具有不确定性产出库存系统的随机比较

200577:100026788(2005)0720105208(,100022):,..,;,.:;;;;:O227:AStochasticComparisononaInventorySystemwithYieldUncertaintyYUHai2bo(EconomicsandManagementSchool,BeijingUniversityofTechnology,Beijing100022,China)Abstract:Inthispaperwediscusstheoptimalproblemonadiscrete2time,singleitem,periodicreviewinventorysystemwithyieldanddemanduncertainty,themainobjectiveistoanalyzehowtheuncertaintyofyieldrateanddemandaffecttheoptimalpolicyandcostinthesystem.Forthesingleperiodcase,byusingthepropertiesofstochasticordering,thestochasticcomparisonresultsontheoptimalinventorylevelandoptimalexpectedcostareobtained.Forthesystemwithmulti-periodandwithindependentyieldratesanddemandsineachperiod,byusingthedynamicprogrammingmethod,weshowthattheoptimalinventorylevelisstochasticmonotoneintimeperiodunderacertaincondition.Thestochasticcomparisonresultsontheoptimalcostfortheinventorysystemwithdependentyieldratesanddemandsarealsoprovided.Keywords:uncertainyield;periodicreview;dynamicprogramming;optimalpolicy;stochasticcomparison:2003204226:(:04BJDJG143):(1965-),,,.:.E2mail:[e-mail].1,(),.,.(yield),./,,,.,.,,()/,:,;,,(base2stockpolicy)(Zipkin(2000)).Karlin(1958),MoinzadehLee(1987),Yao(1988).,(),©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,.,HenigGerchak(1990),.LiZheng(2003).()().Mann(1947).Stoyan(1983)ShakedShanthikumar(1994).,.Karlin(1960),Song(1983).Sethi,YanZhang(2005).CooperGupta(2003),.;;,;,;,.2(ordinarystochasticordering)(convexordering)(supermodularordering),Stoyan(1983)ShakedShanthikumar(1994),LiXu(2000).1D^DR,(i)D^D(Dst^D),P{Dt}P{^Dt}(1)tR.stDst^DE[f(D)]E[f(^D)](2)Rf().(ii)D^D(Dcx^D),(2)Rf().stcx(Stoyan(1983)):(a)Dcx^D,E[D]=E[^D]Var[D]Var[^D],Var[D]D;(b)D^DG()^G(),D^D:(i)E[D]=E[^D];(ii)G(t)^G(t)t;G(t)^G(t)t,(3)Dcx^D.2R2g(u,u)((submodular)),:u1^u1u2^u2,ui,^uiR,i=1,2,g(u1,u2)+g(^u1,^u2)()g(u1,^u2)+g(^u1,u2).3(U1,U2)(^U1,^U2)R2,(U1,U2)(^U1,^U2)((U1,U2)sm(^U1,^U2)E[g(U1,U2)]E[g(^U1,^U2)]R2g(u1,u2),u1,u2R.,g(u,v)=uv,,(U1,U2)sm(^U1,^U2)E[U1U2]E[^U1^U2],(^U1,^U2)(U1,U2).Topkis(1978).60120057©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.3.,Yq=qU,q,U,[0,1],F(),.,U=1,.,D,G().c,hp.xR(x,q)=E[E(g(x,q,U,D))],(4)g(x,q,u,d)=cqu+h[x+qu-d]++p[d-x-qu]+=-px+pd+(c-p)qu+(p+h)[x+qu-d]+,(5)[z]+=max{z,0}.qR(x,q),q.(5)g(x,q,u,d)(x,q),R(x,q)(x,q),R(x,q)q,5R(x,q)5q.I=inf{x:5R(x,q)5q|q=00},(6)q3(x)=[I-x]+(7)I,q3(x)x.5R(x,q)5qq=0=E(U)[c-p+(p+h)G(x)],(8)I=G-1(p-cp+h),G-1(x)G(x).(6)I:xIq3(x)0,xIq3(x)=0.,:I,q3(x)(I),xI.