Mathematical Models of Competition with Explicit Cost Considerations in Supply Chains

MISSING IMAGE

Material Information

Title:
Mathematical Models of Competition with Explicit Cost Considerations in Supply Chains
Physical Description:
1 online resource (175 p.)
Language:
english
Creator:
Konur,Dincer
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Geunes, Joseph P
Committee Members:
Pardalos, Panagote M
Smith, Jonathan
Hamilton, Jonathan H

Subjects

Subjects / Keywords:
chain -- congestion -- considerations -- cost -- explicit -- facility -- game -- location -- supply -- traffic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In this dissertation, we first consider a competitive facility location problem when the competitive firms are identical. Competing firms simultaneously determine their facility location and distribution quantity decisions on a congested distribution network. Each firm may locate more than one facility simultaneously. We adopt a two-stage solution approach to analyze the resulting symmetric competitive multi-facility location problem. First, the firms? supply quantity decisions are solved, given that the firms have chosen identical facility locations. Then, we focus on the firms? facility location decisions, and explain why the firms choose identical facility locations. A heuristic solution method is proposed for determining high-quality solution for the first stage. Numerical studies are conducted to illustrate the efficiency of our heuristic method. We then use the aforementioned setting and analyses of the symmetric competitive multi-facility location game to analyze the effects of traffic congestion on supply chain activities. We utilize the model to provide analytical characterization of the effects of traffic congestion costs on equilibrium distribution flows. We present the results of extensive numerical studies to further illustrate the effects of traffic congestion costs on location, market supply quantity, and distribution decisions. Next, we study the competitive multi-facility location game with traffic congestion costs when the firms are non-identical. Similar to the symmetric case, we utilize a two-stage solution approach. However, heterogeneity of the competing firms requires distinct solution approaches in each stage. Particularly, firms? market-supply decisions for given facility locations are characterized using a variational inequality formulation. Then, a heuristic search method is provided for finding equilibrium locations, and its computational efficiency is compared to a random search method. Finally, we consider pricing decisions for a supplier who sells his/her product via a distributor who, in turn, serves small, localized retailers. The retailers? orders may be horizontally decentralized or centrally managed by the distributor, depending on the distributor?s procurement strategy. We model this problem as a Stackelberg game and determine its solution using backward induction: we first analyze the distributor?s procurement strategy and the retailers? order quantity decisions, then, the results of the retailers? quantity decisions are used to determine the supplier?s wholesale price.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Dincer Konur.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Geunes, Joseph P.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-02-29

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043158:00001


This item is only available as the following downloads:


Full Text

PAGE 1

MATHEMATICALMODELSOFCOMPETITIONWITHEXPLICITCOSTCONSIDERATIONSINSUPPLYCHAINSByDINCERKONURADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

PAGE 2

c2011DincerKonur 2

PAGE 3

Tomymom,RekaKonuranddad,CengizKonur 3

PAGE 4

ACKNOWLEDGMENTS First,Iwouldliketothankmyadvisor,Dr.JosephGeunes,forallhishelp,endlessmotivation,perfectguidance,andnon-stopsupport.Igratefullyappreciatehissupportanddetailededitsthatareineachandeverylineofthisdissertation.Hehasbeenagreatadvisor,mentor,reviewer,andteachernotonlyinthewritingofthisdissertationbutalsoineveryacademicexperienceIhavehad.Ifeelprivilegedtohaveworkedwithhimduringmydoctoralstudies.IwouldalsoliketothankDr.AysegulToptalforintroducingmeintothejoyofresearchandmotivatingmetopursueadoctoraldegree.Shehasbeenagreatadvisorandfriendthroughoutmydoctoralstudies.IwouldliketothankDr.EdwinRomeijnforgivingmetheopportunitytostudywithhimduringhistimeattheUniversityofFloridaandwritingrecommendationlettersforme.Iwouldalsoliketoacknowledgemydissertationcommitteemembers,Drs.J.ColeSmith,PanosPardalos,andJonathanH.Hamiltonfortheirinvaluablesuggestions.IthankDrs.J.ColeSmithandPanosPardalosforwritingreferencelettersduringmyjobsearchaswell.Myspecialthanksaretomyfriends,SezginAyabakan,AslhanKaratas,GulverKaramemis,GoncaYldrm,NailTanroven,andAtayKzlarslan,whohaveplayedarole,onewayoranother,inmylifeduringmytimeinGainesville.Finally,IwanttoexpressmysinceregratitudetomyparentsRekaandCengiz,mysiblingsAlperandBuse,andmygrandmotherSaizer.Theirlove,prayers,andsupporthavealwaysbeenwithme.Theirpresencewasthebiggestmotivationforme. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2ASYMMETRICCOMPETITIVEMULTI-FACILITYLOCATIONGAMEWITHCONVEXCONGESTIONCOSTS ......................... 17 2.1MotivationandLiteratureReview ....................... 17 2.2ProblemFormulationandSolutionApproach ................ 22 2.3Stage-TwoDecisions:Market-SupplyGame ................. 24 2.4Stage-OneDecisions:FacilityLocations ................... 31 2.4.1IdenticalLocationDecisions ...................... 31 2.4.2HeuristicMethodforIdentifyingLocationMatrix ........... 36 2.5NumericalStudies ............................... 39 2.5.1EfciencyofTheHeuristicMethod .................. 39 2.5.2AccountingforCongestioninDecisionMaking ............ 41 3ANALYSISOFTRAFFICCONGESTIONCOSTSINACOMPETITIVESUPPLYCHAIN ......................................... 46 3.1MotivationandLiteratureReview ....................... 46 3.2ModelandAnalysis .............................. 50 3.2.1Stage-OneDecisions .......................... 52 3.2.2Stage-TwoDecisions .......................... 53 3.3EffectsofTrafcCongestiononEquilibriumSupplyQuantities ....... 55 3.3.1ImplicationsforASpecialCase:FacilitiesLocatedwithinMarketAreas .................................. 59 3.4NumericalStudies ............................... 62 4COMPETITIVEMULTI-FACILITYLOCATIONGAMESWITHNON-IDENTICALFIRMSANDCONVEXTRAFFICCONGESTIONCOSTS ............ 73 4.1MotivationandLiteratureReview ....................... 73 4.2ProblemFormulationandSolutionApproach ................ 76 4.3Stage-TwoDecisions .............................. 79 4.4Stage-OneDecisions ............................. 84 4.4.1SearchingforAnEquilibriumLocationMatrix ............ 85 5

PAGE 6

4.4.2GeneratingAViableLocationDecision ................ 87 4.4.3EquilibriumCheck ........................... 89 4.4.4HeuristicAlgorithmforFindingAnEquilibriumLocationDecision 92 4.5Extensions:Multi-ProductandMulti-EchelonChannels .......... 93 4.6NumericalStudy ................................ 96 5SUPPLIERWHOLESALEPRICING:IMPLICATIONSOFDECENTRALIZEDVS.CENTRALIZEDPROCUREMENTUNDERQUANTITYCOMPETITION .. 101 5.1Motivation .................................... 101 5.2LiteratureReview ................................ 106 5.3ProblemFormulationandMethodology ................... 111 5.4RetailStage:SupplyQuantitiesandProcurementStrategy ........ 114 5.4.1DecentralizedRetailing ......................... 115 5.4.2CentralizedRetailing .......................... 117 5.4.3PartiallyCentralizedRetailing ..................... 119 5.4.4ComparisonofProcurementStrategies ................ 120 5.5TheSupplier'sProblem:OptimalWholesalePrice ............. 122 5.5.1WholesalePricingforDecentralizedRetailing ............ 122 5.5.2WholesalePricingforCentralizedandPartiallyCentralizedRetailing 128 5.6Extensions:MultipleMarketsandDiscountPricing ............. 128 5.7NumericalStudy ................................ 130 6CONCLUSIONANDFUTURERESEARCHDIRECTIONS ............ 141 6.1CompetitiveMulti-FacilityLocationProblemswithCongestionCosts ... 141 6.2TrafcCongestionandSupplyChainManagement ............. 144 6.3PricingforCompetitiveRetailers ....................... 145 APPENDIX ASYMMETRYOFEQUILIBRIUMSUPPLYQUANTITIESGIVENIDENTICALFACILITYLOCATIONS ............................... 149 BMIXEDSTRATEGYNASHEQUILIBRIUMFORSYMMETRICLOCATIONGAME ......................................... 154 CSOLUTIONOFDECENTRALIZEDRETAILINGUNDERGENERALIZEDMARKETPRICEANDOPERATINGCOSTFUNCTIONS .................. 156 DCOMPARISONOFTOTALORDERQUANTITIESUNDERDIFFERENTRETAILINGSTRATEGIES ..................................... 161 REFERENCES ....................................... 163 BIOGRAPHICALSKETCH ................................ 175 6

PAGE 7

LISTOFTABLES Table page 2-1Datarangesforproblemclasses1-8foranalysis1 ................ 40 2-2Comparisonoftotalenumerationand2-phaseheuristicmethod ......... 40 2-3Comparisonoftotalenumerationand2-phaseheuristicmethodforeachm .. 41 2-4Datacategoriesforproblemclasses1and2 ................... 43 2-5Statisticsofcases(i)and(ii)forproblemclass1 ................. 44 2-6Statisticsofcases(i)and(ii)forproblemclass2 ................. 44 3-1Dataintervalsforproblemclasses1-4 ....................... 62 3-2Averagestatisticsoverproblemclasses1-4foreachinterval .......... 63 3-3Dataintervalsforproblemclasses1-4 ....................... 66 3-4Averagestatisticsoverproblemclasses1-4foreachinterval .......... 67 3-5Averagenumberoftimesrmslocatedfacilitiesinmarkets ........... 69 3-6Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=3(M:market) 70 3-7Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=5(M:market) 70 3-8Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=7(M:market) 71 3-9Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=10(M:market) 71 4-1Dataintervalsforproblemclasses1-8 ....................... 98 4-2Comparisonofheuristicmethodwithrandomsearchmethod .......... 99 4-3Comparisonofheuristicmethodwithrandomsearchmethodforeachproblemclass .......................................... 100 5-1Supplier'sprot .................................... 132 5-2Retailers'totalprot ................................. 133 5-3Payoffmatrix ..................................... 134 5-4Channelprot ..................................... 134 7

PAGE 8

LISTOFFIGURES Figure page 3-1Patternsofeachcolumnintable 3-2 ........................ 64 3-2Patternsofeachcolumnintable 3-4 ........................ 68 5-1Illustrationsof`(c)andQ(c) ............................ 125 5-2Effectsofretailparameters ............................. 137 5-3Effectsofmarketparameters ............................ 138 5-4Effectsofsupplierparameters ............................ 139 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMATHEMATICALMODELSOFCOMPETITIONWITHEXPLICITCOSTCONSIDERATIONSINSUPPLYCHAINSByDincerKonurAugust2011Chair:JosephGeunesMajor:IndustrialandSystemsEngineering Inthisdissertation,werstconsideracompetitivefacilitylocationproblemwhenthecompetitivermsareidentical.Competingrmssimultaneouslydeterminetheirfacilitylocationanddistributionquantitydecisionsonacongesteddistributionnetwork.Eachrmmaylocatemorethanonefacilitysimultaneously.Weadoptatwo-stagesolutionapproachtoanalyzetheresultingsymmetriccompetitivemulti-facilitylocationproblem.First,therms'supplyquantitydecisionsaresolved,giventhatthermshavechosenidenticalfacilitylocations.Then,wefocusontherms'facilitylocationdecisions,andexplainwhythermschooseidenticalfacilitylocations.Aheuristicsolutionmethodisproposedfordetermininghigh-qualitysolutionfortherststage.Numericalstudiesareconductedtoillustratetheefciencyofourheuristicmethod. Wethenusetheaforementionedsettingandanalysesofthesymmetriccompetitivemulti-facilitylocationgametoanalyzetheeffectsoftrafccongestiononsupplychainactivities.Weutilizethemodeltoprovideanalyticalcharacterizationoftheeffectsoftrafccongestioncostsonequilibriumdistributionows.Wepresenttheresultsofextensivenumericalstudiestofurtherillustratetheeffectsoftrafccongestioncostsonlocation,marketsupplyquantity,anddistributiondecisions. Next,westudythecompetitivemulti-facilitylocationgamewithtrafccongestioncostswhenthermsarenon-identical.Similartothesymmetriccase,weutilizeatwo-stagesolutionapproach.However,heterogeneityofthecompetingrmsrequires 9

PAGE 10

distinctsolutionapproachesineachstage.Particularly,rms'market-supplydecisionsforgivenfacilitylocationsarecharacterizedusingavariationalinequalityformulation.Then,aheuristicsearchmethodisprovidedforndingequilibriumlocations,anditscomputationalefciencyiscomparedtoarandomsearchmethod. Finally,weconsiderpricingdecisionsforasupplierwhosellshis/herproductviaadistributorwho,inturn,servessmall,localizedretailers.Theretailers'ordersmaybehorizontallydecentralizedorcentrallymanagedbythedistributor,dependingonthedistributor'sprocurementstrategy.WemodelthisproblemasaStackelberggameanddetermineitssolutionusingbackwardinduction:werstanalyzethedistributor'sprocurementstrategyandtheretailers'orderquantitydecisions,then,theresultsoftheretailers'quantitydecisionsareusedtodeterminethesupplier'swholesaleprice. 10

PAGE 11

CHAPTER1INTRODUCTION Efcientsupplychainmanagementrequireseffectivedecisionmakinginthreemainstages:SupplyChainDesign,SupplyChainPlanning,andSupplyChainOperations.IntheSupplyChainDesignphase,rmsdecideonthebasicstructureoftheirsupplychainsandsetstrategicgoalsthatwillbeineffectoverarelativelylongtime.TheSupplyChainPlanningphasefocusesonutilizingthegivencongurationstoincreaseprotabilitywhileSupplyChainOperationsensurethattheprocessesrequiredtofunctioninthechain,giventhedesignandplans,areperformedeffectively.Intoday'scompetitiveindustry,itiscrucialforrmstodesign,plan,andoperatetheirsupplychainsasefcientlyaspossible.Anidealsupplychainmanagementpolicywouldtakeintoaccounteveryfactorthatcanaffectthesuccessofthesupplychain. Competitionisafactofthebusinessworld,anditisakeyfactorthatshouldberegardedinsupplychaindecisions.Modelingacompetitivermasamonopolyisoftenunrealisticandcanbefatalifusedtomakedecisions.Inthisdissertation,weconsidercompetitivesupplychainmanagementproblems,whichexplicitlyaccountfortheinherentcompetitionwithinthebusinessworld.Competitionamongrmsmainlyprevailsinsituationswhenthereexistsoneormoreresourcesthatmustbesharedbyasetofnoncooperativerms.Anendcustomermarketforaproductorservice,forinstance,isaresourcesharedbyrmsprovidingthesame(orperfectlysubstitutable)productorservicetothismarket.Thelimitedconsumptioninendcustomermarketsisacommondriverofcompetition;however,itisnottheonlysourceofcompetition.Transportationnetworksandlimitedsuppliesareoftensharedbynoncooperativermsand,hence,motivateadditionalcompetition.Firmsshouldthusdesign,plan,andoperatetheirsupplychainswhileexplicitlyconsideringanypossiblecompetitivefactorspresentinthesystem. 11

PAGE 12

Inadditiontoweighingcompetitionintheirdecisions,itisalsoimportantforrmstoaccountforexplicitcosttermsintheirdecisionmakingprocessesfortrulyoptimizingtheirsupplychainsviamoreaccuratereectionofreal-lifescenarios.InChapters 2 4 ,wemodelexplicittransportationcosts,whicharecompoundedbytrafccongestioncosts.Recentstudiesattheintersectionoftrafccongestionandsupplychainmanagementempiricallydemonstratethenegativeeffectstrafccongestionhasonsupplychainperformance.Specically,giventhatanunderlyingdistributionnetworkissharedbymultiplerms,recognizingtrafccongestion,whichhinderslogisticalefciency,isessential.Furthermore,trafccongestioncostmodelingenablesrepresentationoftheaforementionedcompetitiononthetransportationnetwork.Therefore,weexplicitlymodeltrafccongestioncostsintheproblemsofinterestinChapters 2 4 Facilitylocationandsupplyquantity(orequivalentlyproductionlevel)decisionsareimportantinthedesignandplanningofaourishingsupplychain.Moreover,asnotedabove,competitionandexplicitcostmodelingshouldnotbedisregardedinanydecisionphaseofsupplychains.InChapters 2 4 ,weanalyzeasetofcompetitivefacilitylocationproblemswithtrafccongestioncosts.Classicalfacilitylocationproblemsfocusonanalysesofasingledecisionmaker'sfacilitylocationchoicesintheabsenceofcompetitivefactors.Competitivefacilitylocationproblemsextendtheclassicalfacilitylocationproblemsbydirectlyaccountingfortheeffectsofcompetitionpresentinthesystem.Inparticular,theseproblemsformulatefacilitylocationandassociatedsupplyquantityorpricedecisionsofasetofrms,whocompeteinordertoserveasetofendcustomermarkets.Chapters 2 4 modelcompetitivefacilitylocationproblems(alsoreferredtoasafacilitylocationgameorsimplylocationgame),inwhichasetofrmsisengagedinCournottypequantitycompetitionwithinmultipleendcustomermarkets.Thepotentialfacilitylocationandcustomermarketsarerepresentedbyanitenumberofnodesofaconnectednetwork;thatis,weconsiderlocationgamesonadiscrete 12

PAGE 13

network.Thereisalimitednumberofstudiesthatexaminelocationgamesondiscretenetworks(mostofthecompetitivelocationproblemsassumespatialcompetition).Moreover,thesestudieshavetherestrictiveassumptionthatthecompetingrmslocateatmostonefacility.Fromapracticalpointofview,economiesordiseconomiesofscalemayforcermstolocatemorethanonefacilityeventosupplyasinglecustomermarket.Thus,inouranalyses,thermsareallowedtolocatemultiplefacilitiessimultaneously.Additionally,weconsidernonlineartrafccongestioncostsintheclassoflocationgamesanalyzedinthisdissertation,whichmotivatesrmstolocatemultiplefacilities. Inparticular,werstanalyzeasymmetriccompetitivemulti-facilitylocationgamewithtrafccongestioncostsinChapter 2 .Thecompetingrmsarerecognizedasidentical(orhomogeneous),andthatiswhywedenethisproblemclassassym-metric.Eachrm'sobjectiveistomaximizetheirprotsbydeterminingtheirfacilitylocationsandsupplyquantitiesfromeachfacilitytoeachmarketwhiletakingintoaccountthecompetitioninthemarketsaswellasthecompetitiononthecommondistributionnetworkduetotrafccongestion.Weuseatwo-stagesolutionapproachinouranalyses.Atwo-stagesolutionapproachrstdeterminestheequilibriumsupplyquantities(secondstagedecisions)giventherms'facilitylocationchoices.Then,thesolutionofthesecondstageisutilizedinndingtheequilibriumfacilitylocations(rststagedecisions).Inapplyingthetwo-stagesolutionapproachtothesymmetricmulti-facilitylocationgameunderconsiderationinChapter 2 ,werstsolveforrms'equilibriumsupplyquantitydecisionsundertheassumptionthatthermshavechosenidenticalfacilitylocations.Itisshownthattheequilibriumsolutionofthesecondstageissymmetric,i.e.,rmswillsupplythequantitiestothemarketsinequilibrium,giventhattheyhavelocatedfacilitiesatthesamelocations.Wethenturnourattentiontotherststagedecisions,andinvestigatetherationalebehindidenticalofrmlocationchoices.Followingthis,aheuristicmethodisproposedtodetermineasetoflocationswhereallrmswilllocateafacility.Thisheuristicmethodrankslocationswithrespectto 13

PAGE 14

specicparametersandisintendedtomimicanindividualrm'sapproachtothefacilitylocationdecisionproblem.Wedemonstratetheefciencyoftheheuristicmethodviaournumericalstudiesanddiscussacounter-intuitiveobservation. Includingtrafccongestioncostsinthesymmetriccompetitivemulti-facilitylocationproblemofChapter 2 notonlyallowssubstantiationofcompetitiononthedistributionnetworkbutalsoenablesanalysesoftheeffectsoftrafccongestiononsupplychainperformance.Therearepaststudiescombiningtrafccongestionandsupplychainmanagementintheliterature.Nevertheless,thesestudieshavethefollowingthreemajordrawbacks:(i)theanalysesaremostlybasedonempiricaldataand,hence,lacktheoreticalresults,(ii)congestioncostsareconsideredexogenously,and(iii)thecompetitivenatureoftrafccongestionisignored.InChapter 3 ,ouraimistoutilizethemodeldescribedinChapter 2 toprovideacompleteanalysisontheeffectsoftrafccongestiononsupplychainactivities.Ourapproachovercomesthepreviouslymentioneddrawbacksaswemodeltrafccongestioncostsendogenouslyinsuchawaythatthecompetitivenatureoftrafccongestioniscaptured.Furthermore,wemanagetoanalyticallycharacterizehowchangesintrafccongestioncostsaffectequilibriumowdecisions.Theseanalyticalresultsfurthergrantqualitativecharacterizationoftheeffectsoftrafccongestionlevelsonrms'locationanddistributiondecisions.Chapter 3 alsosummarizesextensivenumericalstudiesconductedtofurtherillustratetheeffectsoftrafccongestioncostsonlocation,marketsupplyquantity,anddistributiondecisions. Whileourworkonthesymmetricmulti-facilitylocationproblemwithtrafccongestioncostscontributestothetheoryoflocationgamesbystudyingadiscretecompetitivemulti-facilitylocationgame(andtothesupplychainmanagementandtransportationliteraturesbyenablingdetailedanalysesoftheeffectsoftrafccongestioncostsonsupplychainmanagement)wearenaturallyinterestedinthenuancesofasymmetriccompetitivemulti-facilitylocationproblems.Specically,itmaybethecasethatcompetingrmsutilizedifferenttechnologiesandvaluecongestiondifferently, 14

PAGE 15

whichresultsinheterogeneityoftherms.Chapter 4 studiesaclassofasymmetriccompetitivemulti-facilitylocationproblemswithtrafccongestioncosts.Similartothesymmetriccase,weadoptatwo-stagesolutionapproach;nevertheless,theasymmetryofthermsnecessitatesdiverseanalysesineachstage.Inparticular,weutilizethewellknownresultthatstatesthatnoncooperativegamescanbeformulatedasvariationalinequalityproblemsundercertainconcavityconditions.Firms'supplyquantitydecisionscanbeformulatedasanasymmetriclinearvariationalinequalityproblem,andwediscussaself-adaptiveprojectionmethodasasolutiontoolforthesecondstageoftheasymmetriccompetitivemulti-facilitylocationproblem.Thechallengingpartliesintheanalysisoftherststagegame,i.e.,searchingforequilibriumfacilitylocationchoices.Werstfocusondeningpropertiesofequilibriumfacilitylocationchoicesandproposeroutinesthatareintendedtoeasethesearchprocess.Then,theseroutinesareembeddedinaheuristicsearchmethod.Finally,wecomparetheefciencyoftheheuristicmethodtoarandomsearchalgorithm. Analysesofthesymmetricandasymmetriccompetitivemulti-facilitylocationproblemscompleteourstudyonmulti-facilitylocationgamesondiscretenetworks.Theseproblemsdealwithcompetitionwithinasingleechelonofasupplychain.InChapter 5 ,weturnourattentiontoamulti-echeloncompetitivesupplychain.Inparticular,anagentinanupperlevelechelonofasupplychainmustacknowledgethecompetitionamongthepartiesinalowerlevelechelon.Forinstance,itisacommonpracticethatasuppliersellshis/herproducttoasetofretailers,whothenselltheproductintheendcustomermarket.Theseretailerscompetewithintheendcustomermarket,anditisimportantforthesuppliertoacknowledgethiscompetitionamongtheretailersindetermininghis/herwholesaleprice,i.e.,sellingpricetotheretailers. Chapter 5 investigatesatwo-echeloncompetitivesupplychainproblem.Specically,weareinterestedinthewholesalepricesettingproblemofasupplierwhosellshis/herproductviaadistributorwho,inturn,suppliesasetofretailers.Theretailersare 15

PAGE 16

competitiveintheendcustomermarket,however,theycanbecentralizedbythedistributordependingonthedistributor'sprocurementstrategy.InChapter 5 ,wemodelaStackelberggametoanalyzethesupplier'swholesalepricesettingproblem,thedistributor'sprocurementstrategy,andtheretailers'orderquantitydecisions.ThesupplieristheleaderoftheStackelberggameandthedistributorandtheretailersarethefollowers.WeusebackwardinductiontosolveforaStackelbergequilibrium:rstthedistributor'sprocurementstrategyandtheretailers'orderquantitydecisionsareanalyzed;then,theseanalysesareusedtosolvethesupplier'swholesalepricesettingproblem.Thesupplierdoesnothavecontroloverthedistributor'sprocurementstrategy,whichaffectsthetotalorderquantitydemandedfromthesupplier.Hence,thesuppliercanachievesubstantialsavingsifs/hecancontrolthedistributor'sprocurementstrategy.Wedenethesupplier'spotentialsavingswhens/hehascontroloverthedistributor'sprocurementstrategyasthevalueofthecontrolforthesupplier.Weconductnumericalstudiestoquantifythevalueofcontrol. Finally,inChapter 6 ,weconcludethisdissertationbysummarizingourcontributions.WefurtherdiscussfutureresearchdirectionsrelatedtoChapters 2 5 .Inthisdissertation,weintroducecompetitivemulti-facilitylocationproblemswithtrafccongestioncostsandprovidedetailedanalysesoftheseproblems.Mathematicalformulationoftrafccongestionproblemswithexplicitconsiderationofcompetitionhelpsusanalyticallycharacterizetheeffectsoftrafccongestiononrms'facilitylocationandsupplyowdecisionsunderrealisticsettings.Furthermore,westudyatwo-echeloncompetitivesupplychainanddiscusstheimplicationsofcentralizedvs.decentralizedretailingonthechannelperformance.Distinctmanagerialinsightsaregainedthroughouranalyses. 16

PAGE 17

CHAPTER2ASYMMETRICCOMPETITIVEMULTI-FACILITYLOCATIONGAMEWITHCONVEXCONGESTIONCOSTS 2.1MotivationandLiteratureReview Facilitylocationproblemshavebeenextensivelystudiedintheliterature.Mostofthepastoperationsresearchstudiesonfacilitylocationtheoryfocusonformulatingasingledecisionmaker'sproblemintheabsenceofcompetitivefactors.Thisstreamofresearchisdiscussedinthefacilitylocationbooksby Drezner ( 1995 )and DreznerandHamacher ( 2002 ),andthereviewpapersby HaleandMoberg ( 2004 ), OwenandDaskin ( 1998 ),and Tanseletal. ( 1983 ),aswellasthereferencescontainedtherein.Asnotedby Plastria ( 2001 ),anassumptionofnocompetitionisoftenimpractical. Rhimetal. ( 2003 )alsoobservethatlocationcompetitionisanimportantfactorincompetitivesupplychains.Asaresult,anotherstreamofresearchfocusesonfacilitylocationproblemsundercompetition.Theproblemsstudiedwithinthisresearchstreamcomprisethefundamentalsofcompetitivelocationtheory.Inthisproblemclass,rms'locationdecisions(alongwithotherstrategicdecisions,suchaspricingdecisions,supplyquantitydecisions,orcapacitydecisions)arestudiedbyapplyingcompetitiveequilibriumtoolsandconcepts. Theclassicalstudyof Hotelling ( 1929 )introducestherstcompetitivelocationproblem.Inthisstudy,twormscompeteinamarketandeachwishestomaximizeitsmarketshareunderademandinelasticityassumption. Smithies ( 1941 )considersthesameproblemwithdemandelasticity. Teitz ( 1968 )extendsHotelling'sproblembyallowingrmstolocatemorethanonefacility.Followingthesebasicstudies,competitivelocationproblemshavebeenstudiedunderdifferentsettingsintheliterature.Thesesettingsdifferintheirassumptionsonthenumberofcompetingrms(twormsversusmoregeneralmultiplermproblems),thenumberofstrategicdecisions(facilitylocations,productpricing,supplyquantitiesandfacilitycapacities),andthenatureofthecompetitionandstrategicgame(sequentialfacilitylocationdecisions,simultaneous 17

PAGE 18

locationdecisions,anddecisionswhenfacilitiesalreadyexistatsomelocations).Thereadermayreferto EiseltandLaporte ( 1996 ), Eiseltetal. ( 1993 ),and Plastria ( 2001 )forreviewsofcompetitivefacilitylocationproblemsunderdifferentassumptions. Inthischapter,weaddressacompetitivefacilitylocationproblemthatconsidersfacilitylocationandmarket-supplyquantitydecisionsforasetofhomogeneousrmssellingaproduct.Inparticular,theproblemwestudyisasymmetriccompetitivefacilitylocationgame.Multiplecompetitivermssellingacommonproducttypearenoncooperativeandhomogeneous,i.e.,theyincuridenticalmarginaldeliverycostsinservingamarketfromthesamelocation(i.e.,fromsupplyfacilitieslocatedwithinthesamecityortown).Firmsaresubjecttotransportationandcongestioncostsasaresultofdeliveriestomarkets,aswellasxedfacilitylocationcostsformaintainingsupplyfacilities.Weassumethatthemarketsandthepotentialfacilitylocationsarerepresentedasverticesofaconnectednetwork.Firmsmustsimultaneouslydeterminetheirsupplyfacilitylocations(stage-onedecisions)andthequantitiestheywillshipfromthesefacilitiestoeachmarket(stage-twodecisions).Wethusadoptatwo-stagesolution:werststudythestage-twodecisions,giventhatthermshaveidenticalfacilitylocationsand,then,wefocusonthestage-onedecisionsandcharacterizetherms'facilitylocationdecisions.Similartwo-stagesolutionapproachesareusedintheanalysisofcompetitivefacilitylocationproblemsby LedererandThisse ( 1990 ), LabbeandHakimi ( 1991 ), Sarkaretal. ( 1997 ), PalandSarkar ( 2002 ), Rhimetal. ( 2003 ),and SaizandHendrix ( 2008 ). Duringthestage-onedecisions,rmsareallowedtolocatemorethanonefacility,andeachfacilityisassumeduncapacitated.Inthestage-twodecisions,weassumethatthermsareengagedinCournotcompetitioninmultiplemarkets.CournotcompetitionandBertrandcompetitionarethemostcommonconceptsusedinmodelingcompetitivemarkets.InCournotmodels,thecompetitionisquantitybased,whereasBertrandmodelsarebasedonpricecompetition.Quantitycompetitionisjustiedforindustries 18

PAGE 19

wheretheproductiondecisionsaremadebeforeactualsalesbegin.Aspointedoutby Hamiltonetal. ( 1994 ),technologyisonereasonwhyrmsmayexperiencealagbetweenproductionandsales.Insuchcases(e.g.,whenrmschooseproductioncapacitiesbeforeactualproduction),whenoutputlevelsarenotadjustableintheshortrun,thecurrentquantityinthemarketwilldeterminetheprice. Hamiltonetal. ( 1994 )notethat,givenproductiondecisions,market-clearingpricescanbeset.Ontheotherhand,inBertrandmodels,competitivermsshouldhaverobustnessintheirproductionsystemsinordertoadjustoutputlevelsinresponsetoprice-sensitivedemand.Furthermore,whenrmscompeteforsalesofidenticalproducts,Cournotcompetitionmayprovideabetterrepresentationofthemarket,sincermswillfocusondeterminingproductionoutputratherthansettingpricesforhomogeneousproducts(perfectsubstitutes).Onemayreferto KrepsandScheinkman ( 1983 ), AndersonandNeven ( 1991 ),and Hamiltonetal. ( 1994 )forfurtherdiscussionsonthejusticationofCournotcompetition.Inthischapter,rmsareassumedtobehomogeneousandtheysellhomogeneousproducts;hence,weuseCournotcompetitiontomodelthemarkets.Inparticular,theCournotmodeldenedin Hamiltonetal. ( 1994 )assumesthatrmssetmillprices,i.e.,spatiallyseparatedmarketshavethesameprice,andcustomerspayfortransportationcosts.Ontheotherhand,weassumethatthetransportationcostsarepaidbycompetingrmsanddifferentmarketsmayobservedifferentprices,i.e.,theCournotmodelofinterestinthischapteristheso-calleddeliveredCournotcompetition,denedsimilartotheCournotmodelof Hamiltonetal. ( 1989 ).WeusethiscompetitionmodelinChapters 3 4 ,and 5 aswell. WenotethatcompetitivelocationproblemsarecommonlymodeledusingCournotcompetitionintheliterature.SpatialcompetitionoftwormsunderCournotcompetitionisstudiedby LabbeandHakimi ( 1991 ).Thisstudywasextendedtomultiplermsby Sarkaretal. ( 1997 ).Bothofthesestudiesassumethatrmslocateasinglefacility.Conversely, PalandSarkar ( 2002 )considerspatialcompetitioninaCournotduopoly 19

PAGE 20

wherethecompetingrmsmaylocatemorethanonefacility.Thedistinguishingassumptionofthesestudiesisthatcompetingrmsentereachmarketbysupplyingapositivequantitytoeachmarket.Theworkin Rhimetal. ( 2003 )and SaizandHendrix ( 2008 )relaxesthisassumptionandconsidersthecaseoffreeentry.Inbothofthesestudies,Cournotcompetitionexistsandrmschoosethelocationoftheirsinglefacilityandthequantitytheywillsupplyfromthisfacilitytoeachmarket,iftheychoosetoenteranymarket.Marketsandpotentialfacilitylocationsarelocatedontheverticesofanetwork.Itshouldbenotedthatdeningpotentialfacilitylocationsasverticesofanetworkismorepracticalandparallelstheresultsof LabbeandHakimi ( 1991 ), LedererandThisse ( 1990 ),and Sarkaretal. ( 1997 ),whichstatethatequilibriumfacilitylocationstendtobeontheverticesofanunderlyingnetworkunderspatialcompetition.Whileahomogeneouscoststructureisassumedby Rhimetal. ( 2003 ), SaizandHendrix ( 2008 )studyaheterogeneouscoststructure.Theproblemconsideredinthischapterappliessimilarassumptionsasthoseof Rhimetal. ( 2003 )and SaizandHendrix ( 2008 );however,rmsareallowedtolocatemorethanonefacility,andrmsaresubjecttononlineartrafccongestioncosts(foradiscussionontheeffectsoftrafccongestiononsupplychainactivitiessee,e.g., McKinnon 1999 Raoetal. 1991 Sankaranetal. 2005 Weisbrodetal. 2001 ,and KonurandGeunes 2011 ). Werstfocusonthestage-twodecisions,giventhatthermschooseidenticalfacilitylocations.Inthisstage,werstshowthattherms'supplyquantitydecisionscorrespondtoasymmetricpurestrategyNashequilibrium,orsimplyPureNashEquilibrium(PNE)solution,giventhatthermschooseidenticalfacilitylocations.Usingthisresult,weshowthatthegameamongthermsactuallyreducestoagameofthelocations.Inparticular,thegameamongthermsissimilartothegamedenedin Rhimetal. ( 2003 ),whichassumeshomogeneousrms;however,inourmodel,rmsmaylocatemorethanonefacility.Ourresultsthenimplythatthisgamecanalsobereducedtoagameoflocations,resultinginanasymmetricgamesimilartotheone 20

PAGE 21

studiedin SaizandHendrix ( 2008 ),whichdoesnotconsidernonlineartrafccongestioncostsaswedointhischapter.Therefore,wecharacterizethepropertiesofthisreducedgameandprovideanexactalgorithmtondthePNEsolution.Then,westudytherms'stage-onedecisionsandexplaintherationalebehindtheassumptionthatrmschooseidenticalfacilitylocations.WenotethatrmschooseidenticalfacilitylocationsinthecaseofauniquePNElocationdecision.However,whenauniquePNEsolutiondoesnotexist,sincetheequilibriumconceptdoesnotcharacterizewhatrmswillactuallydo,weusethemaximizationofexpectedprotsasanobjective,assumingthatanylocationdecisionisequallylikelyforeachrm.WeshowthatamixedstrategyNashEquilibrium(MSNE)impliesthatitisequallylikelyforanyrmtochooseanygivenlocationdecision.Thus,whenrmsarehomogeneous,theywillendupwithidenticalfacilitylocations,andtherefore,westudytheoptimallocationdecisionsetfortheindividualrm. Particularly,aheuristicsolutionmethod,basedonhowanindividualrmmightapproachthelocationdecisionproblem,isproposedfordetermininghigh-qualitylocationdecisionsfortherms.Themethodisatwophasemethodandisintendedtomimichowanindividualrmmightplanitsfacilitylocationstrategyinpractice.Wediscussresultsofextensivenumericalstudiestoshowtheefciencyoftheheuristicmethod.Moreover,inournumericalstudies,weobserveacounter-intuitiveexample,wherethermsmaybebetteroffwhentheyignorecongestioncostsintheirdecisions. Ourworkcontributestotheliteraturebyextendingtheworkof Rhimetal. ( 2003 )and SaizandHendrix ( 2008 )toallowcompetitivermstolocatemorethanonesupplyfacility,aswellasbyexplicitlyconsideringnonlineartrafccongestioncosts,asrmsshareacommondistributionnetwork.Weestablishimportantresultsforcharacterizingthestage-two(marketsupply)decisionsthatwillbeusedinanalyzingtrafccongestioneffectsinacompetitivesupplychaininChapter 3 Therestofthischapterisorganizedasfollows.Section 2.2 describesthedetailedproblemsettingandsolutionapproach.InSection 2.3 ,wecharacterizethepropertiesof 21

PAGE 22

theequilibriumsupplyquantitiesandprovideasolutionalgorithm,giventhatrmsmakeidenticalfacilitylocationdecisions.Section 2.4 focusesonfacilitylocationdecisions.Weexplaintherationalebehindtheassumptionthatrmschooseidenticalfacilitylocations,andprovideatotalenumerationschemeandaheuristicmethodtocharacterizefacilitylocationdecisions.InSection 2.5 ,theresultsofextensivenumericalstudiesarediscussedtocharacterizetheefciencyoftheheuristicmethod,andtheimpactsofaccountingforcongestioninthedecisionmakingprocess. 2.2ProblemFormulationandSolutionApproach Considerasetofkcompetitiverms,indexedbyr2R=f1,2,,kg,whowishtodeterminethelocationsoffacilitiesatmpossiblelocations,indexedbyi2I=f1,2,,mg,aswellasthesupplyquantitiesfromthesefacilitiestoeachofncustomermarkets,indexedbyj2J=f1,2,,ng.Thesupplyrmsincurlineartransportationandconvexcongestioncostsasaresultoftheirsupplyquantitydecisions.Additionally,rmsaresubjecttoxedfacilitylocationcoststhatdependontheirlocationdecisions.Weassumethatrmsmaylocatemorethanonefacility;however,anyfacilityultimatelylocatedisassumedtobeuncapacitated.Therefore,noneofthermswilllocatemorethanonefacilityataspeciclocation. FirmsareengagedinCournotcompetitioninmultiplemarkets.Theunitpriceinanymarketisdeterminedbythetotalquantitysuppliedtothatmarket.Inparticular,theunitpriceinmarketjisdenedasalinearanddecreasingfunctionofthetotalsupplyquantityintomarketj,qj.Letpjdenotethepriceinmarketj.Then pj(qj)=aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj,(2) whereaj0andbj>0.Theparameterajcanbeinterpretedasthemaximumdemandortheconsumptioncapacityinmarketj.Theparameterbjcanbeinterpretedasthepricesensitivityofmarketj.(Inparticular,alllineardemandcurvescanberepresentedasp=1)]TJ /F3 11.955 Tf 12.37 0 Td[(q,wherepdenotesthedeliveredprice,qdenotesthedemand(orquantity 22

PAGE 23

supplied),andbisascalingparameter.)Wenotethatthistypeoffunctionalformiscommonlyusedtomodelthebehaviorofmarketpriceunderquantitycompetition(see,e.g., Rhimetal. 2003 SaizandHendrix 2008 ). Weassumethatthetransportationcostincurredbyarmonthelinkconnectinglocationitomarketjisalinearfunctionofthequantityowonthatlink,andweletcij0denotetheperunittransportationcostonlink(i,j)foranyrm.Itshouldbenotedthatcijcanbeassumedtoincludeperunitproductioncostsaswell.Thatis,aparametervi>0specictolocationicanbeincludedwithincijtoaccountforperunitproductioncostatlocationi.Inadditiontotransportationcosts,weassumethatrmsaresubjecttotrafccongestioncostsasaresultoftheirdistributionvolumedecisions.Wenotethattrafccongestionaffectssupplychainactivities,andrecentstudiesdocumentthenegativeeffectstrafccongestionhasonsupplychainperformance(see,e.g., McKinnon 1999 Raoetal. 1991 Sankaranetal. 2005 Weisbrodetal. 2001 ,and KonurandGeunes 2011 ).Therefore,weconsidercongestioncostsexplicitlyinourmodel.Inparticular,thecongestioncostarmincursonlink(i,j)amountstoqijrgij,whereqijrdenotesthequantityofowonlink(i,j)byrmrandgijisdenedtobeafunctionofthetotalquantityofowonlink(i,j),qij(qij=Pr2Rqijr),andreads gij(qij)=ijqij.(2) InEquation( 2 ),ijdenestrafccongestioncostfactorspecictolink(i,j)andweassumethatij>0.Thus,anyrm'scongestioncostisanondecreasingconvexfunctionofthequantitysentbythermonalink.InChapter 3 ,weprovidejusticationforwhythisfunctionalformischosentomodeltrafccongestioncosts.Finally,armincursaxedfacilitylocationcost,fi,ifitlocatesafacilityatlocationi. Theprotfunctionofarmconsistsoffourterms:totalrevenuesgainedfromsupplyingmarkets,lesstransportation,congestion,andfacilitylocationcosts.Explicitly, 23

PAGE 24

theprotfunctionofrmrreadsas r(Q,X)=Xj2Jpj Xi2IXr2Rqijr!Xi2Iqijr)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xj2JXi2Icijqijr)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xj2JXi2Iqijrgij Xr2Rqijr!)]TJ /F3 11.955 Tf 11.96 0 Td[(fr(xr),(2) whereQiskmnmatrixofqijrvaluesandXismkbinarymatrixrepresentingrms'locationdecisions.Inparticular,xrisanm)]TJ /F1 11.955 Tf 9.3 0 Td[(vectorrepresentinglocationdecisionsofrmrwithentriesxirsuchthatxir=1ifrmrlocatesafacilityatlocationi,xir=0otherwise.Hence,fr(xr)=Pi2Ixirfidenotesthetotalfacilitylocationcostforrmr.Fornotationalsimplicity,wefurtherdeneqjrasthetotalquantityshippedtomarketjbyrmr(qjr=Pi2Iqijr). Asnotedintheprevioussection,weadoptatwo-stagesolutionapproachconsideringthefollowingsequenceofdecisions:rst,rmsmustdeterminetheirfacilitylocationsand,then,theymustdecideontheirsupplyquantitiesateachmarket.Asaresultofcompetition,anyrm'sprotdependsonthedecisionsoftheotherrms.Notethat,unlikethepreviousstudiesby Rhimetal. ( 2003 )and SaizandHendrix ( 2008 ),rmsnotonlycompetebasedonmarketprice,butalsoasaresultofthecongestioncostfunctionsonsupplylinks.Ourtwo-stagesolutionapproachrstsolvesstage-twodecisionsgiventhatthermshaveidenticalfacilitylocations.Weemploytheequilibriumconceptof Nash ( 1951 )todeterminetherms'supplyquantitydecisionsandproposeamethodthatsolvestheassociatedCournotoligopoly.Then,wefocusonthesolutionofstage-onedecisionsandcharacterizetherms'facilitylocationdecisions. 2.3Stage-TwoDecisions:Market-SupplyGame Inthissection,westudythesecondstageofthegame,whichdeterminestherms'supplyquantitydecisionsforagivenlocationdecision.ThisrestrictedgameofdeterminingequilibriumquantitydecisionsisreferredtoastheMarket-SupplyGame.TheMarket-SupplyGameisanon-cooperativegameinwhichthesupplyrmsaretheplayers.Firmssimultaneouslydeterminehowmuchtosendfromfacilitiestomarkets. 24

PAGE 25

Todeterminetherms'ows,weusethePNEconcept,i.e.,normwillbebetteroffbyalteringitssupplyquantitydecisionsunderthegivenlocationdecisions. Nowletusassumethatthelocationdecisionforeachrm,i.e.,thevectorxrforeachr=1,2,,k,ispre-determined.Thatis,Xisxed.Sincefr(xr)isxedforthegivenX=X0,itcanbeomittedfromEquation( 2 )fortheanalysisoftheMarket-SupplyGame.UsingthenotationintroducedintheprevioussectionandEquations( 2 ),( 2 ),and( 2 ),theprotfunctionofrmrforthegivenX=X0canberewrittenas r(QjX=X0)=Xj2J"(aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj)qjr)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xi2Icijqijr)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xi2Iijqijrqij#.(2) ThefunctioninEquation( 2 )isstrictlyconcaveineachqijr0,asbj>0andij>0.Notethatqijr=0forallj2J,i=2I0,whereI0denotesthelocationswherermshavefacilitiesforthegivenX=X0.ItfurtherfollowsfromEquation( 2 )thatanyrm'sstage-twodecisionscanbeanalyzedseparatelyforeachmarket.Therefore,wefocusontheMarket-SupplyGameformarketjintherestofthissection.Thediscussionthatfollowsontherms'Market-SupplyGameatmarketjisvalidfortheMarket-SupplyGameacrossallmarkets. Giventherms'locationdecisionsX=X0,theprotfunctionofrmratmarketj,jr(QjjX=X0),is jr(QjjX=X0)=pj(qj)qjr)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xi2Icijqijr)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xi2Iijqijrqij,(2) whereQjisthevectorofrms'supplyquantitiesatmarketj.Letqijrdenotetheequilibriumquantities.TodeterminetheequilibriumsolutionfortheMarket-SupplyGameatmarketjgiventhatX=X0,oneshouldndthesetoflocationsandthesetofrmssuchthatqijr>0,andsolvetherstorderconditions,@jr(QjX=X0)=@qijr=0,foreachqijr>0simultaneouslyduetoconcavityofEquation( 2 ).Explicitly,therstorder 25

