multiplicative consistency of hesitant fuzzy preference relation and its

International Journal of Information Technology & Decision Making Vol. 13, No. 1 (2014) 47–76 c World Scienti¯c Publishing Company ° DOI: 10.1142/S0219622014500035

MULTIPLICATIVE CONSISTENCY OF HESITANT FUZZY PREFERENCE RELATION AND ITS APPLICATION IN GROUP DECISION MAKING

HUCHANG LIAO Antai College of Economics and Management Shanghai Jiao Tong University Shanghai 200052, China [email protected] ZESHUI XU* Business School, Sichuan University Chengdu 610064, China [email protected] MEIMEI XIA School of Economics and Management Tsinghua University Beijing 100084, China [email protected] Published 6 December 2013 As we may have a set of possible values when comparing alternatives (or criteria), the hesitant fuzzy preference relation becomes a suitable and powerful technique to deal with this case. This paper mainly focuses on the multiplicative consistency of the hesitant fuzzy preference relation. First of all, we explore some properties of the hesitant fuzzy preference relation and develop some new aggregation operators. Then we introduce the concepts of multiplicative consistency, perfect multiplicative consistency and acceptable multiplicative consistency for a hesitant fuzzy preference relation, based on which, two algorithms are given to improve the inconsistency level of a hesitant fuzzy preference relation. Furthermore, the consensus of group decision making is studied based on the hesitant fuzzy preference relations. Finally, several illustrative examples are given to demonstrate the practicality of our algorithms. Keywords: Hesitant fuzzy preference relation; hesitant fuzzy set; consensus; multiplicative consistency; group decision making. 2000 Mathematics Subject Classi¯cation: 90B50, 91B06

*Corresponding

author. 47

48

H. Liao, Z. Xu & M. Xia

1. Introduction Since originally introduced by Zadeh,1 the fuzzy set (FS) has been extended into several di®erent forms, such as the intuitionistic fuzzy set,2–4 the interval-valued intuitionistic fuzzy set,5 the type 2 fuzzy set,6,7 the type n fuzzy set,7 the fuzzy multisets (also named the fuzzy bags),8 and the intuitionistic multiplicative set.9 All these extensions are based on the same rationale that it is not clear to assign the membership degree of an element to a ¯xed set.10,11 The intuitionistic fuzzy set and the intuitionistic multiplicative set overcome this problem by allowing the degree of an element to a ¯xed set represented by two functions: the membership function and the nonmembership function. The interval-valued intuitionistic fuzzy set is the generalization of the intuitionistic fuzzy set, which represents the membership degree and the nonmembership degree in terms of interval-values. The type 2 fuzzy set solves this problem by permitting the membership of an element to a ¯xed set in terms of a FS and the type n fuzzy set extends the type 2 fuzzy set permitting the membership degree to be type n
1 fuzzy set. The fuzzy multisets are the generalization of the fuzzy set where the membership degree of an element to the multisets is not Boolean but fuzzy. Recently, on the basis of the above extensional forms of fuzzy set, Torra10 proposed a new generalized type of fuzzy set named hesitant fuzzy set (HFS), which opens new perspectives for further research on decision making under hesitant environments. As we may have a set of possible values when determining the membership of an element to a given set, the HFS, whose membership is represented by a set of possible values, can represent such case perfectly, while the above mentioned extensions are invalid. Therefore, it is more suitable and powerful to describe the uncertain evaluation information in HFS.10–16 In the process of decision making, the decision maker may feel comfortable to express his/her preferences by comparing each pair of objects and then constructs a preference relation. The preference relation, as the most common representation of information, has attracted great attention from scholars and has been widely applied in multiple criteria decision making.17–19 Up to now, many di®erent types of preference relations have been proposed, such as the fuzzy preference relation,20 the multiplicative preference relation,21,22 the linguistic preference relation,23–25 the intuitionistic fuzzy preference relation,26,27 and the interval-valued intuitionistic preference relation.28,29 However, all these preference relations do not consider the hesitant fuzzy information, they cannot provide all the possible evaluation values of the decision makers when comparing pairwise alternatives (or criteria), which is a common situation in our daily life. To solve this drawback, inspired by HFS, in this paper, we ¯rst give the de¯nition of hesitant fuzzy preference relation (HFPR) and then investigate its distinctive properties. The fuzzy preference relation has been applied extensively in several ¯elds, such as knowledge management,30 partnership selection,31 electronic learning,32 ¯nancial mergers.33 In the decision-making processes, the lack of consistency on a preference

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

49

relation may lead to an unreasonable result. However, in practical application, a prefect consistent preference relation is too hard for the decision makers to obtain due to the di®erent backgrounds, personal habits, the nature of human judgment, or vague knowledge about the preference degree of one alternative over another, especially if the number of alternatives is too large.34 Sometimes, we may even get some incomplete fuzzy preference relation.35,36 Many investigations have been conducted on the consistency of the fuzzy preference relation,37–42 which can be performed to ensure pairwise comparisons logical rather than random. Nonetheless, up to now, as far as we know, no work has been done regarding the consistency of HFPR. Thus, it is necessary and urgent to give some studies on this issue, which is the focus of this paper. The investigation on the consistency of a preference relation generally involves two phases: (1) how to judge whether the preference relation is perfectly/acceptably consistent or not; (2) how to adjust or repair the inconsistent preference relation until it is with acceptable consistency. As for the ¯rst phase, the concepts of consistency has been de¯ned in terms of transitivity, such as weak transitivity, max– max transitivity, max–min transitivity, restricted max–min transitivity, restricted max–max transitivity, additive transitivity, and multiplicative transitivity.37 Based on the above transitivity properties, some methods for measuring the consistency of a preference relation have been developed.37–42 Saaty21 ¯rst derived a consistency ratio in analytic hierarchy process (AHP), and also ¯rst developed the concept of perfect consistency and acceptable consistency, He pointed out that the preference relation is of acceptable consistency if its consistency ratio is less than 0.1. However, the more common situation in practice is the preference relation possessing unacceptable consistency, which may mislead the ranking results. Therefore, we need to repair the inconsistent preference relation, which consists the second phase of our discussion. To improve the consistency ratio, Ergu et al.43 proposed a method which combines the theorem of matrix multiplication, vectors dot product and the de¯nition of consistent pairwise comparison matrix to identify the inconsistent elements. They44 also developed a maximum eigenvalue threshold as the consistency index for the ANP in risk assessment. Associated with the study of the transitivity property, Herrera-Viedma et al.38 ¯rst proposed the additive transitivity property of fuzzy preference relations as a new characterization of the consistency property, based on which the consistent fuzzy preference relations from a set of n
1 preference data were constructed. Gong et al.41 investigated the properties of additive consistent intuitionistic fuzzy preference relations. Ma et al.39 presented two methods derived from graph theory to judge whether a fuzzy preference relation has weak transitivity or not, and then, via a synthesis matrix which re°ects the relationship between the fuzzy preference relation with additive consistency and the original one given by the decision maker, an algorithm was developed to repair the inconsistent fuzzy preference relation. Many people have applied the additive transitivity property of fuzzy preference relations to practice, such as, knowledge management,30 partnership selection,31

50

H. Liao, Z. Xu & M. Xia

forecasting advanced manufacturing technology.45 But as additive transitivity property of a fuzzy preference relation P ¼ ðp ij Þ nn is represented as p ij ¼ p ik þ p kj
0:5, i; j; k ¼ 1; 2; . . . ; n, where p ij ; p ik and p kj are the preference information about the alternatives x i ; x j and x k , if we take p ik ¼ 0:8 and p kj ¼ 0:9 as an example, then, p ij ¼ 1:2 > 1, which does not belong to the unit closed interval [0,1], thus it is unreasonable. Since the preference relations constructed by the additive transitivity are not in the ¯xed scope, it is necessary to transform them to the de¯ned scope. Nonetheless, it may lead to lose original preference information. To solve this problem, based on the multiplicative consistency37 of the fuzzy preference relation P ¼ ðp ij Þ nn , which is represented as p ij p jk p ki ¼ p ik p kj p ji , i; j; k ¼ 1; 2; . . . ; n, where p ij ; p ik and p kj are the preference values given by the decision maker, Chiclana et al.46 developed a method to construct the consistent fuzzy preference relation from a set of n
1 preference values based on multiplicative consistency. Xia and Xu47 proposed a new method which can get the complete consistent fuzzy preference relation quickly without any transformation based on the multiplicative consistency of a fuzzy preference relation, and then they applied it to fuzzy analytic hierarchy process. Later, Xia and Xu48 also developed methods to get the perfect multiplicative consistent interval reciprocal relation from the inconsistent one and to estimate the missing values from an incomplete interval reciprocal relation. In this paper, we shall utilize the multiplicative consistency to propose some novel methods for adjusting or repairing the inconsistent HFPRs. In practical application, in order to choose the most desirable and reasonable solution(s) for a decision-making problem, a group of experts (or decision makers), coming from di®erent aspects, are gathered together to evaluate the alternatives over the criteria. Di®erent experts (or decision makers) may have distinct preferences and then it is needed to propose some consensus reaching methods. There are many achievements on consensus of preference relations.22,24,26,34,35,37,42 However, no work has been done on HFPRs. Based on the multiplicative consensus, we also develop a consensus improving procedure for HFPRs in group decision making. The remainder of this paper is set out as follows. Section 2 gives some basic knowledge on HFS, including the concept of HFS, the operational laws, the aggregation operators, and two types of distance measures for HFSs. Section 3 derives some properties of the HFPR and gives the concepts of multiplicative consistent HFPR, perfect multiplicative consistent HFPR, acceptable multiplicative consistent HFPR, and then two algorithms are proposed to improve the consistency level of the HFPR. Another algorithm is developed to improve the consensus levels of individual hesitant preference relations and an illustrative example is given in Sec. 4. The paper ¯nishes with some conclusions in Sec. 5. 2. Preliminaries HFS, which permits the membership of an element to a set represented by several possible values between 0 and 1, is powerful to determine the membership degree

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

51

especially when we have several di®erent values on it. Torra10 ¯rst introduced this new extension of fuzzy set and gave some basic operations on it. De¯nition 1.10 Let X be a ¯xed set, a HFS on X is in terms of a function h that when applied to X returns a subset of ½0; 1. To be easily understood, Xia and Xu12 express the HFS by a mathematical symbol E ¼ f< x; h E ðxÞ > jx 2 Xg;

ð2:1Þ

where h E ðxÞ is a set of values in ½0; 1, denoting the possible membership degrees of the element x 2 X to the set E. For convenience, Xia and Xu12 called h E ðxÞ a hesitant fuzzy element (HFE). Torra and Narukawa11 de¯ned the complement, union and intersection about HFSs, which are represented as follows: De¯nition 2.11 For three HFEs h, h 1 and h 2 , the following operations are de¯ned: (1) (2) (3) (4) (5)

Lower bound: h
ðxÞ ¼ min hðxÞ, Upper bound: h þ ðxÞ ¼ max hðxÞ, h c ¼ [
2h f1

g, h 1 [ h 2 ¼ fh 2 h 1 [ h 2 jh  maxðh 1
; h 2
Þg, h 1 \ h 2 ¼ fh 2 h 1 [ h 2 jh  minðh 1þ ; h 2þ Þg.

