A Fuzzy-Set Approach to Treat Determinacy and Consistency of Linguistic Terms in Multi-Criteria Decision Making Q

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International Journal of Approximate Reasoning 44 (2007) 165–181 www.elsevier.com/locate/ijar

A fuzzy-set approach to treat determinacy and consistency of linguistic terms in multi-criteria decision making q

J. Ma a,d,*, D. Ruan b,c,Y.Xua, G. Zhang d

a Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China b Belgian Nuclear Research Centre (SCK-CEN), Boeretang 200, Mol 2400, Belgium c Department of Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), Gent 9000, Belgium d Faculty of Information Technology, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australia

Received 28 August 2005; received in revised form 16 June 2006; accepted 18 July 2006 Available online 18 August 2006

Abstract

A fuzzy-set-based approach is presented to describe linguistic information in multi-criteria decision making. After having introduced concepts of determinacy and consistency of linguistic terms, the understandable degree and consistence degree of linguistic terms are illustrated by these two concepts. A case study is demonstrated for the proposed decision-making model with an analytical con- clusion of both advantages and disadvantages. 2006 Elsevier Inc. All rights reserved.

Keywords: Multi-criteria decision making; Linguistic terms; Fuzzy sets; Determinacy of linguistic terms; Consistency of linguistic terms

q This work is partly supported by the National Natural Science Foundation of China with Grant Number 60474022, China-Flanders Bilateral Scientiﬁc Cooperation Joint Project proposal with Grant Number 011S1105, and the Australia Research Council (ARC) under discovery grant DP0559123. * Corresponding author. Tel.: +86 28 87600764. E-mail addresses: [email protected], [email protected] (J. Ma), [email protected] (D. Ruan), xuyang@ home.swjtu.edu.cn (Y. Xu), [email protected] (G. Zhang).

1. Introduction

When we work with uncertain information, we cannot estimate it with an exact numerical value but with natural languages [6]. Experience told us that sometimes we must make a decision in an environment with a great deal of uncertain information. Hence, a realistic strategy is to describe uncertain information by natural languages. So, linguistic approaches have been so far widely studied and applied to information retrieval [2], clin- ical diagnosis [27], risk evaluation in software development [22], education evaluation [21], environment sciences [9], manufacturing systems [19], especially on decision making [10– 15,26,30–32,38,39]. A multi-criteria decision-making problem or group decision-making problem can be seen as a trade-off among a set of alternatives according to several given evaluation factors, in which (1) there are two or more consultancy experts, each of them evaluates the alternatives by their own perception, attitude, and expertise; (2) the decision-maker makes the selection according to the collected comments from the consultancy experts [6]. Since the information provided by the consultancy experts is often in the form of natural languages, two related issues have gained considerable attention in linguistic approach for decision-making problems, i.e., (1) how to describe and handle linguistic information and (2) how to fusion linguistic information. Fuzzy sets and linguistic variables [35–37] are widely used in describing linguistic information because they can efficiently represent the gradually changing of people’s recogni- tion of an object or a concept. Hence, fuzzy-sets based linguistic approaches have drawn considerable attention and have been used in ‘‘supply chain management [4,3]’’, ‘‘information aggregation [1,30–32]’’, ‘‘data warehouse system selection [23]’’, ‘‘investments evaluation [5]’’, and other application fields. Except for the aforementioned fuzzy-sets based methods, we noticed that Ho [16–18] presented the hedge algebra to describe linguistic information and linguistic modifiers. How- ever, due to the lack of efficient process mechanism for inference and computation, it is difficult to apply the hedge algebra directly to develop processing systems. To describe the hierarchical structure of uncertainties, Xu [29] proposed the lattice implication algebra by combining lattice and implication algebra and established the lattice-valued logic by taking lattice implication algebra as a truth-value field. Because the lattice-valued logic can describe incomparable information, it is a potential alternative tool to treat linguistic information. Roughly speaking, two kinds of techniques, namely, numeric and symbolic approaches, are widely used in integrating linguistic information [10]. In the numeric approach, each linguistic term is represented by a number (or a fuzzy number), and all numbers must be placed symmetrically in totally order. The process is based on the algorithms for those numbers, such as arithmetic operations. Numeric approaches can propose effective computation mechanism and be programmed in computer easily. But, its requirements on all linguistic terms that they must be placed symmetrically and ordered in total-order do not accord with people’s experience. Moreover, the linguistic representation of final decision is obtained by selecting the one according to a certain distance, such as the close- ness of fuzzy sets. Obviously, numerical approaches over rely on the computation and omit the formal inference in the people’s decision-making process. In the symbolic approach, each linguistic term is attached to a numeric index and the real processing is based on the operation on those indexes. Hence, such a symbolic approach is also computation-based numeric approach in a sense. J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 167

