Ranking fuzzy numbers in the setting of possibility theory

INFORMATION

SCIENCES

30,183-224

Ranking Fuzzy Numbers in the Setting of Possibility DIDIER

183

(1983)

Theory

DUBOIS

Department of Automatics (DERA), C.E.R.T.,

2, au. E. Belin, 31055 Toulouse, France

and HENRI

PRADE

L.S. I., Umuersit~ Paul Sabatier, 31062 Toulouse, France

Communicated

by K. S. Fu

ABSTRACT

The arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood. Equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals, which is addressed in this paper. A complete set of comparison indices is proposed in the framework of Zadeh’s possibility theory. It is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers. Moreover, this approach is related to previous ones, and its possible extension to the ranking of n fuzzy numbers is discussed at length. Finally, it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities.

INTRODUCTION Fuzzy numbers, or more generally fuzzy sets of the real line, are a convenient concept for the representation and arithmetic manipulation of ill-known numerical quantities (e.g. Nahmias [21], Mizumoto and Tanaka [19, 201, Dubois and Prade [B, 10-121). Apart from combining fuzzy numbers, another crucial issue is that of being able to compare them, namely to decide to what extent one is greater or smaller than another. A consistent and complete approach to the comparison of fuzzy numbers or intervals may have useful consequences in such fields as decision analysis, when the worth of decisions is only approximately known (see discussions in Watson and Weiss [30], Freeling [16], Dubois and Prade [15]). However, existing approaches [3, 17, 1, 28, 30, 321 are not satisfactory. Some are counterintuitive, and most of them consider only one point of view on comparing fuzzy quantities. QElsevier Science Publishing Co., Inc. 1983 52 Vanderbilt Ave., New York, NY 10017

0020-0255/83/$03.00

DIDIER

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DUBOIS AND HENRI PRADE

This paper proposes a complete set of comparison indices in the framework of Zadeh’s possibility theory [34]. It encompasses Baas and Kwakemaak’s [l] method, as well as that of Watson et al. [30], and suggests new scalar comparison indices. The links between the possibilistic indices and a fuzzy one proposed by Zadeh [35] for the purpose of truth qualification of simple fuzzy assertions are exhibited. The presentation is organized as follows. First, a possibility-theory refresher is provided, in which the concept of necessity is stressed; canonical expressions for calculating the possibility and necessity of fuzzy events are established. A second section is devoted to the definition of “closed” and “open” fuzzy intervals which represent the sets of numbers greater (or smaller) than some fuzzily restricted variable. These sets are useful for the definition of the comparison indices. The completeness of this set of indices and the relationships existing between them are laid bare. In Section 3, previous works on this topic are discussed, assuming the proposed framework as a reference; numerical examples are provided to demonstrate the power of discrimination of the indices. In Section 4 these are extended to the problem of simultaneously comparing n fuzzy numbers. Lastly the concept of compatibility of two fuzzy sets (Zadeh [35]) is recalled, and it is pointed out that the possibilistic (scalar) indices can be simply recovered from the (fuzzy) compatibility index, as by a feature-extraction-like procedure.

1.

POSSIBILITY

1.1.

EVALUATION

AND NECESSITY

OF EVENTS

OF POSSIBILITY

Let U be a set of elementary events. Any subset of U is called an event. An event A c fJ is said to occur when some elementary event in A occurs. A possibility meusure (Zadeh [34]) on U is a set function lI from P(U), the set of crisp subsets of U, to the unit interval [O,l], such that

II(u)

II(0)=0,

VA,BE~‘(U), Given a normalized the quantity II,(A)

=l,

II(AuB)=max(II(A),II(B)).

fuzzy set F (i.e., there is some u E U such that pF( u) = l), derived from the membership function p, by

n,(A) =

sup ,+(u) UGA

VACU

(3)

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NUMBERS

185

defines a possibility measure. pF is the possibility distribution underlying II F and is often denoted TV. Equation (3) is easily interpreted as the possibility 01 realizing event A when the possibility of elementary events is known. Reciprocally, when U is finite, the possibility distribution underlying H is given by n(u) 0 II<{ u}). However, when I/ is not finite there exist some possibility measures which do not underlie a possibility distribution. Note that when rF is crisp [i.e., T~( u) E (0, l}], it defines a crisp set and

II,(A)

=l

*

AnF#0.

(4)

When both A and F are fuzzy, (3) can be readily extended, intersection, into H,(A)

using fuzzy set

= supmu&(u)+

(5)

Such an extension is the only possible one if we require (5) to be interpreted in terms of the intersection of the level cuts of F and A. More specifically, (5) is equivalent to (cf. Prade [23])

when

Clearly, II,(A) = II,(F), i.e., the possibility of a fuzzy event with fuzzy set of elementary events is a symmetrical concept expressing (Zadeh [35]) or partial matching between fuzzy sets (Dubois and This is not surprising, because the concept of possibility refers to tion. 1.2.

THE EVALUATION

OF NECESSITY

A necessity measure (Dubois JV gD(V) + [0, l] such that

and Prade [ll],

J-(0)=0,

J-(u)

Shafer [26]) is a set function

=l,

.N(A~B)=~~~(JV(A),JV(B)) Such functions

respect to a consistency Prade [13]). set intersec-

are termed “consonant

(7a) QA,BclJ

belief functions”

by Shafer [26].

(7b)

DIDIER

186

DUBOIS AND HENRI PRADE

Let Abe the complementary set of A, and II be a possibility is easy to check that the set function Jdefined by

measure. Then it

.A’“(A)Ll-II(z)

VAcU, is a necessity measure. Conversely,

given a necessity measure X, n(A)%l-N(r)

VAGU,

(8)

then (9)

defines a possibility measure. Equations (8) and (9) motivate the name “necessity measure” (also called “certainty measure” by Zadeh [36]): the grade of necessity of an event A is the grade of impossibility of the opposite event. Equations (8) and (9) are numerical translations of basic identities in modal logic. If II derives from a normalized membership function pF, then it is obvious that

QA,

~~(A)Pl-n,(A)=.:“61-~~(u)

(10)

When A and F are crisp we have

Hence, while possibility is related to intersection, necessity refers to set inclusion. When A and Fare fuzzy, (10) readily extends, consistently with (5), to

NF(A)=l-

sup~n(pF(u),l-pA(u)) u

=

i~fmax(l-~,(u),~,(u)).

(12)

The logical connective max(1 - a, b) seen in (12) is a multivalent implication defined by the identity P + Q = 7 P V Q, where negation -, and disjunction v are expressed by 1 - a and max(a, b) respectively. Then (12) is consistent with Dienes-Rescher logic [ll, Chapter 111.11.Of course, J$(A) # NA( F). The latter refers to the inclusion of A into F. When A is crisp, (12) yields

4(F)

= uizfAPF(U).

