Linguistic Measures Based on Fuzzy Coincidence for Reaching Consensus in Group Decision Making

NoK'rlt - HOI/AND Linguistic Measures Based on Fuzzy Coincidence for Reaching Consensus in Group Decision Making F. Herrera, E. Herrera-Viedma, and J. L. Verdegay Department of Computer Science and Artificial Intelligence, University of Granada, Spain ABSTRACT Assuming a linguistic framework, a model for the consensus reaching problem in heterogeneous group decision making is proposed. This model contains two types of linguistic consensus measures: linguistic consensus degrees and linguistic proximities to guide the consensus reaching process. These measures evaluate the current consensus state on three levels of action: level of the pairs of alternatives, level of the alternatives, and level of the relation. They are based on a fuzzy characterization of the concept of coincidence, and they are obtained by means of several conjunction functions for handling linguistic weighted information, the LOWA operator for aggregating linguistic information, and linguistic quantifiers representing the concept of fuzzy majority. © 1997 Elsevier Science Inc. KEYWORDS: Linguistic modeling, group decision making, linguistic preference relations, consensus degrees. 1. INTRODUCTION Consensus or synthesis consists in combining a data set provided by different information sources with a view to obtaining more elaborate Address correspondence to F. Herrera, Department of Computer Science and A.I., ETS de Ingenieda Inform~tica, University of Granada, 18071 Granada, Spain. Received May 1, 1996; accepted October 1, 1996. International Journal of Approximate Reasoning 1997; 16:309-334 @ 1997 Elsevier Science Inc. 0888-613X/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PII S0888-613X(96)00121-1 310 F. Herrera et al. information [31, 32]. When the information sources provide imprecise information, the use of fuzzy set theory to deal with this type of information is most advisable. A usual situation, in the real world, which presents the appropriate characteristics to apply consensus theory and fuzzy set theory together, is the group decision making (GDM) situation. In a classical GDM situation there is a problem to solve, a set of possible solution alternatives, and a group of two or more experts, who express their opinions about the set of solution alternatives and attempt to reach a collective decision with the maximum possible consensus on this question: what is/are the best solution alternative(s) to the problem?. Many papers on consensus theory applied to GDM make use of Arrow's work [1] as a starting point and a basic guide. Arrow proposed a qualitative setting composed by a set of axioms, which any acceptable consensus tool for GDM should satisfy. Arrow's impossibility theorem was an important result thereof. According to this theorem, it is impossible to aggregate individual preferences into group preference in a completely rational way. This is a problem that disappears in a cardinal setting in a fuzzy context, on introducing preference intensities, which provide additional degrees of freedom to any aggregation model [13, 9]. In a fuzzy context, the application of consensus theory to GDM problems presents two ways to relate to different decision schemata [6]. The first way, called algebraic consensus, consists in establishing a group choice process which obtains a decision scheme as a solution to the GDM problem. The second way, called topologic consensus, consists in establishing a group consensus reaching process, which, guided by means of a measure of closeness among different decision schemata, called the consensus measure, attempts to achieve the maximum possible degree of consensus on solution alternative(s). Both consensus types may be combined in a resolution scheme (see Figure 1). Given that the set of experts initially have diverging opinions, firstly, topologic consensus is applied, and in each step, the degree of existing consensus among experts' opinions is measured. If the moderator thinks that the consensus degree is satisfactory, then algebraic consensus is applied in order to obtain a solution; otherwise, the experts are persuaded to update their opinions. In this way, a GDM process may be defined as a dynamic and iterative process, in which the experts, via the exchange of information and rational arguments, agree to update their opinions until they become sufficiently similar, and then the solution alternative(s) is/are obtained. Here, we shall focus our research