Weighted continuous triangular fuzzy conflict analysis and its three-way decision method 1

Abstract

In the existing conflict analysis models, they used a triangular fuzzy number on [0, 1] to describe the range of an agent’s attitude towards an issue, but there are still some shortcomings in describing the specific attitude and degree of conflict represented by the triangular fuzzy number. In this paper, the conflict analysis model is extended, improved and perfected. Firstly, the expectation of triangular fuzzy number is used in the [-1, 1] triangular fuzzy information system to reasonably express the specific attitudes represented by a triangular fuzzy number. Secondly, the weights of each issue are obtained by using the Sugeno measure, which determines the total attitude of the agent towards all issues. Thirdly, the relationship between agents is obtained with the help of the weighted distance of triangular fuzzy numbers. Finally, the thresholds α and β are calculated by means of triangular fuzzy decision theory rough sets.

1 Introduction

Conflict exists in all aspects of our lives, from small individuals to large countries and societies. Conflict analysis [13, 46] is to build corresponding mathematical models with the existing conflict information, and then study the conflict and provide some useful guidance for conflict resolution. In recent years, the research of conflict analysis has attracted more and more attention from scholars. Since 1998, Polish mathematician Pawlak [34] first proposed a mathematical model for conflict analysis based on rough set theory, which greatly facilitated conflict resolution and laid down the research framework of conflict analysis. Subsequently, many scholars have extended and improved Pawlak’s conflict analysis model from different perspectives to find effective methods to facilitate conflict resolution. For example, Sun et al. [39] generalized the Pawlak conflict analysis model by proposing a rough set-based conflict analysis model over two universes. Lang et al. [23] applied a three-way decision-making method based on decision rough sets to establish a conflict analysis model, which uses decision-theoretic rough sets to calculate thresholds, providing a scientific and effective method for solving thresholds. Z. Bashir et al. [6] proposed a conflict resolution model using rough sets and game theory. Zhi et al. [46] studied the conflict analysis of one-vote veto based on approximate three-way concept lattice. Gong et al. [23] illustrated that the Pawlak conflict analysis model did not express the degree of positive, neutral, and negative of issues of agents and the different importance of different issues. By extending the attitude assignment of Pawlak’s conflict analysis model from {-1, 0, + 1} to the continuous interval [-1, 1], a weighted continuous conflict analysis model and its three-way decision-making method were proposed, which made the application of conflict analysis more widely.

The three-way decision theory was proposed by Yao [42] to facilitate problem solving from three perspectives, three levels and three regions. The rapid development of this idea in theory [14, 41] and practice [8, 33] since its formulation. Li et al. [33] applied the three-way decision theory to information filtering and proposed a network information filtering model and its application to employment agents. Liu et al. [20] discussed three-way decision making under incomplete information systems. Qian et al. [36] studied attribute approximation for sequential three-way decision making under dynamic granulation conditions. Liang et al. [28] applied the three-way decision theory to multi-attribute decision making proposing a tripartite decision making method in the ideal topsis solution of Pythagorean fuzzy information. It is worth mentioning that Yao [42] proposed a three-way conflict analysis [1–5] model by combining the three-way decision theory and conflict analysis, which promoted the rapid development of conflict analysis. Lang et al. [21] investigated a unified model for three-way conflict analysis based on rough set and formal conceptual analysis. Feng et al. [11] proposed a three-way conflict analysis model in dual hesitant fuzzy situation table. Li et al. [24] discussed a three-way conflict analysis and resolution model based on q-rung orthopair fuzzy information.

Fuzzy set theory was introduced by Zede [45] in 1965 as an effective tool for dealing with uncertain and imprecise information. Since the fuzzy sets are proposed, much related research will be introduced [1, 15]. Especially, the triangular fuzzy numbers [4, 47] have received the attention of many scholars in theory and practice for their simple structure and flexible application.For example, K.N. Abu-Bakr et al. [5] discussed the error analysis of two different fuzzy multiplications of operations on triangular fuzzy numbers. Li et al. [32] studied triangular fuzzy interactive multi-attribute decision making based on distance measure. Zhang et al. [47] investigated a triangular fuzzy number multi-attribute decision-making method based on regret theory and mutation development method. Qu et al. [37] applied the extended ITL-VIKOR model with triangular fuzzy numbers to evaluate water abundance.

The combination of conflict analysis, three-way decision and fuzzy set theory not only truly reflects the actual conflict situation agent’s cautious attitude towards the issue, but also helps to promote the development of three-way conflict analysis [6, 44]. Lang et al. [22] studied three-way group conflict analysis under Pythagorean fuzzy set theory. Yi et al. [44] discussed three-way conflict analysis under the hesitant fuzzy information system. Lin et al. [29] investigated three-way conflict analysis under q-rung orthopair fuzzy set theory. However, we note that Li et al. [26] proposed a three-way conflict analysis model on the triangular fuzzy information system of [0, 1] considering the advantage of triangular fuzzy numbers in expressing the uncertainty of the data, but the model is flawed in describing the agent’s attitude and the degree of the attitude towards the issue. Therefore we generalize, improve and perfect the model. The main contributions of this paper are as follows:

(i) We establish a conflict analysis model on the triangular fuzzy conflict information system of [-1, 1], which makes the model more widely applicable;

(ii) On account of the different importance of each issue, we calculate the weight of the corresponding issue with the help of Sugeno measure;

(iii) By revising the triangular fuzzy number distances, we define a weighted distance function to discuss the relationship between the agents.

The rest of this paper is organized as follows. Section 2 reviews existing models for conflict analysis and some basis concepts of triangular fuzzy number. In Section 3, the concrete attitude of triangular fuzzy number is proposed and the conflict analysis models are compared. The total attitude of the agent to all issues is discussed in the section 4. In Section 5, the relationship between agents is obtained based on the weighted distance between two triangular fuzzy numbers. The thresholds α and β are obtained by triangular fuzzy decision theory rough sets in the section 6. In Section 7, the conclusions and remarks of the paper are given.

2 Preliminaries

In this section, we first review existing models for conflict analysis and some basis notions of triangular fuzzy number.

Definition 2.1. (Pawlak’s model)  [34] An information system is a quadruple S = (U, A, V, f), where U = {x₁, x₂, ⋯ , x _n } is a nonempty and finite set of agents, A = {c₁, c₂, ⋯ , c _m } is a nonempty and finite set of issues, V = {-1, 0, + 1}, f is a function from U × A into V.

The function f means the following $\begin{array}{cc}\hfill & f(x_i,c_j)=\hfill \\ \hfill & \{\begin{array}{cc}-1,\mathit{the}\mathit{agent}{x}_i\mathit{negative}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ 0,\mathit{the}\mathit{agent}{x}_i\mathit{neutral}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ +1,\mathit{the}\mathit{agent}{x}_i\mathit{positive}\mathit{about}\mathit{the}\mathit{issue}{c}_j.\hfill & \hfill \\ \hfill & \hfill \end{array}\hfill \end{array}$ Definition 2.2.   [26] Let S = (U, A, V, f) be a triangular fuzzy information system on [0, 1], where U = {x₁, x₂, ⋯ , x _n } is a nonempty and finite set of agents, A = {c₁, c₂, ⋯ , c _m } is a nonempty and finite set of issues, V = {(l, m, u) |0 ≤ l ≤ m ≤ u ≤ 1}, f is a function from U × A into V. The relative area is denoted as ▵S to represent the specific attitude of a triangular fuzzy number.

