An evidence theory based approach for group decision making under uncertainty

Abstract

With the changing business environment and the active participation of various stakeholders in the decision making process, it plays an increasingly important role to the weight of decision makers and the preference information given by decision makers. This paper presents a novel approach for group decision making under uncertainty with the involvement of the third-party evaluator in the decision making process. Recognizing the challenge in adequately determining the weight of decision makers in group decision making, the evidence theory is appropriately used with the involvement of the third-party evaluator. To effectively model the uncertainty and imprecision in the decision making process, fuzzy preference relations are used for better representing the subjective assessment of individual decision makers. To adequately determine the ranking of available alternatives, the logarithmic least square method is applied for appropriately aggregating the fuzzy preference relation of individual decision makers. A group consensus index is developed for facilitating consensus building in group decision making. This leads to better group decisions being made. A real-world application is presented that shows the proposed approach is effective in solving group decision making problems under uncertainty.

Keywords

Group decision making uncertainty modeling fuzzy numbers

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: $m (φ) = 0 and \sum_{A \subseteq 2^{Θ}} m (A) = 1,$ (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 $m (A) = {\begin{matrix} \frac{1}{1 - K} \sum_{I \cap J = A} m_{1} (I) m_{2} (J), A \neq \emptyset \\ 0, A = \emptyset \end{matrix},$ (2) with $K = \sum_{I \cap J = \emptyset} m_{1} (I) m_{2} (J)$ 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.

3 A novel approach for uncertain GDM

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 $\sum_{γ = 1}^{t} η_{γ} = 1$ . 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.

Table 1
Linguistic terms for evaluation

Linguistic terms Symbolic representation Description

VP s₁ Very poor

P s₂ poor

MP s₃ Medium poor

M s₄ Medium

MG s₅ Medium good

G s₆ good

VG s₇ Very good

Linguistic terms	Symbolic representation	Description
VP	s₁	Very poor
P	s₂	poor
MP	s₃	Medium poor
M	s₄	Medium
MG	s₅	Medium good
G	s₆	good
VG	s₇	Very good

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

Table 2

Utility values for linguistic terms

Linguistic terms (s_g)	VP	P	MP	M	MG	G	VG
Utility values u (s_g)	0.0	0.1	0.3	0.5	0.7	0.9	1.0

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 $\sum_{g = 1}^{7} m_{k}^{γ} (s_{g}) = 1$ . 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.

Table 3

Decision makers’ evaluations provided by the third-party evaluator under different criteria

D	C	The evaluations provided by the third-party evaluator
D ₁	C ₁	$m_{1}^{1} (s_{g})$
	...	...
	C _γ	$m_{1}^{γ} (s_{g})$
	...	...
	C_t	$m_{1}^{t} (s_{g})$
...	...	...
D_k	C ₁	$m_{k}^{1} (s_{g})$
	...	...
	C _γ	$m_{k}^{γ} (s_{g})$
	...	...
	C_t	$m_{k}^{t} (s_{g})$
...	...	...
D_m	C ₁	$m_{m}^{1} (s_{g})$
	...	...
	C _γ	$m_{m}^{γ} (s_{g})$
	...	...
	C_t	$m_{m}^{t} (s_{g})$

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 $R^{(k)} = {(r_{ij}^{(k)})}_{n \times n}, k = 1, 2, \dots, m$ . 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. $R^{(k)} = (\begin{matrix} r_{11}^{(k)} r_{12}^{(k)} . . . r_{1 n}^{(k)} \\ r_{21}^{(k)} r_{22}^{(k)} . . . r_{2 n}^{(k)} \\ . . . . . . . . . . . . \\ r_{n 1}^{(k)} r_{n 2}^{(k)} . . . r_{nn}^{(k)} \end{matrix})$

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: ${\bar{m}}_{k} (s_{g}) = \sum_{γ = 1}^{t} (η_{γ} \times m_{k}^{γ} (s_{g})), (g = 1, 2, . . ., 7) .$ (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: ${\bar{m}}_{k} (s_{g}) = {\bar{m}}_{k}^{1} (s_{g}) = {\bar{m}}_{k}^{2} (s_{g}) = . . . = {\bar{m}}_{k}^{t} (s_{g}) .$ (4)

Based on Equation (4), the D_k’s final belief structure can be obtained by $m_{k}^{f} (s_{g}) = [. . . [{\bar{m}}_{k}^{1} (s_{g}) \oplus {\bar{m}}_{k}^{2} (s_{g})] \oplus . . . \oplus {\bar{m}}_{k}^{t} (s_{g})]$ (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.

