DECSAI Department of Computer Scienceand Arti�cial IntelligenceDirect Approach Processesin Group Decision MakingUsing Linguistic OWA OperatorsF. Herrera, E. Herrera-Viedma, J.L. Verdegay

Technical Report #DECSAI-94111.November, 1994To appear in Fuzzy Sets and SystemsETS de Ingenier��a Inform�atica. Universidad de Granada.18071 Granada - Spain - Phone +34.58.244019, Fax +34.58.243317

Direct Approach Processesin Group Decision MakingUsing Linguistic OWA OperatorsF. Herrera, E. Herrera-Viedma, J.L. VerdegayDept. of Computer Science and Arti�cial IntelligenceUniversity of Granada, 18071 - Granada, Spaine-mail: herrera,viedma,verdegay@robinson.ugr.esAbstractIn a linguistic framework, several group decision making processes by directapproach are presented. These processes are designed using the linguistic orderedweighted averaging (LOWA) operator. To do so, �rst a study is made of theproperties and the axiomatic of LOWA operator, showing the rationality of itsaggregation way. And secondly, we present the use of LOWA operator to solvegroup decision making problems from individual linguistic preference relations.Keywords: Group decision making, linguistic preference relation, fuzzy linguisticquanti�er, OWA operators.1 IntroductionDecision making is a usual task in human activities. It consists of �nding the bestoption from a feasible set. Many decision making processes, in the real world, takeplace in an environment in which the goals, contraints and consequences of possibleactions are not precisely known. In these cases, probability theory has always allowedto deal quantitatively with that lack of precision. However, when the lack of precisionis of a qualitative nature too, the use of other techniques is necessary.Fuzzy set theory applied to decision making allows a more exible framework, whereby it is possible to simulate humans'ability to deal with the fuzziness of human judg-ments quantitatively, and therefore to incorporate more human consistency or "humanintelligence" in decision making models. Di�erent fuzzy decision making models havebeen proposed. A classi�cation for all of them, depending on the number of stagesbefore the decision is reached, is shown in [20]. We are interested in one fuzzy modelin one-stage decision making, i.e., a fuzzy multi-person decision making model appliedin group decision theory.A group decision making process may be de�ned as a decision situation in which(i) there are two or more individuals, each of them characterized by his or her ownperceptions, attitudes, motivations, and personalities, (ii) who recognize the existenceof a common problem, and (iii) attempt to reach a collective decision.In a fuzzy environment, a group decision problem is taken out as follows. It isassumed that there exists a �nite set of alternatives X = fx1; : : : ; xng as well as a �niteset of experts E = fe1; : : : ; emg, and each expert ek 2 E provides his preference relationon X , i.e., pk � XxX, and �Pk (xi; xj) denotes the degree of preference of alternativexi over xj , �Pk (xi; xj) 2 [0; 1].

Sometimes, an individual may have vague knowledge about the preference degree ofthe alternative xi over xj and cannot estimate his preference with an exact numericalvalue. Then, a more realistic approach may be to use linguistic assessments instead ofnumerical values, i.e., supposing that the variables which participate in the problemare assessed by means of linguistic terms [6, 11, 21, 29, 31]. A scale of certaintyexpressions (linguistically assessed) is presented to the individuals, who could then useit to describe their degrees of certainty in a preference. In this environment we havelinguistic preference relations for providing individuals' opinions.On the other hand, assuming a set of alternatives or decisions, two ways to relate todi�erent decision schemata are known. The �rst way, called the algebraic way, consistsin establishing a group selection process which obtains a decision scheme as solutionto group decision making problem. The second one, called the topological way, consistsin establishing a measure of distance between di�erent decision schemata. Severalautors have dealt with both proposals in fuzzy environments [3, 16, 17, 18, 19, 23].We have proposed various models for both possibilities under linguistic assessmentsin [11, 14] and [15], respectively. Here, we will focus on the �rst possibility, and wedevelop various group decision making processess under linguistic preferences, based onthe linguistic information aggregation operation carried out by the Linguistic OrderedWeighted Averaging (LOWA) operator [10]. To do this, we study the properties andthe preference aggregations axiomatic of the LOWA operator, showing its rationalaggregation way. Then we present how to use the LOWA operator for solving a groupdecision making problem from individual linguistic preference relations.The paper is structured as follows: Section 2 shows the use of linguistic preferencerelation and some properties; Section 3 analizes the properties and axiomatic of LOWAoperator; Section 4 presents the proposed group decision making processes; Section5 illustrates its application with some examples; and �nally, Section 6 presents ourconclusions.2 Linguistic Preference Relations in Group DecisionMa-kingLet X be a set of alternatives over which the fuzzy preference attitude of a decision-maker is de�ned. Then, according to Tanino [27, 28], the fuzzy preference may berepresented as:1. A fuzzy choice set to represent his total preference attitude. It is described bya fuzzy subset of X , i.e., by a membership function � on X , whose value �(x)denotes the preference degree of x, or degree to which x is chosen as a desirablealternative.2. A fuzzy utility function. It is described as fuzzy mapping �, which associates afuzzy subset of the utility values space (usually the space of real numbers R) witheach alternative x, � : XxR �! [0; 1]; where �(x; t) denotes the degree to whichthe utility value of the alternative x is equal to t.3. A fuzzy preference relation. It is described by a fuzzy binary relation R on X , i.e.,a fuzzy set on the product set XxX , characterized by a membership function �R :XxX �! [0; 1], where �R(xi; xj) denotes the preference degree of the alternativexi over xj .

