由某些函数构造的状态变权

c I | :1001-7402(2005)04-0119-06 t f / M null侯海军1,2,谷云东1,3,王加银1(1. = S v , 100875;2. m = S “ , 2 m 476000;3. = S v 5 , 100875)K 1 :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。1 o M :变权综合;状态变权构造;多元函数 m s | :O159 D S M :A1 引言 8 , 1a2a34 M X ,i M M l # 1 M b M Z E % 1 o / M ,7 M M 8 r T “ 1 Y by N , M / M 8 1= B 4- 10,12- 14b “ - D M / Z E 1 V s :B X M / M ; B / M b B / Z E “ - X , = / Z E ,M 1 T b B ) M / 5 , f / M Z E , / M 5 ( / M Z E V A T f / M + b) M / Z L ,/ M 1 Q | b l 1.14 S:0,1m (0,)m,X S( X)(S1(X),S2(X),l,S m(X) B m M , T H q :(SI) : xi xj,5 S i(X) S j(X);(SII) M :Si(X) 1 x i h , i jF kiw kSk(X) 1 xi 9 F b , W = (w1,w2,l,w m) b 7 l1.1 P i Z L , L = T / N H q b 19 4 2005 M 12 “ d Fuzzy Systems andMathematicsVol.19, No.4Dec.,2005nullnull l :2004-07-14 “ :S E 1 S “ (60174023); p V “ (20020027013); S / “ (03184);973S E v $ 9 “ (2002CB312200)T e : Z + (1966-), 3 , 2 m , q , Z _ : ;! (1976-), 3 , ,p V , =S v 5 , Z _ : e ? , ! , * , M V U ; F (1974-), 3 , + ,p V , = S v = , Z _ : “ d ? e b 1. 16 S:0,1m (0,)m,X S(X)(S1(X),S2(X),l,S m(X) B m M , O H q :(SI) : xi xj,5 S i(X) S j(X);(SII) M :S i(X) 1 xi h , i Sk(X)(k i) 1 xi 9 F b M M N H q b l 1.211 ! X = (x1,x2,l,x m), Y = (y1,y2,l,ym) 0,1mb X Y, xi yi(i = 1,2,l,m)b l 1.3 ! nullm(X) 0,1m 0,1 f b nullm m f , null i,j 1,2,l,m nullm(x1,l, xi,l,xj,l,xm) = nullm(x1,l,xj,l,xi,l,xm); nullm(x) 9 f , (X Y)null (nullm( X) nullm(Y); nullm h f , (X Y)null (nullm(X) nullm(Y)b l 1.411 T m:0,1m 0,1 m (e ), T / H q :(t.1) Tm(1,l,1,xi,1,l,1) = xi;(t.2) Tm(x1,l,xi,l,xj,l,xm) = Tm(x 1,l,xj,l,xi,l,xm);(t.3) X Ynull T m(X) T m(Y), X = (x1,x2,l,xm),Y = (y1,y2,l,ym);(t.4) Tm(T m(x1,l,x m),xm+ 1,l,x2m- 1) = T m(x1,l,xm- 1,Tm(x m,l,x2m- 1)b l 1.511 Tm: 0,1m 0,1 m (e ), (t.1)a(t.2)a( t.3)a(t.4)b (t.1):Tm(0,l,0,xi,0,l,0) = xi.A m T m T *m m O 9 f b l 1.611 m T m T*m , T m T *m M 1 , T m(X) + T*m (Im -X) = 1b Im - X = (1- x1,1- x2,l,1- xm)b 1.211 B m T m,i B m T *m M 1 ;Q g b2 一种由多元函数与已知状态变权构造新状态变权的方法 2.1 ! S(t)( X) = (S(t)1 (X),S(t)2 (X),l,S(t)m (X)(t1,2,l,n) B F m ( ) M ,nulln n 9 f , S j(X) = nulln(S(1)j (X),S(2)j (X),l, S(n)j (X),5 S(X) ( ) M b S(t)(X)(t 1,2,l,n) M , i xixj,S(t)i (X) S(t)j (X)( t 1,2,l,n), nulln 9 f :S i(X) = nulln(S(1)i (X),S(2)i (X),l, S(n)i (X) nulln(S(1)j (X),S(2)j (X),l,S(n)j (X) = S j(X)b 1.1 (SI) b y S(t)i (X) 1 x ih , Si(X) = nulln(S(1)i (X),S(2)i (X),l,S(n)i (X) 1 xih ,y S(t)i (X) 1 xj(j i) 9 F , S i(X) = nulln(S(1)i (X),S(2)i (X),l,S(n)i (X) 1 xj(j i) 9 F b 1.1 (SII) b# S(X) M b8 bw 2.1 ! T m T *m m , S(t)(X) = (S(t)1 (X),S(t)2 (X),l,S(t)m (X)(t= 1,2,l,n) B F ( ) M , S j(X) = T n(S(1)j (X),l,S(n)j (X)Sj(X) = T *n ( S(1)j (X),l,S(n)j (X)5 S(X) S(X) M b 2.1 nulln h f ,5 V 1 ( ) M 2.1M b 2.2 S(k)(X)(k 1,2,l,n) M r M , 1 / M S(X) M S(k)(X)(k1,2,l,n) r s 5,6b Y L , 6 120 “ d 2005 M | 0 opaop o+ p,5 S(X) S(k)(X)(k1,2,l,n) r ; | 0 op,5 S(X) S(k)(X)(k 1,2,l,n) A r b 2. 1 | S(X) = (x- 11 ,x- 12 ,l,x- 1m )S(X) = (1- x1,1- x2,l,1- xm),null2(t1,t2) =et1sint2, SJ(x) = ex- 1j sin(1- xj)b 2.1 S(X) M b 2. 2 | T 2(t1,t2) = max(t1 + t2 - 1,0),S(X) = (1- x 1,l,1- xm),S(X) = (x - 11 ,l,x- 1m )b S j(X) = T 2(Sj(X),Sj(X) = max(x- 1j - xj,0) = x- 1j - xj. w 2.1 V S(X) M b | T *2 (t1,t2) = min(t1 + t2,1),S(x1,x2) = (x2 - x1,x1 - x2),S(x1,x 2) =(x- 11 - x 1,x- 12 - x2), w 2.1S(x1,x2) = (min(x2 - 2x 1+ x- 11 ,1),min(x1 - 2x2 + x- 12 ,1) M b3 利用函数直接构造状态变权3. 1 m - 1 f / m M 3. 1 ! nullm- 1 9 (h ) f , S i(X) = nullm- 1(x1,l,xi- 1,xi+ 1,l,xm)( i= 1,2,l,m),5 S(X) ( ) M b i xi xj, nullm- 1 # V Si(X) = nullm- 1(x1,l,xi- 1,xi+ 1,l,xm) = nullm- 1(x 1,l,xi- 1, xj,xi+ 1,l,xj - 1,xj + 1,l,xm) nullm- 1(x1,l,xi- 1,xi,xi+ 1,l,xj- 1,xj + 1,l,xm) = S j(X)b 1.1 (SI) b S i(X) = nullm- 1(x1,l,xi- 1,xi+ 1,l,xm) x i 1 Onullm- 19 , S i(X) 1 xi h , 1 xj(j i) 9 b 1.1 H q (SII) b# S(X) M b8 bw 3.1 ! nullm- 1 9 (h ) f , S i(X) = nullm- 1(1- x1,l,1- xi- 1,1- xi+ 1,l,1- xm), 5 S(X) ( ) M bw 3.2 T m- 1 T*m- 1 s Y m - 1 , Si(x1,x 2,l,xm) =T m- 1(x1,l,xi- 1,xi+ 1,l,xm), Sj(x1,x2,l,xm) = T *m- 1(xi,l,xi- 1,xi+ 1,l,xm)b5 S(X)aS(X) m M bw 3.3 ! T m- 1aT *m- 1s Y m- 1 , Si(X) = Tm- 1(1- x1,l,1- xi- 1,1- xi+ 1,l,1- x m)aSi(X) = T *m- 1(1- x1,l,1- xi- 1,1- xi+ 1,l,1- xm), 5S(X)aS(X) M b 3.1 | f nullm- 1(t1,l,tm- 1) = 1(m - 1)2 m- 1i,j= 1titj, A nullm- 1 9 f b Sj(X) =nullm- 1(x1,l,xj - 1,xj + 1,l,xm) = 1(m- 1)2 k,ijxkxi, 3.1 S(X) M b 3.2 | T m- 1 = ,T *m- 1= Sj(X) = kjxk, Sj(X) = kj(1- xk) = 1- kjxk, w 3.2 w 3.3 S(X) M , S(x) M b3. 