DECSAI Department of Computer Scienceand Arti�cial IntelligenceChoice Processesfor Non-Homogeneous Group Decision Makingin Linguistic SettingF. Herrera, E. Herrera-Viedma, J.L. Verdegay

Technical Report #DECSAI-95121.October, 1995ETS de Ingenier��a Inform�atica. Universidad de Granada.18071 Granada - Spain - Phone +34.58.244019, Fax +34.58.243317

Choice Processesfor Non-Homogeneous Group Decision Makingin Linguistic SettingF. Herrera, E. Herrera-Viedma, J.L. VerdegayDept. of Computer Science and Arti�cial IntelligenceUniversity of Granada, 18071 - Granada, Spaine-mail: herrera,viedma,verdegay@decsai.ugr.esAbstractSeveral choice processes of alternatives for non-homogeneous group decision making problems,assuming linguistic preference relations for expressing the opinions of individuals and linguistic val-ues for expressing their respective power or importance degrees, are presented. These processes aredesigned using two choice degrees of alternatives based on the concept of fuzzy majority: quanti-�er guided linguistic dominance degree and quanti�er guided linguistic non-dominance degree, beingthe last one a generalization of Orlovski's non-dominated alternative concept. In order to deal withnon-weighted linguistic information, the linguistic ordered weighted averaging (LOWA) operator isapplied. To deal with weighted linguistic information, three operators of linguistic weighted informa-tion aggregation are used: the linguistic weighted disjunction (LWD) operator, the linguistic weightedconjunction (LWC) operator and the linguistic weighted averaging (LWA) operator.Keywords: Linguistic modelling, group decision making, linguistic preference relation, fuzzy linguisticquanti�er.1 IntroductionHuman beings are constantly making decisions in most of their day-to-day activities. So the study ofdecision making processes has always been a �eld of great interest to many researchers. A brief historicalsurvey covering the study of decision making processess may be divided into three distinct periods [34]:(i) optimization period, where the problem is de�ned exactly, (ii) multiple criteria period, that is as astrict optimization period, and more recently, (iii) fuzzy decision period, that is based in application ofFuzzy Set Theory.Frequently, real world decision making problems are ill de�ned, i.e., their objetives and parametersare not precisely known. These obstacles of lack of precision have been dealt with using the probabilisticapproach. But, due to the fact that the requirements on the data and on the environment are very highand that many real world problems are fuzzy by nature and not random, the probability applicationshave not been very satisfactory in a lot of cases. On the other hand, the application of Fuzzy Set Theoryin real world decision making problems has given very good results. Its main feature is that it providesa more exible framework, where it is possible to solve satisfactorily many of the obstacles of lack ofprecision.In some human activities it should be very suitable to have the help of a computerized decisionsupport system in the decision making processes, e.g. in medical diagnosis, �nancial or political worldprocesses. The study of real world decision making problems to design computerized decision supportsystems implies two steps: (1) establishing appropriate mathematical decision models close to real worldproblems and, (2) implementation of the �xed models. Applications on both senses have been dealt withusing fuzzy techniques in [3, 6, 5, 15, 29]. Here, we are interested in providing new decision models forgroup decision making problems.Group decision making problems may be de�ned as a decision situation in which (i) there are two ormore individuals, each of them characterized by his or her own perceptions, attitudes, motivations, and

personalities, (ii) who recognize the existence of a common problem, and (iii) attempt to reach a collectivedecision. In such a situation, two classical ways to relate to di�erent decision schemes are known [3].The �rst way, called the algebraic way, consists of establishing a group choice process which obtains adecision scheme as solution to group decision making problem. The second one, called the topologicalway, consists of establishing a group consensus process for di�erent decision schemata until achieving thepossible maximum consensus degree about solution alternative(s) set. Both processes may be combinedin a resolution scheme: �rstly, consensus process is applied, in each step, the degree of existing consensusamong experts' opinions is measured; if the consensus degree is satisfactory, then choice process is appliedin order to obtain a solution, otherwise the experts are persuaded to update their opinions. In this way,a group decision making process may be de�ned as a dynamic and iterative process where the experts,via exchange of information and rational arguments, update their opinions until they become su�cientlysimilar. This is represented in Figure 1.GC

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Figure 1: Group Decision Making ProcessSometimes, one may admit that the various individuals that give the opinions are not equally impor-tant (e.g. reliable). In such a case, it is called non-homogeneous group decision making problem and,otherwise, homogeneous group decision making problem. One way of modelling this aspect is to considerthe existence of a manager that assigns a weight to each expert. The weights may be interpreted in atleast two di�erent ways [14]:1. Each individual is viewed as a subgroup and the weight re ects the relative size of this subgroup.2. The weight may re ect the relevance of the individual in the group. This level of relevance may actas a constraint on the opinions that an individual can express.Furthermore, usually, experts express their opinions by means of numerical values (numerical setting).When experts are not able to give exact numerical values to express their opinions, then, a more realisticalternative option is using linguistic assessments instead of numerical values [9, 11, 20, 31, 35, 36, 38].In such a situation, for each variable of problem domain an appropriate linguistic label set is chosenand used by individuals who participate in the decision making process to express their opinions. Thissetting is known as the linguistic setting. In [3, 26, 27, 28, 29] choice processes and consensus modelsfor non-homogeneous and homogeneous group decision making problems are studied and proposed in anumerical setting. We provided several consensus models for non-homogeneous and homogeneous groupdecision making problems in [21, 23] and several choice processess for homogeneous group decision makingproblems in [24, 20, 25, 18, 19, 22].This paper concerns non-homogeneous group decision making problems, where a manager assigns a