(13)

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N.B. .&(A) is not consistent with Zadeh’s usual definition of fuzzy set inclusion, i.e. F G A iff pF d pA. Zadeh’s definition is related to the multivalent implication min(l,l - u + b). On the contrary we have J$(A)=l

iff

FcA,,

whereA,isthepeakofA,i.e.A,={~(~~(u)=l}.Thecondition~~(A)=lis thus stronger than the condition F c A in the sense of Zadeh. See Dubois and Prade [ll] for a more extensive discussion of these points.

1.3. RELATIONSHIP

1.3.1.

7vEcEssrTYmD

BETWEEN

PossrmrTY

Crisp Events

Contrasting with probability theory, II(A) and II(A> do not convey the same amount of information and cannot be calculated from one another. However, they are loosely related via the identity [expressing II(U) = l] (A crisp).

max(II(A),II(A))=l

(14)

The meaning of (14) is “at least one of two contrary events must be possible.” Consequently II(A) and JV( A) are two nonredundant pieces of information which must be considered simultaneously. The following relationships hold between them:

J’-(A)0 II(A)

~1

a

J’(A)=0

*

II(A)=1

VAGU, e

(15) lI(A)=l,

(16)

N(A)=O.

(17)

The relation (15) is quite consistent with commonsense observation: what is necessary must be possible, but not conversely. The relations (16) and (17) make explicit the complementarity of indices N(A) and II(A): when one of them is informative (e.g., JV( A) > 0, II(A) < 1) the other one provides no useful information (respectively, II(A) = 1, JV( A) = 0). This remark will prove crucial when comparing fuzzy numbers: all already known comparison indices take advantage of possibilistic-like indices, but none of them makes explicit use of necessity. This fact explains their relative lack of discrimination power, as will be indicated in Section 4.

188

1.3.2.

DIDIER

DWBOIS AND HENRI PRADE

Fuzzy Events

If A is a fuzzy set, then it is well known that A U x# U, using “max” for defining fuzzy set union. As a consequence max( II F( A), II F( A)) + 1 in general. However we have SUp/.LA”~-(U)>0.5. UCU Hence the following result: 2 0.5

max(II.(A),n.(A)) if and only if

(18)

sup /.bF( 24) > 0.5. u=u

Proof.

=,a[

sup min(pF(u),p,(u)), UGlJ

sup mGF(u),lrl(u))] UCU

= ~~pumin(p,(u),max(p,(u),I-~,(u)))

= IIF(A

ux)

E [-n(@5,

However, if F is normalized,

;z;P,(u)).

;E;pF(u)].

W

then we still have:

II.(A)>l-II.(A)%/I$(A).

(19)

Proof.

n,(A)+ h(z)

2 ~~~[~n(p,(u),p,(u))+~n(l-B,(u)~p,(u))l 2

sup PF( u) =l. u=u

But this is not a necessary condition.

n

RANKING 2.

FUZZY

RANKING

NUMBERS

TWO FUZZY

189 NUMBERS

OR INTERVALS

This section first deals with the simpler problem of comparing a fuzzy number and a crisp (usual) one. The obtained results are then extended to the case of two fuzzy numbers or intervals. A set of comparison indices is derived and proved to be complete in the sense that it is necessary and sufficient for characterizing all respective configurations of two crisp intervals. Relationships between the proposed indices are then exhibited. As a consequence four basic possibilistic comparison indices emerge and can be interpreted as being optimistic or pessimistic. 2.1.

COMPARING

A REAL

NUMBER

AND A FUZZY

INTERVAL

A fuzz_y interval is a convex fuzzy set of the real line, with a normalized membership function, i.e. a fuzzy set M of the real line Iw such that

Qu,v,

vwE[u,vl, 3rnER

~IM(w)amin(CLM(u),CLM(v)) (convexity), P#+f(Iyf) -1

(norm~ation)

.

(20)

When m is unique, M is referred to as a fuzzy number; a fuzzy interval encompasses all sorts of crisp intervals, including real numbers. Given a fuzzy interval M viewed as a fuzzy restriction on the values of some real variable x, several fuzzy sets of numbers having M as a fuzzy bound can be introduced: (a) The set of numbers possibly greater than or equal to x, denoted [ M, + co), such that

in the notation of Section 1. In other words, it is the grade of possibility of the event x d r when x is restricted by M. The notation [M, + co) stems from the fact that if M is a crisp number m, then (21) defines the characteristic function of [m, + cc). (See Figure 1.) (b) Similarly, the set ( - m, M] of numbers of possibly smaller than or equal to x is defined by

DIDIER

190

DUBOIS AND HENRI

PRADE

Now the set of numbers necessarily greater than M, denoted ] M, + co), can be defined by means of necessity grades consistently with (22). Namely, ] M, + co) has a membership function

VrTCLIM. +mj(r> =-TM((-w74 = inf l-pM(u). rgu

(23)

In other words, it is the grade of necessity of the event x < r when x is restricted by M. The notation ] M, + co) stems from the fact that if M is a crisp number m, (23) then defines the characteristic function of the interval ]m, + cc), which is open at m. [M, + 00) and ] M, + 00) are pictured on Figure 1. The reader is referred to [14] for a topologically oriented study of such fuzzily bounded domains of the real line, Similarly the fuzzy set (- w, M[ of numbers necessarily smaller than x is defined by

Vr,

Pc-m,M[(r)=~~l-~M(U)=~M(lr,+W)).

The following relationships

obviously

hold:

[M,+w) =(-w,M[,

(-oo,M] =]M,+m),

where F is the fuzzy set complementary

(24)

(25)

to F, such that PF = 1 - pF;

(-=df[c((-mM]),,

]W+~)c([M,+~)),,

where Fl is the peak of F [the latter relationship

Fig. 1

(26)

stems directly from (16) and

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FUZZY

191

NUMBERS

(17)l; ad [M,+w)n(-ao,M]=M. Indeed the membership en(

sw~,(u),

u&r

value of the left-hand UNPILE) 0> r

=

(27)

side is for any r E R

sup ~~~cL~(u),cL~(u)), u t;“< ”

i.e. the membership function of the convex hull (Lowen [IS]) of M, i.e. M itself, since it is convex. Equation (27) does not hold when M is not convex. M is then only included in the left-hand side of (27). N.B. Given the crisp number r, we should strictly speaking consider two more quantities which naturally parallel H,((m,r]) and .A’+,((- co, r[), namely II,(( - cc, r[) and A”‘(( - cc, r]). However, since the latter quantities are obtained by changing < into < in both (21) and (23), it is easy to see that if pLMis left-continuous in r, then

n,((-oo,rl)=~M((-w,r[); if pM is right-continuous

in r, then

~~((-,,r[)=~~((--,rl>. Similar remarks hold for H,,,(]r,

+ co)) and h”,,([r,

Fig. 2

+ 00)).

