab:

PAGE 28

Therefore,optimalsolutionofCLSPwithnonstationaryproductioncapacitiesisminimumbsuchthatFT(b)0: 2{3 and 2{4 dependsonBpolynomially.Thenextstepistolimitthenumberofpossiblebudgetvalues.Tothataim,apositiveintegerKisselectedandthesetofpossiblebudgetsisreducedtothesetf0;K;2K;:::;(bB=Kc+T)Kg.Afterthisupdate,runningtimeofthedynamicprogrammingformulationdependsonB=Kpolynomially.Moreover,theyprovethatrestricteddynamicprogrammingformulationgivesasolutionvaluethatdeviatesfromoptimalsolutionatmostbyTK. Inthenextstep,theyshowhowtocomputeanupperboundonzwhichisatmost2Tz.ThisupperboundisusedasBinthedynamicprogrammingformulations.Withgivenrelativeerrorwithrespecttotheoptimalsolution,,andK=maxfbB=2T2c;1g: 1.zaz+TK=z+TmaxfbB=2T2c;1g(1+)z, 2.runningtimeispolynomialinB=K,whichisO(T2=). therefore,thisalgorithmbecomesanFTPAS. Recently,Chubanovetal.(2006)presentsanFPTASforCLSPwithmonotonecoststructurebyroundingatraditionaldynamicprogrammingalgorithmandexploitingthecombinatorialpropertiesoftherecursivefunction.Therefore,they

PAGE 29

disprovethestatementbyvanHoeselandWagelmans(2001)thatatraditionaldynamicprogrammingapproachisunlikelyyieldanFPTAS. Theysolvearoundedproblemwhoseoptimalsolutionisdenotedbyzrwiththeobjectivefunction:TXt=1(bpt(xt)=c+bht(It)=c); subjectto (2{6) Foranyt2[1;T]functiont(:)isevaluatedrecursivelyas: whereat1=Pti=1di,at2=Pti=1(Cidi),andfors2[at1;at2].Insteadofsolvingthisdynamicprogrammingformulationinthetraditionalway,theyprovethatt

PAGE 30

hasatmostO(t2V2)nonstablepoints(V=bU=c,whereUisaknownupperboundforz). 2{11 inoverthesetofthenonstablepoints: wheresabisanarbitrarypointbelongingtoaspeciedintervaldependsonaandb.Theyshowthatallnonstablepointscanbefoundinpolynomialtimeandrecursion 2{12 canbesolvableintimepolynomialin1=,T,andloglog(U=L),whereLisaknownlowerboundforz.TherunningtimeoftheresultingFPTASisO(T11(1 L)+T(1 L)TXt=1logtXi=1Ci): 2.3.1RequirementsPlanningwithPricingFlexibility

PAGE 31

functionRt,wherewithoutlossofgeneralityweassumethatRt(0)=0.Furthermore,weassumethattherevenuefunctionisnondecreasingon0dtDt,atwhichpointitattainsitsmaximum. wherehtisanonincreasingfunction,h+tisanondecreasingfunction,andclearlyht(0)=h+t(0)=0. Inaddition,wewillassumethattheproblemparametersCtandDt(t=1;:::;T)aswellasthedemand,production,andinventorylevelsareintegral.Wethenformulatetherequirementsplanningproblemwithpricingexibility(RPP)as:maximizeTXt=1(Rt(dt)pt(xt)ht(It)) subjectto (2{17) (2{18) Constraints 2{13 ensurethebalancebetweeninventory,backlog,productioninow,anddemandoutowatthemanufacturer.Constraints 2{14 aretheproductioncapacityconstraintsandconstraints 2{15 areimposedwithoutloss

PAGE 32

ofoptimality,indicatingthatsatisfyinganydemandinperiodtbeyondDtdoesnotyieldadditionalrevenueandwillnotdecreasecosts.Constraints 2{16 arethenonnegativityconstraintsonproductionanddemandsatisfaction,whileconstraints 2{17 and 2{18 indicatethatbothinventoryandbackloggingareallowed,butthatnopositivebackloggedamountisallowedtoremainattheendoftheplanninghorizon.Finally,constraints 2{19 ensurestheintegralityofthedecisionvariables.NotethattheinitialinventoryorbacklogI0isassumedgiven. NotethattherangeofprotvaluesthatcanbeachievedinthisformulationoftheRPPcontainsbothpositiveandnegativevalues.Zemel(1981)statesthatforsuchproblemstherelativeerrorwithrespecttotheoptimalsolutionvalueisnotameaningfulperformancemeasureofaheuristic.SinceoneofourgoalsistoobtainanecientapproximationschemefortheRPP,wewillnextshowhowwecan,undermildconditions,reformulateourproblemasapurecostminimizationproblem. Tothisend,wedeneasetofnewdecisionvariablesyt(t=1;:::;T)denotingtheamountofunsatiseddemandinperiodtthatispotentiallyprotable,i.e.,yt=Dtdt.Next,weintroducethelossfunctionLt,representingtherevenueshortfallasafunctionofthequantityofunsatiseddemand:Lt(yt)=Rt(Dt)Rt(Dtyt)=Rt(Dt)Rt(dt):

PAGE 33

Withthesemodications,weobtainthefollowingreformulationoftheRPPwhich,withaslightabuseofnotation,wewillstillrefertoastheRPP:minimizeTXt=1(pt(xt)+Lt(yt)+ht(It)) subjecttoxt+yt+It1=Dt+Itt=1;:::;TxtCtt=1;:::;TytDtt=1;:::;Txt;yt0t=1;:::;TItfreet=1;:::;T1IT0xt;yt;It2Zt=1;:::;T: ThemaindierencebetweentheRPPandtheCLSPisthatthedemandisadecisionvariableintheformerwhileitisxedinthelatter.However,thecostminimizationformulationfromSection 2.3 givesrisetoanalternativeinterpretationoftheRPP.Inparticular,wecouldviewtheRPPastheproblemof\satisfying"(orprocuring)thetotalpotentialdemandsDtthroughtwosources:atruesource(production,viatheproductionvariablesxt)andavirtualsource(viathevariablesyt).ThisisillustratedinFigure 3-2 .Furthermore,wemaynownotethatthedecisionvariablesxtandytbetweendierentperiodsareonlylinkedthrough

PAGE 34

Figure2-2: Demand(a)asadecisionvariable;(b)deterministic. theinventorybalanceconstraintsand,inaddition,theseonlydependonthetwovariablesthroughtheirsum.Therefore,wecandenenewdecisionvariableszt=xt+yt(t=1;:::;T)anddecideonthedecompositionofztintoxtandytonaperiod-by-periodbasisbasedoncosts.Inparticular,thecostofprocuringztunitsinperiodt,saykt,canbedeterminedasfollows:kt(zt)minxt;yt2Zfpt(xt)+Lt(yt):xt+yt=zt;0xtCt;0ytDtg:

PAGE 35

subjecttoIt1+zt=Dt+Itt=1;:::;TztCt+Dtt=1;:::;Tzt0t=1;:::;TItfreet=1;:::;T1IT0zt;It2Zt=1;:::;T: Intheremainderofthischapter,wewillstudytheDPapproachproposedbyFlorianetal.(1980)andtheapproximationschemeproposedbyvanHoeselandWagelmans(2001)inthecontextoftheRPP.Thecoreissueisthatthesuccessofknownapproachessuchasthesedependscriticallyontheirrunningtimes{whicharetypicallyderivedbasedontheassumptionthatthecostfunctionscanbeevaluatedinconstanttime.ThisisclearlynotthecasefortheRPPduetothefactthatevaluatingtheprocurementcostfunctionktinvolvessolvingapotentiallyveryhardglobaloptimizationproblem. Inthenextsection,wewilldiscusssomepropertiesofthenewprocurementcostfunction.

PAGE 37

(j=1;:::;n)as Usingthisdenition,weimmediatelyobtainthedesiredresult.2 (WerecallthatconvexityofafunctionfdenedonZmeansthatf(x+1)f(x)isnondecreasinginxandconcavityofafunctionfmeansthatf(x+1)f(x)isnonincreasinginx.) Thenextpropositionsderivetherunningtimerequiredtoevaluatetheprocurementcostfunctionkforthepiecewiseconvexandpiecewiseconcavecases.

PAGE 38

2{20 askij(z)=minx;y2Zfp(x)+L(y):x+y=z;Ci1x
PAGE 39

2.4.3 .ItiseasytoseethatifbothpandLareconcaveoneachsegmentspeciedbyagiveniandj,p(x)+L(zx)isaconcavefunctionofxonthatsegmentaswell.Sincetheminimumofaconcavefunctionoveranintervalisattainedatoneoftheendpointsoftheinterval,wecanndthevalueofkij(z)inO(1)time.Thus,byProposition 2.4.2 thevalueofk(z)canbefoundinO(mn)time. Itremainstoshowthatkijispiecewiseconcavewithnomorethanthreesegments.Notethatthefeasibleregionoftheoptimizationproblemdeningkij(z)hasfourcandidateendpoints:x=Ci1;zDj+1;Ci1;zDj1,twoofwhichwillbefeasibleforanygivenz.Wecanwritekijasfollows:kij(z)=min8>>>>>>><>>>>>>>:p(Ci1)+L(zCi1)whenCi1+Dj1z>>>>>>>>>>>>><>>>>>>>>>>>>>>:minfp(Ci1)+L(zCi1);p(Ci1)+L(zCi+1)gforCi1+Dj1z
PAGE 40

Itisinterestingtonotethat,whileinsomeoftheaboveresultsweassumethatpandLhaveapiecewisestructurewithaxednumberofpiecesthatappearsintherunningtimerequiredforevaluatingk,thefunctionkitselfmayinfacthaveapiecewisestructurewithanumberofsegmentsthatisexponentialintheinputsize.Instead,ourcomplexityresultsfollowbythefactthatthefunctionkcanbewrittenastheminimumofapolynomialnumberoffunctionskijwithaspecicstructure.WerefertoTsevendorj(2001)formoreinformationonthepiecewisenatureoffunctionsandoptimizationproblems. 2.5.1DynamicProgramming

PAGE 41

backlogging,withacumulativeprocurementquantityofZ.TheDPrecursionreadsG0(Z)=8><>:0ifZ=I01ifZ6=I0Gt(Z)=minz2Pt(Z)Gt1(Zz)+kt(z)+htZ

PAGE 42

2.4.3 and 2.4.4 .2 TheFPTASfortheCLSPisbasedonanon-traditionalDPformulationoftheproblemthatemploysanupperboundonthetotalcost,sayB.Inparticular,for0bBandt=1;:::;T,deneFt(b)tobethemaximumendinginventorylevelinperiodtwhenthetotalcostincurreduptoandincludingperiodtisnomorethanabudgetb.Theoptimalsolutionvaluetotheproblemisgivenbytheminimumbudgetfortheentireplanninghorizonthatyieldsanonnegativeendinginventory,i.e.,c=minb=0;:::;BfbjFT(b)0g.VanHoeselandWagelmans(2001)showthatthevaluesFt(b)(b=0;:::;B)satisfythefollowingDPrecursion:

PAGE 43

Animportantcomponentoftherunningtimeofthisrecursionisthetimerequiredtond maxfztjkt(zt)+ht(Ft1(a)+ztDt)ba;0ztCt+Dtg(2{22) which,ifallcostfunctionscanbeevaluatedinconstanttime,isO(log(C+D)),yieldingarunningtimeofO(B2Tlog(C+D)+BTlog(T(C+D)))fortheentirerecursion.FortheCLSPformulationoftheRPP,however,weneedtoaccountforthefactthattheprocurementcostfunctionsktcannotbeevaluatedinconstanttime.Thefollowingpropositionstudiesproblem 2{22 underseveralconditionsonthecostfunctions. 2{22 canbesolvedinO(mnlog(C)log(C+D))time. 2{22 canbesolvedinO(mnlog(C+D))time.

PAGE 44

Withrespecttothevaluezt,notethatitisequaltozt=maxztjmini=1;:::;m;j=1;:::;nkijt(zt)+ht(Ft1(a)+ztDt)ba;0ztmin(Ct+Dt;DtFt1(a)): 2.4.3 saysthatkijisintegralconvex,zijtcanbefoundinO(log(C+D))iterationsofthebinarysearchmethod.Sincezt=maxi;jzijtthevalueofztcanbefoundinO(mnlog(C+D))functionevaluations. CombiningtheseresultswiththeresultsofPropositions 2.4.3 and 2.4.4 ,thedesiredresultnowfollows.2 2.5.2 caneasilybegeneralizedtothecasewherehtispiecewiseconvexorpiecewiseconcavewithnomorethan`segmentsbywritingzt=maxztjmini=1;:::;m;j=1;:::;n;=1;:::;`kijt(zt)+ht(Ft1(a)+ztDt)ba;0ztmin(Ct+Dt;DtFt1(a)) WearenowreadytodeterminetherunningtimeoftheDPrecursion 2{21 fortheRPP.

PAGE 45

2.5.2 .2 2{21 togetherwithanupperboundBontheoptimalcostscthatsatisescB2Tc.VanHoeselandWagelmans(2001)describeaDPalgorithmtondsuchabound.ThefollowingpropositionderivestherunningtimeofthisalgorithmwhenappliedtotheRPP.

PAGE 46

2.5.4 Forcases(i)and(ii),therunningtimeofa.followsfromProposition 2.5.2 andtherunningtimeofb.followsfromPropositions 2.4.3 and 2.4.4 .ThenalresultnowfollowsfromtheanalysisofthealgorithminvanHoeselandWagelmans(2001).2 1.DetermineanupperboundBoncthatsatisesB2Tc. 2.ApplytheDP-recursion 2{21 afterroundingtheproblemdatatobemultiplesofK=maxfb"B=2T2c;1g. Combiningallresultsabove,wecannowconcludewiththemainresultofthissection:

PAGE 47

ofthisreformulation,andusedthisanalysistoderiverunningtimesforaDPalgorithmaswellasaFPTASfortheRPP.

PAGE 48

Thisproblemmayoccur,forexample,incertainsettingswheretheplanningofprocurement,production,andinventoryholdingofrawmaterialsandnalproductisintegrated.Thistwo-levellot-sizingmodelhasbeenstudiedextensivelyforthecasewhereallcostfunctionsareconcaveandcapacitiesareabsent(see,e.g.,Zangwill1969),orprocurementissubjecttostationarycapacities(see,e.g.,KaminskyandSimchi-Levi2003)andvanHoeseletal.2005).AlthoughinthepresenceofgeneralprocurementcapacitiestheproblemisknowntobeNP-hardevenwhenallcostfunctionsareconcave(seeFlorianetal.1980),KaminskyandSimchi-Levi(2003)derivecertainconditionsonthecostfunctionsunderwhichtheproblemissolvableinpolynomialtime.SargutandRomeijn(2006c)showthatthisproblemreducestotheLSP-CCwhenthetotalcostofprocurementandinventoryofrawmaterialsareexogenoustothemodelandstudythisproblemundergeneralandconcavecostfunctions. 36

PAGE 49

InSection 3.2 ,weformulatethetwo-levellot-sizingproblemand,underourassumptionsonthecoststructure,reformulateitasasingle-levelmodelwithcumulativecapacities.TheninSection 3.3 ,weprovethattheproblemisNP-hardforgeneralcostfunctionsandprovideafullypolynomialtimeapproximationschemeincaseallcostfunctionsarenondecreasing.InSection 3.4 wethendevelopadynamicprogrammingapproachthatsolvestheprobleminpolynomialtimewhenallcostfunctionsareconcave.Finally,inSection 3.5 webrieydiscussanextensionwherebackloggingisallowed. 3.2.1Two-LevelLot-SizingwithNon-stationaryProductionCapacities

PAGE 50

Withoutlossofgenerality,allcostfunctionsareassumedtobeequaltozerowhentheirargumentiszero.Moreover,weassumethattheinitialinventoryatbothlevelsisgiven.Thenwecanformulatethetwo-levellot-sizingproblemwithprocurementcapacities(2LSP-PC)asfollows:minimizeTXt=1(pt(yt)+ct(xt)+gt(Jt)+ht(It)) subjectto 3{1 enforcetheprocurementcapacitieswhileconstraints 3{2 and 3{3 )modelthebalancebetweeninow,outow,andstorageattherstandsecondlevel,respectively.Forconvenience,wewilldenecumulativedemandsandcapacitiesasdst=tX=s+1ds=0;:::;t;t=1;:::;TC0t=tX=1Ct=1;:::;T:

PAGE 51

ThismodelwasintroducedearlierbyZangwill(1969),KaminskyandSimchi-Levi(2003),andvanHoeseletal.(2005).Zangwill(1969)developsadynamicprogrammingalgorithmfortheuncapacitatedcase,whichrunsinO(T3)time(seevanHoeseletal.2005).Morerecently,KaminskyandSimchi-Levi(2003)developpolynomial-timealgorithmsforcertaininstancesofthisproblemwithlinearprocurementandinventoryholdingcostfunctions(pt,gt,andht)andconcaveproductioncostfunctions(ct).Inparticular,theyshowthatwhentheproductioncostfunctionshaveaxed-chargestructureandcostfunctionsatbothlevelssatisfyanon-speculativemotivesassumption,the2LSP-PCissolvableinO(T4)time.VanHoeseletal.(2005)showedthat,whentheprocurementcapacitiesarestationary,theproblemcanbesolvedinO(T7)timeifallcostfunctionsareconcave;inO(T6)timeiftheprocurementcostfunctionsareconcave,inventoryholdingcostsarelinear,productioncostshaveaxed-chargestructure,andvariableholdingandproductioncostssatisfyanon-speculativemotivesassumption;andinO(T5)timeiftheprocurementcostfunctionsareconcaveandallothercostfunctionsarelinear.Ontheotherhand,the2LSP-PCisNP-hardunderconcavecostfunctionsandnonstationarycapacitiessinceitgeneralizesthesingle-levelcapacitatedlot-sizingproblem(seeFlorianetal.1980). Weconsiderseveralnewsubclassesofthe2LSP-PC.Allourmodelsapplytosituationswhere,whileprocurementcapacitiesareimportant,thetotalcostofprocurementandinventoryofrawmaterialsisexogenoustothemodel.Thiswouldoccur,forexample,ifprocurementcostsarelinearandstationaryandtherearenocostsassociatedwithholdingrawmaterialsininventory.Moregenerally,weessentiallyassumethatprocurementcostsandinventoryholdingcostsattherstlevelarelinearandsatisfy:

PAGE 52

Undertheseassumptionsthe2LSP-PCbecomesasasingle-levellot-sizingproblemwithwhatwewillcallcumulativecapacityconstraints.Ourcontributionsinthischapterwithrespecttothisnewlot-sizingproblemarethefollowing(wherewerefertothecostfunctionsinthereformulatedmodel). 3{2 : Thisleadstothefollowingobjectivefunction:TXt=1(ptyt+ct(xt)+gtJt+ht(It))=TXt=1gtJ0+pt+TX=tg!yt+ct(xt)TX=tgxt+ht(It)!=J0TXt=1gt+(pT+gT)TXt=1dt+TXt=1(~ct(xt)+ht(It)) whereJ0PTt=1gt+(pT+gT)PTt=1dtisaconstantand~ct(x)=ct(x)TX=tgxt=1;:::;T:

PAGE 53

Thisinfactshowsthat,withoutlossofgeneralityandforconvenience,wemayassumethatpt=gt=0sothatwehave~ct=ct(t=1;:::;T). UponeliminatingthedecisionvariablesJt(t=1;:::;T)thecorrespondingnonnegativityconstraintsbecometX=1xtX=1y+J0t=1;:::;T: 3{1 bytX=1xtX=1C+J0=C0t+J0t=1;:::;T: Thisleadstothesingle-levellot-sizingproblemwithcumulativecapacities(LSP-CC)thatisthefocusofthischapter:minimizeTXt=1(ct(xt)+ht(It)) subjectto(LSP-CC)xt+It1=dt+Itt=1;:::;TtX=1xC0t=1;:::;Txt;It0t=1;:::;T:

PAGE 54

withrelativeerrornomorethan">0inanamountoftimethatispolynomialintheinputsizeoftheproblemand1=". SUBSETSUM:Givenpositiveintegersa1;a2;:::;aTandA,doesthereexistasetSf1;:::;TgsuchthatPi2Sai=A? ThefollowingtheoremshowsthattheLSP-CCisNP-hardingeneralbyreducingtheSUBSETSUMproblemtoit. 3-1 .Firstobservethat,sincetheonlynon-zerodemandisinperiodT,eachproductionquantitywillbeusedtosatisfythedemandinthatperiodandthusthecumulativeproductionovertheplanninghorizonisequaltoA,i.e.,PT=1x=A.Moreover,theproductioncostin

PAGE 55

Figure3-1: Productioncostfunctioninperiodt. periodtsatisesct(x)=xifx2f0;atgct(x)>xotherwise:

PAGE 56

whenallcostfunctionsarenondecreasingandallproblemdata(i.e.,demands,capacities,andcosts)areintegral.Weassumept=pfort=1;:::;Tandgt=0fort=1;:::;T.Therefore,thesingle-levelmodelreducedfromthetwo-levelmodelhasnondecreasingcostfunctions.Inparticular,wewilldrawheavilyonthepaperbyvanHoeselandWagelmans(2001)andmodifytheFPTASthattheydevelopedforthestandardcapacitatedlotsizingproblem(CLSP)toourproblem. FollowingvanHoeselandWagelmans(2001),westartbyintroducinganon-traditionaldynamicprogrammingformulationoftheproblemthatrunsinpseudo-polynomialtime.LetBbeanyintegerupperboundontheoptimalsolutionvalue,sayz,totheLSP-CC.Fort=1;:::;Tandb=0;1;:::;B,wethendeneFt(b)tobethemaximalendinginventoryinperiodt(It)thatcanbeachievedwithabudgetofbinthersttperiods,i.e.,thetotalproductionandinventoryholdingcostsinperiods1;:::;tdonotexceedb.Sincefort=1thecumulativecapacityconstraintisequaltothetraditionalcapacityconstraintweimmediatelyobtainthatF1(b)=maxd1x1C1fx1d1jc1(x1)+h1(x1d1)bgforb=0;:::;B:

PAGE 57

WewillshowthatthesetwopropertiesalsoholdfortheLSP-CC.First,notethattheowbalanceconstraintsimplythatt1X=1x=d0;t1+It1 3{5 aswellastheboundconstraintonxtwillstillbesatised.Moreover,theresultingvalueofItremainsunchanged.Therefore,wecanwithoutlossofoptimalityinthedynamicprogrammingrecursionincreaseIt1anddecreasextuntileitherIt1becomesequaltoFt1(a)orxtbecomesequaltozero,whicheverhappensearlier.Intheformercase,therstclaimfollows.Inthelattercase,thesecondclaimfollows.2

PAGE 58

AsaresultoftheProposition 3.3.2 thefollowingdynamicprogrammingrecursionisobtained:Ft(b)=max0abmax8>>>>>><>>>>>>:maxmaxf0;dtFt1(a)gxtC0td0;t1Ft1(a)fFt1(a)+xtdtjct(xt)+ht(Ft1(a)+xtdt)bag;max0It
PAGE 59

inventorylevel:xtmaxfx:0xC0t;ct(x)`gItmaxfI:I0;ht(I)`g: 3.4.1Introduction 3-2 .Nodes(D;1);(D;2);:::;(D;T)aredemandnodes,where(D;t)representsthedemandinperiodt,dt.Nodes(C;1);(C;2);:::;(C;T)aretransshipmentnodesandnode(C;T+1)isasupplynodehavingsupplyequaltothecumulativedemandovertheplanninghorizon,i.e.,itsdemandisequaltothenegativethereof:d0T.Asintraditionallot-sizingmodels,anarcfromnode(D;t)tonode(D;t+1)(t=1;:::;T1)representstheamountheldininventoryattheendofperiodt,It,andthecostfunctionoftheowonthatarcisht.(Notethat,

PAGE 60

withoutlossofoptimality,theinventoryattheendofperiodTisequaltozero.)Inaddition,anarcfromnode(C;t)tonode(D;t)representstheamountproducedinperiodt,xt,andthecostfunctionoftheowonthatarcisct.Finally,anarcfromnode(C;t+1)tonode(C;t)(t=1;:::;T)representsthecumulativeproductionamountinperiods1;:::;t.Anarcofthisformhasnocostbutanitecapacity,Pt=1C.(Therearenocapacitiesontheotherarcs.)Productionandinventorycostfunctionsarecommonlyconcave,representingtheeconomiesofscalethatareoftenfoundduetothepresenceof,forexample,xedsetupcostsforproductionandxedstoragecosts.Whenthefunctionshtandct(t=1;:::;T)areconcave,theLSP-CCthusbecomesaminimumconcavecostnetworkowproblem.Despitethefactthatanextremepointoptimalsolutiontosuchproblemsexists,theyaregenerallyNP-hard(seeBitranandYanasse1982).However,polynomial-timealgorithmshavebeenfoundformanylot-sizingproblemswithconcavecostsbyemployingthestructureofextremepointsolutionsfortheseproblems. Figure3-2: NetworkowrepresentationofLSP-CC. Inparticular,itiswellknownthatthestandarduncapacitatedeconomiclot-sizingmodelissolvableinO(TlogT)time(seeAggarwalandPark1990,Wagelmansetal.1992,andFedergruenandTzur1991).Understationary(time-invariant)productioncapacitiesthismodelissolvableinO(T3)timewheninventorycostsarelinear(seevanHoeselandWagelmans1996)andinO(T4)time

PAGE 61

withgeneralconcaveinventorycostfunctions(seeFlorianandKlein1971).Ontheotherhand,theeconomiclot-sizingmodelwithnonstationary(time-varying)capacitiesisNP-hard,eveniftheproductioncostfunctionshaveaxed-chargestructureandtheholdingcostfunctionsarelinear(seeFlorianetal.1980).Intheremainderofthissectionwewilldevelopapolynomial-timealgorithmfortheLSP-CCundernonstationarycapacities. ConsiderthesetoffreearcsinanextremepointsolutiontothenetworkowformulationoftheLSP-CC.Sinceallinventoryandproductionarcsareuncapacitated,whenweignorethecumulativeproductionarcsthesetofremainingfreearcsdecomposesintodisjointconnectedandacycliccomponents.Eachofthesecomponentsischaracterizedbyaconsecutivesetofdemandperiods,sayt+1;:::;(where0t0s=t+1;:::;1I=0:

PAGE 62

independentlyoftheothersubplans.Thisisperhapssurprisingsincethesubplansseemtobeintimatelyrelatedthroughthe(cumulative)productioncapacities. Nowdenotethecostofsubplan(t;),i.e.,theminimumcostofsupplyingthedemandsinthatsubplan,bytandletF(t)betheminimumtotalcostofsatisfyingthedemandsinperiodst+1;:::;T.Clearly,F(T)=0andtheoptimalsolutionvalueoftheLSP-CCisgivenbyF(0).ThisthenimmediatelyleadstothefollowingdynamicprogrammingformulationoftheLSP-CC:F(t)=min=t+1;:::;Tft+F()gfort=1;:::;T1F(T)=0:

PAGE 63

productionarcfromnode(C;s+1)tonode(C;s)tobeC0tsC0sd0tfors=t+1;:::;:

PAGE 64

Ifsisnotthelastproductionperiodinthesubplan,weusethefactthatweonlyneedtoconsiderextremepointsolutions.Therefore,thecumulativeproductionarcenteringnodes(C;^s)and(C;s)areatfullcapacity.Therefore,thequantityproducedinperiodsshouldbeequaltothedierencebetweenthetwocapacities,C0tsC0t^s.(Notethatif^s=tweobtainthatproductioninperiodsisequaltoC0ts.)2 3-3 illustratesanextremepointsolutionwith2subplansandtheassociatedarcows Figure3-3: Extremepointsolutionwith2subplansandtheassociatedarcows. Wewillnextformulatetheproblemofcomputingtheminimumcostofasubplan(t;)asadynamicprogrammingproblem.Tothisend,wedenestatesoftheform(s;^s;b)wheret^ss+1andb2f0;1g.Therstelement,s,denotesthecurrentperiodwhilethesecondelement,^sdenotesthepreviousproductionperiodinthesubplan.Thethirdelementindicateswhetherproductionhasbeencompletedforthesubplan(b=1)ornot(b=0).Thesourcenodeis(t;t;0)whilethesinknodeis(+1;+1;1).Foreachstate(s;^s;b)withts1thereareuptothreepotentialdecisionstobemade:(i)donotproduceinperiods+1,(ii)produceinperiods+1butdonotcompleteallproduction,and(iii)completeallproductioninperiods+1.Thismeansthatfromstate(s;^s;0)thefollowingstatescanbereached:

PAGE 65

sincetheproductioninthepreviousproductionperiod^swasuptocapacity.Notethatthisdecisionisonlyfeasibleifthedemandinperiods+1canbesatisedusingpastproduction,i.e.,ifC0t^sdt;s+1. Fromstate(s;^s;1)withts1,ontheotherhand,theonlystatethatcanbereachedis:

PAGE 66

3.3.2 andvanHoeselandWagelmans(2001)canbeusedtodevelopaFPTASforthisgeneralizationundertheadditionalassumptionthattheproductionandbackloggingcostfunctionsareconcave(buttheinventoryholdingcostfunctionsareonlyrequiredtobenondecreasing).InthissectionweshowthatthedynamicprogrammingalgorithmdevelopedinSection 3.4 canbeappliedwithminormodications,leadingtoanO(T4)algorithmfortheLSP-CCwithbacklogging. Weletutbethequantitybackloggedinperiodtwithassociatedcostfunctionbt(t=1;:::;T).Weassumethesebackloggingcostfunctionsareconcaveand,withoutlossofgenerality,equaltozerowhentheirargumentiszero.Inthatcase,themathematicalprogrammingformulationoftheproblembecomes:minimizeTXt=1(ct(xt)+ht(It)+bt(ut))

PAGE 67

subjecttoIt1+xt+ut=dt+It+ut1t=1;:::;TtX=1xC0t=1;:::;Txt;It;ut0t=1;:::;TuT=0 wherewehaveassumedwithoutlossofgeneralitythatI0=u0=0. Wecanagainvieweachextremepointsolutionasdecomposedintoanumberofsubplans,albeitwithaslightmodicationofthedenitionofasubplan.Inparticular,wenowdeneasubplan(t;)tobetheconsecutivesetofdemandperiodst+1;:::;withthepropertythatIt=0andut=0Is>0orus>0s=t+1;:::;1I=0andu=0:

PAGE 68

capacitiesattherst(procurement)level.WeshowedthattheproblemisNP-hardingeneralandprovidedafullypolynomialtimeapproximationschemeundermildconditionsonthecostfunctions.Moreover,wederivedadynamicprogrammingalgorithmthatrunsinpolynomialtimewhenallcostfunctionsareconcave.Thisisincontrastwithtraditionalsingleandtwo-levelcapacitatedeconomiclot-sizingproblemsthatareNP-hardevenunderxed-chargecostfunctionsandprovidesanalternativesetofconditionsonthecostfunctionsofthetwo-levellot-sizingproblemunderwhichtheproblemispolynomiallysolvablethantheonesderivedbyKaminskyandSimchi-Levi(2003).InterestingdirectionsforfutureresearchincludethedevelopmentofaFPTASforalargerclassoftwo-levellot-sizingproblemsaswellasthederivationofotherclassesofcostfunctionsunderwhichthisproblemissolvableinpolynomialtime.Moreover,wewanttoconsiderthecasewheretherawmaterialsareperishableandhaveshelflifeof1`
PAGE 69

Ourmodelfallswithinthegeneralclassofmulti-echelonlot-sizingmodelsthathasbeenanactiveareaofresearcheversincetheseminalpaperofWagnerandWhitin(1958).Thatpaperconsidersanuncapacitatedsingle-echelonmodel 57

PAGE 70

forwhichanalgorithmwasproposedthatrunsinpolynomialtime,O(T2),inthelengthoftheplanninghorizon,T.Moreecientalgorithms,runninginO(TlogT)orevenO(T)time(dependingonthecoststructure)forthisbasiclot-sizingmodelweredevelopedbyAggarwalandPark(1990),FedergruenandTzur(1991),andWagelmansetal.(1992). Florianetal.(1980)andBitranandYanasse(1982)showthatincorporatingnonstationaryproductioncapacitiesintothismodelmakesitNP-hardingeneral,evenwhenproductioncostshaveaxed-chargestructureandholdingcostsarelinear.However,FlorianandKlein(1971)andvanHoeselandWagelmans(1996)proposealgorithmsthatruninO(T4)andO(T3)time,respectively,forthecasewheretheproductioncapacitiesarestationaryandinventoryholdingcostsareconcaveandlinear,respectively.ChungandLin(1988)andvandenHeuvelandWagelmans(2006)consideracapacitatedlot-sizingproblemwithaparticularcostparameterandcapacitystructureandshowthatthisproblemissolvableinO(T2)time. Zangwill(1968,1969)wasthersttostudymulti-echelonlot-sizingproblems.Hecharacterizesthesolutionofsuchproblemswhencostsareconcaveandusesthisstructuretodevelopadynamicprogrammingalgorithmtosolvesuchproblems.AsshownbyvanHoeseletal.(2005),therunningtimeofthisalgorithmisO(T3)forthecaseoftwoechelons.KaminskyandSimchi-Levi(2003)consideratwo-echelonserialsupplychaininwhichproductionattherstecheloniscapacitated.Theydeveloppolynomial-timedynamicprogrammingalgorithmsunderseveralcostandcapacitystructures.VanHoeseletal.(2005)developedanimprovedalgorithmforthecaseofstationarycapacitiesthatallowsforgeneralconcavecostfunctionsandrunsinO(T7)time.Inaddition,theyshowthattherunningtimecanbefurtherimprovedwheninventoryholdingcostfunctionsarelinearandtransportationcostfunctionshaveeitheraxedchargestructuresatisfyinganon-speculative

PAGE 71

motivesassumption(yieldingarunningtimeofO(T6))orarelinear(forarunningtimeofO(T5)).Now,weintroducetheliteratureonmulti-levellot-sizingmodels.Wewanttogivetwoexamplesforthissituation.Wemayhaveadistributionsystem,inwhichproductionoccursattherstlevelandthenitemsareshippedtothenextlevels.Wecanalsohaveamanufacturingsysteminwhichproductsaremanufacturedinaseriesofproductionfacilities,eachofwhichaddsadditionalvaluetotheproduct.InZangwill(1969),adynamicprogrammingalgorithmforamulti-levelsystemwithoutanycapacitiesisproposedwhenbackloggingisnotallowed.VanHoeseletal.(2005)showedthatthisalgorithmrunsinO(T3+(L2)T4),whereListhenumberoflevelsinthesupplychain. KaminskyandSimchi-Levi(2003)proposeathree-levelmodelinwhichtherstandthirdlevelsareproductionstages,andthesecondlevelisatransportationstage.Bothproductionstagesarecapacitated,whilethetransportationstageisuncapacitated.Theyconsiderlinearinventoryholdingcoststhatincreasewiththelevelofthesupplychain,andlinearproductioncostsatbothlevels1and3thatsatisfyatraditionalnon-speculativemotivescondition.Thetransportationcostsatthesecondlevelareofthexed-chargeorgeneralconcaveformandareassumedtosatisfyarestrictiveandnontraditionalnonspeculativemotivescondition.Theyeliminatethethird-levelproductiondecisionsandreducetheproblemtoatwo-levelmodel.Forthexed-chargetransportationcosts,theyprovideapolynomialtimealgorithmthatrunsinO(T4)timefornonstationaryproductioncapacities.Forconcavetransportationcosts,theiralgorithmrunsinexponentialtime.However,theyconsiderstationaryproductioncapacitiesandtheiralgorithmrunsinO(T8)time. Forstationaryproductioncapacitiescase,vanHoeseletal.(2005)proposesapolynomialtimealgorithmthatrunsinO(T7)timeevenwhenallcostfunctionsareconcave.Theyalsoconsidermulti-levelmodelsandproposeanalgorithmthat

PAGE 72

runsinO(LT2L+3)timewhenallcostfunctionsareconcaveandwhereListhenumberoflevelsintheproblem.Notethatthisrunningtimeispolynomialinthenumberofperiodsandexponentialinthenumberoflevels.However,wheninventoryholdingcostsarelinearandtransportationcostseitherarelinearorhavexed-chargestructure,theyproposealgorithmswithrunningtimespolynomialinthenumberofperiodsandinsensitivetothenumberoflevels.TheiralgorithmrunsinO(T7+LT4)(O(T6)whenL=2)timewhentransportationcostshavexed-chargestructureandsatisfyaformofnon-speculativemotivesandinO(T5+LT2)timewhentransportationcostsarelinear. Theydenesubplanasthesetofconsecutiveperiodsateachlevel,inwhichdemandatthelastlevelaresatisedviatheproductionattherstlevelusingtheperiodsatintermediatelevelsasthetransportationperiods.Intheirsolutionprocedure,theyemploythepropertythatanoptimalsolutioncanbedecomposedintoasequenceofconsecutivesubplans.Therefore,optimalsolutionvalueissumofsomeconsecutivesubplancosts.Inthecalculationofthesubplancosts,weneedpossibletransportationandproductionquantities.TheygeneralizetheresultofFlorianandKlein(1971)forproductionquantities.Morovertheyshowthatthetransportedquantitybetweenlevels`and`+1insomeperiodeithermakesthecumulativetransportedquantitiessofarinthesubplanequaltothecumulativeproductionquantitiesofaninitialsequenceofconsecutiveproductionperiodsinthesubplan,ortothecumulativedemandofaninitialsequanceofdemandperiodsinthesubplan. Now,weintroducetheliteratureonmulti-levellot-sizingmodels.Wewanttogivetwoexamplesforthissituation.Wemayhaveadistributionsystem,inwhichproductionoccursattherstlevelandthenitemsareshippedtothenextlevels.Wecanalsohaveamanufacturingsysteminwhichproductsaremanufacturedinaseriesofproductionfacilities,eachofwhichaddsadditionalvaluetotheproduct.