Table 4

The final aggregated belief structure for each decision maker

D	Final aggregated belief structure
D ₁	$m_{1}^{f} (s_{g})$
⋮	⋮
D_k	$m_{k}^{f} (s_{g})$
⋮	⋮
D_m	$m_{m}^{f} (s_{g})$

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. $u_{k} = \sum_{g = 1}^{7} u (s_{g}) m_{k}^{f} (s_{g}), (k = 1, 2, . . ., m) .$ (6)

The weight p_k of D_k (k = 1, 2, ... , m) can be determined by normalizing the results in Equation (6) as follows: $p_{k} = \frac{u_{k}}{\sum_{k = 1}^{m} u_{k}}, (k = 1, 2, . . ., m) .$ (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 $\sum_{γ = 1}^{t} η_{γ} = 1$ , 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 $\sum_{g = 1}^{7} m_{k}^{γ} (s_{g}) = 1$ .

Output: The weight p_k of D_k (k = 1, 2, ... , m).

Step 1: Use Equation (3) to calculate the weighted average belief structure ${\bar{m}}_{k} (s_{g})$ of s_g(g = 1, 2, ... , 7), then use Equations (4) and (5) to obtain the D_k’s final belief structure $m_{k}^{f} (s_{g})$ .

Step 2: Calculate the utility value u_k via Equation (6).

Step 3: Obtain the weight p_k of D_k (k = 1, 2, ... , m) via Equation (7).

Step 4: End

#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 $R^{(k)} = {(r_{ij}^{(k)})}_{n \times n}$ , 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: $Z_{n - 1}^{(k)} = (\begin{matrix} \sum_{j = 2}^{n} σ_{1 j}^{(k)} δ_{1 j}^{(k)} - σ_{12}^{(k)} δ_{12}^{(k)} . . . - σ_{1, n - 1}^{(k)} δ_{1, n - 1}^{(k)} \\ - σ_{21}^{(k)} δ_{21}^{(k)} \sum_{j = 1, j \neq 2}^{n} σ_{2 j}^{(k)} δ_{2 j}^{(k)} . . . - σ_{2, n - 1}^{(k)} δ_{2, n - 1}^{(k)} \\ . . . . . . . . . . . . \\ - σ_{n - 1, 1}^{(k)} δ_{n - 1, 1}^{(k)} - σ_{n - 1, 2}^{(k)} δ_{n - 1, 2}^{(k)} . . . \sum_{j = 1, j \neq n - 1}^{n} σ_{n - 1, j}^{(k)} δ_{n - 1 . j}^{(k)} \end{matrix})$ (9) $Y_{n - 1}^{(k)} = (\begin{matrix} \sum_{j = 1}^{n} σ_{1 j}^{(k)} δ_{1 j}^{(k)} (ln r_{1 j}^{(k)} - ln r_{j 1}^{(k)}) \\ \sum_{j = 1}^{n} σ_{2 j}^{(k)} δ_{2 j}^{(k)} (ln r_{2 j}^{(k)} - ln r_{j 2}^{(k)}) \\ . . . \\ \sum_{j = 1}^{n} σ_{n - 1, j}^{(k)} δ_{n - 1, j}^{(k)} (ln r_{n - 1, j}^{(k)} - ln r_{j, n - 1}^{(k)}) \end{matrix})$ (10) where $σ_{ij}^{(k)}$ and $δ_{ij}^{(k)}$ are both a zero or one integer variable which can be respectively defined as: $σ_{ij}^{(k)} = {\begin{matrix} 0, if r_{ij}^{(k)} = 1 or 0 \\ 1, otherwise \end{matrix}, i, j \in N, k \in M,$ and $δ_{ij}^{(k)} = {\begin{matrix} 0, if r_{ij}^{(k)} = a \\ 1, if r_{ij}^{(k)} \neq a \end{matrix}, i, j \in N, k \in M,$ 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:

$w_{i}^{(k)} = {\begin{matrix} w_{i}^{(k)} = \frac{exp (W_{i}^{(k)})}{\sum_{j = 1}^{n - 1} exp (W_{i}^{(k)}) + 1}, i = 1, 2, . . ., n - 1, \\ w_{i}^{(k)} = \frac{1}{\sum_{j = 1}^{n - 1} exp (W_{i}^{(k)}) + 1}, i = n . \end{matrix}$ (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 $\sum_{i = 1}^{n} w_{i}^{(k)} = 1$ 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:

${\begin{matrix} {FCI}^{(k)} = \frac{1}{n (n - 1)} \sum_{1 ⩽ i ⩽ j ⩽ n} σ_{ij}^{(k)} δ_{ij}^{(k)} (\frac{r_{ij}^{(k)}}{r_{ji}^{(k)}} \frac{w_{j}^{(k)}}{w_{i}^{(k)}} + \frac{r_{ji}^{(k)}}{r_{ij}^{(k)}} \frac{w_{i}^{(k)}}{w_{j}^{(k)}} - 2) \\ {FCR}^{(k)} = \frac{{FCI}^{(k)}}{RI} \end{matrix}$ (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].

Table 5

The mean consistency index of randomly generated matrices

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
RI	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46	1.49	1.52	1.54	1.56	1.58	1.59

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 $w_{i}^{f} = \sum_{k = 1}^{m} p_{k} w_{i}^{(k)}, i = 1, 2, . . ., n .$ (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 $R^{(k)} = {(r_{ij}^{(k)})}_{n \times n}$ provided by D_k for X, and the given consistency threshold 0.1.

Output: The final integration weight $w_{i}^{f}$ and the temporal overall ranking of alternatives.

Step 1: Calculate the

$W_{n - 1}^{(k)} = (ln w_{1}^{(k)}, ln w_{2}^{(k)}, . . ., ln w_{n - 1}^{(k)})^{T}$ via Equations (8)–(10).

Step 2: Obtain the weight

$w^{(k)} = (w_{1}^{(k)}, w_{2}^{(k)}, . . ., w_{n}^{(k)})$ (k = 1, 2, ... , m) of alternatives via Equation (11), which satisfies the conditions $\sum_{i = 1}^{n} w_{i}^{(k)} = 1$ and $w_{i}^{(k)} > 0$ for i = 1, 2,..., n.

Step 3: Calculate the fuzzy consistency ratio FCR^(k) (k = 1, 2, ... , m)via Equation (12).

Step 4: If FCR^(k) < 0.1 (k = 1, 2, ... , m), then go to the next step; Otherwise, a new matrix provided by D_k is required to restructures until at a satisfactory consistency.

Step 5: Calculate alternative’s final integration weight $w_{i}^{f}$ on the alternative X_i via Equation (13), and then obtain the temporal overall ranking of alternatives.

Step 6: End

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 $GCD = \sum_{k = 1}^{m} p_{k} \frac{1}{1 + \sqrt{\sum_{i = 1}^{n} {(w_{i}^{(k)} - w_{i}^{f})}^{2}}} .$ (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.

Output: The final ranking of alternatives.

Step 1: Calculate the GCD via Equation (14) based on the vector $w^{(k)} = (w_{1}^{(k)}, w_{2}^{(k)}, . . ., w_{n}^{(k)})$ and the final integration weight $w_{i}^{f}$ .

Step 2: If GCD≥τ, then go to the next step; Otherwise, decision makers need to adjust their preference relations according to the adjustment rules: R.1∼R.3 until GCD≥τ.

Step 3: The temporal overall ranking is the final ranking of alternatives.

Step 4: End.

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

Fig. 1

The detail framework of the novel approach.

4 Case illustration and comparative analysis

4.1 Case illustration

In this section, an application of the developed approach is illustrated by an example of scientific foundation project selection. Such a scientific foundation project evaluation is usually completed by the cooperation of several evaluation experts by comprehensively incorporating their evaluation opinions. The evaluations of different experts, however, is often inconsistent. In general, the results of the expert review are usually influenced by the individual characteristics of the expert review and the review process [55, 56]. This shows that the assessment should be conducted by considering the differences among evaluation experts rather than using the average value only.

