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Resource Constrained Assignment Problems with Flexible Customer Demand

Permanent Link: http://ufdc.ufl.edu/UFE0024847/00001

Material Information

Title: Resource Constrained Assignment Problems with Flexible Customer Demand
Physical Description: 1 online resource (198 p.)
Language: english
Creator: Rainwater, Chase
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: 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 considers classes of problems that seek to make profitable demand fulfillment decisions with limited available resources. This general problem scenario has been given much consideration over recent decades. In this work, we add to this body of research by considering less explored problem variants that allow decision makers to exploit demand flexibility to increase profit. We first consider a generalization of the capacitated facility location with single-sourcing constraints. Each customer must be assigned to a procured facility, and the level at which the customer's demand is fulfilled (a decision variable) must be determined, subject to falling within pre-specified limits. A customer's revenue is nondecreasing in its resource consumption, according to a general revenue function, and a fixed cost is incurred for each resource procured. We provide an exact branch-and-price algorithm that solves both this problem and a special case in which resource procurement is not considered. Our approach identifies an equally interesting class of pricing subproblems. We discuss how this class of problems can be solved with generalized revenue functions and offer efficient algorithms for solving instances with specially structured revenue functions that correspond to common pricing structures. Our extensive computational study compares the performance of our exact algorithm to that of well-known commercial solvers and demonstrates the advantages of our algorithmic approach for various categories of problem instances. Since real-world scenarios often result in large-scale problem sizes, we consider novel heuristic approaches for both the generalization of the capacitated facility location problem and a particular special case, which can be viewed as an extension of the well-known Generalized Assignment Problem (GAP). We first develop a class of heuristic solution methods for the variant without resource procurement decisions. Our approach is motivated by a rigorous study of the linear relaxation of the model. We show that our class of heuristics is asymptotic optimality in a probabilistic sense under a broad stochastic model. Improvement procedures are discussed and a thorough computational study confirms our theoretical results. We then provide fast and practically implementable optimization-based heuristic solution methods for the generalized class of facility location problems with resource procurement decisiosn. Our procedure is designed for very large-scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems. Finally, we consider a separate class of assignment problems with non-linear resource consumption and non-traditional capacity constraints. The model is applicable to manufacturing scenarios in which products with common production characteristics share setup times or some element of fixed resource consumption. The additional capacity constraints account for real-world restrictions that may result from environmental guidelines, transportation resource limitations, or limited warehouse storage space. We propose a branch-and-price algorithm for this class of problems that requires a unique reformulation of our problem, as well as a study of a new class of knapsack problems. A computational study demonstrates the appeal of our approach over commercial solvers for various problem instances.
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 Chase Rainwater.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Geunes, Joseph P.
Local: Co-adviser: Romeijn, Hilbrand E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024847:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024847/00001

Material Information

Title: Resource Constrained Assignment Problems with Flexible Customer Demand
Physical Description: 1 online resource (198 p.)
Language: english
Creator: Rainwater, Chase
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: 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 considers classes of problems that seek to make profitable demand fulfillment decisions with limited available resources. This general problem scenario has been given much consideration over recent decades. In this work, we add to this body of research by considering less explored problem variants that allow decision makers to exploit demand flexibility to increase profit. We first consider a generalization of the capacitated facility location with single-sourcing constraints. Each customer must be assigned to a procured facility, and the level at which the customer's demand is fulfilled (a decision variable) must be determined, subject to falling within pre-specified limits. A customer's revenue is nondecreasing in its resource consumption, according to a general revenue function, and a fixed cost is incurred for each resource procured. We provide an exact branch-and-price algorithm that solves both this problem and a special case in which resource procurement is not considered. Our approach identifies an equally interesting class of pricing subproblems. We discuss how this class of problems can be solved with generalized revenue functions and offer efficient algorithms for solving instances with specially structured revenue functions that correspond to common pricing structures. Our extensive computational study compares the performance of our exact algorithm to that of well-known commercial solvers and demonstrates the advantages of our algorithmic approach for various categories of problem instances. Since real-world scenarios often result in large-scale problem sizes, we consider novel heuristic approaches for both the generalization of the capacitated facility location problem and a particular special case, which can be viewed as an extension of the well-known Generalized Assignment Problem (GAP). We first develop a class of heuristic solution methods for the variant without resource procurement decisions. Our approach is motivated by a rigorous study of the linear relaxation of the model. We show that our class of heuristics is asymptotic optimality in a probabilistic sense under a broad stochastic model. Improvement procedures are discussed and a thorough computational study confirms our theoretical results. We then provide fast and practically implementable optimization-based heuristic solution methods for the generalized class of facility location problems with resource procurement decisiosn. Our procedure is designed for very large-scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems. Finally, we consider a separate class of assignment problems with non-linear resource consumption and non-traditional capacity constraints. The model is applicable to manufacturing scenarios in which products with common production characteristics share setup times or some element of fixed resource consumption. The additional capacity constraints account for real-world restrictions that may result from environmental guidelines, transportation resource limitations, or limited warehouse storage space. We propose a branch-and-price algorithm for this class of problems that requires a unique reformulation of our problem, as well as a study of a new class of knapsack problems. A computational study demonstrates the appeal of our approach over commercial solvers for various problem instances.
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 Chase Rainwater.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Geunes, Joseph P.
Local: Co-adviser: Romeijn, Hilbrand E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024847:00001


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RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHFLEXIBLE CUSTOMERDEMAND By CHASEE.RAINWATER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c 2009ChaseE.Rainwater 2

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Tomywife,myparentsandmybrother 3

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ACKNOWLEDGMENTS First,tomybeautifulwife.Thisdissertationrepresentsanothermilestoneinour adventuretogether.Forseeingmethroughthefrustrationsandtrialsthatundoubtedly accompaniedthedevelopmentofthisdissertation,Iamforevergrateful.Havingan unwaveringsourceofhappinesstogohometoisthesecrettomysuccess.Tothatend,I shareanysuccessthatI'vehadwithyou. Tomyfamily,yourloveandsupporthavebeeninvaluable.Specicallytomyparents, thankyouforallyoudidtoprepareforthisexperience. ToJoeGeunesandEdwinRomeijn,Icannotbegintodescribetheimpactyouhave hadonmeacademically,professionallyandpersonally.Thechancetoworkforyouranks asoneofthegreatestprivilegesofmylife.Iowetheopportunitytopursueanacademic careertothetwoofyou.IonlyhopethatIcanbehalfthementortoothers,asyou've beentome. ToColeSmith,Ithankyouforbeingsuchanunbelievableteacherandmentor throughoutgraduateschool.Also,thankyouforthedog. Lastly,toCaner,Semraandtherestofmygraduatestudentcolleagues,thankyoufor yourfriendshipthroughoutthesefouryears.Iremainamazednotonlybyyourtalentsand abilities,but,moreimportantly,bythesincerekindnessyou'veconsistentlyshownme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................8 LISTOFFIGURES....................................11 ABSTRACT........................................12 CHAPTER 1INTRODUCTION..................................14 2LITERATUREREVIEW..............................19 2.1CapacitatedFacilityLocationProblemwithSingle-SourceConstraints...19 2.2GeneralizedAssignmentProblem.......................22 2.30-1KnapsackProblem.............................26 2.4FlexibleDemandAssignmentProblems....................28 3CAPACITATEDFACILITYLOCATIONWITHSINGLE-SOURCECONSTRAINTS ANDFLEXIBLEDEMAND.............................30 4EXACTALGORITHMFORCFLFDANDGAPFD...............35 4.1AlternativeRepresentationoftheCFLFD..................36 4.1.1Set-PartitioningFormulation......................36 4.1.2PricingProblem.............................38 4.2KnapsackProblemwithExpandableItems..................39 4.2.1CFLFDwithSpeciallyStructuredRevenueFunctions........43 4.2.2ConvexandLinearRevenueFunctions................44 4.3Branch-and-PriceAlgorithmImplementation.................48 4.3.1InitialFeasibleSolution.........................48 4.3.2SolvingLPRSP............................49 4.3.3NodeandVariableSelection......................51 4.3.4OptimalColumnCost..........................52 4.4ComputationalResults.............................53 4.4.1ExperimentalData...........................53 4.4.2CFLFDResults.............................56 4.4.2.1Nonlinearrevenuefunctions:comparisonwithBARON..56 4.4.2.2Piecewiselinearandlinearrevenuefunctions:comparison withCPLEX.........................58 4.4.3GAPFJResults.............................59 4.5Conclusions...................................61 5

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5GAPFDHEURISTICWITHASYMPTOTICPERFORMANCEGUARANTEES71 5.1ModelAnalysis.................................72 5.2AnAsymptoticallyOptimalHeuristic.....................80 5.2.1DevelopmentoftheHeuristic......................80 5.2.2AverageCaseAnalysisoftheHeuristic................85 5.2.2.1Facility-independentrequirements..............88 5.2.2.2Facility-dependentrequirements...............91 5.2.3ModelExtension............................94 5.3HeuristicImprovementIssues.........................94 5.3.1SolutionImprovement..........................94 5.3.1.1Improvementphase......................95 5.3.1.2Post-processingphase....................95 5.3.2CapacityPerturbationScheme.....................96 5.4ComputationalResults.............................97 5.4.1ExperimentalDesign..........................97 5.4.2Facility-IndependentRequirements..................99 5.4.3Facility-DependentRequirements...................101 5.4.4EectofPost-ProcessingPhase....................102 5.5Conclusions...................................103 6LARGE-SCALEMULTI-EXCHANGEHEURISTICFORFIXED-CHARGE RESOURCECONSTRAINEDASSIGNMENTPROBLEMS...........108 6.1OptimizationandModelFormulation.....................111 6.2HeuristicFramework..............................111 6.2.1FacilityNeighborhoodSearch.....................112 6.2.2Single-CustomerVLSN.........................115 6.3SearchHeuristicImplementation.......................117 6.3.1InitialFeasibleSolution.........................118 6.3.2FSMoveChoice.............................119 6.4ComputationalStudy..............................123 6.4.1ExperimentalData...........................123 6.4.2Results..................................125 6.5CFLFDHeuristicApplicationsandConclusions...............127 7RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHSHARED RESOURCECONSUMPTION...........................131 7.1Introduction...................................131 7.2ModelFormulation...............................134 7.3ExactAlgorithmforRCAS...........................137 7.4SharedConsumptionKnapsackProblem...................139 7.5FlexibleCustomerDemandGeneralization..................150 7.6SharedConsumptionKnapsackproblemwithFlexibleCustomerDemand.157 7.7Branch-and-PriceAlgorithmImplementation.................164 6

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7.7.1InitialFeasibleSolution.........................164 7.7.2HeuristicsforPP i -F...........................165 7.7.3SolvingLPRMP-F...........................167 7.7.4NodeandVariableSelection......................168 7.8Computationalstudy..............................169 7.8.1ExperimentalDesign..........................169 7.8.2BaseResults...............................171 7.8.3ExtendedResults............................173 7.9ConclusionsandFutureResearch.......................175 8CONCLUSION....................................184 APPENDIX AGAPFDASYMPTOTICPROPERTY.......................187 BCFLFDPRICINGPROBLEMPROPERTY....................189 REFERENCES.......................................191 BIOGRAPHICALSKETCH................................198 7

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LISTOFTABLES Table page 4-1CFLFDwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2..61 4-2CFLFDwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2..63 4-3CFLFDwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2..64 4-4BranchingrulecomparisonfortheCFLFDwithnonlinearrevenuefunctions:5 facilities,10customers, =1 : 2...........................64 4-5CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,10customers, =1 : 2.........................................65 4-6CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,15customers, =1 : 2.........................................65 4-7CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,25customers, =1 : 2.........................................66 4-8CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,10customers, =1 : 2.........................................66 4-9CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,15customers, =1 : 2.........................................67 4-10CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,25customers, =1 : 2.........................................67 4-11CFLFDwithlinearrevenuefunctions:30facilities,60customers, =1 : 2....68 4-12GAPFJwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2..68 4-13GAPFJwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2..69 4-14GAPFJwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2..69 4-15GAPFJwithlinearrevenuefunctions:30facilities,90customers, =1 : 2....70 5-1Facility-independentrequirements:15facilities, =1 : 1..............105 5-2Facility-independentrequirements:15facilities, =1 : 2..............105 5-3Facility-independentrequirements:15facilities =1 : 3..............105 5-4Facility-independentrequirements:30facilities, =1 : 1..............105 5-5Facility-independentrequirements:30facilities, =1 : 2..............105 5-6Facility-independentrequirements:30facilities, =1 : 3..............106 8

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5-7Facility-dependentrequirements:15facilities, =1 : 1...............106 5-8Facility-dependentrequirements:15facilities, =1 : 2...............106 5-9Facility-dependentrequirements:15facilities, =1 : 3...............106 5-10Facility-dependentrequirements:30facilities, =1 : 1...............106 5-11Facility-dependentrequirements:30facilities, =1 : 2...............107 5-12Facility-dependentrequirements:30facilities, =1 : 3...............107 5-13Post-processingeectonheuristicwithgreedyandimprovementphase;facility-independent requirements: =1 : 2.................................107 5-14Post-processingeect;facility-dependentrequirements: =1 : 2..........107 6-1FNSresults:15facilities...............................129 6-2RNSresults:15facilities...............................129 6-3FNSresults:30facilities...............................129 6-4RNSresults:30facilities...............................130 7-1FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................178 7-2FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................178 7-3FASR:30facilities,60customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................179 7-4FASR:30facilities,90customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................179 7-5FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=25.......................................180 7-6FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.2 i 2 I ,=5.......................................180 7-7FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.2 i 2 I ,=5.......................................181 7-8FASR:15facilities,75customers,3customertypes, a = t =1 : 2, i =.9 i 2 I ,=5.......................................181 7-9FASR:15facilities,30customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5.....................................182 9

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7-10FASR:15facilities,45customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5.....................................182 7-11FASR:15facilities,30customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q ................................183 7-12FASR:15facilities,45customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q ................................183 10

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LISTOFFIGURES Figure page 4-1Illustrationof j and j forageneralrevenuefunction..............62 4-2Concaveenvelope:convexrevenuefunction.....................62 4-3Concaveenvelope:linearrevenuefunction.....................63 7-1Illustrationof r q and q ...............................177 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHFLEXIBLE CUSTOMERDEMAND By ChaseE.Rainwater August2009 Chair:JosephP.Geunes Cochair:H.EdwinRomeijn Major:IndustrialandSystemsEngineering Thisdissertationconsidersclassesofproblemsthatseektomakeprotabledemand fulllmentdecisionswithlimitedavailableresources.Thisgeneralproblemscenariohas beengivenmuchconsiderationoverrecentdecades.Inthiswork,weaddtothisbody ofresearchbyconsideringlessexploredproblemvariantsthatallowdecisionmakersto exploitdemandexibilitytoincreaseprot.Werstconsiderageneralizationofthe capacitatedfacilitylocationwithsingle-sourcingconstraints.Eachcustomermustbe assignedtoaprocuredfacility,andthelevelatwhichthecustomer'sdemandisfullled adecisionvariablemustbedetermined,subjecttofallingwithinpre-speciedlimits.A customer'srevenueisnondecreasinginitsresourceconsumption,accordingtoageneral revenuefunction,andaxedcostisincurredforeachresourceprocured.Weprovide anexactbranch-and-pricealgorithmthatsolvesboththisproblemandaspecialcase inwhichresourceprocurementisnotconsidered.Ourapproachidentiesanequally interestingclassofpricingsubproblems.Wediscusshowthisclassofproblemscan besolvedwithgeneralizedrevenuefunctionsandoerecientalgorithmsforsolving instanceswithspeciallystructuredrevenuefunctionsthatcorrespondtocommon pricingstructures.Ourextensivecomputationalstudycomparestheperformanceof ourexactalgorithmtothatofwell-knowncommercialsolversanddemonstratesthe advantagesofouralgorithmicapproachforvariouscategoriesofprobleminstances.Since 12

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real-worldscenariosoftenresultinlarge-scaleproblemsizes,weconsidernovelheuristic approachesforboththegeneralizationofthecapacitatedfacilitylocationproblemanda particularspecialcase,whichcanbeviewedasanextensionofthewell-knownGeneralized AssignmentProblemGAP.Werstdevelopaclassofheuristicsolutionmethods forthevariantwithoutresourceprocurementdecisions.Ourapproachismotivated byarigorousstudyofthelinearrelaxationofthemodel.Weshowthatourclassof heuristicsisasymptoticoptimalityinaprobabilisticsenseunderabroadstochasticmodel. Improvementproceduresarediscussedandathoroughcomputationalstudyconrmsour theoreticalresults.Wethenprovidefastandpracticallyimplementableoptimization-based heuristicsolutionmethodsforthegeneralizedclassoffacilitylocationproblemswith resourceprocurementdecisiosn.Ourprocedureisdesignedforverylarge-scaleproblem instances.Weoerauniqueapproachthatutilizesahigh-qualityecientheuristicwithin aneighborhoodsearchtoaddressthecombinedassignmentandxed-chargestructureof theunderlyingoptimizationproblem.Wealsostudythepotentialbenetsofcombining ourapproachwithaso-calledverylarge-scaleneighborhoodsearchVLSNmethod.As ourcomputationaltestresultsindicate,ourworkoersanattractivesolutionapproach thatcanbetailoredtosuccessfullysolveabroadclassofprobleminstancesforfacility locationandsimilarxed-chargeproblems.Finally,weconsideraseparateclassof assignmentproblemswithnon-linearresourceconsumptionandnon-traditionalcapacity constraints.Themodelisapplicabletomanufacturingscenariosinwhichproductswith commonproductioncharacteristicssharesetuptimesorsomeelementofxedresource consumption.Theadditionalcapacityconstraintsaccountforreal-worldrestrictions thatmayresultfromenvironmentalguidelines,transportationresourcelimitations,or limitedwarehousestoragespace.Weproposeabranch-and-pricealgorithmforthisclass ofproblemsthatrequiresauniquereformulationofourproblem,aswellasastudyofa newclassofknapsackproblems.Acomputationalstudydemonstratestheappealofour approachovercommercialsolversforvariousprobleminstances. 13

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CHAPTER1 INTRODUCTION Optimizationmodelswhichdeterminethemostprotablemannertofulllcustomer demandhavebeenwidelyexploredintheoperationsresearchliteratureforover50years. Manyoftheclassicalproblemsstudiedinoureld,suchasthe AssignmentProblem Kuhn[55],the GeneralizedAssignmentproblem RossandSoland[84],the CapacitatedFacilityLocationProblem Nauss[71],the TravelingSalesmanProblem Lin andKernighan[61],the VehicleRoutingProblem Laporte[58],andthenumerous Fixed-ChargeTransportationProblems AdlakhaandKowalski[1]haveconsideredthe assignmentofcustomerstoresourcesundervaryingreal-worldscenarios.Morerecently, theoperationsliteraturehasemphasizedwaystoexploitsourcesofsupplyanddemand exibilitytoincreaseprotmarginsthroughdemandandrevenuemanagementsee,e.g., TalluriandVanRyzin[94].Thishasledtoanumberofnewmodelsthatfocusonprot maximizationbyaccountingforboththecostsandrevenueimplicationsassociatedwith operationsdecisions.Forexample,ChenandHall[22]introduceseveralnewmaximum protscheduling"modelsthatimplicitlyaccountforthefactthatoperationsscheduling decisionscanaectdemandandthereforerevenue.Anotherstreamofliteratureconsiders optimalinventorymanagementwhendemandlevelsandhencerevenuesdependon inventorylevelse.g.,BakerandUrban[11,12],GerchakandWang[43],andBalakrishnan etal.[14]and/orshelfspaceallocationWangandGerchak[99],bothofwhichimpact operationscosts.Severalpapershavealsoconsideredmaximizingprotinproduction planningcontextswithprice-dependentdemand,whereproductionandinventorycosts aredeterminedbysolvinganoptimizationproblemcontainingalot-sizingstructuree.g., Thomas[95],KunreutherandSchrage[56],Gilbert[46],Billeretal.[18],DengandYano [28],Geunes,Romeijn,andTaae[44],andvandenHeuvelandWagelmans[98]. Theworkinthisdissertationaddstothisbroadbodyofresearch.Inthemostgeneral terms,weconsiderasetofcustomers,eachwiththeirowndemandrequirements,and 14

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asetofresourceconstrainedfacilities.Ourmodelsseektoassigneachcustomertoa singlecapacitatedfacilityresourceinamannerwhichmaximizestotalprot.Aswe willdiscussinChapter2,thisstandardresourceconstrainedassignmentproblemisitself diculttosolve,andnumerousexactandheuristicsolutionmethodologieshavealready beenproposed.Theworkinthisdissertationexploresvariantsofthistraditionalproblem whichaccountforuniqueplanningcharacteristicsthatmaybeavailabletoadecision maker.Specically,werstconsideranextensioninwhichthemodelallowsforso-called exiblecustomerdemand.Thisexibilityallowsasuppliertobettermatchcustomer demandswithoperationsresources,thereforeincreasingprots.Real-worldscenariosthat allowfordemandexibilityareoftenfoundintheproductionofconstructionmaterials, suchassteelandwood.Intheseenvironments,distributorsofthesematerialswill acceptdeliveriesfromsuppliersinarangeofsizesBalakrishnanandGeunes[13].The distributorspermitthisexibilitybecausetheyoftenperformfurthercustomizedcutting andnishingoperationsfortheirowncustomers,whoseexactsizespecicationsarenot knowntothedistributorinadvance.Supplierstosuchdistributorsareoftencompensated basedontotalweightdeliveredtothedistributorwithincertainlimitsdeemedacceptable tothedistributor.Clearly,ifthesupplierhasunlimitedresources,theycanmaximize protbydeliveringattheupperlimitofthedistributor'sstatedacceptablesizerange.If, however,thesupplierfacesresourceconstraintse.g.,intermsofitsquantityandsizes ofrawmaterialsandmustmeeteachelementofacollectionofcustomerdemands,the problemofassigningthesedemandstoavailableresourcesinordertomaximizenetprot isnon-trivial.Therstmodelconsideredinthisdissertationrequiresthatthefacilities utilizedtofullldemandmustbedeterminedbythedecisionsmaker.Werefertothis problemasthe CapacitatedFacilityLocationProblemwithSingle-SourceConstraintsand FlexibleDemand CFLFD.Inaddition,westudythespecialcaseofCFLFDinwhich demandisfullledbyaxedsetoffacilitiesi.e.procurementdecisionsareomitted. 15

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Werefertothisproblemasthe GeneralizedAssignmentProblemwithFlexibleDemand GAPFD.Foreachproblem,weproposebothexactandheuristicmethodologies. TheexactapproachutilizedfortheCFLFDandtheGAPFDreformulatesthe problemasaset-partitioningproblem.Theresultingsubproblemtobesolvedtakesthe formofaninterestingclassofnon-linearknapsackproblems.Wederivestructuralresults foranimportantrelaxationofthisclassofknapsackproblems.Theseresultssuggest ecientheuristicandexactapproachesforsolvingtheseknapsackproblemswithvarious revenuefunctions.Byconsideringthesealternativerevenuestructures,ourmodelaccounts forquantitydiscountsandeconomiesofscale,aswellasrevenuesforspecializedgoods. Lastly,weprovideadetaileddiscussionofhowthecustomerdemandfulllmentlevelsare determinedinthisapproach,aswellashowdicultiesthatarisewhenconsideringfacility procurementcanbeovercome. Solvinglarge-scaleinstancesofassignment-basedproblemsisanissuethatisactively consideredintheliterature.Therefore,aportionofthisdissertationfocusesonheuristic proceduresthatcanbeusedtosolvereal-wordsizeinstances.Whiletheexactapproach fortheexibledemandproblemstudiedisapplicabletoproblemswithandwithout facilityprocurement,ourheuristicapproachesrequireseparateconsideration.Forthe GAPFD,wepursueagreedyalgorithmthatismotivatedbyananalysisofthelinear relaxationofourmodel.Importantly,theheuristicpresentedisshowntohaveasymptotic performanceguaranteesunderaverygeneralstochasticmodel.Forverylargeproblems, thecomputationalresultssuggestthattheheuristicproducesnear-optimalsolutionsin dramaticallyreducedtimewhencomparedtothatrequiredbywell-knowncommercial solvers. Aswediscussindetailthroughoutthedissertation,xed-chargeheuristicsrequire carefuldesigntoassurethatthexed-chargedecisionsarefullyconsidered.Therefore, fortheCFLFD,wedevelopaheuristicframeworkthatsearchesaseparatefacility neighborhood,whilerelyingonconstructiveheuristicstodeterminethecustomer 16

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assignmentsanddemandfulllmentlevels.Wepresentthecomputationalsavingsof thisapproachoverotherlarge-scalesearchheuristicsanddiscussthepotentialimpacts thattheframeworkmighthaveonanumberofotherxed-chargeproblems. Clearlytheconsiderationofexibledemandhasnotablepracticalsignicance. Moreover,solvingproblemswiththiselementisachallengethatrequiresnovelsolution approaches.However,inreal-worldproductionenvironments,thetotalresourcesrequired tosatisfycustomerdemandexible,orotherwisemaynotbelimitedtothecumulative capacityconsumedbytheindividualcustomers.Aswewilldiscuss,moreoften,certain productionrequirementsaresharedamongstsubsetsofcustomers.Tomodelthis,weallow foreachcustomertobelongtoasingletype.Then,customersofeachtypeconsumea sharedamountofresourceinadditiontotheindividualconsumptionrequiredtofulll demand.Inadditiontothisconsideration,wearealsointerestedindierentformsof capacityconstraints.Typically,capacitatedassignmentproblemsconsideronlythe capacitylimitationsofindividualfacilities.Aswewillestablishinthereviewofliterature inthisarea,assignmentproblemswithcapacityrestrictionsoncustomersofaparticular typeregardlessofwhichfacilitytheyareassignedhavereceivedmuchlessattention. Accountingforbothoftheseproblemelementsrequiresaseparatemodelfromthose thathavebeenpreviouslydeveloped.Whileassignmentproblemswiththeseadditional characteristicscanstillbeviewedasdeterminingafeasiblepartitionofcustomersamongst asetofavailableresourcesfacilities,theimpactoftheadditionalcapacityconstraints requiresspecialconsiderationintheexactalgorithmthatisproposed.Theresultis aneectivealgorithmforanotherdicultclassofoptimizationproblemswithstrong practicalimplications. Theremainderofthedissertationisorganizedasfollows.InChapter2,wediscuss extensivelytherelevantliterature.InChapter3,weformallypresenttherstclassof problemstobestudiedandintroducetheconceptofexibledemand.Chapter4discusses anexactapproachfortheCFLFDandtheGAFPD.Anewclassofknapsackproblems 17

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thatariseinourapproachispresented.Bothaformalstudyofthestructureofthese problemsandecientsolutionproceduresareprovided.InChapter5,aclassofheuristics isproposedfortheGAPFD.Aprobabilisticanalysisdemonstratesthatourclassof heuristicsisasymptoticallyfeasibleandoptimalunderaverygeneralstochasticmodel. Athoroughcomputationalstudytoassesstheheuristic'sperformanceiscompletedand comparedagainstthetheoreticalclaimsmadeinthechapter.InChapter6,aheuristic frameworkfortheCFLFDwithlinearrevenuefunctionsispresented.Wedevelopa neighborhoodsearchheuristicthatdecomposestheCFLFDandsolvesthecorresponding subproblemwiththeheuristicproposedinChapter5.Wediscussthebenetsofour approachversusthosecommonlyappliedforxed-chargeoptimizationproblems.The motivationbehindthechosenimplementationisprovidedalongwithlessonslearned withrespecttolesssuccessfulimplementations.Then,inChapter7,anewclassof problemsisintroducedwhichaccountsforsharedresourceconsumptionamongsetsof customers,aswellasadditionalcapacityconstraints.Anexactapproachisproposedbased onaset-partitioningformulationofthemodelwithcomplicatingcapacityconstraints. Importantdierencesbetweenthisreformulationandthereformulationpresentedin Chapter4arediscussed.Anotherinterestingclassofknapsackproblemsisstudiedand ecientsolutionapproachespresented.Finally,Chapter8oersconcludingremarks,as wellasadiscussionoffutureresearch. 18

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CHAPTER2 LITERATUREREVIEW Theproblemsstudiedinthisdissertationsharecharacteristicswithnumerousclassical optimizationproblems.Thischapterprovidesanextensivereviewoftheliteraturerelated toeachoftheseproblems.Thediscussionofworkineachareafocusesontheexactand heuristicmethodsthathavebeenpursued.Thepresentationofthesesuccessfulapproaches willservetomotivateeachofthesolutionmethodschosenthroughouttheremainderof thedissertation. 2.1CapacitatedFacilityLocationProblemwithSingle-SourceConstraints Theproblemsstudiedinthisdissertationcanbeviewedasgeneralizationsof well-knownoptimizationproblems.Ofparticularrelevancetotheproblemsintroduced inChapter3andstudiedinChapters4and6isthe CapacitatedFacilityLocationProblem withSingle-SourceConstraints CFLP. maximize X i 2I X j 2J p ij x ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X i 2I f i y i {1 subjecttoCFLP X j 2J a ij x ij b i y i i 2I {2 X i 2I x ij =1 j 2J {3 x ij 2f 0 ; 1 g i 2I ; j 2J {4 y i 2f 0 ; 1 g i 2I : {5 Inthismodel,demandforcustomer j j 2J mustbesatisedbyasingleprocured capacitatedfacility i i 2I ,asenforcedinconstraints2{3.Acustomer j j 2J assignedtofacility i 2I ,resultsinaprotamount p ij andconsumes a ij unitsof facility i 'scapacity.Constraints2{2ensurethatcustomerdemandisexecutedsolely byprocuredfacilitiesandthateachfacility'sresourceavailabilityissatised.Constraints 19

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2{4and2{5placebinaryrestrictionsontheassignmentvariables x ij i 2I ; j 2J andfacilityprocurementvariables y i i 2I TheCFLPwithsingle-sourceconstraintsfallsintothedicultclassof NP -Hard optimizationproblemsseeGareyandJohnson[39],implyingthatitisunlikelythat apolynomial-timesolutionmethodexistsforsolvingproblemsinthisclassunless P = NP .NotethatthisnegativecomplexityresultevenholdsfortheclassicalCFLP inwhicheachcustomer'sdemandmaybesplitbetweentheacquiredfacilitiesaslong asallcustomerdemandsareallocated.Inpractice,asingle-sourcingrestriction,which requiresthatanycustomer'sdemandmustbeallocatedinitsentiretytoexactlyone oftheopenfacilities,isoftenimposedforavarietyofreasons.Forexample,inthe facilitylocationcontext,singlesourcingreducescoordinationcomplexity,reducesthe numberofdeliveriesrequiredtoacustomer,ensuresconsistencyofdeliveriesreceivedby customers,andprovidescustomerswithasinglepointofcontactforsupply.Moreover,a customer'sdemandmaydependonthefacilitytowhichitisassigned.Thisgeneralization isparticularlyrelevantifthefacilities"representmachinesorpeople,eachwithdierent processingcapabilities,andisthereforeoftenfoundinproductionenvironments.Withthe single-sourcingconstraintandfacility-dependentdemandsweobtainaproblemthatis,in general,atleastasdicultasthecaseinwhichacustomer'sdemandmaybesplit,and whichcontainsamoresubstantialadditionalcombinatorialcomponent. EarlyexactalgorithmsforCFLPfocusonbranch-and-boundapproaches.Many eortsutilizedlinearrelaxationsofCFLPtoobtainupperboundsseeSa[87]andAkinc andKumawala[8].DavisandRay[25]improveduponthisworkbygeneratingupper boundsfromrelaxationswiththeadditionalconstraints y i x ij i 2I ; j 2J : {6 Cornueojolsetal.[23]formallyshowedthatboundsobtainedusingthisalternative relaxationarestrongerthanthoseobtainedfromthestandardLPrelaxationofCFLP. 20

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AseparatebodyofresearchutilizedLagrangianrelaxationtoobtainupperbounds. GeorionandMcBride[42],Nauss[71]andCornuejolsetal.[23]allstudiedLagrangian relaxationsinwhichsingle-sourceconstraints2{3arerelaxed.VanRoy[86]and Cornuejolsetal.[23]consideredthealternativeLagrangianrelaxationthatrelaxesthe capacityconstraints2{2.Interestingly,Cornueojolsetal.showedthatthebound obtainedbyrelaxingthecapacityconstraintsisstrongerthanthatobtainedbyrelaxing theassignmentconstraints.Numerousadditionalmethodshavebeenproposedtosolve CFLPtooptimality.TheseincludeGeorionandGraves'[41]implementationofBenders decompositionandErlenkotter's[32]studyofadualascentmethod.Morerecently, Hombergetal.[49]proposedanexactmethodologythatsuccessfullycombinesLagrangian heuristicswitharepeatedmatchingalgorithmtoproducehighqualitysolutionswhile usingLagrangianrelaxationsintheboundingprocedureateachnode.Lastly,Neebeand Rao[72]consideredabranch-and-priceapproachwhichiscommonforassignment-based problemsandisdiscussedingreaterdetailinSection2.2.Theynotethedicultyof solvingproblemswithlargexed-chargesusingthisapproach,whichisachallengethatwe specicallyconfrontinChapter4. NumerousheuristicsfortheCFLPexist.Delmaireetal.[26]consideredawide assortmentofheuristicapproaches,includingevolutionaryalgorithms,tabusearchTS, simulatedannealing,andagreedyrandomizedadaptivesearchprocedureGRASP. Later,Delmaireetal.[27]improvedonthepromisingTSandGRASPheuristicsand proposeddierenthybridizationschemesthatcombinedthesetwoprocedures,yielding qualityresultsthatrequireonlyasmallamountoftime.Anevengreaternumber ofLagrangian-basedheuristicshavebeenproposedfortheCFLPe.g.,Barceloand Casanovas[15],HindiandPienkosz[48],KlincewiczandLuss[53],andBeasley[17]. Theseseparateeortsconsiderdierentrelaxationalternativesforboundingpurposes andoeruniquetechniquesforgeneratingfeasiblesolutions.Ofspecicrelevancetothe methodologyproposedinthiswork,BarceloandCasanova[15]proposedamulti-stage 21

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procedurethatusesdualinformationfromthelinearrelaxationoftheCFLPtoselect asetofopenplantsbeforeproceedingtoapenalty-basedreassignmentprocedurethat, ineect,solvesaspecializedinstanceofthegeneralizedassignmentproblem.Aswill beevidentthroughoutChapter6,separatingtheresourceprocurementanddemand assignmentdecisionsintoindividualphasescanleadtoasuccessfulheuristicframework. Morerecently,Ahujaetal.[6]proposedasearchheuristicwhichexploredaverylarge solutionspacemadepossiblebytheconsiderationofmultipleneighborhoods.Twoof theseneighborhoodsarerepresentedintheformofagraphandsubsequentlysolvedvia a VeryLarge-ScaleNeighborhood VLSNsearchprocedure.VLSNisasearchheuristic procedureshowntobeextremelysuccessfulonproblemswithanassignmentstructure. Forexample,theQuadraticAssignmentProblemAhujaet.al[4],theFleet-Assignment ProblemAhujaet.al[2]andtheVehicleRoutingProblemErgun[31]haveallbeen solvedviaVLSNoverthelastdecade.ForCFLP,Ahujaetal.[6]showedthattheir approachsolved63outof71testedinstancestooptimalityinlessthanone-minuteusing CPLEXtocertifyoptimality.Thesetestsincludedproblemswithupto30facilitiesand 200customers.ThesuccessofthisworkmotivatestheheuristicproposedinChapter6. Therefore,inthatchapteramorespecicdescriptionofAhujaetal.'s[6]approachis provided. 2.2GeneralizedAssignmentProblem InChapters4{5weproposeexactandheuristicapproachesforanextensionofthe GeneralizedAssignmentProblem GAP.TheGAPisaspecialcaseofCFLP,inwhich facilityprocurementisnotconsidered.Thatis,customerdemandisfullledbyaxedset ofcapacitatedresourcesfacilities.TheGAPcanberepresentedas maximize X i 2I X j 2J p ij x ij {7 22

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subjecttoGAP X j 2J a ij x ij b i i 2I {8 X i 2I x ij =1 j 2J {9 x ij 2f 0 ; 1 g i 2I ; j 2J : {10 withparametersandvariablesdenedinthesamemannerastheCFLP.Ofcourse,the GAPis NP -Hard,asshownbyFisheretal.[35].Furthermore,thefeasibilityproblem associatedwiththeGAPis NP -Complete,seeMartelloandToth[64].Theproblem wasoriginallystudiedbyRossandSoland[84],whoproposedanabranch-and-bound algorithmtosolvetheproblemtooptimality.Intheirwork,assignmentconstraints2{8 aredeletedandtheremainingassignmentproblemissolvedtoobtainavalidupperbound. Then,asecondarypenaltyproblemissolvedtocorrectviolatedcapacityrestrictions. Sincethen,alargenumberofadditionalbranch-and-boundapproachesfortheGAPhave beenproposed.Theseworksaredierentiatedbythevaryingapproachesusedtobound thesolution.Fisher[34]consideredthestrengthofboundsobtainedbysolvingithe Lagrangianrelaxationformedbyrelaxingcapacityconstraints2{8iitheLagrangian relaxationobtainedbyrelaxingassignmentconstraints2{9oriiisolvingtheLP relaxationformedbyrelaxingbinaryconstraints2{10.Thisworkdiscussesinteresting trade-osbetweensolvingcomputationallydicultrelaxationsthatprovidedsharper bounds,asshowntobethecasewiththerelaxationgivenbyii,versusweakerbounds obtainedinlesstime.WhilethisdissertationdoesnotutilizethetechniqueofLagrangian relaxation,trade-ossuchasthesearehighlyrelevanttootherdecompositionapproaches thatweconsider. Inadditiontothewell-studiedbranch-and-boundprocedure,anumberofdecomposition basedapproacheshavebeenproposedfortheGAP.BuildingontheLagrangianrelaxation eortsdiscussedpreviously,JornstenandNasberg[52]proposedaLagrangiandecomposition 23

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methodologythatcombinedthetworelaxationsformedbyrelaxingeithertheassignment 2{9orcapacityconstraints2{8.Theyshowedthattheboundobtainedbytheresulting relaxationsolutionisatleastasstrongaseitheroftheboundsobtainedbytheindividual Lagrangianrelaxationalternatives.Whiletheirtestingislimitedtoonly10instances, resultssuggestedthattheapproachisaneectivealternativetothetraditionalLagrangian relaxationsoftheGAP. Ineachoftheproblemsstudiedinthisdissertation,anequivalentpartition-based representationisproposedwithasubsequentsolutionmethodology.Therefore,of particularrelevancetotheworkinthedissertationisSalvesbergh's[88]branch-and-price algorithmfortheGAP.Inthisapproach,theGAPisrepresentedasapartitionoftheset ofcustomers, J ,into jIj disjointandpossiblyemptysubsets,eachofwhichisassignedto exactlyonefacility.Thatis,theformulationGAP-SPequivalentlyrepresentstheGAP. maximize X i 2I D i X d =1 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(p ij x d ij d i subjecttoGAP-SP X i 2I D i X d =1 x d ij d i =1 j 2J {11 D i X d =1 d i =1 i 2I {12 d i 2f 0 ; 1 g d =1 ;:::D i ; i 2I ; where x d i = x d i 1 ;:::;x d i jJj isabinaryvectorrepresentingthe d th subsetofcustomers thatcanbeassignedtofacility i ,and D i isthetotalnumberofsubsetsofcustomersthat canbeassignedtofacility i .Thevariable d i takesthevalueofoneifthe d th column associatedwithfacility i chosen,andzerootherwise.Ingeneral,thenumberofvariablesin GAP-SPisexponentiallylargeinthedimensionoftheunderlyingassignmentproblem. Thebranch-and-priceapproachthereforesolvestheLP-relaxationofGAP-SPbya columngenerationprocedure,wherethecolumnsareaddediterativelyasneeded,and 24