Xia and Xu12 gave other forms of (4) and (5) as follows: (6) h 1 [ h 2 ¼ [
1 2h 1 ;
2 2h 2 maxf
1 ;
2 g, (7) h 1 \ h 2 ¼ [
1 2h;
2 2h 2 minf
1 ;
2 g, and gave some operational laws on the HFEs h, h 1 and h 2 : De¯nition 3.12 Let h, h 1 and h 2 be three HFEs, and  be a positive real number, then (1) (2) (3) (4)

h  ¼ [
2h f
 g, h ¼ [
2h f1
ð1

Þ g, h 1  h 2 ¼ [
1 2h 1 ;
2 2h 2 f
1 þ
2

1
2 g, h 1  h 2 ¼ [
1 2h 1 ;
2 2h 2 f
1
2 g.

Let h j ð j ¼ 1; 2; . . . ; nÞ be a collection of HFEs. We can easily generalize (3) and (4) of De¯nition 3 to the following forms: Q (5)  nj¼1 h j ¼ [
j 2h j f1
nj¼1 ð1

j Þg, Q (6)  nj¼1 h j ¼ [
j 2h j f nj¼1
j g. It is noted that the number of values in di®erent HFEs may be di®erent. Let l h E ðxÞ be the number of values in h E ðxÞ. For two HFEs h 1 and h 2 , let l ¼ maxfl h 1 ; l h 2 g. To operate correctly, Xu and Xia13 gave the following regulation, which is based on the assumption that all the decision makers are pessimistic: If l h 1 < l h 2 , then h 1 should be extended by adding the minimum value in it until it has the same length

52

H. Liao, Z. Xu & M. Xia

with h 2 . If l h 1 > l h 2 , then h 2 should be extended by adding the minimum value in it until it has the same length with h 1 . In this paper, we extend the shorter one by adding the value of 0:5 in it, that is to say, we assume that all the decision makers are neutral. Since there are same values in a HFE according to De¯nition 1, based on the above operational laws, the following theorem is held: Theorem 1. Suppose h 1 and h 2 are two HFEs, then l h 1 h 2 ¼ l h 1  l h 2 ;

l h 1 h 2 ¼ l h 1  l h 2 :

ð2:2Þ

Similarly, it also holds when there are n di®erent HFEs, i.e., l n h ¼ i¼1

i

n Y i¼1

lhi ;

l n h ¼ i¼1

i

n Y

lhi :

ð2:3Þ

i¼1

From Theorem 1, we can see that the dimensions of the derived HFE may increase as the addition or multiplication operations are done, which may increase the complexity of the calculation. Thus, we need to develop some new methods to decrease the dimensions of the derived HFE when we operate the HFEs. In order to export operations on fuzzy sets to HFSs, Torra and Narukawa11 proposed an aggregation principle for HFEs: De¯nition 4.11 Let E ¼ fh 1 ; h 2 ; . . . ; h n g be a set of n HFEs,
be a function on E,
: ½0; 1N ! ½0; 1, then
E ¼ [
2fh 1 h 2 h n g f
ð
Þg:

ð2:4Þ

Based on the above extension principle, Xia and Xu12 developed a series of speci¯c aggregation operators for HFEs. Here, we just review the most common operators which may be used in the next part of our paper. De¯nition 5.12 Let h j ðj ¼ 1; 2; . . . ; nÞ be a collection of HFEs. A hesitant fuzzy weighted averaging (HFWA) operator is a mapping F n ! F such that ( ) n Y n wj HFWAðh 1 ; h 2 ; . . . ; h n Þ ¼  ðw j h j Þ ¼ [
1 2h 1 ;
2 2h 2 ;...;
n 2h n 1
ð1

j Þ ; j¼1

j¼1

ð2:5Þ where w ¼ ðw 1 ; w 2 ; . . . ; w n ÞT is the weight vector of h j ðj ¼ 1; 2; . . . ; nÞ with w j 2 ½0; 1 P and nj¼1 w j ¼ 1. Especially, if w ¼ ð1=n; 1=n; . . . ; 1=nÞT , then the HFWA operator reduces to the hesitant fuzzy averaging (HFA) operator: ( )
 n Y n 1 h j ¼ [
1 2h 1 ;
2 2h 2 ;...;
n 2h n 1
HFAðh 1 ; h 2 ; . . . ; h n Þ ¼  ð1

j Þ1=n : j¼1 n j¼1 ð2:6Þ

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

53

De¯nition 6.12 Let h j ðj ¼ 1; 2; . . . ; nÞ be a collection of HFEs and HFWG: F n ! F. If ( ) n Y n wj wj HFWGðh 1 ; h 2 ; . . . ; h n Þ ¼  h j ¼ [
1 2h 1 ;
2 2h 2 ;...;
n 2h n
j ð2:7Þ j¼1

j¼1

then HFWG is called a hesitant fuzzy weighted geometric (HFWG) operator, where w ¼ ðw 1 ; w 2 ; . . . ; w n ÞT is the weight vector of h j ðj ¼ 1; 2; . . . ; nÞ, with w j 2 ½0; 1 and Pn T j¼1 w j ¼ 1. In the case where w ¼ ð1=n; 1=n; . . . ; 1=nÞ , the HFWA operator reduces to the hesitant fuzzy geometric (HFG) operator: ( ) n Y n 1=n 1=n HFGðh 1 ; h 2 ; . . . ; h n Þ ¼  h j ¼ [
1 2h 1 ;
2 2h 2 ;...;
n 2h n
j : ð2:8Þ j¼1

j¼1

As pointed out in Theorem 1, taking Eq. (2.5) as an example, we can obtain l HFWAðh 1 ;h 2 ;...;h n Þ ¼ l h 1  l h 2      l h n , which is also same as Eqs. (2.6)–(2.8). Therefore, in order not to increase the dimensions of the derived HFEs in the process of calculating, we ¯rst adjust the operational laws in De¯nition 3 into the following forms: De¯nition 7. Let h j ðj ¼ 1; 2; . . . ; nÞ be a collection of HFEs, and  be a positive real number, then (1) (2) (3) (4) (5) (6)

h  ¼ fðh ðtÞ Þ ; t ¼ 1; 2; . . . ; lg, h ¼ f1
ð1
h ðtÞ Þ ; t ¼ 1; 2; . . . ; lg, ðtÞ ðtÞ ðtÞ ðtÞ h 1  h 2 ¼ fh 1 þ h 2
h 1 h 2 ; t ¼ 1; 2; . . . ; lg, ðtÞ ðtÞ h 1  h 2 ¼ fh 1 h 2 ; t ¼ 1; 2; . . . ; lg, Q ðtÞ  nj¼1 h j ¼ f1
nj¼1 ð1
h j Þ; t ¼ 1; 2; . . . ; lg, Q ðtÞ n n  j¼1 h j ¼ f j¼1 h j ; t ¼ 1; 2; . . . ; lg, ðtÞ

where h j

is the tth smallest value in h j .

According to De¯nition 7, we can adjust the above aggregation operators into the following forms: De¯nition 8. Let h j ðj ¼ 1; 2; . . . ; nÞ be a collection of HFEs. An adjusted hesitant fuzzy weighted averaging (AHFWA) operator is a mapping F n ! F such that ( ) n Y n ðtÞ w j AHFWAðh 1 ; h 2 ; . . . ; h n Þ ¼  ðw j h j Þ ¼ 1
ð1
h j Þ j t ¼ 1; 2; . . . ; l ; j¼1

j¼1

ð2:9Þ ðtÞ

where h j is the tth smallest value in h j , and w ¼ ðw 1 ; w 2 ; . . . ; w n ÞT is the weight Pn vector of h j ðj ¼ 1; 2; . . . ; nÞ with w j 2 ½0; 1, j¼1 w j ¼ 1. Especially, if w ¼ ð1=n; 1=n; . . . ; 1=nÞT , then the AHFWA operator reduces to the adjusted hesitant

54

H. Liao, Z. Xu & M. Xia

fuzzy averaging (AHFA) operator: n

AHFAðh 1 ; h 2 ; . . . ; h n Þ ¼ 

j¼1




1 h n j

(

 ¼

1


n Y

) ð1


ðtÞ h j Þ1=n

j t ¼ 1; 2; . . . ; l :

j¼1

ð2:10Þ De¯nition 9. Let h j ðj ¼ 1; 2; . . . ; nÞ be a collection of HFEs and let AHFWG: F n ! F, if ( ) n Y n ðtÞ AHFWGðh 1 ; h 2 ; . . . ; h n Þ ¼  ðh j Þw j ¼ ðh j Þw j j t ¼ 1; 2; . . . ; l ð2:11Þ j¼1

j¼1

then AHFWG is called an adjusted hesitant fuzzy weighted geometric (AHFWG) ðtÞ operator, where h j is the tth smallest value in h j , and w ¼ ðw 1 ; w 2 ; . . . ; w n ÞT is the Pn weight vector of h j ðj ¼ 1; 2; . . . ; nÞ, with w j 2 ½0; 1 and j¼1 w j ¼ 1. In the case T where w ¼ ð1=n; 1=n; . . . ; 1=nÞ , the AHFWA operator reduces to the adjusted hesitant fuzzy geometric (AHFG) operator: ( n

AHFGðh 1 ; h 2 ; . . . ; h n Þ ¼  ðh j Þ j¼1

1=n

¼

n Y

) ðtÞ ðh j Þ1=n

j t ¼ 1; 2; . . . ; l :

ð2:12Þ

j¼1

Xia and Xu12 also de¯ned the score function of HFEs: P De¯nition 10.12 For a HFE h, sðhÞ ¼ l1h
2h
is called the score function of h, where l h is the number of values in h. For two HFEs h 1 and h 2 , if sðh 1 Þ > sðh 2 Þ, then h 1 > h 2 ; if sðh 1 Þ ¼ sðh 2 Þ, then h 1 ¼ h 2 . However, in some special cases, this comparison law cannot be used to distinguish two HFEs: Example 1. Let h 1 ¼ ð0:1; 0:1; 0:7Þ and h 2 ¼ ð0:2; 0:4Þ be two HFEs, then by De¯nition 6, we have sðh 1 Þ ¼

0:1 þ 0:1 þ 0:7 ¼ 0:3; 3

sðh 2 Þ ¼

0:2 þ 0:4 ¼ 0:3: 2

We take sðhÞ as the score degree of h. Since sðh 1 Þ ¼ sðh 2 Þ, then we cannot tell the di®erence between h 1 and h 2 by using De¯nition 10. Actually, such a case is usually common in practice. Hence, below we propose a new function to solve this problem: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 De¯nition 11. For a HFE h, vðhÞ ¼ l1h
i ;
j 2h ð
i

j Þ is called the variance function of h, where l h is the number of values in h, and vðhÞ is called the variance degree of h. For two HFEs h 1 and h 2 , if vðh 1 Þ > vðh 2 Þ, then h 1 < h 2 ; if vðh 1 Þ ¼ vðh 2 Þ, then h 1 ¼ h 2 .

