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Extensions to the Economic Lot Sizing Problem

Permanent Link: http://ufdc.ufl.edu/UFE0024781/00001

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Title: Extensions to the Economic Lot Sizing Problem
Physical Description: 1 online resource (127 p.)
Language: english
Creator: Onal, Mehmet
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: optimization, pricing, production
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We present several variants of the basic economic lot sizing model. In one of our models, we integrate pricing, procurement, transportation, and inventory decisions for a two echelon supply chain. We propose polynomial time dynamic programming algorithms to the problem under various procurement and inventory holding cost functions. In another model, we incorporate pricing decisions to a capacitated production environment where production setups consume a fixed amount of available resource time. We propose Dantzig-Wolfe formulations and a branch-and-price algorithm to solve the problem. We also consider economic lot sizing models where items can be procured from a number of suppliers in each period and they perish after a certain number of periods. We propose solution algorithms to solve some special cases of the problem.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mehmet Onal.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Romeijn, Hilbrand E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024781:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024781/00001

Material Information

Title: Extensions to the Economic Lot Sizing Problem
Physical Description: 1 online resource (127 p.)
Language: english
Creator: Onal, Mehmet
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: optimization, pricing, production
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We present several variants of the basic economic lot sizing model. In one of our models, we integrate pricing, procurement, transportation, and inventory decisions for a two echelon supply chain. We propose polynomial time dynamic programming algorithms to the problem under various procurement and inventory holding cost functions. In another model, we incorporate pricing decisions to a capacitated production environment where production setups consume a fixed amount of available resource time. We propose Dantzig-Wolfe formulations and a branch-and-price algorithm to solve the problem. We also consider economic lot sizing models where items can be procured from a number of suppliers in each period and they perish after a certain number of periods. We propose solution algorithms to solve some special cases of the problem.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mehmet Onal.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Romeijn, Hilbrand E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024781:00001


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First,Iwouldliketothankmyparents,myuncle_IsmetK.Bilgin,andmyauntHamiyetDincer.MyunclewasalwayswithmewheneverIneededhelp,andmyaunthadalwaysabowlofsouptooerwhenIwashungry.Second,IwouldliketothankmysupervisorycommitteechairDr.H.EdwinRomeijnforhisguidanceandtolerance.Third,Iwouldliketothankmydearfriends,AshwinArulselwan,AltannarChinchuluun,AlexGrasas,GudbjortGylfadottir,ArniOlafurJonsson,SibelB.Sonuc,VeraTomainoandPetrosXanthopoulos,whowerewithmeinmystressfuldaysandwithwhomIhadwonderful\Wing-Stop"parties. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2TWO-ECHELONREQUIREMENTSPLANNINGWITHPRICINGDECISIONS 15 2.1Introduction ................................... 15 2.2ModelFormulationandAlgorithmicFramework ............... 16 2.2.1ModelFormulation ........................... 17 2.2.2SolutionStructure ............................ 19 2.2.3DynamicProgrammingApproach ................... 20 2.2.4DynamicProgrammingSubproblems ................. 22 2.3GeneralizedSingle-LevelRequirementsPlanningProblem .......... 23 2.3.1LinearProcurementandHoldingCostFunctions ........... 24 2.3.2Fixed-ChargeProcurementandInventoryHoldingCostFunctions 25 2.3.3RunningTimeAnalysisforthe(RPP) ................. 27 2.3.3.1Piecewise-linearconcavetransportationcostfunctions ... 27 2.3.3.2Piecewise-linearconcaverevenuefunctions ......... 28 2.4RunningTimeAnalysisforthe(RPP-2L) .................. 29 2.5LinearCostFunctions ............................. 31 2.5.1LinearProcurementandSupplierInventoryHoldingCostFunctions 31 2.5.2LinearProcurement,Transportation,HoldingCostFunctions .... 32 2.5.3LinearTransportationandInventoryHoldingCostFunctions .... 33 2.6ConclusionandFutureResearch ........................ 34 3MULTI-ITEMCAPACITATEDLOT-SIZINGPROBLEMWITHSETUPTIMESANDPRICINGDECISIONS ............................ 35 3.1Introduction ................................... 35 3.2TheCLSTPModelandAssumptions ..................... 39 3.3Dantzig-WolfeDecompositionsoftheCLSTP ................ 42 3.3.1FrameworkDevelopment ........................ 42 3.3.2PricingProblem ............................. 44 3.3.2.1Finiteformulation ...................... 44 3.3.2.2Inniteformulation ...................... 46 3.4SolvingthePricingProblem .......................... 48 3.4.1DynamicPricingStrategy ....................... 49 5

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....................... 52 3.5Branch-and-PriceAlgorithm .......................... 54 3.5.1InitialColumnsandConvergence ................... 54 3.5.1.1Dynamicpricingstrategy .................. 54 3.5.1.2Constantpricingstrategy .................. 55 3.5.2Bounding ................................. 56 3.5.3Branching ................................ 56 3.5.4SearchStrategyandHeuristics ..................... 57 3.6ComputationalResults ............................. 57 3.6.1CreatingProblemInstances ...................... 57 3.6.2ComputationalTests .......................... 58 3.7ConclusionandFutureResearch ........................ 62 4THEECONOMICLOTSIZINGPROBLEMWITHPERISHABLEITEMS .. 68 4.1Introduction ................................... 68 4.2ModelFormulationandAnalysis ....................... 70 4.2.1TheModel ................................ 70 4.2.2StructuralPropertiesofOptimalSolutions .............. 74 4.3PolynomialTimeAlgorithms .......................... 85 4.3.1LEFOandFIFOConsumptionOrders ................ 85 4.3.2FEFOandLIFOConsumptionOrders ................ 88 4.4TheELS-PIwithProcurementCapacities .................. 91 4.5FurtherExtensions ............................... 98 4.5.1Backlogging ............................... 98 4.5.1.1LEFOandFIFOconsumptionorders ............ 99 4.5.1.2FEFOandLIFOconsumptionorders ............ 101 4.5.2Pricing .................................. 102 4.6ConclusionandFutureResearch ........................ 105 5INVENTORYMANIPULATIONUNDERCONSUMERPREFERENCE .... 106 5.1Introduction ................................... 106 5.2TheItemHoldBackProblem ......................... 107 5.3TheItemInsertionProblem .......................... 115 5.4SolutionAlgorithms .............................. 117 5.4.1TheItemHoldBackProblem ..................... 117 5.4.2TheItemInsertionProblem ...................... 119 5.4.2.1ConstantInsertionCosts ................... 119 5.4.2.2ConcaveInsertionCostswithFixedCharges ........ 121 5.5ConclusionandFutureResearch ........................ 123 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 127 6

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Table page 3-1AveragepercenterrorgapsobtainedwiththeFiniteFormulationusingitsownupperbound(andusingtheupperboundfoundwiththeInniteFormulation). 59 3-2AveragepercenterrorgapsobtainedwiththeInniteFormulationusingitsownupperbound. ..................................... 60 3-3AveragepercenterrorgapsobtainedwithCPLEXusingitsownupperbounds(andusingtheupperboundfoundwiththeInniteFormulation). ........ 60 3-4Percentageincreaseinprotwhenadynamicpricingstrategyisusedinsteadofaconstantpricingstrategy. ............................. 67 4-1Dataforexample 4.1 ................................. 74 7

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Figure page 2-1NetworkrepresentationofRPP-2Lwith4periods. ................. 18 2-2Networkfor(RPP-2L)withdashedlinesindicatingsubplan(2;5;2;6). ..... 19 3-1Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith10items,lowTBO,andconstantpricing. ....... 63 3-2Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith30items,highTBO,andconstantpricing. ....... 64 3-3Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith10items,highTBO,anddynamicpricing. ....... 65 3-4Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith30items,lowTBO,anddynamicpricing. ....... 66 4-1NetworkrepresentationoftheELS-PIfora3{periodproblem .......... 75 4-2AnoptimalsolutiontotheproblemwithFEFOconsumptionorderconstraints 89 5-1Atwolevelnetworkfora3periodproblem. .................... 111 5-2Anoptimalsolutionfora7periodproblem ..................... 118 8

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39 )isasfollows.TherearedemandsforanitemoveraniteanddiscreteplanninghorizonofTperiods.Thedemandshavetobesatisedbyproducinginafacilitywithnocapacityrestrictions.Anitemproducedinaperiodcanonlysatisfydemandsinthatperiodandthefollowingperiods.Wheneverthereispositiveproduction,asetuphastotakeplace,whichentailsaxedsetupcost.Anyitemproducedincursaunitproductioncostandanyitemcarriedtothenextperiodasinventoriesincursaunitinventoryholdingcost.Theobjectiveofthe(ELS)istondtheminimumcostproductionplangiventheproductionandinventoryholdingcostsineachperiod.WagnerandWhitinshowedthatthereexistsanoptimalsolutiontothe(ELS)thatpossessesthezeroinventoryorderproperty(ZIO),whereifthereispositiveproductioninaperiod,thenthereisnoinventorycarriedtothatperiod.ThisbasicmodelproposedbyWagnerandWhitinhasbeengeneralizedinseveralways.Forinstance,Zangwill( 40 )allowedbackloggingandassumedgeneralconcavecostfunctions.Bakeretal.( 1 ),BitranandYanasse( 3 )andFlorianetal.( 11 )extendedthe(ELS)byintegratingproductioncapacities.Insomecases,itemsmaygothroughseveralintermediatestepsbeforesatisfyinganydemand.Thishappensinserialsupplychainswherevalueisaddedindierentproductionfacilitiestoacertainproduct.Inthatcaseitemsthathavebeenprocessedinonefacilityneedtobetransportedtoanotherfacilitytogothroughotherprocesses.Inanotherexample,itemscanbeprocuredfromthesuppliersandtransportedtoandstoredinseveralwarehousesbeforetheyarebeingconsumedbythecustomers.Itemsthatareprocuredarerstsenttotherstlevelwarehousesandthentosecondandthirdlevelwarehouses(iftheyexist)untiltheyarenallytransportedtoretailers.Thistypeofserialsupplychainscanberepresentedbyatwolevellotsizingmodel.Suchmultiplelevellot 10

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41 ),KaminskyandSimchi-Levi( 27 ),VanHoeseletal.( 23 ).Theabovemodelsassumethatdemandsaregiveninadvanceandhencetheyareexogenoustothemodel.However,throughpricingdecisions,demandlevelsmaybeaectivelycontrolledandoptimaldemandlevelsmaybechosenineachperiod.Thomas( 36 ),KunreutherandScrage( 28 )andvandenHeuvelandWagelmans( 20 )developedmodelsthatintegratepricingdecisionstothe(ELS).Later,Geunesetal.(( 16 ),( 14 ))extendedthesepricingmodelsbyincorporatingproductioncapacities.Therearetwopricingstrategiesthathavebeenstudiedextensivelyintheliterature.Oneisthedynamicpricingstrategy,whereitisassumedthatdierentpricescanbesettoitemsineachperiod.Theotheroneistheconstantpricingstrategy,whereitisassumedthatasinglepriceshouldbesetfortheitemovertheplanninghorizon.Perishabilityoftheitemsisanotherissuetoconsiderwhilemakingproductionplans.Forinstance,dairyproductsandagriculturalproductscannotbeheldintheinventoriesindenitely,whichimpliesthatnotallproductionplansarefeasibleiftheitemsareperishable.Thereisarichliteratureinproductionplanningproblemswithperishableitems.Majorityofthisworkiscontinuoustimemodelsordiscretetimemodelswitheitherstochasticdemandsorstochasticlifetimes.Nahmias( 33 )hasanextensivereviewonsuchinventoryplanningmodels.FriedmanandHoch( 12 )andHsu( 25 )considereconomiclotsizingmodelswithdeterministicdemandswhereaknownfractionofitemsdeteriorateineachperiodastheitemsgetolder.Inthisdissertation,weinvestigatefurtherextensionsofthebasiclotsizingmodels.InChapter2,weconsideranuncapacitatedlotsizingproblemwhereproduction,inventorycarrying,transportation,andpricingdecisionsareintegratedtomaximizetotalprotsinatwoechelonsupplychain.Itemsareproduced(orprocured)inthestechelonandthentransportedtothesecondechelontosatisfythedemands.Weassumethatasdemandsaresatised,revenuesarerealized.Weshowhowthisproblem,undermanydierentrevenue 11