DIDIER

192 J..?.

PAIR WISE

COMPARISON

OF FUZZY

DUBOIS AND HENRI PRADE

NUMBERS

Extending the above approach, we can consider the following quantities in order to assess the position of a fuzzy number N relative to that of a fuzzy number M taken as a reference:

assessing to what extent M is greater than N. The values of these indices are pictured in Figure 2 for a given set of linear M and N.

2.2.1.

Expressions and Interpretations

The four indices have the following expressions:

F,,(lN,+~))=

wkn(PM(u), infl-p,(u)) us

u

u

= sup inf min(~M(u),l-~N(u)), ” “ZU

(29)

-6,([N,+w))= = if

supmax(l-~,(u),Cc,(u)), LIL u

Equation (28) [respectively, (29)] refers to the degree of nonemptiness (or partial matching [13]) of the fuzzy set M fl[ N, + CO) [respectively, M n]N, + oc)]of numbers greater than or equal to [respectively, strictly greater than] N, given

RANKING

FUZZY

193

NUMBERS

that they are restricted by M. Equation (28) [respectively (29)] yields the grade of possibility of the proposition “x is greater than or equal to N ” [respectively, strictly greater than N] given that “x is M”. Briefly,

H,([ N, + 00)) = POSS(x II,(]

> NIX is M),

N, C w)) = POSS(x > N]x is M).

Similarly, (30) [respectively, (31)] refers to the degree of inclusion of the fuzzy set M in [N, + cc) [respectively, IN, + cc)]. Equation (30) [respectively, (31)] yields the grade of necessity (“certainty” according to Zadeh) of the proposition “x is greater than or equal to N” [respectively, “strictly greater than N”] given that “x is M”. Briefly,

A(,,,([ N, + co)) = Ness(x > Nix is M), JM(]

N, + m)) = Ness(x > N]x is M).

Hence the introduced indices can easily be interpreted as special cases of test scores (Zadeh [37]). Now given the four basic indices (28)-(31), we can build three other sets of indices, each containing four, by changing + cc) into - co, changing M into N, or making both changes simultaneously. Before laying bare the relationships between all these indices, let us point out the completeness of (28)-(31) in the crisp case. 2.2.2.

Completeness of the Comparison Indices in the Crisp Case

If M and N are closed crisp intervals M=[m,,m2], N=[n,,n,], Table 1 clearly indicates that the four indices are necessary and sufficient to characterize the respective locations of M and N. Indeed, on the right of the table are all six possible configurations of two intervals with respect to inclusion and disjointness. Three of the indices alone cannot discriminate all these configurations. In the table it is assumed that ml f ni and m2 f n2. Otherwise we get pathological cases, which are examined further on; such cases seldom occur with fuzzy numbers. 2.2.3.

Relationships between the Comparison Indices (28) - (31)

Although the values of each index H,([N, + cc)), II,,,(]N, + co)), M,([ N, + co)), &,(I N, + 00)) are independent and cannot be calculated from

194

DIDIER DUBOIS AND HENRI PRADE

RANKING

FUZZY

NUMBERS

195

each other, there exists some natural loose relationships

between them, namely:

n,([N,+oo))bmax(~I,(lN,+oc)),~~([N,+co)))

(32)

min(n,(]N,+oo)),~~([N,+oo)))~~~(lN,+03))

(33)

Proof. n,([N, + CO))a nM(]N, + 00)) because IN, + cc) is included in [TV, + co). Moreover, II,,,,([ N, + cc)) z JV~([N, + co)) from Section 1.3.2, since M is normalized. Equation (33) is obtained similarly.

However, Table 1 clearly indicates that X,,,([ N, + co)) can be greater than the other.

2.2.4.

Indices Related to II,([

either

N, + CO)) and JM(]

of H,(]

N, + 03)) and

N, + CO))

From (28) it is patent that

which reads

=l-JV~((-cc,N[)=l-.A’-N(]M,+oe)). Moreover (34) obviously

(34)

leads to

~~(]N,+w))=l-n,([M,+w))=~~((-w,M[) =l-

II,((

- cc,N]).

(35) (36)

Interpreting (34) in the setting of test-score semantics (Zadeh [37]) results in the following equality: Poss[x>N]xisM]=Poss[x
DIDIER

196 Hence (36) is interpreted

DUBOIS AND HENRI PRADE

as

Ness[x>NjxisM]=l-Poss[x N” is semantically the following loose relationship

equivalent to “x is not < N.” Another result is between H,([ N, + co)) and II,,,(( - co, N]):

max(HI,([N,+w)),n,((-w,Nl))=l. Proof. Recalling that [N, + oo)U( - 03, N] = W, Equation since the two fuzzy events cover the universe.

(37)

(14) can apply, w

What (34)-(36) point out is that all other indices built from

by exchanging + cc and - cc, and/or M and N, are redundant this set. Equation (37) moreover indicates that

n,([N, + a>> -cl

-

4,(]N,+w))=O,

(38)

.&(]N,+cc))>O

*

II,([N,+cc))=l.

(39)

We could as well choose as a set of independent

i.e. define a fuzzy relation between M and N. 2.2.5.

with respect to

expressing

Indices related to II,(]

Using the possibility-necessity

indices

the possibility

of dominance

or equality

N, + 00))

relationship,

it is clear that

IT,(]N,+w))=l-~~((-w,NI),

(40)

H.(]M,+w))=l-Jlr,((-w,Ml).

(41)

Moreover, the following inequality

holds: (42)

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197

Proof. We have to prove Vu,Vv

~~(~,(u),cL~~,+~,(u))~~~(cL(-,,~~(v),~--~(v)). (43) Denote S(M)={u]~M(~)~O}=supportofM, M,={~]~~(u)=l}=peakofM, rni = inf M,, and the same for N. Obviously

u~S(M)n[n,,+w),

m,=supM,,

we may restrict ourselves to

uf

v~["~,+co)Us(~),

v.