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

55

By De¯nition 11, we can obtain in Example 1 that pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ 0:62 þ 0:62 0:22 ¼ 0:2828; vðh 2 Þ ¼ ¼ 0:1: vðh 1 Þ ¼ 3 2 Then, vðh 1 Þ > vðh 2 Þ, i.e., the variance degree of h 1 is higher than that of h 2 , thus, h1 < h2. From the above analysis, we can see that the relationship between score function and variance function is similar to the relationship between mean and variance in statistics. Inspired by this idea, we develop a scheme below to compare any two HFEs. If sðh 1 Þ < sðh 2 Þ; If sðh 1 Þ ¼ sðh 2 Þ;

then h 1 < h 2 ; then

(1) If vðh 1 Þ < vðh 2 Þ; (2) If vðh 1 Þ ¼ vðh 2 Þ;

then h 1 > h 2 ; then h 1 ¼ h 2 .

Subsequently, Xu and Xia13 de¯ned several di®erent types of distance measures for HFSs. Among them, there are two representative ones, which will be used thereafter: The hesitant normalized Hamming distance: " # lxi n 1 X 1 X ðjÞ ðjÞ d hnh ðM ; N Þ ¼ jh ðx i Þ
h N ðx i Þj n i¼1 l x i j¼1 M and the hesitant normalized Euclidean distance: " !#1=2 lxi n 1 X 1 X ðjÞ ðjÞ 2 jh ðx i Þ
h N ðx i Þj ; d hne ðM ; N Þ ¼ n i¼1 l x i j¼1 M ðjÞ

ð2:13Þ

ð2:14Þ

ðjÞ

where h M ðx i Þ and h N ðx i Þ are the jth largest values in h M ðx i Þ and h N ðx i Þ, respectively.

3. Transitivity and Multiplicative Consistency of HFPR In the process of decision making, in order to avoid the in°uence of the limited ability of human thinking and to obtain the best ranking result, people prefer to take the pairwise comparison of one alternative over another and construct a preference relation. Therefore, the preference relations turn out to be the most common representation formats in expressing the decision makers' preferences. Let X ¼ fx 1 ; x 2 ; . . . ; x n g be a set of alternatives, then R ¼ ðr ij Þ nn is called a fuzzy preference relation on X  X with the condition that r ij  0, r ij þ r ji ¼ 1, i; j ¼ 1; 2; . . . ; n, where r ij denotes the degree that the alternative x i is prior to the alternative x j . The values in a fuzzy preference relation are certain values between 0 and 1. However, when people establish the preference degree of one object over another, they may have a set of possible values but not one single value due to the complexity of the

56

H. Liao, Z. Xu & M. Xia

decision-making problem, the lack of knowledge about problem domain, and so on. In such cases, it is very suitable and reasonable to represent the preference information by HFEs, which permit the membership of an element to a set represented by several possible values. Thus, in what follows, we de¯ne a HFPR: De¯nition 12. Let X ¼ fx 1 ; x 2 ; . . . ; x n g be a ¯xed set, a HFPR H on X is presented by a matrix H ¼ ðh ij Þ nn X  X, where h ij ¼ fh ijs ; s ¼ 1; 2; . . . ; l h ij g is a HFE indicating all the possible degrees to which x i is preferred to x j . Moreover, h ij should satisfy the following conditions: ðsÞ

h ij

ðl h ji
sþ1Þ

þ h ji

¼ 1;

h ii ¼ f0:5g;

l h ij ¼ l h ji ;

i; j ¼ 1; 2; . . . ; n:

ð3:1Þ

With De¯nition 12, we can easily derive the following results: Theorem 2. The transpose H T ¼ ðh ijT Þ nn of the HFPR H ¼ ðh ij Þ nn is also a HFPR, where h ijT ¼ h ji , i; j ¼ 1; 2; . . . ; n. Theorem 3. Let H ¼ ðh ij Þ nn be a HFPR, then, if we remove the ith row and the ith column, then the remaining matrix H ¼ ðh ij Þ ðn
1Þðn
1Þ is also a HFPR. When the decision maker evaluates the preference information, he/she may provide inconsistent preference values and thus constructs the inconsistent preference relation due to the complexity of the considered problem or other reasons. The HFPR H ¼ ðh ij Þ nn X  X should satis¯es the following transitivity properties: (1) If h ik  h kj  h ij , for all i; j; k ¼ 1; 2; . . . ; n, then we say H satis¯es the triangle condition. (2) If h ik  f0:5g, h kj  f0:5g, then h ij  f0:5g, for all i; j; k ¼ 1; 2; . . . ; n, then we say H satis¯es the weak transitivity property. (3) If h ij  minfh ik ; h kj g, for all i; j; k ¼ 1; 2; . . . ; n, then we say H satis¯es max–min transitivity property. (4) If h ij  maxfh ik ; h kj g, for all i; j; k ¼ 1; 2; . . . ; n, then we say H satis¯es max–max transitivity property. (5) If h ik  f0:5g, h kj  f0:5g, then h ij  minðh ik ; h kj Þ, for all i; j; k ¼ 1; 2; . . . ; n, then we say H satis¯es the restricted max–min transitivity property. (6) If h ik  f0:5g, h kj  f0:5g, then h ij  maxðh ik ; h kj Þ, for all i; j; k ¼ 1; 2; . . . ; n, then we say H satis¯es the restricted max–max transitivity property. The weak transitivity is the usual and basic property, which can be interpreted as follows: If the alternative x i is preferred to x k , and x k is preferred to x j , then x i should be preferred to x j ; If the person who is logic and consistent does not want to draw inconsistent conclusions, he/she should ¯rst ensure that the preference relation satis¯es the weak transitivity. However, the weak transitivity is the minimum requirement condition to make sure that the HFPR is consistent. There are another two conditions named additive transitivity and multiplicative transitivity which

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

57

are more restrictive than weak transitivity and can imply reciprocity. The additive transitivity can be generalized to accommodate the HFPR in terms of ðh ik
f0:5gÞ  ðh kj
f0:5gÞ ¼ ðh ij
f0:5gÞ, for all i; j; k ¼ 1; 2; . . . ; n. The multiplicative transitivity is an important property of the fuzzy preference relation P ¼ ðp ij Þ nn , which was ¯rst introduced by Tanino37 and shown as: p ji p kj p  ¼ ki ; ð3:2Þ p ij p jk p ik where p ij denotes a ratio of preference intensity for the alternative x i to that for x j , in another words, x i is p ij times as good as x j , and p ij 2 ½0; 1, for all i; j ¼ 1; 2; . . . ; n. As to the multiplicative transitivity proposed by Tanino,37 it might bring some di±culties in meaning; especially some literature assumes an absolute scale for fuzzy sets.49,50 To handle this problem, some related work has been done by Montero et al.,49,50 devoted to rationality measures and dimension of preference. Due to the existence of intermediate states between extreme rationality and extreme irrationality, Montero49 proposed a nonabsolutely irrational aggregation rules. After that, Cutello and Montero50 extended the rationality measures to fuzzy preference relations. Absolute scale and rationality measures are issues to be further studied, which we will focus on in the future. Even though both additive transitivity and multiplicative transitivity can be used to measure the consistency, the additive consistency may produce unreasonable results as discussed in the Introduction. Thus we shall use the multiplicative transitivity to verify the consistency of a HFPR. The condition of multiplicative transitivity can be rewritten as follows: p ij p jk p ki ¼ p ik p kj p ji : In the case where ðp ik ; p kj Þ 62 fð0; 1Þ; ð1; 0Þg, Eq. (3.3) is equivalent to46: p ik p kj p ij ¼ ; p ik p kj þ ð1
p ik Þð1
p kj Þ

ð3:3Þ

ð3:4Þ

and if ðp ik ; p kj Þ 2 fð0; 1Þ; ð1; 0Þg, we stipulate p ij ¼ 0. Inspired by Eq. (3.4), in what follows, we de¯ne the concept of multiplicative consistent HFPR: De¯nition 13. Let H ¼ ðh ij Þ nn be a HFPR on a ¯xed set X ¼ fx 1 ; x 2 ; . . . ; x n g, then H ¼ ðh ij Þ nn is multiplicative consistent if 8 0; ðh ik ; h kj Þ 2 fðf0g; f1gÞ; > > > > > ðf1g; f0gÞg < ðtÞ ðtÞ ðtÞ h ij ¼ h ik ðxÞh kj ðxÞ > > ; otherwise; > > ðtÞ ðtÞ > h ðxÞh ðxÞ þ ð1
h ðtÞ ðxÞÞð1
h ðtÞ ðxÞÞ : ik

kj

ik

kj

for all i  k  j

ð3:5Þ

58

H. Liao, Z. Xu & M. Xia ðtÞ

ðtÞ

where h ik ðxÞ and h kj ðxÞ are the tth smallest values in h ik ðxÞ and h kj ðxÞ, respectively. Theorem 4. Any HFPR H ¼ ðh ij Þ 22 is multiplicative consistent. ð1Þ

ð2Þ

ðnÞ

ðnÞ

ðn
1Þ

Proof. Suppose that h 12 ¼ fh 12 ; h 12 ; . . . ; h 12 g, then, h 21 ¼f1
h 12 ; 1
h 12 ð1Þ . . . ; 1
h 12 g. Thus, ð1Þ

ð1Þ

0:5h 12 ðxÞ ð1Þ

;

