Note on the ranking of fuzzy numbers: conditions for a total order relation

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Note on the ranking of fuzzy numbers: conditions for a total order relation

Claude Ponsard

To cite this version:

Claude Ponsard. Note on the ranking of fuzzy numbers: conditions for a total order relation. [Research Report] Institut de mathématiques économiques (IME). 1988, 6 p., ref. bib. : 5 ref. �hal-01534191�

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EQUIPE DE RECHERCHE ASSOCIEE AU C.N.R.S.

DOCUMENT DE TRAVAIL

INSTITUT DE MATHEMATIQUES ECONOMIQUES

UNIVERSITE DE DIJON

FACULTE DE SCIENCE ECON OMIQUE ET DE GESTION 4, BOULEVARD GABRIEL - 21000 DIJON

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NOTE ON THE RANKING OF FUZZY NUMBERS ; CONDITIONS FOR A TOTAL ORDER RELATION

Claude PONSARD February 1988

Paper submitted to Fasciculi Mathematici, (Politechnika Poznanska)

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Paper dedicated to Professor Jerzy Albrycht

Abstract : 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.

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

tion.

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From pioneer-work on fuzzy numbers theory made by Dubois and I’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).

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( 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}, tf a £ ]0, 1].

ot 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

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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], tf a ’ G ]0, 1], with > a1 > a > a^, we have :

{x./u (x.) > a ’} c {x./y (x.) > a } . In other words, [a., a .] . c La., a .]

i n i \ n i i’ j a' i j a

and the intervals which are defined by the oucuts of n are nested,Q

Coro)lary. A triangular fuzzy number is a strict fuzzy number such that the intervals defined on (R by its ot-cuts are nested,

Evident. A triangular fuzzy number is continuous on the right and on the left.^

Now,we name j e the subset of strict and continuous fuzzy numbers, c

lot n. and n. be two elements of j p and [a , a 1 and [b , b

]

^,

1 Jj ■ C I 4» d 1 a

a £ JO, 1], their a-cuts respectively.

Definitions : Inclusion ; n. £. n . «s=> [a„, a„]_ ^ [b„, b„]^ , \) a 6 ]Q,1],

— ■ ■- -■ . ■ 1 J I ZOt 1 £ ™

Equality : n. = n . <=> n. £ n . and n . & n. .

---

¹

• l j j j j

^l

Theorem 2. The set

JP

^, endowed with the inclusion and equality relations, c

is totally ordered.

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Proof' : A c c 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 jec are nested. So, it s u f f i c e s to a r g u e a bo ut

Lheir r e s p e c t i v e sup p ort s . To s i m p l i f y the notation, w i l l be om itted .

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^ > b^

or 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 t wo

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 ei 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 t h i r d e l ement of C)f>c > d e n o t e d by n^, and its support, su p p n^ = ^ c -j’ C2^*

We have : n. c n . <=> a„ i b 0 a n d a„ > b.

i - j 2 2 1 1

or a 1 = b 1 a n d a 2 < b 2

n £ n K <=> b 2 C c 2 a n d b /| >

or b 1 = c 1 a nd b 2 < c 2

Six c a s e s are p o s s i b l e :

(1) (a2 < b 2 a nd a.) > ^ ) a n d ( b 2 ^ c 2 an d > c 1 )

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(2) (»2 < b 2 a nd ) a n d ( b 1 = c 1 and b 2 < c^)

(3) (a2 = b 2 and a^ > b ^ ) a nd (b2 <: c 2 and b^ > ) (4) (a2 = b 2 a n d a,! > b ^ ) a n d ( b 1 = ^ and b 2 < c 2 >

(5) (a^ = b^ and a2 < b2 ) and (b2 C c2 and b^ > c ) (6) (a1 = b 1 and a 2 < b 2 ) a n d ( b /| = c 1 and b 2 < c )

In all the cases, w e 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,

— ---L i .1 c 2 i

or : b 2 < a . If a = b 2< we c an c o m p a r e a^ to b^

F i nally, the set o f 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 b est th i n g we c an do is to m a k e use of stric t 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 w h e n the e x i s t e n c e o f a total o r d e r relati on on the set o f fuzzy nu m bers is import ant 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 more 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 in g 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 g dal, 1987). Uut, the s o l u t i o n is d e p e n d e n t on the r u 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 e ach m e t h o d a nd so, it is not unique.

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References.

B O R T O L A N G. a n d Dt'GANI R., 1985, A rev i ew o f some m e t h o d s for r a n k i n g fuzzy subsets, F u z z y S et s a n d S y s t e m s , 15, 1-19.

DU B O I S D. a nd PR A D E H . , 1978, O p e r a t i o n s on fuzzy n u m bers, Internat. J.

S ys t e m s S c i e n c e , 9, 6 1 3-626.

DU B O I S D. a nd PRA DC H., 1980, F u z z y S e t s and Sy s t e m s : T h e o r y a nd A p p l i c a t i o n s , A c a d e m i c Press, N ew York.

K A U F M A N N A., 1983, L e p r o b l è m e du c l a s s e m e n t des n o m b r e s f lo u s en u n o r d r e total, N o t e de t r a v a i 1 n° 112.

O V C H I N N I K O V S. a nd M I G D A L M . , 1987, O n r a nki ng fuzzy sets, F u z z y S e t s a n d S y s t e m s , 24, 113-116.