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39 )andZangwill( 41 )fortheearlieststudiesofsingle-levelandmulti-echelonlot-sizingproblems,andKaminskyandSimchi-Levi( 27 ),VanHoeseletal.( 23 ),andSargutandRomeijn( 35 )forextensionsthatallowforniteprocurementcapacities.Ingeneral,costsareassumedtobeconcaveasaresultofeconomiesofscale,whichmakestheproblemofwhentoprocureandtransportnontrivial.Classicallot-sizingproblemssuchastheonesmentionedaboveassumethatdemandsareknowninadvance.Thisisareasonableassumptionifrmsmakepricingdecisionsfortheitemsbeforemakingprocurement/transportationplansforthoseitems.However,sincethepriceofanitemmayaectthedemandforthatitem,signicantprotincreasescanbeobservedifwecanintegratepricingandprocurementplanningdecisions.Thomas( 36 )wasthersttoincorporatepricingdecisionsintoasingle-echelonlot-sizingmodel.Morerecently,thisareaofresearchhasgainedsignicantattentionfromresearchersviaextensionsthatincorporateaniteprocurementcapacity(see,e.g.,Billeretal.( 2 ),DengandYano( 8 ),andGeunesetal.( 16 ))ortheconstraintthatasinglepricemustbesetforthedurationoftheplanninghorizon(see,e.g.,KunreutherandSchrage( 28 ),Gilbert( 18 ; 19 ),VandenHeuvelandWagelmans( 20 ),andGeunesetal.( 14 )).Inthischapter, 15

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2.2 wepresentamathematicalformulationofourintegratedrequirementplanningmodelanddescribeourdynamicprogrammingapproach.Inparticular,weshowthatthisapproachrequirestherepeatedsolutionofgeneralizedsingle-echelonrequirementsplanningproblemswithpricingdecisions.InSection 2.3 ,wethereforedeveloppolynomial-timealgorithmsforsolvingsuchproblemsundervariousformsofthecostandrevenuefunctions.Section 2.4 thenanalyzestheresultingrunningtimeofthealgorithmforthetwo-echelonproblemaswellasdevelopssignicanteciencyimprovementsthatareobtainedbyconsideringthesimilaritybetweenthesubproblemsthatneedtobesolved.InSection 2.5 weanalyzefurthereciencyimprovementsthatcanbeobtainedifsomeorallofthecostfunctionsarelinear.Finally,weconcludethechapterinSection 2.6 withadiscussionoffutureresearchopportunities. 16

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(2{3)yt;xt;Dt0t=1;:::;TI(`)t0t=1;:::;T;`=1;2:Theobjectivefunctionrepresentsthedierencebetweenrevenuesandcosts.Constraints( 2{1 )and( 2{2 )aresimplytheowbalanceconstraintsatthemanufacturerandretailerlevelsineachperiod.Constraints( 2{3 )ensurethattheinitialinventoryateachleveliszero.Unfortunately,sincethefeasibleregionisunboundeditisnotguaranteedthatanoptimalsolutionto(RPP-2L)exists.However,existenceofanoptimalsolutionmaybeguaranteedundermildadditionalconditions,e.g.,theexistenceofnitedemandvaluesDtbeyondwhichrevenuesnolongerincrease,i.e.,withthepropertythatRt(Dt)=Rt(Dt)forallDtDt.Wewilltherefore,intheremainderofthischapter,simplyassumethatanoptimalsolutionto(RPP-2L)exists. 17

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2-1 .Inthisnetwork,node0isthesourcenode,nodes(1;t) Figure2-1: NetworkrepresentationofRPP-2Lwith4periods. (t=1;:::;T)representthenodesatthemanufacturerlevel,andnodes(2;t)(t=1;:::;T)representthenodesattheretailerlevel.Thedirectedarcsconnectingthesourcenode0tothenodes(1;t)representprocurement,whilethedirectedarcsconnectingnodes(1;t)and(2;t)representtransportation.Thedirectedarcsconnectingnodes(1;t)to(1;t+1)andnodes(2;t)to(2;t+1)representinventoryholding.Finally,thedirectedarcsconnectingnodes(2;t)tothesourcenode0representdemand.SincethecostassociatedwiththedemandarcscorrespondingtoperiodtaregivenbyRt(Dt),whichisaconvexfunctionoftheowDtonthisarcs,weunfortunatelydonothaveaconcavecostnetworkowproblem.Thismeans,inparticular,thatwecannotconcludethatanextremepointoptimalsolutionexiststotheproblem.However,ifthedemandvaluesDtweregivenwecouldeliminatethedemandarcsandassociateasupplyofPTt=1DtwiththesourcenodeandademandofDtwithnode(2;t)(t=1;:::;T).Theresultingproblemisthenthestandardtwo-levellot-sizingproblemstudiedbyZangwill( 41 ),anditiswell-knownthatanextremepointoptimalsolutionexists.Suchanextremepointcanbecharacterizedbyaspanningtreeinthenetwork,whereonlyarcsinthespanningtreemaycarryapositive 18

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39 )forsingle-levellot-sizingproblems,andwasalsousedbyVanHoeseletal.( 23 )inthecontextofmulti-levellot-sizingproblemswithprocurementcapacities.Figure 2-2 showsanexampleofaspanningtreesolutiondecomposedintosubplans.Eachsubplancanbe Figure2-2: Networkfor(RPP-2L)withdashedlinesindicatingsubplan(2;5;2;6). characterizedbytherstandlastperiodsineachofthetwolevels.Thatis,asubplancanberepresentedas(t1;t2;1;2),where(t1;t2)aretherstandlastperiodsofthesubplan 19

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Thisresultnowimpliesthat,ifthevaluesofaregiven,theproblemcanbesolvedtooptimalityinO(T2)time,andtheoverallrunningtimeofthealgorithmdependsonthetimerequiredtondtheoptimalprotvaluesforeachoftheO(T2)majorregenerationintervals.Muchoftheremainderofourworkwillthereforefocusonthesesubproblems. 21

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36 )andGeunesetal.( 16 ).Thegeneralizationslieintheincorporationofaprocurementfunctionpandinventoryholdingcostfunctionsgt(t=1;:::;T),aswellasallowingfornonlinearityoftheholdingcostfunctionsht(t=1;:::;T).Notethat,inthiscontext,pandgtcouldbeinterpretedasprocurementandholdingcostfunctionsforrawmaterials,respectively.Ifallrevenueandcostfunctionsaregeneralconcavefunctions,the(RPP)isaglobaloptimizationproblem(see,e.g.,HorstandTuy( 24 )),sothatwecannotexpecttobeabletoecientlysolveitingeneral.Inthischapter,wewillfocusonseveralparticularstructuresfortherevenueandcostfunctionsthatallowforsolutionalgorithmsforthesubproblemsthathavearunningtimethatispolynomialintheplanninghorizonT. 23

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maxD1;:::;D22Xt=1"Rt(Dt)t1Xt0=1t0!Dt#c12Xt0=1Dt0!:(2{4) 24

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2.3.1 .Notethatagaintheoptimalsetofdemandsinminorregenerationinterval[1;2]ispositiveif[1;2]>0.Thesituationwithrespectanyxedchargesatthesupplierlevelismorecomplicated.Ifweassumethatthetotaldemandsatisedinthelastminorregenerationintervalisstrictlypositive,wehavethatthexedinventoryholdingcostchargesatthesupplierlevelareindependentofthemagnitudesofthedemands,sothatwecanincorporatethesebydening:[1;2]=8><>:maxn0;[1;2]P21t=1H(2)toif2
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15 ; 16 )(andalsoinThomas( 36 )forthespecialcaseofxedchargetransportationcosts).Inparticular,considertheexpressionofeachofthesecostfunctionsastheminimumofnomorethanKxed-chargefunctionsasfollows:ct(x)=8><>:0ifx=0mink=1;:::;KfCtk+tkxgifx>0t=1;:::;T:

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2{4 )thenbecomesmaxk=1;:::;KmaxD1;:::;D22Xt=1"Rt(Dt)1k+t1Xt0=1t0!Dt#:Solvingthisproblemthusreducestosolvingthefollowingcollectionofindependentone-dimensionalconcavemaximizationproblemsmaxDtRt(Dt)1k+t1Xt0=1t0!Dtt=1;:::;2;k=1;:::;K:IfweassumethatallO(KT)oftheseproblemscanbesolvedinconstanttime,thevaluesof[1;2]canbefoundinO(KT2)time.Thisimpliesthatthecorrespondingvariantofthe(RPP)issolvableinO(KT2+T2)=O(KT2)time,whichgeneralizestheresultfromThomas( 36 )tothecaseofK>1. 2.3.3.1 .IftherevenuefunctionshavenomorethanJ(positive-slope)segments,theycanbewrittenasRt(D)=k1Xj=1rjtdjt+rktDk1Xj=1djt!k1Xj=1djtD
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2{4 2{4 8 )andGeunesetal.( 15 ; 16 ),whostudiedavariantofthisproblemwithtime-invarianttransportationcapacities,noprocurementcosts,xed-chargetransportationcostfunctions,andlinearinventoryholdingcostfunctions. (i) 29

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(i) (ii) 2.3.1 ,weletpt(y)=tyt=1;:::;Th(1)t(I)=tIt=1;:::;T:Inthatcase,giventhataunitofproductistransportedin,say,period,wecandeterminetheoptimalprocurementperiodforthatunitindependentlyoftheprocurementquantitiesinanyperiod.Inparticular,theprocurementandinventoryholdingcostsatthe 31

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2.3 thatthefactthatpandgt(t=1;:::;T)areidenticallyzerodoesnotimpacttherunningtimeofourapproach.Moreover,thetransportationcostfunctionscanbetransformedinO(T)time,sothatthisinstanceofthe(RPP-2L)isofthesamedicultastheassociatedinstanceofthe(RPP). 2-1 inO(T)timesincethenetworkisacyclicandhasO(T)arcs.Denotingthelengthsoftheseshortestpathsbyft(t=1;:::;T),the 32

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2.3.3.2 thisproblemcanbeformulatedasmaxzjt2[0;1]:j=1;:::;J;t=1;:::;22Xt=1JXj=1(rjtf(t1;t))djtzjtpt12Xt0=1JXj=1djt0zjt!andcanthereforebesolvedinO(JT)timegiventheorderingofsegment/periodpairs(j;t)innonincreasingorderofRjt;1(rjtf(t1;t))djt 33

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39 ).Thomas( 36 )wasthersttoincorporatepricingdecisionsinanuncapacitatedlot-sizingmodel,assumingadynamicpricingstrategy(i.e.,pricescanvaryovertime).KunreutherandSchrage( 28 )developedaheuristicapproachtosolvethisproblemforthecasewhereaconstantpriceshouldbesetfortheitemovertheplanninghorizonbyrestrictingtheformofthedemandorrevenuefunctions.WhileGilbert( 18 )wasthersttodevelopanexactalgorithmforthisproblemthatrunsinpolynomialtimeunderfurtherrestricteddemandfunctions,VandenHeuvelandWagelmans(2006)proposedapolynomial-timealgorithmtotheoriginalproblemposedbyKunreutherandSchrage( 28 ).Single-itemlot-sizingproblemswithnitebutstationaryproductioncapacitiesunderconstantanddynamicpricingstrategieswerestudiedbyGeunesetal.( 14 ; 16 ).Finally,Gilbert( 19 )consideredamulti-productplanningproblemwith 35

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minimizenXi=1TXt=1(Sityit+citxit+hitIit)(3{1)subjecttoNXi=1(bityit+aitxit)Ctt=1;:::;T 36

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3{1 )minimizesthetotalproductionandinventorycarryingcosts.Constraints( 3{2 )arethecapacityconstraints;constraints( 3{3 )aretheowbalanceconstraints;constraints( 3{4 )arethesetupforcingconstraints(whereMitisanupperboundontheproductionofitemiinperiodtthatissatisedinwithoutlossofoptimality);andconstraints( 3{5 )and( 3{6 )ensureintegralityandnonnegativityofthedecisionvariables.Constraints( 3{7 )ensurethatbothinitialandnalinventoriesarezero.Notethatnonzeroxedinitialandterminalinventorylevelscanbeincorporatedbyappropriatelymodifyingthedemandpatternsince,withoutlossofgenerality,wecanassumethatdemandwillbesatisedaccordingtoarst-inrst-outpolicy;moreover,iftheterminalinventorylevelsaredecisionvariablestheycanalsobeassumedequaltozerowithoutlossofgeneralityduetotheproblem'scoststructure.Muchresearchhasbeendonetoreducetheintegralitygapofformulation( 3{1 ){( 3{7 )fortheCLST.LagrangeanrelaxationandDantzig-Wolfedecompositionaretwoofthetechniquesthathavebeenstudiedtondimprovedlowerbounds.Thetwomethodsareequivalentinthatoneisthedualoftheotherandtheybothexploitthestructureoftheproblemsuchthatwhentyingcapacityconstraints( 3{2 )areremoved,weareleftwithindependentsingleitemlot-sizeproblems.Trigeiroetal.( 37 )andHindietal.( 21 )aretwoexamplesofheuristicsolutiontechniquesthatrelyontheLagrangeanrelaxationandsubgradientoptimizationoftheproblemobtainedbyrelaxingthecapacityconstraints.TheideabehindapplyingDantzig-WolfedecompositiontotheCLSTistowritefeasiblesolutionsasaconvexcombinationofextremepointsofconvexhullsoflot-sizingpolytopes,whichresultsinatighterformulationsincethesubproblems,whicharesingle-itemlot-sizingproblems,donothavetheintegralityproperty.Inanearlyattempt,Manne( 30 )proposedanLPformulationoftheCLSTbydeningtheconceptofdominantproductionscheduleforindividualitems,whichareotherwiseknownasschedulesthathavetheso-calledzero-inventoryordering(ZIO)property:Ii;t1xit=0fori=1;:::;Nandt=1;:::;T(seeWagnerandWhitin( 39 )).Inotherwords,dominant 37