The relative area ▵S means the following

$\begin{array}{cc}\hfill & ▵S=\hfill \\ \hfill & \{\begin{array}{cc}{S}_R-S_L<0,\mathit{the}\mathit{agent}{x}_i\mathit{negative}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ S_R-S_L=0,\mathit{the}\mathit{agent}{x}_i\mathit{neutral}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ S_R-S_L>0,\mathit{the}\mathit{agent}{x}_i\mathit{positive}\mathit{about}\mathit{the}\mathit{issue}{c}_j,\hfill & \hfill \\ \hfill & \hfill \end{array}\hfill \end{array}$

where the straight line x = 0.5 divides a triangular fuzzy number on [0, 1] into two parts, the area of the left part is S _L and the area of the right part is S _R . Remark The following defects exist when the relative area ▵S is used to determine the specific attitude and conflict degree.

(1) When the triangular fuzzy number is (l, m, u) (0 < l = m = u < 0.5), it obviously represents a negative attitude, but the relative area ▵S = 0 is a neutral attitude; when the triangular fuzzy number is (l, m, u) (0.5 ≤ l = m = u < 1), it obviously represents a positive attitude, but the relative area ▵S = 0 is a neutral attitude, this does not correspond to the actual situation.

(2) When any two triangular fuzzy numbers are (l₁, m₁, u₁) and (l₂, m₂, u₂) satisfy 0 < l _i  < m _i  < u _i  < 0.5, (i = 1, 2) and l₁ - u₁ = l₂ - u₂, the relative area ▵S₁ = ▵ S₂, that is to say, the negative degree is the same; when any two triangular fuzzy numbers are (l₁, m₁, u₁) and (l₂, m₂, u₂) satisfy 0.5 < l _i  < m _i  < u _i  < 1, (i = 1, 2) and l₁ - u₁ = l₂ - u₂, the relative area ▵S₁ = ▵ S₂, that is to say the positive degree is the same, this does not correspond for the actual situation.

Definition 2.3.   [13] Let quadruple S = (U, A, V, f) be an information system on [-1, 1], where U = {x₁, x₂, ⋯ , x _n } is a nonempty and finite set of agents, A = {c₁, c₂, ⋯ , c _m } is a nonempty and finite set of issues, V ∈ [-1, 1], f is a function from U × A into V.

The function f means the following $\begin{array}{cc}\hfill & f(x_i,c_j)\hfill \\ \hfill & \{\begin{array}{cc}<0,\mathit{the}\mathit{agent}{x}_i\mathit{negative}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ =0,\mathit{the}\mathit{agent}{x}_i\mathit{neutral}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ >0,\mathit{the}\mathit{agent}{x}_i\mathit{positive}\mathit{about}\mathit{the}\mathit{issue}{c}_j.\hfill & \hfill \\ \hfill & \hfill \end{array}\hfill \end{array}$ Definition 2.4.  [13] Let S = (U, A, V, f) be an information system for conflict analysis, α and β are two thresholds and 0 ≤ α < β ≤ 1, for any x, y ∈ U, then conflict, neutral, and alliance sets of x are defined as follows

CO_(α,β) (x) = {y ∈ U|ρ _A  (x, y) > β};

NE_(α,β) (x) = {y ∈ U|α ≤ ρ _A  (x, y) ≤ β};

AL_(α,β) (x) = {y ∈ U|ρ _A  (x, y) < α},

where ${\rho }_A(x,y)={\sum }_{c_i\in A}{\omega }_i.\frac{\left|c_i\right(x)-c_i(y\left)\right|}{2}$ is the weighted distance function. Definition 2.5.  [30] A fuzzy number $\tilde{A}$ is called a triangular fuzzy number, denoted by $\tilde{A}=(l,m,u)$ , if ${\mu }_{\tilde{A}\left(x\right)}$ is given by $\begin{array}{cc}\hfill {\mu }_{\tilde{A}\left(x\right)}=\{\begin{array}{cc}\frac{x-l}{m-l},\hfill & l\le x\le m,\hfill \\ \frac{x-u}{m-u},\hfill & m\le x\le u,\hfill \\ 0,\hfill & \mathit{otherewise};\hfill \\ \hfill & \hfill \end{array}& \hfill \end{array}$ where l, m, u ∈ R, and l ≤ m ≤ u. And l and u are called the lower and upper bounds.

Definition 2.6.  [30] Let $\widetilde{A_1}=(l_1,m_1,u_1)$ and $\widetilde{A_2}=(l_2,m_2,u_2)$ be two triangular fuzzy numbers, k a real number. Then the addition, subtraction, multiplication, and scalar multiplication operations of $\widetilde{A_1}$ and $\widetilde{A_2}$ are defined as follows

$\widetilde{A_1}\oplus \widetilde{A_2}$ =(l₁, m₁, u₁) ⊕ (l₂, m₂, u₂)=(l₁ + l₂, m₁ + m₂, u₁ + u₂);

$\widetilde{A_1}⊝\widetilde{A_2}$ =(l₁, m₁, u₁) ⊝ (l₂, m₂, u₂)=(l₁ - u₂, m₁ - m₂, u₁ - l₂);

$\widetilde{A_1}\otimes \widetilde{A_2}=(l_1,m_1,u_1)\otimes (l_2,m_2,u_2)=(l_1\times l_2,m_1\times m_2,u_1\times u_2)$ ;

$k\otimes \widetilde{A_1}=k\otimes (l_1,m_1,u_1)=\{\begin{array}{cc}({\mathit{kl}}_1,{\mathit{km}}_1,{\mathit{ku}}_1),\hfill & k\ge 0;\hfill \\ ({\mathit{ku}}_1,{\mathit{km}}_1,{\mathit{kl}}_1),\hfill & k<0.\hfill \end{array}$

3 Conflict analysis of a single issue on triangular fuzzy information system

In this section, based on [-1, 1] the triangular fuzzy information system, we use the expectation of triangular fuzzy number to transform a triangular fuzzy number on [-1, 1] into a real number [-1, 1] to represent the agent’s specific attitude to the issue and compare four conflict analysis models.

Definition 3.1. Let a quadruple S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], where U = {x₁, x₂, ⋯ , x _n } is a nonempty and finite set of agents, A = {c₁, c₂, ⋯ , c _m } is a nonempty and finite set of issues, V = {(l, m, u) |-1 ≤ l ≤ m ≤ u ≤ 1}, f is a function from U × A into V. We shall divide V into three disjoint parts V _N , V _C and V _P . V _N indicates that the agent has a negative attitude towards the issue, V _C indicates that the agent has a neutral attitude towards the issue, and V _P indicates that the agent has a positive attitude towards the issue.

The interpretation of f is as follow $\begin{array}{cc}\hfill f(x_i,c_j)=\{\begin{array}{cc}(l,m,u)\mathit{and}(l,m,u)\in V_N,\mathit{the}\mathit{agent}{x}_i\mathit{negative}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ (l,m,u)\mathit{and}(l,m,u)\in V_C,\mathit{the}\mathit{agent}{x}_i\mathit{neutral}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ (l,m,u)\mathit{and}(l,m,u)\in V_P,\mathit{the}\mathit{agent}{x}_i\mathit{positive}\mathit{about}\mathit{the}\mathit{issue}{c}_j.\hfill & \hfill \\ \hfill & \hfill \end{array}& \hfill \end{array}$ Remark We limit the triangular fuzzy number to [-1, 1] in the paper, the specific meanings of l, m and u in triangular fuzzy number (l, m, u) are shown in Table 1.