The use of fuzzy preference relations in decision making situations to voice experts'opinions about an alternative set, with respect to certain criteria, appears to be a usefultool in modelling decision processes. Among others, they appear in a very natural waywhen we want to aggregate experts' preferences into group ones, that is, in the processesof group decision making.As we have mentioned above, in many cases an expert is not able to estimatehis preference degrees with exact numerical values. Then, another possibility is to uselinguistic labels, that is, to voice his opinions about alternatives by means of a linguisticpreference relation. Therefore, to �x previously a label set it is absolutely essencial tovoice the experts' preferences.In [2] the use of label sets with odd cardinals was studied, the middle label repre-senting a probability of "approximately 0.5", the remaining labels being placed sym-metrically around it and the limit of granularity is 11 or no more than 13. The semanticof the labels is given by fuzzy numbers de�ned on the [0,1] interval, which are describedby membership functions. As the linguistic assessments are merely approximate onesgiven by the individuals, we can consider that linear trapezoidal membership functionsare good enough to capture the vagueness of those linguistic assessments, since ob-taining more accurate values may be impossible or unnecessary. This representation isachieved by the 4-tuple (ai; bi; �i; �i) (the �rst two paremeters indicate the interval inwhich the membership value is 1.0; the third and fourth parameters indicate the leftand right widths of the distribution).We shall consider a �nite and totally ordered label set S = fsig; i 2 H = f0; : : : ; Tg,in the usual sense and with odd cardinality as in [2], where each label si represents apossible value for a linguistic real variable, i.e., a vague property or constraint on [0,1].We shall require the following properties: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 .Assuming a linguistic framework and a �nite set of alternativesX = fx1; x2; : : : ; xng,the experts' preference attitude over X can be de�ned, as a nxn linguistic preferencerelation R, such that, R = (rij); i; j = 1; : : : ; n, where rij 2 S denotes the preferencedegree of alternative xi over xj , linguistically assessed, withs0 � rij � sT ; (i; j = 1; : : : ; n);and where:1. rij = sT indicates the maximum degree of preference of xi over xj .2. sT=2 < rij < sT indicates a de�nite preference of xi over xj .3. rij = sT=2 indicates the indi�erence between xi and xj .So that the linguistic relation re ects a preference, it would be desirable to satisfysome of the following properties, proposed by Tanino in fuzzy environment [27, 28],and interpreted here in a linguistic environment:1. Reciprocity: rij = Neg(rji); and rii = s0 8i; j.

2. Max-Min Transitivity: rik �Min(rij; rjk); 8i; j; k.3. Max-Max Transitivity: rik �Max(rij; rjk); 8i; j; k.4. Restricted Max-Min Transitivity: rij � sT=2; rjk � sT=2;) rik �Min(rij ; rjk); 8i; j; k.5. Restricted Max-Max Transitivity: rij � sT=2; rjk � sT=2;) rik �Max(rij ; rjk); 8i; j; k.In order to make good use of the linguistic preference relations for aggregating ex-perts' preferences, an aggregation operator of linguistic information is needed. Variousoperators have been proposed [2, 10, 29, 32]. It is important that these operators satisfya well de�ned axiomatic. In the next section, we study some postulated axiomatics ofan intuituvely acceptable fuzzy aggregation operator, and more concretely, we anal-ize the properties and axioms that the linguistic aggregation operator, as de�ned byHerrera and Verdegay in [10], veri�es.3 Preferences Aggregation AxiomaticThe main problem in fuzzy logic-based group decision making is how to aggregate theexperts' opinions to obtain a group decision in such a way that some rational criteria aresatis�ed. The problem may be discussed as a special case of information aggregationin multi-person and multi-criteria decision making or social choice theories (see, forinstance, the surveys in books [8, 19, 25] and papers [7, 26, 30]).Many papers on `classical' theory of group decision make use of Arrow's work [1] asa starting point and a basic guide. Arrow proposed a qualitative setting composed by aset of axioms, which any acceptable aggregation tool of group decision making shouldsatisfy. Arrow's impossibility theorem was an important result thereof. According tothis theorem, it is impossible to aggregate individual preferences into group preferencein a completely rational way. This is a problem that dissappears in cardinal setting ina fuzzy context, introducing preference intensities, which provides additional degreesof freedom to any aggregation model [7, 5].3.1 Preferences Aggregation Axiomatic in Fuzzy EnvironmentTherefore an axiomatic to model the aggregation processes in fuzzy set area is needed.Some axiomatic approaches have been partially taken by Fung and Fu [9], Montero [22]and Cholewa [4]. Fung and Fu presented a set of axioms to rational fuzzy group decisionmaking in order to justify the minimum and maximum operations. Montero introducesfuzzy counterparts of veto and dictatorship, and Cholewa o�ers a collection of axiomsfor the aggregation of fuzzy weighted opinions and indicates that the weighted meansatis�es these axioms.In [7], a very detailed analysis about proposed axiomatic approaches to rationalgroup fuzzy decision making is presented. A complete set of axioms in the fuzzy setsetting for homogeneous groups is reviewed. These axioms are natural properties ofa voting procedure that include the ones proposed by Arrow. Some of these are:unrestricted domain, unanimity, neutrality,... A three-group classi�cation has beenestablished:� Imperative axioms, whose violation leads to counterintuitive aggregation modes,e.g.: neutrality.

