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Fuzzy Sets and Systems 216 (2013) 108–133www.elsevier.com/locate/fss

Algorithms for improving consistency or consensus of reciprocal[0,1]-valued preference relations�

Meimei Xiaa,∗, Zeshui Xub, Jian Chena,

a School of Economics and Management, Tsinghua University, Beijing 100084, Chinab Institute of Sciences, PLA University of Science and Technology, Nanjing 210007, China

Received 6 February 2012; received in revised form 12 September 2012; accepted 21 September 2012Available online 9 October 2012

Abstract

We investigate the consistency and consensus of reciprocal [0,1]-valued preference relations (also called fuzzy preference relationsby many authors) based on the multiplicative consistency property, which is an important issue in fuzzy set theory. An algorithm isfirst developed to improve the consistency level of a reciprocal [0,1]-valued preference relation, and the corresponding algorithm forthe incomplete reciprocal [0,1]-valued preference relation is also developed. We further propose the consensus improving algorithmsfor individual reciprocal [0,1]-valued preference relations or incomplete ones. The convergence and robustness of the algorithms areproven and some important conclusions are obtained. In addition, the proposed algorithms can improve the consistency or consensusof reciprocal [0,1]-valued preference relations with less interactions with the decision makers, which can save a lot of time andobtain the results quickly.© 2012 Elsevier B.V. All rights reserved.

Keywords: Fuzzy set; Group decision making; Reciprocal [0,1]-valued preference relation; Multiplicative consistency; Consensus

1. Introduction

Since fuzzy set [40] was introduced, it has been applied in many areas, especially in decision making. Group decisionmaking refers to two or more decision makers, who are characterized by their own ideas, attributes, motivations andknowledge, and tries to achieve a common solution to a problem [21]. In order to avoid a misleading solution, thereare two important problems to be addressed before making the final decision: consistency and consensus, while theformer is used to measure the degree of agreement among the preference values provided by the individual decisionmakers, and the latter is to measure the degree of agreement among the decision makers on the solution of the problem.Upon these two issues, a lot of work has been done in the last decades, and mainly focused on reciprocal [0,1]-valuedpreference relations (also called fuzzy preference relation) [8–10,27] and multiplicative preference relations [29].

� The work was supported in part by the National Natural Science Foundation of China (Nos. 71071161 and 61273209) and the China PostdoctoralScience Foundation (No. 2012M520311).

∗ Corresponding author. Tel.: +86 18810491647.E-mail address: meimxia@163.com (M. Xia).

0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.fss.2012.09.016

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 109

Xu and Wei [39] proposed an algorithm to improve the consistency of a multiplicative preference relation and provedthat the algorithm is convergent. Then Ergu et al. [14] proposed a method about the test of consistency for a multiplicativepreference relation, which combines the theorem of matrix multiplication, vectors dot product, and the definition ofconsistent multiplicative preference relation, to identify the inconsistent elements. Siraj et al. [31] developed a heuristicalgorithm to improve ordinal consistency by identifying and eliminating intransitivities in multiplicative preferencerelation matrices. Ma et al. [24] presented an analysis method to identify the inconsistency and weak transitivity ofa reciprocal [0,1]-valued preference relation and to repair its inconsistency so as to reach weak transitivity based onadditive consistency property. De Baets and De Meyer [8] illustrated that for reciprocal relations the concept of cycle-transitivity provides a framework that can cover more types of transitivity than does the concept of FG-transitivity.Rademaker and De Baets [28] considered the aggregation of monotone reciprocal relations, and showed the relevanceof the concepts in group decision making problems. De Loof et al. [11] proposed an algorithm to compute the averageranks based on the so-called lattice of ideals representation of a poset that avoids enumerating all linear extensions.

Herrera-Viedma et al. [19] gave a consensus model for group decision making problems with different preferencestructures, preference orderings, utility values, reciprocal [0,1]-valued preference relations, and multiplicative prefer-ence relations. Chiclana et al. [3] developed a consensus model for group decision making problems that proceeds fromconsistency to consensus. This model integrates a novel consistency reaching module based on the additive consistencyproperty, and generates advice on how the decision makers should change their preferences in order to increase theirconsistency. Inspired by Xu and Wei’s individual consistency improving method [39] and Chiclana et al.’s consensusframework [3], Dong et al. [12] proposed two new consensus models (i.e., cardinal consensus model and ordinal con-sensus model) under the row geometric mean prioritization method. Dong et al. [13] presented a consensus operatorunder the continuous linguistic environment based on the use of the ordered weighted averaging operator and thedeviation measures. Fu and Yang [16] extended the evidential reasoning approach to group consensus situations formultiple attribute group decision problems under various kinds of uncertainties with pivotal group consensus require-ments. Mata et al. [25] proposed an adaptive consensus support system model to support consensus processes in groupdecision making problems with multigranular linguistic information, which improves the consensus reaching processby adapting the search for preferences in disagreement to the current level of consensus at each round.

In some cases, a decision maker cannot be able to efficiently express any kind of preference degree between two ormore of the available options due to the decision maker not possessing a precise or sufficient level of knowledge of partof the problem, or because that the decision maker is unable to discriminate the degree to which some options are betterthan others [23]. That is to say, if some values in the multiplicative preference relation or the reciprocal [0,1]-valuedpreference relation cannot be provided for some reasons mentioned above, then they can be called an incompletemultiplicative preference relation or an incomplete reciprocal [0,1]-valued preference relation, respectively. Manymethods [5,17] have been developed to estimate the missing values in the preference relations. Herrera-Viedma et al.[23] presented a group decision making model with incomplete reciprocal [0,1]-valued preference relations based on theadditive consistency property. Alonso et al. [2] presented a web consensus support system to deal with group decisionmaking problems with different kinds of incomplete preference relations (fuzzy, linguistic and multigranular linguisticpreference relations) in which the consistency is modeled via the multiplicative consistency property used to estimatethe unknown values of incomplete preference relations as well as to compute the needed consistency measures.

In this paper, we aim to apply the multiplicative consistency property to improve the consistency of a reciprocal[0,1]-valued preference relation and to improve the consensus of individual reciprocal [0,1]-valued preference relationswith less interactions of the decision makers, which can save a lot of time and can give a quick response to the urgentsituations. To do this, the paper is organized as follows. In Section 2, we introduce the geometric consistency indexfor the reciprocal [0,1]-valued preference relation, based on which, we propose a convergent algorithm to improve theconsistency of a reciprocal [0,1]-valued preference relation or an incomplete one. Section 3 develops two convergentiterative algorithms to improve the consensus levels of the individual reciprocal [0,1]-valued preference relations orthe incomplete ones. Section 4 gives some concluding remarks.

2. Consistency model for a reciprocal [0,1]-valued preference relation

In this section, we mainly focus on the multiplicative consistency of the reciprocal [0,1]-valued preference relation,and propose some methods to measure and improve the consistency level of a reciprocal [0,1]-valued preferencerelation. Before doing this, the following definitions are given:

110 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Definition 1 (Cutello and Montero [7], De Baets and De Meyer [8], De Baets et al. [9,10], Orlovsky [27]). Let X ={x1, x2, . . . , xn} be a set of alternatives, then B = (bi j )n×n is called a reciprocal [0,1]-valued preference relation(also called fuzzy preference relation by many authors) on X × X with the conditions: bi j ≥ 0, bi j + b ji = 1,i, j = 1, 2, . . . , n, where bi j denotes the degree that the alternative xi is prior to the alternative x j . If some valuesin B cannot be provided for some reasons mentioned in the Introduction, then B is called an incomplete reciprocal[0,1]-valued preference relation. In this paper, we assume that if bi j is known, then b ji can be obtained by b ji = 1−bi j .

Theorem 1 (Herrera-Viedma et al. [23]). The incomplete reciprocal [0,1]-valued preference relation can be com-pleted if a set of n −1 nonleading diagonal preference values are known, and where the comparison of each alternativeappears at least once. That is, 1, 2, . . . , n appears at least once in the subscript of the known elements bi j (i� j).

This issue has been discussed by Gong [18], Ma et al. [24] and Liu [26] from other different views. Unless otherwisenoted, we assume that all the incomplete reciprocal [0,1]-valued preference relations mentioned in this paper can becompleted.

Definition 2 (Ma et al. [24], Liu [26], Saaty [29]). Let B be a reciprocal [0,1]-valued preference relation, then B iscalled an additive consistent reciprocal [0,1]-valued preference relation if it satisfies the additive transitivity property:

bi j = bi j + b jk − 0.5, i, j, k = 1, 2, . . . , n (1)

which can also be written as

bi j = 12 (wi + w j − 1), i, j = 1, 2, . . . , n (2)

where w = (w1, w2, . . . , wn)T is the priority vector of B and∑n

i=1 wi = 1, wi > 0, i = 1, 2, . . . , n. It is clearthat the additive consistency property has some disadvantages, for example, if b12 = 0.8 and b23 = 0.9, then b13 =0.8 + 0.9 − 0.5 = 1.2 > 1, which is not reasonable. Although it can be transformed into the value in [0, 1] by usingHerrera-Viedma et al.’s method [20], some preference information will be lost. If we use multiplicative consistencyproperty, such a situation will never happen.

Definition 3 (Tanino [32]). Let B be a reciprocal [0,1]-valued preference relation, then B is called a multiplicativeconsistent reciprocal [0,1]-valued preference relation if it satisfies the multiplicative transitivity property:

bi j b jkbki = b ji bk j bik, bi j > 0, i, j, k = 1, 2, . . . , n (3)

It is noted that Definition 3 is given based on the assumptions: bi j�0, bi j�1, i, j = 1, 2, . . . , n and n > 3 which alsowill be used in the following results.

By the simple algebraic manipulation, Eq. (3) can be expressed as [4]

bi j b jkbki = b ji bk j bik ⇔ bi j b jk(1 − bik) = (1 − bi j )(1 − b jk)bik

⇔ bi j b jk − bi j b jkbik = (1 − bi j )(1 − b jk)bik

⇔ bi j b jk = ((1 − bi j )(1 − b jk) + bi j b jk)bik

⇔ bik = bi j b jk

bi j b jk + (1 − bi j )(b jk), i, j, k = 1, 2, . . . , n (4)

Let b12 = 0.8 and b23 = 0.9, then by Eq. (4), we have b13 = (0.8 × 0.9)/(0.8 × 0.9 + (1 − 0.8)(1 − 0.9)) = 0.9730,which can solve the problem as mentioned in Definition 2, therefore multiplicative consistency has more advantagesthan additive consistency. In addition, a multiplicative consistent reciprocal [0,1]-valued preference relation can alsobe given by

bi j = wi

wi + w j, i, j = 1, 2, . . . , n (5)

where w = (w1, w2, . . . , wn)T is the priority vector of B and∑n

i=1 wi = 1, wi > 0, i = 1, 2, . . . , n.

