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Optimizing integrated production, inventory and distribution

University of Florida Institutional Repository

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Iwouldliketothankthepeoplewhohelpedmecompletetheworkcontainedinthisdissertation.ThehelpofmysupervisorsPanosPardalosandEdwinRomeijnwasofgreatvalue.IwouldliketothankPanosPardalosforhistechnicaladvice,encouragementandinsightfulcommentsthroughoutmydissertationwork.IwouldliketothankEdwinRomeijnforworkingcloselywithme.Hisunconditionalsupportinsolvingmanydetailssurroundingthisdissertationandhisvaluablefeedbacksaredeeplyappreciated. IextendmythankstothemembersofmycommitteeSelcukErengucandJosephGeunesfortheirconstructivecriticismconcerningthematerialofthisdissertation.IalsowouldliketoexpressmyappreciationtoallmyfriendsattheISEdepartmentandinGainesvilleforlighteningupmylifeeverydayandmakinggraduateschoolmorefunthanIamsureitissupposedtobe.InparticularIwouldliketothankAdrianaandJorgeJimenez,MirelaandIlirBejleri,EbruandDenizErdo~gmus,PaveenaChaovalitwongse,BayramYildirim,SeviyeYoruk,MaryandKevinTaae,OlgaPerdikaki,HulyaEmir,SergiyButenkoandLihuiBai. IwouldliketoexpressmyspecialthankstomyparentsLeonoraandPerikliDuniandmybrotherDhimitraqDuni.Theirunderstandingandfaithinmeandmycapabilities,theirlove,encouragement,andeternalsupporthavemotivatedmeallthetime.Lastbutnotleast,IwouldliketothankmyhusbandBurakforhislove,patienceandcontinuoussupportthroughoutallmyyearshereattheUniversityofFlorida. iii

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T ABL E O F CONTENTS page ACKNOWLEDGMENTS............................... iii ABSTRACT...................................... vi CHAPTER1INTRODUCTION................................ 1 1.1SupplyChainManagement........................ 1 1.2FrameworkofThisStudy......................... 3 1.3ObjectivesandSummary......................... 6 2MULTI-FACILITYLOT-SIZINGPROBLEM................. 12 2.1Introduction................................ 12 2.2LiteratureReview............................. 13 2.3ProblemDescription........................... 16 2.4ExtendedProblemFormulation..................... 21 2.5Primal-DualBasedAlgorithm...................... 26 2.5.1IntuitiveUnderstandingoftheDualProblem.......... 27 2.5.2DescriptionoftheAlgorithm................... 28 2.5.3RunningTimeoftheAlgorithm................. 30 2.6CuttingPlaneAlgorithm......................... 31 2.6.1ValidInequalities......................... 31 2.6.2SeparationAlgorithm....................... 32 2.6.3FacetsofMulti-FacilityLot-SizingProblem........... 33 2.7DynamicProgrammingBasedHeuristic................. 35 2.7.1Introduction............................ 35 2.7.2DescriptionoftheAlgorithm................... 37 2.7.3RunningTimeoftheAlgorithm................. 40 2.8ComputationalResults.......................... 42 2.9Conclusions................................ 52 3EXTENSIONSOFTHEMULTI-FACILITYLOT-SIZINGPROBLEM... 54 3.1Multi-Commodity,Multi-FacilityLot-SizingProblem......... 54 3.1.1ProblemFormulation....................... 56 3.1.2LinearProgrammingRelaxation................. 59 3.1.3ValidInequalities......................... 61 3.1.4LagrangeanDecompositionHeuristic............... 62 3.1.5OutlineoftheAlgorithm..................... 67 iv

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74 3.1.7ComputationalResults...................... 75 3.2Single-Commodity,Multi-Retailer,Multi-FacilityLot-SizingProblem 82 3.2.1ProblemFormulation....................... 83 3.2.2Primal-DualAlgorithm...................... 86 3.2.3IntuitiveUnderstandingoftheDualProblem.......... 87 3.2.4OutlineofthePrimal-DualAlgorithm.............. 88 3.2.5ComputationalResults...................... 90 3.3MultiFacilityLot-SizingProblemwithFixedChargeTransportationCosts.................................. 93 3.4Conclusions................................ 95 4PRODUCTION-DISTRIBUTIONPROBLEM................. 97 4.1Introduction................................ 97 4.2ProblemFormulation........................... 98 4.3DynamicSlopeScalingProcedure.................... 100 4.3.1Multi-CommodityNetworkFlowProblemwithFixedChargeCostFunction.......................... 100 4.3.2Single-CommodityCase...................... 104 4.3.3Multi-CommodityCase...................... 104 4.3.4Production-DistributionProblem................. 108 4.3.5ExtendedProblemFormulation................. 109 4.4ALagrangeanDecompositionProcedure................ 111 4.5ComputationalResults.......................... 114 4.6Conclusions................................ 126 5CONCLUDINGREMARKS........................... 127 APPENDICES..................................... 130 REFERENCES..................................... 138 BIOGRAPHICALSKETCH.............................. 146 v

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Thegoalofthisdissertationisthestudyofoptimizationmodelsthatintegrateproduction,inventoryandtransportationdecisions,insearchofopportunitiestoimprovetheperformanceofasupplychainnetwork.Weestimatethetotalcostsofagivendesignofageneralsupplychainnetwork,includingproduction,inventoryandtransportationcosts.Weconsiderproductionandtransportationcoststobeofxedchargetype.Fixedchargecostfunctionsarelinearfunctionswithadiscontinuityattheorigin. Themainfocusofthisdissertationisthedevelopmentofsolutionproceduresfortheseoptimizationmodels.Theircomputationalcomplexitymakestheuseofheuristicssolutionproceduresadvisable.OneoftheheuristicsweproposeisaMulti-CommodityDynamicSlopeScalingProcedure(MCDSSP).Thisheuristicmakesuseofthefactthatwhenminimizingaconcavefunctionoveraconvexset,anextremepointoptimalsolutionexists.Thesameholdstrueforlinearprograms.Therefore,theconcavecostfunctionisapproximatedbyalinearfunctionandthecorrespondinglinearprogramissolved.Theslopeofthelinearfunctionisupdatediterativelyuntil vi

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WealsodevelopaLagrangeandecompositionbasedheuristic.Thesubproblemsfromthedecompositionhaveaspecialstructure.Oneofthesubproblemsisthemulti-facilitylot-sizingproblemthatwestudyindetailinChapter2.Themulti-facilitylot-sizingproblemisanextensionoftheeconomiclot-sizingproblem.Weaddanewdimensiontotheclassicalproblem,thefacilityselectiondecision.Weprovidethefollowingheuristicapproachestosolvethisproblem:dynamicprogramming,aprimal-dualmethod,acuttingplanemethodandalinearprogrammingbasedalgorithm.Weproposeasetofvalidinequalitiesandshowthattheyarefacetdening.Wetestedtheperformanceoftheheuristicsonawiderangeofrandomlygeneratedproblems. Wealsostudiedotherextensionsofthemulti-facilitylot-sizingproblem.InChapter3weanalyzeandprovidesolutionapproachestothemulti-commodityandmulti-retailer(single-commodity)versionsoftheproblem. vii

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1.1 Supply Chain Management Mostcompaniesnowadaysareorganizedintonetworksofmanufacturinganddistributionsitesthatprocurerawmaterials,processthemintonishedgoods,anddistributethenishedgoodstocustomers.Thegoalistodelivertherightproducttotherightplaceattherighttimefortherightprice.Theseproduction-distributionnetworksarewhatwecall\supplychains." SupplyChainManagementisagrowingareaofinterestforbothcompaniesandresearchers.Itrstattractedtheattentionofcompaniesinthe1990sastheystartedtorealizethepotentialcostbenetsofintegratingdecisionswithothermembersoftheirsupplychain.Theprimarycostfactorswithinasupplychaincanbeputintothecategoriesofproduction,transportationandinventory.Thesignatureofsupplychainmanagementistheintegrationofactivities.Eectivesupplychainmembersinvariablyintegratethewishesandconcernsoftheirdownstreammembersintotheiroperationswhilesimultaneouslyensuringintegrationwiththeirupstreammembers.Weconcentrateondevelopingoptimizationtoolstoenablecompaniestotakeadvantageofopportunitiestoimprovetheirsupplychain. Formanyyearscompaniesandresearchersfailedtotakeanintegratedviewoftheentiresupplychain.Theyconsideredonlyonepieceoftheoverallproblem,suchasproductionordistributionsubmodels.Thesesubmodelswereoptimizedseparatelyandthesolutionswerethenjoinedtogethertoestablishoperatingpolicies. Anumberofnewdevelopmentshavehadanimpactonmanycompanies.Forexample,increasedmarketresponsivenesshasintensiedtheinter-dependencieswithinthesupplychain(Erengucet.al[30]);technologicalinnovationshaveshortened 1

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thelifespanofmanufacturingequipment,whichinturnincreasesthecostofmanufacturingcapacity;internethasoeredhighspeedcommunication(Geuneset.al[50]).Thesedevelopmentscombinedwithincreasedproductvarietyanddecreasedproductvolumespromptcompaniestoexplorenewwaysofrunningtheirbusiness.Experiencehasshownthatarm'sabilitytomanageitssupplychainisamajorsourceofcompetitiveadvantage.Thisrealizationisthesinglemostimportantreasonfortherecentemphasisonsupplychainmanagementinindustryandacademia.Toexploitthesenewopportunitiestoimprovetheirprotability,thecompaniesneeddecisionsupporttoolsthatprovideevaluationofalternativesusingoptimizationmodels. Severalexamplescanbefoundintheliteratureprovingthatmodelscoordinatingatleasttwostagesofthesupplychaincandetectnewopportunitiesforimprovingtheeciencyofthesupplychain.ChandraandFisher[21]investigatedtheeectofcoordinatingproductionanddistributiononasingle-plant,multi-commodity,multi-periodscenario.Inthisscenario,theplantproducesandstorestheproductsuntiltheyaredeliveredtothecustomersusingaeetoftrucks.Theyproposedtwosolutionapproaches.Therstapproachsolvestheproductionschedulingandroutingproblemsseparatelyandthesecondapproachconsidersboth,productionandroutingdecisionstobeincorporatedintothemodel.Theircomputationalstudyshowedthatthecoordinatedapproachcanyieldupto20%incostssavings. AnilyandFedergruen[4]consideredintegratinginventorycontrolandtransportationplanningdecisionsmotivatedbythetrade-obetweenthesizeandthefrequencyofdelivery.Theirmodelconsideredasinglewarehouseandmultipleretailerswithinventoriesheldonlyattheretailerswhofaceconstantdemand.Burnsetal.[18]investigateddistributionstrategiesthatminimizetransportationandinventorycosts.SuccessfulapplicationsofsupplychaindecisioncoordinationwerereportedatXerox[94]andHewlettPackard[44].Asaresultofcoordinatingthe

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decisionsoninventoriesthroughoutthesupplychain,theywereabletoreducetheirinventorylevelsby25%. 1.2 Framework of This Study Companiesdeliverproductstotheircustomersusingalogisticsdistributionnetwork.Suchnetworkstypicallyconsistofproductowsfromtheproducerstothecustomersthroughdistributioncenters(warehouses)andretailers.Companiesgenerallyneedtomakedecisionsonproductionplanning,inventorylevels,andtransportationineachlevelofthelogisticsdistributionnetworkinsuchawaythatcustomer'sdemandissatisedatminimumcost. Coordinatingdecisionswithothermemberswiththeaimofbiggerprotsandbettercustomerserviceisadistinctivefeatureofsupplychainmanagement.Bhatnagar,ChandraandGoyal[13]distinguishedbetweentwobroadlevelsofcoordination.Atthemostgenerallevel,coordinationcanbeseenintermsofintegratingdecisionsofdierentfunctions(forexample,facilitylocation,inventoryplanning,productionplanning,distributionplanning,etc.).Theyrefertothislevelofcoordinationas\generalcoordination."Atanotherlevel,theproblemofcoordinationmaybeaddressedbylinkingdecisionswithinthesamefunctionatdierentechelonsintheorganization. Theyclassiedtheresearchongeneralcoordinationintothreecategories.Thesecategoriesrepresentintegrationofdecision-makingrelatedto (i) Supplyandproductionplanning (ii) Inventoryanddistributionplanning (iii) Productionanddistributionplanning Studiesoncoordinationbetweensupplierandbuyerfocusedondeterminingtheorderquantitythatisjointlyoptimalforboth.Thesecondcategoryaddressestheproblemofcoordinatinginventoryplanningwithdistributionplanning.Thisproblememergeswhenanumberofcustomersmustbesuppliedfromoneormorewarehouses.Theinventoryanddistributionplanningproblemdecidesonthereplenishmentpolicy

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atthewarehouseandthedistributionscheduleforthecustomers,sothatthetotalofinventoryanddistributioncostsareminimized.Thetrade-oisreducingtheinventorycostsversusanincreaseinthetransportationcosts.Forexample,shippinginsmallerquantitiesandwithhighfrequencyreducestheinventorylevelatthewarehouse,butcauseshighertransportationcosts. Thethirdcategoryofresearchconcentratesonintegratingproductionplanninganddistributionplanning.Theproductionplannerisconcernedwithdeterminingoptimalproduction-inventorylevelsforeachproductineveryperiodsothatthetotalcostofproductionandinventoryholdingisminimized.Ontheotherhand,thedistributionplannermustdetermineschedulesfordistributionofproductstocustomerssothatthetotaltransportationcostisminimized.Thesetwoactivitiescanfunctionindependentlyifthereisasucientlylargeinventorybuerthatcompletelydecouplesthetwo.However,thiswouldleadtoincreasedholdingcostsandlongerleadtimes,sinceononesidethedistributionplaner,inordertominimizetransportationcosts,wouldpreferfulltruckshipmentsandminimumnumberofstops;andontheothersidetheproductionplannerwouldpreferlessnumberofmachinesetups.Thepressureofreducinginventoryandleadtimesinthesupplychainforcedcompaniestoexploretheissueofclosercoordinationbetweenproductionanddistribution. Ourworkcontributestothislastresearchcategory.Weconsidercomplexsupplychainswithcentralizeddecisionsonproduction-inventoryandtransportation.Wemodelthesesupplychainsasmulti-commoditynetworkowproblemswithnon-convexcostfunctions.Inparticularweusexedchargecostfunctionstomodelproductionortransportationcosts.Wepresentseveraloptimizationtechniquestosolvetheseproblems. RelatedworkistheresearchofKingandLove[67]onthecoordinationofproductionanddistributionsystemsatKellySpringeld,amajortiremanufacturerwithfourfactoriesandninemajordistributioncenterslocatedthroughoutthe

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UnitedStates.Theauthorsdescribedacoordinatedsystemformanufacturingplantsanddistributioncenters.Implementationofthissystemresultedinsubstantialimprovementsinoverallleadtimes,customerserviceandaverageinventorylevels.Annualcostswerereducedbyalmost$8million. Blumenfeldetal.[16]reportedonthesuccessfulimplementationofanoptimizationmodelthatsynchronizesschedulingofproductionanddistributionattheDelcoelectronicsdivisionofGeneralmotors.Implementationresultedina26%reductioninlogisticscosts. Brownetal.[17]presentedasuccessfulimplementationofanoptimizationmodelthatcoordinatesdecisionsonproduction,transportationanddistributionatKelloggCompany.KelloggoperatesveplantsintheUnitedStatesandCanada,andithassevencoredistributioncenters.Themodelhasbeenusedfortacticalandoperationaldecisionssince1990.Theoperationalversionofthemodeldetermineseverydayproductionandshippingquantities.Thetacticalversionhelpstoestablishbudgetandmakecapacityexpansionandconsolidationdecisions.Theoperationalmodelreducedproduction,inventoryanddistributioncostsbyapproximately$4:5millionin1995.Thetacticalmodelrecentlyguidedaconsolidationofproductioncapacitywithaprojectedsavingsof$35to$40millionperyear. Modelscoordinatingdierentstagesofthesupplychaincanbeclassiedasstrategic,tacticaloroperational(HaxandCandea[55]).Themodelwestudyhelpsthecompaniestomakestrategicdecisionsrelatedtoproduction,inventoryandtransportation.Severalsurveysintheliteratureaddresscoordinationissues.VidalandGoetschalckx[102]addressedtheissueofstrategicproduction-distributionplanningwithemphasisonglobalsupplychains.Beamon[12]presentedmodelsonmulti-stagesupplychaindesignandanalysis.Erengucetal.[30]surveyedmodelsthatintegrateproductionanddistributionplanning.ThomasandGrin[97]surveyedcoordinationmodelsonstrategicandoperationalplanning.

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1.3 Objectives and Summary Weproposeaclassofoptimizationmodelsthatconsidercoordinationofproduction,transportationandinventorydecisionsinaparticularsupplychain.Thesupplychainweconsiderconsistsofanumberoffacilitiesandretailers.Thismodelhelpstoestimatethetotalcostofagivenlogisticsdistributionnetwork,includingproduction,inventoryholdingandtransportationcosts.Theevaluationtakesplaceoveraxed,niteplanninghorizonofTperiods. TheparticularscenariopresentedconsidersasetoffacilitieswhereKdierentproducttypescanbeproduced.Productsarestoredatthefacilitiesuntildemandoccurs.Moreover,retailersaresuppliedbythefacilitiesandkeepnoinventories.Wedonotallowfortransportationbetweenfacilities.Anexampleofthisparticularscenarioisthesupplychainforveryexpensiveitems(suchascars).Often,itisnotaordablefortheretailerstokeepinventoriesofexpensiveitems,thereforetheyorderthecommoditiesasdemandoccurs(forexample,ForddealerskeepnoinventoriesforCorvette). Weconsiderproductionandtransportationcostfunctionstobeofthexedchargeform.Oftenintheliteratureproductioncostsaremodeledusingthistypeofcostfunction.Thisisbecauseofthespecialnatureoftheproblem.Thus,whenevertheproductionofacommoditytakesplace,axedchargeispaidtosetupthemachines,plusavariablecostforeveryunitproduced.Transportationcostsalsocanbemodelledusingxedchargecosts,sinceusuallythereisaxedcharge(forexample,thecostofthepaperworknecessary)toinitiateashipmentplusothercoststhatdependontheamountshipped. Theobjectiveistondtheproduction,inventory,andtransportationquantitiesthatsatisfydemandatminimumcost.Weformulatethisproblemasamulti-commoditynetworkowproblemonadirected,singlesourcegraphconsistingofTlayers(Figure1{1).Eachlayerofthegraphrepresentsatimeperiod.Ineachlayer,

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abipartitegraphrepresentsthetransportationnetworkbetweenthefacilitiesandretailers. ThesetofnodesofthenetworkconsistsofTcopiesoftheFfacilitiesandRretailers,aswellasthesourcenode.Thesetofarcsconsistsofproductionarcs(betweenthesourcenodeandafacilityataparticulartimeperiod),transportationarcs(betweenafacilityandaretailerataparticulartimeperiod),andinventoryarcs(betweentwonodescorrespondingtoaparticularfacilityinconsecutivetimeperiods).Therearebundlecapacitiesonthetransportationandproductionarcs,andnocapacitiesontheinventoryarcs.Inmulti-commoditynetworkowproblems,thebundlecapacitiestietogethercommoditiesbyrestrictingthetotalow(ofallcommodities)onanarc. OurproblemisrelatedtotheonesstudiedbyWuandGolbasi[106];andFrelingetal.[43];andRomeijnandRomeroMorales[90,91];andRomeroMorales[83];andBalakrishnanandGeunes[6].Incontrasttoourmodel,WuandGolbasi[106]assumeaxed-chargecoststructureforproduction,butassumelineartransportationcosts.Also,theyconsiderintheirmodeltheproductstructureofeachend-item.Ontheotherhand,Frelingetal.[43],RomeijnandRomeroMorales[90,91]andRomeroMorales[83]considerthecaseofasinglecommodityonly,butaccountforthepresenceofso-calledsingle-sourcingconstraints,whereeachretailershouldbesuppliedfromasinglefacilityonly.BalakrishnanandGeunes[6]addressedadynamicrequirements-planningproblemfortwo-stagemulti-productmanufacturingsystemswithbill-of-materialexibility(i.e.,withoptionstousesubstitutecomponentsorsubassembliesproducedbyanupstreamstagetomeetdemandineachperiodatthedownstreamstage).Theirmodelissimilartothemulti-retailerandmulti-facilitylot-sizingproblemwediscussinChapter3.However,theirformulationismoregeneralandconsistsofextraconstraints(similartothesingle-sourcingconstraints)andextracomponentsinthecostfunction(penaltiesforviolatingthesingle-sourcingconstraints).

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Figure1{1:Two-period,two-facility,three-retailersupplychainoptimizationproblem ThesupplychainoptimizationmodelwediscussisanNP-hardproblem.Eventhespecialcaseofthemulti-commoditynetworkowproblemwithxedchargecosts,thesingle-periodandsingle-commodityproblemisNP-hard(GareyandJohnson[45]).Thecomplexityofthisproblemledustoconsidermainlyheuristicapproaches. Thedicultyinsolvingthisproblemmotivatedustolookintosomespecialcasesofthemodel.Theknowledgewegainedfromanalyzingthesimplerproblemsgaveusinsightsabouthowtoapproachthegeneralmodel.Belowwedescribesomeofthespecialcasesweidentied. Thesingle-retailer,single-facilityproblemwithonlyxedchargeproductioncosts,reducestotheclassicallot-sizingproblem(Figure1{2).Theretailerineachperiodhastoshiphisdemandfromthefacility.Therefore,theonlydecisiontobemadeinthisproblemisforthefacilitytodecideonitsproductionschedule,whenandhowmuchtoproduceineveryperiod(Figure1{3).Thesameholdstruewhenthemodelconsidersasinglefacilitywithmultipleretailers.

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Figure1{2:Multi-period,single-facility,single-retailerproblem Figure1{3:Multi-period,single-facility,single-retailerproblem

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Thesingle-commodity,multi-facility,single-retailerproblemwithonlyproductionsetupcostsisdiscussedinChapter2(Figure1{4).Werefertothisproblemasthemulti-facilitylot-sizingproblem.WuandGolbasi[106]showthattheuncapacitatedversionofthisproblemwithholdingcostsnotrestrictedinsignisNP-complete.Weproposeanumberofheuristicapproachestosolvetheproblemsuchasadynamicprogrammingbasedalgorithm,aprimal-dualheuristicandacuttingplanealgorithm.Weproposeasetofvalidinequalitiesandshowthattheyarefacetsoftheconvexhullofthefeasibleregion.Wepresentadierentformulationoftheproblemthatwerefertoastheextendedproblemformulation.Thelinearprogrammingrelaxationoftheextendedformulationgiveslowerboundsthatareatleastasgoodastheboundsfromthelinearprogrammingrelaxationoftheoriginalformulation. Figure1{4:Multi-period,multi-facility,single-retailerproblem InChapter3wediscusstwoextensionsofthemulti-facilitylot-sizingproblem.First,wediscussthemulti-commodityversionoftheproblem.Laterwepresentamodelforthesingle-commodity,multi-retailer,multi-facilityproblem.WeproposeaLagrangeandecompositionschemetosolvethemulti-commodityproblemandaprimal-dualalgorithmtosolvethemulti-retailerproblem.

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Finally,inChapter4wediscussthegeneralproduction-distributionmodelwepresentedabove.Weproposeaheuristicapproachcalledthemulti-commoditydynamicslopescalingprocedure(MCDSSP).Thisisageneralheuristicthatcanbeusedtosolveanymulti-commoditynetworkowproblemwithxedchargecostfunction.TheMCDSSPisalinearprogrammingbasedheuristic.WesolvethelinearprogrammingrelaxationoftheextendedformulationtogeneratelowerboundsandcomparethequalityofthesolutionsfromMCDSSPtothesebounds.Fortheproduction-distributionproblemwealsopresentaLagrangeandecompositionbasedalgorithm.Thisalgorithmdecomposestheproblemintotwosubproblems.TherstsubproblemfurtherdecomposesbycommodityintoKsingle-commodity,multi-facility,multi-retailerproblems,thatwediscussinChapter3.Theperformanceofthesealgorithmsistestedonalargevarietyoftestproblems.Wepresentextensivecomputationalresults.

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2.1 Introduction Inthischapterweanalyzeandprovidealgorithmstosolvethemulti-facilityeconomiclot-sizingproblem.Thisisanextensionofthewell-knownEconomicLot-Sizingproblem.Theeconomiclot-sizingproblemcanbedescribedasfollows:GivenanitetimehorizonTandpositivedemandsforasingleitemineachproductionperiod,ndaproductionschedulesuchthatthetotalcostsareminimized.Thecustomers'demandmustbesatisedfromproductioninthecurrentperiodorbyinventoryfromthepreviousperiods(thatisnobackloggingisallowed).Thetwokindsofcostsconsideredareproductioncostsandholdingcosts. Dierentfromtheclassicaleconomiclot-sizingproblem,themulti-facilitylot-sizingproblemconsidersthatdemandcanbesatisedviamultiplefacilities.Thisaddsanextradimensiontotheclassicalproblem,thefacilityselectiondecision.Transportationcosts,togetherwithproductionandinventorycostsarethebiggestcomponentsoftotalcosts.Thus,incontrastwiththeclassicallot-sizingproblem,thismodel,indecidingonaproductionschedule,considersnotonlyproductionandinventorycosts,butalsotransportationcosts. Themulti-facilitylot-sizingproblemndsanoptimalproduction,inventoryandtransportationschedulethatsatisesdemandatminimumcost.Thisproblemhasmanypracticalapplications.Forexample,amanufacturingcompanyoftenhasmultiplefacilitieswithsimilarproductioncapacities.Theneedforcrossfacilitycapacitymanagementismostevidentinhigh-techindustriesthathavecapitalintensiveequipmentandashorttechnologylifecycle.Thesecompaniesoftenstrugglewiththeirproductionplanningproblemintheircomplexandrapidlychanging 12

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environment.Tobetterutilizetheircapital-intensiveequipmentstheyarepressuredtoproduceavarietyofproductsineachoftheirproductionfacilities.Coordinatingthedecisionsonproduction,inventoryandtransportationamongallthefacilitiesreducesthecostsrelativetohavingeachfacilitymakeitsowndecisionsindependently. Wepresentvarioussolutionapproachestosolvethisproblem.Theimportanceofthealgorithmsweproposeisinthefactthatthesealgorithmscanbeusedassubroutinestosolvemorecomplexsupplychainoptimizationproblems.InSection2.2wepresentaliteraturereviewofeconomiclot-sizingandrelatedproblems.Section2.3givestheproblemformulationandSection2.4discussesanextendedformulationofthemulti-facilitylot-sizingproblem.Itslinearprogrammingrelaxationgivesclose-to-optimalsolutionsandthecorrespondingdualproblemhasaspecialstructure.InSection2.5wediscussaprimal-dualbasedalgorithm.Wepresentasetofvalidinequalitiesfortheproblemandshowthattheyarefacetsdeninginequalities.ThevalidinequalitiesareusedinthecuttingplanealgorithmdiscussedinSection2.6.InSection2.7wediscussadynamicprogrammingbasedalgorithm.Finally,Section2.8presentssomeofthecomputationalresultsandSection2.9concludesthischapter. 2.2 Literature Review Thissectionpresentsaliteraturereviewonthesingle-itemeconomiclot-sizingproblem.TherstcontributionistheEconomicOrderQuantity(EOQ)modelproposedbyHarris[54]in1913.Thismodelconsidersasinglecommoditywithaconstantdemandrate,productiontakingplacecontinuouslyovertime,anddoesnotincorporatecapacitylimits. Amajorlimitationoftheabovemodelistherestrictionthatthedemandiscontinuousovertimeandhasconstantrate.Manne[81]andWagnerandWhitin[104]consideredthelot-sizingproblemwithanitetimehorizonconsistingofanumberofdiscreteperiods,eachwithitsowndeterministicandindependentdemand.Thisistheclassiceconomiclot-sizingproblem.WagnerandWhitindevelopedadynamic

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programmingalgorithmforthesingle-commodity,uncapacitatedversionoftheproblem.TheiralgorithmrunsinO(T2). Thisproblem(apparentlywellsolvedsince1958)recentlyrevealedavarietyofnewresults.Practicalreasonsexistfortheinterestinthismodel.AlmostthirtyyearslaterWagelmansetal.[103],AggarwalandPark[1],andFedergruenandTzur[39]showedthattherunningtimeofthedynamicprogrammingalgorithmcouldbereducedtoO(TlogT)inthegeneralcaseandtoO(T)whenthecostshaveaspecialstructure(ht+ptpt+1),alsoreferredtoastheabsenceofspeculativemotives. Thecapacitatedlot-sizingproblemisNP-hardevenformanyspecialcases(Florianetal.[41]andBitranandYanasse[15]).In1971,FlorianandKleinpresentedaremarkableresult.TheydevelopedanO(T4)algorithmforsolvingthecapacitatedlot-sizingproblemwithequalcapacitiesinallperiods.Thisresultusesadynamicprogrammingapproachcombinedwithsomeimportantpropertiesofoptimalsolutionstotheseproblems.Recently,vanHoeselandWagelmans[99]showedthatthisalgorithmcanbeimprovedtoO(T3)ifbackloggingisnotallowedandtheholdingcostfunctionsarelinear. SeveralsolutionapproacheshavebeenproposedforNP-hardspecialcasesofthecapacitatedlot-sizingproblem.Thesemethodsaretypicallybasedonbranch-and-bound(seeforinstance,Bakeretal.[5]andErengucandAksoy[29]),dynamicprogramming(seeforinstance,Kirca[68]andChenandLee[23])oracombinationofthetwo(seeforinstance,ChungandLin[25]andLoftiandYoon[75]). Themulti-commodityversionoftheproblemhasattractedmuchattention.Manne[81]usedthezeroinventory(ZIO)propertytodevelopacolumngenerationapproachtosolvethisproblem.Baranyetal.[10]solvedthemulti-commoditycapacitatedlot-sizingproblemwithoutset-uptimesoptimallyusingacuttingplaneprocedurefollowedbybranchandbound. Otherextensionstotheclassiceconomiclot-sizingproblemconsiderset-uptimes,backordersandotherfactors.Zangwill[107]extendedWagnerandWhitin's

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modeltoallowforbackloggingandconcavecostfunctions.Veinott[101]studiedanuncapacitatedmodelwithconvexcoststructures.Trigeiroetal.[98]showedthatcapacitatedlot-sizingproblemwithset-uptimesismuchhardertosolvethancapacitatedlot-sizingproblemwithoutset-uptimes.Itiseasytocheckifthecapacitatedlot-sizingproblemproblemwithoutset-uptimeshasafeasiblesolutionornot.Thiscanbedonebycomputingcumulativedemandandcumulativecapacity.Whenset-uptimesareconsidered,thefeasibilityproblemisNP-complete.Thebinpackingproblemisaspecialcaseofcapacitatedlot-sizingproblemwithset-uptimes(GareyandJohnson[45]p.226). Muchresearchonlot-sizingproblemsfocusedondetermininga(partial)polyhedraldescriptionofthesetofthefeasiblesolutionsandapplyingbranch-and-cutmethods(Pochetetal.[89],Leungetal.[74],Baranyetal.[10]).Themainmotivationforstudyingthepolyhedralstructureofthesingleitemlot-sizingproblemistousetheresultstodevelopecientalgorithmsforproblemssuchasthemulti-commodityeconomiclot-sizingproblemthatcontainsthismodelasasubstructure.However,thebranch-and-cutapproachhasnot(yet)resultedincompetitivealgorithmsforthesingle-itemlot-sizingproblemitself.Thereasonisthatgeneratingasinglecutcouldbeastimeconsumingassolvingthewholeproblem. Baranyet.al[9,10]providedasetofvalidinequalitiesforthesingle-commoditylot-sizingproblem,showedthattheseinequalitiesarefacetsoftheconvexhullofthefeasibleregionandfurthermore,theyshowedthattheinequalitiesfullydescribetheconvexhullofthefeasibleregion. PereiraandWolsey[88]studiedafamilyofunboundedpolyhedraarisinginanuncapacitatedlot-sizingproblemwithWagner-Whitincosts.Theycompletelycharacterizedtheboundedfacesofmaximaldimensionandshowedthattheyareintegral.ForaproblemwithTperiodstheyderivedanO(T2)algorithmtoexpressanypointwithinthepolyhedronasaconvexcombinationofextremepointsand

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extremerays.TheyobservedthatforagivenobjectivefunctionthefaceofoptimalsolutionscanbefoundinO(T2). ShawandWagelmans[93]consideredthecapacitatedlot-sizingproblemwithpiecewiselinearproductioncostsandgeneralholdingcosts.TheyshowedthatthisisanNP-hardproblemandpresentedanalgorithmthatrunsinpseudo-polynomialtime. WuandGolbasi[106]consideredamulti-facilityproductionmodelwhereasetofitemsistobeproducedinmultiplefacilitiesovermultipleperiods.Theyanalyzedindepththeproduct-level(single-commodity,multi-facility)subproblem.Theyprovethatgeneral-costversionofthisuncapacitatedsubproblemisNP-complete.Theydevelopedashortestpathalgorithmandshowedthatitachievesoptimalityunderspecialcoststructures. Ourstudyiscloselyrelatedtoalltheworkwementioninthissection.Themulti-facilitylot-sizingproblemisanextensionofthelot-sizingproblem.Weaddtotheclassicalmodelthefacilityselectiondecisionandotherthanproductionandinventorycosts,weconsidertransportationcostsaswell.Theresultsfoundandthealgorithmsdevelopedforthesingle-commoditylot-sizingproblemenlightenusinoursearchforsolutionapproachestothemulti-facilitylot-sizingproblem.ThesetofvalidinequalitiesthatweproposeinSection2.6.1areageneralizationofthevalidinequalitiesproposedbyBaranyet.al[10].TheextendedproblemformulationpresentedinSection2.4isinspiredbyasimilarformulationproposedbyvanHoesel[100]forthelot-sizingproblem.However,ourstudyiscloselyrelatedtotheworkofWuandGolbasi[106].Themulti-facilitylot-sizingproblemisthesamemodelastheoneinWuandGolbasi,howeverweproposeawider(anddierent)rangeofsolutionapproaches. 2.3 Problem Description Themulti-facilitylot-sizingproblemstudiedinthischaptercanbeformulatedusingthefollowingnotation.

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sitproductionset-upcostatfacilityiinperiodt hitinventoryunitcostatfacilityiinperiodt cittransportationunitcostatfacilityiinperiodt Qit(qit)productioncostfunctionatfacilityiinperiodt btdemandinperiodperiodt xitnumberofitemstransportedfromfacilityiinperiodt Iitnumberofitemsintheinventoryatfacilityiinthe endofperiodt Thisproblemcanbeformulatedasanetworkowproblemasfollows:minimizeFXi=1TXt=1(Qit(qit)+citxit+hitIit) subjectto(MF)Iit1+qit=xit+Iiti=1;:::;F;t=1;:::;T Equation(2.1)presentstheowconservationconstraintsforeachfacilityineachtimeperiod,and(2.2)presentstheowconservationconstraintsforthedemandineverytimeperiod.Theowconservationconstraintforthesource,PFi=1PTt=1qit=PTt=1bt,isnotincludedintheformulationbecauseitisimpliedby(2.1)and(2.2).

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ThestructureofthisnetworkisillustratedinFigure2{1.ThesetofnodesinthisnetworkconsistsofTcopiesofeachoftheFfacilitiesandthedemandnode,aswellasasourcenode.Eachlayerofthenetworkrepresentsatimeperiod.ThesetofarcsconsistsofFTproductionarcs,FTtransportationarcsandF(T1)inventoryarcs.Thesourcenodessuppliesthetotaldemandfortheplanninghorizon.Theproductionarcsconnectthesourcetoeachfacility,ineachtimeperiod.Theinventoryarcsconnectthesamefacilityinsuccessiveperiods.Transportationarcsconnectfacilitiestothedemandnodeineachperiod.Theproductioncostfunction Figure2{1:Networkrepresentationofatwo-period,three-facilitylot-sizingproblem isaxedchargecostfunction,thus,ifproductioninatimeperiodisinitiated,theset-upcostplusproductioncostforeachunitproducedhastobepaid.Qit(qit)=8><>:pitqit+sitifqit>00otherwisefori=1;:::;F;t=1;:::;T:

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TheMILPformulationofthemulti-facilitylot-sizingproblemthenreadsasfollows:minimizeFXi=1TXt=1(pitqit+sityit+citxit+hitIit) subjectto(MF)Iit1+qit=xit+Iiti=1;:::;F;t=1;:::;T where,btTpresentsthedemandduringthetimeperiodst(t=1;:::;T)toT,(i.e.btT=PT=tb).StandardsolverssuchasCPLEXcanbeusedtosolveformulation(MF)ofthemulti-facilitylot-sizingproblem. Thesetofconstraints(2.5)showthatthenumberofitemsproducedinthecurrentperiodtogetherwiththeinventoryfrompreviousperiods,shouldbeequaltotheendinginventoryplustheamountshippedtotheretailer.Togetherwith(2.8)theyimplyIit=tX=1(qixi)fori=1;:::;F;t=1;:::;T;

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Usingthesefacts,theinventoryvariablescanbeeliminatedfromtheformulation,reducingthesizeoftheproblem.ThefollowingistheMILPformulationofourproblemwithoutinventoryvariables.minimizeFXi=1TXt=1(p0itqit+sityit+c0itxit) subjectto(R-MF)FXi=1tX=1qib1tt=1;:::;T wherep0it=pit+PT=thiandc0it=citPT=thi.

