A new approach for group decision making method with hesitant fuzzy preference relations

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A new approach for group decision making method with hesitant fuzzy preference relations Fanyong Meng a,b, Qingxian An a,c,∗ a

School of Business, Central South University, Changsha, 410083, China. Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing 210044, China. c Industrial Systems Optimization Laboratory, Charles Delaunay Institute and UMR CNRS 6281, University of Technology of Troyes, Troyes, 10004, France. b

a r t i c l e

i n f o

Article history: Received 18 July 2016 Revised 25 November 2016 Accepted 13 March 2017 Available online xxx Keywords: Group decision making Hesitant fuzzy preference relation Multiplicative consistency Consensus Distance measure

a b s t r a c t This paper focuses on group decision making with hesitant fuzzy preference relations (HFPRs). To derive the consistent ranking order, a new multiplicative consistency concept for HFPRs is introduced that considers all information offered by the decision makers. The main feature is that this concept neither adds values to hesitant fuzzy elements nor disregards any information provided by the decision makers. To judge the multiplicative consistency of HFPRs, 0–1 mixed programming models are constructed. According to the assumption that there is an independent uniform distribution on values in hesitant fuzzy elements, the hesitant fuzzy priority weight vector is derived from multiplicative consistent reciprocal preference relations and their probabilities. Meanwhile, several consistency based 0–1 mixed models to estimate missing values in incomplete HFPRs are constructed that can address the situation where ignored objects exist. Considering the consensus in group decision making, a distance measure based consensus index is defined, and a method for improving the group consensus is provided to address the situation where the consensus requirement is unsatisfied. Then, a distance measure between any two HFPRs is introduced that is used to define the weights of the decision makers. Furthermore, a multiplicative consistency and consensus based interactive algorithm for group decision making with HFPRs is developed. Finally, a multi-criteria group decision making problem with HFPRs is offered to show the concrete application of the procedure, and comparison analysis is also made. © 2017 Published by Elsevier B.V.

1. Introduction With the constant complexity of socioeconomic decisionmaking problems, preference relations with precise judgments [34,37] are suffering more and more restrictions. To extend the application of preference relations, researchers introduced fuzzy sets given by Zadeh [55] to preference relations and proposed several types of fuzzy preference relations, such as interval fuzzy preference relations [38,50], triangular fuzzy preference relations [42,51], trapezoidal fuzzy preference relations [8] and linguistic fuzzy preference relations [21,52]. However, all these types of fuzzy preference relations only give the decision makers (DMs)’ preferred judgments. To both denote the preferred and non-preferred opinions on objects, Szmidt and Kacprzyk [39] introduced the concept of intuitionistic fuzzy preference relations, where the decision makers’ judgments are denoted by using intuitionistic fuzzy values [6]. Later, Xu [54] further provided interval-valued intuitionistic fuzzy preference relations that permit the DMs to apply intervals rather ∗

Corresponding author. E-mail address: [email protected] (Q. An).

than real numbers to express their preferred and non-preferred judgments. Taking the advantages of linguistic varibles and intuitionistic fuzzy values, Meng et al. [32] presented intuitionistic linguistic fuzzy preference relations (ILFPRs) and offered an additive consistency and consensus analysis based group decision making with ILFPRs. Recently, Torra [41] noted that there might be several values for a pair of compared objects and provided hesitant fuzzy sets (HFSs). To denote the hesitancy of the DMs, Xia and Xu [48] introduced the concept of hesitant fuzzy elements (HFEs), by which the authors provided hesitant fuzzy preference relations (HFPRs) that can be seen as an extension of reciprocal preference relations (RPRs) [34]. According to Tanino’s additive consistency concept, Zhu and Xu [64] applied the defined operations on HFEs to derive reduced RPRs that have the highest consistency level with respect to HFPRs. Considering weak consistency, the authors provided hesitant preference relations (HPRs) and hesitant reachability matrix (HRM), by which an algorithm is developed to derive the reduced RPRs. Using Tanino’s multiplicative consistency concept, Zhu et al. [65] proposed two procedures: one procedure builds programming models to derive the crisp priority weight vector (called the α -

http://dx.doi.org/10.1016/j.knosys.2017.03.010 0950-7051/© 2017 Published by Elsevier B.V.

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normalization based method), which is derived from one of RPRs in HFPRs; the other extends all HFEs to the length of the one with the most number of possible values (called the β -normalization based method). Then, the authors defined the consistent HFPRs by considering the consistency of the ordered RPRs. According to the defined distance measure, the authors studied the threshold of consistency. Finally, the authors used the aggregation operators in [48] to calculate the ranking HFEs and applied the score function [48] to derive the ranking orders of objects. As Meng et al. [29,31] noted, different HFEs are derived when we add values to HFEs. Thus, the Hamming distance in [48] needs to be further studied. Zhu et al. [65] did not explain why we should use the ordered RPRs obtained from HFPRs to judge the consistency. Just as the authors said, the α -normalization based method removes elements from HFEs, and the β -normalization based method adds values to HFEs. These two procedures both derive different HFPRs because the elements in HFEs are changed. This means that the priority weight vector is not obtained from the original HFPRs. Zhang et al. [56,57] adopted Tanino’s additive consistency concept and β -normalization based method to develop two methods for decision making with HFPRs. Xu et al. [49] developed two programming model based methods to calculate the crisp priority weight vector from incomplete HFPRs that are based on Tanino’s additive and multiplicative consistency concepts. Although these models are based on the consistency analysis, they fail to address inconsistent case. Considering the incomplete case, following the work of Zhu et al. [65], Zhang [61] introduced two methods to group decision making with incomplete HFPRs: a α -normalization based method and a β -normalization based method. Note that the β normalization based method requires all HFEs offered by the DMs to have the same length. From the above analysis, one can find that all previous researches about HFPRs are based on RPRs and use Tanino’s additive and multiplicative consistency concepts. The α -normalization based approaches [49,61,65] derive the crisp priority weight vector that cannot reflect the hesitancy of the DMs at all. Although the β -normalization based approaches [56,57,61,65] can obtain the hesitant fuzzy priority weight vector, they need add values to HFEs that derives different HFPRs with respect to the original ones. Furthermore, it is unsuitable to judge the consistency of HFPRs by only considering the ordered RPRs. To address these limitations, this paper defines a new multiplicative concept for HFPRs that uses Tanino’s multiplicative consistency concept. In contrast to the previous consistency concepts, the new concept need not add or remove values provided by the DMs. To judge the consistency of HFPRs, 0–1 mixed programming models are established. Based on the assumption of uniform distribution, the probability of each RPR is determined. Then, we can derive the hesitant fuzzy priority weight vector from multiplicative consistent RPRs. After that, 0–1 mixed programming models to determine the missing values in incomplete HFPRs are constructed. Considering group decision making with HFPRs, a consistency and consensus based interactive algorithm is performed. To do this, the rest is organized as follows: Section 2 reviews several related basic concepts, such as HFSs, HFPRs and the operational laws on HFEs. Then, it recalls several previous methods to derive the priority weight vector from HFPRs. Section 3 defines a new consistency concept and compares with several previous ones. Then, 0–1 mixed programming models are constructed to judge the consistency of HFPRs. Meanwhile, an algorithm to derive the hesitant fuzzy priority weight vector from HFPRs is provided. Subsequently, 0–1 mixed programing models to estimate missing values in incomplete HFPRs are established, which have the highest consistency level with respect to the known judgments. Section 4 offers a group consensus index using the defined distance measure. Then, a method to improve the group consensus is introduced. Based on the consistency and con-

sensus analysis, an interactive algorithm to group decision making with HFPRs is developed. Section 5 applies a practical example to show the concrete application of the new procedure. Meanwhile, comparison analysis with several previous methods is performed. Section 6 lists the final conclusions and future remarks. 2. Basic concepts Different from multiplicative preference relations, Orlovsky [34] proposed the concept of RPRs to give the relationship between each pair of objects. Definition 1. [34]. A RPR R on a set of objects X = {x1 , x2 , . . . , xn } is defined by R = (ri j )n×n such that ri j +r ji = 1 with rij ∈ [0, 1], where i, j = 1, 2, …, n. Considering the consistency of RPRs, Tanino [40] introduced the following multiplicative consistency concept. Definition 2. [40]. A RPR R = (ri j )n×n is multiplicatively consistent if the following is true:

ri j rki r jk = r ji rik rk j

(1)

for all i, k, j = 1, 2, ..., n. Property 1. A RPR R = (ri j )n×n is multiplicatively consistent if and only if ri j rki r jk = r ji rik rk j for all i, k, j = 1, 2, ..., n with i < k < j. A multiplicative consistent RPR R = (ri j )n×n can be denoted as follows:

ri j =

wi wi + w j

(2)

for all i, j = 1, 2, ..., n, where w = (w1 , w2 , …, wn ) such that wi ≥ 0  for all i = 1, 2, ..., n and ni=1 wi = 1 [49]. Property 2. Let R = (ri j )n×n be a multiplicative consistent RPR, and let w = (w1 , w2 , …, wn ) be a weight vector such that wi ≥ 0 for all  i = 1, 2, ..., n and ni=1 wi = 1. Then,

1

w i = n

1 j=1 ri j

(3)

−n

for all i = 1, 2, …, n. Proof. From formula (2), we have the following:

wi + w j wj 1 = =1+ . ri j wi wi Thus,

n

1 j=1 ri j

=n+

1 wi ,

by which we derive formula (3).

When a RPR R = (ri j )n×n is multiplicatively consistent, we can apply formula (3) to derive the priority weight vector. Note that when ri j = 0 for some j or ri j = 1 for all j, then formula (3) makes no sense. To address this issue, we can replace ri j = 0 with ri j = 0.0 0 01, ri j = 0.0 0 0 01, ..., and apply ri j = 0.9999, ri j = 0.99999, ... instead of ri j = 1.  Property 3. Let R = (ri j )n×n be a RPR, if its elements satisfy ri j =  n

 n n r r l=1 il l j for all i, k, j = 1, 2, ..., n with i < k < j, nl=1 ril rl j + n nl=1 (1−ril )(1−rl j )

then R is multiplicatively consistent. Proof. For

each

triple

of

 n n r r  l=1 il l j we have n n r r + n n (1−r )(1−r ) il l j il lj l=1 l=1

(i,

k,

j),

from

ri j =

 n

ri j rki r jk =

nl=1 ril rl j   n nl=1 ril rl j + n nl=1 (1 − ril )(1 − rl j )  n nl=1 rkl rli ×  n nl=1 rkl rli + n nl=1 (1 − rkl )(1 − rli )

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 n

nl=1 r jl rlk  n nl=1 r jl rlk + n nl=1 (1 − r jl )(1 − rlk )  n nl=1 r jl rli =   n nl=1 r jl rli + n nl=1 (1 − r jl )(1 − rl i )  n nl=1 ril rlk ×  n nl=1 ril rlk + n nl=1 (1 − ril )(1 − rlk )  n nl=1 rkl rl j ×  n nl=1 rkl rl j + n nl=1 (1 − rkl )(1 − rl j ) ×

= r ji rik rk j From Definition 2 and Property 1, we derive the conclusion. Because of the complexity of decision-making problems and the subjective hesitancy of the DMs, there might be several values for the comparison between each pair of objects. To address this situation, Torra [41] introduced the following concept of HFSs.  Definition 3. [41]. Let X = {x1 , x2 , …, xn } be a finite set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0,1]. To be understood easily, HFS E is expressed by a mathematical symbol:

E = (< xi , hE (xi ) > |xi IX ), where hE (xi ) is a set of several values in [0, 1] denoting the possible membership degrees of the element xi ∈ X to the set E. For convenience, Xia and Xu [48] called h = hE (xi ) a hesitant fuzzy element (HFE) that is a basic construction of HFS E. In the early period of HFSs, HFSs are mainly applied to denote the attribute values of alternatives. Recently, Xia and Xu [48] introduced HFEs to preference relations and proposed HFPRs as follows: Definition 4. [48]. A HFPR H on X is defined by H = (hi j )n×n , m

where hi j = (h1i j , h2i j , ..., hi j i j ) is a HFE denoting the possible preferred degrees of object xi over xj . Elements in H have the following characteristics:

⎧ hl < hl+1 , l = 1, 2, ..., mi j − 1 ⎪ ij ⎪ ⎨ il j m ji +1−l hi j + h ji

⎪ ⎪ ⎩mi j = m ji

= 1, l = 1, 2, ..., mi j

based on Tanino’s multiplicative consistency concept, while Zhang et al. [56] adopted Tanino’s additive consistency concept. However, the β -normalization based method has three common features: (i) It extends all HFEs to the length of HFEs that have the biggest number of elements; (ii) It uses the ordered RPRs to judge the consistency of associated HFPRs; (iii) Generally speaking, it is difficult or even impossible to determine the optimized parameter ϛ because we cannot discriminate the size of ϛ with respect to the known ones. For instance, let h1 = {0.2, 0.5, 0.7} and h2 = {0.3, 0.5, 0.7, 0.8} be two HFEs, then we need add one value to h1 to make them have the same length. According to the principle of the β -normalization based method, the added value can be denoted by v = 0.2ς + 0.7(1 − ς ). However, we cannot sure v ≥ 0.5 or v ≤ 0.5, which will influence the added value. According to model (29) in [65] and model (20) in [56], one can find that this issue will inevitably influence the added values as well as the final ranking results of objects. Let us consider the following HFPR H on X = {x1 , x2 , x3 }, where

⎛

{ 12 }

11 7 { 103 , 20 , 10 }

9 7 { 103 , 20 , 10 }

{ 12 }

{ 103 , 104 , 105 , 107 }

{ 102 , 103 , 106 , 108 }

⎜

H=⎝

hii = 0.5

for all i, j = 1, 2, …, n with i = j, and mij is the number of elements in hij . Definition 4 shows when there is one element in each HFE, then HFPR H degenerates to a RPR. To derive the priority weight vector from HFPRs, Zhu et al. [65] constructed a multiplicative consistency based hesitant goal programming model to derive the crisp priority weight vector, which obtains from one RPR with respect to the associated HFPR. It means that there exists information loss in Zhu et al.’s method [65]. Furthermore, Xu et al. [49] applied Zhu et al.’s hesitant goal programming model to develop a method to address incomplete HFPRs. There are two limitations in such procedure: (i) It cannot denote the hesitancy of the MDs’ opinions; and (ii) it fails to address the inconsistency case. Sometimes, it will be inevitable to adjust the DMs’ original preferences to derive a consistent ranking order. In contrast to the above procedure, Zhu et al. [65] further developed a β -normalization based method to calculate the hesitant fuzzy priority weight vector that needs to add values in HFEs. To do this, the authors built a programming model to determine the optimized parameter ϛ (0 ≤ ϛ ≤ 1). Zhang et al. [56] applied the same principle to provide a group decision making method with HFPRs. Their difference is that Zhu et al.’s method [65] is

⎞ { 103 , 105 , 106 , 107 } ⎟ { 102 , 104 , 107 , 108 }⎠. { 12 }

According to the β -normalization based method in [65], we need to add a value v = 0.3ς + 0.7(1 − ς ) to h12 . When we apply model (29) in [65] to determine the optimized parameter ϛ, 3 11 the range scope of v should be first determined. When v ∈ [ 10 , 20 ], according to Algorithm II and the hesitant fuzzy geometric (HFG) operator [65] the scores are S(x1 ) = 0.4856, S(x2 ) = 0.4942, and S(x3 ) = 0.4704, by which the ranking order is x2 x1 x3 . When we suppose v ∈ [0.55, 0.7], the scores are S(x1 ) = 0.4931, S(x2 ) = 0.4856, S(x3 ) = 0.4725, and the ranking order is x1 x2 x3 . Because the associated NHFPRs are both acceptably consistent, it is difficult to decide which one should be chosen. One can check that the same issue exists in Zhang et al.’s method [56]. Next, let us apply a numerical example to show that these two types of procedures are insufficient to address HFPRs. Example 1. Let X = {x1 , x2 , x3 , x4 }. Suppose that HFPR H is defined as follows:

⎛

(4)

3

⎞ 4 1 1 7 { 25 , 5 , 3 , 9 } { 10 , 2 , 27 , 27 } 17 3 35 31 ⎜ {2, 2, 7} { 71 , 14 , 47 , 78 } { 139 , 45 , 119 , 10 }⎟ ⎜ 5 3 8 11 ⎟ H =⎜ ⎟. 21 2 ⎝ { 23 , 45 , 25 , 9} { 18 , 37 , 34 , 67 } { 21 } { 114 , 15 , 27 } ⎠ 17 29 8 1 7 1 2 1 4 2 2 7 { 314 , 35 , 3 , 17 } { 11 , 11 , 5 , 13 } { 29 , 17 , 11 ,} { 12 } { 21 }

{ 81 , 13 , 35 } { 12 }

Using model (15) in [65] or model (M-4) in [49], we derive the following crisp priority weight vector ωXZ = (0.3528, 0.0228, 0.0345 ⎛ , 0.5899 ), which⎞ is obtained from con1 2

sistent RPR RXZ

1 8 1 2 3 7 1 11

⎜ 78 = ⎜ 21 ⎝ 25 7 17

4 25 4 7 1 2 2 17

10 17 10 11 15 17 1 2

⎟ ⎟. According to ωXZ , the ⎠

ranking order is x4 x1 x3 x2 . Using model (M-8) in [49], we get the crisp priority weight vector ωXu = .3712, 0.6036 ) that is derived from ⎛(0.0 0 0 0, 0.0252, 0⎞

⎜ ⎝

RPR RXu = ⎜

1 2 2 5 2 9 1 3

3 5 1 2 1 8 4 13

7 9 7 8 1 2 2 29

2 3 9 13 27 29 1 2

⎟ ⎟. According to ωXu , we obtain ⎠

x4 x3 x2 x1 . One can check that RXu is inconsistent for r12 = r13 − r23 + 12 . The ranking values and the associated ranking orders obtained from these two methods can be seen in Table 1. These two models can be both considered as the α normalization based method. Next, let us apply two β -

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F. Meng, Q. An / Knowledge-Based Systems 000 (2017) 1–15 Table 1 The score values and the ranking results using methods in [49,65]. Methods

S(x1 )

S(x2 )

S(x3 )

S(x4 )

The ranking order

Model (15) in [65] or model (M-4) in [49]: 1 Model (M-8) in [49]: 2

0.3528 0.0 0 0 0

0.0228 0.0252

0.0345 0.3712

0.5899 0.6036

x4 x1 x3 x2 x4 x3 x2 x1

Table 2 The score values and the ranking results using Algorithm II in [65].

Using Using Using Using

the the the the

hesitant hesitant hesitant hesitant

fuzzy fuzzy fuzzy fuzzy

averaging (HFA) operator for cases (i) and (iii): 3 averaging (HFA) operator for case (v): 4 geometric (HFG) operator for cases (i) and (iii): 5 geometric (HFG) operator for case (v): 6

S(x1 )

S(x2 )

S(x3 )

S(x4 )

The ranking order

0.5405 0.5269 0.4502 0.4363

0.6374 0.6519 0.5508 0.5648

0.6656 0.6413 0.5570 0.5442

0.3198 0.3355 0.2536 0.2717

x3 x2 x1 x4 x2 x3 x1 x4 x3 x2 x1 x4 x2 x3 x1 x4

Table 3 The score values and the ranking results using method in [56].

In cases (i) and (iii): 7 In case (v): 8

S(x1 )

S(x2 )

S(x3 )

S(x4 )

The ranking order

0.4991 0.4850

0.5901 0.6042

0.6093 0.5924

0.3015 0.3183

x3 x2 x1 x4 x2 x3 x1 x4

normalization based methods [56,65] to show the ranking order of the objects listed in Example 1. When the multiplicative consistency and β -normalization based method in [65] is applied in Example 1, there are four cases to add values to HFEs h12 and h34 :

 3  2 h = 18 ς + 35 (1 − ς ) h12 = 18 ς + 35 (1 − ς ) , ii , (i ) 12 ( ) 4 27 4 27 3 h34 = 11 ς + 29 (1 − ς ) h334 = 11 ς + 29 (1 − ς )  3  2 h12 = 18 ς + 35 (1 − ς ) h12 = 18 ς + 35 (1 − ς ) and v (iii ) 2 ( ) 4 27 4 27 h34 = 11 ς + 29 (1 − ς ) h234 = 11 ς + 29 (1 − ς )



For each case, using model (29) in [65] we have the following:

min CI (H1 ) = min CI (H3 ) = 0.0864 and ς1 = ς3 = 0.0858



min CI (H4 ) = 0.0989 ς4 = 0.5614

where Hi , i = 1, 3, 4, correspond to one of the above associated cases, respectively. Note that there is no feasible solution for the second case. According to the given consistency threshold in [65], we know that HFPR H is acceptably consistent. Using Algorithm II in [65], the score values and the ranking results are obtained as shown in Table 2. Table 2 clearly shows that different ranking orders are derived with respect to the HFA operator or the HFG operator. When the additive consistency and β -normalization based method in [56] is adopted in Example 1, there are also four cases as shown above. With respect to each case, using model (20) in [56], the objective function values and the optimized parameters are derived as follows:







minD H1 , H˜ 1 = 0.2040 , ς = 0.0858



and











minD H3 , H˜ 3 = 0.2407 ς = 0.0858

minD H4 , H˜ 4 = 0.1919 ς = 0.5614

where Hi , i = 1, 3, 4, correspond to one of the above associated cases, respectively . It is interesting that the same values of the optimized parameter ϛ are obtained according to these two β normalization based methods. It needs to be noted that there is no feasible solution for the second case too. Let α = 0.9 be the consistency threshold, one can check that the first RPR H(1) and the fourth RPR H(4) are unacceptably consistent. Based on the acceptable consistency analysis, the score values and the ranking results are derived as listed in Table 3.