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 111

Obviously, from Eq. (5), we have

bi j (wi + w j ) = wi ⇔ bi jw j = (1 − bi j )wi ⇔ bi jw j = bi jwi

⇔ ln bi j + ln w j = ln b ji + ln wi , i, j = 1, 2, . . . , n (6)

Then the absolute values of ln bi j + ln w j − ln b ji − ln wi (i, j = 1, 2, . . . , n) can be used to measure the consistencylevel of the reciprocal [0,1]-valued preference relation B and should be kept as small as possible. Motivated by Aguarónand Moreno-Jiménez [1] and Crawford and Williams [6], we give the following definition:

Definition 4. Let B = (bi j )n×n be a reciprocal [0,1]-valued preference relation, and w = (w1, w2, . . . , wn)T thepriority vector derived from B satisfying

∑ni=1 wi = 1 and wi > 0, i = 1, 2, . . . , n, then the geometric consistency

index of B is given by

GC I (B) = 2

(n − 1)(n − 2)

∑i< j

(ln bi j − ln b ji − ln wi + ln w j )2 (7)

Especially, if GC I (B) = 0, then B is a consistent reciprocal [0,1]-valued preference relation. The smaller the value ofGCI, the better the consistency B has.

However, it is very difficult to obtain a consistent reciprocal [0,1]-valued preference relation in practical problems,then we provide the threshold GC I for GCI: if GC I (B) < GC I , then the matrix B is called of acceptable consistency.The value of GC I can be determined according to the decision makers’ preferences and the practical situations whichis an issue to be further studied. Such issues have been investigated intensively by Cutello and Montero [7], and otherrelated work has been done by Saaty [29,30] and Aguarón and Moreno-Jiménez [1].

Wang and Fan [33] proposed a logarithmic least squares model to derive the priority vector from a reciprocal[0,1]-valued preference relation:

(MOD 1) Min J =n∑

i=1

n∑j=1

(ln bi j − ln b ji − ln wi + ln w j )2

s.t.n∑

i=1

wi = 1, wi ≥ 0, i = 1, . . . , n

In fact,

n∑i=1

n∑j=1

(ln bi j − ln b ji − ln wi + ln w j )2

=n∑

i< j

(ln bi j − ln b ji − ln wi + ln w j )2 +

n∑i> j

(ln bi j − ln b ji − ln wi + ln w j )2

=n∑

i< j

(ln bi j − ln b ji − ln wi + ln w j )2 +

n∑j<i

(−(ln b ji − ln bi j − ln w j + ln wi ))2

=n∑

i< j

(ln bi j − ln b ji − ln wi + ln w j )2 +

n∑i< j

(ln bi j − ln b ji − ln wi + ln w j )2

= 2n∑

i< j

(ln bi j − ln b ji − ln wi + ln w j )2 (8)

112 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Therefore, (MOD 1) can be written as

(MOD 1)′ Min J = 2n∑

i< j

(ln bi j − ln b ji − ln wi + ln w j )2

s.t.n∑

i=1

wi = 1, wi ≥ 0, i = 1, . . . , n

Equivalently,

(MOD 1)′′ Min J =n∑

i< j

(ln bi j − ln b ji − ln wi + ln w j )2

s.t.n∑

i=1

wi = 1, wi ≥ 0, i = 1, . . . , n

The following corollary can be obtained:

Corollary 1 (Wang and Fan [33]). The optimal solution to (MOD 1) is

wi =

(∏nj=1

bi j

b ji

)1/n

∑ni=1

(∏nj=1

bi j

b ji

)1/n , i = 1, . . . , n (9)

Based on the above analysis, we give an algorithm to improve the consistency of the reciprocal [0,1]-valued preferencerelation B:

Algorithm 1. Step 1. Let B(p) = (b(p)i j )n×n = B = (bi j )n×n and p = 0, and give the threshold GC I and the controlling

parameter � (0 ≤ � ≤ 1), which can reflect the decision makers’ preferences.Step 2. Obtain the priority vector of B(p) by Eq. (9): w(p) = (w(p)

1 , w(p)2 , . . . , w(p)

n )T.Step 3. Calculate the geometric consistency index of B(p):

GC I (B(p)) = 2

(n − 1)(n − 2)

∑i< j

(ln b(p)i j − ln b(p)

j i − ln w(p)i + ln w

(p)j )2 (10)

Step 4. If GC I (B(p)) ≤ GC I , then go to Step 7; Otherwise, continue with the next step.Step 5. Let B(p+1) = (b(p+1)

i j )n×n , where

b(p+1)i j =

(b(p)i j )1−�

(w

(p)i

w(p)i + w

(p)j

)�

(b(p)i j )1−�

(w

(p)i

w(p)i + w

(p)j

)�

+ (1 − b(p)i j )1−�

(1 − w

(p)i

w(p)i + w

(p)j

)�

=(b(p)

i j )1−�(w(p)i )�

(b(p)i j )1−�(w(p)

i )� + (1 − b(p)i j )1−�(w(p)

j )�, i, j = 1, 2, . . . , n (11)

Step 6. Let p = p + 1, then go to Step 2.Step 7. Let B̄ = B(p). Output the adjusted reciprocal [0,1]-valued preference relation B̄, its geometric consistency

index GC I (B̄) and the number of the iteration p.

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 113

From Eq. (11), we can find that b(p+1)i j is the combination of b(p)

i j and w(p)i /(w(p)

i + w(p)j ) which can be controlled

by the parameter �. Especially, if � = 0, then b(p+1)i j = b(p)

i j ; if � = 1, then b(p+1)i j = w

(p)i /(w(p)

i + w(p)j ). Actually,

in many real-life decision making problems, it is very time consuming, expensive and impracticable to interact withthe decision makers frequently although the method is reliable and accurate. To deal with this situation, Algorithm 1is very suitable because it needs less interactions with the decision makers in the decision making process. It shouldbe noted this does not indicate that the decision makers do not participate in the decision making process, but just doit with less interactions. In fact, the main factors of the decision making process, such as the reciprocal [0,1]-valuedpreference relations, the values of the threshold GC I and � are all provided by the decision makers, which is the coreof the decision making process. Algorithm 1 can also be described by using Fig. A.1.

Example 1. For a decision making problem, there are four alternatives xi (i = 1, 2, 3, 4). The decision maker provideshis/her preferences over these four alternatives, and gives a reciprocal [0,1]-valued preference relation as follows:

B =

⎡⎢⎢⎢⎣

0.5 0.4 0.7 0.3

0.6 0.5 0.6 0.8

0.3 0.4 0.5 0.3

0.7 0.2 0.7 0.5

⎤⎥⎥⎥⎦

Let p = 0, B(p) = B, GC I = 0.4 and � = 0.1. By Eq. (9), we can obtain the priority vector of B:

w = (0.2098, 0.4021, 0.1373, 0.2508)T

and the geometric consistency index of B(0): GC I (B(0)) = 0.6767 >GC I . Then we can use Algorithm 1 to improvethe consistency of the incomplete reciprocal [0,1]-valued preference relation B. By Eq. (11), we have

B(1) =

⎡⎢⎢⎢⎣

0.5000 0.3941 0.6910 0.3142

0.6059 0.5000 0.6159 0.7850

0.3090 0.3841 0.5000 0.3052

0.6858 0.2150 0.6948 0.5000

⎤⎥⎥⎥⎦

and GC I (C (1)) = 0.5481 >GC I . Then, let p = 1, and by Eq. (11), we have

B(2) =

⎡⎢⎢⎢⎣

0.5000 0.3889 0.6828 0.3273

0.6111 0.5000 0.6301 0.7708

0.3172 0.3699 0.5000 0.3099

0.6727 0.2292 0.6901 0.5000

⎤⎥⎥⎥⎦

and GC I (B(2)) = 0.4440 >GC I . Then, let p = 2, and by Eq. (11), we have

B(3) =

⎡⎢⎢⎢⎣

0.5000 0.3842 0.6754 0.3394

0.6158 0.5000 0.6426 0.7574

0.3246 0.3574 0.5000 0.3141

0.6606 0.2426 0.6859 0.5000

⎤⎥⎥⎥⎦

and GC I (B(3)) = 0.3596 <GC I . Therefore, the iteration stops.If we randomly generate a reciprocal [0,1]-valued preference relations, we can obtain the acceptable preference rela-

tion after several iterations by Algorithm 1. If we randomly generate 1000 reciprocal [0,1]-valued preference relations,then we can calculate the average iterations to obtain the preference relations with acceptable consistency. Table 1gives the average iteration by assigning different values to the input parameters. To give an intuitive understanding, weuse Fig. A.2 (see Appendix A) to express the results obtained in Table 1. From Table 1 and Fig. A.2, we can find thatAlgorithm 1 can obtain the acceptable reciprocal [0,1]-valued preference relation after some iterations; as the value ofn increases, the number of the iterations increases; the smaller the values of � and GC I , the bigger the number of theiterations.

114 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Table 1Average values of iterations in Algorithm 1.

n GC I p

� = 0.1 � = 0.2 � = 0.3 � = 0.5

3 0.100 12.297 5.879 3.810 2.2200.200 9.557 4.664 3.072 1.7740.400 6.854 3.421 2.241 1.346

4 0.100 15.247 7.408 4.848 2.7210.200 11.528 5.965 3.770 2.2030.400 8.376 4.344 2.934 1.734

5 0.100 15.938 7.895 5.140 2.8570.200 12.723 6.192 4.125 2.3560.400 9.333 4.675 3.180 1.873

6 0.100 16.445 8.010 5.173 2.9280.200 13.275 6.462 4.232 2.4060.400 9.796 4.901 3.269 1.925

7 0.100 16.551 8.084 5.277 2.9630.200 13.312 6.635 4.324 2.4200.400 10.165 5.021 3.309 1.961

8 0.100 16.714 8.219 5.300 2.9810.200 13.444 6.651 4.329 2.4470.400 10.230 5.097 3.326 1.973

9 0.100 16.834 8.184 5.297 2.9980.200 13.521 6.661 4.334 2.4670.400 10.278 5.074 3.371 1.990

Next we investigate the convergence of Algorithm 1 by comparing the consistency levels of the original reciprocal[0,1]-valued preference relation B and its revised one B̄ by using Algorithm 1, and the following theorem is given:

Theorem 2. Let B be a reciprocal [0,1]-valued preference relation, and B̄ the adjusted one by Algorithm 1, thenGC I (B̄) < GC I (B).