2 m - 1 B f / m M 3.2 ! f (t) (- , + ) (0, + ) h (9 ) f ,nullm- 1 m - 1 9f , Si(X) = f (xi - nullm- 1(x1,l,xi- 1,xi+ 1,l, xm), 5 S(X) m ( ) Mb xi xj, nullm- 1 xi - nullm- 1(x1,l,xi- 1,xi+ 1,l,xm) xj - nullm- 1(x1,l,xi- 1, xj,xi+ 1,l,xj - 1,x j+ 1,l,x m) xj - nullm- 1(x1,l,x i- 1,xi,xi+ 1,l,xj - 1,xj + 1,l,xm)b f (t) h , S i(X) = f (xi - nullm- 1(x1,l,xi- 1,xi+ 1,l,xm) f (xj - nullm- 1(x1,l,xj- 1,xj + 1,121 4 Z + ,! : t f / M l,xm)= Sj( X)b 1.1 H q (SI) b xi- nullm- 1(x1,l,xi- 1,xi+ 1,l,xm) 1 xi9 ,1 xj(j i) 1 b# S i(X) 1 x ih , 1 xj(j i) 9 b 1.1 H q (SII) by N S(X) m M b8 bw 3.4 ! f (t) (- , + ) (0, + ) V f , O f (t) (f (t) 0), nullm- 1 m - 1 9 f , Si(X) = f (x i- nullm- 1(x1,l,x i- 1,xi+ 1,l,x m), 5 S(X) m ( ) M bw 3.5 ! T m- 1 T *m- 1m- 1 ,f (t) (- , + ) (0, + )h (9 ) f , Si(X) = f (x i - Tm- 1(x 1,l,xi- 1,x i+ 1,l,xm),S(X) = f (xi - T*m- 1(x 1,l,xi- 1,xi+ 1,l,xm),5 S(X) S(X) m ( ) M b 3. 1 nullm- 1 h f , 5 3.2# w 3.4 b , | null1(t) = t- 1 h f , f (t) = t- 2h f , S(X) = (x - x- 12 )- 2,(x2 - x- 11 )- 2, A S i(X)(i =1,2) 1 xk(k = 1,2) ( 9 , S(X) i M b 3.3 | f (t) = arcsint- 1,1 9 f , nullm- 1(t1,l,tm- 1) = 1m - 1 m- 1k= 1tk 9 f , Si( X) = arcsin(x i - 1m - 1 kixk)b 3.2 S(X) M _ b 3.4 | Tm- 1(t1,l,tm- 1) = max( m- 1k= 1tk - m+ 2,0),T *m- 1(t1,l,tm- 1) = min( m- 1k= 1tk,1) 11, Sj(X) = emax( kj xk- m+ 2,0)- xj, Sj(X) = arccot(xj - min kjxk,1)b w 3.5 S(X) S(X) M b3. 3 m B f / m M 3.3 ! nullm 9 f O : i xixi,nullm(x1,l,xi,l, xm)- nullm(x1,l,xi,l,xm) xi - xi,f (t) (- , + ) (0, + ) h (9 ) f , Si(X) = f (x i- nullm(X),5 S(X) ( ) M b f (t) h f b i xixj, xi- nullm(X) x j - nullm( X), f (t) h :Si(X) Sj(X)b 1.1 (SI) b xi xi, y xi- nullm(X) xi- nullm(X) O f (t) h , Si(X) S i(X)b Si(X) 1 xi h by xj - nullm(X) xj - nullm(X),V 7 Sj(X) Sj(X), Sj(X) 1 x i(i j) 9 b 1.1 (SII) b# S(X) M b8 bw 3.6 ! nullm 9 f O : i xixi,nullm(x1,l,xi,l, xm)- nullm(x1,l,xi,l,xm) xi - xi,f (t) (- , + ) (0, + ) V f , O f (t) 0(0), S i(X) = f (xi- nullm(X), 5 S(X) ( ) M bw 3.7 ! T m O i xixi,T m(x1,l, xi,l,xm) - T m(x1,l,xi,l,xm) T m(x1, l,xi - xi,l,xm),f (t) h (9 ) f , Si(X) = f (xi - T m(X),5 S(X) ( ) M b y T m(x1, x2,l,xm) Tm(1,l,1,xi,1,l,xm) = xi, i xi xi Tm(x 1,l,xi,l,xm) - T m(x1,l,xi,l,xm) Tm(x1,l,xi - xi,l, xm) xi - xi 3.3S(X) ( ) M bw 3.8 ! T *m O i xi xi,T *m( x1,l,xi,l,xm) - T *m (x1,l,xi,l,xm) 1- T*m (1- x1,l,1- xi+ xi,l,1- xm)bf (t) h (9 ) f , Si(X) = f (xi -T *m(X),5 S(X) ( ) M b122 “ d 2005 M 3.2 h f ,i 3.3 b | f (x) = x2,null2(x1,x2) = x- 11 +x- 12 ,A null2 h f , S(X) = (x1 - (x- 11 + x- 12 )2,(x2 - ( x- 11 + x- 12 )2) i Mb 3. 