DECSAI Department of Computer Scienceand Arti�cial IntelligenceAggregation OperatorsforLinguistic Weighted InformationF. Herrera, E. Herrera-Viedma

Technical Report #DECSAI-95120.October, 1995ETS de Ingenier��a Inform�atica. Universidad de Granada.18071 Granada - Spain - Phone +34.58.244019, Fax +34.58.243317

Aggregation OperatorsforLinguistic Weighted InformationF. Herrera, E. Herrera-ViedmaDept. of Computer Science and Arti�cial IntelligenceUniversity of Granada, 18071 - Granada, Spaine-mail: herrera,viedma@decsai.ugr.esAbstractThe aim of this paper is to model the processes of the aggregation of weighted informa-tion in a linguistic framework. Three aggregation operators of weighted linguistic informationare presented: linguistic weighted disjunction (LWD) operator, linguistic weighted conjunc-tion (LWC) operator and linguistic weighted averaging (LWA) operator. A study of theiraxiomatic is presented to demonstrate their rational aggregation.Keywords: Linguistic modelling, aggregation operators, fuzzy linguistic quanti�er.1 IntroductionSome situations present precise and rigorous quantitative aspects as well as fuzzy and unrigorousqualitative aspects. Therefore, phenomena must be de�ned using quantitative concepts as well asqualitative concepts. Dealing with quantitative concepts is easy, and it may be done by means ofthe numerical variables. The problem is how to deal with qualitative concepts. The use of FuzzySet Theory, proposed and developed by Prof. L. A. Zadeh in [34], has given very good resultsmodelling the qualitative aspects [35]. Fuzzy Set Theory provides a exible framework, where itis possible to satisfactorily solve many of the obstacles of lack of precision. It is an approximatetechnique in its nature, which represents the qualitative aspects in qualitative terms (linguisticterms) by means of linguistic variables, that is, variables whose values are not numbers butwords or sentences in a natural or arti�cial language. The use of words or sentences rather thannumbers is, in general, a less speci�c, more exible, direct, realistic and adequate form to expressthe qualitative aspects and it is very widespread as may be seen in [1, 4, 6, 13, 22, 23, 24, 31, 33].On the other hand, we can �nd situations where the information handled is not equallyimportant, i.e., the framework is heterogeneous. For example, when a group of medical expertsexpress their opinions on the possible illness that a determinated patient presents, (i) on the onehand, their diagnostics must not be considered with equal relevance, given that, there will bemedical experts with more experience or with more study years than others, and therefore, allthe opinions shall not be equally reliable; and (ii) on the other hand, a �nal and global diagnosticmust be made using the initial and individual diagnostics. This heterogeneous framework hasbeen considered by various authors in opinions aggregation operators [2, 20, 10, 32], consensusmodels in group decision making [14, 16], in fuzzy pattern matching [9], and in knowledgesystems [21].One way of modelling the �rst aspect is by assigning a weight to each medical expert. Theweights are quantitative or qualitative values, which may be interpreted in at least two di�erentways [9, 10]:

1. Each medical expert is viewed as a subgroup and the weight re ects the relative size ofthis subgroup.2. The weight may re ect the relevance of the medical expert in the group. This level ofrelevance may act as a constraint on the opinions that a medical expert may express.One way of dealing with the second aspect, in general, is to use adequate operators for combininginformation before reaching a �nal decision or action, usually called information aggregationoperators. Issues of weighted aggregation operators have been studied in a quantitative settingin [2, 8, 9, 20, 21, 25, 27, 32], and in a qualitative setting in [3, 31, 32, 33].In short, we can �nd situations where the information handled is imprecise by nature andis not equally important, and where some appropriate aggregation operators of weighted infor-mation are required. According to this idea, in this paper, we shall present three aggregationoperators for linguistic1 weighted information (linguistic variables for expressing experts' opin-ions and linguistic weights on the experts):1. Linguistic Weighted Disjunction (LWD).2. Linguistic Weighted Conjunction (LWC).3. Linguistic Weighted Averaging (LWA).They are de�ned using the LOWA operator [12, 15], the weighted minimum and maximumoperators [8], two families of connectives [11], and the concept of fuzzy majority represented by afuzzy linguistic quanti�er [36]. In order to demonstrate the good performance of these operatorswe shall study some postulated axiomatics and properties of an intuitively acceptable weightedaggregation operator.In order to do so, the paper is structured as follows: Section 2 shows the considered linguisticframework and the LOWA operator; Section 3 presents the linguistic weighted aggregationoperators and some of their properties; Section 4 contains an example of the application ofthese operators in decision making for non-homogeneous groups; and �nally some conclusionsare discussed.2 PreliminariesHere, regarding to de�ne our aggregation operators, we shall present the work hypothesis. Weare going to speci�c a concrete linguistic model to represent the information and the LOWAoperator to aggregate linguistic information.2.1 Linguistic approachUsually, in a quantitative setting the information is expressed by means of numerical values.However, when we work in a qualitative setting, that is, with vague or imprecise knowledge, thiscannot be estimated with an exact numerical value. Then, a more realistic approach may be touse linguistic assessments instead of numerical values [35], that is, to suppose that the variableswhich participate in the problem are assessed by means of linguistic terms [5, 6, 12, 13, 22, 31,1The word "linguistic" is related to the concept of "linguistic variables" in a formal way, and it does not implysome connections to linguistics.

