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Optimal Supply Chain Planning Problems with Nonlinear Revenue and Cost Functions

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Title:
Optimal Supply Chain Planning Problems with Nonlinear Revenue and Cost Functions
Creator:
Agrali, Semra
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (119 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Geunes, Joseph P.
Committee Members:
Romeijn, Hilbrand E.
Smith, Jonathan
Srinivasan, Sivaramakrishnan
Graduation Date:
8/8/2009

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Cost allocation ( jstor )
Customers ( jstor )
Fixed costs ( jstor )
Integers ( jstor )
Linear programming ( jstor )
Objective functions ( jstor )
Optimal solutions ( jstor )
Safety stock ( jstor )
Supply ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
minlp, nonlinear, operations, supply
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
This dissertation studies problems arising in certain stages of a supply chain. We specifically focus on problems that have nonlinearity in revenue or cost functions, and problems that can be written as mixed-integer linear programming problems. There are four main chapters that provide contributions to the supply chain operations literature. We first consider the allocation of a limited budget to a set of investments in order to maximize net return from investment. In a number of practical contexts, the net return from investment in an activity is effectively modeled using an S-Curve, where increasing returns to scale exist at small investment levels, and decreasing returns to scale occur at high investment levels. We formulate the problem as a knapsack problem with S-Curve return functions and demonstrate that it is NP-Hard. We provide a pseudo-polynomial time algorithm for the integer variable version of the problem, and develop efficient solution methods for special cases of the problem. We also discuss a fully-polynomial-time approximation algorithm for the integer variable version of the problem. Then, we consider a stochastic knapsack problem with random item weights that follow a Poisson distribution. We assume that a penalty cost is incurred when the sum of realized weights exceeds capacity. Our aim is to select the items that maximize expected profit. We provide an effective solution method and illustrate the advantages of this approach. We then consider a supply chain setting where a set of customers with a single product are assigned to multiple uncapacitated facilities. The majority of literature on such problems requires assigning all of any given customer's demand to a single facility. While this single-sourcing strategy is optimal under certain cost structures, it will often be suboptimal under the nonlinear costs that arise in the presence of safety stock costs. Our primary goal is to characterize the incremental costs that result from a single-sourcing strategy. We propose a general model that uses a cardinality constraint on the number of supply facilities that may serve a customer. The result is a complex mixed-integer nonlinear programming problem. We provide a generalized Benders decomposition algorithm to solve the model. Computational results for the model permit characterizing the costs that arise from a single-sourcing strategy. Finally, we consider a multi-period component procurement-planning and product-line design problem with product substitutions and multiple customer segments. Each customer segment has a preferred product and a set of alternative products. If a customer's preferred product is not made available, demand can be satisfied using an alternative product at a substitution cost. We assume each product is assembled-to-order from a set of components, and inventory is held at the component level. Our aim is to determine a product portfolio, substitution plan, and procurement plan in order to maximize profit. We develop a large-scale mixed-integer linear programming formulation, prove that the problem is NP-Hard and propose a Benders decomposition-based exact algorithm. We provide computational tests that compare our algorithm with a commercial mixed-integer linear programming solver, CPLEX, and show that our algorithm can handle large problem sizes, while CPLEX runs out of memory for medium-sized problems. We conclude the dissertation by discussing our contributions to the literature, and provide some future research directions based on our results. ( 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: Geunes, Joseph P.
Statement of Responsibility:
by Semra Agrali.

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Source Institution:
UFRGP
Rights Management:
Copyright Agrali, Semra. 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:
489240948 ( OCLC )
Classification:
LD1780 2009 ( lcc )

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whohavealwaysencouragedmetopursuemydreams 3

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First,Iwouldliketothankmyadvisor,Dr.JoeGeunes,forallhishelp,guidanceandcontinuoussupport.Hepaidattentiontoeverystepofthisdissertationandhelpedmeimprovemywriting.Ifeelfortunatetohavehimasmyadvisor.IwouldliketoacknowledgeDr.ColeSmith,Dr.EdwinRomeijnandDr.SivaSrinivasanforparticipatinginmydissertationcommitteeandtheirinsightfulcomments.Ialsothankmyrecommendationletterwriters,Dr.EdwinRomeijn,Dr.ColeSmithandDr.JoeHartman.SpecialthanksgotoDr.ColeSmithforhiscontinuoussupportandinvaluableadviceduringmyjobsearchprocess.Iwouldliketoacknowledgemyfriendswho,directlyorindirectly,helpedme.IespeciallythankGoncaYldrmforbeingsuchawonderfulfriend.ShelistenedtoallmycomplaintsandgavethesupportthatIneeded.Iwillmissheralot.IthankFadimeUneyYuksektepeforherinvaluablefriendshipandallherhelpespeciallyduringmyjobsearchprocess.Iwouldliketothankmydearcolleague,ChaseRainwaterfornotonlybeingagreatocematebutalsobeingagreatfriend.IalsothankCandaceRainwaterforthesongsthatshesangforus.Iwillmissour\AmeriTurks"band.Moreover,Ithankmyparents,Niluferand_Ilhan,andmysisters,_Ilknur,SemanurandGulforalwaysbelievinginmeandfortheirsupportateverystepofmylife.Iamveryluckytobepartofsuchawonderfulfamily.Icouldnothavedoneanythingwithouttheirpatience,encouragementandlove.Last,butnotleast,Ithankmybelovedhusband,Caner.Heismybestfriend,agreatcolleague,andtheonethatIadmiremost.IthankhimforgivingmebrilliantideaswhenIfeellostinmyresearch,debuggingmycodewithmeandeverythingthathehasbroughttomylife.Withouthissupportandencouragement,thisdissertationwouldnotbecomplete. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2ALGORITHMSFORKNAPSACKPROBLEMSWITHS-CURVERETURNFUNCTIONS ..................................... 16 2.1IntroductionandMotivation .......................... 16 2.2LiteratureReview ................................ 18 2.3ProblemDescription,Formulation,andSolutionProperties ......... 22 2.4PolynomiallySolvableSpecialCase ...................... 28 2.5ModelwithIntegerVariableRestrictions ................... 32 2.5.1Pseudo-PolynomialTimeAlgorithm .................. 34 2.5.2FullyPolynomialTimeApproximationAlgorithm .......... 35 3ASINGLE-RESOURCEALLOCATIONPROBLEMWITHPOISSONRESOURCEREQUIREMENTS .................................. 39 3.1Introduction ................................... 39 3.2LiteratureReview ................................ 42 3.3ProblemAnalysisandaSolutionMethodforaSpecialCase ........ 43 3.4SolutionApproachforProblem[SKPP] .................... 45 3.5ComputationalStudy .............................. 49 4AFACILITYLOCATIONMODELWITHSAFETYSTOCKCOSTS ..... 55 4.1IntroductionandMotivation .......................... 55 4.2LiteratureReview ................................ 58 4.3ProblemDenitionandMathematicalModel ................. 62 4.3.1IdenticalSupplyCostsandCustomerVariances ........... 65 4.3.2SpeciallyStructuredAssignmentandHoldingCosts ......... 68 4.4AGeneralizedBendersDecompositionApproachfor(ILP) ......... 71 4.5ComputationalResults ............................. 75 5COMPONENTPROCUREMENTPLANNINGANDPRODUCTPORTFOLIODESIGNPROBLEM ................................. 85 5.1IntroductionandMotivation .......................... 85 5

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................................ 87 5.3ProblemDescriptionandMathematicalFormulation ............. 90 5.4SolutionMethodology ............................. 96 6CONCLUSIONANDFUTURERESEARCHDIRECTIONS ........... 104 6.1AlgorithmsforSolvingaKnapsackProblemwithS-CurveReturnFunction 104 6.2AFacilityLocationModelwithSafetyStockCosts ............. 105 6.3ProcurementPlanningandProductPortfolioDesignProblem ....... 106 APPENDIX AGENERALIZEDKARUSHKUHN-TUCKEROPTIMALITYCONDITIONS 108 BPROOFOFPROPOSITION3 ........................... 109 REFERENCES ....................................... 112 BIOGRAPHICALSKETCH ................................ 119 6

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Table page 3-1Parametervaluesfortestinstances ......................... 53 3-2Computationaltestresults .............................. 54 4-1Dataparametersettings ............................... 83 4-2Themax,min,andaveragevaluesofZ5fordierentvaluesofE[^c=h] ..... 83 4-3Themax,min,andaveragevaluesofZ5fordierentvaluesofCoV ...... 84 4-4Themax,min,andaveragevaluesofZ5fordierentvaluesofxedcost ... 84 7

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Figure page 2-1AnS-curveresponsefunction ............................ 38 2-2Iterativeboundingfunctionsforthemodel ..................... 38 3-1Gams/BaronPerformanceAnalysis ......................... 53 4-1Costincreasemultiplierforsingle-sourcingasafunctionof 80 4-2RatioofcostsavingsfromsplittingtominimumcostasafunctionofE[c=h] .. 80 4-3TheeectofE[^c=h]onZ5 81 4-4TheeectofNionZkfordierentvaluesofE[^c=h] ............... 81 4-5TheeectofNionZkfordierentvaluesofCoV ................ 82 4-6TheeectofNionZkfordierentvaluesofxedcost ............. 82 4-7CputimesfordierentvaluesofNi 83 8

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Thisdissertationstudiesproblemsarisingincertainstagesofasupplychain.Wespecicallyfocusonproblemsthathavenonlinearityinrevenueorcostfunctions,andproblemsthatcanbewrittenasmixed-integerlinearprogrammingproblems.Therearefourmainchaptersthatprovidecontributionstothesupplychainoperationsliterature. Werstconsidertheallocationofalimitedbudgettoasetofinvestmentsinordertomaximizenetreturnfrominvestment.Inanumberofpracticalcontexts,thenetreturnfrominvestmentinanactivityiseectivelymodeledusinganS-Curve,whereincreasingreturnstoscaleexistatsmallinvestmentlevels,anddecreasingreturnstoscaleoccurathighinvestmentlevels.WeformulatetheproblemasaknapsackproblemwithS-CurvereturnfunctionsanddemonstratethatitisNP-Hard.Weprovideapseudo-polynomialtimealgorithmfortheintegervariableversionoftheproblem,anddevelopecientsolutionmethodsforspecialcasesoftheproblem.Wealsodiscussafully-polynomial-timeapproximationalgorithmfortheintegervariableversionoftheproblem.Then,weconsiderastochasticknapsackproblemwithrandomitemweightsthatfollowaPoissondistribution.Weassumethatapenaltycostisincurredwhenthesumofrealizedweightsexceedscapacity.Ouraimistoselecttheitemsthatmaximizeexpectedprot.Weprovideaneectivesolutionmethodandillustratetheadvantagesofthisapproach. Wethenconsiderasupplychainsettingwhereasetofcustomerswithasingleproductareassignedtomultipleuncapacitatedfacilities.Themajorityofliteratureon 9

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Finally,weconsideramulti-periodcomponentprocurement-planningandproduct-linedesignproblemwithproductsubstitutionsandmultiplecustomersegments.Eachcustomersegmenthasapreferredproductandasetofalternativeproducts.Ifacustomer'spreferredproductisnotmadeavailable,demandcanbesatisedusinganalternativeproductatasubstitutioncost.Weassumeeachproductisassembled-to-orderfromasetofcomponents,andinventoryisheldatthecomponentlevel.Ouraimistodetermineaproductportfolio,substitutionplan,andprocurementplaninordertomaximizeprot.Wedevelopalarge-scalemixed-integerlinearprogrammingformulation,provethattheproblemisNP-HardandproposeaBendersdecomposition-basedexactalgorithm.Weconcludethedissertationbydiscussingourcontributionstotheliterature,andprovidesomefutureresearchdirectionsbasedonourresults. 10

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Intoday'scompetitivebusinessenvironment,companiesthatusetheirsupplychainsystemseectivelygaineconomicadvantagesovercompetitors.Theprimarypurposeofasupplychainistosatisfycustomerneedswhilemaximizingtheoverallvaluegenerated.Thevaluegeneratedbyasupplychainismeasuredbythedierencebetweenthenalproductvalueandtheoverallcostincurredacrossthesupplychain. Ateachstageofasupplychaintwometricsareusedtoassesstheoverallvaluegenerated:revenueobtainedandcostincurred.Thesemetricsusuallycannotbeexpressedaslinearfunctionsoftheactivitylevelsinreallifeproblems;insteadtheyareusuallyexpressedascomplexnonlinearfunctions.Inthisdissertation,weconsidercomplexitiesthatariseinparticularfunctionaldecisionareasofasupplychain,suchasmanufacturingoperationsanddistribution,wherenonlinearrevenueorcostfunctionsarise. Stagesofasupplychainareoftenprovidedwithalimitedbudgetthatcanbeinvestedincompetingactivities.Whenthebudgetlimitistheonlyconstrainingfactor,thentheresultingproblemfallsinthewell-knownclassofknapsackproblems.Thegoalistomaximizetheoverallreturnobtainedfromactivityinvestments.Inanumberofpracticalcontexts(e.g.,advertising),thenetreturnfrominvestmentinanactivityiseectivelymodeledusinganS-Curve,whereincreasingreturnstoscaleexistatsmallinvestmentlevels,anddecreasingreturnstoscaleoccurathighinvestmentlevels.InChapter 2 ,weanalyzeknapsackproblemswithS-Curvereturnfunctionsthatconsidertheallocationofalimitedbudgettoasetofactivitiesorinvestmentsinordertomaximizenetreturnfrominvestment. Anothertypeofknapsackproblemthatwestudyconsidersstochasticitemsizesanddeterministiccapacity.Suchproblemsariseinavarietyofresource-allocationcontextswhentheresourcecapacitymustbeallocatedtotaskswithnon-deterministiccapacityconsumption.InChapter 3 ,weassumethatitemweightsarerandomandfollowaPoisson 11

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Transportationisanotherimportantfunctionofasupplychain.Oneaimintransportationproblemsistodeterminetheassignmentofcustomerstosupplyfacilitiesthatminimizestotaltransportationcostwhileobeyingsupplylimitsandmeetingcustomerdemands.Transportationcostsdependonthelocationsoffacilities.Whilesomeplannersdecideonthefacilitylocationsandtransportationfromtheselocationstocustomersseparately,whentheyareconsideredtogether,theoverallvaluegeneratedcanbeincreased.Moreover,whenweconsidercontextswithuncertaindemands,itisimportanttoconsidertheimpactsofsafetystockcosts.Therefore,inventoryrelatedcostsatsupplyfacilitiesshouldbeaccountedforwhenmakingdecisionsonlocationandallocation.InChapter 4 ,westudyasupplychainsettingwheremultipleuncapacitatedfacilitiesserveasetofcustomerswithstochasticdemands.Sincethedemandisstochastic,someamountofsafetystockisheldatfacilities.Themajorityofliteratureonsuchproblemsrequiresassigningallofanygivencustomer'sdemandtoasinglefacility.Whilethissingle-sourcingstrategyisoptimalunderlinear(orconcave)coststructures,itwilloftenbesuboptimalunderthenonlinearcoststhatariseinthepresenceofsafetystockcosts.Ourprimarygoalistocharacterizetheincrementalcoststhatresultfromasingle-sourcingstrategy. Animportantstageofanysupplychainistheproductionofitems.Inthisstage,oneshoulddecideonwhichproductstooertothemarket,andhowtosatisfythedemand.Oeringasmallnumberofproductswillreducethecostassociatedwitheachproduct, 12

