A consistency and consensus based decision support model for group decision making with multiplicative preference relations

Decision Support Systems 52 (2012) 757–767

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A consistency and consensus based decision support model for group decision making with multiplicative preference relations Zhibin Wu, Jiuping Xu ⁎ Uncertainty Decision-making Laboratory, Sichuan University, Chengdu 610064, PR China

a r t i c l e

i n f o

Article history: Received 26 December 2010 Received in revised form 11 September 2011 Accepted 24 November 2011 Available online 2 December 2011 Keywords: Group decision making Multiplicative preference relation Consistency Consensus Decision support model

a b s t r a c t In group decision making (GDM) with multiplicative preference relations (also known as pairwise comparison matrices in the Analytical Hierarchy Process), to come to a meaningful and reliable solution, it is preferable to consider individual consistency and group consensus in the decision process. This paper provides a decision support model to aid the group consensus process while keeping an acceptable individual consistency for each decision maker. The concept of an individual consistency index and a group consensus index is introduced based on the Hadamard product of two matrices. Two algorithms are presented in the designed support model. The first algorithm is utilized to convert an unacceptable preference relation to an acceptable one. The second algorithm is designed to assist the group in achieving a predefined consensus level. The main characteristics of our model are that: (1) it is independent of the prioritization method used in the consensus process; (2) it ensures that each individual multiplicative preference relation is of acceptable consistency when the predefined consensus level is achieved. Finally, some numerical examples are given to verify the effectiveness of our model. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Over the past few decades, a number of multiple criteria decision making theories, methods, and applications have been developed in the fields of management science, operational research, and industrial engineering [6,8,22,30,34,37,52,54]. Because of the increasing complexity of the socio-economic environment nowadays, many decision-making processes in the real world take place in group settings. Preference relations are popular and powerful techniques to model experts' preferences in group decision making (GDM). The three commonly used preference relations are multiplicative preference relations (also known as pairwise comparison matrix in the Analytical Hierarchy Process) [34,38,40,41,44], fuzzy preference relations [33,42], and linguistic preference relations [1,15,47]. Pairwise comparison focuses exclusively on two alternatives at a time and facilitates experts in expressing their preferences. However, this way of providing preferences limits the experts' global perception of the alternatives [11]. As a consequence, the provided preference relations may lead to irrational or inconsistent conclusions. Therefore, it is important to study the conditions under which consistency is satisfied. For GDM using preference relations, the consistency measure itself includes two sub-problems [17,26]: (1) When can a decision maker, considered individually, be said to be consistent? (2) When can the aggregated group as a whole be considered consistent and in consensus? It is important to

⁎ Corresponding author. Tel.: + 86 28 85418191; fax: + 86 28 85415143. E-mail addresses: [email protected] (Z. Wu), [email protected] (J. Xu). 0167-9236/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2011.11.022

differentiate group consistency from group consensus. Group consistency explores the rationality of the aggregated group preference relation itself and group consensus measures the degree of agreement between individual opinions and the group opinion. Therefore, in essence there are three problems: an individual consistency problem, a collective consistency problem and a consensus problem. When carrying out rational decision making, consistent information is more appropriate than information containing some contradictions. If we were to secure consensus and only thereafter consistency, we would destroy the consensus in favor of individual consistency and the final solution might not be acceptable to the decision makers [12]. Clearly, it is preferable that the set of decision makers reach a high degree of individual consistency and group consensus before applying the selection process. In a rational GDM process, both consensus and consistency should be pursued and sought after. A solution with a high level of consensus is desirable, but also that solution should be derived from information that is consistent enough [2,12]. There are cases where each individual is consistent, but the group solution is not in accordance with any one of the group. We assume that the decision makers expect an internal logic to their judgments, but at the same time they expect a consensus in the group. The aim of this paper is to propose a consistency and consensus based support model to support the consensus reaching process in GDM with multiplicative preference relations. The remainder of this paper is structured as follows. Section 2 briefly reviews the related work on consistency and consensus of preference relations. Section 3 develops a consistency measure of multiplicative preference relation and supplies a method to deal with inconsistency. Section 4

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Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

discusses the consensus measure and the consensus reaching process. Section 5 is devoted to presenting a complete support model for GDM. In Section 6, three illustrative examples are provided. Concluding remarks are made in Section 7. 2. Literature review In this section, we first review the pertinent literature on the consensus models of GDM with preference relations. Then we discuss the basic methodology and motivation of our work.

which is based on the use of a fuzzy majority represented by means of a linguistic quantifier. The model is guided by some linguistic consensus and linguistic consistency measures. Herrera-Viedma et al. developed further a consensus support system model for GDM with multi-granular linguistic preference relations. Cabrerizo et al. [5] discussed a consensus process for linguistic preference relations in GDM with an unbalanced fuzzy linguistic context with incomplete information. There are also a few papers that discuss the consensus reaching process for multiple attribute group decision making (MAGDM) problems [20,36,45,48]. 2.3. The proposed work

2.1. Consistency measures Preference relations are a common format for expressing preferences by using pairwise comparisons. The decision maker makes a direct choice of one object over another when comparing two objects. However, inconsistencies are not unexpected, as making value judgments can be difficult in some situations [34]. Hitherto, many authors have paid attention to the individual consistency problems of preference relations [7,11,15,33,38,46]. For multiplicative preference relations, there are two well established methods to create a consistency index. One of these is the consistency ratio based on Saaty's eigenvector method [38]. The other method proposed by Crawford and Williams [14] is the geometric consistency index based on a row geometric mean prioritization method (RGMM). Furthermore, many methods have been shown to modify multiplicative preference relations with unacceptable consistency to ones of acceptable consistency (see for example, [7,46]). The first issue addressed in this study provides an alternative method to help decision makers improve the consistency of their multiplicative preference relations. 2.2. Consensus models The consensus measure is used to measure the difference among decision makers and is a vital element of consensus models. A consensus process can be viewed as a dynamic and iterative group discussion process with several consensus rounds, in which the decision makers agree to change their preferences following advice given by a moderator. The moderator knows the agreement at each moment of the consensus process by means of the computation of various consensus measures and is in charge of supervising and moving the consensus process towards success [5]. Different models that guide the moderator and the decision makers in achieving the consensus process have been developed for GDM problems (see for example, [3,13,20,24,32,36,48]). For preference relations, the well-known consensus models are the ones for multiplicative preference relations [4,17,39,53], fuzzy preference relations [2,25,31,49], and linguistic preference relations [5,16,21,27,35]. These studies have made great progress in GDM consensus models. Saaty [39] developed a metric for the compatibility of two multiplicative preference relations. Bryson [4] and Yeh et al. [53] presented two indicators to estimate the level of group consensus: the group strong agreement quotient and the group strong disagreement quotient. Dong et al. [17] proposed interesting consensus indexes to measure consensus degree among multiplicative preference relations for AHP group decision making using RGMM. Kacprzyk et al. [31] presented ‘soft’ degrees of consensus for fuzzy preference relations and employed fuzzy linguistic quantifiers to represent a fuzzy majority. Xu [49] presented a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus to determine the importance weight of each preference relation. For incomplete fuzzy preference relations, Herrera-Viedma et al. [25] developed a feedback mechanism to generate advice on how experts should change or complete their preferences in order to reach a solution with high consensus and consistency degrees. For GDM using linguistic preference relations, Herrera et al. [21] introduced a consensus model,

