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199877a(,710049),Lagrangian,,LagrangianBatchChemicalProcessesSchedulingwithFuzzyConstraintsWangZhaohuiGanWenquanChenHaoxunHuBaosheng(SystemEngineeringInstitute,Xi’anJiaotongUniversity,710049)AbstractAfuzzyoptimization2basedmethodisdevelopedtosolvebatchchemicalpro2cessesschedulingproblemwithfuzzybufferstorageconstraints.Theproblemisfirstconvertedtoageneraloptimizationproblem.Lagrangianmultipliersarethenusedtode2composetheproblemintoanumberofunit2wisesubproblemsandamembershipsub2problem.Thesesubproblemscanbeefficientlysolved,andthemultipliersareupdatedbyusingasubgradientmethod.Heuristicsarethenusedtoconstructafeasibleschedule.Anexampleisgiventodemonstratetheefficiencyandeffectivenessoftheapproach.Keywordsbatchchemicalprocesses;scheduling;bufferstorage1,,,;,[1]Kondili[2]2(STN),MILPShah[3],4,Lagrangian,[4][5,6][7]Lagrangian,,,,,,()()[8,9],[8],,,,(),,,,a1996122269504008©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,,,,Zadeh1965,[10],,,Lagrangian,,,Lagrangian2STNSTNrecipenetworks[3,11]:;,1STN1(STN)STN:1)iSi:i;Sqii;Qis:isi,sSi;6sSiQis=1;Qlis:isi,sSqi;6sSqiQlis=1;Pis:is,sSqi;Pi:i,PimaxsSqiPis;Ki:i2)sTs:s();Tvs:s();Csmax:s(3)Ij:j;Vmaxij:ji;Vminij:ji367©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.33.1:(P):STN;,H(1,2,3,,H),(,),t(1FtFH)t,1H+1;::Bijttij;Dstts;Rstts();Css(s);Sss;Sstts;Csbuf,s:I;U;Oijttj;EjjEj={(Oijt,Oijt,Pi,iIj};Tjijjii,setup;Cijt01,Bijt01,0P:max{Bijt},{Cijt}J(1)J=--(1a)=6sCsSs,H+1+6Ht=1Dst(1b)=6sCsSs,0+6Ht=1Rst(1c)=6s6Ht=1SsõSst(1d):1):Sst=Ss,t-1+6iTvsQlis6jkiBi,j,t-Pis-6iTsQis6jkiBijt+Rst-DstPs,t(2)2):CsbufFSstFCsmax(3)CijfVminijFBijtFCijtVmaxij(4)(3)3)Cijt:Cijt{0,1}(5)4)setup:6t+Pi+Siij-1S=t(Cijt+Cijt)F16t+Pi+Siij-1S=t(Cijt+Cijt)F1(6)Pi,i,j,t,t,s.t.(Qijt,Oijt)Ej4619987©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.312Y,A:A={(y,LA(y))ûyY}0FLA(y)F1(7)LA(y),yA,(3):SstFCsmax(8)SstECsbuf(9)E:LS(s,t)=1,(Ss+Ds-Csbuf)öDs0,SstECsbufCsbufSstECsbuf-DsSstCsbuf-Ds(10)SstSstECsbuf,Ds0,sCsbuf(1)(2),(46),(89)4.4.13.1J0((10)1),Csbuf-Ds(3)Csbuf,J0+d,d0,(9),J0J0+d(9),(1):JEJ0(11):JEJ0(12):LJ=1,(J+DJ-J0)öDJ,0,JEJ0J0-DJFJJ0JJ0-DJ(13)DJ312,LS(s,t)=1,LJ=0J0,LS(s,t),z=min{LJ,LS(s,t)},Ps,t(14):maxz(15)s.tzFLJ(16)zFLS(s,t),Ps,t(17)0FzF1(18)(2),(4-6),(8)(13)(10),(16)(17)J0+DJõzFJ+DJ(19)Csbuf+DsõzFSst+Ds(20)567©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(19)412Lagrangian(2):Sst=Ss,0+6iTvsrlis6jki6t-PisS=1BijS-6iTsris6jki6tS=1BijS+6tS=1(RsS-DsS)(21)W0(s,t)=Ss,0+6tS=1(RsS-DsS)(22)W(s,t