Table 3 indicates that different ranking orders are derived according to the β -normalization based method in [56]. From Tables 1–3, one can see that the ranking orders are different that are obtained from the α -normalization and β -normalization based methods. Note that it is unsuitable to disregard information or add values because these two processes change the original information offered by the DMs. 3. An approach to derive the hesitant fuzzy priority weight vector from HFPRs In contrast to the previous methods for decision making with HFPRs, this section provides a new approach for decision making with HFPRs. To do this, this section includes three subsections. SubSection 3.1 introduces a new multiplicative consistency concept for HFPRs. SubSection 3.2 constructs several 0–1 mixed programming models to judge the consistency of HFPRs, by which the associated RPRs and their probabilities are derived. Meanwhile, an approach for decision making with HFPRs is offered. SubSection 3.3 focuses on incomplete HFPRs. To determine the missing values, several consistency based 0–1 mixed programming models are established that address the situation where ignored objects exist. 3.1. A new multiplicative consistency concept Considering the current consistency concepts for HFPRs that disregard some information in HFEs or add information to HFEs, the section gives a new multiplicative consistency concept, which fully considers the DMs’ preferences and needn’t change the HFEs offered by the DMs. As we know, the reason that the DMs apply HFEs to denote their judgments rather than real or fuzzy numbers is that there are several preferred values between the comparison objects. Thus, it is unsuitable to use the β -normalization based method that restricts to consider the consistency of the ordered RPRs. Let us fur ther consider Example 1, when we take

 more reasonable to take

h323 = h324 =

4 7 10 11

 than

h112 = h114

h123 = h124 =

=

1 7 9 13

1 4 1 8 , h13 = 25 10 15 1 17 , h34 = 17

, it is

. Because the RPR

composed by the former values is completely multiplicative consistent, while the latter cannot derive this conclusion. This concretely shows that it is unreasonable to only consider the consistency of ordered RPRs. In crisp RPRs, each element is chosen to judge the

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multiplicative consistency, and they have the same influence on the consistency. Following this point of view, we present the following multiplicative consistency concept for HFPRs. Definition 5. Let H be a HFPR on X. H is called hij -multiplicative consistency if, for any hli j ∈ hi j , l = 1, 2, …, mij , there is a multiplicative consistent RPR R = (ri j )n×n such that ri j = hli j .

Following Definition 5, one can check that the RPR R = (ri j )n×n is multiplicatively consistent if and only if it is rij -multiplicative consistent for any i, j = 1, 2, …, n with i = j. Following this point of view, we introduce the following concept for HFPRs Definition 6. Let H be a HFPR on X. H is multiplicatively consistent if H is hij -multiplicative consistent for all i, j = 1, 2, …, n with i = j. Definition 6 is an extension of the multiplicative consistency concept for RPRs. Note that the HFPR H given in Example 1 is multiplicatively consistent according to Definition 6. There are several strengths of Definition 6: (i) It defines the multiplicative consistency of HFPRs using every element provided by the DMs; (ii) It does not disregard any value in HFEs as the α -normalization based method or add values to HFEs as the β -normalization based method; (iii) It gives the multiplicative consistency of HFPRs according to the derived RPRs with respect to each fixed element, while the α -normalization based method only uses one RPR, and the β -normalization based method restricts to consider the consistency of ordered RPRs. Property 4. Let H be a HFPR on X. H is hesitant multiplicative consistent if and only if H is hij -hesitant multiplicative consistent for all i, j = 1, 2, …, n with i < j. Next, using the new consistency concept for HFPRs, we construct models to judge the multiplicative consistency of HFPRs and to determine the missing values in incomplete HFPRs.

5

judge the multiplicative consistency of H:

   ϕil0 j0 ∗ = min i, j=1,2,...,n,i= j dil,j+ + dil,j− ⎧ α l  l αkil  α l  m  α l m  m  ki ⎪ l=1i j hli j i j × m hki × l=1jk hljk jk − l=1ji hlji ji × ⎪ l=1 ⎪  ⎪  l αikl ⎪ mk j  l αkl j ⎪ ik ⎪m h ×  h − dil,j+ + dil,j− = 0, i, k, ⎪ ik kj l=1 l=1 ⎪ ⎪ ⎪ j = 1, 2, ..., n, i = j = k ⎪ ⎪ ⎨ l l mi j  l αi j mi j  l αi j hi j + l=1 hi j = 1, i, j = 1, 2, ..., n, i = j s.t. l=1   ⎪ mi j m ji l l ⎪ ⎪ α = α = 1 , i, j = 1, 2, ..., n, i = j ⎪ l=1 i j l=1 ji ⎪ ⎪ l ⎪ α = 0 ∨ 1 , i, j = 1 , 2 , ..., n, i = j, l = 1, 2, ..., mi j ⎪ ij ⎪ ⎪ l,+ l,− ⎪ d , d ≥ 0, i, j = 1, 2, ..., n, i = j ⎪ ⎪ ⎩ ilj i j αi0 j0 = 1 (6) for each pair of (i0 , j0 ) with i0 = j0 and each l = 1, 2, …, mi0 j0 . The first constraint is derived from formula (5), and the second constraint guarantees the reciprocity of elements in RPRs. With respect to each l = 1, 2, …, mi0 j0 , take the logarithm for formula (5). Similar to model (6), we derive the following model to judge the multiplicative consistency of H:

   Jil0 j0 ∗ = min i, j=1,2,...,n,i= j dil,j+ + dil,j−    ki l   m   ⎧mi j l α log hli j + m α log hlki + l=1jk α ljk log hljk ⎪ l=1 i j l=1 ki ⎪   ⎪ ⎪− m ji α l log hl ⎪ ⎪ l=1 ji ⎪  lji  mk j l   mik l ⎪ ⎪ − α log hik − l=1 αk j log hlk j − dil,j+ + dil,j− = 0, ⎪ l=1 ik ⎪

⎪ ⎪ ⎪ i, k, j = 1, 2, ..., n, i = j = k ⎪ ⎨  ,  α l  α l  s.t. log mi j hli j i j + m ji hlji ji = 0, i, j = 1, 2, ..., n, i = j l=1 l=1 ⎪  ij l  ji l ⎪ ⎪ ⎪ m α = m α =1 ⎪ l=1 i j l=1 ji ⎪ ⎪ ⎪α l = 0 ∨ 1, i, j = 1, 2, ..., n, i = j, l = 1, 2, ..., mi j ⎪ ij ⎪ ⎪ ⎪ ⎪ dil,j+ , dil,j− ≥ 0, i, j = 1, 2, ..., n, i = j ⎪ ⎪ ⎩ l αi0 j0 = 1 (7)

3.2. Models to judge the multiplicative consistency of HFPRs Generally speaking, it is difficult to directly apply Definition 6 to judge the multiplicative consistency of HFPRs because it needs to judge the consistency of ni, j=1,i< j mi j different RPRs. To address this issue, we use 0–1 indicator variables to construct 0–1 mixed programming models to judge the multiplicative consistency of  HFPRs. Let H be a HFPR, and let αil j =

1 if hli j ∈ hi j is chosen be a 0–1 0 otherwise

indicator variable for element hli j in HFE hij , where i, j = 1, 2, …, n with i = j, and l = 1, 2, …, mij . Then, each element in hij can be dem i j l mi j αl noted by l=1 (hli j ) i j with l=1 αi j = 1. When H is multiplicatively consistent, from Definition 6 we derive the following:

α l  l αkil α l m  m  ki l=1i j hli j i j × m hki × l=1jk hljk jk l=1   α l  l αikl mk j  l αkl j m  mik = l=1ji hlji ji × l=1 hik ×l=1 hk j , where

m i j

m +1−l α ji ji

l=1

(5)

αil j = 1 for all i, j = 1, 2, …, n with i = j, and αil j =

= 1 for each l = 1, 2, …, mij . When we cannot guarantee the consistency of the HFPR H, formula (5) will not necessarily hold. In this case, we introduce the deviation values dil,j+ and dil,j− such that dil,j+ , dil,j− ≥ 0 into formula (5), and construct the following 0–1 mixed programming model to

where i0 , j0 = 1, 2, …, n with i0 = j0 . Solving model (7), when Jil j ∗ = 0 for each pair of (i0 , j0 ) with 0 0

i0 = j0 and each l = 1, 2, …, mi0 j0 , then H is multiplicatively consistent; otherwise, it is inconsistent. However, the derived RPR has the highest consistent level for the element hli j . 0 0

By symmetry, model (7) can be further equivalently converted to the following model:

   φil0 j0 ∗ = min i, j=1,2,...,n,i< j dil,j+ + dil,j−    ki l   m   ⎧mi j l α log hli j + m α log hlki + l=1jk α ljk log hljk ⎪ l=1 i j l=1 ki ⎪   ⎪ m ⎪ ⎪ − l=1ji α lji log hlji ⎪ ⎪   m   ⎪ mik l ⎪ ⎪ − l=1 αik log hlik − l=1k j αkl j log hlk j − dil,j+ + dil,j− = 0 ⎪ ⎪ ⎪ ⎪ ⎪ i, k, j = 1, 2, ..., n, i < k < j ⎪ ⎨   α l  α l  s.t. log mi j hl i j + m ji hl ji = 0, i, j = 1, 2, ..., n, i = j ij ji l=1 l=1 ⎪ ⎪ m i j l m ji l ⎪ ⎪ α = α = 1 , i, j = 1, 2, ..., n, i < j ⎪ i j ji l=1 l=1 ⎪ ⎪ ⎪ l ⎪ α = 0 ∨ 1 , i, j = 1 , 2 , ..., n, i = j, l = 1, 2, ..., mi j ⎪ ij ⎪ ⎪ ⎪ l, + l, − ⎪ d , d ≥ 0, i, j = 1, 2, ..., n, i < j ⎪ ⎪ ⎩ ilj i j αi0 j0 = 1 (8) for each pair of (i0 , j0 ) with i0 < j0 and each l = 1, 2, …, mi0 j0 .

Please cite this article as: F. Meng, Q. An, A new approach for group decision making method with hesitant fuzzy preference relations, Knowledge-Based Systems (2017), http://dx.doi.org/10.1016/j.knosys.2017.03.010

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F. Meng, Q. An / Knowledge-Based Systems 000 (2017) 1–15 Table 4 Multiplicative consistent RPR and its probability for each element. Multiplicative consistent RPRs

⎧ 1 h = 1 , h1 = ⎪ ⎨ 12 8 13 h114

⎪ ⎩

h424

=

10 , h323 17

=

10 , h234 11

⎛

4 25

=

4 7

=

15 17

⎧ 2 h = 1 , h4 = ⎪ ⎨ 12 3 13

7 9

=

7 8

h224 =

4 , h134 5

=

4 11

h414 =

⎪ ⎩

h324

=

27 , h223 31

=

h123 =

⎪ ⎩

h334

=

1 4

1 , h124 7

27 29

=

9 13

1/2

4/7

3/7

1/2

1/11

2/17

3/7

1/2

⎟ ⎟ ⎟ 4/11⎠

1/3

1/5

7/11

1/2

4/5 ⎟

⎞

1/3

1/2

1/4

3/4

1/2

⎟ ⎟ ⎟ 27/29⎠

4/31

2/11

2/29

1/2

1/2

3/5

1/5

27/35

⎜2/5 Rp = ⎜ ⎜4/5 ⎝

1/2

1/7

6/7

1/2

⎟ ⎟ ⎟ 27/29⎠

4/13

2/29

1/2

8/35

27/31

9/11 ⎟

9/13 ⎟

5 22

Θ1

Θ2

Θ3

Θ4

Θ5

Θ6

Θ7

Θ8

Our method

0.7 0.6 0.5 0.4 0.3

0

(9)

φil j ∗ = 0, skip to Step 3; otherwise, turn to next step;

Step 2: Use Property 3 to derive the associated multiplicative consistent RPRs. If there is no fear of confusion, we still apply Rp to denote it; Step 3: With respect to each multiplicative consistent RPR R p = p (ri j )n×n , we apply formula (3) to calculate the associated priority  p p p weight vector W p = (w1 , w1 , ..., wn ), p = 1, 2, ..., ni, j=1,i< j mi j ; Step 4: According to the proportion of each RPR Rp , we derive the hesitant fuzzy priority weight vectorW = (w1 , w2 , ..., wn ),  p where wi = {(wi , ρ (R p )), p ∈ {1, 2, ..., ni, j=1,i< j mi j }} is the hesitant fuzzy priority weight of object xi , i = 1, 2, …, n, and ρ (R p ) = is the probability of RPR Rp .

