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A satisfactory-oriented approach to multiexpert decision-making with linguistic assessments

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This paper proposes a multiexpert decision-making (MEDM) method with linguistic assessments, making use of the notion of random preferences and a so-called satisfactory principle. It is well known that decision-making problems that manage preferences
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  184 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 35, NO. 2, APRIL 2005 A Satisfactory-Oriented Approach to MultiexpertDecision-Making With Linguistic Assessments Van-Nam Huynh and Yoshiteru Nakamori  , Member, IEEE   Abstract— This paper proposes a multiexpert decision-making(MEDM) method with linguistic assessments, making use of thenotion of random preferences and a so-called  satisfactory principle .It is well known that decision-making problems that manage pref-erences from different experts follow a common resolution schemecomposed of two phases: an  aggregation phase  that combines theindividual preferences to obtain a collective preference value foreach alternative; and an  exploitation phase  that orders the collec-tive preferences according to a given criterion, to select the bestalternative/s. For our method, instead of using an aggregation op-erator to obtain a collective preference value, a random prefer-ence is defined for each alternative in the aggregation phase. Then,based on a  satisfactory principle  defined in this paper, thatsays thatit is perfectly satisfactory to select an alternative as the best if itsperformance is as at least “good” as all the others under the sameevaluationscheme,weproposealinguisticchoicefunctiontoestab-lish a rank ordering among the alternatives.Moreover,we also dis-cuss how this linguistic decision rule can be applied to the MEDMproblem in multigranular linguistic contexts. Two application ex-amples taken from the literature are used to illuminate the pro-posed techniques.  Index Terms— Decision making, linguistic hierarchies, linguisticvariables, multigranular linguistic contexts, randomly linguisticpreferences, satisfactory principle. I. I NTRODUCTION T HE mathematical model of fuzzy concepts was first in-troduced in [55] by using the notion of partial degreesof membership, in connection with the automatic representa-tion and manipulation of human knowledge. Since then, mathe-matical foundations as well as successful applications of fuzzyset theory have been developed. In particular, the application of fuzzy set theory to decision-making problems when only quali-tative or uncertain information is available has been the subjectofmuchresearchoverthelast decades,e.g.,[6],[30],[31],[39], [45], [46], [50], and many others (see, e.g., [17] and [40] for a recent review).In practice, there are many decision situations in whichthe information cannot be assessed precisely in a quantitativeform but may be in a qualitative one, and thus, the use of a linguistic approach  is necessary [17]. For example, in multiex-pert decision-making (MEDM) situations, experts’ judgementsincluding preferences are often vaguely qualitative and cannot Manuscript received April 15, 2004; revised October 26, 2004. This paperwas supported by the Monbukagakusho 21st COE Program and the JAIST In-ternational Joint Research Grant. This paper was recommended by AssociateEditor M. Berthhold.The authors are with the Japan Advanced Institute of Science and Tech-nology, Tatsunokuchi, Ishikawa 923-1292, Japan (e-mail: huynh@jaist.ac.jp;nakamori@jaist.ac.jp).Digital Object Identifier 10.1109/TSMCB.2004.842248 be estimated by exact numerical values. Therefore, a morerealistic approach may be to use linguistic assessments insteadof numerical values by means of linguistic variables [11], [21], [23], [25], [29], that is, variables whose values are not numbers but words or sentences in a natural or artificial language. Eachlinguisticvalueis characterizedbya syntacticvalue orlabelanda semantic value or meaning. The label is a word or sentencebelonging to a linguistic term set and the meaning is a fuzzysubset in a universe of discourse [56]–[58]. In linguistic decision analysis, a solution scheme mustcomply with the following three steps [17].1)  Choice of the linguistic term set:  Basically, one hasto choose the granularity of the linguistic term set, itslabels, and their associated semantics.2)  Choiceoftheaggregationoperatorforlinguisticinfor-mation:  It consists of establishing an appropriate ag-gregation operator for aggregating and combining theprovided linguistic preference values.3)  Choice of the best alternatives, carried out in two phases: a)  Aggregation phase:  Obtaining collective lin-guistic preferences on the alternatives by aggre-gating the individual linguistic preferences bymeans of the chosen aggregation operator.b)  Exploitation phase:  Establishing a rank orderingamong the alternatives according to the collec-tive linguistic preferences for choosing the bestone(s).Essentially, the first two steps serve the aggregation phase inthe third step, while the exploitation phase is determined de-pending on the choice of the semantic description of the lin-guistic term set. Roughly speaking, if the linguistic term set issemantically represented, for example, by the space of parame-terized fuzzy numbers, many