)=6iTvsQlis6jki6t-PisS=1BijS-6iTsQis6jki6tS=1BijS(23)Sst=W0(s,t)+W(s,t)(24)LagrangianA(s,t)B(s,t),(Ps,t)(8)(20),AB,C(19),(15):z=z+6s,tA(s,t)(Csmax-Sst)+6s,tB(s,t)(Sst+Ds-Csbuf-Dsõz)+Cõ(J+DJ-J0-DJõz)(25)311:J=6sCsW(s,H+1)-6s6Ht=1(W0(s,t)+W(s,t))Ss(26)RP:max(z)=maxzZ(z)+max{Bijt},{Cijt}(P0+P({Bijt}))(27)Z(z)=z-6s,tB(s,t)Dsõz-CõDJõzP({Bijt})=6s,t(B(s,t)-A(s,t)-CõWs)W(s,t)+6sCõCsW(s,H+1)P0=6s,t(B(s,t)-A(s,t)-CõSs)W0(s,t)+6s,tA(s,t)Csmax+6s,tB(s,t)(Ds-Csbuf)+C(DJ-J0)PLagrangianDP:maxAE0,BE0,CE0(P0+maxzZ(z)+max{Bijt},{Cijt}P({Bijt})(28)s.t.(4)(5)(6)413h(Bijt)=6s6HS=Pis+t(B(s,S)-A(s,S)-CõSsQis+6sCõCsQis)Bijt+6s6HS=t(A(s,S)-B(s,S)+CõSsQis-6sCõCsQis)Bijt,1FtFH-Pis6s6HS=t(A(s,S)-B(s,S)+CõSsQis-6sCõCsQis)Bijt,H-Pis+1tFH(6sCõCsQlis)Bijt+6s6HS=t(A(s,S)-B(s,S)+CõSsQis-6sCõCsQis)Bijt,t=H-Pis+1(29)6619987©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(27):P0+maxzZ(z)+max{Bijt},{Cijt}6j6iIj6th(Bijt)(30)s.t.(4)(5)(6)(30)P0z,{Bijt},{Cijt},(30)RP{SPj}SZ:SPj:max{Bijt},{Cijt}6iIj6th(Bijt),PjU(31)s.t.(4)(5)(6)SZ:maxzZ(z)=maxzz+6s,tB(s,t)Dsõz-CõDJõz(32)s.t.(18)SPj(PjU)[6]SZ4.4DPLagrangian,[5]4.5,,,,Z,,()1,2,,H,H+1,(1AB),0J,(19),,z:z=(J+DJ-J0)öDJ(33)5,21,4,14,2,31,2,3H=10,AABBCE50kg,Csmax150kg,250kg,200kg,150kg,(,[2])(3)J0=2683,136kg1141kg2(2),3.4à,6.7,,DJ=0.02J0,Ds=0.1Csbuf2714,136kg1144kg2(3),3.1à,7.1,1.2à,71PC486ö66TurboCSTN6,,Lagrangian767©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.2Gantt3Gantt1PeknyJF,MillerDL,KudvaGK.AExactAlgorithmforResourceConstrainedSquencingofAppli2cationtoProductionSchedulingunderanAggregateDeadline.ComputersChem.Engng.1993,17(7):6716822KondiliEPantelidesGC,SargentRWH.AGeneralAlgorithmforShort2termSchedulingofBatchOperations-1.MILPFormulation.ComputersChem.Engng.1993,17(2):2112273ShahNPantelidesCC,SargentRWH.AGeneralAlgorithmforShort2termSchedulingofBatchOp2erations-2.ComputationalIssues.ComputersChem.Engng.1993,17(2):2112274GuanXiaohong,PBLuh,ZhangLan.NonliearApproximationMethodinLagrangianRelaxation2BasedAlgorithmsforHydrothermalScheduling.1994,IEEETrans.onPowerSystems.10(2):7727785Luh,PB,HoitompDJ,MaxE,PattipatiKR.SchedulingGenerationandReconfigurationforParallelMachines.IEEE.Trans.onRoboticsandAutomation,19906(6):687696(144)8619987©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.0012301360132001290137015100134014101520115013401440152011801350149015301,1,,,11,19812,11,1982311,1984(68)6ChenHaoxun,ChuChengbin.Porth,J-M.AMoreEfficientLagrangianRelaxationApproachtoJob2shopSchedulingProblems.ProceedingsofIEEEInt.Conf.onRobo