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200033(,650093):,L,,,L,L:;;;:F224.0aTheOrdinaryVariableWeightPrincipleandMultiobjectiveDecision2MakingLIUWen2qi(SchoolofScience,KunmingUniversityofScienceandTechnology,Kunming650093)Abstract:Havingintroducedthetentativeideaofcompromisevariableweightindocu2ment[5],wefirstlypresentthegeneralizeddefinitionofcompromisevariableweightinthispaper,beingfollowedbydiscussingrelationsamongvariousvariableweights.Itisoneofimportantpointofourworktostudythenatureofvariableweightinducedbythefamilyofimpellentutilityfunctionforrevealingtherelationbetweenvariableweightanalysisandutilitytheory.Asaresultofthisstudy,sometypesofimpellentutilityfunctionfamilyfittingtovariableweightanalysisisgiveninthispaper.Finally,thecompatibilityprincipleofvariableweightisstatedforstructuringsomeparticularlyop2eratablevariableweighttorestricttheparameterinapplication,andthemethodofvari2ableweightformultiobjectivedecision2makingisgivenasanexample.Keywords:variableweight;familyofimpellentutilityfunction;compatibilityprinci2ple,multiobjectivedecision2making180[1]L[2][2],Weber2Fechner[3]L[4]L[5],,[5]L[6]L[7,8]L,LLa:1998212223:(98A006Q)©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.Im={(x1,x2,,xm)Tû0xj1,j=1,2,,m}Rm,I+mIm,0=(0,0,,0)T,e=(1,1,,1)T,x=(x1,x2,,xm)T,p=(p1,p2,,pm)Tm;w(x)=(w1(x),w2(x),,wm(x))TmZabab,Z111pIm,w(x)C1(I+m).w(x):1)w(x)02)eTw(x)=13)Pj{1,2,,m},5wj5xj0xjpj0xjpj4)Pj{1,2,,m},6mk=15wk5xjxk+wj0w(x)p,;pw(x);Vp(x)=xTw(x)Zw(x)1)4)5)limxj0+wj(x)=1,j=1,2,,mw(x),(s,w)Zw(x)1)4)6)limxj1-wj(x)=1,j=1,2,,mw(x),(w,s)Zw(x)p=e,w(x),imp;w(x)p=0,w(x),pen,p{0,1}m,w(x),mixZ[4]w(x):wj(x1,x2,,xm)=w(0)jxA-1j6mk=1w(0)kxA-1k(1)(0A1)imp(s,w),,55xl[xTw(x)]=6mj=15wj5xlxj+wl=55xl6mj=1w(0)jxAj6mk=1w(0)kxA-1k=Aw(0)lxA-1l6mk=1w(0)kxA-1k-6mj=1w(0)jxAjõw(0)l(A-1)xA-2l6mk=1w(0)kxA-1k2=Aw(0)lxA-1l6mj=1w(0)jxA-1j+(1-A)w(0)lxA-2l6mj=1w(0)jxAj6mk=1w(0)kxA-1k204)Z,[4]p=exj0+limxj0+wj=limxj0+w(0)jxA-1j6kjw(0)kxA-1k+w(0)jxA-1j=1w(x)(s,w)Z111w(x)p,w{(x)=w(e-x)e-p;220003©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.w(x)(s,w),w{(x)(w,s)Zw{(x)=(w{1(x1,x2,,xm),w{2(x1,x2,,xm),,w{m(x1,x2,,xm))T=(w{1(1-x1,1-x2,,1-xm),w{2(1-x1,1-x2,,1-xm),w{m(1-x1,1-x2,,1-xm))T.,w{(x)1)2)Zw(x)p,5wj5uj0ujpj0ujpjuj=1-xj,5w{j5xj=6mk=15wj5uk5uk5xj=-5wj5uj0ujpj0ujpj5w{5xj0xj1-pj0xj1-pj3),Vpq(x)=xTw{(x)=xTw(e-x)=1-[1-xTw(e-x)]=1-[eTw(e-x)-xTw(e-x)]=1-(e-x)Tw(e-x)=1-Vp(e-x)(pq=e-p),5Vpq5xj=6mk=15w{k5xj+w{j(x)=-5Vp5ujõ(-1)=5Vp5uj=6mk=15wk5uj+wj(u)04)Zw{(x)e-pZw(x)(s,