35]. This approach is appropriate for a lot of problems, since it allows a representation of theinformation in a more direct and adequate form if we are unable of expressing it with precision.A linguistic variable di�ers from a numerical one in that its values are not numbers, butwords or sentences in a natural or arti�cial language. Since words, in general, are less precisethan numbers, the concept of a linguistic variable serves the purpose of providing a means ofapproximated characterization of phenomena, which are too complex, or too ill-de�ned to beamenable to their description in conventional quantitative terms.De�nition [35].- A linguistic variable is characterized by a quintuple (H,T(H),U,G,M) inwhich H is the name of the variable; T(H) (or simply T) denotes the term set of H, i.e., the setof names of linguistic values of H, with each value being a fuzzy variable denoted generically byX and ranging across a universe of discourse U which is associated with the base variable u; G isa syntactic rule (which usually takes the form of a grammar) for generating the names of valuesof H; and M is a semantic rule for associating its meaning with each H, M(X), which is a fuzzysubset of U.Usually, depending on the problem domain, an appropriate linguistic term set is chosen andused to describe the vague or imprecise knowledge. The elements in the term set will determinethe granularity of the uncertainty, that is the level of distinction among di�erent countings ofuncertainty. In [1] the use of term sets with an odd cardinal was studied, representing the midterm an assess of "approximately 0.5", with the rest of the terms being placed symmetricallyaround it and the limit of granularity 11 or no more than 13.For instance, Figure 1 shows a hierarchical structure of linguistic values or labels. Clearly,level 1 provides a granularity containing three labels, level 2 a granularity with nine labels, andof course, di�erent granularity levels could be presented. In fact, in this �gure level 4 presentsthe �nest granularity in a decision process, the numerical values.MEDIUM

MEDIUM

fuzzy set approx. 0.5

0.5

NONE

LOW

VERY-LOWLOWHIGH

HIGH

VERY-HIGHPERFECT

[0,0.5]fuzzy fuzzy [0.5,1]onsets sets on

[0,0.5]numbers onReal Real numbers on [0.5,1]

LEVEL 2

LEVEL 1

LEVEL 3

LEVEL 4 Fig. 1. Hierarchy of labels.On the other hand, the semantic of the elements in the term set is given by fuzzy numbersde�ned on the [0,1] interval, which are described by membership functions. Because the linguisticassessments are just approximate ones given by the individuals, we can consider that lineartrapezoidal membership functions are good enough to capture the vagueness of those linguisticassessments, since it may be impossible or unnecessary to obtain more accurate values. Thisrepresentation is achieved by the 4-tuple (ai; bi; �i; �i), the �rst two paremeters indicate theinterval in which the membership value is 1; the third and fourth parameters indicate the left

and right width. Formally speaking, it seems di�cult to accept that all individuals should agreeon the same membership function associated to linguistic terms, and therefore, there are notany universality distribution concepts.It is well known and accepted that the tuning of membership functions is a crucial issuein control processes with linguistic rules. In our context, we consider an environment whereindividuals can discriminate perfectly the same term set under a similar conception, taking intoaccount that the concept of a linguistic variable serves the purpose of providing a means ofapproximated characterization of imprecise preference information. Moreover, in our develop-ment we don't use the membership functions for aggregating the labels, we de�ne aggregationoperators for linguistic labels by direct computation on labels.2.2 Characterization of the linguistic label setAccordingly, to establish what kind of label set to use ought to be the �rst priority. Then, letS = fsig; i 2 H = f0; : : : ; Tg, be a �nite and totally ordered term set on [0,1] in the usualsense [1, 4]. Any label, si; represents a possible value for a linguistic variable, that is, a vagueproperty or constraint on [0,1]. We consider a term set, S, as in [1] with its semantic given bylinear trapezoidal membership functions. Moreover, it must have the following characteristics:1) The set is ordered: si � sj if i � j.2) There is the negation operator: Neg(si) = sj such that j = T -i.3) Maximization operator: Max(si; sj) = si if si � sj .4) Minimization operator: Min(si; sj) = si if si � sj .For example, this is the case of the following term set of the level 2 of the Figure 1:P = Perfect = (1; 1; :25; 0) V H = V ery High = (:75; :75; :15; :25)H = High = (:6; :6; :1; :15) M =Medium = (:5; :5; :1; :1)L = Low = (:4; :4; :15; :1) V L = V ery Low = (:25; :25; :25; :15)N = None = (0; 0; 0; :25)2.3 The LOWA operatorAssuming the proposed linguistic approach, two main di�erent approaches can be found in orderto aggregate linguistic values: the �rst acts by direct computation on labels [5], and the seconduses the associated membership functions [1, 22, 35].Most available techniques belong to the latter kind. However, the �nal results of thosemethods are fuzzy sets which do not correspond to any label in the original term set. If one�nally wants to have a label, then a "linguistic approximation" is needed [1, 22, 23, 35]. Theprocess of linguistic approximation consists of �nding a label whose meaning is the same orthe closest (according to some metric) to the meaning of an unlabelled membership functiongenerated by some computational model.In this context, to manipulate the linguistic information, we shall work with operators forcombining the linguistic values (non-weighted and weighted) by direct computation on labels.Speci�cally, in this Section, we shall present the non-weighted operator of combination of thelinguistic values based on direct computation, the LOWA operator [12, 15], which will be used

latter in the de�nition of the three weighted operators of combination of linguistic values bydirect computation that we propose here.The linguistic ordered weighted averaging (LOWA) operator, de�ned in [12, 15], is basedon the ordered weighted averaging (OWA) operator de�ned by Yager [28], and on the convexcombination of linguistic labels de�ned by Delgado et al. [5].De�nition 1. Let A = fa1; : : : ; amg be a set of labels to be aggregated, then the LOWA operator,�, is de�ned as �(a1; : : : ; am) = W �BT = Cmfwk; bk; k = 1; : : : ; mg == w1 � b1 � (1� w1)�Cm�1f�h; bh; h = 2; : : : ; mgwhere W = [w1; : : : ; wm], is a weighting vector, such that,1. wi 2 [0; 1] and,2. �iwi = 1;.�h = wh=�m2 wk; h = 2; : : : ; m, and B = fb1; : : : ; bmg is a vector associated to A, such that,B = �(A) = fa�(1); : : : ; a�(n)gwhere, a�(j) � a�(i) 8 i � j;with � being a permutation over the set of labels A. Cm is the convex combination operator ofm labels and if m=2, then it is de�ned asC2fwi; bi; i = 1; 2g = w1 � sj � (1� w1)� si = sk; sj ; si 2 S; (j � i)such that k = minfT; i + round(w1 � (j � i))gwhere "round" is the usual round operation, and b1 = sj ; b2 = si:If wj = 1 and wi = 0 with i 6= j 8i, then the convex combination is de�ned as:Cmfwi; bi; i = 1; : : : ; mg = bj :In [15], we demonstrated that the LOWA operator presents some evidence of rational aggre-gation, because, on the one hand, it veri�es these properties:� The LOWA operator is increasing monotonous with respect to the argument values.� The LOWA operator is commutative.� The LOWA operator is an "orand" operator.And on the other hand, it veri�es these axioms: unrestricted domain, unanimity or idempotence,positive association of social and individual values, independence of irrelevant alternatives, citi-zen sovereignty, neutrality.Here we present an extension of the LOWA operator, an Inverse LOWA operator, that willbe used in the de�nition of some weighted operators.