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Thus,zeroinventorypolicyholdsforthisproblem(qitIi;t1=0,fori=1;:::;F,t=1;:::;T).Thiscompletestheproofthatinanoptimalsolutiontoourproblemdemandissatisedfromeitherproductionortheinventoryofexactlyoneofthefacilities.2 Forthemulti-facilitylot-sizingproblemthereexistsanexactalgorithmthatispolynomialinthenumberoffacilitiesandexponentialinthenumberofperiods(WuandGolbasi[106]). WuandGolbasi[106]showthatforthespecialcasewheninventoryholdingcostsarenotrestrictedinsign,themulti-facilitylot-sizingproblemisanNP-completeproblem.Theyuseareductionfromtheuncapacitatedfacilitylocationproblem. 2.4 Extended Problem Formulation Thelinearprogrammingrelaxationof(MF)isnotverytight.ThisisduetotheconstraintsqitbtTyit.Inthelinearprogrammingrelaxationof(MF),thevariableyitdeterminesthefractionofthedemandfromperiodsttoTsatisedfrom

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productionatfacilityiinperiodt.btTisusuallyaveryhighupperbound,sincetheproductioninaperiodrarelyequalsthisamount.Onewaytotightentheformulationistosplittheproductionvariablesqitbydestinationintovariablesqit(=t;:::;T),wheredenotestheperiodforwhichproductiontakesplace(vanHoesel[100]).Forthenewvariables,atrivialandtightupperboundisthedemandinperiod(i.e.b).Thesplitoftheproductionvariablesleadstothefollowingidentities:qit=TX=tqit(2.16)xit=tXs=1qist(2.17)Iit=tXs=1TX=tqistXs=1qist(2.18) Replacingtheproduction,transportationandinventorydecisionvariables,andafterre-arrangingofterms,theobjectivefunctionbecomesminimizeFXi=1TXt=1[TX=t(pit+ci+Xs=t+1his)qit+sityit] Replacingthedecisionvariablesintheconstraintsoftheformulation(MF)withthenewvariables,weobtainthefollowing:

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subjectto(Ex-MF)FXi=1Xt=1qit=b=1;:::;T where

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Thelinearprogrammingrelaxationof(Ex-MF)replacesconstraintsy2f0;1gwithy0:Thebinaryvariablesyappearonlyinconstraints(2.20),thereforeyitqit Formulation(LP-MF)isthelinearprogrammingrelaxationof(Ex-MF).minimizeFXi=1TXt=1[TX=t subjectto(LP-MF)PFi=1Pt=1qit=b=1;:::;Tqitbyit0i=1;:::;F;t=1;:::;T;tTqit0i=1;:::;F;t=1;:::;T;tTyit0i=1;:::;F;t=1;:::;T:

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formulationofthisproblem(MF)(Proposition2.4.1).AnetworkrepresentationoftheextendedformulationisgiveninFigure2{2. Figure2{2: Networkrepresentationofextendedformulationofatwo-period,three-facilitylot-sizingproblem Formulations(MF)and(Ex-MF)ofmulti-facilitylot-sizingproblemareequivalenttoeachother.Thenotionofequivalencerequiresthattheoptimalsolutiontobothformulationsisthesame(e.g.oncetheproblem(Ex-MF)issolvedintermsofqitvariables,thissolutionshouldyieldtheoptimalsolutiontoproblem(MF)).Forthistoholdtrue,twoconditionsmustbesatised:(i)everysolutionof(Ex-MF)mustcorrespondtoasolutionof(MF)(i.e.nonewsolutionswerecreatedbyredeningthedecisionvariablesof(MF)),and(ii)theremustbeasolutioninthefeasibleregionof(Ex-MF)thatcorrespondstoeveryextremepointintheconvexhullofthesetoffeasiblesolutionsto(MF)(i.e.nosolutionstoformulation(MF)werelostbyredeningthevariables).Condition(i)followsdirectlyfromthedenitionofqit.Asolutionto(Ex-MF)canbedirectlytranslatedintoasolutionto(MF)usingequations(2.16),(2.17)and(2.18).Condition(ii)ismorediculttoargue.Anextremepointoftheconvexhullofthefeasiblesolutionsto(MF)issuchthat(a)itsatisesthezeroinventoryproperty(qitIi;t1=0fori=1;:::;Fandt=1;:::;T);

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(b)demandinperiodt(t=1;:::;T)issatisedfromexactlyonefacility(xitxjt=0fori;j=1;:::;F,i6=j);(c)inperiodt(t=1;:::;T)afacilityeitherdoesnotproduce,orproducesthedemandforanumberofperiods,theperiodsdonotneedtobesuccessive(Proposition2.3.1).Onecaneasilysee(Figure2{2)thatanextremepointto(Ex-MF)satisesexactlythesameconditions.Thecorrespondencebetweentheextremepointsofthetwoformulationsshowsthatthereisanextremeow(treesolution)intheformulation(Ex-MF)foreveryextremeowoftheformulation(MF). Primal-Dual Based Algorithm Thedualof(LP-MF)hasaspecialstructurethatallowsustodevelopaprimal-dualbasedalgorithm.Thefollowingistheformulationofthedualproblem:maximizeTXt=1btvtsubjectto(D-MF)PT=tbwitsiti=1;:::;F;t=1;:::;Tvwit

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replaceitwithwit=max(0;v 2.5.1 Intuitive Understanding of the Dual Problem Inthissectionwegiveanintuitiveinterpretationoftherelationshipbetweentheprimal-dualsolutionsof(Ex-MF).Supposethelinearprogrammingrelaxationofextendedformulation(LP-MF)hasanoptimalsolution(q;y)thatisintegral.Let=f(i;t)jyit=1gandlet(v;w)denoteanoptimaldualsolution. Thecomplementaryslacknessconditionsforthisproblemare(C1)yit[sitPT=tbwit]=0fori=1;:::;F;t=1;:::;T(C2)qit[

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Byconditions(C1),ifafacilityproducesinaparticulartimeperiod,theset-upcostmustbefullypaid(i.e.if(i;t)2,thensit=PT=tbwit).Considerconditions(C3).Now,iffacilityiproducesinperiodt,butdemandinthatperiodissatisedfrominventoryfromapreviousperiod(qitt=0andqittbtyit6=0),thenwitt=0,whichimpliesthatthepricepaidfortheproductinatimeperiodwillcontributetotheset-upcostofonlytheperiodinwhichtheproductisproduced. Byconditions(C2),ifqit>0,thenv= 2.5.2 Description of the Algorithm Thesimplestructureofthedualproblemcanbeexploitedtoobtainnearoptimalfeasiblesolutionsbyinspection.Supposethattheoptimalvaluesoftherstk1dualvariablesv1;:::;vk1areknown.Then,tobefeasiblevkmustsatisfythefollowingconstraints:bkmax(0;vk foralli=1;:::;Fandt=1;:::;k.Inordertomaximizethedualproblemweshouldassigntovkthelargestvaluesatisfyingtheseconstraints.Whenbk>0,thisvalueisvk=mini;tkf NotethatMit;k10impliesvk

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ifb=0thenv=0 fori=1toFdo enddo enddo andwit=wit+1=:::=witT=0:

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fori=1toFdo repeatt=t+1 gotoStep3 forl=ttojdo if Step3: ifP6=?thengotostart solutionisoptimalisbycomparingtheobjectivefunctionvaluesfromtheprimalanddualalgorithms.Sincethedualalgorithmgivesadualfeasiblesolutionandtheprimalalgorithmgivesaprimalfeasiblesolution,atoptimality,thetwoobjectivefunctionsshouldbeequal. 2.5.3 Running Time of the Algorithm Herewediscusstherunningtimeoftheprimal-dualalgorithm.Thetotalnumberoflogicalandarithmeticaloperationsperformedinthedualalgorithmis Thetotalnumberoflogicalandarithmeticaloperationsintheprimalalgorithmis Thus,therunningtimeoftheprimal-dualalgorithmisO(FT2).

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ifyisintegral, else Figure2{5:Cuttingplanealgorithm 2.6 Cutting Plane Algorithm Inthissectionwederiveasetofvalidinequalitiesforthemulti-facilityeconomiclot-sizingproblem.Weshowthattheseinequalitiesarefacetsofthefeasibleregion.Theinequalitiesarethenusedinacuttingplanealgorithmthatndstightlowerbounds. Figure2{5presentsthestepsofthecuttingplanealgorithm.Thealgorithmstopswheneither(i)theoptimalsolutionto(MF)isfound,or(ii)ifnomorevalidinequalitiescanbegenerated.Nextwediscussindetailthevalidinequalities,theseparationalgorithm,andweshowthattheseinequalitiesarefacetsofthefeasibleregionof(MF). 2.6.1 Valid Inequalities

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2.6.2 Separation Algorithm Let(q;x;I;y)bethesolutiontothelinearprogrammingrelaxationof(MF).Ifthissolutionsatisestheintegralityconstraints(y2f0;1g),thisisthesolutionto(MF)aswell.However,ifitdoesnot,wehavetoidentifyavalidinequalitythatcuts

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othissolutionfromthesetoffeasiblesolutionstolinearprogrammingrelaxationof(MF).AchallengingproblemistoidentifythesetsSandLnSforwhichthevalidinequality(2.25)isviolated.AnexponentialnumberofsetsSandLnSexists,however,thefollowingseparationalgorithmthatrunsinpolynomialtime(O(FT2))takesusthroughthestepsneededtoidentifythesetsSandLnS.Notethat,foragiventimeperiodl(l=1;:::;T),thisprocedureidentiesthevalidinequalitythatisthemostviolatedbythecurrentsolution. Forl=1;:::;T1.fori=1;:::;FndSlL=f1;:::;lgwheret2SlifFXi=1qitFXi=1btlyitandt2LnSlifFXi=1qit>FXi=1btlyit2.checkifFXi=1(Xt2Slqit+Xt2LnSlbtlyit)
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Thefollowingisaprocedurethatwillhelpustoseewhetherinequalities(2.25)arefacetsofco().Thismeanstheseinequalitiesarenecessaryifwewishtodescribeco()byasystemoflinearinequalities. Considerformulation(MF): Thisshowsthatthedimensionofthefeasibleregiontoformulation(MF)isatmostequalto3TFFT1. Considerformulation(R-MF):

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Thedimensionofthefeasibleregionofformulation(R-MF)isatmostequalto3TFFT1.However,inTheorem2.6.2weshowthatthereexist3TFFTanelyindependentpointsintheconvexhullofthefeasibleregionofthemulti-facilitylot-sizingproblemthatsatisfythevalidinequalities(2.25)toequality.Thisindicatesthatthereexist3TFFTanelyindependentpointsinthefeasibleregionofthemulti-facilitylot-sizingproblem.Thisconcludesourproofthatthedimensionofthefeasibleregionofthemulti-facilitylot-sizingproblemis3TFFT1.2 Thenextstepistoshowthatthereare3TFFT1anelyindependentpointsinthatsatisfythevalidinequalitytoequality. 2.7 Dynamic Programming Based Heuristic 2.7.1 Introduction Dynamicprogrammingprovidesaframeworkfordecomposingcertainoptimizationproblemsintoanestedfamilyofsubproblems.Thenestedsubstructuresuggestsarecursiveapproachforsolvingtheoriginalproblemfromthesolutionsofthesubproblems.Therecursionexpressesanintuitiveprincipleofoptimalityforsequentialdecisionprocesses;thatis,oncewehavereachedaparticularstate,anecessaryconditionforoptimalityisthattheremainingdecisionsmustbechosenoptimallywithrespecttothatstate(NemhauserandWolsey[86]).

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Dynamicprogrammingwasoriginallydevelopedfortheoptimizationofsequentialdecisionprocesses.Atypicalexamplethatisusedintheliteraturetoexplainhowthedynamicprogrammingalgorithmworksistheeconomiclot-sizingproblem(NemhauserandWolsey[86]andWolsey[105]).Considerthelot-sizingproblemwithTtimeperiods(t=1,...,T).Atthebeginningofperiodt,theprocessisinstatest1,whichdependsonlyontheinitialstates0(initialinventoryI0)andthedecisionvariablesytandqtfort=1;:::;t1.ThecontributionofthecurrentstatettotheobjectivefunctiondependsonIt1.Letusdenotebyvtthevalueoftheoptimaldecisionsinperiodst;:::;Tvt(It1)=minqt;yt(ptqt+styt+ht(It1+qtbt)+vt+1(It1+qtbt))(2.26) Thedicultywiththisalgorithmisthatsincedemandinperiodtissatisedbyproductioninperiodst,itfollowsthatthelevelofinventoryintheendofperiodt1canbeaslargeasbtT,anditappearsthatalargenumberofcombinationsof(qt;It1)mustbeconsideredtosolvetheproblem. Fortunately,thefollowingpropertiesofanoptimalsolutiontotheeconomiclot-sizingproblemmaketheproblemeasier.

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2.7.2 Description of the Algorithm Inthissectionwepresentadynamicprogrammingproceduretosolvethemulti-facilitylot-sizingproblem.Thisprocedureisaheuristic,andthereforedoesnotcaptureallsolutions,possiblyincludingtheoptimalsolutiontotheproblem. InSection2.3wediscussedthenon-splittingpropertyofoptimalsolutionstothemulti-facilitylot-sizingproblem.Itisimportanttonotethataproduction,inventoryandtransportationplanisoptimalifandonlyifthecorrespondingarcswithpositiveowformanarborescence(rootedtree)inthenetwork.Importantimplicationsofthisresultarethefollowing:inanoptimalproductionplan,thedemandbtforagivenperiod(t=1;:::;T)willbeproducedinasinglefacilityinasingletimeperiod;ineverytimeperiodafacilityeitherwillnotproduce,orwillproducethedemandforanumberofperiods,andthesetimeperiodsneednottobesuccessive.Lettbethesetofallperiodscoveredbyproductionatfacilityiinperiodtinanoptimalarborescence.Thentheoptimalproductionplanis Figure2{6:Networkrepresentationofmulti-facilitylot-sizingproblemwithT=4

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Usingthisinformationwecannowsimplifythemulti-facilitylot-sizingproblemtoashortestpathprobleminanacyclicnetwork,sayG0.WebuildG0inthefollowingway:letthetotalnumberofnodesinG0beequalto(T+1),oneforeachtimeperiodalongwithadummynodeT+1.Traversingarc(;0)2G0representsthechoiceofproducinginasinglefacilityinasingletimeperiodt=1;:::;tosatisfythedemandinperiods;+1;:::;01.Thecostofarc(;0)iscalculatedusingthefollowingcostfunction:g;0=mini=1;:::;F;1tsit+citb+cit;+1b+1+:::+cit;01b01:(2.27)Letvbetheminimumcostofasolutionforperiod=1;:::;T1and0.The Figure2{7:Sequentialextremeow recursionfunctionforthemulti-facilitylot-sizingproblemis andvT(FXi=1Ii;T1)=gT;T+1(2.29)

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Figure2{8:Non-sequentialextremeow ThetotalnumberofarcsinG0isequaltoT(T+1)=2.Giventhecostsg;0foreveryarc(;0)2G0,therecursivefunctions(2.28)and(2.29)willprovidetheoptimalsolutioninO(T2).Every(directed)pathinG0thatconnectsnode1toT+1,correspondstoafeasiblesolutiontotheoriginalproblem.ThenetworkG0fora4-periodproblemispresentedinFigure2{6. WuandGolbasi[106]proposeasimilarshortestpathalgorithmtosolvethemulti-facilitylot-sizingproblem.Theyshowedthattheiralgorithmgivestheoptimalsolutiontotheproblemifthefollowingconditionshold:(i)nosimultaneousproductionovermorethanonefacilitycantakeplaceinagivenperiod.Inotherwordsqitqjt=0fori;j=1;:::;F,i6=jandt=1;:::;T.(ii)noproductionwillbe

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scheduledatallifthereisinventorycarriedoverfrompreviousperiodinoneofthefacilities.InotherwordsqitIj;t1=0fori;j=1;:::;Fandt=1;:::;T. Figure2{9:Non-sequentialow:Case1 Theseconditionsobviouslyrestrictthesearchforasolutiontoonlysequentialextremeows.Furthermore,theyinvestigateonlypartofthesequentialextremeows,theonesthatsatisfytheaboveconditions.DierentfromWuandGolbasi,ourprocedureconsidersawiderrangeofextremeows.Weconsiderallthesequentialextremeows,althoughsomeofthemmaynotsatisfyconditions(i)and(ii).Figure2{9presentsasequentialextremeowthatviolatescondition(i)andFigure2{10presentsasequentialextremeowthatviolatescondition(ii). 2.7.3 Running Time of the Algorithm TheabovedynamicprogrammingalgorithmtondtheshortestpathintheacyclicgraphG0hasrunningtimeofO(m),wheremisthenumberofarcsinG0.Sincethegraphiscomplete, 2:

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Therefore,therunningtimeofouralgorithmwillbeO(c+T2),wherecpresentsthetimeittakestocalculatethecostsofallarcsinG0. Figure2{10:Non-sequentialow:Case2 Incalculatingthecostofaparticulararc(;0)2G0,weneedtoperformacertainnumberofcomparisons,additionsandmultiplications. thus,O(FT3).

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ThetotalnumberofadditionsisAdd:=TX=1T+1X0=+1(0+1)F ThisshowsthatthetimecomplexitytosolvetheproblemusingtheabovedynamicprogrammingalgorithmisO(FT4). 2.8 Computational Results TotesttheperformanceofthealgorithmsdiscussedinthischapterwerandomlygeneratedasetoftestproblemsandcomparedthecomputationtimesandsolutionqualitytothegeneralpurposesolverCPLEX.ThealgorithmswerecompiledandexecutedonanIBMcomputerwith2Power3PCprocessors,200MhzCPUseach. Thescopeofourexperiments,otherthancomparingthealgorithmsistoseehowdierentfactors(suchastheratioofset-uptovariablecost,numberoffacilities,etc.)aecttheirperformance.Werstgenerateanominalcaseproblemasfollows: MostoftheaboveparametersarethesameastheonesusedinWuandGolbasi[106]forarelatedproblem.Togeneratemeaningfultransportationvariablecosts,we

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randomlygeneratedfromuniformlydistributedpointsona[0;10]2squarethefacilityanddemandpointlocations,andcalculatedcorrespondingEuclideandistances.WeassumedonetoonecorrespondencebetweentheEuclideandistancesandtheunittransportationcosts. Varyingoneormorefactorsfromthenominalcase,wegeneratedvegroupsoftestproblems.Intherstgroupofproblemswechangethelevelofproductionset-upcostsfromthenominalcasetothefollowing:sitU[200;300],sitU[200;900],sitU[600;900],sitU[900;1500],sitU[1500;2000],sitU[2000;3000],sitU[3000;6000],sitU[5000;10000],andsitU[10000;20000].Thesetogetherwiththenominalcaseproblemgiveatotalof10problemclasses(problemclasses1to10). Inthesecondgroupofproblemswechangethelengthofthetimehorizonto5,10,15,20,25,35,40(problemclasses11to17).Inthethirdgroupwechangethenumberoffacilitiesto120,130,140,160,170,180,190,and200(problemclasses18to25).InthefourthgroupofproblemsthelevelofdemandischangedtobtU[20;50],btU[50;100],btU[100;200],btU[200;400],andbtU[400;1000](problemclasses26to30).Finally,inthefthgroupthelevelofholdingcostsischangedtohtU[20;10],htU[10;10],htU[10;20],htU[20;40],andhtU[40;100](problemclasses31to35). Foreachproblemclasswegenerate20instances.Theerrorsandrunningtimeswepresentforeachproblemclassaretheaveragesoverthe20probleminstances.AsummaryoftheresultsfromtheexperimentsarepresentedinTables2{2to2{5.Wedonotpresenttheresultsfromimplementingthecuttingplanealgorithm,sinceinalmostalloftheproblemsCPLEXoutperformedouralgorithmintermsofsolutionqualityandrunningtimes. Wewouldliketoemphasizethatthelinearprogrammingrelaxationoftheextendedformulationanddualalgorithmgivelowerbounds,whiledynamicprogrammingandprimalalgorithmsgivefeasiblesolutionstotheproblems.The

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Table2{1:Problemcharacteristics Problems Nodes Arcs 4,531 13,350 74,250 465 11 756 2,100 3,000 15 12 1,511 4,350 9,750 55 13 2,266 6,600 20,250 120 14 3,021 8,850 34,500 210 15 3,776 11,100 52,500 325 16 5,286 15,600 99,750 630 17 6,041 17,850 129,000 820 18 3,631 10,680 59,400 465 19 3,931 11,570 64,350 465 20 4,231 12,460 69,300 465 21 4,831 14,240 79,200 465 22 5,131 15,130 84,150 465 23 5,431 16,020 89,100 465 24 5,731 16,910 94,050 465 25 6,031 17,800 99,000 465 26,...,35 4,531 13,350 74,250 465 36,...,45 4,531 13,350 74,250 465

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Table2{2:ResultsfromupperboundproceduresforproblemGroups1and2 DynamicProg. PrimalAlgor. CPLEX Error Time Error Time Time Problem (%) (sec) (%) (sec) (sec) 1 0.00 1.39 1.15 0.17 29.55 2 0.00 1.39 0.85 0.17 31.57 3 0.00 1.39 5.34 0.17 64.56 4 0.00 1.39 6.25 0.17 86.63 5 0.00 1.39 10.26 0.17 99.35 6 0.00 1.39 15.14 0.17 120.37 7 0.00 1.39 15.15 0.16 143.32 8 0.00 1.39 14.03 0.17 192.07 9 0.00 1.39 16.44 0.16 255.40 10 0.00 1.39 33.06 0.16 376.69 11 0.00 0.01 47.37 0.01 0.93 12 0.00 0.03 30.61 0.02 6.30 13 0.00 0.13 17.72 0.04 16.88 14 0.00 0.34 15.25 0.05 36.66 15 0.00 0.75 12.25 0.07 65.33 16 0.00 2.63 11.17 0.13 144.59 17 0.00 4.31 10.58 0.17 203.35 errorsthatwepresentgivethedeviationoftheheuristicsandlowerboundsfromtheoptimalsolutionfoundfromsolvingtheMILPformulationusingCPLEX.ThelinearprogrammingrelaxationofextendedformulationissolvedusingCPLEXcallablelibrariesaswell. Wemeasurethetightnessofthelowerboundsasfollows: CPLEX100; CPLEX100:

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well,astheerrorswentupto52%forproblemclass32.However,itsrunningtimeforallproblemswaslessthan1cpusecond.BothalgorithmsaremuchfasterthanCPLEX. Table2{3:ResultsfromupperboundproceduresforproblemGroups3,4and5 DynamicProg. PrimalAlgor. CPLEX Error Time Error Time Time Problem (%) (sec) (%) (sec) (sec) 18 0.00 1.17 11.04 0.13 64.06 19 0.00 1.27 12.59 0.14 75.04 20 0.00 1.36 8.34 0.16 89.39 21 0.00 1.56 12.42 0.17 116.56 22 0.00 1.67 12.93 0.18 136.75 23 0.00 1.75 10.37 0.19 146.82 24 0.00 1.85 11.03 0.19 174.79 25 0.00 1.95 15.34 0.23 189.03 26 0.00 1.39 2.66 0.16 45.65 27 0.00 1.39 0.77 0.16 26.21 28 0.00 1.39 0.00 0.15 12.19 29 0.00 1.39 0.00 0.16 6.24 30 0.00 1.39 0.00 0.15 4.13 31 0.00 1.39 6.44 0.16 4.72 32 1.24 1.39 51.84 0.16 7.12 33 0.00 1.39 9.35 0.16 91.37 34 0.00 1.39 1.92 0.15 64.41 35 0.00 1.39 0.44 0.16 42.06 Thelinearprogrammingrelaxationoftheextendedformulationgeneratedsolutionsthatarelessthan0:1%fromoptimalforallproblemclasses.Thedualalgorithmprovidedtightlowerboundsaswell.Forallproblemclassesexceptproblemclass32,itgeneratedboundsthatarelessthan2%fromoptimal.However,thelinearprogrammingrelaxationoftheextendedformulationtookalmostasmuchtimeassolvingMILPformulation.Therunningtimeofthedualalgorithmforallproblemclasseswaslessthen1cpusecond. BasedontheresultspresentedinTables2{2to2{5,wecanseethataneectivealgorithmtosolveourproblemwouldcombinethedynamicprogrammingalgorithmtogenerateupperboundsandthedualalgorithmtogeneratelowerbounds.

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Table2{4:ResultsfromlowerboundproceduresforproblemGroups1and2 LinearProg. DualAlgor. Error Time Error Time Problem (%) (sec) (%) (sec) 1 0.01 24.88 0.09 0.10 2 0.01 26.32 0.07 0.09 3 0.00 65.41 0.20 0.09 5 0.00 138.10 0.31 0.10 6 0.00 175.31 0.30 0.10 7 0.00 217.25 0.36 0.10 8 0.00 312.41 0.22 0.10 9 0.00 484.41 0.28 0.10 10 0.00 688.55 0.38 0.10 11 0.00 0.49 1.72 0.00 12 0.00 6.00 1.07 0.01 13 0.00 19.39 0.37 0.02 14 0.02 45.46 0.63 0.04 15 0.01 87.04 0.34 0.06 16 0.00 193.17 0.41 0.14 17 0.00 282.01 0.31 0.18 Now,wewanttoprovidemoreinsightsintotheproblemcharacteristicsthataecttheperformanceoftheheuristics.Ithasbeenshowninotherstudies(HochbaumandSegev[60],Eksiogluetal.[34])thattheproblemdicultydependsonthevaluesofthexedcosts.Inproblemclasses1to10thelevelofproductionset-upcostsisincreased(whileeverythingelseisthesame).Thus,thetimeittooksolvingtheMILPformulationoftheproblem,aswellasthetimeittooksolvingthelinearprogrammingrelaxationofextendedformulation,increasedasthelevelofxedchargecostsincreased.However,thecomputationaltimeofthedual-primalanddynamicprogrammingalgorithmswerenotaectedbythechangeinsetupcosts.Thiscanbeexplainedbynotingthattheirrunningtimedependsonlyonthenumberoffacilitiesandthelengthofthetimeperiod. Fromtheresultsforproblemgroups2and3,onecanseethatasthenumberoftimeperiodsandthenumberoffacilitiesincrease,therunningtimeofthedynamicprogrammingalgorithmincreased.Problemclasses17and25havealmostthesame

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Table2{5:ResultsfromlowerboundproceduresforproblemGroups3,4and5 LinearProg. DualAlgor. Error Time Error Time Problem (%) (sec) (%) (sec) 18 0.00 85.30 0.37 0.07 19 0.00 92.83 0.31 0.08 20 0.00 113.25 0.35 0.09 21 0.01 153.80 0.32 0.10 22 0.00 164.49 0.39 0.11 23 0.00 190.94 0.29 0.12 24 0.01 213.88 0.42 0.13 25 0.01 248.09 0.32 0.14 26 0.01 38.39 0.12 0.09 27 0.01 23.56 0.04 0.09 28 0.00 16.75 0.00 0.10 29 0.00 14.13 0.00 0.10 30 0.00 12.57 0.00 0.10 31 0.01 17.31 1.48 0.09 32 0.16 19.55 13.19 0.09 33 0.00 99.44 0.19 0.10 34 0.00 50.44 0.04 0.10 35 0.00 25.14 0.00 0.09 numberofarcs;howeverthetimeittooktosolveproblemclass17isalmosttwicethetimeittooktosolveproblemclass25.Thereasonisthatthenumberoftimeperiodsinproblemclass17ishigher(therunningtimeofthisalgorithmisO(FT4)). Anotherinterestingobservationisthatallalgorithmsperformedverypoorlyinsolvingproblemclass32.Inthisproblemclasswehavegeneratedtheholdingcostsintheinterval[10;10].Forthisproblemclass,thequalityofbothlowerandupperboundsgeneratedisworsewhencomparedtotherestoftheproblems.Thedynamicprogrammingalgorithmsgavesolutionsthatare1:24%fromoptimalandprimalalgorithmgavesolutions13:19%fromoptimal.Thelowerboundsgeneratedusinglinearprogrammingrelaxationoftheextendedformulationwere0:16%fromoptimalandthedualalgorithmgaveanaverageerrorof51:84%.Thereasonforsuchaperformanceistheholdingcostsbeingnotrestrictedinsign.WuandGolbasi[106]

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Table2{6:ResultsfromupperboundproceduresforproblemGroup6 DynamicProg. PrimalAlgor. CPLEX Error Time Error Time Time Problem (%) (sec) (%) (sec) (sec) 36 1.17 1.41 3.31 0.17 3.68 37 1.05 1.40 4.42 0.15 3.77 38 1.46 1.40 20.21 0.17 4.27 39 1.54 1.40 30.90 0.16 4.79 40 1.24 1.40 51.84 0.17 7.07 41 0.84 1.39 66.16 0.17 8.68 42 0.40 1.40 83.05 0.17 7.13 43 0.41 1.39 147.85 0.16 10.47 44 0.00 1.39 684.71 0.16 17.26 45 0.00 1.39 719.46 0.16 23.31 showthatthemulti-facilitylot-sizingproblemisNP-hardincasethattheholdingcostsarenotrestrictedinsign. Wewantedtofurtherinvestigatetheperformanceofthealgorithmsforthecasewhentheholdingcostsarenotrestrictedinsign.Wegeneratedasixthgroupofproblemsthathaveholdingcostsuniformlydistributedintheinterval[10;10].Were-ranproblemclasses1to10creating10newclassproblems(classproblems36to45). Table2{7:ResultsfromlowerboundproceduresforproblemGroup6 LinearProg. DualAlgor. Error Time Error Time Problem (%) (sec) (%) (sec) 36 0.00 12.33 3.11 0.10 37 0.00 13.67 2.96 0.09 38 0.05 15.68 6.73 0.10 39 0.00 18.03 8.42 0.09 40 0.16 18.98 13.19 0.10 41 0.10 18.04 14.78 0.09 42 0.16 17.18 15.81 0.10 43 0.39 22.82 17.11 0.10 44 0.23 47.38 54.89 0.10 45 0.00 83.41 26.53 0.10

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Tables2{6and2{7presenttheresultsforthesixthgroupofproblems.Forthissetofproblems,thedynamicprogrammingalgorithmperformedwell.Itsmaximumerrorwaslessthan1:54%fromoptimalandtherunningtimeaveraged1:40CPUseconds.Theprimalalgorithmperformedverypoorlyforthisgroupofproblems. Thelinearprogrammingrelaxationoftheextendedformulationgavetightlowerbounds.Themaximumerrorpresentedforthisgroupofproblemsis0:39%.However,therunningtimeofthisalgorithmiscomparabletothetimeittookCPLEXtosolvethecorrespondingMILPformulationoftheproblem.Thedualalgorithmalsoperformedpoorly. Inournalgroupofproblems(problemgroupseven),weconsiderthedemandtoshowseasonalitypattern.Demandsaregeneratedasfollows:bt=200+zt+sin[2 d(t+d=4)] where, ThesedemandsaregeneratedinthesamewayasinBakeretal.[5]andChenetal.[22].Inourtestproblemswetake=67,=125andd=12.Table2{8presentsthecharacteristicsoftheproblemsgenerated.TheerrorspresentedinTable2{9arecalculatedasfollows:Gap(%)=UpperBoundLowerBoundfromDualAlg. LowerBoundfromDualAlg.100:

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Table2{8:CharacteristicsofproblemGroup7 Problem Facilities Periods Nodes Arcs 20 24 505 6,480 300 47 30 24 745 9,720 300 48 40 24 985 12,960 300 49 20 48 1,009 24,480 1,176 50 30 48 1,489 36,720 1,176 51 40 48 1,969 48,960 1,176 52 20 96 2,017 95,040 4,656 53 30 96 2,977 142,560 4,656 54 40 96 3,937 190,080 4,656 55 20 192 4,033 374,400 18,528 56 30 192 5,953 561,600 18,528 57 40 192 7,873 748,800 18,528 58 20 384 8,065 1,486,080 73,920 59 30 384 11,905 2,229,120 73,920 60 40 384 15,745 2,972,160 73,920 61 20 768 16,129 5,921,280 295,296 62 30 768 23,809 8,881,920 295,296 63 40 768 31,489 11,842,560 295,296 torunproblemclass56,andforthesameproblemittookthedynamicprogrammingalgorithmonaverage354cpuseconds.Theprimalalgorithmalsoperformedverywell.Themaximumerrorpresentedis0:352%.Therunningtimeoftheprimalalgorithmwaslessthen1cpusecondforthissetofproblems. Forproblemclasses57to63wedonothavetheoptimalsolutions.CPLEXfailedtosolvetheseproblems,becauseoftheirsize.Therefore,weusethelowerboundsgeneratedfromdualalgorithmtocalculatetheerrorgaps.Thedynamicprogrammingalgorithmgavesverygoodsolutions.Themaximumerrorgapwas0:129%,buttherunningtimeofthealgorithmwentashighas115;453cpuseconds.Theprimalalgorithmgaveamaximumgapof0:268%andmaximumrunningtime14:05cpuseconds.

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Table2{9:ResultsforproblemGroup7 DynamicProg. PrimalAlgor. CPLEX Problem Error Gap Time Error Gap Time Time (%) (%) (sec) (%) (%) (sec) (sec) 46 0.000 0.028 0.08 0.290 0.318 0.02 0.43 47 0.000 0.016 0.12 0.163 0.180 0.03 0.68 48 0.004 0.005 0.17 0.155 0.156 0.03 0.87 49 0.001 0.018 1.08 0.167 0.184 0.05 1.49 50 0.002 0.013 1.63 0.190 0.200 0.06 2.74 51 0.008 0.009 2.17 0.106 0.107 0.07 3.58 52 0.004 0.005 15.49 0.097 0.099 0.13 6.64 53 0.004 0.021 23.25 0.274 0.290 0.18 11.73 54 0.002 0.027 31.16 0.352 0.377 0.23 20.48 55 0.065 0.067 235.32 0.090 0.093 0.44 30.25 56 0.048 0.054 353.44 0.127 0.133 0.64 52.58 57 N/A 0.066 473.03 N/A 0.246 0.82 N/A 58 N/A 0.121 3,654.24 N/A 0.117 1.69 N/A 59 N/A 0.129 5,502.14 N/A 0.151 2.51 N/A 60 N/A 0.096 7,359.58 N/A 0.268 3.31 N/A 61 N/A 0.081 57,550.00 N/A 0.227 6.84 N/A 62 N/A 0.093 86,604.00 N/A 0.163 10.42 N/A 63 N/A 0.108 115,453.00 N/A 0.206 14.05 N/A TheresultsofTable2{9showthatthequalityofthesolutionsgeneratedandthequalityoflowerboundsisverygood.Inparticulartherunningtimesofprimalalgorithmareverysmall. 2.9 Conclusions Inthischapterwediscussthemulti-facilitylot-sizingproblem.Weproposethefollowingheuristicapproachestosolvetheproblem:dynamicprogrammingalgorithm,acuttingplanealgorithmandaprimal-dualalgorithm.Forthisproblemwealsogiveadierentformulationthatwerefertoastheextendedproblemformulation.Thelinearprogrammingrelaxationofextendedformulationgiveslowerboundsthatareatleastashighasthelowerboundsfromlinearprogrammingrelaxationof\original"problemformulation.Wepresentasetofvalidinequalitiesforthemulti-facilitylot-sizingproblemandshowthattheyarefacetdeninginequalities.

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Wetestedtheperformanceoftheheuristicsonawiderangeofrandomlygeneratedproblems.Thedynamicprogrammingalgorithmgavegoodqualitysolutionsforallprobleminstances.Theerror(calculatedwithrespecttotheoptimalsolutionorwithrespecttoalowerboundincasethattheoptimalsolutiondoesnotexists)reportedislessthan1:6%.TherunningtimeofthedynamicprogrammingalgorithmisO(FT4).Thisexplainstherelativelyhighrunningtimesofthisalgorithmfortheseventhgroupofproblems.Linearprogrammingrelaxationofextendedformulationgavehighqualitylowerbounds.Themaximumerrorreportedforproblemclasses1to45is0:39%fromoptimal. Ithasbeenshownthatthemulti-facilitylot-sizingproblemisNP-hardwhenholdingcostsarenotrestrictedinsign(WuandGolbasi[106]).Thisexplainsthefactthatallalgorithmsgavetheirworstresultsforproblemclasses32and36to45.Inparticularprimal-dualalgorithmperformedpoorlyforthesetypeofproblems. Forproblems57to63CPLEXranoutofmemoryandfailedtoprovideanintegersolution.Primal-dualalgorithmhowevergavesolutionswithin0:377%errorgapinlessthan14CPUseconds.

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3.1 Multi-Commodity, Multi-Facility Lot-Sizing Problem Inthepreviouschapterwediscussedthesingle-commodity,multi-facilitylot-sizingproblem.Inpractice,managementofproduction,inventoryandtransportationinaplanttypicallyinvolvescoordinatingdecisionsforanumberofcommodities.Inthissectionweanalyzeandproposesolutionapproachesforthemulti-commodity,multi-facilitylot-sizingproblem. Themulti-commodity,multi-facilitylot-sizingproblemconsistsofndingaproductionandtransportationscheduleforanumberofcommoditiesoveranitetimehorizontosatisfyknowndemandrequirementswithoutallowingbacklogging.Thescheduleissuchthatthetotalproduction,inventoryholding,transportationandset-upcostsareminimized.Thesecostsmayvarybyproduct,facilityandtimeperiod.Productioncapacitiesandjointordercoststietogetherdierentcommoditiesandnecessitatecarefulcoordinationoftheirproductionschedules.Itiseasytoseethatwithoutthepresenceofjointordercostsorproductioncapacities,thisproblemcanbehandledbysolvingeachcommoditysubproblemseparately.Inthissectionweconsiderthemulti-commodityproblemwithonlyproductioncapacities(notransportationcapacitiesandnojointordercosts)andrefertoitasthecapacitatedmulti-commodity,multi-facilitylot-sizingproblem. Alargeamountofworkhasbeendevotedtothecapacitatedmulti-commodity,single-facilitylot-sizingproblemsinceitisthecoreproblemintheAggregatedProductionPlanningmodels.Solutionstolot-sizingproblemsareofteninputstoMasterProductionSchedulesandsubsequentlytoMaterialsRequirementsPlanning 54

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(MRP)systemsina\push"typemanufacturingenvironment(seeBhatnagaretal.[13]andNahmias[85]forareviewofthesemodels). Ithasbeenshownthat,eveninthesingle-commoditycase,thecapacitatedlot-sizingproblemisNP-hard[42].BitranandYanasse[15]showedthatthecapacitatedmulti-commodityproblemisNP-hardandChenandThizy[24]showedthattheproblemisstronglyNP-hard.HeuristicapproachestosolvetheproblemweredevelopedbyLambrechtandVanderVeken[71],DixonandSilver[31]andMaesandvanWassenhove[77,78].ThemethodproposedbyBaranyetal.[10]solvesthesingle-facility,capacitatedproblemoptimallyusingacuttingplanealgorithmfollowedbyabranchandboundprocedure.Thesealgorithmsdonotconsidersetuptimes.Threegroupsofresearcherspioneeredworkonthecapacitatedeconomiclot-sizingproblemwithsetuptimes.Manne[81]usedalinearprogrammingmodel;DzielinskiandGomory[33]usedDantzig-Wolfedecomposition;andLasdonandTerjung[72]usedageneralizedupperboundingprocedure. EppenandMartin[36]providedanalternativeformulationofthecapacitated,multi-commoditylot-sizingproblemknownastheshortestpathformulation.Theyshowedthatthelinearprogrammingrelaxationoftheshortestpathformulationisveryeectiveingeneratinglowerbounds,andtheboundsareequaltothosethatcouldbegeneratedusingLagrangeanrelaxationorcolumngeneration. Sofar,promisingheuristicapproachestosolvethecapacitatedmulti-commoditylot-sizingproblemseemtobethosebasedonLagrangeanrelaxation.ThizyandvanWassenhove[96]andTrigeiroetal.[98]proposedaLagrangeanrelaxationofthecapacityconstraints.TheyupdatedtheLagrangeanmultipliersusingasubgradientapproachandproposedaheuristictoobtainfeasiblesolutions.Merleetal.[32]usedaLagrangeanrelaxationapproachaswell;however,theyupdatedtheLagrangeanmultipliersusingacuttingplanemethod.ChenandThizy[24]analyzedandcomparedthequalityofdierentlowerboundscalculatedusingrelaxationmethodssuchasLagrangeanrelaxationwithrespecttodierentsetsofconstraints

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andlinearprogrammingrelaxation.MillarandYang[82]proposedaLagrangeandecompositionproceduretosolvethecapacitatedmulti-commoditylot-sizingproblem.TheirapproachdecomposestheproblemintoatransportationproblemandKindependentsingle-commoditylot-sizingproblems.Thizy[95]analyzedthequalityofsolutionsfromLagrangeandecompositiononoriginalproblemformulationandshortestpathformulationusingpolyhedralarguments. 3.1.1 Problem Formulation Inmanypracticalsituations,coordinationofproduction,inventoryandtransportationdecisionsinvolvesdierentcommodities.Thiscomplicatestheproblemconsiderably.Inthissectionwediscussthemulti-commodity,multi-facilitylot-sizingproblem.Thisisageneralizationoftheclassicalcapacitated,multi-commoditylot-sizingproblem.Weaddtotheclassicalproblemanewdimension,thefacilityselectionproblem.Inaddition,weconsidertransportationcostsandtheireectonlot-sizingdecisions. AssumethatthereareKcommoditiesthatneedtobeproduced.EachcommodityfacesaknowndemandduringeachofthenextTperiods.Notethatcommoditiesshareacommonproductionresourcewithitemspecicsetupcosts.Thegoalistodecideontheproductionscheduleforeachcommodity,suchthatproduction,transportationandinventorycostsinallthefacilitiesareminimized,demandissatisedandcapacityconstraintsarenotviolated.Foreachcommodityk,wedenethefollowinginputdata: sitkproductionset-upcostforcommoditykatfacilityiinperiodt hitkinventoryunitcostforcommoditykatfacilityiinperiodt citktransportationunitcostforcommoditykatfacilityiinperiodt btkdemandinperiodtforcommodityk vitproductioncapacityatfacilityiinperiodt xitkamountofcommodityktransportedfromfacilityiinperiodt

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endofperiodt yitkabinaryvariablethatequals1ifthereisaproductionset-upfor commoditykatthefacilityiinperiodt. AnMILPformulationoftheproblemisthefollowing:minimizeFXi=1TXt=1KXk=1(pitkqitk+sitkyitk+citkxitk+hitkIitk) subjectto(MC)Ii;t1;k+qitk=xitk+Iitki=1;:::;F;t=1;:::;T;k=1;:::;K NotethatbtTkisthetotaldemandforcommoditykfromperiodttoT.Constraints(3.1)and(3.2)aretheowconservationconstraintsattheproductionanddemandpointsrespectively.Constraints(3.3)aretheproductioncapacityconstraints.Theseconstraintsreectthemulti-commoditynatureofourproblem.Iftheyareabsent,theproblemcanbedecomposedintoKsinglecommodityproblems. Weassumethatinitialandendinginventoriesarezeroforallitems.Thereisnolossofgeneralityinthisassumption,sincewecanresetforeachcommoditythegivenlevelofinitial(ending)inventoryIi1k(IiTk)atzerobyremovingIi1kfromthedemandoftherstperiods(addingIiTktothedemandofthelastperiod),thusobtainingtheadjusteddemandsfort=1;:::;Tandk=1;:::;K. WeproposeaLagrangeandecomposition-basedheuristictosolvethemulti-commodityproblem.Thedecompositionisperformedontheextendedproblem

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formulation.Theextendedformulation,similartothesinglecommodityproblem,requiressplittingtheproductionvariablesqitkbydestinationintovariablesqitk(=t;:::;T),wheredenotestheperiodforwhichproductiontakesplace.Giventhis,wehaveqitk=TX=tqitkxitk=tXs=1qistkIitk=tXs=1TX=tqisktXs=1qistkqitkdenotestheproductionquantityforcommoditykfromfacilityi,intimeperiodtforperiod.Theextendedformulationof(MC)isthefollowing:minimizeFXi=1TXt=1KXk=1(TX=t subjectto(Ex-MC)FXi=1Xt=1qitk=dk=1;:::;T;k=1;:::;K

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3.1.2 Linear Programming Relaxation Thelinearprogrammingrelaxation(LP)of(MC)isobtainedbyreplacingconstraints(3.6)withyitk0fori=1;:::;F;t=1;:::;T;k=1;:::;K: thetotalset-upcostofthissolutionwillbegreaterthanKXk=1mini=1;:::;F;t=1;:::;Tsitk: Inordertoprove(3.13),let(q;I;x;y)beanoptimalsolutionto(LP)and(q;I;x;dye)afeasiblesolutionto(MC)obtainedbyxingthefractionalcomponentsofyto1.Then,wehavethefollowingrelationship:

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Nowwepresentaclassofproblemsforwhichtheboundsderivedin(3.12)and(3.13)areobtainedasymptoticallywithrespectto.Theclassofproblemsweconsiderhasthefollowingproperties: Demands:btk=;bTk=1=;fort=1;:::;T1andk=1;:::;K:

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Thus,y1tk=8><>:1ift=T;k=1;:::;K Valid Inequalities Considertheformulation(MC)ofthemulti-commodityproblem.Letkdenotethesetoffeasiblesolutionstothek-thsingle-commodityproblemandletLPkdenotethesetofthefeasiblesolutionstothelinearprogrammingrelaxationofthek-thsingle-commodityproblem.Wecanre-statethecapacitatedmulti-commodityproblemasfollows:minimizeFXi=1TXt=1KXk=1(pitkqitk+sitkyitk+citkxitk+hitkIitk)subjectto(qk;xk;Ik;yk)2k;k=1;:::;KPKk=1qitkvitt=1;:::;T;i=1;:::;F

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subjectto(MC)(qk;xk;Ik;yk)2LPkk=1;:::;K whereLPk\(y2f0;1g)=k. Givenafractionalpoint(q1;x1;I1;y1);:::;(qK;xK;IK;yK),thegoalistondavalidinequalitythatcutsothisnonintegerpointfromthefeasibleregionoflinearprogrammingrelaxationof(MC).Ignoringconstraints(3.16),theproblemdecomposesintoKsingle-commodity,multi-facilitylot-sizingproblems.Forthesingle-commodityproblem,wehaveproposedinSection2.6.1asetofvalidinequalities.Letbethefeasibleregionofthemulti-commodityowproblem,thenco()\Kk=1co(k) Thisimpliesthatthevalidinequalitiesforthesinglecommodityproblem,arevalidforthemulti-commodityproblemaswell.Thus,onecanusetheseinequalitiestocheckforeachcommoditykifpoint(qk;xk;Ik;yk)canbecutofrom. 3.1.4 Lagrangean Decomposition Heuristic InthissectionwediscussaLagrangeandecomposition-basedheuristicthatweusedtosolvethecapacitatedmulti-commodity,multi-facilitylot-sizingproblem.Lagrangeanrelaxationisaclassicalmethodforsolvingintegerprogrammingproblems(Georion[49],Wolsey[105]).Thismethodhasbeenusedtosolvevariousnetworkowproblems.HeldandKarp[56,57]successfullyusedLagrangeanrelaxationtosolvethetravelingsalesmanproblem;Fisher[40]usedthismethodtosolveamachineschedulingproblem;RossandSoland[92]appliedthismethodtothegeneralassignmentproblem.HolmbergandYuan[62],HolmbergandHellstrand[61],andBalakrishnanetal.[8]usedLagrangeanrelaxationbasedapproachesfornetworkdesignproblems.

