On consistency measures of linguistic preference relations

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European Journal of Operational Research 189 (2008) 430–444 www.elsevier.com/locate/ejor

Decision Support

On consistency measures of linguistic preference relations Yucheng Dong a

a,b,*

, Yinfeng Xu

a,c

, Hongyi Li

b

Department of Management Science, Management School, Xi’an Jiaotong University, Xi’an 710049, PR China b Faculty of Business Administration, The Chinese University of Hong Kong, Shatin, NT, Hong Kong c State Key Lab for Manufacturing Systems Engineering, Xi’an 710049, PR China Received 16 October 2006; accepted 11 June 2007 Available online 16 June 2007

Abstract Inspired by the concept of deviation measure between two linguistic preference relations, this paper further defines the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations. Based on this, we present a consistency index of linguistic preference relations and develop a consistency measure method for linguistic preference relations. This method is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons using the linguistic label set. Using this consistency measure, we discuss how to deal with inconsistency in linguistic preference relations, and also investigate the consistency properties of collective linguistic preference relations. These results are of vital importance for group decision making with linguistic preference relations.  2007 Elsevier B.V. All rights reserved. Keywords: Decision analysis; Linguistic preference relation; Consistency

1. Introduction Everybody makes decisions all the time, educated or uneducated, young or old, with ease or with difficulty. In the multiple attribute decision making (MADM), the decision makers supply their preferences on alternatives in different preference representation structures. Among these structures, the preference relation is the most common one. According to element forms in preference relations, there are often two kinds of preference relations: linguistic preference relations [7,8,10,11,13,23,27] and numerical preference relations (i.e., multiplicative preference relations and fuzzy preference relations) [1–3,18,20,21]. Recently, many researchers [4,5,7–13,23–28] pay attention to group decision making (GDM) using linguistic preference relations, following a common resolution scheme composed by two phases: the aggregation phase and the exploitation phase. Herrera et al. [8,10,12] presented the linguistic ordered weighted averaging (LOWA) operators to aggregate linguistic preference relations. Herrera and Herrera-Viedma [11] analyzed the problem of finding a solution set of alternatives from a collective linguistic preference relation, following two * Corresponding author. Address: Department of Management Science, Management School, Xi’an Jiaotong University, Xi’an 710049, PR China. Tel.: +86 2982673492. E-mail address: [email protected] (Y. Dong).

0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.06.013

Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

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research directions: the choice functions and the mechanisms. Herrera et al. [9] introduced a framework to reach consensus in GDM under linguistic assessments. Xu [24] proposed the linguistic order weighted geometric (LOWG) operators. Xu [25,28] also studied the uncertain linguistic aggregation operators. The consistency measure is a very important problem in decision making using preference relations. It is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons. The lack of consistency in decision making with preference relations can lead to inconsistent conclusions. There have been a lot of consistency measure methods [14,15,19–21] on numerical preference relations. At the same time, some techniques [16,17,22,23] are also presented to deal with the inconsistency in them. Similar to the numerical preference relations [6,14], some traditional definitions to characterize consistency of linguistic preference relations are to use transitivity such as the three-way transitivity, the max–min transitivity and the additive transitivity. However, these consistency definitions using transitivity are unable to measure the consistency degree of an inconsistency linguistic preference relation, and are also unable to identify whether the consistency degree is acceptable. The main aim of this paper is to present a consistency index of linguistic preference relations, and develop a more flexible consistency measure method. Using this consistency measure, we also plan to discuss how to deal with inconsistency in linguistic preference relations, and investigate the consistency properties of collective linguistic preference relations. The rest of this paper is organized as follows. Section 2 introduces some basic notations and operational laws of linguistic variables. Section 3 develops a consistency measure of linguistic preference relations. In Section 4, on the basis of this consistency measure, we discuss how to deal with inconsistency in linguistic preference relations. Section 5 shows the consistency properties of collective linguistic preference relations. In Section 6, two illustrative examples are provided. Concluding remarks are made in Section 7. Finally, all the proofs are presented in Appendix A. 2. Preliminary knowledge: Linguistic variables Xu introduced some basic notations and operational laws of linguistic variables in [23,26,27]. Let S ¼ fsa ja ¼ t; . . . ; 1; 0; 1; . . . ; tg be a linguistic label set with odd cardinality. The label sa represents a possible value for a linguistic variable, and it is required that the linguistic label set should satisfy the following characteristics: (1) The set is ordered: sa > sb if and only if a > b. (2) There is a negation operator: neg(sa) = sa. We call this linguistic label set S as the linguistic scale. For example, S can be defined as: 9 8 > = < s4 ¼ extremely poor; s3 ¼ very poor; s2 ¼ poor > : S ¼ s1 ¼ slightly poor; s0 ¼ fair; s1 ¼ slightly good > > ; : s2 ¼ good; s3 ¼ very good; s4 ¼ extremely good To preserve all the given information, we extend the discrete linguistic label set S to a continuous linguistic label set S ¼ fsa ja 2 ½q; qg, where q (q P t) is a sufficiently large positive integer. If sa 2 S, then we call sa the original linguistic label; otherwise, we call it the virtual linguistic label. In general, the decision maker uses the original linguistic labels to evaluate alternatives, and the virtual linguistic labels can only appear in operations. Consider any two linguistic terms sa ; sb 2 S, and l; l1 ; l2 2 ½0; 1 , Xu introduced some operational laws as follows: (1) (2) (3) (4) (5)

sa  sb = sa+b; s a  s b = s b  s a; lsa = sla; (l1 + l2)sa = l1sa  l2sa; l(sa  sb) = lsa  lsb.

