DECSAI Department of Computer Scienceand Arti�cial IntelligenceA Sequential Selection Processin Group Decision Makingwith a Linguistic Assessment ApproachF. Herrera, E. Herrera-Viedma, J.L. Verdegay

Technical Report #DECSAI-94103February, 1994International Journal of Information Sciences 80 (1995) 1-17.ETS de Ingenier��a Inform�atica. Universidad de Granada.18071 Granada - Spain - Phone +34.58.244019, Fax +34.58.243317

A Sequential Selection Processin Group Decision Makingwith a Linguistic Assessment ApproachF. Herrera, E. Herrera-Viedma, J.L. VerdegayDept. of Computer Science and Arti�cial IntelligenceUniversity of Granada, 18071 - Granada, Spaine-mail: herrera,viedma,verdegay@robinson.ugr.esAbstractIn this paper a Sequential Selection Process in Group Decision Making underlinguistic assessments is presented, where a set of linguistic preference relationsrepresents individuals preferences. A collective linguistic preference is obtained bymeans of a de�ned linguistic ordered weighted averaging operator whose weightsare chosen according to the concept of fuzzy majority, speci�ed by a fuzzy linguisticquanti�er. Then we de�ne the concepts of linguistic nondominance, linguistic dom-inance and strict dominance degrees as parts of the Sequential Selection Process.The solution alternative(s) is obtained by applying these concepts.Keywords: Group decision making, linguistic labels, linguistic preferences, fuzzy ma-jority, fuzzy linguistic quanti�ers, linguistic nondominance degree, linguistic dominancedegree, strict dominance degree.1. IntroductionA group decision making process can be de�ned as a decision situation in which(i) there are two or more individuals that di�er in their preferences (value systems)but have the same acces to information, each of them characterized by his or her ownperceptions, attitudes, motivations, and personalities, (ii) who recognize the existenceof a common problem, and (iii) attempt to reach a collective decision [2].The use of preference relations is normal in group decision making. Moreover, sincehuman judgments including preferences are often vague, fuzzy sets plays an importantrole in decision making. Several authors have provided interesting results on groupdecision making or social choice theory with the help of fuzzy sets. They have proventhat fuzzy sets provided a more exible framework for the discussion of group decisionmaking [8, 10, 11, 12, 13, 14, 16, 18].

In a fuzzy environment it is supposed there exists a �nite set of alternatives X =fx1; : : : ; xng as well as a �nite set of individuals N = f1; : : : ; mg, and each individualk 2 N provides his fuzzy preference relation on X , i.e., pk � XxX, and �Pk (xi; xj)denotes the degree of preference of alternative xi over xj , �Pk (xi; xj) 2 [0; 1]. In thisframework to make decisions consists to choice one or more alternatives of mentionedalternatives set according to individuals' fuzzy preference relations.Sometimes, however, an individual could have vague information about the prefer-ence degree of alternative xi over xj and cannot estimate his preference with an exactnumerical value. Then a more realistic approach may be to use linguistic assessmentsinstead of numerical values, i.e., by supposing that the variables (preference relations)which participate in the problem are assessed by means of linguistic terms [7, 20, 4, 6,9, 15]. A scale of certainty expressions (linguistically assessed) would be presented tothe individual, who could then use it to describe his degree of certainty in a preference.Assuming a set of alternatives or decisions, the basic question is how to relate to dif-ferent decision schemata. According to [3] there are (at least) two possibilities, a groupselection process and a consensus process. The �rst, a calculation of some mean valuedecision schema of a set of decisions, would imply the choice of an algebraic consensusas a mapping l : DxD ! D, whereas the second, the measure of distance betweenschemata, could be called topological consensus involving a mapping k : DxD ! L,where L is a complete lattice.Here, we shall focus on the �rst possibility, for developing a Group Choice Processunder linguistic preferences.Assuming a set of linguistic preferences representing individuals preferences, thegroup choice process develops according to the following scheme: a linguistic orderedweighted averaging (LOWA) operator is de�ned for linguistic labels, based on the or-dered weighted averaging operator [19], and the convex combination of linguistic labels[5]. The concepts of fuzzy majority, represented by means of linguistic quanti�ers andthe LOWA operator, are used in order to obtain a collective linguistic preference. Fi-nally, a sequential selection process acting on the collective preference relation is de�nedaccording to the following two steps:1.- Using the concept of nondominated alternatives [17] for de�ning a nondominancelinguistic degree and obtaining the set of maximal nondominated alternatives from thecollective linguistic preference.2.- De�ning the concepts of linguistic dominance degree and strict dominance de-gree with linguistic labels, and applying them to the set of maximal nondominatedalternatives for obtaining the best alternatives.