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WestartourdiscussionwithareviewoftheLagrangeanrelaxationapproachanditsextension,Lagrangeandecomposition.WecontinuethenwithadetaileddescriptionoftheLagrangeandecompositionbasedheuristicwehaveusedtogenerateupperandlowerboundsforthemulti-commodity,multi-facilitylot-sizingproblem. Review of the method .Georion[49]formallydenesarelaxationofanoptimizationproblem(P)asfollows:minff(x)jx2Xg; Lagrangeanrelaxationhasshowntobeapowerfulfamilyoftoolsforsolvingintegerprogrammingproblemsapproximately.Assumethatproblem(P)isoftheformminxff(x)jAxb;Cxd;x2Xg;

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complicated,inthesensethatproblem(P)withoutthissetofconstraintswouldbemucheasiertosolve. LetbeanonnegativevectorofweightscalledLagrangeanmultipliers.TheLagrangeanrelaxationLR(x;)of(P)istheproblemminxff(x)+(bAx)jCxd;x2Xg; ofndingthehighestlowerboundiscalledtheLagrangeandual(D)of(P)withrespecttothecomplicatingconstraintsAxb. Letv()denotetheoptimalobjectivefunctionvalueofproblem(*).ThefollowingtheorembyGeorionisanimportantresult: (i) IfCofxjCxd;x2Xg=fxjCxdg;thenv(P)v(PR)=v(D)=v(LP):

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InthiscasetheLagrangeanrelaxationhastheintegralityproperty,andthe(D)boundisequaltothe(LP)bound. (ii) IfCofxjCxd;x2XgfxjCxdg;thenv(P)v(PR)=v(D)v(LP); ThissuggeststhatindecidingonhowtoimplementtheLagrangeanrelaxationmethod,oneshouldconsiderthefollowingpropertiesoftheset=fxjCxdg:(i)shouldbesimpleenoughthattheresultingoptimizationsubproblemsarenotcomputationallyintractable(usuallydecomposesintosimplersubsets,=j2Jj),(ii)shouldbecomplexenough,suchthatthesubsetsjdonothavetheintegralityproperty.Otherwise,thelowerboundsgeneratedwouldbeequaltothelowerboundsfromthecorrespondinglinearprogrammingrelaxation. TheLagrangeanfunctionz()=v(LR(x;))isanimplicitfunctionof,andz(),thelowerenvelopeofafamilyoflinearfunctionsof,isaconcavefunctionof,withbreak-pointswhereitisnotdierentiable. AnextensiontoLagrangeanrelaxationistheLagrangeandecompositionmethodintroducedbyGuignardandKim[52].DierentthanLagrangeanrelaxation,Lagrangeandecompositiondoesnotremovethecomplicatingconstraints,butdecomposestheproblemintotwosubproblemsthatcollectivelysharetheconstraintsoftheoriginalproblem.Thisisachievedbyintroducinganewsetofvariablesz,suchthatx=z.Then,problem(P)readsminx;zff(x)jAxb;Czd;x2X;z=x;z2Xg:

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=minxff(x)xjAxb;x2Xg+minzfzjCzd;z2Xg: Signicantforthedecompositionisthattheinnerminimizationproblemiseasiertosolvethantheoriginalproblem(P).RecallthatLagrangeanrelaxation/decompositionisaniterativemethod,theinnerminimizationproblemneedingtopossiblybesolvedseveraltimes.Inthecasethattheinnerminimizationproblemisstilldicultandrequiresaconsiderableamountofcomputationaleortstosolvetooptimality,apracticethatcanbefollowedisndinggoodlowerboundsinstead.ObviouslythiswillaectthequalityofthelowerboundfromtheLagrangean

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relaxation/decomposition.Lettingv(P)betheoptimalsolutiontoproblem(P),thenv(P)v(D). Let!()denotealowerboundand()anupperboundforproblem().If,foreverywedonotsolvetheLagrangeandecompositionproblemLD(x;z;)optimally,butweratherprovidealowerbound,thefollowingholds:v(P)v(D)=maxv(LD(x;z;))max!(LD(x;z;)):max!(LD(x;z;))willstillbealowerboundforproblem(P),howevernotasgoodboundasv(D).Incasethatweuseaheuristicproceduretondafeasiblesolutiontotheinneroptimizationproblem,thenv(D)max(LD(x;z;)).However,wearenotsureanymoreifmax(LD(x;z;))givesalowerboundforproblem(P)sinceoneofthefollowingmayhappen:v(P)max(LD(x;z;))orv(P)max(LD(x;z;)). Inthecasewhenalowerboundingprocedureisusedinsteadofsolvingtheinnerminimizationproblem,itisimportanttoidentifythequalityoftheseboundscomparedtothelowerboundsfromthelinearprogrammingrelaxation.Ifthelowerboundguaranteesthatmax!(LD(x;z;))v(LP) andtherunningtimeofthisprocedureoutperformslinearprogrammingrelaxation,oneisbetterousingtheLagrangeanrelaxation/decompositionmethodtondlowerboundstoproblem(P). InthenextsectionwedescribetheLagrangeandecompositionalgorithmweusedtosolvethemulti-commodity,multi-facilitylot-sizingproblem. 3.1.5 Outline of the Algorithm Considertheextendedproblemformulation(Ex-MC).Thebasicideaofourdecompositionistoseparatethecapacitated,multi-commodityproblemintosubproblemsthatarecomputationallyeasiertosolvethantheoriginalproblem.

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Therearemanywaysonecandothat;howeverweaimtodecomposetheprobleminsuchawaythatithasinterestingmanagerialimplicationsaswell. Wedecomposetheproblemintotwosubproblems.Therstsubproblemconsistsoftheowconservationconstraintsandtheintegralityconstraints.Thissubproblemcanbefurtherdecomposedbycommodity.Thesinglecommoditysub-subproblemshavethespecialstructureofthesingle-commodity,multi-facilitylot-sizingproblemanalyzedinChapter2.Thesecondsubproblemconsistsoftheowconservationconstraintsandthecapacityconstraints.Inthisdecomposition,therstsubproblemconsistsofacollectionofMILPsandthesecondsubproblemisalinearprogram. Belowwegiveanequivalentformulationofthecapacitatedmulti-commodity,multi-facilitylot-sizingproblem.Weintroducethecontinuousvariableszitkthataresimply\copies"oftheproductionvariablesqitk.Thisallowsfortheduplicationofsomeoftheconstraints.minimizeFXi=1TXt=1KXk=1[TX=t subjectto (3.7),(3.9),3.10),and(3.11) Itisclearthattheaboveformulationisequivalenttoformulation(Ex-MC).Relaxingthe\copy"constraints(3.18)andmovingthemtotheobjectivefunctionyieldsthefollowingLagrangeandecompositionproblem:minimizeFXi=1TXt=1KXk=1[TX=t(

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subjectto(LD(x,z,)) (3.7),(3.9),(3.10),(3.11),(3.19),(3.20),and(3.21). TheLagrangeandecomposition(LD(x,z,))problemcannowbeseparatedintothefollowingtwosubproblems:minimizeFXi=1TXt=1KXk=1[TX=t( subjectto(SP1) (3.7),(3.9),(3.10),(3.11), andminimizeFXi=1TXt=1KXk=1TX=titkzitk (3.19),(3.20),and(3.21). ThecorrespondingLagrangeandual(LD)problemisthefollowing:maxv(LD(x;z;)) NotethatunderthegeneralframeworkofLagrangeandecompositionmethod,constraints(3.7)donotneedtobeduplicatedforbothsubproblems.However,computationalexperienceindicatesthataslongastheaddedconstraintsdonotaddcomputationalburdentothesubproblems,constraintduplicationimprovesthespeedoftheconvergenceandyieldsbetterlowerbounds(GuignardandKim[52]). ThepotentialcomputationalsavingsfromusingthedualalgorithmissignicantsincethesubgradientsearchrequiressolvingKsinglecommoditysubproblemsineachiteration.Ifaprobleminstanceconsiders30commoditiesforexample,thisresultsinupto15;000callstothesubproblem(themaximumnumberofiterationsweuseis500).InSection3.1.7wecomparetheperformanceoftheLagrangeandecompositionapproachwhenthesubproblemsaresolvedtooptimalityversusthecasewhenthedualalgorithmisused.

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Subproblem(SP1)canbedecomposedbycommodity.Eachsub-subproblemkhasthefollowingMILPformulation:minimizeFXi=1TXt=1(TX=t( subjectto(SP1k)PFi=1Pt=1qit=d=1;:::;Tqitbyit0i=1;:::;F;t=1;:::;T;tqit0i=1;:::;F;t=1;:::;T;tyit2f0;1gi=1;:::;F;t=1;:::;T Ontheotherhand,subproblem(SP2)issimplyalinearprogram.WeuseCPLEXcallablelibrariestosolvethisproblem.Thesolutionof(SP2)isfeasibleforthe(Ex-MC)problem.However,sincetheset-upcostsarenotconsideredintheformulation,weuseasimpleproceduretocalculateanupperbound(Figure3{1).

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Letzsbetheoptimalsolutiontosubproblem(SP2)initerations. InitializeUBs=0;sumitk=0fori=1;:::;F;t=1;:::;T;k=1;:::;K. ifzsitk0thenUBs=UBs+ Figure3{1:Upperboundprocedure feasiblesolutionscorrespondingtolinearprogrammingrelaxationofsubproblem(SP1)and2bethesetoffeasiblesolutionstolinearprogrammingrelaxationofsubproblem(SP2). GuignardandKim[52]showthat\OptimizingtheLagrangeandecompositiondualisequivalenttooptimizingtheprimalobjectivefunctionontheintersectionoftheconvexhullsoftheconstraintsets"(Theorem3.1.3).Therefore,optimizing(LD)isthesameasoptimizingminimizeFXi=1TXt=1KXk=1[TX=t subjecttoq;y2Cofq;yjq;y21\y2f0;1ggTq2Cofqjq22g subjectto(LP-Ex)q;y21\q22:

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SinceqarecontinuousvariablesCofqjq22g=2;

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ThelowerboundsgeneratedfromtheLagrangeandecompositionheuristicarethesameastheboundsfromLagrangeanrelaxationwithrespecttothecapacityconstraints,howeverwechosetoimplementtheLagrangeandecompositionapproach.Therearetworeasonswechoosetodoso(i)Thedecompositionschemeprovidesfeasiblesolutionstoproblem(Ex-MC)ateveryiteration(ii)Thedecompositionconvergesfaster. Subgradient optimization algorithm .Itiswell-known(NemhauserandWolsey[86])thattheLagrangeandualfunctionisconcaveandnondierentiable.TomaximizeitandconsequentlyderivethebestLagrangeanlowerboundweuseasubgradientoptimizationmethod.FormoredetailsonthesubgradientoptimizationmethodseeHeld,WolfeandCrowder[58]andCrowder[28],andforasurveyofnondierentiableoptimizationtechniquesseeLemarechal[73]. SubgradientoptimizationisaniterativemethodinwhichstepsaretakenalongthenegativeofthesubgradientoftheLagrangeanfunctionz()(z()=v(LD(x;z;)).Ateachiterations,wecalculatetheLagrangeanmultipliersitkusingthefollowingequation:s+1itk=sitk+us(qitkzitk);(3.22) whereus=s(minUBmaxLB) InordertondasubgradientdirectionateachstepoftheLagrangeandecompositionprocedure,weneedtondafeasiblesolutiontosubproblem(SP1).

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If,thenSTOP Step4: OtherwisegotoStep2 Thedualalgorithmprovidesonlyalowerboundto(Ex-MC),butdoesnotprovideafeasibleprimalsolution.Therefore,weusetheprimalalgorithm(Section2.5)tondafeasiblesolutiontosubproblems(SP1k). Inourcomputationalexperiments,weterminatethealgorithmifoneofthefollowinghappens:(i)thebestlowerboundisequaltothebestupperbound(theoptimalsolutionisfound),(ii)thenumberofiterationsreachesaprespeciedbound,(iii)thescalarsislessthanorequalto(asmallnumberclosetozero).Figure3{2presentsthestepsoftheLagrangeandecompositionalgorithm. 3.1.6 Managerial Interpretation of the Decomposition InthissectionwediscussthemanagerialinsightsofthedecompositionprocedureproposedinSection3.1.5.ThechoiceoftheLagrangeandecompositionschemewepresentismotivatednotonlybyitscomputationcapability,butalsobyitsinterestingmanagerialimplications.Severalstudies(BurtonandObel[19]andJornstenandLeisten[64])haverecognizedthatmathematicaldecompositionoftenleadstoinsightsforgeneralmodellingstrategiesandevennewdecisionstructures.Inthisdiscussion

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werefertosubproblem(SP1)astheproduct(commodity)subproblemand(SP2)astheresourcesubproblem. TheLagrangeandecompositionweproposehelpsunderstandingandsolvingmanagerialissuesthatariseinmulti-facilitymanufacturingplanning.Supposeweconsidertheresourcesubproblemasadecisionproblemforaproductionmanagerwhosupervisesmultiplefacilities,andeachproductsubproblemasadecisionproblemforaproductmanager.Therefore,thedecompositioncanbeviewedasadecisionsystemwhereproductmanagerscompeteforresourcecapacityavailablefrommanufacturingfacilities.Theproductionmanager,ontheotherside,representstheinterestsofecientlyallocatingresourcesfrommultiplefacilitiestotheproducts.Oftenthesolutionsproposedbytheproductionmanagerwillnotagreewiththeindividualsolutionsofproductmanagers.AsearchbasedontheLagrangeanmultipliersbasicallypenalizestheirdierences,whileadjustingthepenaltyvectoriteratively.Thisprocessstopswhenthedegreeofdisagreement(thedualitygap)isacceptablylow,orwhenfurtherimprovementisunlikely.SeeWuandGolbasi[106]forasimilardiscussiononarelatedproblem. 3.1.7 Computational Results InthissectionwehavetestedtheperformanceoftheLagrangeanDecompositionalgorithmonalargegroupofrandomlygeneratedproblems.WeusetheCPLEXcallablelibrariestosolvetheMILPformulation(Ex-MC).TheCPLEXrunswerestoppedwheneveraguaranteederrorboundof1%orlesswasachieved,allowingforamaximumCPUtimeof5;000seconds(or10;000secondsdependingonproblemsize).WeuseCPLEXtosolvethelinearprogrammingrelaxationof(Ex-MC)andsubproblem(SP2).Oneofthefactorsthataectstheproblemcomplexityisthetightnessoftheupperboundsonproductionarcs.Ifthearcsareverytight,thereexistonlyafewfeasiblesolutions,andthismakesthesearchfortheoptimalsolutioneasy.Ontheotherhand,ifthearccapacitiesaresoloosewecouldremovethemfrom

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Table3{1:Problemcharacteristics Problem Nodes Arcs Problem Nodes Arcs 1 1,680 6,000 14 2,280 8,400 2 2,240 8,000 15 2,430 9,000 3 2,800 10,000 16 3,330 19,500 4 3,360 12,000 17 4,980 40,500 5 3,920 14,000 18 6,630 69,000 6 4,480 16,000 19 8,280 105,000 7;:::;10 1,680 6,000 20 9,930 148,500 11 1,830 6,600 21 11,580 199,500 12 1,980 7,200 22;:::;25 1,680 6,000 13 2,130 7,800 theproblemformulation,theproblemlosesitsmulti-commoditynature,anditcanbedecomposedintoKsingle-commodityproblems. Inordertocreatechallengingtestproblems,wegeneratedtheupperboundsinthefollowingway:AnecessaryconditionforfeasibilityoftheproblemisFXi=1tX=1viKXk=1tX=1bk8t=1;:::;T: Undertheassumptionthatallfacilitiesinanytimeperiodhavethesameproductioncapacity v; FmaxtPKk=1Pt=1bk

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Usingthisprocedurewegeneratedchallenging(butfeasible)testproblemswithtightcapacityconstraints.Westartourcomputationalexperimentsbygeneratinganominalcaseproblemwiththefollowingcharacteristics: Wealterthenominalcasebychangingproblemcharacteristicstogenerateadditionalproblems.Foreachproblemclasswegenerate20instances.Firstwechangethenumberofcommoditiesfrom30to40,50,60,70,and80generatingsixproblemclasses(problemclasses1to6;problemclass1isthenominalcase). Inthesecondgroupofproblemswechangethecapacitytightnesscoecient()to1:1,1:2,1:4,and1:5(problemclasses7to10).Inthethirdgroupofproblemswechangethenumberoffacilitiesfrom10(thenominalcase)to11,12,13,14and15(problemclasses11to15). Wealsorantheprogramforproblemswithdierentlengthsofthetimehorizon.Inproblemclasses16to21wechangethenumberofperiodsto10,15,20,25,30,and35.Finally,wechangethelevelofthexedchargecosttositU[200;300],U[600;900],U[900;1500]andU[1200;1500](problemclasses22to25). InimplementingtheLagrangeandecompositionalgorithm,forallprobleminstancesweset=1:8.isreducedby20%ifthereisnoimprovementinthelast5iterations.Wesetalimitof500iterationsfortheLagrangeandecompositionalgorithm.Aswehavementionedearlierinthissection,weareinterestedintheeectivenessoftheprimal-dualalgorithmasaheuristicinthecontextofthesubgradientsearchalgorithm.Therefore,wepresentresultsfromtheLagrangeandecompositionalgorithmwhere

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Table3{2:Qualityofupperbounds(in%)fromLagrangeandecomposition Problem CPLEX Scheme1 Scheme2 Scheme3 1 0.06 1.92 1.79 1.82 2 0.05 1.22 1.14 1.16 3 0.01 0.83 0.76 0.76 4 0.01 0.62 0.50 0.52 5 0.01 0.57 0.40 0.42 6 0.01 0.53 0.32 0.32 7 1.44 3.67 3.61 3.61 8 0.38 2.61 2.60 2.53 9 0.01 1.36 1.25 1.24 10 0.01 1.01 0.83 0.83 11 0.08 2.54 2.49 2.49 12 0.17 2.86 2.84 2.79 13 0.21 3.53 3.49 3.50 14 0.34 4.56 4.52 4.49 15 0.50 4.86 4.76 4.76 16 0.24 2.03 1.93 1.95 17 0.37 2.20 2.12 2.12 18 0.48 2.28 2.20 2.22 19 0.53 2.28 2.21 2.20 20 0.57 2.34 2.28 2.29 21 0.55 2.41 2.35 2.32 22 0.02 1.25 1.20 1.20 23 0.04 3.37 3.13 3.21 24 0.22 4.96 4.85 4.75 25 0.77 6.54 6.38 6.40

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Theerrorboundsfromthethreeschemesaswellasthelinearprogrammingrelaxationof(Ex-MC),arepresentedinTables3{2and3{3.ThequalityoftheupperboundswegeneratedusingtheLagrangeandecompositionalgorithmismeasuredasfollows:Error(%)=UpperBoundCPLEXLowerBound CPLEXLowerBound100; CPLEXUpperBound100: TheresultsfromSection2.8showedthatprimal-dualalgorithmtooklittletimetosolveeachsingle-commodityproblem;however,wewereinterestedingaugingtherealsavingsforthemulti-commodity,multi-retailerlot-sizingproblem.Table3{4presentstheCPUrunningtimesofthethreealgorithms.TherunningtimeofScheme1weremuchsmallerinallcases.Thesesavingsareduetotheperformanceofthedualalgorithm. ThequalityofthelowerboundsgeneratedusingeitherschemeoftheLagrangeandecompositionalgorithmorthelinearprogrammingrelaxationofextendedproblem

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Table3{3:Qualityoflowerbounds(in%)fromLagrangeandecomposition Problem LP-Ex Scheme1 Scheme2 Scheme3 1 0.67 0.68 0.67 0.67 2 0.38 0.39 0.38 0.38 3 0.19 0.19 0.19 0.19 4 0.11 0.11 0.11 0.11 5 0.07 0.09 0.07 0.07 6 0.06 0.10 0.06 0.06 7 2.47 2.50 2.48 2.48 8 1.18 1.20 1.18 1.18 9 0.46 0.46 0.46 0.46 10 0.30 0.31 0.31 0.31 11 1.19 1.20 1.19 1.19 12 2.24 2.24 2.24 2.24 13 2.36 2.36 2.36 2.36 14 2.97 2.97 2.97 2.97 15 3.18 3.19 3.18 3.18 16 0.75 0.76 0.75 0.75 17 0.81 0.82 0.81 0.81 18 0.87 0.89 0.87 0.87 19 0.88 0.90 0.88 0.88 20 0.92 0.93 0.92 0.92 21 0.86 0.88 0.86 0.86 22 0.56 0.57 0.56 0.56 23 0.58 0.59 0.58 0.58 24 0.85 0.87 0.85 0.85 25 1.34 1.39 1.34 1.35

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formulationwasalmostidentical.Theresultsindicatedthatasthenumberofcommoditiesincreased,theerrorreportedfromCPLEXandLagrangeandecompositiondecreased(problemclasses1to10).Fortheseproblemclassesthe Table3{4:CPUrunningtimes(inseconds) Problem CPLEX Scheme1 Scheme2 Scheme3 LP-Ex 1 461.57 9.99 176.89 27.26 0.58 2 352.48 12.75 256.41 44.59 0.72 3 145.56 17.53 325.31 48.18 0.90 4 78.75 20.11 398.31 54.64 1.31 5 57.25 36.16 470.40 64.79 1.50 6 58.48 31.31 552.63 74.40 1.72 7 5,000.00 12.35 189.88 32.87 0.76 8 903.27 10.87 183.54 29.44 0.64 9 159.99 9.29 174.81 24.75 0.54 10 16.20 8.50 173.16 23.39 0.50 11 2,000.22 11.39 202.30 31.34 0.62 12 3,782.75 13.33 225.64 35.16 0.68 13 3,294.43 13.77 247.51 39.92 0.75 14 3,906.78 16.89 269.98 42.99 0.81 15 4,722.86 16.48 291.59 56.20 0.86 16 10,000.00 46.25 740.34 157.85 2.60 17 10,000.00 223.17 1,780.03 477.18 8.10 18 10,000.00 542.24 3,321.64 1,089.19 17.78 19 10,000.00 836.31 5,591.35 1,908.96 30.66 20 10,000.00 1,474.48 8,696.25 3,087.81 49.46 21 10,000.00 2,555.11 12,407.64 4,895.43 76.96 22 1,554.23 9.69 160.02 26.50 0.53 23 1,166.33 11.67 208.39 33.82 0.66 24 4,269.34 13.24 302.17 38.28 0.69 25 4,755.97 14.77 431.51 41.68 0.82 solutionfromLagrangeandecompositionwaslessthan2%fromoptimalandtherunningtimesofScheme1weresmallercomparedtoCPLEX.However,theincreaseinthenumberofcommoditiesaectedtherunningtimeofLagrangeandecompositionalgorithmbecauseoftheincreaseinthenumberofsubproblems(SP1k)tobesolvedateveryiteration. Increasingthecapacitytightnesscoecient(problemclasses7to10)madetheproblemseasier.Itiswell-knownthattheuncapacitatedlot-sizingproblemiseasier

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tosolve(capacitatedlot-sizingproblem,dierentthantheuncapacitatedproblemisNP-hard). Theresultsshowedthatproblemclasses11to15werequitechallenging.Asthenumberoffacilitiesincreasesweobservedamonotonicincreaseinthedualitygap.Theresultssuggestedthataddingthefacilityselectiondimensiontotheclassicalmulti-commoditylot-sizingproblemhasquiteaneectonproblemcomplexity. Therunningtimesofallalgorithmsforproblemclasses16to21werethehighest.Oneofthereasonsforthistohappenisthesizeofthenetworkfortheseproblems(Table3{1).Setupcostsappearedtohaveasignicantimpactonthedualitygap(problemclasses22to25).Problemclass22presentedanaverageerrorof1:20%comparedto6:44%fortheproblemclass25.Thisresultisnotsurprisingsinceincreasedsetupcostswidenthegapbetween(SP2),whichignoresthesetupcostsand(SP1).Moreover,sinceouroriginalproblemisaMILPwithbinarysetupvariables,asthesetupcostsincreasetheproblembehavesclosertoacombinatorialproblemthanalinearprogrammingproblem. 3.2 Single-Commodity, Multi-Retailer, Multi-Facility Lot-Sizing Problem Thesatisfactionofthedemandforproductsofasetofcustomersinvolvesseveralcomplexprocesses.Inthepast,thisoftenforcedpractitionersandresearcherstoinvestigatetheseprocessesseparately.AsmentionedbyErengucetal.[30],thetraditionalwayofmanagingoperationsinacompetitivemarketplacesuggestedthatcompaniescompetingonpricewillsacricetheirexibilityinoeringnewproductsorsatisfyingnewdemandsfromtheircustomers.Thecompetitionandtheevolutionofhardwareandsoftwarecapabilitieshasoeredcompaniesthepossibilityofconsideringcoordinateddecisionsbetweenprocessesinthesupplychain. Inthissectionweproposeaclassofoptimizationmodelsthatconsidertheintegrationofdecisionsonproduction,transportationandinventoryinadynamicsupplychainconsistingofanumberofretailersandfacilities.Wecallthismodelthesingle-commodity,multi-retailer,multi-facilitylot-sizingproblem.Thismodel

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estimatesthetotalcostofagivenlogisticsdistributionnetwork,includingproduction,inventoryholding,andtransportationcosts.Theevaluationisperformedforatypicalplanningperiodinthefuture. Thisproblemconsidersasetofplantswhereasingleproducttypeisproduced.ThisisaspecialcaseofthesupplychainoptimizationproblemswediscussedinChapter1.Chapter4considersthemoregeneralproblem,whereanumberofcommoditiesareproducedintheplantsandtheplantsfaceproductionandtransportationcapacityconstraints. 3.2.1 Problem Formulation Inthissectionweconsideraclassofmulti-facility,multi-retailerproduction-distributionproblems.LetRdenotethenumberofretailers.Demandofretailerjinperiodtisgivenbybjt.Theunittransportationcostsfromfacilityitoretailerjinperiodtarecijt.Inthisdiscussionweassumethattransportationandinventorycostfunctionsarelinear,andtheproductioncostfunctionisofthexedchargetype. Themulti-facility,multi-retailerproblemcanbeformulatedasfollows:minimizeFXi=1TXt=1(pitqit+sityit+hitIit+RXj=1cijtxijt) subjectto(MR)qit+Iit1=RXj=1xijt+Iiti=1;:::F;t=1;:::;T

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Decisionvariablesxijtrepresentsthequantitytransportedfromfacilityitoretailerjintimeperiodt.Constraints(3.24)modelthebalancebetweentheinow,storage,andoutowatfacilityiinperiodt.Constraints(3.25)makesurethatretailer'sdemandissatised.Constraints(3.26)relatethexedandvariableproductioncosts(theproductioncanbeinitiatedoncethesetupcostispaid).Figure1{1givesanetworkrepresentationofthisproblem. Themulti-retailer,multi-facilitymodelweproposehelpsmanagerstoanswerquestionsthatariseinmanagingtheproductionanddistributionnetwork.Obviously,themostaccurateanswersaregivenwhentheproblemissolvedtooptimality.However,thisisadiculttasksincetheproblemwepresentisNP-hard.Thisproblemcanbeclassiedasanetworkowproblemwithxedchargecostfunctions.Severalspecialcasesofthesingle-commoditynetworkowproblemhavebeenshowntobeNP-hard:forbipartitenetworks(Johnsonetal.[63]),forsingle-sourcenetworksandconstantxed-to-variablecostratio(HochbaumandSegev[60]),andthecaseofzerovariablecosts(Lozovanu[76]).Forthespecialcaseofthismodelwhenthereisonlyoneretailer,WuandGolbasi[106]showthattheproblemisNP-hardwhentheholdingcostsarenotrestrictedinsign. Wepresenttheextendedformulationofthisproblemaswell.Wedothisbysplittingthevariablesqitbydestinationintovariablesqijt(=t;:::;T),wheredenotestheperiodandjrepresentsthefacilityforwhichproductiontakesplace.Thesplitoftheproductionvariablesimpliesthefollowing:qit=RXj=1TX=tqijt(3.29)xijt=tXs=1qijst(3.30)Iit=tX=1qiRXj=1tX=1xij=RXj=1(tXs=1TX=tqijstXs=1qijst)(3.31)

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Wecanre-formulateproblem(MR)asfollows:minimizeFXi=1TXt=1[RXj=1TX=t where

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3.2.2 Primal-Dual Algorithm Thedualofthelinearprogrammingrelaxationof(Ex-MR)hasaspecialstructure.Belowwepresentthelinearprogrammingrelaxationandthecorrespondingdualoftheformulation(Ex-MR). minimizeFXi=1TXt=1[RXj=1TX=t subjectto(LP-MR)(3:32);(3:33);(3:34)yit0i=1;:::;F;t=1;:::T ThedualproblemreadsmaximizeTXt=1RXj=1bjtvjt

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subjectto(D-MR)TX=tRXj=1bjmax(0;vj Intuitive Understanding of the Dual Problem Inthissectionwegiveanintuitiveinterpretationoftherelationshipbetweenprimal-dualsolutionsof(Ex-MR).Supposethelinearprogrammingrelaxationof(Ex-MR)hasanoptimalsolution(q;y)thatisintegral.Let=f(i;t)jyit=1gandlet(v;w)denoteanoptimaldualsolution.Thecomplimentaryslacknessconditionsforthisproblemareasfollows:(C1)yit[sitPRj=1PT=tbjwijt]=0fori=1;:::;F;t=1;:::;T(C2)qijt[ Byconditions(C2),ifqijt>0,thenvj=

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forj=1toRdo ifbj=0thenvj=0 fori=1toFdowijt=maxf0;vj enddo enddo enddo goestopayforproductionandinventoryholdingcosts,andwijtisthecontributiontotheproductionset-upcost. 3.2.4 Outline of the Primal-Dual Algorithm Supposethattheoptimalvaluesoftherstk1dualvariablesof(D-MR)areknown.Lettheindexkbesuchthatk=(1)R+l,where=1;:::;Tandl=1;:::;R.Then,tobefeasible,thek-thdualvariable(vl)mustsatisfythefollowingconstraints:blmax(0;vl (3.37) foralli=1;:::;Fandt=1;:::;.Inordertomaximizethedualproblemweshouldassignvlthelargestvaluesatisfyingtheseconstraints.When,bl>0,thisvalueisvl=mini=1;:::;F;tf NotethatifMilt10impliesvl

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P=f(j;k)jbjk>0;forj=1;:::;R;k=1;:::;Tg forj=1toRdo repeatk=k+1 enddo gotoStep3 forj=0toRdo if enddo Step3 Figure3{4:Primalalgorithm variablessequentially(Figure3{3).Abackwardconstructionalgorithmcanthenbeusedtogenerateprimalfeasiblesolutions(Figure3{4).Fortheprimal-dualsetofsolutionstobeoptimal,thecomplimentaryslacknessconditionsshouldbesatised.

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andwijt=wijt;+1=:::=wiRtT=0 Theprimalalgorithmsetsyit=1onlywhenMijt=0;thisimpliesthatconditions(C1)willalwaysbesatised.2 3.2.5 Computational Results Inordertotesttheperformanceofouralgorithm,asinprevioussections,werandomlygeneratedasetoftestproblemsandcomparedtheircomputationtimesandsolutionqualitytothegeneralpurposesolverCPLEX.Wegeneratedfeasiblesolutionstoourproblemusingtheprimalalgorithmandlowerboundsusingthedualalgorithm. Thenominalcaseproblemhasthefollowingcharacteristics: ThetransportationvariablecostsaregeneratedthesamewayasdescribedinSection2.8.SeasonaldemandsarerandomlygeneratedinthesamewayaspresentedinBakeretal.[5]andChenetal.[22].bt=200+zt+sin[2 d(t+d=4)] Inourtestproblemswetake=67,=125andd=12. ThecharacteristicsoftheproblemclassesarepresentedinTable3{5.Foreachproblemclasswegenerate20probleminstancesandwereportontheaverageerror

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boundsandaveragerunningtimes.Theerrorboundforeachprobleminstanceiscalculatedasfollows:Error(%)=PrimalSolutionDualSolution DualSolution100: Table3{5:Problemcharacteristics Problem Facilities Periods Nodes Arcs 1 20 24 1,921 360,480 2 30 24 2,161 540,720 3 40 24 2,401 720,960 4 20 48 3,841 1,412,160 5 30 48 4,321 2,118,240 6 40 48 4,801 2,824,320 7 20 96 7,681 5,589,120 8 30 96 8,641 8,383,680 9 40 96 9,601 11,178,240 10 20 192 15,361 22,237,440 11 30 192 17,281 33,356,160 12 40 192 19,201 44,474,880 costincreased,theproblemsbecamemoredicult.Thisaectedtheperformanceoftheprimal-dualalgorithm. Forallproblemclasses,whenset-upcostsareuniformlydistributedontheintervals[200;300]and[200;900],theerrorgapwaslessthan0:60%andtherunningtimeslessthan141cpuseconds.Themaximumerrorreportedis4:073%.Thiscorrespondstoproblemclass3withset-upcostsuniformlydistributedintheinterval[1200;1500].ResultsinTables3{6and3{7indicatethatincreasingthenumberoffacilitiesandthelengthofthetimehorizonaectedtheperformanceoftheprimalanddualalgorithm.