The proof of Theorem 2 is provided in Appendix B.From Theorem 2, we can find that Algorithm 1 can improve the consistency level of a reciprocal [0,1]-valued

preference relation by producing a series of revised reciprocal [0,1]-valued preference relations with the monotonedecreasing consistency levels. By giving a further study, an interesting conclusion can be obtained:

Theorem 3. Let w = (w1, w2, . . . , wn)T be the priority vector of the reciprocal [0,1]-valued preference relation B, andB̄ the adjusted reciprocal [0,1]-valued preference relation associated with the priority vector w̄ = (w̄1, w̄2, . . . , w̄n)T

in Algorithm 1, then w = w̄.

The proof of Theorem 3 is provided in Appendix B.Theorem 3 indicates that the weight vectors of these revised reciprocal [0,1]-valued preference relations obtained

by Algorithm 1 are always the same, which can keep as much original information as possible in the adjusted process.Actually, after the decision makers provide the original information, they are very reluctant to see that their originalinformation is changed, especially, when the original priority is changed without their interactions. Algorithm 1 canimprove the consistency level under the assumption that the priority of the original reciprocal [0,1]-valued preferencerelation is constant, which can fully respect the decision makers’ opinions and motivate the decision makers to takepart in the next decision making. The principle has been used by Xu and Wei [39], Ma et al. [24] and Dong et al. [12].Although we can obtain the priority vectors at the beginning, the decision making should not only focus on the results,but also focus on the process that provides us many rich experiences for the next decision, which is also the core of theconsistency improving process.

Ma et al. [24], Wu and Xu [34] proposed two methods to improve the consistency level of a reciprocal [0,1]-valuedpreference relation, which are all based on the additive consistency. As mentioned in Definition 2, additive consistencyhas some advantages, and we have to use the conversion steps while some unreasonable results are produced in theconsistency improving process, which not only need many time and energy, but also may lose some original decision

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 115

information. The proposed method considers both the priority vector and the preference elements, while the ones givenby Ma et al. [24] and Wu and Xu [34] are only based on the preference elements. Especially, when some elements ofthe reciprocal [0,1]-valued preference relations are unknown, the methods given by Ma et al. [24] and Wu and Xu [34]will be invalid, but such cases would be solved by using our method with minor revision, which will be discussed inthe following.

Similarly, an algorithm can be developed to improve the consistency level of an incomplete reciprocal [0,1]-valuedpreference relation. For an incomplete reciprocal [0,1]-valued preference relation C, Xu et al. [36] proposed thefollowing model to derive the priority vector of C:

(MOD 2) Min J =n∑

j=1

n∑i=1

�i j (ln ci j − ln c ji − ln wi + ln w j )2

s.t.n∑

i=1

wi = 1, wi ≥ 0, i = 1, . . . , n

where �i j is a zero or one integer variable defined as

�i j ={

0 if ci j is unknown,

1 if ci j is known,i, j = 1, . . . , n (12)

Theorem 4 (Xu et al. [36]). The optimal solution to (MOD 2) is

wi =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

exp(x j )∑n−1j=1 exp(x j ) + 1

, i = 1, 2, . . . , n − 1

1∑n−1j=1 exp(x j ) + 1

, i = n(13)

where exp(xi ) is the exponential function of the ith element of a logarithmic weight vector X = (x1, x1, . . . , xn−1)T

determined by X = M−1Y , where

M =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

n∑j=2

�1 j −�12 · · · −�1,n−1

−�21

n∑j=2, j�2

�2 j · · · −�2,n−1

...... · · · ...

−�n−1,1 −�n−2,2 · · ·n∑

j=1, j�n−1�n−1, j

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Y =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

n∑j=1

�1 j (ln c1 j − ln c j1)

n∑j=1

�2 j (ln c2 j − ln c j2)

...n∑

j=1�1 j (ln cn−1, j − ln c j,n−1)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

where M−1 is the inverse matrix of M, and �i j is defined as before.

Definition 5. Let C be an incomplete reciprocal [0,1]-valued preference relation, then the geometric consistency indexof C can be given as

GC I (C) = 1

q

n∑i=1

n∑j=1

�i j (ln(ci j ) − ln(c ji ) − ln(wi ) + ln(w j ))2 (16)

116 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

where q = ∑ni=1

∑nj=1 �i j denotes the number of the known preference values in C. Especially, if GC I (C) = 0, then

C is a consistent incomplete reciprocal [0,1]-valued preference relation.

As mentioned above, the consistent incomplete reciprocal [0,1]-valued preference relation is hard to be obtained inpractical problems, therefore we give a method to improve the consistency level of an inconsistent incomplete reciprocal[0,1]-valued preference relation motivated by Algorithm 1 as follows:

Algorithm 2. Step 1. Let C (p) = (C (p)i j )n×n = (ci j )n×n and p = 0, and give the threshold GC I and the controlling

parameter � (0 ≤ � ≤ 1).Step 2. Let w(p) = (w(p)

1 , w(p)2 , . . . , w(p)

n )T be the priority vector derived from C (p) by (MOD 2).Step 3. Calculate the geometric consistency index of C (p):

GC I (C (p)) = 1

q

n∑i=1

n∑j=1

�i j (ln c(p)i j − ln c(p)

j i − ln w(p)i + ln w

(p)j )2 (17)

where q and �i j are defined as before.

Step 4. If GC I (C (p)) ≤ GC I , then go to Step 6; Otherwise, let C (p+1) = (c(p+1)i j )n×n , where

c(p+1)i j =

(c(p)i j )1−�(w(p)

i )�

(c(p)i j )1−�(w(p)

i )� + (1 − c(p)i j )1−�(w(p)

j )�, �i j = 1 (18)

It should be noted that c(p+1)i j is the combination of c(p)

i j and w(p)i , which can be controlled by the parameter �. Especially,

if � = 0, then c(p+1)i j = c(p)

i j ; if � = 1, then c(p+1)i j = w

(p)i /(w(p)

i + w(p)j ).

Step 5. Let p = p + 1, then go to Step 2.Step 6. Let C̄ = C̄ (p). Output the adjusted reciprocal [0,1]-valued preference relation C̄ , its geometric consistency

index GC I (C̄) and the number of iteration p.

Theorem 5. Let C be a reciprocal [0,1]-valued preference relation, and C̄ the adjusted one by Algorithm 1, thenGC I (C̄) ≤ GC I (C).

The proof of Theorem 5 is provided in Appendix B.To illustrate Algorithm 2, the following example is given:

Example 2. For a decision making problem, there are six alternatives xi (i = 1, 2, . . ., 6). The decision maker provideshis/her preferences over these six alternatives, and gives an incomplete reciprocal [0,1]-valued preference relation asfollows:

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0.5 0.4 x 0.3 0.8 0.3

0.6 0.5 0.6 0.5 x 0.4

x 0.4 0.5 0.3 0.6 x

0.7 0.5 0.7 0.5 0.4 0.8

0.2 x 0.4 0.6 0.5 0.7

0.7 0.6 x 0.2 0.3 0.5

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

The necessary step in Ma et al. [24] and Wu and Xu [34]’s method is to construct the additive consistent reciprocal[0,1]-valued preference relation, however, while the preference relation is incomplete, it is impossible to construct aconsistent one, which can be solved by using Algorithm 2:

Let p = 0, C (p) = C , GC I = 0.4 and � = 0.1. By (MOD 2) and Eq. (17), we can obtain the priority vector of C:

w = (0.1448, 0.1807, 0.1410, 0.2536, 0.1427, 0.1373)T

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 117

and the geometric consistency index of C (0): GC I (C (0)) = 0.9775 >GC I . Then we can use Algorithm 2 to improvethe consistency of the incomplete reciprocal [0,1]-valued preference relation C. By Eq. (18), we have

C (1) =

⎡⎢⎢⎢⎢⎢⎢⎣

0.5 0.4044 x 0.3061 0.7771 0.31920.5956 0.5 0.5962 0.4915 x 0.4164

x 0.4038 0.5 0.3055 0.5899 x0.6939 0.5085 0.6945 0.5 0.4237 0.78730.2227 x 0.4101 0.5763 0.5 0.68270.6808 0.5836 x 0.2127 0.3173 0.5

⎤⎥⎥⎥⎥⎥⎥⎦

and GC I (C (1)) = 0.7917 >GC I . Then, let p = 1, and by Eq. (18), we have

C (2) =

⎡⎢⎢⎢⎢⎢⎢⎣

0.5 0.4084 x 0.3116 0.7550 0.33710.5916 0.5 0.5928 0.4839 x 0.4314

x 0.4072 0.5 0.3105 0.5808 x0.6884 0.5161 0.6895 0.5 0.4454 0.7750.2450 x 0.4192 0.5546 0.5 0.66680.6629 0.5686 x 0.2245 0.3332 0.5

⎤⎥⎥⎥⎥⎥⎥⎦

and GC I (C (2)) = 0.6413 >GC I . Then let p = 2, and by Eq. (18), we have

C (3) =

⎡⎢⎢⎢⎢⎢⎢⎣

0.5 0.4120 x 0.3166 0.7339 0.35360.5880 0.5 0.5897 0.4771 x 0.4449

x 0.4103 0.5 0.3150 0.5726 x0.6834 0.5229 0.6850 0.5 0.4651 0.76440.2661 x 0.4274 0.5349 0.5 0.65210.6464 0.5551 x 0.2356 0.3479 0.5

⎤⎥⎥⎥⎥⎥⎥⎦

and GC I (C (3)) = 0.5196 >GC I . Then let p = 3, and by Eq. (18), we have

C (4) =

⎡⎢⎢⎢⎢⎢⎢⎣

0.5 0.4153 x 0.3211 0.7139 0.36870.5847 0.5 0.5869 0.4709 x 0.4572

x 0.4131 0.5 0.3191 0.5651 x0.6789 0.5338 0.6809 0.5 0.4829 0.75410.2861 x 0.4349 0.5171 0.5 0.63860.6313 0.5428 x 0.2459 0.3614 0.5

⎤⎥⎥⎥⎥⎥⎥⎦

and GC I (C (4)) = 0.4209 >GC I . Then let p = 4, and by Eq. (18), we have

C (5) =

⎡⎢⎢⎢⎢⎢⎢⎣

0.5 0.4182 x 0.3252 0.6952 0.38260.5818 0.5 0.5844 0.4654 x 0.4683

x 0.4156 0.5 0.3229 0.5584 x0.6748 0.5346 0.6771 0.5 0.4990 0.74460.3048 x 0.4416 0.5010 0.5 0.62620.6174 0.5317 x 0.2554 0.3738 0.5

⎤⎥⎥⎥⎥⎥⎥⎦

and GC I (C (5)) = 0.3409 <GC I . Therefore, the iteration stops.