As mentioned earlier, in GDM with preference relations both consistency and consensus should be addressed. There are a few papers that address both of these, particularly for multiplicative preference relations [2,12,17,25]. Dong et al. presented consensus models dealt with multiplicative preference relations [17]. Their consensus models are concise and can be applied to GDM based on the row geometric prioritization method. In [2,12,25], the authors managed consistency and consensus criteria simultaneously for GDM with fuzzy preference relations. They defined a new measure called consistency/consensus level (CCL), CCL = (1− δ) ⋅ CL+ δ ⋅ CR, where CL is the global consistency level, CR is the global consensus degree and δ is the tradeoff parameter between consistency and consensus. These methods are very useful and interesting yet they do not explicitly show if the final individual consistency is limited to an acceptable level. We note that it is possible to convert a multiplicative preference relation into a fuzzy preference relation and vice versa, by means of adequate transformation functions such as proposed in [9,10,23]. In general, research progress in GDM with one kind of preference relations can benefit research in another kind. Nevertheless, the multiplicative preference relation as a typical preference relation has been widely used and accepted in the field of decision making. It is very important in the decision process to look for consistency and consensus models which directly facilitate decision makers. In addition, it is necessary to develop new simple yet effective consensus models for GDM with preference relations. Previous studies have made significant contributions to the consensus models of GDM problems. However, a detailed survey of the literature revealed that consensus modeling of multiplicative preference relations has not been adequately considered. Therefore, based on the fundamental methodology for GDM with preference relations, we focus on a consistency and consensus based support model for multiplicative preference relations. In the designed model, the group multiplicative preference relation is selected as a reference point to modify the decision maker's preference relation which has a maximum group consensus index in each round. Our model has the following characteristics: (1) it is independent of the prioritization method used in the consensus process; (2) it makes each individual multiplicative preference relation still be of acceptable consistency when the predefined consensus level is achieved. Thus, a more flexible and reliable GDM model based on multiplicative preference relations is obtained. 3. Consistency measure In this section, in order for this paper to be as self-contained as possible, we first introduce the concept of multiplicative preference relations and their aggregation in the context of GDM, then we focus our discussion on consistency measure. 3.1. Concepts and properties of the consistency measure Denote N = {1,2,⋯, n} and M = {1,2,⋯, m}. Let X = {x1, x2, ⋯, xn} be a finite set of alternatives, where xi denotes the ith alternative. In a preference relation, an expert provides judgments for every pair of alternatives which reflect the degree of preference of the first alternative over

Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

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the second one. The concept of the multiplicative preference relation (pairwise comparison matrix) is given below.

In order to define the consistency index, we introduce the Hadamard product of two matrices.

Definition 1. ([38]) A multiplicative preference relation on a set of alternatives X is represented by a matrix, A = (aij)n × n ⊂ X × X, being aij belonged precisely to the Saaty 1–9 scale and is interpreted as the ratio of the preference intensity of alternative xi to that of xj, and multiplicative reciprocity is assumed, i.e., aij ⋅ aji = 1, ∀ i, j ∈ N.

Definition 4. ([28]) The Hadamard product of A = (aij)n × n and B = (bij)n × n is defined by C = A ∘ B = (cij)n × n, where cij = aijbij.

Consider a GDM problem with multiplicative preference relations. Let D = {d1, d2, ⋯, dm} be the set of decision makers, and let λ = {λ1, λ2, ⋯, λm} be the weight vector of decision makers, where λk > 0, k ∈ M, ∑ km= 1λk = 1. GDM involves aggregating individual preferences into a single collective preference. Individual judgments can be aggregated in different ways by aggradation operators such as the OWAlike operators proposed by Yager [50,51]. For multiplicative preference relation in AHP, the methods that have been found to be most useful are the aggregation of individual judgments (AIJ) and the aggregation of priorities (AIP) [18,19]. Since the group is assumed to act together as a unit, AIJ is appropriate for our study. In this paper, we choose a geometric average operator to obtain the group preference relation.

Definition 5. Let A = (Aij)n × n and B = (bij)n × n be two multiplicative preference relations, then we define the deviation degree between A and B as follows dðA; BÞ ¼

ð1Þ

k¼1

is called the group multiplicative preference relation. The test of consistency measures is a critical step in decision making using preference relations. Consistent information which does not imply any kind of contradiction is more relevant or important than information containing some contradictions. When a multiplicative preference relation fails to satisfy the consistency requirement, it is necessary to make revisions. In this section, we introduce a consistency measure based on the Hadamard product.

Definition 3. ([38]) A multiplicative preference relation A = (aij)n × n is consistent if aij ¼ aik akj ;∀i; j; k∈N:

ð2Þ

From (2), it follows that aij ¼

1 1 a ¼ aik ; ∀k∈N: aki kj ajk

ð3Þ

Expression (3) shows that for a consistent multiplicative preference relation, any element in the preference relation can be obtained by any row or any column of that preference relation. Multiplying both sides of (3) for all k, we can get that n

aij ¼

!1=n

∏ aik akj k¼1

1=n n
¼ ∏ aik akj :

ð4Þ

k¼1

If A is consistent, then (2) and (4) are equivalent. We note that for a multiplicative preference relation, a matrix can be constructed by (4). That is, for a multiplicative preference relation A = (aij)n × n, we can construct a corresponding matrix G = (gij)n × n, where n




g ij ¼ ∏ aik akj k¼1

1=n

:

ð5Þ

ð6Þ

Remark 1. From the above definition, we have the following property. Let A and B be two multiplicative preference relations with the same dimension, then: (1) d(A, B) ≥ 1. Especially, d(A, B) = 1 if and only if A = B. (2) d(A, B) = d(B, A). From the reciprocity of A and B it follows that −1 X n
n 
X 1 nX 2 aij bji þ aji bij þ 1=n aii bii 2 n i¼1 j¼iþ1 i¼1 −1 X n
 1 1 nX ¼ 2 aij bji þ aji bij þ : n n i¼1 j¼iþ1

dðA; BÞ ¼

Definition 2. Let A1, A2, ⋯, Am be m multiplicative preference relations, where Ak = (aij(k))n × n then A c = (aijc)n × n, where  m
c ðkÞ λk ; aij ¼ ∏ aij

n X n 1 T 1X T T e A∘B e ¼ 2 aij bji ; where e ¼ ð1; 1; ⋯; 1Þn1 : 2 n n i¼1 j¼1

ð7Þ

Definition 6. Let A = (aij)n × n be a multiplicative preference relation. The corresponding consistent preference relation obtained by (5) is denoted by G. Given a threshold value CI, if the consistency index satisfies the following, CI H ðAÞ ¼ dðA; GÞ≤CI;