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solvesGAP-SPitselfbybranch-and-bound.Theso-called pricingproblem solvedto identifyattractivecolumnsisthewell-studied0-1knapsackproblem,discussedindetailin Section2.3.Salvesbergh[88]showedthattheLP-relaxationofGAP-SPprovidesabound atleastastightasthatobtainedbysolvingtheLP-relaxationofGAP,sincethefeasible spaceofLPGAP-SPislimitedtoconvexcombinationsofsolutionstoa0-1knapsack problem.Importantly,theworkdemonstratedthatthebranch-and-priceapproachis particularlysuccessfulwhentheratioofcustomerstofacilitiesissmalli.e.nomorethan 5.ThisphenomenonisthecombinedresultofitheLPrelaxationofthe0-1knapsack problembeingweakerwhenthisratioissmall,thussolvingtheknapsackproblemsto optimalityyieldsstrongerboundsandiithefactthatnumberoffeasible0-1knapsack solutionsbecomesquitelargeastheratioofcustomerstofacilitiesincreases,thusthe columngenerationprocedurebecomeshighlycomputationallyintensiveasthenumberof customersperfacilityincreases. EvenwiththeadvancesofexactalgorithmsfortheGAP,itremainscomputationally impracticaltosolveverylargeinstances.Forthisreason,agreatdealoftheliterature isdevotedtometa-heuristicsfortheGAP.AlargenumberarementionedinRomero RomalesandRomeijn[83]summaryofresearchpursuedfortheGAP.Notableamongst thesearetabusearchYagiuraet.al[101],geneticalgorithmsWilson[100]and simulatedannealingalgorithmsOsman[73].Othersuccessesweredocumentedin AminiandRacer's[10]improvedimplementationofthevariabledepthsearchheuristic thatbenetsfromthegreedyheuristicsproposedbyMartelloandToth[64].Cattrysse etal.[20]proposedaheuristicwhichsolvesthelinearrelaxationoftheset-partitioning representationoftheGAPi.e.GAP-SPandsearchesamongstthecolumnsgeneratedto obtainafeasiblesolution.Theresultsofacomputationalstudyempericallysuggestthat solutionsobtainedinthismannerareoftenwithinlessthan1%ofoptimality. OfparticularsignicancetoChapter5istheclassofheuristicsproposedbyMartello andToth[64]andRomeijnandRomeroMorales[79].Aweightfunction, f i;j ,isdened 25

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tomeasurethepseudo-protofassigningcustomer j j 2J tofacility i i 2I .This functionisusedtodeterminetheorderinwhichtoassignthecustomersandthefacility towhicheachcustomershouldbeassigned.InMartelloandToth[64],weightfunctions arerepresentedbyeitherithexedprotcost p ij i 2I ; j 2J iitheamountof resource i consumedbycustomer j a ij i 2I ; j 2J oriiitheratio a ij b i .Romeijnand Morales[79]proposedaweightfunctionthatseekstoassignacustomertoafacilitywith maximumprotandminimalcapacityconsumption.Toaccomplishthistheychose f i;j = p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij forsomevector .Theyshowedthatif i i 2I istakentobetheoptimaldualvalues associatedwithconstraints2{8thentheirgreedyalgorithmisoptimalwithprobability oneasthenumberofcustomersgoestoinnityunderaverygeneralstochasticmodel.It isthisworkthatmotivatesthegreedyheuristicdevelopedinChapter5forwhichwealso seekasymptoticperformanceguarantees. 2.30-1KnapsackProblem Throughoutthisdissertationnumerouswestudyvariantsof0-1knapsackproblems. Whilealargesegmentofourworkconsiders0-1knapsackproblemswithnon-linearprot functions,eachofoursolutionapproachesreliesheavilyonthepropertiesofthelinear0-1 knapsackproblemandthesubsequentalgorithmsusedtosolveit.Thisproblem,whichwe refertoastheKP-01,ispresentedas maximize X j 2J p j x j {13 subjecttoKP-01 X j 2J a j x j b {14 x ij 2f 0 ; 1 g j 2J : {15 26

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Thisoptimizationproblemrequiresthemostprotablesubsetofcustomerstobechosen withoutviolatingthecapacityconstraint2{14.Martelloetal.[62]providedanexcellent overviewofexactalgorithmsproposedforKP-01.Asstatedinthissurvey,amajority ofexactalgorithmsarebasedoneitherbranch-and-boundordynamicprograming approaches.Notsurprisingly,thevariationsinthebranch-and-boundapproachesconsider varyingproceduresforobtaininggoodupperboundstoKP-01.Theoriginalboundon KP-01wasestablishedinDantzig[24].Theboundisdeterminedbysolvingthelinear relaxationofKP-01inwhichthebinaryrestrictions2{15arerelaxedsayLPKP-01. LPKP-01issolvedbysortingitems j 2J innon-increasingorderof p j a j andincluding itemsinthesolutiontoLPKP-01untileitheriallcapacity b isconsumedorii allcustomers j 2J ; forwhich p j > 0,areincludedinthesolutiontoLPKP-01. Importantly,theoptimalsolutiontothisrelaxationcontainsatmostoneitemwhich violatesthebinaryrestrictions2{15.Thus,aheuristicapproachtosolvingKP-01 issimplytoremovethefractionalcustomerintheoptimalsolutiontoLPKP-01. Dantzig'sboundwaslaterimprovedbyMartelloandTothin[63]andthenagainin[67]. Inthelatterwork,MartelloandToth[67]showedthatstrongerboundscanbeobtained byaddingmaximumcardinalityconstraintstoKP-01andsolvingthecorresponding relaxationviaLagrangiantechniques.Itisimportanttonotethatineachofthesuccessful branch-and-boundimplementationsfortheGAPseealsoHorowitzandSahni[50]and Nauss[70],adepth-rstenumerationschemewaschosen. Asmentionedpreviously,theothercommonapproachusedtosolveKP-01to optimalityisdynamicprogramming.Pisinger[75]proposedadynamicprogramming approachthatreliesonboundsdeterminedbyLPKP-01.Theirapproachwassuccessful inlimitingtheenumerationtobeconsidered,thusyieldinggoodlowerboundsina reasonableamountoftime.Toimproveuponthisapproach,MartelloandToth[65] developedahybridprocedurewhichcombinedtheexpanding-corealgorithmproposed 27

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inPisinger[75]withtheimprovedboundsobtainedin[67].Thisapproachwasshownto solvealmostalltestinstanceswithupto100,000variablesinlessthan.5seconds. 2.4FlexibleDemandAssignmentProblems Eachoftheproblemsstudiedinthisdissertationcanbepresentedasgeneralizations oftheGAP.Specically,eachproblemgeneralizesthenotionofxeddemandconsumption. WemathematicallyintroducetheconceptofexibledemandinChapter3.Thissection discussesseparateworksintheliteraturethathaveconsideredtheconceptofvariable demandfullledbetweencustomer-speciedlimits.Themostrelevantworkinthisarea wasoeredbyBalakrishnanandGeunes[13]whoconsideredaproductionplanning problemwithexibleproductspecication.Theirmodelwasmotivatedbythesteel industry,inwhichcustomerswillacceptsteelplatescutwithinspecieddimensions.They formulatethisproblemasthe FlexibleDemandAssignment problemFDA,givenby maximize X i 2I X j 2J r j + v ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X i 2I f i + b i y i subjecttoFDA X j 2J v ij b i y i i 2I {16 X i 2I x ij =1 j 2J {17 v ij ` j x ij i 2I ; j 2J {18 v ij u j x ij i 2I ; j 2J {19 x ij 2f 0 ; 1 g i 2I ; j 2J {20 y i 2f 0 ; 1 g i 2I {21 wherecustomer j 0 s demandlevel j 2J v ij i 2I ; j 2J mustbesettoavalue within[ ` j ;u j ].Furthermorearevenueof r j isaccruedperunitofcapacityconsumption.It shouldbenotedthattherevenueaccruedasafunctionofthelevelatwhichthedemand 28

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issatisedisindependentofthefacilityusedtosatisfythisdemand.Similarly,thebounds onthedemandfulllmentlevelareindependentofthefacilitytowhichthecustomeris assigned.Ifresource i isusedtofulllanycustomerdemandaxedcost, f i isincurred. Unusedcapacityofprocuredresource i isrecycledatacostof perunit.Inadditionto therecognizablecapacityandassignmentconstraints2{16and2{17,constraints2{18 and2{19ensurethatdemandisfullledbetweentheappropriatebounds.Balakrishnan andGeunes[13]proposedmultipleclassesofstrongvalidinequalitiesforFDA,aswellas aLagrangian-basedupperboundingprocedure.Lagrangian-based,bin-packingandlinear programroundingheuristicsarepresented.Theresolutionprocedureobtainedaninitial lowerboundheuristicallyandutilizedtightenedrelaxationstoobtainaninitialupper bound.TheseboundsareprovidedtoCPLEX'sstandardbranch-and-boundprocedure. Athoroughcomputationalstudyshowedthatthiscompositeapproachsuccessfully solvesbothrealandrandomprobleminstancesupto12resourcesand60customersto optimalityortowithinasmallgap. BeyondtheFDAandtheworkcontainedinthisdissertation,themostrelatedwork toexibledemandfoundintheliteratureisthatdoneontheMulti-levelGeneralized AssignmentProblem[57].InthisextensionoftheGAP,eachcustomerisassignedtoa singleresourceatoneofa xed numberoflevels.However,themodeldoesnotconsider acontinuousrangeoflevelsinwhichacustomer'sdemandmaybefullled,asallowedin FDAandtheproblemsconsideredinthisdissertation. 29

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CHAPTER3 CAPACITATEDFACILITYLOCATIONWITHSINGLE-SOURCECONSTRAINTS ANDFLEXIBLEDEMAND Problemsthatrequiretheallocationoflimitedresourcestodemandsforthose resourcesariseinnearlyallcontexts.Inindustrialcontexts,thecostsassociatedwith acquiringrelevantresourcescanoftenbequantied,ascanthecostsassociatedwithusing theseresourcestosatisfycorrespondingdemands.Insuchcases,optimizationmodels serveasapowerfultoolfordeterminingthebestmixofresourceacquisitionandallocation ofresourcestodemands.TheCapacitatedFacilityLocationProblemCFLPprovides anexampleofawell-knownoptimizationmodelthathasbeensuccessfullyutilizedto determineanoptimalsubsetoffacilitiesfromamongasetofcandidatelocationsaswell astheallocationoftheoutputofthesefacilitiestoasetofknowncustomerdemands. Thisproblemassumesthateachcandidatefacilityhasanassociatedxedoperatingcost, aknowncapacitylimitonoutput,andacostforsatisfyingacustomer'sdemandthatis proportionaltotheamountofthecustomer'sdemandsatisedfromthefacility.Beyond thefacilitylocationcontext,thismodelndsapplicationinawidevarietyofsettingsin whichindividualcapacitatedresourcesmustbeacquiredtosatisfydemandsforresource output.Inthischapter,weintroduceaproblemthatcombineskeyelementsofseveral well-studiedvariantsofthisproblemintoanewclassofgeneralizedcapacitatedfacility locationproblemsthatwewillrefertoasthe CapacitatedFacilityLocationProblemwith Single-SourceConstraintsandFlexibleDemand CFLFD. AsmentionedinChapter2,theCFLPwithsingle-sourceconstraintsaspecialcaseof theCFLFDfallsintothedicultclassof NP -Hardoptimizationproblems,implyingthat itisunlikelythatapolynomial-timesolutionmethodexistsforsolvingproblemsinthis classunless P = NP .ThespecialcaseoftheCFLPinwhichtherearenoxedcosts associatedwiththeacquisitionoffacilitiesconstitutesawell-knownproblemclassknown astheGeneralizedAssignmentProblemGAPstudiedbyRossandSoland[84],amongst others.Thismodelisitselfwidelyapplicableinnumerousproblemcontextssuchasjob 30

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scheduling,locationmodels[85],andtransportationplanning[92].Theclassicalvariants oftheCFLPandtheGAPseektomeetknowncustomerdemandlevelsatminimum cost.Thus,therequiredlevelofresourceconsumptionforeachjoborcustomerdemandis assumedtobeaxedquantity.Morerecently,problemswhichconsiderexiblejobsizes orexiblecustomerdemandquantitieshavebeenproposedbyBalakrishnanandGeunes [13].Accountingforsuchexibledemandsallowsforscenarioscommontoboththesteel [13]andforestryindustries,forexample,wherecustomersmaypermitarangeofdelivery quantitiesorarangeofacceptableproductsizes.Insuchcases,therevenuereceived bythesuppliermayoftenincreaseinthedeliveredquantityorproductsize.Thus,the supplierhasincentivetoincreaserevenuebysatisfyingdemandsattheupperlimitsof acceptableranges.Suchastrategy,however,increasesresourcecostsandmay,therefore, notresultinprotmaximization. ThenewclassoftheCFLFDproblemsthatweintroduceandanalyzeinthis dissertationcombinesalloftheaspectsdescribedabove.Inparticular,thisproblemseeks aprot-maximizingsolutionbasedondecisionsinvolvingtheprocurementofcapacitated resources,theassignmentofcustomerstotheseresources,andthedeterminationof correspondingdemandfulllmentlevels.Let I denotethesetoffacilitiesavailablefor theexecutionofthesetofcustomers J .Eachcustomer j 2J mustbeassignedtoa singlefacility.However,eachfacilitymayonlybeabletoprocesscertaincustomers.That is,onlycustomersintheset J i J maybeassignedtofacility i 2I or,equivalently, customer j 2J mayonlybeprocessedbyfacilitiesintheset I j I ,whereofcourse i 2I j ifandonlyif j 2J i .Ifcustomer j 2J isassignedtofacility i 2I j ,axedprot of p ij isincurredandaxedamountofcapacity a ij isconsumed.Thecorresponding customerdemandfulllmentlevelmustbeselectedfromtheinterval[ ` ij ;u ij ] : An additionalprotisaccruedasafunctionofdemandfulllmentlevel,determinedby thenon-decreasingfunction r ij i 2I ; j 2J i .Lastly,iffacility i 2I isusedtosatisfy anycustomerdemand,axedprocurementcost f i isincurred.Thecapacityoffacility 31

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i 2I isdenotedby b i i 2I .Theobjectiveistodetermineanassignmentofcustomers toprocuredfacilities,aswellasthecorrespondingdemandfulllmentlevels,inorderto maximizetotalprotwhilesatisfyingthecapacityconstraintsofthefacilities. Usingtheprecedingnotation,theCFLFDcanbeformulatedasamixed-integerlinear programmingproblemasfollows: maximize X i 2I X j 2J i r ij v ij + X i 2I X j 2J i p ij x ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X i 2I f i y i subjecttoCFLFD X j 2J i a ij x ij + v ij b i y i i 2I {1 X i 2I j x ij =1 j 2J {2 v ij ` ij x ij i 2I ; j 2J i {3 v ij u ij x ij i 2I ; j 2J i {4 x ij 2f 0 ; 1 g i 2I ; j 2J i {5 y i 2f 0 ; 1 g i 2I : {6 Constraints3{1ensurethatcustomerdemandissatisedsolelybyprocuredfacilities andthatthelevelsatwhichdemandissatisedsatiseseachfacility'sresourceavailability. Constraints3{2requiretheassignmentofeachcustomertoasinglefacility.Inaddition, 3{3and3{4ensurethatifcustomer j isassignedtofacility i i 2I ;j 2J i ,its demandisfullledatalevelwithinitsrespectivebounds.Lastly,3{5and3{6enforce binaryrestrictionsontheassignmentvariables x ij i 2I ; j 2J i andfacilityprocurement variables y i i 2I Observethatthetotalrevenuereceivedwhencustomer j isassignedtofacility i ata levelof v ij 2 [ ` ij ;u ij ]isequalto p ij + r ij v ij ,whichcorrespondstoaper-unitassociated priceof p ij v ij + r ij .Themodelthusallowsforaformofquantitydiscountsthatmayprovide anincentivetocustomerstoacceptaexiblerangeofdemandfulllmentlevels.Similarly, 32

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thetotalquantityoftheresourcethatisconsumedwhencustomer j isassignedtofacility i atalevelof v ij 2 [ ` ij ;u ij ]isequalto a ij + v ij ,whichcorrespondstoaper-unitassociated resourceconsumptionof a ij v ij +1.Ourmodelcanthusaccountforthepresenceof,for example,xedcustomersetuptimesorotherlossofresourcesatthestartofaproduction run. Nofacilityprocurementcosts .InChapters4and5westudyaspecialcaseof theCFLFDinwhichtherearenoxedfacilityprocurementcosts,i.e., f i =0forall i 2I sothatwithoutlossofoptimalitywecanset y i =1forall i 2I .Werefertothisproblem asthe GeneralizedAssignmentProblemwithFlexibleDemand maximize X i 2I X j 2J i p ij x ij + X i 2I X j 2J i r ij v ij {7 subjecttoGAPFD X j 2J i a ij x ij + X j 2J v ij b i i 2I {8 X i 2I x ij =1 j 2J i v ij ` ij x ij i 2I ; j 2J i v ij u ij x ij i 2I ; j 2J i x ij 2f 0 ; 1 g i 2I ; j 2J i : Noticethatthefacilityprocurementvariables y i i 2I havebeenomittedfrom3{7 and3{8. Dynamicproblems .ItisnoteworthythattheCFLFDencompassesmanufacturing problemswithatemporalelement.Specically,consideramulti-periodexibledemand assignmentprobleminwhichcustomershaveaduedate,butcanbeexecutedpriorto thatattheexpenseofaholdingcost.Thisproblemcanbeformulatedasaninstance oftheCFLFDwherethefacilitiesaretime-expanded;i.e.,afacility i;t representsa 33

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singleresource i 2I availableintimeperiod t t =1 ;:::;T ,where T isthelength oftheplanninghorizon.Clearly,customerscanonlybeassignedtofacilitiesrepresenting resourcesthatareavailableonorbeforetheirduedate;i.e.,theset J i;t containsonly customersthatdonothaveaduedatebeforetimeperiod t InthefollowingchapterweproposeanexactsolutionapproachfortheCFLFDand theGAPFD.Then,inChapters5and6wedeveloplarge-scaleheuristicapproachestothe CFLFDandtheGAPFD. 34

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CHAPTER4 EXACTALGORITHMFORCFLFDANDGAPFD TheclassofCFLFDproblemsintroducedinChapter3seeksaprot-maximizing solutionbasedondecisionsinvolvingtheprocurementofcapacitatedresources,the assignmentofcustomerstotheseresources,andthedeterminationofcorresponding customerdemandfulllmentlevels.Inthischapter,wefocusondevelopinganexact approachforsolvingthisproblem.Overthepastdecade,manynonlinearassignment problemsarisingin,forexample,supplychainoptimization,havebeenreformulated asset-partitioningproblems,leadingtobranch-and-pricesolutionapproachestosuch problems.Barnhartetal.[16]provideathoroughdiscussionofhowbranch-and-pricecan beappliedtosolvelargeintegerprogrammingmodels.Applicationsinclude,forexample, theGeneralizedAssignmentProblemSavelsbergh[88],theFixed-ChargeAssigningUsers toSourcesProblemNeebeandRao[72],theMulti-PeriodSingle-SourcingProblem Frelingetal.[37],theContinuous-timeSingle-SourcingProblemHuangetal.[51], jointlocation-inventorymodelsShenetal.[89],andwarehouse-retailernetworkdesign problemswithjointreplenishmentcostsTeoandShu[90],Romeijnetal.[82].Itis thereforenotsurprisingthatthisapproachcanbeeectivelyappliedtotheCFLFD problemclassaswell.However,aswithanybranch-and-pricealgorithm,agreatdeal ofconsiderationmustbegiventotheso-calledpricingsubproblemthatarisesaswell astothebranchingstrategyused.IntheCFLFDproblem,thepricingsubproblem takesaninterestingform,resultinginageneralizationofthe KnapsackProblemwith FlexibleItems [13]orthe KnapsackProblemwithVariableItemSizes [81].Weprovidean ecientapproachforsolvingthisclassofproblemsundereitherconvexorconcaverevenue functions,whichleadstoaneectivebranch-and-priceapproachfortheCFLFDproblem. Thischapterisorganizedasfollows.Section4.1reformulatestheCFLFDasa set-partitoningproblemandintroducethepricingsubproblem.Section4.2proposes methodologiesforsolvingaclassofgeneralizedknapsackproblemswhichincludesthe 35

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relevantpricingsubproblem.Section4.3detailstheimplementationofourbranch-and-price algorithm.InSection4.4weperformanextensivecomputationalstudyoftheproposed proceduretosolveboththeCFLFDproblemandtheGAPFD.Weincludeabroad collectionofexperimentswithrevenuefunctionsthatmodelavarietyofcommonpricing conditions,andshowthatourbranch-and-pricealgorithmsignicantlyoutperformsa state-of-the-artcommercialmixed-integernonlinearprogrammingsolver.Finally,Section 4.5providessomeconcludingremarksandoersdirectionsforfutureresearch. 4.1AlternativeRepresentationoftheCFLFD 4.1.1Set-PartitioningFormulation WecanequivalentlyviewtheCFLFDasaproblemofpartitioningthesetof customers, J ,into jIj disjointandpossiblyemptysubsets,eachofwhichisassigned toexactlyonefacility.Moreformally,wecanwritetheset-partitioningformulationofthe CFLFDas maximize X i 2I D i X d =1 i x d i d i subjecttoSP X i 2I D i X d =1 x d ij d i =1 j 2J {1 D i X d =1 d i =1 i 2I {2 d i 2f 0 ; 1 g d =1 ;:::D i ; i 2I ; where x d i = x d i 1 ;:::;x d i jJ i j isabinaryvectorrepresentingthe d th subsetofcustomers thatcanbeassignedtofacility i ,and D i isthetotalnumberofsubsetsofcustomersthat canbeassignedtofacility i .Furthermore, i isafunctionthatdeterminestherevenue obtainedbyfacility i whensubset x d i isassignedtoit.Inparticular, i x d i istheoptimal valueofthefollowingoptimizationproblem,inwhichthesizesoftheassignedcustomers arechosentomaximizetherevenueofthefacility: 36

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X j 2J i p ij x d ij +maximize X j 2J i r ij v j )]TJ/F21 11.9552 Tf 11.955 0 Td [(f i y i subjecttoCC d i X j 2J i v j b i y i )]TJ/F26 11.9552 Tf 12.536 11.358 Td [(X j 2J i a ij x d ij v j 2 [ ` ij x d ij ;u ij x d ij ] j 2J i y i 2f 0 ; 1 g : Sinceitiseasytoseethat i 0 =0wecan,withoutlossofoptimality,relaxconstraint 4{2to D i X d =1 d i 1 i 2I : 4{2 0 Wechoosethelattersetofconstraintsforconvenience,sincethisimmediatelyimpliesthat theassociateddualvariablesarenonnegative. Generally,thenumberofvariablescolumnsinSPisexponentiallylargeinthe dimensionoftheunderlyingassignmentproblem.Thebranch-and-priceapproachtherefore solvestheLP-relaxationofSPbyacolumngenerationprocedure,wherethecolumnsare addediterativelyasneeded,andsolvesSPitselfbybranch-and-bound.Inthecolumn generationprocedure,theso-called pricingproblem determineswhetherthesolutionto theLP-relaxationofarestrictedversionofSPinwhichonlyasubsetofthecolumnsis considered,sayLPRSPisindeedoptimalor,otherwise,identiesoneormorecolumns thatpriceoutandarethereforeaddedtotherestrictedproblem.Notethatitissucient toeitheridentifyafeasiblesolutiontothepricingproblemthatpricesout or showthat theoptimalsolutiontothepricingproblemdoesnotpriceout.Thatis,itisnotstrictly necessarytosolvethepricingproblemtooptimalityateachiterationofthecolumn generationmethod.Inthefollowingsectionweformallypresentthepricingproblem andinthesubsequentsectionwedevelopapproachestosolvethispricingproblemunder variousformsoftherevenuefunctions r ij i 2I ; j 2J i 37

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4.1.2PricingProblem Thefollowingoptimizationproblemdescribesthepricingproblemassociatedwith facility i 2I : maximize X j 2J i p ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( j x j + X j 2J i r ij v j )]TJ/F15 11.9552 Tf 11.956 0 Td [( i + f i y subjecttoPP i X j 2J i a ij x j + v j b i y v j 2 [ ` ij x j ;u ij x j ] j 2J i {3 x j 2f 0 ; 1 g j 2J i y 2f 0 ; 1 g ; where j j 2J and i 2I aretheoptimaldualvariablesassociatedwiththe assignmentconstraints4{1andthecolumnselectionconstraints4{2inLPRSP. Observethatwhen y =0theoptimalsolutionvalueofPP i istriviallyseentobeequal to0,sothatwecanlimitourselvestosolvingtheproblemundertheassumptionthat facility i isprocuredi.e., y =1.Wethensimplyreplacetheoptimalvaluetothis restrictedproblembyzeroifitisnegative.Tosimplifythedevelopmentofoursolution methods,inSection4.2westudyanequivalentformulationoftherestrictionofPP i to y =1.Inthisreformulation,wereplacethedemandfulllmentlevelvariables, v j ,with decisionvariables w j = v j + a ij x j j 2J i ,whichrepresentthetotalamountofresource i.e.,bothxedandvariableconsumedbycustomer j .Thealternativeformulation, PP 0 i ,iswrittenas maximize X j 2J i p ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( j x j + X j 2J i r ij w j )]TJ/F21 11.9552 Tf 11.955 0 Td [(a ij x j )]TJ/F15 11.9552 Tf 11.955 0 Td [( i + f i subjecttoPP 0 i X j 2J i w j b i {4 38

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w j 2 [ ` 0 ij x j ;u 0 ij x j ] j 2J i {5 x j 2f 0 ; 1 g j 2J i ; {6 where ` 0 ij = ` ij + a ij and u 0 ij = u ij + a ij i 2I ; j 2J i .Itiseasytoseethat4{5assures that4{3remainssatised. ThisproblemisageneralizationoftheKnapsackProblemwithExpandableItems thatallowsfornonlinearrevenuefunctions.BalakrishnanandGeunes[13]proposea dynamicprogrammingalgorithmforthisproblemforthecasewheretherevenuefunctions arelinearandtheproblemdataisinteger.Thisapproachhasarunningtimethat ispseudo-polynomialintheprobleminputs,whichtendstomakeittime-consuming inpractice.Additionally,theirworkdoesnotconsidernon-linearrevenuefunctions. Therefore,inthenextsectionweconsiderapproachesforsolvingourpricingproblem underdierentclassesofrevenuefunctions.Wedevelopaheuristicapproachaswellasa customizedbranch-and-boundapproachtosolvetheproblemtooptimality. 4.2KnapsackProblemwithExpandableItems Inthissectionwefocusonecientmethodologiestosolvetheclassofproblemsgiven by maximize X j 2 ~ J p j x j + X j 2 ~ J r j w j )]TJ/F21 11.9552 Tf 11.955 0 Td [(a j x j subjecttoKPEI X j 2 ~ J w j b {7 w j 2 [ ` 0 j x j ;u 0 j x j ] j 2 ~ J {8 x j 2f 0 ; 1 g j 2 ~ J ; {9 where ~ J isthesetofcustomerstobeconsidered.WecanrepresentKPEIinitsmost generalformas maximize X j 2 ~ J j w j x j 39

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subjecttoKPEI 0 X j 2 ~ J w j x j b {10 w j 0 j 2 ~ J {11 x j 2f 0 ; 1 g j 2 ~ J ; {12 wherethefunctions j aredenedas j w j = 8 > > > > > > > < > > > > > > > : =0 w j =0 = 0
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wherethefunction j j 2 ~ J correspondstothenon-decreasingconcaveenvelope encompassingtheorigin,thefunction j andthepoint b; j u 0 j whichisillustratedin Figure4-1.WewillshowthatfortheforKPEI 0 withconcave,convex,orlinearrevenue functions,theenvelopes j j 2 ~ J canbeobtainedexplicitly. RomeijnandSargut[81]proposeanalgorithmtosolveRKPbasedonabinary searchforobtainingtheoptimalLagrangemultipliersatisfyingtheKKTconditions. However,notethatinthecontextofKPEI 0 ,RKPallowssolutionsinwhich u 0 j < w RKP j b j 2 ~ J .This,ofcourse,correspondstoaninfeasiblesolution.Lemma1ensures thananalternativefeasiblesolutionexistswithanequivalentobjectivevalue. Lemma1. Supposetheoptimalsolutionto RKP containsanon-emptysetofcustomers J forwhich u 0 j
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iicustomer j isfullyassigned,butitsnetprotwasoverestimatedi.e., j w j > w j and ` 0 j w j u 0 j Proof. TheresultfollowsfromTheorem4.2of[81]andthedenitionof j Thispropertycanbeusedtodevelopaneectiveheuristicroundingstrategyasis oftendoneforthetraditionalknapsackproblem.Inparticular,iftheoptimalsolutionto RKPisindeedfractional,wecansimplyremovethatfractionalcustomertogeneratea feasibletosolutiontotheKPEI 0 .Otherwise,ifthesolutionisfeasibletoKPEI 0 ,but j w RKP j 6 = j w RKP j forasinglecustomer j ,wesimplyupdatetheobjectivefunction accordinglytocorrectfortheapproximationusedintherelaxation. ThispropertyalsoimpliesthatKPEI 0 canbesolvedtooptimalityquiteeectively usingbranch-and-boundevenforlargeproblemsizessee,e.g.,MartelloandToth [66],despitethefactthatitisNP-hard.Theimplementationofthebranch-and-bound approachisasfollows.AsolutiontoRKPisobtainedateachnodeofourbranch-and-bound-tree, say~ w RKP .Webranchonthecustomerforwhich j ~ w RKP j 6 = j ~ w RKP j : FromLemma2, thiscorrespondstoacustomerthatiseitherifractionallyassigned,i.e.,0 < ~ w RKP <` 0 j orii~ w RKP ` 0 j ,but j ~ w RKP < j ~ w RKP .Incaseiwedeneasinglebranchwiththe constraint w j ` 0 ij {14 andasecondbranchwithconstraint w j =0 : {15 Inthecaseofiiwecreateabranchwiththeconstraint w j ~ w RKP j {16 andanotherbranchwiththeconstraint w j ~ w RKP j : {17 42

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Constraint4{14canbeaccommodatedbyrstreducing b by ` 0 j andaddingtheconstant ` 0 j totheobjectivevalue.Wethenredenetheboundsofcustomer j tobeintherange [0, u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ]withthemodiedobjectivefunctioncomponent ~ j = j w j + ` 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [( j ` 0 j 0 w j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j : Constraints4{16and4{17areaccountedforbysimplemodicationofthe j suggestedin[81].Thealgorithmgivenin[81]canthenbeappliedateachnodeofour searchtreewithoutfurthermodication. 4.2.1CFLFDwithSpeciallyStructuredRevenueFunctions Aspreviouslydiscussed,theheuristicandexactbranch-and-boundprocedures discussedinthissectionsolveKPEI 0 withanygeneralrevenuefunctions r j .However,the concaveenvelope j maybediculttocharacterizeexplicitly.Therefore,inthefollowing sectionsconsiderimportantpracticalcasesforwhichthiscanbedone.Inparticular,we studythreeclassesofrevenuefunctionswhichmodelproductpricinginbothmassand specializedproductionenvironments.Inadditiontoconsideringthesefunctionsfortheir real-worldappeal,weshowthatwhentherevenuefunctionsarelinearorconvex,KPEI 0 canbemoreecientlysolvedusinganalternativeproceduretosolveRKP. Concaverevenuefunctions .First,weconsidernon-decreasingconcaverevenue functions.Thischoiceoffunctionmodelsthescenarioinwhichmarginaldiscountsfor largercustomerdemandfulllmentlevelsareavailabletothecustomer.Thisscenariois applicabletomanufacturersofproductsoftenpurchasedinbulksizes,suchasdurable goods. 43

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NowletusapplythealgorithmpresentedinSection4.2.Inthiscase,theconcave envelope j j 2 ~ J isgivenby j w j = 8 > > > > > < > > > > > : = j j j w j 0 w j j = j w j j @ + j ` 0 j then j = ` 0 j ,andif j u 0 j u 0 j <@ )]TJ/F22 7.9701 Tf 0 -8.011 Td [(j u 0 j then j = u 0 j .Therefore theconcaveenvelopeconsistsofalinearsegmentextendingfromtheoriginto j ,thenthe truefunctionvalue, j w j ontheinterval j ;u 0 j ],andthenalinearfunctionwithslope0 ontheinterval u 0 j ;b ]. Inthenextsection,westudyingreaterdetailtwoclassesofrevenuefunctionsthat areofequalinterestinmodelingspecicrevenuestructures.Ingeneral,themethod describedinthissectionworksfortheserevenuefunctionsaswell.However,forthese specialcaseswewillproposeamoreecientalgorithm. 4.2.2ConvexandLinearRevenueFunctions ThissectionstudiesKPEIwhentherevenuefunctionsareeitherconvexorlinear. WeproposeamoreecientalgorithmtosolveRKPthatisapplicabletoeachofthese cases.Thealgorithmismotivatedbythesimpliedstructureoftheconcaveenvelopeused inRKP.Thatis,areformulationofRKPcanbesolvedinasimilarmannertothe continuousknapsackproblem. First,weassumeconvexrevenuefunctionswhichareconsistentwithaproduction scenariothatplacesincreasedpremiumsonlargercustomerdemandfulllmentlevels, 44

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orchargesanincreasingamountastheamountofmanufacturingtimerequiredfora customerincreases.Thiscaseisapplicabletohighlycustomizedgoodssuchashigh-end electronicsorspecializedautomobilemanufacturing. Ofcourse,themethodsofSection4.2areapplicableinthiscaseaswell.Asis illustratedinFigure4-2,theconcaveenvelope j j 2 ~ J thencorrespondstoapiecewise linearfunction.Inparticular,let j = j ` 0 j ` 0 j denotetheslopeofthelinearsegmentconnectingtheorigintothefunctionvalue evaluatedat ` 0 j and j = j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [( j ` 0 j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j betheslopeofthelinearsegmentconnectingthepoints )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(` 0 j ; j ` 0 j and )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(u 0 j ; j u 0 j Then,if j > j wehavethat j w j = 8 > > > > > < > > > > > : = j w j 0 w j ` 0 j = j ` 0 j + j w j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ` 0 j < > : = h j u 0 j u 0 j i w j 0 w j u 0 j = j u 0 j u 0 j j g : Wespliteachcustomer j 2 J + intotwoparts.Therstparthasdemandfulllmentlevel w j 1 2 [0 ;` 0 j ]andaprot functiongivenby j w j 1 .Thesecondparthasdemandfulllmentlevel w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 12.405 0 Td [(` 0 j ] andarevenuefunctiongivenby j w j 2 .Since j > j wewillalwaysfullyutilizetheentire 45

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rangecorrespondingtotherstpartofcustomer j beforeutilizinganypartoftherange correspondingtothesecondpartofcustomer j .RKPcanthenbereformulatedas maximize X j 2 J )]TJ/F26 11.9552 Tf 8.247 29.244 Td [( j u 0 j u 0 j w j + X j 2 J + j w j 1 + X j 2 J + j w j 2 subjecttoRKP 0 X j 2 J )]TJ/F21 11.9552 Tf 8.247 12.387 Td [(w j + X j 2 J + w j 1 + X j 2 J + w j 2 b w j 2 [0 ;u 0 j ] j 2 J )]TJ/F21 11.9552 Tf -157.55 -31.833 Td [(w j 1 2 [0 ;` 0 j ] j 2 J + w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ] j 2 J + : Theoptimalsolutiontothisproblemcanbedeterminedbysimplysortingthe customers j 2 J )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(and j 1and j 2for j 2 J + innonincreasingorderoftheircoecientin theobjectivefunction.Weimmediatelyobtainthatatmostoneelementoftheoptimal solution, w RKP ,hasavaluethatisstrictlybetweenitsbounds.Thisimpliesthatatmost onecustomeriseitherifractionaloriiexecutedbetweenitslowerandupperbounds. Formally,thecorrespondingsolutiontotheRKPisconstructedby w RKP j = w RKP 0 j 1 + w RKP 0 j 2 j 2 J + w RKP j = w RKP 0 j j 2 J )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(: Thisalgorithmisalsoapplicablewhentherevenuefunctionsarelinear.Thatis,a unitrevenueof r j isaccruedforeachunitofresourceconsumed;i.e., r j v j = r j v j j 2 ~ J ItiseasytoseefromFigure4-3thatagaintheconcaveenvelopesarepiecewiselinearin thiscase.Formallyif p j > 0theconcaveenvelopeisgivenby4{19whileif p j 0, j isgivenby4{20.MoreinterestingistherelationshipidentiedinTheorem1between RKP 0 andthelinearrelaxationofKPEI 0 givenby 46

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maximize X j 2 ~ J p j x j + X j 2 ~ J r j w j subjecttoKPEI 0 -R X j 2J w j b w j 2 [ ` 0 j x j ;u 0 j x j ] j 2 ~ J {21 x j 2 [0 ; 1] j 2 ~ J {22 where p j = p j )]TJ/F21 11.9552 Tf 11.955 0 Td [(r j a j j 2 ~ J Theorem1. Theoptimizationproblems RKP 0 and KPEI 0 -R areequivalentwhenthe revenuefunctions r j arelinearforall j 2 ~ J Proof. SeetheAppendix. Theorem1impliesthatthealgorithmproposedinSection4.2.2solvestheLP-relaxation ofKPEI.ItshouldbenotedthatKPEI 0 -R 0 canbethoughtofastheLP-relaxationto thetraditionalknapsackproblem,whichhasatmostasinglefractionalelement.This, ofcourse,coincideswiththeresultsinLemma2.Moreinterestingly,notethatweneed onlytobranchoncustomerswhichcorrespondtofractionalassignmentswhentherevenue functionsarelinear.ThisisbestseenbyrevisitingFigure4-3.Letcustomer^ | bethe customerinwhich ^ | 6 = ^ | .IfthiscoincideswithFigure4-3b,thenclearly w RKP 0 ^ | <` 0 ^ | whichcorrespondstoafractionalassignment.ThesituationrepresentedinFigure4-3a suggeststhat w RKP 0 ^ | couldtakeavalueanywhereintherange ;u 0 j .However,from Theorem1thesolutiontoRKP 0 isequivalentlythesolutiontoKPEI 0 -R.Fromthe proofof1, x KPEI 0 -R = w KPEI 0 -R u 0 j forcustomersinwhich j isgivenby4{20i.e.,whenthe caseinFigure4-3aholds.Thus,clearly,0
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4.3Branch-and-PriceAlgorithmImplementation ThusfarwehavefocusedprimarilyonhowtosolveKPEI 0 .Inthissectionwe discussmorespecicdetailsregardingtheimplementationofourbranch-and-price algorithm. 4.3.1InitialFeasibleSolution Ourrstconcernliesinprovidinganinitialsetofcolumnsthatwillensurethata feasiblesolutionexiststotheLPrelaxationoftherestrictedset-partitioningproblem, LPRSP.Ifpossible,weinitializeRSPwithfeasiblesolutionsthatassign all customers toasinglefacility.Thatis,wetrytoinitializeRSPwithacolumnofones 1 foreach facility i 2I forwhichthecorrespondingvalueof i 1 isnitei.e.,forwhichthe correspondingoptimizationproblemdenedinSection4.3.4isfeasible.Ifthisyieldsa feasiblecolumnforatleastonefacility i 2I ,thenwehaveaninitialfeasiblesolution since,implicitly,weincludeacolumnofzeroes 0 forallfacilities i 2I byusingthe relaxedconvexityconstraint4{2 0 If,aswilltypicallybethecase,itisnotfeasibletoassignallcustomerstoasingle facility,weimplementatwo-phaseprocedureforsolvingSP,wherePhase1generatesa feasiblesolutionforLPRSP.Tothisend,weincludeanonnegativeslackvariablefor eachassignmentconstraint4{1.OurPhase1objectiveisthentominimizethesumof onlytheseslackvariables.TheresultingPhase1problemisthusgivenby minimize X j 2J s j subjecttoSP-Phase1 X i 2I D i X d =1 x d ij d i + s j =1 j 2J {23 D i X d =1 d i =1 i 2I {24 d i 2f 0 ; 1 g d =1 ;:::D i ; i 2I : 48

