Cybernetics and Systems Research R. Trapplfed.)
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C O M M E N T S ON THE y-MODEL
J . D o m b i and P . Zysno
Research Group on T h e o r y of the A u t o m a t a , S z e g e d , Hungary RWTH A a c h e n , Operations Research FRG
R e c e n t l y , a class of operators has been suggested, called y - m o d e l , w h i c h is equipped w i t h several suitable formal properties and additionally satisfies empirical
a s p i r a t i o n s . F i r s t , its formal d e v e l o p m e n t from a more abstract p o i n t of v i e w is p r e s e n t e d . Second, a dual form and some p o s s i b l e interpretations are o f f e r e d . T h i r d , it is shown how to evaluate the c l a s s i f i c a t o r y power of each operator w i t h i n the y - m o d e l .
I . INTRODUCTION . If it is intended to d e s i g n an operator for the c o n n e c t i o n of fuzzy truth v a l u e s or for the a g g r e g a t i o n of fuzzy s e t s , it is r e a s o n a b l e to r e q u i r e m a t h e m a t i c a l properties such as conti-
* . ^ ' . . - . n u i t y , strict m o n o t o n i c i t y , injectivity
(implied by continuity and strict m o n o t o n i c i t y ) , c o m m u t a t i v i t y , and a s s o c i a t i v i t y . C o m m u t a t i v i t y is g i v e n if the operator is infective and a s s o c i a t i v e [1]. M o r e o v e r , it should be in a c c o r d a n c e w i t h the truth tables of dual l o g i c . Several operators have been suggested w h i c h satisfy these requirements: M i n i m u m / M a x i m u m [7], P r o d u c t / A l g e b r a i c Sum [2], Hamacher's
D ( x , y ) / K ( x , y ) [5], Yager's Cp( x ) / Dp( x ) [6], and o t h e r s .
It has b e e n shown [3] that operators with the above p r o p e r t i e s result in membership grades b e t w e e n zero and m i n i m u m in the case of inter- section (and), and between maximum and one in the case of union (or), respectively:
C(iiA(x),uB(x)) = min ( pA( x ) , pB( x ) ) (1) D ( yA( x ) , yB( x ) ) = m a x ( yA( x ) , PB( x ) ) (2) M i n i m u m and maximum can be obtained by the
limes of a series of o p e r a t o r s .
H o w e v e r , empirical investigations [8, 10] h a v e shown that human aggregation usually provides m e m b e r s h i p v a l u e s b e t w e e n m i n i m u m and m a x i m u m .
Presumably,, operators based on the above prop- erties are too r e s t r i c t i v e for judgmental
b e h a v i o u r . In order to d e s i g n an averaging m o d e l , at least one of the assumptions w i l l have to be w e a k e n e d . The slightest loss of generality seems to be entailed by the abandonment of associ- a t i v i t y , as the only averaging operator satisfy- ing this property is the m e d i a n [4].
R e c e n t l y , a class of operators [9] has been suggested w h i c h is equipped w i t h the above formal properties (except a s s o c i a t i v i t y ) and additional- ly fulfils empirical aspirations [8]:
6. , 6 . y = (np (x) V Y( l - n( ! - yi( x ) ) V 0<6S1 (3)
1 0<y51
Starting from the convex combination [7] of an intersecting and a unifying o p e r a t o r , its model- ling quality w i t h i n a theory of concept integra- tion has been examined [10]. Our first intention is to present its formal development from a m o r e abstract point of v i e w . A second notion concerns the dual form of the y - m o d e l and some possible interpretations.
2 . The general structure of the y-model Let X be the universe of discourse w i t h the elements x . A,B and T are fuzzy sets in X . T h e n , the convex combination of A,B and T can b e denoted by (A,B;T) and is definedasfev the relation
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712 J. Dombi and P. Zysno
( A , B ; D = TA + TB (4) w h e r e r is the c o m p l e m e n t of r . W r i t t e n in terms
of m e m b e r s h i p f u n c t i o n s , (4) r e a d s
f( A , B ; r ) ° ( ' " frW ) - fA( x )+frW - fB ( x ) <5>
A basic p r o p e r t y of the a b o v e - d e f i n e d c o n v e x c o m b i n a t i o n is e x p r e s s e d by (6):
A 0 B C ( A , B ; D C A U B (6) O b v i o u s l y , the c o n v e x c o m b i n a t i o n is a p a r t i a l
fuzzy set b e t w e e n the i n t e r s e c t i o n and the u n i o n of two fuzzy sets A and B .
