On aggregation operations of linguistic labels

On Aggregation Operations of Linguistic Labels M. Delgado, J. L. Verdegay, and M. A. Vila Departamento de Ciencias de la Computacion e lnteligencia Artificial, Facultad de Ciencias, Universidad de Granada, Granada, Spain

This article is devoted to defining some aggregation operations between linguistic labels. First, from some remarks about the meaning of label addition, a formal and general definition of a label space is introduced. After, addition, difference, and product by a positive real number are formally defined on that space. The more important properties of these operations are studied, paying special attention to the convex combination of labels. The article concludes with some numerical examples. 0 1993 John Wiley & Sons, Inc.

I. INTRODUCTION

Since the concept was introduced by Zadeh in 1975,’ linguistic variables have been widely used. The following two main application fields can be found in the literature:

(a) Those cases of Knowledge Representation Systems where imprecise assessments appear in some rules and/or facts and we must infer a conclusion by using them. This is the generalized modus ponens problem. The most important way to deal with this is the Compositional Rule of Inference,’ which has been developed and improved by several

(b) Those decision-making or optimization problems with imprecise assessments given in a linguistic way for some of its elements (time, money, et~.).~-’O In this case the labels will represent linguistic assessments of utility or probability, that is, the semantic of these values,’ will be fuzzy numbers.

Combining (in a general sense) linguistic values (labels) is needed in both cases, and just in the last one it is requested to make “arithmetic” operations (addition or product). Two approaches may be used to cope with this task. The first one is based on the Extension Principle, which allows us to aggregate and compare labels through computations on the associated membership functions, but it is well known that by using extended arithmetic operations to handle fuzzy numbers, the vagueness of results increases step by step and the shape of membership functions does not keep when the linguistic variables are inter-

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 8, 35 1-370 (1993) 0 1993 John Wiley & Sons, Inc. CCC 0884-8173/93/030351-20

352 DELGADO, VERDEGAY, AND VILA

active. Thus the final results of those methods are fuzzy sets which do not correspond to any label in the original term set. If one wants to finally have a label, a “linguistic appr~ximation”’~~~’’ is needed.

A second kind of method is the symbolic one, which acts by direct computations on labels, only taking into account the meaning and properties of such linguistic assessments. Anyway, it is always possible to construct hybrid algorithms that perform like a “black box.” Inputs are labels, and within the “box” membership functions are used to make the operations and after a linguistic approximation is carried out (if needed). Thus the outputs are also labels, and from the user’s point of view in most cases the results may be summarized in one table, which will be the only tool that the user finally knows and handles.6-8,’2

From our own point of view, symbolic methods seem to be the more “natural” ones when the “linguistic approach” (see Ref. 1) is used, because the linguistic assessments are just approximations which are given and handled if obtaining more accurate values is impossible or unnecessary. On the other hand, it is obviously necessary for these operations to be computationally simple and quick because they are to be used in user-driven algorithms or decision- making processes.

According to these comments in this article we develop symbolic algorithms for addition and multiplication (by a positive real number). In Sec. I1 we analyze some key features of the linguistic numerical labels which leads to a formal definition of the label space (the value set) to be used in assessments. After that we study addition, difference, and product by a positive number on this label space (Secs. 111-V). Section VI deals with the convex combination of labels. The article concludes with some numerical examples.

11. THE GENERALIZED LABEL SET

To establish what kind of label sets will be used should be the first obvious task. Roughly speaking, we will place us in a simple but simultaneously general context characterized by a certain semantic representation of the labels according to Hypotheses 1 and 2 below.

Hypothesis 1. Let X be the interval [ x l , x 2 ] C R, where x l or x 2 may be --x or m, respectively, and X = {pi} i E M C Z (h stands for integer number set) be a finite and totally ordered term set on X in the usual sense (see Refs. 12-14). Any label pi will represent a possible value for a linguistic real variable, that is, a vague property or constraint on X . According to Zadeh,’*” each label has a semantic representation as fuzzy set of X.

We will consider M = (0, 1, . . . , m} if x l 2 0, and M = { - n , -n + 1, . . . , 0, 2, . . . , n} otherwise. The first case corresponds to those magnitudes being positive in nature (time, distance, etc.), which are usually qualified by term sets including ‘‘low,’’ “high,” etc. The second type of index set will be used when positive and negative values could be possible, for example, if the variable is profit and loss. In this case words like “bad,” “good,” and so forth will appear in the term set. Any case M reflects the order of X and we will denote by i,, and i, its first and last elements, respectively.

LINGUISTIC LABELS 353

Figure 1.