PAGE 26

conditionwhenqijr>0reads aj)]TJ /F3 11.955 Tf 11.95 0 Td[(bj[qj+qjr])]TJ /F3 11.955 Tf 11.95 0 Td[(cij)]TJ /F5 11.955 Tf 11.96 0 Td[(ij[qijr+qij]=0.(2) Atthispoint,weassumethatX0consistsofidenticalcolumns,wheretherthcolumncorrespondstothelocationdecisionofrmr.Thatis,xr=x08r2R,wherex0denotesanycolumnofX0.Thus,thenumberoffacilitiesatanycandidatelocationiseitherkor0forsomepositivek.Notethatwedonotneedtoconsiderlocationswherenormhaslocatedafacility.Therefore,weonlystudyquantitydecisionsatsupplylocationswithkfacilities.Thatis,I0denotesthesetoflocationswithkfacilitiesassociatedwithX0.Inthenextproposition,weshowthatthequantitysuppliedfromlocationitomarketjisthesameforeachrm. Proposition2.1. Giventhatxr=x08r2R,qijr=Qij=k8r2R,whereQijisthetotalequilibriumowonlink(i,j). Proof:PleaseseeAppendix A ItfollowsfromProposition 2.1 that,whenthetotalequilibriumquantitysuppliedfromlocationitomarketj,i.e.,Qijisknown,wecanreadilyobtaintheassociatedqijrvalues.ForthegivenX0,withxr=x08r2R,itfollowsfromProposition 2.1 that(i)qijr=Qij=k,(ii)qij=Qij,(iii)qjr=Pi2I0Qij=kand(iv)qj=Pi2I0Qij.RecallthatEquation( 2 )statestherstorderequilibriumconditionforquantitiessuchthatqijr>0;thatis,itgivestherstorderconditionwhenQij>0.Substituting(i)-(iv)intoEquation( 2 ),weget ij)]TJ /F3 11.955 Tf 11.96 0 Td[(bj Xi2I0Qij+Xi2I0Qij k!)]TJ /F5 11.955 Tf 11.95 0 Td[(ijQij+Qij k=ij)]TJ /F5 11.955 Tf 11.95 0 Td[(bjXi2I0Qij)]TJ /F5 11.955 Tf 11.95 0 Td[(ijQij=0,(2) whereij=aj)]TJ /F3 11.955 Tf 12.23 0 Td[(cijand=(k+1)=k.Notethatwemayhaveatmostkmsuchrstorderconditionsdenedformarketj.Nevertheless,therstorderconditionsassociatedwithalocationusethesameequationforeachrm,givenbyEquation( 2 ).Thisactuallyreducesthegameofthermstoagameoflocationsbyreducingthenumberofdecisionvariablesfromkmtom.Therefore,wefocusonsimultaneoussolutionof 26

PAGE 27

atmostmrstorderconditions,oneforeachlocation,denedbyEquation( 2 ).Inthenextproposition,westateconditionsthatmustbesatisedbyQijvalues. Proposition2.2. Theequilibriumquantitiesmustsatisfythefollowingconditions: (a)Qij>0ifandonlyifij>bjPi2I0Qij,(b)Qij=0ifandonlyifijbjPi2I0Qij. Proof:ConsideringProposition 2.1 andEquation( 2 ),theKKTconditionsforanyrmatlocationi,i2I0,canbewrittenasfollows: ij)]TJ /F5 11.955 Tf 11.96 0 Td[(bjPi2I0Qij)]TJ /F5 11.955 Tf 11.96 0 Td[(ijQij+ui=0,uiQij=0,ui0. WerstproveStatement(a).SupposeQij>0,thenitimpliesthatui=0,hence,wehaveij=bjPi2I0Qij+ijQij.SinceijQij>0asij>0andQij>0,itfollowsthatij>bjPi2I0Qij.Nowsupposethatij>bjPi2I0QijandQij=0,thenitimpliesthatui=bjPi2I0Qij)]TJ /F5 11.955 Tf 12.45 0 Td[(ij<0whichisacontradictionsinceui0.Hence,Qij>0.WenowproveStatement(b).SupposeQij=0andij>bjPi2I0Qij,thenitimpliesthatui=bjPi2I0Qij)]TJ /F5 11.955 Tf 10.29 0 Td[(ij<0whichisacontradictionsinceui0.Hence,ijbjPi2I0Qij.NowsupposethatijbjPi2I0QijandQij>0,thenitimpliesthatui=0,hence,wehaveij)]TJ /F5 11.955 Tf 11.95 0 Td[(bjPi2I0Qij=ijQij>0,whichisacontradictionsinceij>0. ItfollowsfromProposition 2.2 thatthesetofactivelocationsatmarketjwillbedeterminedbyijvalues.WerefertoanylocationiasactiveatmarketjifQij>0.Similarly,werefertoanyrmrasactiveatmarketjwheneverqijr>0.ThenextpropositionisadirectresultofProposition 2.2 andstatestheactivenessrelationsbetweentwolocations. Proposition2.3. Ifi1ji2jforlocationsi1,i22I0,theninanequilibriumsolution (a)ifQi2j>0,thenQi1j>0,(b)ifQi1j=0,thenQi2j=0. Proof:Supposethati1ji2jforlocationsi1,i22I0.NowconsiderthatQi2j>0.ThenitfollowsfromProposition 2.2 thati2j>bjPi2I0Qij.Thisimpliesthati1j>bjPi2I0Qij. 27

PAGE 28

Thus,itfollowsfromProposition 2.2 thatQi1j>0,whichprovesStatement(a).NowconsiderthatQi1j=0.ThenitfollowsfromProposition 2.2 thati1jbjPi2I0Qij.Thisimpliesthati2jbjPi2I0Qij.Thus,itfollowsfromProposition 2.2 thatQi2j=0,whichprovesStatement(b). Proposition 2.3 highlightstheimportanceofsortinglocationsaccordingtotheirijvalueswhich,foragivenmarketj,isequivalenttosortinglocationsbasedoncijvalues.Wenotethatincaseoftiedijvalues,Proposition 2.3 impliesactivenessofeitherallornoneofthelocationswithtiedijvalues.Nevertheless,theequilibriumsolutionoftheMarket-SupplyGameatmarketjdoesnotchangewiththesortingorderoftiedvaluesofij,asthesamesetofrstorderconditionsdenedinEquation( 2 )willbesolved.Thisholdstruewhenij>0.WenotethatitispossibletohavemultiplePNEquantitydecisionswhenij=0formorethanonefacility.Insuchacase,someofthelocationswithtiedvaluescanbeactivewhiletheothersareinactive.Furthermore,thesetofactivelocationscanbedenedbyanycombinationofthelocationswithtiedijvalues,whichresultsinmultipleequilibria. Nowletussortlocationsaccordingtotheirijvalues,andwithoutlossofgenerality,letusassumethatij(i+1)j.Therefore;if`locationsareactiveatmarketj,theselocationsare1,2,...,`with`jI0j,wherejI0jdenotesthecardinalityofthesetI0.Thenforanyrmatanylocationi,i`(sinceqijr>0asQij>0)thefollowingrstorderconditionmustbesatised:ij)]TJ /F5 11.955 Tf 11.95 0 Td[(bj(Q1j+Q2j++Q`j))]TJ /F5 11.955 Tf 11.95 0 Td[(ijQij=08i`. Inmatrixnotation,therstorderconditionscanberepresentedas2666666641j2j...`j377777775=2666666641j+bjbjbjbj2j+bj............bjbjbj`j+bj377777775266666664Q1jQ2j...Q`j377777775. 28

PAGE 29

ItfollowsfromtheaboverepresentationthatwecanndtheQijvalueseasilybyinvertingthe``matrixforagivensetofactivelocations.Notethatinvertingthismatrixbasicallyinvolvessolvingtherstorderconditionsforlocations1,2,...,`together.However,ouraimistodeterminethesetofactivelocationsandthenndtheequilibriumquantitiestosolvetheMarket-SupplyGameatmarketj.Inthenextproposition,weprovideanalgorithmthatdeterminesthesolutionoftheMarket-SupplyGameatmarketj.ThealgorithmisbasedonPropositions 2.2 and 2.3 Proposition2.4. Supposethatxr=x08r2R.ThenAlgorithm1,statedbelow,determinesthenumberoftheactivelocationsandthecorrespondingequilibriumowquantities. Algorithm1. Givenxr=x08r2R,thenumberofrms,bj,ijandijvaluesformarketj: Step0.SetQij=08i=2I0.Sorttheremaininglocationssuchthatlocation1hasthegreatestijvalue.If1j>0,set`=2andgotoStep1;otherwiseQij=08i2I0. Step1.Forlocation`,ndQ(`)`jbysolvingthefollowingsetofequationsrepresentedinmatrixform 2666666641j2j...`j377777775=2666666641j+bjbjbjbj2j+bj............bjbjbj`j+bj377777775266666664Q(`)1jQ(`)2j...Q(`)`j377777775.(2) Step2.IfQ(`)`j>0and`0and`=jI0j,stop,locations1,2,...,`areactiveandQij=Q(`)ij8i2I0.Elseif,Q(`)`j0,stop;locations1,2,...,`)]TJ /F4 11.955 Tf 12.16 0 Td[(1areactiveatmarketj.Qij=Q(`)]TJ /F11 7.97 Tf 6.59 0 Td[(1)ijfori`)]TJ /F4 11.955 Tf 11.96 0 Td[(1andQij=0fori`. 29

PAGE 30

Proof:WhenAlgorithm 1 stopsat(`)]TJ /F4 11.955 Tf 12.29 0 Td[(1)thiteration(aniterationreferstoanexecutionofStep2),itmeansthatQ(`)`j<0.Wenowshowthatifwecontinuethealgorithmonestepfurther,thatis,ifweassumethatrst`+1locationsareactive,weshouldhaveQ(`+1)`+1<0.Hence,thismeansthat`+1locationscannotbeactive.Moreover,sinceQ(`)`j<0,`locationscannotbeactive.NowsupposethatQ(`)`j<0.NotethatQ(`)`jisdeterminedbythesolutionoftheEquation( 2 ),ofwhichsolutionshouldsatisfy(i)ij)]TJ /F5 11.955 Tf 11.22 0 Td[(ijQ(`)ij=bj(Q(`)1j+Q(`)2j++Q(`)`j)8i`and(ii)1j)]TJ /F5 11.955 Tf 11.22 0 Td[(1jQ(`)1j=2j)]TJ /F5 11.955 Tf 11.22 0 Td[(2jQ(`)2j==`j)]TJ /F5 11.955 Tf 11.96 0 Td[(`jQ(`)`j.Itfollowsfrom(i)and(ii)thatQ(`)`j=`j+bj`Xi=1(`j)]TJ /F5 11.955 Tf 11.96 0 Td[(ij) ij `j+bj`Xi=1`j ij!, whichmeans,ifQ(`)`j<0,then`j+bjP`i=1(`j)]TJ /F15 7.97 Tf 6.59 0 Td[(ij) ij<0.Similarly,Q(`+1)(`+1)j=(`+1)j+bj(`+1)Xi=1((`+1)j)]TJ /F5 11.955 Tf 11.96 0 Td[(ij) ij (`+1)j+bj`+1Xi=1(`+1)j ij!. NowsupposethatQ(`+1)(`+1)j>0,then(`+1)j+bjP(`+1)i=1((`+1)j)]TJ /F15 7.97 Tf 6.59 0 Td[(ij) ij=(`+1)j+bjP`i=1((`+1)j)]TJ /F15 7.97 Tf 6.58 0 Td[(ij) ij>0.Ontheotherhand,(`+1)j+bjP`i=1((`+1)j)]TJ /F15 7.97 Tf 6.59 0 Td[(ij) ij<`j+bjP`i=1(`j)]TJ /F15 7.97 Tf 6.59 0 Td[(ij) ijas(`+1)<`,whichimplies(`+1)j+bjP(`+1)i=1((`+1)j)]TJ /F15 7.97 Tf 6.59 0 Td[(ij) ij<0.Thisisacontradiction,thus,Q(`+1)(`+1)j<0. Wenotethatsimilaralgorithmswereproposedfortheproblemsstudiedby Rhimetal. ( 2003 )and SaizandHendrix ( 2008 ),inwhicheachrmmayopenatmostonefacility.Thealgorithmproposedby SaizandHendrix ( 2008 )followsasimilarsortingprocedureasinAlgorithm 1 .However,Algorithm 1 issubstantivelydifferentbecausewedirectlyconsidertheimpactsofcongestioncosts(and,therefore,theassociatedrst-orderconditionsaredifferent).Moreover,althoughwesortsupplylocationsbasedoncijvalues,asdo SaizandHendrix ( 2008 ),theinterpretationofcijvaluesisdifferent 30

PAGE 31

inourcase.Thatis,in SaizandHendrix ( 2008 ),cijisaconstantmarginaldeliverycostfromrmitomarketj.Inourmodel,cijonlycorrespondstothemarginalcostfromlocationitomarketjwhenowfromitojequalszero(thatis,cijistheinterceptofthemarginalcostasafunctionofowonlink(i,j)).Undernonlinearcongestioncosts,themarginalcostfordeliveryofanadditionalunitofowdependsontheexistingowlevelbecauseofcongestioncosts,andisthereforeanondecreasingfunctionoftheowonalink. Algorithm 1 capturessomeinterestingpropertiesofequilibriumsolutionswhennonlinearcongestioncostsareconsidered.Inparticular,theconditionQ(`)`j>0inStep2holdsifonlyifij>bjQ(`)1j+Q(`)2j+...+Q(`)(`)]TJ /F11 7.97 Tf 6.58 0 Td[(1)j,whichfollowsfromEquation( 2 ).Thisimpliesthattheparameter`jhasnoinuenceonwhetherornotalocation`isactiveinservingmarketj(althoughthisparameterdoesaffectthequantityofowfromsupplylocation`tomarketjwhensupplypoint`isactiveinservingmarketj).However,`jdoesinuencewhetherornotlocation`+1isactiveinservingmarketj,throughthedependenceofQ(`)ijon`jfori<`+1. Then,applyingAlgorithm 1 separatelyforeachmarket,onecansolvefortheequilibriumsupplyquantities.Next,westudythestage-onedecisions. 2.4Stage-OneDecisions:FacilityLocations Thissectionstudiestherms'supplyfacilitylocationdecisions.Werstdiscusstherationalebehindourpriorassumptionthatallrmsmakeidenticallocationdecisions.Then,weseekthebestlocationdecisionforasinglerm,assumingthatitwillalsobethebestlocationdecisionfortheotherrms. 2.4.1IdenticalLocationDecisions SupposethatweareabletodeterminetheoptimalsupplyquantitydecisionsforanygivenlocationdecisionmatrixX0,whichimpliesthatwecandeterminethetotalprot,includingthefacilitylocationcosts,foranygivenX0.Inthenextproposition,weshow 31

PAGE 32

thatifthereexistsauniquePNElocationdecision,theneachrmchoosesthesamefacilitylocationsinequilibrium. Proposition2.5. SupposethatthereexistsauniquePNElocationmatrix,X.Then,xr=x8r2R,wherexdenotesthecolumnvectordecisionforeachrminX. Proof:SupposethatXistheuniquePNElocationdecisionsuchthatxr16=xr2foranytwormsr1andr2.Thenthereexistsatleastonelocation,saylocationi,suchthatrmr1doesnothaveafacilitywhilermr2hasafacilityatlocationi,thatis,xir1=1andxir2=0.Now,ifwemakexir1=0andxir2=1inXandconstructX,thenXisalsoaPNElocationdecisionsincermsarehomogeneouswithrespecttotransportation,trafccongestionandfacilitylocationcosts.ThiscontradictsthatXistheuniquePNElocationdecision. Proposition 2.5 alsofollowsfromthefactthattherms'locationdecisionsformamulti-playersymmetric(stronglysymmetric; Brantetal. 2009 )gamewithanitenumberofstrategies( Nash 1951 ).Forsymmetricgames,itiswellknownthatasymmetricequilibriumexists,eitherunderpurestrategiesormixedstrategies( Nash 1951 ).Therefore,whenthereexistsauniquePNElocationmatrix,itwillbeasymmetricPNE,i.e.,eachrmmakesthesamelocationdecisions. ItisimportanttonotethatProposition 2.5 parallelstheagglomerationresultdiscussedin AndersonandNeven ( 1991 ).Inparticular, AndersonandNeven ( 1991 )showthattheuniqueequilibriumwithinspatialcompetitionoccurswhentwocompetingrmsspatiallyagglomeratebylocatingtheirsinglefacilityatthecenterofalinearmarket(i.e.,thecustomersareuniformlylocatedonalinesegment).Asimilarresultisgivenin Hamiltonetal. ( 1989 ).Proposition 2.5 canbeconsideredasafurthergeneralizationoftheagglomerationresultforthecaseofthecompetitionmodelstudiedinthischapteronadiscretenetworkgiventhatauniquePNElocationdecisionexists. Recallthatthemethoddescribedintheprevioussectioncharacterizestheequilibriumquantitydecisionsgiventhatthermshaveidenticalfacilitylocations. 32

PAGE 33

Thus,Proposition 2.5 impliesthatwecanusethismethodtodeterminetheprotsassociatedwithanylocationmatrix,whichconsistsofidenticalcolumns,andchoosethebestamongallsuchsolutionstodeterminetheuniquePNE.Nevertheless,theuniquenessofaPNElocationdecisionisnotguaranteed. WhileuniquenessofPNElocationdecisionsimpliesexistenceofasymmetricPNE(whichistheuniquePNElocationmatrixitselfasimpliedbyProposition 2.5 ),inthecaseofmultiplePNElocationdecisions,itispossiblethatnoneoftheequilibriumpointsunderpurestrategiesissymmetric. Chengetal. ( 2004 )discusstheexistenceofatleastonePNEforsymmetricgameswithmultipleplayerssuchthateachplayerhastwostrategies.Thatis,ifrmsonlyhaveoneoptionforlocatingtheirfacilities,thegamecorrespondingtothelocationdecisionsofthermshasatleastonePNEsolution.Wenotethatthesinglelocationcasecanbesolvedbyconsidering2ksolutionswitheachrmeitherlocatingornotlocatingafacilityatthesinglelocation.Iteasilyfollowsfromthediscussionintheprevioussectionthatforanysuchconguration,thequantitydecisionsofthermswithafacilitywillbeidentical.Moreover, Rhimetal. ( 2003 )provetheexistenceofaPNEinacompetitivefacilitylocationgameinwhichrmsareallowedtolocateatmostonefacility,bynotingthatthegamecanbemodeledasacongestiongameundertheassumptionthateachmarketwillbesuppliedfromasinglelocation.Thentheexistenceresultfollows,ascongestiongameshavePNEsolutions( Rosenthal 1973 ).Nevertheless,thegamewestudycannotbemodeledasacongestiongameduetothefactthatrmsmaylocatemorethanonefacility.Furthermore,foragivensetoffacilitylocations,itispossiblethatarmchoosestosupplyagivenmarketfrommultiplefacilitiesduetononlineartrafccongestioncosts. Chengetal. ( 2004 )notethatevenforsymmetricgameswithtwostrategies,theexistenceoruniquenessofasymmetricPNE(i.e.,wheneachplayerchoosesthesamestrategy)isnotguaranteed. Amiretal. ( 2008 )showthataParetodominantsymmetricPNEexistsforsupermodular,doubly 33

PAGE 34

symmetricgames.However,thelocationdecisionsforourproblemdonotconstituteadoublysymmetricgame. WhenasymmetricPNEsolutiondoesnotexist,thisimpliesthateithermultiplePNEsolutionsexistornoPNElocationexists.Forbothofthesecases,aspreviouslynoted,thecorrespondingmixedstrategyNashequilibrium(MSNE)willbesymmetric.Next,westudyMSNEforsuchcasesunderthefollowingassumptions: Assumption2.1. Giventhelocationdecisionsofotherrms,armwillneverlocateanadditionalfacilityiflocatingthisfacilityreducesprot. Assumption2.2. Giventhelocationdecisionsofotherrms,iflocatinganadditionalfacilitydoesnotchangetherm'stotalprot,thermwilladdthisfacility. Assumption2.3. Giventhelocationdecisionsofotherrms,theredonotexistmultipledistinctlocationdecisionscontaininganidenticalnumberoffacilitiesthatresultinthesameprotlevelforanyrm. NotethatAssumptions 2.1 2.3 implythat,giventhelocationdecisionsofotherrms,armwillhaveauniquechoiceoflocationvector.Inthenextproposition,weshowthat,underAssumptions 2.1 2.3 ,aMSNEexistssuchthattheprobabilityofarmchoosinganyparticularlocationvectorxiseither0orequaltosomevaluesuchthat1>0. Proposition2.6. SupposethatAssumptions 2.1 2.3 holdandthatnormwillchoosealocationdecisionthatisweaklyorstrictlydominated.Then,thereexistsamixedstrategyNashequilibriumwithr(x)=orr(x)=0foranylocationvectorx,forallr2R,wherer(x)denotestheprobabilitythatanyrmrwillchooselocationvectorxand1>0. Proof:PleaseseeAppendix B ItfollowsfromtheproofofProposition 2.6 thatwhenauniquesymmetricPNElocationdecisiondoesnotexist,rmswillassignthesameprobabilitiestolocationvectorsthatarenotdominatedinamixedstrategy,anddominatedlocationvectors 34

PAGE 35

willbeassignedprobability0.Moreover,duetothesymmetryofthemixedstrategyequilibrium,rmswillassignthesameprobabilitytoeachparticularlocationvector. Theproblemwithusingtheequilibriumconceptasasolutiontoolforthestage-onedecisionsisthatitfailstodeterminehowrmswillchoosetheirfacilitylocationsincaseswheremultiplePNEsolutionsexistornoPNElocationdecisionexists.WealreadyknowfromProposition 2.5 thatwhenthePNEisunique,allrmswillchoosethesamelocationsand,hence,wecansearchoveronerm'sdecisionstondanequilibriumsolution,astheprotsofthermswillbethesamewhenthelocationdecisionsarethesame.Nevertheless,whenmultiplePNEsolutionsexistorwhennoPNEsolutionexists,wecannotcharacterizetherms'actionsusingthePNEconcept.AsnotedinProposition 2.6 ,inthecaseofmultipleornoPNEsolutions,whenrmsdeterminetheprobabilityofchoosingalocationvector,theywillassignthesameprobabilities,andtheprobabilitiesassociatedwithlocationvectorsthatarenotweaklyorstrictlydominatedarethesameforeachrm.Therefore,whenrms'objectivesaretomaximizeexpectedprots,theywillchoosethesamefacilitylocationsduethesymmetryassumption.Thus,rms'locationdecisionswillcorrespondtoalocationmatrixwithidenticalcolumns,whichproducesthemaximumprots.Therefore,fromthispointon,wefocusondeterminingthebestlocationdecisionofasinglerm,assumingthatotherrmswillchoosethesamelocations.WenotethatthecorrespondingsolutionisaPNEwhenauniquePNEdecisionexists,anditisthebestsymmetricPNEwhenmultiplesymmetricPNEpointsexists.Inbothofthesecases,theresultingsolutionwillbeaSubgamePerfectNashequilibrium( Selten 1975 ). Nowsupposethateitherx1orx2isthebestlocationdecisionforrmr.Todeterminewhichoftheseisbetterforrmr,weneedtocomparetheprotsofrmrgivenX0=X1andX0=X2,whereeachcolumnofX1equalsx1andeachcolumnofX2equalsx2.NotethatwecanndthetotalprotforrmrassociatedwithX1andX2bydeterminingtheprotfromsupplyingmarketsusingthemethoddescribedinthe 35

PAGE 36

previoussection,andthensubtractingthefacilitylocationcostsassociatedwithx1andx2.AtotalenumerationschemewoulddeterminetheprotforeachX0suchthatX0hasidenticalcolumns,andpicktheonewithmaximumprot.Incaseofalternativeoptimalsolutions,Assumptions 2.1 2.2 canbeusedasaselectiontool. TheresultingmatrixXwillgivethebestlocationdecisionforrmraswellasforallotherrms.However,totalenumerationrequiresevaluatingexponentiallymanylocationdecisionsforarm.Inparticular,armmustdeterminetheprotfor2mlocationdecisions,andchoosetheonewiththemaximumprot.Astotalenumerationiscomputationallyburdensome,wenextprovideaheuristicmethodintendedtoberepresentativeofhowindividualrmsmayapproachsimultaneouslocationdecisionsinpractice.Ourheuristicmethodrstchoosesthenumberoffacilitiestobelocatedbasedonarankingoflocationsderivedfromtheproblemparametersandthen,choosesthebestlocationsforthesefacilities.ThecomparisonoftheheuristicmethodwithtotalenumerationthatwelaterprovideinSection 2.5 willcharacterizeconditionsunderwhichthemethodofanalyzinglocationdecisionsintwostepsleadstooptimalornear-optimalperformance. 2.4.2HeuristicMethodforIdentifyingLocationMatrix Becauseweconsiderasimultaneousgameinwhichaplayermaynotpossessallrelevantinformationassociatedwiththeotherplayers,itisimpossibletoprovideageneralcharacterizationofhowanindividualrmwillapproachthedecisionproblem(andto,therefore,characterizethesolutionthatwillresult).Inanattempttoemulateareasonableapproachthatmightbetakenbyanindividualrmundersuchconditions,wehaveconstructedaranking-basedheuristicapproachinwhichpotentiallocationsarerankedinapreferenceorderbasedonproblemdata.Theheuristicmethodweprovideisthusbasedonassigningweightstocandidatelocations.Inparticular,theweightofalocationisdeterminedbyconsideringthedistancefromthelocationtomarkets,thecongestioncostfactorsonthelinksconnectingthelocationtomarketsandthefacility 36

PAGE 37

locationcostatthelocation.RegardingtheprotfunctionofarmgiveninEquation( 2 ),armhastransportation,congestionandfacilitylocationcosts.Furthermore,armgainsrevenuebysupplyingthemarkets.Hence,whenassigningweightstolocations,weconsidercosttermsrelatedtothelocationaswellaspriceinformationfromthemarkets.Forinstance,alocationthatisclosetoamarketwithalowinitialpriceorhighpricesensitivitymaybelessappealingthanalocationthatisfurtherfromamarketwithhighinitialpriceorlowpricesensitivity.Toaccountformarketpriceinformation,weconsidertheratioaj=bjforeachmarketasthemarketpotential;thatis,agreateraj=bjvaluemeansthatarmwillgetmorerevenuebysupplyingthemarketthansupplyingthesameamounttoamarketwithaloweraj=bjvalue. Specically,theweightoflocationiisdeterminedbytheexpression !i=Xj2Jcij (aj=bj) Xi2IXj2Jcij (aj=bj)+Xj2Jij (aj=bj) Xi2IXj2Jij (aj=bj)+fi Xi2Ifi(2) Theweightoflocationi,!i,containsthreenormalizedterms.Thersttermaccountsforthetransportationcostsfromthelocationtomarkets,aswellasthemarketpotentials.Thatis,alocationclosetomarketswithhigherpotentialswillhaveasmallerweightthanalocationclosetomarketswithlowerpotentials.Thesecondtermaccountsforcongestioncostsassociatedwiththelocationandfollowsthesamelogicastherstterm.Thelasttermisthenormalizedfacilitylocationcostatthelocation.BasedonEquation( 2 ),alocationwithlowerweightismorefavorabletorms. Theheuristicmethodhastwophases.Intherstphase,armdecidesonthenumberoffacilitiestolocateasfollows.Supposethatarmisplanningtolocate`facilities,`m.Weassumethatthelocationsofthese`facilitieswillbethe`locationswiththelowestweights,andwecomputetheprotassociatedwithsuchalocationdecision.Werepeatthisprocessforeach0`m,andassumethatthermchoosesthenumberoffacilitiesthatprovidesthemaximumprot.Inthesecondphase 37

PAGE 38

oftheheuristicmethod,armdeterminesthebestlocationsforthenumberoffacilitiesdeterminedintherstphase.Below,weprovideastep-by-stepdescriptionofthealgorithm. Algorithm2. 2-PhaseHeuristicmethod: PhaseI:Determiningthenumberoffacilitiestobelocated Step0.CalculatethelocationweightsusingEquation( 2 ).Sortlocationsinnon-decreasingorderofweight,i.e.,wi
PAGE 39

2.5NumericalStudies Ournumericalstudiesfocusontwokindsofanalysis.Werstconsidertheefciencyoftheheuristicmethodprovidedintheprevioussection.Followingthis,wecomparetherms'bestdecisions(i)whenrmsconsidertrafccongestioncostsindecisionmakingand(ii)whenrmsdisregardtrafccongestioncostsindecisionmaking. 2.5.1EfciencyofTheHeuristicMethod Ourrstanalysisisaimedatcharacterizingtheefciencyoftheheuristicmethodprovidedintheprevioussection.Wegeneratedataforourcomputationaltestsinthefollowingway.Weconsidereightproblemclasses,whereeachproblemclassdiffersincongestioncostfactors,ij,transportationcosts,cij,andfacilitylocationcosts,fi.Byconsideringdifferentproblemclasses,thegoalistoprovideamoreconclusiveanalysis(ratherthansolvingaspecicclassofproblemforwhichtheheuristicmethodisquiteefcient).Foreachoftheclasses,weuseallcombinationsofk2f3,5g,n2f3,5,7gandm2f3,5,7,10,15g,resultingin30combinationsofthevaluesofk,n,andm.Foreachofthesecombinations,wegenerate10probleminstances.Foreveryproblem,weletajU[50,150]andbjU[1,2],whereU[l,u]denotestheuniformdistributionon[l,u].Table 2-1 givesthedistributionrangeforij,cijandfivaluesineachproblemclass.Ineachproblemclass,wesolve300probleminstancesandeachprobleminstanceissolvedusingtotalenumerationandtheheuristicmethodstatedinAlgorithm 2 Table 2-2 comparestotalenumerationwithAlgorithm 2 foragivenrm'snumberoffacilities(#offac.),totalquantitysuppliedtomarkets(SupplyQuant.),totalprot,andCPUtimeinseconds,anddocumentstheoptimalitygap(Opt.gap).AscanbeseenfromTable 2-2 ,the2-Phaseheuristicmethodisofcoursefasterthantotalenumeration,andtheaveragesolutionobtainedbythe2-Phaseheuristicmethodhasanaverageoptimalitygapof2.23%overthe2400probleminstancessolved.Moreover,the2-Phase 39

PAGE 40

Table2-1. Datarangesforproblemclasses1-8foranalysis1 ijcijfiClass1(0,4](0,50][75,125]Class2(0,4](0,50][100,150]Class3(0,4][25,75][75,125]Class4(0,4][25,75][100,150]Class5[4,8](0,50][75,125]Class6[4,8](0,50][100,150]Class7[4,8][25,75][75,125]Class8[4,8][25,75][100,150] heuristicsolutionhasparallelresultswithtotalenumerationfortheaveragenumberoffacilitieslocatedandtheaverageoftotalquantitiessuppliedtomarkets. Table2-2. Comparisonoftotalenumerationand2-phaseheuristicmethod TotalEnumeration2-PhaseHeuristic#ofSupplyTotalCPU#ofSupplyTotalCPUOpt.fac.Quant.Prottimefac.Quant.Prottimegap Class14.2450.812885.3737.534.9251.172854.414.221.47%Class23.8650.122725.2952.764.4450.482687.255.451.51%Class33.3235.361478.1051.704.0435.831441.295.092.96%Class42.9534.681404.4786.333.6435.231356.596.943.86%Class55.8133.371125.4790.085.9533.521121.387.951.64%Class65.1832.381009.3838.885.3532.601005.284.090.63%Class73.9520.69445.1454.394.0820.74439.426.103.23%Class83.1618.41336.5841.343.2518.47330.204.022.57%Avg.4.0634.481426.2256.634.4634.761404.485.482.23% InTable 2-3 ,wecomparethetotalenumerationand2-Phaseheuristicmethodsolutionsforprobleminstanceswiththesamenumberofpotentialfacilitylocations,i.e.,forproblemswithm=3,m=5,m=7,m=10andm=15.Wenotethatasthenumberofpotentiallocationsincreases,thecomputationtimeadvantageofthe2-Phaseheuristicmethodincreasesaswell.Ontheotherhand,theoptimalitygapdoesnotshowaclearincreasingordecreasingtrendasthenumberoflocationsincreases.Therefore,wecansaythat2-PhaseheuristicmethodisrobustandthesolutionqualityofAlgorithm 2 isnotclearlydecreasingastheproblemsizeincreases,althoughAlgorithm 2 becomessubstantiallymoreefcientcomputationally. 40

PAGE 41

Table2-3. Comparisonoftotalenumerationand2-phaseheuristicmethodforeachm TotalEnumeration2-PhaseHeuristic#ofSupplyTotalCPU#ofSupplyTotalCPUOpt.mfac.Quant.Prottimefac.Quant.Prottimegap 32.3925.99982.300.022.4326.04980.280.010.83%53.4030.421200.240.083.6230.661189.570.022.08%74.0333.251366.700.374.4133.571347.370.082.64%104.6635.741503.704.055.1535.961473.260.522.85%155.8046.982078.18278.626.7047.542031.9126.792.77%Avg.4.0634.481426.2256.634.4634.761404.485.482.23% FromtheanalysisofTables 2-2 and 2-3 ,weconcludethatwhenarmdeterminesitsfacilitylocationsusingatwo-phaseapproach(suchthatintherstphase,thenumberoffacilitiesisdeterminedbysortingpotentialfacilitylocationswithrespecttoweights;Equation( 2 )inourcase),theresultingsolutionapproachiscomputationallyefcient,andtherelativeperformanceasmeasuredbytheoptimalitygapisrelativelystrong.Furthermore,thenumberofpotentiallocationsdoesnotheavilyinuencetheoptimalitygap.Thissuggeststhatthestrategyofdecidinglocationsintwophasesmakessense.Thisalsosuggestsafutureresearchdirectionbeyondthescopeofthischapter,inwhichthegameofthermscorrespondstoathree-stagegame.Intherststage,thenumberoffacilitiestobelocatedisdetermined;then,inthesecondstage,facilitylocationsarechosenand,nally,inthethirdstage,thesupplyquantitiesaredetermined. Basedontheanalysisoftheheuristicmethod,wecanalsoarguethatrankingpotentialfacilitylocationsbasedonaweightdenedsimilartoEquation( 2 ),consideringthelowoptimalitygapoftheheuristicmethod,suggeststhatrmswillprefertolocatefacilitiesatlocationsthatareconnectedtomarketswithhighpotentials,i.e.,highaj=bjvalues,viashorterandlesscongestedlinksandthathavelowerfacilitylocationcosts. 2.5.2AccountingforCongestioninDecisionMaking Thissectioncomparesthedecisionsoftherms(i)whenallofthermsexplicitlyconsidertrafccongestioncostsand(ii)whenallrmsdisregardtrafccongestion 41

PAGE 42

costsintheirlocationandsupplyquantitydecisionsinthegeneralcase.Inparticular,wecomparetwocases:(i)whenallofthermsareawareofcongestioninthenetworkandaccountforcongestioncostsintheirdecisions(i.e.,theyareindeedmaximizingtheprotfunctiondenedinEquation( 2 ))and(ii)whenallofthermsdisregardcongestioncostsintheirdecisions(i.e.,theyaremaximizingr(Q,X)=Pj2Jpj)]TJ 5.48 -.72 Td[(Pi2IPr2RqijrPi2Iqijr)]TJ /F10 11.955 Tf 12.42 8.96 Td[(Pj2JPi2Icijqijr)]TJ /F3 11.955 Tf 12.43 0 Td[(fr(xr)insteadofEquation( 2 )),butstillfacecongestioncostsaftertheyimplementtheirdecisions(i.e.,theypaytrafccongestioncostsPj2JPi2Iqijrgij)]TJ 5.48 -.71 Td[(Pr2Rqijrasaresultoftheirdecisionsaftertheymaketheirdecisions).FirmsinCase(i)willdeterminetheirquantitydecisionsusingAlgorithm 1 ,anddeterminefacilitylocationdecisionsusingtotalenumeration.FirmsinCase(ii)donotconsidertrafccongestioncostsintheirdecisionsand,hence,wecannotuseAlgorithm 1 directlytodetermineequilibriumquantitydecisions.Ontheotherhand,usingthenextproposition,weshowthatwhenrmsarenotawareofcongestion,theywillsupplyamarketfromtheclosestfacilitytothemarket,andeachrmwillsupplythesamequantity. Proposition2.7. Supposethatij=08i2I,j2J.GivenX0suchthatX0consistsofidenticalcolumns,qijr=bjij=(k+1)fori=iandqijr=0fori6=i8r2R,wherei=argmaxi2I0fijg. Proposition 2.7 providesasolutionmethodtondtheequilibriumquantitiesforgivenlocationdecisionsX0suchthatX0consistsofidenticalcolumnsforCase(ii).Regardingthediscussionintheprevioussection,totalenumerationcanstillbeusedforCase(ii)todeterminethelocationdecisions. Wegeneratedataforourcomputationaltestsinthefollowingway.Weconsidertwoproblemclasses,whereeachproblemclasshas8parameterdistributionsettings.Thatis,foreachproblemclass,andforeachofthethreeparametersofinterest(cij,fi,andij),wehavetwouniformdistributionsfromwhichparametervaluesaredrawn(resultingin23=8combinationsofdistributionsettings).Foreachofthese8 42

PAGE 43

combinationswithinaclass,weuseallcombinationsofk2f3,5g,n2f3,5,7gandm2f3,5,7,10g,resultingin24combinationsofthevaluesofk,n,andm.Foreachofthesecombinations,wegenerate25probleminstances.Foreveryproblem,weletajU[50,150]andbjU[1,2].Table 2-4 givesthedistributionrangefortheij,cijandfivaluesineachdatacategory,whereBidenotesdatacategoryi.WesolveeachprobleminstanceforrmsinCases(i)and(ii).IfthetotalprotofanysinglerminCase(ii)isnegative,weexcludethisinstancefromouranalysis,sinceweassumethatrmswillnotparticipatewhentheyhavenegativeprots.Inparticular,thisresultsinmorethan15probleminstancesineachofthe24setsforeachofthe8categoriesforProblemClasses1and2. Table2-4. Datacategoriesforproblemclasses1and2 Class1Class2ijcijfiijcijfiB1(0,0.25][25,75][50,100][0.5,1][25,75][50,100]B2(0,0.25][25,75][75,125][0.5,1][25,75][75,125]B3(0,0.25][50,100][50,100][0.5,1][50,100][50,100]B4(0,0.25][50,100][75,125][0.5,1][50,100][75,125]B5[0.25,0.5][25,75][50,100][0.75,1.25][25,75][50,100]B6[0.25,0.5][25,75][75,125][0.75,1.25][25,75][75,125]B7[0.25,0.5][50,100][50,100][0.75,1.25][50,100][50,100]B8[0.25,0.5][50,100][75,125][0.75,1.25][50,100][75,125] Intuitively,wewouldexpectthatrmsinCase(i)havehigherprotssincetheyconsidertrafccongestionintheirdecisions,whereasrmsinCase(ii)disregardtrafccongestionintheirdecisions,butpayforcongestionaftertheirdecisionsareimplemented.However,ournumericalresultsimplythattheoppositeisalsopossible.Tables 2-5 and 2-6 compareCases(i)and(ii)foreachProblemClass.ForProblemClass1,weseethattheaveragetotalprotforasinglerminCase(ii)ishigherthantheaveragetotalprotofasinglerminCase(i),whereas,wehavetheoppositeforProblemClass2.ThisresultforProblemClass1impliesthatrmsmayactuallyincreasetheirprotsiftheydonotconsidertrafccongestionintheirdecisions.Thisphenomenoncanbeexplainedasfollows.Forourproblem,rmsarecompetingon 43

PAGE 44

twodimensions:thepriceinamarketandthecongestiononlinksconnectingsupplylocationsandmarkets.ForCase(ii),sincethecongestioncostisdisregardedinthedecisionmakingprocess,rmscompeteonlyonmarketprice.Sowhentheimpactofcongestioncostisrelativelysmallandwhenrmscompeteonlyonmarketprice,theymayactuallyendupwithhigherprot. Table2-5. Statisticsofcases(i)and(ii)forproblemclass1 #ofSupplyTrans.Cong.Loc.Totalfac.Quant.CostCostCostProtCase(i)2.4734.381467.66201.54210.561628.94Case(ii)2.2439.081642.61415.57191.851661.44 Table2-6. Statisticsofcases(i)and(ii)forproblemclass2 #ofSupplyTrans.Cong.Loc.Totalfac.Quant.CostCostCostProtCase(i)3.0528.631275.48365.74257.761107.56Case(ii)2.2639.301656.181427.80193.18662.54 Next,weprovideasimpleexampletoillustratethephenomenoninwhichCase(ii)resultsinhigherprot. Example2.1. Considertwormscompetinginasinglemarket,market1.Therearetwopotentiallocations,1and2,atwhichthermsmaylocatefacilities.Supposethatfacilitylocationcostsare0atbothlocations,i.e.,f1=f2=0.Letc11=80,c21=90,and11=0.25,21=0.5.Themarketparametersarea1=100andb1=1.ThetotalquantitysuppliedtothemarketandthecorrespondingtotalprotforasinglermforCase(i)are5.33and64.00,respectively.ThetotalquantitysuppliedtothemarketandthecorrespondingtotalprotforasinglermforCase(ii)are6.67and66.67,respectively.Inbothofthecases,onlythefacilitiesatlocation1supplymarket1. AscanbeobservedfromthesolutionofExample 2.1 ,armismoreprotableunderCase(ii),i.e.,disregardingcongestioncostsindecisionmakingmayresult,insomecases,inhigherprots,evenunderaPNEsolutionforbothquantityandfacility 44

PAGE 45

locationdecisions(notethatwhenbothrmslocatefacilitiesatbothlocations,thiscorrespondstoaPNElocationdecision,sincefacilitylocationcostsare0). 45

PAGE 46

CHAPTER3ANALYSISOFTRAFFICCONGESTIONCOSTSINACOMPETITIVESUPPLYCHAIN 3.1MotivationandLiteratureReview Researchontrafcnetworkequilibriumproblems,tollpricing(congestionpricing),andmethodstomitigatetrafccongestionhavetypicallyfocusedonthewelfareofindividualroadusers.However,recentstudiesidentifythenegativeimpactstrafccongestionhasonsupplychainoperations.Inparticular,theperformanceoflogisticsoperationsisaffectedbytrafccongestion,andtheseimpactsaremoredrasticinJust-in-Time(JIT)productionsystems.Despitethefactthattrafccongestionaffectssupplychainoperations,mostofthestudiescombiningtrafccongestionandsupplychainsarebasedonempiricaldataandlacktheoreticalresults.Pastliteraturealsotypicallyassumesthattrafccongestioneffectsareexogenous,andtheseeffectsareanalyzedindirectlybyassumingthatincreasedcongestioneitherimpliesincreasedtraveltimesordecreasedtraveltimereliability.Moreimportantly,trafccongestioneffectsareonlystudiedinsofarastheyaffectthedistributionnetworkofasinglerm.Inthischapter,wefocusontheeffectsoftrafccongestiononsupplychainoperationsbymodelingtrafccongestioncostsendogenously. Westudytwoprimarysupplychaindecisions:facilitylocationdecisionsandsupplyquantitydecisions. McKinnonetal. ( 2008 )notethatcompaniesmayrestructuretheirdistributionsystemsduetoincreasedtrafccongestion.Moreover, Raoetal. ( 1991 )notethatchangesinfacilitylocationsareoftenalong-termreactiontoincreasedtrafccongestion.Forexample, Lee ( 2004 )pointsoutthatwhen7-11Japan(SEJ),aconvenience-storecompany,locatedstoresinkeylocations,SEJwassubjecttomoredramaticeffectsoftrafccongestion.Itisalsoworthnotingthattheeffectsoftrafccongestionsometimesdependonthefacilitylocationchoicesofacompany(see,e.g., Sankaranetal. 2005 ).Therefore,itisimportanttogainabetterunderstandingoftheeffectsoftrafccongestiononfacilitylocationanddistributionowdecisions. 46

