Price and Capacity Competition of Make-To-Order Fi

PrieandCapaityCompetitionofMake-To-OrderFirms1HongChen2FaultyofCommereandBusinessAdministration,UBC,CanadaYat-wahWan3DepartmentofIndustrialEngineeringandEngineeringManagementHKUST,ClearWaterBay,Kowloon,HongKongAbstratInthispaper,weonsiderthe(non-ooperative)ompetitionofaduopolyofmake-to-orderrms,eahmodeledbyanM/M/1queue.Thermspostthepriestheyhargeforservie.Knowingtheexpetedrewardandtheexpetedwaitingtimeateahrm,potentialustomershooseeithertoreeiveservieatoneofthermsortoseekanalternativeopportunitybymaximizingtheirexpetednetrewards.Weonsiderboththeshort-runompetitionandthelong-runompetition.Intheshortrun,theservieapaitiesofbothrmsaregiven,andthermsmaximizetheirexpetedrevenuesbyhoosingpriesonly.Inthelongrun,thermsmaximizetheirexpetedprotsbyhoosingbothpriesandservieapaities.Intheshortrun,aNashequilibriummaynotexist,andwhenitexits,itmaynotbeunique.Givenallelsebeingthesame,armwithalargerapaity,orahighervalueofservieoralowerostofwaitinganhargeapriepremiumandtakealargermarketshare.Thermsmayeitherraiseoruttheiroptimalpries,inresponsetoaninreaseintheirservieapaity,eventhoughahigherservieapaityalwaysleadstoashorterexpetedwaitingtime.Thermsmayhaveeitherahigheroraloweroptimalprieforahigherunitostofwaiting.Inthespeialaseofhomogeneous(oridential)rms,theequilibriumalwaysexists.Intheshortrun,notallpotentialustomersareserved,butinthelongrun,allpotentialustomerswillbeservedunlessnoneofthermshoosetoenterthemarket.Currentversion:November20001WeareindebtedtoMurrayFrankforthehelpfulonversationandomments.ApreliminaryversionofthispaperhasappearedasChen,etal(2000).2SupportedinpartbyanNSERC(Canada)researhgrantandaRGC(HongKong)researhgrant.3SupportedinpartbyaRGC(HongKong)researhgrant.1IntrodutionWiththeadvanementininformationinfrastruture,theollaborationofdierentlevelsinsupplyhains,andtheemergeneofe-ommere,moreandmorermsareadoptingthemake-to-order(MTO)mode.Firmsoperatinginsuhamodestarttoprodueonlyafteranorderisplaed.Anewlyplaedorderanbelledonlyaftertheurrentorderandallpreviousordersareprodued(ifarst-in-rst-outdisiplineisinplae).Consequently,thereisalwaysalead-timebetweentheplaementofanorderandthereeiptofgoods.Intheerelyompetitivebusinessworld,whenmakingpriedeisions,MTOrmsmustantiipatethe(expeted)ostofdelaystheirustomerswouldusetovaluetheprodutorservie.Theinterationbetweenrmsandustomersgivesrisetoamulti-levelmulti-playergame:Customersshoparoundinamarkettondthermthatprovidesthebestexpetednetreward(whihisthevalueofprodutorservieminustheprieandtheexpetedwaitingost).Atthesametime,tomaximizetheirownbenets,rmsreattothebehaviorofustomersandtheationsoftheirompetitors.Intheshortrun,apaitiesofrmsarerelativelyxedandhenermsompetebypostingattrativepriestoinreasetheir(expeted)revenue.Inthelongrun,rmsanmakebothservie(orprodution)apaityandpriedeisions.Whenmakingthepriedeision,thermhastobalanebetweenattratingmoreustomerswithalongerexpetedwaitingtimeandattratinglessustomerswithashorterexpetedwaitingtime.Inthispaper,weonsidertheompetitionbetweentwoheterogeneousmake-to-orderrms.EahrmismodeledbyanM/M/1queue.Thermspostthepriestheyhargeforservie.PotentialustomerarrivalsfollowaPoissonproess.Knowingtheprieandtheexpetedwaitingtimeateahrm,eahpotentialustomermaximizeshisexpetednetrewardbyhoosingarmtoreeiveservie(ifatall)orbytakinganalternativeopportunity.Intheshortrun,theservieapaitiesofbothrmsaregivenandthermsmaximizetheirexpetedrevenuesbyhoosingpriesonly.Inthelongrun,thermshoosebothservieapaitiesandpries.WeareinterestedinestablishingtheexisteneofaNashequilibriumandstudyinghowtheequilibrium(whenexists)varieswiththermharateristis.Thequeueingeetinprie-timeompetitionwasrststudiedintheseminalworkofNaor(1969).Heonsideredamonopolyinwhihustomersouldobservetheexistingqueuelengthbeforedeidingtojoinorbalk.Heshowedthatthesoialoptimality,theindividualoptimalityofustomers,andthemaximalbenetofthermdonotalignwitheahother,andtoregulateustomersbehavior,themonopolistirmwouldgenerallysettoohighaprie.Naor’sworkhasmotivatedotherresearherstostudyonsumer’sbehaviorandthesoialeÆienyofamonopoly;see,forexample,Knudsen(1972),EdelsonandHildebrand(1975),DeVany(1976),LippmanandStidham(1977),Hassin(1986),andChenandFrank(1994).Theintegrityofustomersaddsinanewdimensionwhentherearemultiplelassesofustomersandthermannothekthelassidentityofustomers.MendelsonandWhang(1990)onsideredmultiplelassesofustomersinwhihustomersmightheattheirlassidentities.Theygavesoiallyoptimalprioritypriingshemes(namelyinentive-ompatibleshemes)thatinduedustomerstorevealtheirtruelassidentiation.ExtensionsofthisworkinludeRaoandPetersen(1996),Bradford(1996)andVanMieghem(1999).Naturallyduopolyisonsideredinprie-timeompetition.Luski(1976)onsideredPoisson1arrivalstoamarketoftwoidentialexponentialserviermsandanoutsideopportunity.Cus-tomerspaidaprieandthewaitingostandearnedarewardiftheyjoinedtheservieofeitherrm.Hefoundthattheequilibriumpriesofthermsouldbedierentiftheyoperatedintheooperativemode.Noompletesolutionwasfoundwhenthermsworkednon-ooperatively;heouldonlyndneessaryonditionsfor