Hypothesis 2 . The semantic of any label p i E X is given by an LR fuzzy interval (a i , b, , ci, dJLR ,15 and

(a) for any i , i # io, i # i,, p i is symmetric, that is L(.) = R(*); ci = di = k , ( b ) a i o = x , , c . = 0 , d . = k , b i m = x 2 , d i m = 0 , c i m = k , I f x , # ‘0 10 - - m o r x 2 # m t h e n

(bj, - ai,) = k’ or (bi, - aim) = k’, respectively,

(c) for any i, (d) e = k’ + k“ z k. (See Fig. 1).

- bi) = k“. Additionally if i # io and i # i,, (b, - ai) = k ‘ ,

Let us observe that the parameters k and k‘ can be seen as measures of width, whereas k could represent a gap.

Additionally, it should be noted that most of the usual semantic representa- tions have this format.

Before going further, wondering about the meaning of the aggregation operations between linguistic labels and their relations with the semantic representation seems to be necessary, as the following example shows.

Example 1 . Let us consider the variable “time of carrying out something” to be linguistic and suppose the associate term set is

X = {very-low (vl), low (I), intermediate (in), high (h), very high (vh)}

with semantic for the labels given in Fig. 2. Now let us wonder about the global time to consecutively do two activities

A , and A, with times t (A, ) = I and t(A2) = h. Roughly speaking, the total time is the addition of I and h.

If the extended addition of fuzzy numbers is used on the above representation of the labels, the time I @ h is to be semantically associated as shown in Fig. 3.

Obviously I G3 h is not an element of X . To fulfill this condition a linguistic approximation is needed, and it is intuitively reasonable to obtain ‘‘very high” as such an approximation. However, no expert will qualify as very high a time


L=R=linear, k = 2, k'= 0, k"= 2.5

Figure 2.

around 10 (with an upper bound equal to 14)) when he is talking about the time necessary to carry out two activities. To be qualified with this label the time of carrying out A , and A , should be greater (for instance, around 18 or 20).

This contradiction may be solved if we accept that something changes in the semantic representation of labels after adding two terms. Actually, we may think X does not change but X enlarges (as the result must be evaluated according to another scale) because we are considering time for two tasks. In fact, if we place the fuzzy number given by Fig. 4 on the semantic representation corresponding to the same label set with a referential set twice than the one before, we obtain the result I @ h to be approximated by the label intermediate. This is coherent with common-sense knowledge, because if one spends a low time in one task and a high time in another one, the whole time to do both together will be usually qualified as intermediate.

A similar situation arises when labels are to be multiplied by a real number. For instance, the time to complete one-half of A2 (one-half of the time to complete A,) could be qualified as h, too, in many cases.

A lot of examples about money, risk, and so forth can be found to confirm that when labels are aggregated, the referential set changes and thus, i f we assume the term set keeps, then the semantic representation of labels changes.

According to the above ideas, to achieve a good definition for aggregation operations, a new flexible semantic for labels is needed being adaptive to the number of operations already made.

DEFINITION 2.1. Let X be the label set with basic semantic given b y Hypotheses 1 and 2. The generalized label space based on 3t is the Cartesian product:

where Z f stands for the non-negative integer set. Any ( p i , k ) E % will be called a generalized label and must be interpreted as the label p , with semantic given by: k G3 (a , , bi) c i , di)LR, where the product is the classical one of a real positive number b y a fuzzy number, that is,



Figure 4.

V x E %; x = ( p i , k ) = (ka,, kh; , kci , kdJLR.

Let us observe k represent the number of times that the scale changes that will correspond with the number of aggregations which have been made. Remark. In some cases magnitudes such that “ h times greater” does not

exactly imply “ h times wider” may exist. For them a scale factor cp E 3, cp > 0 could be considered to act as a subjective enlargement factor which allows us to adjust the semantic of labels. In these cases the generalized labels may be represented as [ ( p i , k ) , cp] with semantic k cp G3 (a,, h,, c,, d J L R , that is,

Obviously cp = 1 will represent “ h times greater, h times wider,” cp < 1 reduces this proportion and cp > 1 amplifies it.

The basic label set is X = { ( p i , l)} , iE M } and we will consider ( p i , h ) , h > 0, when we are using the linguistic term p i in some situation where the magnitude to be qualified appears h times “greater” than the initial one. For example: time to carry out h tasks, money to be obtained from h investments, and so forth.

Generalized labels with h = 0 will appear in relation with the difference between labels in the same scale, or to aggregate labels without changing the scale, in particular, when we need to linguistically express the comparison between ( p j , 4 and ( p i , 4, i , j E M , d > 0.