Assume cxA HM(] N, + 00)) < 1; then clearly m, < sup S( N). Thus, we can restrict ourselves to u, v E [ %,,sup S( N)]; indeed, on this interval CL,,,, = p( m, M1 decreases from 1 to some value /? < 1, and pi = pJN, + mj increases from some value y/max(fi,y). For u supS( N), pi(u) = 1. Hence we come down to the following inequality to be proved instead of (43):

mi44&)~1-h(4)


Vu,vG

[m,,supS(N)]. (44)

Note that on this interval p,,, is nonincreasing

and 1 - pN is nondecreasing.

with

Note that inf 62+ = fi,, sup CI_ = sup S( N). Furthermore,

V(x,y,z)EQ+X3,xD-,

x-=y
Let

198

DIDIER

Also,Vu~8+,Vu~9+, because

Equation(44)

DUBOIS AND HENRI PRADE

readsl-~,(u)<~,,,(u),whichholds,

if

u
I-ccN(u)~I-cLN(u)
if

u>u,

l-cLN(u)
Equation (44) holds too for (u, u) E (D _)‘, by the same reasoning, and trivially for(u,u)E(&)‘. If we choose UEQ+, u~i&,~i2_, then (44) reads 1-~N(u)61-~,,,(u), which holds because u < u. The same goes for u E 9, U Q,, u E Cl_. Lastly, if we choose UE~&, uEi&,Ufi+, then (44) reads p,,,,(u) d pM(u), which holds because u > u. The same goes for u E D _ U a,, u E 8,. Now note that when (Y= 1, then

w=(-oo,M]U]N,-too)=(-co,M]UN, and so.

There are however many instances when (42) holds with the equality. This is true wheneuer one of the following conditions is fulflled (using the notation in the proof aboue): Cl. C2. C3.

�, 8, = 0 but pM is continuous at the point sup 52, = inf 52_ L o, &, = 0 but pN is continuous at the point sup D + = inf D _ L w.

Proof. If 3, + 0, let u E !&,; then

Hence the maximum of min( k,,,( u), 1 - pN( u)) and the minimum of max( pM( u), 1 - pN (u)) are attained for u E at,, from which it is obvious that

RANKING If Q2,=0,

FUZZY

199

NUMBERS

it is obvious that

~~((-m,MI)=min(p,(w),~~l-IIN(u)). If pM is continuous H,(lN,

in w, then + a))

