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155Vol.15,No.5200610OPERATIONSRESEARCHANDMANAGEMENTSCIENCEOct.2006:2006202224:(19792),,,,1,2,3(11,430072;21,78712;31,100080):,,,,(,),,;,,,:,,,,,:0227:A:100723221(2006)0520001207RiskAllocationofCapacityInvestmentinSupplyChainsYANGLei1,YUGang,ZHANGHan2qin(11SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China;21De2partmentofManagementScienceandInformationSystems,TheUniversityofTexasAustin,Texas78712,U.S.A.;3.AcademyofMathematicsandSystemsScience,TheChineseAcade2myofSciences;Beijing100080,China)Abstract:Inasupplychain,heachmanagerwillfacekindsofsupplyanddemanduncertaintieswhichmayin2ducesignificantcosts.Buttneymaynotbearcapacityinvestmentrisk.Inthispaper,weconsideraone2retail2ersupplychain.Weanalyzetheimpactofdifferentcontracts(pushcontract,pullcontractandoptioncon2tract)ontheirprofitandriskofcapacityinvestment.Withapushcontract,theretailerpaysahighwholesalepricetomakethesupplierconstructacapacity.Insellingseason,thesupplierisgivenanorderbytheretail2er,sotheretailerbearsallsupplychainscapacityinvestmentrisk.Withapullcontract,thesupplierdecideshowmuchcapacitytoconstruct,andtheretailerordersaccordingtoitscustomerssdemand.Sothesupplierbearsallsupplychainscapacityinvestmentrisk.Withapullcontract,thesupplierdecideshowmuchcapacitytocostruct,andtheretailerordersaccordingtoitscustomerssdemand.Sothesupplierbearsallsupplychainscapacityinvestmentrisk.Theparetosetsforpushandpullcontractsaregiven.Tocoordinatethesup2plychain,finally,weconstructtheoptioncontractinwhichthesupplierandtheretailersharetherisk.WecompareourresultswithCachon(2004)sandgetsimilaritiesanddefferences.Keywords:supplychainmanagement;Paretoset;contract;coordination;capacity;riskallocation0,,;,,,,,,,,,,,([13])([912][1416])(CapacityAllocation)(CapacitatedProduction),(InvestmentRisk),[4]([58]),[5],,,([9,12,14,15]),[12],([1012],)[10][11][10],,,[11],(ComplianceRegime),,,,,,,,,,,(Advance2PurchaseDiscountcontracts),1;2;31,,,:F(x)f(x),FF(0)=0()g(x)g(x)=xh(x),h(x),h(x)=f(x)1-F(x)g(x):g(x)0,,:,,,:K;ck;cp;r;w;we(exerciseprice);wo(optionprice);rcp+ck2200615,,,,:^w(r)K,,K,min{D,k},:K,w,min{D,K},,Kwo,,K,,min{D,K},we,,we=;wo=0,,,,,,,22.12.22.32.42.52.1,K:(k)=rS(K)-ckK-cpS(K)=(r-cp)S(K)-ckK(1)S(K)=E[min{D,K}]=K-K0F(x)dx(2),,,S(K):(K)K,K0K0,K:F(K0)=r-cp-ckr-cp(3)2.2,,K^w,:^r(K,^w)=rS(K)-^wK(4)^wKF(K)=r-^wr(5),^wF(K)K,^wK(5)^wK:^w(K)=r(1-F(K))(6),K:35,:^r(K)=^r(K,^w(K))=r(1-F(K))(j(K)-k)(7)j(K)=S(K)1-F(K)(8)K,:^r(K)=rf(K)K0(9)(9),,,,,^s(K,^w)=(^w-ck)K-cpS(K),K^s(K)=^s(K,^w(K))=[r(1-F(K))-ck]K-cpS(K)(10)(3),^s=(r-cp)[F(K0)-F(K)]K+cp[1-F(K)](K-j(K))(11)(11)^s(K)=(1-F(K))[r-cp-rg(K)-ck1-F(K)](12)ck1-F(K)g(K),(12)K,^s(K)^s(0)=0,,^K3,^K3=argmax^s(K),^w(^K3)2.1.