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Approaches to Nonlinear and Infinite-Dimensional Network Design Problems in Supply Chain Optimization

Permanent Link: http://ufdc.ufl.edu/UFE0022311/00001

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Title: Approaches to Nonlinear and Infinite-Dimensional Network Design Problems in Supply Chain Optimization
Physical Description: 1 online resource (250 p.)
Language: english
Creator: Sharkey, Thomas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: inventory, location, networks, nonlinear, optimization, selection
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: Network design and network flow problems constitute an extremely important class of optimization problems in supply chain management. The design of the infrastructure of a company, the location of the facilities or warehouses of a company, and the management of the inventory of a facility are all examples of network design and network flow problems in supply chain management. Due to the complexity of the real world, many of these network design and network flow problems make assumptions on the problem in order for the resulting optimization model to be effectively studied. However, these assumptions limit the applicability of the model due to the assumptions placed on the real world problem. In this dissertation, we study several new classes of network design problems that more explicitly capture real world supply chain optimization problems. These classes of problems often lead to nonlinear or infinite-dimensional extensions of classic optimization problems. However, unlike most nonlinear or infinite-dimensional optimization problems, the supply chain foundations of these problems allow for the development of effective approximate, heuristic, and exact methods to solve them. In this dissertation, we examine three new classes of network design and network flow problems that have previously received little attention in the literature. The first problem that we examine is concerned with assigning a set of customers to a set of facilities where the facilities are then responsible for producing the amount of demand assigned to the facility. This leads to a new class of nonlinear generalized assignment problems. In many practical situations, the resulting nonlinear functions in these problems are ill-structured (e.g., non-differentiable or discontinuous) so that standard optimization techniques are no longer applicable. We overcome this challenge through novel applications of the theory of linear programming. The second problem that we consider is a problem where the customers have dynamic demand over a finite horizon. We must assign each customer to an open facility and manage production and inventory levels at each open facility to meet the demand of the customers assigned to the facility. We develop approximation algorithms and results for this class of problems as well as setting the theoretical foundation for an algorithm to solve these problems to optimality. The third problem that we consider is a class of minimum-cost flow problems in infinite networks. This class of problems encompasses many supply chain planning problems where a sequence of decisions needs to be made over an infinite horizon. We are able to overcome the typical mathematical difficulties associated with linear programming in infinite-dimensional spaces and extend the well-known network simplex method to solve this class of problems in infinite networks. We do so in a nonstandard but intuitively appealing way that, to the greatest extent possible, employs concepts from finite-dimensional linear programming.
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 Thomas Sharkey.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Romeijn, Hilbrand E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022311:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022311/00001

Material Information

Title: Approaches to Nonlinear and Infinite-Dimensional Network Design Problems in Supply Chain Optimization
Physical Description: 1 online resource (250 p.)
Language: english
Creator: Sharkey, Thomas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: inventory, location, networks, nonlinear, optimization, selection
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: Network design and network flow problems constitute an extremely important class of optimization problems in supply chain management. The design of the infrastructure of a company, the location of the facilities or warehouses of a company, and the management of the inventory of a facility are all examples of network design and network flow problems in supply chain management. Due to the complexity of the real world, many of these network design and network flow problems make assumptions on the problem in order for the resulting optimization model to be effectively studied. However, these assumptions limit the applicability of the model due to the assumptions placed on the real world problem. In this dissertation, we study several new classes of network design problems that more explicitly capture real world supply chain optimization problems. These classes of problems often lead to nonlinear or infinite-dimensional extensions of classic optimization problems. However, unlike most nonlinear or infinite-dimensional optimization problems, the supply chain foundations of these problems allow for the development of effective approximate, heuristic, and exact methods to solve them. In this dissertation, we examine three new classes of network design and network flow problems that have previously received little attention in the literature. The first problem that we examine is concerned with assigning a set of customers to a set of facilities where the facilities are then responsible for producing the amount of demand assigned to the facility. This leads to a new class of nonlinear generalized assignment problems. In many practical situations, the resulting nonlinear functions in these problems are ill-structured (e.g., non-differentiable or discontinuous) so that standard optimization techniques are no longer applicable. We overcome this challenge through novel applications of the theory of linear programming. The second problem that we consider is a problem where the customers have dynamic demand over a finite horizon. We must assign each customer to an open facility and manage production and inventory levels at each open facility to meet the demand of the customers assigned to the facility. We develop approximation algorithms and results for this class of problems as well as setting the theoretical foundation for an algorithm to solve these problems to optimality. The third problem that we consider is a class of minimum-cost flow problems in infinite networks. This class of problems encompasses many supply chain planning problems where a sequence of decisions needs to be made over an infinite horizon. We are able to overcome the typical mathematical difficulties associated with linear programming in infinite-dimensional spaces and extend the well-known network simplex method to solve this class of problems in infinite networks. We do so in a nonstandard but intuitively appealing way that, to the greatest extent possible, employs concepts from finite-dimensional linear programming.
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 Thomas Sharkey.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Romeijn, Hilbrand E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022311:00001


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Dr.H.EdwinRomeijn,myadvisor,hasbeenatremendousmentortomeduringmygraduatecareer.Iwouldliketothankhimforallthetime,eort,andpatiencehehasputintoworkingwithme.HeisagreatrolemodelforacareerinacademiaandIhopetoemulatehissuccessinthefuture.Iwouldalsoliketothankmycommittee,Dr.RavindraAhuja,Dr.JosephGeunes,Dr.WilliamHager,Dr.PanosPardalos,andDr.ColeSmith,fortheirthoughtfulandconstructivefeedbackthroughoutmygraduatestudies.IespeciallythankDr.GeunesandDr.Smithfortheirhelpandguidanceinmyjobsearch.IwouldalsoliketoacknowledgethatthisworkhasbeensupportedbyaNationalScienceFoundationGraduateResearchFellowship.Myfamilyandfriendshaveplayedanimportantroleinmysuccessupuntilthispoint.Iwouldliketoespeciallythankmymother,Andrea,mybrother,Stephen,andmysister,Annemarie.Whetherwewereunderthesamerooforathousandmilesapart,theirunconditionalloveandsupporthasbeenahugemotivationforme.Forallofmyotherfriendsandfamily,toonumeroustolisthere,thathaveshownmenothingbutsupportthroughtheyears,pleaseknowthatIamforevergrateful.Finally,andmostimportantly,mybeautifulwifeMelissahasbeenamazingthroughouttheentiregraduateschoolprocess.DespiteherforcingourpoordogTuckertolistentomepracticemytalksandpresentations,shewasalwaystheretosupportandencouragemeineverywaypossible.Ihopetoonedayndawaytopayherback. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 1INTRODUCTION .................................. 12 2LITERATUREREVIEW .............................. 19 2.1TheGeneralizedAssignmentProblem ..................... 19 2.2TheFacilityLocationProblem ......................... 24 2.3CustomerAssignmentProblems ........................ 27 2.4DemandManagementModels ......................... 29 2.5NonlinearKnapsackProblems ......................... 30 2.6Innite-DimensionalNetwork-FlowProblems ................. 32 3ACLASSOFNONLINEARGENERALIZEDASSIGNMENTPROBLEMS .. 35 4AGREEDYPROCEDUREFORTHENL-GAP ................. 42 4.1PropertiesoftheRelaxation .......................... 43 4.1.1RelationshipBetweenLNL-GAPRandtheKKTConditions .... 46 4.1.2RelationshipBetweenLNL-GAPRandaLagrangianRelaxation .. 49 4.2AGreedyHeuristic ............................... 51 4.3StochasticModelsoftheNL-GAP ....................... 53 4.3.1Facility-IndependentParameters .................... 57 4.3.2Facility-DependentParameters ..................... 58 4.4ChapterSummaryandFutureResearchDirections ............. 62 5SIMPLEX-INSPIREDALGORITHMSFORSOLVINGACLASSOFCONVEXPROGRAMMINGPROBLEMS ........................... 63 5.1MathematicalPreliminaries .......................... 65 5.2ASimplexAlgorithmwithSimplePivots ................... 68 5.2.1IntuitiveDevelopmentandDescriptionoftheAlgorithm ....... 68 5.2.2ProofofCorrectness ........................... 73 5.2.2.1TheCaseK=1 ....................... 75 5.2.2.2TheGeneralCase ....................... 78 5.3ASimplexAlgorithmwithGeneralizedPivots ................ 85 5.4ComputationalExperiments .......................... 89 5.5Extensions .................................... 96 5.5.1Non-DierentiableObjectiveFunction ................. 96 5

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...................... 97 6ACLASSOFNONLINEARNONSEPARABLECONTINUOUSKNAPSACKANDMULTIPLE-CHOICEKNAPSACKPROBLEMS .............. 99 6.1Single-KnapsackProblems ........................... 102 6.1.1StructureofCandidateSolutions .................... 103 6.1.2ConstructingFullCandidateSolutions ................ 106 6.1.3SolvingaSpecialCaseofKP ...................... 108 6.1.4SolutionMethodsandRuntimeAnalysis ............... 109 6.1.5SolutionwithSmallestNumberofFractionalComponents ...... 114 6.2TheMulti-KnapsackProblem ......................... 115 6.3AMultiple-ChoiceKnapsackProblem ..................... 121 6.3.1StructureofCandidateSolutions .................... 122 6.3.2ConstructingFullCandidateSolutions ................ 125 6.3.3SolutionMethodsandRuntimeAnalysis ............... 127 6.3.3.1ImprovedAlgorithmUnderanAssumption ......... 128 6.3.3.2GeneralizingtheImprovedAlgorithm ............ 132 6.4ComputationalTesting ............................. 134 6.5ChapterSummaryandFutureResearchDirections ............. 139 7INTEGRATINGFACILITYLOCATIONANDPRODUCTIONPLANNINGDECISIONS ...................................... 141 8APPROXIMATIONRESULTSFORINTEGRATEDFACILITYLOCATIONANDPRODUCTIONPLANNINGPROBLEMS ................. 146 8.1ApproximatingtheUFLPPwithGeneralDemands ............. 147 8.2ApproximatingtheUFLPPwithSeasonalDemands ............. 151 8.2.1ApproximationAlgorithmsfortheCCFLP .............. 154 8.2.2ResultsonGeneralizationsoftheCCFLP ............... 168 8.3ApproximatingtheUFLPP-DA ........................ 169 8.4ChapterSummaryandFutureResearchDirections ............. 174 9EXACTALGORITHMSFORINTEGRATEDFACILITYLOCATIONANDPRODUCTIONPLANNINGPROBLEMS ..................... 176 9.1ASet-PartitioningFormulationoftheUFLPP ................ 178 9.1.1TightnessoftheSPFfortheUFLPPwithLot-SizingCosts ..... 179 9.1.2TightnessoftheSPFfortheUFLPP ................. 183 9.2TheProductionPlanningandCustomerSelectionProblem ......... 185 9.2.1LinearProductionandInventoryCosts ................ 185 9.2.2SeasonalDemandPatterns ....................... 186 9.2.3Customer-SpecicPrices ........................ 187 9.2.4CustomersHaveExactlyOneNon-ZeroDemandPeriod ....... 192 9.3ChapterSummaryandFutureResearchDirections ............. 192 6

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..................... 194 10.1ProblemDenitionandMathematicalFoundations ............. 195 10.1.1AnInnite-DimensionalMinimum-CostNetwork-FlowProblem ... 195 10.1.2ProblemFormulation .......................... 198 10.1.3Duality .................................. 198 10.2AnInniteNetworkSimplexMethod ..................... 205 10.2.1ComponentsoftheSimplexMethod .................. 205 10.2.1.1ExtremePointsandBasicPrimalSolutions ........ 205 10.2.1.2ComplementaryDualSolutions ............... 206 10.2.1.3ReducedCosts ........................ 209 10.2.1.4PivotOperations ....................... 210 10.2.2SimplexMethod ............................. 210 10.2.2.1ValueConvergence ...................... 211 10.2.2.2SolutionConvergence ..................... 213 10.3AClassofProblemswithFinite-timePivots ................. 216 10.3.1ProblemDenition ........................... 218 10.3.2Finite-timePivotsandtheFinitePartitionProperty ......... 221 10.3.3FindingaFeasibleSolution:PhaseI .................. 223 10.3.4AHybridPhaseI/IISimplexAlgorithm ................ 227 10.4ChapterSummaryandFutureResearchDirections ............. 233 11CONCLUSION .................................... 235 REFERENCES ....................................... 238 BIOGRAPHICALSKETCH ................................ 250 7

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Table page 2-1Relevantapproximationalgorithmsforfacilitylocationproblems. ........ 27 6-1ComparisonofrunningtimesobtainedwiththealgorithmforKPandBARON. 138 6-2ComparisonofrunningtimesobtainedwiththeimprovedalgorithmforMCKPandBARON. ..................................... 139 8

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Figure page 5-1Anillustrationofthepathsin(x1;x2)-spacefollowedby(a)theCSMand(b)ouralgorithmwhensolvingE1. ........................... 73 5-2Scatterplotandbesttfunctionforg(y)=y2. .................. 92 5-3Scatterplotandbesttfunctionforg(y)=ln(+y=). ............ 92 5-4Scatterplotsandbesttfunctionsforg(y)=ey=for=10(solidline),=50(dottedline),and=100(dashedline). .................... 93 5-5Scatterplotandbesttfunctionsforg(y)=20y2andm=1(solidline),m=2(blackdottedline),m=5(blackdashedline),m=10(blackdotted/dashedline),m=20(greydottedline),m=50(greydashedline),andm=100(greydotted/dashedline). ................................. 93 5-6Scatterplotsandbesttfunctionforgk(y)=20y2andK=2(dottedline),K=5(dashedline)andK=10(dotted/dashedline). .............. 95 5-7Scatterplotsandbesttfunctionforgk(y)=ln(+y 95 5-8Scatterplotsandbesttfunctionforgk(y)=ey=100andK=1(solidline),K=2(dottedline),K=5(dashedline)andK=10(dotted/dashedline). .. 96 10-1Anillustrationoflayers. ............................... 197 10-2Anexamplewithmultipleoptimalextremepointsolutions. ............ 217 10-3Areformulatednetwork-owproblem. ....................... 219 9

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1{1 )ensurethatweassigneachcustomertoafacility.Wenotethatthefacilitycostfunction,Hi(xi),intheSCNDproblemmayrequiredeterminingtheoptimalcosttoanoptimizationproblembasedonthesetofcustomersassignedtothefacility.Thefacilitycostfunctionmayalsobeinniteforcertainxitorepresentthatfacilityicannotfeasiblyservethissetofcustomers.GivenanoptimizationproblemthatbelongstotheSCNDframework,therearetwomainresearchapproachesthatcanbeusedtostudytheproblem.First,onecanexamineheuristicand/orapproximatemethodstoreturngoodsolutionstotheproblemquickly.Typically,thesemethodsexploitthestructureoftheSCNDproblem(inparticular,thestructureofHi(xi)).Alternatively,wecandevelopexactapproachestotheSCNDproblemwhich,althoughtheymayrequiremoreeortthantheheuristicandapproximatemethods,determinetheoptimalsolutiontotheproblem.BranchandpricealgorithmshavebeenextremelyeectivetosolvealargenumberofproblemsthatbelongtotheSCNDframework.Interestinglyenough,thepricingproblemthatarisesinthesebranch 13

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2 .InChapters 3 6 ,wemotivate,formulate,andstudyanewclassofnonlineargeneralizedassignmentproblems.InChapter 3 ,weintroducetheclassofproblems,thepricingproblemassociatedwiththem,anddiscusstheapplicationsofthesetwoclassesofproblems.InChapter 4 ,wediscussagreedyprocedurefortheclassofnonlineargeneralizedassignmentproblemsbasedontheoptimalsolutiontoitscontinuousrelaxation.InChapter 5 ,wediscusssolutionmethodstosolveaclassofconvexprogrammingproblems,whichincludesthecontinuousrelaxationofthenonlineargeneralizedassignmentproblemwithconvexfunctions.InChapter 6 ,wediscusstherelaxationofthepricingproblemforthisclassofnonlineargeneralizedassignmentproblems,i.e.,acontinuousnonlinearknapsackproblem.Wedevelopalgorithmstosolvethenonlinearknapsackproblemandvariantsofitthatrequirevirtuallynoassumptionsonthestructuralpropertiesofthenonlinearfunction.InChapters 7 9 ,wemotivate,formulate,andstudytheintegratedfacilitylocationandproductionplanningproblem.InChapter 7 ,weintroducetheclassofproblemsandtheintegratedproductionplanningandcustomerselectionproblemthatarisesasthepricingproblem.InChapter 8 ,weexamineconstantfactorapproximationalgorithmsfortheclass 17

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9 ,wediscussissuesarounddevelopingexactalgorithmsfortheintegratedfacilitylocationandproductionplanningproblems.Theintegratedproductionplanningandcustomerselectionproblemisdiscussedinthischapteraswell.InChapter 10 ,wediscussinnite-dimensionalnetwork-owproblems.Weextendthenetworksimplexmethodtoinnite-dimensionalspacesanddiscussanimportantclassofproblemswheretheinnite-dimensionalnetworksimplexmethodcanbeimplementedinsuchawaythateachpivotrequiresaniteamountoftimeandinformation.WeconcludethedissertationinChapter 11 withasummaryandconcludingremarks. 18

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2.1 andrelevantstreamsofliteratureforthefacilitylocationprobleminSection 2.2 .WediscussextensionsoftheseproblemsandothercustomerassignmentproblemsinSection 2.3 .AsdiscussedinChapter 1 ,supplychainplanningproblemswithcustomerselectionareimportantintheirownrightbutalsoariseassubproblemsinsolvingproblemsintheSCNDframework.InSections 2.4 and 2.5 ,wediscussmodelsandalgorithmsthatarerelatedtotheseplanningproblemswithcustomerselection.WeconcludethischapterinSection 2.6 ,whichdiscussesnetwork-owformulationsofplanningproblemsandapproachestosolveinnite-dimensionaloptimizationproblems.

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132 ].Ithasmanyapplicationsincluding:assigningjobstocomputernetworks(Balachandran[ 11 ]),xed-chargeplantlocation(RossandSoland[ 133 ]),routingproblems(FisherandJaikumar[ 52 ]),andthesingle-sourcingproblem(DeMaioandRoveda[ 40 ]).Therearemanyexistingsolutionmethods(bothexactandheuristic)fortheGAP(seeCattrysseandvanWassenhove[ 31 ]andOsman[ 109 ]).TheGAPisNP-hard(foradenitionofNP-hardness,seeGareyandJohnson[ 62 ]);infact,itisNP-hardtoevendetermineiftheGAPhasafeasibleassignmentofcustomerstofacilities(seeMartelloandToth[ 101 ]).Therefore,itisappropriatetodevelopheuristicsfortheGAP,i.e.,algorithmsthatreturna`good'solutiontotheGAPquickly.TheclassofgreedyheuristicsintroducedbyRomeijnandRomeroMorales[ 124 ]fortheGAParerelatedtoasolutionprocedureforanonlinearversionoftheGAPstudiedinChapter 4 ofthisdissertation.Theseheuristicsdeneaweight,orpseudo-cost,function,f(i;j)(seeMartelloandToth[ 100 ]),foreachfacility/customerpair.Thisfunctionrepresentsameasureofthecostoftheassignmentofcustomerjtofacilityi.Wethendenethe 20

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124 ]isf(i;j)=cij+iaij.ThisheuristicpossessesseveraldesirablepropertieswheniissetequaltotheoptimaldualmultiplierofthecapacityconstraintoffacilityiinthelinearprogrammingrelaxationoftheGAP.Inparticular,theheuristiccanbeshowntobebothasymptoticallyfeasibleandasymptoticallyoptimalunderaverygeneralstochasticmodeloftheGAP.Probabilisticanalysisofcombinatorialoptimizationproblemsandheuristicsforthemhavequiteapresenceintheliterature.Themedianlocationproblem(RheeandTalagrand[ 119 ],themulti-knapsackproblem(VandeGreerandStougie[ 63 ]andMeantietal.[ 102 ]),theset-coveringproblem(Piersma[ 114 ]),andtheparallelmachineschedulingproblem(PiersmaandRomeijn[ 115 ])haveallbeenstudiedunderprobabilisticmodels.Themainideaofalloftheseworksistolinktheproblemtoanempiricalprocess(seeAlexander[ 5 ]andTalagrand[ 150 ])toshowthatthecostoftheoptimalsolutiongrowslinearlyasafunctionofthesizeoftheproblem.However,sincetheGAPmaynothaveafeasiblesolution,complicationsariseinanyprobabilisticanalysisforit.DyerandFrieze[ 46 ]examineastochasticmodeloftheGAPundertheassumptionthatallinstancesarefeasiblewithprobability1.RomeijnandPiersma[ 123 ]provideastochasticmodeloftheGAPandderiveatightconditionunderwhichtheresultingGAPisasymptoticallyfeasiblewithprobability1andthenexaminetheoptimalsolutionvalueoftheGAPunderthestochasticmodel.Further,RomeijnandRomeroMorales[ 125 ]generalizetheseresultstotheso-calledmulti-periodsingle-sourcingproblem,whichcanbeformulatedasanonlinearversionoftheGAP.Theprobabilisticanalysisofheuristicsforcombinatorial 21

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89 ]analyzeaheuristicforthemulti-knapsackproblemunderastochasticmodel.RomeijnandRomeroMorales[ 124 ]analyzeaheuristicfortheGAPunderastochasticmodel.RomeijnandRomeroMorales[ 126 127 ]analyzeheuristicsforthemulti-periodsingle-sourcingproblemwithcyclicandacyclicdemands,respectively,understochasticmodels.Ahujaetal.[ 3 ]analyzeaheuristicforthemulti-periodsingle-sourcingproblemwithperishableinventories.Rainwateretal.[ 118 ]examineaheuristicfortheGAPwithexiblejobsunderastochasticmodel.ItisalsoofinteresttobeabletosolvetheGAPtooptimalitywithanexactalgorithm.SincetheGAPisanintegerprogram,standardintegerprogrammingtechniques(seeNemhauserandWolsey[ 107 ])canbeappliedtosolveit.However,manyexactalgorithmsthatemploythestructureoftheproblemhavebeendevelopedtosolvetheGAP.ThebranchandpricealgorithmofSavelsbergh[ 136 ]isanimportantapproachthatwillbeutilizedinthisdissertation.Thisalgorithmsolvestheset-partitioningformulationoftheGAPusingcolumngenerationandabranchandboundscheme.Inparticular,anyassignmentofcustomerstofacilitiesintheGAPcanbethoughtofasapartitionofthesetofcustomersintomsetsandthentheassignmentofeachofthesesetstoafacility.Wewillnextdescribetheset-partitioningformulationoftheGAPindetail.DenotethenumberofdistinctsubsetsthatcanbefeasiblyassignedtofacilityibyLi.Letthebinaryvector`irepresentthe`thsubsetassociatedwithfacilityi,where`ij=1ifcustomerjbelongstothesubsetand0otherwise.(Wealsoreferto`iasthe`thcolumnassociatedwithfacilityi.)Furthermore,lethi(`i)=Pnj=1cij`ijdenotethecostassociatedwithservingthesetofcustomersgivenby`iatfacilityi.Ifwethendenethedecisionvariabley`itobeequaltooneiffacilityiservesthe`thassociatedsubsetand0otherwise,theset-partitioningformulation(SPF)oftheGAPis 22

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2{1 )ensurethatcustomerjisassignedtoafacilityandconstraints( 2{2 )ensurethatweassignexactlyonesubsetofthecustomerstoeachfacility.ItisnotdiculttoextendthisformulationtothatofageneralSCNDproblem;wesimplydenehi(`i)=Pnj=1cij`ij+Hi(`i):Unfortunately,theSPFhasanexponentialnumberofvariables.Toaddressthisproblem,weusethecolumngenerationapproachofGilmoreandGomory[ 71 ]tosolvetherelaxationofSPF(SPFR).ThisprocedurewillbeusedtocalculatetheLPboundsofSPFRateachnodeofabranch-and-boundtree,leadingtoabranchandpricealgorithmtosolveSPF.Oneofthemostvitalpartstothesuccessofabranch-and-pricealgorithmistheabilitytoquicklysolvethepricingproblem.SupposethatNisthecurrentsetofcolumnsinthereducedLPrelaxationofSPFR.Let((N);(N))betheoptimaldualmultipliersassociatedwiththereducedLPrelaxationofSPFR.Thecolumnswillbesearchedforeachfacilityi=1;:::;m.Inparticular,weneedtosolvethefollowingpricingproblemforfacilityi,minxi2f0;1gnnXj=1cijxij+Hi(xi)nXj=1j(N)xij+i(N):

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2.4 .First,wewillrevisitthepricingproblemassociatedwiththeGAP.RecallthatHi(xi)=0iftheselectedcustomersinxirespectthecapacityconstraintoffacilityianditisinniteotherwise.Therefore,thepricingproblemthatarisesfromtheSPFoftheGAPcanbeformulatedasmaxa>ixibi;xi2f0;1gnnXj=1rijxij;whichissimplythetraditionalknapsackproblem.WewillconsideranonlinearextensionoftheGAPinChapter 3 .Theresultingpricingproblemisthusanonlinearversionoftheknapsackproblem,sowewillreviewtheliteratureonnonlinearknapsackproblemsinSection 2.5 24

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80 ])forthem.Wesaythatanalgorithmisa-approximationalgorithmforaminimizationproblemifthealgorithmrunsinpolynomialtimeandreturnsasolutiontotheproblemwhosecostisnomorethantimesthecostoftheoptimalsolutiontotheproblem.TheeldofdesigningconstantfactorapproximationalgorithmsforfacilitylocationproblemsandvariantshasbeenextremelyactivesinceShmoysetal.[ 143 ]gavetherstconstantfactorguaranteeforthemetricUFLP.Theiralgorithmusedtheideaofroundingtheoptimalsolutiontoalinearprogrammingrelaxationofthefacilitylocationproblem.Sviridenko[ 148 ]usedanLP-roundingtechniquetodevelopanapproximationalgorithm 25

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29 ]usedLP-rounding,thealgorithmofChudakandShmoys[ 37 ],andthealgorithmofMahdianetal.[ 98 ]toachieveanalgorithmwithafactorof1:50.Thesealgorithmshavehighrunningtimessincetheymustsolvealinearprogram.Primal-dualalgorithmsfortheUFLParecomputationallyattractiveandcanachievesimilarapproximationguarantees.Jainetal.[ 87 ]oerprimal-dualalgorithmswithapproximationguaranteesof1:861and1:61.Thecurrentbest-knownapproximationfactorforthemetricUFLPusingaprimal-dualalgorithmachievesaguaranteeof1:52andisduetoMahdianetal.[ 98 ].Thisalgorithmusesthe1:61-approximationalgorithmofJainetal.[ 87 ]andtheideaofscaling(seeCharikarandGuha[ 33 ]).InChapter 8 ,wewilldiscussageneralizationoftheUFLPwherethefacilitycostsaregeneralconcavefunctionsoftheamountofdemandassignedtothefacility.Wewillrefertothisproblemastheconcavecostfacilitylocationproblem(CCFLP).Hajiaghayietal.[ 75 ]generalizethe1:861-approximationalgorithmofJainetal.[ 87 ]fortheUFLPtothespecialcaseoftheCCFLPinwhicheachcustomerhasunitdemand;thisalgorithmrunsinO(n3logn)time.Inaddition,theyshowthattheCCFLPwithunitdemandscanbeconvertedtoaUFLPwithncustomersandnmfacilitiesandthenusethealgorithmofMahdianetal.[ 98 ]toobtaina1:52-approximationalgorithmthatrunsinO(n6)time.FortheCCFLPwithgeneralintegraldemand,thisapproachhasapseudo-polynomialrunningtime.Inparticular,theUFLPwouldhaveO(nD)customersandO(nmD)facilitieswhereDisequaltothesumofthedemandsofthecustomers.Further,fortheCCFLPwithgeneralintegraldemand,wecandevelopa(1:52+)-approximationalgorithmbyapproximatingtheconcavefunctionswithpiecewiselinearfunctionsandusingasimilarreductionasHajiaghayietal.[ 75 ]totheUFLP.However,thisapproachhasarunningtimeofO(n3(lnD 8 ,weprovideageneralizationofthe1:61-approximationalgorithmofJainetal.[ 87 ]fortheUFLPtotheCCFLPwithgeneralintegral(or,rational)demandsthatrunsinO(n4logn)time.Wethenusethisalgorithmandascalingideatogeneralizethe1:52-approximationalgorithm 26