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9 )describedaDantzig-Wolfedecompositionapproachandacolumngenerationmethodtogeneratedominantproductionplanswhentheyareneeded.ThissolvedtheproblemofpriorgenerationoflargenumberofproductionschedulestoconsiderinManne'sformulation.Unfortunately,however,Manne'sformulationisnotequivalenttotheCLSTsince,byconstruction,anyintegralsolutionwillconsistonlyofZIOproductionplansforeachitem,whileitiswell-knownthat,inthepresenceofcapacityconstraints,anoptimalproductionplanmayproduceeveninperiodsinwhichthestartinginventoryisnonzero(seeFlorianandKlein( 10 )).Nevertheless,ManneshowsthattheLP-relaxationofhisformulationwillgiveagoodlowerboundtotheCLSTsince,inanybasicfeasiblesolution,thenumberofdominantproductionscheduleswithfractionalweightsisnomorethanthenumberofcapacityconstraints.Recently,DegraeveandJans( 7 )developedacorrectDantzig-WolfeformulationoftheCLST(alongwithacorrespondingbranch-and-pricealgorithm)byshowingthatitissucienttoextendthecollectionofdominantproductionplanswithperiodsinwhichasetuptakesplacebutnoproduction,leadingtoatotalofO(3T)productionplansforeachitem.Inthischapter,wedeveloptwoalternativeDantzig-WolfedecompositionformulationsfortheCLSTPalongwithcorrespondingbranch-and-pricealgorithms.Thecolumngenerationapproachthatisusedtosolvearelaxationoftheproblemformulationateachnodeofthebranch-and-boundtreerequiresthesolutionofapricingproblemwhichisshowntodecomposeintoappropriateuncapacitatedsingle-itemlot-sizingproblemswithpricingdecisions.Asmentionedabove,ecientalgorithmstosolvesuchproblemstooptimalityexistintheabsenceofinitialinventories.However,theybecomemore 38

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3.2 ,westateourassumptionsandpresenttheformulationoftheCLSTP.InSection 3.3 wedevelopourtwoDantzig-Wolfeformulationsandtheassociatedcolumngenerationmethods.InSection 3.4 wediscussalgorithmsforsolvingtheresultingpricingproblemsunderdynamicandstaticpricingstrategies.InSection 3.5 weprovideimplementationdetailsofourbranch-and-pricealgorithm.WepresentcomputationalresultsinSection 3.6 andconcludethechapterinSection 3.7

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maximizeNXi=1TXt=1(Rit(it)SityitcitxithitIit)(3{8)subjecttoNXi=1(bityit+aitxit)Ctt=1;:::;T 3{8 )maximizesthedierencebetweenthetotalrevenuesandthetotalcosts.IncontrastwiththeCLST,wecannownotassumewithoutlossofgeneralitythattheiteminventoriesareequaltozeroatthestartoftheplanningperiodandrepresenttheinitialinventoriesbyIi00inconstraint( 3{15 ).AsintheCLST,Mitinconstraint( 3{11 )isanupperboundontheproductionofitemiinperiodtthatissatisedinwithoutlossofoptimality;forexample,wemaysetMit=minnPTs=t(isis0is);(Ctbit)=aito. 40

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3{12 )representsthedesiredpricingstrategy,andinthischapterwestudytwoparticularpricingstrategiesindetail:adynamicandaconstantpricingstrategy.Therststrategyisonewherethepriceofanitemcanvaryfromperiodtoperiod.Inthatcase,iisdenedbylowerandupperboundconstraintsonly:(i1;:::;it):LititUit;t=1;:::;TwherewemaychooseLit=1it(0)toensurethatthepriceofanitemisalwaysnonnegative,andUit=it=ittoensurethattheresultingdemandisalwaysnonnegative.Infact,intheabsenceofinitialinventories(i.e.,Ii0=0),wecanpotentiallytightenthelowerboundconstraintsbyrecallingthattherevenuefunctionattainsitmaximumat0it,andthatdemandsarenonincreasinginthepriceeect.Therefore,whenit<0itanyincreaseincostresultingfromtheincreaseddemandswhenitisfurtherdecreasedisnotosetbyalargeenoughincreaseinrevenues,sothatsuchvaluesofitarenotprotable.ThismeansthatwecanthensetLit=maxf1it(0);0itg.Inthesecondpricingstrategythatweconsider,thepriceofanitemshouldbeconstantovertheplanninghorizon.Wefollowearlierstudiesthatconsiderthispricingstrategybyrestrictingourselvestodemandfunctionsforwhichthefunctionsitarestationary,i.e.,it=ifort=1;:::;Tandi=1;:::;N.Inthatcase,iisgivenby:(i1;:::;it):it=i1;t=2;:::;T;Li1i1Ui1where,similarlytothedynamicpricingcase,wemaysetLi1=1i(0)toensurethatthepriceofanitemisalwaysnonnegative;andUi1=min1tTfit=itgtoensurethattheresultingdemandsarenonnegativeineachperiod.Interestingly,underthisstrategyanddemandrelationwehavethatit=i1fort=2;:::;T,sothatwemayhavethat,intheoptimalsolution,it<0i1inoneormoreperiodstevenwhentherearenoinitialinventories.However,whentherearenoinitialinventoriesforitemiwecanstillndapotentiallytighterlowerboundoni1byconsideringthetotalrevenue 41

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3.3.1FrameworkDevelopmentInthissection,wewilldeveloptwoalternativeDantzig-WolfedecompositionsoftheCLSTP.AsinthetraditionalCLST,thecapacityconstraints( 3{9 )arethethetyingconstraintsandiftheyareremoved,theproblemreducestoacollectionofindependentuncapacitatedsingleitemlot-sizingproblemswithpricingdecisionsforeachitemi.NowletXibetheregiondenedbyconstraints( 3{10 ){( 3{15 )foritemi.Notethatsinceiisbounded,Xiisapolytopeaswell.WecanthereforebuildaDantzig-WolfedecompositionformulationoftheCLSTPbywritinganyfeasiblesolutiontotheproblemasaconvexcombinationofthenitesetofextremepointsEiofconv(Xi).Foreaseofexposition,wewillrefertoatypicalelementofEiaseither(yji1;:::;yjiT;xji1;:::;xjiT;Iji1;:::;IjiT;ji1;:::;jiT)or,forshort,j,withassociatedcostji=PTt=1Sityjit+citxjit+hitIjitandcapacityconsumptionjit=aityjit+bitxjit.Lettingthedecisionvariablejirepresenttheweightofproductionanddemandplanj2Eiintheconvexcombination,weobtainthefollowingformulationoftheCLSTP:maximizeNXi=1TXt=1RitXj2Eijitji!NXi=1Xj2EijijisubjecttoNXi=1Xj2EijitjiCtt=1;:::;T 42

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3{16 )ensurethattheweightedaverageofthecapacityrequirementsoftheextremeplansinperiodtshouldnotexceedtheavailablecapacityinthatperiod,whileconstraints( 3{17 )relatetotheoriginalbinarysetupindicatorvariablesyitandstatethattheweightedaveragesoftheextremepointsetupvariablesshouldrepresentavalidproductionplanandthereforebebinary.Constraints( 3{18 )and( 3{19 )ensurethatweindeedconsiderconvexcombinationsofextremepointplansonly.Thisreformulationleadstoabranch-and-pricealgorithminwhichweusethecontinuousrelaxationofthisformulationtodetermineupperbounds.Thiscontinuousrelaxationitselfwillbesolvedthroughacolumngenerationapproachinwhichwerepeatedlysolveaso-calledrestrictedmasterproblemcontainingonlya(relativelysmall)subsetofthedecisionvariables(columns)anddeterminewhetheradditionalcolumnsshouldbeaddedthroughanassociatedpricingproblem.SincetheformulationoftheCLSTPabovecontainsonlyanitenumberofdecisionvariablesthiscolumngenerationalgorithmwillconvergenitely.WewillthereforerefertothisformulationastheFiniteFormulation.Althoughthenitenessoftheformulationiscertainlyamajoradvantage,thisapproachsuersfromamajordrawbackaswell:themasterproblemstobesolvedinthecolumngenerationphasesareconcavemaximizationproblems.Whileeectivealgorithmsforsolvingsuchnonlinearoptimizationproblemsexist,itisneverthelessquestionablewhethertheneedtorepeatedlysolvesuchaproblemwillyieldaneectivesolutionapproachtotheCLSTP.OuralternativeapproachisbasedonthefollowingreformulationoftheCLSTP: maximizeNXi=1TXt=1(ritSityitcitxithitIit)(3{20) 43

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3{22 )willbesatisedatequality.Letting^Xibetheregiondenedbyconstraints( 3{22 ){( 3{23 )foritemi,itisnothardtoseethatanyoptimalsolutiontotheCLSTPcanbeexpressedasaconvexcombinationofanitenumberofextremepointsofconv(^Xi).However,thissetofextremepoints,say^Ei,maycontainanuncountablyinnitenumberofpoints,makingadirectanalogonoftheearlierapproachimpossiblesincethefullmasterproblemwouldthencontainanuncountablyinnitenumberofdecisionvariables.However,anynitesubsetofpointsfrom^Eiwoulddenearestrictedmasterproblemwhosecontinuousrelaxationisalinearprogram.Wewillshowthatthisleadstoabranch-and-pricealgorithminwhichwesolveacontinuousrelaxationoftheCLSTPusingcolumngeneration,wheretherestrictedmasterproblemsarelinearprograms.However,thepricewehavetopayforthisisthatthecolumngenerationphasesmaynotterminatenitely.WewillthereforerefertothisformulationastheInniteFormulation.Intheremainderofthissectionwewilldeveloppricingproblemsthatareusedtogeneratepromisingcolumnsinthecolumngenerationphaseofboththeniteandinniteformulations. 3.3.2.1FiniteformulationAlthoughwehavenotyetexplicitlycharacterizedthesetofextremepointsEi,wearenowabletoderivethegeneralformofthepricingproblemthat,inthecolumngenerationphaseofouralgorithm,identiesoneormoreadditionalcolumnsthatshouldbeadded 44

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3{17 ):Xj2Eiyjitji1i=1;:::;N;t=1;:::;T:Sinceinanyproductionschedulej2Eiwehavethatyjit2f0;1gforallt=1;:::;T,constraints( 3{18 )and( 3{19 )implythatthisconstraintisredundantandwemayremoveitfromtheproblemtoobtainacontinuousrelaxationoftheCLSTP.Sincetheobjectivefunctionisconcaveandthefeasibleregionisapolytope,afeasiblesolutionvectorisoptimaltotherelaxationproblemifandonlyifthereexistsasolutiontothefollowingKKTconditions,wherethedualvariablestareassociatedwithconstraints( 3{16 ),iwithconstraints( 3{18 ),andjiwithconstraints( 3{19 ):TXt=1@RitXj2Eijitji!+ji+TXt=1tjitji+i30j2Ei;i=1;:::;NtNXi=1Xj2EijitjiCt!=0t=1;:::;TiXj2Eiji1!=0i=1;:::;Njiji=0j2Ei;i=1;:::;Nji0j2Ei;i=1;:::;Nt0t=1;:::;T:Usingthedenitionsofjiandjit,therstsetofKKTconditionsisequivalenttoTXt=1uitjit+TXt=1S0ityjit+c0itxjit+hitIjit+i=ji0j2Ei

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3{24 )happenstobesatisedthenwecanconcludethatwehavefoundtheoptimalsolutiontotherelaxation.Otherwise,wecanaddtothemasterproblemanyextremeplansj2Eiforwhich( 3{24 )isviolated.Inparticular,wemaychoosetoadd,foreachitemi,theextremeplanforwhich( 3{24 )ismostviolated(ifany).Wecanndthatplanbyndinganextremepointoptimalsolutiontothefollowingpricingproblemforeachitemi:maximizeTXt=1uititTXt=1S0ityit+c0itxit+hitIitisubjectto(PP-F)(xi;yi;Ii;i)2XiwhereS0it=Sit+tbitandc0it=cit+tait(t=1;:::;T;i=1;:::;N).Clearly,iftheoptimalsolutiontothisproblemisnegativetheoptimalsolutioncorrespondstotheconstraintin( 3{24 )thatismostviolated.Now(PP-F)isanuncapacitatedsingle-itemlot-sizingproblemwithpricingdecisions,linearrevenuefunctions,andinitialinventories,andwewillfocusonsolutionapproachestothisprobleminSection 3.4 3.3.2.1 comesattheexpenseofamaster'sproblemthatisaconcaveoptimizationproblem.Wecouldaddressthisfactbyrestrictingourselvestopiecewise-linearrevenuefunctions,whichwouldallowforthereformulationoftherelaxationoftheniteformulationasalinear 46