p = 5, 10, 12, 16, 22

0.1

for each pair of (i0 , j0 ) with i0 < j0 and each l = 1, 2, …, mi0 j0 . Solving model (9) with respect to each element in hij for all i, j = 1, 2, …, n with i < j, then the associated RPRs are derived. When the probability distribution on elements in HFEs is not provided by the DMs, it is reasonable to assume that all of them have the same probability, namely, there is a uniform distribution. From this point of view, we can conclude that the probability of each obtained RPR is equal to its proportion. Based on the above analysis, we introduce the following algorithm to derive the hesitant fuzzy priority weight vector from HFPRs: Step 1: With respect to HFPR H, use model (9) to derive RPRs,  p denoted by R p = (ri j )n×n , where p = 1, 2, ..., ni, j=1,i< j mi j . When

mi j

5 22

0.2

⎪ α = 1, i, j = 1, 2, ..., n, i < j ⎪ l=1 ⎪ ⎪ ⎪ l ⎪ αi j = 0 ∨ 1, i, j = 1, 2, ..., n, i < j, l = 1, 2, ..., mi j ⎪ ⎪ ⎪ ⎪ ⎪ dl,+ , dl,− ≥ 0, i, j = 1, 2, ..., n, i < j ⎪ ⎪ ⎩ ilj i j αi0 j0 = 1

i, j=1,i< j

p = 3, 6, 11, 13, 18

⎞

x1

x2

x3

x4

Objects

l ij

| p ={ p1 ,..., pl |R pu =R p ,u=1,...,l }|

6 22

p = 2, 7, 9, 15, 17, 20

3/5

   φil0 j0 ∗ = min i, j=1,2,...,n,i< j dil,j+ + dil,j−      ik l ⎧mi j l αi j log hli j + m αik log 1 − hlik ⎪ l=1 l=1 ⎪   m i j l   ⎪ m k j l ⎪ ⎪ + l=1 αk j log 1 − hlk j − l=1 αi j log 1 − hli j ⎪ ⎪   m   ⎪ mik l ⎪ ⎪ − l=1 αik log hlik − l=1k j αkl j log hlk j − dil,j+ ⎪ ⎪ ⎪ ⎨+dl,− = 0, i, k, j = 1, 2, ..., n, i < k < j

n

2/3

⎞

7/8

⎜

6 22

p = 1, 4, 8, 14, 19, 21

1/2

1/2

According to the construction of elements in RPRs, model (8) can be further transformed to the following model:

s.t. mi j ij

⎟ ⎟ ⎟ 5/17 ⎠ 10/11⎟

⎜2/3 Rp = ⎜ ⎜2/9 ⎝

⎜ ⎜2/5 Rp = ⎜ ⎜2/3 ⎝

Probabilities

⎞

10/17

7/9

⎛ 27 35

4/25

1/3

1/2

1 3

1/8

1/2

⎛

9 11

⎧ 2 h = 1 , h3 = ⎪ ⎨ 13 5 14

7/17

⎜

2 , h423 3

⎧ 3 h = 3 , h3 = ⎪ ⎨ 12 5 13

⎜

⎜7/8 Rp = ⎜ ⎜21/25 ⎝ ⎛

h214 =

⎪ ⎩

1/2

The ranking values

Elements

Fig. 1. The ranking values with respect to different methods.

Step 5: Calculate the scores of objects xi , i = 1, 2, …, n, we use the following formula

S ( xi ) =



wip ∈wi

ρ (R p )wip.

(10)

Step 6: According to S(xi ), i = 1, 2, …, n, we rank objects x1 , x2 , …, xn . In Example 1, when we apply the above Algorithm to rank objects x1 , x2 , x3 and x4 . One can check that HFPR H is multiplicatively consistent. With respect to each element, associated multiplicative consistent RPR and its probability are obtained as listed in Table 4. According to Table 4, the hesitant fuzzy priority weight vector is derived as follows:

W =





 5 5 (0.0717, 113 ), (0.2642, 113 ), (0.2621, 22 ), (0.1677, 22 ) ,   5 5 (0.5018, 113 ), (0.5283, 113 ), (0.1748, 22 ), (0.1118, 22 ) ,

 5 5 (0.3763, 113 ), (0.1319, 113 ), (0.5243, 22 ), (0.6708, 22 ) ,   5 5 (0.0502, 113 ), (0.1321, 113 ), (0.0388, 22 ), (0.0497, 22 ) .

Using formula (10), the probability scores are S(x1 ) = 0.1893, S(x2 ) = 0.3461,S(x3 ) = 0.4102,S(x4 ) = 0.0698, by which the ranking order is x3 x2 x1 x4 . With respect to different methods, the ranking values are intuitionally shown in Fig. 1, by which we find that the same ranking order is obtained from our method and the β -normalization based methods [56,65] for the case (i) and (iii). However, their ranking values are different.

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3.3. Models to determine missing values in incomplete HFPRs

the missing values:

Because of various types of reasons, preference relations offered by the DMs might be incomplete [1–3,9,12,22,23,25-27,30,53]. With respect to incomplete HFPRs, Zhang et al. [57] developed a β normalization based method to derive missing values that uses Tanino’s additive consistency concept; Zhu et al. [65] constructed several models to address incomplete HFPRs that disregard missing value and only apply the known values to calculate the priority weight vector. Furthermore, Zhang [61] developed two multiplicative consistency based group decision making methods with incomplete HFPRs that are similar to methods in [57,65]. Different from previous approaches to incomplete HFPRs, this subsection establishes several 0–1 mixed programming models to estimate the missing values in complete HFPRs that are based on the multiplicative consistency analysis. Definition 7. Let H be a HFPR on X. H is called an incomplete HFPR when there are unknown values in H. Let H be an incomplete HFPR, and let U = {hij is an unknown HFE, where i, j = 1, 2, …, n with i < j}. When there are HFEs for all unknown HFEs in H that make the incomplete HFPR H be multiplicatively consistent. According to Property 1, for any fixed hli j , i, j = 1, 2, …, n, l = 1, 2, …, mij , we have the following:



mi j l=1





 l αil j n hi j

×

n   k=1

m

 αlji n

= l=1ji hlji

×

m jk l=1



n  



hljk

αljk



mki l=1

 l αkil  hki

 l αikl mk j  l αkl j  ik m hik l=1 hk j , l=1

(11)

k=1

where

m i j

m +1−l α ji ji

l=1

αil j = 1 for all i, j = 1, 2, …, n with i = j, and αil j =

= 1 for each l = 1, 2, …, mij . According to formula (11), we obtain the following:  mi j n m jk mki l l l n α log(hli j ) + α log(hljk ) + α log(hlki ) i j jk ki l=1 k=1 l=1 l=1  m  n mk j mik ji l − n α ji log(hlji )+ αkl j log(hlk j )+ αikl log(hlik ) k=1

l=1

l=1

l=1

(12)

= 0.

   ψil∗0 j0 = min ni, j=1,i= j εil,j− + εil,j+ m    m k j l ⎧ (n − 2 ) l=1i j αil j log(hli j ) + nk=1,k=i, j α log(1 − hlk j ) ⎪ l=1 k j  ⎪ mik l ⎪ ⎪ + l=1 αik log(1 − hlik ) − ⎪ ⎪  m    m k j l ⎪ ⎪ ⎪ (n−2 ) l=1i j αil j log(1−hli j ) + nk=1,k=i, j α log(hlk j ) ⎪ l=1 k j ⎪  mik l ⎪ ⎪ + l=1 αik log(hlik ) + εil,j− − εil,j+ = 0, ⎪ ⎪ ⎪ ⎪ ⎨i, j = 1, 2, ..., n, i = j , s.t. mi j

α l = 1, i, j = 1, 2, ..., n, i = j ⎪ l=1 i j ⎪ ⎪ ⎪ ⎪ αil j = 1 ∨ 0, i, j = 1, 2, ..., n, i = j, l = 1, 2, ..., mi j ⎪ ⎪ ⎪ ⎪ ⎪ hli j ∈ (0, 1 ), hli j ∈ hi j ∈ U ⎪ ⎪ ⎪ ⎪ ⎪ ε l,− , ε l,+ ≥ 0, i, j = 1, 2, ..., n ⎪ ⎪ ⎩ ilj i j αi0 j0 = 1 (15)

Similar to model (9), we can only determine the missing values in the upper triangular part to derive the complete HFPRs. Thus, we further construct the following 0–1 mixed programming model:

ϑil∗0 j0 = min



n

i, j=1,i< j

 n m ij kj l l (n − 2 ) α log ( h ) + α l log(1 − hlk j ) ij l=1 i j k=1,k=i, j l=1 k j   m  mik ij l l + α log(1 − hlik ) − (n − 2 ) α log(1 − hli j ) i j ik l=1 l=1 m  n mik kj l l l l + α log ( h ) + α log ( h ) = 0. kj ik k=1,k=i, j l=1 k j l=1 ik (13) However, when the incomplete HFPR H is inconsistent, formula (13) will not hold. In this case, we introduce the deviation variables εil,j− and εil,j+ to formula (13), where εil,j− , εil,j+ ≥ 0, i, j = 1, 2, …, n, l = 1, 2, …, mij . Then, we have:

 n m kj l α log(hli j ) + α l log(1 − hlk j ) i j l=1 k=1,k=i, j l=1 k j   m  mik ij l l + α log(1 − hlik ) − (n − 2 ) α log(1 − hli j ) i j ik l=1 l=1 m  n mik kj l l l l + α log ( h ) + α log ( h ) kj ik k=1,k=i, j l=1 k j l=1 ik

+ε

l,− ij

−ε

l,+ ij

ij

= 0.

(14)

Because the higher consistency, the better it is. For any known HFE hi0 j0 , i0 , j0 = 1, 2, …, n, we fix hli j , l = 1, 2, …, mi0 j0 and con0 0



s.t. mi j

α l = 1, i, j = 1, 2, ..., n, i < j ⎪ l=1 i j ⎪ ⎪ ⎪ l ⎪ αi j = 1 ∨ 0, i, j = 1, 2, ..., n, l = 1, 2, ..., mi j , i < j ⎪ ⎪ ⎪ ⎪ ⎪hli j ∈ (0, 1 ), hli j ∈ hi j ∈ U, i < j ⎪ ⎪ ⎪ ⎪ ⎪εil,j− , εil,j+ ≥ 0, i, j = 1, 2, ..., n, i < j ⎪ ⎪ ⎩ l αi0 j0 = 1

.