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Hereagainthenumberofcolumnsassociatedwitheachfacilitymaybeverylarge. Therefore,wesolvethelinearrelaxationofSP-Phase1usingcolumngeneration.The pricingproblemissimilartoPP i ,exceptforthefactthatwemustaccountforthe alteredobjectiveinSP-Phase1.ThepricingprobleminPhase1isthusgivenby maximize )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X j 2J j x j )]TJ/F21 11.9552 Tf 11.955 0 Td [(f i y )]TJ/F21 11.9552 Tf 11.956 0 Td [( i subjecttoPP i -Phase1 X j 2J w j b i y w j 2 [ ` 0 ij x j ;u 0 ij x j ] j 2J x j 2f 0 ; 1 g j 2J y 2f 0 ; 1 g : ThisproblemisaKPEI 0 withrevenuefunctions r ij 0 i 2I ; j 2J .Therefore,itcan besolvedwiththeapproachdiscussedinSection4.2.2. ItiseasytoseethatiftheoptimalvalueofLPSP-Phase1equals0,anyoptimal solutiontothisproblemisfeasibleforLPRSP;otherwise,theprobleminstance isinfeasible.Intheformercaseweusethisfeasiblesolutiontoinitializethecolumn generationprocedureforsolvingSP. 4.3.2SolvingLPRSP Atanynodeinourbranch-and-boundtreewemustsolvearelaxationofSP.As previouslydescribed,thisrequiressolvingapricingproblemviatheheuristicandexact methodsproposedinSection4.2.However,becauseourpricingproblemdecomposesby facility,thereare jIj potentialpricingproblemstoconsider.Itisvalidto:isolvethem individuallyinanyorderandentertherstcolumnthatpricesout;iisolveall jIj problemsandenterthecolumnthatpricesoutthehighest;iiisolveall jIj problemsand enterallcolumnsthatpriceout.Inouralgorithm,wechooseaslightlymodiedversion 49

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ofoptionii.Ateachiterationofourcolumngenerationprocedurewesolveallpricing problemsviatheheuristicmethoddescribedinSection4.2.Allcolumnsthatpriceout favorablyareaddedtoRSPandthecolumngenerationprocedurecontinues.Weonly callonourbranch-and-boundprocedureifallheuristicsolutionsindicatethatnoneof thecolumnsareattractive.Inthiscase,weorderthepricingproblemsinnon-increasing orderoftheobjectivevaluesdeterminedbytheheuristic.Wecontinuetosolvethepricing problemsviabranch-and-bounduntileitherasinglecolumnpricesout,oritisdetermined thatnocolumnpricesout.Thiscompromiseenablesaddingmultiplequalitycolumnsat eachiterationviaanecientheuristicwhileminimizingrelianceonatime-consuming branch-and-boundproceduretosolveeachpricingproblem. Thisimplementationisintendedtoacceleratetheconvergenceofourcolumn generationprocedureusedtosolveLPRSPateachnode.However,asiscommon withcolumngeneration,therateofconvergenceisoftenreducedastheoptimalLPRSP solutionisapproached.Toavoidthisissue,weterminateourcolumngenerationprocedure whenourcurrentLPRSPsolutionvalueisprovablywithin10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 oftheoptimalsolution toLPRSP.Thisallowsustocontinuewithourbranch-and-priceinatimelymanner withacomparablystrongupperboundwithrespecttothatobtainedifLPRSPwas solvedtooptimality.Ofcourse,todetermineavalidupperboundonLPRSP,each pricingproblemmustbesolvedtooptimality.Toensurethatpricingproblemsarenot solvedtooptimalitytoooften,weonlyupdatetheLPRSPupperboundaftersolving jIjjJj pricingproblemseitherheuristicallyorexactly.Thisimplementationchoice anticipatesthatthenumberofcolumnsrequiredtosolveLPRSPisafactorofboththe numberoffacilitiesandthenumberofcustomers.Therefore,thefrequencyofupdating theupperboundforLPRSPshoulddecreaseaseitherthenumberofcustomersorthe numberoffacilitiesincreases. ObtainingqualityfeasiblesolutionstoSPisofequalvaluetoourbranch-and-price implementation.Therefore,asaheuristictoobtainfeasiblesolutionstoSP,wesolveRSP 50

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asanintegerprogramusingcolumnsgeneratedinsolvingLPRSPattherootnode. Forahighpercentageofourtests,thetimerequiredtosolvethisMIPissmall.This implementationprovidedqualitylowerboundsearlyinouralgorithmwhichprovedtobe benecialinpruningourbranch-and-pricetree. 4.3.3NodeandVariableSelection Inourbranch-and-pricealgorithm,weinitiallydeterminetheorderinwhichnodes shouldbeconsideredbyusingadepth-rstrule.Then,onceafeasiblesolutiontoSPis obtained,weexplorethetreeusingthewell-knownbest-boundrule.Thisnodeselection policyisalsoimplementedinthebranch-and-boundprocedureusedtosolveourpricing problemtooptimality. Wefoundthebranchingdecisiontobeofparticularsignicancetoourproblem.Itis commonintheliteraturenottobranchonthe valuesthemselvesinLPRSPinorder topreservethestructureofthepricingproblem.NeebeandRao[72]proposebranching on x variablesthathaveavalueof1inacolumnassociatedwithafractional .This branchingschemeiseasilyaccommodatedinthepricingproblembygeneratingcolumns adheringtoanyassignmentsxedatpreviousnodes.However,asnotedbyCeselliand Righini[21],intuitionsuggeststhattheimpactofbranchingonfractionalprocurement variableswillbegreater.NotethatinthecontextofLPRSP,afractionalprocurement variable, y i ,correspondstoeither0 < P K i k =1 k i < 1or0 < 0 i < 1,where 0 i isacolumn inwhichnocustomersareassignedtofacility i withcost 0 i =0.Thoughthemodel consideredintheirworkmakesprocurementdecisionswithnoexplicitprocurementcost, oneoftheirbranchingstrategiesplacesabranchingpriorityontheprocurementvariables. Similarly,weproposeanimplementationforbranchingonfacilityprocurementvariables xedat1thatpreservesthestructureofthepricingproblembutrequiresnoadditional constraintsbeaddedtoLPRSP.Thatis,if y i isxedto1ataparticularnode,facility i 'sprocurementcostissimplytreatedasaconstantintheLPRSPobjectiveandthe costofcolumnsassociatedwiththatfacilityareappropriatelyreducedbythatamount. 51

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If y i isxedtozeroataparticularnode,thenallcolumnswithcustomersassignedto facility i areomittedandnocolumnsassociatedwithfacility i aregeneratedatthatnode. Consistentwithintuition,ourtestingindicatedthatbranchingpriorityshouldbegiven toprocurementvariablesoverassignmentvariables.Indeterminingwhichofthese x 'sor y 'sonwhichtobranch,weassessthedegreeoffractionalityofeachvariableinasolution toLPRSP.Thevariablewhichisleastfractionali.e.,thatvariablewhichisclosestto0 or1,wheretiesarebrokenarbitrarilyischosenforbranching.Amost-fractionalvariable selectionimplementationisalsousedinthepricingproblembranch-and-boundalgorithm. 4.3.4OptimalColumnCost RecallfromSection4.1thatthecostofthe d th columnassociatedwithfacility i d =1 ;:::;D i ; i 2I ,isdeterminedasafunctionoftheoptimalsizesofthose customersassignedinthatparticularcolumn.Ofcourse,whenthepricingproblemis solvedtooptimalitytheoptimalcustomerdemandfulllmentlevelsareimmediately available.Whenacolumnisgeneratedusingourheuristic,thesizeofthecustomers maynotnecessarilyrepresenttheoptimalsizesforthecorrespondingsetofcustomer assignments.Therefore,CC d i mustbesolvedtooptimality.However,notethatCC d i canbeequivalentlyrepresentedby C +maximize X j 2 J ~ j w j x j subjecttoCC d i X j 2 J w j x j b w j 0 x j 2f 0 ; 1 g 52

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where J isthesetofcustomersassignedtothe d th columnassociatedwithfacility i ,with functions j j 2 ~ J denedas ~ j w j = 8 > > > > > < > > > > > : =0 w j =0 = r ij w j + ` ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(r ij ` ij 0 < w j u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij : = r ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(r ij ` ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij < w j b; andwithconstants C = P j 2 J r ij ` ij + p ij )]TJ/F21 11.9552 Tf 11.211 0 Td [(f i and b = b i )]TJ/F26 11.9552 Tf 11.211 8.966 Td [(P j 2 J a ij + ` ij .Represented inthisform,CC d i simplytakestheformofKPEI 0 andcanbesolveddirectlybythe approachesdiscussedinsection4.2.GivenanoptimalsolutiontoCC d i x ; w ,the optimalcustomerdemandfulllmentlevelsintermsoftheoriginaldecisionvariablesare determinedbysetting w j = ` j + w j for j 2 J 4.4ComputationalResults Inthissectionwediscusstheperformanceofourbranch-and-priceprocedureona randomlydeterminedsetoftestinstances.Weseparatelyconsiderresultsforboththe GAPFDandtheCFLFDwithvaryingrevenuefunctioncharacterizations. 4.4.1ExperimentalData Intestingthenon-linearrepresentationofboththeCFLFDandGAPFDweconsider instanceswith5facilities,whilethedecreaseddicultyofthelinearinstancesallows ustoconsiderinstanceswith30facilities.Ineithercase,instanceswiththenumberof customersequalto jJj =2 jIj ,3 jIj ,and5 jIj arestudied,andforthepurposesofthis studyweassumethatallcustomerscanbeassignedtoallfacilities.Foreachcustomer, wegeneratetherandomvectorsofxedprotparameters P j fromuniformdistributions on[30 ; 50].Thecustomerrequirements A j L j and D j aregeneratedfromuniform distributionson[10 ; 20],[75 ; 125],and[15 ; 35],respectively.Here, A j and L j arethe randomvectorsofxedcapacityconsumptionandcustomerlowerbounds,respectively, and D j isarandomvectorcontainingvaluesrepresentativeofthedierencebetweenupper andlowerboundsofacustomer.Ineachofourtestswefocusoninstancesinwhichthe 53

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facilitycapacitiesareidentical.ForGAPFD,weset b i = jJj i 2I whileforthe CFLFDwegeneratecapacitiessothat b i = jJj i 2I ,where = E min i 2I A i 1 + L i 1 jIj : {25 Theparameter isusedingeneratingGAPFDinstancestocontrolthelevelofexibility availablewhendeterminingthesizeofeachcustomer.Inthesetestsweconsidera moderateexibilitylevelbysetting =1 : 2.Theparameter > 1,usedinthegeneration oftheCFLFDexperiments,inatesthecapacityofanfacilitytoensurethatnotall facilitiesarerequiredinafeasiblesolutiontotheCFLFD.Withoutthisconsideration, thefacilityprocurementdecisionsmaybetrivial.Ineachoftheexperimentsconsidered inthissection,facilitycapacitiesweregeneratedwith =2.Furthermore,inthecase oftheCFLFD,thecostofprocuringanfacilityisdirectlyproportionaltothesizeof thefacilityitself.Thatis,thecostofprocuringfacility i i 2I isgivenby F i = b i C i where C i representstheunitcostofprocurementgeneratedfromauniformdistributionon [0 : 75 ; 1 : 5]. Inourexperiments,weconsiderthethreeclassesofrevenuefunctionsdiscussedin Section4.2.Forprobleminstanceswithlinearrevenuefunctionswegeneratetheelements ofvectors R j ofunitrevenuesusingauniformdistributionon[2 ; 5].Theconvexand concaverevenuefunctionsthatweusedinourexperimentsareoftheform r ij v ij = S ij v ij 2 {26 and r ij v ij = S ij p v ij ; {27 respectively.Aninitialvalueoftheelementsofthevectorofcoecients S j foreach customerisrandomlygeneratedfromauniformdistributionontheinterval[0 : 5 ; 1 : 5]. However,toinsurethattheprobleminstancesarecomparablewescaleeachofthese coecientssothat r ij v ij = p ij + r ij v ij for v ij 2f ` ij ;u ij g where r ij istheunitrevenue 54

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inthecorrespondingprobleminstancewithlinearrevenuefunctions.Thepiecewiselinear functionsconsideredareobtainedbyapproximatingtheconcaveandconvexfunctions generatedin4{26and4{27.Toassesstheeectofvaryingthenumberofsegments usedtoapproximatedthenon-linearfunctions,weconsiderinstanceswith5,10and50 segments. Eachofourinstanceswasrununtileitherasolutionvaluewithin10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 oftheoptimal solutionwasobtainedoratimelimitofonehourwasreached.Ourtablespresentresults for10randomlygeneratedinstancesforeachcombinationofparametersettings.All experimentswereperformedonaDellpoweredge2600withtwoPentiumIV3.2Ghz processorsand6GBofRAM.Themixed-integerprogrammingproblemsaswellasthe relaxedmasterproblemsweresolvedusingCPLEX11.0.Wecomparedtheresultsofour approachforproblemswithnon-linearrevenuefunctionswiththoseobtainedbyBARON 8.1.1. Inourexperimentation,wesoughtbothtoassesstheeectivenessofourbranch-and-price approachandtogaininsightonthedierenceindicultyinthetwonon-linearcases considered.Forthisreason,eachofourrandomlygeneratedinstanceswastestedunder bothaconcaveandconvexrevenuefunction.Therefore,ourtablesassociatedwiththe resultsofproblemswithnon-linearrevenuefunctionsprovidecombinedresultsforeach ofthesecases.Duetothedierenceinthesizeofinstancesconsideredinthelinearcase, theseresultsarepresentedseparately.Specically,eachtablereports ithenumberofcolumnsgeneratedintheentirebranch-and-pricealgorithm, iithenumberofnodesconsideredinthebranch-and-pricetree iiitheamountoftimerequiredtosolvetherelaxedmasterproblemattherootnode ivthetotaltimerequiredbythebranch-and-pricealgorithm vthetotaltimerequiredbythecommercialsolveri.e.,BARONorCPLEX,withthe followingadditionalinformationwhereappropriate: 55

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{instancesunsolvedinonehourcontainadditionalinformationinthesuperscript ofthecommercialsolvertimecolumn; {therstsuperscriptindicatestherelativesolutionerrorcalculatedbyusingthe solversbestlowerandupperbound, z UB S z LB S ;i.e. error 1 = z UB S )]TJ/F21 11.9552 Tf 11.955 0 Td [(z LB S z UB S 100%; {thesecondsuperscriptindicatestherelativesolutionerrorcalculatedusingthe solutionobtainedbybranch-and-pricealgorithmandthebestsolutionobtained bythecommercialsolver, z BP z LB S ;i.e., error 2 = z BP )]TJ/F21 11.9552 Tf 11.955 0 Td [(z LB S z BP 100% : 4.4.2CFLFDResults Inthissectionwecomparetheperformanceourbranch-and-pricealgorithmagainst twowell-knowncommercialsolversonabroadsetofCFLFDinstances.Section4.4.2.1 presentsresultsforCFLFDinstanceswithdierentiablenonlinearfunctionssolvedvia boththeapproachdevelopedinthischapterandBARON.InSection4.4.2.2weperform anadditionalcomparisonforpiecewiselinearandlinearinstancessolvedwithboth branch-and-priceandILOG'sCPLEX11.0. 4.4.2.1Nonlinearrevenuefunctions:comparisonwithBARON Tables4-1{4-3presentresultsforthedierentiablenonlinearconcaveandconvex CFLFDinstanceswithfunctionsgeneratedby4{26and4{27.Ineachofthesethree tablesitisevidentthatbranch-and-priceoutperformsthecommercialsolverinevery instance.Ourcomputationalstudyfoundthatourchoiceofbranchingstrategy,discussed inSection4.3.3,was,inpart,acontributingfactortothissuccess.Asanexample,for theinstancesconsideredinTable4-4,placingabranchingpriorityontheprocurement variablesovertheassignmentvariablesreducedouraveragebranch-and-pricetimebya factorof50fortheconvexinstances.Thisislikelyadirectresultofthenumberofnodes beingreducedbyafactorofmorethan10. 56

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AnanalysisoftheperformanceofthecommercialsolverindicatesthatBARON isunabletosolveanyoftheconcaveinstancestowithinthedesiredtolerancelevels withinonehour.Whenconvexrevenuefunctionsareconsidered,BARON'sperformance improves,solvingthesmallersetofinstancesshowninTable4-1withinanaverage ofapproximatelyoneminute.Evenforthelargestnumberofcustomersconsidered, BARONsolves7of10convexinstancesinanaverageofapproximately32minutes. Notably,ourbranch-and-pricealgorithmrequiresonaveragenomorethantwoseconds tosolveproblemswith jJj 3 jIj .Table4-3indicatesthatthebranch-and-pricetime requiredgrowsnotablywhen jJj =5 jIj .However,eachoftheseinstancescouldstill besolvedwithinthedesiredoptimalitytoleranceswithinanhour.Ourtestingrevealed thattheextensivetimesrequiredforexperimentsshowninTable4-3werenotaresult ofthecolumngenerationprocedureattherootnodeorthetimespentsolvingthe MIPusingcolumnsgeneratedattherootnode.Unfortunately,thetimespentsolving pricingproblemstooptimalityinthesubsequentnodeswasthedirectcauseofthe increasedtimerequirements.Instanceswith jJj =10 jIj customerswereconsideredinthe experimentation,butneitherbranch-and-price,norBARONcouldsolvetheseinstances consistentlywithinonehour.However,branch-and-priceonlyrequiredsecondstosolve instanceswith10facilitieswhencustomerratioswerelimitedto jJj 3 jIj ForthoseinstanceswhichBARONfailedtosolvetooptimalitywithinonehour,the qualityofthebestfoundsolutionswasnotablypoor.Forsmallerconcaveinstancesi.e., jJj 3 jIj therelativeerrorbetweenthebestupperboundandincumbentsolution rangesbetween55%and74%.However,foreachoftheseinstances,branch-and-price solvedtheproblemstooptimalityinamatterofseconds.Evenwhenthetrueoptimal solutionobtainedbybranch-and-priceisusedtoassessthequalityoftheBARON solution,theaveragerelativeerroronlydecreasestobetween8%and34%.Theseresults weresucienttodeterminethatrunningconcaveexperimentswith jJj =5 jIj onthe commercialsolverwasunnecessary;therefore,thecorrespondingcolumnisemptyinTable 57

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4-3.Asnotedpreviously,BARONwasmoresuccessfulinsolvingtheconvexinstances weconsidered.Therefore,evenforthefourinstanceswith jJj =5 jIj thatcouldnotbe solvedinonehour,thecorrespondingerrorsaresubstantiallyreducedfromthoseobtained inanyoftheconcaveinstances.Somewhatunexpectedly,theresultsfromthecommercial solverindicatethatprobleminstanceswithconcaverevenuefunctionswerefarmore dicultthantheoneswithconvexrevenuefunctions.Thedierencebetweenthesetwo typesofrevenuefunctionsislesspronouncedwhencomparingtheperformanceofthe branch-and-pricealgorithm.Tables4-1{4-3indicatethatthenumberofcolumns,number ofnodes,andsolutiontimeinbothcaseswereconsistentineachproblemsetconsidered. Inadditiontothecontributionofthebranchingrulediscussedpreviously,thesuccess ofourbranch-and-pricealgorithmfornonlinearrevenuefunctionscanbeattributedtothe combinedeectivenessoftheproposedmethodologiestosolveourpricingproblemandthe tightnessoftheset-partitioningformulation.Thelatterisevidentbythelimitednumber ofnodesexploredinthesearchtree.EachoftheTables4-1{4-3indicatesthatnomore than30nodeswereconsideredforanysingleinstance.Interestingly,5ofthe10instances presentedinTable4-1weresolvedattherootnode. 4.4.2.2Piecewiselinearandlinearrevenuefunctions:comparisonwith CPLEX Tables4-5{4-10comparetheperformanceofourbranch-and-priceapproachforconvex andconcavepiecewiselinearfunctionsagainstlinearizedformulationssolvedinCPLEX. TheresultsshowninTables4-5and4-6indicatethatbranch-and-pricesolvesallpiecewise linearconvexinstanceswithacustomer-to-facilityratiolessthanorequaltothree, regardlessofthenumberofsegmentscomprisingthepiecewiselinearfunction,inless thanonesecond.However,CPLEXrequiresatleast14timestheaveragecomputational timeofbranch-and-priceforinstanceswith jJj 3 jIj .Forpiecewiselinearconvex instanceswith25customers,thebranch-and-pricetimesgrownotably.However,Table 4-7showsouralgorithm,onaverage,outperformsCPLEXforthisscenario,aswell.In 58

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addition,itisclearthatasthenumberofsegmentsincreases,CPLEXrequiresnotably morecomputationaltime,whichisaresultoftheadditionalbinaryvariablesrequiredto modelpiecewiselinearconvexfunctions.Tothecontrary,Tables4-5and4-6suggestthat thenumberofsegmentshaslittleeectonthetimerequiredbybranch-and-price.While Table4-7doesindicateanincreasingtrendin average timerequiredbybranch-and-price asthenumberofsegmentsincreases,ananalysisofindividualinstancessuggeststhis trendisfoundonlyinexperiments2and6.Tables4-8{4-10showthatbranch-and-priceis competitivewithCPLEXonpiecewiselinearconcaveinstanceswith jJj 3 jIj .However, CPLEXperformssignicantlybetteronconcavepiecewiselinearinstanceswhencompared tothepiecewiselinearconvexresults.Thisresultisexpected,sincepiecewiselinear concaverevenuefunctionscanbemodeledwithadditionalcontinuousdecisionvariables andlinearconstraints,insteadofthebinaryvariablesrequiredforpiecewiselinearconvex functions.Table4-10demonstratesthatCPLEXisabetteralternativeastheratioof customerstofacilitiesincreasesforpiecewiselinearconcaveinstances. Wealsoincludedlinearinstancesinourcomputationaltesting.Ourtestingindicated thatCPLEXwasmoresuccessfulthanbranch-and-priceonproblemswithlessthan30 facilitiesandacustomer-to-facilityratioof2,3,or5.However,asshowninTable4-11, CPLEXisunabletosolveanyofthe10instanceswith30facilitiesand60customers withinanhour,whileourbranch-and-pricemethodologysolveseachoftheseinstancesin anaverageoflessthan15seconds.Unfortunately,ourtestsonlargerinstancesi.e.,30 facilitieswith90and150customersrevealedthanneitherCPLEXnorbranch-and-price wassuccessfulwithinthespeciedtimelimit. 4.4.3GAPFJResults InthissectionwepresentresultsfortheGAPFJ,whichisaspecialcaseofthe CFLFDwhenfacilityprocurementdecisionsareomittedfromthemodel.Forthese problems,wechosetofocusondierentiablenonlinearandpurelylinearrevenue functions.ThenonlinearresultsshowninTables4-12{4-14indicatethatthebranch-and-price 59

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algorithmperformsequallyaswellfortheGAPFJasitdidfortheCFLFD.However,as indicatedbyTable4-15,CPLEXoutperformsourbranch-and-priceforlinearinstancesof theGAPFJ.ThesuccessofCPLEXwasevidentforinstanceswithcustomer-to-facility ratiosof2 jIj and5 jIj aswell.AcomparisonofTable4-12toTable4-1suggests thatBARONismoresuccessfulonconcaveinstanceswhichomitthebinaryfacility procurementvariables.Incontrast,theaveragetimetosolveconvexGAPFJinstances with jJj =2 jIj ismorethanthatrequiredfortheCFLFDinstanceswiththesame numberoffacilitiesandcustomers.Notethatadierentsetofdatawasusedinour GAPFJexperiments,socautionshouldbetakenincomparingresultsfromthegeneral CFLFDandtheGAPFJspecialcase.However,theGAPFJconcaveexperimentswere stillclearlydicultforBARON.Only9ofthe30experimentsconsideredweresolved withintheallottedtime.In3ofthe10experimentsshowninTable4-14,BARONwas unabletoevenndafeasiblesolutionwithinanhour.Incontrast,whileonly11ofthe 30convexinstancesweresolvedbythecommercialsolver,theerrorsofthebestsolution obtainedweremuchbetterthanthoseobtainedintheconcavetests.Infact,in12ofthe 19instancesunsolvedinTables4-13and4-14,thebestsolutionsobtainedwereactually optimal,butBARONhadyettoproveoptimalitywhenthetimelimitwasreached. Thebranch-and-pricetimesinTables4-12{4-14indicatethesuccessoftheproposed methodologyonthisimportantspecialcase.Forexample,theresultscorrespondingto jJj 3 jIj showthattheaveragebranch-and-pricetimewasonlyafractionofasecond. Thesamesetoftablesrevealsthatthenumberofnodesconsideredwasminimal.Infact, in14ofthe15instancesconsideredinthesethreetables,theproblemsweresolvedatthe rootnode.Theincreaseinbranch-and-pricetimefor jJj =5 jIj islesssignicantthanthe resultsobtainedfortheCFLFD. Testingonlargerinstancesindicatedthatproblemswith jJj =10 jIj customers remainsolvableinunderanhour.Inaddition,ifthecustomer-to-facilityratioislessthan three,thenbranch-and-priceissuccessfulinsolvinginstanceswith10facilitiesinless 60

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than15minutes.Thissuggeststhatthecustomer-to-facilityratioisthekeyfactorin determiningthesizeofprobleminstancesinwhichbranch-and-priceissuccessful. 4.5Conclusions Inthischapterweconsideredageneralizationofthecapacitatedfacilitylocation problemthatincludesconsiderationsforexibledemand.Weproposedanexact branch-and-pricealgorithmtosolvetheresultingCFLFDproblembasedonaset-partitioning representationofourmodel.Tosolvetheresultingpricingproblem,westudiedan interestinggeneralizationofnonlinearknapsackproblemswithexibleitemsizes. Motivatedbyarelevantrelaxationwithanoptimalsolutionshowntopossessattractive structuralproperties,wediscussedhowboththeheuristicandexactapproachesused tosolvetheresultingKP[81]canbeappliedtosolveourpricingprobleminitsmost generalform.Then,forrevenuestructurescommontoreal-worldpricingconditions,we proposedanalternativepricingproblemsolutionmethodology,whichismoreecient thanthatrequiredinthemostgeneralcase.Ourcomputationalstudysuggeststhatthe branch-and-priceapproachproposedinthisworkconsistentlyoutperformsapopular commercialnonlinearsolverinourtestingofnonlinearCFLFDandGAPFJinstances. Table4-1.CFLFDwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 1140130.20.83600 ; 12 316130.10.329.5 28590.10.23600 ; 21 8390.10.146.2 3198330.11.83600 ; 17 474330.10.5116.2 4156210.11.03600 ; 13 230210.10.455.6 510770.10.43600 ; 18 10570.10.221.7 6141210.10.93600 ; 8 140210.10.370.6 7118130.20.63600 ; 13 134130.10.222.7 810690.30.73600 ; 9 11290.10.341.3 9188290.11.63600 ; 13 471290.10.5124.8 10170230.11.23600 ; 20 271210.10.499.6 Avg140.917.80.10.93600 ; 14 233.617.60.10.362.8 hello 61

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Figure4-1.Illustrationof j and j forageneralrevenuefunction Figure4-2.Concaveenvelope:convexrevenuefunction 62

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Figure4-3.Concaveenvelope:linearrevenuefunction Table4-2.CFLFDwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 113210.20.23600 ; 29 15410.10.127.9 214111.11.13600 ; 31 15410.10.175.8 313910.20.23600 ; 31 15610.10.196.0 4348190.44.43600 ; 34 363190.21.2621.9 513910.20.23600 ; 25 12910.10.151.9 613510.30.33600 ; 23 13910.10.1101.1 7278150.31.73600 ; 23 313150.10.7232.9 822490.32.23600 ; 28 26690.31.175.1 913730.30.33600 ; 21 17750.10.2154.5 10276150.55.03600 ; 23 344170.21.5324.2 Avg194.96.60.41.63600 ; 27 219.570.10.5176.1 63

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Table4-3.CFLFDwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 1863172.3177.3-623111.319.8656.9 27309418.41393.9-672917.3166.2600.9 3109051.21128.3-1627250.630.33600 ; 2 41299234.2847.6-1554231.5634.73600 ; 1 51083233.1183.4-1212230.755.42931.6 61217234.2417.5-1358270.787.32878.7 71080192.9285.3-1358191.117.13600 ; 3 811362512.4586.3-1207213.51229.82716.7 9727135.936.2-9301518.8698.2913.2 10837174.9108.7-1133172.4335.63041.1 Averages1006.219.446.0516.4-1163.018.84.9327.42088.9 Table4-4.BranchingrulecomparisonfortheCFLFDwithnonlinearrevenuefunctions:5 facilities,10customers, =1 : 2 ConcaveConvex branchingon x branchingon y branchingon x branchingon y BPtotalBPtotalBPtotalBPtotal ExpNodestimeNodestimeNodestimeNodestime secsecsecsec 1612.8130.5811.8130.1 230.290.130.290.1 32917.9330.93216.8330.1 4992.5210.6912.6210.1 5110.570.3140.670.1 65558.2210.650111.9210.1 7691.7130.3852.9130.1 870.890.591.190.1 966510.9290.976518.7290.1 101092.3230.81073.0210.1 Averages1873.7817.80.6197.74.9617.60.1 hello hello hello hello hello hello 64

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Table4-5.CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,10 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.51.40.40.70.42.3 20.11.10.11.60.16.1 30.94.00.84.11.110.7 40.82.50.52.40.68.1 50.31.50.20.70.33.9 60.713.60.512.90.7289.5 70.54.20.43.60.515.5 80.75.40.43.60.624.9 91.147.20.820.10.9241.4 100.89.90.46.00.9330.0 Avg0.69.10.45.60.693.2 Table4-6.CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,15 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.20.80.10.80.32.0 20.37.50.14.70.351.2 30.11.10.12.50.23.2 41.95.71.05.42.022.7 50.11.10.10.70.22.9 60.23.30.22.00.36.9 71.014.90.89.81.249.8 81.243.30.6241.41.11052.0 90.34.40.33.00.516.0 102.817.32.115.22.862.2 Avg0.89.90.528.50.9126.9 hello hello hello hello hello hello 65

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Table4-7.CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,25 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 143.8103.926.2172.619.0464.0 224.78.9131.245.61798.53600.0 326.3100.476.2120.323.8324.3 4307.5257.8977.8505.4517.01360.6 554.694.8158.4381.759.5357.7 6177.4104.3261.0155.8899.1562.2 731.5510.519.01400.225.53600.0 871.1223.788.6248.544.1861.1 942.47.842.016.228.03600.0 1069.755.326.588.551.53600.0 Avg84.9146.7180.7313.5346.61833.0 Table4-8.CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,10 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.20.70.20.60.41.1 20.10.70.10.70.11.1 30.72.90.73.21.36.7 40.41.40.41.00.82.6 50.20.60.20.60.41.0 60.42.60.52.50.84.6 70.21.10.30.60.51.6 80.31.70.31.60.83.6 90.74.40.82.81.67.2 100.62.30.62.61.46.0 Avg0.41.80.41.60.83.5 hello hello hello hello hello hello 66

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Table4-9.CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,15 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.10.50.10.60.11.1 20.20.80.20.50.61.8 30.10.60.10.60.21.3 41.01.31.52.23.52.7 50.10.60.10.40.10.8 60.11.00.11.40.32.5 70.61.40.81.41.74.5 80.70.91.20.81.21.6 90.11.20.12.00.33.4 102.04.71.74.44.510.1 Avg0.51.30.61.41.23.0 Table4-10.CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,25 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 144.47.292.89.0313.556.1 245.51.4145.42.5614.93.8 388.12.0132.37.9472.96.2 4375.029.0293.560.1548.8182.4 5172.88.458.210.7507.731.5 6136.927.3410.638.7322.659.0 737.23.242.74.1201.06.7 873.77.085.313.8506.333.6 963.24.0120.35.6183.212.3 1035.75.260.83.4120.34.7 Avg107.39.5145.515.6379.139.6 hello hello hello hello hello hello 67

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Table4-11.CFLFDwithlinearrevenuefunctions:30facilities,60customers, =1 : 2 Linear RootBPtotalCPLEX ExpColsNodestimetimetime secsecsec 1107119.99.93600 : 6 ;: 2 296479.515.13600 : 7 ; 0 398816.96.93600 : 6 ;: 1 4928110.310.33600 : 5 ; 0 51007112.712.73600 ;: 2 6100819.39.33600 : 8 ;: 1 7104418.08.03600 : 4 ;: 2 892535.65.03600 : 6 ;: 1 91176318.937.23600 : 8 ;: 2 101116710.013.23600 : 3 ;: 2 Avg1022.75.49.112.83600 : 7 ;: 1 Table4-12.GAPFJwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 15510.10.11123.35410.10.1100.0 25810.10.11864.65710.10.1112.2 34910.10.11036.65010.10.186.1 46510.10.13600 ; 1 6410.10.163.7 56210.10.12089.36310.20.277.9 66210.10.11636.06310.10.1101.9 76410.10.11744.56610.10.1245.1 86310.10.12539.76510.10.165.8 96610.10.11741.16010.10.1182.8 106410.10.1897.86710.10.155.6 Avg60.810.10.11827.3 ;: 1 60.910.10.1109.1 hello hello hello hello hello hello hello 68

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Table4-13.GAPFJwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 110110.20.23600 ; 8 9610.30.32994.8 210610.30.33600 ; 12 11210.40.43600 ; 0 312510.50.53600 ; 12 12110.50.53600 ; 0 412510.60.63600 ; 10 11810.30.33600 ; 0 511910.40.43600 ; 12 11510.30.33600 ; 0 612110.20.23600 ; 11 12410.30.33600 ; 0 711010.30.33600 ; 15 11210.20.23600 ; 0 810910.30.33600 ; 12 11410.50.53600 ; 0 911410.50.53600 ; 11 11510.60.63600 ; 0 109810.20.23600 ; 10 11710.30.33600 ; 0 Avg112.810.40.43600 ; 11 114.410.40.43539.5 ; 0 Table4-14.GAPFJwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 126716.06.03600 ; 33 25511.11.13600 ; 1 2274186.786.73600 ; 09 286166.866.83600 ; 4 3246113.713.73600 ; 35 26112.22.23600 ; 1 4293164.964.93600 ; 25 25818.48.43600 ; 0 53331118.237.73600 ; 37 345731.143.13600 ; 1 624814.24.23600 )]TJ/F22 7.9701 Tf 6.587 0 Td [(; )]TJ/F20 7.9701 Tf 6.587 0 Td [( 272138.338.33600 ; 0 726216.16.13600 ; 18 25211.91.93600 ; 2 823512.82.83600 ; 29 23711.21.23600 ; 0 922213.83.83600 )]TJ/F22 7.9701 Tf 6.587 0 Td [(; )]TJ/F20 7.9701 Tf 6.587 0 Td [( 22112.92.93600 ; 1 10256121.121.13600 )]TJ/F22 7.9701 Tf 6.587 0 Td [(; )]TJ/F20 7.9701 Tf 6.587 0 Td [( 262111.111.13600 ; 1 Avg263.62.022.824.73600 ; 27 264.91.616.517.73600 ; 1 hello hello hello hello hello hello hello 69

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Table4-15.GAPFJwithlinearrevenuefunctions:30facilities,90customers, =1 : 2 Linear RootBPtotalCPLEX ExpColsNodestimetimeTime secsecsec 11130179.089.410.0 21052399.4160.160.1 31109189.0104.815.5 410573116.1193.176.7 51053391.497.86.0 61075339.143.03.6 711243108.5129.520.6 81053333.876.241.9 91101155.1107.852.3 1011201133.2192.358.4 Avg1087.42.284.5119.434.5 hello hello hello hello hello hello hello hello 70

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CHAPTER5 GAPFDHEURISTICWITHASYMPTOTICPERFORMANCEGUARANTEES Inthischapterweconsidertheprot-maximizingGAPFDthatrequiresthe assignmentofcustomerswithexibledemandtoavailablecapacitatedresources facilities.AsdiscussedinChapter3,thisclassofproblemsndsapplicationina widerangeofpracticalsettings.Relatedproblemsinsalesandadvertisingplanning involvetradeosbetweenrevenuegenerationandresourceconstraintsandcosts.Insales forceplanningcontexts,forexample,thesalesforceservesasasetofresources,where eachsalespersonhasalimitedamountoftimeand/oreortthattheycanallocateto customers.Itisoftenthecasethatthegreatertheamountofeortasalespersonallocates toagivencustomer,thegreaterthereturnfromthatcustomerintermsofsales.The planningphasethereforeinvolvesdeterminingtheassignmentofsalesforcetocustomers andthedegreeofeortasalespersonshoulddevotetoeachassignedcustomerinorder tomaximizethetotalreturnfromcustomersorexpectedreturn,whentherelationship betweeneortandsalesisnotdeterministic.Thissalessettingmaybeinterpretedmore generallyasapplyingtoasetofavailablemarketinginstruments,whereanallocationof capacity-constrainedmarketinginstrumentstocustomersmustbedeterminedinorderto maximizeprot. SincetheGAPFDgeneralizestheGAP,itisclearly NP -Hard.Furthermore,since thefeasibilityproblemassociatedwiththeGAPis NP -Complete,itisclearthatthe feasibilityproblemassociatedwiththeGAPFDis NP -Completeaswell.Wetherefore developacustomizedfamilyofheuristics,andshowthatthisclassofheuristicsis asymptoticallyfeasibleandoptimalwithprobabilityoneasthenumberofcustomers goestoinnityunderaverybroadprobabilisticmodelfortheproblemparameters.Our heuristicsareinthesamespiritascertainheuristicsthathavebeendevelopedforthe GAPbyMartelloandToth[64]andRomeijnandRomeroMorales[79].Inparticular, givenavectorofmultiplierseachcorrespondingtoafacility,aweightfunctionisdened 71

