DSpace at VNU: An evolutionary approach to fuzzy relation equations with constraints

AN E V O L Ư T Ĩ O N A R Y A P P R O A C H T O FU ZZY R E L A T I O N

E Q U A T I O N S W IT H 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 s tr a c t Fưzz y rclation cquations p la ỵ an ìmỊTortant rolc in arcas stjch CIS

fu z z v s y s te m a n a ly s is , dcsÌỊỊn o f fu z z y co n tro lỉers, a n d f u z zV p a tte r n r c c o g n itio n

hì th is paper, ICC dcfin c th c fu z z y rclation c q ua tion ivith c o n stra in ts 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 r m 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

1 I n t r o d u c t i o n

T h e n o t io n of fu z z y r e l a t i o n e q u a t io n w a s íìr s t stu d ied by S a n c h e z (1 9 7 6 )

S i n c e th en , m a n y í u r t h e r s t u d i e s h a v e been do n e by o t h e r r e s e a r c h e r s (s e c [õ, 6 , 7,

8]) Fu zzv r e la t io n e q u a t i o n s p la y a n im p o rta n t role in a r e a s s u c h a s fu zz y s y s t e m

a n a l y s i s , d e s i g n o f fu z z y c o n t r o lle r s , d e c isio n m a k i n g p r o c e s s e s , a n d fu zz v p a t t e r n

r e c o g n it io n

T h e n o tio n o f fu z z y r e l a t i o n e q u a t io n s is a s s o c i a t e d w it h t h e c o n c e p t o f c.om position o f fu zzy r e la t io n s L et A be a fu zzy s e t in t h e i n p u t s p a c e ư a n d R be a

fu zzy r e la tio n in t h e i n 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 i t i o n o f fu zzy s e t

A a n d fu zzv r e la t io n R, d e n o t e d by AoR, is d e f in e d a s a fu z z y s e t B in t h e o u t p u t

s p a c e V.

w h o s e m e m b e r s h ip f u n c t io n is

x«- u

w h e r e * is t h e t-n o r m o p e r a to r B e c a u s e th e t- n o r m c a n t a k e a v a r i e t y o f ĩo r m u la s , for ea ch t-norm w e o b t a in a p a r t ic u l a r c c m p o s it io n T h e t w o m o s t c o m m o n ly usecỉ

c o m p o s it io n s in n u m e r o u s a p p lic a t io n s are th e s o - c a lle d m a x - m in c o m p o s it io n a n d

m a x - p r o d u c t comỊ)osition, w h i c h a r e de fi ned as follows:

T h e m a x - m in c o m p o s it io n

Ị.iB ( y ) = m a x m i n | n A ( \ ) ^ K ( x y ) l

Xk V

T h e m a x - p r o d u c t c o m p o s it io n

56 D i n h M a n h Tuong

X € Ư

T h e e q u a t i o n A o R = B is a s o -c a lle d fu zzy r e l a t i o n s e q u a t i o n I f we v ie w R

a s a fu z z y 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 t p u t B by (2) T h e t w o b a s is p r o b le m s c o n c e r n in g t h e f u z z y r e la t io n

e q u a t io n are a s follow s:

Problem P l: Given the input fuzzy set A in u and the output fuzzy set B in

V, determ ine th e fuzzy relation R such th a t A o R = B.

Problem P2: Given the fuzzy relation R and the output B, determ ine the

i n p u t A s u c h t h a t A o R = B.

T h e r e ío r e , s o l v i n g t h e fu z z y r e la t io n e q u a t io 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 i s p a p e r we are on ly i n t e r e s t e d in t h e p r o b le m P l S in ce

t h e s o l u t i o n s for t h e p r o b le m P l m a y n o t e x ist , w e í ì r s t n e e d to c h e c k t h e so lv a b ilit y

o f t h e s e e q u a t io n s or t h e e x i s t e n c e o f th e ir s o lu tio n s

T h e o r e m 1.1 Problem P l has solutions if and only if the height of the fuzzy set A is greater th a n or equal to the height of the fuzzy set B, th a t is

T h e p ro o f o f t h i s t h e o r e m c a n s e e in [2].

