Vague multi-attribute group decision-making method based on evidence theory with a new aspect to solve weights

Abstract

In order to solve the problems of weight solving and information aggregation in the Vague multi-attribute group decision-making, this paper first solves the weight of Vague evaluation value, and then fuses the information of Vague sets through evidence theory, and obtains an information aggregation algorithm for Vague multi-attribute group decision-making. Firstly, The algorithm draws on the idea of solving the weight of evidence in the improved evidence theory algorithm, and calculates the weight of Vague evaluation value, and revises the original evaluation information after obtaining the weight of each Vague evaluation value. Secondly, this algorithm analyzes the mathematical relationship between the Vague sets and the evidence theory, and uses the evidence theory to fuse the evaluation information to obtain the final Vague evaluation value of each alternative. Finally, this algorithm uses a score function to calculate the score of each alternative to determine the best alternative. The algorithm given in the paper enables decision-makers to make rational decisions in uncertain environments, and then select the best alternative.

Keywords

Vague sets evidence theory multi-attribute group decision-making score function

1 Introduction

Multi-attribute group decision-making refers to the process in which multiple experts in related fields score each alternative on each attribute, and select the best alternative through the information aggregation algorithm. However, due to the complexity of environment, it is difficult for experts to express evaluation information with accurate real values. To solve this problem, the researchers introduced different theories into the decision-making process, e.g. Pythagorean fuzzy set [1], interval neutrosophic hesitant fuzzy set [2], linguistic neutrosophic number [3], hesitant fuzzy sets [4], hesitant fuzzy linguistic term set [5], distributed linguistic representation [6], proportional interval type-2 hesitant fuzzy set [7]. The emergence of Vague sets theory provides a new way to solve this problem. Bustince and Burillo [8] have pointed out that the concept of intuitionistic fuzzy sets and that of Vague sets coincide with each other. Therefore, this paper no longer distinguishes the two concepts.

As a generalization of Fuzzy sets theory [9], Vague sets theory can express information on three aspects of truth-membership, false-membership and unknown degree at the same time, and is more expressive in capturing vagueness of data. Vague sets theory has been applied in decision-making in many areas,e.g. hotel selection [10]; design concept evaluation [11]; supplier selection [12]; Urban water security evaluation [13]. Considering the advantage of Vague sets in expressing information, scholars introduce Vague sets into multi-attribute group decision making model and get the Vague multi-attribute group decision making model. In the Vague multi-attribute group decision-making model, the Vague evaluation values of each alternative are given by the experts in the form of Vague value, which includes the support degree, opposition degree and unknown degree of the experts to the alternative. Therefore, the evaluation result is more reasonable.

At present, the academic researches on Vague multi-attribute group decision-making mainly focus on the ranking of alternatives, the aggregation of expert evaluation information and the solution of weights. In the researches of ranking of alternatives, Zeng [14] taken into consideration both the hesitation degrees of IFVs and the differences between the membership degrees and the non-membership degrees of IFVs and gave the score function of intuitionistic fuzzy set. Lin [15] solved the problem of information loss caused by the insufficient information of the unknown part and gave the score function of Vague sets to sort the alternatives. The larger the score is, the better the alternative is. Zhou [16] calculated the weights of the experts for different criteria and used TOPSIS method to calculate the distances between alternatives and the ideal solution to sort the alternatives.

In the researches of aggregation of expert evaluation information, scholars have defined the the product algorithm of Vague values and real values [17], the product algorithm between Vague values [18], and the union and intersection algorithm between Vague values [19] to aggregate the evaluation information of each alternative and obtain the final evaluation value of each alternative. Geng [11] aggregated the evaluation information based on minimizing the total deviation degree between each judgment. The unknown degree in Vague evaluation value includes the tendency of expert to support or oppose the alternative. However, these algorithms didn’t consider the reasonable redistribution of the unknown degree, so the aggregated result may lost part information of the original evaluation values. Niu [20] transformed the Vague decision matrix into real decision matrix by vague combination rule, and obtained the final decision result based on grey correlation analysis and evidence theory. Wang [21] transformed Vague values into Fuzzy values to solve the best alternative. However, these methods may lost information when converting Vague values into real values or Fuzzy values. Although Fu [22] used evidence theory to represent intuitionistic fuzzy set, he just solved the weight of expert and weight of attribute and didn’t solve the weight of evaluation value, so he didn’t solve the problem that evidence theory produces anti-intuitive results when dealing with high conflict evidences.

In the researches of solution of weights, Zhang [23] calculated the similarity between the Vague weight value of each expert and the accuracy of the Vague weight value of each expert as the objective weight value and the subjective weight value respectively, and then obtained the weight value of each expert. Elzarka [17] first calculated the weight of each expert according to the mutual score among experts, and then calculated the weight value of each attribute according to the weight matrix given by experts and expert weights. Robinson [18] proposed two methods to solve the expert weights based on RIM Linguistic Quantifiers and Gaussian distribution. At present, most of the methods to solve the expert weight or attribute weight will give the relevant information first, and there is less researches on not giving the information related to the weight at all. In addition, most of the current methods take the expert or attribute as the unit to solve the weight. However, Vague evaluation value is taken as the unit in the process of information fusion. Therefore, taking the Vague evaluation value as the unit to solve the weight can better reflect the state of each Vague evaluation value. However, there are few studies on this aspect.