Table 1

The meanings of l, m and u

	∈ [-1, 0)	= 0	∈(0, 1]
l	possible maxium negative degree	neutral	possible minimum positive degree
m	the most possible negative degree	neutral	the most possible positive degree
u	possible minimum negative degree	neutral	possible maxium positive degree

Definition 3.2.  [19] Let $\tilde{A}=(l,m,u)$ be a triangular fuzzy number, then the expectation of $\tilde{A}=(l,m,u)$ is defined as $E\left(\tilde{A}\right)=\frac{l+2m+u}{4}$ .

Definition 3.3. Let $\tilde{A}=(l,m,u)$ be a triangular fuzzy number, then the variance of $\tilde{A}=(l,m,u)$ is defined as $D\left(\tilde{A}\right)=\frac{(l-E(x){)}^2+(m-E\left(x\right){)}^2+(u-E(x){)}^2}{3}$ .

Theorem 3.1. Let $\tilde{A}=(l,m,u)$ be a triangular fuzzy number and $E\left(\tilde{A}\right)$ its expectation, then the following results hold

$-1\le E\left(\tilde{A}\right)\le 1$ ;

if $\tilde{A}$ is a symmetric triangular fuzzy number, that is to say u - m = m - l, then $E\left(\tilde{A}\right)=m$ ;

if u - m > m - l, then $E\left(\tilde{A}\right)>m$ ;

if u - m < m - l, then $E\left(\tilde{A}\right)<m$ .

Proof. Let $\tilde{A}$ =(l, m, u) be a triangular fuzzy number satisfying -1 ≤ l ≤ m ≤ u ≤ 1.

(1) $-1\le l=\frac{l+2l+l}{4}\le E\left(\tilde{A}\right)$ = $\frac{l+2m+u}{4}\le \frac{u+2u+u}{4}=u\le 1$ .

(2) By knowing condition u - m = m - l, we have 2m = u + l, then $E\left(\tilde{A}\right)$ = $\frac{l+2m+u}{4}$ = $\frac{4m}{4}$ =m.

(3) By knowing condition u - m > m - l, we have 2m < u + l, then $E\left(\tilde{A}\right)$ = $\frac{l+2m+u}{4}>\frac{4m}{4}$ = m.

(4) By knowing condition u - m < m - l, we have 2m > u + l, then $E\left(\tilde{A}\right)$ = $\frac{l+2m+u}{4}<\frac{4m}{4}$ = m.

Theorem 3.2. Let $\tilde{A}$ =(l, m, u) be a triangular fuzzy number, $E\left(\tilde{A}\right)$ is the expectation of $\tilde{A}$ , then the following results hold

$E(\tilde{A}\oplus e)=E\left(\tilde{A}\right)+e$ , where e is real number, which can be regarded as a special triangular fuzzy number e = (e, e, e);

$E(k\otimes \tilde{A})=k·E\left(\tilde{A}\right)$ , where k is a positive real number.

Proof. Let triangular fuzzy number $\tilde{A}$ =(l, m, u) satisfy -1 ≤ l ≤ m ≤ u ≤ 1.

(1) According to Definition 2.6, we have $\tilde{A}\oplus e$ =(l, m, u) ⊕ (e, e, e)=(l + e, m + e, u + e), by Definition 3.2 we get $E(\tilde{A}\oplus e)$ = $\frac{(l+e)+2(m+e)+(u+e)}{4}$ = $\frac{l+2m+u}{4}+e$ = $E\left(\tilde{A}\right)+e$ .

(2) According to Definition 2.6, we have $k\otimes \tilde{A}$ =(kl,km,ku), by Definition 3.2 we obtain

$E(k\otimes \tilde{A})$ = $\frac{\left(\mathit{kl}\right)+2\left(\mathit{km}\right)+\left(\mathit{ku}\right)}{4}$ = $k·\frac{l+2m+u}{4}$ = $k·E\left(\tilde{A}\right)$ .

Definition 3.4. Let S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], where U = {x₁, x₂, ⋯ , x _n } is a nonempty and finite set of agents, A = {c₁, c₂, ⋯ , c _m } is a nonempty and finite set of issues, V = {(l, m, u) |-1 ≤ l ≤ m ≤ u ≤ 1}, f is a function from U × A into V. The expectation value E (c _j  (x _i )) to represent the specific attitude of a triangular fuzzy number f (x _i , c _j ) = (l, m, u).

The expectation value E (c _j  (x _i )) means the following $\begin{array}{cc}\hfill & E\left(c_j\right(x_i\left)\right)\hfill \\ \hfill & \{\begin{array}{cc}<0,\mathit{the}\mathit{agent}{x}_i\mathit{negative}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ =0,\mathit{the}\mathit{agent}{x}_i\mathit{neutral}\mathit{about}\mathit{the}\mathit{issue}{c}_j;\hfill & \hfill \\ >0,\mathit{the}\mathit{agent}{x}_i\mathit{positive}\mathit{about}\mathit{the}\mathit{issue}{c}_j,\hfill & \hfill \\ \hfill & \hfill \end{array}\hfill \end{array}$ When E (c _j  (x _i )) ∈ (0, 1], the larger value of E (c _j  (x _i )) means that the agent x _i is greater positive degree about the issue c _j ; when E (c _j  (x _i )) ∈ [-1, 0), the smaller value of E (c _j  (x _i )) means that the agent x _i is greater negative degree about the issue c _j ; when the expectations are the same, the smaller the variance D (c _j  (x _i )) the greater the degree to which agent x _i positive or negative issue c _j .

Remark (1) When c _j  (x _i )=(1, 1, 1), E (c _j  (x _i )) = +1 means that the agent x _i expresses a positive attitude towards the issue c _j ; when c _j  (x _i )=(0, 0, 0), E (c _j  (x _i )) =0 means that the agent x _i expresses a neutral attitude towards the issue c _j ; when c _j  (x _i )=(-1, - 1, - 1), E (c _j  (x _i )) = -1 means that the agent x _i expresses a negative attitude towards the issue c _j . We define the conflict analysis model of triangular fuzzy information system on [-1, 1], which will degenerate into Pawlak’s conflict analysis model.

(2) If the triangular fuzzy information system is taken as the interval [-1, 1], then E (c _j  (x _i )) ∈ [-1, 1], when E (c _j  (x _i )) <0 the agent x _i shows a negative attitude towards issues c _j ; when E (c _j  (x _i )) =0 the agent x _i shows a neutral attitude towards issues c _j ; when E (c _j  (x _i )) >0 the agent x _i shows a positive attitude towards issues c _j . We define the conflict analysis model of triangular fuzzy information system on [-1, 1], which will degenerate into Gong’s conflict analysis model.

(3) If the triangular fuzzy information system on [-1, 1], then E (c _j  (x _i )) ∈ [-1, 1], when E (c _j  (x _i )) <0 the agent x _i shows a negative attitude towards issues c _j ; when E (c _j  (x _i )) =0 the agent x _i shows a neutral attitude towards issues c _j ; when E (c _j  (x _i )) >0 the agent x _i shows a positive attitude towards issues c _j . We define the conflict analysis model of triangular fuzzy information system on [-1, 1], which will degenerate into Li’s conflict analysis model.

Definition 3.5. Let E (c _j  (x _i )) be the specific attitude of the agent x _i to the issue c _j , to maintain uniformity with the thresholds in Section 5, we define the following mapping $\overline{E}\left(c_j\right(x_i\left)\right)$ = $\frac{1}{2}\left(E\right(c_j\left(x_i\right))+1)$ .

Remark The mapping transforms real numbers on [-1, 1] into real numbers on [0, 1], so that the threshold we need to classify the set of agents into conflict, neutral, and coalitional sets is the same threshold as the one we need to discuss the relationships between agents in Section 5.