This section considers a GDM situation of selecting scientific foundation projects involved four evaluation experts reviewed by the academic committee. In order to determine the weights of four evaluation experts, the academic committee estimates four evaluation experts D₁, D₂, D₃ and D₄ under three evaluation indicators including C₁: review performance, C₂: science credibility and C₃: review ability which constitute three indispensable parts of the evaluation expert selection indicator system [57]. The indicators’ weights are assumed to be η= {0.4, 0.25, 0.35} defined by the academic committee based on different industry backgrounds. The academic committee gives his evaluations of each evaluation expert with respect to each indicator shown as in Table 6.

Table 6
Evaluations of the academic committee with respect to each indicator

Evaluation experts Indicator The evaluations

D ₁ C ₁ ({M},0.9; {MG},0.1)

C ₂ ({P},0.1; {MP},0.1; {M},0.1; {MG},0.7)

C ₃ ({MP},0.1; {M},0.8; {G},0.1)

D ₂ C ₁ ({M},1.0)

C ₂ ({VP},0.1; {M},0.6; {MG},0.2; {G},0.1)

C ₃ ({M},0.1; {MG},0.3; {G},0.6)

D ₃ C ₁ ({P},0.1; {MP},0.1; {M},0.1; {MG},0.6; {G},0.1)

C ₂ ({MP},0.1; {M},0.8; {MG},0.1)

C ₃ ({M},0.1; {MG},0.1; {G},0.8)

D ₄ C ₁ ({VP},0.1; {M},0.1; {MG},0.7; {G},0.1)

C ₂ ({M},0.9; {MG},0.1)

C ₃ ({M},1.0)

Evaluation experts	Indicator	The evaluations
D ₁	C ₁	({M},0.9; {MG},0.1)
	C ₂	({P},0.1; {MP},0.1; {M},0.1; {MG},0.7)
	C ₃	({MP},0.1; {M},0.8; {G},0.1)
D ₂	C ₁	({M},1.0)
	C ₂	({VP},0.1; {M},0.6; {MG},0.2; {G},0.1)
	C ₃	({M},0.1; {MG},0.3; {G},0.6)
D ₃	C ₁	({P},0.1; {MP},0.1; {M},0.1; {MG},0.6; {G},0.1)
	C ₂	({MP},0.1; {M},0.8; {MG},0.1)
	C ₃	({M},0.1; {MG},0.1; {G},0.8)
D ₄	C ₁	({VP},0.1; {M},0.1; {MG},0.7; {G},0.1)
	C ₂	({M},0.9; {MG},0.1)
	C ₃	({M},1.0)

The four evaluation experts D₁, D₂, D₃ and D₄ provide their personal preference relations on four projects X₁, X₂, X₃ and X₄. The four preference relations are denoted by R⁽¹⁾, R⁽²⁾, R⁽³⁾ and R⁽⁴⁾ shown as:

$\begin{matrix} R^{(1)} = (\begin{matrix} 0.5 a 0.6 0.8 \\ a 0.5 a 0.4 \\ 0.4 a 0.5 0.7 \\ 0.2 0.6 0.3 0.5 \end{matrix}); \\ R^{(2)} = (\begin{matrix} 0.5 0.7 0.6 0.4 \\ 0.3 0.5 0.4 0.6 \\ 0.4 0.6 0.5 0.8 \\ 0.6 0.4 0.2 0.5 \end{matrix}); \end{matrix}$

$\begin{matrix} R^{(3)} = (\begin{matrix} 0.5 aa 0.6 \\ a 0.5 0.4 a \\ a 0.6 0.5 0.7 \\ 0.4 a 0.3 0.5 \end{matrix}); \\ R^{(4)} = (\begin{matrix} 0.5 0.6 aa \\ 0.4 0.5 0.3 0.6 \\ a 0.7 0.5 a \\ a 0.4 a 0.5 \end{matrix}) . \end{matrix}$

Phase I: The selection process

Algorithm 1:

Step 1: For each evaluation expert, the aggregation of the belief structures of different indicators can be done based on the indicators’ weight vector η and Table 6. Take D₁ in Table 6 as an example. There are three belief structures from three indicators. $\begin{matrix} m_{1}^{1} ({M}) = 0.9, m_{1}^{1} ({MG}) = 0.1; \\ m_{1}^{2} ({P}) = 0.1, m_{1}^{2} ({MP}) = 0.1, \\ m_{1}^{2} ({M}) = 0.1, m_{1}^{2} ({MG}) = 0.7; \\ m_{1}^{3} ({MP}) = 0.1, m_{1}^{3} ({M}) = 0.8, m_{1}^{3} ({G}) = 0.1 . \end{matrix}$

The weight of each indicator is shown as follows: $η_{1} = 0.4, η_{2} = 0.25, η_{3} = 0.35 .$

According to Equations (3)–(5), the final aggregated evaluation for D₁ can be obtained, namely.

$\begin{matrix} m_{1}^{f} ({MP}) = 0.0007; \\ m_{1}^{f} ({M}) = 0.9664; \\ m_{1}^{f} ({MG}) = 0.0327; \\ m_{1}^{f} ({G}) = 0.0002 . \end{matrix}$

The values of $m_{1}^{f} ({P})$ is less than 0.0001 which can be ignored due to its small impact on the final result. As a result, all aggregated evaluations can be obtained as shown in Table 7.

Table 7

The final aggregated belief structure for each evaluation expert

Evaluation experts	Final aggregated belief structure
D ₁	({MP},0.0007; {M},0.9664; {MG},0.0327; {G},0.0002)
D ₂	({M},0.9230; {MG},0.0172; {G},0.0598)
D ₃	({P},0.0008; {MP},0.0034; {M},0.2571; {MG},0.3337; {G}0.4050)
D ₄	({VP},0.0002; {M},0.8909; {MG},0.1087; {G},0.0002)

Step 2∼Step 3: According to Tables 2 and 7, the utility value and the weight of each evaluation expert can be determined based on Equations (6) and (7). The utility value of D₁ can be calculated as follows: $\begin{matrix} u_{1} = u (MP) m_{1} (MP) + u (M) m_{1} (M) \\ + u (MG) m_{1} (MG) + u (G) m_{1} (G) \\ = 0.3 \times 0.0007 + 0.5 \times 0.9664 + 0.7 \times 0.0327 \\ + 0.9 \times 0.0002 \\ = 0.5065 . \end{matrix}$

Similarly, other utility values can be calculated as u2 = 0.5274, u3 = 0.7277, u4 = 0.5217.

This leads to the normalized weights of evaluation experts calculated respectively as follows:

p1 = 0.2218, p2 = 0.2310, p3 = 0.3187, p4 = 0.2285.

Algorithm 2:

Step 1: On the basis of the given preference relations R⁽¹⁾, R⁽²⁾, R⁽³⁾ and R⁽⁴⁾, the projects’ weights of evaluation experts can be determined according to Equations (8)–(10). For D₁, $Z_{3}^{(1)}$ and $Y_{3}^{(1)}$ can be obtained as follows: $\begin{matrix} Z_{3}^{(1)} = (\begin{matrix} 20 - 1 \\ 010 \\ - 102 \end{matrix}) and \\ Y_{3}^{(1)} = (\begin{matrix} ln \frac{0.6}{0.4} + ln \frac{0.8}{0.2} \\ ln \frac{0.4}{0.6} \\ ln \frac{0.4}{0.6} + ln \frac{0.7}{0.3} \end{matrix}) . \end{matrix}$

Step 2∼Step 4: As a result, $W_{3}^{(1)} = (ln w_{1}^{(1)}$ , $ln w_{2}^{(1)}, ln w_{3}^{(1)})^{T} = (1.3418, - 0.4055, 0.8918)^{T}$ .

