<%BANNER%>

Optimal Supply Chain Planning Problems with Nonlinear Revenue and Cost Functions

Permanent Link: http://ufdc.ufl.edu/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.
Local: Adviser: Geunes, Joseph P.

Record Information

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

Permanent Link: http://ufdc.ufl.edu/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.
Local: Adviser: Geunes, Joseph P.

Record Information

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


This item has the following downloads:


Full Text

PAGE 1

1

PAGE 2

2

PAGE 3

whohavealwaysencouragedmetopursuemydreams 3

PAGE 4

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

PAGE 5

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

PAGE 6

................................ 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

PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

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

PAGE 11

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

PAGE 12

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

PAGE 13

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

PAGE 14

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

PAGE 15

15

PAGE 16

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

PAGE 17

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

PAGE 18

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

PAGE 19

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

PAGE 20

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

PAGE 21

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

PAGE 22

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

PAGE 23

23

PAGE 24

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

PAGE 25

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

PAGE 26

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

PAGE 27

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

PAGE 28

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

PAGE 29

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)

PAGE 30

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

PAGE 31

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

PAGE 32

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.

PAGE 33

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

PAGE 34

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

PAGE 35

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
PAGE 36

Thedynamicprogramrstsolvesthesingle-itemproblem:F1()=minfa1j1(a1);0a11g;=0;:::;: Wecansetasuitablevalueofasfollows.Lettingmax=maxi2Ifi(i)g,then=Nmaxprovidesanupperboundonz,theoptimalsolutionvalue.Notealsothat 36

PAGE 38

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

PAGE 39

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

PAGE 40

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

PAGE 41

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

PAGE 42

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

PAGE 43

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.

PAGE 44

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

PAGE 45

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

PAGE 46

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

PAGE 47

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

PAGE 48

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

PAGE 49

2 isnotstrictlyvalidforallitems,althoughitstillholdsamongvariableswhosevaluesarenotxed.Atanynodethatndsafeasibleintegersolutiontotheproblemwefathomthenodeandupdatethelowerboundifitexceedsthebestlowerbound.WeterminatethealgorithmwhenwendaKKTpointthatsatisesthebinaryrestrictionsorwhenallopennodeshavebeenfathomed. 49

PAGE 50

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

PAGE 51

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

PAGE 52

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

PAGE 53

Gams/BaronPerformanceAnalysis Table3-1. Parametervaluesfortestinstances 53

PAGE 54

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

PAGE 55

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

PAGE 56

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

PAGE 57

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

PAGE 58

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

PAGE 59

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

PAGE 60

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

PAGE 61

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

PAGE 62

62

PAGE 63

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

PAGE 64

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

PAGE 65

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

PAGE 66

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

PAGE 67

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

PAGE 68

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

PAGE 69

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

PAGE 70

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

PAGE 71

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

PAGE 72

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

PAGE 73

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

PAGE 74

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

PAGE 75

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

PAGE 76

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

PAGE 77

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

PAGE 78

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

PAGE 79

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

PAGE 80

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

PAGE 81

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

PAGE 82

TheeectofNionZkfordierentvaluesofCoV Figure4-6. TheeectofNionZkfordierentvaluesofxedcost hello gfgfgfg gfgfgfg 82

PAGE 83

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

PAGE 84

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

PAGE 85

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

PAGE 86

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

PAGE 87

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

PAGE 88

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

PAGE 89

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

PAGE 90

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

PAGE 91

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

PAGE 92

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

PAGE 93

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

PAGE 94

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

PAGE 95

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

PAGE 96

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

PAGE 97

(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:

PAGE 98

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

PAGE 99

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

PAGE 100

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

PAGE 101

WecanusethismodeltoprovidealowerboundfortheBendersdecompositionalgorithmgivenabove.Letusrewritethemodel(MILP-2)withthenewsubproblemasfollows: 101

PAGE 102

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

PAGE 103

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

PAGE 104

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

PAGE 105

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

PAGE 106

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

PAGE 107

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

PAGE 108

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:

PAGE 109

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

PAGE 111

where()=min21maxf;1g 111

PAGE 112

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

PAGE 113

Carraway,R.L.,R.L.Schmidt,L.R.Weatherford.1993.Analgorithmformaximizingtargetachievementinthestochasticknapsackproblemwithnormalreturns.NavalResearchLogistics40161{173. Chand,S.,J.E.Ward,Z.K.Weng.1994.Apartsselectionmodelwithone-waysubstitution.EuropeanJournalofOperationalResearch7365{69. Chopra,S.,P.Meindl.2007.SupplyChainManagement:Strategy,Planning,andOperations.3rded.Prentice-Hall,NewJersey. Daskin,MarkS.,L.V.Snyder,R.T.Berger.2005.Facilitylocationinsupplychaindesign,chap.2.Springer,NewYork,39{66. Daskin,M.S.,C.R.Coullard,Z-J.M.Shen.2002.Aninventory-locationmodel:Formulation,solutionalgorithmandcomputationalresults.AnnalsofOperationsResearch11083{106. Daskin,M.S.,S.H.Owen.199.LocationModelsinTransportationinHandbookofTransportationScience.KluwerAcademicPublishers,Boston. Dean,B.C.,M.X.Goemans,J.Vondrak.2004.Approximatingthestochasticknapsackproblem:thebenettofadaptivity.Proceedingsofthe45thAnnualIEEESymposiumonFoundationsofComputerScience. Erlebacher,S.J.,R.D.Meller.2000.Theinteractionoflocationandinventoryindesigningdistributionsystems.IIETransactions32155{166. Erlenkotter,D.1978.Adual-basedprocedureforuncapacitatedfacilitylocation.Opera-tionsResearch26(6)992{1009. Ervolina,T.R.,M.Ettl,Y.M.Lee,D.J.Peters.2009.Managingproductavailabilityinanassemble-to-ordersupplychainwithmultiplecustomersegments.ORSpectrum31257{280. Franca,P.M.,H.P.L.Luna.1982.Solvingstochastictransportation-locationproblemsbygeneralizedbendersdecomposition.TransportationScience16(2)113{126. Freeland,J.R.,C.B.Weinberg.1980.S-shapedresponsefunctions:Implicationsfordecisionmodels.JournaloftheOperationalResearchSociety31(11)1001{1007. Georion,A.M.1972.Generalizedbendersdecomposition.JournalofOptimizationTheoryandApplications10(4)237{260. Georion,A.M.1974.Lagrangeanrelaxationforintegerprogramming.MathematicalProgrammingStudy282{114. 113

PAGE 114

Gerchak,Y.,M.J.Magazine,A.B.Gamble.1988.Componentcommonalitywithservicelevelrequirements.ManagementScience34(6)753{760. Ginsberg,W.1974.Themultiplantrmwithincreasingreturnstoscale.JournalofEconomicTheory9283{292. Goel,A.,P.Indyk.1999.Stochasticloadbalancingandrelatedproblems.Proceedingsofthe40thAnnualSymposiumonFoundationsofComputerScience. Hale,W.W.,D.F.Pyke,N.Rudi.2000.Anassemble-to-ordersystemwithcomponentsubstitution.Proceedingsofthe4thMSOMConference. Henig,M.1990.Riskcriteriainastochasticknapsackproblem.OperationsResearch38820{825. Hillier,M.S.2000.Componentcommonalityinmulti-periodassemble-to-ordersystems.IIETransactions32755{766. Hiriart-Urruty,J.B.1978.Onoptimalityconditionsinnondierentiableprogramming.MathematicalProgramming1473{86. Hoc,H.H.1982.Topologicaloptimizationofnetworks:Anonlinearmixedintegermodelemployinggeneralizedbendersdecomposition.IEEETransactionsonAutomaticControl27(1)164{169. Hochbaum,D.S.1995.Anonlinearknapsackproblem.OperationsResearchLetters17103{110. Holthausen,D.M.,Jr.,G.Assmus.1982.Advertisingbudgetallocationunderuncertainty.ManagementScience28(5)487{499. Horst,R.,P.M.Pardalos,N.V.Thoai.1995.IntroductiontoGlobalOptimization.KluwerAcademicPublishers,London,UK. Hsu,V.N.,C-L.Li,W-Q.Xiao.2005.Dynamiclotsizeproblemswithone-wayproductsubstitution.IIETransactions37201{215. Johansson,J.1979.Advertisingandthes-curve:Anewapproach.JournalofMarketingResearch16346{354. Kleywegt,A.,J.D.Papastavrou.1999.Thedynamicandstochasticknapsackproblem.OperationsResearch4617{35. Kleywegt,A.,J.D.Papastavrou.2001.Thedynamicandstochasticknapsackproblemwithrandomsizeditems.OperationsResearch4926{41. 114