Definition 3.6. Let S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], α and β are two thresholds and 0 ≤ α < β ≤ 1, then the conflict, neutral, and alliance sets of U about the issues c _j are defined as follows

CO (c _j ) (α,β) (U)= $\left\{x_i\in U\left|\overline{E}\right(c_j\left(x_i\right))<\alpha \right\}$ ;

NE (c _j )(α,β) (U)= $\left\{x_i\in U|\alpha \le \overline{E}(c_j\left(x_i\right))\le \beta \right\}$ ;

AL_{(c j )}(α,β) (U)= $\left\{x_i\in U\left|\overline{E}\right(c_j\left(x_i\right))>\beta \right\}$ .

Example 3.1. Table 2 shows a triangular fuzzy information system on [-1, 1], we obtain the agent’s specific attitude towards the issue through Definition 3.2 in Table 3. Let α = 0.45, β = 0.55, we obtain the three coalitions of agents for each issue according to Definition 3.6 in Table 4.

Table 2

Triangular fuzzy information system

	c 1	c 2	c 3	c 4	c 5
x 1	(-0.4, - 0.2, - 0, 1)	(-0.3, 0.1, 0.4)	(0, 0, 0)	(0.3, 0.6, 0.7)	(-0.8, - 0.3, 0.5)
x 2	(-0.8, - 0.6, - 0.2)	(0.9, 0.9, 0.9)	(0.4, 0.8, 0.9)	(-0.6, - 0.4, 0.3)	(-0.3, 0, 0.2)
x 3	(0.2, 0.5, 0.6)	(-0.3, - 0.3, - 0.3)	(-0.9, - 0.6, 0.1)	(1, 1, 1)	(-0.5, - 0.2, 0.3)
x 4	(-0.4, 0.1, 0.7)	(-0.6, - 0.4, 0.3)	(0.5, 0.5, 0.5)	(0.2, 0.4, 0.6)	(-0.9, - 0.7, - 0.1)
x 5	(-1, - 1, - 1)	(0.2, 0.5, 0.7)	(-0.4, 0, 0.3)	(-0.8, - 0.5, 0)	(-0.5, 0, 0.6)

Table 3

The specific attitude of triangular fuzzy information system

	c 1	c 2	c 3	c 4	c 5
x 1	-0.225	0.075	0	0.55	-0.225
x 2	-0.55	0.9	0.725	-0.275	-0.025
x 3	0.45	-0.3	-0.5	1	-0.15
x 4	0.125	-0.275	-0.5	0.4	-0.6
x 5	-1	0.475	-0.025	-0.45	0.025

Table 4

Three alliances for each issue

		AL(0.45,0.55) (x i )	NE(0.45,0.55) (x i )	CO(0.45,0.55) (x i )
c 1	{x3, x4}	∅	{x1, x2, x5}
c 2	{x2, x5}	{x1}	{x3, x4}
c 3	{x2, x4}	{x1, x5}	{x3}
c 4	{x1, x3, x4}	∅	{x2, x5}
c 5	∅	{x2, x5}	{x1, x3, x4}

Comparisons of conflict analysis models

We compare the four conflict analysis models in Table 5. Here N, C, and P represent a negative attitude, a neutral attitude and a positive attitude of the agent x _i about the issue c _j , respectively. PD and ND represent positive degree and negative degree of the agent x _i about the issues c _j . \ represents the Pawlak conflict analysis model does not involve the degree of positive and negative.

Table 5

Comparisons of conflict analysis models

model	IS	N	C	P	ND	PD
Pawlak	{-1, 0, + 1}	-1	0	+1	∖	∖
Gong	[-1, 1]	f (x i , c j ) <0	f (x i , c j ) =0	f (x i , c j ) >0	f (x i , c j ) ∈ [-1, 0)	f (x i , c j ) ∈ (0, 1]
Li	(l, m, u) ∈ [0, 1]	▵S (c j  (x i )) <0.5	▵S (c j  (x i )) =0.5	▵S (c j  (x i )) >0.5	▵S (c j  (x i )) ∈ [0, 0.5)	▵S (c j  (x i )) ∈ (0.5, 1]
Our	(l, m, u) ∈ [-1, 1]	E (c j  (x i )) <0	E (c j  (x i )) =0	E (c j  (x i )) >0	(E (c j  (x i )) , D (c j  (x i )))	(E (c j  (x i )) , D (c j  (x i )))

By comparison, we summarize the connections and differences of the four conflict analysis models in Table 5.

(1) The four conflict analysis studies different information systems. Pawlak’s conflict analysis model is studied on the three-valued information system {-1,0,+1}. Gong’s conflict analysis model is studied on the real numbers of [-1, 1]. Li’s conflict analysis model is studied on the triangular fuzzy information system of [0, 1]. Our conflict analysis model is studied on the triangular fuzzy information system of [-1, 1].

(2) The four conflict analysis models can describe the agent’s attitude towards an issue. Pawlak’s conflict analysis model used -1, 0, +1 to represent the negative, neutral, and positive attitude of the agent x _i to the issue c _j , respectively. Gong’s conflict analysis model used f (x _i , c _j ) <0, f (x _i , c _j ) =0, f (x _i , c _j ) >0 to represent the negative, neutral and positive attitude of the agent x _i to the issue c _j , respectively. Li’s conflict analysis model used the relative area △S (c _j  (x _i )) <0, △S (c _j  (x _i )) =0, △S (c _j  (x _i )) >0 to represent the negative, neutral, and positive specific attitude of the agent x _i to the issue c _j , respectively. Our conflict analysis model used the expectation E (c _j  (x _i )) <0, E (c _j  (x _i )) =0, E (c _j  (x _i )) >0 to represent the negative, neutral, and positive attitude of the agent x _i to the issue c _j , respectively. But Li’s model has some flaws in describing specific attitudes.

For example, f (x _i , c _j ) = (0.9, 0.9, 0.9) obviously the agent x _i is positive attitude of the issue c _j , however △S (c _j  (x _i )) =0, the specific attitude of the agent x _i to the issue c _j is a neutral attitude. f (x _i , c _j ) = (0.1, 0.1, 0.1) obviously the agent x _i is negative attitude of the issue c _j , however △S (c _j  (x _i )) =0, the specific attitude of the agent x _i to the issue c _j is a neutral attitude. This is not the case either.

(3) Agent has a different degree of conflict over issue. Pawlak’s conflict analysis model does not involve the degree of positive and negative. Gong’s conflict analysis model used f (x _i , c _j ) <0, f (x _i , c _j ) >0 to represent the negative and positive degree of the agent x _i to the issue c _j , respectively. When f (x _i , c _j ) ∈ [-1, 0), the smaller the value of f (x _i , c _j ), the greater the negative degree of the agent x _i to the issue c _j ; when f (x _i , c _j ) ∈ (0, 1], the larger the value of f (x _i , c _j ), the greater the positive degree of the agent x _i to the issue c _j . Li’s conflict analysis model used △S (c _j  (x _i )) <0, △S (c _j  (x _i )) >0 to represent the negative and positive degree of the agent x _i to the issue c _j , respectively. When △S (c _j  (x _i )) ∈ [-1, 0), the smaller the value of △S (c _j  (x _i )), the greater the negative degree of the agent x _i to the issue c _j , when △S (c _j  (x _i )) ∈ (0, 1], the larger the value of △S (c _j  (x _i )), the greater the positive degree of the agent x _i to the issue c _j . Our conflict analysis model used E (c _j  (x _i )) <0, E (c _j  (x _i )) >0 to represent the negative and positive degree of the agent x _i to the issue c _j , respectively. When E (c _j  (x _i )) ∈ (0, 1], the larger value of E (c _j  (x _i )) means that the agent x _i is greater positive degree about the issue c _j ; when E (c _j  (x _i )) ∈ [-1, 0), the smaller value of E (c _j  (x _i )) means that the agent x _i is greater negative degree about the issue c _j ; when the expectations are the same, the smaller the variance D (c _j  (x _i )) the greater the degree to which agent x _i positive or negative issue c _j , but Li’s model is somewhat flawed in describing the degree of conflict.