weight set to expert group. The weights are interpreted according to the second way, i.e., they can beviewed as the "power" (importance degree, competence or ability) of experts' opinions. Exactly, and asa complement to our work started in [20], we present various choice processes for a non-homogeneousgroup developed using several aggregation operators of linguistic information and guided by several choicedegrees of alternatives.To do so, the paper is structured as follows: Section 2 presents the linguistic setting of the groupdecision making problem to solve; Section 3 shows the di�erent aggregation operators to be used; Section4 de�nes the alternative choice degrees to apply; Section 5 provides the choice processes; Section 6contains the development of an example for the sake of illustrating the choice processes; and �nally someconclusions are pointed out.2 Problem SettingIn this section we are going to establish a speci�c mathematicmodel to represent a usual non-homogeneousgroup decision making problem, in such a way that it will be possible to develop our choice processes.Usually, in a classical fuzzy framework, a non-homogeneous group decision problem is considered asfollows. Let X = fx1; : : : ; xng be a non-empty and �nite set of alternatives to be analyzed by a non-empty and �nite set of experts E = fe1; : : : ; emg. Each expert ek 2 E provides his/her opinions on X asa fuzzy preference relation, P k � XxX, with membership function�Pk : XxX ! [0; 1];where �Pk(xi; xj) = pkij denotes the preference degree of the alternative xi over xj . The existence of adistinguished person is assumed, called manager, which assigns a weight �E(k) to each expert ek, de�ningon the expert set, E, a fuzzy set with membership function,�E : E ! [0; 1]:Given an expert ek, his weight �E(k) is interpreted as the degree in which that expert is really a decision-maker in relation to the decision problem, i.e., as the power or importance degree of his opinion.As we work in a linguistic setting, �rst, we have to �x the types of linguistic term sets that willbe used to express individuals' opinions. Here, we use the same linguistic setting de�ned in our earlierworks [24, 20, 25, 19, 21, 22, 23], i.e., according to [2], where a study of label sets with odd cardinals waspresented, we shall consider a �nite and totally ordered label set S = fsig; i 2 H = f0; : : : ; Tg, in theusual sense and with odd cardinality, where each label si represents a possible value for a linguistic realvariable, i.e., a vague property or constraint on [0,1]. The following properties are required:1) The set is ordered: si � sj if i � j.2) There is a 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 .The semantic of each label si is given by a linear trapezoidal membership function represented bythe 4-tuple (ai; bi; �i; �i) (the �rst two paremeters indicate the interval in which the membership value is1.0; the third and fourth parameters indicate the left and right widths of the distribution). For example,consider the following nine linguistic term set with the associated semantic, [2]:V H V ery High (1; 1; 0; 0)H High (:98; :99; :05; :01)MH Moreorless High (:78; :92; :06; :05)FFMH From Fair to Moreorless High (:63; :80; :05; :06)F Fair (:41; :58; :09; :07)FFML From Fair to Moreorless Low (:22; :36; :05; :06)ML Moreorless Low (:1; :18; :06; :05)L Low (:01; :02; :01; :05)V L V ery Low (0; 0; 0; 0)

and graphically:0.0 1.00.5

VHHMHFFMHFFFMLMLVL L

Figure 2: Distribution of the Nine Linguistic Term SetFormally speaking, it seems di�cult to accept that all individuals would agree on the same membershipfunction associated to linguistic terms, and therefore there are not any universality distribution concepts.For example, the two perceptions showed in Figure 3 for the evaluation could be considered. However, inour context, we consider an environment where experts can discriminate perfectly the same term set undera similar conception, taking into account that the concept of a linguistic variable serves the purpose ofproviding a means of approximated characterization of imprecise preference information. Morever, as maybe observed in the next section, we de�ne aggregation operators of linguistic labels by direct computationon labels, i.e., independently of the semantic of the label set, and thus, considering a similar discriminationof the experts.excellentgood excellentgood

10

(1) (2)

0 1Figure 3: Di�erent Distribution ConceptsTherefore, after �xing an appropriate label set L = flig; i 2 J = f0; : : : ; Ug, to express importancedegree of expert set and an appropriate label set S to express experts' opinions, a non-homogeneousgroup decision making problem in a linguistic setting may be represented as follows. As we said at thebeginning, we are assuming a non-empty and �nite set of alternatives X = fx1; : : : ; xng as well as anon-empty and �nite set of experts E = fe1; : : : ; emg. For each expert ek 2 E, we assume an importancedegree, �E(k), assigned by the manager and linguistically assessed in the label set, L,�E : E ! L;from l0, standing for 'de�nitely irrelevant' and lJ , standing for 'de�nitely relevant', through all interme-diate values. Each expert ek provides his/her opinions on X by mean of a linguistic preference relation,P k, assessed in the label set, S, �Pk : XxX ! S;where �Pk (xi; xj) = pkij represents the linguistically assessed preference degree of the alternative xi overxj, with s0 � pkij � sT ; (i; j = 1; : : : ; n);and:1. pkij = sT indicates the maximum degree of preference of xi over xj .2. sT=2 < pkij < sT indicates a de�nite preference of xi over xj .

3. pkij = sT=2 indicates indi�erence between xi and xj.As in our rational consensus model presented in [23], we assume that P k holds the following properties:1. Soft-reciprocity, in the following sense [1],(a) By de�nition pkii = s0 8xi 2 X (the minimum label in S).(b) If pkij � sT=2; then pkji � sT=2.Condition (a) is a convection; if xi is considered by itself, no preference is assigned. Condition(b) seems plausible, because when pkij � sT=2, according to our de�nition of linguistic preferencerelations, it is reasonable to think that the complementary preference, pkji, should automaticallysatisfy pkji � sT=2, since, otherwise, we would have a contradiction. In [1, 19, 21], the secondcondition was established as pkij = Neg(pkij), but here we have relaxed it in order to allow morefreedom in the experts' opinions.2. Completeness, in the following sense [8],pkij � Neg(pkji); 8(xi; xj):This property is required in order to ensure that all the experts consider the alternative set, aboutwhich they are expressing their opinions, as feasible and comprehensive.It is clear that the aggregation of information plays a central role in the choice processes of the groupdecision making problems, and more speci�cally, in our linguistic context, an aggregation operator oflinguistic labels is needed. Various approaches have been proposed, some use direct computation onlabels [10, 24, 38, 39, 42] and, others use computation on associated membership functions [2, 35]. Wework following the �rst approach, which is independent of the semantics in the term set, and as we saidearlier, considering a similar discrimination by the experts, and so, in the next section, we present variousaggregation operators of linguistic labels by direct computation.3 Linguistic Aggregation OperatorsFirstly, we are going to analyze the information to be aggregated in the choice processes of non-homogeneous group decision making problem in a linguistic setting. Clearly, there are two types oflinguistic information:1. Non-weighted linguistic information. This is the situation in which we have only one set of linguisticvalues to aggregate.2. Weighted linguistic information. This is the situation in which we have a set of linguistic values toaggregate, for example opinions and each value is characterized by an importance degree, indicatingits weight in the overall set of values.In both cases, linguistic aggregation operators are needed that combine appropriately the information,in such a way, that the �nal aggregation is the "best" representation of the overall opinions. In thefollowing section, we shall present the operators that we are going to consider in both cases.3.1 Non-weighted linguistic informationIn the literature various aggregation operators of linguistic information have been proposed [2, 10, 24, 22,17, 35, 38, 39, 42]. Some are based on the use of the associated membership functions of the labels [2, 35],and others act by direct computation on labels [10, 24, 22, 17, 38, 39, 42]. Here we work with the latter.We consider only two operators, linguistic ordered weighted averaging (LOWA) operator, presented in[24, 22], and used in linguistic aggregations in [20, 25, 18, 19, 21, 22, 23, 17], and the inverse-linguisticordered weighted averaging (I-LOWA) operator presented in [17].