(8),ID,U.1D.166^,,c,hp.6U,D;6^^U,^D.x,q.66^R(x,q)^R(x,q),R(x,q)(4),^R(x,q)(4)UD^U^D.66^q3(x)^q3(x),I^I.(i)U^U,Dst^D,I^I.(9)(ii)U^U,Dcx^D,p-cp+h,(9);p-cp+h,I^I.:(i)(6)(7)(8)7017©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.5R(x,q)5q|q=00G(x)p-cp+h,(10),5^R(x,q)5q|q=00^G(x)p-cp+h.(11)^G()^D.Dst^D,st(10)(11)x:5^R(x,q)5q|q=00Ax:5R(x,q)5qq=00,(12)I^I.(ii)Dcx^D,cx,p-cp+h,G(x)^G(x)p-cp+h,(13)(12),,I^I.p-cp+h,(12),,I^I.g(x,q,u,d)(5),1.1xq,(i)g(x,q,u,d)(u,d);(ii)g(x,q,u,d)(u,d).(i)f(u,d)=(x+qu-d)+,xqf(u,d)(u,d).,01,uuddf(u+(1-)u,d+(1-)d)=[x+q(u+(1-)u)-(d+(1-)d)]+=[(x+qu-d)+(1-)(x+qu-d)]+f(u,d)+(1-)f(u,d),(14)(14)[x]+.(ii)uu,dd,xqf(u,d)(u,d),,f(u,d)+f(u,d)f(u,d)+f(u,d)(15).f(u,d)-f(u,d)=0,x+qux+qudx+qu-d,x+qudx+quq(u-u),dx+qux+qud,(15),g(x,q,u,d)(u,d).R3(x)=R(x,q3(x))x,q3(x)(7).,:1g(x,q,u,d),.2166^,UD^D^U,6R(x,q)R3(x),6^^R(x,q)^R3(x);(D,U)cx(^D,^U),(i)xqR(x,q)^R(x,q);(ii)xR3(x)^R3(x).3166^,6R(x,q)R3(x),6^^R(x,q)^R3(x);80120057©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(U,D)sm(^U,^D),(i)xqR(x,q)^R(x,q);(ii)xR3(x)^R3(x).4,.N,n=N,N-1,,2,1,nYq,n=qUn,Un,,q,UnFn().nDn,Gn().n.,:01.,c,hp.,nx,q,nRn(x,q)=E[E(gn(x,q,Un,Dn))](16)gn(x,q,u,d)=cqun+h[x+qun-dn]++p[dn-x-qun]+=-px+pdn+(c-p)qun+(p+h)[x+qun-dn]+.(17),{Un}Nn=1{Dn}Nn=1:10:{Un}Nn=1{Dn}Nn=1;20:{Un}Nn=130:{Dn}Nn=1;40:U1stU2ststUND1stD2ststDN.Jn(x,q)nx,qnN,qJn(x,q),q.Vn(x)=infq0Jn(x,q),n=N,N-1,,2,1.(18)nxnN,VN+1(x)Nx.,Vn(x):Vn(x)=infq0[Rn(x,q)+E{E[Vn+1(x+qUn-Dn)]}],n=N,N-1,,2,1.(19)Rn(x,q)(16),VN+1(x),,VN+1(x)=-~h[x]++~p[x]-,(20)~h~p:~p~h.Jn(x,q)=Rn(x,q)+E{E[Vn+1(x+qUn-Dn)]},n=N,N-1,,2,1.(21)HenigGerchak(1990)n=N,N-1,,1,Jn(x,q)(x,q),Vn(x)x,Jn(x,q)q,Vn(x)Vn(x).q3n(x)5Jn(x,q)5q=0(22)9017©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,In=inf{x:5Jn(x,q)5q|q=00},(23)q3n(x)Inn,:nxInq3(x)n=0,xInq3n(x)0.Vn(x)(2),1,2HenigGerchak(1990).2n=1,2,,N,nxInVn(x)-c;xInVn(x)-c.1.1N,c,hp.nUn,q,Dn,x.nq3n(x)In,(22)(23).{Un}Nn=1{Dn}Nn=110,20,3040.I1I2IN.:.(i)IN-1INxIN-1VN-1(x)VN(x).,(21)n=NJN(x,q),(22)(23)IN=G-1N(p+~p-cp+h+(~p-~h))0.2xINVN(x)-c,(20)xIN0VN+1(x)=-~h-c,xINVN(x)VN+1(x)(24)(21)Jn(x,q),n=N,N-1,,1q,q=05Jn(x,q)5qq=0=E[Un][c-p+(p+h)Gn(x)+E(Vn+1(x-Dn))],n=N,N-1,,2,1.(25)UN-1stUNDN-1stDN,(25)n=N,N-1,5JN(x,q)5qq=05JN-1(x,q)5qq=0,(24)xIN5JN-1(x,q)5qq=05JN(x,q)5qq=00,5JN-1(x,q)5qq=00,xIN-1.IN-1IN.(26)IN,xIN,q3N=0,(21),xINVN(x)=(p+h)GN(x)+E[VN+1(x-DN)],(27),(26),xINxIN-1,(27)VN-1(x)=(p+h)GN-1(x)+E[VN(x-DN-1)],(28)DN-1stDN,(27)(28)xINVN-1(x)VN(x),2IN-1xINVN-1(x)VN(x),,xIN-1VN-1(x)VN(x).(29)(ii)In-1In,xIn-1,Vn-1(x)Vn(x),n=N,N-1,,2.I