PAGE 47

McKinnon ( 1999 )presentssurveyresultsonthenegativeeffectsoftrafccongestionontheefciencyoflogisticsoperations.Inasimilarstudy, McKinnonetal. ( 2008 )notethat,onaverage,trafccongestionaccountsfor23%ofthetotaldelaytimesinshipmentsofthecompaniescompletingthesurvey.Thisratecanbehigher(upto34%)insomeindustries( McKinnonetal. 2008 ).Forinstance, Fernieetal. ( 2000 )pointoutthattrafccongestionisoneofthemostimportantfactorsaffectingcostandserviceingroceryretailingintheUK. Sankaranetal. ( 2005 )alsodocumenttheresultsofasurveyanddiscusstheeffectsoftrafccongestiononsupplychainoperations.Asystematicreviewofstudiesattheintersectionoftrafccongestionandsupplychainscanbefoundin Weisbrodetal. ( 2001 ),whichalsodiscusseshowtrafccongestionaffectscostsandproductivity.AnotherstreamofresearchstudiestrafccongestioninJITsystems.BecauseJITsystemsrequiresmalllotsizes, Raoetal. ( 1991 )notethatthisresultsinincreasedtrafccongestion.Moreover,theirsurveyresultsindicatethatcompaniesareawareoftheassociatedcongestionimpactsandtheyproposeshort-andlong-termmethodstomitigatetheeffectsofcongestion. Moinzadehetal. ( 1997 )studytherelationshipbetweensmalllotsizesandtrafccongestionforacompany'sdistributionsystem,withmultipleretailersusingacommon,congestedroad. RaoandGrenoble ( 1991 )alsostudytheeffectsofJITreplenishmentandtheresultingtrafccongestionondistributioncosts.Oneothereldofresearchthatcombinestrafccongestionandsupplychainsfocusesonfreightdistribution.Forexample, Figliozzi ( 2009 )studiestheeffectsoftrafccongestiononthecostsassociatedwithcommercialvehicletours,while Figliozzi ( 2006 )and Figliozzietal. ( 2007 )analyzefreighttoursincongestedurbanareas.Similarly, GolobandRegan ( 2001 2003 )alsostudytheimpactsoftrafccongestiononfreightoperations.Westudytheeffectsoftrafccongestiononrms'facilitylocationanddistributionowdecisionsinacompetitiveenvironment.Thisstudyismotivatedbytheobservationthattrafccongestiononashareddistributionnetworkconstitutesaformofcompetitionforcommondistribution 47

PAGE 48

resources.Furthermore,weassumethatthermsalsocompetewithincommonmarketsintheirsalesvolumes. Themodelunderconsiderationinthischapterconsidersfacilitylocationsandsupplyquantitydecisionsinacompetitiveenvironmentonacongesteddistributionnetwork.Inparticular,weutilizethecompetitivelocationgamedenedinChapter 2 .Thatis,thefollowingsettingisassumed.Competingrmsarenoncooperativeandtheymustsimultaneouslydeterminetheirfacilitylocations(rststagedecisions);then,thesupplyquantitiesowingoutofthesefacilitiesintoeachmarket(secondstagedecisions)mustbedetermined.Firmsmaylocatemorethanonefacility.Inpractice,rmsmayprefertolocatemorethanonefacilityratherthanasinglefacilityeventosupplyasinglefacilityduetoeconomiesordiseconomiesofscale.Specically, Weisbrodetal. ( 2001 )pointoutthathigherlevelofcongestionrelatedcostsarecausedbyhighershippinglevels.Hence,inourformulationoftrafccongestioncosts,rmsaresubjecttodiseconomiesofscale,whichmotivatesrmstolocatemorethanonefacilitytoavoidseveretrafccongestionrelatedcosts.Marketsandpossiblefacilitylocationsarerepresentedasverticesinanetwork.Firmsareassumedtocompeteunderahomogeneouscoststructure;thatis,theyhaveidenticalcostparameters.Forthisreason,weassumethatrmsmakethesamefacilitylocationdecisionswhenmaximizingexpectedprots(whenauniquePureNashEquilibriumdoesnotexist).WeassumeoligopolisticCournotcompetitioninthesecondstage,i.e.,thetotalsupplytoamarketdeterminesthepriceinthatmarket. AsnotedinChapter 2 ,Cournotcompetitionisoneofthemostcommonconceptsusedinmodelingcompetitivemarkets.ACournotoligopolycanbeusedtorepresentenergymarkets( Salant 1982 Oren 1997 Ventosaetal. 2005 ),agriculturalmarkets( KlempererandMeyer 1986 ),groceryretailingmarkets( Mazzarotto 2001 Ellickson 2006 Arnadeetal. 2007 Colangelo 2008 ),andairlineindustries( BranderandZhang 1990 1993 Oumetal. 1995 ParkandZhang 1998 PelsandVerhoef 2004 ). 48

PAGE 49

Moreover,congestionispresentinsuchmarkets( Oren 1997 Fernieetal. 2000 PelsandVerhoef 2004 BassoandZhang 2008 ).Therefore,studyingacompetitivelocationproblemwithCournotcompetitionandcongestioncostsisimportantforamorecompleteanalysisofsuchmarkets. Thischaptercontributestotheliteraturebyprovidingananalyticalcharacterizationoftheeffectsoftrafccongestiononcompetitiverms'equilibriumfacilitylocationandsupplyquantitydecisions.Consideringthecasewithidenticalsupplyrmsenablesustoexplicitlyanalyzeandcharacterizehowcongestioncostsaffectthestructureofequilibriumdecisions,andtousethisanalysistoprovideinsightsintohowequilibriumsolutionschangeinresponsetochangesincongestionlevelsandcosts.WerstsummarizetheresultsofChapter 2 .Then,weprovideanalyticalresultsontheeffectsoftrafccongestioncostsontheequilibriumquantitiesowingfromsupplypointstomarkets.Wealsodiscussresultsforaspecialcaseoftheproblemwhenthepotentialfacilitylocationsarewithinmarketareas.Wenotethatthendingsofthisstudywillhaveanalogousresultsinthecasewithoutcompetition,i.e.,thestructuralandqualitativeresultsareanalogoustotheanalysisoftrafccongestioncostsonasinglerm'ssupplyquantityandfacilitylocationdecisions,asthermswillendupwiththesamedecisions.Nevertheless,asaforementioned,weassumecompetitionamongthermsinthedistributionnetworkinordertohighlightthewayinwhichcompetitors'decisionsaffectarm.Weperformextensivenumericalstudiesthatillustratetheeffectsoftrafccongestiononarm'slocationandquantitydecisions.Furthernumericalstudiesareconductedforthespecialcaseoftheprobleminwhichmarketareasserveastheonlypotentialsitesforfacilitiesaswell. Therestofthischapterisorganizedasfollows.InSection 3.2 ,wedenethemodelusedfortheanalysisoftrafccongestioneffects.Moreover,webrieysummarizetheresultsofChapter 2 thatarerequiredinthischapter.Section 3.3 analyzestheeffectsofincreasedtrafccongestiononequilibriumsupplyquantitiesanddiscussesthe 49

PAGE 50

implicationsoftheresultsforaspecialcase.InSection 3.4 ,weprovidetheresultsofextensivenumericalstudiesthatcharacterizetheeffectsoftrafccongestiononfacilitylocationandsupplyquantitydecisions,andtheeffectsofcongestionforthespecialcase. 3.2ModelandAnalysis Inthissection,wesummarizethemodelusedtoanalyzetheeffectsoftrafccongestiononfacilitylocationandequilibriumsupplyquantitiesalongwithitsanalysis.Asnoted,thesymmetriccompetitivemulti-facilitylocationgamewithtrafccongestioncostsdenedinChapter 2 isutilizedinthischapter.Thatis,weconsiderasetofkcompetitivermsconsideringthelocationoffacilitiesatmpossiblelocationsinordertoservecustomermarketsatnlocations.Eachrmincurstransportation,trafccongestion,andfacilitylocationcostsasaresultoftheirlocationanddistributionvolumedecisions.Moreexplicitly,rmsaresubjecttolineartransportationcostsinthequantityshippedfromafacilitytoamarketandthetrafccongestioncostincurredisconvexinthetotalquantityshippedfromafacilitytoamarket.Axedfacilitylocationcostexistsforeachlocationi.Moreover,weassumethatanyopenfacilityiseffectivelyuncapacitatedand,hence,armwillnotopenmorethanonefacilityatalocation.ThenotationofChapter 2 isusedthroughoutthischapteraswell.Wedeneadditionalnotationasneeded. WeassumetheunitpriceineachmarketisdenedbyEquation( 2 ).Weassumethatthetransportationcostislinearinthequantityofowonlink(i,j)andcij0representsaperunittransportationcost.ThefunctiongijdenedinEquation( 2 ),whichisafunctionofthetotalquantityofowonlink(i,j),determinesthetrafccongestioncostonlink(i,j).Thatis,gij(qij)=ijqij,whereij>0denotesthetrafccongestioncostfactor.Hence,thecongestioncostincurredbyarmusinglink(i,j)increaseswiththetotalquantityofowonthelinkaswellaswiththequantitysentbythermonthatlink.Inparticular,thecongestioncostforrmrisijqijrqij=0 50

PAGE 51

whenqijr=0.Ontheotherhand,whenqijr>0,thecongestioncostofrmrequalsijqijrqij>0andisconvexandincreasinginqijrwhenthequantitiessentbyotherrmsonthelinkarexed.Thus,therm'scongestioncostisanondecreasingconvexfunctionofthequantitysentbythermonthelink.ItshouldbenotedthatthefunctionalformofEquation( 2 )doesnotconsideraxedcapacitylimitonthedistributionnetworkow.Nevertheless,thischoiceoffunctionalformreectsthenatureoftrafccongestion,ascongestioncostsincreaseinvolumeatanincreasingrate.Thisiscompatiblewiththenotein Weisbrodetal. ( 2001 ),whichemphasizesthatcompanieswithhighershippinglevelsaresubjecttoahigherlevelofcongestionrelatedcosts.Furthermore,thisfunctionisdifferentthanthefunctionalformsusedtoformulatecongestionrelatedcostsinqueueingmodels.Queueingmodelsarestudiedtoanalyzetrafcowproblemsforindividualroadusers( Heidemann 1994 HeidemannandWegmann 1997 Vandaeleetal. 2000 WoenselandVandaele 2006 WoenselandVandaele 2007 ).Generally,congestioncostsareformulatedasafunctionoftheowonalink,speedatthecurrentow,andthevalueoftimeforatraveler,alongwiththeroadcharacteristicssuchastrafcdensityandfreeowspeed.Thistypeofcongestionmodelingalsoimpliesthattrafccongestioncostsincreaseatanincreasingratewiththeowonalink,i.e.,convexityofthecongestioncosts( Li 2002 WoenselandCruz 2009 ).Therefore,thecongestionmodelingapproachusedinChapters 2 and 3 isconsistentwithqueueingmodelsstudiedfortrafcowproblems. Recallthat,theprotfunctionofrmrreadsasr(Q,X)=Xj2J"(aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj)qjr)]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xi2Icijqijr)]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xi2Iijqijrqij#)]TJ /F3 11.955 Tf 11.96 0 Td[(fr(xr), wherethersttermistherevenuegainedbyservingmarkets,thesecondtermisthetotaltransportationcost,thethirdtermisthetotaltrafccongestioncost,andthelasttermisthetotalfacilitylocationcost. 51

PAGE 52

InChapter 2 ,weadoptedatwo-stagesolutionapproach,whichrstemploystheNashEquilibriumconceptof Nash ( 1951 )todeterminetherms'supplyquantitydecisionsforaxedsetoflocationdecisions,andthenfocusesonthesolutionfortheStage-onedecisions.Next,wesummarizetheresultsassociatedwithStage-oneandStage-twodecisions. 3.2.1Stage-OneDecisions RecallfromChapter 2 thattherms'locationdecisionsformamulti-playersymmetric(stronglysymmetric; Brantetal. 2009 )gamewithanitenumberofstrategies.ItthusfollowsthattheuniquenessofthePNElocationdecisionimpliesasymmetricPNElocationmatrix( Nash 1951 ).Furthermore,whenthereexistsauniquePNElocationmatrix,thesearchforanequilibriumlocationmatrixcanberestrictedtolocationdecisionssuchthateachrmchoosesthesamefacilitylocations.WecanthususetheAlgorithm 1 ,whichdeterminesthequantitydecisionsunderidenticalfacilitylocationdecisionstocharacterizetheprotofeachsuchlocationmatrix.Thus,choosingthebestamongallsolutionswithidenticalcolumnsdeterminestheuniquePNE.Ontheotherhand,itispossiblethatmultiplePNElocationdecisionsexist,orthataPNElocationdecisiondoesnotexist.Theproblemwithusingtheequilibriumconceptasadecisionmechanismforlocationdecisionsinthiscaseisthatitfailstoexplainandcharacterizerms'actualdecisionsinsuchcases( HarsanyiandSelten 1988 ).Thus,ifrmsdeterminefacilitylocationspurelybasedonexpectedprots(assumingthatanylocationvectorisequallylikelyforanyrm),thensincermsarehomogeneous,theywillmakethesamedecisions.Wecanthereforedeterminerms'locationdecisionsbychoosingthebestamongalllocationmatriceswithidenticalcolumns.Moreover,inthecaseofmultipleornoPNEsolutions,whenrmsdeterminetheprobabilityofchoosingalocationvector,itisshowninChapter 2 thatrmswillassignthesameprobabilities,andtheprobabilityofchoosingeachlocationvectorwillbethesameinamixed-strategyNashEquilibrium. 52

PAGE 53

AtotalenumerationschemewoulddeterminetheprotforeachlocationvectorX0suchthatX0hasidenticalcolumns,andpicktheonewithmaximumprot.Althoughtotalenumerationiscomputationallyburdensome,inournumericalstudiesweusetotalenumerationbecauseourgoalistoanalyzetheeffectsofcongestioncostsonthebestdecisionsoftherms. 3.2.2Stage-TwoDecisions Thesecond-stagedecisionsconstituteanon-cooperativegameamongtherms,whosimultaneouslydeterminehowmuchtosendfromfacilitiestomarketsgiventhelocationdecisionforeachrm.Notethatunlikethepreviousstudiesby Rhimetal. ( 2003 )and SaizandHendrix ( 2008 ),thermsnotonlycompetebasedonprice,butalsoasaresultoftrafccongestioncostfunctionsonsupplylinks. AsisdiscussedinSection 2.3 inChapter 2 ,givenfacilitylocationdecisions,theStage-twodecisionscanbeanalyzedseparatelyforeachmarket.Algorithm 1 determinesthenumberoftheactivelocationsandthecorrespondingequilibriumowquantities,giventhatxr=x08r2R,forasinglemarket.Notethatwestudythemultiple-marketscenarioinouranalysisoftrafccongestionasrms'problemsarenotseparableforeachmarketwhenthefacilitylocationcostsareincluded(i.e.,therst-stagedecisions).Thisisbecauseafacilitythatmightnotbeprotableinthecaseofasinglemarketmaybeprotablewhenitservesseveralmarkets.Furthermore,foraspecialcaseofourproblemthatwelaterdiscuss,weanalyzethecaseinwhichrms'facilitylocationchoicesareconnedtoasubsetofthemarkets.Inwhatfollows,weprovideapropertyofAlgorithm 1 thatwillbeutilizedintheanalysisofeffectsoftrafccongestiononequilibriumsupplyquantities. Supposethatthereare`activelocationsatmarketj.ConsiderthewthiterationofStep2inAlgorithm 1 .LetQ(w)ijbethetentativequantitiescalculatedatthewthiterationusingEquation( 2 ).(NotethatQ(`)ij=Qij.)Inthenextproposition,weshowthatthequantitysuppliedfromanactivelocationdecreasesasthenumberofactivelocations 53

PAGE 54

increasesateachiterationofAlgorithm 1 ,whereasthetotalquantitysuppliedtothemarketincreases. Proposition3.1. (a)Q(w)sj>Q(w+1)sjforlocations,swandw+1`.(b)wXi=1Q(w)ij
PAGE 55

whereA=bjwXi=1(sj)]TJ /F5 11.955 Tf 11.96 0 Td[(ij) ijandB=bjwXi=1sj ij.Thus,consideringEquation( 3 ),theaboveinequalityreadsas(w+1)jsj)]TJ /F5 11.955 Tf 11.95 0 Td[(sjQ(w)sj. Equation( 3 )impliesthattheaboveinequalitycanbewrittenas(w+1)jbj(Q(w)1j+Q(w)2j++Q(w)wj).Furthermore,itfollowsfromEquation( 3 ),whenQ(w)sjQ(w+1)sj,wehavebjPwi=1Q(w)ijbjPwi=1Q(w+1)ij.Thisimpliesthat(w+1)jbj(Q(w+1)1j+Q(w+1)2j++Q(w+1)wj).Wecanwritethelastinequalityas(w+1)jbj(Q(w+1)1j+Q(w+1)2j++Q(w+1)wj+Q(w+1)(w+1)j))]TJ /F5 11.955 Tf 12.82 0 Td[(bjQ(w+1)(w+1)j.Moreover,fromEquation( 3 ),wehavethatbj(Q(w+1)1j+Q(w+1)2j++Q(w+1)wj+Q(w+1)(w+1)j)=(w+1)j)]TJ /F5 11.955 Tf 11.74 0 Td[((w+1)jQ(w+1)(w+1)j.Thus,wehave(w+1)j(w+1)j)]TJ /F5 11.955 Tf 11.46 0 Td[((w+1)jQ(w+1)(w+1)j)]TJ /F5 11.955 Tf 11.46 0 Td[(bjQ(w+1)(w+1)j,whichmeans((w+1)j+bj)Q(w+1)(w+1)j0,whichisacontradictionsinceatthe(w+1)thiterationofthealgorithmwecheckthatQ(w+1)(w+1)j>0andthendeneQ(w+1)sjvalues.ThiscontradictionprovesStatement(a).Statement(b)isadirectresultofStatement(a)andEquation( 3 ). Thenextsectioncharacterizestheeffectsoftrafccongestioncostsontheequilibriumquantitydecisions,giventhatrms'facilitylocationdecisionsareidentical. 3.3EffectsofTrafcCongestiononEquilibriumSupplyQuantities Inthissection,weanalyzethechangesintheequilibriumsupplyquantitieswhenthecongestioncostfactorononeofthelinksconnectingasupplylocationtoamarketincreases.Notethatweassumethefacilitylocationdecisionsofthermsarexedandidentical.Supposethatlocationsaresortedsuchthatlocation1hasthegreatestijvalue.Hence,whenthereare`locationsactiveinamarket,theselocationswillbetherst`locations. Werstnotethatwhenthereare`locationsactiveatamarketinitially,anincreaseinthetrafccongestioncostfactorforoneoftheseactivelocationswillnotresultinanyoftheinitiallyactivelocationsatthatmarketbecominginactiveatthemarket.Thenextpropositionformalizesthisresult. 55

PAGE 56

Proposition3.2. Consider1sjand2sjsuchthat1sj<2sj,andsupposethatlocations1to`areactiveatmarketjunderthe1sjvalue,1sk.Thenlocations1to`arealsoactiveatmarketjunderthe2sjvalue. Proof:WerstshowthatQ1(`)zj0,wehaveQ2(`)zj>0,i.e.,locationzisstillactiveatmarketj.Moreover,consideringEquation( 3 ),Q1(`)zjP`i=1Q2(`)ij.Since,sj>bjP`i=1Q1(`)ijwehavesj>bjP`i=1Q2(`)ij,i.e.,wehaveQ2(`)sj>0andlocationsisstillactiveatmarketj. Proposition 3.2 impliesthatwhenthetrafccongestioncostfactorforoneoftheinitiallyactivelocationsatamarketincreases,itispossiblethatthetotalnumberofactivelocationsatthatmarketmayincrease.Moreover,theinitiallyactivelocationsatthemarketwillcontinuetobeactiveatthemarket.Nextwestudythecases(i)whenthenumberofactivelocationsatthemarketremainsthesameand(ii)whenthenumberofactivelocationsatthemarketincreases. (i)Whenthenumberofactivelocationsatamarketremainsthesame,weknowthatalloftheinitiallyactivelocationswillremainactiveatthatmarket.Thatis,thesetofactivelocationsatthemarketremainsthesame.Thiscasealsocapturesthesituationwhenallofthelocationsareinitiallyactiveatthemarket.Inthiscase,thequantitysuppliedtothemarketfromthelocationforwhichthetrafccongestioncostfactor 56

PAGE 57

increased,willdecrease.Ontheotherhand,thequantitysuppliedtothemarketfromtheotherlocationswillincrease.Moreover,thetotalquantitysuppliedtothemarketdecreases.Weformalizethisdiscussioninthenextproposition. Proposition3.3. Supposethat1sjand2sjaresuchthat1sj<2sj,andthatlocations1to`areactiveatmarketjunder1sjand2sj,1s`;thatis,thenumberofactivelocationsandthesetofactivelocationsatmarketjremainthesame.Then(a)Q1ij>Q2ijfori=sand,Q1ijP`i=1Q2ij,whereQ1ijandQ2ijdenotetheequilibriumquantitiessuppliedfromlocationitomarketjunderthe1sjand2sjvalues,respectively. Proof:Sincethenumberofactivelocationsandthesetofactivelocationsatmarketjremainthesame,Q1ij=Q1(`)ijandQ2ij=Q2(`)ij.NowitdirectlyfollowsfromEquation( 3 )thatQ1ij>Q2ijfori=sand,itfollowsfromtheproofofProposition 3.2 thatQ1ij
PAGE 58

fromtheotherlocationsthatwereinitiallyactiveatthemarketmayincreaseordecrease.However,ifthetotalquantitysuppliedtothemarketfromoneoftheinitiallyactivelocationsatthemarket(forwhichthetrafccongestioncostfactorremainsthesame)increases(decreases),thetotalquantitysuppliedtothemarketfromtheotherinitiallyactivelocationsatthemarket(withunchangedtrafccongestioncostfactors)alsoincreases(decreases).Thenextpropositionformalizesthisdiscussion. Proposition3.4. Supposethat1sjand2sjaresuchthat1sj<2sj,andsupposethatlocations1to`areactiveatmarketjunder1sj,s`,andlocations1to`+}areactiveatmarketjunder2sj.Then(a)Q1ij>Q2ijfori=s.Moreover,(b)ifQ1ijP`+}i=1Q2ij,whereQ1ijandQ2ijdenotetheequilibriumquantitiessuppliedfromlocationitomarketjunder1sjand2sj,respectively. Proof:Notethat,Q1ij=Q1(`)ijandQ2ij=Q2(`+})ij.WeknowfromProposition 3.3 ,thatQ1(`)sj>Q2(`)sj.Moreover,weknowfromProposition 3.1 thatQ2(`)sj>Q2(`+1)sj.ThusitfollowsthatQ1(`)sj>Q2(`+})sj,whichprovesStatement(a).Statement(b)directlyfollowsfromEquation( 3 ).Inparticular,supposeQ1tjP`+}i=1Q2ij.ThenitagainfollowsfromEquation( 3 )thatQ1ij
PAGE 59

theowonlink(i,j),i6=s,thetotalquantitysuppliedtomarketjandthetotalquantitysuppliedtomarketjbyanyrmdecreases(increases). Whenthetotalquantitysuppliedtomarketjbyarmdecreases,thisimpliesthatallofthermsdecreasesupplytomarketj,increasingthepriceinmarketjtobalancetheincreaseinthetrafccongestioncosts.Nevertheless,whenthetotalquantitysuppliedtomarketjbyarmincreasesandthenumberofsupplypointsincreases,thisillustrateshowrmsmaychoosetodivertowtomarketjusinglinksthatarenotasclosetomarketjbutarelesscongested. OurdiscussionofPropositions 3.3 and 3.4 impliesthatincreasedcongestionhampersefcientplanningofsupplychainactivities,becauseitpushesrmstosupplymarketsusingeithermorecongestedlinksorlinksthatarenotclosetothemarket.InSection 3.4 ,weprovidetheresultsofextensivenumericalstudiestocharacterizetheeffectsofincreasedcongestiononrms'facilitylocationdecisionsaswellassupplyquantitydecisions.Next,wediscusstheStage-twodecisionsforapracticalspecialcaseoftheproblemstatedinSection 3.2 ,inwhichmarketlocationsmayalsoserveassupplyfacilitylocations. 3.3.1ImplicationsforASpecialCase:FacilitiesLocatedwithinMarketAreas Acommoncaseinpracticeoccurswhenpotentialfacilitylocationsarewithinthemarketareas.Forinstance,ifthemarketsareconsideredtobeasetofspatiallyseparatedcities,rmsmaylocatetheirfacilitieswithinthesecities.Therefore,westudytherelevantcaseinwhichIJ.Inthiscase,thetransportationcostfromafacilitywithinamarketareatothatmarketwillbeverysmallandthuscanbeapproximatedbysettingcij=0ifi=j.InChapter 2 ,wenotedthatcijmayincludealocation-specicperunitproductioncost,vi.Hence,assumingcjj=0impliesanassumptionthatvj=0aswell.Nevertheless,thisassumptionisnotrestrictiveifweconsiderthefollowingtransformationwhenvj>0.Forvj>0,wecandeneaparameteraj:=aj)]TJ /F3 11.955 Tf 12.92 0 Td[(vj,withcij:=cij)]TJ /F3 11.955 Tf 12.56 0 Td[(vjfori6=j.Thistransformationapproximatestransportationcoststo 59

PAGE 60

marketjfromafacilitywithinthemarketasequalto0,whileaccountingforvariableproductioncosts.However,forsimplicity,weassumethatcjj=0.Anyprobleminstancewithvj>0canbetransformedtoanequivalentprobleminstancewithcjj=0usingthenotedtransformation.Therefore,forthespecialcaseofinteresthere,trafccongestioncostswillbethemaincostdriverwhensupplyingamarketfromalocalfacilitythatisactiveinthemarket.Inthissubsection,wedocumenttheimplicationsoftheresultspresentedearlierforthisspecialcaseandanalyzetheeffectsoftrafccongestioncostsonequilibriumsupplyquantitieswhenthesupplyrmshavexedandidenticalfacilitylocations. SupposethatIJandrmshavemadeidenticallocationdecisions;thatis,ifoneofthermshasafacilityinamarket,thenallrmshaveafacilityinthatmarket.Notethat,asinthepreviousdiscussionofthetheStage-twodecisionsforthegeneralcase,wecananalyzeeachmarketseparately.Furthermore,thesupplyquantitydecisionsforanymarketwithoutanysupplyfacilitieswillcorrespondtotheStage-twodecisionsofthegeneralcase.Therefore,weonlyfocusontherms'supplyquantitydecisionsforamarketinwhicheachrmhasafacility.Supposethateachrmhasafacilityinmarketjand,thus,theunittransportationcostfromthesefacilitiestocustomersinmarketjis0.ThenitfollowsfromAlgorithm 1 thatthefacilitiesinmarketjwillbeactiveinmarketjifthereisanypositivesupplytomarketj.Next,weshowthattherewillbeapositivesupplytomarketjwhenrmshavefacilitiesinthatmarketarea. Proposition3.5. SupposethatIJandrmsmakeidenticalfacilitylocationdecisions.Wheneachrmlocatesafacilityinmarketjandaj>0,thenqj>0,whereqjdenotesthetotalquantitysuppliedtomarketjinequilibrium. Proof:WeknowfromAlgorithm 1 thatthefacilitiesatmarketjwillsupplyapositiveamounttomarketjifqj>0.Moreover,sinceaj>0,itfollowsfromStep1ofAlgorithm 1 thatthetotalquantitysuppliedfromthefacilitiesatmarketjispositive,i.e.,Qjj>0.Thisimpliesthatqj>0. 60

PAGE 61

Notethatforthegeneralcase,itispossiblethatamarketwillnotbesuppliedbyanyofthermsduetohightransportationcosts.Ontheotherhand,forthecasewithIJ,ifrmslocatefacilitiesinamarket,thenthismarketwillnecessarilyreceivesomepositivesupply.Itshouldbenotedthatwhenperunitproductioncostatmarketjisaccountedfor,theconditionaj>0readsasaj)]TJ /F3 11.955 Tf 12.16 0 Td[(vj>0consideringtheaforementionedtransformation.Hence,marketjwillbesuppliedfromthefacilitieswithinthemarketaslongastheperunitproductioncostinmarketjislessthanthemarketpriceatzerosupply,i.e.,aj>vj.However,itisnotnecessarythatthemarketwillbesuppliedstrictlyfromthefacilitieswithinthemarketarea;thatis,rmsmayalsosupplythemarketfromthefacilitiesoutsidethemarketarea,ifcongestioncostswithinthemarketarehigh.Next,wediscusstheeffectsofvariationinjj,thecongestioncostfactorinmarketj,onequilibriumsupplyquantities. Propositions 3.2 3.4 remainvalidforthisspecialcase.Furthermore,thediscussionfollowingPropositions 3.2 3.4 continuestohold.Inparticular,considerascenarioinwhichamarketisonlysuppliedfromfacilitieslocatedinthatmarket.Whencongestioncostsincreasewithinthemarketarea,rmswillreacttothisincreasebyeithercontinuingtosupplythemarketfromthesamefacilitiesorusingadditionalfacilitieslocatedoutsidethemarketarea.Ifrmsstillsupplythemarketonlyusingfacilitieswithinthemarketarea,thetotalsupplytocustomersinthemarketwilldecrease.Fromapracticalpointofview,thisimpliesthatrmswillchoosetosupplylesswithinthemarketbecausetheywillnotbeabletodeliverwithinaspecieddeliverytimeframeorwithahighenoughservicelevelinthecaseofhighcongestioncosts.Furthermore,indoingso,thedecreasedsupplyinthemarketwillleadtoapriceincrease(or,equivalently,rmswillincreaseprice,resultinginareducedmarketdemand).Ifsuppliersdecidetouseadditionalfacilitiesoutsidethemarketareatosupplythemarket,theywillstillreducetheirsupplycomingfromfacilitieswithinthemarket.Thiswillavoidhighcongestioncosts;however,theynowpayhigherunittransportationcostsbysupplying 61

PAGE 62

themarketfromfacilitiesoutsidethemarketarea.Fromapracticalperspective,thisimpliesthatrmswillavoidcongestioncostsandmaintainaspecieddeliverytimeorservicelevelcommitmentsbysupplyingaportionofthemarketdemandfromoutsidethemarket.InSection 3.4 ,wepresenttheresultsofnumericalstudiesthatillustratehowcongestioncostsaffectrms'facilitylocationdecisionsforthisspecialcase. 3.4NumericalStudies Ournumericalstudiesfocusoncharacterizingtheeffectsoftrafccongestioncostsontherms'bestdecisionsforthegeneralcase.WethenpresentournumericalstudiesforthespecialcasedenedinSection 3.3.1 Wenextdiscusstheresultsofourcomputationalstudiesonthegeneralcase.Wegeneratedataforourcomputationalstudiesonthegeneralcaseinthefollowingway.Weconsiderfourproblemclasses,whereeachproblemclassdiffersintransportationcosts,cij,andfacilitylocationcosts,fi.Foreachoftheclasses,weuseallcombinationsofk2f3,5g,n2f3,5,7gandm2f3,5,7,10g,resultingin24combinationsofthevaluesofk,n,andm.Foreachofthesecombinations,wegenerate10probleminstancesandeachprobleminstanceissolvedfor16differentintervalsoftrafccongestioncostfactor,ij,startingfrom0andincreasingto8inincrementsof0.5.Thiswaywecananalyzetheeffectsofincreasingcongestioncostonthefacilitylocationandsupplyquantitydecisionsoftherms.Foreveryproblem,weletajU[50,150]andbjU[1,2].Table 3-1 givesthedistributionintervalofcijandfivaluesineachproblemclass. Table3-1. Dataintervalsforproblemclasses1-4 cijfiClass1(0,50][75,125]Class2(0,50][100,150]Class3[25,75][75,125]Class4[25,75][100,150] Ineachproblemclass,wesolve240probleminstances,andeachinstanceissolved16times,onceforeachintervalofijvalues.Foreachprobleminstance,we 62

PAGE 63

determinethebestlocationdecision(usingtotalenumeration)andthecorrespondingequilibriumquantitydecisionsforasinglerm.Wedocumentthefollowingaveragestatisticsover960probleminstances(240ineachineachProblemClass)foreachintervalofijvaluesinTable 3-2 :agivenrm'snumberoffacilities(#offac.),totalquantitysuppliedtomarkets(SupplyQuant.),totaltransportationcosts(Trans.Cost),totaltrafccongestioncosts(Cong.Cost),totalfacilitylocationcosts(Loc.Cost)andtotalprot. Table3-2. Averagestatisticsoverproblemclasses1-4foreachinterval Rangeof#ofSupplyTrans.Cong.Loc.Totalijfac.Quant.CostCostCostProtInterval1(0,0.5]2.7850.26967.00297.91308.613089.29Interval2[0.5,1]3.5443.22902.27576.97391.982261.97Interval3[1,1.5]3.8639.37879.07665.10427.381855.16Interval4[1.5,2]4.0036.42853.04714.36442.611574.12Interval5[2,2.5]4.1234.09827.08743.01455.171361.77Interval6[2.5,3]4.1932.12797.90761.66462.931194.12Interval7[3,3.5]4.2130.34768.56772.38465.251054.87Interval8[3.5,4]4.2028.70736.29777.83463.28936.21Interval9[4,4.5]4.1727.19705.62777.58458.74835.39Interval10[4.5,5]4.1625.90678.09775.77456.82748.39Interval11[5,5.5]4.0924.59647.55769.48449.28671.05Interval12[5.5,6]4.0423.45620.06761.90442.83603.85Interval13[6,6.5]3.9422.23587.91747.70431.71544.33Interval14[6.5,7]3.8321.06556.68730.90419.15491.24Interval15[7,7.5]3.7119.94526.62712.79405.20444.18Interval16[7.5,8]3.6118.93499.51694.84394.05402.12 Figure 3-1 illustrateshowdifferentperformancemeasuresbehaveasthecongestioncostfactorincreases(increasingintervalonthehorizontalaxiscorrespondstoincreasingcongestioncostfactor).ThefollowingconclusionscanbedrawnbyanalysisofFigure 3-1 (andtheunderlyingdatainTable 3-2 ). 1. ThenumberoffacilitieslocatedincreaseswiththecongestioncostparametervaluesuptoInterval8.AfterInterval8,itdecreases.Thatis,rmswilllocatemorefacilitieswhencongestioncostparametervaluesincreaseuptoapoint.However,afterapoint,rmswilllocatefewerfacilitieswiththeincreaseincongestioncostparametervalues.Notethatthefacilitylocationcostfollowsthesamepattern. 63

PAGE 64

ANumberoffacilitiesvs.interval&totalquantityvs.interval BTransportationcostvs.interval&congestioncostvs.interval CFacilitylocationcostvs.interval&totalprotvs.interval Figure3-1. Patternsofeachcolumnintable 3-2 64

PAGE 65

2. Thetotalquantitysuppliedbyarmdecreasesasthecongestioncostparametervaluesincrease.ThisresultisconsistentwithPropositions 3.3 and 3.4 .Totaltransportationcostalsofollowsthesamepattern. 3. ThetotaltrafccongestioncostincreaseswiththecongestioncostparametervaluesuptoInterval8.AfterInterval8,itdecreases. 4. Thetotalprotdecreasesasthecongestioncostparametervaluesincrease. WenotethatthepatternsobservedinTable 3-2 werealsoobservedwithineachproblemclassstudiedindividually.Consideringthepointsnotedabove,arm'sreactiontoanincreaseincongestioncostcanbeexplainedasfollows.Uptoapoint,armwilllocatemorefacilitiesandsupplylesstomarkets,inordertomaximizeprotbyincreasingthemarketpriceanddecreasingtransportationcoststocompensatefortheincreaseincongestioncosts.However,whenthecongestioncostbecomessignicantlyhigh,thermwillsendlesssupplytomarketsfromfewersupplypointstoavoidcongestioncostsinordertoretainprotability. Next,wediscussnumericalstudiesforthecasewhenthepotentialfacilitylocationsarewithinthemarketareas.Wegeneratedataforourcomputationalstudiesinthefollowingway.Similartopreviousnumericalstudies,weconsiderfourproblemclasses.Table 3-3 givesthedistributionintervalforcijandfivaluesineachproblemclass.Wenotethattherangesofcijandfivaluesarenarrowerwhencomparedtotherangesusedforthefourproblemclassesstudiedpreviously.Thereasonbehindthisisthatwealsointendtoanalyzetherms'locationdecisionsacrossmarketswithsimilartransportationandfacilitylocationcostcharacteristicsbut,withdifferentmarketparameters.Toaccountfordifferentmarketcharacteristics,wedenethemarketpotentialastheratioaj=bjformarketj.Agreateraj=bjmeansthatarmwillgetmorerevenuebysupplyingthemarketthansupplyingthesameamounttoamarketwithloweraj=bjvalue.Asaresult,itispossibletoobservehowthemarketpotentials,i.e.,aj=bjvalues,affectrms'locationdecisions.Foreachoftheproblemclasses,weuseallcombinationsofk2f3,5gandn2f3,5,7,10g,resultingin8combinationsofkandn.Foreachcombination,welet 65

PAGE 66

themaximumnumberoffacilitiesthatcanbelocatedequalthenumberofmarkets,i.e.,n=m.Furthermore,wedenecij=0wheni=j,i.e.,ifrmslocatefacilitiesinmarketj,theywillhavenounittransportationcostfromthesefacilitiestomarketj.Foreachcombination,wegenerate10probleminstancesandeachprobleminstanceissolvedfor16differentintervalsoftrafccongestioncostfactor,ij,startingfrom0andincreasingto8inincrementsof0.5toanalyzetheeffectsofincreasingcongestioncostonrms'facilitylocationandsupplyquantitydecisions.Moreover,toanalyzetherms'locationchoicesascongestionincreases,weconsiderdifferentajandbjvaluesforeachmarket.Inparticular,weassignthelargestajandthelowestbjvaluestotherstmarket,andthelowestajandthehighestbjvaluestothelastmarket,bylettingaj=50+(100=n)(j)]TJ /F4 11.955 Tf 10.17 0 Td[(1)+(100=n)uandbj=2)]TJ /F4 11.955 Tf 10.17 0 Td[((1=n)(j)]TJ /F4 11.955 Tf 10.17 0 Td[(1))]TJ /F4 11.955 Tf 10.17 0 Td[((1=n)u,whereuisuniformlydistributedover(0,1].Forinstance,forproblemswith5markets,a5U[130,150]andb5[1,1.2]whilea4U[110,130]andb4[1.2,1.4].Ineachproblemclass,wesolve80probleminstances,andeachprobleminstanceissolved16times,foreachintervalofijvalues.Foreachprobleminstance,wedeterminethebestlocationdecision(usingtotalenumeration)andthecorrespondingequilibriumquantitydecisionsforasinglerm,aswellasthemarketsinwhichrmslocatefacilities.Table 3-4 documentsthefollowingstatisticsforeachintervalofijvalues:agivenrm'snumberoffacilities(#offac.),totalquantitysuppliedtomarkets(SupplyQuant.),totaltransportationcosts(Trans.Cost),totaltrafccongestioncosts(Cong.Cost),totalfacilitylocationcosts(Loc.Cost),andtotalprot. Table3-3. Dataintervalsforproblemclasses1-4 cijfiClass1[50,75][75,100]Class2[50,75][100,125]Class3[75,100][75,100]Class4[75,100][100,125] AgraphofeachstatisticinTable 3-4 isshowninFigure 3-2 .ThefollowingconclusionscanbedrawnbyanalysisofTable 3-4 66

PAGE 67

Table3-4. Averagestatisticsoverproblemclasses1-4foreachinterval Rangeof#ofSupplyTrans.Cong.Loc.Totalijfac.Quant.CostCostCostProtInterval1(0,0.5]6.2580.990.001060.42624.405765.99Interval2[0.5,1]6.2061.6333.981899.23618.343390.28Interval3[1,1.5]6.0650.56159.241960.67603.252313.11Interval4[1.5,2]5.8943.75293.481841.49585.981745.82Interval5[2,2.5]5.7338.99408.511694.78569.511389.53Interval6[2.5,3]5.6035.46488.721565.77555.201144.15Interval7[3,3.5]5.4632.63544.611453.36541.46960.31Interval8[3.5,4]5.3330.27579.771358.20527.44818.32Interval9[4,4.5]5.1428.12599.851270.58508.43704.50Interval10[4.5,5]5.0326.41609.811200.56496.85610.32Interval11[5,5.5]4.8524.67609.851129.61478.24529.01Interval12[5.5,6]4.7123.23605.851071.15464.04461.50Interval13[6,6.5]4.5421.80596.251012.03447.07402.49Interval14[6.5,7]4.3520.38580.06950.14426.32352.30Interval15[7,7.5]4.1819.12563.77894.02408.74307.95Interval16[7.5,8]3.9717.85543.13834.78387.15269.76 1. Thenumberoffacilitieslocateddecreasesasthecongestioncostparametervaluesincreaseforthespecialcase.Thisisbecausewhencongestioncostsarelow,rmsprefertolocatefacilitiesineachmarketarea;hence,theydonotpaytransportationcosts.Ascongestionincreases,rmsprefertolocatefewerfacilitiessothattheycanavoidhighcongestioncostsineachmarketareabypayingtransportationcostsforshipmentsfromfacilitiesinmarketareastomarketareaswithoutfacilities.Notethatfacilitylocationcostfollowsthesamepattern. 2. Thetotalquantitysuppliedbyarmdecreasesasthecongestioncostparametervaluesincrease.Ontheotherhand,totaltransportationcostincreasesuptoapointandthendecreasesafterthatpoint.Thisisduethefactthat,ascongestioncostsincreasewithinmarkets,rmsprefertopaytransportationcostsfromoutsidethemarkets,eveniftheydecreasetheirtotalquantitiessuppliedtothemarkets;whencongestioncostsarehighenough,thesuppliersreducesuppliesandtendtosupplymarketsfromfacilitieswithinthemarket(andthusdonotpaytransportationcosts). 3. Thetotaltrafccongestioncostincreaseswiththecongestioncostparametervaluesinitiallyandthenitdecreases.Thisisbecause,uptoanintervalcongestioncostparametervalues,rmsagreetopaycongestioncosts;butafterthispoint,rmsnowtradetransportationcostsforhighercongestioncosts. 4. Thetotalprotdecreasesasthecongestioncostparametervaluesincrease. 67

PAGE 68

ANumberoffacilitiesvs.interval&totalquantityvs.interval BTransportationcostvs.interval&congestioncostvs.interval CFacilitylocationcostvs.interval&totalprotvs.interval Figure3-2. Patternsofeachcolumnintable 3-4 68

PAGE 69

Consideringthepointsnotedabove,arm'sreactiontoanincreaseincongestioncostwhenIJcanbeexplainedasfollows.Forlowlevelsofcongestioncosts,rmslocatefacilitiesinmoremarketareassothattheydonotpaytransportationcosts.Ascongestioncostsincrease,however,rmsprefertolocatefacilitiesinfewermarketareas,sothattheycanpreventpayinghighcongestioncosts;however,inthiscase,theypayhighertransportationcosts.Moreover,thetotalquantitiessuppliedtomarketsdecrease,whichresultsinanincreaseinmarketprices.Next,weanalyzetherms'choiceofmarketareastolocatefacilitiesforthespecialcase. Toaccountfordifferentmarketcharacteristics,wedenethemarketpotentialastheratioaj=bjformarketj.Ahigheraj=bjvalueimpliesthatarmwillobtainmorerevenuebysupplyingthismarketthansupplyingthesameamounttoamarketwithaloweraj=bjvalue.Asaresult,itispossibletoobservehowthemarketpotentials,i.e.,aj=bjvalues,affectrms'locationdecisionsthroughournumericalstudies.InTable 3-5 ,weshowtheaveragenumberoftimesthatrmslocatefacilitiesineachmarketareaoveralloftheprobleminstances.ItfollowsfromTable 3-5 thatrmsprefertolocatefacilitiesinmarketareaswithhighmarketpotentials,i.e.,marketswithgreateraj=bjvalues. Table3-5. Averagenumberoftimesrmslocatedfacilitiesinmarkets n=10n=7n=5n=3Market11.000.990.980.85Market21.000.990.970.75Market31.000.960.920.49Market40.980.930.73-Market50.950.830.52-Market60.900.71--Market70.820.47--Market80.76---Market90.61---Market100.45--Furthermore,Tables 3-6 through 3-9 showtheaveragenumberoftimesrmslocateineachmarketareaforeachintervalofcongestioncostfactoroverallprobleminstanceswiththesamenumberofmarkets.Theseresultsindicatethatascongestioncosts 69

PAGE 70

Table3-6. Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=3(M:market) M1M2M3Interval11.001.001.00Interval21.001.001.00Interval31.001.000.93Interval41.001.000.85Interval51.001.000.78Interval61.001.000.73Interval71.000.990.66Interval81.000.950.60Interval91.000.910.51Interval101.000.860.44Interval110.990.790.36Interval120.960.730.28Interval130.910.660.19Interval140.840.600.19Interval150.750.530.15Interval160.690.490.13Average0.850.750.49 Table3-7. Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=5(M:market) M1M2M3M4M5Interval11.001.001.001.001.00Interval21.001.001.001.000.95Interval31.001.001.001.000.84Interval41.001.001.000.980.70Interval51.001.001.000.910.64Interval61.001.001.000.910.56Interval71.001.001.000.860.55Interval81.001.000.990.810.50Interval91.001.000.960.730.46Interval101.001.000.980.680.41Interval111.000.990.900.610.34Interval121.000.990.880.560.31Interval131.000.980.810.480.30Interval140.940.910.780.410.29Interval150.940.860.740.380.25Interval160.860.810.650.360.23Average0.980.970.920.730.52 70