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Table3{6:Errorbounds(in%)ofprimal-dualheuristic Set-UpCosts Problem 200-300 200-900 600-900 900-1,500 1,200-1,500 1 0.249 0.418 1.105 1.936 2.553 2 0.343 0.511 1.576 2.519 3.388 3 0.423 0.609 1.789 3.023 4.073 4 0.246 0.415 1.119 1.920 2.573 5 0.321 0.485 1.502 2.443 3.320 6 0.395 0.556 1.754 2.872 3.959 7 0.248 0.399 1.150 1.953 2.611 8 0.333 0.490 1.487 2.544 3.412 9 0.419 0.580 1.810 2.987 4.026 10 0.250 0.411 1.152 N/A 2.607 11 0.341 0.522 1.533 N/A N/A 12 0.413 0.577 1.832 N/A N/A Wenextrandomlygeneratedasecondgroupofproblemsforwhichdemandisuniformlydistributedinthefollowingintervals:[20;40],[40;100],[100;200],[200;400]and[400;1000].Wererunproblemclasses4;5and6foreachdemanddistribution(problemclasses13to27).Forexample,forproblemclasses13,14and15,demandisuniformlydistributedin[20;40]andtheproblemcharacteristicsarethesameasthecharacteristicsofproblems4,5and6.Thisgivesatotalof15problemclassesand300probleminstances.CPLEXfailedtosolveproblemclasses13to27.Table3{8 Table3{7:Runningtimes(inseconds)ofprimal-dualheuristic Set-UpCosts Problem 200-300 200-900 600-900 900-1,500 1,200-1,500 1 1.16 1.34 1.18 1.32 1.14 2 1.58 1.87 1.63 1.82 1.56 3 2.03 2.40 2.09 2.34 2.00 4 4.15 5.18 4.26 4.98 4.23 5 6.15 7.38 6.14 7.06 5.90 6 8.09 9.63 7.89 9.14 7.69 7 16.78 20.45 16.95 19.29 16.35 8 25.15 29.58 25.02 28.00 24.41 9 33.68 39.00 33.41 36.82 32.07 10 74.42 80.38 74.33 N/A 70.42 11 109.83 104.30 108.93 N/A N/A 12 140.55 139.35 139.57 N/A N/A

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presentstheerrorandrunningtimesoftheprimal-dualheuristicfortheseproblems.Theaveragerunningtimeoftheprimalalgorithmwaslessthan5:22cpusecondsforallproblemclassesandlessthan3:23forthedualalgorithm.Noticethattheerrordecreasedastheaveragevalueofdemandincreased.Fortheseproblemclasses,everythingelsekeptthesame,anincreaseindemandaectedtheratiooftotalvariablecosttototalxedcost.Increasingthisratiogenerallymakestheproblemseasier(HochbaumandSegev[60]). Table3{8:Resultsofprimal-dualheuristic Error Time(sec) Problem (%) Primal Dual 13 3.94 2.54 1.63 14 5.08 3.88 2.41 15 5.98 5.20 3.18 16 1.34 2.54 1.64 17 1.78 3.87 2.42 18 2.18 5.22 3.21 19 0.43 2.54 1.66 20 0.59 3.88 2.42 21 0.73 5.20 3.20 22 0.14 2.55 1.68 23 0.19 3.87 2.45 24 0.25 5.22 3.22 25 0.03 2.55 1.67 26 0.04 3.88 2.45 27 0.06 5.22 3.23 3.3 Multi Facility Lot-Sizing Problem with Fixed Charge Transportation Costs Inthissectionwediscusstheuncapacitatedmulti-facility,multi-retailerproblemwithxedchargetransportationcostsandlinearproductionandinventorycosts.Usually,whenshipmentsaresentfromafacilitytoaretailer,axedchargeispaid(forexample,thecostofthepaperworknecessary)toinitiatetheshipmentplusavariablecostforeveryunittransported.Therefore,modellingthetransportationcostfunctionasaxedchargecostfunctionmakessense.ThefollowingistheMILPformulationoftheproblem:

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minimizeFXi=1TXt=1(pitqit+RXj=1(cijtxijt+sijtyijt)+hitIit) subjectto(MR-T)Ii;t1+qit=PRj=1xijt+Ii;ti=1;:::;F;t=1;:::;TPFi=1xijt=bjtj=1;:::;R;t=1;:::;TqitPRj=1bjtTi=1;:::;F;t=1;:::;Txijtbjtyijti=1;:::;F;j=1;:::;R;t=1;:::;Tqit;Iit;xijt0i=1;:::;F;j=1;:::;R;t=1;:::;Tyit2f0;1gi=1;:::;F;t=1;:::;Tsijtisthetransportationxedchargecostfromfacilityitoretailerjinperiodt. Anicepropertyofthisproblemisthatitslinearprogrammingrelaxationgivestheoptimalsolution.Anoptimalsolutionto(MR-T)hasthefollowingproperties:(i)inafacility,productionwilltakeplaceiftherearenoinventories(thezeroinventorypolicy)(ii)thedemandataretailerwillbesatisedbyproductionorinventoryfromexactlyonefacility(iii)thequantityproducedinafacilityisequaltothedemandofatleastoneretailer,foratleastonetimeperiod(thesetimeperiodsdonotneedtobesuccessive).Thesepropertiesimplythatinanoptimalsolution,ifatransportationarcisused,thetotaldemandisshipped.Thetransportationcostfunctionthenconsistsofonlytwopoints,xijt=0andxijt=bjt. Thetransportationcostfunctionisseparablebyarc.Foreachtransportationarc,thelinearapproximationofthexedchargecostfunctionpassesthroughxijt=0andxijt=bjt.Thisshowsthatthelinearapproximationexactlyrepresentsthetransportationcostfunction.Awell-knownresultoflinearprogrammingisthefollowing:whenalinearcostfunctionisminimizedoveraconvexset,thesolutioncorrespondstoanextremepoint.Inthiscase,theextremepointscorrespondto

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Figure3{5:Fixedchargetransportationcostfunction Theextendedformulationof(MR-T)isthefollowing:minimizeFXi=1TXt=1[RXj=1TX=t subjectto(LP-MR-T)PFi=1Pt=1qijt=bjj=1;:::;R;=1;:::;TPt=1qijtbjyiji=1;:::;F;j=1;:::;R;t=1;:::;T;tTqijt0i=1;:::;F;j=1;:::;R;t=1;:::;T;Tyijt2f0;1gi=1;:::;F;j=1;:::;R;t=1;:::;T; where Forthesamereason,thesameresultholdstrueforthelinearprogrammingrelaxationofextendedformulation,itslinearprogrammingrelaxationgivesintegersolutions. 3.4 Conclusions Inthischapterwediscussthecapacitatedmulti-commodity,multi-facilitylot-sizingproblem,thesingle-commodity,multi-facility,multi-retailerlot-sizingproblemaswellasaspecialcaseofthislastproblemwithxedchargetransportationcostsandlinearproductioncosts.

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Themulti-commodityandmulti-facilityproblemissolvedusingaLagrangeandecompositionbasedheuristic.Thedecompositionseparatestheproblemintotwosubproblemsthatarecomputationallyeasiertosolve.Thedecompositionisperformedinsuchawaythatitprovidesinterestingmanagerialinsights. Themulti-facilityandmulti-retailerproblemissolvedusingaprimal-dualalgorithm.Theperformanceofthealgorithmistestedontwogroupsofrandomlygeneratedproblems.Intherstgroupofproblemsdemandshowsseasonality,andinthesecondgroupdemandisuniformlydistributed.Resultsindicatethattheperformanceoftheprimal-dualalgorithmdependsonthevalueofset-upcosts,thevalueofdemand,thenumberoffacilitiesandthenumberofperiods. Inthischapterwealsodiscussedaclassofuncapacitated,multi-facilityandmulti-retailerproblemswithlinearproductionandinventorycostsandxedchargetransportationcosts.Weshowthatthisclassofproblemshasaniceproperty,thelinearprogrammingrelaxationoftheMILPformulationsgivesintegersolutions.

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4.1 Introduction Thesupplychainoptimizationproblemwediscussinthischapterconsidersproduction,inventory,andtransportationdecisionsinadynamicenvironment.Inparticular,wemodelatwo-stagesupplychain,coordinatingproduction,inventory,anddistributiondecisions.ThissupplychainconsistsofFfacilitiesandRretailers.ThefacilitiesproduceandstoreKdierentcommoditiesforwhichthereisademandoveraplanninghorizonoflengthT.Theproblemistondtheproduction,inventory,andtransportationquantitiesthatsatisfydemandatminimumcost. Weformulatetheproblemasamulti-commoditynetworkowproblemwithxedchargecostfunction.Wepresentaheuristicprocedurethatcanbeusedtosolveanymulti-commoditynetworkowproblemwithxedchargecostfunctions.Thismethod,calledthemulti-commodity,dynamicslopescalingprocedure(MCDSSP)isanextensionofaprocedurethatwasproposedbyKimandPardalos[66]forthesingle-commodityxed-chargenetworkowproblem.MCDSSPapproximatesthexedchargecostfunctionbyalinearcostfunction,anditerativelyupdatesthecoecientsofthelinearapproximationuntilnobettersolutionisfound. WealsoproposeaLagrangeandecompositionbasedalgorithmtosolvetheproduction-distributionproblem.Thealgorithmdecomposestheproblemintotwosubproblems.OneofthesubproblemscanbefurtherdecomposedintoKsingle-commodity,multi-facility,multi-retailerlot-sizingproblems.Thesesingle-commoditysubproblemsaresolvedusingtheprimal-dualalgorithmproposedinSection3.2.2.InSection4.2wegiveamathematicalformulationoftheproduction-distributionproblem.InSection4.3weprovideadetaileddescriptionoftheMCDSSP,inSection 97

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4.4wediscusstheLagrangeandecompositionalgorithmandinSection4.5wepresentresultsfromimplementingtheMCDSSPonasetofrandomlygeneratedproblems.Section4.6concludesthischapter. 4.2 Problem Formulation Inthischapterwenallydiscussandproposesolutionalgorithmsfortheproduction-distributionproblemweintroducedinSection1.3.Theproduction-distributionplanningproblemisformulatedasamulti-commoditynetworkowproblemonadirected,single-sourcegraphconsistingofTlayers(Figure1{1).Recallthateachlayerofthegraphrepresentsatimeperiod.Ineachlayer,abipartitegraphrepresentsthetransportationnetworkbetweenfacilitiesandretailers.Facilitiesinsuccessivetimeperiodsareconnectedthroughinventoryarcs.Thereisadummysourcenodewithsupplyforeachcommodityequaltothecorrespondingtotaldemand.Productionarcsconnectthedummysourcenodetoeachfacilityineverytimeperiod.ThetotalnumberofnodesinthenetworkisjNj=(R+F)T+1,andthetotalnumberofarcsisjAj=FT+FRT+F(T1).Thereareunitcostsassociatedwitheacharcandeachcommodity.Inaddition,therearesetupcostsassociatedwitheachproductionandtransportationarc. Recallthatwehavemadethefollowingassumptionsinourmodel: Theproduction-distributionmodelweproposecaneasilybeextendedtocountforbackordersandinventoriesattheretailersbyaddingextraarcsinthenetwork.Mostofthedecisionvariablesandcostdataarealreadydenedinthepreviouschapters.Thenewvariablesthatweintroduceare

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ThisproblemisNP-hardeveninthesimplercasewhenthereisonlyonetimeperiodandasinglecommodityowinginthenetwork(GareyandJohnson[45]). 4.3 Dynamic Slope Scaling Procedure KimandPardalos[66]developedaheuristiccalledtheDynamicSlopeScalingProcedure(DSSP)forthexed-chargenetworkowproblemwithasinglecommodity,whichwasrenedandtestedinEksiogluetal.[34].Inthissection,weextendthisheuristictothemulti-commoditycase. SinceMCDSSPisageneralheuristicthatcanbeusedtosolveanymulti-commodityproblemwithxedchargecostfunctions,wegiveatrstadescriptionoftheMCDSSPinthiscontext. 4.3.1 Multi-Commodity Network Flow Problem with Fixed Charge Cost Function Themulti-commoditynetworkowproblemwithxed-chargearccostfunctionshasabroadareaofapplications,suchasproductionanddistributionofgoodsinasupplychain,orthedistributionofmessagesinacommunicationnetwork(seeforinstance,MagnantiandWong[80],Gavish[47],Balakrishnanetal.[7],andAhujaetal.[2,3]).Thisproblemisageneralizationoftheclassicalsingle-commoditynetworkowproblemwithxed-chargecosts.Severalspecialcasesofthissingle-commoditynetworkowproblemwereshowntobeNP-hardsuchasbipartitenetworks(Johnsonetal.[63]),single-sourcenetworksandconstantxed-to-variablecostratio(HochbaumandSegev[60]),andthecaseofzerovariablecosts(Lozovanu[76]).Inaddition,thesinglecommoditynetworkowproblemwithgeneralconcavearccostswasstudiedbyGareyandJohnson[45]andGuisewiteandPardalos[53]. Sincethemulti-commoditynetworkowproblemwithxed-chargearccostsisaconcaveminimizationproblem,anyexactgeneral-purposesolutionmethodforsolvingsuchproblemscanbeusedtosolvethemulti-commoditynetworkowproblemwithxed-chargecosts.Examplesofsuchmethodsarebranch-and-bound(HirschandDantzig[59],Gray[51],KenningtonandUnger[65],Barretal.[11],Cabotand

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Erenguc[20],Palekaretal.[87],andLamarandWallace[70]),vertexenumeration(Murty[84]),anddynamicprogramming(Ericksonetal.[37]).However,suchgeneralpurposealgorithmsareoftennotadequatetoolsforsolvinglarge-scaleinstancesofourproblemeciently.Forinstance,thelinearprogrammingrelaxationofthetraditionalMILPformulationoftheproblemdoesnotprovideatightlowerbound.Therefore,standardsimplex-basedbranch-and-boundmethodsthatdonotincludecuttingplaneorcolumngenerationproceduresarenotlikelytosolvelargeinstancesoftheprobleminreasonabletime.Therefore,moreecientspecialpurposealgorithmshavebeenproposedaswell. BienstockandGunluk[14]usedasimplex-basedcuttingplaneapproach.Thisapproachoersanopportunityforcontinuousimprovementoflowerboundsthroughvalidinequalities.Gavish[46]proposedLagrangeanrelaxation,whichnotonlyexploitsthestructureoftheproblem,butfacilitatesthedesignofheuristicsaswell. Cranic,FrangioniandGendron[26]comparedlowerboundsgeneratedusingdierentLagrangeanrelaxationsofthecapacitatedmulti-commoditynetworkowproblemwithxed-chargecosts.TheyshowedthatbundlemethodsusedtooptimizetheLagrangeandualsaresuperiortosubgradientmethods,becausetheyconvergefasterandaremorerobustwithrespecttoproblemcharacteristics.GendronandCrainic[48]usedaboundingproceduretosolvetheproblem.TheirprocedurewasbasedongeneratinglowerboundsusingLagrangeanrelaxation(relaxingthebundlingconstraints)andgeneratingupperboundsusingaResourceDecompositionapproach.Crainicetal.[27]solvedthecapacitatedmulti-commoditynetworkowproblemwithxed-chargecostsusingacuttingplanealgorithmcombinedwithaLagrangeanrelaxation. Theuncapacitatedmulti-commoditynetworkowproblemwithxed-chargecostsisnotasdicultasthecapacitatedproblem.Magnantietal.[79]proposedamethodologytoimprovetheperformanceofBendersdecompositionwhenusedtosolvetheuncapacitatedproblem.HolmbergandHellstrand[61]presenteda

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Lagrangeanheuristicwithinabranch-and-boundframeworkasamethodforndingtheexactoptimalsolutionoftheuncapacitatedproblemwithsingleoriginsanddestinationsforeachcommodity. Problem description and formulation .ConsideraconnectedgraphG(N;A),whereN=f1;:::;NgisthesetofnodesandANNisthesetofarcs.ThenumberofcommoditiesthatneedtoberoutedthroughthisnetworkisgivenbyK,andthedemandforcommoditykatnodeiisdenotedbybik.Eacharc(i;j)2Ahasacapacityforeachindividualcommoditydenotedbyuijk.Inaddition,eacharc(i;j)2Ahasabundlingcapacityjointlyforallcommodities,denotedbyvij. Thedecisionvariablesarethequantitiesofowofeachcommodityalongeacharc,andaredenotedbyxijk((i;j)2A;k=1;:::;K).Weassumethatthetotalcostofowisseparableinthearcs,andthecostofowalongarc(i;j)isgivenbyfij(xij1;:::;xijK). Thegeneralminimumcostmulti-commoditynetworkowproblemcanthenbeformulatedasfollows:minimizeX(i;j)2Afij(xij1;:::;xijK) subjectto(P)Xj:(j;i)2AxjikXj:(i;j)2Axijk=biki=1;:::;N;k=1;:::;K

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AlthoughnotessentialfortheapplicabilityoftheheuristicwewillproposeinSection4.3,wewillassumethatthearccostfunctionshaveaxed-chargestructure.Inparticular,letsijrepresentthexed-chargecostthatisincurredwheneverarc(i;j)2Aisused.Inaddition,letcijkrepresentthevariableperunitcostofmovingcommoditykalongarc(i;j).Moreformally,thisyieldsfij(xij1;:::;xijK)=8><>:0ifPKk=1xijk=0sij+PKk=1cijkxijkifPKk=1xijk>0. ThestandardMILPreformulationofthisproblemcanbeobtainedbyintroducingabinarysetupvariableyijcorrespondingtoeacharc(i;j)2A.Thearccostfunctionscanthenbereplacedbyfij(xij1;:::;xijK)=sijyij+KXk=1cijkxijk TheMILPformulationofthexed-chargemulti-commoditynetworkowproblemthenreadsminimizeX(i;j)2Asijyij+X(i;j)2AKXk=1cijkxijk

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Inprinciple,standardsolverssuchasCPLEXcanbeusedtosolveformulation(Q)ofthexed-chargemulti-commoditynetworkowproblem. 4.3.2 Single-Commodity Case TheDSSPisaprocedurethatiterativelyapproximatesthexed-chargecostfunctionbyalinearfunction,andsolvesthecorrespondinglinearprogrammingproblem.Notethateachoftheapproximatinglinearprogramshasexactlythesamesetofconstraints,anddiersonlywithrespecttotheobjectivefunctioncoecients.ThemotivationbehindtheDSSPisthefactthataconcavefunction(suchasthexed-chargecostfunction),whenminimizedoverasetoflinearconstraints,willhaveanextremepointsolution.Therefore,thereexistsalinearcostfunctionthatyieldsthesameoptimalsolutionastheconcavecostfunction.Theproceduredoesnotguaranteethattheoptimalsolutiontothexed-chargenetworkowproblemisindeedfound.However,substantialexperimentalanalysisindicatesthattheprocedureyieldshighqualitysolutionsthatareclosetotheoptimalsolution. 4.3.3 Multi-Commodity Case Asforthesinglecommoditycase,themulti-commodityvariantoftheDSSP,whichwewillcallMCDSSP,consistsofaninitializationphaseandanupdatephase.Intheformer,weneedtoinitializethelinearapproximationofthexed-chargecostfunction,andinthelatterweneedtoupdatethelinearapproximation. Initialization scheme for MCDSSP .ConsidertheMILPformulation(Q)ofthexed-chargemulti-commoditynetworkowproblem.Thelinearprogrammingrelaxationofthisformulationrelaxesthebinaryconstraintsonthevariablesyij,allowingthemtoassumeanyvariablein[0;1].Fromequation(4.4),wecanseethatyijPKk=1xijk

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Makingthissubstitutionyieldsthefollowingformulationofthelinearprogrammingrelaxationof(P):minimizeX(i;j)2AKXk=1sij Figure4{1:Initiallinearapproximationofthexed-chargecostfunction Update scheme for the MCDSSP .InthesinglecommodityvariantoftheDSSP,thesolutionofalinearapproximationisusedasfollowstoconstructanewlinearobjectivefunction.Forallarcsthatareusedinthesolution,thenewlinearcostcoecientischosentobetheaveragecostperunitshippedonthatarc,measuredusingthetrue,xed-chargearccostfunction.If,inthemulti-commoditycase,thevariableunitcostshappentobecommodity-independent(i.e.,cijk=cijforallk=1;:::;Kandforall(i;j)2A),wecouldusethesameapproachasinthesinglecommoditycase.Supposethat,inthe`thiterationoftheprocedure,wehaveusedthecoecients

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However,ingeneralweneedtosomehowdistributethearccostsamongtheKcommodities.Inotherwords,wewouldliketondslopesc(`+1)ijkthatsatisfysij+KXk=1cijkx(`)ijk=KXk=1 wheneverPKk=1x(`)ijk>0.Thus,whenarc(i;j)isused,weneedtondawaytodistributethexed-chargecostsijamongthecommoditiesowingonarc(i;j).Itisclearthatthereisnouniquewayofaccomplishingthis.Tocharacterizethepossiblewaysofdistributingthecosts,weintroduceasetofweightsw(`)ijk,andlet

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where,jxj+=8><>:1ifx>00otherwise. Thisschemeimplicitlyassumesthatthecommoditiesowingonarc(i;j)havesimilarimpactsonthetotalcosts. Inthe`thiterationoftheMCDSSP,thefollowinglinearprogrammingproblemissolved:minimizeX(i;j)2AKXk=1 criterion .Theheuristicwillstopifoneofthefollowingconditionsismet:

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Asinthesingle-commoditycase,theMCDSSPdoesnotguaranteeconvergence,orevenmonotonicity.Therefore,aftereachiterationoftheupdatingprocedure,wesavethebestsolutionfoundsofar. 4.3.4 Production-Distribution Problem ApplicationofMCDSSPtosolvetheproduction-distributionproblem(PD)isstraightforward.Theonlyslightdierencewithrespecttothegeneralcasediscussedintheprevioussectionsisthatsomearcshavelinearcosts.Sinceinventoryarcshavelinearcosts,theonlycostcoecientsthatneedtobeinitializedandupdatedareproductionandtransportationcostcoecients. Theupdatingschemesforproductionandtransportationcostsaredierent.WeuseDSSPtoupdatethecostcoecientsfortheproductionarcsandMCDSSPforthetransportationarcs.Thereasonisthatintheproductioncostfunction,eachcommodityhasitsownsetupandvariableunitcosts,dierentfromthetransportationarcswherethesetupcostissharedamongallthecommoditiesusingthatarc. ThetotalcostonaproductionarcisequaltoKXk=1(sitk+pitkqitk)fori=1;:::;F;t=1;:::;T:

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Theinitialcostcoecientsfortheproductionarcsare Fortransportationarcs,thelinearapproximationofthexedchargecostfunctionissuchthatsijt+KXk=1cijtkx(`)ijtk=KXk=1p(`+1)itkq(`)ijtkfori=1;:::;F;j=1;:::;R;t=1;:::;T: weightsw(`)ijtkarecalculatedasdescribedinSection4.3.3. 4.3.5 Extended Problem Formulation Inthissectionweprovideanextendedformulationfortheproduction-distributionproblem(PD).MCDSSPprovidesfeasiblesolutionstothe(PD)problem.InordertoevaluatethequalityofthesolutionsfromMCDSSP,wegeneratelowerboundsusingthelinearprogrammingrelaxationoftheextendedformulation.Experimentalresultsfrompreviouschaptershasshownthatlinearprogrammingrelaxationoftheextended

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problemformulationgivestighterboundsthanthelinearprogrammingrelaxationofthe\original"MILPformulation. Letusdenotebyqijtktheproductionatfacilityiinperiodtforcommoditykatretailerjinperiod.Splittingtheproductionvariablesqitkbydestinationintoqijtkallowsforthefollowingsubstitutions:qitk=RXj=1TX=tqijtk(4.5)xijtk=tXs=1qijstk(4.6)Iitk=RXj=1tXs=1TX=tqijsktXs=1qijstk(4.7) Theextendedformulationisthefollowing:minimizeFXi=1TXt=1KXk=1[RXj=1TX=t

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Thevariableunitcostinthearcsoftheextendednetworkisequalto cijtkconsistsofunitproductioncost,unittransportationcostandtotalunitinventorycostfromthetimeproductionoccurstothemomenttheproductisshippedtotheretailer. 4.4 A Lagrangean Decomposition Procedure InthissectionwediscussaLagrangeandecompositionproceduretosolvetheproduction,inventoryandtransportationproblemintrudicedinSection4.2.WeapplytheLagrangeandecompositionprocedureontheextendedproblemformulation(Ex-PD). InordertoapplyLagrangeandecomposition,werstduplicatevariablesqijtk.Wedothisbyintroducingthe\copy"variableszijtksuchthatqijtk=zijtki=1;:::;F;j=1;:::;R;t=1;:::;T;tT;k=1;:::;K(4.14) Thefollowingisanequivalentformulationof(Ex-PD).minimizeFXi=1TXt=1KXk=1[RXj=1TX=t

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Letzsbetheoptimalsolutiontosubproblem(SP2)initerations ifzsijtk0thenUBs=UBs+ Initializesumijt=0fori=1;:::;F;j=1;:::;R;t=1;:::;T Relaxingconstraints(4.14)decomposestheproblemintothefollowingtwosubproblems:minimizeFXi=1TXt=1KXk=1[RXj=1TX=t( subjectto(SP1)(4:8);(4:11);(4:12)and(4:13) andminimizeFXi=1RXj=1TXt=1(KXk=1TX=tijtkzijtk+sijtyijt) subjectto(SP2)(4:15);(4:16);(4:17);(4:18)and(4:13) Subproblem(SP1)canfurtherbedecomposedbycommodity.ThisgivesKsingle-commodity,multi-facility,multi-retailerproblemswithonlyxedchargeproductioncosts(SP1k).Theproductionset-upcostdoesnotallowtheproblemtodecomposefurtherbyretailer.Subproblem(SP2)isanintegerprogrammingproblem.

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If,thenSTOP Step4: OtherwisegotoStep2 Asolutiontothissubproblemisafeasiblesolutionto(Ex-PD).Weusethesesolutionstoobtainupperbounds.Figure4{2presentstheupperboundprocedure. Bothsubproblems(SP1)and(SP2)donothavetheintegralityproperty,thereforethevalueoftheLagrangeandual(inthecasethatthesubproblemsaresolvedoptimally)dominatestheoptimalvalueoftheLagrangeandualobtainedbyrelaxingonesetofconstraints(seeGuignardandKim[52]). Wesolvethesub-subproblem(SP1k)usingthedualalgorithmdiscussedinSection3.2.2.Thisalgorithmgiveshighqualitylowerboundsinreasonableamountoftime.Thedualalgorithmdoesnotguaranteetheoptimalsolutiontothesubproblem.However,usingthedualalgorithmtoobtainalowerbound,ratherthansolvingtheproblemtooptimality,savesinthecomputationaltime.Subproblem(SP2)isnotsolvedtooptimality.Weprovidealowerboundtothesesubproblemsbysolvingthecorrespondinglinearprogrammingrelaxation. SubgradientoptimizationisusedtomaximizetheLagrangeandualfunctionz()(z()=v(LD(x;z;)).Ateachiterations,wecalculatetheLagrangeanmultipliersijtkusingthefollowingequation:

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fori=1;:::;F;j=1;:::;R;t=1;:::;T;tT;k=1;:::;K;andus=s(minUBmaxLB) InordertondasubgradientdirectionateachstepoftheLagrangeandecompositionprocedureusingequations4.19,weneedtondafeasiblesolutiontosubproblem(SP1).Thedualalgorithmprovidesonlyalowerboundto(Ex-PD),butdoesnotprovideafeasibleprimalsolution.Therefore,weusetheprimalalgorithm(Section3.2.2)tondafeasiblesolutiontothesesubproblems. Inourcomputationalexperiments,weterminatethealgorithmifoneofthefollowinghappens:(i)thebestlowerboundisequaltothebestupperbound(theoptimalsolutionisfound),(ii)thenumberofiterationsreachesaprespeciedbound,(iii)thescalarsislessthanorequalto(asmallnumberclosetozero).Figure4{3presentsthestepsoftheLagrangeandecompositionalgorithm.Inthenextsection,wetestthroughcomputationalexperimentstheperformanceoftheLagrangeandecompositionscheme. 4.5 Computational Results InthissectionweillustratetheperformanceoftheMCDSSPandLagrangeandecompositionalgorithmonlarge-scaleinstancesoftheproduction-distributionproblemintroducedinSection4.2.Werandomlygeneratedtestproblems,andcomparedtherunningtimesandsolutionqualitytothegeneralpurposesolverCPLEX.TheCPLEXrunswerestoppedwheneveraguaranteederrorboundof1%

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orlesswasachieved,allowingforamaximumCPUtimeof1;000seconds(or5;000secondsdependingonthesizeoftheproblem). Forvariousproblemclassescharacterizedbythenumberoffacilities,retailers,periods,andcommodities,werandomlygeneratedtheretailer'sdemands;unitproduction,inventoryandtransportationcosts;xedproductionandtransportationsetupcosts;andproductionandtransportationcapacities.Thevariabletransportationcostsdependonthelengthoftheroutefromthefacilitytotheretailer(Section2.8). Thetransportationarcsaresubjecttobundlecapacityconstraints.Thebundlecapacitiesshouldbechosenlargeenoughtomaketheproblemfeasible,butnotsolargethattheyareeectivelyabsent.Wehaveusedthefollowingapproachtogenerate\good"bundlecapacities:foreverytimeperiod,anecessaryconditionforfeasibilityoftheproposedmodelisFXi=1vijtKXk=1bjtkforj=1;:::;R;t=1;:::;T:(4.20) Nowifalltrucksthatmayserveretailerjinperiodthavethesamecapacity

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otherwords,theretailerdoesnothaveachoice.Inordertosatisfythedemand,fullloadshipmentsshouldbereceivedfromallthefacilities.Sincethisimpliesthatalltransportationsetupvariablesneedtobeequaltoone,theproblemreducestoalinearprogrammingproblem.Similarly,forvaluesofonlyslightlylargerthan1,fewornoneofthesetupvariableswillbeallowedtohaveavalueofzero,stillmakingtheproblemrelativelyeasytosolvetooptimality.Ontheotherhand,if=F,thebundlecapacitiesareredundant. Inourtestproblemsweconsidernotonlytransportationarcstobesubjecttobundlecapacities,butproductionarcsaswell.InordertogeneratebundlecapacitiesforproductionarcsweusethesameprocedureasdescribedinSection3.1.7.Theproductionbundlecapacitiesarecalculatedasfollows: FmaxtPRj=1Pt=1PKk=1bjk Varyingoneormorefactorsfromthenominalcase,wegeneratedsevengroupsofproblems.Intherstgroupofproblemsweincreasedthenumberofcommoditiesfrom10to15,20,25,30and35(problemclasses1to6;problemclass1corresponds

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Table4{1:Problemcharacteristics Problem Nodes Arcs 760 2,950 7,750 2 1,140 4,425 11,625 3 1,520 5,900 15,500 4 1,900 7,375 19,375 5 2,280 8,850 23,250 6 2,660 10,325 27,125 7 1,010 4,200 11,500 8 1,260 5,450 15,250 9 1,510 6,700 19,000 10 1,760 7,950 22,750 11 2,010 9,200 26,500 12 2,260 10,450 30,250 13 1,510 5,950 28,000 14 2,260 8,950 60,750 15 3,010 11,950 106,000 16 3,760 14,950 163,750 17 4,510 17,950 234,000 18 5,260 20,950 316,750 19 810 3,540 9,300 20 860 4,130 10,850 21 910 4,720 12,400 22 960 5,310 13,950 23 1,010 5,900 15,500 24{43 760 2,950 7,750

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tothenominalcase).Inthesecondgroupweincreasedthenumberofretailersfrom10(nominalcase)to15,20,25,30,35and40(problemclasses7to12).Inthethirdgroupofproblemsweincreasedthelengthoftimehorizonto10,15,20,25,30and35(problemclasses13to18).Inthefourthgroupweincreasedthenumberoffacilitiesto6,7,8,9and10(problemclasses19to23).Inthefthgroupwechangedthelevelofproductionset-upcoststositkU[200;900],sitkU[600;900],sitkU[900;1500],andsitkU[1200;1500](problemclasses24to27)andinthesixthgroupofproblems,theleveloftransportationset-upcostsischangedtosijtU[200;900],sijtU[600;900],sijtU[900;1500]andsijtU[1200;1500](problemclasses28to31). Withtheseventhgroupofproblemswewanttotesttheeectofthecapacitytightnessontheperformanceofthealgorithm.Sinceweconsidercapacitiesandxedcharges,wecannotclaimthatthetighterthearccapacity,themorediculttheproblembecomes.Thetwoextremecases,arccapacitiesbeingtootightortooloose,maketheproblemeasier.Thisisthereasonthatwewanttoanalyzetheeectofarccapacitiesonthedicultyoftheproblem.Wechangedthevalueofto1;1:1;1:2;1:4;1:5;1:6(problemclasses32to37)andto1;1:1;1:2;1:4;1:5;1:6(problemclasses38to43).InimplementingtheLagrangeandecompositionalgorithm,forallprobleminstancesweset=1:8.isreducedby20%ifthereisnoimprovementinthelast5iterations.Wesetalimitof300iterationsfortheLagrangeandecompositionalgorithm. Table4{1presentsthecharacteristicsoftheproblemsgeneratedinthissection.Tables4{2and4{3presenttheerrorsandrunningtimesofthethreevariantsofMCDSSP.Tables4{5and4{6presenttheerrorsandrunningtimesoftheLagrangeandecompositionalgorithm.TheerrorpresentedfortheheuristicsiswithrespecttothebestlowerboundfoundfromCPLEX.

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Table4{2:ResultsofMCDSSPonproblemGroups1,2,3and4 CPLEX MCDSSP1 MCDSSP2 MCDSSP3 Problem Error Time Error Time Error Time Error Time (%) (sec) (%) (sec) (%) (sec) (%) (sec) 1 0.01 540.01 0.87 1.05 0.87 0.98 0.78 0.91 2 0.01 137.93 0.77 4.03 0.77 2.60 0.74 2.09 3 0.01 85.12 0.70 4.05 0.70 3.95 0.69 2.93 4 0.01 36.98 0.68 8.08 0.68 4.66 0.64 3.27 5 0.01 35.45 0.65 5.88 0.65 4.96 0.58 4.11 6 0.01 41.27 0.61 22.63 0.61 7.01 0.56 4.44 7 0.08 997.66 0.99 1.05 0.98 0.95 0.92 0.96 8 0.10 1,000.00 0.82 3.58 0.82 3.11 0.80 1.79 9 0.11 1,000.00 0.83 3.02 0.83 2.68 0.81 2.55 10 0.11 1,000.00 0.75 3.85 0.75 3.42 0.75 3.20 11 0.10 1,000.00 0.75 5.26 0.75 4.22 0.75 4.11 12 0.12 1,000.00 0.71 5.15 0.71 4.47 0.71 4.36 13 0.29 1,000.00 1.37 2.09 1.35 1.98 1.31 2.06 14 N/A 5,000.00 1.37 4.05 1.34 4.61 1.29 3.95 15 N/A 5,000.00 1.48 6.09 1.46 6.28 1.36 5.61 16 N/A 5,000.00 1.47 7.48 1.47 7.09 1.36 7.33 17 N/A 5,000.00 1.65 19.30 1.63 9.59 1.59 9.71 18 N/A 5,000.00 2.54 12.59 2.54 11.68 2.45 11.87 19 0.23 5,000.00 1.82 0.86 1.79 0.76 1.76 0.75 20 0.67 5,000.00 2.36 1.30 2.34 1.13 2.35 1.11 21 1.39 5,000.00 2.83 3.30 2.82 2.19 2.83 1.68 22 0.72 5,000.00 3.45 2.78 3.39 2.99 3.42 2.12 23 0.74 5,000.00 3.81 4.39 3.71 3.73 3.81 3.02

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CPLEXLowerBound100: Tables4{2and4{3indicatethatMCDSSP3consistentlygavethebestresultsintermsoftimeandsolutionquality.ThisisduetothefactthattheupdatingschemeofMCDSSP3distributesthexedchargecostamongthecommoditiesusingthesamearc,byconsideringmoreinformationthanMCDSSP1andMCDSSP2.MCDSSP3 Table4{3:ResultsofMCDSSPonproblemGroups5,6,7and8 CPLEX MCDSSP1 MCDSSP2 MCDSSP3 Problem Error Time Error Time Error Time Error Time (%) (sec) (%) (sec) (%) (sec) (%) (sec) 24 0.10 958.95 1.28 0.58 1.25 0.54 1.20 0.54 25 0.14 1,000.00 3.12 0.57 2.95 0.67 1.60 0.63 26 0.30 1,000.00 4.38 0.60 4.25 0.55 2.40 0.63 27 0.49 1,000.00 7.31 0.61 7.03 0.61 3.40 0.75 28 0.10 959.18 1.28 0.57 1.25 0.53 1.20 0.54 29 0.26 1,000.00 2.80 0.62 2.66 0.55 2.58 0.56 30 0.36 1,000.00 3.36 0.65 3.19 0.54 3.24 0.54 31 0.57 1,000.00 5.51 0.72 5.26 0.58 5.04 0.57 32 0.26 1,000.00 1.33 1.40 1.22 1.02 1.23 1.19 33 0.18 1,000.00 1.27 0.78 1.20 0.74 1.14 0.77 34 0.12 962.10 1.30 0.66 1.28 0.59 1.17 0.66 35 0.09 944.15 1.30 0.54 1.25 0.49 1.19 0.52 36 0.10 946.46 1.30 0.52 1.24 0.48 1.17 0.50 37 0.10 949.81 1.29 0.51 1.24 0.46 1.19 0.49 38 0.01 506.53 1.17 0.66 1.15 0.59 0.76 0.62 39 0.03 812.30 1.10 0.56 1.03 0.51 0.93 0.53 40 0.12 965.23 1.08 0.58 1.05 0.52 1.04 0.50 41 0.05 859.45 1.25 0.54 1.23 0.51 1.17 0.53 42 0.03 693.57 1.17 0.54 1.13 0.50 1.11 0.51 43 0.02 537.20 1.09 0.52 1.05 0.48 1.03 0.50 assignstoeverycommoditythatusesthearcapartofthesetupcosts,consideringtheunitcostaswellastheamountshipped.DierentfromMCDSSP3,MCDSSP1equallydistributesthesetupcostsamongallthecommoditiesowingonthearc,andMCDSSP2assignsthesetupcoststothecommoditiesconsideringonlytheamountshipped.

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Table4{4presentsthequalityofthelowerboundsascomparedtoCPLEXsolutions.TheerrorpresentediswithrespecttothebestupperboundfoundbyCPLEX.Error(%)=CPLEXUpperBoundLPLowerBound CPLEXUpperBound100: Table4{4:Resultsfromlinearprogrammingrelaxation Problem Error Time Problem Error Time (%) (sec) (%) (sec) 1 1.82 1.53 24 1.98 1.62 2 1.40 2.69 25 2.16 1.83 3 1.11 3.56 26 2.43 2.19 4 0.90 4.53 27 2.57 2.23 5 0.75 5.67 28 2.60 1.61 6 0.65 6.86 29 2.85 1.56 7 1.70 2.41 30 2.95 1.62 8 1.70 3.48 31 3.04 1.57 9 1.63 5.30 32 2.02 2.92 10 1.70 5.95 33 1.97 1.97 11 1.56 7.62 34 1.85 1.68 12 1.58 9.52 35 1.81 1.52 13 1.86 10.11 36 1.81 1.50 14 2.84 32.1 37 1.82 1.40 15 2.85 70.77 38 0.49 2.69 16 2.84 144.41 39 1.14 1.81 17 2.91 301.96 40 1.77 1.60 18 2.97 608.08 41 2.09 1.56 19 2.00 2.23 42 2.37 1.53 20 2.68 3.13 43 2.65 1.52 21 3.72 4.29 22 3.02 5.89 23 3.04 7.99 however,increased,sinceasthenumberofcommoditiesincreases,thesizeofthelinearprogramstobesolvedineveryiterationofMCDSSPincreases.Therunning

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timesofLagrangeandecompositionincreasedaswell,sincebyincreasingthenumberofcommoditiesweareincreasingthenumberofsubproblems(SP1k)thataresolvedineveryiteration.TheseresultsareconsistentwithourndingsinSection3.1.7andotherstudiesonxedchargemulti-commoditynetworkowproblems(WuandGolbasi[106],Eksiogluetal.[35]). Thesizeoftheproblemsincreaseswiththenumberofretailers(thesecondgroupofproblems).ThisexplainstheincreaseintherunningtimesofCPLEX,MCDSSPandLagrangeandecomposition.Everythingelsekeptthesame,moreretailersimplyhigherdemand,higherproductionquantities,smallratiosoftotalxedchargetototalvariablecosts.TheseratiosaecttheperformanceofMCDSSPandLagrangeandecomposition.Inproblemswithsmallratiosoftotalxedchargetototalvariablecosts,thelinearprogrammingrelaxationofthexedchargecostfunctionsgivesbetterapproximations.Recallthat,intheLagrangeandecompositionalgorithm,subproblem(SP2)isnotsolvedtooptimality.Weprovidelowerboundsto(SP2)bysolvingitslinearprogrammingrelaxation. Forproblemclasses14to18,CPLEXwasn'tabletondafeasiblesolutionwithin5;000CPUseconds.ThesolutionsfromthethreeMCDSSPschemeswerewithin2:54%ofoptimality,andthesolutionsfromtheLagrangeandecompositionwerewithin1:64%ofoptimality. Increasingthenumberoffacilitieswhilekeepingeverythingelsethesame(problemclasses19to23)madetheproblemsdicult.Recallthatweassigntoeachfacilityequalproductioncapacity.Theassignmentissuchthatthetotalproductioncapacityuptoperiodtisequaltotimestotaldemanduptoperiodt.Increasingthenumberoffacilities(whilekeepingtotaldemandxed),decreasesthebundlecapacityassignedtoeachfacility.Thisisequivalenttodecreasing.Sofar,wehaveanalyzedtheeectofasinglefactoratatimeontheperformanceoftheheuristic.However,itseemsthatwhenincreasingthenumberoffacilities,wearechangingtwoparameters,sincewearealsodecreasingthebundlecapacitiesineachfacility.