ð8Þ

then we call A a multiplicative preference relation with acceptable consistency. Here, the consistency measure of A is concerned with the compatibility of the consistent preference relation derived from A with A itself. As was suggested by [39,43], using the Hadamard product is more reliable to measure the compatibility of two matrices constructed by ratio scales. Note that CIH(A) = 1 if and only if A is a consistent multiplicative preference relation. By using the above deviation degrees based on the Hadamard product, the representation of individual consistency and group consensus is unified. Thus we use the threshold that was used to test the compatibility of two multiplicative preference relations to set the acceptable level of consistency. Both Saaty [39] and Wang [43] suggested that the admissible bounds for checking the consistency of A can be set at 1.1. Lemma 1. Let xi > 0, λi > 0, i ∈ N and ∑ in= 1 λi = 1, then n

λ

∏ ðxi Þ i ≤

i¼1

n X

λi xi :

ð9Þ

i¼1

Proof. This is the weighted arithmetic–geometric mean inequality, see ([28], P535). Lemma 2. Suppose x > 0, 0 b θ b 1, then  θ 1 θ 1 þ x ≤ þ x: x x The equality holds if and only if x = 1.

ð10Þ

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Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

Step 4. Apply the following strategy to update the last matrix Ah = (aij, h)n × n.

Proof. The proof of Lemma 2 is straightforward. Theorem 1. Let A1, A2, ⋯, Am be m multiplicative preference relations provided by m decision makers. A c is the group multiplicative preference relation utilizing a geometric average operator. Then,  c CIH A ≤ max fCIH ðAk Þg:

ð11Þ

k

θ

1−θ

Ahþ1 ¼ ðAh Þ ∘ðGh Þ

;

ð13Þ

where θ ∈ (0, 1). Let h = h + 1,
and  return to Step 2. Step 5. Let A ¼ Ah . Output A and CIH A . Step 6. End. From Algorithm 1, we have the following theorem.

Proof. By Definition 5 and Lemma 1, we can complete the proof of Theorem 1. Corollary 1. Let A1, A2, ⋯, Am and A c be as before. Then, CIH(A c) ≤ α under the condition that CIH(Ak) ≤ α, ∀ k ∈ M. Corollary 2. If CIH(Ak) = 1, then CIH(A c) = 1. Under this measure, Theorem 1 proves that the inconsistency of the group is smaller than the largest individual inconsistency. Corollary 1 guarantees that if individual multiplicative preference relations are of acceptable consistency, then the group multiplicative preference relation is also of acceptable consistency. Corollary 2 implies that if every multiplicative preference relation is consistent, then the group multiplicative preference relation is also consistent. 3.2. Consistency improving method When the individual multiplicative preference relation A is not of acceptable consistency, we need to return A to the decision maker to reconsider constructing a new relation. A basic procedure for the consistency improving process is depicted in Fig. 1. Generally, this process is guided by a moderator who helps the decision makers alter their preferences to move toward more consistency. The following algorithm can support the moderator in the consistency improving process.

Algorithm 1. Input: The original individual multiplicative preference relation A = (aij)n × n, the parameter θ ∈ (0, 1) and the threshold CI ¼ α. Output: The adjusted
multiplicative preference relation A and the consistency index CI H A . Step 1. Let A0 = (aij, 0)n × n = (aij)n × n and h = 0. Step 2. Compute Gh by (5) and the consistency index CIH(Ah), where CIH ðAh Þ ¼ dðAh ; Gh Þ:

ð12Þ

Step 3. If CIH(Ah) ≤ α, then go to Step 5; otherwise, go to the next step.

Decision makers

Theorem 2. Algorithm 1 is convergent. That is, we have CIH(Ah + 1) b CIH(Ah) for each h, and lim CIH ðAh Þ ≤ α; ∀α > 1. h→∞

Proof. The proof of Theorem 2 is provided in Appendix A. For an inconsistent multiplicative preference relation A = (aij)n × n, Saaty defined a consistency index in terms of the principal eigenvalue λmax of A as follows [38]: CI ðAÞ ¼

λmax ðAÞ−n : n−1

ð14Þ

The consistency test involves the use of a consistency ratio: CR = CI/RI, where RI is a random index. If CR > 0.1, the decision maker is asked to revise his judgment until an acceptable level of consistency is reached. In the consistency improvement method, it is required that λmax(Ah + 1) b λmax(Ah). In the following, we show that the proposed consistency improvement method in this section meets such a requirement. The following lemma is useful, see P361-362 of [29]. Lemma 3. Suppose that A, B ∈ Mn(R), where Mn is a set of n-dimensional nonnegative matrices on the real number set, and 0 b α b 1. Then,
 α 1−α α 1−α ≤ðρðAÞÞ ðρðBÞÞ ρ A ∘B ;

ð15Þ

where ρ(A) = λmax(A) denotes the spectral radium. Theorem 3. In Algorithm 1, we have   λmax Ahþ1 bλmax ðAh Þ:

ð16Þ

Proof. The proof of Theorem 3 is provided in Appendix A. Theorem 3 reveals the relationship of our consistency measure and consistency improvement method with the Saaty's traditional consistency concepts. As the modified matrix has a reduced maximum eigenvalue, our improvement method will converge to an acceptable CR in the sense of Saaty's consistency measure.

Recommendations

Problem description Set of alternatives

Advice rules Multiplicative preference relations

Computation of consistency indexes

Moderator

No

Enough consistency?

Fig. 1. Consistency control process.

Yes

Output

Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

4. Consensus measure In group context, consensus decision making is often considered a desirable outcome. When consensus schemes are utilized, the experts involved are supposed to participate in the discussions towards a consensus solution. In this section, we introduce a consensus index to measure the consensus level between the group members. How to reach a predefined consensus level is described and some theoretical foundations are also given. 4.1. Definition and property The following definition is used to measure the closeness amongst experts' opinions in order to obtain the consensus level. Definition 7. Let A1, A2, ⋯, Am be m multiplicative preference relations provided by m decision makers. Suppose Ac is the group multiplicative preference relation utilizing the geometric average operator. Then the group consensus index of Al is defined by n X n  1X c ðkÞ c a aji ; GCIH ðAl Þ ¼ d Al ; A ¼ 2 n i¼1 j¼1 ij

ð17Þ

c

preference relation Aτ has the largest consensus index. The main step for the proposed consensus model is constructing a new multiplicative preference relation A τ according to Aτ. When establishing the new preference relation, we adopt the following strategy

