A LSGDM method based on social network and IVIFN’s geometric characteristics for evaluating the collaborative innovation problem 1

Abstract

During the development of regional economy, introducing collaborative innovation is an important policy. Constructing a scientific and effective measurement for evaluating the collaborative innovation degree is essential to determine an optimum collaborative innovation plan. As this problem is complex and has a long-lasting impact, this paper will propose a novel large scale group decision making (LSGDM) method both considering decision makers’ social network and their evaluation quality. Firstly, the decision makers will be detected based on their social connections and aggregated into different subgroups by an optimization algorithm. Secondly, decision makers are weighted according to their important degree and decision information, where the information is carried by interval valued intuitionistic fuzzy number (IVIFN). During the information processing, IVIFN is put in rectangular coordinate system considering its geometric meaning. And some related novel concept are given based on the barycenter of rectangle region determined by IVIFN. Meanwhile, the criteria’s weights are calculated by the accurate degree and deviation degree. A classical example is used to illustrate the effect of weighting methods. In summary, a large scale group decision making method based on the geometry characteristics of IVIFN (GIVIFN-LSGDM) is proposed. The scientific and practicability of GIVIFN-LSGDM method is illustrated through evaluating four different projects based on the constructed criteria system. Comparisons with the other methods are discussed, followed by conclusions and further research.

Keywords

Keywords: Large scale group decision making intuitionistic fuzzy number social network analysis interval valued intuitionistic fuzzy number Barycenter

1 Introduction

With the development of economy and society, the group decision making problems have become more and more complex, for example, some problems’ long execution cycles and wide range of influence, resulting a reasonable increase of the decision makers’ number. The traditional group decision making (GDM) methods maybe valid. As the collaborative innovation problem is an important issue related to regional people’s livelihood, a scientific decision making mechanism is needed. How to determine the optimum development plan for coordinated regions is a complicated GDM problem. In order to deal with this kind of problems, a large scale group decision making (LSGDM) methodology is proposed by scholars, whose number of decision makers is more than 20 [1]. Studies on LSGDM mainly contain two aspects: one is how to aggregate the large scale decision makers to small groups; the other one is how to reach a reasonable consensus status. Scholars utilize different tools to solve the aggregating problem, such as c-means clustering method [3], k-means clustering method [4]. Consistency-driven methodology is also popular to present uncertain preferences in LSGDM process [5 –7]. In recent years, information exchange is convenient and fast. People’s social network has also been more complex. Especially for the expert decision making problem, the experts may have some social relationships. The traditional large scale group decision making methods treat decision makers independently, without taking decision makers social network into consideration. Meanwhile, some articles only consider the initial decision-making information when dealing with the clustering problem of large groups [47]. Most of these clustering methods depend on the similarity degree of initial decision results. Due to the large group and heterogeneity of decision makers, connections between them have some uncertainty. The uncertainty in the relationship will lead to deviations of the evaluation results. Decision results will not effectively collect the decision maker’s preference for alternatives during the integration process, if do not consider the influence of the decision maker’s network structure. The positive effects of social analysis have been confirmed by some researches [24 , 37]. Large group decision-making method considering social network, takes the social relationships of decision makers into account, making decisions closer to reality. Coping with the reality problems, it is necessary to analyze the key elements of the social network structure, including role, status, reputation and so on. For example, some decision makers occupy important positions in the network structure, then his/her decision results may have more important influence on the implementation of decision results [25], whose reputation may have an effect on the other decision makers. Especially, some models [48] proposed the concept of leadership during social network analysis. Researches on LSGDM problems referring to decision makers relations, from the complete decision information aspect and incomplete decision information aspect. In the former regard, decision making models are carried out around the full social network structure, referring to the simple graph [45] and directed graph [25]. In the latter respect, discussions are about calculating the missing information firstly [24] and then the best alternative choosing process. In reference [45], the communities are detected totaly by the nodes’ relationship with its neighbours based on their important degree. This paper will consider the importance of nodes and the influence of modularity from the overall structure of the network. The weights of decision makers and subgroups are calculated by social network and decision information quality both.

In this article, authors will discuss the LSGDM problem under the environment of fuzzy set. To be exact, apply the IVIFN to be the decision information carrier. Fuzzy set theory proposed by Zadeh in 1965, has attracted a lot of attentions from different fields of researchers. Then intuitionistic fuzzy set theory [26] and interval-valued intuitionistic fuzzy set theory [27] are constructed successively, which became popular in dealing with practical problem. Depending on the fuzzy set theories, kinds of group decision making methods are proposed [6 , 22]. For the reason that, IVIFN describes the decision maker’s attitude from the positive, negative and hesitant side, and each side is expressed by interval valued fuzzy number, allowing the decision maker to be hesitant between a certain range. IVIFN is more effective than the other fuzzy numbers for describing decision maker’s psychological thoughts. Many researchers use IVIFN to describe decision makers’ preference degree [8 –11]. Facing with fuzzy decision making problems under the environment of interval valued intuitionistic fuzzy set, how to compare different IVIFNs is a key issue. Known that an IVIFN is a pair of intervals, the comparison between intervals is more complex than real numbers. A score function and an accuracy function are defined for ranking IVIFNs [50] initially. However, most of the published papers are focused on the arithmetic meaning. Few methods are based on the geometric meaning of the fuzzy numbers. Wan and Dong [41] introduced a possibility degree method for discussing IVIFNs, from the probability viewpoint, defining the possibility degree of comparison between two IVIFNs. Wan and Dong considered the location relations between the two intervals of IVIFNs, which referred to the geometric meaning. However, they have not discussed the region determined by end points of intervals and the meaning of possibility region. This paper will study more on the aspect.

Scholars proposed different GDM methods to evaluate the performance problem [12 , 34]. This research applies a novel LSGDM method to evaluate the coordinated and sustainable development degree with IVIFNs. In order to analysis the evaluation results for different management plans, researches always propose some data processing operators. Distance measures are useful to compute the distance between two IVIFNs. Some methods based on distance measures are published recently [18 , 46]. Apart from the classical Hamming distance, the Euclidean distance proposed firstly in 2000 [17], and the normalized Hamming distance between two IVIFNs defined followed [42], more distance measures based on intuitionistic fuzzy set for group decision making problem are used to rank the alternatives [15 , 50]. The distance functions are usually used to calculated similarity degree for decision results. Just like F. Shen et al. [20] pointed, distance function may not satisfy some properties of distance measure. And he utilized some examples to show that Chen’s [43] distance function could not distinguish some different decision results. Joshi and Kumar’s distance measure also has the drawback of "division by zero problem" [16]. The IVIFN will be put in rectangular coordinate system, considering its geometry characters. Then some previous distances’ drawbacks would be avoided during the calculating process. Divided by zero is a shortcoming of reference [23] and reference [36], noted by Chen et al. [39]. This paper also pay attention to the "division by zero problem". Except for the structure of decision makers’ network, their decision information’s quality is also important to analysis. More study will be worked on the decision makers’ social network and decision information under the environment of IVIFN.