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5 ,weconsideramulti-periodcomponentprocurement-planningandproduct-linedesignproblemwithproductsubstitutionsandmultiplecustomersegments.Wehaveasetofproducts,eachwithaxeddesigncostandmultiplecustomersegments,whereeachhasdemandsforidealproducts,andsetsofalternativeproductsthatincludeallsubstitutesfortheiridealproducts.Ifacustomer'sidealproductisnotmadeavailable,demandcanbesatisedinmanycasesusinganalternativeproductatasubstitutioncost.Weassumethedemandislostifacustomerleavesthesystemwithoutbuyinganything.Eachproducthasaprotmarginthatiscustomer-segmentdependent.Moreover,eachproductisassembled-to-orderfromasetofcomponents,andinventoryisheldatthecomponentlevel.Hence,ouraimistodetermineaproductportfolio,substitutionplan,andprocurementplaninordertomaximizeprot. Intheremainderofthisdissertation,werstpresentouralgorithmsforknapsackproblemsthatoftenariseinadvertisingbudgetallocationinChapter 2 .WedemonstratethattheresultingknapsackproblemwithS-CurvereturnfunctionsisNP-Hard,provideapseudo-polynomialtimealgorithmfortheintegervariableversionoftheproblem,anddevelopecientsolutionmethodsforspecialcasesoftheproblem.Wealsodiscussafully-polynomial-timeapproximationalgorithmfortheintegervariableversionoftheproblem. 13

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3 ,westudyaclassofstochasticknapsackproblemswithPoissonresourcerequirements.Weprovideapolynomial-timesolutionmethodforthecontinuousrelaxationofthisproblem,acustomizedbranch-and-boundalgorithmforitsexactsolution,andillustratetheadvantagesofthissolutionapproachviaasetofrandomlygeneratedprobleminstances. InChapter 4 ,weanalyzeasupplychainsettingwheremultipleuncapacitatedfacilitiesserveasetofcustomerswithasingleproduct.Weproposeageneralmodelthatusesacardinalityconstraintonthenumberofsupplyfacilitiesthatmayserveacustomer.Theresultisacomplexmixed-integernonlinearprogrammingproblem.WeprovideageneralizedBendersdecompositionalgorithmforthecaseinwhichacustomer'sdemandmaybesplitamonganarbitrarynumberofsupplyfacilities.TheBenderssubproblemtakestheformofanuncapacitated,nonlineartransportationproblem,arelevantandinterestingprobleminitsownright.Weprovideanalysisandinsightonthissubproblem,aswellascomputationalresultsforthegeneralmodelthatpermitcharacterizingthecoststhatarisefromasingle-sourcingstrategy. InChapter 5 ,westudyamulti-periodcomponentprocurement-planningandproduct-linedesignproblemwithproductsubstitutions.Wedevelopalarge-scalemixed-integerlinearprogrammingformulation,provethattheproblemisNP-HardandproposeaBendersdecomposition-basedexactalgorithm. Chapter 6 concludesthisdissertationbydiscussingtherstfourchaptersandprovidingfutureresearchdirectionsrelatedtothesechapters. Inthisdissertation,weprovidesolutionalgorithmstoproblemsthatariseincertaindecisionprocesseswithinasupplychain.Contributionstotheliteratureareasfollows:(1)weshowthatthecontinuousknapsackproblemwithnon-identicalS-curvereturnfunctionsisNP-hard,providepotentialglobaloptimizationapproachesforsolvingthisdicultproblem,andprovidebothapseudo-polynomialtimealgorithmandafullypolynomialtimeapproximationschemeforthediscreteversionoftheproblem;(2)weprovidean 14

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MartelloandToth ( 1990 )).Incertaincontexts(e.g.,investmentinvariousnancialinstruments)theeectiveweightofanitemmayitselfbeadecisionvariable.Thatis,ifwearefreetoinvestanynonnegativeamountuptosomeupperlimitineachelementofasetofinvestmentinstruments,thenwehaveacontinuousversionoftheknapsackproblem.Whenthevalueoftheinstrumentislinearintheamountinvested,thentheresultingproblemisacontinuousknapsackproblemthatcanbesolvedbyinspection:simplysortinstrumentsinnonincreasingorderofper-unitrevenue,andinsertitemsintotheknapsackuntilthecapacityisexhausted(themoregeneralstandardcontinuousknapsackproblemmayemployacapacityconsumptionfactorper-unitofthedecisionvariablevalue,inwhichcasewesimplysortitemsinnonincreasingorderoftheratioofper-unitvaluetoper-unitcapacityconsumption). Ifthevalueoftheinvestmentinstrumentisnotalinearfunctionoftheinvestmentlevel,thentheresultingnonlinearknapsackproblemisnotnecessarilyeasilysolved(see BretthauerandShetty ( 2002a )foracomprehensivereviewoftheliteratureonnonlinearknapsackproblems).Inanumberofpracticalapplications,includingportfolioselectionandadvertisingbudgetallocation,thereturnoninvestmentfunctionmaytakeanonlinearformleadingtocomplexclassesofnonlinearknapsackproblems.Therelationshipbetweenadvertisingbudgetallocationandsalesresponsehasservedasthetopicofmanystudies 16

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SimonandArndt ( 1980 )surveyedthecharacteristicsofsales-advertisingresponsefunctions.Theirsurveyoftheliteratureshowedthatthemajorityofresearchonadvertisingresponsesubscribestooneoftwoproposedshapesoftheresponsefunction:(1)anonnegativeconcave-downwardcurveand(2)anS-curve.Thus,ifasupplier'ssalesresponsetoadvertisingineachmemberofasetofmarketsfollowsoneoftheseforms,thesupplierfacesthechallengeofdeterminingtheamountofalimitedbudgettoallottoeachmarketinordertomaximizesales.Insales-forcetime-managementcontexts,asimilarphenomenonoccurs,wherethefrequencyofsalescallstoaclientaectsthesalesresponse. Lodish ( 1971 )characterizesthisresponseasfollowinganS-curveshapeasafunctionofsalesvisitfrequency.Thesalespersonmustthereforeallocatethenumberofavailablevisitsduringaplanninghorizontoeachmemberofasetofclientsinordertomaximizesalesrevenue. Burkeetal. ( 2008 )analyzedarelatedproblemthatconsidersthecaseofconcave-downwardresponsefunctions(whicharecharacterizedbynonnegativeconcavefunctionswithzeroreturnattheorigin).Theyfocusonasettinginwhichabuyermustpurchaseaxedquantityfromanumberofcapacitatedsuppliers,andwhereeachsupplieroersa(concave)quantitydiscountstructure.Incontrast,wefocusonthecommonlyemployedS-curvereturnfunctionswhereincreasingreturnstoscaleexistatsmallinvestmentlevels,anddecreasingreturnstoscaleoccurathighinvestmentlevels.Figure 2-1 illustratesanexampleoftheshapeoftheS-curvereturnfunctionsweconsider. WeexamineabudgetallocationproblemrequiringthebestallocationofanavailablebudgetAamongNindependentinstruments.Thereturnofinstrumentiisgivenbythefunction~i(ai)whereaiistheinvestmentlevelallocatedtoinstrumenti.Theobjectiveistomaximizetotalnetreturnfrombudgetallocationtothedierentinstrumentswhilenotexceedingthelimitedbudget(welaterdenethetermnetreturnmorepreciselyinSection 2.3 ).Werecognizethepotentialuncertaintiesexistinginsuchapplicationareas,andourmodelcanbeemployedinsuchcontextswheneachfunction~i(ai)representstheexpected 17

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Theremainderofthischapterisorganizedasfollows.InSection 2.2 ,wereviewrelatedpastliteratureonbudgetallocationproblemsandapplications.WedenetheproblemandmodelformulationinSection 2.3 ,anddiscussapolynomiallysolvablespecialcaseinSection 2.4 .InSection 2.5 ,weconsidertheintegervariableversionoftheproblem,providingapseudo-polynomialtimealgorithmaswellasafullypolynomialtimeapproximationscheme. ZoltnersandSinha ( 1980 )providealiteraturereviewandaconceptualframeworkforsalesresourceallocationmodeling.Theydevelopageneralmodelforsalesresourceallocationwhichsimultaneouslyaccountsformultiplesalesresources,multipletimeperiodsandcarryovereects,non-separability,andrisk.Moreover,theydiscussseveralactualapplicationsofthemodelinpractice,whichillustratesthepracticalvalueoftheirintegerprogrammingmodels. Whenthesalesresponseorcostsarenotknownwithcertainty,theyareoftencharacterizedusingprobabilitydistributions. HolthausenandAssmus ( 1982 )discussa 18

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Norkinetal. ( 1998 )proposeageneralstochasticsearchprocedurefortheoptimalallocationofindivisibleresources,whichisposedasastochasticoptimizationprobleminvolvingdiscretedecisionvariables.Thesearchproceduredevelopsabranch-and-boundmethodforthisstochasticoptimizationproblem. Theproblemofresourceallocationamongdierentactivities,suchasallocatingamarketingbudgetamongsalesterritoriesisanalyzedby LussandGupta ( 1975 ).Theyassumethatthereturnfunctionforeachterritoryusesdierentparameters,andderivesingle-passalgorithmsfordierentconcavepayofunctions(basedontheKarush-Kuhn-Tucker,orKKT,conditions)inordertomaximizetotalreturnsforagivenamountofeort.Anumberofecientprocedureshavebeendevelopedsubsequenttothisforsolvingsingle-resource-allocationproblemsunderobjectivefunctionandconstraintassumptionsthatleadtoconvexprogrammingproblems,including Zipkin ( 1980 ), BitranandHax ( 1981 ), BretthauerandShetty ( 1995 2002b ),and KodialamandLuss ( 1998 ).Inaddition,severalpapershavefocusedonnonlinearknapsackproblemssatisfyingtheseconvexityassumptions,whenthevariablesmusttakeintegervalues,including Hochbaum ( 1995 ), Mathuretal. ( 1983 ),and BretthauerandShetty ( 1995 2002b ). Surprisinglylittleliteratureexistsoncontinuousknapsackproblemsinvolvingtheminimizationofaconcaveobjectivefunction(wheretheKKTconditionsarenotsucientforoptimality). MoreandVavasis ( 1990 )provideanecientmethodforndinglocallyoptimalsolutionsforthisclassofproblemsassumingobjectivefunctionseparability. Burkeetal. ( 2008 )consideraprobleminwhichaproducermustprocureaquantityofrawmaterialsfromasetofcapacitatedsuppliers.Theproducerseekstoobtainitsrequiredmaterialsatminimumcost,whereeachsupplierprovidesaconcavequantitydiscount 19

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Sunetal. ( 2005 )provideapartitioningmethodfortheintegerversionofthisproblemthatusesalinearunderestimationoftheobjectivefunctiontoprovidelowerboundsateachiteration. Romeijnetal. ( 2007 )considertheminimizationofaspeciallystructurednonseparableconcavefunctionoveraknapsackconstraint,andprovideanecientalgorithmforsolvingthisproblem. Theliteratureonknapsackproblemsinwhichtheobjectivefunctionisnonconvex(andnonconcave)issomewhatlimited. Ginsberg ( 1974 )wasthersttoconsideraknapsackproblemwithS-curvereturnfunctions,whichhereferredtoas\nicelyconvex-concaveproductionfunctions".Hecharacterizedstructuralpropertiesofoptimalsolutionsassumingdierentiabilityofthereturnfunctions,andpredominantlyassumingidenticalreturnfunctions. Lodish ( 1971 )consideredanonlinearnonconvexknapsackprobleminasalesforceplanningcontextinwhichtheresponsefunctionisdenedatdiscretelevelsofsalesforcetimeinvestment.Heapproximatedthisproblemusingtheupperpiecewiselinearconcaveenvelopeofeachfunction,andprovidedagreedyalgorithmforsolvingthisproblem(thisgreedyalgorithmprovidesanoptimalsolutionforcertaindiscreteknapsacksizes,butnotforanarbitraryknapsacksize). FreelandandWeinberg ( 1980 )addressedthecontinuousversionofthisproblemandproposedsolvingtheapproximationobtainedbyusingtheupperconcaveenvelopeofeachcontinuousreturnfunction. Zoltnersetal. ( 1979 )considergeneralresponsefunctionsandalsoproposeanupperconcaveenvelopeapproximation,alongwithabranchandboundprocedure,thatpermitssuccessivelyprovidingbetterapproximationsofthecontinuousfunctionsateachbranch.WediscussasimilarmethodforsolvingthecontinuousversionoftheproblemwithS-curvereturnfunctionsthattakesadvantageofthespecializedstructureofthesereturnfunctions. MorinandMarsten ( 1976 )devisedadynamicprogrammingapproach 20

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RomeijnandSargut ( 2009 )recentlyconsideredanonconvex,continuous,andseparableknapsackproblem,whichresultsasapricingsubprobleminacolumngenerationapproachforastochastictransportationproblem.Theyuseasequenceofupperboundingfunctionsthatpermitssolvingasequenceofspecially-structuredconvexprogramssuchthat,ingeneral,theprocedureconvergestoanoptimalsolutioninthelimit(wediscussasimilarapproachforsolvingthecontinuousversionofourprobleminthenextsection). Knapsackproblemswithnon-convex(andnon-concave)objectivefunctions,suchasthosementionedinthepreviousparagraph,fallintothedicultclassofglobaloptimizationproblems(see Horstetal. ( 1995 )),whichrequirespecializedsearchalgorithmsthatoftencannotguaranteeniteconvergencetoagloballyoptimalsolution.TheS-curvefunctionsweconsiderfallintothiscategory,althoughweareabletoexploitthespecialstructureofthesefunctionstoprovideeectivemethodsforsolvingthediscreteversionofthisproblem.Aswelaterdiscussingreaterdetail,thecontinuousversionoftheproblemweconsiderfallsintotheclassofmonotonicoptimizationproblems( Tuy ( 2000 )),andspecializedmethodsdevelopedforthisclassofglobaloptimizationproblemsthusprovideaviableoptionforprovidinggoodsolutions. Ourprimarycontributionsrelativetothisbodyofpreviousresearchincludeshowingthatthecontinuousknapsackproblemwithnon-identicalS-curvereturnfunctionsisNP-Hard,providingpotentialglobaloptimizationapproachesforsolvingthisdicultproblem,andinprovidingbothapseudo-polynomialtimealgorithmandafullypolynomialtimeapproximationschemeforthediscreteversionoftheproblem. 21

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WewishtoallocateabudgetofAdollarstothemarketinginstrumentsinordertomaximizetotalexpectedreturn.WeformulatethisknapsackproblemwithS-curvereturnfunctions(KPS)asfollows. Notethatwecanapplyanonnegativeweightcitoanyitemiintheobjectivefunction(e.g.,i(ciai))bysimplyredeningourifunction(i.e.,^i(ai)=i(ciai)),andtheresultingfunctionsretaintheS-curveshape(wethenneedtoredeneouriandivaluesaccordingly).Wecanalsoaccommodateanonnegativeweightciintheconstraint(e.g.,PNi=1ciaiA)usingthevariablesubstitutiona0i=ciaiandredeningthenetreturnfunctionusing^i(a0i)=i(a0i=ci). 22