ð1Þ

0:5h 12 ðxÞ þ 0:5ð1
h 12 ðxÞÞ

¼

0:5h 12 ðxÞ ð1Þ ¼ h 12 ðxÞ: 0:5

¼

0:5h 12 ðxÞ ð2Þ ¼ h 12 ðxÞ 0:5

¼

0:5h 12 ðxÞ ðnÞ ¼ h 12 ðxÞ 0:5

Similarly, ð2Þ

ð2Þ

0:5h 12 ðxÞ ð2Þ

ð2Þ

0:5h 12 ðxÞ þ 0:5ð1
h 12 ðxÞÞ .. . ðnÞ

ðnÞ

0:5h 12 ðxÞ ðnÞ

ðnÞ

0:5h 12 ðxÞ þ 0:5ð1
h 12 ðxÞÞ

which satis¯es Eq. (3.5). Additionally, when h 12 ¼ f0g, Eq. (3.5) also holds. Thus, H ¼ ðh ij Þ 22 is multiplicative consistent, which completes the proof of Theorem 4. Based on De¯nition 13 and Theorems 1 and 4, in order not to increase the dimensions of the derived HFEs in the process of calculating, we can give the following de¯nition: De¯nition 14. Let H ¼ ðh ij Þ nn be a HFPR on a ¯xed set X ¼ fx 1 ; x 2 ; . . . ; x n g, then we call H ¼ ðh ij Þ nn a prefect multiplicative consistent HFPR, where ðtÞ h ij ðxÞ 8 ðtÞ ðtÞ j
1 X > h ik ðxÞh kj ðxÞ 1 > > > ; > > j
i
1 k¼iþ1 h ðtÞ ðxÞh ðtÞ ðxÞ þ ð1
h ðtÞ ðxÞÞð1
h ðtÞ ðxÞÞ > > ik kj ik kj > < ðtÞ ¼ h ij ; > > > > f0:5g; > > > > > ðtÞ : 1
h ji ðxÞ;

iþ1<j iþ1¼j i¼j i>j ð3:6Þ

ðtÞ h ij ðxÞ,

and and are the tth smallest values in h ij ðxÞ, h ik ðxÞ and h kj ðxÞ, respectively, and t ¼ 1; 2; . . . ; l, l ¼ maxfl h ik ; l h kj g. ðtÞ h ik ðxÞ

ðtÞ h kj ðxÞ

De¯nition 15. Let H ¼ ðh ij Þ nn be a HFPR on a ¯xed set X ¼ fx 1 ; x 2 ; . . . ; x n g, then we call H ¼ ðh ij Þ nn an acceptable multiplicative consistent HFPR, if dðH ; H Þ < ;

ð3:7Þ

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

59

where dðH ; H Þ is the distance measure between the given HFPR H and its corresponding prefect multiplicative consistent HFPR H which can be calculated by Eqs. (2.13) or (2.14), and  is the consistency level. Without loss of generality, we usually let  ¼ 0:1 in practice. The HFPR H constructed by the decision maker is generally with unacceptable multiplicative consistency, which means dðH ; H Þ > . Thus we need to adjust the elements in the hesitant preference relation to improve the consistency. Below we propose an iterative algorithm to repair the consistency level of the HFPR: Algorithm 1. Step 1. Suppose that p is the number of iterations,  is the step size, 0   ¼ p  1 and  is the consistency level. Let p ¼ 1, and construct the prefect ðpÞ multiplicative consistent HFPR H ¼ ðh ij Þ nn from H ðpÞ ¼ ðh ij Þ nn by Eq. (3.6). Step 2. Calculate the deviation dðH ðpÞ ; H Þ between H and H ðpÞ by using: 2 3 l h ij n X n X X 1 ðtÞ ðpÞðtÞ 41 d hnh ðH ðpÞ ; H Þ ¼ jh
h ij j5 ð3:8Þ ðn
1Þðn
2Þ i¼1 j¼1 l h ij t¼1 ij or 2

0 131=2 l h ij n X n X X 1 1 ðtÞ ðpÞðtÞ @ d hne ðH ðpÞ ; H Þ ¼ 4 jh
h ij j2 A5 ; ðn
1Þðn
2Þ i¼1 j¼1 l h ij t¼1 ij ð3:9Þ where h ij and h ij are the tth smallest values in h ij and h ij , respectively. If dðH ðpÞ ; H Þ < , then output H ðpÞ , otherwise, go to the next step. a ðpÞ Step 3. Repair the inconsistent multiplicative HFPR H ðpÞ to H ðpÞ ¼ ðh ij Þ nn by using the following equations: ðtÞ

ðpÞðtÞ

a

ðpÞðtÞ

h ij

ðpÞ

ðpÞðtÞ 1


¼

ðh ij ðpÞðtÞ 1


ðh ij

Þ

Þ

ðh ij Þ ðtÞ

ðh ij Þ þ ð1
hðp ij ðtÞ

ðpÞðtÞ 1


Þ

ðtÞ ð1
h ij Þ

;

i; j ¼ 1; 2; . . . ; n; a

ð3:10Þ a

ðtÞ ðpÞðtÞ ðpÞðtÞ ðpÞ ðpÞ where h ij , h ij and h ij are the tth smallest values in h ij , h ij a and h ij , respectively. Let H ðpþ1Þ ¼ H ðpÞ and p ¼ p þ 1, then go to Step 2.

60

H. Liao, Z. Xu & M. Xia

Example 2. Suppose that a decision maker provides his/her preference information over a set of alternatives x 1 ; x 2 ; x 3 ; x 4 in HFEs and thus constructs the following HFPR: 1 0 f0:5g f0:1; 0:4g f0:1; 0:2g f0:4; 0:5; 0:6g C B f0:5g f0:3; 0:8g f0:3; 0:6g C B f0:6; 0:9g C: H ¼B C B f0:2; 0:7g f0:5g f0:2; 0:7g A @ f0:8; 0:9g f0:4; 0:5; 0:6g f0:4; 0:7g f0:3; 0:8g

f0:5g

First, let p ¼ 1 and H ð1Þ ¼ H , then we construct the prefect multiplicative HFPR  H ¼ ðh ij Þ nn from H ð1Þ by Eq. (3.6). We take h 14 as an example, i.e., ! ð1Þ ð1Þ ð1Þ ð1Þ 1 h h h h ð1Þ 12 24 13 34 h 14 ¼ þ 2 h ð1Þ h ð1Þ þ ð1
h ð1Þ Þð1
h ð1Þ Þ h ð1Þ h ð1Þ þ ð1
h ð1Þ Þð1
h ð1Þ Þ 12 24 12 24 13 34 13 34
 1 0:1  0:3 0:1  0:2 þ ¼ 2 0:1  0:3 þ ð1
0:1Þð1
0:3Þ 0:1  0:2 þ ð1
0:1Þð1
0:2Þ ¼ 0:036; 1 ð2Þ h 14 ¼ 2

ð2Þ ð2Þ

ð2Þ ð2Þ

ð2Þ

!

ð2Þ ð2Þ

h 12 h 24

ð2Þ

h 12 h 24 þ ð1
h 12 Þð1
h 24 Þ

þ

h 13 h 34 ð2Þ ð2Þ

ð2Þ

ð2Þ

h 13 h 34 þ ð1
h 13 Þð1
h 34 Þ
 1 0:4  0:6 0:2  0:7 þ ¼ 2 0:4  0:6 þ ð1
0:4Þð1
0:6Þ 0:2  0:7 þ ð1
0:2Þð1
0:7Þ ¼ 0:434:

Hence, in the similar way, we can obtain 1 0 f0:5g f0:1; 0:4g f0:046; 0:727g f0:036; 0:434g B C f0:5g f0:3; 0:8g f0:097; 0:903g C B f0:6; 0:9g  B C: H ¼B C f0:2; 0:7g f0:5g f0:2; 0:7g A @ f0:273; 0:954g f0:566; 0:964g f0:097; 0:903g

f0:3; 0:8g

f0:5g

Then, we use Eq. (3.8) to calculate the hesitant normalized Hamming distance between H ð1Þ and H : 2 3 l x ij 4 X 4 X X 1 1 ðtÞ ðtÞ 4 d hnh ðH ð1Þ ; H Þ ¼ jh ðx Þ
h H ðx ij Þj5 6 i¼1 j¼1 l x ij t¼1 H ð1Þ ij ¼

 1 1 1 ðj0:1
0:046j þ j0:2
0:727jÞ þ ðj0:4
0:036j 6 2 3

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

61

1 þ j0:5
0:434j þ j0:6
0:5jÞ þ ðj0:3
0:097j þ j0:6
0:903jÞ 2 1 1 þ ðj0:8
0:273j þ j0:9
0:954jÞ þ ðj0:4
0:5j þ j0:5
0:566j 2 3  1 þ j0:6
0:964jÞ þ ðj0:4
0:097j þ j0:7
0:903jÞ ¼ 0:2071: 2 Without loss of generality, let  ¼ 0:1, then d hnh ðH ð1Þ ; H Þ ¼ 0:2071 > , which means that H ð1Þ is not a multiplicative consistent HFPR. Therefore, it is needed to a repair the inconsistent multiplicative HFPR H ð1Þ according to H ð1Þ by Eq. (3.10). We hereby let  ¼ 0:8, then 0 1 f0:5g f0:1; 0:4g f0:054; 0:624g f0:062; 0:447; 0:52g B C f0:6; 0:9g f0:5g f0:3; 0:8g f0:124; 0:866g C B a ð1Þ B C: H ¼B C f0:2; 0:7g f0:5g f0:2; 0:7g @ f0:376; 0:946g A f0:48; 0:553; 0:938g f0:134; 0:876g

f0:3; 0:8g

f0:5g

a

If we let H ð2Þ ¼ H ð1Þ and p ¼ 2, then the hesitant normalized Hamming distance between H ð2Þ and H can be calculated, i.e., d hnh ðH ð2Þ ; H Þ ¼ 0:039 < 0:1. Since the hesitant normalized Hamming distance is less than the consistency level, then we can draw a conclusion that H ð2Þ is the repaired multiplicative consistent HFPR of H . In this example, we can also use Eq. (3.9) to calculate the hesitant normalized Euclidean distance instead of the hesitant normalized Hamming distance. Both of them can get the same result. Beside Algorithm 1, the most direct method for repairing the inconsistency is returning the inconsistent multiplicative hesitant preference relation to the decision maker to reconsider and construct a new hesitant preference relation according to his/her new comparisons till it has acceptable consistency. This algorithm can be described as follows: Algorithm 2. Step 1. See Algorithm 1. Step 2. See Algorithm 1. Step 3. Return the inconsistent multiplicative HFPR H ðpÞ to the decision maker to reconsider constructing a new HFPR H ðpþ1Þ according to the new judgments. Let p ¼ p þ 1, then go to Step 2. Example 3. Suppose the analyst does not repair the inconsistent multiplicative HFPR H ð1Þ itself by using Eq. (3.10) in Example 1, but returns it to the decision maker to reconsider their opinions with reference to the prefect multiplicative HFPR H .