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Table4{5:ResultsfromLagrangeandecomposition(ProblemClasses1to23) CPLEX Lag.Decomp. Problem Error Time Error Time (%) (sec) (%) (sec) 1 0.01 540.01 0.73 47.47 2 0.01 137.93 0.62 65.37 3 0.01 85.12 0.57 95.39 4 0.01 36.98 0.54 102.66 5 0.01 35.45 0.52 130.33 6 0.01 41.27 0.49 155.26 7 0.08 997.66 0.95 53.37 8 0.10 1,000.00 0.85 87.23 9 0.11 1,000.00 0.92 109.29 10 0.11 1,000.00 0.84 142.89 11 0.10 1,000.00 0.88 180.61 12 0.12 1,000.00 0.83 224.17 13 0.29 1,000.00 1.17 467.06 14 N/A 5,000.00 1.18 1,093.59 15 N/A 5,000.00 1.28 2,318.55 16 N/A 5,000.00 1.24 4,943.88 17 N/A 5,000.00 1.32 7,676.56 18 N/A 5,000.00 1.64 10,963.00 19 0.23 5,000.00 1.76 57.31 20 0.67 5,000.00 2.13 316.61 21 1.39 5,000.00 2.56 88.50 22 0.72 5,000.00 3.25 114.22 23 0.74 5,000.00 4.10 413.89

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Analternativeapproachistoaddnewfacilitiesthathavethesameproductioncapacityastheonesthatalreadyexist.Ifwedoso,thetotalavailableproductioncapacitywillincrease.Thisisequivalenttoincreasing.Experimentalresults(problemgroupseven)showthatincreasingmakesproblemseasier.Inourtestproblemswedecidedtogowiththerstapproach,ofaddingnewfacilities,andthenupdatethecapacitiesofallfacilitiessuchthattotalavailablecapacityuptotimeperiodtisequaltomultipliedbytotaldemanduptotimeperiodt. Furthermore,MCDSSP,LagrangeandecompositionandCPLEXbenetedfromsmallxedcosts(problemgroupsveandsix).Finally,thetightnessofthebundlecapacitiesforproductionarcs()andtransportationarcs()aectedtheperformanceofCPLEXandoftheheuristics(problemgroupseven).When()increased,andthusthecapacityconstraintsleftmoreroomforchoice,theproblembecamemoredicultquiterapidly.BundlecapacitiesfortheproductionarcsshowedtoaectmorethequalityofthesolutionsfromCPLEXandtheheuristics,thanthetransportationbundlecapacities.Resultsshowedhighererrorboundsforproblemclasses32to37ascomparedtoproblemclasses38to43.TheresultsofTable4{4indicatethatthequalityofthelowerboundsfromthelinearprogrammingrelaxationofextendedformulationwereatmost3:72%fromoptimalandtherunningtimeswerequitesmall. ThemajorobservationfromthetablesisthatCPLEXwasgenerallyabletondabettersolutionthanMCDSSPandLagrangeandecomposition,butattheexpenseofalargeamountofCPUtime.Giventheoperationalnatureofourproblem,itmaynotbefeasibletospendthetimetakenbyCPLEXtosolvetheproblem.Forthelargerproblems(problemclasses14to18),CPLEXwasnotabletondafeasiblesolutionwithin5;000CPUseconds,whereasMCDSSPandLagrangeandecomposition,bytheirnature,wereabletondafeasiblesolutionforallprobleminstances. WeconcludethatMCDSSPcanbeaveryusefultool,giventhespeedatwhichreasonablefeasiblesolutionsarefound.Theerrorboundsofthesolutionobtained

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Table4{6:ResultsfromLagrangeandecomposition(ProblemClasses24to43) CPLEX Lag.Decomp. Problem Error Time Error Time (%) (sec) (%) (sec) 24 0.14 1,000.00 2.00 75.02 25 0.30 1,000.00 3.26 86.56 26 0.49 1,000.00 5.30 93.43 27 0.63 1,000.00 6.41 91.77 28 0.26 1,000.00 2.68 62.74 29 0.36 1,000.00 3.16 63.47 30 0.57 1,000.00 5.57 70.35 31 0.66 1,000.00 5.91 69.11 32 0.26 1,000.00 1.20 104.16 33 0.18 1,000.00 1.01 76.71 34 0.12 962.10 1.02 56.42 35 0.09 944.15 1.00 52.59 36 0.10 946.46 0.98 53.02 37 0.10 949.81 1.01 53.02 38 0.01 506.53 0.81 63.38 39 0.03 812.30 0.83 59.66 40 0.12 965.23 0.78 55.75 41 0.05 859.45 0.98 52.18 42 0.03 693.57 0.96 50.67 43 0.02 537.20 0.87 49.80

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byMCDSSPwereatmostashighas5%forthehardestproblems.TheerrorsfromMCDSSPandLagrangeandecompositionwerecomparable. 4.6 Conclusions Inthischapterwestudiedaclassofproduction-distributionproblemsarisinginsupplychains.Ourproblemsconsiderxed-chargeproductionandtransportationcostsandjointproductionandtransportationcapacities.WerstproposedaheuristiccalledDynamicSlopeScalingProcedure(MCDSSP)tosolvetheseproblems.Thisprocedurecanbeusedtosolveanymulti-commoditynetworkowproblemwithxedchargecostfunctions.WeprovidedthreealternativeimplementationsoftheMCDSSP.Weidentiedoneofthethreealternativeimplementationsofourheuristicthatseemstoconsistentlyprovidethebestsolutionqualityandcomputationtime. WealsodiscussedaLagrangeandecompositionbasedalgorithm.TheLagrangeandecompositionalgorithmdecomposedtheproblemintotwosubproblems(SP1)and(SP2).Subproblem(SP1)wasfurtherdecomposedbycommodityintoKsingle-commodityproblems(SP1k).Theproblems(SP1k)aresimilartothemulti-facility,multi-retailerlot-sizingproblemwediscussedinSection3.2,thereforeweusetheprimal-dualalgorithmtosolvethem.Weprovidelowerboundstosubproblem(SP2)bysolvingthecorrespondinglinearprogrammingrelaxation. ComparingourheuristicstoCPLEX,weconcludethatCPLEXwasusuallyabletondabettersolutionthantheheuristics,attheexpenseofmuchmorecomputationtime.Giventheoperationalnatureofourproblem,andthelimitedavailabilityoftimeavailabletosolvetheprobleminthesesituations,ourheuristicsmaybeanattractivealternative. Itiswell-knownthatndingagoodlowerboundfornetworkowproblemswithxed-chargecostsisdicult,duetothefactthatthelinearprogrammingrelaxationofthetraditionalMILPformulationisnotverytight.However,thelowerboundsgeneratedusingthelinearprogrammingrelaxationoftheextendedformulationgavelowerboundsthatarewithin3:72%ofoptimality.

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Inthisdissertationwestudyaclassofsupplychainoptimizationproblems.Severalfactorsmotivatedthisresearch.First,increasedcompetitionandmarketresponsivenesshavemadesupplychainssubjecttomoreinter-relateddynamicsthaninthepast.Asaresultcompaniesfrequentlyneedtoreconsidertheirdistributionnetworks.Secondly,becauseofthewidevarietyofissuesinvolvedinmanagingasupplychain,mostoftheoptimizationmodelshaveconcentratedonspecicareasofresearchsuchastransportationorinventory,withoutbeingabletotakeaglobalviewofalltheprocessesinvolved.Finally,therecentdevelopmentsincomputingandsolutionalgorithmshavemadeitpossibletoinvestigaterichermodelsthaninthepast. Inparticular,thesupplychainoptimizationproblemwestudyconsidersasetoffacilitiesandasetofretailers.Retailersfacepositivedemandsforanumberofcommodities.Facilitiesproduceandstoretheproductsuntilretailersdemandsoccur.Productionatafacilityisconstrainedandtruckcapacitylimitsthetotalamountofproductsthatcanbeshippedtoaretailerinaperiod.Wedonotallowfortransportationbetweenfacilitiesorbetweenretailers.Thegoalistondthemostecientway(e.g.,thecheapestway)tosatisfythisdemand.Productionandtransportationcostsaremodelledusingxedchargecostfunctions.Thedecisionstobemadeare(i)theselectionofaproductionfacilityandquantities,(ii)theassignmentofretailerstofacilitiesand(iii)thelocationandsizeofinventories. Forabetterunderstanding,wehavefollowedacontractiveapproach,wherethesupplychainmodelswestudyhavebeenenrichedgradually.First,weanalyzethesinglecommodityandmulti-facilitylotsizingproblem.Thisisanextensionofthe 127

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economiclot-sizingproblem.Weaddanewdimensiontotheclassicalproblem,thefacilityselectiondecision.Weconsidertransportationcoststogetherwithproductionandinventorycost,andtheirimpactsonlot-sizingdecisions.Second,westudyaclassofmulti-commodityandmulti-retailerlot-sizingproblems.Weevaluatetheperformanceofthesupplychainwhenmultipleproductshavetobeproducedanddistributed.Inthismodelonlytheproductioncapacityhasbeenrestricted.Thethirdmodelweproposeconsidersadynamicsupplychainconsistingofanumberoffacilitiesandretailers.Finally,wediscussaclassofproduction-distributionproblemsthatconsidersmultiplecommodities,xedchargetransportationcosts,aswellasproductionandtransportationcapacities. Themainfocusofthisdissertationistodevelopsolutionprocedurestosolvetheseoptimizationmodels.Conclusionsabouttheirperformancearedrawnbytestingthealgorithmsonawidecollectionofprobleminstances.Forthemulti-retailerlot-sizingproblemweproposeadierentformulationthatwerefertoastheextendedproblemformulation.Thelinearprogrammingrelaxationoftheextendedformulationgivestighterlowerboundsascomparedtothelinearprogrammingrelaxationofthe\original"problemformulation.Wedevelopaprimal-dualalgorithm,acuttingplanealgorithm,andadynamicprogrammingbasedalgorithmforthisproblem.Wesolvethecapacitated,multi-commodityversionoftheproblemusingaLagrangeandecompositionalgorithm.Thesubproblemsfromthedecompositionaresolvedusingaprimal-dualalgorithm.Aprimal-dualalgorithmisusedtosolvethemulti-retailer,multi-facilitylot-sizingproblemaswell. InChapter4wediscussaslopescalingproceduretosolvetheproduction-distributionproblem.Thisprocedurethatwecallthemulti-commoditydynamicslopescalingprocedure(MCDSSP)canbeusedtosolveanymulti-commoditynetworkowproblemwithxedchargecostfunctions.Themotivationbehindthisheuristicsisthefactthataconcavecostfunction(suchasaxedchargefunction),whenminimizedoveraconvexset,givesanextremepointsolution.Thesameholdstrueforlinear

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programs.Thus,MCDSSPiterativelyapproximatesthexedchargecostfunctionbyalinearfunctionandsolvesthecorrespondinglinearprograminsearchofalinearprogramthatwouldbeminimizedonthesamevertexofthefeasibleregionasthexedchargefunction.

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Givena(l;S)suchthatl1.Therstsetoffeasiblesolutionsthatwediscussisthefollowing:Considerallfeasiblesolutions(q;y;x)2suchthatqit=yit=0,fort=k;:::;landi=1;:::;F.Thesesolutionssatisfythevalidinequality(l;S)atequality. Inthisparticularcase,ourproblemcanbedecomposedintotwomulti-facilitylotsizingproblems,oneconsistingofperiods1;:::;k1,withb0t=bt,fort=1;:::;k2,andb0k1=bk1;l,andtheotherconsistingofperiodsl+1;:::;T.Therstsub-problemhas3(k1)FF(k1)1linearlyindependentsolutions(qpA;ypA;xpA)2R3(k1)F.Thesecondsub-problemhas3(Tl)F(Tl)1linearlyindependentsolutions(qqB;yqB;xqB)2R3(Tl)F. Combiningthesevectorsandinsertingqit=yit=0,fort=k;:::;l,andi=1;:::;F,gives(3(k1)FF(k1))+(3(Tl)FF(Tl))1=(3FTFT1)(3F1)(lk+1)F Nowwewillpresentafewmoreanelyindependentsolutions.Bydenition,n+1solutions0;:::;nareaneindependentifdirections10;20;:::;n0

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arelinearlyindependent.Intherestofourdiscussionwerstpresentasetofsolutionsthatsatisfythe(l;S)inequalityatequality,thenwewillgenerate(3F1)(kl+1)+Fdirectionsandshowthattheyareanelyindependent.Ingeneratingthedirectionswerefertothesolution(0)thatissubtractedfromagivensolution(ifori=1;:::;n)withd. Fixt02fk;:::;lg,andi2f1;:::;Fg. Solutions2 Solutions3 Thefourthsetofsolutionsthatsatisesthe(l;S)atequalityisthefollowing: Foreacht0=k;:::;l,andi=1;:::;F1:

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Solutions4 Analsetofsolutionsisthefollowing:lett0=kandi=1;:::;F,andchoosethefollowingsolutions: Solutions5 Now,wewillconstructdirections(thatwewillshowtobelinearlyindependent)bysubtractingdfromthesetofsolutionspresentedabove.ForSolutions2and3,letd1ibethefollowingsolutionsfori=1;:::;F. Directionsd1i: Directions2 Directions3

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Directions2 Directions3 Wegeneratethefourthsetofdirectionssubtractingdt0Ffort0=k;:::;lfromthefourthsetofsolutions. Directionsdt0F: Directions5

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Thelaststepistoshowthattheindividualdirectionsfromthevesetsofdirectionsareinfactlinearlyindependent.Therstsetofsolutionswehavepresentedarelinearlyindependentbyconstruction.Therstsetofsolutionsissuchthat:qit0=yit0=xit0=0,fort0=k;:::;landi=1;:::;F. Directions2and3giveatotalof2F(lk+1)directions.Considervariablesqit0andyit0fort0=k+1;:::;landi=1;:::;F.ThesevariablesaredierentthanzeroonDirections2and3only. ConsiderCase1forDirections2and3: ConsiderCase2forDirections2and3: Onecanseethatforanyi=1;:::;Fandt0=k+1;:::;l,directions2and3cannotbeexpressedaslinearcombinationsofeachother(i.e.,bt0land1isnotamultipleofbt0;l+1and1).Thisshowsthatinthesetofdirectionswepresent,thereare2F(lk)linearlyindependentdirections. Fort=k,qikandyikisdierentthanzeroindirections2,3and5.Directions2:qik=bkl;yik=1fori=1;:::;F:

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Directions3:qik=bk;l+1;yik=1fori=1;:::;F:Directions5:qik=bkl;yik=1;qFk=bkl;yFk=1;fori=1;:::;F1: Thefourthsethas(F1)(lk+1)directions.Fort0=k;:::;landi=1;:::;F1,wehavexit=btforeacht=k1;:::;t01,thatcreatesalowertriangularmatrix.Thesedirectionsadd(F1)(lk+1)linearlyindependentdirections. Inthefthsetofsolutionsxilisdierentthanzerofori=1;:::;F1inonlyonedirection.xilisequaltozeroinalltheotherdirections.ThisgivesF1extralinearlyindependentdirections. Intotalwehavepresented(consideringDirections2to5):2F(kl)+2F+(F1)(kl+1)+(F1)=(3F1)(kl+1)+F1 linearlyindependentdirections,or:(3F1)(kl+1)+F anelyindependentdirections. Wepresentedasetofsolutionsthatsatisfythevalidinequalityasequalityandshowedthatthesesolutionsgavedim(co())anelyindependentdirections.Wehaveprovedthatthe(l;S)inequalitiesarefacetdening.2

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denition,apathconsistsofasequenceofnodesandarcssuchthatthereisnorepetitionofthenodesinthepathandforanytwoconsecutivenodesanarcexistsconnectingthem.WerstwanttoshowthateverypathinG0thatconnectsnode1toT+1,correspondstoanextremepointsolutionintheextendedformulationofthemulti-retailerlot-sizingproblem. LetPbethesetofallpathsinG0thatconnectnode1toT+1,andp2P.SincepisapathinG0,therewillbeexactlyonearcinpstartingatnode1andexactlyonearcendingatnodeT+1.Thisimpliesthatdemandinperiods1andTwillbesatised.Let(;0)2p.Byconstructionweknowthatanarc(;0)2pimpliesthatdemandinperiods;+1;:::;01issatisedfromproductioninasinglefacilityinasingletimeperiodt(t=1;:::;).Since(;0)2p,thantherewillbenootherarcinthepaththatstartsatoneofthenodes;+1;:::;01andnootherarcthatendsatnodes+1;:::;0.Thisimpliesthatdemandinperiods;+1;:::;0issatisedfromasinglefacilityinasingletimeperiodpriortoorbeginningat. Bydenition,thereisexactlyonearcenteringandexactlyonearcleavingeverynodeinapath.Therefore,therewillbeexactlyonearcenteringnodeandexactlyonearcleaving0(since(;0)2p),whichimpliesthatdemandsintimeperiodsbeforeandafter0aresatisedbyproductionin(only)oneperiodatexactlyonefacility.WeconcludethateverypathinG0thatconnectsnodes1toT+1representsaproduction,inventoryandtransportationschedulethatsatisesdemandineveryperiod.Thisscheduleissuchthatdemandsaresatisedfromproductioninasinglefacilityinasingletimeperiod.Suchascheduletstherequirementsofanextremepointsolutiontothemulti-facilitylot-sizingproblem.Therefore,foreverypathinG0thatconnectsnodes1toT+1correspondsanextremepointsolutionofthemulti-facilitylot-sizingproblem. Thenextstepistoshowthattoeverysequentialextremepointsolutionofthefeasibleregionintheextendedformulationofthemulti-facilitylot-sizingproblemcorrespondsapathinG0thatconnectsnode1withT+1.LetRrepresentthe

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setofallsequentialextremepointsolutionsandr2R.Assuch,rrepresentsaproductionschedulethatsatisesdemandsduringperiods1;:::;T.Twoimportantcharacteristicsofrarethefollowing:(i)demandineverytimeperiodissatisedfromproductioninasinglefacilityinasingletimeperiod(ii)afacilityattimeperiodteitherdoesnotproduce,orproducesthedemandforoneormoreperiods(theseperiodsshouldbesequential).Demandsinperiodst=1;:::;TaresatisedfromproductionatatmostFdierentfacilitiesinatmostTperiods.Considerthefollowingcases: Thisshowsthattoeverysequentialextremeowsolutionto(MF)correspondsapathinG0.Thisconcludesourproof.

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SandraDuniEksio~gluwasbornonSeptember10,1972inTirana,Albania.In1994,shewasawardedabachelor'sdegreeinBusinessAdministrationfromtheSchoolofBusiness,UniversityofTiranainTirana,Albania.Sheearnedhermaster'sdegreeinManagementSciencesfromtheMediterraneanAgronomicInstituteofChania,Greecein1996.ThenshereturnedtoAlbaniaandworkedasinstructorintheBusinessschoolattheUniversityofTirana.ShebeganherdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFloridainAugust1997andreceivedherPh.D.inDecember2002. 146


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OPTIMIZING INTEGRATED PRODUCTION, INVENTORY AND
DISTRIBUTION PROBLEMS IN SUPPLY CHAINS















By

SANDRA DUNI EKSIOGLU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2002
















This work is dedicated to my family.















ACKNOWLEDGMENTS

I would like to thank the people who helped me complete the work contained

in this dissertation. The help of my supervisors Panos Pardalos and Edwin Romeijn

was of great value. I would like to thank Panos Pardalos for his technical advice,

encouragement and insightful comments throughout my dissertation work. I would

like to thank Edwin Romeijn for working closely with me. His unconditional support

in solving in ii: details surrounding this dissertation and his valuable feedbacks are

deeply appreciated.

I extend my thanks to the members of my committee Selcuk Erengiic and Joseph

Geunes for their constructive criticism concerning the material of this dissertation. I

also would like to express my appreciation to all my friends at the ISE department

and in Gainesville for lightening up my life every d4i and making graduate school

more fun than I am sure it is supposed to be. In particular I would like to thank

Adriana and Jorge Jimenez, Mirela and Ilir Bejleri, Ebru and Deniz Erdogmus,

Paveena (' ,,valitwongse, Bayram Yildirim, Seviye Yoriik, Mary and Kevin Taaffe,

Olga Perdikaki, Hiilya Emir, Sergiy Butenko and Lihui Bai.

I would like to express my special thanks to my parents Leonora and Perikli

Duni and my brother Dhimitraq Duni. Their understanding and faith in me and my

capabilities, their love, encouragement, and eternal support have motivated me all

the time. Last but not least, I would like to thank my husband Burak for his love,

patience and continuous support throughout all my years here at the University of

Florida.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ........ iii

ABSTRACT ...................... ................ vi

CHAPTER

1 INTRODUCTION ................................ 1

1.1 Supply C('!, i Management ........... ............ 1
1.2 Framework of This Study ......................... 3
1.3 Objectives and Summary ........... .............. 6

2 MULTI-FACILITY LOT-SIZING PROBLEM ................ 12

2.1 Introduction ................... ......... 12
2.2 Literature Review ................... ....... 13
2.3 Problem Description ................... ..... 16
2.4 Extended Problem Formulation ............... .. ... 21
2.5 Primal-Dual Based Algorithm . . ........ . 26
2.5.1 Intuitive Understanding of the Dual Problem . ... 27
2.5.2 Description of the Algorithm .................. .. 28
2.5.3 Running Time of the Algorithm ................ .. 30
2.6 Cutting Plane Algorithm .................. ..... 31
2.6.1 Valid Inequalities .................. ..... .. 31
2.6.2 Separation Algorithm . . .......... .. 32
2.6.3 Facets of Multi-Facility Lot-Sizing Problem . .... 33
2.7 Dynamic Programming Based Heuristic . . . 35
2.7.1 Introduction .................. ......... .. 35
2.7.2 Description of the Algorithm .................. .. 37
2.7.3 Running Time of the Algorithm ................ .. 40
2.8 Computational Results .................. ..... .. 42
2.9 Conclusions .................. ............. .. 52

3 EXTENSIONS OF THE MULTI-FACILITY LOT-SIZING PROBLEM ... 54

3.1 Multi-Commodity, Multi-Facility Lot-Sizing Problem . ... 54
3.1.1 Problem Formulation ................ .... .. 56
3.1.2 Linear Programming Relaxation ................ .. 59
3.1.3 Valid Inequalities .................. ..... .. 61
3.1.4 Lagrangean Decomposition Heuristic . . ..... 62
3.1.5 Outline of the Algorithm ................... ... .. 67










3.1.6 Managerial Interpretation of the Decomposition .. ......
3.1.7 Computational Results .. ..................
3.2 Single-Commodity, Multi-Retailer, Multi-Facility Lot-Sizing Problem
3.2.1 Problem Formulation .. ...................
3.2.2 Primal-Dual Algorithm .. ..................
3.2.3 Intuitive Understanding of the Dual Problem .. ........
3.2.4 Outline of the Primal-Dual Algorithm .. ............
3.2.5 Computational Results .. ..................
3.3 Multi Facility Lot-Sizing Problem with Fixed Charge Transportation
C o sts . . . . . . . . .
3.4 C conclusions . . . . . . . .

4 PRODUCTION-DISTRIBUTION PROBLEM .. ..............


4.1 Introduction . . . . .
4.2 Problem Formulation .. ............
4.3 Dynamic Slope Scaling Procedure . ..
4.3.1 Multi-Commodity Network Flow Problem
Cost Function . ........
4.3.2 Single-Commodity Case . ....
4.3.3 Multi-Commodity Case . ....
4.3.4 Production-Distribution Problem . .
4.3.5 Extended Problem Formulation . .
4.4 A Lagrangean Decomposition Procedure .


4.5 Computational Results .
4.6 Conclusions . ....

5 CONCLUDING REMARKS .

APPENDICES .. ...........


with Fixed C('i ige
.. . 100
. .. . 04
. .. . 04
... . 08
. .. . 09
.. . 111
.. . 114
. .. . 26


. . ... . 2 7


REFERENCES ......................... . . 138


BIOGRAPHICAL SKETCH .......















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

OPTIMIZING INTEGRATED PRODUCTION, INVENTORY AND
DISTRIBUTION PROBLEMS IN SUPPLY CHAINS

By

Sandra Duni Eksioglu

December 2002

Chair: Panagote M. Pardalos
Co-Chair: H. Edwin Romeijn
Major Department: Industrial and Systems Engineering

The goal of this dissertation is the study of optimization models that integrate

production, inventory and transportation decisions, in search of opportunities to

improve the performance of a supply chain network. We estimate the total costs of a

given design of a general supply chain network, including production, inventory and

transportation costs. We consider production and transportation costs to be of fixed

charge type. Fixed charge cost functions are linear functions with a discontinuity at

the origin.

The main focus of this dissertation is the development of solution procedures

for these optimization models. Their computational complexity makes the use of

heuristics solution procedures advisable. One of the heuristics we propose is a Multi-

Commodity Dynamic Slope Scaling Procedure (\I CDSSP). This heuristic makes use

of the fact that when minimizing a concave function over a convex set, an extreme

point optimal solution exists. The same holds true for linear programs. Therefore,

the concave cost function is approximated by a linear function and the corresponding

linear program is solved. The slope of the linear function is updated iteratively until










no better solution is found. The MCDSSP can be used to solve any multi-commodity

network flow problem with fixed charge cost functions.

We also develop a Lagrangean decomposition based heuristic. The subproblems

from the decomposition have a special structure. One of the subproblems is the multi-

facility lot-sizing problem that we study in detail in C'! Ilpter 2. The multi-facility

lot-sizing problem is an extension of the economic lot-sizing problem. We add a new

dimension to the classical problem, the facility selection decision. We provide the

following heuristic approaches to solve this problem: dynamic programming, a primal-

dual method, a cutting plane method and a linear programming based algorithm. We

propose a set of valid inequalities and show that '1!, ', are facet defining. We tested

the performance of the heuristics on a wide range of randomly generated problems.

We also studied other extensions of the multi-facility lot-sizing problem. In

Chapter 3 we analyze and provide solution approaches to the multi-commodity and

multi-retailer (single-commodity) versions of the problem.















CHAPTER 1
INTRODUCTION

1.1 Supply C(' iii Management

Most companies 1i I. ,1 ivs are organized into networks of manufacturing and

distribution sites that procure raw materials, process them into finished goods, and

distribute the finished goods to customers. The goal is to deliver the right product

to the right place at the right time for the right price. These production-distribution

networks are what we call "supply chains."

Supply C'!i ,i, Management is a growing area of interest for both companies and

researchers. It first attracted the attention of companies in the 1990s as '1i vi, started

to realize the potential cost benefits of integrating decisions with other members

of their supply chain. The primary cost factors within a supply chain can be put

into the categories of production, transportation and inventory. The signature of

supply chain management is the integration of activities. Effective supply chain

members invariably integrate the wishes and concerns of their downstream members

into their operations while simultaneously ensuring integration with their upstream

members. We concentrate on developing optimization tools to enable companies to

take advantage of opportunities to improve their supply chain.

For many years companies and researchers failed to take an integrated view of

the entire supply chain. They considered only one piece of the overall problem, such

as production or distribution submodels. These submodels were optimized separately

and the solutions were then joined together to establish operating policies.

A number of new developments have had an impact on many companies. For

example, increased market responsiveness has intensified the inter-dependencies within

the supply chain (Erengiic et. al [30]); technological innovations have shortened









the life span of manufacturing equipment, which in turn increases the cost of

manufacturing capacity; internet has offered high speed communication (Geunes et.

al [50]). These developments combined with increased product variety and decreased

product volumes prompt companies to explore new v--- of running their business.

Experience has shown that a firm's ability to manage its supply chain is a 1,i i ,i

source of competitive advantage. This realization is the single most important reason

for the recent emphasis on supply chain management in industry and academia. To

exploit these new opportunities to improve their profitability, the companies need

decision support tools that provide evaluation of alternatives using optimization

models.

Several examples can be found in the literature proving that models coordinating

at least two stages of the supply chain can detect new opportunities for improving

the efficiency of the supply chain. C'!i ,i i i and Fisher [21] investigated the effect

of coordinating production and distribution on a single-plant, multi-commodity,

multi-period scenario. In this scenario, the plant produces and stores the products

until '!, i,- are delivered to the customers using a fleet of trucks. They proposed two

solution approaches. The first approach solves the production scheduling and routing

problems separately and the second approach considers both, production and routing

decisions to be incorporated into the model. Their computational study showed that

the coordinated approach can yield up to 211' in costs savings.

Anily and Federgruen [4] considered integrating inventory control and

transportation planning decisions motivated by the trade-off between the size and

the frequency of delivery. Their model considered a single warehouse and multiple

retailers with inventories held only at the retailers who face constant demand. Burns

et al. [18] investigated distribution strategies that minimize transportation and

inventory costs. Successful applications of supply chain decision coordination were

reported at Xerox [94] and Hewlett Packard [44]. As a result of coordinating the










decisions on inventories throughout the supply chain, '!. :, were able to reduce their

inventory levels by 25'.

1.2 Framework of This Study

Companies deliver products to their customers using a logistics distribution

network. Such networks typically consist of product flows from the producers to

the customers through distribution centers (warehouses) and retailers. Companies

generally need to make decisions on production planning, inventory levels, and

transportation in each level of the logistics distribution network in such a way that

customer's demand is satisfied at minimum cost.

Coordinating decisions with other members with the aim of 'i. -. -r profits

and better customer service is a distinctive feature of supply chain management.

Bhatnagar, ('C! ,1,i i. and Goyal [13] distinguished between two broad levels of

coordination. At the most general level, coordination can be seen in terms of

integrating decisions of different functions (for example, facility location, inventory

planning, production pl1 '!'i:i- distribution pl1 'iw!''- etc.). They refer to this level of

coordination as ;, i, I I coordination." At another level, the problem of coordination

may be addressed by linking decisions within the same function at different echelons

in the organization.

They classified the research on general coordination into three categories. These

categories represent integration of decision-making related to

(i) Supply and production planning

(ii) Inventory and distribution planning

(iii) Production and distribution planning

Studies on coordination between supplier and buyer focused on determining the

order quantity that is jointly optimal for both. The second category addresses the

problem of coordinating inventory planning with distribution planning. This problem

emerges when a number of customers must be supplied from one or more warehouses.

The inventory and distribution planning problem decides on the replenishment policy










at the warehouse and the distribution schedule for the customers, so that the total

of inventory and distribution costs are minimized. The trade-off is reducing the

inventory costs versus an increase in the transportation costs. For example, shipping

in smaller quantities and with high frequency reduces the inventory level at the

warehouse, but causes higher transportation costs.

The third category of research concentrates on integrating production planning

and distribution planning. The production planner is concerned with determining

optimal production-inventory levels for each product in every period so that the

total cost of production and inventory holding is minimized. On the other hand,

the distribution planner must determine schedules for distribution of products to

customers so that the total transportation cost is minimized. These two activities

can function independently if there is a sufficiently large inventory buffer that

completely decouples the two. However, this would lead to increased holding costs

and longer lead times, since on one side the distribution planer, in order to minimize

transportation costs, would prefer full truck shipments and minimum number of

stops; and on the other side the production planner would prefer less number of

machine setups. The pressure of reducing inventory and lead times in the supply

chain forced companies to explore the issue of closer coordination between production

and distribution.

Our work contributes to this last research category. We consider complex supply

chains with centralized decisions on production-inventory and transportation. We

model these supply chains as multi-commodity network flow problems with non-

convex cost functions. In particular we use fixed charge cost functions to model

production or transportation costs. We present several optimization techniques to

solve these problems.

Related work is the research of King and Love [67] on the coordination of

production and distribution systems at Kelly Springfield, a 1n ii "r tire manufacturer

with four factories and nine in, ii distribution centers located throughout the










United States. The authors described a coordinated system for manufacturing plants

and distribution centers. Implementation of this system resulted in substantial

improvements in overall lead times, customer service and average inventory levels.

Annual costs were reduced by almost $8 million.

Blumenfeld et al. [16] reported on the successful implementation of an

optimization model that synchronizes scheduling of production and distribution

at the Delco electronics division of General motors. Implementation resulted in a 2'-.

reduction in logistics costs.

Brown et al. [17] presented a successful implementation of an optimization model

that coordinates decisions on production, transportation and distribution at K. 11..-:-

Company. K. 11..-:-; operates five plants in the United States and Canada, and it has

seven core distribution centers. The model has been used for tactical and operational

decisions since 1990. The operational version of the model determines everyd-v

production and shipping quantities. The tactical version helps to establish budget

and make capacity expansion and consolidation decisions. The operational model

reduced production, inventory and distribution costs by approximately $4.5 million in

1995. The tactical model recently guided a consolidation of production capacity with

a projected savings of % ;: to $40 million per year.

Models coordinating different stages of the supply chain can be classified as

strategic, tactical or operational (Hax and Candea [55]). The model we study helps

the companies to make strategic decisions related to production, inventory and

transportation. Several surveys in the literature address coordination issues. Vidal

and Goetschalckx [102] addressed the issue of strategic production-distribution

planning with emphasis on global supply chains. Beamon [12] presented models on

multi-stage supply chain design and analysis. Erengiic et al. [30] surv, v, d models that

integrate production and distribution planning. Thomas and Griffin [97] surveyed

coordination models on strategic and operational planning.










1.3 Objectives and Summary

We propose a class of optimization models that consider coordination of

production, transportation and inventory decisions in a particular supply chain. The

supply chain we consider consists of a number of facilities and retailers. This model

helps to estimate the total cost of a given logistics distribution network, including

production, inventory holding and transportation costs. The evaluation takes place

over a fixed, finite planning horizon of T periods.

The particular scenario presented considers a set of facilities where K different

product types can be produced. Products are stored at the facilities until demand

occurs. Moreover, retailers are supplied by the facilities and keep no inventories.

We do not allow for transportation between facilities. An example of this particular

scenario is the supply chain for very expensive items (such as cars). Often, it is not

affordable for the retailers to keep inventories of expensive items, therefore '! .:, order

the commodities as demand occurs (for example, Ford dealers keep no inventories for

Corvette).

We consider production and transportation cost functions to be of the fixed

charge form. Often in the literature production costs are modeled using this type of

cost function. This is because of the special nature of the problem. Thus, whenever

the production of a commodity takes place, a fixed charge is paid to set up the

machines, plus a variable cost for every unit produced. Transportation costs also

can be modelled using fixed charge costs, since usually there is a fixed charge (for

example, the cost of the paperwork necessary) to initiate a shipment plus other costs

that depend on the amount shipped.

The objective is to find the production, inventory, and transportation quantities

that satisfy demand at minimum cost. We formulate this problem as a multi-

commodity network flow problem on a directed, single source graph consisting of

T -1v-, rs (Figure 1 1). Each l-1v- r of the graph represents a time period. In each IlV--r,










a bipartite graph represents the transportation network between the facilities and

retailers.

The set of nodes of the network consists of T copies of the F facilities and R

retailers, as well as the source node. The set of arcs consists of production arcs

(between the source node and a facility at a particular time period), transportation

arcs (between a facility and a retailer at a particular time period), and inventory

arcs (between two nodes corresponding to a particular facility in consecutive time

periods). There are bundle capacities on the transportation and production arcs,

and no capacities on the inventory arcs. In multi-commodity network flow problems,

the bundle capacities tie together commodities by restricting the total flow (of all

commodities) on an arc.

Our problem is related to the ones studied by Wu and Golbasi [106]; and Freling

et al. [43]; and Romeijn and Romero Morales [90, 91]; and Romero Morales [83];

and Balakrishnan and Geunes [6]. In contrast to our model, Wu and Golbasi [106]

assume a fixed-charge cost structure for production, but assume linear transportation

costs. Also, '! :,- consider in their model the product structure of each end-item.

On the other hand, Freling et al. [43], Romeijn and Romero Morales [90, 91] and

Romero Morales [83] consider the case of a single commodity only, but account for

the presence of so-called single-sourcing constraints, where each retailer should be

supplied from a single facility only. Balakrishnan and Geunes [6] addressed a dynamic

requirements-planning problem for two-stage multi-product manufacturing systems

with bill-of-material flexibility (i. e., with options to use substitute components or

subassemblies produced by an upstream stage to meet demand in each period at the

downstream stage). Their model is similar to the multi-retailer and multi-facility

lot-sizing problem we discuss in Chapter 3. However, their formulation is more general

and consists of extra constraints (similar to the single-sourcing constraints) and

extra components in the cost function (penalties for violating the single-sourcing

constraints).






























Figure 1 1: Two 1I. w y, .hrcc-retailcr 1'y chain ( '' i )em

The supply chain optimization model we discuss is an NP-hard problem. Even

the special case of the multi-commodity network flow problem with fixed charge costs,

the single-period and single-commodity problem is NP-hard (Garey and Johnson [45]).

The complexity of this problem led us to consider mainly heuristic approaches.

The difficulty in solving this problem motivated us to look into some special cases

of the model. The knowledge we gained from analyzing the simpler problems gave

us insights about how to approach the general model. Below we describe some of the

special cases we identified.

The single-retailer, single-facility problem with only fixed charge production costs,

reduces to the classical lot-sizing problem (Figure 1 2). The retailer in each period

has to ship his demand from the facility. Therefore, the only decision to be made in

this problem is for the facility to decide on its production schedule, when and how

much to produce in every period (Figure 1-3). The same holds true when the model

considers a single facility with multiple retailers.



























Figure 1-2: Muli-perioS
Figure~ 1-2: N/\ ii perl()


Si


VO




A 1 5
K




A6


-R


P-T


F






F
A6


-*QD


Jingle-facility, single-retailcr problem


i-r,

r,


Figure 1- Multi-period, single-fcacility, single-rtailcr problem









The single-commodity, multi-facility, single-retailer problem with only production

setup costs is discussed in C'!i lpter 2 (Figure 1-4). We refer to this problem as the

multi-facility lot-sizing problem. Wu and Golbasi [106] show that the uncapacitated

version of this problem with holding costs not restricted in sign is NP-complete. We

propose a number of heuristic approaches to solve the problem such as a dynamic

programming based algorithm, a primal-dual heuristic and a cutting plane algorithm.

We propose a set of valid inequalities and show that '1!, ', are facets of the convex hull

of the feasible region. We present a different formulation of the problem that we refer

to as the extended problem formulation. The linear programming relaxation of the

extended formulation gives lower bounds that are at least as good as the bounds from

the linear programming relaxation of the original formulation.











\--





Figure 1-4: Multi-period, single-retailer problem


In ('!i lpter 3 we discuss two extensions of the multi-facility lot-sizing problem.

First, we discuss the multi-commodity version of the problem. Later we present a

model for the single-commodity, multi-retailer, multi-facility problem. We propose

a Lagrangean decomposition scheme to solve the multi-commodity problem and a

primal-dual algorithm to solve the multi-retailer problem.










Finally, in Chapter 4 we discuss the general production-distribution model

we presented above. We propose a heuristic approach called the multi-commodity

dynamic slope scaling procedure (i!CDSSP). This is a general heuristic that can

be used to solve any multi-commodity network flow problem with fixed charge cost

function. The MCDSSP is a linear programming based heuristic. We solve the

linear programming relaxation of the extended formulation to generate lower bounds

and compare the quality of the solutions from MCDSSP to these bounds. For the

production-distribution problem we also present a Lagrangean decomposition based

algorithm. This algorithm decomposes the problem into two subproblems. The first

subproblem further decomposes by commodity into K single-commodity, multi-

facility, multi-retailer problems, that we discuss in C(i lpter 3. The performance of

these algorithms is tested on a large variety of test problems. We present extensive

computational results.