The rest of the paper is organized as follows. Section 2 introduces definitions about the fuzzy numbers and social analysis. In section 3, two algorithms are proposed to cope with the network analysis problems. The concept of new score function based on IVIFN’s geometric meaning will be given. In Section 4, some models are constructed based on the IVIFN’s geometry characters. A flowchart for the proposed GIVIFN-LSGDM method is given in Section 5. Section 6 shows a reality case about reginal collaborate sustainable development, applying the constructed method of Section 5. This paper concludes in section 7 with summarizing advantages and drawbacks of our developments and pointing out some research in the future.

2 Basic definitions

2.1 Definitions about fuzzy sets

In this research, the first question discussed is determining a tool to describe the decision-making information. One priority selection is introducing the fuzzy numbers. Definitions about IFS and IVIFS are given below. which is better to reflecting DMs opinions.

Definition 1. [26] An intuitionistic fuzzy set (IFS) $R = {< x, μ_{R}^{x}, ν_{R}^{x} > | x \in R}$ is defined on a fixed set R, satisfying μ_R (x) : X → [0, 1] , x ∈ X → μ_R (x) ∈ [0, 1], ν_R (x) : X → [0, 1] , x ∈ X → ν_R (x) ∈ [0, 1], where μ_R (x) and ν_R (x) stand for the membership degree and non-membership degree, respectively, with μ_R (x) + ν_R (x) ≤1, ∀x ∈ R.

Furthermore, let $\bar{μ_{A}} (x) = [{\bar{μ_{A}}}^{L} (x), {\bar{μ_{A}}}^{R} (x)]$ , $\bar{ν_{A}} (x) = [{\bar{ν_{A}}}^{L} (x), {\bar{ν_{A}}}^{R} (x)]$ . Then the function $\bar{π_{A}} (x) = [{\bar{π_{A}}}^{L} (x), {\bar{π_{A}}}^{R} (x)]$ stands for the uncertainty degree of x to X, satisfying ${\bar{π_{A}}}^{L} (x) = 1 - {\bar{μ_{A}}}^{R} (x) - {\bar{ν_{A}}}^{R} (x)$ and ${\bar{π_{A}}}^{R} (x) = 1 - {\bar{μ_{A}}}^{L} (x) - {\bar{ν_{A}}}^{L} (x)$ . $〈 [{\bar{μ_{A}}}^{L} (x), {\bar{μ_{A}}}^{R} (x)], [{\bar{ν_{A}}}^{L} (x), {\bar{ν_{A}}}^{R} (x)] 〉$ is called interval valued intuitionistic fuzzy number (IVIFN) for simple. The IVIFN will degenerate to intuitionistic fuzzy number (IFN) when ${\bar{μ_{A}}}^{L} (x) = {\bar{μ_{A}}}^{R} (x)$ and ${\bar{ν_{A}}}^{L} (x), {\bar{ν_{A}}}^{R} (x)$ , whose precise definition is as below. For the reason that, IFN has better comparability, in Section 3 constructs a transform function to connect the IVIFN and IFN.

Definition 2. [27] An interval-valued intuitionistic fuzzy set (IVIFS) $A = {〈 x, \bar{μ_{A}} (x), \bar{ν_{A}} (x) 〉 | x \in X}$ is defined on a fixed set X, satisfying $\bar{μ_{A}} (x) : X \to L [0, 1], \forall x \in X$ ; $\bar{ν_{A}} (x) : X \to L [0, 1], \forall x \in X$ ; supμ_A (x) + supν_A (x) ≤1, ∀x ∈ X, where $\bar{μ_{A}} (x)$ and $\bar{ν_{A}} (x)$ stand for the membership degree and non-membership degree, respectively.

2.2 Definitions about social net constructing

In this section, some basic definitions about social analysis are given. Suppose T decision makers D = {DM₁, DM₂, . . . , DM_T}, who will be regarded as T nodes, belongs to set V = {v₁, v₂, . . . , v_T}. Edge e (d_p, d_q) exists when decision maker d_p and d_q has a certain connection, and all connections compose a set E = {e_pq, p, q ∈ {1, 2, . . . , T}}. The value of e (d_p, d_q) is calculated by Equation (1). The number of edges adjacent to node v_t, t ∈ {1, 2, . . . , T} is called its degree, written as d_t. This paper considers simple graph, that is to say, the direction and weight of the edges are not distinguished. Then the network is constructed by the edges and nodes.

$e_{pq} = {\begin{matrix} 1 & if d_{p} and d_{q} has a connection; \\ 0 & otherwise . \end{matrix}$ (1)

Centrality is usually used to describe the position of a node. One of the most direct concepts is degree centrality. Suppose a network with T nodes, whose biggest degree is T - 1. For convenience of comparison, the centrality index is normalized.

Definition 3. [44] The degree centrality of node t is defined by Equation (1) whose degree is d_t. ${DC}_{t} = \frac{d_{t}}{T - 1}$ (2)

Eigenvector centrality’s basic idea is about the importance of a node. It depends on both number of its neighboring nodes and importance of its neighboring nodes.

Definition 4. [44] Let EC_t stands for the importance degree of node t, then ${EC}_{t} = c \sum_{j = 1}^{N} a_{tj} x_{j}$ (3) where c is a constant number, A = (a_tj) is the adjacent matrix, j ∈ {1, . . . , N}.

Denote x = [x₁, x₂, . . . , x_T]^′, then Equation (3) could be written as below. $x = c Ax$ (4) where x means the eigenvector of matrix A and eigenvalue c^-1. That is why call it Eigenvector-centrality.

3 Network analysis in large scale group decision making (LSGDM) problem

In this section, an algorithm is constructed for decreasing the dimensions of decision makers for the LSGDM problem. The large scale group decision makers will be departed to some subgroups by applying the following Algorithm 1. Then the communities’ weights will be calculated by Algorithm 2.