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Let s 2 S, we denote I(s) as the lower index of s, and call it the gradation of s in S. For example, if s = sa, then I(s) = a. 3. Consistency measures of linguistic preference relations There is a finite set of alternatives, X ¼ fx1 ; x2 ; . . . ; xn gðn P 2Þ, as well as a linguistic pre-establish label set, S ¼ fsa ja ¼ t; . . . ; 1; 0; 1; . . . ; tg. The decision maker provides his/her opinions on X as a linguistic preference relation, A ¼ ðaij Þnn  X  X , with a membership function uA : X · X ! S, where uA ðxi ; xj Þ ¼ aij denotes the linguistic preference degree of the alternative xi over xj. We assume, without loss of generality, that A is reciprocal in the sense that aij  aji ¼ s0 (s0 is the median label in S). Similar to GDM using numerical preference relations [14], the consistency problem of linguistic preference relations also has two problems: (i) When can a decision maker, considered individually, be said to be consistent and, (ii) When can a whole group of decision makers be considered consistent. In this paper we focus on the first problem. Furthermore, we tackle the first problem from the following two aspects: (1) When can a linguistic preference relation be said to be perfectly consistent. (2) For an inconsistent linguistic preference relation, how to measure the degree of consistency, and when can its consistency degree be considered acceptable. In the following subsections, we firstly introduce the concepts of transitive linguistic preference relations and the consistent linguistic preference relations. Then we present the consistency index of linguistic preference relations for measuring consistency degree of linguistic preference relations. Finally we establish the thresholds of the consistency index to identify whether the linguistic preference relations provided by decision makers are of acceptable consistency. 3.1. The consistency index of linguistic preference relations Definition 1. A ¼ ðaij Þnn is called a transitive linguistic preference relation if there exists aik > s0 and akj > s0 , then aij > s0 for i; j; k ¼ 1; 2; . . . ; n. Consider a linguistic label set S. We say S is the arithmetic progression linguistic label set in the sense that if the alternative xi is ‘‘sa’’ over the alternative xj, and xj is ‘‘sb’’ over the alternative xk, then xi is ‘‘sa+b’’ over xk for any sa, sb 2 S. In other words, if the preference intensities of all gradations in S form an arithmetic progression, we call S the arithmetic progression linguistic label set. In this paper, we assume that S is an arithmetic progression linguistic label set. Under this assumption, we define the consistency of linguistic preference relations as follows. Definition 2. A = (aij)n·n is called a consistent linguistic preference relation if there exists aik  akj = aij for i, j, k = 1, 2, . . . , n. We also call this consistency definition as the additive transitivity of linguistic preference relations. We can easily obtain Theorem 1. Theorem 1. A = (aij)n·n is a transitive linguistic preference relation under the condition that A is a consistent linguistic preference relation. The Proof of Theorem 1 is provided in Appendix A. Theorem 1 shows that consistency is a stronger condition than transitivity. However, it is hard to obtain perfect consistency using the linguistic preference relations, especially when the number of alternatives is large. In the rest of the subsection, we will present a consistency index of linguistic preference relations, by defining the distance (i.e., the deviation degree) of a linguistic preference relation to the set of consistent linguistic preference relations. This consistency index can be used to measure consistency degree of linguistic preference relations.

Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

Let sa, sb 2 S be two linguistic variables, Xu [27] defines the distance between sa and sb as follows: ja  bj ; dðsa ; sb Þ ¼ T where T is the number of linguistic terms in the set S.

433

ð1Þ

Definition 3. Let A ¼ ðaij Þnn and B ¼ ðbij Þnn be two linguistic preference relations, then we define the distance between A and B as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n n X X u 2 2 ð2Þ dðA; BÞ ¼ t ðdðaij ; bij ÞÞ : nðn  1Þ j¼iþ1 i¼1 In [27], Xu defined a different distance metric (i.e., deviation degree) between two linguistic preference relations, and obtained a very important property on GDM using linguistic preference relations (see Corollary 1 in [27]). When using our distance metric, we can obtain the same property (see Theorem 6). Moreover, this distance metric will bring us the additional advantages in establishing the consistency thresholds (see Theorem 3). This is the reason that we adopt the new distance metric in this paper. Definition 4. Let A = (aij)n·n be a linguistic preference relation and M pn be the set of n · n consistent linguistic preference relations, then we define the distance between A and M pn as follows: dðA; M pn Þ ¼ min dðA; P Þ: ð3Þ P 2M pn

We set dðA; M pn Þ as the consistency index (CI) of the linguistic preference relation A, namely CIðAÞ ¼ dðA; M pn Þ:

ð4Þ

The above consistency index has a definite physical implication and reflects the deviation degree between the linguistic preference relation A and the consistent linguistic preference relations. Obviously, the smaller the value of CI(A), the more consistent the linguistic preference relation A. If CI(A) = 0, then A is a consistent linguistic preference relation. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pn Pn Pn 2 2 1 Lemma 1. dðA; M pn Þ P T1 nðn1Þ j¼iþ1 i¼1 ðIðaij Þ  n k¼1 ðIðaik Þ þ Iðakj ÞÞÞ . The Proof of Lemma 1 is provided in Appendix A. P Lemma 2. Let P ¼ ðpij Þnn , where pij ¼ 1n nc¼1 ðaic  acj Þ, then P is a consistent linguistic preference relation. The Proof of Lemma 2 is provided in Appendix A. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pn Pn Pn 2 2 1 Theorem 2. dðA; M pn Þ ¼ T1 nðn1Þ . ðIða Þ  ðIða Þ þ Iða ÞÞÞ ij ik kj j¼iþ1 i¼1 k¼1 n The Proof of Theorem 2 is provided in Appendix A. By Theorem 2, Definition 3 and Eq. (4), we can easily obtain Corollary 1. Corollary 1. CIðAÞ ¼ dðA; P Þ: 3.2. Establishing the consistency thresholds By Corollary 1, we find that CI(A) virtually reflects the deviation degree between A and P ¼ ðpij Þnn . We may approximately consider P as the impersonal linguistic preference relation. Let eij ¼ Iðaij Þ  Iðpij Þ, then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn 2 2 we have CIðAÞ ¼ T1 nðn1Þ j¼iþ1 i¼1 ðeij Þ . The decision makers often have certain consistency tendency in making pairwise comparisons, which is discussed by Jong [15]. The values of ij relatively centralizes the domain close to zero. Thus, we assume that eij (i < j) is independent normally distributed with mean 0 and standard deviation r. Theorem 3. nðn1Þ ðT  r1  CIðAÞÞ2 is a chi-square distribution with nðn1Þ degrees of freedom, namely 2 2 nðn1Þ 2 1 2 nðn1Þ ðT  r  CIðAÞÞ  v ð 2 Þ, on the condition that eij (i < j) is independent normally distributed with 2 mean 0 and standard deviation r, namely eij  N(0,r2).

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Table 1 The values of CI for different n and T when a = 0.1 and r0 = 2 T=5 T=9 T = 17

n=3

n=4

n=5

n=6

n=7

n=8

n=9

0.1765 0.0980 0.0519

0.2424 0.1347 0.0713

0.2790 0.1550 0.0821

0.3019 0.1677 0.0888

0.3176 0.1765 0.0934

0.3290 0.1828 0.0968

0.3376 0.1876 0.0993

The Proof of Theorem 3 is provided in Appendix A. If we further assume that r2 ¼ r20 , namely eij  N ð0; r20 Þ, then the consistency measure is to test hypothesis H0 versus hypothesis H1: H0: r2 6 r20 ; H1: r2 < r20 . P P e The degrees of freedom of the estimator v2 ¼ nj¼iþ1 ni¼1 ðrij0 Þ2 is nðn1Þ . This is a one-sided right-tailed test, 2 2 and we can get the critical value ka of the v distribution at the significance level a. In this way, we have that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 2 ka : CI ¼ ð5Þ T nðn  1Þ If CIðAÞ 6 CI, we conclude that A is of acceptable consistency; otherwise, we conclude that A is of unacceptable consistency. According to the actual situation, the decision makers can set different values for a and r0. Table 1 shows the values of CI for different n and T when setting a = 0.1 and r0 = 2. 4. How to deal with inconsistency in linguistic preference relations When the linguistic preference relations provided by decision makers are of unacceptable consistency (CIðAÞ > CI), the decision makers often want to adjust the elements in linguistic preference relations in order to improve the consistency. In this section, we will present two techniques to deal with the inconsistency in linguistic preference relations. 4.1. An optimization method Let A be a linguistic preference relation with unacceptable consistency. In fact, the main work of dealing with inconsistency in A = (aij)n·n is to find a suitable linguistic preference relation with acceptable consistency A ¼ ðaij Þnn , where aij ¼ aij  xij . In order to preserve the information in A as much as possible, we hope that the distance (or the deviation degree) between A and A is minimal, namely ! n n X X 2 2 min ðIðxij ÞÞ : ð6Þ x nðn  1Þ j¼iþ1 i¼1 At the same time, A must be reciprocal and have the acceptable consistency, namely Iðxij Þ þ Iðxji Þ ¼ 0 and

ð7Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u n n n X X X 1u 2 1 CIðAÞ ¼ t Iðaij þ xij Þ  ðIðaik þ xik Þ þ Iðakj þ xkj ÞÞ T nðn  1Þ j¼iþ1 i¼1 n k¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u n n n X X X 1u 2 1 ¼ t Iðaij Þ þ Iðxij Þ  ðIðaik Þ þ Iðxik Þ þ Iðakj Þ þ Iðxkj ÞÞ T nðn  1Þ j¼iþ1 i¼1 n k¼1 6 CI:

ð8Þ

Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

Therefore, an optimization model to deal with inconsistency in A can be constructed as follows: 8 n n P P > 2 2 > min ð ðIðxij ÞÞ Þ > > x nðn1Þ > j¼iþ1 i¼1 < > s:t: > > > > :

Iðxij Þ þ Iðxji Þ ¼ 0

435

ð9Þ

CIðAÞ 6 CI:

By solving this optimization model, we can obtain the adjusted linguistic preference relation A ¼ ðaij Þnn , where aij ¼ aij  xij . Remark 1. The advantage of this optimization method is to preserve the information in the original preference relations as much as possible. In our future research, we will focus on how to solve the model. However, this method is an artificial consistency improving method, and we do not consider the interventions of the decision makers. We should stress that it is the decision makers, not the mathematical methods that determine which judgements should be adjusted. 4.2. An iterative algorithm In this subsection, we present an iterative algorithm that allows the decision makers to participate in dealing with inconsistency in linguistic preference relations. This algorithm is inspired by the DELPHI approach and the works of Ma [16] and Xu and Wei [22]. The framework of this algorithm can be roughly reflected in Fig. 1. Next, we describe the algorithm in details. Let A be a linguistic preference relation with unacceptable conPn sistency. And let P ¼ ðpij Þnn , where pij ¼ 1n c¼1 ðaic  acj Þ. The main step of this algorithm is to return A to the decision maker to reconsider constructing a new linguistic preference relation A according to his/her new judgements. When structuring A, we suggest that aij 2 ½minfaij ; pij g; maxfaij ; pij g. Follow this procedure until the linguistic preference relation with acceptable consistency is obtained. The algorithm is listed below. Algorithm Step Step Step Step Step

ð0Þ

1: Let Að0Þ ¼ ðaij Þnn ¼ ðaij Þnn and k = 0; Pn ðkÞ ðkÞ ðkÞ ðkÞ 2: Let P ðkÞ ¼ 1n c¼1 ðaic þ acj Þ; nn ¼ ðp ij Þnn , where p ij 3: Calculate the consistency index CI(A(k)) of A(k), where CIðAðkÞ Þ ¼ dðAðkÞ ; P ðkÞ Þ; 4: If CIðAðkÞ Þ 6 CI, then go to Step 6; otherwise, continue with the next step; 5: Return A(k) to the decision maker to reconsider constructing a new linguistic preference relation ðkþ1Þ Aðkþ1Þ ¼ ðaij Þ according to his new judgements. When structuring A(k+1), it is required that ðkþ1Þ

ðkÞ

ðkÞ

ðkþ1Þ

aij 2 ½minfaij ; pij ðkÞ g; maxfaij ; pij ðkÞ g, aij return to Step 2;

¼ s0 , and A(k+1)5A(k). Let k = k + 1 and

Adjustments

Alternatives Linguistic preference relations with unacceptable consistency Linguistic label set

ðkþ1Þ

 aji

Decision makers Suggestions

Linguistic preference relations with acceptable consistency

DECISION MAKING ENVIRONMENT

Fig. 1. The iterative algorithm to deal with inconsistency in linguistic preference relations.

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Step 6: Let A ¼ AðkÞ . Output the adjusted linguistic preference relation with acceptable consistency A, and its consistency index CIðAÞ. Theorem 4. Let A be a linguistic preference relation with unacceptable consistency, and CI be the corresponding consistency threshold. Let {A(k)} be linguistic preference relation sequence in Algorithm and CI(A(k)) be the consistency index of A(k). Then, we have that CI(A(k+1)) < CI(A(k)) for each k, and lim ðCIðAðkÞ ÞÞ 6 CI. k!1

The Proof of Theorem 4 is provided in Appendix A. Theorem 4 guarantees that any linguistic preference relation with unacceptable consistency can be transformed into one with acceptable consistency. Remark 2. In Step 5 of the above algorithm, we require that the decision maker construct A(k+1) in the ðkþ1Þ ðkÞ ðkÞ suggested interval, namely aij 2 ½minfaij ; pij ðkÞ g, maxfaij ; pij ðkÞ g. But, the decision maker may be unwilling to adjust the linguistic preference relation in this interval. In order to avoid the limitation, we may also allow the decision maker to adjust the linguistic preference relation more freely. If CI(A(k+1)) 6 CI(A(k)), continue the algorithm; otherwise we think the decision maker’s adjustment is not successful. 5. Consistency properties of collective linguistic preference relations Consider a group decision making problem with linguistic preference relations. Let D = {d1,d2, . . . ,dm} be the set of decision makers, and k = {k1,k2, . . . ,km} be the weight vector of decision makers, where Pm kk > 0; k ¼ 1; 2; . . . ; m; k¼1 kk ¼ 1. Let A1, A2, . . . ,Am be the linguistic preference relations provided by m decision makers dk (k = 1,2, . . . ,m), where Ak ¼ ðakij Þnn , akij 2 S (k = 1, 2, . . . , m; i, j = 1, 2, . . . , n). Then, denote A ¼ ðaij Þnn ¼ k1 A1  k2 A2  . . .  km Am as the collective linguistic preference relation of A1, A2, . . . ,Am, where aij ¼ k1 a1ij  k2 a2ij  . . .  km amij ;

i; j ¼ 1; 2; . . . ; n:

ð10Þ

Xu [27] proved that A is a reciprocal linguistic preference relation. Now, we further discuss the property on the consistency of A. Theorem 5. CIðAÞ 6 CI under the condition that CIðAk Þ 6 CIðk ¼ 1; 2; . . . ; mÞ. The Proof of Theorem 5 is provided in Appendix A. By Theorem 5, we can easily obtain Corollary 2. Corollary 2. A is a consistent linguistic preference relation on the condition that all of A1, A2, . . . , Am are consistent linguistic preference relations. When using our distance metric, we can also obtain the same property to Xu [27]. The proof of this property (see Theorem 6) is similar to that of Theorem 5. Theorem 6. Let A1, A2, . . . , Am and B be m + 1 linguistic preference relations. If d(Ak, B) < a(k = 1, 2, . . . , m), then dðA; BÞ < a. The Proof of Theorem 6 is provided in Appendix A. 6. Illustrative examples In order to show how these theoretical results work in practice, let us consider the following two examples. Let 9 8 s4 ¼ extremely poor; s3 ¼ very poor; s2 ¼ poor > > > > = < Example S ¼ s1 ¼ slightly poor; s0 ¼ fair; s1 ¼ slightly good > > > > ; : s2 ¼ good; s3 ¼ very good; s4 ¼ extremely good be the linguistic pre-establish label set used in these examples.

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6.1. Example 1 In this example, a decision maker supplies a linguistic preference relation A: 0 1 s0 s2 s1 s1 B s2 s0 s4 s2 C C A¼B @ s1 s4 s0 s4 A: s1 s2 s4 s0 Obviously, A is a transitive linguistic preference relation, but not a consistent linguistic preference relation. By using our consistency measure method, we find CIðAÞ ¼ 0:2103 > CI ¼ 0:1347 (here we set a = 0.1 and r0 = 2), which shows that the decision maker needs to adjust the elements in A in order to improve the consistency. In this subsection, we use two techniques to deal with the inconsistency in A. (1) Dealing with the inconsistency in A using the optimization model. Let yij = I(xij). The optimization model to deal with inconsistency in A, which is introduced in Section 4.1, is as follows: ! 8 n n P P > 2 > 1 > min 6 ðy ij Þ > > > j¼iþ1 i¼1 > x > > 4 4 4 4 > P P P P > 1 1 > Iðaij Þ  4 ðIðaik Þ þ Iðakj ÞÞ þ y ij  4 ðy ik þ y kj Þ > > > j¼iþ1 i¼1 k¼1 k¼1 > > : 2 6 486CI : By solving the above model using MATLAB software, the decision maker obtains the values of yij (i,j = 1,2, . . . ,4). Let A ¼ ðaij Þ be the adjusted linguistic preference relation. Since aij ¼ aij  I 1 ðy ij Þ, we have that 0 1 s0 s1:4 s1 s1:6 Bs s0 s3 s2:4 C B 1:4 C A¼B C @ s1 s3 s0 s3 A s1:6

s2:4

s3

s0

and CIðAÞ ¼ 1:344 < CI. (2) Dealing with the inconsistency in A using the iterative algorithm. ðkÞ

ðkÞ

ðkÞ

ðkÞ

In order to clearly depict this method, let V ðkÞ ¼ ðvij Þnn , where vij ¼ ½minfaij ; pij ðkÞ g; maxfaij ; pij ðkÞ g. Using the iterative algorithm, the decision maker adjusts A in the following way: (i) In the first iteration, the decision maker firstly let A(0) = A, and then calculates P ð0Þ (where P4 ð0Þ ð0Þ ð0Þ pij ¼ 14 c¼1 ðaic þ acj Þ) and V(0): 0 1 s0 s0 s1:25 s2:75 B s s0 s1:25 s2:75 C B 0 C P ð0Þ ¼ B C; @ s1:25 s1:25 s0 s1:5 A s2:75 ð0Þ

V ð0Þ ¼ ðvij Þ44

s2:75 0

s1:5 s0 s0 ½s0 ; s2  B ½s ; s  s0 B 2 0 ¼B @ ½s1:25 ; s1  ½s4 ; s1:25 

½s1 ; s1:25  ½s1:25 ; s4  s0

½s2:75 ; s1  ½s2:75 ; s2  ½s4 ; s1:5 

1 ½s1 ; s2:75  ½s2 ; s2:75  C C C: ½s1:5 ; s4  A s0

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In succession, the decision maker constructs a new linguistic preference relation Að1Þ ¼ ðaij Þ44 (where ð1Þ ð0Þ ð1Þ ð1Þ aij 2 vij and aij  aji ¼ s0 ): 0 1 s0 s2 s1 s2 Bs s3 s2 C B 2 s0 C Að1Þ ¼ B C: @ s1 s3 s0 s4 A s2

s2

s4

s0

Because CIðAð1Þ Þ ¼ 0:1667 > CI, we continue this algorithm. (ii) In the second iteration, the decision maker firstly calculates P ð1Þ and V (1): 0 1 s0 s0:5 s1:25 s3:25 Bs s0 s0:75 s2:75 C B 0:5 C P ð1Þ ¼ B C; @ s1:25 s0:75 s0 s2 A s3:25 ð1Þ

s2:75 0

V ð1Þ ¼ ðvij Þ44

s2 s0 s0 ½s0:5 ; s2  B ½s ; s  s0 B 2 0:5 ¼B @ ½s1:25 ; s1  ½s3 ; s0:75  ½s3:25 ; s2  ½s2:75 ; s2 