(16)

m

m

εil,j− + εil,j+

m    m k j l ⎧ (n − 2 ) l=1i j αil j log(hli j ) + nk=1,k=i, j α log(1 − hlk j ) ⎪ l=1 k j  ⎪  mik ⎪ l l ⎪ + l=1 αik log(1 − hik ) − ⎪ ⎪  m k j l ⎪ ⎪(n − 2 )mi j α l log(1−hl )+ n ⎪ α log(hlk j ) ⎪ k=1,k=i, j ij l=1 i j  l=1 k j ⎪  ⎪ m l, − l, + l l ik ⎪ + l=1 αik log(hik ) + εi j − εi j = 0, ⎪ ⎪ ⎪ ⎪ ⎨i, j = 1, 2, ..., n, i < j

From formula (12), we get the following:

(n − 2 )

7

struct the following 0–1 mixed programming model to determine

With respect to the first constraint in model (16), we have the following:

 n m kj l α log(hli j ) + α l log(1 − hlk j ) i j l=1 k=1,k=i, j l=1 k j     mik m i j l l l l + α log ( 1 − h ) − ( n − 2 ) α log ( 1 − h ) ij ik l=1 ik l=1 i j m  n mik kj + α l log(hlk j )+ l=1 αikl log(hlik ) +εil,j− −εil,j+ k=1,k=i, j l=1 k j

(n − 2 )

m

ij

 i−1  j−1  n αil j log(hli j ) + + + k=1 k=i+1 k= j+1  m  mik kj l l l l α log ( 1 − h ) + α log ( 1 − h ) kj ik l=1 k j l=1 ik = (n − 2 )



m

ij

l=1

 i−1  j−1  n l α log(1−hli j ) + + + i j l=1 k=1 k=i+1 k= j+1 m  mik kj l,− l,+ l l l l × α log(hk j ) + l=1 αik log(hik ) + εi j − εi j l=1 k j − (n−2 )

m

ij

 i−1 m kj l α log(hli j ) + α l log(1 − hlk j ) i j l=1 k=1 l=1 k j   mki  j−1 m k j + α l log(hlki ) + k=i+1 α l log(1 − hlk j ) l=1 ki l=1 k j  n  m mik jk l + α log(1 − hlik ) + α l log(hljk ) ik l=1 k= j+1 l=1 jk

= (n − 2 )

m

ij

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  m  ij l l α log(1 − hlik ) − (n − 2 ) α log(1 − hli j ) i j ik l=1 l=1  i−1 mk j mki l l l l + α log ( h ) + α log ( 1 − h ) kj ki k=1 l=1 k j l=1 ki + +

mik

 j−1

m

k=i+1

n

kj

l=1

m

k= j+1

αkl j log(hlk j ) +

mik l=1

α l log(1 − hljk ) + l=1 jk jk

αikl log(hlik )

Method in [54] Model (17) in [58]



mik

α l log(hlik ) l=1 ik

+εil,j− − εil,j+ = 0



2.5 2 1.5 1

for each pair of (i, j) with i < j. Formula (17) shows that we can only apply the upper triangular elements to determine the missing values. Because when all HFEs in the ith row and in the jth column are all known, the first constraint in model (17) is a constant, which does not influence determining missing values. Thus, we further introduce 0–1 indicator variable δi j = 0 otherwise

Model (M-8) in [46] Our method

0.5

(17)

1 hik ∨ hk j ∈ U for any k= 1, 2,...,n

Model (M-4) in [46] Algorithm 3 in [58]

3

The ranking values

+



[m5G;March 22, 2017;10:2]

for all i, j = 1, 2, …, n with i < j.

Then,

0 x1

x2

x3

x4

Objects

Fig. 2. Ranking values and orders with respect to different methods.

hesitant fuzzy priority weight vector is derived as follows:

W = ({(0.0909,

5 21

4 1 5 ), (0.1667, 21 ), (0.1667, 21 ), (0.1887, 21 ),

1 2 1 1 (0.0774, 21 ), (0.2086, 21 ), (0.3066, 21 ), (0.2544, 21 )},

{(0.1818, 215 ), (0.1667, 214 ), (0.1435, 211 ), (0.3775, 215 ), 1 2 1 1 (0.1547, 21 ), (0.4171, 21 ), (0.3489, 21 ), (0.5088, 21 )}, 5 4 1 5 {(0.3636, 21 ), (0.3333, 21 ), (0.3544, 21 ), (0.1559, 21 ), 1 2 1 1 (0.4023, 21 ), (0.1741, 21 ), (0.1760, 21 ), (0.1770, 21 )}, 5 4 1 {(0.3636, 21 ), (0.3333, 21 ), (0.3347, 21 ), (0.2779, 215 ), 1 2 1 1 (0.3657, 21 ), (0.2002, 21 ), (0.1686, 21 ), (0.0599, 21 )} )

   θil∗0 j0 = min ni, j=1,i< j εil,j− + εil,j+ ⎧   i j l    m k j l δi j (n − 2 ) m α log(hli j ) + ik−1 α log(1 − hlk j ) ⎪ =1 l=1 i j l=1 k j ⎪ ⎪   ⎪ m ki ⎪ + l=1 αkil log(hlki ) ⎪ ⎪ Thus, the probability scores are S(x1 ) = 0.1687, S(x2 ) = ⎪   j−1 mk j l  ik l ⎪ ⎪ ⎪ + k=i+1 αk j log(1 − hlk j ) + m αik log(1 − hlik ) 0.2839,S(x3 ) = 0.2650,S(x4 ) = 0.2824, and the ranking order ⎪ l=1 l=1 ⎪ m jk l  ⎪   ik l is x2 x4 x3 x1 . ⎪ ⎪ + nk= j+1 α jk log(hljk ) + m αik log(1 − hlik ) ⎪ l=1 l=1 ⎪ In this example, when the β -normalization based method in     ⎪    mk j l mi j ⎪ ⎪ − (n − 2 ) α l log(1 − hli j ) + ik−1 α log(hlk j ) [57] is used to rank objects, the ranking scores are ⎪ =1 l=1 i j l=1 k j ⎪  ⎪ mki l ⎪ ⎪ + l=1 αki log(1 − hlki ) S(x1 ) = 0.1928, S(x2 ) = 0.2312,S(x3 ) = 0.2736,S(x4 ) = 0.3024, ⎪ ⎪ ⎪ ⎨+  j−1 mk j α l log(hl ) + mik α l log(hl ) . by which the ranking order is x x x x , where the optimized kj kj ik k=i+1 l=1 ik 4 3 2 1 s.t. l=1  n mik l m jk l l l parameter ς = 0. ⎪ + α log ( 1 − h ) + α log ( h ) ⎪ k = j+1 jk ik l=1 ik ⎪ ⎪ Furthermore, when two α -normalization based methods in l=1 jk ⎪ l,− l,+ ⎪ + ε − ε = 0 , i, j = 1 , 2 , ..., n, i < j ⎪ [49] are adopted to give the ranking order of objects, we obi j i j ⎪ ⎪ m i j l ⎪ ⎪ tain w1 = 0.2429, w2 = 0.0896,w3 = w4 = 0.3337 using model (Mα = 1, i, j = 1, 2, ..., n, i < j ⎪ l=1 i j ⎪ ⎪ 4) [49]. Thus, the ranking order is x3 = x4 . x1 x2 . Meanwhile, ⎪ l ⎪ αi j = 1 ∨ 0, i, j = 1, 2, ..., n, l = 1, 2, ..., mi j , i < j ⎪ according to model (M-8) in [49], the priority weights are w1 = ⎪ ⎪ ⎪ l l ⎪ w2 = 0, w3 = 0.1023,w4 = 0.8977, and the ranking order is x4  h ∈ ( 0 , 1 ) , h ∈ h ∈ U, i < j ⎪ i j ij ij ⎪ ⎪ x3  x1 = x2 . ⎪ l,− l,+ ⎪ εi j , εi j ≥ 0, i, j = 1, 2, ..., n, i < j ⎪ ⎪ Moreover, when we apply the α -normalization based model ⎪ ⎩α l = 1 (17) in [61], the priority weights are derived as follows: i0 j0 (18) Solving model (18) for each known element hli

0 j0

, l = 1, 2, …,

mi0 j0 with hi0 j0 ∈ / U, i0 , j0 = 1, 2, …, n, and i0 < j0 , we derive all missing HFEs. Then, we can apply the algorithm in Section 3.2 to obtain the ranking order of objects. Example 2. Let X = {x1 , x2 , x3 , x4 }. Suppose that the incomplete HFPR H is defined as follows:

⎛

{ 21 }

⎜{ 1 , 2 } ⎜ 2 3 H=⎜ ⎝ x31 x41

{ 13 , 12 } { 21 } { 72 , 23 , 34 } x42

x13

{

x14

1 1 5 , , 4 3 7 1 2 1 5 , 2 8

{ }

{

}

}

⎞

⎟ ⎟ ⎟. { 83 , 12 }⎠ { 12 } x24

Using model (18), the missing HFEs are obtained as h13 =

{ 17 , 15 , 13 , 59 }, h14 = { 15 , 13 , 25 , 12 , 23 , 56 } and h24 = { 13 , 47 , 23 , 10 11 }. For the complete HFPR H, using the algorithm listed in Section 3.2, the

w1 = w2 = 0.3261, w3 = 0.1304, w4 = 0.2174, by which we have x1 = x2  x4  x3 . In this example, when the β -normalization based Algorithm 3 in [61] is adopted. The interval priority weights are obtained as follows:

¯ = ([0.3976, 1.2574], [0.6043, 1.5811], [0.6325, 1.8128], W [0.7953, 2.1957] ). Thus, the ranking values are p1 = 1.3578, p2 = 1.9014, p3 = 2.1335, p4 = 2.6074, by which we get x4 x3 x2 x1 . Fig. 2 intuitively shows the ranking values and the ranking orders with respect to different methods. Note that the ranking orders obtained from all previous methods are different from ours. Remark 1. Note that the β -normalization based methods [57,61] needs to add values in the known HFEs. But the authors did not explain how to determine the optimized parameter. The α -normalization based methods [49,61] disregard the missing values, and the priority weight vector is obtained by only considering the known HFEs. From this example, one can check that

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it is unsuitable to only apply the known HFEs to calculate the priority weight vector. For instance, in Example 2 according to the known HFEs, one can find that object x2 is better than object x1 , and object x4 is better than object x3 . However, we cannot derive these conclusions according to methods in [49,61]. Furthermore, methods in [57,61] cannot address incomplete HFPRs with ignored objects. For example, let the incomplete HFPR H be defined as follows:

⎛

{ 12 }

⎜{ 1 , 1 } ⎜ 3 2 ⎝ x31

H=⎜

{ 12 , 23 } { 21 }

x23

x32

{ }

x42

{ 47 , 34 }

x41

x13 1 2

x14

⎞

⎟ ⎟ ⎟. { 41 , 37 }⎠ { 12 }

9

for all i, j = 1, 2, …, n, where ωk is the weight of the DM ek , ωk q p k k = 1, 2, …, q, k=1 (ri j,k ) is obtained using the weighted geometωk  p pq p p k ric mean (WGM) operator WGM(ri j,11 , ri j,22 , ..., ri j,q ) = qk=1 (ri j,k ) p

p

k with Rk k = (ri j,k )n×n being the pk th multiplicative consistent RPR

for individual HFPR Hk , and nk is the number of multiplicative consistent RPRs derived from individual HFPR Hk . p

Property 5. Let R p = (ri j )n×n be any q multiplicative consistent RPRs. R = (ri j )n×n is the associated comprehensive RPR, where ri j = q

ω (r p ) p p=1 i j p ω p q (r ) + p=1 p=1 i j

x24

for all i, j = 1, 2, …, n, and ωp is the weight q of RPR Rp such that ωp ≥ 0 and ω = 1. Then, R is multip=1 p plicatively consistent. q

One can check that Theorem 3.1 or Theorem 3.2 in [57] and Algorithm 3 in [61] are helpless to determine missing HFEs in H. When we apply models (M-4) and (M-8) in [49], the following crisp priority weight vectors are derived, where W1 = (0.5, 0.5, 0, 0) and W2 = (0, 0.1018, 0, 0.8982). Furthermore, when model (17) in [61] is used, we get W3 = (0.5, 0.5, 0, 0). According to the obtained crisp priority weight vectors, one can conclude that the α -normalization based methods in [49,61] are insufficient to address this incomplete HFPR. However, when model (18) and the algorithm in Section 3.2 are used to address this incomplete HFPR, the probability scores are obtained as follows:

ωp

(r pji )

q

ω (r p ) p p=1 i j p ω p q (r ) + p=1 p=1 i j

Proof. From ri j =

q

ωp

(r pji )

for all i, j = 1, 2, …, n,

one can easily check that R is a RPR. Furthermore, from the mulp tiplicative consistency of RPRs R p = (ri j )n×n , p = 1, 2, …, q, we dep p

p

p p p

rive ri j r jk rki = r ji rik rk j for all i, k, j = 1, 2, …, n and all p = 1, 2, ..., q. Thus,

 p ω p q

q p=1

=

ri j

 p ω p  q r jk

p=1

 p ω p q

q

p=1

r ji

 p ω p

p=1

rki

 p ω p q

p=1

rik

 p ω p rk j

p=1

(20)

for all i, k, j = 1, 2, …, n. According to formula (20), we conclude:

S(x1 ) = 0.2985, S(x2 ) = 0.1805,S(x3 ) = 0.17,S(x4 ) = 0.351,

q  p ω p r jk ri j p=1 q  p ω p q  p ω p q  p ω p q  p ω p ri j + p=1 r ji r jk + p=1 rk j p=1 p=1 q  p  ω p rki p=1 × q  p ω p q  p ω p rki + p=1 rik p=1

 p ω p

q

by which the ranking order is x4 x1 x2 x3 .

p=1

4. A method for group decision making with HFPRs In practical decision-making problems, it usually requires a group of decision makers to make decisions because of the complexity of decision-making problems, the time pressure, and the limited expertise of decision makers, namely, the group decision making [1–3,10,13,15-20,22,23,26-28,39,40,47,56,57,65]. With respect to group decision making with HFPRs, Zhang et al. [56] developed a consistency and consensus based group decision-making method. Based on the additive consistency analysis, Zhang et al. [57] provided an approach for group decision making with incomplete HFPRs. Meanwhile, Xu et al. [49] presented two methods to derive the crisp priority weight vector for group decision making with incomplete HFPRs that are based on the multiplicative and additive consistency analysis. After review previous researches about group decision making with HFPRs [49,56,57,61], we find that there are limitations including: (i) they do not consider the consensus; (ii) they add values to HFEs; (iii) they do not discuss how to determine the weights of the DMs, and (iv) they disregard the missing values. Different from previous discussions about group decision making with HFPRs, this section introduces a new method for group decision making with HFPRs that addresses the limitations in previous methods. Without loss of generality, assume that there are n objects X = {x1 , x2 , …, xn } that are evaluated by q DMs, where E = {e1 , e2 , …, eq }. Let Hk = (hi j,k )n×n (k = 1, 2, …, q) be the HFPR provided by the DM ek , where hij, k is a HFE denoting the hesitancy preferred degree of object xi over xj , i, j = 1, 2, …, n. Definition 8. Let Hk = (hi j,k )n×n be any q HFPRs, k = 1, 2, …, q. Then, the comprehensive HFPR H = (hi j )n×n is defined as follows:

q  p ω p r ji rik p=1 = q  p ω p q  p ω p q  p ω p q  p ω p r ji + p=1 ri j rik + p=1 rki p=1 p=1 q  p ω p rk j p=1 × q  p ω p q  p ω p rk j + p=1 r jk p=1 p=1

for all i, k, j = 1, 2, …, n. Thus, R is multiplicatively consistent. Consensus analysis is a main procedure in group decision making with preference relations [3,13,15-22,24,26-28,56]. Considering the consensus of group decision making with HFPRs, we introduce the following group consensus index. Definition 9. Let Hk = (hi j,k )n×n be any q HFPRs, k = 1, 2, …, q, and H = (hi j )n×n be the associated comprehensive HFPR. Then, the group consensus index (GCI) of Hk is defined as follows:

GCI (Hk , H )

hi j =

q q k=1



k=1

p



p

k ri j,k

ω k

ω k  q k

ri j,k

+

k=1



p

k r ji,k

max

   pk  p  log(ri j,k )− log(ri j ) , i, j=1,i< j

n

max

1≤pk ≤nk R p ∈C (R pk ) k

(21) p

where C (Rk k ) is the set of judgement matrices that are formed p

by RPR Rk k , denoted by

C ( Rk k ) =





2 = max n(n−1 )

p



 p ω p

q



 

R p = ripj

n×n



, p = 1, 2, ..., ql =1,l =k nl , p

k where ripj = ri j,k

ωk , k = 1, 2, ..., q, pk = 1, 2, ..., nk

(19)



ωk q l =1,l =k p

p

ri j,ll

ω l

p

i, j = 1, 2, ..., n, and Rl l = (ri j,ll )

n×n

for all

, l = 1, 2, ..., q,

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pl = 1, 2, ..., nl with l = k}

follows:

D(Hk , Hz )

and ω = {ω1 , ω2 , ..., ωq } is the weight vector on the DM set. Property 6. Let Hk = (hi j,k )n×n be any q HFPRs, k = 1, 2, …, q, and H = (hi j )n×n be the associated comprehensive HFPR obtained in p

p

k Definition 8. Let Rk k = (ri j,k )n×n be a RRP in HFPR Hk such that

  n pk   log(ri j,k ) − log(ripj0 )  i, j=1,i< j   n log(r pt ) − log(r p ). = max maxp ij i j,k i, j=1,i< j p

p

pk pk Rk k = (ri j,k )n×n asRˆk k = (rˆi j,k )n×n ,

pk α p0 (1−α ) (ri j,k ) ( ri j )

(22) p

k rˆi j,k =

where

for all i, j = 1, 2, …, n, and α ∈ (0, 1). Then,

GCI (Hk , H ) ≥ GCI (Hˆ k , Hˆ ), where

GCI (Hˆ k , Hˆ )



n

p

i, j=1,i< j

k |log(rˆi j,k ) − log(rˆipj0 )| = max

max

(23)

n

1≤pt ≤nk Rˆ p ∈C (Rˆ pt ) p

α

t

k t |log(rˆipj,k ) − log(rˆipj )|, Rˆ p = (rˆipj )n×n and rˆipj = ( (ri j,k ) (ripj )

q

p

l =1,l =k

ωl

(ri j,ll )

i, j=1,i< j

(1−α ) ω ) k

for all i, j = 1, 2, …, n.

Proof. From formulae (22) and (23), to show GCI (Hk , H ) ≥ GCI (Hˆ k , Hˆ ), we only need to prove the following:

     pk   log(rˆpk ) − log(rˆp0 ) ≤ n log(ri j,k ) − log(ripj0 ).  i j i j,k i, j=1,i< j i, j=1,i< j

n

For each pair of (i, j) with i < j, we have the following:

  log(rˆpk ) − log(rˆp0 ) ij i j,k   α    ( 1 −α ) pk  = log ri j,k ripj0  α  p0 (1−α ) ωek q pk

ωe  l  − log ri j,k ri j ri j,l l =1,l =k   p   p0   p0   k = α log ri j,k + (1 − α ) log ri j − ωek (1 − α ) log ri j  p    p ωe  l  q k −ωek α log ri j,k − log ri j,ll  l =1,l =k  p       k = α log ri j,k + (1 − α ) log ripj0 − ωek (1 − α ) log ripj0  p   p     k k −ωek α log ri j,k − log ripj0 + ωek log ri j,k   p       k = α log ri j,k − α log ripj0 − ωek (1 − α ) log ripj0  p   k +ωek (1 − α ) log ri j,k             pk pk    ≤ α log ri j,k − log ripj0  + ωek (1 − α )log ri j,k − log ripj0       pk  ≤ log ri j,k − log ripj0  

pz =1

ρ (Rzpz )

min

1≤pz ≤nz

n

|

p r k i, j=1,i< j i j,k

−

z ripj,z

pl

Thus, the conclusion holds. Definitions 8 and 9 both show that the weights of the DMs are needed to calculate the collective HFPRs and to measure the consensus of individual HFPRs. Next, we apply a distance measure between HFPRs to give a method to determine the weights of the DMs. Definition 10. Let Hk = (hi j,k )n×n and Hz = (hi j,z )n×n be any two HFPRs. Then, the distance between Hk and Hz is defined as

 |

 n pk pz min |r − ri j,k | , i, j=1,i< j i j,z

(24)

1≤pk ≤nk

p

where ρ (Rk k ) and ρ (Rz z ) are the probabilities of RPRs Rk k and and nk and nz are the numbers of RPRs obtained from Hk and Hz , respectively. According to formula (24), one can check: (i) D(Hk , Hz ) = 0 if and only if Hk = Hz ; (ii) D(Hk , Hz ) = D(Hz , Hk ); (iii) 0 ≤ D(Hk , Hz ) ≤ 1. From the point of view that the higher the group consensus level is, the bigger weight will be, we define the weight of the DM ek , k = 1, 2, …, q, using the following formula: p

p Rz z ,

ω k = q

n

Rˆ p ∈C (Rˆk k )

with

nz



q 1 1 / , q t=1 D ( H , H ) D(Ht , Hz ) z k z=1,z=t z=1,z=k

 pl  max maxp ) log(ri j,k i, j=1,i< j 1≤pl ≤nk , pl = pk R p ∈C (R l ) k     n pk p  p   − log(ri j ) , maxp log(rˆi j,k ) − log(rˆi j ) i, j=1,i< j

2 = max n (n − 1 )

+



p

1≤pt ≤nk R p ∈C (R t ) t

Adjust



nk 1 = ρ (Rkpk ) pk =1 n (n − 1 )

(25)

where Hk is the HFPR offered by the DM ek , and D(Hk , Hz ) is the distance between Hk and Hz as shown in formula (24). When D(Hk , Hz ) = 0 for all z = 1, 2, …, q with z = k. Then, we consider that all DMs have the same importance, and their weights equal to 1/q. Now, let us introduce a new consistency and consensus based group decision-making method with HFPRs that can address the inconsistent and incomplete cases. γ Step 1: Let Hk , k = 1, 2, …, q, be the HFPR offered by the DM ek , where γ = 0; γ Step 2: When Hk is complete, skip to Step 4; otherwise, go to Step 3; γ Step 3: Use model (18) to estimate missing HFEs in Hk , k = 1, 2, γ …, m. If there is no fear of confusion, we still applyHk to denote the associated complete HFPR; p ,γ γ Step 4: With respect to each complete HFPR Hk , we let Rk k , pk = 1, 2, …, nk , be the multiplicative consistent RPRs with ρ (Rkpk ,γ ) being its probability, which are derived from model (9) and Property 3; Step 5: Use formula (25) to determine the weights of the DMs; Step 6: Apply the WGM operator to calculate the collective RPRs p,γ B p,γ = (bi j )n×n , p = 1, 2, ..., m n; l=1 l Step 7: Let GCI be the threshold of the group consensus index. We apply formula (21) to calculate the group consensus index. If GCI (Hk , H ) = min GCI (Hl , H ) > GCI, then skip to Step 9; otherwise, 1≤l≤q

turn to the next step; Step 8: For the individual HFPR Hk , we use Property 6 to improve its consensus, and let γ = γ + 1. Then, return to Step 6; Step 9: For each consistent collective preference relation B p,γ = p,γ (bi j )n×n , p = 1, 2, ..., m n , we use Definition 8 to calculate l=1 l p,γ

the associated multiplicative consistent RPR R p,γ = (ri j )n×n , p = 1, 2, ..., m n; l=1 l Step 10: Adopt formula (3) to calculate the priority weight vecp,γ p,γ p,γ tor w p,γ = (w1 , w2 , ..., wn ) from each multiplicative consisp,γ

tent RPR R p,γ = (ri j )n×n , p = 1, 2, ..., m n , and compute the asl=1 l p ,γ

sociated probability P (R p,γ ) = m P ( Rk k ); k=1 Step 11: According to the derived hesitant fuzzy priority weight vector in Step 10, we apply formula (10) to calculate the ranking scores; Step 12: According to the ranking scores, rank objects x1 , x2 , …, xn . To help the DMs give the threshold of the group consensus, we offer the following simulation method to obtain the average values of the group consensus index using the stochastic formed RPRs.

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11

Table 5 Average values of GCI with the different input parameters. The number of the DMs

The orders of RPRs 3

4

5

6

7

8

9

3 4 5 6 7

0.3590 0.4238 0.4683 0.5051 0.5386

0.3314 0.3779 0.4141 0.4391 0.4598

0.3209 0.3574 0.3842 0.4031 0.4219

0.3155 0.3425 0.3640 0.3791 0.3955

0.3139 0.3323 0.3504 0.3650 0.3797

0.3151 0.3265 0.3409 0.3540 0.3674

0.3187 0.3235 0.3355 0.3455 0.3588

Table 6 HFPRs on the criteria set.

DM e1 c1 : Return c2 : Risk c3 : Liquidity DM e2 c1 : Return c2 : Risk c3 : Liquidity DM e3 c1 : Return c2 : Risk c3 : Liquidity

c1 : Return

c2 : Risk

c3 : Liquidity

{0.5} {0.1, 0.2, 0.3, 0.4, 0.5} {0.2, 0.3}

{0.5, 0.6, 0.7, 0.8, 0.9} {0.5} x32, 1

{0.7, 0.8} x23, 1 {0.5}

{0.5} x21, 2 {0.7, 0.9}

x12, 2 {0.5} {0.2, 0.3, 0.4, 0.5, 0.6}

{0.1, 0.3} {0.4, 0.5, 0.6, 0.7, 0.8} {0.5}

{0.5} {0.3, 0.5} x31, 3

{0.5, 0.7} {0.5} {0.7, 0.8, 0.9}

x13, 3 {0.1, 0.2, 0.3} {0.5}

4.1. Simulation method

Investment Choice

Input: p0 , m, n, t, where p0 is the simulating times, m is the number of the DMs, and n is the order of RPRs. Output: GCI. p p p Step 1: Let p = 1. Generate m RPRs R1 , R2 , …, Rm with n orders, which are uniform randomly selected from [0, 1]; Step 2: Use the following formula to calculate the weights of the DMs, where

Return

Bank Deposit

m 1 1 , q p p / l=1 D ( R , R ) D(Rlp , Rtp ) t =1,t =k t k t =1,t =l

ω k = m

k = 1, 2, . . . , m, with

p p D(Rk , Rt )

=

2 n (n−1 )

Debentures

Government Bonds

Shares

Fig. 3. The hierarchy structure.

(26) n

p i, j=1,i< j |ri j,k

p − ri j,t |

for all k, t = 1, 2, …, m

and k = t; Step 3: Use the WGM operator to calculate the collective RPR R. Then, we apply the following formula to calculate the group consensus level

GCI (Rkp , R p ) =

Liquidity

Risk

  n 2 log10 (r p ) − log10 (r p ). ij i j,k i, j=1,i< j n (n − 1 )

(27)

Step 4: Let p = p + 1, and return to Step 1 until p = p0 ;  p0

p=1

p k

max GCI (R ,R p )

1≤k≤m Step 5: Output GCI, where GCI = . p0 When setting different input parameters, this simulation method runs 10,0 0 0 times, the obtained the average values of GCIare shown as in Table 5. Generally speaking, the threshold of the group consensus index should be smaller than the average values listed in Table 5. Table 5 shows that the thresholds of the group consensus index increases with the numbers of the DMs for the fixed order of PRRs, while it decreases with the orders of PRRs for the fixed number of the DMs.

5. Numerical example and analysis To show the application of the new method to group decision making with HFPRs as well as to compare with the previous methods, this section offers a practical example about the investing problem [24,57]. Meanwhile, we further analyze the main researches about decision making with HFPRs.

5.1. A practical example A person who interests in investing his money to any one of the four portfolios: bank deposit x1 ; debentures x2 ; government bonds x3 ; and Shares x4 . Out of these portfolios he has to choose only one based upon three criteria: c1 : return, c2 : risk and c3 : liquidity. A hierarchy structure is shown in Fig. 3. To obtain a reasonable decision result, three DMs E = {e1 , e2 , e3 } are invited to compare these four portfolios with respect to each criterion. The associated HFPRs are listed in Tables 6–9 [24,57]. Take the HFPR given by the DM e1 on the criteria set for example; DM e1 considers that the preference degrees are 0.5, 0.6, 0.7, 0.8, and 0.9 on criterion c1 over c2 . Furthermore, because of the limited expertise of this DM or the time pressure, he/she does give the preference degree between criteria c2 and c3 . In a similar way, we obtain the other preferences that form the associated HFPRs. To derive the best choice as well as to rank the possible objects, the following steps are required: Step 1: Because all HFPRs are incomplete, we apply model (18) to determine the missing values that are listed as shown in Tables 10 and 11. Step 2: With respect to each complete HFPR obtained in Step 1, we can obtain their associated multiplicative consistent RPRs using model (9) and Property 3. Take HFPR H2 for example, the multiplicative consistent RPRs and their probabilities are obtained as shown in Table 12.

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F. Meng, Q. An / Knowledge-Based Systems 000 (2017) 1–15 Table 7 HFPRs with respect to criterion c1 .

DM e1 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4 DM e2 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4 DM e3 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4

Bank deposit x1

Bebentures x2

Government bonds x3

Shares x4

{0.5} {0.2, 0.3} {0.5, 0.6, 0.7, 0.8, 0.9} xc411 ,1

{0.7, 0.8} {0.5} {0.5, 0.6, 0.7} xc421 ,1

{0.1, 0.2, 0.3, 0.4, 0.5} {0.3, 0.4, 0.5} {0.5} {0.3, 0.5, 0.6, 0.7, 0.8}

xc141 ,1 xc241 ,1 {0.2, 0.3, 0.4, 0.5, 0.7} {0.5}

{0.5} {0.5, 0.7} xc311 ,2 xc411 ,2

{0.3, 0.5} {0.5} {0.3, 0.4, 0.5} xc421 ,2

xc131 ,2 {0.5, 0.6, 0.7} {0.5} {0.6, 0.7, 0.8}

xc141 ,2 xc241 ,2 {0.2, 0.3, 0.4} {0.5}

{0.5} {0.2, 0.3, 0.4, 0.5, 0.6} xc311 ,3 {0.1, 0.3, 0.5, 0.7, 0.9}

{0.4, 0.5, 0.6, 0.7, 0.8} {0.5} {0.1, 0.2, 0.3} xc421 ,3

xc131 ,3 {0.7, 0.8, 0.9} {0.5} {0.6, 0.7, 0.8}

{0.1, 0.3, 0.5, 0.7, 0.9} xc241 ,3 {0.2, 0.3, 0.4} {0.5}

Bank deposit x1

Bebentures x2

Government bonds x3

Shares x4

{0.5} xc212 ,1 {0.5, 0.6, 0.7} xc412 ,1

xc122 ,1 {0.5} {0.1, 0.3} {0.7, 0.9}

{0.3, 0.4, 0.5} {0.7, 0.9} {0.5} {0.6, 0.7, 0.8}

xc142 ,1 {0.1, 0.3} {0.2, 0.3, 0.4} {0.5}

{0.5} {0.6, 0.7} {0.1, 0.3, 0.4, 0.5, 0.6} {0.3, 0.5}

{0.3, 0.4} {0.5} xc322 ,2 {0.1, 0.2, 0.3, 0.4, 0.5}

{0.4, 0.5, 0.6, 0.7, 0.9} xc232 ,2 {0.5} {0.2, 0.3, 0.4}

{0.5, 0.7} {0.5, 0.6, 0.7, 0.8, 0.9} {0.6, 0.7, 0.8} {0.5}

{0.5} xc212 ,3 {0.3, 0.5, 0.6, 0.7, 0.9} xc412 ,3

xc122 ,3 {0.5} {0.1, 0.2, 0.3} xc422 ,3

{0.1, 0.3, 0.4, 0.5, 0.7} {0.7, 0.8, 0.9} {0.5} {0.5, 0.7, 0.8}

xc142 ,3 xc242 ,3 {0.2, 0.3, 0.5} {0.5}

Bank deposit x1

Bebentures x2

Government bonds x3

Shares x4

{0.5} {0.4, 0.5, 0.6} {0.3, 0.5} {0.2, 0.3, 0.4}

{0.4, 0.5, 0.6} {0.5} xc323 ,1 xc423 ,1

{0.5, 0.7} xc233 ,1 {0.5} {0.6, 0.7, 0.8}

{0.6, 0.7, 0.8} xc243 ,1 {0.2, 0.3, 0.4} {0.5}

{0.5} {0.2, 0.3, 0.4} {0.7, 0.8, 0.9} {0.1, 0.3, 0.5, 0.7, 0.8}

{0.6, 0.7, 0.8} {0.5} {0.5, 0.7} {0.3}

{0.1, 0.2, 0.3} {0.3, 0.5} {0.5} xc433 ,2

{0.2, 0.3, 0.5, 0.7, 0.9} {0.7} xc343 ,2 {0.5}

{0.5} {0.1, 0.2, 0.3, 0.4, 0.5} xc313 ,3 xc413 ,3

{0.5, 0.6, 0.7, 0.8, 0.9} {0.5} {0.3, 0.5, 0.7, 0.8, 0.9} {0.1, 0.2, 0.4, 0.5, 0.6}

xc133 ,3 {0.1, 0.2, 0.3, 0.5, 0.7} {0.5} xc433 ,3

xc143 ,3 {0.4, 0.5, 0.6, 0.8, 0.9} xc343 ,3 {0.5}

Table 8 HFPRs with respect to criterion c2 .

DM e1 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4 DM e2 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4 DM e3 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4

Table 9 HFPRs with respect to criterion c3 .

DM e1 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4 DM e2 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4 DM e3 Bank deposit x1 Bebentures x2 Government bonds x3 Shares x4

Table 10 Determined missing values in incomplete HFPRs on the criteria set. Incomplete HFPRs

Determined missing values

Incomplete HFPR H1 Incomplete HFPR H2 Incomplete HFPR H3

h23, 1 = 0.3077, 0.5, 0.6087, 0.6316, 0.7273, 0.8 h12, 2 = 0.0968, 0.1429, 0.1552, 0.2222, 0.3, 0.3913 h13, 3 = 0.2059, 0.3, 0.3684, 0.5

Step 3: According to the obtained multiplicative consistent RPRs and formulae (24)-(25), the weights of the DMs are obtained as listed in Table 13.