3. Consensus model for group reciprocal [0,1]-valued preference relations

In this section, based on the multiplicative consistency property, we utilize the relationship between the priorityvector and the preference values in a reciprocal [0,1]-valued preference relation to develop a convergence algorithm tohelp the decision makers reach consensus quickly. Occasionally, this process may require an excessive amount of timeto complete.

118 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Theorem 6. Let B(k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relationsprovided by decision makers dk (k = 1, 2, . . . , m), whose weight vector is �k (k = 1, 2, . . . , m), (w(k)

1 , w(k)2 , . . . , w(k)

n )T

the corresponding priority vectors derived from B(k) by Eq. (9), and B(c) = (b(c)i j )n×n the collective reciprocal [0,1]-

valued preference relation by using the fusion method given by Xia and Xu [35], where

b(c)i j =

∏mk=1 (b(k)

i j )�k∏mk=1 (b(k)

i j )�k +∏mk=1 (1 − b(k)

i j )�k, i, j = 1, 2, . . . , n (19)

then

w(c)i =

∏mk=1 (w(k)

i )�k∑ni=1

∏mk=1 (w(k)

i )�k, i = 1, 2, . . . , n (20)

The proof of Theorem 6 is provided in Appendix B.Theorem 6 indicates that the weight vector of the collective reciprocal [0,1]-valued preference relation B(c) obtained

by Eq. (19) is a weighted combination of the weight vectors of the individual reciprocal [0,1]-valued preferencerelations.

Inspired by Dong et al. [12], we define a geometric cardinal consensus index as follows:

Definition 6. Let B(k) (k = 1, 2, . . . , m) be a collection of reciprocal [0,1]-valued preference relations provided bythe decision makers k (k = 1, 2, . . . , m), and wc = (w(c)

1 , w(c)2 , . . . , w(c)

n )T be the collective priority vector obtained byusing Eq. (20). Then, the geometric cardinal consensus indices of B(k) (k = 1, 2, . . . , m) are defined by

GCC I (B(k)) = 2

(n − 1)(n − 2)

∑i< j

(ln b(k)i j − ln b(k)

i j − ln wci + ln wc

j )2, k = 1, 2, . . . , m (21)

Especially, if GCC I (B(k)) = 0, k = 1, 2, . . . , m, then all the individual B(k) (k = 1, 2, . . . , m) reach consensus, whichseldom happens in the practical problems. The smaller the value of GCC I (B(k)), the better the consensus of B(k),then next we will give an algorithm to improve the consensus levels of individual reciprocal [0,1]-valued preferencerelations:

Algorithm 3. Step 1. Let B(k)p = (b(k)

i j p)n×n = (b(k)i j )n×n (k = 1, 2, . . . , m), p = 0, � = (�1, �2, . . ., �m)T the weight

vector of decision makers, and give the threshold GCC I , the maximum number pmax > 1 and the controlling parameter� (0 ≤ � ≤ 1).

Step 2. Let w(c)p = (w(c)

1,p, w(c)2,p, . . . , w(c)

n,p)T be the collective priority vector derived from the collective matrix

B(c)p = (b(c)

i j,p)n×n = (b(c)i j )n×n , where

bci j,p =

∏mk=1 (b(k)

i j,p)�k∏mk=1 (b(k)

i j,p)�k +∏mk=1 (1 − b(k)

i j,p)�k, i, j = 1, 2, . . . , n (22)

Step 3. Calculate the geometric cardinal consensus indices of B(k)p (k = 1, 2, . . . , m):

GCC I (B(k)p ) = 2

(n − 1)(n − 2)

∑i< j

(ln b(k)i j,p − ln b(k)

i j,p − ln w(c)i + ln w

(c)j )2, k = 1, 2, . . . , m (23)

If GCC I (B(k)p ) ≤ GCC I (k = 1, 2, . . . , m) or p > pmax, then go to Step 6; Otherwise, continue with the next step.

Step 4. Let B(k)p+1 = (b(k)

i j,p+1)n×n (k = 1, 2, . . . , m), where

b(k)i j,p+1 =

(b(k)i j,p)1−�(w(c)

i,p)�

(b(k)i j,p)1−�(w(c)

i,p)� + (1 − b(k)i j,p)1−�(w(c)

j,p)�, i, j = 1, 2, . . . , n, k = 1, 2, . . . , m (24)

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 119

which is the fusion of b(k)i j,p and w

(c)i,p, and can be changed by assigning the parameter � different values. Especially, if

� = 0, then b(k)i j,p+1 = b(k)

i j,p; if � = 1, then b(k)i j,p+1 = w

(c)i,p/(w(c)

i,p + w(c)j,p).

Step 5. Let p = p + 1, then go to Step 2.Step 6. Let B̄(k) = B(k)

p (k = 1, 2, . . . , m). Output the adjusted reciprocal [0,1]-valued preference relations B̄(k) (k =1, 2, . . . , m), their geometric cardinal consensus indices GCC I (B̄(k)) (k = 1, 2, . . . , m), the collective priority vectorw̄c = (w̄(c)

1 , w̄(c)2 , . . . , w̄(c)

n )T, and the number of the iteration p.

Algorithm 3 can also be described by using Fig. A.3.

Example 3. Suppose that there are four decision makers k (k = 1, 2, 3), whose weight vector is � = (0.5, 0.2, 0.3)T.The decision makers provide their preferences about the four alternatives xi (i = 1, 2, 3, 4) by the following threereciprocal [0,1]-valued preference relations B(k) (k = 1, 2, 3):

B(1) =

⎡⎢⎢⎣

0.5 0.3 0.6 0.50.7 0.5 0.6 0.70.6 0.4 0.5 0.40.3 0.7 0.6 0.5

⎤⎥⎥⎦ , B(2) =

⎡⎢⎢⎣

0.5 0.6 0.5 0.30.4 0.5 0.4 0.60.5 0.6 0.5 0.60.7 0.4 0.4 0.5

⎤⎥⎥⎦ , B(3) =

⎡⎢⎢⎣

0.5 0.4 0.5 0.50.6 0.5 0.7 0.60.5 0.3 0.5 0.70.5 0.4 0.3 0.5

⎤⎥⎥⎦

Let GC I = 0.4, we first calculate the geometric consistency indices of C (k) (k = 1, 2, 3, 4):

GC I (C (1)) = 0.1197, GC I (C (2)) = 0.3736, GC I (C (3)) = 0.2146

Then we use Algorithm 3 to improve the consensus of group incomplete reciprocal [0,1]-valued preference relations:Let B(1)

0 = B(1), B(2)0 = B(2), B(3)

0 = B(3) and B(4)0 = B(4), we have

w(1)0 = (0.2122, 0.4007, 0.1749, 0.2122)T, w

(2)0 = (0.2220, 0.2241, 0.3037, 0.2502)T

w(3)0 = (0.2178, 0.3649, 0.2411, 0.1762)T, w

(c)0 = (0.2191, 0.3521, 0.2183, 0.2106)T

GCC I (B(1)) = 0.2044, GCC I (B(2)) = 0.8299, GCC I (B(3)) = 0.2709

Suppose GCC I = 0.35, then GCC I (B(2)) > GCC I , thus let � = 0.4, we have

B(1)1 =

⎡⎢⎢⎣

0.5 0.3322 0.5609 0.50400.6678 0.5 0.6070 0.67130.4391 0.3930 0.5 0.44300.4960 0.3287 0.5570 0.5

⎤⎥⎥⎦ , B(2)

1 =

⎡⎢⎢⎣

0.5 0.5134 0.5004 0.37930.4866 0.5 0.4870 0.61040.4996 0.5130 0.5 0.56410.6207 0.3896 0.4359 0.5

⎤⎥⎥⎦

B(3)1 =

⎡⎢⎢⎣

0.5 0.3934 0.5004 0.50400.6066 0.5 0.6681 0.61040.4996 0.3319 0.5 0.62780.4960 0.3896 0.3722 0.5

⎤⎥⎥⎦

and

GC I (B(1)1 ) = 0.0431, GC I (B(2)

1 ) = 0.1346, GC I (B(3)1 ) = 0.0772

GCC I (B(1)1 ) = 0.0736, GCC I (B(2)

1 ) = 0.2988, GCC I (B(3)1 ) = 0.0975

w(1)1 = (0.2154, 0.3812, 0.1914, 0.2120)T, w

(2)1 = (0.2233, 0.2715, 0.2691, 0.2361)T

w(3)1 = (0.2186, 0.3601, 0.2319, 0.1894)T

w(c)1 = w

(c)0 = (0.2191, 0.3521, 0.2183, 0.2106)T

Therefore, all GCC I (B(k)) < GCC I , k = 1, 2, 3, 4, and thus, the decision makers reach a consensus.