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tomeasurethepseudo-protofassigningacustomertoafacility.Thisweightfunction isthenusedtojudiciouslydetermineitheorderinwhichtoassignthecustomers,ii thefacilitytowhicheachcustomershouldbeassigned,andiiianappropriatecustomer demandfulllmentlevel.Inaddition,thesefunctionsmotivateimprovementheuristics thatareessentialinordertobeabletoderiveattractiveperformanceguaranteesfor theheuristic.Themainresultofthischapteristhedevelopmentofaheuristicthatis asymptoticallyfeasibleandoptimalwithprobabilityoneunderaverygeneralstochastic modeloftheproblemparameters.DuetothenatureoftheGAPFD,ourapproachfor obtainingsuchguaranteesis,particularlyforthemostgeneralversionofourmodel, signicantlydierentfromapproachesusedincaseoftheGAP.Specically,werely heavilybothonthesolutiontoasuitableperturbationoftheGAPFDandoncarefully designedsolutionimprovementtechniques.Thus,inadditiontocontributingtothe literatureonappliedoptimizationinoperations,wealsoprovidenewtechniquesfor algorithmdevelopmentandasymptoticanalysisforcombinatorialoptimizationproblems. Asourcomputationaltestsshow,ourheuristicsolutionapproachisabletondoptimalor near-optimalsolutionswithverylimitedcomputationaleortforabroadrangeofproblem dimensions. Theremainderofthischapterisorganizedasfollows.Section5.1providesanumber ofimportantstructuralresults;theseresultsbothmotivateaclassofheuristicsand enableustoderiveassociatedperformanceguaranteesinSection5.2.Section5.3discusses approachestofurtherimprovetheheuristicmethodswepropose,andinSection5.4we presenttheresultsofourcomputationalstudy,whichvalidatetheeectivenessofour proposedmethods. 5.1ModelAnalysis TheheuristicproposedinthischapterisdevelopedwithrespecttotheGAPFD presentedinChapter3withlinearrevenuefunctions.Thatis,perunitofcustomer demandfulllmentarevenueof r ij isaccrued.Moreover,weassumethat J i = J i 2I 72

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Specically,thevariantoftheGAPFDintroducedinChapter3studiedinthischapteris givenby: maximize X i 2I X j 2J p ij x ij + X i 2I X j 2J r ij v ij {1 subjecttoP X j 2J a ij x ij + X j 2J v ij b i i 2I {2 X i 2I x ij =1 j 2J {3 v ij ` ij x ij i 2I ; j 2J {4 v ij u ij x ij i 2I ; j 2J {5 x ij 2f 0 ; 1 g i 2I ; j 2J : {6 Notethattheassumptionthattheunitresourceconsumptioncoecientsareequalto onecanbemadewithoutlossofgenerality.Moreover,wewillassumewithoutlossof generalitythattheunitrevenuesarenonnegative.Inprinciple,weallowthexedprot andresourceconsumptioncoecientstobeeitherpositiveornegative.However,inmost real-lifeapplicationsweshouldexpectthesecoecientstobenonnegative.Finally,note thatwithoutlossofgeneralitywecouldassumethat ` ij =0forall i 2I and j 2J by appropriatelymodifyingthexedprotandresourceconsumptioncoecients.However, forclarityofinterpretationofourmodel,algorithms,andresultswewillallowforpositive valuesoftheselowerboundsonthecustomerdemandfulllmentlevels. Thesolutionapproachthatwewilldevelopandanalyzeinthischapterisaclass ofheuristicsthatisinspiredbytheLagrangerelaxationofareformulationofthe LP-relaxationofP.Inparticular,aswewillshowbelow,theoptimizationproblemLP thatisobtainedbyreplacingthebinaryconstraints5{6bynonnegativityconstraints x ij 0 i 2I ; j 2J 5{6 0 73

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isequivalenttotheproblem maximize X i 2I X j 2J p ij + r ij u ij s ij + X i 2I X j 2J p ij + r ij ` ij t ij {7 subjecttoLP 0 X j 2J a ij + u ij s ij + X j 2J a ij + ` ij t ij b i i 2I {8 X i 2I s ij + t ij =1 j 2J {9 s ij ;t ij 0 i 2I ; j 2J : {10 Theorem2. TheoptimizationproblemsLPandLP 0 areequivalent. Proof. FirstnotethatwemaymodifyLPbyexplicitlyintroducingnonnegativesurplus andslackvariablestoconstraints5{4and5{5.Forconvenience,wewillscaletheseso thattheyareexpressedasafractionofthewidthofthesizerangeofthecorresponding assignment.Inotherwords,constraints5{4and5{5arereplacedby v ij )]TJ/F15 11.9552 Tf 11.955 0 Td [( u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij s ij = ` ij x ij i 2I ; j 2J 5{4 0 v ij + u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij t ij = u ij x ij i 2I ; j 2J 5{5 0 s ij ;t ij 0 i 2I ; j 2J : {11 Itiseasytoseethatthisreformulationisvalidevenifthewidthofthesizerangeofan assignmentis0,i.e.,if ` ij = u ij .Nowsubtractingconstraints5{4 0 from5{5 0 yields u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij x ij = u ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij s ij + t ij i 2I ; j 2J sothatwecanset,withoutlossofgenerality, x ij = s ij + t ij i 2I ; j 2J : {12 Moreover,multiplyingconstraints5{4 0 and5{5 0 by u ij and ` ij respectivelyyields 74

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` ij u ij x ij + u ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij s ij = u ij v ij i 2I ; j 2J 5{4 00 ` ij u ij x ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij t ij = ` ij v ij i 2I ; j 2J : 5{5 00 Subtracting5{5 00 from5{4 00 ,weobtain u ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij u ij s ij + ` ij t ij = u ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij v ij i 2I ; j 2J sothatwecanset v ij = u ij s ij + ` ij t ij i 2I ; j 2J : {13 Noticethatthenon-negativityoftheslackandsurplusvariablesby5{11alongwith 5{12and5{13impliesthat5{10issucienttoensureallnon-negativityconditions inLParealsosatisedinLP 0 .Finally,substituting5{12and5{13intothe objective5{1andconstraints5{2and5{3ofPyieldstheobjective5{7aswellas constraints5{8and5{9ofLP 0 Next,denotethenonnegativedualmultipliersofthecapacityconstraints5{8by i i 2I andthefreedualmultipliersoftheassignmentconstraints5{9by j j 2J ThedualD 0 ofLP 0 isthengivenby minimize X i 2I i b i + X j 2J j subjecttoD 0 j p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij i 2I ; j 2J {14 j p ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i u ij i 2I ; j 2J {15 i 0 i 2I j free j 2J : 75

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Thefollowingtheoremderivesaconvenientandinsightfulexpressionfortheoptimalvalue tobothLP 0 andD 0 asafunctionofthedualmultipliers i i 2I ofthecapacity constraints5{8only. Theorem3. ThecommonoptimalvalueofLP 0 andD 0 canbeexpressedas min 0 L where L = X j 2J max i 2I f i;j + X i 2I i b i andwhere f i;j = 8 > < > : p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i u ij if i r ij p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij if i >r ij : Proof. Fromconstraints5{14and5{15weobtainthat,withoutlossofoptimality,the dualvariables j canbechosenas j =max i 2I max f p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij ;p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i u ij g j 2J : {16 Nownotethattheinnermaximumin5{16isattainedbytherstargumentif i r ij andbythesecondargumentif i r ij .Thisimpliesthatweinfacthave j =max i 2I f i;j j 2J {17 whichyieldsthedesiredresult. Itisusefultointroducesometerminologywithrespecttoafeasiblesolution s;t to LP 0 .Considersomecustomer j .If x ij = s ij + t ij =1forsomefacility i wesaythat customer j isassignedtofacility i andfurthermore,customer j isreferredtoasa non-split customer.Similarly,if0
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Moreformally,wedenetheset F = f i;j :0 0and t ij =0isexecutedatitsupperbound,whileanassignment i;j suchthat s ij =0and t ij > 0isexecutedatitslowerbound.Theset Q = f i;j : s ij > 0and t ij > 0 g thenconsistsofthefacility/customerpairsexecutedstrictlybetweentheirbounds.Finally, C = i : X j 2J a ij x ij + X j 2J v ij = b i = i : X j 2J a ij + u ij s ij + X j 2J a ij + ` ij t ij = b i isthesetoffacilitiesthatoperateatfull-capacity. Thefollowingtheoremestablishesacloserelationshipbetweenanoptimalsolutionto D 0 andthecorrespondingprimaloptimalsolution,providedthatthelatterisunique. Theorem4. SupposethatLP 0 isfeasibleandthattheoptimalbasicsolutionto LP 0 ,say s ;t ,isunique.Furthermore,let beanassociatedcomplementaryoptimal solutiontoD 0 .Theprimalanddualsolutionsthensatisfythefollowingproperties. iLet j 2S beasplitcustomer.Thenthereexistsafacility i 0 suchthat f i 0 ;j =max i 2I i 6 = i 0 f i;j : iiLet j 62S beanon-splitcustomer.Thenitisassignedtofacility i 0 ifandonlyif f i 0 ;j =max i 2I f i;j and f i 0 ;j > max i 2I i 6 = i 0 f i;j : 77

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iiiLet j 62S beanon-splitcustomerthatisassignedtofacility i 0 .Then s i 0 j =1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(t i 0 j 8 < : =0 if i 0 >r i 0 j 2 [0 ; 1] if i 0 = r i 0 j =1 if i 0 0.Bythedenitionof x ,complementary slacknessnowimpliesthatforboth i 0 and i 00 ,atleastoneofthecorresponding dualconstraints5{14and5{15isbinding.This,inturn,impliesthatforthis customerthemaximuminequation5{17isattainedforbothfacilities i 0 and i 00 yieldingclaimi. iiLet j 62S beanon-splitcustomer.Thisimpliesthatthereexistsonlyasingle facility,say i 0 ,suchthat x i 0 j > 0infact, x i 0 j =1.Bycomplementaryslackness andthenondegeneracyofthedualsolution,thismeansthatforallfacilitiesexcept i 0 bothcorrespondingdualconstraints5{14and5{15arenonbinding.This,inturn, impliesthatforthiscustomerthemaximuminequation5{17isattainedforonly facility i 0 ,yieldingclaimii. iiiLet j 62S beanon-splitcustomerthatisassignedtofacility i 0 ,sothat x i 0 j =1. First,recallthat s ij istheprimalvariableassociatedwith5{14and t ij isthe primalvariableassociatedwith5{15.Nowbycomplementaryslacknessanddual nondegeneracywehavethat s i 0 j > 0and t i 0 j =0ifandonlyif i 0 >r i 0 j s i 0 j =0 and t i 0 j > 0ifandonlyif i 0 0and t i 0 j > 0ifandonlyif i 0 = r i 0 j Togetherwith5{12thisyieldsthedesiredresult. ItisinterestingtoseewhatTheorem4impliesintermsofthe x;v variablesinour originalLPformulation. Corollary1. SupposethatLP 0 isfeasibleandthattheoptimalbasicsolutionto LP 0 ,say s ;t ,isunique.Furthermore,let beanassociatedcomplementaryoptimal solutiontoD 0 .Thenthereexistsanoptimalsolution x ;v toLPthatsatisesthe followingproperty.If j 62S isanon-splitcustomerthatisassignedtofacility i inthe 78

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optimalsolutiontoLP 0 ,then v i 0 j 8 > > > > < > > > > : = ` i 0 j if i 0 >r i 0 j 2 [ ` i 0 j ;u i 0 j ] if i 0 = r i 0 j = u i 0 j if i 0
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Proof. TheoptimizationproblemLP 0 has2 jIjjJj variablesand jIj + jJj equality constraints.Thetotalnumberofvariableswhicharenonzeroinabasicfeasiblesolutionis thereforenolargerthan jIj + jJj .Nowobservethatthereare jJj)-222(jSj + jFj + jQj non-zerocomponentsin s;t ; jIj)-222(jCj non-zeroslackvariablesassociatedwithconstraints5{8; Thisyields jIj + jJj jJj)-222(jSj + jFj + jQj + jIj)-222(jCj whichyieldsthedesiredresult. Corollary2. Let s;t beabasicfeasiblesolutiontoLP 0 .Thenthetotalnumberof customersthatareeithersplitorexecutedstrictlybetweentheirboundsisboundedbythe numberoffacilities,i.e., jSj + jQjjIj : Proof. Notethateachsplitcustomerhasatleasttwocorrespondingfractionalassignment variables,sothat jFj 2 jSj .TheresultthenfollowsdirectlyfromTheorem5andthefact that jCjjIj 5.2AnAsymptoticallyOptimalHeuristic 5.2.1DevelopmentoftheHeuristic ThereisanattractiveintuitiveinterpretationoftheresultofTheorem3bynoting thatwecaninterpretthevalueofthedualvariable i asaunitcostofcapacityoffacility i .Then,notethatwecanview p ij )]TJ/F21 11.9552 Tf 12.953 0 Td [( i a ij asaxedpseudo-protthatisreceivedif customer j isassignedtofacility i ,regardlessofitslevel.Next,wecanviewthedierence betweentheactualcorrespondingunitrevenue r ij andthecost i ofusingaunitof capacityoffacility i asaunitpseudo-protthatisreceivedifcustomer j isassignedto facility i .Thesignofthepseudo-protthenindicatesthelevelatwhichacustomershould beassigned:iftheunitpseudo-protispositive,customer j ifassignedtofacility i is executedatitsupperbound u ij ,yieldingatotalpseudo-protof p ij )]TJ/F21 11.9552 Tf 12.026 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 12.026 0 Td [( i u ij 80

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Similarly,iftheunitpseudo-protisnegative,customer j ifassignedtofacility i is executedatitslowerbound ` ij ,yieldingatotalpseudo-protof p ij )]TJ/F21 11.9552 Tf 12.182 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 12.181 0 Td [( i ` ij Insummary,thefunction f i;j canbeviewedasa pseudo-prot associatedwiththe assignmentofcustomer j tofacility i foragivenvectorofdualprices WewillusethisinterpretationtoproposeaheuristicfortheGAPFD.Thatis,our heuristicwill,toalargeextent,assigncustomersaccordingtothepseudo-protfunction. Notethatanynonnegativevector denesadistinctpseudo-protfunction,sothat wewillinfactobtainafamilyofheuristics.However,welatershowthattheheuristic enjoysanattractiveperformanceguaranteeifweuseanoptimaldualsolutiontoeither theoriginalproblemoraperturbationthereof.Inparticular,wewillattempttoassign eachcustomertothefacilitythatmaximizesitspseudo-protfunctionandselectthe correspondingcustomerdemandfulllmentlevelaccordingly.Morespecically,themost protablefacilityforcustomer j isgivenby i j =argmax i 2I j f i;j where I j I isthesetoffacilitiescurrentlyunderconsiderationforcustomer j .Itiseasy toseethat,ingeneral,assigningallcustomers j totheirmostprotablefacility i j atasize asdescribedintheprecedingparagraphcannotbeexpectedtoyieldafeasiblesolution totheGAPFD.Wethereforeselecttheorderinwhichthecustomersareassignedby consideringnotonlythemaximumpseudo-protbutalsothesecondlargestpseudo-prot foreachcustomer.Wedenethedierencebetweenthesetwovalues: j = f i j ;j )]TJ/F15 11.9552 Tf 20.697 0 Td [(max i 0 2I j nf i j g f i 0 ;j tobethe desirability ofassigningcustomer j toitsmostprotablefacility.Wethenassign thecustomerstotheirmostprotablefacilityinnonincreasingorderofdesirability,aslong asitisfeasibletodoso. 81

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Ourheuristicproceedsintwophases.Intherst,greedyphaseoftheheuristic,the set U keepstrackofthesetofcustomersthatremaintobeassigned.Duringthecourse ofthisphaseSteps1{3,customersmaybeidentiedthatcannolongerbefeasibly assigned.Theset ~ J ofsuchcustomerswillbehandledinthesecond,improvementphase ofthealgorithmSteps4{7.Throughouttheheuristic, b 0 i denotesthecapacityremaining forfacility i i 2I Wenowformallypresentourheuristicasfollows: Heuristic{Greedyphase Step0. Set U = J ~ J =, b 0 i = b i for i 2I ,and I j = I .Set x G ij = v G ij =0for i 2I ; j 2J Step1. Let i j 2 argmax i 2I j f i;j for j 2U j = f i j ;j )]TJ/F15 11.9552 Tf 20.697 0 Td [(max i 0 2I j nf i j g f i 0 ;j for j 2U : Step2. Select^ | 2 argmax j 2U j ,i.e.,^ | isthecustomertobeassignednexttofacility i ^ | If a i ^ | j + ` i ^ | b 0 i ^ | continuetoStep3.Otherwise, a i ^ | j + ` i ^ | >b 0 i ^ | ,whichmeansthis assignmentisnotfeasible;let I ^ | = f i : a i ^ | + ` i ^ | b 0 i g .If I ^ | =,set ~ J = U and STOP. Step3. Set x G i ^ | ^ | =1 v G i ^ | ^ | = 8 > < > : min f u i ^ | ^ | ;b 0 i ^ | )]TJ/F21 11.9552 Tf 11.955 0 Td [(a i ^ | ^ | g if r i ^ | ^ | > i ^ | ` i ^ | ^ | if r i ^ | ^ | i ^ | b 0 i ^ | = b 0 i ^ | )]TJ/F15 11.9552 Tf 11.955 0 Td [( v G i ^ | ^ | + a i ^ | ^ | : Let U = Unf ^ | g .If U6 =,returntoStep1;otherwise,STOP. 82

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Ifthegreedyphaseoftheheuristicendswith ~ J =,then x G ;v G isafeasible solutiontotheGAPFD.Otherwise,wewillcontinuetheheuristicwithanimprovement phase.Todistinguishthepartialsolutionobtainedattheendofthegreedyphasefrom thesolutiondeliveredbytheimprovementphaseweset x H ;v H = x G ;v G Intheimprovementphase,wereducethesizeofsomepreviouslyassignedcustomers totheircorrespondinglowerboundstofreeupcapacitythatcanbeusedtoassignany ofthecustomersin ~ J .Realizingthattheminimumamountofcapacitythatisrequired toassigncustomer j tofacility i is a 0 ij a ij + ` ij ,let a 0 beanupperboundonthisvalue amongallunassignedcustomers.Then,atleast X i 2I b 0 i a 0 1 a 0 X i 2I b 0 i )-222(jIj customerscanbeaccommodatedwithintheremainingfacilitycapacities.Thus,all customersin ~ J canbeassignedifthefacilitieshavecumulativeavailablecapacity P i 2I b 0 i j ~ Jj + jIj a 0 .Notethatsuchanassignmentcanbefoundbyarbitrarily assigningthecustomersin ~ J toanyfacilitythatcanfeasiblyaccommodateit. Heuristic{Improvementphase Step4. Let A = f i;j : x H ij =1and v H ij >` ij g andset a 0 =max i;j 2I ~ J ` ij + a ij Step5. Identifyaset A 0 A withthepropertythat X i 2I b 0 i + X i;j 2A 0 v H ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij j ~ Jj + jIj a 0 {18 and,inaddition, A 0 isminimalinthesensethatremovinganyelementfromit causes5{18tobeviolated.Ifsuchasetdoesnotexist,set A 0 = A Step6. Set b 0 i = b 0 i + X j : i;j 2A 0 v H ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij for i 2I v H ij = ` ij for i;j 2A 0 : 83

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Step7. AttempttoidentifyafeasiblesolutiontotheGAPFDbyiassigningand determiningcustomersandcustomerdemandfulllmentlevelsfor j 2 ~ J ,and iiincreasingcustomerdemanfulllmentlevelsforassignments i;j 2A 0 Ifthisissuccessful,returnthesolutionas x H ;v H .Otherwise,theheuristicis unabletondafeasiblesolutiontotheGAPFD. Intheremainderofthissectionweanalyzeabasicimplementationoftheimprovement phaseoftheheuristicwhereweidentifyanarbitraryset A 0 inStep5,andtrytoassign customersin ~ J inarbitraryordertofacilitiesthatcanaccommodatetheminStep7.In Section5.4wewillproposeamoresophisticatedimplementationwithguaranteedsuperior behavior. Thefollowingtheoremestablishesacloserelationshipbetweenthesolutionthatis obtainedbythegreedyphaseoftheheuristicandabasicoptimalsolutiontoLP 0 Theorem6. SupposethatLP 0 isfeasibleandthattheoptimalbasicsolutiontoLP 0 say s ;t ,isunique.Ifwechoose intheheuristicequaltoanassociatedoptimaldual vector ,wehaveforallnon-splitcustomers j 62S thatthegreedyphaseoftheheuristic iassignsthiscustomertothesamefacility,say i j ,asLP 0 ; iiexecutesthiscustomeratthesamesamelevelasLP 0 provided i j ;j 62Q Proof. Theorem4i{iiimpliesthat,inthegreedyphaseoftheheuristic, j > 0forall j 62S and j =0for j 2S .Thus,Step2guaranteesthatallnon-splitcustomersare consideredbeforeanysplitcustomersaslongasthesets I j remainunchanged.Claimi thenimmediatelyfollowsfromthefactthatthepreferredassignmentsofthenon-split customersareallfeasible.Next,Theorem4iiiimpliesthat,inStep3ofthegreedyphase oftheheuristic,allcustomers j forwhich i;j 62Q areexecutedatthesamelevelas inLP 0 .Notethatthegreedyphasedoesnotnecessarilyexecutecustomersforwhich i;j 2Q atthesamelevelasinLP 0 .Thisfollowssince,bycomplementaryslackness, i;j 2Q isequivalentto j = f i;j = p ij )]TJ/F21 11.9552 Tf 12.447 0 Td [( i a ij ,i.e.theunitrevenue r ij )]TJ/F21 11.9552 Tf 12.446 0 Td [( i =0, 84

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whichdoesnotnecessarilyimplythatthesolutiontoLP 0 makesthisassignmentatits lowerbound,astheheuristicdoes. Theorem6statesthat,ifwechoose intheheuristicequaltoanoptimaldualvector ofLP 0 ,thegreedyphaseoftheheuristicstartsbymakingtheassignmentsofnon-split customersinthesolutiontoLP 0 ,withtheonlypossibledeviationbeingthecustomer demandfulllmentlevelsforthosethatarestrictlybetweentheirboundsinthatsolution. This,togetherwithCorollary2,thenimpliesthatthetotalnumberofcustomersthat areunassignedinthegreedyphaseorforwhichtheassignmentordemandfulllment leveldiersfromthesolutiontoLP 0 isnolargerthanthenumberoffacilities, jIj .The improvementphaseoftheheuristicisaimedatcreatingsucientspacetoallowthe assignmentofanyunassignedcustomers.Ourgoalinthenextsectionistousethisresult toderivestrategiesforchoosing fortwoverygeneralstochasticmodelsontheproblem parameterssuchthattheheuristicisasymptoticallyfeasibleandoptimalwithprobability oneasthenumberofcustomersincreases.Thatis,asthenumberofcustomersincreases, itisveryunlikelythattheheuristicisunabletondafeasiblesolutionand,moreover,ifit ndsafeasiblesolution,itsrelativeerrorwilldeclineasthenumberofcustomersincreases. 5.2.2AverageCaseAnalysisoftheHeuristic Thissectionprovidesananalysisoftheasymptoticbehaviorofourheuristicunder aprobabilisticmodelontheproblemparametersthatkeepsthenumberoffacilities, jIj xedandletsthenumberofcustomers, jJj ,approachinnity.Weproposeastochastic modelfortheGAPFDthatissimilartotheonescommonlyusedfortheGAPandits extensionssee,e.g.,DyerandFrieze[30],RomeijnandPiersma[78],andRomeijnand RomeroMorales[80] 1 .Inparticular,weassumethateachcustomerischaracterized byarandomvectorofparameters P j ;R j ;A j ;L j ;D j ,where P j = P 1 j ;:::;P jIj j R j = 1 Asaconvention,wewilldenoteparametersandsolutionsthatarerandomvariablesby capitalletters,whilerealizationswillbedenotedbylowercaseletters. 85

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R 1 j ;:::;R jIj j A j = A 1 j ;:::;A jIj j L j = L 1 j ;:::;L jIj j ,and D j = D 1 j ;:::;D jIj j .Here P j and R j arethevectorsofxedprotsandunitrevenuesforcustomer j ,respectively. Furthermore, A j isthevectorofxedresourceconsumptionsforcustomer j ,while L j is thevectoroflowerboundsand U j L j + D j isthevectorofupperboundsonthesize ofcustomer j .Thevectors P j ;R j ;A j ;L j ;D j areassumedtobei.i.d.onthecompactset [ P ; P ] jIj [ R ; R ] jIj [ A ; A ] jIj [ L ; L ] jIj [ D ; D ] jIj ,wheretheconditionaldistributionsof P j ;R j j A j ;L j ;D j areabsolutelycontinuous.Inaddition,wehave R ;L 0sothatboth thedemandrequirementandtheunitrevenueaccruedarenon-negative.Furthermore, thedierencebetweentheupperandlowerboundparametersistakentobestrictly positive, D > 0,toensurethereisadecisiontobemadewithregardtothelevelat whichacustomer'sdemandissatised.Forconvenience,weassumethat R L > )]TJ/F21 11.9552 Tf 9.298 0 Td [(P sothatthetotalprotassociatedwith any feasibleassignmentisnonnegative.Note thatthisassumptionismildsinceitisautomaticallysatisedif,forexample, P > 0. Moreover,itcanbemadewithoutlossofgeneralitysincewemayaddorsubtracta constantvaluefromallxedprotcoecientswithoutimpactingtheprotrankingofthe solutions.Furthermore,asiscommoninprobabilisticmodelsofthistype,weallowfor theaccommodationofanincreasingnumberofcustomerswhilethenumberoffacilities remainsconstantbylettingthecapacityoffacility i growlinearlywiththenumberof customers,i.e.,welet b i = i jJj where i isapositiveconstant i 2I Finally,wewishtofocusonprobleminstancesthatadmitafeasiblesolution.Note thataninstanceoftheGAPFDhasafeasiblesolutionifandonlyiftheassociatedGAP withallrequirementssettotheirminimumvalue a ij + ` ij i 2I ; j 2J isfeasible.This leadstothefollowingassumptionthatwewillimposeonourprobabilisticmodel: Assumption1. min 0; > e =1 X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [(E min i 2I i A i 1 + L i 1 > 0 : 86

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ByRomeijnandPiersma[78,Theorem3.2],thisassumptionensuresthataninstance randomlygeneratedaccordingtoourstochasticmodelisfeasiblewithprobabilityone as jJj!1 .Notethatthisassumptionismild,sincetheyalsoshowthatinstances generatedareasymptoticallyinfeasiblewithprobabilityoneas jJj!1 if < 0. Intheremainderofthissection,wewillshowthat,forasuitablychosenstrategyfor theparameter ,theheuristicwillprovideafeasibleandoptimalsolutiontotheGAPFD withprobabilityoneas jJj!1 .Inparticular,consideraninstanceoftheGAPFD generatedfromtheprobabilisticmodeldescribedabove,andlet Z jJj Z LP jJj Z G jJj ,and Z H jJj denoteitsoptimalsolutionvalue,thevalueofitsLP-relaxation,andthevalueofthe solutionobtainedbythegreedyandtheimprovementphasesoftheheuristic,respectively. Notethatthesevaluesarerandomvariables,andthatwehaveexplicitlyrecognizedthat theyareafunctionofthenumberofcustomers, jJj .Wethensaythattheheuristicis asymptoticallyfeasibleandoptimal if ithesolution X H ;V H producedbytheheuristicisasymptoticallyfeasiblewith probabilityone; iilim jJj!1 Z jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj =Z jJj =0withprobabilityone. Since,underourassumptionsontheproblemparameters,wehavethat,foranyfeasible instanceoftheGAPFD, Z jJj R L + P jJj with R L + P > 0,thelatterisequivalentto ii 0 lim jJj!1 1 jJj Z jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj =0withprobabilityone whichisthecharacterizationofasymptoticoptimalitythatwewilluseintheremainderof thischapter. WewilldistinguishbetweentwoclassesofinstancesoftheGAPFD.Therstclass ofinstancesischaracterizedbycustomerrequirementsthatare facility-independent ,i.e., A ij = A 1 j L ij = L 1 j ,and D ij = D 1 j i 2I ; j 2J .Forthisclass,wewillshowthatthe heuristicisasymptoticallyfeasibleandoptimalwithprobabilityoneifwechoose equal toanassociatedoptimaldualvector .Thesecondclassofinstancesfollowsthegeneral probabilisticmodeldiscussedearlierinthissection.Forthisclass,wewillshowthatthe 87

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heuristicisasymptoticallyfeasibleandoptimalwithprobabilityoneifwechoose equal toanoptimaldualvectorofanappropriatelyperturbedinstanceoftheGAPFD. 5.2.2.1Facility-independentrequirements RecallthatTheorem6establishesastrongconnectionbetweenanoptimalsolution toLP 0 andthesolutionobtainedbythegreedyphaseoftheheuristiciftheformeris uniquewithrespecttonon-splitcustomersandifwechoose equaltoanassociated optimaldualvector ofLP 0 .Thisconnectionisemployedtoobtainasymptotic feasibilityandoptimality.Beforeweformallyprovethisresult,however,wewillrst derivetwousefulpreliminaryresults.Therstresultprovidesanintuitivecharacterization ofanoptimalsolutiontoLP 0 undertheproposedstochasticmodel.Thisresultis commonformodelsinwhichparametersaregeneratedfromabsolutelycontinuous distributionssee[29,30,79].Itispresentedhereformallyduetoitssignicanceinour asymptoticanalysis. Lemma3. Underourstochasticmodel,ifLP 0 isfeasible,itsoptimalsolutionisunique withprobabilityone. Proof. Anon-uniqueoptimalsolutiontoLP 0 existsonlyifthehyperplanerepresentative ofsolutionswithoptimalprot X i 2I X j 2J p ij + r ij u ij s ij + X i 2I X j 2J p ij + r ij ` ij t ij intersectsthefeasibleregionatmultiplepoints.Recallthattheunitrevenues r ij i 2 I ; j 2J andthexedprots p ij i 2I ; j 2J aregeneratedfromajointdistribution that, conditional onthevaluesoftherequirementsparametersintheconstraints,is absolutelycontinuous.Thustheprobabilitythatdatageneratedbyourstochasticmodel allowsformultipleoptimalsolutionstoLP 0 iszero,sothattheoptimalsolutiontoLP 0 isuniquewithprobabilityone. 88

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Lemma4. Whencustomerrequirementsarefacility-independent,theaggregatecapacity thatiseitherunusedorusedforcustomerlevelsinexcessoftheirlowerboundincreases linearlyin jJj withprobabilityoneas jJj!1 ,inanyfeasiblesolutiontoLP. Proof. Forconvenience,denotetheeectivelowerboundonanassignmentby A 0 ij = A ij + L ij .Sincethecustomerrequirementsarefacility-independentwehavethat A 0 ij = A 0 1 j i 2I .Then,foranyassignmentvector x thatisfeasibletotheLP-relaxationofthe correspondingGAPwehave 1 jJj X i 2I b i )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X j 2J A 0 ij x ij = X i 2I i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X i 2I X j 2J A 0 1 j x ij = X i 2I i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X j 2J A 0 1 j X i 2I x ij = X i 2I i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X j 2J A 0 1 j : Forthecaseoffacility-independentrequirements,RomeijnandPiersma[78]showthat Assumption1isequivalenttothecondition E A 0 1 j < X i 2I i sothat,bytheCentralLimitTheorem, X i 2I i )]TJ/F15 11.9552 Tf 18.701 8.087 Td [(1 jJj X i 2I X j 2J A 0 ij x ij > 0withprobabilityoneif jJj!1 foranyfeasiblerelaxedassignmentvector x .Thisyieldsthedesiredresult. Wearenowreadytoformallyproveourrstasymptoticfeasibilityandoptimality result. Theorem7. Considerprobleminstancesgeneratedaccordingtoourstochasticmodel with,inaddition,facility-independentcustomerrequirements.Moreover,choose inthe heuristicequaltoanassociatedoptimaldualvector ofLP 0 .Thentheheuristicis asymptoticallyfeasibleandoptimal. 89

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Proof. Sinceweareonlyinterestedinaprobabilisticandasymptoticfeasibilityguarantee, wemaybyLemma3assumethatthesolutiontoLP 0 isuniquewithrespecttonon-split customers.ThenTheorem6saysthatthegreedyphaseoftheheuristicassignsnomore than jSj customerstoadierentfacilityortonofacilityatall,andnomorethan jQj customerstothesamefacilitybutatadierentlevelthantheoptimalsolutiontoLP 0 Againdenotingtheeectivelowerboundofacustomerby A 0 = A + L withcorresponding upperbound A 0 = A + L ,wehavethateachoftheunassignedcustomersrequiresno morethan A 0 unitsofcapacity,sothatitwouldsuceifan aggregate of jSj + jIj A 0 unitsofcapacityamongallfacilitieswereavailable.Nowlet b 0 = P i 2I b 0 i < jSj + jIj A 0 denotetheaggregateremainingcapacityattheendofthegreedyphaseoftheheuristic, andrecallthat,byCorollary2,weknowthat jSj + jQjjIj .Sucientcapacitycan thereforebemadeavailableif,intheimprovementphase,weareabletoreducetotheir lowerboundsthelevelof )]TJ/F15 11.9552 Tf 11.125 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 =D customersthatthegreedyphaseof theheuristicassignedattheirupperbounds.Sincethisnumberisindependentofthe numberofcustomers,Lemma4impliesthat,forlargeenough jJj ,thiscanindeedbe done,implyingthattheheuristicisasymptoticallyfeasible. Finally,notethattheobjectivefunctionvalueofafeasiblesolutionthatisobtained bytheheuristicsatises Z H Z LP )]TJ/F26 11.9552 Tf 11.955 9.684 Td [(\006)]TJ/F15 11.9552 Tf 16.604 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 =D + jSj + jQj D R )-222(jSj P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L : {19 Intherighthandsideofthisinequality, )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 D R isthelostrevenuefrom reducingcustomerstotheirlowerboundsinordertoassureenoughaggregatecapacityfor unassignedcustomersinthegreedyphaseoftheheuristic.Theterm jSj + jQj D R islost revenuefromcustomerseithernotassignedtothesamefacilityasinLP 0 ,orcustomers executedatadierentlevelthanLP 0 .Thequantity jSj P )]TJ/F21 11.9552 Tf 12.44 0 Td [(P + R )]TJ/F21 11.9552 Tf 12.44 0 Td [(R L accounts 90

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forthelossofxedprotresultingfromcustomersnotassignedtothesamefacilityasin LP 0 Clearly Z LP Z jJj Z H implies lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z LP jJj )]TJ/F21 11.9552 Tf 11.956 0 Td [(Z H jJj : Providedthattheheuristicsolutionisfeasiblewethenhave lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(Z LP jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj lim jJj!1 1 jJj \000)]TJ/F15 11.9552 Tf 22.084 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 D + jSj + jQj D R + jSj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.956 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L by5{19 lim jJj!1 1 jJj \000 jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 =D + jIj D R + jIj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L byCorollary2 =0 : Sincetheheuristicisasymptoticallyfeasible,thisimpliesthattheheuristicisasymptotically optimalaswell. 5.2.2.2Facility-dependentrequirements TheresultofLemma4canunfortunatelynotbeextendedtothegeneralcasewhere customerrequirementsarefacility-dependent,preventingusfromextendingtheapproach intheprevioussectiontoshowasymptoticfeasibilityoftheheuristicifwechoose equaltoanoptimaldualvector ofLP 0 .Ingeneral,wethereforetakeadierent approach:wechoose equaltotheoptimaldualvectorofaninstanceofLP 0 inwhich thecapacitieshavebeenreducedbyanappropriatelychosensmallamount.Note,however, thatwewillstillapplythetwophasesoftheheuristicusingtheoriginalcapacities.By ensuringthatthetemporarycapacityreductionsarelargeenoughtoensurethatthe customersthatarefractionallyassignedintheLP-relaxationcanbeassignedfeasiblybut, 91

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atthesametime,smallenoughforthecorrespondingsolutiontobeclosetooptimal,we willbeabletoshowthatthegreedyphaseoftheheuristicaloneisasymptoticallyfeasible andoptimal. Moreformally,consideraninstanceoftheGAPFDwith jJj customers.Then associatewiththisinstanceaperturbedinstanceofLP 0 inwhichallofthenormalized capacities i i 2I arereducedby jJj ,where lim jJj!1 jJj =0{20 lim jJj!1 jJj jJj = 1 {21 and 0 < jJj < min i 2I i : WewilldenotetheperturbedproblembyLP 0 jJj anditsoptimalvalueby Z LP 0 jJj jJj Thefollowingpreliminaryresultshowsthattheoptimalvaluesoftheoriginaland perturbedproblemsareveryclose. Lemma5. TheoptimalvaluesofLP 0 andLP 0 jJj arecloseinthesensethat,with probabilityone, lim jJj!1 1 jJj Z LP 0 jJj jJj =lim jJj!1 1 jJj Z LP 0 jJj : Proof. SeeAppendixA. Thefollowingtheorememploysthisresulttoprovethatwehaveaheuristicforthe GAPFDthatisasymptoticallyfeasibleandoptimal. Theorem8. Considerprobleminstancesgeneratedaccordingtoourgeneralstochastic model.Moreover,choose intheheuristicequaltoanoptimaldualvectortoLP 0 jJj Thentheheuristicisasymptoticallyfeasibleandoptimal. Proof. AsintheproofofTheorem7,wemayassumethatthesolutiontotheperturbed instanceofLP 0 isuniquewithrespecttonon-splitcustomers,sinceweareonly 92

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interestedinaprobabilisticandasymptoticfeasibilityguarantee.ThenTheorem6says thatthegreedyphaseoftheheuristicassignsnomorethan jSj customerstoadierent facilityortonofacilityatall,andnomorethan jQj customerstothesamefacilitybut atadierentlevelthantheoptimalsolutiontoLP 0 .Since,byCorollary2,weknow that jSj + jQjjIj ,itiseasytoseethattheadditionalamountofaggregatecapacity overtheamountusedintheperturbedinstanceofLP 0 requiredforthesecustomers isindependentof jJj .By5{21wecanthereforeconcludethat,withprobabilityone, thegreedyphaseoftheheuristicyieldsafeasiblesolutiontotheGAPFD.Moreover,the objectivefunctionvalueofafeasiblesolutionthatisobtainedbythegreedyphaseofthe heuristicsatises Z G Z LP 0 jJj )-222(jSj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L )-222(jQj D R )-222(jSj P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L {22 sothat,inthatcase, lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z jJj )]TJ/F21 11.9552 Tf 11.956 0 Td [(Z H jJj lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z jJj )]TJ/F21 11.9552 Tf 11.956 0 Td [(Z G jJj since Z G Z H lim jJj!1 1 jJj Z LP 0 jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z G jJj since Z Z LP 0 =lim jJj!1 1 jJj Z LP 0 jJj jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z G jJj withprobabilityoneas jJj!1 ,byLemma5 lim jJj!1 1 jJj )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(jSj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L + jQj D R + jSj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.956 0 Td [(R L by5{22 lim jJj!1 1 jJj )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( jSj + jQj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.956 0 Td [(R L + jSj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.956 0 Td [(R L byCorollary2 93