De�nition 2. An I-LOWA (Inverse-Linguistic Ordered Weighted Averaging) operator, �I , is atype of LOWA operator, in whichB = �I(A) = fa�(1); : : : ; a�(n)gwhere, a�(i) � a�(j) 8 i � j:If m=2, then it is de�ned asC2fwi; bi; i = 1; 2g = w1 � sj � (1� w1)� si = sk; sj ; si 2 S; (j � i)such that k = minfT; i + round(w1 � (j � i))g:If the de�nition of the LOWA operator is compared to the de�nition of the I-LOWA operatorit is possible to observe that in the �rst one the large values are more estimated than low values,unlike in the second one. Therefore, from this view point, the LOWA operator presents char-acteristics belonging to the maximum aggregation operator, and the I-LOWA operator presentscharacteristics belonging to the minimum aggregation operator. This peculiarity will be usedlatter in the de�nition of one of our weighted aggregation operators.Clearly, an I-LOWA operator veri�es too the mentioned properties and axioms of the LOWAoperator.2.3.1 The LOWA operator guided by fuzzy majorityHow to calculate the weighting vector of the LOWA operator, W , is a basic question to decide.Yager proposed in [28, 30] two ways to do so. The �rst approach is to use some kind of learningmechanism using sample data; and the second approach is to try to give some semantics ormeaning to the weights. We consider the latter approach, because our idea is to show theconcept of fuzzy majority by means of the weighting vector in the LOWA operator aggregations.Traditionally, the majority is de�ned as a threshold number of individuals. Fuzzy majorityis a soft majority concept, which is manipulated via a fuzzy logic based calculus of linguisticallyquanti�ed propositions. In [17] Kacprzyk speci�ed fuzzy majority rule by means of a linguisticquanti�er to derive various solutions concepts for group decision making problems in a numericalsetting. Here, we shall work in a similar way, but in the �eld of quanti�er guiaded aggregations.Before showing how do so, we brie y shall introduce the concept of fuzzy linguistic quanti�er.2.3.2 Fuzzy linguistic quanti�erHuman discourse is very rich and diverse in its quanti�ers, e.g, about 5, almost all, a few, many,most, as many as possible, nearly half, at least half. Zadeh, using Fuzzy logic, introduced theconcept of linguistic quanti�er to represent the large number of possible quanti�ers [36]. Zadehsuggested that the semantic of a linguistic quanti�er can be captured by using fuzzy subsetsfor its representation. He distinguished between two types of linguistic quanti�ers, absolute andproportional. Absolute quanti�ers are used to represent amounts that are absolute in naturesuch as about 2 or more than 5. These absolute linguistic quanti�ers are closely related to theconcept of the counting or number of elements. Proportional quanti�ers are used to representamounts that are relative in nature such as such as most, at least half. A proportional quanti�ercan be represented by a fuzzy subset, Q, in the unit interval, [0,1], such that for any r 2 [0; 1],

Q(r) indicates the degree to which the proportion r is compatible with the meaning of thequanti�er it represents.A proportional quanti�er, Q : [0; 1]! [0; 1]; satis�es:Q(0) = 0; and 9r 2 [0; 1] such that Q(r) = 1:A non-decreasing quanti�er satis�es:8a; b if a > b then Q(a) � Q(b):The membership function of a non-decreasing proportional quanti�er can be represented asQ(r) = 8><>: 0 if r < ar�ab�a if a � r < b1 if r � bwith a; b; r 2 [0; 1].Some examples of proportional quanti�ers are shown in Figure 2, where the parameters,(a; b) are (0:3; 0:8), (0; 0:5) and (0:5; 1), respectively.1

0 0.3 0 0.5 0 0.5 1 xxx0.8

1 1

"Most" "At least half" "As many as possible"Fig. 2. Proportional fuzzy linguistic quanti�ersIn [28, 30], Yager suggested an interesting way to compute the weights of the OWA aggre-gation operator using linguistic quanti�ers, which, in the case of a non-decreasing proportionalquanti�er Q, is given by this expression:wi = Q(i=n)�Q((i� 1)=n); i= 1; : : : ; n:When a fuzzy linguistic quanti�er, Q, is used to compute the weights of LOWA operator, �, itis symbolized by �Q: Therefore, when a fuzzy linguistic quanti�er, Q, is used to compute theweights of the I-LOWA operator, �I , it is symbolized by �IQ:Clearly, depending on the fuzzy linguistic quanti�er that is choosen to calculate the weights,it is possible to observe the following properties:1

0 x1Fig. 3. Fuzzy linguistic quanti�er "All".

1. If the fuzzy linguistic quanti�er is "All", as is shown in Figure 3, whose membershipfunction is, Q(r) = ( 0 if 0 � r < 11 otherwisethen �Q(a1; : : : ; am) =MIN(a1; : : : ; am) and �IQ(a1; : : : ; am) =MAX(a1; : : : ; am):2. If the fuzzy linguistic quanti�er is "At least m" (m 2 N), as is shown in Figure 4,1

