多目标规划的LINGO求解法-吴有平

1.4120072.410004Multi-ObjectiveProgrammingwithLINGOWuYouping1,2LiuJie1HeJie11.SchoolofCivilEngineeringHunanUniversityofTechnologyZhuzhouHunan412007China2.HunanConstructionEngineeringGroupCorporationChangsha410004ChinaAbstractAsthesolutiononmulti-objectiveprogrammingisexcessivelycumbersomeandtheresultisunsatisfac-torybythemethodofMakingItFeweratpresent,proposesasolutionwhichdivertingtheidealobjectiveintotherealisticobjectiveorconstraints,andthensolvingitbyLINGOsoftware.Andgivestwoexamplesofanalysisandsolvingprocess.Itshowsthatthemethodissimpleandoptimalcomparingtoconventionalmethod.Keywordsmulti-objectiveprogrammingLING[e-mail]ityofTechnologyVol.26No.3May201226320125doi:10.3969/j.issn.16739833.2012.03.002multipleobjectivesprogramming19V.Pareto[1]J.V.Neumann[2]22T.L.Saaty[3]2070LINGOlinearinteractiveandgeneraloptimizerLINDOLINGO102012LINGO9.0x=(x1,x2,,xn)r1s2at,1x1+at,2x2++at,nxn*bt3x1,x2,,xn041~3r=0,1,s=0,1,t=0,1,*=4[4]b1x=(x1,x2,,xn)2maxf(x)minf(x)g(x)=bg(x)bg(x)b3LINGO16000Demodemonstration1504LINGO1maxf(x)minf(x)g(x)=b,g(x)bg(x)bx1,x2,,xn013M1M2M33D1D2D31500LM16.002100LM24.50950LM33.00D110%M250%M16.00D260%M310%M25.50D350%M350%M25.002000LD1[5]1D1,D2,D3xijMiDj23M1D1066D20.55.56.0D315.06.0M2M31.5,1,0.53,2.5,2max0.5x12x13+1.5x21+x22+0.5x23+3x31+2.5x32+2x33x210.1x11+x21+x31x110.5x11+x21+x31x320.6x12+x22+x32x220.1x12+x22+x32x330.5x13+x23+x33x230.5x13+x23+x3332000LD1x11+x21+x31=2000311LINGOx11+x12+x131500x21+x22+x232100x31+x32+x33950xij04max0.5x12x13+1.5x21+x22+0.5x23+3x31+2.5x32+2x33x210.1x11+x21+x31x110.5x11+x21+x31x320.6x12+x22+x32x220.1x12+x22+x32x330.5x13+x23+x33x230.5x13+x23+x33x11+x21+x31=2000x11+x12+x131500x21+x22+x232100x31+x32+x33950xij05LINGOLINGOmax=-0.5*x12-1*x13+1.5*x21+1*x22+0.5*x23+3*x31+2.5*x32+2*x33;0.1*x11-0.9*x21+0.1*x310;0.5*x11-0.5*x21-0.5*x310;0.6*x12+0.6*x22-0.4*x320;-0.1*x12+0.9*x22-0.1*x320;0.5*x13+0.5*x23-0.5*x330;-0.5*x13+0.5*x23-0.5*x330;x11+x21+x31=2000;x11+x12+x13=1500;x21+x22+x23=2100;x31+x32+x33=950.LINGOGlobaloptimalsolutionfound.Objectivevalue:4975.000Totalsolveriterations1VariableValueReducedCostx111000.0000.000000x120.0000000.5000000x130.0000001.500000x21200.00000.000000x221900.0000.000000x230.0000000.000000x31800.00000.000000x32150.00000.000000x330.0000001.0000001000LM1,200LM2800LM3D11900LM2150LM3D2D34975[6]4708.3D12000L[6]LINDO2A10B8A3hB2h120h1230[5]1A,Bx1Ax2Ax3Bx4B2max10x1+9x2+8x3+7x120hA,B3020h8003x1+2x31203x2+2x42010x1+9x2+8x3+7x48003230x1+x2=30x3+x4=30x1,x2,x3,x404max10x1+9x2+8x3+7x43x1+2x31203x2+2x42012201210x1+9x2+8x3+7x4800x1+x2=30x3+x4=30x1,x2,x3,x405LINGOLINGOmax=10*x1+9*x2+8*x3+7*x4;3*x1+2*x3=120;3*x2+2*x4=20;10*x1+9*x2+8*x3+7*x4=800;x1+x2=30;x3+x4=30.220h800LINGOmax=10*x1+9*x2+8*x3+7*x4;3*x1+2*x3=120;x1+x2=30;x3+x4=30.LINGOGlobaloptimalsolutionfound.Objectivevalue:530.0000Totalsolveriterations:1VariableValueReducedCostx120.000000.000000x210.000000.000000x330.000000.000000x40.0000000.333333320A30B10A53010A30800[7]LINDOLINGO2[67]n1LINGO[1]ParetoV.ManualofPoliticalEconomy[M].NewJerseyScholarsBookShelf19719298.[2]NeumannJVMorgensternO.TheoryofGamesandEconomicBehavior[M].LondonPrincetonUniversityPress1953271272.[3]SaatyTL.TheAnalyticHierarchyProcess[M].NewYorkMcGraw-HillInternational1980241243.[4]IgnizioJP.LinearProgramminginSingle-Multiple-ObjectiveSystems[M].NewJerseyPrentice-Hall198237.[5].[M].1999242244.LuKaicheng.Single&MultipleObjectivesandIntegerProgramming[M].BeijingTsinghuaUniversityPress1999242244.[6].LINDO[J].200125(2)15.HongWen.SolveObjectProgrammingwithLINDO[J].JournalofAnhuiUniversityNaturalScienceEdition200125(2)15.[7].LINDO[J].200421(2)108110.LuoGanghuiYeYanmei.SolveMulti-ObjectiveGoalProgrammingWithLINDO[M].ComputerApplicationsandSoftware200421(2)108110.