on the topologic consensus. As was mentioned earlier, the topologic consensus is guided by means of a consensus measure. Assuming numerical preference relations for providing the experts' opinions, several authors introduced hard consensus measures varying between 0 (no consensus or partial agreement) and 1 (full Consensus in Group Decision Making 311 QUESTION SET OF ALTERNATNE$ TOPOLOGIC CONSENSUS GROUP OF E)~PERTS recommendQtions consensus 1 opinions I MODERAIOR [ COI~nsus measut'e oplnions 1 ALGEBRAIC CO NSENSUS Figure 1. Group decision making process. CONSENSUS $OLUT~N consensus or complete agreement) [2, 3, 28, 29]. However, consensus as a full and unanimous agreement is far from being achieved in real situations, and even if it is, in such a situation, the consensus reaching process could be unacceptably costly. So, in practice, a more realistic approach is to use "softer consensus measures" [24], which assess the consensus degree in a more flexible way, and therefore reflect the large spectrum of possible partial agreements, and guide topologic consensus until widespread agreement (not always full) is achieved among experts. Along this line of reasoning, but in different fuzzy GDM contexts, several alternative consensus measures have been proposed: in a numerical context, i.e., with numerical assessments on the unit interval [0, 1], by Kacprzyk [24], Kacprzyk and Fedrizzi [25, 26], and Fedrizzi, Kacprzyk, and Nurmi [15]; and in a linguistic context, i.e., with linguistic assessments on a preestablished label set S, by Fedrizzi and Mich [14], Mich, Gaio, and Fedrizzi [27], Herrera, Herrera-Viedma, and Verdegay [21, 23], and Bordogna, Fedrizzi, and Pasi [5]. In all these cases, the authors have based their consensus measures on the concept of coincidence, i.e., observing the existing coincidence among experts' opinions. Different coincidence meanings have been considered, some based on strict coincidence, i.e., accepting only the total coincidence or null coincidence cases [24-26, 15, 27, 21, 23], and others on less strict coincidence, i.e., accepting different coincidence degrees [24-26, 15, 14, 5]. Here, in a linguistic context, we propose to use a more flexible idea of the concept of coincidence, i.e., using it as a fuzzy concept. We present fuzzy coincidence as a fuzzy set defined on the set of expert pairs and characterized by closeness observed among their respective opinions. In particular, we assume a heterogeneous linguistic context to introduce the new fuzzy coincidence concept, i.e., we define the gradation of the coincidence degree existing among two experts from a label set, S, used to 312 F. Herrera et al. express the experts' opinions, to a new and more appropriate (preestablished) label set, in order to express the coincidence degrees, G. In this way, we present various ways to measure the closeness observed among experts' opinions. Moreover, we study the fuzzy coincidence among experts on three levels of action: the level of the pairs of alternatives, the level of the alternatives, and the level of the relation. Then, using this new idea of fuzzy coincidence, we further advance our previous consensus models [21, 23] for deriving some new softer linguistic consensus measures, which are applied on the three coincidence levels. All consensus measures are obtained using different conjunction functions to manipulate weighted linguistic information [16], the linguistic ordered weighting averaging (LOWA) operator [18, 22] to aggregate linguistic information, and the linguistic quantifiers [42] representing the fuzzy majority concept. In order to do so, in the next section we present some prior considerations on some consensus measures proposed in the literature with a view to clarifying the contributions in this paper. In Section 3, we present briefly the linguistic setting of the GDM problem considered. In Section 4, we present the new linguistic consensus measures. In Section 5, 6, and 7, we show the derivation model of the consensus measures, and finally, some conclusions are pointed out. 2. BACKGROUND ON CONSENSUS MEASURES As we said at the beginning, in a fuzzy context, several alternative softer consensus measures have been proposed. In this section, we briefly analyze these measures with a view to better clarifying the new developments proposed in this paper. In a numerical context, Kacprzyk [24] presented three numerical consensus measures, which are: • assessed on unit interval, [0, 1]; • developed in a simple GDM context with a homogeneous