状态变权向量的性质与构造

*(69974006,60174013):20020322*李德清李洪兴(,100875,37,,);,.,.,.;;;O1590,Mm(ASMmfunc)[13],[1,3].:Mm(x1,,xm)=!mj=1wjxj.,,.,,.X=(x1,,xm),W=(w1,,wm),W∀#X∀#,([4]).,,,,:1)W(X)=(W1(X),,Wm(X))WS(X)Hadamard;2)S(X)([4]).,,.[4],[5].,[68].,.11[4]mWj(j=1,,m).20028384()JournalofBeijingNormalUniversity(NaturalScience)Aug.2002Vol.38No.4Wj:[0,1]m∃[0,1](x1,,xm)∃Wj(x1,,xm)3:W1):!mj=1Wj(x1,,xm)=1;W2):Wj(x1,,xm)(j=1,,m);W3):Wj(x1,,xm)xj.W(X)=(W1(X),,Wm(X)),.W3),W(X).W3)W3%):W3%):Wj(x1,,xm)xj.W(X).[4],[6],1,,:2S:[0,1]m∃[0,1]m,X∃S(X)=(S1(X),,Sm(X)),S(m),:S1)xi&xjSi(X)∋Sj(X);S2)Sj(X)(j=1,,m);S3)W=(w1,,wm),1W1),W2),W3):W(X)=(w1S1(X),,wmSm(X))!mj=1(wjSj(X))=W(S(X)!mj=1(wjSj(X)),(1)S.,S1).,S1):S1%)xi&xjSi(X)&Sj(X),W3)W3%),;S.1[6][4],S1)S2),3:S3%)Si(X)xi,!k)iwkSk(X)xi.S3%)S3),[4]..,[5]1.3:Sj(X)=x1xj-1x-1jxj+1xm(0),j=1,2,,m.(2)23,S3%).,S3%).[6].1S:,X∃S(X)=(S1(X),,Sm(X)),:1):xi&xjSi(X)∋Sj(X);2):Sj(X)(j=1,,m);3):Si(X)xi,!k)iwkSk(X)xi,456()38S(X)m,[6].2S%:[0,1]m∃(0,+∗)m,X∃S%(X)=(S1%(X),,Sm%(X))23,Sj(X)=Sj%(X)!mj=1Sj%(X),j=1,2,,m,S(X)=(S1(X),S2(X),,Sm(X))m..22,,Sj(X)[0,+∗).21S(i):[0,1]m∃[0,1]m,X∃S(i)(X)=(S(i)1(X),S(i)2,,S(i)m(X)),i=1,2,,n1,!ni=1kiS(i)(X)(ki&0)m.1.,1)2),3).S(X)=(S1(X),S2(X),,Sm(X))!ni=1kiS(i)(X)(ki&0).S(i)jxj,Sj(X)=!ni=1kiS(i)j(X)xj.!l)jwlSl(X)=!ni=1ki!l)jwlS(i)l(X),!l)jwlS(i)l(X)xj,!ni=1ki!j)lwlSil(X)xj.3)..2S%(X)=(S1%(X),S2%(X),,Sm%(X)),S+(X)=(S1+(X),S2+(X),,Sm+(X)),1,HadamardS(X)=S%(X)(S+(X)(S1%(X)(S1+(X),S2%(X)(S2+(X),,Sm%(X)(Sm+(X))m.S(X)13.,1)2),3).Sj(x1,x2,,xm)=Sj%(x1,x2,,xm)(Sj+(x1,x2,,xm).Sj%(x1,x2,,xm)Sj+(x1,x2,,xm)xj,Sj(x1,x2,,xm)xj.Si%(x1,x2,,xm)Si+(x1,x2,,xm)xj(i)j),Si(x1,x2,,xm)xj.!i)jwiSi(x1,x2,,xm)xj,3)..3S(X)=(S1(X),S2(X),,Sm(X))1,1)f:[0,1]∃[0,1],t∃f(t),Sf(X)=(f(S1(X)),f(S2(X)),,f(Sm(X)))m;2)f:[0,1]∃[0,1],t∃f(t),Sf(X)=(f(S1(X)),f(S2(X)),,f(Sm(X)))m.,.4:457.3S%(X)=(S1%(X),S2%(X),,Sm%(X)),S+(X)=(S1+(X),S2+(X),,Sm+(X))m,W=(w1wm),S%(X)S+(X)W%(X)=W(S%(X)!mj=1(wjSj%(X)),W+(X)=W(S+(X)!mj=1(wjSj+(X)).W%(X)=W+(X),S%(X)S+(X),S%(X),S+(X).∀,#.,.,.3S%(X)=(S1%(X),S2%(X),,Sm%(X)),S+(X)=(S1+(X),S2+(X),,Sm+(X))m,S1%(X)S1+(X)=S2%(X)S2+(X)==Sm%(X)Sm+(X),S%(X),S+(X).,.1S%(X)=(S1%(X),S2%(X),,Sm%(X))S+(X)=(S1+(X),S2+(X),,Sm+(X))2:Sj%(x1,x2,,xm)=−i)jxi,Sj+(x1,x2,,xm)=x-1j−i)jxi,j=1,2,,m.Sj%(x1,x2,,xm)Sj+(x1,x2,,xm)=1−mi=1x1-ij,3S%(X),S+(X).B(X)[45],(t)B(X),%(t)0,B(X)(B(X)).(B(X))[8].Sj(X)=!B(X)!xj,Sj%(X)=!