0 x1/mFig. 4. Fuzzy linguistic quanti�ers "At least m".whose membership function is,Qm(r) = ( r(1=m) if r < (1=m); r 2 [0; 1]1 if r � (1=m)then �Qm(a1; : : : ; am) =MAX(a1; : : : ; am) and �IQm(a1; : : : ; am) =MIN(a1; : : : ; am):Where "MAX" stands for maximum operator and "MIN" stands for minimum operator.3 Aggregation operators for linguistic weighted informationIn this Section, three aggregation operators for linguistic weighted information based on thedirect computation on labels are presented:1. Linguistic Weighted Disjunction (LWD).2. Linguistic Weighted Conjunction (LWC).3. Linguistic Weighted Averaging (LWA).Following Cholewa's studies [2] and Montero's aggregationmodel [20], if we want to aggregateweighted information we have to de�ne two aggregations:� the aggregation of importance degrees (weights) of information, and� the aggregation of weighted information (information combined with weights).The �rst aspect consists of obtaining a collective importance degree from individual impor-tance degrees that characterizes the �nal result of aggregation operator. In the three operators,as the importance degrees are linguistic values, this is solved using the LOWA operator guidedby the concept of fuzzy majority.The aggregation of weighted information involves the transformation of the weighted in-formation under the importance degrees. The transformation form depends upon the type ofaggregation of weighted information being performed [32]. In [26, 27] Yager discussed the ef-fect of the importance degrees in the types of aggregation "MAX" and "MIN" and suggested

a class of functions for importance transformation in both types of aggregation. For MIN typeaggregation he suggested a family of t-conorms acting on the weighted information and thenegation of the weights, which presents the non-increasing monotonic property in the weights.For MAX type aggregation he suggested a family of t-norms acting on weighted informationand the weight, which presents the non-decreasing monotonic property in the weights. In [32]Yager proposed a general speci�cation of the requirements that any importance transformationfunction, g, must satisfy for any type of the aggregation operator. The function, g; must havethe following properties:1. if a > b then g(w; a) � g(w; b)2. g(w; a) is monotone in w3. g(0; a) = ID4. g(1; a) = a:with a; b 2 [0; 1] expressing the satisfaction with regards to a criterion, w 2 [0; 1] the weightassociated to the criterion, and "ID" an identity element, which is such that if we add it to ouraggregations it doesn't change the aggregated value. Condition one means that the function gis monotonically non-decreasing in the second argument, that is, if the satisfaction with regardsto the criteria is increased the overall satisfaction shouldn't decrease. The second condition maybe viewed as a requirement that the e�ect of the importance be consistent. It doesn't specifywhether g is monotonically non-increasing or non-decreasing in the �rst argument, but must beone of these. It should be noted that conditions three and four actually determine the type ofmonotonicity obtained from two. If a > ID, the g(w; a) is monotonically non-decreasing in w,while if a < ID, then it is monotonically non-increasing. The third condition is a manifestationof the imperative that zero importance items don't e�ect the aggregation process. The �nalcondition is essentially a boundary condition which states that the assumption of all importancesequal to one e�ectively is like not including importances at all [32].Considering the aforementioned ideas and assuming a linguistic framework, that is a labelset, S, to express the information and a label set, L; to express the weights, we propose using thefollowing aggregations of weighted information for the three aggregation operators, with theirrespective aggregation operators and transformation functions:� Linguistic weighted disjunction1. Aggregation operator: MAX linguistic aggregation.2. Transformation function: g = MIN(w; a).� Linguistic weighted conjunction1. Aggregation operator: MIN linguistic aggregation.2. Transformation function: g = MAX(Neg(w); a).� Linguistic weighted averaging1. Aggregation operator: LOWA or I � LOWA.2. Transformation function: g(LOWA) = LC!(w; a) or g(I-LOWA) = LI!(w; a).

The �rst two aggregations are based on canonical generalizations of weighted disjunction andconjunction of fuzzy goals, de�ned by Dubois and Prade in a possibility theory setting [7, 8]. Thelatter one is based on the combination of the LOWA and I-LOWA operator with several linguisticconjunction functions (LC!) and several linguistic implication functions (LI!), respectively.Therefore, the LWA operator is a type of fuzzy majority guided weighted aggregation operator.In the next subsections, we present each aggregation operator of linguistic weighted infor-mation in detail. In order to complete the presentation, in the �nal subsection we provide someevidence of the rationality of their aggregation, checking some of the axioms that they verify.We shall demonstrate that all the operators proposed combine appropriately the weighted in-formation, in such a way, that the �nal aggregation is the "best" representation of the overallindividual information.3.1 Linguistic weighted disjunction and conjunctionLet f(c1; a1); : : : ; (cm; am)g be a set of weighted opinions expressed by a set of experts, E =fe1; : : : ; emg; to evaluate an alternative, xj , where ai shows the opinion of expert ei, assessedlinguistically on the label set, S, ai 2 S, and ci the relevance degree of expert ei, assessedlinguistically on the label set L, ci 2 L.De�nition 1. The aggregation of the set of weighted individual opinions, f(c1; a1); : : : ; (cm; am; )g;according to the Linguistic Weighted Disjunction (LWD) operator is de�ned as(cE; aE) = LWD[(c1; a1); : : : ; (cm; am)];where the opinion of the group, aE, is obtained asaE = MAXi=1;:::;m MIN(ci; ai);and the importance degree of the opinion of the group, cE, is obtained ascE = �Q(c1; : : : ; cm):De�nition 2. The aggregation of the set of weighted individual opinions, f(a1; c1); : : : ; (am; cm)g;according to the Linguistic Weighted Conjuntion (LWC) operator is de�ned as(cE; aE) = LWC[(c1; a1); : : : ; (cm; am)];where the opinion of the group, aE, is obtained asaE = MINi=1;:::;m MAX(Neg(ci); ai);and the importance degree of the opinion of the group, cE, is obtained ascE = �Q(c1; : : : ; cm):Remark: It is clear that both de�nitions always require the following condition: S = L.In the de�nition of the LWD operator the transformation function is the "MIN" function, thatis, one of the t-norms proposed by Yager in [26, 27] for the "MAX" type aggregation operator, butde�ned linguistically, and it satis�es the properties proposed for any g [32]. Something similarhappens in the de�nition of the LWC operator. In both operators it should be possible to chooseany other function of the families proposed by Yager in [26, 27], but always de�ned linguistically.