group of experts (all experts' opinions have the same importance degree) and a homogeneous set of alternatives (all the alternatives have the same relevance degree); • calculated across the global set of the alternatives in a hierarchical pooling process from the experts' opinions, provided by means of the numerical preference relations, and using the fuzzy majority concept represented by a linguistic quantifier [42]; and finally • obtained: (1) the first measure, using a strict idea of the concept of coincidence, that is, establishing a particular pair of alternatives: if the opinions of two experts are equal then they are in agreement (value Consensus in Group Decision Making 313 1), and otherwise they are in disagreement (value 0); (2) the second one, using a less strict idea of the concept of coincidence, that is, establishing a particular alternative pair: if the opinions of two experts are more or less equal according to a degree a (preestablished), then they are in agreement (value 1), and otherwise, they are in disagreement (value 0); and (3) the third one, using another less strict idea of the concept of coincidence represented by a function, s : [0, 1] ~ [0, 1] defined on the closeness between experts' opinions. Kacprzyk and Fedrizzi [25, 26] extended Kacprzyk's measures of GDM contexts with a heterogeneous set of alternatives and a heterogeneous group of experts, respectively. Fedrizzi, Kacprzyk, and Nurmi [15] modified the definition of Kacprzyk and Fedrizzi's measures and calculated them using the ordered weighted averaging (OWA) operator [34]. On the other hand, in a linguistic context, Fedrizzi and Mich [14] presented a new numerical consensus measure, which is: • developed in a homogeneous GDM context with multiple criteria; • calculated for each alternative, independently, from the experts' opinions provided by linguistic labels (not preference relations) by means of computation on a fuzzy representation of linguistic labels (trapezoidal membership functions); and • obtained using a less strict coincidence concept represented by means of a euclidean distance d, which implements the linguistic approximation [30]. Mich, Gaio, and Fedrizzi [27] modified this measure and obtained it by applying a strict coincidence concept, which divided the expert group into subsets according to their evaluations. Herrera, Herrera-Viedma, and Verdegay [21] presented two types of linguistic consensus measures, one to measure the consensus degree and another to measure the closeness between experts opinions. Both are: • assessed on the same label set, S, used to express experts' opinions; • developed in a GDM context with a heterogeneous group of experts and a heterogeneous set of the alternatives with importance and relevance degrees assessed on [0, 1]; • calculated from the experts' opinions, provided by linguistic preference relations, using linguistic quantifiers and a linguistic aggregation operator by direct computation on the labels (the LOWA operator [18, 22]) on three levels of action: preference on the pairs of alternatives, preference on the individual alternatives, and preference on the global set of the alternatives; and • obtained by applying a strict coincidence concept, similar to Mich, Gaio, and Fedrizzi's concept, but according to an average consensus policy, that is, using every subset of experts with over two experts. 314 F. Herrera et al. In their second paper [23], Herrera, Herrera-Viedma, and Verdegay modified their measures to work in a GDM context with heterogeneous groups of experts with importance degrees assessed on S, and homogeneous sets of alternatives. The consensus measures were obtained according to a strict coincidence concept but by means of a strict consensus policy, that is, considering only the subset of experts with maximum cardinality. This consensus model incorporated a new development: it integrated two types of linguistic rationality measures to achieve less distorted consensus solutions. Finally, Bordogna, Fedrizzi, and Pasi [5] presented a linguistic consensus measure, which is: • assessed on the same label set, S, used to express the experts' opinions; • developed in a linguistic GDM context, similar to Herrera, HerreraViedma, and Verdegay's, but with a heterogeneous set of criteria and having linguistic importance degrees assessed on S; • calculated for each alternative independently, from the experts' opinions, provided by linguistic labels, by means of the linguistic version of the OWA operator [35] and considering