Graphically, FormalRepresentation

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ConceptFuzzy Majority LinguisticFuzzy

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SEQUENTIAL SELECTION P R O C E S S

Aggregation of

Non-dominated

Non-dominatedFig. 1. Group Choice ProcessThe aim of this paper is to present the group choice process using a sequentialselection process. To do so, Section 2 shows the linguistic approach in group decisionmaking, Section 3 presents the linguistic ordered weighted averaging operator, Section 4shows how to obtain the collective linguistic preference relation under a fuzzy majority,Section 5 is devoted to developing the sequential selection process, Section 6 presentsexamples, and at the end, some conclusions shall be pointed out.2. The Linguistic Approach in Group Decision MakingThe linguistic approach considers the variables which participate in the problemassessed by means of linguistic terms instead of numerical values, [21]. This approachis appropriate for a lot of problems, since it allows a representation of the experts'information in a more direct and adequate form, whether they are unable of expressingthat with precision.A linguistic variable di�ers from a numerical variable in that its values are notnumbers, they are words or sentences in a natural or arti�cial language. Since words,in general, are less precise than numbers, the concept of a linguistic variable serves thepurpose of providing a means of approximated characterization of phenomena which aretoo complex or too ill-de�ned to be amenable to description in conventional quantitativeterms.A linguistic variable is associated with two rules:1. a syntactic rule, which may take the form of a grammar for generating the namesof the values of the variable, and2. a semantic rule, which de�nes an algorithmic procedure for computing the mean-ing of each value.

De�nition [21].- A linguistic variable is characterized by a quintuple (H,T(H),U,G,M)in which H is the name of the variable; T(H) (or simply T) denotes the term set ofH, i.e.,the set of names of linguistic values of H, with each value being a fuzzy variable denotedgenerically by X and ranging across a universe of discourse U which is associated withthe base variable u; G is a syntactic rule (which usually takes the form of a grammar)for generating the names of values of H; and M is a semantic rule for associating itsmeaning with each H, M(X), which is a fuzzy subset of U. Usually, the semantic of theelements of the term set is given by fuzzy numbers de�ned on the [0,1] interval, whichare described by membership functions.Provided that the linguistic assessments are just approximate ones given by theexperts or decision-makers, we can consider that linear trapezoidal membership func-tions are good enough to capture the vagueness of those linguistic assessments, sinceobtaining more accurate values may be impossible or unnecessary.Therefore, we need a term set de�ning the uncertainty granularity, i.e. the �nestlevel of distinction among di�erent quanti�cations of uncertainty. The elements of theterm set determine the granularity of the uncertainty. In [1] the use of term sets wasstudied with odd cardinals, representing the middle term a probability of "approxi-mately 0.5", the remaining being terms placed symmetrically around it and the limitof granularity 11 or with no more than 13.Let S = fsig; i 2 H = f0; : : : ; Tg be a �nite and totally ordered term set on [0,1] inthe usual sense [1, 4 , 21]. Any label si represents a possible value for a linguistic realvariable, i.e., a vague property or constraint on [0,1]. We consider the term set with anodd cardinal, where the middle label represents an uncertainty of "approximately 0.5"and the remaining terms are placed symmetrically around it. Moreover, we require thefollowing properties for the term set:1) The set is ordered: si � sj if i � j.2) The negation operator is de�ned: 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 .Since aggregation of uncertainty information is a recurrent need in the decisionprocess, combinations of linguistic values are needed. Two main di�erent approachescan be found in order to aggregate and compare linguistic values: The �rst acts by directcomputation on labels [5], and the second uses the associated membership functions[21, 1].Most available techniques belong to the latter kind, however, the �nal results ofthose methods are fuzzy sets which do not correspond to any label in the original