Evidence theory is a method to deal with uncertain information. It can not only flexibly express the unknown degree, but also redistribute the unknown degree in the fusion process of evidences. It does not need prior probability in the process of information fusion [24], and provides a powerful tool for the representation and fusion of uncertain information, which is widely used in fault diagnosis, decision-making, risk assessment, pattern recognition and other fields [25]. In addition, evidence theory is widely used in group decision making in combination with other theories, for example the hesitant fuzzy linguistic term sets [26], linguistic intuitionistic fuzzy numbers [27], double hierarchy hesitant fuzzy linguistic term set [28], fuzzy soft set [29].

In the Vague multi-attribute group decision-making, there are few researches to study the weights of Vague evaluation value. Therefore, in order to reflect the state of each Vague evaluation value and solve the problem that evidence theory produces anti-intuitive results when dealing with high conflict evidences, this paper uses the idea of solving evidence weight in the improved evidence theory algorithms [24, 25] to solve the weight of each Vague evaluation value. In the aggregation of evaluation information, in order to solve the problem of unreasonable distribution of unknown degree and the problem of information loss when transforming Vague values into other types of data, this paper uses evidence theory to aggregate evaluation information. To sum up, the research of Vague multi-attribute group decision-making in this paper mainly focuses on two aspects: the first is to solve the weight of Vague evaluation value under the background of completely unknown information related to weight; the second is the aggregation of evaluation information based on evidence theory.

The rest of this paper are introduced around the Vague multi-attribute group decision-making model. In Sect. 2, this paper gives the Vague multi-attribute group decision-making algorithm. In Sect. 3, this paper gives a example to introduce how to use this algorithm. In Sect. 4, this paper illustrates the feasibility and effectiveness of this algorithm by comparing with other algorithms. In Sect. 5, this paper gives some application backgrounds about this algorithms. Some conclusions of this paper and future research directions are given in Sect. 6.

2 Information aggregation model based on evidence theory

2.1 Background

In the Vague multi-attribute group decision-making, experts evaluate each alternative according to their experience. A={A₁, A₂, …, A_m} is a expert set,C={C₁, C₂, …, C_n} is a attribute set, and D={D₁, D₂, …, D_L} is a expert set. Experts evaluate each alternative according to the performance of each alternative on each attribute, and give the Vague evaluation value of each alternative on each attribute. According to the experts’ evaluation information of m alternatives on n attributes, the decision matrix D^k is constructed. $D^{k} = (d_{ij}^{k})_{m \times n} = [\begin{matrix} d_{11}^{k} & \dots & d_{1 n}^{k} \\ ⋮ & ⋱ & ⋮ \\ d_{m 1}^{k} & \dots & d_{mn}^{k} \end{matrix}], k = 1, . . ., L$ (1)

In the above matrix, $d_{ij}^{k} = [t_{ij}^{k}, 1 - f_{ij}^{k}]$ is the Vague evaluation value of alternative A_i given by the expert D_k after considering attribute C_j, where $t_{ij}^{k}$ , $f_{ij}^{k}$ , $s_{ij}^{k} = (1 - f_{ij}^{k}) - t_{ij}^{k}$ respectively represent the support degree, opposition degree and unknown degree of alternative A_i given by expert D_k on attribute C_j.

2.2 Evidence information aggregation based on evidence theory

In the process of fusion of Vague evaluation values, if a Vague evaluation value is supported by other Vague evaluation values, then the Vague evaluation value is more reliable, its weight is larger, and it has a greater impact on the final fusion conclusion; on the contrary, if the conflict between a Vague evaluation value and other Vague evaluation values is larger, then the credibility of the Vague evaluation value is lower, and its weight is lower, and it has little influence on the final fusion conclusion. This is the idea of solving weight of Vague evaluation value in this paper.

When using evidence theory to fuse evaluation information, this paper starts from two aspects. First, the evidence theory is used to fuse the Vague evaluation values given by all experts to get the comprehensive Vague evaluation value of each alternative on each attribute. Then the evidence theory is used to fuse the comprehensive Vague evaluation values of each alternative on all attributes to get the final Vague evaluation value of each alternative. Combined with evidence theory, this paper studies the information aggregation problem of Vague multi-attribute group decision-making from the following two aspects.

2.2.1 Evidence information aggregation of expert set

When applying evidence theory to describe and solve uncertain problems, the set of all possible results of an uncertain problem is called a frame of discernment Θ which can be expressed by the mathematical formula Θ = {θ₁, θ₂, θ₃, ⋯ , θ_N}. The elements in Θ are independent and mutually exclusive. 2^Θ = {φ, {θ₁} , {θ₂} , ⋯ , {θ₁ ∪ θ₂} , ⋯ , {θ₁ ∪ θ_N} , ⋯ , Θ} is the set of all the possible subsets in Θ. A mass function m : 2^Θ → [0, 1], which satisfies the condition ∑_A∈2^Θm (A) =1, is called a basic probability assignment(BPA). m (A) is the basic probability number of proposition A. If m (A) >0, A will be called as a focal element. There are k basic probability assignments m₁, m₂,..., m_k, and the corresponding focal elements are A₁, A₂,..., A_k. Dempster’s rule of combination can be defined as follows: $m (A) = \frac{\sum_{A_{1} \cap . . . \cap A_{k} = A} m_{1} (A_{1}) \times \dots \times m_{k} (A_{k})}{1 - K}$ (2)

K = ∑_{A₁∩...∩A_k=φ}m₁ (A₁) × ⋯ × m_k (A_k) is the conflict coefficient, which is used to measure the conflict degree of between k bodies of evidence.