We can also find that in most existing linguistic approaches for decision making, consultancy experts have no other choice but use the linguistic terms given by the decision- maker. Although using the predefined linguistic terms will facilitate the decision-maker to process linguistic information, it badly restricts consultancy experts to present their opinions freely. On the other hand, it is common in most existing linguistic approaches that consultancy experts use only one linguistic term to represent their comment on an alternative. Obviously, the used one linguistic term cannot exactly reflect the expert’s opinion, because (1) it is selected from the predefined linguistic term set, which may not coin- cident with the expert’s preference; and (2) it may be the result of balancing among several ones. Hence, the collected comments from consultancy experts will be attached with factitious uncertainties. In our view, to reduce such factitious uncertainties, experts should be allowed to use various linguistic representations or at least be allowed to use more than one linguistic term. Naturally, two questions arise: (1) how to judge the information consistency in several linguistic terms obtained from an expert; and (2) how to process those linguistic terms. In this paper, we introduce the concepts of determinacy and consistency of linguistic terms to resolve these two problems. We shall use them in a typical multi-criteria decision- making problem and discuss a model for dealing with linguistic terms directly. In this model, all experts can select linguistic terms they prefer to use and there is no requirement on all linguistic terms that they must be placed symmetrically in totally order. Concretely, the rest of the paper is organized as follows: in Section 2, we introduce some basic concepts in fuzzy sets; in Section 3, we discuss concepts of determinacy, consistency of linguistic terms on the basis of normal triangular fuzzy sets; in Sections 4 and 5, we illustrate how to handle linguistic comments by the determinacy and consistency through a typical decision-making problem; in Section 6, we discuss both advantages and disadvantages of the model and further work on this topic.

2. Preliminaries

In this section, we introduce some basic concepts of fuzzy sets as the basis for further discussion on the concepts of determinacy and consistency of linguistic terms.

Deﬁnition 2.1 [34]. Let X be a space of points (objects). A fuzzy set (class) A in X is characterized by a membership (characteristic) function fA(x), which associates with each point in X a real number in the interval [0,1], with the value of fA(x)atx representing the ‘‘grade of membership’’ of x in A.

Deﬁnition 2.2 [34]. The complement of a fuzzy set A is denoted by A0 and is deﬁned by fA0 ¼ 1 fA.

Deﬁnition 2.3 [20]. A fuzzy set A is called normal when h(A) = 1, where h(A) is the largest membership grade obtained by any element in that set.

Deﬁnition 2.4 [34]. A fuzzy set A is convex if and only if the sets Ca, deﬁned by Ca ={xjfA(x) P a}, are convex for all a in the interval [0,1]. 168 J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181

Young Middle age Old

0 10 20 30 40 50 60 70 80

Fig. 1. Membership functions representing concepts of a young, middle-aged, and old person.

Fuzzy set is an efficient tool to illustrate concepts with uncertainties, for example the three fuzzy sets in Fig. 1 represent the concepts of a young, middle-aged, and old person [20]. Notice that the words ‘‘young’’, ‘‘middle-aged’’, and ‘‘old’’ are all linguistic terms, fuzzy sets is also an efficient tool for representing linguistic terms. Because it is easy to know that different person will have various definitions for one linguistic term in the same context and also the same linguistic term will have different connotations, there is not absolutely a cor- respondence between one fuzzy set and one linguistic term. That is to say, the fuzzy set corresponding to a linguistic term may change with the context of the problem and the usage of the user. But what we know is that a fuzzy set can be selected to represent a linguistic term. In the following discussion, we shall not limit our discussion to a particular context and let (1) U be the universe of discourse, (2) F be the set of all fuzzy sets on U, (3) Ft be the fuzzy set selected from F corresponding to a linguistic term t. For convenience, we shall suppose that all used fuzzy sets are convex, normal, and integrable.