methods for the total ordering of fuzzy numbers that have been suggested in the literature can beused in the exploitation phase. When the semantics of the lin-guistic term set is based on a predefined ordered structure, tech-niquesof linguisticapproximation arenecessary [9],[45].More importantly, irrespective of the membership function based se-mantics or ordered structure based semantics of the linguisticterm set, one has to face the problem of weighted aggregationoflinguisticinformation.The issueof weightedaggregationhasbeen studied extensively in, e.g., [4], [10], [12], [22]–[24], [48], [51], and [52]. Again, the linguistic aggregation process is de- termineddependingonthesemanticdescriptionofthelinguisticterm set. While several authors perform direct computation ona finite and totally ordered term set, the others use the member-shipfunctionrepresentationtoaggregatelinguisticvaluesbased 1083-4419/$20.00 © 2005 IEEE  HUYNH AND NAKAMORI: SATISFACTORY-ORIENTED APPROACH 185 on the extension principle [56] – [58]. As mentioned in [16], in both approaches the results usually do not exactly match any of the initial linguistic terms, so a process of linguistic approxima-tion must be applied. This process causes loss of informationand hence a lack of precision.In thispaper,we focus ontheMEDM problemwithlinguisticinformation. Usually, a group decision environment is charac-terized by a  fi nite set of experts (actors or decision makers)who are called to express their preference values on a prede- fi ned set of alternatives (or options). The MEDM problem isthen to  fi rst aggregate preferences individually expressed to ob-tain collective preferences, and second, rank the alternatives inorder to select the best one(s). Conventionally, there are sev-eral techniques used to linguistically evaluate the alternativesbased, for example, on the speci fi cation of linguistic preferencerelations or linguistic assessments. This paper assumes the in-formation is given in the form of linguistic assessments [16],[18]. To avoid the limitation mentioned above, we propose aprobability-based approach with the computation solely basedon the order-based semantics of the linguistic terms. It is worthnotingthatbyperformingdirectcomputationonlinguistictermsintheproposedapproach,theburdenofquantifyingaqualitativeconcept is eliminated. Furthermore, as illustrated by applicationexamples, the results yielded by this method are comparable toprevious work.The main contributions of this paper are as follows: First,we propose a new linguistic decision rule for MEDM prob-lems which is based on a probability-based interpretation of weights and a so-called satisfactory principle (described in Sec-tion III and followed in Section IV by an experimental/com-parative study). Second, we introduce a formal notion of lin-guistichierarchiesintermsoforderedstructure-basedsemanticsof the linguistic term sets and simultaneously present a methodof transformation of a linguistic hierarchy de fi ned in the senseof  [18] to that de fi ned in the sense of this paper. Then we showhow the proposed approach can be applied to MEDM problemsde fi ned in multigranular linguistic contexts. As such, in a sense,the proposed approach can be considered as a possible exten-sion of the proposal developed in [18] for MEDM with multi-granular linguistic contexts. However, it should also be men-tioned that, while the multigranular hierarchial linguistic ap-proach with two-tuple linguistic representation in [18] resultsin a linguistic evaluation at the end of the decision process,which consequently, allows us to consider different aggregationschemes and different selection models, the approach based onthe satisfactory principle in this paper introduces a real-valuedchoice function that induces a ranking order among alternativesbut not a linguistic evaluation.The paper is organized as follows. Section II begins with abriefreviewofdescriptionsofthelinguistictermsetinlinguisticdecision analysis and follows by presenting a general scheme of MEDM problems. Section III introduces a new MEDM methodresulted in a satisfactory-oriented linguistic decision rule andSection IV applies the proposed method to an MEDM problemde fi ned over the same linguistic term set. In Section V, after in-troducing the notion of a linguistic hierarchy, we describe howthis method can be applied to solve an MEDM problem de- fi ned in a multigranular linguistic context. Finally, Section VIpresents some concluding remarks.II. P RELIMINARIES In this section, we  fi rst brie fl y recall different approaches todescription of the linguistic term set with its associated seman-tics in linguistic decision analysis (a comprehensive overviewon this given in [17]). Then we shall reformulate a generalscheme for MEDM problems with linguistic information (seealso [18]).  