One easily i m a g i n e s A and B b e i n g themselves r e s u l t s of set o p e r a t i o n s . F o r r e a s o n s of clearness A is r e p l a c e d b y Ln and B b y L„ : A =
= Lu s B = L „ . D e f i n i n g
Ln = C fl D . (7)
Lu = C U D (8)
(6) can b e r e w r i t t e n :
L„ n Lu c (L„ ,L„;r) C L„ U L0 (9)
A s Ln C fl Lu and Lu 3 U L „ , r e l a t i o n s h i p (9) c a n be s i m p l i f i e d :
L0 C ( L „ , LU ;R ) C L U ( 1 0 )
The c o n v e x c o m b i n a t i o n of the i n t e r s e c t i o n and the u n i o n of several sets is a p a r t i a l fuzzy set b e t w e e n i n t e r s e c t i o n and u n i o n .
The c o n v e x c o m b i n a t i o n of , Lu and r can n o w b e d e f i n e d similarly to (4). In order to repre- sent r e l a t i o n (9) b y a g e n e r a l a l g e b r a i c f o r m , w e continue the i n i t i a l o p e r a t o r n o t a t i o n instead of m e m b e r s h i p terms:
e ( X )( UA( x) , YB( x ) ) = ( 1 -Y ( X ) ) C ( PA( X ) , PB( X ) ) +
+ Y ( x ) D ( Pa( X ) , Pb( X ) > (11) If it is assumed that the r e s u l t of the c o n v e x
c o m b i n a t i o n m e r e l y d e p e n d s o n the b a s i c o p e r a - tions C and D , then the v a r i a b l e Y(X) b e c o m e s c o n s t a n t :
6 ( PA( x) , UB( x ) ) = ( 1 - Y ) C ( VA( X ) , PB( X ) ) +
+ D ( pA( x ) , pB( x ) ) (12) iFormula (12) r e p r e s e n t s the general form of the
c o n v e x c o m b i n a t i o n .
If C and D a r e c o n t i n u o u s , s t r i c t l y m o n o t o n i c , and a s s o c i a t i v e , a s d e m a n d e d in the b e g i n n i n g , then their c o n v e x c o m b i n a t i o n is c o n t i n u o u s , s t r i c t l y m o n o t o n i c , i n j e c t i v e , and c o m m u t a t i v e : - If C and D a r e c o n t i n u o u s , then their s u m w i l l
b e c o n t i n u o u s , t o o .
- If C and D a r e s t r i c t l y m o n o t o n i c , t h e n t h e i r sum w i l l b e strictly m o n o t o n i c , t o o .
- If the c o n v e x c o m b i n a t i o n is c o n t i n u o u s a n d s t r i c t l y m o n o t o n i c , then it is i n j e c t i v e . - T h e c o m m u t a t i v i t y is g i v e n a s
0 ( uA( x ) , PB( x ) ) = ( l - Y ) - C ( pA( x ) , pB( x ) ) + + Y " D ( pA( x ) , pB( x ) ) =
= ( l - Y ) . C ( pB( x ) , pA( x ) ) + (13) + Y - D ( P g ( x ) , PA( x ) ) = .
= e( pB( x ) , yA( x ) )
N o w , in order to get a l a r g e r s c o p e for a p p l i c a - tions and e m p i r i c a l r e s e a r c h w e w i l l a d m i t c o n t i n u o u s and m o n o t o n i c t r a n s f o r m a t i o n s o n the e l e m e n t a r y o p e r a t i o n s g i v e n in (12):
y (x) = f ( 0 ( yA( x ) , uB( x ) ) =
= ( l- Y) . f ( C ( yA( x ) , yB( x ) ) + (14) + yf ( ( D ( yA( x ) , yB( x ) ) )
T h e t r a n s f o r m a t i o n can be c h o s e n w i t h r e s p e c t to s c a l i n g a s p e c t s , p s y c h o l o g i c a l i n t e r p r e t a t i o n s , t h e o r e t i c a l c o n d i t i o n s , e m p i r i c a l f a c t s , m o d e l - ling i n t e r e s t s , and so f o r t h . F o r m u l a (14) is the g e n e r a l Y- m o d e l .
3 . T h e d u a l f o r m
T h e m o s t s i m p l e s p e c i f i c a t i o n is o b t a i n e d b y r e p r e s e n t i n g the c o n j u n c t i o n b y the p r o d u c t and the d i s j u n c t i o n b y the a l g e b r a i c s u m . T h e s e two o p e r a t o r s a r e the only p o l y n o m i a l s o l u t i o n s if the a b o v e a x i o m s a r e s a t i s f i e d .
