二级价格控制问题满意解的基于LP与NLP过程的算法

32120021MATHEMATICSINPRACTICEANDTHEORYVol132No11Jan.,2002LPNLP(,529020):SASöORLPNLP,,LPNLP,SAS..:;;LP;;;NLP1:2001207216[5][68],,.[8][13],.SASöORLPNLP,LPNLP,SAS.[8],[8][3],(BLP)21:maxf1(x,y)=cx+dymaxf2(x,y)=xty(BLP)2Ax+ByFbxE0,yE02:maxxf1(x,y)=cx+dy,ymaxyf2(x,y)=xty(BLP)2s.t.,Ax+ByFbxE0,yE0.(BLP)2,x,yRnn,c,dn,A,Bmn,X={(x,y,z)ûAx+By+Iz=b,xE0,yE0,zE0}(BLP)2,z.(BLP)2,x,y(p(x)):maxyf2(x,y)=xtys.t.,ByFb-AxyE0.2LPNLP[8],:X={(x,y)RnRnûAx+ByFb,xE0,yE0},F={(x-,y-)Xûf2(x-,y-)=maxyf2(x-,y)},(BLP)2.(BLP)2,k,:f3k=max(x,y)Xfk(x,y)(k=1,2).(xk3,yk3),f3k=fk(xk3,yk3),(k=1,2).(BLP)2Z=R1R1S.E=RnRn,XE,aifi(x,y)(i=1,2),bifi(x,y)(i=1,2),i{1,2},fi(x0,y0)Fbi,(x0,y0);i=1,2,fi(x-,y-)Eai,(x-,y-)X,(x-,y-)(BLP)2;(x-,y-)F,(x-,y-).LPNLP:1)NLP()[4]X,[8]3,(x23,y23)X.2);2)(x23,y23)FLPP(x23),f2(x23,y23)=maxyX(x32)f2(x23,y).,3),5);3).x23f32=f2(x23,y23),f31=f1(x23,y23),,(x23,y23),(x-,y-)=(x23,y23).4),6).4)(x-,y-),,(x-,y-).5)[8],,,,F,SASöORLP.k(xk,yk)F,(xk,yk),ai(i=1,2),6);6)ai(i=1,2),7),7)ai(i=1,2),(BLP)2NLP[8]:v=min2i=1di,p(4.1)s.t.,fi(x,y)-ai+diE0331:LPNLP(x,y)X,diE0,i=1,2.[8]4,(BLP)2p(4.1)0.P(411)0,,(x,y),8);P(411)0,ai(i=1,2),7),,8).8)LPF.(x-,y-)4);,5),7),,,4).31([8]1)(p1)maxf1=x1+2x2+y1-y2x,y(P2)maxf2=x1y1+x2y2s.t.x1+x2+y1+y2F6x1+y1F3x2-y1-y2F-1x1,x2,y1,y2E0.[8]a1=4,a2=4,32,32,32,32..1)X.NLP,SAS:procnlp;maxz;parmsx1x2y1y2;Bounds0Ax1,0Ax2,0Ay1,0Ay2;z1=x1+x2+y1+y2;z2=x1+y1;z3=-x2+y1+y2;z4=x13y1;z5=x23y2;Run;,:x1=1.500000,x2=1.500000,y1=1.500000,y2=1.500000,z=415.x1+x2+y1+y2=6.2);2)F.NLPSAS:procnlp;maxz6;4332parmsy1y2=-1;Bounds0Ay1,0Ay2;z1=1.5+1.5+y1+y2;z2=1.5+y1;z3=-1.5+y1+y2;z4=1.53y1;z5=1.53y2;Nlincon0Az1A6,0Az2A3,0Az3A6;z6=z4+z5;Run;,,x1=115,x2=115,:y1=115,y2=115.F,(115,115,115,115).3);5);3).,:x1=115,x2=115,,:f1(1.5,1.5,1.5,1.5)=1.5+21.5+1.5-1.5=4.5f2(1.5,1.5,1.5,1.5)=1.51.5+1.51.5=4.5,(x23,y23)=(115,115,115,115),(x-,y-)(x-,y-)=(x23,y23),4),6).4)(x-,y-),,(x-,y-).5)[8],,,,F,SASöORLP.k(xk,yk)F,(xk,yk),ai(i=1,2),6);6)ai(i=1,2),7),7)ai(i=1,2),(BLP)2NLP[8]:minv=2i=1dip(4.1)s.t.,fi(x,y)-ai+diE0(x,y)XxE0,yE0,diE0[8]4,(BLP)2p(411)0.8)LP.(x-,y-)4)..6)f1=415,,a1=418,f2=415,a2=415,7).7):531:LPNLPv=min2i=1dis.t.x1+2x2+y1-y2-4.8+d1E0x1y1+x2y2-4.5+d2E0p(4.1)x1+x2+y1+y2F6x1+y1F3x2-y1-y2F-1x1,x2,y1,y2E0NLPSAS:procnlp;maxv;parmsx1x2y1y2d1d2;Bounds0Ax1,0Ax2,0Ay1,0Ay2,0Ad1,0Ad2;z1=x1+23x2+y1-y2+d1;z2=x13y1;z3=x23y2;z4=z2+z3+d2;z5=x1+x2+y1+y2;z6=x1+y1;z7=-x2+y1+y2;Nlincon4.8Az1,0Az2,0Az3,4.5Az4,0Az5A6,0Az6A3,1Az7A6;v=d1+d2;Run;,v=010099999970,[8]4,a1=418,(BLP)2.,a1=4151,0,.(x-,y-)=(1.5,1.5,1.5,1.5)4).5)(3,1,0,2)f1(3,1,0,2)=3,f2(3,1,0,2)=2,,6).6)a11=315,a12=3p(411),NLPSAS,,v1=0,[8]4,(BLP)2.a21=4,a22=4,a31=415,a32=415p(411),v2=0,v3=0,.(x-,y-),.,,SAS(x-,y-)=(115,115,115,115),8).8)(x-,y-)F,4).633241)1,,,.,,,,,,,.2).:[1],,1[J]1,1996,16(12):3843.[2],1[J]1,1999,(4):141143.[3]BilasM,KarwanH.Two2levellinearprogramming[J].ManagementScience,1984,(30):10041020.[4]1SAS[M]1,1998,71[5]KornalJ,LiptakT.Two2levelplanning[J].Econometrica,1965,(33):141169.[6]FortunyJ,McCarlB.Arepresentationandeconomicinterpretationofatwo2levelprogrammingproblem[J].JOperResSoc,1981,32(9):783792.[7]CandlerW,TownsleyR.Alineartwo2levelprogrammingproblem[J].ComputersandOperRes,1982,(9):5972.[8]1[J]1,2001,(2)1AAlgorithmtoSearchfortheSatisficingSolutionofPrice-ControlProblemWANGQi(DepartmentofMathematics&Physics,Wu2yiUniversityJiangmen,Guangdong529020)Abstract:Inthispaper,amethodofprice2controlbysearchingforsolvingsatisficingsolutionderectlyispresented.Thealgorithm,basedontheLPProcessandtheNLPProcess,unitingtwoimethodsbysearchingderectlyandbysearchingonboundaryispropposed.AnexampleisgiventoillustratetheSASprogrammeofimportantsteps.Atlast,wepresentsomequestions,whichareworththinkingabout.Keywords:price2control;satisfictingsolution;LPProcess;linearcoinstraint;nonlinearconstraint;NLPProcess731:LPNLP