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

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

1. 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 do n e by o th e r r e s e a r c h e r s (se c [õ, 6 , 7,
8 ]). 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 ư

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 tp 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 tp 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 do 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 [ 1 0 ]).
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].

2.

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 ư 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 = 1 ,..., 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 > 0 , 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 = 1 o n e h a s th e H a m m in g d is ta n c e an d p = 2 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.

3. 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 ư 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 = 1 ,

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 =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 = 1 .

on
k),

w h e r e w e c a n e m p lo y th e m ax- min 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 2 i, w ith e q u a l probability a n d if ru = rk,j th e n ơ tỊ = a k,j (k = 1 , 2 ).
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 = 1 ,

m; j = 1 ,

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
1. 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 , 1 ], 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 .

2 . (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 1 , 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 = ơ 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 = 1 ......m; j = 1 ,

n),

60

Dinh M anh T uoìiịị

w h e r e N ( 0 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
1

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.

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- 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.
4

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.
5

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.

6

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 , 5 9 (1 9 9 3 ),
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, 3 0 (1 9 7 6 ), 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 , 1 2(19 83 ), 237- 248.

9

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.
2 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

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\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 à N ộ i
Khái n iệm ph ư ơ ng tr ìn h q u a n hệ lần đầu tiê n được đê xuất và n g h iê n cứu bcii
S a n c h e z (xem [7]). P h ư ơ 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 ạ n p h â n tích các hệ mờ, t h iế t kê các hệ đ iể u k h iển mờ, n h ậ n d ạ n g
m ẫ u mờ. Trong bài báo n à y c h ú n g tôi xác định khái n iệ m phương trìn h q u a n hệ mò
vỏi các rà n g buộc và đê x u ấ t m ột th u ậ t to á n tiế n hoá để tìm n g h iệ m xấp xỉ của
phương trìn h này.