= max(I-CLN(w),~LM(w))=IIM(w),

= If l.~~ is continuous

Similar reasoning

~~~+duLl-

h(u)) =hAw).

in w, then

n

for w E Ll~ results in the same finding.

Thus, we have proved that whenever II M(] N, + co)) = 1, or II,(] under one of conditions Cl-C3, we have

N, + 00)) < 1

I-I,(lN,+oo))=~~((-d~,Ml).

(45)

It is patent that Cl through C3 are very weak conditions, and that cases when a strict inequality occurs in (45) are pathological and rarely encountered in practice, since they may occur only when pM and p,,, are discontinuous simultaneously at point w. For instance, M=

[a,,b],

h’= [a,,b],

Mn]N+co)=O,

(- w,M]=(-~$1, Fu(-oo,M]=lR

n,(]N,+m))=O,

However,

if M=[a,,b],then(-co,M]=(-co,b[

]N,+oo)=]b,+oo), sothat

w=b,

JV~((-c~,M])=l. and NU(-co,M]-=R-

DIDIER

200

{ b}, and (45) is valid. In Figure 2, Equation (41) combine into

DUBOIS AND HENRI PRADE (45) is valid. Equations

n,(]N,+oo))+~,(fM,+oo))~I.

(42) and

(46)

Thus the index II &] N, + to)), which can be interpreted

as

Poss( x > Nix is M), can be used to define a fuzzy relation of possibility of strict dominance between M and N, which can also be expressed as fuzzy relation of necessity of dominance or indifference, since (40) reads Poss(x>

N]xisM)=l-Ness(xG

N]xisM).

Note that when (45) holds, which is bound to occur quite often, then the fuzzy relation II.{]., + co)) satisfies (46) with equality; it is then called a tournament fuzzy relation.

2.2.6. Indices Related to .A’&([N, + m)) This section is very similar to the previous one, and the proofs of exhibited relationships are thus omitted for brevity. The relationships are the following:

~~([N,+oo))=l-II,t(-oo,N[),

(47)

Jr/-,([M,+co))=l-II.(t-oo,M[),

(48)

~~flN,+cx,))~n.((-oo,Mt).

(49)

Under one of conditions

Inte~retively

Cl-C3,

(49) holds with equality;

using (48), it reads

we can write XM([N,+oo))-Ness(x>,N]xisM), II,,.,((-oe,N[)=Poss(x
Whenever relation.

(50) holds, then the fuzzy relation .K([ -, + 00)) is also a tou~~ent

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Overview

Table 2 summarizes all the equalities or inequalities which have been proved to hold between the 16 indices we can build in a possibilistic setting when we want to assess to what extent one of two fuzzy intervals or numbers, M and N, is greater than the other one. Except in pathological cases which are discussed at length in Section 2.25, among these 16 indices only 4 have independent values, and thus we only need 4 index values for pairwise comparisons of fuzzy intervals or numbers. This explains why we found 4 indices were sufficient to describe the respective locations of two crisp intervals having distinct bounds [when some of the bounds are equal, we may have to calculate 6 values, since (45) and (50) may no longer hold]. Using the set of indices shown in Table 2, it can be easily checked that M and N play symmetrical roles and that it is equivalent to calculate to what extent M is greater than N (i.e. comparing M with [N, + co) and ] N, + m) and alternatively comparing N with (- co, M] and (- cc, M[) or to calculate what extent N is smaller than M (i.e. comparing N with (- cc, M] and (- cc, M[ and alternatively comparing M with [N, + co) and IN, + co)). The indices in the first line of Table 2 evaluate the possibility of inequality in a broad sense (M 2 N or N < M). However, in lines 2 and 3 we notice that M,(( - cc, M]) and JV~([ N, + co)) can take distinct values, while we have H.((-w,M])=H,([N,+oo));thisisbecauseMn[N,+w)and(-w,M]n N have the same height while the degree of inclusion of M in [N, + 00) may be different from the degree of inclusion of N in (- 00, M]. Thus lines 2 and 3 together correspond to the evaluation of the possibility of strict inequality and of the necessity of inequality in a broad sense. The last line of Table 2 evaluates the necessity of strict inequality. It is important to keep in mind that the indices of line 4 are equal to zero as soon as the indices of line 1 are strictly less than 1 and that the indices of line 1 are equal to 1 as soon as the indices of line 4 are strictly positive. Bearing Table 1 in mind, we can provide an interpretation of the indices in terms of respective locations of fuzzy numbers M and N, as is done in Table 3. (N.B. See [8], [lo], and especially [ll], or Section 4.1, for the definition and properties of G.) In the table, LARGE means “close to 1.” SMALL means “close

TABLE 2

aThe equalityholds except in some “pathological”

situations

described

in the text.

LARGE

LARGE

SMALL

LARGE

LARGE

SMALL

SMALL

SMALL

1

1

1

1

SMALL

SMALL

SMALL

LARGE

LARGE

1

0

SMALL

SMALL

SMALL

SMALL

LARGE

TABLE 3

RANKING

FUZZY

203

NUMBERS

to 0.” Note that only six cases need to be considered as far as we use only values; indeed, the inequalities (32) and (33) together with max( II (A), II (A)) = 1, forbid considering other configurations” For instance, “H,([ N, + co)) is SMALL" implies that each of the following statements is wrong: “&$([N, + co)) iS LARGE" n,(]N, + m)) is LARGE," "II,([hi,+ co))is SMALL." Note that Table 3 is really a fuzzy version of Table 1. However, since we consider fuzzy numbers, it is useful to introduce more than two levels such as SMALL and LARGE. If we introduce one more level, say MEDIUM, which means “close to 0.5,” then Table 3 can be completed as shown in Table 4. (N.B.Other rows could beadded, thus introducing more refinements.) Another way of interpreting the indices is the following: LARGE and SMALL as reference

(a) H,([ N, + 00)) is large when the least values compatible with N (left side of N) are smaller than or equal to the greatest values compatible with A4 (right side of M). (b) H,(] N, + co)) depends upon the relative locations of the right sides of M and N (it is large when the greatest values compatible with N are smaller or equal to the greatest values compatible with M). (c) A”‘([ N, + m)) depends upon the relative locations of the left sides of M and N (it is large when the least values compatible with N are smaller than the least values compatible with M). (d) N,,,(] N, + 00)) is large if the least values compatible with M are greater than the greatest values compatible with N. 3.

RELATIONSHIPS

WITH PAST WORK

A consistent and extensive framework has been proposed for the ranking of fuzzy numbers, in the setting of possibility theory. This section gives a review of past work on this topic, in order to demonstrate the relevance of our framework. Numerous approaches for ranking fuzzy numbers have been proposed in the literature. Jain [17] proposes to match each fuzzy number N, (i = 1,. . . , n) with a fuzzy goal G, a fuzzy set of the considered rating scale whose meaning is “as great as possible.” If for instance the rating scale is [0, a], then G is such that

Vx=[O,al,

PC(x)=

x ’ (a1

forsomep>O.

The rated objects 0, (i = 1 , . . . , n) are then ranked according of N, with G: c, = supmin(p,(x),pN,(x)). X

(51)

to the consistency

204

DIDIER

DUBOIS AND HENRI PRADE

RANKING

FUZZY

205

NUMBERS

a Fig. 3