1−γ ðτ Þ ðτ Þ γ c aij ; 0bγb1: a ij ¼ aij

ð20Þ

Follow this strategy until all the multiplicative preference relations reach a predefined level of acceptable consensus or the maximum number of iterations is obtained. Details of our consensus model are depicted in Algorithm 2. Algorithm 2. Input: Individual multiplicative preference relations {A1, A2, ⋯, Am}, the weight vector of decision makers λ = {λ1, λ2, ⋯, λm}, the predefined threshold GCI, the maximum number of iterative times hmax ≥ 1 and 0 b γ b 1. n o Output: Modified multiplicative preference relations A 1 ; A 2 ; ⋯; A m ,
 the consensus index of each preference relation GCI H A k , k=1,2,⋯,m, and the number of iterations h. (k) (k) Step 1. Set h = 0 and Ak, 0 = (aij, 0)n × n = Ak = (aij )n × n. Step 2. Calculate the group multiplicative preference relation

where m

761




ðkÞ

aij ¼ ∏ aij

λ

k

c Ahc = (aij, h)n × n corresponding to {A1, h, A2, h, ⋯ Am, h}, where

;

i; j∈N:

ð18Þ

k¼1

In this way, consensus can be somehow understood as closeness between the individual opinions and the group opinion. If GCIH(Al) =1, then the lth decision maker has full consensus with the group preference. Otherwise, the smaller the value of GCIH(Al), the closer that decision maker is to the group. According to the actual situation, the decision makers establish the threshold GCI for the deviation degree between the individual preference relation and the group preference relation. If for all l, GCIH ðAl Þ≤GCI, we conclude that an acceptable level of consensus is achieved among the decision makers. From the above definition, we can get the following properties. Theorem 4. Let A1, A2, ⋯, Am be m multiplicative preference relations provided by m decision makers. A c is the group multiplicative preference relation utilizing a geometric average operator. Then,  c  d A ; Ak ≤ max fdðAl ; Ak Þg: l

(1)

(2)

(m)

aij, h = (aij, h) λ1(aij, h) λ1 ⋯ (aij, h ) λm. Step 3. Calculate  the  group consensus index GCIH(Ak, h), k = 1, 2, ⋯, m. GCIH Ak;h , ∀ k ∈ M or h ≥ hmax, then go to step 5; otherwise, If ≤GCI continue with the next step.      Step 4. Suppose that GCIH Aτ;h ¼ maxk GCIH Ak;h . Let Ak, h + 1 = (k)

(aij, h + 1)n × n, where ðkÞ aij;hþ1

¼

8

 < aðkÞ γ ac 1−γ

k¼τ

:

k≠τ

ij;h

ij;h ðkÞ aij;h

:

ð21Þ

Set h = h + 1 and go to Step 2. Step 5. Let Ak ¼ Ak;h . Output the modified multiplicative preference relationAk , k = 1, 2, ⋯, m, the group consensus index for each preference relation GCIH(Āk), k = 1, 2, ⋯, m, and the number of iterations h. Step 6. End.

ð19Þ

Proof. The proof of Theorem 4 is provided in Appendix A. Theorem 4 shows that the deviation degree between multiplicative preference relation Ak of A1, A2, ⋯, Am and their group multiplicative preference relation Ac is no greater than the largest deviation degree between any two of the multiplicative preference relations in A1, A2, ⋯ Am. 4.2. Consensus reaching process As consensus is the major goal of GDM, in the GDM process, before aggregation we need to apply consensus models to help decision makers reach consensus. Inspired by [27], a rational consensus model for a GDM problem based on multiplicative preference relations can be shown in Fig. 2. In the model, there is often a moderator, via the exchange of information and rational arguments, trying to persuade the experts to alter their opinions to bring these opinions closer. In the following, we give an iterative model to achieve consensus. Let {A1, A2, ⋯, Am} and λ = {λ1, λ2, ⋯, λm} be as before. A c is the group multiplicative preference relation utilizing the geometric average operator. Without loss of generality, suppose that multiplicative

Remark 2. The main advantage of our proposed consensus scheme is that it does not use any priority methods in the process. In Algorithm 2, the group multiplicative preference relation Ahc is considered a reference point in the improvement process in the hth round We modify only one preference relation that has a maximum group consensus level in each round. If there is more than one decision maker simultaneously obtaining the maximum group level, then we can change their preferences in the same round. As mentioned in the first section, the decision makers are assumed to change their preferences following the moderator's advice. All team members agree to support the decision. What if there are extreme outliers still outside the consensus level after the given maximal rounds (rather unlikely, but possible)? If this is the case, as it was recommended by [4,36,43], the moderator may prefer as an arbiter and ignore the preferences of the outliers in order to achieve the decision goal. On the one hand, the proposed model can be considered an interactive method, and the decision makers have the right to participate in the consensus reaching process in each round. On the other hand, since the proposed consensus model can run automatically, a minimum requirement for decision makers is that they only need to provide their preferences in the first round. In the following, we discuss

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Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

Recommendations

Advice rules

Decision makers Input

Multiplicative preference relations

Moderator

No Yes

Computation of consensus degrees

Enough consensus?

Output

Fig. 2. GDM problem consensus process.

the properties of the above consensus model. We show that Algorithm 2 is convergent.

Theorem 5. In Algorithm 2, suppose the kth decision maker has to change his?/her multiplicative preference relation in the (h + 1)th iteration. Let Ak, h + 1 be the preference relation generated by Algorithm 2 for the decision maker k. Then, we have

 GCIH Ak;hþ1 bGCI H Ak;h :

ð22Þ

Proof. The proof of Theorem 5 is provided in Appendix A. From Theorem 5, we know that in each step, a decision maker who modifies their multiplicative preference relation has a better group consensus index. Generally, the group consensus indexes of those who don't modify their preference relations at this time will not exceed the maximum group consensus index of the last time. Thus, the proposed algorithm is convergent in a general sense. At the same time, the modified multiplicative preference relation has an acceptable individual consistency index under the condition that the consensus reaching process starts with multiplicative preference relations of acceptable consistency. Theorem 6. Let A1, A2,⋯,Am and Ac be m multiplicative preference relations and the group multiplicative preference relation, respectively. Let {Al, h} and {Ahc} be the sequences generated by Algorithm 2. If maxl fCIH ðAl Þg≤CI, ∀l ∈M, then we have n
o n
o max CIH Al;hþ1 ≤ max CI H Al;h ≤CI: l

l

ð23Þ

Proof. The proof of Theorem 6 is provided in Appendix A. Therefore, the consistency index for every multiplicative preference relation is still acceptable after the implementation of Algorithm 2. It implies that when we start with preference relations that are acceptable, we end with modified preference relations which not only achieve the predefined consensus level but are also of acceptable individual consistency. These theorems provide theoretical foundations and are of vital importance for GDM problems based on multiplicative preference relations. 5. A decision support model for the GDM process The complete support model for a GDM problem with multiplicative preference relations is depicted as a flowchart in Fig. 3. Following the procedure in Fig. 3, the optimum alternative or the rating order of n alternatives will be regarded as a final consensus solution to the given GDM problem. Both in the consistency control process and the consensus reaching process, there are advice rules generated from Algorithm 1 and Algorithm 2. The guidance advice system integrated in the support model acts as a feedback mechanism. In some situations, the moderator can be replaced by the guidance advice system. However, the decision makers are responsible for the final decision [27], so they decide whether or not to follow the advice generated by the support model. Thus, our