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98 ]totheCCFLPthatalsorunsinO(n4logn)time.Independentoftheworkinthisdissertation,MagnantiandStratila[ 97 ]provideda1:61-approximationalgorithmfortheCCFLPwithgeneralintegraldemandswitharunningtimeofO(n4).AsummaryofpastapproximationalgorithmsforfacilitylocationproblemsthatarerelevanttotheworkinChapter 8 appearsinTable 2-1 Table2-1. Relevantapproximationalgorithmsforfacilitylocationproblems. Reference Problem Factor RunningTime Jainetal.[ 87 ] UFLP 1.861 Jainetal.[ 87 ] UFLP 1.61 Mahdianetal.[ 98 ] UFLP 1.52 Quasi-linear Hajiaghayietal.[ 75 ] CCFLPwithunitdemands 1.861 Hajiaghayietal.[ 75 ] CCFLPwithunitdemands 1.52 Consequenceof[ 75 ] CCFLPwithgeneraldemands 1.52 Pseudo-polynomial Consequenceof[ 75 ] CCFLPwithgeneraldemands 1:52+ O(n3(lnD Chapter 8 CCFLPwithgeneraldemands 1.52 64 ],Bendersetal.[ 17 ],andFleischmann[ 54 ].However,thesemodelshavelimitedapplicabilityduetotheassumptionofstaticdemandandthefactthattheydonotexplicitlymodelinventoryproblems.Huangetal.[ 85 ]considerextensionsofthesingle-sourcingproblemwhereeachcustomerhasaconstantdemandrateandeachfacilitypayscostsinanassociatedEconomicOrderQuantity(EOQ)modelwithademandrateequaltothecumulativedemandratesofcustomersassignedtothatfacility.Duran[ 43 ], 27

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125 ],andFrelingetal.[ 57 ]allconsidersingle-sourcingproblemswherethedemandofthecustomersistime-varying.RomeijnandRomeroMorales[ 125 ]considerthemulti-periodsingle-sourcingproblem(MPSSP)inwhichwemustmeetthedemandsofthesetofcustomersoveradiscrete,nitehorizon.IntheMPSSP,thedemandofeachcustomerineachtimeperiodmustbemetbyasinglefacilityandproductionateachfacilityineachtimeperiodiscapacitated.Frelingetal.[ 57 ]consideravariantoftheMPSSPwiththeadditionalrestrictionthateachcustomermustbeassignedtothesamefacilityandthedemandpatternofeachcustomersharethesameseasonalitypattern,i.e.,thedemandintimeperiodtofcustomerjcanbedecomposedintotheproductofaseasonalfactortoftimeperiodtandthebasedemandleveldjofcustomerj.Daskinetal.[ 39 ]andShenetal.[ 141 ]considerajoint-inventorylocationmodel.Inthisproblem,whichisanextensionoftheUFLP,eachcustomerhasanuncertaindemand.Wearetheninterestedindetermininganassignmentofthecustomerstoopenfacilities,howoftentoproduce(orreorder)atopenfacilities(ordistributioncenters),andwhatlevelofsafetystocktomaintainateachopenfacilityinordertominimizeallrelevantcostswhilemaintainingaspeciedservicelevelthroughoutthesystem.Shenetal.[ 141 ]examineabranchandpricealgorithmtosolvetheset-partitioningformulationofthisproblemfortwospecialcases.Inparticular,theydiscusshowtoeciently(boththeoreticallyandcomputationally)solvethepricingproblemwhen(i)theratioofthevarianceandthemeanofthedemandofeachcustomerisconstantand(ii)thevarianceofthedemandofeachcustomeriszero.Shuetal.[ 144 ]showhowtosolvethegeneralpricingproblemeectivelyforthegeneralpricingproblemthatarisesinShenetal.[ 141 ]anddiscussitsimplicationsinthebranchandpricealgorithmforthegeneraljoint-inventorylocationproblem.Sourirajanetal.[ 147 ]consideranextensionofthisproblemthataccountsforleadtimeinproducingtheproductatthefacilities(ordeliveringtheproducttodistributioncenters). 28

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7 ,wediscussacustomerassignmentproblemwhereeachfacilitymustmanagetheproductionandinventorydecisionsatthefacilityinordertoensurewemeetthecumulativedemandofthesetofcustomersassignedtothefacilityineachtimeperiod.Theproductionplanningproblemthatisfacedbyafacilityisageneralizationoftheclassicaleconomiclot-sizingproblem(seeWagnerandWhitin[ 157 ])whereconcaveproductioncostfunctionsreplacexed-chargepluslinearproductioncostsandconcaveholdingcostfunctionsreplacelinearholdingcosts.ThisproductionplanningproblemcanbesolvedinO(T2)time(seeWagner[ 156 ]andVeinott[ 154 ]).Theeconomiclot-sizingproblemcanbesolvedinO(TlogT)andcanbesolvedinO(T)timeinthecaseofnon-speculativemotives(seeAggarwalandPark[ 1 ],FedergruenandTzur[ 47 ],andWagelmansetal.[ 155 ]).KrarupandBilde[ 92 ]usedaprimal-dualalgorithmtoshowthatthefacilitylocationformulationoftheeconomiclot-sizingproblemyieldedanintegralsolution.Levietal.[ 95 ]developedprimal-dualalgorithmsforthreeimportantclassesofdeterministicinventoryproblems,includingtheeconomiclot-sizingproblem.Theirprimal-dualalgorithmsolvedtheeconomiclot-sizingproblemtooptimality.Theyalsodevelopeda2-approximationalgorithmforthejoint-replenishmentproblem(seeZangwill[ 159 ],Veinott[ 154 ],andArkinetal.[ 9 ])anda2-approximationalgorithmforthemultistageassemblyproblem(seeRoundy[ 134 ]). 29

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152 153 ],FlorianandKlein[ 55 ],andKunreutherandSchrage[ 93 94 ].Therehavebeenseveralpapersthatconsiderpricingdecisionsinthecontextofinventoryplanningproblemsthathaveastructurerelatedtotheeconomiclot-sizingproblem.ThesepapersincludeGeunesetal.[ 66 ],vandenHeuvelandWagelmans[ 78 ],DengandYano[ 41 ],Ahnetal.[ 2 ],Geunesetal.[ 65 ]andMerzifonluogluetal.[ 103 ].Thesepapersassumethatdemandisknownforagivenpricelevel.ExamplesofinventorymanagementandpricingdecisionswheredemandisarandomvariablebaseduponthepricecanbefoundinGallegoandvanRyzin[ 60 ],PetruzziandDada[ 113 ],ChenandSimchi-Levi[ 35 36 ],andBakaletal.[ 10 ].Supplychainplanningproblemswithcustomerselectionarenotonlyimportantfortheirroleinshapingthedemandofthesupplierbutcanalso,asdiscussedinSection 2.1 ,ariseassubproblemsinalgorithms(suchasbranchandprice)tosolveSCNDproblems.Severalclassicinventorycontrolproblemshavebeenstudiedwithcustomerselectiondecisions.Geunesetal.[ 67 ]considercustomerselectioninvariouseconomicorderquantitysettings.Taaeetal.[ 149 ]considerintegratednewsvendorandcustomerselectionproblems.VandenHeuveletal.[ 77 ]discusscustomerselectiondecisionsinthecontextoftheeconomiclot-sizingproblem. 2.1 ,wediscussedthatthepricingproblemthatarisesinabranchandpricealgorithmfortheGAPistheknapsackproblem.InChapter 3 ofthisdissertation,weconsideranonlinearversionofthegeneralizedassignmentproblemwhichleadstoapricingproblemwhichisanonlinearknapsackproblem.Thereisextensiveliterature 30

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101 ]isanexcellentresourceforlinearknapsackproblemsandIbarakiandKatoh[ 86 ]oeracomprehensivetreatmentoftheproblemwherewemaximizeaconvexandseparablefunctionsubjecttoalinearconstraint.BretthauerandShetty[ 26 ]oerathoroughreviewoftheliteratureonnonlinearknapsackproblems.Muchofthepreviousworkonthenonlinearextensionsofthecontinuousknapsackproblemhavefocusedonseparableversionsoftheproblem.MoreandVavasis[ 104 ]showedthatdeterminingtheglobalsolutiontoaconcave,separableminimizationproblemwithaknapsackconstraintisNP-hardandthenfocusedonobtaininglocallyoptimalsolutions.Brucker[ 28 ]developedalinear-timealgorithmforthecaseofaseparable,convex,quadraticobjectivefunctionthroughabreakpointsearchoftheLagrangianfunction.Thisideahasbeenusedseveralothertimes,forexamplebyPardalosandKovoor[ 111 ],HochbaumandHong[ 81 ],andKiwiel[ 90 ].BitranandHax[ 22 ]proposeaso-calledpeggingalgorithmforaseparable,convexobjectivefunctionsubjecttoalinearknapsackconstraintandboundsonthevariables.Robinsonetal.[ 120 ]developapeggingalgorithmforaseparable,quadraticobjectivefunctionsubjecttoalinearknapsackconstraintwhichreliesonprojectingtheoriginontoarelaxedproblem.BretthauerandShetty[ 27 ]developapeggingalgorithmforaclassofproblemswithaseparable,convexobjectivefunctionsubjecttoageneralizedknapsackconstraintthatboundsaseparable,convexfunctionofthedecisionvariables.Themainfocusoftheliteraturewhentheobjectivefunctionisnonseparablehasbeenonthequadraticknapsackproblem.Pang[ 110 ]andPardalosetal.[ 112 ]consideraquadraticobjectivefunctionsubjecttoalinearconstraintandboundsonthevariables.Dussaultetal.[ 44 ]andKlastorin[ 91 ]solvetheproblemthroughaseriesofseparablesubproblems.Capraraetal.[ 30 ]exploreaLagrangianrelaxationinordertosolvethequadraticknapsackproblemexactly.CeselliandRighini[ 32 ]studyaclassofnonseparable,nonlinearknapsackproblemsthattheycallpenalizedknapsackproblemsandthathave 31

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122 ]considertheproblemofmaximizingthedierenceofalinearfunctionandaconcavefunctionwhoseargumentisalinearfunctionofthevariablessubjecttoaknapsackconstraint.Ecientalgorithmsforcontinuousknapsackproblemsalongwithanobservationthatanoptimalsolutionexiststotheproblemwithasmallnumberoffractionalcomponentsformanimportantsteptowardsdiscreteknapsackproblemsusingabranch-and-boundalgorithm.SomeapproachesfordiscreteknapsackproblemsarealgorithmsdevelopedbyHorowitzandSahni[ 83 ],MartelloandToth[ 99 ],andNauss[ 106 ]forthetraditionallinearknapsackproblem;thealgorithmsdevelopedbyBretthauerandShetty[ 25 27 ]foranonlinearintegerknapsackprobleminwhichboththeobjectiveandconstraintfunctionareconvexandseparableinthedecisionvariables;andthealgorithmdevelopedbyGalloetal.[ 61 ]forsolvingabinaryquadratic,nonseparableknapsackproblem. 4 ]oeranin-depthoverviewofthiseld.Moreover,dynamicminimum-costnetwork-owproblems,inwhichatime-delayisassociatedwithowoneacharcinthenetwork,havebeenstudiedaswell.Forexample,HoppeandTardos[ 82 ]studysuchproblemswithadiscretetimeparameter,whileAndersonetal.[ 7 ],AndersonandPhilpott[ 8 ],andFleischerandTardos[ 53 ]dealwithcontinuous-timedynamicnetwork-owproblems.Incontrast,verylittleworkhasbeendevotedtothesolutionofnetworkproblemsoncountablyinnitenetworks.Intheareaofinnite-horizonoptimization,muchattentionhasbeendevotedtothesolutionofplanningproblemsoveraninnitehorizonviaaninnitesequenceofnite-dimensionaltruncations(see,e.g.,Grinold[ 73 ],BeanandSmith[ 15 ],Beanetal.[ 14 ],BesandSethi[ 21 ],SchochetmanandSmith[ 137 138 ],FedergruenandTzur[ 48 { 50 ]).Oneoftheimportantgoalsisthentoestablishtheexistenceofplanningandforecasthorizons,i.e.,nite-dimensionaltruncationsthathavethedesirablepropertythat 32

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10 ,wewillinsteademployaso-calledstrategyhorizonapproachinwhichweattempttodirectlysolvetheinnitehorizonproblem(seealsoGhateandSmith[ 70 ]).Thisimplicitlymeansthatweapproximatedierentdecisionsequenceswithdierentapproximatinghorizons.Thesimplexmethodforlinearprogrammingwasgeneralizedtoclassesofso-calledseparatedcontinuousinnite-dimensionallinearprogrammingproblems,whichgeneralizethecontinuous-timenetwork-owproblems,inPullan[ 116 ]andWeiss[ 158 ].AndersonandNash[ 6 ]studiedlinearoptimizationproblemsingeneralinnite-dimensionalspaceswiththegoalofgeneralizingthesimplexmethod.Theirresultswerelimitedduetothedicultyincharacterizingextremepointsolutionsandbasicsolutionsthroughadecompositionofthedecisionvariablesintosetsofbasicandnonbasicvariables.However,forinnitenetwork-owproblems,Romeijnetal.[ 128 ]recentlydevelopedextremepointcharacterizationswhichledtoacharacterizationofbasicsolutionsthroughbasicandnonbasicvariablesfornetwork-owproblemswithintegraldata.GhateandSmith[ 68 ]extendedtheseresultstoaclassofdoublyinnitelinearprogrammingproblems.Anothercomplicationistheestablishmentofweakandstrongduality,whichisobviouslyofgreatimportancetothedevelopmentofasimplexmethodandtheeconomicinterpretationofoptimaldualsolutions.Byposingtheoptimizationprobleminappropriateinnite-dimensionalspaces,weakdualitycanusuallybeobtainedinarelativelystraightforwardmanner.However,thesespacesthenoftenprovetoorestrictivetoalsoobtaingeneralstrongdualityresults(see,e.g.,Luenberger[ 96 ]andAndersonandNash[ 6 ]).Dualityissuesinsemi-innitelinearprogramswerestudiedbyCharnesetal.[ 34 ],Dunetal.[ 42 ],andBorwein[ 24 ].Grinold[ 72 ]establishedconditionsfortheexistenceofoptimaldualsolutionstoaspecialclassofdoubly-innitelinearprograms.Jonesetal.[ 88 ]appliedthetheorydevelopedbyGrinoldandHopkins[ 74 ]toaninnite-horizonequipmentreplacementproblem.Pullan[ 117 ]derivesdualityresults 33

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38 ]studiesageneralclassofinnite-dimensionallinearprogrammingproblemsandisabletoobtainstrongdualityresultsunderasetofassumptionsontheproblemrelatedtodegeneracy;theseassumptions,however,arequiterestrictiveinaninnite-dimensionalsetting,andinparticularinminimum-costnetwork-owproblemsininnitenetworks.Finally,GhateandSmith[ 69 ]developadualitytheoryforcountablyinnitelinearprograms. 34

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4 6 ,wewillexamineproblemsrelatedtosolvingthefollowingclassofnonlineargeneralizedassignmentproblems,minimizemXi=1nXj=1cijxij+ginXj=1sijxij!!subjectto(NL-GAP) IntheNL-GAP,weareinterestedinassigningthesetofcustomerstothesetoffacilitieswhilerespectingthecapacityconstraintsofeachfacility.Constraints( 3{1 )representthecapacityofeachfacility.NotethattheNL-GAPtsintotheSCNDframeworkbydeningthefunctionHi(xi)asfollows:Hi(xi)=8><>:giPnj=1sijxijifPnj=1aijxijbi1ifPnj=1aijxij>bi:ThemotivationforexaminingthisparticularnonlinearextensionoftheGAPisduetoitsmanyapplicationsinsupplychainmanagementandparallelmachinescheduling.Inthecontextofsupplychainmanagement,considerthefollowinggeneralframeworktomotivatetheNL-GAP.Eachcustomerwillhaveacertaindemandlevel(orrate)foraproduct.Ifcustomerjisassignedtofacilityi,thedemandlevelofcustomerjfortheproductisequaltosij.Further,inmeetingthedemandofcustomerjatfacilityi,itisnecessarytousesomelevel,aij,ofascarceresource(suchasstoragespace,laborhours,ormaterials).Facilityionlyhasacertainlevel,bi,oftheresource.Thefunction,gi(S),representsthecostofproducingSunitsoftheproductatfacilityi.Inthissituation,we 35

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3{1 )canbereplacedbytheconstraintsnXj=1sijxijbi 85 ]considercontinuoustimesinglesourcingproblemswherethevaluessij=sjrepresentthedemandrateofcustomerjperunittimeandwheretheproductioncostfunctiongi(S)=p 141 ]isalsoanNL-GAPwheresij=jisthemeandemandofcustomerjandthecostfunctiongi(S)representstheoperating,inventory,andsafetystockcostsassociatedwithassigningademandlevelwithmeanSandvarianceStofacilityi.Notethatboththeseproblemsleadtoconcavefacilityproductioncostfunctions.Inmanysituations,theproductioncostfunctionsexhibiteconomiesofscaleinproduction(i.e.,gi(S)isconcave).Ifthereareeconomiesofscaleinproductionbutthefacilityihasaninternalproductioncapacityandasubcontractingoption,thenthefunctiongi(S)willbeconcaveupuntiltheinternalcapacity(representingtheinternalproductioncosts)andwilltypicallybeconvexaftertheinternalcapacity(representingthesubcontractingcosts).Attheinternalcapacity,the 36

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57 ]assumethatproductionineachperiodiscapacitatedandproductionandinventoryholdingcostsarelinear,whichleadstoacostfunctiongi(S)thatispiecewiselinearandconvex.If,alternatively,therearexedsetupcostsassociatedwithproductionbutproductionisuncapacitated,apiecewiselinearconcavecostfunctiongi(S)isobtained.Underotherassumptionsonproductionandinventorycostfunctionsandcapacitiesthefunctiongi(S)maybenonconvexand/ornonconcave.Inthesesettings,notethatthefunctionsgi(S)arenon-dierentiable.TheNL-GAPalsohasapplicationsinparallelmachinescheduling.IncontextofthismotivationfortheNL-GAP,weareinterestedinassigningthesetofjobs(indexedbyj=1;:::;n)tothemachines(indexedbyi=1;:::;m).Thevaluesijrepresentsthespanrequiredtoproducejobjonmachinei.Inthisproblem,thefunctiongi(S)isthecostofoperatingmachineiforSunitsofspan.Whenmachineideterioratesovertime,i.e.,themachineprocessesjobsataslowerratethelongerthemachineisrun,thenthefunctiongi(S)isconvex.Whenthemachineirequiresawarmupperiod,i.e.,themachinedoesnotprocessjobsatitsfullcapabilitiesuntilitwarmsup,thenthefunctiongi(S)isconcave.Whenmachineineedsanoperator,whomwemustpayovertimewagesifhe/sheworksoveracertainamountoftime,thenthefunctiongi(S)isapiecewiselinearconvexfunction.TherelaxationoftheNL-GAPandrelatedproblemsthatwillbestudiedinthisdissertationbelongtoamuchlargerclassofglobaloptimizationproblems.Inparticular, 37

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Furthermore,dualitytheoryfromlinearprogrammingwillplayanimportantroleindevelopingmethodstosolveproblemsoftheformGP.WenotethatthefeasibleregionofthedualofLGP,D=f!2Rm;2RK:A>!+S>cg;isindependentofthevalueof.WehavethefollowingcorollarytoTheorem 3.0.1 Proof. 3.0.1 and(!;)beits(optimal)complementarydualsolutioninLGP. ArecurringthemeinthisdissertationwillbetoapplyTheorem 3.0.1 andCorollary 3.0.2 toaproblemoftheformGPandusetheresultstodevelopspecializedalgorithmstosolvetheproblemathand.Forcertainproblems,theseresultswillleadtoextremelypowerfulspecializedalgorithmstosolvetheproblem.InChapter 4 ,wewilldevelopaproceduretosolvetheNL-GAPthatisbasedonsolvingthecontinuousrelaxationofit.WewilluseTheorem 3.0.1 toshowthatthereexistsonlyasmallnumber(independentofn)ofcustomersthatareassignedfractionallytoanyfacilityintheoptimalsolutiontotherelaxationofNL-GAP.OurprocedurewillthenconverttheoptimalsolutionoftherelaxationtoafeasiblesolutiontotheNL-GAP 39

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5 ,wedevelopamethodtosolvetheclassofGPwheregisaconvexfunction.ThismethodisinspiredbythesimplexmethodforlinearprogrammingandreliesonageometricinterpretationofTheorem 3.0.1 .Itgeneratesasequenceofsolutionsthatisnon-decreasingwithrespecttotheobjectivefunctionwhereeachsolutionliesonafaceoftheconstraintpolyhedronofatmostKdimensions.Weshowthatanimplementationofthemethodwherewesolve1-dimensionalconvexoptimizationproblemsconvergesincostasthenumberofiterationstendstoinnityandanimplementationofthemethodwherewesolveK-dimensionalconvexoptimizationproblemsterminatesinanitenumberofiterations.Further,acomputationalstudydemonstratestheeciencyofthealgorithmandsuggeststhattheaverage-caseperformanceofthesealgorithmsisapolynomialofloworderinthenumberofdecisionvariables.RecallthatifweweretosolveaSCNDproblemthroughabranchandpricealgorithm,thatthepricingproblemthatwouldariseisoftheformmaxx2f0;1gnr>xH(x):Therefore,thepricingproblemthatwouldariseinsolvingtheNL-GAPisoftheformmaxr>xg(s>x)subjectto(DKP)a>xbx2f0;1gn:InChapter 6 ,wewilldevelopmethodstosolvethecontinuousrelaxationofDKPandrelatedextensionsoftheknapsackproblem(suchasthemultipleknapsackproblemand 40

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3.0.1 andCorollary 3.0.2 .Usingtheseproperties,weprovideasolutionmethodthatrunsinpolynomialtimeinthenumberofdecisionvariables,whilealsodependingonthetimerequiredtosolveaparticularone-dimensionaloptimizationproblemthatisbasedonthefunctiong.Thus,forthemanyapplicationsinwhichthisone-dimensionalfunctionisreasonablywellbehaved(e.g.,unimodal),theresultingalgorithmrunsinpolynomialtime.ThismethodtosolvethecontinuousrelaxationofDKPwillplayanimportantroleinsolvingDKPsince(i)thealgorithmistheoreticallyandcomputationallyecientand(ii)thenumberoffractionalvariablesintheoptimalsolutiontotherelaxationissmall.Weextendthissolutionmethodtosolveproblemswheretheknapsackconstraintsarereplacedbymultipleknapsackconstraintsormultiple-choiceknapsackconstraints.Computationaltestingdemonstratesthepoweroftheproposedalgorithmsoveracommercialglobaloptimizationsoftwarepackage. 41

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3.0.2 )andusethesedualmultipliersinagreedyheuristic(whichresultsinthegreedyprocedure).ThisheuristicissimilarinspirittotheclassofgreedyheuristicsthatRomeijnandRomeroMorales[ 124 ]examinefortheGAP.Thegreedyprocedureitselfcanbeinterpretedasattemptingtoconverttheoptimalsolutionoftherelaxationtoafeasibleintegersolutionwithoutincreasingthecosttoomuch.Theremainderofthischapterisorganizedasfollows.InSection 4.1 ,weexaminepropertiesofanoptimalsolutiontothecontinuousrelaxationoftheNL-GAP.WewilldevelopacertainsetofdualmultipliersassociatedwiththisoptimalsolutionbasedonCorollary 3.0.2 .InSection 4.1.1 ,weexploretherelationshipbetweenthesedualmultipliersandthegeneralizedKKTconditionsappliedtoacertainformulationoftherelaxationoftheNL-GAP.Further,inSection 4.1.2 ,weshowthatforthecasewhenthefunctionsgi,i=1;:::;m,areconvex,thesedualmultiplierscanbedeterminedbysolvingacertainLagrangiandualproblem.WeproposethegreedyheuristicinSection 4.2 anddiscussthepropertiesoftheheuristicwhenitisappliedwiththedualmultipliersdevelopedinSection 4.1 .WeproposeastochasticmodeloftheNL-GAPinSection 4.3 anddiscusstheaveragecaseperformanceofthegreedyprocedure.WeshowthatthegreedyprocedureisasymptoticallyoptimalandfeasiblewhentherequirementsoftheNL-GAParefacility-independent,i.e.,aij=aj,cij=cj,andsij=sjfori=1;:::;mandj=1;:::;nforanysetofcontinuousfunctions,gi,i=1;:::;m.Forthecaseoffacility-dependentrequirements,weshowthatthereexistsasetofdualmultipliersthat 42

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4.4 withsomefutureresearchdirections.

