yNU JOURNAL QF SCIENCE, Nat Sa ■& Tech , T XIX, Npl 2003 AN E V O L Ư T Ĩ O N A R Y A P P R O A C H T O FUZZY R E L A T I O N E Q U A T I O N S WITH C O N S T R A I N T S D in h M a n h T u o n g F n i ' u ì t y n f T c c h TìCìlogy, V N l ĩ Á b str a ct Fưzzy rclation cquations plaỵ an ìmỊTortant rolc in arcas stjch CIS fu z z v sy ste m a n a ly sis, dcsÌỊỊn o f fu z z y co ntrolỉers, a n d f u z z V p a tte r n rcco g n itio n hì th is paper, ICC dcfinc thc fu z z y rclation cquation ivith constraints a n d proposc an c v o lu tio n a r y a lg o r ith m fo r d e tc rm in in g a n a p p r o x im a tc s o lu tio n of th is cq u a tỉo n I n tr o d u c t i o n T he n o tio n of fu zzy r e la t io n e q u a tio n w a s íìr st stu died by S a n c h e z (1 97 6) S in c e th en , m a n y íu r th e r s t u d i e s h a v e been n e by o th e r r e s e a r c h e r s (se c [õ, , 7, ]) Fuzzv r e la tio n e q u a t io n s play an im p ortan t role in a r e a s su c h a s fu zzy s y s te m a n a ly s is , d e s ig n o f fu zzy c o n tr o lle r s , d ecision m a k in g p r o c e s s e s , a n d fu zzv p a tte r n reco g n itio n T he n otio n o f fu zz y r e la tio n e q u a tio n s is a s s o c ia t e d w ith th e c o n c e p t o f c.omposition o f fuzzy r e la tio n s L et A be a fuzzy s e t in t h e in p u t s p a c e a n d R be a fuzzy r ela tion in th e in p u t - o u t p u t product s p a c e U xV T h e c o m p o s itio n o f fuzzy se t A and fuzzv r e la tio n R, d e n o t e d by AoR, is d e fin e d a s a fu zz y s e t B in th e o u u t sp a c e V AoR = B, (1) HH(y) = max|iA( x ) * n R( x ,y ) , x«-u (2 ) w h o s e m e m b er sh ip fu n c tio n is w h e r e * is th e t-norm o p era to r B e c a u s e th e t-n o rm c a n t a k e a v a r ie t y o f ĩo r m u la s, for each t-norm w e o b ta in a p a r tic u la r c c m p o sitio n T h e tw o m o s t c o m m o n ly usecỉ c o m p o sitio n s in n u m e r o u s a p p lic a tio n s are th e so -c a lle d m a x -m in c o m p o s itio n and m a x - p r o d u c t comỊ)osition, w h ic h a r e de fined as follows: T h e m a x -m in c o m p o s itio n Ị.iB ( y ) = m a x m i n | n A ( \ ) ^ K ( x y ) l XkV T h e m a x -p ro d u ct c o m p o s itio n D in h M a n h Tuong 56 Htì (y) = max X €Ư ( x )n R(* y) • T he e q u a t io n A o R = B is a so -ca lled fuzzy r e la t io n s e q u a t io n I f we v iew R as a fu zzy s y s t e m , t h e n g iv e n a fuzzy s e t A to a fuzzy system R, we can com pute the systerrTs o u u t B by (2) T h e tw o b asis problem s c o n c e r n in g th e fu z z y rela tio n e q u a tio n are as follow s: Problem Pl: Given the input fuzzy set A in u and the output fuzzy set B in V, determine the fuzzy relation R such th a t A o R = B Problem P2: Given the fuzzy relation R and the output B, determine the in p u t A su c h t h a t A o R = B T h ereío re, s o lv in g t h e fu zz y rela tio n e q u a tio n A o R = B m e a n s s o lv in g th e a b o v e tvvo p r o b le m s In t h is p a p er we are only in t e r e s t e d in t h e p rob lem P l Sin ce th e s o lu tio n s for t h e p r o b le m