62

H. Liao, Z. Xu & M. Xia

After re-evaluation, the decision maker provides a new HFPR H ð2Þ as below: 0 1 f0:5g f0:1; 0:4g f0:1; 0:7g f0:1; 0:4g B f0:6; 0:9g f0:5g f0:3; 0:8g f0:1; 0:9g C B C H ð2Þ ¼ B C: @ f0:3; 0:9g f0:2; 0:7g f0:5g f0:2; 0:7g A f0:6; 0:9g f0:1; 0:9g f0:3; 0:8g

f0:5g

Then we use Eq. (3.6) to construct the prefect multiplicative HFPR H 0 from H ð2Þ . It is also the same as H : 0 1 f0:5g f0:1; 0:4g f0:046; 0:727g f0:036; 0:434g B f0:6; 0:9g f0:5g f0:3; 0:8g f0:097; 0:903g C B C H ¼ B C: @ f0:273; 0:954g f0:2; 0:7g f0:5g f0:2; 0:7g A f0:566; 0:964g f0:097; 0:903g

f0:3; 0:8g

f0:5g

Then, we use Eq. (3.8) to calculate the hesitant normalized Hamming distance between H ð2Þ and H 0 : 2 3 l x ij 4 X 4 X X 1 1 ðtÞ ðtÞ 4 d hnh ðH ð2Þ ; H 0 Þ ¼ jh ðx Þ
h H 0 ðx ij Þj5 6 i¼1 j¼1 l x ij t¼1 H ð2Þ ij  1 1 1 ðj0:1
0:046j þ j0:7
0:727jÞ þ ðj0:1
0:036j ¼ 6 2 2 1 þ j0:4
0:434jÞ þ ðj0:1
0:097j þ j0:9
0:903jÞ 2 1 1 þ ðj0:3
0:273j þ j0:9
0:954jÞ þ ðj0:6
0:566j 2 2  1 þ j0:9
0:964jÞ þ ðj0:1
0:097j þ j0:9
0:903jÞ ¼ 0:0308: 2 Since the hesitant normalized Hamming distance is less than the consistency level, i.e., d hnh ðH ð2Þ ; H 0 Þ ¼ 0:0308 < 0:1, then we can draw a conclusion that H ð2Þ is the multiplicative consistent HFPR of H . Both of the above two algorithms can guarantee that any multiplicative inconsistent hesitant preference relation can be transformed into a hesitant preference relation with acceptable consistency level. But in practice, we usually use the former procedure because the latter one wastes a lot of time and resources.

4. Approaches to Group Decision Making Based on Multiplicative Consensus of HFPRs In our daily life, in order to choose the most desirable and reasonable solution(s) for a decision-making problem, people prefer to form a commitment or organization constructed by several decision makers who may sometimes come from

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

63

di®erent aspects instead of just the single decision maker for the sake of avoiding the limited knowledge, personal background, private emotion, and so on. As mentioned above, people may provide the preference information by pairwise comparison and thus construct their preference relations. If people in the commitment or organization express their preference values in HFEs, then some HFPRs can be constructed. The group decision-making problem in hesitant fuzzy circumstance can be described as follows: Suppose that G ¼ fG 1 ; G 2 ; . . . ; G n g is a discrete set of alternatives; d k ðk ¼ 1; 2; . . . ; mÞ are the decision organizations (each of which contains a collection of decision makers), and w ¼ ðw 1 ; w 2 ; . . . ; w m ÞT is the weight vector Pm of the decision organizations with k¼1 w k ¼ 1, ! k 2 ½0; 1, k ¼ 1; 2; . . . ; m. The decision organization d k provides all the possible preference values for each pair of ðkÞ alternatives, and constructs a HFPR H ðkÞ ¼ ðh ij Þ nn ðk ¼ 1; 2; . . . ; mÞ. ðkÞ

De¯nition 16. Let H k ¼ ðh ij Þ nn ðk ¼ 1; 2; . . . ; mÞ be a collection of m HFPRs on a ¯xed set X ¼ fx 1 ; x 2 ; . . . ; x n g, and w ¼ ðw 1 ; w 2 ; . . . ; w m ÞT be the weight vector of P H k ðk ¼ 1; 2; . . . ; mÞ where m k¼1 w k ¼ 1 and 0  w k  1. Then we call H ¼ ðh ij Þ nn a collective HFPR of H k ¼ ðh ij Þ nn ðk ¼ 1; 2; . . . ; mÞ, where h ij can be obtained by the AHFWA or AHFWG operator, i.e., ( ) m Y m ðkÞ ðkÞðtÞ w k h ij ¼  ðw k h ij Þ ¼ 1
ð1
h ij Þ jt ¼ 1; 2; . . . ; l ; i; j ¼ 1; 2; . . . ; n; k¼1

k¼1

ð4:1Þ and

( m

h ij ¼ 

k¼1

ðkÞ ðh ij Þw k

¼

m Y

) ðkÞðtÞ w k ðh ij Þ

j t ¼ 1; 2; . . . ; l ;

i; j ¼ 1; 2; . . . ; n;

k¼1

ð4:2Þ ðkÞðtÞ

ðkÞ

where h ij are the tth smallest values in h ij . In the following, we propose two consensus reaching procedures for HFPRs in group decision making: 4.1. An automatic consensus reaching process As to the other preference relations, many scholars proposed some automatic consensus improving procedures.51,52 Based on two soft consensus criteria


a consensus measure and a proximity measure, Tapia García et al.51 presented a consensus model for group decision-making problems with interval fuzzy preference relations. They also designed an automatic feedback mechanism to help decision makers in consensus reaching process. Furthermore, Cabrerizo et al.52 developed a consensus model for group decision-making problems with unbalanced fuzzy linguistic information based on the above two soft consensus criteria. In a multigranular fuzzy linguistic context, Mata et al.53 also proposed an adaptive consensus support model for group

64

H. Liao, Z. Xu & M. Xia

decision-making problems, which increases the convergence toward the consensus and reduces the number of rounds to reach it. These works are all automatic which transform the experts' opinions themselves, without the experts' interactivity. From this point of view and motivated by the above section, we propose an automatic consensus reaching procedure for HFPRs, which can be clari¯ed as follows: Algorithm 3. ðkÞ

Step 1. Let ðH ðkÞ ÞðpÞ ¼ ððh ij Þ nn ÞðpÞ ðk ¼ 1; 2; . . . ; mÞ and p ¼ 1. Construct the ðkÞ prefect multiplicative consistent HFPRs ðH ðkÞ ÞðpÞ ¼ ððh ij Þ nn ÞðpÞ from ðkÞ ðpÞ ðpÞ ðkÞ ðH Þ ¼ ððh ij Þ nn Þ by Algorithm 1 (or Algorithm 2). Step 2. Aggregate all the individual prefect multiplicative consistent HFPRs ðkÞ ðH ðkÞ ÞðpÞ ¼ ððh ij Þ nn ÞðpÞ into a collective HFPR ðH ÞðpÞ ¼ ððh ij Þ nn ÞðpÞ by the AHFWA or AHFWG operator, where ( ) m Y m ðkÞ ðkÞðtÞ w k    h ij ¼  ðw k h Þ ¼ 1
ð1
h Þ j t ¼ 1; 2; . . . ; l ; ij

k¼1

ij

k¼1

i; j ¼ 1; 2; . . . ; n and

( m

h ij ¼ 

k¼1

ðkÞ ðh ij Þw k

¼

m Y

ð4:3Þ

) ðkÞðtÞ w k ðh ij Þ

j t ¼ 1; 2; . . . ; l ;

k¼1

i; j ¼ 1; 2; . . . ; n

ð4:4Þ ðkÞðtÞ ðkÞ   is the tth smallest value in h ij . where h ij Step 3. Calculate the deviation between each individual HFPR ðH ðkÞ ÞðpÞ ¼ ðkÞ ððh ij Þ nn ÞðpÞ and the collective HFPR H ðpÞ ¼ ððh ij Þ nn ÞðpÞ , i.e., d hnh ððH ðkÞ ÞðpÞ ; H ðpÞ Þ ¼

1 ðn
1Þðn
2Þ 2 3 l h ij n X n X X 1 ðkÞðpÞðtÞ ðpÞðtÞ 5 4  jh ij
h ij j l  h ij t¼1 i¼1 j¼1

ð4:5Þ

or d hne ððH ðkÞ ÞðpÞ ; H ðpÞ Þ 2 0 131=2 l h ij n X n X X 1 1 ðkÞðpÞðtÞ ðpÞðtÞ 2 A5 @ jh
h ij j : ð4:6Þ ¼4 ðn
1Þðn
2Þ i¼1 j¼1 l h ij t¼1 ij If dððH ðkÞ ÞðpÞ ; H ðpÞ Þ   , for all k ¼ 1; 2; . . . ; m, where  is the consensus level, then go to Step 5; Otherwise, go to the next step.

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

65

Step 4. Let ðH ðkÞ Þðpþ1Þ ¼ ððh ij Þ nn Þðpþ1Þ , where ðkÞ

ðkÞðpþ1ÞðtÞ ¼ h ij

ðh ij

ðkÞðpÞðtÞ 1


ðkÞðpÞðtÞ 1
  ðpÞðtÞ  ðh ij Þ ðh ij Þ

Þ

þ ð1


ðh ij

ðpÞðtÞ 

Þ

ðkÞðpÞðtÞ 1
 hðp ij Þ ð1


h ij

ðpÞðtÞ 

Þ

;

i; j ¼ 1; 2; . . . ; n; ð4:7Þ ðkÞðpþ1ÞðtÞ  ðkÞðpÞðtÞ ðpÞðtÞ where h ij , h ij and h ij are the tth smallest values in ðkÞðpþ1Þ ðkÞðpÞ ðpÞ h ij , h ij and h ij , respectively. Let p ¼ p þ 1, then go to Step 2. ðpÞ Step 5. Let H ¼ H , and employ the AHFA or AHFG operator to fuse all the hesitant preference values h ij ðj ¼ 1; 2; . . . ; nÞ corresponding to the object x i into the overall hesitant preference value h i , i.e.,
 n 1 h ij h i ¼ AHFA ðh i1 ; h i2 ; . . . ; h in Þ ¼  j¼1 n ( ) n Y ðtÞ 1=n ¼ 1
ð1
h ij Þ j t ¼ 1; 2; . . . ; l ð4:8Þ j¼1

or n

h i ¼ AHFG ðh i1 ; h i2 ; . . . ; h in Þ ¼  ðh ij Þ1=n j¼1 ( ) n Y ðtÞ 1=n ¼ ðh ij Þ j t ¼ 1; 2; . . . ; l ;