111. AGGREGATING LABELS BY ADDITION

Let ( p i , t ) , ( p j , h) be arbitrary elements of % such that t , h > 0. From their semantic and by using extended addition it is easy to obtain

but S = [(ta; + haj), (tbi + hb,), ( tc; + he,), ( td; + hdj)lLR is not a generalized label, that is, % is not closed with respect to the extended addition. To achieve this key condition the most direct way is to carry out a linguistic approximation


of S, and thus we may establish ( p i , t ) 63 ( p j , h) = ( p s , t + h), ( p s , t + h) being the linguistic approximation of S [on the (t + h) scale] that is, the generalized t + h label “nearest” (according to the used distance criterion) to the fuzzy number S.

Obviously this characterization must be specified, because the linguistic approximation procedure (“nearest” concept) depends upon the used distance, which in turn depends upon the context. The following lemmas allow us to give a more precise definition of ps (i.e., to the index s E M which determine it) being appropriated to our data framework.

LEMMA 3.1. For any i , j E M and positive t and h (t + h)a, 5 (ta, + haj) 5 ( f + h)ar+l ( t + h)br 5 (tb, + hbj) 5 ( t + h)b,+,

where r = [(ti + hj]/(t + h)] (1 ] standsfor the greatest integer less or equal than). Proof. According to Hypotheses 1 and 2 we have:

a, = x1 + (i - io)e; aj = x , + (j - io)e; a, = x1 + (r - io)e.

On the other hand, by definition

[(ti + hj)/(t + h)] 5 (ti + hj)/(t + h) 5 [(ti + hj9/(t + h)J + 1,

and so

x1 + ([(ti + hj]l(t + h)] - io)e I x1 + (((ti + hj]/(t + h)) - io)e

I x1 + ([(ti + hj)l(t + h)J + 1 - io)e,

which implies

The first inequality is proved. The proof for the second one is quite similar. LEMMA 3.2. For any i, j E M and positive h and t

(ta, + haj) - (tc, + hcj) 2 ( t + h)a, - ( t + h)c, (tb; + hbj) + (td; + hdj) 5 (t + h)b,+l + (t + h)d,+,

where r is given in Lemma 3.1 Proof. When both i, j are different to io and i,,, , r is different to io and i,,, -

1 and then:

and the property obviously follows from the Lemma 3.1. Thus we only need to prove the first inequality for r = io, and the second


one for r = i , - 1 . We shall only analyze the first case as the second is to be similarly proved.

If r = io then i = io and may simultaneously b e j = io. Obviously if both indices are equal to io we have ci = cj = c, = 0 and the inequality holds. Alternatively, let us suppose i = io and j # io. In this case, the first inequality becomes

fx , + hx, + hG - io)e - hk r ( t + h)x, 0‘ - io)e z k

and this last relation always holds becausej - io > 1 and e 2 k by Hypothesis 2(d).

Lemmas 3.1 and 3.2 guarantee S is “between” ( p r , t + h) or ( p r + , , t + h), where r is given by Lemma 5.1 Lie., the mode interval of S is neither lower nor upper than the ones of ( p r , t + h) and (P,+~, t + h) and its support is contained in the union of the supports of ( p r , t + h) and ( p , + I , t + h)]. Thus the linguistic approximation to S may only be one of these two generalized labels, whichever the used distance criterion may be. The specific choice of such criterion and the relative position of both the mode interval and the support of S with respect to the ones of ( p , , t + h) and t + h) will determine the final ( p , , t + h).

Taking into account the topology of the problem (all generalized labels have the same shape) a good criterion for the linguistic approximation is the one based in the gravity center of the fuzzy numbers. That is, s is to be obtained as

where gv(.) stands for gravity center. From Hypotheses 1 and 2 is quite easy to obtain

DEFINITION 3.1. For any two ( p i , t ) , ( p j , h) E 94 with positive t and h ,

s = round((ti + hj)/(t + h)) L(ti + hJ3/(t + h)] [(ti + hj)/(t + h)J + 1

$[((ti + hj)/(t + h) + [(ti + hj)/(t + h)ll 5 0.5, otherwise.

With this definition, the addition has some interesting properties: PROPERTY 3.1.

Proof. PROPERTY 3.2. Proof. Obvious from Definition 3.2. PROPERTY 3.3.

For any index i and posirive t and h, ( p i t ) @ ( p i , h) =

( P i , r + h). Obvious, because round ((ti + hi)& + h)) = i.

The addition ($or positive t and h) is commutative.

Let (pi, t ) , ( p j , h), ( p k r r> E % be and assume


ti + hj + I k I s - t + h + l

[(pi t ) @ ( p j h ) l@ ( ~ k 9 0 = ( ~ . v 9 t + h + I ) ; ( P i , t ) @ [ ( P j , h ) @ ( ~ k , I ) l = ( P s , t + h + I )

then 1s - s’I I 1. Proof. Let us note q = round((ti + hj)/(t + h). Obviously s = round

( [ ( t + h)q + lk]/(t + h + I ) ) and according to the above definition

5 0.5 + 0.5 ( t + h)/(t + h + I ) .