For example, f (x _i , c _j ) = (0.51, 0.52, 0.53), f (x₂, c _j ) = (0.61, 0.62, 0.63) and f (x₃, c _j ) = (0.91, 0.92, 0.93) are agents x₁, x₂, x₃ attitudes towards issue c _j , respectively. Then agents x₁, x₂, x₃ concrete attitudes towards issue c _j is △S (c (x₁)) = △ S (c (x₂)) = △ S (c (x₃)) =0.01, obviously the agent x₃ has more positive for the issue c _j than the agent x₂ does for agent x₁, but the positive is the same. This is also not the case.

In summary, our conflict analysis model successfully addresses the shortcomings of Li’s conflict analysis model and is also a generalization of Li’s conflict analysis model.

4 Conflict analysis of multiple issues on triangular fuzzy information system

Definition 4.1.  [12] Let C be a nonempty set, A be an σ-algebra formed by subsets of C, and call the set function μ : A → [0, ∞) a regular fuzzy measure that satisfies the following conditions

μ (∅) =0;

μ (C) =1;

for A₁, A₂ ∈ A, if A₁ ⊆ A₂, then μ (A₁) ≤ μ (A₂).

Definition 4.2.  [12] The σ - λ law is that g _λ satisfies $g_{\lambda }\left({\bigcup }_{i=1}^{\infty }{A}_i\right)=\{\begin{array}{cc}\frac{1}{\lambda }\left({\prod }_{i=1}^{\infty }\right(1+\lambda g_{\lambda }\left(A_i\right))-1),\hfill & \lambda \ne 0,\hfill \\ {\sum }_{i=1}^{\infty }{g}_{\lambda }\left(A_i\right),\hfill & \lambda =0,\hfill \\ \hfill & \hfill \end{array}$

where $\lambda \in (-\frac{1}{\mathit{sup}\mu },\infty )\cup \left\{0\right\}$ , {A _i } ⊂ A, A _i  ∩ A _j  = ∅ , i ≠ j, (i, j = 1, 2, ⋯). In particular, when λ = 0, g _λ degenerates to the classical probability measure. Remark (1) A regular fuzzy measure that satisfies the σ - λ law is called Sugeno measure.

(2) For the sugeno measure, when C is a finite set, then for any subset A of C, we have $g_{\lambda }\left(A\right)=\{\begin{array}{cc}\frac{1}{\lambda }\left(\prod _{c\in A}\right(1+\lambda g_{\lambda }\left(c\right))-1),\hfill & \lambda \ne 0,\hfill \\ {\sum }_{c\in A}{g}_{\lambda }\left(c\right),\hfill & \lambda =0.\hfill \\ \hfill & \hfill \end{array}$

(3) For the Sugeno measure λ can be calculated by the following equation $1+\lambda ={\Pi }_1^m(1+\lambda g_{\lambda }(c_j\left)\right)$ .

Definition 4.3. Let g _λ (c _j ), (j = 1, 2, ⋯ , m) satisfy σ - λ law of fuzzy measures, A _j  = {c₁, c₂, ⋯ , c _j }, the weights are defined as follows w (c _j ) = g _λ (A _j ) - g _λ (A_j-1), (j = 1, 2, ⋯ , m), where g _λ (A₁) = g _λ (c₁), g _λ (A₀) =0, g _λ (A _m ) =1.

Definition 4.4. The total attitude of the agent x _i to all issues A is denoted by $\tilde{E}\left(A\right(x_i\left)\right)$ and is defined as follows $\tilde{E}\left(A\right(x_i\left)\right)$ = ${\sum }_{j=1}^m\left(w\right(c_j)\otimes c_j(x_i\left)\right)$ , (i = 1, 2, ⋯ , n), where c _j  (x _i ) represents the attitude of the agent x _i to the issues c _j , w (c _j ) represents the weight of the issue c _j .

Remark We note that $\tilde{E}\left(A\right(x_i\left)\right)$ is a triangular fuzzy number by the operation of triangular fuzzy numbers, and we express the specific attitude of the agent x _i to all issues by the expectation to transform $\tilde{E}\left(A\right(x_i\left)\right)$ into a real number E (A (x _i )).

Definition 4.5. Let S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], α and β are two thresholds and 0 ≤ α < β ≤ 1, then the conflict, neutral, and alliance sets of U about each issues c _j are defined as follows

= $\left\{x\in U\left|\overline{E}\right(A\left(x_i\right))<\alpha \right\}$ ;

= $\left\{x\in U|\alpha \le \overline{E}(A\left(x_i\right))\le \beta \right\}$ ;

= $\left\{x\in U\left|\overline{E}\right(A\left(x_i\right))>\beta \right\}$ .

Example 4.1. (Continued from Example 3.1) According to Definition 4.2 and Definition 4.3, if g _λ (c₁) =0.1, g _λ (c₂) =0.2, g _λ (c₃) =0.2, g _λ (c₄) =0.1, g _λ (c₅) =0.1, then λ = 1.4067 and we obtain that the weight of issues are w (c₁) =0.1, w (c₂) =0.23, w (c₃) =0.29, w (c₄) =0.19, w (c₅) =0.19.

According to Definition 4.4 the specific attitudes of agent x _i toward all issues c are calculated in Table 6. $\begin{array}{cc}\hfill \tilde{E}\left(A\right(x_1\left)\right)& =\left(w\right(c_1)\otimes c_1(x_1\left)\right)\oplus \left(w\right(c_2)\otimes c_2(x_1\left)\right)\hfill \\ \hfill & \oplus \left(w\right(c_3)\otimes c_3(x_1\left)\right)\oplus \left(w\right(c_4)\otimes c_4(x_1\left)\right)\hfill \\ \hfill & \oplus \left(w\right(c_5)\otimes c_5(x_1\left)\right)\hfill \\ \hfill & =(0.1\otimes (-0.4,-0.2,-0.1\left)\right)\oplus (0.23\hfill \\ \hfill & \otimes (-0.3,0.1,0.4))\oplus (0.29\otimes (0,0,0))\hfill \\ \hfill & \oplus (0.19\otimes (0.3,0.6,0.7\left)\right)\hfill \\ \hfill & \oplus (0.19\otimes (-0.8,-0.3,0.5\left)\right)\hfill \\ \hfill & =(-0.154,0.1,0.289).\hfill \end{array}$ E (A (x₁)) = $\frac{-0.204+2\times 0.06+0.31}{4}$ =0.057, $\overline{E}\left(A\right(x_1\left)\right)$ = $\frac{1}{2}\left(E\right(c\left(x_1\right))+1)$ = $\frac{1}{2}(0.057+1)$ =0.529, so the agent x₁ is positive on all issues.

Similarly, we can get E (A (x₂))=0.306, E (A (x₃))=-0.008, E (A (x₄))=0.056, E (A (x₅))=-0.076, hence agents x₂, x₃ and x₄ are positive on all issues, the agent x₅ is negative on all issues.

