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Mixed Integer Programming Approaches to Lot-Sizing and Asset Replacement Problems

Permanent Link: http://ufdc.ufl.edu/UFE0025002/00001

Material Information

Title: Mixed Integer Programming Approaches to Lot-Sizing and Asset Replacement Problems
Physical Description: 1 online resource (136 p.)
Language: english
Creator: Buyuktahtakin, Ismet
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dynamic, integer, optimization, production, replacement
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: MIXED INTEGER PROGRAMMING APPROACHES TO LOT-SIZING AND ASSET REPLACEMENT PROBLEMS In this dissertation, we develop mixed integer programming approaches for solving capacitated lot-sizing and parallel asset replacement problems. For capacitated lot-sizing, we analyze the use of dynamic programming in mixed integer programming frameworks. Specifically, this research aims to make contributions to the polyhedral characterization of the capacitated lot-sizing problem by defining a new set of valid inequalities derived from the end-of stage solutions of a dynamic programming algorithm. The end-of-stage solutions of the dynamic program provide valid bounds on the partial objective function values of the problem. We then define the stage value function according to the state values for a given level of inventory in a given stage and approximate it by its convex envelope. These inequalities can then be lifted by investigating potential state information at future stages. We test several possible implementations of these inequalities on randomly generated instances and demonstrate that our approach is more efficient than other integer programming based algorithms. We also consider a generalization of the capacitated lot-sizing problem called the multi-item capacitated lot-sizing problem (MCLSP). We study a mixed integer programming model for solving the MCLSP, which incorporates shared capacity on the production of items for each period throughout a planning horizon. We derive valid bounds on the partial objective function of the MCLSP formulation by solving the first t periods of the problem over a subset of all items, using dynamic programming and integer programming techniques. We then develop algorithms for strengthening these valid inequalities by employing lifting and back-lifting procedures. These inequalities can be utilized in a cutting-plane strategy, in which we perturb the partial objective function coefficients to identify violated inequalities for the MCLSP polytope. We test the effectiveness of the proposed valid inequalities on randomly generated instances, and demonstrate that they are promising for solving MCLSP instances. Our study of the parallel replacement problem is motivated by similar characteristics between the parallel replacement problem and lot-sizing problem. The parallel replacement problem under economies of scale (PRES) determines minimum cost replacement schedules for each individual asset in a group of assets that operate in parallel and are economically interdependent as a result of economies of scale. Economies of scale are due to the presence of the fixed charge in any period in which an asset is purchased. Both the parallel replacement and the lot-sizing problems have periodic demands that must be satisfied throughout the planning horizon. In the lot-sizing problem, production or purchases are made by trading off a fixed charge (set-up cost) against inventory holding and production/purchase costs. In the parallel replacement problem under economies of scale, additional fixed charges are incurred if assets are not replaced simultaneously. We prove that PRES is NP-hard. We then derive cutting plane approaches for the integer programming formulation of PRES. These cutting planes are motivated by the optimal replacement strategies implied by the no-splitting rule in the literature, which states that an optimal solution exists such that assets of the same age in the same time period are kept or replaced as a group. As a result of the no-splitting rule and constant demand, a purchase is enforced by a salvage in any optimal solution. We model PRES such that flow conservation constraints require a purchase whenever an asset is salvaged. We then use this property to generate inequalities for strengthening the PRES formulation. In addition, our inequalities have some similar characteristics with the flow cover inequalities derived for capacitated fixed charge networks. We present a set of experiments to illustrate the computational efficiency of the inequalities with respect to solving the mixed integer programs in a cut-and-branch framework. We also study the integer programming formulation of the PRES under technological change and deterioration. We provide optimal solution characteristics and insights about the economics of the problem. We propose cutting planes for strengthening the problem formulation and effective solution algorithms based on these cutting planes for the PRES under technological change. Finally, we present some computational results to illustrate the effectiveness of the proposed methods.
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 Ismet Buyuktahtakin.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hartman, Joseph C.
Local: Co-adviser: Smith, Jonathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025002:00001

Permanent Link: http://ufdc.ufl.edu/UFE0025002/00001

Material Information

Title: Mixed Integer Programming Approaches to Lot-Sizing and Asset Replacement Problems
Physical Description: 1 online resource (136 p.)
Language: english
Creator: Buyuktahtakin, Ismet
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dynamic, integer, optimization, production, replacement
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: MIXED INTEGER PROGRAMMING APPROACHES TO LOT-SIZING AND ASSET REPLACEMENT PROBLEMS In this dissertation, we develop mixed integer programming approaches for solving capacitated lot-sizing and parallel asset replacement problems. For capacitated lot-sizing, we analyze the use of dynamic programming in mixed integer programming frameworks. Specifically, this research aims to make contributions to the polyhedral characterization of the capacitated lot-sizing problem by defining a new set of valid inequalities derived from the end-of stage solutions of a dynamic programming algorithm. The end-of-stage solutions of the dynamic program provide valid bounds on the partial objective function values of the problem. We then define the stage value function according to the state values for a given level of inventory in a given stage and approximate it by its convex envelope. These inequalities can then be lifted by investigating potential state information at future stages. We test several possible implementations of these inequalities on randomly generated instances and demonstrate that our approach is more efficient than other integer programming based algorithms. We also consider a generalization of the capacitated lot-sizing problem called the multi-item capacitated lot-sizing problem (MCLSP). We study a mixed integer programming model for solving the MCLSP, which incorporates shared capacity on the production of items for each period throughout a planning horizon. We derive valid bounds on the partial objective function of the MCLSP formulation by solving the first t periods of the problem over a subset of all items, using dynamic programming and integer programming techniques. We then develop algorithms for strengthening these valid inequalities by employing lifting and back-lifting procedures. These inequalities can be utilized in a cutting-plane strategy, in which we perturb the partial objective function coefficients to identify violated inequalities for the MCLSP polytope. We test the effectiveness of the proposed valid inequalities on randomly generated instances, and demonstrate that they are promising for solving MCLSP instances. Our study of the parallel replacement problem is motivated by similar characteristics between the parallel replacement problem and lot-sizing problem. The parallel replacement problem under economies of scale (PRES) determines minimum cost replacement schedules for each individual asset in a group of assets that operate in parallel and are economically interdependent as a result of economies of scale. Economies of scale are due to the presence of the fixed charge in any period in which an asset is purchased. Both the parallel replacement and the lot-sizing problems have periodic demands that must be satisfied throughout the planning horizon. In the lot-sizing problem, production or purchases are made by trading off a fixed charge (set-up cost) against inventory holding and production/purchase costs. In the parallel replacement problem under economies of scale, additional fixed charges are incurred if assets are not replaced simultaneously. We prove that PRES is NP-hard. We then derive cutting plane approaches for the integer programming formulation of PRES. These cutting planes are motivated by the optimal replacement strategies implied by the no-splitting rule in the literature, which states that an optimal solution exists such that assets of the same age in the same time period are kept or replaced as a group. As a result of the no-splitting rule and constant demand, a purchase is enforced by a salvage in any optimal solution. We model PRES such that flow conservation constraints require a purchase whenever an asset is salvaged. We then use this property to generate inequalities for strengthening the PRES formulation. In addition, our inequalities have some similar characteristics with the flow cover inequalities derived for capacitated fixed charge networks. We present a set of experiments to illustrate the computational efficiency of the inequalities with respect to solving the mixed integer programs in a cut-and-branch framework. We also study the integer programming formulation of the PRES under technological change and deterioration. We provide optimal solution characteristics and insights about the economics of the problem. We propose cutting planes for strengthening the problem formulation and effective solution algorithms based on these cutting planes for the PRES under technological change. Finally, we present some computational results to illustrate the effectiveness of the proposed methods.
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 Ismet Buyuktahtakin.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hartman, Joseph C.
Local: Co-adviser: Smith, Jonathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025002:00001


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IwouldliketoexpressmysincereappreciationtomyadvisorsDr.JosephC.HartmanandDr.J.ColeSmithfortheirassistanceindevelopingtheideasinthisdissertation,theirconstantsupportandenormouspatiencethroughoutmydoctoralstudies.Ithasbeenaprivilegetoworkwiththem.IamgratefultoDr.JosephC.Hartmanforthenancialsupportneededtocompletemystudies,andforgivingmethefreedomtoworkontheproblemsthatIaminterestedin.Iowealottohisencouragement,understandingandguidance.IamindebtedtoDr.J.ColeSmithforhisattentiontodetail,dedicationtoperfection,insightfulcommentsandinvaluablecounseling,whichhavesignicantlycontributedtomydevelopmentasaresearcher.MyFloridaadventurewouldhavenotbeensogreatwithouthim.IamalsogratefultoDr.JosephGeunes,Dr.ArunavaBanerjee,andDr.FazilT.Najafortheirwillingnesstobeonmydissertationcommitteeaswellastheirvaluablesuggestions.IalsowouldliketotakethisopportunitytothanktoDr.FazilT.Najaforhisconstantmoralsupportduringmygraduatestudies.IalsowouldliketorecognizetheprofessorsatUFandLehighUniversityfromwhomIhavelearnedalotduringmygraduatestudy.SpecialthankstoDr.TedRalphsandDr.JeLinderoth,whoservedasreferencesduringmyjobsearch.IwouldliketoexpressmyappreciationtoDr.Jean-PhilippeRichard,whoselecturesinspiredmetodevelopsomeoftheideasinthisthesis,andtoDr.AlperAtamturkforhispreciousfeedbacksregardingmyresearch.IamalsothankfultoDr.TubaYavuz-KahveciandDr.TamerKahvecifortheirconstantsupportandmentoring.Iwouldliketothanktoeveryonewhohelpedmetorealizethisdissertation.Inparticular,IwouldliketothanktomyocematesSemraAgralandCanerTasknnotonlyfortheirfriendshipandsupportbutalsoformakingGainesvilleamoreenjoyableplaceforme.IamindebtedtoCanerforinsistingmetouseasoftwarepackage,whichhelpedmetosavesubstantialamountoftimewhilecodingmyalgorithms.Ialsowould 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 13 1.1BackgroundandMotivation .......................... 13 1.2TheCapacitatedLot-SizingProblem ..................... 15 1.2.1MixedIntegerProgrammingApproachestoLot-SizingProblems .. 16 1.2.2DynamicProgrammingApproachestoLot-SizingProblems ..... 18 1.3ParallelEquipmentReplacementProblem .................. 19 1.3.1SerialReplacementAnalysis ...................... 20 1.3.2ParallelReplacementAnalysis ..................... 20 1.4ContributionsandOverview .......................... 21 2SINGLE-ITEMCAPACITATEDLOT-SIZINGPROBLEM ........... 25 2.1Introduction ................................... 25 2.2DynamicProgrammingApproach ....................... 26 2.3ValidInequalities ................................ 29 2.4ComputationalExperiments .......................... 40 2.4.1InstanceGeneration ........................... 40 2.4.2ImplementationandExperimentalDesign ............... 40 2.4.3SummaryofExperimentalResults ................... 42 2.5Summary .................................... 52 3MULTI-ITEMCAPACITATEDLOT-SIZINGPROBLEM ............ 53 3.1Introduction ................................... 53 3.2ValidInequalities ................................ 55 3.2.1Single-ItemPartialObjectiveInequalities ............... 55 3.2.2Multi-ItemPartialObjectiveInequalities ............... 57 3.3LiftingandSeparation ............................. 64 3.3.1Lifting .................................. 64 3.3.1.1Back-liftingbinaryvariables ................. 64 3.3.1.2Back-liftingcontinuousvariables .............. 67 3.3.1.3Back-liftingbydynamicprogramming ........... 74 3.3.1.4Forward-liftingbinaryvariables ............... 76 3.3.2SeparationAlgorithm .......................... 76 6

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............................. 78 3.5Summary .................................... 82 4PARALLELEQUIPMENTREPLACEMENTPROBLEMUNDERECONOMIESOFSCALE(PRES) ................................. 83 4.1Introduction ................................... 83 4.2PRESunderConstantDemand ........................ 85 4.3ComplexityofPRES .............................. 87 4.4InequalitiesforPRES .............................. 93 4.5ComputationalExperiments .......................... 98 4.5.1InstanceGeneration ........................... 98 4.5.2ImplementationandExperimentalDesign ............... 99 4.5.3SummaryofExperimentalResults ................... 100 4.6Summary .................................... 102 5PARALLELREPLACEMENTPROBLEMUNDERTECHNOLOGICALCHANGEANDDETERIORATION .............................. 104 5.1Introduction ................................... 104 5.2Model ...................................... 105 5.3AnalysisandInsights .............................. 108 5.4OptimizationApproachestoPRESunderTechnologicalChange ...... 112 5.4.1OptimalSolutionCharacteristics .................... 113 5.4.2Inequalities ................................ 115 5.5ComputationalExperiments .......................... 120 5.5.1InstanceGeneration ........................... 120 5.5.2ImplementationandExperimentalDesign ............... 121 5.5.3SummaryofExperimentalResults ................... 122 5.6Summary .................................... 123 6CONCLUSIONSANDFUTURERESEARCHDIRECTIONS .......... 127 REFERENCES ....................................... 130 BIOGRAPHICALSKETCH ................................ 136 7

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Table page 2-1SummaryofexperimentsforT=90,c=2;3. ................... 44 2-2SummaryofexperimentsforT=90,c=4;5. ................... 45 2-3SummaryofexperimentsforT=120,c=2;3. ................... 46 2-4SummaryofexperimentsforT=120,c=4;5. ................... 47 2-5SummaryofexperimentsforT=150,c=2;3. ................... 48 2-6SummaryofexperimentsforT=150,c=4;5. ................... 49 2-7ExperimentsforT=90andf=10000. ...................... 51 3-1SummaryofexperimentsforT=60,M=2and!=2:5. ............. 80 3-2SummaryofexperimentsforT=60,M=2and!=3. .............. 81 3-3SummaryofexperimentsforT=18andM=8. .................. 82 4-1DatagenerationforPRES. .............................. 99 4-2SummaryofexperimentsforT=500,=0:2. ................... 102 4-3SummaryofexperimentsforT=500,=0:6. ................... 103 5-1DatagenerationforPRESundertechnologicalchange. .............. 121 5-2SummaryofexperimentsforT=100,=0:02and=0:2. .......... 123 5-3SummaryofexperimentsforT=100,=0:02and=0:6. .......... 124 5-4SummaryofexperimentsforT=100,=0:03and=0:2. .......... 125 5-5SummaryofexperimentsforT=100,=0:03and=0:6. .......... 126 8

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Figure page 2-1Graphsforforward(left)andbackward(right)dynamicprogrammingrecursions. 37 2-2GraphrepresentationofFt(it)valuesandassociatedconvexenvelopeinequalities. 38 3-1NetworkrepresentationoftheDPformulationofMCLSPforT=4andM=2. 62 3-2Convexhulldeningthefunctionalvaluesfort=3. ................ 62 4-1NetworkrepresentationofPRESwithowrepresentingassetsinusewithN=4. 86 4-2Transformationof3SATtoDpres 89 4-3Solutionnetworkforrst13periodsoftheexamplewithIIC. .......... 95 4-4Solutionnetworkforrst13periodswithIICandtwoNSRCinequalities. .... 97 5-1RepresentationofPRESundertechnologyanddeteriorationasanetworkwithowrepresentingpurchase(B),utilization(X),storage(Y),andsalvage(S)variables,initialinventorysupplynandtechnologicalchangeanddeteriorationparameter(a). ..................................... 109 5-2Averagereplacementagevs.valueforthetechnologicalchangecase ...... 111 5-3Averagereplacementagevs.valueforthedeteriorationcase .......... 111 5-4Averagereplacementagevs.forboththedeteriorationandtechnologicalchangecase .......................................... 112 9

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Inthisdissertation,wedevelopmixedintegerprogrammingapproachesforsolvingcapacitatedlot-sizingandparallelassetreplacementproblems.Forcapacitatedlot-sizing,weanalyzetheuseofdynamicprogramminginmixedintegerprogrammingframeworks.Specically,thisresearchaimstomakecontributionstothepolyhedralcharacterizationofthecapacitatedlot-sizingproblembydeninganewsetofvalidinequalitiesderivedfromtheend-ofstagesolutionsofadynamicprogrammingalgorithm.Theend-of-stagesolutionsofthedynamicprogramprovidevalidboundsonthepartialobjectivefunctionvaluesoftheproblem.Wethendenethestagevaluefunctionaccordingtothestatevaluesforagivenlevelofinventoryinagivenstageandapproximateitbyitsconvexenvelope.Theseinequalitiescanthenbeliftedbyinvestigatingpotentialstateinformationatfuturestages.Wetestseveralpossibleimplementationsoftheseinequalitiesonrandomlygeneratedinstancesanddemonstratethatourapproachismoreecientthanotherintegerprogrammingbasedalgorithms. Wealsoconsiderageneralizationofthecapacitatedlot-sizingproblemcalledthemulti-itemcapacitatedlot-sizingproblem(MCLSP).WestudyamixedintegerprogrammingmodelforsolvingtheMCLSP,whichincorporatessharedcapacityontheproductionofitemsforeachperiodthroughoutaplanninghorizon.WederivevalidboundsonthepartialobjectivefunctionoftheMCLSPformulationbysolvingtherst 10

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Ourstudyoftheparallelreplacementproblemismotivatedbysimilarcharacteristicsbetweentheparallelreplacementproblemandlot-sizingproblem.Theparallelreplacementproblemundereconomiesofscale(PRES)determinesminimumcostreplacementschedulesforeachindividualassetinagroupofassetsthatoperateinparallelandareeconomicallyinterdependentasaresultofeconomiesofscale.Economiesofscaleareduetothepresenceofthexedchargeinanyperiodinwhichanassetispurchased.Boththeparallelreplacementandthelot-sizingproblemshaveperiodicdemandsthatmustbesatisedthroughouttheplanninghorizon.Inthelot-sizingproblem,productionorpurchasesaremadebytradingoaxedcharge(set-upcost)againstinventoryholdingandproduction/purchasecosts.Intheparallelreplacementproblemundereconomiesofscale,additionalxedchargesareincurredifassetsarenotreplacedsimultaneously.WeprovethatPRESisNP-hard.WethenderivecuttingplaneapproachesfortheintegerprogrammingformulationofPRES.Thesecuttingplanesaremotivatedbytheoptimalreplacementstrategiesimpliedbytheno-splittingruleintheliterature,whichstatesthatanoptimalsolutionexistssuchthatassetsofthesameageinthesametimeperiodarekeptorreplacedasagroup.Asaresultoftheno-splittingruleandconstantdemand,apurchaseisenforcedbyasalvageinanyoptimalsolution.WemodelPRESsuchthatowconservationconstraintsrequireapurchasewheneveranassetissalvaged.WethenusethispropertytogenerateinequalitiesforstrengtheningthePRESformulation.Inaddition,ourinequalitieshavesomesimilarcharacteristicswiththeowcoverinequalities 11

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WealsostudytheintegerprogrammingformulationofthePRESundertechnologicalchangeanddeterioration.Weprovideoptimalsolutioncharacteristicsandinsightsabouttheeconomicsoftheproblem.WeproposecuttingplanesforstrengtheningtheproblemformulationandeectivesolutionalgorithmsbasedonthesecuttingplanesforthePRESundertechnologicalchange.Finally,wepresentsomecomputationalresultstoillustratetheeectivenessoftheproposedmethods. 12

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Productionplanningisoneimportantarea,whichrequiresstrategicplanningoftheacquisitionandallocationofresourcessuchasparts,rawmaterials,machinesandlabor,aswellasplanningofproductionactivitiestotransformresourcesorrawmaterialsintonishedgoodsinordertomeetcustomerdemand.Thegoalofproductionplanningistomakeoptimaldecisionswiththetypicalobjectiveofminimizingcostssuchaspurchase,set-up(xed)andinventoryholding,ormaximizingprot.Toachievethisgoal,industrialenterprisesneedtousequantitivetoolsinordertoincreaseproductivitywhilereducingcostsundercapacityrestrictions.Restrictionsonproductionariseduetolimitationsonthemachineand/orlabororotherresourcecapacitiesand,ingeneral,maketheproblemhardertosolve. Parallelreplacementisanotherimportantdecisionproblem,whichrequireseectiveutilizationandtimelyreplacementofcapitalassets.Replacementanalysisaimstoprovidedecisionsupportoptimizingthetrade-obetweenkeepingassetslonger,athigheroperatingcosts,versusreplacingwithnewerassetsathighercapitalcosts.Similartoproductionplanning,solvingparallelreplacementproblemsrequiresanalyticaltoolstomakesequentialdecisionsforthemanagementofthecapitalassets.Theobjectiveistominimizecostsbydeterminingwhethertokeeporreplaceanexistingassetamongagroupofassets,theamountoffutureassetsthataregoingtobepurchased,thetimingofthe 13