½s1 ; s1:25  ½s0:75 ; s3  s0 ½s4 ; s2 

1 ½s2 ; s3:25  ½s2 ; s2:75  C C C: ½s2 ; s4  A s0 ð2Þ

Successively, the decision maker constructs a new linguistic preference relation Að2Þ ¼ ðaij Þ44 (where ð2Þ ð1Þ ð2Þ ð2Þ aij 2 vij and aij  aji ¼ s0 ): 0 1 s0 s2 s1 s3 Bs s3 s2 C B 2 s0 C Að2Þ ¼ B C: @ s1 s3 s0 s3 A s2

s2

s3

s0

Because CIðAð2Þ Þ ¼ 0:1322 < CI, let A ¼ Að2Þ and the algorithm ends. Denote A as the adjusted linguistic preference relation using the iterative algorithm. Obviously, we can find that dðA; AÞ ¼ 0:0642 < dðA; AÞ ¼ 0:1111, which shows that the optimization model can better preserve the information in the original preference relation. On the other hand, we also show that the iterative algorithm pays more attention to the interventions of the decision makers.

6.2. Example 2 Let us consider the example used by Xu [27]. In the example, there are five decision makers d k ðk ¼ 1; 2; . . . ; 5Þ. The decision makers compare five alternatives with respect to certain criterion by using SExample and construct, respectively, the linguistic preference relations. Suppose that the linguistic preference relation B is given by a leading decision maker, and the linguistic preference relations A1 ; A2 ; A3 and A4 are given by the other four decision makers, respectively. They are listed as follows: Table 2 The results of the illustrative example

d(•,B) CI(•)

A1

A2

A3

A4

A

A

0.1648 0.1176

0.1757 0.1030

0.1648 0.1277

0.2018 0.0981

0.1438 0.0885

0.1421 0.0965

Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

0

s0 B s0 B B B ¼ B s2 B @ s1 s4 0 s 0 B s1 B B A2 ¼ B s2 B @ s0 s4 0 s 0 B s2 B B A4 ¼ B s 0 B @ s1 s2

s0 s0 s1 s0 s3 s1 s0 s1 s0 s0 s2 s0 s1 s1 s0

s2 s1 s0 s2 s1 s2 s1 s0 s1 s3 s0 s1 s0 s1 s2

s1 s0 s2 s0 s2 s0 s0 s1 s0 s1 s1 s1 s1 s0 s1

1 s4 s3 C C C s 1 C; C s2 A s0 s4 1 s0 C C C s 3 C; C s1 A s0 s2 1 s0 C C C s 2 C: C s1 A

0

s0 s1 s3 B s1 s0 s1 B B A1 ¼ B s3 s1 s0 B @ s1 s0 s1 s3 s2 s2 0 s s0 s3 0 B s0 s0 s2 B B A3 ¼ B s3 s2 s0 B @ s1 s2 s1 s3 s2 s1

s1 s0 s1 s0 s0 s1 s2 s1 s0 s1

439

1 s3 s2 C C C s 2 C; C s0 A s0 s3 1 s2 C C C s 1 C; C s1 A s0

s0

Note. When using the original example presented in [27], we find that A4 does not satisfy the reciprocity property of linguistic preference relations. Thus, we are afraid there maybe minor typing errors in it. Instead, in this paper, we use the elements of the upper triangle and the reciprocity property to reconstruct a new A4. Without loss of generality, suppose k1 ¼ k2 ¼ k3 ¼ k4 ¼ 14, and the collective linguistic preference relation of A1, A2, A3 and A4 is A. We also consider the case of k1 = 0.4, k2 = 0.3, k3 = 0.2, k4 = 0.1. Then the collective linguistic preference relation of A1, A2, A3 and A4 is A. 0

s0

B B s0:5 B A¼B B s2 B @ s0:25 s3

s0:5

s2

s0:25

s0 s0:75

s0:75 s0

s0:75 s0:5

s0:75 s1

s0:5 s2

s0 s0:25

s3

1

0

s0

C B C B s0:1 C B C; A ¼ B s2:4 C B C B s0:25 A @ s0:3 s0 s3:2 s1 s2

s0:1

s2:4

s0:3

s0 s0:4

s0:4 s0

s0:5 s0:6

s0:5 s1:2

s0:6 s2:1

s0 s0:2

s3:2

1

C s1:2 C C s2:1 C C: C s0:2 A s0

Table 2 shows the results. We can find that the results in this example are in accordance with Theorems 5 and 6. 7. Conclusions The consistency measure is a basic problem in GDM using preference relations. This paper mainly contribute to developing a more flexible consistency measure method for individual linguistic preference relations. This consistency measure method is based on the additive transitivity [14] of preference relations, and the deviation measure of linguistic preference relations presented in Xu [27]. By defining the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations, and setting it as the consistency index of linguistic preference relations, we propose the consistency measure method. Compared with the traditional consistency definitions using transitivity, this consistency measure is more flexible. It can further measure the consistency degree of an inconsistency linguistic preference relation and test whether the inconsistency linguistic preference relation is of acceptable consistency. On the basis of this consistency measure method, we propose two techniques to deal with the inconsistency in linguistic preference relations, and also discuss the consistency properties of collective linguistic preference relations. The results in this paper are very important for the application of linguistic preference relations in GDM. In our future research, we plan to discuss how to solve the optimization model introduced in Section 4.1. At the same time, more simulation and analysis will be done to show the advantage of our measure method.