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Asmentionedintheprevioussection,problem[KPS]fallsintotheclassofmonotonicglobaloptimizationproblems( Tuy ( 2000 )),becausewearemaximizinganondecreasingfunctionsubjecttoanondecreasingconstraintlimitedbyanupperbound(andwherethevariablesarenonnegative). Tuy ( 2000 )demonstratestheintuitiveresultthat,forsuchproblems,anoptimalsolutionexistsontheboundaryofthefeasibleregion.Heproposesaso-calledpolyblockalgorithm,whichperformsasearchoverasequenceofhyper-rectangles.Wenextbrieydescribetheapplicationofthisapproachforsolving[KPS].Letadenotethevectorofaivalues(i=1;:::;N),andletaLandaUdenotelowerandupperboundvectorsona(initiallywehaveaL0=0andaU0isthevectorofivalues,wherethesubscript0correspondstoaniterationcounter).DeneAasthesetofalla2RNthatsatisfythebudgetconstraint( 2{1 ).Beginningwiththeinitialinterval(orpolyblock)P0=[aL0;aU0],itisclearthat(a)ifaU02A,thenthissolutionisoptimal(becauseofthemonotonicityandboundarysolutionproperties),and(b)ifaL0=2A,thentheproblemisinfeasible.Assumingthatneitheroftheseholds,wewishthentobisectthis 24

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GivenanypolyblockPk=[aLk;aUk],thenclearlyifaLk=2A,wecaneliminate(prune)thepolyblock;ontheotherhand,ifaUk2Athenthissolutionprovidesbothanupperandlowerboundforthebestpossiblesolutioninthepolyblock.Ifneitheroftheseholds,thenaLkservesasalowerboundonthebestsolutioninthepolyblock,andweutilizeanupperboundingmethodforthebestsolutioninthepolyblock(thiscanbeobtained,forexample,byestablishingtheupperconcaveenvelopeofeachofthefunctionsi(ai)in[KPS],replacingthesefunctionswiththisupperconcaveenvelopefunctionin[KPS],andsolvingtheresultingconvexprogram;todothis,wesimplydeterminethesmallestpointontheconcaveportionofi(ai)suchthati(ai)=ai2@i(ai),andconnectalinefromtheorigintothispoint).Wethereforehavealloftheelementsweneedforabranch-and-boundtypeofalgorithm,wherebranchingcorrespondstobisectingavariable(andthussplittingapolyblockintwo),andfathomingapolyblockwithindexkisdonebyeither(a)verifyingthataUkisfeasibleandthereforethebestpossiblesolutionforthepolyblock;(b)verifyingthataLkisinfeasible,andthuspruningthepolyblock,or(c)verifyingthatthepolyblock'supperboundsolutionvalueisinferiortothebestknownsolutionvalue.Thispolyblockalgorithmicapproachwilleitherterminatewithan-optimalsolution(whereisapredeterminedoptimalitytolerance),orwillconvergetoanoptimalsolutionvalueinthelimit( Tuy ( 2000 )). Whilethepolyblockalgorithmhasbeenshowntobeeectiveformonotonicoptimizationproblems,theS-curvefunctionsweconsiderhaveaspecialstructurethatwemayexploittoprovidealternativeglobaloptimizationapproachesfor[KPS].Thefollowing 25

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2 and 3 )provideimportantpropertiesthatwewillutilizeindevelopinganadditionalglobaloptimizationsolutionapproachaswellassolutionmethodsforvariousspecialcasesof[KPS].Inparticular,Theorem 3 demonstratesthatanoptimalsolutionalwaysexistssuchthatatmostoneinstrumentiwillexistwithpositiveinvestmentatalevellessthani(i.e.,intheconvexportionofthei(ai)function).Thistheoremgeneralizesasimilarresultprovidedby Ginsberg ( 1974 )whoconsideredthedierentiablecasewithnonzerosecondderivatives(i.e.,strictconcavityintheconcaveportionandstrictconvexityintheconvexportionofthefunction). 2 wemusthave@i(ai)\@j(aj)6=;.Considerasolutionwithak=akforallk2Infi;jg,aj=aj+,andai=aiforsomeminfai;jajg,denotetheobjectivefunctionvalueofthisnewsolutionbyzn,andletdenoteanelementof@i(ai)\@j(aj).Bytheconvexityofi(ai)for0aii(andofj(aj)for0ajj),wehavej(aj+)j(aj)+;i(ai)i(ai): 26

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3 allowsustoeliminatethepartofthefeasibleregionwheremultipleitemsmaytakepositivevaluesstrictlybetween0andiintheconvexportionofthenetreturnfunction.Thispropertybecomesparticularlyusefulinprovidingsolutionmethodsforapracticalspecialcaseofproblem[KPS]inSection 2.4 .Itcanalsoaidinamoreecientapplicationofglobaloptimizationtechniquesfor[KPS].Wenextdiscusssuchaglobaloptimizationapproach(similarapproachesweresuggestedby Zoltnersetal. ( 1979 )foranonlinearsalesresourceallocationproblem,andby RomeijnandSargut ( 2009 )forsolvingasingly-constrainednonlinearpricingproblemembeddedinastochastictransportationproblem). Recognizingthatatmostoneinstrumentexistswithanoptimalvalueintheconvexportionofthereturnfunction,wecanthussolveasetofNsubproblems,wheretheithsubproblemrequires0aiiandjajjforallj6=i.Observethatfortheithsubproblem,eachofthefunctionsj(aj)isconcaveonthefeasibleregion,withtheexceptionofitemi.Forthisitem,weinitiallyapproximatei(ai)usingalinewithslopei(i)=i(seethepictureontheleftinFigure 2-2 ).Theresultingconvexprogrammingproblemservesasarootnodeproblemforabranchandboundsolutionapproachfortheithsubproblem,andthesolutionprovidesanupperboundontheoptimalsolutionoftheithsubproblem.Supposethattheoptimalvalueofaiinthisinitialupperbounding 27

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2-2 Observethateachtimewebranch,weprovideacloser(piecewiselinear)approximationoftheconvexportionofi(ai)fortheithsubproblem,andthateachproblemconsideredatanodeinthebranchandboundtreeisaconvexprogram(andcanthereforebeecientlysolvedusingacommercialsolver,forexample).Moreover,ateachnode,weobtainbothupperandlowerboundsontheithsubproblemsolution.Wecanthereforeusethisbranchandboundprocedureinsearchofan-optimalsolutionforeachoftheNsubproblems(wewouldproposeusingabreadth-rststrategy,iteratingbetweenthedierentsubproblems,inordertofathomasmanyofthedierentsubproblemsasquicklyaspossible).Aswiththepolyblockalgorithm,givenavalueof,thismethodwillndan-optimalsolutioninanitenumberofsteps,andisguaranteedtoconvergetoaglobaloptimalsolutioninthelimit,althoughnotnitely.However,witheitherapproach,wecangenerateamultitudeoffeasiblesolutionsinreasonabletime,withboundsonthedeviationofeachsolutionfromoptimality. Thefollowingsectiondiscussesaspecialcaseinwhichtheresponsefunctionsobeycertainstrictrelationships.Theseassumedrelationshipsleadtoapolynomial-timesolution,andalsoallowustoexplorethegeneralizedKKTconditionsfor[KPS],whicharenecessaryforlocaloptimalityofasolution(see,e.g., Hiriart-Urruty ( 1978 )). 28

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Underidenticalrevenuecurves,Theorem 2 impliesthatallinstrumentswhoseinvestmentlevelispositiveandfallsintheconcavepartofthecurvewillhaveidenticalvaluesofaiatoptimality.Moreover,Theorem 3 allowsustoarbitrarilyselectanyinstrumentasonewhoseaivaluemaybepositiveandfallintheinterval(0;).WeemploythenecessaryKKTconditions(seeAppendixA)toanalyzethisproblem.WerstsupposethattheKKTmultiplierassociatedwiththeknapsackconstraint,denotedbyw,iszero.Inthiscasewehavethati=d(ai)=daiandiai=0foralli2I,whereiisaKKTmultiplierassociatedwiththeithnonnegativityconstraint.Thusifaiispositive,wehavethatd(ai)=dai=0ataKKTpointwhenw=0.Becauseweassume(withoutlossofgenerality)thatthereturnfunctionsarenondecreasing,anyzeroderivativepointintheconvexportionofthecurvemusthaveareturnfunctionequalto(0),andwecanthusignorestationarypointsintheconvexportionofthecurve.Notingthatd()=da=0,andlettingn=bA=c,wehavethatanysolutionsuchthatnoftheaivaluesaresettoservesasacandidateforanoptimalsolution(becauseeachofthesehasobjectivefunctionvaluen(),weneedonlyconsideronesuchsolution). Wenextconsiderthecaseinwhichw>0,whichimpliesthattheknapsackconstraintmustbetightatanyassociatedKKTpoint.SuchaKKTpointmustsatisfythefollowingsystemofequations:w=d(ai)

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Thedicultyofndingasolutiontothissystemofequationsdependsonthefunctionalformofthederivativefunction.Incaseswheretheequationdal=dald(au)=dau=0intersectsthelineal+nau=Aonlyonceintheinterval0al,forthegivenvalueofnwecanperformalinesearchtodeterminetheuniquesolutionsatisfyingtheabovesystemofequations.Whenthe(a)functiontakesasecond-degreepolynomialformonboththeconvexandconcaveintervals,thenthisprovidesasucientconditionforhavingatmostonesolutiontotheabovesystem(notethatiftheequationdal=dald(au)=dau=0islinear,thenitcannotbecollinearwiththeequational+nau=A

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Assumingthepreviouslystatedconditionsforauniquesolutiontotheabovesystemofequations,wethenperformalinesearchforeachpossiblevalueofnalongtheline0al=Anautodetermine(atmost)Nadditionalcandidatesolutions.Notethatif(a)isnotstrictlypositiveforalla2(0;),thenletting~adenotethelargestvalueofasuchthat(a)=0,wecanlimitoursearchtotheinterval(~a;).ThecomplexityofthislinesearchisO(log).TheoverallcomplexityofthisapproachisthereforeO(Nlog).Thefollowingalgorithmsummarizesourapproachforsolving[KPS]withidenticalresponsefunctions,assumingthesystemofequations( 2{3 )hasatmostonesolutionforanyvalueofn. InitializeLB=n();wheren=jA k n Solvesystemofequations( 2{3 ) 2{3 )withal=aln,au=aun,and(aln)+n(aun)>LBthen OptimalSolutionValuez=LB 31

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Formoregeneralversionsoftheproblem,wherethei(ai)functionshavenorelationship,weshowedintheprevioussectionthattheproblemtakestheformofadicultglobaloptimizationproblem.AlthoughtheKKTconditionsarenecessaryforoptimality,itisimpracticalinthegeneralcasetotrytoenumerateallKKTpointsinordertoaccountforalllocalminima,inanattempttondaglobalminimum.Wenextillustratethiscomplexityforthedierentiablecase.ForthecaseinwhichtheKKTmultiplierfortheknapsackconstraintiszero(i.e.,w=0),westillrequiredi(ai)=dai=0forallai>0.Becausedi(i)=dai=0,werequirendingasubsetIoff1;2;:::;NgsuchthatPi2IiAwiththemaximumvalueofPi2Ii(i).Thisproblemisitselfa01knapsackproblemand,therefore,identifyingcandidatesolutionsusingtheKKTconditionsdoesnotleadtoapolynomial-timesolutionapproach.Additionally,thecaseinwhichw>0requiresndingallsolutionsofasetofN(possiblynonlinear)equalityconstraints(whilealsosatisfying2Nnonnegativityconditions),foreachoftheNchoicesofthevariablewhichmaytakeavalueintheconvexportionofthecurve.Wenoteherethatnonlinearprogrammingmethodsusedinanumberofcommercialsolvers(suchasconjugategradientmethods;see,e.g., Bazaraaetal. ( 2006 ))canbeutilizedinanattempttoidentifyalocallyoptimalpoint,althoughthesemethodscannotguaranteendingagloballyoptimalsolutionforglobaloptimizationproblems. Wenextfocusonthepracticalcasewhereallaivariablesmusttakeintegervalues,wherewecanemployourpreviouspropertiesofoptimalsolutionsandprovidealgorithmsofpracticalusethatleadtosolutionswithprovableboundsonperformance.

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Let[KPSPL]denotetherestrictedversionof[KPS]inwhichallofthei(ai)functionsarepiecewise-linearfunctionswithintegerbreakpoints. 33

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3 and 4 togetherimplythatwecansolvetheintegervariableversionoftheproblemusingthecontinuouspiecewise-linearfunctionobtainedbyconnectingsuccessivevaluesofi(ai)atintegervaluesofaiwithalinesegmentforalli,andanoptimalsolutionwillexistwithatmostonevalueofaistrictlybetween0andi.Thispermitstheconstructionofapseudo-polynomialtimealgorithmforsolving[KPSI]aswenextdiscuss. (2{4)Subjectto:Xi2Infjgai whereZ+isthesetofnonnegativeintegers. BalakrishnanandGeunes ( 2003 )providedapseudo-polynomialtimealgorithmfortheaboveproblemwheneachi(ai)hasaxedpluslinearstructure(i.e.,axedrewardforincludingitemi,plusavariablecontributiontoprotperunitweight).Theyreferredtothisproblemasaknapsackproblemwith 34

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2{5 )servesasasimpleknapsackconstraint.Constraintset( 2{6 )forcesanitem'sweighttozeroiftheitemisnotincludedintheknapsack(whenxi=0)andrequirestheitem'sweighttofallbetweensomeprespeciedupperandlowerboundsiftheitemisincluded.Theobjectivefunction( 2{4 )maximizesthenetreturnfromllingtheknapsack.Thedynamicprogramusedtosolve[KPEI]in BalakrishnanandGeunes ( 2003 )isastraightforwardgeneralizationofthestandarddynamicprogramusedforsolvingknapsackproblems,whereallintegerfeasiblevaluesofeachaiareimplicitlyenumerated.Theworst-caserunningtimeforthisdynamicprogrammingapproachforagiveninstrumentjassumedtohaveaninvestmentlevelbetween0andjandagivenvalueofajisO(NAT),whereT=maxi2Ifiig. UsingTheorems 3 and 4 ,wecansolve[KPSI]byusingthisdynamicprogrammingapproachtosolve[KPEI]foreachpossiblevalueofaisuchthat0
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Thedynamicprogramrstsolvesthesingle-itemproblem:F1()=minfa1j1(a1);0a11g;=0;:::;: Wecansetasuitablevalueofasfollows.Lettingmax=maxi2Ifi(i)g,then=Nmaxprovidesanupperboundonz,theoptimalsolutionvalue.Notealsothat 36

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1 ,wehavealowerboundonthesolutionvaluefromtheroundingprocedureofzNK.BecauseNK=N 2N2;if 2N2,whichimplies Figure2-1. AnS-curveresponsefunction Figure2-2. Iterativeboundingfunctionsforthemodel 38

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Toformalizethismodel,letxjequal1ifitemjisassignedtotheresource,andletV(x)denotetherandomvariablefortheaggregatesizeofitemsassignedtotheresource,wherexdenotestheN-vectorofxjvalues.NotethatV(x)isPoissondistributedwithparameter=PNj=1jxj(wewillnditconvenienttousethecontinuousvariableintheproblemformulation,althoughthisvariablecan,ofcourse,besubstitutedoutoftheformulation). WecanformulatethestaticstochasticknapsackproblemwithPoissondistributeditemsizesas(SKPP).Observethatifrjisanexpectedunitrevenue,thenourformulation 39

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Problem[SKPP]hasmanyapplicationsinoperationsplanningandassignmentproblems.Forexample,forajob-to-machineassignmentproblem,theknapsackcapacitymightcorrespondtotheregularworkingtimeofthemachine,theweightoftheitemtotheprocessingtimeofthejobonthemachine,andthepenaltytotheovertimecostassociatedwithusingthemachine.Anotherexamplewouldbeacustomer-package-pickup-to-vehicleassignmentproblem.Ifthesizeofeachcustomer'spickuprequirementsisa 40