3. Determinacy and consistency of linguistic terms

The concept of determinacy of linguistic terms is based on the following idea. Suppose an expert e presents a set of linguistic terms to evaluate an alternative. We hope to learn the real opinion of the expert that implies in those terms, in other words, we hope to find out how much available information we can obtain from those terms. In a general case, if the expert uses one term to present his/her opinion, then it is rational to think that the expert is very confident about the usage of the term. While if the expert uses more than one terms to present his/her opinion, then it is likely to say that he/she cannot definitely select one from them. In our opinion, the determinacy of linguistic terms indicates the understandable degree of linguistic terms.

Deﬁnition 3.1 (Determinacy). The determinacy of a linguistic term t presented by an expert e is deﬁned as Z Z

DeteðtÞ¼1 Ft dU dU; ð3:1Þ R U U where U FtdU is the fuzzy integral of Ft on U. J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 169

Example 3.2. Suppose the fuzzy sets corresponding to three linguistic terms ta, tb, tc are

Fta , Ftb and Ftc , respectively and they are shown in Fig. 2. Suppose 8 6 > 0; 0 x < 0:25; <> x 0:25; 0:25 6 x < 0:5; ðaÞ Ft ¼ a > 0:75 x; 0:5 6 x < 0:75; :> 0; 0:75 6 x 6 1: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 6 0:5 q0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi:25 x ; 0 x < 0:5; ðbÞ ¼ Ftb : 2 0:5 þ 0:25 ðx 1Þ ; 0:5 6 x 6 1: 1; 0:5 6 x 6 0:65; ðcÞ Ft ¼ c 0; otherwise; then Dete(ta) = 0.75, Dete(tb) = 0.5, and Dete(tc) = 0.85.

Proposition 3.3. Let t be a linguistic term, to which the corresponding fuzzy set is Ft. Then Z Z c DeteðtÞ¼ Ft dU dU; ð3:2Þ U U c where Ft is the complement of the fuzzy set Ft. Extending the determinacy for one linguistic term, we can deﬁne the determinacy for a set of linguistic terms as follows. Let T ={t1,t2,...,tn} be a set of linguistic terms. Then the determinacy of T is deﬁned as Z !!, Z [n DeteðT Þ¼1 Fi dU dU: ð3:3Þ U i¼1 U It is obvious that Z !, Z \n c DeteðT Þ¼ Fi dU dU ð3:4Þ U i¼1 U by the property and operation of fuzzy sets.

1 1 1

0 1 0 1 0 1 U U U

Fig. 2. Fuzzy sets corresponding to linguistic terms ta, tb, and tc. 170 J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181

1 t1 t2 t3 t4 t5 t6 t7

0 0.1 0.25 0.3 0.375 0.50.625 0.7 0.75 0.9 1

Fig. 3. Fuzzy sets corresponding to a set of linguistic terms.

Example 3.4. Suppose a set of linguistic terms are shown in Fig. 3, and the corresponding fuzzy sets are deﬁned as follows:

Ft1 ¼ð0; 0; 0:1Þ; Ft2 ¼ð0:1; 0:2; 0:3Þ; Ft3 ¼ð0:25; 0:375; 0:5Þ;

Ft4 ¼ð0:375; 0:5; 0:625Þ; Ft5 ¼ð0:5; 0:625; 0:75Þ; Ft6 ¼ð0:7; 0:8; 0:9Þ;

Ft7 ¼ð0:9; 1:0; 1:0Þ; where F ¼ða; b; cÞ is the triangular fuzzy set shown in Fig. 4. Then

Deteðt1Þ¼0:95; Deteðt2Þ¼0:90; Deteðt3Þ¼0:875; Deteðt4Þ¼0:875;

Deteðt5Þ¼0:875; Deteðt6Þ¼0:90; Deteðt7Þ¼0:95:

Letting T ={t3,t4,t5}, we have Dete(T) = 0.6875. Example 3.4 indicates when the expert uses more than one term to present his/her comment on an alternative, the determinacy of a set of linguistic terms will be smaller than that of each linguistic term and uncertainty in the comment will increase. This phenomena is true in accordance with our experience. For the determinacy of linguistic terms, the following conclusions hold.