On the other hand q 5 ( t i + hj)/(t + h)) + 0.5 and therefore

1s’ - ti + hj + Ik t + h + l

5 0.5. ( t + h)[(ti + hj’)/(t + h)) + 0.51 + lk I s - r + h + l

I 0.5 + 0.5 (h + I)/(t + h + I ) ,

Thus

Similarly we can obtain

and finally

IS - s’I I 1 + 0.5 ( I + h/(t + h + I ) ) .

Since h, t , 1 are positive, h/(h + t + I ) < 1 and thus 0.5 (1 + h/(t + h + 0) < 0.5. By definition, both s and s’ are integers and therefore 1s - s’1 I 1.

This property may be seen as some kind of associativity for the addition. According to it and Lemma 3.2 the only difference between [(pi, t ) CB ( p j , h)] @ ( p k , I ) and (pi, f ) C3 [ ( p j , h) @ ( p k , I ) ] ought to be the final linguistic approximation which may produce in each case the left or the right possible generalized label.

Let us remark Definition 3.2 does not apply for the cases h = 0 or t = 0. From a constructive process quite similar to the above one, we obtain the following characterization.

DEFINITION 3.2. Let (pi, t ) , ( p j , h) E 93 be such that h = 0 or t = 0. Their addition [(pi, t ) @ ( p j , h)] is the generalized label ( p , t + h), where

i,ifi + j < i , , i,ifi + j>i,,,, i + j otherwise.


In the next section we will see the generalized labels with h = 0 will appear in relation with the difference between labels in the same scale. On the other hand, they ought to be used to aggregate labels without changing the scale [let us remember ( p i , h) @ ( p j , 0) = (ps , h)l.

Remark. Under Definition 3.2, Properties 3.1 and 3.3 do not keep but Property 3.2 remains true (the proof is trivial).

IV. DIFFERENCE OF GENERALIZED LABELS

Once the addition has been defined, it seems natural to ask for an inverse operation, a difference between (generalized) labels. From the idea of “inverse” we may give the following general characterization.

DEFINITION 4.1. The difference o f ( p i , t ) , ( p j , h) E % [denotedby ( p i , t ) 8 ( p j , h)] is the generalized label ( p , , r ) such that ( p s , r ) 63 ( p j , h) =

Like the case of addition, this initial characterization must be tuned in order to obtain an operative symbolic algorithm. The following remarks allow us to do it.

( P i , t ) *

Remark 4.1. The crude application of Definition 4.1 implies

1 = t - h; i = round[((t - h) s + hj)/(t - h ) ] , (4.1)

which acts as an equation to obtain s. It is obvious t must be greater or equal to h and there is no case €or which

t < h. In fact this should be inconsistent with the meaning of generalized label. Therefore we impose

Remark 4.2. Equation (4.1) has different solutions for s. For instance, if t # h (which implies t # l ) , s’ = [( t i - hj)/(t - h)] and s2 = [(ti - hj)/(t - h)] + 1 could be considered as possible solutions. Actually, in the case o f t # h the best choice is s = round((ti - hj)/(t - h)) as the following reasoning shows.

By definition (ti - hj)/(t - h) = s’ + a1 and (ti - hj)/(t - h) = s2 - a2, where a’ + a* = 1 and a’ or a* must be less than 0.5. Thus it is easy to obtain

i - al(t - h)/t = ( ( t - h)s’ + hj)/(t - h),

i + a2(t - h)/t = ( ( t - h)s2 + hj)(t - h).

Only when a‘ < 0.5, i = 1, 2 then al(t - h)/t < 0.5 and a2(t - h)/t < 0.5, and therefore for t # h we should take

s = round((ti - hj)/(t - h)).

In the special case h = t we will use s = (i - j )

LINGUISTIC LABELS 36 1

Remark 4.3. By definition s must belong to M , therefore for t # h,

(ti - hj)/(t - h) 2 io 3 r(i - io) L h ( j - i,), (4.3)

(ti - hj)/(t - h) 5 i, 3 t(i, - i ) 2 h(i, - j ) . (4.4)

If t = h we have

Under the sets of conditions (4.2, 4.3, 4.4}, or (4.2, 4.5) the difference, as defined in Definition 4.1, acts as a true operation on the set of generalized labels. All these considerations may be summarized in the following definition.