� Technical axioms, that facilitate the representation and the calculation of theaggregation operator, e.g.: unrestricted domain.� Facultative axioms, that are applied in special circumstances but are not univer-sally accepted, e.g.: unanimity.Obviously a particular aggregation operator � doesn't have to satisfy all axioms to-gether, it must satisfy those that its special application circumstances require. Formore information about the axiomatic of fuzzy aggregation, see [7].In the next sections, we study some properties and axioms that the aggregationoperator of linguistic opinions, LOWA, veri�es.3.2 The Linguistic Ordered Weighted Averaging OperatorThe LOWA operator is based on the ordered weighted averaging (OWA) operatorde�ned by Yager [30], and on the convex combination of linguistic labels de�ned byDelgado et al. [6].Let fa1; : : : ; amg be a set of labels to aggregate, then the LOWA operator � isde�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, wi 2 [0; 1] and �iwi = 1;;�h = wh=�m2 wk; h = 2; : : : ; m, and B is the associated ordered label vector. Eachelement bi 2 B is the i-th largest label in the collection a1; : : : ; am. Cm is the convexcombination operator of m 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 :3.2.1 How to calculate the weights of LOWA operatorQuanti�ers can be used to represent the amount of items satisfying a given predicate.Classic logic is restricted to the use of the two quanti�ers, there exists and for all, thatare closely related respectively to the or and and connectives. Human discourse is muchricher and more diverse in its quanti�ers, e.g. about 5, almost all, a few, many, most,as many as possible, nearly half, at least half. In an attempt to bridge the gap betweenformal systems and natural discourse and, in turn, to provide a more exible knowledgerepresentation tool, Zadeh introduced the concept of linguistic quanti�ers [34].Zadeh suggested that the semantic of a linguistic quanti�er can be captured byusing fuzzy subsets for its representation. He distinguished between two types of lin-guistic quanti�ers, absolute and proportional. Absolute quanti�ers are used to representamounts that are absolute in nature such as about 2 or more than 5. These absolutelinguistic quanti�ers are closely related to the concept of the count or number of ele-ments. He de�ned these quanti�ers as fuzzy subsets of the non-negative real numbers,

R+. In this approach, an absolute quanti�er can be represented by a fuzzy subset Q,such that for any r 2 R+ the membership degree of r in Q, Q(r), indicates the degreeto which the amount r is compatible with the quanti�er represented by Q. Proportionalquanti�ers, such as most, at least half, can be represented by fuzzy subsets of the unitinterval, [0,1]. For any r 2 [0; 1], Q(r) indicates the degree to which the proportion r iscompatible with the meaning of the quanti�er it represents. Any quanti�er of naturallanguage can be represented as a proportional quanti�er or given the cardinality of theelements considered, as an absolute quanti�er. Functionally, linguistic quanti�ers areusually of one of three types, increasing, decreasing, and unimodal. An increasing typequanti�er is characterized by the relationshipQ(r1) � Q(r2) if r1 > r2:These quanti�ers are charecterized by values such as most, at least half. A decreasingtype quanti�er is characterized by the relationshipQ(r1) � Q(r2) if r1 < r2:The quanti�ers characterize terms such as a few, at most �. Unimodal type quanti�ershave the property that Q(a) � Q(b) � Q(c) = 1 � Q(d)for some a � b � c � d: These are useful for representing terms like about q.A natural question in the de�nition of the LOWA operator is how to obtain theassociated weighting vector. In [30, 33], Yager proposed two ways to obtain it. The �rstapproach is to use some kind of learning mechanism using some sample data; and thesecond approach is to try to give some semantics or meaning to the weights. The �nalpossibility has allowed multiple applications on areas of fuzzy and multi-valued logics,evidence theory, designing of fuzzy controllers, and the quanti�er guided aggregations.We are interested in the area of quanti�er guided aggregations, because our idea isto calculate weights using linguistic quanti�ers for representing the concept of fuzzymajority in the aggregations that are made in our group decision making processes.Therefore, in the aggregations of LOWA operator the concept of fuzzy majority isunderlying by means of the weights.In [30, 33], Yager suggested an interesting way to compute the weights of the OWAaggregation operator using linguistic quanti�ers, which, in the case of a non-decreasingproportional quanti�er Q, is given by this expression:wi = Q(i=n)�Q((i� 1)=n); i= 1; : : : ; nbeing the membership function of a non-decreasing proportional quanti�er Q, as follows:Q(r) = 8><>: 0 if r < ar�ab�a if a � r � b1 if r > bwith a; b; r 2 [0; 1]. When it is used a fuzzy linguistic quanti�er Q to compute theweights of LOWA operator �, it is symbolized by �Q:In Section 4, we explain how to use the LOWA operator for solving a group decisionmaking problem from individual linguistic preference relations by the direct approach,according to two types of fuzzy majority:

� Fuzzy majority of dominance, used to quantify the dominance that one alternativehas over all the others, according to one expert's opinions.� Fuzzy majority of experts, used to quantify the dominance that one alternativehas over all the others, according to the experts' opinions, considered as a whole.3.3 Properties of the LOWA OperatorThe LOWA operator has some properties of the OWA operators investigated by Yagerin [30], i.e.: monotonicity property, the commutativity property, and the property to bean "orand" operator. Before demonstrating the properties, we are going to demonstratethe following theorem.Theorem : Let A=[a1; a2; : : : ; an] be an ordered labels vector (a1 � a2 � : : : � an)and ai 2 S, then an � �(a1; a2; : : : ; an) � a1Proof: In [6] the following property was demonstrated: if k is the resulting labelfrom the convex combination of two labels, i and j (i � j), then i � k � j.When observing that property, obviously the theorem is a consequence thereof.Property 1.- The LOWA operator is increasing monotonous with respect to theargument values, in the following sense:LetA = [a1; a2; : : : ; an] be an ordered argument vector, let B = [b1; b2; : : : ; bn]be a second ordered argument vector, such that 8 j; aj � bj then�(A) � �(B)Proof: By induction over the number of arguments to aggregate.-For n=2Let sj ; si; sp; sq, be the ordered labels from S, corresponding to a1; a2; b1; b2,respectively. Clearly j � p and i � q, and therefore given any w1 2 [0; 1]then j � w1 � p �w1;and i � (1 � w1) � q � (1 � w1);and thus j � w1 + i � (1 � w1) � p �w1 + q � (1 � w1):As round is an increasing monotonous function, thenround(j �w1 + i � (1 � w1)) � round(p � w1 + q � (1 � w1)) )) round((j � i) � w1) + i)) � round((p � q) � w1) + q));and as i 2 Z+ and ((j � i) � w1) > 0 theni + round((j � i)w1) � q + round((p � q)w1) )) �(a1; a2) � �(b1; b2):

-Suppose that it is true for n-1, i.e.,�(a1; a2; : : : ; an�1) � �(b1; b2; : : : ; bn�1):-For n�(a1; a2; : : : ; an) = w1 � a1 � (1� w1)�Cn�1f�h; ah; h = 2; : : : ; ng;and�(b1; b2; : : : ; bn) = w1 � b1 � (1� w1)�Cn�1f�h; bh; h = 2; : : : ; ng:As Cn�1f�h; ah; h = 2; : : : ; ng = �(a2; a3; : : : ; an);and Cn�1f�h; bh; h = 2; : : : ; ng = �(b2; b3; : : : ; bn);then by induction hypothesis�(a2; a3; : : : ; an) � �(b2; b3; : : : ; bn):Let sj ; si; be the labels corresponding to �(a2; a3; : : : ; an) and �(b2; b3; : : : ; bn)respectively, then using induction hypothesis sj � si and as a1 � b1, andas the theorem a1 � sj and b1 � si, then�(a1; a2; : : : ; an) = �(a1; sj);and �(b1; b2; : : : ; bn) = �(b1; si):We know that �(a1; sj) � �(b1; si);as it is proven for n=2, and therefore�(a1; a2; : : : ; an) � �(b1; b2; : : : ; bn):Property 2.- The LOWA operator is commutative, i.e.,�(a1; a2; : : : ; an) = �(�(a1); �(a2); : : : ; �(an))where � is a permutation over the set of arguments.Proof: Clearly the commutativity property is veri�ed, because we use an"ordered" weighted average of the arguments.Property 3.- The LOWA operator is an "orand" operator. That is, for any weightingvector W and ordered labels vector A=[a1; a2; : : : ; an], thenMin(a1; a2; : : : ; an) � �(a1; a2; : : : ; an) � Max(a1; a2; : : : ; an)Proof: Clearly it is a consequence of the aforementioned theorem.