120 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Table 2Average values of iterations in Algorithm 3.

n m GCC I p

� = 0.1 � = 0.2 � = 0.5 � = 0.7

3 3 0.050 24.140 11.549 4.076 2.5730.100 20.491 10.090 3.539 2.2350.300 15.036 7.500 2.792 1.823

3 4 0.050 25.368 12.207 4.250 2.7340.100 21.980 10.659 3.749 2.3690.300 17.065 8.239 2.993 1.951

3 5 0.050 26.257 12.594 4.390 2.8180.100 22.839 11.213 3.896 2.4740.300 17.780 8.671 3.107 1.988

4 5 0.050 22.983 11.089 3.927 2.4070.100 19.636 9.570 3.397 2.0540.300 14.391 7.054 2.602 1.831

4 6 0.050 23.500 11.319 3.984 2.4910.100 20.061 9.764 3.473 2.0830.300 14.815 7.285 2.712 1.903

4 7 0.050 23.754 11.511 4.044 2.5590.100 20.482 9.974 3.553 2.0980.300 15.269 7.455 2.775 1.926

5 6 0.050 21.386 10.312 3.735 2.1520.100 18.160 8.762 3.135 2.0050.300 12.845 6.337 2.335 1.653

5 7 0.050 21.710 10.495 3.783 2.1780.100 18.353 8.989 3.161 2.0050.300 13.208 6.519 2.389 1.729

6 7 0.050 20.236 9.738 3.473 2.0290.100 16.905 8.275 3.014 2.0000.300 11.596 5.800 2.119 1.426

7 7 0.050 19.093 9.307 3.205 2.0020.100 15.841 7.741 2.947 1.9970.300 10.607 5.289 2.025 1.178

7 8 0.050 19.329 9.368 3.222 2.0050.100 15.961 7.815 2.965 1.9970.300 10.751 5.375 2.028 1.196

7 9 0.050 19.458 9.464 3.266 2.0080.100 16.159 7.889 2.970 2.0000.300 10.994 5.472 2.039 1.231

8 9 0.050 18.548 8.997 3.091 2.0000.100 15.177 7.462 2.907 2.0000.300 10.040 4.976 2.002 1.070

9 10 0.050 17.846 8.697 3.027 2.0000.100 14.625 7.182 2.816 1.9920.300 9.335 4.681 1.997 1.019

9 11 0.050 18.002 8.793 3.018 2.0000.100 14.708 7.215 2.847 1.9980.300 9.483 4.737 1.994 1.018

Similar to Algorithm 1, we randomly generate 1000 groups of reciprocal [0,1]-valued preference relations, andcalculate the average iterations to obtain the preference relations with acceptable consensus levels. Table 2 gives theaverage iteration by assigning different values to the input parameters. Fig. A.4 describe the date in Table 2 intuitively.From Table 2 and Fig. A.4, we find that Algorithm 3 can help the decision makers reach consensus after a certainiteration; given the value of n, the number of the iterations increases as the value of m increases; given the value of m,the number of the iterations decreases as the value of n increases; the smaller the values of � and GC I , the bigger thenumber of the iterations.

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 121

In the following, the convergence of Algorithm 3 is studied, and the following lemma is given first:

Lemma 1 (Escobar et al. [15]). For any a, b ∈ Rn , it holds that

n∑i=1

ai bi ≤ max

{n∑

i=1

a2i ,

n∑i=1

b2i

}(25)

We first study the relationship between the consistency level of the collective reciprocal [0,1]-valued preferencerelation and the consistency levels of the individual reciprocal [0,1]-valued preference relations and obtain the followingtheorem:

Theorem 7. Let B(k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relationsassociated with the weight vector � = (�1, �2, . . . , �m)T, and B(c) the aggregation of the individual reciprocal[0,1]-valued preference relations B(k) (k = 1, 2, . . . , m), derived by Eq. (19), then

GC I (B(c)) ≤ maxk

{GC I (B(k))} (26)

The proof of Theorem 7 is provided in Appendix B.Theorem 7 indicates that the consistency levels of the collective reciprocal [0,1]-valued preference relation are better

than any individual reciprocal [0,1]-valued preference relations. Furthermore, the following corollary can be obtainedeasily.

Corollary 2. Let B(k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relationsassociated with the weight vector � = (�1, �2, . . . , �m)T, and B(c) the aggregation of the individual reciprocal[0,1]-valued preference relations B(k) (k = 1, 2, . . . , m), derived by Eq. (24), then

GC I (B(k)) ≤ �, k = 1, 2, . . . , m ⇒ GC I (B(c)) ≤ � (27)

which implies that if the geometric consistency indexes of individual reciprocal [0,1]-valued preference relations aresmaller than a value, then the geometric consistency index of the collective reciprocal [0,1]-valued preference relationis always smaller than this value.

Then the consistency levels of the individual reciprocal [0,1]-valued preference relations are investigated before andafter Algorithm 3 being used.

Theorem 8. Let B(k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relationsassociated with the weight vector � = (�1, �2, . . . , �m)T, and let B̄(k) (k = 1, 2, . . . , m) be the adjusted reciprocal[0,1]-valued preference relations by Algorithm 3. Then

GC I (B̄(k)) ≤ GC I (B(k)) (28)

The proof of Theorem 8 is provided in Appendix B.Theorem 8 tells us that the geometric consistency index of the original individual reciprocal [0,1]-valued pref-

erence relation is bigger than that of the revised individual reciprocal [0,1]-valued preference relation obtained byAlgorithm 3.

By analyzing the weight vector of the original collective reciprocal [0,1]-valued preference relation and the revisedone by Algorithm 3, an interesting result can be obtained:

Theorem 9. Let B(k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relationsassociated with the weight vector � = (�1, �2, . . . , �m)T, and w(c) = (w(c)

1 , w(c)2 , . . . , w(c)

n )T the collective priorityvector of B(k) (k = 1, 2, . . . , m), and B̄(k) (k = 1, 2, . . . , m) the adjusted reciprocal [0,1]-valued preference relationsby Algorithm 2, and w̄(c) = (w̄(c)

1 , w̄(c)2 , . . . , w̄(c)

n )T the collective priority vector of B̄(k) (k = 1, 2, . . . , m). Thenw(c) = w̄(c).

122 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

The proof of Theorem 9 is provided in Appendix B.Theorem 9 implies that the weight vector of the original collective reciprocal [0,1]-valued preference relation is always

the same as the revised one obtained by Algorithm 3, which can preserve as much original information as possible,and make the revised individual preference relations obtained by the proposed algorithms to be easily accepted bythe decision makers. Interacting with the decision makers frequently during the consensus process is very reliable andaccurate but impracticable because of the too large amount and the too long period of work needed. In cases whereconsensus must be urgently obtained and the decision makers cannot or are unwilling to modify their preferences,we can make a promise that we only improve the consensus level under the assumption that the weight vector of thedecision makers’ original reciprocal [0,1]-valued preference relations cannot be changed, which is reasonable and canalso be supported by the decision makers. Algorithm 3 can help the decision makers reach consensus among groupopinions quickly with less interactions, which also can avoid forcing the decision makers to modify their opinions.This issue has been discussed in detail by many authors [12,37,38].

Based on the above analysis, the following theorem is given:

Theorem 10. Let B(k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relations, andB̄(k) (k = 1, 2, . . . , m) the adjusted reciprocal [0,1]-valued preference relations by Algorithm 3, then

GCC I (B̄(k)) ≤ GCC I (B(k)) (29)

The proof of Theorem 10 is provided in Appendix B.Theorem 10 indicates that the consensus levels of the original individual reciprocal [0,1]-valued preference relations

are improved by Algorithm 3.Next, we give an algorithm to improve the consensus levels of individual incomplete preference relations. Let

C (k) (k = 1, 2, . . . , m) be the incomplete reciprocal [0,1]-valued preference relations provided by the decision makersk (k = 1, 2, . . . , m), then we first utilize (MOD 2) to derive the priority vectors of C (k) : w(k) = (w(k)

1 , w(k)2 , . . . , w(k)

n )T

(k = 1, 2, . . . , m) and further give the following definition:

Definition 7. Let C (k) (k = 1, 2, . . . , m) be the incomplete reciprocal [0,1]-valued preference relations provided bythe decision makers k (k = 1, 2, . . . , m), and w(k) = (w(k)

1 , w(k)2 , . . . , w(k)

n )T (k = 1, 2, . . . , m) the weight vectors

derived from C (k) (k = 1, 2, . . . , m) by using (MOD 2), wc = (w(c)1 , w

(c)2 , . . . , w(c)

n )T the collective priority vector

with w(c)i = ∏m

k=1 (w(k)i )�k /

∑ni=1

∏mk=1 (w(k)

i )�k , k = 1, 2, . . . , m. Then the geometric cardinal consensus indices ofC (k) (k = 1, 2, . . . , m) are defined by

GCC I (C (k)) = 1

q(k)

n∑i=1

n∑j=1

�(k)i j (ln c(k)

i j − ln c(k)i j − ln wc

i + ln wcj )

2, k = 1, 2, . . . , m (30)

where q(k) = ∑ni=1

∑nj=1 �i j denotes the number of the known preference values in C (k), and �(k)

i j is a zero or oneinteger variable defined as

�(k)i j =

{0 if c(k)

i j is unknown,

1 if c(k)i j is known,

i, j = 1, 2, . . . , n, k = 1, 2, . . . , m (31)

Next, we give an algorithm to improve the consensus levels of individual incomplete reciprocal [0,1]-valued prefer-ence relations:

Algorithm 4. Step 1. Let C (k)p = (c(k)

i j p)n×n = (c(k)i j )n×n (k = 1, 2, . . . , m) and p = 0, � = (�1, �2, . . . , �m)T the

weight vector of decision makers, and give the threshold GCC I , the maximum number pmax > 1 and the controllingparameter � (0 ≤ � ≤ 1).

Step 2. Let w(k)p = (w(k)

1,p, w(k)2,p, . . . , w(k)

n,p)T (k = 1, 2, . . . , m) be the priority vectors derived from the

reciprocal [0,1]-valued preference relations C (k)p = (c(k)

i j p)n×n = (c(k)i j )n×n , and w

(c)p = (w(c)

1,p, w(c)2,p, . . . , w(c)

n,p)T

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 123

the collective priority vector, where

w(c)i =

∏mk=1 (w(k)

i )�k∑ni=1

∏mk=1 (w(k)

i )�k, i = 1, 2, . . . , n (32)

Step 3. Calculate the geometric cardinal consensus indices of C (k) (k = 1, 2, . . . , m):

GCC I (C (k)p ) = 1

q(k)

n∑i=1

n∑j=1

�(k)i j,p(ln c(k)

i j,p − ln c(k)i j,p − ln wc

i + ln wcj )

2 (33)

where q(k) and �(k)i j are defined as before.

If all GCC I (C (k)p ) ≤ GCC I (k = 1, 2, . . . , m) or p > pmax, then go to Step 6; Otherwise, continue with the next

step.Step 4. Let C (k)

p+1 = (c(k)i j,p+1)n×n (k = 1, 2, . . . , m), where

c(k)i j,p+1 =

(c(k)i j,p)1−�(w(c)

i,p)�

(c(k)i j,p)1−�(w(c)

i,p)� + (1 − c(k)i j,p)1−�(w(c)

j,p)�, �(k)

i j = 1 (34)

which is the combination of c(k)i j,p and w

(c)i,p, and can be changed by assigning the parameter � different values. Especially,

if � = 0, then c(k)i j,p+1 = c(k)

i j,p; if � = 1, then c(k)i j,p+1 = w

(c)i,p/(w(c)

i,p + w(c)j,p).

