|
PAGE 1
RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHFLEXIBLE CUSTOMERDEMAND By CHASEE.RAINWATER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1
PAGE 2
c 2009ChaseE.Rainwater 2
PAGE 3
Tomywife,myparentsandmybrother 3
PAGE 4
ACKNOWLEDGMENTS First,tomybeautifulwife.Thisdissertationrepresentsanothermilestoneinour adventuretogether.Forseeingmethroughthefrustrationsandtrialsthatundoubtedly accompaniedthedevelopmentofthisdissertation,Iamforevergrateful.Havingan unwaveringsourceofhappinesstogohometoisthesecrettomysuccess.Tothatend,I shareanysuccessthatI'vehadwithyou. Tomyfamily,yourloveandsupporthavebeeninvaluable.Specicallytomyparents, thankyouforallyoudidtoprepareforthisexperience. ToJoeGeunesandEdwinRomeijn,Icannotbegintodescribetheimpactyouhave hadonmeacademically,professionallyandpersonally.Thechancetoworkforyouranks asoneofthegreatestprivilegesofmylife.Iowetheopportunitytopursueanacademic careertothetwoofyou.IonlyhopethatIcanbehalfthementortoothers,asyou've beentome. ToColeSmith,Ithankyouforbeingsuchanunbelievableteacherandmentor throughoutgraduateschool.Also,thankyouforthedog. Lastly,toCaner,Semraandtherestofmygraduatestudentcolleagues,thankyoufor yourfriendshipthroughoutthesefouryears.Iremainamazednotonlybyyourtalentsand abilities,but,moreimportantly,bythesincerekindnessyou'veconsistentlyshownme. 4
PAGE 5
TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................8 LISTOFFIGURES....................................11 ABSTRACT........................................12 CHAPTER 1INTRODUCTION..................................14 2LITERATUREREVIEW..............................19 2.1CapacitatedFacilityLocationProblemwithSingle-SourceConstraints...19 2.2GeneralizedAssignmentProblem.......................22 2.30-1KnapsackProblem.............................26 2.4FlexibleDemandAssignmentProblems....................28 3CAPACITATEDFACILITYLOCATIONWITHSINGLE-SOURCECONSTRAINTS ANDFLEXIBLEDEMAND.............................30 4EXACTALGORITHMFORCFLFDANDGAPFD...............35 4.1AlternativeRepresentationoftheCFLFD..................36 4.1.1Set-PartitioningFormulation......................36 4.1.2PricingProblem.............................38 4.2KnapsackProblemwithExpandableItems..................39 4.2.1CFLFDwithSpeciallyStructuredRevenueFunctions........43 4.2.2ConvexandLinearRevenueFunctions................44 4.3Branch-and-PriceAlgorithmImplementation.................48 4.3.1InitialFeasibleSolution.........................48 4.3.2SolvingLPRSP............................49 4.3.3NodeandVariableSelection......................51 4.3.4OptimalColumnCost..........................52 4.4ComputationalResults.............................53 4.4.1ExperimentalData...........................53 4.4.2CFLFDResults.............................56 4.4.2.1Nonlinearrevenuefunctions:comparisonwithBARON..56 4.4.2.2Piecewiselinearandlinearrevenuefunctions:comparison withCPLEX.........................58 4.4.3GAPFJResults.............................59 4.5Conclusions...................................61 5
PAGE 6
5GAPFDHEURISTICWITHASYMPTOTICPERFORMANCEGUARANTEES71 5.1ModelAnalysis.................................72 5.2AnAsymptoticallyOptimalHeuristic.....................80 5.2.1DevelopmentoftheHeuristic......................80 5.2.2AverageCaseAnalysisoftheHeuristic................85 5.2.2.1Facility-independentrequirements..............88 5.2.2.2Facility-dependentrequirements...............91 5.2.3ModelExtension............................94 5.3HeuristicImprovementIssues.........................94 5.3.1SolutionImprovement..........................94 5.3.1.1Improvementphase......................95 5.3.1.2Post-processingphase....................95 5.3.2CapacityPerturbationScheme.....................96 5.4ComputationalResults.............................97 5.4.1ExperimentalDesign..........................97 5.4.2Facility-IndependentRequirements..................99 5.4.3Facility-DependentRequirements...................101 5.4.4EectofPost-ProcessingPhase....................102 5.5Conclusions...................................103 6LARGE-SCALEMULTI-EXCHANGEHEURISTICFORFIXED-CHARGE RESOURCECONSTRAINEDASSIGNMENTPROBLEMS...........108 6.1OptimizationandModelFormulation.....................111 6.2HeuristicFramework..............................111 6.2.1FacilityNeighborhoodSearch.....................112 6.2.2Single-CustomerVLSN.........................115 6.3SearchHeuristicImplementation.......................117 6.3.1InitialFeasibleSolution.........................118 6.3.2FSMoveChoice.............................119 6.4ComputationalStudy..............................123 6.4.1ExperimentalData...........................123 6.4.2Results..................................125 6.5CFLFDHeuristicApplicationsandConclusions...............127 7RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHSHARED RESOURCECONSUMPTION...........................131 7.1Introduction...................................131 7.2ModelFormulation...............................134 7.3ExactAlgorithmforRCAS...........................137 7.4SharedConsumptionKnapsackProblem...................139 7.5FlexibleCustomerDemandGeneralization..................150 7.6SharedConsumptionKnapsackproblemwithFlexibleCustomerDemand.157 7.7Branch-and-PriceAlgorithmImplementation.................164 6
PAGE 7
7.7.1InitialFeasibleSolution.........................164 7.7.2HeuristicsforPP i -F...........................165 7.7.3SolvingLPRMP-F...........................167 7.7.4NodeandVariableSelection......................168 7.8Computationalstudy..............................169 7.8.1ExperimentalDesign..........................169 7.8.2BaseResults...............................171 7.8.3ExtendedResults............................173 7.9ConclusionsandFutureResearch.......................175 8CONCLUSION....................................184 APPENDIX AGAPFDASYMPTOTICPROPERTY.......................187 BCFLFDPRICINGPROBLEMPROPERTY....................189 REFERENCES.......................................191 BIOGRAPHICALSKETCH................................198 7
PAGE 8
LISTOFTABLES Table page 4-1CFLFDwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2..61 4-2CFLFDwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2..63 4-3CFLFDwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2..64 4-4BranchingrulecomparisonfortheCFLFDwithnonlinearrevenuefunctions:5 facilities,10customers, =1 : 2...........................64 4-5CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,10customers, =1 : 2.........................................65 4-6CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,15customers, =1 : 2.........................................65 4-7CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,25customers, =1 : 2.........................................66 4-8CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,10customers, =1 : 2.........................................66 4-9CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,15customers, =1 : 2.........................................67 4-10CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,25customers, =1 : 2.........................................67 4-11CFLFDwithlinearrevenuefunctions:30facilities,60customers, =1 : 2....68 4-12GAPFJwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2..68 4-13GAPFJwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2..69 4-14GAPFJwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2..69 4-15GAPFJwithlinearrevenuefunctions:30facilities,90customers, =1 : 2....70 5-1Facility-independentrequirements:15facilities, =1 : 1..............105 5-2Facility-independentrequirements:15facilities, =1 : 2..............105 5-3Facility-independentrequirements:15facilities =1 : 3..............105 5-4Facility-independentrequirements:30facilities, =1 : 1..............105 5-5Facility-independentrequirements:30facilities, =1 : 2..............105 5-6Facility-independentrequirements:30facilities, =1 : 3..............106 8
PAGE 9
5-7Facility-dependentrequirements:15facilities, =1 : 1...............106 5-8Facility-dependentrequirements:15facilities, =1 : 2...............106 5-9Facility-dependentrequirements:15facilities, =1 : 3...............106 5-10Facility-dependentrequirements:30facilities, =1 : 1...............106 5-11Facility-dependentrequirements:30facilities, =1 : 2...............107 5-12Facility-dependentrequirements:30facilities, =1 : 3...............107 5-13Post-processingeectonheuristicwithgreedyandimprovementphase;facility-independent requirements: =1 : 2.................................107 5-14Post-processingeect;facility-dependentrequirements: =1 : 2..........107 6-1FNSresults:15facilities...............................129 6-2RNSresults:15facilities...............................129 6-3FNSresults:30facilities...............................129 6-4RNSresults:30facilities...............................130 7-1FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................178 7-2FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................178 7-3FASR:30facilities,60customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................179 7-4FASR:30facilities,90customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=5.......................................179 7-5FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2 I ,=25.......................................180 7-6FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.2 i 2 I ,=5.......................................180 7-7FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.2 i 2 I ,=5.......................................181 7-8FASR:15facilities,75customers,3customertypes, a = t =1 : 2, i =.9 i 2 I ,=5.......................................181 7-9FASR:15facilities,30customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5.....................................182 9
PAGE 10
7-10FASR:15facilities,45customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5.....................................182 7-11FASR:15facilities,30customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q ................................183 7-12FASR:15facilities,45customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q ................................183 10
PAGE 11
LISTOFFIGURES Figure page 4-1Illustrationof j and j forageneralrevenuefunction..............62 4-2Concaveenvelope:convexrevenuefunction.....................62 4-3Concaveenvelope:linearrevenuefunction.....................63 7-1Illustrationof r q and q ...............................177 11
PAGE 12
AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHFLEXIBLE CUSTOMERDEMAND By ChaseE.Rainwater August2009 Chair:JosephP.Geunes Cochair:H.EdwinRomeijn Major:IndustrialandSystemsEngineering Thisdissertationconsidersclassesofproblemsthatseektomakeprotabledemand fulllmentdecisionswithlimitedavailableresources.Thisgeneralproblemscenariohas beengivenmuchconsiderationoverrecentdecades.Inthiswork,weaddtothisbody ofresearchbyconsideringlessexploredproblemvariantsthatallowdecisionmakersto exploitdemandexibilitytoincreaseprot.Werstconsiderageneralizationofthe capacitatedfacilitylocationwithsingle-sourcingconstraints.Eachcustomermustbe assignedtoaprocuredfacility,andthelevelatwhichthecustomer'sdemandisfullled adecisionvariablemustbedetermined,subjecttofallingwithinpre-speciedlimits.A customer'srevenueisnondecreasinginitsresourceconsumption,accordingtoageneral revenuefunction,andaxedcostisincurredforeachresourceprocured.Weprovide anexactbranch-and-pricealgorithmthatsolvesboththisproblemandaspecialcase inwhichresourceprocurementisnotconsidered.Ourapproachidentiesanequally interestingclassofpricingsubproblems.Wediscusshowthisclassofproblemscan besolvedwithgeneralizedrevenuefunctionsandoerecientalgorithmsforsolving instanceswithspeciallystructuredrevenuefunctionsthatcorrespondtocommon pricingstructures.Ourextensivecomputationalstudycomparestheperformanceof ourexactalgorithmtothatofwell-knowncommercialsolversanddemonstratesthe advantagesofouralgorithmicapproachforvariouscategoriesofprobleminstances.Since 12
PAGE 13
real-worldscenariosoftenresultinlarge-scaleproblemsizes,weconsidernovelheuristic approachesforboththegeneralizationofthecapacitatedfacilitylocationproblemanda particularspecialcase,whichcanbeviewedasanextensionofthewell-knownGeneralized AssignmentProblemGAP.Werstdevelopaclassofheuristicsolutionmethods forthevariantwithoutresourceprocurementdecisions.Ourapproachismotivated byarigorousstudyofthelinearrelaxationofthemodel.Weshowthatourclassof heuristicsisasymptoticoptimalityinaprobabilisticsenseunderabroadstochasticmodel. Improvementproceduresarediscussedandathoroughcomputationalstudyconrmsour theoreticalresults.Wethenprovidefastandpracticallyimplementableoptimization-based heuristicsolutionmethodsforthegeneralizedclassoffacilitylocationproblemswith resourceprocurementdecisiosn.Ourprocedureisdesignedforverylarge-scaleproblem instances.Weoerauniqueapproachthatutilizesahigh-qualityecientheuristicwithin aneighborhoodsearchtoaddressthecombinedassignmentandxed-chargestructureof theunderlyingoptimizationproblem.Wealsostudythepotentialbenetsofcombining ourapproachwithaso-calledverylarge-scaleneighborhoodsearchVLSNmethod.As ourcomputationaltestresultsindicate,ourworkoersanattractivesolutionapproach thatcanbetailoredtosuccessfullysolveabroadclassofprobleminstancesforfacility locationandsimilarxed-chargeproblems.Finally,weconsideraseparateclassof assignmentproblemswithnon-linearresourceconsumptionandnon-traditionalcapacity constraints.Themodelisapplicabletomanufacturingscenariosinwhichproductswith commonproductioncharacteristicssharesetuptimesorsomeelementofxedresource consumption.Theadditionalcapacityconstraintsaccountforreal-worldrestrictions thatmayresultfromenvironmentalguidelines,transportationresourcelimitations,or limitedwarehousestoragespace.Weproposeabranch-and-pricealgorithmforthisclass ofproblemsthatrequiresauniquereformulationofourproblem,aswellasastudyofa newclassofknapsackproblems.Acomputationalstudydemonstratestheappealofour approachovercommercialsolversforvariousprobleminstances. 13
PAGE 14
CHAPTER1 INTRODUCTION Optimizationmodelswhichdeterminethemostprotablemannertofulllcustomer demandhavebeenwidelyexploredintheoperationsresearchliteratureforover50years. Manyoftheclassicalproblemsstudiedinoureld,suchasthe AssignmentProblem Kuhn[55],the GeneralizedAssignmentproblem RossandSoland[84],the CapacitatedFacilityLocationProblem Nauss[71],the TravelingSalesmanProblem Lin andKernighan[61],the VehicleRoutingProblem Laporte[58],andthenumerous Fixed-ChargeTransportationProblems AdlakhaandKowalski[1]haveconsideredthe assignmentofcustomerstoresourcesundervaryingreal-worldscenarios.Morerecently, theoperationsliteraturehasemphasizedwaystoexploitsourcesofsupplyanddemand exibilitytoincreaseprotmarginsthroughdemandandrevenuemanagementsee,e.g., TalluriandVanRyzin[94].Thishasledtoanumberofnewmodelsthatfocusonprot maximizationbyaccountingforboththecostsandrevenueimplicationsassociatedwith operationsdecisions.Forexample,ChenandHall[22]introduceseveralnewmaximum protscheduling"modelsthatimplicitlyaccountforthefactthatoperationsscheduling decisionscanaectdemandandthereforerevenue.Anotherstreamofliteratureconsiders optimalinventorymanagementwhendemandlevelsandhencerevenuesdependon inventorylevelse.g.,BakerandUrban[11,12],GerchakandWang[43],andBalakrishnan etal.[14]and/orshelfspaceallocationWangandGerchak[99],bothofwhichimpact operationscosts.Severalpapershavealsoconsideredmaximizingprotinproduction planningcontextswithprice-dependentdemand,whereproductionandinventorycosts aredeterminedbysolvinganoptimizationproblemcontainingalot-sizingstructuree.g., Thomas[95],KunreutherandSchrage[56],Gilbert[46],Billeretal.[18],DengandYano [28],Geunes,Romeijn,andTaae[44],andvandenHeuvelandWagelmans[98]. Theworkinthisdissertationaddstothisbroadbodyofresearch.Inthemostgeneral terms,weconsiderasetofcustomers,eachwiththeirowndemandrequirements,and 14
PAGE 15
asetofresourceconstrainedfacilities.Ourmodelsseektoassigneachcustomertoa singlecapacitatedfacilityresourceinamannerwhichmaximizestotalprot.Aswe willdiscussinChapter2,thisstandardresourceconstrainedassignmentproblemisitself diculttosolve,andnumerousexactandheuristicsolutionmethodologieshavealready beenproposed.Theworkinthisdissertationexploresvariantsofthistraditionalproblem whichaccountforuniqueplanningcharacteristicsthatmaybeavailabletoadecision maker.Specically,werstconsideranextensioninwhichthemodelallowsforso-called exiblecustomerdemand.Thisexibilityallowsasuppliertobettermatchcustomer demandswithoperationsresources,thereforeincreasingprots.Real-worldscenariosthat allowfordemandexibilityareoftenfoundintheproductionofconstructionmaterials, suchassteelandwood.Intheseenvironments,distributorsofthesematerialswill acceptdeliveriesfromsuppliersinarangeofsizesBalakrishnanandGeunes[13].The distributorspermitthisexibilitybecausetheyoftenperformfurthercustomizedcutting andnishingoperationsfortheirowncustomers,whoseexactsizespecicationsarenot knowntothedistributorinadvance.Supplierstosuchdistributorsareoftencompensated basedontotalweightdeliveredtothedistributorwithincertainlimitsdeemedacceptable tothedistributor.Clearly,ifthesupplierhasunlimitedresources,theycanmaximize protbydeliveringattheupperlimitofthedistributor'sstatedacceptablesizerange.If, however,thesupplierfacesresourceconstraintse.g.,intermsofitsquantityandsizes ofrawmaterialsandmustmeeteachelementofacollectionofcustomerdemands,the problemofassigningthesedemandstoavailableresourcesinordertomaximizenetprot isnon-trivial.Therstmodelconsideredinthisdissertationrequiresthatthefacilities utilizedtofullldemandmustbedeterminedbythedecisionsmaker.Werefertothis problemasthe CapacitatedFacilityLocationProblemwithSingle-SourceConstraintsand FlexibleDemand CFLFD.Inaddition,westudythespecialcaseofCFLFDinwhich demandisfullledbyaxedsetoffacilitiesi.e.procurementdecisionsareomitted. 15
PAGE 16
Werefertothisproblemasthe GeneralizedAssignmentProblemwithFlexibleDemand GAPFD.Foreachproblem,weproposebothexactandheuristicmethodologies. TheexactapproachutilizedfortheCFLFDandtheGAPFDreformulatesthe problemasaset-partitioningproblem.Theresultingsubproblemtobesolvedtakesthe formofaninterestingclassofnon-linearknapsackproblems.Wederivestructuralresults foranimportantrelaxationofthisclassofknapsackproblems.Theseresultssuggest ecientheuristicandexactapproachesforsolvingtheseknapsackproblemswithvarious revenuefunctions.Byconsideringthesealternativerevenuestructures,ourmodelaccounts forquantitydiscountsandeconomiesofscale,aswellasrevenuesforspecializedgoods. Lastly,weprovideadetaileddiscussionofhowthecustomerdemandfulllmentlevelsare determinedinthisapproach,aswellashowdicultiesthatarisewhenconsideringfacility procurementcanbeovercome. Solvinglarge-scaleinstancesofassignment-basedproblemsisanissuethatisactively consideredintheliterature.Therefore,aportionofthisdissertationfocusesonheuristic proceduresthatcanbeusedtosolvereal-wordsizeinstances.Whiletheexactapproach fortheexibledemandproblemstudiedisapplicabletoproblemswithandwithout facilityprocurement,ourheuristicapproachesrequireseparateconsideration.Forthe GAPFD,wepursueagreedyalgorithmthatismotivatedbyananalysisofthelinear relaxationofourmodel.Importantly,theheuristicpresentedisshowntohaveasymptotic performanceguaranteesunderaverygeneralstochasticmodel.Forverylargeproblems, thecomputationalresultssuggestthattheheuristicproducesnear-optimalsolutionsin dramaticallyreducedtimewhencomparedtothatrequiredbywell-knowncommercial solvers. Aswediscussindetailthroughoutthedissertation,xed-chargeheuristicsrequire carefuldesigntoassurethatthexed-chargedecisionsarefullyconsidered.Therefore, fortheCFLFD,wedevelopaheuristicframeworkthatsearchesaseparatefacility neighborhood,whilerelyingonconstructiveheuristicstodeterminethecustomer 16
PAGE 17
assignmentsanddemandfulllmentlevels.Wepresentthecomputationalsavingsof thisapproachoverotherlarge-scalesearchheuristicsanddiscussthepotentialimpacts thattheframeworkmighthaveonanumberofotherxed-chargeproblems. Clearlytheconsiderationofexibledemandhasnotablepracticalsignicance. Moreover,solvingproblemswiththiselementisachallengethatrequiresnovelsolution approaches.However,inreal-worldproductionenvironments,thetotalresourcesrequired tosatisfycustomerdemandexible,orotherwisemaynotbelimitedtothecumulative capacityconsumedbytheindividualcustomers.Aswewilldiscuss,moreoften,certain productionrequirementsaresharedamongstsubsetsofcustomers.Tomodelthis,weallow foreachcustomertobelongtoasingletype.Then,customersofeachtypeconsumea sharedamountofresourceinadditiontotheindividualconsumptionrequiredtofulll demand.Inadditiontothisconsideration,wearealsointerestedindierentformsof capacityconstraints.Typically,capacitatedassignmentproblemsconsideronlythe capacitylimitationsofindividualfacilities.Aswewillestablishinthereviewofliterature inthisarea,assignmentproblemswithcapacityrestrictionsoncustomersofaparticular typeregardlessofwhichfacilitytheyareassignedhavereceivedmuchlessattention. Accountingforbothoftheseproblemelementsrequiresaseparatemodelfromthose thathavebeenpreviouslydeveloped.Whileassignmentproblemswiththeseadditional characteristicscanstillbeviewedasdeterminingafeasiblepartitionofcustomersamongst asetofavailableresourcesfacilities,theimpactoftheadditionalcapacityconstraints requiresspecialconsiderationintheexactalgorithmthatisproposed.Theresultis aneectivealgorithmforanotherdicultclassofoptimizationproblemswithstrong practicalimplications. Theremainderofthedissertationisorganizedasfollows.InChapter2,wediscuss extensivelytherelevantliterature.InChapter3,weformallypresenttherstclassof problemstobestudiedandintroducetheconceptofexibledemand.Chapter4discusses anexactapproachfortheCFLFDandtheGAFPD.Anewclassofknapsackproblems 17
PAGE 18
thatariseinourapproachispresented.Bothaformalstudyofthestructureofthese problemsandecientsolutionproceduresareprovided.InChapter5,aclassofheuristics isproposedfortheGAPFD.Aprobabilisticanalysisdemonstratesthatourclassof heuristicsisasymptoticallyfeasibleandoptimalunderaverygeneralstochasticmodel. Athoroughcomputationalstudytoassesstheheuristic'sperformanceiscompletedand comparedagainstthetheoreticalclaimsmadeinthechapter.InChapter6,aheuristic frameworkfortheCFLFDwithlinearrevenuefunctionsispresented.Wedevelopa neighborhoodsearchheuristicthatdecomposestheCFLFDandsolvesthecorresponding subproblemwiththeheuristicproposedinChapter5.Wediscussthebenetsofour approachversusthosecommonlyappliedforxed-chargeoptimizationproblems.The motivationbehindthechosenimplementationisprovidedalongwithlessonslearned withrespecttolesssuccessfulimplementations.Then,inChapter7,anewclassof problemsisintroducedwhichaccountsforsharedresourceconsumptionamongsetsof customers,aswellasadditionalcapacityconstraints.Anexactapproachisproposedbased onaset-partitioningformulationofthemodelwithcomplicatingcapacityconstraints. Importantdierencesbetweenthisreformulationandthereformulationpresentedin Chapter4arediscussed.Anotherinterestingclassofknapsackproblemsisstudiedand ecientsolutionapproachespresented.Finally,Chapter8oersconcludingremarks,as wellasadiscussionoffutureresearch. 18
PAGE 19
CHAPTER2 LITERATUREREVIEW Theproblemsstudiedinthisdissertationsharecharacteristicswithnumerousclassical optimizationproblems.Thischapterprovidesanextensivereviewoftheliteraturerelated toeachoftheseproblems.Thediscussionofworkineachareafocusesontheexactand heuristicmethodsthathavebeenpursued.Thepresentationofthesesuccessfulapproaches willservetomotivateeachofthesolutionmethodschosenthroughouttheremainderof thedissertation. 2.1CapacitatedFacilityLocationProblemwithSingle-SourceConstraints Theproblemsstudiedinthisdissertationcanbeviewedasgeneralizationsof well-knownoptimizationproblems.Ofparticularrelevancetotheproblemsintroduced inChapter3andstudiedinChapters4and6isthe CapacitatedFacilityLocationProblem withSingle-SourceConstraints CFLP. maximize X i 2I X j 2J p ij x ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X i 2I f i y i {1 subjecttoCFLP X j 2J a ij x ij b i y i i 2I {2 X i 2I x ij =1 j 2J {3 x ij 2f 0 ; 1 g i 2I ; j 2J {4 y i 2f 0 ; 1 g i 2I : {5 Inthismodel,demandforcustomer j j 2J mustbesatisedbyasingleprocured capacitatedfacility i i 2I ,asenforcedinconstraints2{3.Acustomer j j 2J assignedtofacility i 2I ,resultsinaprotamount p ij andconsumes a ij unitsof facility i 'scapacity.Constraints2{2ensurethatcustomerdemandisexecutedsolely byprocuredfacilitiesandthateachfacility'sresourceavailabilityissatised.Constraints 19
PAGE 20
2{4and2{5placebinaryrestrictionsontheassignmentvariables x ij i 2I ; j 2J andfacilityprocurementvariables y i i 2I TheCFLPwithsingle-sourceconstraintsfallsintothedicultclassof NP -Hard optimizationproblemsseeGareyandJohnson[39],implyingthatitisunlikelythat apolynomial-timesolutionmethodexistsforsolvingproblemsinthisclassunless P = NP .NotethatthisnegativecomplexityresultevenholdsfortheclassicalCFLP inwhicheachcustomer'sdemandmaybesplitbetweentheacquiredfacilitiesaslong asallcustomerdemandsareallocated.Inpractice,asingle-sourcingrestriction,which requiresthatanycustomer'sdemandmustbeallocatedinitsentiretytoexactlyone oftheopenfacilities,isoftenimposedforavarietyofreasons.Forexample,inthe facilitylocationcontext,singlesourcingreducescoordinationcomplexity,reducesthe numberofdeliveriesrequiredtoacustomer,ensuresconsistencyofdeliveriesreceivedby customers,andprovidescustomerswithasinglepointofcontactforsupply.Moreover,a customer'sdemandmaydependonthefacilitytowhichitisassigned.Thisgeneralization isparticularlyrelevantifthefacilities"representmachinesorpeople,eachwithdierent processingcapabilities,andisthereforeoftenfoundinproductionenvironments.Withthe single-sourcingconstraintandfacility-dependentdemandsweobtainaproblemthatis,in general,atleastasdicultasthecaseinwhichacustomer'sdemandmaybesplit,and whichcontainsamoresubstantialadditionalcombinatorialcomponent. EarlyexactalgorithmsforCFLPfocusonbranch-and-boundapproaches.Many eortsutilizedlinearrelaxationsofCFLPtoobtainupperboundsseeSa[87]andAkinc andKumawala[8].DavisandRay[25]improveduponthisworkbygeneratingupper boundsfromrelaxationswiththeadditionalconstraints y i x ij i 2I ; j 2J : {6 Cornueojolsetal.[23]formallyshowedthatboundsobtainedusingthisalternative relaxationarestrongerthanthoseobtainedfromthestandardLPrelaxationofCFLP. 20
PAGE 21
AseparatebodyofresearchutilizedLagrangianrelaxationtoobtainupperbounds. GeorionandMcBride[42],Nauss[71]andCornuejolsetal.[23]allstudiedLagrangian relaxationsinwhichsingle-sourceconstraints2{3arerelaxed.VanRoy[86]and Cornuejolsetal.[23]consideredthealternativeLagrangianrelaxationthatrelaxesthe capacityconstraints2{2.Interestingly,Cornueojolsetal.showedthatthebound obtainedbyrelaxingthecapacityconstraintsisstrongerthanthatobtainedbyrelaxing theassignmentconstraints.Numerousadditionalmethodshavebeenproposedtosolve CFLPtooptimality.TheseincludeGeorionandGraves'[41]implementationofBenders decompositionandErlenkotter's[32]studyofadualascentmethod.Morerecently, Hombergetal.[49]proposedanexactmethodologythatsuccessfullycombinesLagrangian heuristicswitharepeatedmatchingalgorithmtoproducehighqualitysolutionswhile usingLagrangianrelaxationsintheboundingprocedureateachnode.Lastly,Neebeand Rao[72]consideredabranch-and-priceapproachwhichiscommonforassignment-based problemsandisdiscussedingreaterdetailinSection2.2.Theynotethedicultyof solvingproblemswithlargexed-chargesusingthisapproach,whichisachallengethatwe specicallyconfrontinChapter4. NumerousheuristicsfortheCFLPexist.Delmaireetal.[26]consideredawide assortmentofheuristicapproaches,includingevolutionaryalgorithms,tabusearchTS, simulatedannealing,andagreedyrandomizedadaptivesearchprocedureGRASP. Later,Delmaireetal.[27]improvedonthepromisingTSandGRASPheuristicsand proposeddierenthybridizationschemesthatcombinedthesetwoprocedures,yielding qualityresultsthatrequireonlyasmallamountoftime.Anevengreaternumber ofLagrangian-basedheuristicshavebeenproposedfortheCFLPe.g.,Barceloand Casanovas[15],HindiandPienkosz[48],KlincewiczandLuss[53],andBeasley[17]. Theseseparateeortsconsiderdierentrelaxationalternativesforboundingpurposes andoeruniquetechniquesforgeneratingfeasiblesolutions.Ofspecicrelevancetothe methodologyproposedinthiswork,BarceloandCasanova[15]proposedamulti-stage 21
PAGE 22
procedurethatusesdualinformationfromthelinearrelaxationoftheCFLPtoselect asetofopenplantsbeforeproceedingtoapenalty-basedreassignmentprocedurethat, ineect,solvesaspecializedinstanceofthegeneralizedassignmentproblem.Aswill beevidentthroughoutChapter6,separatingtheresourceprocurementanddemand assignmentdecisionsintoindividualphasescanleadtoasuccessfulheuristicframework. Morerecently,Ahujaetal.[6]proposedasearchheuristicwhichexploredaverylarge solutionspacemadepossiblebytheconsiderationofmultipleneighborhoods.Twoof theseneighborhoodsarerepresentedintheformofagraphandsubsequentlysolvedvia a VeryLarge-ScaleNeighborhood VLSNsearchprocedure.VLSNisasearchheuristic procedureshowntobeextremelysuccessfulonproblemswithanassignmentstructure. Forexample,theQuadraticAssignmentProblemAhujaet.al[4],theFleet-Assignment ProblemAhujaet.al[2]andtheVehicleRoutingProblemErgun[31]haveallbeen solvedviaVLSNoverthelastdecade.ForCFLP,Ahujaetal.[6]showedthattheir approachsolved63outof71testedinstancestooptimalityinlessthanone-minuteusing CPLEXtocertifyoptimality.Thesetestsincludedproblemswithupto30facilitiesand 200customers.ThesuccessofthisworkmotivatestheheuristicproposedinChapter6. Therefore,inthatchapteramorespecicdescriptionofAhujaetal.'s[6]approachis provided. 2.2GeneralizedAssignmentProblem InChapters4{5weproposeexactandheuristicapproachesforanextensionofthe GeneralizedAssignmentProblem GAP.TheGAPisaspecialcaseofCFLP,inwhich facilityprocurementisnotconsidered.Thatis,customerdemandisfullledbyaxedset ofcapacitatedresourcesfacilities.TheGAPcanberepresentedas maximize X i 2I X j 2J p ij x ij {7 22
PAGE 23
subjecttoGAP X j 2J a ij x ij b i i 2I {8 X i 2I x ij =1 j 2J {9 x ij 2f 0 ; 1 g i 2I ; j 2J : {10 withparametersandvariablesdenedinthesamemannerastheCFLP.Ofcourse,the GAPis NP -Hard,asshownbyFisheretal.[35].Furthermore,thefeasibilityproblem associatedwiththeGAPis NP -Complete,seeMartelloandToth[64].Theproblem wasoriginallystudiedbyRossandSoland[84],whoproposedanabranch-and-bound algorithmtosolvetheproblemtooptimality.Intheirwork,assignmentconstraints2{8 aredeletedandtheremainingassignmentproblemissolvedtoobtainavalidupperbound. Then,asecondarypenaltyproblemissolvedtocorrectviolatedcapacityrestrictions. Sincethen,alargenumberofadditionalbranch-and-boundapproachesfortheGAPhave beenproposed.Theseworksaredierentiatedbythevaryingapproachesusedtobound thesolution.Fisher[34]consideredthestrengthofboundsobtainedbysolvingithe Lagrangianrelaxationformedbyrelaxingcapacityconstraints2{8iitheLagrangian relaxationobtainedbyrelaxingassignmentconstraints2{9oriiisolvingtheLP relaxationformedbyrelaxingbinaryconstraints2{10.Thisworkdiscussesinteresting trade-osbetweensolvingcomputationallydicultrelaxationsthatprovidedsharper bounds,asshowntobethecasewiththerelaxationgivenbyii,versusweakerbounds obtainedinlesstime.WhilethisdissertationdoesnotutilizethetechniqueofLagrangian relaxation,trade-ossuchasthesearehighlyrelevanttootherdecompositionapproaches thatweconsider. Inadditiontothewell-studiedbranch-and-boundprocedure,anumberofdecomposition basedapproacheshavebeenproposedfortheGAP.BuildingontheLagrangianrelaxation eortsdiscussedpreviously,JornstenandNasberg[52]proposedaLagrangiandecomposition 23
PAGE 24
methodologythatcombinedthetworelaxationsformedbyrelaxingeithertheassignment 2{9orcapacityconstraints2{8.Theyshowedthattheboundobtainedbytheresulting relaxationsolutionisatleastasstrongaseitheroftheboundsobtainedbytheindividual Lagrangianrelaxationalternatives.Whiletheirtestingislimitedtoonly10instances, resultssuggestedthattheapproachisaneectivealternativetothetraditionalLagrangian relaxationsoftheGAP. Ineachoftheproblemsstudiedinthisdissertation,anequivalentpartition-based representationisproposedwithasubsequentsolutionmethodology.Therefore,of particularrelevancetotheworkinthedissertationisSalvesbergh's[88]branch-and-price algorithmfortheGAP.Inthisapproach,theGAPisrepresentedasapartitionoftheset ofcustomers, J ,into jIj disjointandpossiblyemptysubsets,eachofwhichisassignedto exactlyonefacility.Thatis,theformulationGAP-SPequivalentlyrepresentstheGAP. maximize X i 2I D i X d =1 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(p ij x d ij d i subjecttoGAP-SP X i 2I D i X d =1 x d ij d i =1 j 2J {11 D i X d =1 d i =1 i 2I {12 d i 2f 0 ; 1 g d =1 ;:::D i ; i 2I ; where x d i = x d i 1 ;:::;x d i jJj isabinaryvectorrepresentingthe d th subsetofcustomers thatcanbeassignedtofacility i ,and D i isthetotalnumberofsubsetsofcustomersthat canbeassignedtofacility i .Thevariable d i takesthevalueofoneifthe d th column associatedwithfacility i chosen,andzerootherwise.Ingeneral,thenumberofvariablesin GAP-SPisexponentiallylargeinthedimensionoftheunderlyingassignmentproblem. Thebranch-and-priceapproachthereforesolvestheLP-relaxationofGAP-SPbya columngenerationprocedure,wherethecolumnsareaddediterativelyasneeded,and 24
PAGE 25
solvesGAP-SPitselfbybranch-and-bound.Theso-called pricingproblem solvedto identifyattractivecolumnsisthewell-studied0-1knapsackproblem,discussedindetailin Section2.3.Salvesbergh[88]showedthattheLP-relaxationofGAP-SPprovidesabound atleastastightasthatobtainedbysolvingtheLP-relaxationofGAP,sincethefeasible spaceofLPGAP-SPislimitedtoconvexcombinationsofsolutionstoa0-1knapsack problem.Importantly,theworkdemonstratedthatthebranch-and-priceapproachis particularlysuccessfulwhentheratioofcustomerstofacilitiesissmalli.e.nomorethan 5.ThisphenomenonisthecombinedresultofitheLPrelaxationofthe0-1knapsack problembeingweakerwhenthisratioissmall,thussolvingtheknapsackproblemsto optimalityyieldsstrongerboundsandiithefactthatnumberoffeasible0-1knapsack solutionsbecomesquitelargeastheratioofcustomerstofacilitiesincreases,thusthe columngenerationprocedurebecomeshighlycomputationallyintensiveasthenumberof customersperfacilityincreases. EvenwiththeadvancesofexactalgorithmsfortheGAP,itremainscomputationally impracticaltosolveverylargeinstances.Forthisreason,agreatdealoftheliterature isdevotedtometa-heuristicsfortheGAP.AlargenumberarementionedinRomero RomalesandRomeijn[83]summaryofresearchpursuedfortheGAP.Notableamongst thesearetabusearchYagiuraet.al[101],geneticalgorithmsWilson[100]and simulatedannealingalgorithmsOsman[73].Othersuccessesweredocumentedin AminiandRacer's[10]improvedimplementationofthevariabledepthsearchheuristic thatbenetsfromthegreedyheuristicsproposedbyMartelloandToth[64].Cattrysse etal.[20]proposedaheuristicwhichsolvesthelinearrelaxationoftheset-partitioning representationoftheGAPi.e.GAP-SPandsearchesamongstthecolumnsgeneratedto obtainafeasiblesolution.Theresultsofacomputationalstudyempericallysuggestthat solutionsobtainedinthismannerareoftenwithinlessthan1%ofoptimality. OfparticularsignicancetoChapter5istheclassofheuristicsproposedbyMartello andToth[64]andRomeijnandRomeroMorales[79].Aweightfunction, f i;j ,isdened 25