In o rd er to s o l v e p r o b le m P l , one introduces th e cp-operator T he (p-operator

is a n o p e r a to r 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 t e s t- n o r m o p e r a t o r

If t h e t- n o r m o p e r a t o r is s p e c if ie d a s m in im u m , th e (p-operator becomes the

s o - c a lle d a -o p er a to r :

For fu z z y s e t s A in u a n d B in V, u s in g t h e cp-operator w e c a n d e fin e th e

fu z z y r e la tio n R* in U x V w h ic h is d efin ed as

-supnA(x) ^ ^ B(x) for all y eB.

if a < b

if a > b

W e d e n o t e t h i s f u z z y r e l a t i o n by A(pB.

T h e íbllovving t h e o r e m is d e m o n s t r a t e d (s e e [2,3]).

T h e o r e 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 e x a c t s o l u t i o n o f p ro b lem P l m ay n o t e x is t

T h e r e 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 t io n d o e s n o t e x is t , w h a t

w e c a n do is to d e t e r m i n e a p p r o x im a t e s o lu t io n s W a n g L X p r o p o sed t h e m e th o d

o f d e t e r m i n i n g a p p r o x im a t e s o lu t io n th ro u g h n e u r a l netvvork t r a i n i n g (s e e [1 0 ]).

T h e íu r t h e r d e t a i l s o f fu zz y r e la t io n e q u a t i o n s c a n be fou nd i n [2, 3].

2 F u z z y r e la t io n e q u a t i o n s w ith c o n s t r a in t s

T h e a p p r o x i m a t e r e a s o n i n g in fu zzy s y s t e m s is b a se d o n t h e r u le o f

g e n e r a li z e d M o d u s P o n e n s T h is in f e r e n c e r u le s t a t e s t h a t g iv e n tvvo fu z z y

p r o p o s itio n s “if X is A t h e n y is B ” a n d “x is A ”’ w e s h o u l d in fer a n e w p r o p o s itio n “y

is B m such t h a t t h e c lo s e r t h e A ' to A, t h e c lo s e r t h e B* to B, vvhere A a n d A ’ a r e

fu zzy s e t s in s p a c e u , B a n d B ’ a r e fu zzy s e t s in s p a c e V T h e fu zz y p r o p o s itio n “if X

is A t h e n y is B” is i n t e r p r e t e d a s a fu zzy r e la t io n R in U x V T h e fu zz y s e t B ’ in t h e

c o n c lu s io n o f g e n e r a l i z e d M o d u s P o n e n s ru le is d e t e r m i n e d as B ’ = A ’ o R In t h e

l i t e r a t u r e , m a n y d if f e r e n t in t e r p r e t a t i o n s o f fu zz y i f - t h e n r u le s a r e p ro p o sed , for

e x a m p l e , L u k a s i e w i c z im p lic a t io n , Z adeh i m p lic a t io n , M a m d a n i im p lic a t io n , etc

W e w is h d e t e r m in e t h e fu zz r e la t io n R in t e r p r e t in g fu zz y p r o p o s it io n “if X is A t h e n

y is B" su ch t h a t t h e c lo s e r t h e A ’ to A, t h e c lo s e r t h e B ’ = A ’ o K to B.

T h e n o tio n o f fu zz y r e la t io n e q u a t io n vvith c o n s t r a i n t s is s t a t e d as fo llo w s

G iv e n t h e fu zzy s e t s A a n d A, (i = 1, .,k ) i n s p a c e ư a n d t h e fuzzy s e t B in s p a c e V,

w e s h o u ld d e t e r m i n e a fu zzy r e la t io n R* in p r o d u ct U x V s u c h t h a t t h e f o llo w in g

r e q u ir e m e n t s are s a t is f ie d :