linguistic quantifiers; and • obtained using a less strict coincidence concept represented by means of a usual distance function d defined directly on S and proposed initially by Herrera, Herrera-Viedma, and Verdegay [21]. Now, we present a consensus model with a structure similar to [21, 23], i.e., with two types of linguistic consensus measures calculated on three levels of action, but with the following peculiarities: • it is designed for GDM situations with heterogenous groups of experts and heterogeneous sets of alternatives 'using linguistic weighting degrees; • it is developed in a heterogeneous linguistic context, i.e., using different linguistic domains to express the opinions, the importance, and relevance degrees, as well as the consensus measures; • its consensus measures are obtained using a fuzzy formulation of the concept of coincidence. 3. LINGUISTIC SETI'ING OF THE GDM PROBLEM As was mentioned earlier, we assume a GDM problem developed in a linguistic context, i.e., the experts use linguistic terms instead of numerical values to express their preferences [10, 12, 18, 22, 30, 35, 40]. We consider finite and totally ordered term sets on [0, 1], S = {si}, i ~ H = {0 ..... T}, with an odd cardinal, in which the middle label represents an uncertainty of "approximately 0.5" and the remaining terms are placed around it symmetrically, as in [4]. Moreover, the term set must have the following Consensus in Group Decision Making 315 characteristics: 1. The set is ordered: s i >_ sj if i >_ j. 2. There is the negative operator: Neg(s i) -- sj such that j -- T - i. 3. Maximization operator: Max(s i, sj) = si if s i >_ sj. 4. Minimization operator: Min(s i, sy) = s i if s i <_ sj. We consider that the semantic of the elements in the term set is given by fuzzy numbers defined on the interval [0, 1], which are described by linear trapezoidal membership functions. This representation is achieved by the 4-tuple (ai, bi, ai, fli). The first two parameters indicate the interval in which the membership value is 1; the third and fourth parameters indicate the left and right width. Example 3.1. The following seven label set, S, verifies the aforementioned properties: MA Maximum (1, 1, .25, 0) 1/34 Very_Much (.75, .75, .15, .25) Mu Much (.6, .6, .1, .15) M Medium (.5, .5, .1, .1) L Little (.4, .4,. 15,. 1) VL Very_Little (.25, .25, .25, .15) MI Minimum (0, 0, 0, .25) In this linguistic context, the mathematical model of the GDM problem considered is the following. Let X = {xl,...,x n} be a heterogeneous, nonempty, and finite set of alternatives to be analyzed by a heterogeneous, nonempty, and finite set of experts E = {el,..., era}. Assuming a label set, V = {vi}, i ~ I = {0 ..... M}, to express importance and relevance degrees, for each alternative, x i ~ X, we suppose that a linguistic relevance degree is defined, la.R(i)E V, from t30 standing for "definitely irrelevant" to v M standing for "definitely relevant," across all the intermediate values. Similarly, for each expert e k E E, we assume that a linguistic importance degree is known, /.rE(k)~ V, assigned by a distinguished person, called the moderator, to each expert e k. Then, each expert e k provides his/her opinions on X as a linguistic preference relation, pk c X >( X, with membership function /zek : X × X ~ S, where /zek(xi, xj) = pk denotes the linguistic preference degree of the alternative x i over xj. We assume, without loss of generality, that pk is reciprocal in the sense that p~ = Neg(pk), and by definition piki = s o (the minimum label in S). Given an expert e k, his importance degree, /zE(k), is interpreted as the degree to which the expert is really a decision maker in relation to the 316 F. Herrera et al. decision problem. And given an alternative xi, its relevance degree, ~R(i), is interpreted as the degree to which the alternative is really an option in relation to the problem domain. EXAMPLE 3.2 Assume the following nine label set V to express the importance and relevance degrees: T Total (1, 1, 0, 0) EH Extremely_High (.98, .99, .05, .01) VH Very_High (.78, .92, .06, .05) H High (.63, .80, .05, .06) M Medium (.41, .58, .09, .07) L Low (.22, .36, .05, .06) VL Very_Low (.1, .18, .06, .05) EL Extremely_Low (.01, .02, .01, .05) N Null (0, O, O, O) Let X = {xl, x2, x3, X 4} be a heterogeneous set of four alternatives, for which the respective linguistic relevance degrees are /ZR(1) = EH, /ZR(2) = M, /xR(3) = VH, /xR(4) = VL. Let E = {el, e 2, e3, e 4} be a heterogeneous group of four experts, for