PAGE 73

InZangwill(1969),adynamicprogrammingalgorithmforamulti-levelsystemwithoutanycapacitiesisproposedwhenbackloggingisnotallowed.VanHoeseletal.(2005)showedthatthisalgorithmrunsinO(T3+(L2)T4),whereListhenumberoflevelsinthesupplychain. KaminskyandSimchi-Levi(2003)proposeathree-levelmodelinwhichtherstandthirdlevelsareproductionstages,andthesecondlevelisatransportationstage.Bothproductionstagesarecapacitated,whilethetransportationstageisuncapacitated.Theyconsiderlinearinventoryholdingcoststhatincreasewiththelevelofthesupplychain,andlinearproductioncostsatbothlevels1and3thatsatisfyatraditionalnon-speculativemotivescondition.Thetransportationcostsatthesecondlevelareofthexed-chargeorgeneralconcaveformandareassumedtosatisfyarestrictiveandnontraditionalnonspeculativemotivescondition.Theyeliminatethethird-levelproductiondecisionsandreducetheproblemtoatwo-levelmodel.Forthexed-chargetransportationcosts,theyprovideapolynomialtimealgorithmthatrunsinO(T4)timefornonstationaryproductioncapacities.Forconcavetransportationcosts,theiralgorithmrunsinexponentialtime.However,theyconsiderstationaryproductioncapacitiesandtheiralgorithmrunsinO(T8)time. Forstationaryproductioncapacitiescase,vanHoeseletal.(2005)proposesapolynomialtimealgorithmthatrunsinO(T7)timeevenwhenallcostfunctionsareconcave.Theyalsoconsidermulti-levelmodelsandproposeanalgorithmthatrunsinO(LT2L+3)timewhenallcostfunctionsareconcaveandwhereListhenumberoflevelsintheproblem.Notethatthisrunningtimeispolynomialinthenumberofperiodsandexponentialinthenumberoflevels.However,wheninventoryholdingcostsarelinearandtransportationcostseitherarelinearorhavexed-chargestructure,theyproposealgorithmswithrunningtimespolynomialinthenumberofperiodsandinsensitivetothenumberoflevels.Theiralgorithm

PAGE 74

runsinO(T7+LT4)(O(T6)whenL=2)timewhentransportationcostshavexed-chargestructureandsatisfyaformofnon-speculativemotivesandinO(T5+LT2)timewhentransportationcostsarelinear. Theydenesubplanasthesetofconsecutiveperiodsateachlevel,inwhichdemandatthelastlevelaresatisedviatheproductionattherstlevelusingtheperiodsatintermediatelevelsasthetransportationperiods.Intheirsolutionprocedure,theyemploythepropertythatanoptimalsolutioncanbedecomposedintoasequenceofconsecutivesubplans.Therefore,optimalsolutionvalueissumofsomeconsecutivesubplancosts.Inthecalculationofthesubplancosts,weneedpossibletransportationandproductionquantities.TheygeneralizetheresultofFlorianandKlein(1971)forproductionquantities.Morovertheyshowthatthetransportedquantitybetweenlevels`and`+1insomeperiodeithermakesthecumulativetransportedquantitiessofarinthesubplanequaltothecumulativeproductionquantitiesofaninitialsequenceofconsecutiveproductionperiodsinthesubplan,ortothecumulativedemandofaninitialsequanceofdemandperiodsinthesubplan. Returningtosingle-echelonlot-sizingproblems,Zangwill(1969)studiedeconomiclot-sizingproblemswithbackloggingandsolvedthisprobleminO(T3)timewhencostsareconcave,andinO(T2)timewhentheprocurementcostfunctionshaveanon-stationaryxed-chargecomponentandastationarylinearcomponent.Wheninventoryholdingandbackloggingcostsarelinearandthelinearcomponentoftheprocurementcostsarenonstationary,vanHoeseletal.(1994)showthattherunningtimeofZangwill'sapproachisstillO(T2)and,inaddition,applygeometrictechniquestosolvethisprobleminO(TlogT)timeor,underadditionalconditionsonthecostcomponents,inO(T)time.Furthermore,AtamturkandHochbaum(2001)considerasingle-echelonlot-sizingmodelwithproductioncapacitiesanduncapacitatedsubcontractingundervarious

PAGE 75

coststructures,wheretherunningtimeforthemodelwithstationaryproductioncapacitiesandconcavecostsisO(T5). Themaincontributionofthischapteriswithrespecttoourknowledgeonthecomplexityofandsolutionmethodsforconcave-costlot-sizingproblems,inparticulartwo-levellot-sizingproblemswithstationarycapacitiesattherstlevel.Whereastodatetheonlymodelsinthisclassforwhichpolynomial-timesolutionmethodswereknownonlyallowedasinglesupplysourceandnobacklogging,inSargutandRomeijn(2006b),wederivesuchmethodsinthepresenceofbackloggingatthesecondlevelaswellascapacitatedanduncapacitatedoutsourcingorovertimeproductionopportunities.Inaddition,weshowhowimprovedrunningtimescanbeobtainedforseveralclassesofmorerestrictivecostfunctions,includingcaseswheresomeofthecostfunctionsarelinear,haveaxed-chargestructure,orsatisfycertainnon-speculativemotivesconditions. Thischapterisorganizedasfollows.InSection 4.2 ,weprovideamathematicalprogrammingandanetworkowformulationofourproblemandoutlineouroverallsolutionapproach.InSections 4.3 and 4.4 wethendeveloppolynomial-timealgorithmsforsubproblemsthatneedtobesolvedrepeatedlyinourapproach.Section 4.5 dealswithuncapacitatedmodels,andweendthechapterinsection 4.6 withasummaryandsomesuggestionsforfutureresearch. 4.2.1ProblemFormulation

PAGE 76

Weassumethatthesupplierhastheoptiontosatisfyeachperiod'sdemandearly(whichleadstoinventoryholdingatthemanufacturer)orlate(whichcorrespondstobackloggingofmanufacturerdemand)atacost.Wewillallowforsubcontractingofdemands,wherethesubcontractordeliversdirectlytothesupplier.Thesuppliercombinessubcontractedandin-houseproductionintoshipmentstothemanufacturer.Thismeansthatwhetherthesuppliersubcontractsornotistransparenttothemanufacturer.ThequantitysubcontractedineachperiodislimitedbyBunits. Ourdecisionvariablesandcostfunctionsare: Ourobjectiveistominimizesystem-wideproduction,subcontracting,transportation,inventoryholding,andbackloggingcostsovertheentireplanninghorizon.Wemaythenformulatetheproblemasamathematicalprogrammingproblemasfollows:minTXt=1st(zt)+pt(yt)+ct(xt)+h(1)t(I(1)t)+h(2)t(I(2)t)+bt(ut)

PAGE 77

subjectto(P) (4{5) 4{1 and 4{2 enforcetheproductionandsubcontractingcapacityconstraintsineachperiod.Constraints 4{3 and 4{4 representtheowbalanceconstraintsatthesupplierandmanufacturerlevel,respectively.Constraints 4{5 and 4{6 modeltheinitialinventoryandterminalbackloggingconditions.Notethatthelatterensuresthatallmanufacturerdemandissatisedduringtheplanninghorizon.AllbuttheassumptionthatI(1)0=0aremadewithoutlossofgenerality(andcanbehandledinastraightforwardmannerbyupdatingthedemandsequenceasinsingle-levellot-sizingproblems).WerefertovanHoeseletal.(2005)foradiscussiononhowtodealwithinitialinventoriesatthesupplierlevel.Thisapproachcan,usingthealgorithmsdevelopedinthispaper,beextendedtoourmodelsinastraightforwardmanner. Throughoutthispaper,wewillassumethatallcostfunctionsarenonnegative,concave,nondecreasing,andequaltozerowhentheirargumentiszero.However,inadditiontothecaseofgeneralconcavecostfunctions,wewillalsostudythealgorithmicimplicationsoftwoothercoststructures.

PAGE 78

NotethatourmodelcanbeformulatedasaconcavecostnetworkowproblemasillustratedinFigure 4-1 .Inthisnetwork,wehaveasourcethatsuppliesthetotalmanufacturerdemandPTt=1dt.Costlessarcsdistributethissupplyoveraproduction(nodeP)andasubcontracting(nodeS)source.Thenodesareindicatedby(`;t),where`denotesthelevelinthechainandttheperiod.Thisformulationimmediatelyimpliesthatthereexistsanextremepointsolutiontoourproblem,i.e.,asolutionthatisavertexofthepolytopethatformsthefeasibleregionof(P)aslongasafeasiblesolutionexists,i.e.,aslongasTXt=1dtT(C+B): Figure4-1: Networkrepresentationwithinventoryholdingandbacklogging.

PAGE 79

casesofourproblemcontainingonlyproductiondecisions(seeFlorianandKlein1971),productionandsubcontractingdecisions(seeAtamturkandHochbaum2001),orproductionandtransportationdecisionswithoutbacklogging(seeKaminskyandSimchi-Levi2003andVanHoeseletal.2005)decomposesintoasequenceofconsecutiveso-calledsubplansthatcontainatmostonefreeproductionarc,i.e.,atmostoneproductionarcwithowstrictlybetweenitslowerboundof0andupperboundofC.Thiscorrespondstothefactthattwofreeproductionarcsinasubplanwouldformacycleoffreearcs,whichcontradictstheextremepointnatureofthesolution. Wewillextendthischaracterizationofextremepointsolutionstoourmoregeneralclassofproblems.Considerallfreearcsinagivenextremepointsolution.Sinceallinventory,backlogging,andtransportationarcsareuncapacitated,whenweignoreallproductionandsubcontractingarcsthesetofremainingfreearcsdecomposesintodisjointconnectedandacycliccomponents.Thefactthatwehaveanextremepointsolutionthenimmediatelyimpliesthateachcomponentmaybeconnectedtoatmostonefreeproductionorsubcontractingarc.Itiseasytoseethateachcomponentthatisnotasingletonnodeatthesupplierlevelcontainsatleastonenodeatthemanufacturerlevel.Inagivencomponent,wedenotethetherstandlastnodesatthemanufacturerlevelby(2;1+1)and(2;2),indicatingthatinthiscomponentthedemandsofperiods1+1;:::;2aresatised.Inaddition,let(1;t1+1)and(1;t2)denotetherstandlastnodesatthesupplierlevel,sothatthecomponentcanbecharacterizedby(t1;t2;1;2).Bydenition,wehavethatt1
PAGE 80

period1.However,thiswouldcontradictthefactthatthecomponents(whendisregardingproductionandsubcontractingarcs)aredisjoint.Therefore,wecanincludethenodes(1;1+1);:::;(1;t11)inthecurrentcomponent,whichmeansthatwithoutlossofgeneralitywemayassumethatt11.Similarly,supposethat2
PAGE 81

ItiseasytoseethatthedynamicprogrammingrecursioncanbesolvedinO(T4)timeifthecostsofallsubplansareknown.SincethereareO(T4)subplans,thecomputationofthesecostscanbeexpectedtobethebottleneckandthechallengeisthereforetodevelopalgorithmstoecientlyndthecostsofthesubplans.

PAGE 82

andthenetworkrepresentationisgiveninFigure 4-2 Figure4-2: Networkrepresentationwithbackloggingonlyatthemanufacturer. Ofcourseinthiscasetheextremepointsdecomposeagainintosubplansoftheform(t1;t2;1;2)witht11
PAGE 83

istheminimumcostrequiredtosatisfythedemandsinperiods1+1;:::;2usingproductionorsubcontractingonlyinperiods1+1;:::;2andusingatmostoneproductionorsubcontractingquantitythatisstrictlybetweenitslowerboundofzeroanditscapacity.TheoptimalsolutiontotheproblemisgivenbyF0(0). ThisdynamicprogrammingrecursioncanbesolvedinO(T2)time,againassumingthatthecostsofallsubplansareknown.Wewilldevelopecientalgorithmsforcomputingthesubplancostsforthiscase. 4.3.1SubplanProperties Since,aswehaveseenabove,eachsubplanhasatmostonefreeproductionarc,wemaygeneralizearesultofFlorianandKlein(1971)anddeterminethenumberofperiodsinasubplanwhereproductionisatfullcapacityas andthequantityproducedintheremainingproductionperiodas=d1+1;2KC: Notethat,byconstruction,0
PAGE 84

inperiodtinasubplan(t1;t2;1;2),thequantityshippedsatisesoneofthefollowingconditions: sucienttofullymeetthedemandofperiods1;:::;sforsomes=t;:::;2. Thisfollowssinceotherwisetherewouldeitherbeaperiod(torlater)whosedemandissatisedpartiallyfromthetransportedquantityinperiodtandpartiallyfromtheendinginventoryatthesupplierlevelinperiodtorboththetransportedquantityinperiodtandtheendinginventoryatthesupplierlevelwouldbeusedtosupplydemandsinperiodtorearlier.Eventhoughinthelattercasethetransportedquantityandtheendinginventorydonotnecessarilysatisfydemandinthesameperiod,bothcasesleadtoacycleoffreearcsandthereforecontradictthatthesolutionisanextremepoint.

PAGE 85

Thedecisionsthatcanbemadeinstate(t;yc;xc)leadingtostate(t+1;yc;xc)are: Thetotalcostofthedecisiontotransitionfromstate(t;yc;xc)tostate(t+1;yc;xc)isequaltopt+1(ycyc)+ct+1(xcxc)+h(1)t+1(ycxc)+h(2)t+1(xcd1+1;t+1)++bt(d1+1;t+1xc)+: Since,ineachstate,thereareatmostO(T)candidatedecisionsandthecostofeachdecisioncanbecalculatedinconstanttimethetotaltimerequiredtondthecostsofagivensubplanisO(T4)(whereweassumethatwehavecomputedthecumulativedemandsdtsinapreprocessingsteptakingO(T2)time).ThecostofallO(T4)subplanscanthereforebedeterminedinO(T8)time.However,solvingthedynamicprogramforsubplan(0;t2;1;2)usingabackwardrecursionweinfactobtainthecostsofallsubplans(t1;t2;1;2),therebyreducingthetotalrunningtimetoO(T7).(SeealsoKaminskyandSimchi-Levi2003andVanHoeseletal.2001forasimilarsavingstechnique.) Inthenexttwosubsections,wewilldiscusstwosetsofcostfunctionsunderwhichthecostsofthesubplanscanbedeterminedmoreeciently.Inneithercasewillwefurtherrestricttheproductioncostfunctions.However,wewillassumethattheholdingandbackloggingcostfunctionsarelinearfunctions.

PAGE 86

Furthermore,wewillassumethatthetransportationcostfunctionsconsistofaxed-chargeandalinearcomponentand,inaddition,thatthelineartransportationcostcomponenttogetherwiththeholdingandbackloggingcostfunctionssatisfyaparticulartypeofnon-speculativemotivescondition.Wewillalsoassumethatthetransportationcostfunctionsarelinearbutdonotnecessarilysatisfythatnon-speculativemotivescondition.Withaslightabuseofnotationwewillrepresenttheunittransportation,inventoryholding,andbackloggingcostsbyct,h(1)t,h(2)t,andbt(t=1;:::;T),respectively. Underthiscondition,thesubplansolutionswillenjoyanattractivestructurethatcanbeemployedtoecientlyndtheoptimalsubplancosts.Inparticular,wehavethefollowingresult:

PAGE 87

However,summingcondition 4{9 overperiodst;:::;t01yields:t01X=tc+h(2)t01X=th(1)+c+1whichimpliesthatct+t01X=th(2)t01X=th(1)+ct0: Weagaincalculatetheminimumcostforeachsubplanusingdynamicprogramming.However,wemaynowchoosethestatetobeoftheform(t;yc)wheretisthecurrentperiodandyc2fkC+v:k=0;:::;K;v=0;1gisthe

PAGE 88

cumulativeproductionquantityinthecurrentsubplan.SinceKTthereareO(T2)states.Thisreductionofthestatespacefollowsfromthelinearityoftheinventoryholding,backlogging,andtransportationcosts,whichmakeitpossibletodeterminetheoptimalcostassociatedwiththequantityproducedinagivenperiodbasedononlythecumulativeproductionquantitybeforethecurrentperiod. Thedecisionsthatcanbemadeinstate(t;yc)are:produceorCunits,wheretheformerisonlyallowedifyc=kCforsomeintegerkandthelatterisonlyallowedifyc
PAGE 89

Notealsothatthearccostsmay,insomecases,bebasedonshortestpathsthatdonotfullyliewithinthesubplanunderconsideration.However,thisisnoproblemduetothelinearityofthecosts.Withthisinformation,itisclearthatthereareconstantnumberofdecisionsandthecostsofeachdecisioncanbedeterminedinconstanttime,sothatitimmediatelyfollowsthatthecostsofasubplancanbefoundinO(T2)timebyndingtheminimumcostrequiredtogofromstate(t1;0)tostate(t2;d1+1;2).SinceonceagainitsucestoconsiderO(T3)subplans,thetimerequiredtondallsubplancostsisO(T5). Whenallcostfunctionsareconcavewecanagainapplyadynamicprogrammingapproachwithstatesoftheform(t;yc;xc),wheretandycareasbefore,butwenowhavethatxc2fkC+v:k=0;:::;K;v=0;1g[fd1+1;tgandxcd1+1;tsinceashipmentinperiodtcannotsatisfythedemandofanyperiodsbeyondt.Similarly,adecisiontotransportxcxcunitscanonlybemadefortransportationquantitiesequaltozero,thetotalavailablequantityycxc,ord1+1;t+1xc,wherethelatterisonlyallowedifd1+1;t+1>xc.Since,ineachstate,therearenowO(1)candidatedecisionsthetotaltimerequiredtondthecostsofagivensubplanisO(T3)time.(Notethatthisimmediatelyimpliesthatwewillnotacomputationalsavingshere,sincethenumberofcandidatedecisionswillstillbeO(1).)ThecostofallO(T2)subplanscanthereforebedeterminedinO(T5)time. Incasethetransportation,inventoryholding,andbackloggingcostsarelinear,theapproachofholdingandbackloggingcaseholdswithoutmodication.Thus,thecostofallO(T2)subplanscanbecomputedinO(T4)timeinthiscase.

PAGE 90

4.6 .Intheremainderofthissection,wewillcomparethecausesofthedierencesbetweentherunningtimesforthedierentmodels.IdentifyingthesedierencesthenenablesustofocusonalimitedsetofmodelsintheremainderoftheChapter. Letusrstcomparetherunningtimesofthemodelswithandwithoutinventoryholdingatthemanufacturerdiscussedabovewhenallcostfunctionsareconcave.Whenholdinginventoryatthemanufacturerisnotallowed,therunningtimereducesbecauseoftwodierenteects.Firstly,thenumberofsubplansthatneedtobeconsideredreducesfromO(T3)toO(T2).Secondly,inthedynamicprogrammingapproachtocomputingsubplancoststhenumberofpotentialshipmentquantitiesineachperiodreducesfromO(T)toO(1).Together,thiscausesareducinginrunningtimeofO(T2).However,fortheothertwocoststructuresonlythersteectapplies,i.e.,theonlysavingsareduetothereductioninthenumberofsubplansthatneedtobeconsideredbyO(T). Finally,letuscomparethemodelswithrespecttothedierentcoststructures.Whenconsideringmodelswithlineartransportation,inventoryholding,andbackloggingcoststotheothers,weachieveasavingsintherunningtimesince,whencomputingsubplancostsintheformercase,onlythecumulativeproductionuptotimetisimportanttodeterminehowbesttoproceed.ThismeansareductioninthesizeofthestatespacebyO(T).Moreover,whenbothinventoryholdingandbackloggingareallowedatthemanufacturerlevel,thenumberofdecisionsineachstatereducesfromO(T)toO(1)whenwemove

PAGE 91

fromgeneralconcavecostfunctionstooneoftheothercoststructures,foratotalsavingsofO(T2). Theseobservationsextendtothemorecomplexmodelsthatwewillstudynext.Therefore,intheremainderofthischapterwewillonlyconsidermodelswherebothbackloggingandinventoryholdingisallowedatthemanufacturerlevelandallcostfunctionsareconcave,andsimplyinfertherunningtimesfortheothercasesfromthediscussionabove. Wewillheresummarizethemostimportantstructuralinsightsthathavebeenobtained:

PAGE 92

4.5 ,whilethemodelinwhichproductionisuncapacitatedandsubcontractingcapacitatedcanbehandledbysimplyinterchangingtheroleofproductionandsubcontracting. 4.2.2 weknowthateachextremepointdecomposesintoanumberofsubplans,eachofwhichhasatmostonefreeproductionarcoratmostonefreesubcontractingarc,butnotboth.Ifwehaveafreeproductionarcinthesubplan,itiseasytoseethatallsubcontractedquantitiesareequalto0orB.Similarly,ifwehaveafreesubcontractingarcinthesubplanthenallproductionquantitiesareequalto0orC. Nowconsiderasubplanwithdemandperiods1+1;:::;2.WeknowthatthereareatmostKp=d1+1;2 First,supposethatweknowthatthereareexactlyk(wherek=0;:::;Kp)periodsinwhichweproducetofullcapacityandthereisnofractionalproductionperiod.Thenthenumberofperiodsinwhichwesubcontracttofullcapacityand

PAGE 93

thequantitysubcontractedinthefractionalsupplyperiodareequaltosk=d1+1;2kC Bsk=d1+1;2kCskBrespectively.Notethat,incaseofuncapacitatedsubcontracting(B=1),thesereducetosk=0sk=d1+1;2kC: Cpk0=d1+1;2k0Bpk0C: 4.6 Capacitatedsubcontracting

PAGE 94

withsmallestcostthenyieldstheactualsubplancosts.Inthissection,wewillonlydescribetheformerprocedureindetail.Thelatterprocedurecanbeobtainedbysimplyinterchangingtheroleoftheproductionandsubcontractingoptions. Considersubplan(t1;t2;1;2).Similartotheproblemwithoutsubcontractingweusestatesoftheform(t;yzc;xc),wheret2ft1;:::;t2gisthecurrentperiod,yzc2fk1C+k2B+vk:k1=0;:::;k;k2=0;:::;k;v=0;1g Thedecisionsthatcanbemadeinstate(t;yzc;xc)leadingtostate(t+1;yzc;xc)are: Thetotaltransportation,inventoryholding,andsubcontractingcostofthedecisiontotransitionfromstate(t;yzc;xc)tostate(t+1;yzc;xc)isequaltoct+1(xcxc)+h(1)t+1(yzcxc)+h(2)t+1(xcd1+1;t+1)++bt(d1+1;t+1xc)+:Ifyzcyzc=C,aproductioncostofpt+1(C)isadded;ifyzcyzc=k,asubcontractingcostofst+1(k)isadded;ifyzcyzc=B,asubcontractingcostofst+1(B)isadded;ifyzcyzc=C+k,aproductionandsubcontracting

PAGE 95

costofpt+1(C)+st+1(k)isadded,andifyzcyzc=C+Baproductionandsubcontractingcostofpt+1(C)+st+1(B)isadded.(Notethatwehaveassumed,foreaseofexposition,thatB6=Candk6=C.Ifthisisnotthecase,wemayeitherexpandthestatespacewithavariableindicatingwhetherproductionand/orsubcontractingtookplaceinperiodt+1orsimplychoosetheassignmentofquantitiestoproductionandsubcontractingthatleadtominimumcost,neitherofwhichwouldincreasetherunningtimeofthealgorithm.) Theminimumcostofsubplan(t1;t2;1;2)(forthegivenk)isnowgivenbytheminimumcostrequiredto,startingatstate(t1;0;0),reachstate(t2;d1+1;2;d1+1;2)(atwhichpointalldemandsinthesubplanhavebeensatised). Since,ineachstate,thereareatmostO(T)candidatedecisionsandthecostofeachdecisioncanbecalculatedinconstanttimethetotaltimerequiredtondthecostsofagivensubplanforagivenkisO(T6).SincethereareO(T)choicesforkandsincethetimerequiredtondtheminimumsubplancostsforthecaseofafractionalproductionperiodisofthesameorder,thecostofallO(T3)(relevant)subplanscanbedeterminedinO(T10)time. 4.3.2 .Ifwethenalsocomputetheminimumcostwithsubcontractingthecostofthesubplanisobviouslygivenbytheleastcostlysolutionamongthesetwocandidates. Notethat,ifthereisaperiodinwhichsubcontractingtakesplace,therewillbenofreeproductionperiod.Tocomputetheminimumcostofasubplan(t1;t2;1;2)underthisconstraint,weagainconsideradynamicprogrammingapproachwithstatesoftheform(t;yzc;xc).DeningKandasinEquations 4{7 and 4{8 ,thisstatecontainsthecurrentperiodt2ft1;:::;t2g,thecumulative

PAGE 96

productionandsubcontractedquantityinthecurrentsubplanyzc2fkC+v:k=0;:::;K;v=0;1g,andthecumulativetransportedquantityinthesubplanxc2fkC+v:k=0;:::;K;v=0;1g[fd1+1;s:s=t;:::;2g.Asbefore,weshouldhavexcyzcsinceproductscannotbetransportedbeforetheyareproduced.ItiseasytoseethatthereareO(T3)states. Thedecisionsthatcanbemadeinstate(t;yzc;xc)leadingtostate(t+1;yzc;xc)are: Thetotaltransportation,inventoryholding,andsubcontractingcostofthedecisiontotransitionfromstate(t;yzc;xc)tostate(t+1;yzc;xc)isequaltoct+1(xcxc)+h(1)t+1(yzcxc)+h(2)t+1(xcd1+1;t+1)++bt(d1+1;t+1xc)+: SincethereareO(T3)statesandO(T2)potentialdecisionsperstate,wecandeterminethecostofasubplaninO(T5)time.ThecostsofallsubplanscanthusbefoundinO(T8)time.

PAGE 97

anadditionalcost.Intermsoftheoptimizationmodel(P),thiscanbehandledbyapplyingtheproductioncostfunctionpttoallunitsprocuredinperiodtandinterpretingthesubcontractingcostfunctionstasanincrementalcostfunction.Thatis,thetermpt(yt)intheobjectivefunctionisreplacedbypt(yt+zt).AnappropriatenetworkrepresentationofthissituationisgiveninFigure 4-3 .Herethearcsoftheform(P;t)arecostless,arcsoftheform(S;t)havecostfunctionst,andarcsoftheform(t;(1;t))havecostfunctionpt.Fromthisnetworkrepresentation,wecanimmediatelyconcludethat,withoutlossofoptimality,wemayrestrictourselvestosolutionsinwhichzt>0impliesthatyt=C.Inaddition,ananalysisoftheextremepointstructureofoptimalsolutionsyieldsthatweagainhavethatineachsubplanthereiseitheratmostoneperiodsuchthat00. Figure4-3: Networkrepresentationwithovertimeproduction. Nowconsiderasubplanwithdemandperiods1+1;:::;2.WeknowthatthereareatmostKp=d1+1;2

PAGE 98

periodsinwhichweonlyproduceorproduceandsubcontracttofullcapacity,respectively.However,thefractionalproduction(orsubcontracted)quantitydependsontheactualnumberofperiodsinwhichwesubcontract(orproduce)tofullcapacity. First,supposethatweknowthatthereareexactlykperiodsinwhichweproducetoatleastfullcapacityandthereisnofractionalproductionperiod.Sincewecanonlyusesubcontractinginperiodsinwhichproductionisatfullcapacity,weonlyneedtoconsiderk=Kp+s;:::;Kp.Thenthenumberofperiodsinwhichwebothproduceandsubcontracttofullcapacityandthequantitysubcontractedinthefractionalsupplyperiodareequaltop+sk=d1+1;2kC Bp+sk=d1+1;2kCp+skBrespectively.Notethat,incaseofuncapacitatedsubcontracting(B=1),thesereducetop+sk=0p+sk=d1+1;2kC: Similarly,wecanconsidercaseswherethereareexactlyperiodsinwhichweproduceandsubcontracttofullcapacityandthereisnofractionalsubcontractingperiod.Sincewecanonlyusesubcontractinginperiodsinwhichproductionisatfullcapacity,weonlyneedtoconsiderk0=0;:::;Kp+s.Thenumberofperiodsinwhichweproducetofullcapacityandthequantityproducedinthefractional

PAGE 99

supplyperiodareequaltopk0=d1+1;2k0B Cpk0=d1+1;2k0Bpk0C:

PAGE 100

ForeachofthethreelevelswewilldeveloparecursionthatrelatesthevaluesCt`(s1;s2)andshouldbesolvedbackwardsintime(fort=T;:::;1)andupthechain(fromlevel2tolevel0).

PAGE 101

ThissetofrecursionstakeO(T4)time,wherethebottleneckoperationsoccuratthesupplierlevel. Underthexedcoststructurewithspeculativemotives,weknowthat,ineachperiodwheretransportationtakesplace,wemayassumethattheendinginventoryatthesupplieriszero.Inthatcase,therecursionatthesupplierlevelsimpliesto:

PAGE 102

Since,atthesupplierlevel,thereareonlytwopossibleshipmentsizes,therunningtimeofthedynamicprogrammingalgorithmreducestoO(T3).NotethatthisisthesamerunningtimeaswasobtainedinVanHoeseletal.(2001)forthemodelwithonlyinventoryholdingatthemanufacturer.

PAGE 103

4-1 and 4-2 below. Inalltables,h,b,andhbindicatetheinventorystructureatthemanufacturer:inventoryholdingonly,backloggingonly,orinventoryholdingandbacklogging.Inallourmodelstheproductioncostfunctionsareessentiallygeneralconcavefunctions.Thecaptionslinear,fc/nsm(xed-charge/non-speculativemotives),andconcaveinthetablesrefertothestructureoftheremainingcostfunctions. Table 4-1 containstherunningtimesformodelswithoutsubcontractingopportunitiesandincludesmodelswithandwithoutproductioncapacities.Table 4-2 containstherunningtimesformodelswithsubcontractingopportunities,whereweassumethatproductioniscapacitated.AsshowninSection 4.5 ,therunningtimesformodelswithuncapacitatedproductionanduncapacitatedsubcontractingopportunitiesareequivalenttothoseformodelswithuncapacitatedproductionandnosubcontractingopportunities. Table4-1: Runningtimesformodelswithoutsubcontractingopportunities. bO(T3)O(T3)O(T4)O(T5)O(T5)hO(T3);yO(T3);yO(T5)*O(T6)O(T7)*hbO(T3)O(T4)O(T5)O(T6)O(T7) *SeeVanHoeseletal.(2001).ySeeZangwill(1969). Interestingandpromisingfutureresearchdirectionswillextendthemodelsinthischapterby,forexample,addingpricingordemandselectiondecisions;or

PAGE 104

Table4-2: Runningtimesformodelswithsubcontractingopportunities. bO(T5)O(T6)O(T6)O(T7)O(T8)O(T8)hO(T6)O(T7)O(T8)O(T8)O(T9)O(T10)hbO(T6)O(T7)O(T8)O(T8)O(T9)O(T10) allowingfortransportationcapacitiesornonstationaryproductionorsubcontractingcapacities.Finally,itwillbeinterestingtoexploremodelsinwhichnotsystem-widecostsbut,however,theinterplaybetweentwodistinctparties(supplierandmanufacturer)areexplored.