P l m a y not e x ist, w e íìr s t n e e d to c h eck t h e so lv a b ility o f t h e s e e q u a tio n s or t h e e x is t e n c e o f th eir so lu tio n s T h e o re m 1.1 Problem P l has solutions if and only if the height of the fuzzy set A is greater th an or equal to the height of the fuzzy set B, th a t is supnA(x) ^ ^ B(x) for all y eB T h e proof o f t h is t h e o r e m c a n s e e in [2] In order to s o lv e p ro b le m P l , one introduces the cp-operator The (p-operator is a n op erator cp: [0,1] X [0,1] -> [0,1] d e íìn e d by acpb = sup {: € | 0,1 ||a • c £ b vvhere * d e n o te s t-n o r m o p e r a to r If th e t-n orm o p e r a to r is sp ecified as m in im u m , the (p-operator becomes the so -c a lle d a-o p era tor: if if For fu zzy s e t s A in u a < b a > b a n d B in V, u s in g t h e cp-operator w e c a n d efin e th e fu zzy rela tion R* in U x V w h ic h is defin ed as H r (x*y) = ^ A( x ) W B( y) We d e n o te t h is fu z z y r e la tio n by A(pB T he íbllovving t h e o r e m is d e m o n str a te d (se e [2,3]) An Evolutionary Approach To Fuzzy Rclation Equations 57 T h e o re n i 1.2 If the solution of problem P l exists, then the largest R (in the sense of fuzzy set theoretic inclusion) th a t satisĩies the fuzzy relation equation AoR = B is R* = AọB Hovvever, in m a n y c a s e s , an ex a ct s o lu tio n o f problem P l m ay not e x ist T h e re ío r e, R* = A(pB m a y n o t bc so lu tio n If an e x a c t s o lu tio n d o e s not e x ist, w h a t w e can is to d e te r m in e a p p r o x im a te s o lu tio n s W a n g L X proposed th e m e th o d o f d e te r m in in g a p p r o x im a te s o lu tio n th rou gh n e u r a l netvvork tr a in in g (se e [ ]) T h e íu r th e r d e ta ils o f fu zzy rela tio n e q u a t io n s c a n be found in [2, 3] F u z z y r e la t io n e q u a t io n s w ith c o n s t r a in t s The a p p r o x im a te g e n e r a liz e d M odus r e a s o n in g P o n e n s T h is in fuzzy system s in fer e n c e r u le is states b a sed on th a t g iv e n th e r u le of tvvo fu zzy p ro p o sitio n s “if X is A t h e n y is B ” and “x is A ”’ w e s h o u ld infer a n e w p r o p o sitio n “y is B m such t h a t t h e c lo se r th e A' to A, th e c lo se r th e B* to B, vvhere A an d A ’ are fuzzy s e t s in sp a c e u , B a n d B ’ a re fuzzy s e ts in sp a c e V T h e fu zzy p r o p o sitio n “if X is A th e n y is B” is in t e r p r e te d as a fuzzy r e la tio n R in U x V T he fu zzy s e t B ’ in th e c o n c lu s io n o f g e n e r a liz e d M o d u s P o n en s rule is d e t e r m in e d as B ’ = A ’ o R In th e lite r a tu r e , m a n y d iffe r e n t in te r p r e ta tio n s of fu zzy if-th e n ru les are proposed, for e x a m p le , L u k a s ie w ic z im p lic a tio n , Zadeh im p lic a tio n , M a m d a n i im p lic a tio n , etc W e w ish d e te r m in e th e fuzz rela tio n R in te r p r e tin g fu zzy p r o p o sitio n “if X is A th e n y is B" such t h a t th e c lo se r th e A ’ to A, th e c lo se r th e B ’ = A ’ o K to B T h e notion o f fu zzy r e la tio n eq u a tio n vvith