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3.0.1 totheNL-GAPRyieldsthefollowingresult. 4.1.1 .Inanalyzingthepropertiesofx,itwillbeusefultodeneseveralsetsassociatedwithit.GivenasolutionxthatisabasicfeasiblesolutiontotheLNL-GAPRforsomeS2Rm,wedeneFx=f(i;j):0
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Therefore,thereexistsanoptimalsolutiontotheNL-GAPRsuchthatthenumberofsplitcustomersisatmost2m,independentonthenumberofcustomersn.TheideabehindthegreedyprocedurefortheNL-GAPistoreassignonlythesplit(orsplitbasic)customersfromx.Inordertoaccomplishthis,wewillnextdevelopasetofdualmultipliersassociatedwithxthathelpinguaranteeingthatallnon-splitjobsareproperlyassignedinthegreedyprocedure.AswasthecasewiththefeasibleregionofthedualofLGP,thefeasibleregionofthedualofLNL-GAPRisindependentofS.Inparticular,itisgivenbyD=8><>:vjcij+aijisijifori=1;:::;m;j=1;:::;n2Rm+;2Rm;v2Rn9>=>;:Thefollowingresulthelpscharacterizetherelationshipbetweenxandits(optimal)complementarydualsolutionfromCorollary 3.0.2 (i) Foreachj62SBx,xij=1impliesthatvj=minicij+aijisijiandforallk6=ivj
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3.0.2 ,thereexistsafeasibledualsolution,(;;v)thatsatisesthecomplementaryslacknessconditionswithx.Ifj2SBx,theremustexisttwoindicesi;ksuchthatarebasic.Bycomplementaryslackness,vj=cij+aijkisij=ckj+akjkskjk.Since(;;v)isdualfeasible,wehavethatvjci0j+ai0ji0si0ji0foralli0.Thisproves(ii).Wewillnowfocusonproving(i).Bycomplementaryslacknessandthefeasibilityofx,foreveryj,theremustexistisuchthatvj=cij+aijisijiandbydualfeasibilitywehavevjcij+aijisijiforalli.Therefore,vj=minicij+aijisiji.Itremainstobeshownthatforj62SBx,thereisasinglefacilityithatachievesthisminimumor,equivalently,hasvj=cij+aijisiji.Sincej62SBx,thereisasingleisuchthat(i;j)2Bx.NotethatabasisofDcanbecharacterizedbyaselectionof2m+nactiveconstraints.Thechoices(i;j)2Bxandi2Mxformsuchaselectionduetothecomplementaryslacknessprinciple.SinceDisnon-degenerate,wehavethatonlythehyperplanesvj=cij+aijisijifor(i;j)2Bxandi=0fori2Mxmaybeactiveat(;;v).Therefore,ifj62SBx,thereexistsasingleisuchthatvj=cij+aijisijiwhichproves(i). ThegreedyprocedurethatwillbedevelopedinSection 4.2 willusethedualsolution(;)thatsatisesLemma 4.1.3 .Inprinciple,determiningtheoptimalsolutiontoNL-GAPRmaybeadicultproblem.Forexample,theNL-GAPRwithconcavefunctionsgi,i=1;:::;m,andnocapacityconstraintsisNP-hard.However,foracertainclassoffunctionsgi,i=1;:::;m,wewillshowthatobtaining(;)canbedonebysolvingacertainLagrangianrelaxationofNL-GAPR.Intheremainderofthissection,wewillexploretherelationshipbetweensomefundamentalconcepts(suchasKKTconditionsandLagrangianrelaxation)innonlinearprogrammingandtheproblemLNL-GAPR. 46

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wherethemaindierencebetweenNL-GAPR0andLNL-GAPRisthatSisnowavectorofdecisionvariablesinNL-GAPR0.ItisclearthattheproblemsNL-GAPRandNL-GAPR0areequivalent.Assumingthefunctionsgi,i=1;:::;m,arelocallyLipschitzcontinuous,thegeneralizedKKTconditions(seeHiriart-Urruty[ 79 ]),whicharenecessarybuttypicallynotsucientforoptimality,forNL-GAPR0are 47

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WewillnowshowanintuitivelyappealingrelationshipbetweenanysolutiontothegeneralizedKKTconditionstoNL-GAPR0andtheLNL-GAPR. 4{8 )-( 4{19 ),thenxisoptimaltotheLNL-GAPRwithvaluesSand(;;v)isoptimaltothedualofLNL-GAPR. Proof. 4{16 )-( 4{19 ),xisfeasibletotheLNL-GAPRwithvaluesS.Wewillrstshowthat(,,v)isfeasibletothedualofLNL-GAPR.RecallthatthefeasibleregionofthedualofLNL-GAPRisgivenby Bycondition( 4{8 ),wehavethatcij+aijisijivj=ij0;wheretheinequalityfollowsfromcondition( 4{10 ).Thisimpliesthat( 4{20 )issatised.Further,( 4{21 )followsfromcondition( 4{11 ).Therefore,(,,v)isdualfeasible.Ifweshowthatxand(,,v)satisfythecomplementaryslacknessconditionsappliedtoLNL-GAPR,thenourdesiredresultfollows.Notethatifxij>0,itfollowsfromcondition( 4{12 )thatij=0.Bycondition( 4{8 ),wehavethatcij+aijisijivj=0:

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4{13 )-( 4{15 ).Therefore,xsatisesthecomplementaryslacknessconditionsofLNL-GAPRwith(,,v).Thedesiredresultfollowssince(,,v)isdualfeasible. ThisresultsaysthatifwehaveasolutiontothegeneralizedKKTconditions,thentheprimalsolutionisoptimaltotheassociatedLNL-GAPR.Althoughthisisaninterestingresult,itmayonlybeusefulforthecasewhenthefunctionsgi,i=1;:::;mareconvex,sincetheKKTconditionsaresucientforoptimalitytoNL-GAPR0inthissituation. 4{4 )andthecapacityconstraints( 4{5 )fromtheNL-GAPR0.Foraxed;,wedenethefunctionL(;)tobeequaltotheoptimalsolutionvalueoftheproblemminx2X;S2RmmXi=1nXj=1cijxij+gi(Si)!+mXi=1inXj=1aijxijbi!+iSinXj=1sijxij!!:Forany2R+mand2Rm,L(;)isalowerboundontheoptimalsolutionvalueofNL-GAPR0.Notethatthisproblemdecomposesbyeachcustomerandeachfacility,i.e.,itcanbewrittenas minx2X;S2RmmXi=1nXj=1(cij+aijisiji)xij+gi(Si)+iSi+ibi!=nXj=1mini=1;:::;m(cij+aijisiji)+mXi=1ibi+mXi=1minSi2R(gi(Si)+iSi): 49

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max2R+m;2RmL(;):(4{23)TherearewellknownmethodstosolvetheLagrangiandualproblem.Thesemethodsareespeciallyattractiveiftheinnerproblem,i.e.,L(;)canbesolvedeciently.Formostfunctionsgi(forexample,convex,unimodal,orconcavefunctions),wewillbeabletosolveL(;)eciently.Forcertainclassesoffunctions,wealsohavethedesirablepropertythattheoptimalsolutionto( 4{23 )alsosatisesLemma 4.1.3 withtheoptimalsolutionxtoNL-GAPR. 4{23 ),thentheoptimaldualsolution,(;),to( 4{23 )ispartofthecomplementarydualsolutionoftheoptimalsolutionx. Proof. 18 ],wehavethat(x,S)and(;)satisfythecomplementaryslacknesswithrespecttothedualizedconstraints, 50

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(x;S)=argminx2X;S2RmmXi=1nXj=1(cij+aijisiji)xij+gi(Si)+iSi+ibi!:(4{26)Wenowmustdenethedualvariablesvtohaveacompletedualsolution(;;v).Inparticular,wedenevj=mini=1;:::;m(cij+aijisiji)forj=1;:::;n:Bythedenitionofv,thesolution(;;v)2D,i.e.,itisfeasibletothedualofLNL-GAPR.Further,ifxij>0,equations( 4{22 )and( 4{26 )implythatcij+aijisiji=minkckj+akjkskjk=vj:Therefore,foranyiandj,wehavethat 4{24 ),( 4{25 ),( 4{27 )andprimalfeasibilityofximplythecomplementaryslacknessconditionsoftheLNL-GAPRdenedwithS.Notethedualfeasibilityof(;;v)impliesthatthissolutionisthecomplementarydualsolutiontox. Thisresultimpliesthatdetermining(;)satisfyingtheconditionsofLemma 4.1.3 isequivalenttosolvingtheLagrangiandualproblemwhenthefunctionsgi,i=1;:::;m,areconvex.ThismeansthatforthisspecialclassoftheNL-GAP,wesimplyneedtofocusontheLagrangiandualproblemtoobtain(;)anddonotnecessarilyneedtodeterminetheoptimalsolutionxtotheNL-GAPR. 124 ]fortheGAP.Wedeneaweightfunction,f(i;j),representingapseudo-costofassigningcustomerjtofacilityi.Wedenethedesirabilityofacustomer 51

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124 ]):GreedyAlgorithm Step0. 4.1.3 asthegreedyprocedurefortheNL-GAP.Thereasonthatwerefertothisasa`procedure'ratherthan`heuristic'isthatdetermining(;)maybeadicult 52

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Proof. 4.1.3 .Further,foranyj02SBx,wehavethatj=0.Therefore,wewillconsiderjbeforeanyj02SBxinthegreedyalgorithm.Sincexdoesnotviolateanyofthecapacityconstraints,thepartialsolutiongivenbyxijforj2NSxwillnotviolatethecapacityconstraintseither.Thisimpliesthatwewillassignallj2NSxtoitsmostdesirablefacility,i.eij=argmini=1;:::mcijaijisiji.ByLemma 4.1.3 ,wehavethatijwillbethefacilitywhichcustomeriisassignedtoinx. IfwedeneC=max(i;j)cijandS=max(i;j)sij,wecanprovideaperformanceguaranteeonthegreedyprocedure.Thefollowingresultwillbequiteusefulinanalyzingthegreedyprocedureunderastochasticmodel. (i) (ii) Foranyfacilityi,Pnj=1sijxGij2hPnj=1sijxij2mS;Pnj=1sijxij+2mSi. Proof. 4.2.1 saysthatwecanreassignupto2mcustomersinobtainingxGfromx.Therefore,atworst,wecanreassigneachofthesecustomerstoafacilitysuchthatcij=C.Thisimplies(i).Foraparticularfacility,wecanfullyassign(orfullyremove)upto2mnewcustomersinobtainingxGfromx.Thisimpliesthat(ii)holds. 53

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123 ].Further,wewillletbidependlinearlyonn,i.e.bi=inforsomei>0,whichwasdoneinDyerandFrieze[ 46 ]andRomeijnandPiersma[ 123 ].Inotherwords,constraints( 4{1 )canbewrittenaseither 4{28 ),thecapacity(suchasstoragespaceorlaborhours)ofthefacilitygrowsasthecompanybecomeslarger(i.e.,ithasmoreandmorecustomers).Insomeways,wecanviewthisasthesizeofthefacilitiesofthecompanygrowingasthenumberofcustomersbecomeslarger.Therefore,theprocessingcapabilities(suchasthenumberofproductionlinesand/orthenumberofworkers)ofthefacilitiesalsogrowsasthenumberofjobsbecomeslarger,sothatthefacilitycanproducemoredemandforsimilarprices. 54

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4{28 )isstandardinasymptoticanalysisoftheGAPandthatnormalizingtheargumentscanbethoughtofasgeneralizingequation( 4{28 )tothenonlinearportionoftheNL-GAP.Wearenowinapositiontoexaminetheasymptoticfeasibility/optimalityofthegreedyprocedure.WesaythatthegreedyprocedureisasymptoticallyfeasibleifxGisfeasibletoNA(n)withprobability1asn!1.Wewillsaythattheprocedureisasymptoticallyoptimaliflimn!1fn(xG)fn(x)=0:Ouranalysisoftheasymptoticfeasibilityandoptimalityofthegreedyprocedurewillbeorganizedasfollows.Wewillbeginbyassumingtheasymptoticfeasibilityofthegreedyprocedure,i.e.,weassumethatthefollowingconditionholds: 4.3.1 holdsismoredicultthanprovingthatthegreedyprocedureisasymptoticallyoptimalunderCondition 4.3.1 .Therefore,wewillrstproveasymptoticoptimalityofthegreedyprocedureunderCondition 4.3.1 andthendiscusssituationsunderwhichtheconditiondoesindeedhold.OurrstpreliminaryresultestablishesthatLemma 4.2.1 canbeappliedwithprobability1. Proof. ThislemmaisusefulinprovingthegreedyprocedureisasymptoticallyoptimalunderCondition 4.3.1 4.3.1 andcontinuousfunctionsgi,i=1;:::;m,thegreedyprocedureisasymptoticallyoptimal.

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4.2.2 and 4.3.2 ,weknowthatfn(xG)2mCU n;2mSU n]giz+1 n;2mSU n]giz+1 105 ]).Thisimpliesthatforany>0,thereexistssomen(;i)suchthatforeverynn(;i)and&2[0;SU]thatmaxz2[2mSU n;2mSU n]gi(z+&)gi(&)<:Therefore,forany>0,thereexistssomen()=maxi=1;:::;mn(;i)suchthatforeverynn(),&2[0;SU],andi=1;:::;m,maxz2[2mSU n;2mSU n]gi(z+&)gi(&)<:Thisimpliesthatforany>0,thereexistsn0suchthatforeverynn0,fn(xG)fn(x)2mCU n;2mSU n]giz+1 4.3.1 holds.RomeijnandPiersma[ 123 ]showthattheproblemsNA(n)andNAR(n)arenotnecessarilyfeasibleforthestochasticmodel 56

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123 ])Asn!1,NA(n)andNAR(n)arefeasiblewithprobability1if>0andisinfeasiblewithprobability1if<0.Therefore,inourprobabilisticanalysisofNA(n),itisappropriatetoassumethatAssumption 4.3.4 holds.Inthefollowingsections,weanalyzethefeasibilityoftheprocedureundertwoseparatemodels,facility-independentparametersandfacility-dependentparameters. 4.3.3 provesthatthegreedyprocedurewith(;)=(;)isasymptoticallyfeasibleandoptimal.WerstbeginwithapreliminaryresultthatdealswiththeunusedcapacityofanyfeasiblesolutiontoNAR(n). Proof. 123 ]showthatAssumption 4.3.4 isequivalenttothecondition 57

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4{29 )andtheCentralLimitTheorem. WecannowproveourmainresultaboutthegreedyprocedureappliedtothestochasticmodeloftheNL-GAP,NA(n),withfacility-independentparameters. Proof. 4.3.2 showsthatDisnon-degenerate.Therefore,Lemma 4.2.1 holdsandweonlyreassignsplitbasicjobstoobtainxGfromx.Thenumberofthesereassignments,i.e.,thenumberofsplitbasicjobs,isindependentofnsinceLemma 4.1.2 showsthatjSBxj2m.ByLemma 4.3.6 ,theunusedaggregatecapacityofnon-splitfacilitiesgrowslinearlyinnwithprobability1asn!1andtherefore,growssucientlylargetoaccommodatethereassignmentsthatneedtobemadetoobtainxG(sincethisnumberisindependentofn).Therefore,thegreedyprocedureisasymptoticallyfeasibleandCondition 4.3.1 holds.Theorem 4.3.3 showsthatthegreedyprocedureisalsooptimal. 4.3.1 sincewe 58

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4.2 ,wewillexploreanalternativeprocedurethatisbasedonadierentsetofdualmultipliers.Todoso,weexamineaproblemwhereweslightlyperturbthecapacityconstraints,minimizefn(x)=mXi=11 (i) limn!1n=0, (ii) limn!1nn=1,and (iii) 0
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n(;n)=minx2XmXi=1nXj=11 andXisthesetofcontinuousassignmentconstraints.Wewillshowthatwemayrestrictourselvestosolutionsthatlieinacompactsetinmaximizingn(;n).First,notethatwehave max0n(;)n(0;)CL+mXi=1minSi2[SL;SU]gi(Si):(4{31)Further,forany,wehavethat n(;)=minx2XmXi=1nXj=11 withprobability1asn!1byRomeijnandPiersma[ 123 ].Therefore,forany,( 4{31 )and( 4{32 )implythatCL+mXi=1minSi2[SL;SU]gi(Si)CU+mXi=1maxSi2[SL;SU]gi(Si)+()mXi=1i:

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;withprobability1asn!1.Sincen(;n)n(;),thisfunctionalsoachievesitsmaximumonthesamecompactset.Therefore,wehavethat n(;n)n(;0)+n(4{33)withprobability1asn!1.Thisimpliesthat,fn(x)fn(x(n))=max0n(;n)max0n(;0)+n=fn(x)+n!fn(x)asn!1: Proof. 4.2.1 statesthatonlysplitjobsinx(n)arereassignedbythegreedyproceduretoobtainxG(n).SincetheamountofadditionalcapacityintheproblemNA(n)overtheamountinNAR(n,n)goestoinnity(i.e.,limn!1nn)andthenumberofreassignmentsisindependentofn(seeLemma 4.1.2 ),xG(n)isfeasibletoNA(n)withprobability1asngoestoinnity.ThisimpliesthatxG(n)satisesCondition 4.3.1 .ByTheorem 4.3.3 andLemma 4.3.8 ,wehavethat,withprobability1asn!1,limn!1fn(xG(n))=limn!1fn(x(n))=limn!1fn(x):Therefore,xG(n)isasymptoticallyoptimaltoNA(n). 61

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4.3.2 canbegeneralizedtootherclassesofnonlinearfunctions.Wearemainlyfocusingonthecasewhenthefunctionsgi,i=1;:::;m,arenon-decreasingsinceinalltheapplicationsthatserveasmotivationforstudyingtheNL-GAP,thefunctionsarenon-decreasing.ItalsomaybeinterestingtoexaminethepropertiesofthegreedyheuristicifweweretousetheoptimalsolutiontotheLagrangiandualproblemstudiedinSection 4.1.2 .ThisisduetothefactthatwemayeectivelydeterminetheoptimalsolutiontotheLagrangiandualproblem,sincethesubproblemsforaxeddualsolutioncanbedeterminedeciently. 62

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5.5.1 toallowfornon-dierentiableconvexfunctions).Inaddition,weassumethatthefeasibleregionofCPisnonemptyandbounded(wewillrelaxtheboundedassumptioninSection 5.5.2 ),sothatanoptimalsolutiontoCPexists.Forconvenience,wedenotethefeasibleregionofCPbyP.WewilldeveloptwosolutionmethodsthatsolveCPthroughsolvingasequenceofeither1-dimensionalorK-dimensionalconvexprogrammingproblems.ThelatteristhereforeparticularlyattractiveifKn.Themethodsthatwedeveloparepartiallyinspiredbythesimplexmethodsforlinearandconvexprogramming,whereweemploythefactthatwecanshowthatCPhasanoptimalsolutionthatliesonafaceofPofdimensionK.Notethatthisgeneralizesthewell-knownresultthatalinearprogrammingproblemhasanextremepointoptimalsolution(providedanoptimalsolutionexists).OurproposedalgorithmsgenerateasequenceofsolutionsonfacesofPofdimensionnomorethanK.Zangwill[ 160 ]proposedaso-calledconvexsimplexmethod(CSM)thatcan,inprinciple,beappliedtoCP.However,thatalgorithmisonlyguaranteedtoasymptoticallyconvergeinvaluetotheoptimalsolutionvalue.Ofcourse,anyotheralgorithmdeveloped 63

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13 ]orBertsekas[ 19 ])aswellasinteriorpointmethodsfornonlinearprogramming(seeForsgenetal.[ 56 ]forareview).However,sincethesemethodsareforgeneralproblems,theywillnottakeadvantageofthespecialstructureofCPandsome,liketheCSM,onlyguaranteeasymptoticconvergence.Itisourgoaltodevelopanalgorithmthatachievesconvergenceinanitenumberofiterationsbyfullyemployingthestructureofourproblem.OurmotivationforstudyingCPstemsfromitsmanypracticalapplications,inparticularinsupplychainoptimization,aseitheraproblemofindependentinterestorasubprobleminanalgorithmtosolvealargerproblem.TherelaxationoftheNL-GAPwithconvexfunctionsgicanbeformulatedasaCP.Inparticular,wedenetheconvexfunctionginCPasg(Sx)=mXi=1ginXj=1sijxij!:Asanotherexample,consideramarketselectionproblemwhereacompanycanchoosetoservethedemandforaproductineachofnmarkets.Theobjectivefunctionparametercithencorrespondstothenegativeoftherevenueandsitothedemandrateassociatedwithmarketi,whilexiindicatesthefractionofmarketithatthecompanychoosestoserve.Thefunctiong(s>x)wouldrepresentthecostofacquiring(orproducing)s>xunitsofdemand.Undercertaincostandcapacitystructures,thefunctiongisconvex(see,forexample,Frelingetal.[ 57 ]).ThisproblemthenbelongstotheclassCPwithK=1,wheretheobjectivefunctionoftheproblemisc>x+g(s>x),i.e.,weareminimizingtotalnetcost.ThismodelcanbeextendedtoasettingwhereKproductsareoeredandthedemandrateinmarketiforproductkissik.Sharkeyetal.[ 140 ]discussthisapplicationandextensionsthereofinmoredetail.Theremainderofthischapterisorganizedasfollows.InSection 5.1 ,weformallyprovethatthereexistsanoptimalsolutiontoCPonafaceofPofdimensionKand 64

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5.2.1 ,wedescribeanalgorithmwithsimplepivots(i.e,pivotsbasedonasinglenonbasicvariable)tosolveCP.InSection 5.2.2 ,weprovethecorrectnessofthisalgorithm.Inparticular,itisshownthatforK=1,thisalgorithmterminatesinanitenumberofiterations.ForgeneralK,itisshownthatthealgorithmasymptoticallyconvergesincosttotheoptimalsolutionvalue.InSection 5.3 ,weuseaninterestinginsightintothealgorithmwithsimplepivotsforK=1todevelopanalgorithmwithgeneralizedpivots(i.e.,pivotsbasedonacollectionofnonbasicvariables).ItisshownthatthealgorithmwithgeneralizedpivotssolvesCPforgeneralKinanitenumberofiterations.WeperformacomputationalstudyofthealgorithmsinSection 5.4 ,anddiscusssomeextensionsofouralgorithmsandresultsinSection 5.5 Proof. 3.0.1 saysthatthereisanoptimalsolution,xtoCPthatisalsoanextremepointofthelinearprogram:minimizec>xsubjectto(LCP)Sx=SxAx=bx0:

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Thisresultmotivatesouralgorithms,whichwillrestrictattentiontosolutionstoCPonfacesofPofdimensionK.Clearly,ifthenonlinearfunctiongwereabsenttheproblemsCPandLCPwouldbeidenticalandwecouldapplythestandardsimplexmethodforlinearprogrammingtoCP,whichwouldmovebetweenbasicfeasiblesolutionsalongedgesofthefeasibleregion.Inordertodevelopouralgorithm,weneedtoestablishacharacterizationofallsolutionsonfacesofPofdimensionK.Tothisend,considerthefollowingreformulationofCP:minimizec>x+g(y)subjectto(ACP)Sxy=0Ax=bx0:Notethatapointx2PisonafaceofdimensionKifandonlyifnomorethanm+Kofitselementsarestrictlypositive.WewillusethispropertytocharacterizesuchsolutionsintermsofthefeasibleregionofACP,whichwewilldenotebyQ.Formally,let(xB;yB)denoteasubvectorof(x;y)withexactlym+Kelements(wherexBoryBmaybeanemptyvector) 66

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67

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5.2.1IntuitiveDevelopmentandDescriptionoftheAlgorithmItisstandardinsimplex-typealgorithmstoexpressthebasicvariablesintermsofthenonbasicvariables: NotethateachcolumnofB1Ncanbeusedtoconstructasearchdirection,eachofwhichcorrespondstoaparticularnonbasicvariable.Inparticular,thissearchdirectionisobtainedbyaddingzeroelementscorrespondingtoallnonbasicvariablesexceptfortheoneassociatedwiththecolumn,whichissettoone.Letd=(dx;dy)denoteasearchdirectioncorrespondingtoaparticularnonbasicvariable(wheredxanddydenotethecomponentscorrespondingtoxandy,respectively).Notethat,fornonbasicvariablesiny,wewillexplicitlyconsidertwodistinctsearchdirections,associatedwithincreasinganddecreasingthatvariable.Wethenconsideraone-dimensionalconvexoptimizationproblemofthefollowingform: minimize2c>(x+dx)+g(y+dy)(5{3)wheredenotesthequantitybywhichtheassociatednonbasicvariableischangedandthefeasiblerangeofvaluesforisimpliedbythenonnegativityconstraints:=f2R: 68

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5{3 )asafunctionofisnegativeatthepoint=0.Nowconsidertheoptimizationproblem( 5{3 )correspondingtoaneligiblenonbasicvariable(anddirectionofchange).(Notethathereandintheremainderofthedevelopmentofouralgorithmwedonotlimitourselvestoaspecicpivotrule.)Ifthesolutioncorrespondingtothenonzeroboundarypointofisoptimal,thenfurtherimprovementoftheobjectivefunctioninthedirectiondispreventedbythenonnegativityconstraints.Wethenpivottheselectednonbasicvariableintothebasisandreplaceitby(oneof)thebasicvariable(s)inxBthatreachedthevaluezero.Alternatively,wemayhaveaninterioroptimalsolutiontoproblem( 5{3 ).WethendistinguishbetweenthecasewheretheselectedeligiblenonbasicvariableisinxNorinyN.Intheformercase,wepivottheselectednonbasicvariableintothebasis,replacingavariableinyB(whichisguaranteedtoexist,sinceotherwisetheinteriorpointwouldnotbeoptimal).Inthelattercase,wecouldleavethebasisunchangedandmerelychangethevalueoftheselectednonbasicvariableinyN.However,wemayalternativelyreplaceavariableinyBbypivotingtheselectednonbasicvariableinyNintothebasis,providedsuchapivotexists.Wewillnowformallystateourgeneralalgorithmwithsimplepivots.Simplexalgorithmwithsimplepivots Step0. 69

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5{3 ).Performapivotontheselectedeligiblenonbasicvariableandwritethenewsetofbasicvariablesintermsofthenewsetofnonbasicvariables.ReturntoStep1.Clearly,tocompletelyspecifythealgorithmitisnecessarytoprovideapivotrule,inaccordancewiththerestrictionsdependingonthesolutionof( 5{3 ),thatdetermines(i)whicheligiblevariablewillbepivotedoninStep1and(ii)whichbasicvariablewillbepivotedoutofthebasis(ifany)inStep2.WewillnowbrieycompareouralgorithmtotheCSMofZangwill[ 160 ].TheCSMminimizesaconvexobjectivefunctionofnvariablesoverapolyhedrondenedbymequalityandnnonnegativityconstraints.Notethat,withoutanyadditionalstructureintheobjectivefunctionbesidesconvexity,allnvariablescouldpotentiallybenonzerointheoptimalsolution.TheCSMmethod,similarlytoouralgorithm,decomposesthevectorofdecisionvariablesintoacollectionofmbasicandnmnonbasicvariables.Ineachiteration,aone-dimensionaloptimizationproblemisdenedbasedonpivotingonaneligiblenonbasicvariable.However,thiseligiblenonbasicvariableonlyentersthebasisifthenonnegativityconstraintononeofthebasicvariablesisbindingintheone-dimensionaloptimizationproblem;thebasicvariablethatreachesthevaluezerothenbecomesanonbasicvariable.ItisclearthattheCSMcouldbeappliedtoCPortoamodicationofACPinwhichthevariablesyarereplacedbynonnegativevariablesrepresentingtheirpositiveandnegativeparts.Alternatively,aslightmodicationoftheCSMcouldbeapplieddirectlytoACP,wheretheunconstrainedvariablesycouldbeeitherbasicornonbasicbut,perhapsinterestingly,wouldneverleavethebasisiftheyeverenterit.Itiseasytoseethat,ingeneral,thesequenceofiterationpointsgeneratedbytheCSMcouldbeverydierentfromthesequenceofiterationpointsgeneratedbyouralgorithm,particularlyifK
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2(x1x2)2subjectto(E1)x1+x3=1x2+x4=1x1;x2;x3;x40where1=issomepositiveinteger.NowapplytheCSMfromthestartingpoint(0;0;1;1).Inthisstartingpoint,variablesx1andx2arenonbasicandx3andx4arebasic.x1isaneligiblenonbasicvariable,theassociateddirectionis(1;0;1;0),andthecorrespondingone-dimensionaloptimizationproblemreadsminimize01+1 22whoseoptimalsolutionisgivenby=.Thenextiterateis(;0;1;1)withthebasisunchanged.Thenx2isaneligiblenonbasicvariable,theassociateddirectionis(0;1;0;1),andthecorrespondingone-dimensionaloptimizationproblemreadsminimize01+1 2()2whoseoptimalsolutionisgivenby=.Thenextiterateisthen(;;1;1)withthebasisagainunchanged.ItiseasytoseethattheCSMcontinuesbyalternatingx1andx2aseligiblenonbasicvariables,sothatthealgorithmalternatesbetweensolutionsoftheform(k;k;1k;1k)and((k+1);k;1(k+1);1k)untiltheoptimalsolution(1;1;0;0)isreachedafter2=iterations.Incontrast,ouralgorithmproceedsbyrstreformulatingtheproblemas:minimizex1+1 2y21

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22whoseoptimalsolutionisgivenby=1.Thenextiterateis(1;1;0;;)andthevariablex3inthebasisisreplacedbyx2.Theny1iseligibletoenterthebasiswithassociateddirection(0;1;0;1;1)anditiseasytoseethatthenextiterateistheoptimalsolution(1;1;0;0;0),reachedafter3iterations.Figure 5-1 illustratesthepathsthattheCSMandouralgorithmwithsimplepivotstakeinsolvingE1.TheaboveexampledoesnotonlyshowthatthesequenceofpointsvisitedbyouralgorithmmaybeverydierentfromthesequenceofpointsvisitedbytheCSM,butalsothattheCSMmayvisitmanymorepointsthanouralgorithm.Inparticular,fortheclassofproblemsabove,theratiobetweenthenumberofiterationsrequiredbytheCSMandthenumberofiterationsrequiredbyouralgorithmcanbearbitrarilylarge,dependingonthevalueoftheconstant.InSection 5.2.2.1 wewillexplainthisbehaviorbyshowing 72

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Anillustrationofthepathsin(x1;x2)-spacefollowedby(a)theCSMand(b)ouralgorithmwhensolvingE1. that,whenK=1,themaximumnumberofiterationsrequiredbyouralgorithmdependsonlyonthestructureofP,whiletheexampleaboveclearlydemonstratesthatthenumberofiterationsrequiredbytheCSMmaydependonthenonlinearcomponentgoftheobjectivefunction.Beforewereturntothisissue,however,wewillrstprovethecorrectnessofouralgorithm. 160 ]. 73

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Proof. 74

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5{5 )and( 5{7 )aresatised.Inaddition,weletvji=0fori2Nxandvjn+k=0fork2Ny.Fori2Nx,wethenhavethatthesearchdirectioncorrespondingtoji,(dx;dy),hasthat 5{4 ),( 5{9 ),( 5{10 ),andthefactthattheredoesnotexistaneligiblenonbasicvariablethatconditions( 5{6 )and( 5{8 )aresatised,whichyieldsthedesiredresult.