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Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

Acknowledgements We are very grateful to the editor and the anonymous referees for their valuable comments and suggestions, which have been very helpful in improving the paper. Moreover, Yucheng Dong and Yinfeng Xu acknowledge the financial support of grants (Nos. 70121001, 70525004 and 70471035) from NSF of China. And, Hongyi Li acknowledge the financial support of a grant (No. CUHK4443/04H) from the Research Grants Council of the Hong Kong Special Administrative Region, China. Appendix A Proof of Theorem 1. Since A = (aij)n·n is a consistent linguistic preference relation, it follows that aij ¼ aik  akj ;

i; j; k ¼ 1; 2; . . . ; n:

ð11Þ

Without loss of generality, assume that aik > s0

ð12Þ

akj > s0 :

ð13Þ

and

By (11)–(13), we have Iðaij Þ ¼ Iðaik Þ þ Iðakj Þ > 0

ð14Þ

aij > s0 ;

ð15Þ

i.e.

which completes the proof of Theorem 1. h Proof of Lemma 1. Since vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n n X X u 2 1 dðA; M pn Þ ¼ min t ðIðaij Þ  Iðpij ÞÞ2 : T P 2M pn nðn  1Þ j¼iþ1 i¼1

ð16Þ

Let c > 1, then we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n n X X u 2 1 2 min t ðlogc ðcIðaij Þ Þ  logc ðcIðpij Þ ÞÞ : dðA; M pn Þ ¼ T P n 2M pn nðn  1Þ j¼iþ1 i¼1

ð17Þ

Let A0 ¼ ða0ij Þnn ¼ ðcIðaij Þ Þnn and P 0 ¼ ðp0ij Þnn ¼ ðcIðpij Þ Þnn . Since A is a linguistic preference relation and P is a consistent preference relation, we obtain that a0ij  a0ji ¼ cIðaij Þ  cIðaji Þ ¼ cIðaij ÞþIðaji Þ ¼ 1;

i; j ¼ 1; 2; . . . ; n

ð18Þ

i; j; k ¼ 1; 2; . . . ; n:

ð19Þ

and p0ik  p0kj ¼ cIðpik Þ  cIðpkj Þ ¼ cIðpij Þ ¼ p0ij ;

By (18) and (19), we have that A 0 is a multiplicative preference relation and P 0 is a consistent multiplicative preference relation. Let M 0pn ¼ fP 0 ¼ ðp0ij Þnn jp0ij ¼ cIðpij Þ and P ¼ ðpij Þnn 2 M pn g, and M Rþn be the set of n · n consistent multiplicative preference relations. Obviously we have M 0pn  M Rþn :

ð20Þ

Let E = (eij)n·n and F = (fij)n·n be two multiplicative preference relations, we define the distance between E and F as follows:

Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

441

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n n X X u 2 t DðE; F Þ ¼ t ðlogc ðeij Þ  logc ðfij ÞÞ2 : nðn  1Þ j¼iþ1 i¼1

ð21Þ

By (17), (20) and (21), it can be shown that 1 1 min min DðA0 ; W Þ: DðA0 ; P 0 Þ P dðA; M pn Þ ¼ min dðA; P Þ ¼ 0 0 P 2M pn T P 2M pn T W 2M Rþn

ð22Þ

By the theory of logarithmic least squares prioritization method used in AHP [20], we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u n n n u 2 X X 1X t 0 min DðA ; W Þ ¼ Iðaij Þ  ðIðaik Þ þ Iðakj ÞÞ : W 2M Rþ nðn  1Þ j¼iþ1 i¼1 n k¼1 n Thus

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u n n n X X X 1u 2 1 dðA; M pn Þ P t Iðaij Þ  ðIðaik Þ þ Iðakj ÞÞ ; T nðn  1Þ j¼iþ1 i¼1 n k¼1

which completes the proof of Lemma 1. Proof of Lemma 2. Since pik  pkj ¼ ¼

n 1X ðaic  ack Þ n c¼1

! 

ð23Þ

ð24Þ

h

n 1X ðakc  acj Þ n c¼1

! ¼

n 1X ðaic  ack  akc  acj Þ n c¼1

n n 1X 1X ðaic  s0  acj Þ ¼ ðaic  acj Þpij ; n c¼1 n c¼1

we complete the proof of Lemma 2. h Pn Proof of Theorem 2. Since pij ¼ 1n k¼1 ðaik  akj Þ, from Definition 3, we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u n n n X X 1u 2 1X t dðA; P Þ ¼ Iðaij Þ  ðIðaik Þ þ Iðakj ÞÞ : T nðn  1Þ j¼iþ1 i¼1 n k¼1

!

ð25Þ

ð26Þ

Since P is a consistent linguistic preference relation, it can be shown that dðA; P Þ P min dðA; P Þ ¼ dðA; M pn Þ:

ð27Þ

P 2M pn

By Lemma 1 and (26), we also have dðA; P Þ 6 dðA; M pn Þ:

ð28Þ

dðA; M pn Þ ¼ dðA; P Þ;

ð29Þ

So

i.e.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u n n n X X 1u 2 1X t Iðaij Þ  ðIðaik Þ þ Iðakj ÞÞ ; dðA; M pn Þ ¼ T nðn  1Þ j¼iþ1 i¼1 n k¼1

which completes the proof of Theorem 2.