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Inthersttwomotivatingexamples,capacityviolationsarepermittedatacost.Inthesemotivatingexamples,observethatinpractice,theoverowcapacityitselfmayhaveahardlimit(intherstexample,overtimeavailabilityislimitedbythetotalhoursinthedaylessregulartime,andinthesecondexample,anextradeliveryvehicleitselfhasacapacity).BecausethePoissondistributionisdenedfromzerotoinnity(asaremanycommonlyemployedprobabilitydistributions),thisimpliesthatourmodelcannotensureobeyingsuchahardcapacitylimitwithprobabilityone.Bysettingtheoverowcostappropriately,however,themodelcanensurethattheprobabilityofexceedingthiscapacityisnegligible(whenoverowcapacityislimited,ourmodelis,therefore,moreappropriateforsituationsinwhichthispenaltycostisrelativelyhigh,leadingtoalowprobabilityofexhaustingoverowcapacity).Notethatinthethirdsales/inventoryexample,nosuchhardlimitexistsoneectiveoverowcapacity,becausedemandoverowscorrespondtolostsales. Theremainderofthischapterisorganizedasfollows.InSection 3.2 ,wereviewrelatedpastliteratureonthestochasticknapsackproblems.WedenetheproblemandformulateitsmathematicalmodelinSection 3.3 ,anddiscussaspecialcase.WeproposeasolutionapproachtotheprobleminSection 3.4 andprovidethecomputationalstudyinSection 3.5 41

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Deanetal. ( 2004 )consideraversionofthestochasticknapsackprobleminwhichitemsizesareindependentrandomvariables,whilethevaluesofitemsarexed.Anitem'ssizeisrevealedimmediatelyupondeterminingwhetherornottoallocateittotheknapsack.Thegoalistodesignanalgorithmthatselectstheitems,oneatatime,untiltheknapsackcapacityisexceeded. GoelandIndyk ( 1999 )studyknapsackproblemswithPoissonitemsizes,wheretheobjectiveisthemaximizationofthesumofvaluesofitemsincludedintheknapsack,subjecttoaconstraintonthemaximumprobabilityofoverow.Theyprovideapolynomialtimeapproximationschemeforthisproblemviaasimplereductiontothedeterministiccase. Severalpastpapershaveconsideredthecaseinwhichitemsizesarexedbutitemvaluesarerandom(see,e.g., Henig ( 1990 ), Carrawayetal. ( 1993 ),and SteinbergandParks ( 1979 ));incontrast,weconsiderprobabilisticitemsizes. Anadditionalproblemclassworthmentioningistheclassofdynamicstochasticknapsackproblems,whereitemsarrivedynamicallyovertime.Thevaluesandthesizesoftheitemsarerandomandbecomeknownatthetimeofthearrival.Thegoalistondacontrolpolicyforacceptingorrejectingarrivingobjects(asafunctionofthecurrentstateofthesystem)inordertomaximizethetotalvalueofitemsacceptedintheknapsack.( KleywegtandPapastavrou ( 1999 ), KleywegtandPapastavrou ( 2001 )), Papastavrouetal. ( 1996 ),and RossandTsang ( 1989 )provideexamplesofproblemsfallinginthisclass.Inthischapter,however,westudyastaticknapsackproblemwithoutatimedimensionandwhererandomitemrealizationsarenotrevealeduntilafterallassignmentsaremade. 42

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BarnhartandCohn ( 1998 )identiedspecialcasesofthegeneralproblemthatleadtosimplesolutionmethods,andprovideddominancerulesforuseinanimplicitenumerationapproach. Kleywegtetal. ( 2001 )providedasample-averageapproximationschemeforsolvingthisproblemapproximately.Morerecently, Merzifonluogluetal. ( 2009 )providedapolynomial-timesolutionalgorithmforthecontinuousrelaxationofthisproblemundernormalitemsizes,andusedthissolutiontoprovidestrongupperboundsinabranch-and-boundscheme.OurconsiderationofPoissondistributeditemsizesbroadensthesetoftoolsavailableforthisproblemclass.Moreover,thePoissondistributionhastheaddedbenetofbeingdenedonlyfornonnegativevalues,incontrasttothenormaldistribution,whichisoftenusedasanapproximationforrandomvariablesthatcannottakenegativevalues(notethataPoissondistributionwithparameterisequivalenttothesumofindependentPoissonrandomvariables,eachwithparameter1;bythecentrallimittheorem,thePoissondistributiontendstoanormaldistributionasincreases).Tothebestofourknowledge,nopriormethodexistsintheliteratureforaddressingthisrelevantproblemclassunderanassumptionofPoissondistributeditemweights.

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(3{1)Subjectto:=NXj=1jxj Lettingg()=BPi=0(Bi)ei 44

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3{2 )and( 3{3 ),respectively,andletjandjdenoteKKTmultipliersfortheconstraintsxj1andxj0,respectively,forj=1;:::;N.TheassociatedKKTconditionscanbewrittenas: ^rjjj+j=jPB1i=0ei (3{9) 0xj1j=1;:::;N Beforeprovidingthesolutionalgorithmforthegeneralcase,werstanalyzeaspecialcase. 45

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=1. Nextconsiderthecaseinwhichanoptimalsolutionexiststhatselectsmorethanoneitem.Thisimplies> 3{5 )as^rj0j+0j=XB1i=0ei 3{5 )and( 3{6 ))that^rjPB1i=0e00i 46

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3{7 )andcontradictstheoptimalityofthissolution.Conversely,supposewehaveanoptimalsolutionwith=0suchthatxj<1andxk>0.Thenwemusthave^rjPB1i=0e00i 3{8 ),whichcontradictstheoptimalityoftheproposedsolution.2 2 impliesarankedorderingoftheattractivenessofitemsbasedonthe^rj(andthereforerj)values.Ifweassumethattherearenotanytiesinrjvalues,thenthis,togetherwiththefactthatanoptimalsolutionexistswithatmostonefractionalitem,suggestsasolutionmethodfortherelaxation(althoughthisneednotnecessarilybetheuniqueoptimalsolution,ourmethodologythatfollowsensuresndingthisoptimalsolution).Thatis,supposethatitemktakesafractionalvalueinanoptimalsolution.Thenthisimpliesthat^rk=PB1i=0ei 2 wemustalsohavexj=1forj=1;:::;k1andxj=0forj=k+1;:::;N.Let1(k)=Pk1j=1j.Tondthefractionalvalueofxkwemustthensolvethefollowingone-dimensionaloptimizationproblem:[1D]Maximize^rkkxkBXi=0(Bi)e(1(k)+kxk)(1(k)+kxk)i 47

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3{12 )willalwaystakeavaluebetween0and1.SupposethattheparameterofthePoissonrandomvariablethatsatises( 3{12 )equals^k.Thiscorrespondstoasolutioninthevariablexksuchthat ~xk=^k1(k) Tosummarizeoursolutionalgorithm,assumingitemsareindexedinnonincreasingorderof^rjvalues,werstsetxj=1forallj:^rj>0andsetxj=0forallj:^rj<.Denotejminandjmax,respectively,asthelowestandhighestitemindicessuchthat^rj0.Then,inincreasingorderfromk=jmintok=jmax,wesolve[1D]forxkandsetxj=1forallindiceslowerthankandxj=0forallindiceshigherthank.Notethatbecausethecontinuousrelaxationisaconvexprogramwithalinearconstraintset(andthereforetheKKTconditionsarenecessaryandsucientforoptimality),wecanimmediatelyterminatethealgorithmuponidentifyingaKKTpoint.Recallthatwehaveassumedthusfarthatnotiesexistin^rjvalues.Forthecontinuousrelaxationof[SKPP],thisassumptionismadewithoutlossofgenerality.Thatis,ifanytwoitems,sayj1andj2,haveatieintheir^rjvalues,thenwemaybreaktiesarbitrarilyorcombinethetwoitemsintoasingleitemwithparametervaluej1+j2whensolvingthecontinuousrelaxation(althoughwecannotcombinethevariablesintoonewhensolvingthebinaryversionoftheproblem). 48

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2 isnotstrictlyvalidforallitems,althoughitstillholdsamongvariableswhosevaluesarenotxed.Atanynodethatndsafeasibleintegersolutiontotheproblemwefathomthenodeandupdatethelowerboundifitexceedsthebestlowerbound.WeterminatethealgorithmwhenwendaKKTpointthatsatisesthebinaryrestrictionsorwhenallopennodeshavebeenfathomed. 49

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3-1 summarizesthevaluesoftheparametersthatweusedingeneratingrandomprobleminstances.Wegenerated24datasetsinordertoanalyzetheeectofthenumberofitems,N,theknapsackcapacity,B,andtheunitpenaltyvalue,.Foreachdataset,wesampledthevalueoftheexpectedweight,j,fromauniformdistributionontheinterval[0;],for=f0:5;1;5;10g,resultinginatotalof96datasets.Werandomlygenerated10instancesbysamplingthevalueoftheitemperunitsize,rj,fromauniformdistributionontheinterval[5;25]foreachof96datasets(foratotalof960probleminstances).Theseprobleminstancesandparametervalueswerecreatedtoavoidtrivialsolutions.Thatis,theseparametervaluesweusedensuredthegenerationofprobleminstancessuchthatanumberofitemsexistedsuchthat^rj0,and,therefore,suchthat(a)wecouldnotdeterminetheoptimalsolutionofthecontinuousrelaxationbyinspection,and(b)theoptimalsolutiontothecontinuousrelaxationwasnotbinary.WeprovideresultsfrombothourcustomizedalgorithmimplementationandaGAMS/BARONimplementationinTable 3-2 .NotethatforeachdatasetgiveninTable 3-1 ,wegenerated40testinstances.WelimitedtherunningtimeofGAMS/BARONto900CPUseconds.WeobservedthatineachcaseGAMS/BARONterminatedwithoneofthefollowingthreeconditions( Rosenthal ( 2007 )): (1)Terminationwithanoptimalsolution:Thesolutionfoundwasprovablyoptimal. (2)Terminationwitharesourceinterruption:GAMS/BARONcouldnotndaprovablyoptimalsolutionwithinthegiventimelimit.Theoutputprovidedlowerandupperboundsforthemodelattheendofthetimelimit. (3)Terminationwithanintegersolution:Themodelterminatedwithanintegersolutionbeforethetimelimitwasreached,presumablyduetonumericaldicultieswiththePoissondistributionfunction.Whenthemodelterminatedprematurelywithanintegersolution,theabsolutegapbetweenthelowerandupperboundreportedbyGAMS/BARONwasinnity.Inotherwords,GAMS/BARONterminatedwithanintegersolutionwhenitcouldnotndanupperbound,andthesolutionwas,therefore,not 50

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3-2 ).TheterminatingintegersolutionreportedwasthelowerboundfoundbyGAMS/BARON.AlthoughthesesolutionswerenotprovablyoptimalbyGAMS/BARON,wewereabletocomparetheirsolutionvaluestotheoptimalsolutionfoundviaouralgorithm.Wefoundthatthesesolutionsweretypicallyoptimalorveryclosetooptimal,withamaximumdeviationfromoptimalityof0.9%amongsuchprobleminstances. ForeachdatasetfromTable 3-1 ,Table 3-2 showsthepercentageofinstances(acrossallvaluesof)thatweresolvedtooptimalitybyGAMS/BARONwithin900CPUsecondsandthepercentageofinstancesthatterminatedwithintegersolutions(notprovablyoptimal).Inaddition,thetableshowstheaverageCPUtimesoftheinstancesthatweresolvedtooptimalityandterminatedwith(notprovablyoptimal)integersolutions(alltimesareinseconds).Wecomputedtheoptimalitygapofaninstanceasthedierencebetweenupperandlowerboundsasapercentageofthebestlowerboundattheendof900CPUsecondsfortheinstancesthatcouldnotbesolved.Weprovidethisaveragegapinthecolumnlabeled\GapforUnsolvedInstances."Wecomputedthegapofaninstancethatterminatedwithanintegersolution(notprovablyoptimal)asthepercentagedierencebetweentheobjectivefunctionvalueoftheintegersolutionfoundbyGAMS/BARONandtheoptimalsolutionwefoundusingouralgorithm(sinceGAMS/BARONwasunabletoprovideaniteupperbound).Weprovidethisgapinthecolumnlabeled\GapforIntegerSolutions(NPO)."Theaveragetimeforourcustomizedbranch-and-boundalgorithmisprovidedinthecolumnlabeled\Averagetime"underthe\OurAlgorithm"column.Thetableshowsthatallinstancesweresolvedtooptimalitywithinafractionofsecondwhenusingouralgorithm. AllinstancesintherstdatasetweresolvedbyGAMS/BARONtooptimalityinabout85CPUseconds(onaverage)whileouralgorithmfoundanoptimalsolutionwithinafractionofsecond.Infourofthe24datasets,noneoftheinstancesweresolvedtooptimalitywithin900secondsusingGAMS/BARON.Theaverageoptimalitygapsofthese 51

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ForprobleminstanceswhereGAMS/BARONterminatedwithcondition(3),thenumberofinstancesthatterminatedwithanintegersolutionnotonlydependedonthedatasetfromTable 3-2 ,butalsoonthevalueof.Figure 3-1 showsthepercentageofinstancesthatterminatedwithcondition(3)forthefourdierentvalues.Asillustratedinthegure,fortheinstanceswhere=0:5,noneoftheinstancesterminatedwithanintegersolution,andGAMS/BARONeitherterminatedwithanoptimalsolutionorranfor900secondsandcouldnotndaprovablyoptimalsolution.However,when=1,atotalof42%oftheinstancesterminatedwithintegersolutions.Thispercentagedecreasedasthevalueofincreasedto5and10,asshowninthegure.Asthevalueofincreased,weobservedthatthepercentageofinstancesGAMSsolvedoptimallyincreasedandtheoptimalitygapfortheinstancesthatwerenotsolvedalsodecreased(aswenotedpreviously,asthevalueoftheparameterincreases,thePoissondistributiontendstoanormaldistribution,and,insuchcases,itmaybemoreeectivetomodeltheproblemunderanormaldistributionassumption;see Merzifonluogluetal. ( 2009 )).Ontheotherhand,allinstancesweresolvedoptimallywithinafractionofsecondusingouralgorithm.AsillustratedinTable 3-2 ,ouralgorithmoutperformedGAMS/BARONacrossthe960randomlygeneratedprobleminstances.Moreover,ouralgorithmalsoconsistentlyensuresndingaprovablyoptimalsolution,whereasGAMS/BARONranthefull900secondsorwasunabletodetermineaniteupperboundinasubstantialpercentageofinstances. 52

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Gams/BaronPerformanceAnalysis Table3-1. Parametervaluesfortestinstances 53

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Computationaltestresults 1100.0%0.0%84.450-0.065272.5%27.5%112.424-0.038335.0%35.0%0.64814.20%0.064410.0%37.5%0.66225.44%0.08650.0%55.0%-42.43%0.11265.0%57.5%0.46773.52%0.115772.5%0.0%11.3236.64%0.115872.5%10.0%3.2421.09%0.118950.0%20.0%0.45612.26%0.1471037.5%15.0%0.67022.46%0.148110.0%50.0%-41.26%0.2581215.0%47.5%0.71237.91%0.2241377.5%0.0%8.4164.66%0.1521457.5%22.5%0.5130.48%0.1321562.5%5.0%8.8071.67%0.1711650.0%7.5%0.57412.98%0.210170.0%50.0%-29.81%0.3321842.5%32.5%0.61719.58%0.2701987.5%0.0%19.8543.79%0.1462067.5%7.5%0.7440.50%0.1342167.5%20.0%0.6160.46%0.1892250.0%25.0%0.64511.63%0.218230.0%50.0%-23.02%0.3502460.0%30.0%1.09610.70%0.312 hellobusalakseynedenenbasacikmiyor..... 54