which the respective linguistic importance degrees are ~e(1) =M, ~e(2) = VH, ~E(3) =M, ~e(4) = L. Then, following Example 3.1, linguistic preference relations over X, in this linguistic context, may be considered as: -- VL VM VL] pl= VM -- M M VL L -- VL ' VM L VM -- p2 ~ p3 = M -- VM p4 = _ M VM VL -- VM M VM -- L VM VL] M -- L VL VL M -- VL " VM VM VM -- In the GDM problem, in order to aggregate linguistic labels, we use the LOWA operator [18, 22], which allows us to represent the concept of fuzzy majority in the aggregation processes. The LOWA operator is based on the ordered weighted averaging (OWA) operator defined by Yager [34], and on the convex combination of linguistic labels defined by Delgado et al. [11]. Consensus in Group Decision Making 317 DEFINITION 3.1 Let A = {al,... , am} be a set of labels to be aggregated. Then the LOWA operator d~ is defined as t~(a 1 ..... a m ) = ~'B T = ~m{wk, bk, k = 1,..., m} = WlQ)b 1 ~) (1 - w 1) Q)~C~rn1{ flh, bh, h = 2 .... , m} where W = [w 1 ..... win], is a weighting vector such that (i) w i ~ [0, 1] and (ii) Ei = 1, flh = Wh/E~Wk, h = 2,..., m, and B = {b 1 ..... b m} is a vector associated with A such that B = tr(A) = {a~(1) .... , a,~(,)}, where a~( h < a~(i) Vi < j, with tr being a permutation over the set of labels A. ~m is the convex combination operator of m labels, and if m = 2, then it is defined as ~2{wi,bi, i=l,2}=WlQ)Sj~(1-Wl)Q)si=sk, sj,siES (j>i) such that, k = MIN{T, i + round(w 1 • (j - i))}, where round is the usual rounding operation, and b I = sj, b 2 = s i. If wj = 1 and w i = 0 with i ~ j Vi, then the convex combination is defined as ~m {wi, bi ' i = 1 ..... m} = bj. Other approaches to aggregation of linguistic labels may be found in [4, 11, 30, 35, 36, 38-40]. How to calculate the weighting vector of the LOWA operator, W, is a basic question to decide. Yager proposed in [34, 37] an interesting way to compute the weights of the OWA aggregation operator using linguistic quantifiers [42], representing the concept of fuzzy majority. In our case, we use two types of fuzzy majority: • Fuzzy majority of alternatives, used to quantify the different fuzzy coincidence degrees according to one pair of experts' opinions. • Fuzzy majority of experts, used to quantify the different consensus measures according to every pair of experts' opinions. According to Yager [34, 37] the weights can be obtained by means of the following expression: Q(i) Q(i~nl ) w i--- - - , i= 1,...,n, n where Q is a nondecreasing proportional quantifer represented by the following membership function: ! if r < a, --a Q(r)= if a<r<b, a if r>b with a, b, r ~ [0, 1]. 324 F. Herrera et al. 5.2. Computing Process In this first step of the computing process, for each pair of alternatives, (x i, xj), the different pair linguistic consensus measures are calculated according to the following definitions: DEFINITION 5.2 The pair linguistic consensus degree, PCq, is defined according to this expression: PCij=dgQl(LC-'(tZcq(ekl),rkl),k= l ..... m-1, l=k +1 ..... m), and rkt = dp( lzE(k) , lxe(l)) , with weighting vector of the LOWA operator, w = [0.5, 0.5]. rkl is an averaging importance degree, which represents the importance degree of the coincidence degree of the pair of experts, e~l. It is obtained by means of the LOWA operator th with that weighting vector in order to achieve a mean aggregation of the importance degrees. LC -~ represents a family of connectives, i.e., linguistic conjunction functions [17]. We shall use as linguistic conjunction functions the following t-norms, which are monotonically nonincreasing in the weights w, and satisfy the properties required for any transformation function of the weighted information (a, w) [16, 17]: 1. The classical Min operator: LC( (a, w) = Min(a, w). 2. The nilpotent Min operator: Min(a,w) if w > Neg(a), LCZ" (a, w) = ~ go otherwise. 