3.4 Axiomatic of the LOWA OperatorIn what follows we are going to study some of the proposed axioms in fuzzy settingconsidering the LOWA operator which works with linguistically valued preferences.Before this, we include the following linguistic notation that we shall use.Let A = fx1; : : : ; xng be a �nite non-empty set of alternatives.Let E = fe1; : : : ; emg be a panel of experts.Let S = fsi : i = 0; ::; Tg be a label set to voice experts' opinions.Let xij 2 S be the linguistic rating of alternative xi by expert ej .Let Fj be the linguistic rating set over alternatives by expert ej .Let �Fj be the linguistic membership function of Fj such that xij = �Fj (xi).Let F be the linguistic rating set such that F = �(F1; : : : ; Fn).� Axiom I: Unrestricted domain. For any set of individual preference patternsfFj ; j = 1; : : : ; mg there is a social preference pattern F , which may be con-structed, 8F1; : : : ; Fm 2 Sn; 9F 2 Sn such that F = �(F1; : : : ; Fm):It is basically technical, and clearly it is satis�ed in accordance with the LOWAoperator de�nition.� Axiom II: Unanimity or Idempotence. If everyone agrees on a preference pattern,it must be chosen as the social choice pattern,Fj = F; 8j ) F = �(F; F; : : : ; F ):Following this deni�tion, the LOWA operator can immediately be veri�ed.� Axiom III: Positive association of social and individual values. If an individ-ual increases his linguistic preference intensity for xi then the social linguisticpreference for xi cannot decrease. This means that if F 0j and Fj are such that�Fj � �F 0j , then if �(F1; : : : ; Fj; : : : ; Fm) = F and �(F1; : : : ; F 0j ; : : : ; Fm) = F 0 ,then �F � �F0 :Clearly it is satis�ed, because it is a consequence of increasing monotonicityproperty of the LOWA operator.� Axiom IV: Independence of irrelevant alternatives. The social preference intensityfor xi only depends on the individual preference intensity for xi, and not for xk,k 6= i, ��(F1;:::;Fm)(xi) = '(xi1; : : : ; xim):It is basically technical, and is satis�ed by the de�nition of the LOWA operator.Clearly this axiom does not extend strictly speacking, since for preference rela-tions the independence of irrelevant alternatives deals with pairs of alternatives.

� Axiom V: Citizen sovereignty. It means that any social preference pattern canbe expressed by the society of individuals; in other words8F; 9F1; : : : ; Fm such that F = �(F1; : : : ; Fm)A weaker form of citizen sovereignty called Non-Dictatorship [22], is as follows:there is no individual ej such that�(F1; : : : ; Fj; : : : ; Fm) = Fj :This requirement prohibits any individual from acting as a veto or dictator underany circumstances.Obviously, this axiom is satis�ed in its general form, because it is a consequenceof axiom II (unanimity). Clearly, as the LOWA operator is commutative, then italso satis�es the weaker form of the axiom.� AxiomVI: Decomposability of the voting procedure. This means that it is possibleto split the set of individuals into disjoint subgroups, build the social preferencepattern for each subgroup and then combine the local social preference patternsto obtain the result. Some forms of this axiom are:1. Strongest form, associativity:�(�(F1; F2); F3) = �(F1; �(F2; F3))2. Weaker form, auto-distributivity:�((F1; �(F2; F3)) = �(�(F1; F2); �(F1; F3))This axiom is not veri�ed for any of its forms. Thus, the LOWA operator is notassociative, example:Suppose, there is a set of nine labels, and we want to aggregate the labelss7; s6; s5. If we �x two weights w1 = 0:3; w2 = 0:7 then�(�(s7; s6); s5) = s5 6= s6 = �(s7; �(s6; s5)):And the LOWA operator is not auto-distributive, example:Let s1; s2; s4 be the opinions to be aggregated. Using the same weights as before,�(s1; �(s2; s4)) = s2 6= s1 = �(�(s1; s2); �(s1; s4)):� Axiom VII: Neutrality. The neutrality axiom refers to the invariance propertiesof the voting procedure. There are three types:1. Neutrality with respect to alternatives. If xi and xk are such that xij =xkj ; 8j; then ��(F1 ;:::;Fm)(xi) = ��(F1;:::;Fm)(xk)2. Neutrality with respect to voters. In a homogeneous group, this is theanonymity property, i.e., the commutativity of �.3. Neutrality with respect to the intensity scale or Neutrality of Complement.If F cj is the complement to Fj , such that F cj=Neg(Fj), the social pattern�(F c1 ; : : : ; F cm) should be the complement fo the social preference pattern,�(F1; : : : ; Fm)c = �(F c1 ; : : : ; F cm)