(B(X))!xj,j=1,2,,m,S(X)=(S1(X),,Sm(X))S%(X)=(S%1(X),,Sm%(X))B(X)(B(X)).Sj(X)Sj%(X)=!B(X)!xj(%(B(X))!B(X)!xj)=1%(B(X)),j=1,2,,m,j,3,S(X)S%(X),B(X)(B(X))..3,.,.,(2).458()382mS(X)=(S1(X),S2(X),,Sm(X)):Sj(x1,x2,,xm)=xj/∀x,j=1,2,,m,∀x=1m!mi=1xi,S(X).,.4f:(-∗,+∗)∃(0,1),X=(x1,x2,,xm),∀x=1m!mi=1xi,(.)Sj(x1,x2,,xm)=f(xj-∀x)(&1/m)(j=1,2,,m)mS(X)=(S1(X),S2(X),,Sm(X));(/)x1=x2==xm,S(X)(1)W=(w1,w2,,wm).(.)S(X)13.,,1)3).1)xi&xj,xi-∀x&xj-∀x,f(t)f(xi-∀x)∋f(xj-∀x),Si(X)∋Sj(X).3)xj-∀x=xj-xjm-1m!k)jxkxj,f(t),Sj(x1,x2,,xm)=f(xj-∀x)xj.!i)jwiSi(x1,x2,,xm)xj.!i)jwiSi(x1,x2,,xm)=!i)jwif(xi-∀x)=!i)jwif(xi-xjm-1m!k)jxk),xi-xjm-1m!k)jxkxj;f(t),xj,!i)jwiSi(x1,x2,,xm)xj,13).(/)x1=x2==xm,Sj(X)=f(xi-∀x)=f(xj-∀x)(j=1,2,,m),j,(1)..f:(-∗,+∗)∃(0,1),t∃f(t),f%(t)∋0.X=(x1,x2,,xm),∀x=1m!mi=1xi,Sj(x1,x2,,xm)=f(xj-∀x)(&1/m)(j=1,2,,m)mS(X)=(S1(X),S2(X),,Sm(X)).3Sj(X)&1,j=1,2,,m,2.3f(t)=1+tp,-1∋0,p,4m:Sj(x1,x2,,xm)=1+(xj-∀x)p,j=1,2,,m.4f(t)=e-t(&0),4mS(X):4:459Sj(x1,x2,,xm)=e-(xj-∀x),j=1,2,,m.5f(t)=1+ln(2+t)(&0),4mSj(x1,x2,,xm)=1+ln(2+xj-∀x),j=1,2,,m.6f(t)=!2-arcsin(t),4mSj(x1,x2,,xm)=!2-arcsin(xj-∀x),j=1,2,,m.=1.,,.X=(x1,x2,,xm),xj(j=1,,m)..5mX=(x1,x2,,xm),Y=(y1,y2,,ym),yj=lnxj(j=1,2,,m);∀yY.f:(-∗,+∗)∃(0,+∗),,:(.)Sj(x1,x2,,xm)=f(yj-∀y)(&1/m)(j=1,2,,m)mS(X)=(S1(X),S2(X),,Sm(X));(/)x1=x2==xm,S(X)(1)W=(w1,w2,,wm).(.)S(X)13.,2),1)3).1)xi&xj,yi&yj,yi-∀y&yj-∀y.f(t)Si(x1,x2,,xm)∋Sj(x1,x2,,xm).3)Sj(x1,x2,,xm)=f(lnxj-1m!mi=1lnxi)=f((-1m)lnxj-1m!i)jlnxi),(-1m)lnxj-1m!i)jlnxixj,f(t),Sj(x1,x2,,xm)xj.i)jSj(x1,x2,,xm)xj,3).(/)(1)..7f(t)=e-t,5:Sj(x1,x2,,xm)=e-(yj-∀y),j=1,2,,m.&0,&1m.,=m,=1m,Sj(X)=−i)jxj,[4].8f(t)=1+2!arctant,5:Sj(X)=1+2!arctan(yj-∀y),j=1,2,,m.4,,Hadamard,;.,.460()38,,,.,.5[1].(0)[J].,1998,13(1):12[2].(1)[J].,1999,14(1):1[3],.[M].:,1996[4].(2)[J].,1995,9(3):1[5].(3)[J].,1996,10(2):12[6],.[J].,1999,19(7):116[7],.[J].,2000,20(1):106[8].[J].,1997,17(4):58[9],YenVC.[J].(),1994,30(1):41[10].(4)[J].(),1996,32(4):470[11].(5)[J].(),1997,33(2):151THEPROPERTIESANDCONSTRUCTIONOFSTATEVARIABLEWEIGHTVECTORSLiDeqingLiHongxing(DepartmentofMathematics,BeijingNormalUniversity,100875,Beijing,China)AbstractSomepropertiesonvariableweightvectorsarediscussedindetail.Anewkindofstatevariableweightvectorscanbeconstructedbymeansofthepropertiesundersomeconditions.Thedefinitionofanequaleffectbetweenstatevariableweightvectorsisgiven.Itispointedoutthattheequaleffectisanequivalentrelation.Amethodforconstructingstatevariableweightvectorsbyusingtheaverageofgivenstatevariableweightvectorsisgivenandanewkindofstatevariableweightvectorsisgained.Keywordsvariableweightvectors;statevariableweightvectors;balancefunctions;decisionandfuzzydecisionmaking4:461