If w e d e n o t e A, o R* = B, (i = 1 , , k) t h e n

w h e r e a is c o n s t a n t , a > 0 , a n d d(.,.) is t h e d i s t a n c e b e t w e e n fu zzy s e t s T h e d i s t a n c e

d(C, D) b e t w e e n t h e fu z z y s e t c a n d th e fuzzy s e t D 19 d e ĩ i n e d as fo llo w s

d(C, D) = (f ||IC (x) - ^ 1 D (y)|r dx ) p , p ỉ l

For p = 1 o n e h a s t h e H a m m i n g d i s t a n c e a n d p = 2 y i e l d s t h e E u c lid e a n

d is t a n c e In t h e c a s e s t h e s p a c e u is fin it e w e c a n s i m p l y d e íìn e

d ( C , D ) = i V c ( x ) - n D( x) |

58 D i n h Ma n h TIiong

H en ce, o u r p roblem is to d e te r m in e th e fuzzy r ela tio n R* w h ic h s a t i s í ì e s (3) and (4), gi v e n th e fu zz y s e t s A, A, (i = 1, .,k) in space u and t h e fu zzy s e t in sp ace V.

3 An e v o l u t i o n a r y a p p r o a c h to fu zz y r e l a t i o n e q ư a t i o n s vvith c o n s t r a i n t s

It is v e r y d if fic u lt to d e t e r m in e th e e x a c t s o lu t io n o f fu z z y r e la t io n e q u a t io n s

w i t h c o n s t r a i n t s In t h i s s e c tio n , w e p rop ose a n e v o l u t i o n a r y s c h e m e for

d e t e r m i n i n g t h e a p p r o x im a t e so lu tio n of fu zzy r e la t io n e q u a t i o n vvith c o n s t r a in t s by

u s i n g a n e v o lu t i o n s t r a t e g y E v o lu tio n s t r a t e g i e s a re a l g o r i t h m s w hich i m i t a t e th e

p r in c ip le s o f n a t u r a l e v o lu t i o n a s m eth o d to so lv e p a r a m e t e r o p t im iz a t io n p ro b lem s ( s e e [1, 4, 9]) W e r e f o r m u la t e our problem in form o f an o p t im iz a t io n problem Ciiven t h e fu zz y s e t s A a n d A, (i = 1, k) in u a n d t h e fu zz y s e t B in V A s s u m e

t h a t R is a fu z z y r e la t io n in UxV D e n o te

A o R = B\

A,o R = B/ (i = 1 , k).

For e a ch f u z z y r e la t io n R, w e d e fin e a rea] v a l u e f(R) a s fo llo w s

r ( R ) = d ( B ' B ) + £ | d ( B , \ B ) - a d ( A , A ) |

1=1

O u r p r o b le m n o w is to d e te r m in e th e fu zzv r e la t io n R such t h a t f(R) is

m in im u m

To a p p ly t h e e v o lu t i o n s t r a t e g y to t h e a b o v e p r o b le m , w e íìr st n e e d to h a v e

s u i t a b l e r e p r e s e n t a t i o n s for fu zz y s e t s and fuzzy r e la t io n s A s s u m e t h a t th e s p a c e s

u a n d V c o n s i s t C)f fin it e n u m b e r of e le m e n t s , u = {Uj, u rn}, V = {Vj, v n} T h e n ,

e a ch fu z z y s e t A in u is r e p r e s e n t e d as a v e c t o r A = ( a lf a m), vvhere a, is

m e m b e r s h ip d e g r e e o f u, to t h e fu zz y s e t A, th a t is a, = nA(a >)* A n a lo g ic a lly , t h e fu zzy

set B in V has the representation B = (bj bn) Each fuzzy relation R is

r e p r e s e n t e d a s a m a t r ix of ord er mxn R = (r,j), w h e r e r„ = hk ( uì , v ,) Ư nd er t h e s e

a s s u m p t i ơ n s , w h e n g iv e n t h e fu zzy se t s A, A, (i = 1, k), B a n d th e fu zzy re], on

R w e c a n e a s i l y comp^lte t h e fuzzy s e t s B ’ = A o R a n d B / = A, o R (i = 1 k),

w h e r e w e c a n e m p lo y t h e m ax- m in c o m p o s itio n or th e m a x -p r o d u c t co m p o sitio n

H e n c e, w e c a n c o m p u t e t h e v a lu e o f cb jective íu n c t io n f(K).