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5.2.1 .Thisimpliesthatthealgorithmterminateswithanoptimalsolution. AsabyproductoftheproofofTheorem 5.2.2 weobtainthatthenumberofiterationsthatouralgorithmrequiresforK=1isboundedbyanumberthatonlydependsonthestructureofthepolyhedronPandnotonthenonlinearcomponentg(Sx)oftheobjectivefunctionofCP.ThisresulthighlightsanimportantadvantageofourmethodovertheCSMwhenK=1.

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Nowletusconsiderthefollowinggeneralizationoftheclassofproblems(E1):minimizex1+g(x1x2)subjectto(E01)x1+x3=1x2+x4=1x1;x2;x3;x40 77

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2y2. 20 ]),i.e.,morethanonebasicvariablereachzerointhepivot,thebasicvariableamongthesewhoserowinthesimplextableau(afterscaling 78

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20 ]).Nownote,however,thatastraightforwardimplementationofthislexicographicpivotrulewillnotworkforouralgorithm,sincewedonotwanttopreventrepeatingabasisaslongasthecorrespondingvectoryNisdistinct.Inotherwords,weareonlyinterestedinmakingsurethatwedonotrepeatabasisalongwiththesamecorrespondingvalueofyN.Wethereforemodifythelexicographicpivotruletoasfollows:wheneverwearriveatabasisafterperforminganondegeneratepivot,wereindexthevariablessothatthem+Kbasicvariablesappearbeforethenonbasicvariables;wethenapplythelexicographicpivotruleasdescribedabove.Wewillrefertothisruleasthemodiedlexicographicrule. Proof. 79

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Thislemmaisimportantbecauseithelpsshowthatifthealgorithmgeneratesaninnitesequenceofsolutions,thennoneoftheiterationpointsofthealgorithmcanbeidenticaltoaclusterpointofthesequenceofgeneratedsolutions.Thisinfactismadeformalandusedinprovingthenextresult. Proof. 80

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limsupi!1c>dx;ki+rg(yki)>dy;ki<0(5{11)(seeBertsekas[ 18 ]).Wethenusethisresult,andtheno-cyclingresultfromLemma 5.2.4 ,toshowthatanyclusterpointofthesequenceofsolutionsgeneratedbythealgorithmisanoptimalsolutiontoACP.Itiseasytoseethat,sincethereareonlyanitenumberofcandidatedirections,thesequenceofdirectionsisbounded.Nowconsiderasequencef(xki;yki)g1i=1thatconvergestoanonstationarypoint(x;y).Inordertoshow( 5{11 ),wewillbreakthesequencef(xki;yki)g1i=1intoanitesetofinnitesequencesasfollows.Wedecomposethesequencef(xki;yki)g1i=1intodistinctsubsequences,eachofwhichcorrespondstosolutionswithanidenticalbasisandanidenticalnonbasicvariableandsignselectedforentryintothebasisinthenextiteration(i.e.,whenanelementofyisthenonbasicenteringvariable,wealsospecifywhetherthatvariableisincreasedordecreased).Sincethereareonlyanitenumberofbasesandanitenumberofvariables,theremustbeatleastonechoiceofbasis,nonbasicenteringvariable,andsignwhosecorrespondingsubsequenceisinnite;denotetheindicesofthissubsequencebyfk0ig1i=1.Itiseasytoseethat(dx;k0i;dy;k0i)isconstantfori=1;2;:::,say(dx;dy).Since(x;y)isanonstationarypoint,thereexistsaneligiblenonbasicvariable,i.e.,Theorem 5.2.1 saysthatthereexistssome(dx;dy)suchthatc>dx+rg(y)>dy<0:However,notethatc>dx+rg(yk0i)dyc>dx+rg(yk0i)>dyforalli=1;2;:::sinceweusethesteepestdescentpivotrule.Nowrecallingthat(dx;k0i;dy;k0i)=(dx;dy)fori=1;2;:::andusingthatthepartialderivativesofgare 81

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5{11 )holdsandthusthesequenceofdirectionsgeneratedbyouralgorithmisgradientrelated.Sincenoiterationofthealgorithmcanmovetoapointwithaworseobjectivefunctionvalue,thesequenceofgeneratedsolutionvalues(c>xk+g(yk))1k=1isnon-increasing.ByLemma 5.2.4 ,thealgorithmdoesnotcycle,sothatnoneofthesolutions(xk;yk)canitselfbeaclusterpointofthesequenceofsolutions(xk;yk)1k=1.Let(xk0i0;yk0i0)1i=1betheinnitesequenceofsolutionsfromwhichanondegeneratepivotwasperformed,i.e.,solutionsiniterationsksuchthat(xk;yk)6=(xk+1;yk+1).BythefactthatPisbounded,thereexistsaniteupperboundonthelengthofanylinesegmentcontainedinP.Moreover,thereareonlyanitenumberofpotentialsearchdirections,sothatthesetsin( 5{3 )areuniformlyboundedoverallnondegeneratepivotoperations.Thisresult,togetherwiththefactthatthesequenceofdirectionsisgradientrelated,impliesbyProposition2.2.1inBertsekas[ 18 ]thatanyclusterpointofthesubsequenceofsolutions(xk0i0;yk0i0)1i=1generatedbyouralgorithmisastationarypointofACP.SincetheobjectivefunctionofACPisconvex,thisimpliesthatanyclusterpointof(xk0i0;yk0i0)1i=1isanoptimalsolutiontoACP.Thedesiredresultfollowsbyrealizingthatthesetofclusterpointsof 82

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Wearenowreadytoproveourmainconvergenceresult. Proof. 5.2.1 saysthatthissolutionisanoptimalsolutiontoACP.Nowsupposethatthealgorithmdoesnotterminatenitely.SincePisbounded,thesequenceofsolutionsgeneratedbythealgorithmmusthaveaclusterpoint.Lemma 5.2.5 saysthatthisclusterpointisoptimaltoACP.Sinceouralgorithmgeneratesasequenceofsolutionsthatisnon-increasingwithrespecttotheobjectivefunction,thisimpliesthatthecostsoftheentiresequenceofsolutionsconvergestotheoptimalcosttoACP. Thisimmediatelyleadstothefollowingresult: 83

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2k;11 2k1;1 2k;1 2k1;11 2k;11 2k1;1 2k(5{12)wherex1,x2,x3,x4,andy3arebasic.Thedirectionimpliedbyraisingy2isd=(0;1;0;1;0;1;1),whichiseligible.Inordertodeterminethestepsize,wesolvetheproblemminimize01 2k11 2k12++1 2k2whoseoptimalsolutionis=3 2k+1.Wethenarriveatthesolution11 2k;11 2k+1;1 2k;1 2k+1;11 2k;11 2k+1;1 2k+1

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2k1 2k2+1 2k+12:Theoptimalsolutiontothisproblemis=3 2k+2.Wethenarriveatasolutionoftheform( 5{12 )withkreplacedbyk+2.Therefore,althoughthesequenceofsolutionsgeneratedbythealgorithmapproachestheoptimalsolution(1;1;0;0;1;1;0),thissolutionisnotreachedinanitenumberofiterations.ItisinterestingthatouralgorithmwithsimplepivotsisguaranteedtogenerateonlyanitenumberofsolutionsifK=1,butmayingeneralrequireaninnitenumberofiterations.AcloselookattheconvergenceproofforK=1revealsthatanimportantpropertyofouralgorithmforthatcaseisthatweareguaranteedto,afteranitenumberofiterations,optimizetheobjectiveoveranedgeofP,i.e.,overafaceofPofdimensionK=1.Thissuggeststhatifwe,insteadofoptimizingovera1-dimensionalsubsetofPineachiteration,optimizeoverasubsetofdimensionK(orlower),wemaybeabletoconstructanalgorithmthatterminatesinanitenumberofiterations.InSection 5.3 weshowthatthisisindeedthecase. 5{3 ),i.e.,aone-dimensionalconvexoptimizationproblemoveraboundedinterval,thismodiedalgorithmwillrelyontheabilitytoecientlysolveaconvexoptimizationproblemoverapolytopeofdimensionatmostK.Thisalgorithmcanbeexpectedtobeparticularly 85

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5{3 )andperformapivotoperation.However,iftheselectednonbasicvariableisinyN,wewillactuallyconsiderchangingallelementsofyN.Putdierently,ratherthanviewingeachelementofyNindividually,weconsiderthevectoryNinitsentirety.Thissubproblemthen,infact,optimizestheobjectivefunctionofACPoverthefaceofthepolytopePthatischaracterizedbyxN=0.Moreformally,thissubproblemcanbewrittenas minimize2c>(x+Dx)+g(y+Dy):(5{13)Heretheelementsofthevectorarethecoecientsofthecorrespondingsearchdirections,i.e.,representsthechangeinthenonbasicvariablesinyN.Moreover,therowsofDxandDycorrespondingtothebasicvariablesxBandyB,respectively,arethepartialrowsofthematrixB1Nin( 5{2 )thatcharacterizehowthebasicvariableschangewhenthevectorofnonbasicvariablesinyNchanges.TheremainingrowsofDxareequaltozero(sincenoelementofxNchangesduringthepivot)whiletheremainingrowsofDyareunitvectors.Finally,theset=f2Rk:x+Dx0g(wherekisthedimensionofyNor)characterizesthesetoffeasiblesearchdirectionmultipliers.Aftersolvingthissubproblem,weproceedinasimilarfashionasinouralgorithmwithsimplepivots.Thatis,iftheoptimalsolutionofthesubproblemliesontheboundaryofthesetwepivotoneofthevariablesinyNintothebasis,removingoneofthevariablesinxBthatreachedthevalue0.Ontheotherhand,iftheoptimalsolutionofthesubproblemliesintheinteriorof 86

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Step0. 5{3 ).Performapivotontheselectedeligiblenonbasicvariableandwritethenewsetofbasicvariablesintermsofthenewsetofnonbasicvariables.ReturntoStep1. 5{13 ).PerformapivotonthenonbasicsubvectoryNandwritethenewsetofbasicvariablesintermsofthenewsetofnonbasicvariables.ReturntoStep1.Inordertocompletelyspecifythealgorithm,itisnecessarytoprovideapivotruleusedinStep1todeterminetheeligiblenonbasicvariableandinStep2todeterminethebasicvariableleavingthebasis.Asthefollowingtheoremshows,thisalgorithmenjoysasimilarconvergenceresultforgeneralKasthealgorithmwithsimplepivotsdoesforthecaseK=1. Proof. 5.2.2 .First,wewillshowthatouralgorithmwillonlyperformanitenumberofpivot 87

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5.2.2 Theorem 5.2.2 provesthatouralgorithmwithsimplepivotswhenappliedtoproblemswithK=1terminatesinanitenumberofiterationsaslongasweuseananti-cyclingrule.ToestablishananalogousresultforgeneralKandforouralgorithmwithgeneralizedpivotsinTheorem 5.3.1 ,weneedto,inaddition,alwaysgivepreferencetoyNinordertoguaranteethatthealgorithmwithgeneralizedpivotsterminatesinanitenumberofiterations.ThereasonforthisdierenceagainhighlightsthespecialpropertiesofthealgorithmforK=1.InthecaseofK=1,weoptimizeoveranedgeofPinanyiterationinwhicheitheryisselectedastheeligiblenonbasicvariableoryisbasic.However,inouralgorithmwithgeneralizedpivotsweonlyoptimizeoverafaceofPinaniteration 88

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23 ]andSmale[ 146 ]).Thegoalofourcomputationaltestsinthissectionistoevaluatewhetherthealgorithmsdevelopedinthischapterinherittheeciencyinpracticeofthesimplexmethodforlinearprogramming.Inthisstudy,wewillfocusonexaminingtheaveragenumberofsolutionsthatthealgorithmsgenerateinsolvingthreedierentclassesofproblems.Inparticular,theproblemsthatwewillconsiderareoftheformminimizec>x+g(Sx)subjectto(TP)Ax+z=bx1x;z0:WehaveimplementedaslightmodicationofouralgorithmstosolvethereformulationATP,whichisobtainedfromTPinthesamewayasACPisobtainedfromCP.Inparticular,wehavechosentohandletheupperboundsimplicitly(asinthesimplexmethodforlinearprogramming).WeuseBland'sruleasthepivotselectionrule,wherethevariablesinyareviewedashavingsmallerindicesthanthevariablesinxandz.We 89

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140 ]orChapter 6 )thatexploitthestructureofCPwithK=1andarethuscompetitorsofthealgorithmwithsimplepivotsdevelopedinthischapter.ForK=1,weconsidered3dierentclassesofconvexfunctions,eachparameterizedbyasingleparameterforwhichwechose3dierentvalues,foratotalof9classesofproblems.Foreachclassofproblems,werandomlygenerated25instancesoftheproblemform=1;2;5;10;20;50;100andn=50;100;200;500;1000.(Thisisconsistentwithstandardlinearknapsackproblems,wherethenumberofitemsnistypically(much)largerthanthenumberofknapsackconstraintsm.)Wethenappliedthealgorithmwithsimplepivotstoeachinstanceand,foreachclassofproblemsandproblemsizen,werecordedtheaveragenumberofsolutionsgeneratedtosolvethe25instances.Totestthealgorithmwithgeneralizedpivots,wealsoconsideredvaluesofK=2;5;10.InChapter 6 ,weconsiderageneralproblemclassoftheformCPwheregisnotrequiredtobeconvexandK=1.Wedevelopanalgorithmthathasbothaworst-caseandanaverage-caserunningtimeofO(n2maxflogn;g)form=1andO(nm+1maxflogn;m3;S(n;m);g)form2,whereisthetimerequiredtosolveaone-dimensionaloptimizationproblemoftheform( 5{3 )andS(n;m)isthetimerequiredtosolvealinearprogramwithnvariablesandmconstraints.IfweletC(n)betheaver-agenumberofsolutionsthatouralgorithmwithsimplepivotsgeneratesforK=1,then 90

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5-2 and 5-3 .TheR2valuesoftheregressionswere0.999and0.998,respectively,sotheseresultssuggestthatC(n)=O(nlogn)fortheseclassesoffunctions.Forthethirdclassoffunctions,weobservedforthisclassofproblemsthattheaveragecasebehaviorofouralgorithmdoesdependon,despitethefactthatCorollary 5.2.3 statesthattheworst-casenumberofiterationsrequiredbythemethodshouldbeindependentofg(andtherefore).Basedonavisualinspectionofthedata,whichsuggestedthattherateofincreaseintheaveragenumberofiterationsasafunctionofnisslowerthanlinear,weperformedregressionoftheformC(n)=0n1(or,equivalently,logC(n)=log0+1logn)foreachindividualvalueof.Thescatterplotoftheobservations(n;C(n))foreachandtheresultoftheregressionappearinFigure 91

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.TheR2valueoftheregressionswerealllargerthan0.99.Moreover,inallcasestheestimateof1wassignicantlysmallerthan1,suggestingthatC(n)=O(n)forthisclassoffunctions. Figure5-2. Scatterplotandbesttfunctionforg(y)=y2. Figure5-3. Scatterplotandbesttfunctionforg(y)=ln(+y=). Wenextconsideredproblemswithmultipleconstraints,inparticular,m=2,5,10,20,50,and100.Verysurprisingly,weconsistentlyobservedthatthenumberofiterationsrequireddecreasedasmwasincreased.Figure 5-5 illustratesthisbehaviorforthefunctiong(y)=20y2class.Foreachvalueofm,weperformedaregressionoftheformC(n)=nlogn.TheR2valuesofeachoftheseregressionsweregreaterthan:99.ThissuggeststhatC(n)=O(nlogn),i.e.,therateofgrowthofthenumberiterationsisupperboundedbyafunctionthatisindependentofm,forthisclassoffunctionsand,therefore, 92

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Scatterplotsandbesttfunctionsforg(y)=ey=for=10(solidline),=50(dottedline),and=100(dashedline). hasamuchimprovedaverage-timecomplexityoverthealgorithmofSharkeyetal.[ 140 ]forproblemswithm2.Foreachoftheotherclassesoffunctionstestedform=1,similarresultswereobservedastheclassg(y)=20y2whenwevariedm. Figure5-5. Scatterplotandbesttfunctionsforg(y)=20y2andm=1(solidline),m=2(blackdottedline),m=5(blackdashedline),m=10(blackdotted/dashedline),m=20(greydottedline),m=50(greydashedline),andm=100(greydotted/dashedline). Finally,westudiedtheaveragecaseperformanceofthealgorithmwithgeneralizedpivots.Werstexaminedthealgorithmwithgeneralizedpivotsunderasinglelinearconstraint,i.e.,m=1,alongwiththeupperboundsonthevariables.Weperformed 93

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5-6 .Forthesecondandthirdclassesoftestproblemsweincludedallparis(n;K)inthelogregression.Forthelogarithmicfunction,theR2valuewasover:97andindicatedthatC(n;K)=O(n1:26K:39).ThescatterplotandbesttfunctionforthelogarithmicfunctionappearinFigure 5-7 .Fortheexponentialfunction,theR2valueoftheregressionwasover:96andindicatedthatC(n;K)=O(n1:18K1:45).ThescatterplotandbesttfunctionfortheexponentialfunctionappearinFigure 5-8 .BasedonthestandarderrorsoftheestimatedparameterswesafelyconcludethatthetestresultssupportthatC(n;K)=O(n2K2)forthealgorithmwithgeneralizedpivotsonproblemswithm=1;infact,forparticularclassesofproblemsourresultsindicatethatamuchstrongerresultmayinfacthold.Finally,anexaminationofresultsforthealgorithmwithgeneralizedpivots 94

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Figure5-6. Scatterplotsandbesttfunctionforgk(y)=20y2andK=2(dottedline),K=5(dashedline)andK=10(dotted/dashedline). Figure5-7. Scatterplotsandbesttfunctionforgk(y)=ln(+y 95

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Scatterplotsandbesttfunctionforgk(y)=ey=100andK=1(solidline),K=2(dottedline),K=5(dashedline)andK=10(dotted/dashedline). 5.5.1Non-DierentiableObjectiveFunctionWehavethusfarassumedthatthefunctiongisconvexwithcontinuouspartialderivatives.Inmanycases,nondierentiabilitycanbedealtwithbyintroducingadditionaldecisionvariablesandconstraints.Forexample,inthecaseofK=1,ifgispiecewiselinearCPcanbereformulatedasapurelinearprogram.However,ingeneralthismayleadtoalargeincreaseinthedimensionalityoftheproblem.Therefore,wewillshowinthissectionthatwecanextendouralgorithmwithgeneralizedpivotstothecasewherethefunctiongiscontinuousbutnotnecessarilydierentiable.Themainmodicationthatisrequiredisthedenitionofthetermeligiblenonbasicvariable.Tothisend,wewilllet@g(y)denotetheusualsetofsubgradientsofgaty.Furthermore,wedenethesetofdirectionalsubgradientsofgatasolutionyinthedirectiond6=0tobe:@dg(y)=24limh"0gy+hd

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5{4 ).Notethat,asbefore,weshouldactuallyconsidertwoopposingsearchdirectionsfornonbasicvariablesinyN.WenowextendTheorem 5.2.1 tothenon-dierentiablecase. Proof. 5.2.1 byemployingthegeneralizedKKTconditions(seeHiriart-Urruty[ 79 ]).Inparticular,thesegeneralizedconditionsreplaceequations( 5{7 )and( 5{8 )byk=vjn+kfork2Byk=v>B1Njn+kfork2Ny02@g(y)+: 5.2.2 andTheorem 5.3.1 ,invokingTheorem 5.5.1 wherenecessary. 97

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5{3 )isunbounded,wecannotguaranteethatunboundednessoftheproblemcanbedetectedinnitetime. 98

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145 ]).ThesolutionmethodstosolvePanditsextensionsthatwedevelopinthischapterwillrelyontheanalysisofafamilyofcloselyrelatedlinearprograms.Wewillusethecomplementaryslacknessconditionsoflinearprogrammingtocharacterizeasetofsolutionswhichwillcontainanoptimalsolution.Thisisincontrasttomostapproachestosolvenonlinearoptimizationproblemswhereknownoptimalityconditions(suchastheKKTconditions)arefundamentaltodevelopingalgorithmsfortheseproblems.Thismeansthatourapproachdoesnotrequireanyrestrictionsonthefunctiong;thatis,wedonotevenrequirethisfunctiontobedierentiableorcontinuous.Infact,theonlyassumptionthatweplaceuponPistheexistenceofanoptimalsolutiontotheproblem.Forthecaseofknapsackconstraints,wewillshowthatanoptimalsolutiontoPcanbefoundbysolvinganumberofone-dimensionaloptimizationproblemsoftheformmaxz2[`;u]zg(z)thatispolynomialinthenumberofdecisionvariables,n.Inthecase 99

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67 ]studytheproblemwhereg(z)=p 141 ],Frelingetal.[ 57 ],TeoandShu[ 151 ],andHuangetal.[ 85 ]. 100

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149 ]consideraclassofselectivenewsvendorproblemswhoserelaxationbelongstoP.Inthatpaper,themodelconsidersacompanythatfacesnormallydistributeddemandineachofnavailablemarketsandwishestoselectasubsetofthesemarketstoservetomaximizetotalexpectedprot.Inaddition,ineachselectedmarket,theadvertisinglevelmaybesettodeterminetheparametersofthedemanddistribution.Undervariousassumptionsontheresponseofthedemanddistributiontoadvertisinglevelsandthepresenceorabsenceofabudgetconstraint,thismodelfallswithintheproblemclassPaseitheranonlinearknapsackproblemoranonlinearmultiple-choiceknapsackproblem.Recently,Romeijnetal.[ 122 ]consideredaclassofnonseparableknapsackproblemsthatisoftheformPwithXgivenby( 6{1 ),buttheyrestrictthemselvestothecasewheregisconcaveandlocallyLipschitzcontinuousandthecoecientssandbarenonnegative.Sincetheydealwithaconvexmaximizationproblemsubjecttoasinglelinearknapsackconstraint,theymayrestrictthemselvestosearchingtheextremepointsofthefeasibleregion(seeHorstetal.[ 84 ])fortheoptimalsolution.ThealternativesolutionmethodthatwedevelopinthischapterisnotonlymuchmoregenerallyapplicablethanthealgorithmdevelopedinRomeijnetal.[ 122 ],butitalsosubstantiallyimprovestherunningtimeofthetheiralgorithmwhenappliedtothecasewheregisconcave,fromO(n3)timetoO(n2logn).Theremainderofthischapterisorganizedasfollows.Section 6.1 beginsbyfocusingonthecontinuousnonlinearnonseparableknapsackproblem.InSection 6.1.1 ,weexamineafamilyoflinearprogramsrelatedtothisknapsackproblemtodevelopthestructureofpotentialoptimalsolutionstotheproblem.InSection 6.1.2 ,wediscussoursolutionapproach,andaspecialcaseoftheproblem,whichplaysanimportantroleinthealgorithmforthegeneralcase,isstudiedinSection 6.1.3 .WeanalyzeandimprovetherunningtimeofthealgorithminSection 6.1.4 .Wediscussthecomplexityofdeterminingtheoptimalsolutiontotheproblemwiththefewestnumberoffractionalvariablesin 101

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6.1.5 .Section 6.2 dealswiththecaseofmultipleknapsackproblemsandSection 6.3 withmultiple-choiceknapsackproblems.Section 6.4 discussesacomputationalcomparisonofthealgorithmsproposedinthischapterwithastate-of-the-artcommercialglobaloptimizationsoftwarepackage. 6{1 ),andrefertothisclassofproblemsasKP.Wewillassume,withoutlossofgenerality,thattheredoesnotexistanitemisuchthatbi=si=0.Otherwise,wecouldsetxi=1ifri>0andxi=0otherwiseandsimplyeliminatethatitemfromconsideration.Furthermore,weassumethatthematrix(bs)hasranktwo;ifnot,theproblemreducestoeitheralinearcontinuousknapsackproblemoraninstanceofKPinwhichtheequalityconstraintisabsent.TheformercanbesolvedwiththeclassicalgreedymethodinO(nlogn)timeorusingamedian-ndingalgorithm(see,forexample,Zemel[ 161 ]whichgeneralizedtheworkofBalasandZemel[ 12 ])inO(n)time.ThelattercanbesolvedinO(nlogn+n)time,whereisthetimerequiredtosolveanoptimizationproblemwithobjectivefunctionfromthesameclassasg(seeHuangetal.[ 85 ]).VariantsofKPinwhichtheequalityconstraintisreplacedbyinequalityconstraintsoftheformB0b>xBwithB0xBafeasiblesolutionwiththesameobjectivefunctionvalueforthetransformedproblemisobtainedbylettingxn+1=(Bb>x)=(BB0)2[0;1].Similarly,thevectorcontainingtherstncomponentsofanyfeasiblesolutiontothetransformedproblemisfeasiblefortheoriginalproblem.Finally,ifwehaveaone-sidedinequality,b>xB(orB0b>x),replacingtheequalityconstraint,wemayaddintheimplicitlowerboundB0=(b)>eb>xwherebi=minfbi;0gande2Rnisavectorconsistingofallones(or 102

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whereSisaconstantthatparameterizesthefamilyofproblems.Clearly,ifweweretoknowthevalueofs>xinanyoptimalsolutiontoKPwecouldsolvetheproblembysolvingtheassociatedLKP.ThisobservationcanbeusedtoshowthatthereexistsanoptimalsolutiontoKPwithnomorethantwofractionalcomponents.Thatis,ifwedenotethesetofindicescorrespondingtofractionalcomponentsofafeasiblesolutionxtoKPbyFx=fi=1;:::;n:0
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3.0.1 showsthatthereexistsanoptimalsolutionxtoKPthatisalsoanextremepointoptimalsolutiontoLKP.SincexisabasicsolutiontoLKP,onlyitstwobasicvariablescanbefractional,whichprovesthedesiredresult. WewilldevelopasolutionapproachbystudyingthestructureofbasicfeasibleandoptimalsolutionstoproblemsoftheformLKP.WewillrstemploythecomplementaryslacknessconditionsassociatedwithLKP,whicharenecessaryforoptimality,tocharacterizeapartialprimalsolutionthatsatisesthecomplementaryslacknessconditionswithagivenfeasibledualsolution.Denotingthedualmultiplierscorrespondingtoconstraints( 6{3 )and( 6{2 )byand,respectively,andtheonescorrespondingtotheupperboundsin( 6{4 )byi(i=1;:::;n),thecomplementaryslacknessconditionsincludexi(bi+si+iri)=0i=1;:::;ni(1xi)=0i=1;:::;n:Theseconditionsmotivatethefollowingforcingrulethatdeterminesthevalueofsomeprimalvariablesforagivendualsolution: 104

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6.1.2 )with((i;j);(i;j))forsome(i;j)2.SincethesepartialcandidatesolutionstoLKPareindependentofthevalueS,tosolveKPwemayinfactrestrictourselvestosuchsolutionsaswell.Wehavethusobtainedtheresultsummarizedinthefollowingtheorem.