PAGE 105

93

PAGE 106

Thentheexpectedtotalcostisgivenby:I(Q)=cQ+pZ1Q(Q)f()dsZQ0(Q)f()d; ps: ForsingleproductandsinglefacilitynewsvendorproblemsLau(1997)presentsclosed-formexpressionsforcomputingtheexpectedtotalcosts,whicharepresentedforvariousdemanddistributions.Intheliterature,manystudiesextendthesingleproductsinglefacilitynewsvendorproblembyconsideringaxed-costassociatedwiththereplenishment(seeMoonandSilver2000),quantitydiscounts(seeJuckerandRosenblatt1985,PantumsinchaiandKnowles1991),pricedependentdemand(seeLauandLau1988),andtheuseofadierentobjectivesuchasmaximizationoftheprobabilitythatatleastacertainprotlevelisachieved(seeSankarasubramanianandKumanraswamy1983). Eppen(1979),ChenandLin(1989),andChangandLin(1991)considermulti-locationnewsvendorproblems.LauandLau(1996)considersamultiple-productmodelwithsingleandmultiplecapacityconstraints.SolutionofthemodelwithsinglecapacityconstraintsextendsHadleyandWhitin(1963)(pp.306-307)byallowingallkindsofcontinuousdemanddistributions.Tosolvethemodelwithmultipleconstraints,theyemployactivesetmethods,whichisproposedinLuenberger(1973).Theyalsoconsiderthecasewheredemanddistributionhasa

PAGE 107

nonzerolowerbound.Theotheranalysisofmulti-commoditytypesystems,withorwithoutcapacityconstraints,canbefoundinSivazlianandStanfel(1975),MoonandSilver(2000),andLauandLau(1995).AnexcellentreviewonthenewsvendorproblemhasbeenrecentlyprovidedbyKhouja(1999). Inthischapter,weextendtheclassicalnewsvendorproblemtoacapacitatedmulti-warehousemulti-retailersetting.Thisproblemintegratesinventoryandtransportationdecisionsinthepresenceofcapacitiesthatconstrainthequantityofproductthatcanbesuppliedfromeachwarehouse.Asintheclassicalnewsvendorproblem,weconsiderthecaseofasingleproduct.Wewillassumethattheretaileroutletsarespacedapart,sothateachendcustomerconsidersonlyasingleretailer,andifthatretailerdoesnothavetheproductinstock,thenalostsaleoccurs. Theretailerpaysaprocurementcostforeachunithebuys.Iftheactualdemandattheretailoutletislessthanthetotalquantitysuppliedtotheoutlet,remainingitemsaresoldatasalvagevaluewhichisstrictlylessthantheprocurementcost.Iftheactualdemandismorethantheshippedquantity,theretailerpayspenaltycostforeachunsatiseddemand.Penaltycostisacostvalueassignedtothelossofgoodwillassociatedwithalostsale.Weassumethattransportationisoutsourcedtoathirdpartylogisticsprovider,which,forthepurposeofevaluatingthetransportationcost,allowsustouseindividualtransportationcostforeachwarehouseandretailerpair. Weassumethatthereisasinglecentralownerofallwarehousesandtheretaileroutlets,suchasinthecaseoflargechainstoressuchasWalMart,whichfacesboththetransportationcostsaswellastheinventoryrisk.Eveninthecaseswheretheretailersdonothavethesameownerasthewarehouses,theproblemwillberelevantifthewarehousewishestocarrytheinventoryriskattheretaileraspartofanegotiatedcontract,inreturnfortheretailercarryingthewarehouse'sproduct.Theoverallobjectiveoftheproblemistodetermineasetofshipment

PAGE 108

sizesthatminimizesthetotalexpectedoverageandunderagecostsandthetotalshipmentcosts,therebyintegratingthetransportationandinventorydecisions.Weconsidertwoversionsoftheproblem:eachretailercanreceiveshipmentsfrommorethanonewarehouseandeachretailercanreceiveshipmentfromatmostonewarehouse. Arecentpaper,Benjaafaretal.(2004),considersamultiple-product,multiple-facilitymake-to-stocksystem.Theproblemsettingisdierentinthewaythedemandforeachproductoccursoneunitatatimeaccordingtoanindependentpoissonprocessandalsoeachfacilityhasaniteproductionrate.Toensurefeasibility,theyassumethattotaldemandrateislessthanorequaltototalproductionrate.Thepapertriestodeterminethebasestocklevelsofproductiatfacilityj,sij,andalsothefractionofthedemandofproductisuppliedbyfacilityj,ij,foralliandjtominimizethelong-runexpectedtotalcostperunittime.Theyconsiderthreetypesofcosts: Theyconsiderthreemodels:

PAGE 109

Ourmodelissimilartothedemandallocationproblemwithacentralizedinventory,sincetheinventorytypecostsareincurredattheretailers.AscomparedtoBenjaafaretal.(2004),ouruncertaintyisatthedemandside.Inotherwords,wedecidehowmuchtoshipeachretailerandhowtoallocatethisamountamongthewarehouses,butweknowthecapacitiesofthewarehouses.HoweverinBenjaafaretal.(2004),theydeterminethesupply(basestocklevel)foreachproductateachfacility.Intermsofcomputations,theydealwiththeproblemswithasmallnumberofproductsandfacilities(numberofproductsisatmost9,numberoffacilitiesisatmost4).However,wesolvelargeinstancestooptimalityinareasonabletimeusingabranch-and-priceapproach. Ourproblemmayhavetwointerpretations.Intherstone,itisageneralizationofthestandardnewsvendorproblemto(i)multiplewarehousesandmultipleretailers,(ii)capacitiesatthewarehouses,and(iii)transportationcosts.Inthesecondinterpretation,ourproblemisgeneralizationofthestandardtransportationproblemtoaccountfor(i)uncertaindemandsand(ii)moregeneraltransportationcosts.InSection 5.2 ,weworkwiththemodelwhereagivenretailercanreceiveshipmentsfrommorethanonewarehouse.However,inSection 5.3 ourmodelhasarestrictionthateachretailerreceivesreplenishmentfromatmostonewarehouse.Moreover,weintroducethenonlinearintegerknapsackproblemwhoseitemsizescanbeinuencedandpresentasolutionmethod.InSection 5.4 ,wepresentextensionsforasingle-sourcingproblemandhowthesolutionmethodchanges.InSection 5.5 ,wediscusstheimplicationsofthesingle-sourcingrestriction.WestatethecomputationalresultsinSection 5.6 .Lastly,weconcludethechapterandstatefutureresearchdirectionsinSection 5.7

PAGE 110

5-1 Figure5-1: Warehouse-retailernetwork,with3warehousesand4retailers. Ourdecisionvariableqij(i=1;:::;m;j=1;:::;n)istheamounttransportedfromwarehouseitoretailerj.Ciisthestoragecapacityofwarehousei.Theunitprocurementcostforwarehouseiisdenotedbyci.Theunitpenaltycostof

PAGE 111

retailerjisdenotedbypj,andthesalvagevalueofretailerjisdenotedbysj.LetIjdenotetheexpectednewsvendor-typecosts(salvageandpenalty)atretailerjasafunctionofQj=Pmi=1qij,andletTijdenotethetransportationcostfunctionfromwarehouseitoretailerjasafunctionofqij.Theexpectedtotalsalvageandpenaltycostfunctionatretailerjisequaltothecostofthestandardnewsvendorproblem,andreads 5.2.6 .Itisjustiabletoassumethatpj>sj,sinceotherwise,backloggingandsalvagingmayoccuratthesametime.Weassumethatwesatisfythedemandifwehavetheitem.

PAGE 112

5.2.5 .Now,wecanformulatethemulti-sourcingtransportationproblem(MSTP)as: Notethattheobjectivefunctionofthisoptimizationproblemisnotseparableinthevariablesqij.WesolveMSTPbyusingLagrangeanrelaxation.Beforeexplainingthesolutionprocedurewewanttodepictthepropertiesoftheoptimalsolution.

PAGE 113

constraintsareremoved,wehaveanoptimalsolutionwhereeachretailerissuppliedbyonlyitscostminimizingwarehouse. Sincethecostsarelinear,wehaveanextremepointoptimalsolutionwhichcontainsatmostn+mnon-zerovariables.Letthenumberofthenon-bindingcapacityconstraintsbem1;thenthenumberofthebindingconstraintsbecomesmm1.Wedenotenumberofnon-splitretailersbyn1;thenthenumberofsplitretailersisnn1.isdenedastheaveragenumberofwarehousestowhichasplitretailerisassignedand2.Now,thenumberofnon-zerovariablesis:n1+(nn1)+m1

PAGE 114

5{1 .Letbethevectorofifori=1;:::;m.WehavetheuncapacitatedproblemP:L()minimizenXj=1Ij(mXi=1qij)+nXj=1mXi=1(ci+tij)qijmXi=1i(CinXj=1qij) subjectto(P)qij0i=1;:::;m;j=1;:::;n:

PAGE 115

Now,wewillexplainthesubgradientalgorithmwhichisproposedinPolyak(1969)thatndstheoptimalvector.Wedene(k)iasthevalueofiatthekthstep.Listhecurrentbestobjectivefunctionvalueandbzisanupperboundvalueforz. SubgradientAlgorithm 1.FindL((k))andcorrespondingqijvalues.IfL((k))>L,setLtoL((k)).Iftheobjectivevalueisnotimprovingfor100iterationsk+1=k=2. 2.Setithcomponentofthesubgradientvector()toi=nXj=1qijCi;i=1;:::;m: Theresultingistheoptimaldualvariablevectoroftheconstraint 5{1 ofMSTP.Ifwesatisfycapacityconstraintswiththesolutionattheoptimalvalue,thissolutionisalsooptimalforMSTP.Otherwise,welookforalternativesolutions.Supposethatattheresultingsolutionofthesubgradientalgorithmretailerjissuppliedviawarehousek.Withoutchangingtheobjectivefunctionvalue,wecansharequantitysuppliedtoretailerjwithanotherwarehouseisuchthatci+tij+i=ck+tkj+k.Tojustifythisstatement,weuseKarush-KuhnTucker(KKT)conditions.SinceMSTPisaconvexproblem,anyKKTpointisoptimal.Wedeneiasthemultiplierforthecapacityconstraintofwarehouseiandijasthemultiplierforthenonnegativityconstraintsforqij,wewriteKKT

PAGE 116

conditionsforMSTPas: (pjsj)Fj(Qj)pj=i+ijtijcii=1;:::;m;j=1;:::;n whereQj=Pmi=1qij.Equation 5{3 canberewrittenasij=Fj(Qj)(pjsj)pj+i+tij+cii=1;:::;m;j=1;:::;n 5{5 ,qij=0foralli.Therefore,itisimpossibletogetQj=Pmi=1qijunlessQj=0.Thenforgivenjwecanwrite Inotherwords,retailerjissuppliedfromwarehouseithathasminimumci+tij+ivalue.Therefore,QjthatisthesolutionofEquation 5{8 isequivalenttoQjinthesolutionofL().Tondtheoptimalqijvaluesweneedtondafeasiblesolutiontothefollowingsetof(in)equalitieswhereiistheoptimalivalue:

PAGE 117

(5{9) (5{10) Inequalities 5{9 and 5{10 followfromtheconstraints 5{4 and 5{6 .Theequalities 5{11 and 5{12 followfromtheconstraints 5{3 5{5 ,and 5{6 Wecangetanoptimalprimalsolutionfromtheoptimaldualsolution.Attheendofthesubgradientalgorithm,wegettheoptimaldualvariablesi.Letussaythatatthesolutionoftheoptimaldualsolutionretailerjissuppliedviawarehousei.Fromconditions 5{11 5{12 ,and 5{13 ,weknowthatwecanalternativelysatisfyretailerjwithwarehousekifci+tij+i=ck+tkj+k.Therefore,ifcurrentqijvaluesdonotsatisfy 5{9 and 5{10 ,wemayreallocatetheretailersdemandintopossiblewarehousesinawaythattheseconditionsaresatised. 5-2 ,werepresentthemulti-sourcingproblemasaconvex-costnetworkowproblemthathasacostfunctionthatisseparableinthearcowcosts.Wehavemsupplynodessuchthateachnode(warehouse)ihascapacityCi.Moreover,wehavenretailernodesdenotedbyj0,andthedummydemandnode,D.Anarcfromwarehouseitoretailerj0correspondstothetotaltransportationandpurchasecost,whichislinearintheow.Thecostofthearcfromretailerj0tothesinknode,S,isIj.Theboldedarcshavezerocosts.

PAGE 118

Figure5-2: Convexcostnetworkowrepresentationforn=m=3. Intheliterature,manymethodsaresuggestedtosolveconvexnetworkowproblems.LetUbethelargestcapacityofthearcs.AccordingtoAhujaetal.(1993)ifsupply,demand,andcapacitiesareintegervaluedandwerestrictthefeasiblesolutionstointegers,theproblemcanbesolvedin:O(Alog(U)S(N;A;C) timeusingacapacityscalingalgorithm,whereS(N;A;C)isthetimerequiredforsolvingashortestpathproblemonanetworkwithNnodes,Aarcs,andwithCasthelargestcostarc.Forourproblem,sincethereisonlyonedemandnode,S,ndingtheshortestpathfromasupplynodetothedemandnoderequiresncomparisons.Therefore,computationtimeofthisalgorithmbecomesO((n+1)2(m+1)log(mXi=1Ci)):

PAGE 119

forthecontinuousowcase.Therunningtimeoftheiralgorithmispolynomialinlog1 Zangwill(1967)presentsaconvexsimplexalgorithm,whichisageneralizationofthelinearsimplexalgorithmandcanalsobeusedtosolveproblemswithnonseparableconvexobjectivefunctions.Itusesgradientatthecurrentsolutionasthecostvector.Thistechniquediersfromthelinearmethodinthatthenonbasicvariablescantakenonzerovaluesbetweentheupperandlowerboundsandwhenthenonbasicvariablehasbeenselectedforachangeinvalue,aonedimensionalsearchisrequiredtodeterminethemagnitudeofthechange.However,duetodegeneracyitispossiblethatthisalgorithmmaynotconverge. TheFrank-WolfemethodisaNewton-typemethod.Itstartsfromaninitialsolutionandsolveslinearprogrammingsubproblems.Iflinearprogrammingsubproblemsareeasilysolved,thismethodisveryecient,butitmayexperienceslowconvergence. Weintraub(1971)considerstheproblemofdeterminingcontinuousowsofminimumcostinanetworkwithconvexcostfunctions.Heproposesanegativecyclecancellingalgorithm,inwhichthecircuitowwiththemostnegativeincrementalcostisaddedtothecurrentow.Heshowsthatalgorithmhaslinearconvergencetooptimalsolutionandconvergewithin1percentoftheoptimalsolutioninO(AN3)calculations,whichisO((n+1)(m+1)(n+m+2)3)forourproblem.

PAGE 120

WeagainuseLagrangeanrelaxationandrelaxthecapacityconstraints 5{14 and 5{15 .Letbethevectorofi,dualvariablesoftheconstraints 5{14 fori=1;:::;mandbethematrixofijfori=1;:::;m;j=1;:::;n,dualvariablesoftheconstraints 5{15 .Forgivenvectorandmatrix,P;is:L(;)minnXj=1Ij(mXi=1qij)+nXj=1mXi=1(ci+tij)qij+nXj=1mXi=1fijzijmXi=1i(CinXj=1qij)nXj=1mXi=1ij(Cizijqij) subjectto(P;)qij0i=1;:::;m;j=1;:::;n;zij2f0;1gi=1;:::;m;j=1;:::;n:

PAGE 121

WhiledeterminingL(;),wecaneasilyassignzijvalues.zij=8>><>>:1iffijijCi<0;0otherwise. Andtheremainingproblemisanuncapacitatedmulti-warehousemulti-retailernewsvendorproblem,inwhicheachretailerisassignedtoitscostminimizingwarehouseiandsuppliedQj,whereF(Qj)=pjcitijiij

PAGE 122

IfRx0fj()disconstantintheinterval[Q Qj],thentherstexpressionofIjisconstantforQ 5.2.3 ifthefollowingconditionistrue: maxi=1;:::;mfpjitijci because,FjisinvertiblewhenQjisnotintheinterval[Q Qj].Ifwehaveatleastoneretailerwithmaxi=1;:::;mfpjitijci thenourfeasiblesetforretailerjenlargestoQ 5-3 .Wedrawthenetworkforthecasethatallretailerssatisfytheconstraint 5{18 .Ifretailerjdoesnotsatisfythiscondition,node(`;j)haszerosupply.Notethatallarcsareuncapacitatedandhavezerocost.

PAGE 123

Figure5-3: Transportationnetworkinthecaseofatspots. 5.3.1ModelFormulation

PAGE 124

Notethat,inthecaseoflinearprocurementandtransportationcosts,wegetthesameoptimalobjectivevalueifwesolveSSTPwiththerestrictionthatweneedtoassigneachretailertoawarehousewiththeconstraint:mXi=1zij=1j=1;:::;n: Now,wecanformulateSSTPasbelow:minimizenXj=1mXi=1(Ij(qij)Ij(0)+(ci+tij)qij)zij+nXj=1Ij(0) subjectto(SSTP) Constraints 5{19 ensurethatthecapacityconstraintsofthewarehousei=1;:::;maresatised.Inthiscase,zijandqijarenotlinkedviatheconstraints.

PAGE 125

However,theobjectivefunctionandthecapacityconstraintensurethatifzij=0thenqij=0.However,forthecurrentcoststructures,zij=0andqij=0isnotdierentfromzij=1andqij=0.Constraints 5{20 assurethataretailerissuppliedbyatmostonewarehouse. Inthecomingsections,insteadofIj(qij)Ij(0),wewilluseI0j(qij).Now,weshowthatSSTPisNP-completeinthestrongsensebyreducingthe3-partitonproblemtoaspecialcaseofSSTP.The3-partitionproblemisdenedasfollows(seeGareyandJohnson1979): 3-PARTITION:LetsetAhas3melementsandBbeapositiveinteger.Thesizesoftheelementsarea1;a2:::;a3msuchthatB=4aiB=2fori=1;:::;3mandPi2Aai=mB.CanAbepartitionedintomdisjointsetsA1;A2;:::;AmsuchthatforanysetAi,Pj2Aiaj=B? Fromtherestrictionsonaj,eachsubsetcontainsexactlythreeelementsfromA.Givenanyinstanceofa3-partitionproblem,wedenethefollowinginstanceoftheSSTPproblemof3mretailersandmwarehouses:ci=c>sji=1;:::;m;j=1;:::;n;tij=0i=1;:::;m;j=1;:::;n;Ci=Bi=1;:::;mI0j(x)=pjZ1x(x)deFj()sjZx0(x)deFj()Ij(0)j=1;:::;n pjsj:

PAGE 126

5.3.1 saysorderquantityforretailerjisajforj=1;:::;n.SincetotaldemandsatisedisPnj=1aj=mBandeachwarehousehascapacityB.TheonlypossiblescenarioistopartitionretailersintomgroupsthathavetotaldemandB. Now,supposethatwehaveasolutiontothe3-partitionproblem.ThatmeanswecanpartitiontheelementsintomgroupsthattotalsizeoftheelementsinagroupisexactlyB.Therefore,totalcostoftheproductionplanbecomesPnj=1I0j(aj)+cPnj=1ajandcapacityconstraintsofwarehousesaresatised.2 InafeasiblesolutionofSSTP,nretailersarepartitionedintom+1subsets.Lastsubsetistheretailersthatarenotassignedtoanyofthewarehouses.

PAGE 127

Therefore,wecanreformulateSSTPasaset-partitioningproblem,saySP.ToformulateSP,weneedtodenethefollowingnotation: Wethenobtainthefollowingset-partitioningformulationofSSTP: minimizemXi=1KiXk=1g(zi:k)ik+nXj=1Ij(0) subjectto(SP) wherefori=1;:::;m;k=1;:::;Kig(zi:k)=minimizenXj=1I0j(qijk)+(ci+tij)qijkzijk

PAGE 128

subjectto pjsj)zijkCi: pjsj)zijkCi; Thereasonwhyweemployaset-partitioningformulationisthefactthatSPisalinearintegerprogrammingproblemascomparedtoanonlinearmixedintegerproblem(SSTP).Moreover,LPrelaxationofSPgivesatighterboundthenLPrelaxationofSSTP.

PAGE 129

GivenafeasiblesolutionofSP,wecangetafeasiblesolutionofSSTPbyusingthefollowingrelations:zij=KiXk=1zijkik;qij=KiXk=1qijkik: ThecolumngenerationprocedurestartswithafeasiblesolutiontoSP.Forexample,anypartitionofretailerstothewarehousesisfeasible.Westartwiththesolutionthatallwarehousesdonotshiptoanyoftheretailers.Wedenotethesetofcolumnsconsideredforwarehouseiby

PAGE 130

1.WesolveLPSP,withtheadditionalconstraintsthatki=0forallk=2 2.WesolvethepricingproblemassociatedwithLPSPforeachwarehousei.Iftheoptimalsolutionvaluetothepricingproblemforwarehouseiisnegative,thecorrespondingcolumnisaddedto IfwegetanintegersolutiontoLPSP,thenthisistheoptimalsolutionforSP.WecaneasilytransformthesolutionofSPtoSSTP.Ifwegetafractionalsolution,webranchandcreatetwocomplementaryproblems.Wegodepth-rstinthesearchtree.Wecontinueuntilweprunealltheleafnodes. ThesuccessofcolumngenerationdependsonhowecientlywecansolvethepricingproblemofLPSP.Inthenextsection,weexplainthepricingproblem. Lettheoptimaldualvariablesforconstraints 5{23 and 5{24 bejandi,respectively.Notethatj0forallj.ThenthepricingproblemforLPSPforwarehouseicanbestatedas:minimizeg(z)nXj=1jzji

PAGE 131

minimizenXj=1I0j(Qj)+(ci+tij)Qjjzji Thisproblemisanonlinearknapsackproblemwhoseitemsizescanbeinuenced.InSection 5.3.4 ,wedescribeoursolutionapproachtoamoregeneralprobleminthemaximizationform. maximizenXj=1j(Qj)zj NotethatPisaspecialcaseofGP.BeforedescribingthesolutionprocedureforGP,wedescribeaprocedurethatmayreducetheproblemsizebyeliminatingsome

PAGE 132

retailers.Forexample,ifj(Qj)0forallQjthenwithoutlossofoptimalitywecansetQj=0andzj=0.Ifj(0)0thenzj=1andwecanhavetwocases: Inthatcase,wecanassignzj=1andreducej(Qj)byj(0).InFigure 5-4 ,weshowthetypeofjfunctionwhenwecannotdeterminetheoptimalzjvaluebeforesolvingtheproblem(thatiswhywestillhaveanintegerknapsackproblem)andthejfunctionwhenzj=1.Wesolvethepricingproblembybranch-and-boundalgorithmandforeachnodeofthebranch-and-boundtreewesolvetheLPrelaxationofthepricingproblem.WeexplainthesolutionmethodologyfortheLPrelaxationofGPinthefollowingsection. Figure5-4: 5-4 a.Foreachofthoseretailers,let

PAGE 133

andwedenethefunctionj(Qj)=8>><>>:j( 5-5 .FortheretailersthathavejasinFigure 5-4 b,wedenej(Qj)=j(Qj)forallQj: WewriteourrelaxationofRGP(GP')as: maximizenXj=1j(Qj) subjectto(GP')

PAGE 134

qjQj=Q0j;zj=1otherwise thenj(Q0j)=j(Qj)zj:

PAGE 135

When0j(0)<0,Qj=0from 5{35 .Otherwise,Qjisthesolutionof0j(Qj)=.Moreover,Qj2[0; qj]when=0j( pjsj)g:

PAGE 136

Sincethecumulativedistributionfunctionsofthedemandareinvertible,theonlypotentialissuemaybethelinearsegmentsofthefunctionsj.Foreaseofexplanation,wedepictPnj=1QjversusinFigure 5-6 .TheverticallinesinFigure 5-6 correspondtothelinearsegmentsofj,when Figure5-6: Ifwendaatoneoftheverticallinearsegmentsthenwehaveafractionalsolution.Otherwise,wehaveanintegersolution.Verticallinescorrespondstovalueswheretheslopeofthejfunctionisequaltotheaveragej:=j(Qj) Incomputations,weneedtocalculate

PAGE 137

5.3.5 .2 Intheprevioussection,wediscussedhowwesolvetheLPrelaxationofthepricingproblem.Moreover,weshowthatthesolutionhasatmostonefractional

PAGE 138

variable,forexamplezj.Wehaveanintegerfeasiblesolutionifwesetzj=0.Therefore,wegetanapproximatesolutionforpricing.Intheimplementation,wesolvepricingapproximately.Ifwegetanegativesolutionvaluewecreatethecorrespondingcolumn;otherwisewesolvethepricingproblembyapplyingabranch-and-boundalgorithm.Thereasonbehindthischoiceisthatsolvingpricingapproximatelyrequireslesscomputationtimethansolvingpricingoptimally.InSection 5.6 ,wecomparetheCPUtimesoftwopricingstrategies.Intherstpricingstrategy,wesolvethepricingproblemapproximately;ifwecannotndanegativesolutionthenwesolveitoptimally.Inthesecondstrategy,wealwayssolvethepricingproblemoptimally. iisfractionalthenthereexistsatleastonefractionalz k iandKiXk=1k i=1: i>0andtogetz i>0.Therefore,tohaveallz

PAGE 139

tohavesomecolumnstobeequal,whichisnotallowedinacolumngenerationprocedure.2 1=8>><>>:1ifistrue;0otherwise.

PAGE 140

Fixed-chargeprocurementcosts

PAGE 141

minimizenXj=1I0j(Qj)+(ci+tij)Qjjzji+Biyi Thistypeofprocurementcostdoesnotchangethepricingproblem.Weonlyneedtoconsidertwocases.Whenyi=0,itmeansnocapacitybywarehouseiisallocatedtoanyoftheretailersandobjectivevalueofthepricingproblemisiandzj=0forallj.Whenyi=1,wecansolvethisproblemasweexplaininSection 5.3.4

PAGE 142

Forthiscase,wecanwritethepricingproblemforwarehouseias:minimizenXj=1I0j(Qj)+tijQjjzji+mint=1;:::;TifBti+ctinXj=1Qjgyi=mint=1;:::;TifminimizenXj=1I0j(Qj)+tijQjjzji+Btiyi+ctinXj=1Qjyi!g Therefore,weneedtosolveTipricingproblems(andforeachunderliningxed-chargeprocurementcostfunction)forwarehouseiandchoosetheleastcostone.Again,yi=0caseistrivialandthesolutionvalueisi. minnXj=1I0j(Qj)+tijQjjzj+ci(nXj=1Qjzj)i

PAGE 143

Weusethisproblembyusingthefollowingapproach.IfweknowPnj=1Qjzjthenitremainstondthebestallocationofthisamounttotheretailers.Forthisaim,wedene(w)=minnXj=1I0j(Qj)+tijQjjzj+ci(w)i 0zj1j=1;:::;n Weproposetoperformakindofsearchfortheoptimalwoverthe[0;Ci]interval.However,westilldonotknowthestructureof(w). minimizenXj=1mXi=1I0j(qij)+(ci+tij)qijzij+mXi=1i(ei)+nXj=1Ij(0)

PAGE 144

subjectto(SSTP') Now,wecanwritethepricingproblemwithcapacityexpansionopportunityas: minimizenXj=1I0j(Qj)+(ci+tij)Qjjzj+i(ei)i WecannotusethesamemethodinSection 5.3.4 tosolvethispricingproblembecauseoftheconstraint 5{57 .WeexplainhowtoconvertthisproblemtothestructureinSection 5.3.4 .Tothatextent,supposethatthereisamaximumamountofcapacityexpansionforwarehouseigivenby

PAGE 145

ornot.Therefore,wecanwrite:Qn+1zn+1= 5{57 )withcapacityexpansionopportunitiesas:nXj=1Qjzj+Qn+1zn+1Ci+ Notethatifweassumeiasanon-decreasingfunctiontheneiisanon-increasingfunctionandei(0)=0ascanbeseeninFigure 5-7 .Moreover,itiseasytoshowthatifiisconvex(concave)theneiisalsoconvex(concave).Now,wecanrestatethepricingproblemwithcapacityexpansionopportunitiesas:

PAGE 146

minimizenXj=1I0j(Qj)+(ci+tij)Qjjzj+ei(Qn+1)zn+1+i( (5{61) (5{62) Foreaseofexplanation,weremovetheconstanttermi( 5.3.4 .Now,wewanttosolve maximizen+1Xj=1j(Qj) subjectton+1Xj=1QjCi+ Now,supposethatthecapacityexpansionhasaxed-chargestructure:Liyi+i(ei)wherei(0)=0andLi>0; minimizenXj=1mXi=1I0j(qij)+(ci+tij)qijzij+mXi=1(Liyi+i(ei))

PAGE 147

subjecttonXj=1qijeiyiCii=1;:::;mQj0j=1;:::;nei0i=1;:::;mzj2f0;1gj=1;:::;nyi2f0;1gi=1;:::;m

PAGE 148

Weknowthat@Ij(Qj) Combining 5{63 and 5{64 ,wegetIj(Qj+j)Ij(Qj)@Ij(Qj) 5.5.2 ,wecanconcludethat:zMSTPzSSTP=nXj=1(Ij(Qj+j)Ij(Qj)+tj(Qj+j;Qj))nXj=1(pj+tminj+cmin)maxfj;0g+(sj+tmaxj+cmax)minfj;0g;

PAGE 149

Thenweconcludethat,zSSTPzMSTPnXj=1pjmaxfj;0gnXj=1(tmaxj+cmax)minfj;0g:2 InSection 5.6 ,weanalyzethefactorsthatcreatedierencebetweenthequantitiesshippedtoaretailerinMSTPandSSTPsolutions. 5.6.1ExperimentalDesign

PAGE 150

instanceswithn=m=10,n=m=5,andn=m=2,whilen=50;80,and100.Weobservethatcomputationtimeincreasesasnincreases.Eventhoughthesizesoftheproblemsarebig,thebranch-and-pricealgorithmtakeszerotofourminutes.Foreachnandmpair,wegenerateverandominstancesinthefollowingway: Weassumethatdemandofretailerjisdistributeduniformlyintheinterval[0;eDj],whereeDjisgenerateduniformlyintheinterval[LD;UD].WechooseLD=0andUD=40forourcomputations. Weconservetherelationshipbetweenthesalvage,penaltyandthepurchasecostsas:pj>ci>sji=1;:::;m;j=1;:::;n: Locationsofthewarehousesandtheretailersareuniformlygeneratedinthesquare[0;5]2.Sincethetransportationcostisproportionaltothedistance,thedistancebetweentheretailersandwarehousesaretakenasthetransportationcosts. WedeneDastheexpecteddemandperwarehouseandD=(LD+UD)n Therefore,totalcapacity,Pmi=1Ci,isintheinterval(0:3(LD+UD)n

PAGE 151

Wedenethetransportationcostofshippingxunitsfromwarehouseitoretailerjas:fij1fx>0g+tijx: Foreachn;mpairwegenerateveprobleminstancesforthesesixcombinationsofcapacityandxed-cost. Webranchbasedonavariabledichotomy.Thismeansthatwechooseafractionalzijandattheleftbranchweequateittoone,andattherightbranchequateittozero.Wetesttwoselectionrulesforthebranchingvariable. Therstrule(B1)choosesthefractionalvariableclosestto0.5;iftherearetiesthelowestindexedoneischosen.Inthesecondpricingrule(B2),wecalculateapseudo-cost,egij,forallfractionalzijvalues,whereegij=KiXk=1g(zi:k)zijkki

PAGE 152

requiresanamountofmemorythatisalinearfunctionoftheproblemsize,anditiseasytoimplement. Westopifwendasolutionvaluelessthanorequalto0.998ofthesolutionvalueoftherootsolution;otherwise,thealgorithmcontinuesuntilitndstheoptimalsolution.Weputthisconditionsinceweobservedthatafterbranching,theobjectivefunctionvaluedoesnotchangetoomuch.Therefore,wedonotwanttospendalotoftimetondabettersolutionthan0.998percentofthesolutionvalueoftherootnode. Whilesolvingthepricingproblemwefollowtwoapproaches.Intherstrule(P1),wesolvethepricingproblemapproximately(seeSection 5.3.5 )andifwecannotgetanegativeobjectivevaluethenwesolvepricingoptimally.Thisstrategyisappliedtoreducetherunningtimesincesolvingpricingoptimallymayrequireapplyingthebranch-and-boundalgorithmifthesolutionisnotintegral. Inthesecondrule(P2),wesolvepricingoptimallyallthetime.Thisstrategyaimsatreducingthenumberofpricingproblemswesolve,byaddingmostnegativecolumnsallthetime. Ourperformancemeasurestocompareourstrategiesare: numberofnodesgenerated, Wecollecttheseperformancemeasurestocomparestrategiesandtoseehowtheyaecttherunningtime.Wealsosolvethemulti-sourcingversionsofeach

PAGE 153

instancegenerated.LetzMSTPbetheoptimalobjectivevalueofthemulti-sourcingproblemandzSSTPbetheoptimalobjectivevalueofthesingle-sourcingproblem.Wecomputethefollowingmetric: 5.3.5 )andgeneratetheresultingcolumns.Wecontinueuntilnocolumnpricesout.Attheendofthecolumngeneration,ifwedonotgetanintegersolution,wesolvetheproblemundertherestriction:ki2f0;1gi=1;:::;m;k=1;:::;Ki: Formostofthetestproblemswegenerated,inthebranch-and-pricetreetheLP-relaxationvaluesoftheparentnodeandthechildrennodesdonotdiermorethan0.2percent.ThismaybebecauseMIPhasasymmetricstructure,thatmeanswehavemanysolutionswithobjectivefunctionvaluesclosertotheoptimalsolution.Inotherwords,foragiveni,g(zi:k)valuesthatarecoecientsofki

PAGE 154

haveclosevalues.Thereasonisthatcosttermsarecreateduniformlyinthesameintervalsandthismayresultincloseg(zi:k1)andg(zi:k2)valuesifsubsetsk1andk2havethesamenumberofretailers.Asanexample,thebranch-andpricetreeofaproblemfromgroupn=100,m=20isshowninFigure 5-8 Figure5-8: Branch-and-pricetreeofaproblem. Floatingnumberarithmeticincursrandomizedround-oerrors.Inordertondcandidatecolumnsweperformsomenumericcalculationswithoatingpointnumbers.Insomeinstances,whenthepricingvaluesapproachzero,CPLEXdoesnotaddthecolumnsthatwegenerate.Therefore,theprogramgoesintoaninniteloopbycreatingthesamecolumns. Weobservealotofdegenerateiterations.Thereasonisthatwhenwehaveanintegralsolutionwiththecurrentcolumnsthesolutionhasadegeneratebasis.Inthiscase,wehavemnonzerobasicvariablessinceforeachi,onekitakesvalue1.Atmostn1oftheassignmentconstraints 5{23 havenonzeroslackvariable.Because,tohavennonzeroslackvariables,weneedtohavemcolumnswithallzeros,whichwehaveattheinitialsolutionbutnotinthelaterphasesofthecolumngenerationalgorithm.Therefore,wehaveatmostm+n1nonzerobasicvariables. TheaverageCPUinseconds,numberofcolumnsgenerated,andnumberofnodesgeneratedfortherootnode(B&Proot)andthewholeproblem(B&P)in