decision support model plays a role in assisting the decision makers and the moderator in the decision process. The main features of the above support model are emphasized as follows. (1) The consensus reaching process proposed in the model is independent of which priority method is used in the process. (2) It automatically adjusts each multiplicative preference relation to one meeting the predefined consensus requirements. (3) Before conducting the consensus reaching process, the consistency control process is carried out to ensure the rationality of each decision maker. (4) Some theoretical results serve as a foundation for the whole group decision model. When the group achieves the predefined level of consensus, the consistency of each decision maker is also guaranteed. After obtaining the final multiplicative group preference relation, either the eigenvector method or the row geometric mean prioritization method can be used in the selection process. As a result, a general decision support model for solving GDM problems with multiplicative preference relations considering consistency and consensus is presented. 6. Illustrative examples This section presents several examples to show the implementation process and the validity of the proposed approach in practice.

Example 1. Consider the numerical example which was discussed by Dong et al. [17]. Suppose we have a set of five decision makers providing the following multiplicative preference relations {A1, A2, A3, A4, A5} on a set of four alternatives {X1, X2, X3, X4} which need to be ranked from the best to the worst. Let w (k) = (w1(k), w2(k), w3(k), w4(k)) T be the individual priority vector derived from multiplicative preference relation Ak using the eigenvector method. It should be noted that other prioritization methods can also be used in the computation process. Srdjevic [40] presented a multicriteria preference synthesis (MPS) procedure across a Euclidean distance (ED) and minimum violations (MV) criterion to choose the most appropriate method for a given multiplicative preference relation. The main conclusion is that there is no prioritization method that is superior to the others in all cases [40]. Ak and w (k) (k = 1, 2, 3, 4, 5) are given below. 0

1 B 1=4 A1 ¼ B @ 1=6 1=7 ð1Þ

w

w

ð2Þ

6 3 1 1=2

1 0 7 1 B 4C C; A2 ¼ B 1=5 @ 1=7 2A 1 1=9

1 7 9 4 6C C; 1 2A 1=2 1

T

T

¼ ð0:6526; 0:2247; 0:0762; 0:0465Þ ;

1 B 1=3 A3 ¼ B @ 1=5 1=8

3 1 1=4 1=5

5 4 1 1=2

1 8 5C C; 2A 1

0

1 4 B 1=4 1 A4 ¼ B @ 1=5 1=3 1=6 1=3

ð3Þ

¼ ð0:5705; 0:2771; 0:0959; 0:0565Þ ;

ð4Þ

¼ ð0:5970; 0:2217; 0:1084; 0:0728Þ ;

w

5 1 1=4 1=6

¼ ð0:6168; 0:2238; 0:0972; 0:0621Þ ; 0

w

4 1 1=3 1=4

T

T

5 3 1 1=2

1 6 3C C; 2A 1

Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

Set of decision makers Environment Decision Goal Information Source

763

Preferences

Set of alternatives

Multiplicative preference relations

Problem description

Consistency control process (Algorithm 1) Recommendations

Decision makers Multiplicative preference relations

Advice rules

Computation of consistency indexes

No Enough consistency?

Yes

Consensus reaching process (Algorithm 2) Decision makers

Recommendations

Advice rules No

Multiplicative preference relations with acceptable consistency

Computation of consensus degrees

Enough consensus?

Yes

Selection process Preference relations with acceptable consistency and consensus

Aggregation

Group preference relations

Exploitation

Solution based on consistency and consensus

Fig. 3. A GDM process framework with multiplicative preference relations.

0

1 1=2 1 B 2 1 2 A5 ¼ B @ 1 1=2 1 1=2 1=3 1=4 ð5Þ w ¼ ð0:2185; 0:4123;

1 2 3C C; 4A 1 T 0:2679; 0:1013Þ :

which also shows that the given preference relations are of acceptable consistency. Stage 2: Consensus reaching process. The group consensus indexes for each decision makers are GCI H ðA1 Þ ¼ 1:0502; GCI H ðA2 Þ ¼ 1:1144; GCI H ðA3 Þ ¼ 1:0382;

Let λ = {0.1, 0.3, 0.1,0.2, 0.3} be the weights of decision makers. From the above multiplicative preference relations, we calculate the group multiplicative preference relation Ac using (1) and the corresponding priority vector wc = (w1c , w2c , w3c , w4c ) T which are listed below. We can see that the ranking order of alternatives in wc is the same as that of w1, w2, w3 and w4 but is very different from that of w5. 0

1:0000 B 0:4490 c B A ¼@ 0:2877 0:1963 c

1 2:2270 3:4756 5:0937 1:0000 2:9804 4:0005 C C; 0:3355 1:0000 2:4623 A 0:2500 0:4061 1:0000 T

w ¼ ð0:4909; 0:2988; 0:1368; 0:0734Þ : In the following, we show how to apply the GDM support model described in Section 4 to obtain a solution based on consistency and consensus. Stage 1: Consistency control process. By consistency measure, the initial consistency indexes are CI H ðA1 Þ ¼ 1:0255; CI H ðA2 Þ ¼ 1:0450; CIH ðA3 Þ ¼ 1:0226;  c CI H ðA4 Þ ¼ 1:0314; CI H ðA5 Þ ¼ 1:0241; CIH A ¼ 1:0215:

GCI H ðA4 Þ ¼ 1:0400; GCI H ðA5 Þ ¼ 1:3666: Here, the consensus level is set at GCI ¼ 1:1. As A2 and A5 do not reach the consensus level, we continue to carry out Algorithm 2. Set γ = 0.9, the algorithm is terminated after ten steps. The consistency and consensus indexes for each decision maker at each step are listed in Table 1. From Table 1, we find only A5 is modified. The consistency indexes of the modified multiplicative preference relation are decreasing in the first nine steps but increase a little in the tenth step. The consistency index of the group multiplicative preference relation is no greater than the largest consistency index of the individual multiplicative preference relation at every step. The group consensus indexes for every decision maker at each step are also decreasing.
If we use Saaty's
consistency ratio (CR), we have CR A5 ¼  0:0265, CR Ac ¼ 0:0388. We find that the results in this example are in accordance with the theorems in Sections 2 and 3. The modified multiplicative preference relations A5 are as follows. 0

If we fix a minimum threshold value CI ¼ 1:1, we see that all the multiplicative preference relations are of acceptable consistency. Using Saaty's consistency ratio (CR), we have CRðA1 Þ ¼ 0:0379; CRðA2 Þ ¼ 0:0671; CRðA3 Þ ¼ 0:0335;  c CRðA4 Þ ¼ 0:0466; CRðA5 Þ ¼ 0:0359; CR A ¼ 0:0319;

