Professor Jerzy Albrycht

Keywords: fuzzy numbers, ranking of fuzzy numbers, total order rela tion

It is well-known that the set of fuzzy numbers is not orde red. The author states the conditions which have to be satisfied so that the problem of ranking fuzzy numbers has a solution. In fact, the subset of fuzzy numbers, the elements of which are strictly defined and continuous,is endowed with a total order relation.

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From pioneer-work on fuzzy numbers theory made by Dubois andI'rade (1978, 1980), the problem of their ranking has been studied by many authors.

Usual methods are applied together to the problem of ranking fuzzy numbers and to the more general problem of ranking whatever fuzzy subsets (for a review, see Bortolan and Degani, 1985). However, by defi nition, a fuzzy number is a normalized and convex fuzzy subset of the real line II? . These conditions are often understood in a broad sense : the normality property is not imposed in one and only qne point (flat fuzzy numbers) ; the convexity property is replaced by a less restricti ve piecewise convexity property.

The scope of this paper is to state that the subset of the fuzzy numbers which are strictly defined is endowed with a total order relation if its elements are continuous.

Definition ; strict fuzzy nujnber, A strict fuzzy number, denoted by n, is defined by a membership function from \R to [0 , 1] such that, y x. 6 IR, ^n^Xi^ ^ ^ the three following conditions :

(

1)3 ! x^ G IP such that M (x_) = 1 : strict normality. 0 n U (2 ) V (x., x ) G IR2 , V A > 0 , V X' > 0 , A + A' = 1, M (Ax. + A'x.
) >y Min(u (x.), u (x,)) : convexity (quasi-concavity of n i j n i n j the membership function).

( 5) ruipp n i:; bounded and closed, where supp n is the support of n.

By definition, supp n = {x./ y (x.) > 0}. j n l

According to the theorem on the decomposition of a fuzzy subset by its a-cuts, we have :

n = Max(a . (n) ) where (n) = {x. / U (x.) > / a}, t f a £ ]0, 1]. o t ot ot ] n j
As {x, / u (x.) > a} = [a., a.] where [a., a.l are the closed intervals i n i i j a a j a of /P ssociated with each value of a, we have (n) = [a., a i j a

V a e JO, 1].

Theorem 1. A strict fuzzy number is continuous iff the intervals which are defined on IR by its a-cuts are nested.

Proof : Consider IP endowed with its natural metric structure and let n be a strict fuzzy number. If the closed intervals which are defined by the a-cuts of n are nested, the a-cuts are got by a mapping o from ]0 , 1] to cPda., a.]) the power-set of the closed intervals of /R*, such that :

(a1 > a) => o(a') c a( a), W a' > a , i.e. [a., a-L, c [a., a,]. 1 J Ot 1 J Ot
By virtue of the normality and convexity properties of a strict fuzzy number, the series of its a-cuts is monotonous and decreasing for increasing values of a. We have :

[a., a . ] , = n [a., a . ] , M a' > a.

i J a a 1 J a llui:;,lh(' in;i|>|>in<j a is non! ¡minus ;is I In' int (msimI ion of any family of closed subsets of a metric space is closed.

Reciprocally, as n is strictly normalized, we have :

[a., a ] c [a., a.] a < a .
Consider the support of n : we have

1 J i J oi 1 supp n = [a , a ] ^ [a , a 1 , tfa > a . J ug 1 J u u
As the function yn is quasi-concave and continuous,

\ / a G ]0 , 1 ], t f a ' G ]0 , 1 ], with > a 1 > a >
]_ ^ [b", b"]^ , \ ) a 6 ]Q,1], -■ ■ --■ . ■ 1 J I ZOt 1 £

  The set JP , endowed with the inclusion and equality relations, c is totally ordered.

  o r d i n g to t h e o r e m 1, Lhe .intervals d e f ined on it? by the cx-cuts of Lhe e l e m e n t s of j e c are nested. So, it s u f f i c e s to a r g u e a bout Lheir r e s p e c t i v e supports. To s i m p l i f y the notation, w i l l be omitted. R e f l e x i v i t y : n. C n. o [a^, a^] s [a^, a ^] . E v i d e n t . A n t i s y m m e t r y : n. 5 n^ <=> a^ b 2 and a^ > bô r a 1 = b 1 and a 2 < b 2 n £. n^ <=> b 2 < a 2 and b^ > or a 1 = b a nd b 2 < a 2 As it is inpossible to ha v e b o t h : a" < b a nd b" < a , only o n e o f the two I L L £ inclusion r e l a t i o n s holds. In the p a r t i c u l a r case w h e r e a 2 = b 2> we h a v e e i t he r a^ > b^ or b^ > a ^. T r a n s i t i v i t y : c o n s i d e r a th i r d e lement of C) f > c > d e n o t e d by n^, and its support, su p p n^ = ^c -j' C 2 ^* W e have : n. c n . <=> a" i b 0 a nd a" (a2 < b 2 a nd a.) > ^) a nd ( b 2 ^ c 2 an d > a1 = b 1 and a 2 < b 2 ) a nd ( b / | = c 1 and b 2 < c ) In all the cases, we h a v e n. £ a,. l K To t a l i t y : tf (n., n.) € , we have e i t h e r : a_ < b, a . If a = b 2< we c an c o m p a r e a^ to bF i nally, the set of fuzzy n u m b e r s is not o r d e r e d w h e n at least one of its e l e m e n t s is not s t r i c t l y d e f i n e d and not c o n t i n u o u s . In a p p l i cations, the best th i n g we c an do is to m a k e use of strict a n d c o n t i n u o u s fuzzy n u m b e r s as o f t e n as the r e p r e s e n t a t i o n of an e m p i r i c a l m o d e l e n a b l e s il and wh e n the e x i s t e n c e of a total o r d e r relation on the set o f fuzzy nu mbers is important for the m a t h e m a t i c a l modelling. If not, w e find a g a i n the m ore g e n e r a l p r o b l e m of r a n k i n g any fuzzy n u m b e r s (e.g. K a u f m a n n , 1903) and rank ing w h a t e v e r fuzzy se t s (e.g. O v c h i n n i k o v and M i gdal, 1987). Uut, the s o l u t i o n is d e p e n d e n t on the ru l e s w h i c h are c h o s e n in the f r a m e w o r k of each m e t h o d a nd so, it is not unique.