Table 6

The relationships between $\tilde{E}\left(A\right(x_i\left)\right)$ , E (A (x _i ))

and $\overline{E}\left(A\right(x_i\left)\right)$

	$\tilde{E}\left(A\right(x_i\left)\right)$	E (A (x i ))	$\overline{E}\left(A\right(x_i\left)\right)$
x 1	(-0.204, 0.06, 0.31)	0.057	0.529
x 2	(0.072, 0.303, 0.547)	0.306	0.653
x 3	(-0.215, - 0.041, 0.267)	-0.008	0.496
x 4	(-0.166, 0.006, 0.379)	0.056	0.528
x 5	(-0.417, - 0.08, 0.262)	-0.079	0.461

Let α = 0.45, β = 0.55, we get that the agent has three coalitions for all issues A by Definition 3.5 and Definition 4.5

1. =∅;

2. ={x₁, x₃, x₄, x₅};

3. ={x₂}.

Next, we summarize the process of computing the three coalitions in Algorithm 1.

Algorithm 1. Computer three collations on each issue and on all issues

Input :  A triangular fuzzy information system for conflict analysis.

Output :  Three collation of conflict analysis on each issue and on all issues.

1 :  U = {x₁, x₂, ⋯ , x _n }, A = {c₁, c₂, ⋯ , c _m };

2 :  for each x _i  ∈ U do

3 :    for each c _j  ∈ A do

4 :      Compute E (c _j  (x _i ));

5 :  Convert E (c _j  (x _i )) to $\overline{E}\left(c_j\right(x_i\left)\right)$ ;

6 :    end for

7 :  end for

8 :  Provide single point set measures by experts;

9 :  Compute w (c _j )

10 :  Integrate the attitude of x _i on all into one, denote by $\tilde{E}\left(A\right(x_i\left)\right)$ ;

11 :  Compute the specific attitude of x _i to all issues, denote by E (A (x _i ));

12 :  Convert E (A (x _i )) to $\overline{E}\left(A\right(x_i\left)\right)$ ;

13 :  end for

14 :  Provide triangular fuzzy loss functions by experts;

15 :  Computer thresholds α, β;

16 :  Determine three collations on each issue and on all issues.

We explain the following for Algorithm 1. From step 1 to step 6, we transform the agent’s attitude towards the issue c _j  (x _i ) = (l, m, u) into a [-1, 1] real number to get the agent’s specific attitude towards the issue, and transform the [-1, 1] real number to [0, 1] by mapping $\overline{E}\left(c\right(x_i\left)\right)$ . From step 7 to step 12, based on the Sugeno measure, we obtained the weights for each issue, thus determining the agent’s total attitude towards all issues. From step 13 to step 15, we compute the thresholds, thus obtaining the three coalitions for each issue and for all issues.

We analyze the time complexity of Algorithm 1 as follows. We assume that the computer takes the same amount of time to perform one computation, denoted as T₀. From step 1 to step 2, we need the time for (2mn) ∗ T₀. From step 7 to step 12, we need the time for (3n + 1) ∗ T₀. From step 13 to step 14, we need the time for T₀. So to execute the algorithm once we need the time for (2mn + 3n + 2) ∗ T₀. Furthermore, the complexity of the algorithm is O (mn). Therefore, the algorithm is effective in practical conflict analysis.

5 Relationship between agents on issues sets

Definition 5.1. Let $\widetilde{A_1}=(l_1,m_1,u_1)$ and $\widetilde{A_2}=(l_2,m_2,u_2)$ be two triangular fuzzy numbers, then the distance from $\widetilde{A_1}$ to $\widetilde{A_2}$ is defined as $d(\widetilde{A_1},\widetilde{A_2})=\sqrt[2]{\frac{(l_1-l_2{)}^2+(m_1-m_2{)}^2+(u_1-u_2{)}^2}{12}}.$

Remark In reference 30, the denominator of the distance between the two triangular fuzzy numbers is 3, we have changed 3 to 12 here, they are not affected in the ordering, but 12 can drop the distance at [0, 1].

Definition 5.2. Let a quadruple S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], w (c _j ) is the weight of issue c _j and ${\sum }_{cj\in A}w\left(c_j\right)=1$ , then the weight distance D _A between any two agents x _{i 1} and x _{i 2} are defined as follows D _A  (x _{i 1} , x _{i 2} )=∑ _{c j ϵA} w (c _j ) d _{c j}  (x _{i 1} , x _{i 2} ) .

Theorem 5.1. Let a quadruple S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], for x _i  ∈ A, (i = 1, 2, 3), then the weight distance D _A has the following results

0 ≤ D _A  (x _{i 1} , x _{i 2} ) ≤1, if D _A  (x _{i 1} , x _{i 2} )) =0 ⇔ x _{i 1}  = x _{i 2} ;

D _A  (x _{i 1} , x _{i 2} )= D _A  (x _{i 2} , x _{i 1} );

D _A  (x _{i 1} , x _{i 3} ) ≤ D _A  (x _{i 1} , x _{i 2} ) + D _A  (x _{i 2} , x _{i 3} ).

Definition 5.3. Let S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], (U, D _A ) is a conflict space, α and β are two thresholds and 0 ≤ α < β ≤ 1, then the conflict, neutral and alliance sets of x ∈ U about all issues A are defined as follow

1. $C{O}_A{}^{(\alpha ,\beta )}(x)=\{\in U|D_A\left(x,y\right)>\beta \}$ ;

2. $N{E}_A{}^{(\alpha ,\beta )}(x)=\{\in U|\beta \ge D_A\left(x,y\right)\ge \alpha \}$ ;

3. $A{L}_A{}^{(\alpha ,\beta )}(x)=\{\in U|\beta \ge D_A\left(x,y\right)\ge \alpha \}$ .

Theorem 5.2. Let S = (U, A, V, f) be a triangular fuzzy information system on [-1, 1], (U, D _A ) is a conflict space, α and β are two thresholds and 0 ≤ α < β ≤ 1, then the following conclusions hold

1. $y\in C{O}_A{}^{(\alpha ,\beta )}(x)\iff x\in C{O}_A{}^{(\alpha ,\beta )}(y);$

2. $y\in N{E}_A{}^{(\alpha ,\beta )}(x)\iff x\in N{E}_A{}^{(\alpha ,\beta )}(y);$

3. $y\in A{L}_A{}^{(\alpha ,\beta )}(x)\iff x\in A{L}_A{}^{(\alpha ,\beta )}(y).$

Proof. (1) According to Definition 5.3, ={y ∈ U|D _A  (x, y) > β} and ={z ∈ U|D _A  (y, z) > β}, if , then D _A  (x, y) > β. From D _A  (x, y) = D _A  (y, x) and D _A  (y, x) > β, so , vice versa. Hence, .

(2) According to Definition 5.3, ={y ∈ U|β ≥ D _A  (x, y) ≥ α} and ={z ∈ U|β ≥ D _A  (z, y) ≥ α}, if , then β ≥ D _A  (x, y) ≥ α. From D _A  (x, y) = D _A  (y, x) and β ≥ D _A  (y, x) ≥ α, so , vice versa. Hence, .

(3) According to Definition 5.3, ={y ∈ U|D _A  (x, y) < α} and ={z ∈ U|D _A  (y, z) < α}, if , then D _A  (x, y) > α. From D _A  (x, y) = D _A  (y, x) and D _A  (y, x) < α, so , vice versa. Hence, .

Example 5.1. (Continued from Example 4.1) According to Definition 5.1 and Definition 5.2, the weighted distance calculation based on triangular fuzzy information system on [-1, 1] is shown in Table 7. Let α = 0.45 and β = 0.55, the relationship between agents can be obtained by Definition 5.3 as shown in Table 8.