Evidence information aggregation of expert set based on evidence theory is to fuse Vague evaluation values of alternative A_i given by different experts on attribute C_j to obtain comprehensive Vague evaluation value d_ij. Before the Vague evaluation values of alternative A_i given by different experts on attribute C_j are fused, the weight value of Vague evaluation value of alternative A_i given by expert D_k on attribute C_j should be obtained first according to the evaluation information given by each expert. When solving the weight value of Vague evaluation value, the similarity between Vague evaluation values should be solved first. When a Vague evaluation value is more similar to other Vague evaluation values, the more it is supported by other Vague evaluation values, then the Vague evaluation value should be given a larger weight, on the contrary, it should be given a smaller weight.

At present, there are many methods to solve the similarity of Vague values, e.g. Lu [30], Li [31], Zeng [32], Wang [33]. After comparison, this paper chooses the method given in paper [33] to solve the similarity of Vague values. There are two Vague values x = [t_x, 1 - f_x] and y = [t_y, 1 - f_y], then the similarity M (x, y) between x and y is defined as below: $\begin{matrix} M (x, y) = 1 - \frac{| t_{x} - t_{y} | + | f_{x} - f_{y} |}{3} \\ - \frac{| π_{x} - π_{y} | + π_{x} + π_{y}}{3} \end{matrix}$ (3)

In the above equation, π_x = 1 - t_x - f_x,π_y = 1 - t_y - f_y.

Then the similarity matrix sim of Vague evaluation values of alternative A_i given by different experts on attribute C_j is defined as: $\begin{matrix} sim = (M (d_{ij}^{k_{1}}, d_{ij}^{k_{2}}))_{L \times L} \\ = [\begin{matrix} M (d_{ij}^{1}, d_{ij}^{1}) & M (d_{ij}^{1}, d_{ij}^{2}) & \dots & M (d_{ij}^{1}, d_{ij}^{L}) \\ M (d_{ij}^{2}, d_{ij}^{1}) & M (d_{ij}^{2}, d_{ij}^{2}) & \dots & M (d_{ij}^{2}, d_{ij}^{L}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ M (d_{ij}^{L}, d_{ij}^{1}) & M (d_{ij}^{L}, d_{ij}^{2}) & \dots & M (d_{ij}^{L}, d_{ij}^{L}) \end{matrix}] \\ i = 1, 2, . . ., m, j = 1, 2, . . ., n \end{matrix}$ (4)

The similarity between Vague evaluation values represents mutual support degree, and the total support $\sup (d_{ij}^{k})$ of Vague evaluation value $d_{ij}^{k}$ is defined as: $\begin{matrix} \sup (d_{ij}^{k}) = \sum_{h = 1, h \neq k}^{L} M (d_{ij}^{k}, d_{ij}^{h}), k = 1, 2, . . ., L \end{matrix}$ (5)

From this, the support matrix support of Vague evaluation values of alternative A_i given by different experts on attribute C_j is defined as: $\begin{matrix} support & = (\sup (d_{ij}^{k}))_{1 \times L} \\ = [\sup (d_{ij}^{1}), \sup (d_{ij}^{2}), . . ., \sup (d_{ij}^{L})] \end{matrix}$ (6)

After calculating the total support of a Vague evaluation value $d_{ij}^{k}$ , the weight $w_{ij}^{k}$ of Vague evaluation value $d_{ij}^{k}$ is defined as: $w_{ij}^{k} = \frac{\sup (d_{ij}^{k})}{\sum_{h = 1}^{L} \sup (d_{ij}^{h})}, k = 1, 2, . . ., L$ (7)

The weight matrix weight of Vague evaluation values of alternative A_i given by different experts on attribute C_j is defined as: $weight = (w_{ij}^{k})_{1 \times L} = [w_{ij}^{1}, w_{ij}^{2}, . . ., w_{ij}^{L}]$ (8)

After obtaining the weight of each Vague evaluation value, evidence theory can be used to fuse the evidence information of expert set. Firstly, according to the weight of each Vague evaluation value obtained, the Vague evaluation values of alternative A_i given by different experts on attribute C_j are weighted, and the weighted average Vague evaluation value ${\bar{d}}_{ij} = [{\bar{t}}_{ij}, 1 - {\bar{f}}_{ij}]$ of alternative A_i on attribute C_j is obtained, where ${\bar{t}}_{ij} = \sum_{k = 1}^{L} w_{ij}^{k} t_{ij}^{k}$ , ${\bar{f}}_{ij} = \sum_{k = 1}^{L} w_{ij}^{k} f_{ij}^{k}$ .