The projects’ weights of D₁ can be determined by Equation (11) as $w^{(1)} = (0.4823, 0.0840, 0.3076, 0.1261) .$

The consistency index of D₁ is calculated by Equation (12) as ${FCI}^{(1)} = 0.0005, {FCR}^{(1)} = 0.0006 .$

Matrix R⁽¹⁾ is an acceptable consistent preference relation because FCR⁽¹⁾ = 0.0006 < 0.1. Similarly, the projects’ weights with respect to D₂, D₃ and D₄ are determined as: $\begin{matrix} w^{(2)} = (0.2967, 0.1942, 0.3394, 0.1697); \\ w^{(3)} = (0.2348, 0.2435, 0.3652, 0.1565); \\ w^{(4)} = (0.2727, 0.1818, 0.4243, 0.1212) . \end{matrix}$

A consistency check is conducted for D₂, D₃ and D₄. Similarly, the consistency index of D₂, D₃ and D₄ can be calculated as follows: ${FCR}^{(2)} = 0.1949; {FCR}^{(3)} = 0; {FCR}^{(4)} = 0 .$

The matrices R⁽³⁾ and R⁽⁴⁾ are acceptable consistent preference relations because FCR⁽³⁾ and FCR⁽⁴⁾ are less than 0.1. However, the matrix R⁽²⁾ is not acceptable due to FCR⁽²⁾ = 0.1949 > 0.1. This matrix is returned to the D₂ to reconsider. A new matrix is developed as follows (where r₁₄ = 0.4 → 0.6, r₄₁ = 0.6 → 0.4): $R^{(2) *} = (\begin{matrix} 0.5 0.7 0.6 0.6 \\ 0.3 0.5 0.4 0.6 \\ 0.4 0.6 0.5 0.8 \\ 0.4 0.4 0.2 0.5 \end{matrix}) .$

In the similar manner, $\begin{matrix} w^{(2) *} = (0.3509, 0.1875, 0.3278, 0.1338); \\ {FCR}^{(2) *} = 0.0709 . \end{matrix}$ The adjusted matrix R^(2)* is acceptable because FCR^(2)* = 0.0709 < 0.1. As a result, the projects’ weights can be obtained for D₁, D₂, D₃ and D₄.

Step 5: Combined with the weights of evaluation decision makers, the final integration weights of projects X₁, X₂, X₃, X₄ can be respectively calculated with respect to Equation (13) as follows. $\begin{matrix} \begin{matrix} {w_{1}^{f}, w_{2}^{f}, w_{3}^{f}, w_{4}^{f}} \\ = (p_{1}, p_{2}, p_{3}, p_{4}) (\begin{matrix} w_{1}^{(1)} w_{2}^{(1)} w_{3}^{(1)} w_{4}^{(1)} \\ w_{1}^{(2)} w_{2}^{(2)} w_{3}^{(2)} w_{4}^{(2)} \\ w_{1}^{(3)} w_{2}^{(3)} w_{3}^{(3)} w_{4}^{(3)} \\ w_{1}^{(4)} w_{2}^{(4)} w_{3}^{(4)} w_{4}^{(4)} \end{matrix}) \\ = (0.2218, 0.2310, 0.3187, 0.2285) \\ (\begin{matrix} 0.4823 0.0840 0.3076 0.1261 \\ 0.3509 0.1875 0.3278 0.1338 \\ 0.2348 0.2435 0.3652 0.1565 \\ 0.2727 0.1818 0.4243 0.1212 \end{matrix}) \\ = (0.3252, 0.1811, 0.3573, 0.1364) . \end{matrix} \end{matrix}$

The temporal overall ranking of the four projects can then be obtained as follows: $X_{3} > X_{1} > X_{2} > X_{4} .$

The final project approved is project X₃.

Phase II: The consensus process

Based on Equation (14), the group consensus degree (GCD) in this situation can be calculated as follows: $GCD = 90.52$ With the discussion above and the relevant literature [54], the acceptable consensus degree can be set at 80% due to the almost incomplete preference information. This shows that the final ranking derived above is acceptable.

4.2 Comparative analysis and discussion

To further demonstrate the effectiveness of the proposed approach in solving GDM problems under uncertainty, a comparative analysis between the proposed approach and the logarithmic least squares approach [39] is conducted. The differences between these two approaches is whether the weights of decision makers are considered and whether the group consensus is reached.