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6.1.2 weshowhowwecan,nevertheless,constructanitecollectionoffullcandidatesolutions. 6.1.3 .Considersomexed(i;j)2anddene,forconvenience,thesetsofindicesthatdecomposethesetofvariablesintothreecategoriesaccordingtotheForcingRule:I(i;j)0=fk:rk<(i;j)bk+(i;j)skg;I(i;j)=fk:rk=(i;j)bk+(i;j)skg;andI(i;j)1=fk:rk>(i;j)bk+(i;j)skg:Aswementionedbefore,i;j2I(i;j)andusually,butnotnecessarily,wewillhaveI(i;j)=fi;jg.TheForcingRulesaysthatxk=0forallk2I(i;j)0andxk=1forallk2I(i;j)1.Wenowdeterminethevaluesofx(i;j)kfork2I(i;j)bysolvingthefollowingoptimizationsubproblem:maximizeR(i;j)+Xk2I(i;j)rkxkg0@S(i;j)+Xk2I(i;j)skxk1A

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6{5 )intotheobjectivefunction,weseethattheobjectivefunctionoftheproblemSP(i;j)canbereformulatedas:maximizeR(i;j)+(i;j)(BB(i;j))+(i;j)Xk2I(i;j)skxkg0@S(i;j)+Xk2I(i;j)skxk1A:ThisproblemisthereforeaninstanceoftheclassofproblemsKPwherer=s(inthecaseofSP(i;j),=(i;j)).WewilldevelopanecientmethodtosolvethisclassofproblemsinSection 6.1.3 .But,rstweprovethatoneofthefullcandidatesolutionsisoptimaltoKP. Proof. 107

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whereX=fx2[0;1]n:b>x=Bg:DenoteoptimalsolutionstotheseproblemsbyxLandxU,respectively.Next,wesolvethefollowingone-dimensionaloptimizationproblem: maxLyUyg(y)(6{8)anddenoteanoptimalsolutiontothisproblembyy.Itiseasytoseethatyg(y)s>xg(s>x)forallx2X.Therefore,anysolutionx2Xsuchthats>x=yisanoptimalsolutiontoKP0.Nownotethatsuchasolutionisgivenbyx=axL+(1a)xU,wherea2[0;1]denotestheuniquevalueforwhichy=aL+(1a)U.TheproceduretosolveKP0dependsbothonthestructureofXandthepropertiesofg.SinceXisthefeasibleregionofacontinuousknapsackproblem,problems( 6{6 )and( 6{7 )canbesolvedinO(n)timeusingthealgorithmofZemel[ 161 ].Italsodependsonthetimerequiredtosolve( 6{8 ),whichisdenedasfollows. 6{8 )by.

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6{8 )canbesolvedinconstanttimebysimplyevaluatingandcomparingLg(L)andUg(U).Ifgisconvexthen( 6{8 )canbesolvedinO(log(UL))timebybinarysearchovertheinterval[L;U].Therearemanyotherinstancesof( 6{8 )thatcanbesolvedeciently,forexampleiftheobjectivefunctionof( 6{8 )isapolynomialofdegreethreeoranyunimodalfunction.Although( 6{8 )mayingeneralbeahardglobaloptimizationproblem,itisimportanttonotethatthetimerequiredtosolveitisindependentofthenumberofvariablesnintheknapsackproblemKP0.RecallthatintheobjectivefunctionofSP(i;j),theconstantS(i;j)inthefunctiongcorrespondstoaslightredenitionofthefunctiongwhentransitioningfromSP(i;j)toKP0.However,thisredenitionsimplytranslatesintoashiftoftheboundsinthesubproblem( 6{8 ).Therefore,weconcludethatwecansolvetheproblemSP(i;j)inO(jI(i;j)j+)time. 6.1.4 thatabasicalgorithmwouldsimplydeterminethesolutionsx(i;j)forall(i;j)2.WewillrstanalyzetherunningtimeofthisapproachandthendiscussanimprovementtothealgorithmtoultimatelyobtainarunningtimeofO(n2maxflogn;g).Todeterminex(i;j)forsome(i;j)2,weneedto(i)nd(i;j)and(i;j),thepartialsolutionasdictatedbytheForcingRule,andtheparametersoftheproblemSP(i;j)inO(n)time;(ii)solvetherstphaseoftheassociatedinstanceofSP(i;j)inO(jI(i;j)j)time;and(iii)solvethesecondphaseofSP(i;j)inO()time.Sincejj=O(n2)andjI(i;j)j=O(n),thisimmediatelyleadstoarunningtimeofO(n2maxfn;g)forthealgorithm.Wecanreducetherunningtimeofthealgorithmbyemployingarelationshipbetweenthesubproblemsthatincludeaparticularitemi.Foragivenitemileti=fj:(i;j)2gcharacterizetheitemsthatremaintobeconsideredtogetherwithitemi.(Notethatit 109

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110

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6{9 ).Similarly,ifbi6=0thissetcanbecharacterizedasDi=fj:j>i;bjsi=bisj;rjbi=ribjg.AnalternativecharacterizationofthesetDiisthatitisthesetofitemsjwiththepropertythatri=bi+siimpliesthatrj=bj+sj.Thischaracterizationimmediatelyleadstotheconclusionthat,oncewehaveconsideredallsolutionscorrespondingtoitemi,wedonotneedtoconsiderthesolutionscorrespondingtoanyitemj2Dinfig,leadingtoasignicantreductionincomputationaleort.Putdierently,thesetsDieectivelyformapartitionofthesetofitems,whichwecanindexbyasetIf1;:::;ngwiththepropertythatDi\Dj=;foralli;j2Iand[i2IDi=f1;:::;ng:WearenowreadytopresentouralgorithmforsolvingKP.KPAlgorithm Step0. 111

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Proof. 112

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6{6 )and( 6{7 ),andO()timetosolvetheone-dimensionaloptimizationproblem.SinceI(i;j`)\I(i;j`1)=Diforall`=2;:::;jij(i.e.,eachitemk62Diappearsinatmostonesubproblem),thisimpliesthatthetimerequiredtosolveallsubproblemsencounteredinSteps2and3isOPj2i(jI(i;j)j+)=O(n+njDij+n).Thetotaltimerequiredforagiveni2IisthusO(nlogn+njDij+n).OncewehavedeterminedtheoptimalobjectivefunctionvaluesofSP(i;j)for(i;j)2,wecandeterminetheoptimalsolutionxtoKPinO(n+)timebyresolvingSP(i;j)forthepair(i;j)achievingthemaximumobjectivefunctionvalue.ThisnowyieldsthatthetotaltimerequiredtondanoptimalsolutiontoKPcanbefoundinOn++Xi2I(nlogn+njDij+n)!=O(n2logn+n2+n2)=O(n2maxflogn;g)time. Sincethefunctiongisnotassumedtopossessanyparticularstructure,thisresultsuggeststhatthisalgorithmachievesanear-optimumrunningtimefortheproblemofdeterminingtheobjectivefunctionvalueforallx(i;j),(i;j)2,sincethereareO(n2)pairsofvariablesthatneedtobeconsidered.Romeijnetal.[ 122 ]studiedasubclassofKPinwhichthefunctiongisconcaveandlocallyLipschitzcontinuousandsandbarenonnegativevectors.(Notethatthenonnegativityofoneofthesevectorscanbeassumedwithoutlossofgenerality,butnotthenonnegativityofboth.)Concavityofgimpliesthatthereexistsanextremepointoptimalsolution,i.e.,asolutioninwhichatmostonevariableisfractional.TheydevelopedasolutionapproachbasedongeneralizedKKT-conditionsforKPthatndsanoptimalsolutionwiththatpropertyinO(n3)time.Itiseasytoseethatouralgorithm,forthecasewheregisconcave,runsinO(n2logn)timesinceisaconstant.Moreover,sinceeachcandidatesolutionx(i;j)isgivenbyanextremepointofthefeasibleregionofSP(i;j)togetherwiththebinarycomponentsasdictatedbytheForcingRule,thealgorithmpresentedinthischapterwillndasolutionwithatmostonefractionalvariable.Wecan 113

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122 ]inreducedworst-caserunningtime. 6.1.1 guaranteesthatthereexistsanoptimalsolutionwithnomorethantwofractionalcomponents.Ifthealgorithmdoesnotreturnsuchasolution,wecanofcourserecoveronebysolvinganinstanceofLKPwiththevalueofSobtainedintheoptimalsolutionfound.Asolutionwithfewerfractionalvalues,ifoneexists,couldinprinciplebefoundasfollows.Afterdeterminingx(i;j)forsome(i;j)2,wemayattempttondacorrespondingcandidatesolutionwithnomorethanonefractionalvariableby,inturn,xingeachofthevariablesinI(i;j)andndingoptimalsolutionstoSP(i;j)foreachpossiblechoiceofbinaryvaluesfortheremainingvariablesinI(i;j).Inaddition,wecouldconsidereachpossiblechoiceofbinaryvaluesforallvariablesinI(i;j).Clearly,thisprocedurewillidentifyanoptimalsolutionwiththesmallestnumberoffractionalvariablesattheexpenseofconsideringO(2jI(i;j)j)additionalsolutionsforeach(i;j)2.Inthe,relativelycommon,situationwhereI(i;j)=fi;jgandthusjI(i;j)j=2,thisdoesnotchangetheorderoftherunningtimeofthealgorithm.However,wewillnextshowthatthisapproachdoesnotleadtoapolynomialtimealgorithmingeneralunlessP=NP. Proof. 62 ]).The 114

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6.1 canbeextendedtoproblemswithmultiplelinearconstraints.Wewillrefertothisclassofproblemsasnonseparable,nonlinearmulti-knapsackproblems,thatmaybeformulatedasfollows:maxr>xg(s>x)subjectto(MKP) 115

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6{10 ).ItisstraightforwardtoextendtheresultofTheorem 6.1.1 toMKP: 6.1.1 fromastudyofthestructureofextremepointsolutionstothelinearprogrammaximizer>xsubjectto(LMKP)s>x=Sb>jx=Bjj=1;:::;mx2[0;1]n:

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6.1.3 :

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6{6 )or( 6{7 )withnvariablesandmconstraintsbyS(n;m).Thesecondstepstillconsistsofaone-dimensionaloptimizationproblemoftheform( 6{8 ). Proof. 6.1.4 thatxisanoptimalsolutiontoMKP. Letusnextdiscusstheruntimeofthesolutionmethod.Inastraightforwardmanner,wecanconcludethatthetimerequiredbythealgorithmisO(nm+1maxfn;m3;S(n;m);g)sincethesizeofisO(nm+1)andwerequire:(i)O(m3)timetond(;);(ii)O(n)timetoapplytheForcingrule;(iii)O(S(n;m)+)timetosolvethesubproblem.Wecan,however,improvetherunningtimeofthealgorithminasimilarmanneraswedid 118

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6{6 )and( 6{7 )andO()timetosolvetheone-dimensionaloptimizationproblem.ThisimpliesthatthetimerequiredtosolveallsubproblemsencounteredinSteps2and3isOPj2(S(n;m)+)=O(nS(n;m)+n),sothatthetotaltimerequiredforagivenisindeedOnm3+nlogn+nS(n;m)+n:Dening=ff1;:::ng:jj=mg,thisnowyieldsthatthetotaltimerequiredtondanoptimalsolutiontoKPcanbefoundinOX2(nm3+nlogn+nS(n;m)+n)!=Onm(nm3+nlogn+nS(n;m)+n)=Onm+1maxfm3;logn;S(n;m);gtime. Thisimprovementdoesnotseemtohavemucheectontherunningtime,sincetypicallyS(n;m)willdominatebothm3andn.Typically,though,S(n;m)willbealooseupperboundonthetimerequiredtosolvethelinearprograms.TherunningtimeforagiveninstanceofMKPwillreplaceS(n;m)withS(max2jIj;m).Inmostcases, 120

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6{11 )isreplacedbyaconstraintoftheformLnXi=1viXj=1bijxijB:ThisfollowsfromthefactthatwemayconvertsuchaproblemtoaproblemoftheformMCKPbyaddinganotheritem,sayn+1,withtwovariants,say1and2.Wesetrn+1;1=rn+1;2=sn+1;1=sn+1;2=bn+1;2=0andbn+1;1=BL.Theseproblemsareequivalentsince,foranyfeasiblesolutiontotheinequalityconstrainedproblem,wemaysetxn+1;1=BPni=1Pvij=1bijxij 121

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6.2 ,namelywithPni=1vi=O(nv)itemsandn+1constraints.However,applyingtheproceduredevelopedforthisproblemclasswouldleadtoasolutionprocedurethatrequiresthesolutionofanumberofone-dimensionaloptimizationproblemsthatisexponentialinthenumberofitemsn.Inthissection,wewilldevelopatailoredalgorithmforMCKPthatrequiresthesolutionofanumberofsuchproblemsthatispolynomialinboththenumberofitemsnandthenumberofvariantsperitemv. 122

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(i) thereareatmosttwotwo-waysplititems,i.e.thereexists(i;j;k)and(i0;j0;k0)suchthatxij;xik>0andxi0j0;xi0k0>0; (ii) thereisatmostonethree-waysplititem,i.e.thereexists(i;j;k;`)suchthatxij;xik;xi`>0. Proof. 3.0.1 showsthatthereexistsanoptimalsolution,x,toMCKPthatisalsoanoptimalextremepointsolutiontoLMCKP.Sincetherearen+2constraintsinLMCKP,itfollowsthatx,sinceitisabasicsolution,willhavenomorethann+2positivecomponents.Byconstraints( 6{15 )atleastonevariableassociatedwitheachitemi=1;:::;nshouldbestrictlypositive,foratotalofnpositivevariables.Theremainingtwopositivevariablesmaycorrespondtodistinctitems,leadingtoasolutionoftheform(i),ortothesameitem,leadingtoasolutionoftheform(ii). AsfortheLKP,wedenotethedualmultipliersof( 6{13 )and( 6{14 )byand.Moreover,wedenotethedualmultipliersfortheconstraints( 6{15 )byi(i=1;:::;n).Givenadualfeasiblesolution(;;)tothedualofLMCKP,thecomplementaryslacknessconditionsinclude:xij(bij+sij+irij)=0j=1;:::;vi;i=1;:::n:Theseconditionsmotivatethefollowingforcingrulethatdeterminesthevalueofsomeprimalvariablesforagivendualsolution: 123

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6.3.1 ,anybasicfeasiblesolutiontoLMCKPcanbecharacterizedbyeithertwotwo-waysplititemsoronethree-waysplititem.Wewillrstconsiderthersttypeofbasicfeasiblesolutiondenedbyitemisplitbetweenvariantsjandk,anditemi0splitbetweenvariantsj0andk0.Let(1)bethecollectionofindices(i;j;k;i0;j0;k0)thatindeedcorrespondtoabasicsolution.This,inparticular,meansthatforall=(i;j;k;i0;j0;k0)2(1)thefollowingsystemofequationshasauniquesolution(;;i;i0):rij=bij+sij+irik=bik+sik+iri0j0=bi0j0+si0j0+i0ri0k0=bi0k0+si0k0+i0:Notethatdualfeasibilityrequiresthat^{=max`=1;:::;v^{r^{`b^{`s^{`^{=i;i0:Ifthisisviolated,cannotcharacterizeapotentialoptimalbasicfeasiblesolutiontosomeLMCKPandwemaydisregardit.Wemaydeterminetheremainingvaluesoffromdualfeasibilityvia^{=max`=1;:::;v^{r^{`b^{`s^{`^{=1;:::;n;^{6=i;i0:Similarly,wemayconsiderthesecondtypeofbasicfeasiblesolutionsthatisdenedbyanitemisplitbetweenvariantsj,k,and`.Let(2)bethecollectionofindices(i;j;k;`)thatindeedcorrespondtoabasicsolution.This,inparticular,meansthatforall=(i;j;k;`)2(2)thefollowingsystemofequationshasauniquesolution(;;i):rij=bij+sij+i

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6.3.1 thatmayoccurinabasicfeasiblesolutiontoLMCKP,thereexistsauniquechoiceofadualmultipliers(;;)thatwillsatisfythecomplementaryslacknessconditionsrelatedtoLMCKP.ThisfactwillallowustorestrictattentiontoarelativelysmallnumberofcandidatesolutionstoMCKP(oneforeachpotentialsolutionsplit)ascomparedtoapplyingthealgorithmthatwedevelopedforMKPinSection 6.2 tothisclassofproblems,inwhichcasewewouldhavetoconsiderO((nv)n1)candidatesolutions.

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6.3.2 )weknowthatxij=0ifj62Ii.Todeterminetheremainingcomponentsofxwesolvethefollowingsubproblem:maximizenXi=1Xj2Iirijxijg0@nXi=1Xj2Iisijxij1Asubjectto(SP) 6.1.3 ).NotethatsolvingSPthusinvolvessolvingtwolinearmultiple-choiceknapsackproblemstondappropriatelowerandupperboundsusing( 6{6 )and( 6{7 )as 126

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6{8 ).WenowshowthatthesetindeedcontainsanoptimalsolutiontoMCKP. Proof. 45 ]orZemel[ 161 ].Finally,wemustsolvethecorrespondingone-dimensionalglobaloptimizationprobleminO()time.ThisimpliesthattheruntimetodeterminethecandidatesolutionforagivenisO(nv+).ThereareO(n2v4)numberofpairstwo-waysplitsandO(n2v3)three-waysplitsin,sothisyieldsastraightforwardalgorithmwitharuntimeofO(n2v4maxfnv;g).Itturnsout,foragivensplit(j;k)ofitemi,itisonlynecessarytoexaminethissplitwithO(v)splitsofitemi0.ThestraightforwardalgorithmexaminesO(v2)splitsofitem 127

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wherethevectors(bij;sij;1)and(bik;sik;1)areassumedtobelinearlyindependent(sinceotherwisethissplitofitemicannotbepartofabasicsolution).Thisimpliesthateitherbij6=bikorsij6=sik.Forthisdiscussion,weassumethatbij6=bikalthoughanequivalentargumentcanbemadeforthecasewhenbij=bikandsij6=sik.SimilartotheimprovedalgorithmforKP,wewillsolveforandiasfunctionsofinorderforEquations( 6{18 )-( 6{19 )tohold.Wewilldenotethesefunctionsbyijk()andijki().Inparticular,wemusthavethat 128

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(6{22) inorderfor(ijk();;ijki())tobedualfeasible.WewilldenethesetDijk=f`=1;:::vi:ri`=ijk()bi`+si`+ijki()forallgwhichimpliesthatif`2Dijkthenxi`willappearasavariableinallsubproblemswhenwesplititemibetweenvariantsjandk.Foranyvariant`62Dijkofitemi,wemaydeterminethevalue`suchthatequalityholdsin( 6{22 ).Sincetheright-handsideof( 6{22 )isalinearfunctionintermsof,wehavethateither( 6{22 )holdsforall`orforall`.InO(vi)time,wecandeterminetherangeof,[ ijk],inorderfor(ijk();;ijki())tobedualfeasible.Notethatforall2( ijk)andthereexistssome`62Dijkthatri`>ijk()bi`+si`+ijki().Therefore,weneedtoonlyconsiderathree-waysplitofitemi,=(i;j;k;`),for`62Dijkand= (i) forsomej0=1;:::;vi0,ri0j0=ijk()bi0j0+si0j0+ijki0();and (ii) forallj0=1;:::;vi0,ri0j0ijk()bi0j0+si0j0+ijki0().Thisimpliesthat 129

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6{23 )isattainedforbothj0andk0sinceotherwisethesolution(;;)(where=(i;j;k;i0;j0;k0)2)isnotdualfeasible.Sincewehave(fornow)assumedthatAssumption 6.3.5 holds,thissituationcanonlyoccuratthebreakpointsofthefunctionijki0().Typically,exactlytwovariantsofitemi0willachievethemaximumin( 6{23 ).Notethatitispossiblethatmorethantwovariantsofitemi0toachievethemaximumin( 6{23 )atsome0.Foranychoiceofvariantsj0;k0achievingthemaximum,wehavethatIi0(where=(i;j;k;i0;j0;k0))isexactlythesetofvariantsachievingthemaximumin( 6{23 )for=0.Thismeansthatforanychoiceofvariantsj0;k0achievingthemaximumin( 6{23 ),thesubproblemsthatarisewhensolvingforx(with=(i;j;k;i0;j0;k0))willbethesame.Therefore,weneedtoonlyconsiderasinglepairofvariantsofitemi0atanybreakpointofijki0().Thisimpliesthatweneedtoconsideratmostvi0ways(eachcorrespondingtoabreakpointofijki0())tosplititemi0.WecandeterminethesebreakpointsinO(v2)timebyrstsortingtheslopesofthevi0linearfunctionsdeningijki0()andthendeterminingtheintervalswhereeachlinearfunctionobtainsthemaximum.Wethensortallthevaluesofcorrespondingtoabreakpointofanyofthefunctionsijki0().Notethatifanyofthesevalues,(=(i;j;k;i0;j0;k0)),areoutsidetherange[ ijk],thenwedonotneedtoconsiderthesplitsince(;;)isnotdualfeasible. 130

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131

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6.3.5 ,wecandetermineanoptimalsolutiontoMCKPinO(n2v3maxflogn;v;g)time. 6.3.5 isviolated.WewillshowthattherunningtimeisstillO(n2v3maxflogn;v;g).In 132

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;],wherethevariantsj0;k0achievethemaximumin( 6{23 )ofijki0().Considersome=(i;j;k;^{;^|;^k)suchthat2( ;)and(;;)isdualfeasible.ThisimpliesthatwewillneedtosolvethesubproblemSP.WehavethatIi0=fj;kg,soxi0j0andxi0k0appearasvariablesinSP.Sinceisnotabreakpointofijki0(),thesevariablesareunaccountedforinourcomplexityanalysisabove.AlthoughtheappearanceofthesevariablesrequireadditionalcomputationaleortinsolvingSP,wenowdetailthatwesaveatleastthesameamountwhenconsideringsplittingitemi0betweenj0andk0.ThiswouldimplythatthecomplexityofouralgorithmisstillO(n2v3maxflogn;v;g).First,sinceijk^{()=i0j0k0^{(),wehavethatwillbeconsideredwhenwesplititemi0betweenvariantsj0andk0.Also,whenweconsider0=(i0;j0;k0;^{;^|;^k),wewillhavethatI0{=I{forall{6=i;i0.Further,since(;;)isdualfeasible,wehavethatforall`=1;:::vi:rijijk()bijsij=ijki()rilijk()bilsilwhichimpliesthatriji0j0k0()bijsijandriki0j0k0()biksikachievethemaximuminthedenitionofi0j0k0i(0).Thisimpliesthatbothj;kareinthesetIi.Foranyvariant 133

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;),itfollowsthatIi0=fj0;k0g.Thismeansthatforanyvariantl06=j0;k0ofitemi0that:i0j0k0i0()=ri0j0i0j0k0()bi0j0si0j0>ri0`0i0j0k0()bi0`0si0l0andthereforeI0i0=Ii0.ThisimpliesthatthesubproblemsSPandSP0areidentical.Therefore,weneedtoonlysolveitonce.Sincex^{^|andx^{^kappearasvariablesinthesesubproblems,thecomputationalsavingsofneedingonlytosolveasingleoneofthesubproblemsexceedstheadditionalcomputationaleorttoaccountforxi0j0andxi0k0appearingasvariablesinSP.Thiscanbeseenbecauseinourcomplexityanalysisofdeterminingallsplitsinvolvingvariantsj0;k0ofitemi0,weincludetheadditionaltimerequiredtosolvethelinearmultiple-choiceknapsackproblemswhenx^{^|andx^{^kappearinSP0butwedonotactuallysolvetheseproblems,sinceSPandSP0areequivalent.Therefore,therunningtimeoftheimprovedalgorithmremainsthesame,eveniftheinstanceofMCKPviolatesAssumption 6.3.5 .Thisimpliesthefollowingresult: 108 ]said:\Amongthecurrentlyavailableglobalsolvers,BARONisthefastestandmostrobustone".Forourexperiments,wehaveusedtheBARONsolverthroughitsGAMSinterface. 134

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135

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1. Therstproblemclassthatweconsiderismotivatedbyeconomiesofscaleinproductionplanning,i.e.,weassumethatthetotaldemandisproduced,andtheproductionfunctionisaconcavefunctionoftheproductionquantity.Inparticular,weconsiderthefunctiong(S)=p 76 ]andGeunesetal.[ 67 ]),whereC=2KhifKisaxedproductioncostandhaholdingcostrate.Interestinglythough,thesamefunctionalformofgarisesinthepresenceofstochasticdemandsandriskpoolingundernewsvendorcosts(seeTaaeetal.[ 149 ]);inthiscase,however,sirepresentsthevarianceofdemandinmarketi.SinceCisascalingparameteronly,wehaveinourtestssimplyusedC=1. 2. Thesecondproblemclassthatweconsidercorrespondstosituationsinwhichthereareagaineconomiesofscaleinprocurementcosts,butthisformonlyholdsuptosomeproductionlevel,whichmayrepresentacapacity.Beyondtheselevels,weassumethatthecompanyisabletooutsourceadditionalunitsoftheproduct,buttheoutsourcingcostsareaconvexfunctionoftheamountoutsourced.Tomodelthissituation,wehaveconsideredtheconsiderthefunctiong(S)=(S)3+3forsomeconstants;>0.Thisfunctionisconcaveontheinterval(;]andconvexontheinterval[;1).Inourtests,wehaveusedthevalues=1=n2and=n,wheretheparticulardependenceonnensuresproperscalingamongproblemdimensions. 136

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Theformofginthesecondproblemclasswaschosenforbothitsgeneralshape(concaveatsmallvaluesoftheargumentandconvexatlargevalues)aswellassinceitcanberepresentedbyasingleexpression.ThisisparticularlyimportantforBARONwhenusedthroughitsGAMSinterface.However,fromapracticalpointofviewamorerealisticmodelwouldhavetheconcavepartofthesameformasintherstproblemclass,followedbyaconvexcostcomponentforoutsourcing.Wethereforealsoconsideredthefollowingfunction:g(S)=8><>:p 137

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6-1 showstheresultsofthecomputationaltestsforKP,wherethecomputationtimesareallinsecondsonaDellPowerEdge2600withtwoPentium43.2Ghzprocessors,6gigabytesofmemory,andthreeUltra32015KRPMSCSIdrives.Thetableclearlyillustratestheinsensitivityofthealgorithmpresentedinthischaptertothefunctionalformofg.Inaddition,itshowsthatthealgorithmperformscompetitivelywithBARONfortheeasiestproblemclass,butastheformofthefunctiongbecomesmorecomplextheresultsveryclearlyshowthepowerofourapproach.Infact,forsomeinstancesBARONwasnotabletosolvetheproblem(orverifythatthesolutionfoundwasgloballyoptimal)withinthetimelimitof3600seconds(onehour)thatweset.Inparticular,wheneverthecomputationtimehasasuperscript,thecorrespondingnumberrepresentsthenumberofinstancesforwhichBARONwasnotsuccessful(outofatotalof10instances).Inthesecases,thetimesprovidedareaveragesoverthesuccessfulinstancesonly. Table6-1. ComparisonofrunningtimesobtainedwiththealgorithmforKPandBARON. Problemclass1 Problemclass2 Problemclass3 B BARON thischapter BARON thischapter BARON 1,000 .24 .60 5.74 .39 3.98 4n .20 .49 3.732 4.20 5,000 5.25 11.06 266.20 10.90 112.69 4n 4.90 10.89 203.021 114.89 10,000 36.06 56.31 911.90 46.79 470.891 39.83 48.77 540.30 46.87 467.482 470.13 220.16 1914.85 213.89 2209.99 4n 408.24 210.73 2188.88 207.92 1951.00 Next,westudiedtheperformanceoftheimprovedalgorithmforMCKPtoBARON.TofacilitateacomparisonwiththeresultsobtainedforKPaswellasamongdierentvaluesofv,wechose,foreachvalueofv,thenumberofmarketsnsothatn(v1)isapproximatelyequaltoacorrespondingvalueofnusedforKP(wherewenotethattheKPimplicitlycontainsv=2variants).Forexample,forn(v1)=10,000andv=4,wesetn=3,333.Inaddition,againforconsistencywiththeKPinstances, 138

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6-2 showstheresultsofthecomputationaltestsforMCKP.Althoughtheperformanceofbothapproachesisstillcomparablefortheeasiestproblemclass,BARONwasnotabletosolveevenasingleprobleminstanceinthetwoharderclasseswith5,000variablesandv=3or10,000variablesormorewithintheCPUtimelimitofonehour.ThisisinstarkcontrasttotheimprovedalgorithmforMCKP,whichwasabletosolvetheprobleminstancesofallproblemclassesandalldimensionstestedinaveryreasonableamountofCPUtime. Table6-2. ComparisonofrunningtimesobtainedwiththeimprovedalgorithmforMCKPandBARON. Problemclass1 Problemclass2 Problemclass3 BARON thischapter BARON thischapter BARON 1,000 3 .82 .81 .99 150.99 .98 191.86 4 .48 .77 .38 61.41 .20 38.21 5,000 3 24.05 23.98 25.41 | 20.85 | 4 14.27 19.49 19.39 1212.14 5.34 1872.54 10,000 3 127.21 109.88 106.28 | 41.79 | 4 73.26 146.66 69.30 | 25.00 | 20,000 3 195.21 253.27 469.45 | 185.91 | 4 95.77 215.89 213.99 | 94.51 | 139