3.1 Aggregating algorithm constructed with network analysis

An important definition in this section is Modularity, which is used to deal with the aggregating problem, defined by Equation (5):

$\begin{matrix} Δ Q = [\frac{W_{c} + s_{i, in}}{2 W} - (\frac{S_{c} + s_{i}}{2 W})^{2}] \\ - [\frac{W_{c}}{2 W} - (\frac{S_{c}}{2 W})^{2} - (\frac{S_{i}}{2 W})^{2}] \end{matrix}$ (5) where s_i,in is the edge numbers of node i and the other nodes in community C. Aggregating results will be calculated by fast greedy method based on the modularity gain degree. The fast greedy method’s core idea is proposed by Newman [32].

A principle of network analysis in the LSGDM problem is that let each node be a separate community initially. In order to maximize modularity of the whole community, calculate the biggest local contribution of each node community. The basic algorithm process contains four steps.

Algorithm 1

Step 1. Let each nodes to be an original community.

Step 2. Based on the modularity, some neighbors are determined to be merged, after one iteration.

Step 3. Each community is regarded as one new node. And calculate its degree and connect information, that is the basic for next iteration.

Step 4. Repeat step 3 until the modularity of the community no longer increases.

3.2 A method for calculating communities’ weight vector

In order to prevent the dense inside and sparse connection to the outside, determine the subgroups’ weight vector based on two aspects. One aspect is considering the nodes degree and eigenvector centralities; the other aspect is the influence of a subgroup to the others. In a subgroup, one node has two kinds of edges, an inside one and an outside one. The inside one connects nodes belong to the same subgroup. And the outside one connects two kinds of nodes, where one belongs to the subgroup and the other one belongs to the outside. A subgroup may get high weight, for the nodes contained in this subgroup with high degree. But the connections of this subgroup and the outside nodes may be not as dense as the inside. Depending on the aggregating method, all T nodes will be separated to S parts, each subgroup has Q_s nodes, s ∈ {1, 2, . . . , S}. The main progresses for determining the subgroups’ weight are as follows.

Algorithm 2

Step 1. Calculate each node’s degree centrality (DC) and eigenvector centrality (EC). Combine the two centralities to get a node weight λ_node,t for node t, by Equation (6). The normalized weight is calculated by Equation (7).

$λ_{node, t} = \sqrt{{DC}_{t} \cdot {EC}_{i}}$ (6)

$\bar{λ_{node, t}} = \frac{λ_{node, t}}{\sum_{t = 1}^{T} λ_{node, t}}$ (7)

Based on all nodes weight, the network’s average centrality λ_net is computed by Equation (8) and the subgroup’s average centrality $λ_{comm}^{s}$ is computed by Equation (9):

$λ_{net} = \frac{1}{T} \sum_{t = 1}^{T} λ_{node, t}$ (8)

$λ_{comm}^{s} = \frac{1}{S_{t}} \sum_{t = 1}^{S_{t}} λ_{node, t}$ (9)

A subgroup’s inside weigh λ_in,s is determined by the distance of this subgroup’s average centrality and the network’s average centrality, computed by Equation (10).

$λ_{in, s} = \frac{1}{λ_{comm}^{s} - λ_{net}}$ (10)

And normalize λ_in,t by Equation (11), obtaining $\bar{λ_{in, s}}$ .

$\bar{λ_{in, s}} = \frac{λ_{in, s}}{\sum_{s = 1}^{Q_{s}} λ_{in, s}}$ (11)

Step 2. Suppose there are Q_s nodes in subgroup C_s. Calculating the number of edges for all Q_s nodes, which connects the inside nodes and outside nodes, denoted as λ_out,s. Normalize λ_out,s and get $\bar{λ_{out, s}}$ , a outside weight λ_out,s, by Equation (12).

$\bar{λ_{out, s}} = \frac{λ_{out, s}}{\sum_{s = 1}^{S} λ_{out, s}}$ (12)

Step 3. Combine the inside weight and outside weight of the network together, by Equation (13), obtaining subgroup’s weight λ. α is moderator variables which can adjust the importance degree of $\bar{λ_{in, s}}$ and $\bar{λ_{out, s}}$ .

$λ = α \bar{λ_{in, s}} + (1 - α) \bar{λ_{out, s}}$ (13)

4 Model constructing for LSGDM problem with IVIFN

4.1 A new score function based on IVIFNs

This paper proposes distance measure from the perspective of geometric center of gravity, a novel score function defined after it, considering the IVIFN’s geometry characteristics. As IVIFN could be understood as an extension of IFN, give a transform function to connect the IFN and IVIFN. Let α = 〈 [μ^-, μ⁺] , [ν^-, ν⁺] 〉 be an IVIFN. l_PQ is a line defined by x + y = 1. P and Q are the intersection of l_PQ and coordinate axis. Coordinates of A, B, C and D are A = (μ^-, ν^-), B = (μ⁺, ν^-), C = (μ⁺, ν⁺) and D = (μ^-, ν⁺), respectively. Lengths of L_AB, L_CD, L_AD and L_BC are L_AB = L_CD = μ⁺ - μ^- and L_AD = L_BC = ν⁺ - ν^-. Obviously, the quadrilateral determined by points A, B, C and D is a rectangle. So, its barycentric coordinate is $G = (μ^{-} + \frac{(μ^{+} - μ^{-})}{2}, ν^{-} + \frac{(ν^{+} - ν^{-})}{2})$ , which could be regard as the most representative point of the decision maker’s preference. For the reason that different rectangles may obtain the same barycenter, some other parts contained in triangle determined by points O, P and Q.

Name the region determined by point A, B, C and D “possible region”. Name the region determined by point C, E and F “impossible region”. And name the region determined by point O, N, C and T “possible fuzzy region”. There regions are shown in Figure 1. Next part introduces a ratio which measures the uncertainty degree of the decision making results.

Fig. 1

The barycenter of IVIFN shown in rectangular coordinate system.

Definition 5. Let α = 〈 [μ^-, μ⁺] , [ν^-, ν⁺] 〉 be an IVIFN, written as α = 〈 [a, b] , [c, d] 〉 for simple. Define $R = \frac{(b - a) \times (d - c) + \frac{1}{2} \times (1 - b - d)^{2}}{b \times d + \frac{1}{2} \times (1 - b - d)^{2}}$ (14) IVIFN’s Certainty Ratio.

