A review of some methods for ranking fuzzy subsets

Fuzzy Sets and Systems 15 (1985) 1-19 North-Holland

1

A R E V I E W OF S O M E M E T H O D S F O R R A N K I N G FUZZY SUBSETS G. B O R T O L A N and R. D E G A N I Institute for Research on System Dynamics and Bioengineering (LADSEB)-CNR, Corso Stati Unifi 4, 35020 Padova, Italy Received August 1983 Revised January 1984 This paper deals with the problem of ranking n fuzzy subsets of the unit interval. A number of methods suggested in the literature is reviewed and tested on a group of selected examples, where the fuzzy sets can be nonnormal and/or nonconvex. The ranking is obtained from: (i) the index of strict preference defined by Watson, (ii) three indexes proposed by Yager, (iii) the algorithm used by Chang, (iv) three versions of the a-preference index suggested by Adamo, (v) the index defined by Baas and Kwakernaak, (vi) three modified versions used by Baldwin and Guild, (vii) the method proposed by Kerre, (viii) three forms of the index suggested by Jain, (ix) the four grades of dominance studied by Dubois and Prade. In simple cases the results are good for all the methods, with some exceptions. In questionable cases, where the decision must be probably modelled in accordance with the context in which it is imbedded, the best indexes seem to be the dominances suggested by Dubois and Prade. These indexes do not force any particular choice, but clearly describe the situation, hence allowing the decision-maker himself to make his 'best' choice.

Keywords: Ranking function, Normal/nonnormal fuzzy set, Convex/nonconvex fuzzy set.

1. Introduction The problem treated here deals with the ordering of n fuzzy subsets of the unit interval I. The restriction of the domain of definition of the fuzzy subsets from R to I does not introduce any loss of generality. The importance of this problem lies in the fact that these subsets can be obtained in a decision-making problem (see Degani and Pacini [7], Watson [21], Efstathiou and Rajkovifi [10], Tong and Bonissone [19]), to represent the overall utilities (or values or suitabilities) of a set of alternatives. A comparison between these subsets is therefore a comparison between the alternatives, i.e. a decisionmaking procedure. The way the subsets can be obtained is not treated in this paper, whose aim is to show how different methods select an alternative, given a set of fuzzy subsets that properly represent a group of them. All the approaches proposed in the literature seem to suffer from some pathological examples. For this reason a large collection of cases is used here to test in practice the robustness of each method under particular hypotheses (overlapping/nonoverlapping, convex/nonconvex, normal/nonnormal fuzzy sets). 0165-0114/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

2

G. Bortolan, R. Degani

i

..

A 3 I A2

I

/---J i/ /""/ i

Baas-Kwakernaak

AI = A2 : A3

Jain

A I = A2 : A3

Baldwin-Guild

A I > A2 = A3

Chang

AI <

A2 = A 3

."

g

%50o'

I

!

I

0.~0

I

I

I

1.~]0

Fig. 1. Comparison of three fuzzy sets.

The main criticism to the methods of direct comparison reviewed in this paper is that they tend to defuzzify an intrinsically fuzzy rating into a crisp rating. For this reason a not yet well developed linguistic approach is also suggested (Freeling [12], Efstathiou and Tong [11]) where the subjective aspect of ranking fuzzy sets is maintained either in the verbal outputs describing the situation and in the man/machine interaction used to set up a linguistic preference relation. However, the lack of a well accepted 'golden choice' (even a fuzzy one) in all cases prevents the identification of a general mathematical or linguistic procedure that is able to interpret any situation correctly, unless a verbal sentence like 'indefinite choice' or an equivalent numerical output is used in all difficult cases. One example is shown in Fig. 1, which is used by Baldwin and Guild [4] to illustrate their point that already existing methods of comparison (Baas and Kwakernaak [2] and Jain [13, 14]) are not able to discriminate otherwise discriminable alternatives. In this example their method prefers A1 to A2 and A3, the last two resulting at the same level. To quote them: " . . . our procedure gives an equal membership to alternatives A2 and A3, while A 2 is clearly preferable to A 3 : but should a decision between A2 and A3 be required (e.g. because A1 is rejected on some other grounds) the procedure should be repeated for this pair only, when a clear difference will result." The same situation is discussed by Chang [6], whose method however prefers both A2 and A3, which again are not distinguishable, to AI: " . . . Baldwin and Guild prefer A1 to A2 and A3, which is clearly not a satisfied result. Our procedure gives the same ranking to alternatives A2 and A3, though A 3 is intuitively pre[erable to A 2 . " Obviously two opposite intuitions are applied to this example, perhaps suggesting that Baas-Kwakernaak's and Jain's methods, which present a completly confused situation, are more realistic. We don't have a rule to offer for the solution of this problem, but we suggest verifying that each situation is judged in the context of an evolving process, where decisions change smoothly and are consequently more or less predictable [5].