T h e id e a C)f e v o lu t i o n s t r a t e g y for ou r p rob lem is a s fo llo w s Each in d iv id u a l

is r e p r e s e n t e d a s a p a ir (R, Z)» vvhere R = (r,,) is a m a t r ix o f ord er 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 a ch p o p u la t i o n c o n s i s t s o f N in d iv id u a ls , all i n d i v i d u a l s in th e p o p u la tio n

h a v e t h e s a m e m a t i n g p r o b a b ilitie s In ea ch it e r a t iv e s t e p , tvvo ra n d om ly s e le c t e d

p a r e n ts:

a nđ

produ ce a n o ffsp r in g

(R ,E ) = ((r„ ) ( o „ ) ) ,

w h e r e r,, = r 1tJ ar rtJ = r2 i, w ith e q u a l p rob ab ility a n d if ru = r k,j t h e n ơ tỊ = a k,j (k = 1 ,2 ).

T h e m u t a t io n o p e r a to r is períbrm ed on t h e o ffsp r in g (R, £ ) vvhich as

g e n e r a t e d by th e a b o v e c r o s s o v e r op erator A p p ly in g t h e m u t a t i o n to t h e o ff s p r in g (H, £)» w e o b t a in t h e n e w o ffsp r in g ( R \ X):

R' = (r’ti), r’„ = r(J + N(0, CT„ ), (i = 1 , m; j = 1 , n),

vvhere N (0 , n l() is a n o r m a ll y d is tr ib u te d ra n d o m v a l u e w ith e x p e c t a t i o n z er o and

S ta n d a rd deviation C7,r

W e now r e p r e s e n t t h e s c h e m e o f e v o lu tio n a r y a lg o r ith m for d e t e r m i n i n g th e

a p p r o x im a t e s o lu t io n o f fu z z y r e la t io n e q u a t io n w ith c o n s t r a in t s

Algorithm

1 G e n e r a t e a p o p u la t i o n o f N in d iv id u a ls (R, Z), w h e r e R = (rtJ) is a m a t r ix of

o r d e r m x n , ea ch r,j is r a n d o m ly t a k e n from th e in t e r v a l [0 , 1 ], ỵ = ( a lf) is a m x n -

m atrix o f S t a n d a r d d e v i a t i o n s

2 (Iterative step)

Randomly s e l e c t tw o p a r e n t s from N in d iv id u a ls

( R1, Z |) = ((r1,,), ( a 1,,)) a n d

(Hi I 2 ) = ((r2.,) (ơ2,,)).

T h e s e p a r e n t s p ro d u ce a n o ffsp r in g

(R, I ) = ((r,,), (ơ,,)),

w h e r e r,j = r l,j or r tJ = r2(J w i t h e q u a l p robab ility a n d if r,} = r 1,, t h e n cTif a CTlij if r tỊ = r 2tJ

t h e n Gtj = ơ 2,,.

Applying the m u tation to the oíĩspring (R, Z), w e obtain the n e w oíĩsp rin g (R \ Z)

R’ = (r\ị),

r ’„ = r ẽ, + N(0, Ơ(J), (i = 1 m; j = 1 , n),

60 D in h M a n h Tuoìiịị

w h e r e N ( 0 t o tJ) is a n o r m a lly d is tr ib u te d ra n d o m v a lu e w ith e x p e c t a t io n z e r o and

S t a n d a r d d e v i a ti o n ơ,, If all r ’,j s t a y w i t h 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 a d d ed to t h e p o p u la tio n

E lim in a t e t h e vveakest in d iv id u a l from N + l in d iv ic ỉu a ls (o r ig in a l N

i n d iv id u a ls p lu s o n e o ffsp rin g).