CHAPTER 2
MULTI-FACILITY LOT-SIZING PROBLEM

2.1 Introduction

In this chapter we analyze and provide algorithms to solve the multi-facility

economic lot-sizing problem. This is an extension of the well-known Economic Lot-

Sizing problem. The economic lot-sizing problem can be described as follows: Given

a finite time horizon T and positive demands for a single item in each production

period, find a production schedule such that the total costs are minimized. The

customers' demand must be satisfied from production in the current period or by

inventory from the previous periods (that is no backlogging is allowed). The two kinds

of costs considered are production costs and holding costs.

Different from the classical economic lot-sizing problem, the multi-facility lot-

sizing problem considers that demand can be satisfied via multiple facilities. This

adds an extra dimension to the classical problem, the facility selection decision.

Transportation costs, together with production and inventory costs are the bhi-.-- -1

components of total costs. Thus, in contrast with the classical lot-sizing problem,

this model, in deciding on a production schedule, considers not only production and

inventory costs, but also transportation costs.

The multi-facility lot-sizing problem finds an optimal production, inventory

and transportation schedule that satisfies demand at minimum cost. This problem

has many practical applications. For example, a manufacturing company often has

multiple facilities with similar production capacities. The need for cross facility

capacity management is most evident in high-tech industries that have capital

intensive equipment and a short technology life cycle. These companies often -li, i--1,.

with their production planning problem in their complex and rapidly changing










environment. To better utilize their capital-intensive equipment '! i:, are pressured to

produce a variety of products in each of their production facilities. Coordinating the

decisions on production, inventory and transportation among all the facilities reduces

the costs relative to having each facility make its own decisions independently.

We present various solution approaches to solve this problem. The importance

of the algorithms we propose is in the fact that these algorithms can be used as

subroutines to solve more complex supply chain optimization problems. In Section 2.2

we present a literature review of economic lot-sizing and related problems. Section

2.3 gives the problem formulation and Section 2.4 discusses an extended formulation

of the multi-facility lot-sizing problem. Its linear programming relaxation gives close-

to-optimal solutions and the corresponding dual problem has a special structure.

In Section 2.5 we discuss a primal-dual based algorithm. We present a set of valid

inequalities for the problem and show that ', i, are facets defining inequalities. The

valid inequalities are used in the cutting plane algorithm discussed in Section 2.6. In

Section 2.7 we discuss a dynamic programming based algorithm. Finally, Section 2.8

presents some of the computational results and Section 2.9 concludes this chapter.

2.2 Literature Review

This section presents a literature review on the single-item economic lot-sizing

problem. The first contribution is the Economic Order Quantity (EOQ) model

proposed by Harris [54] in 1913. This model considers a single commodity with a

constant demand rate, production taking place continuously over time, and does not

incorporate capacity limits.

A major limitation of the above model is the restriction that the demand is

continuous over time and has constant rate. Manne [81] and Wagner and Whitin [104]

considered the lot-sizing problem with a finite time horizon consisting of a number

of discrete periods, each with its own deterministic and independent demand. This

is the classic economic lot-sizing problem. Wagner and Whitin developed a dynamic









programming algorithm for the single-commodity, uncapacitated version of the

problem. Their algorithm runs in 0(T2).

This problem (apparently well solved since 1958) recently revealed a variety of

new results. Practical reasons exist for the interest in this model. Almost thirty years

later Wagelmans et al. [103], Aggarwal and Park [1], and Federgruen and Tzur [39]

showed that the running time of the dynamic programming algorithm could be

reduced to O(T log T) in the general case and to O(T) when the costs have a special

structure (ht + pt > pt+l), also referred to as the absence of speculative motives.

The capacitated lot-sizing problem is NP-hard even for many special cases

(Florian et al. [41] and Bitran and Yanasse [15]). In 1971, Florian and Klein presented

a remarkable result. They developed an O(T4) algorithm for solving the capacitated

lot-sizing problem with equal capacities in all periods. This result uses a dynamic

programming approach combined with some important properties of optimal solutions

to these problems. Recently, van Hoesel and Wagelmans [99] showed that this

algorithm can be improved to O(T3) if backlogging is not allowed and the holding cost

functions are linear.

Several solution approaches have been proposed for NP-hard special cases of

the capacitated lot-sizing problem. These methods are typically based on branch-

and-bound (see for instance, Baker et al. [5] and Erengiic and Aksoy [29]), dynamic

programming (see for instance, Kirca [68] and Chen and Lee [23]) or a combination of

the two (see for instance, C('!hu and Lin [25] and Lofti and Yoon [75]).

The multi-commodity version of the problem has attracted much attention.

Manne [81] used the zero inventory (ZIO) property to develop a column generation

approach to solve this problem. B ,i 1-11 et al. [10] solved the multi-commodity

capacitated lot-sizing problem without set-up times optimally using a cutting plane

procedure followed by branch and bound.

Other extensions to the classic economic lot-sizing problem consider set-up

times, backorders and other factors. Zangwill [107] extended Wagner and Whitin's









model to allow for backlogging and concave cost functions. Veinott [101] studied

an uncapacitated model with convex cost structures. Trigeiro et al. [98] showed

that capacitated lot-sizing problem with set-up times is much harder to solve

than capacitated lot-sizing problem without set-up times. It is easy to check if the

capacitated lot-sizing problem problem without set-up times has a feasible solution

or not. This can be done by computing cumulative demand and cumulative capacity.

When set-up times are considered, the feasibility problem is NP-complete. The bin

packing problem is a special case of capacitated lot-sizing problem with set-up times

(Garey and Johnson [45] p.226).

Much research on lot-sizing problems focused on determining a (partial)

polyhedral description of the set of the feasible solutions and applying branch-and-cut

methods (Pochet et al. [89], Leung et al. [74], Barany et al. [10]). The main motivation

for studying the polyhedral structure of the single item lot-sizing problem is to use

the results to develop efficient algorithms for problems such as the multi-commodity

economic lot-sizing problem that contains this model as a substructure. However,

the branch-and-cut approach has not (yet) resulted in competitive algorithms for the

single-item lot-sizing problem itself. The reason is that generating a single cut could

be as time consuming as solving the whole problem.

Barany et. al [9, 10] provided a set of valid inequalities for the single-commodity

lot-sizing problem showed that these inequalities are facets of the convex hull of the

feasible region and furthermore, '!, i, showed that the inequalities fully describe the

convex hull of the feasible region.

Pereira and Wolsey [88] studied a family of unbounded polyhedra arising in

an uncapacitated lot-sizing problem with Wagner-Whitin costs. They completely

characterized the bounded faces of maximal dimension and showed that '!, i, are

integral. For a problem with T periods '! i:, derived an O(T2) algorithm to express

any point within the polyhedron as a convex combination of extreme points and










extreme rays. They observed that for a given objective function the face of optimal

solutions can be found in O(T2).

Shaw and Wagelmans [93] considered the capacitated lot-sizing problem with

piecewise linear production costs and general holding costs. They showed that this

is an NP-hard problem and presented an algorithm that runs in pseudo-polynomial

time.

Wu and Golbasi [106] considered a multi-facility production model where a set

of items is to be produced in multiple facilities over multiple periods. They analyzed

in depth the product-level (single-commodity, multi-facility) subproblem. They prove

that general-cost version of this uncapacitated subproblem is NP-complete. They

developed a shortest path algorithm and showed that it achieves optimality under

special cost structures.

Our study is closely related to all the work we mention in this section. The

multi-facility lot-sizing problem is an extension of the lot-sizing problem. We add

to the classical model the facility selection decision and other than production and

inventory costs, we consider transportation costs as well. The results found and

the algorithms developed for the single-commodity lot-sizing problem enlighten

us in our search for solution approaches to the multi-facility lot-sizing problem.

The set of valid inequalities that we propose in Section 2.6.1 are a generalization

of the valid inequalities proposed by Barany et. al [10]. The extended problem

formulation presented in Section 2.4 is inspired by a similar formulation proposed by

van Hoesel [100] for the lot-sizing problem. However, our study is closely related to

the work of Wu and Golbasi [106]. The multi-facility lot-sizing problem is the same

model as the one in Wu and Golbasi, however we propose a wider (and different)

range of solution approaches.

2.3 Problem Description

The multi-facility lot-sizing problem studied in this chapter can be formulated

using the following notation.










Problem Data:
T number of periods in the planning horizon
F number of facilities
pit production unit cost at facility i in period t
sit production set-up cost at facility i in period t
hit inventory unit cost at facility i in period t
it transportation unit cost at facility i in period t
Qit(qit) production cost function at facility i in period t
bt demand in period period t
Decision Variables:

qit number of items produced at facility i in period t
xit number of items transported from facility i in period t
lit number of items in the inventory at facility i in the
end of period t

The objective is to find a minimum cost production, inventory and transportation

plan to fulfill demand when the set-up costs and production, inventory, transportation

unit costs can vary from period to period, from facility to facility. The model assumes

no starting or ending inventory.

This problem can be formulated as a network flow problem as follows:

F T
minimize (Qit(qit) + c~tx~t + hitt)
i= t 1

subject to (i\ F)


lit- +qit = xit + lit i = ,...,F;t = ,...,T (2.1)
F
.xit bt t= ,...,T (2.2)
i= 1
lio = 0 i 1,...,F (2.3)

it, xit, it > 0 i ,... ,F;t 1,...,T (2.4)


Equation (2.1) presents the flow conservation constraints for each facility in each

time period, and (2.2) presents the flow conservation constraints for the demand in

every time period. The flow conservation constraint for the source, 1 t1 qiit

E 1 bt, is not included in the formulation because it is implied by (2.1) and (2.2).










The structure of this network is illustrated in Figure 2-1. The set of nodes in

this network consists of T copies of each of the F facilities and the demand node, as

well as a source node. Each 1--,.r of the network represents a time period. The set of

arcs consists of F x T production arcs, F x T transportation arcs and F x (T 1)

inventory arcs. The source node s supplies the total demand for the planning horizon.

The production arcs connect the source to each facility, in each time period. The

inventory arcs connect the same facility in successive periods. Transportation arcs

connect facilities to the demand node in each period. The production cost function








T / F31









Figure 2-1: Network representation of a two-period, three-facility lot-sizing problem


is a fixed charge cost function, thus, if production in a time period is initiated, the

set-up cost plus production cost for each unit produced has to be paid.
t pitqit + Sit if qit > 0


t)itit it i t > 0 fori 1 ,... ,F;t 1,... ,T.
0 otherwise

The standard mixed-integer linear programming (M! I.P) formulation of this function

can be obtained by introducing a binary set-up variable yit corresponding to each

production arc. The production cost function can then be replaced by















where,

Y 1 if qit > 0
Yit [
0 otherwise.

The MILP formulation of the multi-facility lot-sizing problem then reads as follows:

F T
minimize Y y(pitqit + sityit + citXit + hitlit)
i= 1 t 1


subject to


Iit-1 +
F

i= 1



qit, xit,


qit


xit + lit


, .. .,T


i= 1,...,F;t

t= 1,...,T

i= 1,...,F;t

it= 1,...,F; t

i 1,...,F ; t


xit = bt

qit < btTYit

lit > 0

Yit e {0,1}


,... T

, ,

1 ,... ,


where, btT presents the demand during the time periods t (t = 1,..., T) to T, (i.e.

btT = 7=t b,). Standard solvers such as CPLEX can be used to solve formulation

(P\!F*) of the multi-facility lot-sizing problem.

The set of constraints (2.5) show that the number of items produced in the

current period together with the inventory from previous periods, should be equal

to the ending inventory plus the amount shipped to the retailer. Together with (2.8)

'! i:,- imply


lit = (qi, i,)
T1


and using (2.6),


F t
= qi bit
i= 1 1


fort 1,...,T.


(2.5)

(

(2.7)

(2.8)

(2.9)


F

i 1


Qit(qit) = Pitqit + sityit


for i = 1,... ,F; t = 1,... ,T,


(\!F*)


for i= 1,... F; t = ,... T,










Using these facts, the inventory variables can be eliminated from the formulation,

reducing the size of the problem. The following is the MILP formulation of our

problem without inventory variables.

F T
minimize Y i(p tqt + sitit + c$,xit)
i= t=1

subject to (R-MF*)

F t
qir > bit t = ,... ,T (2.10)
i= 1 r= 1
t t
Y qi, > xi, i =l ,...,F; t ,...,T (2.11)
T=l T=1
F
xit = bt t= ,...,T (2.12)
i=1
qit < btT it i 1,...,F;t 1,...,T (2.13)

xit, qit > 0 t=l,...,F;t= ,...,T (2.14)

it E {0,1} i= ,...,F;t 1,...,T (2.15)
T t it Cit
where p't = it + 3 it hi, and ci cu E t hir.

Proposition 2.3.1 (Non-splitting p1**/'p ,I;): There exists an optimal solution to the

uncapacitated, single-,..c- t iiit..; multi-f'i. .:.:1;i lot-- ..:,." problem such that the demand

in period t is .'I.:/7., / from either production or the inventory of '. i,. /;/ one of the

facilities.

Proof: The multi-facility lot-sizing problem minimizes a concave cost function

over a bounded convex set, therefore its optimal solution corresponds to a vertex of

the feasible region. Let (q*, y*, x*, I*) be an optimal solution to the multi-facility lot-

sizing problem. In an uncapacitated network flow problem, a vertex is represented by

a tree solution. The tree representation of the optimal solution implies that demand

in every time period will be satisfied by exactly one of the facilities. In other words,

x*x = 0, for i / j and t = 1, 2,..., T. Furthermore, for each facility, in each time

period if the inventory level is positive, there will be no production, and vice versa.










Thus, zero inventory policy holds for this problem (q^it-1 = 0, for i = 1,... F,

t =1,..., T). This completes the proof that in an optimal solution to our problem

demand is satisfied from either production or the inventory of exactly one of the

facilities. o

Another characteristic of an optimal solution to the multi-facility lot-sizing

problem is the following: Every facility in a given time period either does not produce,

or produces the demand for a number of periods (the periods do not need to be

successive). This property can easily be derived from Proposition 2.3.1 and the tree

representation of an optimal solution. Figure 2-8 presents an example of such an

extreme flow solution, where demands bl and b3 are satisfied from production at

facility 1 in period 1 and demand b2 is satisfied from production at facility 2 in period

2. Such a production schedule happens in case that the setup cost at facility 2 in

period 2 is very small (as compared to the setup cost at facility 1 in periods 1, 2, 3 and

setup cost at facility 2 in periods 1 and 3), but inventory holding cost at facility 2 in

period 2 is very high.

For the multi-facility lot-sizing problem there exists an exact algorithm that is

polynomial in the number of facilities and exponential in the number of periods (Wu

and Golbasi [106]).

Assign demands bl,...,bT to facilities i 1,..., F. This takes O(FT).

Given the assignment, solve for each facility an uncapacitated, single item, single

facility lot-sizing problem. This takes F O(T log T).

Wu and Golbasi [106] show that for the special case when inventory holding costs are

not restricted in sign, the multi-facility lot-sizing problem is an NP-complete problem.

They use a reduction from the uncapacitated facility location problem.

2.4 Extended Problem Formulation

The linear programming relaxation of ( \! *) is not very tight. This is due to

the constraints qi < btTu-t. In the linear programming relaxation of (\!P*), the

variable /it determines the fraction of the demand from periods t to T satisfied from










production at facility i in period t. btT is usually a very high upper bound, since the

production in a period rarely equals this amount. One way to tighten the formulation

is to split the production variables qi by destination into variables qit, (r = t,..., T),

where T denotes the period for which production takes place (van Hoesel [100]). For

the new variables, a trivial and tight upper bound is the demand in period 7 (i.e. b,).

The split of the production variables leads to the following identities:

T
qiit itT (2.16)
-r=t
t
Xit= qist (2.17)
s=l
t T t
,it =Y Y qisr :qst (2.18)
s=l T=t s=l
Replacing the production, transportation and inventory decision variables, and after

re-arranging of terms, the objective function becomes

F T T
minimize > Y[ {(pit + CiT + his)8qitr + Sitit]
i=1 t=1 T=t s=t+l

Replacing the decision variables in the constraints of the formulation (! F*) with the

new variables, we obtain the following:

Constraints (2.5) transform to

t-1 T t-l T t t T t
is qist- + itr qist + is S qist
s= T=t-l s=l T=t s=1 s=1 T=t s=1

for i = 1,...,F; t 1,...,T.

After re-arranging terms, it follows that the constraints (2.5) are redundant.

Constraints (2.6) transform to

F t
S qist bt for t = ,... T.
i= s=l 1










* Constraints (2.7) transform to


qit, < bryit for i = 1,..., F; t = 1,..., T; t < < T.


Finally, constraints (2.8) transform to


qur > 0, fori 1,...,F;t 1,...,T;t

The extended problem formulation is the following:

F T T
minimize [ citgqit + sityit]
i=1 t= r=t

subject to (Ex-MF)

F T
Qit = bT r 1,...,T (2.19)
i= 1 t= 1
qit, bryt < 0 i = ,...,F;t 1,...,T; t T (: )

qit, > 0 i 1,...,F;t 1l,...,T; t <
Yit {0,1} i 1,...,F;t 1,...,T, (2.22)


where cit = Pit + ci, + Z t+l his. The above formulation resembles the model for

the uncapacitated facility location problem. The relationship with the uncapacitated

facility location problem is the following: consider W x T facilities from which

customers can be served. The decision to be made is whether or not to open (whether

to get a shipment from) a certain facility it. The cost for opening a facility (set-up

the machines) is denoted by sit. The cost of serving customer (satisfying demand

in period) r (r > t) is cit, per unit of demand. Our problem resembles the facility

location problem, however it is not exactly the same problem. The fact that the costs

assigned to the same facility in different periods are related to each other, makes the

multi-facility lot-sizing problem a special case of the facility location problem.










The linear programming relaxation of (Ex-MF) replaces constraints y E {0, 1}

with y > 0. The binary variables y appear only in constraints (2.20), therefore


Yit > fori 1,...,F;t 1,...,T;t I bT

Since set-up costs sit > 0, and (Ex-MF) is a minimization problem, in an optimal

solution we will have


yi-t = it for i = 1,... F; t = 1,... T; t r < T.
S bT

We know that the most qit can be is br, and in this case y = 1; the least qit can be is

zero and in this case y = 0. This result makes the constraint yit < 1 redundant.

Formulation (LP-MF) is the linear programming relaxation of (Ex-MF).

F T T
minimize [I 6ditqitq + Sityit]
i= t=1 r=t

subject to (LP-MF)


ES 1 qitr = b r= 1,...,T

qit bryit < 0 i =1,...,F;t 1,...,T;t
qit, > 0 i= ,...,F;t= l,...,T;t< r< T

Yit > 0 i 1,...,F;t 1,...,T.


In the special case of only one facility, our problem reduces to the economic

lot-sizing problem. The linear programming relaxation of the extended formulation

of this special case ak---,- has an integer solution. This result is due to Krarup and

Bilde [69]. Although this does not hold true for the multi-facility lot-sizing problem,

the lower bounds generated solving (LP-MF) are very good and the solutions are close

to optimal. In fact, the lower bounds found solving (LP-MF) are much tighter than

the lower bounds generated solving the linear programming relaxation of the original










formulation of this problem (\! F*) (Proposition 2.4.1). A network representation of

the extended formulation is given in Figure 2-2.






F 2/ \
q/ q211

/ /
/F /2,1 b





'~......... ... "9, _
q022

2 q322



Figure 2-2: Network representation of extended formulation of a two-period, three-
facility lot-sizing problem


Formulations (\!F *) and (Ex-MF) of multi-facility lot-sizing problem are

equivalent to each other. The notion of equivalence requires that the optimal solution

to both formulations is the same (e. g. once the problem (Ex-MF) is solved in terms

of qit, variables, this solution should yield the optimal solution to problem (\ I *)).

For this to hold true, two conditions must be satisfied: (i) every solution of (Ex-

MF) must correspond to a solution of (ill *) (i. e. no new solutions were created by

redefining the decision variables of (ill *)), and (ii) there must be a solution in the

feasible region of (Ex-MF) that corresponds to every extreme point in the convex hull

of the set of feasible solutions to (ill *) (i. e. no solutions to formulation (\!F*) were

lost by redefining the variables). Condition (i) follows directly from the definition

of qit-. A solution to (Ex-MF) can be directly translated into a solution to (il F*)

using equations (2.16), (2.17) and (2.18). Condition (ii) is more difficult to argue. An

extreme point of the convex hull of the feasible solutions to (ill *) is such that (a) it

satisfies the zero inventory property (qiti,t1 = 0 for i = 1,..., F and t = 1,..., T);










(b) demand in period t (t = 1,..., T) is satisfied from exactly one facility (xitjt = 0

for i,j = 1,..., F, i / j); (c) in period t (t = 1,..., T) a facility either does not

produce, or produces the demand for a number of periods, the periods do not need

to be successive (Proposition 2.3.1). One can easily see (Figure 2-2) that an extreme

point to (Ex-MF) satisfies exactly the same conditions. The correspondence between

the extreme points of the two formulations shows that there is an extreme flow (tree

solution) in the formulation (Ex-MF) for every extreme flow of the formulation (\! I *).

Proposition 2.4.1 The optimal cost of linear 1"''',i';.nun':i. relaxation of the extended

formulation of multi-f', ..7;'.:/ lot- ...:.:, problem (Ex-MF) is at least as high as the

optimal cost of linear i" 'pi-'"'r.1 relaxation of (.' ..': ri formulation (_FlP).

Proof: Every feasible solution to the linear programming relaxation of extended

formulation of multi-facility lot-sizing problem (Ex-MF) can be transformed to a

solution to linear programming relaxation of original problem formulation (\ I *)

using equations (2.16), (2.17) and (2.18). It follows that the optimal solution of linear

programming relaxation of (Ex-MF) can be transformed to a feasible solution (not

necessary the optimal solution) to linear programming relaxation of (\! [*). D

2.5 Primal-Dual Based Algorithm

The dual of (LP-MF) has a special structure that allows us to develop a primal-

dual based algorithm. The following is the formulation of the dual problem:
T
maximize 1,,,,
t= 1

subject to (D-MF)

^ tl-"'.,,- S it i= 1,...,F;t 1,...,T

v,-wit- <_ cit, i 1, ... F;t = ,..., T;t < T < T

wit, > 0 i= 1,...,F;t 1,...,T;t
In an optimal solution to (D-MF) both constraints witr > 0 and witT > v. cit

should be satisfied. Since wit, is not in the objective function of (D-MF), we can









replace it with wit = max(0, v, cit,). This leads to the following condensed dual

formulation:
T
maximize 7Y I,,
t=1
subject to (D-MF*)


ZTLtbmax(0,v--itr) < si i 1,...,F;t 1,...,T.

Recall that the extended formulation of the multi-retailer lot-sizing problem is a

special case of the facility location problem. The primal-dual scheme we discuss, in

principle, is similar to the primal-dual scheme proposed by Erlenkotter [38] for the

facility location problem. However, the implementation of the algorithm is different.

Wagelmans et al. [103] consider the extended formulation of lot-sizing problem.

They solve the corresponding dual problem in O(T log T). They show that the dual

variables have the following property: ', > ', t for t = 1,..., T. This property of

the dual variables is used to show the dual ascent algorithm that '1 :, propose gives

the optimal solution to the economic lot-sizing problem.

2.5.1 Intuitive Understanding of the Dual Problem

In this section we give an intuitive interpretation of the relationship between

the primal-dual solutions of (Ex-MF). Suppose the linear programming relaxation of

extended formulation (LP-MF) has an optimal solution (q*, y*) that is integral. Let

0 {(i, t), = 1} and let (v*, w*) denote an optimal dual solution.

The complementary slackness conditions for this problem are
(CI) *It[Sit tl,l, 0 !i 1,
(C i) y E th-"'t-] 0 for i 1,... ,F;t 1,...,T

(C2) q*[ it- v+',] = 0 fori =,...,F;t =,...,T;t
(C3) "', -[,- '-1 = 0 fori 1,...,F;t 1,...,T;t < r< T

(C4) v[bt EEt q,] 0 for t ,...,T.










By conditions (C1), if a facility produces in a particular time period, the set-up

cost must be fully paid (i. e. if (i, t) E 0, then sit _= T-'-). Consider conditions

(C3). Now, if facility i produces in period t, but demand in that period is satisfied

from inventory from a previous period (qit 0 and qitt btYt / 0), then i ,, =0,

which implies that the price paid for the product in a time period will contribute to

the set-up cost of only the period in which the product is produced.

By conditions (C2), if qT > 0, then v* = Cita + Thus, we can think of v* as

the total cost (per unit of demand) in time T; of this, cit goes to pi', for production

and inventory holding costs, and wi*, is the contribution to the production set-up cost.

2.5.2 Description of the Algorithm

The simple structure of the dual problem can be exploited to obtain near optimal

feasible solutions by inspection. Suppose that the optimal values of the first k 1 dual

variables vl, ... v_ are known. Then, to be feasible must satisfy the following

constraints:
k-1
bk max(0, k Citk) < -1 Sit b, max(0, v ci,) (2.23)
T=t

for all i = 1,..., F and t = 1,..., k. In order to maximize the dual problem we should

assign to vk the largest value satisfying these constraints. When bk > 0, this value is


i, = min + itk + } (2.24)
i,t<_k bk

Note that .i, _1 > 0 implies vk > citk. A dual feasible solution can be obtained

simply by calculating the value of the dual variables sequentially using equation (2.24)

(Figure 2-3), and a backward construction algorithm can then be used to generate

primal feasible solutions (Figure 2-4). For the primal-dual set of solutions to be

optimal, the complimentary slackness conditions should be satisfied. However, the

primal and dual solutions found do not necessarily satisfy the complimentary slackness

conditions.










S_ ,= ..., F, =12,..,T
for r=1 toTdo
if by = 0 then v, = 0
else
r<71 Cit. {rt+Miui /b.}
for t 1 to 7 do
for i 1 to F do
mlax-{0, v, c, }
S- nmax {0, }
enddo
enddo
enddo

Figure 2-3: Dual algoritihmn


Proposition 2.5.1 The solutions obtained using the primal and dual i'l'.:thms are

feasible and they il;,.i ;, .'i>;.fy the complimentary slackness conditions (C1) and (C2).

Proof: It is clear that primal and dual solutions generated are feasible by

construction. Since the primal algorithm sets qit, > 0 only when Cit V itr = 0,

the solution satisfies conditions (C2). The dual algorithm constructs solutions by

making sure that equations (2.23) are satisfied. Therefore, the dual solutions are

ahlv--- such that br+lwitr+l < ., -. If 3/ ,- = 0, then witr+ = 0, and since the dual

algorithm sets 3 1,-+1 = i 1, b+tlwitr+1, we also have f1, +1 = 0. Continuing

this way it is clear that if at some point in the calculation we get 31, = 0, we

subsequently obtain the following:


[,= m,.+ = ....= ir 0


and

Witr = Witr+1 I = = 0.

The primal algorithm sets yit = 1 only when i1,- = 0; this implies that conditions

(C1) will alh--,v- be satisfied. o

Hence, in order to determine if the solution obtained using the primal-dual

algorithm is optimal, one can check conditions (C3). Another way to check if the










yi = 0, q. 7 = 0, i = 1, ... F, 1 .... ,T; r > I
P= {'" >0.t= 1,...,T}
Start: j maxt E P, t 0
1: for i 1 to do
repeat t t 1
until 0 and c, 0
1, i t* t, go to 2
enddo
go to S' 3
S 2: for = 1* to j do
if czti Vi =0
P = P- {}, qi. =
enddo
Step 3: if P / then go to start
else Stop

Figure 2-4: Primal algorithm


solution is optimal is by comparing the objective function values from the primal and

dual algorithms. Since the dual algorithm gives a dual feasible solution and the primal

algorithm gives a primal feasible solution, at optimality, the two objective functions

should be equal.

2.5.3 Running Time of the Algorithm

Here we discuss the running time of the primal-dual algorithm. The total number

of logical and arithmetical operations performed in the dual algorithm is

FT assignments during initialization
2T + FT + FT2 comparisons
FT + FT2 multiplications
FT2 additions
FT2 + T assignments
2FT2 subtractions

The total number of logical and arithmetical operations in the primal algorithm is

FT + FT2 assignments during initialization
T + 2FT comparisons
T + 2FT assignments
2T + FT2 subtractions

Thus, the running time of the primal-dual algorithm is O(FT2).










1: Solve Lt relaxation of
2:
if y* is integral,
then STOP, the solution is <
else
solve the '" n problem
if no violated 'y is found
:)P
else
add the v to (LP-MF), go to Step 1.

Figure 2-5: Cutting algorithm


2.6 Cutting Plane Algorithm

In this section we derive a set of valid inequalities for the multi-facility economic

lot-sizing problem. We show that these inequalities are facets of the feasible region.

The inequalities are then used in a cutting plane algorithm that finds tight lower

bounds.

Figure 2-5 presents the steps of the cutting plane algorithm. The algorithm

stops when either (i) the optimal solution to (\!F *) is found, or (ii) if no more valid

inequalities can be generated. Next we discuss in detail the valid inequalities, the

separation algorithm, and we show that these inequalities are facets of the feasible

region of (i\IF*).

2.6.1 Valid Inequalities

Theorem 2.6.1 : For any 1 < I
F
Y(Eqd + byt) > biC )
i=1 teS tEL\S

are valid inequalities for the multi fr .1.:1 ,; economic lot--...::,' problem.

Proof: In order to prove that (2.25) are valid inequalities we should prove that

'1 :,i are satisfied by all feasible solutions with integer y. Consider formulation (\! I*).

Given a solution (q, x, I, y), let yi = 0 for t E L \ S. Inequality qit < btTYit implies that

if yi = 0, then qit = 0. Thus,











F F F I
S( q + > t itu) qit >
i-1 tES tEL\S i=1 tES i=1 t=1

Consider constraints (2.5). Summing up these constraints over all i = 1,..., F we get

F I F I
E E ^ E(it (E Xjt+ ) v i, ...,r.
i= 1 t= 1 i 1 t= 1

Since I > 0 for all 1 = 1,..., T, and F x = bL for all t = 1,...,T

F I
SI qtl > bit for = 1,..., T.
i= 1 t= 1

As a results, solutions such that yt = 0 for t E L \ S, satisfy the inequality (2.25). Let

t' = argmint{t E L \ S, yit 1}. Then, yt = qi = 0 for t E (L \ S) n 1,..., t' 1}.

Hence,

F F t'- 1
( qtit + byit) > ) q L + b6i > bit-l + b6l b11.
i=1 teS tEL\S i=1 t=l1

This completes the proof that every feasible solution with integer y will satisfy the

valid inequalities (2.25). O

The intuition behind the inequalities (2.25) is as follows: Assume that no

production takes place in the periods in L \ S. Then the full demand bll has to be

produced in the periods in S, giving ES 1 EtS qit > blu. Now, suppose that we do

produce in some of the periods in L \ S, and let period k be the first such period. The

production for periods 1,..., k 1 then has to be done in periods in S. It is possible

that the remaining demands bkl is produced in a single period in L \ S, which explains

the coefficients of the y variables.

2.6.2 Separation Algorithm

Let (q*, x* y*) be the solution to the linear programming relaxation of (\!F*).

If this solution satisfies the integrality constraints (y E {0, 1}), this is the solution to

(\! F*) as well. However, if it does not, we have to identify a valid inequality that cuts










off this solution from the set of feasible solutions to linear programming relaxation

of ( IP*). A challenging problem is to identify the sets S and L \ S for which the

valid inequality (2.25) is violated. An exponential number of sets S and L \ S exists,

however, the following separation algorithm that runs in polynomial time (O(FT2))

takes us through the steps needed to identify the sets S and L \ S. Note that, for a

given time period I (1 1,..., T), this procedure identifies the valid inequality that is

the most violated by the current solution.

For 1 1,...,T


i=i i=i
F F
and t eL \ Si if iqu > buyt
F i= 1 i= 1
2. check if ( qt + > bi < b1l.
i=1 teS1 tEL\SI
If so, the inequality is violated; otherwise for each 1, the following holds:
F F
miny(y + b>1Yi4) -b11} ( q + > btlt) bl > 0
i1 tES tEL\S i=1 tESi tEL\SI

If no violation can be found, all valid inequalities of the form (2.25) are satisfied

by the current solution (q*, x*, *, y*). Therefore, either this solution is already an

integer solution, or if this is not the case, the valid inequalities are not able to cut-off

this non-integer solution from the feasible region of formulation ( \!F*). There are two

reasons for that to happen, either our valid inequalities are not facets of the convex

hull of the feasible region, or in case that '!, i,- are facets, I! i,- may not describe the

whole feasible region, and other facets are needed as well. In Section 2.6.3 we will

show that the valid inequalities (2.25) are indeed facets of the convex hull of the

feasible region.

2.6.3 Facets of Multi-Facility Lot-Sizing Problem

Let ) be the feasible region of the multi-facility lot-sizing problem, let co(J)

be the corresponding convex hull, and let G {q, y E co(K) : I (s5Es fit +










tEL\s bytlit) = bl}. G consists of all the points in the convex hull of the feasible
region that satisfy the valid inequality as equality. In other words G consists of all

the points in the convex hull of the feasible region that lie on the plane defined by the

inequality (2.25). A common approach to show whether an inequality defines a facet

of co()) is to show that there exist precisely dim(co(l)) vectors on the boundary of

G that are affinely independent. Note that if the boundary of G does not contain

zero, this is equivalent to showing that there exist precisely dim(co())) vectors

on the boundary of G that are linearly independent (N. in! -1, ri and Wolsey [86],

Wolsey [105]). In our problem zero is not a feasible solution.

The following is a procedure that will help us to see whether inequalities (2.25)

are facets of co(4). This means these inequalities are necessary if we wish to describe

co(K) by a system of linear inequalities.

Proposition 2.6.1 : If bt > t = 1,... ,T, the dimension of the feasible set of

multi-f,'. i.:. economic lot-- .:..' problem is dim(() = 3TF F T 1.

Proof: We show that the dimension of the feasible region of the multi-facility

lot-sizing problem is 3TF F T 1 in both cases, when formulations (\! *) or

(R-MF*) are considered.

Consider formulation (\F! *):
A feasible solution in K( has a total of 3TF of q, x, y variables and F(T 1) of I
variables.
The assumption that bt > 0 implies that b1 > 0 and yu = 1 for at least one
i 1= ,..., F. This reduces the dimension of ) by one.
The equations i, xit = bt for t 1,..., T reduce the dimension of ) by T.
The equations li,t-1 + qit = xit + it for i = 1,..., F, and t = 1,..., T reduce the
dimension of K by TF.

This shows that the dimension of the feasible region to formulation (F\!*) is at

most equal to 3TF F T 1.

Consider formulation (R-MF*):
A feasible solution in K( has a total of 3TF q, x, and y variables.
The assumption that bt > 0, for t = 1,..., T implies that b1 > 0 and yi = 1 for
at least one i 1,..., F. Therefore, the dimension of + is reduced by one.










The equations i xi = bt for t 1,..., T reduce the dimension of ) by T.
The equations 7 i qit = iT xi for i 1,..., F reduce dimension of ) by F.

The dimension of the feasible region of formulation (R-MF*) is at most equal to

3TF F T 1. However, in Theorem 2.6.2 we show that there exist 3TF F T

affinely independent points in the convex hull of the feasible region of the multi-

facility lot-sizing problem that satisfy the valid inequalities (2.25) to equality. This

indicates that there exist 3TF-F-T affinely independent points in the feasible region

of the multi-facility lot-sizing problem. This concludes our proof that the dimension of

the feasible region of the multi-facility lot-sizing problem is 3TF F T 1. o

To illustrate that the above result is correct, we consider the following simple

example. B ,n 111i et al. [10] show that the dimension of the feasible region to the

economic lot-sizing problem is 2T 2. The economic lot-sizing problem is a special

case of the multi-facility problem with F = 1; therefore its dimension is 3T-1-T-1 =

2T- 2.

The next step is to show that there are 3TF F T 1 affinely independent

points in K) that satisfy the valid inequality to equality.

Theorem 2.6.2 If bt > 0, for t = 1,...,T, the (1, S) :,., ,,rl.:// /1, /I,,, 1 a facet of +

whenever 1 < T, 1 E S and L \ S / 0.

Proof: See Appendix.

2.7 Dynamic Programming Based Heuristic

2.7.1 Introduction

Dynamic programming provides a framework for decomposing certain

optimization problems into a nested family of subproblems. The nested substructure

,il-.-1 -1-; a recursive approach for solving the original problem from the solutions

of the subproblems. The recursion expresses an intuitive principle of optimality

for sequential decision processes; that is, once we have reached a particular state, a

necessary condition for optimality is that the remaining decisions must be chosen

optimally with respect to that state (N. i!, 111"- r and Wolsey [861).










Dynamic programming was originally developed for the optimization of sequential

decision processes. A typical example that is used in the literature to explain how the

dynamic programming algorithm works is the economic lot-sizing problem (N. i!i111l- r

and Wolsey [86] and Wolsey [105]). Consider the lot-sizing problem with T time

periods (t=1,...,T). At the beginning of period t, the process is in state st-1, which

depends only on the initial state so (initial inventory lo) and the decision variables yt

and qt for t = 1,..., t 1. The contribution of the current state t to the objective

function depends on It-1. Let us denote by ', the value of the optimal decisions in

periods t,...,T


(It-i) = min(ptt + styt + ht(It-i + qt bt) + ', i(It-1 + qt be)) (2.26)
qt,yt

The difficulty with this algorithm is that since demand in period t is satisfied by

production in periods r < t, it follows that the level of inventory in the end of period

t 1 can be as large as btT, and it appears that a large number of combinations of

(qt, It-l) must be considered to solve the problem.

Fortunately, the following properties of an optimal solution to the economic

lot-sizing problem make the problem easier.

Theorem 2.7.1 (Nemhauser and W .1-, ; [861) An optimal solution to economic

1. I- ..:, problem .'/.:-i. the following:

qtIt_= 0 for t 1,...,T

If qt > 0, then qt = C, t b for t < < T

If It-1 > 0, then It- = Et,, br for 1 < 7' < t

From Theorem 2.7.1 it follows that 2(T t) combinations of (qt, It-i) must be

considered to solve the recursive function (2.26). Thus the overall running time

is O(T2), and recursive optimization yields a polynomial-time algorithm for the

uncapacitated economic lot-sizing problem.










2.7.2 Description of the Algorithm

In this section we present a dynamic programming procedure to solve the multi-

facility lot-sizing problem. This procedure is a heuristic, and therefore does not

capture all solutions, possibly including the optimal solution to the problem.

In Section 2.3 we discussed the non-splitting property of optimal solutions to the

multi-facility lot-sizing problem. It is important to note that a production, inventory

and transportation plan is optimal if and only if the corresponding arcs with positive

flow form an arborescence (rooted tree) in the network. Important implications of

this result are the following: in an optimal production plan, the demand bt for a given

period (t = 1,..., T) will be produced in a single facility in a single time period;

in every time period a facility either will not produce, or will produce the demand

for a number of periods, and these time periods need not to be successive. Let Tt

be the set of all periods covered by production at facility i in period t in an optimal

arborescence. Then the optimal production plan is

T
qit = qitr=- bi 1,...,F;t 1,...,T.
T-=t T-ETt
The bold arcs in Figures 2-7 to 2-10 represent the extreme flows. As illustrated in

Figures 2-7 and 2-8, an extreme flow to the multi-facility lot-sizing problem is not

necessarily sequential. The example in Figure 2-8 shows an extreme point solution in

which demands b1 and b3 are satisfied from the production at facility 1 in period 1,

and the demand b2 is satisfied from production at facility 2 in period 2.