PAGE 26
tomeasurethepseudo-protofassigningcustomer j j 2J tofacility i i 2I .This functionisusedtodeterminetheorderinwhichtoassignthecustomersandthefacility towhicheachcustomershouldbeassigned.InMartelloandToth[64],weightfunctions arerepresentedbyeitherithexedprotcost p ij i 2I ; j 2J iitheamountof resource i consumedbycustomer j a ij i 2I ; j 2J oriiitheratio a ij b i .Romeijnand Morales[79]proposedaweightfunctionthatseekstoassignacustomertoafacilitywith maximumprotandminimalcapacityconsumption.Toaccomplishthistheychose f i;j = p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij forsomevector .Theyshowedthatif i i 2I istakentobetheoptimaldualvalues associatedwithconstraints2{8thentheirgreedyalgorithmisoptimalwithprobability oneasthenumberofcustomersgoestoinnityunderaverygeneralstochasticmodel.It isthisworkthatmotivatesthegreedyheuristicdevelopedinChapter5forwhichwealso seekasymptoticperformanceguarantees. 2.30-1KnapsackProblem Throughoutthisdissertationnumerouswestudyvariantsof0-1knapsackproblems. Whilealargesegmentofourworkconsiders0-1knapsackproblemswithnon-linearprot functions,eachofoursolutionapproachesreliesheavilyonthepropertiesofthelinear0-1 knapsackproblemandthesubsequentalgorithmsusedtosolveit.Thisproblem,whichwe refertoastheKP-01,ispresentedas maximize X j 2J p j x j {13 subjecttoKP-01 X j 2J a j x j b {14 x ij 2f 0 ; 1 g j 2J : {15 26
PAGE 27
Thisoptimizationproblemrequiresthemostprotablesubsetofcustomerstobechosen withoutviolatingthecapacityconstraint2{14.Martelloetal.[62]providedanexcellent overviewofexactalgorithmsproposedforKP-01.Asstatedinthissurvey,amajority ofexactalgorithmsarebasedoneitherbranch-and-boundordynamicprograming approaches.Notsurprisingly,thevariationsinthebranch-and-boundapproachesconsider varyingproceduresforobtaininggoodupperboundstoKP-01.Theoriginalboundon KP-01wasestablishedinDantzig[24].Theboundisdeterminedbysolvingthelinear relaxationofKP-01inwhichthebinaryrestrictions2{15arerelaxedsayLPKP-01. LPKP-01issolvedbysortingitems j 2J innon-increasingorderof p j a j andincluding itemsinthesolutiontoLPKP-01untileitheriallcapacity b isconsumedorii allcustomers j 2J ; forwhich p j > 0,areincludedinthesolutiontoLPKP-01. Importantly,theoptimalsolutiontothisrelaxationcontainsatmostoneitemwhich violatesthebinaryrestrictions2{15.Thus,aheuristicapproachtosolvingKP-01 issimplytoremovethefractionalcustomerintheoptimalsolutiontoLPKP-01. Dantzig'sboundwaslaterimprovedbyMartelloandTothin[63]andthenagainin[67]. Inthelatterwork,MartelloandToth[67]showedthatstrongerboundscanbeobtained byaddingmaximumcardinalityconstraintstoKP-01andsolvingthecorresponding relaxationviaLagrangiantechniques.Itisimportanttonotethatineachofthesuccessful branch-and-boundimplementationsfortheGAPseealsoHorowitzandSahni[50]and Nauss[70],adepth-rstenumerationschemewaschosen. Asmentionedpreviously,theothercommonapproachusedtosolveKP-01to optimalityisdynamicprogramming.Pisinger[75]proposedadynamicprogramming approachthatreliesonboundsdeterminedbyLPKP-01.Theirapproachwassuccessful inlimitingtheenumerationtobeconsidered,thusyieldinggoodlowerboundsina reasonableamountoftime.Toimproveuponthisapproach,MartelloandToth[65] developedahybridprocedurewhichcombinedtheexpanding-corealgorithmproposed 27
PAGE 28
inPisinger[75]withtheimprovedboundsobtainedin[67].Thisapproachwasshownto solvealmostalltestinstanceswithupto100,000variablesinlessthan.5seconds. 2.4FlexibleDemandAssignmentProblems Eachoftheproblemsstudiedinthisdissertationcanbepresentedasgeneralizations oftheGAP.Specically,eachproblemgeneralizesthenotionofxeddemandconsumption. WemathematicallyintroducetheconceptofexibledemandinChapter3.Thissection discussesseparateworksintheliteraturethathaveconsideredtheconceptofvariable demandfullledbetweencustomer-speciedlimits.Themostrelevantworkinthisarea wasoeredbyBalakrishnanandGeunes[13]whoconsideredaproductionplanning problemwithexibleproductspecication.Theirmodelwasmotivatedbythesteel industry,inwhichcustomerswillacceptsteelplatescutwithinspecieddimensions.They formulatethisproblemasthe FlexibleDemandAssignment problemFDA,givenby maximize X i 2I X j 2J r j + v ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X i 2I f i + b i y i subjecttoFDA X j 2J v ij b i y i i 2I {16 X i 2I x ij =1 j 2J {17 v ij ` j x ij i 2I ; j 2J {18 v ij u j x ij i 2I ; j 2J {19 x ij 2f 0 ; 1 g i 2I ; j 2J {20 y i 2f 0 ; 1 g i 2I {21 wherecustomer j 0 s demandlevel j 2J v ij i 2I ; j 2J mustbesettoavalue within[ ` j ;u j ].Furthermorearevenueof r j isaccruedperunitofcapacityconsumption.It shouldbenotedthattherevenueaccruedasafunctionofthelevelatwhichthedemand 28
PAGE 29
issatisedisindependentofthefacilityusedtosatisfythisdemand.Similarly,thebounds onthedemandfulllmentlevelareindependentofthefacilitytowhichthecustomeris assigned.Ifresource i isusedtofulllanycustomerdemandaxedcost, f i isincurred. Unusedcapacityofprocuredresource i isrecycledatacostof perunit.Inadditionto therecognizablecapacityandassignmentconstraints2{16and2{17,constraints2{18 and2{19ensurethatdemandisfullledbetweentheappropriatebounds.Balakrishnan andGeunes[13]proposedmultipleclassesofstrongvalidinequalitiesforFDA,aswellas aLagrangian-basedupperboundingprocedure.Lagrangian-based,bin-packingandlinear programroundingheuristicsarepresented.Theresolutionprocedureobtainedaninitial lowerboundheuristicallyandutilizedtightenedrelaxationstoobtainaninitialupper bound.TheseboundsareprovidedtoCPLEX'sstandardbranch-and-boundprocedure. Athoroughcomputationalstudyshowedthatthiscompositeapproachsuccessfully solvesbothrealandrandomprobleminstancesupto12resourcesand60customersto optimalityortowithinasmallgap. BeyondtheFDAandtheworkcontainedinthisdissertation,themostrelatedwork toexibledemandfoundintheliteratureisthatdoneontheMulti-levelGeneralized AssignmentProblem[57].InthisextensionoftheGAP,eachcustomerisassignedtoa singleresourceatoneofa xed numberoflevels.However,themodeldoesnotconsider acontinuousrangeoflevelsinwhichacustomer'sdemandmaybefullled,asallowedin FDAandtheproblemsconsideredinthisdissertation. 29
PAGE 30
CHAPTER3 CAPACITATEDFACILITYLOCATIONWITHSINGLE-SOURCECONSTRAINTS ANDFLEXIBLEDEMAND Problemsthatrequiretheallocationoflimitedresourcestodemandsforthose resourcesariseinnearlyallcontexts.Inindustrialcontexts,thecostsassociatedwith acquiringrelevantresourcescanoftenbequantied,ascanthecostsassociatedwithusing theseresourcestosatisfycorrespondingdemands.Insuchcases,optimizationmodels serveasapowerfultoolfordeterminingthebestmixofresourceacquisitionandallocation ofresourcestodemands.TheCapacitatedFacilityLocationProblemCFLPprovides anexampleofawell-knownoptimizationmodelthathasbeensuccessfullyutilizedto determineanoptimalsubsetoffacilitiesfromamongasetofcandidatelocationsaswell astheallocationoftheoutputofthesefacilitiestoasetofknowncustomerdemands. Thisproblemassumesthateachcandidatefacilityhasanassociatedxedoperatingcost, aknowncapacitylimitonoutput,andacostforsatisfyingacustomer'sdemandthatis proportionaltotheamountofthecustomer'sdemandsatisedfromthefacility.Beyond thefacilitylocationcontext,thismodelndsapplicationinawidevarietyofsettingsin whichindividualcapacitatedresourcesmustbeacquiredtosatisfydemandsforresource output.Inthischapter,weintroduceaproblemthatcombineskeyelementsofseveral well-studiedvariantsofthisproblemintoanewclassofgeneralizedcapacitatedfacility locationproblemsthatwewillrefertoasthe CapacitatedFacilityLocationProblemwith Single-SourceConstraintsandFlexibleDemand CFLFD. AsmentionedinChapter2,theCFLPwithsingle-sourceconstraintsaspecialcaseof theCFLFDfallsintothedicultclassof NP -Hardoptimizationproblems,implyingthat itisunlikelythatapolynomial-timesolutionmethodexistsforsolvingproblemsinthis classunless P = NP .ThespecialcaseoftheCFLPinwhichtherearenoxedcosts associatedwiththeacquisitionoffacilitiesconstitutesawell-knownproblemclassknown astheGeneralizedAssignmentProblemGAPstudiedbyRossandSoland[84],amongst others.Thismodelisitselfwidelyapplicableinnumerousproblemcontextssuchasjob 30
PAGE 31
scheduling,locationmodels[85],andtransportationplanning[92].Theclassicalvariants oftheCFLPandtheGAPseektomeetknowncustomerdemandlevelsatminimum cost.Thus,therequiredlevelofresourceconsumptionforeachjoborcustomerdemandis assumedtobeaxedquantity.Morerecently,problemswhichconsiderexiblejobsizes orexiblecustomerdemandquantitieshavebeenproposedbyBalakrishnanandGeunes [13].Accountingforsuchexibledemandsallowsforscenarioscommontoboththesteel [13]andforestryindustries,forexample,wherecustomersmaypermitarangeofdelivery quantitiesorarangeofacceptableproductsizes.Insuchcases,therevenuereceived bythesuppliermayoftenincreaseinthedeliveredquantityorproductsize.Thus,the supplierhasincentivetoincreaserevenuebysatisfyingdemandsattheupperlimitsof acceptableranges.Suchastrategy,however,increasesresourcecostsandmay,therefore, notresultinprotmaximization. ThenewclassoftheCFLFDproblemsthatweintroduceandanalyzeinthis dissertationcombinesalloftheaspectsdescribedabove.Inparticular,thisproblemseeks aprot-maximizingsolutionbasedondecisionsinvolvingtheprocurementofcapacitated resources,theassignmentofcustomerstotheseresources,andthedeterminationof correspondingdemandfulllmentlevels.Let I denotethesetoffacilitiesavailablefor theexecutionofthesetofcustomers J .Eachcustomer j 2J mustbeassignedtoa singlefacility.However,eachfacilitymayonlybeabletoprocesscertaincustomers.That is,onlycustomersintheset J i J maybeassignedtofacility i 2I or,equivalently, customer j 2J mayonlybeprocessedbyfacilitiesintheset I j I ,whereofcourse i 2I j ifandonlyif j 2J i .Ifcustomer j 2J isassignedtofacility i 2I j ,axedprot of p ij isincurredandaxedamountofcapacity a ij isconsumed.Thecorresponding customerdemandfulllmentlevelmustbeselectedfromtheinterval[ ` ij ;u ij ] : An additionalprotisaccruedasafunctionofdemandfulllmentlevel,determinedby thenon-decreasingfunction r ij i 2I ; j 2J i .Lastly,iffacility i 2I isusedtosatisfy anycustomerdemand,axedprocurementcost f i isincurred.Thecapacityoffacility 31
PAGE 32
i 2I isdenotedby b i i 2I .Theobjectiveistodetermineanassignmentofcustomers toprocuredfacilities,aswellasthecorrespondingdemandfulllmentlevels,inorderto maximizetotalprotwhilesatisfyingthecapacityconstraintsofthefacilities. Usingtheprecedingnotation,theCFLFDcanbeformulatedasamixed-integerlinear programmingproblemasfollows: maximize X i 2I X j 2J i r ij v ij + X i 2I X j 2J i p ij x ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X i 2I f i y i subjecttoCFLFD X j 2J i a ij x ij + v ij b i y i i 2I {1 X i 2I j x ij =1 j 2J {2 v ij ` ij x ij i 2I ; j 2J i {3 v ij u ij x ij i 2I ; j 2J i {4 x ij 2f 0 ; 1 g i 2I ; j 2J i {5 y i 2f 0 ; 1 g i 2I : {6 Constraints3{1ensurethatcustomerdemandissatisedsolelybyprocuredfacilities andthatthelevelsatwhichdemandissatisedsatiseseachfacility'sresourceavailability. Constraints3{2requiretheassignmentofeachcustomertoasinglefacility.Inaddition, 3{3and3{4ensurethatifcustomer j isassignedtofacility i i 2I ;j 2J i ,its demandisfullledatalevelwithinitsrespectivebounds.Lastly,3{5and3{6enforce binaryrestrictionsontheassignmentvariables x ij i 2I ; j 2J i andfacilityprocurement variables y i i 2I Observethatthetotalrevenuereceivedwhencustomer j isassignedtofacility i ata levelof v ij 2 [ ` ij ;u ij ]isequalto p ij + r ij v ij ,whichcorrespondstoaper-unitassociated priceof p ij v ij + r ij .Themodelthusallowsforaformofquantitydiscountsthatmayprovide anincentivetocustomerstoacceptaexiblerangeofdemandfulllmentlevels.Similarly, 32
PAGE 33
thetotalquantityoftheresourcethatisconsumedwhencustomer j isassignedtofacility i atalevelof v ij 2 [ ` ij ;u ij ]isequalto a ij + v ij ,whichcorrespondstoaper-unitassociated resourceconsumptionof a ij v ij +1.Ourmodelcanthusaccountforthepresenceof,for example,xedcustomersetuptimesorotherlossofresourcesatthestartofaproduction run. Nofacilityprocurementcosts .InChapters4and5westudyaspecialcaseof theCFLFDinwhichtherearenoxedfacilityprocurementcosts,i.e., f i =0forall i 2I sothatwithoutlossofoptimalitywecanset y i =1forall i 2I .Werefertothisproblem asthe GeneralizedAssignmentProblemwithFlexibleDemand maximize X i 2I X j 2J i p ij x ij + X i 2I X j 2J i r ij v ij {7 subjecttoGAPFD X j 2J i a ij x ij + X j 2J v ij b i i 2I {8 X i 2I x ij =1 j 2J i v ij ` ij x ij i 2I ; j 2J i v ij u ij x ij i 2I ; j 2J i x ij 2f 0 ; 1 g i 2I ; j 2J i : Noticethatthefacilityprocurementvariables y i i 2I havebeenomittedfrom3{7 and3{8. Dynamicproblems .ItisnoteworthythattheCFLFDencompassesmanufacturing problemswithatemporalelement.Specically,consideramulti-periodexibledemand assignmentprobleminwhichcustomershaveaduedate,butcanbeexecutedpriorto thatattheexpenseofaholdingcost.Thisproblemcanbeformulatedasaninstance oftheCFLFDwherethefacilitiesaretime-expanded;i.e.,afacility i;t representsa 33
PAGE 34
singleresource i 2I availableintimeperiod t t =1 ;:::;T ,where T isthelength oftheplanninghorizon.Clearly,customerscanonlybeassignedtofacilitiesrepresenting resourcesthatareavailableonorbeforetheirduedate;i.e.,theset J i;t containsonly customersthatdonothaveaduedatebeforetimeperiod t InthefollowingchapterweproposeanexactsolutionapproachfortheCFLFDand theGAPFD.Then,inChapters5and6wedeveloplarge-scaleheuristicapproachestothe CFLFDandtheGAPFD. 34
PAGE 35
CHAPTER4 EXACTALGORITHMFORCFLFDANDGAPFD TheclassofCFLFDproblemsintroducedinChapter3seeksaprot-maximizing solutionbasedondecisionsinvolvingtheprocurementofcapacitatedresources,the assignmentofcustomerstotheseresources,andthedeterminationofcorresponding customerdemandfulllmentlevels.Inthischapter,wefocusondevelopinganexact approachforsolvingthisproblem.Overthepastdecade,manynonlinearassignment problemsarisingin,forexample,supplychainoptimization,havebeenreformulated asset-partitioningproblems,leadingtobranch-and-pricesolutionapproachestosuch problems.Barnhartetal.[16]provideathoroughdiscussionofhowbranch-and-pricecan beappliedtosolvelargeintegerprogrammingmodels.Applicationsinclude,forexample, theGeneralizedAssignmentProblemSavelsbergh[88],theFixed-ChargeAssigningUsers toSourcesProblemNeebeandRao[72],theMulti-PeriodSingle-SourcingProblem Frelingetal.[37],theContinuous-timeSingle-SourcingProblemHuangetal.[51], jointlocation-inventorymodelsShenetal.[89],andwarehouse-retailernetworkdesign problemswithjointreplenishmentcostsTeoandShu[90],Romeijnetal.[82].Itis thereforenotsurprisingthatthisapproachcanbeeectivelyappliedtotheCFLFD problemclassaswell.However,aswithanybranch-and-pricealgorithm,agreatdeal ofconsiderationmustbegiventotheso-calledpricingsubproblemthatarisesaswell astothebranchingstrategyused.IntheCFLFDproblem,thepricingsubproblem takesaninterestingform,resultinginageneralizationofthe KnapsackProblemwith FlexibleItems [13]orthe KnapsackProblemwithVariableItemSizes [81].Weprovidean ecientapproachforsolvingthisclassofproblemsundereitherconvexorconcaverevenue functions,whichleadstoaneectivebranch-and-priceapproachfortheCFLFDproblem. Thischapterisorganizedasfollows.Section4.1reformulatestheCFLFDasa set-partitoningproblemandintroducethepricingsubproblem.Section4.2proposes methodologiesforsolvingaclassofgeneralizedknapsackproblemswhichincludesthe 35
PAGE 36
relevantpricingsubproblem.Section4.3detailstheimplementationofourbranch-and-price algorithm.InSection4.4weperformanextensivecomputationalstudyoftheproposed proceduretosolveboththeCFLFDproblemandtheGAPFD.Weincludeabroad collectionofexperimentswithrevenuefunctionsthatmodelavarietyofcommonpricing conditions,andshowthatourbranch-and-pricealgorithmsignicantlyoutperformsa state-of-the-artcommercialmixed-integernonlinearprogrammingsolver.Finally,Section 4.5providessomeconcludingremarksandoersdirectionsforfutureresearch. 4.1AlternativeRepresentationoftheCFLFD 4.1.1Set-PartitioningFormulation WecanequivalentlyviewtheCFLFDasaproblemofpartitioningthesetof customers, J ,into jIj disjointandpossiblyemptysubsets,eachofwhichisassigned toexactlyonefacility.Moreformally,wecanwritetheset-partitioningformulationofthe CFLFDas maximize X i 2I D i X d =1 i x d i d i subjecttoSP X i 2I D i X d =1 x d ij d i =1 j 2J {1 D i X d =1 d i =1 i 2I {2 d i 2f 0 ; 1 g d =1 ;:::D i ; i 2I ; where x d i = x d i 1 ;:::;x d i jJ i j isabinaryvectorrepresentingthe d th subsetofcustomers thatcanbeassignedtofacility i ,and D i isthetotalnumberofsubsetsofcustomersthat canbeassignedtofacility i .Furthermore, i isafunctionthatdeterminestherevenue obtainedbyfacility i whensubset x d i isassignedtoit.Inparticular, i x d i istheoptimal valueofthefollowingoptimizationproblem,inwhichthesizesoftheassignedcustomers arechosentomaximizetherevenueofthefacility: 36
PAGE 37
X j 2J i p ij x d ij +maximize X j 2J i r ij v j )]TJ/F21 11.9552 Tf 11.955 0 Td [(f i y i subjecttoCC d i X j 2J i v j b i y i )]TJ/F26 11.9552 Tf 12.536 11.358 Td [(X j 2J i a ij x d ij v j 2 [ ` ij x d ij ;u ij x d ij ] j 2J i y i 2f 0 ; 1 g : Sinceitiseasytoseethat i 0 =0wecan,withoutlossofoptimality,relaxconstraint 4{2to D i X d =1 d i 1 i 2I : 4{2 0 Wechoosethelattersetofconstraintsforconvenience,sincethisimmediatelyimpliesthat theassociateddualvariablesarenonnegative. Generally,thenumberofvariablescolumnsinSPisexponentiallylargeinthe dimensionoftheunderlyingassignmentproblem.Thebranch-and-priceapproachtherefore solvestheLP-relaxationofSPbyacolumngenerationprocedure,wherethecolumnsare addediterativelyasneeded,andsolvesSPitselfbybranch-and-bound.Inthecolumn generationprocedure,theso-called pricingproblem determineswhetherthesolutionto theLP-relaxationofarestrictedversionofSPinwhichonlyasubsetofthecolumnsis considered,sayLPRSPisindeedoptimalor,otherwise,identiesoneormorecolumns thatpriceoutandarethereforeaddedtotherestrictedproblem.Notethatitissucient toeitheridentifyafeasiblesolutiontothepricingproblemthatpricesout or showthat theoptimalsolutiontothepricingproblemdoesnotpriceout.Thatis,itisnotstrictly necessarytosolvethepricingproblemtooptimalityateachiterationofthecolumn generationmethod.Inthefollowingsectionweformallypresentthepricingproblem andinthesubsequentsectionwedevelopapproachestosolvethispricingproblemunder variousformsoftherevenuefunctions r ij i 2I ; j 2J i 37
PAGE 38
4.1.2PricingProblem Thefollowingoptimizationproblemdescribesthepricingproblemassociatedwith facility i 2I : maximize X j 2J i p ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( j x j + X j 2J i r ij v j )]TJ/F15 11.9552 Tf 11.956 0 Td [( i + f i y subjecttoPP i X j 2J i a ij x j + v j b i y v j 2 [ ` ij x j ;u ij x j ] j 2J i {3 x j 2f 0 ; 1 g j 2J i y 2f 0 ; 1 g ; where j j 2J and i 2I aretheoptimaldualvariablesassociatedwiththe assignmentconstraints4{1andthecolumnselectionconstraints4{2inLPRSP. Observethatwhen y =0theoptimalsolutionvalueofPP i istriviallyseentobeequal to0,sothatwecanlimitourselvestosolvingtheproblemundertheassumptionthat facility i isprocuredi.e., y =1.Wethensimplyreplacetheoptimalvaluetothis restrictedproblembyzeroifitisnegative.Tosimplifythedevelopmentofoursolution methods,inSection4.2westudyanequivalentformulationoftherestrictionofPP i to y =1.Inthisreformulation,wereplacethedemandfulllmentlevelvariables, v j ,with decisionvariables w j = v j + a ij x j j 2J i ,whichrepresentthetotalamountofresource i.e.,bothxedandvariableconsumedbycustomer j .Thealternativeformulation, PP 0 i ,iswrittenas maximize X j 2J i p ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( j x j + X j 2J i r ij w j )]TJ/F21 11.9552 Tf 11.955 0 Td [(a ij x j )]TJ/F15 11.9552 Tf 11.955 0 Td [( i + f i subjecttoPP 0 i X j 2J i w j b i {4 38
PAGE 39
w j 2 [ ` 0 ij x j ;u 0 ij x j ] j 2J i {5 x j 2f 0 ; 1 g j 2J i ; {6 where ` 0 ij = ` ij + a ij and u 0 ij = u ij + a ij i 2I ; j 2J i .Itiseasytoseethat4{5assures that4{3remainssatised. ThisproblemisageneralizationoftheKnapsackProblemwithExpandableItems thatallowsfornonlinearrevenuefunctions.BalakrishnanandGeunes[13]proposea dynamicprogrammingalgorithmforthisproblemforthecasewheretherevenuefunctions arelinearandtheproblemdataisinteger.Thisapproachhasarunningtimethat ispseudo-polynomialintheprobleminputs,whichtendstomakeittime-consuming inpractice.Additionally,theirworkdoesnotconsidernon-linearrevenuefunctions. Therefore,inthenextsectionweconsiderapproachesforsolvingourpricingproblem underdierentclassesofrevenuefunctions.Wedevelopaheuristicapproachaswellasa customizedbranch-and-boundapproachtosolvetheproblemtooptimality. 4.2KnapsackProblemwithExpandableItems Inthissectionwefocusonecientmethodologiestosolvetheclassofproblemsgiven by maximize X j 2 ~ J p j x j + X j 2 ~ J r j w j )]TJ/F21 11.9552 Tf 11.955 0 Td [(a j x j subjecttoKPEI X j 2 ~ J w j b {7 w j 2 [ ` 0 j x j ;u 0 j x j ] j 2 ~ J {8 x j 2f 0 ; 1 g j 2 ~ J ; {9 where ~ J isthesetofcustomerstobeconsidered.WecanrepresentKPEIinitsmost generalformas maximize X j 2 ~ J j w j x j 39
PAGE 40
subjecttoKPEI 0 X j 2 ~ J w j x j b {10 w j 0 j 2 ~ J {11 x j 2f 0 ; 1 g j 2 ~ J ; {12 wherethefunctions j aredenedas j w j = 8 > > > > > > > < > > > > > > > : =0 w j =0 = 0
PAGE 41
wherethefunction j j 2 ~ J correspondstothenon-decreasingconcaveenvelope encompassingtheorigin,thefunction j andthepoint b; j u 0 j whichisillustratedin Figure4-1.WewillshowthatfortheforKPEI 0 withconcave,convex,orlinearrevenue functions,theenvelopes j j 2 ~ J canbeobtainedexplicitly. RomeijnandSargut[81]proposeanalgorithmtosolveRKPbasedonabinary searchforobtainingtheoptimalLagrangemultipliersatisfyingtheKKTconditions. However,notethatinthecontextofKPEI 0 ,RKPallowssolutionsinwhich u 0 j < w RKP j b j 2 ~ J .This,ofcourse,correspondstoaninfeasiblesolution.Lemma1ensures thananalternativefeasiblesolutionexistswithanequivalentobjectivevalue. Lemma1. Supposetheoptimalsolutionto RKP containsanon-emptysetofcustomers J forwhich u 0 j
PAGE 42
iicustomer j isfullyassigned,butitsnetprotwasoverestimatedi.e., j w j > w j and ` 0 j w j u 0 j Proof. TheresultfollowsfromTheorem4.2of[81]andthedenitionof j Thispropertycanbeusedtodevelopaneectiveheuristicroundingstrategyasis oftendoneforthetraditionalknapsackproblem.Inparticular,iftheoptimalsolutionto RKPisindeedfractional,wecansimplyremovethatfractionalcustomertogeneratea feasibletosolutiontotheKPEI 0 .Otherwise,ifthesolutionisfeasibletoKPEI 0 ,but j w RKP j 6 = j w RKP j forasinglecustomer j ,wesimplyupdatetheobjectivefunction accordinglytocorrectfortheapproximationusedintherelaxation. ThispropertyalsoimpliesthatKPEI 0 canbesolvedtooptimalityquiteeectively usingbranch-and-boundevenforlargeproblemsizessee,e.g.,MartelloandToth [66],despitethefactthatitisNP-hard.Theimplementationofthebranch-and-bound approachisasfollows.AsolutiontoRKPisobtainedateachnodeofourbranch-and-bound-tree, say~ w RKP .Webranchonthecustomerforwhich j ~ w RKP j 6 = j ~ w RKP j : FromLemma2, thiscorrespondstoacustomerthatiseitherifractionallyassigned,i.e.,0 < ~ w RKP <` 0 j orii~ w RKP ` 0 j ,but j ~ w RKP < j ~ w RKP .Incaseiwedeneasinglebranchwiththe constraint w j ` 0 ij {14 andasecondbranchwithconstraint w j =0 : {15 Inthecaseofiiwecreateabranchwiththeconstraint w j ~ w RKP j {16 andanotherbranchwiththeconstraint w j ~ w RKP j : {17 42
PAGE 43
Constraint4{14canbeaccommodatedbyrstreducing b by ` 0 j andaddingtheconstant ` 0 j totheobjectivevalue.Wethenredenetheboundsofcustomer j tobeintherange [0, u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ]withthemodiedobjectivefunctioncomponent ~ j = j w j + ` 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [( j ` 0 j 0 w j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j : Constraints4{16and4{17areaccountedforbysimplemodicationofthe j suggestedin[81].Thealgorithmgivenin[81]canthenbeappliedateachnodeofour searchtreewithoutfurthermodication. 4.2.1CFLFDwithSpeciallyStructuredRevenueFunctions Aspreviouslydiscussed,theheuristicandexactbranch-and-boundprocedures discussedinthissectionsolveKPEI 0 withanygeneralrevenuefunctions r j .However,the concaveenvelope j maybediculttocharacterizeexplicitly.Therefore,inthefollowing sectionsconsiderimportantpracticalcasesforwhichthiscanbedone.Inparticular,we studythreeclassesofrevenuefunctionswhichmodelproductpricinginbothmassand specializedproductionenvironments.Inadditiontoconsideringthesefunctionsfortheir real-worldappeal,weshowthatwhentherevenuefunctionsarelinearorconvex,KPEI 0 canbemoreecientlysolvedusinganalternativeproceduretosolveRKP. Concaverevenuefunctions .First,weconsidernon-decreasingconcaverevenue functions.Thischoiceoffunctionmodelsthescenarioinwhichmarginaldiscountsfor largercustomerdemandfulllmentlevelsareavailabletothecustomer.Thisscenariois applicabletomanufacturersofproductsoftenpurchasedinbulksizes,suchasdurable goods. 43
PAGE 44
NowletusapplythealgorithmpresentedinSection4.2.Inthiscase,theconcave envelope j j 2 ~ J isgivenby j w j = 8 > > > > > < > > > > > : = j j j w j 0 w j j = j w j j @ + j ` 0 j then j = ` 0 j ,andif j u 0 j u 0 j <@ )]TJ/F22 7.9701 Tf 0 -8.011 Td [(j u 0 j then j = u 0 j .Therefore theconcaveenvelopeconsistsofalinearsegmentextendingfromtheoriginto j ,thenthe truefunctionvalue, j w j ontheinterval j ;u 0 j ],andthenalinearfunctionwithslope0 ontheinterval u 0 j ;b ]. Inthenextsection,westudyingreaterdetailtwoclassesofrevenuefunctionsthat areofequalinterestinmodelingspecicrevenuestructures.Ingeneral,themethod describedinthissectionworksfortheserevenuefunctionsaswell.However,forthese specialcaseswewillproposeamoreecientalgorithm. 4.2.2ConvexandLinearRevenueFunctions ThissectionstudiesKPEIwhentherevenuefunctionsareeitherconvexorlinear. WeproposeamoreecientalgorithmtosolveRKPthatisapplicabletoeachofthese cases.Thealgorithmismotivatedbythesimpliedstructureoftheconcaveenvelopeused inRKP.Thatis,areformulationofRKPcanbesolvedinasimilarmannertothe continuousknapsackproblem. First,weassumeconvexrevenuefunctionswhichareconsistentwithaproduction scenariothatplacesincreasedpremiumsonlargercustomerdemandfulllmentlevels, 44
PAGE 45
orchargesanincreasingamountastheamountofmanufacturingtimerequiredfora customerincreases.Thiscaseisapplicabletohighlycustomizedgoodssuchashigh-end electronicsorspecializedautomobilemanufacturing. Ofcourse,themethodsofSection4.2areapplicableinthiscaseaswell.Asis illustratedinFigure4-2,theconcaveenvelope j j 2 ~ J thencorrespondstoapiecewise linearfunction.Inparticular,let j = j ` 0 j ` 0 j denotetheslopeofthelinearsegmentconnectingtheorigintothefunctionvalue evaluatedat ` 0 j and j = j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [( j ` 0 j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j betheslopeofthelinearsegmentconnectingthepoints )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(` 0 j ; j ` 0 j and )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(u 0 j ; j u 0 j Then,if j > j wehavethat j w j = 8 > > > > > < > > > > > : = j w j 0 w j ` 0 j = j ` 0 j + j w j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ` 0 j < > : = h j u 0 j u 0 j i w j 0 w j u 0 j = j u 0 j u 0 j j g : Wespliteachcustomer j 2 J + intotwoparts.Therstparthasdemandfulllmentlevel w j 1 2 [0 ;` 0 j ]andaprot functiongivenby j w j 1 .Thesecondparthasdemandfulllmentlevel w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 12.405 0 Td [(` 0 j ] andarevenuefunctiongivenby j w j 2 .Since j > j wewillalwaysfullyutilizetheentire 45
PAGE 46
rangecorrespondingtotherstpartofcustomer j beforeutilizinganypartoftherange correspondingtothesecondpartofcustomer j .RKPcanthenbereformulatedas maximize X j 2 J )]TJ/F26 11.9552 Tf 8.247 29.244 Td [( j u 0 j u 0 j w j + X j 2 J + j w j 1 + X j 2 J + j w j 2 subjecttoRKP 0 X j 2 J )]TJ/F21 11.9552 Tf 8.247 12.387 Td [(w j + X j 2 J + w j 1 + X j 2 J + w j 2 b w j 2 [0 ;u 0 j ] j 2 J )]TJ/F21 11.9552 Tf -157.55 -31.833 Td [(w j 1 2 [0 ;` 0 j ] j 2 J + w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ] j 2 J + : Theoptimalsolutiontothisproblemcanbedeterminedbysimplysortingthe customers j 2 J )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(and j 1and j 2for j 2 J + innonincreasingorderoftheircoecientin theobjectivefunction.Weimmediatelyobtainthatatmostoneelementoftheoptimal solution, w RKP ,hasavaluethatisstrictlybetweenitsbounds.Thisimpliesthatatmost onecustomeriseitherifractionaloriiexecutedbetweenitslowerandupperbounds. Formally,thecorrespondingsolutiontotheRKPisconstructedby w RKP j = w RKP 0 j 1 + w RKP 0 j 2 j 2 J + w RKP j = w RKP 0 j j 2 J )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(: Thisalgorithmisalsoapplicablewhentherevenuefunctionsarelinear.Thatis,a unitrevenueof r j isaccruedforeachunitofresourceconsumed;i.e., r j v j = r j v j j 2 ~ J ItiseasytoseefromFigure4-3thatagaintheconcaveenvelopesarepiecewiselinearin thiscase.Formallyif p j > 0theconcaveenvelopeisgivenby4{19whileif p j 0, j isgivenby4{20.MoreinterestingistherelationshipidentiedinTheorem1between RKP 0 andthelinearrelaxationofKPEI 0 givenby 46
PAGE 47
maximize X j 2 ~ J p j x j + X j 2 ~ J r j w j subjecttoKPEI 0 -R X j 2J w j b w j 2 [ ` 0 j x j ;u 0 j x j ] j 2 ~ J {21 x j 2 [0 ; 1] j 2 ~ J {22 where p j = p j )]TJ/F21 11.9552 Tf 11.955 0 Td [(r j a j j 2 ~ J Theorem1. Theoptimizationproblems RKP 0 and KPEI 0 -R areequivalentwhenthe revenuefunctions r j arelinearforall j 2 ~ J Proof. SeetheAppendix. Theorem1impliesthatthealgorithmproposedinSection4.2.2solvestheLP-relaxation ofKPEI.ItshouldbenotedthatKPEI 0 -R 0 canbethoughtofastheLP-relaxationto thetraditionalknapsackproblem,whichhasatmostasinglefractionalelement.This, ofcourse,coincideswiththeresultsinLemma2.Moreinterestingly,notethatweneed onlytobranchoncustomerswhichcorrespondtofractionalassignmentswhentherevenue functionsarelinear.ThisisbestseenbyrevisitingFigure4-3.Letcustomer^ | bethe customerinwhich ^ | 6 = ^ | .IfthiscoincideswithFigure4-3b,thenclearly w RKP 0 ^ | <` 0 ^ | whichcorrespondstoafractionalassignment.ThesituationrepresentedinFigure4-3a suggeststhat w RKP 0 ^ | couldtakeavalueanywhereintherange ;u 0 j .However,from Theorem1thesolutiontoRKP 0 isequivalentlythesolutiontoKPEI 0 -R.Fromthe proofof1, x KPEI 0 -R = w KPEI 0 -R u 0 j forcustomersinwhich j isgivenby4{20i.e.,whenthe caseinFigure4-3aholds.Thus,clearly,0
PAGE 48
4.3Branch-and-PriceAlgorithmImplementation ThusfarwehavefocusedprimarilyonhowtosolveKPEI 0 .Inthissectionwe discussmorespecicdetailsregardingtheimplementationofourbranch-and-price algorithm. 4.3.1InitialFeasibleSolution Ourrstconcernliesinprovidinganinitialsetofcolumnsthatwillensurethata feasiblesolutionexiststotheLPrelaxationoftherestrictedset-partitioningproblem, LPRSP.Ifpossible,weinitializeRSPwithfeasiblesolutionsthatassign all customers toasinglefacility.Thatis,wetrytoinitializeRSPwithacolumnofones 1 foreach facility i 2I forwhichthecorrespondingvalueof i 1 isnitei.e.,forwhichthe correspondingoptimizationproblemdenedinSection4.3.4isfeasible.Ifthisyieldsa feasiblecolumnforatleastonefacility i 2I ,thenwehaveaninitialfeasiblesolution since,implicitly,weincludeacolumnofzeroes 0 forallfacilities i 2I byusingthe relaxedconvexityconstraint4{2 0 If,aswilltypicallybethecase,itisnotfeasibletoassignallcustomerstoasingle facility,weimplementatwo-phaseprocedureforsolvingSP,wherePhase1generatesa feasiblesolutionforLPRSP.Tothisend,weincludeanonnegativeslackvariablefor eachassignmentconstraint4{1.OurPhase1objectiveisthentominimizethesumof onlytheseslackvariables.TheresultingPhase1problemisthusgivenby minimize X j 2J s j subjecttoSP-Phase1 X i 2I D i X d =1 x d ij d i + s j =1 j 2J {23 D i X d =1 d i =1 i 2I {24 d i 2f 0 ; 1 g d =1 ;:::D i ; i 2I : 48
PAGE 49
Hereagainthenumberofcolumnsassociatedwitheachfacilitymaybeverylarge. Therefore,wesolvethelinearrelaxationofSP-Phase1usingcolumngeneration.The pricingproblemissimilartoPP i ,exceptforthefactthatwemustaccountforthe alteredobjectiveinSP-Phase1.ThepricingprobleminPhase1isthusgivenby maximize )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X j 2J j x j )]TJ/F21 11.9552 Tf 11.955 0 Td [(f i y )]TJ/F21 11.9552 Tf 11.956 0 Td [( i subjecttoPP i -Phase1 X j 2J w j b i y w j 2 [ ` 0 ij x j ;u 0 ij x j ] j 2J x j 2f 0 ; 1 g j 2J y 2f 0 ; 1 g : ThisproblemisaKPEI 0 withrevenuefunctions r ij 0 i 2I ; j 2J .Therefore,itcan besolvedwiththeapproachdiscussedinSection4.2.2. ItiseasytoseethatiftheoptimalvalueofLPSP-Phase1equals0,anyoptimal solutiontothisproblemisfeasibleforLPRSP;otherwise,theprobleminstance isinfeasible.Intheformercaseweusethisfeasiblesolutiontoinitializethecolumn generationprocedureforsolvingSP. 4.3.2SolvingLPRSP Atanynodeinourbranch-and-boundtreewemustsolvearelaxationofSP.As previouslydescribed,thisrequiressolvingapricingproblemviatheheuristicandexact methodsproposedinSection4.2.However,becauseourpricingproblemdecomposesby facility,thereare jIj potentialpricingproblemstoconsider.Itisvalidto:isolvethem individuallyinanyorderandentertherstcolumnthatpricesout;iisolveall jIj problemsandenterthecolumnthatpricesoutthehighest;iiisolveall jIj problemsand enterallcolumnsthatpriceout.Inouralgorithm,wechooseaslightlymodiedversion 49
PAGE 50
ofoptionii.Ateachiterationofourcolumngenerationprocedurewesolveallpricing problemsviatheheuristicmethoddescribedinSection4.2.Allcolumnsthatpriceout favorablyareaddedtoRSPandthecolumngenerationprocedurecontinues.Weonly callonourbranch-and-boundprocedureifallheuristicsolutionsindicatethatnoneof thecolumnsareattractive.Inthiscase,weorderthepricingproblemsinnon-increasing orderoftheobjectivevaluesdeterminedbytheheuristic.Wecontinuetosolvethepricing problemsviabranch-and-bounduntileitherasinglecolumnpricesout,oritisdetermined thatnocolumnpricesout.Thiscompromiseenablesaddingmultiplequalitycolumnsat eachiterationviaanecientheuristicwhileminimizingrelianceonatime-consuming branch-and-boundproceduretosolveeachpricingproblem. Thisimplementationisintendedtoacceleratetheconvergenceofourcolumn generationprocedureusedtosolveLPRSPateachnode.However,asiscommon withcolumngeneration,therateofconvergenceisoftenreducedastheoptimalLPRSP solutionisapproached.Toavoidthisissue,weterminateourcolumngenerationprocedure whenourcurrentLPRSPsolutionvalueisprovablywithin10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 oftheoptimalsolution toLPRSP.Thisallowsustocontinuewithourbranch-and-priceinatimelymanner withacomparablystrongupperboundwithrespecttothatobtainedifLPRSPwas solvedtooptimality.Ofcourse,todetermineavalidupperboundonLPRSP,each pricingproblemmustbesolvedtooptimality.Toensurethatpricingproblemsarenot solvedtooptimalitytoooften,weonlyupdatetheLPRSPupperboundaftersolving jIjjJj pricingproblemseitherheuristicallyorexactly.Thisimplementationchoice anticipatesthatthenumberofcolumnsrequiredtosolveLPRSPisafactorofboththe numberoffacilitiesandthenumberofcustomers.Therefore,thefrequencyofupdating theupperboundforLPRSPshoulddecreaseaseitherthenumberofcustomersorthe numberoffacilitiesincreases. ObtainingqualityfeasiblesolutionstoSPisofequalvaluetoourbranch-and-price implementation.Therefore,asaheuristictoobtainfeasiblesolutionstoSP,wesolveRSP 50