3. The weakest conjunction: [ Min(a, w) if Max(a, w) = gM, LCS (a, W) [ go otherwise. And ~Q1 is the LOWA operator for which the weighting vector is obtained by means of the linguistic quantifier, 01 , used to represent the concept of fuzzy majority of experts. REMARK 5.1 Note that Definition 5.2 explicitly requires this restriction, G = IT, i.e., that the linguistic domain used to express consensus measures is the same one used to express importance and relevance degrees. This limitation may be bridged if we use a method to transform labels among different linguistic domains, but this is not our goal in this paper. Consensus in Group Decision Making 325 EXAMPLE 5.3 Continuing with the GDM context given in Example 3.2, from importance degrees of experts, for each pair of experts, eke, the averaging importance degrees, rkl ~ V, are calculated, {r12 = H, r13 = M, r14 = M, r23 = H, r24 ----- H, r34 = M}, in which, for example, as /Ze(1) = M = u 4 and /ze(2) = VH = v 6, then r12 --- H = v 5, since 5 = MIN{8, 4 + round((6 - 4) × 0.5)}. Here and in the next examples, we assume the nilpotent Min operator, LC~, to manipulate linguistic weighted information, and as the linguistic quantifier Q1 the quantifier given in Example 3.3, "As many as possible," with the pair (0.5, 1). Then, in this context, from the fuzzy coincidence sets obtained in Example 5.2 and from the previous averaging importance degrees, on each pair of alternatives, (xi, xj), i 4= j, we calculate the pair linguistic consensus degree, PCij, by means of the conjunction function, LCZ', and of the LOWA operator, ~bQ,, with the weighting vector, W = [0, 0, 0, 0.32, 0.35, 0.33]: {PC12 = EL, PCI3 = M, PC14 = M,}, {PC21 --- EL, PC23 = N, PC24 = EL,}, {PC31 = M, PC32 = EL, PC34 = M,}, { PC41 = L, PCa2 = N, PC43 = M,}, in which, for example, PC41 is obtained as PC41 = ~bQ,(LC 2 (T, H), LC~' (H, M), LC Z' (N, M), LC2 (H, H), LC 2 (r, H), LC Z' (r, M)) = L. COMMENT 5.1 In general, the consensus degrees are low. For example, on the pairs of alternatives (x2, x3) and (x 4, x2), there is no consensus among the experts' opinions, and on the set of pairs of alternatives {(Xl, x2), (x2, xl), (Xa, x4), (x3, Xe)} the consensus is too low. Only a maximum consensus degree with a value M is achieved on some pairs of alternatives. However, if we observe the sets of fuzzy coincidences obtained in Example 5.2, their membership functions present, in general, values above the middle value, M, which should result in high consensus degrees. So, from this viewpoint, apparently, there is a contradiction. However, we must not forget that we are working implicitly in a heterogeneous GDM context with different meanings of fuzzy majority. So, this situation sometimes is due to the influence of the chosen conjunction function, and in others, it is due to the influence of the chosen linguistic quantifier. In our case, both the conjunc- 326 F. Herrera et al. tion function, LCS" and the linguistic quantifier "As many as possible" induce a pessimistic influence of the consensus state. So, for example, if LC( ~ is chosen as a conjunction function, then PC12 = M, and similarly, other consensus degrees will be higher. In the same way, if "At least half" is chosen as a linguistic quantifier, maintaining LCf, then PC12 = H. Therefore, in short, we must choose both appropriate conjunction functions and linguistic quantifiers in tune with our consensus idea. Now, similarly, from the fuzzy coincidence on the pairs of alternatives, we define another pair linguistic consensus measure. DEFINITION 5.3 The pair linguistic proximity PPi~ of an expert e k is defined according to this expression: = 4'o,(LC-'(lZc,j(ek,),lx•(,)),, =1 ..... m,, ¢ k), PP, knowing that when tZc,j(e~t) ¢~ Cij then tZc,(e~t) = iXc,(et~). REMARK 5.2 Note that in this definition, as in Definition 5.2, the consensus measure is defined by means of the LOWA operator 4'0' and the conjunction function LC-*, and using the sets of fuzzy coincidence of Definition 5.1, but in this case considering only the importance degrees of the remaining experts and not the averaging importance degrees. Therefore, in this sense, /xe(1) is used as the importance degree given to the coincidence degree observed between the expert analyzed and another expert e~ in the group. EXAMPLE 5.4 As in Example 5.3, but this time assuming the importance degrees given in Example 3.2 instead of averaging importance degrees, on each pair of alternatives, (x/, xj), and for each expert e k, we calculate pair linguistic proximities PPi~ by means of the conjunction function LCf, and the LOWA operator 4'Q, with the weighting vector W = [0, 0.32, 0.68]: 1. Expert el: {PP~e = L, PP13 = M, PP14 = M, PP11 = N, eel 3 = N, PPI 4 = N, PP11 = M, PP~2 = M, PP134 = M, PP~I = M, PPg2 = N, PP~3 = M} 2. Expert e2: {pp22 = L, PP~3 = N, PP214 = M, PP21 = L, PP~3 = N, PP~4 = L, PP21 = M, PPf2 = M, PPf4 = M, pp2 = M, PP422 = N, PP423 = M} 3. Expert e3: { pp32 = L, pp33 = M, pp34 = M, Pp32, = L, pp33 = N, pp34 = L, pp31 = M, pp32 = M, pp34 = M, pp3 = M, PP32 = N, pP433 = M} Consensus in Group Decision Making 327 4. Expert e4: { PP42 = M, pea3 = M, PP44 = M, PP41 = M, PP43 = N, PP44 = M, pp4 = M, PP~2 = N, pp4, = M, pp4 = M, PP442 = N, pp4 = M} For example, pp4 is obtained as pp4 = 4ao2(LC~ (L, M), LC~' (M, VII), LC~ (M, M)) = N. COMMENT 5.2 Logically, as in Example 5.3 and for the same reasons, here, all the experts present very low proximities--in any case, never higher than the middle value M. 6. PHASE 2: WORKING ON THE ALTERNATIVES 6.1. Coincidence Process In this step the concept of fuzzy coincidence among experts' opinions is defined, working on the level of the alternatives. DEFINITION 6.1 The fuzzy coincidence on an alternative, x i, is defined as a fuzzy set C i in the nonfuzzy set of pairs of experts, E 2, namely Ci = {(ekt, tzG(ekt))}, characterized by a membership function iZc~ : E 2 --~ G indicating the coincidence degree between experts e k and et's opinions on pairs of alternatives in which the alternative x i appears: iZci(ekt) = ~bQ2(LC~ ( tZcij(ekt),ri'j),LC-' ( tZcji(ekt),r'ij), j~i,j= 1,...,n), and r;j = c~( tzR(i) , lzR( j)) , with the weighting vector of the LOWA operator given by W = [0.5, 0.5]. r;y is an averaging relevance degree, which represents the relevance degree of the coincidence degree achieved on the pair of altematives (x i, xi). It is obtained in the same way as rkt in Definition 5.2. In this case, the coincidence degree iZc~(ekt) is obtained by means of a LOWA operator for which the weighting vector is calculated by means of the linguistic quantifier QZ used to represent the concept of fuzzy majority of alternatives. The restriction pointed out in Remark 5.1 is applied here too and in the next definitions. 328 F. Herrera et al. EXAMPLE 6.1 Continuing with the GDM context given in Example 3.2, from the relevance degrees of the alternatives, for each pair of alternatives, (xi, x:), the averaging relevance degrees r~: ~ V are ' =VII, ' =EH, ' =H, ' =H, ' =L, ' =M}. {r12 r13 r14 r23 r24 r34 in which, for example, as /zR(1)= EH = v 7 and p~R(2)= VH = v6, we have rt12 = EH = v 7, since 7 = MIN{8, 6 + round((7 - 6) × 0.5)}. Assuming, like Q2, the linguistic quantifier given in Example 3.3, i.e., "At least half," with the pair (0, 0.5), then, from the fuzzy coincidence sets calculated in Example 5.2 and from the above averaging relevance degrees, by means of the LOWA operator ~bO2 with W = [0.33, 0.35, 0.32, 0, 0, 0], and the conjunction function LC~' for each alternative x i its fuzzy coincidence set C i in E 2, is obtained, resulting in C 1 = {(e12 , VH), (el3 , V/-/), (e14 , EH), (e23 , VH), (e24 , VH), (e34 , EH)}, C2 = {(e12, H), (el3 , M), (el4, H), (e23, H), (e24, H), (e34 , H)}, C 3 = {(el2 , n), (el3 , VH), (el4 , H), (e23 , H), (e24 , M), (e34 , H)}, C 4 ~-- {(e12 , n), (el3 , H), (el4 , M), (e23 , n), (e24 , n), (e34 , H)}. For example, /Xc,(e14) is obtained as tZc,(el,) = 6Qz(LC~ (N, H), LC~" (M, L), LC Z" (M, M), LC~ (n, n), LC~" (N, L), LC~" (N, M)) = M. 6.2. Computing Process In this second step, on each alternative, Xg, the different alternative linguistic consensus measures are calculated according to the following definitions: DEFINITION 6.2 The alternative linguistic consensus degree, ACg, is defined according to this expression: AC i = ¢kQl(LC-~(txC(ekl),rkl); k= 1 .... ,m1,/= k + 1 .... ,m). EXAMPLE 6.2 In the same way as we did in Example 5.3, on each alternative, xg, we calculate the alternative linguistic consensus degree, ACg, but this time, considering the aforementioned fuzzy coincidence sets {AC 1 = M, AC 2 = M, AC 3 = M, AC 4 --- M}, Consensus in Group Decision Making 329 in which, for example, AC 4 is obtained as hC 4 = dpQ~(LC-~ (H, H), LC-" (n, M), LC-~ (M, M), LC-~ (H, H), LC-~ (H, H), LC -~ (H, M)) = M COMMENT 6.1 On this level, the effect pointed out in Comment 5.1 also appears, since there is a maximum consensus degree with value M on any alternative, in spite of the fact that some observed coincidence degrees are high. Besides, here the effect of the averaging relevance degrees, used to calculate the coincidence degrees, is included. DEFINITION 6.3 The alternative linguistic proximity APi k of an expert e k is defined according to this expression: APi k = dpQ~(LC-" ( tZc,(ekt),lze(l)),l : 1,...,m,l --/: k), knowing that when i.tc(ekt) ~ C i, then tXci(ekt) = ~c(etk). EXAMPLE 6.3 In the same way we did in Example 5.4, here, on each alternative x i for each expert e k, we calculate the alternative linguistic proximity AP~ k, but this time, considering the aforementioned fuzzy coincidence sets {AP~ = L, AP~ = N, AP 1 = EL, AP41 = EL}, (AP? = L, = EL, AP 3 = EL, = EL}, {AP3a = L, AP~ = N, AP~ = EL, AP 