When using the evidence theory to fuse the evidence information of expert set, the frame of discernment is Θ = {S, N}, where S stands for support and N stands for opposition. From this, the basic probability numbers ${\bar{m}}_{ij} (S) = {\bar{t}}_{ij}$ , ${\bar{m}}_{ij} (N) = {\bar{f}}_{ij}$ , ${\bar{m}}_{ij} (Θ) = (1 - {\bar{f}}_{ij}) - {\bar{t}}_{ij}$ are obtained, where ${\bar{m}}_{ij} (S)$ , ${\bar{m}}_{ij} (N)$ , ${\bar{m}}_{ij} (Θ)$ respectively represent the weighted average support degree, weighted average opposition degree and weighted average unknown degree of alternative A_i on attribute C_j according to the evaluation information of L experts. Then, according to the Dempster’s rule of combination (2), the evidence information of expert set is aggregated to obtain the comprehensive Vague evaluation value d_ij of the alternative A_i on the attribute C_j, and the comprehensive decision matrix D is defined as: $D = (d_{ij})_{m \times n} = [\begin{matrix} d_{11} & \dots & d_{1 n} \\ ⋮ & ⋱ & ⋮ \\ d_{m 1} & \dots & d_{mn} \end{matrix}]$ (9)

In the above matrix, d_ij = [t_ij, 1 - f_ij], where $\begin{matrix} t_{ij} & = m_{ij} (S) \\ = \frac{\sum_{A_{1} \cap . . . \cap A_{L} = S} {\bar{m}}_{ij} (A_{1}) \times \dots \times {\bar{m}}_{ij} (A_{L})}{1 - K} \end{matrix}$ (10) $\begin{matrix} f_{ij} & = m_{ij} (N) \\ = \frac{\sum_{A_{1} \cap . . . \cap A_{L} = N} {\bar{m}}_{ij} (A_{1}) \times \dots \times {\bar{m}}_{ij} (A_{L})}{1 - K} \end{matrix}$ (11) $K = \sum_{A_{1} \cap . . . \cap A_{L} = φ} {\bar{m}}_{ij} (A_{1}) \times \dots \times {\bar{m}}_{ij} (A_{L})$ (12)

2.2.2 Evidence information aggregation of attribute set

After the comprehensive decision matrix D is obtained, the comprehensive Vague evaluation values of alternative A_i on different attributes can be fused by using evidence theory. Before the fusion of information, it is necessary to obtain the weight of the comprehensive Vague evaluation value of alternative A_i on attribute C_j according to the comprehensive evaluation information.

First of all, the similarity matrix sim of the comprehensive Vague evaluation values of the alternative A_i on different attributes is defined as: $\begin{matrix} sim = (M (d_{{ij}_{1}}, d_{{ij}_{2}}))_{n \times n} \\ = [\begin{matrix} M (d_{i 1}, d_{i 1}) & M (d_{i 1}, d_{i 2}) & \dots & M (d_{i 1}, d_{in}) \\ M (d_{i 2}, d_{i 1}) & M (d_{i 2}, d_{i 2}) & \dots & M (d_{i 2}, d_{in}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ M (d_{in}, d_{i 1}) & M (d_{in}, d_{i 2}) & \dots & M (d_{in}, d_{in}) \end{matrix}] \\ i = 1, 2, . . ., m \end{matrix}$ (13)

the total support sup (d_ij) of comprehensive Vague evaluation value d_ij is defined as: $\begin{matrix} \sup (d_{ij}) = \sum_{p = 1, p \neq j}^{n} M (d_{ij}, d_{ip}), j = 1, 2, . . ., n \end{matrix}$ (14)

From this, the support matrix support of comprehensive Vague evaluation values of alternative A_i on different attributes is defined as: $\begin{matrix} support & = (\sup (d_{ij}))_{1 \times n} \\ = [\sup (d_{i 1}), \sup (d_{i 2}), . . ., \sup (d_{in})] \end{matrix}$ (15)

Then the weight w_ij of comprehensive Vague evaluation value d_ij is defined as: $w_{ij} = \frac{\sup (d_{ij})}{\sum_{p = 1}^{n} \sup (d_{ip})}, j = 1, 2, . . ., n$ (16)

The weight matrix weight of the comprehensive Vague evaluation values of the alternative A_i on different attributes is defined as: $weight = (w_{ij})_{1 \times n} = [w_{i 1}, w_{i 2}, . . ., w_{in}]$ (17)

Then, according to the weight of each comprehensive Vague evaluation value obtained, the comprehensive Vague evaluation values of alternative A_i on different attributes are weighted, and the weighted average comprehensive Vague evaluation value ${\bar{d}}_{i} = [{\bar{t}}_{i}, 1 - {\bar{f}}_{i}]$ is obtained, where ${\bar{t}}_{i} = \sum_{j = 1}^{n} w_{ij} t_{ij}$ , ${\bar{f}}_{i} = \sum_{j = 1}^{n} w_{ij} f_{ij}$ .