Such a method has been criticized by Dubois and Prade [7] and Baldwin and Guild [3] as likely to provide counterintuitive rankings (cf. Figure 3). Dubois and Prade [7] and Freeling [16] have suggested the use of the extended maximum and minimum operators [lo]. However, as Baldwin and Guild [2] point out, z( N1, N,) may be neither N1 nor N2, so that in some instances, no discrimination is achieved where intuition would discriminate [see Figure 4(a)]. Besides, N1 and N, may be very close to each other while &( N,, N2) = N2 and intuition are not so definite [see Figure 4(b)]. Yager [32] bases his ranking procedure on the computation of the Hamming distance between fuzzy sets, say for bounded-supports fuzzy numbers N1 and N2:

Yager’s purpose is to rank fuzzy truth values, viewed as fuzzy sets of the unit interval, in terms of their proximity to the standard fuzzy truth value “true”

#

----

mTa”x (N,,&)

Fig. 4.

DIDIER

206

DUBOIS AND HENRI PRADE

defined by plrue( u) = u Vu E [O,11. This proximity is expressed by means of the Hamming distance (52). Clearly Yager’s purpose is very specific and is not relevant to the framework of this paper, although (52) may be viewed as a probabilistic comparison index between fuzzy numbers. Baas and Kwakemaak [l] have proposed to extend the inequality ui > u2 in a canonical way to fuzzy numbers:

It is clear that this index has been considered in the previous section under the name IINL([ N,, + cc)). These authors readily extend (53) to n fuzzy numbers as follows: the grade of membership of N, to the set I of greatest fuzzy numbers among N,,...,N, is

P&> =

Vi,

sup rr,.ma~, I

~n~*,(~,). i

Although this approach is very natural in the framework of fuzzy set theory, Baldwin and Guild [3] have stressed its lack of discrimination power in situations such as the one pictured in Figure 5, where int~tion would favor N2 as being greater than Ni. Baas and Kwakemaak [l] also define a fuzzy preferability index P, for object 0, rated by Ni; however, ultimately the P,‘s have still to be tanked and the difficulty remains unsolved. Watson et al. [30] suggested viewing the ranking problem as modeling the implication 0, is Ni and 0, is N2

Fig.

*

0, is preferred to Oz.

5. d(~%‘,>&)=d(&~~~,)=~

RANKING

FUZZY

The membership

and the right-hand

207

NUMBERS

grade of the left-hand

side is

side is the usual order of numbers, 1 Pv(%,

The multivalent implication index d’ is defined by

%) =

t0

is proposed

if

denoted

Y:

ui>uz,

otherwise. to be a 4 b = max(l-

a, b). So an

inf max(l-~,(u,,u,>,~Ly(U1,U2))

d’(N,>N,)=

u11 u2

= ~,i~t~[l-min(r,,(u,),p,*(“*))l

=l-

%*([N,, +4).

(55)

Thus this index d’ is nothing new, but represents another calculation on the same approach as Baas and Kwakemaak’s. By the way, this result clearly indicates that the implication max(1 - a, b) is consistent with possibility theory. It is patent that Baas and Kwakemaak’s ranking index, despite the mentioned drawback, is the most natural one in accordance with the basic principles of possibility theory. Its lack of discrimination power does not make it irrelevant, but indicates that some other indices should be used simultaneously. Namely, it is clearly an optimistic index, since in Figure 5, N1 and Nz are not discriminated, because their locations do not prevent 0, from having an ultimately better rating than 0,. Baldwin and Guild [3] modify (53), replacing ui > u2 by a fuzzy ordering relation R so as to improve the discrimination power of the index, i.e.

Although this idea is very much worthwhile, d, is still an optimistic index. The relation Y could be similarly softened in Watson et al’s framework. Very recently, Tsukamoto et al. [29] have proposed a set of ranking indices very much related to the ones described in the previous section. Namely, a fuzzy interval is viewed as the intersection of two S-shaped fuzzy sets: M = MI. n M,,

208

DIDIER

DUBOIS AND HENRI

PRADE

where

MrA (-w,M], M$ [M,+w). If M is the referential fuzzy set, the comparison means of the following indices: t, =

of M and N is performed

by

supmin(l-~,,.(u),CLN(u)) k n,(lM, + w>>, ld

t2 =

swfin(puM(u),pN(u))= n,(M) = n,(N), u

t3=supmin(l-ll~R(u),CLN(u))~~N((-w,M[). u

It is clear that t, is an index for the assessment of the possibility authors also mention the dual indices

t, and -t, can be related to already introduced t2=rIN((-a,M]n[M,+a))= =1-max(

of e+&ity.

The

indices if we notice that

min(~,((-w,MI),~I,([M,+oo)))

t,, t3). _-

This is because M and N are convex normalized fuzzy sets and (- 03, M]U [M,+co)=R. Let m andn be such that pw(m)=pN(n)=l, andm
RANKING Moreover,

FUZZY since

209

NUMBERS

[ M, + co)n a = ] M, + co),

and similarly It, =l-max(t,,tZ), so that (t,,t,,r,) contain redundant information with respect to (I~, t,,t,). _-Hence Tsukamoto et al. [29] develop basically the same approach as ours: however, they do not interpret it in the setting of possibility theory, nor do they demonstrate the completeness of the set of indices. 4.

EXTENSIONS

TO THE RANKING

OF n FUZZY

NUMBERS

Let N,,..., N,, be n fuzzy quantities having the shapes of fuzzy numbers or fuzzy intervals. The problem is now to be able to rank them in, say, decreasing order, through the use of the introduced indices. There are at least two ways of achieving this purpose: (a) to extend the pairwise indices to nary versions, (b) to process the fuzzy relations obtained by pairwise comparison obtain some final rankings. The following is a discussion 4.1.

so as to

of these two approaches,

GLOBAL RANKlNG INDICES

Grades of dominance of a fuzzy number N, over all other Nj, j # obtained if we state the problem in the following form: Is N, greater equal to) the greatest of the N,‘s (j + i)? If xi,. . . ,x, denote variables by Ni,...,N,, the variable Y=max(x,,...,x,) is restricted by the max(N,,...,

i, can be than (or restricted fuzzy set

N,) defined by (see Dubois and Prade [ll])

VW,

Pzz(N,.....iv”)W =

sup U,,...,“”

r=p.n

7 . ..n

PLN,CUi).

(57)

w=max(u,,...,u,)

This fuzzy set is very simply obtained (57) can be rewritten as

from the shapes of N,, . . . , N,,; namely,

(58)

DIDIER

210

DUBOIS AND HENRI

PRADE

The dual operator z can be similarly defined. The problem of finding grades of dominance of N, over the other N,‘s (j # i) comes down to one of pairwise comparison of N, and max, + , N,. Based on the results of Section 2, the framework of possibility theory provides four indices for the dominance of N, in the set N,, . . . , N,,: (a) a grade of possibility

of dominance,

(this is the index first proposed by Baas and Kwakemaak (b) a grade of possibility of strict dominance,

[l, Equation

(54)]);

i ‘sN,,+mjj;

PSD(N,)=&v,

(c) a grade of necessity of dominance,

(d) a grade of necessity of strict dominance, NSD(N,)=l-II;;;;;;,

,,+

([

N,,+cc)).

The set Ni,..., N, can then be ranked in terms of decreasing values of each index; we thus obtain four linear orderings, which may or may not be consistent. If they are consistent, the corresponding ranking is then validated.

‘123456

)

t

Fig. 6

T

9

40

RANKING

FUZZY

NUMBERS

211 TABLE 5

,~

Let us consider a numerical example with three fuzzy numbers having linear membership functions (Figure 6). The calculation of the values of the indices is a simple matter of intersecting straight lines (Table 5). Due to the overlapping of the fuzzy numbers, none among them is necessarily strictly dominant. However, Ns is clearly the greatest in terms of the possibility of strict dominance, which refers to the right-hand sides of the fuzzy numbers. The grade of necessity of dominance clearly rules out Nr (left sides). However, it is possible to choose precise values ur, uz, us such that u1 = us > u2, which is expressed by the index PD.