With the use of the logarithmic least squares approach [39], the overall ranking of the four projects can be determined in the following. On the basis of the given preference relations R⁽¹⁾, R⁽²⁾, R⁽³⁾, R⁽⁴⁾ and Equations (28) and (30) in [39], the following can be determined as (where n = 4): $D_{3} = (\begin{matrix} 7 - 2 - 2 \\ - 28 - 3 \\ - 2 - 38 \end{matrix}) and Y_{3} = (\begin{matrix} 3.4500 \\ - 2.5055 \\ 3.9282 \end{matrix}) .$ Using Equations (31)–(34) in [39], W₃ = (ln w₁, ln w₂, ln w₃) ^T = (0.7443, 0.1475, 0.7324) ^T. As a result, w = (0.3318, 0.1827, 0.3279, 0.1576).

The consistency index of D₁, D₂, D₃ and D₄ are determined respectively by Equation (51) in [39], $\begin{matrix} {FCR}^{(1)} = 0.0852; {FCR}^{(2)} = 0.2094; \\ {FCR}^{(3)} = 0.0151; {FCR}^{(4)} = 0.0162 . \end{matrix}$ The matrices R⁽¹⁾, R⁽³⁾ and R⁽⁴⁾ are all acceptable consistent preference relations due to FCR⁽¹⁾, FCR⁽³⁾ and FCR⁽⁴⁾ are all less than 0.1. However, R⁽²⁾ is not an acceptable consistent preference relation because FCR⁽²⁾ = 0.2094 > 0.1. This matrix is returned to D₂ to reconsider structuring new matrix as follows (where r₁₄ = 0.4 → 0.6, r₄₁ = 0.6 → 0.4): $R^{(2)} = (\begin{matrix} 0.5 0.7 0.6 0.6 \\ 0.3 0.5 0.4 0.6 \\ 0.4 0.6 0.5 0.8 \\ 0.4 0.4 0.2 0.5 \end{matrix}) .$ This leads to the following $D_{3} = (\begin{matrix} 7 - 2 - 2 \\ - 28 - 3 \\ - 2 - 38 \end{matrix}) and Y_{3} = (\begin{matrix} 4.2609 \\ - 2.5055 \\ 3.9282 \end{matrix}) .$ Thus, $\begin{matrix} W_{3} = (ln w_{1}, ln w_{2}, ln w_{3})^{T} \\ = (0.8944, 0.2076, 0.7925)^{T}; \\ w = (0.3552, 0.1787, 0.3208, 0.1452) . \end{matrix}$ Applying Equation (51) in [39], $\begin{matrix} {FCR}^{(1)} = 0.0852; \\ {FCR}^{(2)} = 0.0746; \\ {FCR}^{(3)} = 0.0261; \\ {FCR}^{(4)} = 0.0162 . \end{matrix}$ The values of FCR⁽¹⁾, FCR⁽²⁾, FCR⁽³⁾ and FCR⁽⁴⁾ are all less than 0.1. This means that all four fuzzy preference relations are perfectly multiplicative consistent. The overall ranking of the four projects can be determined as $X_{1} > X_{3} > X_{2} > X_{4} .$ Table 8 has summarized the comparative analysis result. It shows that the best project X₁ in [39] is different from the result X₃ calculated by the proposed approach. This demonstrates the importance of effectively considering the weights of decision makers in GDM. With the use of the proposed approach to GDM, the use of the weights of evaluation experts identified by the evidence theory leads to better representing the uncertainty and imprecision simultaneously in GDM.

Table 8
The results of the comparative analysis

Methods Ranking values Ranking

The proposed approach w = (0.3252, 0.1811, 0.3573, 0.1364) X₃ > X₁ > X₂ > X₄

Method in [39] w = (0.3552, 0.1787, 0.3208, 0.1452) X₁ > X₃ > X₂ > X₄

Methods	Ranking values	Ranking
The proposed approach	w = (0.3252, 0.1811, 0.3573, 0.1364)	X₃ > X₁ > X₂ > X₄
Method in [39]	w = (0.3552, 0.1787, 0.3208, 0.1452)	X₁ > X₃ > X₂ > X₄

In order to further illustrate the important role of decision makers’ weights in group decision making. Refers to the literature [23], it is assumed that all decision makers are of equal importance as the weight vectors of the decision makers are not given. Here, five different weight assignments (the sum is 1) are randomly given, and then the change trend of alternative ranking results and group consensus degree is observed. The detailed results are shown in Table 9.