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BildeandKrarup ( 1977 ),and Erlenkotter ( 1978 )).Becausethisproblemhasaconcavecostobjectivefunction(suchthatanextremepointoptimalsolutionexists),weagainndthatanoptimalsolutionfortheUFLPexistssuchthatagivencustomer'sdemandisentirelyassignedtoasinglesupplyfacility.Morerecentworkconsiderspracticalgeneralizationsofthisclassofproblemsthataccountnotonlyforxedoperatingandvariableassignmentcosts,butalsoforinventory-relatedcostsatfacilities.Inparticular,whenweconsidercontextswithuncertaindemands,itisimportanttoconsidertheimpactsofsafetystockcosts. ChopraandMeindl ( 2007 )provideillustrationsofgeneraltrendsinsupplychaincostsasafunctionofthenumberoffacilities.Forexample,itisclearthatanincreaseinthenumberoffacilitiesinasupplychainnetworkresultsinacorrespondingincreaseinfacilitycosts.Reducingthenumberoffacilities,however,tendstoincreaseoutboundtransportationcosts,whichmustbebalancedagainstfacilityandinventorycosts.Similarly, ChopraandMeindl ( 2007 )notethatanincreaseinthenumberoffacilitiestendstoincreasetotalsupplychaininventorycostsduetotheneedtoincreasetotalsystem-widesafetystockcostsinordertomeetcustomerservicelevelexpectations.Conversely,areductioninthenumberoffacilitiesthatholdsafetystockpermitsareductionintotalsafetystockcostasaresultoftherisk-poolingbenetsfromaggregating 55

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Becausesafetystockcostsrepresentanon-trivialcomponentofoverallfacility-relatedcosts,recentliteraturehasrecognizedtheneedtoaccountforsafetystockcostswhenmakingfacilitylocationdecisions(e.g., Shenetal. ( 2003 )).Themajorityofthiswork,however,continuestoenforcesingle-sourcingrestrictions,whichareoptimalfortheUFLPanduncapacitatedtransportationproblemsembeddedintheselargerinventory-locationproblems.Unfortunately,safetystockcostscannotberepresented,ingeneral,asalinearorconcavefunctionoftheassignmentdecisionvariables.Thus,imposingsingle-sourcingrequirementsonsuchinventory-locationproblemsmaybesuboptimalwhencomparedtotheproblemintheabsenceofthisrequirement.Ourprimarygoalinthischapteris,therefore,toimproveourunderstandingofthedegreeoflossthatmayresultfromenforcingasingle-sourcingrequirement. Clearlytherearesomebenetstoenforcingsingle-sourcerequirements,althoughthesebenetsaretypicallydiculttoquantify.Fromapracticalstandpoint,customersoftenpreferhavingasinglepointofcontactfordeliveryandproblemresolution.Similarly,suppliersfacelowercoordinationcomplexityunderasingle-sourcingarrangement.Algorithmically,heuristicsolutionapproachesareofteneasiertoconstructbecauseofthecombinatorialnatureofsolutionstoproblemsthatusesingle-sourcingrequirements.Incontrast,intheabsenceofsinglesourcing,acustomerhasabuilt-inbackupplanwhentheirdemandissplitamongmultiplesources,andoneofthesourcesisunabletodeliver.Withourgoalofunderstandingthecostsofsingle-sourcinginmind,weaddressthefollowingproblem: 56

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Notethatwhenthelimitonthenumberoffacilitiesthatcansupplyanygivencustomerequalsone,wehavethesingle-sourcingconstraint.WhenthislimitequalsN(whereNisthenumberoffacilities),weeectivelyhavenolimitonthenumberofsuppliersthatcanserveacustomer.Thisproblemfallsintheclassofmixed-integernonlinearprogrammingproblemsandisNP-hard(byvirtueofgeneralizingtheUFLP). Shenetal. ( 2003 )considerasimilarjointlocation-inventoryproblemwithasingle-sourcingrequirementthatminimizesthecostoffacilitylocation,transportation,andholdingworkingprocessinventoryandsafetystock.Theirmodelissimilartoours,exceptthatwedonotrequiresinglesourcingandourmodelincludesacardinalityconstraintonthenumberofsourcesthatcansupplyacustomer.Interestingly,whensingle-sourcingisrequiredandcustomerdemandsarenormallydistributed,theexpressiontypicallyusedforsafetystockcostisconcaveintheassignmentdecisionvariables(whenweconsiderthecontinuousrelaxationoftheseassignmentvariables).Whensinglesourcingisnotrequired,however,thisexpressionisinsteadconvex,destroyingtheconcavityoftheobjectivefunction.Thus,theproblemstudiedbyShenetal.(2003)containsstructuralpropertiesthatarelostwhenthesingle-sourcingrequirementisdropped.FrancaandLuna(1982)alsostudyasimilarproblemwheredemandsplittingisallowed(i.e.,whenacustomer'sdemandmaybesplitamongmultiplesupplyfacilities).Insteadofconsideringinventory-relatedcostsatthesupplierechelon,however,theyconsiderinventoryholdingandshortagecostsatthecustomerstage,andprovideageneralizedBendersdecompositionalgorithmtosolvetheproblem. Inthischapter,werstdeneandformulateageneralmodelforassigningcustomerstosupplyfacilitieswhensuppliersafetystockcostsareconsidered,demandsplittingis 57

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Shenetal. ( 2003 ).Weanalyzethespecialcasewithzeroxedfacilitycosts,whichresultsinaninterestingandpracticallyrelevanttransportationproblemwithsafetystockcosts.Wedemonstrateimportantpropertiesofoptimalsolutionsforspecialcasesofthisclassoftransportationproblemsthat,insomecases,leadtoclosed-formsolutions.Moreover,theseoptimalsolutionpropertiesprovideinsightoneectivewaystomanageriskduetouncertaindemandinsupplychains.WeprovideageneralizedBendersDecompositionalgorithmtosolvethegeneralproblemwithxedsupply-facilityoperatingcosts.Wethendiscusstheresultsofanempiricalstudyintendedtocharacterizethecostofsingle-sourcingrequirements. Therestofthischapterisorganizedasfollows.Section 4.2 nextreviewsrelatedliteratureonlocation-inventoryproblems.WedenethegeneralproblemandmodelformulationinSection 4.3 ,anddiscusssolutionmethodsforspecialcasesinwhichnoxedcostcomponentexists.ThenwepresentthegeneralizedBendersdecompositionalgorithminSection 4.4 .Section 4.5 discussestheresultsofourcomputationalstudy. DaskinandOwen ( 199 ), Meloetal. ( 2007 ), OwenandDaskin ( 1998 ), Daskinetal. ( 2005 ),and Snyder ( 2006 )foracomprehensivereviewoffacilitylocationproblems.Ontheotherhand,inventorytheoryliteratureassumesthatlocationdecisionshavebeenmadebeforehand,and,basedonthisassumption,itevaluatestheinventoryrelateddecisions.Theaimistondthebest 58

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Recently,jointlocation-inventorymodelshavegainedattention(see Shenetal. ( 2003 ), Vidyarthietal. ( 2007 ), Ozsenetal. ( 2008a ), Ozsenetal. ( 2008b ), ShenandDaskin ( 2005 ), Shen ( 2005 ), NozickandTurnquist ( 1998 ), NozickandTurnquist ( 2001b ), NozickandTurnquist ( 2001a )).Theproblemanalyzedby Shenetal. ( 2003 )isthemostcloselyrelatedtoourwork.Inparticular, Shenetal. ( 2003 )considerajointlocation-inventoryproblem,wheremultipleretailerseachwithstochasticdemandareassignedtodistributioncenters(DCs).Becauseofuncertaindemand,someamountofsafetystockmustbecarriedatdistributioncenters.Intheirmodel,theyenforceasingle-sourcingrequirement,i.e.,eachcustomer'sdemandmustbeassignedtoasingleDC. Shuetal. ( 2005 )studyasimilarproblemwithonesupplierandmultipleretailers.Eachretailercanserveasadistributioncentertoachieveriskpoolingbenets. Thesolutionmethodsappliedtotheselocation-inventorymodelstypicallydependontheformoftheobjectivefunction.Theformoftheobjectivefunction,inturn,dependsonthedecisionvariablerestrictions.Forinstance,ifwehavebinaryassignmentvariablesandanobjectivefunctionthatusesthesquaredvaluesofthesebinaryvariables,thenthesesquaredtermscanbelinearizedbysimplyreplacingthemwiththeiroriginalbinaryvalues(sincex=x2forbinaryvariables).Thisaectstheconvexityofthesafetystockcostcomponentoftheobjectivefunctionand,therefore,thesolutiontechniquesthatcanbesuccessfullyapplied.Wemodelourproblemasamixed-integernonlinearprogrammingproblemwithcontinuousassignmentvariables.We,therefore,needtoconsidersolutiontechniquesrelevanttomixed-integernonlinearprogrammingproblemsingeneral,andlocation-inventoryproblemsinparticular. Themajorityofpastresearchonlocation-inventorytheoryemphasizesthebenetsofriskpoolingthroughcentralizationofinventory,andthusrequiressuchsingle-sourcingconstraints.Recently, Ozsenetal. ( 2008b )studyalogisticssystemwithasingleplant,a 59

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Ozsenetal. ( 2008b )arethattheirmodelhasonemorestagethanourmodel,atwhichDCsorderproductsfromasingleplantandtheyproposeaLagrangianrelaxationalgorithm,whilewedonotaccountfortheordercostsfromaplantandproposeanexactalgorithmthatusesgeneralizedBendersdecomposition. Lagrangianrelaxationbasedalgorithmshavebeenwidelyusedinthelocation-inventoryliteratureforproblemsthatrequiresinglesourcing. Daskinetal. ( 2002 )consideraproblemsimilartotheoneaddressedin Shenetal. ( 2003 ),wheretheyaccountforbothworkinginventoryandsafetystockcostterms.Theymodelthisproblemasanonlinearintegerprogrammingproblemwithbinaryassignmentvariables,andproposeaLagrangianrelaxationsolutionalgorithm.Similarly, Sourirajanetal. ( 2007 )applyLagrangianrelaxationtoaprobleminwhichaproductionfacilityreplenishesasingleproductatmultipleretailers.TheirmodeldeterminestheDClocationsthatminimizetotallocationandinventorycosts. Snyderetal. ( 2007 ), Ozsenetal. ( 2008a )and MirandaandGarrido ( 2006 )alsoproposesolutionmethodsbasedonLagrangianrelaxationformixed-integernonlinearmodels.However,eachofthesepapersassumesthatsinglesourcingisrequired.Moreover,Lagrangianrelaxationbasedsolutionmethodsdonotprovidestrictlybettersolutionsthanthecontinuousrelaxationforseveralimportantspecialcasesoftheproblem 60

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Georion ( 1974 )). Severalheuristicsolutionmethodshavealsobeenproposedintheliteratureforlocation-inventoryproblems. ErlebacherandMeller ( 2000 )consideraproblemwhereproductsaredistributedfromplantstoDCsandfromDCstoretailers.TheiraimistominimizethesumofthexedoperatingcostsofopenDCs,inventoryholdingcostsatDCs,totaltransportationcostsfromplantstoDCs,andtransportationcostsfromDCstocustomers.DCsandcustomersarelocatedonagrid,andeachcustomermustbeassignedtoasingleDC;thusdemandsplittingisnotallowed.Theyproposealocation-allocationheuristicthatusesthebettersolutionobtainedusingtwodierentapproaches.TherstapproachassignseachcustomertoitsclosestDCandthenreducesthenumberofDCsbygreedilyreassigningcustomerstootherDCs,untilreachingapredeterminednumberofopenDCs.ThesecondapproachstartsbyassigningonecustomertoeachopenDC(wherethenumberofopenDCsequalsapredeterminednumber),andthenaddstheremaining(unassigned)customerstoDCsuntilallcustomersareassigned. ThesolutionmethodweproposeusesgeneralizedBendersdecomposition(see Georion ( 1972 )),whichhasbeenusedeectivelyforcertainclassesofmixed-integernonlinearprogrammingproblems.Forexample, Hoc ( 1982 )consideredatransportationandcomputercommunicationnetworkdesignproblemwithabudgetconstraint. Hoc ( 1982 )formulatedthisproblemasamixed-integernonlinearprogrammingmodelandproposedanapproachusinggeneralizedBendersdecomposition. FrancaandLuna ( 1982 )alsoproposedasimilaralgorithmforalocation-inventoryproblemthatiscloselyrelatedtoourwork.Intheirmodel,theyallowbackorderingwithanassociatedpenaltyfunction.Theirmodelconsidersinventoryholdingcostattheretaillevel.Incontrast,ourmodelconsidersinventorycostsatthesupplierlevel.Thenextsectionformallydenesourproblem,providesthemathematicalmodelandanalyzestwospecialcases. 61

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62

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Ifweassignthefractionxijofcustomeri'sdemandtosupplyfacilityj,thentheexpectedassignmentcostequalscijxij,wherecij=^ciji.Weassumethatallcustomerdemandsareindependentandnormallydistributed.Notethatthedemandseenbysupplyfacilityjinatimeperiodhasmean(j)=Pi2Iixijandvariance2(j)=Pi2I2ix2ij,i.e.,DjN((j);2(j)).Weassumethatsupplyfacilityjfollowsaperiodicreviewinventorypolicy,andordersuptoastocklevelSjatthebeginningofeveryperiod,suchthatPrfDjSjg=j;letzj=Sj(j) Pi2I2ix2ij. WewishtodecidewhichsupplyfacilitiestoopenandhowtoallocatethedemandofeachcustomeritoatmostNioftheseopensupplyfacilitiesinordertominimizethetotalexpectedcost.Weformulatethislocation-inventoryproblem(ILP)asfollows:(ILP)Z=MinimizeXj2JFjyj+Xi2IXj2Jcijxij+Xj2Jhjzjs Xi2I2ix2ij Theobjectivefunction( 4{1 )minimizesthesumofthexedcostoflocatingsupplyfacilities,theassignmentandvariablecostfromsupplyfacilitiestocustomers,andthesafetystockcosts.Constraintset( 4{2 )ensuresthateachcustomer'sdemandisfullyassignedtosupplyfacilities.Notethatthisconstraintwillbesatisedatequalityinanoptimalsolution.Constraintset( 4{3 )limitsthenumberofsupplyfacilitiesthatcanservecustomeritoatmostNi.Constraintset( 4{4 )permitsassigningcustomerdemandonlyto 63