In any way, both operators try to reduce the e�ect of elements with low importance. To do so,in the �rst operator, the elements with low importance are transformed into small values andin the second one into large values.Since ci expresses the degree of importance of the opinion of expert ei in the overall opinion,then:� When ci = sT , the opinion of ei has a direct in uence on the acceptance (rejection) ofalternative xj .� When ci = s0, the opinion of ei has no in uence on the acceptance (rejection) of alternativexj .As the LOWA operator, �, is an "orand" operator, the importance degree of opinion of thegroup, cE , veri�es the following expression:MIN(c1; c2; : : : ; cm) � cE � MAX(c1; c2; : : : ; cm)3.2 Linguistic weighted averagingBefore de�ning the linguistic weighted aggregation (LWA) operator, and assuming S = L,consider the following two families of connectives:1. Linguistic conjunction functions (LC!).The linguistic conjunction functions that we shall use are the following t-norms, which aremonotonically non-decreasing in the weights and satisfy the properties required for anytransformation function, g; [11]:(a) The classical MIN operator: LC!1 (w; a) =MIN(w; a):(b) The nilpotent MIN operator:LC!2 (w; a) = ( MIN(w; a) if w > Neg(a)s0 otherwise.(c) The weakest conjunction:LC!3 (w; a) = ( MIN(w; a) if MAX(w; a) = sTs0 otherwise.2. Linguistic implication functions (LI!).The linguistic implication functions that we shall use are monotonically non-increasing inthe weights and satisfy the properties required for any transformation function g [11]:(a) Kleene-Dienes's implication function:LI!1 (w; a) =MAX(Neg(w); a):(b) G�odel's implication function:LI!2 (w; a) = ( sT if w � aa otherwise.

(c) Fodor's implication function:LI!3 (w; a) = ( sT if w � aMAX(Neg(w); a) otherwise.De�nition 3. The aggregation of the set of weighted individual opinions, f(c1; a1); : : : ; (cm; am)g;according to the Linguistic Weighted Averaging (LWA) operator is de�ned as(cE; aE) = LWA[(c1; a1); : : : ; (cm; am)];where the importance degree of the group opinion, cE, is obtained ascE = �Q(c1; : : : ; cm):and, the opinion of the group, aE, is obtained asaE = f [g(c1; a1); : : : ; g(cm; am)];where f 2 f�Q; �IQg; g 2 fLC!1 ; LC!2 ; LC!3 g if f = �Q, and g 2 fLI!1 ; LI!2 ; LI!3 g if f = �IQ.It should be observed that according to the class of transformation functions proposed byYager in [26, 27] for MIN type aggregation, when the aggregation operator, f; is the I-LOWAoperator, �IQ, and given that �IQ is an aggregation operator with characteristics of a MIN typeaggregation operator (as was seen in the presentation of the LOWA operator), then we havedecided to use the linguistic implications functions, LI!; as the transformation function type.Something similar happens when f is the LOWA operator �Q.Lemma 1. The linguistic weighted disjunction operator, LWD; is a particular weightedaggregation operator of the LWA operator type.Proof:Suppose that we have a group ofm experts. If a linguistic non-decreasing relative quanti�er,Qm, "At least m", is chosen, as is shown in Figure 3, the LOWA operator, �Qm , as an aggregationoperator, and the following linguistic conjunction function as a transformation function,LC!1 (w; a) = MIN(w; a);then since the weighting vector is W = [w1 = 1; w2 = 0; : : : ; wm = 0],aE = �Qm [MIN(c1; a1); : : : ;MIN(cm; am)] =MAXi=1;:::;m MIN(ci; ai);is veri�ed, and therefore,LWA[(c1; a1); : : : ; (cm; am)] = LWD[(c1; a1); : : : ; (cm; am)]:Lemma 2. The linguistic weighted conjunction operator, LWC; is a particular weightedaggregation operator of the LWA operator type.

Proof:Assuming the above linguistic quanti�er, if it is choosen the I-LOWA operator, �IQm , as ag-gregation operator, and the following linguistic implication operator as transformation function,LI!(w; a) = MAX(Neg(w); a);then as the weighting vector is W = [w1 = 1; w2 = 0; : : : ; wm = 0], it is veri�edaE = �IQm [MAX(Neg(c1); a1); : : : ;MAX(Neg(cm); am)] =MINi=1;:::;m MAX(Neg(ci); ai);and therefore, LWA[(c1; a1); : : : ; (cm; am)] = LWC[(c1; a1); : : : ; (cm; am)]:3.3 Axiomatic of the aggregation operators for linguistic weighted informa-tionPrevious works on the aggregation of fuzzy weighted opinions, developed in a numerical setting,are those by Cholewa, Montero and Dubois and Koning. Cholewa [2] o�ers a collection ofaxioms that weighted aggregations should follow, and proposes the weighted arithmetic mean asa typical aggregation operator that satis�es these axioms, Montero [20] characterizes the fuzzymajority rule and studies the existence of absolutely decisive groups, and Dubois and Koning[10] analize brie y the di�erent axiomatic approaches existing for weighted aggregation.As was aforementioned earlier, in [2] a complete set of axioms in the fuzzy set setting forheterogeneous groups is given. Some of these axioms are: independence of alternatives, com-mutativity, : : :. Obviously a particular weighted aggregation operator doesn't have to satisfyall the axioms together, it must satisfy those that its special application circumstances require.Bellows, we are going to postulate an axiomatic approach with ten axioms, and we shall checkwhich axioms our weighted aggregation operators verify. Speci�cally, Axioms I,II,III,IV,V,VIare obtained directly from those proposed by Cholewa in [2], but de�ned in linguistic setting,and others are proposed by ourselves.As has been shown in the subsection above, if we chooce the linguistic quanti�er, Q, "Atleast m", and an appropriate transformation function, the LWA operator is a generalization ofthe LWD and LWC operators. Therefore, here, we shall only study the axiomatic of the LWAoperator, and in those cases where the axiom not be veri�ed, then, we shall study what happenswith the LWD and LWC operators.Assume the following framework:Let X = fx1; : : : ; xng (n � 2) be a �nite non-empty set of alternatives to be evaluated.Let E = fe1; : : : ; emg (m � 2) be a group of the experts to analyze X.Let S = fsi : i = 0; ::; Tg be a label set to voice experts' opinions and their respective importancedegrees.Axiom I : Independence of alternatives xi 2 X. The collective opinion for xi, (cE; aE), only de-pends on the individual opinions for xi, [(c1; a1); : : : ; (cm; am)]. This means that linguisticfunctions v and w exist for the aggregation of linguistic weighted opinions,v : (SxS)m ! S;