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140

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8 and 9 canbedescribedasfollows.WearegivenasetofncustomersandTtimeperiodswherethedemandofcustomerjinperiodtisgivenbydjt(j=1;:::;n;t=1;:::;T).Itwillbeconvenienttoalsodenethecumulativedemandofcustomerjasdj=PTt=1djt.Wewishtoassignacustomerjtoanopenfacilityiandmeetthedemandofthecustomerthroughproductionandinventorydecisionsatthefacility.Thereisaconnectioncost,cij,associatedwithfacilityiandcustomerj,whichisexpressedasacostperunitofdemand.Eachfacilityhasanopeningcostoffi,whichwemustpayifweassignanycustomerstothefacility.Eachfacilityihasaconcavefunctionrepresentingthecostofproducingpunitsintimeperiodt,Pit(p),andaconcavefunctionrepresentingthecostofholdingIunitsintimeperiodt,Hit(I).Ouruncapacitatedfacilitylocationandproductionplanningproblem(UFLPP)(wherewehave 141

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Constraints( 7{1 )-( 7{4 )aretraditionalfacilitylocationproblemsandconstraints( 7{5 )-( 7{7 )areproductionplanningconstraintsateachfacilitytoensurethat,ineachtimeperiod,wemeetthedemandofallcustomersassignedtothefacility.Ifweknowthesetofcustomersassignedtoagivenfacility,thenwesimplyneedtosolveanuncapacitatedproductionplanningproblematthatfacility.Thisproductionplanningproblemisageneralizationoftheclassiceconomiclot-sizingproblem(seeWagnerandWhitin[ 157 ])whereconcaveproductionandinventorycostfunctionshavereplacedxed-chargepluslinearproductioncostsandlinearinventorycosts.TheUFLPPtsintotheSCNDframeworkbydeningthefacilitycostfunction,Hi(xi)tobezeroifxiistheemptysetandotherwisesetitequaltotheoptimalsolutionvalueoftheproductionplanningproblemminimizefi+TXt=1(Pit(pit)+Hit(Iit)) 142

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143

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125 ]).Themaindierencesbetweenthetwoproblemsare:(i)theabsenceofproductioncapacitiesintheUFLPP-DA,(ii)thepresenceoffacilityopeningdecisionsintheUFLPP-DA,and(iii)thepresenceofconcaveproductionandinventorycostsintheUFLPP-DAratherthanlinearproductionandinventorycostsinthemulti-periodsingle-sourcingproblem.ItisnotdiculttoseethatanyinstanceoftheUFLPP-DAcanbeconvertedtoanequivalentinstanceoftheUFLPP.Inparticular,ratherthanviewcustomerjasasinglecustomerwithdemandstream(dj1;dj2;:::;djT)intheUFLPP-DA,weviewitasasetofcustomers,jtfort=1;:::;T,wherethedemandstreamofcustomerjtbeinggivenbydjtt0=0ift06=tanddjtt=djtift0=tintheUFLPP.ThismeansthatwecanconverttheUFLPP-DAtoaUFLPPwithO(nT)customers.Further,notethatifwearegivenaninstanceoftheUFLPPsuchthateverycustomerhasexactlyonenon-zerodemandperiod,thisproblemcanbeconvertedtoanequivalentinstanceoftheUFLPP-DA.Apopularstreamofworkforfacilitylocationproblemsistodevelopapproximationresultsforthem.InChapter 8 ,weexamineapproximationresultsfortheUFLPPandtheUFLPP-DA.Weshowthatboththeseproblemsareashardasthesetcoverproblemand,therefore,itisunlikelythatthereexistsconstantfactorapproximationalgorithmsforthegeneralproblem.Therefore,itisappropriatetofocusonapproximationalgorithmsforspecialcasesoftheproblem.Thesespecialcasescomeintwoforms:(i)specializetheproductionandinventorycoststructureand(ii)specializethedemandpatternofthecustomers.Weshowthatforseveralproduction/coststructuresboththeUFLPPandtheUFLPP-DAcanbeapproximatedwithinaconstantfactor.Further,weshowthataspecialclassoftheUFLPPgivesrisetoaclassofmetricuncapacitatedfacilitylocationproblemswherethefacilitycostfunctionisaconcavefunctionintheamountofdemandassignedtothefacility.Wedevelopagreedyalgorithmforthisproblemwithanapproximationguaranteeof1:61.Wethenusethegreedyalgorithmtogetherwiththeidea 144

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9 ,wefocusonissuesarounddevelopingexactalgorithmstosolvetheUFLPPandtheUFLPP-DA.FortheUFLPP,weshowthatforthespecialcasewhentheproductionandinventorycostsarethatofaneconomiclot-sizingproblem,reformulatingtheproblemsusingtheplant-locationformulationofthelot-sizingproblem(seeKrarupandBilde[ 92 ])yieldsatighterrelaxation.Wealsoshowthattheset-partitioningformulationoftheproblem(seeSection 2.1 foradiscussion)yieldsaneventighterrelaxationontheproblem.Therefore,wecanexpectthatabranchandpricealgorithmwillperformwellonthisinstanceoftheUFLPP.FortheUFLPPwithgeneralconcaveproductionandinventorycosts,wediscussthat,ingeneral,wecannotdetermineiftherelaxationoftheset-partitioningformulationistighterorweakerthanthecontinuousrelaxationoftheUFLPP.Thisdoesnotnecessarilymeanthatthebranchandpricealgorithmwillbeineective;therearemanyexamplesintheliterature(Shenetal.[ 141 ],Shuetal.[ 144 ],andHuangetal.[ 85 ])wherewecannotsayanythingaboutthetightnessoftheset-partitioningformulationyetthebranchandpricealgorithmisstillhighlyeective.Intheseexamples,oneofthemoreimportantelementsinthebranchandpricealgorithmistheabilitytoeectivelysolvethepricingproblem.ThepricingproblemthatarisesfromtheUFLPPisaninterestingproductionplanningproblemwithcustomerselection.Althoughitwasrecentlyshownthatevenwitheconomiclot-sizingcoststhisproblemisNP-hard,wediscusssomepolynomially-solvablesubclassesofit. 145

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7{3 )(orconstraints( 7{10 ))arerelaxedtoallowforsplittabledemands.Inthischapter,wefocusontheformerclassofproblems. 146

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8.1 ofthischapterweshowthattheUFLPPisashardasthesetcoverproblemandconcludethat,ingeneral,itishighlyunlikelythataconstantfactorapproximationalgorithmexists.Therefore,wewillfocusonapproximatingspecialcasesoftheUFLPP.Thecasesthatwewillstudyfallintotwocategories:(i)specializingtheproductionandinventorycoststructureatthefacilities(Section 8.1 );and(ii)specializingthedemandpatternofthecustomers(Section 8.2 ).Forproblemsbelongingto(i),wewilloerreductionsfromtheUFLPPtopreviouslystudiedvariantsofthemetricUFLP.Fortheproblemsin(ii),wewillshowthattheybelongtoaspecialclassoffacilitylocationproblemswithfacilitycostfunctionsthatareconcavefunctionsoftheamountofdemandassignedtothefacility.Wewilldevelopa1:52-approximationalgorithmforthisconcavecostfacilitylocationproblem(CCFLP).InSection 8.3 ,wediscussapproximationresultsfortheUFLPP-DA.WeshowthatthetheUFLPP-DAisashardasthesetcoverproblemanddevelopanalogousresultsofSection 8.1 forproblemswithspecializedcoststructures.WealsodiscussanewclassoffacilitylocationproblemsthatarisefromaspecialcaseoftheUFLPP-DA.WeconcludethechapterinSection 8.4 withasummaryandsomefutureresearchdirections. Proof. 147

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SinceFeige[ 51 ]showedahardnessresultaboutapproximatingthesetcoverproblem,Theorem 8.1.1 impliesthatwecannotdevelopanapproximationalgorithmwithaguaranteeofbetterthan(1)lognunlessNPTIME[O(nloglogn)].Therefore,itisinterestingtoidentifyspecialcasesoftheUFLPPthatcanbeapproximatedtowithinaconstantfactor.ThefollowingtwotheoremsdealwithtwodierentcoststructuresfortheUFLPPforwhichthisisthecase.BothresultsassumethattheproductioncostfunctionsandinventorycostfunctionsarelinearandcanbewrittenasPit(pit)=bitpit(i=1;:::;m;t=1;:::;T)andHit(Iit)=hitIit(i=1;:::;m;t=1;:::;T)respectively.Therstresult,inaddition,assumesthattheproductionandinventoryholdingcostsdonotdependonthefacility. 148

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Proof.

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Thesecondresultagaindealswithinstancesforwhichtheproductioncostandholdingcostfunctionsarelinear.Howeverwenowassumethatanorderingofthefacilitiesexistssuchthat,ineverytimeperiod,itisascost-eectivetoproduce(orhold)aunitofdemandatfacilityithanitisatfacilityi0>i.Notethatwedonotmakeanyassumptionsregardingthefacilityopeningcosts,soitmaybeveryexpensivetoopenafacilitywithcheapproduction/inventorycostsandverycheaptoopenafacilitywithexpensiveproduction/inventorycosts. Proof. 142 ]oeraprimal-dualapproximationalgorithmforthisproblemwithanapproximationguaranteeof6,whenthereexistsanorderingofthefacilitiessuchthatifi
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8.1.2 above.Itisclearthatifi
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Proof. Wewillrefertotheproblemclass(P)withconcavefunctionsfiastheconcavecostfacilitylocationproblem(CCFLP).Beforestudyingthisprobleminmoredetail,notethatithasotherapplicationsbesidestheUFLPPwithseasonaldemands.Forexample,Daskinetal.[ 39 ]andShenetal.[ 141 ]considerajointinventory-locationproblem,specialcasesofwhichbelongtotheclassoftheCCFLP.InSection 8.2.1 ,wewilldevelopa1:61-approximationalgorithmfortheCCFLPandusethisalgorithmandtheideaof 153

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8.2.2 ,weexaminetheproblem(P)forotherstructuresofthefunctionsfi,i=1;:::;m. 87 ].Wenowinformallydescribethisalgorithm.Atanypointinthealgorithmwewillhavetwosetsofcustomers:connectedcustomersandunconnectedcustomers.Eachcustomerwillthenmakeanoertoeachfacility.Theoerofaconnectedcustomertoafacilityisequaltotheamountthecustomerwouldsaveinpayingtheconnectiontothisfacilityasopposedtopayingtheconnectioncosttothefacilityitiscurrentlyassigned.Theoerofanunconnectedcustomertoafacilitywillbebasedonthecustomer'sbudgetandthe 154

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87 ]:(i)theideaof\contributionwithdrawal",i.e.,ifacustomerswitchesfacilities,itwithdrawssomeofitscontributiontothefacilityitisinitiallyconnectedtoandoersittothefacilityitswitchestoand(ii)ecientlysolvinganonlinearfractionalbinaryprogrammingproblemthatisnecessaryateachiterationofthealgorithm.Wethenusethisalgorithmtogeneralizethetwo-phasealgorithmofMahdianetal.[ 98 ]totheCCFLP,derivinganapproximationalgorithmfortheCCFLPwithaguaranteeof1:52.WewillnowdescribethenotationusedinouralgorithmtoapproximatetheCCFLP.LetD=f1;:::;ngbethesetofcustomers.Atanypointinthealgorithm,Uf1;:::;ngwilldenotethesetofunconnectedcustomers.Moreover,foranyfacility,i,Aiwilldenotethesetofcustomersassignedtoitbythealgorithmsofar,andTi=Pj2Aidjwillbethecorrespondingtotaldemandcurrentlyassigned.Thereisanotionoftime,,associatedwiththealgorithm.Foranyj2U,wesetthebudgetofcustomerjattimeequaltoj=.Eachcustomerwilloersomemoneyfromitsbudgettoafacilityi,whichwedenoteoji.Theoerofcustomerjtofacilityi(wherej62Ai)dependsonwhethercustomerjhasbeenassignedtoafacilityearlierthantime.Inparticular,attime,ifj2U,thenoji=djmaxfjcij;0g;ifj2Ai0,thenoji=djmaxfwji0+ci0jcij;0gwherewji0=(fi0(Ti0)fi0(Ti0dj))=dj.Ifj2Ai0,thenwji0istheamountofcustomerj'scontributiontofacilityi0thatwouldbewithdrawnfromfacilityi0ifcustomerjswitchesfacilities.Notethattheintroductionofwji0issignicantlydierentthanthealgorithmofJainetal.[ 87 ],whichdoesnotallowacustomertowithdrawsomeofitscontributionfromthefacilityitisassigned.Theideabehindthealgorithmisto,astimeprogresses,assigna 155

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Step0. Proof. 156

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(8{1) 8{1 )ensuresthattheamountoeredtoafacilityisequaltotheadditionalamountincurredbythefacilityinservingcustomersinS1[S2.Sinceweareminimizing,wecandisregardsolutionstothisproblemwhereweselectacustomerthatdoesnotoeranythingtofacilityi.Inotherwords,forpotentiallyoptimalsolutionstoSP(i),( 8{1 )canberewrittenasXj2S1dj(wji0+ci0jcij)+Xj2S2dj(cij)=fiTi+Xj2S1[S2dj!fi(Ti):Ifweletaj=(wji0+ci0jcij)ifj2Dn(Ai[U)andaj=cijifj2U,thenweseethat( 8{1 )canbewrittenasXj2S2dj=fiTi+Xj2S1[S2dj!fi(Ti)+Xj2S1[S2ajdj:ThismeansthatwehaveaclosedformexpressionforgivenS1andS2bydividingthepreviousequationbyPj2S2dj.Therefore,wecanwriteSP(i)asaproblemFP(i):minS12Dn(Ai[U);S22Ufi(Ti+Pj2S1[S2dj)fi(Ti)+Pj2S1[S2djaj 157

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Lemma 8.2.3 showsthatitsucestosolvetheproblemsFP(i)fori=1;:::;minordertoimplementtheGreedyAlgorithm.OurnextresultdealswiththecomplexityofsolvingaproblemoftheformFP(i). Proof. 141 ]).Itcanalsobeshownthatifaj()=aj0()forcustomersj;j0,thenthereexistsanoptimalsolutiontoKP()whereweselectbothjandj0orweselectneitherandthereforewecanviewj;j0asasingleentity.InordertosolveFP(i),itissucienttoknowtheorderingofthecustomersbasedonaj(),sincewemaythenevaluatenpotentialsolutionstoFP(i)todeterminetheoptimalsolution.Therefore,weturnourattentiontodetermining 158

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159

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160

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8.2.4 impliesarunningtimeofO(mn3maxflogn;g).Wewillnowfocusondeterminingtheapproximationguaranteeofthealgorithm.Thisanalysisgeneralizesthealgorithmforthemetricuncapacitatedfacilitylocationproblemwithanapproximationguaranteeof1.61presentedinJainetal.[ 87 ].Webeginwithapropertyofthevariablesj. Proof. 161

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ConsidertheoptimalsolutiontotheCCFLPandletAidenotethesetofcustomersassignedtofacilityi.Ifwecanshowtheexistenceofapairofconstants(Rf;Rc)suchthatforeveryiXj2AidjjRffi0@Xj2Aidj1A+RcXj2AidjcijthentheGreedyAlgorithmisan(Rf;Rc)-approximationalgorithm.Wewillnowfocusonafacilityiandthecustomersindexed1;:::;ksuchthat12k.Wewillletejdenotetheconnectioncostofthej-thcustomerintheorderingtofacilityiandf=fi(Pkj=1dj).Foreachj2Ai,denethecriticaltimeofjasthetimerightbeforejisconnectedtoafacility.Foranycustomer`
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8{2 )and( 8{3 )yieldsjci0j+r`jci0`:Applyingthetriangleinequalitytothisequationprovesourdesiredresult. Lemma 8.2.6 relatesthetriangleinequalitytoouralgorithm.Thefollowinglemmaderivesasetofinequalitiesinvolvingthefacilityopeningcostbasedupontheconcavityofthefacilitycostfunction. 163

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Ifwecombineequations( 8{4 )and( 8{5 ),ityieldsthat: P`2D1\Aid`:Itfollowsthat Therefore,bycombiningequations( 8{6 )and( 8{7 ),wehaveX`2D1d`maxfr`;je`;0g+X`2D2d`maxfr`;je`;0gfikX`=1d`!=f:

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8.2.6 andLemma 8.2.7 leadtoourmainresultabouttheGreedyAlgorithm. 87 ]).ItwasshownbyJainetal.[ 87 ]that,forthecaseofunitdemands:dj=1forj=1;:::;k,ifRf=1:61thenRc=1:61andifRf=1thenRc=2.Further,Mahdianetal.[ 98 ]showthatifRf=1:11thenRc=1:78,whichwillbeimportantindevelopinga1:52-approximationalgorithmfortheCCFLP.Theseresultsstillholdforgeneralintegraldjbyreplicatingeachjbydjcopiesandeachr`jbyd`djcopies.ItcaneasilybeseenthatthereplicatedcopiesstillsatisfytheconstraintsofFLP.Thisleadstothefollowingresult. 165

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8.2.8 togeneralizethealgorithmofMahdianetal.[ 98 ]. Proof. 33 ].Specically,wescaleupthefacilityopeningcostsbyafactorof,i.e.fi(Pnj=1djxij)=fi(Pnj=1djxij),andapplytheGreedyAlgorithm.Giventhesolutionreturnedbytherstphase,wescaledownthefacilitycostfunctionsbacktotheoriginalfacilitycostfunctionsatthesamerate.Ifatanypointinthisphase,wecanreassignasetofcustomerstoadierentfacilitywithoutincreasingthecostofthesolution,weperformthereassignment.Ifweareatapointinthesecondphaseofthealgorithm,saythefacilitycostfunctionsarescaledupbyafactorofc,wherenoreassignmentscanbeperformed,thenitcanbeshownthatdeterminingthenextfactor,,whereareassignmentcanbeperformedisequivalenttodeterminingthemaximumvalueofthatsatisesXj2Sdjmaxfwijj+cijjcij;0g=fi(Ti+Xj2Sdj)fi(Ti)forsomefacilityi=1;:::;mandsetSDnAi.Forafacilityi,denetheproblemKi(0)=maxSDnAiXj2Sdjmaxfwijj+cijjcij;0g0fi(Ti+Xj2Sdj)f0fi(Ti)whereijisthefacilityjiscurrentlyassignedtoandwijjisdenedasabove.Foraxed0,thisproblembelongstothesameclassastheproblemKP()intheproofofLemma 8.2.4 .IfwesortthecustomersinDnAiaccordingtodjmaxfwijj+cijjcij;0ginnon-increasingorder,thenanoptimalsolutiontoKi(0)containstherstkcustomersintheordering.Notethattheorderingofthecustomersisindependentof0.Dene0kto 166

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98 ].Mahdianetal.[ 98 ]showedthattheirtwo-phasealgorithmisa(Rf+ln+;1+Rc1 8.2.8 .ThisanalysisreliedonderivingafactorrevealinglinearprogrambyanalyzinganalgorithmthatscalesdowninLdiscretestepsrathercontinuously.Wecanapplyasimilaranalysisasthistoshowthatourtwo-phasealgorithmisa(Rf+ln+;1+Rc1 1:504)=(1:5181+;1:518)-approximationalgorithmfortheCCFLP.Therefore,thetwo-phasealgorithmhasaguaranteeof1:52. ThisimmediatelyleadstothefollowingresultfortheUFLPPwithseasonaldemands. Proof. 156 ]orVeinott[ 154 ]toevaluatethefunctionfi(z)inO(T2)time.Forxed-chargepluslinearproductioncostsandlinearholdingcosts,wecanusethe 167

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1 ],FedergruenandTzur[ 47 ],orWagelmansetal.[ 155 ]toevaluatethefunctionfi(z)inO(TlogT)time. 59 ].Theydiscussthreeexamplesoffacilitylocationproblemswithstochasticdemandsthatbelongtothisclassofproblems.OtherexamplesoffacilitylocationproblemswithsubadditivefacilitycostscanbefoundinGaborandvanOmmeren[ 58 ]andRodolakisetal.[ 121 ].GaborandvanOmmeren[ 59 ]developanapproximationalgorithmwithaguaranteeof2(1+),forany>0.Also,foraspecialclassofsubadditivecostfunctions,theydevelopanapproximationalgorithmwithaguaranteeof2.ItwasobservedinbothGaborandvanOmmeren[ 59 ]andRodolakisetal.[ 121 ]thattheconcaveenvelopeofasubadditivefunctionisa2-approximationofthefunction.SincetheGreedyAlgorithmthatwedevelopedaboveisa(1;2)-approximationalgorithmfortheCCFLP,thisyieldsa(2;2)-approximationalgorithmforthefacilitylocationproblemwithdiscretesubadditivecostfunctions,giventhatwecanevaluatetheconcaveenvelopeofasubadditivefunctionatasinglepointinpolynomialtime.Notethatthisislessrestrictivethanneedingtoconstructtheentireconcaveenvelopeofasubadditivefunctioninpolynomialtime.Hajiaghayietal.[ 75 ]alsoconsideredtheproblem(P)withunitdemandsbutwherethefunctionsfiareconvex.Theyoeredapolynomial-timeexactalgorithmforthisproblem.Thenexttheoremshowsthat,fornon-unitdemands,therecannotexistanapproximationalgorithmforthevariantof(P)wherethecostfunctionsareconvexfunctionsunlessP=NP. 168

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Proof. 2Pni=1aj?Wedeneaninstanceof(P)withncustomershavingdemandsdj=aj(j=1;:::;n)and2facilitieswithallconnectioncostsequalto0,i.e.cij=0fori=1;2andj=1;:::;n.Wedenethepiecewiselinearconvexcostfunctionsfifori=1;2as:fi(y)=8><>:1if0y1 2Pnj=1dj(2(n;2)+1)y1 2Pnj=1djify1 2Pnj=1dj:Ifthereexistsapartition,saySf1;:::;ng,thenwemayassignallcustomersinStofacility1andallcustomersinf1;:::;ngnStofacility2atatotalcostof2.Similarly,iftheoptimalsolutiontothe(P)is2,thenthesetofcustomersassignedtofacility1formapartition.Otherwise,atleastoneunitofdemandwillpaythecostofthesecondsegmentoffi,sotheoptimalcostisatleast2(n;2)+2.Nowsupposethatwehaveapolynomialtime(n;2)-approximationalgorithmthatweapplytothisproblem.Ifitreturnsasolutionwithcostnomorethan2(n;2),thenthereexistsapartition.Ifitdoesnot,thentheredoesnotexistapartition.Thisimpliesthedesiredresult. 8.1 fortheUFLPP-DA.WethendiscussanotherspecialclassoftheUFLPP-DAandshowthatitcanbeformulatedasaproblembelongingtoanewclassoffacilitylocationproblems,whichisageneralizationofthefacilitylocationproblemwithservice-installationcosts(seeShmoys 169

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142 ]).WerstbeginwithseveralresultsthatwillhelpdeterminethecomplexityofapproximatingthegeneralUFLPP-DA. Proof. Proof. 8.3.1 andthefactthatinproofofTheorem 8.1.1 ,wereducedtheSetCoverproblemtoaUFLPPwhereeachcustomerhasatmostonenon-zerodemandperiod. Therefore,usingtheresultofFeige[ 51 ],wecannotdevelopanapproximationalgorithmwithaguaranteeofbetterthan(1)lognunlessNPTIME[O(nloglogn)]forthegeneralUFLPP-DA.ItisappropriatetothenfocusonspecialcasesoftheUFLPP-DAthatcanbeapproximatedwithinaconstantfactor.Thefactthatwecanconvertany 170

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8.1.2 and 8.1.3 Proof. 8.3.3 ,wecanconverttheUFLPP-DAproblemintoametricUFLPinO(nT2+nmT)timesincewehaveO(nT)customersintheUFLPP.However,ifwenotethatthetimeO(nT2)reectsthetimerequiredtocomputePTt=1Ctd|tforeachcustomer|intheUFLPP,weseethatthiscanbereducedtoO(nT)timesinceeachcustomerhasexactlyonepositivedemand.Therefore,wecanconverttheUFLPP-DAproblemintoametricUFLPprobleminO(nmT)time. Proof. 8.3.3 ,wecanconverttheUFLPP-DAproblemintoaclassoffacilitylocationproblemwithserviceinstallations(withaknown6-approximationalgorithm)inO(nmT2)timesincewehaveO(nT)customersintheUFLPP.However,wemay(again)improvethetimerequiredtoperformthisconversionbynotingthatO(nmT2)reectsthetimerequiredtocomputefji=PTt=1Citd|tforeachcustomer|andfacilityiintheUFLPP.NotingthatforeachcustomerinthisparticularUFLPPinstancehasexactlyonepositivedemand,thistimecanbereducedtoO(nmT)time. 171

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8.2 totheUFLPP-DAwithseasonaldemands,i.e.,djt=djtforj=1;:::;nandt=1;:::;T.ThisisduetothefactthatifwearegivenaninstanceoftheUFLPP-DAwithseasonaldemands,theconversiontoaUFLPPdestroystheseasonality.ItisstillanopenquestionwheretheUFLPP-DAwithseasonaldemandsadmitsaconstantfactorapproximationalgorithm.WewillnowdiscussanotherspecialclassoftheUFLPP-DA,inparticular,weexaminetheUFLPP-DAwitheconomiclot-sizingcosts.Inotherwords,theproductioncostfunctionsareequaltoPit(pit)=ait+bitpitandHit(Iit)=hitIit.NotethatwealreadyknowfromLemma 8.3.2 ,thatthisproblemwithoutproductionsetupcostsisasdicultasthesetcoverproblemtoapproximate.However,wewillpresentageneralframeworktoformulatethisproblemasanewclassoffacilitylocationproblemsandthendiscussspecialcasesthatcanpossiblybeapproximatedwithinaconstantfactor.ItwillbeconvenienttodeneHisttobethevariablecostassociatedwithproducingaunitatfacilityiintimeperiodsandholdituntiltimeperiodt,i.e.,Hist=bis+Pt1=shi.Bydeningthevariablexijst=1ifwemeetthedemandofcustomerjintimeperiodtwithproductionatfacilityiintimeperiods,thentheUFLPP-DAwitheconomiclot-sizingcostscanbeformulatedasminmXi=1fiyi+mXi=1TXt=1aitzit+mXi=1nXj=1TXt=1tXs=1djt(cij+Hist)xijstsubjecttomXi=1tXs=1xijst=1forj=1;:::;n;t=1;:::;Txijstyifori=1;:::;m;j=1;:::;n;s=1;:::;t;t=1;:::;Txijstzisfori=1;:::;m;j=1;:::;n;s=1;:::;t;t=1;:::;Txijst2f0;1gfori=1;:::;m;j=1;:::;n;s=1;:::;t;t=1;:::;Tyi2f0;1gfori=1;:::;mzit2f0;1gfori=1;:::;m;t=1;:::;T:

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142 ]).Itisindeedageneralizationofthefacilitylocationwithserviceinstallationcostswithtwoadditionalconsiderations:(i)ifacustomerisassignedtoafacility,thefacilitycanselectoneservicefromasetofservicestoinstalltoservethecustomerand(ii)wepayacertainamountforeachcustomerdependingontheservicechosenforthecustomeratthefacility.InthecaseoftheUFLPP-DAwitheconomiclot-sizingcosts,thesetofcustomersisgivenbythepairs(j;t)forj=1;:::;nandt=1;:::;T.Foragivencustomer(j;t),thechoicetodeterminewhichservicetoinstallatthefacilitytowhichitisassignedisequivalenttodeterminingtheperiodinwhichwewillproducethedemandofthecustomer.Wemustpaythevariableproductionandinventorycostsassociatedwiththecustomerandthetimeperiodwewillservethecustomer,i.e.,ifwewillservecustomer(j;t)inperiodsatfacilityi,thenwepaydjtHist.Ingeneral,thenewfacilitylocationproblemwithcustomerserviceexibility(FLP-CSF)canbeformulatedasfollows.Wearegivenasetoffacilitiesi=1;:::;m,asetofservices,k=1;:::;K,andasetofcustomers`=1;:::;L.Thecostofopeningfacilityiisfi,thecostofinstallingservicekatfacilityiisfki,andtheper-unitcostofservingcustomer`withservicekatfacilityiisSik`.Thesetofservicesthatmayservecustomer`isdenotedbyK`.TheFLP-CSFisthusformulatedasminmXi=1fiyi+mXi=1KXk=1fkizik+mXi=1LX`=1Xk2K`d`Sik`xik`+mXi=1LX`=1d`ci`xik`subjecttomXi=1xi`=1for`=1;:::;Lxi`Xk2K`zikfori=1;:::;m;`=1;:::;Lxi`yifori=1;:::;m;`=1;:::;Lxi`2f0;1gfori=1;:::;m;`=1;:::;L