c o n s t r a in t s is s t a t e d as follow s G iv e n th e fuzzy s e t s A a n d A, (i = 1, ,k ) in sp a c e a n d th e fuzzy s e t B in s p a c e V, w e s h o u ld d e te r m in e a fuzzy r e la tio n R* in p rod u ct U x V such t h a t t h e fo llo w in g r e q u ir e m e n ts are sa tisfie d : A o R* = B (3) If w e d e n o te A, o R* = B, (i = , , k) th e n d(Bif B) = a d(A„ A), (4) w h e r e a is c o n s ta n t, a > , an d d(.,.) is th e d is ta n c e b e t w e e n fuzzy s e t s T he d is ta n c e d(C, D) b e tw e e n th e fu zz y s e t c and th e fuzzy s e t D 19 d e ĩin e d as follow s d(C, D) = (f ||IC(x) - ^1 D(y)|r dx ) p , pỉl For p = o n e h a s th e H a m m in g d is ta n c e an d p = y ie ld s th e E u c lid e a n d ista n c e In th e c a s e s th e sp a c e u is fin ite w e c a n s im p ly d e íìn e d ( C , D ) = i V c ( x ) - n D( x) | X t u 58 D in h Ma n h TIiong H ence, our problem is to determ ine th e fuzzy relation R* w h ich s a tis íìe s (3) and (4), gi v en th e fu zzy s e t s A, A, (i = 1, ,k) in space u and th e fuzzy s e t in space V An e v o l u t i o n a r y a p p r o a c h to fuzzy r e l a t i o n e q a t i o n s vvith c o n s t r a i n t s It is very d ifficu lt to d e te r m in e th e ex a ct s o lu tio n o f fu zz y rela tio n e q u a tio n s w ith c o n s tr a in ts In t h is section , we propose an e v o lu tio n a r y sch em e for d e te r m in in g t h e a p p r o x im a te solu tio n of fuzzy r e la tio n e q u a t io n vvith c o n s tr a in ts by u s in g a n e v o lu tio n s tr a te g y E volution s tr a te g ie s are a lg o r ith m s which im ita te th e p r in c ip les o f n a t u r a l e v o lu tio n as m ethod to solve p a r a m e te r o p tim iz a tio n problem s (s e e [1, 4, 9]) W e r e fo r m u la te our problem in form o f an o p tim iz a tio n problem Ciiven th e fu zzy s e t s A a n d A, (i = 1, k) in u a n d th e fu zzy s e t B in V A s s u m e th a t R is a fu zz y r e la tio n in UxV D en ote A o R = B\ A,o R = B/ (i = , k) For ea ch fu z z y r e la tio n R, w e defin e a rea] v a lu e f(R) a s follow s r(R) = d(B'.B) + £ | d ( B , \ B ) - a d ( A , A ) | =1 O u r p ro b lem n ow is to d e te r m in e th e fuzzv r e la tio n R such th a t f(R) is m in im u m To a p p ly th e e v o lu tio n str a teg y to th e above p rob lem , w e íìrst n e e d to h a v e s u it a b le r e p r e s e n t a t io n s for fu zzy s e ts and fuzzy r e la tio n s A s s u m e th a t th e sp a c e s u and V c o n s is t C)f fin ite n u m b e r of e le m e n ts , u = {Uj, ea ch fu zzy s e t A in u u rn}, V = {Vj, is r e p re sen ted as a v ecto r A = ( a lf v n} T h en , a m), vvhere a, is m e m b e r sh ip d e g r e e o f u, to th e fu zzy s e t A, th a t is a, = nA(a >)* A n alo g ica lly , th e fuzzy set B in V has the representation B = (bj bn) Each fuzzy relation R is r e p r e s e n te d a s a m a trix of order mxn R = (r,j), w h e r e r„ = hk( uì ,v ,) Ưnder th e s e a s s u m p tiơ n s , w h e n g iv e n th e fuzzy sets A, A, (i = 1, k), B a n d th e fuzzy re], R w e c a n e a s ily comp^lte th e fuzzy se