2 = EL), {AP~ = M, AP 4 = M, AP 4 = M, AP 4 = EL}. Here, for example, AP 4 is obtained as AP 4 = d~Q~(LC -~ (EH, U), LC " (VH, VII), LC -' (EH, U)) = U. 7. PHASE 3: WORKING ON THE RELATION 7.1. Coincidence Process In this last phase, the concept of fuzzy coincidence among experts' opinions is defined, working on the level of the relation. DEFInrrtON 7.1 The fuzzy coincidence on the relation is defined as a fuzzy set C = {(ekt, lxC(ekt))} in the nonfuzzy set of pairs of experts, E 2, characterized by a membership function, ix c : E 2 ~ G, indicating the coincidence 330 F. Herrera et al. degree between experts e k and el's opinions on all the pairs of alternatives: I~C(ek,) = q~a2(LC-~(lXc,(ek,), lxn(i)),i= 1 ..... n). REMARK 7.1 In this definition, the coincidence degree, IxC(ekl), is obtained as in Definition 6.1, i.e., by means of the LOWA operator ~bo2 and of the conjunction function LC -~ , but in this case, using relevance degrees tZn(i) instead of averaging relevance degrees r~j representing the relevance degree of the coincidence degree achieved on each alternative X i • EXAMPLE 7.1 Assuming the relevance degrees given in Example 3.2, as was done in Example 6.1, for overall alternatives X, its set of fuzzy coincidence, C, in E 2 is obtained from relevance degrees and the fuzzy coincidence sets obtained in Example 6.1, by means of the LOWA operator the2 with the weighting vector W = [0.5, 0.5, 0, 0] and the same conjunction function, LC{, resulting in C = {(el2, I/H), (el3,1,'7-/), (e14 , VH), (e23 , l/H), (e24 , H), (e34 , VH)}. For example, /.tc(el4) is obtained as /zc(e14) = ~bQ2(LC-~ (EH, EH), LC ~ (H, M), LC -~ (H, VH), LC -" (M, VL)) = VH. 7.2. Computing Process In this last computing process, on overall opinions, i.e., on the relation, the relation linguistic consensus measures are calculated according to the following definitions: DEFINITION 7.2 The relation of linguistic consensus degree, RC, is defined according to this expression: RC = q~QI(LC -~ (i~c(ekt), rkl), k = 1,..., m - 1, l = k + 1 ..... m). DEFINITION 7.3 The relation of linguistic proximity Rpk of an expert e k is defined according to this expression: Re k = dpQI(LC -~ (t~c(ekl), I~E(I)), l = 1,..., m, I :g k), knowing that when tzc(ekt) q~ C then lzc(ekt) = I~c(etk). EXAMPLE 7.2 Working as in Examples 6.2 and 6.3, then RC = M, Consensus in Group Decision Making 331 and respectively. {~1 =L,~2=EL,~3=L,~4=M}, COMMENT 7.1 In view of the resulting consensus measures, which indicate a medium consensus current state, the consensus reaching process can stop or continue. In the second case, then, the moderator has to make some of the following considerations: • Advise all the experts to change their opinions on pairs of alternatives, and in particular on the set of pairs of alternatives {(X 1, X2),(X 2, Xl),(X 2,x3),(x 2, X4),(X 3,x2), (X 4,xl), (X 4, X2)}. • Advise the experts {el, e2, e3} to diminish their disagreement among them. • Decide on the usefulness of maintaining the pessimistic effect of the chosen linguistic quantifier to calculate the linguistic consensus measures and of the chosen conjunction function in the following step of the consensus measuring process. 8. CONCLUSIONS A consensus model is proposed in order to develop a consensus reaching process in a GDM context with heterogeneous groups of experts and a heterogeneous set of alternatives in a heterogeneous linguistic framework. This model contains two types of consensus measures to guide the consensus process from two different perspectives. The first type, called linguistic consensus degrees, studies the consensus state from a global perspective, considering all the experts, and the second type, called linguistic consensus proximity, studies the consensus state from a particular perspective, i.e., considering particular experts. Furthermore, all the types of measures are applied on three level of actions for representing the current consensus state. Therefore, the consensus model presents three consensus measures of each type. The main features of the consensus model are the following ones: • its measures are based on a fuzzy characterization of the concept of coincidence defined from an ad hoc closeness table; • it uses different linguistic domains to express the opinions and the consensus measures; • its measures are calculated by means of the LOWA operator, several conjunction functions, and linguistic quantifiers representing the concept of fuzzy majority; 332 F. 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