and for the aggregation of the powers of aggregated opinionsw : Sm ! S;such that (cE; aE) = (w[c1; : : : ; cm]; v[(c1; a1); : : : ; (cm; am)]):It is basically technical, and is satis�ed by the de�nition of the LWA operator, with theLOWA operator being the function w and the composition of the I-LOWA operator witha linguistic implication operator (LI!) or the composition of the LOWA operator with alinguistic conjunction function (LC!) the function v.Axiom II : Commutativity. Having �xed an alternative, xi 2 X , thenLWA[(c1; a1); : : : ; (cm; am)] = LWA[�(c1; a1); : : : ; �(cm; am)]where � is a permutation over the set of weighted opinions.Clearly it is satis�ed, because the I-LOWA operator as well as the LOWA operator use"ordered" weighted average of the arguments.Axiom III : Associativity. Having �xed an alternative xi 2 X , thenLWA[(c1; a1); : : : ; (cm; am)] = LWA[LWA[(c1; a1); : : : ; (cm�1; am�1)]; (cm; am)]This axiom is not veri�ed by the LWA operator, because neither the I-LOWA operatornor the LOWA operator satisfy the associativity property as was demonstrated in [15].So, it is not veri�ed by the LWD and LWC operators because the aggregated weights areobtained by means of the LOWA operator.Axiom IV : Quasiequivalence and increasingness of power. If everyone agrees on an opinion,"a", about an alternativ, xi 2 X; and thus, a1 = a2 = : : := am = a, then if(cE ; aE) = LWA[(c1; a); : : : ; (cm; a)];the collective opinion, (cE; aE), must satisfy the following conditions:1. aE = a (quasiequivalence), and2. cE > max(c1; : : : ; cm) (increasingness of power).This axiom is not veri�ed for any of its conditions.� The LWA operator is not quasiequivalent. For example, suppose a set of nine labels.Then, if we want to aggregate the linguistic weighted opinions [(s0; s6); (s1; s6)] of twoexperts, having �xed the linguistic quanti�er, Q2, "At least 2", then the two weightsare w1 = 1; w2 = 0. Thus,Case 1: If f = �IQ2 and g = LI!(w; a) =MAX(Neg(w); a), then�IQ2(MAX(Neg(s0); s6);MAX(Neg(s1); s6)) = s7 6= s6:Case 2: f = �Q2 and g = LC!(w; a) =MIN(w; a);�Q2(MIN(s0; s6);MIN(s1; s6)) = s1 6= s6:

Since this example of the LWA operator is the case in which it works as the LWDand LWC operators then these do not verify this property either.� The LWA operator does not verify the increasingness of power. Clearly it is a con-sequence of the property of the LOWA operator of being an "orand" operator, andtherefore, MIN(c1; : : : ; cm) � cE � MAX(c1; : : : ; cm):Axiom V : Positive sensitivity in its strongest form. A weighted collective opinion is increasedif and only if any weighted individual opinion is increased. This means that if (aE; cE) isthe weighted collective opinion obtained for xi as(cE; aE) = LWA[(c1; a1); : : : ; (cm; am)];and (cE; bE) is the weighted collective opinion obtained for xi as(cE; bE) = LWA[(c1; b1); : : : ; (cm; bm)];with aj � bj , thenaE > bE if and only if 9ek 2 E such that ak > bk:Clearly this axiom is not veri�ed by the LWA operator. Example: suppose a label setwith eight elements. Let [(s1; s4); (s2; s5)] be two weighted opinions to be aggregated.Considering the linguistic quanti�er, Q2, "At least 2", then the two weights are w1 =1; w2 = 0 and then,Case 1: if f = �IQ2 and g =MAX(Neg(w); a), then(cE; aE) = LWA[(s1; s4); (s2; s5)] = (s2; s5); andCase 2: if f = �Q2 and g =MIN(w; a); then(cE; aE) = LWA[(s1; s4); (s2; s5)] = (s2; s2):If the �rst expert changes his opinion by (s1; s5) thenCase 1: (cE; bE) = LWA[(s1; s5); (s2; s5)] = (s2; s5);Case 2: (cE; bE) = LWA[(s1; s5); (s2; s5)] = (s2; s2);and therefore, although the opinion of an expert has been increased, however, the collectiveopinion has not been increased, independently of the aggregation operator of the weightedopinions considered in the LWA operator. The LWD and LWC operators don't verifythis axiom as also happens in the axiom above.Axiom VI : Neutrality of complement. If (cj; aj)c is the complement of weighted opinion(cj ; aj), such that (cj ; aj)c = (cj; Neg(aj)), then, having �xed an alternative, xi 2 X ,LWA[(c1; a); : : : ; (cm; a)]c = LWA[(c1; a1)c; : : : ; (cm; am)c]:This axiom is not veri�ed by the LWA operator. Example: consider a label set witheight elements. Let [(s4; s6); (s1; s5)] be the weighted opinions to be aggregated and itscomplement weighted opinions, [(s4; s1); (s1; s2)], then having �xed the linguistic quanti�er,Q2, "At least 2", then the two weights are w1 = 1; w2 = 0; and then