PAGE 71

Table3-8. Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=7(M:market) M1M2M3M4M5M6M7Interval11.001.001.001.001.001.001.00Interval21.001.001.001.001.001.000.95Interval31.001.001.001.001.001.000.84Interval41.001.001.001.001.000.990.63Interval51.001.001.001.001.000.940.50Interval61.001.001.001.000.950.850.43Interval71.001.001.001.000.940.780.40Interval81.001.001.000.990.900.710.39Interval91.001.000.990.980.800.630.38Interval101.001.000.990.950.760.600.36Interval111.001.000.960.910.760.510.34Interval121.001.000.930.910.740.530.30Interval131.001.000.930.850.710.490.28Interval141.001.000.890.810.610.450.26Interval150.980.960.860.780.560.440.25Interval160.910.900.810.750.560.410.23Average0.990.990.960.930.830.710.47 Table3-9. Averagenumberoftimesrmslocatedfacilitiesinmarkets,n=10(M:market) M1M2M3M4M5M6M7M8M9M10Interval11.001.001.001.001.001.001.001.001.001.00Interval21.001.001.001.001.001.001.001.001.000.89Interval31.001.001.001.001.001.001.001.000.940.69Interval41.001.001.001.001.001.001.001.000.840.59Interval51.001.001.001.001.001.000.990.950.780.46Interval61.001.001.001.001.000.990.960.890.680.45Interval71.001.001.001.000.981.000.890.830.590.40Interval81.001.001.000.990.980.960.890.780.500.39Interval91.001.001.000.990.980.890.850.730.460.34Interval101.001.001.000.990.960.880.800.660.450.36Interval111.001.001.000.990.950.840.730.660.450.33Interval121.001.000.990.980.910.810.680.640.450.30Interval131.001.000.990.980.900.790.650.580.450.28Interval141.001.001.000.950.890.750.590.540.430.28Interval151.001.000.990.930.860.750.590.510.410.26Interval161.001.000.980.880.850.730.590.490.380.23Average1.001.001.000.980.950.900.820.760.610.45 71

PAGE 72

increase,theaveragenumberoftimesthatrmslocatefacilitiesinaspecicmarketareadecreases.Thisresultisconsistentwiththerstpointabove.Furthermore,itcanbeseenthatwithinanyintervalofcongestioncostparametervalues,rmsprefertolocatefacilitiesinmarketswithhighermarketpotentialascongestioncostsincrease. 72

PAGE 73

CHAPTER4COMPETITIVEMULTI-FACILITYLOCATIONGAMESWITHNON-IDENTICALFIRMSANDCONVEXTRAFFICCONGESTIONCOSTS 4.1MotivationandLiteratureReview Theproblemofinterestinthischapterassumescompetitionbetweenmultiple,non-identicalrmssupplyingasingleproducttomultiplemarkets.Eachrmmustdetermineitssupplyfacilitylocationsandthequantitiesitwillsupplyfromeachfacilitytoeverymarket.Firmsarenoncooperativeandmustmakesimultaneousdecisions.Potentialfacilitylocationsandmarketsarelocatedonanitenumberofverticesofanetwork.ThecompetitionbaseisthatofCournot,i.e.,thepriceinamarketisdeterminedbythetotalquantitysuppliedtothemarket.Inparticular,weextendtheproblemsstudiedinChapters 2 and 3 byconsideringheterogeneousrms.Thatis,anasymmetriccompetitivemulti-facilitylocationproblemwithconvextrafccongestioncostsisanalyzed.OnemayrefertoChapters 2 and 3 forfurtherdiscussiononcompetitivefacilitylocationproblems,motivationtoconsidermulti-facilitylocationgameswithtrafccongestioncosts. SimilartoChapter 2 ,atwo-stagesolutionapproachisadopted:rst,PureNashEquilibrium(PNE)supplyquantitiesforgivenfacilitylocations(theStage-twogame)aredeterminedandthesearethenusedtosearchforequilibriumfacilitylocations(theStage-onegame).Themaincontributioninthisworkliesintheanalysisofthecaseofheterogenoussuppliers,whichrequiressubstantiallydifferenttechniques.Inparticular,avariationalinequalityapproachisutilizedforsolvingtheStage-twogameandtheresultingsolutiontechniqueisembeddedwithinaheuristicsearchmethodfortheStage-onegame. GabayandMoulin ( 1980 )suggestvariationalinequalitiesasamechanismtodetermineequilibriumsolutionsinnoncooperativegames.Onemayreferto FacchineiandPang ( 2003 ), HarkerandPang ( 1990 )and KinderlehrerandStampacchia ( 1980 )foranintroductiontovariationalinequalities,solutionapproachesandtheproblemsstudied 73

PAGE 74

invariationalinequalitytheory.Applicationsofvariationalinequalitiesonequilibriumproblemscanbeseenin Konnov ( 2007 )and Nagurney ( 1999 ). Dongetal. ( 2004 )and Nagurneyetal. ( 2002 )providerepresentativeexamplesofvariationalinequalityformulationsofequilibriumproblemsincompetitivesupplychains.Intheliterature,differentsolutionapproacheshavebeenproposedfordifferenttypesofvariationalinequalityproblems(VIP). HanandLo ( 2002 ), He ( 1997 ), HeandLiao ( 2002 )and Wangetal. ( 2001 )considernonlinearVIPs,whereas Andreanietal. ( 1997 ), HeandZhou ( 2000 )and LiaoandWang ( 2002 )focusonlinearVIPs. Inparticular,spatialnetworkequilibriumproblemshavebeenstudiedusingassociatedvariationalinequalityformulationsintheliterature.Spatialnetworkequilibriumproblemsfocusonpricecompetition(orCournot-typequantitycompetition)forasetofrmsonanetwork. Frieszetal. ( 1983 ), Frieszetal. ( 1984 ), ChaoandFriesz ( 1984 ), Harker ( 1984 1986 ), Tobin ( 1987 ), DafermosandNagurney ( 1984 1987 1989 ), Nagurney ( 1987 1988 ),and Milleretal. ( 1991 )studyspatialnetworkequilibriumproblemsunderdifferentassumptions,andstudyvariationalinequalityformulationsfortheseequilibriumproblems. TobinandFriesz ( 1986 )introducefacilitylocationdecisionswithinspatialnetworkequilibriumproblems,andtheydenespatiallycompetitivenetworkfacilitylocationproblems.Inspatiallycompetitivenetworkfacilitylocationproblems,anenteringrm'sfacilitylocationandproductiondecisionsareanalyzedbyanticipatingthereactionsofcompetingrmswhoalreadyhaveexistingfacilitiesonthegivennetwork.Asnotedby Frieszetal. ( 1988b ),thisproblemcorrespondstoaStackelberggame,wheretheenteringrmistheleaderandthermswithexistingfacilitiesarethefollowers. TobinandFriesz ( 1986 ), Tobinetal. ( 1995 ), Frieszetal. ( 1988a ), Frieszetal. ( 1988b 1989 ), Milleretal. ( 1992a 1996 ),and Milleretal. ( 1992b ),alsostudyspatiallycompetitivenetworkfacilitylocationproblemsundervariousassumptions.Thischapterstudiesthesimultaneousfacilitylocationdecisionsofasetofnon-identicalcompetingrms,i.e.,thegamedenedinthisstudyisnota 74

PAGE 75

Stackelberggame.ItisworthnothingthatStackelberggamesareoftenmodeledasmathematicalprogramswithequilibriumconstraints(MPECs).MPECsconsideranoptimizationproblemofaleaderwithconstraintsdeningthefollowers'equilibriumconditions.Onemayreferto Luoetal. ( 1996 )fordetaileddiscussiononMPECs.Thechapterdoesnotusealeader-followersetting. FindingasolutiontotheStage-onegameisimportantforunderstandingandcharacterizingthestructuralpropertiesofequilibriumfacilitylocations.Governmentagencies,landuseplanners,andsupplierstocompetingrmsmaybenetfromunderstandingthelocationsprivatedecisionmakerswillchooseinequilibrium.Ontheotherhand,ndingequilibriumfacilitylocationscanbecomputationallychallenging.Heuristicmethodsarediscussedfortheproblemsstudiedby Rhimetal. ( 2003 )and SaizandHendrix ( 2008 ). Rhim ( 1997 )proposesageneticalgorithmtondaPNElocationdecisionforthemodeldescribedin Rhimetal. ( 2003 ).Ontheotherhand, SaizandHendrix ( 2008 )provideamulti-startsearchalgorithm.Inthecaseofidenticalrms,itisnotedinChapter 2 thatrmswillchooseidenticalfacilitylocationsinequilibrium.Thus,whenrmsarehomogeneous,theywillultimatelychooseidenticalfacilitylocationsinequilibrium,whichsubstantiallyreducestherequiredsearchspace.Intheanalysisoftheheterogenouscase,however,thesearchcannotberestrictedtothecaseofidenticalfacilitylocationchoicesforeachrm;hence,aheuristicmethodisdiscussedintheanalysesoftheStage-onegameforthemoregeneralcaseofheterogenousrms.Inparticular,conditionsthatmustbesatisedbyaPNElocationdecisionaredenedandaheuristicmethodisdesignedthatenablesafastsearchforasolutionthatsatisestheseconditions. Therestofthischapterisorganizedasfollows.InSection 4.2 ,theproblemunderconsiderationisformulatedanddetailsoftheproblemsettingandsolutionapproacharepresented.Section 4.3 discussesthesolutionofequilibriumsupplyowsforgivenlocationdecisionsoftherms.InSection 4.4 ,theStage-onegameisstudied.Sections 75

PAGE 76

4.4.1 4.4.4 analyzetheconditionsthatanequilibriumlocationdecisionmustsatisfyandprovideaheuristicsearchmethodfortheStage-onegame.Themodelisthenextendedtothemulti-productcaseandimplicationsformulti-echelonsupplychannelsarediscussedinSection 4.5 .InSection 4.6 ,numericalstudiesontheefciencyoftheheuristicmethodaredocumented. 4.2ProblemFormulationandSolutionApproach ThemodelofinterestinthischapterisageneralizationoftheonediscussedinChapter 2 ,whichconsidersidenticalrms.Thatis,asetofknon-identicalrms,indexedbyr2R=f1,2,,kg,whowishtosupplyasetofncustomermarkets,indexedbyj2J=f1,2,,ng,isconsidered.Thermscompetewitheachotherinthemarketsforthesalesofasingleproduct(thissettingisextendedtoaccountformultipleproductsinSection 4.5 ).Firmsmaylocatesupplyfacilitiesatmpotentiallocations,indexedbyi2I=f1,2,,mg,inordertosupplythemarkets.Thecostsincurredbysupplyrmsincludetransportation(linearinthequantityshippedfromfacilitiestomarkets),trafccongestion(convexandnon-decreasinginthequantitysuppliedfromafacilitytoamarket),andxedfacilitylocationcosts.Amarket'spriceforthegoodisalinear,decreasingfunctionofthetotalquantitysuppliedtothemarketfromallrms,andeachrmwishestomaximizeitsownprot.Moreover,itisassumedthatarmwillnotopenmorethanonefacilityatagivenlocation,implyingthatthermwillcreatesufcientcapacityatthelocationtoaccommodatethequantitysuppliedbythefacilitytoallmarketsinequilibrium. Inparticular,letqijr0denotethequantityshippedfromthefacilityofrmratlocationitomarketj,qjrdenotethetotalquantityshippedtomarketjbyrmr(i.e.,qjr=Pi2Iqijr),qirdenotethetotalquantityshippedfromlocationibyrmr(i.e.,qir=Pj2Jqijr),qijdenotethetotalquantityshippedfromlocationitomarketj(i.e.,qij=Pr2Rqijr),andqjdenotethetotalquantityshippedtomarketj(i.e.,qir=Pr2RPi2Iqijr).Furthermore,letusdeneQaskmnmatrixofqijrvalues, 76

PAGE 77

Xasmkbinarymatrixrepresentingrms'locationdecisions,xrasanm)]TJ /F1 11.955 Tf 9.3 0 Td[(vectorrepresentinglocationdecisionsofrmr,andxirsuchthatxir=1ifrmrlocatesafacilityatlocationi,xir=0otherwise. Thepriceinmarketj,pj,isdenedidenticallywithEquation( 2 ),i.e., pj(qj)=aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj,(4) whereaj0andbj>0denotethepriceatzerodemandandthepricesensitivityformarketj(bothparametersareassumedtobenitenumbers).NotethatEquation( 4 )istheinversedemandfunctionassociatedwithCournotcompetition.Asillustratedbytheprotfunction,thetransportationcostislinearinthequantitysentfromfacilityitomarketjforanyrmrwithmarginalcostcijr0.Itshouldberemarkedthatcijrcanbeeasilyadjustedtoaccountforanyper-unitproductioncostswithoutlossofgenerality.Thatis,alocation-specicparametervir0denotingtheper-unitproductioncostatlocationiforrmrcanbeaddedtocijr.Thetrafccongestioncostcoefcientforlink(i,j)forrmrisdenedasgijr(whichisafunctionofthetotalquantityofowonthelink),where gijr(qij)=ijrqij.(4) Theparameterijr>0isatrafccongestioncostmultiplierforowonlink(i,j)forrmr.Hence,thecongestioncostincurredbyarmusinglink(i,j)increasesinthetotalowonthelink.Chapter 3 providesajusticationofthisfunctionalform,whichassumesthatthecongestioncostincurredbyrmronlink(i,j)isnondecreasingandconvexinqijr.Furthermore,Equation( 4 )introducesanothertypeofcompetition,competitiononthedistributionnetwork,inadditiontothecompetitionwithinthemarketsimpliedbyEquation( 4 ). Theprotfunctionofarmconsistsofthesupplyrm'stotalrevenue,lessvariabletransportationcosts,trafccongestioncosts,andfacilitylocationcosts(fr(xr)=Pi2Ixirfirdenotesthetotalfacilitylocationcostforrmr,wherefirdenotesthexedcost 77

PAGE 78

ofopeningafacilityatlocationiforrmr).Then,theprotfunctionforeachrmrcanthenbeformulatedasfollows: Hr(Q,X)=Xj2Jpj Xi2IXr2Rqijr!Xi2Iqijr)]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xj2JXi2Icijrqijr)]TJ /F10 11.955 Tf 11.96 11.35 Td[(Xj2JXi2Iqijrgijr Xr2Rqijr!)]TJ /F3 11.955 Tf 11.96 0 Td[(fr(xr).(4) Asistypicalinpractice,armwilldeterminethelocationsoffacilitiespriortodeterminingthequantitiestosupplyfromfacilitiestomarkets;thus,itisassumedthatrmssimultaneouslydeterminetheirfacilitylocations(therststage)andthentheirsupplyquantities(thesecondstage).Thefollowingtwo-stagesolutionapproachisadopted.First,givenfacilitylocationdecisions,theStage-twodecisionsareanalyzed.Then,usingthesolutionoftheStage-twogame,theStage-onedecisionsareanalyzed. Inparticular,theStage-onegamecorrespondstoak-matrixgame(eachrmisaplayer),whereeachplayerhasthesamesetof2mstrategies(eachstrategyofrmrisabinaryvector,xr,denotingrmr'sfacilitylocationdecisionsatmlocations).Therefore,checkingwhetheralocationmatrixXisanequilibriumrequiresk(2m)]TJ /F4 11.955 Tf 12.25 0 Td[(1)comparisons(asonemustcheckwhetherxristhebestoptionforrmr,whichrequires2m)]TJ /F4 11.955 Tf 13.02 0 Td[(1comparisons,andthiswillberepeatedforallplayers).Furthermore,thereare2mkalternativeoutcomesoftheStage-onegame.Nevertheless,asisdiscussedinSection 4.4 ,notallalternativelocationmatricesarecandidatesforanequilibriumsolution.Section 4.4 introducespropertiesofanequilibriumlocationmatrix,denotedbyX,tonarrowthesearchforX,andtoapplythesepropertiesinasearchheuristicthatreturnsX,ifoneexists. Ontheotherhand,oneneedstodeterminetheequilibriumsupplyquantitiesforagivenlocationmatrixtocheckwhetherthegivenlocationmatrixisanequilibriumsolution.Section 4.3 formulatestheproblemofndingequilibriumsupplyquantitiesasavariationalinequalityproblem(VIP)anddescribesaproceduretodetermineequilibriumsupplyquantitiesforanygivenlocationmatrixX,denotedbyQ(X).Thisprocedureis 78

PAGE 79

alsoutilizedwithinthesearchheuristicdescribedinSection 4.4 ;hence,anequilibriumsolutionX,ifoneexists,andacorrespondingQ(X),areultimatelygenerated. Thesolutiongeneratedbythetwo-stagesolutionapproach,i.e.,theX,Q(X)pair,iscalledaSubgamePerfectNashEquilibrium( Selten 1975 ),astheproblemofndingequilibriumlocationandsupplyquantitiesissolvedintwostages(eachstagerepresentsasubgame).However,itshouldbenotedthatthetwo-stagesolutionapproachsolvestheintegratedequilibriumproblem,whichseeksequilibriumfacilitylocationsandsupplyquantitiessimultaneously.ThisfollowsfromtheuniquenessofQ(X)foranyX(theuniquenessresultwillbediscussedinSection 4.3 );therefore,thesetofSubgamePerfectNashEquilibriumsolutionsachievedbythetwo-stagesolutionapproachisequaltothesetofNashEquilibriumsolutionsoftheintegratedequilibriumproblem. 4.3Stage-TwoDecisions ThissectionstudiestheStage-twogameforagivenX,i.e.,whenxir2f0,1g8i=1,2,...,m,r=1,2,...,khavebeenpre-determined.Thisimpliesthatfr(xr)isxed,andcanbeignoredwhenanalyzingtheStage-twogame.Basedontheprotfunction( 4 )anddenitionsoftheprice( 4 )andcongestion( 4 )functions,rmr'soptimizationproblem,giventhefacilitylocations,canthenbewrittenas maxXj2Jh(aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj)qjr)]TJ /F10 11.955 Tf 11.96 11.35 Td[(Xi2Icijrqijr)]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xi2Iijrqijrqijis.t.qjrWxir8i2I,qijr08i2I,j2J, (4) whereWisalargenumber.Therstsetofconstraintsensuresthatrmrcanonlysupplyamarketfromanopenfacility,andthesecondsetofconstraintsimposesthenonnegativityonthesupplyquantities.Duetothemarketpriceandtrafccongestioncostfunctions,anyrm'squantitydecisionswillbeaffectedbytheotherrms'quantitydecisions;hence,aNashEquilibriumsolutionissoughttocharacterizetherms'quantitydecisions.ItisstraightforwardtoshowthattheobjectivefunctioninEquation 79

PAGE 80

( 4 )isastrictlyconcavefunctionineachvariableqijr0,becausebj>0andijr>0.ItfurtherfollowsfromEquation( 4 )thattheobjectivefunctionofanyrmisseparableinthemarkets,i.e.,theequilibriumconditionsformarketjcanbeanalyzedindependentlyoftheothermarkets(similarresultsaregivenin Rhimetal. 2003 SaizandHendrix 2008 ,andChapter 2 ).Therefore,inwhatfollows,theStage-twogameforanarbitrarymarketjisexamined,giventhelocationchoicesoftherms. LetIrdenotethesetoflocationsatwhichrmrhasopenedafacilityandjIrjdenoteitscardinality(inthecorrespondinglocationmatrixX).Observethatqijr=0forallj2J,i=2Ir.Therefore,supplyquantitiesformarketj,giventhelocationmatrixX,canberepresentedbya-vector,where=Pr2RjIrj.LetQjdenotethisvectorsuchthatQj2R+.Then,theprotfunctionforrmrinmarketjunderthesesupplyquantities,denotedbyjr(Qj),canbewrittenas jr(Qj)=pj(qj)qjr)]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xi2Ircijrqijr)]TJ /F10 11.955 Tf 11.96 11.35 Td[(Xi2Irijrqijrqij.(4) DuetotheconcavityofEquation( 4 ),therst-orderconditions(@r(Qj)=@qijr=0,forqijrvaluessuchthatqijr>0)mustbesatisedataNashequilibriumsolutionfortheStage-twogame( Nash 1951 ).Inparticular,aNashequilibriumsolutionmustsatisfytheconditions aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bj[qj+qjr])]TJ /F3 11.955 Tf 11.95 0 Td[(cijr)]TJ /F5 11.955 Tf 11.95 0 Td[(ijr[qijr+qij]=0ifqijr>0, (4) aj)]TJ /F3 11.955 Tf 11.96 0 Td[(bj[qj+qjr])]TJ /F3 11.955 Tf 11.95 0 Td[(cijr)]TJ /F5 11.955 Tf 11.95 0 Td[(ijr[qijr+qij]0ifqijr=0. (4) Then,tondanequilibriumsolution,onemustdetermineqijrvaluesthatsolvetheassociatedrst-orderconditionssimultaneously. Thesolutionapproachforthecaseofhomogenousrms,discussedinChapter 2 ,isarelativelysimplesortingbasediterativeapproachsimilartotheonesproposedby Rhimetal. ( 2003 )and SaizandHendrix ( 2008 ).Specically,thelocationsatwhichrmshaveopenfacilitiesaresortedbasedoncostparameters,andthenthetotalquantities 80

PAGE 81

sentfromtheselocationsaredetermined.However,asortingmethodwillnotworkfortheStage-twogameofthecurrentmodel,asEquation( 4 )isnotseparableini(thefacilitylocationchoices)duetotheconvextrafccongestioncosts(neitherdoesthissortingmethodworkforthesymmetriccasewhenrmschoosedistinctlocationdecisions).Next,avariationalinequalityformulationfortheStage-twogameformarketjfortheheterogenouscaseispresented.Itshouldbenotedthatvariationalinequalityformulationshavebeendevelopedforspatialnetworkequilibriumproblems(see,e.g., Frieszetal. 1983 Frieszetal. 1984 Harker 1984 1986 DafermosandNagurney 1984 1987 1989 ChaoandFriesz 1984 Tobin 1987 Nagurney 1987 1988 ).TheassociatedvariationalinequalityformulationfortheStage-twogameformarketjisanasymmetricvariationalinequalityproblem(VIP)and,hence,onecanuseefcientsolutionalgorithmsdesignedforasymmetricVIPsfortheStage-twogameformarketj. TheNashequilibriumquantitiesmustbeoptimalforeachrm,giventheoptimaldecisionsofallotherrms.Astheprotfunctionofeachrmisstrictlyconcaveineveryqijr,theoptimalityconditionsforeachrmcanbewritteninvariationalinequalityform.Itfollowsfrom GabayandMoulin ( 1980 ), Nagurney ( 1999 ),and Nagurneyetal. ( 2002 )thatQjisaNashequilibriumifitsatisesthefollowingvariationalinequality: )]TJ /F10 11.955 Tf 11.96 11.35 Td[(Xr2RXi2Ir@jr(Qj) @qijr(qijr)]TJ /F3 11.955 Tf 11.96 0 Td[(qijr)0,8Qj2R+.(4) ThevariationalinequalityinEquation( 4 )formarketjthentakesthefollowingexplicitform Xr2RXi2Ir[cijr)]TJ /F3 11.955 Tf 11.96 0 Td[(aj+bj(qj+qjr)+ijr(qijr+qij)](qijr)]TJ /F3 11.955 Tf 11.95 0 Td[(qijr)0,8Qj2R+.(4) TostudythepropertiesofEquation( 4 ),aclassicalvariationalinequalityrepresentationisgivennext.Inparticular,letQrj2Srdenotethevectorofsuppliesfromrmrfacilitiestomarketj;thatis,Qrj=(q1jr,...,qjIrjjr)T8r2R,whereSrdenotesthestrategysetofrmr.ThenQj=((Q1j)T,...,(Qkj)T)T2S,whereS=S1...Sk,isaNash 81

PAGE 82

equilibriumsolutionifitsatises jr(Qrj,Q[)]TJ /F9 7.97 Tf 6.59 0 Td[(r]j)jr(Qrj,Q[)]TJ /F9 7.97 Tf 6.59 0 Td[(r]j)8Qrj2Sr,8r2R,(4) whereQ[)]TJ /F9 7.97 Tf 6.58 0 Td[(r]j=(Q1j)T,...,(Q(r)]TJ /F11 7.97 Tf 6.58 0 Td[(1)j)T,(Q(r+1)j)T,...,(Qkj)TT.ItcanbeshownthatQj2SsolvestheStage-twoGameatmarketjifitsolvesthefollowingVIPforthegivenlocationdecisionX: hF(Qj),Qj)]TJ /F7 11.955 Tf 11.96 0 Td[(Qji0,8Qj2S,(4) whereF(Qj)=(rQ1jj1(Qj),...,rQkjjk(Qj))isa-rowvectorfunctionandrQrjjr(Qj)=(@jr(Qj)=@q1jr,...,@jr(Qj)=@qjIrjjr)8r2R(see,e.g., Nagurney 1999 ).NotethatF(Qj)isalinear,continuous,anddifferentiablefunction. Observethatasrmswillnotsupplyagainstanegativeprice,supplyquantitiescanbeconsideredtohaveupperbounds(inparticular,onecansetW=aj=bjinEquation( 4 )aspj>0andqijr(aj)]TJ /F3 11.955 Tf 12.48 0 Td[(cijr)=bj,asrmswillnotendupwithnegativeprots).Theselinearupperboundsonthesupplyquantities,togetherwithnonnegativityofsupplyquantities,implyacompactandconvexsubsetofthestrategyset,withinwhichEquation( 4 )admitsatleastonesolution.Moreover,Equation( 4 )indicatesthatrQrjjr(Qj)=([c1jr)]TJ /F3 11.955 Tf 12.6 0 Td[(aj+bj(qj+qjr)+1jr(q1jr+q1j)],...,[cjIrjjr)]TJ /F3 11.955 Tf 12.6 0 Td[(aj+bj(qj+qjr)+jIrjjr(qjIrjjr+qjIrjj)]).ThentheJacobianmatrixofF(Qj),rF(Qj),consistsofthefollowingvalues:2bj+2ijr,2bj,bj+ijrorbj.Notingthatbj>0andijr>0,itfollowsthateachcomponentoftheJacobianmatrixispositive.Thus,foranyQj6=0,QTjrF(Qj)Qj>0.Thenitfollowsfromthemid-valuetheoremasnotedin Nagurney ( 1999 )thatF(Qj)isstrictlymonotoneonS.Inparticular,F(Qj)isstrictlymonotoneontheentirespaceR+.Asadirectresultofthestrictmonotonicity,Equation( 4 )hasauniquesolution.(Onecanreferto Nagurney ( 1999 )forproofsofthediscussionontheexistenceanduniquenessoftheequilibriumsolutionoftheStage-twogame.)BeforepresentingasolutionmethodforEquation( 4 ),itisworthnotingthattheequilibrium 82

PAGE 83

problemoftheStage-twogamecanequivalentlybestatedasthemaximizationofaquadraticconcavefunctionaswell. Next,analgorithmthatsolvestheVIPstatedinEquation( 4 )isdiscussed.BecauseEquation( 4 )impliesanasymmetriclinearVIP,analgorithmforasymmetriclinearVIPswillbegiven.However,itshouldbenotedthatthealgorithmsstudiedforspatialnetworkequilibriumproblemscanalsobeusedtosolvetheStage-twogame(see,e.g., Frieszetal. 1983 Frieszetal. 1984 Harker 1984 1986 DafermosandNagurney 1984 1987 1989 ChaoandFriesz 1984 Tobin 1987 Nagurney 1987 1988 ).Thealgorithmstatedbelowistheself-adaptiveprojectionmethodproposedby Han ( 2006 )forsolvinglinearvariationalinequalitiesofthefollowingform:(My+z)T(y)]TJ /F7 11.955 Tf 11.96 0 Td[(y)0,8y2K, whereyisthevectorofdecisionvariables(andydenotesasolution),Kisanonempty,convex,andclosedsubsetofRn,M2Rnnisagivenmatrix,andz2Rnisagivenvector.Asmentionedbefore,theresultinglinearVIPoftheStage-twogameforthegivenXisasymmetric,forwhichMisdenedbythepartialderivativesasstatedinEquation( 4 ),andthevectorzconsistsofthecijr)]TJ /F3 11.955 Tf 12.52 0 Td[(ajvalues.Moreover,K=S=R+.Thealgorithmcanbeformalizedasfollows. Algorithm3. Self-adaptiveProjectionmethodfortheVIPoftheStage-twogameatmarketj. Step0.StartwithaQ0j2R.Set`:=0.Set0<<2,0>0,0,andasequencef`g[0,1)withP1`=0`<1.Set`:=0. Step1.Determinee(Q`j,)=Q`j)]TJ /F3 11.955 Tf 12.54 0 Td[(PS[Q`j)]TJ /F5 11.955 Tf 12.55 0 Td[((MQ`j+z)],PS[]beingtheorthogonalprojectionfromRontoS.Ifke(Q`j,`)k1,stop. Step2.ComputethenextiterateusingQ`+1j=Q`j)]TJ /F5 11.955 Tf 11.96 0 Td[((I+`M))]TJ /F11 7.97 Tf 6.58 0 Td[(1e(Q`j,`). Step3.Choosethenextparameter`+1fromtheinterval1 1+```+1(1+`)`.Set`:=`+1andgotoStep1. 83

PAGE 84

ThealgorithmrequirescalculatingtheinverseofamatrixandtakingtheprojectionofapointontothesetS.InEquation( 4 ),Sisthenonnegativeorthantand,hence,projectioniseasilycarriedout.Inparticular,asnotedby Han ( 2006 )aswell,projectionontoSusingtheEuclidean-normisdenedcomponent-wiseforeachelementofthevectortobeprojected.Explicitly,PS[y]j=yjifyj0,and,PS[y]j=0ifyj<0.TheSelf-adaptiveProjectionmethod,statedinAlgorithm1,convergestoasolutionofthevariationalinequalityformulationinEquation( 4 )asMispositivedeniteforthevariationalinequalityformulationinEquation( 4 ).Moreover,thereexistsasolutionforEquation( 4 )whenSisthenonnegativeorthant.Then,itfollowsfrom Han ( 2006 )thatthealgorithmconvergestoasolutionofEquation( 4 ). 4.4Stage-OneDecisions Beforeprovidingamethodforndinganequilibriumlocationsolution,itisrstimportanttodiscussexistenceresultsforPNEsolutionsoftheStage-onegame. Rhimetal. ( 2003 )demonstratetheexistenceofaPNElocationdecisionunderidenticalrmsbyshowingthattheassociatedStage-onegamecanbemodeledasacongestiongamewheneachmarketisonlysuppliedfromfacilitiesatasinglelocation.Itisawell-knownresultthatcongestiongameshavePNEpoints( Rosenthal 1973 ).Theconditionthatamarketissuppliedfromasinglelocationispossiblewhenthemarginaldeliverycosttothemarketisthelowestfromthatsupplylocation.As Rhimetal. ( 2003 )modeldeliverycostasalinearfunctionofthesupplyquantity,themarginaldeliverycosttoamarketisconstantforeachlocation;thus,onecanverifythestatedexistencecondition.However,theproblemunderconsiderationinthisstudyappliesnonlinearcosts,andbecauseoftheconvexcongestioncosts,rmsmayprefertosupplyamarketfrommorethanonelocation,andtheselocationsmaybedistinctforeachrm. SaizandHendrix ( 2008 )extendthemodelof Rhimetal. ( 2003 )byrelaxingtheassumptionofidenticalrms.Themodelofinterestin SaizandHendrix ( 2008 )considersrm-andlocation-speciclineardeliverycosts,andtheexistenceofaPNElocationdecisionisnotguaranteed.The 84

PAGE 85

currentstudyisafurthergeneralizationof SaizandHendrix ( 2008 ),andtheexistenceofaPNElocationdecisionisnotguaranteed.However,theproposedsearchmethoddiscussednextndsanequilibriumlocationmatrix,ifoneexists,orconcludesthatthereisnotanequilibriumsolution. 4.4.1SearchingforAnEquilibriumLocationMatrix Todetermineanequilibriumlocationdecisionmatrix,ifoneexists,theinitialfocusliesindeningdominantstrategies,whicharecandidatesforapossibleequilibriumlocationmatrix.Later,agivenlocationmatrixinadominantstrategyshouldbecheckedtodeterminewhetheritisanequilibriumlocationmatrix. RecallthatQ(X)denestheequilibriumquantitiesforagivenlocationmatrixX,andtheprevioussectionshowedhowtondQ(X)foranygivenX.Letr(X)=Hr(Q(X),X)(i.e.,r(X)includesfacilitylocationcostsfr(X)=fr(xr)=Pi2Ifirxir).Then,similartoEquation( 4 ),theconditionrequiredforalocationmatrixXtocorrespondtoanequilibriumdecisionreads rxr,X[)]TJ /F9 7.97 Tf 6.59 0 Td[(r]rxr,X[)]TJ /F9 7.97 Tf 6.58 0 Td[(r]8xr,8r2R,(4) wherexrdenotestheequilibriumlocationdecisionofrmr,X[)]TJ /F9 7.97 Tf 6.59 0 Td[(r]denotestheequilibriumdecisionsofallotherrms,i.e.,X[)]TJ /F9 7.97 Tf 6.59 0 Td[(r]=[x1,...,xr)]TJ /F11 7.97 Tf 6.59 0 Td[(1,xr+1,...,xk],andXdenotesanequilibriumlocationmatrix,ifoneexists.ThefollowingcorollaryisadirectimplicationofEquation( 4 )andcharacterizesanon-equilibriumlocationmatrix. Corollary1. If9r2Rsuchthatr(X)<0,thenXisnotanequilibriumlocationmatrix. Corollary 1 followsfromthefactthatifr(X)<0,rmrwillbebetteroffbynotlocatinganyfacility,hence,Equation( 4 )impliesthatXisnotanequilibriumlocationmatrix.Thatis,Xcanbeanequilibriumdecisionifeachoneofitsnon-zerocolumnsproducespositiveprotforthecorrespondingrm.Thisconditionissimilartotheviabilityconditionusedin Rhimetal. ( 2003 ).Atthispoint,itisimportanttomentiontheStableSetconceptintroducedin DobsonandKarmarkar ( 1987 )andusedby Rhim 85

PAGE 86

( 1997 )and Rhimetal. ( 2003 )tostudylocationdecisionsinasimilarcompetitivefacilitylocationsetting. DobsonandKarmarkar ( 1987 )denestabilitywithrespecttothreedifferentcompetitivefactors:viabilityoflocations,conditionsofentry,andsurvival.In Rhimetal. ( 2003 ),eachrmmayopenatmostonefacility,andasetoffacilitylocationsisdenedtobestableaslongasthosermswithafacilitymakeapositiveprot(viabilitycondition)andthermswithoutafacilitycannotmakeapositiveprotbyopeningafacility(survivalcondition).Armisreferredtoasanentrantwheneverthermhasafacility.However,intheproblemofinterestinthisstudy,armmayopenmorethanonefacility.Thus,armisanentrantifthermopensatleastonefacility.Corollary 1 impliesthatanentrantrmshouldmakepositiveprotasaresultofitslocationandcorrespondingquantitydecisions,whichcanbereferredtoastheviabilitycondition.Itshouldbenotedthatwhileviabilityisnecessaryforalocationdecisiontobeanequilibrium,itisnotasufcientcondition.Thatis,aviablelocationdecisionisnotnecessarilyanequilibriumlocationdecision.Ontheotherhand,deningasurvivalconditioncanbeambiguous.Itshouldbenotedthatthesurvivalconditiondenedin Rhimetal. ( 2003 )doesnotimplythatanentrantrmmustchooseeachlocationthatisindividuallyprotable.Thisfollowsfromthefactthatthefacilitylocationdecisionsofanentrantrmarenotindependent.Inparticular,supposethatanentrantrmmaymakepositiveprotbylocatingafacilityatalocationwherethermhasnofacility.Itispossiblethatlocatingafacilityatthatlocationmaydecreasetheoverallprotoftheentrantrm.However,anon-entrantrm,bydenition,cannotmakeapositiveprotbylocatingasetoffacilitiesatanysubsetofthelocationsandthisdiscussionisalreadyindicatedbyCorollary 1 Inwhatfollows,aheuristicroutineisstatedtomovetoaviablelocationdecisionfromarandomlygivenlocationdecisionX.Priortothis,however,anotherroutineisdenedtoeasetheprocessofgeneratingaviablelocationdecision. 86

PAGE 87

Inparticular,itshouldbenotedthatwhilecheckingwhetherXisanequilibriumlocationdecision,oneshouldsetxir=0ifqir(X)=0,andthenchecktheconditionsinEquation( 4 ).Thissimplyfollowsfromthefactthatnormwilllocateanullfacility,i.e.,afacilitythatdoesnotsupplyanymarket.Hence,thefollowingroutineisdenedthatwillbeusedintheheuristicapproach.AlistofmatricesdenotedbyLisintroducedtokeeptrackofthelocationmatricesthatareshowntobenon-equilibrium. Routine0:GivenXandL,thefollowingproceduregeneratesadominatinglocationmatrix: 1:CheckwhetherX2L 2:IfX2L,setcontinue=0,stopandreturncontinue=0 3:Else,setcontinue=1,L=L[fXg,determineQ(X) 4:Ifqir(X)=0,setxir=0 5:ReturnX0=Xandcontinue=1. ThesetLconsistsofthelocationmatriceswhicharenotinequilibriumand,hence,ifthegivenlocationmatrixisinL,checkingwhetheritsatisestheequilibriumconditionsisnotrequired.However,whenX=2L,theoutputofRoutine0isthelocationmatrixX0,whichdominatestheoriginallocationmatrixXforatleastonerm(unlessX0=X),whichiswhythetermdominatinglocationmatrixisused.ThisroutineisnextusedingeneratingaviablelocationmatrixfromarandomlygivenlocationdecisionX. 4.4.2GeneratingAViableLocationDecision LetX0begeneratedfromarandomlygivenXbyusingRoutine0.IfX0isnotviable,thenthereexistsatleastonermwithnegativetotalprot.Thisfurtherimpliesthatthereexistsatleastonefacilityofthatrmwithnegativeprot,i.e.,forwhichthefacilitylocationcostexceedsthetotalprotofthermgainedbysupplyingmarketsfromthefacility.Thisdiscussiondoesnotimplythateachfacilitymustbeprotableinaviablelocationmatrix.Instead,itimpliesthattheremustbeafacilitywithnegativeprotinalocationmatrixthatisnotviable.Forsuchmatricesthatarenotviable,itthus 87

PAGE 88

makessensetosetxir=0whenir(X0)<0forarmrsuchthatr(X0)<0,whereir(X0)denotetherm'sprotatlocationi.Then,theequilibriumquantitiesforthemodiedmatrixXcanbedeterminedandthecorrespondingmodiedmatrixX0canbegenerated.Repeatingthisprocess,aviablelocationmatrixcanbefound.Specically,thisroutineisdenedasfollows. Routine1:GivenXandL,thefollowingproceduregeneratesaviablelocationmatrix: 1:ApplyRoutine0 2:Ifcontinue=0,stopandreturncontinue=0 3:Else,deneR)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(asthesetofrmssuchthatr(X0)<0,8r2R)]TJ ET BT /F1 11.955 Tf 29.26 -251.03 Td[(4:IfR)]TJ /F4 11.955 Tf 10.41 -4.33 Td[(=;,X0isviable;stopandreturnX1=X0 5:Else,let(^,^r)=argminfir(X0):i2I,r2R)]TJ /F2 11.955 Tf 7.09 -4.34 Td[(gandsetx^^r=0inX 6:GotoStep1. NotethatthequantitydecisionsforXandX0arethesame,i.e.,qir(X)=qir(X0).Hence,their(X0)valuescaneasilybecalculatedfromtheir(X)valuesbysimplylettingir(X0)=ir(X)whenx0ir=1orxir=0and,ir(X0)=0whenx0ir=0andxir=1.InStep5ofRoutine1,theinitiallocationmatrixXismodied;thatis,Routine1doesnotclosethefacilitieswithnegativeprotsunderX0andtrytogetaviablelocationmatrixthathasfewerfacilitieslocatedthanX0.ThereasonforsuchamodicationistocapturethepossibilitythatthenullfacilitiesclosedwithRoutine0canhavepositivesupplyafterthefacilitieswithpositivesupplybutwithnegativeprotsareclosed.Thus,aviablelocationmatrixwithpotentiallymoreopenfacilitiesisgenerated.Atthispoint,itshouldbenotedthatthematrixXenteringStep1ofRoutine1,andthemodiedmatrixgeneratedattheendofStep5ofRoutine1,differinonlyoneentry.Inparticular,letX(i)andX(i+1)aretwoconsecutivelocationmatricesenteringStep1ofRule1andletX0(i)andX0(i+1)bethematricesgeneratedbyRoutine0inStep1ofRule1correspondingtoX(i)andX(i+1),respectively.ThenX(i)andX(i+1)differinonlyoneentry.However, 88

PAGE 89

X0(i)andX0(i+1)maydifferinmorethanoneentry.ObservethatthefacilitywiththemostnegativeprotisclosedinStep5ofRoutine1,assuchafacilityislesslikelytobeopeninanequilibriumsolution.Itshouldberemarkedthatinaviablelocationmatrix,theremaybesomefacilitieswithnegativeprotforsomerms,althoughnormwillhavenegativetotalprot.Routine1willalwaysndaviablelocationmatrixafterstartingwitharandomlocationmatrix.ThisfollowsbecauseStep5ofRoutine1canberepeatedatmostmktimes,andafteratmostmkrepetitions,thereturnisX=0,whichisviable,intheworstcase.AviablelocationmatrixgeneratedattheendofRoutine1correspondingtothegivenlocationmatrixXisdenotedbyX1. 4.4.3EquilibriumCheck NowsupposethatX1isaviablelocationmatrixgeneratedfromXbyusingRoutine1.TodeterminewhetherX1isanequilibriumlocationmatrix,oneneedstocheckifx1r,therthcolumnofX1,isthebestresponseofrmr,8r2R.Todoso,itisrequiredtocheckallpossiblelocationvectorsforrmrwhilethelocationdecisionsoftheotherrmsarexed.Notethatthereare2mdifferentlocationdecisionsforeachrmand,hence,theprotofrmrshouldbeevaluatedfor2m)]TJ /F4 11.955 Tf 12.16 0 Td[(1locationvectors,whilekeepingtheotherrms'locationdecisionsunchanged.Ifx1risshownnottobethebestresponseofrmrforsomer2R,thenX1isnotanequilibriumlocationmatrix.Asaresult,anotherviablelocationmatrixshouldbeconsideredasapotentialequilibriumlocationmatrix.NotethatitissufcienttoshowthatthereexistsabetterlocationdecisionforatleastonerminX1toconcludethatX1isnotanequilibriumlocationmatrix.Tothisend,twoadditionalroutines,referredtoasRoutine2andRoutine3,aredenedtocheckwhetherabetterlocationdecisionexistsforrmrunderX1.TheintuitionbehindRoutine2isasfollows.Ifthereexistsarmrfacilitywithnegativeprot,itisdeterminedwhetherclosingthisfacilitywillincreasermr'stotalprot.Aspreviouslynoted,theremayexistfacilitieswithnegativeprotsforaviablelocationmatrix.Ifthetotalprotincreasesbyclosingthisfacility,thenthecurrentlocationvectorforthermunderX1isnotthebest 89

PAGE 90

responseofthermand,hence,X1isnotanequilibriumlocationmatrix.Therefore,anotherviablelocationmatrixisconsideredasacandidateequilibriummatrix.ThisroutineproceedsinthesamewayasRoutine1,whichisusedasasubroutineinRoutine2,butisappliedonlytoonecolumnofX1eachtime. Routine2:GivenXandL,thefollowingproceduresearchesforanimprovinglocationmatrix: 1:ApplyRoutine1 2:Ifcontinue=0,stopandreturncontinue=0 3:Else,setr=1andX2=X1 4:DeneI)]TJ /F9 7.97 Tf -1.59 -7.29 Td[(rsuchthatir(X2)<0,8i2I)]TJ /F9 7.97 Tf -1.59 -7.29 Td[(r 5:IfI)]TJ /F9 7.97 Tf -1.59 -7.29 Td[(r=;,setr=r+1 6:Ifr>k,stopandreturnX2 7:Else,gotoStep4 8:Else,let(^,r)=argminfir(X1):i2I)]TJ /F9 7.97 Tf -1.59 -7.29 Td[(rgandx2^r=0 9:Ifr(X1)
PAGE 91

withnegativeprotdoesnotincreasetheprotofthecorrespondingrm.TheoutputofRoutine2iseithercontinue=0orX2,whichisaviablelocationmatrixandsatises(i)or(ii).Whencontinue=0,thismeansthatifcontinued,onewillendupwithamatrixX2thathasalreadybeenanalyzedand,hence,Routine2shouldbestartedwithanotherlocationmatrixX.NowsupposethatX2isgeneratedusingRoutine2.ItisstillnotknownwhetherX2isanequilibriumlocationmatrix.ThenextstepistodeterminewhetherX2containsthebestresponsesforeachrm.Forthispurpose,afullneighborhoodsearchforeachrm,asexplainedinRoutine3,isperformed. Routine3:thefollowingproceduredetermineswhetherXisequilibrium: 1:Setr=1 2:Ifrk,ndthebestresponseofrmr,denotedbyxr(X),viatotalenumeration 3:Ifxr(X)=xr,setr=r+1andgotoStep2 4:Else,stopandreturnequilibrium=0 5:Else,returnX=Xandequilibrium=1. TotalenumerationinRoutine3generatesallpossiblexrvectorsandthendeterminesthebestresponseofrmrwhentheotherrms'locationdecisionsarexedbycomparingthetotalprotofrmrforeachmatrix X(whichdiffersfromXonlyintherthcolumn).ThepurposeofRoutines2and3istodetermineifagivenviablelocationmatrixisnotinequilibriumasquicklyaspossible.IfRoutine2cannotguaranteethattheviablelocationmatrixisnotanequilibriumlocationmatrix,thenRoutine3completesthecheckbyconsideringallotheroptionsforeachrm.Hence,attheconclusionofRoutine3,eitheranimprovedlocationdecisionforarmisspotted,whichimpliesthatthelocationmatrixisnotinequilibrium,oranequilibriumlocationmatrixisreached.Inwhatfollows,aheuristicmethodtosearchforanequilibriumlocationmatrixisexplained. 91