:ork representation of multi-facility lot-sizing


uire 2 6


I I with 4











Using this information we can now simplify the multi-facility lot-sizing problem

to a shortest path problem in an .. i, ii: network, zv G'. We build G' in the following

way: let the total number of nodes in G' be equal to (T + 1), one for each time period

along with a dummy node T + 1. Traversing arc (r, r') E G' represents the choice of

producing in a single facility in a single time period t = 1,..., T to satisfy the demand

in periods 7, 7 + 1,..., -' 1. The cost of arc (r, T') is calculated using the following

cost function:


(2.27)


g,' = min sit + citTrb + cit,T+lb,+l + ... + cit,_'-lbT -1.
i=1,...,F;1

Let v, be the minimum cost of a solution for period r 1,..., T 1 and r' > r. The


",, < -. \



F \ \
N?
F,


Figure 2-7: Sequential extreme flow


recursion function for the multi-facility lot-sizing problem is


and


(2.28)


V(- I'i,-1) min{gI + V( I'i,'-l)}
i= 1 i= 1

F
Vr( Ii,T-1) g= T,T+
i 1


(2.29)


T

t-I












F


F, F




bt ^\

/" \\\A \\\\F









Figure 2-8: Non-sequential extreme flow


The total number of arcs in G' is equal to T(T + 1)/2. Given the costs g,,'

for every arc (7, 7') e G', the recursive functions (2.28) and (2.29) will provide the

optimal solution in O(T2). Every (directed) path in G' that connects node 1 to T + 1,

corresponds to a feasible solution to the original problem. The network G' for a

4-period problem is presented in Figure 2-6.

Theorem 2.7.2 For every path on G' that connects node 1 to T + 1, there exists a

corresponding an extreme point solution in the extended problem formulation (Ex-MF),

and every sequential extreme point of (Ex-MF) is represented by a path on G'.

Proof: See Appendix.

Wu and Golbasi [106] propose a similar shortest path algorithm to solve

the multi-facility lot-sizing problem. They showed that their algorithm gives the

optimal solution to the problem if the following conditions hold: (i) no simultaneous

production over more than one facility can take place in a given period. In other

words qitqjt = 0 for i,j = 1,..., F, i / j and t = 1,..., T. (ii) no production will be










scheduled at all if there is inventory carried over from previous period in one of the

facilities. In other words qitla,t-_ = 0 for i,j = 1,..., F and t = 1,..., T.








\F, ,










Ebli

Figure 29: Non-sequential flow: Case 1


These conditions obviously restrict the search for a solution to only sequential

extreme flows. Furthermore, i,- investigate only part of the sequential extreme

flows, the ones that satisfy the above conditions. Different from Wu and Golbasi, our

procedure considers a wider range of extreme flows. We consider all the sequential

extreme flows, although some of them may not satisfy conditions (i) and (ii). Figure

2 9 presents a sequential extreme flow that violates condition (i) and Figure 2 10

presents a sequential extreme flow that violates condition (ii).

2.7.3 Running Time of the Algorithm

The above dynamic programming algorithm to find the shortest path in the

i.. ,, 1 graph G' has running time of O(m), where m is the number of arcs in G'.

Since the graph is complete,


S T(T + 1)
t= 2
tLi










Therefore, the running time of our algorithm will be O(c + T2), where c presents the

time it takes to calculate the costs of all arcs in G'.

F,













""-- N,,
b 0^''. ^^-N

F,,









Figure 2-10: Non-sequential flow: Case 2


In calculating the cost of a particular arc (r, r') E G', we need to perform a

certain number of comparisons, additions and multiplications.

For an arc (r, 7') (where 7 = 1,..., T and r' = r,..., T + 1), the number of

comparisons to be made in order to calculate its cost is equal to


rF 1.


The total number of comparisons needed to generate all the arc costs for a

problem with T time periods is

T
Comp. = r(rF 1)
r=i

thus, O(FT3).

The number of additions required to calculate the cost for arc (r, r') is


(7' 7 + 1)7F.










The total number of additions is
T T+1
Add. = (' r- 7 + 1)rF
T =l1T'=T+1

thus, O(FT4).

The number of multiplications required to calculate the cost for arc (7, 7') is


(r' 7)rF.


The total number of multiplications is

T T+1
Mltp. (T' 7)7F
T =l1T'= T+1

thus, O(FT4).

This shows that the time complexity to solve the problem using the above

dynamic programming algorithm is O(FT4).

2.8 Computational Results

To test the performance of the algorithms discussed in this chapter we randomly

generated a set of test problems and compared the computation times and solution

quality to the general purpose solver CPLEX. The algorithms were compiled and

executed on an IBM computer with 2 Power3 PC processors, 200 Mhz CPUs each.

The scope of our experiments, other than comparing the algorithms is to see how

different factors (such as the ratio of set-up to variable cost, number of facilities, etc.)

affect their performance. We first generate a nominal case problem as follows:
Production set-up costs sit ~ U[1200, 1500]
Production variable costs pit U[5, 15]
Holding costs hit ~ U[5, 15]
Demand bt ~ U[5, 15]
Number of facilities F = 150
Number of periods T = 30

Most of the above parameters are the same as the ones used in Wu and Golbasi

[106] for a related problem. To generate meaningful transportation variable costs, we










randomly generated from uniformly distributed points on a [0, 10]2 square the facility

and demand point locations, and calculated corresponding Euclidean distances. We

assumed one to one correspondence between the Euclidean distances and the unit

transportation costs.

Varying one or more factors from the nominal case, we generated five groups of

test problems. In the first group of problems we change the level of production set-up

costs from the nominal case to the following: sit ~ U[200, 300], sit ~ U[200, 900],

sit ~ U[600,900], sit ~ U[900,1500], sit ~ U[1500, 2000], sit ~ U[2000, 3000],

sit ~ U[3000, 6000], sit ~ U[5000, 10000], and sit ~ U[10000, 20000]. These together

with the nominal case problem give a total of 10 problem classes (problem classes 1 to

10).

In the second group of problems we change the length of the time horizon to

5, 10, 15, 20, 25, 35, 40 (problem classes 11 to 17). In the third group we change

the number of facilities to 120, 130, 140, 160, 170, 180, 190, and 200 (problem

classes 18 to 25). In the fourth group of problems the level of demand is changed to

bt ~ U[20, 50], bt ~ U[50, 100], bt ~ U[100, 200], bt ~ U[200, 400], and bt ~ U[400, 1000]

(problem classes 26 to 30). Finally, in the fifth group the level of holding costs is

changed to ht ~ U[-20, -10], ht U[-10, 10], ht ~ U[10, 20], ht ~ U[20, 40], and

ht ~ U[40, 100] (problem classes 31 to 35).

For each problem class we generate 20 instances. The errors and running times

we present for each problem class are the averages over the 20 problem instances. A

summary of the results from the experiments are presented in Tables 2-2 to 2-5. We

do not present the results from implementing the cutting plane algorithm, since in

almost all of the problems CPLEX outperformed our algorithm in terms of solution

quality and running times.

We would like to emphasize that the linear programming relaxation of the

extended formulation and dual algorithm give lower bounds, while dynamic

programming and primal algorithms give feasible solutions to the problems. The























Table 2 1: Problem characteristics

Problems Nodes Arcs' Arcs" Arcs"*
1,...,10 4 1 13 7
11 7- 2,1 3, 15
12 1511 4, 9,750 55
13 2 6i
14 3.021 8,. 34 210
15 3. 11,1 52,5
16 5, 15, ,7
17 6.041 17,, 1 :
18 3 1 10, 465
19 3 1 11, 64I 4(65
4 1 12, 65
21 4. 1 1 4,2.
22 5,131 15,1 84,150
5. 1 16 ,100
24 5,731 16, 0 9/
25 6 1 17, ,
.35 41, 1 7
45 4 531 13,. 7/
SOriinal formula tion


'c -ogrammini
C-o~isrrit


armulation


Ext cncic
SF









bound procedures for problem G


errors that we present give the deviation of the heuristics and lower bounds from the

optimal solution found from solving the MILP formulation using CPLEX. The linear

programming relaxation of extended formulation is solved using CPLEX callable

libraries as well.

We measure the tightness of the lower bounds as follows:


CPLEX Lower bound
Error( ) CPLEX 100,
CPLEX

and the quality of the heuristics using the following:


Upper bound CPLEX
Error( ) CPLEX 100.
CPLEX

For all problem classes, except problem class 32, the dynamic programming algorithm

gave the optimal solution in all 20 instances. The running time of this algorithm was

at most 4.31 cpu seconds. On the other hand, the primal algorithm did not perform


F "c Prog. Primal ';or. CPLEX
'or rror
Problem () (sec) ( ) (sec) (sec)
1 1 1.15 0.17 29.55
2 1. 0. 0.17 31.57
3 1 5.34 0.17 6
4 1. 0.17
5 1 1 0.17
6 0.00 1. 15.14 0.17
7 0. 1., 15.15 0.16 14
8 0.00 1.. 14. 0.17 07
9 0.' 1. 1 44 0.16 255.,
10 0.00 1. C 0.16
1 0. 0.01 47.37 0.01
12 0. 61 0.02
13 i 0.13 7.72 .7 0. 6.
14 .4 15 0. 5
15 0.75 1 0. 7
16 11.17 0.13 147
17 i 4.31 11 0.17


Table 2 2:


uilts from


1 and 2










well, as the errors went up to 52' for problem class 32. However, its running time

for all problems was less than 1 cpu second. Both algorithms are much faster than


CPLEX.

Table 2


Results >m I


bound ::cdurcs for


'icm CGroups 3,4 and 5


The linear programming relaxation of the extended formulation generated

solutions that are less than 0.1 from optimal for all problem classes. The dual

algorithm provided tight lower bounds as well. For all problem classes except problem

class 32, it generated bounds that are less than 2' from optimal. However, the linear

programming relaxation of the extended formulation took almost as much time as

solving MILP formulation. The running time of the dual algorithm for all problem

classes was less then 1 cpu second.

Based on the results presented in Tables 2-2 to 2-5, we can see that an effective

algorithm to solve our problem would combine the dynamic programming algorithm

to generate upper bounds and the dual algorithm to generate lower bounds.


S "c Prog. Primal Algor. CPLEX
Error Error
Problem ( ) (sec) (, ) (sec) (sec)
18 0.1 1.17 11.04 0.13 64.C
19 0. 1.27 12.59 0.14 75.04
20 0. 1. 8.34 0.16
21 0. 1.56 12.42 0.17 11 6
22 I 1. 1 0.18 1 75
1.75 1 7 0.19 146.82
24 .I I 11. 0.19 174.79
1 15.34 0. 189.
I 2.66 0.16 45.65
S 1. 77 0.16
1. Q 0.15 12.19
0. 1.. 0.00 0.16 6.24
0. 1.. 0. 0.15 4.13
31 0.00 1. 44 0.16 4.72
1.24 51.84 0.16 7.12
0.00 1.. 9.35 0.16 91. -
34 0.' 1. 1 0.15 4.41
0.00 1.. 0.44 0.16 C









Table 2 4: Results from lower bound procedures : i problem G 1 and 2

Linear Prog. Dual or.
'Or 'Or
Problem () (sec) ( ) (sec)
1 ii 24. 10
2 l 1 7 i
3 65. 41
5 I 1 10 .1 0.10
6 175 1 10
7 0.00 217.25 0. 0.10
8 0. 312.41 0.22 0.10
9 0.00 484.41 I 0.10
10 0.i ( 5 0.: 0.10
1 0. 0.49 1.72 0.
12 0.1 6.00 1.07 0.
13 0. 19. 0. 0.
14 I 45. 14
15 .4
16 l 1 .17 41 0.14
S17 1 18


Now, we want to provide more insights into the problem characteristics

that affect the performance of the heuristics. It has been shown in other studies

(Hochbaum and Segev [60], Eksioglu et al. [34]) that the problem difficulty depends on

the values of the fixed costs. In problem classes 1 to 10 the level of production set-up

costs is increased (while everything else is the same). Thus, the time it took solving

the MILP formulation of the problem, as well as the time it took solving the linear

programming relaxation of extended formulation, increased as the level of fixed charge

costs increased. However, the computational time of the dual-primal and dynamic

programming algorithms were not affected by the change in setup costs. This can be

explained by noting that their running time depends only on the number of facilities

and the length of the time period.

From the results for problem groups 2 and 3, one can see that as the number of

time periods and the number of facilities increase, the running time of the dynamic

programming algorithm increased. Problem classes 17 and 25 have almost the same









lem Groups 3, 4 and 5


number of arcs; however the time it took to solve problem class 17 is almost twice the

time it took to solve problem class 25. The reason is that the number of time periods

in problem class 17 is higher (the running time of this algorithm is O(FT4)).

Another interesting observation is that all algorithms performed very poorly

in solving problem class 32. In this problem class we have generated the holding

costs in the interval [-10, 10]. For this problem class, the quality of both lower and

upper bounds generated is worse when compared to the rest of the problems. The

dynamic programming algorithms gave solutions that are 1.2!' from optimal and

primal algorithm gave solutions 13.19'. from optimal. The lower bounds generated

using linear programming relaxation of the extended formulation were 0.1,' from

optimal and the dual algorithm gave an average error of 51.S !' The reason for such

a performance is the holding costs being not restricted in sign. Wu and Golbasi [106]


Linear Prog. Dual or.
'or '(or
Problem () (sec) ( ) (sec)
18 7
19 l 92. .1
113
21 i 1 153. : 0.10
22 164. 11
0.00 1 .94 i 0.12
24 0.01 213.l 0.42 0.13
0.01 2z 0.32 0.14
S0.01 0.12 0.
27 0.01 0.04 0.
0. 16.75 0.1 0.10
0. 14.13 0.00 0.10
12.57 0.10
31 0.01 17.31 1.48
32 0.16 19.55 13.19
S*.44 19 10

14


Table 2 5: Results


lower bounti


*dures for









es for problem Group 6


F c Prog. Primal _or. CPLEX
'or ITOr
Problem ( ) (sec) ( ) (sec) (sec)
1.17 1.41 3 17
1.1 140 4. 015 3.77
1. 1. 17 4.27
1.54 1.40 0.16 4.79
1.24 1., 51.84 17 7.
41 0.84 1." 16 0.17 8.
42 0. 1. 8 0.17 7.13
0.41 1. 147.- 0.16 10.47
44 1 0.16 17.
45 0.00 1. 719. 0.16 31


show that the multi-facility lot-sizing problem is NP-hard in case that the holding

costs are not restricted in sign.

We wanted to further investigate the performance of the algorithms for the case

when the holding costs are not restricted in sign. We generated a sixth group of

problems that have holding costs uniformly distributed in the interval [-10, 10]. We

re-ran problem classes 1 to 10 creating 10 new class problems (class problems 36 to

45).


Table 2-7: BRsult


:edurcs for oblcm Group G


Linear P Dal Algor.
'or 'Or
Problem ) (sec) ( ) (se)
0.1 1 3.11 0.10
0. 13.67 O
0.05 15. 0 0.10
0 1 8.42 0.1
0.16 18.98 13.19 0.10
41 0.10 18.04 14.78 0.1
42 0.16 17.18 15.81 0.10
43 0. 22.82 17.11 0.10
44 0. 47 5 0.10
45 0. 83.41 ,3 0.10


.ble 2 6: Resulth


bound


I>m lower bound










Tables 2-6 and 2-7 present the results for the sixth group of problems. For this

set of problems, the dynamic programming algorithm performed well. Its maximum

error was less than 1.5 !' from optimal and the running time averaged 1.40 CPU

seconds. The primal algorithm performed very poorly for this group of problems.

The linear programming relaxation of the extended formulation gave tight lower

bounds. The maximum error presented for this group of problems is 0.3' However,

the running time of this algorithm is comparable to the time it took CPLEX to

solve the corresponding MILP formulation of the problem. The dual algorithm also

performed poorly.

In our final group of problems (problem group seven), we consider the demand to

show seasonality pattern. Demands are generated as follows:

2xr
bt = 200 + azt + a sin [2 (t + d/4)]

where,
a is the standard error of demand
zt i.i.d. standard normal random variable
a amplitude of the seasonal component
d is the length of a seasonal cycle in periods

These demands are generated in the same way as in Baker et al. [5] and Chen et

al. [22]. In our test problems we take a = 67, a = 125 and d = 12. Table 2-8 presents

the characteristics of the problems generated. The errors presented in Table 2-9 are

calculated as follows:

Upper Bound Lower Bound from Dual Alg.
Gap (,) = 100.
Lower Bound from Dual Alg.

For problem classes 46 to 56 we compare the solutions from the dynamic

programming and primal algorithms with the corresponding optimal solutions

from CPLEX. For these problems, dynamic programming performed very well, as the

maximum error presented is 0.0,,' However, the running time of this algorithm is

higher than the running time of CPLEX. It took CPLEX on average 53 cpu seconds










eteristics # problem Group 7


Extendcedl orimuiaon
F 'cc programming,


to run problem class 56, and for the same problem it took the dynamic programming

algorithm on average 354 cpu seconds. The primal algorithm also performed very well.

The maximum error presented is 0.35"' The running time of the primal algorithm

was less then 1 cpu second for this set of problems.

For problem classes 57 to 63 we do not have the optimal solutions. CPLEX failed

to solve these problems, because of their size. Therefore, we use the lower bounds

generated from dual algorithm to calculate the error gaps. The dynamic programming

algorithm gaves very good solutions. The maximum error gap was 0.12' ", but the

running time of the algorithm went as high as 115, 453 cpu seconds. The primal

algorithm gave a maximum gap of 0 2-<' and maximum running time 14.05 cpu

seconds.


Problem F periods Nodes Arcs* Arcs"
24 5 6,4
47 24 745 9,
48 241 1
48 1 24,4- 1,1'
S 1,' 1,176
51 48 1 148, 1,1
52 17 3 1
77 1 4
54 3 1 4
55 t1 374, 18

57_ t 7,873 748, 18
58 1 1.- 73
11 i 73
S15,745
16. 5 1 i
1,9
31t, 11,8


Table 2 8: C


formulation









Table 2 9: Results for problem G 7

D 'c Prog. Primal or. C
Problem _rror C 'or C
( ) ( ) (se) ( ) ( ) (sec) (sec)
46 0 0. 0. 0.318 0.
47 0.C 0. .6 0.12 0.1 0.1.
48 0 0. 0.17 0.155 0.156 7
S 0. 0. 8 1. 0.1 0184 1.
0 0.013 1. 0.1 0. 2.74
51 0 8 0. 2.17 0.1 0.107 0.07 3.58
52 0 0. 15.49 0.C 0. 0.13 6.
40 24 0. 2 0.274 0 0.18 11.-
54 0 0. 31.16 0. 0.377 0.23 .48
55 0 0.( 0 .C 0 0.44 .25
6 0. 0. 3.44 0.127 0.133 ;4 52.58
57 N/A O.C N/A 0.246 C N/A
58 N/A 0.121 3654.24 '/A 0.117 1. /A
N/A 0.129 2.14 N/A 0.151 2.51 N/A
N/A 0. 7, .58 -A 0 J //A
61 N/A 0. .1 57 N/A 0.227 4 N/A
N/A 0. /A 0.1 1 /A
N/A 0.108 115.453. N/A 0 1 N/A


The results of Table 2-9 show that the quality of the solutions generated and

the quality of lower bounds is very good. In particular the running times of primal

algorithm are very small.

2.9 Conclusions

In this chapter we discuss the multi-facility lot-sizing problem. We propose the

following heuristic approaches to solve the problem: dynamic programming algorithm,

a cutting plane algorithm and a primal-dual algorithm. For this problem we also give

a different formulation that we refer to as the extended problem formulation. The

linear programming relaxation of extended formulation gives lower bounds that are

at least as high as the lower bounds from linear programming relaxation of 1_,1Ih1 ,!"

problem formulation. We present a set of valid inequalities for the multi-facility

lot-sizing problem and show that I, i,- are facet defining inequalities.










We tested the performance of the heuristics on a wide range of randomly

generated problems. The dynamic programming algorithm gave good quality solutions

for all problem instances. The error (calculated with respect to the optimal solution

or with respect to a lower bound in case that the optimal solution does not exists)

reported is less than 1.,.' The running time of the dynamic programming algorithm

is O(FT4). This explains the relatively high running times of this algorithm for the

seventh group of problems. Linear programming relaxation of extended formulation

gave high quality lower bounds. The maximum error reported for problem classes 1 to

45 is 0.3'' ,. from optimal.

It has been shown that the multi-facility lot-sizing problem is NP-hard when

holding costs are not restricted in sign (Wu and Golbasi [106]). This explains the fact

that all algorithms gave their worst results for problem classes 32 and 36 to 45. In

particular primal-dual algorithm performed poorly for these type of problems.

For problems 57 to 63 CPLEX ran out of memory and failed to provide an

integer solution. Primal-dual algorithm however gave solutions within 0.;:77' error

gap in less than 14 CPU seconds.















CHAPTER 3
EXTENSIONS OF THE MULTI-FACILITY LOT-SIZING PROBLEM

3.1 Multi-Commodity, Multi-Facility Lot-Sizing Problem

In the previous chapter we discussed the single-commodity, multi-facility lot-

sizing problem. In practice, management of production, inventory and transportation

in a plant typically involves coordinating decisions for a number of commodities. In

this section we analyze and propose solution approaches for the multi-commodity,

multi-facility lot-sizing problem.

The multi-commodity, multi-facility lot-sizing problem consists of finding a

production and transportation schedule for a number of commodities over a finite

time horizon to satisfy known demand requirements without allowing backlogging.

The schedule is such that the total production, inventory holding, transportation

and set-up costs are minimized. These costs may vary by product, facility and time

period. Production capacities and joint order costs tie together different commodities

and necessitate careful coordination of their production schedules. It is easy to see

that without the presence of joint order costs or production capacities, this problem

can be handled by solving each commodity subproblem separately. In this section

we consider the multi-commodity problem with only production capacities (no

transportation capacities and no joint order costs) and refer to it as the capacitated

multi-commodity, multi-facility lot-sizing problem.

A large amount of work has been devoted to the capacitated multi-commodity,

single-facility lot-sizing problem since it is the core problem in the Aggregated

Production Planning models. Solutions to lot-sizing problems are often inputs to

Master Production Schedules and subsequently to Materials Requirements Planning









(\I RP) systems in a "push" type manufacturing environment (see Bhatnagar et

al. [13] and N iliiii s [85] for a review of these models).

It has been shown that, even in the single-commodity case, the capacitated lot-

sizing problem is NP-hard [42]. Bitran and Yanasse [15] showed that the capacitated

multi-commodity problem is NP-hard and Chen and Thizy [24] showed that the

problem is strongly NP-hard. Heuristic approaches to solve the problem were

developed by Lambrecht and VanderVeken [71], Dixon and Silver [31] and Maes

and van Wassenhove [77, 78]. The method proposed by Barany et al. [10] solves the

single-facility, capacitated problem optimally using a cutting plane algorithm followed

by a branch and bound procedure. These algorithms do not consider setup times.

Three groups of researchers pioneered work on the capacitated economic lot-sizing

problem with setup times. Manne [81] used a linear programming model; Dzielinski

and Gomory [33] used Dantzig-Wolfe decomposition; and Lasdon and Terjung [72]

used a generalized upper bounding procedure.

Eppen and Martin [36] provided an alternative formulation of the capacitated,

multi-commodity lot-sizing problem known as the shortest path formulation. They

showed that the linear programming relaxation of the shortest path formulation is

very effective in generating lower bounds, and the bounds are equal to those that

could be generated using Lagrangean relaxation or column generation.

So far, promising heuristic approaches to solve the capacitated multi-commodity

lot-sizing problem seem to be those based on Lagrangean relaxation. Thizy and

van Wassenhove [96] and Trigeiro et al. [98] proposed a Lagrangean relaxation

of the capacity constraints. They updated the Lagrangean multipliers using a

subgradient approach and proposed a heuristic to obtain feasible solutions. Merle

et al. [32] used a Lagrangean relaxation approach as well; however, '. :,- updated the

Lagrangean multipliers using a cutting plane method. Chen and Thizy [24] analyzed

and compared the quality of different lower bounds calculated using relaxation

methods such as Lagrangean relaxation with respect to different sets of constraints










and linear programming relaxation. Millar and Yang [82] proposed a Lagrangean

decomposition procedure to solve the capacitated multi-commodity lot-sizing problem.

Their approach decomposes the problem into a transportation problem and K

independent single-commodity lot-sizing problems. Thizy [95] analyzed the quality

of solutions from Lagrangean decomposition on original problem formulation and

shortest path formulation using polyhedral arguments.

3.1.1 Problem Formulation

In many practical situations, coordination of production, inventory and

transportation decisions involves different commodities. This complicates the problem

considerably. In this section we discuss the multi-commodity, multi-facility lot-sizing

problem. This is a generalization of the classical capacitated, multi-commodity

lot-sizing problem. We add to the classical problem a new dimension, the facility

selection problem. In addition, we consider transportation costs and their effect on

lot-sizing decisions.

Assume that there are K commodities that need to be produced. Each

commodity faces a known demand during each of the next T periods. Note that

commodities share a common production resource with item specific setup costs.

The goal is to decide on the production schedule for each commodity, such that

production, transportation and inventory costs in all the facilities are minimized,

demand is satisfied and capacity constraints are not violated. For each commodity k,

we define the following input data:

Pitk production unit cost for commodity k at facility i in period t
Sitk production set-up cost for commodity k at facility i in period t
hitk inventory unit cost for commodity k at facility i in period t
itk transportation unit cost for commodity k at facility i in period t
btk demand in period t for commodity k
I production capacity at facility i in period t

We introduce the following decision variables:

qitk production quantity for commodity k at facility i in period t
Xitk amount of commodity k transported from facility i in period t










litk amount of commodity k in the inventory at the facility i in the
end of period t
yitk a binary variable that equals 1 if there is a production set-up for
commodity k at the facility i in period t.

An MILP formulation of the problem is the following:

F T K
minimize 1 (Pitkqitk + SiitYitk + CitkXtk + hitkitk)
i=- t=l k=

subject to (MiC)


Ii,t1,k + qitk xitk+ Iitk i 1 ,...,F;t 1,...,T; k 1,...,K (3.1)
F
Xitk = btk t 1,...,T;k 1,...,K (3.2)
i= 1
K
Y qitk < t 1,...,F;t ,...,T (3.3)
k=l
qitk < btTkYitk i = 1,..., F; t = 1,..., T; k = 1,..., K (3.4)

qitk,Iitk,Xitk > 0 i 1,... ,F;t 1,...,T;k= 1,...,K (' )

yitk e {0,1} i= ,...,F;t= ,...,T;k= 1,...,K (3. )

Note that btTk is the total demand for commodity k from period t to T. Constraints

(3.1) and (3.2) are the flow conservation constraints at the production and demand

points respectively. Constraints (3.3) are the production capacity constraints. These

constraints reflect the multi-commodity nature of our problem. If' i,- are absent, the

problem can be decomposed into K single commodity problems.

We assume that initial and ending inventories are zero for all items. There is no

loss of generality in this assumption, since we can reset for each commodity the given

level of initial (ending) inventory ilk (liTk) at zero by removing lilk from the demand

of the first periods (adding liTk to the demand of the last period), thus obtaining the

adjusted demands for t 1,..., T and k 1,..., K.

We propose a Lagrangean decomposition-based heuristic to solve the multi-

commodity problem. The decomposition is performed on the extended problem










formulation. The extended formulation, similar to the single commodity problem,

requires splitting the production variables qitk by destination into variables qitrk

(r = t, ..., T), where r denotes the period for which production takes place. Given

this, we have
T
qitk Y qitrk



Xitk qistk
s=l
tT t
litk = qisrk qistk
s=l r=t s=l
qitTk denotes the production quantity for commodity k from facility i, in time period t

for period 7. The extended formulation of (! C) is the following:

F T K T
minimize > > I(> CitrkQitrk + Sitkitk)
i= t=l k= r=t

subject to (Ex-MC)

F T
> qitTk drk t 1,... ,T; k 1,..., K (3.7)
i= t 1
K T

k Qtrk k=l T=t
qitrk- bTkitk < 0 i 1,...,F;t 1,...,T;t
qitrk > 0 i 1,...,F;t= 1,...,T;t < 7
itk e {0,1} i 1,...,F;t= 1,...,T; k= 1,...,K (3.11)


citrk consists of the unit production cost of commodity k at facility i in period t, the

unit transportation cost from facility i in period T, as well as the total unit cost of

holding the item from period t to 7 at facility i (Ctrk =itk + Cik + s=t+ hisk)"










3.1.2 Linear Programming Relaxation

The linear programming relaxation (LP) of (\ MC) is obtained by replacing

constraints (3.6) with


yitk > 0 for i= 1,...,F;t= 1,...,T;k= 1,...,K.


The solution to this relaxation is generally far from optimal. We want to give a

measure of the quality of the linear relaxation, and specifically disphyli a worst case

for the error bound. Note that v(LP) denotes the optimal objective function value of

(LP) and v(MC) denotes the optimal objective function value of (M\C).

Theorem 3.1.1
K
v(LP) > mm Sit (3.12)
i 1,...,F ;t 1,...,T
k 1
FT K K
v(MC) v(LP) < S Sitk i mm 1 itk (3.13)
i=l t=l k=l k=l
We present a class of problems for which these bounds are attained i'inpl/.. .l'l. with

respect to e.

Proof: The fixed charge production cost Sitk is assumed to be positive, thus there

exists an optimal value of problem (LP) such that yitk itk/btTk. Since,

qitk > 1 F T (3.14)
btTk blTk
the total set-up costt iof this solution will be greater than

the total set-up cost of this solution will be greater than

K

i=1,...,F;t 1,...,T
k=1

This proves (3.12).

In order to prove (3.13), let (q, I, x, y) be an optimal solution to (LP) and

(q, I, x, [y]) a feasible solution to (\I C) obtained by fixing the fractional components

of y to 1. Then, we have the following relationship:











F T K
v(MC) v(LP) < (Sitk Sitk
i-1 t=I k=1 tTk
F T K K F T
S k i=1,...,F;t 1,...,T
i=l t 1 k 1 k 1 i=- t= 1
FT K K
< Sitik i ..itk (due to (3.14))
i=l t=l k=l k=l
Now we present a class of problems for which the bounds derived in (3.12) and (3.13)

are obtained ..i-mptotically with respect to e. The class of problems we consider has

the following properties:

Demands:


btk = ,bTk = 1/e, for t = 1,...,T-1 and k 1,...,K.


Costs:


1tk 1,Pltk = hltk > 1, Ctk 0 for t =,...,T; k =,...,K.


Sitk > l,ik > hitk > itk >0 for i =2,..., F; t =,..., T; k =,..., K.

Capacities:


vit = K, for t = 1,... ,T-1 and VlT = K/e.


The optimal solution for this class of problems is such that


qltk =btk, ltk 1, ltk btk for t = 1,..., T; k = 1,..., K,


and an optimal solution to the corresponding linear programming relaxation is


q1tk = btk, Ytk btk for t= t T;k= 1, t..., K.
btTk










Thus,

Y 1 if t T;k= 1,...,K
Y1tk =
1/+cT-t)c for t 1,...,T 1;k 1,...,K.
As c approaches to zero, we have the following equation:

1 if t T;k= 1,...,K
Yltk S
0 for t 1,...,T 1;k 1,...,K.

Therefore,
K K
lim (LP) 81Tk = Min Sitk
e-->O i 1,...,F;t=1,...,K
k=1 k=1
and
F T K
v(MC)- Sitk.
i= t=l k= 1
We have showed that, for a class of multi-retailer, multi-facility lot-sizing problems the

right hand sides of (3.12) and (3.13) are attained ..I-., !nil i. lly with respect to e. O

3.1.3 Valid Inequalities

Consider the formulation (\ C) of the multi-commodity problem. Let 4k denote

the set of feasible solutions to the k-th single-commodity problem and let LPk denote

the set of the feasible solutions to the linear programming relaxation of the k-th

single-commodity problem. We can re-state the capacitated multi-commodity problem

as follows:
F T K
minimize > 1 (Pitkqitk + itkYitk + CitkXtk + hitklitk)
i=- t=l k=
subject to

(qk,,Ikyk) e k, k=l 1,...,K

E lIqitr t' t,..., T;i1,...,F

Or equivalently,

F T K
minimize ( > (pitkqitk + SitkYitkk CtkXitk+ htklitk)
i=l t=l k=









subject to (\IC*)

(qk, Xk,Ik,k) e LPk k 1,... ,K (3.5)
K

k=1
itk e {0,1} i= ,...,F;t= 1,...,T;k= 1,...,K (3.7)

where LPk n (y e {0,1}) -= k.

Given a fractional point (ql, xi, i, yi),.. (qK, XK, K, KYK), the goal is to find

a valid inequality that cuts off this non integer point from the feasible region of

linear programming relaxation of (\ C*). Ignoring constraints (3.16), the problem

decomposes into K single-commodity, multi-facility lot-sizing problems. For

the single-commodity problem, we have proposed in Section 2.6.1 a set of valid

inequalities. Let ) be the feasible region of the multi-commodity flow problem, then


co()) C n i=co()k)


This implies that the valid inequalities for the single commodity problem, are valid for

the multi-commodity problem as well. Thus, one can use these inequalities to check

for each commodity k if point (qk, Xk, Ik, yk) can be cut off from K).

3.1.4 Lagrangean Decomposition Heuristic

In this section we discuss a Lagrangean decomposition-based heuristic that we

used to solve the capacitated multi-commodity, multi-facility lot-sizing problem.

Lagrangean relaxation is a classical method for solving integer programming problems

(Geoffrion [49], Wolsey [105]). This method has been used to solve various network

flow problems. Held and Karp [56, 57] successfully used Lagrangean relaxation

to solve the traveling salesman problem; Fisher [40] used this method to solve a

machine scheduling problem; Ross and Soland [92] applied this method to the general

assignment problem. Holmberg and Yuan [62], Holmberg and Hellstrand [61], and

Balakrishnan et al. [8] used Lagrangean relaxation based approaches for network

design problems.









We start our discussion with a review of the Lagrangean relaxation approach

and its extension, Lagrangean decomposition. We continue then with a detailed

description of the Lagrangean decomposition based heuristic we have used to generate

upper and lower bounds for the multi-commodity, multi-facility lot-sizing problem.

Review of the method. Geoffrion [49] formally defines a relaxation of an

optimization problem (P) as follows:


min {f(x) x E X},

as a problem (RP) over the same decision variable x:


min{g(x) x E Y}

such that (i) the feasible set of (RP) contains that of (P), that is X C Y, and (ii)

over the feasible set of (P), the objective function of (RP) dominates (is at least as

good as) that of (P), that is


g(x) < f(x), Vx X.

The importance of a relaxation is the fact that it provides bounds on the

optimal value of difficult problems. The solution of the relaxation, although usually

infeasible for the original problem, can often be used as a starting point for specialized

heuristics.

Lagrangean relaxation has shown to be a powerful family of tools for solving

integer programming problems approximately. Assume that problem (P) is of the

form

min{f(x) lAx > b, Cx > d, x E X},

where X contains only the integrality constraints. The reason for distinguishing

between the two types of constraints is that the first of these (Ax > b) is typically









complicated, in the sense that problem (P) without this set of constraints would be

much easier to solve.

Let A be a nonnegative vector of weights called Lagrangean multipliers. The

Lagrangean relaxation LR(x, A) of (P) is the problem


min{f(x) + A(b Ax) Cx > d, x e X},

in which the slack values of the complicating constraints Ax > b have been added to

the objective function with weights A, and the constraints Ax > b have been dropped.

LR(x, A) is a relaxation of (P), since (i) the feasible region of LR(x, A) contains the

feasible region of (P), and (ii) for any x feasible for (P), f(x) + A(b Ax) is less than

or equal to f(x). Thus for all A > 0, the optimal objective function value of LR(x, A),

which depends on A, is a lower bound on the optimal value of (P). The problem


max v(LR(x, A))
A>0

of finding the highest lower bound is called the Lagrangean dual (D) of (P) with

respect to the complicating constraints Ax > b.

Let v(*) denote the optimal objective function value of problem (*). The

following theorem by Geoffrion is an important result:

Theorem 3.1.2 The Lagri,.', .,; dual (D) is equivalent to the following primal

relaxation (PR):

min{f(x)lAx > b,x e Co{x e X|Cx > d}},

in the sense that v(D) = v(PR).

This result is based on linear programming duality and properties of optimal

solutions of linear programs. Let denote by (LP) the linear programming relaxation of

problem (P). Then the following holds:

(i) If Co{xlCx > d,x X} = {xlCx > d}, then v(P) > v(PR) v(D) = (LP).










In this case the Lagrangean relaxation has the integrality property, and the (D)

bound is equal to the (LP) bound.

(ii) If Co{xlCx > d,x e X} C {xlCx > d}, then v(P) > v(PR) = v(D) > v(LP),

and it is possible that the Lagrangean bound is strictly better than the (LP) bound.

This i-l--. -I-; that in deciding on how to implement the Lagrangean relaxation

method, one should consider the following properties of the set E = {x|Cx > d}:

(i) B should be simple enough that the resulting optimization subproblems are not

computationally intractable (usually E decomposes into simpler subsets, E = IljEjj),

(ii) E should be complex enough, such that the subsets Ej do not have the integrality

property. Otherwise, the lower bounds generated would be equal to the lower bounds

from the corresponding linear programming relaxation.

The Lagrangean function z(A) = v(LR(x, A)) is an implicit function of A, and

z(A), the lower envelope of a family of linear functions of A, is a concave function of A,

with break-points where it is not differentiable.

An extension to Lagrangean relaxation is the Lagrangean decomposition method

introduced by Guignard and Kim [52]. Different than Lagrangean relaxation,

Lagrangean decomposition does not remove the complicating constraints, but

decomposes the problem into two subproblems that collectively share the constraints

of the original problem. This is achieved by introducing a new set of variables z, such

that x = z. Then, problem (P) reads


min{f(x) Ax > b, Cz > d, x E X,z = x,z E X}.


Relaxing the "< ..i- constraints z = x yields a decomposable problem, ju-1IF i'i,-; the

name "Lagrangean decompc.-i i.. (LD(x, z, A)) (Lagrangean relaxation refers to the

case when only one subset of constraints is relaxed), which is given by


min{f(x) + A(z x) Ax > b, Cz > d, x e X,z e X}
x,z










min{f(x) AxlAx > b, x e X} + min{AzlCz > d,z c X}.
x z

Guignard and Kim [52] show that Lagrangean decomposition can in some cases

yield bounds substantially better than Ii Il1 lil 1 I" Lagrangean relaxation. The

Lagrangean dual problem (D) is the following:


max{min LD(x, z, A)}.
A xz

The following theorem by Guignard and Kim is an important result:

Theorem 3.1.3


v(D) = min{f(x)x e Co{xlAx > b, x X} n Co{xlCx > d,x e X}}.


If one of the subproblems has the integrality property, then v(D) is equal to the

better Lagrangean relaxation bound. If both have the integrality property, then

v(D) = v(LP). Note that finding a lower bound for problem (P) using a Lagrangean

relaxation/decomposition algorithm requires optimally solving the inner minimization

problem and the outer maximization problem. Since the Lagrangean function z(A) =

v(LD(x, z, A)) is a piecewise concave function, we use a subgradient optimization

algorithm to maximize it. In implementing the subgradient algorithm, an important

issue is choosing a step size to move along the subgradient direction that guarantees

convergence to the A that maximizes the dual function (D).

Significant for the decomposition is that the inner minimization problem

is easier to solve than the original problem (P). Recall that Lagrangean

relaxation/decomposition is an iterative method, the inner minimization problem

needing to possibly be solved several times. In the case that the inner minimization

problem is still difficult and requires a considerable amount of computational efforts

to solve to optimality, a practice that can be followed is finding good lower bounds

instead. Obviously this will affect the quality of the lower bound from the Lagrangean










relaxation/decomposition. Letting v(P) be the optimal solution to problem (P), then

v(P) > v(D).