1 B 0:7187 A5 ¼ B @ 0:4259 0:2635 0 1 B 0:3303 Ac ¼ B @ 0:2227 0:1620

1:3914 1 0:3804 0:2737 3:0273 1 0:3091 0:2356

2:3479 2:6286 1 0:3486 4:4899 3:2350 1 0:4487

1 3:7948 3:6539 C C; 2:8687 A 1 1 6:1728 4:2442 C C; 2:2286 A 1

764

Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

Table 1 Consistency and consensus indexes for Example 1. Step

Consistency indexes

Consensus indexes

CIH(Ac)

k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k = 10

(1.0255,1.0450,1.0226,1.0314,1.0241) (1.0255,1.0450,1.0226,1.0314,1.0220) (1.0255,1.0450,1.0226,1.0314,1.0204) (1.0255,1.0450,1.0226,1.0314,1.0192) (1.0255,1.0450,1.0226,1.0314,1.0184) (1.0255,1.0450,1.0226,1.0314,1.0179) (1.0255,1.0450,1.0226,1.0314,1.0176) (1.0255,1.0450,1.0226,1.0314,1.0176) (1.0255,1.0450,1.0226,1.0314,1.0176) (1.0255,1.0450,1.0226,1.0314,1.0179)

(1.0502,1.1144,1.0382,1.0400,1.3666) (1.0429,1.1028,1.0324,1.0345,1.3115) (1.0366,1.0927,1.0275,1.0299,1.2652) (1.0313,1.0838,1.0234,1.0261,1.2263) (1.0268,1.0760,1.0200,1.0230,1.1935) (1.0229,1.0691,1.0172,1.0204,1.1657) (1.0196,1.0630,1.0149,1.0184,1.1421) (1.0168,1.0577,1.0130,1.0167,1.1219) (1.0145,1.0529,1.0114,1.0154,1.1048) (1.0125,1.0487,1.0102,1.0143,1.0901)

1.0215 1.0221 1.0227 1.0232 1.0237 1.0243 1.0248 1.0253 1.0257 1.0252

T

wð5Þ ¼ ð0:3951; 0:3440; 0:1764; 0:0846Þ ; T

w ¼ ð0:5544; 0:2667; 0:1134; 0:0656Þ : c

Stage 3: Selection process. The final group multiplicative preference relation is shown above. Thus the ranking of the alternatives is X1 >X2 >X3 >X4, which indicates that X1 is the best option. Our ranking order of the alternatives is the same as Dong et al.'s paper but with different degrees of one alternative over another [17]. The consensus model is different from previous papers. The consistency and consensus measures are based on the compatibility of two ratio matrices based on the Hadamard product proposed by Saaty. The consistency and consensus based model is more simple and straightforward. We use the group multiplicative preference relation as the reference point matrix at every step. It is not necessary to compute any priority vector in the consensus reaching process. In [17], both the consensus measure and the consensus model depend on a row geometric mean prioritization method.

Example 2. As we have mentioned, the two aggregation methods (AIJ and AIP) discussed in [19] are widely used for comparisons in literature. For AIP, a consensus degree was proposed by comparing the position of each alternative in the decision makers' individual priorities and in the group priorities [17]. Let w (k) and w c be the individual and collective priority vector respectively. Let v (k) = (v1(k), v2(k), ⋯, vn(k)), where vi(k) is the position of the ith alternative in w(k). Similarly, let v c = (v1c , v2c , ⋯, vnc), where vic is the position of the ith alternative in w c. Then, the ordinal consensus of Ak is defined by OCI(Ak) = (1/
index  n) ∑ in= 1|vi(k) − vic|. If OCI AðkÞ ≤OCI,∀ k, where OCI is a predefined threshold, we conclude that an acceptable ordinal consensus is reached. A similar algorithm to Algorithm 2 can be established based on these indicators. To show the proposed decision support model works under these indicators, we still use the data from Example 1. When setting OCI ¼ 0 and γ = 0.9, we apply the proposed model to adjust the given multiplicative preference relations. The algorithm ends after 8 steps. Resulting indicators are listed in Table 2. We find only A5 is modified. The modified preference relation for the 5th decision maker and the final group preference relation are denoted as A5 and Ac . The corresponding priority vectors w5 and wc are also listed below. The results show that the proposed model is suitable for dealing with ordinal consensus. 0

1:0000 B 0:8549 B ¼ A5 @ 0:4922 0:2937 0 1:0000 B 0:3480 Ac ¼ B @ 0:2326 0:1674

1:1698 1:0000 0:3985 0:2830 2:8738 1:0000 0:3134 0:2380

2:0316 2:5096 1:0000 0:3295 4:2992 3:1904 1:0000 0:4412

1 3:4044 3:5338 C C; 3:0350 A 1:0000 1 5:9750 4:2019 C C; 2:2666 A 1:0000

T

w5 ¼ ð0:3613; 0:3588; 0:1917; 0:0882Þ ; wc ¼ ð0:5437;

0:2722;

T

0:1172; 0:0669Þ :

Example 3. Consider the following problem [53]. There are three individuals, who are managers from the design, manufacturing and marketing departments, participating in a group decision about new product development strategy through AHP. The five decision criteria for new product development are cost, manufacturability, quality, technological improvement and market share, denoted as C1, C2, C3, C4, C5 respectively. The three managers give their preferences over the five criteria by multiplicative preference relations. The multiplicative preference relations Ak and the corresponding weight vector w k (k = 1, 2, 3) for criteria are calculated accordingly. 0

0

1 1=3 1=5 C C 1=9 C C; 1=3 A 1

1 5 7 3 B 1=5 1 3 1=3 B A1 ¼ B B 1=7 1=3 1 1=7 @ 1=3 3 7 1 3 5 9 3

1 1 1=3 7 1=2 3 B 3 1 3 1 5C B C C; A2 ¼ B 1 1=3 1 1=3 3 B C @ 2 1 3 1 5A 1=3 1=5 1=3 1=5 1

ð1Þ

T

w ¼ ð0:2813; 0:0695; 0:0321; 0:1590; 0:4581Þ ; ð2Þ T w ¼ ð0:1418; 0:3497; 0:1312; 0:3217; 0:0555Þ 0

1 7 B 1=7 1 B A3 ¼ B B 1=5 3 @ 1=4 4 1=3 5 0 1:0000 B 0:4409 B c A ¼B B 0:3057 @ 0:5503 0:6934 ð3Þ

w

c

5 1=3 1 3 4

4 1=4 1=3 1 1

2:2680 1:0000 0:6934 2:2894 1:7100

1 3 1=5 C C 1=4 C C; 1 A 1

3:2711 1:4422 1:0000 3:9791 2:2894

1:8171 0:4368 0:2513 1:0000 0:8434

1 1:4422 0:5848 C C 0:4368 C C; 1:1856 A 1:0000 T

¼ ð0:4878; 0:0436; 0:0809; 0:1780; 0:2098Þ ; T

w ¼ ð0:3264; 0:1232; 0:0841; 0:2574; 0:2088Þ

Without loss of generality, each manager is assumed to have equal importance and let λ = (1/3, 1/3, 1/3) be their weight vector. From the above multiplicative preference relations, we calculate the group multiplicative preference relation A c using (1) and the corresponding priority vector w c which are listed above. In the following, we show how to apply the GDM support model described in Section 4 to obtain a solution based on consistency and consensus. Stage 1: Consistency control process. The initial consistency indexes are shown in Table 2. If we fix a minimum threshold value CI ¼ 1:1, we see that all the multiplicative preference relations are of acceptable consistency,

Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767 Table 2 Consistency and consensus indexes for Example 2.