PAGE 92

4.4.4HeuristicAlgorithmforFindingAnEquilibriumLocationDecision Theheuristicalgorithmstartswitharandomlocationmatrixandrstmovestoaviablelocationmatrix.Then,itcheckswhethertheequilibriumconditionsaresatisedbythisviablematrix.Duringthemovefromarandomlocationmatrixtoaviablelocationmatrix,Routines0and1areutilized.Routines2and3areusedtocheckforequilibriumconditions.Routines2and3aremainlyaimedatsimplifyingtheprocessofcheckingequilibriumconditionsbyeasilyshowingwhethertheequilibriumconditionsarenotsatised,whenthecurrentviablematrixisnotanequilibriumlocationmatrix.However,afullsearchisneededtodetermineanequilibriumlocationmatrix.Itshouldbeemphasizedthatthealgorithmdoesnotperformafullsearchforeachnon-equilibriumlocationmatrix,whicheasesthecomputationalburden,asacompletesearchisburdensome.Inparticular,atotalenumerationschemetondalloftheequilibriumlocationdecisions,ortondoutthatnoequilibriumlocationdecisionexists,requirescheckingtheequilibriumconditionsfor2mklocationdecisions.Moreover,checkingtheequilibriumconditionsforanygivenlocationdecisionrequiresanalyzingk(2m)]TJ /F4 11.955 Tf 12.29 0 Td[(1)otheroptions.Thenitfollowsthatatotalenumerationschemewouldrequiresolvingforequilibriumquantitiesk2mk(2m)]TJ /F4 11.955 Tf 12.52 0 Td[(1)times,whichisexponentialinbothmandk.Hence,thefollowingheuristicmethodthatutilizesthepreviouslydenedroutinesisproposed. Algorithm4. Heuristicmethodtondanequilibriumlocationmatrix,ifoneexists. Step0.LetL=;. Step1.IfjLj=2mk,stop;theredoesnotexistanequilibriumlocationmatrix.Else,generatearandomlocationmatrix,X,suchthatX=2L. Step2.ApplyRoutine2withXandL.Ifcontinue=0,gotoStep1.Else,generateX2. Step3.ApplyRule3toX2.Ifequilibrium=0,gotoStep1.Else,equilibriumisfound,stopandreturnX. 92

PAGE 93

Duringthealgorithm,thesetLkeepstrackofthelocationmatricesthathavebeenprocessed.NotethatifthealgorithmdoesnotstopatStep3,thenthelocationsinLarenotequilibriumlocationdecisions,andonecannotgenerateanequilibriumlocationdecisionfromtheselocationmatricesusingRoutines0,1,2and3.Hence,anewlocationmatrixthatisnotinLisgenerated.Moreover,sincethereare2mkpossiblelocationdecisions,itisconcludedthattheredoesnotexistanequilibriumlocationdecisionwhenjLj=2mk. Theefciencyoftheheuristicmethodfollowsfromthefactthatitreducesthenumberoffullequilibriumchecks.Algorithm 4 doesnotperformafullequilibriumcheckforanynon-viablelocationmatrices,andevenavoidsthisforsomeviablelocationmatrices(thosethatarenon-equilibrium).Algorithm 4 ndsanequilibriumforanygivenproblemifanequilibriumsolutionexistsfortheparticularproblem.Ifnoequilibriumlocationdecisionexistsforthegivenproblem,thenAlgorithm 4 outputsthenon-existenceofanequilibrium.Furthermore,ifmultipleequilibriaexistfortheStage-onegame,thenAlgorithm 4 ndsoneofthem,althoughitcanbeeasilymodiedtoreturnalloftheequilibria.InSection 4.6 ,Algorithm 4 iscomparedwitharandomsearchalgorithmandnumericalresultsarepresentedthatdemonstratetheefciencyofAlgorithm 4 inndinganequilibriumlocationmatrix. 4.5Extensions:Multi-ProductandMulti-EchelonChannels Inthissection,themodelstudiedintheprevioussectionsisrstextendedformultipleproducts,i.e.,whenrmssupplyasetofdifferentitemstothemarketsusingthesamedistributionnetwork.Then,implicationsofthepreviouslydocumentedmethodsarediscussedformultipleechelonsupplychannels. Itisacommonpracticethatrmssupplyavarietyofproductstoendcustomermarketssimultaneously.Theshipmentrequirementsofvariousitems,ofcourse,willresultinhigherlevelsofcongestionontheunderlyingdistributionnetwork.Theanalysisthusfarofthelocation-supplygamewithtrafccongestioncostsforasingleproduct 93

PAGE 94

notonlycontributetothecurrentliteratureoncompetitivefacilitylocationgames,butalsopermitgeneralizationtomorerealisticmultipleproductcases.Therefore,inwhatfollows,thesettingdenedinSection 4.2 isextendedtoamultipleproductcaseandtheimplicationsontheresultsinSections 4.3 and 4.4 arediscussedforthiscase. Inparticular,letZbethesetoflproductsthatanyrmrsuppliestoanymarketj.Furthermore,superscriptzisusedtoindexthepreviousnotationforproducttype,z=1,2,...,l.Forinstance,qzijrdenotesthequantityofproductzshippedfromthefacilityofrmratlocationitomarketj.Itisassumedthatthesaleslevelofaproducttypeisindependentofotherproducts,andthatCournotcompetitionexistsamongrmsforeachproducttypeineverymarket.Specically,similartoEquation( 4 ),thepriceforproductzatmarketjisdenedbypzj(qzj)=azj)]TJ /F3 11.955 Tf 11.99 0 Td[(bzjqzj.Thecongestioncostfunctionisasfollows: gzijr Xz2Zqzij!=zijrXz2Zqzij.(4) Equation( 4 )denesproduct-specictrafccongestioncosts,thatis,thetotaltrafccongestioncostpaidbyrmrforshippingonlink(i,j)ischaracterizedasPz2ZzijrqzijrPz2Zqzij.Ontheotherhand,lettingzijr=ijr8z2Z,thetotaltrafccongestioncostincurredbyrmrforshippingproductsonlink(i,j)amountstoijrPz2ZqzijrPz2Zqzij.Thentheprotfunctionforeachrmrreads Hr(Q,X)=Xj2JXz2Zpzj Xi2IXr2Rqzijr!Xi2Iqzijr)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xj2JXz2ZXi2Iczijrqzijr)]TJ /F10 11.955 Tf 11.29 11.36 Td[(Xj2JXz2ZXi2Iqzijrgzijr Xr2RXz2Zqzijr!)]TJ /F3 11.955 Tf 11.95 0 Td[(fr(xr),(4) whichequalsrevenuelesstransportation,trafccongestion,andfacilitylocationcosts.Thetwo-stagesolutionapproachcanbeappliedtothemultipleproductcaseaswell.First,thesecondstageproblemdeterminesrms'equilibriumsupplyquantitiesatanymarketforeachproducttype,andthentherststageproblemsolvesforequilibriumlocationchoices.Givenrms'facilitylocations,thesecondstagegamecanbeshown 94

PAGE 95

tobeseparableinmarkets;however,itisnotseparablebyproducttype.Nevertheless,theconcavityresultsandthefollowingpropertiesofthecorrespondingVIPfortheequilibriumproblemwithmultipleproductsstillhold.Therefore,theassociatedsolutionmethodforthesingleproductcasediscussedinSection 4.3 canbeeasilyadjustedtohandlemultipleproducts,anditcanbeefcientlyusedtodetermineequilibriumsupplyquantities.(Inthemultipleproductcase,Qjwouldcorrespondtoal-vector,whichdenotesthesupplyquantitiesofeachproducttypeformarketj.)Analysisoftherststagegamerequiresminormodications(theonlymodicationrequiredisinStep4ofRoutine0:xirshouldbesetto0whenPz2Zqz()ijr=0,whereqz()ijrdenotesqzijrinequilibrium).Next,somepossiblegeneralizationstoapplytomulti-echelonsupplychainsarediscussed. Atwo-echelonsupplychainisstudiedin Nagurneyetal. ( 2002 ).Inparticular,theyconsiderasupplychainnetworkwithmanufacturers,retailers,andconsumermarkets;however,thisstudyassumesthatfacilitylocationsarepredetermined.Inanattempttostudycompetitivefacilitylocationproblemsinmulti-echelonchannels,rstthesettingofthegameshouldbedened.Supposethattherearekepartiesattheethechelonofat-echelonsupplychannel.Ifpartiesatdifferentechelonsdecideontheircontrolvariablessimultaneously,onegamecanbesolved,assumingconcavityconditionsaresatised,todetermineequilibriumoutcomesofthewholechannel.Atwo-stagesolutionapproachcanbeutilizedwhentheparties'controlvariablesaresupplyquantitiesand/orpriceandfacilitylocationchoices:rst,thesupplyquantityandpricedecisionscanbesolvedviavariationalinequalityformulations(see,e.g., Nagurneyetal. 2002 ),andthenequilibriumlocationsolutionscanbesearched. Ontheotherhand,ifthereareprioritiesinthetimingorsequenceofdecisionsatdifferentechelons,thendistinctanalyseswouldberequired.Thisscenariocorrespondstoasequentialentrygame,anddecisionmakerswithprioritiesintakingactionwouldanticipateandconsiderthefollowers'reactionsintheirdecisions.Forinstance,we 95

PAGE 96

mightconsideratwo-echelonsupplychaininwhichasetofcompetitivemanufacturers,whosimultaneouslydeterminetheirplantlocationsandproductionlevels(orwholesaleprices),makedecisionspriortoasetofcompetitiveretailers,whosimultaneouslydeterminestorelocationsandorderquantitiesfromthemanufacturers.TheunderlyinggamewouldcorrespondtoaStackelberggamewithasetofcompetitiveleaders(manufacturers)andasetofcompetitivefollowers(retailers).WhileStackelberggameswithasingleleaderandasetofcompetitivefollowershavebeenstudiedtodeterminetheleader'swholesalepriceandthefollowers'orderquantitydecisionsintheliterature(see,e.g., BernsteinandFedergruen 2003 2005 ),tothebestofourknowledge,therearenostudiesthatconsiderthewholesalepricedecisionsofmultiplecompetitiveleadersandthequantitydecisionsofmultiplecompetitivefollowersintheliterature.Furthermore,includinglocationdecisions,whichintroducesbinaryvariables,wouldresultinachallengingproblemclass.Themodelsandanalysespresentedinthischaptermaythusserveasastartingpointforanalysesofmoregeneralmulti-echeloncompetitivefacilitylocationproblems. 4.6NumericalStudy Inthissection,theheuristicsearchmethodstatedinAlgorithm 4 iscomparedwitharandomsearchmethod.Itshouldbenotedthatinordertosolvefortheequilibriumowquantitiesforagivenlocationmatrix,Algorithm 3 isusedwiththeparametersettingsprovidedin Han ( 2006 )(theefciencyofAlgorithm 3 isdiscussedin Han 2006 ).ThissectionaimsatdemonstratingtheefciencyandbenetsofAlgorithm 4 .However,becausethemodelunderconsiderationisnewtotheliterature,nobenchmarkalgorithmexistsforcomparison.Thus,thenumericalstudiesintendtodemonstratethepotentialbenetsoftheproposedheuristicalgorithmwhencomparedtoanaveorrandomsearchalgorithmthatmightbeappliedinpracticeintheabsenceofanalternativeapproach.Asaresult,Algorithm 4 iscomparedwiththefollowingrandomsearchalgorithm. 96

PAGE 97

Algorithm5. Randomsearchmethodtondanequilibriumlocationmatrix,ifoneexists: Step0.LetL=;.GotoStep1. Step1.IfjLj=2mk,stop;theredoesnotexistanequilibriumlocationmatrix.Else,generatearandomlocationmatrix,X,suchthatX=2L.GotoStep2. Step2.ApplyRoutine3toX.Ifequilibrium=0,gotoStep1.Else,equilib-riumisfound,stopandreturnX. NotethattheonlydifferencebetweenAlgorithm 5 andAlgorithm 4 isthatRoutine2isnotusedinAlgorithm 5 .ThisalsomeansthatRoutine1(embeddedinRoutine2)and,hence,Routine0(embeddedinRoutine1)arenotusedinAlgorithm 5 aswell.Algorithm 5 appliesafullequilibriumchecktothegivenrandomlocationmatrixandrepeatsSteps1and2untileitheranequilibriumlocationmatrixisfoundorallofthelocationmatricesaredeterminedtobenon-equilibrium.ItisworthpointingoutthatthelistofmatricesinAlgorithm 5 increasesby1ateachoccurrenceofStep1.Ontheotherhand,thelistofmatricesinAlgorithm 4 mayincreasebymorethan1ineachoccurrenceofStep1.Furthermore,Algorithm 5 appliesRoutine3toeachelementofthelist,whereasAlgorithm 4 appliesRoutine3onlytothelocationmatricesgeneratedbyRoutine2. Forcomparisonpurposes,thesamesequenceofrandomlocationmatriceswasusedwithinAlgorithm 5 andAlgorithm 4 foreachprobleminstance.Atotalof27differentcombinationsofk=f2,3,4g,m=f2,3,4g,andn=f2,3,4gwereconsideredforeightdifferentproblemclasseswithvaryingper-unittransportationcost,congestioncostfactor,andfacilitylocationcostdistributions.Table 4-1 givesthedistributionintervalforcirj,ijr,andfirvaluesineachclass.Tenrandomlygeneratedprobleminstancesweresolvedwithineachproblemclassforeachproblemcombination.Foreachclassof 97

PAGE 98

problems,ajU[50,100]andbjU[1,2],whereU[l,u]denotestheuniformdistributionon[l,u]. Table4-1. Dataintervalsforproblemclasses1-8 cijrijrfirClass1[0,50][0,0.75][50,125]Class2[0,50][0,0.75][125,250]Class3[0,50][0.75,1.5][50,125]Class4[0,50][0.75,1.5][125,250]Class5[50,100][0,0.75][50,125]Class6[50,100][0,0.75][125,250]Class7[50,100][0.75,1.5][50,125]Class8[50,100][0.75,1.5][125,250] Anequilibriumlocationdecisionwasfoundforeveryprobleminstance.EachrowinTable 4-2 summarizestheaverageof80probleminstances(10fromeachproblemclass),foreachcombinationofk,mandn(resultingin2160totalinstances)forthefollowingdata:lengthofthelistattermination(listlength),numberoffullequilibriumchecks(#ofchecks)andCPUtimeinseconds.AscanbeseeninTable 4-2 ,theheuristicmethodismuchfasterthantherandomsearchmethod.Thisisduetothefollowingtwopoints:(i)theheuristicmethoddoesnotperformafullequilibriumcheck(whichiscomputationallyburdensome)foreachelementwithinthelistand,(ii)itmovestoaviablelocationmatrixfromthegivenrandomlocationmatrixandmaydeterminethattheviablematrixisnotanequilibriumlocationmatrixwithoutafullequilibriumcheck.NoticethatCPUtimestendtobelinearwithn.Itshouldalsobenotedthatintheworstcase,bothalgorithmswouldrequire2kmequilibriumchecks(orexecutingAlgorithm 4 k2km(2m)]TJ /F4 11.955 Tf 12.6 0 Td[(1)times,whichwouldberequiredfortotalenumerationofthelocationmatrices).Nonetheless,whenthelistlengthsatterminationarecompared,Algorithm 4 analyzesfewermatricesandperformsfullequilibriumchecksforaround40%ofthesematriceswhencomparedtoAlgorithm 5 ,whichperformsfullequilibriumchecksforallofthelocationmatricesitanalyzes.Table 4-3 comparestheaveragevaluesof270probleminstanceswithineachproblemclass. 98

PAGE 99

Table4-2. Comparisonofheuristicmethodwithrandomsearchmethod Algorithm2Algorithm3list#ofCPUlist#ofCPUkmnlengthcheckstimelengthcheckstime2226.605.550.167.907.900.142236.936.280.238.888.880.242247.036.480.358.008.000.3023212.207.780.4325.8325.831.0723318.1814.131.0737.4337.432.3323424.7819.482.1432.2532.252.7624228.9018.851.9697.1397.139.3124342.1331.734.46125.33125.3318.6124481.7068.8811.70134.70134.7025.3232216.9512.680.4231.3831.380.6932320.1515.850.7933.7533.751.1732419.2315.781.0729.3029.301.4233234.0020.851.40271.33271.3314.7633383.1058.285.36266.80266.8022.3733478.8551.736.25289.48289.4828.9334294.3840.805.151764.381764.38215.50343251.08133.7324.742257.532257.53380.39344682.28480.35111.862023.582023.58470.0642220.7011.800.51120.20120.203.1242330.3517.701.31104.18104.184.4642447.6037.753.00105.98105.986.03432149.0028.389.562169.652169.65155.44433176.90101.2514.371646.431646.43178.07434240.20168.8532.491854.901854.90272.254421767.73146.85803.8429366.9529366.9510784.854431797.20789.25600.5627287.4827287.4812388.534441663.70682.85312.4226259.4026259.4010954.59average274.14110.8872.503568.893568.891331.21 ItcanbeobservedfromTable 4-3 thatforprobleminstanceswithhighervaluesofcijr,Algorithm 4 ismoreefcient(i.e.,shorterlistlength,fewernumberofchecks,andlesstime),whileAlgorithm 4 isslightlylessefcientforprobleminstanceswithhighercongestioncostfactors.ThereisnotanysignicantrelationshipbetweentheefciencyofAlgorithm 4 andfacilitylocationcosts.Nevertheless,theresultspresentedclearlyillustratethatAlgorithm 4 outperformsAlgorithm 5 inaveragecomputationaltime,as 99

PAGE 100

Table4-3. Comparisonofheuristicmethodwithrandomsearchmethodforeachproblemclass Algorithm2Algorithm3list#ofCPUlist#ofCPUlengthcheckstimelengthcheckstimeClass1242.14151.6531.373106.273106.27684.64Class2397.53179.13135.994039.974039.971577.19Class3522.75318.56111.463061.443061.441009.38Class4782.01149.55269.943722.403722.401208.88Class565.6823.959.414294.394294.392206.29Class644.2412.134.753114.653114.651369.39Class7103.2543.2414.603506.113506.111253.74Class835.538.862.503705.903705.901340.18average274.14110.8872.503568.893568.891331.21 wellasinaveragelistsizeandtheaveragenumberoffullequilibriumchecks,foreachproblemclass. 100

PAGE 101

CHAPTER5SUPPLIERWHOLESALEPRICING:IMPLICATIONSOFDECENTRALIZEDVS.CENTRALIZEDPROCUREMENTUNDERQUANTITYCOMPETITION 5.1Motivation Thisworkinthischapterismotivatedbythestructureandoperationofanair-conditioningproductssupplychannelinFlorida.Inparticular,thechanneloperatesasfollows.Asuppliermanufacturesairconditioningproducts,whichitsellstolocalretailersviaadistributioncompanyinFlorida.Theretailerscompetewitheachotherandindependentlydeterminetheirequipmentorderquantitiesandinformthedistributor;thedistributorthentransmitstheseorderstothesupplier.Theretailers'ordersareshippedtothedistributorandthedistributorthenshipstheorderstoretailers.Thedistributor'slivelihoodis,therefore,dependentonmeetingtheneedsofboththesupplierandtheretailers.Initsroleasintermediary,thedistributordirectlyaffectsthesystem'seconomicperformance.Currently,thedistributordoesnotengageinprocurementquantitydecisions,andtheretailerordersarethereforeindividuallytransmittedtothesupplier.Thiscontextraisesseveralinterestingresearchquestionsabouttheroleadistributorcanplayincentralizingretailerprocurement,andhowthisroleinuencesthecompetitionforchannelprotbetweenasupplieranditscompetitiveretailers.Inparticular,thischapterisinterestedinhowthedistributor'sprocurementstrategycanaffectsupplierwholesalepricingdecisions,aswellasthevaluetothesupplierofbeingabletocontrolthisstrategy. Asupplier'swholesalepricingstrategyrepresentsoneofthemostcrucialdecisionsinuencingtheprotabilityandefciencyofasupplychainforaproductorservice.Thesupplychainmanagementliteraturehas,therefore,focusedintenselyonpricingproblemsinrecentyears.Inthischapter,westudyasupplier'spricingproblemforagoodsoldtomultipleretailersfromaverticallydecentralizedperspective,anddiscusstheimplicationsfromaverticallycentralizedperspective.Underaverticallydecentralizedapproach,partiesatdifferentstagesofadistributionchannelmake 101

PAGE 102

decisionsbasedonlocalobjectivesonly.Conversely,averticallycentralizedapproachassumesexistenceofacentraldecisionmakerwhosegoalistoachieveachannel-wideoptimalsolution.TheproblemwestudycorrespondstoaStackelberggameinaverticallydecentralizedsetting,wherethesupplieractsastheleaderbysettingawholesaleprice,andtheretailersactasfollowersbydetermininghowmuchtheywillsupplytothemarket,i.e.,theirorderquantities.Weconsiderthecasewhenretailers'ordersaretransmittedviaasinglecompany(e.g.,adistributor)andcanbeoperatedusingoneofthreedifferentstrategies:decentralized,centralized,andpartiallycentralized.Itshouldbenotedthatcentralizationordecentralizationoftheretailersdoesnotimplycentralizationordecentralizationofthechannel. Thechannelwestudy,asnotedpreviously,currentlyoperatesinacompletelyverticallydecentralizedmanner;however,inprinciple,thedecisionsandactionsattheretailstagemaybehorizontallycentralizedordecentralized.Inparticular,whentheretailersarehorizontallydecentralized(asiscurrentlythecase),eachretailerdetermineshis/herorderquantityfromthesupplierindependentlyofotherretailers.ThiscasecorrespondstoaCournotOligopolyamongtheretailers,whichassumesthatthemarketpriceisdeterminedbythetotalsupplytothemarket.Underahorizontallycentralizedprocurementapproach,however,retailerorderquantitieswouldbedeterminedbyacentraldecisionmakeronbehalfoftheretailstage.Inthesettingwehavedescribed,thiscentraldecisionmakerwouldcorrespondtothedistributor.Inparticular,becauseofthedistributor'srelativechannelpower,itsrelationshipwiththesupplier,andthefragmentednatureofthelocalizedretailers,thedistributorcanpotentiallyassumetheprocurementfunctionfortheretailersitserveswithlittleornorisk.Asthedistributorisinterestedinthenancialviabilityofbothitssupplierandtheretailersitserves,thiscontextraisesinterestingquestionsabouthowprocurementcentralization(ordecentralization)impactstheperformanceofthedistributor'supstreamanddownstreampartners,aswellastheentirechannel.Wewillalsoconsiderthe 102

PAGE 103

potentialforapplyingaso-calledpartiallycentralizedapproach,underwhichweassumethatindividualretailerspermitthedistributortoprocureontheirbehalf,providedthateachretailerretainsitscurrentmarketshareunderindependentordering.Althoughtheproblemwediscussismotivatedbyaparticularchannel,themodelandanalysisweprovidemayapplymorebroadly,notonlytootherchannelsinvolvingadistributorthatservesmultiple,separateretailers,butalsotosettingsinwhichachainofretailstoreswishestoconsiderthepotentialbenetsofcentralizedprocurement. Whentheretailersarehorizontallydecentralized,theycompeteintheend-customermarkettosellthesupplier'sproduct.Whencompetitionoverhomogeneousproductsexists(i.e.,underlowproductdifferentiation),asnotedinpreviouschapters,themarketcanoftenberepresentedusingCournotcompetition,asquantitydecisionswillbeofgreaterinterestthanpricingdecisions.Hence,becauseweassumethattheretailerssellthesameproduct,aCournotcompetitionassumptionreadilyappliestotheproblemclasswestudy.Furthermore, Iyer ( 1998 )notesthatdecentralizeddecisionmakingisefcientformarketswhereproductdifferentiationislow(lowproductdifferentiationimpliesthatmultipleproductsserveasdemandsubstitutes).Inthiscase,thesuppliersetshis/herwholesalepriceforasetofcompetitiveretailers.Forthiscase,wediscussthesupplier'swholesalepricesettingproblemaswellasthecompetingretailers'orderquantitydecisionsfromthesupplier. Ontheotherhand,retailerorderscanbecentrallymanagedbythedistributor.Problemsaccountingforcentralizedcontrolofretailershavebeenanalyzedinthesupplychainmanagementliterature( DongandRudi 2004 YangandZhou 2006 Shaoetal. 2009 ).Throughhorizontalcentralization,asnotedby Shaoetal. ( 2009 ),thedistributorcanbenetfromreducedretailercompetition.Inourmodel,aswelatershow,adistributorwouldprefertooperatethroughasingleretailerforagivenproduct,i.e.,theonewiththelowestoperatingcostinthemarketforthatproductunderhorizontalcentralization.Inpractice,thiscasecorrespondstothescenariowherethemostefcient 103

PAGE 104

retailertakesoverthedistributor'sroleandinterfacesdirectlywiththesupplier.Wewilldiscusshowthesuppliersetsitwholesalepriceforthiscase.Althoughhorizontallycentralizedcontrolofretailersisultimatelythemostprotablesolutionfortheretailstageforasingleproduct,individualretailersmaybemoreefcientatsellingdifferentproducts.Asaresult,individualretailersmaywishtopreservethemarketsharetheygainedunderhorizontaldecentralizationforagivenproductforanumberofreasons(e.g.,incaseswheretheproduct'savailabilitymaybringcustomersintotheretailstoreand,therefore,affecttheretailer'sprotfromsalesofotherproducts).Recentstudiesinassortmentplanningpointoutthefactthataproduct'sdemanddependsnotonlyonthepriceoftheproduct,butalsoontheinventorylevelsoftheotherproductsintheretailshop( Bitranetal. 2010 ).Therefore,asanalternativetostricthorizontalcentralizationofretailers,weconsiderthecaseinwhichretailerstransitionfromdecentralizedcontroltocentrallycontrolledordering,butthemarketsharegainedunderdecentralizedcontrolmustbemaintainedascontroloforderingistransitionedtoacentraldecisionmaker.Werefertothisstrategyasthepartialcentralizationofretailprocurementanddiscussthedistributor'sdecisionsunderthisscenario.Furthermore,weillustratehowthesupplierdetermineshis/heroptimalwholesalepriceunderpartiallycentralizedretailerprocurement. IntheStackelberggameweconsider,thesupplieristheleaderandthedistributorandretailersarefollowers.Thesupplierdeterminesitswholesalepriceandtheretailersanddistributorreactbydeterminingdownstreamorderquantities(or,equivalently,marketsupplyquantities)basedonthedistributor'sprocurementstrategy.WeusebackwardinductiontodeterminetheStackelbergequilibriumofthisgame;thatis,werstcharacterizethedecisionsatthedownstream(distributorandretail)stagesandusethissolutiontosolvethesupplier'sproblem.Inparticular,thesequenceofeventsisasfollows:(i)thesuppliersetshis/herwholesaleprice,(ii)thedistributordetermineswhetherretailerordersaredecentralized,centralized,orpartiallycentralized,(iii)the 104

PAGE 105

distributorannouncesretailerorderquantities,and(iv)thesuppliershipstheretailers'orderquantities,andamarketclearingpriceisdetermined.Itshouldbenotedthatthesupplierdoesnotcontrolthedistributor'sprocurementstrategy.Itisobviousthatthesupplier'swholesalepricedecisionunderanassumptionofdecentralizedretailerswillbesuboptimalifthedistributorhorizontallycentralizesretailerorders.However,wewillshowthatcentralizedorpartiallycentralizedcontrolofretailordersisbenecialtotheretailstageforanygivensupplierwholesaleprice;hence,thedistributorwillchooseeithercentralizedorpartiallycentralizedcontrolifretailersarewillingtorelinquishthiscontrol.ItwillthenfollowthattheStackelbergequilibriumsolutionisachievedwhenthesuppliersetshis/herwholesalepriceforcentralizedorpartiallycentralizedretailers.Nevertheless,wewillshowthatthisStackelbergequilibriumisnotnecessarilyanoptimalsolutionforthesystem(e.g.,whenthesupplier'sproductioncostsarelinearintheproductionquantity).Inparticular,thesuppliermaybebetteroffunderdecentralizedretailordering;thatis,thesuppliercanachievesubstantialsavingsifs/heisabletodictatethedistributor'sprocurementstrategy.Thisintroducesanimportantconceptwhichwerefertoasthevalueofcontrol.Thevalueofcontroldeneshowmuchthesuppliercansaveifs/hecontrolsthedistributor'sprocurementstrategy,andwequantifythevalueofthiscontrolviaournumericalstudies. Thischaptercontributestotheliteraturebymodelingasupplier'spricingdecisionforasetofcompetitiveretailerswhoseorderingprocessescanbecontrolledbyadistributorunderdifferentprocurementstrategies.Inouranalysis,weallowanarbitrarynumberofnon-identicalretailers.Furthermore,weconsidergeneralcostfunctionsinourmodels.Tothebestofourknowledge,thevalueofcontrolforasupplierontheretailprocurementstrategyinacompetitivemarkethasnotbeenstudiedintheliterature.Thus,anothercontributionofthischapterliesinintroducingandquantifyingthevalueofthiscontrolforthesupplier.Wecharacterizepropertiesofthesupplier'sprotfunction,andgainimportantmanagerialinsightsthroughouranalyticaland 105

PAGE 106

numericalstudiesofthechannel.Wearguehowasuppliermayencouragecentralizedordecentralizedmanagementoftheretailers,dependingonwhethers/hefaceseconomiesordiseconomiesofscaleinproductioncosts.Ournumericalstudiesindicatethatwhenthesupplier'sproductioncostislinearinproductionquantity,equilibriumisachievedwhentheretailersarecentralizedbythedistributorandthesuppliersetshis/herwholesalepriceaccordingly.However,system-wideprotismaximizedwhenretailersaredecentralizedandthesuppliersetsthewholesalepricefordecentralizedretailers.Thisobservationindicatesthataneffectivecoordinationmechanismshouldprecludehorizontalcentralization,i.e.,verticalcentralizationmayrequirehorizontaldecentralization.Basedontheseobservations,wediscusswhyacoordinationmechanismmayrequirerestrictionsonretailprocurementstrategy. Therestofthischapterisorganizedasfollows.Section 5.2 discussesrelevantworkintheliterature.Section 5.3 denestheproblemsettingandoursolutionapproach.InSection 5.4 ,foragivenwholesaleprice,weanalyzetheretailers'orderquantitydecisionsunderdifferentprocurementstrategiesimposedbythedistributorandthedistributor'sbestchoiceofprocurementstrategy.Then,inSection 5.5 ,werstderiveanexplicitexpressionfortheretailers'totalorderquantityasafunctionofthewholesaleprice.Then,weformulatethesupplier'sproblemintermsofthewholesalepriceandprovideasolutionalgorithmforthisproblem.Section 5.6 extendsourresultstoamulti-market(ormulti-product)settinganddiscussesgeneralizationsthatpermitthesuppliertoofferquantitydiscountpricing.Section 5.7 proceedswithournumericalstudiesthatanalyzethevalueofcontrol.Inaddition,wedeterminetheStackelbergequilibriumandarguethatchannel-wideprotsarenotmaximizedatthisequilibrium. 5.2LiteratureReview Inthesupplychainliterature,asupplier'spricingstrategyinatwo-echelonchannelisoftenconsideredwithinthebroadercontextofchannelcoordination.Channelcoordinationfocusesonmechanismsthatensureprotlevelsatorneartheoptimal 106

PAGE 107

centralizedsolutionwhileusingdecentralizeddecisionmaking.Ifacoordinationmechanismexiststhatresultsinanoptimalcentralizedsolutionunderdecentralizeddecisions,thenperfectcoordinationisachieved.Theapplicationofpricingasachannelcoordinationmechanismappearsinmanyinventorycontrolproblems. Monahan ( 1984 )showsthatanall-unitsquantitydiscountschedulecanbeusedtoincreaseasupplier'sprotwithoutreducingthebuyer'sprotwithrespecttothatunderdecentralizeddecisionsinasingle-supplier,single-buyersystem(underdemandandcostassumptionssimilartothoseintheeconomicorderquantity,orEOQ,model).Thestudiesby LalandStaelin ( 1984 ), Banerjee ( 1986 ), LeeandRosenblatt ( 1986 ),and Toptaletal. ( 2003 )focusoncoordinatedpricingdecisionsformoregeneralmodelsunderdeterministicdemand.Whenstochasticdemandisconsidered,thecoordinationmechanismsproposedincluderevenue-sharingcontracts( Cachon 2003 GiannoccaroandPontrandolfo 2004 CachonandLariviere 2005 ),buy-backpolicies( Pasternack 1985 EmmonsandGilbert 1998 ),returnspolicies( Taylor 2001 ),andrebatepolicies( Taylor 2002 ).Broaderclassesofpricingproblemsunderdifferentproblemsettingsandcoordinationmechanismshavebeenstudiedintheoperationsresearchliterature,see,e.g.,thereviewsby Tsayetal. ( 2000 ), Cachon ( 2003 ),and Sarmahetal. ( 2006 ). Thestudieswehavecitedthusfar,whetherconsideringdeterministicorstochasticdemand,assumethattheretailer'spriceisexogenouslydetermined.Incontrast, JeulandandShugan ( 1983 )considerthepricingdecisionofamanufacturerinachannelcoordinationcontextwhendemandisafunctionoftheretailer'sprice. Weng ( 1995a b )considersasupplier'squantitydiscountpricingproblemunderprice-sensitivedemand,wherethesinglesupplierandasingleretailer(ormultiple,identicalretailers)operateundertheclassicalEOQmodel.Additionalworkonpricingdecisionsofasupplierforcoordinatedchannelswithprice-sensitivedemandcanbefoundin Moorthy ( 1987 ), JeulandandShugan ( 1988 ), Chenetal. ( 2001 ), BoyacandGallego ( 2002 ), ViswanathanandWang ( 2003 ),and Qinetal. ( 2007 ).Whilethesepapersrecognizea 107

PAGE 108

supplier'spricingdecisionasaStackelberggame(involvingthesupplierandbuyers),theydonotconsidercompetitionamongthebuyers.Wenotethat,beyondStackelberggames,variouscooperative,bargaining,andcoalitiongamesbetweenasupplieranditsbuyershavebeenanalyzedintheliterature.Forliteratureongame-theoreticanalysesoftheseproblems,onemayreferto KohliandPark ( 1989 ), Abad ( 1994 ), IyerandPadmanabhan ( 2005 ), NagarajanandSosic ( 2008 ), Sarmahetal. ( 2006 ), XieandWei ( 2009 ),andthereferencestherein.Competitionattheretail-levelofthesupplychainisdisregardedinthepaperscitedthusfar. Thisstudyconsidersasupplierwhochargesaxedwholesalepricetoasetofretailers,whosellthesupplier'sproductinanend-customermarket.Thechannelisassumedtobeverticallydecentralized.Decentralizeddecisionmakingoftenprevailsinsituationsinvolvingadominantparty( ToptalandCetinkaya 2006 ),whenthereiscompetitionatthesupplierlevel( Moorthy 1988 ),andinothersituations,suchassupplychainswithmultipleinventorysites( LeeandBillington 1993 LeeandWhang 1999 ). LeeandBillington ( 1993 )notethatevenwhenthesemultiplesitesarecontrolledbyasinglerm,thermmaystillpreferdecentralizeddecisionmaking.Asnotedpreviously,verticaldecentralizationispreferredformarketswithlowproductdifferentiation( Iyer 1998 ).Furthermore,thedecentralizeddecisionsettingservesasabenchmarkforcomparingtheeffectivenessofcoordinationmechanismsthatcanincreaseoverallchannelprots.Whilethechannelisverticallydecentralized,weconsiderbothhorizontallydecentralizedandcentralizeddecisionsattheretailechelon.Iftheretailstageishorizontallydecentralized,theretailerscompeteintheend-customermarket.Inparticular,wemodeltheretailers'competitionusingquantityorCournotcompetition,asthistypeofcompetitionoftenappliesincaseswithlowproductdifferentiation. Asignicantamountofliteratureexistsoncoordinationproblemsunderdownstream(e.g.,retail-level)competitioninasupplychain. IngeneandParry ( 1995 1998 2000 ) 108

PAGE 109

studyacoordinationproblembetweenamanufacturerandtworetailerscompetingonprice. Iyer ( 1998 )studieschannelcoordinationwhenretailersareengagedinpriceandnon-pricecompetition. BernsteinandFedergruen ( 2003 )focusoncoordinatedinventoryreplenishmentandpricingdecisionsinatwo-echelonsupplychainconsistingofasinglemanufacturerandmultiplecompetitiveretailers(underbothprice-andquantity-basedcompetition).Theymodelthemanufacturer'sandtheretailers'protsunderdeterministicdemand.Followingthis, BernsteinandFedergruen ( 2005 )analyzecoordinationmechanismsbetweenamanufacturerandasetofcompetitiveretailerswhoobserverandomdemandinasingle-periodsetting.Thisstudyisthenextendedby BernsteinandFedergruen ( 2004 )toaninniteplanninghorizon.Otherstudiesfocusingonchannelcoordinationwithretailercompetitionconsiderreturnspolicies( PadmanabhanandPng 1997 ),two-parttariffs( TsayandAgrawal 2000 ),quantitydiscounts( BalachanderandSrinivasan 1998 XiaoandQi 2008 ),vendormanagedinventory( Bernsteinetal. 2006 ),andrevenuesharingcontracts( Yaoetal. 2008 ).Additionalreferencesinthisresearchstreamcanbefoundin Cachon ( 2003 ). TheunderlyingStackelberggamewestudyisdenedsimilarlytothegamesstudiedby YangandZhou ( 2006 )and KeskinocakandSavasaneril ( 2008 ). YangandZhou ( 2006 )analyzeaStackelberggamewhereamanufacturerleadsbydeterminingaxedwholesaleprice,andtwonon-identicalcompetitiveretailersfollowbydeterminingtheirorderquantities.Themanufacturer'soptimalpricingdecisionisconsideredwhentheretailersarecompetitive(i.e.,decentralized),united(i.e.,centralized),andtakeactionssequentially.However,inthisstudy, YangandZhou ( 2006 )donotconsiderthemarketentrycondition.(Thecompetitionbaseforthetworetailersisprice;hence,bothretailersenterthemarketatanymanufacturerwholesaleprice,andtheirsellingpricesdifferinthemarket.However,inCournotcompetition,aretailermayrefusetoenterthemarketbasedonhis/heroperatingcostsaswellasthesupplier'swholesaleprice;thatis,thesupplieraffectsthenumberofretailersenteringthemarket).Inourstudy,weconsider 109

PAGE 110

Cournot-typequantitycompetition;hence,enteringthemarketisaretailer'schoiceandthesuppliercaneffectivelycontrolretailerentrydecisions.AswediscussinSection 5.5 ,thesupplierconsidersretailers'marketentrydecisionswhilesettinghis/herwholesaleprice. KeskinocakandSavasaneril ( 2008 )studyasupplier'spricingproblemwhenanarbitrarynumberofidenticalretailerscompeteonsalesvolumes.Theyanalyzethedecisionsofcompetitiveandcooperativeretailers.Whilethecompetitivecasecorrespondstowhatwecallhorizontaldecentralization,thecooperativecaseisdifferentfromthehorizontalcentralizationweapply.Theretailerscooperateviacollaborativepurchasing,givenasupplier'slineardiscountpricingfunction.However,intheirstudy,retailers'cooperationlevelsaredeterminedbythesupplier,i.e.,thesupplierdecideswhethertheretailersarecooperativebysettingaspilloverfactor,whichdenestheadditionaldiscountaretailerachievesfromthesupplierasaresultofotherretailers'orderquantities.Ourstudyextendsthesettingof KeskinocakandSavasaneril ( 2008 )tothecaseofnon-identicalretailers.Further,weconsiderhorizontalcentralizationviaanintermediary(thedistributor)insteadofcooperation. Inparticular,weconsidertwodifferentcentralizationscenariosforretailers:centralizationandpartialcentralization.Centralizationoccurswhenindividualretailstores'decisionsarecontrolledbyasinglerm,aswouldoccurinthecaseofachainstore( Shaoetal. 2009 )orwhenadistributorordersonbehalfofmultipleretailers( AnupindiandBassok 1999 ).Centralizedcontrolofretailershasbeenanalyzedindifferentformsinthesupplychainmanagementliterature( DongandRudi 2004 YangandZhou 2006 Shaoetal. 2009 ). DongandRudi ( 2004 )consideramanufacturerandmultiple,centralized,identical-costretailerswithafocusontranshipmentandwholesalepricedecisions.Asmentionedpreviously, YangandZhou ( 2006 )considerthecasewhentworetailersuniteandbehaveinacentralizedmanner. Shaoetal. ( 2009 )consideramanufacturer'swholesalepricedecisionfortwoidenticalretailerswhocanbecentralizedordecentralized.Thecentralizationoftheretailstageweapplyin 110

PAGE 111

thischapterissimilartotheaforementionedliterature.Whenretailersarecentralized,theobjectiveisforthedistributor,whoactsasacentraldecisionmakerfortheretailstage,tomaximizethetotalprotattheretailstage.Aswelaterdemonstrate,whilecentralizedretailprocurementleadstothehighestretail-stageprot,itmaybedifcultorimpossibletoimplementbecauseanoptimalcentralizedpolicyrequiresthattheend-customermarketbesuppliedbyasingleretailerforagivenproduct.Whenaretailersellsmultipleproducts,however,thestockofagivenproductcanaffectaretailer'sprotfromotherproducts.Forinstance, Bitranetal. ( 2010 )notethatalongwiththepriceofaparticularproduct,theavailabilityofotherproductsinastoreaffectsthedemandoftheproduct.Recentstudiesonretailerassortmentplanninganalyzetheeffectsofproductavailabilityonretailprots.Forinstance,abasketshopper(aconsumerintendingtobuymultipleproductsfromaretailshop)mayprefertopurchasehis/herentirebasketfromanothershopwhenagivenshopdoesnothaveoneoftheitemsinhis/herbasket( BellandLattin 1998 ). VanRyzinandMahajan ( 1999 )notethatincreasingthevarietywithinanassortmentincreasesthepossibilityacustomerwillbuysomethingfromaretailer.Werefertheinterestedreaderto SmithandAgrawal ( 2000 ), CachonandKok ( 2007 ),and Koketal. ( 2009 )forfurtherdiscussionsonassortmentplanning.Becauseoftheseassortmentplanningissues,weconsiderpartialcentralizationasanalternativetocompletecentralization.Theobjectiveofpartialcentralizationisthesameascentralization,exceptthatthedistributormustmaintaineachretailer'smarketshareinordertoavoidthedisadvantagescentralizationintroduceswithrespecttoassortmentplanningrequirements. 5.3ProblemFormulationandMethodology Considerasupplier(manufacturer)whosellsaproducttoanendcustomermarketviaasetofnretailers,indexedbyi2f1,2,...,ng.Weassumethattheretailers'ordersforthesupplier'sproductsareplacedthroughadistributor.Thedistributormaychoosedecentralized,centralized,orpartiallycentralizedcontroloftheretailerorders.The 111

PAGE 112

marketprice(i.e.,theretailers'sellingprice)isdeterminedbythetotalquantitysuppliedtothemarket.Thus,whentheretailersaredecentralized,theyareengagedinCournot(quantity-orsales-volume-based)competition.Inparticular,letqidenotethequantitythatretaileri,i2f1,2,...,ng,suppliestotheendcustomermarket.Thenp,themarketprice,isdeterminedbythefunctionp(Q),whereQdenotesthetotalquantitysuppliedtothemarket,i.e.,Q=Pni=1qi.Inparticular,similarlytoChapters 2 4 ,weassumethatp(Q)isalineardecreasingfunctionofQsuchthat p(Q)=a)]TJ /F3 11.955 Tf 11.96 0 Td[(bQ=a)]TJ /F3 11.955 Tf 11.96 0 Td[(bnXi=1qi,(5) wheretheparametersa0andb>0denotethepriceatzerodemandandthemarketpricesensitivity,respectively(andbothparametersareassumedtobereal-valuednitenumbers).Themarketmagnitudeanddemandelasticitycanalsoberepresentedbytheseparameters,respectively.Wenotethatp(Q),representstheinversedemandfunctionassociatedwithaCournotsetting,andiswidelyusedtomodelsituationsinwhichmarketpriceisdeterminedbythetotalsupplytothemarket(see,e.g., BernsteinandFedergruen 2003 KeskinocakandSavasaneril 2008 ). Thesuppliermustdelivertheentireorderforeachretailer;thus,thesupplier'sproductionquantityequalsthesumoftheretailers'orderquantities.Aseachretailerwillorderthequantitythats/hewillsupplytothemarket,thesupplier'sproductionquantityisQ=Pni=1qi.Thesupplierincursproductioncosts,whichareafunctionofQand,hence,thesupplier'sprotequalstotalrevenuegainedfromsellingtoretailerslessproductioncosts.Nowletcdenotethesupplier'swholesaleprice(i.e.,thesupplier'ssellingpricetoretailers).Weassumethatthewholesalepricesetbythesupplieristhesameforeachretailer.ThisisconsistentwiththeRobinson-PatmanAct,whichprohibitspricediscrimination.Furthermore,asuppliermaychoosetoutilizeasimplewholesalepricingscheme( IngeneandParry 1995 1998 ).Thenthesupplier'sprotasafunction 112