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8.1.3 andtheUFLPP-DAinTheorem 8.3.4 .Inparticular,wewillconsidertheFLP-CSFwherethereexistsanorderingofthefacilitiessuchthatifi
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8.3 thattheUFLPP-DAwitheconomiclot-sizingcostscanbeformulatedasanewclassoffacilitylocationproblemswithservice-installationcosts.Wearecurrentlyinvestigatingwhetherconstantfactorapproximationalgorithmsexistforspecialcasesofthisnewfacilitylocationproblem(theFLP-CSF)whichwouldincludegeneralizationsofthetwoproblemclassesoftheUFLPP-DAdescribedinTheorems 8.3.3 and 8.3.4 withproductionsetupcosts(i.e.,theUFLPP-DAwitheconomiclot-sizingcosts).ItwillalsobeinterestingtoinvestigatewhetherotherspecialclassesoftheUFLPPhaveconstantfactorapproximationalgorithmsincludinggeneralizationsoftheproblemsdescribedinTheorem 8.1.2 andTheorem 8.1.3 thatincludeproductionsetupcosts. 175

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2.1 ,Savelsbergh[ 136 ]usesabranchandpriceapproachforsolvingtheGeneralizedAssignmentProblem(GAP)byformulatingitasaset-partitioningproblem.Subsequently,setpartitioningformulationshaveledtoeectivesolutionmethodsforseveralnonlinearassignmentandfacilitylocationproblems;seeFrelingetal.[ 57 ],Shenetal.[ 141 ],Huangetal.[ 85 ],Shuetal.[ 144 ],orRomeijnetal.[ 129 ].Inalloftheseapplications,thecriticalfactorthatdeterminestheecacyoftheapproachistheabilitytosolvetheassociatedpricingproblem.Despitethefactthatthisproblemis,inmanycases,NP-hard,theydooftenallowforecientsolutionapproachesthatallowinstancesofthesetpartitioningproblemofsignicantdimensiontobesolvedinreasonabletime. 176

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77 ]evenforthecasewherePtandHtarethatoftheclassiceconomiclot-sizingproblem.However,anumberofspecialclassesofthisproblemarepolynomiallysolvable.WewilldiscussthePPCSPinmoredetailinSection 9.2 .Theremainderofthesectionisorganizedasfollows.InSection 9.1 weremindthereaderoftheset-partitioningformulation(SPF)oftheUFLPP.WediscussanimportantspecialcaseoftheUFLPP,inparticular,thecasewhentheproductionandinventorycostsarethatoftheclassiceconomiclot-sizingproblem.Forthisspecialcase,weshowthattheSPFyieldsatighterrelaxationthantheUFLPPitself.WeuseinsightfromthisresulttodiscusstherelationshipoftheSPFandtheUFLPPingeneral.InSection 9.2 ,wediscussthePPCSPinmoredetail.Inparticular,wediscussthecomplexityoftheproblemalongwithseveralpolynomiallysolvablesubclasses.WeconcludethechapterinSection 9.3 withasummaryandfutureresearchdirections. 177

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178

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9.1.1 ).Wethenuseinsightfromtheseresultstodiscusstherelationshipbetweenv(SPFR)andv(UFLPPR)ingeneral(Section 9.1.2 ). 179

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92 ].Therelaxationofthisformulationisthereforeexpectedtoprovideabetterlowerboundontheoptimalsolutionoftheproblem.Inparticular,wedeneabinarydecisionvariablezitthatisequalto1ifthereisaproductionsetupatfacilityiinperiodt,andabinarydecisionvariablexijstthatisequalto1ifwemeetthedemandofcustomerjintimeperiodtusingproductionintimeperiodsatfacilityi.Furthermore,wedeneHisttobethevariablecostassociatedwithproducingaunitatfacilityiintimeperiodsandholdituntiltimeperiodt,i.e.,Hist=bis+Pt1=shi.Wethencanformulateouruncapacitatedfacilitylocationandlot-sizingproblem(UFLLSP)asminimizemXi=1fiyi+mXi=1nXj=1djcijxij+mXi=1TXs=1aiszis+TXt=sHistnXj=1djtxijst!!subjectto 9{4 )ensurethatwemeetthedemandofeachcustomerthroughproductionandinventorydecisionsatthefacilitytowhichitisassigned,whileconstraints( 9{5 )ensurethataproductionsetuptakesplaceatafacilityinanytimeperiodthatproductiontakesplace.Intheset-partitioningformulationoftheUFLLSP,recallthatfor`= 180

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Proof.

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9{1 ),sothatoursolutiontoUFLLSPRsatisesconstraints( 9{3 ).Fori=1;:::;m,j=1;:::;n,andt=1;:::;T:tXs=1xijst=tXs=1LiX`=2x`ist`ijy`i=LiX`=2`ijy`itXs=1x`ist=LiX`=2`ijy`i=xijimplyingthatoursolutiontoUFLLSPRsatisesconstraint( 9{4 ).Fori=1;:::;m,j=1;:::;n,andt=s;:::;T,ands=1;:::;T:xijst=LiX`=2x`ist`ijy`iLiX`=2x`isty`iLiX`=2z`isy`i=zissothatconstraints( 9{5 )aresatised.Itisclearthatallvariablesdenedarenon-negativeandthatfori=1;:::;m,yi1bythedenitionofyiandconstraints( 9{2 )ofSPFR.Foraxedi=1;:::;mandj=1;:::;n,consider:xij=LiX`=2`ijy`iLiX`=2y`i1wheretheinequalityholdsbyconstraints( 9{2 )ofSPFR.Thisimpliesthatforanyi=1;:::;m,j=1;:::;n,s=1;:::;t,t=1;:::;Tthatxijandxijstarelessthanorequalto1.Forxedi=1;:::;mandt=1;:::;T,consider:zit=LiX`=2z`ity`iLiX`=2y`i1:Therefore,thecorrespondingsolutionisfeasibletoUFLLPSR.NowobservethatthesolutionvalueofywithrespecttofacilityiinSPFRisequaltoLiX`=1hi(`i)y`i=LiX`=2hi(`i)y`i=LiX`=2fi+nXj=1djcij`ij+gi(`i)!y`i

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9.1.1 todiscusstherelationshipbetweenv(SPFR)andv(UFLPPR)formoregeneralproductionandinventorycostfunctions. 9.1.1 willconvertanyfeasiblesolutiontoSPFRtoafeasiblesolutiontoUFLPPR.Inparticular,letp`it;I`itbetheproductionandinventorylevelsinanoptimalsolutionoftheoptimizationproblemcorrespondingtoHi(`i).Then,consideranyfeasiblesolutionytoSPFRanddeneacorrespondingsolutiontoUFLPPRasfollows:yi=LiX`=2y`ii=1;:::;mxij=LiX`=2`ijy`ii=1;:::;m;j=1;:::;n

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9.1.1 ,thissolutionsatisesconstraints( 7{1 )and( 7{2 ).Wealsohavethatforalli=1;:::;mandt=1;:::;T,Ii;t1+pit=LiX`=2I`i;t1y`i+LiX`=2p`ity`i=LiX`=2(I`i;t1+p`it)y`i=LiX`=2nXj=1djt`ij+Iit!y`i=LiX`=2nXj=1djt`ijy`i+LiX`=2Iity`i=nXj=1djtLiX`=2`ijy`i!+Iit=nXj=1djtxij+Iit;sothatthissolutionsatisesconstraints( 7{5 ).Therefore,thissolutionisfeasibletoUFLPPR.NowobservethatthesolutionvalueofywithrespecttofacilityiinSPFRisequaltoLiX`=1hi(`i)y`i=LiX`=2hi(`i)y`i=LiX`=2fi+nXj=1djcij`ij+gi(`i)!y`i=LiX`=2fi+nXj=1djcij`ij+TXt=1(Pit(p`it)+Hit(I`it))!y`i=fi(LiX`=2y`i)+nXj=1djcijLiX`=2`ijy`i!+TXt=1LiX`=2Pit(p`it)y`i!+TXt=1LiX`=2Hit(I`it)y`i!=fiyi+nXj=1djcijxij+TXt=1LiX`=2Pit(p`it)y`i!+TXt=1LiX`=2Hit(I`it)y`i!fiyi+nXj=1djcijxij+TXt=1(Pit(pit)+Hit(Iit));wherethelastinequalityholdssincethefunctionsPitandHitareconcaveandconstraints( 9{2 ).Therefore,anysolutiontotheSPFRcanbeconvertedtoasolutiontothe 184

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77 ]provethatthePPCSPisNP-hardevenwhentheproductionandinventorycoststructuresarethatoftheclassiceconomiclot-sizingproblem. 185

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8.2 ,i.e.,g(z)istheoptimalsolutionvaluetotheproblemminimizeTXt=1(Pt(pt)+Ht(It))subjecttoIt1+pt=tz+Itfort=1;:::;TI0=0pt;It0fort=1;:::;T:

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85 ]showthatforanyconcavefunctiong,theoptimalsolutiontothisproblemcanbefoundinpolynomialtime.SinceLemma 8.2.1 provesthatourfunctiongisconcave,wemayapplythealgorithmofHuangetal.[ 85 ]tosolvethePPCSPwithseasonaldemands.Inparticular,wereorderthecustomersinnonincreasingorderoftheratiorj 85 ]showthatanoptimalsolutionexiststothePPCSPwithseasonaldemandsthatselectstherstjcustomersinthisordering.TheorderingofthecustomersrequireO(nlogn)timeandweneedtosolvenproductionplanningproblems,sowecansolvethePPCSPwithseasonaldemandsinO(nmaxflogn;T2g).Whentheproductionandinventorycostsarethatofaneconomiclot-sizingproblem,theruntimeoftheapproachisO(nmaxflogn;TlogTg). 187

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8.2.1 ,itiseasytoseethatg(k;)isaconcavefunction.SincetheextremepointsoftherelaxationofCSP(k)areintegralandthealgorithmfromSection 6.1 returnsanextremepointsolutionifthefunctionisconcave,wecansolveCSP(k)inO(n2maxflogn;T2g)timeforgeneralconcaveproductionandinventorycostfunctionsandO(n2maxflogn;TlogTg)foreconomiclot-sizingcosts.Thisimmediatelyimpliesthat,bysolvingCSP(k)fork=1;:::;n,wecanndanoptimalsolutiontothePPCSPwithcustomer-specicpricesinO(n3maxflogn;T2g)andO(n3maxflogn;TlogTg)forgeneralconcavecostfunctionsandeconomiclot-sizingcosts,respectively.However,wecanreducetherunningtimeofthisalgorithmbyafactorofnbyemployingsimilaritiesbetweenthenproblemsoftheformCSP(k)thatneedtobesolved. 188

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6.1 tosolvethePPCSPwithcustomer-specicprices.Thisresultissummarizedinthefollowingtheorem: Proof. 6.1 .First,notethatallextremepointsofthecontinuousrelaxationofthefeasibleregionofeachoftheseproblemsisintegral,andapplyingtheapproachofSection 6.1 totherelaxationofCSP(k)forsomexedkyieldsanintegralsolution.Therefore,wecan,withoutlossofoptimality,relaxtheintegralityconstraints;wewill,forthesakeofconvenience,refertotherelaxationoftheproblemasCSP(k)also.Recallthat,intheapproachofSection 6.1 ,weconstructacollectionofcandidatesolutionsindexedby=f(i;j):i
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6.1 ),thetimerequiredtodeterminethesetsofI(i;j)0,I(i;j),andI(i;j)1forall(i;j)2inO(n2logn).ItnowremainstocompletethecandidatesolutionsforeachvalueofkbydeterminingthevaluesofthevariablesinI(i;j);wewillrefertothecorrespondingsolutionsasx(i;j)(k).ThesesolutionscanbefoundbysolvingthesubproblemsmaxX`2I(i;j)1r`+X`2I(i;j)r`x`g0B@k;X`2I(i;j)1d`+X`2I(i;j)d`x`1CAsubjectto(SP(i;j)(k)) 0x`1`2I(i;j)RecallthatSP(i;j)(k)canbesolvedbyrstsolvingtwolinearknapsackproblems,namely,max(ormin)X`2I(i;j)d`x`subjectto(KP(i;j)(k))X`2I(i;j)x`=kjI(i;j)1j0x`1`2I(i;j)andselectingthesolutionofthesetwoproblemswiththebestobjectivefunctionvaluetoSP(i;j)(k).ItisimportanttonotethatSP(i;j)(k)onlyhasafeasiblesolution,andthusonlyneedstobeconsidered,forjI(i;j)1jkjI(i;j)1j+jI(i;j)j.WewillproceedbyrstadaptingthealgorithmfromSection 6.1 by,foreach(i;j)2,solvingallrelevantsolutionsx(i;j)(k)consecutively,fork=jI(i;j)1j;:::;jI(i;j)1j+jI(i;j)j.Supposenowthatwehavefoundasolution 190

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6.1 showsthat,foragiveni,theonlyvariablesthatcanoccurinmorethanoneofthesetsI(i;j)aretheonesforwhichtherevenue/demandpairisidenticaltothatofcustomeri.Denoting,foreachi,thesetofsuchcustomersbyDi,weobtainPj2ijI(i;j)j=O(n+njDij)and,sinceitiseasytoseethatthesetsDiaredisjoint,weobtainthedesiredresult:OnXj=1Xj2i(jI(i;j)jlogjI(i;j)j+jI(i;j)j)!=O(logn+)nXj=1Xj2ijI(i;j)j!=O(n2maxflogn;g):

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66 ]consideredajointpricingandlot-sizingmodelwherethedemandtobesatiseddependsonthepricelevel.Theyassumethattherevenuefunctionineachperiodispiecewiselinearandconcaveandshowthattheproblemcanbeinterpretedasanorderselectionproblem.ThismeansthatthismodelisaspecialcaseofthePPCSPwhereeachcustomerhasexactlyoneperiodwithpositivedemand.Geunesetal.[ 66 ]showthatiftheproductionandinventorycostsareeconomiclot-sizingcosts,thisproblemcanbesolvedinO(JT2)timewhereJisthemaximumnumberofcustomershavingapositivedemandinanysingletimeperiod.Therefore,forthepricingproblemthatarisesinsolvingtheUFLPP-DA,wecansolvetheresultingPPCSPinO(nT2)time.WearecurrentlyinvestigatingthecomplexityofthePPCSPwhereeachcustomerhasexactlyoneperiodwithpositivedemandundermoregeneralconcaveproductionandinventorycostfunctions. 92 ].Although,ingeneral,wecannotconcludethatarelationshipbetweenthetightnessoftheset-partitioningformulationandthecontinuousrelaxationoftheUFLPP,solvingtheUFLPPthroughabranchandpricealgorithmhastheinherentadvantagethatwereducethesizeofthenonlinearoptimizationproblems 192

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193

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131 ]andRomeijnandSmith[ 130 ]yieldsbothweakandstrongduality.Thiscircumventsthenecessityofdealingexplicitlywithabstractdualspaces.Wethendevelopasimplexmethodbycharacterizingbasicprimalsolutions,constructingcomplementarydualsolutions,computingreducedcosts,andusingtheseresultstodeneaninnite-dimensionalpivotoperation.Wederiveconvergenceresultsand,asaconsequence,showtheexistenceofapairofprimalanddualsolutionsthatsatisfystrongduality.Withthissoundtheoreticalbasis,weanalyzeaclassofnetwork-ow 194

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10.1 weformallydeneourclassofminimum-costnetwork-owproblemsandlaythemathematicalfoundationsfortheinnite-dimensionalsimplexmethodbyderivingdualityresults.InSection 10.2 wedeveloptheinnite-dimensionalsimplexmethodandanalyzeitsconvergenceproperties.InSection 10.3 weanalyzeaclassofproblemsforwhichthesimplexmethodcanbeimplementedinsuchawaythatallpivotstakeonlyaniteamountoftime.InSection 10.4 weconcludethechapterbydiscussingfutureresearchdirections. 10.1.1AnInnite-DimensionalMinimum-CostNetwork-FlowProblemConsideraninnitedirectednetworkG=(N;A),whereN=f1;2;3;:::gdenotesthesetofnodeswhileANNdenotesthesetofdirectedarcs.Wemaketheregularityassumptionthateachnodeinthenetworkhasnitein-andout-degree.Inthischapter,westudyalargeclassofminimum-costnetwork-owproblemsdenedonsuchnetworks.Werstdeneavectorofcostsc2RjAj(withtypicalelementcij)thatrepresentstheunitcostsofashipmentalongthearcsofthenetwork.Furthermore,wedeneavectorofintegralsuppliesb2ZjNj(withtypicalelementbi)thatrepresentsthenetamountstobeshippedfromthenodesinthenetwork.Inparticular,thismeansthat(i)ifbi>0wesaythatiisasupplynodeandweneedtoshipthesupplyfromnodei; 195

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10-1 showsanexampleofanetworkdecomposedintolayers. Figure10-1. Anillustrationoflayers. 197

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10{1 )aretheusualow-balanceconstraintswhileconstraints( 10{2 )boundtheowonindividualarcs.Forconvenience,wewilldenotethefeasibleregionoftheprimalproblembyP.Ourregularityassumptionsonthestructureofthenetworkensurethattheow-balanceconstraintsarewelldened.Underourassumptionsontheproblemdata,SchochetmanandSmith[ 137 ]showthatthefunctionCiswelldened,i.e.,C(x)=limn!1X(i;j)2Ancijxijexistsforallx2PandiscontinuouswithrespecttotheproducttopologyonP. 131 ]andRomeijnandSmith[ 130 ]denethenaturaldualofinequality-constrainedlinearprogrammingproblems.Extendingthisideatoourproblemwouldyieldthefollowingdualproblem:maximizeB(;)limsupn!10@Xi2LnbiiX(i;j)2Anuijij1A

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10{1 )whilethevectorcontainsthedualvariables(ormultipliers)associatedwiththeconstraintset( 10{2 ).NotethatthedualobjectivefunctionBcanbewrittenasB(;)=Xi2NbiiX(i;j)2Auijijiftheinnitesumsconverge.Romeijnetal.[ 131 ]andRomeijnandSmith[ 130 ]provideasucientconditionunderwhichthereexistsaweakdualityrelationshipbetween(P)anditsnaturaldualproblem.Unfortunately,rewritingouroptimizationproblemusinginequalityconstraintsdoesnotsatisfythiscondition.Asanexample,consideraminimum-costnetwork-owprobleminanetworkwithnodesi=1;2;:::,arcs(i;i+1)fori=1;2;:::,anddemandsb1=1andbi=0fori=2;3;::::minimize1Xi=11 2ixi;i+1subjecttox1;2=1xi;i+1xi1;i=0fori=2;3;:::xi;i+11fori=1;2;:::xi;i+10fori=2;3;:::whosecandidatedualwouldbe:maximize1liminfn!1nXi=1i;i+1

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2ifori=1;2;:::i;i+10fori=1;2;:::Thereisonlyonefeasiblesolutiontotheprimalproblem:xi;i+1=1fori=1;2;:::withcost1.Moreover,anysolutionoftheformi=andi;i+1=0fori=1;2;:::isafeasiblesolutiontothecandidatedualwithcost.Soforany<1wehaveafeasiblesolutiontothecandidatedualwhosecostexceedsthatoftheoptimalprimalsolution.Infact,wehavethatthecandidatedualisunbounded.Weconcludethattheprimalandcandidatedualproblemdonotsatisfyweakduality.Weovercomethisproblemusingaconceptoftransversality(seealsoRomeijnetal.[ 131 ]andRomeijnandSmith[ 130 ]): 200

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Forconvenience,wewilldenotethefeasibleregionofthedualproblembyD.Thefollowingtheoremthenshowsthatthepairofproblems(P)and(D)satisfyweakduality. 10{2 ),=limsupn!10@Xi2Ln0@X(i;j)2AxijX(j;i)2Axji1AiX(i;j)2Anxijij1Abyconstraint( 10{1 ),=limsupn!10@X(i;j)2AnnABn;n+1xijiX(j;i)2AnnAFn;n+1xjiiX(i;j)2Anxijij1A=limsupn!10@X(i;j)2An(ijij)xij+X(i;j)2AFn;n+1xijjX(i;j)2ABn;n+1xiji1Alimsupn!10@X(i;j)2Ancijxij+X(i;j)2AFn;n+1xijjX(i;j)2ABn;n+1xiji1Abyconstraint( 10{3 ),limsupn!1X(i;j)2Ancijxij+limsupn!10@X(i;j)2AFn;n+1xijjX(i;j)2ABn;n+1xiji1A

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10{5 ),=C(x)whichprovesthedesiredresult. Wenextextendthenotionofcomplementaryslacknesstoourclassofproblems. Thisenablesustoshowthatifaprimalfeasiblesolutionandadualfeasiblesolutionsatisfycomplementaryslackness,theyareoptimaltotheirrespectiveproblems. Proof. 10{6 ), 202

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10{7 )and( 10{8 ),=limsupn!10@Xi2LnbiiX(i;j)2Anuijij1Abyconstraint( 10{5 ),=B(;):CombiningthiswiththeweakdualityresultofTheorem 10.1.2 weobtainthatxand(;)areoptimalsolutionsto(P)and(D),respectively. Constraint( 10{5 )seems,atrstsight,somewhatrestrictiveanddenitelyinconvenienttodealwith.Thefollowingpropositionandcorollarycharacterizetwosetsofmildsucientconditionsunderwhichthisconstraintcanbereplacedbyasomewhatstrongerbuteasierandmoreintuitiveone.Moreover,theresultsinSection 10.3 illustratethatevenweakerconditionssuceforalargeclassofminimum-costnetwork-owproblems. 10{5 )inthedualproblem(D)canbereplacedby 203

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10{3 ),( 10{4 ),and( 10{9 ).Wewillshowthat2Tsothatconstraint( 10{5 )issatisedaswell.First,notethatlimsupn!1X(i;j)2AFn;n+1xijjX(i;j)2ABn;n+1xijilimsupn!1X(i;j)2AFn;n+1xijj+X(i;j)2ABn;n+1xijilimsupn!10@X(i;j)2An;n+1xij1Amaxfjij:i2Ln+1g:Theresultnowfollowssincelimsupn!1X(i;j)2An;n+1xijS<1: 10.1.5 isensured. 10{5 )inthedualproblem(D)canbereplacedbyconstraint( 10{9 ). Proof. 10.1.5 applieswithS=MU,ortheprimalproblem(P)isinfeasible. Intheremainderofthischapterwewilldealwithsituationswhereconstraint( 10{5 )canindeedbereplacedbyconstraint( 10{9 ).Notethatsinceconstraint( 10{9 )is,in 204

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10{5 ),theresultsofTheorem 10.1.2 andTheorem 10.1.4 stillholdwiththemodieddualconstraintinsuchcases.Theresultsobtainedthusfardonotimplythatstrongdualityholds,i.e.,theexistenceofanx2Pand(;)2DsuchthatC(x)=B(;).Inthenextsectionwewilladdressthisissuethroughthedevelopmentofourinnitenetworksimplexmethod.Inparticular,wewillshowthatthereexistsapairofoptimalprimalanddualsolutionsthatsatisfystrongduality. 128 ]showthatthefeasibleregionPof(P)containsanextremepoint.However,itdoesnotimmediatelyfollowthattheoptimizationproblem(P)hasanextremepointoptimalsolution.Ingeneral,morerestrictiveassumptionsonthespacethattheprimal(anddual)optimizationproblemsareposedinarerequiredtoconcludethisdirectly.However,intheremainderofthissectionwewillneverthelessdevelopasimplexmethodthatmovesthroughasequenceofextremepointsofP.Analogoustothenite-dimensionalcase,weconstructacomplementarydualsolutiontoagivenbasicfeasible(extremepoint)solutionto(P).Weusethisdualsolutionandthestructureofprimalbasicfeasiblesolutionstocharacterizereducedcostsandthepivotstep.Finally,weshowthatthesequenceofextremepointsgeneratedbythealgorithmconvergesinvaluetotheoptimalsolutionvalueof(P)andalsostudytheissueofsolutionconvergence. 10.2.1.1ExtremePointsandBasicPrimalSolutionsInniteminimum-costnetwork-owproblems,anextremepointsolutioncanbecharacterizedbyadecompositionoftheowvariablesintobasicandnonbasicvariableswherethe(basic)arcscorrespondingtothebasicvariablesformaspanningtreeinthegraph.Wheneachofthenonbasicvariablesisxedateitheritsupperoritslowerbound, 205

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128 ]providesuchacharacterizationfortheclassofnetwork-owproblemsthatwestudyinthischapter.Theyshowthatafeasiblesolutionxisanextremepointsolutionifandonlyifthegraphinducedbytheso-calledfreearcs,i.e.,thearcscorrespondingtovariablesforwhichtheboundconstraintsarestrictlysatised,hasthepropertythateachnodeliesonnomorethanoneinnitepath.Inaddition,theyshowedthatthefreearcgraphforeachextremepointcanbeextendedsuchthateachnodeiliesonexactlyoneinnitepathintheextendedgraph.Wemaythenviewthisextendedfreearcgraphasabasiccharacterizationoftheextremepoint,wherethevariablescorrespondingtoarcsintheextendedfreearcgrapharebasicandtheothervariablesarenonbasic.Itwillbeusefultoformanintuitiveanalogybetweenthebasicarcgraphintheinnitenetworkandthebasicarcgraphinanitenetwork.Inparticular,wecouldinterpretthebasicarcgraphintheinnitenetworkasaspanningtree\rooted"atavirtualnodeatinnity.Notethatwecould,inprinciple,alsochooseanynodei2Nastherootofthistree,aslongaswetheninterpretinnitepathsasbeingconnectedtooneanotherthroughthevirtualnodeatinnity. 206

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10{9 )issatised.Itremainstobeshownthatthischoiceuniquelydeterminesthevaluesofalltruedualmultipliers.Wewilldothisasfollows:werstconstructadualsolutionwiththepropertythatthedualmultipliersalonganinnitepathinthebasicarcgraphconverge,insomesense,tozero;wenextshowthatthissolutionindeedsatisesconstraint( 10{9 ).Recallthat,inthebasicarcgraph,eachnodeinthenetworkhasexactlyoneinnitepath.However,thesepathsarenotnecessarilydirected.LetBbeasetofbasicarcscorrespondingtoabasicfeasiblesolutionandlettheinnitepathinthebasicarcgraphfromsomenodei02Nvisitthesequenceofuniquenodesi0i1i2,wherek=0if(ik;ik+1)2Bandk=1if(ik+1;ik)2B(k=0;1;2;:::).Wecanthendenethefollowingcandidatesetofdualmultipliers~forthenodesonthispath.Westartbysetting~i0=0.Next,weiterativelyset~ik+1=8><>:~ikcikik+1if(ik;ik+1)2B~ik+cik+1ikif(ik+1;ik)2Bfork=0;1;2;::::Equivalently,thismeansthatweset~ik=k1X`=0`ci`+1i`(1`)ci`i`+1fork=0;1;2;::::Nownotethat1X`=0`ci`+1i`(1`)ci`i`+1X(i;j)2Ajcijj<1