Step 5. Let p = p + 1, then go to Step 2.Step 6. Let C̄ (k) = C (k)

p (k = 1, 2, . . . , m). Output the adjusted reciprocal [0,1]-valued preference relations C̄ (k)

(k = 1, 2, . . . , m), their geometric cardinal consensus indices GCC I (C̄ (k)) (k = 1, 2, . . . , m), and the number of theiteration p.

Theorem 11. Let C (k) (k = 1, 2, . . . , m) be a collection of individual reciprocal [0,1]-valued preference relations, andC̄ (k) (k = 1, 2, . . . , m) the adjusted reciprocal [0,1]-valued preference relations by Algorithm 3, then

GCC I (C(k)

) � GCC I (C (k)) (35)

The proof of Theorem 11 is provided in Appendix B.

Example 4. Suppose that there are four decision makers k (k=1, 2, 3, 4), whose weight vector is�=(0.1, 0.3, 0.2, 0.4)T.The decision makers provide their preferences about the four alternatives xi (i = 1, 2, 3, 4) by the following four in-complete reciprocal [0,1]-valued preference relations C (k) (k = 1, 2, 3, 4):

C (1) =

⎡⎢⎢⎢⎢⎣

0.5 0.6 x 0.7

0.4 0.5 0.2 0.8

x 0.8 0.5 0.4

0.3 0.2 0.6 0.5

⎤⎥⎥⎥⎥⎦ , C (2) =

⎡⎢⎢⎢⎢⎣

0.5 0.8 0.4 x

0.2 0.5 0.3 0.6

0.6 0.7 0.5 0.3

x 0.4 0.7 0.5

⎤⎥⎥⎥⎥⎦

C (3) =

⎡⎢⎢⎢⎢⎣

0.5 0.3 0.4 0.6

0.7 0.5 x 0.5

0.6 x 0.5 0.7

0.4 0.5 0.3 0.5

⎤⎥⎥⎥⎥⎦ , C (4) =

⎡⎢⎢⎢⎢⎣

0.5 0.6 0.6 0.3

0.4 0.5 0.4 0.5

0.4 0.6 0.5 x

0.7 0.5 x 0.5

⎤⎥⎥⎥⎥⎦

Let GC I = 0.4, we first calculate the geometric consistency indices of C (k) (k = 1, 2, 3, 4):

GC I (C (1)) = 1.3486 > GC I , GC I (C (2)) = 0.9937 > GC I

GC I (C (3)) = 0.2401, GC I (C (4)) = 0.3109

124 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Then we first use Algorithm 2 to improve the consistency of C (1) and C (2). There are too many subscript and superscript,to avoid confusion, here we will not repeat the process, and only give the final adjusted ones:

D(1) =

⎡⎢⎢⎣

0.5 0.6052 x 0.69550.3948 0.5 0.2995 0.7050

x 0.7005 0.5 0.53280.3045 0.2950 0.4672 0.5

⎤⎥⎥⎦ , D(2) =

⎡⎢⎢⎣

0.5 0.7461 0.4757 x0.2539 0.5 0.3312 0.48820.5243 0.6688 0.5 0.4026

x 0.5118 0.5974 0.5

⎤⎥⎥⎦

Thus, GC I (D(1)) = 0.3374 < GC I and GC I (D(2)) = 0.2485 < GC I .Then we use Algorithm 4 to improve the consensus of the individual incomplete reciprocal [0,1]-valued preference

relations:Let D(1)

0 = D(1), D(2)0 = D(2), D(3)

0 = C (3) and D(4)0 = C (4), we have

w(1)0 = (0.3380, 0.2158, 0.2950, 0.1512)T, w

(2)0 = (0.3293, 0.1525, 0.2666, 0.2515)T

w(3)0 = (0.1948, 0.2828, 0.3464, 0.1760)T, w

(4)0 = (0.2428, 0.2001, 0.2204, 0.3367)T

w(c)0 = (0.2698, 0.2042, 0.2696, 0.2564)T

GCC I (D(1)0 ) = 0.9091, GCC I (D(2)

0 ) = 0.6428, GCC I (D(3)0 ) = 0.7390, GCC I (D(4)

0 ) = 0.4556

Suppose GCC I = 0.35, then GCC I (D(k)) > GCC I , k = 1, 2, 3, 4, thus let � = 0.5, we have

D(1)1 =

⎡⎢⎢⎣

0.5 0.5873 x 0.60790.4127 0.5 0.3627 0.5798

x 0.6373 0.5 0.52270.3921 0.4202 0.4773 0.5

⎤⎥⎥⎦ , D(2)

1 =

⎡⎢⎢⎣

0.5 0.6633 0.4879 x0.3367 0.5 0.3798 0.46570.5121 0.6202 0.5 0.4571

x 0.5343 0.5429 0.5

⎤⎥⎥⎦

D(3)1 =

⎡⎢⎢⎣

0.5 0.4294 0.4496 0.55680.5706 0.5 x 0.47160.5504 x 0.5 0.61030.4432 0.5284 0.3897 0.5

⎤⎥⎥⎦ , D(4)

1 =

⎡⎢⎢⎣

0.5 0.5847 0.5506 0.40180.4153 0.5 0.4154 0.47160.4494 0.5846 0.5 x0.5982 0.5284 x 0.5

⎤⎥⎥⎦

and

GC I (D(1)1 ) = 0.0939, GC I (D(2)

1 ) = 0.0621, GC I (D(3)1 ) = 0.0843, GC I (D(4)

1 ) = 0.0777

GCC I (D(1)1 ) = 0.2630, GCC I (D(2)

1 ) = 0.2398, GCC I (D(3)1 ) = 0.1661, GCC I (D(4)

1 ) = 0.1468

w(1)1 = (0.3047, 0.2119, 0.2846, 0.1987)T, w

(2)1 = (0.2990, 0.1771, 0.2690, 0.2548)T

w(3)1 = (0.2321, 0.2434, 0.3094, 0.2151)T, w

(4)1 = (0.2571, 0.2030, 0.2448, 0.2951)T

w(c)1 = w(c) = (0.2698, 0.2042, 0.2696, 0.2564)T

Therefore, all GCC I (D(k)) < GCC I , k = 1, 2, 3, 4, and thus, the decision makers reach a consensus.

Many consensus improving methods [22,34,38] have been proposed recently. Wu and Xu [34] have compared theirproposed method with the ones given by Herrera-Viedma et al. [22] and Xu and Cai [38], and find some advantagesof their proposed ones. We can give a further analysis and find some advantages of our proposed method: Firstly, theexisting methods [22,34,38] only use the preference values to improve the consensus levels of the individual preferencerelations, while our method uses both the weight vector and the preference values, especially, when the individualreciprocal [0,1]-valued preference relations are incomplete ones, the existing ones [22,38] may be invalid, which willnot happen in our methods. Secondly, some existing methods [22,34] is based on additive consistency, while theproposed method is based on multiplicative consistency and can avoid unreasonable results which may be producedby using the methods based on additive consistency. Thirdly, in the proposed methods, the priority collective priority

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 125

is always constant, which not only can preserve the original information as much as possible, but also can fully respectthe decision makers’ opinions and encourage them to get involved in the next decision making, while the priority vectormay be changed in some existing methods [22,34,38], which may not be accepted by the decision makers, becausetheir original decision has been changed with the automatic revisions but not with their interactions.

4. Concluding remarks

In this paper, we have focused on the consistency of a reciprocal [0,1]-valued preference relation and the consensusof individual reciprocal [0,1]-valued preference relations based on multiplicative consistency. Two convergent algo-rithms have been developed to deal with the corresponding situations. We have also investigated the robustness of thealgorithms and have obtained some interesting correlations between the input and output parameters. After that, wehave generalized the proposed algorithms to improve the consistency and consensus levels for incomplete reciprocal[0,1]-valued preference relations. The proposed algorithms can obtain the desirable results quickly with less interactionof the decision makers which can save a lot of time and avoid forcing the decision makers to revise their preferenceinformation. We have also found that the priority vector of a reciprocal [0,1]-valued preference relation is invariant inthe consistency model, and the collective priority vector of group reciprocal [0,1]-valued preference relations is alsoinvariable in the consensus model, which can avoid missing the original preference information as much as possible andfully respect the decision makers’ opinions. Several examples have been given to illustrate advantages of the developedmethods.

Acknowledgments

The authors are very grateful to the editors and the anonymous reviewers for their insightful and constructivecomments and suggestions that have led to an improved version of this paper.

Appendix A

See Figs. A.1–A.4.

Obtain the priority vector of the preference relation

Calculate the geometric consistency index of the preference relation

Acceptable consistency?

End

yesno

Revise the preference

relation

Construct the preference relation

Fig. A.1. The consistency improving process in Algorithm 1.

126 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Fig. A.2. In (a)–(g), the horizontal axis shows the average iteration p, the vertical axis the values of controlling parameter �, and the different colorsthe threshold GC I in Algorithm 1. (For interpretation of the references to color in this figure caption, the reader is referred to the web version ofthis article.)

Appendix B

Proof of Theorem 2. Suppose that {B(p)} is the reciprocal [0,1]-valued preference relation sequence in the iteration pin Algorithm 1, then we only prove GC I (B(p+1)) ≤ GC I (B(p)). Based on (MOD 1), we have

GC I (B(p+1)) = 2

(n − 1)(n − 2)

∑i< j

(ln b(p+1)i j − ln b(p+1)

j i − ln w(p+1)i + ln w

(p+1)j )2

≤ 2

(n − 1)(n − 2)

∑i< j

(ln b(p+1)i j − ln b(p+1)

j i − ln w(p)i + ln w

(p)j )2 (36)

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 127

Calculate the priority vector of the collective preference relation

Calculate the geometric cardinal consensus indices of the individu al preference relations

Acceptable consensus?

End

Revise the individual preference relations

Construct the individual preference relations

Obtain the collective preference relation

no yes

yesno

Acceptable consistency?Consistency improving

process

Fig. A.3. The consensus improving process of Algorithm 3.