term set. If one �nally wants to have a label, then a "linguistic approximation" isneeded [21, 22, 1, 7]. The process of linguistic approximation consists of �nding a labelwhose meaning is the same or the closest (according to some metric) to the meaningof an unlabelled membership function generated by some computational model. Asimpli�ed solution is the following. For each element of the term set and for theunlabelled membership function representing the result of some arithmetic operation,two features are extracted: the �rst moment of the distribution and the area beneaththe curve. A weighted Euclidean distance, where the weights re ect the relevance ofthe two parameters in determining semantic similarity, provides the metric required toselect the element of the term set that most closely represents the result. There areneither a general criterion to evaluate the goodness of an approximation nor a generalmethod for associating a label to a fuzzy set, so that speci�c problems may require todevelop taylored methods.Consider the following nine linguistic term set with the associated semantic, [1],(the �rst two paremeters indicate the interval in which the membership value is 1.0;the third and fourth parameters indicate the left and right width of the distribution):C Certain (1; 1; 0; 0)EL Extremely Likely (:98; :99; :05; :01)ML Most Likely (:78; :92; :06; :05)MC Meaningful Chance (:63; :80; :05; :06)IM It May (:41; :58; :09; :07)SC Small Chance (:22; :36; :05; :06)V LC V ery Low Chance (:1; :18; :06; :05)EU Extremely Unlikely (:01; :02; :01; :05)I Impossible (0; 0; 0; 0)and graphically:0.0 1.00.5Fig. 2. Distribution of the nine linguistic term setFormally speaking, it seems di�cult to accept that all individuals should agree on thesame membership function associated to linguistic terms, and therefore there are not

any universality distribution concepts. For example, the two following close perceptionsfor the evaluationexcellentgood excellentgood

10

(1) (2)

0 1Fig. 3. Di�erent Distribution Concepts.could be considered.It is well known and accepted that the tuning of membership functions is a crucialissue in process control. In our context, we consider an environment where expertscan discriminate perfectly the same term set under a similar conception, taking intoaccount that the concept of a linguistic variable serves the purpose of providing a meansof approximated characterization of imprecise preference information.On the other hand, we shall focus on the second approach, which is independent ofthe semantic of the term set, considering a similar discrimination of the experts. Wepresent an aggregation operator of linguistic labels by direct computation on labels,which is based on the ordered weighted averaging (OWA) operator, [19], and the convexcombination of linguistic labels de�ned by Delgado et al. [5]. We call it a linguisticordered weighted averaging (LOWA) operator.3. The Linguistic Ordered Weighted Averaging OperatorA mapping F , F : [0; 1]n ! [0; 1]is called an OWA operator of dimension n if associated with F is a weighting vectorW;W = [w1; : : : ; wn], such that, i)wi 2 [0; 1], ii)�iwi = 1; and F (a1; : : : ; an) = w1 � b1+w2 � b2 + : : :+ wn � bn, where bi is the i-th largest element in the collection a1; : : : ; an.Denoting B as the vector consisting of the arguments of F put in descending order,F (a1; : : : ; an) = W �BTprovides an aggregation type operator that always lies between the "and" and the "or"aggregation. Its properties are presented in [19] and its �rst use in group decisionmaking in [8].This operator can be extended to linguistic arguments using the convex combinationof linguistic labels de�ned in [5]. In fact, let M be a collection of linguistic labels,