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3{20 ){( 3{23 )oftheCLSTP,intuitionsuggeststhatthepricingproblemforgeneratingadditionalcolumnsshouldbemaximizeTXt=1Rit(it)TXt=1S0ityit+c0itxit+hitIitisubjectto(PP-I)(xi;yi;Ii;i)2Xi:Now(PP-I)isanuncapacitatedsingle-itemlot-sizingproblemwithpricingdecisions,concaverevenuefunctions,andinitialinventories,andwewillfocusonsolutionapproachestothisprobleminSection 3.4 .Itisclearthatthisapproachsuersfromtwopotentialcomplications:(i)thecolumngenerationalgorithmcouldpotentiallygenerateaninnitenumberofcolumns;and(ii)itisnotobviousthattheprocedureconvergestoanoptimalsolutiontotherelaxation.However,thecorrectnessoftheabovealgorithmfollowsfromthefactthatitisanapplicationofthealgorithmproposedbyDantzig( 6 )forconvexprogrammingproblems.Dantzig( 5 )provedthatthiscolumngenerationalgorithmeitherndstheoptimuminanitenumberofiterationsorconvergestotheoptimalsolutionaslongasthereexistsanon-degeneratebasicsolutiontothemasterproblem.(InSection 3.5 wewilladdresstheproblemofndinganon-degeneratebasicsolutiontothemasterproblem.)Sinceeachiterationofthisalgorithmaddsanewbreakpointtoapiecewise-linearconcaveunder-approximationtotheconcaverevenuefunctions,thisalgorithmissometimesalsocalledgridlinearization(SeeLasdon( 29 )).Thismethodforcreatingapiecewise-linearapproximationismoreecaciousthancreatingalinearapproximationtothefunctionin 47

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3{25 ):Iit=Ii0+tXj=1xittXj=1(ititit)t=1;:::;T:Then,ifwedene~Rt(t)=Rt(t)ttTX=tht=1;:::;T~ct=ct+TX=tht=1;:::;Twecanformulate(PP)asmaximizeTXt=1~Rt(t)TXt=1(Styt+~ctxt) 48

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36 )proposesadynamicprogrammingalgorithmthatsolvestheprobleminO(T2)timewhennoinitialinventoryispresentandtheinverseofthederivativeoftherevenuefunctionscanbeevaluatedinconstanttime.Thisalgorithmisbasedontheobservationthatthereexistsanoptimalproductionandpricingplanthatpossessesthezero-inventory-ordering(ZIO)propertyandcantherefore,likeforthestandardeconomiclot-sizingproblemwithxeddemands(seeWagnerandWhitin( 39 )),bedecomposedintoasequenceofregenerationintervals.(Wewill,withaslightabuseofnotation,refertosuchsolutionsasextremepointsolutions.)Theproblemcanthenbesolvedbydeterminingtheoptimalpriceeects(or,equivalently,demandsorprices)fortheperiodsineachregenerationinterval.Lettingf(s;t)bethemaximumprotobtainableintheregenerationinterval(s;t)(i.e.,inperiodss;:::;t)andF(t)themaximumprotobtainableinperiods1throught,thefollowingbackwarddynamicrecursionsolvestheproblem:F(s)=maxt:tsff(s;t)+F(t+1)gs=1;:::;TF(T+1)=0:

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(i) Ifthereisnoproductionintheregenerationinterval(1;t),thetotaldemandsatisedinperiods1throughtisequaltotheinitialinventoryI0ift
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14 )forthelot-sizingproblemwithconstantcapacitiesanddynamicpricingdecisionsinaregenerationinterval,wheretherstperiodistheonlyproductionperiodandtheproductioncapacityisequaltoIi0.TheresultsinthatpaperthenimplythatthissubproblemcanbesolvedinO((J+1+Rlog)T)time,where=maxs=1;:::;t~R0+s(Us)mins=1;:::;t~R0s(Ls)andO(R)isthetimerequiredtondavalueofssatisfying( 3{26 )forsomevalueofr.Incaset=Twealsohavetoaccountforthepossibilitythatsomeoftheinitialinventoriesremainattheendoftheplanninghorizon.Thiswillonlyhappenifthereisnomarginalrevenueassociatedwithsatisfyinganydemandfromtheseremaininginventories.Inotherwords,thissolutionshouldsatisfy~R0+s(s)=0fors=1;:::;TandPst=1(sss)Ii0.Itiseasytoseethatsuchasolution,ifoneexists,canbefoundinO(RT)time. (ii) Next,supposethatproductioninregenerationinterval(1;t)takesplaceinperiod.Weagainignorethenonnegativityconstraintsontheinventorylevelsasdiscussedabove.Moreover,wewillhavethattheproductionquantityinperiodsatisesx=I0tXs=1(sss)sothatweshouldsolvethefollowingoptimizationproblem:maximizetXs=1~Rs(s)~css

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14 )forregenerationintervalswithasinglefractionalprocurementperiodinthelot-sizingproblemwithconstantcapacitiesanddynamicpricingdecisions.NowobservethatthereareO(T2)regenerationintervalswiths>1,andtheprotsofthesecanbefoundinO(RT2)time.Moreover,fortheO(T)initialregenerationintervals,wecanndtheprotincase(i)inO((J+1+Rlog)T),andtheprotincase(ii)foreachoftheO(T)xedproductionperiodsinO(RT)time.Thismeansthat(PP)underadynamicpricingstrategycanbesolvedinO((J+1+Rlog)T2)time.Inparticular,thisimmediatelyimpliesthat,forthespecialcasewheretherevenuefunctionsarelinear(whichisrelevantinouralgorithmfortheFiniteFormulation),(PP)underadynamicpricingstrategycanbesolvedinO((1+log)T2)time. 20 )studythisprobleminthecaseofzeroinitialinventoriesandundertheassumptionthatperiod1isaproductionperiodanddevelopanexactalgorithmwitharunningtimeofO(T3logT).Ingeneral,supposethatistherstproductionperiod.Notingthatwewillagainrestrictourselvestoextremepointsolutions,thisperiodiseither(i)therstperiodofthesecondregenerationinterval(anddemandintherstregenerationintervalissatisedpreciselybyinitialinventoryonly);or(ii)intherstregenerationinterval(anddemand 52

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3.4.1 ): (i) Inthiscase,itiseasytoseethatthiscanonlyhappenif>1andI0=1Xs=1(ss)or,equivalently,=()P1s=1sI0 38 ). (ii) Inthiscase,aparticularpriceeectisonlyvalidiftheinitialinventoriesaresucientlyhightosatisfyalldemandsuptotherstproductionperiod,sothatweshouldensurethatI01Xs=1(ss)or,equivalently,weshouldrestrictourselvestovalues2maxfL;()g;U:WenextfollowVandenHeuvelandWagelmans( 20 )andwritethetotallot-sizingcostsasafunctionofthepriceeectforagivensetSf;:::;Tgofproductionperiods(with,ofcourse,2S).Lettingt(S)denotetherstproductionperiodinS\ft;:::;Tg,thiscostfunctionreadsasfollows:C(;S)()=Xt2SSt~cI0+TXt=~ct(S)(tt)=Xt2SSt~cI0+TXt=~ct(S)tTXt=~ct(S)t=A(;S)B(;S)whereA(;S)andB(;S)aredenedappropriately.ThismeansthatC(;S)ispiecewise-linearandconvexand,moreover,hasthesamestructureasthecostfunctioninVandenHeuvelandWagelmans( 20 ).WecanthereforendtheoptimalpriceeectforthiscaseinO(T3logT)time. 53

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5 ; 6 )showsthatatleastonenon-degeneratebasicfeasiblesolutiontothemasterproblematanodeshouldexist.Inourimplementationofthismethod,wewillmakesurethatwestartthealgorithmwithaninitialnon-degeneratebasicfeasiblesolution. 54

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3{16 )andtheivariablescorrespondingtotheinitialproductionplansforitemsi=1;:::;N.Sincecapacitiesarestrictlypositivethisbasisisnon-degeneratesothatconvergenceofthecolumngenerationalgorithmforboththeniteandtheinniteformulationisguaranteed. 3.5.1.1 .However,ifanydemandsremainunsatisedwesolveanauxiliaryproblemwherewerelaxthecapacityconstraintsandmaximizetheminimumamountofunusedcapacity(whichmaybenegative!)overperiodst=1;:::;T.Iftheoptimalobjectivefunctionvalueofthisproblemisnegativeweconcludethatthereisnofeasiblesolutiontoourproblem(atthecurrentnodeinthebranch-and-boundtree).Otherwise, 55

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6 )andLasdon( 29 )).Especiallywhenusingtheinniteformulationwestopwheneverthedierencebetweentheseupperandlowerboundsisbelowthetolerance>0. 56

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7 ),weempiricallyndthattheobjectivevaluesobtainedbythisrepeatedroundingheuristicappeartofollowaunimodalpatternasthecutovalueisincreased.Therefore,westopthisheuristicassoonastheobjectivevaluedecreases. 3.6.1CreatingProblemInstancesForourcomputationaltests,wemodiedacollectionof540widelyusedprobleminstancescreatedbyTrigeiroetal.( 37 )fortheCLST.TheseprobleminstancesfortheCLSTweregeneratedusingafullfactorialdesignonveproblemcharacteristics:numberofitems,coecientofvariationofdemandacrossperiods,timebetweenorders,averagesetuptimesandcapacityutilization,whilethenumberofperiodswasxedtoT=20.Since,inourprobleminstances,wedonothaveaxeddemandpatternthesecondcharacteristicwasnotutilized.Instead,wecreatedparametervaluesforthedemandfunctionsasfollows:itxedat250anditrandomlygeneratedfromtheuniformdistributiononthesetf2:0;2:5;3:0g.Thisyieldsarangeofdemandvaluesfrom0to250.Furthermore,wechoseit(pit)=pit,correspondingtoacommonlyusedlinearrelationshipbetweendemandsandprices.This,inturn,meansthattherevenue 57

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3.2 ofeachinstancedirectlyusingCPLEXaswell.Allexperimentswereperformedona3.4GhzPentiumIVSystem(MEMORY)underWindowsXP.Duetothedicultyofsolvingespeciallylarge-scaleinstancestooptimalityweimposedaglobalupperboundonthesolutiontimeof1200secondsforallinstancesandallsolutionapproaches.Wegrouptheresultsofallofourresultsbypricingstrategy(constant,dynamic),capacityvalues(low,medium,high),setuptimevalues(low,high),andTBO(low,medium,high)inordertoassesstheeectofthecorrespondingparametersonthealgorithms'performance.Wedenetheerrorgapas(UBLB)=LB,whereLBisalowerbound,i.e.,thevalueofthebestintegralsolutionfoundbythealgorithmwithinthegiventimelimit,andUBisthebestupperboundontheoptimalintegersolutionvaluetothe 58

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3-1 and 3-2 showtheaveragepercenterrorgapsusingtheFiniteandInniteFormulation,respectively,whileTable 3-3 showstheaveragepercenterrorgapusingCPLEX,whereineachcasethebestLBandUBfoundwiththecorrespondingmethodwasusedtodeterminethegap.Weobservedthat,consistently,theUBobtainedbytheInniteFormulationwastighterthanthatobtainedwiththeothermethods.WethereforealsoreportanimprovedboundontheactualgapachievedwiththeFiniteFormulationandCPLEXthatusesthebestUBfoundusingtheInniteFormulation;theseareprovidedinparenthesesinTables 3-1 and 3-3 ). Table3-1: AveragepercenterrorgapsobtainedwiththeFiniteFormulationusingitsownupperbound(andusingtheupperboundfoundwiththeInniteFormulation). constant dynamic #items 102030 102030 low 5.575.956.72 (1.81)(1.47)(1.32) (1.61)(1.81)(2.55) 4.254.375.17 (1.51)(1.28)(1.33) (1.20)(1.31)(2.04) 2.973.073.55 (1.07)(0.85)(0.92) (0.77)(0.91)(1.37) 3.093.203.33 (0.68)(0.73)(0.86) 5.445.726.96 (2.12)(1.78)(1.79) (1.70)(1.96)(3.11) 2.803.113.24 (1.07)(0.91)(1.01) (0.71)(0.99)(1.14) 3.924.084.84 (1.35)(1.13)(1.16) (1.06)(1.21)(1.94) 6.076.217.28 (1.98)(1.56)(1.40) (1.80)(1.83)(2.84) 59