w),limxj0+wj(x)=1.limxj1-w{j(x)=lim1-xj0+wj(e-x)=limuj0+wj(u)=1,w{(x)(w,s)Z1V2(s,w)Vpw{(x)=w(e-x)w(x)Zw(x)=w(x)111,w{,Z33©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,1.14),Vp(x),5Vp5xj0(j=1,2,,m),4)Z3(w,s)Vp4Vp22w(0)=(w(0)1,w(0)2,,w(0)m)T,6mj=1w(0)j=1w(0)j0(j=1,2,,m);u(x)=(u1(x1),u2(x2),,um(xm))TC2(I+m)(uj(t)C2(0,1)),u(x)=(u1(x1),u2(x2),,um(xm))T.211{uj(t)}mj=1C2[0,1],u(x)=(u1(x1),u2(x2),,u(xm))T,w(0),pIm,1)u(x)02)uj(t)0tpj0tpj(j=1,2,,m)3)j{1,2,,m}xI+m,6mk=1w(0)kuk(xk)[uj(xj)(xj-xk)+uj(xj)]0wj(x)=w(0)juj(xj)6mk=1w(0)kuk(xk)j=1,2,,m(2)w(x)=(w1(x),w2(x),,wm(x))TpZw(x)1)2)Z5wj5xj=w(0)juj(xj)6mk=1w(0)kuk(xk)-[w(0)j]2uj(xj)uj(xj)6mk=1w(0)kuk(xk)2=w(0)j6kjw(0)kuk(xk)6w(0)kuk(xk)2uj(xj)0xjpj0xjpj3)Z,P1jm,5Vp5xj=55xj(xTw(x))=55xj6mk=1w(0)kuk(xk)6mk=1w(0)kuk(xk)xk420003©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.=[w(0)juj(xj)xj+w(0)juj(xj)]6mk=1w(0)kuk(xk)-6mk=1w(0)kuk(xk)xkw(0)juj(xj)6mk=1w(0)kuk(xk)2=w(0)juj(xj)6mk=1w(0)kuk(xk)(xj-xk)-w(0)juj(xj)6mk=1w(0)kuk(xk)6mk=1w(0)ku(xk)2=w(0)j6mk=1w(0)kuk(xk)[uj(xj)(xj-xk)+uj(xj)]6mk=1w(0)kuk(xk)206mk=15wk5xjxk+wj04)Z.{uj(t)}mj=1211,(2)w(x){uj(t)}mj=1,{uj(t)}mj=1w(x)2,Z{uj(t)}mj=1:w(0)j();uj(t)(),Z,()()Z{uj(t)}mj=10uj(t)1(t(0,1),j=1,2,,m),{uj(t)}mj=1Z,{uj(t)}mj=1uj(t)=16t3-12pjt2+4t,j=1,2,,m(3){uj(t)}mj=1Z212{uj(t)}mj=1w(x),{M-uj(1-t)}mj=1w{(x)Z(M)Zvj(t)=M-uj(1-t)(j=1,2,,m).vj(t)(j=1,2,,m).vj(t)=[M-uj(1-t)]=-uj(1-t)(-1)=uj(1-t)0,v(x)0.vj(t)=-uj(1-t),vj(t)01-tpj01-tpjvj(t)0t1-pj0t1-pj,6mk=1w(0)kvk(xk)[vj(xj)(xj-xk)+vj(x)]=6mk=1w(0)kuk(1-xk)[-uj(1-xj)(xj-xk)+uj(1-xj)]=6mk=1w(0)kuk(1-xk)[uj(1-xj)((1-xj)-(1-xk))+uj(1-xj)]0w(0)q=e-p,{vj(t)}mj=1211Z53©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.w3j(x)=w(0)jvj(xj)6mk=1w(0)kvk(xk)=w(0)juj(1-xj)6mk=1w(0)kuk(1-xk)=wj(e-x),j=1,2,,mw3(x)=(w31(x),w32(x),,w3m(x))T=(w1(e-x),w2(e-x),,wm(e-x))T=w{(x)(w(x){uj(t)}mj=1)Z.,,5(a),(b),(c)Z5(a)(pj=0)5(b)(pj=0)5(c)(pj(0,1))5w(x)w{(x)6Z6w(x)w{(x)213w(x){uj(t)},w(x)(s,w)Pj{1,2,,m}limxj0+uj(xj)=+.w(x){uj(t)}mj=1,w(x)wj(x)=w(0)juj(xj)6mk=1w(0)kuk(xk)j=1,2,,mlimxj0+uj(xj)=+,limxj0+wj(xj)=limxj0+w(0)juj(xj)6kjw(0)kuk(xk)+w(0)juj(xj)=limxj0+w(0)juj(xj)c+w(0)juj(xj)=1,j=1,2,,mw(x)(s,w)Z,j{1,2,,m}limxj0+wj=1,E0,D0,0xjD,w(0)juj(xj)6mk=1w(0)kuj(xk)-1=w(0)juj(xj)c+w(0)juj(xj)-1E1-Ew(0)juj(xj)c+w(0)juj(xj)1+E620003©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.uj(xj)1E1w(0)jc(1-E)+,E0limxj0+uj(xj)=+..211w(x){uj(t)}mj=1,w(x)(w,s)1jmlimxj1-wj(x)=+,5wj5xk0xkpk0xkpk214uj(t)=1A(A+1)tA+1-pjA(A+1)tA+cjt,j=