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lim jJj!1 jIj jJj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L + P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L =0 : Sincetheheuristicisasymptoticallyfeasible,thisimpliesthattheheuristicisasymptotically optimalaswell. ItisinterestingtonotethatTheorem8actuallyshowsthatthegreedyphaseofthe heuristicaloneisasymptoticallyfeasibleandoptimal. 5.2.3ModelExtension Itisinterestingtonotethatourheuristiccanstillbeappliedandretainthe associatedtheoreticalpropertiesifthevariablerevenuefunctionisconvexratherthan linear,i.e.,iftheterm r ij v ij intheobjectivefunctionisreplacedby~ r ij v ij where~ r ij is aconvexandnondecreasingfunction.Sucharevenuefunctionmayberelevantfroma practicalpointofviewbyrealizingthatacustomermaybewillingtopayanincreasing amountperunitofproductsuppliedwithintheacceptablerange.Infact,inlightofthe discussioninSection5.1,ourmodelcouldaccommodateasituationthatexhibitsboth economiesofscaleforthesupplierthroughthexedprottermandalargermarginal valuetocustomerswhoreceiveadditionalunitsofproduct.Toapplyourheuristics tothismodelgeneralization,wecansimplylinearizetherevenuefunctionbydening r ij =~ r ij u ij )]TJ/F15 11.9552 Tf 12.849 0 Td [(~ r ij ` ij = u ij )]TJ/F21 11.9552 Tf 12.325 0 Td [(` ij .Theasymptoticperformanceguaranteesthenfollow inarelativelystraightforwardmannerbyrealizingthatthesolutiontotheLP-relaxation overestimatesthecustomerrevenuefornomorethan jIj customers. 5.3HeuristicImprovementIssues 5.3.1SolutionImprovement Recallthatourstatementoftheheuristicapproachleftsomeexibility,inparticular inSteps5and7oftheimprovementphase.Althoughabasicimplementationwas sucienttoobtainasymptoticperformanceguarantees,wewillinthissectiondiscuss 94

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amoresophisticatedimplementationwhichisguaranteedtoyieldsuperiorresultsin practice. 5.3.1.1Improvementphase First,considertheselectionofaset A 0 ofcustomerswhosecustomerlevelswillbe reducedtotheirlowerboundinStep5.Ratherthanidentifyinganarbitrarysetthat satisesthepropertiesspeciedintheheuristic,wesequentiallyaddassignmentsfrom A to A 0 inthereverseoftheorderinwhichtheywereassignedinthegreedyphaseofthe heuristic,untiltheset A 0 satisesthedesiredproperties.Next,ratherthanattempting toarbitrarilyassigncustomersin ~ J tofacilitiesinStep7weusethemodiedgreedy algorithmproposedbyRomeijnandRomeroMorales[79]tosolvethefollowinginstanceof theGAP: maximize X i 2I X j 2 ~ J p 0 ij x ij subjecttoI X j 2 ~ J a 0 ij x ij b 0 i i 2I X i 2I x ij =1 j 2 ~ J x ij 2f 0 ; 1 g i 2I ; j 2 ~ J where p 0 ij = p ij + r ij ` ij and a 0 ij = a ij + ` ij i 2I ; j 2J .Denotetheoptimalsolutionto Iby x I .Theheuristicsolutionisupdatedbysetting x H ij = x I ij and v H ij = x I ij ` ij i 2I ; j 2 ~ J : {23 5.3.1.2Post-processingphase BoththegreedyphaseoftheheuristicinSection5.2.1andtheimprovementphase describedinSection5.3.1.1aredesignedtoprovidehighqualityfeasiblesolutions. However,itmaystillbepossibletoimprovethequalityofthesolution.Inparticular, 95

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given thatafeasibleassignment x H hasbeenobtainedineitherthegreedyorimprovement phase,weproposeto optimally determinecorrespondingcustomerdemandfulllment levels v P wherethesuperscriptPdenotesthesolutionafterthepost-processingphase. Infact,if B i = f j : x H ij =1 g isthesetofcustomersassignedtofacility i ,theoptimal customerdemandfulllmentlevelscanbedeterminedasfollows.First,solvethefollowing continuousknapsackproblemsfor i 2I : maximize X j 2B i r ij w ij subjecttoKP i X j 2B i w ij b i )]TJ/F26 11.9552 Tf 12.435 11.358 Td [(X j 2B i a ij 0 w ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij j 2B i : Then,if w denotestheoptimalsolutionstotheseproblems,set v P ij = w ij .Itiseasyto seethattheproblemsKP i canbesolvedbychoosingtheexiblecomponentofthesizes ofthecustomersin B i aslargeaspossibleinnonincreasingorderof r ij aslongasfacility capacityallows. 5.3.2CapacityPerturbationScheme Recallthat,forthegeneralcasewhererequirementsareallowedtobefacility-dependent, wereducethenormalizedfacilitycapacities i bysomeamount jJj withintheframework ofSection5.2.2.2.DespitetheasymptoticfeasibilityresultofTheorem8,itisof coursestillpossiblethattheheuristicfailstondafeasiblesolution.Inparticular,an inappropriateperturbationmayleadtoinfeasibilityforoneofthefollowingtworeasons: iIf jJj istoolarge,theresultingperturbedcapacitiesmaybesuchthattheinstance ofLP 0 jJj isinfeasiblesothatwecannotperformthegreedyphaseofthe heuristic. iiIf jJj istoosmall,thenwefailtoreserveenoughcapacitytoaccommodatethe customersthatremainunassignedinthegreedyphase. 96

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Weproposetousethisinformationtoiterativelymodifythecapacityperturbationas needed.Notethat,toensurethatnoperturbedfacilitycapacitiesarenonpositive,we shouldinitiallyhave0 < jJj < min i 2I i .Incasetheheuristicisunsuccessful duetoi,weupdate = jJj anddecrease jJj .If,ontheotherhand,itisunsuccessful duetoii,weupdate = jJj andincrease jJj .Ineithercase,wesetthenewcapacity perturbationto jJj = + )]TJ/F21 11.9552 Tf 11.955 0 Td [( where 2 ; 1andreapplytheheuristic. Forinstanceswithfacility-independentrequirements,noperturbationisrequiredto obtainasymptoticperformanceguarantees.However,asforinstanceswithfacility-dependent requirements,itisofcoursepossiblethattheheuristicdoesnotndafeasiblesolution. Inthatcase,wecanapplythesameiterativescheme,recognizingthatweinitiallyhave jJj =0. 5.4ComputationalResults Inthissectionwetesttheperformanceofourheuristicsonalargesetofrandomly generatedtestproblems.Followingthetheoreticalresults,weseparatelyconsiderproblems withfacility-independentandfacility-dependentrequirements. 5.4.1ExperimentalDesign WeusethestochasticmodelgiveninSection5.2.2asthebasisforgeneratingproblem instances.Weconsiderinstanceswith jIj =15and jIj =30facilities,and jJj = 5 jIj ; 10 jIj ; 25 jIj ; 50 jIj ,and100 jIj customers.Foreachcustomer,wegeneratethevectors ofrevenueparameters R j and P j independentlyfromuniformdistributionson[1 ; 2] and[30 ; 50],respectively.Thecustomerrequirements A j L j and D j aregenerated fromuniformdistributionson[10 ; 20],[75 ; 125],and[15 ; 35],respectively.Notethat,for instanceswithfacility-independentrequirements,onlyasinglevalueforeachofthese parametersisgenerated,whileforinstanceswithfacility-dependentrequirements,we generate jIj valuesoneforeachfacilityindependentlyfromthespecieddistributions. 97

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Wefocusedoninstancesinwhichthefacility-capacitieswereidentical,thatis,weset b i = jJj i 2I .Forinstanceswithfacility-independentrequirements,thevaluein Assumption1thenreducesto = )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(E A 1 j + L 1 j jIj {24 andwethereforeconsidercapacitiesoftheform = E A 1 j + L 1 j jIj : {25 Forinstanceswithfacility-dependentrequirements,RomeijnandRomeroMorales[80] showedthatthevalueinAssumption1reducesto = )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(E min i 2I A i 1 + L i 1 jIj {26 providedthatthelowerboundsofthecustomerrequirements A i 1 + L i 1 areindependent andhaveanincreasingfailureratedistribution,asisthecaseinourexperiments.We thereforeconsidercapacitiesoftheform = E min i 2I A i 1 + L i 1 jIj : {27 Itiseasytoseethatinboth5{25and5{27, > 1isequivalenttoAssumption1being satised.Moreover,thesechoicesensurethatthetightnessoftheinstancesacrossdierent valuesof jIj iscomparableforagivenvalueof .Inourexperiments,wehaveconsidered valuesof =1 : 1,1 : 2,and1 : 3.Theformertwovaluescorrespondtocaseswherethe capacityconstraintsareexpectedtohaveastronglimitingeectonthecustomerdemand fulllmentlevelsthatcanbeaccommodated.Thethirdvaluecorrespondstoloosely capacitatedinstancessincetheyyieldthattheexpressionsin5{24and5{26are positiveevenwhentheranges D i 1 areaddedtothecustomerrequirements. Weapplyboththegreedyphaseandtheimprovementphasetoeachproblem instance.WeusetheiterativecapacityperturbationschemedescribedinSection5.3.2 98

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toimprovetheabilityoftheheuristictondfeasiblesolutionsforsmallervaluesof jJj ; herewesimplysettheupdateparameterequalto = 1 2 .Moreover,unlessotherwise noted,thepost-processingphasedescribedinSection5.3.1.2isappliedaswell.Finally, forameaningfulassessmentoftheheuristicperformancewerunCPLEXuntilasolution isobtainedwhoseobjectivefunctionvalueisatleastasgoodastheonefoundbythe heuristicoruntil15minutesofCPUtimehavebeenused.Foreachproblemclass, wepresentaverageresultsof25randomlygeneratedinstances.Allexperimentswere performedonaPCwitha3.40GHzPentiumIVprocessorand2GBofRAM,andall mixed-integerandlinearprogrammingproblemsweresolvedusingCPLEX10.1.The tablesreport ithenumberofinstancesinwhichtheheuristicfoundafeasiblesolution, iianupperboundontherelativesolutionerrorasmeasuredby error= z LP 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(z H z H 100% ; where,sincetheerrorismeaninglessifnofeasiblesolutionisfound,theaverage errorisdeterminedwithrespecttotheinstancesforwhichtheheuristicisableto ndafeasiblesolutiononly, iiitheaverageCPUtimeusedbytheheuristic,forboththegreedyphaseandthe greedyphasefollowedbytheimprovementphaseoveralliterations, ivtheaveragenumberofcapacityperturbationiterationsperformed, vtheCPUtimerequiredbyCPLEX,and vithenumberofinstancesforwhichCPLEXfailedtoobtainasolutionofthedesired qualitywithintheallottedtimeindicatedbyasuperscript. 5.4.2Facility-IndependentRequirements Tables5-1{5-6summarizetheresultsobtainedwithourheuristicswhenapplied toinstancesgeneratedaccordingtothemodeldescribedintheprevioussectionwith facility-independentrequirements.RecallthatTheorem7saysthattheheuristicformed bythegreedyandimprovementphasesisasymptoticallyfeasibleandoptimal.However, 99

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asimilarguaranteecannotbegivenforthegreedyphasealone;inparticular,applying thegreedyphasealonedoesnotguaranteeasymptoticfeasibility.Thecomputational resultsconrmthis:forthetwoclassesinwhichthecapacityconstraintsaretightest =1 : 1and =1 : 2,thegreedyphaseisnotabletondafeasiblesolutioninanyof theinstancesgenerated.Incontrast,whenthegreedyphaseisconsideredinconjunction withtheimprovementphaseafeasiblesolutionisobtainedforalmostallinstances,with theexceptionofinstanceswith jJj =5 jIj and =1 : 1forboth jIj =15and jIj =30. However,ourresultsshowthat,forsuchinstances,performingtheiterativecapacity perturbationschemeofSection5.3.2yieldsafeasiblesolutioninallbutasingleinstance with jIj =15, jJj =75,and =1 : 1.Notethat,when =1 : 3,thegreedyphasealone isabletondafeasiblesolutionforallinstances.Thiscanpartlybeexplainedbythefact that,forlarge jJj ,itcanbeexpectedthatallcustomerscanbeperformedattheirupper bound,makingtheimprovementphaseunnecessaryevenfromatheoreticalpointofview. Itisnoteworthy,however,thatthegreedyphasealonestillperformsverywellforinstances withsmallervaluesof jJj when =1 : 3,despitethelackofanytheoreticalfeasibility guarantee. Theresultsclearlyshowthattheaverageerrorapproacheszeroasthenumberof customersincreases.Itisinterestingtonotethat,forbothvaluesof jIj ,theaverageerrors arelargestwhen =1 : 2.Thisbehaviorisaconsequenceofthenatureoftheimprovement phase,inwhichtheheuristiccreatescapacityforunassignedcustomersbydecreasing thesizeofalreadyassignedcustomerstotheirlowerbounds.When =1 : 2,customers cangenerallybeperformedathigherlevelsthanwhen =1 : 1,sothattheneteect oftheimprovementphaseonsolutionqualityisunderstandablylargerfor =1 : 2than for =1 : 1However,thispatterndoesnotcontinueto =1 : 3since,asweconcluded above,theimprovementphaseisnotrequiredfortheseinstances.Wealsoremarkthatthe solutionerrorsdependmainlyontheratio jJj = jIj betweenthenumberofcustomersand thenumberoffacilities. 100

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Theheuristiciscomputationallyveryecient,onaveragetakingonlyslightlymore than1secondofCPUtimewhen jIj =15andabout4secondsofCPUtimewhen jIj =30.Forsmallerprobleminstances,CPLEXisabletondsolutionsofthesame qualityveryrapidlyaswell.However,forlargerinstancesandasthecapacityincreases CPLEXisuptoapproximately10timesslowerfor jIj =15anduptoapproximately50 timesslowerfor jIj =30.Ourheuristicisthereforeespeciallypromisingforlargeinstances andcaseswheretheGAPFDneedstobesolvedrepeatedly,forexampleunderdierent scenariosorwhenitisasubprobleminamorecomplexstrategicoptimizationproblem. Perhapssurprisingly,despitethefactthataninstanceofLP 0 hastobesolved,thetime requiredbytheheuristicincreasesonlymodestlyapproximatelylinearlyinthesizeof theproblem. 5.4.3Facility-DependentRequirements Recallthat,forinstanceswithfacility-dependentrequirements,theheuristicemploys thedualsolutiontotheLP-relaxationofaninstanceoftheGAPFDinwhichthe normalizedcapacities arereducedbyaquantity jJj satisfying5{20and5{21 toensureasymptoticfeasibilityandoptimality.Toensurethatnoperturbedfacility capacitiesarenonpositiveforanyvalueof jJj ,weproposetochoose jJj = p jJj {28 where0 << 1,andwherethemagnitudeof representsatradeobetweenfeasibility andsolutionquality.Inourcomputationalexperimentswesimplyuse = 1 2 Tables5-7{5-12summarizetheresultsobtainedwithourheuristicswhenappliedto instancesgeneratedaccordingtothemodeldescribedinSection5.4.1withfacility-dependent requirements.RecallthatTheorem8saysthatthegreedyphaseoftheheuristicis asymptoticallyfeasibleandoptimal.Althoughthegreedyphasealonefailstonda feasiblesolutiontoasubstantialnumberofprobleminstancesforsmallerratios jJj = jIj thegreedyphaseisuniformlysuccessfulforlargerratios.Moreover,thepatternofaverage 101

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errorsafterthegreedyphasealoneshowsconditionallyonndingafeasiblesolutiona decreasingtrend,illustratingthetheoreticalasymptoticoptimalityresult. Thecontributiontofeasibilityoftheiterativecapacityperturbationschemeis apparentforinstanceswith jJj 25 jIj :afeasiblesolutionwasobtainedinallinstances. Itshouldnotbesurprisingthatthisresultsinanincreaseinaverageerror,sincethe instancesforwhichasingleiterationofthegreedyphaseisnotabletondafeasible solutionareclearlytheharderones.However,thepatternofaverageerrorsstillexhibitsa stronglydecreasingtrend,againclearlyillustratingthetheoreticalasymptoticoptimality result. TheaverageCPUtimerequiredbytheheuristicissubstantiallylargerforinstances withfacility-dependentrequirementsthanforinstanceswithfacility-independent requirements,particularlywhen jIj =30.CPLEXisabletosolveinstanceswithloose capacities =1 : 3inabouttwicethetimerequiredbytheheuristicandeveninonly about50%moretimeforthesmallestinstances.Theinstanceswithtightercapacities clearlyillustratethestrengthoftheheuristic.When =1 : 1and jJj 25 jIj ,CPLEX wasnotabletondafeasiblesolutionwithin15minutesofCPUtimeforanyinstance, andwassuccessfulforonly2instanceswith jJj =10 jIj and jIj =30.When =1 : 2 weseeasimilarbehaviorfor jJj 10 jIj and jIj =30.For jIj =15and jJj =10 jIj thetimerequiredbyCPLEXexceedsthatoftheheuristicbyafactorof100,whilefor jJj 25 jIj CPLEXisagainnotsuccessfulinasubstantialnumberofprobleminstances. NotethatthecomputationtimesforCPLEXareaveragedoveronlythoseinstancesin whichitwassuccessfulandthereforedonotincludetheinstancesforwhichCPLEXwas unsuccessfulwithin15minutes.Moreover,foreachinstanceinwhichtheCPUtimelimit expired,CPLEXhadnotyetfoundafeasiblesolution. 5.4.4EectofPost-ProcessingPhase Theasymptoticfeasibilityandoptimalityguaranteesoftheheuristicforinstances withfacility-independentaswellasinstanceswithfacility-dependentrequirementshold 102

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evenintheabsenceofthepost-processingproceduredescribedinSection5.3.1.2.However, asmentionedabove,theresultspresentedthusfarpertaintosolutionsthathavebeen improvedbythispost-processingphase.Therefore,wewillinthissectionstudytheeect ofthepost-processingphaseonthequalityofthesolutions,illustratingsimultaneously theasymptoticperformanceguaranteeswithoutthepost-processingphaseaswellasthe practicalimportanceofapplyingthisphase. Tables5-13and5-14summarizetheseresults.Forbrevity,wehaveonlyfocusedon instanceswithintermediatecapacities =1 : 2;however,theresultsforothervalues of arequalitativelysimilar.Forthecaseoffacility-independentrequirementswehave omittedtheresultsofthegreedyphasealone,sincethisphasewasneverabletond afeasiblesolutionandthusthepost-processingphaseisirrelevant.Inbothtables,the columnslabeledbefore"containthesolutionerrorswithoutpost-processing,whilethe columnslabeledafter"containthesolutionerrorswithpost-processing.Weseethat, inallcasesandforbothtypesofprobleminstances,theresultswithoutpost-processing phaseareconsistentwiththeasymptoticoptimalityguarantees.However,wealsoseethat thepost-processingphasesubstantiallyreducessolutionerror,particularlyforproblem instanceswithasmallratioof jJj = jIj 5.5Conclusions ThischapterconsideredtheGAPFD,whichgeneralizestheclassicalGAP.Our extensionappliestosituationsinwhich,alongwiththeassignmentofcustomersto facilities,aexibledegreeofresourceconsumptionmustbedeterminedforeachof theseassignments.TosolvetheGAPFD,weproposeaclassofheuristicsmotivated byattractivepropertiesoftheoptimalsolutionoftheLP-relaxationtotheGAPFD anditscorrespondingdual.Fortwoclassesofcustomerrequirementsweshowthatan implementationoftheheuristicexiststhatisasymptoticallyfeasibleandoptimalwith probabilityoneunderaverybroadstochasticmodelontheproblemparameters.Our computationalstudydemonstratesthattheheuristicperformsverywell,particularly 103

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forlargeratiosofthenumberofcustomerstothenumberoffacilities.Whenadditional improvementstrategiesthatweproposeinthischapterarealsoconsidered,theheuristic issuccessfuloninstanceswithsmallerratiosaswell.Weobservethatthetimerequired toobtainsolutionsofcomparablequalityisconsiderablylessforourheuristicthanforthe commercialsolverCPLEX.Thefactthatourheuristicobtainsqualitysolutionssoquickly isencouragingforfurtherresearchdirections.Specically,webelievethattheheuristic maybeveryvaluablewhensolvingmoregeneralrelatedoptimizationproblemsforwhich theGAPFDarisesasasubproblemthatneedstobesolvedrepeatedly. 104

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Table5-1.Facility-independentrequirements:15facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 750|0.04241.120.335.840.24 1500|0.09250.380.1310.29 3750|0.20250.090.2910.85 7500|0.35250.050.5111.74 1,5000|0.72250.031.0314.18 Table5-2.Facility-independentrequirements:15facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 750|0.04253.200.0710.20 1500|0.07251.820.1110.41 3750|0.18250.470.2710.94 7500|0.33250.110.4911.93 1,5000|0.68250.020.9814.85 Table5-3.Facility-independentrequirements:15facilities =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 75250.790.06250.790.0611.30 150250.240.10250.240.1012.00 375250.060.24250.060.2414.93 750250.020.45250.020.4516.28 1,500250.000.91250.000.91110.82 Table5-4.Facility-independentrequirements:30facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.18251.050.351.561.74 3000|0.27250.360.3911.51 7500|0.68250.090.9514.26 1,5000|1.40250.041.9016.76 3,0000|2.95250.023.95113.34 Table5-5.Facility-independentrequirements:30facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.15253.560.2110.83 3000|0.26251.600.3812.11 7500|0.66250.380.9215.82 1,5000|1.34250.121.8319.79 3,0000|2.73250.023.71119.44 105

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Table5-6.Facility-independentrequirements:30facilities, =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 150250.670.18250.670.1818.21 300250.290.34250.290.34117.62 750250.040.84250.040.84128.24 1,500250.011.63250.011.63150.19 3,000250.003.25250.003.251163.96 Table5-7.Facility-dependentrequirements:15facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 75105.060.08259.320.394.843.04 15062.650.18259.320.924.88144.55 375251.280.36251.280.361| 750250.800.78250.800.781| 1,500250.521.89250.521.891| Table5-8.Facility-dependentrequirements:15facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 7541.550.07257.170.344.361.56 150150.610.17252.680.482.6050.30 375250.210.31250.210.311341.90 750250.100.65250.100.651478.38 1,500250.051.41250.051.411333.57 Table5-9.Facility-dependentrequirements:15facilities, =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 7590.970.07258.900.222.680.38 150190.290.16250.660.201.120.47 375250.020.29250.020.2910.81 750250.010.57250.010.5711.72 1,500250.001.17250.001.1714.50 Table5-10.Facility-dependentrequirements:30facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 15012.250.402513.383.107.72139.74 3000|0.50258.035.377.00440.51 750140.791.76252.6010.833.64| 1,500250.484.86250.484.861| 3,000250.3114.11250.3114.111| 106

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Table5-11.Facility-dependentrequirements:30facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.34255.012.21655.25 3000|0.42253.954.216374.52 750200.221.44250.674.682| 1,500250.093.54250.093.541| 3,000250.068.93250.068.931| Table5-12.Facility-dependentrequirements:30facilities, =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.322511.481.694.842.97 30040.350.38251.911.151.964.02 750210.041.20250.041.2516.16 1,500250.012.75250.012.7518.53 3,000250.006.38250.006.38113.70 Table5-13.Post-processingeectonheuristicwithgreedyandimprovementphase; facility-independentrequirements: =1 : 2 jIj =15 jIj =30 error%error% jJj beforeafterbeforeafter 5 jIj 19.423.2019.473.56 10 jIj 10.621.8210.901.60 25 jIj 4.050.474.150.38 50 jIj 2.000.112.030.12 100 jIj 0.980.021.000.02 Table5-14.Post-processingeect;facility-dependentrequirements: =1 : 2 jIj =15 jIj =30 GreedyphaseImprovementphaseGreedyphaseImprovementphase error%error%error%error% jJj beforeafterbeforeafterbeforeafterbeforeafter 5 jIj 7.921.5521.307.17||18.805.01 10 jIj 4.910.6110.482.68||15.843.95 25 jIj 2.780.212.780.212.070.224.400.67 50 jIj 1.910.101.910.101.360.091.360.09 100 jIj 1.320.051.320.050.930.060.930.06 107

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CHAPTER6 LARGE-SCALEMULTI-EXCHANGEHEURISTICFORFIXED-CHARGERESOURCE CONSTRAINEDASSIGNMENTPROBLEMS Inthischapterweproposeheuristicproceduresforageneralizationofthewell-known CapacitatedFacilityLocationProblemwithSingle-SourcingconstraintsCFLP,known asthe CapacitatedFacilityLocationProblemwithFlexibleDemand CFLFD.Inaddition tothestandardproblemscopeoftheCFLP,theCFLFDpermitsexiblecustomer demandspecications.Thatis,forameasurableproductcharacteristice.g.,weight, length,volume,unitsdelivered,acustomerspeciesanallowablerangefordemand fulllment.Thisassignment-basedoptimizationproblemfallsintotheclassofchallenging mixed-integerprogramsthatbecomeverydiculttosolveasthenumberofcustomers perfacilityincreases.Becausemanypracticalapplicationsofthisproblemclassrequire obtaining/updatingsolutionsveryquickly,itisimportanttoidentifyfastheuristicsolution methodsthat,onaverage,providenear-optimalsolutions. AsmentionedinChapter2,theCFLPisaspecial-caseoftheCFLFD,ourproblem clearlybelongstotheclassof NP -Hardoptimizationproblems.Thedicultyofsuch problemsrequiresconsideringbothexactandheuristicsolutionmethodologies.Recent theoreticaladvancesinintegerprogramminghaveresultedinexactsolutionmethodologies thathaveprovensuccessfulonpreviouslyunsolvedprobleminstances.Forexample,the decomposition-basedseparationalgorithmfortheCapacitatedVehicleRoutingProblem CVRP,proposedbyRalphsetal.[76],solvesthreeofthepreviouslyunsolvedVRP instancesfromtheTSPLIBrepositorypresentedbyReinelt[77].Similarly,astabilized branch-and-cut-and-pricealgorithmfortheGeneralizedAssignmentProblemGAP, introducedbyPigattietal.[74],wasabletosolvethreepreviouslyunsolvedinstancesfrom theOR-Library.Unfortunately,evenwiththeseinnovativetechniques,manyreal-world sizeproblemscannotbesolvedwithinpractitioners'timerequirements.Infact,for numerousintegerprogrammingmodelsthatconsidertheassignmentofcustomersto resourcesi.e.,theGAP,Savelsbergh[88]andPigattietal.[74];theCVRP,Fukasaw 108

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[38];andtheMultiperiodSingle-SourcingProblemMPSSP,Frelingetal.[37],exact proceduresareoftensuccessfulonlyoninstanceswithasmallratioofcustomersto resources,limitingthenumberofproblemsthatcanbesolvedwithinacceptabletolerance levels.Moreover,thenumberofscenariosforwhichdecisionmakersmustrepeatedlyrevise theirstrategicplansisgrowingrapidlyasinformationbecomesavailableinreal-time.For example,productionplannersnowhaveaccesstochangesininventorylevels,demandrates andresourcelevels,astheyoccur.Tomakeuseofthisinformation,aplanningschedule mayrequireupdatesnumeroustimesaday.Insuchcases,itisnecessarytodesignan ecientheuristictoserveaseitherasupplementtoanexactalgorithmorasastand-alone procedurethatprovidesqualitysolutionswithlimitedcomputationaleort. InChapter4weproposedanexactalgorithmfortheCFLFD.Asiscommonwhen applyingbranch-and-pricetoassignmentproblems,thesuccessoftheapproachwas limitedtoproblemswitharelativelysmallratioofcustomerstofacilitiesusuallyup toabout10.Inthischapter,weproposetherstheuristicmethodologytargetedat solvingthebroadclassofprobleminstanceswithalargeratioofthenumberofcustomers tothenumberoffacilities.Theheuristicapproachweproposeemploysacombined facilityneighborhoodsearchmethodandafastheuristicsolutionmethodforsolving ageneralizationoftheGAP.Wediscussspecicimplementationissuesrelatedtothis methodology,includingmethodsforobtaininginitialfeasiblesolutions,eectivewaysto searchalargeneighborhoodofsolutionsandecientwaystodevelophybridapproaches thatcombinesuccessfulindividualheuristicmethodologies.Whilesuccessfulheuristics havebeendevelopedforrelevantproblems,suchasAhujaetal.'s[6]multi-exchange heuristicCFLP,thenotionofexibledemandprovidesadditionalchallengesthataremet throughtheapproachproposedinthiswork.Computationaltestsillustratethebenetsof ourproposedapproachforsolvingproblemsinthisclass. BalakrishnanandGeunes[13]proposedLagrangian-based,bin-packing-based,and LP-roundingheuristicsforthecloselyrelatedFlexibleDemandAssignmentProblem 109

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FDA.However,theircomputationaltestingfocusedoninstanceswithasmallratioof customerstofacilities.Aswillbeevidentinthefollowingsections,areassignmentof evenasinglecustomerintheCFLFDrequiresasubsequentassessmentoftheamount ofdemandfullledforeachcustomerassignedtothatparticularfacility.Therefore,the reassignmentofcustomersinourapproachwilldierdistinctlyfromthatofprevious work.InthecaseoftheapproachinAhujaetal.[6],ifthelimitedreassignmentfailsto ndanimprovingorfeasiblesolution,theauthorsconsideracompletereassignmentof customers,determinedbyheuristicallysolvinganinstanceofthegeneralizedassignment problem.ItisimportanttoemphasizethatthemethodestablishedinAhujaetal.[6] allowsfortotalcustomerreassignmentonlyasalastresort.However,animportant aspectofourapproachismotivated,inpart,bythislessfrequentlyemployed complete reassignmentstep.Wecontendthatwiththeavailabilityofaquick,eectiveheuristicto solvethenecessaryassignmentsubproblemandinlightoftheincreasedcomputational eortrequiredtofullyassessthepartialreassignmentofcustomers,theeortspent consideringadditionallargeneighborhoodssuchasthoseproposedinAhujaetal.[4]is notnecessaryforasuccessfulheuristicapproachfortheCFLFD.Moreover,sincecertain verylargeneighborhoodsconsideredinpreviousworkgrowquadraticallyinthenumber ofcustomers,thesemethodsarenotapplicableforsolvinglargerprobleminstances. TheworkofAhujaetal.[6],considersinstanceswithafacilitytocustomerratioofno morethan10.Therefore,theapproachoeredinthisworkservesastherstheuristicto considerafacilitylocationproblemwithdemandexibilityandoersageneralsolution frameworkthatisparticularlyapplicabletolarge-scaledecomposableassignmentproblems thathavereceivedlittleattentionintheliterature. Theremainderofthispaperisorganizedasfollows.Section6.1formallyintroduces theversionoftheCFLFDstudiedinthischapter.Section6.2proposestwoseparate searchtechniquesforsolvingtheCFLFD.Implementationdetailsoftheheuristic approachesarepresentedinSection6.3.InSection6.4,weperformacomputational 110

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studyofthevariousimplementationsofourheuristiconabroadcollectionofexperiments. Finally,inSection6.5,wediscusstheapplicationofourheuristicframeworktosimilar xed-chargeproblemsandoersomeconcludingremarks. 6.1OptimizationandModelFormulation TheoptimizationmodeldiscussedinthischapteristheCFLFDpresentedinChapter 3withlinearrevenuefunctionsi.e r ij v ij = r ij v ij .Furthermore,asinChapter5welet J i = J i 2I .Specically,themodelconsideredinthissectionisgivenbyproblemas follows: maximize X i 2I X j 2J r ij v ij + X i 2I X j 2J p ij x ij )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X i 2I f i y i subjecttoCFLFD-L X j 2J a ij x ij + v ij b i y i i 2I {1 X i 2I : j 2J x ij =1 j 2J {2 v ij ` ij x ij i 2I ; j 2J {3 v ij u ij x ij i 2I ; j 2J {4 x ij 2f 0 ; 1 g i 2I ; j 2J {5 y i 2f 0 ; 1 g i 2I : {6 ForthenotationalsimplicitywewillrefertoCFLFD-LasCFLFDthroughoutthis chapter,notinghoweverthatthemostgeneralversionoftheCFLFDisgiveninChapter 3.Inthenextsection,weproposeasearchheuristicthatexploitsthestructureofthe CFLFDinawaythatenablesustoconsideraverylargeneighborhood. 6.2HeuristicFramework Thissectiondescribesthecomponentsofageneralsearchheuristicframeworkfor theCFLFD.Weconsidertechniquesforexploringtwoclassesofsearchneighborhoods. Section6.2.1describesthecoreofourheuristic.Thisapproachsearchesaneighborhood denedbymanipulatingthesetofopenfacilities.Inthissearchprocedure,allcustomers 111

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arereassignedateachiteration.Whileahighlyecientmethodologyisusedtodetermine thereassignmentofallcustomersamongasetofopenfacilities,thenumberofopen facilitysetstoconsiderisstillverylarge.InSection6.2.2,weintroduceaspecialized implementationofaso-calledvery-large-neighborhood-searchVLSNheuristic.This techniqueallowsformultiplecustomerexchangesinanymove;however,nomorethanone customeramongthoseassignedtoaparticularfacilitymaybereassignedinasinglemove. Thissecondaryprocedureisconsideredinanadditionalhybridimplementationdiscussed inSection6.3.2. 6.2.1FacilityNeighborhoodSearch Ourprimarysearchapproachisdesignedtoeasilyallowforimprovingsolutions thatcorrespondtoopeningorclosingafacility.WhiletheVLSNproceduredescribedin Section6.2.2iseectiveinidentifyingalternativecustomerassignmentsamongagiven setofopenfacilities,itrarelyidentiessolutionsthatalterthesetofopenfacilities.This limitationisconrmedbyAhujaetal.'s[6]inclusionofthefacilityneighborhoodstructure intheirapproachforCFLP.Inthisneighborhood,theopening,closing,ortransferringof facilitiesisconsidered.Toestimatetheimpactofeachofthesemoves,theyattempted toidentifya subset ofcustomerswhosereassignmenttoasingledierentfacility resultedinacostsavings.Asmentionedintheprevioussection,theidenticationof potentiallyimprovingmovesinAhujaetal.'s[6]approachrequiredthedirectcomparison ofassignmentcostparameters.However,inthecaseoftheCFLFD,determiningthe changeinprotfromthereassignmentofevenasubsetofcustomersrequiresthatthe levelthatdemandissatisedmustbereassessedforeachcustomerassignedtoany facilitywhichaddsorlosesacustomerinthereassignment.Therefore,sinceconsidering reassignmentsofsmallsubsetsofcustomersisalreadymorecomputationallyintensive inthecaseoftheCFLFDthanCFLP,wefocusimmediatelyondeterminingacomplete reassignmentofallcustomersviaanecientheuristicwhenconsideringthevarious optionswithrespecttoopeningandclosingfacilities.Tocontrastthisprocedurewith 112

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previouswork,Ahujaetal.'s[6]approachonlyconsideredtotalreassignmentofcustomers whentheirpartialreassignmentprocedurewasunabletondanyimprovingfeasible solution.Ourstrategyeectivelyallowsustoconsidersinglefacilitymovesopen,close, orswapandmulti-exchangemultiplecustomersbeingreassignedamong numerous facilitiesinasinglesearchstep.Clearlythisheuristicsearchesaverylargeneighborhood, andifimplementedwithasuccessfulsubproblemmethodology,islikelytooerhigh qualitysolutions.Ofcourse,reassigningallcustomersateachstepofoursearchcanbe computationallyintensive,aswell.Tocompensateforthisextraeort,wecontendthat, forproblemswithalargeratioofcustomerstofacilities,relyingonanecientlyobtained subproblemsolutiontodeterminecustomerassignmentsisatleastaseectiveasVLSN. Alternatively,wefocusonanintelligentimplementationthatecientlysearchesthisvery largefacilityneighborhoodanddeterminesasetofopenfacilitiesandsubsequentcustomer assignmentswhichcorrespondtoahighqualitysolution.Intheremainderofthissection wepresentthesearchframeworkandintroducetherelevantsubproblemtobesolvedin thesecondarystageofoursearch. Ourframeworkallowsforasearchofaverylargeneighborhood.Ineachpotential move,oursearchevaluatesthebenetofmanipulatingthesetofopenfacilitiesand reassigningcustomersamongthesefacilities.Specically,foranypartialfeasiblesolution toCFLFD, x N ;v N ;y N ,let O bethesetofopenfacilitiesinthissolution,i.e., O = f i 2I : y N i =1 g and C bethesetofclosedfacilities,i.e., C = f i 2I : y N i =0 g : Ourheuristiccanbedescribedbythreeseparatemoves. Close :closea single facilityin O andreassignallcustomers; Open :opena single facilityin C andreassignallcustomers; 113

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Swap :closea single facilityin O andreplaceitbyopeningaclosedfacilityin C ,and thenreassignallcustomers. Foreachneighborhoodmovedescribedabove,reassigningallcustomersamongagiven setofopenfacilities, ~ I ,anddeterminingthecorrespondingdemandfulllmentlevels,is accomplishedbysolvingthefollowingmixed-integerprogram maximize X i 2 ~ I X j 2J p ij x ij + X i 2 ~ I X j 2J r ij v ij subjecttoGAPFD X j 2J a ij x ij + v ij b i i 2 ~ I {7 X i 2 ~ I x ij =1 j 2J {8 ` ij x ij v ij u ij x ij i 2 ~ I j 2J {9 x ij 2f 0 ; 1 g i 2 ~ I j 2J : {10 ThisproblemistheGeneralizedAssignmentProblemwithFlexibleJobswithlinear revenuefunctions,whichwasstudiedinChapter5.Thisclassofoptimizationproblems isclearly NP -HardsincetheGAPisanimportantspecialcase.However,intheprevious chapterwedevelopedanecientconstructiveheuristicfortheGAPFD,whichwasshown tobeasymptoticallyfeasibleandoptimalunderaverygeneralstochasticmodel. Thisframeworkcanclearlybeimplementedinavarietyofways.Toproducean ecientimplementation,importantdesigndecisionsmustbemade.Section6.3discusses eachoftheseissuesandpresentsthemostsuccessfulimplementationdeterminedthrough ourcomputationalstudy.However,beforeconsideringtheseimplementationissueswe introduceanadditionallarge-scaleneighborhoodsearchtechniquethathasbeenappliedto numerousset-partitioningoptimizationproblems. 114