Proposition 3.5. Suppose the corresponding fuzzy set Ft of a linguistic term t is a single point u 2 U, i.e., FtðuÞ¼1 and FtðxÞ¼0 for any x 5 u. Then Dete(t) = 1. Similarly, if FtðxÞ¼1 for any x 2 U, then Dete(t) = 0.

Proposition 3.6. For any two linguistic terms t1 and t2, if the corresponding fuzzy sets Ft1 and Ft2 satisfy Ft1 Ft2 . Then Dete(t1) P Dete(t2).

0 a b c

Fig. 4. Triangular fuzzy set F ¼ða; b; cÞ. J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 171

Proposition 3.7. For any linguistic term t, 0 6 Dete(t) 6 1.

Theorem 3.8. Let T = {t1,t2,...,tn} be a set of linguistic terms. Then

DeteðT Þ 6 DeteðSÞð3:5Þ for any S T. Besides the determinacy of linguistic terms, another important property is the consistency of them, which means the shared meaning in them. Suppose an expert presents a set of linguistic terms, which have nothing to be shared by all terms, then it is rational to say there is inconsistency among them and in turn the expert’s comment cannot be accepted. We expect those linguistic terms obtained from one expert should be consistent enough in real applications.

Deﬁnition 3.9. Let T ={t1,t2,...,tn} be a set of linguistic terms, and Fti , i =1,2,...,n, be the corresponding fuzzy sets of ti. Then the consistency of T is () _ \n

ConeðT Þ¼ a : ðFti Þa 6¼; ; ð3:6Þ i¼1 where ðFti Þa is the a-cut of Fti for i =1,2,...,n.

Example 3.10. Suppose the corresponding fuzzy sets of linguistic terms are shown in Fig. 3. Consider the following four cases:

(1) if T ={t4,t5}, then Cone(T) = 0.5; (2) if T ={t5,t6}, then Cone(T) = 0.22; (3) if T ={t4,t5,t6}, then Cone(T)=0; (4) if T ={t3,t4,t5}, then Cone(T) = 0.5.

Proposition 3.11. Let T1 and T2 be two sets of linguistic terms and T1 T2. Then Cone(T1) P Cone(T2).

Proposition 3.12. Let T = {t1,t2} and Ft1 Ft2 . Then Cone(T) = 1.

4. Model illustration

4.1. Overview

In this section, we introduce the determinacy and consistency of linguistic terms to a typical multi-criteria decision-making problem. In this model, our discussion is based the following assumptions:

(1) Linguistic terms can be presented by convex, normal and integrable fuzzy sets. (2) Experts can select preferred linguistic terms to evaluate alternatives. 172 J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181

(3) Experts can deﬁne the fuzzy set corresponding to linguistic term by themselves. (4) Comments for an alternative from one expert can include two or more linguistic terms.

Without loss of generality, let Q be a decision-making problem, A ={a1,a2,...,an} the set of alternatives, E ={e1,e2,...,em} the set of consultancy experts, and T k ¼ftk;1; tk;2; ...; tk;lk g the set of linguistic terms of which the consultancy expert ek selected for the question Q, k =1,2,...,m. The present model includes the following steps:

Step 1: Experts evaluating: experts select linguistic terms, deﬁne corresponding fuzzy sets, and evaluate each alternative. Step 2: Synthesizing individual expert’s comments: the decision maker synthesizes each expert’s comment for each alternative. Step 3: Synthesizing group experts’ comments: the decision maker synthesizes all experts’ comment for each alternative. Step 4: Making decision: the decision maker forms the ﬁnal decision according to synthesized group experts’ comment for each alternative.

4.2. Experts evaluating

Because of the difference in experience and in evaluation factors, the linguistic terms selected by experts may different from others. Therefore, experts should define the corresponding fuzzy set for each selected linguistic term. For example, experts can select seven linguistic terms and define them as shown in Fig. 3, or select five linguistic terms and define them as shown in Fig. 5. For each expert, whose comments on all alternatives form Table 1, where vk,j(as) takes the value of 1 if the expert assigns the corresponding linguistic term tk,j to the alternative

1 t1 t2 t3 t4 t5

0 0.1 0.2 0.375 0.5 0.6250.8 0.9 1

Fig. 5. Example of ﬁve linguistic terms.