PAGE 115

Kodialam,M.S.,H.Luss.1998.Algorithmsforseparablenonlinearresourceallocationproblems.OperationsResearch46(2)272{284. Krarup,J.,O.Bilde.1977.PlantLocation,SetCoveringandEconomicLotSize:AnO(nm)-AlgorithmforStructuredProblems,chap.NumerischeMethodenbeiOptimierungsaufgaben,Band3:OptimierungbeiGraph-TheoretischenundGanzzahligenProblemen.BirkhauserVerlag,Basel,Switzerland,155{180. Lodish,L.M.1971.Callplan:Aninteractivesalesman'scallplanningsystem.ManagementScience18(4)25{40. Lu,Y.,J-S.Song,D.D.Yao.2003.Orderllrate,leadtimevariability,andadvancedemandinformationinanassemble-to-ordersystem.OperationsResearch51(2)292{308. Luss,H.,S.K.Gupta.1975.Allocationofeortresourcesamongcompetingactivities.OperationsResearch23(2)360{366. Martello,S.,P.Toth.1990.KnapsackProblems-AlgorithmsandComputerImplementa-tions.Wiley,NewYork,NY. Mathur,K.,H.M.Salkin,S.Morito.1983.Abranchandsearchalgorithmforaclassofnonlinearknapsackproblems.OperationsResearchLetters2(4)155{160. Melo,T.,S.Nickel,F.SaldanhadaGama.2007.FacilityLocationandSupplyChainManagement-AComprehensiveReview,chap.BerichtedesFraunhoferITWM,Nr.130. Merzifonluoglu,Y.,J.Geunes,H.E.Romeijn.2009.Thestaticstochasticknapsackproblemwithnormallydistributeditemsizes.Workingpaper,UniversityofFlorida,DepartmentofIndustrialandSystemsEngineering,Gainesville,FL,2008. Miranda,P.A.,R.A.Garrido.2006.Asimultaneousinventorycontrolandfacilitylocationmodelwithstochasticcapacityconstraint.NetworksandSpatialEconomics639{53. More,J.J.,S.A.Vavasis.1990.Onthesolutionofconcaveknapsackproblems.Mathemati-calProgramming49(3)397{411. Morin,T.L.,R.E.Marsten.1976.Analgorithmfornonlinearknapsackproblems.ManagementScience22(10)1147{1158. Norkin,V.I.,Y.M.Ermoliev,A.Ruszczynski.1998.Onoptimalallocationofindivisiblesunderuncertainty.OperationsResearch46(3)381{395. Nozick,L.K.,M.A.Turnquist.1998.Integratinginventoryimpactsintoaxed-chargemodelforlocatingdistributioncenters.TransportationResearchPartE34(3)173{186. 115

