An evidence theory based approach for group decision making under uncertainty

1 Introduction

Group decision making (GDM) is about selecting the best alternative from available alternatives in the presence of several decision makers [1–3]. It is becoming increasingly important due to the changing business environment and the active participation of various stakeholders in the decision making process. GDM usually consists of two phases including (a) the consensus process and (b) the selection process [4–8]. The consensus process is about the pursuit of the maximum degree of agreement between decision makers. The selection process is related to a comprehensive assessment of the overall performance of available alternatives with respect to the preference of individual decision makers. Effective GDM requires the development of appropriate approaches for selecting the best alternative from available alternatives through adequate consensus building with respect to the preference of individual decision makers [9].

Much researches have been done, and numerous consensus models have been developed for facilitating consensus building in solving the GDM problem [10–13]. Overall these models can be grouped into four groups in solving the GDM problem. The first group focuses on the representation of the preference of individual decision makers in consensus building [14–17]. The second group concentrates on the use of the minimum adjustment for achieving a group agreement [18–20]. The third group is related to the adoption of the individual consistency measure and the group consensus measure in consensus building [21, 22]. The fourth group considers the attitude of individual decision makers in consensus building [23, 24]. Such models have demonstrated their applicability in solving various GDM problems under different circumstances. These models, however, have not adequately incorporated the relative importance of individual decision makers represented by their respective weights in consensus building for GDM although such weights have a significant impact on the GDM outcome [23, 25].

Numerous approaches have been developed for determining the weights of decision makers in GDM including the subjective approach and the objective approach. The subjective approach focuses on an appropriate consideration of the decision maker’s preference in the decision making process. The adoption of such an approach allows decision makers to express their preferences in a specific manner. Such preferences are then aggregated for determining the weights of decision makers in a given situation. Saaty [26], for example, develops a pairwise comparison based approach for determining the weights of decision makers. Ramanathan and Ganesh [27] propose a subjective approach for better determining the weights of decision makers in GDM. Such approaches are useful in considering the subjective opinions of the decision maker. They are, however, often criticized for the presence of inconsistent subjective assessments and the challenging nature of the GDM process when there are many decision makers present.

The objective approach concentrates on the utilization of mathematical theories for determining the weights of decision makers. Yue [28] develops an objective approach based on the concept of ideal solutions for determining the weights of decision makers. Yue [29] proposes an approach for ranking the decision makers in determining the weights of decision makers. Wang et al. [30] construct a mathematical programming model for obtaining the weights of decision makers by minimizing the square logarithm compatibility in GDM. Tao et al. [31] provide a Shapley value based approach for assessing the weights of decision makers in GDM. Dong et al. [24] present a self-management mechanism for determining the weights of decision makers in a dynamical manner in GDM. Such approaches have demonstrated their respective merits in various GDM situations. They are, however, often questioned by the inconsistent weights obtained and the computational effort required.

There are attempts to combine the subjective approach and the objective approach together for better determining the weights of decision makers in GDM. Song and Zou [32], for example, introduce an approach for determining the weights of decision makers based on a convex combination of the subjective weight and the objective weight. Jabeur et al. [33] propose a hybrid approach for assessing the weights of decision makers with the adoption of a distance measure. Xu and Cai [34] develop an approach for deriving the weights of decision makers based on the interval multiplicative preference relation. Efe [35] proposes an integrated approach for deriving the weights of decision makers. Such approaches try to determine the weights of decision makers indirectly based on decision makers’ preferences about alternatives. They are often criticized due to their capacity in adequately handling the uncertainty and imprecision in the GDM process.

More and more decision makers express their opinions over alternatives by using fuzzy preference relations under the condition of the uncertainty and fuzziness of human thinking [36, 37]. In some situations, the decision makers may only provide incomplete preference relations due to the diversity of reasons, such as the lack of knowledge or data, time pressure, and their limited expertise related to the problem domain. Recently, many useful decision making methods have been developed to deal with group decision making (GDM) problems within incomplete fuzzy preference relation [38–40]. For example, Xu [39] presented a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. These existing methods have made significant contributions to the GDM problems with incomplete fuzzy preference relations.

All the existing studies have made considerable progress on GDM studies. However, there are still some research gaps:

First, the reliability of the preference of the decision maker with respect to the available alternatives in assessing their relative importance in GDM is often questioned due to various backgrounds of the decision makers. Existing method [23] takes the weight vectors of decision makers into consideration, but it is not reasonable for equal weights assumed when weights are not given. Very often, the involvement of the third-party evaluator to assess the knowledge, skills, and experience of the decision makers for determining their relative importance in the GDM process is critical.

Second, in the process of preference elicitation, decision makers may provide uncertain preference information even preference relations with some values unknown instead of crisp preference information due to the lack of knowledge. Therefore, it is also necessary to take incomplete preference information into account in GDM problems. Existing methods have made significant contributions to the GDM problems with incomplete fuzzy preference relations. However, they rarely analyze the individual consistency and the group consensus at the same time.

Finally, the weight vectors of decision makers usually play an important role in GDM. However, current researches do not take the weights of decision makers into consideration for the measurement of group consensus degree.

To fill the gap mentioned above, a novel approach for GDM is presented with incomplete fuzzy preference relations, which not only takes the decision makers’ weights into consideration but also manages the individual consistency and the group consensus simultaneously. In order to do so, evidence theory is extensively used [41,42, 41,42] to identify the weights of decision makers due to the belief structure which can express the uncertainty involving fuzzy and probability simultaneously. To adequately determine the ranking of available alternatives, the logarithmic least square method is applied for appropriately aggregating the incomplete fuzzy preference relation better representing the subjective assessment of individual decision makers. The consistency check is also conducted. A group consensus index considering the weight vectors of decision makers is developed for facilitating consensus building in group decision making. This leads to better group decisions being made. A real-world application is given to show the applicability of the proposed approach in solving GDM problems under uncertainty.

The rest of this paper is organized as follows. Section 2 presents a brief introduction to the evidence theory and fuzzy preference relations. Section 3 presents a novel approach for GDM under uncertainty. Afterwards, an example of a scientific foundation project selection and comparative discussion are given for illustrating the applicability and effectiveness of the proposed approach in Section 4. Finally, conclusion is presented in Section 5.

2 Preliminaries

In this section we briefly review some basic concepts of the belief structure and the Dempster’s rule of combination of the evidence theory and fuzzy preference relations.

The evidence theory provides a powerful tool for decision making through the representation and integration of uncertain information [43, 44]. With the use of this theory, the belief structure which can be used to better represent the subjective assessment on the decision maker provided by the third-party evaluator can be defined as follows.

Definition 1. [44] Let Θ be a set of mutually exclusive and collectively exhaustive events, often referred to as a frame of discernment. A basic probability assignment (BPA) is a mapping m: 2 ^Θ  → [0, 1], which satisfies the following conditions: m ( φ ) = 0 and ∑ A ⊆ 2 Θ m ( A ) = 1 ,(1) where m(A) represented how strongly the evidence supports A is called the basic probability assignment of A. It is called the focal element for all A when it satisfies the condition of m(A)>0.

When there are more than two evidences, a combination rule is adopted for fusing them. Considering two evidences indicated by m₁ and m₂ and these belief structures are mutually independent, the Dempster’s rule of combination can be used, denoted by m(A) =m₁(I) ⊕ m₂(J) (I∩J = A, where I, J ⊂ Θ) as follows m ( A ) = { 1 1 - K ∑ I ∩ J = A m 1 ( I ) m 2 ( J ) , A ≠ ∅ 0 , A = ∅ ,(2) with K = ∑ I ∩ J = ∅ m 1 ( I ) m 2 ( J ) where K is a normalization constant. When K = 1, it reflects an absence of conflicts between m₁ and m₂. K = 0 implies a complete contradiction between m₁ and m₂. This shows that the Dempster’s rule of combination is only applicable to such belief structures that satisfy the condition of K < 1. This rule satisfies the commutative and associative properties, i.e., (i) m_1⊕ m₂=m_2⊕ m₁ and (ii) (m_1⊕m₂) ⊕m₃=m_1⊕ m_2⊕ m_3⊕.

Decision makers express their opinions over alternatives by using fuzzy preference relations under uncertainty [37]. A fuzzy preference relation can usually be represented as follows.

Definition 2. [45] A fuzzy preference relation R on a set of objects X = {X₁, X₂, ...  , X _n } is a fuzzy set on the product set X×X. It is characterized by a membership function μ _R : X × X ∈ [0, 1].

A fuzzy preference relation R on X can be conveniently expressed by an n×n matrix R = (r _ij ) _{n  ×n}, where r_ij =μ _R (X _i , X _j ) (i, j = 1, 2, ...  , n) is interpreted as the preference degree or intensity of the alternative X _i over X _j . When r _ij  = 0.5, it indicates indifference between X _i and X _j (X _i ∼X _j ); r _ij  > 0.5 indicates that X _i is preferred to X _j (X_i > X _j ); r_ij = 1, denotes that X _i is definitely preferred to X _j . In general, R = (r _ij ) _{n  ×n} satisfies the additive reciprocity property, namely, r _ij +r_ji = 1 and r_ii = 0.5 for all i, j = 1, 2,..., n.

In complex GDM problems, the preference relations provided by DMs may be irrational and may result in inconsistent conclusions [46]. The consistency in fuzzy preference relations given by decision makers has a direct influence on the final ranking results [39]. Considering that the values in the fuzzy preference relations based on additive consistency may not be in the fixed scope. For example, if we let r₁₂ = 0.8 and r₂₃ = 0.8, it is unreasonable for r₁₃ = r₁₂ + r₂₃–0.5 = 1.1 > 1. The multiplicative consistency on fuzzy preference relations is of considerable importance in GDM [47, 48]. It avoids the drawback of additive consistency and has no transformation which can keep more original information.

In real situations, decision makers may only provide incomplete preference relations. Such incomplete preferences can usually be defined as follows.

Definition 3. [49] Let R = (r _ij ) _{n  ×n} be an incomplete fuzzy preference relation. R is a multiplicative consistent incomplete fuzzy preference relation if all the known elements satisfy multiplicative transitivity i.e. r _il r _lj r_ji = r _li r _jl r _ij , i, j, l∈N.

In a group decision situation, there are m (m≥2) decision makers and n (n≥2) alternatives denoted by D = {D₁, D₂, ...  , D _m } and X = {X₁, X₂, ...  , X _n }, respectively. A third-party evaluator is invited to determine the weights of decision makers with respect to the criteria C = {C₁, C₂, ...  , C _t } (t≥2). The criteria’s weight vector can be expressed by η= {η₁, ...  , η _γ , ...  , η _t }, determined by the decision maker based on the research background in advance, where η _γ satisfying η _γ >0 (γ= 1, 2, ...  , t) and ∑ γ = 1 t η γ = 1 . In order to obtain the alternatives’ final ranking, it is extremely necessary to adequately consider the weights of individual decision makers and their own corresponding ranking of alternatives. In addition, the group consensus should be also considered in solving the GDM problem. The weights of decision makers has a significant impact on the GDM outcome. This is because such weights can change the group consensus degree and directly influence the overall ranking of the available alternatives in a situation.

The evaluation of the decision maker provided by the third-party evaluator is the prerequisite in determining the weights of decision makers. To facilitate the making of the subjective assessment in the evaluations of decision makers by the third-party in GDM, linguistic terms [50] shown as in Table 1 can be used.

To facilitate the computation process in GDM, each linguistic term (s _g ) is approximated by its corresponding utility value defined as in Table 2.

In an uncertain environment, a piece of evidences provided by the third-party evaluator can be represented as the belief structure m k γ ( s g ) (k = 1, 2, ...  , m; γ= 1, 2, ...  , t; g = 1, 2, ...  , 7), shown as in Table 3. The belief structure m k γ ( s g ) represents the evaluation of D _k (k = 1, 2, ...  , m) with respect to C _γ (γ= 1, 2, ...  , t), which can directly express the uncertainty involving fuzzy and probabilistic information simultaneously, and also shows how strongly the evidence supports the linguistic term s _g , satisfying ∑ g = 1 7 m k γ ( s g ) = 1 . For instance, under C₁, D₁’s evaluation provided by a third-party evaluator shows as “it is 0.6 sure that the level is good (G) and 0.4 sure that the level is middle (M)”. Accordingly, D₁’s belief structure m 1 1 ( s g ) can be expressed by “({G}, 0.6; {M}, 0.4)”. Such functions can then be synthesized with respect to the Dempster’s evidence combination rules under different criteria. The combination of evidence can significantly reduce the uncertainty of decision making, make the final decision more clear. As a result, the evaluation of the decision maker by the third-party evaluator in GDM can be better conducted, leading to more reliable weights of individual decision makers in GDM.

In order to rank the alternatives X = {X₁, X₂, ...  , X _n }, these m decision makers provide their preference information of the pros and cons of the two alternatives, which can be represented by the following fuzzy preference relation R ( k ) = ( r ij ( k ) ) n × n , k = 1 , 2 , … , m . R is an incomplete fuzzy preference relation if some of its elements cannot be provided by the decision maker, which can be denoted as the unknown number a. R ( k ) = ( r 11 ( k ) r 12 ( k ) . . . r 1 n ( k ) r 21 ( k ) r 22 ( k ) . . . r 2 n ( k ) . . . . . . . . . . . . r n 1 ( k ) r n 2 ( k ) . . . r nn ( k ) )

With the formulation of the GDM problem as above, a novel approach can be developed for effectively solving such a problem, detailed as follows.

Phase I: The selection process

#1 Calculate the weights of decision makers

The belief structure can directly express the uncertainty involving fuzzy and probabilistic information simultaneously, and also shows how strongly the evidence supports the linguistic term. With the use of the belief structure m k γ ( s g ) representing D _k ’s evaluation with respect to C _γ and the criteria weight vector η, D _k ’s (k = 1, 2, ...  , m) comprehensive evaluation, namely the integrated belief structure, can be conducted by integrating their belief structures under different criteria. To avoid the problem that the Dempster’s rule cannot directly handle the combination of highly conflicting evidences, the weighted average method [51] can be adopted. As a result, the weighted average belief structure of s _g for D _k can be obtained as follows: m ¯ k ( s g ) = ∑ γ = 1 t ( η γ × m k γ ( s g ) ) , ( g = 1 , 2 , . . . , 7 ) .(3)

On the basis of Equation (3), the weighted average belief structure of s _g under each criterion for D _k can be identified as follows: m ¯ k ( s g ) = m ¯ k 1 ( s g ) = m ¯ k 2 ( s g ) = . . . = m ¯ k t ( s g ) .(4)

Based on Equation (4), the D _k ’s final belief structure can be obtained by m k f ( s g ) = [ . . . [ m ¯ k 1 ( s g ) ⊕ m ¯ k 2 ( s g ) ] ⊕ . . . ⊕ m ¯ k t ( s g ) ](5) where ⊕ is the Dempster’s rule of combination.

As a consequence, the overall aggregated belief structure for each decision maker can be obtained as in Table 4.

The utility value of each decision maker reflects the credibility of the opinions expressed by each decision maker in GDM problem. The larger the utility value of each decision maker, the higher the credibility of their provided opinions. With the determination of each decision maker’s integrated belief structure in Table 4 and the utility value u (s _g ) of each linguistic term s _g in Table 2, the utility value of each decision maker can be determined as follow. u k = ∑ g = 1 7 u ( s g ) m k f ( s g ) , ( k = 1 , 2 , . . . , m ) .(6)

The weight p _k of D _k (k = 1, 2, ...  , m) can be determined by normalizing the results in Equation (6) as follows: p k = u k ∑ k = 1 m u k , ( k = 1 , 2 , . . . , m ) .(7)

Based on the above analyses, the following algorithm is offered.

Algorithm 1

Input: The predefined decision makers set D = {D₁, D₂, ...  , D _m }, the criteria set C = {C₁, C₂, ...  , C _t } (t≥2), The criteria’s weight vector η= {η₁, ...  , η _γ , ...  , η _t }, where η _γ satisfying η _γ >0 (γ= 1, 2, ...  , t) and ∑ γ = 1 t η γ = 1 , the utility value u (s _g ) of the linguistic term s _g (g = 1, 2, ...  , 7), and the evaluation m k γ ( s g ) of D _k (k = 1, 2, ...  , m) with respect to C _γ (γ= 1, 2, ...  , t) by the third-party evaluator, satisfying ∑ g = 1 7 m k γ ( s g ) = 1 .

#2 Derive the temporal overall ranking of alternatives

In order to obtain the overall ranking of alternatives, the corresponding ranking of alternatives for D _k (k = 1, 2, ...  , m) should be assessed by combining with the given fuzzy preference relations after determining the D _k ’s weight in Equation (7). Based on the logarithmic least square method [39] and fuzzy preference relation R ( k ) = ( r ij ( k ) ) n × n , D _k ’s ranking of alternatives is determined by W n - 1 ( k ) = Z n - 1 ( k ) - 1 Y n - 1 ( k ) ,(8) where Z n - 1 ( k ) and Y n - 1 ( k ) can be described as follows: Z n - 1 ( k ) = ( ∑ j = 2 n σ 1 j ( k ) δ 1 j ( k ) - σ 12 ( k ) δ 12 ( k ) . . . - σ 1 , n - 1 ( k ) δ 1 , n - 1 ( k ) - σ 21 ( k ) δ 21 ( k ) ∑ j = 1 , j ≠ 2 n σ 2 j ( k ) δ 2 j ( k ) . . . - σ 2 , n - 1 ( k ) δ 2 , n - 1 ( k ) . . . . . . . . . . . . - σ n - 1 , 1 ( k ) δ n - 1 , 1 ( k ) - σ n - 1 , 2 ( k ) δ n - 1 , 2 ( k ) . . . ∑ j = 1 , j ≠ n - 1 n σ n - 1 , j ( k ) δ n - 1 . j ( k ) )(9) Y n - 1 ( k ) = ( ∑ j = 1 n σ 1 j ( k ) δ 1 j ( k ) ( ln r 1 j ( k ) - ln r j 1 ( k ) ) ∑ j = 1 n σ 2 j ( k ) δ 2 j ( k ) ( ln r 2 j ( k ) - ln r j 2 ( k ) ) . . . ∑ j = 1 n σ n - 1 , j ( k ) δ n - 1 , j ( k ) ( ln r n - 1 , j ( k ) - ln r j , n - 1 ( k ) ) )(10) where σ ij ( k ) and δ ij ( k ) are both a zero or one integer variable which can be respectively defined as: σ ij ( k ) = { 0 , if r ij ( k ) = 1 or 0 1 , otherwise , i , j ∈ N , k ∈ M , and δ ij ( k ) = { 0 , if r ij ( k ) = a 1 , if r ij ( k ) ≠ a , i , j ∈ N , k ∈ M , where a denotes the unknown number in an incomplete fuzzy preference relation, and M = {1, 2, ...  , m}, N = {1, 2, ...  , n}. In this situation, W n - 1 ( k ) = ( ln w 1 ( k ) , ln w 2 ( k ) , . . . , ln w n - 1 ( k ) ) T . For each decision maker D _k , the weight of each alternative can be expressed as:

w i ( k ) = { w i ( k ) = exp ( W i ( k ) ) ∑ j = 1 n - 1 exp ( W i ( k ) ) + 1 , i = 1 , 2 , . . . , n - 1 , w i ( k ) = 1 ∑ j = 1 n - 1 exp ( W i ( k ) ) + 1 , i = n .(11)

For D _k , the weights of alternatives can be expressed by the vector w ( k ) = ( w 1 ( k ) , w 2 ( k ) , . . . , w n ( k ) ) (k = 1, 2, ...  , m) which satisfies the conditions ∑ i = 1 n w i ( k ) = 1 and w i ( k ) > 0 for i = 1, 2,..., n. Thus, individual alternatives’ ranking can be obtained.

In real situation, it is often difficult if not impossible for decision makers to give consistent assessments in assessing their preference of individual alternatives in GMD. To ensure that the GDM outcome can be accepted by all the decision makers, it is necessary to verify the reliability of the rankings based on the multiplicative consistency. On the basis of the consistency check [39], the measurement of consistency of the pairwise comparison matrix, namely fuzzy preference relation, provided by D _k can be calculated as follows:

{ FCI ( k ) = 1 n ( n - 1 ) ∑ 1 ⩽ i ⩽ j ⩽ n σ ij ( k ) δ ij ( k ) ( r ij ( k ) r ji ( k ) w j ( k ) w i ( k ) + r ji ( k ) r ij ( k ) w i ( k ) w j ( k ) - 2 ) FCR ( k ) = FCI ( k ) RI(12)

The fuzzy consistency ratio (FCR) is obtained by comparing the fuzzy consistency index (FCI) with the RI, where RI (random index) is the mean consistency index shown in Table 5 [52].

FCR^(k) < 0.1 shows that the fuzzy preference relation R is at the acceptable consistency. If an inconsistent fuzzy preference relation R is present, the decision maker is required to restructure a new matrix according to their new judgments until the fuzzy preference relation R achieves at a satisfactory consistency.

Based on the D _k ’s weight p _k and its corresponding alternatives’ weights vector w ( k ) = ( w 1 ( k ) , w 2 ( k ) , . . . , w n ( k ) ) (k = 1, 2, ...  , m), the final integration weight w i f on the alternative X _i can be determined by w i f = ∑ k = 1 m p k w i ( k ) , i = 1 , 2 , . . . , n .(13)

According to the calculated final integration weight w i f , the overall ranking of alternatives can be obtained. The larger the integration weight w i f , the better the alternative.

From the above analyses, the algorithm 2 is introduced as follows.

Algorithm 2

Input: The predefined decision maker D _k (k = 1, 2, ...  , m), the alternatives set X = {X₁, X₂, ...  , X _n }, individual fuzzy preference relation R ( k ) = ( r ij ( k ) ) n × n provided by D _k for X, and the given consistency threshold 0.1.

Phase II: The consensus process

Achieving a high level of consensus is essential in GDM. Classically, consensus is about the full and unanimous agreement among all the decision makers on all the possible alternatives [53]. It is necessary for measuring the consensus degree according to fuzzy preference relations of decision makers. In this study, a new consensus index is defined by taking the weights of the decision makers into consideration. This leads to the development of a group consensus degree (GCD) for measuring the similarity between individual values and collective values.

Definition 4. Let w i ( k ) and w i f be as earlier, GCD is determined by GCD = ∑ k = 1 m p k 1 1 + ∑ i = 1 n ( w i ( k ) - w i f ) 2 .(14)

The GCD value is normalized between [0, 1]. The larger GCD value shows the higher consensus degree among all decision makers. If GCD = 1, that means a full and unanimous agreement of all decision makers.

When the GCD value reaches the consensus level threshold τ which can be determined in advance, the decision making process is completed, and the ranking outcome can be determined. Otherwise, decision makers need to adjust their preference relations and recalculate related parameters until GCD≥τ. This is called the feedback adjustment which aims to give adjustment suggestions for assisting decision makers to improve the consensus. Similar to [5], the adjustment rules can be listed as follows:

R.1. If w i ( k ) > w i f , D _k should decrease the preference value of X _i ;

R.2. If w i ( k ) = w i f , D _k should not change the preference value of X _i ;

R.3. If w i ( k ) < w i f , D _k should increase the preference value of X _i .

There is not a uniform method [54] to choose the threshold value τ in GDM. When the decision is of much importance, the threshold for the consensus level should logically be as large as possible. Particularly, when the preference relations are almost incomplete, a relatively small threshold could be applied.

From the above analyses, the algorithm 3 is introduced as follows.

Algorithm 3

Input: The predefined decision maker D _k (k = 1, 2, ...  , m), the calculated D _k ’s weight p _k and alternative’s weight vector w ( k ) = ( w 1 ( k ) , w 2 ( k ) , . . . , w n ( k ) ) (k = 1, 2, ...  , m), the final integration weight w i f , and the given consensus level threshold τ.

To sum up, the detail framework of the novel approach can be drawn as Fig. 1.

OPEN IN VIEWER

Fig. 1

The detail framework of the novel approach.

This paper presents a novel approach for GDM with fuzzy preference relations based on the evidence theory and the logarithmic least squares approach. Such an approach is capable of adequately considering the weights of decision makers while appropriately handling the uncertainty and imprecision in an effective manner. Such a way of identifying decision makers’ weights is more in accordance with the decision maker’s way of thinking. Furthermore, the proposed approach can be used under the condition in which the preference information is partially unknown. It is closer to the actual GDM situation in which both complete and incomplete fuzzy preference relations are often present. In addition, a consensus index is developed for facilitate the consensus building in GDM under uncertainty. More importantly it can be used to facilitate the interaction between the decision maker and the decision making process so that the final group decision can be better accepted. An example is given for illustrating the applicability of the proposed approach in solving similar GDM problems in real world. A comparative study shows that the proposed approach is favorable in solving real world GDM problems due to various advantages that it has over comparable approach.

However, there are still some limitations which need to be further studied:

With the development of information technology, more and more people participate in GDM problems, which makes large-scale group decision making (LSGDM) problems become a research hotspot [58, 59]. In addition, in the actual decision making process, decision makers are often bounded rational. Self-confidence, as one of the human psychological behaviors, has an important impact on decision making [60]. Thus, in future work, we will address to extend the study to LSGDM problems and discuss the influence of decision makers’ self-confidence levels on decision making.