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3312Vol.33,No.12200712ACTAAUTOMATICASINICADecember,20071112,,,.,,..,,Markov,TP273+.1TheProductionandMaintenanceControlProblemofHybridSystemswithDemandUncertaintyLIUJun1RUIZhi-Yuan1WEIYao-Bing1CHENJi-Ming2AbstractUnderthecircumstancesthatthedemandisuncer-tain,furthermore,thedemandsometimescannotbesatis¯edbyproduction,thepaperstudiestheproductionandmaintenancecontrolproblemoffailureproneproductionsystems.Aso-calledcomplextrinalhedgingpointcontrolpolicyispresented,whichconsiderstheproblempurchasingextraproductioncapacity.Byutilizingthedecompositionmethodthatdecomposestheprob-lemformoverthe¯nitehorizontothein¯nitehorizon,thecor-respondinghedgingpointsareobtained.Meanwhile,anear-optimalsinglehedgingpointcontrolpolicyisalsopresented.Thenumericalresultshavedemonstratedthecontrolpoliciesandtesti¯edthemethod.KeywordsHybridsystem,maintenance,Markovprocess,hedgingpointcontrolpolicy1,[1]..,,.2006-7-242006-11-20ReceivedJuly24,2006;inrevisedformNovember20,2006(60604029),(2GS063-A52-005-01),(3ZS062-B25-034),(0703-06)SupportedbyNationalNaturalScienceFoundationofChina(60604029),theProjectoftheGreatTechnologyInnovationofGansuProvince(2GS063-A52-005-01),theNaturalScienceFunda-tionofGansuProvince(3ZS062-B25-034),ResearchItemofEduca-tionDepartmentofGansuProvince(0703-06)1.7300502.3100271.SchoolofMechanicalandElectronicalEngineering,LanzhouUniversityofTechnology,Lanzhou7300502.StateKeyLab-oratoryofIndustrialControlTechnology,ZhejiangUniversity,Hangzhou310027DOI:10.1360/aas-007-1331133233.,.,,,.,,.,,,Gershwin[2]..,AkellaKumar[3]..,,,.[4],.Boukas[5].[6],Boukas,,,,.,.,,,d.,,,.,.2,1.1Fig.1Systemmodelx(t)u(t)d(t)t(x)(u)(d).s(t)2f0;1gMarkov,t,1,0.s(t)=1,0·u(t)·l;s(t)=0;u(t)=0.l.,r¹m·rm(t)·¹rmt,r¹m,¹rm.rm(t)Markov,Q(t)=ápmpmrm(t)¡rm(t)!,pm,1=pm;1=rm(t).,.,d,.pvrd.t//(®(t);¯(t)),®(t);¯(t)2fup(U);down(D)g,UUUDDUDD.em=rm=(pm+rm)ev=rv=(pv+rv).Em=lem,Ev=dev.v(t)t.C=h0;¹ki.¹k.l+kd.,,._x(t)=(u(t)+v(t)¡z(t);s(t)=1v(t)¡z(t);s(t)=0(1),z(t),¯(t)=up,z(t)=d¯(t)=down,z(t)=0.,J(x;u;rm;v)=minlimT!11TE[(G(t)+h(t))dt](2),GG(t)=c+x++c¡x¡+cps+crrm¹s.c+c¡x+x¡.cp,crrm.s=0,¹s=1;s=1,¹s=0.h(t)=cvvv.cv.J(x;u;rm;v)(1).,Hu,VakiliYu[7]LiberopoulosCaramanis[8],.,[9],.,.33.1Bellman266666664pmpv¡pm¡pvpm(1¡pv)(1¡pm)pvpmpvrm(1¡pv)rmpv¡rm¡pvrmpv(1¡rm)pv(1¡pm)rvpmrvpmrv¡rv¡pmpm(1¡rv)rmrv(1¡rm)rvrm(1¡rv)rmrv¡rv¡rm377777775Vij(x)(i2f0;1g,j2f1;0g),,,Bellman1213338:J¤=min0·u·lfdV11(x)dx(u¡d)g+(pmpv¡pm¡pv)V11(x)+pm(1¡pv)V01(x)+(1¡pm)pvV10(x)+pmpvV00(x)+c+x++c¡x¡+cpJ¤=¡dV01(x)dxd+c+x++c¡x¡+minr¹m·rm·¹rm½rm(1¡pv)V11(x)+(rmpv¡rm¡pv)V01(x)+rmpvV10(x)+(1¡rm)pvV00(x)+crrm¾J¤=min0·u·lfdV10(x)dxug+(1¡pm)rvV11(x)+pmrvV01(x)+(pmrv¡rv¡pm)V10(x)+pm(1¡rv)V00(x)+c+x++c¡x¡+cpJ¤=minr¹m·rm·¹rm(rmrvV11(x)+(1¡rm)rvV01(x)+rm(1¡rv)V10(x)+(rmrv¡rv¡rm)£V00(x)+crrm)+c+x++c¡x¡Bellman.