Ranking fuzzy subsets

2. ~sanking methods Let us suppose that we have n nom'm] convex fuzzy subsets ~-, i ~ N = { 1 , 2 , . . . , n}: ~ = {z, ~ ( z ) } ,

z ~ S, ~ / .

A simple method to order the ~.'s consists in the definition of a ranking function F mapping each fuzzy set into the real line, where a natural order exists. This approach has been followed by Yager [22], Chang [6] and Adamo [1]. Hence F : ~(I)---~ R where ~ (I) is the set of fuzzy subsets of L F is such that F(~.)
implies

~.
F(~.) = F(f~)

implies

~. = fit,

F(~)>F(f~)

implies

~'>fii-

Yager [22, 24] has proposed several ranking functions, where he does not assume any hypothesis of normality or convexity. The first function is

,InI g(z)~a~(z) dz fl(l~.) So1 t ~ (z) dz where the weight g(z) is a measure of the importance of the value z. If we assume linear weights, that is g(z)~z, then FI(~) represents the center of gravity of the fuzzy set ~ (Fig. 2a). o~.

~_

d o °050d

i

'O.SO

(a)

F

'"~l

1

1.00

"'"i I.O0

1I O.SO

(b)

Fig.

2. Yagerindexes.

•

i) O

!

0 O0

0 SO

(e)

i

1.00

4

(3. Bortolan, R. Degani A second index is suggested by possibility theory: F2(~.) = max min(z, t~(z)).

z~Si

In this case F2(~) measures the consistency of ~. with the linear fuzzy set defined by tz~(z)= z (Fig. 2b). The third ranking function proposed by Yager is the following. If U? is the a-level set of ~- and if M(U'~) is the mean value of the elements of U~, then F3(~.)=

M(U~)da

I0cxm~

where am~=hgt(~-).

Graphically F3(~-) can be represented by the area between the dotted line of Fig. 2c and the membership axis. For each value of the grade of membership the dotted line represents the average value of the elements having at least that grade of membership. Chang [6] uses the index

F(f%)= [ ztzc.(z)dz aS i

which reduces to

F(f%.) -

( b - a)(a + b + c) 6

for a triangular fuzzy set ~ with support S~ = [a, b] and height hgti = ~a,(z) Iz=¢Adamo [1] uses the concept of a-level set to obtain an a-preference index which is given by

F , , ( ~ ) = m a x {z l ~ . ~ ( z ) ~ a } for a given threshold ~ e I. Fig. 3 shows an example for a = 0.9. Other authors consider a different formalization of the problem. The aim is to obtain a fuzzy set of optimal alternatives 0 = {i,/~0(i)},

..: ~..--.....9. u1

~0.00'

'

ieN,

/ i ,I

O.SO

FO.9(~ I) : 0.29 FO.9(~ 2) : 0.62

1.00 Fig. 3. 0.9-preference index.

Ranking fuzzy subsets

5

where/x6(i) is the degree to which the i-th alternative may be considered the best alternative. One of the first indexes defined in these terms has been proposed by Baas and Kwakernaak [2]. They first define the conditional fuzzy set O / R = { i , p ~ 6 T R ( i l z l , . . . , z i . . . . . z,)},

i~N, ziaI,

with membership function zn) = {10

tz57k(i I Z l , . . . ,

if zi ~ zi~ V j e N , otherwise

The membership in the optimal set is given by ~z6(i) = = sup=. [tz67k(i [ zl . . . . . Z,)A (r~n ~c, (z~))] =

sup z 1. . . . .

min/xc~(z~) zn

z i ~ms.x I z i

j*~i

zl~maxl~l

z~

A simple graphical method to evaluate / ~ ( i ) is shown in Fig. 4. The subset whose peak value is greatest is the preferred one. The membership value of the point where the membership of the other subset intersects the membership function of the maximal set gives its degree in the optimal set. Baldwin and Guild [4] also propose a similar index, but in order to obtain more sensitivity, for example in the case of Fig. 1, they use a two-dimensional preference relation

P, = {(z,, zi), ~,,(z,, z~)} indicating how much zi is greater than zi, Yj ~ i. In particular they suggest

tz~,,(z~, zi) = zi - zi.

~2 FO(1) = 0.22 Po(2) = 1.oo

-i:Jo, :,L'

i

1.00

Fig. 4. Baas-Kwakernaakmethod.