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4{5 )reectstheintegralityrequirements. Letting(x)=Pi2IPj2Jcijxij+Pj2Jhjzjq Pi2I2ix2ij,thefollowinglemmahelpsincharacterizingthestructureoftheobjectivefunctionof(ILP). Pi2I[fij(xij)]2.NowweneedtoshowthatF(x)isconvex.Let~F(x)=[f11(x11);:::;fij(xij)].ThenF(x)isthel2normof~F(x),i.e.,F(x)=~F(x).F(x1+(1)x2)=~F(x1+(1)x2)=~F(x1)+(1)~F(x2)(because~F(x)islinearinxij)~F(x1)+(1)~F(x2)(triangularinequality)=F(x1)+(1)F(x2): Pi2I[fij(xij)]2isalsoconvex.Moreover,sincethersttermof(x)islinearandthesecondtermisthesummationofconvexfunctions,(x)isconvexinx.2 1 impliesthat(ILP)becomesaconvexprogram,forgivenyjandtijvariables.WewillusethisfactlaterwhenconstructingaBendersdecompositionalgorithm.Beforediscussingasolutiontechniqueforthegeneralmodel,wewouldliketoanalyzetwospecialcasesof(ILP).Bothofthesespecialcasesassumethatlocationsarexed,orequivalently,axedvalueofthevectorofyjvariables,whichwedenoteby~y(notethatthisisequivalenttotheassumptionofzeroxedcosts).ThesespecialcasesalsoassumethatNi=Nforalli=1;:::;N,whichpermitsdroppingconstraintset( 4{3 )fromtheformulation.Theresultingproblemisanuncapacitatedtransportationproblemwithsafetystockcostswhich,tothebestofourknowledge,hasnotbeenconsideredinthe 64

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ByLemma 1 weknowthattheobjectivefunctionofthisspecialcaseisaconvexfunctionofx.Sincealloftheconstraintsof(ILP)arelinearinx,theproblemwithzeroxedcostsforfacilitiesisaconvexprogrammingproblemsuchthattheKKTconditionsarenecessaryandsucientforoptimalityforthisspecialcase(notethatanyfeasiblesolutionsuchthatPi2Ixij=0violatesthedierentiabilityassumptionrequiredforapplicationoftheKKTconditionsattheassociatedpoint;however,weareabletoconsidersuchsolutionsseparatelyinouranalysis). Forthisspecialcase,weassumetheassignmentcostiscustomer-specicandequaltociforcustomeri,i.e.,cij=ciforallj2Jandforeachcustomeri.Wewillrefertocasesinwhichtransportationcostsarefacilityinvariantascaseswithsymmetrictransportationcosts.Wealsoassumethatthesupplyfacilityunitholdingcostsandrequiredcycleservice 65

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4{2 )andnonnegativityconstraintsonthexijvariables,wenextanalyzetheKKTconditionsforthisspecialcase,whichcanbewrittenasfollows.ci+hzxij Pi2Ix2ijiij=0;8i2I;j2J; Givenasolutionandanysupplyfacilityj,letI(j)denotethesetofcustomerssuchthatxij>0.Similarly,denoteJ(i)asthesetoffacilitiessuchthatxij>0.Thefollowingtheoremcharacterizesthestructureofoptimalsolutionsforthisspecialcase. 1. 2. 4{8 )wesetij=0foralli2I(j).Fromcondition( 4{6 ),werequire 66

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4{6 )holdsforalli2Iandj2J.Wehavethereforeconstructedasolutionsatisfying( 4{6 ),( 4{8 ),and( 4{10 ).ByassumptionwehavePj2J(i)1 4{7 )and( 4{9 )hold,andallKKTconditionsaresatisedbythesolutionwehaveconstructed.2 5 impliesthatanybalancedsolutionisoptimalunderidenticalsupplycostsandidenticalcustomervariancevalues.Thatis,providedthatallcustomersassignedtoasupplyfacilityhaveanequalfractionoftheirexpecteddemandallocatedtothesupplyfacility,thesolutionisoptimal.Thus,forexample,anoptimalsolutionexistssuchthatallcustomersareassignedtoasinglesupplyfacility,whichisconsistentwiththewellknownuseofinventoryaggregationtoobtainsafetystockriskpoolingbenets.Theorem 5 illustratesthatwecanobtainthesamedegreeofriskpoolingbenetsinanumberofdierentways,withoutrequiringinventoryaggregation.Thatis,givenaproblemwithNfacilitiesandNcustomers,forexample,asolutionsuchthatallNfacilitiesareopen,and1 67

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whereg()=(12) 2wehaveasymmetrictransportationcostinstancewithc11=c12andc22=c21,whichresultsinthespecialcaseinwhichassignmentcostsarefacilityindependent(asinthespecialcasediscussedintheprevioussubsection). Forthetwo-by-twospecialcaseinwhichfacilityholdingcostsandcustomervariancesareequal,andassignmentcostsobey( 4{11 )and( 4{12 ),wehavethefollowingproposition. 4{11 )and( 4{12 ),anoptimalsolutionexistssuchthatx11=x22=andx12=x21=1,withminimumcostc11+c22+2H Observethatwhen=1 2,thesymmetriccostcase,theoptimalcostequalsc12+c22+p 2;1 2;1 2;1 2;(x11;x12;x21;x22)=(0;1;0;1);(x11;x12;x21;x22)=(1;0;1;0).Thiscaseisconsistent 68

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2;1anoptimalsingle-sourcingsolutionexists,thefollowingcorollaryshowsthatthisisnotthecasefortheremainingvaluesofontheinterval[0;1]. Figure 4-1 illustratesthevalueof()for2[0;1].Wecanshowthatthepeakvaluesoccuratthevaluesofsuchthatthetermsintheminimumoperatorgiveninthecorollaryareequal.Thisoccursat=0:2725and=0:7275,where()=12:7%.Ateitherofthesevaluesoftheminimumcostsingle-sourcingsolutionexceedstheminimumpossiblecostby0:127H,whiletheactualpercentagecostincreaseassociatedwithsinglesourcingdependsonthetransportationandholdingcostparametervalues.Thisanalysisillustratesthefactthatsingle-sourcingsolutionsareeitheroptimalorclose-to-optimalwhentransportationcostsaresymmetric(asisthecasewhen=1 2)orseverelyasymmetric(asisthecasewhen=0or1).Intheformercase,multipleoptimalsolutionsexist(usingeitheroneortwofacilities)whileinthelattercase,asingleoptimalsolutionexiststhatusesthedominantfacility(intermsoflowertransportationcosts).Forintermediatecases,however(whentransportationcostsareneithersymmetricnorgrosslyasymmetric),weseethatthecostperformanceofasingle-sourcingstrategycanbeworsethanademandsplittingstrategybyanon-trivialamount.Ourcomputationaltestsonthe 69

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Nextconsideranasymmetrictransportationcostcaseinwhichc12=c21=~c,suchthatc11=c22=~c+Hg().Notethatinthiscase,theaveragevalueofcij,whichwedenotebyE[c],equals~c+Hg() 2.Weareinterestedinhowthemaximumpercentagecostsavingsfromdemandsplitting(relativetosinglesourcing)behavesasafunctionoftheratiooftheaverageassignmentcosttoholdingcost,whichwedenotebyE[c=h].Foraxedvalueofh,wethenhaveE[c=h]=~c h+Hg() 2h.Letusconsideravalueofsuchthattheoptimalsinglesourcingsolutionsetsx12=x21=1andx11=x22=0,whichwecanshowoccursforanvalueintheinterval[0;0:2725].Wethereforeassume=0:25.Notethatforthiscase,theminimumcostsolutiongivesanobjectivefunctionvalueofzopt=c12+c21+2H(1) ~c+Hf(),wheref()=1 ~c+Hf()=z(1f()) 2p 4-2 illustratesthebehaviorof 2).Figure 4-2 illustratesthefollowing.Foraxedvalueofh,asweincreasecijvalues,theassignmentcostsdominate,andtheproblemapproachesthestandarduncapacitatedfacilitylocationproblem(inthiscase,asingle-sourcingsolution 70

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Notehowever,thatwhenwepermitE[c=h]togotozero(aswedoinourcomputationaltests),cijvaluesbecomeincreasinglysymmetric,and 1 ),weknowthattheremainingproblemisaconvexprogram.Letustemporarilyxthelocationvectorat~yandthebinaryassignmentvectorat~t,suchthatconstraints( 4{3 ),( 4{4 )and( 4{5 )admitafeasiblesolutioninthexijvariables.Thentheassociatedrestrictedproblembecomes(ILP(~t,~y))MinimizeXj2JFj~yj+Xi2IXj2Jcijxij+Xj2Jhjzjs Xi2I2ix2ijSubjecttoXj2Jxij1;8i2I;0xij~tij;8i2I;j2J: Notethatthexed-chargecomponent,Pj2JFj~yj,intheobjectivefunctionisaconstantforagivenvector~y.Similarly,theright-hand-sidevalueofeachconstraintinset( 4{13 )iseither0or1,dependingonthevalueof~tij.Wealsonotethat(ILP(~t,~y))isfeasibleifandonlyifPj2J~tij1foralli2I. 71

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Xi2I2ix2ijSubjecttoXj2Jxij1;8i2I;0xijtij;8i2I;j2J: 72

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where(x)=Pi2IPj2Jcijxij+Pj2Jhjzjq Pi2I2ix2ij. Problem(ILP)isthereforeequivalenttothefollowingMasterProblem(MP):(MP)MinimizeXj2JFjyj+Subjecttominx0[(x)+Xi2Ii(1Xj2Jxij)+Xi2IXj2Jij(xijtij)];80;0; 4{15 )forallpossiblevaluesofand.Wethereforegeneratevalidcutssuccessivelythatcorrespondtospecicvaluesofthevectorsandandaddthemtotheformulationinaniterativefashion(suchcutsaregenerallyreferredtoasBenderscuts).Givenaparticularbinaryvectortkwecansolvetheconvexprogrammingproblem(ILSP)andrecovercorrespondingoptimaldualmultipliervectorskandk.Wecanthenwritev(tk)=minx024Xi2IXj2Jcijxij+Xj2Jhjzjs Xi2I2ix2ij+Xi2Iki(1Xj2Jxij)+Xi2IXj2Jkij(xijtkij)35

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Xi2I2ix2ij35: Wethereforehavethatminx0hPi2IPj2J(cij+kijki)xij+Pj2Jhjzjq Pi2I2ix2iji=v(tk)Pi2Iki+Pi2IPj2Jkijtkij.Substitutingthisin( 4{15 )providesthefollowingBenderscutfor(MP)correspondingtothedualmultiplierskandkv(tk)Xi2IXj2Jkij(tijtkij): OurRelaxedMasterProblem(RMP)thenbecomes(RMP)MinimizeXj2JFjyj+Subjecttov(tk)Xi2IXj2Jkij(tijtkij);8k=1;:::;K;Xj2JtijNi;8i2I;Xj2Jtij1;8i2I;tijyj;8i2I;j2J;tij;yj2f0;1g;8i2I;j2J;0; 4{15 )(forallpossibleand),becausekandkmaximizev(tk)overalland.Notethatthe(RMP)formulationisa0-1integerprogram.Ateachiteration,wesolvetheRMPtoobtaina(possibly)newtkvector.Giventhistkvector,wethensolvethesubproblem(ILSP)todeterminethecorrespondingoptimaldual(KKT)multipliervalues.Wethen 74

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4{18 )totheRMPformulation.Ifthevalueofatthepreviousiterationdoesnotviolatethisnewcutattheprevioustk,thenthecurrentsolutionisoptimal.Otherwisewere-solveRMPandrepeatthisprocedureuntilthesametkvectorisoptimalinsuccessiveiterations.Intheworstcase,ifweweretogenerateaconstraintoftheformof( 4{18 )forallpossibletvectors,theresultingRMPformulationwouldbeequivalenttoMP.Inpractice,however,arelativelysmallnumberofsuchcutsareneededtondanoptimalsolution.Wenextformalizethealgorithmasfollows. 4{18 )totheRMPformulationandreturntoStep2. 75

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WeimplementedourBendersdecompositionalgorithmusingGAMS22.6runningonaUnixmachinewithtwoPentium4,3.2Ghzprocessors(with1Mcache)and6GBofRAM.WeusedCPLEX11forsolvingthe0-1integerprogrammingmasterproblem(RMP)andCONOPT2version2.071K-010-061forsolvingtheconvexprogrammingsubproblems(ILSP).AllofourtestproblemsusedM=10customersandN=5supplyfacilities,whichisthemaximumproblemsizethatwecouldconsistentlysolvewithin1200secondsinGAMS. Thelimitonthenumberofsupplyfacilitiesthatcanserveeachcustomer,i.e.,Niforcustomeri2I,isanimportantparameterforourmodel.Sincethemaximumnumberofsupplyfacilitiesforallinstanceswas5,weparametricallyvariedNibetween1and5foreachprobleminstance(andusedthesamevalueofNiforeachcustomer).Obviously,whenwesetNito1foreachcustomeri2I,weobtainanoptimalsolutionfortheproblemwithsingle-sourcingrequirements.LetZkbetheoptimalobjectivefunctionvaluewhenNi=k.Ourmaingoalistoanalyzetheeectofdierentparametersonthepercentagedierencebetweentheminimumcostwhendemandsplittingisallowedandwhensinglesourcingisimposed.Wethereforecalculatedthepercentagedierence,Zk,asZk=(Z1Zk)=Zkfork=1;:::;5andforeachsetofparametervalues.NotethatZ5characterizesthepercentagecostdierencebetweenthesingle-sourcingcaseandthecaseinwhichdemandsplittingisunrestricted. 76

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4-1 providestheuniformdistributionparametersforeachsettingofE[^c=h].OurchoiceofvaluesofE[^c]wasbasedonthefactthatinpractice,theholdingcostisoftenapercentageofthetotalvalueofanitem.Thatis,supposeh=ic0,whereiisapercentageholdingcostrate(oftenbetween15%and25%)andc0istheitem'svalue.Next,suppose^c=^{c0,i.e.,where^{reectsthepercentageoftotalvaluethatconstitutestransportationcost.Then,forexample,if^{=10%,andi=20%,wehave^c=0:5. Theratioofthestandarddeviationtothemeandemandisanotherimportantparameterthataectsthecostperformanceofsingle-sourcingrelativetodemandsplitting.Sincethestandarddeviationaectsthemagnitudeofsafetystockholdingcostandthemeanaectsthemagnitudeofassignmentcosts,insteadofanalyzingtheeectsofthesetwoparametersseparately,weanalyzedtheirratio,i.e.,thecoecientofvariation(CoV==)ofdemand.Werandomlygeneratedmeandemandsbetween4000and6000andused3dierentvaluesforCoV,0:35,0:40,and0:45,todeterminetheassociatedstandarddeviationvalues. Theotherimportantparameteraectingcostperformanceisthexedcostofasupplyfacility.Ahighxedcostdecreasesthenumberofopensupplyfacilities,whichinturnaectstheassignmentofcustomerstosupplyfacilities.WerandomlygeneratedfourdierentdatasetsforFjvaluesfromtheuniformdistributionsshowninTable 4-1 77

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Byusingthecrosscombinationsofthesethreeparametersettings,i.e.,E[^c=h],CoV,andFj,wegenerated600(543)dierentdatasets.Foreachdatasetwegenerated10randomtestinstances,resulting6000testinstancesintotal.Wesettheservicelevelto97.5%(z=1:96)foralltestinstances. First,weanalyzedtheeectofE[^c=h].Table 4-2 summarizestheresultsfordierentvaluesofE[^c=h].WeprovidethemaximumandminimumvaluesofZ5fromamongthe6000instancesinthecolumnslabeledmaxandmin,respectively,withtheaveragevalueinthecolumnlabeledaverage. Thehighestpercentagedierenceobtainedamong6000instancesequals6:82%.Theminimumpercentagedierenceis0%,whichmeansthatinsomeofthecasesasingle-sourcingsolutionisoptimaleventhoughsinglesourcingisnotenforced.ThemostremarkablerowinTable 4-2 istheonecorrespondingtoE[^c=h]=0:5.Theminimumpercentagedierenceamongthe1200testinstanceswithE[^c=h]=0:5is2:39%.Thismeansthatinnoneofthese1200instanceswassingle-sourcingoptimal.TheeectofE[^c=h]onthepercentagegapisinteresting.AsseeninFigure 4-3 ,bothlowandhighlevelsofE[^c=h]leadtotheoptimalityofsingle-sourcingsolutions. AthigherlevelsofE[^c=h],theproblembecomessimilartoanuncapacitatedfacilitylocationproblem,wheresinglesourcingisoptimal.Also,atlowerlevelsofE[c=h],thefacilityandsafetystockcostsdominatetheobjectivefunction.Inthepresenceofxedfacilitylocationcosts,themodelreducesthenumberoffacilitiesandusesaggregationtoobtainriskpoolingbenets.However,atintermediatevaluesoftheratioofthetransportationcosttotheholdingcost,themodelseekstoreducetransportationcosts 78