DEFINITION 4.2. Let ( p , , t ) , ( p j , h) E % be and suppose either {t > h ; t(i - i,) 2 h(j - i,); t(i, - i ) 2 h(i, - j ) } or (t = h ; io 5 i - j 5 i,} holds. The difference between (pi, t ) and ( p j , h ) ((pi, t ) 8 ( p j , h)) is the generalized label ( p , , 1) where

1 = r - hands = round((ti - hj)/(t - h)) if t > h,

1 = O a n d s = i - j otherwise.

V. PRODUCT BY A POSITIVE REAL NUMBER

From applying Definitions 3.1 or 3.2 repeatedly it is easy to obtain: DEFINITION 5.1. Let (pi, t ) E % be and u a positive integer number. The product u CB (pi, t ) is given by (pi, t ) @ ..".. CB (pi, t ) , that is,

u @ (pi, t ) = (pi, ut ) if t > 0,

i, if ui< io ,

ui otherwise. u @ (pi, 0) = ( p , , 0) with s =

From a constructive process quite similar to the ones in the above sections

DEFINITION 5.2. Let (pi, t ) E 93 be and a E (0, 1) . The product a CB (pi, t ) is the generalized label ( p s , t ) where s = round(ai). When t > 0, ( p , , t ) is the label whose semantic representation is the nearest

From these definitions we can give the general product by any positive real

DEFINITION 5.3. Let be ( p , t ) E % be and, y E R y > 0. The product, y

we arrive at:

one to the fuzzy number ((pa ta,, (pa rbi , (pa tc, , (pa tdJLR.

number as follows:

@ ( P , t ) is


where u = [yJ and a = y - u. The following property can be considered as a kind of distributivity. PROPERTY 5.1. Let ( p i , t ) , ( p j , h ) E 93 with t , h > 0 be, and y a positive real number. I f w e denote ( p , , 0 = y @ ( ( p i , t ) 0 (p , , h)) , ( p , , , , 1’) = Y 0 ( p i , t ) CT3 y @ ( p j , h ) , then I = 1’ and 1s - s’I 5 1. Proof. By definition 1 = u(t + h) + ( t + h) and I’ = ut + t + uh + h.

Trivially 1 is equal to 1 ’ . To prove the second thesis we must consider two different cases according to Definition 5.3.

(A) u = [yl > 0. Let note k, = round((ti + hj)/(t + h)) and k2 = round(ak,) (a = y - u). Then,

((ti + hj)/(t + h)) - 0.5 5 k, 5 ((ti + hj)l(t + h)) + 0.5 a k , - 0.5 5 k, S a k , + 0.5 (5.1)

and thus

a((ti + hj)/(t + h)) - (1 + a)0.5 5 kz 5 a ((ti + hj)/(t + h)) + (1 + a) 0.5.

(5 .a Since (by definition)

s = round ( t u(r + h)kl + h) + + ( t t + + h)k, h ) = round(%)

it is obvious that

vk, + k, uk, + k2 0.5 5 s I ~ + 0.5. ___-

u + 1 u + l

Let replace k, and k, by their lowest bounds [given in (5.1) and (5.2), respectively], in the left-hand inequality before. We obtain

ti + hj ti + hj t + h t + h U- - 0.5 u + a- - (1 + a)0.5

u + l 5 s ,

that is,

u + a t i + hj a0.5 --- (1 +-+. u + l t + h

Similarly, from the right-hand side of the inequality,


v + at i + hj a0.5 S S - - + 1 +-

v + l t + h u 1-1'

and therefore

v + at i + hj v + l t + h v + 1'

a0.5 (5.3)

Now, let us introduce

k; = round(ai); k; = round(aj1

and

K,' = round ( vti t (v + + tk; = round (2) vi + k' ; v + l

Since (by definition)

(5.4) t (v + 1)K; + h(v + I)K;

(v + l ) ( t + h) tK,' + hK; ) = ( t + h )'

it is obvious that ai - 0.5 5 k; 5 ai + 0.5 and (vi + k;) / (v + 1) - 0.5 5 k ' , ' ~ (vi + k ; ) + 0.5.

Let us replace k; by its upper and lower bounds. After doing some simplifi- cations we obtain

v + a v + 2 u+*<v<i - + 0.5- .v + a

I - - 0.5- v + l v + l ' - v + l v + 1 '

and similarly,

.v + a v + 2 5 k 2 5 J - + 0.5- v f a u + 2 j - - 0.5 - v + l v + l v + 1 v + 1 '

Finally, from Eq. (5.4)


u + a t i + hj u + 2 5 s I <-- + 0.5- u + a t i + hj v + 2 0.5 - u + l t + h u + l u + l t + h u + 1 ’

that is

$ 1 - -~ u + a t i + hj 1 5 0 S z . v + 2 u + l t + h (5.5)

Now we may combine Eqs. (5.3) and (5.5) to obtain

IS - s’I I I + O . ~ / ( U + 1) + 0.5 (U + 2)/(v + 1) = I + 0.5 (v + 2 + a)/(u + 1).