It should be noted that in the proof process of Theorem 2, w(p+1) = (w(p+1)1 , w

(p+1)2 , . . . , w(p+1)

n )T is the weightvector of B(p+1) which is obtained by solving (MOD 1) in the iteration p+1, which can also be written as

Min J =∑i< j

(ln b(p+1)i j − ln b(p+1)

j i − ln wi + ln w j )2

s.t.n∑

i=1

wi = 1, wi ≥ 0, i = 1, 2, . . . , n

That is to say, when wi = w(p+1)i , then J obtain the minimum value Jw(p+1) , and when wi�w

(p+1)i , then Jw(p+1) < J ,

therefore, Eq. (36) holds.Since

b(p+1)i j =

(b(p)i j )1−�(w(p)

i )�

(b(p)i j )1−�(w(p)

i )� + (1 − b(p)i j )1−�(w(p)

j )�, i, j = 1, 2, . . . , n (37)

we have

GC I (B(p+1)) ≤ 2

(n − 1)(n − 2)

∑i< j

(ln((b(p)i j )1−�(w(p)

i )�) − ln((b(p)j i )1−�(w(p)

j )�) − ln w(p)i + ln w

(p)j )2

= 2(1 − �)2

(n − 1)(n − 2)

∑i< j

(ln b(p)i j − ln b(p)

j i − ln w(p)i + ln w

(p)j )2

= (1 − �)2GC I (B(p)) ≤ GC I (B(p)) (38)

128 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

Fig. A.4. In (a)–(o), the horizontal axis shows the average iteration p, the vertical axis the values of controlling parameter �, and the different colorsthe threshold GCC I in Algorithm 3. (For interpretation of the references to color in this figure caption, the reader is referred to the web version ofthis article.)

which completes the proof. Moreover, GC I (B(p)) ≥ 0, for each p. Thus, the sequence {GC I (B(p))} is monotonenon-increasing and has lower bounds. �

Proof of Theorem 3. Suppose {B(p+1)} is the reciprocal [0,1]-valued preference relation sequence in the iteration p+1in Algorithm 1, then its priority vector w(p+1) can be written as

w(p+1)i =

(∏nj=1

b(p+1)i j

b(p+1)j i

)1/n

∑ni=1

(∏nj=1

b(p+1)i j

b(p+1)j i

)1/n , i = 1, . . . , n (39)

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 129

Since

b(p+1)i j =

(b(p)i j )1−�(w(p)

i )�

(b(p)i j )1−�(w(p)

i )� + (1 − b(p)i j )1−�(w(p)

j )�, i, j = 1, 2, . . . , n (40)

we have

w(p+1)i =

(∏nj=1

(b(p)i j )1−�(w(p)

i )�

(b(p)j i )1−�(w(p)

j )�

)1/n

∑ni=1

(∏nj=1

(b(p)i j )1−�(w(p)

i )�

(b(p)j i )1−�(w(p)

j )�

)1/n =

⎛⎝(∏n

j=1

b(p)i j

b(p)j i

)1/n⎞⎠1−� (∏n

j=1(w(p)

i )�

(w(p)j )1−�

)1/n

∑ni=1

⎛⎜⎝⎛⎝(∏n

j=1

b(p)i j

b(p)j i

)1/n⎞⎠1−� (∏n

j=1(w(p)

i )�

(w(p)j )1−�

)1/n⎞⎟⎠

= (w(p)i )1−�(w(p)

i )�∑ni=1 (w(p)

i )1−�(w(p)i )�

= w(p)i , i, j = 1, 2, . . . , n (41)

therefore, w = w̄. �

Proof of Theorem 5. Suppose that {C (p)} is the reciprocal [0,1]-valued preference relation sequence in the iteration pin Algorithm 3, then we only prove GC I (C (p+1)) < GC I (C (p)). Based on (MOD 2), we have

GC I (C (p+1)) =n∑

i=1

n∑j=1

�i j (ln c(p+1)i j − ln c(p+1)

j i − ln w(p+1)i + ln w

(p+1)j )2

≤n∑

i=1

n∑j=1

�i j (ln c(p+1)i j − ln c(p+1)

j i − ln w(p)i + ln w

(p)j )2 (42)

Since

c(p+1)i j =

(c(p)i j )1−�(w(p)

i )�

(c(p)i j )1−�(w(p)

i )� + (1 − c(p)i j )1−�(w(p)

j )�, i, j = 1, 2, . . . , n (43)

we have

GC I (C (p+1)) ≤n∑

j=1

n∑i=1

�i j (ln((c(p)i j )1−�(w(p)

i )�) − ln((c(p)j i )1−�(w(p)

j )�) − ln w(p)i + ln w

(p)j )2

= (1 − �)2n∑

i=1

n∑j=1

(ln c(p)i j − ln c(p)

i j − ln w(p)i + ln w

(p)j )2

= (1 − �)2GC I (C (p)) ≤ GC I (C (p)) (44)

which completes the proof. Moreover, GC I (C (p)) ≥ 0, for each p. Thus, the sequence {GC I (C (p))} is monotonenon-increasing and has lower bounds. �

Proof of Theorem 6. By Eq. (19), we have

b(c)i j

b(c)j i

=

∏mk=1 (b(k)

i j )�k∏mk=1 (b(k)

i j )�k +∏mk=1 (1 − b(k)

i j )�k∏mk=1 (b(k)

j i )�k∏mk=1 (b(k)

j i )�k +∏mk=1 (1 − b(k)

j i )�k

=∏m

k=1 (b(k)i j )�k∏m

k=1 (b(k)j i )�k

, i, j = 1, 2, . . . , n (45)

130 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

then

w(c)i =

(∏nj=1

b(c)i j

b(c)j i

)1/n

∑ni=1

(∏nj=1

b(c)i j

b(c)j i

)1/n =

(∏nj=1

∏mk=1 (b(k)

i j )�k∏mk=1 (b(k)

j i )�k

)1/n

∑ni=1

(∏nj=1

∏mk=1 (b(k)

i j )�k∏mk=1 (b(k)

j i )�k

)1/n =

∏mk=1

⎛⎜⎜⎜⎜⎜⎜⎝

(∏nj=1

b(k)i j

b(k)j i

)1/n

∑ni=1

(∏nj=1

b(k)i j

b(k)j i

)1/n

⎞⎟⎟⎟⎟⎟⎟⎠

�k

∑ni=1

∏mk=1

⎛⎜⎜⎜⎜⎜⎜⎝

(∏nj=1

b(k)i j

b(k)j i

)1/n

∑ni=1

(∏nj=1

b(k)i j

b(k)j i

)1/n

⎞⎟⎟⎟⎟⎟⎟⎠

�k

=∏m

k=1 (w(k)i )�k∑n

i=1∏m

k=1 (w(k)i )�k

i = 1, 2, . . . , n (46)

which completes the proof of the theorem. �

Proof of Theorem 7. Let �(k)i j = ln b(k)

i j − ln b(k)j i − ln w

(k)i + ln w

(k)j . Then for all k = 1, 2, . . . , m, we have

maxk

{GC I (B(k))} = 2

(n − 1)(n − 2)max

k

⎧⎨⎩∑i< j

(ln b(k)i j − ln b(k)

j i − ln w(k)i + ln w

(k)j )2

⎫⎬⎭

= 2

(n − 1)(n − 2)max

k

⎧⎨⎩∑i< j

(�(k)i j )2

⎫⎬⎭ (47)

and

GC I (B(c)) = 2

(n − 1)(n − 2)

∑i< j

(ln

m∏k=1

(b(k)i j )�k − ln

m∏k=1

(b(k)j i )�k − ln

m∏k=1

(w(k)i )�k + ln

m∏k=1

(w(k)j )�k

)2

= 2

(n − 1)(n − 2)

∑i< j

(�k

m∑k=1

(ln b(k)i j − ln b(k)

j i − ln w(k)i + ln w

(k)j )

)2

= 2

(n − 1)(n − 2)

∑i< j

(�k

m∑k=1

�(k)i j

)2

= 2

(n − 1)(n − 2)

⎛⎝ m∑

k=1

�2k

∑i< j

(�(k)i j )2 + 2

∑k<l

�k�l

∑i< j

(�(k)i j �(l)

i j )

⎞⎠

≤ 2

(n − 1)(n − 2)

⎛⎝ m∑

k=1

�2k

∑i< j

(�(k)i j )2 + 2

∑k<l

�k�l max

⎧⎨⎩∑i< j

(�(k)i j )2,

∑i< j

(�(l)i j )2

⎫⎬⎭⎞⎠

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 131

≤ 2

(n − 1)(n − 2)

(m∑

k=1

�k

)2

maxk

⎧⎨⎩∑i< j

(�(k)i j )2

⎫⎬⎭

= 2

(n − 1)(n − 2)max

k

⎧⎨⎩∑i< j

(�(k)i j )2

⎫⎬⎭ = max

k{GC I (B(k))} (48)

Therefore, the proof of the theorem is completed. �

Proof of Theorem 8. Let {B(k)p+1 = (b(k)

i j,p+1)n×n} (k = 1, 2, . . . , m) be the reciprocal [0,1]-valued preference relationsequence in the iteration p+1 in Algorithm 3, and

b(k)i j,p+1 =

(b(k)i j,p)�(w(c)

i,p)1−�

(b(k)i j,p)�(w(c)

i,p)1−� + (1 − b(k)i j,p)�(w(c)

j,p)1−�, i, j = 1, 2, . . . , n (49)

where w(c)p = (w(c)

1,p, w(c)2,p, . . . , w(c)

n,p)T is the collective priority vector defined by Eq. (20) in the iteration p, and let

G p = (gi j,p)n×n =(

w(c)i,p

w(c)i,p + w

(c)j,p

)n×n

(50)

then by Theorem 7, we have known that GC I (B(c)) ≤ maxk{GC I (B(k))}, where B(c) the aggregation of the in-dividual reciprocal [0,1]-valued preference relations B(k) (k = 1, 2, . . . , m), derived by Eq. (19). By Eq. (49),we know that B(k)

p+1 is the combination of B(k)p and G p according to Eq. (48), therefore, we have GC I (B(k)

p+1) ≤max{GC I (B(k)

p ), GC I (G p)} = GC I (B(k)p ), k = 1, 2, . . . , m. Consequently, GC I (B̄(k)) ≤ GC I (B(k)

0 ) = GC I (B(k)),for k = 1, 2, . . . , m. �

Proof of Theorem 9. Based on Theorem 6, we have

w(k)i,p+1 =

(w(k)i,p)1−�(w(c)

i,p)�∑ni=1((w(k)

i,p)1−�(w(c)i,p)�)