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AveragepercenterrorgapsobtainedwiththeInniteFormulationusingitsownupperbound. constant dynamic #items 102030 102030 low 0.530.180.11 0.440.160.08 0.330.130.07 0.250.090.05 0.620.220.12 0.280.100.06 0.370.150.08 0.640.220.13 AveragepercenterrorgapsobtainedwithCPLEXusingitsownupperbounds(andusingtheupperboundfoundwiththeInniteFormulation). constant dynamic #items 102030 102030 low 6.828.849.25 (1.50)(1.80)(2.13) (0.77)(0.80)(0.75) 6.477.858.31 (1.29)(1.39)(1.64) (0.67)(0.65)(0.65) 5.836.777.12 (1.07)(1.10)(1.16) (0.61)(0.54)(0.57) 4.896.016.30 (0.47)(0.48)(0.49) 7.859.6310.15 (1.85)(2.03)(2.32) (0.89)(0.85)(0.82) 5.145.906.09 (0.53)(0.49)(0.50) 5.907.137.53 (1.10)(1.24)(1.47) (0.61)(0.59)(0.59) 8.0810.4211.05 (1.89)(2.14)(2.38) (0.90)(0.91)(0.88) 60

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7 )andTrigeiroetal.( 37 ).WhilefortheCLSTthatisexpectedsinceManne( 30 )showedthatthelinearrelaxationoftheDantzig-WolfeformulationoftheCLSTisagoodapproximationtotheintegerproblemwheneverthenumberofitemsislargeascomparedtothenumberofcapacityconstraints,itisinterestingthatthisresult,atleastempirically,extendstotheCLSTP.Ingeneral,theerrorgapincreasesascapacitiesgettighter,averagesetuptimesincrease,andTBOincreases.Wealsoobservethattheeectsofcapacity,setuptime,andTBOaremorepronouncedwhenthenumberofitemsissmall.ItisalsointerestingtonotethattheerrorgapsarelowerunderdynamicpricesthanunderconstantpriceswhenusingtheInniteFormulationwhilethereverseistruewhenusingtheFiniteFormulation.Finally,weobservethattheadvantageofourbranch-and-pricealgorithmwiththeInniteFormulationoverCPLEXincreasesasproblemsbecomemorediculttosolve(i.e.,ascapacitiesgettighter,averagesetuptimesincrease,andTBOincreases).Thisisparticularlyapparentunderaconstantpricingstrategy,highTBOvalues,andalargernumberofitems.Theresultsaboveallusedaxedrunningtimeof1200seconds.Toassesstherateofconvergenceofourbranch-and-pricealgorithmsandCPLEXwehave,forseveralrepresentativeinstances,createdaplotthattrackstheupperandlowerboundsfoundbythealgorithmsastimeprogresses.Figures 3-1 { 3-4 showtheplotsonarepresentativesetoftestinstanceswith10and30itemsrespectively.ThetestinstancesforthegureswereselectedtoshowthebehaviorofthesolutionapproachesonarangeofcombinationsofTBO,setuptime,andcapacityvalues.Thecaptionwitheachtableindicatesthenumberofitems,low/highTBO,andthepricingstrategyused.Moreover,eachgureislabeledwiththeremainingproblemcharacteristics;e.g.,\lowST,highC"representsaninstancewithlowsetuptimesandhighcapacities.Inallgures,thesolidlinesrepresentlowerboundsandthedashedlinesupperbounds.Theboundsforthebranch-and-price 61

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3-4 breaksthisprotincreasedownbythedierentcharacteristicsoftheprobleminstances.Inparticular,weconcludethatthedynamicpricingstrategyisparticularlyprotablewhencapacitiesarelowandsetuptimesandTBOarehigh.Thiscorrespondswithwhatonemightintuitivelyexpect:foranyxeddemandvector,totalcoststendtobelargerwhensetuptimesarelargerbecausesomeoftheavailablelimitedcapacityisusedforsetups.Moreover,duetothisuseofcapacityforsetuptimereducesthenumberoffeasibleproductionplans.AsimilarargumentcanbemadeforthecostlinessofreducedcapacitiesandlargerTBO.Inthesecases,theabilitytocontrolpricesandhenceaectingthedemandsprovidesagreatdealofexibilitytotheproducer.Particularlydynamicpricescanservetobettermatchdemandstovariableandlimitedcapacities.SinceconstantpricesprovideasmallerdegreeofexibilitythedierencebetweentheprotunderthedierentpricingstrategiesisampliedwhencapacityissmallerandsetuptimesandTBOarelarger. 62

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Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith10items,lowTBO,andconstantpricing. 63

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Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith30items,highTBO,andconstantpricing. 64

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Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith10items,highTBO,anddynamicpricing. 65

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Comparisonofbranch-and-priceandCPLEXperformanceonarepresentativesetoftestproblemswith30items,lowTBO,anddynamicpricing. 66

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Percentageincreaseinprotwhenadynamicpricingstrategyisusedinsteadofaconstantpricingstrategy. 102030 low 67

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33 )hasanextensivereviewonproductionandinventorymodelsthatdealwithsuchperishableitems.ContinuoustimeEOQtypemodelswithdeterministicdemandsincludetheworksofGhareandSchrader( 17 )andCohen( 4 ).Ontheotherhand,thereisarichstreamofliteratureondiscretetimemodelswithstochasticdemandandxedlifetimes.Forinstance,NahmiasandPierskalla( 34 ),Fries( 13 ),Nahmias( 31 ; 32 )developmodelswheredemandisstochasticandtheitemsdeteriorateafteraxedlifetime.FriedmanandHoch( 12 )consideranextensionofthe(ELS)wheretheinventoriesspoilineachperiodatarateasafunctionoftheirages.TheyassumethatifthereareIitunitsofitemsofageiattheendofperiodt,thentotheperiodt+1onlyriIitofthoseitemswillbetransferred.Here,(1ri)isthedeteriorationrateofanitematageiwhere0ri1.FriedmanandHochshowthatiftheitemsareperishable,optimalsolutionsmaynotsatisfythezeroinventoryordering(ZIO)property.However,theydemonstratethat,ifthedeteriorationrateisassumedtoincreaseastheitemsgetolder,thatis,ifriri+1,thenthereexistsanoptimalsolutionwhereproductioninaperiodsatisessomesetofconsecutivedemandperiods.ThedeteriorationrateinFriedmanandHoch( 12 )dependsonlyontheageoftheitemandisindependentoftheperiodtheitemisproduced.Thismaynotberealisticconsideringthatfoodproductstendtodeterioratefasterinsummerthaninwinter.Giventhis,Hsu( 25 )proposesan(ELS)modelwherestockdeteriorationratesdependbothontheageandtheperiodtheitemisprocured.Inhismodel,ifinperiodtthereareyititems 68

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4.2 ,wepresenttheELS-PImodelanddiscussoptimalsolutionstructuresunderdierentconsumptionorderconstraints.Theseobservationsthenleadustopolynomialtimedynamicprogrammingalgorithms,whichwediscussinSection 4.3 .InSection 4.4 weinvestigatetheproblemunderprocurementcapacities.Insection 4.5 ,weconsidertwoextensionstotheproblem:backloggingandpricing.Section 4.6 concludesthischapterbystatingpossiblefurtherresearchdirections. 4.2.1TheModelWepresentthe(ELS-PI)undervariousitemconsumptionorderconstraints.Particularlywefocusonconsumptionordersthatarecausedbyeithertheconsumerbehaviororthephysicalconstraintsoftheinventorysystem.Theconsumerbehaviormaybecomeimportantwhenitemsoftwodierentagesareexhibitedtogetherontheshelves.Wheneverthishappens,consumersusuallyhaveapreferencefortheitemsthathavealongerliferemainingforobviousreasons.Whethertheyareabletoexercisetheirpreferencesornot,however,dependsontheallocativemechanism.Iftheconsumersareallowedtochoosetheitemsthemselves,whichisusuallythecaseinretailstores,they 70

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4{1 )aretheinventorybalanceconstraints.Constraints( 4{2 )and( 4{3 )guaranteethatitemsprocuredfromacertainsuppliercanonlybeallocatedtosatisfythedemandsofperiodsbeforetheexpirationdateoftheitems.Ontheotherhand,constraints( 4{4 )guaranteethattheitemallocationvariablesxtkiobeytheassumeditemconsumptionorderi(ICOi)wherei2fLEFO;FEFO;LIFO;FIFOg. 73

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4.1 showshowsuchanoptimalsolutionmightberealized. Table4-1: Dataforexample 4.1 t 4-1 listsrelevantdatafortheexample.Weassumethattherearenoconstraintsontheinventoryconsumptionorder.Thatis,theinventorymanagerisfreetodistributeanyitemintheinventoriestotheconsumers.Inthatcase,hewouldprocure40unitsinperiods1and2and20unitsinperiod4.Hewoulddistributetheitemsprocuredinperiod1tosatisfythedemandsinperiods1and2;theitemsprocuredinperiod2tosatisfythedemandsinperiods3and5;theitemsprocuredinperiod4tosatisfythedemandinperiod4.Thetotalcostofthisprocurementplanis350.TherststructuralpropertywepresentrelatestothedistributionofitemstodemandsandisageneralizationoftheZIOpropertythatincorporatestheperishabilityoftheitemsandmultiplesuppliersavailableineachperiod. 74

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Proof. Figure4-1. NetworkrepresentationoftheELS-PIfora3{periodproblem Now,assumethatinanoptimalsolution,therearetwosuppliers(t1;k1)and(t2;k2)suchthat1=P2F(t1;k1)\F(t2;k2)xt1k1>0and2=P2F(t1;k1)\F(t2;k2)xt2k2>0.Thatis,supplier(t1;k1)satisesatotaldemandof1andsupplier(t2;k2)satisesatotaldemandof2inperiodsF(t1;k1)\F(t2;k2).LetthissolutioncorrespondtotheowX=(xtk;It)inthenetworkandconsidergeneratingtwoothersolutionsandcorrespondingnetworkowsasfollows: 75

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Decreaseprocurementfromthesupplier(t2;k2)by2andincreaseprocurementfromthesupplier(t1;k1)bythesameamounttoobtainX0=(x0tk;I0t),wherex0t1k1=xt1k1+2,x0t2k2=xt2k22.Assumingnoitemsdeterioratebeforebeingconsumed,afterthischangewehaveI0t=It+2fort=minft1;t2g;:::;maxft1;t2g1whileamountofitemsprocuredfromtheothersuppliersandinventorycarriedinotherperiodsremainthesameasinX. (ii) Decreaseprocurementfromthesupplier(t1;k1)by1andincreaseprocurementfromthesupplier(t2;k2)bythesameamounttoobtainX00=(x00tk;I00t),wherex00t1k1=xt1k11,x00t2k2=xt2k2+1.Assumingnoitemsdeterioratebeforebeingconsumed,afterthischangewehaveI0t=It1fort=minft1;t2g;:::;maxft1;t2g1whiletheamountofitemsprocuredfromtheothersuppliersandinventorycarriedinotherperiodsremainthesameasinX.ObservethatX=X0+(1)X00forsome2(0;1).Sincethecostsareconcave,X0andX00shouldalsobeoptimalandinneitherofthesolutions,suppliers(t1;k1)and(t2;k2)bothsatisfythedemandsofperiodsinF(t1;k1)\F(t2;k2).Continuinginthismanner,anoptimalsolutioncanbereachedwherethepropertyissatised.WhatremainsistoshowthatafterthechangesinX,orderofconsumptiondoesnotcauseanyitemtoperishbeforebeingconsumed.ThiswillthenimplythatbothX0andX00arefeasible.ThechangesinXleadtofeasiblesolutionsiftherearenoconstraintsontheitemconsumptionordersinceanyallocationschemeispossible.ToshowthatX0andX00arebothfeasibleunderanyofthefour(i.e.,FIFO,LIFO,FEFOandLEFO)itemconsumptionorders,wewillinvestigateeachconsumptionorderseparately.basically,wewillshowthattheabovechangesintheprocurementplandonotdelayconsumptionofanyitembeyonditsexpirationdate.FIFOcase:Choose(t1;k1)<(t2;k2)tobetwoconsecutivesuppliersthatviolatestheproperty.Thischoiceofsupplierswillguaranteethatitemsprocuredfromthesupplier(t2;k2)areplacedtothequeuerightaftertheitemsprocuredfromthesupplier(t1;k1).Then,adecreaseof2>0inprocurementfromthesupplier(t2;k2)canbeimmediatelycompensatedbyanequivalentincreaseinprocurementfromthesupplier(t1;k1).Likewise,adecreaseof1>0inprocurementfromthesupplier(t1;k1)canbecompensatedbyanequivalentincreaseinprocurementfromthesupplier(t2;k2).Becausenoother 76