ts B’ = A o R an d B / = A, o R (i = on k), w h e r e w e c a n e m p lo y th e m ax- c o m p o sitio n or th e m a x -p ro d u ct com position H ence, w e c a n c o m p u te th e v a lu e o f cbjective íu n ctio n f(K) T h e id e a C)f e v o lu tio n s tr a te g y for our problem is a s fo llo w s Each individual is r e p r e s e n te d a s a pair (R, Z)» vvhere R = (r,,) is a m a tr ix o f order m xn w ith rtJ e [0,1] (i = 1, m ; j = 1, n ), y = ( n tJ) is a ( m x n ) - m a t r i x o f S t a n d a r d d e v i a t i o n s a,j E ach p o p u la tio n c o n s is t s of N in d iv id u a ls , all in d iv id u a ls in th e population h a v e th e s a m e m a t in g p rob a b ilities In each ite r a tiv e s te p , tvvo random ly se le c te d parents: An Kvolutionary Approach To Fuzzy Rclation Equcitions 59 a nđ produce a n offsprin g (R ,E ) = ((r„ ) ( o „ ) ) , w h e r e r,, = r 1tJ ar rtJ = r i, w ith e q u a l probability a n d if ru = rk,j th e n tỊ = a k,j (k = , ) T h e m u ta tio n o p e r a to r is períbrm ed on th e o ffsp rin g (R, £ ) vvhich as g e n e r a te d by th e a b o v e c r o sso v e r operator A p p lyin g th e m u ta tio n to th e o ffsp r in g (H, £)» w e o b ta in th e n e w o ffsp rin g (R \ X): R' = (r’ti), r’„ = r(J + N(0, CT„ ), (i = , m; j = , n), vvhere N (0, n l() is a n o r m a lly d istrib u ted random v a lu e w ith e x p e c t a t io n zero and S ta n d ard deviation C7,r W e now r e p r e s e n t th e s c h e m e o f e vo lu tion a ry a lgo rith m for d e t e r m in in g th e a p p r o x im a te s o lu tio n o f fu zzy r e la tio n e q u a tio n w ith c o n s tr a in ts Algorithm G e n e r a te a p o p u la tio n o f N in d iv id u a ls (R, Z), w h e r e R = (rtJ) is a m a trix of o rder m xn , each r,j is r a n d o m ly ta k e n from th e in ter v a l [0 , ], ỵ = ( a lf) is a m xn - m atrix o f S ta n d a r d d e v ia tio n s (Iterative step) Randomly s e le c t tw o p a r e n ts from N in d iv id u a ls ( R , Z |) = ((r1,,), ( a 1,,)) and (Hi I 2) = ((r2.,) (ơ2,,)) T h e s e p a r e n ts produce an offsprin g (R, I ) = ((r,,), (ơ,,)), w h e r e r,j = r l,j or rtJ = r2(J w ith eq u al probability and if r,} = r 1,, th e n cTif a CTlij if rtỊ = r2tJ t h e n Gtj = 2,, Applying the mutation to the oíĩspring (R, Z), we obtain the new oíĩspring (R \ Z) R’ = (r\ị), r ’„ = rẽ, + N(0, Ơ(J), (i = m; j = , n), 60 Dinh M anh T uoìiịị w h e r e N ( t o tJ) is a n o r m a lly d istr ib u te d random v a lu e w ith e x p e c ta tio n z e r o and S t a n d a r d dev iati o n ơ,, If all r ’,j s ta y w ith i n t h e i n t e r v a l [0, 1], t h e nevv i n d i v i d u a l ( R \ Z) is add ed to th e pop u latio n E lim in a te th e vveakest in d iv id u a l from N+l in d ivicỉuals (o r ig in a l N in d iv id u a ls p lu s o ne offspring) C o n clu sio n W e h a v e d efin ed th e notion of fuzzy rela tio n e q u a tio n vvith c o n s tr a in ts , and prop osed th e e v o lu tio n a r y a lg o r ith m for d e te r m in in g an a p p r o x im a te s o lu tio n of t h is e q u a tio n T h is e v o lu tio n a r y algorith m can be a p