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6.2.2Single-CustomerVLSN InthissectionwediscusauniqueVLSNimplementationfortheCFLFDwhich wewillconsiderasapost-processingphaseforourmainfacilityneighborhoodsearch procedure.Inrecentliterature,avarietyofassignment-basedoptimizationproblemshave beensolvedusingVLSN,e.g.,theMulti-PeriodSingle-SourcingProblemAhujaetal.[3], theSingleSourceCapacitatedFacilityLocationProblemAhujaetal.[6],andVehicle RoutingandSchedulingProblemsThompsonandPsaraftis[97].DetailsoftheVLSN procedurearewelldocumented.Therefore,thissectiononlyoersdetailsofconsiderations madetoaccommodatetheuniquedemandexibilitycomponentofourproblem.For adetailedsurveyoftheVLSNtechnique,thereaderisreferredtoAhujaetal.[5].For aspecicdiscussionofsingle-customermulti-exchangeVLSNappliedtotheCFLP,a comprehensivediscussionisprovidedinAhujaetal.[6,Section4]. Ingeneral,thesingle-customermulti-exchangesearchneighborhoodisexploredby constructingaso-calledimprovementgraph.Theimprovementgraphconsistsofanode foreachcustomer,separatenodesforeachfacility,andanoriginnode.Arcsconnecteach pairof`customernodes',providedthatthecustomersareassignedtodierentfacilities. Inaddition,arcsconnecteach`customernode'toeach`facilitynode',excludingthenode representingthefacilitytowhichthecustomeriscurrentlyassigned.Lastly,thegraph includesarcsfromeach`customernode'totheoriginnode,aswellasanarcfromeach `facilitynode'backtotheorigin.Theinclusionof`facilitynodes'andtheoriginnodes allowsforexchangesinwhichacustomerisaddedtoremovedfromafacility,butno customerisrelinquishedfromaddedtothatfacility.Usingthisrepresentation,an improvingmoveisobtainedbyidentifyinganegativesubset-disjointcycleinthenetwork. Acomprehensivediscussionofdisjointcyclesandtheoptimizationeortrequiredto identifythemisfoundinThompsonandOrlin[96].Forthepurposesofthiswork,we applytheeectiveheuristicproposedbyAhujaetal.[7]tothe NP -Hardproblemof identifyinganegativesubset-disjointcycle. 115

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Thusfar,theimplementationofVLSNonourproblemisnodierentthanwhen appliedtoanysimilarproblemwithaset-partitioningstructure.However,unlikeany modelpreviouslysolvedwithVLSN,theexibledemandcomponentoftheCFLFDmust beaccountedforwhendeterminingthecostofthearcscontainedintheimprovement graph.Ingeneral,thecostofanyarcintheimprovementgraphissimplythedierence intotalprotresultingfromassignmentchangesrepresentedbythatarc.IntheCFLP, aswellasothermodelswithxedassignmentprotsorcosts,thisdierenceisdependent solelyonthecostprotparametersintheproblem.InthecaseoftheCFLFD,each assignmentexchangemustbeaccompaniedbyacorrespondingdemandfulllmentlevel decision.Therefore,tocalculatethecostofanarc,asubproblemmustbesolvedto determinetheappropriatechangeincustomerdemandfulllmentlevelsassociatedwith eachexchange. Toillustrate,considertwo`customernodes', j 1 and j 2 .Let i j 2 bethefacilitytowhich customer j 2 iscurrentlyassignedand ~ J thesetofcustomerscurrentlyassignedto i j 2 The costofthearcconnecting`customernodes', j 1 and j 2 is z SP ~ J )]TJ/F21 11.9552 Tf 11.955 0 Td [(z SP f ~ Jn j 2 g[f j 1 g ; where,foranysetofcustomers J assignedto { ,thevalue z SP J isobtainedbysolvingthe followingoptimizationproblem,SP, +maximize X j 2 J r {j w {j subjecttoSP X j 2 J w {j b { )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X j 2 J a {j + ` {j 0 w {j u {j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` {j j 2 J ; 116

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and isaconstantequalto P j 2 J p {j + r {j ` {j )]TJ/F21 11.9552 Tf 12.82 0 Td [(f { .Inthisformulation,thedecision variable w {j representsthetotalamountofresourcei.e.,bothxedandvariable consumedbycustomer j whenassignedtofacility { .ItiseasytoseethatSPcanbe treatedinthesamemannerasa0/1continuousknapsackproblem.Theproblemcanbe solvedbychoosingtheexiblecomponentofthesizesofthedemandfulllmentaslargeas possibleinnonincreasingorderof r {j aslongascapacityallows.Inasimilarmanner,SP canbeusedtodeterminetheappropriatecostsofarcsconnectingtheremainingnodes. Theabilitytoecientlycalculatetruearccostsisanotableresultthatisuniqueto thedevelopmentofaheuristicfortheCFLFD.Withtheavailabilityofaprocedureto solveSPecientlytooptimality,weavoidthereducedimpactofVLSNthatisgenerally causedbyhavingtorelyonarccostestimates.Thus,evenwiththeadditionaldemand exibilitydecisioncomponent,alargeneighborhoodcanbeecientlyexploredforproblem sizessimilartothoseconsideredinrelatedwork.Whiletheliteraturehasshownthat thistechniqueiseectiveindeterminingcustomerassignmentsforaxedsetoffacilities, establishinganappropriatesetofopenfacilitiesisaweaknessoftheimplementation describedinthissection.Therefore,thisapproachwillbeutilizedonlyinthelimited manneroutlinedattheendofthefollowingsection. 6.3SearchHeuristicImplementation Thissectionwillprimarilyfocusonimplementationchoicesregardingtheneighborhood searchheuristicproposedinSection6.2.1.However,attheendofthesection,the motivationforahybridapproachwhichmeldsthefacilityneighborhoodsearchFS approachwiththeVLSNinSection6.2.2ispresented.WithregardtoFS,anumber ofkeyconsiderationsarenecessary.First,sinceFSfallsintheclassofimprovement heuristics,wemustdeterminehowaninitialfeasiblesolutionisobtained.Second,the eortrequiredtoconsiderafullsetoffacilitymovesisextensive.Therefore,consideration shouldbegiventointelligentlyconsiderasubsetofpotentiallyattractivemoves.Then,of course,thecriteriausedtodeterminesearchterminationmustbespecied.Thechoices 117

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madewithrespecttoeachoftheseissueshasaconsiderableeectonthesuccessofour heuristic.Inthissectionwetreateachoftheseissuesseparately.Weoeralternatives foreachissue,discussourndingswithrespecttoeachoption,andprovidethebest implementationencountered. 6.3.1InitialFeasibleSolution SincealternativeheuristicsfortheCFLFDdonotexistthatmayserveasasource foraninitialfeasiblesolution,westudytwoalternativestodetermineastartingsolution. TherstismotivatedbytheavailabilityofanecientheuristictosolvetheGAPFD. Inthisalternativeweassumethatallfacilitiesareopen,andthensolveaGAPFDvia theconstructiveheuristicproposedinChapter5todeterminethecustomerassignments andthelevelatwhicheachcustomer'sdemandissatised.Thisalternativehasclear advantages.First,withthecapacityofallfacilitiesavailableforcustomerassignments, itisrelativelyeasytondafeasiblesolutiontothecorrespondingGAPFD.Theresults inChapter5indicatethatsolutionsobtainedbytheirGAPFDheuristicforinstances withlargeamountsofavailablefacilitycapacityareveryclosetooptimalforaxed setoffacilities,andcanbedeterminedwithaminimalamountofcomputationaleort. Unfortunately,thedisadvantageofthisalternativeisthatthefacilityprocurementcosts areneglected.Therefore,whileeasytoimplementandintuitivetoconsider,thequalityof thesolutionsobtainedusingthisapproachwereofpoorqualityandultimatelyresultedin prolongeddurationofoursearchheuristic. Thesecondalternativeisamodiedrandomsolutiongenerationapproach.This methodforgeneratinganinitialfeasiblesolutioniscommonlyusedtostartanimprovement heuristicsearche.g.,theTravelingSalesmanProblem,LinandKerninghan[61];andthe ResourceConstrainedProjectSchedulingProblem,LeeandKim[59].Whenrandomly generatingafeasiblesolutiontotheCFLFD,weexplicitlyattempttominimizethe numberoffacilitieswhichare`opened'inthesolution.Let J and I bethesetofof unassignedcustomersandunusedfacilities,respectively.Furthermore,let b 0 i bethe 118

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remainingcapacityoffacility i i 2I .Theprocedureforrandomlygeneratingafeasible solutiontotheCFLFDisasfollows. RandomlygenerateCFLFDsolution Step0. Set J = J I = I and b 0 i = b i for i 2I Step1. Randomlychooseafacility^ { 2 I .Set y I ^ { =1remove^ { from I .ProceedtoStep 2. Step2. If J =,STOPwithfeasiblesolution.Else,randomlychooseacustomer^ | 2 J If a ^ { ^ | + ` ^ { ^ | b ^ { set x I ^ { ^ | =1, v I ^ { ^ | = ` ^ { ^ | b 0 ^ { = b 0 ^ { )]TJ/F15 11.9552 Tf 12.492 0 Td [( a ^ { ^ | + ` ^ { ^ | andrepeatStep2. Otherwise,if I6 =,returntoStep1;elsereturntoStep0. Thisprocedurerandomlyassignscustomerstoasinglerandomlychosenfacilityaslongas capacityallows.Whensucientcapacitynolongerexists,anewfacilityisopenedandthe procedurecontinues.Thisapproachtypicallyyieldsasolutionwithfeweropenfacilities thantherstapproachproposed.Randomlyassigningcustomerstoopenfacilities individually,ontheotherhand,appearstobelessdesirablethantakingadvantageof theGAPFDheuristic.However,inourcomputationaltesting,thesolutionsobtainedvia randomgenerationcontainedanumberofopenfacilitiesmoreconsistentwiththenumber foundinthenalsolutiongeneratedbytheneighborhoodsearchheuristic.Therefore,the durationoftheoverallsearchprocedurewasreducedbychoosingtherandommethod. Sincethequalityoftheultimatesolutionfoundbythesearchheuristicwasunchangedby themethodusedtoobtaintheinitialfeasiblesolution,weusetherandomprocedurein ourcomputationaltesting. 6.3.2FSMoveChoice Inthissectionwedeterminethebestsetofmovestoconsiderinoursearchand theorderinwhichtheyshouldbeconsidered.Anobviousimplementationconsiders openingoneatatime all facilitiesin C calledan O -move,closingoneatatime all facilitiesin O calleda C -move,andswapping all pairsoffacilitiesin O and C during anysingleiterationofthesearchcalledan S -move.Werefertothisimplementationas 119

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the FullNeighborhoodsearch FNS.Inthisimplementation,theorderinwhichmoves areconsideredisirrelevant.Intuitively,sincetheentireneighborhoodisexplored,a highlydesirableheuristicsolutionislikelytoresult.However,theeorttosearchthefull neighborhoodrequiresextensivecomputationaleort.Inourcomputationalstudy,we willanalyzetheadvantagesofsolutionsobtainedfromthisimplementationaswellasthe associatedcomputingtime. Alternatively,wemayconsideronlyasubsetofthefullneighborhoodateach iteration.Thismayleadtoareductioninthetimeneededtocompletethesearch heuristic.However,thecorrespondingreductioninthesizeoftheneighborhood exploredhasthepotentialforconvergencetosolutionsoflesserquality.Inthealternative implementationtofollowwecontendthatthetimesavedthroughthesearchofareduced neighborhoodisnotattheexpenseofsolutionquality.Thedirectionforoursecond implementationresultsfromofacarefulstudyoftheprogressoftheFNSimplementation. AcloseanalysisofthebestmovesateachiterationofFNSconsistentlyrevealedthatthe heuristicbeginsbychoosingasequenceof O -moves.Then,atsomeiterationinthesearch procedure,thebestmovebecomesan S -move.The S -moveiscontinuouslyidentiedas thebestmovechosenuntilthesearchterminateswithoutndingfurtherimprovement. Therefore,considerationof C -movesand S -movesduringtherstphaseoftheFNS-search equatestowastedtime.Similarly,theconsiderationof O -movesand C -movesinthe secondphaseoftheFNSimplementationistypicallynotbenecial. Basedontheprecedingdiscussion,wedivideoursecondimplementationintotwo phases.Initially,weconsideronly O -moves.Sincethenumberof O -movestoconsider atanyiterationisontheorderofthenumberoffacilities, jIj ,weconsidereachofthese alternatives.Recallthat,toassessthebenetofeachmove,wesolveaGAPFDto determinethecorrespondingcustomerassignmentsandthelevelatwhichtheirdemand isfullled.Thebestincumbentisidentiedasthesolutionwiththelargestnetprot. Werepeattheopenneighborhoodsearchonthissolution.Phase1continuesuntilno 120

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improvementisattainedthroughan O -move.Atthispoint,weproceedbysearching theneighborhoodofthebestsolutionobtainedinPhase1,denedbypotentialswaps ofopenandclosedfacilities.Clearly,thenumberofpotential S -movestoconsiderat eachiterationisontheorderof jIj 2 .Consideringeachofthesemovesistimeprohibitive. Ahujaetal.[6]citethisconcernaswell,andidentifytworulesforassessingtheexpected impactofpotential S -moves.Ourruleisanextensionoftheruleproposedinthatwork, whichreassignsallcustomerstothesamenewfacility,whileleavingtheothercustomer assignmentsunchanged.Thatis,weconsidereachpairofcurrentlyopen/closedsets o;c 8 o 2 O c 2 C .Allcustomerscurrentlyassignedtofacility o aretemporarilyassignedto facility c .Allcustomersassignedtofacilities i 2I ; i 6 = o retaintheircurrentassignments. Fortheupdatedcollectionofassignments,let i j denotethefacilitytowhichcustomer j isassigned.Giventhissetoftemporaryassignments,wesolvethefollowingLPSP 0 to determinetheoptimallevelsatwhichtofulllthosecustomers'demands: +maximize X j 2J r i j j w i j j subjecttoSP 0 X j : i j = i w ij b i )]TJ/F26 11.9552 Tf 14.567 11.357 Td [(X j : i j = i a ij + ` ij i 2 O nf o g [f c g 0 w i j j u i j j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` i j j j 2J ; where isaconstantequalto P j 2J )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(p i j + r i j j ` i j j )]TJ/F26 11.9552 Tf 12.612 8.966 Td [(P i 2 O nf o g [f c g f i .Itiseasytosee thatthisproblemdecomposesinto j O nf o g [f c gj continuousknapsackproblemsofthe formSP,presentedinSection6.2.2.Theoptimalobjectivevalueofthisproblem, z SP 0 determinesthe`swappriority'ofaspecicopen/closepair, o;c .Specically,afterSP 0 is solvedforeach o;c o 2 O c 2 C ,thecollectionofpairsissortedinnon-increasingorder ofthecorrespondingvalue, z SP 0 121

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Ateachiterationofoursearchweconsidermovesintheorderdeterminedbythe precedingprocedureuntileitherianimprovingmoveisfoundoriinoimprovingmoves arefoundamongtherst jIj swapcandidatesconsideredinthelastiteration.Inthecase ofi,weproceedtothenextiterationwiththeredenedneighborhood,while,inthecase ofii,theentiresearchprocedureterminates.Comparisonofthisprioritymeasureagainst thetruedesirabilityofmovesdeterminedbysearchingthefullswapneighborhoodrevealed thatourmethodologywaseectiveinidentifyingpromising S -moveseciently.Wewill addresstheeectivenessofourprioritymeasuremoredirectlyinSection6.4. Wehavenowcompletelydescribedourfacilityneighborhoodsearchimplementation. However,wehavenotexplicitlyexplainedouromissionof C -moves.Again,thebasis forthisimplementationwasthatthefullneighborhood very rarelyfounda C -moveto bethebestoptionatanyiteration.Webelievethisresultisduetothenatureofthe initialfeasiblesolutionprovided.Sincetherandomlygeneratedfeasiblesolutionaimsto containasmallnumberofopenedfacilities,a C -moveisintuitivelyunnecessary.Onthe contrary,whenourinitialsolutionwasobtainedbyopeningallfaciltiesandsolvingthe correspondingGAPFD,thenonly C -moveswerechosenintheinitialphaseofFNS,and O -moveswererarelyutilized.Therefore,whether O -movesand/or C -movesshouldbe consideredinPhase1dependsuponthecharacteristicsoftheinitialsolutionprovidedto thesearchheuristicbythedecisionmaker. Lastly,weareinterestedinthepotentialimpactoftheVLSNprocedure,discussedin Section6.2.2,whenusedinconjunctionwiththefacilityneighborhoodsearch.Therefore, toimproveuponthecustomerassignmentsassociatedwiththesetofprocuredfacilities determinedinthefacilityneighborhoodsearchprocedure,wewillalsotestahybrid implementation.Thishybridizationperformsasingle-customermultiple-exchangeVLSN onthebestsolutionfoundinthefacilityneighborhoodsearchprocedure.Ofparticular interestwillbewhethertheimprovedsolutionmeritstheaddedcomputationalexpense. 122

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Withthisconsiderationandthedetailsoftheproposedheuristicinplace,weare readytodiscussacomputationalstudydesignedtotesttheeectivenessofthevariantsof ourheuristicwhencomparedwiththeperformanceofawell-knowncommercialsolver. 6.4ComputationalStudy Ourcomputationalstudyconsiderstwelveseparateproblemscenarios.Foreach problemscenario,weconsider10separateinstances.Wevarytheratioofcustomers/facilities inordertoexploretheeectonbothcomputationaltimerequirementsandsolution quality.Ourstudyassessestheperformanceofthethreeimplementationsdiscussed inSection6.3.2.Additionally,wedeterminethetimerequiredforastate-of-the-art commercialsolvertondsolutionsofequivalentqualitytothoseobtainedbyourvarious implementations.TheexperimentswereperformedonaPCwitha3.40GHzPentiumIV processorand2GBofRAM.Allmixed-integerprogrammingproblemsweresolvedusing CPLEX11.0. 6.4.1ExperimentalData InourtestingoftheCFLFD,weconsiderinstanceswith15and30facilitiesanda varyingnumberofcustomersequalto jJj =3 jIj ,5 jI j 10 jIj ,25 jIj ,50 jIj ,and100 jIj .For eachcustomer,wegeneratetherandomvectorsofxedprotparameters P j fromuniform distributionson[30 ; 50]andtheelementsoftherandomvectorofrevenues, R j ,froma uniformdistributionon[2 ; 5].Thevectorsofcustomerrequirements A j L j and D j are generatedfromuniformdistributionson[10 ; 20],[75 ; 125],and[15 ; 35],respectively.Here D j isarandomvectorwhoseelementsrepresenttherangeofacceptablesizestofulll customerdemand,i.e.,theupperboundonthedemandforcustomer j ,whenassigned tofacility i ,isequalto L ij + D ij .SimilartotheGAPFD,wegenerateidenticalfacility capacitiessuchthat b i = jJj i 2I ,where = E min i 2I A i 1 + L i 1 jIj ; {11 123

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with E referingtotheexpectedvalueofthegivenexpression.Theparameter > 1, inatesthecapacityofafacilitytoensurethatnotallfacilitiesarerequiredinafeasible solutiontotheCFLFD.Intheseexperiments,thefacilitycapacitiesweregeneratedusing =2.Theparameter controlsthelevelofexibilityavailablewhendeterminingthe levelatwhicheachcustomer'sdemandisfullled.Thesecomputationaltestsconsider amoderateexibilitylevelbysetting =1 : 2.Weassumethatthecostofprocuringa facilityisdirectlyproportionaltothecapacityoftheindividualfacility.Therefore,the costofprocuringfacility i i 2I isgivenby F i = b i C i ,where C i representstheunitcost ofprocurementgeneratedfromauniformdistributionon[0 : 75 ; 1 : 5]. Asasidenote,recallthatateachstepoftheneighborhoodsearchweeectivelysolve aGAPFD.IntheheuristicproposedinChapter5perturbationofresourcecapacitiesis necessarywhencustomer-requirementparametersarefacilitydependent.Forthisreason, weperturboursetcapacitiesusingtheprocedureandparameterproposedinthatchapter. Thetablesinthefollowingsectionassesstheperformanceofthethreeimplementations describedinSection6.3.2.Eachrowinthetablesrepresentstheaverageresultscollected amongst10instancesgeneratedforthatparticularscenario.Thefollowingmeasuresare reportedinTables6-1{6-4.Acolumnlabeled FNS indicatesthatthefullneighborhood i.e.,allswap,open,closemoveswasconsideredateachiterationoftheprocedure.A columnlabeled RNS indicatesthatthetwo-phaseimplementationwasused.Thatis,we consideredonly O -movesinPhase1untilnoadditionalimprovementwasfound,then S -movesinPhase2,untiltheprocedureterminated.Lastly,acolumnincludingaheading of hybrid indicatesresultsobtainedbyrunningasingle-customermulti-exchangeVLSN onthesolutionobtainedbyeithertheFNSorRNSprocedure.Inaddition,weusethe followingnotation: UBError :Theupperboundontheerrorassociatedwiththeobjectivevalueofthe solutionobtainedfromthespeciedprocedure.Forexample,inthecaseofFNS, UBerror= z UB )]TJ/F21 11.9552 Tf 11.955 0 Td [(z FNS z FNS : 124

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Theupperboundontheoptimalsolution, z UB ,wasobtainedfromCPLEX.The valuewastakentobethebestupperboundavailableaftersolvingtheCFLFDfor15 minutes. HeuristicProcedureTime :Computationaltimeinsecondsrequiredbythe speciedprocedurei.e.FNS,RNS,orHybrid. CPLEXTime :ComputationaltimeinsecondsforCPLEXtoobtainthesameor bettersolutionthanthatfoundbythespeciedprocedure. 6.4.2Results AssuggestedinSection6.2.1,thehighestqualityheuristicresultsarelikelyto resultfromthefullneighborhoodsearchimplementationFNS.Tables6-1and6-3 provideaverageresultsforinstanceswith15and30facilities,fortheFNS,bothwith andwithoutthesupplementaryVLSNstep.Thesetablesindicatethatsolutionswith anaverageerrornomorethan4%wereobtainedforeachoftheproblemsizestested. TheadditionalVLSNstepi.e.,FNShybridreducedtheerroronlyasmallamountin eachsetofexperiments.Themostextremeimprovementsoccurforexperimentswitha customer-to-facilityratiolessthan10forinstanceswith15facilities,whilethebenetof theVLSNstepextendstoinstanceswithacustomer-to-facilityratioupto25forinstances with30facilities.ThetimerequiredforFNSwithoutVLSNrangedfromapproximately 2to90secondsfor15facilityinstancesand1to25minutesfor30facilityinstances. Table6-1showsthatCPLEXrequiredupto10timestheamountofcomputationaltime toobtainthesamesolutionsfor15facilityinstances,whilefromTable6-3weseethat CPLEXconsistentlyoutperformedtheFNSimplementationwithouttheVLSNstepfor instanceswith30facilities.Infact,forinstanceswith30facilitiesand3000customers, theFNSimplementation,aswellasCPLEX,requiredmorethantheallotedtimeof30 minutesandthereforetheseresultsarenotreported.TheFNShybridtimeinTables6-1 and6-3indicatesthattheadditionalVLSNeortonlymarginallyincreasedthetotaltime forexperimentswitharatioofcustomerstofacilitiesnogreaterthan10.Unfortunately,as shownintheresultsforthe750customerexperiments,inTable6-1,forlargerinstances, 125

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theadditionalVLSNstepbecomestimeprohibitive.Infact,thehybridimplementation forinstanceswith1500customersormoretakesmorethantheallottedruntimeto terminate;therefore,theseresultsareomittedfromTables6-1and6-3.Interestingly,while theimprovementinsolutionqualityinthehybridimplementationofFNSisminimal,the timerequiredbyCPLEXtoobtaintheseslightlybettersolutionsisconsistentlymorethan doublethetimerequiredtoobtainthesolutionsproducedbyFNSalone.Therefore,for instanceswithasmallcustomer-to-facilityratio,thehybridimplementationisattractive. Fortheseinstances,theFNShybridruntimeremainssmall,butthequalityofsolutionsis improvedtoadegreethatthecommercialsolverhasdicultyduplicatingtheresultina comparableamountoftime. Thevalueofthetwo-phasereducedneighborhoodsearchimplementationi.e.,RNS ispresentedinTables6-2and6-4,whichoeraverageresultscollectedoverthesame setof15and30facilityinstancesconsideredinTables6-1and6-3.Theaveragetimefor RNS,withoutVLSN,isatleast6timeslessthanFNSfor15facilityinstancesandatleast 15timeslessfor30facilityinstances.Moresignicantly,theaverageerrorsofthesolutions obtainedfromRNSareonlyslightlyhigherthanthoseobtainedfromFNS.Infact,for instanceswith15facilitiesand150customersorgreater,theaverageerroriswithinone one-hundredthofapercentofthatobtainedthroughFNS.Themostextremeincreasein errorcorrespondsto30facility,90customerinstances,wheretheerrorisapproximately twicethatobtainedfromtheFNSimplementation.However,forinstancesofthissize, CPLEXobtainshighqualitysolutionsinasmallamountoftime;therefore,theneedfor theheuristicislesssignicant.TheimpactofVLSNappliedtosolutionsobtainedfrom RNSissimilartothatseenwhenVLSNiscombinedwithFNS.Themostsignicant improvementinsolutionqualityisseenwithcustomer-to-facilityratiosof10orless.As withtheFNShybrid,theRNShybridisexceedinglytimeconsumingforbothsetsof instanceswithacustomer-to-facilityratiogreaterthan10. 126

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BecausetheRNSresultsprovidecomparablyhighqualitysolutionsinlesstime, theratiooftimerequiredforCPLEXandRNStondthesamesolutionisnotably higherthanthesameratiowithrespecttoFNS.Inaddition,ourstudyfoundthatthe averagenumberofsearchiterationsrequiredbyboththeFNSandRNSimplementations wasremarkablysimilarineachsetofexperiments.Thissuggeststhatour2-Phase RNSimplementationdoesnotterminateprematurelyasaresultofsearchingalimited neighborhood.Furthermore,recallthatFNSconsiders all S -movesateachiteration, whilenomorethan jIj swapsareconsideredintheRNSimplementation.Sincethenal solutionobtainedbyeachimplementationis,onaverage,remarkablysimilar,weconclude thatthemostdesirableswapswereconsideredinphase2ofourRNSimplementation. ThisisastrongindicationthattheswaporderingruleproposedinSection6.3.2is eective. Lastly,Tables6-1and6-2suggestasmallbutnoticeableincreaseinaverage heuristicerrorasthenumberofcustomersincreases.Itshouldbenotedthaterrors werecalculatedusingthebestupperboundobtainedbyCPLEXaftersolvingtheCFLFD asamixed-integerprogramfor15minutes.Itisexpectedthatthedicultyofsolvingthe CFLFDincreaseswithanincreaseinthenumberofcustomers.Therefore,itislikelythat theupperboundobtainedafter15minutesforinstanceswith1500customersisweaker thantheupperboundobtainedforaninstancewithonly150customers.Thissuggests thaterrorsmaybeinatedasthenumberofcustomersincreases,whichispreciselywhat weobserveinTables6-1and6-2. 6.5CFLFDHeuristicApplicationsandConclusions Thesuccessoftheheuristicframeworkproposedinthischapterispromisingfor optimizationproblemswithasimilarstructure.Specically,thereareanumberoffamiliar problemswithaxed-chargecomponentthatcanbesolvedwithourgeneralframework. Asmentionedpreviously,theCFLPclearlytsintoourframeworkandanecient methodforsolvingtheGAPsubproblemisreadilyavailable.Ourheuristicsearchesthe 127

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neighborhoodofaCFLPsolutioninadierentmanner,withalternativemovechoicesto thoseproposedbyAhujaetal.[6].Specically,themannerinwhichcustomerassignments aredeterminedwhenevaluatingpotentialmovesreliespredominantlyonthereassignment ofallcustomersthroughasecondaryheuristic.Thisapproachispursuedinlieuof searchingthemulti-customerexchangefacilityimprovementgraphandconsideringpartial reassignmentsofcustomersinintermediarystepsofthefacilityneighborhoodsearch,as donefortheCFLPinAhujaetal.[6].Ourresultssuggestthatthisimplementationchoice providesqualitysolutionsinareasonableamountoftimeforproblemsizesmuchlarger thanthoseconsideredinapproachesforrelatedproblems.Thereforethismethodology oersauniquealternativetosolvingthisclassofproblems.Inaddition,thewell-studied UncapacitatedFacilityLocationProblemUFLPErlenkotter[32],Sun[93]tswellinto ourframework.FortheUFLP,thesubproblemtodeterminetheassignmentsassociated withasetofprocuredfacilitiescanbetriviallysolvedtooptimality.Inthiscase,our heuristicsimpliestoarelatedapproachproposedbyGhosh[45].However,theorder andsubsetoftheneighborhoodmovessearchedinourimplementationisdistinctly dierent.Analclassofoptimizationproblemswhichtsintoourheuristicframework istheFixed-ChargeTransportationProblem.Variousapproacheshavebeenproposed tosolvethisproblembothheuristicallyi.e.,AdlakhaandKowalski[1]andexactly i.e.,Gray[47].Interestingly,ifplacedinourframework,theunderlyingsubproblem solvedtodeterminethevalueofeachmoveissimplyalineartransportationproblem. Thetransportationproblemitselfhasbeenwellstudiedandsolutionmethodshavebeen presentedinworkoriginatingwithFordandFulkerson[36]. Thelessonslearnedinthisworkwithregardtoimplementationofourneighborhood search,canbeapplieddirectlytoeachoftheadditionaloptimizationproblemsmentioned inthissection.Ourcomputationalstudyillustratesthananintelligentsearchofareduced facilityneighborhoodoershighqualitysolutionswithasmallamountofcomputational eort.Furthermore,forproblemswithasmallratioofcustomerstofacilities,the 128

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inclusionofaVLSNprocedureimprovesthequalityofthesolutionsobtainedwith minimaladditionaleort.However,ourstudyalsodemonstratesthatasingle-customer multi-exchangeheuristicisnotpracticalforproblemswithalargenumberofcustomers. Fortheseinstances,ourfacilityneighborhoodsearchperformsverywellasastand-alone heuristic.Therefore,ourworkoersanattractiveheuristicwhichcanbetailoredto successfullysolveabroadclassofprobleminstancesforboththeCFLFDandsimilar xed-chargeproblems. Table6-1.FNSresults:15facilities FNSFNSCPLEXFNSFNShybridFNShybridCPLEXFNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 452.41.581.62.40.909.1 753.90.8145.84.10.39722.7 1509.20.6634.810.90.52292.7 37533.90.4090.078.90.37267.3 75035.01.42160.9576.21.41301.6 150082.51.56640.5--Table6-2.RNSresults:15facilities RNSRNSCPLEXRNSRNShybridRNShybridCPLEXRNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 450.42.000.90.41.54183.5 750.50.8745.70.70.43541.8 1501.00.6634.82.80.53259.2 3753.30.4090.647.50.37265.9 7504.41.43161.1530.01.42301.8 15009.91.56642.0--Table6-3.FNSresults:30facilities FNSFNSCPLEXFNSFNShybridFNShybridCPLEXFNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 9074.73.863.075.22.7325.5 150153.12.204.3155.91.22230.6 300219.41.3310.02810.631281.4 750620.20.6351.417940.281416.9 15001407.70.39188.4--3000-----129

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Table6-4.RNSresults:30facilities RNSRNSCPLEXRNSRNShybridRNShybridCPLEXRNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 904.28.290.95.05.121.0 1508.72.403.011.71.7311.5 30012.51.489.963.00.97775.9 75033.40.6251.41261.50.281344.0 150076.30.39185.5--3000207.70.291660.3--hello 130

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CHAPTER7 RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHSHARED RESOURCECONSUMPTION 7.1Introduction InChapters4{6westudiedCFLFDanditsspecialcase,GAPFD.InbothGAPFD andCFLFDthetotalamountofresourceconsumption,aswellasrevenue,isequal totheaccumulationofeachindividualcustomer'scontributiontothesetwometrics. However,inalmostanymanufacturingscenario,certainproductgroupssharesimilar productionrequirements.Forexample,fulllingcertaincustomers'demandsmayrequire thesamemachinesettingsduetocommoncomponents.Therefore,producingforthese commoncustomersonthesamemachineminimizesthetotaltimerequiredtosatisfy theirdemands.Theoptionofassigningcustomerstocommonfacilities,atanadded benet,isconsideredinjoint-costassignmentproblemsShubik[91].Workinthisarea modelstheimpactofjointrewardsobtainedfromseparateindividualswhoarewilling toworkasagroup,asseenintheMixed-IntegerSetupKnapsackProblemproposedby Altayetal.[9].Separately,problemsthatconsiderjointcostsacrossanumberofitems arefrequentlyfoundinmulti-iteminventorysettingssuchastheJointReplenishment ProblemFedergruenandZheng[33].However,thechoiceofhowtoutilizecapacitysaved byassigningcustomerswithsimilarproductionrequirementstothesamemachineisa separatefeatureoftheoptimizationmodelandonethathasreceivedmuchlessattention inthesestudies.Morerelevantworkisfoundinschedulingproblemsthatexploittheuse ofcommonresourceswhendeterminingthereleasetimeofcustomers'jobs.Forexample, Li[60]consideredaresourceconstrainedschedulingprobleminwhichtheamountof resourceconsumedisafunctionofthetimeinwhichthecustomers'jobisreleased. Conceivably,customersofsimilartypes,releasedinuninterruptedsequence,requirea lesseramountofresource.Forproblemswithanassignment-basedstructure,Mazzola [68]consideredageneralizationoftheGAPwithnonlinearcapacityinteraction.Rather thanaccountingforsomesharedsetupcomponentofresourceconsumption,thiswork 131

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modeledtheinteractionbetweencostumersassignedtothesamefacility.Incontrast totheapproachproposedinthischapter,Mazzola[68]providedabranch-and-bound algorithmthatwasshowntosolveproblemswithup20customers. Anotherinterestingextensiontomoretraditionalassignmentproblemsistheinclusion ofvariedformsofdemandfulllmentconstraints.Inlightofnewrestrictionsin21 st centuryproduction,newformsofconstraintsneedbeconsideredinconjunctionwith traditionalplanningmodels.Thatis,ratherthanconsideringonlylimitsonphysical resourcesortime,theplannermustmakedecisionsthatadheretoadditionallimits placedonthesatisfactionofcustomerdemand.Forexample,asmentionedpreviously, customersmaybegroupedintotypes,witheachtypeinterpretedaseitheriaspecic client,iicommonproductionrequirements,iiiacommonshipmentdestinationoriv commonmanufacturingbyproducts.Giventhesepossibleinterpretations,thelevelat whichcustomerdemandisfullledmaybelimitedbyeithertransportationresourcesor physicalspaceallocatedinthewarehousefacility.Alternatively,withgrowingconcern onmanufacturers'impactsontheenvironment,onemustconsiderhowproduction associatedwithfulllingcustomerdemandmayresultineitherpollutionorhazardous materials.Thatis,thetotalamountofdemandsatisedwithinaparticulargroupor typemaybelimitedbyoutsideparties,suchastheEnvironmentalProtectionAgency ortheOccupationalSafetyandHealthAdministration.Thesearespecicexamplesthat arecommonacrossanumberindustries.Thus,itisclearthatitisimportanttoinclude constraintsthatlimitproductionacrossallresourcesforagivensubsetofcustomers. Themodelsstudiedintheprecedingchaptersdonotaccountforthisadditionalform ofproductionlimitations.Itisimportanttonotethatfewotherworksintheliterature considerthisproblemelementeither.Looselyrelatedproblemsthathavereceivedmore attentionintheliteraturearethosewhichconsidermultipleresourceconsumption. AspecicexampleisfoundintheMulti-resourceGeneralizedAssignmentProblem, consideredbyGavishandPirkul[40],amongothers.Thismodelassumesthateachfacility 132

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consumesmultipleresourcestosatisfyacustomer'sdemand.However,limitsoncapacity consumptiondospancustomersassignedtodierentagents.Separately,Mazzolaand Neebe[69]studytheResource-ConstrainedAssignmentSchedulingProblemRCAS.This extensionofthe pure assignmentproblemconsidersasideconstraintthatrestrictsthe totalconsumptionofaresourceamongstdecentralizedcustomerandfacilitysets.Itisthe limitationofdemandfulllmentforcustomersassignedtonumerousfacilitiesthatwewish toaccountforinthemodelconsideredinthischapter. Asbrieyalludedtointhediscussionofrelevantliterature,thischapterconsidersa newclassofproblemsthat,amongotherthings,modelproductionenvironmentsinwhich aportionofcapacityconsumptionissharedamongcommoncustomerssatisedbythat resource.AsintheGAPFDandCFLFD,thenewoptimizationmodelassignscustomers tofacilitiesanddeterminesthecorrespondingdemandfulllmentlevels.However,dierent considerationsmustbetakenintoaccountwhenmakingthesedecisions.First,customers arenowgroupedbytype.Customersofthesametype,assignedtothesamefacility, consumeaxedamountofresourceinadditiontotheirindividualconsumption.Also,in additiontotheresourcelimitationsofindividualfacilities,aggregateresourceconsumption amongcustomersofaparticulartypeissubjecttoaseparatesetofrestrictions.This modeladdsalevelofcomplexitytothedecision-makingprocess.Aplannermustnow considertheimpactofsavingcapacitybyassigningcustomersofthesametypetothe samefacility.Furthermore,theadditionallimitationofproductionassociatedwith aparticularcustomersetmayaectthelevelsatwhichdemandsaresatised.Since thisproblemstillconsiderstheassignmentofcustomerstofacilities,weagainpursue abranch-and-priceapproachbasedonareformulationofourmodel.However,unlike Chapter4,theso-calledmasterproblemisnolongerintheformoftheset-partitioning problem.Duetotheadditionalcapacityconstraints,themasterproblem,pricingproblem, andcolumnrepresentationsmustbecarefullydeveloped.Weshowthatthepricing 133

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problembelongstoanunstudiedclassofknapsackproblemsforwhichweproposeecient solutionapproaches. Theremainderofthischapterisorganizedasfollows.Section7.2presentsthe optimizationmodelwhichweconsider.Sections7.3and7.4developanexactalgorithmfor aspecialcaseofthemodelwithxedcustomerdemand.Then,Section7.5considersthe variantofthemodelwithexibledemandandderivesanimportantequivalentformulation oftheproblemforwhichanexactalgorithmisderived.Section7.7providesdetailsof theimplementationofourbranch-and-pricealgorithm.Finally,Section7.8discussesa computationalstudyofourapproach. 7.2ModelFormulation Weconsiderasetofcustomers J .Eachcustomer'sdemand j j 2J mustbe satisedbyasinglefacility i i 2I .Ifcustomer j j 2J isassignedtofacility i i 2I ,thenaxedprot, p ij ,isaccruedandaxedamountofcapacity a ij isconsumed. Furthermore,thecorrespondingcustomerdemandfulllmentlevelmustbeselectedfrom theinterval[ ` ij ;u ij ]andanadditionalprot r ij isaccruedperunitofdemandfulllment. Eachfacility i 2I hascapacity b i i 2I .Inadditiontotheseconsiderations,wehavea setofcustomertypes Q ,whereeachcustomerisassociatedwithasingletype.Customers oftype q q 2Q belongtotheset J q .Therstdistinguishingcharacteristicofthismodel isdenedbythemannerinwhichresourceconsumptionmaybeshared.If any customer oftype q q 2Q isassignedtofacility i ,axedamountofresource, f iq ,isconsumed. Furthermore,thecollectivecapacityconsumedbycustomersoftype q islimitedby g q q 2Q .Theobjectiveistodeterminetheassignmentofcustomerstofacilities,aswellas thecorrespondingcustomersdemandfulllmentlevels,inordertomaximizetotalprot, whilesatisfyingthecapacityconstraintsofthefacilitiesandtheindividualcustomertypes. Themodelweconsideristhengivenby maximize X i 2I X j 2J p ij x ij + r ij v ij 134