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WeillustratetheaveragevalueofZkfordierentvaluesofk(wherek=Niforeachi2I)inFigure 4-4 .AscanbeseenfromFigure 4-4 ,whenthereisnolimitonthenumberoffacilitiesthatcansupplyanycustomer,i.e.,whenNi=5,anoptimalsolutionassignscustomerstoatmost3dierentsupplyfacilities.Inthemajorityofcases,assigningeachcustomertoatmost2supplyfacilitiesisoptimal.Thegapbetweentheperformanceofthesinglesourcingandmultiplesourcingsolutionsissignicant.However,thedierencewhenweincreaseNifrom2to3isnotsignicant. Next,weanalyzetheeectofCoV.Table 4-3 summarizestheresults.AswecanseeinbothTable 4-3 andFigure 4-5 ,asthecoecientofvariationincreasesfrom0:35to0:45,thepercentagecostdierencebetweenoptimalsinglesourcinganddemandsplittingsolutionsdecreases.ThemainreasonforthisisthatastheCoVincreases,thestandarddeviationofdemandincreases.Inturn,thisleadstohighersafetystockholdingcosts.Themodeltendstoopenfewersupplyfacilitiesandbenetsfromriskpoolingbyassigningmorecustomerstofewersupplyfacilities.Similarly,weexpectadecreaseinthepercentagecostdierenceastheCoVapproachestheoriginbecause,inthiscase,thesafetystockholdingcostbecomessosmallthattheproblembecomessimilartoanuncapacitatedfacilitylocationproblem. Wenextanalyzetheeectofthexedfacilityopeningcost.ThiseectisshowninTable 4-4 .Aswewouldexpect,asthexedcostincreases,fewerlocationsareopened,andcustomersarethereforeassignedtofewerlocations.Thus,thebenetsofdemandsplittingdecreaseasthexedfacilitycostsincrease. 79

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4-7 illustratestheseresults.AsFigure 4-7 shows,thegreatestCPUtimeisneededwhenNi=2.Inmostoftheinstanceswhenthereisnolimitonthenumberoffacilitiesthatcansupplyacustomer,theoptimalsolutionassignsacustomertoatmost3supplyfacilities.WhenwelimitthenumberofsupplyfacilitiestoNi=2,thecorrespondingconstraintbecomestightandtherequiredCPUtimeincreases.ThisincreaseinCPUtimecomesasaresultoftheincreasedtimeCPLEXmustspendsolvingthe0-1integermasterproblem(RMP).However,whenNi=5,theconstraintislooseinalmostallinstances,andtherequiredCPUtimeissignicantlylower. hello Figure4-1. Costincreasemultiplierforsingle-sourcingasafunctionof RatioofcostsavingsfromsplittingtominimumcostasafunctionofE[c=h] 80

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TheeectofE[^c=h]onZ5 TheeectofNionZkfordierentvaluesofE[^c=h] hello gfgfgfg gfgfgfg 81

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TheeectofNionZkfordierentvaluesofCoV Figure4-6. TheeectofNionZkfordierentvaluesofxedcost hello gfgfgfg gfgfgfg 82

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CputimesfordierentvaluesofNi Dataparametersettings Table4-2. Themax,min,andaveragevaluesofZ5fordierentvaluesofE[^c=h] Z5 hello gfgfgfg gfgfgfg 83

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Themax,min,andaveragevaluesofZ5fordierentvaluesofCoV Z5 Table4-4. Themax,min,andaveragevaluesofZ5fordierentvaluesofxedcost Z5 hello gfgfgfg gfgfgfg 84

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Dependingonthecustomer'sproleandheridealproduct,analternativeproductsetcanoftenbeidentied(whichincludesallsubstituteproductsthatthiscustomermaybewillingtobuyinsteadofheridealproduct).Forexample,ifacustomer'sidealproductisacomputerwitha2GBRAM,shemaybewillingtobuyacomputerwiththesamecongurationwith3GBRAMinstead,ifanincentiveisoered.Then,thisnewproductwillserveasoneofthesubstitutesforheridealproductandwillbeamemberofthealternativeproductsetforthiscustomer.Anotherexamplecanbefoundinproblemsettingswhereone-waysubstitutionispossible.Inthesecontexts,aproductcanbesubstitutedforhigher/lowerlevelqualityorperformanceproducts.Insuchcases,thealternativesetforanidealproductwillincludeallproductswhosequality/performancelevelsarehigher/lowerthanthatoftheidealproduct.Acustomermayleavethesystem 85

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Severalissuescomplicatetheoptimizationoftheproductportfoliointhisenvironment.Oneoftheseissuesistheproductarchitecture.Ifaproductisassembledfromanumberofcomponentsthatmaybeorderedfromanoutsidesupplierormanufacturedinhouse,thentheproblemcontainsanembeddedcomponentprocurement/productionplanningproblem.Inthistypeofenvironment,severaldierentproductscanoftendrawfromthesamesetofcomponents.Thepossibilityofsubstitutionacrossproductsimpactscomponentinventorylevels.Therefore,whileoptimizingtheassortmentofproducts,wealsoneedtoconsiderthecomponentprocurement/manufacturingplanthatminimizesthecomponentorder/manufacturingandinventoryholdingcosts. Inthischapterweconsideramulti-periodcomponentprocurementplanningandproductportfoliodesignproblemwithproductsubstitutions.Weassumethatthereisasetofpotentialor\ideal"productsforwhichdemandsoccurfromasetofcustomersegments.Eachcustomersegment-idealproductpairhasanalternativeproductsetthatincludesallpotentiallyacceptablesubstitutesfortheidealproductforthatcustomersegment.Iftheidealproductofacustomerisnotavailable,thesubstituteproductsareoeredinapredeterminedorrankedorderatsomediscount.Forexample,inthepreviousexample,whereone-waysubstitutionispossible,higher/lowerquality/performanceproductsmayberankedandoeredtothecustomerinascending/descendingquality/performanceorder.Thecustomerhasthreechoicesatanypoint:(i)acceptasubstituteproduct,(ii)declinethecurrentoeredsubstituteproductandconsiderthenextavailableone,or(iii)leavethesystemwithoutbuyinganyproduct.Productsareassembledfromasetofcomponents,whichareprocuredfromanoutsidesupplierataxedplusvariablecostineachprocurementperiod.Wemodelthisproblemasalargescalemixed-integerlinearprogrammingproblem.WerstshowthatthisproblemisNP{Hard,evenfor 86

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Theremainderofthischapterisorganizedasfollows.WegiveabriefoverwievoftheliteraturerelatedtoourworkinSection 5.2 .InSection 5.3 ,wedenetheproblemandformulatethemathematicalmodel,andinSection 5.4 weprovidetheBendersdecomposition-basedalgorithmthatwedevelopedtosolvethemodel. Themostcloselyrelatedbranchoftheassemble-to-orderliteraturestudiessystemswhereproductsrequirecommoncomponentsandanalyzestheeectsofcommonalityoncomponentinventorylevels(see Bakeretal. ( 1986 ), Gerchaketal. ( 1988 ), GerchakandHenig ( 1989 ), Hillier ( 2000 ), Luetal. ( 2003 )). AkcayandXu ( 2004 )studyanassemble-to-ordersystemwithmultiplecomponentsandmultipleproducts,whereeachproducthasaprespeciedresponsetimewindow.Thesystemreceivesarewardifthedemandisfullledwithinitsresponsetimewindow.Theyformulatethisproblemasatwo-stagestochasticintegerprogramtodeterminetheoptimalbasestocklevelsofcomponentssubjecttoaninvestmentbudget. Afentakisetal. ( 1984 )developabranchandboundalgorithmforoptimallotsizinginmultistageassemblysystems.Theirmethodissuitableforproductswithanassemblystructureonly. AfentakisandGavish ( 1986 )relaxthisrestrictionandexaminethelotsizingproblemforgeneralproductstructuresbytransformingthegeneralproductstructureproblemintoanequivalentandlargerassemblysystem. Rosling ( 1989 )identiestheoptimalpolicyforuncapacitatedmultistagegeneralassemblysystemsunderarestrictionontheinitialstocklevels.Withthiscondition,theassemblysystemcanbeinterpretedasaseriessystem,andhence,canbesolvedoptimally. 87

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Pentico ( 1976 ), Chandetal. ( 1994 ), Bassoketal. ( 1999 ), Raoetal. ( 2004 ), Hsuetal. ( 2005 ), TasknandUnal ( 2009 )). Bassoketal. ( 1999 )studysingleperiodmulti-productinventorymodelwithstochasticdemandandfulldownwardsubstitutionwheretheunsatiseddemandforaproductcanbelledwithaproductwithhigherutility. SmithandAgrawal ( 2000 )developamodelthatdeterminestheeectofsubstitutiononthedemanddistribution,inventorylevelsofitems,andcustomerservicelevels.Theyassumethatdemandoriginatesfromarandomnumberofcustomers,whoselectrandomlywithknownfrequenciesfroma\choiceset"ofitemsthatcontainsallpotentialsubstitutes. Pentico ( 1974 )studiesanassortmentprobleminwhichasetofcandidatesizesofsomeproductisgiven,fromwhichasubsetofsizeswillbeselectedtobestocked.Demandforanunstockedsizeislledfromalargerstockedsizewithanassociatedsubstitutioncost.Heprovidesanoptimalstationarystockingpolicyundercertainassumptions,andextendstheproblembyconsideringanonlinearcostfunctionin Pentico ( 1976 ). BalakrishnanandGeunes ( 2000 )studyadynamic,multi-periodrequirements-planningproblemwithexiblebills-of-materialswithanoptiontosubstitutecomponents.Theymodeltheproblemasanintegerprogramandprovideadynamic-programmingsolutionalgorithmthatgeneralizesthesingle-itemlot-sizingalgorithm. Haleetal. ( 2000 )studyamodelwithtwoproducts,eachcomposedoftwocomponents,oneofwhichcanbedownwardsubstituted.Theyformulatethisproblemasatwo-stagestochasticprogram,theobjectiveofwhichtheyprovedtobejointlyconcaveintheorderquantities,allowingthemtodevelopboundsontheoptimalorderquantities. Yunesetal. ( 2007 )developmarketingandoperationalmethodologiesandtoolsforJohnDeere,oneoftheworld'sleadingproducersofmachinery,reducingcostsbyconcentratingproductlinecongurationswhilemaintaininghighcustomerserviceandprots.Deere'sproducts 88

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Ourworkismostcloselyrelatedtothatof Ervolinaetal. ( 2009 ).Theyproposeaprocessthataimsatndingmarketableproductalternativesthatareassembled-to-orderfromacertainnumberofcomponents,eachhavingalimitedsupply.Theyprovideasingle-periodmathematicalmodelthatdeterminesasubstitutionplanforasystem,wherethedemandisdeterministicandaknownpercentageofcustomersacceptasubstituteproductifitspriceandqualityarewithinacertainrange.Intheirmodel,theydeneacoreproductsetthatincludestheproductsforwhichdemandsmayoccur,andasetofalternativeproductsthatincludestheproductsthatmaybeusedtosatisfythedemandoccurredforcoreproducts.Theysimulatethesystemandprovidecomputationalresults.Nosolutionalgorithmisprovided.Incontrast,weproposeamulti-periodproductionandsubstitutionplaninwhichwedecidewhichproductstooer,howtosatisfydemandsandhowtoprocurethecomponentsthatareusedtoproduceproducts.Ourmodelhavedistinctfeaturesthenthatof Ervolinaetal. ( 2009 ):(i)ourproblemisamulti-periodproblem;(ii)itincludesacomponentprocurementplanthataccountsforeconomiesofscale,whichmakestheproblemmuchcomplexandrealistic;(iii)weassumethatdemandmayoccurforanyproductthatisdecidedtobedesigned(notonlyforacertainsetofproducts);(iv)wedeneanalternativeproductsetforeverycustomersegment-idealproductpair(notjustonesetforallcustomersegmentsandcoreproducts);and(v)weproposeanexactalgorithmtosolvethemodel. Ourcontributiontotheexistingliteratureistwofold.First,wemodelamulti-periodcomplexproduct-linedesignproblemwithproductsubstitutions,inwhichproductsare 89

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SmithandAgrawal ( 2000 ),wherearandomnumberofcustomerschoosesaproductfromthechoicesetthatcontainsallpotentialsubstitutes.Incontrast,weassumethatdemandoccursforaspecicproduct,andthatproducthasanalternativeproductset,whichcorrespondstoa\choiceset"thatisdenedforeverycustomersegment.Weassumethatthealternativeproductsetisknownbothtotheretailerandthecustomer.Ifacustomer'sidealproductisnotmadeavailable,demandmaybesatisedusinganalternativeproductfromthissetatasubstitutioncost,ifthecustomeracceptsasubstitute. Yunesetal. ( 2007 )developanalgorithmwithacustomermigrationcomponent,whichquantitativelycharacterizescustomerbehaviorbypermittingacustomertomigratetoanalternativecongurationifherrstchoiceisunavailable.Foreverycustomer,theycreateamigrationlistthatconsistsofasetofacceptablecongurationsandissortedindecreasingorderofpreference.Inthisstudy,weusealternativeproductsetsthataredenedforeveryproduct-customersegmentpair,likethemigrationlistsdevelopedby Yunesetal. ( 2007 ).Wealsoassumethatthesealternativeproductsetsarerankedinorderofcustomerspreferences.Therefore,ifacustomer'sidealproductisnotmadeavailable,shecaneitherleavethesystemorpurchasethenextsubstituteitemfromthealternativeproductlist.Whenthecustomerleavesthesystemwithoutapurchase,the 90

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Eachproductmhasaprotmargin,pmt,thatisperioddependent.Moreover,eachproductisassembled-to-orderfromasetofcomponents,I,andinventoryisheldatthecomponentlevel.Componentiprocurementcostscontainanonnegativexedcost,ait,plusvariablecost,bit,inperiodt.Eachproducthasanassociatedusagevector,u,whichdeterminesthenumberofcomponentsincludedintheproduct(i.e.,uimisthenumberofrequiredcomponentsoftypeiinproductm).Ifaproductdoesnotcontainacomponent,thecorrespondingrowoftheusagevectoriszero. Theparametersanddecisionvariablesfortheproblemareasfollows: 91

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Theproblem,inamulti-periodsetting,requiresdeterminingwhichproductstooer,howmanycomponentstoprocure,andhowmanycomponentstoholdininventoryineachperiodinordertomaximizeoverallprot.Hence,ouraimistodetermineaproductportfolio,substitutionplan,andprocurementplaninordertomaximizeprot.Weformulatethisproblemasamixed-integerlinearprogrammingproblemasfollows: (5{1)SubjecttoHit=Hi;t1+VitXc2CXm2MXn2Ncm(uinYcmnt);8i2I;t2T 92

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Theobjectivefunction( 5{1 )maximizesthesumofprotmarginslessthesubstitutioncost,xedproductdesigncost,xedandvariableprocurementcostandtheinventoryholdingcostofcomponents.Constraints( 5{2 )and( 5{3 )areinventorybalanceconstraints.Constraints( 5{4 )and( 5{5 )trackthenumberofcustomersremaininginthesystemaftereachsubstituteoer.Constraints( 5{6 )ensurethatweproduceandsellonlytheproductsthatareoered.Constraints( 5{7 )ensurethatweprocurecomponentsonlyifweincurtheassociatedprocurementcost.Theotherconstraintsincludenonnegativityandbinaryrequirements. Letcit=bit+PTl=thilandLit=Pc2CPm2MPn2NcmPTl=tuinDcml.SinceHit=Ptl=1VilPc2CPm2MPn2NcmPtl=1uinYcmnlforalli2Iandt2T,thenwecanrewrite(MILP-1)asfollows: 93

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Observethatwhentheproductsthatwillbeoeredandthenumberofproductsthatwillbeassembledineachperiodareknown,theresultingproblemisactuallyacomponentprocurementplanningproblem,whichisawell-knownuncapacitatedlotsizingproblem.Wewillusethisfactlatertodevelopadecomposition-basedsolutionalgorithm.First,however,weshowthat(MILP-2)isNP-Hard.