Table 1

Comments on alternatives of the expert ek

tk,1 tk,2 tk;lk Synthesized comments a1 vk,1(a1) vk,2(a1) vk;lkða1Þ ck,1 a v (a ) v (a ) v ða Þ c . 2 . k,1 2 . k,2 2 . . k;lk 2 . k,2 ...... an vk,1(am) vk,2(am) vk;lkðamÞ ck,m J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 173

Table 2 Comments on three alternatives of an expert bad not bad medium good very good Synthesized comments a1 01 1 0 0 commonly a2 00 0 1 1 excellent a3 00 1 1 0 good

as, or 0 otherwise. The ‘‘synthesized comments’’, denoted by ck,1,ck,2,...,ck,m in Table 1, are obtained based on the selected linguistic terms Tk.

Example 4.1. For example, if an expert selects ﬁve linguistic terms for the problem, which are ‘‘bad’’, ‘‘not bad’’, ‘‘medium’’, ‘‘good’’, and ‘‘very good’’, and there are three alternatives to be evaluated. Then the comments of the expert may be shown in Table 2.

4.3. Synthesizing individual expert’s comments

The synthesized comments are what an expert feeds back to the decision maker, which have a close relation to the selected linguistic terms. Thus, how to represent it is a key issue in this step. In order to preserve the information collected from experts, this paper adopts the following strategy, which is similar to the voting strategy in data fusion [33]. Let a be an alternative, e an expert, and Te the selected linguistic terms in a real decision-making problem Q. The synthesized comment of e on a is ComeðaÞ¼fðti; DsynceðtiÞÞ : ti 2 T e ; DsynceðtiÞ¼DeteðtiÞDeteðT e ÞConeðT e Þg; ð4:1Þ where T e T e and T e ¼fti 2 T e : vti ðaÞ¼1g.

Example 4.2. Suppose the fuzzy set of linguistic terms in Example 4.1 is shown in Fig. 5. Also suppose

bad ¼ t1 ¼ð0;0;0:1Þ; not bad ¼ t2 ¼ð0:1;0:2;0:5Þ; medium ¼ t3 ¼ð0:375;0:5;0:625Þ;

good ¼ t4 ¼ð0:5;0:8;0:9Þ; very good ¼ t5 ¼ð0:9;1;1Þ: Then we have

Comeða1Þ¼fðt2; 0:235Þ; ðt3; 0:257Þg;

Comeða2Þ¼fðt5; 0:95Þg;

Comeða3Þ¼fðt3; 0:257Þ; ðt4; 0:235Þg: Example 4.2 indicates that we have preserved all linguistic terms from the expert and attached a degree to each for illustrating the reliability of each term.

4.4. Synthesizing group experts’ comments

For each alternative ai, i =1,2,...,n, we can get a set of synthesized comments fComej ðaiÞ : j ¼ 1; 2; ...; mg of all experts. The next step is to aggregate all these 174 J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 synthesized comments. We can use many methods, such as Yager’s [30–33] and Herrera’s [12–15] methods. In this paper, we use the extended weighted sum as the aggregation method. Suppose the weighted vector for all experts is W =(w1,w2,...,wm). Then the ﬁnal synthesized comments can be computed as follows: ]m

ComEðaiÞ¼ wjComej ðaiÞ; ð4:2Þ j¼1 U where i =1,2,...,n,and is an operator such that ] Com a Com a t; a t T T ; a Dsync t Dsync t : 4:3 ei ð Þ ej ð Þ¼fð Þj 2 ei [ ej ¼ ei ð Þþ ej ð Þg ð Þ

Example 4.3. Recall Examples 4.1 and 4.2. Without loss of generality, suppose there are three consultancy experts e1, e2, and e3, and their synthesized comments on the alternative a are

Come1 ðaÞ¼fðte1;2; 0:235Þ; ðte1;3; 0:257Þg;

Come2 ðaÞ¼fðte2;5; 0:95Þg;

Come3 ðaÞ¼fðte3;3; 0:257Þ; ðte3;4; 0:235Þg; respectively. Also suppose the weighted vector is W ¼ð0:30:20:5 Þ. Then the ﬁnal synthesized comment of the three experts on a1 is