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Theorem 4.1 furtherimpliesthat,demandofaperiodisnotsatisedbymorethanonesupplierinanoptimalsolution.Therefore,asacorollarytothistheorem,weclaimthatanoptimalsolutionexistswhereauniquesupplierfullysatisesthedemandofaperiod. Proof. 80

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AlthoughCorollary 4.2 statesthatitisbestforthestoremanagertorstselltheitemsthatexpireearlier,whentheitemsaredisplayedontheshelvesandwhentheconsumersarefreetochooseamongthedisplayeditems,theywillbuytheonethatexpireslaterleadingtoaLEFOconsumptionorder.Insuchacase,inanoptimalprocurementplan,twoitemsfromdierentsuppliersarenotcarriedintheinventoriestogether.ThisimmediatelyimpliesthatitemsareorderedfromatmostonesupplierineachperiodandthatZIOpropertyholdsinoptimalsolutions. Proof. (i) Assumevt1k1=vt2k2.DuetoTheorem 4.1 ,eitherxt1k1=0orxt2k2=0forall2F(t1;k1)\F(t2;k2)=ft2;:::;vt1k1=vt2k2g.Intherstcase,itemsprocuredfrom(t1;k1)arenotcarriedovertoperiodt2andhencetheyarenotintheinventorieswiththeitemsprocuredfrom(t2;k2).Inthelattercase,xt2k2=vt2k2P=t2xt2k2=0,whichimpliesthatthereisnoprocurementfromthesupplier(t2;k2).Inneitherofthecasesitemsprocuredfromtwodierentsuppliersaredisplayedtogetherintheinventories. 81

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Assumevt1k1>vt2k2.DuetoTheorem 4.1 ,eitherxt1k1=0orxt2k2=0forall2F(t1;k1)\F(t2;k2)=ft2;:::;minfvt1k1;vt2k2g=vt2k2g.Ifxt2k2=0forall2F(t1;k1)\F(t2;k2),thenxt2k2=vt2k2P=t2xt2k2=0whichimpliesthatthereisnoprocurementfromthesupplier(t2;k2).Thereforeconsiderthecasewherext1k1=0forall2F(t1;k1)\F(t2;k2).DuetoLEFOconsumptionorderconstraints,wecannothavext2k2>0forany2F(t1;k1)\F(t2;k2)whilext1k1>0forsome>vt2k2.Therefore,noitemprocuredfromthesupplier(t1;k1)iscarriedintheinventoriesalongwiththeitemsprocuredfromthesupplier(t2;k2). (iii) Assumevt1k10,thencustomerswillprefertheitemsprocuredfrom(t2;k2)insteadoftheitemsprocuredfrom(t1;k1)inperiodsF(t1;k1)\F(t2;k2).TogetherwithTheorem 4.1 ,thisimpliesthatifxt2k2>0,xt1k1=0forall2F(t1;k1)\F(t2;k2)=ft2;:::;minfvt1k1;vt2k2g=vt1k1g,whichfurtherimpliesthatnoitemsexistintheinventoriesthatwereprocuredfromthesupplier(t1;k1)whentheitemsfromthesupplier(t2;k2)areprocuredandplacedintheinventories.Thereforeweeitherhavext2k2=0orwedonotcarrytheitemsprocuredfromthesupplier(t1;k1)intheinventoriestoperiodt2.Bothcasesimplythatitemsprocuredfrom(t1;k1)arenotdisplayedtogetherwiththeitemsprocuredfromthesupplier(t2;k2). 4.1 andthroughobservationoftheinventorysystemthatleadstothoseconsumptionorders.ConsiderrsttheELS-PIwiththeLIFOconsumptionorderconstraints.LIFOconsumptionorderisrealizediftheinventorysystemisdesignedasastack,wheretheitemsarealwaysinsertedtoandconsumedfromthefrontofthestack.Recallthat,iftheitemsareprocuredfromseveralsuppliersinasingleperiod,itemsprocuredfromthesupplierwithlowerindexnumberareplacedinthestorageearlierthantheonesprocuredfromthesupplierwithhigherindexnumber.Then,usingTheorem 4.1 ,itiseasytoprovethefollowingcorollary. 82

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Proof. 4.1 .Toprovethatji2fti;vti1;ki1+1;Tg,considertwoconsecutivesupplierswithpositiveprocurement:(ti1;ki1)and(ti;ki).Ifti1
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i 4.3 ,anoptimalsolutionthatobeystheZIOpropertyexistswhentheitemsareconsumedinLEFOmanner.ThereforeZIO=LEFO.AnyprocurementplanthatobeystheZIOpropertyalsoobeysanyofthefourconsumptionordersbynaturesincenoitemsfromdierentsupplieraredisplayedtogetherinasolutionthatobeystheZIOproperty.ThereforeZIOLIFOandZIOFIFO.DuetoCorollary 4.2 thereexistsanoptimalsolutiontotheELS-PIwithnoconsumptionorderconstraintswhichobeystheFEFOconsumptionorder.ThereforeLIFOFEFOandFIFOFEFO.Theaboveargumentdoesnotleaveoutthepossibilitythattheobjectivevalueswillbeequalunderallconsumptionorderconstraints.However,Example 4.1 canbeusedtoshowthatstrictinequalitiescanberealizedunderdierentconsumptionorders.FortheprobleminExample 4.1 ,itiseasytoseethatZIO=LEFO=500,LIFO=450,FIFO=400andFEFO=350.ObservethattheobjectivefunctionvaluewiththeFEFOconsumptionorderconstraintsisequaltotheobjectivefunctionvaluewithoutanyconsumptionorderconstraints.ThatisFEFO=0asstatedinthetheorem. 84

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4.3.1LEFOandFIFOConsumptionOrdersDuetoCorollary 4.3 ,anoptimalsolutiontotheELS-PIwithLEFOconsumptionorderconstraintscanbedecomposedintothesocalledregenerationintervals.Aregenerationinterval(1;2)isasetofconsecutiveperiods1;1+1;:::;2suchthatI11=I2=0andI>0for=1;:::;21andasinglesupplierinperiodperiod1satisesthedemandofperiods1through2.LettingK=maxtfKtg,anO(KT2)algorithmtotheproblemisthenstraightforwardtoachieveevenundergeneralconcavecostfunctionswherewesearchfortheoptimalsupplierforeachpossibleregenerationinterval.IftheconsumptionorderisFIFO,ZIOpropertymightbeviolatedbutoptimalsolutionshavethepropertythatasupplieralwayssatisesagroupofconsecutivedemandperiods.TosolvetheELS-PIwithFIFOconsumptionorderconstraints,wedeneanetworkwheretheshortestdistancebetweenthesourcenodeandoneofthesinknodeswillbeequivalenttothecostoftheoptimalprocurementplanovertheplanninghorizon.Inthisnetwork,everynodecorrespondstoatriple(X;t;k),wheret2f0;:::;Tg,k2f1;:::;KgandXisthecumulativeprocurementuptoandincludingthesupplier(t;k).DuetoTheorem 4.3 ,X2fD11;:::;D1Tg,whereDuv=vPi=uDi.Thesourcenodeis(0;0;0)andthesinknodesareoftheform(D1T;T;k),wherek=1;:::;KT.ObservethatthereareO(KT2)nodesinthisnetwork.Arcsinthenetworkrepresenttheinventoryholdingandprocurementdecisions.Theyexistbetweenthenodesoftheform(X1;t;k1)and(X2;t+1;k2)andbetweenthenodesoftheform(X1;t;k1)and(X2;t;k2)wherek2>k1.Ifthereisanarcbetweenthenodes(X1;t;k1)and(X2;t+1;k2),thisimpliesthattheamountofitemsprocuredfromthesupplier(t+1;k2)isX2X1andtheamountofitemscarriedintheinventoriesbetweenperiodtandt+1isX1D1;t.ThecostofthisarcisgivenbyPt+1;k2(X2X1)+Ht(X1D1t):

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(4{5) Observethat,iftheprocurementandholdingcostsaredenedasabove,wecansubstituteIt=tPi=1KtPk=1xiktPi=1DiintheobjectivefunctionoftheELS-PIanddroptheinventory 86

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(4{6) wherectk=ptk+PTj=thj.Inthismanner,wecangenerateanequivalentproblemwhereinventoryholdingcostsarezero.Then,aslightmodicationofthealgorithmproposedinHsu( 25 )canbeappliedtosolvetheproblem.Followinghisprocedure,wedene(t;k;)tobetheoptimalcostoftheELS-PIwheredemandstobesatisedarerestrictedtobefromperiod1throughperiod(1T)andthelastsupplierwithpositiveprocurementis(t;k),whichcompletelysatisesthedemandofperiod.Weset(1;k;1)=S1k+c1kD1fork=1;:::;K1anddene(t;k;t)=min(i;j):i
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4{7 ).Ontheotherhand,foranyxed(t;k;)triplewith>t,ittakesO(j(t;k;)j)timetocompute(t;k;)giventhatthevalueof(t;k;1)isknown.Then,thecomplexitytocomputeall(t;k;)values,andalsothecomplexityoftheoverallalgorithm,isgivenbyO0@KT+X(t;k)(T;KT)TX=t+1j(t;k;)j1A:Itemsfromthesupplier(t;k)haveauniqueexpirationdatevtk.Therefore,foraxed(t;k)pair,anyparticularsupplier(i;j)isinvestigatedonlyoncewhilecomputing(t;k;)for=t+1;:::;vtk.Asaresult,foraxed(t;k)pair,wehaveTX=t+1j(t;k;)jKT:Therefore,theoverallcomplexityofthealgorithmisO(KT+K2T2). 4{5 ).Figure 4-2 showsanetworkrepresentationofapossibleoptimalsolutiontotheELS-PIwithFEFOconsumptionorderconstraintsoveraplaninghorizonof7periods.Inthegure,itisassumedthatthereisasinglesupplieravailableineachperiod.Demandnodeswiththesamecoloraresatisedbythesamesupplier.Itemsprocuredinperiod1satisfythedemandsofperiods1,2and7;itemsprocuredinperiod3satisfythedemandsofperiods3and4whereasitemsprocuredinperiod3satisfythedemandsofperiods5and6.Wedeneablock(1;2;t;k)tobeasetofconsecutivedemandperiods1;:::;2suchthatasetuphasalreadybeencarriedouttoprocureitemsfromthesupplier(t;k)andthatprocurementfromthatsuppliersatisesthedemandinperiodt2andcansatisfy 88

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AnoptimalsolutiontotheproblemwithFEFOconsumptionorderconstraints thedemandsofperiods1through2.Clearly,fortheblock(1;2;t;k)tobefeasible,weshouldhavethatt12vtk.Notethatthisdenitionallowsforsatisfyingthedemandsofsomeoftheperiodsbetween1and2byprocuringfromsuppliersotherthan(t;k)wherenewsetupshavetobecarriedout,asintheexampledrawninFigure 4-2 .Wedene(1;2;t;k)tobetheminimumcosttosatisfythedemandsofperiodsintheblock(1;2;t;k)excludingthesetupcostofthesupplier(t;k).Weassumethat(1;2;t;k)=1ifthatblockisinfeasible.Tosatisfythedemandsbetweenperiods0and00,where1000<2,thestoremanagereither(i)ordersitemsfromthesupplier(t;k)or(ii)ordersitemsfromanothersupplier.Thelatteroptionincursadditionalsetupcosts.Since,inanoptimalsolution,theblock(1;2;t;k)canbeinanotherblockandablockmaycontainseveralotherblockswithinitself(asinFigure 4-2 ),wecarryoutthefollowingapproach.Werstcomputethecostsofthesmallestblocksthatcontainasingledemandperiodthataresatisedbyasinglesupplier,i.e.,wecompute(;;t;k)for=1;:::;Tand(t;k)2F1()suchthat(;;t;k)=ctkD:Wethencomputethecostsoftheblockswithtwodemandperiods,i.e.,(;+1;t;k)for=1;:::;T1and(t;k)2F1()\F1(+1).Wecontinuethisprocedureby 89

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4{9 )issolvedinO(T2)time,whichimpliesthatthetotalcomplexityofthisalgorithmisO(KT4).ThisalgorithmcanalsobeusedtosolvetheELS-PIwithLIFOconsumptionorderconstraints.WhentheitemsareconsumedinLIFOmanner,ifthereispositive 90

