由某些函数构造的状态变权

:1001-7402(2005)04-0119-06侯海军1,2,谷云东1,3,王加银1(1.,100875;2.,476000;3.,100875):研究状态变权的构造问题,给出一种由多元函数和已知状态变权构造新状态变权和三种由多元函数直接构造状态变权的方法。特别地,证明了文[5]由状态均值构造状态变权的方法都可看作是函数构造状态变权的特例,并进一步给出一种基于几何均值的状态变权构造方法。:变权综合;状态变权构造;多元函数:O159:A1　引言,[1][2][3],,,[4-10,12-14]:;,,,,[5],1.1[4]S:[0,1]m(0,)m,XS(X)@(S1(X),S2(X),,Sm(X))m,:(SI):xixj,Si(X)Sj(X);(SII):Si(X)xi,kiwkSk(X)xi,W=(w1,w2,,wm)1.1,194200512FuzzySystemsandMathematicsVol.19,No.4Dec.,2005:2004-07-14:(60174023);(20020027013);(03184);973(2002CB312200):(1966-),,,,:;(1976-),,,,,:,,,;(1974-),,,,,:1.1[6]S:[0,1]m(0,)m,XS(X)@(S1(X),S2(X),,Sm(X))m,:(SI):xixj,Si(X)Sj(X);(SII):Si(X)xi,Sk(X)(ki)xi1.2[11]X=(x1,x2,,xm),Y=(y1,y2,,ym)[0,1]mXY,xiyi(i=1,2,,m)1.3m(X)[0,1]m[0,1]mm,i,j{1,2,,m}m(x1,,xi,,xj,,xm)=m(x1,,xj,,xi,,xm);m(x),(XY)(m(X)m(Y));m,(XY)(m(X)m(Y))1.4[11]Tm:[0,1]m[0,1]m(),:(t.1)Tm(1,,1,xi,1,,1)=xi;(t.2)Tm(x1,,xi,,xj,,xm)=Tm(x1,,xj,,xi,,xm);(t.3)XYTm(X)Tm(Y),X=(x1,x2,,xm),Y=(y1,y2,,ym);(t.4)Tm(Tm(x1,,xm),xm+1,,x2m-1)=Tm(x1,,xm-1,Tm(xm,,x2m-1))1.5[11]Tm:[0,1]m[0,1]m(),(t.1)(t.2)(t.3)(t.4)(t.1):Tm(0,,0,xi,0,,0)=xi.mTmT*mm1.6[11]mTmT*m,TmT*m,Tm(X)+T*m(Im-X)=1Im-X=(1-x1,1-x2,,1-xm)1.2[11]mTm,mT*m;2　一种由多元函数与已知状态变权构造新状态变权的方法2.1S(t)(X)=(S(t)1(X),S(t)2(X),,S(t)m(X))(t{1,2,,n})m(),nn,Sj(X)=n(S(1)j(X),S(2)j(X),,S(n)j(X)),S(X)()S(t)(X)(t{1,2,,n})),xixj,S(t)i(X)S(t)j(X)(t{1,2,,n}),n:Si(X)=n(S(1)i(X),S(2)i(X),,S(n)i(X))n(S(1)j(X),S(2)j(X),,S(n)j(X))=Sj(X)1.1(SI)S(t)i(X)xi,Si(X)=n(S(1)i(X),S(2)i(X),,S(n)i(X))xi,S(t)i(X)xj(ji),Si(X)=n(S(1)i(X),S(2)i(X),,S(n)i(X))xj(ji)1.1(SII)S(X)2.1TmT*mm,S(t)(X)=(S(t)1(X),S(t)2(X),,S(t)m(X))(t=1,2,,n)(),Sj(X)=Tn(S(1)j(X),,S(n)j(X))Sj(X)=T*n(S(1)j(X),,S(n)j(X))S(X)S(X)2.1n,()2.12.2S(k)(X)(k{1,2,,n}),S(X)S(k)(X)(k{1,2,,n})[5,6],[6]1202005+,S(X)S(k)(X)(k{1,2,,n});,S(X)S(k)(X)(k{1,2,,n})2.1S(X)=(x-11,x-12,,x-1m)S(X)=(1-x1,1-x2,,1-xm),2(t1,t2)=et1sint2,SJ(x)=ex-1jsin(1-xj)2.1S(X)2.2T2(t1,t2)=max(t1+t2-1,0),S(X)=(1-x1,,1-xm),S(X)=(x-11,,x-1m)Sj(X)=T2(Sj(X),Sj(X))=max(x-1j-xj,0)=x-1j-xj.2.1S(X)T*2(t1,t2)=min(t1+t2,1),S(x1,x2)=(x2-x1,x1-x2),S(x1,x2)=(x-11-x1,x-12-x2),2.1S(x1,x2)=(min(x2-2x1+x-11,1),min(x1-2x2+x-12,1))3　利用函数直接构造状态变权3.1m-1m3.1m-1(),Si(X)=m-1(x1,,xi-1,xi+1,,xm)(i=1,2,,m),S(X)()xixj,m-1Si(X)=m-1(x1,,xi-1,xi+1,,xm)=m-1(x1,,xi-1,xj,xi+1,,xj-1,xj+1,,xm)m-1(x1,,xi-1,xi,xi+1,,xj-1,xj+1,,xm)=Sj(X)1.1(SI)Si(X)=m-1(x1,,xi-1,xi+1,,xm)xim-1,Si(X)xi,xj(ji)1.1(SII)S(X)3.1m-1(),Si(X)=m-1(1-x1,,1-xi-1,1-xi+1,,1-xm),S(X)()3.2Tm-1T*m-1m-1,Si(x1,x2,,xm)=Tm-1(x1,,xi-1,xi+1,,xm),Sj(x1,x2,,xm)=T*m-1(xi,,xi-1,xi+1,,xm)S(X)S(X)m3.3Tm-1T*m-1m-1,Si(X)=Tm-1(1-x1,,1-xi-1,1-xi+1,,1-xm)Si(X)=T*m-1(1-x1,,1-xi-1,1-xi+1,,1-xm),S(X)S(X)3.1m-1(t1,,tm-1)=1(m-1)2m-1i,j=1titj,m-1Sj(X)=m-1(x1,,xj-1,xj+1,,xm)=1(m-1)2k,ijxkxi,3.1S(X)3.2Tm-1=,T*m-1=Sj(X)=kjxk,Sj(X)=kj(1-xk)=1-kjxk,3.23.3S(X),S(x)3.2m-1m3.2f(t)(-,+)(0,+)(),m-1m-1,Si(X)=f(xi-m-1(x1,,xi-1,xi+1,,xm)),S(X)m()xixj,m-1xi-m-1(x1,,xi-1,xi+1,,xm)xj-m-1(x1,,xi-1,xj,xi+1,,xj-1,xj+1,,xm)xj-m-1(x1,,xi-1,xi,xi+1,,xj-1,xj+1,,xm)f(t),Si(X)=f(xi-m-1(x1,,xi-1,xi+1,,xm))f(xj-m-1(x1,,xj-1,xj+1,1214,:,xm))=Sj(X)1.1(SI)xi-m-1(x1,,xi-1,xi+1,,xm)xi,xj(ji)Si(X)xi,xj(ji)1.1(SII)S(X)m3.4f(t)(-,+)(0,+),f(t)(f(t)0),m-1m-1,Si(X)=f(xi-m-1(x1,,xi-1,xi+1,,xm)),S(X)m()3.5Tm-1T*m-1m-1,f(t)(-,+)(0,+)(),Si(X)=f(xi-Tm-1(x1,,xi-1,xi+1,,xm)),S(X)=f(xi-T*m-1(x1,,xi-1,xi+1,,xm)),S(X)S(X)m()3.1m-1,3.23.4,1(t)=t-1,f(t)=t-2,S(X)=((x-x-12)-2,(x2-x-11)-2,Si(X)(i=1,2)xk(k=1,2),S(X)3.3f(t)=arcsint[-1,1],m-1(t1,,tm-1)=1m-1m-1k=1tk,Si(X)=arcsin(xi-1m-1kixk)3.2S(X)3.4Tm-1(t1,,tm-1)=max(m-1k=1tk-m+2,0),T*m-1(t1,,tm-1)=min(m-1k=1tk,1)[11],Sj(X)=emax(kjxk-m+2,0)-xj,Sj(X)=arccot(xj-min{kjxk,1})3.5S(X)S(X)3.3mm3.3m:xixi,m(x1,,xi,,xm)-m(x1,,xi,,xm)xi-xi,f(t)(-,+)(0,+)(),Si(X)=f(xi-m(X)),S(X)()f(t)xixj,xi-m(X)xj-m(X),f(t):Si(X)Sj(X)1.1(SI)xixi,xi-m(X)xi-m(X)f(t),Si(X)Si(X)Si(X)xixj-m(X)xj-m(X),Sj(X)Sj(X),Sj(X)xi(ij)1.1(SII)S(X)3.6m:xixi,m(x1,,xi,,xm)-m(x1,,xi,,xm)xi-xi,f(t)(-,+)(0,+),f(t)0(0),Si(X)=f(xi-m(X)),S(X)()3.7Tmxixi,Tm(x1,,xi,,xm)-Tm(x1,,xi,,xm)Tm(x1,,xi-xi,,xm),f(t)(),Si(X)=f(xi-Tm(X)),S(X)()Tm(x1,x2,,xm)Tm(1,,1,xi,1,,xm)=xi,xixiTm(x1,,xi,,xm)-Tm(x1,,xi,,xm)Tm(x1,,xi-xi,,xm)xi-xi3.3S(X)()3.8T*mxixi,T*m(x1,,xi,,xm)-T*m(x1,,xi,,xm)1-T*m(1-x1,,1-xi+xi,,1-xm)f(t)(),Si(X)=f(xi-T*m(X)),S(X)()12220053.2,3.3f(x)=x2,2(x1,x2)=x-11+x-12,2,S(X)=((x1-(x-11+x-12))2,(x2-(x-11+x-12))2)3.5f(t)=e-t,m(X)=sin(1mmk=1xk)xixim(x1,,xi,,xm)-m(x1,,xi,,xm)xi-xi.Si(X)=esin(1mmk=1xk)-xi,3.3S(X)3.6f(t)=arctant,Tm(X)=mk=1xm.xixi,Tm(X)-Tm(X)xi-xi.Sj(X)=arctan(xj-mk=1xk)3.7S(X)3.3:3.4f(t)g(t)(-,+)(0,+)(),mxixi(m(x1,,xi,,,xm)-m(x1,,xi,,xm)xi-xi)Si(X)=f(g(xi)-m(g(X))),S(X)()g(X)=(g(x1),,g(xm))3.7f(t)=e-t,g(xj)=xj(0),Tm(X)=mk=1xk,Sj(X)=emk=1xk-xj.3.4S(X)3.3:3.5[5]f(t)(-,+)(0,+)(),X=(x1,x2,,xm),x=1mmk=1xk,Si(X)=f(xi-x)(1m),S(X)(),,4　利用几何均值和单调函数直接构造状态变权4.1f(t)(-,+)(0,+)(),X=(x1,x2,,xm),xi0,!=mk=1xk1m,Si(X)=fxi!1m(i=1,2,,m),S(X)=(S1(X),S2(X),,Sm(X))()4.1,,4.1f(t)=e-t,Sj(X)=e-xj(mk=1xk)-1m,4.1S(X)4.2f(t)=lnt,Sj(X)=lnx1mimk=1xk-1m=-1mkjlnxk,4.1S(X)5　小结,,[15][16][17][18][19]1234,::[1].[M].:,1985.[2].()[J].,1995,9