It is easy to prove (v + 2 + a)/(u + 1) 5 2, because a < 1. On the other hand both s and s’ are integer numbers, therefore we can assure Is- s’I 5 1.

(B) v = [ y ] = 0. In this case a = y and (from Definition 5.1) it is easy to obtain

+ hj I 0.5 (1 + a) and s’ - a-/ ti + hj 5 0.5 (1 + a). 1 s - a--/ t + h 1 t + h

Therefore, also in this case 1s - s’I 5 1 + a I 1

VI. CONVEX COMBINATION

An interesting application of the above defined operations is the convex combination of labels. This one appears when it is necessary to combine linguistic labels and probability values giving some kind of “average.” It may be useful, for example, in Decision-Making Problems. We first define the convex combination of two labels, and then generalize it to any number of labels by using a recursive approach.

DEFINITION 6.1. Let ( p i , f ) , ( p i , t ) E %, t > 0 be andh E [o, 11. (Without losing generality we shall adm’it j > i). The conuex combination A @ ( p j , t ) C3 (1 - A) (pi, t ) is the generalized label ( p , , t ) given by

This convex combination shows the following property. PROPERTY 6.1. Proof. According to Definition 6.1, c = i + round(X(j - i)) and therefore

On the above hypotheses, i I c 5 j .

X ( j - i) - 0.5 + i I c 5 X ( j - i) + 0.5 + i ,

so that i 5 c holds.


Obviouslyj 2 A ( j - i) + i (as A 5 l), therefore c S j + 0.5 and thus c 5 j (because both c and j are integer numbers).

In order to establish a recursive procedure to generalize the above definition to an arbitrary number of labels we will denote C{Ak, (pi(,), t ) , k = 1, K } the convex combination of K labels with weights { A k , k = 1, 2 , . . . , K } ( A A E

DEFINITION 6.2. Let ( p i ( k ) , t ) E %, k = 1, 2 , . . . , K be, and assume i ( K ) 5 i(K - 1) . . . 5 i(1) without loss of generality. For any set of coeficients { A k E [O, 11, k = 1,2, . . . ,K CA, = l} the convex combination of these K generalized labels is given by

[0, 11, k = 1, 2 , . . . , K , Z A k = 1) .

where

In order to show how to develop this definition we will consider the case K = 3.

Let ( p i , t ) , ( p j , t ) , ( p s , t ) E % be such that s 5.j 5 iand, A, p, y E [0, 13 verifying A + p + y = 1. To obtain the convex combination of these three labels the following steps are to be done:

VII. EXAMPLES

In this section we will consider two examples, corresponding to the two possible cases for the label set:

(1) M = (0, 2 , 3, 4, 5, 6} , that is, a case of positive label set. ( 2 ) M = { - 3 , - 2 , - 1, 0, 1, 2, 3) that is, a part of labels are considered to be

negative.

In both cases, tables of additions and differences are computed for several scales (the second element of generalized labels), that is, we will compute (pi, t ) CB ( p j , h) and ( p i , t ) 8 ( p j , h) i , j E M for several t and h values. Some situations o f t # h, h = 0, and t = h are considered for both cases. The product by several real numbers belonging to [0, 11 and some convex combinations are also computed, restricting ourselves to the initial label space, that is, with seCond component equal to one.

Case 1 . X = {lowest, very low, low, medium, high, very high, highest} Obviously all labels have “positive meaning” and so M = (0, 1, 2 , 3, 4, 5 ,

DELGADO, VERDEGAY, AND VILA

6}, that is p o = lowest, pI = very low, p 2 = low, p 3 = medium, ~4 = high, 1-35 = very high, p6 = highest.

Addition table with t = 3 and h = 1. (Definition 3.1).

lowest lowest lowest v. low v. low v. low v. low low v. low v. low v. low v. low low low low low low low low low low medium medium medium medium low medium medium medium medium high high high medium medium high high high high v. high v. high high high high v. high v. high v. high v. high highest v. high v. high v. high v. high highest highest highest

lowest v. low low medium high v. high highest

Difference table with t = 3 and h = 1. (Definition 4.2).