ComEðaÞ¼fðte1;2; 0:071Þ; ðte1;3; 0:051Þ; ðte2;5; 0:19Þ; ðte3;3; 0:129Þ; ðte3;4; 0:118Þg:

Moreover, if we suppose that te1;3 and te3;3 are the same linguistic terms, then

ComEðaÞ¼fðte1;2; 0:071Þ; ðte1;3; 0:18Þ; ðte2;5; 0:19Þ; ðte3;4; 0:118Þg: ð4:4Þ

The last question is choosing which linguistic term as the ﬁnal aggregated comment to each alternative. To resolve it, there are many methods. Here we select the term(s) with the biggest synthesized value(s). Therefore, in Example 4.3, the ﬁnal aggregated comment on a is the linguistic term t2,5, which means ‘‘very good’’.

4.5. Making decision

To select the best alternative, we must concern the concrete context of the problem at hand. Suppose the problem aims at buying a new laptop, surely we would like to select the one which has lower price and higher performance. So, a predeﬁned order can be presented among all linguistic terms. Without loss of generality, let ^ be the order among all linguistic terms. Notice that the aim of making decision is to order all alternatives and then select the best several alternatives, we shall take the following strategy to make decision. Firstly, we shall order all the alternatives as follows. Specially, we use // to mean that two objects (linguistic terms or alternatives) are incomparable to each other. Suppose (t1,a1),(t2,a2),...,(tn,an) are the synthesized linguistic comments for alternatives a1,a2,...,an, respectively. J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 175

(1) If ti//tj, where i 5 j and i, j 2 {1,...,n}, then ai//aj. (2) If ti ^ tj and ai 6 aj, where i 5 j and i, j 2 {1,...,n}, then ai 6 aj. (3) If ti ^ tj and ai P aj, where i 5 j and i, j 2 {1,...,n}, then ai//aj.

So, all alternatives will be ordered in a partial order. Nextly, we shall select the best alternative(s) according to their partial order. For instance, suppose there are seven linguistic terms N, VL, L, M, H, VH, P and there partial order is N ^ VL ^ L ^ M ^ H ^ VH^ P: ð4:5Þ If (L,0.449), (VL,0.381), (H,0.32), (M,0.698) are synthesized linguistic comments for alternatives a1, a2, a3, and a4, then by the above-mentioned strategy, the partial order among a1, a2, a3, and a4 is

a2 6 a1 6 a3 and a2 6 a1 6 a4 and a3==a4; so, the best alternatives are a3 and a4.

Remark 4.4. Because of the diﬀerence of the background of real problems, the order among linguistic terms may be totally order or partial order. Moreover, the best alternative may be one or more.

5. Example illustration and discussion

In this section, we shall illustrate the model based on the following example which is taken from [15].

5.1. Example illustration

Example 5.1. A distribution company needs to renew its computing system, so it contracts a consulting company to carry out a survey of the diﬀerent possibilities existing on the market, to decide which is the best option for its needs. The alternatives are: LINUX (x1), WINDOWS2000 (x2), VMS (x3), and SOLARIS (x4). The consulting company has a group of four consultancy departments, i.e., cost analysis (p1), system analysis (p2), risk analysis (p3), and technology analysis (p4). Each department provides a performance vector expressing its preferences for each alternative (see Table 3), where Lij is taken from the term set T ={N,VL,L,M,H,VH,P} deﬁned in Fig. 6. For this example, the following steps illustrate how to determine the best alternative(s) by the model.

Table 3 Comments of all consultancy departments for all alternatives

Lij p1 p2 p3 p4 x1 L L VL H x2 M L VL VL x3 MVLML x4 MMMM 176 J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181

N V L L M H VH P 1

0 0.25 0.4 0.5 0.6 0.75 1

Fig. 6. The set T of seven linguistic terms with their semantics, where N = (0,0,0.25), VL = (0,0.2,0.4), L = (0.25,0.4,0.5), M = (0.4,0.5,0.6), H = (0.5,0.6,0.75), VH = (0.6,0.8,1), P = (0.75,1,1).