Table 7

The weighted distance between any two agents

	x 1	x 2	x 3	x 4	x 5
x 1	0	0.337	0.246	0.190	0.245
x 2	0.337	0	0.499	0.333	0.234
x 3	0.246	0.499	0	0.319	0.390
x 4	0.190	0.333	0.319	0	0.372
x 5	0.245	0.234	0.390	0.372	0

Table 8

Relationships between agents

	AL(0.45,0.55) (x i )	NE(0.45,0.55) (x i )	CO(0.45,0.55) (x i )
x 1	{x2, x3, x4, x5}	∅	∅
x 2	{x1, x4x5}	{x3}	∅
x 3	{x1, x4, x5}	{x2}	∅
x 4	{x1, x2, x3, x5}	∅	∅
x 5	{x1, x2, x3, x4}	∅	∅

We have d _{c 1}  (x₁, x₂) =0.166, d _{c 2}  (x₁, x₂) =0.441, d _{c 3}  (x₁, x₂) =0.366, d _{c 4}  (x₁, x₂) =0.405, d _{c 5}  (x₁, x₂) =0.189. w (c₁) =0.1, w (c₂) =0.23, w (c₃) =0.29, w (c₄) =0.19, w (c₅) =0.19,

D _A  (x₁, x₂)=0.166 × 0.1 + 0.441 × 0.23 + 0.336 × 0.29 + 0.405 × 0.19 + 0.189 × 0.19 = 0.337 .

Next, we summarize the whole process of computing the relationship between agents in Algorithm 2.

Algorithm 2. Computer relationship between agents on issues set

Input :  A triangular fuzzy information system for conflict analysis.

Output :  Relationship between agents on issues set.

1 :  U = {x₁, x₂, ⋯ , x _n }, A = {c₁, c₂, ⋯ , c _m };

2 :  for each c _l  ∈ A do

3 :    for each x _i , x _j  ∈ U do

4 :      Compute d _{c l}  (x _i , x _j );

5 :    end for

6 :  Provide single-point set measures by experts;

7 :  Compute w (c _j );

8 :  Compute the weighted distance D _A  (x _i , x _j ) of issues set c;

9 :  end for

10 :  Provide triangular fuzzy loss functions by experts;

11 :  Computer thresholds α, β;

12 :  Determine relationship between agents on issues set.}

We explain the following for Algorithm 2. From step 1 to step 5, we calculated the distance of any two agents to each issue by Definition 5.1. From step 6 to step 9, we obtained the weights of each issue based on the Sugeno measure, which in turn computed the weighted distances according to Definition 5.3. From step 10 to step 12, we compute the thresholds from Theorem 6.3, which gives the relationship between agents.

We analyze the time complexity of Algorithm 2 as follows. We assume that the computer takes the same amount of time to perform one computation, denoted as T. From step 1 to step 5, we need time $\frac{{\mathit{mn}}^2}{2}\ast T$ . From step 6 to step 7, we need time T. In step 8 we need time $\frac{n^2}{2}\ast T$ . From step 10 to step 12, we need time T. So to execute the algorithm once we have time $(\frac{(m+1)n^2}{2}+2)\ast T$ . Moreover, the complexity of the algorithm is $O\left(\frac{(m+1)n^2}{2}\right)$ . Therefore, the proposed algorithm is feasible in practice.

6 Threshold solving method based on triangular fuzzy decision rough set theory

Definition 6.1. Let S = (U, A, V, f) be a triangular fuzzy information system for conflict analysis, then triangular fuzzy loss functions are defined in Table 9.

Table 9

Triangular fuzzy loss functions

action	Ω	¬Ω
a P	${\tilde{\lambda }}_{\mathit{PP}}=(l_{\mathit{PP}},m_{\mathit{PP}},u_{\mathit{PP}})$	${\tilde{\lambda }}_{\mathit{PN}}=(l_{\mathit{PN}},m_{\mathit{PN}},u_{\mathit{PN}})$
a B	${\tilde{\lambda }}_{\mathit{BP}}=(l_{\mathit{BP}},m_{\mathit{BP}},u_{\mathit{BP}})$	${\tilde{\lambda }}_{\mathit{BN}}=(l_{\mathit{BN}},m_{\mathit{BN}},u_{\mathit{BN}})$
a N	${\tilde{\lambda }}_{\mathit{NP}}=(l_{\mathit{NP}},m_{\mathit{NP}},u_{\mathit{NP}})$	${\tilde{\lambda }}_{\mathit{NN}}=(l_{\mathit{NN}},m_{\mathit{NN}},u_{\mathit{NN}})$

Here Ω, ¬Ω denote two states of the agent x, a _P , a _B and a _N denote three actions of agent x, respectively. ${\tilde{\lambda }}_{\mathit{PP}}$ , ${\tilde{\lambda }}_{\mathit{BP}}$ and ${\tilde{\lambda }}_{\mathit{NP}}$ denote the losses of taking action a _P , a _B and a _N when the agent belongs to Ω; ${\tilde{\lambda }}_{\mathit{PN}}$ , ${\tilde{\lambda }}_{\mathit{BN}}$ and ${\tilde{\lambda }}_{\mathit{NN}}$ denote the losses of taking action a _P , a _B and a _N when the agent belongs to ¬Ω. ${\tilde{\lambda }}_{\mathit{PP}}$ , ${\tilde{\lambda }}_{\mathit{BP}}$ , ${\tilde{\lambda }}_{\mathit{NP}}$ , ${\tilde{\lambda }}_{\mathit{PN}}$ , ${\tilde{\lambda }}_{\mathit{BN}}$ and ${\tilde{\lambda }}_{\mathit{NN}}$ are triangular fuzzy numbers between 0 and 1.

Definition 6.2. The value of expected loss $\tilde{R}\left(a_j\right|x)$ , can be defined as follows $\tilde{R}\left(a_j\right|x)$ = $\left(D_A\right(x,y)\otimes {\tilde{\lambda }}_{\mathit{jp}})\oplus \left(\right(1-D_A(x,y))\otimes {\tilde{\lambda }}_{\mathit{jN}}),$ where ${\tilde{\lambda }}_{\mathit{jP}}=(l_{\mathit{jP}},m_{\mathit{jP}},u_{\mathit{jP}})$ , ${\tilde{\lambda }}_{\mathit{jN}}=(l_{\mathit{jN}},m_{\mathit{jN}},u_{\mathit{jN}})$ , (j = P, B, N ; i = 1, 2, ⋯ , n).

Theorem 6.1. Let S = (U, A, V, f) be a triangular fuzzy information system for conflict analysis, α and β two thresholds satisfying 0 ≤ α < β ≤ 1. The the following results hold

if D _A  (x, y) < α, then y ∈ AL^(α,β) (x);

if β ≥ D _A  (x, y) ≥ α, then y ∈ NE^(α,β) (x);

if D _A  (x, y) > β, then y ∈ CO^(α,β) (x),

where

$\alpha =\frac{(l_{\mathit{PN}}-l_{\mathit{BN}})+2(m_{\mathit{PN}}-m_{\mathit{BN}})+(u_{\mathit{PN}}-u_{\mathit{BN}})}{(l_{\mathit{BP}}-l_{\mathit{PP}})+(l_{\mathit{PN}}-l_{\mathit{BN}})+2(m_{\mathit{PP}}-m_{\mathit{BP}})+2(m_{\mathit{PN}}-m_{\mathit{BN}})+(u_{\mathit{PN}}-u_{\mathit{pp}})+(u_{\mathit{PN}}-u_{\mathit{BN}})}$ ;

$\beta =\frac{(l_{\mathit{PN}}-l_{\mathit{NN}})+2(m_{\mathit{PN}}-m_{\mathit{NN}})+(u_{\mathit{PN}}-u_{\mathit{NN}})}{(l_{\mathit{PN}}-l_{\mathit{NN}})+(l_{\mathit{NP}}-l_{\mathit{PP}})+2(m_{\mathit{PN}}-m_{\mathit{NN}})+2(m_{\mathit{NP}}-m_{\mathit{PP}})+(u_{\mathit{NP}}-u_{\mathit{PP}})+(u_{\mathit{PN}}-u_{\mathit{NN}})}$ .