ð4:9Þ

j¼1 ðtÞ

where h ij is the tth smallest values in h ij . Step 6. Rank all the objects corresponding to the methods given in Sec. 2. This consensus improving procedure can be interpreted like this: First, we construct the prefect multiplicative consistent HFPRs for the individual HFPRs given by the di®erent decision organizations. Then a collective HFPR can be obtained by aggregating the constructed prefect multiplicative consistent HFPRs. We can easily calculate the distance between each individual HFPR and the collective HFPR, respectively. If the distance is greater than the given consensus level, we need to improve it, otherwise it is acceptable. To improve the consensus level of each individual HFPR, we fuse it with the collective HFPR by using Eq. (4.7), and then get some new individual HFPRs, thus we can iterate until all the individual HFPRs are acceptable. We now consider a group decision-making problem that concerns the evaluation and ranking of the main factors of electronic learning (adapted from Ref. 32) to illustrate our procedure: Example 4. As the electronic learning (e-learning) not only can provide expediency for learners to study courses and professional knowledge without the

66

H. Liao, Z. Xu & M. Xia

constraint of time and space especially in an asynchronous distance e-learning system, but also may save internal training cost for some enterprises organizations in a long-term strategy, meanwhile, it also can be used as an alternative selftraining for assisting or improving the traditional classroom teaching, the elearning becomes more and more popular along with the advancement of information technology and has played an important role in teaching and learning not only in di®erent levels of schools but also in various commercial or industrial companies. Many schools and businesses invest manpower and money in e-learning to enhance their hardware facilities and software contents. Thus it is meaningful and urgent to determinate which is the most important among the main factors which in°uence the e-learning e®ectiveness. Based on the research of Wang54 and Tzeng et al.,55 there are four key factors (or criteria) to evaluate the e®ectiveness of an e-learning system. These four main factors are:  1:  2:  3:  4:

the the the the

synchronous learning; e-learning material; quality of web learning platform; self-learning.

In order to rank the above four factors, a committee comprising three decision makers d l ðl ¼ 1; 2; 3Þ (whose weight vector is ! ¼ ð0:3; 0:4; 0:3ÞT Þ is found. After comparing pairs of the factors (or criteria)  i ði ¼ 1; 2; 3; 4Þ, the decision makers d l ðl ¼ 1; 2; 3Þ give their preferences using HFEs, and then obtain the hesitant preference relations as follows: 1 0 f0:5g f0:2; 0:3; 0:4g f0:4; 0:5; 0:6g f0:3; 0:7g C B C B f0:5g f0:5; 0:6g f0:3; 0:4g C B f0:6; 0:7; 0:8g C; B H1 ¼ B C f0:4; 0:5g f0:5g f0:4; 0:5g C B f0:4; 0:5; 0:6g A @ f0:3; 0:7g f0:6; 0:7g f0:5; 0:6g f0:5g 0

f0:5g

f0:3; 0:4g f0:5; 0:6; 0:7g f0:3; 0:4; 0:6g

B B f0:5g B f0:6; 0:7g H2 ¼ B B B f0:3; 0:4; 0:5g f0:3; 0:6g @ f0:4; 0:6; 0:7g f0:4; 0:6g 0 B B B H3 ¼ B B B @

f0:5g

f0:4; 0:7g

f0:4; 0:6g

f0:5g

f0:6; 0:7g

f0:3; 0:4g

f0:5g

f0:2; 0:4g f0:4; 0:7g f0:3; 0:6; 0:7g

f0:6; 0:8g

f0:5g

f0:5; 0:7g

f0:3; 0:6g

f0:3; 0:6g

f0:3; 0:5g

f0:5g

f0:4; 0:6g

f0:3; 0:4; 0:7g f0:4; 0:7g f0:4; 0:6g

f0:5g

1 C C C C: C C A

1 C C C C; C C A

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

67

To solve this problem, the following steps are given according to Algorithm 3: Step 1. Let ðH ðkÞ ÞðpÞ ¼ H k and p ¼ 1, we ¯rst construct respectively the prefect multiplicative consistent hesitant preference relations ðH ðkÞ Þð1Þ ¼ ðkÞ ðkÞ ððh ij Þ nn Þð1Þ ðk ¼ 1; 2; 3Þ from ðH ðkÞ Þð1Þ ¼ ððh ij Þ nn Þð1Þ (k ¼ 1; 2; 3Þ by Algorithm 1 (or Algorithm 2): 1 0 f0:5g f0:2; 0:3; 0:4g f0:2; 0:3; 0:5g f0:202; 0:361; 0:5g C B C B f0:6; 0:7; 0:8g f0:5g f0:5; 0:6g f0:4; 0:6g C B C; ðH ð1Þ Þð1Þ ¼ B C B f0:5; 0:7; 0:8g f0:4; 0:5g f0:5g f0:4; 0:5g C B A @ f0:5; 0:639; 0:798g f0:4; 0:6g f0:5; 0:6g f0:5g 0  ð2Þ ð1Þ

ðH

Þ

B B B ¼B B B @ 0

f0:5g

f0:3; 0:4g f0:222; 0:609g f0:361; 0:596; 0:672g

f0:6; 0:7g

f0:5g

f0:4; 0:7g

f0:5; 0:845g

f0:391; 0:778g

f0:3; 0:6g

f0:5g

f0:6; 0:7g

f0:3; 0:4g

f0:5g

f0:328; 0:404; 0:639g f0:155; 0:5g f0:5g

B B f0:6; 0:8g ðH ð3Þ Þð1Þ ¼ B B f0:391; 0:8g @

f0:2; 0:4g f0:2; 0:609g f0:202; 0:639g f0:5g

f0:5; 0:7g

f0:3; 0:5g

f0:5g

f0:361; 0:798g f0:222; 0:6g f0:4; 0:6g

1 C C C C; C C A

1

C f0:4; 0:778g C C: f0:4; 0:6g C A f0:5g

Step 2. Fuse the individual prefect multiplicative consistent hesitant preference ðkÞ relations ðH ðkÞ Þð1Þ ¼ ððh ij Þ nn Þð1Þ into a collective prefect hesitant preference relation ðH Þð1Þ ¼ ððh ij Þ nn Þð1Þ by the AHFWA or AHFWG operator. We hereby take the AHFWA operator, i.e., Eq. (4.3), as an example, and then we obtain 1 f0:5g f0:242; 0:372; 0:472g f0:209; 0:447; 0:579g f0:27; 0:506; 0:617g C B C B f0:532; 0:633; 0:765g f0:5g f0:462; 0:673g f0:442; 0:771g C B C: H ð1Þ ¼ B C B f0:332; 0:543g f0:5g f0:49; 0:619g C B f0:426; 0:571; 0:792g A @ f0:394; 0:514; 0:745g f0:256; 0:563g f0:396; 0:53g f0:5g 0

Step 3. Calculate the deviation between each individual prefect multiplicative ðkÞ consistent hesitant preference relation ðH ðkÞ Þð1Þ ¼ ððh Þ nn Þð1Þ and the ij

collective hesitant preference relation H ð1Þ ¼ ððh ij Þ nn Þð1Þ . In this example, we use Eq. (4.5) i.e., the hesitant normalized Hamming distance, as a

68

H. Liao, Z. Xu & M. Xia

representation, and then we have d hnh ððH ð1Þ Þð1Þ ; H ð1Þ Þ ¼ 0:162; d hnh ððH ð3Þ Þð1Þ ; H ð1Þ Þ ¼ 0:07:

d hnh ððH ð2Þ Þð1Þ ; H ð1Þ Þ ¼ 0:128;

Without loss of generality, we let the consensus level  ¼ 0:1. We can see that both d hnh ððH ð1Þ Þð1Þ ; H ð1Þ Þ and d hnh ððH ð2Þ Þð1Þ ; H ð1Þ Þ are bigger than 0.1, then we need to improve these individual prefect multiplicative consistent HFPRs. Step 4. Let  ¼ 0:7, and by Eq. (4.7), we can construct respectively the new indiðkÞ vidual hesitant preference relations ðH ðkÞ Þð2Þ ¼ ððh ij Þ nn Þð2Þ ðk ¼ 1; 2; 3Þ as below: 1 f0:5g f0:229; 0:35; 0:45g f0:206; 0:401;0:556g f0:248;0:461; 0:583g C B B f0:553; 0:654;0:78g f0:5g f0:473; 0:652g f0:429;0:725g C C ð2Þ B ð1Þ  ðH Þ ¼B C; C B f0:445;0:612;0:794g f0:352; 0:53g f0:5g f0:463;0:584g A @ f0:425; 0:552;0:762g f0:296;0:574g f0:427; 0:551g f0:5g 0

1 f0:5g f0:259; 0:38; 0:48g f0:213; 0:463;0:588g f0:296;0:533; 0:634g C B B f0:522;0:623;0:747g f0:5g f0:443; 0:681g f0:459;0:796g C C ð2Þ B ð2Þ  ðH Þ ¼B C; C B f0:412; 0:55; 0:788g f0:322; 0:56g f0:5g f0:523;0:644g A @ f0:374; 0:481;0:715g f0:222;0:544g f0:366; 0:491g f0:5g 0

1 f0:5g f0:229; 0:38; 0:48g f0:206; 0:463;0:588g f0:248;0:504; 0:624g C B B f0:522; 0:623;0:776g f0:5g f0:473; 0:681g f0:429;0:773g C C ð3Þ ð2Þ B  ðH Þ ¼B C: C B f0:412; 0:55; 0:794g f0:322; 0:53g f0:5g f0:463;0:613g A @ 0

f0:384; 0:51; 0:762g

f0:246;0:574g

f0:397; 0:551g

f0:5g

Let p ¼ 2, then go back to Step 2. We fuse the individual hesitant preference relations ðH ðkÞ Þð2Þ ðk ¼ 1; 2; 3Þ into a collective hesitant preference relation ðH Þð2Þ ¼ ððh ij Þ nn Þð2Þ by the AHFWA operator: 0

f0:5g

B B f0:532; 0:633; 0:766g H ð2Þ ¼ B B f0:422; 0:57; 0:792g @ f0:393; 0:512; 0:744g

f0:241; 0:371; 0:471g f0:209; 0:445; 0:579g f0:268; 0:504; 0:616g f0:5g

f0:461; 0:673g

f0:442; 0:77g

f0:331; 0:542g

f0:5g

f0:488; 0:618g

f0:252; 0:562g

f0:394; 0:528g

f0:5g

1 C C C: C A

Thus, the hesitant normalized Hamming distance between each individual prefect multiplicative consistent hesitant preference relation ðH ðkÞ Þð2Þ and the collective hesitant preference relation H ð2Þ can be calculated as follows: d hnh ððH ð1Þ Þð2Þ ; H ð2Þ Þ ¼ 0:048; d hnh ððH ð3Þ Þð2Þ ; H ð2Þ Þ ¼ 0:02:

d hnh ððH ð2Þ Þð2Þ ; H ð2Þ Þ ¼ 0:039;

Now all d hnh ððH ðkÞ Þð2Þ ; H ð2Þ Þ < 0:1 (k ¼ 1; 2; 3Þ, then go to Step 5.