De�nition of the LOWA operator. Let A = fa1; : : : ; amg be a set of labels to be aggregated, then theLOWA 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: (i) wi 2 [0; 1] and, (ii) �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 of m labels,� is the general product of a label by a positive real number and � is the general addition of labels de�nedin [10]. If m=2, then C2 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 :Example of the LOWA operator, �. Suppose that we want to aggregate by means of the LOWAoperator the following four labels, fL;ML;H; VHg:Assuming this weighting vectorW = [0:3; 0:2; 0:4;0:1]the general expression of the aggregation of labels is:�(L;ML;H; VH) = [0:3; 0:2; 0:4;0:1](VH;H;ML;L) = C4f(0:3; V H); (0:2;H); (0:4;ML); (0:1;L)g:Then, we obtain the �nal result applying the recursive de�nition of the convex combination, C4; asfollows. Firstly, we develop C4 until its simpler expression in the following steps:1. For m = 4;C4f(0:3; V H)(0:2;H)(0:4;ML)(0:1;L)g= 0:3� V H �C3f(0:29;H); (0:57;ML); (0:14; L)g:2. For m = 3;C3f(0:29;H); (0:57;ML); (0:14; L)g= 0:29�H �C2f(0:57;ML); (0:14; L)g:Now, we are going to go back solving the simpler cases until to obtain the �nal result:1. For m = 2; C2f(0:57;ML); (0:14;L)g= 0:57�ML� (1� 0:57)� L =ML (s2);since as ML = s2 and L = s1 thenminf9; 1 + round(0:57 � (2� 1))g = minf8; 1 + round(0:57)g = minf8; 2g = 2:2. For m = 3;C3f(0:29;H); (0:57;ML); (0:14;L)g= 0:29�H �C2f(0:57;ML); (0:14;L)g= FFML (s3);since as H = s7 and ML = s2 thenminf8; 2 + round(0:29 � (7� 2))g = minf8; 2 + round(1:45)g = minf8; 3g = 3:

3. Finally, we obtain the �nal result for m = 4;C4f(0:3; VH)(0:2;H)(0:4;ML)(0:1; L)g=0:3� V H �C3f(0:29;H); (0:57;ML); (0:14;L)g= FFMH (s5);since as V H = s8 and FFML = s3 thenminf8; 3 + round(0:3 � (8� 3))g = minf8; 3 + round(1:5)g = minf8; 5g = 5:De�nition of the I-LOWA operator. An I-LOWA (Inverse-Linguistic Ordered Weighted Averaging)operator, �I , is a type 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:The LOWA and I-LOWA operators are increasing monotonous, commutative, "orand" operators,which verify these axioms: Unrestricted domain, Unanimity or Idempotence, Positive association of socialand individual values, Independence of irrelevant alternatives, Citizen sovereignty, Neutrality [22, 17].3.1.1 Obtaining of the weighting vectorHow to calculate the weighting vector of the LOWA operator, W , is a basic question to be answered.Yager proposed in [37, 40] two ways for doing 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 or meaning tothe weights. We consider the latter approach, because our idea is to use the concept of fuzzy majority bymeans of the weighting vector in the aggregations of the LOWA operator.Traditionally, the majority is de�ned as a threshold number of individuals. Fuzzy majority is asoft majority concept, which is manipulated via a fuzzy logic based calculus of linguistically quanti�edpropositions. In [26], Kacpryzk speci�ed the fuzzy majority rule by means of a linguistic quanti�er toderive various solutions concepts for group decision making problems in a numerical setting. Here, weshall work in a similar way, but in the �eld of quanti�er guided aggregations. Before showing how to dothis, we shall brie y introduce the concept of the fuzzy linguistic quanti�er.Human 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. Classic logic is restricted only to the use of two quanti�ers,there exists and for all. Zadeh, using Fuzzy logic, introduced the concept of linguistic quanti�er torepresent the large number of possible quanti�ers [43]. Zadeh suggested that the semantic of a linguisticquanti�er may be captured by using fuzzy subsets for its representation. He distinguished betweentwo types of linguistic 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 absolute linguistic quanti�ersare closely related to the concept of the count or number of elements. He de�ned these quanti�ers as fuzzysubsets of the non-negative real numbers, R+. In this approach, an absolute quanti�er can be representedby 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. Proportional quanti�ers, suchas most, at least half, may be represented by fuzzy subsets of the unit interval, [0,1]. For any r 2 [0; 1],Q(r) indicates the degree to which the proportion r is compatible with the meaning of the quanti�er itrepresents.

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 relative 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 4, 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"Figure 4: Proportional Fuzzy Linguistic Quanti�ersIn [37, 40], Yager suggested an interesting way to compute the weights of the OWA aggregationoperator using linguistic quanti�ers, which, in the case of a non-decreasing proportional quanti�er Q, isgiven 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 the LOWA operator �, it issymbolized by �Q: In such a way, when a non-decreasing proportional linguistic quanti�er, Q, is used tocompute the weights of the I-LOWA operator, �I , it is symbolized by �IQ:Example of the LOWA operator �Q. Suppose that we want to aggregate by means of the LOWAoperator the same four above labels, fL;ML;H; VHg: If we use the fuzzy linguistic quanti�er "As manyas possible" to calculate the weighting vector, i.e., W = f0; 0; 0:5;0:5g and the aggregation of the labels,working in the similar way, is the following:�(L;ML;H; VH) = [0; 0; 0:5;0:5](VH;H;ML;L) = C4f(0; V H); (0;H); (0:5;ML); (0:5;L)g=ML:In Sections 4 and 5, we explain how to use the LOWA operator for solving a non-homogeneous groupdecision making problem from individual linguistic preference relations according to two types of fuzzymajority:� Fuzzy majority of dominance, used to quantify the dominance that one alternative has over all theothers, according to one expert's opinions.� Fuzzy majority of experts, used to quantify the dominance that one alternative has over all theothers, according to the experts' opinions, considered as a whole.3.2 Weighted linguistic informationIssues of weighted aggregation operators have been studied in a linguistic setting in [4, 17, 38, 42]. Herewe shall use the operators proposed in [17], which act by direct computation on labels. The operatorsare:

1. Linguistic Weighted Disjunction (LWD).2. Linguistic Weighted Conjunction (LWC).3. Linguistic Weighted Averaging (LWA).In the three operators we have on the one hand, the aggregation of power or importance degrees ofopinions, and on the other hand, the aggregation of opinions.Let f(c1; a1); : : : ; (cm; am)g be a set of weighted opinions expressed by a set of experts E = fe1; : : : ; emgto evaluate an alternative xj, where ai shows the opinion of expert ei, assessed linguistically on the labelset S, ai 2 S, and ci the relevance degree of expert ei, assessed linguistically on the label set L, ci 2 L.In the following we consider that S = L for all operators.De�nition of the Linguistic Weighted Disjunction (LWD) operator. The aggregation of set ofweighted individual opinions, f(c1; a1); : : : ; (cm; am; )g; according to the 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 of the Linguistic Weighted Conjunction (LWC) operator. The aggregation of set ofweighted individual opinions, f(c1; a1); : : : ; (cm; am)g; according to the 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):De�nition of the Linguistic Weighted Averaging (LWA) operator. The aggregation of set ofweighted individual opinions, f(c1; a1); : : : ; (cm; am)g; according to the LWA operator is de�ned as(cE ; aE) = LWA[(c1; a1); : : : ; (cm; am)];where the importance degree of the opinion of the group, 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 LC! if f = �Q and g 2 LI! if f = �IQ, being LC! the following linguisticconjunction functions [16, 17]:1. The classical MIN operator: LC!(w; a) =MIN (w; a):