5 | f (t) = e- t,nullm(X) = sin( 1m mk= 1x k) P i xi xi nullm(x1,l,xi,l,xm)- nullm(x 1,l,xi,l, xm) x i- x i. Si(X) = esin( 1m mk= 1xk)- x i, 5 3.3S(X) M b 3. 6 | f (t) = arctant,Tm(X) = mk= 1xm. xi xi H , Tm(X) - T m(X) xi -xi. S j(X) = arctan(xj - mk= 1xk)b w 3.7S(X) M b 3.3 V : 3.4 ! f (t)ag(t) (- , + ) (0, + ) h (9 ) f ,nullm 9 f O i xi xi (nullm(x1,l,xi,l,xm) - nullm(x 1,l,xi,l,x m) xi - xi)b Si(X) = f (g(xi) -nullm(g(X),5 S(X) ( ) M b g(X) = (g(x1),l,g(xm)b 3.7 | f (t) = e- t,g(xj) = xnullj(0null),Tm(X) = mk= 1xk, Sj(X) = emk= 1x nullk- x nullj. 3.4S(X) M b 3.3 V / : 3.55 ! f f (t) (- , + ) (0, + ) h (9 ) f ,X = (x1,x2,l,xm), x = 1m mk= 1xk,Si(X) = f ( x i - x)( 1m),5 S(X) ( ) M b Y L ( / M V A T f / M B + y f b B ,T f / M 6 B + , + ( / M ZE b4 利用几何均值和单调函数直接构造状态变权 4.1 ! f f (t) (- , + ) (0, + ) h (9 ) f , X = (x1,x 2,l,xm) _ , ! y xi0, != mk= 1xk1m,Si(X) = f xi! 1m (i = 1,2,l,m), 5 S(X) = (S1(X),S2(X),l,Sm(X) ( ) M _ b 4.1 B + ( f / M Z E , V B t z L = M , E F y % ! 5 1 b 4. 1 ! f (t) = e- t h ,S j(X) = e- xj(mk= 1xk)- 1m, 5 4.1 S(X) M _ b 4.2f (t)= lnt9 ,Sj(X)= ln x 1mi mk= 1xk- 1m = - 1m kj lnxk , 4.1 S( X) M _ b5 小结 ) M / , f / M Z E , M / 4 B Z E b V I 15a 16a 17a 18a 19 B M123 4 Z + ,! : t f / M “ d y % a M ? C # M 5 b I D :1 . “ “ M. : = S v ,1985.2 . y bW M V U O ( )J. “ d ,1995,9(3): 1j9.3 . y bW M V U O ( )J. “ d ,1996,10(2): 12j19.4 , . M 8 “ ( f / J. “ d L l,1999,19(7):116j118.5 b, . M _ / J. = S v (1 S ),2002,38(4): 455j461.6 Z + ,! , F . M t / Z E J. = S v (1 S ),2005,41(4):334j338.7 b,! , .1 M _ l T J.“ d L l,2004,24(5):97j102.8 . ( f # M 8 J. “ d L l,1997,17(4):58j64.9a ,! , , . 1 M _ 8 “ : J. = S v ,2004,40(1):1j7.10 Z ,! .1 ( M : J. = S v (1 S ),2002,38(6):739j743.11 .y bW M V U O ()J. “ d ,1996,12(4):30j38.12 , . “ d 9 M. : S ,1996.13 , . M. : S ,1994.14 , . M V U M. ? :? S / ,1994.15 , Yen V C. y bW % J. = S v (1 S ),1994,30(1):41j46.16 . y bW M V U O ()J. = S v (1 S ),1996,32(4): 470j475.17 . y bW M V U O ()J. = S v (1 S ),1997,33(2): 151j157.18 . y bW M V U O ()J. “ d ,1998,13(1):