PAGE 155

thecaseofzeroxed-costoftransportationaregiveninTables 5.6.4 and 5-2 .WealsosolveallproblemsetsinthecaseoflowxedcostsandtheresultsaregiveninTables 5-3 and 5-4 .Werepeattheexperimentsforhighxed-costandtheresultscanbeseeninTables 5-5 and 5-6 .Weobservethatexceptforthepairm=25andn=50,thetimespenttosolvethezeroxed-costcaseishigherthanthetimestosolvenonzeroxed-costcases.Thereasoniswhenwehavexedcost,itincreasesthenumberofretailerseliminatedfromthepricingproblems.Asxedcostincreasesmoreretailersmaybeeliminatedfromthepricingproblems.CPUtimesspenttosolveSSTPforallmandnpairscanalsobeseeninFigures 5-9 and 5-10 ,wherewedenotethem;npairasm:n. Table5-1: Zeroxedcostoftransportation(highcapacity). B&ProotB&P m.nCPUcolsnodesCPUcols 5.5012:450932:26:621:8801128:410.509:3401136:42:613:0401214:825.5010:3121883:83:814:1141980:68.8060:3282024:0494:5982300:016.8042:9652532:89:571:9352848:040.8067:5344811:09108:9005121:810.100143:0303121:41:4158:4683175:820.100108:5734064:39:5167:1484356:550.100148:6766811:64:2192:3847043:4 Asnincreases,CPUtimeincreases.Thepercentageofthepricingproblemswherebranch-and-boundisappliedforallcombinationsofn,m,xedcost,andcapacitycanbeseeninTable 5-7 ,whereLCandHCrepresentshighcapacityandlowcapacity,respectively.Forgivenn,wegethighestpercentageswhenwehavehighxedcostandlown=m.Wecanexplainthisresultbyreferringto 5.3.4 ).IfwendthevaluesuchthatPnj=1Qj=Ciataverticalline,thenwehaveafractionalsolution.Asxedcostsincrease,thenumberofverticallines

PAGE 156

Table5-2: Zeroxedcostoftransportation(lowcapacity). B&ProotB&P m.nCPUcolsnodesCPUcols 5.505:332451:81:45:770464:810.505:343728:81:56:168759:325.507:0801116:617:8861134:48.8019:0161108:24:225:2841228:616.8016:4821606:4726:2501763:040.8038:0642842:6555:1923045:610.10043:6361843:46:264:8062010:420.10033:4702268:010:666:5122622:650.10073:9484003:67:8127:7704343:2 Table5-3: Lowxedcostoftransportation(highcapacity). B&ProotB&P m.nCPUcolsnodesCPUcols 5.505:620540:81:46:238552:610.506:260632:41:87:678666:625.505:692733:815:426:8541099:68.8023:396874:42:230:99493716.8028:1421184:22:636:8461234:240.8032:7701428:8554:9941524:810.10071:3881611:04:6102:6121722:220.10054:5061715:46:695:1301875:050.10079:7862090:24:2113:6022179:0 Table5-4: Lowxedcostoftransportation(lowcapacity). B&ProotB&P m.nCPUcolsnodesCPUcols 5.501:762217:41:82:116234:810.502:400289:81:52:780297:825.503:000330:818.809:190436:4116.8011:456534:83:816:344590:640.8020:462646:8110.10024:774690:6120.10021:554681:44:233:076761:250.10053:970911:01 decreasesbutthelengthoftheselinesincreases;therefore,probabilityofhitting

PAGE 157

Table5-5: Highxedcostoftransportation(highcapacity). B&ProotB&P m.nCPUcolsnodesCPUcols 5.503:736358:41:84:532374:410.503:182360:8125.503:010345:234:552370:68.8019:832704:61:422:556720:616.8016:916708:63:824:762760:440.8033:408953:42:644:7381002:610.10042:376970:6120.10046:0201022:4365:8321076:050.10057:4941020:85:8115:3361154:4 Table5-6: Highxedcostoftransportation(lowcapacity). B&ProotB&P m.nCPUcolsnodesCPUcols 5.501:044121:81:41:254126:410.501:063128:7125.503:276255:018.805:840246:61:87:142263:616.807:504245:21:49:224258:640.8020:806457:4110.10013:654350:05:427:222447:420.10013:954332:24:225:878392:450.10050:210596:21 averticallineincreases.Moreover,aproblemwithhighcapacityalwaystakesalongertimethanitslowcapacityversion,sincethefeasibleregionenlarges. Table5-7: Thepercentageofthepricingproblemsthatbranch-and-boundisused. zeroxed-costlowxed-costhighxed-cost m.nLCHCLCHCLCHC 5.5054:0860:1161:7949:5560:0053:6610.5044:7544:6262:1651:9263:6849:4425.5039:2837:4866:0475:1486:9960:258.8057:5953:2359:8051:8766:0552:7716.8042:4344:3462:0748:2067:0255:4240.8043:7138:2863:5153:3284:7094:1710.10052:9942:0459:8555:0577:9150:4420.10048:2438:1960:0256:5468:2254:6150.10043:7632:2168:9559:7187:2665:40

PAGE 158

Figure5-9: SSTPCPUtimes(highcapacity). Figure5-10: SSTPCPUtimes(lowcapacity). Whenwarehouseshavehighcapacities,weobservethatthehighxed-costcasehasthehighestoptimalsolutionvalue.Forsomen,mpairs,wecannotdierentiatebetweenzeroandlowxed-costcasesandforsomecasesthelow

PAGE 159

xed-costcasehasahighercost(seeFigure 5-11 ).Forlowcapacities,allxed-costcaseshavesimilarobjectivefunctionvalues. Figure5-11: SSTPsolutionvalues(highcapacity). TheeectofcapacitiesonSSTPsolutionvaluesforzeroxed-cost,lowxed-cost,andhighxed-costcasescanbeseeninFigures 5-12 5-13 ,and 5-14 ,respectively.Forallxed-costalternatives,SSTPsolutionvaluesfollowasimilarpattern. WealsosolveourproblemsetswiththeheuristicdescribedinSection 5.6.3 .Theheuristicworkswellforthelowcapacitycases.Forhighcapacitiesitalsoworkswellexceptwhenm=10,n=100andm=20,n=100.Inthosecases,heuristictimeexceedsthebranch-and-pricetimebecauseattheendofthecolumngenerationwegetafractionalsolutionandtheheuristicalgorithmneedstosolveanintegerproblemusingCPLEX.ThesummaryofCPUtimesinsecondsforheuristicandbranch-and-pricealgorithmsaregiveninFigures 5-15 and 5-16

PAGE 160

Figure5-12: SSTPsolutionvalues(zeroxedcost). Figure5-13: SSTPsolutionvalues(lowxedcost). Figure 5-17 alsogivesanideaofhowclosetheheuristicsolutionistotheoptimalsolution.Weobservethattherelativeerroroftheheuristicdecreasesasn=mincreasesandthenstartstoincreaseasn=mincreasesmore. WecomparetheCPUtimesofthetwopricingrulesforhighcapacitiesinFigure 5-18 andforlowcapacitiesinFigure 5-19 .Now,considerthelow

PAGE 161

Figure5-14: SSTPsolutionvalues(highxedcost). Figure5-15: Heuristicandbranch-and-pricesolutiontimes(highcapacity). capacitycase.Thenumberofpricingproblemssolvedusingthebranch-and-boundalgorithmandalsothenumberofpricingproblemssolvedarebothhigherwhenweusethesecondpricingrulethanthecasewhenweapplytherstpricingrule(see

PAGE 162

Figure5-16: Heuristicandbranch-and-pricesolutiontimes(lowcapacity). Figure5-17: Heuristicandbranch-and-pricesolutionvalues. Figure 5-20 ).However,whenweusetherstpricingrule,CPUtimesarealittlebitlower.Moreover,thenumberofcolumnsgeneratedarealsoverycloseforthetwopricingstrategies(seeFigure 5-21 ).

PAGE 163

Forthehighcapacitycase,wecannotseetheserelations,andthenthenumberofLPrelaxationssolvedaftercreatingcolumnsalsoaectstherunningtimesofthebranch-and-pricealgorithm(seeFigures 5-22 and 5-23 ). Figure5-18: Timecomparisonforpricingstrategies(highcapacity). Figure5-19: Timecomparisonforpricingstrategies(lowcapacity).

PAGE 164

Figure5-20: Pricingstrategies(lowcapacity). Wecomparetwobranchingrulesbyusingthosedatasetswhosebranch-and-pricetreescontainmorethanonenodeforn=50.AsseenintheTable 5-8 ,branchingrule2isbetterthanbranchingrule1sinceittakeslesstimebyreducingtheaveragenumberofnodescreated. Table5-8: Branchingrulecomparison. rule1rule2 averagenumberofnodes6:5565averagerootcolumns396:111392:778averagetotalcolumns473:778452:889averageB&Ptime4:4393:590averageB&Bapplied385:889337:444averagepricingsolved630:000583:333averageSSTPvalue7045:3827045:321 WealsoobservethefactorsthatcreateahighpercentagedierencebetweentheoptimalsolutionvaluesofMSTP(zMSTP)andSSTP(zSSTP).WedeneRaszSSTPzMSTP

PAGE 165

Figure5-21: Columnsgeneratedfortwopricingrules(lowcapacity). Figure5-22: Pricingstrategies(highcapacity). Actually,Risameasureofhowmuchthesingle-sourcingconstraintdegradesthesolution.Firstly,wetesttheeectofthenumberofwarehouses,m,onR.Tothatextent,wesolveproblemswith5,10,25,and50warehouseswhilekeeping

PAGE 166

Figure5-23: Columnsgeneratedfortwopricingrules(highcapacity). thenumberofretailersas50.Inthesecomputations,wekeeptheexpectedtotalcapacitiesconstant.SSTPandMSTPoptimalsolutionvaluesaregiveninFigure 5-24 .InFigure 5-25 ,weshowthatasmincreasesRincreases. Figure5-24: ComparisonofMSTPandSSTPsolutionvaluesasmchanges.

PAGE 167

Figure5-25: Figure5-26: InFigure 5-26 ,weshowhowRchangesforallpairsofmandn;alsoweseethatforagivenn,Rincreasesasmincreases.Secondly,wedesignatesttoobservetheeectsoftheotherparametersonR.Weperformthistestforn=50andm=10.Wegenerate5probleminstancesasdescribedinSection 5.6.1 andcallthisdatasetthebasecase.Fortheotherdatasets,weonlychangeoneortwoparametersatatime.Table 5-9 showsthechangeintheotherdatasets.WeobservethehighestRwhenwedoublethenumberofwarehouses,distribute

PAGE 168

capacityevenly,anddierentiatethetransportationcosts.Inotherwords,togethighanddierenttransportationcostsforaretailer,weplacethewarehousesonthebaselinewithequalintervalsasseeninFigure 5-27 .Dashedlinesshowthedistancesoftheboldedretailertosomewarehouses.Indataset7,weequallydistributetotalcapacityto20warehouses. Figure5-27: Locationsofthewarehousesandtheretailersindataset6. Table5-9: Datasets. datasetnodescription 1basecase2demandisgenerateduniformlyin[40;80]3penaltyisgenerateduniformlyin[40;60]4capacityisgenerateduniformlyin[1:5D;3D]5foraretailerjwexedpjDj=48006locationsoftheretailersgeneratedinthesquare[0;10]2720warehousesevencapacities,locationsoftheretailersgeneratedinthesquare[0;10]2

PAGE 169

Figure5-28: EectsofparametersonR. Figure5-29: decreasesformostofthecases.Thereasonisthatatthebeginningofthepricingproblem,asxedcostincreaseswemayeliminatemoreretailers.Themaintrendistherunningtimeincreasesasnincreases.Thisresultisexpectedsincewhilesolvingapricingproblem,thenumberofsearchintervalsisO(n). Anobviousobservationisthatanhighercapacityversion(notashighastheuncapacitatedversion),takesmoretimethanalowercapacityversion,sincewesearchalargerfeasibleregion.Wealsotesttheimplicationsofthesingle-sourcingrestriction.WeseethatforaxednthepercentagedeviationofMSTPfromSSTPincreasesasmincreases.Thereasonisthattheexpectedtotalcapacity

PAGE 170

isconstantanddistributedamongmorewarehouses.Therefore,itreducesthemaximumamountthatcanbesuppliedtoaretailerintheSSTPsolution.Anotherfactorthatcreatesahugedierenceisdistributingthecapacityevenlyamongmorewarehouses. ComputationresultsshowthatasweincreasetheexpecteddemandtermsforallretailersthenbothMSTPandSSTPsolutionvaluesincreasedramatically.However,ifthereareonlyafewretailerswithhighpenaltycostanddemandterms,thentheSSTPsolutionvalueincreasesmorethantheMSTPsolutionvalue.Whenpenaltycostsarehigher,computationtimeishigher.AllotherresultsaresummarizedintheAppendix. Weshowthatbranch-and-pricealgorithmcaneasilybeadaptedforxed-chargetransportationcosts.However,themulti-sourcingproblemsubgradientoptimizationalgorithmdepictedonlyprovidesalowerboundfortheoptimalsolutionvalue.Determiningthequalityofthislowerboundisacandidateforafutureresearchtopic. Includingtheopportunityofthereallocationoftheitemsattheretailersafterdemandisobservedisalsoaninterestingfutureresearchdirection.Whilethiscoordinationmechanismdecreasestotalcosts,itmaybehardtomaintainit.Forexample,weneedaninformationsystemthatkeepstrackoftheinventoriesateach

PAGE 171

location.Therefore,wewanttoobservethetradeobetweenthemaintenancecostandthereductionininventorycosts.

PAGE 172

Thisdissertationiscomposedoftwomajorparts.Intherstpart,weconsiderthreedeterministicrequirementsplanningproblems.Inthesecondpart,weconsiderastochasticrequirementsplanningproblem. BitranandYanasse(1982)showsthatdynamiclot-sizingproblemwithnonstationary(time-varying)capacitiesisNP-hard,eveninthesimplestcase,whereweassumexed-chargeproductioncostandzeroholdingcost.Therstmodelinthisdissertationisadynamiclot-sizingproblemwithnonstationaryproductioncapacities,wherewealsodecidehowmuchdemandtosatisfyineachperiod.FormoregeneralcostfunctionsthanthesimplestNP-hardcaseoftheproblem,weproposeafullypolynomialtimeapproximationscheme.Inthesecondmodel,weconsideradierentcapacityconstraint,whichwecallcumulativecapacitiesconstraint.AlthoughforgeneralcostfunctionsthisproblemisNP-hard,weshowthatitispolynomiallysolvablewhenallcostfunctionsareconcaveandproductioncapacitiesarenonstationary.Inthethirdmodel,weconsideratwo-echelonmodelwheretheproduction,transportation,inventory(holdingandbacklogging),subcontracting,andovertimedecisionsareintegrated.Weproposepolynomialtimealgorithmswhenallcostfunctionsareconcaveandwehaveproductionandsubcontractingcapacities.Wealsostatethereductionsinthecomputationtimeifweassumexed-chargeandlinearcoststructures. Inthesecondpart,weconsideramulti-warehousemulti-retailerprobleminastochasticsetting,wherethedemandsobservedattheretailersarerandomvariables.Thisproblemcanbeinterpretedasastochastictransportationproblem.Weconsidertwoversionsoftheproblem:withorwithoutsingle-sourcing 160

PAGE 173

restrictions.Weobservetheeectsoftheparametersonthesingle-sourcingandmulti-sourcingsolutionvalues.Recently,therstmulti-locationmulti-productmodelisstudiedbyBenjaafaretal.(2004),whereeachwarehousedeterminesitsorderorsupplylevelforeachproductandeachretailerissuppliedviaatmostoneorunlimitednumberofwarehouses.However,inourmodeltotalcapacitiesofthewarehousesareknownbutwedecidehowmuchdemandwesatisfy.Wecanoptimallysolvetheproblemwithsingle-sourcingrestrictionbyusingbranch-and-pricealgorithm,andwithoutthisrestrictionbyusingsubgradientalgorithm.Weobservethatwhentheretailershaveverydierentparameters,wehavedieringMSTPandSSTPsolutionvalues.Iftheretailersaresimilarintermsoftheirparameters,thesolutionvaluesofMSTPandSSTPareveryclose.However,thisobservationisnottrueforthesuppliers,becauseaswedistributetotalcapacitybetweenretailersevenly,thisdierenceincreases.

PAGE 175

Zeroxed-costcase. m.n5.5010.5025.508.8016.8040.80 numberofnodes6:62:63:849:5rootcolumns932:21136:41883:82024:02532:84811:0totalcolumns1128:41214:81980:62300:02848:05121:8roottime12:4549:34010:31260:32842:96567:534B&Ptime21:88013:04014:11494:59871:935108:9B&Bapplied767618855139214702300pricingsolved127613862280261433166008SSTPsolution7611:2147372:0967724:31212083:22513125:30012067:120uncap.solution6797:2085799:9225378:4549952:63310044:2138264:656heuristicsolution8396:0827555:9147935:17013391:90013390:45012565:580heuristictime5:2104:6862:65880:85828:45824:696heuristic/z1:1031:0251:0281:1091:0201:042optimal/heuristictime4:2282:9666:1921:1462:6937:146MSTPsolution7607:9027364:8227546:65812078:67513111:0511905:14zSSTPzMSTP

PAGE 176

TableA-1: Continued. m.n10.10020.10050.100 numberofnodes1:49:54:2rootcolumns3121:44064:36811:6totalcolumns3175:84356:57043:4roottime143:030108:573148:676B&Ptime158:468167:148192:384B&Bapplied143518722535pricingsolved341449007870SSTPsolution16148:60015471:35015463:400uncap.solution12430:36011587:40010555:458heuristicsolution17279:26015672:82516298:820heuristictime731:1122986:99525:576heuristic/z1:0701:0131:054optimal/heuristictime0:3200:6287:578MSTPsolution16145:60015456:60015194:78zSSTPzMSTP

PAGE 177

TableA-2: Highxed-costcase. m.n5.5010.5025.508.8016.8040.80 numberofnodes1:8131:43:82:6rootcolumns358:4360:8345:2704:6708:6953:4totalcolumns374:4364:8370:6720:6760:41002:6roottime3:7363:1823:0119:83216:91633:408B&Ptime4:5323:5644:55222:55624:76244:738B&Bapplied228:6212:6343:4441:6533:81273:2pricingsolved426430570836:8963:21352SSTPsolution8345:4528098:4288531:2661332414014:2813345:24uncap.solution7819:0286877:5486486:19411861:1611516:0210046:38percentageofB&B53:66249:44260:24652:77255:41994:172

PAGE 178

TableA-2: Continued. m.n10.10020.10050.100 numberofnodes135:8rootcolumns970:61022:41020:8totalcolumns975:810761154:4roottime42:37646:0257:494B&Ptime44:83265:832115:336B&Bapplied544:87581203:4pricingsolved108013881840SSTPsolution17599:9616434:117044:64uncap.solution14447:1613456:2212772:36percentageofB&B50:44454:61165:402

PAGE 179

TableA-3: Lowxed-costcase. m.n5.5010.5025.508.8016.8040.80 numberofnodes1:41:815:42:22:65rootcolumns540:8632:4733:8874:41184:21428:8totalcolumns552:6666:61099:69371234:21524:8roottime5:626:265:69223:39628:14232:77B&Ptime6:2387:67826:85430:99436:84654:994B&Bapplied298:8416:41596:8567723:41070:6pricingsolved60380221251093:21500:82008SSTPsolution7855:3267622:5027987:912112:0413264:212488:96uncap.solution7124:0386151:335735:6689605:710360:3528841:98percentageofB&B49:55251:92075:14451:86648:20153:317

PAGE 180

TableA-3: Continued. m.n10.10020.10050.100 numberofnodes4:66:64:2rootcolumns16111715:42090:2totalcolumns1722:218752179roottime71:38854:50679:786B&Ptime102:61295:13113:602B&Bapplied1128:61311:81666pricingsolved205023202790SSTPsolution16635:5815506:3415983:24uncap.solution13063:4812100:9811272:4percentageofB&B55:05456:54359:713

PAGE 181

169

PAGE 182

TableB-1: Zeroxed-costcase. m.n5.5010.5025.508.8016.8040.80 numberofnodes1:41:514:275rootcolumns451:8728:81116:61108:21606:42842:6totalcolumns464:8759:31134:41228:61763:03045:6roottime5:3325:3437:08019:01616:48238:064B&Ptime5:776:1687:88625:28426:25055:192B&Bapplied2693775037988531570pricingsolved4978431280138620103592SSTPsolution9978:4549888:4539976:69015397:54016063:44015157:000uncap.solution7377:0006340:0035636:52010524:8049448:0648416:656heuristicsolution10371:3049943:85510471:02415662:06016136:22015964:900heuristictime1:1921:6831:2208:03610:7386:554heuristic/z1:0391:0061:0481:0171:0051:053optimal/heuristictime4:3793:2557:1794:2362:8488:794MSTPsolution9975:5589880:8189850:40415394:70016051:52014940:400zSSTPzMSTP

PAGE 183

TableB-1: Continued. m.n10.10020.10050.100 numberofnodes6:210:67:8rootcolumns1843:42268:04003:6totalcolumns2010:42622:64343:2roottime43:63633:47073:948B&Ptime64:80666:512127:77B&Bapplied118414992249pricingsolved223431085140SSTPsolution19547:10019001:24019138:12uncap.solution12214:58011532:20010579:614heuristicsolution19955:80019078:56020394:46heuristictime22:34012:2546:2heuristic/z1:0211:0041:066optimal/heuristictime3:0368:85622:602MSTPsolution19543:88018984:14018916:800zSSTPzMSTP

PAGE 184

TableB-2: Highxed-costcase. m.n5.5010.5025.508.8016.8040.80 numberofnodes1:4111:81:41rootcolumns121:8128:667255246:6245:2457:4totalcolumns126:4134268:6263:6258:6479:2roottime1:0441:0633:2765:847:50420:806B&Ptime1:2541:273:967:1429:22424:938B&Bapplied93121326:2219:8240:2569:2pricingsolved155190375332:8358:4672SSTPsolution10420:429768:34610593:22816050:516790:916159:94uncap.solution8409:5787245:5236716:90212056:9611192:3810223:246percentageofB&B60:00063:68486:98766:04667:02084:702

PAGE 185

TableB-2: Continued. m.n10.10020.10050.100 numberofnodes5:44:21rootcolumns350332:2596:2totalcolumns447:4392:4623:4roottime13:65413:95450:21B&Ptime27:22225:87859:542B&Bapplied461:2398:4776:6pricingsolved592584890SSTPsolution20403:919853:720366:94uncap.solution14395:813718:9812794:92percentageofB&B77:90568:21987:258

PAGE 186

TableB-3: Lowxed-costcase. m.n5.5010.5025.508.8016.8040.80 numberofnodes1:81:5113:81rootcolumns217:4289:75330:8436:4534:8646:8totalcolumns234:8297:75344:4442:4590:6670:6roottime1:7622:439:1911:45620:462B&Ptime2:1162:783:79:92616:34424:674B&Bapplied165:6230303:8305:2458:8553:8pricingsolved268370460510:4739:2872SSTPsolution10132:57810041:85210174:58215625:7216317:0815483:54uncap.solution7711:0246663:765990:17411025:9410012:8968988:636percentageofB&B61:79162:16266:04359:79662:06763:509

PAGE 187

TableB-3: Continued. m.n10.10020.10050.100 numberofnodes14:21rootcolumns690:6681:4911totalcolumns697:8761:2940:4roottime24:77421:55453:97B&Ptime26:31233:07663:586B&Bapplied471:6578:6841:2pricingsolved7889641220SSTPsolution19848:0219304:0619526:38uncap.solution12904:91221411288:24percentageofB&B59:84860:02168:951

PAGE 188

Aggarwal,A.,J.K.Park.1990.Improvedalgorithmsforeconomiclotsizeproblems.Oper.Res.41415-417. Ahuja,R.K.,T.L.Magnanti,J.B.Orlin.1993.NetworkFlows:Theory,Algo-rithms,andApplications.Prentice-Hall,EngleewoodClis,NewJersey. Atamturk,A.,D.S.Hochbaum.2001.Capacityacquisition,subcontracting,andlotsizing.ManagementSci.471081-1100. Balas,E.,M.Padberg.1976.Setpartitioning:Asurvey.SIAMReview18710-760. Benders,J.F.,J.A.E.E.vanNunen.1983.Apropertyofassignmenttypemixedintegerlinearprogrammingproblems.Oper.Res.Lett.247-52. Benjaafar,S.,M.ElHafsi,andF.deVericourt.2004.Demandallocationinmultiple-product,multiple-facility,make-to-stocksystems.ManagementSci.501431-1448. Biller,S.,L.M.A.Chan,D.Simchi-Levi,J.Swann.2005.Dynamicpricingandthedirect-to-customermodelintheautomotiveindustry.ElectronicCommerceRes.5(April)309-334. Bitran,G.R.,H.H.Yanasse.1982.Computationalcomplexityofthecapacitatedlotsizeproblem.ManagementSci.1812-20. Chan,L.M.A.,A.Muriel,Z.Shen,D.Simchi-Levi.2002a.Approximationalgorithmsfortheeconomiclotsizingmodelwithpiecewiselinearcoststructures.Oper.Res.501058-1067. Chan,L.M.A.,A.Muriel,Z.Shen,D.Simchi-Levi,C.P.Teo.2002b.Eectivenessofzeroinventoryorderingpoliciesforanone-warehousemulti-retailerproblemwithpiecewiselinearcoststructures.ManagementSci.111446-1460. Chang,P.,C.Lin.1991.Ontheeectofcentralizationonexpectedcostsinamultilocationnewsboyproblem.J.Oper.Res.Soc.421025-1030. Chen,M.,C.Lin.1989.Eectsofcentralizationonexpectedcostsinamulti-locationnewsboyproblem.J.Oper.Res.Soc.40597-602. 176

PAGE 189

Chubanov,S.,M.Y.Kovalyov,E.Pesch.2006.Anfptasforasingle-itemcapacitatedlot-sixingroblemwithmonotonecoststructure.Mathemati-calProgramming106453-466. Chung,C.-S.,C.-HM.Lin.1988.AnO(T2)algorithmfortheNI=G=NI=NDcapacitatedlotsizeproblem.ManagementSci.34420-426. Day,J.S.1956.Subcontractingpolicyintheairframeindustry.GraduateSchoolofBusinessAdmininstration,HarvardUniversity,Boston,MA. Deng,S.,C.Yano.2006.Jointproductionandpricingdecisionswithsetupcostsandcapacityconstraints.ManagementSci.52741-756. Eppen,G.D.1979.Eectsofcentralizationonexpectedcostsinamulti-locationnewsboyproblem.ManagementSci.25498-501. Federgruen,A.,M.Tzur.1991.AsimpleforwardalgorithmtosolvegeneraldynamiclotsizingmodelswithnperiodsinO(nlogn))orO(n)time.Man-agementSci.37909-925. Florian,M.,M.Klein.1971.Deterministicproductionplanningwithconcavecostsandcapacityconstraints.ManagementSci.1812-20. Florian,M.,J.K.Lenstra,A.H.G.RinnooyKan.1980.Deterministicproductionplanning:algorithmsandcomplexity.ManagementSci.26669-679. Garey,M.R.,D.S.Johnson.1979.ComputersandIntractability:AGuidetotheTheoryofNP-completeness.W.H.FreemanandCompany,NewYork. Geunes,J.,Y.Merzifonluoglu,H.E.Romeijn.2005.Capacitatedproductionplanningwithpricesensitivedemandandgeneralconcaverevenuefunctions.Workingpaper,UniversityofFlorida,Gainesville,FL. Geunes,J.,H.E.Romeijn,K.Taae.2006.Requirementsplanningwithdynamicpricingandorderselectionexibility.Oper.Res.54(2)394-401. Gilbert,S.M.1999.Coordinationofpricingandmulti-periodproductionforconstantpricedgoods.Eur.J.Oper.Res.114330-337. |.2000.Coordinationofpricingandmultiple-periodproductionacrossmultipleconstantpricedgoods.ManagementSci.461602-1616. Hadley,G.,T.M.Whitin.1963.Analysisofinventorysystems.Prentice-Hall,EnglewoodClis,NewJersey. Hochbaum,D.S.,J.G.Shanthikumar.1990.Convexseparableoptimizationisnotmuchharderthanlinearoptimization.J.Assoc.Comput.Mach.37843-862.

PAGE 190

Ibaraki,T.1976.Theoreticalcomparisonofsearchstrategiesinbranchandboundalgorithms.Int.J.Comput.Inform.Sciences5315-344. Jucker,J.,M.Roesenblatt.1976.Single-periodinventorymodelswithdemanduncertaintyandquantitydiscounts:behavioralimplicationsandanewsolutionprocedure.NavalRes.Logist.32537-550. Kaminsky,P.,D.SimchiLevi.2003.Productionanddistributionlotsizinginatwostagesupplychain.IIETrans.351065-1075. Khouja,M.1999.Thesingle-periodnewsvendorproblem:literaturereviewandsuggestionsforfutureresearch.OMEGA27537-553. KunreutherH.,L.Schrage.1973.Jointprisingandinventorydecisionsforconstantpriceditems.ManagementSci.19732-738. Lau,H.L.1997.Simpleformulasfortheexpectedcostsinthenewsboyproblem:aneducationalnote.Eur.J.Oper.Res.100557-561. Lau,H.L.,H.L.Lau.1988.Newsboyproblemwithprice-dependentdemanddistribution.IIETrans.20168-175. Lau,H.S.,H.L.Lau.1995.Themulti-productmulti-constraintnewsboyproblem:application,formulation,andsolution.J.Oper.Management31153-162. Lau,H.S.,H.L.Lau.1996.Thenewsstandproblem:Acapacitatedmulti-productsingle-periodinventoryproblem.Eur.J.Oper.Res.9429-42. Lee,C.Y.,S.Cetinkaya,W.Jaruphongsa.2003.Adynamicmodelforinventorylotsizingandoutboundshipmentschedulingatathird-partywarehouse.Oper.Res.51735-747. Luenberger,D.G.1973.IntroductiontoLinearandNonlinearProgramming.Addison-Wesley,EnglewoodClis,NewJersey. Merzifonluoglu,Y.,J.Geunes,H.E.Romeijn.2005.Capacity,demand,andproductionplanningwithsubcontractingandovertimeoptions.Workingpaper,UniversityofFlorida,Gainesville,FL. Moon,I.,E.A.Silver.2000.Themulti-itemnewsvendorproblemwithabudgetconstraintandxedorderingcost.J.Oper.Res.Soc.51602-608. Pantumsinchai,P.,T.Knowles.1991.Standardcontainersizediscountsandsingleperiodinventoryproblem.DecisionSciences22612-619 Polyak,B.T.1969.Minimizationofunsmoothfunctionals.U.S.S.R.ComputationalMathematicsandMathematicalPhysics914-29.

PAGE 191

Sankarasubramanian,E.,S.Kumanraswamy.1983.Optimalorderingquantityforpre-determinedlevelofprot.ManagementSci.29512-514. Sargut,F.Z.,H.E.Romeijn.2006a.Capacitatedproductionplanningwithpricesensitivedemandandgeneralconcaverevenuefunctions.ForthcominginJ.IndustrialandManagementOptimization. |.2006b.Capacitatedproductionandsubcontractinginaserialsupplychain.Workingpaper,UniversityofFlorida,Gainesville,FL. |.2006c.Lot-sizingwithnon-stationarycumulativecapacities.Workingpaper,UniversityofFlorida,Gainesville,FL. Savelsbergh,M.P.W.1997.Abranchandpricealgorithmforthegeneralizedassignmentproblem.Oper.Res.45831-841. Sivazlian,B.D.,L.E.Stanfel.1975.AnalysisofSystemsinOperationsResearch.Prentice-Hall,EnglewoodClis,NewJersey. Thomas,L.J.1970.Price-productiondecisionswithdeterministicdemand.ManagementSci.16747-750. Tsevendorj,I.2001.Piecewise-convexmaximizationproblems.J.GlobalOptim.211-14. vandenHeuvel,W.,A.P.M.Wagelmans.AgeometricalgorithmtosolvetheNI=G=NI=NDcapacitatedlot-sizingprobleminO(T2)time.ForthcominginComputersandOper.Res. |.2006.Apolynomialtimealgorithmforadeterministicjointpricingandinventorymodel.Eur.J.Oper.Res.170463-480. vanHoesel,C.P.M.,A.P.M.Wagelmans.1996.AnO(T3)-algorithmfortheeconomiclot-sizingproblemwithconstantcapacities.ManagementSci.42142-150. |.2001.Fullypolynomialapproximationschemesforsingle-itemcapacitatedeconomiclot-sizingproblems.Math.Oper.Res.26339-357. vanHoesel,S.,H.E.Romeijn,D.RomeroMorales,A.P.M.Wagelmans.2005.Integratedlot-sizinginserialsupplychainswithproductioncapacities.ManagementSci.511706-1719. vanHoesel,S.,A.P.M.Wagelmans,B.Moerman.1994.AnO(T2)algorithmfortheNI=G=NI=NDcapacitatedlotsizeproblem.Eur.J.Oper.Res.75312-331.

PAGE 192

VeinottJr.,A.F.1963.Unpublishednotes,StanfordUniversity,Stanford,California. Wagelmans,A.P.M.,S.vanHoesel,A.Kolen.1992.Economiclotsizing:AnO(nlogn)-algorithmthatrunsinlineartimeintheWagner-Whitincase.Oper.Res.40-S1S145-S156. Wagner,H.M.,T.M.Whitin.1958.Dynamicversionoftheeconomiclotsizemodel.ManagementSci.589-96. Weintraub,A.1971.Aprimalalgorithmtosolvenetworkowproblemswithconvexcosts.ManagementSci.2187-97. Woeginger,G.J.2000.Whendoesadynamicprogrammingformulationguaranteetheexistenceofafullypolynomialtimeapproximationscheme.InformsJ.Computing,1257-74. Zangwill,W.I.1967.Theconvexsimplexmethod.ManagementSci.14221-238. |.1968.Minimumconcavecostowsincertainnetworks.ManagementSci.14429-450. |.1969.Abackloggingmodelandadynamiceconomiclotsizeproductionsystem-anetworkapproach.ManagementSci.15506-527. Zemel,E.1981.Measuringthequalityofapproximatesolutionstozero-oneprogramming.Math.Oper.Res.6319-332. Zipkin,P.H.2000.FoundationsofInventoryManagement.McGrawHill,BurrRidge,IL.