(1) Computing the determinacy of each linguistic term: DetðNÞ¼0:875; DetðVLÞ¼0:8; DetðLÞ¼0:875; DetðMÞ¼0:9; DetðHÞ¼0:875 DetðVHÞ¼0:8 DetðPÞ¼0:875:

(2) Notice that each Lij is a linguistic term, hence for any alternative xi, the consistency of all comments is 1. Then we can compute the synthesized comments. For L11, because L11 = L, hence the synthesized comment of the expert e1 on the alternative x1 is (L,0.875 · 1) = (L,0.875). Similarly, we get Table 4. (3) Suppose the weighted vector is W ¼ð0:25 0:25 0:25 0:25 Þ. By using the weighted sum aggregating method, we get the synthesized group experts’ comments on all alternatives shown in Table 5. By Table 5, the best alternative is x4 (this result is similar to that in [15]).

Table 4 Synthesized individual expert’s comments on all alternatives

p1 p2 p3 p4 x1 (L,0.875) (L,0.875) (VL,0.8) (H,0.8) x2 (M,0.9) (L,0.875) (VL,0.8) (VL,0.8) x3 (M,0.9) (VL,0.8) (M,0.9) (L,0.875) x4 (M,0.9) (M,0.9) (M,0.9) (M,0.9)

Table 5 Synthesized group experts’ comments on all alternatives under W ¼ð0:25 0:25 0:25 0:25 Þ Ci ci x1 {(L,0.438),(VL,0.2),(H,0.2)} (L,0.438) x2 {(M,0.225),(L,0.219),(VL,0.4)} (VL,0.4) x3 {(M,0.45),(VL,0.2),(L,0.219)} (M,0.45) x4 {(M,0.9)} (M,0.9) J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 177

Suppose the weighted vector is W ¼ð0:40:30:20:1 Þ and then we get C1 ¼fðL; 0:875 0:4 þ 0:875 0:3Þ; ðVL; 0:8 0:4Þ; ðH; 0:8 0:4Þg ¼fðL; 0:613Þ; ðVL; 0:32Þ; ðH; 0:32Þg: Similarly, we can obtain synthesized group experts’ comments on other alternatives (see Table 6). In this case the best alternative is x4, too.

Now, suppose Table 3 has been revised as Table 7. In this case, the consistency of linguistic terms is no longer equal to 1. Similarly, the determinacy of the used linguistic terms will change accordingly. Hence we have

(1) Computing the determinacies of the used linguistic terms. For L13, we get

ConðL13Þ¼0:429;

DetðL13Þ¼1 0:65 1 0:5 þ 0:15 0:429 0:5 ¼ 0:707; 1 C13 ¼fðL; 0:875 0:429 0:707Þ; ðVL; 0:8 0:429 0:707Þg ¼fðL; 0:265Þ; ðVL; 0:243Þg: Similarly, we get Table 8. (2) Suppose the weighted vector is W ¼ð0:25 0:25 0:25 0:25 Þ and we use the weighted sum to aggregate the comments. Then we get the synthesized group experts’ comments on all alternatives shown in Table 9.ByTable 9, we get c1 ¼ðL; 0:449Þ; c2 ¼ðVL; 0:381Þ; c3 ¼ðH; 0:32Þ; c4 ¼ðM; 0:698Þ;

and hence, the best alternative is x4 or x3.

Table 6 Synthesized group experts’ comments on all alternatives under W ¼ð0:40:30:20:1 Þ Ci ci x1 {(L,0.613),(VL,0.32),(H,0.32)} (L,0.613) x2 {(M,0.36),(L,0.35),(VL,0.56)} (VL,0.56) x3 {(M,0.63),(VL,0.32),(L,0.35)} (M,0.63) x4 {(M,0.9)} (M,0.9)

Table 7 Modiﬁed experts’ comments on all alternatives

Lij p1 p2 p3 p4 x1 LL VL, LH, VH x2 ML, VL VL VL x3 M, HVLM, LL, VL x4 M, LM M M

Table 8 Modiﬁed synthesized individual expert’s comments on all alternatives

p1 p2 p3 p4 x1 (L,0.766) (L,0.766) (L,0.265), (VL,0.243) (H,0.265), (VH,0.243) x2 (M,0.81) (L,0.265), (VL,0.243) (VL,0.64) (VL,0.64) x3 (M,0.36), (H,0.32) (VL,0.64) (M,0.36), (L,0.32) (L,0.265), (VL,0.243) x4 (M,0.36), (L,0.32) (M,0.81) (M,0.81) (M,0.81) 178 J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181