Original Modified

Consistency indexes

Ordinal consensus indexes

CIH(Ac)

(1.0255, 1.0450, 1.0226, 1.0314, 1.0241) (1.0255, 1.0450, 1.0226, 1.0314, 1.0176)

(0, 0, 0, 0, 1) (0, 0, 0, 0, 0)

1.0215 1.0253

which is consistent with the results of Saaty's consistency ratio (CR) [53]. Stage 2: Consensus reaching process. The group consensus indexes for each decision maker are listed in Table 2. Here, the consensus level is set at GCI ¼ 1:1. As all the preference relations do not reach the consensus level, we continue to carry out Algorithm 2. Set γ= 0.9, the algorithm is terminated after 20 steps. We find that all three preference relations are modified. Specifically, A1, A2 and A3 have been modified 4, 14, and 2 times respectively. The final modified individual multiplicative preference relations and group multiplicative preference relation and their corresponding prioritization vectors are given below. 0

1:00 B 0:22 B A1 ¼ B B 0:17 @ 0:35 1:90 0 1:00 B 0:51 B A2 ¼ B B 0:34 @ 0:61 0:64

4:53 1:00 0:47 2:95 4:19

5:97 2:17 1:00 5:87 6:63

2:84 0:34 0:17 1:00 2:19

1:96 1:00 0:67 2:15 1:44

2:97 1:48 1:00 3:85 1:95

1:65 0:46 0:26 1:00 0:74

1 0:53 0:24 C C 0:15 C C; 0:46 A 1:00 1 1:57 0:69 C C 0:51 C C; 1:35 A 1:00

765

In [53], the final weight vector of five criteria is w = (0.3743, 0.1228, 0.0833, 0.1867, 0.2270) T, which leads to the same results as our paper. Our results also verify the consensus criteria based on a group strong agreement quotient (GSAQ) and a group strong disagreement quotient (GSDQ) [53]. Compared to the method in [53], our decision support model is very effective due to its simpler structure and reduced computation complexity. 7. Conclusions In a GDM context, prior to the selection of the best alternative, it would be desirable that decision makers achieve a high degree of consensus while keeping their preferences rational. In this paper we have investigated a decision support model which simultaneously addresses the individual consistency and group consensus. The main work presented in this paper is summarized as follows: (1) We have proposed a new consistency measure for multiplicative preference relations. An algorithm has been presented to make a multiplicative preference relation of acceptable consistency. We have shown that our consistency improvement method is related to Saaty's consistency index. (2) We have also proposed a consensus index to measure the consensus level. For both the consistency and consensus measures, the concept of a deviation measure has been used based on the Hadamard product of two matrices. A consensus reaching process has been put forward to help the group reach a predefined consensus level. (3) A framework for the decision support model has been presented to aid the whole GDM process based on preference relations. The effectiveness of the proposed model has been illustrated by three numerical examples.

T

wð1Þ ¼ ð0:3160; 0:0755; 0:0425; 0:1823; 0:3837Þ ; w

ð2Þ

¼ ð0:3149; 0:1360; 0:0902; 0:2684; 0:1905Þ 0

1:00 6:26 4:89 3:69 B 0:16 1:00 0:42 0:27 B A3 ¼ B B 0:20 2:41 1:00 0:31 @ 0:27 3:77 3:19 1:00 0:39 4:56 3:94 1:03 0 1:0000 3:8171 4:4255 B 0:2620 1:0000 1:0979 B Ac ¼ B B 0:2260 0:9109 1:0000 @ 0:3868 2:8814 4:1605 0:7784 3:0210 3:7074 ð3Þ

w

Our model's main improvement is that it is independent of the prioritization method used in the entire consensus scheme. The model guarantees that each of the individual multiplicative preference relation is still rational and of acceptable consistency. As a consequence, the proposed model allows us to achieve a higher level of consistency and consensus solutions for a GDM with preference relations. The consistency concepts and corresponding properties of our GDM model can be extended to deal with other kinds of preference relations.

T

1 2:57 0:22 C C 0:25 C C; 0:97 A 1:00 2:5851 0:3471 0:2404 1:0000 1:1878

1 1:2847 0:3310 C C 0:2697 C C; 0:8419 A 1:0000

Acknowledgments

T

¼ ð0:4669; 0:0492; 0:0788; 0:1863; 0:2187Þ ; T

wc ¼ ð0:3722; 0:0822; 0:0691; 0:2177; 0:2587Þ

The consistency and consensus indexes are listed in Table 3. Stage 3: Selection process. Thus the ranking of five criteria is C1 > C5 > C4 > C2 > C3, which indicates that C1 is the most important criterion.

The authors are very grateful to the Editor-in-Chief, Professor A.B. Whinston, and the three anonymous referees, for their constructive comments and suggestions that have improved the quality of the paper. This research was supported by the Key Program of National Natural Science Foundation of China (Grant No. 70831005), and Major Bidding Program of National Social Science Foundation of China (Grant No. 08&ZD009), and also supported by Projects of International Cooperation and Exchanges NSFC (Grant No. 71011140076). Appendix A Proof of Theorem 2. From step 4 of Algorithm 1 and (5), we have that

Table 3 Consistency and consensus indexes for Example 3.