ð30Þ

h

Pn Pn e 2 e 2 Proof of Theorem 3. Since nðn1Þ ðT  r1  CIðAÞÞ ¼ j¼iþ1 i¼1 ð rij Þ and rij (i < j) is independent normally 2 2 ðT  r1  CIðAÞÞ  v2 ðnðn1Þ Þ. This comdistributed with mean 0 and standard deviation 1, we have that nðn1Þ 2 2 pletes the proof of Theorem 3. h

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Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

Proof of Theorem 4. From (3) and (4), we have that CIðAðkþ1Þ Þ ¼ min dðAðkþ1Þ ; P Þ 6 dðAðkþ1Þ ; P ðkÞ Þ:

ð31Þ

P 2M pn

ðkþ1Þ

Since aij (

ðkÞ

ðkÞ

2 ½minfaij ; pij ðkÞ g; maxfaij ; pij ðkÞ g and A(k+1)5A(k), we have ðkþ1Þ

ðdðaij

2

ðkÞ

; pij ðkÞ ÞÞ 6 ðdðaij ; pij ðkÞ ÞÞ 2

2 2

ðkþ1Þ ðkÞ ðdðass ; pss ðkÞ ÞÞ < ðdðaðkÞ ss ; p ss ÞÞ

for 8 i; j;

ð32Þ

for 9 s; s:

From (2) and (32), it can be shown that dðAðkþ1Þ ; P ðkÞ Þ < dðAðkÞ ; P ðkÞ Þ:

ð33Þ

By Corollary 1, we have that CIðAðkÞ Þ ¼ dðAðkÞ ; P ðkÞ Þ:

ð34Þ

From (31), (33) and (34), it can be shown that CIðAðkþ1Þ Þ < CIðAðkÞ Þ:

ð35Þ

For each k, CI(A(k)) P 0. Thus, the sequence {CI(A(k))} is monotone decreasing and has lower bounds. Then we have lim ðCIðAðkÞ ÞÞ ¼ inffCIðAðkÞ Þg:

ð36Þ

k!1

Let lim ðAðkÞ Þ ¼ A1 , then k!1

CIðA1 Þ ¼ inffCIðAðkÞ Þg:

ð37Þ

Suppose that CIðA1 Þ > CI:

ð38Þ 1

By applying the above algorithm to continue improving the consistency of A , we can obtain the adjusted linguistic preference relation A1 . Obviously, we have CIðA1 Þ < CIðA1 Þ:

ð39Þ

Thus CIðA1 Þ < inffCIðAðkÞ Þg;

ð40Þ (k)

which contradicts the definition of inf{CI(A )}. This completes the proof of Theorem 4. h Proof of Theorem 5. Let !2 n 1X k k ¼  ðIðaic Þ þ Iðacj ÞÞ ; n c¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn k 2 then CIðAk Þ ¼ T1 nðn1Þ j¼iþ1 i¼1 y ij . Because CIðAk Þ 6 CIðk ¼ 1; 2; . . . ; mÞ, it can be shown that y kij

Iðakij Þ

n n X X j¼iþ1 i¼1

y kij 6

nðn  1Þ 2 ðT  CIÞ : 2

Based on (10) and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u n n n X X 1u 2 1X t CIðAÞ ¼ Iðaij Þ  ðIðaic Þ þ Iðacj ÞÞ ; T nðn  1Þ j¼iþ1 i¼1 n c¼1

ð41Þ

ð42Þ

ð43Þ

Y. Dong et al. / European Journal of Operational Research 189 (2008) 430–444

443

we have that

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !!2 u n n m n X X X 1u 2 1X t k k k : CIðAÞ ¼ kk ðIðaij Þ  ðIðaic Þ þ Iðacj ÞÞÞ T nðn  1Þ j¼iþ1 i¼1 k¼1 n c¼1

By (41) and (44), it can be shown that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u q ffiffiffiffiffiffiffiffiffi ffi n n m   X X X X u 1 2 CIðAÞ ¼ t ðk2 y k Þ þ 2 kk kl y kij y lij T nðn  1Þ j¼iþ1 i¼1 k¼1 k ij k
k
which completes the proof of Theorem 5.

ð44Þ

ð45Þ

ð46Þ

k¼1

h

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn kffi 2 Proof of Theorem 6. Let zkij ¼ ðIðakij Þ  Iðbij ÞÞ2 , then dðAk ; BÞ ¼ T1 nðn1Þ j¼iþ1 i¼1 zij . Because dðAk ; BÞ < Pn Pn k 2 nðn1Þ aðk ¼ 1; 2; . . . ; mÞ, it can be shown that j¼iþ1 i¼1 zij 6 2 ðT  aÞ . Consequently, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv uX n n X 1 2 u t ðIðaij Þ  Iðbij ÞÞ2 dðA; BÞ ¼ T nðn  1Þ j¼iþ1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv !2ffi u n n m u X X X 1 2 t ðkk ðIðakij Þ  Iðbij ÞÞÞ ¼ T nðn  1Þ j¼iþ1 i¼1 k¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv ! u n qffiffiffiffiffiffiffiffiffi n m X X u 1 2 tX X 2 k k l ¼ ðk z Þ þ 2 kk kl zij zij T nðn  1Þ j¼iþ1 i¼1 k¼1 k ij k
k
k¼1

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