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Mixedintegerprogramming(MIP)isanaturalframeworktomodelproductionplanningandparallelreplacementproblemsbecauseoftheproblemcharacteristics(suchasxedcosts)withregardstopurchasedecisionsandcapacityconstraintsonproductionorpurchases.ThemaindrawbackofthisapproachisthatMIPmodelsmaybediculttosolveforlargeinstancesthatareusuallyencounteredinproductionplanningandparallelreplacementsystems.Inparticular,branch-and-boundalgorithms,inwhichlinearprogramming(LP)relaxationsareusedtoprunenodesinthesearchtree,donot,generally,performwellforproblemswithxedchargenetworkowcharacteristicsduetotheweakboundsprovidedbytheLPrelaxation.Toovercomethislimitation,sophisticatedtechniquescanbeusedtotightentheLPrelaxationboundsbytighteningthemathematicalformulations.OnewaytoachievetightformulationsistoaddvalidinequalitiesorcuttingplanestotheLPformulation.Thestrengthenedformulationsmaysubstantiallyreducethecomputationaltimeneededtosolvethem.Furthermore,forharderinstances,thesetechniquesmaymakeitpossibletoincreasethesizeofmodelssolvabletooptimality,orclosetooptimality. Besidesmixedintegerprogrammingapproaches,designingotherecientoptimizationalgorithmsmaybeextremelyusefulforsolvingthesehardproblems.Forinstance,dynamicprogramming(DP)isasequentialoptimizationapproachwhichworkswellforaclassofproductionplanningandparallelreplacementproblems.ThismotivatesustousetheinformationfromDPformulationstoenhancetherelatedMIPformulations. Thisdissertationfocusesonthedevelopmentofmethodstosolveaclassofproductionplanningandparallelreplacementproblemsmoreeciently.Weaddresstheseproblemsbyanalyzingtheirmixedintegerprogrammingformulations.WealsostudythetechniquesthroughwhichwecanutilizeDPtoobtainstrongerMIPformulationsoftheseproblems. 14

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Intheremainderofthischapter,werstintroduceaspecicproductionplanningproblemcalledthelot-sizingproblem,whichisaspecialcasexedchargenetworkowproblem.Wealsodiscusstheearlierpolyhedralcontributionsandoptimizationapproachestolot-sizingproblems.Wethenpresentanotherspecialcasexedchargenetworkowproblemcalledtheparallelequipmentreplacementproblem.Wealsogiveabriefdiscussionoftheearlierstudiesonthisproblem.Finally,wediscussourcontributionsandgiveabriefoutlineofthedissertation. Florianetal. ( 1980 )). Specically,considerasetofproductsoritems,forwhichperiodicdemandisknownforagivenhorizon.Demandineachperiodcanbesatisedeitherthroughproductionorinventoryremainingfrompreviousperiods.Demandsarenotbacklogged.Productionisrestrictedduetomachine,labororresourcecapacity,andallitemstobeproducedcompeteforthislimitedcapacity.Inallthelot-sizingmodelsthatweconsiderinthisdissertation,productioniscapacitated,whichisarealisticconsiderationthatarisesinmanyindustrialsettings. Therelevantcostsareset-up(xed)costs,productionorpurchasecosts,andinventoryholdingcosts.Thexedset-upcostmustbepaidbeforeanyproductioncanoccur,andtotalproductioncostsdependonthequantityproduced.Thereisalsoaunitinventorycostforeachunitofitemthatisheldinstockorinventoryperunitoftime. 15

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AtamturkandKucukyavuz ( 2008 ); AtamturkandMu~noz ( 2004 ); PochetandWolsey ( 1991 ); Wolsey ( 1958 ))isatthecoreofproductionplanningproblemsinvolvingmultipleproductsandlevelsoveranitediscretetimehorizon. Lot-sizinghasbeenstudiedextensivelyinthemathematicalprogrammingliterature,includingdeningvalidinequalitiesandfacetsfortheassociatedpolyhedron. Baranyetal. ( 1984b )rstdenedtheconvexhulloftheuncapacitatedlot-sizingproblem(ULSP)andthenusedthefacetsfortheULSPtoreformulatethemulti-itemcapacitatedlot-sizingproblem( Baranyetal. ( 1984a )). KucukyavuzandPochet ( 2009 )givetheconvexhullofsolutionsfortheULSPwithbacklogging.Fortheconstant-capacitycase, Leungetal. ( 1989 )and PochetandWolsey ( 1993 )derivefacetsforthesingle-itemcapacitatedlot-sizingproblem.However,acompletelineardescriptionoftheconvexhullofsolutionsforthisproblemisunknown. Pochet ( 1988 ), Milleretal. ( 2000 2003b ), Loparicetal. ( 2003 )and AtamturkandMu~noz ( 2004 )presentfacet-deninginequalitiesforthecapacitatedlot-sizingpolytopewherecapacityrestrictingtheproductionisgeneral. AtamturkandKucukyavuz ( 2005 )analyzepolyhedralcharacteristicsoftheULSPinwhichthereexistcapacitiesandxedchargecostsoninventoryineachperiod.Also,forthecaseinwhichperiodicxedchargecostsexistforinventoryaswellasproduction, AtamturkandKucukyavuz ( 2008 )provideanO(T2)optimalalgorithm. BalasandSaxena ( 2008 )providerank-1splitcutsforgeneralMIPproblemsandtestthemonthebenchmarkinstancesof AtamturkandMu~noz ( 2004 )forthegeneralcapacitatedlot-sizingproblem. Thesingle-itemcapacitatedlot-sizingformulationformsasubstructureforthemulti-itemandmulti-levelversionsofthelot-sizingproblem.Thereforetheresultsfound 16

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Miller ( 1999 ); Milleretal. ( 2000 2003b ); Padbergetal. ( 1985 ); PochetandWolsey ( 2006 )). Constantino ( 1996 )providescuttingplanesbasedonsubmodularinequalitiesforthesingleitemcapacitatedlot-sizingproblemwithstart-upcosts,andusesthesecutsforthemulti-itemversionoftheproblem. Constantino ( 2000 )extendsthe(K;l;S;I)inequalitiesof PochetandWolsey ( 1993 )forthemulti-itemlot-sizingproblemandderivesseveralclassesofvalidinequalitiesforthemulti-itemcapacitatedlot-sizingproblem,includingspecialcaseswithbackloggingandstart-upcosts. PochetandWolsey ( 1991 )and BelvauxandWolsey ( 2000 2001 )provideformulationsandvalidinequalitiesforthemulti-itemandmulti-stagelot-sizingproblems. Milleretal. ( 2003a )giveatightformulationforthemulti-itemlot-sizingproblemwithconstantdemandsandset-uptimes.A0-1mixedintegerprogrammingformulationofapracticalcaseofmulti-itemlot-sizingandschedulingisgivenin Smith-DanielsandSmith-Daniels ( 1986 ). JansandDegraeve ( 2004 )proposeadecompositionalgorithmtosolveamulti-item,multi-resourcecapacitatedlot-sizingproblemwithbacklogging.Lagrangean-basedheuristicsanddecompositionalgorithmsformulti-itemcapacitatedlot-sizingproblemarestudiedin Brahimietal. ( 2006 ), ChenandThizy ( 1991 ), Diabyetal. ( 1992 ), TempelmeierandDerstro ( 1996 )and Trigeiroetal. ( 1989 ). 17

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WagnerandWhitin ( 1958 ). Wagelmansetal. ( 1992 ), AggarwalandPark ( 1993 )and FedergruenandTzur ( 1991 )provideO(TlogT)algorithmsforthesameproblem. Love ( 1973 )givesanO(T3)algorithmfortheULSPwithpiecewiseconcavecostsandboundedinventory.Fortheconstantcapacitycase, FlorianandKlein ( 1971 )provideanO(T4)algorithmwhile VanHoeselandWagelmans ( 1996 )presentanO(T3)algorithm. Florianetal. ( 1980 )presentadynamicprogrammingalgorithmwithcomplexityO(DTCT)forthesingle-itemcapacitatedlot-sizingproblem,whereDT=PTt=1dtisthecumulativedemandandCT=PTt=1ctisthecumulativecapacityoverallperiods.Forthecapacitatedlot-sizingproblemwheretherearenorestrictionsonthecostfunctions, Chenetal. ( 1994 )developacontinuousdynamicprogrammingalgorithmthatisexponentialincomplexitybutdemonstratedtobecomputationallyecientinpractice. AtamturkandHochbaum ( 2001 )providepolynomial-timealgorithmsfortheconstant-capacitylot-sizingproblemwithcapacityacquisitionandsubcontracting. VanHoeselandWagelmans ( 2001 )givefullypolynomialapproximationschemesforthesingle-itemcapacitatedlot-sizingproblem.Forthesameproblem, Bakeretal. ( 1978 )provideabranch-and-boundalgorithmwhile Chungetal. ( 1994 )presentahybridbranch-and-boundanddynamicprogrammingalgorithm. Thereareexactapproachesthattiedynamicprogrammingtointegerprogramming. EppenandMartin ( 1987 )providetighterMIPformulationsforthesingleandmulti-itemlot-sizingproblemsusingavariableredenitionapproach.Theyrstdropthecapacityconstraintsfromthetraditionallot-sizingformulationandrepresentthesubproblemwiththedynamicprogrammingnetworkstructure.Thisshortestpathnetworkcanbewrittenasanintegerprogram(IP),withthearcscorrespondingtobinaryvariablesandthenodescorrespondingtoowbalanceconstraints.Theythenrelatethevariablesofthetraditionalmodeltothenewsetofvariablesthroughalineartransformation.By 18

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Martinetal. ( 1990 )formulatepolynomiallysolvableoptimizationproblemsasshortestpathproblemsbyusingdynamicprogramming.TheythenrepresentthedynamicprogramasanLPhavingapolynomialnumberofvariablesandconstraints.TheextremepointsofthisLParerepresentedbythesolutionvectorsoftheDP,andthedualoftheLPprovidestheDPformulation.Theyalsoshowthatwithanappropriatechangeofvariables,theLPformulationobtainedfromtheDPprovidesapolyhedraldescriptionofthemodelconsidered. Oneofthemostimportantmotivationsbehindreplacingassetsisadvancesintechnology.Assetsinusemaybecomeobsolete,astechnologicalimprovementsmakeitpossiblefornewerassetstooperatemoreeciently.Therefore,astechnologyevolves,managersmighttendtokeeplessinventorytotakeadvantageofnewtechnologyandincreasetheperformanceofthesystem.Anothermotivationforreplacementisthedeteriorationoftheassetsastheyareusedovertime.Deteriorationresultsinincreasedoperatingandmaintenancecostsandreducedcapacityduetomachinebreakdownsnecessitatingthereplacementoftheagedassetwithanewerone. 19

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Anumberofpapershaveexaminedtheserialequipmentreplacementproblem.Therstdynamicprogrammingformulationforthisproblemisprovidedby Bellman ( 1955 ).Inthisformulationthestatespacerepresentstheageoftheasset,andthedecisionsincludewhethertokeeporreplaceanassetineachstage.Later, Wagner ( 1975 )providesanotherdynamicprogrammingformulationinwhichthestatesarerepresentedbythetimeperiodsandthedecisionsarehowmanyperiodstokeeptheasset.Thedynamicprogrammingformulationoftheproblemundertechnologicalchangeandmultipleassetsisalsostudiedintheliterature(see,e.g. Beanetal. ( 1985 ), Beanetal. ( 1994 ), Oakfordetal. ( 1984 )).Forothersingleassetreplacementstudiesconsideringtheimpactoftechnologicalchangeoncapitalandoperatingcosts,see HoppandNair ( 1991 ), HritonenkoandYatsenko ( 2007 )and Regnieretal. ( 2004 ),amongothers. Jonesetal. ( 1991 ).Theyprovethatinanoptimalpolicy,groupsofsameagedassetsinthesametimeperiodareeitherkeptorreplacedtogether,assumingconstantdemandandnocapitalbudgetingconstraints.Undermildassumptions,theyalsoprovethatitisneveroptimaltoreplacenewerclustersbeforeolderclusters.Theserulesvastlyreduce 20

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TangandTang ( 1993 )proveastrongerresult,whichstatesthatinanyperiod,anoptimalreplacementpolicyeitherkeepsorreplacesallmachinesregardlessofage. ChildressandDurango-Cohen ( 2005 )considerastochasticversionoftheparallelreplacementproblemwithincreasingfailureratesandanalyzethestructureofoptimalpoliciesundergeneralclassesofreplacementcostfunctions. Chen ( 1998 )providesa0-1integerprogrammingformulationoftheproblemandusesBenders'decompositionwhile McClurgandChand ( 2002 )provideaforwardalgorithmforsolvingtheproblemwithdiscountedcosts. Chandetal. ( 2000 )integratetheparallelreplacementproblemwiththecapacityexpansionproblemanduseenumerativeandheuristicalgorithmstosolvetheproblem. Rajagopalan ( 1998 )providesanintegerprogrammingformulationmodelwithuctuatingdemandandvarioustechnologychoices,andgivesadualbasedsolutionproceduretosolvethisproblem. HartmanandLohmann ( 1997 )presentanintegerprogrammingmodelforsolvingthedemandconstrainednitehorizonparallelreplacementproblemwithhomogeneousassetswherepurchases,leasesandrebuildsareviablereplacementoptions,andanalyzereal-sizedproblemsfromtherailroadindustry. Hartman ( 2000 )providesageneralintegerprogrammingmodelincludingdemandandcapitalbudgetingconstraintsandillustratesthatthelinearprogrammingrelaxationoftheparallelreplacementproblemundereconomiesofscalehasintegerextremepointsifthebinaryvariablesarexed. 1.2.1 ,mostoftheintegerprogrammingbasedapproachesproposedforlot-sizingfocusonthelot-sizingpolyhedronwithoutconsiderationoftheobjectivefunction.Ourcontributioninthisstudyisprovidinganapproachthatusestheobjectivefunctiontoguideusinaddingvalidinequalitiesthattightenthefeasibleregioninthepartofthepolyhedronwhereanoptimalsolutionlies.Werstdescribenewvalidinequalitiesforthesingle-itemcapacitatedlot-sizingproblemthatarederived 21

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Ourresearchontheparallelreplacementproblemshasthefollowingfourcontributions.First,weprovethatPRESisNP-hard,asthishasnotyetbeenshownintheliterature.Second,wedenenewcuttingplanesfortheproblemtostrengthenitsformulation.Thecuttingplanesexploittheoptimalreplacementstrategiesandthenetworkstructureoftheproblem.Computationalresultsshowthattheincorporationofthesecutsintheproblemformulationsignicantlyreducesthetimerequiredtosolveparallelreplacementproblems.Third,weprovideanintegerprogrammingmodelfortheparallelreplacementproblemwherewedirectlyincorporatetheeectoftechnologicalchangeanddeteriorationonthesystemcapacity,andgivesomeinsightsonoptimalassetreplacementdecisionsundertechnologicaladvancesanddeterioration.Fourth,weanalyzeandprovideoptimalsolutioncharacteristicsforPRESundertechnologicalchange,and,usingtheseproperties,weextendtheinequalitiesforPREStothetechnologicalchangecase. Althoughthisresearchaddressessolutionalgorithmsforlot-sizingandparallelreplacementproblems,webelievethattheresultspresentedinthisthesismaygiveinsightsonsolvingotherproblems.Sincebothofthestudiedproblemshavexedchargenetworkowproperties,thedevelopedsolutionapproacheshavethepotentialtobeusedforotherxedchargenetworkowproblems.Itisalsopossibletogeneralizesomeoftheresults 22

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Theremainderofthisthesisisoutlinedasfollows. InChapter 2 ,weconsiderthesingleitemcapacitatedlot-sizingproblem(CLSP).WeuseiterativesolutionsofforwardandbackwarddynamicprogrammingformulationsfortheCLSPtogeneratevalidinequalitiesforanequivalentintegerprogrammingformulation.Theinequalitiesessentiallycaptureconvexandconcaveenvelopesofintermediate-stagevaluefunctions,andcanbeliftedbyexaminingpotentialstateinformationatfuturestages.Wetestseveralpossibleimplementationsthatemploytheseinequalities,anddemonstratethatourapproachismoreecientthanalternativeintegerprogrammingbasedalgorithms.Forcertaindatasets,ouralgorithmalsooutperformsapuredynamicprogrammingalgorithmfortheproblem. InChapter 3 ,westudyamixedintegerprogrammingmodelforsolvingthemulti-itemcapacitatedlot-sizingproblem(MCLSP),whichassumessharedcapacityontheproductionofitemsineachperiodthroughoutaplanninghorizon.WederivevalidboundsonthepartialobjectivefunctionoftheMCLSPformulationbysolvingtherstt-periodrelaxationsoftheproblemusingdynamicprogrammingandintegerprogrammingtechniques.Wethenliftthesevalidinequalitiesbystrengtheningthecutcoecientsviaback-liftingtechniques.Usingthesetechniques,wedevelopeectivesolutionalgorithms,inwhichwechangetheorientationofthepartialobjectivefunctioninequalitiessuchthattheycutothefractionaloptimalsolutions.WetesttheeectivenessoftheproposedvalidinequalitiesonrandomlygeneratedMCLSPinstances. InChapter 4 ,westudyPRES.WeshowthatPRESisNP-hard,andderivecuttingplanesfortheintegerprogrammingformulationoftheparallelreplacementproblemconsideringconstantdemand.Themotivationbehindthecutsisthe\no-splittingrule"intheliterature.Experimentalresultsillustratetheeectivenessofthecutswithrespecttosolvingtheintegerprogramsinacut-and-branchframework. 23

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5 ,weconsidertheparallelreplacementproblemundertechnologicalchangeanddeteriorationandprovideitsintegerprogrammingformulation.Ourmodelincorporatestechnologicalchangeasagainincapacity,whiledeteriorationisconsideredintermsoflossincapacity,increasedoperatingandmaintenancecosts(O&M)costsanddecreasedsalvagevalues.Weillustratehowtechnologyanddeteriorationaecttheoptimalreplacementpolicyandgivesomeinsightsintotheproblemandoptimalsolutioncharacteristics.WealsoextendtheinequalitiesdevelopedforPREStothetechnologicalchangecaseanddemonstratetheireectivenessinthecomputationalexperiments.WeconcludewithChapter 6 ,anddiscusspromisingfuturedirections. 24

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Foreachperiodt=1;:::;T,letpt,standhtdenotetheper-unitproduction,set-upandinventorycostsforperiodt,respectively.Also,denevariablextastheamountproducedinperiodt,ytasabinaryset-upvariableforperiodt(whichequals1ifproductionoccursinperiodtand0otherwise),anditastheamountofinventoryheldattheendofperiodt.TheclassicalintegerprogrammingformulationfortheCLSPisgivenas: minTXt=1(ptxt+styt+htit)(2{1) subjectto: Theobjectivefunction( 2{1 )minimizesthecostsassociatedwithset-up,productionandinventorydecisionvariables.Constraints( 2{2 )enforceowbalanceconditionsthatrequireinventoryremainingattheendoftimettoequalpreviouslyheldinventoryplusnew 25

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2{3 )restrictproductioncapacitytonotexceedctifyt=1,andprohibitproductionifyt=0,ineachperiodt.Constraints( 2{4 )denenonnegativityrestrictionsonthei-andx-variables,and( 2{5 )denebinaryrestrictionsonthey-variables.Notethatwhiletheinitialinventoryi0isfreetotakeanyvalue,weassumethatitequalszerointhisdissertation.Alsonotethattheitandxtvariablesareintegerifytisinteger. Thisstudymakescontributionstothepolyhedralcharacterizationoflot-sizingproblemsbydeninganewsetofvalidinequalitiesfortheCLSPthatarederivedfromtheend-of-stagesolutionsofadynamicprogrammingalgorithm.Wepresentseveralimplementationsthatincorporateinequalitiesintotherootnodeofthebranch-and-boundtree.ComputationaltestsindicatethatouralgorithmismoreecientthanthesolutionofthetraditionalMIPformulationandtheapproachof AtamturkandMu~noz ( 2004 )onarandomlygenerateddataset.Asinmuchofthecitedrelatedliterature,weadvocatetheuseofourapproachonlywhenastraightforwarddynamicprogrammingapproachisintractableduetothesizeoftheinputdata,orwhenthelot-sizingconstraintsareasubsetofalargerproblem. Therestofthischapterisoutlinedasfollows.InSection 2.2 ,wereviewthedynamicprogrammingformulationfortheCLSPandthenderivevalidinequalitiesbasedonforwardandbackwardapproachesinSection 2.3 .Section 2.4 discussesimplementingthevalidinequalitiesandfollowswithcomputationalresultsthatillustratetheeciencyofourapproach. Florianetal. ( 1980 )denesthestateofthesysteminperiodtasthecumulativelevelofproductionthroughtimet.Ourapproachisquitesimilar,althoughwedenethestateofthesysteminperiodtastheinventorylevelattimet. 26