When using the evidence theory to fuse the evidence information of attribute set, the frame of discernment is Θ = {S, N}, where S stands for support and N stands for opposition. From this, the basic probability numbers ${\bar{m}}_{i} (S) = {\bar{t}}_{i}$ , ${\bar{m}}_{i} (N) = {\bar{f}}_{i}$ , ${\bar{m}}_{i} (Θ) = (1 - {\bar{f}}_{i}) - {\bar{t}}_{i}$ are obtained, where ${\bar{m}}_{i} (S)$ , ${\bar{m}}_{i} (N)$ , ${\bar{m}}_{i} (Θ)$ respectively represent the weighted average comprehensive support degree, weighted average comprehensive opposition degree and weighted average comprehensive unknown degree of alternative A_i after considering n attributes. Then, according to the Dempster’s rule of combination (2), the final Vague evaluation value d_i = [t_i, 1 - f_i] of each alternative is obtained, where $\begin{matrix} t_{i} & = m_{i} (S) \\ = \frac{\sum_{A_{1} \cap . . . \cap A_{n} = S} {\bar{m}}_{i} (A_{1}) \times \dots \times {\bar{m}}_{i} (A_{n})}{1 - K} \end{matrix}$ (18) $\begin{matrix} f_{i} & = m_{i} (N) \\ = \frac{\sum_{A_{1} \cap . . . \cap A_{n} = N} {\bar{m}}_{i} (A_{1}) \times \dots \times {\bar{m}}_{i} (A_{n})}{1 - K} \end{matrix}$ (19) $K = \sum_{A_{1} \cap . . . \cap A_{n} = φ} {\bar{m}}_{i} (A_{1}) \times \dots \times {\bar{m}}_{i} (A_{n})$ (20)

2.3 Determination of the best decision alternative

In the Vague multi-attribute decision-making problem, the research of score function has always been a focus. The score function is used to solve the degree of alternative A_i meeting the requirements of the decision-makers. The larger the score is, the better the alternative is. According to the evaluation function E, the Vague evaluation value E (A_i) = [t_{A
_i}, 1 - f_{A
_i}] of the alternative A_i can be obtained. Through analyzing the relationship of t_{A
_i}, f_{A
_i}, and 1 - f_{A
_i} - t_{A
_i}, Lin [15] overcame the problem of information loss caused by the insufficient information of the unknown part and gave the score function formula: $S (E (A_{i})) = \frac{2}{3} t_{A_{i}} + \frac{1}{3} (1 - f_{A_{i}})$ (21)

After getting the score S (E (A_i)) of alternative A_i, the larger the value of S (E (A_i)), the better the alternative A_i.

2.4 The validity proof of this method

To illustrate the effectiveness of this method, this paper will raise three initial hypotheses and prove them by four examples.

Hypothesis 1: Similar to ordering comparison of Vague sets, when the total support degree of two alternatives is the same, the alternative with smaller total opposition degree is better.

Example 1: There are two alternatives, and their Vague evaluation values are as follows:

The three Vague evaluation values of alternative A₁ are: d₁₁ = [0.6, 0.8],d₁₂ = [0.7, 0.8],d₁₃ = [0.5, 0.6].

The three Vague evaluation values of alternative A₂ are: d₂₁ = [0.6, 0.8],d₂₂ = [0.7, 0.7],d₂₃ = [0.5, 0.6].

The total support degree of alternative A₁ is 1.8, and the total opposition degree of alternative A₁ is 0.8. The total support degree of alternative A₂ is 1.8, and the total opposition degree of alternative A₂ is 0.9. The total support degree of two alternatives is the same, and the total opposition degree of alternative A₁ is less than that of alternative A₂.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₁ is [0.3333, 0.3403, 0.3264]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₁ = [0.8616, 0.8668]. According to the formula (21), the score of alternative A₁ is 0.8633.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₂ is [0.3264, 0.3403, 0.3333]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₂ = [0.8427, 0.845]. According to the formula (21), the score of alternative A₂ is 0.8435.

The score of alternative A₁ is greater than that of alternative A₂. This result proves the hypothesis 1.

Hypothesis 2: Similar to ordering comparison of Vague sets, when the total opposition degree of two alternatives is the same, the alternative with greater total support degree is better.

Example 2: There are two alternatives, and their Vague evaluation values are as follows:

The three Vague evaluation values of alternative A₁ are: d₁₁ = [0.5, 0.8],d₁₂ = [0.6, 0.7],d₁₃ = [0.4, 0.6].

The three Vague evaluation values of alternative A₂ are: d₂₁ = [0.6, 0.8],d₂₂ = [0.6, 0.7],d₂₃ = [0.4, 0.6].

The total support degree of alternative A₁ is 1.5, and the total opposition degree of alternative A₁ is 0.9. The total support degree of alternative A₂ is 1.6, and the total opposition degree of alternative A₂ is 0.9. The total opposition degree of two alternatives is the same, and the total support degree of alternative A₂ is greater than that of alternative A₁.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₁ is [0.3257, 0.3409, 0.3334]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₁ = [0.7286, 0.7457]. According to the formula (21), the score of alternative A₁ is 0.7343.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₂ is [0.3357, 0.3429, 0.3214]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₂ = [0.7728, 0.7832]. According to the formula (21), the score of alternative A₂ is 0.7763.

The score of alternative A₂ is greater than that of alternative A₁. This result proves the hypothesis 2.

Hypothesis 3: When the total support degree is same and the total opposition degree is same between two alternatives, the ranking of alternatives depends on the allocation of weights. Among the Vague evaluation values of an alternative, the Vague evaluation value that conflicts with other Vague evaluation values will be given smaller weight.

Example 3: Between an alternative with high degree of conflict between Vague evaluation values and an alternative with low degree of conflict between Vague evaluation values, If the opposition degree of most Vague evaluation values of the alternative with high degree of conflict is greater than the support degree, the alternative with low degree of conflict is better. There are two alternatives, and their Vague evaluation values are as follows:

The three Vague evaluation values of alternative A₁ are: d₁₁ = [0.3, 0.5],d₁₂ = [0.4, 0.4],d₁₃ = [0.8, 0.9].