If, for i n s t a n c e , in v i e w of a p s y c h o l o g i c a l i n t e r p r e t a t i o n [ 1 0 ] , t h e t r a n s f o r m a t i o n is d e f i n e d b y the l o g a r i t h m , then the r e s u l t w i l l b e the y - m o d e l g i v e n b y e q u a t i o n (3). O n the
Comments on the y-Model 713
other h a n d , its dual form can b e derived b y applying the transformation log(l-x):
m . m « . .
p ,(x) = l-(l-np.(x))'"Y(n(l-p,(x)))Y U & Y <' (15)
Y 1 i OSy(x)S1
M o d e l s (3) and (15) are b o t h in a c c o r d a n c e w i t h the truth tables of dual l o g i c . A m o r e d i f f e r e n - tiated aggregation can be provided by introduc- ing w e i g h t s 6 . w i t h £ { . = m . If the n e g a t i o n
i=l
of p(x) is l-p(x) then the de M o r g a n rules are s a t i s f i e d :
yy. ( x ) = l - ( l - n y . ( x ) )1 _ Y( l - ( l - n ( l -y i( x ) ) ) )Y '
= i - ( i - n yi( x ) )1 _ Y( n ( i - ui( x ) ) )Y 0 6 )
C o m p a r i n g the values p ^ ( x ) and p ^ , (x), it can b e stated that '
are s a t i s f i e d . M o r e o v e r , if Y = X then
xY - 1 - XX (20)
(x) s V I (x) (17)
The y - m o d e l has a l r e a d y b e e n studied e m p i r i c a l l y [8, 10] w i t h i n the framework of evaluation t h e o r y . A n application together with its dual f o r m m i g h t , for instance, be found in a d e c i s i o n - m a k i n g typology of optimizers and s a t i s f i e r s . While the first would n e e d , say, a car w h i c h is "quick and c o m f o r t a b l e " , the second w o u l d a c c e p t a v e h i c l e w h i c h is "not slow and not u n c o m f o r t a b l e " . Representing the o p t i m i z e r ' s j u d g m e n t a l concept by the primal and the
satisfier's judgmental concept b y the dual f o r m , g e n e r a l l y higher evaluations are predicted for the latter with r e s p e c t to the specified list of c r i t e r i a .
4 . A s p e c t s of classification
B e s i d e s its role as a class of o p e r a t o r s for the a g g r e g a t i o n of fuzzy sets, the y - m o d e l m a y b e used if the grade of impreciseness of a certain o p e r a t o r w i t h respect to classification is of i m p o r t a n c e .
In crisp set theory two sets X and Y induce a c l a s s i f i c a t i o n on X if the characteristic func- tions
In fuzzy set theory the characteristic function is replaced by the m e m b e r s h i p f u n c t i o n , i . e . (18) and (19) a r e not s a t i s f i e d . H o w e v e r , one may evaluate the grade of classificatory power c associated w i t h a certain operator by the inte- gral of the aggregated functions HA(X) a n <J ^ ( x ) •
(21) 1
C" = ^ A f T A( x ) d x
1 '
c" = '( 1 _ yA U S( x ) ) d x (22) For 0 < y < 1 this m e a s u r e is monotonic and takes the v a l u e s of the interval [0,1], perfect classification b e i n g indicated by zero and indiscriminability by o n e . As an e x a m p l e , the classificatory power of M i n i m u m and Maximum m a y be c o n s i d e r e d . The corresponding membership values for the intersection and the union are given by
PA n s( x ) = m i n (p ( x ) , ^ < x » (23)
PA U A( x ) = m a x ( yA( x ) , u7 C( x ) ) (24)
A Q A
XX U Y " 1 XX n Y = °
(18) (19)
PA( x )
F i g . I: C l a s s i f i c a t o r y power of M i n i m u m and M a x i m u m
Figure 1 shows the corresponding graphs of the membership f u n c t i o n s . The v a l u e s of the respective definite integrals are obviously c . = 1 / 4 and c = 1 / 4 .
min m a x
714 J. Dombi and P. Zysno
The classificatory power of the Y-model is not fixed, since c_ = f(y) (Figure 2 and 3 ) . The
O
-model equals the product and the algebraic sum for y = 0 and y = 1, respectively. By solving the corresponding definite integrals
1 . 1
Jx(l-x) dx and Jx+(l-x)-x(1-x) d x , it is easy to 0 0
verify that
. 1/6 < c < 5/6 (24)
F i g . 2. of the primal form of the y-model
Classificatory aspectc are of importance in m o s t conceptual systems in t e c h n i c a l , e c o n o m i c a l , social, m e d i c a l , and other a r e a s . V e r y o f t e n , the used concepts do not m a k e up d i s j u n c t i v e classes. Technical and biological systems u s u a l l y dispose of partially compensatory subsystems in order to overcome local f e e b l e n e s s . H e n c e , the utility of models and consequently their practi- cability will be increased if fuzzy c l a s s e s and operators are used in representing a given situation, for instance for s i m u l a t i o n s , cybernetic systems, evaluative h i e r a r c h i e s , and o t h e r s .
F i g . 3 . of the dual form of the y-model
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