lowest lowest v. low low v. low v. low lowest low medium medium low low v. low v. low lowest medium v. high high high medium medium low low high highest highest v. high v. high high high medium v. high highest highest v. high v. high highest highest

lowest v. low low medium high v. high highest

Addition table with t = 2 and h = 0. (Definition 3.2). lowest v. low low medium high v. high highest

lowest lowest v. low low medium high v. high highest v. low v. low low medium high v. high highest highest low low medium high v. high highest highest highest medium medium high v. high highest highest highest highest high high v. high highest highest highest highest highest v. high v. high highest highest highest highest highest highest highest highest highest highest highest highest highest highest

Difference table with t = 2 and h = 0. (Definition 4.1). lowest v. low low medium high v. high highest

lowest lowest lowest lowest lowest lowest lowest lowest v. low v. low v. low lowest lowest lowest lowest lowest low low low v. low v. low lowest lowest lowest medium medium medium low low v. low v. low lowest high high high medium medium low low v. low v. high v. high v. high high high medium medium low highest highest highest v. high v. high high high medium


Addition table with t = 2 and h = 2. (Definition 3.11. ~~ ~ ~

lowest v. low low medium high v. high highest lowest lowest v. low v. low low low medium medium v. low v. low v. low low low medium medium high low v. low low low medium medium high high medium low low medium medium high high v. high high low medium medium high high v. high v. high v. high medium medium high high v. high v. high highest highest medium high high v. high v. high highest highest

Difference table with t = 2 and h = 2. (Definition 4.1) ~~~ ~

lowest v. low low medium high v. high highest lowest lowest v. low v. low lowest low low v. low lowest medium medium low v. low lowest high high medium low v. low lowest v. high v. high high medium low v. low lowest highest highest v. high high medium low v. low lowest

Product by several [0 , 11 values. (Definition 5.2). lowest v. low low medium high v. high highest

0.05 lowest lowest lowest lowest lowest lowest lowest 0.25 lowest lowest v. low v. low v. low v . low low 0.45 lowest lowest v. low v. low low low medium 0.65 lowest v. low v. low low medium medium high 0.85 lowest v. low low medium medium high v. high

Some convex combinations. (Definition 6.2). n Coefficients Components Result 2 0.50 0.50 v. low v. high medium 3 0.80 0.10 0.10 v. low low highest low 4 0.30 0.20 0.30 0.20 lowest low high highest medium 4 0.10 0.50 0.10 0.30 lowest low high highest high

Case 2. X = {worst, very bad, bad, medium, good, very good, best}

The labels worst, very bud, and bud have "negative meaning" whereas good, very good, and best have "positive meaning." In its turn medium is a "central value." Thus in this case M = { - 3, - 2, - 1, 0, 1, 2, 3) and p - , = worst, p - 2 = very-bad, = bad, po = medium, p1 = good, p , = very-good, p 3 = best.

DELGADO, VERDEGAY, AND VILA

Addition table with t = 3 and h = 1. (Definition 3.1). worst v. bad bad medium good v. good best

worst worst worst worst v. bad v. bad v. bad v. bad v. bad v. bad v. bad v. bad v. bad bad bad bad bad v. bad bad bad bad bad medium medium medium bad bad medium medium medium good good good medium medium good good good good v. good v. good good good good v. good v. good v. good v. good best v. good v. good v. good v. good best best best

Difference table with t = 3 and h = 1. (Definition 4.1). worst v. bad bad medium good v. good best

worst worst v. bad v. bad v. bad worst worst bad medium bad bad v. bad v. bad worst worst medium v. good good good medium bad bad v. bad good best best v. good v. good good good medium v. good best best v. good v. good best best

Addition table with t = 2 and h = 0. (Definition 3.2). worst v. bad bad medium good v. good best

worst worst worst worst worst v. bad bad medium v. bad worst worst worst v. bad bad medium good bad worst worst v. bad bad medium good v. good medium worst v. bad bad medium good v. good best good v. bad bad medium good v. good best best v. good bad medium good v. good best best best best medium good v. good best best best best

Difference table with t = 2 and h = 0. (Definition 4.1). worst v. bad bad medium good v. good best

worst v. bad v. bad worst worst worst worst worst v. bad bad bad v. bad v. bad worst worst worst bad good medium bad bad v. bad v. bad worst medium v. good good good medium bad bad v. bad good best v. good v. good good good medium bad v. good best best best v. good v. good good good best best best best best best v. good v. good

LINGUISTIC LABELS

Addition table with t = 2 and h = 2. (Definition 3.1).

worst worst worst v. bad v. bad bad bad medium v. bad worst v. bad v. bad bad bad medium good bad v. bad v. bad bad bad medium good good medium v. bad bad bad medium good good v. good good bad bad medium good good v. good v. good v. good bad medium good good v. good v. good best best medium good good v. good b. good best best

worst v. bad bad medium good v. good best

Difference table with t = 2 and h = 2. (Definition 4.2). worst

worst medium v. bad good bad v. good medium best good v. good best

v. bad bad medium good v. good best bad v. bad worst medium bad v. bad worst good medium bad v. bad worst v. good good medium bad v. bad worst best v. good good medium bad v. bad

best v. good good medium bad best v. good good medium

Product by several [0, 11 values (Definition 5.2). worst v. bad bad medium good v. good best