Definition 6. An INIFN α = 〈 [a, b] , [c, d] 〉 can be transformed to an IFN-from $\tilde{β} = (u, v)$ based on the following Equation (15),

$β = (u, v) = (a + (b - a) / 2 \times R, c + (d - c) / 2 \times R)$ (15) where R is the INIFN’s Certainty Ratio.

Next, a theorem is given to prove the novel "IFN" obtained by Definition 4.1 satisfies the formal IFN’s conditions.

Theorem 7. Let α = 〈 [a, b] , [c, d] 〉 be an INIFN. The number $\tilde{β} = (u, v)$ obtained by Equation (15) is an IFN.

Prove

Let α = 〈 [a, b] , [c, d] 〉 be an INIFN. From the Theorem 1,

$R = \frac{(b - a) \times (d - c) + \frac{1}{2} \times (1 - b - d)^{2}}{b \times d + \frac{1}{2} \times (1 - b - d)^{2}}$ ,

$\tilde{β} = (u, v) = (a + (b - a) / 2 \times R, c + (d - c) / 2 \times R)$ .

Based on the definition of INIFN, cases are classified by value of b + d.

i) b + d = 1.

Because b + d = 1, 1 - b - d = 0 and a ≤ b, c ≤ d, a, c ∈ [0, 1], $R = \frac{(b - a) \times (d - c) + 0}{b \times d + 0} = \frac{(b - a) \times (d - c)}{b \times d}$ , obtain 0 ≤ R ≤ 1. In addition, R = 0 if a = b or c = d; and R=1 if a = 0 and c = 0.

ii) b + d < 1.

Because b + d = 1, 1 - b - d = 0 and a ≤ b, c ≤ d, a, c ∈ [0, 1], Then

(b - a) × (d - c) ≤ b × d;

$(b - a) \times (d - c) + \frac{1}{2} \times (1 - b - d)^{2} \leq b \times d + \frac{1}{2} \times (1 - b - d)^{2}$ .

obtaining 0 ≤ R ≤ 1. In addition, R=1 if a = 0 and c = 0.

iii) b + d = 0. The decision result is reduced to a number 0. This condition will not be considered in the following procedure. That is to say, all evaluating results are positive real number.

In all, 0 ≤ R ≤ 1. Next will discuss the ranges of u + v.

Firstly,

$u = a + \frac{b - a}{2} \times R \leq a + \frac{b - a}{2} = \frac{a + b}{2} \leq 1$ ;

$v = c + \frac{d - c}{2} \times R \leq c + \frac{d - c}{2} = \frac{c + c}{2} \leq 1$ ;

then u ∈ [0, 1] and v ∈ [0, 1].

Secondly, calculate the value of u + v.

$u + v = (a + c) + \frac{R}{2} (b + d - a - c) \leq (a + c) + \frac{1}{2} (1 - a - c) = \frac{1 + a + c}{2} \leq \frac{1 + b + d}{2} \leq 1$

Finally, conclusion of Theorem 1 could be proved.

Depending on Definition 4.1 and Definition 4.1, introduce a new score function for INIFNs. And any different INIFNs could be compare by their score values.

In addition, π_R (x) =1 - μ_R (x) - ν_R (x) is defined as the uncertainty degree of x to R. If π_R (x) =0, R is reduced to a fuzzy set which is proposed by Zadeh [29].

For the reason that, two different IFNs cannot be compared directly. In order to measure the difference degree between two different IFNs, some distance measures have been proposed. Researches about this comparison problem are mainly related to calculate the distance of one certain fuzzy numbers.

Recently, You and Yang [47] give a distance function $\bar{h} (α, β)$ to measure any two IFNs’ difference degree, which is more effective for classifying IFNs, shown as below.

$\begin{matrix} \bar{h} (α, β) = | μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | + | S_{A} (x_{i}) - S_{B} (x_{i}) | \end{matrix}$ (16)

Definition 8. Let α be an INIFN, written as α = 〈 [a, b] , [c, d] 〉 without confusion. Its new score function based on gravity center is given by Equation (17).

$f (score) = (a + (b - a) / 2 \times R) + (c + (d - c) / 2 \times R)$ (17) where R is the INIFN’s Certainty Ratio.

In order to show the effectiveness of the novel Score function, give the following Example 1 from [40], which also be used by references [15] and [38].

Example

The weights of Criterion 1, Criterion 2, Criterion 3 are determined after the original papers as "0.10, 0.65, 0.25", and the alternatives’ decision results are shown in Table 1. Comparison results from different score functions are shown in Figure 2, with methods from [49], [40], [15] and [38] and novel score function proposed by this paper. From Figure 2, it is obvious to see that, the novel score function is able to compare the four alternatives. However, the comparison results are different with each other. Then, in order to show the effectiveness of Certainty Ratio, next will do more research.

Table 1

A example to show the effectiveness of novel score function

	Criterion 1	Criterion 2	Criterion 3
Alter-1	〈[0.40, 0.50] , [0.30, 0.40] 〉	〈[0.40, 0.60] , [0.20, 0.40] 〉	〈[0.10, 0.30] , [0.50, 0.60] 〉
Alter-2	〈[0.58, 0.70] , [0.10, 0.30] 〉	〈[0.61, 0.70] , [0.20, 0.30] 〉	〈[0.10, 0.30] , [0.50, 0.60] 〉
Alter-3	〈[0.30, 0.60] , [0.30, 0.40] 〉	〈[0.50, 0.60] , [0.30, 0.40] 〉	〈[0.50, 0.60] , [0.10, 0.30] 〉
Alter-4	〈[0.61, 0.80] , [0.10, 0.20] 〉	〈[0.60, 0.70] , [0.10, 0.30] 〉	〈[0.30, 0.40] , [0.10, 0.20] 〉

Fig. 2

Comparison results of different Score functions.

In order to determine the criteria’s weights and the decision makers weights, studying the characteristics of IVIFNs made by decision makers, when dealing with the multi-criteria group decision making problem. It is easy to understand that the criterion’s weight will be higher when the decision result is more accurate based on it. And the criterion’s weight will be higher when its deviation degree is higher. Depending this idea, the function H (β) = μ + ν means accurate degree for IFN β = (μ, ν). Let C = {C₁, C₂, …, C_n} be a set with n criteria, j ∈ N, N = {1,2, …, n}, X = {X₁, X₂, …, X_m} be a set with m alternatives with i ∈ M, M = {1, 2, …, m}. Decision results will be give in the following form as Equation (18). a_ij stands for decisions of alternative X_i made by decision maker considering criterion C_j. For any criteria C_j, its accurate degree is determined by Equation (19).