The three Vague evaluation values of alternative A₂ are: d₂₁ = [0.5, 0.6],d₂₂ = [0.5, 0.6],d₂₃ = [0.5, 0.6].

The total support degree of alternative A₁ is 1.5, and the total opposition degree of alternative A₁ is 1.2. The total support degree of alternative A₂ is 1.5, and the total opposition degree of alternative A₂ is 1.2. The total opposition degree of two alternatives is the same, and the total support degree of two alternatives is the same.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₁ is [0.3417, 0.3583, 0.3]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₁ = [0.5934, 0.5963]. According to the formula (21), the score of alternative A₁ is 0.5944.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₂ is [0.3333, 0.3333, 0.3333].According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₂ = [0.6324, 0.6353]. According to the formula (21), the score of alternative A₂ is 0.6334.

The score of alternative A₂ is greater than that of alternative A₁.

Example 4: Between an alternative with high degree of conflict between Vague evaluation values and an alternative with low degree of conflict between Vague evaluation values, If the support degree of most Vague evaluation values of the alternative with high degree of conflict is greater than the opposition degree, the alternative with high degree of conflict is better. There are two alternatives, and their Vague evaluation values are as follows:

The three Vague evaluation values of alternative A₁ are: d₁₁ = [0.7, 0.8],d₁₂ = [0.7, 0.8],d₁₃ = [0.1, 0.5].

The three Vague evaluation values of alternative A₂ are: d₂₁ = [0.5, 0.7], d₂₂ = [0.5, 0.7], d₂₃ = [0.5, 0.7].

The total support degree of alternative A₁ is 1.5, and the total opposition degree of alternative A₁ is 0.9. The total support degree of alternative A₂ is 1.5, and the total opposition degree of alternative A₂ is 0.9. The total opposition degree of two alternatives is the same, and the total support degree of two alternatives is the same.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₁ is [0.3796, 0.3796, 0.2408]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₁ = [0.8124, 0.8234]. According to the formula (21), the score of alternative A₁ is 0.8161.

According to the formulas (3), (4), (5), (6),(7) and (8), the weight matrix weight of alternative A₂ is [0.3333, 0.3333, 0.3333]. According to the formulas (10), (11) and (12), the comprehensive Vague evaluation value is d₂ = [0.7283, 0.7457]. According to the formula (21), the score of alternative A₂ is 0.7341.

The score of alternative A₁ is greater than that of alternative A₂. In the method of this paper, the Vague evaluation value that conflicts with other evaluation values is given smaller weight, which reduces its impact on the final result. Therefore, the above two situations occur. These two examples prove the hypothesis 3.

To sum up, when an alternative has higher total support degree, lower total opposition degree, and more Vague evaluation values with support degree greater than opposition degree when there is a high conflict exists between Vague evaluation values, the alternative is better.

3 Example analysis

This paper will illustrate the feasibility and effectiveness of the above algorithm through an example of construction-contractor selection. A = {A₁, A₂, A₃} is the construction-contractor set. C = {C₁, C₂, C₃, C₄} is the attribute set.C₁, C₂, C₃ and C₄ respectively represent the construction-contractor’s technical capacity, management capability, financial stability and experience. D = {D₁, D₂, D₃} is the expert set.

Step 1: Experts D₁, D₂ and D₃ give Vague evaluation value of each alternative on each construction-contractor, and the decision matrices D¹, D² and D³ obtained are as follows: $\begin{matrix} D^{1} = [\begin{matrix} [0.2, 0.3] & [0.2, 0.6] & [0.4, 0.5] & [0.6, 0.8] \\ [0.4, 0.7] & [0.3, 0.6] & [0.2, 0.4] & [0.5, 0.8] \\ [0.2, 0.5] & [0.4, 0.7] & [0.5, 0.6] & [0.3, 0.7] \end{matrix}] \\ D^{2} = [\begin{matrix} [0.6, 0.8] & [0.5, 0.5] & [0.5, 0.7] & [0.5, 0.8] \\ [0.2, 0.6] & [0.4, 0.8] & [0.5, 0.8] & [0.3, 0.9] \\ [0.3, 0.5] & [0.1, 0.4] & [0.4, 0.6] & [0.6, 0.8] \end{matrix}] \\ D^{3} = [\begin{matrix} [0.2, 0.6] & [0.4, 0.8] & [0.5, 0.5] & [0.5, 0.8] \\ [0.6, 0.9] & [0.5, 0.7] & [0.5, 0.8] & [0.2, 0.9] \\ [0.5, 0.8] & [0.7, 0.9] & [0.4, 0.9] & [0.4, 0.8] \end{matrix}] \end{matrix}$

Step 2: The evidence information of expert set is aggregated by evidence theory to obtain the comprehensive Vague evaluation value of each construction-contractor, and then the comprehensive evaluation matrix D is obtained. In this paper, the solution of comprehensive Vague evaluation value d₁₁ is taken as an example, and the solution of other comprehensive Vague evaluation value is the same as that of d₁₁.