C o n c lu s io n

W e h a v e d e fin e d th e n o tio n o f fuzzy r e la t io n e q u a t io n vvith c o n s t r a i n t s , and

p r o p o se d t h e e v o lu t io n a r y a lg o r ith m for d e t e r m in in g an a p p r o x im a t e s o l u t i o n of

t h i s e q u a t io n T h is e v o lu t i o n a r y a lg o r ith m c a n b e a p p lie d to d e t e r m i n e th e

a p p r o x im a t e s o lu t io n o f t h e p rob lem P l in c a s e an e x a c t s o lu tio n o f p roblem P l does

n o t e x is t

R eferences

1 B a ck T., H o f f m e is t e r F, a n d S h w e fe l H F A s u r v e y o f E v o lu t io n S t r a t e g i e s , in

P r o c e e d ỉn g s o f th e f o u r t h I n t e r n a t i o n a l C on feren ce on G e n e tic A l g o r i t h m ,

M o rg a n K a n g m a n n , C a n M atco, 1991.

2 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.

3 Li- X in W a n g , A c o u r se in F u z z y S y s t e m s a n d C o n tr o l P r e n tic e - H a ll, Inc.,

1997.

4 M ic h a le w i c z z , G e n e tic A l g o r i t h m s + D a t a S t r u c t u r e s = E v o l u tio n P r o g r a m s ,

S p r in g e r , 1996.

5 P e d r y c z w , F u zzy r e la t io n a l e q u a t io n s w ith g e n e r a li z e d c o n n e c t iv e s a n d th eir

a p p lic a t io n s Fuzzy s e t s a n d S y s t e m s, 1 0(1983), 1 85-201.

6 P e d r y c z w , s- t F u zz y r e la t io n a l e q u a t io n s F u z z y s e ts a n d S y s t e m s , 5 9 ( 1 9 9 3 ) ,

189- 196.

S a n c h e z E, R e s o lu t io n o f c o m p o s ite fu zz r e la t io n e q u a t io n s I n f o r m a t i o n a n d

C o n tr o l, 3 0 ( 1 9 7 6 ) , 38- 49.

s S a n c h e z E, S o lu t io n o f fuzz>' e q u a t io n s w ith e x t e n d e d o p e r a to r s F u z z y S e i s a n d

S y s t e m s , 1 2 (1 9 8 3 ), 2 37- 248.

9 S c h w e f e l H p, E v o lu tio n S t r a t e g ie s : A F a m ily o f Non- L in e a r O P t im iz a t io n

T e c h n i q u e s B a se d o n I m i t a t i n g S o m e P r in cip les of O rgan ic E v o lu tio n A n n a l s o f

O p e r a t i o n s R e s e a r c h Vol 1(1984), 165- 167.

10 W a n g L D, S o lv i n g fu zz y r e la t io n a l e q u a t io n s th ro u g h netvvork t r a in i n g Proc

2 nti I E E E Inter Conf on F u z z y S y s t e m s S a n F ra n cisco , 1993, 9 5 6 - 960.

TAP CHI K H O A HOC D H Q G H N KH TN & CN t XIX NọỊ, 2003

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VỚI CÁC RÀ N G B U Ộ C

D in h M ạnh T ư ờ n g

K h o a C ô n g nghệ, Đ H Q G H à N ộ i

K hái n iệ m p h ư ơ n g tr ìn h q u a n hệ lần đầu t iê n được đê x u ấ t và n g h iê n cứ u bcii

S a n c h e z (xem [7]) P h ư ơ n g tr ìn h q u a n hệ mò đ ó n g vai trò q u an tr ọ n g t r o n g n h iề u lĩn h vực, c h a n g h ạ n p h â n tích các hệ mờ, t h i ế t kê các h ệ đ iể u k h iể n mờ, n h ậ n d ạ n g

m ẫ u mờ T rong b à i báo n à y c h ú n g tôi xác định khái n iệ m ph ư ơ n g tr ìn h q u a n hệ m ò vỏi các r à n g buộc v à đê x u ấ t m ột t h u ậ t to á n t iế n hoá để tìm n g h iệ m x ấ p xỉ củ a

ph ư ơ n g tr ìn h n ày.