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

69

Step 5. Let H ¼ H ð2Þ , and employ the AHFA or AHFG operator to fuse all the hesitant preference values h ij ðj ¼ 1; 2; . . . ; nÞ corresponding to the object  i into the overall hesitant preference value h i . We hereby use the AHFA operator to fuse the information. By Eq. (4.8), we have h 1 ¼ f0:315; 0:458; 0:545g;

h 2 ¼ f0:485; 0:537; 0:694g;

h 3 ¼ f0:444; 0:519; 0:633g;

h 4 ¼ f0:391; 0:503; 0:597g:

Step 6. Using the score function, we can get sðh 1 Þ ¼ 0:439, sðh 2 Þ ¼ 0:572, sðh 3 Þ ¼ 0:532, and sðh 4 Þ ¼ 0:495. As sðh 2 Þ > sðh 3 Þ > sðh 4 Þ > sðh 1 Þ, we can draw a conclusion that  2   3   4   1 , which denotes that the e-learning material is the most important factor in°uencing the a®ectivity of e-learning. Surely, in this example, we can also use the AHFWG and AHFG operators to fuse the HFEs in Steps 2 and 5, and we can also use the hesitant normalized Euclidean distance to calculate the deviation between each individual prefect multiplicative consistent hesitant preference relation and the collective prefect multiplicative consistent hesitant preference relation in Step 3. In addition, from Step 3, we can see that if we take the consensus level within the interval 0:07    0:128, the result will keep the same. In other words, if the consensus level is ¯xed, small error measurements perhaps do not cause a complete di®erent output, which is to say, our procedures are robust. For example, suppose the third decision maker gives his/her preferences with another hesitant preference relation as: 1 0 f0:5g f0:2; 0:4g f0:4; 0:7g f0:3; 0:6; 0:7g C B B f0:6; 0:8g f0:5g f0:6; 0:8g f0:3; 0:6g C : C B H3 ¼ B C: C B f0:3; 0:6g f0:2; 0:4; g f0:5g f0:4; 0:6g A @ f0:3; 0:4; 0:7g f0:4; 0:7g f0:4; 0:6g f0:5g In the following, we begin to check whether the output will be changed or not. Step 1. The prefect multiplicative consistent hesitant preference relations of the third decision maker can be calculated easily, which is 1 0 f0:5g f0:2; 0:4g f0:2; 0:609g f0:202; 0:639g C B B f0:6; 0:8g : f0:5g f0:6; 0:8g f0:4; 0:778g C C B ð3Þ ð1Þ  ðH Þ ¼ B C: C B f0:391; 0:8g f0:2; 0:4g f0:5g f0:4; 0:6g A @ f0:361; 0:798g f0:222; 0:6g

f0:4; 0:6g

f0:5g

70

H. Liao, Z. Xu & M. Xia

Step 2. The collective prefect hesitant preference relation can be derived by the AHFWA operator as: 1 f0:5g f0:242; 0:372; 0:472g f0:209; 0:447; 0:579g f0:27; 0:506; 0:617g C B C B f0:5g f0:497; 0:710g f0:442; 0:771g C B f0:532; 0:633; 0:765g C: ¼B C B f0:332; 0:543g f0:5g f0:49; 0:619g C B f0:426; 0:571; 0:792g A @ f0:394; 0:514; 0:745g f0:256; 0:563g f0:396; 0:53g f0:5g 0

:

H ð1Þ

Step 3. Calculate the deviation between each individual prefect multiplicative consistent hesitant : preference relation and the collective hesitant preference relation H ð1Þ : :

d hnh ððH ð1Þ Þð1Þ ; H ð1Þ Þ ¼ 0:162; :

:

:

d hnh ððH ð2Þ Þð1Þ ; H ð1Þ Þ ¼ 0:130;

d hnh ððH ð3Þ Þð1Þ ; H ð1Þ Þ ¼ 0:093: Since the consensus level  : ¼ 0:1, then we can see that both d hnh ððH ð1Þ Þð1Þ ; : ð1Þ H Þ and d hnh ððH ð2Þ Þð1Þ ; H ð1Þ Þ are bigger than 0.1. Thus, we need to improve these individual prefect multiplicative consistent HFPRs. Step 4. Let  ¼ 0:7, then we can : construct respectively the new individual hesitant preference relations ðH ðkÞ Þð2Þ ðk ¼ 1; 2; 3Þ as: 0

f0:5g

B B f0:553;0:654;0:78g : B ð2Þ B ð1Þ  ðH Þ ¼B B f0:445;0:612; 0:794g @ f0:425;0:552; 0:762g 0

f0:5g

B B B f0:522;0:623; 0:747g ð2Þ B ð2Þ  ðH Þ ¼B B f0:412;0:55;0:788g B @ :

f0:374;0:481; 0:715g

f0:229; 0:35;0:45g f0:206;0:401;0:556g f0:248; 0:461;0:583g f0:5g

f0:498; 0:679g

f0:429;0:725g

f0:332;0:512g

f0:5g

f0:463;0:584g

f0:296;0:574g

f0:427; 0:551g

f0:5g

f0:259; 0:38;0:48g f0:213;0:463;0:588g f0:296; 0:533;0:634g f0:5g

f0:468; 0:707g

f0:459;0:796g

f0:303;0:542g

f0:5g

f0:523;0:644g

f0:222;0:544g

f0:366; 0:491g

f0:5g

1 C C C C; C C A 1 C C C C C; C C A

1 f0:5g f0:229; 0:38;0:48g f0:206;0:463;0:588g f0:248; 0:504;0:624g C B C B B f0:522;0:623; 0:776g f0:5g f0:528; 0:739g f0:429;0:773g C : C B ðH ð3Þ Þð2Þ¼B C: B f0:412;0:55;0:794g f0:270;0:482g f0:5g f0:463;0:613g C C B A @ 0

f0:384;0:51;0:762g

:

f0:246;0:574g

f0:397; 0:551g

f0:5g

Let p ¼ 2, then go back to Step 2. The individual hesitant preference relations :

ðH ðkÞ Þð2Þ (k ¼ 1; 2; 3Þ can be fused into a collective hesitant preference relation ðH Þ

ð2Þ

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

71

by the AHFWA operator: 0

f0:5g

f0:241;0:371;0:471g f0:209; 0:445;0:579g f0:268; 0:504; 0:616g

B : B f0:532;0:633;0:766g H ð2Þ ¼ B B @ f0:422;0:57; 0:792g

f0:5g

f0:496;0:709g

f0:442; 0:77g

f0:302;0:516g

f0:5g

f0:488;0:618g

f0:393; 0:512;0:744g

f0:252;0:562g

f0:394;0:528g

f0:5g

1 C C C: C A

Then, the hesitant normalized Hamming distance between each individual prefect : multiplicative consistent hesitant preference relation ðH ðkÞ Þð2Þ and the collective :

hesitant preference relation H ð2Þ can be calculated as: :

:

:

:

d hnh ððH ð1Þ Þð2Þ ; H ð2Þ Þ ¼ 0:056;

:

:

d hnh ððH ð2Þ Þð2Þ ; H ð2Þ Þ ¼ 0:039;

d hnh ððH ð3Þ Þð2Þ ; H ð2Þ Þ ¼ 0:023: :

:

Now all d hnh ððH ðkÞ Þð2Þ ; H ð2Þ Þ < 0:1 (k ¼ 1; 2; 3Þ, then go to Step 5. : : Step 5. Let H ¼ H ð2Þ , and employ the AHFA operator to fuse all the hesitant preference values h ij ð j ¼ 1; 2; . . . ; nÞ corresponding to the object  i into the overall hesitant preference value h i . h 1 ¼ f0:315; 0:458; 0:545g; h 3 ¼ f0:433; 0:519; 0:628g;

h 2 ¼ f0:494; 0:537; 0:703g; h 4 ¼ f0:391; 0:503; 0:597g:

Step 6. Using the score function, we can get sðh 1 Þ ¼ 0:439, sðh 2 Þ ¼ 0:578, sðh 3 Þ ¼ 0:527, and sðh 4 Þ ¼ 0:495. As sðh 2 Þ > sðh 3 Þ > sðh 4 Þ > sðh 1 Þ, we can draw a conclusion that  2   3   4   1 , which denotes that the elearning material is also the most important factor in°uencing the a®ectivity of e-learning. From the above example, it can be seen that although the third decision maker slightly changes his/her preference, the output of our procedure also keeps the same. Thus, our algorithm is robust and practicable. 4.2. An interactive consensus reaching process From Example 4, we can see the above mentioned procedure is automatic and thus easy to implement. However, since it does not interact with experts when changing the preference values in Step 4, this method sometimes may reach a false or unrealistic consensus degree due to that the experts could not agree with the changes proposed by the system. Indeed, the most direct method to repair the inconsensus is returning the inconsistent preference relations to the experts to reconsider and then constructing new preference relations according to their new comparisons till all the preference relations are with acceptable consensus. Therefore, it is reasonable for us to investigate some interactive consensus reaching processes for HFPRs. Although Algorithm 3, as a decision-making aid method, does not focus on feedback mechanism, it provides a good way to judge whether the HFPRs are

72

H. Liao, Z. Xu & M. Xia

consensus or not. Inspired by this, we can develop an interactive consensus reaching procedure as below: Algorithm 4. Step Step Step Step

1. 2. 3. 4.

See Algorithm 3. See Algorithm 3. See Algorithm 3. Return the inconsensus multiplicative HFPRs ðH ðkÞ ÞðpÞ to the experts to reconsider constructing a new HFPR ðH ðkÞ Þðpþ1Þ according to their new judgments. In this case, they can refer to our new constructed individual ðkÞ HFPRs ðH ðkÞ ÞðpÞ ¼ ððh ij Þ nn ÞðpÞ ðk ¼ 1; 2; . . . ; mÞ. Let p ¼ p þ 1, then go to Step 2. Step 5. See Algorithm 3. Step 6. See Algorithm 3.

Here we do not want to illustrate this algorithm by numerical examples. The Schematic diagram of this algorithm is provided in Fig. 1. Comparing the automatic consensus reaching process Algorithm 3 with and interactive consensus reaching process Algorithm 3, we can ¯nd both of the two algorithms have advantages and disadvantages: (1) The automatic Algorithm 3 is easy to implement and can save a lot of time. It also can give a quick response to the urgent situations. In some settings, if the decision makers do not want to interact with the experts, or if they cannot ¯nd the initial experts to re-evaluate and alter their preferences, or if consensus must be urgently obtained, the automatic procedure developed in Sec. 4.1 is a good choice to derive a consensus solution for group decision making. Such a method contains most initial information. But, sometime the results may not re°ect the

Fig. 1. Schematic diagram of interactive procedure.