6

G. Bortolan, R. Degani

The optimal set is then given by /~6 (i) = rain sup (t~a,(z,) A p.~(z i) A p,p,,(zi, zi)). i~'i

z . z~

Dubois and Prade [8] first proposed the use of a fuzzified infix operator to solve the ranking problem. However, Baldwin and Guild [3] pointed out that if we must choose one of n different alternatives then mLx can be of no help because it can be different from all the fuzzy sets under consideration. Kerre [17] also suggests to evaluate the fuzzified mLx, but in addition he follows an approach indicated by Yager [23] and computes the Hamming distance between all fuzzy sets and their mLx. The Hamming distance between two fuzzy sets is defined (in the continuous case) by

dist(~, ~)= S, I~m(z)-~m(z)ldz. The fuzzy set whose distance from m~tx is minimum is the best choice. Watson et al. [21] propose an index related to the concept of implication. Particularly, let x: y:

"alternative 1 is fil and alternative 2 is fi2", "1 is strictly preferred to 2",

where t~(z~, z 9 = min[t~,(z3, m~(z2)], {;

iffzl>z2, otherwise,

~ (zl, z2) =

for Zl ~ S~, z2~ $2. Then from the assumption t~(x ~ y) = min max[1 - t~x(zl, z2), txy(zl, z2)] z l , z2

they derive tx(x ~ y) = m i n { 1 - min[tza,(z0, p.a2(Z2)]} ZI~Z2

= 1 - max min[tt,a,(z,), t~(z2)], Z2)Z l

which is called the index of strict preference of fil over u2. In the method suggested by Jain [13, 14] first the support S of fi = U ~ is determined. Let z ~ , = sup S. Then the maximizing set a.~x = {z, tL~(z)}, of S is evaluated, where k ~Zm~t.x ~

z ~ S,

Ranking fuzzy subsets

OOs

~

.

~

PO(1)

0

PO(2)

0

,

%. 00'

'

'

o.~o '

:

0.43

:

0.73

Zmax

'

'

i.~o

Fig. 5. Jain method. Finally /~o(i) = hgt(~, f3 tim=) is obtained. The graphical method to derive t~o is shown in Fig. 5 for k = 1. Note that/~o(i) is not very different from the index/;2 proposed by Yager. This approach has been variously criticized by Dubois and Prade [8, 9], Baldwin and Guild [4], Chang [6] and Kerre [17], both for the strong dependence of the final ranking on the choice of k in the definition of tZm= and for some specific examples. It is obvious that the classification can depend on the definition of the maximizing set, hence on k, but we do not feel that this is a criticism. If the results are good for a fixed choice of k (for instance k = 1), then no further problem arises on this point. Dubois and Prade [9] recognize the inadequacy of mSx in some cases and propose a set of four indices able to completely describe the relative location of two fuzzy numbers ~ and fii- In particular they define: (1) a grade of possibility of dominance (of ~ over fq), P D ( ~ ) = Hc~([fq, +oo))a_--Voss(~ I> fq) = sup min[tza,(zl), sup ~a,(zj)] ZI

Zj :s~Z I

= sup min[tza,(z,), tza~(zi)], Zt, Zj Zl;~Z |

which is clearly the index also proposed by Baas and Kwakernaak; (2) a grade of possibility of strict dominance, PSD(~) = Ha, (] fq, +oo))_--aVoss(~ > fq) =sup Zl

inf min[iza,(zJ, 1-/za~(zi)]; Z I, Z l ~ Z l

(3) a grade of necessity of dominance, ND(fii-) = ,hrc~([fq, +oo)) a__Nec(fii- I> fii) = inf sup m a x [ l - t~a,(z,), ~ ( z i ) ] ; Zl

Z l,

Zl11{Z|

8

G. Bortolan, R. Degani

(4) a grade of necessity of strict dominance, NSD(~.) = No, (]fq, +co)) =a Nec(~ > (q) = 1 - s u p min[/zo` (zi),/x~(zi)] = 1 - P D ( f q ) , zI~ zI

which is the same as the index of strict preference of ~ over ~- defined by Watson. In order to extend these indices to the ranking of n fuzzy numbers t ~ l , . . . , Dubois and Prade define the grades of dominance of a fuzzy number ~. over all other fq, j ~ i, as the grades of dominance of ~. over m~xi, i ~. They then obtain the n-ary versions: (1') a grade of possibility of dominance, PD(~) =

Ho`([mfix f~, + o o ) ) = P o s s ( ~

~>mS_x ~ ) = r a i n Poss(~. I> fq);

(2') a grade of possibility of strict dominance,

(3') a grade of necessity of dominance,

(4') a grade of necessity of strict dominance,

The fuzzy sets fil . . . . . fq can then be ranked in descending order in terms of decreasing values of each index. If the four orderings are consistent then the fuzzy sets can be ordered without any problem. Otherwise the results can be read in such a way that the relative position of the fuzzy numbers can be inferred. An example is shown in Table 1, where the ~ ' s are very much overlapping. The results can be interpreted as follows: no set is necessarily strictly dominant (the supports overlap); t~2 and t~3 are better than ~1 on the left side (ND); t~3 is better than tll and u2 on the right side (PSD); ~ and u3 on the right can both be better than ti2 (PD). The second alternative proposed by Dubois and Prade for ordering n fuzzy sets is to process the fuzzy outranking relations P'D, PSD, I'~D, NSD built on the binary indices PD(i, j) =/-/o` ([~, +oo)),

PSD(i, j) = HO`(](q, +oo)),

ND(i, j) = NO`([(q, +oo)),

NSD(i, j) = NO`(]~, +oo)).