PAGE 51
asanintegerprogramusingcolumnsgeneratedinsolvingLPRSPattherootnode. Forahighpercentageofourtests,thetimerequiredtosolvethisMIPissmall.This implementationprovidedqualitylowerboundsearlyinouralgorithmwhichprovedtobe benecialinpruningourbranch-and-pricetree. 4.3.3NodeandVariableSelection Inourbranch-and-pricealgorithm,weinitiallydeterminetheorderinwhichnodes shouldbeconsideredbyusingadepth-rstrule.Then,onceafeasiblesolutiontoSPis obtained,weexplorethetreeusingthewell-knownbest-boundrule.Thisnodeselection policyisalsoimplementedinthebranch-and-boundprocedureusedtosolveourpricing problemtooptimality. Wefoundthebranchingdecisiontobeofparticularsignicancetoourproblem.Itis commonintheliteraturenottobranchonthe valuesthemselvesinLPRSPinorder topreservethestructureofthepricingproblem.NeebeandRao[72]proposebranching on x variablesthathaveavalueof1inacolumnassociatedwithafractional .This branchingschemeiseasilyaccommodatedinthepricingproblembygeneratingcolumns adheringtoanyassignmentsxedatpreviousnodes.However,asnotedbyCeselliand Righini[21],intuitionsuggeststhattheimpactofbranchingonfractionalprocurement variableswillbegreater.NotethatinthecontextofLPRSP,afractionalprocurement variable, y i ,correspondstoeither0 < P K i k =1 k i < 1or0 < 0 i < 1,where 0 i isacolumn inwhichnocustomersareassignedtofacility i withcost 0 i =0.Thoughthemodel consideredintheirworkmakesprocurementdecisionswithnoexplicitprocurementcost, oneoftheirbranchingstrategiesplacesabranchingpriorityontheprocurementvariables. Similarly,weproposeanimplementationforbranchingonfacilityprocurementvariables xedat1thatpreservesthestructureofthepricingproblembutrequiresnoadditional constraintsbeaddedtoLPRSP.Thatis,if y i isxedto1ataparticularnode,facility i 'sprocurementcostissimplytreatedasaconstantintheLPRSPobjectiveandthe costofcolumnsassociatedwiththatfacilityareappropriatelyreducedbythatamount. 51
PAGE 52
If y i isxedtozeroataparticularnode,thenallcolumnswithcustomersassignedto facility i areomittedandnocolumnsassociatedwithfacility i aregeneratedatthatnode. Consistentwithintuition,ourtestingindicatedthatbranchingpriorityshouldbegiven toprocurementvariablesoverassignmentvariables.Indeterminingwhichofthese x 'sor y 'sonwhichtobranch,weassessthedegreeoffractionalityofeachvariableinasolution toLPRSP.Thevariablewhichisleastfractionali.e.,thatvariablewhichisclosestto0 or1,wheretiesarebrokenarbitrarilyischosenforbranching.Amost-fractionalvariable selectionimplementationisalsousedinthepricingproblembranch-and-boundalgorithm. 4.3.4OptimalColumnCost RecallfromSection4.1thatthecostofthe d th columnassociatedwithfacility i d =1 ;:::;D i ; i 2I ,isdeterminedasafunctionoftheoptimalsizesofthose customersassignedinthatparticularcolumn.Ofcourse,whenthepricingproblemis solvedtooptimalitytheoptimalcustomerdemandfulllmentlevelsareimmediately available.Whenacolumnisgeneratedusingourheuristic,thesizeofthecustomers maynotnecessarilyrepresenttheoptimalsizesforthecorrespondingsetofcustomer assignments.Therefore,CC d i mustbesolvedtooptimality.However,notethatCC d i canbeequivalentlyrepresentedby C +maximize X j 2 J ~ j w j x j subjecttoCC d i X j 2 J w j x j b w j 0 x j 2f 0 ; 1 g 52
PAGE 53
where J isthesetofcustomersassignedtothe d th columnassociatedwithfacility i ,with functions j j 2 ~ J denedas ~ j w j = 8 > > > > > < > > > > > : =0 w j =0 = r ij w j + ` ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(r ij ` ij 0 < w j u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij : = r ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(r ij ` ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij < w j b; andwithconstants C = P j 2 J r ij ` ij + p ij )]TJ/F21 11.9552 Tf 11.211 0 Td [(f i and b = b i )]TJ/F26 11.9552 Tf 11.211 8.966 Td [(P j 2 J a ij + ` ij .Represented inthisform,CC d i simplytakestheformofKPEI 0 andcanbesolveddirectlybythe approachesdiscussedinsection4.2.GivenanoptimalsolutiontoCC d i x ; w ,the optimalcustomerdemandfulllmentlevelsintermsoftheoriginaldecisionvariablesare determinedbysetting w j = ` j + w j for j 2 J 4.4ComputationalResults Inthissectionwediscusstheperformanceofourbranch-and-priceprocedureona randomlydeterminedsetoftestinstances.Weseparatelyconsiderresultsforboththe GAPFDandtheCFLFDwithvaryingrevenuefunctioncharacterizations. 4.4.1ExperimentalData Intestingthenon-linearrepresentationofboththeCFLFDandGAPFDweconsider instanceswith5facilities,whilethedecreaseddicultyofthelinearinstancesallows ustoconsiderinstanceswith30facilities.Ineithercase,instanceswiththenumberof customersequalto jJj =2 jIj ,3 jIj ,and5 jIj arestudied,andforthepurposesofthis studyweassumethatallcustomerscanbeassignedtoallfacilities.Foreachcustomer, wegeneratetherandomvectorsofxedprotparameters P j fromuniformdistributions on[30 ; 50].Thecustomerrequirements A j L j and D j aregeneratedfromuniform distributionson[10 ; 20],[75 ; 125],and[15 ; 35],respectively.Here, A j and L j arethe randomvectorsofxedcapacityconsumptionandcustomerlowerbounds,respectively, and D j isarandomvectorcontainingvaluesrepresentativeofthedierencebetweenupper andlowerboundsofacustomer.Ineachofourtestswefocusoninstancesinwhichthe 53
PAGE 54
facilitycapacitiesareidentical.ForGAPFD,weset b i = jJj i 2I whileforthe CFLFDwegeneratecapacitiessothat b i = jJj i 2I ,where = E min i 2I A i 1 + L i 1 jIj : {25 Theparameter isusedingeneratingGAPFDinstancestocontrolthelevelofexibility availablewhendeterminingthesizeofeachcustomer.Inthesetestsweconsidera moderateexibilitylevelbysetting =1 : 2.Theparameter > 1,usedinthegeneration oftheCFLFDexperiments,inatesthecapacityofanfacilitytoensurethatnotall facilitiesarerequiredinafeasiblesolutiontotheCFLFD.Withoutthisconsideration, thefacilityprocurementdecisionsmaybetrivial.Ineachoftheexperimentsconsidered inthissection,facilitycapacitiesweregeneratedwith =2.Furthermore,inthecase oftheCFLFD,thecostofprocuringanfacilityisdirectlyproportionaltothesizeof thefacilityitself.Thatis,thecostofprocuringfacility i i 2I isgivenby F i = b i C i where C i representstheunitcostofprocurementgeneratedfromauniformdistributionon [0 : 75 ; 1 : 5]. Inourexperiments,weconsiderthethreeclassesofrevenuefunctionsdiscussedin Section4.2.Forprobleminstanceswithlinearrevenuefunctionswegeneratetheelements ofvectors R j ofunitrevenuesusingauniformdistributionon[2 ; 5].Theconvexand concaverevenuefunctionsthatweusedinourexperimentsareoftheform r ij v ij = S ij v ij 2 {26 and r ij v ij = S ij p v ij ; {27 respectively.Aninitialvalueoftheelementsofthevectorofcoecients S j foreach customerisrandomlygeneratedfromauniformdistributionontheinterval[0 : 5 ; 1 : 5]. However,toinsurethattheprobleminstancesarecomparablewescaleeachofthese coecientssothat r ij v ij = p ij + r ij v ij for v ij 2f ` ij ;u ij g where r ij istheunitrevenue 54
PAGE 55
inthecorrespondingprobleminstancewithlinearrevenuefunctions.Thepiecewiselinear functionsconsideredareobtainedbyapproximatingtheconcaveandconvexfunctions generatedin4{26and4{27.Toassesstheeectofvaryingthenumberofsegments usedtoapproximatedthenon-linearfunctions,weconsiderinstanceswith5,10and50 segments. Eachofourinstanceswasrununtileitherasolutionvaluewithin10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 oftheoptimal solutionwasobtainedoratimelimitofonehourwasreached.Ourtablespresentresults for10randomlygeneratedinstancesforeachcombinationofparametersettings.All experimentswereperformedonaDellpoweredge2600withtwoPentiumIV3.2Ghz processorsand6GBofRAM.Themixed-integerprogrammingproblemsaswellasthe relaxedmasterproblemsweresolvedusingCPLEX11.0.Wecomparedtheresultsofour approachforproblemswithnon-linearrevenuefunctionswiththoseobtainedbyBARON 8.1.1. Inourexperimentation,wesoughtbothtoassesstheeectivenessofourbranch-and-price approachandtogaininsightonthedierenceindicultyinthetwonon-linearcases considered.Forthisreason,eachofourrandomlygeneratedinstanceswastestedunder bothaconcaveandconvexrevenuefunction.Therefore,ourtablesassociatedwiththe resultsofproblemswithnon-linearrevenuefunctionsprovidecombinedresultsforeach ofthesecases.Duetothedierenceinthesizeofinstancesconsideredinthelinearcase, theseresultsarepresentedseparately.Specically,eachtablereports ithenumberofcolumnsgeneratedintheentirebranch-and-pricealgorithm, iithenumberofnodesconsideredinthebranch-and-pricetree iiitheamountoftimerequiredtosolvetherelaxedmasterproblemattherootnode ivthetotaltimerequiredbythebranch-and-pricealgorithm vthetotaltimerequiredbythecommercialsolveri.e.,BARONorCPLEX,withthe followingadditionalinformationwhereappropriate: 55
PAGE 56
{instancesunsolvedinonehourcontainadditionalinformationinthesuperscript ofthecommercialsolvertimecolumn; {therstsuperscriptindicatestherelativesolutionerrorcalculatedbyusingthe solversbestlowerandupperbound, z UB S z LB S ;i.e. error 1 = z UB S )]TJ/F21 11.9552 Tf 11.955 0 Td [(z LB S z UB S 100%; {thesecondsuperscriptindicatestherelativesolutionerrorcalculatedusingthe solutionobtainedbybranch-and-pricealgorithmandthebestsolutionobtained bythecommercialsolver, z BP z LB S ;i.e., error 2 = z BP )]TJ/F21 11.9552 Tf 11.955 0 Td [(z LB S z BP 100% : 4.4.2CFLFDResults Inthissectionwecomparetheperformanceourbranch-and-pricealgorithmagainst twowell-knowncommercialsolversonabroadsetofCFLFDinstances.Section4.4.2.1 presentsresultsforCFLFDinstanceswithdierentiablenonlinearfunctionssolvedvia boththeapproachdevelopedinthischapterandBARON.InSection4.4.2.2weperform anadditionalcomparisonforpiecewiselinearandlinearinstancessolvedwithboth branch-and-priceandILOG'sCPLEX11.0. 4.4.2.1Nonlinearrevenuefunctions:comparisonwithBARON Tables4-1{4-3presentresultsforthedierentiablenonlinearconcaveandconvex CFLFDinstanceswithfunctionsgeneratedby4{26and4{27.Ineachofthesethree tablesitisevidentthatbranch-and-priceoutperformsthecommercialsolverinevery instance.Ourcomputationalstudyfoundthatourchoiceofbranchingstrategy,discussed inSection4.3.3,was,inpart,acontributingfactortothissuccess.Asanexample,for theinstancesconsideredinTable4-4,placingabranchingpriorityontheprocurement variablesovertheassignmentvariablesreducedouraveragebranch-and-pricetimebya factorof50fortheconvexinstances.Thisislikelyadirectresultofthenumberofnodes beingreducedbyafactorofmorethan10. 56
PAGE 57
AnanalysisoftheperformanceofthecommercialsolverindicatesthatBARON isunabletosolveanyoftheconcaveinstancestowithinthedesiredtolerancelevels withinonehour.Whenconvexrevenuefunctionsareconsidered,BARON'sperformance improves,solvingthesmallersetofinstancesshowninTable4-1withinanaverage ofapproximatelyoneminute.Evenforthelargestnumberofcustomersconsidered, BARONsolves7of10convexinstancesinanaverageofapproximately32minutes. Notably,ourbranch-and-pricealgorithmrequiresonaveragenomorethantwoseconds tosolveproblemswith jJj 3 jIj .Table4-3indicatesthatthebranch-and-pricetime requiredgrowsnotablywhen jJj =5 jIj .However,eachoftheseinstancescouldstill besolvedwithinthedesiredoptimalitytoleranceswithinanhour.Ourtestingrevealed thattheextensivetimesrequiredforexperimentsshowninTable4-3werenotaresult ofthecolumngenerationprocedureattherootnodeorthetimespentsolvingthe MIPusingcolumnsgeneratedattherootnode.Unfortunately,thetimespentsolving pricingproblemstooptimalityinthesubsequentnodeswasthedirectcauseofthe increasedtimerequirements.Instanceswith jJj =10 jIj customerswereconsideredinthe experimentation,butneitherbranch-and-price,norBARONcouldsolvetheseinstances consistentlywithinonehour.However,branch-and-priceonlyrequiredsecondstosolve instanceswith10facilitieswhencustomerratioswerelimitedto jJj 3 jIj ForthoseinstanceswhichBARONfailedtosolvetooptimalitywithinonehour,the qualityofthebestfoundsolutionswasnotablypoor.Forsmallerconcaveinstancesi.e., jJj 3 jIj therelativeerrorbetweenthebestupperboundandincumbentsolution rangesbetween55%and74%.However,foreachoftheseinstances,branch-and-price solvedtheproblemstooptimalityinamatterofseconds.Evenwhenthetrueoptimal solutionobtainedbybranch-and-priceisusedtoassessthequalityoftheBARON solution,theaveragerelativeerroronlydecreasestobetween8%and34%.Theseresults weresucienttodeterminethatrunningconcaveexperimentswith jJj =5 jIj onthe commercialsolverwasunnecessary;therefore,thecorrespondingcolumnisemptyinTable 57
PAGE 58
4-3.Asnotedpreviously,BARONwasmoresuccessfulinsolvingtheconvexinstances weconsidered.Therefore,evenforthefourinstanceswith jJj =5 jIj thatcouldnotbe solvedinonehour,thecorrespondingerrorsaresubstantiallyreducedfromthoseobtained inanyoftheconcaveinstances.Somewhatunexpectedly,theresultsfromthecommercial solverindicatethatprobleminstanceswithconcaverevenuefunctionswerefarmore dicultthantheoneswithconvexrevenuefunctions.Thedierencebetweenthesetwo typesofrevenuefunctionsislesspronouncedwhencomparingtheperformanceofthe branch-and-pricealgorithm.Tables4-1{4-3indicatethatthenumberofcolumns,number ofnodes,andsolutiontimeinbothcaseswereconsistentineachproblemsetconsidered. Inadditiontothecontributionofthebranchingrulediscussedpreviously,thesuccess ofourbranch-and-pricealgorithmfornonlinearrevenuefunctionscanbeattributedtothe combinedeectivenessoftheproposedmethodologiestosolveourpricingproblemandthe tightnessoftheset-partitioningformulation.Thelatterisevidentbythelimitednumber ofnodesexploredinthesearchtree.EachoftheTables4-1{4-3indicatesthatnomore than30nodeswereconsideredforanysingleinstance.Interestingly,5ofthe10instances presentedinTable4-1weresolvedattherootnode. 4.4.2.2Piecewiselinearandlinearrevenuefunctions:comparisonwith CPLEX Tables4-5{4-10comparetheperformanceofourbranch-and-priceapproachforconvex andconcavepiecewiselinearfunctionsagainstlinearizedformulationssolvedinCPLEX. TheresultsshowninTables4-5and4-6indicatethatbranch-and-pricesolvesallpiecewise linearconvexinstanceswithacustomer-to-facilityratiolessthanorequaltothree, regardlessofthenumberofsegmentscomprisingthepiecewiselinearfunction,inless thanonesecond.However,CPLEXrequiresatleast14timestheaveragecomputational timeofbranch-and-priceforinstanceswith jJj 3 jIj .Forpiecewiselinearconvex instanceswith25customers,thebranch-and-pricetimesgrownotably.However,Table 4-7showsouralgorithm,onaverage,outperformsCPLEXforthisscenario,aswell.In 58
PAGE 59
addition,itisclearthatasthenumberofsegmentsincreases,CPLEXrequiresnotably morecomputationaltime,whichisaresultoftheadditionalbinaryvariablesrequiredto modelpiecewiselinearconvexfunctions.Tothecontrary,Tables4-5and4-6suggestthat thenumberofsegmentshaslittleeectonthetimerequiredbybranch-and-price.While Table4-7doesindicateanincreasingtrendin average timerequiredbybranch-and-price asthenumberofsegmentsincreases,ananalysisofindividualinstancessuggeststhis trendisfoundonlyinexperiments2and6.Tables4-8{4-10showthatbranch-and-priceis competitivewithCPLEXonpiecewiselinearconcaveinstanceswith jJj 3 jIj .However, CPLEXperformssignicantlybetteronconcavepiecewiselinearinstanceswhencompared tothepiecewiselinearconvexresults.Thisresultisexpected,sincepiecewiselinear concaverevenuefunctionscanbemodeledwithadditionalcontinuousdecisionvariables andlinearconstraints,insteadofthebinaryvariablesrequiredforpiecewiselinearconvex functions.Table4-10demonstratesthatCPLEXisabetteralternativeastheratioof customerstofacilitiesincreasesforpiecewiselinearconcaveinstances. Wealsoincludedlinearinstancesinourcomputationaltesting.Ourtestingindicated thatCPLEXwasmoresuccessfulthanbranch-and-priceonproblemswithlessthan30 facilitiesandacustomer-to-facilityratioof2,3,or5.However,asshowninTable4-11, CPLEXisunabletosolveanyofthe10instanceswith30facilitiesand60customers withinanhour,whileourbranch-and-pricemethodologysolveseachoftheseinstancesin anaverageoflessthan15seconds.Unfortunately,ourtestsonlargerinstancesi.e.,30 facilitieswith90and150customersrevealedthanneitherCPLEXnorbranch-and-price wassuccessfulwithinthespeciedtimelimit. 4.4.3GAPFJResults InthissectionwepresentresultsfortheGAPFJ,whichisaspecialcaseofthe CFLFDwhenfacilityprocurementdecisionsareomittedfromthemodel.Forthese problems,wechosetofocusondierentiablenonlinearandpurelylinearrevenue functions.ThenonlinearresultsshowninTables4-12{4-14indicatethatthebranch-and-price 59
PAGE 60
algorithmperformsequallyaswellfortheGAPFJasitdidfortheCFLFD.However,as indicatedbyTable4-15,CPLEXoutperformsourbranch-and-priceforlinearinstancesof theGAPFJ.ThesuccessofCPLEXwasevidentforinstanceswithcustomer-to-facility ratiosof2 jIj and5 jIj aswell.AcomparisonofTable4-12toTable4-1suggests thatBARONismoresuccessfulonconcaveinstanceswhichomitthebinaryfacility procurementvariables.Incontrast,theaveragetimetosolveconvexGAPFJinstances with jJj =2 jIj ismorethanthatrequiredfortheCFLFDinstanceswiththesame numberoffacilitiesandcustomers.Notethatadierentsetofdatawasusedinour GAPFJexperiments,socautionshouldbetakenincomparingresultsfromthegeneral CFLFDandtheGAPFJspecialcase.However,theGAPFJconcaveexperimentswere stillclearlydicultforBARON.Only9ofthe30experimentsconsideredweresolved withintheallottedtime.In3ofthe10experimentsshowninTable4-14,BARONwas unabletoevenndafeasiblesolutionwithinanhour.Incontrast,whileonly11ofthe 30convexinstancesweresolvedbythecommercialsolver,theerrorsofthebestsolution obtainedweremuchbetterthanthoseobtainedintheconcavetests.Infact,in12ofthe 19instancesunsolvedinTables4-13and4-14,thebestsolutionsobtainedwereactually optimal,butBARONhadyettoproveoptimalitywhenthetimelimitwasreached. Thebranch-and-pricetimesinTables4-12{4-14indicatethesuccessoftheproposed methodologyonthisimportantspecialcase.Forexample,theresultscorrespondingto jJj 3 jIj showthattheaveragebranch-and-pricetimewasonlyafractionofasecond. Thesamesetoftablesrevealsthatthenumberofnodesconsideredwasminimal.Infact, in14ofthe15instancesconsideredinthesethreetables,theproblemsweresolvedatthe rootnode.Theincreaseinbranch-and-pricetimefor jJj =5 jIj islesssignicantthanthe resultsobtainedfortheCFLFD. Testingonlargerinstancesindicatedthatproblemswith jJj =10 jIj customers remainsolvableinunderanhour.Inaddition,ifthecustomer-to-facilityratioislessthan three,thenbranch-and-priceissuccessfulinsolvinginstanceswith10facilitiesinless 60
PAGE 61
than15minutes.Thissuggeststhatthecustomer-to-facilityratioisthekeyfactorin determiningthesizeofprobleminstancesinwhichbranch-and-priceissuccessful. 4.5Conclusions Inthischapterweconsideredageneralizationofthecapacitatedfacilitylocation problemthatincludesconsiderationsforexibledemand.Weproposedanexact branch-and-pricealgorithmtosolvetheresultingCFLFDproblembasedonaset-partitioning representationofourmodel.Tosolvetheresultingpricingproblem,westudiedan interestinggeneralizationofnonlinearknapsackproblemswithexibleitemsizes. Motivatedbyarelevantrelaxationwithanoptimalsolutionshowntopossessattractive structuralproperties,wediscussedhowboththeheuristicandexactapproachesused tosolvetheresultingKP[81]canbeappliedtosolveourpricingprobleminitsmost generalform.Then,forrevenuestructurescommontoreal-worldpricingconditions,we proposedanalternativepricingproblemsolutionmethodology,whichismoreecient thanthatrequiredinthemostgeneralcase.Ourcomputationalstudysuggeststhatthe branch-and-priceapproachproposedinthisworkconsistentlyoutperformsapopular commercialnonlinearsolverinourtestingofnonlinearCFLFDandGAPFJinstances. Table4-1.CFLFDwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 1140130.20.83600 ; 12 316130.10.329.5 28590.10.23600 ; 21 8390.10.146.2 3198330.11.83600 ; 17 474330.10.5116.2 4156210.11.03600 ; 13 230210.10.455.6 510770.10.43600 ; 18 10570.10.221.7 6141210.10.93600 ; 8 140210.10.370.6 7118130.20.63600 ; 13 134130.10.222.7 810690.30.73600 ; 9 11290.10.341.3 9188290.11.63600 ; 13 471290.10.5124.8 10170230.11.23600 ; 20 271210.10.499.6 Avg140.917.80.10.93600 ; 14 233.617.60.10.362.8 hello 61
PAGE 62
Figure4-1.Illustrationof j and j forageneralrevenuefunction Figure4-2.Concaveenvelope:convexrevenuefunction 62
PAGE 63
Figure4-3.Concaveenvelope:linearrevenuefunction Table4-2.CFLFDwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 113210.20.23600 ; 29 15410.10.127.9 214111.11.13600 ; 31 15410.10.175.8 313910.20.23600 ; 31 15610.10.196.0 4348190.44.43600 ; 34 363190.21.2621.9 513910.20.23600 ; 25 12910.10.151.9 613510.30.33600 ; 23 13910.10.1101.1 7278150.31.73600 ; 23 313150.10.7232.9 822490.32.23600 ; 28 26690.31.175.1 913730.30.33600 ; 21 17750.10.2154.5 10276150.55.03600 ; 23 344170.21.5324.2 Avg194.96.60.41.63600 ; 27 219.570.10.5176.1 63
PAGE 64
Table4-3.CFLFDwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 1863172.3177.3-623111.319.8656.9 27309418.41393.9-672917.3166.2600.9 3109051.21128.3-1627250.630.33600 ; 2 41299234.2847.6-1554231.5634.73600 ; 1 51083233.1183.4-1212230.755.42931.6 61217234.2417.5-1358270.787.32878.7 71080192.9285.3-1358191.117.13600 ; 3 811362512.4586.3-1207213.51229.82716.7 9727135.936.2-9301518.8698.2913.2 10837174.9108.7-1133172.4335.63041.1 Averages1006.219.446.0516.4-1163.018.84.9327.42088.9 Table4-4.BranchingrulecomparisonfortheCFLFDwithnonlinearrevenuefunctions:5 facilities,10customers, =1 : 2 ConcaveConvex branchingon x branchingon y branchingon x branchingon y BPtotalBPtotalBPtotalBPtotal ExpNodestimeNodestimeNodestimeNodestime secsecsecsec 1612.8130.5811.8130.1 230.290.130.290.1 32917.9330.93216.8330.1 4992.5210.6912.6210.1 5110.570.3140.670.1 65558.2210.650111.9210.1 7691.7130.3852.9130.1 870.890.591.190.1 966510.9290.976518.7290.1 101092.3230.81073.0210.1 Averages1873.7817.80.6197.74.9617.60.1 hello hello hello hello hello hello 64
PAGE 65
Table4-5.CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,10 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.51.40.40.70.42.3 20.11.10.11.60.16.1 30.94.00.84.11.110.7 40.82.50.52.40.68.1 50.31.50.20.70.33.9 60.713.60.512.90.7289.5 70.54.20.43.60.515.5 80.75.40.43.60.624.9 91.147.20.820.10.9241.4 100.89.90.46.00.9330.0 Avg0.69.10.45.60.693.2 Table4-6.CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,15 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.20.80.10.80.32.0 20.37.50.14.70.351.2 30.11.10.12.50.23.2 41.95.71.05.42.022.7 50.11.10.10.70.22.9 60.23.30.22.00.36.9 71.014.90.89.81.249.8 81.243.30.6241.41.11052.0 90.34.40.33.00.516.0 102.817.32.115.22.862.2 Avg0.89.90.528.50.9126.9 hello hello hello hello hello hello 65
PAGE 66
Table4-7.CFLFDwithpiecewiselinearconvexrevenuefunctions:5facilities,25 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 143.8103.926.2172.619.0464.0 224.78.9131.245.61798.53600.0 326.3100.476.2120.323.8324.3 4307.5257.8977.8505.4517.01360.6 554.694.8158.4381.759.5357.7 6177.4104.3261.0155.8899.1562.2 731.5510.519.01400.225.53600.0 871.1223.788.6248.544.1861.1 942.47.842.016.228.03600.0 1069.755.326.588.551.53600.0 Avg84.9146.7180.7313.5346.61833.0 Table4-8.CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,10 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.20.70.20.60.41.1 20.10.70.10.70.11.1 30.72.90.73.21.36.7 40.41.40.41.00.82.6 50.20.60.20.60.41.0 60.42.60.52.50.84.6 70.21.10.30.60.51.6 80.31.70.31.60.83.6 90.74.40.82.81.67.2 100.62.30.62.61.46.0 Avg0.41.80.41.60.83.5 hello hello hello hello hello hello 66
PAGE 67
Table4-9.CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,15 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 10.10.50.10.60.11.1 20.20.80.20.50.61.8 30.10.60.10.60.21.3 41.01.31.52.23.52.7 50.10.60.10.40.10.8 60.11.00.11.40.32.5 70.61.40.81.41.74.5 80.70.91.20.81.21.6 90.11.20.12.00.33.4 102.04.71.74.44.510.1 Avg0.51.30.61.41.23.0 Table4-10.CFLFDwithpiecewiselinearconcaverevenuefunctions:5facilities,25 customers, =1 : 2 5segments10segments50segments BPtotalCPLEXBPtotalCPLEXBPtotalCPLEX Exptimetimetimetimetimetime secsecsecsecsecsec 144.47.292.89.0313.556.1 245.51.4145.42.5614.93.8 388.12.0132.37.9472.96.2 4375.029.0293.560.1548.8182.4 5172.88.458.210.7507.731.5 6136.927.3410.638.7322.659.0 737.23.242.74.1201.06.7 873.77.085.313.8506.333.6 963.24.0120.35.6183.212.3 1035.75.260.83.4120.34.7 Avg107.39.5145.515.6379.139.6 hello hello hello hello hello hello 67
PAGE 68
Table4-11.CFLFDwithlinearrevenuefunctions:30facilities,60customers, =1 : 2 Linear RootBPtotalCPLEX ExpColsNodestimetimetime secsecsec 1107119.99.93600 : 6 ;: 2 296479.515.13600 : 7 ; 0 398816.96.93600 : 6 ;: 1 4928110.310.33600 : 5 ; 0 51007112.712.73600 ;: 2 6100819.39.33600 : 8 ;: 1 7104418.08.03600 : 4 ;: 2 892535.65.03600 : 6 ;: 1 91176318.937.23600 : 8 ;: 2 101116710.013.23600 : 3 ;: 2 Avg1022.75.49.112.83600 : 7 ;: 1 Table4-12.GAPFJwithnonlinearrevenuefunctions:5facilities,10customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 15510.10.11123.35410.10.1100.0 25810.10.11864.65710.10.1112.2 34910.10.11036.65010.10.186.1 46510.10.13600 ; 1 6410.10.163.7 56210.10.12089.36310.20.277.9 66210.10.11636.06310.10.1101.9 76410.10.11744.56610.10.1245.1 86310.10.12539.76510.10.165.8 96610.10.11741.16010.10.1182.8 106410.10.1897.86710.10.155.6 Avg60.810.10.11827.3 ;: 1 60.910.10.1109.1 hello hello hello hello hello hello hello 68
PAGE 69
Table4-13.GAPFJwithnonlinearrevenuefunctions:5facilities,15customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 110110.20.23600 ; 8 9610.30.32994.8 210610.30.33600 ; 12 11210.40.43600 ; 0 312510.50.53600 ; 12 12110.50.53600 ; 0 412510.60.63600 ; 10 11810.30.33600 ; 0 511910.40.43600 ; 12 11510.30.33600 ; 0 612110.20.23600 ; 11 12410.30.33600 ; 0 711010.30.33600 ; 15 11210.20.23600 ; 0 810910.30.33600 ; 12 11410.50.53600 ; 0 911410.50.53600 ; 11 11510.60.63600 ; 0 109810.20.23600 ; 10 11710.30.33600 ; 0 Avg112.810.40.43600 ; 11 114.410.40.43539.5 ; 0 Table4-14.GAPFJwithnonlinearrevenuefunctions:5facilities,25customers, =1 : 2 ConcaveConvex RootBPtotalBARONRootBPtotalBARON ExpColsNodestimetimetimeColsNodestimetimetime secsecsecsecsecsec 126716.06.03600 ; 33 25511.11.13600 ; 1 2274186.786.73600 ; 09 286166.866.83600 ; 4 3246113.713.73600 ; 35 26112.22.23600 ; 1 4293164.964.93600 ; 25 25818.48.43600 ; 0 53331118.237.73600 ; 37 345731.143.13600 ; 1 624814.24.23600 )]TJ/F22 7.9701 Tf 6.587 0 Td [(; )]TJ/F20 7.9701 Tf 6.587 0 Td [( 272138.338.33600 ; 0 726216.16.13600 ; 18 25211.91.93600 ; 2 823512.82.83600 ; 29 23711.21.23600 ; 0 922213.83.83600 )]TJ/F22 7.9701 Tf 6.587 0 Td [(; )]TJ/F20 7.9701 Tf 6.587 0 Td [( 22112.92.93600 ; 1 10256121.121.13600 )]TJ/F22 7.9701 Tf 6.587 0 Td [(; )]TJ/F20 7.9701 Tf 6.587 0 Td [( 262111.111.13600 ; 1 Avg263.62.022.824.73600 ; 27 264.91.616.517.73600 ; 1 hello hello hello hello hello hello hello 69
PAGE 70
Table4-15.GAPFJwithlinearrevenuefunctions:30facilities,90customers, =1 : 2 Linear RootBPtotalCPLEX ExpColsNodestimetimeTime secsecsec 11130179.089.410.0 21052399.4160.160.1 31109189.0104.815.5 410573116.1193.176.7 51053391.497.86.0 61075339.143.03.6 711243108.5129.520.6 81053333.876.241.9 91101155.1107.852.3 1011201133.2192.358.4 Avg1087.42.284.5119.434.5 hello hello hello hello hello hello hello hello 70
PAGE 71
CHAPTER5 GAPFDHEURISTICWITHASYMPTOTICPERFORMANCEGUARANTEES Inthischapterweconsidertheprot-maximizingGAPFDthatrequiresthe assignmentofcustomerswithexibledemandtoavailablecapacitatedresources facilities.AsdiscussedinChapter3,thisclassofproblemsndsapplicationina widerangeofpracticalsettings.Relatedproblemsinsalesandadvertisingplanning involvetradeosbetweenrevenuegenerationandresourceconstraintsandcosts.Insales forceplanningcontexts,forexample,thesalesforceservesasasetofresources,where eachsalespersonhasalimitedamountoftimeand/oreortthattheycanallocateto customers.Itisoftenthecasethatthegreatertheamountofeortasalespersonallocates toagivencustomer,thegreaterthereturnfromthatcustomerintermsofsales.The planningphasethereforeinvolvesdeterminingtheassignmentofsalesforcetocustomers andthedegreeofeortasalespersonshoulddevotetoeachassignedcustomerinorder tomaximizethetotalreturnfromcustomersorexpectedreturn,whentherelationship betweeneortandsalesisnotdeterministic.Thissalessettingmaybeinterpretedmore generallyasapplyingtoasetofavailablemarketinginstruments,whereanallocationof capacity-constrainedmarketinginstrumentstocustomersmustbedeterminedinorderto maximizeprot. SincetheGAPFDgeneralizestheGAP,itisclearly NP -Hard.Furthermore,since thefeasibilityproblemassociatedwiththeGAPis NP -Complete,itisclearthatthe feasibilityproblemassociatedwiththeGAPFDis NP -Completeaswell.Wetherefore developacustomizedfamilyofheuristics,andshowthatthisclassofheuristicsis asymptoticallyfeasibleandoptimalwithprobabilityoneasthenumberofcustomers goestoinnityunderaverybroadprobabilisticmodelfortheproblemparameters.Our heuristicsareinthesamespiritascertainheuristicsthathavebeendevelopedforthe GAPbyMartelloandToth[64]andRomeijnandRomeroMorales[79].Inparticular, givenavectorofmultiplierseachcorrespondingtoafacility,aweightfunctionisdened 71
PAGE 72
tomeasurethepseudo-protofassigningacustomertoafacility.Thisweightfunction isthenusedtojudiciouslydetermineitheorderinwhichtoassignthecustomers,ii thefacilitytowhicheachcustomershouldbeassigned,andiiianappropriatecustomer demandfulllmentlevel.Inaddition,thesefunctionsmotivateimprovementheuristics thatareessentialinordertobeabletoderiveattractiveperformanceguaranteesfor theheuristic.Themainresultofthischapteristhedevelopmentofaheuristicthatis asymptoticallyfeasibleandoptimalwithprobabilityoneunderaverygeneralstochastic modeloftheproblemparameters.DuetothenatureoftheGAPFD,ourapproachfor obtainingsuchguaranteesis,particularlyforthemostgeneralversionofourmodel, signicantlydierentfromapproachesusedincaseoftheGAP.Specically,werely heavilybothonthesolutiontoasuitableperturbationoftheGAPFDandoncarefully designedsolutionimprovementtechniques.Thus,inadditiontocontributingtothe literatureonappliedoptimizationinoperations,wealsoprovidenewtechniquesfor algorithmdevelopmentandasymptoticanalysisforcombinatorialoptimizationproblems. Asourcomputationaltestsshow,ourheuristicsolutionapproachisabletondoptimalor near-optimalsolutionswithverylimitedcomputationaleortforabroadrangeofproblem dimensions. Theremainderofthischapterisorganizedasfollows.Section5.1providesanumber ofimportantstructuralresults;theseresultsbothmotivateaclassofheuristicsand enableustoderiveassociatedperformanceguaranteesinSection5.2.Section5.3discusses approachestofurtherimprovetheheuristicmethodswepropose,andinSection5.4we presenttheresultsofourcomputationalstudy,whichvalidatetheeectivenessofour proposedmethods. 5.1ModelAnalysis TheheuristicproposedinthischapterisdevelopedwithrespecttotheGAPFD presentedinChapter3withlinearrevenuefunctions.Thatis,perunitofcustomer demandfulllmentarevenueof r ij isaccrued.Moreover,weassumethat J i = J i 2I 72
PAGE 73
Specically,thevariantoftheGAPFDintroducedinChapter3studiedinthischapteris givenby: maximize X i 2I X j 2J p ij x ij + X i 2I X j 2J r ij v ij {1 subjecttoP X j 2J a ij x ij + X j 2J v ij b i i 2I {2 X i 2I x ij =1 j 2J {3 v ij ` ij x ij i 2I ; j 2J {4 v ij u ij x ij i 2I ; j 2J {5 x ij 2f 0 ; 1 g i 2I ; j 2J : {6 Notethattheassumptionthattheunitresourceconsumptioncoecientsareequalto onecanbemadewithoutlossofgenerality.Moreover,wewillassumewithoutlossof generalitythattheunitrevenuesarenonnegative.Inprinciple,weallowthexedprot andresourceconsumptioncoecientstobeeitherpositiveornegative.However,inmost real-lifeapplicationsweshouldexpectthesecoecientstobenonnegative.Finally,note thatwithoutlossofgeneralitywecouldassumethat ` ij =0forall i 2I and j 2J by appropriatelymodifyingthexedprotandresourceconsumptioncoecients.However, forclarityofinterpretationofourmodel,algorithms,andresultswewillallowforpositive valuesoftheselowerboundsonthecustomerdemandfulllmentlevels. Thesolutionapproachthatwewilldevelopandanalyzeinthischapterisaclass ofheuristicsthatisinspiredbytheLagrangerelaxationofareformulationofthe LP-relaxationofP.Inparticular,aswewillshowbelow,theoptimizationproblemLP thatisobtainedbyreplacingthebinaryconstraints5{6bynonnegativityconstraints x ij 0 i 2I ; j 2J 5{6 0 73
PAGE 74
isequivalenttotheproblem maximize X i 2I X j 2J p ij + r ij u ij s ij + X i 2I X j 2J p ij + r ij ` ij t ij {7 subjecttoLP 0 X j 2J a ij + u ij s ij + X j 2J a ij + ` ij t ij b i i 2I {8 X i 2I s ij + t ij =1 j 2J {9 s ij ;t ij 0 i 2I ; j 2J : {10 Theorem2. TheoptimizationproblemsLPandLP 0 areequivalent. Proof. FirstnotethatwemaymodifyLPbyexplicitlyintroducingnonnegativesurplus andslackvariablestoconstraints5{4and5{5.Forconvenience,wewillscaletheseso thattheyareexpressedasafractionofthewidthofthesizerangeofthecorresponding assignment.Inotherwords,constraints5{4and5{5arereplacedby v ij )]TJ/F15 11.9552 Tf 11.955 0 Td [( u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij s ij = ` ij x ij i 2I ; j 2J 5{4 0 v ij + u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij t ij = u ij x ij i 2I ; j 2J 5{5 0 s ij ;t ij 0 i 2I ; j 2J : {11 Itiseasytoseethatthisreformulationisvalidevenifthewidthofthesizerangeofan assignmentis0,i.e.,if ` ij = u ij .Nowsubtractingconstraints5{4 0 from5{5 0 yields u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij x ij = u ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij s ij + t ij i 2I ; j 2J sothatwecanset,withoutlossofgenerality, x ij = s ij + t ij i 2I ; j 2J : {12 Moreover,multiplyingconstraints5{4 0 and5{5 0 by u ij and ` ij respectivelyyields 74
PAGE 75
` ij u ij x ij + u ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij s ij = u ij v ij i 2I ; j 2J 5{4 00 ` ij u ij x ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij t ij = ` ij v ij i 2I ; j 2J : 5{5 00 Subtracting5{5 00 from5{4 00 ,weobtain u ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij u ij s ij + ` ij t ij = u ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij v ij i 2I ; j 2J sothatwecanset v ij = u ij s ij + ` ij t ij i 2I ; j 2J : {13 Noticethatthenon-negativityoftheslackandsurplusvariablesby5{11alongwith 5{12and5{13impliesthat5{10issucienttoensureallnon-negativityconditions inLParealsosatisedinLP 0 .Finally,substituting5{12and5{13intothe objective5{1andconstraints5{2and5{3ofPyieldstheobjective5{7aswellas constraints5{8and5{9ofLP 0 Next,denotethenonnegativedualmultipliersofthecapacityconstraints5{8by i i 2I andthefreedualmultipliersoftheassignmentconstraints5{9by j j 2J ThedualD 0 ofLP 0 isthengivenby minimize X i 2I i b i + X j 2J j subjecttoD 0 j p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij i 2I ; j 2J {14 j p ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i u ij i 2I ; j 2J {15 i 0 i 2I j free j 2J : 75
PAGE 76