PAGE 193

ZeynepSargutwasbornonFebruary28,1979,inAnkara,Turkey.SheattendedBilkentUniversityandreceivedaB.S.inIndustrialEngineeringin2000.Afourth-yearprojectoninventorymodelsmakeherwanttopursueanM.S.inindustrialengineering.ShecontinuedhereducationinMiddleEastTechnicalUniversityandreceivedanM.S.inindustrialengineeringin2002.ShepursuedaPh.D.inindustrialandsystemsengineeringattheUniversityofFlorida.ShewasawardedanAlumniFellowshipforthedurationofthedoctoralprogram.Duringhergraduatestudies,sheacquiredexpertiseinproductionandinventorycontrolandsupplychainmanagement.Sheisplanningtobepursueanacademiccareer. 181


Permanent Link: http://ufdc.ufl.edu/UFE0015711/00001

Material Information

Title: Efficient Approaches to Integrated Requirements Planning Problems in Supply Chain Optimization
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015711:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015711/00001

Material Information

Title: Efficient Approaches to Integrated Requirements Planning Problems in Supply Chain Optimization
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015711:00001


This item has the following downloads:


Full Text











EFFICIENT APPROACHES TO INTEGRATED REQUIREMENTS PLANNING
PROBLEMS IN SUPPLY CHAIN OPTIMIZATION
















By

FATMA ZEYNEP SARGUT


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

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Fatma Zeynep Sargut



































To Umut















ACKNOWLEDGMENTS

I owe a debt of gratitude to my supervisor, Edwin Romeijn, for his patience

during my dissertation and all the things I have learned by being his student.

I am very grateful to my mum, F i-. i Alisan, to my father, Miifit Alisan, to

my brother Onur Alisan, and Umut's family for their continuous support. My

father never stopped sending me emails and making me smile. Maybe I have known

him better during these four years than throughout my entire life.

Although Nietzsche saw hope as the latest evil left Pandora's box, hope is still

the greatest thing for the world and me. Umut, I could not do it without you.















TABLE OF CONTENTS
page

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

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

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

ABSTRACT ....................... ........... xi

CHAPTER

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

2 CAPACITATED REQUIREMENTS PLANNING WITH PRICING
FLEXIBILITY AND GENERAL COST AND REVENUE FUNCTIONS 9

2.1 Introduction . . . . . . . 9
2.2 Background Information ..... ................... 10
2.2.1 Dynamic Lot-Sizing Problem ......... ........ 10
2.2.2 Fully Polynomial Time Approximation Schemes for CLSP 14
2.3 Model Formulation ....... ............ ..... 18
2.3.1 Requirements Planning with Pricing Flexibility ....... 18
2.3.2 Capacitated Economic Lot-Sizing Formulation ....... 21
2.4 Properties of the New Procurement Cost Function . ... 24
2.5 Results ...... ....................... .. 28
2.5.1 Dynamic Programming ................ .... 28
2.5.2 Fully Polynomial Time Approximation Scheme . ... 30
2.6 Conclusions ............... .......... .. 34

3 LOT-SIZING WITH NON-STATIONARY CUMULATIVE CAPACITIES 36

3.1 Introduction ............... ......... .. 36
3.2 Model Formulation ......... . . .... 37
3.2.1 Two-Level Lot-Sizing with Non-stationary Production
Capacities ..... ........... ..... .. 37
3.2.2 Single-Level Lot-Sizing with Cumulative Capacities . 40
3.3 General Cost Functions ................ ... ... .. 41
3.3.1 Proof of NP-Hardness ....... . . ...... 42
3.3.2 A Fully Polynomial Time Approximation Scheme . 43
3.4 Concave Costs .................. ......... .. .. 47
3.4.1 Introduction .................. ........ .. 47









3.4.2 Solution Approach. .................. .... 49
3.4.3 Computing the Subplan Costs ................ .. 50
3.5 Allowing for Backlogging .. . . . 54
3.6 Summary and Directions for Future Research . . 55

4 CAPACITATED PRODUCTION AND SUBCONTRACTING IN A
SERIAL SUPPLY CHAIN .................. ..... .. 57

4.1 Introduction ..... . . ....... ........... 57
4.2 Problem Formulation and Solution Approach . . 63
4.2.1 Problem Formulation .................. ..... 63
4.2.2 Solution Approach. .................. .... 66
4.2.3 Only Backlogging at the Manufacturer Level . ... 69
4.3 Models without Subcontracting Opportunities . . 71
4.3.1 Subplan Properties ... . . ..... ... 71
4.3.2 Inventory Holding and Backlogging at the Manufacturer Level 72
4.3.3 Model with only Backlogging at the Manufacturer Level 77
4.3.4 Evaluation .................. ......... .. 78
4.3.5 Managerial Insights ................ .... 79
4.4 Models with Subcontracting Opportunities . . ..... 80
4.4.1 General Properties of the Subplans . . ...... 80
4.4.2 Concave Costs ..... . . . 81
4.4.3 Subcontracting versus Overtime Production Option . 84
4.5 The Uncapacitated Case ..... . . ...... 87
4.5.1 Both Inventory Holding and Backlogging at the Manufacturer 88
4.5.2 Only Backlogging at the Manufacturer . . 90
4.6 Summary and Concluding Remarks ............. .. 91

5 MULTI-WAREHOUSE MULTI-RETAILER NEWSVENDOR PROBLEM 93

5.1 Introduction and Related Literature ............. .. 93
5.2 Multi-Sourcing Transportation Problem . . ..... 98
5.2.1 Model Formulation ... . . ... 98
5.2.2 Properties of the Model and the Optimal Solution ..... ..100
5.2.3 Lagrangean Relaxation Approach . . ..... 102
5.2.4 Minimum Convex Cost Flow Representation ........ 105
5.2.5 Fixed-Charge Transportation Costs . . .... 107
5.2.6 General Demand Distributions . . 109
5.3 A Branch-and-Price Algorithm for Single-Sourcing Transportation
Problem ........ ....... .......... ..... 111
5.3.1 Model Formulation ................ .... .. 111
5.3.2 A Set-Partitioning Formulation . . ..... 114
5.3.3 The Pricing Problem for LPSP . . ..... 118
5.3.4 A General Class of Problems ..... . . ..... 119
5.3.5 Solving Pricing Problem to Optimality . . ... 125
5.3.6 Branching Rule for Branch-and-Price . . 126









5.4 Extensions of the Single-Sourcing Model . . 127
5.4.1 Fixed-Charge Transportation Costs . . .... 127
5.4.2 More General Procurement Cost Functions . .... 128
5.4.3 Capacity Expansion ................ .... .. 131
5.5 Discussion .. .. ... .. .. .. .. ... .. .. .. ....... 135
5.6 Computational Results .................. .... 137
5.6.1 Experimental Design ............ .. . 137
5.6.2 Branch-and-Price Strategies ................. 139
5.6.3 A Heuristic for SSTP ............. .... . 141
5.6.4 Computational Results ............... .. 141
5.6.5 Discussion of the Results .............. .. 156
5.7 Conclusions and Future Research Directions . . 158

6 CONCLUSIONS .................. ............ 160

APPENDIX

A SUMMARY OF COMPUTATIONAL RESULTS FOR HIGH CAPACITY 162

B SUMMARY OF COMPUTATIONAL RESULTS FOR LOW CAPACITY 169

REFERENCES .................. ................ .. 176

BIOGRAPHICAL SKETCH .................. ......... .. 181















LIST OF TABLES
Table page

4-1 Running times for models without subcontracting opportunities ... 91

4-2 Running times for models with subcontracting opportunities. ...... 92

5-1 Zero fixed cost of transportation (high capacity). . . 143

5-2 Zero fixed cost of transportation (low capacity). . . ..... 144

5-3 Low fixed cost of transportation (high capacity). . . ..... 144

5-4 Low fixed cost of transportation (low capacity). . . ..... 144

5-5 High fixed cost of transportation (high capacity). . . 145

5-6 High fixed cost of transportation (low capacity). . . ..... 145

5-7 The percentage of the pricing problems that branch-and-bound is used. 145

5-8 Branching rule comparison. .................. ..... 152

5-9 Data sets .................. ................ .. 156

A-1 Zero fixed-cost case. .................. .......... 163

A-2 High fixed-cost case. .................. .......... 165

A-3 Low fixed-cost case. .................. .......... 167

B-l Zero fixed-cost case. .................. .......... 170

B-2 High fixed-cost case. .................. .......... 172

B-3 Low fixed-cost case. .................. .......... 174















LIST OF FIGURES


Figure

2-1 Network flow representation of DLSP......... . .....

2-2 Demand (a) as a decision variable; (b) deterministic . ...

3-1 Production cost function in period t.......... . .....

3-2 Network flow representation of LSP-CC........ . .....

3-3 Extreme point solution with 2 subplans and the associated arc flows.

4-1 Network representation with inventory holding and backlogging. .

4-2 Network representation with backlogging only at the manufacturer.


4-3 Network representation with overtime production.


Warehouse-retailer network, with 3 warehouses and 4 retailers ..

Convex cost network flow representation for n = m 3.......

Transportation network in the case of flat spots .. ........


5-4 Qj(Qj): (a) zj cannot be determined (b) zj


4j and bj(Qj). .. .. ... .. .. ..

E', Q' versus A: (a) fractional solution
ii versus 7i for two cases ......

Branch-and-price tree of a problem.

SSTP CPU times (high capacity) .

SSTP CPU times (low capacity).....

SSTP solution values (high capacity).

SSTP solution values (zero fixed cost).

SSTP solution values (low fixed cost). .

SSTP solution values (high fixed cost).


i r ..... . .


integer solution.....


page

13

22

43

48

52

66

70


. . 85


5-5

5-6

5-7

5-8

5-9

5-10

5-11

5-12

5-13

5-14









5-15 Heuristic and branch-and-price solution times (high capacity). ..... 149

5-16 Heuristic and branch-and-price solution times (low capacity). ...... ..150

5-17 Heuristic and branch-and-price solution values. ............ ..150

5-18 Time comparison for pricing strategies (high capacity). . ... 151

5-19 Time comparison for pricing strategies (low capacity). . .... 151

5-20 Pricing strategies (low capacity). ................ ..... 152

5-21 Columns generated for two pricing rules (low capacity). . ... 153

5-22 Pricing strategies (high capacity). ................ ..... 153

5-23 Columns generated for two pricing rules (high capacity). . ... 154

5-24 Comparison of MSTP and SSTP solution values as m changes. ....... .154

5-25 R as m changes. .................. .. ..... ...... 155

5-26 R for all m n pairs. .................. ...... ..... 155

5-27 Locations of the warehouses and the retailers in data set 6. . ... 156

5-28 Effects of parameters on R. ................ .. .... 157

5-29 R for all data sets. ............... ........... 157















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

EFFICIENT APPROACHES TO INTEGRATED REQUIREMENTS PLANNING
PROBLEMS IN SUPPLY CHAIN OPTIMIZATION

By

Fatma Zeynep Sargut

August 2006

C'!I in: H. Edwin Romeijn
Major Department: Industrial and Systems Engineering

In standard dynamic economic lot-sizing models, a sequence of demands

for a single good over a finite and discrete planning horizon must be satisfied at

minimum production and inventory holding cost, where the cost parameters and

demands are assumed to be known. In this dissertation, we study variations of

dynamic economic lot-sizing models by integrating more decisions and adding more

constraints. For each model, either we identify polynomially solvable cases and

present solution algorithms or present approximation schemes.

In our first model, we integrate pricing decisions and backlogging under

nonstationary production capacities. We present a fully polynomial time

approximation scheme for this model.

In our second model, we consider a setting where any remaining capacity is

transferred to the next production period. This is in contrast with traditional

capacitated lot-sizing models, where the quantity produced in each period is limited

by some capacity, but any capacity remaining at the end of a period is essentially

lost. We prove that this problem is NP-hard for general cost functions and provide

a fully polynomial time approximation scheme for the case when all cost functions









are nondecreasing. We then develop a dynamic programming approach that solves

the problem in polynomial time when all cost functions are concave.

In our third model, we consider a two-echelon supply chain consisting of a

supplier and a manufacturer, where a sequence of deterministic but nonstationary

demands of the manufacturer for a single good needs to be satisfied over a finite

horizon. We assume stationary production capacities. In particular, we consider

an integrated model that minimizes total system costs, consisting of production,

inventory, transportation, backlogging, and subcontracting costs.

In our last model, we consider a multi-warehouse multi-retailer model in a

stochastic demand setting. In this model, we consider a single period, where we

do not know the exact demand, but its distribution. Moreover, warehouses have

capacities on the amount they can supply. For each retailer, we have newsvendor

type costs ,and we also have the costs for assigning each retailer to each warehouse.

We solve an assignment and capacity allocation problem.














CHAPTER 1
INTRODUCTION

According to Zipkin (2000), every inventory lies between two activities or

processes which we call supply and demand. Supply processes are production,

transportation or any activity that adds to the current inventory, while demand

processes are the activities that subtract material from the current inventory.

In an inventory model, supply and demand processes can be modelled in

various v--,v. Moreover, the time can be modelled as discrete or continuous. In

discrete time models, we model time as a finite sequence of discrete time points

and a period is defined as the interval between two consecutive time points. In

the dynamic lot-sizing problem (DLSP) (introduced by Wagner and Whitin 1958,

demands of a single item are observed and ordering decisions are made at discrete

time points at a single location. In addition, the demands in each period and the

cost parameters are known and vary over time. We can order as much as we want,

and we receive our order immediately. Therefore, DLSP has an uncapacitated

supply process and a time-varying demand process. In DLSP, we try to minimize

total ordering (production) and inventory costs during a finite planning horizon.

The single location variant of the DLSP has received a lot of attention in the

literature and it is solvable in polynomial time in the length of the time horizon

(see Wagner and Whitin 1958). Aggarwal and Park (1990), Federgruen and Tzur

(1991), and Wagelmans et al. (1992) propose more efficient algorithms for the

special cases of DLSP. Zangwill (1969) proposes an O(T3) dynamic programming

algorithm when a backlogging opportunity is present.

When production capacities are present, we obtain the capacitated lot-sizing

problem (CLSP). In contrast to the DLSP, this problem is known to be NP-hard









for many special cases (see Florian et al. 1980 and Bitran and Yanasse 1972). An

interesting and important special case that allows a polynomial time algorithm

arises when production capacities are stationary. The papers that present

polynomial time algorithms in the case of stationary production capacities are

Florian and Klein (1972), Florian et al. (1980), and van Hoesel and Wagelmans

(1996). Chlui and Lin (1988) and van den Heuvel and Wagelmans (2006) present

polynomial time algorithms for another special case of the CLSP when the

fixed-costs of production are non-ini, i. i-i- the holding costs are linear, the

linear part of the production cost functions are non-in, i. i-ii.: and the capacities

are non-decreasing over time.

A more general setting of DLSP considers the determination of optimal

price levels, and thereby demands, for the good in each period jointly with the

production and inventory decisions that together maximize profit. This problem

is studied by Thomas (1970) in the absence of capacities and when the price of

the product in each period is a decision variable. Kunreuther and Schrage (1971)

and Gilbert (1999) consider the uncapacitated problem when a single price must

be used over the entire horizon. Kunreuther and Schrage (1971) provided bounds

on the optimal solution value under time varying production cost assumptions,

while Gilbert (1999) assumed a time-invariant production setup and holding

costs, and provides an exact polynomial time algorithm. A recent paper by van

den Heuvel and Wagelmans presents an optimal polynomial time algorithm for

Kunreuther and Schrage's problem. Geunes et al. (2005) presents polynomial

time algorithms for the uncapacitated and stationary production capacities cases

when the revenue function in each period is a piecewise linear concave function of

the price. Biller et al. (2005) developed a greedy algorithm for the variant of the

problem in which production capacities are nonstationary but all cost functions are

linear. Gilbert (2000) also considers a situation in which pricing and production









decisions must be made for multiple constant-priced products that share the

same production capacity. Deng and Yano (2006) consider both set-up costs and

nonstationary production capacities in their problem and show that taking prices

as decision variables does not change the structure of the optimal production

decisions characterized by Florian and Klein (1971). Geunes et al. (2005) presents

a solution approach for economies of scale in production (fixed-cost structure),

price-sensitive demand (general concave revenue functions), and time-invariant

production capacities. They provide polynomial-time solution methods for this

problem when we may set different prices for the good in each period and we set a

constant price for the good over the entire planning horizon.

Subcontracting is the procurement of an item or service which is normally

capable of economic production in the prime contractor's own facilities and which

requires the prime contractor to make specifications available to the supplier

(Day 1956). However, overtime production occurs at the end of the period and

incurs relatively high production costs. In another setting of DLSP, regular

production capacity levels are also decision variables. Atamtiirk and Hochbaum

(2001) investigates the trade-off between acquiring capacity, subcon il i ,lir:

production, and holding inventory under various cost structures, where the amount

of subcontracting is uncapacitated. Merzifonluoglu et al. (2006) determines

the optimal price, production, inventory, subcoiil i I Ii:.-l overtime, and regular

production capacity levels.

Another research area arising from DLSP is a multi-level dynamic lot-sizing

problem. The 50th anniversary issue of the journal Management Science is devoted

to the most influential papers during these fifty years. One of these papers is

Dynamic Version of the Economic Lot Size Model by Harvey M. Wagner and

Thomas M. Whitin. Harvey M. Wagner writes a comment in their paper and

predicts that a candidate paper for the next 50th anniversary issue of Management









Science is a paper that presents an efficient solution method to a dynamic lot-sizing

model that encompasses a multi-echelon environment. Zangwill (1969) studied

the uncapacitated version of the multi-level lot-sizing problem, and developed a

dynamic programming algorithm that is polynomial in both the planning horizon

and the number of levels. Lee et al. (2003) consider a two-level model where the

transportation costs are nonconcave functions. Kaminsky and Simchi-Levi (2003)

propose a three-level model in which the first and third levels are production

stages, and the second level is a transportation stage. Both production stages

are capacitated, while the transportation stage is uncapacitated. They consider

linear inventory holding costs that increase with the level of the supply chain,

and linear production costs at both levels 1 and 3 that satisfy a traditional

nonspeculative motives condition. The transportation costs at the second level

are of the fixed-charge or general concave form and are assumed to satisfy a

restrictive and nontraditional nonspeculative motives condition. They eliminate the

third-level production decisions and reduce the problem to a two-level model. For

the fixed-charge transportation costs, they provide a polynomial time algorithm.

For concave transportation costs, they provide a polynomial time algorithm in the

presence of stationary production capacities. Van Hoesel et al. (2005) proposes a

polynomial time algorithm which is computationally less expensive even when all

cost functions are concave. They also consider multi-level models and propose an

algorithm that runs in polynomial time in the number of periods and exponential

in the number of levels when all cost functions are concave. However, when

inventory holding costs are linear and transportation costs either are linear or have

fixed-charge structure and satisfy a form of nonspeculative motives, running times

are polynomial in the number of periods and relatively insensitive to the number of

levels.






5


In this thesis, we first address some extensions of the DLSP. In ('!i ipter 2,

we first give background information about DLSP and fully polynomial time

approximation schemes (FPTAS). We then introduce the requirements planning

with pricing flexibility problem (RPP). This problem considers the pricing (or how

much demand to satisfy) and the production decisions in a single location. In the

literature, polynomially solvable cases of RPP are well studied (see Thomas 1970,

Geunes et al. 2005, and Biller et al. 2003, Deng and Yano 2006, van den Heuvel

and Wagelmans 2006, Gilbert 1999, Geunes et al. 2005). However, we consider

RPP with general cost and revenue functions, and also non-stationary production

capacities. We present the first FPTAS for RPP.

In C'!i lpter 3, we again consider a single location model. However, we have a

different capacity constraint, which we call cumulative capacity constraint, which

states that the remaining capacities in the previous periods can be used in the

coming periods. We assume non-stationary production capacities for each period.

We show that this problem is NP-hard with general production cost functions and

zero inventory costs. As opposed to the traditional dynamic lot-sizing problem

with nonstationary production capacities, we show that this problem is solvable in

polynomial time when all cost functions are concave.

In C'! Ilpter 4, we analyze a two-echelon supply chain consisting of a supplier

and a manufacturer, where a sequence of deterministic but nonstationary demands

of the manufacturer for a single item needs to be satisfied over a finite horizon. Our

results generalize van Hoesel et al. (2005) and Atatumtfirk and Hochbaum (2001).

In particular, we consider an integrated model that minimizes total system costs,

consisting of production, inventory (holding and backlogging), transportation,

and subcontracting costs. This is appropriate in cases where, for example, the

supplier is a captive subsidiary of the manufacturer. In each period, the production

quantity is limited by a finite but stationary capacity. Alternatively, units may









be obtained by subcontracting from an outside supplier, where the capacity of

the outside supplier may or may not be limited. Note that this subcontracting

option may be used regardless of whether the production capacity in a given

period is exhausted. However, we will show how our approach can be modified

to allow for an alternative sourcing option that can only be used if production

is at full capacity, and therefore can be interpreted as an overtime production

option. To the best of our knowledge, this study is the first multi-level dynamic

lot-sizing model that encompasses subcontracting and overtime decisions. We

present polynomial time dynamic programming algorithms for the variations of the

model where subcontracting, production, and overtime decisions are capacitated

or uncapacitated, and when backlogging is allowed or not. We consider the case

when all cost functions are concave. Moreover, we present the algorithms with

improved running times when we have linear inventory costs and either fixed-charge

transportation costs with speculative motives or linear transportation costs. We

also consider the case where holding at the retailer is not allowed; in other words,

the retailer prefers to backlog instead of carrying inventory. For this case, we

show that the running time of the dynamic programming algorithm improves for

capacitated production cases.

The second theme this thesis addresses is based on the newsvendor problem.

N. 1.-- idor-based models are used when the demand of the item is a random

variable and the item has a short useful life. In these models, we consider a single

period, in which we order at the beginning of the period and then demand occurs

at the end of the period. First of all, we pl i a unit purchase cost for each item

we order. Moreover, according to the observed demand, we incur some costs. Any

inventory remaining at the end of the period has some salvage value, which is less

than unit purchase cost. However, there is a penalty (for instance, due to loss

of goodwill) associated with any unit of demand that cannot be satisfied. Since









demand is a random variable, the retailer aims to minimize his/her expected cost in

a single period.

For single product and single location newsvendor problems Lau (1997)

presents closed-form expressions for computing the expected total costs for various

demand distributions. In the literature, many studies extend the single product

single facility newsvendor problem by considering a fixed-cost associated with the

replenishment (see Moon and Silver 2000), quantity discounts (see Jucker and

Rosenblatt 1985, Pantumsinchai and Knowles 1991), price dependent demand (see

Lau and Lau 1988), and the use of a different objective such as maximization of the

probability that at least a certain profit level is achieved (see Sankarasubramanian

and Kumanraswamy 1983).

Eppen (1979), ('!C. i and Lin (1989), and ('C!i I, and Lin (1991) consider

multi-location newsvendor problems. Lau and Lau (1996) considers a multiple-product

model with single and multiple capacity constraints. The solution of the model

with single capacity constraints extends Hadley and Whitin (1963) (pp 306-307)

by allowing all kinds of continuous demand distributions. To solve the model

with multiple constraints, they employ active set methods, which is proposed in

Luenberger (1973). They also consider the case where the demand distribution has

a nonzero lower bound. The other analysis of multi-commodity type systems, with

or without capacity constraints, can be found in Sivazlian and Stanfel (1975), Moon

and Silver (2000), and Lau and Lau (1995). An excellent review on the newsvendor

problem has been recently provided by Khouja (1999).

In ('! Ilpter 5, we extend the classical newsvendor problem to a capacitated

multi-warehouse multi-retailer setting. This problem integrates inventory and

transportation decisions in the presence of capacities that constrain the quantity of

product that can be supplied from each warehouse. As in the classical newsvendor

problem, we consider the case of a single product. We consider two versions of






8


the problem: with or without single-sourcing restriction, which enforces that each

retailer can be shipped by at most one warehouse. We compare the solutions of

the single and multi-sourcing versions and report the cases where their optimal

objective values have significant difference. We solve single-sourcing problem using

branch-and-price algorithm and our pricing problem is an interesting problem in

its own right. It is a nonlinear integer knapsack problem whose item sizes can be

influenced and we present two solution methods, one of which gives an approximate

solution and the other an optimal solution.















CHAPTER 2
CAPACITATED REQUIREMENTS PLANNING WITH PRICING FLEXIBILITY
AND GENERAL COST AND REVENUE FUNCTIONS

2.1 Introduction

In this chapter, we study the requirements planning problem with pricing

flexibility (RPP), which can be viewed as an extension of the economic lot-sizing

model that was first introduced by Wagner and Whitin (1958). In this standard

model, a sequence of demands for a single good over a finite and discrete planning

horizon must be satisfied at minimum production and inventory holding cost.

A more general setting considers the determination of optimal price levels, and

thereby demands, for the good in each period jointly with the production and

inventory decisions that together maximize profit. Thomas (1970) studied this

problem in the absence of capacities and with concave revenues, fixed-charge

production costs, and linear inventory holding costs. More recently, Geunes et

al. (2005) developed a polynomial-time dynamic programming algorithm for the

problem under these and some more general cost structures and in the presence of

stationary production capacities. Biller et al. (2003) developed a greedy algorithm

for the variant of the problem in which production capacities are nonstationary but

all cost functions are linear. We extend these studies by allowing for time-varying

production capacities and general cost functions.

Since it is well known that even the standard economic lot-sizing problem with

time-varying production capacities is NP-hard, even when all cost functions are

concave (see Florian et al. 1980), we focus on developing a pseudo-polynomial-time

dynamic programming (DP) algorithm as well as a fully polynomial time

approximation scheme (FPTAS) for the RPP (see Sargut and Romeijn 2006a).









Our approach is based on a reformulation of the integrated pricing and lot-sizing

problem as a pure lot-sizing problem with a complex procurement cost function

(that itself involves solving an optimization problem) in Section 2.3. We next

proceed, in Section 2.4, by analyzing properties of this new procurement cost

function under different cost structures in the original problem. We then draw

heavily upon earlier papers by Florian et al. (1980), in which a DP algorithm for

the economic lot-sizing problem is proposed, as well as van Hoesel and Wagelmans

(2001), in which a FPTAS for the economic lot-sizing problem is developed. In

particular, Section 2.5 is devoted to the analysis of the running time of these two

approaches as applied to the lot-sizing formulation of the RPP. We end the chapter

in Section 2.6 with some concluding remarks.

2.2 Background Information

2.2.1 Dynamic Lot-Sizing Problem

After Wagner and Whitin (1958) introduced DLSP, many variations of DLSP

have been studied. Some of the variations increase the number of constraints and

some of them increase the number of variables that cause tradeoff at the objective

function. It is alv-- i a question if these extensions make the problem harder or

not. In a DLSP, basically:

* we consider a planning horizon with finite number of periods,

* we know the demand and cost parameters in each period,

* demand and cost parameters may vary over time, and

* we try to minimize the total inventory and ordering (production) costs of a
single item during the planning horizon.

We can state the DLSP for a single facility with general cost terms as:

T
min (pt (xt)+ ht(It))
t=1









subject to (DLSP)


dt + It = xt + It- t= ,...,T (2-1)

Xt,It > 0 t ,...,T, (2-2)


where T is the length of the the planning horizon. For each period t = 1,..., T, we

define:

* dt as the demand in period t,

* ht as the inventory holding cost in period t,

* pt as the ordering (production) cost in period t.

Our decision variables are:

* xt, the order (production) size in period t,

* It, the inventory level at the end of the period t. We can express It as a
resulting state of the decision variables x x2,..., Xt,
t t
It o0+ x,- d,
i= 1 i 1

and state DLSP only in terms of x1,..., T.

In the DLSP formulation, constraints 2-1 maintain balance of inflow and

outflow at the facility and constraints 2-2 enforce the nonnegativity of the variables

xt and It.

Wagner and Whitin (1958) develops one of the simplest and earliest dynamic

lot-sizing model in the literature. In this model, ordering cost has fixed-cost

structure, inventory holding costs are linear, backlogging is not allowed, and

production is uncapacitated. Let us denote the fixed-cost of ordering in period t

as st. The size of the decision space of this problem is 0(2T), since in each period

we decide either to order or not. However, Wagner and Whitin (1958) proposes a









forward dynamic programming recursion that works in O(T2) time:

t k
F(t) = mint s + dkj + F(i)I,
k= i= j

where F(t) is the minimum total cost of the periods 1,..., t. The reason we can

use this dynamic programming recursion is that there exists an optimal solution

satisfying zero inventory ordering property:



It-xt = 0 for t= 1,...,T.

In other words, we cannot order (produce) at period t, if we have positive

inventory at the end of the period t 1. Therefore, ordering (production) amount

in period t is dt + + d8 for s > t 1. Aggarwal and Park (1990), Wagelmans et

al. (1992), and Federgruen and Tzur (1991) show that we can solve Wagner-Whitin

case in O(T log T) time. Moreover, we can solve this problem in O(T) time if

non-speculative motives assumption is satisfied. According to non-speculative

motives assumption ordering as late as possible is better. Formally,


pt + ht > pt+l for t = 1,..., T 1.


Veinott (2005) showed that even with general concave production and holding

costs DLSP can be solved in O(T2) time. Zangwill (1969) gives a network

approach and illustrates the problem as a concave cost network flow problem.

This representation can be seen in Figure 2-1, S is the supply node and the other

nodes are the demand nodes.

Zangwill (1969) extends the work of Veinott (2005) by allowing backlogging

in the model and shows that optimal solution is an extreme point solution with

an arborescent structure. In other words, each node in the network has only one

incoming arc that has positive flow. Therefore, production quantity in period t is











+ d4


Figure 2-1: Network flow representation of DLSP.

the sum of demands of some consecutive periods including period t. The choice of

production period in a subplan results in a O(T3) running time.

We obtain capacitated lot-sizing problem (CLSP) by adding the following

constraints:

xt
where Ct is the ordering (production) capacity in period t. Under stationary

(time-invariant) production capacities, Florian and Klein (1971) gives an algorithm

that runs in O(T4) time with general concave inventory cost functions with or

without backlogging. According to Florian and Klein (1971), optimal plans consist

of independent subplans in which

* inventory level is nonzero in every period except the last, where it is zero,

* production level, when positive is at capacity except for at most one period in
which it is less than the capacity.

We can characterize a subplan by its demand periods. In particular, a subplan

(t1, t2) is defined as a consecutive sequence of periods t1 + 1,..., t2, where It

It =0 and I > 0 for i = t +1,..., t2 1.










Therefore, Florian and Klein (1971) concludes that when production capacities

are constant, C, we produce at full capacity for K periods, where

Sdtl + + dt2


S= dt, + + d,2 KC.

Moreover, at only one period we produce e. Van Hoesel and Wagelmans (1996)

gives an O(T3) algorithm when inventory holding costs are linear and backlogging

is not allowed. Bitran and Yanasse (1982) shows that CLSP with nonstationary

(time-varying) capacities is NP-hard, even in the simplest case, where we assume

fixed-charge production cost and zero holding cost with equal demands.

2.2.2 Fully Polynomial Time Approximation Schemes for CLSP

CLSP with many cost structures is proved to be NP-hard (see Bitran and

Yanasse 1982, Florian et al. 1980). For NP-hard cases of DLSP, solution methods

based on dynamic programming, branch-and-bound and their combinations

are proposed. Since solving NP-hard problems optimally is computationally

expensive, we may choose to solve the problem approximately. Let z* be the

optimal solution value and z' be the solution value of the approximation scheme.

For a minimization problem let us define relative deviation from the optimal

solution as
Za
-1.
z*

We list three types of approximation schemes:

c-approximation scheme: It is an algorithm that alv--, returns a near-optimal

solution whose relative deviation from the optimal solution is e. In other

words:


z < (1 + e *.









* Polynomial time approximation scheme: It is an c-approximation scheme that
has a running time polynomial in the problem size.

* Fully polynomial time approximation scheme (FPTAS): It is an e-approximation
scheme that has a running time polynomial in both 1/c and the problem size.

Woeginger (2001) states that an FPTAS is the strongest possible polynomial

time approximation result that we can derive for an NP-hard problem.

Van Hoesel and Wagelmans (2001) provides the first FPTAS for CLSP with

non-stationary production capacities, when we have

* nondecreasing production and holding costs and backlogging is not allowed, or

* (piecewise) concave backlogging and production costs and nondecreasing
holding costs. For piecewise concave case, they assume the number of pieces is
polynomially bounded.

They assume that all demands, capacities, production, and inventory levels are

integer. Moreover, all cost functions can be evaluated in polynomial time at any

value in their domain and are scaled such that they are integer valued. They state

that it is unlikely to get an FPTAS with the traditional dynamic programming

formulations (for example the formulation in Florian et al. 1980) and define a

different pseudo-polynomial dynamic programming formulation. They define Ft(b)

as the maximum inventory amount that we can have at the end of the period t

with budget b. For the first period:


Fi(b) max {xi dipi(ri) + hi(x di) < b} for b 0,..., B (2-3)
Xi
where B is any upper bound on the objective function value.

For periods t = 2,..., T and b = 0,..., B, we condition the budget we spend

in periods 1,..., t 1 (t a) and we spend the remaining amount, b a in period

t. They show that the best ending inventory level of period t 1, when we spend

totally a during periods 1,..., t 1 is Ft_ (a). Therefore, Ft(b) is written in terms

of all Fti(a) such that a < b:












maxo
F,(b) max max < b a},

maxo (2-4)

Therefore, optimal solution of CLSP with nonstationary production capacities

is minimum b such that

FT(b) > 0.

The running time of the dynamic programming algorithm based on formulas 2-3

and 2-4 depends on B polynomially. The next step is to limit the number of

possible budget values. To that aim, a positive integer K is selected and the set

of possible budgets is reduced to the set {0, K, 2K,..., ([B/K] + T)K}. After

this update, running time of the dynamic programming formulation depends on

B/K polynomially. Moreover, they prove that restricted dynamic programming

formulation gives a solution value that deviates from optimal solution at most by

TK.