[10],,.,,.3.2,8:(u;v)=8:(u1;v1)=8:(0;0)xz¤mu(min(l;d);0)x=z¤mu(l;0)z¤vxz¤mu(l;max(d¡l;0))x=z¤v(l;¹k)xz¤vUU(u2;v2)=((l;0)xz¤md(0;0)x¸z¤mdUDrm=8:r1=(¹rmxz¤dur¹mx¸z¤duDUr2=(¹rmxz¤ddr¹mx¸z¤ddDD(3)z¤mu,z¤v,z¤md,z¤du.,z¤muz¤v.UU,z¤mu,z¤muz¤muz¤v,z¤v,z¤v,UD,,z¤md,,z¤md,,DU,z¤du,,z¤du,,DDDU,.,(3)z¤mu¸z¤md,z¤mu¸z¤du,z¤du¸z¤dd..z¤duz¤mu,xz¤mu,x=z¤mu,.,z¤mu¸z¤mdz¤du¸z¤dd,UUDUUDDD,.44.1\\,,d,,z¤muz¤du.1.,z¤muz¤duz¤du0,z¤mu=z¤du+H£ln[W((c++c¡)(lr¹m¡dr¹m¡pmd)£exp(L(d¹rm+pmd¡l¹rm))+c+(¹rm¡r¹m)](4)z¤du0,z¤mu=z¤du+H£ln[W((c++c¡)(l¹rm¡d¹rm¡pmd)£(5)exp(L(dr¹m+pmd¡lr¹m))¡c¡(¹rm¡r¹m)](6),W=¹rmpmc+(l¡d)(pm+r¹m)(l¹rm¡d¹rm¡pmd),L=z¤dud(l¡d),H=dl¡d2lr¹m¡dr¹m¡pmd..[6].,d,z¤du·z¤mu,x(t)·z¤duz¤dux(t)·z¤mu,s¡s+x(t)·z¤duz¤dux(t)·z¤mu,s(t)2f0;1g.[6],fz¤du;z¤mu(x;s),1.,fz¤du;z¤mu(x;s)x(t)=z¤mupz¤du;z¤mu(z¤mu)8:fz¤du;z¤mu(x;1¡)=d2pm¢(z¤du;z¤mu)x2(¡1;z¤du)fz¤du;z¤mu(x;0¡)=dpm(l¡d)¢(z¤du;z¤mu)x2(¡1;z¤du)fz¤du;z¤mu(x;1+)=d2pm¢(z¤du;z¤mu)x2[z¤du;z¤mu)fz¤du;z¤mu(x;0+)=dpm(l¡d)¢(z¤du;z¤mu)x2[z¤du;z¤mu)pz¤du;z¤mu(z¤mu)=d2(l¡d)¢(z¤du;z¤mu)(7)¢(z¤du;z¤mu)=Ã(z¤du;z¤mu)µ(¹rm)µ(r¹m)expf¼(r¹m;z¤du)g,Ã(z¤du;z¤mu)=(d(l¡d)µ(¹rm)(pm+r¹m)expf¡µ(r¹m)(z¤du¡z¤mu)g¡lpm(¹rm¡r¹m))¡1,¼(r¹m;z¤du)=¡µ(r¹m)(z¤du¡z¤mu),µ(rm)=lrm¡drm¡dpmd(l¡d).,x¡1,fz¤du;z¤mu(x;1¡)fz¤du;z¤mu(x;0¡).,z¤du0,,(2)(6),133433J¡=¡c¡[Rz¤du¡1x(fz¤du;z¤mu(x;1¡)+fz¤du;z¤mu(x;0¡))dx+R0z¤dux(fz¤du;z¤mu(x;1+)+fz¤du;z¤mu(x;0+))dx]+c+[Rz¤mu0x(fz¤du;z¤mu(x;1+)+fz¤du;z¤mu(x;0+))dx+z¤mupz¤du;z¤mu(z¤mu)]+cp[Rz¤du¡1fz¤du;z¤mu(x;1¡)dx+Rz¤muz¤dufz¤du;z¤mu(x;1+)dx+pz¤du;z¤mu(z¤mu)]+cr[¹rm£Rz¤du¡1fz¤du;z¤mu(x;0¡)dx+r¹mRz¤muz¤dufz¤du;z¤mu(x;0+)dx](8),z¤du0J+=¡c¡[R0¡1x(fz¤du;z¤mu(x;1¡)+fz¤du;z¤mu(x;0¡))dx]+c+[Rz¤du0x(fz¤du;z¤mu(x;1¡)+fz¤du;z¤mu(x;0¡))dx+Rz¤muz¤dux(fz¤du;z¤mu(x;1+)+fz¤du;z¤mu(x;0+))dx+z¤mupz¤du;z¤mu(z¤mu)]+cp[Rz¤du¡1fz¤du;z¤mu(x;1¡)dx+Rz¤muz¤dufz¤du;z¤mu(x;1+)dx+pz¤du;z¤mu(z¤mu)]+cr[¹rm£Rz¤du¡1fz¤du;z¤mu(x;0¡)dx+r¹mRz¤muz¤dufz¤du;z¤mu(x;0+)dx](9),,@J(zdu;zmu)@zdu¡@J(zdu;zmu)@zmu=0,.¤z¤muz¤du,z¤muz¤du,.2.,z¤duz¤du0,z¤du=d(l¡d)pmd+¹rmd¡l¹rmln·c+(c++c¡)(pmd+r¹md¡lr¹m)£(t(l¹rm¡d¹rm¡pmd)+(l¡d)(¹rm¡r¹m))](10),ttexp(t)=¡(l¡d)(pm+r¹m)