(7)(9),(0,),j(K)K,(11),^s(K0)0K0,^w(K0),,,K0,,,,,[10][10],,(,[10]),,,,,,,,,,,,,,1,,(SalvageValue),[10],,,2.3,Kw,,K,,w[10],s(K,w)=(w-cp)S(K)-ckK,w,F(K)=w-cp-ckw-cp,,Kww(K)=ck1-F(K)+cp,:s(K)=ckS(K)1-F(K)-ckK=ck(j(K)-K)4200615K:s(K)=ckS(K)f(K)(1-F(K))2=ckj(K)h(K)0,j(K)h(K)K,[10]2.1K0,j(K)Kj(K)j(K)h(K)r(K,w)=(r-w)S(K),K:r(K)=r(K,w(K))=(r-cp-ck1-F(K))S(K)=(r-cp)(F(K0)-F(K))j(K),r(K)Kr(K0)=0,K0,w(K0),K3=argmaxr(K),K32.4,,,,,,,,,,,(),KK,,K[0,^K3]K,K^K3,,K^K32.2,^s(K0)0,KK0,K^s(K)^K3,K^s(K)=0,KK^s(K)0,[^K3,K],K[0,^K3)(K,)K,K,Kr(K0)=0,[K3,k0],K,(0,K3)U(K0,)K[10],K02.5,,,^K3,[^K3,K],,,,,,,Kwo,,K,,min{D,K},wewowe,wowe,,55,:or(K,wo,we)=(r-we)S(K)-woK(13)wowe,:F(Kr)=r-wo-wer-we(14)os(K,wo,we)=(we-cp)S(K)-(ck-wo)K(15)wowe,,:F(Ks)=we-cp-ck+wowe-cp(16)(K)=rS(K)-ckK-cpS(K)=(r-cp)S(K)-ckK,K0=F-1(r-cp-ckr-cp),r-wewo=r-cpck,wo=ck,or(K)=(K),os(K)=(1-)(K)(14)Kr(16)Ks,,F-1(r-cp-ckr-cp),[10],,,(Advance2PurchaseDis2countcontracts),K3F-1(r-cp-ckr-cp)2.2^K3K3K0,^K3K3,,r(K3)=(p-c)[(F(K0)-F(K3))j(K3)-j(K3)f(K3)]=0(17)(F(K0)-F(K3)=j(K3)f(K3)j(K3)(18)^s(K3)=(1-F(K3))[r-cp-rg(K3)-ck1-F(K3)]=(1-F(K3))[r-cp-rK3f(K3)1-F(K3)-(r-cp)(1-F(K0))1-F(K3)]=(r-cp)(1-F(K3))-rK3f(K3)-(r-cp)(1-F(K0))=(r-cp)(F(K0)-F(K))-rK3f(K3)=(r-cp)j(K3)f(K3)j(K3)-rK3f(K3)=[(r-cp)j(K3)K3j(K3)-r]K3f(K3)[(r-cp-r)K3f(K3)=-cpK3f(K3)0(19)2.1K3j(K3)j(K3)0,(19),KK,(17)(19),^K3K362006153,,,,,,,:20050509Columbia:[1].[J].,2005,8(1):78287.[2],,.[J].,2005,8(1):42247.[3],.[J].,2005,25(6):87293.[4],.[J].:,2004,32(5):1112113.[5],,.[J].,2005,32(6):1532155.[6].[J].,2005,(6):17219.[7],.[J].,2005,21(6):4512453.[8].[J].,2005,22(4):69271.[9]ArmonyM,PlambeckE.Theimpactofduplicateordersondemandestimationandcapacityinvestment[Z].Workingpaper,NewYorkUni2versitandStanfordUniversity.[10]CachonG.Theallocationofinventoryriskinasupplychin:push,pullandadvance2purchasediscountcontracts[J].ManagementScience,2004,50(2):2222238.[11]CachonG,Larivier,M.Contractingtoassuresupply:howtosharedemandforecastsinasupplychain[J].ManagementScience,2001,47(5):629246.[12]CachonG,LariviereM.Capacitychoiceandallocation:strategicbehaviorandsupplychainperformanco[J].1999,45(8):109121108.[13]DixitA,PindyckRS.Investmentunderuncertainty.princetonuniversitypress,princeton,1994.[14]GlassermanP,SridharT.Asimpleapproximationforamultistagecapacitatedproductioninventorysystem[J].NavalResearchLogistics,1996,