In the next step, they show how to compute an upper bound on z* which

is at most 2Tz*. This upper bound is used as B in the dynamic programming

formulations. With given relative error with respect to the optimal solution, c, and

K =max{ [eB/2T2], :

1. a < z* + TK z* + T max{ B/2T2,1} < (1 +)*,

2. running time is polynomial in B/K, which is O(T2/C).

therefore, this algorithm becomes an FTPAS.

Recently, Chubanov et al. (2006) presents an FPTAS for CLSP with monotone

cost structure by rounding a traditional dynamic programming algorithm and

exploiting the combinatorial properties of the recursive function. Therefore, they










disprove the statement by van Hoesel and Wagelmans (2001) that a traditional

dynamic programming approach is unlikely yield an FPTAS.

They solve a rounded problem whose optimal solution is denoted by z,* with

the objective function:
T
E (Lp(xt)/p + Lht(It)/Pj),
t=1
where p = eL/2T and L is a known lower bound for z*. They proved that


Zr* < + c)z*.


To solve the rounded problem, they use a modification of a traditional dynamic

programming algorithm.


tt(s) = min


t
1(Lp(x)/p] + [h(l)/p])
i=1


subject to


di + Ii

lo

It

xi

xi

i, .i


xi + Ii-


< ci

> 0

C Z


i= 1,... ,t


(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(2-10)


1,... ,t

1, ,t

1, t.


For any t E [1, T] function Qt(.) is evaluated recursively as:


t(s) min {[pt(s s' + dt)/pj + Lht(s)/p + t-1(s') s s' + dc E [0, Ct]},
e[ t-1 t-11
(2-11)

where at E di, a = Z (C, di), and for s E [al, a]. Instead of solving

this dynamic programming formulation in the traditional way, they prove that Qt









has at most O(t2V2) nonstable points (V = LU/p, where U is a known upper

bound for z*).

Definition 2.2.1 s is a nonstable point of the function f, which is 1. I,.1 in the

interval [a, b], if s E [a + 1,b 1] and f(s) / f(s 1), or s E {a, b}.

Bf is the set of nonstable points of f. They write dynamic programming recursion

2-11 in over the set of the nonstable points:


t(s s)h /p] + min {[pt(S sab + dt)/pj + t- (sab)}, (2-12)
aEBpt',bEBt-

where Sab is an arbitrary point belonging to a specified interval depends on a and b.

They show that all nonstable points can be found in polynomial time and recursion

2-12 can be solvable in time polynomial in 1/e, T, and loglog(U/L), where L is a

known lower bound for z*. The running time of the resulting FPTAS is

U T t
O(T l( oglog +T( +ioglog ) log Ci).
t= 1 i= 1

2.3 Model Formulation

2.3.1 Requirements Planning with Pricing Flexibility

We start by formulating our requirements planning model with pricing

flexibility. Let T denote the length of the planning horizon. Our goal is to find the

production, inventory, and demand satisfaction plan that maximizes total profit

while observing the production capacities. We therefore define, for each period

t = 1,..., T, the following decision variables and input data with appropriate

assumptions:

* Production: The production quantity in period t is denoted by the variable xt.
The production capacity is equal to Ct, and the cost of production is given by
the nondecreasing function pt. Without loss of generality, we will assume that
p (0) 0.

* Demand: The demand satisfied in period t is denoted by the nonnegative
variable dt. The revenue as a function of demand satisfied is given by the









function Rt, where without loss of generality we assume that Rt(O) = 0.
Furthermore, we assume that the revenue function is nondecreasing on
0 < dt < Dt, at which point it attains its maximum.

Inventory and backlogging: The quantity in inventory or backl.-:.-: is denoted
by the free variable It, where positive values denote inventory and negative
values denote backlog. The corresponding cost function is denoted by ht, again
with ht(0) = 0. Furthermore, we will often write


h,(,) i ht (I,)
thtw [ hi)


if I < 0
if h > 0


where ht is a nonincreasing function, h+ is a nondecreasing function, and
clearly h- (0) h- (0) 0.

In addition, we will assume that the problem parameters Ct and Dt (t =

1,..., T) as well as the demand, production, and inventory levels are integral. We

then formulate the requirements planning problem with pricing flexibility (RPP)


T
maximize (RL (d,)
t=1


Xt + It-1


dt + It


Xt < Ct


Xt, dt


> 0


It free

IT > 0

Xt, dt, It E Z


pt(xt) ht(It))


t 1,...,T

t 1,... ,T

t 1,... ,T

t 1,...,T

t 1,...,T


t = 1. ... T.


Constraints 2-13 ensure the balance between inventory, backlog, production

inflow, and demand outflow at the manufacturer. Constraints 2-14 are the

production capacity constraints and constraints 2-15 are imposed without loss


subject to


(2-13)

(2-14)

(2-15)

(2-16)

(2-17)

(2-18)

(2-19)









of optimality, indicating that satisfying any demand in period t beyond Dt does

not yield additional revenue and will not decrease costs. Constraints 2-16 are the

nonnegativity constraints on production and demand satisfaction, while constraints

2-17 and 2-18 indicate that both inventory and backlogging are allowed, but that

no positive backlc-_._., 1 amount is allowed to remain at the end of the planning

horizon. Finally, constraints 2-19 ensures the integrality of the decision variables.

Note that the initial inventory or backlog 1o is assumed given.

Note that the range of profit values that can be achieved in this formulation

of the RPP contains both positive and negative values. Zemel (1981) states that

for such problems the relative error with respect to the optimal solution value is

not a meaningful performance measure of a heuristic. Since one of our goals is to

obtain an efficient approximation scheme for the RPP, we will next show how we

can, under mild conditions, reformulate our problem as a pure cost minimization

problem.

To this end, we define a set of new decision variables yt (t 1,... T) denoting

the amount of unsatisfied demand in period t that is potentially profitable, i.e.,

yt = Dt dt. Next, we introduce the loss function Lt, representing the revenue

shortfall as a function of the quantity of unsatisfied demand:


Lt(yt) = Rt(Dt) Rt(Dt t) = Rt(D) Rt(dt).


Note that Lt is nondecreasing and Lt(O) = 0. Our new objective function can

now be found by subtracting the original objective function from the total revenue

associated with satisfying Dt units of demand in period t (t = 1,..., T), and

minimize the resulting function. In other words, our new objective function

becomes:
T T
t 1 t 1 (R pt(xt) ht )) = ( (yt) + pt(t) + ht ) .
t=1 t=1









With these modifications, we obtain the following reformulation of the RPP which,

with a slight abuse of notation, we will still refer to as the RPP:

T
minimize (pt(xt) + Lt(yt) + ht(It))
t=1

subject to


t + yt + It- = Dt+It t = 1,...,T

Xt < Ct t= 1,... ,T

yt < D t t= ,...,T

xt, Yt > 0 t= 1,... .T

It free t= 1,..., T-

IT > 0

Xt, t, It E Z t 1 ,... ,T.


2.3.2 Capacitated Economic Lot-Sizing Formulation

In this section, we will show that the RPP can be further reformulated as a

capacitated economic lot-sizing problem (CLSP). This reformulation will enable us

to apply and adapt known approaches to the CLSP with general cost functions to

the RPP.

The main difference between the RPP and the CLSP is that the demand is

a decision variable in the former while it is fixed in the latter. However, the cost

minimization formulation from Section 2.3 gives rise to an alternative interpretation

of the RPP. In particular, we could view the RPP as the problem of i -ly!ig"

(or procuring) the total potential demands Dt through two sources: a true source

(production, via the production variables Xt) and a virtual source (via the variables

yt). This is illustrated in Figure 3-2. Furthermore, we may now note that the

decision variables Xt and yt between different periods are only linked through









Xt Xt Yt











dt = Dt yt D,

(a) (b)

Figure 2-2: Demand (a) as a decision variable; (b) deterministic.


the inventory balance constraints and, in addition, these only depend on the

two variables through their sum. Therefore, we can define new decision variables

zt = xt + yt (t = 1,... T) and decide on the decomposition of zt into xt and yt on a

period-by-period basis based on costs. In particular, the cost of procuring Zt units

in period t, i- kt, can be determined as follows:


kt(zt) min {pt(xt) + Lt(yt) : xt + yt = Zt, 0 < xt < Ct, 0 < yt < Dt.
x t,yt Z

With the procurement cost functions kt, we obtain the following formulation of the

RPP as a CLSP:
T
minimize (kt(zt) + ht(It))
t=1









subject to


It- +t = Dt + It t= ,...,T

zt < Ct + Dt t= 1,...,T

Zt > 0 t 1,...,T

It free t= 1,..., T-

IT > 0

Zt,It E Z t ,..., T.


Note that this is indeed a CLSP, with the following interpretations:

* the procurement cost function in period t is given by kt;

* the procurement capacity in period t is equal to Ct + Dt;

* the procurement quantity in period t is equal to zt;

* the demand in period t is equal to Dt.

In the remainder of this chapter, we will study the DP approach proposed by

Florian et al. (1980) and the approximation scheme proposed by van Hoesel and

Wagelmans (2001) in the context of the RPP. The core issue is that the success

of known approaches such as these depends critically on their running times

which are typically derived based on the assumption that the cost functions can be

evaluated in constant time. This is clearly not the case for the RPP due to the fact

that evaluating the procurement cost function kt involves solving a potentially very

hard global optimization problem.

In the next section, we will discuss some properties of the new procurement

cost function.









2.4 Properties of the New Procurement Cost Function

In this section, we will for clarity of expression omit the subscript indicating

the period t. In our first proposition we will show that our assumptions imply that

the procurement cost function k is nondecreasing.

Proposition 2.4.1 The procurement cost function k is nondecreasing on its

domain {0,1,..., C+ D}.

Proof: First, note that since the revenue function R is a nondecreasing function of

demand d, it is easy to see that the cost function L is a nondecreasing function of

y. This implies that we can relax the optimization problem defining k(z) so that in

fact

k(z) min {p(x) + L(y) :x + y> z, 0< x < C, 0< x,yEZ

This immediately implies that the domain of k is {0, 1,..., C + D}. Now observe

that the feasible regions of these optimization problems as parameterized by z are

nested and decreasing as z increases, so that k is nondecreasing. o


The following proposition gives a convenient representation of k in case the

cost functions p and L have a so-called piecewise structure (e.g., general piecewise

linear, piecewise concave, or piecewise convex).

Proposition 2.4.2 If p has a piecewise structure with m -y-in, ,.'I/ and L has a

piecewise structure with n segments then k can be expressed as


k (z) min k" (z)
i=1,...,m;j=1,...,n

for i'i.'i' i ..'I :, /, .7,. ,l functions k'U.

Proof: Denote the (integral) breakpoints of the function p by 0 = Co < C1 <

S.. < C" = C + 1 and, similarly, of the function L by 0 = Do < D1 < ... <

D" = D + 1. Then define the minimum cost associated with a supply of z when

restricting ourselves to the ith segment of p (i = ,..., m) and the jth segment of L









(j = ,...,n) as


k'(z) min{p(x) + L(y) : x + y = z, Ci-1 < x < Ci, D-1 < y< DJ}. (2-20)
x,yEZ

Using this definition, we immediately obtain the desired result. o


In the remainder of this chapter, we will focus mainly on two classes of cost

functions L and p that together span many of the cost and revenue structures

proposed in the literature.

* L and p are piecewise convex. Note that (piecewise) convexity of the loss
function L corresponds to (piecewise) concavity of the revenue function R,
which is a commonly made assumption (see, e.g., Geunes et al. 2005). As for
the production cost function p, all piecewise linear functions are piecewise
convex, which means that this class includes, for example, piecewise linear
concave production functions p that represent economies of scale in production
as well as the (modified) all-units discount cost structure proposed by C'!i i: et
al. (2002).

* L and p are piecewise concave. In a similar way as above, this case includes
general concave production cost functions and piecewise linear concave revenue
functions (see, e.g., Geunes et al. 2005 for a situation in which such a revenue
function naturally occurs).

(We recall that convexity of a function f defined on Z means that f(x + 1) -

f(x) is nondecreasing in x and concavity of a function f means that f(x + 1) f(x)

is nonincreasing in x.)

The next propositions derive the running time required to evaluate the

procurement cost function k for the piecewise convex and piecewise concave cases.

Proposition 2.4.3 If p is a piecewise convex function with m segments and L is a

piecewise convex function with n -jr,,, ,/1 then the function k can be evaluated in


O(mn log C) time. Moreover, the function k is piecewise convex.









Proof: Note that we can rewrite the function k"I as defined in Equation 2-20 as


k' (z) = min{p(x) + L(y) : x + y z, Ci-1 < x < Ci, D-1 < y < D}
x,yEZ
Smin{p(x) + L(z x) : C1i- < x < Ci, Dj- < z x < Dj}

Smin{p(x) +L(z -x) : max{Ci-1,z D 1} < x < min{C,z DJ- + 1}}.
xEZ

It is easy to see that if both p and L are convex on each segment specified by a

given i and j, p(x)+L(z-x) is a convex function of x on the corresponding segment

as well. Therefore, we can find the value of k j(z) in O(log(C0 C-1)) = O(log C)

time using, for example, golden section search. Thus, by Proposition 2.4.2 the value

of k(z) can be found in O(mn log C) time.

It remains to be shown that, for all i = 1,..., j = 1,..., n, the function

ki is convex on its domain. To this end, let Ci-1 + Dj-1 < z < Ci + Dj 1 and

suppose that kij(z) = p(x) + L(z x) for some max{Ci-1, Dj + 1} < x <

min{Ci, z Dj-1 + 1}. Convexity of p and L on the relevant domains then implies

that

k'"(z+ 1) min{p(x+ 1) + L(z- x),p(x) + L(z- x + 1)}

(where p(x + 1) is replaced by oo if x = C 1 and L(z x + 1) is replaced by oo if

x = C-1). This implies that


kj(z + 1) kj(z) = min {p(x + 1) p(x), L(z x + 1) L(z x)}


which is nondecreasing in z. D


Proposition 2.4.4 If p is a piecewise concave function with m -, ,,, ,'..1 and L is

a piecewise concave function with n segments then the function k can be evaluated

in O(mn) time. Moreover, the functions k' as 1. I, .. in Equation 2-20 are

piecewise concave with no more than three u,,r ,.









Proof: Let us rewrite the function kV as in the proof of Proposition 2.4.3. It is

easy to see that if both p and L are concave on each segment specified by a given i

and j, p(x) + L(z x) is a concave function of x on that segment as well. Since the

minimum of a concave function over an interval is attained at one of the endpoints

of the interval, we can find the value of kO(z) in 0(1) time. Thus, by Proposition

2.4.2 the value of k(z) can be found in 0(mn) time.

It remains to show that k0" is piecewise concave with no more than three

segments. Note that the feasible region of the optimization problem defining ki(z)

has four candidate end points: x = C-1, z Dj + Ci 1, z Dj-', two of which

will be feasible for any given z. We can write kVi as follows:


kV (z) = min


p(Ct
P(Z -

p(Ct
P(Z -


-1) + L(z Ci-1)

Dj + 1) + L(D 1)

- 1) + L(z Ci + 1)

Dj-1) + L(Dy-l)


when C'-1 + Dj

when C'-1 + Dj

when Ci + Dj-

when C'-1 + Dj


1 < z < Ci-1 + Dj

- 1 < < C +D

- 1 < < C +D

< < Ci + Dj-1


where the condition indicates whether the given function corresponds to a feasible

choice of x and should therefore be taken into account in the minimization. If

Ci-1 + Dj < Ci + Dj-1 we can rewrite kij as:


kV (z2)


min {p(C'-



min {p(z -



min {p(z -


) + L(z Ci-),p(C 1) + L(z C + 1)}

for C-1 + D -1 < z < Ci-1 + Di

Di + 1) + L(D 1 1),p(C' 1) + L(z C' + 1))}

for Ci-1 + D < z < Ci + DJ-1

Dj + 1) + L(Dj 1),p(z Dj-1) + L(DJ-1)}

for Ci + Dj-1 < z < Ci + Dj- 1.


A similar expression can be given for kij if C-1 + Dj > C' + Dj- Noting that the

minimum of two concave functions is again concave, the desired result follows. o









It is interesting to note that, while in some of the above results we assume that

p and L have a piecewise structure with a fixed number of pieces that appears in

the running time required for evaluating k, the function k itself may in fact have

a piecewise structure with a number of segments that is exponential in the input

size. Instead, our complexity results follow by the fact that the function k can be

written as the minimum of a polynomial number of functions k" with a specific

structure. We refer to Tsevendorj (2001) for more information on the piecewise

nature of functions and optimization problems.

2.5 Results

2.5.1 Dynamic Programming

In this section, we apply the classical DP formulation of the CLSP as

developed by Florian et al. (1980) by adapting it to the case where backlogging

is allowed. In this formulation, the time periods t = 0,..., T + 1 denote the

stages, and the state at stage t is given by the cumulative procured quantity up to

and including time t and is denoted by Z. Here the state Z includes past excess

procurement as summarized by the initial inventory level lo. For convenience,

define
t t
Ct= C, and D D, t 0,...,T

and

C = max Ct and D max Dt t = 0,..., T.
t=1,...,T t=1,...,T

Then the set of procurement decisions that can be made from state Z at stage t is

given by


PtZ) : 0 z Ct+Dt,Io

The optimal value function is denoted by Gt(Z) and represents the minimum cost

satisfying all demands in periods 1,..., t through procurement, inventory, and









backlogging, with a cumulative procurement quantity of Z. The DP recursion reads


Go(Z) -= 0 if Z = Io
Go(Z) ifZI0

oo ifZ / Io

Gt(Z) mmi {Gt_(Z- z)+kt(z)+ht(Z-Dt)} for t 1,2,...,T.
zEP~t(Z)

The optimal solution to the CLSP formulation of the RPP is then given by


GT+1 mmin GT(Z).
DT
The total number of states in this formulation is O (T(C + D)), and the total

number of arcs is O (T2(C + D)2). Given that the values Ct and Dt can be

computed in O(T) time, it remains to determine the effort required to determine

the cost associated with all decisions, i.e., the cost required to compute


kt(z) + ht (Z Dt)

for all feasible states Z in stage t and all feasible decisions z E Pt(Z). Clearly,

if all cost functions can be evaluated in constant time we obtain the well known

result that the CLSP problem can be solved in 0 (T2(C + D)2) time. For the

CLSP formulation of the RPP, however, this assumption is clearly violated for the

procurement cost functions kt. The following theorem derives the running time of

the DP approach for solving the RPP under different sets of assumptions:

Theorem 2.5.1

* If p and L are general nondecreasing functions then the DP il,, ,:thm solves
the RPP in 0 (T2(C + D)3) time.

* If p is a piecewise convex function with m -u,,, ,.'1 and L is a piecewise
convex function with n segments then the DP i,1,,rithm solves the RPP in
0 (T2mn(C + D)2 log(C)) time.

* If p is a piecewise concave function with m segments and L is a piecewise
concave function with n 'lim 1 t;1 then the DP i'1j. .:hm solves the RPP in
0 (T2mn(C + D)2) time.









Proof: First, note that a procurement cost function k needs to be evaluated at

most once per candidate decision. Result (i) now follows by observing that, in

general, the procurement cost function kt can be evaluated in O(max(C, D))

O(C + D) time. Moreover, results (ii) and (iii) follow by applying Propositions

2.4.3 and 2.4.4. O


2.5.2 Fully Polynomial Time Approximation Scheme

In this section, we will derive conditions under which a fully polynomial time

approximation scheme (FPTAS) exists for the RPP based on such a scheme for the

CLSP as developed by van Hoesel and Wagelmans (2001). In particular, a FPTAS

finds a feasible solution to the RPP with cost no more than (1 + e)c*, where c*

denotes the optimal cost and a > 0, in an amount of time that is a polynomial

function of the input size and 1/e.

The FPTAS for the CLSP is based on a non-traditional DP formulation of the

problem that employs an upper bound on the total cost, i- B. In particular, for

0 < b < B and t = 1,..., T, define Ft(b) to be the maximum ending inventory

level in period t when the total cost incurred up to and including period t is no

more than a budget b. The optimal solution value to the problem is given by the

minimum budget for the entire planning horizon that yields a nonnegative ending

inventory, i.e., c* = minb o,...,B{b| Fr(b) > 0}. Van Hoesel and Wagelmans (2001)

show that the values Ft(b) (b 0,..., B) satisfy the following DP recursion:


Fi(b) = max {z Di ki(zi) + hi(zi DI) < b, 0 < zi < Ci + Di}

max max Fti(a) + Zt D I kt(z) + ht(Ft_(a) + Zt Dt) < b -a,
a=0,...,b
Ft(b) max 0 < zt < Ct + D, (2-21)

max {It Ia e {0,...,b} : ht(I) b a, 0 < It < Ft-i(a) D }.









An important component of the running time of this recursion is the time required

to find


max {zt I kt(zt) + ht(Ftl-(a) + zt D) < b a, 0 < zt < Ct + Dt} (222)


which, if all cost functions can be evaluated in constant time, is O(log(C + D)),

yielding a running time of O(B2Tlog(C + D) + BTlog(T(C + D))) for the entire

recursion. For the CLSP formulation of the RPP, however, we need to account for

the fact that the procurement cost functions kt cannot be evaluated in constant

time. The following proposition studies problem 2-22 under several conditions on

the cost functions.

Proposition 2.5.2

* If pt is a piecewise convex function with no more than m -,,, ,n.I- Lt is a
piecewise convex function with no more than n -.,,, ,.1n- and hi is a convex
function, then problem 2 22 can be solved in 0 (mn log(C) log(C + D)) time.

* If pt is a piecewise concave function with no more than m segments, Lt is a
piecewise concave function with no more than n segments, and h, is a concave
function, then problem 2 22 can be solved in 0 (mn log(C + D)) time.

Proof: Note that we can decompose the problem into the problems of finding


zj max {t kt(z) + h,(Ft1(a) + Zt Dt) < b a, (Dt F~ i(a))+ < Zt < Ct + Dt}


and


z- max {zt kt(zt) + h h(F,-_(a) + t D) < b- a, 0 < zt < min(Ct + Dt, Dt Ft-i(a))}.


Since kt and h, are nondecreasing, z+ can be found in O(log(C + D)) iterations of

the binary search method (or function evaluations).









With respect to the value z note that it is equal to


z- =max zt min k (zt) + h (Ft_-(a) + Zt Dt) < b a,
i = ,...,m; j= 1,..., n

0 < zt < min(Ct + Dt,Dt Ft- 1(a)) .


To determine this value, let us define


z max {zt | (kt (zt) + h- (Ft-l(a) + zt Dt)) < b a, Dt Ft-(a) < zt < Ct + Dt .


Since h- is convex and Proposition 2.4.3 -,i- that k'j is integral convex, z11

can be found in O(log(C + D)) iterations of the binary search method. Since

Zt = m-:: ; z1' the value of z~ can be found in O(mnlog(C + D)) function

evaluations.

Combining these results with the results of Propositions 2.4.3 and 2.4.4, the

desired result now follows. o


Note that the result of Proposition 2.5.2 can easily be generalized to the case where

ht is piecewise convex or piecewise concave with no more than f segments by

writing


zt = max t mm k3n (zt) + h'-(Ft-i(a) + Zt Dt) < b a,
z 1,...,m ;j 1,...,n;K=1,...,

0 < zt < min(Ct + Dt,Dt- Ft-(a))


at the expense of a factor f in the running times.

We are now ready to determine the running time of the DP recursion 2-21 for

the RPP.

Proposition 2.5.3

If the functions pt, Lt, and ht are piecewise convex with no more than m, n,
and i -. ii,, I ,./,, -.r I.: 1./; the DP formulation of the RPP runs in

O(B2rmnT log Clog(C + D) + BTlogT + BT log(C + D))









time.

* If the functions pt, Lt, and ht are piecewise concave with no more than m, n,
and ul C,, ,' /',. -. /,. 1./; the DP formulation of the RPP runs in

O(B2mnnYTlog(C + D) + BTlogT + BTlog(C + D))

time.

Proof: This follows immediately by the analysis in van Hoesel and Wagelmans

(2001) and Proposition 2.5.2. D


The FPTAS is now based on a suitable discretization of the DP recursion

2-21 together with an upper bound B on the optimal costs c* that satisfies

c* < B < 2Tc*. Van Hoesel and Wagelmans (2001) describe a DP algorithm to find

such a bound. The following proposition derives the running time of this algorithm

when applied to the RPP.

Proposition 2.5.4

* If the functions pt, Lt, and ht are piecewise convex with no more than m, n,
and -f gr, ,nl-. "i'. /, 1,'. ;l then an upper bound B on the optimal costs c*
that -,/l.:-/ c* < B < 2Tc* can be found in 0 (mnTlogT(logC)2 log U) time,
where U is an upper bound on the minimum cost in i,.;, period:


U- m=axr kt(Ct + Dt), ht ZD h D- .
t 1,...,T
T= l Tr=t+l


* If the functions pt, Lt, and ht are piecewise concave with no more than m, n,
and -1 mr,, ,n. -- /,i'. 1.; /; then problem an upper bound B on the optimal
costs c* that -,/.:-/, < c* < B < 2Tc* can be found in (mnTlogTlogClogU)
time.

Proof: The only components of the DP algorithm that involve the procurement

cost function kt are

* The determination of an upper bound on the procurement level in each period
t as a function of an upper bound A on the production costs in that period:

t = max {z < Ct + Dtlkt(z) < A}.









* The determination of the upper bound U from Proposition 2.5.4.

For cases (i) and (ii), the running time of a. follows from Proposition 2.5.2

and the running time of b. follows from Propositions 2.4.3 and 2.4.4. The final

result now follows from the ,in 1,i -; of the algorithm in van Hoesel and Wagelmans

(2001). o


Now recall the FPTAS proposed by van Hoesel and Wagelmans (2001):

1. Determine an upper bound B on c* that satisfies B < 2Tc*.

2. Apply the DP-recursion 2-21 after rounding the problem data to be multiples
of K = max {B/2T2],}.

Combining all results above, we can now conclude with the main result of this

section:

Theorem 2.5.5 The FPTAS has a ru',,i,.:.u, time that is y..''l;.,.:,'l in the size of

the problem instance and 1/E and determines a feasible solution to the RPP with a

value no 7./,, than (1 + E)c* if

* the functions pt, Lt, and ht are piecewise convex with no more than m, n, and
segments, ', /,i;. l; or

* the functions pt, Lt, and ht are piecewise concave with no more than m, n,
and -, ,,, ,/l- i/ /,:' .1;



2.6 Conclusions

In this chapter, we have studied the capacitated requirements planning

problem with pricing flexibility (RPP) by reformulating it as a standard capacitated

economic lot-sizing problem. The reformulation of the RPP as a lot-sizing problem

provides insight into the relationship between the two problems and significantly

simplifies the application and analysis of algorithmic approaches that have been

developed for the ELSP to the RPP. In particular, we have analyzed the properties






35


of this reformulation, and used this analysis to derive running times for a DP

algorithm as well as a FPTAS for the RPP.















CHAPTER 3
LOT-SIZING WITH NON-STATIONARY CUMULATIVE CAPACITIES

3.1 Introduction

In this chapter, we study a new dynamic lot-sizing problem which we will refer

to as the lot-sizing problem with cumulative capacities (LSP-CC). In traditional

capacitated lot-sizing models, the quantity produced in each period is limited by

some capacity, but any capacity remaining at the end of a period is essentially lost.

In contrast, our new model applies to settings where any remaining capacity is

transferred to the next production period. This means that the cumulative quantity

that we can produce up to and including period t is constrained by the cumulative

capacity up to and including period t.

This problem may occur, for example, in certain settings where the planning

of procurement, production, and inventory holding of raw materials and final

product is integrated. This two-level lot-sizing model has been studied extensively

for the case where all cost functions are concave and capacities are absent (see,

e.g., Zangwill 1969), or procurement is subject to stationary capacities (see, e.g.,

Kaminsky and Simchi-Levi 2003) and van Hoesel et al. 2005). Although in the

presence of general procurement capacities the problem is known to be NP-hard

even when all cost functions are concave (see Florian et al. 1980), Kaminsky and

Simchi-Levi (2003) derive certain conditions on the cost functions under which the

problem is solvable in polynomial time. Sargut and Romeijn (2006c) show that this

problem reduces to the LSP-CC when the total cost of procurement and inventory

of raw materials are exogenous to the model and study this problem under general

and concave cost functions.









In Section 3.2, we formulate the two-level lot-sizing problem and, under our

assumptions on the cost structure, reformulate it as a single-level model with

cumulative capacities. Then in Section 3.3, we prove that the problem is NP-hard

for general cost functions and provide a fully polynomial time approximation

scheme in case all cost functions are nondecreasing. In Section 3.4 we then develop

a dynamic programming approach that solves the problem in polynomial time when

all cost functions are concave. Finally, in Section 3.5 we briefly discuss an extension

where backlogging is allowed.

3.2 Model Formulation

3.2.1 Two-Level Lot-Sizing with Non-stationary Production Capacities

We start by considering a two-level production system. This system may

consist, for example, of a supplier of raw materials at the first level and a

manufacturing facility producing a single good at the second level. The goal is

to satisfy a deterministic and dynamic sequence of demands over a finite and

discrete planning horizon at minimum cost through production and inventory. We

denote the planning horizon by T. The procurement capacity faced at the first

level in the system in period t is denoted by Ct while the demand faced at the

second level in period t is denoted by dt (t 1,..., T). Our decision variables and

associated cost functions are given below.

* yt denotes the quantity procured in period t with associated cost function pt
(t 1,. ,T);

* xt denotes the quantity produced in period t with associated cost function ct
(t 1- ,...,T);

* Jt denotes the inventory level at the procurement level at the end of period t
with associated cost function gt (t 1,..., T);

* It denotes the inventory level at the production level at the end of period t
with associated cost function ht (t 1,..., T).










Without loss of generality, all cost functions are assumed to be equal to zero

when their argument is zero. Moreover, we assume that the initial inventory at

both levels is given. Then we can formulate the two-level lot-sizing problem with

procurement capacities (2LSP-PC) as follows:

T
minimize (p,(yt) + cm,(x) + g,(Jt) + h,(I))
t=1


subject to


Yt < Ct

Yt + Jt-1 = Xt + Jt

t + It-1 = dt + It

yt, t, Jt,It I> 0


t 1,...,T

t 1,...,T

t 1,...,T

t 1,..., T.


Constraints 3-1 enforce the procurement capacities while constraints 3-2 and

3-3) model the balance between inflow, outflow, and storage at the first and

second level, respectively. For convenience, we will define cumulative demands and

capacities as


dst = d,
Sr=s+l
t
Cot = C"
Tr=l


= O,...,t; t= 1,...,T


t= ,..., T.


To ensure that the feasible region of the 2LSP-PC is nonempty, we assume that the

demands and capacities satisfy the following condition:


dot < Cot+Jo +


t 1,...,T


i.e., the cumulative production capacity in the first t periods is sufficient to satisfy

the cumulative demand in the same periods (t = 1,..., T).


(3-1)

(3-2)

(3-3)









This model was introduced earlier by Zangwill (1969), Kaminsky and

Simchi-Levi (2003), and van Hoesel et al. (2005). Zangwill (1969) develops a

dynamic programming algorithm for the uncapacitated case, which runs in O(T3)

time (see van Hoesel et al. 2005). More recently, Kaminsky and Simchi-Levi

(2003) develop polynomial-time algorithms for certain instances of this problem

with linear procurement and inventory holding cost functions (pt, gt, and ht) and

concave production cost functions (ct). In particular, they show that when the

production cost functions have a fixed-charge structure and cost functions at

both levels satisfy a non-speculative motives assumption, the 2LSP-PC is solvable

in O(T4) time. Van Hoesel et al. (2005) showed that, when the procurement

capacities are stationary, the problem can be solved in O(T7) time if all cost

functions are concave; in O(T6) time if the procurement cost functions are concave,

inventory holding costs are linear, production costs have a fixed-charge structure,

and variable holding and production costs satisfy a non-speculative motives

assumption; and in O(T5) time if the procurement cost functions are concave and

all other cost functions are linear. On the other hand, the 2LSP-PC is NP-hard

under concave cost functions and nonstationary capacities since it generalizes the

single-level capacitated lot-sizing problem (see Florian et al. 1980).

We consider several new subclasses of the 2LSP-PC. All our models apply

to situations where, while procurement capacities are important, the total cost of

procurement and inventory of raw materials is exogenous to the model. This would

occur, for example, if procurement costs are linear and stationary and there are

no costs associated with holding raw materials in inventory. More generally, we

essentially assume that procurement costs and inventory holding costs at the first

level are linear and satisfy:

* pt(yt) = ptt where pt > 0 (t = 1,..., T);

S gt(J) = gJt where g > 0 (t = 1,...,T);









* pt+ t = Pt+ (t = 1,..., T- 1).

Under these assumptions the 2LSP-PC becomes as a single-level lot-sizing

problem with what we will call cumulative capacity constraints. Our contributions

in this chapter with respect to this new lot-sizing problem are the following (where

we refer to the cost functions in the reformulated model).

* We show that the problem is NP-hard under general cost functions and
develop a fully polynomial time approximation scheme (FPTAS) for the case
where all cost functions are nondecreasing.

* We develop a dynamic programming algorithm that solves the problem in
((T4) time when all cost functions are concave.

* We extend these results to allow for backlogging.

3.2.2 Single-Level Lot-Sizing with Cumulative Capacities

In this section we will reformulate the 2LSP-PC as a single-level lot-sizing

problem under the cost structure given above. To this end, we first eliminate the

inventory variables Jt by using the inventory balance constraints at the first level

3-2:
t t


This leads to the following objective function:

T
S(ptt + Ct(xt) + gtJt + ht(It))
t=1
T T T
T (gtjo + (P T gt") yt +ct(xt) igT
t=1 T=t T=t
T T T
=Jo Zgt+ (PT + gr) dt + (ct(xt) + t())
t=1 t=1 t=1

where Jo 9t gt + (PT + gT) tE 1 dt is a constant and

T
Ct(x) t(x)- gx t ,...,T.
r=t










This in fact shows that, without loss of generality and for convenience, we may

assume that pt = gt 0 so that we have ct = ct (t = 1,..., T).

Upon eliminating the decision variables Jt (t = 1,..., T) the corresponding

nonnegativity constraints become

t t
X, < y + Jo t =l,..., T.
T=l T=l

However, since the decision variables yt (t = 1,..., T) do not appear in the

objective function we can replace these and the capacity constraints 3-1 by

t t
X < C, +Jo Cot + Jo t = 1,..., T.
T=l T= 1

It is now easy to see that, without loss of generality, we may assume that Jo = 0.