p lied to d e te r m in e the a p p r o x im a te s o lu tio n of t h e problem P l in c a se an e x a c t so lu tio n o f problem P l does n o t e x ist R eferences B ack T., H o ffm e ister F, an d S h w efe l H F A s u r v e y of E v o lu tio n S t r a t e g ie s , in P r o c e e d ỉn g s o f the fo u r th I n te r n a tio n a l C onference on G e n e tic A l g o r i t h m , M organ K a n g m a n n , C an M atco, 1991 Chin- Teng Lin, c s George Lee, N e u r a l F u z z y S y s t e m s Prentice- Hall, Inc., 1996 Li- Xin W ang, A c o u rse in F u z z y S y s te m s a n d C o n tro l P ren tice- H a ll, Inc., 1997 M ic h a le w ic z z , G e n e tic A l g o r i th m s + D a ta S tr u c tu r e s = E v o lu tio n P r o g r a m s , S p rin g er , 1996 P ed rycz w , F uzzy r e la tio n a l eq u a tio n s w ith g e n e r a liz e d c o n n e c tiv e s a n d th eir a p p lic a tio n s Fuzzy s e ts and S y s t e m s , 10(1983), 185-201 P e d ry cz w , s- t F u zzy r e la tio n a l e q u a tio n s F u z z y s e ts a n d S y s t e m s , (1 9 ), 189- 196 S a n c h e z E, R e so lu tio n o f c o m p o site fuzz r ela tio n e q u a tio n s I n fo r m a tio n a n d C o n tr o l, (1 ), 38- 49 s S a n c h e z E, S o lu tio n o f fuzz>' e q u a tio n s w ith e x te n d e d op erato rs F u z zy S e is a n d S y s t e m s , 2(19 83 ), 237- 248 S c h w e fe l H p, E v o lu tio n S tr a te g ie s: A F am ily o f Non- L in ea r O P tim iz a tio n T e c h n iq u e s B ased on I m ita tin g S o m e Prin cip les of O rganic E v o lu tio n A n n a l s o f O p e r a tio n s R e s e a rc h Vol 1(1984), 165- 167 10 W a n g L D, S o lv in g fu zzy r ela tio n a l e q u a tio n s th rou gh netvvork tr a in in g Proc nti I E E E Inter Conf on F u z z y S y s te m s S a n Francisco, 1993, 956- 960 An Evolutionciìy Approach To Fuzzy Rclation Equations TAP CHI K H O A HOC D H Q G H N KHTN & CN t XIX NọỊ, 2003 \1ỘT GIẢI P H Á P T I Ê N HOẢ C H O P H Ư Ơ N G T R Ì N H Q U A N H Ệ MỜ VỚI CÁC RÀNG BU Ộ C D in h M ạnh T ường K h o a C ô n g nghệ, Đ H Q G H N ộ i Khái n iệm ph ng tr ìn h q u a n hệ lần đầu tiê n đê xuất n g h iê n cứu bcii S a n c h e z (xem [7]) P h ng trình qu an hệ mò đóng vai trò quan tr ọ n g tr o n g n h iều lĩn h vực, c h a n g h n p h â n tích hệ mờ, t h iế t kê hệ đ iể u k h iển mờ, n h ậ n d n g m ẫ u mờ Trong báo n y c h ú n g xác định khái n iệ m phương trìn h q u a n hệ mò vỏi rà n g buộc đê x u ấ t m ột th u ậ t to n tiế n hố để tìm n g h iệ m xấp xỉ phương trìn h ... Problem Pl: Given the input fuzzy set A in u and the output fuzzy set B in V, determine the fuzzy relation R such th a t A o R = B Problem P2: Given the fuzzy relation R and the output B, determine... An Evolutionary Approach To Fuzzy Rclation Equations 57 T h e o re n i 1.2 If the solution of problem P l exists, then the largest R (in the sense of fuzzy set theoretic inclusion) th a t satisĩies... cp-operator The (p-operator is a n op erator cp: [0,1] X [0,1] -> [0,1] d e íìn e d by acpb = sup {: € | 0,1 ||a • c £ b vvhere * d e n o te s t-n o r m o p e r a to r If th e t-n orm o p e r a to