4.2.

FUZZY

UUTRA NKING R ELA TIONS

Another approach consists of building fuzzy outranking relations on the set Ni,. _. , N,, through pain,ise comparison of the fuzzy numbers. Denoted by PD, PSD, ND, NSD the four fuzzy relations built on the indices

KU*, +m), K(l-9 +cQ>>,Jv. ([-, +cc)), JK (]a, +cc)).

4.2.1.

Relation of ~~~~ibi~~tyofDominance

PD satisfies the following properties: Vi=l,...,n,

if i#j, Qi,j, Equation

p,,(i,i)

=l

fhdirj)
(60) is the dual of the “perfect

-j =I-

(reflexivity);

(59)

kAj,i)=l;

(60)

hs,(j,

~tisymmet~”

i). introduced

(61) by Zadeh

DIDIER

212 [33]; this property

DUBOIS AND HENRI

PRADE

is satisfied by NSD, and reads

However, the lack of transitivity of the fuzzy relations PD and NSD prevents us from using the Hasse-diagram approach to the search for nondominated elements (cf. Zadeh [33], Orlowski ]22]). Other methods are possible. For instance, the technique described by Shimura [27] is well fitted to a fuzzy relation such as PD. This method first collects grades of preference f,(j) and f,(i) for a pair of objects (i, j) E X2; they are normalized in the following way:

V(i,j)EX*,

Pft(i,j) =

f,(i)

m4f,(iM,(.d)

= relative grade of preference of i overj Note that R satisfies (59) and (60), i.e., the relation PD fits Shimura’s framework. The grade of possibility of dominance of TV,over { N, , j f i } is defined by

th(i) = ,=~_y,,nPPD(i,.il; the fuzzy numbers

(63)

can then be ranked in terms of gPD(i).

PROPOSITION. =PD(N,).

=

(

suptinh,(u), 84

sup sup minh,(x,)

wz, w=m~J(\,)

i

1

RANKING

FUZZY

but, denoting

NUMBERS

213

hgt( A) = sup pa, we have

hgt( ,=() A,j=n$?w(A,nA,) (cf.[9,A=x21). . .n

N~~hgt([N,,+~~)n[N,,+ocl))=l PD(N,)

=

Vi,j. mm

/=I

Hence P~~,(U),IL,~, , +,,(u))

. . . ..n

u

=&m(i),

B

Hence Shimura’s method for processing the fuzzy relation of possible dominance is equivalent to the one in Section 4.1. Another method for processing the PD relation can be found in Bonmssone and Tong [28]. The latter use fuzzy arithmetic [ll] for calculating a fuzzy dominance index 8(i) defined by subtracting a weighted sum of N, (j + i) from N,. The weights are chosen as g,,(i). The fuzzy-dominance-index values are then interpreted linguistically.

4.2.2. PSD

Relation of Necessay

Dominance

ND or Possible Strict Dominance

The fuzzy relation ND generally satisfies the following identity:

Vi,j,

cL~O(~~j)+~N,(j,~)=l,

(64)

provided that the N,‘s satisfy condition Cl, C2, or C3, for instance when they have continuous membership functions. Hence ND is a so-called tournament relation, which can be processed by means of a ranking algorithm described by Dubois and Prade [ll, p. 2791: Note that pND(i, i) = 0.5, so that pND( i, j) > 0.5

means

The ranking algorithm first considers dominated by N,, such that

Then inclusion

relationships

between

i dominates j

the fuzzy class

P, (i)

of numbers

the fuzzy classes P, (i) are searched for;

DIDIER

214

DUBOIS AND HENRI PRADE

however, since Zadeh’s inclusion is usually too strong a concept for any result to be obtained, we use a weak inclusion A --
This inclusion such that

is transitive.

As shown in [ll], inconsistencies

P, (i)-
ad

P, (j)-
occur only if 3i, j

(i)

to

3k:

pND(i, k) > 0.5,

pND(k, j) 2 0.5,

3:

pLND(j, I) > 0.5,

p~o(f,

(65) i) > 0.5,

i.e., N, dominates Ni. Before proceeding to the main property of ND let us recall two definitions: The a-cut of N, is the set { uIpN,( u) > CX}= (N,),. The strong a-cut of N, is the set { uJpcLN,( u) > a} h (N,),. PROPOSITION. ND is not inconsistent when the N,‘s have continuous member-

ship functions. Proof. It is sufficient to show that (65) cannot occur, First notice that pND( i, j) > 0.5 means

Qu,

ma ( I-

P~,(u)~+,

. +,,(U,)

> 0.5,

i.e., in terms of a-cuts,

(N,” [N,,+ d),.,

=W =

where (N,),,, is the 0.5~cut of N,, (N,), is the strong 0.5-cut of N,. In other words,

(N,)c u([N,t+d),,,

RANKING

FUZZY

NUMBERS

215

Similarly jlND(irj))0.5

means

(K)*.,C

([N,$+(JrJ))G.

Ilence 3k:

pND(irk)>0.5

reads

(iV,),,C

pND(krj)a0.5

reads

(Nk)~G([~,+~)),,,,

reads

(N,),,,C ([N,,+oo))os.

31: pNo(jrl)>0.5

SincePN,TPAJ,YcL,v,tPN, are continuous, strong a-cuts are open intervals. Let

the a-cuts

([Nk,+co>f~,

are closed intervals

and the

n (0.5) = inf( N,,,), n(0.5) The three inclusion

relationships n,(0.5)

= inf( Nrj3).

imply

> 3,(0.5)

Z n,(O.5) > nt(0.5).

Let u ~]~~(0.5),ni(0.5)[; clearly ~~,(u)=p~~,,.+~~(u)<

0.5 and

/.N,(u) > 0.5.

i.e., ~~~(1, i) < 0.5. It is thus clear that (65) is impossible, ND never occurs.

i.e., inconsistency

of H

It is easy to see that the fuzzy relation PSD is also a tournament relation which can be processed by the same approach without inconsistency in the ranking. FXAMPLE. Table 6.

The values of PSD and ND in the previous example are given in

DIDIER

216

DUBOIS AND HENRI

PRADE

TABLE 6

~

Row i of each matrix contains the membership values of the fuzzy sets of numbers do~nated by N,, for PSD, and ND, respectively. it is easy to see that PSD, --< PSD, , PSD, --< PSD, , i.e., N3 is ranked first, while PSD, Z=-C PSD, (double inclusion, i.e. weak equivalence). Hence N3 is possibly strictly greater than Nt and N2. Moreover ND, --
ND, --
while

ND, >--< ND,, i.e., Ni is ranked last; in other words, N2 and N3 are necessarily

as great as Nr.

N.B. Other methods for turning fuzzy ranking relations into crisp orderings exist in the literature, and could be useful for comparing fuzzy numbers. Let us mention: (a) B/in’s nlgorithm [4], where a crisp linear ordering A* is determined from a fuzzy relation, where A* is consistent with the greatest amount of membership in the fuzzy relation; this consistency is expressed by the choice of A* as maximizing

where

I(h)=

{(i,i)EA,PR(itj)
RANKING

FUZZY

NUMBERS

217

and A is any linear ordering such that

pR(jr j) =1

-

(i,j)EA.

This approach is valid only if a(i, j) such that pa(i,j) =pR(jr i) =l, for instance when R = ND or PSD. (b) Roy’s ulgorithm (ELECTRE III) [25], based on the use of a perception threshold s(i, j), such that i is viewed as significantly dominating j as soon as

where s( i, j) depends upon the membership value pR( i, j). The ranking algorithm is then based on the knowledge of how many elements are significantly dominated by i or dominate i. The procedure provides two crisp partial orderings: one obtained by selecting first those elements which dominate as many elements as possible, the other by selecting those elements which are dominated by as many elements as possible. More details can be found in [25].

5.

RELATIONSHIP

WITH ZADEH’S

COMPATIBILITY

INDEX

A very natural approach to the comparison of two fuzzy sets has been first put forward by Zadeh [35]: Let A and B be two fuzzy sets over a universe ZJ The compatibility of A with respect to B is obtained by applying the extension principle and is a fuzzy set of the unit interval, which we shall denote C( A/B), such that

Pc(a,B)(t)=

SUP clE?(u) u:t=pA(u)

= 0

if

au:

pLa(u)=t.

C( A /B) can be interpreted as the fuzzy set of possible values of the membership grade in A of an element of U fuzzily restricted by B. As such, C( A/B) could be symbolically written pA (B), underlying the idea that an ill-located element of I/ has but a fuzzy membership grade in A. C(A /B) is equated to the fuzzy truth value r qualifying a proposition “X is A,” with respect to a proposition “X is B” taken for granted (Zadeh [35]), i.e., “‘X is A’ is 7” is viewed as equivalent to “X is B.” If Adenote the fuzzy complement of A, it is easy to see that C(z/B)