Let w(*) denote a lower bound and Q(*) an upper bound for problem (*). If, for

every A we do not solve the Lagrangean decomposition problem LD(x, z, A) optimally,

but we rather provide a lower bound, the following holds:


v(P) > v(D) = maxv(LD(x, z, A)) > maxw(LD(x, z, A)).
A A

maxA w(LD(x, z, A)) will still be a lower bound for problem (P), however not as

good bound as v(D). In case that we use a heuristic procedure to find a feasible

solution to the inner optimization problem, then v(D) < maxA Q(LD(x, z, A)).

However, we are not sure anymore if maxA Q(LD(x, z, A)) gives a lower bound for

problem (P) since one of the following may happen: v(P) > maxA Q(LD(x, z, A)) or

v(P) < max A(LD(x, z, )).

In the case when a lower bounding procedure is used instead of solving the

inner minimization problem, it is important to identify the quality of these bounds

compared to the lower bounds from the linear programming relaxation. If the lower

bound guarantees that

max (LD(x, z, A)) > v(LP)
A

and the running time of this procedure outperforms linear programming relaxation,

one is better off using the Lagrangean relaxation/decomposition method to find lower

bounds to problem (P).

In the next section we describe the Lagrangean decomposition algorithm we used

to solve the multi-commodity, multi-facility lot-sizing problem.

3.1.5 Outline of the Algorithm

Consider the extended problem formulation (Ex-MC). The basic idea of our

decomposition is to separate the capacitated, multi-commodity problem into

subproblems that are computationally easier to solve than the original problem.










There are many v-w-, one can do that; however we aim to decompose the problem in

such a way that it has interesting managerial implications as well.

We decompose the problem into two subproblems. The first subproblem consists

of the flow conservation constraints and the integrality constraints. This subproblem

can be further decomposed by commodity. The single commodity sub-subproblems

have the special structure of the single-commodity, multi-facility lot-sizing problem

analyzed in Chapter 2. The second subproblem consists of the flow conservation

constraints and the capacity constraints. In this decomposition, the first subproblem

consists of a collection of MILPs and the second subproblem is a linear program.

Below we give an equivalent formulation of the capacitated multi-commodity,

multi-facility lot-sizing problem. We introduce the continuous variables zitrk that are

simply "colp, of the production variables qitrk. This allows for the duplication of

some of the constraints.

F T K T
minimize it [ citrkqite k + Sitklitkl
i=1 t=l k=-1 =t


subject to


qit-rk
F T
z itk -
i 1 t 1
K T

k-1 7- t


Zitrk

S brk


< I ,


zitrk > 0


(3.7), (3.9), 3.10), and (3.11)

i 1,..., F;t 1,..., T;t < 7 < T;k = 1,...,K (3.18)

S= 1,...,T;k= 1, ..., K (3.19)


(3 )

(3.21t)


It is clear that the above formulation is equivalent to formulation (Ex-MC).

Relaxing the "(<. ,,i- constraints (3.18) and moving them to the objective function

yields the following Lagrangean decomposition problem:

FT K T T
minimize il l[ (critk + Aitk)qitrk Sitkitk AitTkZitrkl
i=1 t=l k=-1 =t 7=t


i= 1,...,F;t 1,..., T

t l,...,F;t t,...,T;k 1,...,K .









subject to (LD(x,z,A))

(3.7), (3.9), (3.10), (3.11), (3.19), (3.20), and (3.21).

The Lagrangean decomposition (LD(x,z,A)) problem can now be separated into

the following two subproblems:
F T K T
minimize > Y [ (ciitrk i+ trk)qitrk + Sitktitkl
i=l t=l k=l ~ t

subject to (SPi)

(3.7), (3.9), (3.10), (3.11),

and
F T K T
minimize 1 > 1 -AitrkZitrk
i 1 t= k= 1= t
subject to (SP2)

(3.19), (3.20), and (3.21).

The corresponding Lagrangean dual (LD) problem is the following:


maxv(LD(x, z, A))
A

Note that under the general framework of Lagrangean decomposition method,

constraints (3.7) do not need to be duplicated for both subproblems. However,

computational experience indicates that as long as the added constraints do not add

computational burden to the subproblems, constraint duplication improves the speed

of the convergence and yields better lower bounds (Guignard and Kim [52]).

The potential computational savings from using the dual algorithm is significant

since the subgradient search requires solving K single commodity subproblems in each

iteration. If a problem instance considers 30 commodities for example, this results in

up to 15, 000 calls to the subproblem (the maximum number of iterations we use is

500). In Section 3.1.7 we compare the performance of the Lagrangean decomposition

approach when the subproblems are solved to optimality versus the case when the

dual algorithm is used.









Subproblem (SP1) can be decomposed by commodity. Each sub-subproblem k

has the following MILP formulation:
F T T
minimize > Z(Z (cit + Ait-)qit- + sityit)
i= 1 t= 1 = t

subject to (SP1k)

il F t1 l qitT d 7 1,... ,T

qit -b-yit < 0 i = ,..., F;t = 1,... ,T;- > t

qitr > 0 i= t,...,F;t -l,...,T;7-> t

Yit {0,1} i 1,...,F;t 1,...,T

One can see that this formulation is the same as the formulation of multi-facility

lot-sizing problem we discussed in the previous chapter. Since for most of the test

problems discussed in C'!h pter 2, the dual algorithm gave close to optimal lower

bounds and its running time was much smaller than the running time of the linear

programming relaxation, we use the dual algorithm to find good lower bounds for the

subproblems (SP1k) for all k = 1,..., K. However, it should be recognized that the

quality of the lower bounds from the decomposition may not al--,v be as good as in

the case that we solve the subproblems optimally.

On the other hand, subproblem (SP2) is simply a linear program. We use

CPLEX callable libraries to solve this problem. The solution of (SP2) is feasible for

the (Ex-MC) problem. However, since the set-up costs are not considered in the

formulation, we use a simple procedure to calculate an upper bound (Figure 3-1).

Proposition 3.1.1 The Lagrir,, ..;,. decomposition of (Ex-MC) gives lower bounds

that are at least as high as the corresponding linear 1'i ',i,,,,,,.:, relaxation of extended

formulation.

Proof: Let 1 denote the set all feasible solutions to the extended formulation

of the multi-commodity, multi-retailer problem (Ex-MC). Let Ii be the set of the










Let z* be the optimal solution to subproblem ( 's) in iteration s.
UB= -0; = 0 fori 1= F = 1... T; k 1k .....K.
for = 1,..., F; I = :...:T; k 1..., K
for r= t...,T do
if zt*k_ > 0 then lU" = UlB I -
I t, i ,
if > 0 then U I +

-ure 3 1: bound 1 e

feasible solutions corresponding to linear programming relaxation of subproblem (SP1)

and 42 be the set of feasible solutions to linear programming relaxation of subproblem

(SP2).

Guignard and Kim [52] show that "Optimizing the Lagrangean decomposition

dual is equivalent to optimizing the primal objective function on the intersection of

the convex hulls of the constraint I (Theorem 3.1.3). Therefore, optimizing (LD)

is the same as optimizing
FT K T
minimize [ >1 ZZ eTitrkitrk + SitkYitkl
i i t=l k=l = t

subject to


q, y Co{q, yq, y e 4 n y E {0,1}} q c Co{qlq e 42}

The linear programming relaxation of (Ex-MC) minimizes the same objective function

over the intersection of 1t and )2.
FT K T
minimize > [ Z Y c itrkqitrk + SitkYitkl
i i t=l k=l = t

subject to (LP-Ex)


q, E I1 i 0 E q 2.









Since q are continuous variables


Co{qlq e )2} 2,

however,

Co{q, y|q, ye ic ny, {0, 1}} C i.

This shows that the feasible region of the dual problem (LD) is smaller than the

feasible region of the linear programming relaxation (LP-Ex). Therefore, the objective

function value we get solving (LD) will be at least as high as the objective function

value of the linear programming relaxation. O

Proposition 3.1.2 The Lagr ,'.; ,, decomposition of (Ex-MC) gives lower bounds

that are equal to the bounds from Lagri,., ,,;,. relaxation of (Ex-MC) with respect to the

I "'.r. :1/; constraints (3.8).

Proof: Let tI be the set of solutions satisfying constraints (3.7),(3.9),(3.10)

and (3.11) and )2 be the set of solutions satisfying constraints (3.8) and (3.11). Let

V* be the intersection of )2 with the convex hull of 1i. Solving the Lagrangean

relaxation with respect to the capacity constraints (3.8) is equivalent to minimizing

the objective function of (Ex-MC) over V* (Theorem 3.1.2). On the other hand,

solving the Lagrangean decomposition of (Ex-MC) is equivalent to minimizing the

objective function over the intersection of VY = )2n (3.7) with the convex hull of tI

(Theorem 3.1.3), or in other words minimizing the objective function of (Ex-MC) over

the intersection of V* with (3.7). Note that all the solutions in V* satisfy constraints

(3.7) since these constraints are contained in )i. Therefore, solving the Lagrangean

decomposition problem is equivalent to minimizing the objective function of (Ex-MC)

over .*. This concludes the proof that both the Lagrangean relaxation with respect

to the capacity constraints and the Lagrangean decomposition scheme we propose give

the same lower bounds for (Ex-MC) D









The lower bounds generated from the Lagrangean decomposition heuristic are

the same as the bounds from Lagrangean relaxation with respect to the capacity

constraints, however we chose to implement the Lagrangean decomposition approach.

There are two reasons we choose to do so (i) The decomposition scheme provides

feasible solutions to problem (Ex-MC) at every iteration (ii) The decomposition

converges faster.

Subgradient optimization algorithm. It is well-known (N. in!i1 m1-, and Wolsey

[86]) that the Lagrangean dual function is concave and nondifferentiable. To maximize

it and consequently derive the best Lagrangean lower bound we use a subgradient

optimization method. For more details on the subgradient optimization method see

Held, Wolfe and Crowder [58] and Crowder [28], and for a survey of nondifferentiable

optimization techniques see Lemarichal [73].

Subgradient optimization is an iterative method in which steps are taken

along the negative of the subgradient of the Lagrangean function z(A) (z(A) =

v(LD(x, z, A)). At each iteration s, we calculate the Lagrangean multipliers Aitrk using

the following equation:

AXSk
where
s 7 (minUB maxLB)
y1 :T IT tK
ti= 1 t -Z T (qitrk Zitrk)2
" is a number greater than 0 and less or equal to 2. 7" is reduced if the lower bound

fails to improve after a fixed number of iterations; maxLB is the best lower bound

found up to iteration s and minuB is the best upper bound. Calculating the step

size A using equation (3.22) is a common heuristic rule that has been used in the

literature. The step size updating using rule (3.22) generally allows convergence to the

A* that maximizes the Lagrangean function z(A) (N. in!, -, r and Wolsey [86]).

In order to find a subgradient direction at each step of the Lagrangean

decomposition procedure, we need to find a feasible solution to subproblem (SP1).










1: Initialize A, mint i
Step 2: Solve the '


Ss, u. 7, e. count,
( 'i) and ( 2). C-


the lower bounty


K

k -


the current iteration s
count = count I 1
If LB'i > maxLB, then -B = LF
Step 3: C< an 1 ,r bound UE
If UTB < mi-- then = UB"
If minu(n> = maxs.j, then STOP
If 7 e, then STOP
Step 4: U Jate the iultipliers using equation (3.22)
5: if the number of iterations reach the
Otherwise e to Step 2

Figure 3-2: Lagrtangean decc :)si


' d limit (count)


tion algorithm


The dual algorithm provides only a lower bound to (Ex-MC), but does not provide a

feasible primal solution. Therefore, we use the primal algorithm (Section 2.5) to find a

feasible solution to subproblems (SPIk).

In our computational experiments, we terminate the algorithm if one of the

following happens: (i) the best lower bound is equal to the best upper bound (the

optimal solution is found), (ii) the number of iterations reaches a prespecified bound,

(iii) the scalar 7y is less than or equal to c (a small number close to zero). Figure 3-2

presents the steps of the Lagrangean decomposition algorithm.

3.1.6 Managerial Interpretation of the Decomposition

In this section we discuss the managerial insights of the decomposition procedure

proposed in Section 3.1.5. The choice of the Lagrangean decomposition scheme we

present is motivated not only by its computation capability, but also by its interesting

managerial implications. Several studies (Burton and Obel [19] and Jornsten and

Leisten [64]) have recognized that mathematical decomposition often leads to insights

for general modelling strategies and even new decision structures. In this discussion










we refer to subproblem (SPI) as the product (commodity) subproblem and (SP2) as

the resource subproblem.

The Lagrangean decomposition we propose helps understanding and solving

managerial issues that arise in multi-facility manufacturing planning. Suppose we

consider the resource subproblem as a decision problem for a production manager who

supervises multiple facilities, and each product subproblem as a decision problem for

a product manager. Therefore, the decomposition can be viewed as a decision system

where product managers compete for resource capacity available from manufacturing

facilities. The production manager, on the other side, represents the interests of

efficiently allocating resources from multiple facilities to the products. Often the

solutions proposed by the production manager will not agree with the individual

solutions of product managers. A search based on the Lagrangean multipliers

basically penalizes their differences, while adjusting the penalty vector iteratively.

This process stops when the degree of disagreement (the duality gap) is acceptably

low, or when further improvement is unlikely. See Wu and Golbasi [106] for a similar

discussion on a related problem.

3.1.7 Computational Results

In this section we have tested the performance of the Lagrangean Decomposition

algorithm on a large group of randomly generated problems. We use the CPLEX

callable libraries to solve the MILP formulation (Ex-MC). The CPLEX runs were

stopped whenever a guaranteed error bound of 1 or less was achieved, allowing for

a maximum CPU time of 5, 000 seconds (or 10, 000 seconds depending on problem

size). We use CPLEX to solve the linear programming relaxation of (Ex-MC) and

subproblem (SP2). One of the factors that affects the problem complexity is the

tightness of the upper bounds on production arcs. If the arcs are very tight, there

exist only a few feasible solutions, and this makes the search for the optimal solution

easy. On the other hand, if the arc capacities are so loose we could remove them from










Table 1: Problem characteristics

Problem Nodes Arcs Problem Nodes Arcs
1 1 6--- 114 8,
2 2,240 15 2. 9,1
3 2,, 10 16 3) 19,..
4 3 1 17 4
5 14 18
6 4, 16 19 105
7,..., 10 1 9 1 .
11 1,; 6 21 11,
12 1 i 7 22,...,25 1, 6
13 2,130 7,


the problem formulation, the problem loses its multi-commodity nature, and it can be

decomposed into K single-commodity problems.

In order to create challenging test problems, we generated the upper bounds in

the following way: A necessary condition for feasibility of the problem is

F t K t
EE > wbE k Vt 1,... ,T. (3 )
i= T= 1 k=lT =1

Under the assumption that all facilities in any time period have the same

production capacity v, we can replace


F t

i 1 T 1


Ftv,


thus,


In fact v should be su


/K t
v > bk Vt 1,
k=1 r=1

ch that the following is satisfied:

K t
v > 1 max K- -1 7bk
-F t t
F t t


.. ,T .


Letting 6(> 1) be the capacity tightness coefficient, then

K t
= k-1 7= 1 b7k
v max
F t t










Using this procedure we generated challenging (but feasible) test problems with

tight capacity constraints. We start our computational experiments by generating a

nominal case problem with the following characteristics:
Production set-up costs sit ~ U[200, 900]
Production variable costs pit U[5, 15]
Holding costs hit U[5, 15]
Demand bt ~ U[5, 15]
Number of facilities F = 10
Number of periods T = 5
Number of commodities K = 30
Capacity tightness 6 = 1.3

We alter the nominal case by changing problem characteristics to generate

additional problems. For each problem class we generate 20 instances. First we

change the number of commodities from 30 to 40, 50, 60, 70, and 80 generating six

problem classes (problem classes 1 to 6; problem class 1 is the nominal case).

In the second group of problems we change the capacity tightness coefficient (6)

to 1.1, 1.2, 1.4, and 1.5 (problem classes 7 to 10). In the third group of problems we

change the number of facilities from 10 (the nominal case) to 11, 12, 13, 14 and 15

(problem classes 11 to 15).

We also ran the program for problems with different lengths of the time horizon.

In problem classes 16 to 21 we change the number of periods to 10, 15, 20, 25, 30,

and 35. Finally, we change the level of the fixed charge cost to sit U[200, 300],

U[600, 900], U[900, 1500] and U[1200, 1500] (problem classes 22 to 25).

In implementing the Lagrangean decomposition algorithm, for all problem

instances we set 7 = 1.8. 7 is reduced by 211'. if there is no improvement in the

last 5 iterations. We set a limit of 500 iterations for the Lagrangean decomposition

algorithm. As we have mentioned earlier in this section, we are interested in the

effectiveness of the primal-dual algorithm as a heuristic in the context of the

subgradient search algorithm. Therefore, we present results from the Lagrangean

decomposition algorithm where




















'y ( upper bounds (in


Problem CPLEX Schemcl Scheimc2 Schemnc
1 0. 1 1.79 1.82
2 0.05 1.22 1.14 1.16
3 0.01 0.83 0.-- 0.-
4 0.01 0.50 0.52
5 0.01 0.57 0.1 0.42
6 0.01 0.53
7 1.44 3. 3.61 11
8 0. 2.61 2.
9 0 1 1., 1 1.24
10 0.01 1.01 0.3 0.
11 2.54 2 2.
12 0.17 2.) 2.84 2.'
13 0.21 3 3. 3
14 0 4. 4.52 4.
15 0. 4 ., 4. 4.
16 0.24 1 1
17 0 2.12 2.12
18 0.48 2.22
19 0.53 2.21
0.57 2.34 2.29
21 0.55 2.41 2.35
22 0.02 1.25 1
0.04 3.13
24 0.22 4.75
0.77 ;4


)


Tablc 3-2:


)osl.tion










Scheme: a lower bound to subproblem (SPlk) is found using the dual algorithm

Scheme2: subproblem (SPIk) is solved to optimality using the exact MILP

formulation

Scheme3: a lower bound to subproblem (SPlk) is found solving its linear

programming relaxation.

The error bounds from the three schemes as well as the linear programming relaxation

of (Ex-MC), are presented in Tables 3-2 and 3-3. The quality of the upper bounds we

generated using the Lagrangean decomposition algorithm is measured as follows:

Errr Upper Bound CPLEX Lower Bound
Error (.) 100,
CPLEX Lower Bound

and the quality of the lower bounds is calculated as follows:

Lower Bound CPLEX Upper Bound
Error (,) = 100.
CPLEX Upper Bound

The results indicated that the quality of the upper bounds generated when we solve

the subproblems to optimality (Scheme2) is almost the same as the quality of the

upper bounds when we use the dual algorithm (Schemel) or the linear programming

relaxation (Scheme3). The difference in the quality of the solutions was no more

than 0.2'". A main incentive for using the primal-dual algorithm to solve the single-

commodity, multi-facility lot-sizing problems (SPlk) is the potential computational

savings when solving the multi-commodity problem.

The results from Section 2.8 showed that primal-dual algorithm took little time

to solve each single-commodity problem; however, we were interested in gauging

the real savings for the multi-commodity, multi-retailer lot-sizing problem. Table

3-4 presents the CPU running times of the three algorithms. The running time of

Schemel were much smaller in all cases. These savings are due to the performance of

the dual algorithm.

The quality of the lower bounds generated using either scheme of the Lagrangean

decomposition algorithm or the linear programming relaxation of extended problem





















Tab' 3-3: y ( lower bounds (in ) m Lagrange an *

Problem, LP-Ex Scheimel Schemc2 Schemc3
1 0. 0. I ;7
2 0.: 0. 0.'. 0.
3 0.19 0.19 0.19 0.19
4 0 0.10.11 0.11 0.11
5 0.07 0.C 0.07 0.07
6 0 0.10 0., O.C
7 2.47 2.50 2.48 2.48
8 1.18 1 1.18 1.18
9 0.46 0. 0.' 46
10 0. 0.31 l 1
11 1.19 1 1.19 1.19
12 2.24 2.24 2.24 2.24
13
14 2.97 7 7
15 3.18 3.19 3.18 3.18
16 0.75 0. 75
17 0.81 0.82 0.81 51
18 0.87 0.. 0.87 0.87
19 0., 0 (
0 0."'
21 0.,- 0.: 0.,
22 0 0.57 0.56 0.56
23 0.58 0.59 0.58 0.58
24 0.8 0.87 I C
25 1.34 1 i 1. 1.


)oSltiIon










formulation was almost identical. The results indicated that as the number

of commodities increased, the error reported from CPLEX and Lagrangean

decomposition decreased (problem classes 1 to 10). For these problem classes the


Table 3-4: CP( running times (in seconds)

Problem LEX Schlcmcl Sclhcmc2 Scheme LP-Ex
1 -.57 9 1 27 0.58
2 48 12. 41 44.59 0.72
3 145 17. "'. 11 48.18
4 78.' 11 1 54. 1.31
5 57.25 16 4- 4. 1.50
6 58.48 31 1 5 74.40 1.72
7 5 i 12. 1, .,:. 32.87 0.
8 7 10.87 1 )4 .44 0.64
9 1 9 174.81 24.75 4
10 i 173.16i
11 .22 11. 31
12 3, 75 13.33 ... 16
13 3,294. 13.77 2 .51 2 0.75
14 3, 7 16.. ;
15 4,7 t 16.48 1 O 51 0.;
16 t10 4 7 4 15 7.
17 10 .( 17 1, .18 8.10
18 10 542.24 3,32 1.4 1 .19 17.78
19 10 8 31 5,591. 1 66
10 1. 4.48 7.81
21 10 i 2,555.11 i 2, : 4 4, .'"
22 1,55- 9. 1 0 (
1,1 11.67 33.82 0.
24 4, 13.24 17 0.
4,75 7 14.77 1.51 41.


solution from Lagrangean decomposition was less than 2'. from optimal and the

running times of Schemel were smaller compared to CPLEX. However, the increase

in the number of commodities affected the running time of Lagrangean decomposition

algorithm because of the increase in the number of subproblems (SP1k) to be solved at

every iteration.

Increasing the capacity tightness coefficient 6 (problem classes 7 to 10) made the

problems easier. It is well-known that the uncapacitated lot-sizing problem is easier










to solve capacitatedd lot-sizing problem, different than the uncapacitated problem is

NP-hard).

The results showed that problem classes 11 to 15 were quite challenging. As the

number of facilities increases we observed a monotonic increase in the duality gap.

The results S.-,--. -1. 1 that adding the facility selection dimension to the classical

multi-commodity lot-sizing problem has quite an effect on problem complexity.

The running times of all algorithms for problem classes 16 to 21 were the highest.

One of the reasons for this to happen is the size of the network for these problems

(Table 3-1). Setup costs appeared to have a significant impact on the duality gap

(problem classes 22 to 25). Problem class 22 presented an average error of 1.211 '.

compared to 6.4 !'. for the problem class 25. This result is not surprising since

increased setup costs widen the gap between (SP2), which ignores the setup costs and

(SP1). Moreover, since our original problem is a MILP with binary setup variables, as

the setup costs increase the problem behaves closer to a combinatorial problem than a

linear programming problem.

3.2 Single-Commodity, Multi-Retailer, Multi-Facility Lot-Sizing Problem

The satisfaction of the demand for products of a set of customers involves several

complex processes. In the past, this often forced practitioners and researchers to

investigate these processes separately. As mentioned by Erengiic et al. [30], the

traditional way of managing operations in a competitive market place l.-L--. -1. I that

companies competing on price will sacrifice their flexibility in offering new products or

satisfying new demands from their customers. The competition and the evolution of

hardware and software capabilities has offered companies the possibility of considering

coordinated decisions between processes in the supply chain.

In this section we propose a class of optimization models that consider the

integration of decisions on production, transportation and inventory in a dynamic

supply chain consisting of a number of retailers and facilities. We call this model

the single-commodity, multi-retailer, multi-facility lot-sizing problem. This model










estimates the total cost of a given logistics distribution network, including production,

inventory I'. lii.- and transportation costs. The evaluation is performed for a typical

planning period in the future.

This problem considers a set of plants where a single product type is produced.

This is a special case of the supply chain optimization problems we discussed

in Chapter 1. Chapter 4 considers the more general problem, where a number

of commodities are produced in the plants and the plants face production and

transportation capacity constraints.

3.2.1 Problem Formulation

In this section we consider a class of multi-facility, multi-retailer production-

distribution problems. Let R denote the number of retailers. Demand of retailer j in

period t is given by bjt. The unit transportation costs from facility i to retailer j in

period t are cijt. In this discussion we assume that transportation and inventory cost

functions are linear, and the production cost function is of the fixed charge type.

The multi-facility, multi-retailer problem can be formulated as follows:

F T R
minimize Y -(pitqit + sityit + hilit + i jtxijt)
i=1 t=1 j 1

subject to (\ilR)

R
qit + lit- = xijt + it i= 1,...F;t= 1,...,T (3.24)
j=1
F
xijt j= 1,...,R;t= 1,...,T (3 )
i=1
R
qit < bjtT-it i 1,..., F; t= 1,... ,T (3
j=1


xijt, lit, qit > 0 i ,. .. F;j = l,..., R; t 1,. T (3.27)

Yit {0,1} i= 1,... ,F; j= 1,...,R;t= 1,...,T (3 )










Decision variables xijt represents the quantity transported from facility i to retailer j

in time period t. Constraints (3.24) model the balance between the inflow, storage,

and outflow at facility i in period t. Constraints (3.25) make sure that retailer's

demand is satisfied. Constraints (3.26) relate the fixed and variable production costs

(the production can be initiated once the setup cost is paid). Figure 1-1 gives a

network representation of this problem.

The multi-retailer, multi-facility model we propose helps managers to answer

questions that arise in managing the production and distribution network. Obviously,

the most accurate answers are given when the problem is solved to optimality.

However, this is a difficult task since the problem we present is NP-hard. This

problem can be classified as a network flow problem with fixed charge cost functions.

Several special cases of the single-commodity network flow problem have been shown

to be NP-hard: for bipartite networks (Johnson et al. [63]), for single-source networks

and constant fixed-to-variable cost ratio (Hochbaum and Segev [60]), and the case of

zero variable costs (Lozovanu [76]). For the special case of this model when there is

only one retailer, Wu and Golbasi [106] show that the problem is NP-hard when the

holding costs are not restricted in sign.

We present the extended formulation of this problem as well. We do this by

splitting the variables qit by destination into variables qjt, (r = t, ..., T), where T

denotes the period and j represents the facility for which production takes place. The

split of the production variables implies the following:

R T
qd >- 3 > qjt (3.29)
j=l T=t

t
tijt (3.30)
s=1
t R t R t T t
ILt 1 j x CC ( qLST 1 jt) (331)
T=l J=l T=l j=l s=l T=t S=i









We can re-formulate problem (\!1) as follows:
F T R T
minimize >I[>I 1 cijtcqijt7r + sityit]
i=1 t=l j=l 7=t

subject to (Ex-MR)
F T-
ijtr =- b 1,...,R;T= 1,...T (3. )
i= 1 t= 1
qijtr < bjyit i 1,...,F;j 1,...R;t 1,...T;t
qijt > 0 i 1,...,F; 1,...,R;t ,...T;t
Et e {0,1} i= ,...,F;t= 1,...T (3 )

where cijt = pit + cij, + Y:=t+l his. The variable unit cost on the arcs of the extended

network consist of the production unit cost at facility i in period t, the transportation

unit cost from facility i to retailer j in period r, and the total unit holding cost at

facility i from production period t to the shipping period r.

Proposition 3.2.1 The optimal cost of linear ",'..i:,,,,,,:iI. relaxation of the extended

formulation of multi-f'a.: .:..; multi-retailer 1 I--..;:,. problem (Ex-MR) is at least as

high as the optimal cost of linear ,'.ull,,,,, .:,i relaxation of (. .:':,..l formulation (_ll:).

Proof: Every feasible solution to the linear programming relaxation of extended

formulation of multi-facility, multi-retailer lot-sizing problem (Ex-MR) can be

transformed to a solution to linear programming relaxation of formulation (\llR)

using equations (3.29), (3.30) and (3.31). It follows that the optimal solution of

linear programming relaxation of (Ex-MR) can be transformed to a feasible solution

(not necessary the optimal solution) to linear programming relaxation of (\!l ).

Computational results show in fact that the linear programming relaxation of (Ex-

MR) gives solutions that are close to optimal. O










3.2.2 Primal-Dual Algorithm

The dual of the linear programming relaxation of (Ex-MR) has a special

structure. Below we present the linear programming relaxation and the corresponding

dual of the formulation (Ex-MR).

F T R T
minimize >[ 1 CijtqijtrLT + sityit]
i= t=l j=l 7=t
subject to (LP-MR)


(3.32),(3.33),(3.34)

yit > O i= 1,...,F;t = 1,...T (3. )


The dual problem reads

T R
maximize Y Y1 bjtvjt
t=1 j=1

subject to (D-MR)


Y T 1'.-"'.,- < sit i= 1,...,F;t= 1,... T

Vj ,'.,- < cijt i =,..., F;t= ,...,T;t
Wi,,t- > 0 i= 1,...,F;j = 1,...,R;t= 1,...,T

t

In an optimal solution to (D-MR), both constraints ti.,t- > 0 and .,- >

vj. cijtT should be satisfied. Since t,- .,- is not in the objective function, we can

replace it with ', ,t- =max(0, vj,- Cjt.). This leads to the following condensed dual

formulation:
T R
maximize bijt1jt
t=1 j=1







87

subject to (D-MR*)
T R
Z bj max(0, vj, cijt) < Sit = 1, ..., F; t= 1, ... T.
T=t j=l

3.2.3 Intuitive Understanding of the Dual Problem

In this section we give an intuitive interpretation of the relationship between

primal-dual solutions of (Ex-MR). Suppose the linear programming relaxation of

(Ex-MR) has an optimal solution (q*,y*) that is integral. Let 0 = {(i, t) yi = 1} and

let (v*, w*) denote an optimal dual solution. The complimentary slackness conditions

for this problem are as follows:

(Ci) /, [.t E t '-" ,- 0 fori 1,...,F;t 1,...,T


(C2) ,7*.-[ + ',- 0 fori 1,...,F;j 1,...,R;

t= 1,...,T;t < < T

(C3) ,, ., ,-' ,, = 0 for i= ,...,F;j= 1,...,R;

t= 1,...,T;t < T < T

(C4) v*[, ., l q1 tTl] 0 for j l,...,R;t 1,...,T.

By conditions (C1), if a facility produces in a particular time period, the set-up cost

must be fully paid (i.e., if (i,t) E 0, then si = E 1 3 E '=ti-', -). Consider

conditions (C3). Now, if facility i produces in period t, but demand of retailer j in

that period is satisfied from the inventory from a previous period or from production

(inventory) at a different facility (qitt = 0 and qijtt '..', / 0), then ,, .,, 0. This

implies that the price paid for the product will contribute to set-up cost of only the

period when the product is produced.

By conditions (C2), ifjtr > 0, then vj- = -ijtr + .- Thus, we can think of v,

as the total cost (per unit of demand) in period 7 for retailer j. Of this amount, Cijt-










T for i 1,..., 1,..., 1,.
for t 1 to T do
for j 1 to -do
if bji, 0 then tiy, 0
else
j7 = 1,...,F:t for = 1 to 7 do
for i = 1 to F do
= max {0, v.j-- }
_= x{mx {,O M ,A-_ }
enddo
enddo
enddo
enddo

.ure 3: Dual algorithm


goes to 1p i for production and inventory holding costs, and i, ._ is the contribution

to the production set-up cost.

3.2.4 Outline of the Primal-Dual Algorithm

Suppose that the optimal values of the first k 1 dual variables of (D-MR)

are known. Let the index k be such that k = (7 1) R + 1, where 7 = 1,... ,T

and I 1,..., R. Then, to be feasible, the k-th dual variable (,.-) must satisfy the

following constraints:

bIT max(0, Ctilr) < Mrilt,-1 = Sit-

Efi EC~ bj max(0, v], cfts) E i bj max(0, j ijtT)

for all i = 1,..., F and t = 1,..., r. In order to maximize the dual problem we should

assign t.- the largest value satisfying these constraints. When, bi, > 0, this value is


min=mm {cit + ilt,- (3.38)
i=1,...,F;T>t blT

Note that if 3..,-_1 > 0 implies ,.- > cilt. The dual solution found using equation

(3.38) may not necessarily satisfy the complimentary slackness conditions. However,

a dual feasible solution can be obtained simply by calculating the value of the dual










0, 0, i 1 ... ; j k 1 T: > k
P {(j )" >0, for j 1..., k 1,...,1}
Start : T max k P, k 0
Step 1 : for i 1 to F do
for j 1 to do
repeat k = k I 1
until = 0 and cajk, 0jl 0
= 1, and i* = i, k' = k, go to Step 3
enddo
enddo
go to Step 3
S. 2 : for t k* to I do
for j 0 to do
if --jk*t VAjt c"t 0
then gi',jkt bj, P P- (j, t)
enddo
enddo
Step 3 : if P / 0 then go to Start

F: 3 4: Primal ;.


variables sequentially (Figure 3-3). A backward construction algorithm can then be

used to generate primal feasible solutions (Figure 3-4). For the primal-dual set of

solutions to be optimal, the complimentary slackness conditions should be satisfied.

Proposition 3.2.2 The solutions obtained with the primal and dual il'. -rithms are

feasible and they il;,.,r;, .'i:;fy the complimentary slackness conditions (C1) and (C2).

Proof: It is clear that the primal and dual solutions generated are feasible by

construction. Since the primal algorithm sets qijt, > 0 only when jtr vj. ., = 0,

the solution satisfies conditions (C2). The dual algorithm constructs solutions by

making sure that equation (2.23) is satisfied. Therefore, the dual solutions are ak--,v

such that bj,.+,, ., r+1 < i ... If if .,. = 0, then i, ., +1 = 0, and, since the dual

algorithm sets 3.1 .,+1 = .,-1 bj,+i, it.~r+1, we also have .i ., +1 = 0. Continuing

this way it is clear that if at some point in the calculation we get i .,. = 0, we

subsequently obtain












S... = .,, r+l ..=MiRtT 0

and

t' = t',t, r+l= WiRtT = 0

The primal algorithm sets yit = 1 only when ., = 0; this implies that conditions

(C1) will al--,-- be satisfied. o

Hence, one can determine whether the solution obtained with the primal and dual

algorithms is optimal by checking if conditions (C3) are satisfied, or if the objective

function values from the primal and dual algorithms are equal.

3.2.5 Computational Results

In order to test the performance of our algorithm, as in previous sections, we

randomly generated a set of test problems and compared their computation times and

solution quality to the general purpose solver CPLEX. We generated feasible solutions

to our problem using the primal algorithm and lower bounds using the dual algorithm.

The nominal case problem has the following characteristics:
Production set-up costs sit ~ U[200, 300]
Production variable costs pit U[5, 15]
Holding costs hit ~ U[5, 15]
Number of retailers R = 60

The transportation variable costs are generated the same way as described in

Section 2.8. Seasonal demands are randomly generated in the same way as presented

in Baker et al. [5] and Chen et al. [22].

2xr
bt = 200 + azt + a sin [ (t + d/4)]


In our test problems we take a = 67, a = 125 and d = 12.

The characteristics of the problem classes are presented in Table 3-5. For each

problem class we generate 20 problem instances and we report on the average error










bounds and average running times. The error bound for each problem instance is

calculated as follows: Error ( ,) Primal Solution-Dual Solution 100.
Dual Solution
Table 3-6 presents the error bounds from the primal-dual algorithm. We were

able to get the optimal solutions only for problem classes 1 and 2. For these two

problem classes linear programming relaxation of the extended formulation gave the

optimal solution. However, because of the the size of the problems (problem classes

3 to 12), CPLEX ran out of memory without providing either an integer feasible

solution or a lower bound. Results on Table 3-6 indicate that as the value of set-up

Table 3-5: Problemi characteristics
Problem Facilities Periods Nodes Arcs
1 24 ,21 ,4"
2 30 24 2.161
3 24 2,401 7
4 48 3,841 1,412,1
5 48 2,118,2.
6 48 4, 2 4,.
7 7i 1 5,
8 8,641 ,
9 9 11,178,2 -
10 192 15 1 ,4 .
11 1 17 1 1
12 192 19 1 44,474,,


cost increased, the problems became more difficult. This affected the performance of

the primal-dual algorithm.

For all problem classes, when set-up costs are uniformly distributed on the

intervals [200, 300] and [200, 900], the error gap was less than 0.n'.- and the running

times less than 141 cpu seconds. The maximum error reported is 4.07 :' This

corresponds to problem class 3 with set-up costs uniformly distributed in the interval

[1200, 1500]. Results in Tables 3-6 and 3-7 indicate that increasing the number of

facilities and the length of the time horizon affected the performance of the primal

and dual algorithm.









: Error bounds


We next randomly generated a second group of problems for which demand is

uniformly distributed in the following intervals: [20, 40], [40, 100], [100, 200], [200, 400]

and [400, 1000]. We rerun problem classes 4, 5 and 6 for each demand distribution

(problem classes 13 to 27). For example, for problem classes 13, 14 and 15, demand is

uniformly distributed in [20, 40] and the problem characteristics are the same as the

characteristics of problems 4, 5 and 6. This gives a total of 15 problem classes and

300 problem instances. CPLEX failed to solve problem classes 13 to 27. Table 3-8


Table 3-7: Running times (in


seconds) < pr


mal-dual heuristic


Set-Up )sts
Problem 1.. 1 ,-1.
1 0.2, 0.418 1.105 1. 2.5
2 0. 0.511 1. 2.519 3.
3 0.423 1.' 3.1 4.
4 0.2 0.415 1.119 1
5 0. 1. 2 2. 3
6 0 1.754 2.872 3
7 0.2 1.150 1 i 2.611
8 0. 0./ 1. 2.544 3.412
9 0.419 0. 1.810 7 4.
10 0 0.411 1.152 N/A 2.
11 0.341 0.522 1.5 N/A N/A
12 0.413 0.577 1. I N/A N/A


Set-Up Costs
Problem -! 1 1.
1 1.16 1." 1.18 1.32 1.14
2 1.58 1.87 1 1. 1.56
3 2 2. 2.34 2.
4 4.15 5.18
5 15 7.." 14 7.C -
6 8.1 9. 7., 9.14 7.
7 1 78 .45 1 19.29 11
8 15 .58 12 24.41
9 33.41 2
10 74. 74.1 N/A 70.42
11 1 1 1 1 N/A N/A
12 140.55 1 .35 1 '.57 N/A N/A


till


Table 3


-dual heuristic










presents the error and running times of the primal-dual heuristic for these problems.

The average running time of the primal algorithm was less than 5.22 cpu seconds

for all problem classes and less than 3.23 for the dual algorithm. Notice that the

error decreased as the average value of demand increased. For these problem classes,

everything else kept the same, an increase in demand affected the ratio of total

variable cost to total fixed cost. Increasing this ratio generally makes the problems

easier (Hochbaum and Segev [60]).

able Results '-dual heuristic

Error Time (sec)
Problem ( ) Primal Dual
13 3 2.54 1.
14 5. 3 2.41
15 5 5 3.18
16 1.34 2.54 1.
17 1.78 3.87 2.
18 2.18 5.22 3.21
19 0.43 2.54 1.
0 0 3 2.
21 0.73 5 3
22 0.14 2.55 1.
0.19 3.87 2.45
24 0 5.22 3.22
0.1 2.55 1.67
0.04 2.45
27 0.1 5.22 -


3.3 Multi Facility Lot-Sizing Problem with Fixed C'I irge Transportation Costs

In this section we discuss the uncapacitated multi-facility, multi-retailer problem

with fixed charge transportation costs and linear production and inventory costs.

Usually, when shipments are sent from a facility to a retailer, a fixed charge is paid

(for example, the cost of the paperwork necessary) to initiate the shipment plus a

variable cost for every unit transported. Therefore, modelling the transportation

cost function as a fixed charge cost function makes sense. The following is the MILP

formulation of the problem: