由三角模构造的状态变权向量

:2005-06-09:(1966-),,,,.22220064JOURNALOFSHANGQIUTEACHERSCOLLEGEVol.22No.2April,2005侯海军,王庆东(,476000):讨论一种特殊多维三角模与余三角模以及它们在状态变权构造中的应用,给出基于三角模与余三角模的状态变权向量的构造模式,为状态变权的构造与应用提供了系统的研究工具.:综合决策;变权;状态变权向量;理想三角模;理想余三角模:O159:A:1672-3600(2006)02-0064-04StatevariableweightvectorsconstructionbasedontriangularnormsHOUHaijun,WANGQingdong(DepartmentofMathematics,ShangqiuTeachersCollege,Shangqiu476000,China)Abstract:Aspecialmultidimrndiontriangularnormandcotriangularnormaregiven.Theapplyinstatevariableweightonthetriangularnormandcotriangularnormarediscussed.Constructionalmodelswhicharebasedontriangularnormandcotriangularnormgiven.Theseshouldprovidesystematicstudytools,forconstructionandapplyofstatevariableweightvectors.Keywords:multifactorsynthesisdecisionmaking;variableweightvector;statevariableweightvector;ideatriangularnorms;ideacotriangualrnorms,(ASMmfunc)MmMm(x1,x2,,xm)=mj=1wjxj.wj![0,1]mj=1wj=1,.,.,,,([1]).,[1],.,[2,3,4],,.:1)Hadamard;2)[1,2].,,,[5,6,7,8],,..,.,,.11,0,.1,.[0,1].11mw=(w1,w2,,wm)m,j=1,2,,m;wj![0,1]mj=1wj=1.12[1](m)mwj(j=1,2,,m)wj:[0,1]m∀[0,1],(x1,x2,,xm)∀wj(x1,x2,xm):w1),mj=1wj(x1,x2,,xm)=1;w2),wj(x1,x2,,xm)xi,(i,j=1,2,,m);w3),wj(x1,x2,,xm)xj(j=1,2,,m)W(X)=(w1(X),w2(X),,wm(X))m.注11定义11与文献[2,3,4]的定义区别在于wj#0,1,因此文献[4]中一些结论不再成立.注12定义12中的W(X)称为惩罚型变权向量,文献[2,3]对w3)的不同形式,给出了不同类型的变权向量.若无特别说明,本文均指惩罚型变权向量.注13变权向量在某状态向量下,某权值允许为0,这样的变权是有实际意义的.注14:易证定义15蕴涵于定义14,从而定义15是定义14的特款;反之若常权允许权值取0和1,定义14与定义15等价.本文假设常权的权值不能取0和1,且都是基于定义15下的状态变权.注15:激励型状态变权向量可以相应的定义.注16:文献[2,3,4,5,6,7]给出的状态变权向量大都是满足定义15的状态变权向量.13[2]S:[0,1]m∀(0,1]m,X∀S(X).S(X)(S1(X),S2(X),Sm(X)),Sj:[0,1]m∀(0,1];X∀Sj(X),j=1,2,mSm,S:s1):S(ij(X))=S(X),ij(X)Xij;s2):xi!xj,Si(X)∀Sj(X).s3)w=(w1,w2,,wm),Wj(X)wjSj(X)mk=1wkSk(X),W(X)=(w1(X),w2(X),,wm(X))12w1),w2),w3).[4]:1.3s1),.14[4]S:[0,1]m∀(0,∃)m,X∀S(X),S(X)(S1(X),S2(X),,Sm(X))m,:S1):xi!xj,Si(X)∀Sj(X);S2):Si(X)xi,k#iwkSk(X)xi,w=(w1,w2,,wm).14S1)();S2),,,,,,k#iwkSk(X)xi,Sk(X)(k#i)xi.14.15S:[0,1]m∀(0,∃)m,X∀S(X).S(X)(S1(X),S2(X),,Sm(X))m,:S1):xi!xj,Si(X)∀Sj(X);S2):Si(X)xi,Sk(X)(k#i)xi.X(1)=(x1,,xi-1,xi(1),xi+1,,xm),X(2)=(x1,,xi-1,x(2)i,xi+1,,xm),xi(1)!xi(2),Si(X(1)∀Si(X(2)),Sj(X(1))!Sj(X(2)).2,,[9],.,.21[9]Tmm(),Tm:[0,1]m∀[0,1]:t1)Tm(1,,1,xi,1,,1)=xi;t2)Tm(x1,,xi,xj,xm)=Tm(x1,,xj,xi,xm)t3)X∀Y,Tm(X)∀Tm(Y),:X=(x1,x2xm),Y=(y1,y2,,ym)t4)Tm(Tm(x1,,xm),xm+1,x2m-1)=Tm(x1,,xm-1,Tm(xm,,x2m-1)).652,:22mTm,Tmt5):xi!x%i,Tm(x1,,xi-1,xi,xi,xm)-Tm(x1,,xi-1,x%i,xi,xm)m)∀Tm(x1,xi-1,xi-x%i,xi+1,,xm).23[9]T*mm(),T*m:[0,1]m∀[0,1]:t%1),t2),t3),t4).t%1)T*m(0,,0,xi,0,,0)=xi.24mT*m,T*m:t%1)xi!x%i,T*m(x1,,xi-1,xi,xi+1,xm)-T*m(x1,xi-1,x%i,xi+1,xm)∀1-Tm*(1-x1,,1-xi-1,1-xi+x%i,1-xi+1,,1-xm).25[9]mTmmT*m,TmT*m,:Tm(X)+T*m(1-X)=121[9]mTm,mT*mTm,Tm=1-T*m(IM-X);.22mTm,mT*m,Tm,.Tmm,Tm,21,mT*mTm,T*m(x1,,xm)=1-Tm(1-x1,,1-xm),T*m.xi!x%i,T*m(x1,,xi-1,xi,xi,xm)-T*m(x1,,xi-1,x%i,xi,xm)=Tm(1-x1,,1-xi-1,1-x%i,1-xi,1-xm)-Tm(1-x1,,1-xi-1,1-xi,1-xi,1-xm)∀Tm(1-x1,,1-xi-1,xi-x%i,1-xi,1-xm)=1-T*m(x1,,xi-1,1-xi+x%i,xi,xm)T*mTmm.[9]&∋,(∋%,.)#.231)Tm,T∗m,1)Tm(x1,x2,,xm)∀xi(i=1,2,,m),T*m(x1,x2,,xm)!xi(i=1,2,,m);2)TmT*m,xi!x%ixi-x%i!Tm(x1,,xi-1,xi,xi+1,,xm)-Tm(x1,,xi-1,x%i,xi+1,,xm)xi-x%i!T*m(x1,,xi-1,xi,xi+1,,xm)-Tm*(x1,,xi-1,x%i,xi+1,,xm).,.3,,[2,3,4],[4].+++.31(1)Tm-1m-1,Si(x1,x2,,xm)=Tm-1(x1,,xi-1,xi+1,,xm),S(X)=(S1(X),S2(X),,Sm(X))m.(2)T*m-1m-1,S%i(x1,x2,,xm)=T*m-1(x1,,xi-1,xi+1,,xm),S%(X)=(S%1(X),S%2(X),,S%m(X))m.(1)xi!xj,Si(X)=Tm-1(xi,xi-1,xi+1,,xm)=Tm-1(x1,xi-1,xj,xi+1,,xj-1,xj+1,xm)∀Tm-1(x1,xi-1,xi,xi+1,,xj-1,xj+1,xm)=Sj(X).xi!xi%Si(X)=Tm-1(x1,,xi-1,xi+1,,xm)=Si(X%),Sj(X)=Tm-1(x1,,xi-1,xi,xi+1,,xj-1,xj+1,,xm)!Tm-1(x1,xi-1,x%i,xi+1,,xj-1,xj+1,xm)=Sj(X).Si(X)xi,xj.S(X)..注若置Si(X)=Tm-1(1-x1,,1-xi-1,1-xi+1,1-xm),S%i(X)=T*m-1(1-x1,,1-xi-1,1-xi+1,,1-xm)则S(X),S%(X)是激励型状态变权向量.32Tm-1T*m-1m-1,f(t)[0,1],Si(x)=f(xi-Tm-1(x1,,xi-1,xi+1,,xm))S%i(X)=f(xi-T*m-1(x1,,xi-1,xi+1,,xm)),S(X)=(S1(X),S2(X),Sm(X))S%(X)=(S%1(X),S%2(X),,S%m(X))m.S(X).xi!xj,xi-Tm-1(x1,xi-1,xi+1,,xm)!xj-Tm-1(x1,xi-1,xj,xi+1,,xj-1,xj+1,xm)!xj-Tm-1(x1,xi-1,xi,xi+1,,xj-1,xj+1,xm)f(t),Si(X)=f(xi-Tm-1(x1,,xi-1,xi+1,,xm))∀f(xi-T*m-1(x1,,xi-1,xi+1,xm))=Sj(X).xi-Tm-1(x1,xi-1,xi+1,,xm)xi,xj,Si(X)xi,xj.662006S(X)=(S1(X),S2(X),,Sm(X))m.Tm-1T*m-1m-1,f(t)[0,1]f%(t)0,Si(X)=f(xi-Tm-1(x1,,xi-1,xi+1,,xm))S%i(x)=f(xi-T*m-1(x1,,xi-1,xi+1,,xm)),S(X)=(S1(X),S2(X),,Sm(X))S%(X)=(S%1(X),S%2(X),,S%m(X))m.32f(t)(f%(t)0).33TmT*mm,(1)f(t)[0,1],Si(X)=f(xi-Tm(X)),S%i(X)=f(xi-T*m(X)),S(X)=(S1(X),S2(X),,Sm(X))S%(X)=(S%1(X),S%2(X),,S%m(X))m.2)f(t)[0,1],Si(X)=f(xi-Tm(X)),Sli(X)=f(xi-T*m(X)),S(X)=(S1(X),S2(X),,Sm(X))S%(X)=(S%1(X),S%2(X),,S%m(X))m.2),,xi!xj,xi-Tm(X)!xj-Tm(X),f(t):Si(X)!Sj(X%);xi!x%i,31:xi-x%i!Tm(X)-Tm(X%),xi-Tm(X)!x%i-Tm(X%),f(t)Si(X)!Si(X%);Tm(X)!Tm(X%)xj-Tm(X)∀xj-Tm(X%),Sj(X)∀Sj(X%),S(X).TmT*mm,f(t)(t)[0,1],(t).Si(X)=f((xi)-Tm((X)))S%i(X)=f((xi)-T*m((X))),1)f(t)[0,1],S(X)S%(X)m.2)f(t)[0,1],S(X)S%(X)m.31Sj(X)=&k#jxk,31S(X)=(S1(X),S2(X),,Sm(X)).S%j(X)=min{k#j1-xk),1},31:S%(X)=(S%1(X),S%2(X),,S%m(X)).32f(t)=arccot(x),Sj(X)=arccot(xj-min{k#jxk,1}),32S(X)=(S1(X),,Sm(X)).331)f(t)=et[0,1],Sj(X)=exj-&xk,3.3S(X)=(S1(X),,Sm(X)).2)g(t)=e-t