When solving the comprehensive Vague evaluation value d₁₁, the similarity between the Vague evaluation values $d_{11}^{1}$ , $d_{11}^{2}$ and $d_{11}^{3}$ is obtained according to formula (3), and then the similarity matrix sim is obtained. $sim = [\begin{matrix} 1 & 0.5667 & 0.6333 \\ 0.5667 & 1 & 0.5333 \\ 0.6333 & 0.5333 & 1 \end{matrix}]$

Due to the unknown degree of Vague evaluation value, the similarity between the same Vague evaluation values is not necessarily 1 when solving the similarity between Vague evaluation values. But considering the background of this paper, that is, in the similarity matrix sim, the diagonal element is the similarity between the Vague evaluation values given by the same expert for the same construction-contractor on the same attribute, that is, the similarity between the same thing, so the diagonal value is directly set to 1.

After the similarity matrix sim is obtained, the total support of Vague evaluation values $d_{11}^{1}$ , $d_{11}^{2}$ and $d_{11}^{3}$ can be obtained according to formula (5), and then the support matrix support is obtained. $support = [1.2, 1.1, 1.1666]$

From this, the weights of the Vague evaluation values $d_{11}^{1}$ , $d_{11}^{2}$ and $d_{11}^{3}$ can be obtained according to formula (7), and then the weight matrix weight is obtained. $weight = [0.3462, 0.3173, 0.3365]$

According to the Vague evaluation values $d_{11}^{1}$ , $d_{11}^{2}$ , $d_{11}^{3}$ and the weight matrix weight, the weighted average Vague evaluation value ${\bar{d}}_{11} = [0.3269, 0.5596]$ can be obtained.

Finally, according to the formulas (10), (11), (12), the comprehensive Vague evaluation value d₁₁ = [0.3478, 0.3748] can be obtained.

Other comprehensive Vague evaluation values can be obtained in the same way, and then the comprehensive evaluation matrix D can be obtained. $\begin{matrix} D = [\begin{matrix} [0.3478, 0.3748] & [0.4806, 0.5194] \\ [0.6259, 0.6895] & [0.5944, 0.6445] \\ [0.3837, 0.4215] & [0.5707, 0.6095] \end{matrix} \\ \begin{matrix} [0.5392, 0.542] & [0.8287, 0.8607] \\ [0.5948, 0.6344] & [0.6266, 0.8194] \\ [0.6298, 0.662] & [0.6942, 0.7568] \end{matrix}] \end{matrix}$

Step 3: The evidence information of attribute set is aggregated by evidence theory to obtain the final Vague evaluation value of each construction-contractor. In this paper, the solution of final Vague evaluation value d₁ of construction-contractor A₁ is taken as an example, and the solution of other final Vague evaluation values is the same as that of d₁.

When solving the final Vague evaluation value d₁, the similarity between the comprehensive Vague evaluation values d₁₁, d₁₂, d₁₃ and d₁₄ is obtained according to formula (3), and then the similarity matrix sim is obtained. $sim = [\begin{matrix} 1 & 0.8817 & 0.8625 & 0.6564 \\ 0.8817 & 1 & 0.9471 & 0.7443 \\ 0.8625 & 0.9471 & 1 & 0.7759 \\ 0.6564 & 0.7443 & 0.7759 & 1 \end{matrix}]$

Because the diagonal values of the similarity matrix sim is the similarity of the comprehensive Vague evaluation values of the same construction-contractor on the same attribute, it is the similarity of the same thing, so the diagonal value is directly set to 1.

From this, the total support of comprehensive Vague evaluation values d₁₁, d₁₂, d₁₃ and d₁₄ can be obtained according to formula (14), and then the support matrix support is obtained. $support = [2.4006, 2.5731, 2.5855, 2.1766]$

After that, the weights of the comprehensive Vague evaluation values d₁₁, d₁₂, d₁₃ and d₁₄ can be obtained according to formula (16), and then the weight matrix weight is obtained. $weight = [0.2466, 0.2643, 0.2656, 0.2235]$

According to the comprehensive Vague evaluation values d₁₁, d₁₂, d₁₃, d₁₄ and the weight matrix weight, the weighted average comprehensive Vague evaluation value ${\bar{d}}_{1} = [0.5413, 0.566]$ can be obtained.

Finally, according to the formulas (18), (19), (20), the final Vague evaluation value d₁ = [0.6986, 0.6986] of construction-contractor A₁ can be obtained.

Similarly, the final Vague evaluation value of construction -contractor A₂ is d₂ = [0.9092, 0.9094], and the final Vague evaluation value of construction-contractor A₃ is d₃ = [0.8109, 0.8109].

Step 4:According to the final Vague evaluation value of each construction-contractor, the score function formula (21) is used to solve the score value of each construction-contractor, so as to select the best construction-contractor. $\begin{matrix} S (d_{1}) = 0.6986, S (d_{2}) = 0.90927, S (d_{3}) = 0.8109 \end{matrix}$

According to the score of each construction-contractor, the optimal construction-contractor is A₂. At this point, the selection process of the construction-contractor is over.

The reasons why A₂ is the best construction-contractor are as follows: The total support degree and the total opposition degree of A₁ are 5.1 and 4.3. The total support degree and the total opposition degree of A₂ are 4.8 and 3.1. The total support degree and the total opposition degree of A₃ are 4.8 and 3.8.

1) Although the total support degree of A₂ and A₃ is the same, the total opposition degree of A₂ is smaller than that of A₃, so A₂ is superior to A₃.