2. The nilpotent MIN operator:LC!(w; a) = � MIN (w; a) if w > Neg(a)s0 otherwise.3. The weakest conjunction:LC!(w; a) = � MIN (w; a) if MAX(w; a) = sTs0 otherwise.and LI! any of the following linguistic implication functions[16, 17]:1. Kleene-Dienes's implication function:LI!(w; a) =MAX(Neg(w); a):2. G�odel's implication function: LI!(w; a) = � sT if w � aa otherwise.3. Fodor's implication function:LI!(w; a) = � sT if w � aMAX(Neg(w); a) otherwise.Where "MAX" stands for maximum operator and "MIN" stands for minimum operator.The LWD operator, the LWC operator, and the LWA operator verify the following axioms: Inde-pendence of alternatives, Commutativity, Positive sensitivity in its weaker form, Neutrality with respectto alternatives, Unrestricted domain, To be an "orand" operator [17]. The ful�lment of these axiomsprovides evidence of rational aggregation of these operators in particular frameworks. In the followingsections, we shall show the use of these linguistic information aggregation operators in the choice processesof alternatives in non-homogeneous group decision making.In the following we shall symbolize as WAO (Weighted Aggregation Operator) any of these operatorsof weighted linguistic information aggregation.4 Choice Degrees of AlternativesAs we said in the introduction, we are going to present several choice processes for non-homogeneousgroup decision making problems, which are guided by several choice degrees. In this section we de�nethe choice degrees of alternatives. There are two:1. Quanti�er Guided Linguistic Dominance Degree.2. Quanti�er Guided Linguistic Non-Dominance Degree.Both express, linguistically, the relevance degree of one alternative in a set of the alternatives and act onthe set of the alternatives to develop the choice processes. They were de�ned in [19] in a more speci�ccontext. Here, we show the canonical generalizations of both degrees, which are based on the conceptof fuzzy majority of the dominance. They are de�ned from a linguistic preference relation, P , as thosepresented in Section 2, using the non-weighted linguistic information aggregation operator LOWA. Weshall apply them in our choice processes on two levels of action:� Individual Level. The linguistic degrees are calculated for each alternative according to the opinionsof one expert, considered individually, i.e., P contains the opinions of one particular expert, andthus, individual degrees are calculated on this level.� Collective Level. Here, the linguistic degrees are calculated for each alternative according to theopinions of the expert group, considered as a whole, i.e., P contains the collective opinions of theexpert group, and thus, collective degrees are calculated on this level.The degrees are described in the following subsections.

4.1 Quanti�er guided linguistic dominance degreeLet X = fx1; : : : ; xng be a set of n alternatives, and let P be a linguistic preference relation assessed inthe label set S, �P : XxX ! S;where �P (xi; xj) = pij 2 S represents the linguistically assessed preference degree of the alternative xiover xj. Then, given an alternative xi 2 X, we de�ne the quanti�er guided linguistic dominance degreeof xi, QGLDDi, according to the following expression:QGLDDi = �Q(pij; j = 1; :::; n; j 6= i);where �Q is the non-weighetd linguistic information aggregation operator LOWA, and Q is a linguisticquanti�er chosen appropriately according to a particular sense of fuzzy majority. In such a way, QGLDDiquanti�es, linguistically, the dominance that one alternative xi has over a fuzzy majority of the remainingalternatives.Once we have obtained the set of the linguistic dominance degrees, fQGLDDig, they may be used toobtain the set of alternatives with maximum linguistic dominance degree, XQGLDD , in this way,XQGLDD = fxijxi 2 X;QGLDDi = maxjfQGLDDjgg;which contains the most relevant alternatives in the set, X.4.2 Quanti�er guided linguistic non-dominance degreeGiven an alternative xi 2 X, we de�ne the quanti�er guided non-dominance degree of xi, QGLNDDi, asfollows, QGLNDDi = �Q(Neg(psji); j = 1; :::; n; j 6= i);where psji represents the degree to which the alternative xi is strictly dominated by the alternative xj,and it is obtained in this way, psji = s0 if pij > pji;or psji = sh 2 S if pji � pij with pji = sl; pij = st and l = t + h:In this case, QGLNDDi quanti�es, linguistically, the degree to which one alternative xi is not dominatedby a fuzzy majority of the alternatives of the set X. This means that if QGLNDDi = � 2 S then thealternative xi is dominated by a fuzzy majority of elements to a degree not higher than �.We should note that when the fuzzy quanti�er represents the statement "all", whose algebraic aggre-gation representation corresponds to the conjunction operator, "min", then this linguistic non-dominancedegree coincides with Orlovski's non-dominated alternative concept, de�ned in a numerical setting in [33]and in a linguistic setting in [25]. So, we say that this quanti�er guided linguistic non-dominance degreeis a generalization of Orlovski's non-dominated alternative concept.Once we have obtained the set of the linguistic non-dominance degrees, fQGLNDDig, they may beused to obtain the set of alternatives with maximum linguistic non-dominance degree, XQGLNDD , in thisway, XQGLNDD = fxijxi 2 X;QGLNDDi = maxjfQGLNDDjgg;which contains the most relevant alternatives in the set, X.4.3 The use of the choice degrees of alternativesThese choice degrees of alternatives are two possible options to rank the alternatives from an expert'sopinions provided by means of a linguistic preference relation. They work from di�erent perspectives.The �rst one works using the dominance that one given alternative has over all ones, and the second oneusing the non-dominance that all alternatives have over one. Both degrees are equally valid and obtainthe best alternatives from the preferences provided by the experts.Given that both degrees are not dual concepts their results do not coincide always. Therefore, some-times, one of them can obtain a more general solution, and then, the application of the another one canbe useful. By this reason, in this paper, we shall look for designing choice process applying both degreestogether, the combined choice processes.