, i = 1, 2, . . . , n (51)

then

w(c)i,p+1 =

∏mk=1(w(k)

i,p+1)�k∑ni=1

∏mk=1(w(k)

i,p+1)�k=

∏mk=1((w(k)

i,p)1−�(w(c)i,p)�)�k∑n

i=1∏m

k=1((w(k)i,p)1−�(w(c)

i,p)�)�k

=∏m

k=1((w(k)i )1−�(

∏mk=1(w(k)

i )�k )�)�k∑ni=1

∏mk=1((w(k)

i )1−�(∏m

k=1(w(k)i )�k )�)�k

=∏m

k=1((w(k)i )�k )1−�(

∏mk=1(w(k)

i )�k )�∑ni=1

∏mk=1((w(k)

i )�k )1−�(∏m

k=1(w(k)i )�k )�

=∏m

k=1(w(k)i )�k∑n

i=1∏m

k=1(w(k)i )�k

= w(c)p , i = 1, 2, . . . , n (52)

Therefore, w(c) = w̄(c). �

Proof of Theorem 10. Since

b(k)i j,p+1 =

(b(k)i j,p)1−�(w(c)

i,p)�

(b(k)i j,p)1−�(w(c)

i,p)� + (1 − b(k)i j,p)1−�(w(c)

j,p)�, i, j = 1, 2, . . . , n (53)

132 M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108 –133

we have

GCC I (B(k)p+1) = 2

(n − 1)(n − 2)

∑i< j

(ln b(k)i j,p+1 − ln b(k)

j i,p+1 − ln w(c)i,p+1 + ln w

(c)j,p+1)2

= 2

(n − 1)(n − 2)

∑i< j

(ln((b(k)i j,p)1−�(w(c)

i,p)�) − ln((b(k)j i,p)1−�(w(c)

j,p)�) − ln w(c)i,p + log w

(c)j,p)2

= 2(1 − �)2

(n − 1)(n − 2)

∑i< j

(ln b(k)i j,p − ln b(k)

j i,p − ln w(c)i,p + ln w

(c)j,p)2

= (1 − �)2GCC I (B(k)p ) ≤ GCC I (B(k)

p ) (54)

which completes the proof. �

Proof of Theorem 11. If we replace the preference value c(k)i j (�(k)

i j = 0) by w(k)i /(w(k)

i + w(k)j ), then all the incomplete

reciprocal [0,1]-valued preference relations can be considered as complete ones, but the priority vectors form them arestill as before, then the collected priority vector is also still as before. Since

c(k)i j,p+1 =

(c(k)i j,p)1−�(w(c)

i,p)�

(c(k)i j,p)1−�(w(c)

i,p)� + (1 − c(k)i j,p)1−�(w(c)

j,p)�, i, j = 1, 2, . . . , n (55)

we have

GCC I (C (k)p+1) =

n∑i=1

n∑j=1

�(k)i j (ln c(k)

i j,p+1 − ln c(k)j i,p+1 − ln w

(c)i,p+1 + ln w

(c)j,p+1)2

=n∑

i=1

n∑j=1

�(k)i j (ln((c(k)

i j,p)1−�(w(c)i,p)�) − ln((c(k)

j i,p)1−�(w(c)j,p)�) − ln w

(c)i,p + ln w

(c)j,p)2

= (1 − �)2n∑

i=1

n∑j=1

(ln c(k)i j,p − ln c(k)

j i,p − ln w(c)i,p + ln w

(c)j,p)2

= (1 − �)2GCC I (C (k)p ) ≤ GCC I (C (k)

p ) (56)

which completes the proof. �

References

[1] J. Aguarón, J.M. Moreno-Jiménez, The geometric consistency index: approximated thresholds, Eur. J. Oper. Res. 147 (2003) 137–145.[2] S. Alonso, E. Herrera-Viedma, F. Chiclana, F. Herrera, A web based consensus support system for group decision making problems and

incomplete preferences, Inf. Sci. 180 (2010) 4477–4495.[3] F. Chiclana, F. Mata, L. Martínez, E. Herrera-Viedma, S. Alonso, Integration of a consistency control module within a consensus decision

making model, Int. J. Uncertainty Fuzziness Knowl Based Syst. 16 (2008) 35–53.[4] F. Chiclana, E. Herrera-Viedma, S. Alonso, F. Herrera, Cardinal consistency of reciprocal preference relation: a characterization of multiplicative

transitivity, IEEE Trans. Fuzzy Syst. 17 (2009) 14–23.[5] F. Chiclana, E. Herrera-Viedma, S. Alonso, A note on two methods for estimating missing pairwise preference values, IEEE Trans. Syst. Man

Cybern. 39 (2009) 1628–1633.[6] G. Crawford, C. Williams, A note on the analysis of subjective judgement matrices, J. Math. Psychol. 29 (1985) 387–405.[7] V. Cutello, J. Montero, Fuzzy rationality measures, Fuzzy Sets Syst. 62 (1994) 39–44.[8] B. De Baets, H. De Meyer, Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity, Fuzzy Sets Syst. 152

(2005) 249–270.[9] B. De Baets, H. De Meyer, B. De Schuymer, S. Jenei, Cyclic evaluation of transitivity of reciprocal relations, Soc. Choice Welfare 26 (2006)

217–238.[10] B. De Baets, H. De Meyer, K. De Loof, On the cycle-transitivity of the mutual rank probability relation of a poset, Fuzzy Sets Syst. 161 (2010)

2695–2708.[11] K. De Loof, B. De Baets, H. De Meyer, R. Brüggemann, A hitchhiker’s guide to poset ranking, Combin. Chem. High Throughput Screening

11 (2008) 734–744.

M. Xia et al. / Fuzzy Sets and Systems 216 (2013) 108–133 133

[12] Y.C. Dong, G.Q. Zhang, W.C. Hong, Y.F. Xu, Consensus models for AHP group decision making under row geometric mean prioritizationmethod, Decision Support Syst. 49 (2010) 281–289.

[13] Y.C. Dong, Y.F. Xu, H.Y. Li, B. Feng, The OWA-based consensus operator under linguistic representation models using position indexes, Eur.J. Oper. Res. 203 (2010) 455–463.

[14] D. Ergu, G. Kou, Y. Peng, Y. Shi, A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP, Eur. J. Oper.Res. 213 (2011) 246–259.

[15] M.T. Escobar, J. Aguarón, J.M. Moreno-Jiménez, A note on AHP group consistency for the row geometric mean priorization procedure, Eur.J. Oper. Res. 153 (2004) 318–322.

[16] C. Fu, S.L. Yang, The group consensus based evidential reasoning approach for multiple attributive group decision analysis, Eur. J. Oper. Res.206 (2010) 601–608.

[17] M. Fedrizzi, S. Giove, Incomplete pairwise comparison and consistency optimization, Eur. J. Oper. Res. 183 (2007) 303–313.[18] Z.W. Gong, Least-square method to priority of the fuzzy preference relations with incomplete information, Int. J. Approx. Reason. 47 (2008)

258–264.[19] E. Herrera-Viedma, F. Herrera, F. Chiclana, M. Luque, A consensus model for multiperson decision making with different preference structures,

IEEE Trans. Syst. Man Cybern. 34 (2002) 394–402.[20] E. Herrera-Viedma, F. Herrera, F. Chiclana, M. Luque, Some issues on consistency of fuzzy preference relations, Eur. J. Oper. Res. 154 (2004)

98–109.[21] E. Herrera-Viedma, L. Martínez, F. Mata, F. Chiclana, A consensus support system model for group decision-making problems with multigranular

linguistic preference relations, IEEE Trans. Fuzzy Syst. 13 (2005) 644–658.[22] E. Herrera-Viedma, S. Alonso, F. Chiclana, F. Herrera, A consensus model for group decision making with incomplete fuzzy preference

relations, IEEE Trans. Fuzzy Syst. 15 (2007) 155–178.[23] E. Herrera-Viedma, F. Chiclana, F. Herrera, S. Alonso, Group decision-making model with incomplete fuzzy preference relations based on

additive consistency, IEEE Trans. Syst. Man Cybern. 37 (2007) 176–189.[24] J. Ma, Z.P. Fan, Y.P. Jiang, J.Y. Mao, L. Ma, A method for repairing the inconsistency of fuzzy preference relations, Fuzzy Sets Syst. 157 (2006)

20–33.[25] F. Mata, L. Martínez, E. Herrera-Viedma, An adaptive consensus support model for group decision-making problems in a multigranular fuzzy

linguistic context, IEEE Trans. Fuzzy Syst. 17 (2009) 279–290.[26] X.W. Liu, Y.W. Pan, Y.J. Xu, S. Yu, Least square completion and inconsistency repair methods for additively consistent fuzzy preference

relations, Fuzzy Sets Syst. 198 (2012) 1–19.[27] S.A. Orlovsky, Decision-making with a fuzzy preference relation, Fuzzy Sets Syst. 1 (1978) 155–167.[28] M. Rademaker, B. De Baets, Aggregation of monotone reciprocal relations with application to group decision making, Fuzzy Sets Syst. 184

(2011) 29–51.[29] T.L. Saaty, Multicriteria Decision Making: The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.[30] T.L. Saaty, Fundamentals of Decision Making, RSW Publications, 1994.[31] S. Siraj, L. Mikhailov, J. Keane, A heuristic method to rectify intransitive judgments in pairwise comparison matrices, Eur. J. Oper. Res. 216

(2012) 420–428.[32] T. Tanino, Fuzzy preference orderings in group decision-making, Fuzzy Sets Syst. 12 (1984) 117–131.[33] Y.M. Wang, Z.P. Fan, Group decision analysis based on fuzzy preference relations: Logarithmic and geometric least squares methods, Appl.

Math. Comput. 194 (2007) 108–119.[34] Z.B. Wu, J.P. Xu, A concise consensus support model for group decision making with reciprocal preference relations based on deviation

measures, Fuzzy Sets Syst. 206 (2012) 58–73.[35] M.M. Xia, Z.S. Xu, On consensus in group decision making based on fuzzy preference relations. In: E. Herrera-Viedma, J.L. García-Lapresta,

J. Kacprzyk, H. Nurmi, M. Fedrizzi, S. Zadrozny (Eds.), Consensual Processes, Springer-Verlag, 2010.[36] Y.J. Xu, R. Patnayakuni, H. Wang, Logarithmic least squares method to priority for group decision making within complete fuzzy preference

relations, Appl. Math. Modelling; doi: http://dx.doi.org/10.1016/j.apm.2012.05.010, in press.[37] Z.S. Xu, An automatic approach to reaching consensus in multiple attribute group decision making, Comput. Ind. Eng. 56 (2009) 1369–1374.[38] Z.S. Xu, X.Q. Cai, Group consensus algorithms based on preference relations, Inf. Sci. 181 (2011) 150–162.[39] Z.S. Xu, C.P. Wei, A consistency improving method in the analytic hierarchy process, Eur. J. Oper. Res. 116 (1999) 443–449.[40] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.