This leads to the single-level lot-sizing problem with cumulative capacities

(LSP-CC) that is the focus of this chapter:


T
minimize > (ct(xt) + ht(It))
t=1


subject to


(LSP-CC)


xt + It- = dt+ It
t
,XT < Cot
T= 1
Xt, It > 0


t= 1,...,T

= 1,...,T

t 1,. ,T.


Finally, note that, in this problem, we may without loss of generality assume that

o0 =0.

3.3 General Cost Functions

We next turn to an analysis of the LSP-CC with general cost functions. In

particular, we will show that this problem is NP-hard in general. In addition, we

provide a fully polynomial time approximation scheme (FPTAS) for the LSP-CC

with nondecreasing cost functions, i.e., an algorithm that finds a feasible solution









with relative error no more than E > 0 in an amount of time that is polynomial in

the input size of the problem and 1/E.

3.3.1 Proof of NP-Hardness

In this section, we will show that LSP-CC with zero holding cost is NP-hard

by reducing the subset sum problem to an instance of LSP-CC. The SUBSET SUM

problem is defined as follows (see Garey and Johnson 1979):


SUBSET SUM: Given positive integers al, a2,..., aT and A, does there exist a set

S C {1,..., T} such that Ees a = A?


The following theorem shows that the LSP-CC is NP-hard in general by reducing

the SUBSET SUM problem to it.

Theorem 3.3.1 The LPS-CC is NP-hard.

Proof: Consider an instance of the SUBSET SUM problem. We then define the

following instance of the LSP-CC:


Ct = at for t = 1,...,T

dt = 0 fort 1,...,T- 1
A for t = T

0 if x = 0

ct(x) 1 + "a x if 0 < x < at for t = 1,...,T
at
x + 1 if at < x < 1YCt a,
ht() 0 if It > 0 for t = 1,..., T.


The production cost function is illustrated in Figure 3-1. First observe that, since

the only non-zero demand is in period T, each production quantity will be used

to satisfy the demand in that period and thus the cumulative production over the

planning horizon is equal to A, i.e., 1 XT, A. Moreover, the production cost in











Ct(X)



at +at-+...+al+l -----------------------




at+



at at +at-l+...+al x



Figure 3-1: Production cost function in period t.


period t satisfies


ct(X) = x if x E {0, at}

ct(x) > x otherwise.


This implies that the cost of any production plan with production in any period t

not equal to either 0 or at exceeds A. In other words, if the LSP-CC has a solution

with cost A the set of periods in which production takes place provides a solution

to the SUBSET SUM problem. On the other hand, if the SUBSET SUM problem

has a solution, w S, producing xt = at in all periods t E S provides a solution to

the LSP-CC with cost A. O


We have shown that the LSP-CC is NP-hard for general production cost

functions, even when the holding costs are zero. In the remainder of this section we

provide a FPTAS for the LSP-CC with general cost functions.

3.3.2 A Fully Polynomial Time Approximation Scheme

Although the LSP-CC is NP-hard for general cost functions, we will show in

this section that a fully polynomial time approximation scheme (FPTAS) exists









when all cost functions are nondecreasing and all problem data (i.e., demands,

capacities, and costs) are integral. We assume pt = p for t = 1,..., T and g = 0 for

t = 1,..., T. Therefore, the single-level model reduced from the two-level model has

nondecreasing cost functions. In particular, we will draw heavily on the paper by

van Hoesel and Wagelmans (2001) and modify the FPTAS that they developed for

the standard capacitated lot sizing problem (CLSP) to our problem.

Following van Hoesel and Wagelmans (2001), we start by introducing a

non-traditional dynamic programming formulation of the problem that runs in

pseudo-polynomial time. Let B be any integer upper bound on the optimal solution

value, w z*, to the LSP-CC. For t 1,..., T and b = 0, 1,..., B, we then define

Ft(b) to be the maximal ending inventory in period t (It) that can be achieved with

a budget of b in the first t periods, i.e., the total production and inventory holding

costs in periods 1,..., t do not exceed b. Since for t = 1 the cumulative capacity

constraint is equal to the traditional capacity constraint we immediately obtain

that


Fi(b) = max {xi di ci((xi) + hi(xi di) < b} for b 0,..., B.
dl<_l
A recursion for the values Ft(b) can then be derived by, for t = 2,..., T, allocating

a budget of 0 < a < b to periods 1,..., t 1 and a budget b a for period t, and

optimizing over the value of a. For the CLSP, van Hoesel and Wagelmans (2001)

show that, for a given value of a,

* if it is possible to feasibly extend a production plan for periods 1,..., t 1 that
ends with the maximal inventory level Ft-_(a) to period t it is optimal to do
so;

* otherwise, without loss of optimality attention may be restricted to Xt = 0 and
dt < It-1 < Ft_-(a).










We will show that these two properties also hold for the LSP-CC. First, note

that the flow balance constraints imply that

t-1
x, = do,t-l + It-1
T=1

so that the cumulative capacity constraint in period t can be written as

t-1
X5t < co x -X Ctt o,t-1 It-1
T=l

Moreover, since the demand in period t needs to be satisfied we need to have


xt > max{0, dt It-i}.


The next proposition now shows that properties (i) and (ii) hold for the LSP-CC.

Proposition 3.3.2 If there exists max{0, d Fti(a)} < xt < Cot do,t-1 Ft-i(a)

such that

Ct(xt) + ht (Ft-_(a) + xt dt) < b a (3-5)

then we only need to consider production plans with It-1 = Fti(a) in the /i.;i,,'. .

,,..ji. 'r,,.:,,j recursion. Otherwise, we only need to consider production plans with

xt = 0 and dt < It-1 < Ftl(a).

Proof: Suppose we have a feasible solution with It-1 < Fli(a) and xt > 0. Then if

we increase It-1 and decrease xt by the same amount while retaining nonnegativity

of xt condition 3-5 as well as the bound constraint on xt will still be satisfied.

Moreover, the resulting value of It remains unchanged. Therefore, we can without

loss of optimality in the dynamic programming recursion increase It-1 and decrease

xt until either It-_ becomes equal to Fli(a) or xt becomes equal to zero, whichever

happens earlier. In the former case, the first claim follows. In the latter case, the

second claim follows. o









As a result of the Proposition 3.3.2 the following dynamic programming

recursion is obtained:

maxmax{0,dt -Ftl (a)}zt
Ft(b) = max max { Ft-(a) + xt dt t(xt) + ht(Ft-(a) + xt d) < b a},

maxo
It is easy to see that this recursion runs in O(B2Tlog(CT)) time, where C

maxt 1,...,T Ct. The FPTAS is now based on a suitable approximation of this

dynamic programming recursion where the state variable representing the budget

is restricted to integer multiples of some value K > 1. Van Hoesel and Wagelmans

(2001) showed that, if we have an upper bound B on z* that satisfies z* < B <

2Tz*, we can use a value K = max{L B/2T2, 1} for some E > 0 to achieve a

solution to with value at most (1 + E)z* in polynomial time, yielding the desired

FPTAS.

It remains to be shown that a cost bound B with the desired property can be

found in polynomial time. Once again we will modify the approach of van Hoesel

and Wagelmans (2001) to apply to the LSP-CC. The idea of their algorithm is to

find the smallest value L with the property that there exists a feasible solution to

the problem such that all cost functions contribute at most L to the total cost.

Hence, such a feasible solution has cost at most 2TL. Clearly, in any optimal

solution of the original problem, each cost function contributes no more than z*

so that L < z*. This implies that B = 2TL is an upper bound on z* such that

B < 2Tz*. To show that L can be found in polynomial time, we first show that it

is possible to determine in polynomial time whether or not there exists a feasible

solution if the contribution of each cost function is at most Given this value,

we define for each period t the following upper bounds on the production and









inventory level:


Xt max{x: 0 < x < Cot, Ct(x) < }

It max { : I > 0, ht(I) < f }.


These bounds can be determined using binary search in a total of O(Tlog(CT))

time. It is easy to see that a feasible solution in which each cost function

contributes at most i exists if and only if there exists a feasible solution that

satisfies the above upper and lower bounds on the production and inventory levels.

We can check this using the following dynamic programming algorithm. Let it

denote the largest ending inventory in period t achievable by a production plan for

the first t periods that satisfies all upper and lower bounds. In particular, we have


mi = min{xi di, 1i}

mt = min {mt- + min{Cot do,t-l mt-l,xt} dt,It} t 2,...,T.


If mT > 0 then there exist a solution in which each cost function contributes at

most Now noting that L is bounded from above by maxt1,...,Tmax{ct(Cot), ht(dtT)}

the value L can be determined in polynomial time using binary search.

3.4 Concave Costs

3.4.1 Introduction

The LSP-CC may be formulated as a minimum cost network flow problem as

illustrated in Figure 3-2. Nodes (D, 1), (D, 2),..., (D, T) are demand nodes, where

(D, t) represents the demand in period t, dt. Nodes (C, 1), (C, 2),..., (C, T) are

transshipment nodes and node (C, T + 1) is a supply node having supply equal to

the cumulative demand over the planning horizon, i.e., its demand is equal to the

negative thereof: -doT. As in traditional lot-sizing models, an arc from node (D, t)

to node (D, t + 1) (t = 1,..., T 1) represents the amount held in inventory at the

end of period t, It, and the cost function of the flow on that arc is ht. (Note that,









without loss of optimality, the inventory at the end of period T is equal to zero.) In

addition, an arc from node (C, t) to node (D, t) represents the amount produced in

period t, xt, and the cost function of the flow on that arc is ct. Finally, an arc from

node (C, t + 1) to node (C, t) (t = 1,..., T) represents the cumulative production

amount in periods 1,..., t. An arc of this form has no cost but a finite capacity,

t'1 CT. (There are no capacities on the other arcs.) Production and inventory

cost functions are commonly concave, representing the economies of scale that are

often found due to the presence of, for example, fixed setup costs for production

and fixed storage costs. When the functions ht and ct (t = 1,..., T) are concave,

the LSP-CC thus becomes a minimum concave cost network flow problem. Despite

the fact that an extreme point optimal solution to such problems exists, they

are generally NP-hard (see Bitran and Yanasse 1982). However, polynomial-time

algorithms have been found for many lot-sizing problems with concave costs by

employing the structure of extreme point solutions for these problems.

-do0
[0,C01] [0,C02] [0,Co3] [0,CO,T-1 [0,Cor]
C,1 C,2 C,3 --- ------- CT ,T+






D,1 \D, 2 D,3---- --- DT
1, 2 13 IT-
di d2 d3 dr

Figure 3-2: Network flow representation of LSP-CC.


In particular, it is well known that the standard uncapacitated economic

lot-sizing model is solvable in O(T log T) time (see Aggarwal and Park 1990,

Wagelmans et al. 1992, and Federgruen and Tzur 1991). Under stationary

(time-invariant) production capacities this model is solvable in O(T3) time when

inventory costs are linear (see van Hoesel and Wagelmans 1996) and in O(T4) time









with general concave inventory cost functions (see Florian and Klein 1971). On

the other hand, the economic lot-sizing model with nonstationary (time-varying)

capacities is NP-hard, even if the production cost functions have a fixed-charge

structure and the holding cost functions are linear (see Florian et al. 1980). In

the remainder of this section we will develop a polynomial-time algorithm for the

LSP-CC under nonstationary capacities.

3.4.2 Solution Approach

As mentioned above, minimum concave cost network flow problems have an

extreme point optimal solution. Extreme point solutions to minimum cost network

flow problems have the property that there does not exist a cycle formed by free

arcs, i.e., arcs whose flow is strictly between its lower bound (usually zero) and

upper bound (see, e.g., A!Li I et al. 1993). In the remainder of this section we will

assume for simplicity that all demands are strictly positive. However, all results can

be easily extended to the general case where periods with zero demand are allowed.

Consider the set of free arcs in an extreme point solution to the network

flow formulation of the LSP-CC. Since all inventory and production arcs are

incapacitated, when we ignore the cumulative production arcs the set of remaining

free arcs decomposes into disjoint connected and ... i-, .i components. Each of these

components is characterized by a consecutive set of demand periods, i- t+ 1...,

(where 0 < t < < T), with the property that


t = 0

I, > 0 s t+1,...,r-1

I, = 0.


We will refer to a component satisfying these properties as a subplan, denoted

by (t, 7). As we will show in the next section, the cost structure of the LSP-CC

allows us to compute the minimum cost of supplying the demands in any subplan









independently of the other subplans. This is perhaps surprising since the subplans

seem to be intimately related through the (cumulative) production capacities.

Now denote the cost of subplan (t, 7), i.e., the minimum cost of supplying

the demands in that subplan, by at, and let F(t) be the minimum total cost of

satisfying the demands in periods t + 1,..., T. Clearly, F(T) = 0 and the optimal

solution value of the LSP-CC is given by F(0). This then immediately leads to the

following dynamic programming formulation of the LSP-CC:


F(t) min {at, +F(r)} fort 1,...,T 1
T=t+l,...,T
F(T) 0.


This recursion is, of course, very similar to the recursion obtained for the economic

lot-sizing problem without capacities or with stationary capacities. It is easy to

see that, if the subplan costs at, were given, the LSP-CC could be solved in O(T2)

time. However, since there are O(T2) subplans the computation of the subplan

costs can be expected to be the bottleneck operation and the challenge is therefore

to develop efficient algorithms for finding these costs.

3.4.3 Computing the Subplan Costs

It is easy to see that all cumulative production arcs carry a positive flow that

is possibly equal to their capacity. Now first note that it is easy to determine

the flow on the cumulative production arcs that connect a pair of subplans in

an extreme point solution. In particular, if (t, r) is a subplan, we know that the

flow from node (C, t + 1) to node (C, t), i.e., the flow from the current subplan to

past subplans, should be equal to dot since It = 0 by the definition of a subplan,

regardless of the actual subplans present in the solution up to time t. Moreover,

since the cost on the cumulative production arcs is costless, it follows that we can

view the subplan (t, r) in isolation by redefining the capacity of the cumulative









production arc from node (C, s + 1) to node (C, s) to be


C Co- dot fors t + 1,..., .


(For convenience, we also define Ct 0.) Furthermore, we know that, in an

extreme point solution, there cannot be a cycle formed by free arcs alone. We

conclude that between each pair of production periods within a given subplan there

must be at least one cumulative production arc that is at capacity. However, since

the capacities of the cumulative production arcs are nondecreasing over time, we

may assume without loss of generality that the first cumulative production arc

between two production periods in a subplan is at capacity. The following theorem

characterizes the solutions that we need to consider for a given subplan.

Theorem 3.4.1 Consider a subplan (t, r) and let s be a production period in that

subplan. Furthermore, let s be the previous production period in the subplan (where

s = t if s is the first production period). The production i;,u,'il: in period s is then

equal to:

* dt Cf if s is the last production period in the subplan (which is only feasible
if do, < Cos);

* C' C' otherwise.

Proof: Let us first consider the case where period s is the last production period

in the subplan. Then the quantity produced in period s should bring the total

production quantity in the subplan to dtT. Since the cumulative production

quantity up to the previous production period s was binding, the quantity that

remains to be produced in period s is dtT Cg. (Note that if s = t we have that

C~ = Ct = 0 by definition so that the production quantity is dtT.) This is only

feasible if the total demand in the subplan can be produced by period s, i.e., if

dtT < Ct, Cos dot or, equivalently, if dor < Cos.









If s is not the last production period in the subplan, we use the fact that

we only need to consider extreme point solutions. Therefore, the cumulative

production arc entering nodes (C, s) and (C, s) are at full capacity. Therefore, the

quantity produced in period s should be equal to the difference between the two

capacities, C'8 Cl,. (Note that if = t we obtain that production in period s is

equal to C~,.) O


Figure 3-3 illustrates an extreme point solution with 2 subplans and the associated

arc flows

S Co Co d3 C04 do5
C, I ------ C,2 C,3 C-4 ------ C,5 C,6

C' d35 Cd 4
C01 dd3 -C01C
C04 d03 d05 C04


D,1 D,2 D,3 D,4 -- ,5
C01 -01 0C1 -02 C34 -d34
dl d2 d3 d4 d5

Figure 3-3: Extreme point solution with 2 subplans and the associated arc flows.


We will next formulate the problem of computing the minimum cost of a

subplan (t, r) as a dynamic programming problem. To this end, we define states

of the form (s, s, b) where t < s < s < -r + 1 and b E {0, 1}. The first element,

s, denotes the current period while the second element, s denotes the previous

production period in the subplan. The third element indicates whether production

has been completed for the subplan (b = 1) or not (b = 0). The source node

is (t, t, 0) while the sink node is (r + 1, 7 + 1, 1). For each state (s, s, b) with

t < s < 1 there are up to three potential decisions to be made: (i) do not

produce in period s + 1, (ii) produce in period s + 1 but do not complete all

production, and (iii) complete all production in period s + 1. This means that from

state (s, s, 0) the following states can be reached:









* (s + 1, s, 0): this means that we do not produce in period s + 1. The cost of
this decision is equal to
hs+l(Ct, dt,,+i)
since the production in the previous production period s was up to capacity.
Note that this decision is only feasible if the demand in period s + 1 can be
satisfied using past production, i.e., if C{, > dt,s+l.

* (s + 1, s + 1, 0): this means that we produce up to capacity in period s + 1. The
cost of this decision is equal to

C+1(t,+1 s) + h +l(C,s+1 dt,s+).

Note that this decision is only feasible if the demand in period s + 1 can be
satisfied using production up to period s + 1, i.e., if C>,+,1 > dt,s+l (since
otherwise ending inventory in period s + 1 would be negative). Moreover, we
only need to consider this decision if Ct,,+, < dt, (since otherwise period s + 1
should be the last production period in the subplan).

* (s + 1, s + 1, 1): this means that we make period s + 1 the last production
period in the subplan. The cost of this decision is equal to

Cs+l(dtT Ct) + hs+l(ds+1,T).

Note that this decision is only feasible if dt, < Ct,s+l, so that all demand of the
subplan can indeed be satisfied using production up to time s + 1.

From state (s, s, 1) with t < s < r 1, on the other hand, the only state that

can be reached is:

* (s + 1, s, 1): this means that we do not produce in period s + 1. The cost of
this decision is equal to
hs+ (ds+1,7).

From states ( 1, s, b), we can only go to states ('r, s, 1) and (r, 7, 1). Since

the period T is the latest period that the last production can occur. Finally, from

state (r, s, 1) (which means that we have satisfied all demands in the subplan with

production up to period s) we can reach state (r + 1, + 1, 1) at no cost. We are

now ready to prove the main result of this section:


Theorem 3.4.2 The LSP-CC can be solved in O(T4) time.









Proof: We first calculate and store the values dt, for 1 < t < r < T in O(T2)

time as well as the values Cot for 1 < t < T in O(T) time in a preprocessing step.

The dynamic programming formulation for computing the costs of a single subplan

has O(T2) states and a constant number of decisions per state. Since, using the

information obtained in the preprocessing step, the costs of all decisions can be

found in constant time, the cost of a single subplan can be found in O(T2) time.

The time required to find all subplan costs is therefore O(T4). Since the high level

dynamic program that employs the subplan costs runs in O(T2) time, the LSP-CC

with concave costs can be solved in O(T4) time. E


3.5 Allowing for Backlogging

In this section, we will generalize the LSP-CC to allow for backlogging. It

immediately follows that this generalization is NP-hard in general. In addition,

the results of Section 3.3.2 and van Hoesel and Wagelmans (2001) can be used to

develop a FPTAS for this generalization under the additional assumption that the

production and backlogging cost functions are concave (but the inventory holding

cost functions are only required to be nondecreasing). In this section we show

that the dynamic programming algorithm developed in Section 3.4 can be applied

with minor modifications, leading to an O(T4) algorithm for the LSP-CC with

backlogging.

We let ut be the quantity backlc.-.-. 1 in period t with associated cost function

bt (t = 1,..., T). We assume these backlogging cost functions are concave and,

without loss of generality, equal to zero when their argument is zero. In that case,

the mathematical programming formulation of the problem becomes:

T
minimize (ct(xt) + ht(It) + b(ut))
t= 1









subject to


It- + xt +ut dt + I + ut- t 1,...,T
t
E XT < Cot 7 1,...,T
T=1
Xt, It, Ut > 0 t= 1,...,T

UT = 0


where we have assumed without loss of generality that Io = uo = 0.

We can again view each extreme point solution as decomposed into a number

of subplans, albeit with a slight modification of the definition of a subplan. In

particular, we now define a subplan (t, 7) to be the consecutive set of demand

periods t + 1,..., T with the property that


It 0 and Ut 0

I > 0 or u > 0 s t+l,...,- 1

I = 0 and UT = 0.


The costs of the subplans can now be determined using a similar dynamic

programming approach as for the case without backlogging. The only modification

is that, whereas a decision was earlier deemed infeasible if it led to a negative

inventory level, such a decision is now feasible where the cost of negative inventory

is interpreted as the cost of the corresponding backl- -.-. 1 amount. We therefore

have the following result.

Theorem 3.5.1 The LSP-CC with backlogging can be solved in O(T4) time.

3.6 Summary and Directions for Future Research

In this chapter, we have studied an economic lot-sizing problem with a new

structure on the procurement capacities. While interesting in its own right, this

model can also be viewed as a special case of the two-level lot-sizing problem with









capacities at the first (procurement) level. We showed that the problem is NP-hard

in general and provided a fully polynomial time approximation scheme under mild

conditions on the cost functions. Moreover, we derived a dynamic programming

algorithm that runs in polynomial time when all cost functions are concave. This

is in contrast with traditional single and two-level capacitated economic lot-sizing

problems that are NP-hard even under fixed-charge cost functions and provides

an alternative set of conditions on the cost functions of the two-level lot-sizing

problem under which the problem is polynomially solvable than the ones derived by

Kaminsky and Simchi-Levi (2003). Interesting directions for future research include

the development of a FPTAS for a larger class of two-level lot-sizing problems as

well as the derivation of other classes of cost functions under which this problem is

solvable in polynomial time. Moreover, we want to consider the case where the raw

materials are perishable and have shelf life of 1 < f < T periods.















CHAPTER 4
CAPACITATED PRODUCTION AND SUBCONTRACTING IN A // SERIAL
SUPPLY CHAIN

4.1 Introduction

This chapter deals with a basic two-echelon supply chain consisting of a

supplier and a manufacturer, where a sequence of deterministic but nonstationary

demands of the manufacturer for a single good needs to be satisfied over a finite

horizon. In particular, we consider an integrated model that minimizes total

system costs, consisting of production, inventory, transportation, backlogging,

and subcontracting costs. This is is appropriate in cases where, for example, the

supplier is a captive subsidiary of the manufacturer. In each period, production

may take place at the supplier. Items are either stored there or transported to

the manufacturer level. Transported items are either used to satisfy outstanding

demands or stored at the manufacturer level. We assume that any unsatisfied

demand at the manufacturer is backl..---- .1 Production in each period is limited

by a finite but stationary capacity. Alternatively, units may be obtained by

subcontracting from an outside supplier, where the capacity of the outside

supplier may or may not be limited. Note that this subcontracting option

may be used regardless of whether the production capacity in a given period is

exhausted. However, we will show how our approach can be modified to allow

for an alternative sourcing option that can only be used if production is at full

capacity, and therefore can be interpreted as an overtime production option.

Our model falls within the general class of multi-echelon lot-sizing models

that has been an active area of research ever since the seminal paper of Wagner

and Whitin (1958). That paper considers an uncapacitated single-echelon model









for which an algorithm was proposed that runs in polynomial time, O(T2), in the

length of the planning horizon, T. More efficient algorithms, running in O(T log T)

or even O(T) time (depending on the cost structure) for this basic lot-sizing model

were developed by Aggarwal and Park (1990), Federgruen and Tzur (1991), and

Wagelmans et al. (1992).

Florian et al. (1980) and Bitran and Yanasse (1982) show that incorporating

nonstationary production capacities into this model makes it NP-hard in general,

even when production costs have a fixed-charge structure and holding costs are

linear. However, Florian and Klein (1971) and van Hoesel and Wagelmans (1996)

propose algorithms that run in O(T4) and O(T3) time, respectively, for the case

where the production capacities are stationary and inventory holding costs are

concave and linear, respectively. Chlun and Lin (1988) and van den Heuvel and

Wagelmans (2006) consider a capacitated lot-sizing problem with a particular cost

parameter and capacity structure and show that this problem is solvable in O(T2)

time.

Zangwill (1968,1969) was the first to study multi-echelon lot-sizing problems.

He characterizes the solution of such problems when costs are concave and uses this

structure to develop a dynamic programming algorithm to solve such problems. As

shown by van Hoesel et al. (2005), the running time of this algorithm is O(T3) for

the case of two echelons. Kaminsky and Simchi-Levi (2003) consider a two-echelon

serial supply chain in which production at the first echelon is capacitated. They

develop polynomial-time dynamic programming algorithms under several cost and

capacity structures. Van Hoesel et al. (2005) developed an improved algorithm for

the case of stationary capacities that allows for general concave cost functions and

runs in O(T7) time. In addition, they show that the running time can be further

improved when inventory holding cost functions are linear and transportation

cost functions have either a fixed charge structure satisfying a non-speculative









motives assumption (yielding a running time of O(T6)) or are linear (for a running

time of O(T5)). Now, we introduce the literature on multi-level lot-sizing models.

We want to give two examples for this situation. We may have a distribution

system, in which production occurs at the first level and then items are shipped

to the next levels. We can also have a manufacturing system in which products

are manufactured in a series of production facilities, each of which adds additional

value to the product. In Zangwill (1969), a dynamic programming algorithm

for a multi-level system without any capacities is proposed when backlogging

is not allowed. Van Hoesel et al. (2005) showed that this algorithm runs in

O(T3 + (L 2)T4), where L is the number of levels in the supply chain.

Kaminsky and Simchi-Levi (2003) propose a three-level model in which the

first and third levels are production stages, and the second level is a transportation

stage. Both production stages are capacitated, while the transportation stage is

incapacitated. They consider linear inventory holding costs that increase with the

level of the supply chain, and linear production costs at both levels 1 and 3 that

satisfy a traditional non-speculative motives condition. The transportation costs at

the second level are of the fixed-charge or general concave form and are assumed

to satisfy a restrictive and nontraditional nonspeculative motives condition. They

eliminate the third-level production decisions and reduce the problem to a two-level

model. For the fixed-charge transportation costs, they provide a polynomial time

algorithm that runs in O(T4) time for nonstationary production capacities. For

concave transportation costs, their algorithm runs in exponential time. However,

they consider stationary production capacities and their algorithm runs in O(Ts)

time.

For stationary production capacities case, van Hoesel et al. (2005) proposes a

polynomial time algorithm that runs in O(T7) time even when all cost functions

are concave. They also consider multi-level models and propose an algorithm that









runs in O(LT2L+3) time when all cost functions are concave and where L is the

number of levels in the problem. Note that this running time is polynomial in

the number of periods and exponential in the number of levels. However, when

inventory holding costs are linear and transportation costs either are linear or have

fixed-charge structure, they propose algorithms with running times polynomial

in the number of periods and insensitive to the number of levels. Their algorithm

runs in O(T7 + LT4) (O(T6) when L = 2) time when transportation costs

have fixed-charge structure and satisfy a form of non-speculative motives and in

O(T5 + LT2) time when transportation costs are linear.

They define subplan as the set of consecutive periods at each level, in which

demand at the last level are satisfied via the production at the first level using

the periods at intermediate levels as the transportation periods. In their solution

procedure, they employ the property that an optimal solution can be decomposed

into a sequence of consecutive subplans. Therefore, optimal solution value is sum

of some consecutive subplan costs. In the calculation of the subplan costs, we need

possible transportation and production quantities. They generalize the result of

Florian and Klein (1971) for production quantities. Morover they show that the

transported quantity between levels f and + 1 in some period either makes the

cumulative transported quantities so far in the subplan equal to the cumulative

production quantities of an initial sequence of consecutive production periods in the

subplan, or to the cumulative demand of an initial sequence of demand periods in

the subplan.

Now, we introduce the literature on multi-level lot-sizing models. We want to

give two examples for this situation. We may have a distribution system, in which

production occurs at the first level and then items are shipped to the next levels.

We can also have a manufacturing system in which products are manufactured in

a series of production facilities, each of which adds additional value to the product.









In Zangwill (1969), a dynamic programming algorithm for a multi-level system

without any capacities is proposed when backlogging is not allowed. Van Hoesel et

al. (2005) showed that this algorithm runs in O(T3 + (L 2)T4), where L is the

number of levels in the supply chain.

Kaminsky and Simchi-Levi (2003) propose a three-level model in which the

first and third levels are production stages, and the second level is a transportation

stage. Both production stages are capacitated, while the transportation stage is

incapacitated. They consider linear inventory holding costs that increase with the

level of the supply chain, and linear production costs at both levels 1 and 3 that

satisfy a traditional non-speculative motives condition. The transportation costs at

the second level are of the fixed-charge or general concave form and are assumed

to satisfy a restrictive and nontraditional nonspeculative motives condition. They

eliminate the third-level production decisions and reduce the problem to a two-level

model. For the fixed-charge transportation costs, they provide a polynomial time

algorithm that runs in O(T4) time for nonstationary production capacities. For

concave transportation costs, their algorithm runs in exponential time. However,

they consider stationary production capacities and their algorithm runs in O(T8)

time.

For stationary production capacities case, van Hoesel et al. (2005) proposes a

polynomial time algorithm that runs in O(T7) time even when all cost functions

are concave. They also consider multi-level models and propose an algorithm that

runs in O(LT2L+3) time when all cost functions are concave and where L is the

number of levels in the problem. Note that this running time is polynomial in

the number of periods and exponential in the number of levels. However, when

inventory holding costs are linear and transportation costs either are linear or have

fixed-charge structure, they propose algorithms with running times polynomial

in the number of periods and insensitive to the number of levels. Their algorithm









runs in O(T7 + LT4) (O(T6) when L = 2) time when transportation costs

have fixed-charge structure and satisfy a form of non-speculative motives and in

O(T5 + LT2) time when transportation costs are linear.

They define subplan as the set of consecutive periods at each level, in which

demand at the last level are satisfied via the production at the first level using

the periods at intermediate levels as the transportation periods. In their solution

procedure, they employ the property that an optimal solution can be decomposed

into a sequence of consecutive subplans. Therefore, optimal solution value is sum

of some consecutive subplan costs. In the calculation of the subplan costs, we need

possible transportation and production quantities. They generalize the result of

Florian and Klein (1971) for production quantities. Morover they show that the

transported quantity between levels and + 1 in some period either makes the

cumulative transported quantities so far in the subplan equal to the cumulative

production quantities of an initial sequence of consecutive production periods in the

subplan, or to the cumulative demand of an initial sequence of demand periods in

the subplan.

Returning to single-echelon lot-sizing problems, Zangwill (1969) studied

economic lot-sizing problems with backlogging and solved this problem in O(T3)

time when costs are concave, and in O(T2) time when the procurement cost

functions have a non-stationary fixed-charge component and a stationary linear

component. When inventory holding and backlogging costs are linear and the

linear component of the procurement costs are nonstationary, van Hoesel et al.

(1994) show that the running time of Zangwill's approach is still O(T2) and,

in addition, apply geometric techniques to solve this problem in O(T log T)

time or, under additional conditions on the cost components, in O(T) time.

Furthermore, Atamtfirk and Hochbaum (2001) consider a single-echelon lot-sizing

model with production capacities and uncapacitated subcontracting under various









cost structures, where the running time for the model with stationary production

capacities and concave costs is O(T5).

The main contribution of this chapter is with respect to our knowledge on

the complexity of and solution methods for concave-cost lot-sizing problems, in

particular two-level lot-sizing problems with stationary capacities at the first level.

Whereas to date the only models in this class for which polynomial-time solution

methods were known only allowed a single supply source and no backlogging,

in Sargut and Romeijn (2006b), we derive such methods in the presence of

backlogging at the second level as well as capacitated and uncapacitated outsourcing

or overtime production opportunities. In addition, we show how improved running

times can be obtained for several classes of more restrictive cost functions,

including cases where some of the cost functions are linear, have a fixed-charge

structure, or satisfy certain non-speculative motives conditions.

This chapter is organized as follows. In Section 4.2, we provide a mathematical

programming and a network flow formulation of our problem and outline our

overall solution approach. In Sections 4.3 and 4.4 we then develop polynomial-time

algorithms for subproblems that need to be solved repeatedly in our approach.

Section 4.5 deals with uncapacitated models, and we end the chapter in section 4.6

with a summary and some -i-i::. -I i..i for future research.

4.2 Problem Formulation and Solution Approach

4.2.1 Problem Formulation

We consider a serial supply chain consisting of a supplier and a manufacturer.

The manufacturer faces a deterministic stream of demands for a single item

over a horizon of T time periods, given by dt (t 1,..., T). The supplier can

satisfy manufacturer demands by production, but it faces a limited but stationary

production capacity in each period denoted by C. The supplier may hold items

in inventory for some periods before sending a shipment to the manufacturer.









We assume that the supplier has the option to satisfy each period's demand

early (which leads to inventory holding at the manufacturer) or late (which

corresponds to backlogging of manufacturer demand) at a cost. We will allow

for subcontracting of demands, where the subcontractor delivers directly to the

supplier. The supplier combines subcontracted and in-house production into

shipments to the manufacturer. This means that whether the supplier subcontracts

or not is transparent to the manufacturer. The quantity subcontracted in each

period is limited by B units.

Our decision variables and cost functions are:

* yt: supplier production in period t with associated cost function pt (t
1,. .. T );

* zt: subcontracting in period t with associated cost function st (t = 1,..., T);

* xt: transportation from supplier to manufacturer in period t with associated
cost function ce (t = 1,..., T);

* I) : inventory level at the end of period t at the supplier level ( = 1) or
manufacturer level ( 2) with associated cost function hg) ( = 1,2;
t= 1,. ,T);

* Ut: quantity backl.. .-: from period t + 1 to period t with associated cost
function bt (t = 1,... T);

Our objective is to minimize system-wide production, subconl I ir_

transportation, inventory holding, and backlogging costs over the entire planning

horizon. We may then formulate the problem as a mathematical programming

problem as follows:
T
min 1 (st(zt) + pt(t) + ct(Xt) + h( )) + h(2)(2)) + bt(ut))
t 1









subject to (P)


it < C t= 1,...,T (4-1)

zt < B t 1,...,T (42)

yt+ zt+lI = xt + t ) t 1,...,T (4-3)

Xt + I)1 + u = dt + It(2) Ut- t ,...,T (44)

I) = 0 1,2 (4-5)

Ut = 0 t 0,T (4-6)

yt, xt, Ut > 0 t= 1,...,T

zt > 0 t 1,...,T

It > 0 1,2; t 1,...,T.