Case 1: if f = �IQ2 and g = LI!(w; a) =MAX(Neg(w); a), then(cE; aE) = LWA[(s4; s6); (s1; s5)]c = (s4; s6) 6= (s4; s3) = LWA[(s4; s1); (s1; s2)]:Case 2: And if f = �Q2 and g = LC!(w; a) =MIN(w; a); then(cE; aE) = LWA[(s4; s6); (s1; s5)]c = (s4; s4) 6= (s4; s1) = LWA[(s4; s1); (s1; s2)]:Thus, the LWD and LWC operators do not verify this axiom either.Axiom VII : Positive sensitivity in its weaker form. If an expert increases his weighted opinionfor xi then the collective weighted opinion for xi cannot decrease. This means that if (aj ; cj)and (bj; cj) are such that, aj � bj , then if(cE; aE) = LWA[(c1; a1); : : : ; (cj; aj); : : : ; (cm; am)];and (cE; bE) = LWA[(c1; a1); : : : ; (cj; bj); : : : ; (cm; am)];then aE � bE:Obviously, this axiom is satis�ed, it is a consequence of the monotonic property of theLOWA and I-LOWA operators.Axiom VIII : Neutrality with respect to alternatives. If we have two alternatives, xi and xk,and the weighted opinions known about both are (cj ; aj) and (cj; a0j); such that, aj =a0j ; 8j; then LWA[(c1; a1); : : : ; (cm; am)] = LWA[(c1; a01); : : : ; (cm; a0m)]:Clearly this axiom is satis�ed too.Axiom IX : Unrestricted domain. Having �xed any alternative, xi 2 X , for any set ofweighted individual opinions, f(c1; a1); : : : ; (cm; am)g; there is a weighted collective opinion,(cE ; aE), which may be constructed, i.e.,8[(c1; a1); : : : ; (cm; am)] 2 (SxS)m; 9(cE; aE) 2 (SxS) such that(cE; aE) = LWA[(c1; a1); : : : ; (cm; am)]:It is satis�ed in accordance with the LWA operator de�nition.Axiom X : The LWA operator is an "orand" operator. This is a property of the LWAoperator presented here in the form of an axiom. This property is postulated in thefollowing sense: having �xed an alternative, xi 2 X , for any set of weighted individualopinions, f(c1; a1); : : : ; (cm; am)g, if (cE; aE) is such that(cE; aE) = LWA[(c1; a1); : : : ; (cm; am)]then1. MIN(a1; : : : ; am; c1; : : : ; cm) � aE � MAX(a1; : : : ; am; c1; : : : ; cm); and

2. MIN(c1; : : : ; cm) � cE � MAX(c1; : : : ; cm)This property is a consequence of the property of the LOWA and I-LOWA operators ofbeing "orand" operators.In conclusion, the LWD operator, the LWC operator, and the LWA operator verify thefollowing axioms: independence of alternatives, commutativity, positive sensitivity in its weakerform, neutrality with respect to alternatives, unrestricted domain, being an "orand" operator.The ful�lment of those axioms provides evidence of rational aggregation of these operators inparticular frameworks. In the following sections we shall show an application of the use of theseaggregation operators for linguistic weighted information in the choice processes for alternativesin heterogeneous groups.4 Example of applicationAssuming the set of seven labels presented in Section 2, that is,S = fs6 = P; s5 = VH; s4 = H; s3 =M; s2 = L; s1 = V L; s0 = Ng;suppose an investment company, which wants to invest a sum of money in the best option.There is a panel with four possibles options where to invest the money:� x1 is a car company,� x2 is a food company,� x3 is a computer company,� x4 is an arms company.The investment company has a group of four consultancy departments:� d1 is the risk analysis department,� d2 is the growth analysis department,� d3 is the social-political impact analysis department, and� d4 is the environmental impact analysis department.In each department there is one expert with di�erent importance degrees (ci for the expert ofthe department di): fc1 = s4; c2 = s5; c3 = s2; c4 = s6g:The assessments of the option set by the experts from each department are the following (aij isthe assessment assigned to the option xi by the expert from department dj):1. For e1: fa11 = s1; a21 = s5; a31 = s4; a41 = s6g:2. For e2: fa12 = s1; a22 = s3; a32 = s3; a42 = s1g:3. For e3: fa13 = s4; a23 = s5; a33 = s3; a43 = s2g:4. For e4: fa14 = s1; a24 = s6; a34 = s1; a44 = s0g:

Thus, using the linguistic weighted conjunction (LWD) operator the issues are the following:(i) the collective assessments on alternatives are the following (ai for collective assessment of thealternative xi):a1 =MAXj=1;:::;4fMIN(cj; a1j)g =MAXfMIN(s4; s1);MIN(s5; s1);MIN(s5; s4);MIN(s6; s1)g = s2a2 =MAXj=1;:::;4fMIN(cj; a2j)g =MAXfMIN(s4; s5);MIN(s5; s3);MIN(s5; s5);MIN(s6; s6)g = s6a3 =MAXj=1;:::;4fMIN(cj; a3j)g =MAXfMIN(s4; s4);MIN(s5; s3);MIN(s5; s3);MIN(s6; s1)g = s4a4 =MAXj=1;:::;4fMIN(cj; a4j)g =MAXfMIN(s4; s6);MIN(s5; s1);MIN(s5; s2);MIN(s6; s0)g = s4(ii) and the collective importance degree, cE , meaning the credibility degree of the solution, withthe linguistic quanti�er, Q, "As many as possible" with the par (0.5,1) and W = [0; 0; 0:5; 0:5]is the following: cE = �Q(c1; c2; c3; c4) = �Q(s4; s5; s5; s6) = s4:Clearly alternative x2 is the best assessed one.5 ConclusionsIn this paper various aggregation operators for the linguistic weighted information are presented.These operators are very useful for modelling those processes in which there are various infor-mation sources and the information is linguistic in nature and is not equally relevant. Theiraggregation has been checked examining some of the axioms that an acceptable weighted aggre-gation operator must verify.References[1] P.P. Bonissone and K.S. Decker, "Selecting Uncertainty Calculi and Granularity: An Exper-iment in Trading-o� Precision and Complexity," in :Uncertainty in Arti�cial Intelligence,L.H. Kanal and J.F. Lemmer, Eds. London: North-Holland, 1986, pp. 217-247.[2] W. Cholewa, "Aggregation of Fuzzy Opinions: An Axiomatic Approach," Fuzzy Sets andSystems, vol. 17, pp. 249-259, 1985.[3] C. Carlsson and R. Full�er, "On Fuzzy Screening Systems," Proc. 3th European Congress onIntelligent Technologies and Soft Computing, Aachen, 1995, pp. 1261-1264.[4] M. Delgado, J.L. Verdegay, and M.A. Vila, "Linguistic Decision Making Models," Int. J.Intelligent Systems, vol. 7, pp. 479-492, 1993.[5] M. Delgado, J.L. Verdegay, and M.A. Vila, "On Aggregation Operations of Linguistic La-bels," Int. J. Intelligent Systems, vol. 8, pp. 351-370, 1993.[6] M. Delgado, J.L. Verdegay, and M.A. Vila, "A Model for Incomplete and Vague Informationin Decision Making Problems," Int. J. Intelligent Systems, vol. 9, pp. 365-378 1994.[7] D. Dubois and H. Prade, "A Review of Fuzzy Set Aggregation Connectives," InformationSciences, vol. 36, pp. 85-121, 1985.[8] D. Dubois and H. Prade, "Weighted Minimum and Maximum Operations in Fuzzy SetTheory," Information Sciences, vol. 39, pp. 205-210, 1986.[9] D. Dubois, H. Prade, and C. Testemale, "Weighted Fuzzy Pattern Matching," Fuzzy Setsand Systems, vol. 28, pp. 313-331, 1988.