R a n k i n g f u z z y subsets

9

Table 1. D u b o i s - P r a d e indexes in a p r o b l e m case

u3 PD

°7

0°7

/

/

CO"O0'

2 '

'-. '0.S0

'

'

PSD

ND

NSD

~1

1.o0

0.30

0.30

0.0o

~2

0.88

0.40

0.50

0.00

~3

1.00

0.60

0.50

o.00

"1.~]0

None of these relations is a fuzzy order, but all can be processed in such a way that some ranking can be obtained (see [9] for details). Inconsistent rankings obviously suggest overlapping sets and indicate the necessity of a direct choice from the decision-maker. Tsukamoto et al. [20] also propose a set of indices for ranking two fuzzy sets ~and ~. The three indexes h, t2, t3 are closely related to those proposed by Dubois and Prade. In particular, tl = Poss(~. >~ (q-),

t2 = Poss(~ = t~),

t3 = Poss(~- ~
These indices are then used to heuristically construct a linguistic description representing to the decision-maker the fuzzy ranking. The linguistic ordering has for example the form or

"~ "~

is not much better than (q" is unknown with fq",

which are left to the interpretation of the user. Quite recently Murakami et al. [18] used two methods, called a-cut and centroid methods, to order triangular and/or trapezoidal normal fuzzy sets. The a-cut method is the same as the A d a m o procedure. The centroid method measures the centroid (z0, N0) of each fuzzy set t~, defined by

z~.(z) dz ZO =

1

z~a(z) dt,.~(z) ,

Io lxc,(z)dz

NO ----

I

I0 z dtLa(z)

We can easily see that Zo is the same as Yager's Ft index, where g(z)--z. As regards No it is constant for equal shape membership functions. In particular /Xo=½ (½) for all triangular (rectangular) normal fuzzy sets, whose ordering through (z0, tLo) is therefore the same which can be obtained through Zo = F~ alone. The authors recognize that the optimal alternative, defined as the one which attains the maximum value on either of the z and /z axes, cannot be unique, i.e. Pareto solutions are possible. In this case they suggest a choice based

I1/2

10

G. Bortolan, R. Degani

/x°2 °

Zo2 > Zol

~2>~ 1

~ol > ~o2

~1>~2

....... i .........

tJ, i,

!

0.00 Zo10.50

Zo21.00 Fig. 6. Centroid method.

upon the importance the decision-maker himself attaches to z and ~ separately. This means that the ordering can be deduced from one parameter only (z0 or t~0) properly indicated. We add that many situations can occur where z 0 seems the only rational index (see for example Fig. 6). The methods reviewed here generally apply to normal convex fuzzy sets. However they can also be extended to general fuzzy sets, which can be nonnormal and/or nonconvex. The extension is straightforward for all the methods. Obviously the general formulation must be used for the index suggested by Chang. It is also obvious that PD(~.) is always equal to the index proposed by Baas and Kwakernaak.

3. Comparative examples Table 2 shows 20 case studies. Some of them depict rather simple situations, some are very difficult to be interpreted. The last example is different from the previous ones, because it refers to a real situation, where the sets t~l, fi2, fi3 to be compared are the overall utilities of three pathological alternatives related to the classification of an electrocardiographic signal. The fuzzy model used to construct these global utilities, first described by Degani and Pacini in [7], is a new approach to the diagnostic interpretation of the electrocardiogram which is usually performed through tree logic or statistical inference (see Jenkins [15] for a review on computerized electrocardiography). In order to easily compare the results in each case we have normalized in some sense all the indices, in such a way that the maximum possible value is 1. To this aim Chang's index has been multiplied by 2, because 0.5 is its maximum value, obtained for ~ defined by ~r~(z) = 1,

Vz e[0, 1].

As regards the index suggested by Kerre, we have changed it from F(~-) = dist(~, m~x ~ )

.

.

.

.

.

.

.

.

.