Original Modified

Consistency indexes

Consensus indexes

CIH(Ac)

(1.0543, 1.0112, 1.0407) (1.0326, 1.0103, 1.0319)

(1.1.4133, 1.9800, 1.2953) (1.0964, 1.0911, 1.0898)

1.0110 1.0138


θ
1−θ aij;hþ1 ¼ aij;h g ij;h ;

1=n θ
1−θ 1=n n
n
¼ ∏ ail;h alj;h g il;h g lj;h ¼ g ij;h : g ij;hþ1 ¼ ∏ ail;hþ1 alj;hþ1 l¼1

l¼1

766

Z. Wu, J. Xu / Decision Support Systems 52 (2012) 757–767

Proof of Theorem 5. From (1) and (21), we have

Consequently, we obtain
θ
1−θ
θ
θ−1 g ij;h g ji;h ¼ aij;h g ji;h g ji;h aij;hþ1 g ji;hþ1 ¼ aij;h
θ ¼ aij;h g ji;h : From Lemma 2, it follows that
θ
θ aij;hþ1 g ji;hþ1 þ aji;hþ1 g ij;hþ1 ¼ aij;h g ji;h þ aji;h g ij;h ≤aij;h g ji;h þ aji;h g ij;h : Since A (h) ≠ G (h), there is at least one pair (i, j) such that the above expression strictly holds. Thus, we have −1 X n
 1 nX aij;hþ1 g ji;hþ1 þ aji;hþ1 g ij;hþ1 2 n i¼1 j¼iþ1 −1 X n
 1 1 1 nX aij;h g ji;h þ aji;h gij;h þ : þ b 2 n n i¼1 j¼iþ1 n

 ∞ CIH A ¼ lim CIH ðAh Þ ¼ inf fCIH ðAh Þg

Using proof by contradiction, it can be shown that lim CIH ðAh Þ≤α . This h→∞ completes the proof of Theorem 2.

Proof of Theorem 3. From step 4 of Algorithm 1, we have θ

1−θ



1−γ
ð1−γÞλ k ðkÞ ðkÞ γ ðkÞ acij;h acji;h aij;h acji;h aij;hþ1 acji;hþ1 ¼ aij;h
γ
ð1−γÞλ
γþð1−γÞλ k k ðkÞ ðkÞ ðkÞ ¼ aij;h acji;h aij;h acji;h ¼ aij;h acji;h :


θ
θ ðkÞ c ðkÞ c ðkÞ c ðkÞ c ðkÞ c ðkÞ c aij;hþ1 aji;hþ1 þ aji;hþ1 aij;hþ1 ¼ aij;h aji;h þ aji;h aij;h b aij;h aji;h þ aji;h aij;h :

It implies that

h→∞

h→∞

Ahþ1 ¼ ðAh Þ ∘ðGh Þ

For the decision maker k, we get the following

Letting θ = γ + (1 − γ)λk, then 0 b θ b 1. From Lemma 2, for i ≠ j, we have

That is, CIH(Ah + 1) b CIH(Ah). From Definition 6, we have CIH(Ah) ≥ 1, ∀ h. Therefore, the sequence {CIH(Ah)} is monotone decreasing and has a lower bound. Applying the limit existence theorem for a sequence, we know that lim CIH ðAh Þ exists. Let A∞ ¼ lim Ah . Assume that h→∞

λ
λ
λ m
m p p k ðpÞ ðpÞ ðkÞ acij;hþ1 ¼ ∏ aij;hþ1 ¼ ∏ aij;hþ1 aij;hþ1 p¼1 p¼1;p≠k
 

1−γ λk m ðpÞ λp ðkÞ γ c ¼ ∏ aij;h aij;h aij;h p¼1;p≠k

 

1−γ λk m ðpÞ λp ðkÞ λk ðkÞ γ−1 c ¼ ∏ aij;h aij;h aij;h aij;h p¼1;p≠k
ð1−γÞλ k ðkÞ ¼ acij;h aji;h aij;h :

n
 XX 1 n−1 ðkÞ c ðkÞ c aij;hþ1 aji;hþ1 þ aji;hþ1 aij;hþ1 2 n i¼1 j¼iþ1 n
 1 XX 1 1 n−1 ðkÞ c ðkÞ c þ b 2 a a þ aji;h aij;h þ : n n i¼1 j¼iþ1 ij;h ji;h n

Hence, we have GCIH(Ak, h + 1) b GCIH(Ak, h). This completes the proof of Theorem 5.

:

Since Gh is a virtual consistent multiplicative preference relation, we have λmax(Gh) = n and λmax(Gh) b λmax(Ah). Therefore, from Lemma 3, we have

Proof of Theorem 6. We suppose the kth decision maker has to change his\her multiplicative preference relation in the hth iteration. We consider two cases. For l = k, since Ak, h + 1 is a weighted combination of Ak, h and Ahc, from Theorem 1, we have

  α 1−α α 1−α bðρðAh ÞÞ ðρðAh ÞÞ ¼ ρðAh Þ: ρ Ahþ1 ≤ðρðAh ÞÞ ðρðGh ÞÞ



 n o CI H Ak;hþ1 ≤max CIH Ak;h ; CIH ðAh Þ : c Then,  from the definition of Ah, we obtain that maxl CI H Al;h . Consequently,

This completes the proof of Theorem 3.

Proof of Theorem 4. From Definition 7, we have n X n  c  1X c ðkÞ aij aji ; d A ; Ak ¼ 2 n i¼1 j¼1

dðAl ; Ak Þ ¼

n X n 1X ðlÞ ðkÞ a a : 2 n i¼1 j¼i ij ji

At the same time, from Lemma 1, we have

c ðkÞ aij aji

m

¼∏ l¼1




m  X ðlÞ λl ðkÞ aij aji ≤ l¼1



 n
o n
o ¼ max CIH Al;h : CI H Ak;hþ1 ≤max CI H Ak;h ; max CIH Al;h l

l

For l ≠ k, we have CIH(Al, h + 1) = CIH(Al, h). Summarizing both cases, we have n
o n
o max CIH Al;hþ1 ≤ max CIH Al;h : l

ðlÞ ðkÞ λl aij aji :

  CIH Ach ≤

l

This completes the proof of Theorem 6. References

It follows that !

n X n m m n X n
 X X  c  1X 1X ðlÞ ðkÞ ðlÞ ðkÞ d A ; Ak ≤ 2 λl aij aji λl a a ¼ 2 n i¼1 j¼1 l¼1 n l¼1 i¼1 j¼1 ij ji m m X X λl dðAl ; Ak Þ≤ λl max fdðAl ; Ak Þg ¼ max fdðAl ; Ak Þ: ¼ l¼1

l¼1

l

This completes the proof of Theorem 4.

l

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Zhibin Wu is a Ph.D. candidate at the School of Management, Sichuan University, Sichuan, China. He received his B.S. degree from the Department of Information and Computation Science, Chongqing University, China, in 2005, and his M.S. degree from the Department of Operational Research and Cybernetics, Chongqing University, China, in 2008. His research results have been published in Fuzzy Sets and Systems and Knowledge-Based Systems. His research interests include group decision making, soft computing, aggregation operators, multi-criteria decision analysis and applications, and decision support systems.

Jiuping Xu is a professor at the School of Management, Sichuan University, Sichuan, China. He obtained his Ph.D. in applied mathematics from Tsinghua University, Beijing, China and Ph.D. in physical chemistry from Sichuan University, Chengdu, China, in 1995 and 1999, respectively. His current research interests include group decision making, project management, uncertain programming, supply chain management, and applied mathematics. His research results have been published in IEEE Transaction on Fuzzy Systems, Information Sciences, Expert Systems with Applications, International Journal of Production Economics, Fuzzy Sets and Systems, Computers & Industrial Engineering, Mathematical Analysis and Applications, among others.