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sinceinventorymustalwaysbenonnegativeandsucienttocoverfuturedemandsifcapacityinfutureperiodsisnotsucienttocoverthesedemands. Similarly,deningCt=Ptj=1cjandDt=Ptj=1dj,8t=1;:::;T,themaximuminventorylevelatperiodt=1;:::;Tinanyoptimalsolutionisgivenby: wherethersttermof( 2{7 )givesthemaximuminventorythatcouldaccumulateaftertperiodsandthesecondtermgivesthecumulativedemandinfutureperiods.Asthisisanitehorizonproblem,noinventoryremainsafterthenalperiodofproductioninanoptimalsolution. Notethatinperiodt=1;:::;T,giveninventorylevelLtitUt,productioninanyoptimalsolutionatperiodtliesinthesetXt;it=fmaxf0;it+dtUt1g,:::,minfct;it+dtLtgg.SettingF0(0)=0(sincewehaveassumedi0=0),theforwarddynamicprogrammingrecursioncannowbewrittenas: whereyt=1ifxt>0andyt=0otherwise.TheoptimalobjectivefunctionvalueisdenedasFT(0). Thisdynamicprogrammingformulationcanberepresentedasanacyclicgraph,asshowninFigure 2-1 (depictedinSection 2.3 ).Thenodesrepresentfeasiblestatesineachperiod,whilethearcsrepresentfeasibledecisionsforeachstate.Thegoalistond 27

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2-1 aremerelylabeledwiththeinventoryattheendoftheperiod.)Here,thearclengthsaredenedbythecostsassociatedwitheachdecisionarc. ThenumberofnodesgeneratedforagiveninstanceisPT1t=1(UtLt)+2,andthemaximumnumberofarcsisPT2t=1(UtLt)(Ut+1Lt+1)+(U1L1)+(UT1LT1).Thenumberofarcsdeterminesthecomplexityofsolvingthisproblembydynamicprogramming,andsinceUtLtDtforeacht=1;:::;T,thecomplexityofthealgorithmisgivenbyO(T+TD2T),asallLtandDtcanbedenedinO(T)time.SinceTD2T>>T,thecomplexityofthealgorithmisessentiallysummarizedbyO(TD2T). Thebackwardrecursion,whichtraversesthenetworkinFigure 2-1 fromnode(T;0)to(0;0),isdenedsimilarlyas: where,Gt(it)representstheminimumcostofmakingoptimalproductiondecisionsthroughperiodt,endingwithinventoryitinabackwardrecursion,andasbefore,yt=1ifxt>0andyt=0otherwise.Inthiscase,theinitialconditionisdenedasGT(0)=0andtheoptimalobjectivefunctionisG0(0). Notethatthestatespaceandfeasibleperiodicdecisionsareidenticalforeachdynamicprogrammingapproach.Thus,thecomplexityofthebackwardrecursionisthesameastheforwardrecursion.However,thebenetofusingbothapproachescanbereadilyillustratedbyexaminingtherepresentativenetworks,whicharealsothesame. Thenumberofnodes(states)inthe\middle"ofthenetworkisgenerallygreaterthanthenumberofnodesateachendofthenetwork.Thisisbecausethenetworkinitiatesandendswithasinglenode(noinventory).Fromeachoftheseendnodes,thenumber 28

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2{8 )usuallyapproachesitsworst-casepseudopolynomialrunningtime,thatalgorithmisrarelyimplementedinfullwhenDTislarge.However,atruncatedversionofthisalgorithmyieldsinformationthattranslatestovalidinequalitiesfortheCLSPmixed-integerprogrammingformulation. Webeginbynotingthatforanyt=1;:::;T,thefollowinginequalityisvalid: sinceFt(i1)
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2{10 )(i.e.,anysolution(^x;^y;^i)thatsatises( 2{12 )mustalsosatisfy( 2{10 )). Proof. 2{11 )mustbevalidforatleastonevalueofi,itissucienttoshowthatFt(Lt)htLtFt(i)htiforeachi=Lt+1;:::;Ut.SupposebycontradictionthatFt(Lt)htLt>Ft(i)htiforsomeLt+1iUt.ModifythesolutionassociatedwithFt(i)byproducingiLtfewerunitsofinventoryatthemostrecentperiod(s)ofproduction.ThenamodiedfeasiblesolutionwouldexisthavingLtunitsofinventory.Letting^Ft(Lt)representtheobjectivefunctionofthemodiedsolution,wehavebFt(Lt)+ht(iLt)
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forparametersmtqandbtq,q=1;:::;Qt,whereQtisthenumberofsegmentsdeningtheconvexenvelope.(Suchinequalitiesmustbevalidbytheassumptionthatinequalities( 2{13 )formtheconvexenvelopeofFt(it).)OurprocedureforderivingtheseinequalitiesisgiveninAlgorithm 1 .Inthisalgorithm,atstaget,givencomputedvaluesFt(Lt);:::;Ft(Ut),weensurethatmt1<
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i.e.,conv(Pt)istheepigraphofthestagetvaluefunction,projectedontothedimensionsyt+1;:::;yT. Supposethatwebeginbyxingyt=1foranyperiodtsuchthatLt>0.Forsimplicityinthefollowingdiscussion,however,assumethatnosuchvariablescanbexedto1,exceptfory1,whichmustequal1ifd1>0.(Thefollowingresultscanreadilybemodiedforthecaseinwhichsomeyt-variablescanbexedto1fort>1.)AfterexecutingAlgorithm 1 above,deneit;q1anditqtobethelowerandupperinventorylevels,respectively,thatdenesegmentqofthevaluefunction,8q=1;:::;Qt. Proof. DeneWtobethe(Tt+1)(Tt+1)matrixwhererow1ofWisgivenbywTt+2wTt+1,andwhererowiofWisgivenbywTt+2wi1,fori=2;:::;Tt+1.Wisabinary-valuedmatrixwith1'sinitsrstcolumnandonitsdiagonal,and0's 32

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2{13 )forstaget,segmentq,denesafacettoconv(Pt). Proof. 2{13 ).DeningeandeiasintheproofofLemma 2 ,considerpointswi=(Ft(itq);(eei))foreachi=1;:::;Tt,andwTt+1=(Ft(it;q1);e).TherstTtpointsmustbelongtoconv(Pt)bytheassumptionthatnovariableyu,foru=t+1;:::;T,mustbexedto1inanyfeasiblesolution,whilethelastpointbelongstoconv(Pt)sinceopeningupproductioninallfutureperiodsmustbefeasiblegivenaninventorylevelofit;q1Lt.Furthermore,thesepointsarebindingon( 2{13 )bydenitionofFt(itq)andFt(it;q1).Considerthe(Tt)(Tt+1)matrixWinwhichrowiisgivenbywTt+1wi,foreachi=1;:::;Tt.Columns2;:::;Tt+1formanidentitymatrix,whichveriesthatWhasfullrowrank,andhencepointswi;8i=1;:::;Tt+1,areanelyindependent. However,supposenowthatgivensomesegmentq2f1;:::;Qtgforperiodt,wehavethatallfeasiblesolutionstotheCLSPmustsetyu=1,forsomet+1uT,giventhattheinventorylevelafterperiodtisitq.Inthiscase,ourderivationofpointswiintheproofofProposition 1 isinvalid,andinfact,( 2{13 )doesnotnecessarilyinduceafacettoconv(Pt).Instead,wenowinvestigatehowtolift( 2{13 )intoafacet-deninginequalitytoconv(Pt)inthiscase. Atthesmallestfeasibleinventorylevelforperiodt,denet1=Lt,anddeneSt1ft+1;:::;Tgasthesetoftimeperiodsuforwhichyu=1ineveryfeasiblesolutionforwhichit=itqinstaget.Incrementthisinventoryleveluntilthereexistsau2St1suchthatyu=0insomefeasiblesolution.Lett2bethisinventorylevel,anddeneSt2St1

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Givenaconstraint( 2{13 )correspondingtostaget,segmentq,denersuchthattritq
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2{16 ). 2{10 )and( 2{12 ),wecanalsogenerateinequalitiesconsideringthemaximumstatevalueineachperiod.Specically, andthestrengthenedinequality: canbothbegeneratedwithoutcuttingoanyoptimalsolutions,althoughtheymaycutofeasiblenon-optimalsolutions.Hence,whilewecanuse( 2{19 )inoptimizingCLSP,theseinequalitiesarenottechnicallyvalid. 2{19 ). Proof. wheretheinequalityisduetosavingcostsfromproducing(Utit)fewerunits,aswellaspotentialsavingsininventoryandset-upcosts.ButassumingFt(it)htit>Ft(Ut)htUt,thenbynoting( 2{20 ),wealsohave:Ft(it)htit>Ft(Ut)htUt>hbFt(it)+ht(Utit)ihtUt)Ft(it)>bFt(it):

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AswiththeconvexenvelopedenedbythelowerboundsonthevaluefunctionFt(it),wecandeneaconcaveenvelopefromtheupperboundsonFt(it)as: forparametersm0tqandb0tq,q=1;:::;Qt,whereQtisthenumberofsegmentsdeningtheconvexenvelope.TheseinequalitiescanbederivedsimilarlytothosefortheconvexenvelopeasinAlgorithm 1 .Notethattheseupper-boundinginequalitieswillnotcutooptimallinearprogrammingrelaxationsolutionsifimplementedinisolationbecauseoptimalityensuresthatthepartialobjectivefunctionvaluesontheleft-hand-sideof( 2{21 )(andof( 2{18 )and( 2{19 ))areminimized.However,giventhepresenceoflower-boundinginequalities,theupper-boundinginequalitiesservetodistributecostsamongstages.Forinstance,supposethatalower-boundinginequalityoftheform( 2{10 )statesthatthepartialobjectivefunctionthroughstage2tTisatleast1,andthatanupper-boundinginequalityoftheform( 2{18 )statesthattheobjectivethroughstage1t0
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2-1 .Productionoccursinperiods1,3and4asxt=(5;0;4;2)atatotalcostof43(F4(0)fortheforwardrecursionandF0(0)forthebackwardrecursion). Figure2-1. Graphsforforward(left)andbackward(right)dynamicprogrammingrecursions. Thefollowingdiscussionfollowsaccordingtotheforwarddynamicprogrammingapproach.Fort=1,theinequalitycorrespondingto( 2{10 )is: andreducesto: accordingto( 2{12 ).ThesingleinequalitythatdenestheentireconvexenvelopeforF1(i1),from( 2{13 ),is: Ingeneral,theperiod1constraintz1(p1+h1)i1+(s1+p1d1)willalwaysdeneexactlythefunctionF1(i1)ifd1>0,sincecost(s1+p1d1)mustbepaidinanysolutioninordertoaccommodateperiod1demand(recallthati0=0),andeachextraunitproducedisplacedintoinventoryatacostp1+h1.Notethat( 2{24 )isastrongerinequalitythan( 2{23 ),whichisstrongerthan( 3{13 ). 37

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2-2 illustratesthatF2(i2)andF3(i3)arenotconvexoveri3.Thus,wedeneconvexenvelopeinequalities,asgiveninthegure. Figure2-2. GraphrepresentationofFt(it)valuesandassociatedconvexenvelopeinequalities. Fort=2,connectinginventorylevels0and3denes: Theconvexenvelopeisdenedbytwoinequalitiesfort=3.Connectinginventorylevels0and1denes: whileconnectinginventorylevels1and3denes: Proposition 1 indicatesthat( 2{25 )and( 2{27 )denefacetstoP2andP3,respectively,sincetheupperinventorylimitofbothinequalitiesis3,andthusnoy-variablesatfuturetimeperiodsmaybexedto1.However,theupperinventorylimitfor( 2{26 )is1;ifonly 38

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2{26 )byaddingthetermt1(1y4)totheright-hand-sideof( 2{26 ),andthenbycomputingt1=462(3)28asprescribedby( 2{16 ).Theresultinginequalityisgivenby Thus,therstsegmentoftheconvexenvelopeforF3(i3)isessentiallyshiftedupby12unitsify4=0. ThegraphicalrepresentationsofFt(it)inFigure 2-2 alsoillustratetheinequalitiesusedtodenetheconcaveenvelopeofFt(it).Fort=1,asnotedearlier,theconvexandconcaveenvelopesarethesamesinceproductionmustoccurintherstperiod.Fort=2,inventorylevelsbetween0and1dene: whileinventorylevelsbetween1and3dene: Fort=3,thesingleinequality: summarizestheconcaveenvelopeoverallinventorylevels.Aswiththeconvexenvelope,theseinequalitiesmaybestrengthenedthroughasimilarliftingprocedure. Finally,ifdesired,wecanrepeatalloftheaboveproceduresusingthebackwarddynamicprogramminginformation.Forexample,fromFigure 2-1 andt=2,therecursiondenes: 39

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AtamturkandMu~noz ( 2004 )whonotedthat(a)thetightnessofthecapacitieswithrespecttodemandand(b)theratiobetweentheset-upcostandtheinventoryholdingcostplaymajorrolesindeningthedicultyofaCLSPinstance.Specically,instancesaregeneratedforcapacity-to-demandratiosc2f2;3;4;5g,set-up-to-holding-costratiosf2f100;200;500;1000g,andnumberofstagesT2f90;120;150g.Fiverandominstancesaregeneratedforeachcombinationoftheseparameters,foratotalof240instances.Theunitproductioncostspt,demandsdt,capacitiesctandset-upcostsstarerandomlygeneratedfromintegeruniformdistributionswithrangesasfollows:pt2[81;119];dt2[1;19];ct20:75cd;1:25cd;st20:90fh;1:10fhwheredandhareaveragesfordemandandholdingcosts.Theholdingcosthtisxedat10foreachperiod. 2{1 ){( 2{5 ),andsolvetheresultingmodelusingCPLEXwithitsdefaultsettings. 40

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Wecomparetheeciencyofsolvingthefollowingmodelsinourcomputationalexperiments. 2{1 ){( 2{5 ),withoutaddinganyuserinequalities. 2{12 )withbase. 2{12 )and( 2{19 )withbase. 2{13 )thatviolatetheinitialLPrelaxationof( 2{1 ){( 2{5 ),withbase. 2{21 )withenvl. 2{13 )thatviolatetheinitialLPrelaxation,withbase. 2{12 )foreachstageconsidered,andprovidestighteningwiththeliftedconvexenvelopeinequalitieswheretheycutothecurrentLPrelaxationsolution. AtamturkandMu~noz ( 2004 ),withoutaddinganyusercuts. AtamturkandMu~noz ( 2004 ),withbc.Weaddtheseinequalitiesbasedontheheuristicseparationapproachof AtamturkandMu~noz ( 2004 )withoneroundtotheinitialLPrelaxationofthebottleneckmodel.ThenwesolvethestrengthenedmodelwithCPLEX. AtamturkandMu~noz ( 2004 )showthatbcliftisequivalenttothe(l;S)inequalityof Baranyetal. ( 1984b )fortheuncapacitatedcase,andatleastasstrongastheowcoverinequalityof Padbergetal. ( 1984 )andthesurrogateowinequalityof Pochet ( 1988 ).Sincebcliftwasalsoshowntobethemosteectivecuttingplaneimplementation,werestrictthecomputationalcomparisonofourprocedurestobcandbclift.Notethatthereareseveralfurtherimplementationsthatcombinetheuseofourinequalities,but 41

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2-1 { 2-6 wereportthefollowingdatabycolumn: { 2{12 ),( 2{19 )). { 2{13 )thatviolatetheinitialLPrelaxation,( 2{21 )). { 2{12 ),liftedinequalities( 2{13 )thatviolatetheinitialLPrelaxation). Tables 2-1 and 2-2 presentresultsforT=90,whereeachtableentrycorrespondstotheaverageperformanceofanalgorithmover20instances(veeachforf=100,200,500and1000).WeobservethatthepresentedDPbasedstrategiesallimproveuponthedirectsolutionoftheCLSPbyCPLEX,eitherwiththebaseorbottleneckformulation.The 42

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2{13 )thatviolatetheinitialLPrelaxation,appearstoprovidethemostbenet,withafactorof2improvementinCPUruntimeoverbase,averagingoverallinstances.Theenvlstrategy,whichissimilartoliftenvbutdoesnotperformtheliftingoperation,alsoperformswellinmostinstances.Thebcliftimplementationimprovesuponbcasexpectedfortheseinstances,buttheliftenvstrategyperformsbetterthanbcliftimplementation.Hence,weomitthebcandbcliftcomputationalresultsfromthispointforward. Tables 2-3 and 2-4 presenttheresultswhenrepeatingtheexperimentsforT=120.Asexpected,theseinstancesaresignicantlymorechallengingthantheT=90instances.AswiththeT=90case,theenvlandliftenvlstrategiesperformedwell,withnearlyafactorof6improvementoverbasewhenaveragingoverallinstances.Theweakl+env(60,60)strategyprovidedthebestcomputationalresults(nearly8timesimprovementoverbase),althoughitrequiredasignicantportionoftheDPtobesolvedingeneratingtheinequalities. Tables 2-5 and 2-6 providecomputationalresultswhenT=150.Fortheseexperiments,somealgorithmsfailedtosolveinstanceswithintheallotted1800CPUseconds.Thenumberofsuchinstancesisdenotedwithasuperscriptinthenalcolumn.Ifaninstancefailstosolvewithinthetimelimit,atimeof1800secondsisrecordedasthecomputationaltimefortheinstance,thusunderestimatingthetrueaveragetimerequiredtooptimizetheseinstances. Ofthe80instancesforT=150,34couldnotbesolvedtooptimalitybybasewithinthetimelimit,while77couldbesolvedbyliftenv(100)and71couldbesolvedbyweakl+env(50).Theliftenv(75)strategyreducedtheaveragebasesolutiontimebyafactorof4,whileliftenv(50)improvedsolutiontimebynearlyafactorof2whenaveragedoveralloftheinstances. 43

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SummaryofexperimentsforT=90,c=2;3. expstage initgapgapimpDpineqCPXineqnodesineqtimetime 2.98-0158590980.0029.44weakl30 2.3121.5830143380300.0324.53weakl45 1.9632.4445124257740.0819.24weakl60 1.5147.1960104319690.1431.01weaklu(30,30) 2.3321.0960138561410.0333.59weaklu(45,45) 1.9831.8290120322100.0825.04weaklu(60,60) 1.5446.4712095215270.1325.93envl30 1.9036.75132125130340.0312.12envl45 1.4849.8822712286390.0823.80envl60 1.0365.6732913171140.1338.63envlu(30,30) 1.9036.75133122122460.0311.65envlu(45,45) 1.4849.8822812384600.0824.04envlu(60,60) 1.0365.6533111682250.1340.69liftenvl30 1.8936.96135110154480.0316.72liftenvl45 1.4650.5223010840460.0812.37liftenvl60 1.0066.3933311563180.1336.75weakl+env(30,30) 1.8936.96165114143590.0314.85weakl+env(45,45) 1.4650.5227510076550.0819.60weakl+env(60,60) 1.0066.3939310927250.1320.83bc2.98-03321587360.00127.82bclift1.5343.63149?170192201.8053.38 5.00-0139372100.0018.15weakl30 3.8323.4130162197610.1013.97weakl45 3.1536.3345140246740.2217.98weakl60 2.4948.9860128178040.2716.31weaklu(30,30) 3.8622.8060149217740.1114.65weaklu(45,45) 3.1935.6090134254550.2220.32weaklu(60,60) 2.5348.1912011997770.2713.96envl30 3.1637.47142117211440.1120.68envl45 2.4651.1124313829430.2110.91envl60 1.7764.4335413118800.2718.52envlu(30,30) 3.1637.47142117211440.1120.81envlu(45,45) 2.4651.1124313429510.2111.04envlu(60,60) 1.7764.4135412819540.2718.79liftenvl30 3.1337.9014414071100.1010.12liftenvl45 2.4451.5824513127350.2211.76liftenvl60 1.7664.7935613317530.2719.77weakl+env(30,30) 3.1337.90174125172350.1017.45weakl+env(45,45) 2.4451.5829012135760.2113.03weakl+env(60,60) 1.7664.7941612419190.2719.31bc4.99-0301416330.0080.84bclift2.2551.65173?25769671.6227.54