5 Di�erent Approaches to Non-Homogeneous Group DecisionMaking in a Linguistic SettingAssuming the framework presented in Section 2, that is, there is a non-empty and �nite set of alternatives,X = fx1; : : : ; xng, to be analyzed by a non-empty and a �nite set of experts E = fe1; : : : ; emg. For eachexpert ek 2 E, we assume an importance degree, �E(k), assigned by a manager and linguistically assessedin the label set, L, �E : E ! L:Each expert ek provides his/her opinions on X by means of a linguistic preference relation, P k, assessedin the label set, S, �Pk : XxX ! S:We assume that P k is soft-reciprocal and complete without any loss of generality.As is well known, basically two approaches may be considered to develop a choice process in groupdecision making problems. A direct approachfP 1; : : : ; Pmg ! solutionaccording to which, on the basis of the individual preference relations, a solution is derived, and anindirect approach fP 1; : : : ; Pmg ! P ! solutionproviding the solution on the basis of a collective preference relation, P , which is a preference relation ofthe group of individuals as a whole.We have developed choice processes for homogeneous group decision making usig both the indirectapproach [20, 25, 19] and the direct approach [22]. Here, we consider both approaches, but for non-homogeneous groups. We present two direct ways and two indirect ways to develop a choice process innon-homogeneous group decision making problems in a linguistic setting:1. Direct approaches:(a) Dominance Guided Direct Choice Process.(b) Non-Dominance Guided Direct Choice Process.2. Indirect approaches:(a) Dominance Guided Indirect Choice Process.(b) Non-Dominance Guided Indirect Choice Process.Direct and indirect processes are based on linguistic dominance and linguistic non-dominance degrees,applied individually on the alternative set. Both types of processes are developed along three activitystates:� Aggregation State. The goal of this state is to aggregate individual linguistic weighted information.These information units, as will be shown, may be the weighted experts' opinions as well as weighteddominance or weighted non-dominance degrees obtained from the experts' opinions (this is thecase in which we aggregate dominance or non-dominance degrees of one alternative consideringdi�erent experts). This activity state is developed using linguistic weighted information aggregationoperators like those presented in Section 3. When the LWA operator is used, then the linguisticquanti�er Q; chosen to calculate the associated weighting vector W; represents the concept of fuzzymajority of experts.� Exploitation State. The goal of this state is to calculate dominance or non-dominance degrees ofeach alternative from linguistic preference relations (individual or collective).� Selection State. The goal of this state is to �nd out the solution of the alternative(s) according todi�erent degrees.

(Step 3)

SOLUTIONALTERNATIVE(S)

STATE

SELECTION

(Step 1) (Step 2)

Individual

IndividualLinguisticPreferenceRelations

kP

EXPLOITATION

STATE

AggregatedDegrees

STATE

AGGREGATIONDegrees

Figure 5: Ordering of the States in the Direct Approach(Step 3)

SOLUTIONALTERNATIVE(S)

STATE

SELECTION

(Step 1) (Step 2)

IndividualLinguisticPreferenceRelations

kP

STATE

Degrees

STATE

AGGREGATION EXPLOITATIONRelationPreferenceCollective

Collective

Figure 6: Ordering of the States in the Indirect ApproachDepending on the approach considered, these states are applied in one order or another. In a choiceprocess by direct approach they are applied as is shown in Figure 5,and in the indirect approach as is shown in Figure 6.We must denote that in the exploitation state as well as in the aggregation state the concept offuzzy majority is used, but with a di�erent meaning. In the �rst one, because the linguistic degreesare calculated, the fuzzy majority of dominance is used. In the second one, as individual opinions orindividual degrees of di�erent experts are aggregated, the fuzzy majority of experts is used. Therefore,we can use di�erent linguistic quanti�ers in each state.As we said previously, sometimes the result of the choice processes based on the application of onlyone choice degree is a general solution. Due to this reason, our �nal goal is to design choice processesbased on the application of both choice degrees together in order to obtain a more speci�c solution.Concretly, we present two di�erent models combining both degrees:3. Combined Choice Processess.(a) Conjunctive Choice Process.(b) Sequential Choice Process.Both processes can be developed under a direct or an indirect approach.The following subsections are devoted to developing each one of the aforementioned processes. Firstly,we shall show the more elemental processes in both approaches (direct and indirect). Then, we shalldevelop how to combine these elemental processes in the conjunctive and sequential model. And, �nally,we shall discuss application conditions of the choice processes and how to integrate the di�erent modelsto achieve a more complete choice process.

5.1 Direct approaches5.1.1 Dominance Guided Direct Choice Process (DDP)This process is based on the use of the concept of the quanti�er guided linguistic dominance degree(QGLDD). The process works with two types of dominance degrees:� Individual Quanti�er Guided Linguistic Dominance Degree (IQGLDD). This is a dominance degreeof each alternative obtained according to the opinion of each individual. Therefore, this degree isguided by a quanti�er that represents the concept of fuzzy majority of dominance of one alternativeover the remaining alternatives.� Aggregated Quanti�er Guided Linguistic Dominance Degree (AQGLDD). This is the dominancedegree of each alternative obtained according to the opinions of a set of individuals and of theirrespective importance degrees. This degree is guided by a quanti�er that represents the concept offuzzy majority of experts.After �xing two labels set, S and L, and the concepts of fuzzy majority of dominance and fuzzymajority of experts by means of two fuzzy quanti�ers, Q1 and Q2 respectively, the process is describedin the following steps:1. Exploitation State.From each linguistic preference relation of each expert, P k, using the LOWA operator �Q1 , �nd outthe individual quanti�er guided linguistic dominance degree of each alternative xi, called IQGLDDki ,according to this expression:IQGLDDki = �Q1(pkij; j = 1; : : : ; n; j 6= i)with k = 1; : : : ;m; i = 1; : : : ; n.2. Aggregation State.For each alternative xi, calculate the aggregated quanti�er guided linguistic dominance degree, madeup by two components, AQGLDD1i , which contains the dominance degree, and , AQGLDD2i , whichexpresses the power degree of aggregated dominance degree. They are calculated as follows(AQGLDD1i ; AQGLDD2i ) =WAO[(�E(k); IQGLDDki ) k = 1; : : : ;m)]with i = 1; : : : ; n:3. Selection State.Obtain the set of alternatives with the maximum linguistic dominance degree XAQGLDDmax , as followsXAQGLDDmax = fxi 2 XjAQGLDD1i = maxjfAQGLDD1j gg;and the solution set, i.e., those alternatives with the maximum degree. It is obvious that thecomponent, AQGLDD2i , of each one of the alternatives in the solution set is such that,AQGLDD21 = AQGLDD22 = : : : = AQGLDD2n:This may be called, AQGLDD2, and expresses the credibility degree of the solution obtained.This process is shown in Figure 7.5.1.2 Non-Dominance Guided Direct Choice Process (NDDP)This process selects the solution of the alternative(s) according to its respective quanti�er guided linguisticnon-dominance degree (QGLNDD). It is calculated on the basis of Orlovski's generalized concept of non-dominated alternatives extended to a linguistic environment. This process works in the same way as theone above. Having �xed two label set, S and L, and some quanti�ers Q1 and Q2, it follows the plandescribed in the Figure 8, and developed in the following steps:

LINGUISTIC

POWER

DEGREES

Q2Q1

LOWA and fuzzy

dominancemajority of

AGGREGATED

DOMINANCE

DEGREES

I N D I V I D U A L S

majority ofWAO and fuzzy

µE

SOLUTION

STATEEXPLOITATION

STATE STATESELECTIONAGGREGATION

Xmax

AQGLDD

1(AQGLDD , AQGLDD )

