多目标线性规划模型的模糊数学解法

233Vol123No13JOURNALOFQINZHOUUNIVERSITY20086June.,2008[]20080428[]广西高等教育改革/十五0立项项目(2005240,2006072);钦州学院课题(2006XJ09,2007XJ12)1[]王远干(1974-),男,广西钟山人,钦州学院数学与计算机科学系讲师1多目标线性规划模型的模糊数学解法(,535000)[]多目标线性规划是优化问题的一种,是大学生数学建模中经常考核的知识点.它的解要求各子目标函数都要同时达到较优的值,造成模型求解比较困难.文章给出一种基于模糊数学规划的求解方法,通过引进伸缩因子,将各子目标模糊化,从而把多目标规划问题转化为单目标问题,利用MatlabLingo等数学工具软件便可容易求得其模糊最优解.最后用一个实例说明了求解过程,该方法简单易行,适合在建模竞赛时采用.[]数学建模;优化问题;模糊数学;伸缩指标[]G642.4[]A[]16738314(2008)03001404,,[1].,.1992,.[2]....,Ma-tLabLingo.,[3].,[4],,,,.1,,::(1)zi=ci1x1+ci2x2+,+cinxn,.第3期王远干:多目标线性规划模型的模糊数学解法:A=(aij)m@n,B=(cij)r@n,b=(b1,b2,,,bm)T,x=(x1,x2,,xm)TZ=(Z1,Z2,,,Zr)T:maxZ=Cx:Ax[bx\0(3),x*,,.,,,[5].,,[6].2,,,:Ax[bx\0Zi,i=1,2,,,r,Z*i:Z*i=maxZi|Zi=Enj=1cij,Ax[b,x\0i=1,2,,,r(4)(4),.Zi,i=1,2,,,r,di(di0).di,:,,[7].Zi,ŽMi,[8]:ŽMi(x)=gi(Enj=1cijxj)=0,Enj=1cijxjZ*i-di1-1di(Z*i-Enj=1cijxj),Z*i-di[Enj=1cijxjZ*i1,Z*i[Enj=1cijxji=1,2,,,r(5):ŽM=Hri=1ŽMi,ŽM.:D=x|Ax[b,x\0D,.[9].:ŽDf=DHŽM:ŽDf(x*)=maxx\0D(x)CŽM(x)=maxxIDŽM(x)x*.,x*ŽM(x)D[10].,,,:K=ŽM(x)=Hrr=1ŽMi(x):maxZ=K1-1di(Z*i-Enj=1cijxj)\K,i=1,2,,,rEnj=1akjxj[bk,k=1,2,,,mK\0,x1,x1,,,xn\0:maxZ=KEnj=1cijxj-diK\Z*i-di,i=1,2,,,rEnj=1akjxj[bk,k=1,2,,,mK\0,x1,x2,,,xn\0(6)(6),MatLabLingo,(6)(x*1,x*2,,,x*n,K*),x*=(x*1,x*2,,,x*n)[11].Z**=Cx*.3,.,,,.,,,.(6)di,di,,Z*i,[12],:Z-iFZiFZ*i(7)15钦州学院学报第23卷:Z*i=minZi(8),di:di=Z*--Z-i(9).:maxZ1=2x1+5x2+7x3+3x4maxZ2=4x1+2x2+3x3+11x4maxZ3=9x1+3x2+x3+2x4minW1=1.5x1+2x2+0.3x3+3x4minW2=0.5x1+3x2+0.7x3+2x4s.t.3x1+4.5x2+1.5x3+7.5x4=200x1\0,x2\0,x3\0,x4\0(10),:max=Z1=2x1+5x2+7x3+3x4Z2=4x1+2x2+3x3+11x4Z3=9x1+3x2+x3+2x4Z4=-1.5x1-2x2-0.3x3-3x4Z5=-0.5x1-3x2-0.7x3-2x4s.t.3x1+4.5x2+1.5x3+7.5x4=200x1\0,x2\0,x3\0,x4\0(11)(11),Zi(i=1,2,3,4,5),:Z*=(Z*1,Z*2,Z*3,Z*4,Z*5)=(933.33,400.00,600.00,-40.00,-33.33)(11),Zi(i=1,2,3,4,5)Z-i,:Z-=(Z-1,Z-2,Z-3,Z-4,Z-5)=(80.00,88.89,53.33,-100.00,-133.33),(9),di:di=Z*i-Z-i=(853.33,311.11,646.67,60.00,100.00)(12)(6),:maxZ=K2x1+5x2+7x3+3x4-853.33K=804x1+2x2+3x3+11x4-311.11K=88.899x1+3x2+x3+2x4-646.67K=53.33-1.5x1-2x2-0.3x3-3x4-60K=-100-0.5x1-3x2-0.7x3-2x4-100K=-133.333x1+4.5x2+1.5x3+7.5x4=200K\0,x1\0,x2\0,x3\0,x4\0(13)(13),,:(x*1,x*2,x*3,x*4,K*)=(27.61,0.00,78.12,0.00,0.61)(14):Z**=C*(x*1,x*2,x*3,x*4)T=(602.06,344.80,326.61,-74.85,-68.48)(15),(10)Z**0=(602.06,344.80,326.61,74.85,68.48).Z**di,di,Z**.,,Z**,di,,.4,.,.,,.MatLabLingo.,.