$A_{ij} = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{n 2} & \dots & a_{mn} \end{matrix}]$ (18)

$f (w_{j}) = \sum_{i = 1}^{m} w_{j} (μ_{ij} + ν_{ij})$ (19)

Model 1 is constructed to ensure the maximum accurate degree.

Model 1 ${\begin{matrix} \max f_{1} (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} w_{j} (μ_{ij} + ν_{ij}) \\ s . t . \sum_{j = 1}^{n} w_{j} = 1 \\ 0 \leq w_{j} \leq 1 . \end{matrix}$ (20)

And the deviation degree is obtained by Equation (21), based on the distance equation (16).

$f (w_{j}) = \sum_{i = 1}^{m} \sum_{1 \leq i < k \leq m} w_{j} d (a_{ij}, a_{kj})$ (21)

Model 2 is constructed to ensure the maximum deviation degree.

Model 2 ${\begin{matrix} \underset{2}{maxf} (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{1 \leq i < k \leq m} w_{j} d (a_{ij}, a_{kj}) \\ s . t . \sum_{j = 1}^{n} w_{j} = 1 \\ 0 \leq w_{j} \leq 1 . \end{matrix}$ (22)

Taking the accurate degree and deviation degree into consideration simultaneously, the following Model 3 is given to calculate the criteria’s weights, where β standing for the weight of accurate degree determined by actual requirement.

Model 3 ${\begin{matrix} \max f_{3} (w) = β f_{1} (w) + (1 - β) f_{2} (w) \\ s . t . \sum_{j = 1}^{n} w_{j} = 1 \\ 0 \leq w_{j} \leq 1 . \end{matrix}$ (23)

Next, the criteria’s weights of Model 3 is calculated by constructing a lagrangian function L (w, λ).

$L (w, λ) = f (w) + λ (\sum_{j = 1}^{n} w_{j} - 1)$ (24)

Take the derivative of w_j and λ simultaneously.

${\begin{matrix} \frac{\partial L}{\partial w_{j}} = \sum_{i = 1}^{m} (β (μ_{ij} + ν_{ij})) + \sum_{1 \leq i < k \leq m} (1 - β) d (a_{ij}, a_{kj})) + λ = 0; \\ \frac{\partial L}{\partial λ} = \sum_{j = 1}^{n} w_{j} - 1 = 0 . \end{matrix}$ (25)

The criteria’s weight is obtained by Equation (26).

$\begin{matrix} w_{j} = \frac{\sum_{i = 1}^{m} (β (μ_{ij} + ν_{ij}) + \sum_{1 \leq i < k \leq m} (1 - β) d (a_{ij}, a_{kj}))}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} (β (μ_{ij} + ν_{ij}) + \sum_{1 \leq i < k \leq m} (1 - β) d (a_{ij}, a_{kj}))} . \end{matrix}$ (26)

Decision maker’s weight is determined by his/her position and evaluation quality. The position aspect could be obtained by the node’s weight from Equation (26). The evaluation quality is represented by the deviation degree of decision result, which is calculated based on Equation (21). The weight of decision maker is computed by Equation (21), introducing the different value ρ.

$w_{DM}^{t} = ρ \bar{λ_{node, i}} + (1 - ρ) \sum_{i = 1}^{m} \sum_{1 \leq i < k \leq m} w_{j} d (a_{ij}, a_{kj})$ (27)

Depending on the criteria’s weight and decision makers’ weights, each community’s weight is calculated by Equation (28), where $a_{ij}^{tl}$ standing for the evaluation of decision maker DM_l from community t with alternative Alter_i under criterion C_j. Let $B^{t} = (b_{i}^{t})_{m \times 1}$ be the community’s representing matrix. Then the alternative A_i could obtain a weighted decision result by Equation (29) where $b_{i}^{s}$ is the definition of an IFN.

$b_{i}^{s} = \sum_{j = 1}^{n} \sum_{l = 1}^{S_{t}} w_{DM}^{l} (a_{ij}^{sl})$ (28)

$b_{i} = \sum_{s = 1}^{S} λ^{s} b_{i}^{s}$ (29)

Let b_i = (μ_i, ν_i), satisfying the definition of an IFN. The alternatives’ will de ranked by values calculated by Equation (30).

$a_{i} = μ_{i} - ν_{i}$ (30)

5 The implementation procedure of the proposed GIVIFN-LSGDM method

Depending on the above analysis, the proposed LSGDM method based on the IVIFN’s geometric characteristics is named GIVIFN-LSGDM method for short, whose main concrete steps are shown in Figure 3. Specifically, the proposed method is divided into the following concrete steps.

Fig. 3

Research framework of the proposed GIVIFN-LSGDM method.

Step 1. Determine the DMs’ network connections by Equation (1). Draw the orginal network structure for decision makers.

Step 2. Apply Algorithm 1 to aggregate the decision makers to T subgroups, according to decision makers’ network structure, where each subgroup has S_t, t ∈ {1, . . . , T} decision makers.

Step 3. Utilize Equation (13) to calculate each subgroup’s weight, where its inside weight is obtained by Equation (11), outside weight by Equation (12) and each node’s weight by Equation (7).

Step 4. Apply Equation (26) to calculate the criteria’s weights, where the decision results’ accurate degree and deviation degree are obtained by Equation (19) and Equation (21), respectively.

Step 5. Use Equation (28) to get each subgroup B_l’s representing matrix $B_{l} = (b_{i}^{t})_{m \times 1}$ , i ∈ {1, . . . , m}.

Step 6. Apply Equation (29) to weight the subgroups’ representing matrix, obtaining a weighted matrix A = (b_i) _m×1.

Step 7. Utilize Equation (30) to get alternatives’ final scores and rankings.

6 Case study for a LSGDM problem

This section discuses a three-regions collaborative innovation to promote regional sustainable development. The GIVIFN-LSGDM method will be applied to determine which combination is better for reality.

6.1 An example about coordinated development for a collaborative innovation problem

In China, innovation is treated as the first driving force leading development. The Chinese government recognizes the importance of innovation from a strategic perspective, believing that innovation supports the construction of a modern economic system. Under the current economic development situation, the competition trend of technological innovation among countries, regions, industries and enterprises is particularly obvious. Due to the constraints of resources, technology, and knowledge, it is difficult to innovate independently by relying on only one region, one industry or one enterprise. Therefore, the innovation goals will be easier to achieve, through integrating resources between regions, industries and enterprises. During the developing process, the strengths will be maximized and weaknesses will be avoided, with discovering advantages and disadvantages and and avoid.