[10] D. Dubois and J.L. Koning, "Social Choice Axioms for Fuzzy Set Aggregation," Fuzzy Setsand Systems, vol. 43, pp. 257-274, 1991.[11] J. Fodor and M. Roubens, Theory and Decision Library Series D: Fuzzy Preference Mod-elling and Multicriteria Decision Support. London: Kluwer Academic Publishers, 1994.[12] F.Herrera and J.L. Verdegay, "Linguistic Assessments in Group Decision," Proc. 1th Euro-pean Congress on Fuzzy and Intelligent Technologies, Aachen, 1993, pp. 941-948.[13] F. Herrera, E. Herrera-Viedma, and J.L. Verdegay, "A Sequential Selection Process inGroup Decision Making with Linguistic Assessment," Information Sciences, vol. 85, pp.223-239, 1995.[14] F. Herrera, E. Herrera-Viedma, and J.L. Verdegay, "A Model of Consensus in Group De-cision Making under Linguistic Assessments," Fuzzy Sets and Systems, vol. 78, pp. 73-87,1996.[15] F. Herrera, E. Herrera-Viedma, and J.L. Verdegay, "Direct Approach Processes in GroupDecision Making Using Linguistic OWA Operators," Fuzzy Sets and Systems, vol. 79, pp.175-190, 1996.[16] F. Herrera, E. Herrera-Viedma, and J.L. Verdegay, "A Rational Consensus Model in GroupDecision Making under Linguistic Assessments," Fuzzy Sets and Systems, to appear, 1996.[17] J. Kacprzyk, "Group Decision Making with a Fuzzy Linguistic Majority," Fuzzy Sets andSystems, vol. 18, pp. 105-118, 1986.[18] J. Kacprzyk and M. Roubens, Lecture Notes in Economics and Mathematical Systems 301:Non-Conventional Preference Relations in Decision Making. Berlin: Springer-Verlag, 1988.[19] J. Kacprzyk and M. Fedrizzi, Multiperson Decision Making Models Using Fuzzy Sets andPossibility Theory. Dordrecht: Kluwer Academic Publishers, 1990.[20] J. Montero, "Aggregation of Fuzzy Opinions in a Non-Homogeneous Group," Fuzzy Setsand Systems, vol. 25, pp. 15-20, 1988.[21] E. Sanchez, "Importance in Knowledge Systems," Information Systems, vol. 14, pp. 455-464, 1989.[22] M. Tong and P. P. Bonissone, "A Linguistic Approach to Decision Making with Fuzzy Sets,2IEEE Transactions on Systems, Man and Cybernetics, vol. 10, pp. 716-723, 1980.[23] M. Tong and P. P. Bonissone, "Linguistic Solutions to Fuzzy Decision Problems," Studiesin the Management Sciences, vol. 20, pp. 323-334, 1984.[24] V. Torra and U. Cortes, "Towards an Automatic Consensus Generator Tool: EGAC," IEEETransactions on Systems, Man, and Cybernetics, vol. 25, pp. 888-894, 1995.[25] R.R. Yager, "Fuzzy Decision Making Using Unequal Objectives," Fuzzy Sets and Systems,vol. 1, pp. 87-95, 1978.[26] R.R. Yager, "A New Methodology for Ordinal Multiple Aspect Decision Based on FuzzySets," Decision Sciences, vol. 12, pp. 589-600, 1981.[27] R.R. Yager, "A Note on Weighted Queries in Information Retrieval Systems," J. of theAmerican Society of Information Sciences, vol. 38, pp. 23-24, 1987.[28] R.R. Yager, "On Ordered Weighted Averaging Aggregation Operators in Multicriteria Deci-sion Making," IEEE Transactions on Systems, Man, and Cybernetics, vol. 18, pp. 183-190,1988.[29] R.R. Yager, "Applications and Extension of OWA Aggregation," Int. J. Man-MachineStudies, vol. 37, pp. 103-132, 1992.

[30] R.R. Yager, "Families of OWA Operators," Fuzzy Sets and Systems, vol. 59, pp. 125-148,1993.[31] R.R. Yager, "Fuzzy Screening Systems," in Fuzzy Logic: State of the Art, R. Lowen, Ed.Kluwer Academic Publishers, 1993, pp. 251-261.[32] R.R. Yager, "On Weighted Median Aggregation," Int. J. Uncertainty, Fuzzines andKnowledge-Based Systems, vol. 2, pp. 101-113, 1994.[33] R.R. Yager, "An Approach to Ordinal Decision Making," Int. J. Approximate Reasoning,vol. 12, pp. 237-261, 1995.[34] L. A. Zadeh, "Fuzzy Sets," Information and Control, vol. 8, pp. 338-353, 1965.[35] L. A. Zadeh, "The Concept of a Linguistic Variable and Its Applications to ApproximateReasoning-I," Information Sciences, vol. 8, pp. 199-249, 1975.[36] L. A. Zadeh, "A Computational Approach to Fuzzy Quanti�ers in Natural Languages,"Computers and Mathematics with Applications, vol. 9, pp. 149-184, 1983.