[][1],.56[J].,2007,22(6):17-25.[2].[J].,2003,19(6):49-51.[3].[J].,2005,35(4):1-5.[4],,.[J].,1999,1:15-21.[5],,.[J].,2001,15(2):85-88.[6],,.[J].,2004,16(1):40-44.[7],,.[J].(),2005,43(3):282-286.[8],,,.[M].:,2005.[9],.[J].(),2007,25(1):74-80.[10].[J].,2001,10(3):13-18.16第3期王远干:多目标线性规划模型的模糊数学解法[11],.[J].(),2004,35(3):514-517.[12],.[J].(),2007,25(5):67-69.OneSolutionofMult-iObjectiveLinearProgrammingModelwithFuzzyMathematicsWANGYuan-gan(Departmentofmathematicsandcomputer,QinzhouUniversity,Qinzhou535000,China)Abstract:Mult-iobjectivelinearprogrammingisoneoftheoptimalproblems,whichhasoftenbeenusedintheChinaUn-dergraduateMathematicalContestinModeling.Itrequiresalltheobjectivefunctionstoachievecomparativelyoptimumvalueatthesametime,soitmakesthesolvingoftheproblemmoredifficultly.Byusingtheflexibleindexesandleteverysub-objectfuzz-ing,asolutionoftheproblembasedonthefuzzymathematicalprogrammingisproposed.Usingthemethod,wecanturnthemult-iobjectiveproblemintoasingleobjectiveproblem,andgettheoptimalsolutioneasilywiththesoftwaretoolssuchas,Ma-tLab,Lingo,etc.Finally,oneexampleisgiventoilluminatehowtogetthefuzzyoptimalsolution,whichshowsthatthemethodisverysimpleandsuitableforusinginmodelingcontest.Keywords:Mathematicalmodeling;Optimizationproblem;Fuzzymathematics;Flexibleindexes[](上接第13页)图9[][1].(())[M].,2007.DiscussiononofInfiniteApproximationUsingGeometricSketchpadLIANGChang-dong,TANDa-yao(Mathematics&computerdepartmentofQinzhouUniversity,Qinzhou535000,China)Abstract:UsingtheiterationofGeometricSketchpadtodemonstratethedynamicprocessofinfiniteapproximationaboutmathematics,Buildingbridgesfromtheimageofthinkingtoabstractthinking,Topromotetheimageofthinkingintoabstractthinking.Keywords:infiniteapproximation;Dynamicdemo;Iteration[]17