PAGE 116

Nozick,L.K.,M.A.Turnquist.2001b.Atwo-echeloninventoryallocationanddistributioncenterlocationanalysis.TransportationResearchPartE37425{441. Owen,S.H.,M.S.Daskin.1998.Strategicfacilitylocation:Areview.EuropeanJournalofOperationalResearch111(3)423{447. Ozsen,L.,C.R.Coullard,M.S.Daskin.2008a.Capacitatedwarehouselocationmodelwithriskpooling.NavalResearchLogistics55295{312. Ozsen,L.,C.R.Coullard,M.S.Daskin.2008b.Facilitylocationmodelingandinventorymanagementwithmulti-sourcing.forthcominginTransportationScience. Papastavrou,J.D.,S.Rajagopalan,A.Kleywegt.1996.Thedynamicandstochasticknapsackproblemwithdeadlines.ManagementScience421706{1718. Pentico,D.1974.Theassortmentproblemwithprobabilisticdemands.ManagementScience21(3)286{290. Pentico,D.1976.Theassortmentproblemwithnonlinearcostfunctions.OperationsResearch24(6)1129{1142. Rao,U.S.,J.M.Swaminathan,J.Zhang.2004.Multi-periodinvenotoryplanningwithdownwardsubstitution,stochasticdemandandsetupcosts.IIETransactions3659{71. Romeijn,H.E.,J.Geunes,K.Taae.2007.Onanonseparableconvexmaximizationproblemwithcontinuousknapsackconstraints.OperationsResearchLetters35(2)172{180. Romeijn,H.E.,F.Z.Sargut.2009.Thestochastictransportationproblemwithsinglesourcing.WorkingPaper,DepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,Gainesville,FL,2007. Rosenthal,R.E.2007.Gams-auser'sguide. Rosling,K._1989.Optimalinventorypoliciesforassemblysystemsunderrandomdemands.OperationsResearch37(4)565{579. Ross,K.W.,D.H.K.Tsang.1989.Thestochasticknapsackproblems.IEEETransactionsonCommunications37740{747. Shen,Z-J.M.2005.Amulti-commoditysupplychaindesignproblem.IIETransactions37753{762. Shen,Z-J.M.,D.Coullard,M.S.Daskin.2003.Ajointlocation-inventorymodel.Trans-portationScience37(1)40{55. 116

PAGE 117

Shu,J.,C-P.Teo,Z-J.M.Shen.2005.Stochastictransportation-inventorynetworkdesignproblem.OperationsResearch53(1)48{60. Simon,J.L.,A.J.Arndt.1980.Theshapeoftheadvertisingresponsefunction.JournalofAdvertisingResearch20(4)11{27. Smith,S.A.,N.Agrawal.2000.Managementofmulti-itemretailinventorysystemwithdemandsubstitution.OperationsResearch48(1)50{64. Snyder,L.V.2006.Facilitylocationunderuncertainty:Areview.IIETransactions38(7)547{564. Snyder,L.V.,M.S.Daskin,C-P.Teo.2007.Thestochasticlocationmodelwithriskpooling.EuropeanJournalofOperationalResearch1791221{1238. Sourirajan,K.,L.Ozsen,R.Uzsoy.2007.Asingle-productnetworkdesignmodelwithleadtimeandsafetystockconsiderations.IIETransactions39411{424. Steinberg,E.,M.S.Parks.1979.Apreferenceorderdynamicprogramforaknapsackproblemwithstochasticrewards.JournaloftheOperationalResearchSociety30141{147. Sun,X.L.,F.L.Wang,D.Li.2005.Exactalgorithmforconcaveknapsackproblems:Linearunderestimationandpartitionmethod.JournalofGlobalOptimization3315{30. Taskn,Z.C.,A.T.Unal.2009.Tacticallevelplanninginoatglassmanufacturingwithco-production,randomyieldsandsubstitutableproducts.EuropeanJournalofOperationalResearch199252{261. Tuy,H.2000.Monotonicoptimization:Problemsandsolutionapproaches.SIAMJournalonOptimization11(2)464{494. VanHoesel,C.P.M.,A.P.M.Wagelmans.2001.Fullypolynomialapproximationschemesforsingle-itemcapacitatedeconomiclot-sizingproblems.MathematicsofOperationsResearch26(2)339{357. Vidyarthi,N.,E.Celebi,S.Elhedhli,E.Jewkes.2007.Integratedproduction-inventory-distributionsystemdesignwithriskpooling:Modelformulationandheuristicsolution.TransportationScience41(3)392{408. Wagelmans,A.,S.V.Hoesel,A.Kolen.1992.Economiclotsizing:Ano(nlogn)algorithmthatrunsinlineartimeinthewagner-whitincase.OperationsResearch40(1)145{156. Yunes,T.H.,D.Napolitano,A.Scheller-Wolf,S.Tayur.2007.Buildingecientproductportfoliosatjohndeereandcompany.OperationsResearch55(4)615{629. 117

PAGE 118

Zoltners,A.A.,P.Sinha.1980.Integerprogrammingmodelsforsalesresourceallocation.ManagementScience26(3)242{260. Zoltners,A.A.,P.Sinha,P.S.C.Chong.1979.Anoptimalalgorithmforsalesrepresentativetimemanagement.ManagementScience25(12)1197{1207. 118

PAGE 119

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