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subjecttoFASR X q 2Q f iq max j 2J q x ij + X j 2J v ij b i i 2I {1 X i 2I X j 2J q v ij g q q 2Q {2 X i 2I x ij =1 j 2J {3 ` ij x ij v ij u ij x ij i 2I ; j 2J {4 x ij 2f 0 ; 1 g i 2I ;j 2J {5 whichwerefertoasthe FlexibleDemandResourceAllocationProblemwithShared ResourceConsiderations .Constraints7{1requirethatthecapacityvariableandshared consumedbycustomersassignedtofacility i isnogreaterthan b i i 2I .Furthermore, constraints7{2ensurethatdemandsatisedforcustomersoftype q doesnotexceed g q q 2Q .Constraints7{3an7{4aretheassignmentandexibilityconstraints introducedinChapter3.Forsakeofsimplicity,andwithoutlossofgenerality,we've assumedthattheindividualcustomerxedcapacityconsumptions a ij i 2I ; j 2J areincludedinthedemandlevelbounds.Thatis,both ` ij and u ij areincreasedbythe amount a ij i 2I ; j 2J .Thisimpliesthatthexedprotparameters a ij arereducedby theamount r ij a ij i 2I ; j 2J NoticethatFASRisamixedintegerprogramwithasetofnon-linearconstraintsand linearobjectivefunction.Thenon-linearconstraints,7{1canbelinearizedasshownin thefollowingformulation. maximize X i 2I X j 2J p ij x ij + r ij v ij subjecttoFASR 0 X q 2Q f iq s iq + X j 2J v ij b i i 2I 135

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X i 2I X j 2J q v ij g q q 2Q X i 2I x ij =1 j 2J ` ij x ij v ij u ij x ij i 2I ; j 2J s iq x ij i 2I ; j 2J q ; q 2Q {6 x ij 2f 0 ; 1 g i 2I ; j 2J Theconstraints7{6ensurethatifacustomeroftype q q 2Q isassignedtofacility i i 2I ,thentheaxedamountofresource, f iq ,isconsumed.Itshouldbenoted thatthealternativeformulationdoesnotpreventsomepercentageofthe f iq unitsof resource i frombeingconsumed,evenwhennocustomersoftype q areassignedtofacility i i 2I ; q 2Q .Inotherwords,noconstraintprevents s iq i 2I ; q 2Q frombeing greaterthanzeroevenwhen x ij =0forall j 2J q .Itisinterestingtonotethatthis scenarioarisesonlyiftheoptimalsolutionissuchthatiallcustomersassignedtofacility i havetheirdemandsatisedattheirupperbounds,oriinocustomersareassignedto facility i i 2I inanoptimalsolution.However,ineachofthesecases,theobjective valueandcorrespondingdecisionvariablesarestilloptimaltotheoriginalFASRmodel. ThereforeFASR 0 isanequivalentrepresentationofFASR. Thepurposeofthischapteristodevelopabranch-and-pricealgorithmforFASR. Asstatedinpreviouschapters,thesuccessofabranch-and-priceapproachreliesstrongly ontheabilitytoeectivelysolvethecorrespondingpricingproblem.Tothisend,we focusmuchofourattentionondevelopingecientsolutionmethodsfortheresulting pricingproblem.Thedevelopmentoftheseapproachesismosteasilypresentedbyrst consideringaspecialcaseofFASR.Therefore,inSection7.3weinitiallydevelopan exactsolutionprocedureforthenon-exiblevariantofFASRinwhich ` ij = u ij = a ij i 2I ; j 2J .WerefertothisspecialcaseoftheFASRastheResourceConstrained AssignmentProblemwithSharedResourceConsumptionRCAS. 136

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maximize X i 2I X j 2J p ij x ij subjecttoRCAS X q 2Q f iq s iq + X j 2J a ij x ij b i i 2I {7 X i 2I X j 2J q a ij x ij g q q 2Q X i 2I x ij =1 j 2J s iq x ij i 2I ; j 2J q ; q 2Q x ij 2f 0 ; 1 g i 2I ; j 2J where p j = p j + r j a j j 2J 7.3ExactAlgorithmforRCAS Inthissectionweproposeabranch-and-pricealgorithmforRCASbasedonan interestingreformulationoftheproblem.Firstwewillneedthefollowingdenitionsand terminology. x d i isabinaryvectorwithelements x d ij j 2 J representingthe d th subsetof customersthatcanbefeasiblyassignedtofacility i withrespecttoconstraint 7{7; D i isthetotalnumberofsubsetsofcustomersthatcanbeassignedtofacility i ; d i isabinaryvariablewithvalue1ifthe d th subsetassociatedwithfacility i isused, and0otherwise; i x d i = P j 2J p ij x d ij ; d =1 :::D i ; i 2I q i x d i = P j 2 T q a d ij x d ij d =1 :::D i ; i 2I ; q 2Q Themasterprobleminthiscasecanbewrittenas maximize X i 2I D i X d =1 i x d i d i 137

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subjecttoMP X i 2I D i X d =1 q i x d i d i g q q 2Q {8 X i 2I D i X d =1 x d ij d i =1 j 2J {9 D i X d =1 d i =1 i 2I {10 d i 2f 0 ; 1 g d =1 ;:::;D i ; i 2I Nextwedenethefollowingnotation: q q 2Q arethedualvariablesassociatedwiththecustomersetcapacity constraint,7{8inMP; j j 2J arethedualvariablesassociatedwithconstraints,7{9inMP; i i 2I arethedualvariablesassociatedwiththeconvexityconstraints,7{10,in MP. Thepricingproblemassociatedwithresource i i =1 ;:::; I isnowwrittenas maximize X j 2J )]TJ/F15 11.9552 Tf 6.466 -9.684 Td [( p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j a ij x j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i subjecttoPP i X q 2Q f q max j 2 T q f x j g + X j 2J a ij x j b i x j 2f 0 ; 1 g j 2J where q j isthecustomersettowhichcustomer j j 2J belongs.Aneectivemethod forsolvingthepricingproblemisinstrumentalindevelopinganeectivebranch-and-price procedure.Therefore,Section7.4studiesaclassofoptimizationproblemsthatincludes PP i 138

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7.4SharedConsumptionKnapsackProblem Inthissection,wedevelopbothheuristicandexactapproachesforsolvingthe followingclassofknapsackproblems. maximize X j 2J p j x j subjecttoSKP X q 2Q f q s q + X j 2J a j x j b {11 s q x j j 2J q ; q 2Q {12 x j 2f 0 ; 1 g j 2J : {13 s q 2f 0 ; 1 g q 2Q : {14 InSKP,thebinaryrestrictions7{14areclearlyredundant.Therefore,theyarenot includedinthefollowingrelaxationofSKP,whichwerefertoasSKPR, maximize X j 2J p j x j subjecttoSKPR X q 2Q f q s q + X j 2J a j x j b s q x j j 2J q ; q 2Q x j 2 [0 ; 1] j 2J : {15 Intheaboverelaxation,werelaxthebinaryrestrictionsontheassignmentvariables x j j 2J byreplacing7{13with7{15.Beforecontinuing,itisworthwhiletonote thatthereclearlyexistsanoptimalsolutiontobothSKPandSKPRforwhich x SKP j =0 and x SKPR j =0if p j 0.Therefore,thefollowingassumptionholdswithoutlossof generality. 139

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Assumption2. Forall j 2J p j > 0 TobegintomotivateanapproachtosolveSKP,consideranequivalentnon-linear formulationofSKP,SKP 0 maximize X j 2J p j y j subjecttoSKP 0 X q 2Q 8 < : f q + X j 2J q a j y j 9 = ; s q b {16 s q y j j 2J q ; q 2Q y j 2f 0 ; 1 g j 2J s q 2f 0 ; 1 g q 2Q Thealternativedecisionvariables, y j j 2J areintroducedtodistinguishbetweenthe assignmentvariablesinthealternativepresentationsofSKP.Clearly,inthecaseofSKP andSKP 0 ,thereexistsanoptimalsolutioninwhich x j = y j j 2J .However,this relationshipdoesnotnecessarilyholdwhencomparingtheoptimalassignmentvalues x j j 2J obtainedinSKPRtothevalues y j j 2J foundinthefollowingrelaxationof SKP 0 ,whichwerefertoasSKPR 0 maximize X j 2J p j y j subjecttoSKPR 0 X q 2Q 8 < : f q + X j 2J q a j y j 9 = ; s q b s q y j j 2J q ; q 2Q y j 2 [0 ; 1] j 2J s q 2f 0 ; 1 g q 2Q : 140

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NotethatinSKPR 0 thebinaryrestrictionsontheassignmentvariables y j j 2J arerelaxed.However,thebinaryrestrictionsonthevariables s q q 2Q remain. Interestingly,SKPR 0 canbereformulatedasaKPEIstudiedinChapter4.Inthis equivalentformulation,exibilityvariables w q q 2Q representthecollectiveamountof capacityconsumedbyallcustomersoftype q q 2Q .Thisequivalentformulationcanbe writtenas maximize X q 2Q ~ r q w q subjecttoKPEI X q 2Q w q b {17 w q u 0 q s q q 2Q {18 w q ` 0 q s q q 2Q {19 s q 2f 0 ; 1 g q 2Q {20 where ` 0 q = f q and u 0 q = f q + P j 2J q a j q 2Q ,and~ r q w q isanoptimalsolutiontothe followingparametricoptimizationproblem,CKP, maximize X j 2J p j y j subjecttoCKP f q + X j 2J q a j y j w q y j 2 [0 ; 1] j 2J q ; if w q > 0,and0otherwise,foranyfeasible w q q 2Q toKPEI.Notethat,without lossofgenerality,wecanalsolet ` 0 q =min f f q ;b g and u 0 q =min n f q + P j 2J q a j ;b o q 2Q .Forconvenience,weretainthislattersetofdenitionsthroughouttheremainder 141

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ofthechapter.Now,noticethatCKPisthecontinuousknapsackproblem.Therefore, thefunction r q w q isreadilyobtained.First,let S q bethesetofcustomers j 2J q with y CKP j > 0inanoptimalsolutiontoCKPwith w q = u 0 q .Itshouldbenotedthatthereisat mostone j 2J q forwith0 > > > < > > > > : =0 w q =0 = 0
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BoththeexactandheuristicmethodologiestosolveSKParemotivatedbya relaxationofKPEIstudiedinChapter4denotedbyRKP.Forthespeciallystructured revenuefunctionsinthischapter,werefertothisrelaxationasKPEIR. maximize X q 2Q q w q subjecttoKPEIR X q 2Q w q b {21 w q 0 q 2Q : Thefunction q q 2Q isthenon-decreasingconcaveenvelopeencompassingtheorigin, thefunction~ r q andthepoint b; ~ r q u 0 q .Theprotfunctions, q w q q 2Q ,aredened inthefollowinglemma.First,let k q =inf k q =0 ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1: P k q k 0 =0 p ^ | qk 0 f q + P k q k 0 =0 a ^ | qk 0 p ^ | k q +1 a ^ | k q +1 {22 bethesmallestindexforwhichthecollectiveperunitprotforcustomersassociatedwith k q k q isnolessthantheperunitprotassociatedwithcustomer^ | k q +1 with p ^ | j ~ S q j a ^ | j ~ S q j =0. Lemma7. InKPEIR,thefunctions q q 2Q aregivenby q w q = 8 > > > > < > > > > : = ~ p k q ~ a k q w q 0 w q ~ a k q =~ r q w q ~ a k q
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q w q = 8 > > > > > < > > > > > : = ~ r q q q w j 0 w j j =~ r q w q q
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~ a k q ; ~ r q ~ a k q q 2Q .Inaddition,let J 0 q = n ^ | k q 2 ~ S q : k q = k q ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 o q 2Q ThenKPEIRisequivalentlyrepresentedbyKPEIR 0 maximize X j 2J o ^ p j ^ a j w j + X q 2Q X j 2J 0 q ^ p j ^ a j w j subjecttoKPEIR 0 w j 2 [0 ; ^ a j ] j 2J o {24 w j 2 [0 ; ^ a j ] j 2J 0 w j 0 j 2J o w j 0 j 2J o where ^ p j = 8 > > < > > : =~ p k ~ q j if j 2J o = p j if j 2J 0 {25 and ^ a j = 8 > > < > > : =~ a k ~ q j if j 2J o = a j if j 2J 0 : {26 and~ q j isthecustomertypethataparticular`dummycustomer' j 2J o represents. Furthermore,theparameter a j representstheamountofcapacityconsumedandprot accruedbycustomer j intheoptimalsolutiontotheparametricoptimizationproblem CKPwith w q = u 0 q .Again,thereisatmostonecustomerwithineachtypeforwhich a j 6 = a j and p j 6 = p j : {27 145

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Then,givenanoptimalsolutiontoKPEIR 0 ,theoptimalsolutiontoKPEIRisgivenby w KPEIR q = 0 @ X j 2J 0 q w KPEIR 0 j + w KPEIR 0 | q 1 A q 2Q where | q = f j 2J o : q j = q g Clearly,KPEIR 0 issolvablebyconsideringcustomers j 2 J o [ [ q 2Q J 0 q in non-increasingorderof ^ p j ^ a j .Therefore,wearepreparedtoformallyprovideanalgorithm forKPEIR. KPEIRAlgorithm Step0. Set w KPEIR q =0for q 2Q .Find k q andestablishsets J o J 0 q q 2Q and ~ J = J o [ [ q 2Q J 0 q Step1. Sortcustomers j 2 ~ J innon-increasingorderof ^ p j ^ a j Step2. Let^ | betherstcustomerin ~ J and^ q = q 2Q :^ | 2J 0 q or~ q ^ | = q theset associatedwith^ | .Set w KPEIR ^ q = w KPEIR ^ q +min f b; ^ a ^ | g b = b )]TJ/F15 11.9552 Tf 11.955 0 Td [(min f b; ^ a ^ | g : Set ~ J = ~ Jnf ^ | g .If b =0or ~ J =,STOP,elserepeatStep2. UsingthisalgorithmwecanecientlysolveKPEIR.Theimportantstructural propertyofKPEIRfromLemma2inChapter4motivatesbothaheuristicandcustomized branch-and-boundtosolveSKP.Thisresultisrestatedinthefollowinglemma. Lemma8. Anoptimalsolutionto KPEIR existswithatmostonecustomertype q such that i w q <` 0 q ,or ii q w q > ~ r q w q and ` 0 q w q u 0 q Proof. SeeproofofLemma2. 146

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TounderstandhowKPEIRanditsstructureallowustoeectivelysolveSKP, considertherelationshipbetweenSKPRandKPEIRgiveninthefollowingtheorem. Theorem9. SKPRandKPEIRareequivalent. Proof. First,let w KPEIR beanoptimalsolutiontoKPEIR.Considerconstructingafeasible solutiontoSKPRinthefollowingmanner,forall q q 2Q If w KPEIR q =0, s SKPR q =0 x SKPR j =0 j 2J q : Elseif0
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Since P q 2Q w KPEIR q b ,clearly X q 2Q f q s SKPR q + X j 2J a j x SKPR j b: Inaddition, s SKPR q x SKPR j j 2J q ; q 2Q and x SKPR j 2 [0 ; 1] j 2J : Therefore,thefeasiblesolution w KPEIR toKPEIRequatestoafeasiblesolutionto SKPR.Likewise,itiseasytoseethatafeasiblesolutiontoSKPRcanbeconstructedfrom afeasiblesolutiontoSKPRbysettingtheexibilityvariable, w KPEIR q ,equaltothetotal capacityconsumedbycustomersoftype q intheoptimalsolutionsolution x SKPR ;s SKPR Lastly,bythemannerinwhichthefunction q q 2Q isdenedabove,asolutionto KPEIR,withobjectivefunction z KPEIR canbeconvertedtoasolutionwithanequivalent objective z SKPR andviceversa. TheequivalenceofSKPRandKPEIR,alongwiththestructuralpropertyinTheorem 9revealagreatdealaboutthestructureofSKPR.Beforecontinuing,itwillbeusefulto introducethefollowingsets.Let P = f q :0
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Thefollowinglemmaboundsthenumberoffractionalsharedresourceconsumption variables s SKPR q q 2Q Lemma9. AnoptimalsolutiontoSKPR x SKPR ;s SKPR existsforwhich jPj 1 : Proof. FromtheproofofTheorem9,afractional s q q 2Q onlyoccurswhen 0
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forsome ^ k q k q ,thenthiscorrespondstoasolutioninwhichasinglecustomer, | ^ k q ,is fractionallyincluded.Inaddition,fromtheproofofTheorem9,ifthisoccurswith 0
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Furthermore,thefulllmentlevels v d ij j 2J correspondto an extremepoint solutiontothefollowinglinearprogram,solvedwithrespecttotheassignments x d i maximize X j 2J r ij v ij + C d i subjecttoSPv i x d i X j 2J v ij b d i {30 ` ij x d ij v ij u ij x d ij j 2J {31 where C d i = P j 2J p ij x d ij and b d i = b i )]TJ/F26 11.9552 Tf 11.955 8.966 Td [(P q 2Q f q max j 2J q x d ij ; S i isthenumberof unique subsetsofcustomersthatcanbeassignedtofacility i whilesatisfying7{29; D i isthesetofindices, d ,forallcolumnsascharacterizedintherstbullet associatedwithfacility i ; D i ispartitionedintosets D is suchthat x d i = x d 0 i whenever d;d 0 2D is and x d i 6 = x d 0 i whenever d 2D is and d 0 2D is 0 with s 6 = s 0 ; s i isabinaryvariablewithvalue1ifthe s th subsetofassignmentsassociatedwith facility i isusedand0otherwise; d i isacontinuousvariablewithvaluein[0 ; 1]representingtheproportionofthe values )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(x d i ;v d i includedinthesolutiontoMP-F; i x d i ;v d i is P j 2J )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(p ij x d ij + r ij v d ij ; q i x d i ;v d i = P j 2J q v d ij Formally,themasterproblem,MP-F,iswrittenasfollows maximize X i 2I X d 2D i i x d i ;v d i d i subjecttoMP-F X i 2I X d 2D i q i x d i ;v d i d i g q q 2Q {32 X i 2I X d 2D i x d ij d i =1 j 2J {33 151

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X d 2D is d i = s i s =1 ;:::;S i ; i 2I {34 X s 2S i s i =1 i 2I {35 d i 0 d 2 D i ; i 2I {36 s i 2f 0 ; 1 g s =1 :::;S i ; i 2I : {37 ThefollowingtheoremensuresthatMP-FisavalidrepresentationofFASR. Theorem10. IntheoptimizationproblemMP-F, iThenumberofcolumns, )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(x d i ;v d i d 2D i isnite; iiMP-FandFASRareequivalent. Proof. Fori,notethatthedimensionofthefeasiblesubsetsofcustomersassociatedwith anyfacilityisclearlyexponential.Furthermore,recallthatMP-Fonlyconsiderscolumns inwhichthedemandfulllmentlevelsforaxedvectorofassignments, x d i ,correspondto anextremepointsolutiontoSPv i x d i .Sincethenumberofextremepointsinthefeasible regionofSPv i isnite,thenumberofcolumnsinMP-Fthatcorrespondtothesameset ofcustomerassignmentstoaparticularfacilityisniteaswell.Thus,sincethenumberof facilitiesinMP-Fisxed,thenumberofcolumnsincludedinMP-Fisnite. Forii:Let ; beasolutiontoMP-F.Furthermore,let x F ;v F beasolutionin termsofdecisionvariablesinFASR.First,ifweset x F ij = X d 2D i d i x d ij i 2I ; j 2J {38 thenby7{33{7{37, X i 2I x F ij =1 i 2I ; j 2J ; {39 satisfyingassignmentconstraint7{3.Nowset v F ij = X d 2D i d i v d ij i 2I ; j 2J : {40 152

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Recallthat,bydenition, x d i v d i isifeasibletobothfacilitycapacityconstraint7{1 andexibilityconstraint7{4andii v d i isanextremepointsolutiontothefeasible regionofSPv i x d i .Since v F ij isrepresentedasaconvexcombinationoftheseextreme points, x F v F isfeasibleto7{1and7{4,aswell.Moreover,by7{32wehavethat X j 2J q v F ij g q i 2I ; q 2Q ; {41 satisfyingcustomertypecapacityconstraint7{2.Lastly, X i 2I X d 2D i i x d i ;v d i d i = X i 2I X d 2D i X j 2J )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(p ij x d ij + r ij v d ij d i {42 = X i 2I X j 2J p ij X d 2D i d i x d ij + r ij X d 2D i d i v d ij {43 = X i 2I X j 2J )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p ij x F ij + r ij v F ij {44 sotheobjectivevalueassociatedwith ; inMP-Fisequivalenttothatofthe constructedsolution x F v F inFASR.Thus,anyfeasiblesolutiontoMP-Fisalso feasibletoFASRandtheirobjectivefunctionvaluesareequivalent. Toshowformulationequivalence,wemustalsobecertainthatanyfeasiblesolution toFASRisalsofeasibletoMP-F,againwithequivalentobjectivevalues.However,given anyfeasiblesolution x F v F ,thedenitionofthecolumnsincludedinMP-Fensures thatwecanrepresentthissolutionasaconvexcombinationofthecolumnscomprising MP-Fbychoosingtheappropriatevaluesfor d i associatedwithcolumnscorresponding tothecustomersubsetsindicatedby x F .Thisimmediatelyensuresthat7{33{7{37 aresatised.Furthermore,since x F v F isfeasibleto7{2,thecustomertypecapacity constraint7{32issatisedaswell.Lastly,byareversepresentationof7{42{7{44, theobjectivevaluesofthesolutionstoeachformulationareequivalent.Thisshowsthe desiredresult. 153

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Inourexactalgorithm,wesolvethelinearrelaxationofMP-Fwhichwedenoteas LPMP-F,ateachnodeofabranch-and-boundtree.However,asdescribedinpartiof theproofofTheorem10,thetotalnumberofcolumnsinMP-Fisexponential.Therefore, wesolveLPMP-FbyaddingcolumnsiterativelytoarestrictedversionofMP-Finwhich asubsetofcolumnsisconsidered.WedenotethisrestrictedrelaxationasLPRMP-F. ItisimportanttonotethatpartiofTheorem10ensuresthatthiscolumngeneration procedurehasniteconvergence.Thederivationoftheso-called pricingproblem ,solved toidentifyattractivecolumns,requiresastudyofthedualofLPRMP-F.However,to simplifythepresentationofthisrelaxationanditscorrespondingdual,rstnoticethat MP-Fcanbeequivalentlyreformulatedas maximize X i 2I X d 2D i i x d i ;v d i d i subjecttoMP-F 0 X i 2I X d 2D i q i x d i ;v d i d i g q q 2Q X i 2I X d 2D i x d ij d i =1 j 2J X d 2D i d i =1 i 2I {45 d i 0 d 2 D i ; i 2I {46 X d 2D is d i 2f 0 ; 1 g s =1 ;:::;S i ; i 2I {47 bysubstituting7{34into7{35andreplacing7{37with7{47.Therefore,the relaxedversionofMP-F,whichincludesonlyarestrictedsetcolumns,LPRMP-F,can bewrittenas maximize X i 2I X d 2 ~ D i i x d i ;v d i d i 154

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subjecttoLPRMP-F X i 2I X d 2 ~ D i q i x d i ;v d i d i g q q 2Q {48 X i 2I X d 2 ~ D i x d ij d i =1 j 2J {49 X d 2 ~ D i d i =1 i 2I {50 d i 0 d 2 ~ D i ; i 2I {51 X d 2 ~ D is d i 1 s =1 ;:::; ~ S i ; i 2I {52 NoticethatinLPRMP-F,thebinaryrestriction7{47isrelaxedto7{52andthe sets ~ D i ~ D is and ~ S i containtheindicesofthesubsetofcolumnsbeingconsideredatany particulariterationofourcolumngenerationprocedure.Furthermore,notethatbecause of7{50,constraint7{52isredundantinLPRMP-F.Thus,thedualofLPRMP-F iscorrectlydenedwithrespecttoconstraints7{48{7{51only.Thepurposeofthe pricingproblemistoidentifyaviolatedconstraintintheinthefollowingoptimization problem,DRMP-F, minimize X q 2Q g q q + X j 2J j + X i 2I i subjecttoDRMP-F X q 2Q q i x d i ;v d i q + X j 2J x d ij j + i i x d i ;v d i d 2D i ; i 2I {53 q 0 q 2Q {54 j free j 2J {55 i free i 2I {56 where 155

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q q 2Q arethedualvariablesassociatedwiththecustomerssetcapacity constraint,7{48inLPRMP-F; j j 2J arethedualvariablesassociatedwithconstraints,7{49inLPRMP-F; i i 2I arethedualvariablesassociatedwiththeconvexityconstraints,7{50,in LPRMP-F. Ifwesubstitutethedenitionsof q i x d i ;v d i and i x d i ;v d i intothisformulation, DRMP-Fisequivalentlyrepresentedby minimize X q 2Q g q q + X j 2J j + X i 2I i subjecttoDRMP-F X q 2Q X j 2J q v d ij q + X j 2J x d ij j + i X j 2J )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p ij x d ij + r ij v d ij d 2D i ; i 2I {57 q 0 q 2Q {58 j free j 2J {59 i free i 2I {60 Ourpricingproblemseekstoidentifyapairofvectors x i ;v i ,whichviolates7{57 withithevector x i satisfying7{29andiithecorrespondingdemandfulllmentlevels v i determinedbySPv i .Avector x i ;v i violates7{57if X j 2J [ p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x ij + r ij v ij ] )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X q 2Q X j 2J q q v ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( i > 0{61 orequivalently X j 2J p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x ij + )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j v ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i > 0{62 where q j isthetypetowhichcustomer j belongs.Therefore,inlightofiandii,along withdualconstraint7{57,thepricingproblemassociatedwithfacility i i 2I isgiven by maximize X j 2J p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x j + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j v j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i 156

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subjecttoPP i -F X q 2Q f iq max j 2 T q f x j g + X j 2J v j b i ` ij x j v j u ij x j j 2J x j 2f 0 ; 1 g j 2J : InthefollowingsectionwestudytheclassofproblemsthatincludesPP i -Fby extendingtheresultsofSection7.4. 7.6SharedConsumptionKnapsackproblemwithFlexibleCustomerDemand Inthissection,westudythefollowingclassofknapsackproblems. maximize X j 2J p j x j + r j v j subjecttoSKFP X q 2Q f q s q + X j 2J v j b v j u j x j j 2J {63 v j ` j x j j 2J {64 s q x j j 2J q ; q 2Q x j 2f 0 ; 1 g j 2J : s q 2f 0 ; 1 g q 2Q : whoselinearrelaxationisgivenby maximize X j 2J p j x j + r j v j subjecttoSKFPR X q 2Q f q s q + X j 2J v j b 157

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v j u j x j j 2J v j ` j x j j 2J s q x j j 2J q ; q 2Q x j 2 [0 ; 1] j 2J : AsintheSKP 0 ,thenon-linearrepresentationofSKFP,SKFP 0 ,isgivenby maximize X j 2J p j x j + r j j subjecttoSKFP 0 X q 2Q 0 @ f q + X j 2J q j 1 A s q b j u j y j j 2J j ` j y j j 2J s q y j j 2J q ; q 2Q y j 2f 0 ; 1 g j 2J {65 s q 2f 0 ; 1 g q 2Q where j j 2J q areusedtorepresentthedemandfulllmentlevelsand y j j 2J q the assignmentsinSKFP 0 .AswithSKP 0 ,weconsiderthefollowingrelaxationofSKFP 0 maximize X j 2J p j y j + r j j subjecttoSKFPR 0 X q 2Q 0 @ f q + X j 2J q j 1 A s q b j u j y j j 2J 158

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j ` j y j j 2J s q y j j 2J q ; q 2Q y j 2 [0 ; 1] j 2J s q 2f 0 ; 1 g q 2Q : inwhichthebinaryrestrictions7{65arerelaxed.Interestingly,SKFPR 0 canalso bereformulatedastheKPEIpresentedinSection7.4with ` 0 q =min f f q ;b g u 0 q = min n f q + P j 2J q u j ;b o q 2Q and~ r q w q theoptimalsolutiontothefollowing parametricoptimizationproblem,FCKP, maximize X j 2J p j y j + r j j subjecttoFCKP f q + X j 2J q j w q ` j y j j u j y j j 2J q y j 2 [0 ; 1] j 2J q : Tomoreexplicitlystate~ r q w q ,wedenethefollowingsetsandnotation.Let J )]TJ/F22 7.9701 Tf -1.107 -7.294 Td [(q = f j 2 J q : p j 0 g bethesetofcustomerswithanon-positivexedprotand J + q = f j 2J q : p j > 0 g thecustomerswithapositivexedprot.Furthermore,spliteach j 2 J + into twocustomers, j 1 and j 2 .Thefollowinglemmacharacterizes~ r q w q q 2Q usingthe denitionprovidedinLemma6. Lemma11. ForSKPFR 0 ,thecorrespondingfunction ~ r q q 2Q isgivenbyLemma6 with ^ J q = J )]TJ/F22 7.9701 Tf -2.208 -7.294 Td [(q [J + q usedinplaceof J q and a j 1 = ` j {66 a j 2 = u j )]TJ/F21 11.9552 Tf 11.956 0 Td [(` j {67 159

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p j 1 = p j + r j ` j {68 p j 2 = r j u j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` j {69 for j 1 and j 2 in J + q and a j = u j {70 p j = p j + r j u j {71 for j 2 J )]TJ/F22 7.9701 Tf -1.107 -7.294 Td [(q Proof. NoticethattheFCKPisitselftheKPEIwithalinearobjective.Section4.2.2 ofChapter4providesaseparatealgorithmforthisproblem.Thisalgorithmsplitseach customer j 2 J + intotwoparts.Therstparthascustomerdemandsize j 1 2 [0 ;` j ] andaperunitprotgivenby j 1 = p j ` j + r j .Thesecondparthascustomerdemandsize j 2 2 [0 ;u j )]TJ/F21 11.9552 Tf 12.161 0 Td [(` j ]andaperunitprot j 2 = r j .Customers j 2 J )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(havecustomerdemand size j 2 [0 ;u j ]andaperunitprotgivenby j = p j u j + r j .InChapter4weshowedthat thelinearversionofFCKPcanbereformulatedasaCKPwithcustomers j 2 J )]TJ/F22 7.9701 Tf -1.107 -7.294 Td [(q and j 1 and j 2 in J + q .AsestablishedinSection7.4,thedenitionof~ r q q 2Q inLemma6 containsasolutiontotheCKP.Therefore,byreplacing J q withaset ^ J q thatincludes allcustomersinboth J )]TJ/F22 7.9701 Tf -1.106 -7.294 Td [(q and J + q withthespeciedprotandresourceconsumption parameters,Lemma6denes~ r q q 2Q withtheadditionalconsiderationofexible demand. OurapproacheswillagainbemotivatedbyKPEIRpresentedintheprevioussection. Hereagain,thefunction q q 2Q isanon-decreasingconcaveenvelopeencompassing theorigin,thefunction~ r q describedinLemma11andthepoint b; ~ r q u 0 q .Similartothe resultofLemma11, q w q q 2Q ,fortheexiblevariantoftheproblem,isdenedby Lemma7with ^ J q usedinlieuof J q andparametersforeachcustomer" j 2 ^ J q given by7{66{7{71.Therefore,thealgorithmpresentedintheprevioussectionstillapplies 160

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usingtheparametersgivenby7{66{7{71todene^ p and^ a fortheexpandedcustomer set. Interestingly,thereisarelationshipbetweenKPEIRandSKFPRwhichisgivenin thefollowingtheorem.Notethat ^ J q mayconsistofcustomersin J )]TJ/F22 7.9701 Tf -2.208 -7.294 Td [(q ,or J + q ,orboth. Thatis,uptotwocustomers"in ^ J q maybeassociatedwiththesametruecustomerin SKFPR.Therefore,let | k q bethecustomerinSKFPRtowhichanindividualcustomer" ^ | k q 2 ^ J q corresponds.Furthermore,let x t beavectoroftemporaryassignmentsusedto simplifytherepresentationofthefollowingtheorem.Usingthisnotation,westatethe relationshipbetweenoptimalsolutionstoKPEIRandSKFPR. Theorem11. GivenanoptimalsolutiontoKPEIR w KPEIR with ~ r q w q denedby Lemma11,anoptimalsolutiontoSKFPR x SKFPR v SKFPR s SKFPR isgivenbythe following. If w KPEIR q =0 s SKPR q =0 x SKPR j =0 j 2J q : v SKPR j =0 j 2J q : If 0
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v SKFPR j =0 j : | k q 6 = j 8 k q =0 ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1; q 2Q : Lastly,if ~ a ^ k w KPEIR q < ~ a ^ k +1 forsome ^ k q k q s SKFPR q =1 x t j k q =1 k q =0 ;:::; ^ k q x t j ^ k q +1 = w KPEIR q )]TJ/F15 11.9552 Tf 12.101 0 Td [(~ a ^ k q a j ^ k q +1 x t j k q =0 k q = ^ k q +2 ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 x t j =0 j 2f ^ J q = S q g x SKFPR j =max j k q : | k q = j n x t j k q o j 2J q v SKFPR j = x SKFPR j u j j 2fJ )]TJ/F22 7.9701 Tf -2.208 -7.892 Td [(q ~ S q g v SKFPR j = x t | k 0 q ` j + x t | k 00 q u j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` j k 0 q ;k 00 q : | k 00 q = | k 00 q = j and k 0 q
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Therefore,thefeasiblesolution w KPEIR toKPEIRequatestoafeasiblesolution toSKPR.Furthermore,afeasiblesolutiontoKPEIRcanbeconstructedfromafeasible solutiontoSKFPRbysettingtheexibilityvariable, w KPEIR q ,equaltothetotalcapacity consumedbycustomersoftype q intheoptimalsolution x SKFPR ;v SKFPR ;s SKFPR .As inTheorem9,bythemannerinwhichthefunction q q 2Q isdened,asolutionto KPEIR,withobjectivefunction z KPEIR ,canberepresentedasasolutiontoSKFPRwith anequivalentobjective z SKFPR andviceversa. FromtheequivalenceofKPEIRandSKFPRandLemma8,agreatdealof informationregardingthestructureofSKFPRcanbeobtained.Firstnotethata fractional s SKFPR q correspondstoasolutiontoKPEIRwith0
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Therefore,aswithSKPR,anoptimalsolutiontoSKFPRcaneasilybeconvertedtoa feasiblesolutiontoSKFP.Wewilldiscussspecicwaystoobtainafeasiblesolutionfrom theoptimalsolutiontoSKFPRinSection7.7.2.Furthermore,SKFPRcanbesolvedat eachnodeofacustomizedbranch-and-boundproceduretosolveSKFPtooptimality. 7.7Branch-and-PriceAlgorithmImplementation Ourimplementationandcomputationaltestingwillfocussolelyontheexiblevariant oftheproblem,FASR,discussinSection7.5.Theimplementationofourbranch-and-price algorithmispartiallymotivatedbythedemonstratedsuccessofthechoicesmadein Section4.3.However,asaresultoftheuniquecharacteristicsofMP-Fandthepricing problemPP i -F,someadditionalconsiderationsmustbegiventotheimplementationofthe branch-and-pricealgorithmfortheFASR. 7.7.1InitialFeasibleSolution ToensurethatafeasiblesolutionexiststoLPRMP-Fweproposeaslightlymodied two-phaseprocedurefromthatusedtosolveLPRSPinChapter4.Phase1ofour approachisusedtogenerateafeasiblesetofcolumnstoLPRMP-F.Tothisend,we includenonnegativeslackvariablesforeachcustomertypecapacityconstraint7{32 andassignmentconstraint7{33.OurPhase1objectiveisthentominimizethesumof theseslackvariables.TheresultingPhase1problemisthusgivenby minimize X j 2J % j + X q 2Q & q subjecttoRMP-F-Phase1 X i 2I X d 2D i q i x d i ;v d i d i )]TJ/F21 11.9552 Tf 11.955 0 Td [(& q g q q 2Q X i 2I X d 2D i x d ij d i + % j =1 j 2J X d 2D i d i =1 i 2I 164

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d i 0 d 2 D i ; i 2I X d 2D is d i 2f 0 ; 1 g s =1 ;:::;S i ; i 2I WesolvethelinearrelaxationofRMP-F-Phase1usingcolumngeneration.The pricingproblemissimilartoPP i -F,withaslightlymodiedobjective. maximize )]TJ/F26 11.9552 Tf 11.956 11.357 Td [(X j 2J )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( j x j + q j v j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i subjecttoPP i -F-Phase1 X q 2Q s q f iq + X j 2J v j b i v j 2 [ ` ij x j ;u ij x j ] j 2J s q x j j 2J q ; q 2Q x j 2f 0 ; 1 g j 2J s q 2f 0 ; 1 g q 2Q : ThisproblemisanSKFPwith p ij =0and r ij =0 i 2I ; j 2J .Therefore,itcanbe solvedasdiscussedinSection7.6.IftheoptimalvalueofLPRMP-F-Phase1equals0, anyoptimalsolutiontothisproblemisfeasibleforLPRMP-F;otherwise,theproblem instanceisinfeasible.Intheformercaseweusethisfeasiblesolutiontoinitializethe columngenerationprocedureforsolvingLPRMP-F. 7.7.2HeuristicsforPP i -F AsmentionedinSection7.6,theresultofLemma10canbeusedtodevelopvarious heuristicroundingstrategies.Inourimplementation,weconsidertwoalternative strategies. Heuristic1 .IftheoptimalsolutiontothelinearrelaxationofPP i -F,sayLPPP i -F, isindeedfractional,thenweknowthatthefractionalvariablesarelimitedtoasingle customertype.Aroundingproceduresimilartothatcommonlyusedforaknapsack 165

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problemisasfollows.Let q = f q 2Q : jF q j 1 g bethecustomertypeassociatedwith partiallyassignedcustomersinthesolutiontoLPPP i -F.Set x PP i -F-H1 j = x LPPP i -F j v PP i -F-H1 j = v LPPP i -F j j 2fJ = F q g ; x PP i -F-H1 j =0 v PP i -F-H1 j =0 j 2F q ; and s PP i -F-H1 q =max j 2J q x PP i -F-H1 j ; q 2Q : Intheaboveprocedure,allfractionalassignmentvariablesandthecorresponding demandfulllmentlevelsintheLP-relaxationofPP i -Faresettozero. Thenextheuristicusesthefractionalvariablestomakeassumptionsaboutwhich customertypesareincludedinthesolution.Then,asecondaryoptimizationproblem issolvedtodeterminethecorrespondingsetofcustomertoincludeandthesubsequent demandfulllmentlevels. Heuristic2 .Ratherthansimplyremovingallpartiallyassignedcustomers,we canalternativelyattempttofullyincludeapartialsetofthesefractionalcustomers.To accomplishthis,weusethesolutiontoLPPP i -Ftodeterminewhichcustomertypesare includedinourheuristicsolution.Thatis,set s PP i -F-H2 q = s LPPP i -F q q 2Q : Then ~ Q = q 2Q : s PP i -F-H2 q =1 isthesetofcustomertypesincludedintheheuristic solution.Set x PP i -F-H2 j =0 v PP i -F-H2 | =0 j 2J q ; q 2 n Q = ~ Q o : Usingthesetofcustomertypes, ~ Q ,wesolvethefollowingoptimizationproblem maximize X q 2 ~ Q X j 2J q p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x j + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j v j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i 166