A. Description of the Linguistic Term Set in Linguistic Decision Analysis In practice, when attempting to qualify phenomena relatedto human perception, we are often led to use words in naturallanguage instead of numerical values. This arises for differentreasons [6]. First, the information may be unquanti fi able due toits nature, and can be stated only in linguistic terms (e.g., whenevaluating the  “ comfort ”  or  “ design ”  of a car [35], terms like “ good, ” “ medium, ” “ bad ”  would be used). In other cases, pre-cise quantitative information may not be stated because eitherit is unavailable or the cost of its computation is too high, soan  “ approximate value ”  may be tolerated (e.g., when evaluatingthespeedofacar,linguistictermslike “ fast, ”“ veryfast, ”“ slow ” may be used instead of numerical values). In such situations, alinguistic approach is necessary and helpful. By scanning theliterature, one can  fi nd an extensive application of linguistic ap-proaches to many different areas of decision analysis, includinggroup decision-making[2],[20] – [22],[27], [29] – [31],multicri-teria decision-making (MCDM) [3], [5], [44], [53], consensus [14], [25], [26], software development [8], [34], [49], subjec- tive assessment of car evaluation [35], material selection [7], personel management [28], environmental assessment [15], etc. In any linguistic approach to solving a problem, the term setof a linguistic variable and its associated semantics must bede fi ned  fi rst to supply the users with an instrument by whichthey can naturally express their information. In accomplishingthis objective, an important aspect to analyze is the granularityof uncertainty, i.e., the level of discrimination among differentcountings of uncertainty or, in the other words, the cardinalityof the linguistic term set used to express the information. Asmentioned in [2], the cardinality of the term set must be smallenough so as not to impose useless precision on the users, andit must be rich enough in order to allow a discrimination of theassessments in a limited number of degrees.Syntactically, there are two main approaches to generatinga linguistic term set. The  fi rst one is based on a context-freegrammar [1], [56] – [58]. This approach may yield an in fi niteterm set. A similar approach is to consider primary linguisticterms (e.g.,  high, low ) as generators, and linguistic hedges (e.g., very, rather, more, or less ) as unary operations. Then the lin-guistic term set can be obtained algebraically [37], [38]. How- ever, according to observations in [36], the generated language does not have to be in fi nite, and in practice human beings canreasonablymanagetokeepaboutseventermsinmind.Asecondapproach is to directly supply a  fi nite term set and consider allterms as primary ones, distributed on a scale on which a totalorder is de fi ned [2], [10], [16], [18], [21], [22], [53], [54]. For instance, a set of seven terms could be given as follows:in which if and only if .  186 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS — PART B: CYBERNETICS, VOL. 35, NO. 2, APRIL 2005 For the semantic aspect, once the mechanism of generatinga linguistic term set has been determined, its associated seman-ticsmustbede fi nedaccordingly.Intheliterature,therearethreemain possibilities for de fi ning the semantics of the linguisticterm set:  semantics based on membership functions and a se-mantic rule, semantics based on the ordered structure of theterm set  , and  mixed semantics . Usually, the  fi rst semantic ap-proach is used when the term set is generated by means of agenerative grammar. This approach consists of two elements:1) the primary fuzzy sets designed as associated semantics of the primary linguistic terms and 2) a semantic rule for gen-erating fuzzy sets semantically associated with nonprimary lin-guistic terms from primary fuzzy sets. Often, while the primaryterms are labels of primary fuzzy sets which are de fi ned sub- jectively and context-dependently, the semantic rule de fi neslinguistichedgesandconnectivesasmathematicaloperationsonfuzzy sets aimed at modifying the meaning of linguistic termsapplied. The second semantic approach is based on a  fi nite lin-guistic term set accompanied with an ordered structure whichintuitively represents the semantical order of linguistic terms.Further, these linguistic terms are assumed to distribute on ascale(e.g., [0,1])either symmetricallyornonsymmetricallyde-pendingonaparticularsituation.Thethirdsemanticapproachisa mixed representation of the previous two approaches, that is,an ordered structure of the primary linguistic terms and a fuzzyset representation of linguistic terms (see [17] for more details).In this paper we adopt the ordered structure based semantics of the linguistic term set.  B. General Scheme of MEDM Problems There are various formulations of fuzzy MEDM problems inthe literature. However, a common characteristic of these prob-lemsisa fi nitesetofexperts,denotedby ,whoare asked to assess another  fi nite set of alternatives (or candi-dates) . The general scheme of MEDM prob-lems considered in this paper follows [18], as shown in Table I,where linguistic assessments can be given either in the samelinguistic term set or in different linguistic term sets of a lin-guistic hierarchy.From the literature on linguistic decision analysis, one can fi nd that there are two general decision models: the  fi rst modelis mainly based on an aggregation-and-ranking scheme, andthe second is based on consensus-reaching oriented solutionschemes. The approach proposed in this paper could be con-sidered as following the  fi rst general model.III. S ATISFACTORY -O RIENTED  L INGUISTIC  D ECISION  R ULE In this section, we shall propose a linguistic decision rulebased on a  satisfactory principle  and a probability-based ap-proach. To this end, we assume a subjective probability distri-bution de fi ned on the set of experts . This assumption es-sentially underlies