Thefollowingtheoremderivesaconvenientandinsightfulexpressionfortheoptimalvalue tobothLP 0 andD 0 asafunctionofthedualmultipliers i i 2I ofthecapacity constraints5{8only. Theorem3. ThecommonoptimalvalueofLP 0 andD 0 canbeexpressedas min 0 L where L = X j 2J max i 2I f i;j + X i 2I i b i andwhere f i;j = 8 > < > : p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i u ij if i r ij p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij if i >r ij : Proof. Fromconstraints5{14and5{15weobtainthat,withoutlossofoptimality,the dualvariables j canbechosenas j =max i 2I max f p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij ;p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i u ij g j 2J : {16 Nownotethattheinnermaximumin5{16isattainedbytherstargumentif i r ij andbythesecondargumentif i r ij .Thisimpliesthatweinfacthave j =max i 2I f i;j j 2J {17 whichyieldsthedesiredresult. Itisusefultointroducesometerminologywithrespecttoafeasiblesolution s;t to LP 0 .Considersomecustomer j .If x ij = s ij + t ij =1forsomefacility i wesaythat customer j isassignedtofacility i andfurthermore,customer j isreferredtoasa non-split customer.Similarly,if0
PAGE 77
Moreformally,wedenetheset F = f i;j :0 0and t ij =0isexecutedatitsupperbound,whileanassignment i;j suchthat s ij =0and t ij > 0isexecutedatitslowerbound.Theset Q = f i;j : s ij > 0and t ij > 0 g thenconsistsofthefacility/customerpairsexecutedstrictlybetweentheirbounds.Finally, C = i : X j 2J a ij x ij + X j 2J v ij = b i = i : X j 2J a ij + u ij s ij + X j 2J a ij + ` ij t ij = b i isthesetoffacilitiesthatoperateatfull-capacity. Thefollowingtheoremestablishesacloserelationshipbetweenanoptimalsolutionto D 0 andthecorrespondingprimaloptimalsolution,providedthatthelatterisunique. Theorem4. SupposethatLP 0 isfeasibleandthattheoptimalbasicsolutionto LP 0 ,say s ;t ,isunique.Furthermore,let beanassociatedcomplementaryoptimal solutiontoD 0 .Theprimalanddualsolutionsthensatisfythefollowingproperties. iLet j 2S beasplitcustomer.Thenthereexistsafacility i 0 suchthat f i 0 ;j =max i 2I i 6 = i 0 f i;j : iiLet j 62S beanon-splitcustomer.Thenitisassignedtofacility i 0 ifandonlyif f i 0 ;j =max i 2I f i;j and f i 0 ;j > max i 2I i 6 = i 0 f i;j : 77
PAGE 78
iiiLet j 62S beanon-splitcustomerthatisassignedtofacility i 0 .Then s i 0 j =1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(t i 0 j 8 < : =0 if i 0 >r i 0 j 2 [0 ; 1] if i 0 = r i 0 j =1 if i 0 0.Bythedenitionof x ,complementary slacknessnowimpliesthatforboth i 0 and i 00 ,atleastoneofthecorresponding dualconstraints5{14and5{15isbinding.This,inturn,impliesthatforthis customerthemaximuminequation5{17isattainedforbothfacilities i 0 and i 00 yieldingclaimi. iiLet j 62S beanon-splitcustomer.Thisimpliesthatthereexistsonlyasingle facility,say i 0 ,suchthat x i 0 j > 0infact, x i 0 j =1.Bycomplementaryslackness andthenondegeneracyofthedualsolution,thismeansthatforallfacilitiesexcept i 0 bothcorrespondingdualconstraints5{14and5{15arenonbinding.This,inturn, impliesthatforthiscustomerthemaximuminequation5{17isattainedforonly facility i 0 ,yieldingclaimii. iiiLet j 62S beanon-splitcustomerthatisassignedtofacility i 0 ,sothat x i 0 j =1. First,recallthat s ij istheprimalvariableassociatedwith5{14and t ij isthe primalvariableassociatedwith5{15.Nowbycomplementaryslacknessanddual nondegeneracywehavethat s i 0 j > 0and t i 0 j =0ifandonlyif i 0 >r i 0 j s i 0 j =0 and t i 0 j > 0ifandonlyif i 0 0and t i 0 j > 0ifandonlyif i 0 = r i 0 j Togetherwith5{12thisyieldsthedesiredresult. ItisinterestingtoseewhatTheorem4impliesintermsofthe x;v variablesinour originalLPformulation. Corollary1. SupposethatLP 0 isfeasibleandthattheoptimalbasicsolutionto LP 0 ,say s ;t ,isunique.Furthermore,let beanassociatedcomplementaryoptimal solutiontoD 0 .Thenthereexistsanoptimalsolution x ;v toLPthatsatisesthe followingproperty.If j 62S isanon-splitcustomerthatisassignedtofacility i inthe 78
PAGE 79
optimalsolutiontoLP 0 ,then v i 0 j 8 > > > > < > > > > : = ` i 0 j if i 0 >r i 0 j 2 [ ` i 0 j ;u i 0 j ] if i 0 = r i 0 j = u i 0 j if i 0
PAGE 80
Proof. TheoptimizationproblemLP 0 has2 jIjjJj variablesand jIj + jJj equality constraints.Thetotalnumberofvariableswhicharenonzeroinabasicfeasiblesolutionis thereforenolargerthan jIj + jJj .Nowobservethatthereare jJj)-222(jSj + jFj + jQj non-zerocomponentsin s;t ; jIj)-222(jCj non-zeroslackvariablesassociatedwithconstraints5{8; Thisyields jIj + jJj jJj)-222(jSj + jFj + jQj + jIj)-222(jCj whichyieldsthedesiredresult. Corollary2. Let s;t beabasicfeasiblesolutiontoLP 0 .Thenthetotalnumberof customersthatareeithersplitorexecutedstrictlybetweentheirboundsisboundedbythe numberoffacilities,i.e., jSj + jQjjIj : Proof. Notethateachsplitcustomerhasatleasttwocorrespondingfractionalassignment variables,sothat jFj 2 jSj .TheresultthenfollowsdirectlyfromTheorem5andthefact that jCjjIj 5.2AnAsymptoticallyOptimalHeuristic 5.2.1DevelopmentoftheHeuristic ThereisanattractiveintuitiveinterpretationoftheresultofTheorem3bynoting thatwecaninterpretthevalueofthedualvariable i asaunitcostofcapacityoffacility i .Then,notethatwecanview p ij )]TJ/F21 11.9552 Tf 12.953 0 Td [( i a ij asaxedpseudo-protthatisreceivedif customer j isassignedtofacility i ,regardlessofitslevel.Next,wecanviewthedierence betweentheactualcorrespondingunitrevenue r ij andthecost i ofusingaunitof capacityoffacility i asaunitpseudo-protthatisreceivedifcustomer j isassignedto facility i .Thesignofthepseudo-protthenindicatesthelevelatwhichacustomershould beassigned:iftheunitpseudo-protispositive,customer j ifassignedtofacility i is executedatitsupperbound u ij ,yieldingatotalpseudo-protof p ij )]TJ/F21 11.9552 Tf 12.026 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 12.026 0 Td [( i u ij 80
PAGE 81
Similarly,iftheunitpseudo-protisnegative,customer j ifassignedtofacility i is executedatitslowerbound ` ij ,yieldingatotalpseudo-protof p ij )]TJ/F21 11.9552 Tf 12.182 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 12.181 0 Td [( i ` ij Insummary,thefunction f i;j canbeviewedasa pseudo-prot associatedwiththe assignmentofcustomer j tofacility i foragivenvectorofdualprices WewillusethisinterpretationtoproposeaheuristicfortheGAPFD.Thatis,our heuristicwill,toalargeextent,assigncustomersaccordingtothepseudo-protfunction. Notethatanynonnegativevector denesadistinctpseudo-protfunction,sothat wewillinfactobtainafamilyofheuristics.However,welatershowthattheheuristic enjoysanattractiveperformanceguaranteeifweuseanoptimaldualsolutiontoeither theoriginalproblemoraperturbationthereof.Inparticular,wewillattempttoassign eachcustomertothefacilitythatmaximizesitspseudo-protfunctionandselectthe correspondingcustomerdemandfulllmentlevelaccordingly.Morespecically,themost protablefacilityforcustomer j isgivenby i j =argmax i 2I j f i;j where I j I isthesetoffacilitiescurrentlyunderconsiderationforcustomer j .Itiseasy toseethat,ingeneral,assigningallcustomers j totheirmostprotablefacility i j atasize asdescribedintheprecedingparagraphcannotbeexpectedtoyieldafeasiblesolution totheGAPFD.Wethereforeselecttheorderinwhichthecustomersareassignedby consideringnotonlythemaximumpseudo-protbutalsothesecondlargestpseudo-prot foreachcustomer.Wedenethedierencebetweenthesetwovalues: j = f i j ;j )]TJ/F15 11.9552 Tf 20.697 0 Td [(max i 0 2I j nf i j g f i 0 ;j tobethe desirability ofassigningcustomer j toitsmostprotablefacility.Wethenassign thecustomerstotheirmostprotablefacilityinnonincreasingorderofdesirability,aslong asitisfeasibletodoso. 81
PAGE 82
Ourheuristicproceedsintwophases.Intherst,greedyphaseoftheheuristic,the set U keepstrackofthesetofcustomersthatremaintobeassigned.Duringthecourse ofthisphaseSteps1{3,customersmaybeidentiedthatcannolongerbefeasibly assigned.Theset ~ J ofsuchcustomerswillbehandledinthesecond,improvementphase ofthealgorithmSteps4{7.Throughouttheheuristic, b 0 i denotesthecapacityremaining forfacility i i 2I Wenowformallypresentourheuristicasfollows: Heuristic{Greedyphase Step0. Set U = J ~ J =, b 0 i = b i for i 2I ,and I j = I .Set x G ij = v G ij =0for i 2I ; j 2J Step1. Let i j 2 argmax i 2I j f i;j for j 2U j = f i j ;j )]TJ/F15 11.9552 Tf 20.697 0 Td [(max i 0 2I j nf i j g f i 0 ;j for j 2U : Step2. Select^ | 2 argmax j 2U j ,i.e.,^ | isthecustomertobeassignednexttofacility i ^ | If a i ^ | j + ` i ^ | b 0 i ^ | continuetoStep3.Otherwise, a i ^ | j + ` i ^ | >b 0 i ^ | ,whichmeansthis assignmentisnotfeasible;let I ^ | = f i : a i ^ | + ` i ^ | b 0 i g .If I ^ | =,set ~ J = U and STOP. Step3. Set x G i ^ | ^ | =1 v G i ^ | ^ | = 8 > < > : min f u i ^ | ^ | ;b 0 i ^ | )]TJ/F21 11.9552 Tf 11.955 0 Td [(a i ^ | ^ | g if r i ^ | ^ | > i ^ | ` i ^ | ^ | if r i ^ | ^ | i ^ | b 0 i ^ | = b 0 i ^ | )]TJ/F15 11.9552 Tf 11.955 0 Td [( v G i ^ | ^ | + a i ^ | ^ | : Let U = Unf ^ | g .If U6 =,returntoStep1;otherwise,STOP. 82
PAGE 83
Ifthegreedyphaseoftheheuristicendswith ~ J =,then x G ;v G isafeasible solutiontotheGAPFD.Otherwise,wewillcontinuetheheuristicwithanimprovement phase.Todistinguishthepartialsolutionobtainedattheendofthegreedyphasefrom thesolutiondeliveredbytheimprovementphaseweset x H ;v H = x G ;v G Intheimprovementphase,wereducethesizeofsomepreviouslyassignedcustomers totheircorrespondinglowerboundstofreeupcapacitythatcanbeusedtoassignany ofthecustomersin ~ J .Realizingthattheminimumamountofcapacitythatisrequired toassigncustomer j tofacility i is a 0 ij a ij + ` ij ,let a 0 beanupperboundonthisvalue amongallunassignedcustomers.Then,atleast X i 2I b 0 i a 0 1 a 0 X i 2I b 0 i )-222(jIj customerscanbeaccommodatedwithintheremainingfacilitycapacities.Thus,all customersin ~ J canbeassignedifthefacilitieshavecumulativeavailablecapacity P i 2I b 0 i j ~ Jj + jIj a 0 .Notethatsuchanassignmentcanbefoundbyarbitrarily assigningthecustomersin ~ J toanyfacilitythatcanfeasiblyaccommodateit. Heuristic{Improvementphase Step4. Let A = f i;j : x H ij =1and v H ij >` ij g andset a 0 =max i;j 2I ~ J ` ij + a ij Step5. Identifyaset A 0 A withthepropertythat X i 2I b 0 i + X i;j 2A 0 v H ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(` ij j ~ Jj + jIj a 0 {18 and,inaddition, A 0 isminimalinthesensethatremovinganyelementfromit causes5{18tobeviolated.Ifsuchasetdoesnotexist,set A 0 = A Step6. Set b 0 i = b 0 i + X j : i;j 2A 0 v H ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij for i 2I v H ij = ` ij for i;j 2A 0 : 83
PAGE 84
Step7. AttempttoidentifyafeasiblesolutiontotheGAPFDbyiassigningand determiningcustomersandcustomerdemandfulllmentlevelsfor j 2 ~ J ,and iiincreasingcustomerdemanfulllmentlevelsforassignments i;j 2A 0 Ifthisissuccessful,returnthesolutionas x H ;v H .Otherwise,theheuristicis unabletondafeasiblesolutiontotheGAPFD. Intheremainderofthissectionweanalyzeabasicimplementationoftheimprovement phaseoftheheuristicwhereweidentifyanarbitraryset A 0 inStep5,andtrytoassign customersin ~ J inarbitraryordertofacilitiesthatcanaccommodatetheminStep7.In Section5.4wewillproposeamoresophisticatedimplementationwithguaranteedsuperior behavior. Thefollowingtheoremestablishesacloserelationshipbetweenthesolutionthatis obtainedbythegreedyphaseoftheheuristicandabasicoptimalsolutiontoLP 0 Theorem6. SupposethatLP 0 isfeasibleandthattheoptimalbasicsolutiontoLP 0 say s ;t ,isunique.Ifwechoose intheheuristicequaltoanassociatedoptimaldual vector ,wehaveforallnon-splitcustomers j 62S thatthegreedyphaseoftheheuristic iassignsthiscustomertothesamefacility,say i j ,asLP 0 ; iiexecutesthiscustomeratthesamesamelevelasLP 0 provided i j ;j 62Q Proof. Theorem4i{iiimpliesthat,inthegreedyphaseoftheheuristic, j > 0forall j 62S and j =0for j 2S .Thus,Step2guaranteesthatallnon-splitcustomersare consideredbeforeanysplitcustomersaslongasthesets I j remainunchanged.Claimi thenimmediatelyfollowsfromthefactthatthepreferredassignmentsofthenon-split customersareallfeasible.Next,Theorem4iiiimpliesthat,inStep3ofthegreedyphase oftheheuristic,allcustomers j forwhich i;j 62Q areexecutedatthesamelevelas inLP 0 .Notethatthegreedyphasedoesnotnecessarilyexecutecustomersforwhich i;j 2Q atthesamelevelasinLP 0 .Thisfollowssince,bycomplementaryslackness, i;j 2Q isequivalentto j = f i;j = p ij )]TJ/F21 11.9552 Tf 12.447 0 Td [( i a ij ,i.e.theunitrevenue r ij )]TJ/F21 11.9552 Tf 12.446 0 Td [( i =0, 84
PAGE 85
whichdoesnotnecessarilyimplythatthesolutiontoLP 0 makesthisassignmentatits lowerbound,astheheuristicdoes. Theorem6statesthat,ifwechoose intheheuristicequaltoanoptimaldualvector ofLP 0 ,thegreedyphaseoftheheuristicstartsbymakingtheassignmentsofnon-split customersinthesolutiontoLP 0 ,withtheonlypossibledeviationbeingthecustomer demandfulllmentlevelsforthosethatarestrictlybetweentheirboundsinthatsolution. This,togetherwithCorollary2,thenimpliesthatthetotalnumberofcustomersthat areunassignedinthegreedyphaseorforwhichtheassignmentordemandfulllment leveldiersfromthesolutiontoLP 0 isnolargerthanthenumberoffacilities, jIj .The improvementphaseoftheheuristicisaimedatcreatingsucientspacetoallowthe assignmentofanyunassignedcustomers.Ourgoalinthenextsectionistousethisresult toderivestrategiesforchoosing fortwoverygeneralstochasticmodelsontheproblem parameterssuchthattheheuristicisasymptoticallyfeasibleandoptimalwithprobability oneasthenumberofcustomersincreases.Thatis,asthenumberofcustomersincreases, itisveryunlikelythattheheuristicisunabletondafeasiblesolutionand,moreover,ifit ndsafeasiblesolution,itsrelativeerrorwilldeclineasthenumberofcustomersincreases. 5.2.2AverageCaseAnalysisoftheHeuristic Thissectionprovidesananalysisoftheasymptoticbehaviorofourheuristicunder aprobabilisticmodelontheproblemparametersthatkeepsthenumberoffacilities, jIj xedandletsthenumberofcustomers, jJj ,approachinnity.Weproposeastochastic modelfortheGAPFDthatissimilartotheonescommonlyusedfortheGAPandits extensionssee,e.g.,DyerandFrieze[30],RomeijnandPiersma[78],andRomeijnand RomeroMorales[80] 1 .Inparticular,weassumethateachcustomerischaracterized byarandomvectorofparameters P j ;R j ;A j ;L j ;D j ,where P j = P 1 j ;:::;P jIj j R j = 1 Asaconvention,wewilldenoteparametersandsolutionsthatarerandomvariablesby capitalletters,whilerealizationswillbedenotedbylowercaseletters. 85
PAGE 86
R 1 j ;:::;R jIj j A j = A 1 j ;:::;A jIj j L j = L 1 j ;:::;L jIj j ,and D j = D 1 j ;:::;D jIj j .Here P j and R j arethevectorsofxedprotsandunitrevenuesforcustomer j ,respectively. Furthermore, A j isthevectorofxedresourceconsumptionsforcustomer j ,while L j is thevectoroflowerboundsand U j L j + D j isthevectorofupperboundsonthesize ofcustomer j .Thevectors P j ;R j ;A j ;L j ;D j areassumedtobei.i.d.onthecompactset [ P ; P ] jIj [ R ; R ] jIj [ A ; A ] jIj [ L ; L ] jIj [ D ; D ] jIj ,wheretheconditionaldistributionsof P j ;R j j A j ;L j ;D j areabsolutelycontinuous.Inaddition,wehave R ;L 0sothatboth thedemandrequirementandtheunitrevenueaccruedarenon-negative.Furthermore, thedierencebetweentheupperandlowerboundparametersistakentobestrictly positive, D > 0,toensurethereisadecisiontobemadewithregardtothelevelat whichacustomer'sdemandissatised.Forconvenience,weassumethat R L > )]TJ/F21 11.9552 Tf 9.298 0 Td [(P sothatthetotalprotassociatedwith any feasibleassignmentisnonnegative.Note thatthisassumptionismildsinceitisautomaticallysatisedif,forexample, P > 0. Moreover,itcanbemadewithoutlossofgeneralitysincewemayaddorsubtracta constantvaluefromallxedprotcoecientswithoutimpactingtheprotrankingofthe solutions.Furthermore,asiscommoninprobabilisticmodelsofthistype,weallowfor theaccommodationofanincreasingnumberofcustomerswhilethenumberoffacilities remainsconstantbylettingthecapacityoffacility i growlinearlywiththenumberof customers,i.e.,welet b i = i jJj where i isapositiveconstant i 2I Finally,wewishtofocusonprobleminstancesthatadmitafeasiblesolution.Note thataninstanceoftheGAPFDhasafeasiblesolutionifandonlyiftheassociatedGAP withallrequirementssettotheirminimumvalue a ij + ` ij i 2I ; j 2J isfeasible.This leadstothefollowingassumptionthatwewillimposeonourprobabilisticmodel: Assumption1. min 0; > e =1 X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [(E min i 2I i A i 1 + L i 1 > 0 : 86
PAGE 87
ByRomeijnandPiersma[78,Theorem3.2],thisassumptionensuresthataninstance randomlygeneratedaccordingtoourstochasticmodelisfeasiblewithprobabilityone as jJj!1 .Notethatthisassumptionismild,sincetheyalsoshowthatinstances generatedareasymptoticallyinfeasiblewithprobabilityoneas jJj!1 if < 0. Intheremainderofthissection,wewillshowthat,forasuitablychosenstrategyfor theparameter ,theheuristicwillprovideafeasibleandoptimalsolutiontotheGAPFD withprobabilityoneas jJj!1 .Inparticular,consideraninstanceoftheGAPFD generatedfromtheprobabilisticmodeldescribedabove,andlet Z jJj Z LP jJj Z G jJj ,and Z H jJj denoteitsoptimalsolutionvalue,thevalueofitsLP-relaxation,andthevalueofthe solutionobtainedbythegreedyandtheimprovementphasesoftheheuristic,respectively. Notethatthesevaluesarerandomvariables,andthatwehaveexplicitlyrecognizedthat theyareafunctionofthenumberofcustomers, jJj .Wethensaythattheheuristicis asymptoticallyfeasibleandoptimal if ithesolution X H ;V H producedbytheheuristicisasymptoticallyfeasiblewith probabilityone; iilim jJj!1 Z jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj =Z jJj =0withprobabilityone. Since,underourassumptionsontheproblemparameters,wehavethat,foranyfeasible instanceoftheGAPFD, Z jJj R L + P jJj with R L + P > 0,thelatterisequivalentto ii 0 lim jJj!1 1 jJj Z jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj =0withprobabilityone whichisthecharacterizationofasymptoticoptimalitythatwewilluseintheremainderof thischapter. WewilldistinguishbetweentwoclassesofinstancesoftheGAPFD.Therstclass ofinstancesischaracterizedbycustomerrequirementsthatare facility-independent ,i.e., A ij = A 1 j L ij = L 1 j ,and D ij = D 1 j i 2I ; j 2J .Forthisclass,wewillshowthatthe heuristicisasymptoticallyfeasibleandoptimalwithprobabilityoneifwechoose equal toanassociatedoptimaldualvector .Thesecondclassofinstancesfollowsthegeneral probabilisticmodeldiscussedearlierinthissection.Forthisclass,wewillshowthatthe 87
PAGE 88
heuristicisasymptoticallyfeasibleandoptimalwithprobabilityoneifwechoose equal toanoptimaldualvectorofanappropriatelyperturbedinstanceoftheGAPFD. 5.2.2.1Facility-independentrequirements RecallthatTheorem6establishesastrongconnectionbetweenanoptimalsolution toLP 0 andthesolutionobtainedbythegreedyphaseoftheheuristiciftheformeris uniquewithrespecttonon-splitcustomersandifwechoose equaltoanassociated optimaldualvector ofLP 0 .Thisconnectionisemployedtoobtainasymptotic feasibilityandoptimality.Beforeweformallyprovethisresult,however,wewillrst derivetwousefulpreliminaryresults.Therstresultprovidesanintuitivecharacterization ofanoptimalsolutiontoLP 0 undertheproposedstochasticmodel.Thisresultis commonformodelsinwhichparametersaregeneratedfromabsolutelycontinuous distributionssee[29,30,79].Itispresentedhereformallyduetoitssignicanceinour asymptoticanalysis. Lemma3. Underourstochasticmodel,ifLP 0 isfeasible,itsoptimalsolutionisunique withprobabilityone. Proof. Anon-uniqueoptimalsolutiontoLP 0 existsonlyifthehyperplanerepresentative ofsolutionswithoptimalprot X i 2I X j 2J p ij + r ij u ij s ij + X i 2I X j 2J p ij + r ij ` ij t ij intersectsthefeasibleregionatmultiplepoints.Recallthattheunitrevenues r ij i 2 I ; j 2J andthexedprots p ij i 2I ; j 2J aregeneratedfromajointdistribution that, conditional onthevaluesoftherequirementsparametersintheconstraints,is absolutelycontinuous.Thustheprobabilitythatdatageneratedbyourstochasticmodel allowsformultipleoptimalsolutionstoLP 0 iszero,sothattheoptimalsolutiontoLP 0 isuniquewithprobabilityone. 88
PAGE 89
Lemma4. Whencustomerrequirementsarefacility-independent,theaggregatecapacity thatiseitherunusedorusedforcustomerlevelsinexcessoftheirlowerboundincreases linearlyin jJj withprobabilityoneas jJj!1 ,inanyfeasiblesolutiontoLP. Proof. Forconvenience,denotetheeectivelowerboundonanassignmentby A 0 ij = A ij + L ij .Sincethecustomerrequirementsarefacility-independentwehavethat A 0 ij = A 0 1 j i 2I .Then,foranyassignmentvector x thatisfeasibletotheLP-relaxationofthe correspondingGAPwehave 1 jJj X i 2I b i )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X j 2J A 0 ij x ij = X i 2I i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X i 2I X j 2J A 0 1 j x ij = X i 2I i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X j 2J A 0 1 j X i 2I x ij = X i 2I i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X j 2J A 0 1 j : Forthecaseoffacility-independentrequirements,RomeijnandPiersma[78]showthat Assumption1isequivalenttothecondition E A 0 1 j < X i 2I i sothat,bytheCentralLimitTheorem, X i 2I i )]TJ/F15 11.9552 Tf 18.701 8.087 Td [(1 jJj X i 2I X j 2J A 0 ij x ij > 0withprobabilityoneif jJj!1 foranyfeasiblerelaxedassignmentvector x .Thisyieldsthedesiredresult. Wearenowreadytoformallyproveourrstasymptoticfeasibilityandoptimality result. Theorem7. Considerprobleminstancesgeneratedaccordingtoourstochasticmodel with,inaddition,facility-independentcustomerrequirements.Moreover,choose inthe heuristicequaltoanassociatedoptimaldualvector ofLP 0 .Thentheheuristicis asymptoticallyfeasibleandoptimal. 89
PAGE 90
Proof. Sinceweareonlyinterestedinaprobabilisticandasymptoticfeasibilityguarantee, wemaybyLemma3assumethatthesolutiontoLP 0 isuniquewithrespecttonon-split customers.ThenTheorem6saysthatthegreedyphaseoftheheuristicassignsnomore than jSj customerstoadierentfacilityortonofacilityatall,andnomorethan jQj customerstothesamefacilitybutatadierentlevelthantheoptimalsolutiontoLP 0 Againdenotingtheeectivelowerboundofacustomerby A 0 = A + L withcorresponding upperbound A 0 = A + L ,wehavethateachoftheunassignedcustomersrequiresno morethan A 0 unitsofcapacity,sothatitwouldsuceifan aggregate of jSj + jIj A 0 unitsofcapacityamongallfacilitieswereavailable.Nowlet b 0 = P i 2I b 0 i < jSj + jIj A 0 denotetheaggregateremainingcapacityattheendofthegreedyphaseoftheheuristic, andrecallthat,byCorollary2,weknowthat jSj + jQjjIj .Sucientcapacitycan thereforebemadeavailableif,intheimprovementphase,weareabletoreducetotheir lowerboundsthelevelof )]TJ/F15 11.9552 Tf 11.125 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 =D customersthatthegreedyphaseof theheuristicassignedattheirupperbounds.Sincethisnumberisindependentofthe numberofcustomers,Lemma4impliesthat,forlargeenough jJj ,thiscanindeedbe done,implyingthattheheuristicisasymptoticallyfeasible. Finally,notethattheobjectivefunctionvalueofafeasiblesolutionthatisobtained bytheheuristicsatises Z H Z LP )]TJ/F26 11.9552 Tf 11.955 9.684 Td [(\006)]TJ/F15 11.9552 Tf 16.604 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 =D + jSj + jQj D R )-222(jSj P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L : {19 Intherighthandsideofthisinequality, )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 D R isthelostrevenuefrom reducingcustomerstotheirlowerboundsinordertoassureenoughaggregatecapacityfor unassignedcustomersinthegreedyphaseoftheheuristic.Theterm jSj + jQj D R islost revenuefromcustomerseithernotassignedtothesamefacilityasinLP 0 ,orcustomers executedatadierentlevelthanLP 0 .Thequantity jSj P )]TJ/F21 11.9552 Tf 12.44 0 Td [(P + R )]TJ/F21 11.9552 Tf 12.44 0 Td [(R L accounts 90
PAGE 91
forthelossofxedprotresultingfromcustomersnotassignedtothesamefacilityasin LP 0 Clearly Z LP Z jJj Z H implies lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z LP jJj )]TJ/F21 11.9552 Tf 11.956 0 Td [(Z H jJj : Providedthattheheuristicsolutionisfeasiblewethenhave lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(Z LP jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z H jJj lim jJj!1 1 jJj \000)]TJ/F15 11.9552 Tf 22.084 -9.684 Td [( jSj + jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 D + jSj + jQj D R + jSj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.956 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L by5{19 lim jJj!1 1 jJj \000 jIj A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 0 =D + jIj D R + jIj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L byCorollary2 =0 : Sincetheheuristicisasymptoticallyfeasible,thisimpliesthattheheuristicisasymptotically optimalaswell. 5.2.2.2Facility-dependentrequirements TheresultofLemma4canunfortunatelynotbeextendedtothegeneralcasewhere customerrequirementsarefacility-dependent,preventingusfromextendingtheapproach intheprevioussectiontoshowasymptoticfeasibilityoftheheuristicifwechoose equaltoanoptimaldualvector ofLP 0 .Ingeneral,wethereforetakeadierent approach:wechoose equaltotheoptimaldualvectorofaninstanceofLP 0 inwhich thecapacitieshavebeenreducedbyanappropriatelychosensmallamount.Note,however, thatwewillstillapplythetwophasesoftheheuristicusingtheoriginalcapacities.By ensuringthatthetemporarycapacityreductionsarelargeenoughtoensurethatthe customersthatarefractionallyassignedintheLP-relaxationcanbeassignedfeasiblybut, 91
PAGE 92
atthesametime,smallenoughforthecorrespondingsolutiontobeclosetooptimal,we willbeabletoshowthatthegreedyphaseoftheheuristicaloneisasymptoticallyfeasible andoptimal. Moreformally,consideraninstanceoftheGAPFDwith jJj customers.Then associatewiththisinstanceaperturbedinstanceofLP 0 inwhichallofthenormalized capacities i i 2I arereducedby jJj ,where lim jJj!1 jJj =0{20 lim jJj!1 jJj jJj = 1 {21 and 0 < jJj < min i 2I i : WewilldenotetheperturbedproblembyLP 0 jJj anditsoptimalvalueby Z LP 0 jJj jJj Thefollowingpreliminaryresultshowsthattheoptimalvaluesoftheoriginaland perturbedproblemsareveryclose. Lemma5. TheoptimalvaluesofLP 0 andLP 0 jJj arecloseinthesensethat,with probabilityone, lim jJj!1 1 jJj Z LP 0 jJj jJj =lim jJj!1 1 jJj Z LP 0 jJj : Proof. SeeAppendixA. Thefollowingtheorememploysthisresulttoprovethatwehaveaheuristicforthe GAPFDthatisasymptoticallyfeasibleandoptimal. Theorem8. Considerprobleminstancesgeneratedaccordingtoourgeneralstochastic model.Moreover,choose intheheuristicequaltoanoptimaldualvectortoLP 0 jJj Thentheheuristicisasymptoticallyfeasibleandoptimal. Proof. AsintheproofofTheorem7,wemayassumethatthesolutiontotheperturbed instanceofLP 0 isuniquewithrespecttonon-splitcustomers,sinceweareonly 92
PAGE 93
interestedinaprobabilisticandasymptoticfeasibilityguarantee.ThenTheorem6says thatthegreedyphaseoftheheuristicassignsnomorethan jSj customerstoadierent facilityortonofacilityatall,andnomorethan jQj customerstothesamefacilitybut atadierentlevelthantheoptimalsolutiontoLP 0 .Since,byCorollary2,weknow that jSj + jQjjIj ,itiseasytoseethattheadditionalamountofaggregatecapacity overtheamountusedintheperturbedinstanceofLP 0 requiredforthesecustomers isindependentof jJj .By5{21wecanthereforeconcludethat,withprobabilityone, thegreedyphaseoftheheuristicyieldsafeasiblesolutiontotheGAPFD.Moreover,the objectivefunctionvalueofafeasiblesolutionthatisobtainedbythegreedyphaseofthe heuristicsatises Z G Z LP 0 jJj )-222(jSj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L )-222(jQj D R )-222(jSj P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L {22 sothat,inthatcase, lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z jJj )]TJ/F21 11.9552 Tf 11.956 0 Td [(Z H jJj lim jJj!1 1 jJj )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Z jJj )]TJ/F21 11.9552 Tf 11.956 0 Td [(Z G jJj since Z G Z H lim jJj!1 1 jJj Z LP 0 jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z G jJj since Z Z LP 0 =lim jJj!1 1 jJj Z LP 0 jJj jJj )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z G jJj withprobabilityoneas jJj!1 ,byLemma5 lim jJj!1 1 jJj )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(jSj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L + jQj D R + jSj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.956 0 Td [(R L by5{22 lim jJj!1 1 jJj )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( jSj + jQj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.956 0 Td [(R L + jSj )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + R )]TJ/F21 11.9552 Tf 11.956 0 Td [(R L byCorollary2 93
PAGE 94
lim jJj!1 jIj jJj )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R L + D )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L + P )]TJ/F21 11.9552 Tf 11.955 0 Td [(P + )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L =0 : Sincetheheuristicisasymptoticallyfeasible,thisimpliesthattheheuristicisasymptotically optimalaswell. ItisinterestingtonotethatTheorem8actuallyshowsthatthegreedyphaseofthe heuristicaloneisasymptoticallyfeasibleandoptimal. 5.2.3ModelExtension Itisinterestingtonotethatourheuristiccanstillbeappliedandretainthe associatedtheoreticalpropertiesifthevariablerevenuefunctionisconvexratherthan linear,i.e.,iftheterm r ij v ij intheobjectivefunctionisreplacedby~ r ij v ij where~ r ij is aconvexandnondecreasingfunction.Sucharevenuefunctionmayberelevantfroma practicalpointofviewbyrealizingthatacustomermaybewillingtopayanincreasing amountperunitofproductsuppliedwithintheacceptablerange.Infact,inlightofthe discussioninSection5.1,ourmodelcouldaccommodateasituationthatexhibitsboth economiesofscaleforthesupplierthroughthexedprottermandalargermarginal valuetocustomerswhoreceiveadditionalunitsofproduct.Toapplyourheuristics tothismodelgeneralization,wecansimplylinearizetherevenuefunctionbydening r ij =~ r ij u ij )]TJ/F15 11.9552 Tf 12.849 0 Td [(~ r ij ` ij = u ij )]TJ/F21 11.9552 Tf 12.325 0 Td [(` ij .Theasymptoticperformanceguaranteesthenfollow inarelativelystraightforwardmannerbyrealizingthatthesolutiontotheLP-relaxation overestimatesthecustomerrevenuefornomorethan jIj customers. 5.3HeuristicImprovementIssues 5.3.1SolutionImprovement Recallthatourstatementoftheheuristicapproachleftsomeexibility,inparticular inSteps5and7oftheimprovementphase.Althoughabasicimplementationwas sucienttoobtainasymptoticperformanceguarantees,wewillinthissectiondiscuss 94
PAGE 95
amoresophisticatedimplementationwhichisguaranteedtoyieldsuperiorresultsin practice. 5.3.1.1Improvementphase First,considertheselectionofaset A 0 ofcustomerswhosecustomerlevelswillbe reducedtotheirlowerboundinStep5.Ratherthanidentifyinganarbitrarysetthat satisesthepropertiesspeciedintheheuristic,wesequentiallyaddassignmentsfrom A to A 0 inthereverseoftheorderinwhichtheywereassignedinthegreedyphaseofthe heuristic,untiltheset A 0 satisesthedesiredproperties.Next,ratherthanattempting toarbitrarilyassigncustomersin ~ J tofacilitiesinStep7weusethemodiedgreedy algorithmproposedbyRomeijnandRomeroMorales[79]tosolvethefollowinginstanceof theGAP: maximize X i 2I X j 2 ~ J p 0 ij x ij subjecttoI X j 2 ~ J a 0 ij x ij b 0 i i 2I X i 2I x ij =1 j 2 ~ J x ij 2f 0 ; 1 g i 2I ; j 2 ~ J where p 0 ij = p ij + r ij ` ij and a 0 ij = a ij + ` ij i 2I ; j 2J .Denotetheoptimalsolutionto Iby x I .Theheuristicsolutionisupdatedbysetting x H ij = x I ij and v H ij = x I ij ` ij i 2I ; j 2 ~ J : {23 5.3.1.2Post-processingphase BoththegreedyphaseoftheheuristicinSection5.2.1andtheimprovementphase describedinSection5.3.1.1aredesignedtoprovidehighqualityfeasiblesolutions. However,itmaystillbepossibletoimprovethequalityofthesolution.Inparticular, 95
PAGE 96
given thatafeasibleassignment x H hasbeenobtainedineitherthegreedyorimprovement phase,weproposeto optimally determinecorrespondingcustomerdemandfulllment levels v P wherethesuperscriptPdenotesthesolutionafterthepost-processingphase. Infact,if B i = f j : x H ij =1 g isthesetofcustomersassignedtofacility i ,theoptimal customerdemandfulllmentlevelscanbedeterminedasfollows.First,solvethefollowing continuousknapsackproblemsfor i 2I : maximize X j 2B i r ij w ij subjecttoKP i X j 2B i w ij b i )]TJ/F26 11.9552 Tf 12.435 11.358 Td [(X j 2B i a ij 0 w ij u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij j 2B i : Then,if w denotestheoptimalsolutionstotheseproblems,set v P ij = w ij .Itiseasyto seethattheproblemsKP i canbesolvedbychoosingtheexiblecomponentofthesizes ofthecustomersin B i aslargeaspossibleinnonincreasingorderof r ij aslongasfacility capacityallows. 5.3.2CapacityPerturbationScheme Recallthat,forthegeneralcasewhererequirementsareallowedtobefacility-dependent, wereducethenormalizedfacilitycapacities i bysomeamount jJj withintheframework ofSection5.2.2.2.DespitetheasymptoticfeasibilityresultofTheorem8,itisof coursestillpossiblethattheheuristicfailstondafeasiblesolution.Inparticular,an inappropriateperturbationmayleadtoinfeasibilityforoneofthefollowingtworeasons: iIf jJj istoolarge,theresultingperturbedcapacitiesmaybesuchthattheinstance ofLP 0 jJj isinfeasiblesothatwecannotperformthegreedyphaseofthe heuristic. iiIf jJj istoosmall,thenwefailtoreserveenoughcapacitytoaccommodatethe customersthatremainunassignedinthegreedyphase. 96
PAGE 97
Weproposetousethisinformationtoiterativelymodifythecapacityperturbationas needed.Notethat,toensurethatnoperturbedfacilitycapacitiesarenonpositive,we shouldinitiallyhave0 < jJj < min i 2I i .Incasetheheuristicisunsuccessful duetoi,weupdate = jJj anddecrease jJj .If,ontheotherhand,itisunsuccessful duetoii,weupdate = jJj andincrease jJj .Ineithercase,wesetthenewcapacity perturbationto jJj = + )]TJ/F21 11.9552 Tf 11.955 0 Td [( where 2 ; 1andreapplytheheuristic. Forinstanceswithfacility-independentrequirements,noperturbationisrequiredto obtainasymptoticperformanceguarantees.However,asforinstanceswithfacility-dependent requirements,itisofcoursepossiblethattheheuristicdoesnotndafeasiblesolution. Inthatcase,wecanapplythesameiterativescheme,recognizingthatweinitiallyhave jJj =0. 5.4ComputationalResults Inthissectionwetesttheperformanceofourheuristicsonalargesetofrandomly generatedtestproblems.Followingthetheoreticalresults,weseparatelyconsiderproblems withfacility-independentandfacility-dependentrequirements. 5.4.1ExperimentalDesign WeusethestochasticmodelgiveninSection5.2.2asthebasisforgeneratingproblem instances.Weconsiderinstanceswith jIj =15and jIj =30facilities,and jJj = 5 jIj ; 10 jIj ; 25 jIj ; 50 jIj ,and100 jIj customers.Foreachcustomer,wegeneratethevectors ofrevenueparameters R j and P j independentlyfromuniformdistributionson[1 ; 2] and[30 ; 50],respectively.Thecustomerrequirements A j L j and D j aregenerated fromuniformdistributionson[10 ; 20],[75 ; 125],and[15 ; 35],respectively.Notethat,for instanceswithfacility-independentrequirements,onlyasinglevalueforeachofthese parametersisgenerated,whileforinstanceswithfacility-dependentrequirements,we generate jIj valuesoneforeachfacilityindependentlyfromthespecieddistributions. 97
PAGE 98
Wefocusedoninstancesinwhichthefacility-capacitieswereidentical,thatis,weset b i = jJj i 2I .Forinstanceswithfacility-independentrequirements,thevaluein Assumption1thenreducesto = )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(E A 1 j + L 1 j jIj {24 andwethereforeconsidercapacitiesoftheform = E A 1 j + L 1 j jIj : {25 Forinstanceswithfacility-dependentrequirements,RomeijnandRomeroMorales[80] showedthatthevalueinAssumption1reducesto = )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(E min i 2I A i 1 + L i 1 jIj {26 providedthatthelowerboundsofthecustomerrequirements A i 1 + L i 1 areindependent andhaveanincreasingfailureratedistribution,asisthecaseinourexperiments.We thereforeconsidercapacitiesoftheform = E min i 2I A i 1 + L i 1 jIj : {27 Itiseasytoseethatinboth5{25and5{27, > 1isequivalenttoAssumption1being satised.Moreover,thesechoicesensurethatthetightnessoftheinstancesacrossdierent valuesof jIj iscomparableforagivenvalueof .Inourexperiments,wehaveconsidered valuesof =1 : 1,1 : 2,and1 : 3.Theformertwovaluescorrespondtocaseswherethe capacityconstraintsareexpectedtohaveastronglimitingeectonthecustomerdemand fulllmentlevelsthatcanbeaccommodated.Thethirdvaluecorrespondstoloosely capacitatedinstancessincetheyyieldthattheexpressionsin5{24and5{26are positiveevenwhentheranges D i 1 areaddedtothecustomerrequirements. Weapplyboththegreedyphaseandtheimprovementphasetoeachproblem instance.WeusetheiterativecapacityperturbationschemedescribedinSection5.3.2 98