Table 9 Modiﬁed synthesized group experts’ comments on all alternatives Ci x1 {(L,0.449),(VL,0.061),(H,0.066),(VH,0.061)} x2 {(M,0.203),(L,0.066),(VL,0.381)} x3 {(M,0.18),(H,0.32),(L,0.146),(VL,0.221)} x4 {(M,0.698),(L,0.08)}

5.2. Discussions

In the aforementioned example, the presented model mainly concerns on the collected linguistic terms and the determinacy and consistency accompanied to them. In other words, the linguistic terms in this model is treated as a whole. Moreover, the determinacy and consistency of linguistic terms are aﬀected by the shape of the terms. In the above model, if we take into account the decision makers’ perception, we can expand it. For instance, if an expert’s comment on an alternative is H, the decision maker always would like to pay more attention on that alternative than others. While, if the expert’s comment is VL, decision makers would not like to pay more attention again. This means, distinguishing the perception of decision makers on each linguistic term will impose on making the decision. In view of it, we have many strategies to improve the model, such as revising the aggregation processing. For example, suppose Wd is the degrees of decision makers’ perception on each linguistic term, then the synthesized comment cj can be computed as follows:

j j d C ¼ Ck W ; W W ; ð5:1Þ where is a composition. Take C1 in Table 5 for instance, suppose W d ¼f0ðNÞ; 1ðVLÞ; 2ðLÞ; 4ðMÞ; 8ðHÞ; 16ðVHÞ; 32ðPÞg; then the aggregated comment is C1 ¼fðL; 2 0:438Þ; ðVL; 1 0:2Þ; ðH; 8 0:2Þg ¼ fðL; 0:876Þ; ðVL; 0:2Þ; ðH; 1:6Þg: Similarly, for i = 2,3,4, we get

C2 ¼fðM; 0:9Þ; ðL; 0:438Þ; ðVL; 0:4Þg; C3 ¼fðM; 1:8Þ; ðL; 0:438Þ; ðVL; 0:2Þg; C4 ¼fðM; 3:6Þg:

Thus the best alternative will be x4 or x1. From the given example, we find an interesting phenomena that there may be more than one alternative which cannot be compared with each other by the model. For instance, we cannot tell clearly that the Linux (x1) must be better than the SOLARIS (x4), and vice versa. We think this phenomena is not completely resulted from the model but reflects an actual essence of a specific decision-making problem. To deal with the incomparability, Xu et al. [29] and Ma [24,25] have studied the lattice-valued logic in order to present a processing framework based on logic for such problems. J. Ma et al. / Internat. J. Approx. Reason. 44 (2007) 165–181 179

6. Conclusions

In this paper, we introduced the determinacy and consistency of linguistic terms to multi-criteria decision-making problems and presented a fuzzy-set based model to deal with linguistic information. This model has the following advantages:

(1) It permits consultancy experts to select linguistic terms for presenting their opinions by their preference. It is not demanded that all consultancy experts must use the same linguistic terms. (2) It does not require all linguistic terms to be placed symmetrically and to have totally order. So consultancy experts and decision makers have more independent right to present their opinions. (3) It treats each linguistic term as a whole and only concerns on the determinacy and consistency of them. This strategy makes it more ﬂexible to freely assign a fuzzy set to a linguistic term according to the context of real problems.

However, due to the complexity of human natural languages, there are still some unsolved problems in the presented model, such as,

(1) how to deal with linguistic terms directly? (2) how to deal with incomparability among linguistic terms? and (3) how to deﬁne related weighted vectors?

Because the classical logic is an alternative tool to human reasoning with certainty information, non-classical logics including many-valued logic, fuzzy logic, lattice-valued logic and so on will play the crucial role in the study of uncertainty reasoning. Researches and practices have proved this [7,8,28]. In the view of the authors, making decision is a kind of uncertainty reasoning. So, it is rational to deal with decision-making problems on the basis of non-classical logics as complementary to traditional methods.

Acknowledgements

The authors appreciated the anonymous referees’ instructive comments and suggestions to improve the presented work.

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