the calculating basis for the proposed choicefunction. Motivations for the assumption of such a probabilitydistribution are as follows.From a practical point of view, given a set of alternatives ,if there is an ideal expert, say , whose evaluation of alterna-tives the decision maker (DM) completely believes in, then itis enough for the DM to use  ’ s assessments to rank alterna-tives and select the best one(s). However, in practice this is not TABLE IG ENERAL  MEDM P ROBLEM generally the case. Thus, numerous experts are called to expresstheir preference values on the alternatives, on the one hand, tocollect enough information for the problem from various pointsof view, and, on the other hand, to reduce the subjectivity of thedecision. In this sense, , for each , may beinterpreted as the probability that DM randomly selects expertfrom the population as a suf  fi cient source of informationforthedecision-makingpurpose.Suchaprobabilitydistributionmay come from DM ’ s knowledge of the experts. Lacking anysuch knowledge, a uniform distribution would be assumed. It isof interest to note that in a different but similar context, such aprobability distribution is also assumed in the voting model forlinguistic modeling [32], [33]. From a theoretical point of view, in traditional decision anal-ysis, MEDM and MCDM methods often involve a measureon ( plays the role of criteria in MCDM) that must be a capacity  on [13], i.e., such that ,, and for any . Important sub-classes of capacities are probability measures (i.e., additive ca-pacities), belief functions, possibility and necessity measures.Although in the following discussion we only deal with the caseofaprobabilitydistributionassumedon ,othercapacitiessuchaspossibilityornecessity measureswouldbe interestingtocon-sider and this is left for further research.In the tradition of linguistic decision analysis, a weightingvectoris also often associated with such that and. Collective preference values for the alternativesmay then be obtained via a linguistic weighted aggregationoperation, for example [17], of the form(1)where and are, respectively, a weighted aggregation opera-tion and a product operation of a number by a label (or its fuzzyset based semantics). Formally, (1) can be seen as a linguisticcounterpart of expected utilities in decision-making under un-certainty [41], where the set of experts plays the role of states of the world, and then the weights play the role of subjectiveprobabilities assigned to the states.  HUYNH AND NAKAMORI: SATISFACTORY-ORIENTED APPROACH 187 Let us return to the general MEDM problem with a proba-bility function de fi ned on . Assume thatis thelinguistic termset accompanied with theordered structuresuch that iff , and .Under such a formulation, the problem induces randompreferences, denoted by , each for an alterna-tive with associated probability distribution de fi ned by(2)for and .Quite importantly, as mentioned in [2], the procedure of asking each expert to linguistically evaluate each alternative interms of its performance adopts an absolute evaluation and isbased on the assumption that the alternatives are independent.Therefore, if we view the collective preference values of alter-natives as random preferences , , we have foreach , which is stochastically independent of all the others.This assumption allows us to easily compute the probabilitiesof comparisons of two independent probability distributionsof the two random preferences. That is, we can work out theprobability that one of the associated random preferences is less than  or  equal to  the other. More particularly, for any ,such that , we have(3)where isthecumulativeprobabilityfunctionde fi nedby(4)The quantity could be interpreted as the proba-bility of   “ the performance of is as at least good as that of   ” under the evaluation scheme . Intuitively,  it is perfectlysatisfactorytoselectanalternativeasthebestifitsperformanceis as at least good as all the others under the same evaluationscheme . We have called this the  satisfactory principle .Now we are ready, based on the satisfactory principle, to pro-pose a choice function de fi ned as follows:(5a)(5b)Then the satisfactory-oriented linguistic decision model forthe MEDM problem is de fi ned by(6)In the following section, we shall illustrate how this modelworks in practice by an application taken from [16]. Fig. 1. Linguistic term set.TABLE IIMEDM P ROBLEM IN  U PGRADING  C OMPUTING  R ESOURCES IV. MEDM P ROBLEM  D EFINED  O VER  AC OMMON  L INGUISTIC  T ERM  S ET  A. Problem Description A distribution company needs to upgrade its computingsystem, so it hires a consulting company to survey the differentpossibilities existing on the market, to decide which is thebest option for its needs. The options (alternatives) are thefollowing:The consulting company has a group of four consultancy de-partmentsEach department in the consulting company provides an evalu-ation vector expressing its assessment of options. These evalu-ations are assessed in the set of seven linguistic terms (graph-ically, shown in Fig. 1)in which if and only if , and are given in Table II.As usual, the selection model used to solve this problem con-sists of two steps.1) Obtain a  collective performance value  for each option.2) Apply a selection process based on the obtained col-lective performance vector.
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