PAGE 99
toimprovetheabilityoftheheuristictondfeasiblesolutionsforsmallervaluesof jJj ; herewesimplysettheupdateparameterequalto = 1 2 .Moreover,unlessotherwise noted,thepost-processingphasedescribedinSection5.3.1.2isappliedaswell.Finally, forameaningfulassessmentoftheheuristicperformancewerunCPLEXuntilasolution isobtainedwhoseobjectivefunctionvalueisatleastasgoodastheonefoundbythe heuristicoruntil15minutesofCPUtimehavebeenused.Foreachproblemclass, wepresentaverageresultsof25randomlygeneratedinstances.Allexperimentswere performedonaPCwitha3.40GHzPentiumIVprocessorand2GBofRAM,andall mixed-integerandlinearprogrammingproblemsweresolvedusingCPLEX10.1.The tablesreport ithenumberofinstancesinwhichtheheuristicfoundafeasiblesolution, iianupperboundontherelativesolutionerrorasmeasuredby error= z LP 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(z H z H 100% ; where,sincetheerrorismeaninglessifnofeasiblesolutionisfound,theaverage errorisdeterminedwithrespecttotheinstancesforwhichtheheuristicisableto ndafeasiblesolutiononly, iiitheaverageCPUtimeusedbytheheuristic,forboththegreedyphaseandthe greedyphasefollowedbytheimprovementphaseoveralliterations, ivtheaveragenumberofcapacityperturbationiterationsperformed, vtheCPUtimerequiredbyCPLEX,and vithenumberofinstancesforwhichCPLEXfailedtoobtainasolutionofthedesired qualitywithintheallottedtimeindicatedbyasuperscript. 5.4.2Facility-IndependentRequirements Tables5-1{5-6summarizetheresultsobtainedwithourheuristicswhenapplied toinstancesgeneratedaccordingtothemodeldescribedintheprevioussectionwith facility-independentrequirements.RecallthatTheorem7saysthattheheuristicformed bythegreedyandimprovementphasesisasymptoticallyfeasibleandoptimal.However, 99
PAGE 100
asimilarguaranteecannotbegivenforthegreedyphasealone;inparticular,applying thegreedyphasealonedoesnotguaranteeasymptoticfeasibility.Thecomputational resultsconrmthis:forthetwoclassesinwhichthecapacityconstraintsaretightest =1 : 1and =1 : 2,thegreedyphaseisnotabletondafeasiblesolutioninanyof theinstancesgenerated.Incontrast,whenthegreedyphaseisconsideredinconjunction withtheimprovementphaseafeasiblesolutionisobtainedforalmostallinstances,with theexceptionofinstanceswith jJj =5 jIj and =1 : 1forboth jIj =15and jIj =30. However,ourresultsshowthat,forsuchinstances,performingtheiterativecapacity perturbationschemeofSection5.3.2yieldsafeasiblesolutioninallbutasingleinstance with jIj =15, jJj =75,and =1 : 1.Notethat,when =1 : 3,thegreedyphasealone isabletondafeasiblesolutionforallinstances.Thiscanpartlybeexplainedbythefact that,forlarge jJj ,itcanbeexpectedthatallcustomerscanbeperformedattheirupper bound,makingtheimprovementphaseunnecessaryevenfromatheoreticalpointofview. Itisnoteworthy,however,thatthegreedyphasealonestillperformsverywellforinstances withsmallervaluesof jJj when =1 : 3,despitethelackofanytheoreticalfeasibility guarantee. Theresultsclearlyshowthattheaverageerrorapproacheszeroasthenumberof customersincreases.Itisinterestingtonotethat,forbothvaluesof jIj ,theaverageerrors arelargestwhen =1 : 2.Thisbehaviorisaconsequenceofthenatureoftheimprovement phase,inwhichtheheuristiccreatescapacityforunassignedcustomersbydecreasing thesizeofalreadyassignedcustomerstotheirlowerbounds.When =1 : 2,customers cangenerallybeperformedathigherlevelsthanwhen =1 : 1,sothattheneteect oftheimprovementphaseonsolutionqualityisunderstandablylargerfor =1 : 2than for =1 : 1However,thispatterndoesnotcontinueto =1 : 3since,asweconcluded above,theimprovementphaseisnotrequiredfortheseinstances.Wealsoremarkthatthe solutionerrorsdependmainlyontheratio jJj = jIj betweenthenumberofcustomersand thenumberoffacilities. 100
PAGE 101
Theheuristiciscomputationallyveryecient,onaveragetakingonlyslightlymore than1secondofCPUtimewhen jIj =15andabout4secondsofCPUtimewhen jIj =30.Forsmallerprobleminstances,CPLEXisabletondsolutionsofthesame qualityveryrapidlyaswell.However,forlargerinstancesandasthecapacityincreases CPLEXisuptoapproximately10timesslowerfor jIj =15anduptoapproximately50 timesslowerfor jIj =30.Ourheuristicisthereforeespeciallypromisingforlargeinstances andcaseswheretheGAPFDneedstobesolvedrepeatedly,forexampleunderdierent scenariosorwhenitisasubprobleminamorecomplexstrategicoptimizationproblem. Perhapssurprisingly,despitethefactthataninstanceofLP 0 hastobesolved,thetime requiredbytheheuristicincreasesonlymodestlyapproximatelylinearlyinthesizeof theproblem. 5.4.3Facility-DependentRequirements Recallthat,forinstanceswithfacility-dependentrequirements,theheuristicemploys thedualsolutiontotheLP-relaxationofaninstanceoftheGAPFDinwhichthe normalizedcapacities arereducedbyaquantity jJj satisfying5{20and5{21 toensureasymptoticfeasibilityandoptimality.Toensurethatnoperturbedfacility capacitiesarenonpositiveforanyvalueof jJj ,weproposetochoose jJj = p jJj {28 where0 << 1,andwherethemagnitudeof representsatradeobetweenfeasibility andsolutionquality.Inourcomputationalexperimentswesimplyuse = 1 2 Tables5-7{5-12summarizetheresultsobtainedwithourheuristicswhenappliedto instancesgeneratedaccordingtothemodeldescribedinSection5.4.1withfacility-dependent requirements.RecallthatTheorem8saysthatthegreedyphaseoftheheuristicis asymptoticallyfeasibleandoptimal.Althoughthegreedyphasealonefailstonda feasiblesolutiontoasubstantialnumberofprobleminstancesforsmallerratios jJj = jIj thegreedyphaseisuniformlysuccessfulforlargerratios.Moreover,thepatternofaverage 101
PAGE 102
errorsafterthegreedyphasealoneshowsconditionallyonndingafeasiblesolutiona decreasingtrend,illustratingthetheoreticalasymptoticoptimalityresult. Thecontributiontofeasibilityoftheiterativecapacityperturbationschemeis apparentforinstanceswith jJj 25 jIj :afeasiblesolutionwasobtainedinallinstances. Itshouldnotbesurprisingthatthisresultsinanincreaseinaverageerror,sincethe instancesforwhichasingleiterationofthegreedyphaseisnotabletondafeasible solutionareclearlytheharderones.However,thepatternofaverageerrorsstillexhibitsa stronglydecreasingtrend,againclearlyillustratingthetheoreticalasymptoticoptimality result. TheaverageCPUtimerequiredbytheheuristicissubstantiallylargerforinstances withfacility-dependentrequirementsthanforinstanceswithfacility-independent requirements,particularlywhen jIj =30.CPLEXisabletosolveinstanceswithloose capacities =1 : 3inabouttwicethetimerequiredbytheheuristicandeveninonly about50%moretimeforthesmallestinstances.Theinstanceswithtightercapacities clearlyillustratethestrengthoftheheuristic.When =1 : 1and jJj 25 jIj ,CPLEX wasnotabletondafeasiblesolutionwithin15minutesofCPUtimeforanyinstance, andwassuccessfulforonly2instanceswith jJj =10 jIj and jIj =30.When =1 : 2 weseeasimilarbehaviorfor jJj 10 jIj and jIj =30.For jIj =15and jJj =10 jIj thetimerequiredbyCPLEXexceedsthatoftheheuristicbyafactorof100,whilefor jJj 25 jIj CPLEXisagainnotsuccessfulinasubstantialnumberofprobleminstances. NotethatthecomputationtimesforCPLEXareaveragedoveronlythoseinstancesin whichitwassuccessfulandthereforedonotincludetheinstancesforwhichCPLEXwas unsuccessfulwithin15minutes.Moreover,foreachinstanceinwhichtheCPUtimelimit expired,CPLEXhadnotyetfoundafeasiblesolution. 5.4.4EectofPost-ProcessingPhase Theasymptoticfeasibilityandoptimalityguaranteesoftheheuristicforinstances withfacility-independentaswellasinstanceswithfacility-dependentrequirementshold 102
PAGE 103
evenintheabsenceofthepost-processingproceduredescribedinSection5.3.1.2.However, asmentionedabove,theresultspresentedthusfarpertaintosolutionsthathavebeen improvedbythispost-processingphase.Therefore,wewillinthissectionstudytheeect ofthepost-processingphaseonthequalityofthesolutions,illustratingsimultaneously theasymptoticperformanceguaranteeswithoutthepost-processingphaseaswellasthe practicalimportanceofapplyingthisphase. Tables5-13and5-14summarizetheseresults.Forbrevity,wehaveonlyfocusedon instanceswithintermediatecapacities =1 : 2;however,theresultsforothervalues of arequalitativelysimilar.Forthecaseoffacility-independentrequirementswehave omittedtheresultsofthegreedyphasealone,sincethisphasewasneverabletond afeasiblesolutionandthusthepost-processingphaseisirrelevant.Inbothtables,the columnslabeledbefore"containthesolutionerrorswithoutpost-processing,whilethe columnslabeledafter"containthesolutionerrorswithpost-processing.Weseethat, inallcasesandforbothtypesofprobleminstances,theresultswithoutpost-processing phaseareconsistentwiththeasymptoticoptimalityguarantees.However,wealsoseethat thepost-processingphasesubstantiallyreducessolutionerror,particularlyforproblem instanceswithasmallratioof jJj = jIj 5.5Conclusions ThischapterconsideredtheGAPFD,whichgeneralizestheclassicalGAP.Our extensionappliestosituationsinwhich,alongwiththeassignmentofcustomersto facilities,aexibledegreeofresourceconsumptionmustbedeterminedforeachof theseassignments.TosolvetheGAPFD,weproposeaclassofheuristicsmotivated byattractivepropertiesoftheoptimalsolutionoftheLP-relaxationtotheGAPFD anditscorrespondingdual.Fortwoclassesofcustomerrequirementsweshowthatan implementationoftheheuristicexiststhatisasymptoticallyfeasibleandoptimalwith probabilityoneunderaverybroadstochasticmodelontheproblemparameters.Our computationalstudydemonstratesthattheheuristicperformsverywell,particularly 103
PAGE 104
forlargeratiosofthenumberofcustomerstothenumberoffacilities.Whenadditional improvementstrategiesthatweproposeinthischapterarealsoconsidered,theheuristic issuccessfuloninstanceswithsmallerratiosaswell.Weobservethatthetimerequired toobtainsolutionsofcomparablequalityisconsiderablylessforourheuristicthanforthe commercialsolverCPLEX.Thefactthatourheuristicobtainsqualitysolutionssoquickly isencouragingforfurtherresearchdirections.Specically,webelievethattheheuristic maybeveryvaluablewhensolvingmoregeneralrelatedoptimizationproblemsforwhich theGAPFDarisesasasubproblemthatneedstobesolvedrepeatedly. 104
PAGE 105
Table5-1.Facility-independentrequirements:15facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 750|0.04241.120.335.840.24 1500|0.09250.380.1310.29 3750|0.20250.090.2910.85 7500|0.35250.050.5111.74 1,5000|0.72250.031.0314.18 Table5-2.Facility-independentrequirements:15facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 750|0.04253.200.0710.20 1500|0.07251.820.1110.41 3750|0.18250.470.2710.94 7500|0.33250.110.4911.93 1,5000|0.68250.020.9814.85 Table5-3.Facility-independentrequirements:15facilities =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 75250.790.06250.790.0611.30 150250.240.10250.240.1012.00 375250.060.24250.060.2414.93 750250.020.45250.020.4516.28 1,500250.000.91250.000.91110.82 Table5-4.Facility-independentrequirements:30facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.18251.050.351.561.74 3000|0.27250.360.3911.51 7500|0.68250.090.9514.26 1,5000|1.40250.041.9016.76 3,0000|2.95250.023.95113.34 Table5-5.Facility-independentrequirements:30facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.15253.560.2110.83 3000|0.26251.600.3812.11 7500|0.66250.380.9215.82 1,5000|1.34250.121.8319.79 3,0000|2.73250.023.71119.44 105
PAGE 106
Table5-6.Facility-independentrequirements:30facilities, =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 150250.670.18250.670.1818.21 300250.290.34250.290.34117.62 750250.040.84250.040.84128.24 1,500250.011.63250.011.63150.19 3,000250.003.25250.003.251163.96 Table5-7.Facility-dependentrequirements:15facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 75105.060.08259.320.394.843.04 15062.650.18259.320.924.88144.55 375251.280.36251.280.361| 750250.800.78250.800.781| 1,500250.521.89250.521.891| Table5-8.Facility-dependentrequirements:15facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 7541.550.07257.170.344.361.56 150150.610.17252.680.482.6050.30 375250.210.31250.210.311341.90 750250.100.65250.100.651478.38 1,500250.051.41250.051.411333.57 Table5-9.Facility-dependentrequirements:15facilities, =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 7590.970.07258.900.222.680.38 150190.290.16250.660.201.120.47 375250.020.29250.020.2910.81 750250.010.57250.010.5711.72 1,500250.001.17250.001.1714.50 Table5-10.Facility-dependentrequirements:30facilities, =1 : 1 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 15012.250.402513.383.107.72139.74 3000|0.50258.035.377.00440.51 750140.791.76252.6010.833.64| 1,500250.484.86250.484.861| 3,000250.3114.11250.3114.111| 106
PAGE 107
Table5-11.Facility-dependentrequirements:30facilities, =1 : 2 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.34255.012.21655.25 3000|0.42253.954.216374.52 750200.221.44250.674.682| 1,500250.093.54250.093.541| 3,000250.068.93250.068.931| Table5-12.Facility-dependentrequirements:30facilities, =1 : 3 GreedyphaseImprovementphase CPLEX jJj feas.error%timesec.feas.error%timesec.it.timesec. 1500|0.322511.481.694.842.97 30040.350.38251.911.151.964.02 750210.041.20250.041.2516.16 1,500250.012.75250.012.7518.53 3,000250.006.38250.006.38113.70 Table5-13.Post-processingeectonheuristicwithgreedyandimprovementphase; facility-independentrequirements: =1 : 2 jIj =15 jIj =30 error%error% jJj beforeafterbeforeafter 5 jIj 19.423.2019.473.56 10 jIj 10.621.8210.901.60 25 jIj 4.050.474.150.38 50 jIj 2.000.112.030.12 100 jIj 0.980.021.000.02 Table5-14.Post-processingeect;facility-dependentrequirements: =1 : 2 jIj =15 jIj =30 GreedyphaseImprovementphaseGreedyphaseImprovementphase error%error%error%error% jJj beforeafterbeforeafterbeforeafterbeforeafter 5 jIj 7.921.5521.307.17||18.805.01 10 jIj 4.910.6110.482.68||15.843.95 25 jIj 2.780.212.780.212.070.224.400.67 50 jIj 1.910.101.910.101.360.091.360.09 100 jIj 1.320.051.320.050.930.060.930.06 107
PAGE 108
CHAPTER6 LARGE-SCALEMULTI-EXCHANGEHEURISTICFORFIXED-CHARGERESOURCE CONSTRAINEDASSIGNMENTPROBLEMS Inthischapterweproposeheuristicproceduresforageneralizationofthewell-known CapacitatedFacilityLocationProblemwithSingle-SourcingconstraintsCFLP,known asthe CapacitatedFacilityLocationProblemwithFlexibleDemand CFLFD.Inaddition tothestandardproblemscopeoftheCFLP,theCFLFDpermitsexiblecustomer demandspecications.Thatis,forameasurableproductcharacteristice.g.,weight, length,volume,unitsdelivered,acustomerspeciesanallowablerangefordemand fulllment.Thisassignment-basedoptimizationproblemfallsintotheclassofchallenging mixed-integerprogramsthatbecomeverydiculttosolveasthenumberofcustomers perfacilityincreases.Becausemanypracticalapplicationsofthisproblemclassrequire obtaining/updatingsolutionsveryquickly,itisimportanttoidentifyfastheuristicsolution methodsthat,onaverage,providenear-optimalsolutions. AsmentionedinChapter2,theCFLPisaspecial-caseoftheCFLFD,ourproblem clearlybelongstotheclassof NP -Hardoptimizationproblems.Thedicultyofsuch problemsrequiresconsideringbothexactandheuristicsolutionmethodologies.Recent theoreticaladvancesinintegerprogramminghaveresultedinexactsolutionmethodologies thathaveprovensuccessfulonpreviouslyunsolvedprobleminstances.Forexample,the decomposition-basedseparationalgorithmfortheCapacitatedVehicleRoutingProblem CVRP,proposedbyRalphsetal.[76],solvesthreeofthepreviouslyunsolvedVRP instancesfromtheTSPLIBrepositorypresentedbyReinelt[77].Similarly,astabilized branch-and-cut-and-pricealgorithmfortheGeneralizedAssignmentProblemGAP, introducedbyPigattietal.[74],wasabletosolvethreepreviouslyunsolvedinstancesfrom theOR-Library.Unfortunately,evenwiththeseinnovativetechniques,manyreal-world sizeproblemscannotbesolvedwithinpractitioners'timerequirements.Infact,for numerousintegerprogrammingmodelsthatconsidertheassignmentofcustomersto resourcesi.e.,theGAP,Savelsbergh[88]andPigattietal.[74];theCVRP,Fukasaw 108
PAGE 109
[38];andtheMultiperiodSingle-SourcingProblemMPSSP,Frelingetal.[37],exact proceduresareoftensuccessfulonlyoninstanceswithasmallratioofcustomersto resources,limitingthenumberofproblemsthatcanbesolvedwithinacceptabletolerance levels.Moreover,thenumberofscenariosforwhichdecisionmakersmustrepeatedlyrevise theirstrategicplansisgrowingrapidlyasinformationbecomesavailableinreal-time.For example,productionplannersnowhaveaccesstochangesininventorylevels,demandrates andresourcelevels,astheyoccur.Tomakeuseofthisinformation,aplanningschedule mayrequireupdatesnumeroustimesaday.Insuchcases,itisnecessarytodesignan ecientheuristictoserveaseitherasupplementtoanexactalgorithmorasastand-alone procedurethatprovidesqualitysolutionswithlimitedcomputationaleort. InChapter4weproposedanexactalgorithmfortheCFLFD.Asiscommonwhen applyingbranch-and-pricetoassignmentproblems,thesuccessoftheapproachwas limitedtoproblemswitharelativelysmallratioofcustomerstofacilitiesusuallyup toabout10.Inthischapter,weproposetherstheuristicmethodologytargetedat solvingthebroadclassofprobleminstanceswithalargeratioofthenumberofcustomers tothenumberoffacilities.Theheuristicapproachweproposeemploysacombined facilityneighborhoodsearchmethodandafastheuristicsolutionmethodforsolving ageneralizationoftheGAP.Wediscussspecicimplementationissuesrelatedtothis methodology,includingmethodsforobtaininginitialfeasiblesolutions,eectivewaysto searchalargeneighborhoodofsolutionsandecientwaystodevelophybridapproaches thatcombinesuccessfulindividualheuristicmethodologies.Whilesuccessfulheuristics havebeendevelopedforrelevantproblems,suchasAhujaetal.'s[6]multi-exchange heuristicCFLP,thenotionofexibledemandprovidesadditionalchallengesthataremet throughtheapproachproposedinthiswork.Computationaltestsillustratethebenetsof ourproposedapproachforsolvingproblemsinthisclass. BalakrishnanandGeunes[13]proposedLagrangian-based,bin-packing-based,and LP-roundingheuristicsforthecloselyrelatedFlexibleDemandAssignmentProblem 109
PAGE 110
FDA.However,theircomputationaltestingfocusedoninstanceswithasmallratioof customerstofacilities.Aswillbeevidentinthefollowingsections,areassignmentof evenasinglecustomerintheCFLFDrequiresasubsequentassessmentoftheamount ofdemandfullledforeachcustomerassignedtothatparticularfacility.Therefore,the reassignmentofcustomersinourapproachwilldierdistinctlyfromthatofprevious work.InthecaseoftheapproachinAhujaetal.[6],ifthelimitedreassignmentfailsto ndanimprovingorfeasiblesolution,theauthorsconsideracompletereassignmentof customers,determinedbyheuristicallysolvinganinstanceofthegeneralizedassignment problem.ItisimportanttoemphasizethatthemethodestablishedinAhujaetal.[6] allowsfortotalcustomerreassignmentonlyasalastresort.However,animportant aspectofourapproachismotivated,inpart,bythislessfrequentlyemployed complete reassignmentstep.Wecontendthatwiththeavailabilityofaquick,eectiveheuristicto solvethenecessaryassignmentsubproblemandinlightoftheincreasedcomputational eortrequiredtofullyassessthepartialreassignmentofcustomers,theeortspent consideringadditionallargeneighborhoodssuchasthoseproposedinAhujaetal.[4]is notnecessaryforasuccessfulheuristicapproachfortheCFLFD.Moreover,sincecertain verylargeneighborhoodsconsideredinpreviousworkgrowquadraticallyinthenumber ofcustomers,thesemethodsarenotapplicableforsolvinglargerprobleminstances. TheworkofAhujaetal.[6],considersinstanceswithafacilitytocustomerratioofno morethan10.Therefore,theapproachoeredinthisworkservesastherstheuristicto considerafacilitylocationproblemwithdemandexibilityandoersageneralsolution frameworkthatisparticularlyapplicabletolarge-scaledecomposableassignmentproblems thathavereceivedlittleattentionintheliterature. Theremainderofthispaperisorganizedasfollows.Section6.1formallyintroduces theversionoftheCFLFDstudiedinthischapter.Section6.2proposestwoseparate searchtechniquesforsolvingtheCFLFD.Implementationdetailsoftheheuristic approachesarepresentedinSection6.3.InSection6.4,weperformacomputational 110
PAGE 111
studyofthevariousimplementationsofourheuristiconabroadcollectionofexperiments. Finally,inSection6.5,wediscusstheapplicationofourheuristicframeworktosimilar xed-chargeproblemsandoersomeconcludingremarks. 6.1OptimizationandModelFormulation TheoptimizationmodeldiscussedinthischapteristheCFLFDpresentedinChapter 3withlinearrevenuefunctionsi.e r ij v ij = r ij v ij .Furthermore,asinChapter5welet J i = J i 2I .Specically,themodelconsideredinthissectionisgivenbyproblemas follows: maximize X i 2I X j 2J r ij v ij + X i 2I X j 2J p ij x ij )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X i 2I f i y i subjecttoCFLFD-L X j 2J a ij x ij + v ij b i y i i 2I {1 X i 2I : j 2J x ij =1 j 2J {2 v ij ` ij x ij i 2I ; j 2J {3 v ij u ij x ij i 2I ; j 2J {4 x ij 2f 0 ; 1 g i 2I ; j 2J {5 y i 2f 0 ; 1 g i 2I : {6 ForthenotationalsimplicitywewillrefertoCFLFD-LasCFLFDthroughoutthis chapter,notinghoweverthatthemostgeneralversionoftheCFLFDisgiveninChapter 3.Inthenextsection,weproposeasearchheuristicthatexploitsthestructureofthe CFLFDinawaythatenablesustoconsideraverylargeneighborhood. 6.2HeuristicFramework Thissectiondescribesthecomponentsofageneralsearchheuristicframeworkfor theCFLFD.Weconsidertechniquesforexploringtwoclassesofsearchneighborhoods. Section6.2.1describesthecoreofourheuristic.Thisapproachsearchesaneighborhood denedbymanipulatingthesetofopenfacilities.Inthissearchprocedure,allcustomers 111
PAGE 112
arereassignedateachiteration.Whileahighlyecientmethodologyisusedtodetermine thereassignmentofallcustomersamongasetofopenfacilities,thenumberofopen facilitysetstoconsiderisstillverylarge.InSection6.2.2,weintroduceaspecialized implementationofaso-calledvery-large-neighborhood-searchVLSNheuristic.This techniqueallowsformultiplecustomerexchangesinanymove;however,nomorethanone customeramongthoseassignedtoaparticularfacilitymaybereassignedinasinglemove. Thissecondaryprocedureisconsideredinanadditionalhybridimplementationdiscussed inSection6.3.2. 6.2.1FacilityNeighborhoodSearch Ourprimarysearchapproachisdesignedtoeasilyallowforimprovingsolutions thatcorrespondtoopeningorclosingafacility.WhiletheVLSNproceduredescribedin Section6.2.2iseectiveinidentifyingalternativecustomerassignmentsamongagiven setofopenfacilities,itrarelyidentiessolutionsthatalterthesetofopenfacilities.This limitationisconrmedbyAhujaetal.'s[6]inclusionofthefacilityneighborhoodstructure intheirapproachforCFLP.Inthisneighborhood,theopening,closing,ortransferringof facilitiesisconsidered.Toestimatetheimpactofeachofthesemoves,theyattempted toidentifya subset ofcustomerswhosereassignmenttoasingledierentfacility resultedinacostsavings.Asmentionedintheprevioussection,theidenticationof potentiallyimprovingmovesinAhujaetal.'s[6]approachrequiredthedirectcomparison ofassignmentcostparameters.However,inthecaseoftheCFLFD,determiningthe changeinprotfromthereassignmentofevenasubsetofcustomersrequiresthatthe levelthatdemandissatisedmustbereassessedforeachcustomerassignedtoany facilitywhichaddsorlosesacustomerinthereassignment.Therefore,sinceconsidering reassignmentsofsmallsubsetsofcustomersisalreadymorecomputationallyintensive inthecaseoftheCFLFDthanCFLP,wefocusimmediatelyondeterminingacomplete reassignmentofallcustomersviaanecientheuristicwhenconsideringthevarious optionswithrespecttoopeningandclosingfacilities.Tocontrastthisprocedurewith 112
PAGE 113
previouswork,Ahujaetal.'s[6]approachonlyconsideredtotalreassignmentofcustomers whentheirpartialreassignmentprocedurewasunabletondanyimprovingfeasible solution.Ourstrategyeectivelyallowsustoconsidersinglefacilitymovesopen,close, orswapandmulti-exchangemultiplecustomersbeingreassignedamong numerous facilitiesinasinglesearchstep.Clearlythisheuristicsearchesaverylargeneighborhood, andifimplementedwithasuccessfulsubproblemmethodology,islikelytooerhigh qualitysolutions.Ofcourse,reassigningallcustomersateachstepofoursearchcanbe computationallyintensive,aswell.Tocompensateforthisextraeort,wecontendthat, forproblemswithalargeratioofcustomerstofacilities,relyingonanecientlyobtained subproblemsolutiontodeterminecustomerassignmentsisatleastaseectiveasVLSN. Alternatively,wefocusonanintelligentimplementationthatecientlysearchesthisvery largefacilityneighborhoodanddeterminesasetofopenfacilitiesandsubsequentcustomer assignmentswhichcorrespondtoahighqualitysolution.Intheremainderofthissection wepresentthesearchframeworkandintroducetherelevantsubproblemtobesolvedin thesecondarystageofoursearch. Ourframeworkallowsforasearchofaverylargeneighborhood.Ineachpotential move,oursearchevaluatesthebenetofmanipulatingthesetofopenfacilitiesand reassigningcustomersamongthesefacilities.Specically,foranypartialfeasiblesolution toCFLFD, x N ;v N ;y N ,let O bethesetofopenfacilitiesinthissolution,i.e., O = f i 2I : y N i =1 g and C bethesetofclosedfacilities,i.e., C = f i 2I : y N i =0 g : Ourheuristiccanbedescribedbythreeseparatemoves. Close :closea single facilityin O andreassignallcustomers; Open :opena single facilityin C andreassignallcustomers; 113
PAGE 114
Swap :closea single facilityin O andreplaceitbyopeningaclosedfacilityin C ,and thenreassignallcustomers. Foreachneighborhoodmovedescribedabove,reassigningallcustomersamongagiven setofopenfacilities, ~ I ,anddeterminingthecorrespondingdemandfulllmentlevels,is accomplishedbysolvingthefollowingmixed-integerprogram maximize X i 2 ~ I X j 2J p ij x ij + X i 2 ~ I X j 2J r ij v ij subjecttoGAPFD X j 2J a ij x ij + v ij b i i 2 ~ I {7 X i 2 ~ I x ij =1 j 2J {8 ` ij x ij v ij u ij x ij i 2 ~ I j 2J {9 x ij 2f 0 ; 1 g i 2 ~ I j 2J : {10 ThisproblemistheGeneralizedAssignmentProblemwithFlexibleJobswithlinear revenuefunctions,whichwasstudiedinChapter5.Thisclassofoptimizationproblems isclearly NP -HardsincetheGAPisanimportantspecialcase.However,intheprevious chapterwedevelopedanecientconstructiveheuristicfortheGAPFD,whichwasshown tobeasymptoticallyfeasibleandoptimalunderaverygeneralstochasticmodel. Thisframeworkcanclearlybeimplementedinavarietyofways.Toproducean ecientimplementation,importantdesigndecisionsmustbemade.Section6.3discusses eachoftheseissuesandpresentsthemostsuccessfulimplementationdeterminedthrough ourcomputationalstudy.However,beforeconsideringtheseimplementationissueswe introduceanadditionallarge-scaleneighborhoodsearchtechniquethathasbeenappliedto numerousset-partitioningoptimizationproblems. 114
PAGE 115
6.2.2Single-CustomerVLSN InthissectionwediscusauniqueVLSNimplementationfortheCFLFDwhich wewillconsiderasapost-processingphaseforourmainfacilityneighborhoodsearch procedure.Inrecentliterature,avarietyofassignment-basedoptimizationproblemshave beensolvedusingVLSN,e.g.,theMulti-PeriodSingle-SourcingProblemAhujaetal.[3], theSingleSourceCapacitatedFacilityLocationProblemAhujaetal.[6],andVehicle RoutingandSchedulingProblemsThompsonandPsaraftis[97].DetailsoftheVLSN procedurearewelldocumented.Therefore,thissectiononlyoersdetailsofconsiderations madetoaccommodatetheuniquedemandexibilitycomponentofourproblem.For adetailedsurveyoftheVLSNtechnique,thereaderisreferredtoAhujaetal.[5].For aspecicdiscussionofsingle-customermulti-exchangeVLSNappliedtotheCFLP,a comprehensivediscussionisprovidedinAhujaetal.[6,Section4]. Ingeneral,thesingle-customermulti-exchangesearchneighborhoodisexploredby constructingaso-calledimprovementgraph.Theimprovementgraphconsistsofanode foreachcustomer,separatenodesforeachfacility,andanoriginnode.Arcsconnecteach pairof`customernodes',providedthatthecustomersareassignedtodierentfacilities. Inaddition,arcsconnecteach`customernode'toeach`facilitynode',excludingthenode representingthefacilitytowhichthecustomeriscurrentlyassigned.Lastly,thegraph includesarcsfromeach`customernode'totheoriginnode,aswellasanarcfromeach `facilitynode'backtotheorigin.Theinclusionof`facilitynodes'andtheoriginnodes allowsforexchangesinwhichacustomerisaddedtoremovedfromafacility,butno customerisrelinquishedfromaddedtothatfacility.Usingthisrepresentation,an improvingmoveisobtainedbyidentifyinganegativesubset-disjointcycleinthenetwork. Acomprehensivediscussionofdisjointcyclesandtheoptimizationeortrequiredto identifythemisfoundinThompsonandOrlin[96].Forthepurposesofthiswork,we applytheeectiveheuristicproposedbyAhujaetal.[7]tothe NP -Hardproblemof identifyinganegativesubset-disjointcycle. 115
PAGE 116
Thusfar,theimplementationofVLSNonourproblemisnodierentthanwhen appliedtoanysimilarproblemwithaset-partitioningstructure.However,unlikeany modelpreviouslysolvedwithVLSN,theexibledemandcomponentoftheCFLFDmust beaccountedforwhendeterminingthecostofthearcscontainedintheimprovement graph.Ingeneral,thecostofanyarcintheimprovementgraphissimplythedierence intotalprotresultingfromassignmentchangesrepresentedbythatarc.IntheCFLP, aswellasothermodelswithxedassignmentprotsorcosts,thisdierenceisdependent solelyonthecostprotparametersintheproblem.InthecaseoftheCFLFD,each assignmentexchangemustbeaccompaniedbyacorrespondingdemandfulllmentlevel decision.Therefore,tocalculatethecostofanarc,asubproblemmustbesolvedto determinetheappropriatechangeincustomerdemandfulllmentlevelsassociatedwith eachexchange. Toillustrate,considertwo`customernodes', j 1 and j 2 .Let i j 2 bethefacilitytowhich customer j 2 iscurrentlyassignedand ~ J thesetofcustomerscurrentlyassignedto i j 2 The costofthearcconnecting`customernodes', j 1 and j 2 is z SP ~ J )]TJ/F21 11.9552 Tf 11.955 0 Td [(z SP f ~ Jn j 2 g[f j 1 g ; where,foranysetofcustomers J assignedto { ,thevalue z SP J isobtainedbysolvingthe followingoptimizationproblem,SP, +maximize X j 2 J r {j w {j subjecttoSP X j 2 J w {j b { )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X j 2 J a {j + ` {j 0 w {j u {j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` {j j 2 J ; 116
PAGE 117
and isaconstantequalto P j 2 J p {j + r {j ` {j )]TJ/F21 11.9552 Tf 12.82 0 Td [(f { .Inthisformulation,thedecision variable w {j representsthetotalamountofresourcei.e.,bothxedandvariable consumedbycustomer j whenassignedtofacility { .ItiseasytoseethatSPcanbe treatedinthesamemannerasa0/1continuousknapsackproblem.Theproblemcanbe solvedbychoosingtheexiblecomponentofthesizesofthedemandfulllmentaslargeas possibleinnonincreasingorderof r {j aslongascapacityallows.Inasimilarmanner,SP canbeusedtodeterminetheappropriatecostsofarcsconnectingtheremainingnodes. Theabilitytoecientlycalculatetruearccostsisanotableresultthatisuniqueto thedevelopmentofaheuristicfortheCFLFD.Withtheavailabilityofaprocedureto solveSPecientlytooptimality,weavoidthereducedimpactofVLSNthatisgenerally causedbyhavingtorelyonarccostestimates.Thus,evenwiththeadditionaldemand exibilitydecisioncomponent,alargeneighborhoodcanbeecientlyexploredforproblem sizessimilartothoseconsideredinrelatedwork.Whiletheliteraturehasshownthat thistechniqueiseectiveindeterminingcustomerassignmentsforaxedsetoffacilities, establishinganappropriatesetofopenfacilitiesisaweaknessoftheimplementation describedinthissection.Therefore,thisapproachwillbeutilizedonlyinthelimited manneroutlinedattheendofthefollowingsection. 6.3SearchHeuristicImplementation Thissectionwillprimarilyfocusonimplementationchoicesregardingtheneighborhood searchheuristicproposedinSection6.2.1.However,attheendofthesection,the motivationforahybridapproachwhichmeldsthefacilityneighborhoodsearchFS approachwiththeVLSNinSection6.2.2ispresented.WithregardtoFS,anumber ofkeyconsiderationsarenecessary.First,sinceFSfallsintheclassofimprovement heuristics,wemustdeterminehowaninitialfeasiblesolutionisobtained.Second,the eortrequiredtoconsiderafullsetoffacilitymovesisextensive.Therefore,consideration shouldbegiventointelligentlyconsiderasubsetofpotentiallyattractivemoves.Then,of course,thecriteriausedtodeterminesearchterminationmustbespecied.Thechoices 117
PAGE 118
madewithrespecttoeachoftheseissueshasaconsiderableeectonthesuccessofour heuristic.Inthissectionwetreateachoftheseissuesseparately.Weoeralternatives foreachissue,discussourndingswithrespecttoeachoption,andprovidethebest implementationencountered. 6.3.1InitialFeasibleSolution SincealternativeheuristicsfortheCFLFDdonotexistthatmayserveasasource foraninitialfeasiblesolution,westudytwoalternativestodetermineastartingsolution. TherstismotivatedbytheavailabilityofanecientheuristictosolvetheGAPFD. Inthisalternativeweassumethatallfacilitiesareopen,andthensolveaGAPFDvia theconstructiveheuristicproposedinChapter5todeterminethecustomerassignments andthelevelatwhicheachcustomer'sdemandissatised.Thisalternativehasclear advantages.First,withthecapacityofallfacilitiesavailableforcustomerassignments, itisrelativelyeasytondafeasiblesolutiontothecorrespondingGAPFD.Theresults inChapter5indicatethatsolutionsobtainedbytheirGAPFDheuristicforinstances withlargeamountsofavailablefacilitycapacityareveryclosetooptimalforaxed setoffacilities,andcanbedeterminedwithaminimalamountofcomputationaleort. Unfortunately,thedisadvantageofthisalternativeisthatthefacilityprocurementcosts areneglected.Therefore,whileeasytoimplementandintuitivetoconsider,thequalityof thesolutionsobtainedusingthisapproachwereofpoorqualityandultimatelyresultedin prolongeddurationofoursearchheuristic. Thesecondalternativeisamodiedrandomsolutiongenerationapproach.This methodforgeneratinganinitialfeasiblesolutioniscommonlyusedtostartanimprovement heuristicsearche.g.,theTravelingSalesmanProblem,LinandKerninghan[61];andthe ResourceConstrainedProjectSchedulingProblem,LeeandKim[59].Whenrandomly generatingafeasiblesolutiontotheCFLFD,weexplicitlyattempttominimizethe numberoffacilitieswhichare`opened'inthesolution.Let J and I bethesetofof unassignedcustomersandunusedfacilities,respectively.Furthermore,let b 0 i bethe 118
PAGE 119
remainingcapacityoffacility i i 2I .Theprocedureforrandomlygeneratingafeasible solutiontotheCFLFDisasfollows. RandomlygenerateCFLFDsolution Step0. Set J = J I = I and b 0 i = b i for i 2I Step1. Randomlychooseafacility^ { 2 I .Set y I ^ { =1remove^ { from I .ProceedtoStep 2. Step2. If J =,STOPwithfeasiblesolution.Else,randomlychooseacustomer^ | 2 J If a ^ { ^ | + ` ^ { ^ | b ^ { set x I ^ { ^ | =1, v I ^ { ^ | = ` ^ { ^ | b 0 ^ { = b 0 ^ { )]TJ/F15 11.9552 Tf 12.492 0 Td [( a ^ { ^ | + ` ^ { ^ | andrepeatStep2. Otherwise,if I6 =,returntoStep1;elsereturntoStep0. Thisprocedurerandomlyassignscustomerstoasinglerandomlychosenfacilityaslongas capacityallows.Whensucientcapacitynolongerexists,anewfacilityisopenedandthe procedurecontinues.Thisapproachtypicallyyieldsasolutionwithfeweropenfacilities thantherstapproachproposed.Randomlyassigningcustomerstoopenfacilities individually,ontheotherhand,appearstobelessdesirablethantakingadvantageof theGAPFDheuristic.However,inourcomputationaltesting,thesolutionsobtainedvia randomgenerationcontainedanumberofopenfacilitiesmoreconsistentwiththenumber foundinthenalsolutiongeneratedbytheneighborhoodsearchheuristic.Therefore,the durationoftheoverallsearchprocedurewasreducedbychoosingtherandommethod. Sincethequalityoftheultimatesolutionfoundbythesearchheuristicwasunchangedby themethodusedtoobtaintheinitialfeasiblesolution,weusetherandomprocedurein ourcomputationaltesting. 6.3.2FSMoveChoice Inthissectionwedeterminethebestsetofmovestoconsiderinoursearchand theorderinwhichtheyshouldbeconsidered.Anobviousimplementationconsiders openingoneatatime all facilitiesin C calledan O -move,closingoneatatime all facilitiesin O calleda C -move,andswapping all pairsoffacilitiesin O and C during anysingleiterationofthesearchcalledan S -move.Werefertothisimplementationas 119