~3

0

0

NSD (Watson)

.10 .18 .I0 .02 .11 .11 .15 0 0 0 0 .80 .18 .03 .40 0 0

(a)

I

I

I

1

.90 .84 .95

1

.90 .90 .90 .18 .91 .91 .95 I .82 .82 .69

~2

, , !,

~1

ND

0.gM 0.gm 0.5 Bass-K~akerrmak Baldwin-Guild l.p. g. r.a. Kerre Jain k=l k:2 k=I/2 Dubois-Prade PD PSI)

Chang Adamo

FI F2 F3

Methods

.

Yager

.

.95 I I I I

.84 0 0 0 0

0 0 0 .80 .90 .84

I .32 .47 .20 I

0

.72 .55

.9O .90 .90 .18 .91 .91 .95

~2

.7O .72 .70 .14 .71 .71 .75

~1

(b)

/

Table 2 Case studies

.90 0 0 0 0

.81 .70

0 0 0 .85

0

.8O .81 .80 .16 .81 .81 .85

~1

(e)

.97 I I I I

.9~ .92

.25 .4O .14 I

I

.95 .95 .95 .09 .95 .95 .97

~2 •20 •27 •20 .04 •21 •21 •25 0 0 0 0 .80 •32 .12 .55 0 0 0 0

~1 .40 .45 .40 .08 .41 .41 .45 0 0 0 0 .80 .55 .33 .72 0 0 0 0

~2

(d)

! .70 .72 .70 .14 .71 .71 .75 I .40 .44 .30 1 .8Q .80 .94 I I I I

~3

.03 .09 .02 0 .01 .01 .05 0 0 0 0 .8g .09 0 .26 0 0 0 0

~1

.60 .63 .60 .12 .61 .61 .65 0 0 0 0 .85 .63 .42 .78 0 0 0 0

~2

(e)

.97 I -97 .10 I I I I .45 .67 .27 1 I I I I I I I

, , w ,

N

FI F2 F3

0.9M 0.9m 0.5 Baas-Kwakernaak Baldwin-Guild l.p. g. r.a. Kerre Jaln k=1 k=2 k=I/2 Dubois-Prade PD PSI) ND NSD (Watson)

Chang Adamo

Yager

.....................7 2

.61 .66 .58 .40 .55 .55 .75 .84 .42 .44 .34 .96 .66 .53 .78 .84 .54 .54 0

(f)

.53 .69 .56 .34 .66 .66 .72 1 .33 .37 .24 .8g .69 .51 .81 I .46 .46 .16

.41 .66 .40 .58 .55 .55 .75 .82 .30 .36 .21 .51 .66 .53 .78 .82 .66 0 0

(g)

.60 .63 .60 .12 .61 .61 .65 1 .58 .42 .55 .89 .63 .42 .78 I .32 I .18

Table 2. (Continued)

.41 .66 .40 .58 .55 .55 .75 .66 .24 .30 .16 .42 .66 .53 .78 .66 .50 0 0

(h)

.70 .72 .70 .14 .71 .71 .75 I .66 .54 .60 .95 .72 .54 .84 I .50 I .34

.76 .90 .80 .46 .91 .91 .95 1 .42 .55 .28 I .90 .84 .95 I .74 .63 .26

.70 .76 .70 .41 .73 .73 .85 .74 .33 .40 .23 .86 .76 .65 .86 .74 .23 .38 0

(1)

.63 .66 .60 .38 ,55 .55 .75 .60 .30 .34 .22 .76 .66 .54 .78 .60 .16 .18 0

1 .70 .75 .70 .28 .72 .72 .80 1 .37 .42 .27 I .82 .71 .89 I .50 .67 0

:.

.63 .75 .65 .37 .72 .72 .80 1 .27 .35 .19 .91 .82 .71 .89 I .50 .35 0

(j)

ij

i

.57 .75 .57 .52 .72 .72 .80 1 .27 .35 .19 .75 .82 .71 .89 I .50 0 0

I(--1

NSD (Watson)

k=2 k=I/2 Dubois-Prade PD PSI) ND

Kerre Jain

0

0

.37 .40 .2B .85 .69 .56 .80 I .20 ,50

1

l.p. g. r.a.

.45 .53 .31 I .90 .82 .94 I .80 .50

1

Baldwin-Guild

Baas-Kwakernaak

.56 .64 .54 .33 .54 .54 .70

.62 .81 .62 .56 .81 .81 .85

Yager FI F2 F3 Chang Adamo 0.9M 0.gm 0.5

~2

~1

Methods

(k)

L".

0

.27 .28 .21 .75 .64 .45 .77 I 0 .50

1

.50 .58 .50 .20 .52 .52 .60

~3

0

.27 .30 .20 .91 .73 .60 .83 I .73 ,27

1

.50 .61 .50 .29 .53 .53 .65

~1

/ (1)

0

.28 .24 .23 .91 .67 .48 .80 I .24 .76

1

.50 .54 .50 .I0 .51 .51 .55

~2

0

.40 .40 .30 .76 .73 .58 .84 I .30 .30

1

.44 .66 .45 .43 .62 .62 .70

~1

Table 2. (Continued)

0

.42 .42 .34 .92 .69 .56 .80 .88 .40 .50

.88

.53 .64 .52 .37 .54 .54 .70

~2

(m)

0

.42 .44 .32 .96 .80 .67 .89 I .60 .50

1

.52 .72 .55 .42 .71 .71 .75

~3

.20

0 0 0 .64 .33 .13 .54 0 .20 0

0

.20 .33 .20 .08 .22 .25 .30

~1

(n)

1

.68 .66 .57 I .80 .68 .80 .80 .80 I

.80

.80 .80 .64 .25 0 .82 .87

~2

, :-

,

.

.80

0 0 0 .78 .66 .49 .80 0 .80 0

0

.60 .66 .60 .24 .62 .76 .70

~1

(o)

/1 / .20

1

.20 .20 .20 I .20 .20 .20 .20 .20 I

.90 .20 .18 .03 0 .91 0

~2

.'°:..

N N

.

.

.

.

.

FI F2 F3

.

.

.

.

.

~3

0.9M 0.9m 0.5 Baas-Kwakernaak Baldwin-Guild l.p. g. r.a. Kerre Jain k=1 k=2 k=I/2 Dubois-Prade PD PSI) ND NSD (Watson)

Chang Adamo

Yager

.

..................... ~2

.20 .20 .04 .01 0 .22 0 0 0 0 0 .89 .20 .09 .20 0 0 .80 .80

(P)

.80 .83 .80 .32 .82 .96 .90 .20 .20 .20 .20 .88 .83 .73 .90 .20 I .20 I

:.

..

.60 .71 .60 .47 .64 .92 .80 .20 .20 .20 .19 .72 .71 .58 .82 .20 .83 .17 .80

.60 .20 .12 .09 0 .64 0 .20 .20 .20 .20 .97 .20 .20 .20 .20 .16 .84 .80

(q)

Table 2. (Continued)

.87 I .90 .35 I I I .20 .07 .12 .04 .82 I I I .20 I .16 .80

.95 .20 .19 .01 0 .95 0 .20 .20 .20 .16 I .20 .20 .20 .20 0 .85 .80

(r)

.50 I .50 .21 I I I I .58 .78 .38 .70 I I I I I 0 0

(s)

.50 .58 .50 .20 .52 .52 .60 I .58 .38 .74 .90 .58 .38 .74 I 0 I 0

I

.15 .34 .15 .09 .23 .23 .31 .50 .34 .14 .40 .44 .34 .14 .54 .50 0 .40 0

.28 .40 ,23 .20 .22 .22 .30 .50 .40 .40 .40 .50 .40 .40 .53 .50 .40 .40 0

(t)

t

\.-

....

.46 .66 .47 .40 .62 .62 .70 I .66 .49 .80 .88 .66 .48 .80 I .60 I .50

% : t : |:

\

I I I

I

/'--\

_=.

5-

Ranking fuzzy subsets

15

tO

F'(~) = 1-F(~)=

1 - d i s t ( ~ . , m~x ~ ) i

simply because it should otherwise indicate an inverse ordering with respect to all the other methods. The preference relation P~ used by Baldwin and Guild can have different definitions, which could affect the final results. We have used, as suggested by the authors themselves, the following preferences: tz~,(z~, zj) = z ~ - z i

izp,,(zl, z i) = zi_zi2

linear preference (1.p.), 2

lzo,(zl, zi) = ",/-zzi- ~ z i

gambler (g.), risk-averse (r.a.).

For the same reason we have applied three different values to the parameter k used by Jain. In particular we have chosen k=l,

k=2,

k=½.

Another kind of problems derives from the application of Adamo's method, if the fuzzy sets are not normalized. We have used a = 0.9 max(hgt t~i) = 0.9M, i

in order to have at least one prevailing set. Another choice has been a = 0.9 min(hgt fii) = 0.9 m, which necessarily takes into account all the fuzzy sets under examination. Finally a =0.5 has also been tested, which can be interpreted as a sufficient safeguard against neglecting (nonnormal) fuzzy sets whose height is relatively high (~>0.5). The extended grades of dominance defined by Dubois and Prade have also been computed, whereas the fuzzy relations obtained from pairwise comparisons have not been considered. Cases (a) and (b) represent quite simple situations, where all the methods give the same ordering. However, some methods (Kerre, Baas-Kwakernaak, DuboisPrade) are unable to differentiate the two cases, i.e. they are not sensitive to shifts along the z axis, as long as the two sets are not overlapping. Another simple situation is shown in case (c), where ul and fi2 are not very different from those in case (b). However Chang prefers uz to u2, which seems to contradict intuition. The example of case (d) shows three fuzzy sets, whose ordering is straightforward. All the methods prefer u3, but some are unable to distinguish fil from u2 (Baas-Kwakernaak, Baldwin-Guild, Kerre, Dubois-Prade). This problem is irrelevant if the goal is the choice of the best alternative. In addition it is possible to

16

G. Bortolan, R. Degani

exclude fi3 and to consider fix and fi2 only. In this case the ranking procedures result in a correct preference of fi2 over fix. Case (e) shows another simple situation. All the methods behave correctly (if fix = f2 is acceptable), except Kerre who obtains fil > fi2This inconsistency disappears if a left-right compounded Hamming distance is evaluated, as recently suggested by Kauffrnan in [16]. The ordering obtained with this new method is in fact the same as the ordering obtained with the index F3 proposed by Yager, at least in the comparison of convex fuzzy sets. The example of case (f) is more complex for the partial overlap of supp fix and supp fi2. Some methods prefer fix, some prefer fi2- Jain changes ranking as k changes, whereas there is no change in the three indexes used by Baldwin and Guild. However, the small differences shown by Jain indexes indicate a confused (overlapped) situation, hence suggest a difficult ranking. This difficulty is also expressed by the inconsistent orderings indicated by the grades of dominance. The examples of cases (g) and (h) are quite similar. As in case (f), fi2 is the set with the rightmost peak and fix expands to the right of fi2- However, fi~ also expands very much to the left of fi2- We could then decide for fi2 > fi~. This ordering is given by all the methods in case (g), with the only exception of Chang. However in case (h) four methods (Yager's F2, A d a m o 0.5, Jain k = 1, 2) change from fi2 > fix to fix > fi2- PSD also, which indicates equal strict dominances in case (g), in case (h) gives a greater possibility of strict dominance for fix than for fi2 (right-hand sides). ND (left sides) clearly prefers fi2 to fix in both cases. Case (i) is an example where three fuzzy sets have the same support, but the situation is clear. All the methods give the same ranking. The example of case (j) is the same already seen in Fig. 1. It shows the typical inability of distinguishing fuzzy sets which only differ on their left sides, when only the right part of their membership functions is considered (Yager's F2, Adamo, Jain). Baas and Kwakernaak are also unable to discriminate this situation, whereas Baldwin and Guild cannot distinguish fi2 from fi3. Dubois and Prade indicate a preference based on the left parts of the memberships (ND). The indexes reported for Chang are obtained, as already said, by means of the general expression. For this reason fi3 results greater than fi2, whereas no difference is reported when the simplified version for triangular fuzzy sets is used [6]. The sets in ease (k) represent a dual situation with respect to ease (j). All the methods rank first fix. The only exception is the Baas-Kwakernaak index, which does not discriminate. In case (1) fix includes fi2- Some methods prefer fix to fiz (gamblers), some are unable to discriminate, some prefer fiz to fil (risk-averse). Case (m), which represents a very confused situation, is the same as in Table 1. Almost all possible results can be seen: preference for fix (Chang), preference for fi2 (Yager's F1, Baldwin-Guild r.a.), preference for fi3 (Yager's F 2 / F 3 , Adamo, Baldwin-Guild g., Kerre, Jain), equal preference for fil and fi3 (BaasKwakernaak), equal preference for fi2 and fi3 (Baldwin-Guild 1.p.). Dubois-Prade

Ranking fiuzzysubsets

17

correctly summarize the situation as follows: fi3 is preferable on the right side, fi2 and fi3 are equally preferable on the left side. In case (n) fi2 is nonnormal. All the methods prefer fi2 to fil. The only exception is Adamo 0.9M, which gives f i l > fi2 because /~a, does not reach the prescribed confidence level. Example (o) is similar to case (n), in the sense that fi2 is again nonnormalized and its support is to the right of supp fil. However, hgt fiz<
18

G. Bortolan, R. Degani

4. Condusions All the methods of direct comparison between fuzzy subsets reviewed here behave reasonably well in non-questionable situations (cases (a) to (e), (i) and (p)). H o w e v e r , in some instances Chang (examples (c) and (e)) and K e r r e (examples (e) and (p)) give a result inconsistent with intuition. Five other methods ( B a a s - K w a k e r n a a k , Baldwin-Guild (1.p., g., r.a.), D u b o i s Prade) with a single step can only choose the first of three fuzzy sets in case (d), whereas N D seems to suggest an incorrect preference for set fil on the left side in case (p). In difficult cases, i.e. when intuition is not evident, the results are mainly scattered. S o m e exceptions can be seen for example in case (t), where all the methods give the same ranking, and in case (j) where only Chang prefers fi3 instead of ill. In these cases the four indices p r o p o s e d by Dubois and Prade correctly describe the relative position of the sets, cases (o) and (t) excepted where an incorrect description occurs. O t h e r methods can be acceptable (for example Y a g e r ' s F 1 or F 3 , which in addition behave consistently in the simple cases) but the choice of one of t h e m is subjective and probably context dependent. In addition a single index can be troublesome because it can show firm preferences also when the decision is not evident (see for example the values of F3 in case (r)). T h e tendency towards the use of multiple numerical indexes is quite clear in the literature (Freeling [12], T s u k a m o t o , Nikiforuk and G u p t a [20], Dubois and Prade [9], Murakami, M a e d a and I m a m u r a [18]) and is supported by the results presented here. In general the four indices PD, PSD, ND, N S D can be conveniently used. T h e y do not suggest wrong consistent orderings and leave the choice to the decision-maker himself when inconsistent rankings only d e s c r i b e the relative position of the sets. This description is generally correct, but exceptions can be due to the presence of n o n n o r m a l fuzzy sets which need particular attention.

References [1] J.M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems 4 (1980) 207-219. [2] S.M. Baas and H. Kwakernaak, Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica 13 (1977) 47-58. [3] J.F. Baldwin and N.C.F. Guild, Comments on the fuzzy max operator of Dubois and Prade, Internat. J. Systems Sci. 10 (1979) 1063-1064. [4] J.F. Baldwin and N.C.F. Guild, Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems 2 (1979) 213-233. [5] G. Bortolan and R. Degani, Ranking of fuzzy alternatives in electrocardiography, Preprints IFAC Conf. on Fuzzy Information, Knowledge Representation and Decision Analysis (1983) 397--402. [6] W. Chang, Ranking of fuzzy utilities with triangular membership functions, Proc. Int. Conf. on Policy Anal. and Inf. Systems (1981) 263-272. [7] R. Degani and G. Pacini, Linguistic pattern recognition algorithms for computer analysis of ECG, Proc. BIOSIGMA 78 (1978) 18-26.

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[8] 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 (1978) 357-360. [9] D. Dubois and H. Prade, Ranking of fuzzy numbers in the setting of possibility theory, Inform. Sei. 30 (1983) 183-224. [10] J. EIstathiou and V. Rajkovi~, Multiattribute deeisionmaking using a fuzzy heuristic approach, IEEE Trans. Systems Man Cybernet. 9 (1979) 326-333. [11] J. Efstathiou and R.M. Tong, Ranking fuzzy sets: a decision theoretic approach, IEEE Trans. Systems Man Cybernet, 12 (1982) 655-659. [12] A.N.S. Freeling, Fuzzy sets and decision analysis, IEEE Trans. SystemsMan Cybernet. 10 (1980) 341-354. [13] R. Jain, Decision-making in the presence of fuzzy variables, IEEE Trans. Systems Man Cybernet. 6 (1976) 698-703. [14] R. Jaln, A procedure for multiple-aspect decisionTmaking using fuzzy sets, Internat. J. Systems Sci. 8 (1977) 1-7. [15] J.M. Jenkins, Computerized electrocardiography, CRC Critical Reviews in Bioengineering 6 (1981) 307-350. [16] A. Kaufmann, Le probl~me du classement des nombres flous en un ordre total, Note de travail n. 112 (1983). [17] E.E. Kerre, The use of fuzzy set theory in electrocardiological diagnostics, in: M.M. Gupta and E. Sanchez, Eds., Approximate Reasoning in Decision Analysis (North-Holland, Amsterdam, 1982) 277-282. [18] S. Murakami, H. Maeda and S. lmamura, Fuzzy decision analysis on the development of centralized regional energy control system, Preprints IFAC Conf. on Fuzzy Information, Knowledge Representation and Decision Analysis (1983) 353-358. [19] R.M. Tong and P.P. Bonissone, A linguistic approach to decisionmaking with fuzzy sets, IEEE Trans. Systems Man Cybernet. 10 (1980) 716--723. [20] Y. Tsukamoto, P.N. Nikiforuk and M.M. Gupta, On the comparison of fuzzy sets using fuzzy chopping, Proc. 8th Triennial IFAC World Congress 5 (1981) 46--52. [21] S.R. Watson, J.J. Weiss and M.L. Donnell, Fuzzy decision analysis, IEEE Trans. Systems Man Cybernet. 9 (1979) 1-9. [22] R.R. Yager, Ranking fuzzy subsets over the unit interval, Proc. 1978 CDC (1978) 1435-1437. [23] R.R. Yager, On choosing between fuzzy subsets, Kybernetes 9 (1980) 151-154. [24] R.R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Inform Sci. 24 (1981) 143-161.