Clearly it is veri�ed for the form of neutrality respect to alternatives. As theLOWA operator is commutative, then it also veri�es the neutrality with respectto voters. However it does not verify the neutrality with respect to intensity scale.Example:Consider a label set with eight elements S. Let s3; s2 2 S be the labels to beaggregated and its complement labels s4; s5, and w1 = 0:1, thenNeg(�(s3; s2)) = s5 6= s4 = �(s4; s5):In conclusion, the LOWA operator veri�es the following axioms: Unrestricted do-main, Unanimity or Idempotence, Positive association of social and individual values,Independence of irrelevant alternatives, Citizen sovereignty, Neutrality. The ful�lmentof those axioms provides evidence of rational aggregation using the LOWA operator inparticular frameworks.4 Direct Approach to Group Decision Making underLinguistic AssessmentsSuppose we have a set of n alternatives X = fx1; : : : ; xng and a set of experts E =fe1; : : : ; emg. Each expert ek 2 E provides a preference relation linguistically assessedinto the label set S, !P k : XxX ! S;where !P k (xi; xj) = pkij 2 S represents the linguistically assessed preference degree ofthe alternative xi over xj . We assume that P k is reciprocal without loss of generality.As is known, basically two approaches may be considered. A direct approachfP 1; : : : ; Pmg ! solutionaccording to which, on the basis of the individual preference relations, a solution isderived, and an indirect approachfP 1; : : : ; Pmg ! P ! solutionproviding the solution on the basis of a collective preference relation, P , which is apreference relation of the group of individuals as a whole.In [14], we considered a group decision making process by indirect approach. Here,we consider the direct approach. We present three direct ways to solve a group decisionmaking process in a linguistic framework: (i) Dominance Process (DP), (ii) Strict-Dominance Process (SDP), (iii) Non-Dominance Process (NDP).These processes are based on linguistic dominance, linguistic strict-dominance, andlinguistic non-dominance degrees respectively, as de�ned in [13]. The direct approachprocesses are calculated on two levels of action:� Level of Preference or Level of Individual. The di�erent linguistic degrees arecalculated for each alternative according to the opinions of each expert, consideredindividually, i.e., individual degrees are calculated on this level .� Level of Degrees or Level of Group. Here, the di�erent linguistic degrees arecalculated for each alternative according to the opinions of the expert group,considered as a whole, i.e., social degrees are calculated on this level.

Note that between both levels the concept of fuzzy majority is di�erent, dominancemajority and experts' majority respectively, and therefore we can use di�erent fuzzylinguistic quanti�ers on each level.The following subsections are devoted to developing each one of the aforementionedmodels.4.1 Dominance ProcessThis is a resolution model that uses the linguistic dominance degree of each alternativeto decide which of them to choose as a solution to the group decision making problem.The linguistic dominance degree of one alternative gives a measure of the averagingpreference degree that the overall ones in function of the experts' opinions.After �xing a label set S and the concepts of fuzzy majority of dominance andfuzzy majority of experts by means of two fuzzy quanti�ers, Q1 and Q2 respectively,the process is described in the following steps:1. For each linguistic preference relation of each expert, P k , using the LOWA oper-ator �Q1 , �nd out the individual linguistic dominance degree of each alternativexi, called IDki , according to this expresion:IDki = �Q1(pkij ; j = 1; : : : ; n; j 6= i)with k = 1; : : : ; m; i = 1; : : : ; n.2. For each alternative xi, calculate the social linguistic dominance degree, calledSDi, as follows SDi = �Q2(IDki ; k = 1; : : : ; m)with i = 1; : : : ; n:3. Obtain the set of alternatives with maximum linguistic dominance degree Xdmax,as follows Xdmax = fxi 2 X=SDi = Maxj(SDj)gand the solution set, i.e., those alternatives with maximum degree.This process is shown in Figure 1.P

P

P

e

e

e

experts

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relations

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linguistic

ID , ID ,....,ID

ID , ID ,....,ID

ID , ID ,....,ID

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[

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DIRECT APPROACH BASED ON LINGUISTIC DOMINANCE DEGREE

LOWA and LOWA and

d

Q2

......

......

......

....

......

......

......

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....Fig. 1. Dominance Process

4.2 Strict-Dominance ProcessThis proposal of direct approach selects the alternative(s) solution according to itsrespective linguistic strict-dominance degree. The linguistic strict-dominance degreegives a linguistic measure of the number of times that one alternative is preferredoverall ones according to all experts' opinions.In this process, as we shall see, fuzzy majority of dominance is not underlying in theweights of an LOWA operator. Here, it is used to quantify linguistically the numberof times that one alternative is preferred to overall ones. Therefore, we need to de�nea fuzzy quanti�er of the Q0:[0,1]! L type, being L a label set. This can be done bymeans of this expresion: Q0(r) = 8><>: l0 if r < ali if a � r � blU if r > bl0 and lU are the minimum and maximum labels in L, respectively, andli = Suplq2Mflqg;with M = flq 2 L : �lq(r) = Supt2J f�lt(r� ab� a )gg, with a; b; r 2 [0; 1]. Another de�nition of Q0 can be found in [32].Fixed two label sets, S and L, and the concepts of fuzzy majority of alternativesand experts by means of two fuzzy quanti�ers, Q01 and Q2, respectively, the process isshown in Figure 2,P