2

i i

P IQGLDDi

kk

PREFERENCE

RELATIONS

LINGUISTIC

experts

DEGREES

INDIVIDUAL

DOMINANCE

(k)

GROUP OF EXPERTS

NON-HOMOGENEOUS

M A N A G E R

DIRECT APPROACH BASED ON LINGUISTIC DOMINANCE DEGREEFigure 7: Dominance Guided Direct Choice Process1. Exploitation State.(a) For each linguistic preference relation of each expert, P k, �nd its respective linguistic strictpreference relation, P s;(k).(b) From each linguistic strict preference relation of each expert, P s;(k), using the LOWA oper-ator �Q1 , �nd out the individual quanti�er guided linguistic non-dominance degree of eachalternative xi, called IQGLNDDki , according to this expression:IQGLNDDki = �Q1(Neg(ps;(k)ji ); j = 1; : : : ; n; j 6= i);with k = 1; : : : ;m; i = 1; : : : ; n.2. Aggregation State.For each alternative xi, calculate its aggregated quanti�er guided linguistic dominance degree, madeup by two components, AQGLNDD1i , which contains the aggregated dominance degree, andAQGLNDD2i , which expresses the power degree of the aggregated dominance degree. They arecalculated as follows(AQGLDD1i ; AQGLNDD2i ) =WAO[(�E(k); IQGLNDDki ) k = 1; : : : ;m)]with i = 1; : : : ; n:3. Selection State.Obtain the set of alternatives with the maximum linguistic non-dominance degree XAQGLNDDmax , asfollows XAQGLNDDmax = fxi 2 XjAQGLNDD1i = maxjfAQGLNDD1jggand therefore, the solution set are those alternatives with the maximum degree.5.2 Indirect approaches5.2.1 Dominance Guided Indirect Choice Process (DIP)It is based on the use of a quanti�er guided linguistic dominance degree on the collective preferencerelation which is called Collective Quanti�er Guided Linguistic Dominance Degree (CQGLDD). Before

LINGUISTIC

POWER

DEGREES

Q2Q1

LOWA and fuzzy

dominancemajority of

i

kIQGLNDD

I N D I V I D U A L S

majority ofWAO and fuzzy

µE

SOLUTION

STATEEXPLOITATION

STATE STATESELECTIONAGGREGATION

Xmax2

i i

Pk

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RELATIONS

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experts

(k)

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DIRECT APPROACH BASED ON LINGUISTIC NON-DOMINANCE DEGREE

NONDOMINANCE

DEGREES

INDIVIDUAL

DEGREES

NONDOMINANCE

AGGREGATED

(AQGLNDD , AQGLNDD )1

AQGLNDDFigure 8: Non-Dominance Guided Direct Choice Processcalculating this degree for each alternative, we aggregate all the experts' opinions (all their linguisticpreference relations) in one collective opinion (a collective linguistic preference relation), which representsthe social opinion of the group. From this collective opinion, we obtain the dominance degree for eachalternative xi, CQGLDDi, and by applying it to the alternative set, we obtain the solution set of thealternative(s). This process is shown in Figure 9 and is described in the following steps:1. Aggregation State.From the set of weighted linguistic preference relations, f(�E(1); P 1); : : : ; (�E(m); Pm)g; by meansof the concept of fuzzy majority of experts and the use of a WAO; a weighted collective linguisticpreference relation, (�CE ; PC), is obtained as(�CE ; PC) = WAO[(�E(1); P 1); : : : ; (�E(m); Pm)];where �CE expresses the importance degree of the collective opinion obtained, and it is equal forall preferences values, pcij 2 PC ; with each pcij being obtained by means of the component ofopinions aggregation of the WAO operator chosen. For example, if the LWD (linguistic weighteddisjunction) operator is chosen as WAO, thenpcij =MAX[MIN (�E(1); p1ij); : : : ;MIN (�E(m); pmij )]:2. Exploitation State.From the collective linguistic preference relation, PC, using the LOWA operator �Q1 , �nd out thecollective quanti�er guided linguistic dominance degree for each alternative xi, called CQGLDDi,according to this expression:CQGLDDi = �Q1(pcij ; j = 1; : : : ; n; j 6= i)with i = 1; : : : ; n.3. Selection State.Obtain the set of alternatives with the maximum linguistic dominance degree XCQGLDDmax , as followsXCQGLDDmax = fxi 2 XjCQGLDDi = maxjfCQGLDDjggand the solution set is made up by those alternatives with the maximum degree.

LINGUISTIC

PREFERENCE

RELATIONS

Pk

Q2

majority ofexperts

WAO and fuzzy

Q1

LINGUISTIC

PREFERENCE

RELATIONS

LINGUISTIC

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µcE

INDIRECT APPROACH BASED ON LINGUISTIC DOMINANCE DEGREE

Xmax

SOLUTION

STATEAGGREGATION

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LOWA and fuzzy

dominancemajority of

DEGREES

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COLLECTIVE

(CQGLDD )i

µE(k)

(Pc

)Figure 9: Dominance Guided Indirect Choice Process5.2.2 Non-Dominance Guided Indirect Choice Process (NDIP)This process is based on the use of a quanti�er guided linguistic non-dominance degree, using the collectivequanti�er guided linguistic non-dominance degree of each alternative xi, CQGLNDDi. It is shown inFigure 10 and developed in the following way:1. Aggregation State.Obtain the weighted collective linguistic preference relation, (�CE ; PC), as(�CE ; PC) = WAO[(�E(1); P 1); : : : ; (�E(m); Pm)]:2. Exploitation State.(a) From the collective linguistic preference relation, PC, �nd its respective linguistic strict pref-erence relation, P s.(b) From the linguistic strict preference relation, P s, using the LOWA operator �Q1 , �nd outthe collective quanti�er guided linguistic non-dominance degree for each alternative xi, calledCQGLNDDi, according to this expression:CQGLNDDi = �Q1 (Neg(psji); j = 1; : : : ; n; j 6= i)with i = 1; : : : ; n.3. Selection State.Obtain the set of alternatives with the maximum linguistic non-dominance degree XCQGLNDDmax , asfollows XCQGLNDDmax = fxi 2 XjCQGLNDDi = maxj(CQGLNDDj)gand therefore, the solution set is made up by those alternatives with the maximum degree.5.3 Combined Choice ProcessAs we have commented, when the solution sets found in the elemental direct (indirect) choice processesare formed by various alternatives, it may be suitable to combine the di�erent elemental models in orderto obtain a more speci�c solution set. That is, we can develop processes based on both choice processesof alternatives. We have considered two ways of doing that.