pk 2 M; k = 1; : : : ; m, and assume pm � pm�1 � : : : � p1 without loss of generality.For any set of coe�cients f�k 2 [0; 1]; k = 1; 2; ::;m;��k = 1g the convex combinationof these m generalized labels is the one given byCf�k; pk; k = 1; : : : ; mg = �1 � p1 � (1� �1)�Cf�h; ph; h = 2; : : : ; mg�h = �h= nX2 �k; h = 2; : : : ; m:In [5] the aggregation of labels was de�ned by addition, the di�erence of generalizedlabels, and the product by a positive real number over a generalized label space S,based on S, i.e., the Cartesian product S = SxZ+, with the basic label set S =f(si; 1); i 2 Hg. In our context, all the operations are carried out over the basic setS. Brie y, the result of the expression � � sj � (1� �) � si; j � i, is the sk such thatk = minfT; i + round(� � (j � i))g.An example to clarify of this operation is the following. Suppose the term set:S = fs8 = C; s7 = EL; s6 =ML; s5 =MC;s4 = IM; s3 = SC; s2 = V LC; s1 = EU; s0 = Igand � = 0:4, 1� � = 0:6SC VLC C ELML IM IM EL EL� = 0:4 I VLC EU MC IMIM SC SC ML MLVLC SC VLC ML MCwhere, for example:k11 = minf8; 3+ round(0:4 � (6� 3))g = 4 (IM)k21 = minf8; 0+ round(0:6 � (3� 0))g = 2 (VLC)Therefore, the LOWA operator can be de�ned asF (a1; : : : ; am) = W �BT = Cfwk; bk; k = 1; : : : ; mg == w1 � b1 � (1� w1)�Cf�h; bh; h = 2; : : : ; mgwhere �h = wh=�m2 wk; h = 2; : : : ; m, and B is the associated ordered label vector. Eachelement bi 2 B is the i-th largest label in the collection a1; : : : ; am.

4. The Collective Linguistic Preference Relation Under aFuzzy MajoritySuppose we have a set of n alternatives X = fx1; : : : ; xng and a set of individualsN = f1; : : : ; mg. Each individual k 2 N provides a preference relation linguisticallyassessed in the term set S, �P k : XxX ! S;where �P k (xi; xj) = pkij 2 S represents the linguistically assessed preference degree ofthe alternative xi over xj . We assume that pk is reciprocal in the sense, pkij = Neg(pkji),and by the de�nition pkii = None (the minimum label in S).As is now known, basically two approaches may be considered. A direct approachfP 1; : : : ; Pmg ! solutionaccording to which, on the basis of the individual preference relations, a solution isderived, and an indirect approachfP 1; : : : ; Pmg ! P ! solutionproviding the solution on the basis of a collective preference relation, P , which is apreference relation of the group of individuals as a whole.Here we consider the indirect derivation, and hence we must derive a collectivelinguistic preference relation.First, we introduce the concept of a fuzzy quanti�er [23], used in order to specifythe fuzzy majority concept as proposed Kacprzyk [10]. The fuzzy linguistic quanti�erswere introduced by Zadeh (Zadeh 1983). Linguistic quanti�ers are typi�ed by termssuch as most, at least half, all, as many as possible and assuming a quanti�er Q is afuzzy set in [0,1]. Zadeh distinguished between two types of quanti�ers, absolute andproportional or relative. Absolute quanti�ers are used to represent amounts that areabsolute in nature. These quanti�ers are closely related to the concepts of the countingof the number of elements. Zadeh suggested that these absolute quanti�ers values canbe represented as fuzzy subsets of the non-negative real numbers, R+. In particular,he suggested that an absolute quanti�er can be represented by a fuzzy subset Q, wherefor any r 2 R+, Q(r) indicates the degree to which the value r satis�es the conceptrepresented byQ. And, relative quanti�ers represent proportion type statements. Thus,if Q is a relative quanti�er, then Q can be represented as a fuzzy subset of [0; 1] suchthat for each r 2 [0; 1], Q(r) indicates the degree to which r portion of objects satis�esthe concept denoted by Q.