PAGE 113

ofQ,S(Q),reads S(Q)=cQ)]TJ /F3 11.955 Tf 11.96 0 Td[(f(Q),(5) wherecQisthetotalrevenueandf(Q)representstheproduction-relatedcostfunction.Weassumethatf(Q)isacontinuousanddifferentiablefunctionofQ.Inparticular,f(Q)isanondecreasingfunction,sinceproductioncostisexpectedtoincreasewithproductionquantity.Furthercharacteristicsoff(Q)willbediscussedinmoredetailwhenweanalyzethesupplier'sproblem. Weassumethateachretailerissubjecttooperatingcosts,andletvi(qi)denotetheoperatingcostfunctionofanyretaileri.Inparticular,letwi>0denotetheper-unitoperatingcostofretaileri.Observethatwicanbeinterpretedasaper-unitshippingcostfromretaileritothemarket(andretailerslocatedindifferentareaswillthereforehavedifferentwivalues)orsimplyaper-unitoperating(e.g.,materialhandling)costofretaileri(whenretailersutilizedifferenttechnologies,thismayalsoleadtodifferentwivalues).Consideringthepracticeofretailers'ordersbeingshippedtothedistributorandthentotheretailers,wicanbeinterpretedaswi=w+ri,wherewwoulddenoteshippingcostfromthesuppliertothedistributorwarehouseandriwoulddenotetheshippingcostfromthedistributortoretaileriplustheshippingcostfromretaileritothemarket.Thetotaloperatingcostofretaileri,then,amountstovi(qi)=wiqi.(Similarretailercostfunctionsareusedin KeskinocakandSavasaneril 2008 .)Weassumethatretailersarenon-identical;thus,wi16=wi2fori16=i2,i1,i22f1,2,...,ng.Theprotofretaileriequalsthetotalrevenuegainedfromsupplyingthemarket,lesspurchaseandoperatingcosts.Then,consideringEquation( 5 ),theprotfunctionofretailerireads i()778(!Q)=p(Q)qi)]TJ /F3 11.955 Tf 11.96 0 Td[(cqi)]TJ /F3 11.955 Tf 11.95 0 Td[(vi(qi)= a)]TJ /F3 11.955 Tf 11.95 0 Td[(bnXi=1qi!qi)]TJ /F3 11.955 Tf 11.95 0 Td[(cqi)]TJ /F3 11.955 Tf 11.96 0 Td[(wiqi,(5) where)778(!Qdenotesthen-vectorofretailersupplyquantities.ThersttermofEquation( 5 )denotesthetotalrevenue,thesecondtermisthetotalpurchasecost,andthelasttermrepresentstheoperatingcosts. 113

PAGE 114

Thesupplierandtheretailers,whenhorizontallydecentralized,areprotmaximizingagents;thus,thesupplier'sproblem,PS,andtheproblemofretaileri,Pi,canbeformulatedasfollows: (PS)maxS(Q)(Pi)maxi()777(!Q)s.t.Q0,s.t.qi0. Thedistributorcanaffecttheprocurementstrategyunderwhichtheretailersoperate.Ifthedistributorcentralizesretailerprocurement(anddeterminesthemarketsupply),thentheretailerorderquantitiescanbecollectivelysettomaximizethesumoftheretailers'prots.Ontheotherhand,ifthedistributordoesnotcentralizetheretailers,i.e.,whenretailersarehorizontallydecentralized,andthedecisionofanyindividualretailerisaffectedbythedecisionsoftheotherretailers.Therefore,weusetheequilibriumconceptof Nash ( 1951 )tocharacterizethequantitydecisionsoftheretailersincaseofhorizontaldecentralization. Werstfocusonsolvingfortheretailers'orderquantitiesgiventhedistributor'sprocurementstrategy,and,thenthesupplier'swholesaleprice-settingproblemissolved.Here,giventhesupplier'swholesaleprice,theretailers'orderquantitydecisionsaredeterminedunderthreedifferentdistributorprocurementstrategies.Then,foreachsuchstrategy,thesupplier'swholesalepricesettingproblemissolved. 5.4RetailStage:SupplyQuantitiesandProcurementStrategy Inthissection,foragivensupplierwholesaleprice,weprovideasolutionmethodfortheretailers'orderquantitydecisionsunderthreedifferentdistributorprocurementstrategies.Specically,wecharacterizeretailerorderquantitieswhenretailerprocurementisdecentralized,centralized,andpartiallycentralized.Attheendofthissection,wecomparethesequantitydecisionsunderdifferentprocurementstrategies.Furthermore,wediscussthedistributor'schoiceofprocurementstrategyandimplicationsonthesupplier'spreference,ifs/hecancontrolorinuencethedistributor'sprocurementstrategy. 114

PAGE 115

5.4.1DecentralizedRetailing Whenretailerprocurementisdecentralized,thedistributorallowsretailerstoindependentlydeterminetheirorderquantities.Inthiscase,thedistributor'sroleistoprovidecommunicationbetweenthesupplierandtheretailerswhowanttosellthesupplier'sproduct.(Thedistributorcanchargeretailersonper-unitbasis,inparticular,whenordersareshippedtothedistributor'swarehouserst,andthentoretailers.Thisper-unitchargecanbeconsideredaspartoftheretailers'per-unitoperatingcosts;hence,thiswillnotchangetheanalyticalresults.Nevertheless,asthedistributoristhecentraldecisionmakerfortheretailstage,thisper-unitchargeisnotnecessary.)Horizontaldecentralizationattheretailstagehasbeenanalyzedintheliterature(see,e.g., DongandRudi 2004 YangandZhou 2006 Shaoetal. 2009 ).Underhorizontaldecentralization,eachretailer'sobjectiveistomaximizehis/herprotbydeterminingitsmarketsupplyquantity,whichisaffectedbythesupplyquantitydecisionsofallotherretailers.Inparticular,retailerdecisionscorrespondtoaCournotoligopolyandouraimistodeterminetheNashequilibriumsolutionofthisgame.Recallthatretaileri'sproblemisdenedbyPi.Itcanbeshownthati()777(!Q)isstrictlyconcave,giventheorderquantitiesoftheotherretailers.(Thisfollows,as@2i()777(!Q)=@q2i=)]TJ /F4 11.955 Tf 9.3 0 Td[(2b<0andasb>0.)Thisimpliesthattherst-orderconditions(@i()778(!Q)=@qi=0,forqi>0)mustbesatisedattheuniqueNashequilibriumsolution.Theuniquenessfollowsfrom(i)strictconcavityoftheprotfunctions,(ii)theassumptionofnon-identicalretailers,and(iii)boundedretailerquantitydecisions,aandbarereal-valuednumbers,i.e.,theretailerswillnotsupplyagainstanegativemarketprice,and,thus,theirsupplyquantitieswillbeboundedfromabove,which,togetherwithqi0,8i,impliesthecompactnessoftheretailers'strategysets.ItfollowsfromEquation( 5 )thatifqi>0,thentheNashequilibriumsolutionmustsatisfythecondition a)]TJ /F3 11.955 Tf 11.96 0 Td[(bnXi=1qi)]TJ /F3 11.955 Tf 11.96 0 Td[(bqi)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.95 0 Td[(wi=0.(5) 115

PAGE 116

Letqidenotethequantitysuppliedbyretaileriatequilibrium.Thenretaileriisdenedasactiveifqi>0.Inwhatfollows,wediscussimportantcharacteristicsoftheequilibriumsupplyquantities.TheproofofthefollowingpropositionandthecorrectnessofthealgorithmthatfollowsareprovidedinAppendix C undergeneralmarketpriceandretaileroperatingcostfunctions(Appendix C showsthatProposition 5.1 andAlgorithm 6 holdwhenp(Q)isadecreasingconcavefunctionofQandvi(qi)isanincreasingconvexfunctionofqi.Astheseconditionsaresatisedforlinearmarketpriceandoperatingcostfunctions,wedonotrepeattheproofsforthecasewhenp(Q)=a)]TJ /F3 11.955 Tf 12.19 0 Td[(bQandvi(qi)=wiqi).Thefollowingpropositionprovidesimportantcharacteristicsoftheequilibriumsupplyquantities. Proposition5.1. (i)qi>0ifandonlyifc+wi0,thenqi1>0,and(b)ifqi1=0,thenqi2=0. Proof:PleaseseeAppendix C Condition(i)ofProposition 5.1 simplystatesthataretailerisactiveifandonlyifitsmarginalcost(c+wi)islessthantheequilibriummarketprice.Condition(ii)indicatesthatsortingretailerswithrespecttotheirper-unitoperatingcostsisimportantincharacterizingequilibriumsupplyquantities.Inparticular,ifweknowthatthereare`nretailersactiveatequilibrium,itthenensuesthattheseretailersaretherst`retailerswiththelowestwivalues.Now,withoutlossofgenerality,supposethatretailersaresortedinincreasingorderofwivaluesandletusassumethat`retailersareactiveforagivenwholesalepricec.Thatis,qi>0forretailersi=1,2,...,`andqi=0forretailersi=`+1,`+2,...,n.ThentheequilibriumsupplyquantitiesaredeterminedbythesimultaneoussolutionofEquation( 5 )foralli`.However,wedonotknowthenumberofactiveretailersapriori.ThenextalgorithmisarealizationofthealgorithmstatedinAppendix C ,whichsolvesfortheequilibriumquantitiesundergeneralizedmarketpriceandoperatingcostfunctions,determiningthenumberofactiveretailers 116

PAGE 117

aswellastheassociatedequilibriumsupplyquantitieswhenp(Q)=a)]TJ /F3 11.955 Tf 12.99 0 Td[(bQandvi(qi)=wiqi. Algorithm6. Withoutlossofgenerality,supposethatretailersaresortedinincreasingorderofwivalues.Givena,b,candwi8i2f1,2,...,ng; Step0.Ifa)]TJ /F4 11.955 Tf 12.14 0 Td[((c+w1)0,setqi=08i2f1,2,...,ngand`=0.Else,set`=1andgotoStep1. Step1.Determineq(`)ifori`bysolvingthefollowingsystemofequations.GotoStep2.a)]TJ /F3 11.955 Tf 11.95 0 Td[(b`Xi=1q(`)i)]TJ /F3 11.955 Tf 11.95 0 Td[(bq(`)i)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.95 0 Td[(wi=08i`. Step2.If`=n,stop.Alloftheretailersareactive;qi=q(`)iforinand`=n.Else,if`
PAGE 118

ConsideringEquation( 5 ),thesolutionundercentralizedretailingisobtainedbysolvingthefollowingproblem: PcmaxPni=1i()777(!Q)s.t.qi0,i=1,...,n. OnecaneasilyseethatthesolutionofthedecentralizedprocurementstrategyisfeasibleforPc,andcentralizedprocurement,therefore,cannotreducethetotalprotoftheretailers.ThefollowingpropositioncharacterizesthesolutiontoPc.AlloftheproofsarepresentedintheAppendix. Proposition5.2. qc1=a)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F9 7.97 Tf 6.59 0 Td[(w1 2b0andqci=0fori=2,3,...,n,whereqcidenotesthesupplyquantityofretaileriundercentralizedretailing. Proof:WerstshowthattheobjectivefunctionofPcisstrictlyconcavebyshowingthattheHessianmatrixoftheobjectivefunction,H,ispositivedeniteforall)777(!Q2Rn.NotethatPni=1i()778(!Q)=Pni=1(a)]TJ /F3 11.955 Tf 11.95 0 Td[(bPni=1qi)qi)]TJ /F3 11.955 Tf 11.95 0 Td[(cqi)]TJ /F3 11.955 Tf 11.96 0 Td[(wiqi=)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(bPni=1qiPni=1qi)]TJ /F10 11.955 Tf -435.72 -14.94 Td[(Pni=1wiqiandonecanshowthatH=)]TJ /F4 11.955 Tf 9.3 0 Td[(2b1where1isannnmatrixof1's,i.e.,Hissymmetric.Asb>0,wehaveH<0,whichimpliesstrictconcavityofPni=1i()778(!Q).ThentheKKTconditionsarenecessaryandsufcientforproblemPcaswehavelinearconstraints.TheKKTconditionsread (a)a)]TJ /F4 11.955 Tf 11.95 0 Td[(2bPni=1qi)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 11.95 0 Td[(wi+ui=0i=1,2,...,n,(b)uiqi=0i=1,2,...,n,(c)ui0i=1,2,...,n. Werstarguethatonlyoneoftheqivaluescanbenon-zeroToestablishacontradiction,supposethatqr>0andqs>0,whichimpliesur=us=0by(b).Thenitfollowsfrom(a)forretailersrandsthata)]TJ /F4 11.955 Tf 12.18 0 Td[(2bPni=1qi=c+wranda)]TJ /F4 11.955 Tf 12.18 0 Td[(2bPni=1qi=c+ws,whichmeanswr=ws.Thisestablishesacontradictionaswr6=ws;hence,onlyoneoftheqivaluescanbenon-zero.Furthermore,onlyq10asotherwise(a)impliesu1<0whichcontradicts(c).Inthiscase,itfollowsfrom(a)thatqc1=a)]TJ /F9 7.97 Tf 6.58 0 Td[(c)]TJ /F9 7.97 Tf 6.59 0 Td[(w1 2b0. Proposition 5.2 impliesthatonlytheretailerwiththelowestper-unitoperatingcost,i.e.,onlyretailer1willsupplytheproducttotheendcustomermarket.Inpractice, 118

PAGE 119

thiswouldindicatethattheretailerwiththelowestper-unitoperatingcostactsasthedistributor,or,equivalently,thedistributorverticallyintegrateswiththemostefcientretailer.However,thisdegreeofcentralizationbasedonasingleproductorproductlinecanbeimpractical.Furthermore,whenthedistributoralsocarriesotherproductssoldviaretailers,orretailersselladditionalproducts,centralizationoftheretailersforthesupplier'sproductmaynotbepreferred.Therefore,wenextstudyaprocurementstrategythatmaintainsthemarketshareunderdecentralizeddecisions,whiletargetingincreasedprotlevels. 5.4.3PartiallyCentralizedRetailing Theideaofpartialcentralizationofretailersistousethesameobjectiveasundercentralization,buttoapplyconstraintsthatpreserveeachretailer'smarketshare.Letkidenoteretaileri'sfractionofmarketsupplyatequilibriumunderdecentralizedprocurementforthegivenc(thatis,ki=qi=Pni=1qiandcanbedeterminedusingAlgorithm 6 foranygivenc).Foragivenc,kiisaconstant.Ontheotherhand,whenwestudythesupplier'sproblemunderpartiallycentralizedretailers,kiwillbeafunctionofc.NotethatPni=1ki=1.Now,ifweletqpidenethesupplyquantityofretaileri,in,underpartialcentralization,weshouldhaveqpi=(Pni=1qpi)=ki.Thesolutiontothepartiallycentralizedprocurementproblemwillthenbeobtainedbysolvingthefollowingproblem: PpmaxPni=1i()777(!Q)s.t.qi=kiPni=1qi,i=1,...,n. Sincethesolutionoftheretailers'gameunderdecentralizationisfeasibleforPp,partialcentralizationleadstoasolutionthatisatleastasgoodfortheretailstage.ThesolutionofPpisstatednext. Proposition5.3. qpi=kia)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F21 7.97 Tf 6.59 5.98 Td[(Pni=1wiki 2b. 119

PAGE 120

Proof:Underpartialcentralization,theobjectivefunctionofPpreadsasnXi=1i()777(!Q)=nXi=1"(a)]TJ /F3 11.955 Tf 11.96 0 Td[(bnXi=1qi)qi)]TJ /F3 11.955 Tf 11.95 0 Td[(cqi)]TJ /F3 11.955 Tf 11.96 0 Td[(wiqi#=Q a)]TJ /F3 11.955 Tf 11.95 0 Td[(bQ)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F9 7.97 Tf 18.31 14.94 Td[(nXi=1wiki!. NotethattheobjectivefunctionisconcaveinQ;hence,usingtherstordercondition,onecanconcludethatQ=a)]TJ /F9 7.97 Tf 6.58 0 Td[(c)]TJ /F21 7.97 Tf 6.58 5.97 Td[(Pni=1wiki 2bintheoptimalsolution.Therefore,qpi=kia)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F21 7.97 Tf 6.58 5.98 Td[(Pni=1wiki 2b. Unlikecentralization,alloftheretailersmaintainthemarketshareassociatedwithdecentralizedprocurementunderpartialcentralization.Eventhoughcentralizationisstillmoreprotablefortheretailstage(asthesolutionunderpartialcentralizationisfeasibleundercentralization),thedistributormaypreferpartialcentralizationasaresultofthepracticalconcernswehavenoted. 5.4.4ComparisonofProcurementStrategies Itisclearthatthetotalprotoftheretailstageismaximizedunderacentralizedprocurementstrategy.Incaseswherecompletecentralizationisnotpracticalorpossibleandmarketsharemustbemaintained,partialcentralizationmaximizesthetotalprotattheretailstage.Decentralizationasaprocurementstrategyresultsinthelowestlevelofprotattheretailstage.Asaresultwehavethefollowingrelation: nXi=1i()778(!Qc)nXi=1i()777(!Qp)nXi=1i()778(!Q),(5) where)778(!Q,)777(!Qc,and)777(!Qpdenotethequantitydecisionvectorsoftheretailersunderdecentralization,centralization,andpartialcentralization,respectively.Therefore,thedistributorwillprefercentralizationoftheretailersifpossible,orpartialcentralizationwhenhorizontalcentralizationisnotpracticalorpossible. Asnotedpreviously,thesupplierdoesnothavecontroloverthedistributor'sprocurementstrategy.Thesupplier'sprotlevelsforagivenwholesalepricewill,however,bestronglyaffectedbythedistributor'sprocurementstrategy.Specically,thetotalquantityorderedfromthesupplierdiffersundereachofthedistributor's 120

PAGE 121

procurementstrategies.Inwhatfollows,wecomparethetotalquantityorderedfromthesupplierfordifferentprocurementstrategies. Proposition5.4. LetQ=Pni=1qi,Qc=Pni=1qci,andQp=Pni=1qpidenotetheretailers'totalorderquantityunderdecentralization,centralization,andpartialcentralization,respectively.ThenQpQcQ. Proof:PleaseseeAppendix D Proposition 5.4 statesthattheretailers'aggregatedorderquantityismaximizedunderdecentralizedprocurement.Whilecentralizedprocurementresultsinasmallertotalorderquantitythandecentralizedprocurement,itresultsinagreatertotalorderquantitythanpartiallycentralizedprocurement.Hence,thedistributorcanincreasetotalretailprotviacentralizationorpartialcentralization,whiledecreasingthetotalquantityorderedfromthesupplier.Ontheotherhand,thesupplier'spreferenceswilldifferaccordingtotheeconomiesordiseconomiesofscalethesupplierfaces.Foragivenwholesaleprice,wesummarizethesupplier'sbestinterestswithrespecttothedistributor'sprocurementstrategyasfollows. Givenc,ifthesupplierissubjecttoeconomiesofscale,itisinthesupplier'sbestinterestifretailerprocurementisdecentralized.Ifdistributorcentralizesretailerprocurement,itistheninthesupplier'sbestinterestifretailerprocurementiscentralizedratherthanpartiallycentralized. Givenc,ifthesupplierissubjecttodiseconomiesofscale,itisinthesupplier'sbestinterestifretailerprocurementiscentralizedorpartiallycentralized.Ifthedistributorcentralizesretailerprocurement,itistheninthesupplier'sbestinterestiftheretailersarepartiallycentralizedratherthancentralized. Theabovepointsleadtoimportantinsightsoncoordinationandcentralizationofsupplyandretailstages.Itshouldbenotedthatthechannel-wideprotsunderdifferentprocurementstrategiesvarydependingonthesupplier'seconomiesordiseconomiesofscale.Therefore,sincethemaximumchannel-wideprotscanbeachievedunderaspecicprocurementstrategy,acoordinationpolicyorapproachmayneedtoconstraintheretailprocurementstrategy.InSection 5.7 ,wediscussfurtherinsightsonthe 121

PAGE 122

coordinationofthechannelunderdifferentdownstreamcharacteristics.Thenextsectionanalyzesthesupplier'swholesaleprice-settingproblemforeachprocurementstrategy. 5.5TheSupplier'sProblem:OptimalWholesalePrice Inthissection,werstcharacterizethesupplier'sprotasafunctionofthewholesaleprice.RecallfromEquation( 5 )thatthesupplier'sprotfunctiondependsonthetotalorderquantityoftheretailers.Furthermore,itisclearfromAlgorithm 6 andPropositions 5.2 and 5.3 thatthetotalretailerorderquantityunderanyprocurementstrategydependsonthesupplier'swholesaleprice.Hence,wederiveclosed-formexpressionsfortotalretailerorderquantityasafunctionofthewholesaleprice.Thisenablesstatingthesupplier'sproblemintermsofconly,thesupplier'swholesaleprice.Inwhatfollows,wediscussthesupplier'swholesaleprice-settingproblemfordecentralized,centralized,andpartiallycentralizedretailersusingthisapproach. 5.5.1WholesalePricingforDecentralizedRetailing Supposethatthedistributorprefersdecentralizedretailing,i.e.,whenretailersarecompetitiveintheendcustomermarket.Next,wedenotethesumofretailersupplyquantitiesbyQ(c)(tostressits'dependenceonthesupplier'swholesaleprice),andderiveaclosed-formexpressionforthisfunction.ThisenablesstatingPSintermsofconly. Let`(c)denotethenumberofactiveretailersforagivenvalueofc.Assumingwithoutlossofgeneralitythatretailersaresortedinincreasingorderofwivalues,itthenfollowsthatif`(c)=`,thenqi>0forretailersi`,anditcanbeshownthat(seeproofofProposition 5.4 giveninAppendix D ) Q(cj`(c)=`)=`(a)]TJ /F3 11.955 Tf 11.96 0 Td[(c) b(`+1))]TJ /F10 11.955 Tf 14.04 17.11 Td[(P`i=1wi b(`+1),(5) whereQ(cj`(c)=`)denotesthetotalquantityorderedbytheretailersasafunctionofc,giventhat`(c)=`.However,asisclearfromAlgorithm 6 ,`(c)isitselfafunctionofc.Forthisreason,werstcharacterize`(c)andthenuseittoderiveQ(c). 122

PAGE 123

Supposethatc1Q(c2j`(c2)=`).Thatis,whenthenumberofactiveretailersdoesnotchangewithanincreaseinthewholesaleprice,thetotalorderquantitydecreaseswiththeincreaseinthewholesaleprice.Letusdenec`astheminimumwholesalepriceresultingin`)]TJ /F4 11.955 Tf 12.96 0 Td[(1activeretailers,i.e.,c`=minfc:`(c)=`)]TJ /F4 11.955 Tf 12.96 0 Td[(1gfor`1andobservefromEquation( 5 )thatQ(c)isdecreasingincforc2[c`,c`)]TJ /F11 7.97 Tf 6.59 0 Td[(1).Thus,ifc`0g.ConsideringStep2ofAlgorithm 6 ,itfollowsthat`(c)=maxf`:c+w`+1a)]TJ /F3 11.955 Tf 12.72 0 Td[(bP`i=1q(`)ig=maxn`:c+w`+1a)]TJ /F3 11.955 Tf 11.95 0 Td[(b`(a)]TJ /F9 7.97 Tf 6.59 0 Td[(c) b(`+1))]TJ /F21 7.97 Tf 13.15 12.18 Td[(P`i=1wi b(`+1)o;then `(c)=max(`:w`+1a)]TJ /F3 11.955 Tf 11.95 0 Td[(c `+1+P`i=1wi `+1).(5) Equation( 5 )impliesthat`(c)isadecreasingstepfunctionofc,andc`
PAGE 124

Proposition5.5. Q(c)isapiecewiselinearandcontinuousfunctionofc.Furthermore,eachpieceisadecreasinglinearfunctionofcsuchthatQ(cr+2))]TJ /F9 7.97 Tf 6.58 0 Td[(Q(cr+1) cr+1)]TJ /F9 7.97 Tf 6.58 0 Td[(cr+2Q(cr+1))]TJ /F9 7.97 Tf 6.58 0 Td[(Q(cr) cr)]TJ /F9 7.97 Tf 6.59 0 Td[(cr+1foranyr,1r`(0))]TJ /F4 11.955 Tf 11.95 0 Td[(1. Proof:Werstnotethatonlythec`values,`=1,2,...,n,arepossiblevaluesofdiscontinuity.Hence,toprovethatQ(c)isapiecewisecontinuousfunctionofc,weshowthatQ(c)iscontinuousatanysuchpoint,saycr+1.ToshowthatQ(c)iscontinuousatcr+1,weshowthatlimc!c+r+1Q(c)=limc!c)]TJ /F17 5.978 Tf -.7 -6.19 Td[(r+1Q(c).Todoso,werstcharacterizecr+1.Bydenition,cr+1=minfc:`(c)=rg=minfc:maxf`:w`+1a)]TJ /F9 7.97 Tf 6.59 0 Td[(c `+1+P`i=1wi `+1g=rg=minfc:wr+1a)]TJ /F9 7.97 Tf 6.58 0 Td[(c r+1+Pri=1wi r+1g=minfc:ca+Pri=1wi)]TJ /F4 11.955 Tf 13.57 0 Td[((r+1)wr+1g=a+Pri=1wi)]TJ /F4 11.955 Tf 13.57 0 Td[((r+1)wr+1.Now,notethatlimc!c+r+1Q(c)=Q(cr+1)=Q(cr+1j`(cr+1)=r).Thenlimc!c+r+1Q(c)=r(a)]TJ /F9 7.97 Tf 6.59 0 Td[(cr+1) b(r+1))]TJ /F21 7.97 Tf 14.25 12.17 Td[(Pri=1wi b(r+1)=r(a)]TJ /F11 7.97 Tf 6.59 0 Td[((a+Pri=1wi)]TJ /F11 7.97 Tf 6.58 0 Td[((r+1)wr+1)) b(r+1))]TJ /F21 7.97 Tf 14.25 12.17 Td[(Pri=1wi b(r+1)=rwr+1)]TJ /F21 7.97 Tf 6.58 5.97 Td[(Pri=1wi b.Ontheotherhand,limc!c)]TJ /F17 5.978 Tf -.7 -6.19 Td[(r+1Q(c)=Q(limc!c)]TJ /F17 5.978 Tf -.7 -6.19 Td[(r+1c)=Q(cr+1,`(limc!c)]TJ /F17 5.978 Tf -.7 -6.19 Td[(r+1))=Q(cr+1j`(cr+1)=r+1).Thenlimc!c)]TJ /F17 5.978 Tf -.71 -6.19 Td[(r+1Q(c)=(r+1)(a)]TJ /F9 7.97 Tf 6.59 0 Td[(cr+1) b(r+2))]TJ /F21 7.97 Tf 13.79 12.3 Td[(Pr+1i=1wi b(r+2)=(r+1)(a)]TJ /F11 7.97 Tf 6.59 0 Td[((a+Pri=1wi)]TJ /F11 7.97 Tf 6.58 0 Td[((r+1)wr+1)) b(r+2))]TJ /F21 7.97 Tf 13.79 12.3 Td[(Pr+1i=1wi b(r+2)=((r+1)2)]TJ /F11 7.97 Tf 6.59 0 Td[(1)wr+1)]TJ /F11 7.97 Tf 6.59 0 Td[((r+2)Pri=1wi b(r+2)=(r+2)rwr+1)]TJ /F11 7.97 Tf 6.59 0 Td[((r+2)Pri=1wi b(r+2)=rwr+1)]TJ /F21 7.97 Tf 6.59 5.98 Td[(Pri=1wi b.Thuslimc!c+r+1Q(c)=limc!c)]TJ /F17 5.978 Tf -.71 -6.19 Td[(r+1Q(c)foranyr,1r`(0))]TJ /F4 11.955 Tf 12.38 0 Td[(1.Hence,Q(c)iscontinuousinc.RecallthatQ(cr+1)=rwr+1)]TJ /F21 7.97 Tf 6.58 5.97 Td[(Pri=1wi b.ThenitfollowsthatQ(cr+2))]TJ /F3 11.955 Tf 12.08 0 Td[(Q(cr+1)=(r+1)(wr+2)]TJ /F3 11.955 Tf 12.08 0 Td[(wr+1)=b.Similarly,Q(cr+1))]TJ /F3 11.955 Tf 13.04 0 Td[(Q(cr)=r(wr+1)]TJ /F3 11.955 Tf 13.04 0 Td[(wr)=b.Furthermore,weknowthatcr+1=a+Pri=1wi)]TJ /F4 11.955 Tf 11.96 0 Td[((r+1)wr+1.Thisimpliesthatcr+1)]TJ /F3 11.955 Tf 11.97 0 Td[(cr+2=(r+2)(wr+2)]TJ /F3 11.955 Tf 11.96 0 Td[(wr+1).Similarly,cr)]TJ /F3 11.955 Tf 11.96 0 Td[(cr+1=(r+1)(wr+1)]TJ /F3 11.955 Tf 11.95 0 Td[(wr).Thus,Q(cr+2))]TJ /F9 7.97 Tf 6.58 0 Td[(Q(cr+1) cr+1)]TJ /F9 7.97 Tf 6.58 0 Td[(cr+2=r+1 b(r+2)Q(cr+1))]TJ /F9 7.97 Tf 6.59 0 Td[(Q(cr) cr)]TJ /F9 7.97 Tf 6.58 0 Td[(cr+1=r b(r+1). Figure 5-1 illustrates`(c)andQ(c).Proposition 5.5 impliesthatQ(c)isacontinuousdecreasingconvexfunctionofc. NowconsiderproblemPS.Letc2[c`,c`)]TJ /F11 7.97 Tf 6.59 0 Td[(1),i.e.,supposethereare`activeretailers.Withinthissegment,thesupplier'sproblemistomaximizeS(Q)=cQ)]TJ /F3 11.955 Tf 12 0 Td[(f(Q)subjecttoc`cc`)]TJ /F11 7.97 Tf 6.59 0 Td[(1(notethatwecanwritec`cc`)]TJ /F11 7.97 Tf 6.59 0 Td[(1insteadofc`
PAGE 125

Acvs.`(c) Bcvs.Q(c) Figure5-1. Illustrationsof`(c)andQ(c) productioncostfunction,tobeanondecreasingcontinuousanddifferentiablefunctionofQ.Usingthesecondderivativetest,wecanshowthatifd2f(Q) dQ2)]TJ /F4 11.955 Tf 23.01 0 Td[(2b`(0)+1 `(0)thenS(Q(c))isconcavewithrespecttoc.Notethatthisconditioncanbesatisedbyconcaveandconvexproductioncostfunctions.Wenextarguealternativeconditionsforconcaveproductioncostfunctions,whensatised,implytheconcavityofS(Q(c))withrespecttoc.Furthermore,weshowthattheconvexityoff(Q)impliesstrictconcavityofS(Q(c))withrespecttoc. Proposition5.6. Supposethatf(Q)isconcaveinQandconsiderc1,c22[c`,c`)]TJ /F11 7.97 Tf 6.58 0 Td[(1)suchthatc1>c2for1``(0)+1.DeneQ1andQ2asthetotalorderquantityofthenoncooperativebuyerswhenc=c1andc=c2,respectively.Ifc1)]TJ /F3 11.955 Tf 13.01 0 Td[(c2df(Q1)=dQ)]TJ /F9 7.97 Tf 6.59 0 Td[(df(Q2)=dQ 2thenS(Q(c))isconcaveincon[c`,c`)]TJ /F11 7.97 Tf 6.59 0 Td[(1).Furthermore,iff(Q)isconvexinQforQ0thenS(Q(c))isstrictlyconcaveincforc`cc`)]TJ /F11 7.97 Tf 6.58 0 Td[(1suchthat1``(0)+1. Proof:Supposethatf(Q)isconcaveinQandletc1>c2suchthatc1,c22[c`,c`)]TJ /F11 7.97 Tf 6.59 0 Td[(1)for0``(0).Denerxh(x)astherstderivativeofh(x)withrespecttox.LetQ1andQ2bedenedasintheproposition.ConsideringEquation( 5 ),wecanwriteQ(c)=)]TJ /F5 11.955 Tf 12.63 0 Td[(c,where=`a)]TJ /F21 7.97 Tf 6.58 5.97 Td[(P`i=1wi b(`+1)and=` b(`+1).S(Q(c))=S(c)isconcaveif[rcS(c2))-222(rcS(c1)][c2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1]0forallc1,c20( Bazaraaetal. 2006 ).First 125

PAGE 126

notethatS(c)=cQ(c))]TJ /F3 11.955 Tf 12.74 0 Td[(f(Q(c))andrcS(c)=)]TJ /F4 11.955 Tf 9.3 0 Td[(2c++rQf(Q).Thus,[rcS(c2))-222(rcS(c1)][c2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1]=[)]TJ /F4 11.955 Tf 9.3 0 Td[(2(c2)]TJ /F3 11.955 Tf 11.96 0 Td[(c1)+(rQf(Q2))-222(rQf(Q1))][c2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1].Thenif[)]TJ /F4 11.955 Tf 9.3 0 Td[(2(c2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1)+(rQf(Q2))-222(rQf(Q1))]0,wehaveS(c)concaveover[c`,c`)]TJ /F11 7.97 Tf 6.59 0 Td[(1).Thatis,ifc1)]TJ /F3 11.955 Tf 12.29 0 Td[(c2rQf(Q1)rQf(Q2) 2thenS(Q(c))isconcavewithrespecttocintheinterval[c`,c`)]TJ /F11 7.97 Tf 6.58 0 Td[(1).Nowsupposethatf(Q)isconvexinQforQ0.NotethatS(Q(c))consistsoftwoparts:cQ(c)andf(Q(c)).WerstshowthatcQ(c)isstrictlyconcaveforc`cc`)]TJ /F11 7.97 Tf 6.59 0 Td[(1.WeknowfromEquation( 5 )thatQ(c)=`(a)]TJ /F9 7.97 Tf 6.58 0 Td[(c) b(`+1))]TJ /F21 7.97 Tf 13.65 12.17 Td[(P`i=1wi b(`+1)forc`cc`)]TJ /F11 7.97 Tf 6.58 0 Td[(1.TheniteasilyfollowsthatcQ(c)isstrictlyconcaveinc.Nextweshowthatf(Q(c))isconvex.Notethatweassumef(Q)isnondecreasingandconvexfunction.Furthermore,weknowthatQ(c)islinearincforc`cc`)]TJ /F11 7.97 Tf 6.59 0 Td[(1,i.e.,itisconvex.Thenitfollowsthatf(Q(c))isconvex(see, Bazaraaetal. 2006 ).Thus,S(Q(c))isstrictlyconcaveoverc`cc`)]TJ /F11 7.97 Tf 6.59 0 Td[(1suchthat0``(0). Proposition 5.6 statesthatifthedifferenceinthewholesalepriceisgreaterthanthehalfofthedifferenceinthemarginalproductioncostsatthequantitiescorrespondingtothewholesalepricesc1andc2,thenthesupplier'sprotfunctionwillbepiecewiseconcave.Proposition 5.6 furtherimpliesthatwhenthesupplier'sproductioncostisconvexinQ,thesupplier'sprotfunctionisstrictlyconcaveinthewholesalepricecforc2[c`,c`)]TJ /F11 7.97 Tf 6.58 0 Td[(1];hence,itisapiecewiseconcavefunction.Convexproductioncostsaresometimesobservedinpracticeandhavebeenstudiedintheliterature(see,e.g., Klein 1961 Veinott 1964 EliashbergandSteinberg 1987 SmithandZhang 1998 ).Inparticular,whenthemarginalproductioncostincreasesintheproductionquantity,theproductioncostfunctionwillbeconvex.Thiscanbeduetoincreasedcapacityrequirements( JohnsonandMontgomery 1974 )orexpensiveovertimerequirements( SmithandZhang 1998 Ming-huiandCheng-xiu 2005 ). WehenceforthassumethatS(Q(c))=S(c)isapiecewiseconcavefunctionofc.Thesolutionmethodwenextproposeforthesupplier'sproblemisbasedonthispiecewiseconcavestructure;thus,thisapproachcanalsobeusedforanykind 126

PAGE 127

ofconcaveproductioncostfunctionbyusingapiecewiselinearapproximationoftheproductioncostfunction(aslinearpiecesoff(Q)willimplyconcavityofeachpieceofS(c)).Thesupplier'sproblemineachintervalinvolvesthemaximizationofaconcavefunctionsubjecttolinearboundaryconstraints.Letc`denotethewholesalepricethatmaximizesS(c)overc`+1cc`for1``(0)(notethatwedonotconsiderc>c1,becauseforc>c1,`(c)=0andQ(c)=0,hence,wecandeneS(c)=0).ThenextCorollary,whichcharacterizesc`,isadirectresultoftheconcavityofeachpieceofS(c). Corollary2. Let(`)S(c)denotethesupplier'sprotfunctionontheintervalc`+1cc`andletc0`bedenedsuchthatd(`)S(c)=dc=0atc0`.Thenc`=8>>>><>>>>:c`+1ifc0`c`, wherec`=a+P`)]TJ /F11 7.97 Tf 6.59 0 Td[(1i=1wi)]TJ /F5 11.955 Tf 12.28 0 Td[(`w`for1``(0)andc`(0)+1=0(pleaseseetheproofofProposition 5.5 forthederivationofthisequationforc`). Thefollowingcorollarycharacterizesthesolutionofthesupplier'sproblemunderdecentralizedprocurement. Corollary3. Givena,b,andwi8i2f1,2,...,ng,thesupplier'soptimalwholesalepricewhenretailersaredecentralizedis c=argmax1``(0)fS(c`)g.(5) Corollary 3 followsfromCorollary 2 andthepiecewiseconcavestructureofS(c).Notethatmultipleoptimamayexistforthesupplier'sproblem.Inthecaseofalternativeoptima,ifthesupplierpreferstoworkwithfewer(more)retailers,s/hemaychoosethemaximum(minimum)ofthealternativeoptimalwholesaleprices.Next,wediscussthesupplier'sproblemforcentralizedandpartiallycentralizedretailing. 127

PAGE 128

5.5.2WholesalePricingforCentralizedandPartiallyCentralizedRetailing First,supposethatthedistributorcentralizesretailerprocurement.Inthiscase,weknowfromProposition 5.2 thatQ(c)=a)]TJ /F9 7.97 Tf 6.58 0 Td[(c)]TJ /F9 7.97 Tf 6.58 0 Td[(w1 2b.Therefore,Q(c)isindependentofthenumberofactiveretailers,andisalinearlydecreasing,continuousfunctionofc.Thenthesupplier'sprotfunctionisnotapiecewisefunction,i.e.,itisdifferentiableoverc>0.WecaneasilygeneralizethediscussionontheconcavityofS(c)andconcludethattheoptimalwholesaleprice,c,isdeterminedbytherst-orderoptimalityconditionforthesupplier'sconcaveprotfunctionwhenf(Q)isconvexinQ.Next,supposethatthedistributorpartiallycentralizesretailerprocurement.ThenProposition 5.3 impliesthatQ(c)=a)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F21 7.97 Tf 6.59 5.97 Td[(P`(c)i=1wiki 2b.Similarlytothecaseofdecentralizedretailing,underpartialcentralization,Q(c)notonlydependsoncbutalso`(c),whichitselfisafunctionofc.Thus,itfollowsthatQ(c)hasapiecewiseconcavestructureasinthecaseofdecentralizedretailers.Wenotethatalthoughkiisafunctionofc,wecanstillshowthatS(c)ispiecewiseconcaveinciff(Q)isconvexinQ.Thus,wecanuseCorollaries 2 and 3 tosolvethesupplier'sproblemwhenthedistributorpartiallycentralizestheretailersandthesupplier'sproductioncostfunctionisconvex.Thenextsectionextendsourmodeltoconsidermultiplemarketsandsupplierdiscountpricing. 5.6Extensions:MultipleMarketsandDiscountPricing Werstextendthepreviousresultstoallowformultiplemarkets.Furthermore,weshowhowourresultscanbegeneralizedtoasettinginwhichthesupplieroffersaquantitydiscountpricingscheme. Nowsupposethatthesuppliersellshis/herproducttomendcustomermarketsthroughncompetitiveretailers,whoseordersmaybecontrolledbyadistributor(orareunderjointownershipofachainstore).Letjindexmarketsandsupposethepriceinmarketjisdeterminedbytheequationpj(Qj)=aj)]TJ /F3 11.955 Tf 12.35 0 Td[(bjPni=1qij,whereqijdenotesthequantitysuppliedtomarketjbyretaileriandQj=Pni=1qij.Thenretaileri'sproblemunderdecentralizedprocurementistomaximizei(bQ)=Pmj=1)]TJ /F3 11.955 Tf 10.46 -9.68 Td[(aj)]TJ /F3 11.955 Tf 11.95 0 Td[(bjPni=1qijqij)]TJ ET BT /F1 11.955 Tf 224.03 -687.85 Td[(128

PAGE 129

cPmj=1qij)]TJ /F10 11.955 Tf 13.1 8.97 Td[(Pmj=1wijqij,whereweusebQtodenotethenmmatrixofqijvalues,i=1,2,...,n,j=1,2,...,m,andwijdenotestheoperatingcostofretaileriforservingmarketj.Notethattheobjectivefunctionofretaileriisstrictlyconcaveinitsqijvaluesforgivensupplyquantitiesoftheotherretailers,asbj>0.Thus,therst-ordercondition,aj)]TJ /F3 11.955 Tf 12.41 0 Td[(bjPnj=1qij)]TJ /F3 11.955 Tf 12.41 0 Td[(bjqij)]TJ /F3 11.955 Tf 12.4 0 Td[(c)]TJ /F3 11.955 Tf 12.41 0 Td[(vij=0,mustbesatisedataNashequilibriumsolutionifqij>0.Notethattherst-orderconditionassociatedwithretaileri'ssupplyquantityatmarketjdoesnotdependonthesupplyquantitiesforothermarketsoronothermarketparameters.Hence,theretailers'problemsareseparablebymarket,whichcanbeobservedini(bQ)aswell.Then,usingAlgorithm 6 foreachmarketseparately,onecansolvefortheequilibriumsupplyquantitiesatthemarkets.Furthermore,thetotalquantitysuppliedtomarketjasafunctionofcreadsQj(c)=`j(c)(aj)]TJ /F9 7.97 Tf 6.59 0 Td[(c) bj(`j(c)+1))]TJ /F21 7.97 Tf 17.21 12.52 Td[(P`j(c)i=1wij bj(`j(c)+1),where`j(c)denotesthenumberofactiveretailersinmarketjforagivenvalueofc.ItfollowsfromProposition 5.5 thatQj(c)ispiecewisecontinuousanddecreasinginc,whichimpliesthatQ(c)=Pmj=1Qj(c)isapiecewisecontinuousanddecreasingfunctionofc.UsingthebreakpointsoftheQj(c)functions,wecandeterminetherangeofcvaluesforwhicheachpieceofQ(c)isdened.ThenthesolutionmethodproposedinCorollaries 2 and 3 canbeusedtosolvethesupplier'sproblem.Furthermore,whentheretailersarecentralizedorpartiallycentralizedbythedistributorinallofthemarkets,itcanagainbeshownthateachmarketcanbeconsideredseparately,andthesolutionsdiscussedinSections 5.4.2 and 5.4.3 forasinglemarketcanbeusedforeachmarket.Wenotethatcentralizationoftheretailersinthemultiple-marketcasewouldimplythateachmarketisoccupiedbyasingleretailer;inparticular,theonewiththelowestoperatingcostinthatmarket.Finally,itshouldbenotedthatthemultiple-marketcasecanalsoberepresentedasasingle-market,multiple-productscenario,suchthattheproductsaresuppliedbythesamesupplierandaredifferentiated(i.e.,theyarenotdemandsubstitutes;hence,theyhaveindependentinversedemandfunctions). 129