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SummaryofexperimentsforT=90,c=4;5. expstage initgapgapimpDpineqCPXineqnodesineqtimetime 7.21-0116211280.009.70weakl30 5.4624.383013385540.206.47weakl45 4.5137.164511969670.306.62weakl60 3.4751.346011338380.375.97weaklu(30,30) 5.5023.736012794420.197.22weaklu(45,45) 4.5736.409011977520.307.56weaklu(60,60) 3.5450.4912010047150.368.58envl30 4.5736.8014212435440.195.82envl45 3.5550.8424110616940.307.44envl60 2.4066.9434812712050.3613.89envlu(30,30) 4.5736.8014212537310.196.56envlu(45,45) 3.5550.8424110619380.307.94envlu(60,60) 2.4066.9434812511900.3613.80liftenvl30 4.5636.9114310051430.196.66liftenvl45 3.5450.9524210217690.307.78liftenvl60 2.3967.0434911410860.3613.93weakl+env(30,30) 4.5636.9117311941470.196.43weakl+env(45,45) 3.5450.9528710019600.308.03weakl+env(60,60) 2.3967.0440911814340.3614.14bc7.21-0176317440.0024.81bclift2.5163.56216?16730711.8111.29 9.68-0123387800.0016.30weakl30 7.4323.1030127141930.248.64weakl45 6.0637.174511886280.367.26weakl60 4.6052.08609457010.436.61weaklu(30,30) 7.5022.436099156010.259.55weaklu(45,45) 6.1436.359010297850.368.74weaklu(60,60) 4.6951.161209646370.428.26envl30 6.2335.6713210143800.245.84envl45 4.7750.7623211514650.367.32envl60 3.2366.5834413311350.4214.65envlu(30,30) 6.2335.6713210143780.255.94envlu(45,45) 4.7750.7623212113580.367.46envlu(60,60) 3.2366.5834413711510.4315.57liftenvl30 6.2135.8613313447360.257.20liftenvl45 4.7550.9923313616410.388.42liftenvl60 3.2266.7334414010750.4414.93weakl+env(30,30) 6.2135.8616312247580.266.84weakl+env(45,45) 4.7550.9927811316080.388.15weakl+env(60,60) 3.2266.7340413111620.4415.80bc9.68-0265466310.0053.76bclift3.0467.74251?8127541.857.41

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SummaryofexperimentsforT=120,c=2;3. expstage initgapgapimpDpineqCPXineqnodesineqtimetime 2.89-01887512840.00407.74weakl40 2.2122.86401992133890.06188.11weakl60 1.8235.91601651129310.19105.45weakl80 1.4348.8680137681990.3279.10weaklu(40,40) 2.2322.41801992692070.07246.76weaklu(60,60) 1.8435.331201671745210.19177.74weaklu(80,80) 1.4548.151601261464840.32204.85envl40 1.8834.67199153729520.07104.55envl60 1.4550.08328173156550.2079.81envl80 1.0363.9748516083900.3190.81envlu(40,40) 1.8834.67200147679570.0781.40envlu(60,60) 1.4849.33330177122510.1963.42envlu(80,80) 1.0762.57486170113220.31119.10liftenvl40 1.8735.00201159485060.0680.31liftenvl60 1.4550.20332130431820.19144.10liftenvl80 1.0263.8548915284190.3280.97weakl+env(40,40) 1.8735.00241148787740.07117.29weakl+env(60,60) 1.4452.71392142130750.1955.46weakl+env(80,80) 1.0764.58569140284360.32170.14 5.08-02518946230.00516.09weakl40 3.9322.15402404095450.23345.65weakl60 3.1836.15602071605430.46160.68weakl80 2.3951.3080173333390.5835.08weaklu(40,40) 3.9521.65802434425240.23333.01weaklu(60,60) 3.2135.541202122531280.45276.53weaklu(80,80) 2.4350.60160183732920.58116.56envl40 3.3833.39196206615440.2364.40envl60 2.5848.61345226511910.45132.34envl80 1.7564.9850623853080.5871.47envlu(40,40) 3.3833.39196195615710.2264.88envlu(60,60) 2.5848.61345217519250.45133.37envlu(80,80) 1.7564.9850623552420.5866.78liftenvl40 3.3633.79198213737930.2391.37liftenvl60 2.5748.87347218144170.4551.66liftenvl80 1.7365.3050823267240.5893.13weakl+env(40,40) 3.3633.79238210860220.2386.32weakl+env(60,60) 2.5748.87407230203230.4664.53weakl+env(80,80) 1.7365.3058819292230.5787.82

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SummaryofexperimentsforT=120,c=4;5. expstage initgapgapimpDpineqCPXineqnodesineqtimetime 7.29-01886665670.00388.91weakl40 5.5323.59402023702660.42285.68weakl60 4.4538.22601681887710.67151.83weakl80 3.3353.4980158652230.7860.87weaklu(40,40) 5.5723.02801984166990.41294.18weaklu(60,60) 4.5037.571201832014020.66224.51weaklu(80,80) 3.3852.76160151731920.7899.25envl40 4.7833.931911781344690.41144.22envl60 3.6849.21336221240600.6588.54envl80 2.4566.06505224116770.78163.49envlu(40,40) 4.7833.931921661350390.41143.15envlu(60,60) 3.6849.21336207223820.6577.20envlu(80,80) 2.4566.06505226114430.81163.54liftenvl40 4.7634.291921901331030.42152.21liftenvl60 3.6549.61337166177570.6554.94liftenvl80 2.4366.34505160149040.78158.61weakl+env(40,40) 4.7634.282321751499680.41166.42weakl+env(60,60) 3.6549.61397172159780.6652.17weakl+env(80,80) 2.4366.3458521657110.7868.33 9.24-01736966830.00321.65weakl40 6.9624.54401893499710.53216.19weakl60 5.5539.58601761363930.78114.15weakl80 4.1354.98801531214610.91113.13weaklu(40,40) 7.0123.94801673604490.52263.32weaklu(60,60) 5.6138.871201622298830.77184.16weaklu(80,80) 4.2054.251601381124020.90127.53envl40 6.0334.82195171942890.5291.14envl60 4.5151.0734224275710.7735.62envl80 3.1366.2150222540000.9047.37envlu(40,40) 6.0334.82195171942890.5291.20envlu(60,60) 4.5151.0734224275710.7735.92envlu(80,80) 3.1366.2150222540000.9047.75liftenvl40 6.0135.02196206762200.5381.22liftenvl60 4.5051.2234320470790.7735.67liftenvl80 3.1166.4250223836040.9049.11weakl+env(40,40) 6.0135.02236185773990.5278.71weakl+env(60,60) 4.5051.2240324277170.7738.47weakl+env(80,80) 3.1166.4158221549870.9059.59

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SummaryofexperimentsforT=150,c=2;3. expstage initgapgapimpDpineqCPXineqnodesineqtimetime 2.73-023915794840.001064:419weakl50 2.0523.89802598859700.12892:827weakl75 1.6737.011252235678130.38701:654weakl100 1.2451.951721833057530.61434:142weaklu(50,50) 2.0922.491002549654590.12925:257weaklu(75,75) 1.7335.291502045911900.38679:434weaklu(100,100) 1.3049.912001712739830.62451:042envl50 1.8232.902562683160560.12463:542envl75 1.4148.094312081251140.37495:383envl100 0.9663.65636226353990.62368:541envlu(50,50) 1.8232.882572584218500.12554:182envlu(75,75) 1.4148.094321891030540.37360:711envlu(100,100) 0.9663.61637187388300.61383:712liftenvl50 1.8133.352592343285440.12451:231liftenvl75 1.3848.95435215646630.37298:551liftenvl100 0.9564.17640194184530.60251.67weakl+env(50,50) 1.8033.413091853816730.12595:952weakl+env(75,75) 1.3848.97510203851800.37328:281weakl+env(100,100) 0.9564.06740247308200.60408:361 5.06-028314473860.001028:1510weakl50 3.7425.35652968845410.45882:697weakl75 2.9041.211012304868480.91596:344weakl100 2.1555.621392041350591.15206:25weaklu(50,50) 3.7724.5810026810024710.44982:909weaklu(75,75) 2.9839.961502416123710.93760:897weaklu(100,100) 2.2154.532001913070761.15621:784envl50 3.2934.842643063072120.44514:673envl75 2.4251.57463314818710.91321:182envl100 1.7065.60691324331071.15523:213envlu(50,50) 3.2834.882643192923340.45512:973envlu(75,75) 2.4151.62463337489850.91301:191envlu(100,100) 1.6965.68692303285171.16400:371liftenvl50 3.2335.802652943384220.45540:734liftenvl75 2.3852.29464326425490.92237:111liftenvl100 1.6965.75693291236071.16392:94weakl+env(50,50) 3.2335.793153182299840.45465:703weakl+env(75,75) 2.3852.28539278993410.92405:492weakl+env(100,100) 1.7165.53793269313341.15431:942

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SummaryofexperimentsforT=150,c=4;5. expstage initgapgapimpDpineqCPXineqnodesineqtimetime 6.95-02899841510.00736:196weakl50 5.1325.90502774587840.87474:743weakl75 4.0541.13752272769471.49342:511weakl100 2.9856.311002171058541.76161:961weaklu(50,50) 5.1625.471002895135790.79520:913weaklu(75,75) 4.1040.511502342083601.29314:401weaklu(100,100) 3.0255.77200195870281.58197:971envl50 4.5434.692752592173130.79448:193envl75 3.3951.24472285338881.28192:861envl100 2.3266.41703279264631.55367:802envlu(50,50) 4.5434.692752672243670.79445:143envlu(75,75) 3.3951.24472285366641.28205:271envlu(100,100) 2.3266.41703278269471.55367:542liftenvl50 4.5234.932762211962580.80355:012liftenvl75 3.3851.39473321399201.28202:731liftenvl100 2.3266.26705315244241.72294:891weakl+env(50,50) 4.5334.883262261940000.80384:582weakl+env(75,75) 3.3851.38548291386331.32203:241weakl+env(100,100) 2.2767.22805282118961.57252:59 9.11-025212062880.00855:339weakl50 6.6426.81652208677001.02706:585weakl75 5.2941.58991844558751.51443:153weakl100 3.8956.821371901997081.78257:562weaklu(50,50) 6.7425.871002149008261.02796:776weaklu(75,75) 5.3940.621501724596801.51549:014weaklu(100,100) 3.9855.882001651832441.77343:751envl50 5.8735.502653042653721.02479:012envl75 4.5050.45460338479301.51246:40envl100 3.0866.21689368320081.77409:102envlu(50,50) 5.8735.502653042765291.03481:482envlu(75,75) 4.5050.45460327479291.50246:19envlu(100,100) 3.0866.21689365321021.78413:152liftenvl50 5.8735.602652713022031.02491:663liftenvl75 4.5050.50460301622351.51247:341liftenvl100 3.0766.32689275297531.77369:242weakl+env(50,50) 5.8735.583152752501091.01394:902weakl+env(75,75) 4.4950.58535263545531.52227:331weakl+env(100,100) 3.1265.92789330277551.77409:422 49

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WegeneratedteninstanceswithT=90,acapacity-to-demandratioc=3andasetup-to-holding-costratiof=10000.Theunitproductioncostspt,demandsdt,capacitiesctandsetupcostsstwererandomlygeneratedwithuniformdistributionswithrangesasfollows:pt2[1;5];dt2[0;600];ct20:7cd;1:1cd;st20:90fh;1:10fh.NotethatthepossiblespreadindemandmakesexecutingtheDPrecursionchallenging.Theholdingcosthtwasxedat1foreachperiod. TheresultsofthisexperimentaregiveninTable 2-7 .Here,wecomparetheweakl+envmethodwithDPinequalitiesgeneratedatvariousnumbersofstageswiththebasestrategy,andwiththecompletesolutionbyDP.Asbefore,ourweakl+envoutperformedbasewhensolving1/9,1/6,and1/3ofthepossibleDPstages.However,theseinstancesdisplaythepotentialtrade-oincomputationalbenetresultingfromtighteningthebasemodelwithDPbasedinequalitiesandthecomputationalexpenseassociatedwithgeneratingtheseinequalities.Asthenumberofstagesincreases,thetimespentbyCPLEXafterthegenerationoftheDPbasedinequalitiesdecreases.However,factoringinthecomputationaleortrequiredtogeneratetheseinequalities,thebestoverallimplementationtesteduses15stages,whichisroughly34percentfasteronaveragethantheDPapproach(givenincolumn1). 50

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ExperimentsforT=90andf=10000. Instance1DPtime=205base-6.820.000540.00851.14weakl+env(10+10)6.149.9725543.28456.22weakl+env(15+15)5.9412.90342158.80163.54weakl+env(30+30)5.0825.51635359.55134.62 Instance2DPtime=155base-7.680.000610.00172.47weakl+env(10+10)6.3816.93362192.1388.92weakl+env(15+15)6.3217.71452446.6994.39weakl+env(30+30)5.9622.40755353.80156.14 Instance3DPtime=219base-7.360.0002120.0024.73weakl+env(10+10)5.821.20312013.8630.58weakl+env(15+15)5.722.554120810.7348.56weakl+env(30+30)4.7535.467019367.2892.56 Instance4DPtime=229base-6.530.000590.00175.19weakl+env(10+10)5.8610.2631534.11252.22weakl+env(15+15)5.7811.49415610.14363.08weakl+env(30+30)5.0223.12684760.44350.48 Instance5DPtime=225base-9.020.0002580.00219.89weakl+env(10+10)7.8413.0825566.44356.42weakl+env(15+15)7.5716.083522016.63174.37weakl+env(30+30)6.2930.276517993.31204.73 Instance6DPtime=220base-7.220.0002330.00491.48weakl+env(10+10)5.819.6728523.55667.84weakl+env(15+15)5.6821.333822710.11249.03weakl+env(30+30)4.9431.586818363.44219.92 Instance7DPtime=231base-6.730.0002340.00172.83weakl+env(10+10)5.6116.6433532.72389.92weakl+env(15+15)5.3820.0640548.25204.95weakl+env(30+30)4.7529.426920861.92217.70 Instance8DPtime=256base-6.780.0002230.00117.22weakl+env(10+10)5.4120.21352142.8091.56weakl+env(15+15)5.3421.24442087.9985.64weakl+env(30+30)4.6132.017417660.14108.26 Instance9DPtime=241base-7.320.000500.0026.56weakl+env(10+10)5.9618.5835434.0249.06weakl+env(15+15)5.7521.45435010.3829.59weakl+env(30+30)5.5624.04724864.0594.16 Instance10DPtime=235base-5.950.000640.0047.41weakl+env(10+10)5.1213.95242022.8029.33weakl+env(15+15)5.0415.2933568.0652.95weakl+env(30+30)4.6621.686219360.89116.53

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2{12 )tothetraditionalMIPformulationbasedonstageinformationfromthepartialexecutionofaforwardDPrecursionandthenaddsliftedcutting-planeinequalitiesinacut-and-branchfashionattherootnode. Forfutureresearch,wewillexploretheuseoftheinequalitiesdevelopedinthispaperwithinproblemdomainsthatcontaintheCLSPconstraintsasasubstructure.OnenaturalextensionofthisworkwouldinvestigatemorecomplexvariationsoftheCLSP,suchasthemulti-itemCLSP,whichweconsiderinChapter 3 .GiventhesuccessofourapproachontheCLSP,webelievethatasimilarapproachmayproveeectiveforothercombinatorialproblemsaswell. 52

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Florianetal. ( 1980 )),andformsthebasisofmanyproductionplanningandinventoryproblems. ThedataforMCLSPissummarizedbelow. Thedecisionvariablesaregivenasfollows: TheMCLSPcanbeformulatedasthefollowingmixedintegerprogram: minMXi=1TXt=1(pitxit+fityit+hitsit) (3{1a) s.t.si;t1+xitdit=sitt=1;:::;T;i=1;:::;M 53

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Theobjectivefunction( 3{1a )minimizesthesumofproduction,set-up,andinventorycosts.Constraints( 3{1b )denetheremaininginventoryforeachproductaftereachtimeperiod.Constraints( 3{1c )forcethebinaryset-upvariableyittoequal1wheneverxitispositive,andensurethatproductionforaniteminaperiodcannotexceedtheminimumofthecapacityinthatperiodandthecorrespondingremainingdemandthroughtheendofhorizon.Constraints( 3{1d )representthesharedcapacityconstraintlinkingtheproductionofdierentitems.Finally,constraints( 3{1e )and( 3{1f )stateboundsandintegerrestrictionsonthevariables.MCLSPdiersfromCLSPbyincorporatingmultipleitemsandthesharedcapacityrestrictiononthem. Lot-sizingisafundamentalprobleminoptimizationwithimportantapplicationsinproduction/inventoryplanningandsupplychainmanagement.Inparticular,theMCLSPstructure( AtamturkandKucukyavuz ( 2008 ); AtamturkandMu~noz ( 2004 ); PochetandWolsey ( 1991 ); Wolsey ( 1958 ))isatthecoreoftheproductionplanningproblemsinvolvingmultipleproductsandlevelsoveranitediscretetimehorizon. PolyhedralstudiesontheMCLSParedevotedtostrengtheningtheproblemformulationbyderivingvalidinequalitiesviaanalysisofknapsack,singlenodeowandsingleperiodrelaxationsoftheproblem( Miller ( 1999 ); Milleretal. ( 2000 2003b ); Padbergetal. ( 1985 ); PochetandWolsey ( 2006 )).TheseapproachesfocusontheMCLSPpolyhedronwithoutconsiderationoftheobjectivefunction.Ourapproachistousetheobjectivefunctiontoguideusinaddingvalidinequalitiesthatservetotightenthefeasibleregioninthepartofthepolyhedronwhereanoptimalsolutionlies.InChapter 2 wegeneratevalidinequalitiesforthesingle-itemcapacitatedlot-sizingproblemthatarederivedfromtheend-ofstageoptimalsolutionsofadynamicprogramming(DP) 54

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TherestofChapter 3 isoutlinedasfollows.InSection 3.2 weproposepartialobjectivefunctioninequalitiesfortheMCLSP.InSection 3.3 wediscussliftingandseparationproblemsassociatedwiththepartialobjectiveinequalities.InSection 3.4 wepresentcomputationalresultstodemonstratetheeectivenessoftheproposedinequalitiesandtheliftingprocedurewithaseparationalgorithm. 3{1 )intoMsingle-itemproblems,whereeachindividualitemcanusetheremainingcapacityafterallottingenoughcapacitytoeachotheriteminordertomeetdemandineachperiod.WeextendthevalidinequalitiesinChapter 2 forthesingle-itemCLSPstothek-itemcase,wherewedecomposetheproblemintotwo-itemproblemsandutilizedynamicprogrammingformulationstoderivevalidinequalitiesfortheMCLSP.Inaddition,weconsiderintegerprogrammingbasedapproachestosolvek-item,t-periodrelaxationsoftheproblem(kM,tT)toobtainvalidboundsonthepartialMCLSPobjectivefunction. 2 forthesingle-itemcapacitatedlot-sizingproblem.SupposethatwehaveemployedthisalgorithmandhaveobtainedtheoptimalcostFj(sj)tofeasiblyaccumulatesjunitsofinventory,forallstagesj=f1;:::;Tgandallpossiblestatevaluesforsj. Foranyt=1;:::;T,thefollowinginequalityisvalid: 55

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Wecancomputestrongerinequalitiesthan( 3{2 )byexaminingallcostsaccruedthroughtimetasafunctionoftheinventorysatstaget.WhilethestagetvaluefunctionFt(st)isnotnecessarilyaconvexfunctionofst,theinequalitiesthatdenetheconvexenvelopeofFt(st)arevalidfortheCLSP.Suchinequalitieshavetheform: fortheslopeandtheinterceptparametersmandb,respectively. NowsupposethatwedecomposeanMCLSPinstanceintoMsingle-itemCLSPs,whereeachCLSPallowsitsitemtoconsumealloftheallowableproductioncapacityateachtimestage.Theinequalitiesgivenin( 3{2 )and( 3{3 )thatwederivefromanyCLSParevalidfortheMCLSPformulation( 3{1 ),sincetheyarebasedonsingle-itemrelaxationsof( 3{1 ). InthecontextoftheMCLSP,wecanstrengthentheseinequalitiesbyincorporatingnecessaryconditionsforfeasibilitybasedonsharedcapacityconstraints.First,givenasingleitemiandthepartialobjectivefunctionPtj=1(pijxij+fijyij+hijsij)correspondingtotherstt=1;:::;Tperiods,wecangeneratevalidlowerboundsonthisfunctionbyminimizingthispartialobjectivefunctionsubjecttotheconstraints( 3{1b ){( 3{1f ).Inthisproblem,weonlyrestrictyi1;:::;yittobebinary,becauseanoptimalsolutionwillexistinwhichallothery-variablesaresetto1.Inaddition,wecaneitheroptimizethisproblemorusethelowerboundobtainedafterapredeterminedsolutiontimelimit.LettingLbethebestlowerboundwithintheallottedtimelimitoverthersttstages,thefollowinginequalityisvalid: 56