Table 9

The adjusted ranking and GCD value with different p_k

pk	Ranking values	Ranking	GCD
(0.1, 0.25, 0.4, 0.25)	w = (0.2981, 0.1981, 0.3649, 0.1390)	X₃ > X₁ > X₂ > X₄	92.06%
(0.2, 0.25, 0.3, 0.25)	w = (0.3228, 0.1822, 0.3591, 0.1359)	X₃ > X₁ > X₂ > X₄	90.84%
(0.25, 0.25, 0.25, 0.25)	w = (0.3352, 0.1742, 0.3562, 0.1344)	X₃ > X₁ > X₂ > X₄	90.44%
(0.3, 0.25, 0.2, 0.25)	w = (0.3476, 0.1662, 0.3533, 0.1329)	X₃ > X₁ > X₂ > X₄	90.13%
(0.4, 0.25, 0.1, 0.25)	w = (0.3723, 0.1503, 0.3476, 0.1298)	X₁ > X₃ > X₂ > X₄	89.8%

It can be seen from the Table 9 that when the weights of decision makers change in group decision making, the total ranking results of alternatives and the level of group consensus will change accordingly. From the fifth random weight in Table 9, it can be found that the ranking result is consistent with the result in reference [39], but the consensus degree is reduced to less than 90%, which proves the significance of the proposed method to calculate the consensus degree considering the weights of decision makers. It can be found that the weights of decision makers have a certain influence on the decision making results, which demonstrates that it is necessary to consider the weights of decision makers in selection and consensus reaching processes.

Based on the above comparative analysis, the advantages of the proposed approach can be summarized as follows:

In order to get a precise superiority index for each alternative, it is necessary to determine the decision makers’ weights. In this study, it has considerable practical value for us to use evidence theory to identify the weights of decision makers due to the belief structure which can express the uncertainty involving fuzzy and probability simultaneously. The process to the uncertainty problem is more in accordance with people’s way of thinking which bring a new insight to carry out specific process in identifying decision makers’ weights.

The proposed approach can be used under the condition where the preference information is partially unknown. It is closer to the actual situation to consider incomplete fuzzy preference relations owing to the lack of information grasped by decision makers.

The proposed method takes the weights of decision makers into consideration for the measurement of group consensus degree. It is also worth mentioning that the proposed method not only verifies the group consensus degree among all decision makers, also conducts the consistency check. Such mechanisms are essential for obtaining a more reasonable and accurate GDM outcomes. It is worth pointing out that this method can cope with more situations and are more reasonable than some existing methods.

5 Conclusion

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.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos.71971218, 71671188); Talent introduction project of Anhui Science and Technology University (No. GLYJ201902).

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An evidence theory based approach for group decision making under uncertainty

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Linguistic terms for evaluation Linguistic terms Symbolic representation Description VP s1 Very poor P s2 poor MP s3 Medium poor M s4 Medium MG s5 Medium good G s6 good VG s7 Very good

4.1 Case illustration

Table 8 The results of the comparative analysis Methods Ranking values Ranking The proposed approach w = (0.3252, 0.1811, 0.3573, 0.1364) X3 > X1 > X2 > X4 Method in [39] w = (0.3552, 0.1787, 0.3208, 0.1452) X1 > X3 > X2 > X4

Footnotes

Acknowledgments

References

Table 1
Linguistic terms for evaluation

Linguistic terms Symbolic representation Description

VP s₁ Very poor

P s₂ poor

MP s₃ Medium poor

M s₄ Medium

MG s₅ Medium good

G s₆ good

VG s₇ Very good

Table 8
The results of the comparative analysis

Methods Ranking values Ranking

The proposed approach w = (0.3252, 0.1811, 0.3573, 0.1364) X₃ > X₁ > X₂ > X₄

Method in [39] w = (0.3552, 0.1787, 0.3208, 0.1452) X₁ > X₃ > X₂ > X₄