Step 4: Let GCI = 0.15. Applying the WGM operator, the comprehensive judgment matrices can be obtained. Then, we can judge the consensus of individual HFPRs according to formula (21). Let α = 0.8. The worst original consensus, the worst adjusted consensus and the associated interactive times are obtained as listed in Table 14. Step 5: According to the obtained comprehensive judgment matrices, the associated multiplicative consistent RPRs are obtained using Property 5. Then, we apply formulae (3)-(4) to calculate the scores of objects and criteria, shown as in Table 15. Step 6: Using the weighted mathematical average (WMA) operator, the final ranking scores are obtained as follows:

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13

Table 11 Determined missing values in incomplete HFPRs with respect to each criterion. Incomplete HFPRs

Determined missing values

Incomplete HFPRs with respect to criterion c1



Incomplete HFPR H1c1

hc241 ,1 = 0.2863, 0.3333, 0.3374, 0.3463, 0.3717, 0.3956, 0.4299, 0.5

⎧ c1 = 0.4909, 0.5, 0.5809, 0.5946, 0.6, 0.6044 h ⎪ ⎨ 13,2 hc1

Incomplete HFPR H2c1

= 0.3798, 0.3827, 0.4385, 0.4495, 0.5, 0.5045

14,2 ⎪ ⎩ c1 h24,2 = 0.3798, 0.445, 0.4495, 0.5, 0.5091, 0.546  c

h131 ,3 = 0.5091, 0.555, 0.6044, 0.6202, 0.6517, 0.7, 0.7101, 0.7534, 0.8175, 0.84

Incomplete HFPR H3c1

hc241 ,3 = 0.2899, 0.433, 0.4495, 0.4909, 0.5, 0.5045, 0.5091, 0.555, 0.6202, 0.6231, 0.6549

Incomplete HFPRs with respect to criterion c2



Incomplete HFPR H1c2 Incomplete HFPR

hc141 ,1 = 0.3333, 0.3769, 0.3956, 0.4495, 0.5, 0.5701, 0.6044

hc122 ,1 = 0.25, 0.3, 0.3483, 0.3956 hc142 ,1 = 0.16, 0.2016, 0.208, 0.2321, 0.2591, 0.3343, 0.3463

H2c2

hc232 ,2 = 0.4495, 0.5, 0.5505, 0.555, 0.6, 0.6202, 0.6517, 0.6626, 0.7861 hc122 ,3 = 0.125, 0.2343, 0.3038, 0.3374, 0.3956, 0.433, 0.5

⎧ ⎪ ⎨

hc142 ,3 = 0.25, 0.3956, 0.433, 0.4495, 0.5

Incomplete HFPR H3c2

⎪ ⎩

hc242 ,3 = 0.433, 0.5, 0.567, 0.6044, 0.6626

Incomplete HFPRs with respect to criterion c3 Incomplete HFPR

H1c3

Incomplete HFPR

H2c3



hc233 ,1 = 0.5337, 0.555, 0.5701, 0.6, 0.6101, 0.6202, 0.6342 hc243 ,1 = 0.4, 0.4299, 0.4663, 0.4834, 0.5, 0.5091, 0.5505, 0.6, 0.6202

hc343 ,2 = 0.5765, 0.6044, 0.7, 0.715, 0.7948, 0.8018, 0.8131, 0.8209, 0.8785 hc133 ,3 = 0.3451, 0.3798, 0.445, 0.5, 0.5505, 0.567, 0.6044, 0.6549

⎧ ⎪ ⎨

hc143 ,3 = 0.4495, 0.5, 0.5402, 0.5505, 0.6044, 0.6202, 0.6667, 0.7469, 0.75

Incomplete HFPR H3c3

⎪ ⎩

hc343 ,3 = 0.3956, 0.4495, 0.5, 0.5276, 0.5505, 0.555, 0.6202, 0.6667, 0.7237, 0.75

Table 12 Multiplicative consistent RPRs and their probabilities with respect to complete HFPR H2 . Multiplicative consistent RPRs

⎛

0.5

⎜

0.1429

A12 = ⎝0.8571

⎛

0.6 0.3

⎜ A32 = ⎝0.7

0.5

0.7

0.5

⎛

0.5

⎞

⎟

0.7

Multiplicative consistent RPRs

2 13

A22 = ⎝0.6087

0.5

⎟ 0.5⎠ 0.5

⎛

3 13

⎞

0.5

⎟ 0.7⎠

0.3

0.5

2 13

0.3913

⎞ ⎟

0.4⎠

0.7

0.6

0.5

0.5

0.2222

0.3

⎟

0.6⎠

0.7

0.4

0.5

0.5

0.0968

0.3

0.7

2 13

⎞

0.5

⎜ A62 = ⎝0.9032

ρ (A2p2 )

0.3

0.5

⎜ A42 = ⎝0.7778 ⎛

0.3

0.5

⎜ ⎛

0.3

0.1552

⎜ A52 = ⎝0.8448

ρ (A2p2 )

0.1

0.4⎠

0.5

0.9 0.5

⎞

2 13

⎞ ⎟

0.5

0.8⎠

0.2

0.5

2 13

Table 13 The weights of the DMs.

On criteria set With respect to the criterion c1 With respect to the criterion c2 With respect to the criterion c3

The weight of the DM e1

The weights of the DM e2

The weights of the DM e3

0.3059 0.2584 0.2796 0.2853

0.3108 0.4328 0.3010 0.2531

0.3833 0.3088 0.4194 0.4615

Table 14 The worst original consensus, the worst adjusted consensus and the associated interactive times.

On criteria set With respect to the criterion c1 With respect to the criterion c2 With respect to the criterion c3

The worst original consensus

The worst adjusted consensus

The associated interactive times

GOI (H20 , H 0 ) = 0.3591 GOI (H1c1 ,0 , H c1 ) = 0.234 GOI (H2c2 ,0 , H c2 ) = 0.2469 GOI (H2c3 ,0 , H c3 ) = 0.2459

GOI (H364 , H 64 ) = 0.13 GOI (H1c1 ,22 , H c1 ,22 ) = 0.1482 GOI (H1c2 ,22 , H c2 ,22 ) = 0.1483 GOI (H3c3 ,25 , H c3 ,25 ) = 0.1496

64 22 22 25

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F. Meng, Q. An / Knowledge-Based Systems 000 (2017) 1–15 Table 15 The scores of objects and criteria.

The The The The The

scores of object x1 scores of object x2 scores of object x3 scores of object x4 weights of criteria

criterion c1

criterion c2

criterion c3

0.2684 0.2422 0.1980 0.2914 0.3054

0.1959 0.3418 0.1916 0.2707 0.2769

0.2903 0.2351 0.2678 0.2068 0.4176

Table 16 Ranking values and orders with respect to different methods. Methods

Method in [57] Model (M-8) in [49] Model (M-12) in [49] Our method

Ranking values

Ranking orders

S(x1 )

S(x2 )

S(x3 )

S(x4 )

0.2897 0.2286 0.2211 0.2575

0.3018 0.234 0.2596 0.2668

0.2544 0.1912 0.2211 0.2254

0.2829 0.3463 0.2982 0.2503

x2 x1 x4 x3 x4 x2 x1 x3 x4  x2  x1 = x3 x2 x1 x4 x3

S(x1 ) = 0.2575, S(x2 ) = 0.2668, S(x3 ) = 0.2254 and S(x4 ) = 0.2503. Thus, the ranking order is x2 x1 x4 x3 . When different methods are used in this example, the ranking values and the ranking orders are obtained as shown in Table 16. Table 16 shows that the same ranking order is obtained according to our method and method in [57]. However, their ranking values are different. Furthermore, different ranking orders are derived using models (M-8) and (M-12) [49] that are both different from our ranking results. 5.2. Comparison analysis This subsection further analyzes several previous methods from their theoretical aspects. According to their principles, they can be classified into two types. One type is based on the α -normalization method; the other uses the β -normalization method. (i) The α -normalization based methods [49,61,65]: The main limitations of this type of approaches include: a) it can only derive the crisp priority weight vector that cannot reflect the hesitancy of the DMs; b) there exists information loss; c) it disregards the missing values; d) it is insufficient to address inconsistent HFPRs. (ii) The β -normalization based methods [56,57,61,65]: The main shortages of this type of approaches contain: a) it needs to add value to HFEs; b) it restricts to only consider the ordered RPRs; c) it does not the probability of RPRs. d) it requires the DMs to subjectively give the optimized parameter ϛ for incomplete HFPRs. Furthermore, none of them can address incomplete HFPRs with ignored objects or considers how to determine the weights of the DMs. There are several features of our method, including: a) The new consistency concept neither adds values to HFEs nor disregards any information; b) The hesitant fuzzy priority weight vector is derived from multiplicative consistent RPRs that considers their probabilities; c) 0–1 mixed programming models for judging the consistency of HFPRs and determining missing values in incomplete HFPRs are constructed; d) A new consensus index is defined, and an approach to improving the consensus is offered; e) A distance measure based method to determining the weights of the DMs is provided;

f) The new interactive algorithm for group decision making with HFPRs is based on the multiplicative consistency and consensus analysis. Furthermore, it can address the inconsistent and incomplete cases. 6. Conclusion Considering decision making with HFPRs, this paper first analyses the limitations of several previous methods from theoretical and numerical aspects. Then, a new multiplicative consistency concept for HFPRs is introduced that can address the shortages in previous ones. To show the application of the new concept, 0–1 mixed programming models to judge the consistency of HFPRs are constructed. Meanwhile, 0–1 mixed programming models for determining missing values in incomplete HFPRs are established. Based on the constructed programming models, an algorithm to calculate the hesitant fuzzy priority weight vector is offered. Subsequently, a new group consensus index is defined by using the distance measure. When the consensus requirement is unsatisfied, an interactive method for improving the group consensus is offered. Furthermore, the weights of the DMs are determined by applying the defined distance measure between two HFPRs. Using the given consistency concept and the defined group consensus index, an algorithm for group decision making is developed. Note that the new group algorithm does not only consider the consistency and consensus but also can address the inconsistent and incomplete cases. To illustrate the application of the developed theoretical results, a personal investment problem is offered. This paper mainly focuses on HFPRs. In future, we shall continue to study other types of preference relations, such as interval hesitant fuzzy preference relations (IHFPRs) [11,36,44], hesitant fuzzy multiplicative preference relations (HFMPRs) [58,59,62], and hesitant fuzzy linguistic preference relations (HFLPRs) [43,46,60,63]. Furthermore, we will research the application of the developed theoretical results in some other fields, such as the evaluating machine tool [4,7], the water leakage problem in urban water system [9], the propulsion/manoeuvring system selection problem [33], the economic production problem [5,35], the evaluation of airline service quality [14], and the performance of IT department in the manufacturing industry [25], and the evaluation of university teachers [45]. Acknowledgement This work was supported by the State Key Program of National Natural Science of China (No. 71431006), the National Natural Science Foundation of China (Nos. 71571192, and 71501189), the Natural Science Foundation of Hunan Province (2017JJ3397), the Hunan Province Foundation for Distinguished Young Scholars of China (No. 2016JJ1024), and the Innovation-Driven Planning Foundation of Central South University (Nos. 2015CX010, 2016CXS027). References [1] S. Alonso, F.J. Cabrerizo, F. Chiclana, F. Herrera, E. Herrera-Viedma, Group decision making with incomplete fuzzy linguistic preference relations, Int. J. Intell. Syst. 24 (2009) 201–222. [2] S. Alonso, F. Chiclana, F. Herrera, E. Herrera-Viedma, A consistency based procedure to estimate missing pairwise preference values, Int. J. Intell. Syst. 23 (2008) 155–175. [3] S. Alonso, E. Herrera-Viedma, F. Chiclana, F. Herrera, A web cased consensus support system for group decision making problems and incomplete preferences, Inf. Sci. 180 (2010) 4477–4495. [4] Q. An, H. Chen, B. Xiong, J. Wu and L. Liang, Target intermediate products setting in a two-stage system with fairness concern. Omega. doi:10.1016/j.omega. 2016.12.005. [5] Q. An, Y. Wen, B. Xiong, M. Yang, X. Chen, Allocation of carbon dioxide emission permits with the minimum cost for Chinese provinces in big data environment, J. Clean. Prod. 142 (2017) 886–893. [6] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.

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Please cite this article as: F. Meng, Q. An, A new approach for group decision making method with hesitant fuzzy preference relations, Knowledge-Based Systems (2017), http://dx.doi.org/10.1016/j.knosys.2017.03.010