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3{2 )and( 3{3 )generatedfromintermediatestagesolutionstothesingle-itemCLSPsmaybeweakenedduetothefactthatweallotallproductioncapacitytotheitemunderexamination.Wethustighten( 3{2 )and( 3{3 )byreducingthepossibleinventorylevelsallowedforitemiateachstageduetonecessarycapacityutilizationoftheotheritems. Theminimuminventorylevelatperiodt=1;:::;Tforitemiinanyoptimalsolutionisgivenby: sinceinventoryofitemimustalwaysbenonnegativeandlargeenoughtocoverfuturedemandsforitemiifcapacityinfutureperiodsisnotsucienttocoverthesedemands(see,e.g., Constantino ( 1996 ),Chapter 2 ). Tocomputethemaximumpossibleinventorylevelforitemiatperiodt=1;:::;T;letCt=Ptj=1cjandDit=Ptj=1dij.Thenthemaximuminventorylevelforitemiatperiodtinanyoptimalsolutionisgivenby: Thersttermof( 3{6 )givesthemaximuminventoryofitemithatcouldaccumulateaftertperiods.Thistermensuresthatwesatisfythedemandforalltheitemsandallotenoughcapacityforthefutureuncovereddemandcorrespondingtotheotheritems.Thesecondtermof( 3{6 )givesthecumulativedemandforitemithatmustbesatisedinfutureperiods. 57

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Wecalculateminimumandmaximuminventorylevelsformulti-itemproblemsatperiodt=1;:::;Tinanyoptimalsolutionasfollows.GivenK=fi1;:::;ikg,theminimuminventorylevelatperiodt=1;:::;Tinanyoptimalsolutionisgivenby: andthemaximuminventorylevelatperiodt=1;:::;Tinanyoptimalsolutionisgivenby: Notethatwealsocomputethelowerandupperinventoryboundsforeachindividualitemusing( 3{5 )and( 3{6 ),sinceindividualboundsmayalsohelptoreducethestatespace.Forinstance,supposethatthemaximuminventorylevelgivenby( 3{9 )fortwoitemsinastageis3units,whilemaximumindividualinventorylevelsfortheseitemsgivenby( 3{6 )are1and2,respectively.Letthestatesforatwo-itemDPalgorithmbegivenbyapairofinventoryvaluesfortheitems.Thentheindividualinventorybound( 3{6 )eliminatesthestates(0;3),(3;0)and(2;1)fromthestatespace.Similarly,iftheminimuminventorylevelgivenby( 3{8 )forthetwoitemsis2,andtheindividualminimuminventorylevelsare2and0,respectively,thenwecanalsoeliminatestates(1;1)and(0;2). TheDPrecursionforthemulti-itemproblemissimilartotheDPrecursionforsingle-itemproblem( 2{8 )giveninChapter 2 .Forperiodt=1;:::;T,giveninventorylevelLBitsitUBitforeachitemi,theproductioninanyoptimalsolutionatperiodtliesinthesetXt;sit=fmaxf0;sit+ditminfUBi;t1;UBi1;:::;ik;tgg,:::,minfct;sit+ditLBitgg.SettingF0(0;:::;0)=0(sincewehaveassumedsi0=0),theforwarddynamicprogrammingrecursionforthemulti-itemproblemcanbewrittenas: 58

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minxit2Xt;sit;8i2K(Xi2K(pitxit+fityit+hitsit)+Ft1(s1t+d1tx1t;:::;skt+dktxkt));81tT;LBitsitUB0it; whereUB0it=minfUBi;t;UBi1;:::;ik;tg,andyit=1ifxit>0andyit=0otherwise.TheoptimalobjectivefunctionisdenedasFT(0;:::;0). Thisdynamicprogrammingformulationcanberepresentedasanacyclicgraph,asshowninFigure 3-1 .Thenodesrepresentfeasiblestatescorrespondingtothepossibleinventorylevelsoftheitemsineachperiod,whilethearcsrepresentfeasibledecisionsforeachstate.ThenodesinFigure 3-1 arelabeledwiththeinventoryattheendofeachperiodforeachitemi2K.Here,thearclengthsaredenedbythecostsassociatedwitheachdecisionarc.NotethatallthearcsinFigure 3-1 arenotpresentedforclaritybutifthereisanarcfromastatesttoanotherstatest+1,thentheremustalsobeanarcfromsttoeachofallpossibleinventoryvalueslowerthanst+1.Thegoalistondashortestpathconnectingnode(0;:::;0)inperiod0,representingtheinitialstateofthesystem(noinventoryattimezero),tonode(0;:::;0)inperiodT,representingthenalperiodoftheproblemwhennoinventoryisneededastheproblemterminates. ThenumberofnodesgeneratedforagiveninstanceisPT1t=1Qi2K(UB0itLBit)+2,andthemaximumnumberofarcsisPT1t=1Qi2K(UB0itLBit)Qi2K(UB0i;t+1LBi;t+1)+Qi2K(UB0i1LBi;1)+Qi2K(UB0i;T1LBi;T1).Thenumberofarcsdeterminesthecomplexityofsolvingthisproblembydynamicprogramming,andlettingUBLB=UB0itLBitforeacht=1;:::;T,thecomplexityofthealgorithmisgivenbyO(KT+T(UBLB)2K),asallLBitandUBitcanbedenedinO(KT)time.Thecomplexityofthedynamicprogrammingalgorithmisexponentialduetotheterm(UBLB)2K. Since,asjKjgrows,thestatespaceinadynamicprogrammingapproachtosolvetheproblemgrowsexponentially,weconsidercomputingFkt-valuesforthecaseinwhich 59

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3{10 )tocomputetheoptimalstatefunctionvaluesFj(si1j;si2j),foreachpossibletimeperiodandstate.Notethathere,Fj(si1j;si2j)denotestheminimumcosttofeasiblyaccumulatesi1junitsofinventoryforitem1andsi2junitsofinventoryforitem2throughstatej. AftercomputingthestatespaceandcorrespondingfunctionalvalueswithDPforeacht,wederiveinequalitiesthatdenetheentireconvexhullofthepointsdenedby(si1t;si2t;Ft(si1t;si2t))byusingtheGiftWrappingalgorithmbasedonJarvis'smarchalgorithm( Jarvis ( 1973 )).Thisapproachprojectsallpointsontothetwo-dimensionalspacebyreplacing(si1t;si2t;Ft(si1t;si2t))with(si1t;si2t;0).Webeginbydeterminingaone-dimensionaledgeofthethree-dimensionalconvexhull,whichpassesthroughpointsr1andr2.Thisedgecanbefoundbyprojectingthegivensetofpointsontoanytwodimensions,ndingapointq1havingthesmallestvalueinonedimensionoftheprojection,andthenndingaprojectedpointq2suchthatnoprojectedpointsliestrictlyonbothsidesofthelinespanningq1andq2.Wetaker1andr2tobethethree-dimensionalcounterpartsofprojectedpointsq1andq2,respectively.Next,notethatthethree-dimensionalconvexhullcontainsatmosttwofacetsthatpassthroughr1andr2.Wendathirdpointr3suchthattheplanepassingthroughr1,r2andr3inducesafacettotheconvexhullofpoints,againbyexaminingeachoftheotherpointstoensurethattheylieonlyononesideoftheplane(fornow,assumingthattherearenopointsthatareanecombinationsofr1,r2andr3).Then,weputedges(r1;r2),(r1;r3)and(r2;r3)inaqueue.Fromthispoint,thealgorithmselectsandremovesanedgefromthequeueandndstheotherfacetpassingthroughtheedge.Ifanewfacetisidentied,thetwonewedgesdeningthisfacetareaddedtothequeue,unlesstheyarealreadypresent.Ifanedgeispresent,itisremovedfromthequeue,sincethesecondfacetincludingthisedgehasbeenfound.Atanypoint,ifthereexistsmorethanthreepointsthatlieonageneratedplaneoftheconvexhull,wendthetwo-dimensionalconvexhullofthose 60

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ForthejKj=2itemlot-sizingproblem,theinequalitiesdeningthelowerconvexhullofthestagetvaluefunctionhavetheform: forparametersatq,btqandctq,q=1;:::;Qt,whereQtisthenumberoffacetsdeningthelowerconvexhull,andtheinequalitiesdeningtheupperhullhavetheform: forparameters~atq,~btqand~ctq,~q=1;:::;~Qt,where~Qtisthenumberoffacetsdeningtheupperconvexhull. 3-1 givestheDPrepresentationofthisinstance,witheachnoderepresentingafeasiblestate(feasibleinventorylevelsforitem1anditem2,respectively)ineachperiod.TheF-valuesareprovidedaboveeachnodeinthegure. TheoptimalsolutionisrepresentedbytheboldpathinFigure 3-1 .Productionforitem1andproductionforitem2isgivenbyx1t=(2;1;3;0)andx2t=(2;4;0;1),withatotalcostof142. Now,letK=f1;2g.Fort=1,inequality( 3{7 )is: 8x11+7y11+s11+11x21+6y21+s2143;(3{13) 61

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NetworkrepresentationoftheDPformulationofMCLSPforT=4andM=2. Figure3-2. Convexhulldeningthefunctionalvaluesfort=3. 62

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3{7 )is: 8x11+7y11+s11+11x21+6y21+s21+9x12+8y12+s12+5x22+5y22+s2284:(3{14) ThesingleinequalitythatdenestheentireconvexhullforF2(s1;2;s2;2),from( 3{11 ),is: 8x11+7y11+s11+11x21+6y21+s21+9x12+8y12+s12+5x22+5y22+s2284+10s12+6s22: Theconvexhullofthepointsformedbys1;2,s2;2andF2(s1;2;s2;2)isaplane,buttheconvexhullofthepointsformedbys1;3,s2;3andF3(s1;3;s2;3)isthree-dimensionalasillustratedinFigure 3-2 .Let(s1;3;s2;3;F2(s1;2;s2;2))representthepointsofthisconvexhull.Thenthefacetformedbypoints(1;2;135),(2;0;121)and(2;1;128)aspartofthelowerconvexhullfort=3( 3{11 )is: andthefacetformedbypoints(1;2;135),(2;0;121)and(1;0;112)aspartofthelowerconvexhullfort=3( 3{11 )is: Thefacetformedbynodes(2;1;128),(2;0;121)and(0;0;100)aspartofthelowerconcavehullfort=3( 3{12 )is: TheGiftWrappingalgorithmcanbeappliedtopointslyinginanydimensionalspace,sowecanusethisalgorithmtogenerateconvexhullinequalitiesformulti-iteminstanceswiththreeormoreitems.However,adynamicprogrammingalgorithmtypicallyconsumesprohibitivelylargecomputationalresourcesforjKj3.Therefore,weobtaininequalitiesoftheform( 3{11 )onlywhenjKj2.ForjKj3,wedonotattempttoexplorethe 63

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3{7 )byutilizingintegerprogrammingtechniques. 3.3.1 ,weprovideliftingandback-liftingalgorithmsfortheseinequalities.WethendescribeaseparationalgorithminSection 3.3.2 inwhichweorienttheobjectivefunctionsothatthenewpartialobjectiveinequalitycutsothefractionaloptimalsolutionobtainedfromarelaxationofMCLSP. Eastonetal. ( 2003 ))toimprovethecoecientsoftheinequalities( 3{2 ),( 3{3 )and( 3{4 ).Liftingonbinaryvariableshasproventobeusefulforsolving0-1integerprogramsbybranch-and-cutalgorithms(see Balas ( 1975 ); Crowderetal. ( 1983 ); Wolsey ( 1975 )amongothers).Inordertoextendtheideasfor0-1integerprogrammingtothemixedintegerprogrammingcase,itisessentialtostudytheliftingofcontinuousvariables.Whileliftingofthebinaryvariableshasbeenwidelystudied,therearefewerstudiesdealingwithliftingofthecontinuousvariables(see deFariasetal. ( 2000 2002 ); deFariasandNemhauser ( 2001 ); Eastonetal. ( 2003 ); Richardetal. ( 2002 )amongothers).Inthisstudyweanalyzeexactandapproximateapproachesforliftingbothbinaryandcontinuousvariablesinthepartialobjectiveinequality( 3{4 )andtheconvexenvelopeinequalities( 3{3 )and( 3{11 ). 3{4 )toobtainavalidinequalityoftheform: 64

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3{19 ),weminimizeayit0suchthat: forallfeasiblesolutionstotheMCLSP.Ifyit0=0,thenayit0canbesettoanyarbitraryvalue.Else,ifyit0=1,thenayit0mustbeatleastaslargeastheright-hand-sideof( 3{20 )forany(s;x;y)solutionwithyit0=1.Thisleadsustosolvinganintegerprogramwithanobjectivefunctionthroughperiodtexcludingitemisubjecttotheoriginalsetofconstraintsincludingallitemsandallperiodsandtheconstraintforcingyit0tobe1.Thisintegerprogramisgivenasfollows: (3{21a) s.t.sl;j1+xljdlj=sljj=1;:::;T;l=1;:::;M Aftersolving( 3{21 ),wesetait0=Lzifait0
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3{21d ).Wecallthisthemasterproblemandformulateitas: minz (3{22b) Constraints( 3{21b );( 3{21c );( 3{21e )( 3{21g ): Letsk;xk;yk;zkrepresentanoptimalsolutionto( 3{22 ).Wesetthelowerboundequaltozk.Next,weconstructasubproblembyxingthey-variablesintheoriginalproblem( 3{1 )toyk.Wenextsolvethissubproblem,whichisoftheform: mintXj=1(pijxij+hijsij) (3{23a) s.t.sl;j1+xljdlj=sljj=1;:::;T;l=1;:::;M Associatedualvariablesljwith( 3{23b ),ljwith( 3{23c )andjwith( 3{23d ).Denelj=1ifl=iandjt,and0otherwise.Thenweobtainthefollowingdualformulation: maxMXl=1TXj=1dljljminfcj;TXr=jdlrgykljlj!TXj=1cjj s.t.ljljjljpljj=1;:::;T; 66

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3{23 )isinfeasible,then( 3{24 )mustbeunboundedsincethetrivialall-zerosolutionguaranteesthefeasibilityofthedualformulation( 3{24 ).Letting(;;)beanunboundeddualrayto( 3{24 ),wegetthefollowingfeasibilitycut: LetA=PMl=1PTr=1dlrljPTj=1cjj.Thenwecanfurtherimprovetheinequality( 3{25 )by: min(A;MXl=1TXj=1minfcj;TXr=jdlrgykljlj)yljA; sinceminfcj;PTr=jdlrgykljlj0foralllandj. Otherwise,if( 3{23 )hasanoptimalsolution,thensodoes( 3{24 ).Let(;;)beanoptimalsolutionwithobjectivezkSP.SettheupperboundequaltozkSPifzkSP
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3{21b )( 3{21f );sit0maxf1;LBitg:(3{28) Sincesolvingforasit0requiresthesolutionofanonlinearmixedintegerproblemgivenby( 3{28 ),weconsiderthecontinuousrelaxationofthisproblemtoestimatetheliftingcoecientwhileguaranteeingitsvalidity.Wediscussthreedierenttechniquestocomputeasit0inthismanner. Firstsupposethatwesubstitute=LHt;t0(s;x;y) 3{28 ).Weobtainthefollowingnonlinearprogram: max s.t.LHt;t0(s;x;y)sit0 Constraints( 3{21b )( 3{21e ) (3{29c) 0ylj1j=1;:::;t;l=1;:::;M Toobtaintheapproximateliftingcoecientofsit0,wesolveproblem( 3{29 )byxingasaparameterandusethebisectionmethodtodeterminecandidatevaluesofandthensolvethefeasibilityproblemgivenby( 3{29 ).Ifwendafeasiblesolutionto( 3{29 ),wedeterminethelargestvalueofallowedbythesolutionobtainedandletthislargestvaluebeournewlowerboundon.Ifafeasiblesolutionisnotfound,thenwesettheupperboundto,andrepeatthebisectionmethoduntilthedierencebetweentheupperandlowerboundonisatmost,where>0issomeprespeciedparameter.PseudocodeforthisalgorithmisgiveninAlgorithm 2 68

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2 hascomplexityO(log2ul Q),whereQisthecomplexityofsolvingtheLP( 3{29 ).SinceQisapolynomialfunctionoftheinputfor( 3{29 ),Algorithm 2 ispolynomialincomplexity. Interval[l;u]thatcontainsoptimal 3{29 ) 3{29 )isfeasiblethen 3{29 )foragiven 3{29c )( 3{29e )g Dinkelbach ( 1967 ); Ibaraki ( 1983 ))thatG()isconvex,continuousanddecreasingmonotonically(i.e.,G()00
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Afterdeterminingtheintervalfor,weutilizeabinarysearchalgorithmsimilartoAlgorithm 2 .WemodifythisalgorithmbyimprovinglowerandupperboundsineachstepusingtheconvexityofG().Considerthepoints(l;G(l))and(u;G(u))ontheandG()axes,andnotethatthelineconnectingthesepointscrossesthe-axisatunew=l+(ul)G(l)=(G(u)G(l)).SinceG(l)>0,G(u)<0andG()isconvex,cannotbegreaterthanunew.Also,lettingG0()bethederivativeofG()at,considerthetangentlinespassingthroughlanduhavingslopesG0(l)andG0(u),respectively.ThetangentlinepassingthroughG()atlintersectsthe-axisat^l=(LHt;t0(sl;xl;yl))=slit0andthetangentlinepassingthroughG()atuintersectsthe-axisat^u=(LHt;t0(su;xu;yu))=suit0.SinceG()isconvex,maxf^l;^ugcannotbegreaterthan.Thuswecanimprovethelowerboundltomaxf^l;^ug.Wethenupdatelanduasdescribedabove,set=(u+l)=2andcheckwhetherG()=0.IfG()=0,thenisoptimal.IfG()>0,wesetl=,andotherwiseifG()<0,thenu=.Weiterateinthisfashionuntilthedierencebetweenuandlisatmostapredeterminedvalue>0.ThisprocedureisgiveninAlgorithm 3 Athirdapproachtocomputingasit0istodirectlyconsidertheliftingproblemgivenby( 3{28 ),withy-variablesrelaxedtobecontinuousandboundedbetween0and1.Let1beafeasibleparameterlowerboundforsit0,anddeneZ()as: 3{29c )( 3{29d );sit0=g;(3{30) whereZ()=if( 3{30 )isinfeasiblefor.WeseekavalueofthatmaximizesZ()=.NotethatZ()isapiecewise-linearconcavefunction,sinceonlyappearsintherighthandsideoftheconstraintsdeningthefeasibleregionofthelinearprogram( 3{30 ).TomaximizeZ()=,weseekthelastpiecewiselinearsegmentofZ()(from 70

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Interval[l;u]thatcontainstheoptimal 3{30 )forboth=land=l,and(b)theleftmostpointusuchthatthesamebasisoptimizes( 3{30 )forboth=uand=u.IfluorG(l)G(l)=(ll)=1,then=l.Else,if 71

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3{30 )given=mdoesnotchange.Thenwecomputetheslopegivenby[G(um)G(lm)]=(umlm).Iftheslopeis1,thenthealgorithmterminateswith=m.Iftheslopeislargerthan1,thenwesetl=lm,whileifitissmallerthan1,thenwesetr=um.Wecontinuebyrepeatingthealgorithmuntilwereachanoptimal-value.Notethatthisalgorithmnitelyconvergestoanoptimalsolution,incontrasttoAlgorithms 2 and 3 .ThisprocedureispresentedinAlgorithm 4 Interval[l;u]thatcontainsanoptimal 3{30 )forboth=landl;(2)theleftmostpointusuchthatthesamebasisoptimizes( 3{30 )forboth=uandu. 3{30 )given=mdoesnotchange. endifend Example3 72

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3{4 )fort=1;2;3are: 2033y11+109x11+10s112142;(3{31) respectively.ThedimensionsofthefacesoftheMCLSPdenedbytheinequalities( 3{31 ),( 3{32 )and( 3{33 )are8,4and3,respectively. Afterback-liftings1tin( 3{31 ),weobtain: 2033y1t+109x1t109s1t2142:(3{34) Afterback-liftingy12rstandthens12in( 3{32 ),weobtain: Ifweback-liftx12in( 3{32 ),weobtain: Afterback-liftingy12thens13,in( 3{33 ),weobtain: 73