=leC(A,‘B)

DIDIER

218

where 8 denotes the extended subtraction

Pc+J,(4 =

DUBOIS AND HENRI

[8, 111. Indeed, sup

cLB(U)

u:r=l-p,(u)

i.e., C(x/B) is the antonym of C(A/B). However, C(A/B) is generally not related to C(A/B),

k(A,BAd =

PRADE

sup l-/JL(U)=lu:r=p,(u)

since

u: ,&f,,,‘B(“).

Let T and ‘p be the fuzzy sets of [O,l] such that

It is obvious that if N is a continuous that %i is also normalized, then C( N/N)

= r,

normalized

fuzzy set of the real line such

C( w/N) = cp

[the continuity and normalization conditions avoid the possibility of some u such that { u]pN (u) = u } = 0 1. Clearly T and cp can be interpreted as “ true” and “ false” respectively. In the following we indicate that values of introduced scalar indices for comparing fuzzy numbers can be recovered from the knowledge of C( M/ N) and C( N/M). First, the grade of partial matching of M and N defined by

can be expressed as’

n,(N) = n,(M) = F(C(M/N)) ‘This result was first noticed 11:447-463 (1979)].

by Baldwin

and Pilsworth

= k(C(N/M)).

(66)

[ Inrernat. J. Man-Machine Srud.

RANKING

FUZZY

NUMBERS

219

Proof.

=

sup suPmin(a,C1,~(u),~nf(u)) ae10, 1) u

=

sup ~~10,

sup

m+Wv(U))

I] u:p&f(u),a

(67)

Note that the supremum

is attained

for OL= II,(N):

Let u* be such that ~N(~*)=supu:p,(u,>II,(N~~N(~).

indeed, Vcu> PI,(N),

Then

Hence, in (6’3 k,+_,(u) 2 a can be changed into pM( u) = (Y. Moreover, the inclusion index H,,(M)

can be expressed as

A$(M)=l-n,(C(M/‘N)).

Proof. Na,,(M)=l-IIN

=l-

rfJc(

M/N))

=l-II,(lBC(M/N)) =l-II,(C(M/N)) if we notice that Vt, plec(MjN)(t)

= CL~(~,~)(~- t) and

220

DIDIER

Now the following identities

DUBOIS AND HENRI PRADE

hold:

This is because

=min(HI,((-~,MI),H,([M,+oo)))

(see Section 3))

Using identities obtained in Section 2, the compatibility index can be linked to the four basic scalar indices where N is taken as a reference, H,(C(M/N))=min(H,([M,+oo)),l-Jlr,(lM,+co))),

(68)

n&W/N))

(69)

= max(l-~~([M,+OO)),~IN(lM,+bO))),

l-n,(C(N/M))=min(n.(]M,+oo)),l-JV;,([M,+oo))).

(70)

This last identity holds under the same conditions as (49, i.e. Cl or C2 or C3. In other words, the values of all the possibilistic ranking indices can be recovered from the knowledge of C( M/N) and C( N/M), through a featureextraction-like procedure. Note that (68)-(70) are sufficient to recover the values of the four indices, since max( II,([ M, + co)),1 - JV,,(] M, + 00))) =l. Equations (68)-(70) are pictured in Figure 7. CONCLUSION This paper has demonstrated that possibility theory is a natural framework for the derivation of comparison indices aiming at ranking fuzzy numbers. The problem of how to actually compute the index values has not been considered, but it is clearly a matter of finding intersection points between membership

RANKING FUZZY NUMBERS

221

222

DIDIER

DUBOIS AND HENRI PRADE

functions, so that the calculation is obvious for linear and parabolic shapes for instance. If general L-R membership functions are used for fuzzy numbers (cf. [S, ll]), the problem comes down to solving equations of the form

find x such that

find X’ such that

findx”suchthat

L(~)=,-L(~~(=-v,([U,i;a))),

l-R(~)=~~~)(=n,(]M,+a;))).

where

i.e.,

,,(x)=t(v)

for x
R(y)

for xam,

and L, R are membership functions of positive fuzzy numbers such that L(O) = R(0) = 1 (cf. [8, 111). With suitable families of mappings L and R, the above equations can be trivially solved. In other respects, the use of possibilistic indices for comparison purposes does not prevent probabilistic-like ones from being employed. A general framework for partial matching, inclusion, and similarity of fuzzy sets has been proposed in [13]; it encompasses both the introduced possibility indices and probabi~stic ones (such as Yager’s [32], recalled in Section 3). Probabilistic indices are based on evaluating areas limited by membership functions (see also Willmott (311). The main potential app~ca~ons of the proposed approach lie in decisionanalysis problems, where the attractiveness of alternatives must be evaluated and compared. Very often probabilities of consequences have to be assessed, but cannot be precisely elicited. In such instances, it is quite natural to resort to fuzzy linguistic probabilities (cf. Zadeh [38], Dubois and Prade [9, 12, 151). The same sort of situation occurs when the importance of criteria to be aggregated is assessed by fuzzy weights (cf. Baas and Kwakemaak [l], Jain 1171). Examples of app~cation-o~ented papers where fuzxy-number comparison procedures have been needed are those of Dubois [6] (comparison of bus-route modifications in a

RANKING mass

FUZZY

transit

NUMBERS

system),

imprecise

statements

(economic

analysis

Cayrol,

223

Farreny,

in artificial-intelligence of flexible

manufacturing

and

Prade

[5] (pattern

software

systems),

matching

of

and Rizzi

[24]

systems).

REFERENCES 1. S. M. Baas and H. Kwakemaak, Rating and ranking of multiple aspect alternatives using fuzzy sets, Automatica 13(1):47-58 (1977). 2. J. F. Baldwin and N. C. F. Guild, Comments on the fuzzy max operator of Dubois and Prade, Infernat. J. $~~nt.r Sci. 10(9):1063-1064 (1979). 3. J. F. Baldwin and N. C. F. Guild, Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems 2(3):213-233 (1979). 4. J. M. Blin, Fuzzy relations in group decision theory, J. Cybernetics 4(2):17-22 (1974). 5. M. Cayrol, H. Farreny, and H. Prade, Possibility and necessity in a pattern matching process, in IXth International Congress on Cybernetics, Namur (Belgium), 1980, pp. 53-65. 6. D. Dubois, An application of fuzzy sets to bus transportation network modification, in Proceedings of the Joint Automatic Control Conference, Philudeiphiu. Vol. 3, 1978, pp. 53-60. D. Dubois and H. Prade, Comment on tolerance analysis using fuzzy sets “and” a procedure for multiple aspect decision making, Internat. J. Systems Sci. 9:357-360 (1978). D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9:613-626 (1978).

D. Dubois and H. Prade, Decision-making under fuzziness, in Adounces in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. Yager, Eds.), North Holland. 1979, pp. 279-303. 10. D. Dubois and H. Prade, Fuzzy real algebra: Some results, Fuzzy Sets und Systems 2(4):327-348

(1979).

11. D. Dubois and H. Prade, Fuzz_ySets and Systems: Theory of Applications, Academic, New York, 1980. 12. D. Dubois and H. Prade, Additions of interactive fuzzy numbers, IEEE Trans. Automar. Control 26(4):926-936

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Addendum: While this paper was in press, we became aware of the work of Bortolan and Degani (A review of some methods for ranking fuzzy subsets, Int. Rep. 83-01, LADSEB, Padova, Italia, 1983). Their experimental study developed in the framework of electrocardiography clearly demonstrates the reliability of the approach presented here (see also Bortolan G., Degani R. Ranking of fuzzy alternatives in electrocardiography-Proc. IFAC Symposium on Fuzzy Informution, Knowledge Representation, and Decision Analysis, Marseille, July 1983, 397-402). Recen,ed 9 Jub

19%‘; revised IO June 1983