There are three regions named B, T and H for simple. Their economic development levels from high to low are Region-B, Region-T and Region-H. Because independent development of the three regions has encountered a certain of bottlenecks. It is significant to implement coordinated development, that is treating three regions as a whole. The B-T-H region has intensive intellectual resources, strong technological complementarity, a solid manufacturing base, and rapid development of high-tech industries. This combine-region is a core demonstration zone for China to participate in international competition and promote collaborative innovation. It is also a strategically positioned part of China’s three major economic zones. The coordinated development of the B-T-H region is necessary to realize the complementary advantages of the B-T-H region, promoting the development of a bigger economic zone and leading the development of the northern hinterland. It is a major national strategy. Systematically study on the participants’ degree of collaboration, will encourage the further gathering of innovative talents in Region-B, Region-T and Region-H, sharing innovative elements and interconnect technology market. Then, transfer and transformation of scientific and technological achievements will also be improved, increasing the success rate of innovation activities and the quality of transformation. In addition, promote the sustainable development of the economy. However, the economic development levels and industrial structures of Region-B, Region-T and Region-H are different. Although government proposed a policy for the coordinated development of B-T-H. This case hope to show that whether it is suitable to adjust development structure slightly. Depending on the reality, it is hard to develop the three regions in perfect synchronization. Propose four combinations as alternatives: 1. develop Region-B, Region-T and Region-H as Region-B-T-H simultaneously; 2. prioritize the implementation of coordinated development of region-B and region-T as Region-B-T simultaneously, then Region-H; 3. prioritize the implementation of coordinated development of region-B and region-H as Region-B-H simultaneously, then Region-T; 4. prioritize the implementation of coordinated development of region-T and region-H as Region-T-H simultaneously, then Region-B.

In order to analysis which development structure is better, 30 experts are invited to evaluate the criteria, as shown in Figure 4. Build an evaluation standard system, after discussing with the invited experts and our project team staff.

Fig. 4

The criteria considered by decision makers.

The decision makers’ origin social network is shown in Figure 5.

Fig. 5

The original decision makers network.

The original decision results of DM₁ is given by Table 2. And the remaining 29 are given in Appendix A.1, Table A1-A29.

Table 2

The decision result given by DM1

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.55, 0.65] , [0.20, 0.35] 〉	〈[0.55, 0.60] , [0.10, 0.30] 〉	〈[0.45, 0.55] , [0.25, 0.35] 〉	〈[0.55, 0.60] , [0.15, 0.35] 〉	〈[0.35, 0.45] , [0.40, 0.50] 〉
Alter-2	〈[0.70, 0.75] , [0.10, 0.20] 〉	〈[0.65, 0.75] , [0.15, 0.20] 〉	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.40, 0.50] , [0.20, 0.30] 〉
Alter-3	〈[0.45, 0.55] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.55, 0.65] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-4	〈[0.40, 0.50] , [0.40, 0.55] 〉	〈[0.60, 0.65] , [0.10, 0.15] 〉	〈[0.45, 0.50] , [0.35, 0.45] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉

Applying the proposed aggregating method of Section 4, the decision makers are parted to 4 subgroups, shown in Figure 6. Let α = 0.5, meaning the network’s inside weight and outside weight are equal important. Let β = 0.5, meaning the accurate degree and deviation degree with equal importance degree. Let ρ = 0.5, meaning the decision makers’s relationships and evaluation results have the same importance degree. The 4 alternatives’ evaluation results are (0.3430, 0.0867), (0.3739, 0.0657), (0.3221, 0.1097), (0.2897, 0.1356). And the final ranking values are 0.2564, 0.3082, 0.2124, 0.1541, meaning Alter2 ≻ Alter1 ≻ Alter3 ≻ Alter4. From the ranking results, the second alternative is the best one. That is to say, developing Region-B and Region-T first, then Region-H, has the highest creative degree. This result could be explained by the three regions’ economic development levels, where Region-B and Region-T are better than region-H. At the same time, Region-B’s innovation output can be better undertaken by Region-T, the two regions technology industries will have a better expected performance from the perspective of innovation collaboration. Region-H is a traditional agricultural major province. Prioritize the development of B and H, from the perspective of recent economic benefits, which has advantages over other programs. However, from the perspective of long-term economic and social benefits, the second option of simultaneously considering the three regions is more competitive. Because Region-B and Region-T have a smaller area. Many basic needs rely on external support. Especially, relying on the Region-H. Region-B is about to encounter various "big city diseases". In China, the coordinated development of the three places is currently mainly option. At present, certain phased results have been achieved, but the synergy effect needs to be improved.

Fig. 6

The aggregating result of the proposed method.

6.2 Comparisons with the other LSGDM methods

Apply the LSGDM method proposed by T. Wu et al [45], who also considering the social relations of decision makers, obtain 3 subgroups, shown in Figure 7.

Fig. 7

The aggregating result of Lovain method.

Apply the LSGDM method proposed by You and Yang [47], who only discussed the decision results and did not considered the social relations. Figure 8 shows the numbers of aggregating result will be stable after a large number of evolutions. Determine the threshold value 0.76, after trying different threshold values from 0.75, 0.76, 0.77, 0.78, 0.79. The aggregating results will be too large when the threshold value is bigger than 0.79. From Figure 9, find that the aggregating result is 5, except for a single one. The cluster numbers, mean-consistency and std-consistency from evolution number 7501 to 8000 are shown in Figure 10.

Fig. 8

The cluster numbers with threshold value γ = 0.76.

Fig. 9

The aggregating result with threshold value γ = 0.76.

Fig. 10

The consistency degree and cluster results with threshold value γ = 0.76.

Ranking results by different methods are shown in Figure 11. The overall rankings of the three method are the same with each other, that shows the validity and rationality of the proposed method in this paper. Although the specific results obtained by each method are different. You’s [47] method did not take the relations of decision makers into account, only considering evaluation results. Wu’s [45] method weighted the decision makers and subgroups by their connections, without discussing their decision information’s quality. In this paper, the proposed method considers the network of decision makers and the evaluations simultaneously.

Fig. 11

Comparisons of different LSGDM methods.