0.05 medium medium medium medium medium medium medium 0.25 bad bad medium medium medium good good 0.45 bad bad medium medium medium good good 0.65 v. bad bad bad medium good good v. good 0.85 worst v. bad bad medium good v. good best

Some convex combinations. (Definition 6.2). n Coefficients Components Result 2 0.50 0.50 v. bad v. good medium 3 0.80 0.10 0.10 v. bad bad best medium 4 0.30 0.20 0.30 0.20 worst bad good best v. good 4 0.10 0.50 0.10 0.30 worst bad good best good

VIII. CONCLUDING REMARKS

As can be seen, it is possible to define aggregation operations between linguistic labels on the basis of their meanings.

It should be remarked that, once they are defined, these can be performed without any reference to this semantic representation. So, they are very useful from a computational point of view, since they may be implemented as tables or simple procedures.

On the other hand, defined operations are coherent, from an intuitive point of view, and they have some interesting properties, therefore they could be


applied to Decision-Making or Optimization Problems involving linguistic information.

References

1. L.A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning, Parts I, 11, and 111,” Information Sciences 8, 199-249; 8, 301-357; 9,

2. R. Lopez De Mantaras, P. Meseguer, F. Sanz, C. Sierra, and S. Verdaguers, “A fuzzy logic approach to the management of linguistically expressed uncertainty, Proceedings of the Eighteenth International Symposium on Multiple Valued Logic Computer Society Press, Palma de Mallorca, (Spain, 1988), pp. 144-151.

3 . M. Mizumoto and H.-J. Zimmerman, “Comparison of fuzzy reasoning methods,” Fuzzy Sets Syst. 8, 253-283 (1982).

4. M. Delgado, E. Trillas, J.L. Verdegay, and M.A. Vila, “The generalized “modus ponens” with linguistic labels,” Proceedings of the Secondlnternational Conference on Fuzzy Logics and Neural Network (IIzuka, Japan, 1990), pp. 725-729.

5. J.L. Castro, M. Delgado, and J.L. Verdegay, “Using fuzzy expected utilities in decision making problems,” Third World Conference on Mathematics at the Service of the Man, (Barcelona, 1989).

6. M. Delgado, J.L., Verdegay, and M.A. Vila, “Ranking linguistic outcomes under fuzziness and randomness,” Proceeding ofthe Eighteenth International Symposium on Multiple Valued Logic (Computer Society Press, Palma de Mallorca, Spain,

7. M. Delgado, J.L. Verdegay, and M.A. Vila, “On valuation and optimization problems in fuzzy graphs: A general approach and some particular cases,” ORSA .I. Comput. 2 , I , 74-84 (1990).

8 . M. Delgado, J.L. Verdegay, and M. Vila, “Playing matrixgames defined by linguistic labels,” in Multiperson Decision Making Using Fuzzy Sets and Possibility Theory, edited by J. Kacprzyk and M. Fedrizzi (Kluwer, 1990), pp. 298-310.

9. R. Degani and G. Bortolan, “The problems of linguistic approximation in clinical decision making,” Int. J . Approxi. Reason. 2 , 143-161 (1988).

10. M. Tong and P. Bonissone, “Linguistic solution to fuzzy decision problems,” TIMS Stud. Manage. Sci. 20, 323-334 (1984).

1 1 . L.A. Zadeh, “Fuzzy sets and information granularity,” in Advances in Fuzzy Sets Theory and Applications, edited by M.M. Gupta et al. (North-Holland, Amsterdam, 1979), pp. 3-18.

12. P.P. Bonissone and K.S. Decker, “Selecting uncertainty calculi and granularity: An experiment in trading-off precision and complexity,” KBS Working Paper, General Electric Corporate Research and Development Center, Schenectady, New York, 1985.

13. P.P. Bonissone and R.M. Tong, “Editorial: Reasoning with uncertainty in expert systems,” Znt. J . Man-Mach. Stud. 22, 241-250 (1985).

14. P.P. Bonissone, “Reasoning with uncertainty in expert systems: Past, present and future,’’ KBS Working Paper, General Electric Corporate Research and Develop- ment Center, Schenectady, New York, 1985.

15. D. Dubois and H. Prade, Fuzzy and Systems Theory and Applications (Academic Press, New York, 1980).

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1988), pp. 352-356.

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On aggregation operations of linguistic labels