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subjecttoSP-H2 X q 2 ~ Q X j 2J q v j b i )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X q 2 ~ Q f iq ` j x j v j u j x j j 2J q ; q 2 ~ Q x j 2 [0 ; 1] j 2J q ; q 2 ~ Q : SP-H2isaKPEI 0 -RstudiedinChapter4andcanbesolvedveryeciently. Moreover,weknowthereexistsanoptimalsolutiontoSP-H2forwhichthereisatmost onefractionallyassignedcustomer.GiventheoptimalsolutiontoSP-H2,theremaining heuristicsolutionisgivenasfollows.Ifoneexists,let | bethefractionalcustomerincluded intheoptimalsolution x SP-H2 ;v SP-H2 .Set x PP i -F-H2 j = x SP-H2 j v PP i -F-H2 j = v SP-H2 j j 2 0 @ 8 < : [ q 2 ~ Q J q 9 = ; = | 1 A ; x PP i -F-H2 | =0 v PP i -F-H2 | =0 : Ourtestingindicatedthatourheuristicproceduresweremostsuccessfulwhilesolving LPRMP-Fintherootnode.Atsubsequentnodesofthetreewereliedpredominantly ontheexactbranch-and-boundalgorithmtosolvePP i -F.NotethatHeuristic2clearly requiresmorecomputationaleortthanHeuristic1.Therefore,ourimplementationonly utilizesHeuristic2attherootnode.Thatis,intherootnodewerstattempttoidentify anattractivecolumnusingHeuristic1.Ifweareunsuccessful,weconsiderthemore intensiveHeuristic2.However,innon-rootnodes,weonlythesolvethepricingproblems heuristicallyviatheroundingschemeofHeuristic1.Thisimplementationchoicewas showntomostconsistentlyproduceresultsintheleastamountoftime. 7.7.3SolvingLPRMP-F Similartoourdiscussionofthebranch-and-pricealgorithmpresentedinChapter4,at anynodeinourbranch-and-boundtreewemustsolvearelaxationofMP-F.Sinceour 167

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pricingproblemstilldecomposesbyfacility,thereareagain jIj potentialpricingproblems toconsider.OurruleforconsideringthevariouspricingproblemsistakenfromSection 4.3.2ofChapter4.Thatis,ateachiterationofourcolumngenerationprocedurewe solveallpricingproblemsheuristicallyaccordingtotherulesdescribedinSection7.7.2. AllattractivecolumnsareaddedtoLPRMP-Fandthecolumngenerationprocedure continues.Ifnocolumnisfoundtopriceoutviaourheuristics,weorderthepricing problemsinnon-increasingorderoftheobjectivevaluesdeterminedbytheheuristic. Pricingproblemsaresolvedviabranch-and-bounduntileitherasinglecolumnpricesout, oritisdeterminedthatnocolumnpricesout.Toreducetheeectofslowconvergenceas weapproachtheoptimalsolutiontoLPRMP-F,weterminateourcolumngeneration procedurewhenourcurrentLPRMP-Fsolutionvalueisprovablywithin10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 ofthe optimalsolutiontoLPRMP-F.Recallthattheupperboundusedtocalculatethisgap requiressolvingallpricingproblemsexactly.Therefore,weagainonlyupdatetheupper boundaftersolving jIjjJj pricingproblemseitherheuristicallyortooptimality. InChapter4,qualityfeasiblesolutionstotheSPwereobtainedbysolvingRSPas anintegerprogramusingcolumnsgeneratedinsolvingLPRSPattherootnode.We investigatedthisimplementationchoiceforMP-F.Ourtestingshowedthatthetimefor CPLEXtosolvethisMIPwasnotablylargerthaninthecaseofSP.Thisislikelydue tothealternativestructureofMP-FversusSP,specically,theadditionofconstraints 7{32.Moreover,wefoundthatbettersolutionscouldbeobtainedatlowlevelsofthe tree.Therefore,unlikeourimplementationforSP,wedonotsolveanMIPusingthe columnsfoundintherootnodeofthesearchtree. 7.7.4NodeandVariableSelection OurnodeselectionruleismotivatedbythesuccessoftheimplementationinChapter 4.Weinitiallysearchthetreeusingadepth-rstrule.OnceafeasiblesolutiontoMP-F isobtained,weexplorethetreeusingabest-boundrule.Thisnodeselectionpolicyisalso 168

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implementedinthebranch-and-boundprocedureusedtosolveourpricingproblemto optimality. Again,ratherthanbranchingonthe valuesinLPRMP-Fwebranchon x variables thathaveavalueof1inacolumnassociatedwithafractional .Thechoiceof x to branchonisbasedonthedegreeoffractionalityofeachvariableinthesolutionto LPRMP-F.Weexploredbothleastfractionali.e.,thatvariablewhichisclosestto0or 1,wheretiesarebrokenarbitrarilyandmostfractionali.e.,thatvariablewhichisclosest to0.5.Whilethedierenceinperformanceforthetwoapproacheswasveryslight,the mostfractionalruleisusedinthecomputationalresultsshowninSection7.8. 7.8Computationalstudy Inthissectionweprovideacomputationalstudyofourbranch-and-pricealgorithm. Sincethemotivationbehindthischapteristheexiblevariantoftheproblem,wefocuson themoregeneralFASR.Section7.8.1discussestheinstancegenerationschemechosenfor thisstudy.Then,inSections7.8.2and7.8.3wediscusstheperformanceofouralgorithm versusthecommercialsolverCPLEXonabroadrangeoftestinstances. 7.8.1ExperimentalDesign Inourcomputationaltests,ourmainsetofinstancesconsiders15and30facilities withthenumberofcustomersequalto jJj =2 jIj ,3 jIj ,and5 jIj .Foreachfacility/customer combinationwestudyinstanceswith jQj =3customertypes.Foreachcustomer,we generatetherandomvectorsofxedprotparameters P j andunitrevenues R j from uniformdistributionson[30 ; 50]and[2 ; 5],respectively.Furthermore,thecustomer requirements L j and D j aregeneratedfromuniformdistributionson[75 ; 125],and[15 ; 35], respectively.Here, L j isarandomvectorofcustomerlowerboundsand D j isarandom vectorcontainingvaluesrepresentativeofthedierencebetweenupperandlowerbounds ofacustomer.Wealsogeneratesharedcapacityconsumptionparameterssuchthat f iq = iq jJj {72 169

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whereisanon-negativeparameterthatmeasurestheabsolutemagnitudeofthe sharedresourceconsumptionand iq istherelativemagnitudeofthesharedresource consumptionoftype q forfacility i .Moreover, X q 2Q iq =1 : Fortheseexperiments,weset iq = jJ q j jJj .Ineachofourteststhefacilitycapacitiesare givenby b i = + i jJj i 2I : {73 wheretheparameter i 2 [0 ; 1]isameasureofthefractionofcustomertypesthatcanbe assignedtofacility i i 2I andagain = a E min i 2I A i 1 + L i 1 jIj : {74 Theparameter a measuresthecapacityavailableforvariableconsumption.Inthesetests weconsidertheexibilityleveldeterminedbysetting a =1 : 2.However,weconsider alternativevaluesof i .Lastly,ourcustomertypecapacityrestrictionsaregivenby g q = t E min i 2I A i 1 + L i 1 jJ q j q 2Q {75 where t determinestheexibilityavailableforcustomersoftype q withrespectto capacity g q .Inthesetestsweconsidertheexibilityleveldeterminedbysetting t =1 : 2, aswell. Inourexperimentation,wesoughttocomparetheeectivenessofourbranch-and-price approachagainstthecommercialsolverCPLEX.Eachofourinstanceswasrununtileither asolutionvaluewithin.1%oftheoptimalsolutionwasobtainedoratimelimitofone hourwasreached.Ourtablespresentresultsfor10randomlygeneratedinstancesforeach combinationofparametersettings.Specically,eachtablereports ithenumberofcolumnsgeneratedintherootnodeofthebranch-and-pricealgorithm; 170

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iithetotalnumberofcolumnsgeneratedthroughouttheentirebranch-and-price algorithm; iiithenumberofnodesconsideredinthebranch-and-pricetree; ivtheamountoftimerequiredtosolvetherelaxedmasterproblemattherootnode; vthetotaltimerequiredbythebranch-and-pricealgorithm; vithetotaltimerequiredbytheCPLEX,withthefollowingadditionalinformation whereappropriate: {thesuperscriptindicatestherelativesolutionerrorcalculatedbyusingthe solver'sbestlowerandupperbound, z UB S z LB S ;i.e. error= z UB S )]TJ/F21 11.9552 Tf 11.955 0 Td [(z LB S z UB S 100 : AllexperimentswereperformedonaaPCwitha3.40GHzPentiumIVprocessor and2GBofRAM.Themixed-integerprogrammingproblemsaswellastherelaxed masterproblemsweresolvedusingCPLEX11.2.InSection7.8.2wediscussabasesetof resultsforourbranch-and-pricealgorithm.Then,inSection7.8.3wediscusshowthese resultschangewithdierentinstancegenerationparameters. 7.8.2BaseResults Ourmainsetofresultsconsiderstwosetsofinstances:iwith15facilitiesand numberofcustomersequalto30,45and75andiiwith30facilitiesand60,90,and 150customers.Forbothsetsofinstances,thecustomersareseparatedintothreeequal sizesetsi.e.types.Thismainsetofinstancesgeneratessharedresourceconsumption variablesi.e. f iq i 2I ; q 2Q withmagnitudeparameterequal5.Thefacility capacitiesaregeneratedwith i = : 5.Thatis,wecanexpect50%ofthecustomertypes tobeabletobeassignedtoaparticularfacility.Lastly,aspreviouslystated,theexibility allowancesaresetatthemoderatelevels a =1 : 2and t =1 : 2. Tables7-1and7-2showthatthebranch-and-pricealgorithmsolvesthe15facility instanceswith30and60customersinlesstimethanCPLEX,onaverage.Eachofthe optimalsolutionsforthe15facility/30customerinstancesinTable7-1isobtainedin 171

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lessthan100nodesandthetimetosolvetherootrelaxationislessthan10seconds. TheaveragetimeforCPLEXtosolvetheseinstancesismorethan6timesthatof thebranch-and-pricealgorithm.Interestingly,itisclearthatalargeportionofthe columnsconsideredineachoftheinstancesweregeneratedintherootnodeofthe branch-and-pricealgorithm.Eachofthe15facility/45customerinstancesinTable7-2 wasalsosolvedinlesstimewiththebranch-and-pricealgorithm.However,thenumber ofnodesconsideredandcolumnsgeneratedissubstantiallyhigherthanthatseenin30 customerinstancesinTable7-1.Forthe45customerinstances,Table7-2showsthat morethan1000nodeswereoftenrequiredtoobtainanoptimalsolution.Furthermore, afarhigherpercentageofcolumnswasgeneratedoutsideoftherootnode.Whilethe computationalrequirementsforthebranch-and-pricealgorithmwerenotablyincreasedin Table7-2,itshouldbenotedthatCPLEXfailedtosolve4ofthe10instanceswithinthe allottedhour.Eachofthese4instanceswassolvedtooptimalityviabranch-and-pricein lessthan25minutes.Unfortunately,neitherCPLEXorourbranch-and-pricealgorithm wasabletosolveinstanceswith15facilitiesand75customerswiththeparameters speciedatthebeginningofthissectionwithinthe1hourtimelimit. Tables7-3and7-4considerprobleminstanceswithalargernumberoffacilitiesand customers.Specically,30facilityinstanceswith60and90customersareconsidered. Thesamesetofparametersisusedtogeneratethedemandrequirementsandcapacity limitations.Theperformanceofbranch-and-priceoverCPLEXisevenmoreclearly denedintheseresults.With60customers,Table7-3showsthatbranch-and-pricetakes anaverageoflessthan30secondstosolvetheseinstances,whileCPLEXdoesnotsolve anyoftheinstancestothespeciedtolerancelimitsintheallottedtime.Interestingly, eventhoughtheinstancesaremuchlarger,thetimetosolvetherootnodeproblemisstill lessthan10seconds.Similartowhatwasseenwiththe15facilityinstances,Table7-4 showsagainthatthenumberofcolumnsgeneratedgrowssignicantlyasthecustomers perfacilityisincreasedto3.However,thebranch-and-pricealgorithmisstillabletosolve 172

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all10instancesinanaverageoflessthan11minutes,whileagainCPLEXfailstosolve anyoftheseinstanceswithin1hour. Neitherthebranch-and-pricealgorithm,norCPLEXisabletosolveinstanceswith30 facilities/150customerswithinanhour.Therefore,asistypicalwithbranch-and-price approaches,ourmainresultsinTables7-1{7-4suggestthatouralgorithmismost successfulwithacustomer-to-faciltyratiolessthanorequalto3.Interestingly,while GAPFDandCFLFDwithlinearrevenuefunctionscouldbesolvedecientlybyCPLEX forinstanceswithcustomer-to-faciltyratiosof5ormore,thesameisclearlynottrue forFASR.Thisreinforcesthedicultyofthisclassofproblems.Ofcourse,anumberof dierenttypesofinstancescanbeconsideredusingthedatamodelproposedinSection 7.8.Inthefollowingsection,weprovideafewinsightsintohowinstanceswithdierent characteristicsmaybeeasierormorediculttosolvethanthoseconsideredinthis section. 7.8.3ExtendedResults Inthissection,weconsidervariousalternativestotheparametersusedtogenerate instancesinourmainsetofresults.Whileitisimpractical,ofcourse,toconsiderall variations,thissectionstrivestoprovidesomeinsightintohowalternativeinstancesmay impactthecomputationalrequirementsofbranch-and-priceversusCPLEX.First,recall thatthemagnitudeofthesharedresourceconsumptioni.e.waschosentobe5in ourmainresults.Inourcomputationalstudy,wealsoconsideredmagnitudesof1and 25aswell,withallremainingparametersthesame.With=25,Table7-5suggests theproblemsbecomedramaticallyeasierforbothbranch-and-priceandCPLEX.Thisis perhapsduetothefactthatthesharedresourceconsumptionbecomesthedominating componentoftheproblem,withtheexibledemandrequiredbyindividualcustomers requiringlessconsideration.Morelikely,however,isthatgiventhecombinationof extremelylargemagnitudeswithafairlylargevalue =0 : 5,thefacilitycapacityavailable isabundant.AswesawinChapter5,problemswithsimilarstructuretoFASRaremore 173

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easilysolvedbyCPLEXwithexcesscapacity.Interestingly,whenwastakentobe1,the problemsbecamemucheasierforCPLEXthanbranch-and-price.Ofcourse,with=1, thesharedconsumptioncomponentoftheproblemisdrasticallyminimized.Therefore,the problemresemblesaGAPFDwithadditionalsideconstraints.FromChapter4weknow thatCPLEXoutperformsbranch-and-priceforthisclassofproblemsifrevenuefunctions aretakentobelinear. Anotherproblemcomponentthatcanbechangedisthevalues i i 2I ,which canbeinterpretedastheanticipatedfractionofcustomertypesthatmaybeassigned toanyindividualfacility.Inourmainresultswesetthevaluesof i i 2I at0.5. Wealternativelyconsideredtheimpactofchangingthesevaluesto0.2and0.9.Tables 7-6and7-7provideresultsforinstanceswithmuchtighterfacilitycapacitiesresulting from i =0 : 2 i 2I .WhencomparedwithresultsofTables7-1and7-2,the branch-and-pricealgorithmsolvesthesemorecapacity-restrictedinstancesinlesstime, onaverage.However,thetimerequiredbyCPLEXincreasesnotably.Whiletheaverage timeforbranch-and-pricetosolvethe15facility/45customerinstancesdecreasesbymore than4minutes,CPLEXsolvednoneoftheseinstanceswithinonehour.Theimproved performanceofthebranch-and-pricealgorithmwhen i =0 : 2 i 2I isexpectedsincethe numberoffeasiblecolumnsinMP-Fisdecreased.Alternatively,ifweincrease i i 2I to0.9,CPLEXconsistentlyoutperformsourbranch-and-pricealgorithm.Table7-8shows thatCPLEXrequiresanaverageofonly3secondstosolvethese`looser'instances,while branch-and-pricetakesmorethan2minutes. Thelastproblemcomponentthatweconsideristhecustomertypecapacities, g q q 2Q .Inourbaseresults,thesecapacitiesweregeneratedwithparameter t =1 : 2. InTables7-9{7-12weshowtheresultsofmodifyingthisparameter.First,Tables7-9 and7-10provideresultsforinstanceswithcustomertypecapacitiesgeneratedwith t =1 : 1.Bothbranch-and-priceandCPLEXsolvetheseinstancesinlesstimethan thatrequiredforinstancesgeneratedwith t =1 : 2.However,branch-and-pricestill 174

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solvesthe15facility/30customerinstancesshowninTable7-9morethan5timesfaster thanCPLEX.Thisperformancedierenceisevengreaterforthe15facility/45customer instancesinTable7-9.Fortheseinstances,branch-and-pricerequireslessthanaminute, onaverage,whileCPLEXrequiresmorethan13minutes.Interestingly,ifthecustomer typecapacitiesareunbounded,asisthecasefortheinstancesinTables7-11and7-12,the branch-and-pricealgorithmagainrequireslesstimethanthatneededinthebaseresults, whilethetimerequiredbyCPLEXincreases.Infact,CPLEXfailstosolve7outof10of the45customerinstanceswithinanhour.However,branch-and-priceisabletosolveall 10ofthe45customerinstancesandmorethanhalfofthe75customerinstanceswithin theallottedtime.Itshouldbenotedthattheperformanceofbranch-and-priceinTables 7-11and7-12islikelyrelatedtothechangeinthestructureofMP-Fwhencustomer typecapacitiesareunbounded.Inthiscase,constraints7{32areeectivelyomitted. Importantly,thisdrasticallyreducesthenumberofcolumnsrequiredinMP-F.Without thecomplicatingcustomertypecapacityconstraints,atmostonecolumnneedstobe consideredforagivensubsetofcustomerassignments.Thatis,intheabsenceof7{32,the optimalcustomerdemandfulllmentlevelsi.e. v ij i 2I ; j 2J foraxedsubsetof assignmentscaneasilybedetermined. 7.9ConclusionsandFutureResearch Inthischapterweconsideredaclassofassignmentproblemsthatseparatecustomers intodisjointsets.Customersofthesametypei.e.belongingtothesamesetareassumed tosharecommonproductionrequirements.Theproposedmodelconsiderednon-linear resourceconsumptionattributesamongcustomersofthesametype.Inaddition,further capacityrestrictionslimitedtheresourceconsumptionofallcustomersassociatedwitha particulartype,independentofwhatfacilityisusedtosatisfythedemand.Weproposed anexactbranch-and-pricealgorithmtosolvetheresultingFASRproblembasedona reformulationofourmodelasaset-partitioningrepresentationwithsideconstraints. Thisreformulationrequiresuniquecolumnrepresentationstoaccuratelymodelthe 175

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exibledemandcomponentoftheproblem.Tosolvetheresultingpricingproblem, westudiedaclassofknapsackproblemswithanimportantrelationshiptotheclassof knapsackproblemsstudiedinChapter4.Ourcomputationalstudysuggeststhatthe branch-and-priceapproachproposedinthisworkperformswellincomparisontoCPLEX onalargeassortmentofprobleminstances. Infutureresearch,itmaybeadvantageoustoconsideraslightlyalteredreformulation ofFASR.Specically,inthischapterthedemandfulllmentlevelsassociatedwitha columnweredeterminedbyanextremepointsolutiontothefollowingoptimization problem maximize X j 2J r ij v ij + C d i subjecttoSPv i x d i X j 2J v ij b d i ` ij x d ij v ij u ij x d ij j 2J foragivensubsetofassignments.Recallthat C d i = P j 2J p ij x d ij and b d i = b i )]TJ/F26 11.9552 Tf -405.99 -14.941 Td [(P q 2Q f q max j 2J q x d ij .Theorem10establishedthatusingthiscolumnrepresentationin conjunctionwiththeappropriateconstraintsanddecisionvariablesyieldedanequivalent representationofFASR.Alteringthecolumnsofthemasterproblemsothatdemand fulllmentlevelsweredeterminedby maximize X j 2J r ij v ij + C d i subjecttoSPv i x d i X j 2J v ij b d i ` ij x d ij v ij u ij x d ij j 2J 176

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X j 2J q v ij g q q 2Q {76 with C d i and b d i denedabove,wouldyieldanalternativepricingproblemwithan additionalsetconstraints.Theadditionalconstraints7{76wouldpotentiallyeliminate columnsfromMP-Fthatwouldbeallowedintheprocedureproposedinthischapter.An importantquestionwouldbewhetherconsideringthismorehighlyconstrainedpricing problemwouldyieldatighterlinearrelaxationofMP-F.Ifso,canthepricingproblem besolvedaseciently?Acomputationalstudymayrevealinterestingtradeosbetween implementingtheapproachprovidedinthischapterversusanexactapproachthatutilizes thisslightlymodiedcolumnrepresentation. Figure7-1.Illustrationof r q and q 177

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Table7-1.FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1485611212.28.73.7 242042014.84.91.7 3458536156.912.3513.5 4374574295.147.1111.1 5460735893.222.6226.6 649851532.23.114.0 7395578373.013.345.3 841947178.411.540.3 9542694393.316.314.7 1041746091.93.417.5 Avg446.8559.4254.114.399.3 Table7-2.FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 179722336417.8394.52059.0 2106812051312.225.6133.2 3820272710054.8785.53600.0 : 17% 4891263710437.0791.83600.0 : 30% 511781747938.4107.8535.5 677115732415.2126.02038.0 79261441517.359.6582.6 8108218231157.9120.61480.8 9967286614836.71329.73600.0 : 20% 1086316101739.7141.43600.0 : 20% Avg936.31986.2485.87.7388.22124.2 : 09% 178

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Table7-3.FASR:30facilities,60customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 187913221476.074.63600.0 : 18% 28121101975.744.93600.0 : 12% 3817916254.313.33600.0 : 17% 4881109313110.160.33600.0 : 43% 58331022395.518.33600.0 : 20% 685190998.816.43600.0 : 32% 783388997.311.03600.0 : 26% 8794881174.311.13600.0 : 17% 9798849116.012.33600.0 : 21% 10895957115.914.23600.0 : 32% Avg839.3993.949.66.427.73600.0 : 24% Table7-4.FASR:30facilities,90customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 12939352010123.1305.03600.0 : 44% 2280830993323.2111.23600.0 : 46% 3261427991118.453.23600.0 : 17% 4247626811722.470.63600.0 : 25% 52682366020324.9557.93600.0 : 35% 6290531001721.770.03600.0 : 27% 7333736965128.8152.03600.0 : 21% 82602436347323.71215.03600.0 : 39% 9208631939716.2229.43600.0 : 30% 1031775061108319.33600.03600.0 : 41% Avg2762.63517.2208.622.2636.63600.0 : 32% 179

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Table7-5.FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=25 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 155758192.74.031.9 264868293.14.861.0 356956912.32.423.8 4579893672.017.4266.0 5604874253.515.291.1 6621853592.312.7181.3 768168112.42.415.2 863966032.53.115.2 959499151.2.621.0301.8 10603715111.74.570.1 Avg609.5749.923.62.58.7105.7 Table7-6.FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.2 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 131731716.86.887.2 230935199.114.888.2 32863542111.224.0645.2 431031018.88.835.9 5291291110.510.589.5 631935779.816.1739.2 7322339913.517.9404.2 8303303111.911.993.3 9308308111.711.7161.6 10306306110.710.768.4 Avg307.1323.65.210.413.3241.4 180

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Table7-7.FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.2 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1594656330.148.13600.0 % 262496410526.9138.63600.0 % 36056811548.373.73600.0 % 45867232374.7215.23600.0 % 56568723333.9119.13600.0 % 65866922531.080.43600.0 % 7616616141.241.33600.0 % 8592653952.388.73600.0 % 965383337212.6292.23600.0 % 105958604936.0148.33600.0 % Avg610.77553058.7124.53600.0 % Table7-8.FASR:15facilities,75customers,3customertypes, a = t =1 : 2, i =.9 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1370643545525.2201.814.2 2360838921320.573.51.3 3221925271316.351.51.4 4341440785522.5171.01.4 5327639124323.1146.71.9 6339842693725.3169.23.6 7390044452725.0150.72.3 8344740574321.9154.21.4 92325328355.14.5108.71.7 10246630782916.975.31.0 Avg3175.93789.53721.1130.33.0 181

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Table7-9.FASR:15facilities,30customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1350448172.48.26.6 240640611.31.32.3 3440504133.710.9227.0 440342934.15.41.3 546348233.54.627.1 639639613.13.18.9 733836331.73.034.9 841743754.86.15.6 9351582109.2.119.636.1 1042242212.12.111.7 Avg398.6446.915.62.96.536.1 Table7-10.FASR:15facilities,45customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 179484677.511.2231.2 2737888196.317.6117.4 3105214695910.666.8618.0 49311399539.752.01283.9 51144175613111.4124.6846.4 610211356315.030.024.9 79801094176.520.2930.4 88091257778.175.43600.0 : 18% 989114891279.077.9275.8 107721409794.258.7274.2 Avg913.11296.3607.853.4820.2 : 018% 182

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Table7-11.FASR:15facilities,30customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1356356110.910.9618.3 236441657.118.623.6 338738712.92.964.7 435036954.38.3719.7 5376428215.213.3430.3 635136634.110.3206.0 738238218.38.467.1 835237495.29.5750.6 9353431313.616.9906.4 10379415112.17.11011.7 Avg365392.48.85.410.6479.8 Table7-12.FASR:15facilities,45customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 15967244541.2116.03600.0 : 5% 2707741533.444.71278.7 36548142742.8111.73600.0 : 44% 46558273137.987.23600.0 : 85% 56597883348.293.83600.0 : 4% 6572617933.645.83600.0 : 0% 77769112532.769.63473.6 8723723126.626.72461.4 96257983158.8118.23600.0 : 4% 106247683734.583.73600.0 : 79% Avg659.1771.124.439.079.83241.4 : 74% 183

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CHAPTER8 CONCLUSION Inthisdissertationweexplorednumerousvariantsofresourceconstrainedassignment problemsthataccountforreal-worldoperationsdecisions.Ineachoftheproblems considered,theoptimizationmodelsseektoexploitlesserstudiedrelationshipsbetween customersandthemanufacturertoincreasearm'sprot.InChapter3,weintroduced ageneralizationofthecapacitatedfacilitylocationwithsingle-sourcingconstraints.A notablefeatureofthismodelwastheallowanceofexiblecustomerdemand,whichhas receivedlittleattentionintheliterature.Weconsideredvariantsoftheproblemwith andwithouttherequirementthatresourcesmustbeprocuredataxedcosttothe decisionmaker.InChapter4,weprovidedanexactbranch-and-pricealgorithmbasedon areformulationofthemodelthatsolvedbothproblems.Ourapproachrequiredthestudy ofaninterestingclassofknapsackproblemswithexibledemand.Weshowedimportant structuralresultsofarelaxationofthisclassofknapsackproblemsthatledtoanecient solutionapproachforourpricingproblemwithgeneralizedrevenuefunctions.Weoered evenmoreecientalgorithmsforsolvinginstanceswithspeciallystructuredrevenue functionsthatcorrespondtocommonpricingstructures.Thecomputationalstudyofour branch-and-pricealgorithmdemonstratedthevalueofourapproach.Adetaileddiscussion oftheimplementationchoicesthatresultedinreducedsolutiontimeswasprovided. Elementsofthisdiscussionwererelevanttoallxed-chargeproblemssolvedusingcolumn generation. InChapters5and6,wedevelopedheuristicstosolvelarge-scaleinstancesofthe problemvariantswithCFLFDandwithoutGAPFDresourceprocurementdecisions. InChapter5,weproposedaclassofgreedyheuristicsforGAPFDwithlinearrevenue functionsthatwasmotivatedbypropertiesofanoptimalsolutiontothelinearrelaxation ofourmodel.Wepresentedanovelperturbationschemethatguaranteedourclassof heuristicswasasymptoticallyoptimalityunderaverygeneralstochasticmodel.Our 184

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computationalstudydemonstratedthatourheuristicperformedparticularlywellfor instanceswithalargeratioofcustomers-to-facilities.Applyingtheconceptofinstance perturbationindevelopingheuristicsforothercapacitatedassignmentproblemswould beinteresting.Comparingthesuccessofsuchheuristicsonlarge-scaleproblemsversus establishedprocedureswouldbeespeciallyimportantforvalidatingthecontributionof perturbationincaseswhereityieldsstrongperformanceguarantees. InChapter6,wedevelopedalarge-scalesearchheuristicforCFLFDwithlinear revenuefunctions.Ourapproachutilizedthehigh-qualityecientheuristicproposed forGAPFDwithinafacilityneighborhoodsearchtoaddressthecombinedassignment andxed-chargestructureofourunderlyingoptimizationproblem.Wealsoconsidered theadvantagesofdevelopingahybridapproachthatutilizedaso-calledverylarge-scale neighborhoodsearchVLSNmethod.Ourcomputationalresultsindicatedthatour heuristicframeworkwasaneectiveapproachforsolvingCFLFD.Itwouldbeinteresting toapplythisheuristicframeworktootherxed-chargeassignmentproblems.Inaddition, sincetheheuristiccallsforassignmentstobemadeusingasecondaryprocedureinthis case,theGAPFDheuristicitwouldbeinterestingtoconsideradditionalheuristicsand exactapproachesinthisphase. Lastly,Chapter7introducedanadditionalclassofassignmentproblemswith non-linearcapacityconsumptionamongcustomersandcapacityconstraintsthat spannedallavailableresources.Thismodelwasapplicabletoproductionscenarios thatconsiderproductswithsimilarproductionrequirements.Theadditionalcapacity constraintsaccountedforreal-worldlimitationsonhazardemissions,logisticsresources orwarehousespace.Thebranch-and-pricealgorithmdevelopedforthisclassofproblems requiredaninterestingreformulationofourproblemthatincludedcolumnswithaunique representationwhencomparedtothosetypicallyseeninassignmentproblems.The subproblemtobesolvedresultedinastudyofanotherclassofknapsackproblemswith animportantrelationshiptotheknapsackproblemsstudiedinthecaseofCFLFDand 185

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GAPFD.Acomputationalstudydemonstratedtheadvantagesofourapproachovera well-knowncommercialsolver.GeneralizingFASRtoallowforasharedprotassociated witheachcustomertypeisanaturalextension.Furthermore,whilethebranch-and-price approachwasshowntobesuccessfuloninstanceswithlinearrevenuefunctions,theability tosolveinstanceswithgeneralrevenuefunctionsmayresultinagreaterimpact.For thatreason,theclassofknapsackproblemsdenotedbySKFPshouldbestudiedwith additionalnon-linearconsiderations.OfparticularinterestwouldbehowMP-Fmightbe modiedtomaintainitsequivalencewithanon-linearrepresentationofFASR. Lastly,whilethenotionofcustomersetsi.e.typeswasstudiedindepthinthis dissertation,anotherinterestinggeneralizationofCFLFDwouldbetogroupfacilitiesinto disjointsets.Eachofthesesetsmaybelimitedbyitsowncapacityrestriction,inaddition totheindividualcapacityofthefacilitiesbelongingtothesets.Alongwithprocuringeach facility,aprocurementdecisionassociatedwiththesetmustbeconsideredaswell.The additionalconsiderationoffacilitysetsisapplicabletomanyproductionandpersonnel planningscenarios.Forexample,assumethatfacilitiesrepresentindividualmachinesat variousmanufacturinglocations.Eachmachinehasamaximumamountoftimethatit canberuneachday.However,theproductsproducedbythemachinesateachlocation mustbestoredatanon-sitewarehousebeforebeingtransportedtothecentralized distributor.Thespaceavailableineachwarehouseislimitedsuchthatifeachmachineis runforitsmaximumtime,theamountofproductsproducedwillexceedthespaceneed tostorethem.Therefore,anadditionalconstraintonthetotalnumberofhourseachset ofmachinesisproducingisnecessary.Thisverygeneralscenarioisapplicableacrossa varietyofindustries.Therefore,thispotentialclassofproblemsisrichinapplications. Interestingly,itispossiblethattheneighborhoodsearchheuristicusedtosolveCFLFD maybeextendedtothismulti-levelxed-chargeproblem. 186

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APPENDIXA GAPFDASYMPTOTICPROPERTY Lemma5 TheoptimalvaluesofLP 0 andLP 0 jJj arecloseinthesensethat, withprobabilityone, lim jJj!1 1 jJj Z LP 0 jJj jJj =lim jJj!1 1 jJj Z LP 0 jJj : Proof. ThenormalizedoptimalvalueofLP 0 jJj canbeexpressedas 1 jJj Z LP jJj jJj =min 0 jJj ; jJj where jJj ; jJj = 1 jJj X j 2J max i 2I f i;j + X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [( jJj = jJj ;0 )]TJ/F26 11.9552 Tf 11.955 20.443 Td [( X i 2I i jJj : A{1 Wewillshowthatwemayrestrictourselvestovectors inacompactset.First,notethat min 0 jJj ; jJj ; R L + D + P: Furthermore,forany wehave jJj ; = 1 jJj X j 2J max i 2I )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i + u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij + p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [( X i 2I i 1 jJj X j 2J max i 2I p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij + X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [( X i 2I i R L + P + X i 2I i i )]TJ/F15 11.9552 Tf 18.7 8.087 Td [(1 jJj X j 2J min i 2I i a ij + ` ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( X i 2I i R L + P +min 0 0 ; 0> e =1 X i 2I 0 i i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X j 2J min i 2I 0 i a ij + ` ij X i 2I i )]TJ/F21 11.9552 Tf 11.956 0 Td [( X i 2I i R L + P + )]TJ/F21 11.9552 Tf 11.955 0 Td [( X i 2I i 187

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withprobabilityoneas jJj!1 ,byRomeijnandPiersma[78,Theorem3.1].Sincethe value Z LP jJj jJj isnonnegative,thefunction jJj ; thusattainsitsminimumonthe compactset= 0: P i 2I i )]TJ/F26 11.9552 Tf 7.314 9.684 Td [( where)-300(= )]TJ/F15 11.9552 Tf 8.032 -6.662 Td [( R L + D + P )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L )]TJ/F21 11.9552 Tf 11.955 0 Td [(P = )]TJ/F21 11.9552 Tf 12.06 0 Td [( withprobabilityoneas jJj!1 Nownotethat jJj ; jJj jJj ; .Bytheconvexityof jJj ;0in and equationA{1itthenfollowsthatthefunction jJj ; jJj alsoattainsitsminimumon withprobabilityoneas jJj!1 .Thismeansthat jJj ; jJj jJj ;0 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 11.9552 Tf 7.314 0 Td [( jJj withprobabilityoneas jJj!1 sothat 1 jJj Z LP jJj min 0 jJj ; jJj min 0 jJj ;0 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 11.9552 Tf 7.314 0 Td [( jJj = 1 jJj Z LP jJj )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 11.9552 Tf 7.314 0 Td [( jJj withprobabilityoneas jJj!1 : Thedesiredresultnowfollowsbyusing5{20. 188

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APPENDIXB CFLFDPRICINGPROBLEMPROPERTY Theorem1 Theoptimizationproblems RKP 0 and KPEI 0 -R areequivalentwhen therevenuefunction r j arelinearforall j 2 ~ J Proof. Wecanrewritetheintervalconstraints4{21asfollows: x j 2 w j u 0 j ; w j ` 0 j j 2J : 4{21 0 Clearly,wewouldliketochoosethevalueof x j assmallaspossiblei.e., x j = w j =u 0 ij if p j 0andaslargeaspossiblei.e., x j =min f 1 ;w j =` 0 j g if p j > 0.Therefore,ifwedene J )]TJ/F15 11.9552 Tf 10.731 -4.339 Td [(= f j 2J : p j 0 g and J + = f j 2J : p j > 0 g ,wecanformulatetheLP-relaxationof theKPEIas maximize X j 2 J )]TJ/F26 11.9552 Tf 8.247 29.243 Td [( r j + p j u 0 j w j + X j 2 J + min r j w j + p j ; r j + p j ` 0 j w j subjecttoKPEI 0 -R X j 2J w j b w j 2 [0 ;u 0 j ] j 2J : Therevenuefunctionsofitems j 2 J + arenolongerlinear,butinsteadaconcavefunction ofthejobsize w j .Eachjob j 2 J + canbesplitintotwoparts.Therstparthasjob size w j 1 2 [0 ;` 0 ij ]andarevenuefunctiongivenby r j + p 0 j ` 0 j w j 1 .Thesecondparthasjob size w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 12.511 0 Td [(` 0 j ]andarevenuefunctiongivenby r j w j 2 .Therefore,KPEI 0 -Rcan alternativelybewrittenas maximize X j 2 J )]TJ/F26 11.9552 Tf 8.247 29.244 Td [( r j + p j u 0 j w j + X j 2 J + r j + p j ` 0 j w j 1 + X j 2 J + r j w j 2 subjecttoKPEI 0 -R 0 X j 2 J )]TJ/F21 11.9552 Tf 8.246 12.387 Td [(w j + X j 2 J + w j 1 + X j 2 J + w j 2 b i 189

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w j 2 [0 ;u 0 j ] j 2 J )]TJ/F21 11.9552 Tf -157.55 -31.833 Td [(w j 1 2 [0 ;` 0 j ] j 2 J + w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ] j 2 J + : Notethat r ij + p j u 0 j = r j u 0 j + p j u 0 j = j u 0 j u 0 j B{1 and r j + p j ` 0 j = r j ` 0 j + p j ` 0 j = ij ` 0 ij ` 0 ij = ij B{2 and r j = p j + r j u 0 j )]TJ/F15 11.9552 Tf 12.942 0 Td [( p j )]TJ/F21 11.9552 Tf 11.956 0 Td [(r j ` 0 j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j = j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [( j ` 0 j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j = j : B{3 Therefore,theoptimizationproblemKPEI 0 -R 0 ispreciselyRKP 0 presentedinSection 4.2.2,whichyieldsthedesiredresult. 190

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BIOGRAPHICALSKETCH ChaseRainwaterwasborninMountainHome,AR.HegraduatedfromMountain HomeHighSchoolin1999.HeattendedtheUniversityofArkansas,whereheearned aBachelorofScienceinIndustrialEngineeringinMay2004.Afterhisundergraduate studies,hemarriedCandaceZieleniuk-Rainwaterandenrolledinthegraduateprogramat theUniversityofFlorida.HebeganhisdoctoralstudiesinAugust2005intheIndustrial andSystemsEngineeringDepartment.HeearnedhisDoctorofPhilosophyinindustrial andsystemsengineeringinAugust2009.Followinggraduation,hewilljointhefacultyof theDepartmentofIndustrialEngineeringattheUniversityofArkansas. 198