An absolute quanti�er satis�es: Q(0) = 0;9k such that Q(k) = 1A relative quanti�er, Q : [0; 1]! [0; 1];satis�es: Q(0) = 0;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 relative quantifer 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 relative quanti�ers are shown in �gure 4,1

0 0.3 0 0.5 0 0.5 1 xxx0.8

1 1

"Most" "At least half" "As many as possible"Fig. 4. Linguistic quanti�erswhere the parameters, (a; b) are (0:3; 0:8), (0; 0:5) and (0:5; 1), respectively.By means of the concept of fuzzy majority speci�ed by a fuzzy linguistic quanti�erand the use the LOWA operator, the collective preference relation, P , is obtained asP = F (P 1; : : : ; Pm)

with pij = F (p1ij ; : : : ; pmij ) and the weight vector, W , obtained from the non-decreasingfuzzy linguistic quanti�er representing the fuzzy majority over the individuals [19].Yager computes the weights wi of the aggregation from the function Q describingthe quanti�er [19]. In the case of an absolute proportional quanti�erwi = Q(i)�Q(i� 1); i = 1; : : : ; m;and in the case of a relative proportional onewi = Q(i=m)� Q((i� 1)=m); i= 1; : : : ; m:5. The Sequential Selection ProcessAs we said earlier, the sequential selection process acts on the collective preferencerelation in two steps:1. Using the concept of non-dominated alternatives [17] for de�ning the conceptof linguistic non-dominance degree and obtaining the set of maximal non-dominatedalternatives from the collective linguistic preference.2. De�ning the concepts of dominance and strict dominance with linguistic labels,and to applying them to the set of maximal non-dominated alternatives for obtainingthe best alternative.5.1. Linguistic Non-dominace DegreeSuppose the linguistic collective preference P = (pij); i; j = 1; : : : ; n, then let P s bea linguistic strict preference relation �P s(xi; xj) = psij such that,psij = None if pij < pji;or psij = sk 2 S if pij � pji with pij = sl; pji = st and l = t + k:The linguistic non-dominance degree of xi is de�ned as�ND(xi) = Minxj2X [Neg(�P s(xj ; xi))]where the value �ND(xi) is to be meant as a linguistic degree to which the alternativexi is not dominated by any of the elements in X.Finally, a set of maximal non-dominated alternatives, XND � X , is obtained asXND = fx 2 X = �ND(x) = Maxy2X [�ND(y)]g:Therefore, aggregating the knowledge of the experts, XND is selected as the set ofpreferred alternatives in our choice process.

5.2. Linguistic Dominance DegreeWe de�ne a linguistic degree which acts on the alternatives of XND,LDD(xi) = FQi6=j (pij);where FQ is a LOWA operator whose weights are de�ned the using relative quantiferQ, and whose components are the elements of the corresponding row of P , i.e., for xithe set of n � 1 labels fpij = j = 1; : : : ; n and i 6= jg.This measure allows us to de�ne the set of non-dominated alternatives with maxi-mum linguistic dominance degree:XLDD = fx 2 XND = LDD(x) = Maxy2XND [LDD(y)]g:5.3. Strict Dominance DegreeWe de�ne this degree as a real degree which acts on the alternatives of XLDD ,SDD(xi) = Q( rin� 1);where ri = cardfpiq 2 S = piq > pqig:Thus, we obtain the �nal solution to the selection process as the following set ofalternatives, XSDD = fx 2 XLDD = SDD(x) = Maxy2XLLD [SDD(y)]g:Remark: The last two dominance degrees use the concept of fuzzy majority in theirde�nition represented by a fuzzy linguistic quanti�er.6. ExamplesExample 1Consider the above nine linguistic term set (�gure 2) and an associated semantic,and four individuals whose linguistic preferences using the above nine element term setare: P 1 = 2666664 � SC MC V LCMC � IM IMSC IM � V LCML IM ML � 3777775P 2 = 2666664 � IM IM VLCIM � MC IMIM SC � V LCML IM ML � 3777775