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11 )andBitranandYanasse( 3 )).ELS-PIgeneralizestheeconomiclotsizingproblembyincorporatingitemdeteriorationandmultiplesuppliersavailableineachperiodtoprocuretheitems.Then,itimmediatelyfollowsthattheELS-PIwithprocurementcapacitiesisalsoNP-hardunderallthosespecialcases.Ontheotherhand,theeconomiclotsizingproblemwithconstantprocurementcapacitiescanbesolvedinpolynomialtimeundergeneralconcavecostfunctions(seeFlorianandKlein( 10 )andvanHoeselandWagelmans( 22 )).We,therefore,trytoidentifyspecialcasesoftheELS-PIthatcanbesolvedinpolynomialtimeunderconstantprocurementcapacitieswhereweassumethatCtk=Cforeachsupplier(t;k).Werstprovethatalthoughtheeconomiclotsizingproblemwithconstantprocurementcapacitiesispolynomiallysolvable,theELS-PIwithconstantprocurementcapacitiesisNP-hardevenifthereisasinglesupplieravailableineachperiod. Proof. 91

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Thereisexactlyonesupplieravailableineachperiod (ii) Itemsprocuredinperiodtdeteriorateafterperiodtfort=2;:::;Twhiletheitemsprocuredinperiod1canlastuntiltheendofperiodTLet'scallthisproblemtheELS-PI(C1).AssumethatHt(It)=0fort=1;:::;TandthattheprocurementcostfunctionoftheuniquesupplierinperiodtisgivenbyPt(x)=8><>:Stx>00x=0t=1;:::;TLetQ=ftj2tT1;Dt>Ctg,bethesetofperiods2throughT1wheredemandexceedscapacity.NotethatatotaldemandofDtCtinallperiodst2Qshouldbesatisedthroughitemsprocuredinperiod1whetherornotthereisasetuptoprocureitemsinperiodt.ThereforewecangenerateanequivalentproblemwherethereisanewsetofdemandvectorD0suchthatD0t=minfDt;Ctgfort=2;:::;TandD01=D1+Pt2QfDtCtg.InthisproblemD0tCtfort=2;:::;Tandhencewithoutlossofgeneralitywecanassumethattherearenoprocurementcapacitiesinperiods2throughT.Inotherwords,foranyELS-PI(C1)instancewithconstantprocurementcapacitiesineachperiod,thereisanequivalentELS-PI(C1)instancewhereprocurementiscapacitatedonlyintherstperiod.ThenitissucienttoproveNP-hardnessoftheELS-PI(C1)assumingthatC1=CandCt=1fort=2;:::;T.ThedecisionproblemfortheKNAPSACKisasfollows.Givenanitesetf1;:::;Ng,asizes(t)2Z+andavaluev(t)2Z+foreacht2f1;:::;NgandpositiveintegersBandK,isthereisasubsetT0f1;:::;NgsuchthatPt2T0s(t)BandsuchthatPt2T0v(t)K.Inpolynomialtime,wecantransformtheKNAPSACKintoanELS-PI(C1)instancewithprocurementcapacitiesonlyintherstperiod.LettingT=N+1,wesetSt=v(t1),Dt=s(t1)fort=2;:::;TandsetS1=0andD1=0.Wethensettheprocurementcapacityinperiod1toC=PNt=1s(t)B. 92

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SincethespecialcaseoftheELS-PIconstructedintheproofofTheorem 4.5 isalsoaspecialcaseoftheELS-PIwithLIFOandFEFOconstraints,wecanimmediatelyconcludethattheELS-PIwithconstantcapacitiesremainNP-completewiththoseconsumptionorderconstraints.Thisholdsevenwhentheholdingcostsarezeroandproductioncostfunctionshaveaxedchargestructurewherethevariablepartiszero. 93

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Proof. (i) Increaseprocurementfromthesupplier(i`;j`)by1anddecreaseprocurementfromthesupplier(im;jm)bythesameamounttoobtainX0=(x0t;I0t)wherex0i`j`=xi`j`+1andx0imjm=ximjm1whileeveryvariableremainsthesameasinX.Thischangemayresultinthreepossibleconsequences:(i)procurementfromthesupplier(i`;j`)raisestocapacity;(ii)procurementfromthesupplier(im;jm)decreasestozero;and/or(iii)wewillhaveP(i`;j`)(i;j)(i;j)xij=sPj=1Djforsomethatsatises`
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GiventhepropertyinTheorem 4.6 ,weproposethefollowingapproachtosolvetheELS-PIwithconstantprocurementcapacitiesassuminggeneralconcaveprocurementcostsandlinearinventoryholdingcostssuchthatHt(x)=htxfort=1;:::;T.Wedene(1;2;t1;k1;t2;k2)tobetheminimumcostofthesub{plan(1;2;t1;k1;t2;k2).Then,wedeneanetworkwheretheshortestdistancebetweenthesourcenodeandthesinknodewillbeequivalenttotheminimumcostforthesub{plan(1;2;t1;k1;t2;k2).Inthisnetwork,everynodecorrespondstoatriple(X;t;k),wheret2ft1;:::;2g,k2f1;:::;KgandXisthecumulativeprocurementinthesub{planfromthesuppliers(i;j)thatsatisfy(t1;k1)(i;j)(t;k).WedeneDuv=vPi=uDiandassumeDuv=0forvk1.Ifthereisanarcbetweenthenodes(X1;t;k1)and(X2;t+1;k2),thisimpliesthattheamountofitemsprocuredfromthesupplier(t+1;k2)isX2X1andtheamountofitemscarriedintheinventoriesbetween 96

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4.5.1BackloggingInthissection,weinvestigateanextensionoftheELS-PIwherebackloggingisallowed.Inotherwords,weallowthedemandinaperiodtobesatisedlater,throughprocurementinsomeperiodt>.WeletutdenotethetotalamountofdemandthathasbeenbackloggedandletBtbethecostofbacklogginginperiodt.WeassumethatBtisaconcavefunctionofutandformulatetheELS-PIwithbackloggingasfollows.MinimizeTXt=1KtXk=1Ptk(xtk)+Ht(It)+Bt(ut)!subjecttoItuttXi=1KiXk=1xik+tXi=1Di=0t=1;:::;Txtk=Xi2f1;:::;t1g[F(t;k)xtkit=1;:::;T;k=1;:::;KtDi=X(t;k):ivtkgxtkii=1;:::;T

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4.3.1 totheELS-PIwithgeneralconcavecostfunctionscanbemodiedtoaccountforthebackloggingcaseasfollows.Wecreateanetworksuchthattheshortestdistancebetweenthesourcenodeandoneofthesinknodescorrespondstotheminimumcostforaparticularsub{plan.Thenodesofthenetworkarerepresentedbythetriple(X;t;k),suchthattosolveforthesub{plan(1;2),wehavethatt2f11;:::;2g,k2f1;:::;KgandX2fD11;:::;D12g.Thesourcenodeis(0;11;0)andthesinknodesareoftheform(D12;t;k),wheret=1;:::;2andk=1;:::;Kt.Observethatwecansettheexpirationdatesofitemsprocuredfromthesupplier(t;k)to2ifvtk>2becausenoitemsarecarriedafterperiod2inthesub{plan(1;2).Then,duetoTheorem 4.3 ,inthisnetwork,anarcbetweenthenodes(X1;t1;k1)and(X2;t2;k2)ispossibleifX22fX1;D1;vt2k2g:Thereisnoarcfromthenode(X1;t1;k1)to(X2;t2;k2)ifX1D1;vt2;k2andX2>X1,sothatnoitemprocuredfromthesupplier(t2;k2)deterioratesbeforebeingconsumed.IfX2=X1,wehavethatt2=t1+1andk2=1forconvenience.Anarcbetweenthenodes(X1;t;k1)and(X2;t+1;k2)ispossibleifX1D1;t.Thatis,inventorycarriedinaperiodcannotbenegative.UnlikethenetworkdenedinSection 4.3.1 ,therearearcsbetweenthesourcenodeandthenodes(X;t;k)fort=1;:::;2andk=1;:::;Kt.ThesearcsindicatethatatotaldemandofD1;t1werebackloggedfromthesupplier(t;k).Therefore,thecostsofthesearcsaregivenbyt1Xi=1Bi(Di;t1)+Ptk(X):ThecostsoftheremainingarcsarecomputedasinthenetworkinSection 4.3.1 .ThereareO(KT2)nodesinthisnetwork.O(KT)arcsemanatefromthesourcenode.Observethatsincetheprocurementcostsareconcave,iftherearetwosuppliers, 100

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4.1 ,thealgorithmweproposedinSection 4.3.2 canbeeasilyextendedtoincludebackloggingwhereweassumethatproductioncostshaveaxedchargestructureasin( 4{5 ),holdingcostsarelinearsuchthatHt(x)=htxandbackloggingcostsarelinearsuchthatBt(x)=bt(x)fort=1;:::;T.Wecanthen,substituteIt=Pti=1PKik=1xikPti=1Di+utintheformulation.Doingthat,weeliminatetheinventoryvariables,modifytheproductioncostfunctionsasin( 4{6 )andmodifytheunitbackloggingcoststob0t=bt+ht.Toallowforbacklogging,weredenethedenitionoftheblock(1;2;t;k)whereweletthesupplier(t;k)besuchthatt>1.Westillassumethatsupplier(t;k)satisesthedemandinperiod2andhenceweshouldhave(t;k)2F1(2)fortheblock(1;2;t;k)tobefeasible.Calculationof(1;2;t;k)fort1isthesameasinsection 4.3.2 .Fort>1,wemakethefollowingadjustments. 101

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4.3.2 .Observethatthesechangesdonotaddtothecomplexityinthecomputationoftheblockcosts.Therefore,theELS-PIwithbackloggingcanbesolvedinO(KT4)timeaswell.IftheitemsareconsumedinLIFOmanner,Corollary 4.4 stillholds.UnlikeSection 4.3.2 ,duetobacklogging,westillneedtosearchforthebestprocurementperiodforeachblockandhencethecomplexityremainsatO(KT4).

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4{5 ),holdingcostsarelinearsuchthatHt(x)=htxandbackloggingcostsarelinearsuchthatBt(x)=bt(x)fort=1;:::;T.Wecanthen,substituteIt=Pti=1PKik=1xikPti=1Di+utintheformulation.Doingthat,weeliminatetheinventoryvariables,modifytheproductioncostfunctionsasin( 4{6 ),modifytheunitbackloggingcoststob0t=bt+htandchangetherevenuefunctionstoeRt(Dt)=Rt(Dt)DtPTi=thi.Weassumedynamicpricingstrategy,wherepricesareallowedtochangeineachperiod.OurmainobservationisthatthecompositionoftheoptimalsolutionstructurespeciedinSection 4.2 holdsforanydemandvector.Therefore,itshouldholdfortheoptimaldemandvectoraswell.Therefore,wecanstillrestrictoursearchtothesolutionswiththesamecharacteristicsstatedinSection 4.5.1 .LetD(t;k)betheoptimaldemandleveltosatisfyinperiodgiventhatdemandinperiodissatisedbyprocurementfromthesupplier(t;k).Then,D(t;k)satisesthateR0+(D(t;k))ctkeR0(D(t;k))fortvtkeR+0(D(t;k))ctk+t1Xi=b0eR0(D(t;k))for
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4.3.2 ,werstcalculate(;)for(=1;:::;T)andthenwemoveontothelargerblockstocalculate(;+j)foreachpossible(;j)pairtonallysolvethefollowingrecursions:G(1)=max2:21f(1;2)+G(2+1)g1=1;:::;TG(T+1)=0:whereweassumethat(1;2)=ifthereexistsnosupplier(t;k)2F1(2).ObservethatpricingdecisionsaddnoadditionalcomplexitywiththeassumedcostfunctionstructuresunderFEFOconsumptioncase.Hence,thealgorithmrunsinO(KT4)time.Underdierentitemconsumptionorder,thecomplexityofthealgorithmchangesasfollows.IftheitemsconsumptionorderisLEFO,thenasinglesuppliersatisesthedemandinasingleblock.Thereforeforeachsetofconsecutivedemandperiods(1;2),wesearchforasupplier(t;k),(1t2),thatcanfullysatisfydemandsinthoseperiods.NotethatallblockprotscanbefoundinO(KT3)timeandhencethecomplexityisO(KT3).TheabovecomplexityresultalsoholdsiftheitemconsumptionorderisFIFO.Finally,iftheitemconsumptionorderisLIFO,unlikeSection 4.3.2 ,duetobacklogging,westillneedtosearchforthebestprocurementperiodforeachblockandhencethecomplexityremainsatO(KT4). 104

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4.4 andshowedlaterinExample 4.1 ,thisresultsinhigherprocurementcosts.InthischapterweinvestigatetwostrategieswherethestoremanagercaninuencetheinventoryconsumptionordertodevelopproductionplanswithcostsclosetothecostsoftheprocurementplansthatcouldbeachievedunderFEFOconsumptionorderconstraintsalthoughtheconsumersarealwayslookingfortheitemsthatexpirelater,whichenforcesLEFOconsumptionorder.Therststrategyistoseparatethedisplayandthestorageareasoftheitems.Insteadofdisplayingtheitemsassoonastheyareprocured,inventorymanagerholdsbacksomeoftheitemsinaseparatestorageareaandtransfersthemtothedisplayareawhenevertheyareneeded.Withtheexistenceofastoragearea,thestoremanagerdoesnotneedtosticktoaZIOpolicy;hecanpossesitemswithdierentlifetimesintheinventoriesandcanmanipulatetheordertheyareconsumedbydisplayingthemwhennecessary.However,henowhastopayforthetransferofitemsbetweenthestorageandthedisplayareasandforowningaseparatestoragelocation.Thesecondstrategyistomanipulatetheorderinwhichtheitemsareconsumedbydisplayingtheitemsinaqueueandrestrictingtheconsumerstopurchasetheitemsonlyfromthefrontofthequeue.Vendormachinesandmilkracksinthegrocerystoresarethesimplestexamplesforthistypeofqueuedisplays.Althoughtheyareeectivein 106

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5.2 and 5.3 ,weintroduceandanalyzetheitemholdbackandtheiteminsertionproblemsrespectively.InSection 5.4 weproposepolynomialtimesolutionalgorithmstosomespecialcasesofthoseproblemsandinSection 5.5 ,weconcludethischapter. 107

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41 ),KaminskyandSimchi-Levi( 27 )andVanHoeseletal.( 23 ).Inthemultilevellotsizemodels,eachlevelmayrepresentoneofthestepsinthemanufacturinganddistributionofaproduct.Intheitemholdbackproblem,therstlevelrepresentsthestoragearea,i.e.,theareawheretheprocureditemsarerstbrought, 108

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MinimizeTXt=1KtXk=1Ptk(xtk)+Gt(zt)+H(1)tI(1)t+H(2)tI(2)t!(5{1)subjecttoxtk=Xi2F(t;k)iXj=txtkjit=1;:::;T;k=1;:::;Kt 109

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5{1 )minimizesthesumofprocurement,transferandtheinventoryholdingcostsinthedisplayandthestorageareas.Constraints( 5{4 )and( 5{5 )areinventorybalanceequationsinthedisplayandthestorageareasrespectively.Constraints( 5{7 )statesthatitemsareconsumedinLEFOmanner.Constraints( 5{8 )guaranteenonnegativityofthevariablesandstatethatinitialinventoriesinboththedisplayandthestorageareasarezero.Ourstobservationregardingthismodelisthat,inanoptimalsolution,itemsprocuredfromtwodierentsuppliersarenotdisplayedtogetherontheshelves. Proof. 5-1 fora3periodproblem.Intherstlevel,itemsareprocuredandplacedinthestoragearea.Inthesecondlevel,itemsaredisplayedontheshelvesandtheyareconsumed.Flowonarcsbetweenasuppliernodeandanodeintherstlevelrepresentstheamountofprocurementfromaparticularsupplier.Flowonverticalarcsbetweentherstlevelandthesecondlevelrepresentstheamountofitemstransferredinaperiodfromthestorageareatothediplayarea.Theowonthehorizontalarcsrepresentstheamountofinventorycarriedbetweentwoperiodsinthestorageandthedisplayareas.LetX=xtk;zt;I(1)t;I(2)trepresenttheowoftheitemsinthenetworkcorrespondingtoanoptimalsolution.Considertwosuppliers(t1;k1)and(t2;k2)andletbetherstperiodwheretheitemsprocuredfromthosesuppliersappearontheshelvestogether.Let1>0and2>0betheamountofitemsontheshelvesprocuredfromthesuppliers(t1;k1)and(t2;k2)respectivelythatbecomeavailablebeforedemandoccursinperiod.Notethatwedonotexcludethepossibilitythattheremaybeitemsfromotherperiodsavailableontheshelvesaswell. 110

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Atwolevelnetworkfora3periodproblem. Duetotheconcavityofthecosts,wecanassumethat1itemsprocuredfrom(t1;k1)and2itemsprocuredfrom(t2;k2)arealltransferredinauniqueperiod.Ifthisisnotthecase,arecongurationoftheowispossiblewithanequivalentorlowerowcostwherethisassumptionissatised.Letj1t1andj2t2bethetransferperiodsoftheitemsprocuredfromthesuppliers(t1;k1)and(t2;k2)respectively.Wecangeneratetwoothersolutionsandcorrespondingowsasfollows. (i) Decreasethenumberofitemsprocuredfrom(t2;k2)andtransferredinperiodj2by2andincreasethenumberofitemsprocuredfrom(t1;k1)andtransferredinperiodj1bythesameamount.Assumingnoitemsdeteriorateduetothischangeintheprocurementplan,weobtainX0=x0tk;z0t;I(1)0t;I(2)0t,wherex0t2k2=xt2k22x0t1k1=xt1k1+2I(1)0t=(I(1)t+2t=t1;:::;j1I(1)t2t=t2;:::;j2ifj1
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Decreasethenumberofitemsprocuredfrom(t1;k1)andtransferredinperiodj1by1andincreasethenumberofitemsprocuredfrom(t2;k2)andtransferredinperiodj2bythesameamount.Assumingnoitemsdeteriorateafterthischangeintheprocurementplan,weobtainX00=x00tk;z00t;I(1)00t;I(2)00t,wherex00t2k2=xt2k2+1x00t1k1=xt1k11I(1)00t=(I(1)t1t=t1;:::;j1I(1)t+1t=t2;:::;j2ifj1<>:I(1)t1t=t1;:::;t2I(1)tt=t2;:::;j1I(1)t+1t=j1;:::;j2ifj1t2z00j1=zj11j1
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FollowingTheorem 5.1 ,itisstraightforwardtoprovetheUniqueProductionPeriodPropertyfortheitemholdbackproblemasstatedinCorollary 5.1 .ItisalsoeasytoconcludethatitemstransferredinaperiodsharethesamesupplierandhencethesameexpirationdateasstatedinCorollary 5.2 .TheproofofTheorem 5.1 alsoimpliesthatinanoptimalsolutionnotransfertakesplacebeforealltheitemsinthedisplayareaaredepletedasstatedinCorollary 5.3 113

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4.2 ,weproveinTheorem 5.2 thatinanoptimalsolutiontotheitemholdbackproblem,amongtheitemsinthestoragelocation,thestoremanagertransferstothedisplayareatheitemsthatexpireearlierrst. Proof. 114

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4.1 andCorollary 4.1 ofChapter4canbeextendedtothismodel.ThereforeoptimalsolutionstructureisthesameastheELS-PIwithnoconsumptionorderconstraints. 116

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5.2 ,thetransferreditemssharethesamesupplierandduetoTheorem 5.2 ,itemsthatexpireearlierarealwaystransferredearlier.Thesepropertiesresultinacertainoptimalsolutionstructure.AsinChapter4,wedeneablock(1;2;t;k)tobeasetofconsecutivedemandperiods1;:::;2whereasetuphasalreadybeencarriedoutforthesupplier(t;k)suchthatprocurementfromthatsuppliersatisesthedemandinperiod2andcanpotentiallysatisfythedemandofanyperiodbetween1and2.Anoptimalsolutiontotheitemholdbackproblemisdecomposedintosuchblocks,wheretheseblockscansubsumeotherblocks.Figure 5-2 showsapossibleoptimalsolutiontoasevenperiodproblem.Inthegure,dierentcolorsonarcsrepresenttheowofitemsprocuredfromdierentsuppliers.Observethatalthoughthereareitemsthatwereprocuredfromthesuppliers(1,1)and(2,1)inthestorage,newitemsareprocuredfromsupplier(5,1)tosatisfythedemandofperiod5.Therearethreeblocksinthegure;block(5,5;5,1)isinblock(3,6;2,1),whichisinblock(1,7;1,1).WedeneSP(1;2;t;k)tobethecostofblock(1;2;t;k).Itisalsothesolutiontothetwolevellotsizingproblemdenedbetweenperiodstand2suchthat(i)demands 117

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Anoptimalsolutionfora7periodproblem between1and2areallsatisedbyprocurementfromthesupplier(t;k),(ii)asetupforthatsupplierhasalreadybeencarriedoutand,(iii)transferofitemscannotoccurbeforeperiod1.Forxedvaluesof1,2andaxedsupplier(t;k),SP(1;2;t;k)canbecomputedinO(T2)timebecauseitcanbereducedtoasingleleveleconomiclotsizeproblem(seeChapter2andZangwill( 41 )forthesolutionalgorithmstotwolevellotsizeproblemswithgeneralconcavecosts).Moreover,whenwearesolvingforSP(1;T;t;k),wendallthevaluesSP(1;2;t;k)for12T.ThereforethevaluesforallSP(1;2;t;k)canbecomputedinO(KT4)timeintotal.WethenfollowasimilarapproachasinChapter4.Wedene(1;2;t;k)tobetheminimumcosttosatisfydemandoftheblock(1;2;t;k)andcomputeitbythefollowingrecursion.(1;2;t;k)=min8><>:min00<2fSP(1;0;t;k)+(0+1;2;t;k)gmin00<2f(1;0)+(0+1;2;t;k)gwhere,(1;2)=min(t;k)2F1(1)\F1(2)fStk+(1;2;t;k)g:Duetotheorderofprocesses,whencalculating(1;2)foraparticular(1;2)pair,weknowall(1;0),SP(1;0;t;k)andall(0+1;2;t;k)for0<2.Oncethosevaluesareknown,ittakesO(T)timetocalculateasingleblockcost.LettingK=maxtKt, 118

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4.3.2 whereweslightlymodifythedenitionoftheblockcost(1;2;t;k)asfollows(1;2;t;k)=min8>><>>:min00<2f0(1;0)+(0+1;2;t;k)gmin00
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1. Theinsertioncostfunctionsareconcavewithxedchargestructurewithtimeinvariantcontinuouscomponents.Thatis,Gtk(x;y)=Btk+g(x+y)(x;y)>(0;0)0x=0 2. Theinsertionscanbemadeonlytothebeginningofthequeue.Thatis,thequeueisaLIFOqueuebutthereisacostforinsertinganyitemasafunctionoftheitemsalreadyinthequeue(andhadtobemovedtowardstheback)atthetimeofinsertionInthatcase,wesolvetheiteminsertionproblemasfollows.First,wetransformittoaproblemwithasinglesupplierineachperiodasfollows.Foreacht=1;:::;T,weassociateit=Ptj=1KjandderiveanequivalentproblemdenedoveraplanninghorizonofT0=iT=PTj=1KjperiodswiththedemandvectorD0suchthatD0it=Dtfort=1;:::;TD0itj+1=0fort=1;:::;T;j=2;:::;Kt:Sincewehaveexactlyonesupplieravailableineachperiod,wedropthesubscriptforthesupplierinournotationandsetthecostfunctionsandexpirationdatesasfollowsS0itk+1=Stkfort=1;:::;T;k=1;:::;Ktc0itk+1=ctkfort=1;:::;T;k=1;:::;KtB0itk+1=Btkfort=1;:::;T;k=1;:::;Ktv0itk+1=ivtkfort=1;:::;T;j=1;:::;Kt

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S.Nahmias,Thexedchargeperishableinventoryproblem,OperRes26(1978),464{481. [33] S.Nahmias,Perishableinventorytheory:Areview,OperRes30(1982),680{708. [34] S.NahmiasandW.P.Pierskalla,Optimalorderingpoliciesforaproductthatperishesintwoperiodssubjecttostochasticdemand,NavalResLogistQuart20(1973),207{229. [35] F.SargutandH.Romeijn,Capacitatedproductionandsubcontractinginaserialsupplychain,IIETrans39(2007),1031{1043. [36] J.Thomas,Price-productiondecisionswithdeterministicdemand,ManageSci16(1970),747{750. [37] W.Trigeiro,J.Thomas,andJ.McClain,Capacitatedlotsizingwithsetuptimes,ManageSci35(1989),353{366. [38] A.Wagelmans,S.vanHoesel,andA.Kolen,Economiclotsizing:anO(nlogn)algorithmthatrunsinlineartimeintheWagner-Whitincase,OperRes40-S1(1992),S145{S156. [39] H.WagnerandT.Whitin,Dynamicversionoftheeconomiclotsizemodel,ManageSci5(1958),89{96. [40] W.Zangwill,Adeterministicmultiproductmultifacilityproductionandinventorymodel,OperRes14(1966),486{507. [41] W.Zangwill,Abackloggingmodelandamulti-echelonmodelofadynamiceconomiclotsizeproductionsystem{anetworkapproach,ManageSci15(1969),506{527. 126

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MehmetOnalwasbornin1980,inDarende,Turkey.Hereceivedhisbachelor'sdegreeinindustrialengineeringattheMiddleEastTechnicalUniversity,Ankara,Turkeyin2003.Hereceivedhismaster'sdegreeinindustrialengineeringandmanagementsystemsattheUniversityofCentralFlorida,Orlando,Florida,in2005.HehasbeenaPh.D.studentintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridasinceAugust2005. 127