P

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SSD , SSD ,....,SSD

strict-dominance

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ISD , ISD ,....,ISD

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DIRECT APPROACH BASED ON LINGUISTIC STRICT-DOMINANCE DEGREE

LOWA and

Q’

sd

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.....Fig. 2. Strict-Dominance Processand described in the following steps:1. For each linguistic preference relation of each expert P k, �nd out the individuallinguistic strict-dominance degree of each alternative xi, called ISDki , accordingto this expresion: ISDki = Q01( rkin� 1);

with k = 1; : : : ; m; i = 1; : : : ; n, and where,rki = #fxj 2 X such that pij > pji; j = 1; : : : ; n; j 6= ig;and # stands for the cardinal of the term set.2. For each alternative xi, calculate the social linguistic strict-dominance degree,called SSDi, as followsSSDi = �Q2(ISDki ; k = 1; : : : ; m)with i = 1; : : : ; n:3. Obtain the set of alternative with maximum linguistic strict-dominance degreeXsdmax, as follows Xsdmax = fxi 2 X=SSDi = Maxj(SSDj)gthe solution set.4.3 Non-Dominance ProcessHere, we present a third model using the direct approach, which selects the alter-native(s) solution according to its respective linguistic non-dominance degree. Thelinguistic non-dominance degree of one alternative expresses a linguistic measure inwhich that alternative is not dominated by any, according to the experts' opinions. Itis calculated on the basis of the concept of non-dominated alternatives described byOrlovsky [24], extended here to linguistic environment.After �xing a label set, S, and the concept of fuzzy majority of experts by meansof the fuzzy quanti�ers, Q2, the group decision making process is shown in Figure 3,P

P

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nd

n

m

......

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....Fig. 3. Non-Dominance Process.and described in the following steps:

1. For each linguistic preference relation of each expert, P k , �nd its respective lin-guistic strict preference relation, P ks , with !P ks = pks ij , such that,pks ij = s0 if pkij < pkji;or pks ij = sh 2 S if pkij � pkji with pkij = sl; pkji = st and l = t+ h:2. For each linguistic strict preference relation of each expert P ks , �nd the individuallinguistic non-dominance degree of each alternative xi, called INDki , accordingto this expresion: INDki = Minxj2X [Neg(!P ks (xj; xi))]with k = 1; : : : ; m; i = 1; : : : ; n.3. For each alternative xi, calculate the social linguistic non-dominance degree, calledSNDi, as follows SNDi = �Q2(INDki ; k = 1; : : : ; m)with i = 1; : : : ; n:4. Obtain the set of alternatives with maximum linguistic non-dominance degreeXndmax, as follows Xndmax = fxi 2 X=SNDi = Maxj(SNDj)gand the solution set.4.4 Sequential ProcessSometimes, any of above solution set of alternatives can be formed by various alterna-tives. This may happen because of the existence of balance between all the alternatives,or the existence of an inconsistency situation between all the experts' opinions. In thesecases, the combined application of the three models could be very interesting, becauseit can help to avoid the inconsistency situation and to identify the best solution set ofalternatives.ALTERNATIVES

SET OF

ALTERNATIVESOF DOMINANTMAXIMAL SET MAXIMAL SET OF

NON-DOMINATEDALTERNATIVES

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ALTERNATIVES

sociallinguistic

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sd_d X

nd_sd_d max

max #(X )>1 sd_d[ P , P ,...,P ]

Fig. 4. Sequential Process.

In line with the aforementioned, we propose a group decision making process whichcombines the ones described earlier, called The Sequential Process. This model consistsin applying each one of them in sequence, according to a previously established order.There is no criterion to establish an order, e.g., we can establish an order based on theorder of the presentation of our work. From this supposition, the sequential process isshown in Figure 4 and developed in three steps:1. Apply the �rst model, DP over X, and obtain Xdmax. If #(Xdmax) = 1 then End,and this is the solution. Otherwise continue, using the following step.2. Apply the second model, SDP over X, and obtain Xsd dmax � Xdmax. If #(Xsd dmax) = 1then End, and this is the solution. Otherwise continue, using the following step.3. Apply the third model, NDP over X, and obtain Xnd sd dmax � Xsd dmax , and this isthe best solution.5 ExamplesIn this Section the application of aforementioned models is shown in the followingconditions:Let the nine linguistic label set be S:C Certain (1; 1; 0; 0)EL Extremely likely (:98; :99; :05; :01)ML Most likely (:78; :92; :06; :05)MC Meaningful chance (:63; :80; :05; :06)IM It may (:41; :58; :09; :07)SC Small chance (:22; :36; :05; :06)V LC V ery low chance (:1; :18; :06; :05)EU Extremely unlikely (:01; :02; :01; :05)I Impossible (0; 0; 0; 0)Suppose a set of four alternativesX = fx1; x2; x3; x4g, as well as a set of four expertsE = fe1; e2; e3; e4g, whose respective opinions are the following linguistic preferencerelations, expressed over X , using the labels of S:P 1 = 26664 � SC C IMC � EU ELI EL � V LCC EU ML � 37775P 2 = 26664 � IM C EUIM � EU CI EL � V LCEL I ML � 37775P 3 = 26664 � IM EL IIM � I ELEU C � V LCC EU ML � 37775P 4 = 26664 � IM C EUIM � EU CI EL � V LCEL I ML � 37775Using the linguistic quanti�er "As many as possible" with the pair (0.5,1) for all theoperations, and assuming that Q1=Q2 and S = L, then the direct models are appliedas follows:

1. Dominance Process.(a) Step 1. The individual linguistic dominance degrees of each alternativeusing the LOWA operator with W=(0; 0:334; 0:666) are:i. Expert one, (ID11; ID12; ID13; ID14) = [EU; VLC;EU; SC]:ii. Expert two,(ID21; ID22; ID23; ID24) = [V LC; VLC;EU; VLC]:iii. Expert three,(ID31; ID32; ID33; ID34) = [EU;EU;EU;SC]:iv. Expert four,(ID41; ID42; ID43; ID44) = [V LC; VLC;EU; VLC]:(b) Step 2. The social linguistic dominance degree of each alternative usingthe LOWA operator with W=(0; 0; 0:5; 0:5) is:(SD1; SD2; SD3; SD4 = [EU; VLC;EU; VLC]):(c) Step 3. The set of alternatives with maximum linguistic dominance degreeis: Xdmax = fx2; x4g2. Strict-Dominance Process.(a) Step 1. The individual linguistic strict-dominance degrees of each alterna-tive are:i. Expert one,(ISD11; ISD12; ISD13; ISD14) = [Q01(1=3); Q01(2=3); Q01(1=3); Q01(2=3)] == [I; SC; I; SC]:ii. Expert two,(ISD21; ISD22; ISD23; ISD24) = [Q01(1=3); Q01(1=3); Q01(1=3); Q01(2=3)] == [I; I; I; SC]:iii. Expert three,(ISD31; ISD32; ISD33; ISD34) = [Q01(1=3); Q01(1=3); Q01(1=3); Q01(2=3)] == [I; I; I; SC]:iv. Expert four,(ISD41; ISD42; ISD43; ISD44) = [Q01(1=3); Q01(1=3); Q01(1=3); Q01(2=3)] = [I; I; I; SC]:

(b) Step 2. The social linguistic strict-dominance degree of each alternativeusing the LOWA operator with W=(0; 0; 0:5; 0:5) is:(SSD1; SSD2; SSD3; SSD4) = [I; I; I; SC]:(c) Step 3. The set of alternatives with maximum linguistic dominance degreeis: Xsdmax = fx4g3. Non-Dominance Process.(a) Step 1. Find the respective strict linguistic preference relations of eachexpert.P 1s = 26664 � I C IV LC � I MLI ML � IC I IM � 37775P 2s = 26664 � I C II � I CI ML � IML I IM � 37775P 3s = 26664 � I ML II � I MLI C � IC ML IM � 37775P 4s = 26664 � I C II � I CI ML � IML I IM � 37775(b) Step 2. The individual linguistic non-dominance degrees of each alternativeare:i. Expert one,(IND11; IND12; IND13; IND14) = [I; V LC; I; VLC]:ii. Expert two,(IND21; IND22; IND23; IND24) = [V LC; VLC; I; I ]:iii. Expert three,(IND31; IND32; IND33; IND34) = [I; I; VLC; VLC]:iv. Expert four,(IND41; IND42; IND43; IND44) = [V LC; VLC; I; I ]:(c) Step 3. The social linguistic non-dominance degree of each alternative usingthe LOWA operator with W=(0; 0; 0:5; 0:5) is:(SND1; SND2; SND3; SND4 = [I; EU; I; I]):(d) Step 4. The set of alternatives with maximum linguistic non-dominancedegree is: Xdmax = fx2g

4. Sequential Process.(a) Step 1. After applying the DP model we obtain the following set of alter-natives with maximum linguistic dominance degree:Xdmax = fx2; x4g:As #(Xdmax) > 1, then continue,(b) Step 2. Apply the SSP model over X. As Xdmax = fx2; x4g then only itsrespective strict-dominance degrees are considered,(SSD2; SSD4) = [I; SC];and therefore the Xsd dmax is Xsd dmax = fx4g;and as #(Xsd dmax) = 1 then End.In consequence, by applying the proposed models together, it is possible to dis-tinguish better amongst alternatives.Remark: In the sequential process, the order of application of the di�erentprocesses is chosen by the user.6 ConclusionsThe paper has developed basically two ideas:-�nding out evidence of the rationality of the LOWA operator,-showing its usefulness in processes of group decision making in a linguistic envi-ronment.We have checked the �rst goal, expressing the properties of the LOWA operator andexamining some of the axioms of an acceptable aggregation operator that it satis�es.Then we combined the LOWA operator with other fuzzy tools, such as fuzzy linguisticquanti�er, linguistic preference relations, concept of dominance and non-dominance, toshow its use in the �eld of group decision. We presented three models of group decisionmaking based on rational properties of the LOWA operator.This paper shows how to use fuzzy technicals to incorporate more human consis-tency in decision models. In the future, we are interested in developing decision modelsthat allow a closer link between the computer world and human beings world in �elds,such as multi-criteria decision making and multi-stage decision making, using linguisticelements.References[1] K.J. Arrow, Social Choice and Individual Values (Wiley, New York, 1963).[2] P.P. Bonissone and K.S. Decker. Selecting Uncertainty Calculi and Granularity:An Experiment in Trading-o� Precision and Complexity, in: L.H. Kanal and J.F.Lemmer, Eds., Uncertainty in Arti�cial Intelligence (North-Holland, 1986) 217-247.

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