P 3 = 2666664 � IM MC IIM � ML IMSC V LC � V LCC IM ML � 3777775P 4 = 2666664 � IM MC SCIM � IM SCSC IM � V LCMC MC ML � 3777775Using the linguistic quanti�er "At least half" with the pair (0:0; 0:5), and the corre-sponding LOWA operator with W = (0:5; 0:5; 0; 0), the collective linguistic preferenceis: P = 2666664 � IM MC V LCIM � MC IMSC IM � V LCEL IM ML � 3777775Next, we apply the sequential selection process: The linguistic strict preferencerelation is P s = 2666664 � I V LC II � EU II I � IMC I IM � 3777775and the linguistic non-dominance degree�ND(x1; x2; x3; x4) = [SC;C; IM;C] :The set of maximal non-dominated alternatives, XND, is:XND = fx2; x4gAs we have obtained a non-dominated alternative set with more than one element,then we apply the linguistic dominance degree on the alternatives of XND. To do so,we use the same linguistic quanti�er, obtaining:(LDD(x2); LDD(x4)) = [ML;EL] :therefore, XLDD = fx4gSince XLDD has only one alternative, this is the best alternative for the group selectionprocess.

Example 2Consider the following thirteen linguistic term set with an associated semantic, [1]:C Certain (1; 1; 0; 0)EL Extremely Likely (:98; :99; :05; :01)V HC V ery High Chance (:87; :96; :04; :03)ML Most Likely (:78; :92; :06; :05)HC High Chance (:75; :87; :04; :04)MC Meaningful Chance (:63; :80; :05; :06)L Likely (:53; :69; :09; :12)IM It May (:41; :58; :09; :07)SC Small Chance (:22; :36; :05; :06)V LC V ery Low Chance (:1; :18; :06; :05)NL Not Likely (:05; :15; :03; :03)EU Extremely Unlikely (:01; :02; :01; :05)I Impossible (0; 0; 0; 0)and graphically:0.0 1.00.5Fig. 5. Distribution of linguistic term setConsider the same four individuals whose linguistic preferences, using the abovethirteen element term, are:P 1 = 2666664 � SC HC NLHC � MC MCSC IM � V LCVHC IM ML � 3777775P 2 = 2666664 � IM MC V LCMC � HC IMIM SC � NLML MC VHC � 3777775P 3 = 2666664 � IM HC IMC � ML IMSC V LC � V LCC MC ML � 3777775P 4 = 2666664 � IM HC SCMC � IM SCSC MC � NLHC HC VHC � 3777775

Note that these preferences represent the same problem as before (example 1), butwith a �nest granularity in the linguistic term set.As in the earlier example, the linguistic quanti�er "At least half", and the corre-sponding LOWA operator are used in order to obtain the collective linguistic preference:P = 2666664 � IM HC VLCMC � HC LSC L � V LCEL MC VHC � 3777775The linguistic strict preference relation isP s = 2666664 � I SC INL � NL II I � IHC EU MC � 3777775and the linguistic non-dominance degree is:�ND(x1; x2; x3; x4) = [SC;EL; IM;C] :The set of maximal non-dominated alternatives is:XND = fx4gIn the �rst step, the sequential selection process has found the best alternative, i.e., x4.This can be explained by the �nest granularity of the second term set.ConclusionsIn this paper we have presented a sequential decision process in group decisionmaking where the preferences of the individuals are represented by linguistic prefer-ence relations. The sequential decision process has been based on the concepts of fuzzymajority, fuzzy linguistic quanti�ers, linguistic ordered weighted averaging operators,non-dominance and dominance degree.We have developed one of the two possibilities in a decision schema, the groupselection process. The second possibility, a consensus process under a linguistic assess-ment approach ought be developed in order to obtain a linguistic consensus degree andto cooperate in the solution of the group decision problem. This is a problem to bediscussed in future work.Acknowledgments: We would like to express our gratitude to the referees, whosecomments and suggestions helped to improve the previous version of the paper.

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