LINGUISTIC

PREFERENCE

RELATIONS

Pk

Q2

majority ofexperts

WAO and fuzzy

Q1

LINGUISTIC

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DEGREES

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µE

c

Xmax

SOLUTION

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CQGLNDDNONDOMINANCE

INDIRECT APPROACH BASED ON LINGUISTIC NON-DOMINANCE DEGREE

(CQGLNDD )i

µE(k)

c(P )Figure 10: Non-Dominance Guided Indirect Choice Process5.3.1 Conjunctive Choice ProcessThe conjunctive choice process consists of applying, in a parallel way, the processes DDP and NDDP(DIP and NDIP), with the solution set, called XQGC , being the intersection of the solution sets obtainedin the aforementioned processes, i.e.,XQGC = XAQGLDDmax \ XAQGLNDDmax (XCQGLDDmax \ XCQGLNDDmax ):This process is shown in Figure 11.We should point out the existence of a problem when it is veri�ed thatXAQGLDDmax \ XAQGLNDDmax = � (XCQGLDDmax \ XCQGLNDDmax = �);then the process does not obtain any solution alternative(s). In such a situation, it is necessary toapply another process, and we propose the sequential choice process, which is presented in the followingsubsection.5.3.2 Sequential Choice ProcessThis model consists of applying each one of the direct (indirect) choice processes in sequence, accordingto a previously established order. There is no criterion to establish an order. When the dominancedegree is �rstly applied then we have a dominance based sequential choice process, and when it is thenon-dominance degree then we call it the non-dominance based sequential choice process. From thesede�nitions, in Figure 12 a dominance based sequential choice degree is shown. It is developed in twosteps:1. Apply the �rst model, DDP (DIP) over X, and obtain XAQGLDDmax (XCQGLDDmax ). If it is veri�ed that#(XAQGLDDmax ) = 1 (#(XCQGLDDmax ) = 1);then End, and this is the solution (# stands for the cardinality of the set). Otherwise continue,using the following step.2. Apply the second model, NDDP (NDIP) over XAQGLDDmax (XCQGLDDmax ), and obtain the solution.XAQGL NDD DDmax � XAQGLDDmax (XCQGL NDD DDmax � XCQGLDDmax ):

XAQGLNDD

max

XAQGLNDD

max

kP µE(k)

CHOICE PROCESS

GUIDEDDOMINANCE

CHOICE PROCESS

NON-DOMINANCEGUIDED

SET OF ALTERNATIVES

X

CONJUNCTION

X S O L U T I O N

DIRECT (INDIRECT) DIRECT (INDIRECT)

X

X AND

QGC

max

AQGLDD

max

AQGLDD

max

DEGREES

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EXPERTS Q2

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1Q

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M A N A G E RGROUP OF EXPERTS

NON-HOMOGENEOUS

FIRST DIRECT (INDIRECT) APPROACH BASED ON BOTH DEGREES

I N P U T D A T A

Figure 11: Conjunctive Direct (Indirect) Choice Process5.4 On the use of the choice processes5.4.1 Dominance based processes versus non-dominance based processesAs we said in the section devoted to the choice degrees, the dominance and non-dominance propertiesare equally valid to obtain the solution set of alternatives. They work under di�erent perspectives, butalways that the experts express their preferences adequately, that is, preferences without inconsistencies,they obtain the best alternatives. When only one of them is applied we can obtain general solutions, andtherefore, we look for applying both degrees together.5.4.2 Direct processes versus indirect processesBoth approaches have di�erent philosophies of working but perfectely valid and applicable. They obtainthe solution composed by the best alternatives according to the preferences of a majority of experts. Thedirect approach is the derivation of a solution without an explicit use of a social linguistic preferencerelation, and the indirect approach is developed through a social linguistic preference relation.In the �rst one a global ranking of the alternatives is established from individual rankings obtainedfor each expert, and in the second one, it is established from the social preference of all experts.5.4.3 Conjunctive processes versus sequential processesAs we have pointed out, a conjunctive process is more restrictive than a sequential one because it ispossible to obtain an empty solution set. Also, we should point out that when it is veri�ed thatXAQGLDDmax \ XAQGLNDDmax 6= � (XCQGLDDmax \ XCQGLNDDmax 6= �);then XQGC = XAQGL NDD DDmax (XCQGL NDD DDmax );

SOLUTION (step 1)

kP

µE(k)

XAQGLDD

maxAQGLDD

#(X )>1max

I N D I V I D U A L S

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EXPERTS Q

SET OF

ALTERNATIVES

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yes

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CHOICE PROCESS

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AT STEP 1 AND 2

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FUZZY MAJORITY OF

2

Q1

max max

max

I N P U T D A T AAT STEP 1 AND 2

SECOND DIRECT (INDIRECT) APPROACH BASED ON BOTH DEGREESFigure 12: Sequential Direct (Indirect) Choice Processthat is, in this case both processes are equivalent. Therefore, we should use both processes following thisprocedure, which we call "complete choice process":1. First, apply the choice conjunctive process.2. If XQGC = � then apply the sequential choice process, otherwise this is the solution set and End.Logically we can use a complete choice process by a direct approach and another one by an indirectapproach.Remark: When we want to solve a group decision making problem, with a view to achieve speci�csolutions, we must apply any of these two complete choice processes.6 ExampleIn this Section, the application of the aforementioned complete models is shown, considering the followingframework:Consider the nine label set, S, presented in Section 2. Assuming a set of four alternatives, X =fx1; x2; x3; x4g, to be analyzed by a non-homogeneous group of four individuals, E = fe1; e2; e3; e4g.whose respective linguistic importance degrees, assuming L = S are:�E(1) = H �E(2) = V H �E(3) = FFML �E(4) = L;and whose linguistic preferences relations over X using the aforementioned nine label set, S, are:P 1 = 2664 � FFML VH V LFFMH � L HV L H � MLV H L MH � 3775P 2 = 2664 � F V H LF � L VHV L H � MLH V L MH � 3775

P 3 = 2664 � F H V LF � V L HL V H � MLVH L MH � 3775P 4 = 2664 � F VH LF � L V HV L H � MLH V L MH � 3775. Using the linguistic quanti�er "At least half" with the pair (0,0.5) for all the operations, and assumingthat Q1=Q2, then the di�erent models proposed are applied as follows:6.1 Complete choice process by direct approachClearly, to apply any complete process, previously, we have to develop the elemental processes. Therefore,subsequently, we calculate the solution according to the two elemental processes, and �nally we applythe combined choice processes according to the algorithm of the complete choice process.6.1.1 Dominance guided direct choice process1. Exploitation State.From linguistic preference relations and using the LOWA operator with W=(0:666; 0:334;0), weobtain the set of the individual quanti�er guided linguistic dominance degrees of each alternativexi for each expert ek, fIQGLDDki g, presented in Table 1.e

e

e

e

x x x x

A l t e r n a t i v e s

E x p e r t s

1

2

3

4

21 3 4

VH

FFML H FFML H

H

VHFFMH

FFML

FMHFFMH

F MH

IQGLDD

VH FFMH

1

4Table 1: Individual Quanti�er Guided Linguistic Dominance Degree2. Aggregation State.The aggregated quanti�er guided linguistic dominance degree of each alternative using the "Lin-guistic Weighted Disjunctive" (LWD) operator is shown in Table 2.x x x x

A l t e r n a t i v e s

21 3 4

H HFFFMH(E) 4AQGLDD

G r o u p ofE x p e r t sTable 2: Aggregated Quanti�er Guided Linguistic Dominance DegreeRemark: Here as well as in all cases of the example where a linguistic weighted informationaggregation operator is used, the component of aggregation of the weights of operator obtainsthe same result, because it is de�ned with the LOWA operator in every case. This result, as wesaid earlier, is considered as the "credibility degree" of the solution obtained. In this example thecredibility degree is AQGLDD2 = V H.