Advantages of the proposed method.

a. The model constructed takes into account the social relations between decision makers, during dealing with the large scale group decision making problem. Subgroups are detected by every decision maker’s position in the network. Each subgroup’s weight is calculated by its importance degree, which is determined by the nodes’ position and their edges contained in it.

b. Decision maker’s weight is considered by both of the position in whole network and effectiveness of decision information. The importance of eigenvector centrality and degree centrality is considered at the same time, for calculating the subgroups’ weights.

c. Utilize modularity to realize clustering of network nodes, depending on the nodes importance degree of the whole network.

d. Decision maker’s evaluating results are represented by the interval valued intuitionistic fuzzy number. This paper constructs a transform function, considering the barycenter of IVIFN, which is displayed by using rectangular coordinate system.

7 Conclusions and further research

In this article, the collaborative innovation problem is studied by proposing a novel large scale group decision making method with decision makers’ social network under the environment of interval valued intuitionistic fuzzy numbers. The optimum alternative is chosen among all collaborative innovation plans. During the decision making process, the large scale decision makers are aggregated by their important degree to the whole social network structure. And weighted by their important degree and decision information¡¯s quality. Analysis in this paper makes the decision result nearer to the objective reality. Analysis in this paper makes the decision result nearer to the objective reality. As the collaborative innovation mechanism has a long period time effects on the related regions, referring to the economic, people’s livelihood and the natural environment, research on this problem should be more comprehensive and sophisticated. Based on the proposed GIVIFN-LSGDM method for ranking four different alternatives under a constructed criteria system, more focus should be given on this problem.

In the future, there are two aspects particularly noteworthy to pay attention. One is about the method, it should be meaningful carried out the decision makers’ social network in directed graph, or weighted the relationship between different decision makers. The other one is about the coordinated development between regions. Because the margin of the region’s interior economy development level is large, it is necessary to refine the inside collaborative region and apply the collaborative policy gradually.

Footnotes

Appendix

The following tables are given by the remaining 29 decision makers of the illustrative example. Table A1

The decision result given by DM2

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.50, 0.55] , [0.30, 0.40] 〉	〈[0.50, 0.55] , [0.20, 0.30] 〉	〈[0.40, 0.45] , [0.20, 0.25] 〉	〈[0.50, 0.55] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-2	〈[0.75, 0.80] , [0.05, 0.10] 〉	〈[0.35, 0.45] , [0.45, 0.50] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉
Alter-3	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.55, 0.60] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-4	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.50, 0.55] , [0.25, 0.30] 〉	〈[0.50, 0.55] , [0.15, 0.20] 〉	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉

Table A2

The decision result given by DM3

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.65, 0.75] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.60, 0.70] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.10, 0.20] 〉	〈[0.60, 0.65] , [0.20, 0.30] 〉
Alter-2	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.55, 0.65] , [0.25, 0.30] 〉
Alter-3	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉	〈[0.60, 0.65] , [0.20, 0.35] 〉	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.65, 0.75] , [0.20, 0.25] 〉
Alter-4	〈[0.55, 0.60] , [0.20, 0.30] 〉	〈[0.70, 0.75] , [0.10, 0.20] 〉	〈[0.55, 0.65] , [0.10, 0.25] 〉	〈[0.45, 0.50] , [0.40, 0.45] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉

Table A3

The decision result given by DM4

Alter-1	〈[0.60, 0.70] , [0.20, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉
Alter-2	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.85, 0.90] , [0.00, 0.10] 〉	〈[0.55, 0.60] , [0.20, 0.30] 〉
Alter-3	〈[0.60, 0.65] , [0.20, 0.30] 〉	〈[0.65, 0.70] , [0.10, 0.15] 〉	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉
Alter-4	〈[0.55, 0.65] , [0.05, 0.15] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.20, 0.30] 〉	〈[0.55, 0.60] , [0.20, 0.30] 〉	〈[0.60, 0.70] , [0.20, 0.25] 〉

Table A4

The decision result given by DM5

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.25, 0.25] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉
Alter-2	〈[0.80, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉
Alter-3	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉
Alter-4	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.55, 0.65] , [0.30, 0.35] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.65] , [0.25, 0.30] 〉

Table A5

The decision result given by DM6

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.60, 0.65] , [0.20, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉
Alter-2	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉
Alter-3	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.10, 0.15] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.10, 0.15] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉
Alter-4	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.60, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉

Table A6

The decision result given by DM7

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.75, 0.85] , [0.05, 0.15] 〉	〈[0.85, 0.90] , [0.10, 0.10] 〉
Alter-2	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.05, 0.15] 〉
Alter-3	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.15, 0.20] 〉
Alter-4	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.50, 0.55] , [0.40, 0.45] 〉	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉

Table A7

The decision result given by DM8

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.80, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.70, 0.80] , [0.05, 0.15] 〉
Alter-2	〈[0.85, 0.95] , [0.00, 0.05] 〉	〈[0.80, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉
Alter-3	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.70, 0.75] , [0.10, 0.15] 〉	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉
Alter-4	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.60, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.20, 0.35] 〉

Table A8

The decision result given by DM9

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.60, 0.70] , [0.25, 0.25] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-2	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.80, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉
Alter-3	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-4	〈[0.50, 0.55] , [0.40, 0.45] 〉	〈[0.55, 0.65] , [0.25, 0.30] 〉	〈[0.50, 0.55] , [0.40, 0.45] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉

Table A9

The decision result given by DM10

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.75, 0.85] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.85, 0.85] , [0.10, 0.15] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉
Alter-2	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.90, 0.90] , [0.05, 0.10] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-3	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.20, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-4	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.75] , [0.15, 0.20] 〉	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉

Table A10

The decision result given by DM11

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.65, 0.70] , [0.20, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-3	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉
Alter-4	〈[0.55, 0.65] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.65] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉

Table A11

The decision result given by DM12

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉
Alter-2	〈[0.85, 0.95] , [0.00, 0.05] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.90, 0.95] , [0.05, 0.05] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-3	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.35] 〉	〈[0.70, 0.75] , [0.15, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-4	〈[0.60, 0.65] , [0.25, 0.35] 〉	〈[0.55, 0.65] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉

Table A12

The decision result given by DM13

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉
Alter-3	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.20, 0.30] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.15, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-4	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.60, 0.65] , [0.20, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉

Table A13

The decision result given by DM14

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.65, 0.75] , [0.15, 0.25] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-2	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.80] , [0.10, 0.20] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉
Alter-3	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉
Alter-4	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.20, 0.30] 〉

Table A14

The decision result given by DM15

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉
Alter-2	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.05, 0.15] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉
Alter-3	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-4	〈[0.55, 0.65] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉

Table A15

The decision result given by DM16

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.75, 0.80] , [0.20, 0.30] 〉	〈[0.70, 0.75] , [0.05, 0.10] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-2	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.85, 0.80] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉
Alter-3	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.20, 0.25] 〉
Alter-4	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.50, 0.55] , [0.35, 0.40] 〉

Table A16

The decision result given by DM17

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.80] , [0.10, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.70, 0.75] , [0.15, 0.25] 〉	〈[0.70, 0.80] , [0.10, 0.15] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.20] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.05, 0.15] 〉
Alter-3	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-4	〈[0.60, 0.65] , [0.25, 0.35] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉

Table A17

The decision result given by DM18

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.00] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-2	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.95, 0.95] , [0.00, 0.05] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉
Alter-3	〈[0.65, 0.70] , [0.15, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉
Alter-4	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉

Table A18

The decision result given by DM19

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.05, 0.15] 〉
Alter-2	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉
Alter-3	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.65, 0.75] , [0.15, 0.20] 〉	〈[0.70, 0.80] , [0.10, 0.15] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.65, 0.70] , [0.20, 0.30] 〉
Alter-4	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.60, 0.70] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.35] 〉

Table A19

The decision result given by DM20

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉
Alter-2	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.85, 0.95] , [0.00, 0.05] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-3	〈[0.60, 0.70] , [0.30, 0.35] 〉	〈[0.60, 0.70] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉
Alter-4	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉

Table A20

The decision result given by DM21

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.05, 0.10] 〉
Alter-2	〈[0.70, 0.75] , [0.10, 0.15] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.20] 〉
Alter-3	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.65] , [0.25, 0.20] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.15, 0.25] 〉
Alter-4	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.50, 0.55] , [0.40, 0.45] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.55, 0.60] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉

Table A21

The decision result given by DM22

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.85, 0.90] , [0.00, 0.10] 〉	〈[0.80, 0.85] , [0.05, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.20] 〉
Alter-2	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-3	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.70] , [0.25, 0.30] 〉
Alter-4	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.55, 0.65] , [0.25, 0.35] 〉

Table A22

The decision result given by DM23

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.25, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.00, 0.05] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉
Alter-3	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-4	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.55, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉

Table A23

The decision result given by DM24

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.25, 0.25] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-2	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.10, 0.20] 〉	〈[0.80, 0.85] , [0.05, 0.15] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉
Alter-3	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉
Alter-4	〈[0.50, 0.55] , [0.35, 0.40] 〉	〈[0.55, 0.60] , [0.20, 0.30] 〉	〈[0.50, 0.60] , [0.30, 0.40] 〉	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉

Table A24

The decision result given by DM25

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.65, 0.70] , [0.20, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.8, 0.85] , [0.10, 0.15] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.70, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.90] , [0.05, 0.10] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-3	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.15, 0.25] 〉	〈[0.60, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉
Alter-4	〈[0.60, 0.65] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.35, 0.40] 〉

Table A25

The decision result given by DM26

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.15, 0.25] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.80, 0.90] , [0.05, 0.10] 〉
Alter-2	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.70, 0.75] , [0.10, 0.25] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉
Alter-3	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.15, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.65, 0.80] , [0.15, 0.20] 〉
Alter-4	〈[0.55, 0.60] , [0.35, 0.40] 〉	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.70] , [0.20, 0.30] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.50, 0.70] , [0.20, 0.30] 〉

Table A26

The decision result given by DM27

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.85, 0.90] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.75, 0.90] , [0.10, 0.10] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.65, 0.85] , [0.05, 0.15] 〉
Alter-3	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.75] , [0.15, 0.20] 〉
Alter-4	〈[0.55, 0.65] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.50, 0.55] , [0.40, 0.45] 〉	〈[0.55, 0.70] , [0.20, 0.25] 〉

Table A27

The decision result given by DM28

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉	〈[0.85, 0.80] , [0.15, 0.15] 〉
Alter-2	〈[0.65, 0.70] , [0.20, 0.30] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.80, 0.85] , [0.10, 0.15] 〉	〈[0.95, 0.70] , [0.25, 0.30] 〉
Alter-3	〈[0.60, 0.65] , [0.25, 0.35] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.75, 0.65] , [0.60, 0.35] 〉
Alter-4	〈[0.55, 0.60] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.65, 0.70] , [0.15, 0.20] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.60, 0.55] , [0.35, 0.45] 〉

Table A28

The decision result given by DM29

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.65, 0.75] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.80, 0.85] , [0.05, 0.10] 〉	〈[0.80, 0.85] , [0.00, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.15] 〉
Alter-2	〈[0.75, 0.85] , [0.10, 0.15] 〉	〈[0.75, 0.80] , [0.10, 0.20] 〉	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.85, 0.90] , [0.10, 0.05] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉
Alter-3	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.65, 0.70] , [0.25, 0.30] 〉	〈[0.70, 0.75] , [0.15, 0.25] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉
Alter-4	〈[0.60, 0.65] , [0.25, 0.30] 〉	〈[0.55, 0.60] , [0.25, 0.35] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.60, 0.65] , [0.25, 0.35] 〉

Table A29

The decision result given by DM30

	Criterion 1	Criterion 2	Criterion 3	Criterion 4	Criterion 5
Alter-1	〈[0.75, 0.80] , [0.10, 0.20] 〉	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.80, 0.90] , [0.25, 0.10] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.75, 0.85] , [0.10, 0.15] 〉
Alter-2	〈[0.85, 0.90] , [0.05, 0.10] 〉	〈[0.70, 0.80] , [0.10, 0.15] 〉	〈[0.90, 0.95] , [0.00, 0.05] 〉	〈[0.70, 0.75] , [0.10, 0.15] 〉	〈[0.80, 0.80] , [0.05, 0.15] 〉
Alter-3	〈[0.60, 0.65] , [0.30, 0.35] 〉	〈[0.65, 0.70] , [0.20, 0.25] 〉	〈[0.75, 0.80] , [0.15, 0.20] 〉	〈[0.60, 0.65] , [0.20, 0.25] 〉	〈[0.70, 0.70] , [0.20, 0.25] 〉
Alter-4	〈[0.55, 0.60] , [0.30, 0.40] 〉	〈[0.55, 0.60] , [0.30, 0.35] 〉	〈[0.70, 0.75] , [0.20, 0.25] 〉	〈[0.65, 0.70] , [0.15, 0.25] 〉	〈[0.65, 0.80] , [0.15, 0.20] 〉

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