PAGE 120
the FullNeighborhoodsearch FNS.Inthisimplementation,theorderinwhichmoves areconsideredisirrelevant.Intuitively,sincetheentireneighborhoodisexplored,a highlydesirableheuristicsolutionislikelytoresult.However,theeorttosearchthefull neighborhoodrequiresextensivecomputationaleort.Inourcomputationalstudy,we willanalyzetheadvantagesofsolutionsobtainedfromthisimplementationaswellasthe associatedcomputingtime. Alternatively,wemayconsideronlyasubsetofthefullneighborhoodateach iteration.Thismayleadtoareductioninthetimeneededtocompletethesearch heuristic.However,thecorrespondingreductioninthesizeoftheneighborhood exploredhasthepotentialforconvergencetosolutionsoflesserquality.Inthealternative implementationtofollowwecontendthatthetimesavedthroughthesearchofareduced neighborhoodisnotattheexpenseofsolutionquality.Thedirectionforoursecond implementationresultsfromofacarefulstudyoftheprogressoftheFNSimplementation. AcloseanalysisofthebestmovesateachiterationofFNSconsistentlyrevealedthatthe heuristicbeginsbychoosingasequenceof O -moves.Then,atsomeiterationinthesearch procedure,thebestmovebecomesan S -move.The S -moveiscontinuouslyidentiedas thebestmovechosenuntilthesearchterminateswithoutndingfurtherimprovement. Therefore,considerationof C -movesand S -movesduringtherstphaseoftheFNS-search equatestowastedtime.Similarly,theconsiderationof O -movesand C -movesinthe secondphaseoftheFNSimplementationistypicallynotbenecial. Basedontheprecedingdiscussion,wedivideoursecondimplementationintotwo phases.Initially,weconsideronly O -moves.Sincethenumberof O -movestoconsider atanyiterationisontheorderofthenumberoffacilities, jIj ,weconsidereachofthese alternatives.Recallthat,toassessthebenetofeachmove,wesolveaGAPFDto determinethecorrespondingcustomerassignmentsandthelevelatwhichtheirdemand isfullled.Thebestincumbentisidentiedasthesolutionwiththelargestnetprot. Werepeattheopenneighborhoodsearchonthissolution.Phase1continuesuntilno 120
PAGE 121
improvementisattainedthroughan O -move.Atthispoint,weproceedbysearching theneighborhoodofthebestsolutionobtainedinPhase1,denedbypotentialswaps ofopenandclosedfacilities.Clearly,thenumberofpotential S -movestoconsiderat eachiterationisontheorderof jIj 2 .Consideringeachofthesemovesistimeprohibitive. Ahujaetal.[6]citethisconcernaswell,andidentifytworulesforassessingtheexpected impactofpotential S -moves.Ourruleisanextensionoftheruleproposedinthatwork, whichreassignsallcustomerstothesamenewfacility,whileleavingtheothercustomer assignmentsunchanged.Thatis,weconsidereachpairofcurrentlyopen/closedsets o;c 8 o 2 O c 2 C .Allcustomerscurrentlyassignedtofacility o aretemporarilyassignedto facility c .Allcustomersassignedtofacilities i 2I ; i 6 = o retaintheircurrentassignments. Fortheupdatedcollectionofassignments,let i j denotethefacilitytowhichcustomer j isassigned.Giventhissetoftemporaryassignments,wesolvethefollowingLPSP 0 to determinetheoptimallevelsatwhichtofulllthosecustomers'demands: +maximize X j 2J r i j j w i j j subjecttoSP 0 X j : i j = i w ij b i )]TJ/F26 11.9552 Tf 14.567 11.357 Td [(X j : i j = i a ij + ` ij i 2 O nf o g [f c g 0 w i j j u i j j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` i j j j 2J ; where isaconstantequalto P j 2J )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(p i j + r i j j ` i j j )]TJ/F26 11.9552 Tf 12.612 8.966 Td [(P i 2 O nf o g [f c g f i .Itiseasytosee thatthisproblemdecomposesinto j O nf o g [f c gj continuousknapsackproblemsofthe formSP,presentedinSection6.2.2.Theoptimalobjectivevalueofthisproblem, z SP 0 determinesthe`swappriority'ofaspecicopen/closepair, o;c .Specically,afterSP 0 is solvedforeach o;c o 2 O c 2 C ,thecollectionofpairsissortedinnon-increasingorder ofthecorrespondingvalue, z SP 0 121
PAGE 122
Ateachiterationofoursearchweconsidermovesintheorderdeterminedbythe precedingprocedureuntileitherianimprovingmoveisfoundoriinoimprovingmoves arefoundamongtherst jIj swapcandidatesconsideredinthelastiteration.Inthecase ofi,weproceedtothenextiterationwiththeredenedneighborhood,while,inthecase ofii,theentiresearchprocedureterminates.Comparisonofthisprioritymeasureagainst thetruedesirabilityofmovesdeterminedbysearchingthefullswapneighborhoodrevealed thatourmethodologywaseectiveinidentifyingpromising S -moveseciently.Wewill addresstheeectivenessofourprioritymeasuremoredirectlyinSection6.4. Wehavenowcompletelydescribedourfacilityneighborhoodsearchimplementation. However,wehavenotexplicitlyexplainedouromissionof C -moves.Again,thebasis forthisimplementationwasthatthefullneighborhood very rarelyfounda C -moveto bethebestoptionatanyiteration.Webelievethisresultisduetothenatureofthe initialfeasiblesolutionprovided.Sincetherandomlygeneratedfeasiblesolutionaimsto containasmallnumberofopenedfacilities,a C -moveisintuitivelyunnecessary.Onthe contrary,whenourinitialsolutionwasobtainedbyopeningallfaciltiesandsolvingthe correspondingGAPFD,thenonly C -moveswerechosenintheinitialphaseofFNS,and O -moveswererarelyutilized.Therefore,whether O -movesand/or C -movesshouldbe consideredinPhase1dependsuponthecharacteristicsoftheinitialsolutionprovidedto thesearchheuristicbythedecisionmaker. Lastly,weareinterestedinthepotentialimpactoftheVLSNprocedure,discussedin Section6.2.2,whenusedinconjunctionwiththefacilityneighborhoodsearch.Therefore, toimproveuponthecustomerassignmentsassociatedwiththesetofprocuredfacilities determinedinthefacilityneighborhoodsearchprocedure,wewillalsotestahybrid implementation.Thishybridizationperformsasingle-customermultiple-exchangeVLSN onthebestsolutionfoundinthefacilityneighborhoodsearchprocedure.Ofparticular interestwillbewhethertheimprovedsolutionmeritstheaddedcomputationalexpense. 122
PAGE 123
Withthisconsiderationandthedetailsoftheproposedheuristicinplace,weare readytodiscussacomputationalstudydesignedtotesttheeectivenessofthevariantsof ourheuristicwhencomparedwiththeperformanceofawell-knowncommercialsolver. 6.4ComputationalStudy Ourcomputationalstudyconsiderstwelveseparateproblemscenarios.Foreach problemscenario,weconsider10separateinstances.Wevarytheratioofcustomers/facilities inordertoexploretheeectonbothcomputationaltimerequirementsandsolution quality.Ourstudyassessestheperformanceofthethreeimplementationsdiscussed inSection6.3.2.Additionally,wedeterminethetimerequiredforastate-of-the-art commercialsolvertondsolutionsofequivalentqualitytothoseobtainedbyourvarious implementations.TheexperimentswereperformedonaPCwitha3.40GHzPentiumIV processorand2GBofRAM.Allmixed-integerprogrammingproblemsweresolvedusing CPLEX11.0. 6.4.1ExperimentalData InourtestingoftheCFLFD,weconsiderinstanceswith15and30facilitiesanda varyingnumberofcustomersequalto jJj =3 jIj ,5 jI j 10 jIj ,25 jIj ,50 jIj ,and100 jIj .For eachcustomer,wegeneratetherandomvectorsofxedprotparameters P j fromuniform distributionson[30 ; 50]andtheelementsoftherandomvectorofrevenues, R j ,froma uniformdistributionon[2 ; 5].Thevectorsofcustomerrequirements A j L j and D j are generatedfromuniformdistributionson[10 ; 20],[75 ; 125],and[15 ; 35],respectively.Here D j isarandomvectorwhoseelementsrepresenttherangeofacceptablesizestofulll customerdemand,i.e.,theupperboundonthedemandforcustomer j ,whenassigned tofacility i ,isequalto L ij + D ij .SimilartotheGAPFD,wegenerateidenticalfacility capacitiessuchthat b i = jJj i 2I ,where = E min i 2I A i 1 + L i 1 jIj ; {11 123
PAGE 124
with E referingtotheexpectedvalueofthegivenexpression.Theparameter > 1, inatesthecapacityofafacilitytoensurethatnotallfacilitiesarerequiredinafeasible solutiontotheCFLFD.Intheseexperiments,thefacilitycapacitiesweregeneratedusing =2.Theparameter controlsthelevelofexibilityavailablewhendeterminingthe levelatwhicheachcustomer'sdemandisfullled.Thesecomputationaltestsconsider amoderateexibilitylevelbysetting =1 : 2.Weassumethatthecostofprocuringa facilityisdirectlyproportionaltothecapacityoftheindividualfacility.Therefore,the costofprocuringfacility i i 2I isgivenby F i = b i C i ,where C i representstheunitcost ofprocurementgeneratedfromauniformdistributionon[0 : 75 ; 1 : 5]. Asasidenote,recallthatateachstepoftheneighborhoodsearchweeectivelysolve aGAPFD.IntheheuristicproposedinChapter5perturbationofresourcecapacitiesis necessarywhencustomer-requirementparametersarefacilitydependent.Forthisreason, weperturboursetcapacitiesusingtheprocedureandparameterproposedinthatchapter. Thetablesinthefollowingsectionassesstheperformanceofthethreeimplementations describedinSection6.3.2.Eachrowinthetablesrepresentstheaverageresultscollected amongst10instancesgeneratedforthatparticularscenario.Thefollowingmeasuresare reportedinTables6-1{6-4.Acolumnlabeled FNS indicatesthatthefullneighborhood i.e.,allswap,open,closemoveswasconsideredateachiterationoftheprocedure.A columnlabeled RNS indicatesthatthetwo-phaseimplementationwasused.Thatis,we consideredonly O -movesinPhase1untilnoadditionalimprovementwasfound,then S -movesinPhase2,untiltheprocedureterminated.Lastly,acolumnincludingaheading of hybrid indicatesresultsobtainedbyrunningasingle-customermulti-exchangeVLSN onthesolutionobtainedbyeithertheFNSorRNSprocedure.Inaddition,weusethe followingnotation: UBError :Theupperboundontheerrorassociatedwiththeobjectivevalueofthe solutionobtainedfromthespeciedprocedure.Forexample,inthecaseofFNS, UBerror= z UB )]TJ/F21 11.9552 Tf 11.955 0 Td [(z FNS z FNS : 124
PAGE 125
Theupperboundontheoptimalsolution, z UB ,wasobtainedfromCPLEX.The valuewastakentobethebestupperboundavailableaftersolvingtheCFLFDfor15 minutes. HeuristicProcedureTime :Computationaltimeinsecondsrequiredbythe speciedprocedurei.e.FNS,RNS,orHybrid. CPLEXTime :ComputationaltimeinsecondsforCPLEXtoobtainthesameor bettersolutionthanthatfoundbythespeciedprocedure. 6.4.2Results AssuggestedinSection6.2.1,thehighestqualityheuristicresultsarelikelyto resultfromthefullneighborhoodsearchimplementationFNS.Tables6-1and6-3 provideaverageresultsforinstanceswith15and30facilities,fortheFNS,bothwith andwithoutthesupplementaryVLSNstep.Thesetablesindicatethatsolutionswith anaverageerrornomorethan4%wereobtainedforeachoftheproblemsizestested. TheadditionalVLSNstepi.e.,FNShybridreducedtheerroronlyasmallamountin eachsetofexperiments.Themostextremeimprovementsoccurforexperimentswitha customer-to-facilityratiolessthan10forinstanceswith15facilities,whilethebenetof theVLSNstepextendstoinstanceswithacustomer-to-facilityratioupto25forinstances with30facilities.ThetimerequiredforFNSwithoutVLSNrangedfromapproximately 2to90secondsfor15facilityinstancesand1to25minutesfor30facilityinstances. Table6-1showsthatCPLEXrequiredupto10timestheamountofcomputationaltime toobtainthesamesolutionsfor15facilityinstances,whilefromTable6-3weseethat CPLEXconsistentlyoutperformedtheFNSimplementationwithouttheVLSNstepfor instanceswith30facilities.Infact,forinstanceswith30facilitiesand3000customers, theFNSimplementation,aswellasCPLEX,requiredmorethantheallotedtimeof30 minutesandthereforetheseresultsarenotreported.TheFNShybridtimeinTables6-1 and6-3indicatesthattheadditionalVLSNeortonlymarginallyincreasedthetotaltime forexperimentswitharatioofcustomerstofacilitiesnogreaterthan10.Unfortunately,as shownintheresultsforthe750customerexperiments,inTable6-1,forlargerinstances, 125
PAGE 126
theadditionalVLSNstepbecomestimeprohibitive.Infact,thehybridimplementation forinstanceswith1500customersormoretakesmorethantheallottedruntimeto terminate;therefore,theseresultsareomittedfromTables6-1and6-3.Interestingly,while theimprovementinsolutionqualityinthehybridimplementationofFNSisminimal,the timerequiredbyCPLEXtoobtaintheseslightlybettersolutionsisconsistentlymorethan doublethetimerequiredtoobtainthesolutionsproducedbyFNSalone.Therefore,for instanceswithasmallcustomer-to-facilityratio,thehybridimplementationisattractive. Fortheseinstances,theFNShybridruntimeremainssmall,butthequalityofsolutionsis improvedtoadegreethatthecommercialsolverhasdicultyduplicatingtheresultina comparableamountoftime. Thevalueofthetwo-phasereducedneighborhoodsearchimplementationi.e.,RNS ispresentedinTables6-2and6-4,whichoeraverageresultscollectedoverthesame setof15and30facilityinstancesconsideredinTables6-1and6-3.Theaveragetimefor RNS,withoutVLSN,isatleast6timeslessthanFNSfor15facilityinstancesandatleast 15timeslessfor30facilityinstances.Moresignicantly,theaverageerrorsofthesolutions obtainedfromRNSareonlyslightlyhigherthanthoseobtainedfromFNS.Infact,for instanceswith15facilitiesand150customersorgreater,theaverageerroriswithinone one-hundredthofapercentofthatobtainedthroughFNS.Themostextremeincreasein errorcorrespondsto30facility,90customerinstances,wheretheerrorisapproximately twicethatobtainedfromtheFNSimplementation.However,forinstancesofthissize, CPLEXobtainshighqualitysolutionsinasmallamountoftime;therefore,theneedfor theheuristicislesssignicant.TheimpactofVLSNappliedtosolutionsobtainedfrom RNSissimilartothatseenwhenVLSNiscombinedwithFNS.Themostsignicant improvementinsolutionqualityisseenwithcustomer-to-facilityratiosof10orless.As withtheFNShybrid,theRNShybridisexceedinglytimeconsumingforbothsetsof instanceswithacustomer-to-facilityratiogreaterthan10. 126
PAGE 127
BecausetheRNSresultsprovidecomparablyhighqualitysolutionsinlesstime, theratiooftimerequiredforCPLEXandRNStondthesamesolutionisnotably higherthanthesameratiowithrespecttoFNS.Inaddition,ourstudyfoundthatthe averagenumberofsearchiterationsrequiredbyboththeFNSandRNSimplementations wasremarkablysimilarineachsetofexperiments.Thissuggeststhatour2-Phase RNSimplementationdoesnotterminateprematurelyasaresultofsearchingalimited neighborhood.Furthermore,recallthatFNSconsiders all S -movesateachiteration, whilenomorethan jIj swapsareconsideredintheRNSimplementation.Sincethenal solutionobtainedbyeachimplementationis,onaverage,remarkablysimilar,weconclude thatthemostdesirableswapswereconsideredinphase2ofourRNSimplementation. ThisisastrongindicationthattheswaporderingruleproposedinSection6.3.2is eective. Lastly,Tables6-1and6-2suggestasmallbutnoticeableincreaseinaverage heuristicerrorasthenumberofcustomersincreases.Itshouldbenotedthaterrors werecalculatedusingthebestupperboundobtainedbyCPLEXaftersolvingtheCFLFD asamixed-integerprogramfor15minutes.Itisexpectedthatthedicultyofsolvingthe CFLFDincreaseswithanincreaseinthenumberofcustomers.Therefore,itislikelythat theupperboundobtainedafter15minutesforinstanceswith1500customersisweaker thantheupperboundobtainedforaninstancewithonly150customers.Thissuggests thaterrorsmaybeinatedasthenumberofcustomersincreases,whichispreciselywhat weobserveinTables6-1and6-2. 6.5CFLFDHeuristicApplicationsandConclusions Thesuccessoftheheuristicframeworkproposedinthischapterispromisingfor optimizationproblemswithasimilarstructure.Specically,thereareanumberoffamiliar problemswithaxed-chargecomponentthatcanbesolvedwithourgeneralframework. Asmentionedpreviously,theCFLPclearlytsintoourframeworkandanecient methodforsolvingtheGAPsubproblemisreadilyavailable.Ourheuristicsearchesthe 127
PAGE 128
neighborhoodofaCFLPsolutioninadierentmanner,withalternativemovechoicesto thoseproposedbyAhujaetal.[6].Specically,themannerinwhichcustomerassignments aredeterminedwhenevaluatingpotentialmovesreliespredominantlyonthereassignment ofallcustomersthroughasecondaryheuristic.Thisapproachispursuedinlieuof searchingthemulti-customerexchangefacilityimprovementgraphandconsideringpartial reassignmentsofcustomersinintermediarystepsofthefacilityneighborhoodsearch,as donefortheCFLPinAhujaetal.[6].Ourresultssuggestthatthisimplementationchoice providesqualitysolutionsinareasonableamountoftimeforproblemsizesmuchlarger thanthoseconsideredinapproachesforrelatedproblems.Thereforethismethodology oersauniquealternativetosolvingthisclassofproblems.Inaddition,thewell-studied UncapacitatedFacilityLocationProblemUFLPErlenkotter[32],Sun[93]tswellinto ourframework.FortheUFLP,thesubproblemtodeterminetheassignmentsassociated withasetofprocuredfacilitiescanbetriviallysolvedtooptimality.Inthiscase,our heuristicsimpliestoarelatedapproachproposedbyGhosh[45].However,theorder andsubsetoftheneighborhoodmovessearchedinourimplementationisdistinctly dierent.Analclassofoptimizationproblemswhichtsintoourheuristicframework istheFixed-ChargeTransportationProblem.Variousapproacheshavebeenproposed tosolvethisproblembothheuristicallyi.e.,AdlakhaandKowalski[1]andexactly i.e.,Gray[47].Interestingly,ifplacedinourframework,theunderlyingsubproblem solvedtodeterminethevalueofeachmoveissimplyalineartransportationproblem. Thetransportationproblemitselfhasbeenwellstudiedandsolutionmethodshavebeen presentedinworkoriginatingwithFordandFulkerson[36]. Thelessonslearnedinthisworkwithregardtoimplementationofourneighborhood search,canbeapplieddirectlytoeachoftheadditionaloptimizationproblemsmentioned inthissection.Ourcomputationalstudyillustratesthananintelligentsearchofareduced facilityneighborhoodoershighqualitysolutionswithasmallamountofcomputational eort.Furthermore,forproblemswithasmallratioofcustomerstofacilities,the 128
PAGE 129
inclusionofaVLSNprocedureimprovesthequalityofthesolutionsobtainedwith minimaladditionaleort.However,ourstudyalsodemonstratesthatasingle-customer multi-exchangeheuristicisnotpracticalforproblemswithalargenumberofcustomers. Fortheseinstances,ourfacilityneighborhoodsearchperformsverywellasastand-alone heuristic.Therefore,ourworkoersanattractiveheuristicwhichcanbetailoredto successfullysolveabroadclassofprobleminstancesforboththeCFLFDandsimilar xed-chargeproblems. Table6-1.FNSresults:15facilities FNSFNSCPLEXFNSFNShybridFNShybridCPLEXFNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 452.41.581.62.40.909.1 753.90.8145.84.10.39722.7 1509.20.6634.810.90.52292.7 37533.90.4090.078.90.37267.3 75035.01.42160.9576.21.41301.6 150082.51.56640.5--Table6-2.RNSresults:15facilities RNSRNSCPLEXRNSRNShybridRNShybridCPLEXRNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 450.42.000.90.41.54183.5 750.50.8745.70.70.43541.8 1501.00.6634.82.80.53259.2 3753.30.4090.647.50.37265.9 7504.41.43161.1530.01.42301.8 15009.91.56642.0--Table6-3.FNSresults:30facilities FNSFNSCPLEXFNSFNShybridFNShybridCPLEXFNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 9074.73.863.075.22.7325.5 150153.12.204.3155.91.22230.6 300219.41.3310.02810.631281.4 750620.20.6351.417940.281416.9 15001407.70.39188.4--3000-----129
PAGE 130
Table6-4.RNSresults:30facilities RNSRNSCPLEXRNSRNShybridRNShybridCPLEXRNShybrid #timeerrortimetimeerrortime Customerssec%secsec%sec 904.28.290.95.05.121.0 1508.72.403.011.71.7311.5 30012.51.489.963.00.97775.9 75033.40.6251.41261.50.281344.0 150076.30.39185.5--3000207.70.291660.3--hello 130
PAGE 131
CHAPTER7 RESOURCECONSTRAINEDASSIGNMENTPROBLEMSWITHSHARED RESOURCECONSUMPTION 7.1Introduction InChapters4{6westudiedCFLFDanditsspecialcase,GAPFD.InbothGAPFD andCFLFDthetotalamountofresourceconsumption,aswellasrevenue,isequal totheaccumulationofeachindividualcustomer'scontributiontothesetwometrics. However,inalmostanymanufacturingscenario,certainproductgroupssharesimilar productionrequirements.Forexample,fulllingcertaincustomers'demandsmayrequire thesamemachinesettingsduetocommoncomponents.Therefore,producingforthese commoncustomersonthesamemachineminimizesthetotaltimerequiredtosatisfy theirdemands.Theoptionofassigningcustomerstocommonfacilities,atanadded benet,isconsideredinjoint-costassignmentproblemsShubik[91].Workinthisarea modelstheimpactofjointrewardsobtainedfromseparateindividualswhoarewilling toworkasagroup,asseenintheMixed-IntegerSetupKnapsackProblemproposedby Altayetal.[9].Separately,problemsthatconsiderjointcostsacrossanumberofitems arefrequentlyfoundinmulti-iteminventorysettingssuchastheJointReplenishment ProblemFedergruenandZheng[33].However,thechoiceofhowtoutilizecapacitysaved byassigningcustomerswithsimilarproductionrequirementstothesamemachineisa separatefeatureoftheoptimizationmodelandonethathasreceivedmuchlessattention inthesestudies.Morerelevantworkisfoundinschedulingproblemsthatexploittheuse ofcommonresourceswhendeterminingthereleasetimeofcustomers'jobs.Forexample, Li[60]consideredaresourceconstrainedschedulingprobleminwhichtheamountof resourceconsumedisafunctionofthetimeinwhichthecustomers'jobisreleased. Conceivably,customersofsimilartypes,releasedinuninterruptedsequence,requirea lesseramountofresource.Forproblemswithanassignment-basedstructure,Mazzola [68]consideredageneralizationoftheGAPwithnonlinearcapacityinteraction.Rather thanaccountingforsomesharedsetupcomponentofresourceconsumption,thiswork 131
PAGE 132
modeledtheinteractionbetweencostumersassignedtothesamefacility.Incontrast totheapproachproposedinthischapter,Mazzola[68]providedabranch-and-bound algorithmthatwasshowntosolveproblemswithup20customers. Anotherinterestingextensiontomoretraditionalassignmentproblemsistheinclusion ofvariedformsofdemandfulllmentconstraints.Inlightofnewrestrictionsin21 st centuryproduction,newformsofconstraintsneedbeconsideredinconjunctionwith traditionalplanningmodels.Thatis,ratherthanconsideringonlylimitsonphysical resourcesortime,theplannermustmakedecisionsthatadheretoadditionallimits placedonthesatisfactionofcustomerdemand.Forexample,asmentionedpreviously, customersmaybegroupedintotypes,witheachtypeinterpretedaseitheriaspecic client,iicommonproductionrequirements,iiiacommonshipmentdestinationoriv commonmanufacturingbyproducts.Giventhesepossibleinterpretations,thelevelat whichcustomerdemandisfullledmaybelimitedbyeithertransportationresourcesor physicalspaceallocatedinthewarehousefacility.Alternatively,withgrowingconcern onmanufacturers'impactsontheenvironment,onemustconsiderhowproduction associatedwithfulllingcustomerdemandmayresultineitherpollutionorhazardous materials.Thatis,thetotalamountofdemandsatisedwithinaparticulargroupor typemaybelimitedbyoutsideparties,suchastheEnvironmentalProtectionAgency ortheOccupationalSafetyandHealthAdministration.Thesearespecicexamplesthat arecommonacrossanumberindustries.Thus,itisclearthatitisimportanttoinclude constraintsthatlimitproductionacrossallresourcesforagivensubsetofcustomers. Themodelsstudiedintheprecedingchaptersdonotaccountforthisadditionalform ofproductionlimitations.Itisimportanttonotethatfewotherworksintheliterature considerthisproblemelementeither.Looselyrelatedproblemsthathavereceivedmore attentionintheliteraturearethosewhichconsidermultipleresourceconsumption. AspecicexampleisfoundintheMulti-resourceGeneralizedAssignmentProblem, consideredbyGavishandPirkul[40],amongothers.Thismodelassumesthateachfacility 132
PAGE 133
consumesmultipleresourcestosatisfyacustomer'sdemand.However,limitsoncapacity consumptiondospancustomersassignedtodierentagents.Separately,Mazzolaand Neebe[69]studytheResource-ConstrainedAssignmentSchedulingProblemRCAS.This extensionofthe pure assignmentproblemconsidersasideconstraintthatrestrictsthe totalconsumptionofaresourceamongstdecentralizedcustomerandfacilitysets.Itisthe limitationofdemandfulllmentforcustomersassignedtonumerousfacilitiesthatwewish toaccountforinthemodelconsideredinthischapter. Asbrieyalludedtointhediscussionofrelevantliterature,thischapterconsidersa newclassofproblemsthat,amongotherthings,modelproductionenvironmentsinwhich aportionofcapacityconsumptionissharedamongcommoncustomerssatisedbythat resource.AsintheGAPFDandCFLFD,thenewoptimizationmodelassignscustomers tofacilitiesanddeterminesthecorrespondingdemandfulllmentlevels.However,dierent considerationsmustbetakenintoaccountwhenmakingthesedecisions.First,customers arenowgroupedbytype.Customersofthesametype,assignedtothesamefacility, consumeaxedamountofresourceinadditiontotheirindividualconsumption.Also,in additiontotheresourcelimitationsofindividualfacilities,aggregateresourceconsumption amongcustomersofaparticulartypeissubjecttoaseparatesetofrestrictions.This modeladdsalevelofcomplexitytothedecision-makingprocess.Aplannermustnow considertheimpactofsavingcapacitybyassigningcustomersofthesametypetothe samefacility.Furthermore,theadditionallimitationofproductionassociatedwith aparticularcustomersetmayaectthelevelsatwhichdemandsaresatised.Since thisproblemstillconsiderstheassignmentofcustomerstofacilities,weagainpursue abranch-and-priceapproachbasedonareformulationofourmodel.However,unlike Chapter4,theso-calledmasterproblemisnolongerintheformoftheset-partitioning problem.Duetotheadditionalcapacityconstraints,themasterproblem,pricingproblem, andcolumnrepresentationsmustbecarefullydeveloped.Weshowthatthepricing 133
PAGE 134
problembelongstoanunstudiedclassofknapsackproblemsforwhichweproposeecient solutionapproaches. Theremainderofthischapterisorganizedasfollows.Section7.2presentsthe optimizationmodelwhichweconsider.Sections7.3and7.4developanexactalgorithmfor aspecialcaseofthemodelwithxedcustomerdemand.Then,Section7.5considersthe variantofthemodelwithexibledemandandderivesanimportantequivalentformulation oftheproblemforwhichanexactalgorithmisderived.Section7.7providesdetailsof theimplementationofourbranch-and-pricealgorithm.Finally,Section7.8discussesa computationalstudyofourapproach. 7.2ModelFormulation Weconsiderasetofcustomers J .Eachcustomer'sdemand j j 2J mustbe satisedbyasinglefacility i i 2I .Ifcustomer j j 2J isassignedtofacility i i 2I ,thenaxedprot, p ij ,isaccruedandaxedamountofcapacity a ij isconsumed. Furthermore,thecorrespondingcustomerdemandfulllmentlevelmustbeselectedfrom theinterval[ ` ij ;u ij ]andanadditionalprot r ij isaccruedperunitofdemandfulllment. Eachfacility i 2I hascapacity b i i 2I .Inadditiontotheseconsiderations,wehavea setofcustomertypes Q ,whereeachcustomerisassociatedwithasingletype.Customers oftype q q 2Q belongtotheset J q .Therstdistinguishingcharacteristicofthismodel isdenedbythemannerinwhichresourceconsumptionmaybeshared.If any customer oftype q q 2Q isassignedtofacility i ,axedamountofresource, f iq ,isconsumed. Furthermore,thecollectivecapacityconsumedbycustomersoftype q islimitedby g q q 2Q .Theobjectiveistodeterminetheassignmentofcustomerstofacilities,aswellas thecorrespondingcustomersdemandfulllmentlevels,inordertomaximizetotalprot, whilesatisfyingthecapacityconstraintsofthefacilitiesandtheindividualcustomertypes. Themodelweconsideristhengivenby maximize X i 2I X j 2J p ij x ij + r ij v ij 134
PAGE 135
subjecttoFASR X q 2Q f iq max j 2J q x ij + X j 2J v ij b i i 2I {1 X i 2I X j 2J q v ij g q q 2Q {2 X i 2I x ij =1 j 2J {3 ` ij x ij v ij u ij x ij i 2I ; j 2J {4 x ij 2f 0 ; 1 g i 2I ;j 2J {5 whichwerefertoasthe FlexibleDemandResourceAllocationProblemwithShared ResourceConsiderations .Constraints7{1requirethatthecapacityvariableandshared consumedbycustomersassignedtofacility i isnogreaterthan b i i 2I .Furthermore, constraints7{2ensurethatdemandsatisedforcustomersoftype q doesnotexceed g q q 2Q .Constraints7{3an7{4aretheassignmentandexibilityconstraints introducedinChapter3.Forsakeofsimplicity,andwithoutlossofgenerality,we've assumedthattheindividualcustomerxedcapacityconsumptions a ij i 2I ; j 2J areincludedinthedemandlevelbounds.Thatis,both ` ij and u ij areincreasedbythe amount a ij i 2I ; j 2J .Thisimpliesthatthexedprotparameters a ij arereducedby theamount r ij a ij i 2I ; j 2J NoticethatFASRisamixedintegerprogramwithasetofnon-linearconstraintsand linearobjectivefunction.Thenon-linearconstraints,7{1canbelinearizedasshownin thefollowingformulation. maximize X i 2I X j 2J p ij x ij + r ij v ij subjecttoFASR 0 X q 2Q f iq s iq + X j 2J v ij b i i 2I 135
PAGE 136
X i 2I X j 2J q v ij g q q 2Q X i 2I x ij =1 j 2J ` ij x ij v ij u ij x ij i 2I ; j 2J s iq x ij i 2I ; j 2J q ; q 2Q {6 x ij 2f 0 ; 1 g i 2I ; j 2J Theconstraints7{6ensurethatifacustomeroftype q q 2Q isassignedtofacility i i 2I ,thentheaxedamountofresource, f iq ,isconsumed.Itshouldbenoted thatthealternativeformulationdoesnotpreventsomepercentageofthe f iq unitsof resource i frombeingconsumed,evenwhennocustomersoftype q areassignedtofacility i i 2I ; q 2Q .Inotherwords,noconstraintprevents s iq i 2I ; q 2Q frombeing greaterthanzeroevenwhen x ij =0forall j 2J q .Itisinterestingtonotethatthis scenarioarisesonlyiftheoptimalsolutionissuchthatiallcustomersassignedtofacility i havetheirdemandsatisedattheirupperbounds,oriinocustomersareassignedto facility i i 2I inanoptimalsolution.However,ineachofthesecases,theobjective valueandcorrespondingdecisionvariablesarestilloptimaltotheoriginalFASRmodel. ThereforeFASR 0 isanequivalentrepresentationofFASR. Thepurposeofthischapteristodevelopabranch-and-pricealgorithmforFASR. Asstatedinpreviouschapters,thesuccessofabranch-and-priceapproachreliesstrongly ontheabilitytoeectivelysolvethecorrespondingpricingproblem.Tothisend,we focusmuchofourattentionondevelopingecientsolutionmethodsfortheresulting pricingproblem.Thedevelopmentoftheseapproachesismosteasilypresentedbyrst consideringaspecialcaseofFASR.Therefore,inSection7.3weinitiallydevelopan exactsolutionprocedureforthenon-exiblevariantofFASRinwhich ` ij = u ij = a ij i 2I ; j 2J .WerefertothisspecialcaseoftheFASRastheResourceConstrained AssignmentProblemwithSharedResourceConsumptionRCAS. 136
PAGE 137
maximize X i 2I X j 2J p ij x ij subjecttoRCAS X q 2Q f iq s iq + X j 2J a ij x ij b i i 2I {7 X i 2I X j 2J q a ij x ij g q q 2Q X i 2I x ij =1 j 2J s iq x ij i 2I ; j 2J q ; q 2Q x ij 2f 0 ; 1 g i 2I ; j 2J where p j = p j + r j a j j 2J 7.3ExactAlgorithmforRCAS Inthissectionweproposeabranch-and-pricealgorithmforRCASbasedonan interestingreformulationoftheproblem.Firstwewillneedthefollowingdenitionsand terminology. x d i isabinaryvectorwithelements x d ij j 2 J representingthe d th subsetof customersthatcanbefeasiblyassignedtofacility i withrespecttoconstraint 7{7; D i isthetotalnumberofsubsetsofcustomersthatcanbeassignedtofacility i ; d i isabinaryvariablewithvalue1ifthe d th subsetassociatedwithfacility i isused, and0otherwise; i x d i = P j 2J p ij x d ij ; d =1 :::D i ; i 2I q i x d i = P j 2 T q a d ij x d ij d =1 :::D i ; i 2I ; q 2Q Themasterprobleminthiscasecanbewrittenas maximize X i 2I D i X d =1 i x d i d i 137
PAGE 138
subjecttoMP X i 2I D i X d =1 q i x d i d i g q q 2Q {8 X i 2I D i X d =1 x d ij d i =1 j 2J {9 D i X d =1 d i =1 i 2I {10 d i 2f 0 ; 1 g d =1 ;:::;D i ; i 2I Nextwedenethefollowingnotation: q q 2Q arethedualvariablesassociatedwiththecustomersetcapacity constraint,7{8inMP; j j 2J arethedualvariablesassociatedwithconstraints,7{9inMP; i i 2I arethedualvariablesassociatedwiththeconvexityconstraints,7{10,in MP. Thepricingproblemassociatedwithresource i i =1 ;:::; I isnowwrittenas maximize X j 2J )]TJ/F15 11.9552 Tf 6.466 -9.684 Td [( p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j a ij x j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i subjecttoPP i X q 2Q f q max j 2 T q f x j g + X j 2J a ij x j b i x j 2f 0 ; 1 g j 2J where q j isthecustomersettowhichcustomer j j 2J belongs.Aneectivemethod forsolvingthepricingproblemisinstrumentalindevelopinganeectivebranch-and-price procedure.Therefore,Section7.4studiesaclassofoptimizationproblemsthatincludes PP i 138
PAGE 139
7.4SharedConsumptionKnapsackProblem Inthissection,wedevelopbothheuristicandexactapproachesforsolvingthe followingclassofknapsackproblems. maximize X j 2J p j x j subjecttoSKP X q 2Q f q s q + X j 2J a j x j b {11 s q x j j 2J q ; q 2Q {12 x j 2f 0 ; 1 g j 2J : {13 s q 2f 0 ; 1 g q 2Q : {14 InSKP,thebinaryrestrictions7{14areclearlyredundant.Therefore,theyarenot includedinthefollowingrelaxationofSKP,whichwerefertoasSKPR, maximize X j 2J p j x j subjecttoSKPR X q 2Q f q s q + X j 2J a j x j b s q x j j 2J q ; q 2Q x j 2 [0 ; 1] j 2J : {15 Intheaboverelaxation,werelaxthebinaryrestrictionsontheassignmentvariables x j j 2J byreplacing7{13with7{15.Beforecontinuing,itisworthwhiletonote thatthereclearlyexistsanoptimalsolutiontobothSKPandSKPRforwhich x SKP j =0 and x SKPR j =0if p j 0.Therefore,thefollowingassumptionholdswithoutlossof generality. 139
PAGE 140
Assumption2. Forall j 2J p j > 0 TobegintomotivateanapproachtosolveSKP,consideranequivalentnon-linear formulationofSKP,SKP 0 maximize X j 2J p j y j subjecttoSKP 0 X q 2Q 8 < : f q + X j 2J q a j y j 9 = ; s q b {16 s q y j j 2J q ; q 2Q y j 2f 0 ; 1 g j 2J s q 2f 0 ; 1 g q 2Q Thealternativedecisionvariables, y j j 2J areintroducedtodistinguishbetweenthe assignmentvariablesinthealternativepresentationsofSKP.Clearly,inthecaseofSKP andSKP 0 ,thereexistsanoptimalsolutioninwhich x j = y j j 2J .However,this relationshipdoesnotnecessarilyholdwhencomparingtheoptimalassignmentvalues x j j 2J obtainedinSKPRtothevalues y j j 2J foundinthefollowingrelaxationof SKP 0 ,whichwerefertoasSKPR 0 maximize X j 2J p j y j subjecttoSKPR 0 X q 2Q 8 < : f q + X j 2J q a j y j 9 = ; s q b s q y j j 2J q ; q 2Q y j 2 [0 ; 1] j 2J s q 2f 0 ; 1 g q 2Q : 140
PAGE 141
NotethatinSKPR 0 thebinaryrestrictionsontheassignmentvariables y j j 2J arerelaxed.However,thebinaryrestrictionsonthevariables s q q 2Q remain. Interestingly,SKPR 0 canbereformulatedasaKPEIstudiedinChapter4.Inthis equivalentformulation,exibilityvariables w q q 2Q representthecollectiveamountof capacityconsumedbyallcustomersoftype q q 2Q .Thisequivalentformulationcanbe writtenas maximize X q 2Q ~ r q w q subjecttoKPEI X q 2Q w q b {17 w q u 0 q s q q 2Q {18 w q ` 0 q s q q 2Q {19 s q 2f 0 ; 1 g q 2Q {20 where ` 0 q = f q and u 0 q = f q + P j 2J q a j q 2Q ,and~ r q w q isanoptimalsolutiontothe followingparametricoptimizationproblem,CKP, maximize X j 2J p j y j subjecttoCKP f q + X j 2J q a j y j w q y j 2 [0 ; 1] j 2J q ; if w q > 0,and0otherwise,foranyfeasible w q q 2Q toKPEI.Notethat,without lossofgenerality,wecanalsolet ` 0 q =min f f q ;b g and u 0 q =min n f q + P j 2J q a j ;b o q 2Q .Forconvenience,weretainthislattersetofdenitionsthroughouttheremainder 141
PAGE 142
ofthechapter.Now,noticethatCKPisthecontinuousknapsackproblem.Therefore, thefunction r q w q isreadilyobtained.First,let S q bethesetofcustomers j 2J q with y CKP j > 0inanoptimalsolutiontoCKPwith w q = u 0 q .Itshouldbenotedthatthereisat mostone j 2J q forwith0 > > > < > > > > : =0 w q =0 = 0
PAGE 143