3. Selection State.The set of alternatives with the maximumaggregated quanti�er guided linguistic dominance degreeis: XAQGLDDmax = fx2; x4g6.1.2 Non-dominance guided direct choice process1. Exploitation State.(a) From linguistic preference relations we �nd the respective strict preference relations:P s;(1) = 2664 � V L F V LML � FFML FFMLV L V L � FFFMH V L V L � 3775P s;(2) = 2664 � ML V L V LV L � FFML V LV L V L � V LF L FFMH � 3775P s;(3) = 2664 � FFML H V LV L � L V LV L V L � V LVH F F � 3775P s;(4) = 2664 � V L L V LF � V L MLV L ML � V LFFMH V L FFMH � 3775.(b) From strict preference relations and using the LOWA operator with W=(0:666; 0:334; 0), weobtain the set of the individual quanti�er guided linguistic non-dominance degrees of eachalternative xi for each expert ek, fIQGLNDDki g, as presented in Table 3.e

e

e

e

x x x x

A l t e r n a t i v e s

E x p e r t s

1

2

3

4

21 3 4

FFML

F

VL

MH

VH F F

VHFFMLMHF

F L VH

MHFFML

IQGLNDD1

4Table 3: Individual Quanti�er Guided Linguistic Non-Dominance Degree2. Aggregation State.The aggregated quanti�er guided linguistic non-dominance degree of each alternative using the"Linguistic Weighted Disjunctive" (LWD) operator is shown in Table 4, and the power degree ofthe aggregated non-dominance degree AQGLNDD2 = V H as in the process above.x x x x

A l t e r n a t i v e s

21 3 4

E x p e r t sG r o u p of

VHFHF(E) AQGLNDD4Table 4: Aggregated Quanti�er Guided Linguistic Non-Dominance Degree3. Selection State.The set of alternatives with the maximum aggregated quanti�er guided linguistic non-dominancedegree is: XAQGLNDDmax = fx4g

6.1.3 Conjunctive choice processIn this process, the solution set is given by the following expression:XQGC = XAQGLDDmax \XAQGLNDDmax = fx4g:In this case, the complete choice process �nish with the application of the conjunctive choice process,since #(XQGC) = 1; and therefore, we do not need to apply the sequential choice process.6.2 Complete choice process by indirect approachHere, we work as in the above complete process, i.e., �rtsly, we develop the elemental processes and �nallywe apply the procedure of the complete choice process.6.2.1 Dominance guided indirect choice process1. Aggregation State.From linguistic preference relations and using the LWD operator, we obtain the collective linguisticpreference relation PC : PC = 2664 � MH H FFMLMH � MH MHF FFMH � MLVH MH VH � 3775with �CE = V H.2. Exploitation State.The collective quanti�er guided linguistic dominance degree of each alternative using the LOWAoperator is shown in Table 5.x x x x

A l t e r n a t i v e s

21 3 4

(E) 4OpinionCQGLDDCollective

H VHFFMHMHTable 5. Collective Quanti�er Guided Linguistic Dominance Degree3. Selection State.The set of alternatives with the maximum collective quanti�er guided linguistic dominance degreeis: XAQGLDDmax = fx4g6.2.2 Non-dominance guided indirect choice process1. Aggregation State. As in the DIP, we �nd the same collective linguistic preference relation.2. Exploitation State.(a) From the collective linguistic preference relation we �nd the respective strict preference rela-tion: P s = 2664 � V L FFML V LV L � L V LV L V L � V LFFMH V L MH � 3775 :

(b) From the strict preference relation and using the LOWA operator with W=(0:666; 0:334;0),we obtain the set of the collective quanti�er guided linguistic non-dominance degrees for eachalternative, which are presented in Table 6.x x x x

A l t e r n a t i v e s

21 3 4

(E)OpinionCollective

VH CQGLNDD4VH MLFFMLTable 6: Collective Quanti�er Guided Linguistic Non-Dominance Degree3. Selection State.The set of alternatives with the maximum collective quanti�er guided linguistic non-dominancedegree is: XAQGLNDDmax = fx2; x4g6.2.3 Conjunctive choice processIn this process, the solution set is given by the following expression:XQGC = XAQGLDDmax \XAQGLNDDmax = fx4g:As in the above complete process, we �nd the solution in the step of application of the conjunctivechoice process, since #(XQGC) = 1.7 ConclusionsIn this paper, we have presented several choice processes of alternatives for non-homogeneous groups ina linguistic setting. These processes can allow us to model many real world situations where informationhandled is qualitative and is not equally important. They are based on fuzzy tools, such as, linguisticlabels, fuzzy quanti�ers, the LOWA operator, the LWA operator, etc,. They are useful to design intelligentdecision support systems managing human decision processes with linguistic information.References[1] J.C. Bezdek, B. Spillman and R. Spillman, A Fuzzy Relation Space for Group Decision Theory,Fuzzy Sets and Systems 1 (1978) 255-268.[2] P.P. Bonissone and K.S. Decker. Selecting Uncertainty Calculi and Granularity: An Experimentin Trading-o� Precision and Complexity, in: L.H. Kanal and J.F. Lemmer, Eds., Uncertainty inArti�cial Intelligence (North-Holland, 1986) 217-247.[3] C. Carlsson, D. Ehrenberg, P. Eklund, M. Fedrizzi, P. Gustafsson, P. Lindholm, G. Merkuryeva,T. Riissanen and A.G.S. Ventre, Consensus in Distributed Soft Environments, European Journal ofOperational Research 61 (1992) 165-185.[4] C. Carlsson and R. Full�er, On Fuzzy Screening Systems, Proc. of Third European Congress onIntelligent Technologies and Soft Computing, Aachen, (1995) 1261-1264.[5] C. Carlsson and R. Full�er, Active DSS and Approximate Reasoning, Proc. of Third EuropeanCongress on Intelligent Technologies and Soft Computing, Aachen, (1995) 1209-1216.[6] C. Carlsson and H.J. Sebastian, Active DSS: Theory and Methodology for a New DSS Technology,Proc. of Third European Congress on Intelligent Technologies and Soft Computing, Aachen, (1995)1202-1208.[7] W. Cholewa, Aggregation of Fuzzy Opinions: An Axiomatic Approach, Fuzzy Sets and Systems 17(1985) 249-259.

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