Constraints 4-1 and 4-2 enforce the production and subcontracting capacity

constraints in each period. Constraints 4-3 and 4-4 represent the flow balance

constraints at the supplier and manufacturer level, respectively. Constraints 4-5

and 4-6 model the initial inventory and terminal backlogging conditions. Note

that the latter ensures that all manufacturer demand is satisfied during the

planning horizon. All but the assumption that () = 0 are made without loss

of generality (and can be handled in a straightforward manner by updating the

demand sequence as in single-level lot-sizing problems). We refer to van Hoesel et

al. (2005) for a discussion on how to deal with initial inventories at the supplier

level. This approach can, using the algorithms developed in this paper, be extended

to our models in a straightforward manner.

Throughout this paper, we will assume that all cost functions are nonnegative,

concave, ri. iidI. i ,-;i- and equal to zero when their argument is zero. However,

in addition to the case of general concave cost functions, we will also study the

algorithmic implications of two other cost structures.









Note that our model can be formulated as a concave cost network flow problem

as illustrated in Figure 4-1. In this network, we have a source that supplies the

total manufacturer demand t 1 dt. Costless arcs distribute this supply over

a production (node P) and a subcontracting (node S) source. The nodes are

indicated by (f, t), where denotes the level in the chain and t the period. This

formulation immediately implies that there exists an extreme point solution to

our problem, i.e., a solution that is a vertex of the polytope that forms the feasible

region of (P) as long as a feasible solution exists, i.e., as long as

T
dt < T(C+ B).
t=1

Our solution approach, which generalizes the approaches of Kaminsky and

Simchi-Levi (2003) and Van Hoesel et al. (2005), will be based on this result

and is discussed in the next section.




P S




1,1 1,2 1,3 1,4 1,5


2,1 2,2 2,3 2,4 2,5



Figure 4-1: Network representation with inventory holding and backlogging.


4.2.2 Solution Approach

Extreme points in minimum cost network flow problems have the property

that there does not exist a cycle formed by free arcs, i.e., arcs that carry a strictly

positive flow that is also strictly below capacity (see A!mli et al. 1993). Previous

work has used this observation to show that any extreme point solution to special









cases of our problem containing only production decisions (see Florian and Klein

1971), production and subcontracting decisions (see Atamttirk and Hochbaum

2001), or production and transportation decisions without backlogging (see

Kaminsky and Simchi-Levi 2003 and Van Hoesel et al. 2005) decomposes into a

sequence of consecutive so-called subplans that contain at most one free production

arc, i.e., at most one production arc with flow strictly between its lower bound of

0 and upper bound of C. This corresponds to the fact that two free production arcs

in a subplan would form a cycle of free arcs, which contradicts the extreme point

nature of the solution.

We will extend this characterization of extreme point solutions to our more

general class of problems. Consider all free arcs in a given extreme point solution.

Since all inventory, backlogging, and transportation arcs are uncapacitated, when

we ignore all production and subcontracting arcs the set of remaining free arcs

decomposes into disjoint connected and .,. i1 .i components. The fact that we

have an extreme point solution then immediately implies that each component

may be connected to at most one free production or subcontracting arc. It is

easy to see that each component that is not a singleton node at the supplier level

contains at least one node at the manufacturer level. In a given component, we

denote the the first and last nodes at the manufacturer level by (2, r1 + 1) and

(2, 72), indicating that in this component the demands of periods r- + 1,..., 2

are satisfied. In addition, let (1, tl + 1) and (1, t2) denote the first and last nodes

at the supplier level, so that the component can be characterized by (tl, t2a '-, 72).

By definition, we have that tl < t2 and ri < r2. Moreover, the fact that the

component is connected implies that r1 < t2. Now suppose that r1 < ti, i.e., the

demand of period ri + 1 is satisfied, at least partially, through backlogging. If, in

the current solution, production or subcontracting to the supplier takes place in

one of the periods -r + 1,..., tl 1, this should satisfy demands up to and including









period T7. However, this would contradict the fact that the components (when

disregarding production and subcontracting arcs) are di-j Piiil Therefore, we can

include the nodes (1, 7r + 1),... (, t 1) in the current component, which means

that without loss of generality we may assume that tl < Tr. Similarly, suppose

that T2 < t2, i.e., production takes place after the last period whose demand is

satisfied in the current component. However, this means that the demands in

periods 72 + 1,..., t2 should also be satisfied within the current component, so that

without loss of generality we may assume that t2 < 72. Combining both results,

we denote the subplans that comprise an extreme point solution as (tI,, t2,1 72)

with tl < 7- < t2 < 72. Note that, somewhat surprisingly due to the presence of

backlogging in our model, these are precisely the subplans that are obtained when

only considering production decisions (see Van Hoesel et al. 2005).

The characterization of extreme points through subplans immediately leads to

the following backward dynamic programming recursion:


F(ti,Ti 7) = min { (t, 2, 71, 72) + F(t2, 72)
O for all t1 = 0,..., TI; T7 = 0,..., T 1.

F(tI,T) = 0 for all /1= 0,...,T


where F(ti, -i) denotes the minimum cost required to satisfy the demands in

periods -r +1,..., T with production or subcontracting only in periods t +1,..., T,

and Q(tI, t2, 71, 72) denotes the minimum cost associated with subplan (ti, t2, -1, 72).

That is, 1(t1, t2, -1,r72) is the minimum cost required to satisfy the demands in

periods 1 + 1,.... 2 with production or subcontracting only in periods t + 1,... t

using at most one production or subcontracting quantity that is strictly between its

lower bound of zero and its capacity. Clearly, the optimal solution to the problem

is given by F(0, 0).









It is easy to see that the dynamic programming recursion can be solved in

O(T4) time if the costs of all subplans are known. Since there are O(T4) subplans,

the computation of these costs can be expected to be the bottleneck and the

challenge is therefore to develop algorithms to efficiently find the costs of the

subplans.

4.2.3 Only Backlogging at the Manufacturer Level

We have assumed so far that inventory can be held both at the supplier and at

the manufacturer level. However, in certain settings the manufacturer may not be

able to accommodate any early deliveries due to limited storage space, for example

when the manufacturer is a supermarket or restaurant. If, on the other hand, the

manufacturer is willing to accept late deliveries, we retain the backlogging option

in the model and the backlogging costs can then be interpreted to include some

compensation to the manufacturer. In that case, the mathematical programming

formulation of the problem reduces to:

T
min (s,(zL) + pt(yt) + c() + h((It) + b6,(u))
t=1

subject to


Yt < C t= ,...,T

zt < B t 1,...,T

yt + zt+ = ( Xt + ~) t =,... ,T

t + t = dt + ut-1 t 1,...,T

o(1) 0

UT 0

Yt, xt, ut > 0 t 1,... ,T

Zt > 0 t= 1,...,T

() > 0 t= 1,...,T









and the network representation is given in Figure 4-2.




P S




1,1 1,2 1,3 1,4 1,5


2,1 212 2,3 2,4 2,5


Figure 4-2: Network representation with backlogging only at the manufacturer.


Of course in this case the extreme points decompose again into subplans of

the form (ti, t2, -1,) with tl < T1 < t2 < T2. However, since only backlogging

is allowed at the manufacturer, the demand in period tl + 1 can only be satisfied

through products transported in periods t1 + 1 or later. Therefore, without loss of

generality we may assume that demand periods tl + 1,..., -1 1 are included in

the subplan or, in other words, that 71 = tl. Similarly, the demand in period 72

can only be satisfied through products transported in periods 72 or later. Therefore,

without loss of generality we may assume that production and subcontracting

periods t2 + 1,..., T~ are included in the subplan or, in other words, that 72 t2.

Thus, we only need to consider subplans of the form (T-, -2, -1, 72) which, where

convenient, we will simply denote by (-1, 7r).

The reduction in the number of subplans that needs to be considered directly

leads to the following dynamic programming formulation for solving (P')


F'(i) min (T, + F 2) for all1 T 0,..., T 1.
72 =- I+ ,...,T
F'(T) 0


where F'(r-) denotes the minimum cost required to satisfy the demands in periods

T- + 1,... T with production or subcontracting in the same periods, and -'(ri, 72)









is the minimum cost required to satisfy the demands in periods Tr + 1,..., T2 using

production or subcontracting only in periods ri + 1,..., T2 and using at most one

production or subcontracting quantity that is strictly between its lower bound of

zero and its capacity. The optimal solution to the problem is given by F'(0).

This dynamic programming recursion can be solved in O(T2) time, again

assuming that the costs of all subplans are known. We will develop efficient

algorithms for computing the subplan costs for this case.

4.3 Models without Subcontracting Opportunities

4.3.1 Subplan Properties

Consider a subplan with demand periods ri + 1,..., 2. For convenience, we

will in the remainder of this chapter define the cumulative demand in a sequence of

periods t,..., s by dts (where, by convention, we let dts = 0 if t > s), so that the

total demand in the subplan is given by d, +1,2.

Since, as we have seen above, each subplan has at most one free production

arc, we may generalize a result of Florian and Klein (1971) and determine the

number of periods in a subplan where production is at full capacity as


K = d +1, (4-7)

and the quantity produced in the remaining production period as

c = d 1,+, KC. (4-8)


Note that, by construction, 0 < c < C.

The absence of cycles formed by free arcs also enables the characterization of

the potential quantities that can be transported in any period of a subplan. The

result obtained by Van Hoesel et al. (2001) for the case where backlogging is not

allowed applies directly to our model as well. In particular, if there is a shipment









in period t in a subplan (tl, -, 1,-72), the quantity shipped satisfies one of the

following conditions:

* it results in zero ending inventory at the supplier level in period t; or

* the cumulative quantity shipped in the subplan up to and including period t is
sufficient to fully meet the demand of periods r~,,..., s for some s = t,..., -2.

This follows since otherwise there would either be a period (t or later) whose

demand is satisfied partially from the transported quantity in period t and partially

from the ending inventory at the supplier level in period t or both the transported

quantity in period t and the ending inventory at the supplier level would be

used to supply demands in period t or earlier. Even though in the latter case the

transported quantity and the ending inventory do not necessarily satisfy demand

in the same period, both cases lead to a cycle of free arcs and therefore contradict

that the solution is an extreme point.

4.3.2 Inventory Holding and Backlogging at the Manufacturer Level

In this section we will develop efficient algorithms to compute the subplan

costs in the absence of subcontracting but in the presence of both inventory holding

and backlogging at the manufacturer level.

Concave costs. Recall that a typical subplan is of the form (t1, t2, -1, 2)

where tl < <7 < t2 < r2. To calculate the minimum cost of this subplan we

will generalize the dynamic programming approach developed by Van Hoesel

et al. (2001) for the case where backlogging is not allowed. The states in this

formulation are of the form (t, y x), where t E {ti,..., t2} is the current period,

yc E {kC + v : k = 0,..., K; v = 0, 1} is the cumulative production quantity in the

current subplan, and x' E {kC+v : k = 0,...,K; v 0, 1}U{dl+i,,s : s= t,... ,T2}

is the cumulative transported quantity in the subplan. Moreover, we should have

xc < yc since products cannot be transported before they are produced. Since

K < T there are O(T3) states.









The decisions that can be made in state (t, yC, x) leading to state (t + i, y, x)

are:

* produce y' yC units; the production quantity may equal c or C units, where
the former is only allowed if yC = kC for some integer k and the latter is only
allowed if yc < KC, or zero.

* transport xc xc units; the transportation quantity may equal zero, the total
available quantity yc xc, or d,,+1,s xc for some s = t + 1,..., for which
dT-+1, > xC.

The total cost of the decision to transition from state (t, yc, xc) to state

(t+l, y, xc) is equal to


Pt+1 (9c c) +C t+1 _c c_ +1,t+1)+) +bt ((d7-,+It+l x )+)
Pt+i c-y -c+Ct+ .-xc+h(') -xc+h ((c-

The minimum cost of subplan (tl, t2, 7,72) is now given by the minimum cost

required to, starting at state (t, 0, 0), reach state (t2, d7-+1,2, d,-+1,2) (at which

point all demands in the subplan have been satisfied).

Since, in each state, there are at most O(T) candidate decisions and the cost

of each decision can be calculated in constant time the total time required to find

the costs of a given subplan is O(T4) (where we assume that we have computed the

cumulative demands dt, in a preprocessing step taking O(T2) time). The cost of all

O(T4) subplans can therefore be determined in O(Ts) time. However, solving the

dynamic program for subplan (0, t2, T-, -72) using a backward recursion we in fact

obtain the costs of all subplans (t1, t2, r, 72), thereby reducing the total running

time to O(T7). (See also Kaminsky and Simchi-Levi 2003 and Van Hoesel et al.

2001 for a similar savings technique.)

In the next two subsections, we will discuss two sets of cost functions under

which the costs of the subplans can be determined more efficiently. In neither

case will we further restrict the production cost functions. However, we will

assume that the holding and backlogging cost functions are linear functions.









Furthermore, we will assume that the transportation cost functions consist of a

fixed-charge and a linear component and, in addition, that the linear transportation

cost component together with the holding and backlogging cost functions satisfy

a particular type of non-speculative motives condition. We will also assume

that the transportation cost functions are linear but do not necessarily satisfy

that non-speculative motives condition. With a slight abuse of notation we will

represent the unit transportation, inventory holding, and backlogging costs by ct,

h(), h 2), and bt (t 1,..., T), respectively.

Fixed-charge transportation costs under a non-speculative motives

condition. In addition to the above assumptions on the cost functions, we consider

the following non-speculative motives condition:


ct + h(2) < h(l) + t for t 1,..., T- 1. (4-9)


Under this condition, the subplan solutions will enjoy an attractive structure that

can be employ, l to efficiently find the optimal subplan costs. In particular, we

have the following result:

Proposition 4.3.1 In I, :, period in which we transport, we should transport all

available inventory at the supplier level, i.e., we i,,i'; restrict ourselves to solutions

in which xi1) = 0 for t = 1,...,T.

Proof: Suppose that we have a period t such that xt > 0 and I(1) > 0, and

suppose that some of the ending inventory in period t, a quantity 6 > 0, is

transported in period t' > t. Then consider an alternative solution in which we

instead transport this quantity in period t. This leads to a change in total costs

equal to
t- 1 t- 1

c r=t ) =t









However, summing condition 4-9 over periods t,..., t' 1 yields:

t-1 t'- 1
C + h) < (h() + c,+l)

which implies that

ct+Zh( ) < +h(1 c,.
r=t 1 r=t

Therefore, the alternative solution is at least as good as the original solution.

We may repeat the argument until we obtain a solution that satisfies the desired

property. O


Using this result, we may now modify the dynamic programming approach

as follows. First, the note that the property implies that we only need to consider

states of the form (t, y", xc) with x' C {kC + ve : k = 0,..., K; v = 0, 1};

unfortunately, this means that the size of the state space remains O(T3). However,

we now only need to consider transitions between states (t, yc, x) and (t + 1, y, xc)

if xc E {x, yc}. This reduces the number of decisions in each state to 0(1) and

thereby the time required to find the cost of a given subplan to O(T3), which, after

employing the same savings technique as before, yields a total running time of our

algorithm under this cost structure of O(T6).

Linear transportation inventory holding and backlogging costs. In

this section, we will show how we can determine the costs of the subplans more

efficiently when the production costs are concave but all other cost functions

are assumed to be linear. With a slight abuse of notation we will represent the

unit transportation, inventory holding, and backlogging costs by ct, h,), and bt,

respectively.

We again calculate the minimum cost for each subplan using dynamic

programming. However, we may now choose the state to be of the form (t, y)

where t is the current period and ye0 {kC + ve : k = 0,..., K; v = 0, 1} is the









cumulative production quantity in the current subplan. Since K < T there are

O(T2) states. This reduction of the state space follows from the linearity of the

inventory I .1 lii:- backlogging, and transportation costs, which make it possible to

determine the optimal cost associated with the quantity produced in a given period

based on only the cumulative production quantity before the current period.

The decisions that can be made in state (t, yc) are: produce c or C units,

where the former is only allowed if yc = kC for some integer k and the latter is

only allowed if yc < KC, or do not produce at all. Denote the state that is reached

next by (t + 1, yc). The cost of this decision consists of the following components:

* the cost of producing yc yc units in period t + 1 at a cost of pt+((yc y);

* the transportation, inventory I '.1 lii:- and backlogging cost of the yc yc units
produced in period t + 1 to the manufacturer according to the FIFO principle.
In order to compute these costs, let and denote the first period in the
subplan whose demand is not fully satisfied by the first yc and yc units of
production, respectively, and let 6 and 6 denote the amounts in periods and
respectively, that remain to be satisfied. Note that (f, 6) can be determined
for each state in the subplan in a preprocessing step taking O(T2) time. Then,

* if yc yc < 6, i.e., production in period t is fully used to satisfy demand in
period the total costs are Gt+l,(yc gc), where Gt+l,e is the minimum cost
of supplying a unit of demand in period from the supplier level in period
t+1.

* if y0 y0 > 6 the total costs are given by the cost of shipping y0 y0 units to
the manufacturer according to the FIFO principle:

Gt+l,,6 + GDt+l,'+l,_-1 + Gt+1,,(d 6)

where GDte+l,+l,_1 is the minimum cost of supplying the demands of periods
+ 1,... ,- 1 from the supplier level in period t + 1, i.e.,

-1
GDt+l,g+l,- 1 drGt+l,r-
r= +l

Note, first of all, that the quantities Gt+l,e and GDt+i,t+,i-1 can be computed

in a global preprocessing step (i.e., independent of the subplans) in O(T3) time.









Note also that the arc costs may, in some cases, be based on shortest paths that do

not fully lie within the subplan under consideration. However, this is no problem

due to the linearity of the costs. With this information, it is clear that there are

constant number of decisions and the costs of each decision can be determined in

constant time, so that it immediately follows that the costs of a subplan can be

found in O(T2) time by finding the minimum cost required to go from state (ti, 0)

to state (t2, d,1+1,2,). Since once again it suffices to consider O(T3) subplans, the

time required to find all subplan costs is ((Ts).

4.3.3 Model with only Backlogging at the Manufacturer Level

In this section we will develop efficient algorithms to compute the subplan

costs in the absence of subcontracting options and when only backlogging is

allowed at the manufacturer level.

When all cost functions are concave we can again apply a dynamic programming

approach with states of the form (t, yC, xc), where t and y' are as before, but we

now have that x cE {kC + ve : k = 0,..., K; v = 0, 1} U {dr,+l,t} and xc < d-+li,t

since a shipment in period t cannot satisfy the demand of any periods beyond t.

Similarly, a decision to transport xc xc units can only be made for transportation

quantities equal to zero, the total available quantity yc xc, or d,-+i,t+l xc, where

the latter is only allowed if dl+1,t+l > xc. Since, in each state, there are now 0(1)

candidate decisions the total time required to find the costs of a given subplan is

O(T3) time. (Note that this immediately implies that we will not a computational

savings here, since the number of candidate decisions will still be 0(1).) The cost

of all O(T2) subplans can therefore be determined in O(T5) time.

In case the transportation, inventory I'.. lii:- and backlogging costs are linear,

the approach of holding and backlogging case holds without modification. Thus,

the cost of all O(T2) subplans can be computed in O(T4) time in this case.









4.3.4 Evaluation

First, note that in all cases the running time required to find the costs of all

subplans dominates the running time of the higher level dynamic program that

finds the optimal solution to an instance of the problem. The running times of the

complete algorithms for solving all our model variants are summarized in Section

4.6. In the remainder of this section, we will compare the causes of the differences

between the running times for the different models. Identifying these differences

then enables us to focus on a limited set of models in the remainder of the C'! lpter.

Let us first compare the running times of the models with and without

inventory holding at the manufacturer discussed above when all cost functions

are concave. When holding inventory at the manufacturer is not allowed, the

running time reduces because of two different effects. Firstly, the number of

subplans that need to be considered reduces from O(T3) to O(T2). Secondly, in

the dynamic programming approach to computing subplan costs the number of

potential shipment quantities in each period reduces from O(T) to 0(1). Together,

this causes a reducing in running time of O(T2). However, for the other two cost

structures only the first effect applies, i.e., the only savings are due to the reduction

in the number of subplans that need to be considered by O(T).

Finally, let us compare the models with respect to the different cost

structures. When considering models with linear transportation, inventory holding,

and backlogging costs to the others, we achieve a savings in the running time

since, when computing subplan costs in the former case, only the cumulative

production up to time t is important to determine how best to proceed. This

means a reduction in the size of the state space by O(T). Moreover, when both

inventory holding and backlogging are allowed at the manufacturer level, the

number of decisions in each state reduces from O(T) to 0(1) when we move









from general concave cost functions to one of the other cost structures, for a total

savings of O(T2).

These observations extend to the more complex models that we will study

next. Therefore, in the remainder of this chapter we will only consider models

where both backlogging and inventory holding is allowed at the manufacturer level

and all cost functions are concave, and simply infer the running times for the other

cases from the discussion above.

4.3.5 Managerial Insights

The study of multi-echelon lot-sizing problems in this section does not only

show that these problems are solvable in polynomial time, but also provide insights

into the structure of production and transportation plans that lead to high-quality

solutions. These insights results generalize the well known zero-inventory ordering

policy that follows from a study of the solution structure of the well known

uncapacitated single-echelon lot-sizing problem. Besides the practical importance

of such insights, note that these results can sometimes also be used to develop

high-quality efficient heuristics for more complex lot-sizing problems (see, e.g.,

C'!I io et al. 2002a, 2002b.
We will here summarize the most important structural insights that have been
obtained:

* Whenever economies-of-scale in costs are present, any quantity transported
between supplier and manufacturer should either leave no inventory at the
supplier or be sufficient to satisfy demand in an integer number of planning
periods.

* Whenever a non-speculative motives between linear holding and backlogging
cost functions and a linear component in fixed-charge transportation cost
functions are satisfied, any quantity transported between supplier and
manufacturer should leave no inventory at the supplier.

* Whenever I. l1ii1 backl.:_ _:i_ and transportation cost functions are linear,
any unit produced should satisfy demand according to a combination of the
FIFO rule and minimum total unit holding, back. _._il._ and transportation
costs.









4.4 Models with Subcontracting Opportunities

We next turn to models in which uncapacitated or capacitated subcontracting

opportunities exist. In either case, we will assume that regular production is

capacitated. Note that the model in which both production and subcontracting are

uncapacitated are discussed in Section 4.5, while the model in which production

is uncapacitated and subcontracting capacitated can be handled by simply

interchanging the role of production and subcontracting.

4.4.1 General Properties of the Subplans

From the analysis in Section 4.2.2 we know that each extreme point decomposes

into a number of subplans, each of which has at most one free production arc or at

most one free subcontracting arc, but not both. If we have a free production arc in

the subplan, it is easy to see that all subcontracted quantities are equal to 0 or B.

Similarly, if we have a free subcontracting arc in the subplan then all production

quantities are equal to 0 or C.

Now consider a subplan with demand periods ri + 1,..., -2. We know that

there are at most

K dl12 and K'" L d| 2
C c I I B
periods in which we produce or subcontract to full capacity, respectively. However,

the fractional production (or subcontracted) quantity depends on the actual

number of periods in which we subcontract (or produce) to full capacity.

First, suppose that we know that there are exactly k (where k = 0,..., K)

periods in which we produce to full capacity and there is no fractional production

period. Then the number of periods in which we subcontract to full capacity and









the quantity subcontracted in the fractional supply period are equal to


K L d1+1, kCJ
"s d+l,, kC- "B


respectively. Note that, in case of uncapacitated subcontracting (B = oo), these reduce to
K, 0


kf = d7-+1,7 kC.


Similarly, in case there are exactly k' = 0,..., K periods in which we

subcontract to full capacity and there is no fractional subcontracting period, the

number of periods in which we produce to full capacity and the quantity produced

in the fractional supply period are equal to

S d +l1,T2 k'B

c,= dr- +1 ,2 k'B ,k' C.


The minimum subplan cost is then equal to the minimum cost among the solutions

found with a fractional production or a fractional subcontracting period. In the

next sections, we will develop algorithms for the case of a fractional subcontracting

period. Note that, as before, we focus here on the running times required to find

the subplan costs. The running times of the complete algorithms for solving all our

model variants are summarized in Section 4.6.

4.4.2 Concave Costs

Capacitated subcontracting. Our approach is to find the subplan costs

under the condition that there are k full capacity production periods and no

fractional production period. Repeating this procedure for each of the O(T) values

for k as well as executing the analogous procedure for the case of k' full capacity

subcontracting periods and no fractional subcontracting period and finding the one









with smallest cost then yields the actual subplan costs. In this section, we will

only describe the former procedure in detail. The latter procedure can be obtained

by simply interchanging the role of the production and subcontracting options.

Consider subplan (tl, t2, -1, 72). Similar to the problem without subcontracting

we use states of the form (t, yzc, xc), where t E {tl,..., t2} is the current period,


yzc E {kC + k2B + ,, : ki 0,...,k; k2 0,...,k; v 0, 1}


is the cumulative production and subcontracted quantity in the current subplan,

and


xc E {klC+k2B I ., : k= 0,...,k; k2= 0,..., Kk; v= 0, 1}U{d,+l,s: s t,..., r2


is the cumulative transported quantity in the subplan. As before, we should have

xc < yzt since products cannot be transported before they are produced. Since

k, Kk < T there are O(T5) states.

The decisions that can be made in state (t, yzc, xc) leading to state (t +

1, yz, x) are:

* produce and/or subcontract a total of yzc yze E {0, C, Ck, B, C + Ck, C + B}
units, where, in addition, yzc and yzc yield feasible states;

* transport x' xc units; the transportation quantity may equal zero, the total
available quantity yz 7 xc, or d,,+l,s xc for some s = t + 1,..., rT for which
dTl+1, > xC.

The total transportation, inventory holding, and subcontracting cost of the

decision to transition from state (t, yzc, xc) to state (t+l, yz, 7c) is equal to


ct+1( x) + h1(yz + ) ((xc dl+1,t+l)+) + bt ((dl1,t+l xc)+)


If yzc yze = C, a production cost of pt+l(C) is added; if yzc yzc 1k, a

subcontracting cost of st+1(Ck) is added; if yzc yze = B, a subcontracting cost

of st+l(B) is added; if yzc yze = C + Ck, a production and subcontracting









cost of pt+l(C) + st+l(ck) is added, and if yzc yz = C + B a production and

subcontracting cost of pt+,(C) + st+l(B) is added. (Note that we have assumed,

for ease of exposition, that B / C and Ck / C. If this is not the case, we may

either expand the state space with a variable indicating whether production and/or

subcontracting took place in period t + 1 or simply choose the assignment of

quantities to production and subcontracting that lead to minimum cost, neither of

which would increase the running time of the algorithm.)

The minimum cost of subplan (t1, t2, 1 -72) (for the given k) is now given by

the minimum cost required to, starting at state (ti,0, 0), reach state (t2, d-1+1,72, d1+1,72)

(at which point all demands in the subplan have been satisfied).

Since, in each state, there are at most O(T) candidate decisions and the cost

of each decision can be calculated in constant time the total time required to find

the costs of a given subplan for a given k is O(T6). Since there are O(T) choices

for k and since the time required to find the minimum subplan costs for the case of

a fractional production period is of the same order, the cost of all O(T3) (relevant)

subplans can be determined in O(T10) time.

Uncapacitated subcontracting. When there is no subcontracting capacity,

we distinguish between subplans whose cost is attained using production alone and

subplans whose cost is attained using subcontracting. For any subplan, we can of

course compute the minimum cost without using subcontracting as in Section 4.3.2.

If we then also compute the minimum cost with subcontracting the cost of the

subplan is obviously given by the least costly solution among these two candidates.

Note that, if there is a period in which subcontracting takes place, there

will be no free production period. To compute the minimum cost of a subplan

(tl, t2, 7-1 72) under this constraint, we again consider a dynamic programming
approach with states of the form (t, yz', xe). Defining K and c as in Equation s

4-7 and 4-8, this state contains the current period t E {tl,..., t2}, the cumulative









production and subcontracted quantity in the current subplan yz' E {kC + v :

k = ,..., K; v = 0, 1}, and the cumulative transported quantity in the subplan

xc E {kC + ve : k = 0,..., K; v = 0, 1} U {d,+l, :s = t,... 2}. As before,

we should have xc < yzc since products cannot be transported before they are

produced. It is easy to see that there are O(T3) states.

The decisions that can be made in state (t, yzc, xc) leading to state (t +

1, yz, x) are:

* produce and/or subcontract yzc yze E {0, c, C, C + e,..., KC + e} units,
where, in addition, yzc and yzc yield feasible states;

* transport xc xc units; the quantity transported may equal zero, the total
available quantity yzc xc, or d,,+l,s xc for some s = t + 1,..., T2 for which
dT+l1,s > x.
The total transportation, inventory holding, and subcontracting cost of the

decision to transition from state (t, yzt, xc) to state (t+l, yzt, xc) is equal to

ct+( xc) + hQ()yz xc) + h(21 ((xc dl,) + b, ((dl, 1 xc))
Ct+ (1) c c 1( c _7+l,t+l)+) + bt ((d7_+l,t+l )+) .

If yzc yze = c, a subcontracting cost of st+l(c) is added; otherwise, a cost of

min {st+(yz yzc),pt+,(C) + st+(yz yz C)}

is added.

Since there are O(T3) states and O(T2) potential decisions per state, we can

determine the cost of a subplan in O(T5) time. The costs of all subplans can thus

be found in O(Ts) time.

4.4.3 Subcontracting versus Overtime Production Option

As mentioned in the introduction, it may sometimes be appropriate to only

allow the use of the additional sourcing option in a given period if production is

at full capacity. In particular, this would be appropriate if the additional source

is in fact not an outside supplier but a possibility of using overtime production at









an additional cost. In terms of the optimization model (P), this can be handled

by applying the production cost function pt to all units procured in period t and

interpreting the subcontracting cost function st as an incremental cost function.

That is, the term pt(yt) in the objective function is replaced by pt(yt + zt). An

appropriate network representation of this situation is given in Figure 4-3. Here the

arcs of the form (P, t) are costless, arcs of the form (S, t) have cost function st, and

arcs of the form (t, (1, t)) have cost function pt. From this network representation,

we can immediately conclude that, without loss of optimality, we may restrict

ourselves to solutions in which Zt > 0 implies that yt = C. In addition, an analysis

of the extreme point structure of optimal solutions yields that we again have that

in each subplan there is either at most one period such that 0 < yt < C or at most

one period such that zt > 0.




P S




1 2 3 4 5


1,1 1,2 1,3 1,4 1,5


2,1 2,2 2,3 2,4 2,5



Figure 4-3: Network representation with overtime production.


Now consider a subplan with demand periods Tr + 1,..., T2. We know that

there are at most

KP L d= | l and KP+ = I d1
SC C+B B









periods in which we only produce or produce and subcontract to full capacity,

respectively. However, the fractional production (or subcontracted) quantity

depends on the actual number of periods in which we subcontract (or produce) to

full capacity.

First, suppose that we know that there are exactly k periods in which we

produce to at least full capacity and there is no fractional production period. Since

we can only use subcontracting in periods in which production is at full capacity,

we only need to consider k = KP+,..., KP. Then the number of periods in which

we both produce and subcontract to full capacity and the quantity subcontracted

in the fractional supply period are equal to

p+s Ld ,+1, B kCJ

+s8 d ,,d kC lB

respectively. Note that, in case of uncapacitated subcontracting (B = oo), these reduce to

p+s 0
=0

CP+8 7-dl+1,72 W


We may now proceed to determine the subplan costs in a similar way as before,

while ensuring that any state transition only uses subcontracting when production

is at full capacity.

Similarly, we can consider cases where there are exactly periods in which we

produce and subcontract to full capacity and there is no fractional subcontracting

period. Since we can only use subcontracting in periods in which production is at

full capacity, we only need to consider k' = 0,..., KP+. The number of periods

in which we produce to full capacity and the quantity produced in the fractional









supply period are equal to

d,1+1 ,72 k' B
K Lk' k'JC

ep,/ dr-+1l,2 k'B K, C.



The running times of models in which subcontracting can only be used

if production is at full capacity is therefore the same as for models in which

subcontracting may be used regardless of production quantity.

4.5 The Uncapacitated Case

Zangwill (1969) shows that for multi-level lot-sizing problems without

production capacities, subcoill i Iiir. or backlogging an extreme point solution is

arborescent, which means each node has at most one incoming arc with positive

flow. As the consequence of this arborescent structure, a so-called zero inventory

ordering property (ZIO) is conserved in any extreme point solution. The following

theorem extends this result to the uncapacitated two-level lot sizing problem with

backlogging and subcontracting.

Theorem 4.5.1 If production is uncapacitated then an extreme point solution has

the following properties:


I~ t+l4 0, I,)t+ =1 0, UtXt 0 for t 1,...,T- 1


Ztyt 0 for t 1,..., T.

Proof: Any extreme point solution does not include a cycle of free arcs (i.e., arcs

that carry a positive flow). It is easy to see that, if any of the pairs stated in the

theorem has positive value, the corresponding arcs create two paths using free arcs

only from the source to a node in the network and therefore a cycle of free arcs. o


This result immediately leads to the following corollary.









Corollary 4.5.2 In an extreme point solution each arc carries the demand of a

consecutive set of periods.

Now note that in no period can we both produce and subcontract a positive

amount. Therefore, subcontracting can be accounted for in a straightforward

manner by redefining the production cost function as:


Pt(yt) min pt(yt), st(yt)}

and treat the problem as one without subcontracting. Since the minimum of two

concave functions is also concave this does not influence the structural properties of

the problem and we may still limit ourselves to extreme point solutions. Therefore,

in the remainder of this section, we will develop a dynamic programming algorithm

for the two-level lot-sizing problem without subcontracting and with either

backlogging and inventory holding or backlogging only at the manufacturer

level. (Note that the case without backlogging was studied by Zangwill (1969) and

Van Hoesel et al. (2001).)

4.5.1 Both Inventory Holding and Backlogging at the Manufacturer

Motivated by the arborescent structure of extreme point solutions, define

Cte(si, 82) to be the minimum total cost required to satisfy the demand of periods

Si,..., s2 from level i at time t (where, in addition to the supplier level 1 and

manufacturer level 2, we denote the production level by 0). Clearly, we are trying

to find Clo(1, T), i.e., the total cost of satisfying demands of all periods from the

source at time 1. For convenience, we will let Ct2(t, t) = 0 for t = 1,..., T.

For each of the three levels we will develop a recursion that relates the values

Cte(1s, 82) and should be solved backwards in time (for t = T,..., 1) and up the
chain (from level 2 to level 0).