BoththeexactandheuristicmethodologiestosolveSKParemotivatedbya relaxationofKPEIstudiedinChapter4denotedbyRKP.Forthespeciallystructured revenuefunctionsinthischapter,werefertothisrelaxationasKPEIR. maximize X q 2Q q w q subjecttoKPEIR X q 2Q w q b {21 w q 0 q 2Q : Thefunction q q 2Q isthenon-decreasingconcaveenvelopeencompassingtheorigin, thefunction~ r q andthepoint b; ~ r q u 0 q .Theprotfunctions, q w q q 2Q ,aredened inthefollowinglemma.First,let k q =inf k q =0 ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1: P k q k 0 =0 p ^ | qk 0 f q + P k q k 0 =0 a ^ | qk 0 p ^ | k q +1 a ^ | k q +1 {22 bethesmallestindexforwhichthecollectiveperunitprotforcustomersassociatedwith k q k q isnolessthantheperunitprotassociatedwithcustomer^ | k q +1 with p ^ | j ~ S q j a ^ | j ~ S q j =0. Lemma7. InKPEIR,thefunctions q q 2Q aregivenby q w q = 8 > > > > < > > > > : = ~ p k q ~ a k q w q 0 w q ~ a k q =~ r q w q ~ a k q
PAGE 144
q w q = 8 > > > > > < > > > > > : = ~ r q q q w j 0 w j j =~ r q w q q
PAGE 145
~ a k q ; ~ r q ~ a k q q 2Q .Inaddition,let J 0 q = n ^ | k q 2 ~ S q : k q = k q ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 o q 2Q ThenKPEIRisequivalentlyrepresentedbyKPEIR 0 maximize X j 2J o ^ p j ^ a j w j + X q 2Q X j 2J 0 q ^ p j ^ a j w j subjecttoKPEIR 0 w j 2 [0 ; ^ a j ] j 2J o {24 w j 2 [0 ; ^ a j ] j 2J 0 w j 0 j 2J o w j 0 j 2J o where ^ p j = 8 > > < > > : =~ p k ~ q j if j 2J o = p j if j 2J 0 {25 and ^ a j = 8 > > < > > : =~ a k ~ q j if j 2J o = a j if j 2J 0 : {26 and~ q j isthecustomertypethataparticular`dummycustomer' j 2J o represents. Furthermore,theparameter a j representstheamountofcapacityconsumedandprot accruedbycustomer j intheoptimalsolutiontotheparametricoptimizationproblem CKPwith w q = u 0 q .Again,thereisatmostonecustomerwithineachtypeforwhich a j 6 = a j and p j 6 = p j : {27 145
PAGE 146
Then,givenanoptimalsolutiontoKPEIR 0 ,theoptimalsolutiontoKPEIRisgivenby w KPEIR q = 0 @ X j 2J 0 q w KPEIR 0 j + w KPEIR 0 | q 1 A q 2Q where | q = f j 2J o : q j = q g Clearly,KPEIR 0 issolvablebyconsideringcustomers j 2 J o [ [ q 2Q J 0 q in non-increasingorderof ^ p j ^ a j .Therefore,wearepreparedtoformallyprovideanalgorithm forKPEIR. KPEIRAlgorithm Step0. Set w KPEIR q =0for q 2Q .Find k q andestablishsets J o J 0 q q 2Q and ~ J = J o [ [ q 2Q J 0 q Step1. Sortcustomers j 2 ~ J innon-increasingorderof ^ p j ^ a j Step2. Let^ | betherstcustomerin ~ J and^ q = q 2Q :^ | 2J 0 q or~ q ^ | = q theset associatedwith^ | .Set w KPEIR ^ q = w KPEIR ^ q +min f b; ^ a ^ | g b = b )]TJ/F15 11.9552 Tf 11.955 0 Td [(min f b; ^ a ^ | g : Set ~ J = ~ Jnf ^ | g .If b =0or ~ J =,STOP,elserepeatStep2. UsingthisalgorithmwecanecientlysolveKPEIR.Theimportantstructural propertyofKPEIRfromLemma2inChapter4motivatesbothaheuristicandcustomized branch-and-boundtosolveSKP.Thisresultisrestatedinthefollowinglemma. Lemma8. Anoptimalsolutionto KPEIR existswithatmostonecustomertype q such that i w q <` 0 q ,or ii q w q > ~ r q w q and ` 0 q w q u 0 q Proof. SeeproofofLemma2. 146
PAGE 147
TounderstandhowKPEIRanditsstructureallowustoeectivelysolveSKP, considertherelationshipbetweenSKPRandKPEIRgiveninthefollowingtheorem. Theorem9. SKPRandKPEIRareequivalent. Proof. First,let w KPEIR beanoptimalsolutiontoKPEIR.Considerconstructingafeasible solutiontoSKPRinthefollowingmanner,forall q q 2Q If w KPEIR q =0, s SKPR q =0 x SKPR j =0 j 2J q : Elseif0
PAGE 148
Since P q 2Q w KPEIR q b ,clearly X q 2Q f q s SKPR q + X j 2J a j x SKPR j b: Inaddition, s SKPR q x SKPR j j 2J q ; q 2Q and x SKPR j 2 [0 ; 1] j 2J : Therefore,thefeasiblesolution w KPEIR toKPEIRequatestoafeasiblesolutionto SKPR.Likewise,itiseasytoseethatafeasiblesolutiontoSKPRcanbeconstructedfrom afeasiblesolutiontoSKPRbysettingtheexibilityvariable, w KPEIR q ,equaltothetotal capacityconsumedbycustomersoftype q intheoptimalsolutionsolution x SKPR ;s SKPR Lastly,bythemannerinwhichthefunction q q 2Q isdenedabove,asolutionto KPEIR,withobjectivefunction z KPEIR canbeconvertedtoasolutionwithanequivalent objective z SKPR andviceversa. TheequivalenceofSKPRandKPEIR,alongwiththestructuralpropertyinTheorem 9revealagreatdealaboutthestructureofSKPR.Beforecontinuing,itwillbeusefulto introducethefollowingsets.Let P = f q :0
PAGE 149
Thefollowinglemmaboundsthenumberoffractionalsharedresourceconsumption variables s SKPR q q 2Q Lemma9. AnoptimalsolutiontoSKPR x SKPR ;s SKPR existsforwhich jPj 1 : Proof. FromtheproofofTheorem9,afractional s q q 2Q onlyoccurswhen 0
PAGE 150
forsome ^ k q k q ,thenthiscorrespondstoasolutioninwhichasinglecustomer, | ^ k q ,is fractionallyincluded.Inaddition,fromtheproofofTheorem9,ifthisoccurswith 0
PAGE 151
Furthermore,thefulllmentlevels v d ij j 2J correspondto an extremepoint solutiontothefollowinglinearprogram,solvedwithrespecttotheassignments x d i maximize X j 2J r ij v ij + C d i subjecttoSPv i x d i X j 2J v ij b d i {30 ` ij x d ij v ij u ij x d ij j 2J {31 where C d i = P j 2J p ij x d ij and b d i = b i )]TJ/F26 11.9552 Tf 11.955 8.966 Td [(P q 2Q f q max j 2J q x d ij ; S i isthenumberof unique subsetsofcustomersthatcanbeassignedtofacility i whilesatisfying7{29; D i isthesetofindices, d ,forallcolumnsascharacterizedintherstbullet associatedwithfacility i ; D i ispartitionedintosets D is suchthat x d i = x d 0 i whenever d;d 0 2D is and x d i 6 = x d 0 i whenever d 2D is and d 0 2D is 0 with s 6 = s 0 ; s i isabinaryvariablewithvalue1ifthe s th subsetofassignmentsassociatedwith facility i isusedand0otherwise; d i isacontinuousvariablewithvaluein[0 ; 1]representingtheproportionofthe values )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(x d i ;v d i includedinthesolutiontoMP-F; i x d i ;v d i is P j 2J )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(p ij x d ij + r ij v d ij ; q i x d i ;v d i = P j 2J q v d ij Formally,themasterproblem,MP-F,iswrittenasfollows maximize X i 2I X d 2D i i x d i ;v d i d i subjecttoMP-F X i 2I X d 2D i q i x d i ;v d i d i g q q 2Q {32 X i 2I X d 2D i x d ij d i =1 j 2J {33 151
PAGE 152
X d 2D is d i = s i s =1 ;:::;S i ; i 2I {34 X s 2S i s i =1 i 2I {35 d i 0 d 2 D i ; i 2I {36 s i 2f 0 ; 1 g s =1 :::;S i ; i 2I : {37 ThefollowingtheoremensuresthatMP-FisavalidrepresentationofFASR. Theorem10. IntheoptimizationproblemMP-F, iThenumberofcolumns, )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(x d i ;v d i d 2D i isnite; iiMP-FandFASRareequivalent. Proof. Fori,notethatthedimensionofthefeasiblesubsetsofcustomersassociatedwith anyfacilityisclearlyexponential.Furthermore,recallthatMP-Fonlyconsiderscolumns inwhichthedemandfulllmentlevelsforaxedvectorofassignments, x d i ,correspondto anextremepointsolutiontoSPv i x d i .Sincethenumberofextremepointsinthefeasible regionofSPv i isnite,thenumberofcolumnsinMP-Fthatcorrespondtothesameset ofcustomerassignmentstoaparticularfacilityisniteaswell.Thus,sincethenumberof facilitiesinMP-Fisxed,thenumberofcolumnsincludedinMP-Fisnite. Forii:Let ; beasolutiontoMP-F.Furthermore,let x F ;v F beasolutionin termsofdecisionvariablesinFASR.First,ifweset x F ij = X d 2D i d i x d ij i 2I ; j 2J {38 thenby7{33{7{37, X i 2I x F ij =1 i 2I ; j 2J ; {39 satisfyingassignmentconstraint7{3.Nowset v F ij = X d 2D i d i v d ij i 2I ; j 2J : {40 152
PAGE 153
Recallthat,bydenition, x d i v d i isifeasibletobothfacilitycapacityconstraint7{1 andexibilityconstraint7{4andii v d i isanextremepointsolutiontothefeasible regionofSPv i x d i .Since v F ij isrepresentedasaconvexcombinationoftheseextreme points, x F v F isfeasibleto7{1and7{4,aswell.Moreover,by7{32wehavethat X j 2J q v F ij g q i 2I ; q 2Q ; {41 satisfyingcustomertypecapacityconstraint7{2.Lastly, X i 2I X d 2D i i x d i ;v d i d i = X i 2I X d 2D i X j 2J )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(p ij x d ij + r ij v d ij d i {42 = X i 2I X j 2J p ij X d 2D i d i x d ij + r ij X d 2D i d i v d ij {43 = X i 2I X j 2J )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p ij x F ij + r ij v F ij {44 sotheobjectivevalueassociatedwith ; inMP-Fisequivalenttothatofthe constructedsolution x F v F inFASR.Thus,anyfeasiblesolutiontoMP-Fisalso feasibletoFASRandtheirobjectivefunctionvaluesareequivalent. Toshowformulationequivalence,wemustalsobecertainthatanyfeasiblesolution toFASRisalsofeasibletoMP-F,againwithequivalentobjectivevalues.However,given anyfeasiblesolution x F v F ,thedenitionofthecolumnsincludedinMP-Fensures thatwecanrepresentthissolutionasaconvexcombinationofthecolumnscomprising MP-Fbychoosingtheappropriatevaluesfor d i associatedwithcolumnscorresponding tothecustomersubsetsindicatedby x F .Thisimmediatelyensuresthat7{33{7{37 aresatised.Furthermore,since x F v F isfeasibleto7{2,thecustomertypecapacity constraint7{32issatisedaswell.Lastly,byareversepresentationof7{42{7{44, theobjectivevaluesofthesolutionstoeachformulationareequivalent.Thisshowsthe desiredresult. 153
PAGE 154
Inourexactalgorithm,wesolvethelinearrelaxationofMP-Fwhichwedenoteas LPMP-F,ateachnodeofabranch-and-boundtree.However,asdescribedinpartiof theproofofTheorem10,thetotalnumberofcolumnsinMP-Fisexponential.Therefore, wesolveLPMP-FbyaddingcolumnsiterativelytoarestrictedversionofMP-Finwhich asubsetofcolumnsisconsidered.WedenotethisrestrictedrelaxationasLPRMP-F. ItisimportanttonotethatpartiofTheorem10ensuresthatthiscolumngeneration procedurehasniteconvergence.Thederivationoftheso-called pricingproblem ,solved toidentifyattractivecolumns,requiresastudyofthedualofLPRMP-F.However,to simplifythepresentationofthisrelaxationanditscorrespondingdual,rstnoticethat MP-Fcanbeequivalentlyreformulatedas maximize X i 2I X d 2D i i x d i ;v d i d i subjecttoMP-F 0 X i 2I X d 2D i q i x d i ;v d i d i g q q 2Q X i 2I X d 2D i x d ij d i =1 j 2J X d 2D i d i =1 i 2I {45 d i 0 d 2 D i ; i 2I {46 X d 2D is d i 2f 0 ; 1 g s =1 ;:::;S i ; i 2I {47 bysubstituting7{34into7{35andreplacing7{37with7{47.Therefore,the relaxedversionofMP-F,whichincludesonlyarestrictedsetcolumns,LPRMP-F,can bewrittenas maximize X i 2I X d 2 ~ D i i x d i ;v d i d i 154
PAGE 155
subjecttoLPRMP-F X i 2I X d 2 ~ D i q i x d i ;v d i d i g q q 2Q {48 X i 2I X d 2 ~ D i x d ij d i =1 j 2J {49 X d 2 ~ D i d i =1 i 2I {50 d i 0 d 2 ~ D i ; i 2I {51 X d 2 ~ D is d i 1 s =1 ;:::; ~ S i ; i 2I {52 NoticethatinLPRMP-F,thebinaryrestriction7{47isrelaxedto7{52andthe sets ~ D i ~ D is and ~ S i containtheindicesofthesubsetofcolumnsbeingconsideredatany particulariterationofourcolumngenerationprocedure.Furthermore,notethatbecause of7{50,constraint7{52isredundantinLPRMP-F.Thus,thedualofLPRMP-F iscorrectlydenedwithrespecttoconstraints7{48{7{51only.Thepurposeofthe pricingproblemistoidentifyaviolatedconstraintintheinthefollowingoptimization problem,DRMP-F, minimize X q 2Q g q q + X j 2J j + X i 2I i subjecttoDRMP-F X q 2Q q i x d i ;v d i q + X j 2J x d ij j + i i x d i ;v d i d 2D i ; i 2I {53 q 0 q 2Q {54 j free j 2J {55 i free i 2I {56 where 155
PAGE 156
q q 2Q arethedualvariablesassociatedwiththecustomerssetcapacity constraint,7{48inLPRMP-F; j j 2J arethedualvariablesassociatedwithconstraints,7{49inLPRMP-F; i i 2I arethedualvariablesassociatedwiththeconvexityconstraints,7{50,in LPRMP-F. Ifwesubstitutethedenitionsof q i x d i ;v d i and i x d i ;v d i intothisformulation, DRMP-Fisequivalentlyrepresentedby minimize X q 2Q g q q + X j 2J j + X i 2I i subjecttoDRMP-F X q 2Q X j 2J q v d ij q + X j 2J x d ij j + i X j 2J )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p ij x d ij + r ij v d ij d 2D i ; i 2I {57 q 0 q 2Q {58 j free j 2J {59 i free i 2I {60 Ourpricingproblemseekstoidentifyapairofvectors x i ;v i ,whichviolates7{57 withithevector x i satisfying7{29andiithecorrespondingdemandfulllmentlevels v i determinedbySPv i .Avector x i ;v i violates7{57if X j 2J [ p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x ij + r ij v ij ] )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X q 2Q X j 2J q q v ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( i > 0{61 orequivalently X j 2J p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x ij + )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j v ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i > 0{62 where q j isthetypetowhichcustomer j belongs.Therefore,inlightofiandii,along withdualconstraint7{57,thepricingproblemassociatedwithfacility i i 2I isgiven by maximize X j 2J p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x j + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j v j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i 156
PAGE 157
subjecttoPP i -F X q 2Q f iq max j 2 T q f x j g + X j 2J v j b i ` ij x j v j u ij x j j 2J x j 2f 0 ; 1 g j 2J : InthefollowingsectionwestudytheclassofproblemsthatincludesPP i -Fby extendingtheresultsofSection7.4. 7.6SharedConsumptionKnapsackproblemwithFlexibleCustomerDemand Inthissection,westudythefollowingclassofknapsackproblems. maximize X j 2J p j x j + r j v j subjecttoSKFP X q 2Q f q s q + X j 2J v j b v j u j x j j 2J {63 v j ` j x j j 2J {64 s q x j j 2J q ; q 2Q x j 2f 0 ; 1 g j 2J : s q 2f 0 ; 1 g q 2Q : whoselinearrelaxationisgivenby maximize X j 2J p j x j + r j v j subjecttoSKFPR X q 2Q f q s q + X j 2J v j b 157
PAGE 158
v j u j x j j 2J v j ` j x j j 2J s q x j j 2J q ; q 2Q x j 2 [0 ; 1] j 2J : AsintheSKP 0 ,thenon-linearrepresentationofSKFP,SKFP 0 ,isgivenby maximize X j 2J p j x j + r j j subjecttoSKFP 0 X q 2Q 0 @ f q + X j 2J q j 1 A s q b j u j y j j 2J j ` j y j j 2J s q y j j 2J q ; q 2Q y j 2f 0 ; 1 g j 2J {65 s q 2f 0 ; 1 g q 2Q where j j 2J q areusedtorepresentthedemandfulllmentlevelsand y j j 2J q the assignmentsinSKFP 0 .AswithSKP 0 ,weconsiderthefollowingrelaxationofSKFP 0 maximize X j 2J p j y j + r j j subjecttoSKFPR 0 X q 2Q 0 @ f q + X j 2J q j 1 A s q b j u j y j j 2J 158
PAGE 159
j ` j y j j 2J s q y j j 2J q ; q 2Q y j 2 [0 ; 1] j 2J s q 2f 0 ; 1 g q 2Q : inwhichthebinaryrestrictions7{65arerelaxed.Interestingly,SKFPR 0 canalso bereformulatedastheKPEIpresentedinSection7.4with ` 0 q =min f f q ;b g u 0 q = min n f q + P j 2J q u j ;b o q 2Q and~ r q w q theoptimalsolutiontothefollowing parametricoptimizationproblem,FCKP, maximize X j 2J p j y j + r j j subjecttoFCKP f q + X j 2J q j w q ` j y j j u j y j j 2J q y j 2 [0 ; 1] j 2J q : Tomoreexplicitlystate~ r q w q ,wedenethefollowingsetsandnotation.Let J )]TJ/F22 7.9701 Tf -1.107 -7.294 Td [(q = f j 2 J q : p j 0 g bethesetofcustomerswithanon-positivexedprotand J + q = f j 2J q : p j > 0 g thecustomerswithapositivexedprot.Furthermore,spliteach j 2 J + into twocustomers, j 1 and j 2 .Thefollowinglemmacharacterizes~ r q w q q 2Q usingthe denitionprovidedinLemma6. Lemma11. ForSKPFR 0 ,thecorrespondingfunction ~ r q q 2Q isgivenbyLemma6 with ^ J q = J )]TJ/F22 7.9701 Tf -2.208 -7.294 Td [(q [J + q usedinplaceof J q and a j 1 = ` j {66 a j 2 = u j )]TJ/F21 11.9552 Tf 11.956 0 Td [(` j {67 159
PAGE 160
p j 1 = p j + r j ` j {68 p j 2 = r j u j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` j {69 for j 1 and j 2 in J + q and a j = u j {70 p j = p j + r j u j {71 for j 2 J )]TJ/F22 7.9701 Tf -1.107 -7.294 Td [(q Proof. NoticethattheFCKPisitselftheKPEIwithalinearobjective.Section4.2.2 ofChapter4providesaseparatealgorithmforthisproblem.Thisalgorithmsplitseach customer j 2 J + intotwoparts.Therstparthascustomerdemandsize j 1 2 [0 ;` j ] andaperunitprotgivenby j 1 = p j ` j + r j .Thesecondparthascustomerdemandsize j 2 2 [0 ;u j )]TJ/F21 11.9552 Tf 12.161 0 Td [(` j ]andaperunitprot j 2 = r j .Customers j 2 J )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(havecustomerdemand size j 2 [0 ;u j ]andaperunitprotgivenby j = p j u j + r j .InChapter4weshowedthat thelinearversionofFCKPcanbereformulatedasaCKPwithcustomers j 2 J )]TJ/F22 7.9701 Tf -1.107 -7.294 Td [(q and j 1 and j 2 in J + q .AsestablishedinSection7.4,thedenitionof~ r q q 2Q inLemma6 containsasolutiontotheCKP.Therefore,byreplacing J q withaset ^ J q thatincludes allcustomersinboth J )]TJ/F22 7.9701 Tf -1.106 -7.294 Td [(q and J + q withthespeciedprotandresourceconsumption parameters,Lemma6denes~ r q q 2Q withtheadditionalconsiderationofexible demand. OurapproacheswillagainbemotivatedbyKPEIRpresentedintheprevioussection. Hereagain,thefunction q q 2Q isanon-decreasingconcaveenvelopeencompassing theorigin,thefunction~ r q describedinLemma11andthepoint b; ~ r q u 0 q .Similartothe resultofLemma11, q w q q 2Q ,fortheexiblevariantoftheproblem,isdenedby Lemma7with ^ J q usedinlieuof J q andparametersforeachcustomer" j 2 ^ J q given by7{66{7{71.Therefore,thealgorithmpresentedintheprevioussectionstillapplies 160
PAGE 161
usingtheparametersgivenby7{66{7{71todene^ p and^ a fortheexpandedcustomer set. Interestingly,thereisarelationshipbetweenKPEIRandSKFPRwhichisgivenin thefollowingtheorem.Notethat ^ J q mayconsistofcustomersin J )]TJ/F22 7.9701 Tf -2.208 -7.294 Td [(q ,or J + q ,orboth. Thatis,uptotwocustomers"in ^ J q maybeassociatedwiththesametruecustomerin SKFPR.Therefore,let | k q bethecustomerinSKFPRtowhichanindividualcustomer" ^ | k q 2 ^ J q corresponds.Furthermore,let x t beavectoroftemporaryassignmentsusedto simplifytherepresentationofthefollowingtheorem.Usingthisnotation,westatethe relationshipbetweenoptimalsolutionstoKPEIRandSKFPR. Theorem11. GivenanoptimalsolutiontoKPEIR w KPEIR with ~ r q w q denedby Lemma11,anoptimalsolutiontoSKFPR x SKFPR v SKFPR s SKFPR isgivenbythe following. If w KPEIR q =0 s SKPR q =0 x SKPR j =0 j 2J q : v SKPR j =0 j 2J q : If 0
PAGE 162
v SKFPR j =0 j : | k q 6 = j 8 k q =0 ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1; q 2Q : Lastly,if ~ a ^ k w KPEIR q < ~ a ^ k +1 forsome ^ k q k q s SKFPR q =1 x t j k q =1 k q =0 ;:::; ^ k q x t j ^ k q +1 = w KPEIR q )]TJ/F15 11.9552 Tf 12.101 0 Td [(~ a ^ k q a j ^ k q +1 x t j k q =0 k q = ^ k q +2 ;:::; j ~ S q j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 x t j =0 j 2f ^ J q = S q g x SKFPR j =max j k q : | k q = j n x t j k q o j 2J q v SKFPR j = x SKFPR j u j j 2fJ )]TJ/F22 7.9701 Tf -2.208 -7.892 Td [(q ~ S q g v SKFPR j = x t | k 0 q ` j + x t | k 00 q u j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` j k 0 q ;k 00 q : | k 00 q = | k 00 q = j and k 0 q
PAGE 163
Therefore,thefeasiblesolution w KPEIR toKPEIRequatestoafeasiblesolution toSKPR.Furthermore,afeasiblesolutiontoKPEIRcanbeconstructedfromafeasible solutiontoSKFPRbysettingtheexibilityvariable, w KPEIR q ,equaltothetotalcapacity consumedbycustomersoftype q intheoptimalsolution x SKFPR ;v SKFPR ;s SKFPR .As inTheorem9,bythemannerinwhichthefunction q q 2Q isdened,asolutionto KPEIR,withobjectivefunction z KPEIR ,canberepresentedasasolutiontoSKFPRwith anequivalentobjective z SKFPR andviceversa. FromtheequivalenceofKPEIRandSKFPRandLemma8,agreatdealof informationregardingthestructureofSKFPRcanbeobtained.Firstnotethata fractional s SKFPR q correspondstoasolutiontoKPEIRwith0
PAGE 164
Therefore,aswithSKPR,anoptimalsolutiontoSKFPRcaneasilybeconvertedtoa feasiblesolutiontoSKFP.Wewilldiscussspecicwaystoobtainafeasiblesolutionfrom theoptimalsolutiontoSKFPRinSection7.7.2.Furthermore,SKFPRcanbesolvedat eachnodeofacustomizedbranch-and-boundproceduretosolveSKFPtooptimality. 7.7Branch-and-PriceAlgorithmImplementation Ourimplementationandcomputationaltestingwillfocussolelyontheexiblevariant oftheproblem,FASR,discussinSection7.5.Theimplementationofourbranch-and-price algorithmispartiallymotivatedbythedemonstratedsuccessofthechoicesmadein Section4.3.However,asaresultoftheuniquecharacteristicsofMP-Fandthepricing problemPP i -F,someadditionalconsiderationsmustbegiventotheimplementationofthe branch-and-pricealgorithmfortheFASR. 7.7.1InitialFeasibleSolution ToensurethatafeasiblesolutionexiststoLPRMP-Fweproposeaslightlymodied two-phaseprocedurefromthatusedtosolveLPRSPinChapter4.Phase1ofour approachisusedtogenerateafeasiblesetofcolumnstoLPRMP-F.Tothisend,we includenonnegativeslackvariablesforeachcustomertypecapacityconstraint7{32 andassignmentconstraint7{33.OurPhase1objectiveisthentominimizethesumof theseslackvariables.TheresultingPhase1problemisthusgivenby minimize X j 2J % j + X q 2Q & q subjecttoRMP-F-Phase1 X i 2I X d 2D i q i x d i ;v d i d i )]TJ/F21 11.9552 Tf 11.955 0 Td [(& q g q q 2Q X i 2I X d 2D i x d ij d i + % j =1 j 2J X d 2D i d i =1 i 2I 164
PAGE 165
d i 0 d 2 D i ; i 2I X d 2D is d i 2f 0 ; 1 g s =1 ;:::;S i ; i 2I WesolvethelinearrelaxationofRMP-F-Phase1usingcolumngeneration.The pricingproblemissimilartoPP i -F,withaslightlymodiedobjective. maximize )]TJ/F26 11.9552 Tf 11.956 11.357 Td [(X j 2J )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( j x j + q j v j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i subjecttoPP i -F-Phase1 X q 2Q s q f iq + X j 2J v j b i v j 2 [ ` ij x j ;u ij x j ] j 2J s q x j j 2J q ; q 2Q x j 2f 0 ; 1 g j 2J s q 2f 0 ; 1 g q 2Q : ThisproblemisanSKFPwith p ij =0and r ij =0 i 2I ; j 2J .Therefore,itcanbe solvedasdiscussedinSection7.6.IftheoptimalvalueofLPRMP-F-Phase1equals0, anyoptimalsolutiontothisproblemisfeasibleforLPRMP-F;otherwise,theproblem instanceisinfeasible.Intheformercaseweusethisfeasiblesolutiontoinitializethe columngenerationprocedureforsolvingLPRMP-F. 7.7.2HeuristicsforPP i -F AsmentionedinSection7.6,theresultofLemma10canbeusedtodevelopvarious heuristicroundingstrategies.Inourimplementation,weconsidertwoalternative strategies. Heuristic1 .IftheoptimalsolutiontothelinearrelaxationofPP i -F,sayLPPP i -F, isindeedfractional,thenweknowthatthefractionalvariablesarelimitedtoasingle customertype.Aroundingproceduresimilartothatcommonlyusedforaknapsack 165
PAGE 166
problemisasfollows.Let q = f q 2Q : jF q j 1 g bethecustomertypeassociatedwith partiallyassignedcustomersinthesolutiontoLPPP i -F.Set x PP i -F-H1 j = x LPPP i -F j v PP i -F-H1 j = v LPPP i -F j j 2fJ = F q g ; x PP i -F-H1 j =0 v PP i -F-H1 j =0 j 2F q ; and s PP i -F-H1 q =max j 2J q x PP i -F-H1 j ; q 2Q : Intheaboveprocedure,allfractionalassignmentvariablesandthecorresponding demandfulllmentlevelsintheLP-relaxationofPP i -Faresettozero. Thenextheuristicusesthefractionalvariablestomakeassumptionsaboutwhich customertypesareincludedinthesolution.Then,asecondaryoptimizationproblem issolvedtodeterminethecorrespondingsetofcustomertoincludeandthesubsequent demandfulllmentlevels. Heuristic2 .Ratherthansimplyremovingallpartiallyassignedcustomers,we canalternativelyattempttofullyincludeapartialsetofthesefractionalcustomers.To accomplishthis,weusethesolutiontoLPPP i -Ftodeterminewhichcustomertypesare includedinourheuristicsolution.Thatis,set s PP i -F-H2 q = s LPPP i -F q q 2Q : Then ~ Q = q 2Q : s PP i -F-H2 q =1 isthesetofcustomertypesincludedintheheuristic solution.Set x PP i -F-H2 j =0 v PP i -F-H2 | =0 j 2J q ; q 2 n Q = ~ Q o : Usingthesetofcustomertypes, ~ Q ,wesolvethefollowingoptimizationproblem maximize X q 2 ~ Q X j 2J q p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( j x j + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( q j v j )]TJ/F21 11.9552 Tf 11.955 0 Td [( i 166
PAGE 167
subjecttoSP-H2 X q 2 ~ Q X j 2J q v j b i )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X q 2 ~ Q f iq ` j x j v j u j x j j 2J q ; q 2 ~ Q x j 2 [0 ; 1] j 2J q ; q 2 ~ Q : SP-H2isaKPEI 0 -RstudiedinChapter4andcanbesolvedveryeciently. Moreover,weknowthereexistsanoptimalsolutiontoSP-H2forwhichthereisatmost onefractionallyassignedcustomer.GiventheoptimalsolutiontoSP-H2,theremaining heuristicsolutionisgivenasfollows.Ifoneexists,let | bethefractionalcustomerincluded intheoptimalsolution x SP-H2 ;v SP-H2 .Set x PP i -F-H2 j = x SP-H2 j v PP i -F-H2 j = v SP-H2 j j 2 0 @ 8 < : [ q 2 ~ Q J q 9 = ; = | 1 A ; x PP i -F-H2 | =0 v PP i -F-H2 | =0 : Ourtestingindicatedthatourheuristicproceduresweremostsuccessfulwhilesolving LPRMP-Fintherootnode.Atsubsequentnodesofthetreewereliedpredominantly ontheexactbranch-and-boundalgorithmtosolvePP i -F.NotethatHeuristic2clearly requiresmorecomputationaleortthanHeuristic1.Therefore,ourimplementationonly utilizesHeuristic2attherootnode.Thatis,intherootnodewerstattempttoidentify anattractivecolumnusingHeuristic1.Ifweareunsuccessful,weconsiderthemore intensiveHeuristic2.However,innon-rootnodes,weonlythesolvethepricingproblems heuristicallyviatheroundingschemeofHeuristic1.Thisimplementationchoicewas showntomostconsistentlyproduceresultsintheleastamountoftime. 7.7.3SolvingLPRMP-F Similartoourdiscussionofthebranch-and-pricealgorithmpresentedinChapter4,at anynodeinourbranch-and-boundtreewemustsolvearelaxationofMP-F.Sinceour 167
PAGE 168
pricingproblemstilldecomposesbyfacility,thereareagain jIj potentialpricingproblems toconsider.OurruleforconsideringthevariouspricingproblemsistakenfromSection 4.3.2ofChapter4.Thatis,ateachiterationofourcolumngenerationprocedurewe solveallpricingproblemsheuristicallyaccordingtotherulesdescribedinSection7.7.2. AllattractivecolumnsareaddedtoLPRMP-Fandthecolumngenerationprocedure continues.Ifnocolumnisfoundtopriceoutviaourheuristics,weorderthepricing problemsinnon-increasingorderoftheobjectivevaluesdeterminedbytheheuristic. Pricingproblemsaresolvedviabranch-and-bounduntileitherasinglecolumnpricesout, oritisdeterminedthatnocolumnpricesout.Toreducetheeectofslowconvergenceas weapproachtheoptimalsolutiontoLPRMP-F,weterminateourcolumngeneration procedurewhenourcurrentLPRMP-Fsolutionvalueisprovablywithin10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 ofthe optimalsolutiontoLPRMP-F.Recallthattheupperboundusedtocalculatethisgap requiressolvingallpricingproblemsexactly.Therefore,weagainonlyupdatetheupper boundaftersolving jIjjJj pricingproblemseitherheuristicallyortooptimality. InChapter4,qualityfeasiblesolutionstotheSPwereobtainedbysolvingRSPas anintegerprogramusingcolumnsgeneratedinsolvingLPRSPattherootnode.We investigatedthisimplementationchoiceforMP-F.Ourtestingshowedthatthetimefor CPLEXtosolvethisMIPwasnotablylargerthaninthecaseofSP.Thisislikelydue tothealternativestructureofMP-FversusSP,specically,theadditionofconstraints 7{32.Moreover,wefoundthatbettersolutionscouldbeobtainedatlowlevelsofthe tree.Therefore,unlikeourimplementationforSP,wedonotsolveanMIPusingthe columnsfoundintherootnodeofthesearchtree. 7.7.4NodeandVariableSelection OurnodeselectionruleismotivatedbythesuccessoftheimplementationinChapter 4.Weinitiallysearchthetreeusingadepth-rstrule.OnceafeasiblesolutiontoMP-F isobtained,weexplorethetreeusingabest-boundrule.Thisnodeselectionpolicyisalso 168
PAGE 169
implementedinthebranch-and-boundprocedureusedtosolveourpricingproblemto optimality. Again,ratherthanbranchingonthe valuesinLPRMP-Fwebranchon x variables thathaveavalueof1inacolumnassociatedwithafractional .Thechoiceof x to branchonisbasedonthedegreeoffractionalityofeachvariableinthesolutionto LPRMP-F.Weexploredbothleastfractionali.e.,thatvariablewhichisclosestto0or 1,wheretiesarebrokenarbitrarilyandmostfractionali.e.,thatvariablewhichisclosest to0.5.Whilethedierenceinperformanceforthetwoapproacheswasveryslight,the mostfractionalruleisusedinthecomputationalresultsshowninSection7.8. 7.8Computationalstudy Inthissectionweprovideacomputationalstudyofourbranch-and-pricealgorithm. Sincethemotivationbehindthischapteristheexiblevariantoftheproblem,wefocuson themoregeneralFASR.Section7.8.1discussestheinstancegenerationschemechosenfor thisstudy.Then,inSections7.8.2and7.8.3wediscusstheperformanceofouralgorithm versusthecommercialsolverCPLEXonabroadrangeoftestinstances. 7.8.1ExperimentalDesign Inourcomputationaltests,ourmainsetofinstancesconsiders15and30facilities withthenumberofcustomersequalto jJj =2 jIj ,3 jIj ,and5 jIj .Foreachfacility/customer combinationwestudyinstanceswith jQj =3customertypes.Foreachcustomer,we generatetherandomvectorsofxedprotparameters P j andunitrevenues R j from uniformdistributionson[30 ; 50]and[2 ; 5],respectively.Furthermore,thecustomer requirements L j and D j aregeneratedfromuniformdistributionson[75 ; 125],and[15 ; 35], respectively.Here, L j isarandomvectorofcustomerlowerboundsand D j isarandom vectorcontainingvaluesrepresentativeofthedierencebetweenupperandlowerbounds ofacustomer.Wealsogeneratesharedcapacityconsumptionparameterssuchthat f iq = iq jJj {72 169
PAGE 170
whereisanon-negativeparameterthatmeasurestheabsolutemagnitudeofthe sharedresourceconsumptionand iq istherelativemagnitudeofthesharedresource consumptionoftype q forfacility i .Moreover, X q 2Q iq =1 : Fortheseexperiments,weset iq = jJ q j jJj .Ineachofourteststhefacilitycapacitiesare givenby b i = + i jJj i 2I : {73 wheretheparameter i 2 [0 ; 1]isameasureofthefractionofcustomertypesthatcanbe assignedtofacility i i 2I andagain = a E min i 2I A i 1 + L i 1 jIj : {74 Theparameter a measuresthecapacityavailableforvariableconsumption.Inthesetests weconsidertheexibilityleveldeterminedbysetting a =1 : 2.However,weconsider alternativevaluesof i .Lastly,ourcustomertypecapacityrestrictionsaregivenby g q = t E min i 2I A i 1 + L i 1 jJ q j q 2Q {75 where t determinestheexibilityavailableforcustomersoftype q withrespectto capacity g q .Inthesetestsweconsidertheexibilityleveldeterminedbysetting t =1 : 2, aswell. Inourexperimentation,wesoughttocomparetheeectivenessofourbranch-and-price approachagainstthecommercialsolverCPLEX.Eachofourinstanceswasrununtileither asolutionvaluewithin.1%oftheoptimalsolutionwasobtainedoratimelimitofone hourwasreached.Ourtablespresentresultsfor10randomlygeneratedinstancesforeach combinationofparametersettings.Specically,eachtablereports ithenumberofcolumnsgeneratedintherootnodeofthebranch-and-pricealgorithm; 170
PAGE 171
iithetotalnumberofcolumnsgeneratedthroughouttheentirebranch-and-price algorithm; iiithenumberofnodesconsideredinthebranch-and-pricetree; ivtheamountoftimerequiredtosolvetherelaxedmasterproblemattherootnode; vthetotaltimerequiredbythebranch-and-pricealgorithm; vithetotaltimerequiredbytheCPLEX,withthefollowingadditionalinformation whereappropriate: {thesuperscriptindicatestherelativesolutionerrorcalculatedbyusingthe solver'sbestlowerandupperbound, z UB S z LB S ;i.e. error= z UB S )]TJ/F21 11.9552 Tf 11.955 0 Td [(z LB S z UB S 100 : AllexperimentswereperformedonaaPCwitha3.40GHzPentiumIVprocessor and2GBofRAM.Themixed-integerprogrammingproblemsaswellastherelaxed masterproblemsweresolvedusingCPLEX11.2.InSection7.8.2wediscussabasesetof resultsforourbranch-and-pricealgorithm.Then,inSection7.8.3wediscusshowthese resultschangewithdierentinstancegenerationparameters. 7.8.2BaseResults Ourmainsetofresultsconsiderstwosetsofinstances:iwith15facilitiesand numberofcustomersequalto30,45and75andiiwith30facilitiesand60,90,and 150customers.Forbothsetsofinstances,thecustomersareseparatedintothreeequal sizesetsi.e.types.Thismainsetofinstancesgeneratessharedresourceconsumption variablesi.e. f iq i 2I ; q 2Q withmagnitudeparameterequal5.Thefacility capacitiesaregeneratedwith i = : 5.Thatis,wecanexpect50%ofthecustomertypes tobeabletobeassignedtoaparticularfacility.Lastly,aspreviouslystated,theexibility allowancesaresetatthemoderatelevels a =1 : 2and t =1 : 2. Tables7-1and7-2showthatthebranch-and-pricealgorithmsolvesthe15facility instanceswith30and60customersinlesstimethanCPLEX,onaverage.Eachofthe optimalsolutionsforthe15facility/30customerinstancesinTable7-1isobtainedin 171
PAGE 172
lessthan100nodesandthetimetosolvetherootrelaxationislessthan10seconds. TheaveragetimeforCPLEXtosolvetheseinstancesismorethan6timesthatof thebranch-and-pricealgorithm.Interestingly,itisclearthatalargeportionofthe columnsconsideredineachoftheinstancesweregeneratedintherootnodeofthe branch-and-pricealgorithm.Eachofthe15facility/45customerinstancesinTable7-2 wasalsosolvedinlesstimewiththebranch-and-pricealgorithm.However,thenumber ofnodesconsideredandcolumnsgeneratedissubstantiallyhigherthanthatseenin30 customerinstancesinTable7-1.Forthe45customerinstances,Table7-2showsthat morethan1000nodeswereoftenrequiredtoobtainanoptimalsolution.Furthermore, afarhigherpercentageofcolumnswasgeneratedoutsideoftherootnode.Whilethe computationalrequirementsforthebranch-and-pricealgorithmwerenotablyincreasedin Table7-2,itshouldbenotedthatCPLEXfailedtosolve4ofthe10instanceswithinthe allottedhour.Eachofthese4instanceswassolvedtooptimalityviabranch-and-pricein lessthan25minutes.Unfortunately,neitherCPLEXorourbranch-and-pricealgorithm wasabletosolveinstanceswith15facilitiesand75customerswiththeparameters speciedatthebeginningofthissectionwithinthe1hourtimelimit. Tables7-3and7-4considerprobleminstanceswithalargernumberoffacilitiesand customers.Specically,30facilityinstanceswith60and90customersareconsidered. Thesamesetofparametersisusedtogeneratethedemandrequirementsandcapacity limitations.Theperformanceofbranch-and-priceoverCPLEXisevenmoreclearly denedintheseresults.With60customers,Table7-3showsthatbranch-and-pricetakes anaverageoflessthan30secondstosolvetheseinstances,whileCPLEXdoesnotsolve anyoftheinstancestothespeciedtolerancelimitsintheallottedtime.Interestingly, eventhoughtheinstancesaremuchlarger,thetimetosolvetherootnodeproblemisstill lessthan10seconds.Similartowhatwasseenwiththe15facilityinstances,Table7-4 showsagainthatthenumberofcolumnsgeneratedgrowssignicantlyasthecustomers perfacilityisincreasedto3.However,thebranch-and-pricealgorithmisstillabletosolve 172
PAGE 173
all10instancesinanaverageoflessthan11minutes,whileagainCPLEXfailstosolve anyoftheseinstanceswithin1hour. Neitherthebranch-and-pricealgorithm,norCPLEXisabletosolveinstanceswith30 facilities/150customerswithinanhour.Therefore,asistypicalwithbranch-and-price approaches,ourmainresultsinTables7-1{7-4suggestthatouralgorithmismost successfulwithacustomer-to-faciltyratiolessthanorequalto3.Interestingly,while GAPFDandCFLFDwithlinearrevenuefunctionscouldbesolvedecientlybyCPLEX forinstanceswithcustomer-to-faciltyratiosof5ormore,thesameisclearlynottrue forFASR.Thisreinforcesthedicultyofthisclassofproblems.Ofcourse,anumberof dierenttypesofinstancescanbeconsideredusingthedatamodelproposedinSection 7.8.Inthefollowingsection,weprovideafewinsightsintohowinstanceswithdierent characteristicsmaybeeasierormorediculttosolvethanthoseconsideredinthis section. 7.8.3ExtendedResults Inthissection,weconsidervariousalternativestotheparametersusedtogenerate instancesinourmainsetofresults.Whileitisimpractical,ofcourse,toconsiderall variations,thissectionstrivestoprovidesomeinsightintohowalternativeinstancesmay impactthecomputationalrequirementsofbranch-and-priceversusCPLEX.First,recall thatthemagnitudeofthesharedresourceconsumptioni.e.waschosentobe5in ourmainresults.Inourcomputationalstudy,wealsoconsideredmagnitudesof1and 25aswell,withallremainingparametersthesame.With=25,Table7-5suggests theproblemsbecomedramaticallyeasierforbothbranch-and-priceandCPLEX.Thisis perhapsduetothefactthatthesharedresourceconsumptionbecomesthedominating componentoftheproblem,withtheexibledemandrequiredbyindividualcustomers requiringlessconsideration.Morelikely,however,isthatgiventhecombinationof extremelylargemagnitudeswithafairlylargevalue =0 : 5,thefacilitycapacityavailable isabundant.AswesawinChapter5,problemswithsimilarstructuretoFASRaremore 173
PAGE 174