2) Although the total support degree and total opposition degree of A₂ are smaller than that of A₁, the difference of total opposition degree is larger than that of total support degree. In addition there is a large degree of conflict between the Vague evaluation values of A₂ and the number of Vague evaluation values with support degree greater than opposition degree in A₂ is more than A₁. so A₂ is superior to A₁.

So the A₂ is the best. The result is consistent with the hypotheses of this paper.

4 Comparative study

In this part, the feasibility and effectiveness of this algorithm will be illustrated by comparing with other algorithms. Because this paper does not consider the expert weights and attributes weights, this paper also will not consider the influence of expert weights and attribute weights on the final result when using other algorithms. Before using other algorithms for comparison, the first step is to reset the expert weights and the attribute weights in these algorithms to make them conform to the background of this paper. The comparison results are shown in Table 1.

Table 1
The comparison results

the settings of expert weights the settings of attribute weights the ranking order

paper [16] the weight value of each expert on each attribute is set to 1 w_{C
₁} = w_{C
₂} = w_{C
₃} = w_{C
₄} = 0.25 A₂ > A₃ > A₁

paper [17] $w_{D_{1}} = w_{D_{2}} = w_{D_{3}} = \frac{1}{3}$ w_{C
₁} = w_{C
₂} = w_{C
₃} = w_{C
₄} = [0.25, 0.25] A₂ > A₃ > A₁

paper [23] $w_{D_{1}} = w_{D_{2}} = w_{D_{3}} = \frac{1}{3}$ none A₂ > A₃ > A₁

this paper none none A₂ > A₃ > A₁

	the settings of expert weights	the settings of attribute weights	the ranking order
paper [16]	the weight value of each expert on each attribute is set to 1	w_{C ₁} = w_{C ₂} = w_{C ₃} = w_{C ₄} = 0.25	A₂ > A₃ > A₁
paper [17]	$w_{D_{1}} = w_{D_{2}} = w_{D_{3}} = \frac{1}{3}$	w_{C ₁} = w_{C ₂} = w_{C ₃} = w_{C ₄} = [0.25, 0.25]	A₂ > A₃ > A₁
paper [23]	$w_{D_{1}} = w_{D_{2}} = w_{D_{3}} = \frac{1}{3}$	none	A₂ > A₃ > A₁
this paper	none	none	A₂ > A₃ > A₁

From the Table 1, the four algorithms get consistent results, which shows the feasibility and effectiveness of the algorithm of this paper. More specifically, the results of the closeness to ideal solution of A₁, A₂ and A₃ are 0.5128, 0.5413 and 0.5130 in paper [16]; the results of the score of A₁, A₂ and A₃ are 0.5539, 0.5949, 0.5599 in paper [17]; the final results of A₁, A₂ and A₃ are 0.3244, 0.3426 and 0.333 in paper [23]. Although the results of the four algorithms are consistent, the paper [16] and paper [17] do not effectively distinguish A₁ and A₃. So, the paper [23] and this paper are better. The reason why this paper can distinguish A₁ and A₃ effectively is that the weights of Vague evaluation values are considered and the unknown degree of Vague sets are redistributed.

5 Implications

Compared with other algorithms, this algorithm is suitable for decision-making in complex environment, that is, evaluation value can not be expressed with accurate value, and expert weights and attribute weights are difficult to determine. Specifically, the method of this paper is applicable to the selection of emergency plans in emergency situations. This kind of problem has the characteristics of sudden occurrence, so it is impossible to conduct detailed investigation in advance, and it requires decision makers to quickly choose emergency plans to reduce economic losses and casualties. In addition, this algorithm is also suitable for decision-making in various complex environments, such as project investment, product development, partner selection, government management, economic benefit evaluation and so on.

6 Conclusion and further research

The unknown degree in Vague evaluation value includes the tendency of expert to support or oppose the alternative. However, most of algorithms do not consider the reasonable redistribution of the unknown degree of Vague sets. So, I use evidence theory to solve this problem. Although there are researchers studied the relationship between Vague sets and evidence theory to solve multi-attribute group decision-making problem, they don’t consider the problem that evidence can’t deal high conflict evidence. I solve this problem by solving the weight of each Vague evaluation value to reduce the impact of conflict evidence. Starting from the above research ideas, the paper first strengthens the role of the important Vague evaluation value in the information fusion process, and then redistributes the unknown degree contained in the Vague sets during the information aggregation process. The algorithm given in this paper solves the problem of difficult or inaccurate information aggregation in Vague multi-attribute group decision-making to a certain extent, and provides a new solution for decision-making problems in complex environments.

In doing the work of this paper, the limitation is that the calculation of this paper is large. However, the results show that this method is better. Through the research of the paper, some later research directions have also been found. First, one of the later research directions of this paper is to combine evidence theory with current information aggregation algorithms to obtain better algorithms. Second, how to combine the weight of Vague evaluation value with expert weight and attribute weight on the background that expert weight and attribute weight are known is also one of the later research directions.

Footnotes

Acknowledgments

This research is supported by Major Project Plan of Applied Research of Philosophy and Social Sciences in Henan Universities in 2020 (No.2020-YYZD-02), General Project of Humanities and Social Sciences Research of Henan Provincial Department of Education in 2021 (No.2021-ZZJH-020) and 2020 Henan Philosophy and Social Science Planning Project (No.2020BJJ041).

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