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10{9 ).Thisdoesnotimmediatelyfollowfromtheresultssofarsinceinnitepathsinthebasicarcgraphdonothavetopassthroughthelayersinincreasingorder.Therefore,wedenemntobethesmallestindexofalayerwiththepropertythatatleastoneinnitepathfromnodesinlayersLnandbeyondpassthroughit.NotingthatAnAmn1isthesetofarcsentirelycontainedinlayersmn;mn+1;:::thisimpliesthatmaxfjij:i2LngX(i;j)2AnAmn1jcijj:SinceP(i;j)2Ajcijj<1,theresultnowfollowsiflimn!1mn=1.Itiseasytoseethatlimn!1mnexistssincemnisnondecreasingwithrespectton.Nowsupposethatthislimitisnite,saym.Thismeansthat,foralln=m;m+1;:::,theremustexistanodei2LnwiththepropertythattheinnitepathassociatedwithipassesthroughLm.Inotherwords,thereareaninnitenumberofnodeswiththepropertythattheirinnitepathinthebasicarcgraphpassesthroughLm.Ontheotherhand,sincethenumberofnodesinLmisniteandthedegreeofallthesenodesisnitetherecanonlybeanitenumberofdistinctinnitepathsthatpassthroughLm.Therefore,thereexistsanodeinLmthatliesontwodistinctinnitepaths(therebyessentiallyformingwhatcouldbecalledadoubly-innitepath).However,thiscontradictsthenatureofthebasicarcgraph.Wethusconcludethatconstraint( 10{9 )issatised. 208

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10{3 )and( 10{4 )(atequality)forallarcs(i;j)2B.However,consideranonbasicarc(i;j)62B.Weshouldthenconsidertwocases: (i) Supposethatxij=0.Thenconstraint( 10{4 )issatisedforarc(i;j),butconstraint( 10{3 )maybeviolated,i.e.,wemayhavethatciji+j<0. (ii) Supposethatxij=uij.Thenconstraint( 10{3 )issatisedforarc(i;j),butconstraint( 10{4 )maybeviolated,i.e.,wemayhavethatij=cij+ij<0or,equivalently,ciji+j>0.Analogoustominimum-costnetwork-owproblemsinnitenetworks,wenowdenethereducedcostassociatedwitharc(i;j)2Aascij=ciji+j:Itimmediatelyfollowsthatcij=0if(i;j)2B.Moreover,thecomplementarydualsolutionisinfeasibleifandonlyifthereexistssomearc(i;j)62Bsuchthat(i)xij=0andcij<0or(ii)xij=uijandcij>0.Ontheotherhand,ifthecomplementarydualsolutionisfeasible,Theorem 10.1.4 saysthatit,aswellasthecorrespondingprimalsolution,isoptimal. 209

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10.2.1.3 isselected.Ifthisarcisaddedtothebasicarcgraphofthecurrentsolutiononeoftwothingswillhappen: 1. A(nite)cycleiscreatedinthebasicarcgraph.Inthatcase,weproceedexactlyasinthenitenetworksimplexmethodandchangetheowalongthiscycleuntiloneofthebasicarcsinthecyclereachesoneofitsbounds.Thatarcisthenremovedfromthebasis,anewbasicfeasiblesolutionisobtained,andanewdualcomplementisdetermined.Wewillcallthisanitecyclepivot. 2. Twoinnitepathsareconnectedinthebasicarcgraph,creatingadoubly-innitepath.Asintheprevioussituation,wemaynowaugmenttheowalongthisdoubly-innitepathwhileensuringfeasibilityofthesolutionuntiloneofthearcsinthedoubly-innitepathreachesoneofitsbounds.Thisarcisthenremovedfromthebasis,therebybreakingthedoubly-innitepath,anewbasicfeasiblesolutionisobtained,andanewdualcomplementisdetermined.Notethat,usingthepreviouslydiscussedinformalinterpretationofinnitepathsleadingtoavirtualnodeatinnity,wemayinterpretadoubly-innitepathasan\innitecycle".Wewillthereforecallthisaninnitecyclepivot. 210

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Step0. 211

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Proof. 105 ]). Proof. 10.2.1.2 itimmediatelyfollowsthatanycomplementarydualsolutionsatises
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Proof. 10.2.1 and 10.2.2 implythatthereexistsaconvergentsubsequenceoftheprimal/dualpairsf(xn;n;n):n=1;2;:::g,sayindexedbyfnm:m=1;2;:::g.Letlimm!1(xnm;nm;nm)=(x;;):Sinceforallm=1;2;:::,xnmand(nm;nm)satisfycomplementaryslackness,itfollowsthatxand(;)satisfycomplementaryslacknessaswell.Inaddition,fromthedescriptionofthesimplexmethoditcanbeseenthat(;)isdualfeasiblesothatthepairxand(;)satisfystrongdualitybyTheorem 10.1.4 andareoptimaltotheirrespectiveproblems.Furthermore,sincethesequenceofcostsC(xn)isnonincreasingandthereisasubsequenceoffxn:n=1;2;:::gthatconvergestoanoptimalcostsolutionwehavelimn!1C(xn)=C(x)=CbythecontinuityofC. 10.2.3 thatanylimitpointofthesequenceofsolutionsgeneratedbythesimplexmethodisoptimal. 213

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10.2.4 showsthatxisanoptimalsolutionto(P).Thiscontradictstheuniquenessoftheoptimalsolutionsothattheresultfollows. Establishingsucientconditionsforuniquenessoftheoptimalsolutiontoaninnite-horizonoptimizationproblemisnotoriouslyhard,asdiscussedinRyanandBean[ 135 ],andmostoftheliteratureoninnite-horizonoptimizationhasdealtwithderivingresults(suchastheexistenceofsolutionorforecasthorizons)underthisassumption(see,forexample,BeanandSmith[ 15 16 ],BesandSethi[ 21 ],andSchochetmanandSmith[ 139 ]).Theorem 10.2.5 ,whichstatesthattheInniteNetworkSimplexMethodconvergessolution-wisewhentheoptimalsolutionisunique,isafurtherexampleofsucharesult.However,duetotheadditionalnetwork-owstructureinourproblemclass,wecanestablishthefollowingsucientconditionunderwhichtheoptimalsolutionisindeedunique:

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10-2 .Thesuppliesareb1=1andbi=0fori=2;3;:::.Thecostsofthearcsaregivenasfollows:c12=c13=1;c2k;2k+2=c2k+1;2k+3=1 2kfork=1;2;:::;c26=0;c6k+2;6k+6=c6k;6k+2fork=1;2;:::;andc6k1;6k+3=c6k3;6k1fork=1;2;:::.Theupperboundsarechosentobeuij=2forall(i;j)2Asothattheyarenonbindinginanyfeasiblesolution.Thelayerswillthenbedenedasfollows:L1=f1g,L2=f2;3g,L3=f4;5;6g,L4=f7;8;9g,etc.Itiseasytoseethatthisproblemhastwoalternativeoptimalsolutions.Therstsolution,sayx,istosetx1;2=x6k4;6k=x6k;6k+2=1fork=1;2;:::andxij=0forallotherarcs.Thesecondsolution,sayx,istosetx1;3=x6k3;6k1=x6k1;6k+3=1fork=1;2;:::andxij=0forallotherarcs.Thevalueofbothsolutionsis1.Nowsupposethatwestartthesimplexmethodwiththefollowinginitialsolution:thebasicvariablesarex13=x2k+1;2k+3=1fork=1;2;:::andx2k;2k+2=0fork=1;2;:::andtheremainingvariablesarenonbasic.Thecostofthissolutionisequalto2. 215

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4.Therearenownononbasicvariablesinlayers1and2withnegativereducedcostsothecurrentsolutionissavedasx2. 8.Therearenownononbasicvariablesinlayers1{3withnegativereducedcostsothecurrentsolutionissavedasx3. 10.2 wedevelopedanetworksimplexalgorithmforsolvingminimum-costnetwork-owproblemsininnitenetworksdescribedinSection 10.1 .However,therearetwoobstaclesthatmayingeneralpreventthisalgorithmtobeimplementableandusedtoapproximatethesolutiontosuchproblems: (i) Eventhougheachiterationofthesimplexalgorithmconsistsofonlyanitenumberofpivots,eachindividualpivotmayrequireaninnitenumberofoperations. 216

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Anexamplewithmultipleoptimalextremepointsolutions. (ii) Thealgorithmrequiresabasicfeasibleinitialsolution,anditisnotclearhowsuchasolutionmaybeobtained.Inthissection,wewillstudyalargeclassofnetwork-owproblemsforwhichboththeseobstaclescanbeovercome.Inparticular,weconsidergraphsGwiththepropertythatforeachnodei2N,thesetofitspredecessorsPi(i.e.,allnodesjwiththepropertythatthereexistsadirectedpathfromjtoi)isnite.Werefertothisasthenitepredecessorassumption.Notethatthisassumptionwilloftenbesatisedinsequentialdecisionproblemsovertime.Forthisclassofgraphswewillstudyproblemsforwhichtheowbalanceequalityconstraints( 10{1 )arereplacedbyinequalityconstraints,representingthatitisfeasibleto(i)delivermoreunitsthanrequiredtodemandnodes;and(ii)usefewerthanthesuppliedunitsfromsupplynodes.Thisclassofproblemsclearlyencompassesproblemsforwhichthischangecanbemadewithoutlossofoptimalitybutalsoproblemsthatarenaturallyformulatedinthisway.Forexample,inproductionplanning(orlot-sizing)problemsoveraninnitehorizonwecanwithoutlossofoptimalityassumethatthedemandineachperiodwillbesatisedatequality;moreover,thesupplyatthesupplynodescanbeviewedasaproductioncapacityanddoesnotneedtobefullyused.WewillreformulatetheproblemasanequivalentproblemthatfallswithintheproblemclassintroducedinSection 10.1 .Wewillthencharacterizetheextremepointsoftheequivalent 217

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10.1 withtheonlymodicationbeingthattheowbalanceconstraints( 10{1 )arereplacedbyXj2N:(i;j)2AxijXj2N:(j;i)2Axjibiforalli2N:( 10{1 (i) areconnectedbyanarctoanodeinlayern:L0n+1=j62Ln:9(i;j)2Awithi2Ln; orareapredecessorofanodeinL0n+1.Moreformally:Ln+1=L0n+1[[i2L0n+1PinLn:Thisimpliesthatifi2LnthenPiLn,sotheredonotexistanyarcs(j;i)wherej2Ln+1andi2Ln,whichinturnimpliesthatABn;n+1=;.Intheremainderofthissectionwewillassumethatwehavealayeringofthisform. 218

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10.1 .Acommonapproachinnitenetworkswouldbetosimplyaddslackvariablestotheinequalityconstraints.Thesevariableswouldthenbeinterpretedasowstoasinknodethatabsorbsallexcesssupplies.However,inthecaseofaninnitenetworkthissinknodehasinnitein-degreeand,moreover,itsdemandwouldnotbewelldened.Therefore,weinsteaduseaninnitedirectedpathtorepresentthesurplusow:foreverynodeii0weintroducearticial(transshipment)nodesi1;i2;i3;:::andarticial(costless)arcs(i`;i`+1)for`=0;1;2;:::.Wewillrefertothisinnitepathasthepathtoinnity(PTI)correspondingtonodei.Figure 10-3 showsanetworkwherethewhitenodesrepresentthenodesintheoriginalgraphandthegreynodesarethePTIs. Figure10-3. Areformulatednetwork-owproblem. Wethusobtainthefollowingminimum-costnetwork-owproblem(P0):minimizeX(i;j)2Acijxij

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Forconvenience,wewilldenotethegraphcorrespondingtothisreformulatedproblembyG0=(N0;A0).Notethattheupperboundsinconstraints( 10{10 )areredundantbutareincludedtoensurethat(P0)fallswithinourclassofminimum-costnetwork-owproblemsdescribedinSection 10.1 .Inparticular,thepredecessorsetsofthenodesinNhavenotchangedanditiseasytoseethatthepredecessorsetofi`containsonlyanitenumberofelementsmorethanPi(foralli2Nand`=1;2;:::).Sincethearticialarcsarecostlessitimmediatelyfollowsthattheassumptiononthecostfunctionissatisedaswell.Moreover,aswewillseelater,wecanassumewithoutlossofgeneralitythatthedualvariablesi`=0andi`1i`=0fori2Nand`=1;2;:::.Finally,sinceG0satisesthenitepredecessorconditionitiseasytoseethatwecancreatealayeringforwhichallarcsconnectinglayerntolayern+1areforwardarcs,i.e.,ABn;n+1=;.Toensurethatourweakandstrongdualityresultsapplyto(P0),wemakethefollowingassumption: 10{5 )inthedualof(P0)canbereplacedbyconstraint( 10{9 ).NotethatthisassumptionisweakerthantheassumptioninProposition 10.1.5 appliedto(P0)sinceweonlyconsider 220

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10.1.6 appliedtothegraphGbutmuchweakerthanthatassumptionappliedtotheextendedgraphG0). 10.1 and 10.2 .)Withthisconvention,agivensetofbasicarcsB0correspondingtoabasicfeasiblesolutionimpliesanaturaldecompositionoftheoriginalsetofnodesNintoacountablenumberofequivalenceclassesofnodesthatareconnectedthrougharcsinthebasis.Eachoftheseequivalenceclassesisofoneofthefollowingtypes: (i) itisnite,i.e.,allnodesintheclassareconnectedtoaninnitepathconsistingofarcsinA0nA(i.e.,articialarcs); (ii) itisinnite,i.e.,allnodesintheclassareonaninnitepathconsistingofarcsinA(i.e.,originalarcs).Basicfeasiblesolutionsthatonlycontainequivalenceclassesoftype(i)areofparticularimportance.Wewillsaythatsuchsolutionssatisfythenitepartitionproperty: 221

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10.2 wehaveassumedthataninitialbasicfeasiblesolutiontotheminimum-costnetwork-owproblemisavailable.Innitenetworks,wecaneasilyformulateaPhaseIminimum-costnetwork-owprobleminanextendednetwork.Thiscouldbeachieved,forexample,byaddingatransshipmentnodetothenetworkandarcsfromallnodeswithbi>0tothenewnodeandfromthenewnodetoallnodeswithbi<0.Inaddition,anarcfromthenewnodetoasinknodewillaccountforanyexcesssupply.Atrivialbasicsolutionisfoundbylettingallnewarcsbebasicandusedtosupplyalldemands.Thebasisisthenextendedwithoriginalarcshavingow0.Whenthecostsofthenewarcsarepositive(sayequalto1)andthecostsofalloriginalarcsaresettozero,anybasicoptimalsolutiontothePhaseIproblemprovidesabasicfeasiblesolutiontotheoriginalproblem.Intheinnite-dimensionalcasethesituationismorecomplicatedsincethenewtransshipmentnodewouldhaveinnitein-andout-degreeandtransshipaninniteamountofow.Inaddition,weareinterestedinndingabasicfeasiblesolutionthatsatisesthenitepartitionproperty.Wethereforeproposetoconstructanextendednetwork-owproblemthatwillproducesuchasolutionbyemployingthelayeringofthenodesandthearcsingraphG0asdescribedinSection 10.1.1 .NotethatthislayeringcorrespondstoalayeringoftheunderlyinggraphGaswell.Inthefollowing,ifweusethelayeringofGwewillusethenotationofSection 10.1.1 whileifweusethelayeringofG0wewilladda0totherelevantsymbol.TheideaofthePhaseIproblemistoallowthesatisfactionofalldemandsinalayerthroughsuppliesinthesamelayer,sothatwemaydetermineaninitialsolutiontothePhaseIproblemeasily.Tothisend,weextendthenodesetN0byaddinganarticialsupplynodesnwithsupplyS0+Pi2Lnmax(bi;0)toeachlayerLn(n=1;2;:::).Inaddition,wecreateanarcfromnodesntoeachdemandnodei2Ln(i.e.,toeachnodewithbi<0)aswellasaPTIforsn(consistingofnodessn`(`=1;2;:::)andcorrespondingarcs)withappropriateupperbounds.Observethat, 223

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10.3.1 impliesthatafeasiblesolutiontothePhaseIproblemexistsinwhichalldemandsaresuppliedwithinthecorrespondinglayer,bysettingxsn;i=biforalliwherebi<0.Abasisforthissolutionconsistsof,foreachn=1;2;:::,(i)aspanningtreeinthesubnetworkinducedbythesetLn,(ii)thePTIsofthesupplyandtransshipmentnodesinLn,and(iii)thePTIsofthedemandnodesinLnexceptfortheirrstarc.WenextconstructacoststructurefortheextendedgraphwiththepropertythatanybasicoptimalsolutiontothePhaseIproblemisabasicfeasiblesolutionto(P0)thatsatisesthenitepartitionproperty.Inaddition,thiscoststructurewillensurethatapplyingtheInniteNetworkSimplexMethodtothePhaseIproblemyieldsasequenceofsolutionsthatconvergestoitsoptimalsolution.WestartbylettingthecostofallarcsinA0aswellasthePTIofnodessnbeequaltozero.Inaddition,wemakethecostofallarcs(sn;i)foralli2Lnsuchthatbi>0andforalln=1;2;:::equaltoc0sni=1.Anysolutionwithcost0isthenfeasibleto(P0).Tondanoptimalsolutionthatsatisesthenitepartitionproperty,wenextreimposecostsonarcsinAn;n+1foralln=1;2;:::.Tothisend,denotethesearcsbya1;a2;:::andassumethattheyareorderedinsuchawaythatarcsbetweenearlierlayershavelowernumbersthanarcsbetweenlaterlayers:ifaj2An;n+1andak2Am;m+1thenm>nimpliesthatk>j.WethendeneU`=`X`0=1ua`0andsetthecostsofthesearcsequaltoc0a`=U`fora`21[n=1An;n+1where<1 2.Thismeansthatsatisfyingaunitofdemandinlayernfromanoriginalsupplynodecostslessthan1sothatitisalwaysadvantageoustosupplyaunitofdemandthroughtheoriginalnetworkratherthanfromthearticialsupplynodes.Inaddition, 224

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10.1.1 thatensuresthat(i)adualcomplementarysolutiontoanyprimalbasicfeasiblesolutioniswelldened;and(ii)theprimalobjectivefunctioniscontinuous.However,wewillseethattheseobstaclesmaybeovercomeforthePhaseIproblem.First,notethatsincetheinitialbasicfeasiblesolutiontothePhaseIproblemhasthenitepartitionpropertyitsdualcomplementiswelldened.ThismeansthatwecanapplythesimplexalgorithmdevelopedinSection 10.2 .ThefollowingtheoremthennotonlyshowsthatvalueandsolutionconvergenceholdinthesensethattheresultsofTheorem 10.2.3 andCorollary 10.2.4 extendtothePhaseIproblembut,inaddition,thatthesequenceofsolutionsgeneratedbytheinnitenetworksimplexmethodconverges.Further,weassumethatafteriterationn,ifthereexistsanodeisuchthatbi<0andxsn;i>0thenweterminatethealgorithmsincethisimpliesthattheoriginalproblemisinfeasible. Proof. 225

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10.2.2 holdsand,inturn,theresultofCorollary 10.2.4 aswell.Thisprovesthatxisabasicfeasiblesolutionto(P0).Next,supposethatxdoesnotsatisfythenitepartitionproperty.ThismeansthatthereexistsaninnitepathofbasicarcsinA.Moreover,duetothecoststructureofthePhaseIproblemthatensuresthatitisoptimaltosatisfydemandfromlaterlayers,noneofthearcsonthisinnitepathcarryzeroow.Butsincethecostofsendingaunitofowalongthispathisstrictlypositiveandtheowsareintegralthiscontradictstheoptimalityofx.Thisprovesthatxsatisesthenitepartitionproperty.Finally,supposethatthesequenceofsolutionsfxn:n=1;2;:::gdoesnotconverge.Thisimpliesthatthereexistsasubsequencefnm:m=1;2;:::gsuchthatlimm!1xnm=~x6=x:Nowsupposethatthereexistsatleastonearcin[1n=1An;n+1onwhichxand~xdier,andlet`=argminf`0=1;2;::::xa`06=~xa`0gdenotethesmallestindexofsuchanarc.Withoutlossofgeneralitysupposethatxa`0<~xa`0.Then1X`0=1c0a`0xa`01X`0=1c0a`0~xa`0=c0a`(xa`~xa`)+1X`0=`+1c0a`0(xa`0~xe`0):Moreover,1X`0=`+1c0a`0(xa`0~xe`0)1X`0=`+1c0a`0ua`0=1X`0=`+1U`ua`01X=U`+1=U`+1 2,sothat 226

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Intheory,theresultsofthissectionshowthatwecanndabasicfeasiblesolutionthatsatisesthenitepartitionpropertybysolvingaPhaseInetwork-owproblem.Thissolutionmaythenbeusedasthestartingsolutionfortheactualnetwork-owproblemthatweintendtosolve,whichwerefertoasthePhaseIIproblem.However,themajorpitfallofthisapproachisofcoursethefactthatitwilltakeaninniteamountoftimetondthestartingsolutionforthePhaseIIproblem,therebypreventingthepracticalimplementationofthemethod.InthenextsectionwewilldevelopahybridPhaseI/IIsimplexalgorithm.OneoftheresultsobtainedasthebyproductoftheproofofTheorem 10.3.3 willbeimportantinthedevelopmentofthishybridalgorithmandwethereforestateitasacorollary. 10.3.4 ,whichsaysthatthesimplexalgorithmappliedtothePhaseIproblemyieldsasequenceofsolutionswiththepropertythattheowsuptoanylayereventuallylockintotheowsintheoptimalsolutiontotheproblem.Moreprecisely,letusdenotethesequenceofsolutionsthatisgeneratedbythePhaseIalgorithmbyfyn:n=1;2;:::gandlet 227

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10.3.4 weknowthat,afterperformingkqmiterationsofPhaseI,the 229

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Wenextprovideaconvergenceresultonthecostsassociatedwiththematrixofiteratesfxn;m:m=1;:::;n;n=1;2;:::g. Proof. 10.3.5 ,Theorem 10.2.3 ,andthefactthatlimm!1qm=1implythatlimm!1limn!1Cxn;m(Aqm)=limm!1Cxm(Aqm)=limm!1C(xm)=C: 230

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Step0.

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10.3.6 byshowingthatforallnthereexistssomeanwiththepropertythatvalueconvergenceholdsforthesequenceofsolutionsfxn;an:n=1;2;:::g.Forconvenience,letA0=;. 10.3.5 .Then,letan=max0;maxm=1;:::nfm:kqmngforn=1;2;::::Bythedenitionofqmwehavethatthesequencefan:n=1;2;:::gisnondecreasing.Also,sincelimm!1qm=1andlimm!1kqm=1(bytheirrespectivedenitions),itfollowsthatlimn!1an=1.Further,sincexn;0(Aqan)=x0(Aqan)andpivotsperformedobtainingxn;anonlyaectowonarcsuptoqanwehavexn;an(Aan)=xan(Aan):Sincefxan:n=1;2;:::gisasubsequenceoffxm:m=1;2;:::g,thissubsequencecontainsaconvergentsubsequencewhichwewilldenotebyfn:n=1;2;:::g.Thatis,foreachmwehavelimn!1xn(Am[Am;m+1)=x(Am[Am;m+1) 232

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Infact,thisproofimmediatelyimpliesthefollowingsolutionconvergenceresult: (i) anyconvergentsubsequenceconvergestoanoptimalsolution; (ii) iftheoptimalsolutionto(P0)isunique,sayx,thenlimn!1xan(Aan)=x: 233

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3 ofthisdissertation,weexaminedanonlinearextensionofthegeneralizedassignmentproblem(theNL-GAP)wherethereisanonlinearcostfunctionforeachfacilitywhoseargumentisalinearfunctionoftheassignmentstothefacility.Inthesupplychainmanagementcontext,wecaninterpretthisfunctionastheproductioncostoftheamountofdemandassignedtothefacility.Oftentimes,productioncostfunctionsinsupplychainmanagementareill-structured(suchasnon-dierentiableordiscontinuous)sothatstandardoptimizationtechniquesmaynotbeapplicable.WeovercomethisdicultyinChapter 3 byusingtheoryfromlinearprogrammingindevelopingnewstructuralpropertiesaboutsomeoptimalsolutiontoaclassofnonlinearcontinuousoptimizationproblems.Thisapproachrequiresnoassumptionsaboutthenonlinearfunctionsotherthantheexistenceofanoptimalsolutiontotheproblemandleadstoanovelsetofdualmultipliersfortheproblem.WeusethisapproachinexaminingthecontinuousrelaxationoftheNL-GAPinChapter 4 anddevelopingagreedyprocedureforit.Weshowthatforfacility-independentparameters,thisgreedyprocedureisasymptoticallyoptimalandfeasibleforanysetofcontinuousfacilitycostfunctions.Forfacility-dependentparameters, 235

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5 ,weuseageometricalinterpretationoftheresultsinChapter 3 todevelopmethodstosolveaclassofconvexprogrammingproblems.ThisclassofproblemsincludestherelaxationoftheNL-GAPwheretheproductioncostfunctionsareconvex.ThesupplychainplanningproblemwithcustomerselectionthatarisesfromtheNL-GAPisaninterestingnonlinearknapsackproblem.InChapter 6 ,weusethestructuralresultsfromChapter 3 todevelopanecientalgorithmtosolvethecontinuousrelaxationofthisnonlinearknapsackproblem.Ouralgorithmusescomplementaryslacknessconditionsfromlinearprogramminginanovelwaythatrequiresnoassumptionsonthenonlinearportionoftheproblem.Weextendthealgorithmtosimilarclassesofnonlinearmultipleknapsackproblemsandnonlinearmultiple-choiceknapsackproblems.Thesealgorithmsareshowntobeextremelyeectivewhencomparedtoacommercialglobaloptimizationsoftwarepackage.Inthefuture,itwillbeinterestingtoexamineiftheapproachdiscussedinChapter 3 canbesuccessfullyappliedtootherclassesofnonlinearoptimizationproblems.InChapter 7 ofthisdissertation,wehaveexaminedintegratingfacilitylocationandproductionplanningproblems.Intheseproblems,wemustassignthesetofcustomerstoopenfacilitiesanddeterminetheproductionandinventorylevelsofeachopenfacilitytoensurethatwemeetthedemandassignedtothefacilityineachtimeperiod.InChapter 8 ,weshowthatitisunlikelythat,ingeneral,theseclassesofintegratedlocationandproductionplanningproblemscanbeapproximatedwithinaconstantfactor.Therefore,itisappropriatetofocusonapproximationalgorithmsforspecialcasesoftheproblem.Wehaveshownthatseveralspecialcasesoftheproblemcanbeapproximatedwithinaconstantfactor.Oneofthesespecialcasesgivesrisetoametricuncapacitatedfacilitylocationproblemwhereeachfacilitycostfunctionisaconcavefunctionoftheamountofdemandassignedtothefacility.Forthisnewclassoffacilitylocationproblems, 236

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9 ,wesetthetheoreticalfoundationsofabranchandpricealgorithmtosolvetheseintegratedproblems.Thesupplychainplanningproblemwithcustomerselectionthatarisesfromthisclassofproblemsisanintegratedproductionplanningandcustomerselectionproblem.AlthoughthisclassofproblemswasrecentlyshowntobeNP-hard,wediscussseveralpracticallyrelevantspecialcasesoftheproblemthatarepolynomiallysolvable.Itwillbeinterestingtoseeifthetheoreticalfoundationsofthealgorithmleadtoacomputationallyecientexactalgorithmfortheseintegratedlocationandproductionplanningproblems.InChapter 10 ofthisdissertation,weexaminednetworkowproblemsininnitenetworks.Thisclassofproblemsencompassesmanysupplychainplanningproblemswhereasequenceofdecisionsneedstobemadeoveraninnitehorizon.Despitethetypicalmathematicalpathologiesassociatedwithlinearprogrammingininnite-dimensionalspaces,weareabletoextendthewell-knownnetworksimplexmethodtoinnite-dimensionalspaces.Wedidsoinanonstandardbutintuitivelyappealingwaythat,tothegreatestextentpossible,employsknowledgefromthenite-dimensionalnetworksimplexmethod.Further,foracertainclassofnetworkowproblemsininnitenetworks,weshowedthatournetworksimplexmethodcanbeimplementedinsuchawaythateachpivotoperationonlyrequiresaniteamountoftime.Inthefuture,theidenticationofotherclassesofproblemswiththispropertywillincreasethepracticalapplicabilityofourinnite-dimensionalnetworksimplexmethod. 237

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