A special measure of uncertainty

Fuzzy Sets and Systems 38 (1990) 281-287 North-Holland

281

A SPECIAL MEASURE

OF U N C E R T A I N T Y

Hans B A N D E M E R Sektion Mathematik, Bergakademie Freiberg, DDR-9200 Freiberg, G.D.R. Received November 1987 Revised February 1989

Abstract: A new measure of uncertainty is presented, which evaluates the 'hardness' of decision if there are two or more competing decisions. The case of 'dead heat' is starting point for the construction of this measure, which is then generalized to the case of fuzzy decisions and arbitrary universes.

Keywords: Measure of uncertainty; competing decisions; fuzzy decisions.

1. Introduction Functionals to evaluate fuzzy sets with respect to their uncertainty, vagueness, fuzziness, or the like will be referred to as measures of fuzziness in the following. They differ with respect to three properties: the types of sets which assume the minimum and the maximum value, respectively, and the partial ordering with respect to which the measure is monotonic. A special class of such measures applies when a crisp decision x is to be drawn from a fuzzy decision suggested by a fuzzy set A over the set X of possible crisp actions, where the membership value mm(X) means the suitability of x. The 'simplest' case for decision, i.e. where the decision-maker does not have any choice and hence does not feel any scruples, is the case of a singleton: mA(X ) = 1 if x = x0 and 0 elsewhere. T o evaluate decisions from some set A with respect to the 'hardness' of these scruples we may choose the measure of uncertainty U as introduced by Higashi and Klir [4]: 1

VA E F ( X ) :

U(A) =

f0

lOgE(cardA ~) dcr

(1)

provided that X is finite and A is normalized (i.e. maxx mA(X) = 1). A ~ is the re-cut of A and F(X) is the set of all fuzzy sets on X. The measure U assigns the value 0 to each singleton of X and its maximum value log2(card X ) to the universe; U is monotonic with respect to set inclusion. If there exists a basic probability assignment for m, say p(B) for all B of the power set P(X), in the sense of possibility theory, then U(A) can be written as a weighted sum of log2(card B); B ~ P(X), see Dubois and Prade [3]. Other measures of fuzziness of a fuzzy set A can be based on the cardinality of 0165-0114/90/$03.50 © 1990--Elsevier Science Publishers B.V. (North-Holland)

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A itself, for finite universes defined as N

card A = ~] ma(xi)

(2)

i=1

where N is the number of elements in X. Then VA e F(X):

FI(A) = cardA

(3)

F2(A) = cardA - 1

(4)

as well as VA E F(A):

are measures of fuzziness, provided that A is normalized as mentioned under (1) (cf. [5, 1, 2]). Both the measures take their respective maximum for X and they are monotonic with respect to set inclusion. Further, F1 is one for singletons, while F2 is zero for these. Hence F2 shows all properties of U which we want to use.

When turning to arbitrary universes X, the measures U, F1, and F2 are defined, because a specification of cardinality is lacking. In our context assume that, on X, a finite additive measure, say Q, is given. Then we F ( X ) to the set FQ(X) of all fuzzy sets A on X which are integrable with to Q, i.e. for which the integral card A = [x ma(x) dQ (x)

not yet we will restrict respect

(5)

does exist. To give some simple examples: Q may be the counting measure, if X is finite, or the Lebesgue measure if e.g. X is a compact set of some Euclidean space. For our application case it is important to mention that Q can be a combination of those measures.

2. The case of dead heat

When specifying input membership functions, there is, in general, only some ordering on which the experts agree, and additionally, the values mA(X) of the output used for the decision are computed according to a certain compromising procedure. Hence the case of 'dead heat', mA(Xl) = ma(X2)

= 1,

Xl

4:x2,

(6)

("a second gold medal must be taken from the stock") can be qualitatively different from the case where there is some (very small) e > 0 with

1 = ma(x 0 = rot(X2) + e.

(7)

The usual idea in fuzzy set theory is that the two cases, (6) and (7), are nearly equivalent. But it can happen in decision making, that a situation of 'dead heat' may raise our scruples essentially. The measure of uncertainty U according to (1) is continuous with respect to each value mA(xi) and hence the values of U in the two situations (6) and (7) will differ only insignificantly.

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283

For the case where dead h e a t / s of importance, we suggest to use as a measure of fuzziness the modified cardinality: VA • F ( X ) :

F3(A) = ~'cardA LcardA - 1

if c a r d A 1 ¢ 1, if c a r d A ] = 1,

(8)

provided that X is finite and A is normalized; A ] means the core of A, i.e. where ma(x) = 1. /73 is zero for singletons, maximum for X and monotonic with respect to set inclusion, but contrary to the measures U, F1, and F2, F3 signals the case of dead heat by a jump. For the case that supp A is a continuum, e.g. an interval, and Q is the Lebesgue measure, the corresponding measure U does not distinguish singletons from crisp sets of cardinality zero; F3 signals the case where the core of A has a finite but very small diameter as a case of dead heat. Moreover, it seems to be rather questionable whether, for continuum universes and supp A also a continuum, a crisp decision is necessary, desirable or even possible to perform. E.g. even if the fuzzy solution is reduced to a crisp value of a control variable, say a temperature, the decision is performed in a fuzzy manner, by adjusting a thermostat, which keeps the temperature within a sufficiently small but finite interval. Moreover, in some cases, fitting of the fuzzy solution by a 'most suitable' value of a linguistic variable might do. Hence we will deal with this case separately, although guided by the ideas used in the case of a finite universe. Let A be normalized and supp A a continuum. All sets that will occur in the following are assumed to be integrable with respect to the given cardinality measure Q. For decision-making a type of fuzzy decisions D(y), y • X, is given with the membership functions moor) (x), where

mo(y)(y) = 1 and

Vx :/:y: moty) (x) < 1.

(9)

Note that y is not only a location parameter. The decision D(y) generalizes the crisp decision x = y. For our purpose the core A~ of A is of interest. Hence, for all z e A 1, we consider the bounded difference

Zl(z) = A - O(z)

(10)

defined by

ma(z)(X) = max{ma(x)

-

mD(z)(X), O}

(11)

and introduce

G(A) = card zLyJa,A(z).

(12)

This functional will tend to small values if the core of A lies in some D°'(z) with high er, and the functional will show large values if the core of A consists of several disjoint subsets having finite distances from each other such that no D ~(z) can cover them all with high re. This may motivate the choice of this functional. The functional G is a measure of uncertainty, since for every singleton {x0} we

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H. Bandemer

have G({xo}) = 0 and for X and arbitrary A,

mat O, Vx, z ~ X :

1 - mn~z)(X) >1max{ma(x) - mo
sup (1 - mo~z)(X)) >>-sup max{ma(x) - mo~z)(X), 0},

zeA l

zeA 1

sup (1 - motz)(x)) >i su R max{mA(x) - mo¢~)(x), 0}. zEX

z~A

Further, if A2 _~A1 then A 1 _~A] and Vz ~ A] U A~, max{(mA2(x) - mo¢z)(x), 0} i> max{mA,(X) -- mo¢z)(x), 0},

sue, m,,¢..2(x ) >>-sup 1

zeAl

SUR m a ( z ) ; A 2 ( X )

zEA 2

zeA 1

>!

A,(x ),

sup, m a ( z ) ; A l ( X ) .

zEA~

Hence the defining properties are verified. The functional can have jumps, e.g. when an additional maximum of mA is growing and reaches the core in a region separated from the region of the maxima already existing. This, however, is the case where our scruples will rise. Obviously, G ( A ) <~card A, and equality is attained if the possible decisions D ( z ) , z ~ A ~, are disjoint in totality, i.e. if max min {mn~z)(x) } = O. xEX zeA I

(13)

It may also be of interest that G ( A ) = 0 for all A with a one-element core, say {x0} = A ~, and D(xo)~_A. This seems to be a pleasant generalization of the property that a measure of uncertainty vanishes for all singletons.

3. The case of competing decisions In section 2 we considered, in some detail, the case that a dead heat raises our scruples in decision making essentially. O f course, this case may be a very rare event and, hence, its consideration seems to be of only little interest. But the resulting functional G according to (12) can be generalized to the usual case that a multitude of given decisions is to be evaluated with respect to the 'hardness' of our decision scruples. It seems reasonable, however, to maintain assumption (9) for the decisions. When looking through the lines, in which the property of G is verified, we find that the respective cores A 1 can be replaced by suitable a~o-CUtS. Hence we get respective sets of competing decisions { D ( y ) } choosing y e A~ ° or y e A~ °, respectively. Moreover, we may fix the parameter set of D, say W c X , independently of A, this way touching the case that there is a prescribed set of available decisions.

A special measure of uncertainty

285

Finally, the assumption that X is an interval and Q is the Lebesgue measure is a purely technical one. Using the definition of cardinality according to (5) we can generalize the functional G to quite arbitrary universes.

4. A numerical example

To give an impression of our aim we sketch a very simple example from quadratic programming: z = (x - 2) 2 + y2 __ max,

(14a)

x-y>~O,

0~
(14b)

x+y<-T,

0~
(14c)

where T is a family of fuzzy restrictions with parameter t • [0, 6] and membership functions (cf. Figure 1)

mT(s;t)=

1 (6-- S)/(6-- t) 0

for s ~
(15)

For each t the problem is solved according to the well-known approach due to Orlovsky [6]. Given a crisp restriction x + y ~ u with u • [0, 6] we obtain the following solutions 0 ~< u < 4:

(0, 0),

(16a)

u = 4:

(0, 0), (4, 0), (2, 2),

(16b)

4
(4, u - 4), (u - 2, 2),

(10c)

u = 6:

(4, 2).

(16(t)

Hence for t < 4 the point (0, 0) is the only one with a membership value mA(X, y; t ) = 1. The membership function is positive only over the two segments (4,y), 0 ~ y <~2; (x, 2), 2~
m~f !

0

I

I

I

I

I

l

2

3

4

5

Fig. 1.

'i ,

6

t

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H. Bandemer

mA

t

2

3

d

s

Fig. 2. the sense that m A ( 4 , U -- 4; t) = m A ( u - 2; 2; t).

For t = 4 there are t h r e e points: (0, 0); (4, 0); (2, 2 ) ; each with m e m b e r s h i p value 1. For 4 < t < 6 we find two points in the core of A, and for t = 6 again only one. Figure 2 shows the continuous part of the m e m b e r s h i p functions o v e r the segment (x, 2). According to the structure of the solution set we choose a mixed m e a s u r e of cardinality with Q(0, 0) = 1 and d Q = dx on [2, 4] and d Q = dy on [0, 2]. A n explicit specification of the fuzzy decision type, e.g. by mo(u, o)(x, y ) = [1 - 10 Ix - u[ - 1 0

ly -

ol l +

would be of interest only in the n e i g h b o u r h o o d of the point (4, 2) w h e r e t is in the vicinity of 6, causing there a decrease of the uncertainty measure to zero, and is omitted h e r e for brevity. Table 1 shows the decision situation for some t and the corresponding values of G. Table 1 t

core A

G(A)

1 2 3 3.9 4

(0, o) (0, o) (0, o) (0, o) (0, o) (4, o) (2, 2) (4, 0.1) (2.1, 2) (4, 1) (3, 2) (4, 1.5) (3.5, 2)

0.8 1.0 1.3 1.9 3.0

4.1 5 5.5

1.9 1.0 0.5

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287

Acknowledgement The author is indebted to one of the referees for valuable hints, which have improved the presentation of the paper essentially.

References [1] E. Czogala and W. Pedzycz, An approach to the identification of fuzzy systems, Archiwum Automatiki i Telemechaniki 1 (1980) 43-50. [2] E. Czogala, S. Gottwald and W. Pedzycz, Aspects for the evaluation of decision situations, in: M.M. Gupta, E. Sanchez, Eds., Fuzzy Information and Decision Processes (North-Holland, Amsterdam, 1982) 41-49. [3] M. Dubois and H. Prade, A note on measures of specificity for fuzzy sets, BUSEFAL 19 (1984) 83-89. [4] M. Higashi and G.J. Klir, Measures of uncertainty and information based on possibility distributions, lnternat. J. General Systems 9 (1983) 43-58. [5] A. Kaufmann, Introduction h la Theorie des Sou.s-ensembles Flous, 4. Complements et Nouvelle Applications (Masson, Paris, 1977). [6] S. A. Orlovsky, On programming with fuzzy constraint sets, Kybernetes 6 (1977) 197-201.