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

73

realistic opinions of the experts due to that the experts may not agree with the changes proposed by the automatic system. (2) Interacting with the decision makers frequently during the consensus reaching process is very reliable and accurate. However, the feedback mechanism wastes a lot of time and sometimes this ideal consensus is just a utopian consensus which is di±cult to achieve. 5. Conclusion In this paper, we have investigated the consistency and consensus of hesitant fuzzy preference relations. Based on the transitivity properties, we have given the concepts of multiplicative consistency, perfect multiplicative consistency and acceptable multiplicative consistency of hesitant fuzzy preference relation. Then, we have proposed two algorithms, in which the former procedure is used to construct the acceptable multiplicative consistent hesitant fuzzy preference relation without the participation of the decision maker, while the latter is returning the inconsistent multiplicative hesitant fuzzy preference relation to the decision maker to reconsider and then constructing a new hesitant fuzzy preference relation according to his/her new comparisons until it has acceptable consistency. Subsequently, we have also investigated the consensus of group decision-making problems and developed a convergent iterative algorithm to improve the consensus levels of individual hesitant fuzzy preference relations. The numerical examples have shown that our algorithms are very e®ective in solving decision-making problems with hesitant fuzzy preference information. Acknowledgments The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported in part by the National Natural Science Foundation of China (Nos. 71071161 and 61273209). References 1. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. 2. K. Atanassov, Intuitionistic fuzzy sets, in VII ITKR's Session, So¯a, ed. V. Sgurev (June 1983). 3. K. Atanassov, On Intuitionistic Fuzzy Sets Theory (Springer-Verlag, 2012). 4. Z. S. Xu and H. C. Liao, Intuitionistic fuzzy analytic hierarchy process, IEEE Transactions on Fuzzy Systems (2013), doi: 10.1109/TFUZZ.2013.2272585. 5. K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989) 343–349. 6. M. Mizumoto and K. Tanaka, Some properties of fuzzy sets of type 2, Information and Control 31 (1976) 312–340. 7. D. Dubois and H. Prade, Fuzzy Sets and Systems (Academic Press, 1980).

74

H. Liao, Z. Xu & M. Xia

8. G. Meghabghab, Crisp bags, one-dimensional fuzzy bags, multidimensional fuzzy bags: Comparing successful and unsuccessful user's Web behavior, International Journal of Intelligent Systems 24 (2009) 649–676. 9. M. M. Xia, Z. S. Xu and H. C. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Transactions on Fuzzy Systems 21 (2013) 113–133. 10. V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010) 529– 539. 11. V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, the 18th IEEE Int. Conf. Fuzzy Systems, Jeju Island, Korea (2009), pp. 1378–1382. 12. M. M. Xia and Z. S. Xu, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52 (2011) 395–407. 13. Z. S. Xu and M. M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181 (2011) 2128–2138. 14. H. C. Liao and Z. S. Xu, Subtraction and division operations over hesitant fuzzy sets, Journal of Intelligent & Fuzzy Systems (2013), doi: 10.3233/IFS-130978. 15. H. C. Liao and Z. S. Xu, Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment, Journal of intelligent & Fuzzy Systems (2013), doi: 10.3233/IFS-130841. 16. H. C. Liao and Z. S. Xu, A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optimization and Decision Making (2013), doi: 10.1007/s10700013-9162-0. 17. G. Kou, Y. Peng and Y. Shi, Evaluation of classi¯cation algorithms using MCDM and rank correlation, International Journal of Information Technology & Decision Making 11 (2012) 197–225. 18. Y. Peng, G. Kou, G. Wang and Y. Shi, FAMCDM: A fusion approach of MCDM methods to rank multiclass classi¯cation algorithms, Omega 39 (2011) 677–689. 19. Y. Peng, G. Kou, G. Wang and Y. Shi, Ensemble of software defect predictors: An AHPbased evaluation method, International Journal of Information Technology & Decision Making 10 (2011) 187–206. 20. Z. S. Xu and Q. L. Da, A least deviation method to obtain a priority vector of a fuzzy preference relation, European Journal of Operational Research 164 (2005) 206–216. 21. T. L. Saaty, The Analytic Hierarchy Process (McGraw-Hill, New York, NY, 1980). 22. Z. P. Fan, J. Ma, Y. P. Jiang, Y. H. Sun and L. Ma, A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations, European Journal of Operational Research 174 (2006) 311–321. 23. F. Herrera and E. Herrera-Viedma, Choice functions and mechanisms for linguistic preference relations, European Journal of Operational Research 120 (2000) 144–161. 24. Z. S. Xu, An approach to group decision making based on incomplete linguistic preference relations, International Journal of Information Technology & Decision Making 4 (2005) 153–160. 25. Z. S. Xu, Incomplete linguistic preference relations and their fusion, Information Fusion 7 (2006) 331–337. 26. Z. S. Xu, Intuitionistic preference relations and their application in group decision making, Information Sciences 177 (2007) 2363–2379. 27. W. C. Wei and X. J. Tang, An intuitionistic fuzzy group decision-making approach based on entropy and similarity measures, International Journal of Information Technology & Decision Making 10 (2011) 1111–1130. 28. Z. S. Xu and X. Q. Cai, Incomplete interval-valued intuitionistic fuzzy preference relations, International Journal of General Systems 38 (2009) 871–886.

Multiplicative Consistency of Hesitant Fuzzy Preference Relation

75

29. Z. S. Xu and R. R. Yager, Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group, Fuzzy Optimization and Decision Making 8 (2009) 123–139. 30. T. C. Wang and T. H. Chang, Application of consistent fuzzy preference relations in predicting the success of knowledge management implementation, European Journal of Operational Research 182 (2007) 1313–1329. 31. T. C. Wang and Y. H. Chen, Applying consistent fuzzy preference relations to partnership selection, Omega 35 (2007) 384–388. 32. R. J. Chao and Y. H. Chen, Evaluation of the criteria and e®ectiveness of distance elearning with consistent fuzzy preference relations, Expert Systems with Applications 36 (2009) 10657–10662. 33. T. C. Wang and Y. L. Lin, Applying the consistent fuzzy preference relations to select merger strategy for commercial banks in new ¯nancial environments, Expert Systems with Applications 36 (2009) 7019–7026. 34. S. Alonso, E. Herrera-Viedma, F. Chiclana and F. Herrera, Individual and social strategies to deal with ignorance situations in multi-person decision making. International Journal of Information Technology & Decision Making 8 (2009) 313–333. 35. E. Herrera-Viedma, F. Chiclana, F. Herrera and S. Alonso, Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Transactions on Sysems, Man, and Cybernetics


Part B: Cybernetics 37 (2007) 176–189. 36. F. Chiclana, E. Herrera-Viedma and S. Alonso, A note on two methods for estimating missing pairwise preference values. IEEE Transactions on Systems, Man, and Cybernetics


Part B: Cybernetics 39 (2009) 1628–1633. 37. T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems 12 (1984) 117–131. 38. E. Herrera-Viedma, F. Herrera, F. Chiclana and M. Luque, Some issues on consistency of fuzzy preference relations, European Journal of Operational Research 154 (2004) 98–109. 39. J. Ma, Z. P. Fan, Y. P. Jiang, J. Y. Mao and L. Ma, A method for repairing the inconsistency of fuzzy preference relations, Fuzzy Sets and Systems 157 (2006) 20–33. 40. Y. C. Dong, Y. F. Xu and H. Y. Li, On consistency measures of linguistic preference relations, European Journal of Operational Research 189 (2008) 430–444. 41. Z. W. Gong, L. S. Li, J. Cao and F. X. Zhou, On additive consistent properties of the intuitionistic fuzzy preference relation, International Journal of Information Technology & Decision Making 9 (2010) 1009–1025. 42. Z. S. Xu and X. Q. Cai, Group consensus algorithms based on preference relations, Information Sciences 181 (2011) 150–162. 43. D. Ergu, G. Kou, Y. Peng and Y. Shi, A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP, European Journal of Operational Research 213 (2012) 246–259. 44. D. Ergu, G. Kou, Y. Shi and Y. Shi, Analytic network process in risk assessment and decision analysis, Computers & Operations Research (2011), doi: 10.1016/j.cor.2011.03.005. 45. T. H. Chang and T. C. Wang, Measuring the success possibility of implementing advanced manufacturing technology by utilizing the consistent fuzzy preference relations, Expert Systems with Applications 36 (2009) 4313–4320. 46. F. Chiclana, E. Herrera-Viedma, S. Alonso and F. Herrera, Cardinal consistency of reciprocal preference relation: A characterization of multiplicative transitivity, IEEE Transactions on Fuzzy Systems 17 (2009) 14–23. 47. M. M. Xia and Z. S. Xu, Methods for fuzzy complementary preference relations based on multiplicative, Computers & Industrial Engineering 61(4) (2011) 930–935.

76

H. Liao, Z. Xu & M. Xia

48. M. M. Xia and Z. S. Xu, Some issues on multiplicative consistency of interval reciprocal relations, International Journal of Information Technology & Decision Making 10 (2011) 1043–1065. 49. J. Montero, Rational aggregation rules, Fuzzy Sets and Systems 62 (1994) 267–276. 50. V. Cutello and J. Montero, Fuzzy rationality measures, Fuzzy Sets and Systems 62 (1994) 39–54. 51. J. M. Tapia García, M. J. del Moral, M. A. Martinez and E. Herrera-Viedma, A consensus model for group decision making problems with interval fuzzy preference relations, International Journal of Information Technology & Decision Making 11(4) (2012) 709–725. 52. F. J. Cabrerizo, S. Alonso and E. Herrera-Viedma, A consensus model for group decision making problems with unbalanced fuzzy linguistic information, International Journal of Information Technology & Decision Making 8 (2009) 109–131. 53. F. Mata, L. Martínez and E. Herrera-Viedma, An adaptive consensus support model for group decision making problems in a multi-granular fuzzy linguistic context, IEEE Transactions on Fuzzy Systems 17 (2009) 279–290. 54. Y. S. Wang, Assessment of learner satisfaction with asynchronous electronic learning systems, Information & Management 41 (2003) 75–86. 55. G. H. Tzeng, C. H. Chiang and C. W. Li, Evaluating intertwined e®ects in e-learning program: A novel hybrid MCDM model based on factor analysis and DEMATEL, Expert Systems with Applications 32 (2007) 1028–1044.