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

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

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Hartman, Joseph C.
Committee Co-Chair:
Smith, Jonathan
Committee Members:
Geunes, Joseph P.
Najafi, Fazil T.
Banerjee, Arunava
Graduation Date:
8/8/2009

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Assets ( jstor )
Dynamic programming ( jstor )
Fixed charges ( jstor )
Fixed costs ( jstor )
Integer programming ( jstor )
Integers ( jstor )
Objective functions ( jstor )
Optimal solutions ( jstor )
Technological change ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
dynamic, integer, optimization, production, replacement
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Industrial and Systems Engineering thesis, Ph.D.

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. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2009.
Local:
Adviser: Hartman, Joseph C.
Local:
Co-adviser: Smith, Jonathan.
Statement of Responsibility:
by Ismet Buyuktahtakin.

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UFRGP
Rights Management:
Copyright Buyuktahtakin, Ismet. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
489120214 ( OCLC )
Classification:
LD1780 2009 ( lcc )

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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