PAGE 130

Wenextreturntothesingle-marketsetting,butsupposethatthesupplieroffersaquantitydiscountpricingscheme,suchthatthewholesalepriceisalineardecreasingfunctionofthetotalquantityorderedfromthesupplier.Specically,thesupplier'swholesalepriceisgivenbyc(Q)=)]TJ /F5 11.955 Tf 12.39 0 Td[(Pni=1qi,whereisthemaximumwholesalepriceandisaper-unitdiscount.Wenotethatthiskindofdiscountpriceschemehasbeenusedinpastliterature(see,e.g., IngeneandParry 1995 1998 KeskinocakandSavasaneril 2008 ).Whenthesupplierofferssuchaquantitydiscountingschedule,retaileri'sproblemunderdecentralizedprocurementistomaximizei()778(!Q)=(a)]TJ /F3 11.955 Tf -427 -23.91 Td[(bPni=1qi)qi)]TJ /F4 11.955 Tf 9.83 0 Td[(()]TJ /F5 11.955 Tf 9.82 0 Td[(Pni=1qi)qi)]TJ /F3 11.955 Tf 9.82 0 Td[(wiqi.Wecanshowthatwhenb>theobjectivefunctionofretaileriisstrictlyconcaveinqiforgivensupplyquantitiesoftheotherretailers.Inasimilarsetting, KeskinocakandSavasaneril ( 2008 )assumethatb>,asmarginalrevenuesdecreasefasterthanmarginalcosts.Hence,themethodofSection 5.4.1 canbeusedunderthisassumption.Furthermore,whentheretailersarecentralizedorpartiallycentralizedbythedistributor,thediscussioninSections 5.4.2 and 5.4.3 stillapplies,thatis,thetotalorderfromthesupplierdecreasesduetocentralization.Whenthesupplieroffersdiscountpricing,themethodsandalloftheresultsofSection 5.4 canapplybylettingbbeb)]TJ /F5 11.955 Tf 12.42 0 Td[(andlettingcbe.Itshouldalsobenotedthatwhenc=and>0,thesupplierincreaseshis/hersalestotheretailersunderanydistributorprocurementstrategybyofferingquantitydiscounts.Ifthesupplier'sdecisionvariableisonly,i.e.,whenthediscountfactorisaxedconstant,themethoddescribedinSection 5.5 todeterminetheoptimalwholesalepricecanbeusedtodeterminetheoptimalvalueof.Ontheotherhand,whenorthe(,)pairaresupplierdecisionvariables,thesupplier'sproblembecomeschallenging.WeposethesecasesasfutureresearchquestionsinChapter 6 .Thefollowingsectiondocumentsournumericalstudies. 5.7NumericalStudy Inthissection,wediscussasetofnumericalstudiesintendedtoanalyzethevalueofcontrol.Thevalueofcontrolinthiscontextcanbeinterpretedastheamountthe 130

PAGE 131

suppliercansaveifs/hecancontrolorinuencethedistributor'sprocurementstrategy.Recallthattheretailersmaybedecentralized,centralizedorpartiallycentralized.Thatis,thedistributormaychooseoneofthethreedistinctprocurementstrategies:decentralizedretailing(DP),centralizedretailing(CP),orpartiallycentralizedretailing(PC).Asdiscussedpreviously,thesuppliercanbenetfromcontrollingthedistributor'sprocurementstrategy.Toanalyzethevalueofcontrolandtheeffectsofchannelparametersonthevalueofcontrol,weconducttwosetsofnumericalstudiesdetailednext. Throughoutournumericalstudies,weassumethatthesupplier'sproductioncostfunctionconsistsofaxedsetupcostandalineartermintheproductionquantity,thatis,f(Q)=+Q,whereistheproductionsetupcostandistheper-unitproductioncost.Agivenprobleminstanceissolvedfor9differentscenariosconsistingofeachpairofcombinationsofDP,CP,andPC.Anypairofvaluesrepresentsthesupplier'swholesalepricedecisionwhenassumingaspecicprocurementstrategyalongwiththedistributor'sactualprocurementstrategy.Forinstance,(DP,CP)impliesthatthedistributorprefersdecentralizedprocurementandthesuppliersetsthewholesalepriceassumingcentralizedprocurement.Therefore,forthe(DP,CP)scenario,thesupplier'sdecisionissuboptimalass/hesetsthewholesalepriceassuming(incorrectly)centralizedprocurement.Thesupplier'sactionisoptimalonlyinscenarios(DP,DP),(CP,CP),and(PC,PC).Therstsetofnumericalstudiesstudiesthevalueofcontrol.Todoso,wesolve10probleminstancesforeachcombinationofthefollowingproblemparametersinthesingle-marketcase:n2f3,4,5g,a2f100,110,120g,b2f1,1.25,1.5g,W=fU[5,20],U[20,35],U[35,50]g,=f50,75,100g,and=f25,50,75g,whereWdenotesthen-vectorofwivaluesgeneratedfromauniformdistributionoverthespeciedrange.Thus,wesolve729differentproblemclassesand7290probleminstancesintherstsetofnumericalstudies.Table 5-1 providestheaveragesupplierprotoveralloftheprobleminstancesforeachofthe9scenarios. 131

PAGE 132

Table5-1. Supplier'sprot SupplierDPCPPCDP222.74220.97220.05DistributorCP149.20150.08149.84PC137.05137.99138.16 Notethatweconsiderthecasewhenthedistributor'sprocurementstrategyisknowntothesupplier.Inthiscase,thesupplierwillendupincellsonthediagonalofTable 5-1 ,i.e.,thesupplier'sprotlevelsaredenedin(DP,DP)or(CP,CP)or(PC,PC)ifthedistributorappliesdecentralized,centralized,orpartiallycentralizedprocurement,respectively. Observation5.1. Thesupplier'soptimalprotinscenario(DP,DP)isgreaterthanthesupplier'soptimalprotinscenario(CP,CP);and,thesupplier'soptimalprotinscenario(CP,CP)isgreaterthanitsoptimalprotinscenario(PC,PC).Thatis,(cD)]TJ /F5 11.955 Tf 12.38 0 Td[()Q(cD))]TJ /F5 11.955 Tf 12.37 0 Td[((cC)]TJ /F5 11.955 Tf 12.38 0 Td[()Qc(cC))]TJ /F5 11.955 Tf 12.37 0 Td[((cP)]TJ /F5 11.955 Tf 12.38 0 Td[()Qp(cP))]TJ /F5 11.955 Tf 12.37 0 Td[(,wherecD,cC,andcPdenotethesupplier'soptimalwholesalepricesfordecentralized,centralized,andpartiallycentralizedprocurement,respectively. Observation 5.1 followsfromProposition 5.4 .Inparticular,givenanywholesalepricec,thesupplier'sprotequals(c)]TJ /F5 11.955 Tf 9.4 0 Td[()Q(c))]TJ /F5 11.955 Tf 9.4 0 Td[(,(c)]TJ /F5 11.955 Tf 9.4 0 Td[()Qc(c))]TJ /F5 11.955 Tf 9.4 0 Td[(,and(c)]TJ /F5 11.955 Tf 9.4 0 Td[()Qc(p))]TJ /F5 11.955 Tf 9.39 0 Td[(,respectivelywhenretailersaredecentralized,centralized,andpartiallycentralized.AsQ(c)Qc(c)Qp(c),itthenreadilyfollowsthat(c)]TJ /F5 11.955 Tf 10.2 0 Td[()Q(c))]TJ /F5 11.955 Tf 10.21 0 Td[((c)]TJ /F5 11.955 Tf 10.21 0 Td[()Qc(c))]TJ /F5 11.955 Tf 10.21 0 Td[((c)]TJ /F5 11.955 Tf 12.36 0 Td[()Qc(p))]TJ /F5 11.955 Tf 12.36 0 Td[(foranyc.Theaboverelationimpliesthatthevalueofcontrolisalwayspositiveforthesupplierwhenthesupplierissubjecttolinearproductioncosts.Fortheprobleminstancessolved,wecanuseTable 5-1 toquantifythevalueofcontrolforthesupplier. Inparticular,supposethattheretailersarepartiallycentralized.Thesuppliercanincreasehis/herprotby8.63%(100(150.08)]TJ /F4 11.955 Tf 12.96 0 Td[(138.16)=138.16=8.63)bypersuadingthedistributortoapplycentralizedretailingandby61.22%(100(222.74)]TJ /F4 11.955 Tf -452.54 -23.91 Td[(138.16)=138.16=61.22)bypersuadingthedistributortoapplydecentralizedretailing. 132

PAGE 133

Now,supposethattheretailersarecentralized.Inthiscase,thesuppliercanincreasehis/herprotby48.41%(100(222.74)]TJ /F4 11.955 Tf 12.41 0 Td[(150.08)=150.08=48.41)bypersuadingthedistributortoapplydecentralizedretailing.Next,wediscussthedistributor'sprocurementstrategychoice. Table 5-2 providestheaverageretailprotoveralloftheprobleminstancesforeachofthe9scenarios.AscanbeobservedfromTable 5-2 ,thedistributorwillalwaysprefercentralizedretailingintheabsenceofanyincentive.ThisobservationwaspresentedearlierinEquation( 5 ). Table5-2. Retailers'totalprot SupplierDPCPPCDP76.0967.8467.55DistributorCP110.2598.1497.54PC96.3686.1886.23 Observation5.2. Thechannelwillendupinscenario(CP,CP).Thatis,thedistributorwillcentralizetheretailersandthesupplierwillsethis/herwholesalepriceforcentralizedprocurement. Observation 5.2 isreadilyimpliedasthedistributorwillprefercentralizationforanysupplierwholesalepriceand,therefore,thesupplierwillsethis/heroptimalwholesalepriceforcentralizedretailers.The(CP,CP)scenarioisalsotheequilibriumsolutionofthefollowinggame.Supposethatthedistributor(rowplayer)hasthreepossiblestrategies:decentralizedprocurement(DP),centralizedprocurement(CP),orpartiallycentralizedprocurement(PC).Similarly,thesupplier(columnplayer)hasthreestrategiesinsettingthewholesaleprice:assumingdecentralizedprocurement(DP),centralizedprocurement(CP),orpartiallycentralizedprocurement(PC).CombiningTables 5-1 and 5-2 ,oneobtainsthepayoffmatrixgiveninTable 5-3 below.ItisclearfromTable 5-3 thatthe(CP,CP)scenarioistheuniqueequilibriuminpurestrategies. Table 5-4 providesthechannel-wideprotsassociatedwithTable 5-3 .Itfollowsthatthestate(DP,DP)resultsinthehighestaveragechannel-wideprotsoverallproblem 133

PAGE 134

Table5-3. Payoffmatrix SupplierDPCPPCDP76.09,222.7467.84,220.9767.55,220.05DistributorCP110.25,149.2098.14,150.0897.54,149.84PC96.36,137.0586.18,137.9986.23,138.16 instancessolved,i.e.,whenthedistributorchoosesdecentralizedprocurementandthesuppliersetsthewholesalepricefordecentralizedprocurement.Wenotethatthiscaseisobservedinalloftheprobleminstancessolved.However,wecannotprovideaformalproofofthisobservationforanyprobleminstance,asitisnotpossibletogetaclosed-formexpressionforthesupplier'soptimalwholesalepricefordecentralizedprocurement.(Wenotethattoshowthat(DP,DP)ischannel-wideoptimal,weneedtoshowthatchannel-wideprotin(DP,DP)isgreaterthanchannel-wideprotinboth(CP,CP)and(PC,PC).However,wecanshowthatchannel-wideprotin(CP,CP)isgreaterthanthechannel-wideprotin(PC,PC).ThisfollowsfromdenitionofcentralizedprocurementandProposition 5.4 .)Thisimpliesthatthesystem-wideprotscanbeincreasedbymovingtostate(DP,DP)from(CP,CP).Furthermore,thesupplier'sgainisgreaterthanthedistributor'slosswhenbothpartieschoosestrategyDPinsteadofCP.Thus,thechannelcanbecoordinatedbythesupplier,andanycoordinationmechanismshouldforbidcentralizationofretailingprocurement. Table5-4. Channelprot SupplierDPCPPCDP298.83288.81287.60DistributorCP259.45248.22247.38PC233.41224.17224.39 Thenextsetofnumericalstudiesisintendedtoanalyzehowtheaveragepercentageretailloss,valueofcontrol,andincreaseinchannel-wideprotareaffectedbymovingfromstate(CP,CP)to(DP,DP)underdifferentparametervalues.Wedenethepercentageretaillossas100%R(CP,CP))]TJ /F11 7.97 Tf 6.59 0 Td[(R(DP,DP) R(CP,CP),whereR(CP,CP)denotes,for 134

PAGE 135

example,thetotalretail-levelprotwhenthesupplierplansforCPandthedistributorusesCP.Similarly,thepercentagevalueofcontrolis100%S(DP,DP))]TJ /F11 7.97 Tf 6.58 0 Td[(S(CP,CP) S(CP,CP),andpercentageincreaseinchannel-wideprotis100%C(DP,DP))]TJ /F11 7.97 Tf 6.59 0 Td[(C(CP,CP) C(CP,CP),whereS()andC()denotethesupplierandchannel-wideprot,respectively. Weconsidertheeffectsofthenumberofretailers(n),marketpotential(a),sensitivityofmarketpricetototalsupply(b),retailers'per-unitoperationcosts(wi's),andthesupplier'sproductionsetupcost()andper-unitproductioncost().Thenumericalstudiesareconstructedasfollows:wechangeoneparametervalueatatime,leavingtheremainingparametersattheirpreviouslydenedvalues.Forinstance,whenwevaryn,foreachn2f2,3,4,5,6,7,8,9,10g,wesolve10probleminstancesforeachcombinationofthefollowingproblemparametersinthesingle-marketcase:a2f100,110,120g,b2f1,1.25,1.5g,W=fU[5,20],U[20,35],U[35,50]g,=f50,75,100g,and=f25,50,75g.Similarly,whenweanalyzea,foreacha2f100,105,110,115,120,125,130,135,140g,wesolve10probleminstancesforeachcombinationofthefollowingproblemparametersinthesingle-marketcase:n2f3,4,5g,b2f1,1.25,1.5g,W=fU[5,20],U[20,35],U[35,50]g,=f50,75,100g,and=f25,50,75g.Thecombinationsforotherparametersaredenedsimilarly.Wenotethatalloftheobservationsassociatedwiththeprevioussetofnumericalstudiesarevalidforthissetofnumericalstudiesaswell.ThehorizontalaxesofthegraphsshowninFigures 5-2 5-4 denetherangeforeachanalyzedparameter.Theverticalaxisgivetheaveragepercentageretailloss,valueofcontrol,andincreaseinchannel-wideprot.Inwhatfollows,weinterpretourresultsontheeffectsofdifferentchannelparameters. EffectsofRetailParameters Theretailparameterswevariedcorrespondedtothenumberofretailers,n,andretailerunitscosts,orwivalues.BasedonFigure 5-2 ,weobservethat: Asnincreases,theretaillossfrommovingfrom(CP,CP)to(DP,DP)isincreasing.Thisresultisexpectedbecauseasthenumberofretailersincreases,thecompetitionincreasesunderdecentralizedprocurement;hence,theretailstage 135

PAGE 136

canachievesubstantialsavingswhenthedistributorcentralizesprocurement.Furthermore,asnincreases,thevalueofcontrolduetomovingfrom(CP,CP)to(DP,DP)isincreasing.Thisresultfollows,asanincreasednumberofretailersimplieshighercompetition,andthesupplierbenetsfromthis.Finally,asnincreases,theincreaseinchannel-wideprotsfrommovingfrom(CP,CP)to(DP,DP)isincreasing.Thisfollowsbecausemorecompetingretailersleadtogreatermarketprotability. Asthewi'sincrease,theretaillossduetomovingfrom(CP,CP)to(DP,DP)isdecreasing.Thisresultfollowsasfewerretailerssupplythemarketunderdecentralizedprocurementwhenretailersaresubjecttohigheroperatingcosts.Therefore,theretailstageisnotlikelytoachievesubstantialsavingswhenthedistributorcentralizesprocurement.Ontheotherhand,aswi'sincrease,thevalueofcontrolisincreasingandtheincreaseinchannel-wideprotsduetomovingfrom(CP,CP)to(DP,DP)isincreasing. EffectsofMarketParameters Themarketpotential,a,andelasticity,b,serveasthemarketparameters.BasedonFigure 5-3 ,weobservethat: Asmarketpotential(a)increases,theretaillossduetomovingfrom(CP,CP)to(DP,DP)isincreasing.Thisresultisexpectedbecauseasthemarketcapacityincreases,thedistributorwillachievegreatersavingsviacentralization.However,thevalueofcontrolduetomovingfrom(CP,CP)to(DP,DP)decreasesrelativelyslowly.Thisresultfollowsastheinuenceofsupplier'swholesalepriceontheretailers'quantitydecisionsdiminishesasaincreases.Finally,asaincreases,theincreaseinthechannel-wideprotsduetomovingfrom(CP,CP)to(DP,DP)followsrelativelyastablepattern.Thatis,marketcapacitydoesnotaffectthepercentageincreaseinchannel-wideprotsduetomovingfrom(CP,CP)to(DP,DP). Aselasticity(b)increases,theretaillossduetomovingfrom(CP,CP)to(DP,DP)tendstodecrease.However,astraightdecreasingtrendisnotobserved.Ontheotherhand,asbincreases,thevalueofcontrolandtheincreaseinchannel-wideprotsduetomovingfrom(CP,CP)to(DP,DP)areincreasing. EffectsofSupplierParameters Thesupplier'sxedcost()andunitproductioncost()serveasthesupplier'sparametersofinterest.BasedonFigure 5-4 ,weobservethat: Asthexedcost()increases,theretaillossduetomovingfrom(CP,CP)to(DP,DP)followsastablepattern.Thisfollowsasisnotadirectdeterminantofthesupplier'swholesalepricedecision(excludingcaseswherethesupplier'sprot 136

PAGE 137

APercentchangeinmeasuresofinterestvs.n BPercentchangeinmeasuresofinterestvs.wi's Figure5-2. Effectsofretailparameters 137

PAGE 138

APercentchangeinmeasuresofinterestvs.a BPercentchangeinmeasuresofinterestvs.b Figure5-3. Effectsofmarketparameters 138

PAGE 139

APercentchangeinmeasuresofinterestvs. BPercentchangeinmeasuresofinterestvs. Figure5-4. Effectsofsupplierparameters 139

PAGE 140

isnegativeduetohighproductionsetupcosts).Ontheotherhand,asincreases,thevalueofcontrolduetomovingfrom(CP,CP)to(DP,DP)isincreasing.Thisisduetothefactthat,inthecaseofhighproductionsetupcosts,theincreaseinthesupplier'sprotduetodecentralizationishigher.Finally,asincreases,theincreaseinthechannel-wideprotsduetomovingtomovingfrom(CP,CP)to(DP,DP)isincreasing.Thisfollowsfromthediscussionsontheretaillossandvalueofcontrolduetomovingfrom(CP,CP)to(DP,DP). Asthesupplier'sunitproductioncost()increases,theretaillossduetomovingfrom(CP,CP)to(DP,DP)isdecreasing.Thisresultfollowsasfewerretailerssupplythemarketunderdecentralizedprocurementforalargersincethesupplier'swholesalepricewillbehigher.However,asincreases,thevalueofcontrolandtheincreaseinchannel-wideprotsduetomovingfrom(CP,CP)to(DP,DP)arebothincreasing,whichfollowfromasimilardiscussionasinthecaseof. 140

PAGE 141

CHAPTER6CONCLUSIONANDFUTURERESEARCHDIRECTIONS ThischapterconcludesthedissertationbysummarizingourworkinChapters 2 5 ,providingconcludingremarks,andhighlightingourcontributionstotheexistingliterature.WefurtherdiscussfutureresearchdirectionsrelatedtoChapters 2 5 6.1CompetitiveMulti-FacilityLocationProblemswithCongestionCosts WestudiedasymmetriccompetitivefacilitylocationprobleminChapter 2 andanasymmetriccompetitivefacilitylocationprobleminChapter 4 .Bothofthesechaptersextendthecurrentliteraturebyallowingrmstosimultaneouslylocatemorethanonefacilityandbyaccountingfornonlinearcongestioncosts. InChapter 2 ,weprovidedasolutionmethodtodeterminePNEquantitydecisionsofthermsinSection 2.3 .Werstshowedthesymmetryoftheequilibriumsupplyquantitiesgiventhatthermshaveidenticalfacilitylocationdecisions.Thisresultreducedthegameofthermstoasocalledgameofthelocations,forwhichweproposedaniterativesolutionmethod.Section 2.4 discussedthefacilitylocationdecisionsoftherms.Weexplainedwhyitissafetoassumethatrmswillchooseidenticalfacilitylocations.Therefore,thelocationdecisionsofthermscanbedeterminedbyndingthemostprotablelocationmatrixcontainingidenticalcolumns.Asatotalenumerationschemeiscomputationallyburdensome,weproposedaheuristicmethodthatndsagoodlocationdecision,whichmaybeadoptedbyalloftherms.Asimpliedbyournumericalstudies,theheuristicisanefcientmethodthatrankslocationsbasedoncertainproblemparametersintherstphase. TheresultspresentedinChapter 2 areusedintheanalysisoftheimpactsoftrafccongestioncostsonequilibriumsupplyowsby KonurandGeunes ( 2011 ),whichconstitutesChapter 3 .TheanalysisoftheheuristicmethodproposedinChapter 2 suggestsafutureresearchdirection:rms'decisionscanbemodeledasathree-stagegame.Inthisgame,rmsrstdeterminethenumberoffacilities(stage-one),thenthe 141

PAGE 142

locationsofthesefacilities(stage-two),and,nally,thesupplyquantities(stage-three).Furthermore,weobservedacounter-intuitiveresultinournumericalstudies.Itispossiblethatrmsmaybebetteroffwhentheyignorecongestioncostsintheirdecisions.Inparticular,theproblemweformulateassumestwokindsofcompetitionamongtherms:competitioninthemarketsandcompetitiononthedistributionnetwork.Hence,ourcounter-intuitiveresultindicatesthatrmsmayhavesubstantialsavingswhentheyignoreoneofthetypesofcompetition(competitionoverthedistributionnetwork)inournumericalstudies.Wenotethattheanalysisofrms'decisionswhentheycompeteovermorethanoneresourceservesasaninterestingfutureresearcharea. InChapter 4 ,weformulatedacompetitivelocationgamewithnonlinearcostsformultiple,non-identicalrmsinamultiple-marketsettingunderCournotcompetition.Eachrmincursrm-speciclineartransportationcosts,convextrafccongestioncosts,andxedfacilitylocationcosts.Chapter 4 determinesequilibriumowsforanygivenfacilitylocationsviaformulatingtheequilibriumproblemasavariationalinequalityproblem.Theresultingformulationisanasymmetriclinearvariationalinequalityproblemdenedoverthenonnegativeorthant.Projectionmethodscanbeusedasasolutionmethodandaself-adaptiveprojectionmethodproposedin Han ( 2006 )wasutilized.Second,anequilibriumfacilitylocationdecisionwassought.Tondsuchasolution,routinesweredenedbasedonpropertiesofequilibriumsolutionsinordertoeasethesearchforanequilibriumlocationmatrix.Utilizingtheseroutines,aheuristicsearchmethodthatndsanequilibriumlocationdecision,ifoneexists,wasdiscussed.Theresultsofnumericalstudiesimplythattheheuristicmethodforndinganequilibriumlocationdecisionsisquiteefcientwhencomparedtoarandomsearchmethod.Furthermore,potentialgeneralizationsformulti-productandmulti-echelonsupplychainswerediscussed. 142

PAGE 143

Chapter 4 providestoolstodetermineequilibriumfacilitylocationsandsupplyowsfromtheselocationstomultiplemarketsonacongesteddistributionnetwork.Themaincontributionsofthischapterlieinanalyzingthecasewhenheterogeneousrmsareallowedtosimultaneouslylocatemorethanonefacility,aswellastheconsiderationofnonlineartrafccongestioncosts.Notingtherecentstudiesonthenegativeeffectsoftrafccongestioncostsonsupplychains(see,e.g., Raoetal. 1991 McKinnon 1999 Weisbrodetal. 2001 Sankaranetal. 2005 KonurandGeunes 2011 ),thisstudycanbeusedtoanalyzetheeffectsoftrafccongestioncostsonheterogeneousrms'distributionandfacilitylocationdecisionsincompetitivesupplychains. TheproblemsofinterestinChapters 2 and 4 wereanalyzedassumingthatthermsarenon-cooperativeandtheytakesimultaneousactions.Ontheotherhand,studyingcompetitivelocationgameswithnonlinearcosttermswhencooperationisallowedamongtherms,orundersequentialactionsbytherms,remainasfutureresearchdirections.Itshouldbenotedthatstudyingcooperativecompetitivelocationgamesoncongestednetworksisimportantintheanalysisofmethodstomitigateinherenttrafccongestion.TheresultsinChapters 2 and 4 providetoolsthatmaybeusefulinsuchanalysis. Oneadditionalfutureresearchdirectionwouldincludestudyingcompetitivefacilitylocationgamesinmulti-echelonsupplychains(anequilibriumproblemforatwo-echelonsupplychainisstudiedin Nagurneyetal. ( 2002 );however,thisstudyassumesthatfacilitylocationsarepredetermined).Analysisoftwo-echeloncompetitivefacilitylocationproblemswithtrafccongestionisimportantforgovernmentagencies.Theseorganizationshaveasubstantialinterestindevelopingcongestionmitigationpoliciesandmayactastheupperleveldecisionmakersinsuchasetting.Again,theanalysisinChapters 2 and 4 willprovideafoundationforstudyingsuchproblems. 143

PAGE 144

6.2TrafcCongestionandSupplyChainManagement InChapter 3 ,westudiedfacilitylocationandsupplyquantitydecisionsformultiplermsinacompetitiveenvironmentonacongestednetwork.Themaincontributionofthischaptertotheliteratureliesincharacterizingtheeffectsoftrafccongestiononperformanceofacompetitivesupplychainbyincludingtrafccongestioncostsdirectly.WeusedthesymmetriccompetitivefacilitylocationproblemdenedinChapter 2 inouranalysisbecausermstypicallyshareacommondistributionnetworkand,hence,theyinessencecompetefortrafccapacityonthecommondistributionnetwork.Furthermore,tocaptureabroaderpictureofpracticalrealities,weconsideredrms'competitionintheendcustomermarkets.Thismodelingapproachisusefulinprovidingamorecompleteanalysisofdifferentmarketssuchasgroceryretailing,energy,airlineandagriculturalmarkets,asnotedinSection 3.1 AsnotedinChapter 2 ,sincetheresultinggameofthermsissymmetric,thermswillchooseidenticallocationdecisionsinequilibrium.Therefore,ouranalysisinChapter 3 alsoexplainstheeffectsoftrafccongestiononasinglerm'ssupplyquantityandfacilitylocationdecisionsonacongesteddistributionnetworkwithpricesensitivemarkets.Studyingmoregeneralproblemformulationswithexplicittrafccongestioncostsremainsasafutureresearcharea.Onedirectioninthisareawouldbetoanalyzecongestioneffectsbyrelaxingthehomogeneityassumptionoftherms.TheresultsofChapter 4 willhelpinstudyingsuchsettings.Wemodeledtrafccongestioncostsendogenouslyandprovidedanalyticalresultsonhowtrafccongestioncostaffectsequilibriumsupplyquantitydecisions.Section 3.3 ofChapter 3 givesadetaileddiscussionoftheseeffects.Increasedtrafccongestionhindersefcientuseofthedistributionnetwork,asrmsmaychoosetosupplyamarketfrommultiple,distant,anddecentralizedfacilities.Similarresultsareobservedforahighlyrelevantspecialcase:whenthepotentialfacilitylocationsarewithinthemarketareas.Moreover,ournumericalstudiescharacterizetheeffectsoftrafccongestiononfacilitylocationdecisionsas 144

PAGE 145

well.Forthegeneralcaseoftheproblemstudied,weobservedthatrmstendtolocatemorefacilitiesascongestionincreases,uptoacertainlevel.Thisisduetothefactthat,withincreasedcongestion,rmsarenotabletocoveramarketwithinaspecicdeliverytimeoratadesiredserviceleveland,thus,morefacilitiesarelocated.However,beyondalevelofcongestion,rmslocatefewerfacilities,astheirsupplyowsdecreaseinordertoavoidhighcongestioncosts.Inadditiontothegeneralcase,weconductednumericalstudiesforthespecialcase.Numericalstudiesforthespecialcaseindicatedthatrmsprefertolocatefacilitiesinmarketareaswithhigherpotentials,i.e.,higherinitialmarketpriceand/orlowersensitivitytothesupplyquantities.Also,rmswilllocatefewerfacilitiesascongestionincreaseswithinmarketareas.Thereasonforthisisthat,ascongestionincreases,rmstendtolocatefacilitiesinmarketareaswithhighermarketpotentialsandpaytransportationcoststosupplyothermarkets,insteadofpayingcongestioncostswithineachmarketarea. TheresultsofChapter 3 documentthenegativeeffectsoftrafccongestiononcompetitiverms.Asaresult,itispossiblethatrmsmaybewillingtocooperatewithgovernmentagenciestoreducetrafccongestion.Itisevenpossiblethatrmsmaycooperatewitheachothertomitigatetrafccongestion,and,therebyreducethenegativeeffectsoftrafccongestion,asnotedby HensherandPuckett ( 2005 ).Studyingsuchtrafccongestionmitigationpolicies,usingmathematicalmodelingtechniques,remainsasafutureresearcharea. 6.3PricingforCompetitiveRetailers Chapter 5 modelsasupplier'spricingdecisionsforsalesofagoodtoanendcustomermarketviaasetofretailers,whoarejointlyservedbyadistributor.Inparticular,westudiedaStackelberggameforwhichthesupplieristheleaderandthedistributorandretailersactasfollowers.Weconsidereddifferentprocurementstrategiesattheretailstage:decentralized,centralized,andpartiallycentralizedprocurement.Underdecentralizedprocurement,retailersengageinCournotquantitycompetitionin 145

PAGE 146

theendcustomermarket.Forthiscase,todetermineretailers'quantitydecisions(theequilibriumsolutionoftheCournotgame),weproposedaniterativesolutionmethodbasedonsortingretailerswithrespecttocertainparameters(thismethodisgeneralizedtothecasesinwhichthemarketpricefunctionisdecreasingconcaveandtheretaileroperatingcostfunctionisincreasingconvex;and,thisgeneralizationprovidesanalternativeapproachtovariationalinequalitybasedalgorithmsforbroadclassesofequilibriumproblems,referredtoasmarketequilibriumproblems). Wecontributetotheliteratureonsupplierpricingproblemsforcompetitiveretailers(i.e.,whentheretailstageisdecentralized)byconsideringanarbitrarynumberofnon-identicalretailers,andbyconsideringmoregeneralformsofthesupplier'sproductioncostfunction.Wefurthercontributebystudyingtwodifferentcentralizationlevelsofretailprocurement:centralizationandpartialcentralization.Ourndingsimplythatthesuppliermayencourageordiscouragecentralizationattheretailstage,dependingonwhethers/heobserveseconomiesordiseconomiesofscaleinproduction. Consideringdifferentstrategiesattheretailstagepermittedstudyingtheimportantconceptofthevalueofcontrol.Tothebestofourknowledge,thevalueofcontrolinthecontextofChapter 5 hasnotbeenpreviouslyanalyzed.Weshowedthatifthesuppliercontrolstheprocurementstrategyofthedistributor,s/hemayextractsubstantialsavings.Inparticular,whenthesupplier'sproductioncostsarelinearintheproductionquantity,weshowedthatitisinthesupplier'sbestinterestiftheretailstageisdecentralized.However,theequilibriumstateindicatescentralizedprocurementandthesupplier,therefore,setshis/herwholesalepriceforcentralizedretailers.Thisstate,nevertheless,doesnotmaximizechannel-wideprotinournumericalstudies.Thechannel-wideprotismaximizedunderdecentralizedprocurementforalloftheprobleminstancessolved.Therefore,acoordinationmechanismshouldrestrictcentralizationattheretailingstage;thatis,verticalcentralizationislikelytorequirehorizontaldecentralization. 146

PAGE 147

ThesettingofChapter 5 servesasabenchmarkforimportantfutureresearchdirections.Studyingthesupplier'sproblemwithcompetingretailersundermoregeneralmarketpriceandretailercostfunctionsremainsasafutureresearchdirection.WhilegeneralizedStackelberggameshavebeenanalyzedintheliterature(see,e.g., Sheralietal. 1983 Milleretal. 1991 Tobin 1992 ),theseStackelberggamesassumethatretailerscompetewitheachotherandoneoftheretailersactsasaleader.Anotherfutureresearchdirectionwouldinvolveanalyzingcoordinationmechanismsunderdifferentprocurementstrategies.Fordecentralized,competitiveretailers,pricingasacoordinationmechanismhasbeenstudiedintheliterature(see,e.g., IngeneandParry 1995 1998 2000 BernsteinandFedergruen 2003 KeskinocakandSavasaneril 2008 ).OurstudywillbehelpfulintheanalysisofcoordinationmechanismsforthegeneralizedsettingsconsideredinChapter 5 .Furthermore,analysisofsuchcoordinationmechanismsforcentralizedandpartiallycentralizedretailersremainsasafutureresearcharea.Studyingthecaseinwhichthesupplierplaystheroleofthedistributorprovidesanotherfutureresearchdirection.ThiscasewouldcorrespondtoanalyzingtheeffectsofVendor-Managed-Inventoryformultiplecompetitiveretailers,wherethesuppliermaydecideonretailerorderquantitiesaswellasthewholesaleprice. Anotherimportantfutureresearchdirectionconsistsofanalyzingthesupplier'sproblemwhenthecompetitiveretailerscanbecooperative.Intheabsenceofachainstoreordistributor,itisstillpossiblethatthecompetingretailersacttogetherviacoalitions/cooperations.Forinstance, KeskinocakandSavasaneril ( 2008 )studyasupplier'sdiscountpricingforretailerswhopurchasefromthesuppliercollaboratively.Furthermore,retailersorbuyerscanformgrouppurchasingorganizations.Asnotedby Nagarajanetal. ( 2010 ),GroupPurchasingOrganizationsareobservedinvariousindustriesincludinghealthcare,education,andretailing;werefertheinterestedreaderto Nagarajanetal. ( 2010 )forabroaderdiscussiononmodeling,formation,andissues 147

PAGE 148

relatedtoGroupPurchasingOrganizations. Nagarajanetal. ( 2010 )studystabilityofGroupPurchasingOrganizationformation.Intheirsetting,however,thesupplier'sdiscountpricingisxed.Aninterestingresearchquestion,therefore,wouldbetoanalyzeasupplier'sdiscountpricingdecisionswhenretailerscanformcoalitionstotakeadvantageofthesupplier'sdiscountpricing.Inparticular,GroupPurchasingOrganizationsareformedtotakeadvantageofdiscountsofferedbythesupplierinmanycases( Nagarajanetal. 2010 ChenandRoma 2011 ).Thus,whenthesuppliersetsadiscountpolicyassumingnon-cooperativeretailers,discountscanbeadetrimenttoasupplierincaseswithretailercooperation.Thequestionofinterest,then,wouldbehowasuppliershouldsethis/herwholesalepricescheduletoavoidabuseofdiscountsbyretailercooperatives. 148

PAGE 149

APPENDIXASYMMETRYOFEQUILIBRIUMSUPPLYQUANTITIESGIVENIDENTICALFACILITYLOCATIONS Firstnotethatforthelocationswherethereisnofacility,qijr=08r2R.Hence,weonlyfocusonthelocationswheretherearekfacilities.WeprovethestatementusingtheKKTconditionsdenedfortheoptimalqijrvalues.Togetherwithqijr0,theKKTconditionsread ij)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqj)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqjr)]TJ /F5 11.955 Tf 11.96 0 Td[(ijqij)]TJ /F5 11.955 Tf 11.95 0 Td[(ijqijr+uijr=0,uijrqijr=0,uijr0, whereij=aj)]TJ /F3 11.955 Tf 12.72 0 Td[(cij.Nowconsideranytwormsr1andr2.Weshowthatqijr1=qijr28i2I0,whereI0denotesthelocationswithkfacilitiesforthegivenX0,byconsideringCases1and2,denedbelow.Thenitfollowsthatqijr1=qijr28i2I0foranytwormsr1andr2.Thus,weshowthatqijristhesameforalloftherms.Hence,lettingQijdenotethetotalequilibriumquantityowonthelink(i,j),sincethereexistkrmsatanylocationi2I0,itfollowsthatqijr=Qij=k.CaseI:qijr1>0andqijr2>08i2I0 Supposethatqijr1>0andqijr2>08i2I0.Thenqijr1andqijr2mustsatisfytherstorderconditions,i.e.,uijr1=uijr2=08i2I0.Withoutlossofgenerality,weassumethatI0=f1,2,3,...,sgsuchthatsm.Therstorderconditions,then,read ij)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqjr1)]TJ /F5 11.955 Tf 11.95 0 Td[(ijqij)]TJ /F5 11.955 Tf 11.95 0 Td[(ijqijr1=08i2I0, (A) ij)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqj)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqjr2)]TJ /F5 11.955 Tf 11.95 0 Td[(ijqij)]TJ /F5 11.955 Tf 11.96 0 Td[(ijqijr2=08i2I0. (A) Itfollowsfrom( A )and( A )that bjqjr1+ijqijr1=bjqjr2+ijqijr28i2I0, (A) zj(qzjr1)]TJ /F3 11.955 Tf 11.96 0 Td[(qzjr2)=)]TJ /F3 11.955 Tf 9.3 0 Td[(bj(qjr2)]TJ /F3 11.955 Tf 11.95 0 Td[(qjr2)=)]TJ /F3 11.955 Tf 9.3 0 Td[(bjXi2I0(qijr1)]TJ /F3 11.955 Tf 11.95 0 Td[(qijr2)8z2I0. (A) 149

PAGE 150

Subtracting( A )forlocation2from( A )forlocation1,weget 1jq1jr1)]TJ /F5 11.955 Tf 11.96 0 Td[(2jq2jr1=1jq1jr2)]TJ /F5 11.955 Tf 11.96 0 Td[(2jq2jr2. (A) Itfollowsfrom( A )that1j(q1jr1)]TJ /F3 11.955 Tf 12.14 0 Td[(q1jr2)=2j(q2jr1)]TJ /F3 11.955 Tf 12.14 0 Td[(q2jr2).Followingsimilarargument,subtracting( A )forlocationi+1fromexpression( A )forlocationi,is)]TJ /F4 11.955 Tf 11.95 0 Td[(1,weget 1j(q1jr1)]TJ /F3 11.955 Tf 11.96 0 Td[(q1jr2)=2j(q2jr1)]TJ /F3 11.955 Tf 11.95 0 Td[(q2jr2)=3j(q3jr1)]TJ /F3 11.955 Tf 11.95 0 Td[(q3jr2)=...=sj(qsjr1)]TJ /F3 11.955 Tf 11.96 0 Td[(qsjr2). (A) Considering( A ),( A )canbewrittenas 2zj(qzjr1)]TJ /F3 11.955 Tf 11.96 0 Td[(qzjr2)=)]TJ /F3 11.955 Tf 9.3 0 Td[(bjXi2I0ij(qzjr1)]TJ /F3 11.955 Tf 11.95 0 Td[(qzjr2). (A) Sinceij>0,( A )isonlysatisedwhenqzjr1=qzjr2foranylocationz2I0.Thus,itfollowsthatqijr1=qjr28i2I0.CaseII:qijr1=0fori2I0r1I0andqijr2=0fori2I0r2I0 Supposethatqijr1=0forlocationsi2I0r1I0andqijr2=0forlocationsi2I0r2I0.WeconsiderthefollowingthreesubcasesofCaseII,whichcaptureallofthepossibilitiesofCaseII,andprovethestatementforthesethreesubcases.SubcaseI:I0r1=I0r2=I0rI0 SupposethatI0r1=I0r2=I0rI0,i.e.,qijr1=qijr2=0forlocationsi2I0rI0.Forlocationsi=2I0r,thatis,forlocationsi2I0nI0r,wehaveqijr1>0andqijr2>0.Thus,SubcaseIreducestoCaseIwithI0nI0rinsteadofI0,whichmeanswehaveqijr1=qijr2forlocationsi2I0nI0r.Thus,forSubcaseI,wehaveqijr1=qijr28i2I0.SubcaseII:I0r16=I0r2andeitherI0r1=;orI0r2=; SupposethatI0r16=I0r2andeitherI0r1=;orI0r2=;.Withoutlossofgenerality,supposethatI0r2=;. Situation(i):ConsiderI0r1=I0.Situation(i)impliesthatqijr1=08i2I0,thus,qjr1=0.Nowconsideranylocationz2I0andsupposethatqzjr2>0.ItfollowsfromtheKKTconditionsthatuzjr2=0.Moreover,fromtheKKT 150

PAGE 151

conditionsforqzjr1andqzjr2,wehave zj)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqj)]TJ /F5 11.955 Tf 11.96 0 Td[(zjqzj+uzjr1=0, (A) zj)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqjr2)]TJ /F5 11.955 Tf 11.95 0 Td[(zjqzj)]TJ /F5 11.955 Tf 11.95 0 Td[(zjqijr2=0. (A) Itfollowsfrom( A )and( A )that)]TJ /F3 11.955 Tf 9.3 0 Td[(uzjr1=bjqjr2+zjqzj+zjqijr2,whichimplies bjqjr2+zjqzj+zjqzjr2=0 (A) sinceuzjr10,zj>0andbj>0.Moreover,sinceqijr20,( A )isonlysatisedwhenqjr2=qzj=qzjr2=0.Therefore,wehaveacontradictionwithqzjr2>0,thus,qzjr1=qzjr2=0foranylocationz2I0forSituation(i),i.e.,qijr1=qjr28i2I0. Situation(ii):ConsiderI0r1I0.Situation(ii)impliesthatthereisatleastonelocation,saylocationt,t2I0nI0r1suchthatqtjr1>0andqtjr2>0.Weshowbycontradictionthatqijr2=08i2I0r1.Supposethatqzjr2>0foranylocationz2I0r1.ItfollowsfromtheKKTconditionsthatuzjr2=0.Moreover,fromtheKKTconditionsforqzjr1andqzjr2,wehave zj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqjr1)]TJ /F5 11.955 Tf 11.96 0 Td[(zjqzj+uzjr1=0, (A) zj)]TJ /F3 11.955 Tf 11.95 0 Td[(bjqj)]TJ /F3 11.955 Tf 11.96 0 Td[(bjqjr2)]TJ /F5 11.955 Tf 11.95 0 Td[(zjqzj)]TJ /F5 11.955 Tf 11.95 0 Td[(zjqijr2=0. (A) Itfollowsfrom( A )and( A )thatbjqjr1)]TJ /F3 11.955 Tf 12.3 0 Td[(uzjr1=bjqjr2+zjqijr2.Sinceuzjr10,zj>0andqzjr2>0,itimpliesthat bjqjr1>bjqjr2. (A) Nowconsideranylocationt2I0nI0r1suchthatqtjr1>0andqtjr2>0.ThenitfollowsfromKKTconditionsthatutjr1=utjr2=0andbjqjr1+tjqtjr1=bjqjr2+tjqtjr2.Sincebjqjr1>bjqjr2,wehaveqtjr1
PAGE 152

t2I0nI0r1.Thus,itfollowsthat Xt2I0nI0r1qtjr10foranylocationz2I0r1,wehave Xt2I0r1qtjr10andqijr2>0.Since,weknowqzjr1=qzjr2=0foranylocationz2I0r1,wecanignoresuchlocations.Then,Situation(ii)reducestoCase1withI0nI0rinsteadofI0,whichmeanswehaveqijr1=qijr2forlocationsi2I0nI0r.Thus,qijr1=qijr28i2I0forSituation(ii). Situations(i)and(ii)togetherimplythatqijr1=qijr28i2I0forSubcaseII.SubcaseIII:I0r16=;,I0r26=;andI0r16=I0r2 SupposethatI0r16=;,I0r26=;andI0r16=I0r2.Firstnotethatforanylocationi2I0r1\I0r2,wehaveqijr1=qijr2=0,thus,wecandisregardsuchlocationsandonlystudythesituationwhenI0r16=;,I0r26=;andI0r1\I0r2=;.Thissituationimpliesthefollowingconditions: qijr1=0andqijr2>08i2I0r1, (A) qijr1>0andqijr2=08i2I0r2. (A) Wenowshowbycontradictionthatconditions( A )and( A )cannotbesatisedatthesametime.Consideranylocationz2I0r1,thatis,qzjr1=0andqzjr2>0.ItfollowsfromtheKKTconditionsthatuzjr2=0and(i)bjqjr1)]TJ /F3 11.955 Tf 12.47 0 Td[(uzjr1=bjqjr2+zjqzjr2,whichmeansthatbjqjr1>bjqjr2asuzjr20,zj>0andqzjr2>0.Nowconsideranylocationt2I0r2, 152

PAGE 153

thatis,qtjr1>0andqtjr2=0.ItfollowsfromtheKKTconditionsthatutjr2=0and(ii)bjqjr1+tjqtjr1=bjqjr2)]TJ /F3 11.955 Tf 12.09 0 Td[(uzjr2,whichmeansthatbjqjr10andqtjr1>0.(i)and(ii)establishesacontradiction.Thatis,wecannotsatisfytheconditions( A )and( A )atthesametime.Withoutlossofgenerality,supposethatwedonothavecondition( A ),i.e.,qijr1=0andqijr2=08i2I0r1.Hence,wecandisregardanylocationi2I0r1.Then,SubcaseIIIreducestoSubcaseII.Thus,qijr1=qijr28i2I0forSubcaseIII. 153