easilysolvedbyCPLEXwithexcesscapacity.Interestingly,whenwastakentobe1,the problemsbecamemucheasierforCPLEXthanbranch-and-price.Ofcourse,with=1, thesharedconsumptioncomponentoftheproblemisdrasticallyminimized.Therefore,the problemresemblesaGAPFDwithadditionalsideconstraints.FromChapter4weknow thatCPLEXoutperformsbranch-and-priceforthisclassofproblemsifrevenuefunctions aretakentobelinear. Anotherproblemcomponentthatcanbechangedisthevalues i i 2I ,which canbeinterpretedastheanticipatedfractionofcustomertypesthatmaybeassigned toanyindividualfacility.Inourmainresultswesetthevaluesof i i 2I at0.5. Wealternativelyconsideredtheimpactofchangingthesevaluesto0.2and0.9.Tables 7-6and7-7provideresultsforinstanceswithmuchtighterfacilitycapacitiesresulting from i =0 : 2 i 2I .WhencomparedwithresultsofTables7-1and7-2,the branch-and-pricealgorithmsolvesthesemorecapacity-restrictedinstancesinlesstime, onaverage.However,thetimerequiredbyCPLEXincreasesnotably.Whiletheaverage timeforbranch-and-pricetosolvethe15facility/45customerinstancesdecreasesbymore than4minutes,CPLEXsolvednoneoftheseinstanceswithinonehour.Theimproved performanceofthebranch-and-pricealgorithmwhen i =0 : 2 i 2I isexpectedsincethe numberoffeasiblecolumnsinMP-Fisdecreased.Alternatively,ifweincrease i i 2I to0.9,CPLEXconsistentlyoutperformsourbranch-and-pricealgorithm.Table7-8shows thatCPLEXrequiresanaverageofonly3secondstosolvethese`looser'instances,while branch-and-pricetakesmorethan2minutes. Thelastproblemcomponentthatweconsideristhecustomertypecapacities, g q q 2Q .Inourbaseresults,thesecapacitiesweregeneratedwithparameter t =1 : 2. InTables7-9{7-12weshowtheresultsofmodifyingthisparameter.First,Tables7-9 and7-10provideresultsforinstanceswithcustomertypecapacitiesgeneratedwith t =1 : 1.Bothbranch-and-priceandCPLEXsolvetheseinstancesinlesstimethan thatrequiredforinstancesgeneratedwith t =1 : 2.However,branch-and-pricestill 174
PAGE 175
solvesthe15facility/30customerinstancesshowninTable7-9morethan5timesfaster thanCPLEX.Thisperformancedierenceisevengreaterforthe15facility/45customer instancesinTable7-9.Fortheseinstances,branch-and-pricerequireslessthanaminute, onaverage,whileCPLEXrequiresmorethan13minutes.Interestingly,ifthecustomer typecapacitiesareunbounded,asisthecasefortheinstancesinTables7-11and7-12,the branch-and-pricealgorithmagainrequireslesstimethanthatneededinthebaseresults, whilethetimerequiredbyCPLEXincreases.Infact,CPLEXfailstosolve7outof10of the45customerinstanceswithinanhour.However,branch-and-priceisabletosolveall 10ofthe45customerinstancesandmorethanhalfofthe75customerinstanceswithin theallottedtime.Itshouldbenotedthattheperformanceofbranch-and-priceinTables 7-11and7-12islikelyrelatedtothechangeinthestructureofMP-Fwhencustomer typecapacitiesareunbounded.Inthiscase,constraints7{32areeectivelyomitted. Importantly,thisdrasticallyreducesthenumberofcolumnsrequiredinMP-F.Without thecomplicatingcustomertypecapacityconstraints,atmostonecolumnneedstobe consideredforagivensubsetofcustomerassignments.Thatis,intheabsenceof7{32,the optimalcustomerdemandfulllmentlevelsi.e. v ij i 2I ; j 2J foraxedsubsetof assignmentscaneasilybedetermined. 7.9ConclusionsandFutureResearch Inthischapterweconsideredaclassofassignmentproblemsthatseparatecustomers intodisjointsets.Customersofthesametypei.e.belongingtothesamesetareassumed tosharecommonproductionrequirements.Theproposedmodelconsiderednon-linear resourceconsumptionattributesamongcustomersofthesametype.Inaddition,further capacityrestrictionslimitedtheresourceconsumptionofallcustomersassociatedwitha particulartype,independentofwhatfacilityisusedtosatisfythedemand.Weproposed anexactbranch-and-pricealgorithmtosolvetheresultingFASRproblembasedona reformulationofourmodelasaset-partitioningrepresentationwithsideconstraints. Thisreformulationrequiresuniquecolumnrepresentationstoaccuratelymodelthe 175
PAGE 176
exibledemandcomponentoftheproblem.Tosolvetheresultingpricingproblem, westudiedaclassofknapsackproblemswithanimportantrelationshiptotheclassof knapsackproblemsstudiedinChapter4.Ourcomputationalstudysuggeststhatthe branch-and-priceapproachproposedinthisworkperformswellincomparisontoCPLEX onalargeassortmentofprobleminstances. Infutureresearch,itmaybeadvantageoustoconsideraslightlyalteredreformulation ofFASR.Specically,inthischapterthedemandfulllmentlevelsassociatedwitha columnweredeterminedbyanextremepointsolutiontothefollowingoptimization problem maximize X j 2J r ij v ij + C d i subjecttoSPv i x d i X j 2J v ij b d i ` ij x d ij v ij u ij x d ij j 2J foragivensubsetofassignments.Recallthat C d i = P j 2J p ij x d ij and b d i = b i )]TJ/F26 11.9552 Tf -405.99 -14.941 Td [(P q 2Q f q max j 2J q x d ij .Theorem10establishedthatusingthiscolumnrepresentationin conjunctionwiththeappropriateconstraintsanddecisionvariablesyieldedanequivalent representationofFASR.Alteringthecolumnsofthemasterproblemsothatdemand fulllmentlevelsweredeterminedby maximize X j 2J r ij v ij + C d i subjecttoSPv i x d i X j 2J v ij b d i ` ij x d ij v ij u ij x d ij j 2J 176
PAGE 177
X j 2J q v ij g q q 2Q {76 with C d i and b d i denedabove,wouldyieldanalternativepricingproblemwithan additionalsetconstraints.Theadditionalconstraints7{76wouldpotentiallyeliminate columnsfromMP-Fthatwouldbeallowedintheprocedureproposedinthischapter.An importantquestionwouldbewhetherconsideringthismorehighlyconstrainedpricing problemwouldyieldatighterlinearrelaxationofMP-F.Ifso,canthepricingproblem besolvedaseciently?Acomputationalstudymayrevealinterestingtradeosbetween implementingtheapproachprovidedinthischapterversusanexactapproachthatutilizes thisslightlymodiedcolumnrepresentation. Figure7-1.Illustrationof r q and q 177
PAGE 178
Table7-1.FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1485611212.28.73.7 242042014.84.91.7 3458536156.912.3513.5 4374574295.147.1111.1 5460735893.222.6226.6 649851532.23.114.0 7395578373.013.345.3 841947178.411.540.3 9542694393.316.314.7 1041746091.93.417.5 Avg446.8559.4254.114.399.3 Table7-2.FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 179722336417.8394.52059.0 2106812051312.225.6133.2 3820272710054.8785.53600.0 : 17% 4891263710437.0791.83600.0 : 30% 511781747938.4107.8535.5 677115732415.2126.02038.0 79261441517.359.6582.6 8108218231157.9120.61480.8 9967286614836.71329.73600.0 : 20% 1086316101739.7141.43600.0 : 20% Avg936.31986.2485.87.7388.22124.2 : 09% 178
PAGE 179
Table7-3.FASR:30facilities,60customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 187913221476.074.63600.0 : 18% 28121101975.744.93600.0 : 12% 3817916254.313.33600.0 : 17% 4881109313110.160.33600.0 : 43% 58331022395.518.33600.0 : 20% 685190998.816.43600.0 : 32% 783388997.311.03600.0 : 26% 8794881174.311.13600.0 : 17% 9798849116.012.33600.0 : 21% 10895957115.914.23600.0 : 32% Avg839.3993.949.66.427.73600.0 : 24% Table7-4.FASR:30facilities,90customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 12939352010123.1305.03600.0 : 44% 2280830993323.2111.23600.0 : 46% 3261427991118.453.23600.0 : 17% 4247626811722.470.63600.0 : 25% 52682366020324.9557.93600.0 : 35% 6290531001721.770.03600.0 : 27% 7333736965128.8152.03600.0 : 21% 82602436347323.71215.03600.0 : 39% 9208631939716.2229.43600.0 : 30% 1031775061108319.33600.03600.0 : 41% Avg2762.63517.2208.622.2636.63600.0 : 32% 179
PAGE 180
Table7-5.FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.5 i 2I ,=25 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 155758192.74.031.9 264868293.14.861.0 356956912.32.423.8 4579893672.017.4266.0 5604874253.515.291.1 6621853592.312.7181.3 768168112.42.415.2 863966032.53.115.2 959499151.2.621.0301.8 10603715111.74.570.1 Avg609.5749.923.62.58.7105.7 Table7-6.FASR:15facilities,30customers,3customertypes, a = t =1 : 2, i =.2 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 131731716.86.887.2 230935199.114.888.2 32863542111.224.0645.2 431031018.88.835.9 5291291110.510.589.5 631935779.816.1739.2 7322339913.517.9404.2 8303303111.911.993.3 9308308111.711.7161.6 10306306110.710.768.4 Avg307.1323.65.210.413.3241.4 180
PAGE 181
Table7-7.FASR:15facilities,45customers,3customertypes, a = t =1 : 2, i =.2 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1594656330.148.13600.0 % 262496410526.9138.63600.0 % 36056811548.373.73600.0 % 45867232374.7215.23600.0 % 56568723333.9119.13600.0 % 65866922531.080.43600.0 % 7616616141.241.33600.0 % 8592653952.388.73600.0 % 965383337212.6292.23600.0 % 105958604936.0148.33600.0 % Avg610.77553058.7124.53600.0 % Table7-8.FASR:15facilities,75customers,3customertypes, a = t =1 : 2, i =.9 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1370643545525.2201.814.2 2360838921320.573.51.3 3221925271316.351.51.4 4341440785522.5171.01.4 5327639124323.1146.71.9 6339842693725.3169.23.6 7390044452725.0150.72.3 8344740574321.9154.21.4 92325328355.14.5108.71.7 10246630782916.975.31.0 Avg3175.93789.53721.1130.33.0 181
PAGE 182
Table7-9.FASR:15facilities,30customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1350448172.48.26.6 240640611.31.32.3 3440504133.710.9227.0 440342934.15.41.3 546348233.54.627.1 639639613.13.18.9 733836331.73.034.9 841743754.86.15.6 9351582109.2.119.636.1 1042242212.12.111.7 Avg398.6446.915.62.96.536.1 Table7-10.FASR:15facilities,45customers,3customertypes, a =1 : 2, t =1 : 1, i =.5 i 2I ,=5 RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 179484677.511.2231.2 2737888196.317.6117.4 3105214695910.666.8618.0 49311399539.752.01283.9 51144175613111.4124.6846.4 610211356315.030.024.9 79801094176.520.2930.4 88091257778.175.43600.0 : 18% 989114891279.077.9275.8 107721409794.258.7274.2 Avg913.11296.3607.853.4820.2 : 018% 182
PAGE 183
Table7-11.FASR:15facilities,30customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 1356356110.910.9618.3 236441657.118.623.6 338738712.92.964.7 435036954.38.3719.7 5376428215.213.3430.3 635136634.110.3206.0 738238218.38.467.1 835237495.29.5750.6 9353431313.616.9906.4 10379415112.17.11011.7 Avg365392.48.85.410.6479.8 Table7-12.FASR:15facilities,45customers,3customertypes, a =1 : 2, i =.5 i 2I =5, g q = 1 q 2Q RootTotalBPRootBPtotalCPLEX ExpColsColsNodestimetimeTime secsecsec 15967244541.2116.03600.0 : 5% 2707741533.444.71278.7 36548142742.8111.73600.0 : 44% 46558273137.987.23600.0 : 85% 56597883348.293.83600.0 : 4% 6572617933.645.83600.0 : 0% 77769112532.769.63473.6 8723723126.626.72461.4 96257983158.8118.23600.0 : 4% 106247683734.583.73600.0 : 79% Avg659.1771.124.439.079.83241.4 : 74% 183
PAGE 184
CHAPTER8 CONCLUSION Inthisdissertationweexplorednumerousvariantsofresourceconstrainedassignment problemsthataccountforreal-worldoperationsdecisions.Ineachoftheproblems considered,theoptimizationmodelsseektoexploitlesserstudiedrelationshipsbetween customersandthemanufacturertoincreasearm'sprot.InChapter3,weintroduced ageneralizationofthecapacitatedfacilitylocationwithsingle-sourcingconstraints.A notablefeatureofthismodelwastheallowanceofexiblecustomerdemand,whichhas receivedlittleattentionintheliterature.Weconsideredvariantsoftheproblemwith andwithouttherequirementthatresourcesmustbeprocuredataxedcosttothe decisionmaker.InChapter4,weprovidedanexactbranch-and-pricealgorithmbasedon areformulationofthemodelthatsolvedbothproblems.Ourapproachrequiredthestudy ofaninterestingclassofknapsackproblemswithexibledemand.Weshowedimportant structuralresultsofarelaxationofthisclassofknapsackproblemsthatledtoanecient solutionapproachforourpricingproblemwithgeneralizedrevenuefunctions.Weoered evenmoreecientalgorithmsforsolvinginstanceswithspeciallystructuredrevenue functionsthatcorrespondtocommonpricingstructures.Thecomputationalstudyofour branch-and-pricealgorithmdemonstratedthevalueofourapproach.Adetaileddiscussion oftheimplementationchoicesthatresultedinreducedsolutiontimeswasprovided. Elementsofthisdiscussionwererelevanttoallxed-chargeproblemssolvedusingcolumn generation. InChapters5and6,wedevelopedheuristicstosolvelarge-scaleinstancesofthe problemvariantswithCFLFDandwithoutGAPFDresourceprocurementdecisions. InChapter5,weproposedaclassofgreedyheuristicsforGAPFDwithlinearrevenue functionsthatwasmotivatedbypropertiesofanoptimalsolutiontothelinearrelaxation ofourmodel.Wepresentedanovelperturbationschemethatguaranteedourclassof heuristicswasasymptoticallyoptimalityunderaverygeneralstochasticmodel.Our 184
PAGE 185
computationalstudydemonstratedthatourheuristicperformedparticularlywellfor instanceswithalargeratioofcustomers-to-facilities.Applyingtheconceptofinstance perturbationindevelopingheuristicsforothercapacitatedassignmentproblemswould beinteresting.Comparingthesuccessofsuchheuristicsonlarge-scaleproblemsversus establishedprocedureswouldbeespeciallyimportantforvalidatingthecontributionof perturbationincaseswhereityieldsstrongperformanceguarantees. InChapter6,wedevelopedalarge-scalesearchheuristicforCFLFDwithlinear revenuefunctions.Ourapproachutilizedthehigh-qualityecientheuristicproposed forGAPFDwithinafacilityneighborhoodsearchtoaddressthecombinedassignment andxed-chargestructureofourunderlyingoptimizationproblem.Wealsoconsidered theadvantagesofdevelopingahybridapproachthatutilizedaso-calledverylarge-scale neighborhoodsearchVLSNmethod.Ourcomputationalresultsindicatedthatour heuristicframeworkwasaneectiveapproachforsolvingCFLFD.Itwouldbeinteresting toapplythisheuristicframeworktootherxed-chargeassignmentproblems.Inaddition, sincetheheuristiccallsforassignmentstobemadeusingasecondaryprocedureinthis case,theGAPFDheuristicitwouldbeinterestingtoconsideradditionalheuristicsand exactapproachesinthisphase. Lastly,Chapter7introducedanadditionalclassofassignmentproblemswith non-linearcapacityconsumptionamongcustomersandcapacityconstraintsthat spannedallavailableresources.Thismodelwasapplicabletoproductionscenarios thatconsiderproductswithsimilarproductionrequirements.Theadditionalcapacity constraintsaccountedforreal-worldlimitationsonhazardemissions,logisticsresources orwarehousespace.Thebranch-and-pricealgorithmdevelopedforthisclassofproblems requiredaninterestingreformulationofourproblemthatincludedcolumnswithaunique representationwhencomparedtothosetypicallyseeninassignmentproblems.The subproblemtobesolvedresultedinastudyofanotherclassofknapsackproblemswith animportantrelationshiptotheknapsackproblemsstudiedinthecaseofCFLFDand 185
PAGE 186
GAPFD.Acomputationalstudydemonstratedtheadvantagesofourapproachovera well-knowncommercialsolver.GeneralizingFASRtoallowforasharedprotassociated witheachcustomertypeisanaturalextension.Furthermore,whilethebranch-and-price approachwasshowntobesuccessfuloninstanceswithlinearrevenuefunctions,theability tosolveinstanceswithgeneralrevenuefunctionsmayresultinagreaterimpact.For thatreason,theclassofknapsackproblemsdenotedbySKFPshouldbestudiedwith additionalnon-linearconsiderations.OfparticularinterestwouldbehowMP-Fmightbe modiedtomaintainitsequivalencewithanon-linearrepresentationofFASR. Lastly,whilethenotionofcustomersetsi.e.typeswasstudiedindepthinthis dissertation,anotherinterestinggeneralizationofCFLFDwouldbetogroupfacilitiesinto disjointsets.Eachofthesesetsmaybelimitedbyitsowncapacityrestriction,inaddition totheindividualcapacityofthefacilitiesbelongingtothesets.Alongwithprocuringeach facility,aprocurementdecisionassociatedwiththesetmustbeconsideredaswell.The additionalconsiderationoffacilitysetsisapplicabletomanyproductionandpersonnel planningscenarios.Forexample,assumethatfacilitiesrepresentindividualmachinesat variousmanufacturinglocations.Eachmachinehasamaximumamountoftimethatit canberuneachday.However,theproductsproducedbythemachinesateachlocation mustbestoredatanon-sitewarehousebeforebeingtransportedtothecentralized distributor.Thespaceavailableineachwarehouseislimitedsuchthatifeachmachineis runforitsmaximumtime,theamountofproductsproducedwillexceedthespaceneed tostorethem.Therefore,anadditionalconstraintonthetotalnumberofhourseachset ofmachinesisproducingisnecessary.Thisverygeneralscenarioisapplicableacrossa varietyofindustries.Therefore,thispotentialclassofproblemsisrichinapplications. Interestingly,itispossiblethattheneighborhoodsearchheuristicusedtosolveCFLFD maybeextendedtothismulti-levelxed-chargeproblem. 186
PAGE 187
APPENDIXA GAPFDASYMPTOTICPROPERTY Lemma5 TheoptimalvaluesofLP 0 andLP 0 jJj arecloseinthesensethat, withprobabilityone, lim jJj!1 1 jJj Z LP 0 jJj jJj =lim jJj!1 1 jJj Z LP 0 jJj : Proof. ThenormalizedoptimalvalueofLP 0 jJj canbeexpressedas 1 jJj Z LP jJj jJj =min 0 jJj ; jJj where jJj ; jJj = 1 jJj X j 2J max i 2I f i;j + X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [( jJj = jJj ;0 )]TJ/F26 11.9552 Tf 11.955 20.443 Td [( X i 2I i jJj : A{1 Wewillshowthatwemayrestrictourselvestovectors inacompactset.First,notethat min 0 jJj ; jJj ; R L + D + P: Furthermore,forany wehave jJj ; = 1 jJj X j 2J max i 2I )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i + u ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(` ij + p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [( X i 2I i 1 jJj X j 2J max i 2I p ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i a ij + r ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( i ` ij + X i 2I i i )]TJ/F21 11.9552 Tf 11.955 0 Td [( X i 2I i R L + P + X i 2I i i )]TJ/F15 11.9552 Tf 18.7 8.087 Td [(1 jJj X j 2J min i 2I i a ij + ` ij )]TJ/F21 11.9552 Tf 11.956 0 Td [( X i 2I i R L + P +min 0 0 ; 0> e =1 X i 2I 0 i i )]TJ/F15 11.9552 Tf 18.7 8.088 Td [(1 jJj X j 2J min i 2I 0 i a ij + ` ij X i 2I i )]TJ/F21 11.9552 Tf 11.956 0 Td [( X i 2I i R L + P + )]TJ/F21 11.9552 Tf 11.955 0 Td [( X i 2I i 187
PAGE 188
withprobabilityoneas jJj!1 ,byRomeijnandPiersma[78,Theorem3.1].Sincethe value Z LP jJj jJj isnonnegative,thefunction jJj ; thusattainsitsminimumonthe compactset= 0: P i 2I i )]TJ/F26 11.9552 Tf 7.314 9.684 Td [( where)-300(= )]TJ/F15 11.9552 Tf 8.032 -6.662 Td [( R L + D + P )]TJ/F21 11.9552 Tf 11.955 0 Td [(R L )]TJ/F21 11.9552 Tf 11.955 0 Td [(P = )]TJ/F21 11.9552 Tf 12.06 0 Td [( withprobabilityoneas jJj!1 Nownotethat jJj ; jJj jJj ; .Bytheconvexityof jJj ;0in and equationA{1itthenfollowsthatthefunction jJj ; jJj alsoattainsitsminimumon withprobabilityoneas jJj!1 .Thismeansthat jJj ; jJj jJj ;0 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 11.9552 Tf 7.314 0 Td [( jJj withprobabilityoneas jJj!1 sothat 1 jJj Z LP jJj min 0 jJj ; jJj min 0 jJj ;0 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 11.9552 Tf 7.314 0 Td [( jJj = 1 jJj Z LP jJj )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 11.9552 Tf 7.314 0 Td [( jJj withprobabilityoneas jJj!1 : Thedesiredresultnowfollowsbyusing5{20. 188
PAGE 189
APPENDIXB CFLFDPRICINGPROBLEMPROPERTY Theorem1 Theoptimizationproblems RKP 0 and KPEI 0 -R areequivalentwhen therevenuefunction r j arelinearforall j 2 ~ J Proof. Wecanrewritetheintervalconstraints4{21asfollows: x j 2 w j u 0 j ; w j ` 0 j j 2J : 4{21 0 Clearly,wewouldliketochoosethevalueof x j assmallaspossiblei.e., x j = w j =u 0 ij if p j 0andaslargeaspossiblei.e., x j =min f 1 ;w j =` 0 j g if p j > 0.Therefore,ifwedene J )]TJ/F15 11.9552 Tf 10.731 -4.339 Td [(= f j 2J : p j 0 g and J + = f j 2J : p j > 0 g ,wecanformulatetheLP-relaxationof theKPEIas maximize X j 2 J )]TJ/F26 11.9552 Tf 8.247 29.243 Td [( r j + p j u 0 j w j + X j 2 J + min r j w j + p j ; r j + p j ` 0 j w j subjecttoKPEI 0 -R X j 2J w j b w j 2 [0 ;u 0 j ] j 2J : Therevenuefunctionsofitems j 2 J + arenolongerlinear,butinsteadaconcavefunction ofthejobsize w j .Eachjob j 2 J + canbesplitintotwoparts.Therstparthasjob size w j 1 2 [0 ;` 0 ij ]andarevenuefunctiongivenby r j + p 0 j ` 0 j w j 1 .Thesecondparthasjob size w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 12.511 0 Td [(` 0 j ]andarevenuefunctiongivenby r j w j 2 .Therefore,KPEI 0 -Rcan alternativelybewrittenas maximize X j 2 J )]TJ/F26 11.9552 Tf 8.247 29.244 Td [( r j + p j u 0 j w j + X j 2 J + r j + p j ` 0 j w j 1 + X j 2 J + r j w j 2 subjecttoKPEI 0 -R 0 X j 2 J )]TJ/F21 11.9552 Tf 8.246 12.387 Td [(w j + X j 2 J + w j 1 + X j 2 J + w j 2 b i 189
PAGE 190
w j 2 [0 ;u 0 j ] j 2 J )]TJ/F21 11.9552 Tf -157.55 -31.833 Td [(w j 1 2 [0 ;` 0 j ] j 2 J + w j 2 2 [0 ;u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j ] j 2 J + : Notethat r ij + p j u 0 j = r j u 0 j + p j u 0 j = j u 0 j u 0 j B{1 and r j + p j ` 0 j = r j ` 0 j + p j ` 0 j = ij ` 0 ij ` 0 ij = ij B{2 and r j = p j + r j u 0 j )]TJ/F15 11.9552 Tf 12.942 0 Td [( p j )]TJ/F21 11.9552 Tf 11.956 0 Td [(r j ` 0 j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j = j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [( j ` 0 j u 0 j )]TJ/F21 11.9552 Tf 11.955 0 Td [(` 0 j = j : B{3 Therefore,theoptimizationproblemKPEI 0 -R 0 ispreciselyRKP 0 presentedinSection 4.2.2,whichyieldsthedesiredresult. 190
PAGE 191
REFERENCES [1]Adlakha,V.,K.Kowalski.2003.Asimpleheuristicforsolvingsmallxed-charge transportationproblems. OMEGA 31 205{211. [2]Ahuja,R.K.,J.Goodstein,A.Mukherjee,J.B.Orlin,D.Sharma.2007.Very large-scaleneighborhoodsearchalgorithmforthecombinedthrough-eet-assignment model. INFORMSJournalonComputing 19 416{428. [3]Ahuja,R.K.,W.Huang,H.E.Romeijn,D.RomeroMorales.2007.Aheuristic approachtothemulti-periodsingle-sourcingproblemwithproductionandinventory capacitiesandperishabilityconstraints. INFORMSJournalonComputing 19 14{26. [4]Ahuja,R.K.,K.Jha,J.B.Orlin,D.Sharma.2007.Verylarge-scaleneighborhood searchforthequadraticassignmentproblem. INFORMSJournalonComputing 19 646{657. [5]Ahuja,R.K.,J.B.Olrin,A.Punnen.2002.Asurveyofverylargescaleneighborhood searchtecnhiques. DiscreteAppliedMathematics 123 75{102. [6]Ahuja,R.K.,J.B.Orlin,S.Pallottino,M.P.Scaparra,M.G.Scutella.2004.A multi-exchangeheuristicforthesingle-sourcecapacitatedfacilitylocationproblem. ManagementScience 50 749{760. [7]Ahuja,R.K.,J.B.Orlin,D.Sharma.2001.Multi-exchangeneighborhoodsearch algorithmsforthecapacitatedminimumspanningtreeproblem. Mathematical Programming 91 71{97. [8]Akinc,U.,M.Khumawala.1977.Anecientbranchandboundalgorithmforthe capacitatedwarehouselocationproblem. ManagementScience 23 585{594. [9]Altay,N.,P.E.RobinsonJr.,K.M.Bretthauer.2008.Ahybridheuristicforthe generalizedassignmentproblem. DiscreteOptimization 190 598{609. [10]Amini,M.M.,M.Racer.1995.Ahybridheuristicforthegeneralizedassignment problem. EuropeanJournalofOperationalResearch 87 343{348. [11]Baker,R.C.,T.L.Urban.1988.Adeterministicinventorymodelwithan inventory-leveldependentdemand. NavalResearchLogistics 35 99{116. [12]Baker,R.C.,T.L.Urban.1988.Adeterministicinventorysystemwithan inventory-level-dependentdemandrate. JournaloftheOperationalResearch Society 39 823{831. [13]Balakrishnan,A.,J.Geunes.2003.Productionplanningwithexibleproduct specications:anapplicationtospecialtysteelmanufacturing. OperationsResearch 51 94{112. 191
PAGE 192
[14]Balakrishnan,A.,M.S.Pangburn,E.Stavrulaki.2004.Stackthemhigh,let'em y":lotsizingpolicieswheninventoriesstimulatedemand. ManagementScience 50 630{644. [15]Barcelo,J.,J.Casanovas.1984.Aheuristiclagrangeanalgorithmforthecapacitated plantlocationproblem. EuropeanJournalofOperationalResearch 15 212{226. [16]Barnhart,C.,E.L.Johnson,G.L.Nemhauser,M.W.P.Savelsbergh,P.H.Vance. 1998.Branch-and-price:columngenerationforsolvinghugeintegerprograms. OperationsResearch 46 316{329. [17]Beasley,J.E.1993.Lagrangianheuristicsforlocationproblems. EuropeanJournalof OperationalResearch 65 383{399. [18]Biller,S.,L.M.A.Chan,D.Simchi-Levi,J.Swann.2005.Dynamicpricingandthe direct-to-customermodelintheautomotiveindustry. ElectronicCommerceJournal 122 584{601. [19]Bretthauer,K.M.,B.Shetty.2002.Nonlinearknapsackproblem-algorithmsand applications. EuropeanJournalofOperationalResearch 138 459{472. [20]Cattrysse,D.G.,M.Salomon,L.N.VanWassenhove.1994.Asetpartitioning heuristicforthegeneralizedassignmentproblem. EuropeanJournalofOperational Research 72 167{174. [21]Ceselli,A.,G.Righini.2005.Abranch-and-pricealgorithmforthecapacitated p-medianproblem. Networks 45 125{142. [22]Chen,Z.-L.,N.G.Hall.2007.Thecoordinationofpricingandschedulingdecisions. TechnicalReport. [23]Cornuejols,G.,R.Sridharan,J.M.Thizy.1991.Acomparisonofheuristicsand relaxationsforthecapacitatedplantlocationproblem. EuropeanJournalof OperationalResearch 50 280{297. [24]Dantzig,G.B.1957.Discretevariableextremumproblems. OperationsResearch 5 266{277. [25]Davis,P.R.,T.L.Ray.1969.Abranchandboundalgorithmforthecapacitated facilitieslocationproblem. NavalResearchLogistics 16 [26]Delmaire,H.,J.A.Diaz,E.Fernandez,M.Ortega.1997.Comparingnewheuristics forthecapacitatedplantlocationproblem.Tech.Rep.DR97/10,Departmentof StatisticsandOperationsResearch,UniversitatPolitecnicadeCatalunya,Barcelona, Spain. [27]Demaire,H.,J.A.Diaz,E.Fernandez,M.Ortega.1999.Reactivegraspandtabu searchbasedheuristicsforthesinglesourcecapacitatedplantlocationproblem. InformationSystemsandOperationsResearch 37 194{225. 192
PAGE 193
[28]Deng,S.,C.A.Yano.2006.Jointproductionandpricingdecisionswithsetupcosts andcapacityconstraints. ManagementScience 52 741{756. [29]Dyer,M.,A.Frieze.1989.Probabilisticanalysisofthemultidimensionalknapsack problem. MathematicsofOperationsResearch 14 162{176. [30]Dyer,M.,A.Frieze.1992.Probabilisticanalysisofthegeneralisedassignment problem. MathematicalProgramming 55 169{181. [31]Ergun,O.,J.B.Orlin,A.Steele-Feldman.2006.Creatingverylargescale neighborhoodsoutofsmalleronesbycompoundingmoves:astudyonthevehicle routingproblem. JournalofHeuristics 12 1381{1231. [32]Erlenkotter,D.1978.Adual-basedprocedureforuncapacitatedfacilitylocation. OperationsResearch 26 992{1009. [33]Federgruen,A.,Y.Zheng.1992.Thejointreplenishmentproblemwithgeneraljoint coststructures. OperationsResearch 40 384{403. [34]Fisher,M.L.2004.Thelagrangianrelaxationmethodforsolvinginteger programmingproblems. ManagementScience 50 1861{1872. [35]Fisher,M.L.,R.Jaikumar,L.VanWassenhove.1986.Amultiplieradjustment methodforthegeneralizedassignmentproblem. ManagementScience 32 1095{1103. [36]Ford,L.R.,D.R.Fulkerson.1956.Solvingthetransportationproblem. Management Science 3 24{32. [37]Freling,R.,H.E.Romeijn,D.RomeroMorales,A.P.M.Wagelmans.2003.A branch-and-pricealgorithmforthemulti-periodsingle-sourcingproblem. Operations Research 51 922{939. [38]Fukasawa,R.,H.Longo,J.Lysgaard,M.P.deAragao,M.Reis,E.Uchoa, R.Werneck.2006.Robustbranch-and-cut-and-priceforthecapacitatedvehicle routingproblem. MathematicalProgramming 106 491{511. [39]Garey,M.R.,D.S.Johnson.1979. AguidetothetheoryofNP-completeness .W.H. Freeman,SanFrancisco. [40]Gavish,B.,H.Pirkul.1991.Algorithmsforthemulti-resourcegeneralized assignmentproblem. ManagementScience 37 695{713. [41]Georion,A.M.,G.W.Graves.1974.Multicommoditydistributionsystemdesignby bendersdecomposition. MathematicalProgramming 2 82{114. [42]Georion,A.M.,R.McBride.1978.Lagrangeanrelaxationappliedtocapacitated facilitylocationproblems. IIETransactions 40{47. 193
PAGE 194
[43]Gerchak,Y.,Y.Wang.1994.Periodic-reviewinventorymodelswith inventory-level-dependentdemand. NavalResearchLogistics 41 99{116. [44]Geunes,J.,H.E.Romeijn,K.Taae.2005.Requirementsplanningwithpricingand orderselectionexibility. OperationsResearch 54 394{401. [45]Ghosh,D.2003.Neighborhoodsearchheuristicsfortheuncapacitatedfacility locationproblem. EuropeanJournalofOperationalResearch 150 150{162. [46]Gilbert,S.M.1999.Coordinationofpricingandmulti-periodprocurementfor constantpricedgoods. EuropeanJournalofOperationalResearch 114 330{337. [47]Gray,P.1971.Exactsolutionofthexed-chargetransportationproblem. Operations Research 19 1529{1538. [48]Hindi,K.S.,K.Pienkosz.1999.Ecientsolutionoflargescale,single-source, capacitatedplantlocationproblems. JournaloftheOperationalResearchSociety 50 268{274. [49]Holmberg,K.,M.Ronnqvist,D.Yuan.1999.Anexactalgorithmforthecapacitated facilitylocationproblemswithsinglesourcing. EuropeanJournalofOperational Research 113 544{559. [50]Horowitz,E.,S.Sahni.1974.Computingpartitionswithapplicationstothe knapsackproblem. JournaloftheAssociationofComputingMachines 21 277{292. [51]Huang,W.,H.E.Romeijn,J.Geunes.2005.Thecontinuous-timesingle-sourcing problemwithcapacityexpansionopportunities. NavalResearchLogistics 52 193{211. [52]Jornsten,K.,M.Nasberg.1986.Anewlagrangianrelaxationapproachtothe generalizedassignmentproblem. EuropeanJournalofOperationalResearch 27 313{323. [53]Klincewicz,J.G.,HananLuss.1986.Alagrangianrelaxationheuristicforcapacitated facilitylocationwithsingle-sourceconstraints. JournaloftheOperationalResearch Society 37 495{500. [54]Kolesar,P.J.1967.Abranchandboundalgorithmfortheknapsackproblem. ManagementScience 13 723{735. [55]Kuhn,H.W.2005.Thehungarianmethodfortheassignmentproblem. Naval ResearchLogistics 52 7{21. [56]Kunreuther,H.,L.Schrage.1973.Jointpricingandinventorydecisionsforconstant priceditems. ManagementScience 19 732{738. 194
PAGE 195
[57]Laguna,M.,J.P.Kelly,J.L.Gonzalez-Verlade,F.Glover.1995.Tabusearchfor themultilevelgeneralizedassignmentproblem. EuropeanJournalofOperational Research 82 176{189. [58]Laporte,G.1992.Thevehicleroutingproblem:anoverviewofexactand approximatealgorithms. EuropeanJournalofOperationalResearch 59 345{358. [59]Lee,J.K.,Y.D.Kim.1996.Searchheuristicsforresourceconstrainedproject scheduling. TheJournalofRisk 47 678{689. [60]Li,C.1995.Schedulingtominimizethetotalresourceconsumptionwithaconstraint onthesumofcompletiontimes. EuropeanJournalofOperationalResearch 80 381{388. [61]Lin,S.,V.W.Kernighan.1973.Aneectiveheuristicalgorithmforthe traveling-salesmanproblem. OperationsResearch 21 498{516. [62]Martello,S.,D.Pisinger,P.Toth.2000.Newtrendsinexactalgorithmsforthe0-1 knapsackproblem. EuropeanJournalofOperationalResearch 123 325{332. [63]Martello,S.,P.Toth.1977.Anupperboundforthezero-oneknapsackproblem andabranchandboundalgorithm. EuropeanJournalofOperationalResearch 1 169{175. [64]Martello,S.,P.Toth.1981.Analgorithmforthegeneralizedassignmentproblem. J.P.Brans,ed., Operationalresearch .North-Holland,589{603. [65]Martello,S.,P.Toth.1990.Dynamicprogrammingandstrongboundsforthe0-1 knapsackproblem. ManagementScience 45 414{424. [66]Martello,S.,P.Toth.1990. KnapsackProblems,AlgorithmsandComputerImplementations .JohnWiley&Sons,NewYork. [67]Martello,S.,P.Toth.1997.Upperboundsandalgorithmsforhard0-1knapsack problems. OperationsResearch 45 768{778. [68]Mazzola,J.B.1986.Generalizedassignmentproblemwithnonlinearcapacity interaction. ManagementScience 35 923{941. [69]Mazzola,J.B.,A.W.Neebe.1986.Resource-constrainedassignmentscheduling. OperationsResearch 34 560{572. [70]Nauss,R.M.1976.Anecientalgorithmforthe0-1knapsackproblem. Management Science 23 27{31. [71]Nauss,R.M.1978.Animprovedalgorithmforthecapacitatedfacilitylocation problem. JournaloftheOperationalResearchSociety 29 1195{1201. [72]Neebe,A.W.,M.R.Rao.1983.Analgorithmforthexed-chargeassigningusersto sourcesproblem. JournaloftheOperationalResearchSociety 34 1107{1113. 195
PAGE 196
[73]Osman,I.H.1995.Heuristicsforthegeneralisedassignmentproblem:simulated annealingandtabusearchapproaches. ORSpektrum 17 211{225. [74]Pigatti,A.,M.PoggideAragao,E.Uchoa.2005.Stabilized branch-and-cut-and-priceforthegeneralizedassignmentproblem. ElectricNotesin DiscreteMathemematics 19 389{395. [75]Pisinger,D.1995.Anexpanding-corealgorithmfortheexact0-1knapsackproblem. EuropeanJournalofOperationalResearch 87 175{187. [76]Ralphs,T.K.,L.Kopman,W.R.Pulleybank,L.E.Trotter.2003.Onthecapacitated vehicleroutingproblem. MathematicalProgramming 94 343{359. [77]Reinelt,G.1991.TSPLIB-atravelingsalesmanproblemlibrary. ORSAJournalon Computing 3 376{384. [78]Romeijn,H.E.,N.Piersma.2000.Aprobabilisticfeasibilityandvalueanalysisofthe generalizedassignmentproblem. JournalofCombinatorialOptimization 3 325{355. [79]Romeijn,H.E.,D.RomeroMorales.2000.Aclassofgreedyalgorithmsforthe generalizedassignmentproblem. DiscreteAppliedMathematics 103 209{235. [80]Romeijn,H.E.,D.RomeroMorales.2001.Generatingexperimentaldataforthe generalizedassignmentproblem. OperationsResearch 49 866{878. [81]Romeijn,H.E.,F.Z.Sargut.2008.Thestochastictransportationproblemwith single-sourcing.Tech.rep.,DepartmentofIndustrialandSystemsEngineering, UniversityofFlorida. [82]Romeijn,H.E.,J.Shu,C.P.Teo.2007.Designingtwo-echelonsupplynetworks. EuropeanJournalofOperationalResearch 178 449{462. [83]RomeroMorales,D.,H.E.Romeijn.2004. HandbookofCombinatorialOptimization vol.5,chap.Thegeneralizedassignmentproblemandextensions.KluwerAcademic Publishers,259{311. [84]Ross,G.T.,R.M.Soland.1975.Abranchandboundalgorithmforthegeneralized assignmentproblem. MathematicalProgramming 8 91{103. [85]Ross,G.T.,R.M.Soland.1977.Modelingfacilitylocationproblemsasgeneralized assignmentproblems. ManagementScience 24 345{357. [86]Roy,T.J.Van.1986.Acrossdecompositionalgorithmforcapacitatedfacility location. OperationsResearch 34 145{163. [87]Sa,G.1969.Branch-and-boundandapproximatesolutiontothecapacitateplant locationproblem. OperationsResearch 17 1005{1016. [88]Savelsbergh,M.P.W.1997.Abranch-and-pricealgorithmforthegeneralized assignmentproblem. OperationsResearch 45 831{841. 196
PAGE 197
[89]Shen,Z.J.,C.Coullard,M.S.Daskin.2003.Ajointlocation-inventorymodel. TransportationScience 37 40{55. [90]Shu,J.,C.P.Teo.2004.Warehouse-retailernetworkdesignproblem. Operations Research 52 396{408. [91]Shubik,M.1962.Incentives,decentralizedcontrol,theassignmentofjointcostsand internalpricing. ManagementScience 8 325{343. [92]Srinivasan,V.,G.L.Thompson.1973.Analgorithmforassigningusestosourcesin aspecialclassoftransportationproblems. OperationsResearch 21 284{295. [93]Sun,M.2006.Solvingtheuncapacitatedfacilitylocationproblemusingtabusearch. Computers&OperationsResearch 33 2563{2589. [94]Talluri,K.T.,G.J.VanRyzin.2004. TheTheoryandPracticeofRevenueManagement .Springer. [95]Thomas,J.1970.Price-productiondecisionswithdeterministicdemand. ManagementScience 16 747{750. [96]Thompson,P.,J.B.Orlin.1989.Thetheoryofcyclictransfers.Tech.Rep.200-89, OperationsResearchCenter,MassachusettsInstituteofTechnology,CambridgeMA. [97]Thompson,P.M.,H.N.Psaraftis.1993.Cycletransferalgorithmsformulti-vehicle routingandschedulingproblems. OperationsResearch 41 935{946. [98]vandenHeuvel,W.,A.P.M.Wagelmans.2006.Apolynomialtimealgorithmfora deterministicjointpricingandinventorymodel. EuropeanJournalofOperational Research 170 463{480. [99]Wang,Y.,Y.Gerchak.2001.Supplychaincoordinationwhendemandisshelf-space dependent. Manufacturing&ServiceOperationsManagement 3 82{87. [100]Wilson,J.M.1997.Ageneticalgorithmforthegeneralisedassignmentproblem. JournaloftheOperationalResearchSociety 48 804{809. [101]Yagiura,M.,T.Ibaraki,F.Glover.1999.Anejectionchainapproachforthe generalizedassignmentproblem.Tech.rep.,DepartmentofAppliedMathematics andPhysics,GraduateSchoolofInformatics,KyotoUniversity. 197
PAGE 198
BIOGRAPHICALSKETCH ChaseRainwaterwasborninMountainHome,AR.HegraduatedfromMountain HomeHighSchoolin1999.HeattendedtheUniversityofArkansas,whereheearned aBachelorofScienceinIndustrialEngineeringinMay2004.Afterhisundergraduate studies,hemarriedCandaceZieleniuk-Rainwaterandenrolledinthegraduateprogramat theUniversityofFlorida.HebeganhisdoctoralstudiesinAugust2005intheIndustrial andSystemsEngineeringDepartment.HeearnedhisDoctorofPhilosophyinindustrial andsystemsengineeringinAugust2009.Followinggraduation,hewilljointhefacultyof theDepartmentofIndustrialEngineeringattheUniversityofArkansas. 198
|
