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# Optimal Supply Chain Planning Problems with Nonlinear Revenue and Cost Functions

## Material Information

Title: Optimal Supply Chain Planning Problems with Nonlinear Revenue and Cost Functions
Physical Description: 1 online resource (119 p.)
Language: english
Creator: Agrali, Semra
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

## Subjects

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

## Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Semra Agrali.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024786:00001

## Material Information

Title: Optimal Supply Chain Planning Problems with Nonlinear Revenue and Cost Functions
Physical Description: 1 online resource (119 p.)
Language: english
Creator: Agrali, Semra
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

## Subjects

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

## Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Semra Agrali.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024786:00001

Full Text

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whohavealwaysencouragedmetopursuemydreams 3

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

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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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Afentakis,P.,B.Gavish.1986.Optimallot-sizingalgorithmsforcomplexproductstructures.OperationsResearch34(2)237{249. Afentakis,P.,B.Gavish,U.Karmarkar.1984.Computationallyecientoptimalsolutionstothelot-sizingprobleminmultistageassemblysystems.ManagementScience30(2)222{239. Akcay,Y.,S.H.Xu.2004.Jointinventoryreplenishmentandcomponentallocationoptimizationinanassemble-to-ordersystem.ManagementScience50(1)99{116. Baker,K.R.,M.J.Magazine,H.L.W.Nuttle.1986.Theeectofcommonalityonsafetystockinasimpleinventorymodel.ManagementScience32(8)982{988. Balakrishnan,A.,J.Geunes.2000.Requirementsplanningwithsubstitutions:Exploitingbill-of-materialsexibilityinproductionplanning.Manufacturing&ServiceOperationsManagement2(2)166{185. Balakrishnan,A.,J.Geunes.2003.Productionplanningwithexibleproductspecications:Anapplicationtospecialitysteelmanufacturing.OperationsResearch51(1)94{112. Barnhart,C.,A.M.Cohn.1998.Thestochasticknapsackproblemwithrandomweights:aheuristicapproachtorobusttransportationplanning.ProceedingsofTristanIII,PuertoRico. Bassok,Y.,R.Anupindi,R.Akella.1999.Single-periodmultiproductinventorymodelswithsubstitution.OperationsResearch47(4)632{642. Bazaraa,M.S.,H.Sherali,C.M.Shetty.2006.NonlinearProgramming:TheoryandAlgorithms.JohnWiley&Sons,NewYork,NY. Bilde,O.,J.Krarup.1977.Sharplowerboundsforthesimplelocationproblem.AnnalsofDiscreteMathematics179{97. Bitran,G.R.,A.C.Hax.1981.Disaggregationandresourceallocationusingconvexknapsackproblemswithboundedvariables.ManagementScience27(4)431{441. Bretthauer,K.M.,B.Shetty.1995.Thenonlinearresourceallocationproblem.OperationsResearch43(4)670{683. Bretthauer,K.M.,B.Shetty.2002a.Thenonlinearknapsackproblem-algorithmsandapplications.EuropeanJournalofOperationalResearch138459{472. Bretthauer,K.M.,B.Shetty.2002b.Apeggingalgorithmforthenonlinearresourceallocationproblem.Computers&OperationsResearch29505{527. 112

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