PAGE 154

APPENDIXBMIXEDSTRATEGYNASHEQUILIBRIUMFORSYMMETRICLOCATIONGAME Werstnotethat,thelocationdecisionsofthermscorrespondstoasymmetricgame.Itiswellknownthatforsymmetricgames,thereexistasymmetricMixedStrategyNashEquilibrium(MSNE)incasesofmultipleequilibriaortheredoesnotexistaPureNashEquilibrium(PNE).SymmetryofMSNEmeansthattheprobabilityofchoosingaspeciclocationdecisionisthesameforeachrm,hence,ifweknowtheprobabilityassignedtolocationvectorxbyarm,weknowtheprobabilitiesassignedbyeachrmatequilibrium.Now,letusfocusonasinglermandconsideranylocationvectorx.SupposeAssumptions 2.1 2.3 hold.TocapturethepreferencesinAssumptions 2.1 and 2.2 ,weformulateutilityfunctionofarmandusethisfunctionastherm'sobjective.Wecharacterizetheutilityfunctionofrmr,giventhelocationdecisionsofallotherrmsasafunctionofxrasfollows r(xr)=8>>>><>>>>:)]TJ /F3 11.955 Tf 9.3 0 Td[(Mif9xsuchthatr(Q(x),x)>r(Q(xr),xr),)]TJ /F3 11.955 Tf 9.3 0 Td[(Mif9xsuchthatr(Q(x),x)=r(Q(xr),xr)andjxj>jxrj,r(Q(xr),xr)otherwise,(B) whereM!1,r(Q(xr),xr)denotesthetotalprot,includingfacilitylocationcosts,ofrmrwhenxristhelocationvector,andjxjdenotesthenumberoffacilitieslocatedunderlocationvectorx.Notethat,thepurposeofformulatingautilityfunctionasinEquation B andlettingM!1isjusttoreectAssumptions 2.1 and 2.2 mathematically.NowgiventhatanyrmusesEquation( B )asanobjective,wefocusondeterminingtheprobabilityassignedtolocationvectorxbyanyrm,sayrmr1,usingtheutilitiesofanyotherrm,sayrmr2,i.e.,wecomparetworms.SupposethereareTpossiblelocationvectorsandrmr1assignsprobabilityr1ttolocationvectortT.Nowletusfocusonutilitymatrixofrmr2,sayA.DuetoEquation( B ), 154

PAGE 155

eachrowofAconsistsof1nonnegativeandt)]TJ /F4 11.955 Tf 12.91 0 Td[(1of)]TJ /F3 11.955 Tf 9.3 0 Td[(Mvalues.Weconsiderthefollowingtwocases.CaseI:EachcolumnonAhas1nonnegativevalue Inthiscase,nostrategyisweaklyorstrictlydominated,hence,r1t>080tT.Then,weshouldhavear1t)]TJ /F3 11.955 Tf 12.76 0 Td[(M(1)]TJ /F5 11.955 Tf 12.75 0 Td[(r1t)=br1z)]TJ /F3 11.955 Tf 12.75 0 Td[(M(1)]TJ /F5 11.955 Tf 12.76 0 Td[(r1z),wherea>0andb>0foranytandz,1tTand1zT.Thenitfollowsthatr1t r1z=a+M b+M.ThenlimM!1r1t r1z=1,i.e.,r1t=r1zforany1tTand1zT.Moreover,sincetherearenitenumberofstrategiesforanyrm,r1t=r1z>0.Lettingdenotethisprobability,wehaver1(x)=foranylocationvectorx,asM!1.TheniteasilyfollowsfromthesymmetryoftheMSNE,r(x)=foranyrmr2R.CaseII:Therearecolumnswithnononnegativevalues Inthiscase,thelocationvectorscorrespondingtothecolumnswithnonnegativevaluesweaklyorstrictlydominatesthelocationvectorscorrespondingtothecolumnswithoutnonnegativevalues.Hence,wecanassignprobability0tothecolumnswithoutnonnegativevalues.Fortheremainingcolumns,then,CaseIIreducestoCaseI. Wenotethataweaklyorstrictlydominatedstrategy,i.e.,alocationvector,whenutilityfunctionisusedasanobjective,isalsoweaklyorstrictlydominatedwhentheprotfunctionisusedasanobjectivebytherms.ItfollowsfromCasesIandIIthatanyrmwillassignprobability0toweaklyorstrictlydominatedlocationvectorsandanyrmwillassignprobabilitytoanyotherlocationvectorasM!1. 155

PAGE 156

APPENDIXCSOLUTIONOFDECENTRALIZEDRETAILINGUNDERGENERALIZEDMARKETPRICEANDOPERATINGCOSTFUNCTIONS Here,weprovideasolutionmethodfordecentralized,i.e.,competitiveretailers'quantitydecisionsforagivensupplierwholesaleprice.Weassumethatthemarketpricefunction,p(Q),satisesthefollowingconditions. AssumptionC.1. (i)p(Q)isacontinuously,twicedifferentiablefunctionofQ.(ii)p(Q)isadecreasingconcavefunctionofQ,i.e.,dp(Q)=dQ<0andd2p(Q)=dQ20,forQ0,and@p(Q)=@qi<0and@2p(Q)=@q2i0forqi0andforalli=1,...,n. Notethatwhenp(Q)=a)]TJ /F3 11.955 Tf 12.81 0 Td[(bQ,Assumption C.1 issatised.Furthermore,weassumethatvi(qi)satisesthefollowingconditions. AssumptionC.2. (i)vi(qi)isacontinuously,twicedifferentiablefunctionofqi.(ii)vi(qi)isanincreasingconvexfunctionofqi,i.e.,dvi(qi)=dqi>0andd2vi(qi)=dq2i0forqi0andforalli=1,...,n. Againnotethatwhenvi(qi)=wiqi,Assumption C.2 issatised.UnderAssumptions C.1 and C.2 ,theprotfunctionofretailerireads i()777(!Q)=p(Q)qi)]TJ /F3 11.955 Tf 11.95 0 Td[(cqi)]TJ /F3 11.955 Tf 11.96 0 Td[(vi(qi),(C) anditcanbeshownthati()777(!Q)isconcave,giventheorderquantitiesoftheotherretailers.(Thisfollows,as@2i()778(!Q)=@q2i=2@p(Q)=@qi+qi@2p(Q)=@q2i)]TJ /F3 11.955 Tf 10.83 0 Td[(d2vi(qi)=dq2i0forqi0,underAssumptions C.1 and C.2 .)Thisimpliesthattherst-orderconditions(@i()778(!Q)=@qi=0,forqi>0)mustbesatisedataNashequilibriumsolution.Inparticular,ifqi>0thenaNashequilibriumsolutionmustsatisfythecondition p(Q)+qi@p(Q) @qi)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dvi(qi) dqi=0.(C) 156

PAGE 157

Itcanbeshownthat)777(!Q2Rn+solvestheretailers'gameifitsolvesthefollowingvariationalinequalityproblem: hF()777(!Q),)778(!Q)]TJ 11.95 8.86 Td[()777(!Qi0,8)777(!Q2Rn+,(C) whereF()777(!Q)=()]TJ /F5 11.955 Tf 9.29 0 Td[(@1()778(!Q)=@q1,...,)]TJ /F5 11.955 Tf 9.29 0 Td[(@n()777(!Q)=@qn)isann-rowvectorfunction(see,e.g., GabayandMoulin 1980 Frieszetal. 1983 Harker 1984 Nagurney 1999 Milleretal. 1996 ).Hence,onecanuseexistingvariationalinequalitymethodstosolveEquation( C )andthustheretailers'game.Werefertheinterestedreadertothebooksby Milleretal. ( 1996 )and Nagurney ( 1999 ),andthereferencestherein,foralgorithmicsolutionsforvariationalinequalityformulationsofequilibriumproblemssimilartotheretailers'game.ThemethodwediscussnextisaniterativemethodthatrequiressolvingasystemofequationsoftheformH`()777(!Q`)=0,whereH`()777(!Q`)isan`-vectorfunctionand)777(!Q`isan`-vector,with`n. ThefollowinglemmaisgeneralizationofProposition 5.1 Lemma1. (i)qi>0ifandonlyifc+dvi(qi)=dqi0,thenqi1>0,and(b)ifqi1=0,thenqi2=0. Proof:Supposethattheequilibriumsupplyquantitiesareknownandxedatqiforalloftheretailersexceptretaileri.ThentheoptimalsupplyquantityofretaileriisgivenbytheKKTconditions,sinceretaileri'sproblemistomaximizehis/herprot,whichisaconcavefunctionofqi,withtheconstraintqi0,whichisalinearconstraint.TheKKTconditionsforretaileri,then,are p(Q)+qi@p(Q) @qi)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 13.15 8.08 Td[(dvi(qi) dqi+ui=0, (C) uiqi=0, (C) ui0. (C) 157

PAGE 158

Supposethatqi>0,whichmeansui=0.Then,condition( C )impliesthatp(Q)+qi@p(Q)=@qi=c+dvi(qi)=dqi.Furthermore,as@p(Q)=@qi<0fromAssumption C.1 andqi>0,itfollowsthatc+dvi(qi)=dqi0.Thisprovesthecondition(i)ofLemma 1 Nowletdvi1(0)=dqi10.Thenitfollowsfromcondition(i)that c+dvi2(qi2)=dqi20.Thiscontradictionprovesstatement(a)ofcondition(ii)ofLemma 1 .Nowsupposethatqi1=0.Letusassumethatqi2>0.Thenstatement(a)ofcondition(ii)impliesthatqi1>0,whichcontradictsthatqi1=0,hence,qi2=0.Thisprovesstatement(b)ofcondition(ii)ofLemma 1 andcompletestheproofofcondition(ii). ThefollowingalgorithmisgeneralizationofAlgorithm 6 158

PAGE 159

Algorithm7. Withoutlossofgenerality,supposethatretailersaresortedinincreasingorderofdvi(0)=dqivalues.Givena,b,candvi(qi)8i2f1,2,...,ng; Step0.Ifc+dv1(0)=dq1p(0),setqi=08i2f1,2,...,ngand`=0.Else,set`=1andgotoStep1. Step1.Determineq(`)ifori`bysolvingthefollowingsystemofequations: p(Q)+qi@p(Q)=@qi)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.96 0 Td[(dvi(qi)=dqi=0,8i`.(C) DeneQ(`)=P`i=1q(`)i,andgotoStep2. Step2.If`=n,stop.Alloftheretailersareactive;qi=q(`)iforinand`=n.Else,if`08i`andc+dv`+1(0)=dq`+1p(Q(`)),weshouldhaveq`+1=0.Supposethatq(`)i>08i`andc+dv`+1(0)=dq`+1p(Q(`)).Toestablishacontradiction,letusassumethatq`+1>0,whichmeansq(`+1)(`+1)>0.Sinceq`+1>0,itfollowsfromcondition(ii)ofLemma 1 thatqi>08i`,whichmeansq(`+1)i>08i`.Furthermore,asq(`+1)`+1>0,itfollowsfromcondition(i)ofLemma 1 thatc+dv`+1(q(`+1)`+1)=dq`+1
PAGE 160

( C )impliesthat Q(`)>Q(`+1), (C) @p(Q(`))=@qi@p(Q(`+1))=@qi<0 (C) 8i`+1asp(Q)isassumedtobeadecreasingconcavefunctionofQinAssumption C.1 .Nowconsideranyretaileri,i`.ThenconsideringEquation( C )atiterations`and`+1forretaileri,wehave p(Q(`))+q(`)i@p(Q(`)) @qi)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dvi(q(`)i) dqi=0, (C) p(Q(`+1))+q(`+1)i@p(Q(`+1)) @qi)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dvi(q(`+1)i) dqi=0. (C) Combining( C ),( C ),and( C ),onecanconcludethat q(`+1)i@p(Q(`+1)) @qi)]TJ /F3 11.955 Tf 11.96 0 Td[(q(`)i@p(Q(`)) @qi0.However,Assumption C.2 impliesthatdvi(q(`+1)i)=dqi)]TJ /F3 11.955 Tf 10.22 0 Td[(dvi(q(`)i)=dqi0forq(`)iq(`+1)i,whichcontradictsdvi(q(`+1)i)=dqi)]TJ /F3 11.955 Tf 11.96 0 Td[(dvi(q(`)i)=dqi>0.Therefore, q(`)i0.Weknowfrom( C )thatQ(`+1))]TJ /F3 11.955 Tf 12.11 0 Td[(Q(`)<0,i.e.,q(`+1)`+1+P`i=1(q(`+1)i)]TJ /F3 11.955 Tf 11.37 0 Td[(q(`)i)<0.Itthenfollowsthatq(`+1)`+1<0asP`i=1(q(`+1)i)]TJ /F3 11.955 Tf 11.37 0 Td[(q(`)i)>0.Thiscontradictsq(`+1)`+1>0.TheothercasesconsideredinStep2ofAlgorithm 7 areobvious.Furthermore,Step1ofAlgorithm 7 isthesimultaneoussolutionoftherstorderconditionsforeachactiveretailerandStep0isadirectapplicationofProposition 1 .Therefore,Algorithm 7 givesthenumberofactiveretailersaswellasthecorrespondingequilibriumsupplyquantitiesatthemarket. 160

PAGE 161

APPENDIXDCOMPARISONOFTOTALORDERQUANTITIESUNDERDIFFERENTRETAILINGSTRATEGIES WerstshowthatQpQc.WeknowfromPropositions 5.2 and 5.3 thatQc=a)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F9 7.97 Tf 6.59 0 Td[(w1 2bandQp=a)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F21 7.97 Tf 6.59 5.98 Td[(Pni=1ki(Pnj=1wjkj) 2b.Nowletwp=Pnj=1wjkj,i.e.,Qp=a)]TJ /F9 7.97 Tf 6.58 0 Td[(c)]TJ /F9 7.97 Tf 6.58 0 Td[(wp 2b.Fromdenitionofw1,wehavew1wp.Thus,itfollowsthatQpQc. Next,weshowthatQcQ.ThefollowinglemmastatesapropertyofAlgorithm 6 ,and,itindicatesthatQcQ. Lemma3. Supposethattheretailersaredecentralized,andAlgorithm 6 isusedtodeterminetheequilibriumnumberofactiveretailersandtheirorderquantities.Let`=`intheoutputofAlgorithm 6 .Then(i)Pri=1q(r)iq(r+1)iforr<`andir. Proof:Supposethat`=`.Werstprovestatement(i).Forr<`,weknowfromAlgorithm 6 thatbq(r)1+c+w1=bq(r)2+c+w2=...=bq(r)l+c+wrand,usingtheseequalities,onecanderiveq(r)s=q(r)i+(wi)]TJ /F3 11.955 Tf 13.02 0 Td[(ws)=bfori,sl.ThisexpressionandthelinearequationgiveninAlgorithm 6 forretaileritogetherimplythatq(r)i=a)]TJ /F9 7.97 Tf 6.59 0 Td[(c)]TJ /F11 7.97 Tf 6.58 0 Td[((r+1)wi+Pri=1wi b(r+1),whichimpliesthatPri=1q(r)i=r(a)]TJ /F9 7.97 Tf 6.59 0 Td[(c) b(r+1))]TJ /F21 7.97 Tf 12 12.18 Td[(Pri=1wi b(r+1).Similarly,itcanbeshownthatPr+1i=1q(r+1)i=(r+1)(a)]TJ /F9 7.97 Tf 6.59 0 Td[(c) b(r+2))]TJ /F21 7.97 Tf 13.3 12.29 Td[(Pr+1i=1wi b(r+2).Toestablishacontradiction,letusassumethatPri=1q(r)iPr+1i=1q(r+1)i,thatis,r(a)]TJ /F9 7.97 Tf 6.58 0 Td[(c) b(r+1))]TJ /F21 7.97 Tf 13.41 12.17 Td[(Pri=1wi b(r+1)(r+1)(a)]TJ /F9 7.97 Tf 6.58 0 Td[(c) b(r+2))]TJ /F21 7.97 Tf 13.41 12.29 Td[(Pr+1i=1wi b(r+2).Itthenfollowsthat (r+1)w(r+1)a)]TJ /F3 11.955 Tf 11.95 0 Td[(c+brXi=1wi. (D) Nevertheless,since`retailersareactive,q(r+1)(r+1)>0forr<`thus,itfollowsfromStep2ofAlgorithm 6 thatc+w(r+1)
PAGE 162

( D )contradicts( D )andthiscontradictioncompletestheproofofstatement(i).Statement(ii)isadirectimplicationofStatement(i)withrespecttothefollowingrelations:a)]TJ /F3 11.955 Tf 10.17 0 Td[(bPri=1q(r)i)]TJ /F3 11.955 Tf 10.16 0 Td[(bq(r)i)]TJ /F3 11.955 Tf 10.16 0 Td[(c)]TJ /F3 11.955 Tf 10.16 0 Td[(wi=08iranda)]TJ /F3 11.955 Tf 10.16 0 Td[(bPr+1i=1q(r+1)i)]TJ /F3 11.955 Tf 10.16 0 Td[(bq(r+1)i)]TJ /F3 11.955 Tf 10.16 0 Td[(c)]TJ /F3 11.955 Tf 10.17 0 Td[(wi=08ir+1.SincePri=1q(r)iq(r+1)iforr<`andir. 162

PAGE 163

REFERENCES Abad,P.L.,1994.Supplierpricingandlotsizingwhendemandispricesensitive.EuropeanJournalofOperationalResearch78(3),334. Amir,R.,Jakubczyk,M.,Knauff,M.,2008.Symmetricversusasymmetricequilibriainsymmetricsupermodulargames.InternationalJournalofGameTheory37(3),307. Anderson,S.,Neven,D.,1991.Cournotcompetitionyieldsspatialagglomeration.InternationalEconomicReview32(4),793. Andreani,R.,Friedlander,A.,Martnez,J.M.,1997.Solutionofnite-dimensionalvariationalinequalitiesusingsmoothoptimizationwithsimplebounds.JournalofOptimizationTheoryandApplications94(3),635. Anupindi,R.,Bassok,Y.,1999.Centralizationofstocks:retailersvs.manufacturer.ManagementScience45(2),178. Arnade,C.,Gopinath,M.,Pick,D.,2007.Measuringthedegreeofretailcompeitioninu.s.cheesemarkets.JournalofAgriculturalandFoodIndustrialOrganization5,1. Balachander,S.,Srinivasan,K.,1998.Quantitydiscounts,manufacturerandchannelprotmaximization:impactofretailerheterogeneity.MarketingLetters9(2),169. Banerjee,A.,1986.Onaquantitydiscountpricingmodeltoincreasevendorprots.ManagementScience32(11),1513. Basso,L.,Zhang,A.,2008.Ontherelationshipbetweenairportpricingmodels.TransportationResearchPartB42,725. Bazaraa,M.S.,Sherali,H.D.,Shetty,C.M.,2006.Nonlinearprogramming:theoryandalgorithms.JohnWileyandSons,Inc. Bell,D.R.,Lattin,J.M.,1998.Shoppingbehaviorandconsumerpreferenceforstorepriceformat:whylargebasketshopperspreferEDLP.MarketingScience17(1),66. Bernstein,F.,Chen,F.,Federgruen,A.,2006.Coordinatingsupplychainswithsimplepricingschemes:theroleofvendor-managedinventories.ManagementScience52(10),1483. Bernstein,F.,Federgruen,A.,2003.Pricingandreplenishmentstrategiesinadistributionsystemwithcompetingretailers.OperationsResearch51(3),409. Bernstein,F.,Federgruen,A.,2004.Ageneralequilibriummodelforindustrieswithpriceandservicecompetition.OperationsResearch52(6),868. Bernstein,F.,Federgruen,A.,2005.Decentralizedsupplychainswithcompetingretailersunderdemanduncertainty.ManagementScience51(1),18. 163

PAGE 164

Bitran,G.,Caldentey,R.,Vial,R.,2010.Pricingpoliciesforperishableproductswithdemandsubstitution,WorkingPaper. Boyac,T.,Gallego,G.,2002.Coordinatingpricingandinventoryreplenishmentpoliciesforonewholesalerandoneormoregeographicallydispersedretailers.InternationalJournalofProductionEconomics77(2),95. Brander,J.,Zhang,A.,1990.Marketconductintheairlineindustry:anempiricalinvestigation.TheRANDJournalofEconomics21(4),567. Brander,J.,Zhang,A.,1993.Dynamicoligopolybehaviourintheairlineindustry.InternationalJournalofIndustrialOrganization11,407. Brant,F.,Fischer,F.,Holzer,M.,2009.SymmetriesandcomplexityofpureNashequilibrium.JournalofComputerandSystemSciences75(3),163. Cachon,G.P.,2003.Supplychaincoordinationwithcontracts.In:Graves,S.,deKok,T.(Eds.),HandbooksinoperationsResearchandManagementScience:SupplyChainManagement.ElsevierSciencePublishers,North-Holland,Amsterdam,Netherlands,Ch.6. Cachon,G.P.,Kok,A.G.,2007.Categorymanagementandcoordinationinretailassortmentplanninginthepresenceofbasketshoppingconsumers.ManagementScience53(6),934. Cachon,G.P.,Lariviere,M.A.,2005.Supplychaincoordinationwithrevenue-sharingcontracts:strengthsandlimitations.ManagementScience51(1),30. Chao,G.S.,Friesz,T.L.,1984.Spatialpriceequilibriumsensitivityanalysis.TransportationResearchPartB18(6),423. Chen,F.,Federgruen,A.,Zheng,Y.-S.,2001.Coordinationmechanismsforadistributionsystemwithonesupplierandmultipleretailers.ManagementScience47(5),2001. Chen,R.R.,Roma,P.,2011.Groupbuyingofcompetingretailers.ProductionandOperationsManagement20(2),181. Cheng,S.-F.,Reeves,D.,Vorobeychik,Y.,Wellman,M.,2004.Notesonequilibriainsymmetricgames.In:Proceedingsofthe6thInternationalWorkshoponGameTheoreticandDecisionTheoreticAgents. Colangelo,G.,2008.Privatelabelingandcompetitionbetweenretailers.JournalofAgriculturalandFoodIndustrialOrganization6,1. Dafermos,S.,Nagurney,A.,1984.Sensitivityanalysisforthegeneralspatialeconomicequilibriumproblem.OperationsResearch32(5),1069. 164

PAGE 165

Dafermos,S.,Nagurney,A.,1987.Oligopolisticandcompetitivebehaviourofspatiallyseparatedmarkets.RegionalScienceandUrbanEconomics17(2),245. Dafermos,S.,Nagurney,A.,1989.Supplyanddemandequilibrationalgorithmsforaclassofmarketequilibriumproblems.TransportationScience23(2),118. Dobson,G.,Karmarkar,U.S.,1987.Competitivelocationonanetwork.OperationsResearch35(4),565. Dong,J.,Zhang,D.,Nagurney,A.,2004.Asupplychainnetworkequilibriummodelwithrandomdemand.EuropeanJournalofOperationsResearch156(1),194. Dong,L.,Rudi,N.,2004.Whobenetsfromtransshipment?Exogenousvs.endogenouswholesaleprices.ManagementScience50(5),645. Drezner,Z.,1995.Facilitylocation:asurveyofapplicationsandmethods.SpringerVerlag,NewYork,NY. Drezner,Z.,Hamacher,H.W.,2002.Facilitylocation:applicationsandtheory.SpringerVerlag,NewYork,NY. Eiselt,H.A.,Laporte,G.,1996.Sequentiallocationproblems.EuropeanJournalofOperationsResearch96(2),217. Eiselt,H.A.,Laporte,G.,Thisse,J.-F.,1993.Competitivelocationmodels:aframeworkandbibliography.TransportationScience27(1),44. Eliashberg,J.,Steinberg,R.,1987.Marketing-productiondecisionsinanindustrialchannelofdistribution.ManagementScience33(8),981. Ellickson,P.,2006.Qualitycompetitioninretailing:astructuralanalysis.InternationalJournalofIndustrialOrganization24,521. Emmons,H.,Gilbert,S.M.,1998.Theroleofreturnspoliciesinpricingandinventorydecisionsforcataloguegoods.ManagementScience44(2),276. Facchinei,F.,Pang,J.-S.,2003.Finite-dimensionalvariationalinequalitiesandcomplementarityproblems.SpringerVerlag,NewYork,NY. Fernie,J.,Pfab,F.,Marchant,C.,2000.RetailgrocerylogisticsintheUK.InternationalJournalofLogisticsManagement11(2),83. Figliozzi,M.,2006.Modelingtheimpactoftechonologicalchangesonurbancommercialtripsbycommercialactivityroutingtype.TransportationResearchRecord:JournaloftheTransportationResearchBoard1964/2006,118. Figliozzi,M.,2009.Theimpactsofcongestiononcommercialvehicletourcharacteristicsandcosts.TransportationResearchPartE46(4),496. 165

PAGE 166

Figliozzi,M.,Kingdon,L.,Wilkitzki,A.,December2007.Analysisoffreighttoursinacongestedareausingdisaggregateddata:characteristicsanddatacollectionchallenges.In:Proceedingsofthe2ndAnnualNationalUrbanFreightConference.LongBeach,CA. Friesz,T.L.,Harker,P.T.,Tobin,R.L.,1984.Alternativealgorithmsforthegeneralnetworkspatialpriceequilibriumproblem.JournalofRegionalScience24(4),475. Friesz,T.L.,Miller,T.,Tobin,R.L.,1988a.Algorithmsforspatiallycompetitivenetworkfacility-location.EnvironmentandPlanningB:PlanningandDesign15(2),191. Friesz,T.L.,Tobin,R.L.,Miller,T.,1988b.Competitivenetworkfacilitylocationmodels:asurvey.PapersinRegionalScience65(1),47. Friesz,T.L.,Tobin,R.L.,Miller,T.,1989.Existencetheoryforspatiallycompetitivenetworkfacilitylocationmodels.AnnalsofOperationsResearch18(1),267. Friesz,T.L.,Tobin,R.L.,Smith,T.E.,Harker,P.T.,1983.Anonlinearcomplementarityformulationandsolutionprocedureforthegeneralderiveddemandnetworkequilibriumproblem.JournalofRegionalScience23(3),337. Gabay,D.,Moulin,H.,1980.OntheuniquenessandstabilityofNash-equilibriainnoncooperativegames.In:Bensoussan,A.,Kleindorfer,P.,Tapiero,C.(Eds.),AppliedStochasticControlofEconometricsandManagementScience.Amsterdam,North-Holland,pp.271. Giannoccaro,I.,Pontrandolfo,P.,2004.Supplychaincoordinationbyrevenuesharingcontracts.InternationalJournalofProductionEconomics89(2),131. Golob,T.,Regan,A.,2001.Impactsofhighwaycongestiononfreightoperations:perceptionsoftruckingindustrymanagers.TransportationResearchPartA35(7),577. Golob,T.,Regan,A.,2003.Trafccongestionandtruckingmanagers'useofautomatedroutingandscheduling.TransportationResearchPartE39(1),61. Hale,T.S.,Moberg,C.R.,2004.Locationscienceresearch:areview.AnnalsofOperationsResearch123,21. Hamilton,J.,Klein,J.,Sheshinski,E.,Slutsky,S.,1994.Quantitycompetitioninaspatialmodel.TheCanadianJournalofEconomics27(4),903. Hamilton,J.H.,Thisse,J.-F.,Weskamp,A.,1989.Spatialdiscrimination:Bertrandvs.Cournotinamodeloflocationchoice.RegionalScienceandUrbanEconomics19(1),87. Han,D.,2006.Solvinglinearvariationalinequalityproblemsbyaself-adaptiveprojectionmethod.AppliedMathematicsandComputation182(2),1765. 166

PAGE 167

Han,D.,Lo,H.K.,2002.Twonewself-adaptiveprojectionmethodsforvariationalinequalityproblems.ComputersandMathematicswithApplications43(12),1529. Harker,P.T.,1984.Avariationalinequalityapproachforthedeterminationofoligopolisticmarketequilibrium.MathematicalProgramming30(1),105. Harker,P.T.,1986.Alternativemodelsofspatialcompetition.OperationsResearch34(3),410. Harker,P.T.,Pang,J.-S.,1990.Finite-dimensionalvariationalinequalityandnonlinearcomplementarityproblems:asurveyoftheory,algorithmsandapplications.MathematicalProgramming48,161. Harsanyi,J.C.,Selten,R.,1988.Ageneraltheoryofequilibriumingames.Cambridge:MITPress. He,B.,1997.Aclassofprojectionandcontractionmethodsformonotonevariationalinequalities.AppliedMathematicsandOptimization35(1),69. He,B.,Zhou,J.,2000.Amodiedalternatingdirectionmethodforconvexminimizationproblems.AppliedMathematicsLetters13(2),123. He,B.S.,Liao,L.Z.,2002.Improvementsofsomeprojectionmethodsformonotonenonlinearvariationalinequalities.JournalofOptimizationTheoryandApplications112(1),111. Heidemann,D.,1994.Queuelengthanddelaydistributionsattrafcsignals.TransportationResearchPartB28(5),377. Heidemann,D.,Wegmann,H.,1997.Queueingatunsignalizedintersections.TransportationResearchPartB31(3),239. Hensher,D.,Puckett,S.,2005.Refocusingthemodellingoffreightdistribution:developmentofaneconomic-basedframeworktoevaluatesupplychainbehaviourinresponsetocongestioncharging.Transportation32(6),573. Hotelling,H.,1929.Stabilityoncompetition.EconomicJournal39(153),41. Ingene,C.A.,Parry,M.E.,1995.Channelcoordinationwhenretailerscompete.MarketingScience14(4),360. Ingene,C.A.,Parry,M.E.,1998.Manufacturer-optimalwholesalepricingwhenretailerscompete.MarketingLetters9(1),65. Ingene,C.A.,Parry,M.E.,2000.Ischannelcoordinationallitiscrackeduptobe?JournalofRetailing76(4),511. Iyer,G.,1998.Coordinatingchannelsunderpriceandnonpricecompetition.MarketingScience17(4),338. 167

PAGE 168

Iyer,G.,Padmanabhan,V.,2005.Contractualrelationshipsandcoordinationindistributionchannels.In:Chakravarty,A.K.,Eliashberg,J.(Eds.),ManagingBusinessInterfaces.SpringerUS. Jeuland,A.P.,Shugan,S.M.,1983.Managingchannelprots.MarketingScience2(3),239. Jeuland,A.P.,Shugan,S.M.,1988.Replyto:managingchannelprots:comment.MarketingScience7(2),202. Johnson,L.A.,Montgomery,D.C.,1974.Operationsresearchinproductionplanning,schedulingandinventorycontrol.JohnWileyandSons,Inc.,NewYork. Keskinocak,P.,Savasaneril,S.,2008.Collaborativeprocurementamongcompetingbuyers.NavalResearchLogistics55(6),516540. Kinderlehrer,D.,Stampacchia,G.,1980.Anintroductiontovariationalinequalitiesandtheirapplications.AcademicPress,NewYork,NY. Klein,M.,1961.Onproductionsmoothing.ManagementScience7(3),286. Klemperer,P.,Meyer,M.,1986.Pricecompetitionvs.quantitycompeition:theroleofuncertainity.TheRANDJournalofEconomics17(4),618. Kohli,R.,Park,H.,1989.Acooperativegametheorymodelofquantitydiscounts.ManagementScience35(6),693. Kok,A.G.,Fisher,M.L.,Vaidyanathan,R.,2009.Assortmentplanning:reviewofliteratureandindustrypractice.In:Agrawal,N.,Smith,S.A.(Eds.),RetailSupplyChainManagement.SpringerUS,Ch.6. Konnov,I.V.,2007.Equilibriummodelsandvariationalinequalities.Elsevier,Amsterdam,TheNetherlands. Konur,D.,Geunes,J.,2011.Analysisoftrafccongestioncostsinacompetitivesupplychain.TransportationResearchPartE47(1),1. Kreps,D.,Scheinkman,J.,1983.QuantityprecommitmentandBertrandcompetitionyieldCournotoutcomes.TheBellJournalofEconomics14(2),326. Labbe,M.,Hakimi,S.L.,1991.Marketandlocationalequilibriumfortwocompetitors.OperationsResearch39(5),749. Lal,R.,Staelin,R.,1984.Anapproachfordevelopinganoptimaldiscountpricingpolicy.ManagementScience30(12),1524. Lederer,P.J.,Thisse,J.-F.,1990.Competitivelocationonnetworksunderdeliveredpricing.OperationsResearchLetters9(3),147. Lee,H.,2004.Thetriple-asupplychain.HarwardBusinessReview82(10),102. 168

PAGE 169

Lee,H.,Whang,S.,1999.Decentralizedmulti-echelonsupplychains:incentivesandinformation.ManagementScience45(5),633. Lee,H.L.,Billington,C.,1993.Materialmanagementindecentralizedsupplychains.OperationsResearch41(5),835. Lee,H.L.,Rosenblatt,M.J.,1986.Ageneralizedquantitydiscountpricingmodeltoincreasesupplier'sprots.ManagementScience32(9),1177. Li,M.,2002.Theroleofspeed-owrelationshipincongestionpricingimplementationwithanapplicationtoSingapore.TransportationResearchPartB36,731. Liao,L.-Z.,Wang,S.,2002.Aself-adaptiveprojectionandcontractionmethodformonotonesymmetriclinearvariationalinequalities.ComputersandMathematicswithApplications43,41. Luo,Z.-Q.,Pang,J.-S.,Ralph,D.,1996.Mathematicalprogramswithequilibriumconstraints.CambridgeUniversityPress,NewYork,NY. Mazzarotto,N.,2001.Competitionpolicytowardsretailers:size,sellermarketpowerandbuyermarketpower,WorkingPaper01-4.CentreforCompetitionandRegulation. McKinnon,A.,1999.Theeffectoftrafccongestionontheefciencyoflogisticaloperations.InternationalJournalofLogistics:ResearchandApplications2(2),111. McKinnon,A.,Edwards,J.,Piecky,M.,Palmer,A.,2008.Trafccongestion,reliabilityandlogisticalperformance:amulti-sectoralassessment.Tech.rep.,LogisticsResearchCentre,Heriot-WattUniversity,Edinburgh. Miller,T.,Friesz,T.L.,Tobin,R.L.,1992a.Heuristicalgorithmsfordeliveredpricespatiallycompetitivenetworkfacilitylocationproblems.AnnalsofOperationsResearch34(1),177. Miller,T.C.,Friesz,T.L.,Tobin,R.L.,1996.Equilibriumfacilitylocationonnetworks.Springer,Berlin. Miller,T.C.,Tobin,R.L.,Friesz,T.L.,1991.StackelberggamesonanetworkwithCournot-Nasholigopolisticcompetitors.JournalofRegionalScience31(4),435. Miller,T.C.,Tobin,R.L.,Friesz,T.L.,1992b.NetworkfacilitylocationmodelsinStackelberg-Nash-Cournotspatialcompetition.PapersinRegionalScience71(3),27. Ming-hui,X.,Cheng-xiu,G.,2005.Supplychaincoordinationwithdemanddisruptionsunderconvexproductioncostfunction.WuhanUniversityJournalofNaturalSciences10(3),493. 169

PAGE 170

Moinzadeh,K.,Klastorin,T.,Berk,E.,1997.Theimpactofsmalllotorderingontrafccongestioninaphysicaldistributionsystem.IIETransactions29(8),671. Monahan,J.P.,1984.Aquantitydiscountpricingmodeltoincreasevendorprots.ManagementScience30(6),720. Moorthy,K.S.,1987.Managingchannelprots:comment.MarketingScience6(4),375. Moorthy,K.S.,1988.Strategicdecentralizationinchannels.MarketingScience7(4),335. Nagarajan,M.,Sosic,G.,2008.Game-theoreticanalysisofcooperationamongsupplychainagents:reviewandextensions.EuropeanJournalofOperationalResearch187(3),719. Nagarajan,M.,Sosic,G.,Zhang,H.,2010.Stablegrouppurchasingorganizations,Workingpaper. Nagurney,A.,1987.Computationalcomparisonsofspatialpriceequilibriummethods.JournalofRegionalScience27(1),55. Nagurney,A.,1988.Algorithmsforoligopolisticmarketequilibriumproblems.RegionalScienceandUrbanEconomics18,425. Nagurney,A.,1999.Networkeconomics:avariationalinequalityapproach.KluwerAcademicPublishers,Norwell,MA. Nagurney,A.,Dong,J.,Zhang,D.,2002.Asupplychainnetworkequilibriummodel.TransportationResearchPartE38(5),281. Nash,J.,1951.Non-cooperativegames.AnnalsofMathematics54(2),286. Oren,S.,1997.Economicinefciencyofpassivetransmissionrightsincongestedelectricitysystemswithcompeitivegeneration.TheEnergyJournal18,63. Oum,T.,Zhang,A.,Zhang,Y.,1995.Airlinenetworkrivalry.TheCanadianJournalofEconomics28,836. Owen,S.H.,Daskin,M.S.,1998.Strategicfacilitylocation:areview.EuropeanJournalofOperationsResearch111(3),423. Padmanabhan,V.,Png,I.P.L.,1997.Manufacturer'sreturnspoliciesandretailcompetition.MarketingScience16(1),81. Pal,D.,Sarkar,J.,2002.Spatialcompetitionamongmulti-storerms.InternationalJournalofIndustrialOrganization20(2),163. Park,J.-H.,Zhang,A.,1998.Airlinealliancesandpartnerrms'outputs.TransportationResearchPartE34(4),245. 170

PAGE 171

Pasternack,B.A.,1985.Optimalpricingandreturnpoliciesforperishablecommodities.ManagementScience4(2),166. Pels,E.,Verhoef,E.,2004.Theeconomicsofairportcongestionpricing.JournalofUrbanEconomics55,257. Plastria,F.,2001.Staticcompetitivefacilitylocation:anoverviewofoptimisationapproaches.EuropeanJournalofOperationsResearch129(3),461. Qin,Y.,Tang,H.,Guo,C.,2007.Channelcoordinationandvolumediscountswithprice-sensitivedemand.InternationalJournalofProductionEconomics105(1),43. Rao,K.,Grenoble,W.,1991.ModellingtheeffectsoftrafccongestiononJIT.InternationalJournalofPhysicalDistributionandLogisticsManagement21(2),3. Rao,K.,Grenoble,W.,Young,R.,1991.TrafccongestionandJIT.JournalofBusinessLogistics12(1),105. Rhim,H.,1997.Geneticalgorithmsforacompetitivelocationproblem.TheKoreanBusinessJournal31,380. Rhim,H.,Ho,T.H.,Karmarkar,U.S.,2003.Competitivelocation,productionandmarketselection.EuropeanJournalofOperationsResearch149(1),211. Rosenthal,R.,1973.Aclassofgamespossesingpure-strategyNashequilibria.InternationalJournalofGameTheory2(1),65. Saiz,M.E.,Hendrix,E.M.,2008.MethodsforcomputingNashequilibriaofalocation-quantitygame.ComputersandOperationsResearch35(10),3311. Salant,S.,1982.Imperfectcompetitionintheinternationalenergymarket:acomputerizedNash-Cournotmodel.OperationsResearch30(2),252. Sankaran,J.,Gore,A.,Coldwell,B.,2005.Theimpactofroadtrafccongestiononsupplychains:insightsfromAuckland,NewZealand.InternationalJournalofLogistics:ResearchandApplications8(2),159. Sarkar,J.,Gupta,B.,Pal,D.,1997.LocationequilibriumforCournotoligopolyinspatiallyseparatedmarkets.JournalofRegionalScience37(2),195. Sarmah,S.,Acharya,D.,Goyal,S.,2006.Buyervendorcoordinationmodelsinsupplychainmanagement.EuropeanJournalofOperationalResearch175(1),1. Selten,R.,1975.Reexaminationoftheperfectnessconceptforequilibriumpointsinextensivegames.InternationalJournalofGameTheory4(1),25. Shao,J.,Krishnan,H.,McCormick,S.T.,2009.Incentivesfortransshipmentinasupplychainwithdecentralizedretailers,Workingpaper. 171

PAGE 172

Sherali,H.D.,Soyster,A.L.,Murphy,F.H.,1983.Stackelberg-Nash-Cournotequilibria:characterizationsandcomputations.OperationsResearch31(2),253. Smith,R.L.,Zhang,R.Q.,1998.Innitehorizonproductionplanningintime-varyingsystemswithconvexproductionandinventorycosts.ManagementScience44(9),1313. Smith,S.A.,Agrawal,N.,2000.Managementofmulti-itemretailinventorysystemswithdemandsubstitution.OperationsResearch48(1),50. Smithies,A.,1941.Optimumlocationinspatialcompetition.JournalofPoliticalEconomy49(3),423. Tansel,B.C.,Francis,R.L.,Lowe,T.J.,1983.Locationonnetworks:asurvey,partsiandii.ManagementScience29(4),482. Taylor,T.A.,2001.Channelcoordinationunderpriceprotection,midlifereturns,andend-of-lifereturnsindynamicmarkets.ManagementScience47(9),1220. Taylor,T.A.,2002.Supplychaincoordinationunderchannelrebateswithsalesefforteffects.ManagementScience48,992. Teitz,M.B.,1968.Locationalstrategiesforcompetitivesystems.JournalofRegionalScience8(2),135. Tobin,R.L.,1987.Sensitivityanalysisforgeneralspatialpriceequilibria.JournalofRegionalScience27(1),77. Tobin,R.L.,1992.UniquenessresultsandalgorithmforStackelberg-Cournot-Nashequilibria.AnnalsofOperationsResearch34(1),21. Tobin,R.L.,Friesz,T.L.,1986.Spatialcompetitionfacilitylocationmodels:denition,formulationandsolutionapproach.AnnalsofOperationsResearch6(3),47. Tobin,R.L.,Miller,T.,Friesz,T.L.,1995.Incorporatingcompetitors'reactionsinfacilitylocationdecisions:amarketequilibriumapproach.LocationScience3(4),239. Toptal,A.,Cetinkaya,S.,2006.Contractualagreementsforcoordinationandvendor-manageddeliveryunderexplicittransportationconsiderations.NavalResearchLogistics53(5),397. Toptal,A.,Cetinkaya,S.,Lee,C.-Y.,2003.Thebuyer-vendorcoordinationproblem:modelinginboundandoutboundcargocapacityandcosts.IIETransactions35(11),9871002. Tsay,A.A.,Agrawal,N.,2000.Channeldynamicsunderpriceandservicecompetition.ManufacturingandServiceOperationsManagement2(4),372. 172

PAGE 173

Tsay,A.A.,Nahmias,S.,Agrawal,N.,2000.Modelingsupplychaincontracts:areview.In:Tayur,S.,Ganeshan,R.,Magazine,M.(Eds.),Quantitativemodelsforsupplychainmanagement.KluwerAcademic,Norwell,MA,Ch.10,pp.299. VanRyzin,G.,Mahajan,S.,1999.Ontherelationshipbetweeninventorycostsandvarietybenetsinretailassortments.ManagementScience45(11),1496. Vandaele,N.,Woensel,T.V.,Verbruggen,A.,2000.Aqueueingbasedtrafcowmodel.TransportationResearchPartD5,121. Veinott,A.F.,1964.Productionplanningwithconvexcosts:aparametricstudy.ManagementScience10(3),441. Ventosa,M.,Ballo,A.,Ramos,A.,Rivier,M.,2005.Electricitymarketmodelingtrends.EnergyPolicy33,897. Viswanathan,S.,Wang,Q.,2003.Discountpricingdecisionsindistributionchannelswithprice-sensitivedemand.EuropeanJournalofOperationalResearch149(3),571. Wang,Y.,Xiu,N.,Wang,C.,2001.Anewversionofextragradientmethodforvariationalinequalityproblems.ComputersandMathematicswithApplications42,969. Weisbrod,G.,Vary,D.,Treyz,G.,2001.Economicimplicationsofcongestion.NCHRPReport#463.TransportationResearchBoard. Weng,Z.K.,1995a.Channelcoordinationandquantitydiscounts.ManagementScience41(9),1509. Weng,Z.K.,1995b.Modelingquantitydiscountsundergeneralprice-sensitivedemandfunctions:optimalpoliciesandrelationships.EuropeanJournalofOperationalResearch86(2),300. Woensel,T.V.,Cruz,F.,2009.Astochasticapproachtotrafccongestioncosts.ComputersandOperationsResearch36,1731. Woensel,T.V.,Vandaele,N.,2006.Empiricalvalidationofaqueueingapproachtouninterruptedtrafcows.AQuarterlyJournalofOperationsResearch4(1),5972. Woensel,T.V.,Vandaele,N.,2007.Modellingtrafcowswithqueueingmodels:areview.Asia-PacicJournalofOperationalResearch24(4),435461. Xiao,T.,Qi,X.,2008.Pricecompetition,costanddemanddisruptionsandcoordinationofasupplychainwithonemanufacturerandtwocompetingretailers.Omega36(5),741. Xie,J.,Wei,J.C.,2009.Coordinatingadvertisingandpricinginamanufacturerretailerchannel.EuropeanJournalofOperationalResearch197(2),785. 173

PAGE 174

Yang,S.-L.,Zhou,Y.-W.,2006.Two-echelonsupplychainmodels:consideringduopolisticretailersdifferentcompetitivebehaviors.InternationalJournalofProductionEconomics103(1),104. Yao,Z.,Leung,S.C.,Lai,K.,2008.Manufacturersrevenue-sharingcontractandretailcompetition.EuropeanJournalofOperationalResearch186(2),637. 174

PAGE 175

BIOGRAPHICALSKETCH DincerKonurwasborninIstanbul,Turkeyin1984.Hehascompletedhisprimary,secondary,andhighschooleducationinIstanbul.HehasearnedhisB.S.degreeinindustrialengineeringfromBilkentUniversityinAnkara,Turkey,in2007.Upongraduation,hestartedhisdoctoralstudyintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.HehasreceivedhisM.S.degreeinDecember2009duringhisdoctoralstudies.DincerKonurwillcompletehisPh.D.inAugust2011.Upongraduation,hewilljoinIntermodalFreightTransportationInstituteattheUniversityofMemphisasapost-doctoralresearcher. 175