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3{34 ),( 3{35 ),( 3{36 )and( 3{37 )are10,9,8and7,respectively. Back-liftinginequalities( 3{31 )and( 3{32 )resultsinfacet-deninginequalitiesfortheMCLSPinstance,andback-liftinginequality( 3{33 )considerablyincreasesthedimensionofthepolyhedralfaceitinduces. TheLPrelaxationvaluefor( 3{1 )forthisexampleinstanceis5587,whiletheoptimalobjectiveto( 3{1 )is9919.Alltheinequalities( 3{31 ){( 3{37 )cutotheoptimalLPrelaxationsolution.Byaddingliftedinequalities( 3{34 ),( 3{35 ),and( 3{37 )to( 3{1 ),LPrelaxationof( 3{1 )improvesto7747:33,a28%improvement. 3{2 )byallottingthemaximumcapacityineachperiodtoitemisuchthatamplecapacityremainstosatisfydemandfortheotheritems,andsolvingthesingle-itemDPthroughperiodt.Now,givenitemiandperiodt0witht0t,considerback-liftingtheinventoryvariablesit0inthevalidinequality( 3{2 ).Thenaswehaveshown,weback-liftvariablesit0bycomputing: 3{21b )( 3{21f );sit0maxf1;LBitg:(3{38) Theonlydierencebetweenproblems( 3{28 )and( 3{38 )isthatwerelaxourconstraintsettoasingleitemin( 3{38 ).Thereforetocomputeasit0,weneedtosolveadynamicprogramuptoperiodtwithupdatedcostsinordertosettheinventorycostofitemiinperiodt0to0,whilekeepingalltheothercoststhesame.Then,bybacktrackingfromperiodttoperiodt0,wecancomputetheminimumcostvaluesateachstateatperiodtandthecorrespondingstatevaluesinperiodt0.Afterobtainingthestatevaluesinperiodt0withtheirminimumcorrespondingcostsinperiodt,wecancomputetheobjectivefunctiongivenin( 3{38 )byenumeratingallfeasiblesit01andcorrespondingHt;t0(s;x;y)valuesandthencomputingtheoptimalliftingcoecientasit0toback-liftthevariablesit0in( 3{2 ). 74

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3{2 )andtherst(left-most)convexenvelopeinequalityoftheform( 3{3 ).Foragivenitemiandperiodt0weback-liftst0tothepartialobjectiveinequality( 3{4 )toobtainavalidinequalityoftheform: Toprovidethestrongestpossibleinequalityoftheform( 3{19 ),wemaximizeastsuchthat: ~aststXj2f1;:::;tg(fjyj+pjxj+hjsj)L:(3{40) Thisleadsustosolvingthefollowingmixedintegernon-linearprogram: ~ast=min8<:Pj2f1;:::;tg(fjyj+pjxj+hjsj)L st:( 3{1b )( 3{1f );stmaxf1;LBtg9=;:(3{41) Dene: (3{42a) s.t.sj1+xjdj=sjj=1;:::;T; Thenwecanwrite( 3{41 )asfollows: ~ast=minsktmaxf1;LBtg(F(skt)L) Infacttheequation( 3{41 )givestheminimumslopeinourstagetvaluefunctionforitemiwhenwecomputetherstconvexenvelopeinequality( 3{3 ).Thereforeafter 75

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3{2 ),weobtainanequivalentcuttotherstconvexenvelopeinequality( 3{3 ). 3{4 )toobtainavalidinequalityoftheform: Toprovidethestrongestpossibleinequalityoftheform( 3{44 ),wemaximizeayit0suchthat: ~ayit0yit0Xj2f1;:::;tg(fjyj+pjxj+hjsj)L:(3{45) Thisleadsustosolvingthefollowingmixedintegerlinearprogram: ~ayit0=min8<:Xj2f1;:::;tg(fjyj+pjxj+hjsj)L:( 3{1b )( 3{1f );yit0=19=;:(3{46) 3{1a ).Inourseparationalgorithmwebeginbyseekingpartialobjectiveinequalities( 3{4 )overasubsetKofitemsthatareviolatedbythegivenfractionalsolution.Wealsorevisetheobjectivecoecientsusedtogeneratetheinequalitiestoobtainpartialobjectiveinequalities( 3{4 )withdierentorientation.Ourheuristicseparationalgorithmisdescribedbelow. 3{1 ). 76

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minXi2KtXj=1pijxij+f0ijyij+hijsij: LetanoptimalsolutiontoSUBtbe(~st;~xt;~yt)withobjectiveztSUB.Wethentestwhether: where1isapredeterminedparameter.Notethatastheratiogivenintheleft-handsideof( 3{48 )getssmaller,thelowerboundztSUBonthepartialobjectivefunctiongivenin( 3{47 )getstighterandthepotentialofthenewlowerboundztMIPobtainedaftersolvingtheMIPversionofSUBtoviolate(^st;^xt;^yt)increases.If( 3{48 )holdstrue,thengotoStep3.ElsegotoStep5. ProceedtoStep4. 3{49 )isviolatedby(^st;^xt;^yt),thenadd( 3{49 )tobothformulationsLPtandSUBt,re-solveLPtandupdatethesolution(^st;^xt;^yt).Ineithercase,proceedtoStep5.

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3{11 )and( 3{12 )insolvingrandomlygeneratedMCLSPinstances.ForthecomputationalexperimentswesolveallmathematicalprogrammingproblemsusingCPLEX11.0.An1800CPU-secondtimelimitisimposedforalltestinstances. Thedatausedintherstsetofexperimentinstancesaregeneratedusingasimilarschemeasin AtamturkandMu~noz ( 2004 ).Wegeneratedataasfollows:Demandsaregeneratedfromanintegeruniformdistributionbetween1and5foreachitem.WexthenumberofstagesatT=60.Toobservetheeectofdierentcapacitiesandcostparametersonthecomputations,wevaryacapacitymultiplierparameter!2f2:5;3gandset-uptoholdingcostratios2f100;200;500g,andgenerateverandominstancesforeachcombinationforatotalof30instances.Thesharedcapacitiesctaregeneratedfromanintegeruniformdistributionbetween0:75!dand1:25!d,wheredisequalto5(theaveragedemandforanitem).Theset-upcostsstarerandomlygeneratedfromanintegeruniformdistributionwithrange0:9hand1:1h,wherehistheaverageholdingcost.Theunitproductioncostsptaregeneratedfromanintegeruniformdistributionbetween81and119.Theholdingcosthtisxedat10foreachperiod. OurimplementationexecutestheforwardDPforthetwoitemMCLSPforalimitednumberofstagestogenerateourproposedinequalities.WeappendtheseinequalitiestostrengthentheMCLSPformulation( 3{1 )andsolvetheresultingmodelusingCPLEX 78

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3{1 ),withoutaddinganyuserinequalities. 3{11 )and( 3{12 )withbase. Table 3-1 summarizestheresultsofourcomputationalexperiments.InTables 3-1 and 3-2 wereportthefollowingdatabycolumn: Tables 3-1 and 3-2 presentresultsforT=60andM=2,whereeachtableentrycorrespondstotheaverageperformanceofanalgorithmoverveinstances(veeachfor=100,200and500).InTables 3-1 and 3-2 ,wealsoreporttheoverallaveragesforthe15instances.Weobservethattheconvexhullinequalities( 3{11 )and( 3{12 )improvethedirectsolutionoftheMCLSPbyCPLEXonaverage.Outofthe30instances,twoofthemcannotbesolvedbybasewithin1800CPUseconds.Theseinstancescouldbesolvedwiththeimplementationoftheinequalities( 3{11 )and( 3{12 )withstage=10and20.Itisclearthatasthenumberofstagesincreases,theDPsolutiontimerequiredtogeneratetheconvexhullinequalitiesincreases.ThereforeanecientimplementationoftheDPalgorithmtoobtaintheinequalities( 3{11 )and( 3{12 )isacrucialconsiderationineectivelyusingtheseinequalities. 79

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SummaryofexperimentsforT=60,M=2and!=2:5. stageDPineqCPXineqnodesineqtimetime 100base 02782387050.00219.48convexhull 528872502599092.17239.38 10677929311458910.89165.04 1598292956997433.07142.94 20127122023144075.4386.21 200base 01865848080.00498.09convexhull 525162206114231.74505.90 10603521534540310.12411.96 15950515032593829.15433.36 201250212421354269.34376.54 500base 01584522200.00270.53convexhull 524171901576491.71102.39 10630011546804110.37326.89 151016115723703632.26312.26 201356213515270771.87215.41 Averagebase 02074252450.00329.37convexhull 526072203429941.87282.56 10637120830934410.46301.30 15983220121098331.49296.19 201292515413256372.21226.05 Forthesecondsetofexperimentswegeneratedataasfollows:Demandsaregeneratedfromanintegeruniformdistributionbetween1and10foreachitem.WexthenumberofstagesatT=18andnumberofitemsatM=8.Inthisexperimentwevaryset-uptoholdingcostratios2f100;200;500g,andgenerateverandominstancesforeachcombinationforatotalof15instances.Thesharedcapacitiesctaregeneratedfromanintegeruniformdistributionbetween0:75!dand1:25!d,wheredisequalto10(theaveragedemandforanitem)and!=6.Theset-upcostsst,ptandhtaregeneratedexactlyinthesamewayaspresentedintherstexperiment. Oursecondimplementationsolves( 3{1 )withanobjectiveof( 3{47 )forveindividualitemsusingseparationalgorithmtogenerateinequalities( 3{49 ).Eachtimeweobtainaninequality( 3{49 )weapplyanexactliftingofy-ands-variablesusingAlgorithm 2 ,inwhichy-variablesarerestrictedtobinaries.Werestrictourimplementationtoexactliftingsinceapproximateliftingdidnotimprovesolutiontimes.Weappendtheliftedinequalities 80

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SummaryofexperimentsforT=60,M=2and!=3. stageDPineqCPXineqnodesineqtimetime 100base 0330965870.00101.63convexhull 525742082915481.84219.53 1060312688915710.98111.70 1586822545110340.79154.27 201126222830819103.67196.75 200base 01449572990.00614.40convexhull 530611706686321.70442.33 10688318451401813.13408.87 151035821216023049.57295.95 201396015585677128.11246.81 500base 01914816410.00263.89convexhull 527871814428292.03268.53 10676716028788115.53216.57 151023210910195359.37222.71 201410210087925145.63284.62 Averagebase 02225118420.00326.64convexhull 528071864676701.86310.13 10656020429701913.21245.71 15975719210442949.91224.31 201310816168140125.81242.73 ( 3{49 )tostrengthentheMCLSPformulationandsolvetheresultingmodelusingCPLEXwithitsdefaultsettings.Wecomparetheeciencyofthefollowingsolutionapproachesinourcomputationalexperiments. 3{1 ),withoutaddinganyuserinequalities. 3{49 )withbase. Table 3-3 presentsresultsforT=18andM=8,whereeachtableentrycorrespondstotheaverageperformanceofanalgorithmoverveinstances(veeachfor=100,200and500).Weuseveindividualitemsforwhichwegeneratetheinequalities( 3{49 )uptoperiod6.Herethecolumn(objineq)denesthenumberofpartialobjectivefunctioninequalities( 3{49 )thataregeneratedinsepliftstrategy,while(cuttime)givesthecutgenerationtimeinCPUseconds.Weobservethattheliftedinequalities( 3{49 )improvethedirectsolutionoftheMCLSPbyCPLEXonaverage.Theseresultsshowthatlifting 81

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SummaryofexperimentsforT=18andM=8. objineqcpxineqnodescuttimetime 100base -322.71378850.0123.9seplift 2.9323.86220053.398.3200base -264.14394070.0143seplift 5.0271.29367701.4115.3300base -371.571629270.0562.0seplift 3.6374.711170510.2458.6 Averagebase -319.48800730.0276.3seplift 3.8323.29586081.6224.1 andseparationoftheseinequalitiesispromisingintermsofimprovingthecomputationalsolutiontimes. 82

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FraserandPosey ( 1989 ),forexample),parallelreplacementproblemsarecombinatorialasgroupsofassetsmustbeanalyzedsimultaneously. Thischapterpresentsvalidinequalitiesandacut-and-branchsolutionproceduretoanintegerprogrammingformulationforthedeterministic,parallelreplacementproblemundereconomiesofscale(PRES)inwhichaconstantnumberofassetsarerequiredforoperationsineachperiodoveranitehorizonoflengthT.Attheendofeachperiod,anassetmaybesalvagedorretained,assumingithasnotreacheditsmaximumphysicallifeN,atwhichtimeitmustberetired.Assetsmaybereplacedthroughthepurchaseofnewassets.Thepurchaseofassetsissubjecttoaxedcharge,regardlessoftheordersize.Attheendofthenitehorizon,allassetsaresalvaged.ThesolutionconsistsofpurchaseandsalvagedecisionsforeachassetoverthenitehorizonwiththeobjectiveofminimizingdiscountedpurchaseandO&Mcostslesssalvagevalues. Thisstudyismotivatedbytheintegerprogrammingresultsof Hartman ( 2000 )andthesuccessfuluseofcuttingplanesinsolvinglot-sizingproblems( Baranyetal. ( 1984a )).Conceptually,theparallelreplacementproblemandthelot-sizingproblemaresimilar.Inthelot-sizingproblem,inventorypurchasesaremadebytradingoaxedcharge(set-upcost)againstinventorycarrycharges.Intheparallelreplacementproblem 83

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Themodelpresentedisaxed-chargeminimumcostowmodel.Thus,ifthebinaryvariables(requiredforimposingaxedchargewithassetpurchases)arexed,thentheoptimalsolutiontothelinearprogrammingrelaxationoftheresultingformulationisinteger-valued,ifafeasiblesolutionexists.Thus,branch-and-boundproceduresmustonlyfocusontheTbinaryvariables.Inthischapter,weprovidevalidinequalitiesthatfocusontheseTvariablestofurtherreducethedicultyofsolvingPRES. Thevalidinequalitiesareusefulfromacomputationalstandpointbuttheyarealsointerestingastheyarederivedfromaconsequenceofthe\no-splittingrule",whichhasbeenusedtoreducethecomputationtimeinearlierdynamicprogrammingapproachestotheproblem.TherulestatesthatanoptimalsolutiontoPRESexistssuchthatallassetsofthesameageinthesametimeperiodareeitherkeptorreplacedasagroup.\No-splittingrule"leadstothefactthatifanassetissalvagedinaperiod,thentherehastobeapurchaseinthatperiod.Wemodelourproblemusingthispropertysuchthatowconservationconstraintsimposethatasalvagetriggersapurchaseinaperiod.Wethenusethispropertytogeneratevalidinequalitiestighteningconstraints,whichenforcethexedchargeineachperiodofanassetpurchase. Thischaptermakesthreecontributionstothereplacementanalysisliterature.First,weprovethatPRESisNP-hard.Second,asetofvalidinequalitiesisdenedforPRES.Theirrelationshiptotheresultofthe\no-splittingrule"statingthatapurchaseisenforcedbyasalvageismadeclearintheirdevelopment.Third,computationalresultsshowthattheincorporationoftheseinequalitiesintoacut-and-branchproceduredrasticallyimprovesthesolutiontimeofPRES.Thisisespeciallytrueforlargeprobleminstances,suchasthosefromtherailroadindustryanalyzedin HartmanandLohmann ( 1997 ). 84

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Hartman ( 2000 ),wedenedecisionvariablesaccordingtothenumberofperiodsanassetisretainedasopposedtodeningwhetheranassetiskeptorretainedaftereachperiod.(Theapproachesareequivalent,butthisapproachsignicantlyreducesthenumberofvariablesandisakintosingleassetapproaches,asin Oakfordetal. ( 1984 ).)Itisassumedthatthenumberofassetsininventoryattimezeroisequaltodemandandnoassetintheinitialinventoryhasanageequaltoitsmaximumservicelife.Thiseliminatestheautomaticdecisionofhavingtoreplaceanassetattimezero.Assumingconstantdemandandnocapitalbudgetingconstraintseliminatestheneedtostoreassets,asin Hartman ( 2000 ). Anassetisdenedbyitsage,i=0;1;:::;Nattheendoftimeperiodj=0;1;:::;T.AnassetmayberetainedorsalvagedaftereachperiodunlessitreachesageNatwhichtimeitmustbesalvaged.TheproblemissolvedoverTperiods,withpurchasesallowedattheendofperiods0;1;:::;T1.AllassetsaresoldattheendoftimeperiodT.Thedecisionvariablesaresummarizedasfollows: ThecostsassociatedwithXjkincludethepurchasecostattimej,O&Mcostsovertheensuingjthroughkperiods,minusrevenuefromsalvageattimek.CostsassociatedwithSijaresimilaranddenedasrijtoavoidconfusion,butthereisnopurchasecostasthisvariableisconcernedwithassetsalreadyowned.Thexedcostkjisincurredinanyperiodjinwhichapurchaseismade. Otherrelevantparametersinclude: 85

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4-1 .Theinitialinventoryofassetsownedattimezeroisrepresentedassourcenodestothenetwork.Thesearevisuallyrepresentedassquarenodesinthegure,labeledwiththeageoftheinitialassets.Thevalueofnirepresentstheamountofsupplyofageiassetsattimezero. AnadditionalT+1nodesrepresenttheperiodsoftheproblem,0throughT,whereTisasinknodedemandingtoreceiveowofPiniassets.Nodes0throughT1representtransshipmentnodesinwhichthereisnosupplyordemand.Notethatasupplynodeiisconnectedtonodes0throughNi,representingthedecisionstosalvageanassetimmediatelythroughretaininganassetuntilageN. Figure4-1. NetworkrepresentationofPRESwithowrepresentingassetsinusewithN=4. Foragivenproblem,thereareatmostN1supplynodes,1demandnode,andTtransshipmentnodes.Furthermore,thereareatmost(N2+N)=2arcsconnectingthesupplytothetransshipmentnodes,Narcsowingintothedemandnode,andfewerthanNTarcsconnectingtransshipmentnodes. Withthesevariablesandparameters,theintegerprogrammingformulationforPRESfollows: minT1Xj=0kjZj+T1Xi=0minfi+N;TgXj=i+1cijXij+N1Xi=1NiXj=0rijSij 86

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(4{1b) (4{1c) (4{1d) (4{1e) (4{1g) (4{1i) Withconstantdemand,PRESisaxed-chargeminimumcostowproblem.Theobjectivefunction( 4{1a )minimizesdiscountedpurchaseandO&Mcostslesssalvagevaluesaccordingtolifecyclecostscijandrij.Constraints( 4{1b )providePiniassetstothenetworkinwhichowisconservedthroughConstraints( 4{1d )and( 4{1e ).AllowculminatesatthedemandnodeT,asdenedbyConstraint( 4{1f ). Constraint( 4{1g )includesthexedchargevariableZjsuchthatifanyassetsarepurchased,thexedchargeisimposed.Asdemanddisconstantandthenumberofinitialassetsinthesystemisd,themaximumnumberofassetsthatcanbepurchasedinanyperiodisd.TheX-andS-variablesarerestrictedtobegeneralintegers,andtheZjbeingrestrictedasbinary.Notethatconstraints( 4{1h )arenotrequiredasintegralityismaintainedif( 4{1i )isenforced. 87

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Proof. 4-2 Giventhevariablesandclausesfrom3SAT,wesetT=(2)(+)andN=2(+).Thetimeperiodscanbegroupedinto++1\blocks"of2consecutivenodes,plusoneextranoderepresentingthenishattimeT.Eachblockcontainsonenodecorrespondingtoeachpossiblevariablevalue.Indexingthenodesofablockas0;:::;21,atruevalueforvicorrespondstonode2(i1)andafalsevalueforvicorrespondstonode2(i1)+1,foreachi=1;:::;. Therstblocks,whichwecall\variableblocks,"requireustoselectatleastonevalueforeachvariable.Thenextblocks(the\clauseblocks")correspondtothe 88

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Transformationof3SATtoDpres Wenowintroducetheinitialassetinventorynodes.(Werefertothesesimplyas\assetnodes"below,andtoallothernodesas\timenodes"whereambiguityispossible.)Weletn2(`1)+1=1for`=1;:::;+,andni=0forallothervaluesofi.Notethatanarcconnectsn2(`1)+1totimenode2(+`+1)1.Hence,theassetavailablewithage2(`1)+1isassociatedwithblock+`+1,inthesensethatanassetwiththisagecanbereplacedatanytimeperiodinitsassociatedblock(orbefore).Forsimplicity,werefertotheseinitialassetsbytheirassociatedblock. 89

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Torevisethec-costs,foreachpermissiblenodehcorrespondingtoblocki=1;:::;+andvariablevaluej2f0;:::;21g,setch`=(+i)A+(j+1)B,for`=h+1;:::;(2)(+)+j.Notethatwethusrevisethearccostsexitingnodehtoallnodesuptoandincludingnodejofthenalblock.Also,fornodehinthenalblockcorrespondingtovaluej2f0;:::;21g,setch`=A(j+1)B,8`=h+1;:::;T.Notethatallcostsintheproblemarenonnegative,sincethesmallestofthec-valuecostsisA(2)B=1,andallotherr-andk-costsareeither0or1. Toestablishtheequivalencebetweenthe3SATinstanceanditstransformedDpresinstance,werstneedtostateandprovethefollowingthreeclaims.

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Wethushavethattherstarcvisitspermissiblenodej2f0;:::;21ginblocki,andthatthesecondarc'scostis(+i)A+(j+1)B.ThisimpliesthatthecostoftheremainingpathtonodeTmustnotexceedA(j+1)B.Supposebycontradictionthatthesecondarcterminatesinnodehcorrespondingtonode^j2f0;:::;21gofblock^i,whereh6=2(+)+(j+1).First,ifhisgreaterthanthisvalue,thenthecostofthearcisG+B,whichistoolarge.Ifhissmaller,thensupposethat^i=++1isinthenalblock,but^jA(j+1)Bandarethustooexpensivetouse.Now,supposethat^i=+.Allpathsfromnodesinblock+toTrequireatleasttwoarcs,mustvisitanodeinthenalblock,andhaveacostofatleastA,whichistooexpensive.Finally,if^i<+,thenalloutgoingarcsfromnodehhaveacostofatleastA+B,whichisalsotooexpensive.Therefore,thesecondarcmusttraveltonode2(+)+(j+1). Finally,thethirdarcmustconnectdirectlytoT,sincethecostfrom2(+)+(j+1)toanylarger-indexednodeisexactlyA(j+1)B,makingthetotalpathlengthaccumulatedthusfarequalto(+i+1)A,andsincetheonlyarcsexitinganynodesreachablefromhhavepositivecosts. 1 ,thenitscostisatleast(+i+1)A+B. 1 91

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Giventheseclaims,wenowshowthatifthe3SATinstancehasasolution,thenDpresalsohasasolution.Foreachvariablei=1;:::;,ifviistrue,thenlettheassetnodecorrespondingtovariableiusethepathwithintermediatenodesj=2(i1)inblocki,andj=2(i1)inblock++1.Iffalse,itusesintermediatenodesj=2(i1)+1inthesetwoblocksinstead.Foreachclausei=1;:::;,identifyaliteralthatsatisestheclauseandassociateitwithj2f0;:::;21gasdescribedbefore.Fortheassetnodecorrespondingtoblock+i,useintermediatenodesjinblocks+iand++i.Notethatallpathsgivenareshortestpaths.Next,whenestablishingthepathsfromassetnodescorrespondingtovariableblocks,exactlynodesarevisitedinblock++1.Toseethis,observethatwhenpathsareestablishedfromtheassetnodescorrespondingtoclauseblocks,theyuseoneofthenodesinblock++1thatwasvisitedinapathfromanassetnodecorrespondingtoavariableblock(sincethepathsfromtheclause 92

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IftheDpresproblemisfeasible,Claims 1 { 3 demonstratethattheformofthesolutionisexactlyasdescribedinthepreviousparagraph.Hence,a3SATsolutioncanbederivedbychoosingthevariablevaluesaccordingtotherstnodevisitedbyassetnodescorrespondingtovariableblocks.Thesolutionisveriedtobefeasibleforclausei=1;:::;byndingtherstnodejinblock+ivisitedbythepathfromtheassetnodecorrespondingtoblock+i,andnotingthatjcorrespondstoavariablevaluechoseninthe3SATsolution. Finally,notethatallnumericaldatausedintheproblemispolynomiallyboundedintermsofthe3SATinputsize,andsoDpresisstronglyNP-complete.Thiscompletestheproof. ( 2003 )proposesvalidinequalitiesassociatedwiththesalvagesofinitialassetsandtheowconservationforadierentnetworkowformulationofPRESwhereeachnoderepresentstheageofanassetandthecorrespondingtimeperiod.OurinequalitiesfortheminimumcostxedchargenetworkowformulationgiveninSection 4.2 areequivalenttotheinequalitiesof Luo ( 2003 ). Althoughwederiveoneclassofvalidinequalities,wepresentonesubsetseparatelyforclarityasfollows. 4{1a ){( 4{1i ). Proof. 93

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4{2 )arevalidforPRES. Werefertotheseinequalitiesas\InitialInventoryInequalities,"orIIC.Theyareillustratedinthefollowingexample. Weintroducetheinitialinventoryinequalities( 4{2 )now.Forexample,forthesix-yearoldcluster,weinclude: 94

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4-3 Figure4-3. Solutionnetworkforrst13periodsoftheexamplewithIIC. Oursecondclassofvalidinequalitiesaresimilartothetraditionalowcoverinequalitiesderivedforxedchargenetworks( NemhauserandWolsey ( 1988 ))inthattheyarederivedbyisolatingsomeowthroughanode(orsetofnodes).However,theinequalitiesaretailoredtotheparallelreplacementprobleminthattheyusethepropertystatingthateverytimethereisasaleofassets(clusterorclusters),anensuingpurchasemustoccurasaresultoftheno-splittingrule(NSR)of Jonesetal. ( 1991 )andconstantdemand.(Recallthatforconstantdemandandhomogeneousassets,theNSRstatesthatanoptimalsolutiontoPRESexistssuchthatanyclusterofsameagedassetsinthesametimeperiodareeitherkeptorreplacedintheirentirety.)Thus,examiningFigure 4-3 ,wecantracethesaleofanyclusterthroughoutthehorizonTbacktotimeperiod0,whenitwasmerelyaninitialclusterininventory.Thisiscritical,asweknowthenumberofassetsineachclusterattimezero,andthuswehavesomeinformationtoboundowthroughthenetwork.ItistheseboundsthatcanbeusedtotightenConstraints( 4{1g ). Flowcoverinequalities( NemhauserandWolsey ( 1988 ))arederivedwhenthecapacityofinowarcsexceedsthecapacity(demand)atanode.Theseareusedtowritecoverinequalitiesinordertoimprovethelowerboundsoflinearprogrammingrelaxations.Inourapplication,thereisnosituationinwhichthecapacityofinboundarcsexceedthe 95

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4-3 .Thesecorrespondtoparametersn5andn6,respectively.Considernodeslabeled2and10asone\supernode".Ifweonlyconsiderasubsetoftheinowtothissupernode,suchasS5;2andS6;2,then:S5;2+S6;29Xj=3X2;j+18Xj=11X10;j: 4-4 96

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Solutionnetworkforrst13periodswithIICandtwoNSRCinequalities. Asillustratedintheexample,eachinequalityrequiresthreecomponents:thedenitionofthe\supernode,"thedesignatedowintothesupernodeandtheisolatedowfromthesupernode.Tofacilitatethesedenitions,denethefollowing: 1. InventorysupplynodesSdenedbyi2f1;2;:::;N1g. 2. TransshipmentnodessetRdenedbyj2f0;1;:::;T1g. Withthesedenitions,furtherdenethefollowingsets: 1. ThesupernodeisdenedasasetofnodesPR. 2. SetofinownodesIS. 3. SetofoutownodesOP. 4. SetofisolatedarcsAdenedas(i;j)suchthati2Oandj2RnO. Giventhesedenitions,wecanwritethefollowingNSRCinequality. 4{1a ){( 4{1i ). Proof.

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FurthernotethatthesaleofassetsinOresultsinapurchase,suchthat:X(j;k)2AXjkdXj2OZj: Chen ( 1998 ).Specically,wegeneratedataasfollows:WexthehorizonT=500andconsidertwosizesforthenumberofinitialclusters,g=10and20.Themaximumservicelifeoftheassetisgivenby:N=g+[0;(Tg)]+;

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Beanetal. ( 1985 ))toensureoptimaltimezerodecisions(foraninnitehorizonproblem),problemswithmaximumagesof50periods(200quarters)andhorizonsof100periods(400quarters)arenotuncommon,leadingtotheneedtosolvelarge-scaleproblems.OtherparametersusedtogeneratetheproblemdataaregiveninTable 4-1 whereU[a;b]denotesanintegernumberdrawnuniformlyfromtheinterval[a;b].Wealsoconsiderthexedcosttopurchasecostratiof2f10;50;100g.Foreachofthe12possiblecombinationsofg,,f,veinstances,foratotalof60instances,aregenerated. Table4-1. DatagenerationforPRES. ParametersData AgeofassetsinclusteriU[0;N]SizeofclusterniU[2;10]FixedcostKU[100f;500f]UnitpurchasecostpjU[100;500]UnitO&McostcijU[50;100]andincreasing=U[0;10]eachperiodUnitsalvagevaluesij70percentofpjdecreasing30percenteachperiod 4{2 )tostrengthenthePRESformulation( 4{1a ){( 4{1i )whilewegeneratetheinequalities( 4{3 )basedontheinitialoptimalLPrelaxationsolutionandfeedthemtoCPLEXtouseascuttingplanes.WethensolvetheresultingmodelusingCPLEXwithitsdefaultsettings. Weperformednumerousexperiments,asthereisclearlyatradeobetweenthenumberofinequalitiesgeneratedandtheentiresolutiontimeofthealgorithm.Thesolutionsarehighlydynamicaschangestothesolutionintherstfewperiodsofthe 99

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Wecomparetheeciencyofsolvingthefollowingmodelsinourcomputationalexperiments. 4{1a ){( 4{1i ),withoutaddinganyuserinequalities. 4{2 )withbase. 4{3 )withbase. 4{2 )and( 4{3 )withbase. 4-2 and 4-3 summarizetheresultsofourcomputationalexperiments.Averagesover5probleminstancesaregivenfor: 100

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Therequiredtimetogeneratetheproposedcuttingplanesislessthan0:01CPUsecondsforeachinstance. Table 4-2 presentsresultsforT=500and=0:2,whereeachtableentrycorrespondstotheaverageperformanceofanalgorithmover5instances(veeachforf=10,50and100).WeobservethatbothoftheIICandNSRCinequalitiesimproveuponthedirectsolutionofPRESbyCPLEX.Theiic+nsrcstrategyappearstoprovidethemostbenet,withafactorof3.5improvementinCPUruntimeoverbase,averagingoverallinstances. Table 4-3 presentsresultsforT=500and=0:6,whereeachtableentrycorrespondstotheaverageforveinstances.WeobservethatbothoftheIICandNSRCinequalitiesimproveuponthedirectsolutionofPRESbyCPLEX.Amongalltheimplementations,theiic+nsrcstrategyperformsbest,withafactor1.5improvementinthesolutiontimeonaverage. InTables 4-2 and 4-3 ,weobservethatasandg,andthusNvaluesincrease,theinstancesgetharder.Inaddition,theinstancestendtobecomehardertosolveasthexedcosttopurchasecostratiofincreases.Theaveragegapimprovementduetotheinequalities( 4{2 )is46%{50%,whilethegapimprovementduetotheinequalities( 4{3 )is3%.Althoughwegenerateonlybetweenzeroandthreeinequalities( 4{3 )perinstance,weobservethattheyarequiteeectiveatimprovingthesolutiontimes.Inaddition,weobservethatasthemaximumageNdecreases,thenumberandtheeciencyoftheNSRC 101

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SummaryofexperimentsforT=500,=0:2. exp gNfinitgapgapimpnodeineqCPXineqtime base 10108100.030.0000724.6iic 0.0163.33058851.60.2nsrc 0.032.9600.2719.8iic+nsrc 0.0066.2905885.21.60.2base 500.310.003055019.694.6iic 0.2362.57401158851.870.2nsrc 0.3112.39203911962.8iic+nsrc 0.2363.0219835885.61.848.6base 1000.380.00338026.238.6iic 0.2547.7533858852.414.6nsrc 0.380.813421.624.819.4iic+nsrc 0.2548.8833958861.813.8 base 20116100.040.000013.629.4iic 0.0231.56067852.21.6nsrc 0.031.8301.613.47iic+nsrc 0.0232.2006787.42.21.6base 500.220.00156013.834.4iic 0.1737.5616067850.86.6nsrc 0.225.331491.413.87.8iic+nsrc 0.1735.571596786.20.66.6base 1000.160.0016014.830iic 0.1253.892567850.43.2nsrc 0.1612.14161.214.813.8iic+nsrc 0.1155.20256785.60.43 Average base 0.190.00594015.8341.93iic 0.1349.4475663351.5316.07nsrc 0.195.914241.1715.4721.77iic+nsrc 0.1350.1941863361.4012.30 4{3 )increases.Onemayconsiderimplementingtheminabranch-and-cutalgorithm,inwhichtheinequalities( 4{3 )aregeneratedbasedontheLPrelaxationsolutionsateachnodeinthebranch-and-boundtree.Asourexperimentsonlyproducedafewinstancesthatrequiredmorethan30minutesofsolutiontime,weonlyimplementedNSRCattherootnodetosavefromtheoverheadofobtainingLPrelaxationsolutionsatthenodesofthebranch-and-boundtree. 102

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SummaryofexperimentsforT=500,=0:6. exp gNfinitgapgapimpnodeineqCPXineqtime base 10304100.000.00002.259.2iic 0.000.0004635903nsrc 0.000.00002.259.4iic+nsrc 0.000.0004635903base 500.010.0000560iic 0.0040.0004635905nsrc 0.010.0000560.4iic+nsrc 0.0040.0004635905.2base 1000.450.004907021583iic 0.3565.245041463594.4489nsrc 0.450.0348680.621.4557.8iic+nsrc 0.3565.29507446359.46.4516.4 base 20308100.000.0000060.6iic 0.0040.0004758502.8nsrc 0.000.0000062iic+nsrc 0.0040.0004758502.8base 500.260.003924022.4370.8iic 0.2443.303797475854.4289.2nsrc 0.250.2939241.222.4334iic+nsrc 0.2443.69353347585.65.6289.2base 1000.220.003416039358.4iic 0.1851.793676475854367.8nsrc 0.220.293633139339.6iic+nsrc 0.1852.14333247585.62.4293.8 Average base 0.160.002041014.93248.67iic 0.1346.72208646972.002.13192.80nsrc 0.160.1020710.4715.00235.53iic+nsrc 0.1346.85199046972.272.40185.07 103

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4 parallelreplacementproblemsarecombinatorialsincedeterminingtheoptimalpolicyrequiresexaminingthereplacementofgroupsofassetsovertime,greatlycomplicatingtheanalysiswhencomparedtosingleassetproblems. Theliteratureinparallelreplacementanalysisgenerallyconsiderstechnologicalimprovementsanddeteriorationonlyintermsoftheobjectivefunction{costsandrevenues.Here,weconsidertechnologicalchangeanddeteriorationintheparallelreplacementproblem(PRP)bothintermsofcostsandcapacitygainsand/orlosses. Rajagopalan ( 1992 )studiesthisinacapacityexpansionproblem.Wemodelthisproblemasanintegerprogramandanalyzethemodeltodemonstratehowtechnologyanddeteriorationaecttheoptimalpolicyandtheoptimalcost. Thischapterisoutlinedasfollows.InSection 5.2 ,weprovideanintegerprogrammingformulationforthePRESundertechnologicalchangeanddeterioration.InSection 5.3 ,weexperimentallyanalyzetheeectsoftechnologyanddeteriorationontheoptimalreplacementpolicy.InSection 5.4 ,weanalyzetheoptimalsolutioncharacteristicsofPRESundertechnologicalchangeandthenderiveinequalitiesbasedontheoptimalsolutionproperties.Section 5.5 discussesimplementationstrategiesthatemploythese 104

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Hartman ( 2000 )withconstantdemand.Weagainassumethatthecapacityattimezeroissucienttomeetdemandintherstperiod,andthatnoassetintheinitialinventoryisequaltoitsmaximumservicelife.WealsoassumethatO&Mcostsarenon-decreasingandsalvagevaluesarenon-increasingasafunctionofage. Anassetisidentiedbyitsage,i=0;1;:::;Nattheendoftimeperiodj=0;1;:::;T.WemayretainorsalvageanassetaftereachperiodunlessitreachesitsmaximumageN,atwhichtimeitmustbesalvaged.TheproblemissolvedoverTperiods,withpurchasesallowedattheendofperiods0;1;:::;T1.AllassetsaresoldattheendoftimeperiodT. Thedecisionvariablesaregivenasfollows: Thedeterministiccostsassociatedwitheachofthesedecisionsaredenedasfollows: 105

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Withthesevariablesandparameters,theintegerprogrammingformulationforPRESundertechnologicalchangeanddeteriorationfollows. minT1Xj=0(pjBj+kjZj)+N1Xi=0T1Xj=0(cijXij+hijYij)NXi=1TXj=0rijSij s.t.N1Xi=0aijXijd0jT1; wherea0=aN;minfj+N;Tgforj=0;:::;T1. Theobjectivefunction( 5{1a )minimizesdiscountedpurchase,inventoryandO&Mcostslesssalvagerevenues.Constraints( 5{1b )ensurethatwehaveenoughcapacitytosatisfyperiodicdemands.Constraints( 5{1c )through( 5{1e )arereferredtoasowconservationconstraints(seenetworkinterpretationbelow).Constraint( 5{1f )enforcesthatpurchasedassetscaneitherbeusedorplaceddirectlyininventory.Constraint( 5{1g ) 106

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WeillustratetheformulationasanetworkgiveninFigure 5-1 .Thenetworkisdrawnontwoaxeswiththey-axisrepresentingtheageofanasset(i=1;:::;N)andthex-axisrepresentingtheendofthetimeperiod(j=1;:::;T).Figure 5-1 representsaproblemwithN=3andT=5.Thenodesarelabeledaccordingtotheageofanassetandtheendoftimeperiod,(i;j),althoughthelabelshavebeenremovedfromthegureforclarity.Thelabelsa01,a12,a23anda34ontopofeachnodethroughthediagonal(0;1)to(3;4)representthetechnologicalchangeanddeteriorationparametersforani-periodoldassetatperiodj.Notethatweremovedthea-valuelabelsfromtheremainderofthegureforclarity.WecaninterpretaijastheproportionofcapacityprovidedbyXijtowardsmeetingdemand.Weassumewithoutlossofgeneralitythata006=1(sinceotherwisea-valuescanbenormalizedwithdemandmakingthepercentagechangeincapacitymoreobvious).Wedenea-valuesbasedonthetechnologicalchangeanddeteriorationasfollows. 107

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ReferringtoFigure 5-1 ,thenumberofi-periodoldinitialinventoryassets(ni)isrepresentedbythesupplyateachnode(i;0),i>0andi0forcesareplacementattimezero,whichcanbeincorporatedeasily.)Theowintoeachnode(0;j),j0andj0. Flowbetweennodes(i;j)and(i+1;j+1)representsassetsinuse(Xij)orininventory(Yij)fromtheendofperiodjtotheendofperiodj+1,atwhichtimetheassetsarei+1periodsold.UtilizedassetsrepresentedbydashedarcsinFigure 5-1 contributetomeetingdemandwhilestoredassetsarerepresentedbycurvedarcsanddonotcontributetomeetingdemand.Weassumeherethatstoredassets\age"eachperiodinstorage.AsXijandYij-variablesarecontinuous,Xij=(Xij+Yij)representstheutilizationlevelofeachassetinoperationinagivenperiodassumingdemandisspreadequallyovereachcluster. 108

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RepresentationofPRESundertechnologyanddeteriorationasanetworkwithowrepresentingpurchase(B),utilization(X),storage(Y),andsalvage(S)variables,initialinventorysupplynandtechnologicalchangeanddeteriorationparameter(a). notechnologicalchange,technologicalchange,deterioration,andbothdeteriorationandtechnologicalchange. 109

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Case1:NoTechnologicalChangeandNoDeterioration Inthiscasetheaveragetotalcostis$95053:16,theaveragereplacementageis6:18andtheaverageinventoryiszero. Case2:TechnologicalChangeandNoDeterioration Inthiscasetheaveragetotalcostis$74289:98,theaveragereplacementageis6:32andtheaverageinventoryis0:3. Case3:DeteriorationandNoTechnologicalChange Inthiscasetheaveragetotalcostis$74299:46,theaveragereplacementageis5:33andtheaverageinventoryis0:99. Case4:DeteriorationandTechnologicalChange Inthiscasetheaveragetotalcostis$119880,theaveragereplacementageis5:38andtheaverageinventoryis1:39. 110

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5-2 5-3 and 5-4 illustratethechangeintheaveragereplacementageaswechangethevalueforthetechnologicalchangecase,deteriorationcaseandboth,respectively.Asweincrease-valuesfrom1percentto10percent,theaveragereplacementageincreasesinthetechnologicalchangecase.FromFigure 5-3 ,weobservethatwetendtoreplacemorefrequentlyinthedeteriorationcase,thustheaveragereplacementagereducesasweincreasethevalue.InFigure 5-4 ,weobservethattheaveragereplacementagedecreasesas100increasesfrom1percentto10percentatmost. Inthetechnologicalchangecase,asweincreasethechangeina-valuesfrom1percentto10percent,theaverageinventoryreduces,whileunderdeterioration,asweincreasethechangeina-valuesfrom1percentto10percent,theaverageinventoryincreases.Underthecasewherewehavebothdeteriorationandtechnologicalchange,averageinventoryincreasesasweincreasethechangeina-valuesfrom1percentto10percent. Figure5-2. Averagereplacementagevs.valueforthetechnologicalchangecase Figure5-3. Averagereplacementagevs.valueforthedeteriorationcase 111

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Averagereplacementagevs.forboththedeteriorationandtechnologicalchangecase 5.2 .Weobservethatasthechangeinparameteraincreases,theaveragereplacementageofclustersincreasesinthetechnologicalchangecase,whiletheaveragereplacementageofclustersdecreasesunderdeterioration.Underdeterioration,moreassetsareheldinreservetomeetfuturedemandwhenexpectinglossesincapacityasweincreasedeterioration.Ontheotherhand,inthetechnologicalchangecase,fewerassetsareneededininventoryasweincreasetechnologicalchange,implyingthatastechnologyimproves,wetendtokeeplowerinventorywhiletakingadvantageofnewtechnology.Wealsoobservethattechnologicalchangereducescosts,whiledeteriorationincreasesthetotalcosts.Inaddition,incorporatingtechnologicalchangeanddeteriorationintothemodeltendstocomplicatethesolutionprocedure. 112

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Proof. Proof. Consider(1).Inthiscase,notethatwecandeneanewsolution(usingprimenotation)whereS0ij=Sij+minfYij;Si+1;j+1g,I0ij=YijminfYij;Si+1;j+1gandS0i+1;j+1=Si;jminfYij;Si+1;j+1g.Oursavingswiththenewsolutionis(ri+1;j+1rij+hij)minfYij;Si+1;j+1g.Asri+1;j+1rijandcostsassociatedwithinventoryarereduced,andthisconstructedsolutionhasalowercostwhileretainingfeasibility.Thus,theoriginalsolutioncannotbeoptimal. Consider(2).InthiscaseweassumeSi+1;j+1=0.Sk+1;j+1>0asonlyonei;jexistswithYij1.Inthissituation,theremustexistanotheragek6=isuchthatSk+1;j+1>0asthedemandconstraintholdsatequalityandbyassumption,Yl;j+1<1foralll.Assume 113

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ThisconstructioncanbeaccomplishedifthereismorethanoneiandjsuchthatYij1.Therefore,bycontradiction,anoptimalsolutiontoPREScannotcontainaniandjsuchthatYij1. Proof. 5 andLemma 3 Luo ( 2003 )tothePRESundertechnologicalchangeproblem.ByCorollary 1 ,weknowthatinanoptimalsolution,anysalvagerequiresapurchase.Usingthisproperty,wemodifyIICandNSRCcutsforthePRESundertechnologicalcaseformulationwithoutcuttingoanyoptimalsolutions,althoughtheymightcutofeasiblenon-optimalsolutions.Hence,whilewecanemploytheseinequalitiesinoptimizingPRESundertechnologicalchange,thesearenotvalidinequalities. 5 Proof. 5{2 )impliesthatthenumberofassetssalvagedcannotbegreaterthanthesizeoftheinitialinventorycluster.Asthenumberof 115

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1 inanyoptimalsolution,ifSij>0,thenZj=1whensalvagevaluesarenon-increasinginage,O&Mcostsarenon-decreasinginageforaxedperiodandinventorycostsarenonzeroandlessthanO&Mcosts.Thus,constraints( 1 )satisfyalloptimalPRESundertechnologicalchangesolutionsunderthespeciedcostassumptions. Now,weextendtheNSRCinequalitiesgiveninChapter4forthePRESundertechnologicalchange.AsdiscussedinChapter4,eachinequalityrequiresthreecomponents: 1. AsupernodewhichisasetP0ofnodes(0;j)withalloftheirassociated\diagonalnodes"(1;j+1);(2;j+2);:::;(N;N+j)foreachjdenedbyall(0;j)nodes. 2. AsetI0ofinownodeswithatleastone(i;j),i>jforeach(0;j)2P0. 3. AsetO0ofoutownodes(i;j),jiandO0P0. Giventhesedenitions,wecanwritethefollowing. 5 Proof.

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117

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Weintroducetheinitialinventoryinequalities( 5{2 )now.Forexample,forthetwo-yearoldcluster,weinclude: 118

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0:94S2;1+0:90S4;1+0:86S6;1+0:84S7;1S1;2S2;3S8;9(0:98n1+0:94n3+0:90n5+0:88n6) 1:04Z6=23:23Z6: 119

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Chen ( 1998 ).Specically,wegeneratedataasfollows:WexthehorizontoT=100andconsidertwosizesforthenumberofinitialclusters,g=10and20.Thephysicallifeoftheassetisgivenas:N=g+[0;(Tg)]+; 5-1 ,whereU[a;b]denotesanintegernumberdrawnuniformlyfromtheinterval[a;b].Wealsoconsiderthexedcosttopurchasecostratiof2f10;50;100g,andthechangeparameterfora-values2f0:02;0:03g.Foreachofthe24possiblecombinationsofg,,andf,veinstancesarerandomlygeneratedforatotalof120problems. 120

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DatagenerationforPRESundertechnologicalchange. ParametersData AgeofassetsinaninitialclustergiU[0;N]SizeofclusterniU[2;10]UnitpurchasecostpjU[100;500]FixedcostkjU[100f;500f]UnitinventoryholdingcosthijU[1;50]UnitO&McostcijU[50;100]andincreasing=U[0;10]eachperiodUnitsalvagevaluerij70percentofpjdecreasing30percenteachperioda-valuea0;0=1,ai;j=ai1;j1anda0;j=a0;j1+U[0;] 5{2 )andNSRCTech.WeaddinequalitiestothecutpoolofCPLEXandsolvethePRESundertechnologicalchangeformulation( 5{1a ){( 5{1j )usingCPLEXwithitsdefaultsettings.Sincedecisionsearlierinthehorizonhaveasignicanteectonthesolutionslaterinthehorizon,wegeneratevalidinequalitiesand( 5{3 )earlyinthehorizon.Inourimplementation,wegenerateasmanyNSRCTechinequalitiesaswecanthatcutotheinitialoptimalfractionalsolution.TogenerateNSRCTechcuts,wepickafractionalZ-valueasapartoftheinitialLPrelaxationsolution.WethenselectthetimeperiodnodecorrespondingtothefractionalZ-valueasanoutownode,fromwhichtheowcorrespondingtothesalvageisisolated.Bybacktrackingtotheinitialinventoryassets,wedeterminethenodesandtheowsthatweusetogenerateNSRCTechcuts.WeaddbothIICTechandNSRCTechcutstotherootnode,andthensolvetheaugmentedformulationbyCPLEX. Wecomparetheeciencyofthefollowingsolutionapproachesinourcomputationalexperiments. 5{1a ){( 5{1j ),withoutaddinganyuserinequalities. 5{2 )withbase. 5{3 )withbase. 5{2 )and( 5{3 )withbase. 121

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5-2 and 5-5 summarizetheresultsofourcomputationalexperiments.Averagesover5probleminstancesaregivenfor: Tables 5-2 and 5-3 presentsresultsforT=100and=0:02,whereeachtableentrycorrespondstotheaverageperformanceofthealgorithmover5instances(veeachforf=10,50and100).WeobservethatbothoftheiictechandnsrctechstrategiesimproveuponthedirectsolutionofPRESundertechnologicalchangebyCPLEX,whiletheinequalities( 5{2 )andtheircombinationwith( 5{3 )performwell,improvingtheresultsbyafactorof3.5inCPUruntimeoverbase,averagingoverallinstances.Althoughwegenerateonly1:5NSRCTechcutsperinstanceonaverage,theirimplementationimprovesthesolutiontimebyafactorof2:5. WepresentresultsforT=100and=0:03inTables 5-4 and 5-5 .WeobservethatbothoftheiictechandnsrctechstrategiesimprovethesolutionofPRESundertechnologicalchangeoverCPLEXbyafactorof2,onaverage. 122

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SummaryofexperimentsforT=100,=0:02and=0:2. (10,28)base101.490.00188420159.4079.4iictech1.433.826526405.4132.4030.2nsrctech1.490.0133411.4126.6018.4iictech+nsrctech1.433.846526406.8132.4030.2base502.160.0054310151.4045iictech1.987.1713468406117.6041.2nsrctech2.160.34141162118.2037.8iictech+nsrctech1.987.1723188408117.4071.6base1004.480.008700143.4019.2iictech3.6121.02137740610706nsrctech4.460.551377210706iictech+nsrctech3.6021.10137740810706 (20,36)base100.850.0029610150029.8iictech0.814.312251666141.6016nsrctech0.850.0111241.2147.4010.6iictech+nsrctech0.814.312251667.2141.6015.8base500.920.005330133.2018.6iictech0.849.031233666121.407.4nsrctech0.920.013371.4120.204.2iictech+nsrctech0.849.031233667.4121.407.4base1001.720.00511580127.20.2370.6iictech1.4712.544785663.4102.2022.8nsrctech1.720.0335911.8101.8017.2iictech+nsrctech1.4712.544785665.2102.20.222.8 averagebase1.940.0013299.230.00144.100.0393.77iictech1.699.654940.07535.47120.370.0020.60nsrctech1.930.163980.971.63120.200.0015.70iictech+nsrctech1.699.666559.97537.10120.330.0325.63 5-4 and 5-5 weobservethatthedicultyofaninstanceispositivelycorrelatedwithandg,andthus,N,values.Additionally,thexedcosttopurchasecostratiofhasapositiveeectonthedicultyoftheinstances.Theaveragegapimprovementduetotheinequalities( 5{2 )and( 5{3 )isnegativelycorrelatedwithN.Itisalsoclearthatasthechangeina-valueparameterincreasesfrom0:02to0:03,theinstancestendtobecomehardertosolve. 4 tothetechnologicalchange 123

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SummaryofexperimentsforT=100,=0:02and=0:6. (10,64)base101.860.00398600132.40369.6iictech1.850.90321342074.4129.23.2365.8nsrctech1.860.03347111.2145.20339.4iictech+nsrctech1.850.92321342075.6129.23.2365.4base503.110.00426080101.20237.8iictech3.061.362512026.6120.837.6nsrctech3.110.03946530.899.20347.6iictech+nsrctech3.061.362512027.4120.837.4base1003.430.00386320141.60438.2iictech3.332.381632080105.638nsrctech3.420.051381.6105.803.6iictech+nsrctech3.332.381632081.6105.637.8 (20,68)base101.110.003160138.4017.4iictech1.100.95165234611949nsrctech1.110.001461.2119.803.8iictech+nsrctech1.100.961652347.211949base501.940.00787030132.20888.4iictech1.901.7121772337128.43.830.8nsrctech1.940.0866571.6136.2056.8iictech+nsrctech1.901.7221772338.6128.4430.8base1002.020.00427500126.60633.2iictech1.953.43444352337.21263.8373.8nsrctech2.020.11445461.4119.60.6371iictech+nsrctech1.953.43443842338.61263.6373.8 averagebase2.250.0040478.270.00128.730.00430.77iictech2.201.7913220.872200.20121.503.47132.50nsrctech2.240.0530141.831.30120.970.10187.03iictech+nsrctech2.201.7913212.502201.50121.503.47132.37 124

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SummaryofexperimentsforT=100,=0:03and=0:2. (10,28)base101.800.0087270148.2025.2iictech1.666.731394400.6125.205.8nsrctech1.790.0543611.2132.6012.6iictech+nsrctech1.666.731394401.8125.205.8base503.560.001047990138.40491.8iictech3.336.8426290406134.6067nsrctech3.550.07242492.4140.4054.2iictech+nsrctech3.336.8426294408.4134.6067base1003.830.0010180116.6015.2iictech3.0518.19355401.29902.8nsrctech3.810.313582.299.402.8iictech+nsrctech3.0518.23355403.49903 (20,36)base100.880.004500135.8011.6iictech0.817.88410664.813104.8nsrctech0.880.013891.4130.803.8iictech+nsrctech0.817.88410666.213104.8base501.440.001038670112.80.2724.6iictech1.2710.0661626666108.40200nsrctech1.440.061350841.699.20489.4iictech+nsrctech1.2710.0661626667.8108.80.2200.6base1002.450.0013844001170.2726.6iictech2.393.1694198662.2104.40346.6nsrctech2.450.19690031.2106.60238.6iictech+nsrctech2.393.2094198663.4104.40.2349.4 averagebase2.330.0059550.370.00128.130.07332.50iictech2.098.8130712.13533.47117.100.00104.50nsrctech2.320.1238907.401.67118.170.00133.57iictech+nsrctech2.098.8330712.90535.17117.170.07105.10

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SummaryofexperimentsforT=100,=0:03and=0:6. (10,64)base102.360.00310070148.62.4386.6iictech2.303.2633242071.8141.6338.6nsrctech2.360.0547641.4148.2045.8iictech+nsrctech2.303.3333242073.2141.6338.4base502.920.0088642013531455.8iictech2.842.871286962061131.43.21239.8nsrctech2.920.271086121.4132.601252.2iictech+nsrctech2.852.861336102062.4130.431239.8base1004.540.00828410121.21.8859.4iictech4.314.95761882080121.83.4561.8nsrctech4.540.03760621.8125.20572.4iictech+nsrctech4.314.95750332081.8121.83562.4 (20,68)base101.250.0035760152.82.445.4iictech1.232.873842346129.239.6nsrctech1.251.3445821.2131.4024.2iictech+nsrctech1.232.873842347.2129.239.4base502.150.00420940172.42.4416.6iictech2.121.27389002335143.43375.8nsrctech2.150.01385801.2144.80371.4iictech+nsrctech2.121.28387212336.2143.43375.4base1003.030.00335970118.42.41451.2iictech2.913.83673352333.61263.2553.2nsrctech3.030.09826722.2129.60662.8iictech+nsrctech2.913.83774352335.8125.83.2631.4 averagebase2.710.0046959.570.00141.402.40769.17iictech2.623.1752471.102204.57132.233.13463.13nsrctech2.710.3052545.101.53135.300.00488.13iictech+nsrctech2.623.1954751.302206.10132.033.03476.13

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Wehavepresentedmixedintegerprogrammingapproachesforsolvingcapacitatedlot-sizingandparallelassetreplacementproblems.Inparticular,ourstudyhasyieldedcuttingplanesthatareeectiveinsolvingtheseproblemswithinavarietyofcuttingplanealgorithms.Inthisconcludingchapter,wesummarizeourcontributionsanddiscusssomepromisingfutureresearchdirections. InChapter 2 ,westudythesingleitemcapacitatedlot-sizingproblem(CLSP).Weintroduceasetofdynamicprogramming(DP)-basedinequalitiesthatcanbeusedtoaugmenttheCLSPmixedintegerprogrammingformulation.Wetestseveralpossibleimplementationsthatemploytheseinequalities.OurbestimplementationappendsaninitialsetofDPbasedinequalitiestothemixedintegerprogrammingformulationbasedonstageinformationfromthepartialexecutionofaforwardDPrecursion,andthenaddsliftedcutting-planeinequalitiesinacut-and-branchfashionattherootnode.Computationalexperimentsshowthatthesevalidinequalitiesdrasticallyreducetheproblemsolutiontime.Additionally,wedemonstratethatourapproachismoreecientthanalternativeintegerprogramming-basedalgorithms. InChapter 3 ,weextendourresultsinChapter 2 tothemulti-itemcapacitatedlot-sizingproblem(MCLSP).WeuseDPbasedinequalitiesgeneratedonMCLSPrelaxedformulationstotightentheMCLSPformulation.Furthermore,weuseintegerprogrammingtechniquestoprovidepartialobjectiveinequalitiesoversubsetsofitemsandtimeperiods.Weanalyzeliftingtechniquesforimprovingthesevalidinequalitiesandprescribeaseparationalgorithmthatallowstheseinequalitiestobegeneratedinacuttingplanealgorithm. Thecomputationalexperimentswiththepartialobjectiveinequalitiessuggestthattheyareeectiveinsolvingmulti-itemlot-sizingproblemswhenusedascuttingplanes.Itisofinteresttostudythestrengthoftheseinequalities.Inaddition,the 127

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TheresultsinChapters 2 and 3 canbeextendedtotheproblemdomainsthatcontaintheCLSPandMCLSPconstraintsasasubstructure.OurinequalitiesmayalsobeeectiveformoregeneralversionsoftheCLSPandMCLSP,suchasvariationsthatincludeinventoryxedcostsorbounds,andvariablelowerboundsonproductionandbackorders,sincethesefeaturesdonotworsenthecomplexityofthedynamicprogrammingalgorithmsthatareusedtogenerateourproposedinequalities.Furthermore,sincetheCLSPisaspecialtypecapacitatedxed-chargenetworkowproblem,webelievethatabetterunderstandingofhowtheobjectivefunctionisusedtoobtainvalidinequalitiesforlot-sizingproblemswillprovideusabetterunderstandingofhowtoeectivelygeneratevalidinequalitiestostrengthenxed-chargenetworkowformulations. Additionally,thereexistseveralotherproblemsasidefromlot-sizingandxedchargenetworkow,suchasknapsackproblemsandequipmentreplacementproblems,whereonecanemploythetechniquespresentedinChapters 2 and 3 .Particularly,someoftheresultsforthepartialobjectiveinequalitiespresentedinChapter 3 havethepotentialtobeextendedtogeneralmixedintegerprogrammingmodels.FutureresearchincludesgeneralizingDPbasedandpartialobjectiveinequalitiesasfaraspossibletomaximizethebreadthofproblemsthatcanbenetfromtheseapproaches. InChapter 4 ,wedenevalidinequalitiesforanintegerprogrammingformulationfortheparallelreplacementproblemundereconomiesofscale(PRES),whichincludesxedandvariablecosts.PRESisconcernedwiththereplacementschedule(periodickeepandreplacedecisions)foreachindividualassetinagroupofassetsthatoperateinparallelandareeconomicallyinterdependent.Specically,weexaminethecasewhereaxedchargeisincurredineachperiodwhenanassetispurchased,assumingconstantdemand.Thevalidinequalitiesaremotivatedbyanimplicationofthe\no-splitting 128

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Chapter 4 focusesonthehomogeneousassetcaseinwhichallassetsaresimilarovertimeandthereisonlyonetypeofassetavailableforpurchaseineachtimeperiod.Itshouldbeclearthattheproblembecomesmorecomplicatedintheheterogeneousassetcasewheremultipletypesofassetsareavailableineachperiodforreplacementovertime.Thisisclearlyamorerealisticinstanceasmanufacturersandserviceprovidersgenerallyhaveanumberofsuppliersfromwhichtochoosetheirequipment.Futureresearchincludesmodifyingthevalidinequalitiesforthiscase. InChapter 5 ,wepresentanintegerprogrammingformulationforsolvingthePRESundertechnologicalchangeanddeterioration.Wealsoprovideinsightsintotheeectsoftechnologicalchangeanddeteriorationontheoptimalreplacementpolicyandoptimalcosts.WeanalyzetheoptimalsolutioncharacteristicsofPRESundertechnologicalchange,andthenextendthecuttingplanesproposedforPREStothetechnologicalchangecasebasedontheseoptimalsolutioncharacteristics.ThecomputationalexperimentswiththesecuttingplanessuggestthattheyarequiteeectiveinsolvingPRESundertechnologicalchangeproblemsinacut-and-branchalgorithm.Furtherresearchincludesvaryingdemandsanddevelopingsolutionapproachesforthedeteriorationcase. 129

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_I.EsraBuyuktahtaknwasborninKonya,Turkey1980,toAyseandAdemBuyuktahtakn.ShegraduatedfromMeramLyceeofSciencein1998.ShereceivedherBSdegreeinindustrialengineeringfromFatihUniversityin2002andMSdegreeinindustrialengineeringfromBilkentUniversityinTurkey2005.AftergraduationfromBilkentUniversity,sheenrolledintheDepartmentofIndustrialandSystemsEngineeringatLehighUniversity,whereshecompletedherMSdegreeinmanagementsciencein2007.InAugust2007,shejoinedtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridatopursueherPhDdegree.FollowinggraduationinAugust2009,shewilljointheSystemsandIndustrialEngineeringDepartmentattheUniversityofArizonaasaVisitingAssistantProfessor. 136