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5{11 ),withoutlossofoptimality.Additionally,assume=0,whichrequiresthatcustomersstayinthesystemuntilthelastsubstituteproductisoerediftheiridealproductisnotavailable.Then,wecanwritethisspecialcaseof(MILP-2)asfollows:MaximizeXm2MXn2Nm(pnwmn)YnXm2MfmZm Foreveym2M,letkm=maxn2Nm(pnwmn)andcmn=km(pnwmn),i.e.,(pnwmn)=kmcmn.WecanwritetheaboveobjectiveasMaximizeXm2MXn2Nm(kmcmn)YmnXm2MfmZm=Xm2MkmXn2NmYmnXm2MXn2NmcmnYmnXm2MfmZm: 95

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Thenextsectionprovidessuchasolutionalgorithm,whichisaBendersdecomposition-basedexactalgorithmthatwehavedevelopedforsolving(MILP-2). (5{26)SubjecttotXl=1VilXc2CXm2MXn2NcmtXl=1uin~Ycmnl0;8i2I;t2T WecandesignaBendersdecompositionalgorithmthatuses(SP-IP1)asasubproblem.InordertouseaBendersdecompositionalgorithm,weneedtwotypesofinformation:(1) 96

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(5{31)SubjecttotXl=1VilXc2CXm2MXn2NcmtXl=1uin~Ycmnl0;8t2T Notethatsincethisisaminimizationproblem,therestrictionit1intheLP-relaxationgivenabovewillalwaysholdandthereforecanbeomittedfromtheformulation.Letitanditbethedualvariablesassociatedwithconstraints( 5{32 )and( 5{33 ),respectively.Thenthedualofmodel(SP-LP(i))forcomponenti2Icanbewrittenas(D-SP-LP(i))MaximizeXt2TXc2CXm2MXn2NcmtXl=1uin~YcmnlitSubjecttoLititait;8t2T;ititcit;8t2T;it;it0;8t2T:

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Thesolutionto(D-SP-LP(i))isoneoftheextremepoints(p;p).Therefore,themaximumvalueofthesubproblemisitsvalueatoneoftheextremepointsofF.WeaddtheconstraintXi2IXt2TXc2CXm2MXn2NcmtXl=1uin~Ycmnlit forallextremepointsofthedualfeasibleregionasBenderscutstocreateaBendersmasterproblem. SincethefeasiblespaceofthesubproblemisindependentofthechoicemadefortheYvariables,wecanusethedualvariablesobtainedbysolvingtheaboveformulationinourBenderscut.Thenwecanwritethemasterproblemasfollows: 98

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wherePisthenumberofextremepointsassociatedwiththesubproblems(SP-LP(i)).Thereisanexponentialnumberofsuchextremepoints,eachofwhichcorrespondstoaconstraintoftheformof( 5{36 ).However,wecangeneratevalidBenderscutsandsuccessivelyaddthesetotheformulation.Then,therelaxedmasterproblemcanbewrittenas whereKdenotesthenumberofBenderscutswehavegenerated. Notethatthe(RMP)formulationisamixedintegerlinearprogram.Ateachiteration,wesolve(RMP)toobtaina(possibly)newYkvector.GiventhisYkvector,wethensolvethesubproblem(SP-LP(i))foralli2Itodeterminethecorrespondingoptimaldualvalues,.Theobjectivefunctionvalueof(RMP)givesusanupperboundateachiteration.Theobjectivefunctionvalueofthesubproblemswillcombinetoforma 99

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5{36 )to(RMP)formulation,re-solve(RMP)andrepeatthisprocedure. Theobjectivefunctionvalueofarelaxationoftheactualsubproblemmayleadtoaweakboundforouralgorithm.IfwecanrewritethelotsizingmodelsothattheLP-relaxationgivesasolutioninwhichthe-variablestakeintegralvalues,wecanusethenewformulation'soptimalobjectivefunctionvaluetocalculatethelowerbound. DeneVitsasthenumberofunitsofcomponentiprocuredinperiodttosatisfyassemblyrequirementsinperiodst2T.BydisaggregatingtheVitvariablesintoVitsvariables,wecanobtainamodelintheformofthesimpleplantlocationformulationgivenin KrarupandBilde ( 1977 ),whichisknowntohaveanoptimalsolutioninwhichthebinaryvariablesareintegerintheLP-relaxationsolution. Notethat(SP-IP1)isseparableamongcomponents.Therefore,wecansolve(SP-IP1)foreverycomponentseparatelyandthenmergetheresults.SinceweassumetheYvariablesareknownin(SP-IP1),wecanusetheseknownvaluestomakeconstraint( 5{28 )astightaspossible.Letis=Pc2CPm2MPn2Ncmuin~Ycmns.Thenthesubproblemformulationforcomponenti2Icanbewrittenasfollows:(SP-DIP(i))MinimizeXt2Taitit+citTXs=tVits! 100

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WecanusethismodeltoprovidealowerboundfortheBendersdecompositionalgorithmgivenabove.Letusrewritethemodel(MILP-2)withthenewsubproblemasfollows: 101

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Notethat(SP(i))isalinearprogram,andwecansolveitusingthealgorithmprovidedin Wagelmansetal. ( 1992 ),andobtaintheobjectivefunction'svalue.Letisanditsbethedualvariablesassociatedwithconstraints( 5{65 ),and( 5{66 ),respectively.Thenwecanwritethedualof(SP(i))as(D-SP(i))MaximizeXs2TisisSubjecttoXs2T;stitsisait;8t2Tisitscit;8t;s2T;stits;is0;8t;s2T;st: Wagelmansetal. ( 1992 )toobtainafeasiblesolutionwithV0and0withadualobjectivefunctionvalueofSP0z.SetLB=RMP0z+0SP0z.andlet(Y;Z;V;)=(Y0;Z0;V0;0)denotetheinitialincumbentsolution. 102

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5{36 )tothe(RMP)formulationandreturntoStep2. WehaveperformedasetofpreliminarycomputationalteststhatshowthatthisalgorithmisabletosolveprobleminstancesthatcannotbesolvedinCPLEX.FutureresearchwillincludeabroadsetofcomputationalteststocharacterizetheperformanceoftheBendersdecompositionapproachacrossawiderangeofparametersettings. 103

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Inthischapter,weconcludethedissertationbydiscussingtherstfourchapters,providingconcludingremarks,andsummarizingourcontributionstotheexistingliterature.Wealsobrieydiscussfutureresearchdirectionsbasedontheresultsofchapters. Johansson ( 1979 )).AfurtherexplorationofthesystemofequationsdenedbythegeneralizedKKTconditionsmightalsoprovidevalueinthedevelopmentofalgorithmsforthegeneralformoftheproblemwhereinvestmentlevelsmaytakeanyreal-valuednumber. InChapter 3 ,weanalyzedastochasticknapsackproblemwheretheweightsofitemsarePoissondistributedrandomvariablesandapenaltyisassessedwhentheknapsackcapacityisexceeded.Weprovidedapolynomial-timesolutionforthecontinuousrelaxationofthisproblemandacustomizedbranch-and-boundalgorithmtosolvethe 104

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4 ,ourmodeldeterminesthelocationoffacilitiesandtheassignmentofcustomerstosupplyfacilitiesinordertominimizethetotalsupplyfacilityopeningcost,customer-supplyfacilityassignmentcostandthesafetystockcostsatsupplyfacilities.Intheliterature,similarproblemshavebeeninvestigatedwithasingle-sourcingrequirementforeachcustomer.Werelaxthisconstraintandapplyanupperboundonthenumberoffacilitiestowhichacustomercanbeassigned.Clearly,whenthisnumberequalsone,weobtainaspecialcaseoftheproblemthatenforcessingle-sourcing.Ourgoalwastocharacterizethedierencebetweenthecostsofproblemswheredemand-splittingisallowedandthosethatenforcesingle-sourcing. Theresultinglocation-inventoryproblemfallsintoaclassofdicultmixed-integernonlinearprogrammingproblems.Thestructureoftheobjectivefunction,however,leadsustocharacterizethesolutionpropertiesforsomespecialcases.Forthegeneralproblem,weproposedageneralizedBendersdecompositionalgorithm.Weimplementedouralgorithmandconductedabroadsetofcomputationalteststoanalyzetheeectsofkeyparametersonthepercentagedierenceincostswhendemand-splittingisallowedandwhensinglesourcingisrequired.Accordingtoourcomputationalstudy,withtheparametersettingswetested,thispercentagedierencecanbeashighas6.82%. Therelativevaluesofassignmentandholdingcosts,thecoecientofvariationofcustomerdemands,andthexedopeningcostsoffacilitiesarethemostimportantparametersaectingtheoptimalassignmentofcustomers.Therefore,weanalyzedthe 105

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Thisresearchcanbeextendedinanumberofdierentways.Onepossibleextensionwouldconsidertheadditionofnitecapacitiestosupplyfacilities.Anotherextensionmightconsideraddingapenaltycostforassigningacustomertomorethanonefacility,insteadofusingarestrictiononthenumberoffacilitiestowhichacustomercanbeassigned.Anadditionalinterestingextensionconsidersservice-level-dependentassignmentcosts,whichreectcasesinwhichsomefacilitiesmayrequireahigherservicelevelandanincreaseinassociatedassignmentcost.Inthissetting,customersmayacceptreducedservicelevelsinsteadofpayinghighercosts.Thus,insteadofdeningpre-speciedservicelevelsatthesupplyfacilities,wemaytreatfacilityservicelevelsasdecisionvariables. 5 ,weassumethatproductsareassembledfromanumberofcomponentsthatareprocuredfromanoutsidesupplier.Demandsforidealproductsoccurfromasetofcustomersegments,andeachidealproductcanbesubstitutedwithasetofalternativeproductsatasubstitutioncost,subjecttotheavailabilityoftheproductsinthesubstitutionset.Eachcustomermayleavethesystemwithoutbuyinganyproduct 106

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Theproblemincludesanembeddedcomponentprocurementplan,whichisactuallyalotsizingproblem.WeproposeaBendersdecomposition-basedexactalgorithmthatusesthislotsizingproblemasasubproblem.Sincethelotsizingproblemisamixed-integerlinearprogrammingproblem,wecannotuseitdirectlyasasubproblemforthedecompositionalgorithm.We,therefore,useitslinearrelaxationtoobtainthedulasandrewritethemodelbydisaggregatingtheamountofcomponentsprocuredineachperiod,producingatightformulationofthelotsizingproblem,andusethenewdisaggregatedmodel'sobjectivefunctionvaluetoobtainbetterlowerbounds. Wehaveperformedasetofpreliminarycomputationalteststhatshowthatthisalgorithmisabletosolvemediumtolargeprobleminstances.FutureresearchwillincludeabroadsetofcomputationalteststocharacterizetheperformanceoftheBendersdecompositionapproachacrossawiderangeofparametersettings. Thisresearchcanbeextendedinseveraldierentways.Weassumethatdemandisknown.Thisassumptionmightberelaxed,andthedependencyofdemandsonthesetofsubstitutableproductscanbeaccountedforwithinthemodel.Anotherextensionmightaccountforcustomerservicelevelswithineachgroupofcustomersegments.Forexample,wemightrequirethatacertainpercentageofalldemandsreceivetheiridealproduct. 107

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Wheneachi(ai)functionislocallyLipschitzcontinuous,thegeneralizedKKTconditionsfor[KPS]arenecessarybutarenotsucientforoptimality(seeHiriart-Urruty Hiriart-Urruty ( 1978 )).WewillrefertoapointthatsatisesthegeneralizedKKTconditionsasa\KKTpoint",andletwbeaKKTmultiplierforthebudgetconstraintandleti(foralli2I)bemultipliersassociatedwiththelower-bound(nonnegativity)constraints.Dene@i(ai)asthesetofsubgradientsofthefunctioni()atai,with@+i(ai)denotingtherightdirectionalderivativeataiand@i(ai)denotingthecorrespondingleftdirectionalderivative.ForourS-curves,thesetofsubgradientsataiisequaltotheinterval[@i(ai),@+i(ai)]ifailiesintheconvexportionofthefunction,whilethesetofsubgradientsataiequalstheinterval[@+i(ai),@i(ai)]ifailiesintheconcaveportionofthefunction.ThegeneralizedKKTconditionscanbewrittenas:@i(ai)+wi30;8i=1;:::;N;wXni=1aiA=0;iai=0;8i=1;:::;N;nXi=1aiA;w0;i0;8i=1;:::;N:

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FromLemma( 1 )weknowthatthistwo-supplier,two-customerproblemisaconvexprogrammingproblem.ThereforethegeneralizedKKTConditionsarenecessaryandsucientforoptimality.TheKKTconditionsforthisproblemcanbewrittenasfollows: Pi2Ix2ijiij=0fori=1;2andj=1;2 (A-1)i(1Xj2Jxij)=0fori=1;2 (A-2)ijxij=0fori=1;2andj=1;2 (A-3)1Xj2Jxij0fori=1;2 (A-4)xij0fori=1;2andj=1;2 (A-5)i0fori=1;2 (A-6)ij0fori=1;2andj=1;2 (A-7) Forthegivensolution,x11=x22=andx12=x21=1where0<<1,fromcondition( A-3 )wesetij=0fori=1;2andj=1;2.Sincex11+x12=x21+x22=1,condition( A-2 )isalreadysatised.Fromcondition( A-1 ),werequire1=c12+H(1) A-1 )to( A-7 )andisthereforeoptimal.Thevalueoftheobjectivefunction,Zopt,equalsc12+c21+2H(1) 109

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where()=min21maxf;1g 111

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SemraAgralwasborninMalatya,Turkey,in1980.Shegraduatedfromhighschool,MalatyaAnadoluLisesi,in1998.ShereceivedherB.S.degreeinindustrialengineeringfrom_IstanbulTechnicalUniversityin2003.Upongraduation,sheattendedKocUniversity,whereshereceivedhermaster'sdegreeinindustrialengineeringin2005.InAugust2009,shereceivedherPh.D.degreeinindustrialengineeringfromtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.Followinggraduation,shewilljointhefacultyoftheDepartmentofIndustrialEngineeringatBahcesehirUniversity. 119