Evaluation Model of Humanistic Systems by Fuzzy Multiple Integral and Fuzzy Correlation

1. I n t r o d u c t i o n ~ ]e concep t o f fuzzy s e t s d e f i n e d by Zadeh 1 ) g i v e s an i m p o r t a n t m a t h e m a t i c a l c l u e f o r an approach to s t u d i e s o f sys tems w i th no sha rp b o u n d a r i e s . L a t e l y , t h e fuzzy i n t e g r a l 2 ) b a s e d on t h e fuzzy s e t t h e o r y has been proposed and a p p l i e d t o t he measurement o f fuzzy o b j e c t s , e s p e c i a l l y , p a t t e r n r e c o g n i t i o n s . We c o n s i d e r t h e fuzzy i n t e g r a l i s a l s o a v a i l a b l e to be a p p l i e d to t h e e v a l u a t i o n o f fuzzy sys t ems , because t h e fuzzy measure in t h e fuzzy i n t e g r a l w i l l be r e g a r d e d as a p r e f e r e n c e measure o f e v a l u a t i o n and t h e fuzzy i n t e g r a l w i l l r e p r e s e n t a c o n f l i c t i n g p r o c e s s be tween t h e p r e f e r e n c e measure and t h e e v a l u a t e d o b j e c t . When we c o n s i d e r t h e e v a l u a t i o n problem o f complex sys tems composed o f s e v e r a l subsys t ems , e s p e c i a l l y humani s t i c ty]~e ones , i t i s v e r y d i f f i c u l t t o s e t up t h e u t i l i t y model o f o v e r a l l sys tem. Because , i n g e n e r a l , t h e human p r e f e r e n c e has a h i e r a r c h i c a l , s t r o n g l y n o n l i n e a r and q u a l i t a t i v e p r o p e r t y . For such a c a s e , we p ropose an u t i l i t y model o f e v a l u a t i o n by a p p l y i n g a fuzzy m u l t i p l e i n t e g r a l . In t h i s model t h e p r e f e r e n c e measure f o r t h e o v e r a l l sys tem i s g i v e n by a c o m p o s i t i o n o f t h e s u b s y s t e m s ' measure~ ) Apply ing t h e p roposed method t o a p r a c t i c a l c a s e s , t h e p r e f e r e n c e measure has t o be d e t e ~ n i n e d by e x p e r i m e n t . To r e a l i z e such a p r o c e s s , we i n t r o d u c e a fuzzy d i s t r i b u t i o n f u n c t i o n and g ive an a l g o r i t h m to c a l c u l a t e t h e measure . F u r t h e r , we i n t r o d u c e a fuzzy c o r r e l a t i o n among subsys tems i n o r d e r t h a t t h e fuzzy measure w i l l be i d e n t i f i e d e f f e c t i v e l y by e x p e r i m e n t a l d a t a . The proposed method i s s u c c e s s f u l l y a p p l i e d t o t h e s u b j e c t i v e e v a l u a t i o n o f a c l a s s o f f i g u r e s ' l a r g e n e s s .