Proof. By Definition 6.1 and Definition 6.2, we have the expected losses $\tilde{R}\left(a_P\right|y)$ , $\tilde{R}\left(a_B\right|y)$ and $\tilde{R}\left(a_N\right|y)$ associated with taking the individual action for the agent y as follow

$\tilde{R}\left(a_P\right|y)$ = $\left(D_c\right(x,y)\otimes {\tilde{\lambda }}_{\mathit{PP}})\oplus \left(\right(1-D_c(x,y))\otimes {\tilde{\lambda }}_{\mathit{PN}})$ ;

$\tilde{R}\left(a_B\right|y)$ = $\left(D_c\right(x,y)\otimes {\tilde{\lambda }}_{\mathit{BP}})\oplus \left(\right(1-D_c(x,y))\otimes {\tilde{\lambda }}_{\mathit{BN}})$ ;

$\tilde{R}\left(a_N\right|y)$ = $\left(D_c\right(x,y)\otimes {\tilde{\lambda }}_{\mathit{NP}})\oplus \left(\right(1-D_c(x,y))\otimes {\tilde{\lambda }}_{\mathit{NN}})$ .

By Definition 3.2, the value of $\tilde{R}\left(a_P\right|y)$ , $\tilde{R}\left(a_B\right|y)$ and $\tilde{R}\left(a_N\right|y)$ are as follow

$E\left(\tilde{R}\right(a_P\left|y\right))$ ={(l _PN  + 2m _PN  + u _PN ) + [(l _PP  - L _PN ) +2 (m _PP  - m _PN ) + (u _PP  - u _PN )] D _C  (x, y)}/4;

$E\left(\tilde{R}\right(a_B\left|y\right))$ ={(l _BN  + 2m _BN  + u _BN ) + [(l _BP  - L _BN ) +2 (m _BP  - m _BN ) + (u _BP  - u _BN )] D _C  (x, y)}/4;

$E\left(\tilde{R}\right(a_N\left|y\right))$ ={(l _NN  + 2m _NN  + u _NN ) + [(l _PN  - L _NN ) +2 (m _NP  - m _NN ) + (u _NP  - u _NN )] D _C  (x, y)}/4.

The Bayesian decision procedure suggests the following minimum-cost decision rules

if $E\left(\tilde{R}\right(a_P\left|y\right))\le E(\tilde{R}\left(a_B\right|y\left)\right)$ and $E\left(\tilde{R}\right(a_P\left|y\right))\le E(\tilde{R}\left(a_N\right|y\left)\right)$ , then y ∈ AL^(α,β) (x);

if $E\left(\tilde{R}\right(a_B\left|y\right))\le E(\tilde{R}\left(a_P\right|y\left)\right)$ and $E\left(\tilde{R}\right(a_B\left|y\right))\le E(\tilde{R}\left(a_N\right|y\left)\right)$ , then y ∈ NE^(α,β) (x);

if $E\left(\tilde{R}\right(a_N\left|y\right))\le E(\tilde{R}\left(a_P\right|y\left)\right)$ and $E\left(\tilde{R}\right(a_N\left|y\right))\le E(\tilde{R}\left(a_B\right|y\left)\right)$ , then y ∈ CO^(α,β) (x).

Let triangular fuzzy loss functions ${\tilde{\lambda }}_{\mathit{PP}}$ , ${\tilde{\lambda }}_{\mathit{BP}}$ , ${\tilde{\lambda }}_{\mathit{NP}}$ , ${\tilde{\lambda }}_{\mathit{PN}}$ , ${\tilde{\lambda }}_{\mathit{BN}}$ and ${\tilde{\lambda }}_{\mathit{NN}}$ satisfy the following conditions

l _PP  ≤  l _BP  ≤  l _NP , m _PP  ≤  m _BP  ≤  m _NP , u _PP  ≤  u _BP  ≤  u _NP ;

l _NN  ≤  l _BN  ≤  l _PN , m _NN  ≤  m _BN  ≤  m _PN , u _NN  ≤  u _BN  ≤  u _PN .

We have

if D _c  (x, y) < α and D _c  (x, y) < γ, then y ∈ AL^(α,β) (x);

if D _c  (x, y) ≤ β and D _c  (x, y) ≥ α, then y ∈ NE^(α,β) (x);

if D _c  (x, y) > β and D _c  (x, y) > γ, then y ∈ CO^(α,β) (x),

where

$\alpha =\frac{(l_{\mathit{PN}}-l_{\mathit{NN}})+2(m_{\mathit{PN}}-m_{\mathit{NN}})+(u_{\mathit{PN}}-u_{\mathit{NN}})}{(l_{\mathit{PN}}-l_{\mathit{NN}})+(l_{\mathit{NP}}-l_{\mathit{PP}})+2(m_{\mathit{PN}}-m_{\mathit{NN}})+2(m_{\mathit{NP}}-m_{\mathit{PP}})+(u_{\mathit{NP}}-u_{\mathit{PP}})+(u_{\mathit{PN}}-u_{\mathit{NN}})}$ ;

$\gamma =\frac{(l_{\mathit{PN}}-l_{\mathit{BN}})+2(m_{\mathit{PN}}-m_{\mathit{BN}})+(u_{\mathit{PN}}-u_{\mathit{BN}})}{(l_{\mathit{BP}}-l_{\mathit{PP}})+(l_{\mathit{PN}}-l_{\mathit{BN}})+2(m_{\mathit{BP}}-m_{\mathit{PP}})+2(m_{\mathit{PN}}-m_{\mathit{BN}})+(u_{\mathit{BP}}-u_{\mathit{PP}})+(u_{\mathit{PN}}-u_{\mathit{BN}})}$ ;

$\beta =\frac{(l_{\mathit{BN}}-l_{\mathit{NN}})+2(m_{\mathit{BN}}-m_{\mathit{NN}})+(u_{\mathit{BN}}-u_{\mathit{NN}})}{(l_{\mathit{NP}}-l_{\mathit{BP}})+(l_{\mathit{BN}}-l_{\mathit{NN}})+2(m_{\mathit{NP}}-m_{\mathit{BP}})+2(m_{\mathit{BN}}-m_{\mathit{NN}})+(u_{\mathit{NP}}-u_{\mathit{BP}})+(u_{\mathit{BN}}-u_{\mathit{NN}})}$ .

If 0 ≤ α < γ < β ≤ 1, we can get

(1) if D _A  (x, y) < α, then y ∈ AL^(α,β) (x);

(2) if β ≥ D _A  (x, y) ≥ α, then y ∈ NE^(α,β) (x);

(3) if D _A  (x, y) > β, then y ∈ CO^(α,β) (x).

Example 6.1. (Continued from Example 5.1) The triangular fuzzy loss functions are shown in Table 10, the thresholds α, γ and β can be calculated from the Theorem 6.1.

Table 10

The triangular fuzzy loss functions

action	Ω	¬Ω
a P	${\tilde{\lambda }}_{\mathit{PP}}=(0.1,0.15,0.3)$	${\tilde{\lambda }}_{\mathit{PN}}=(0.4,0.5,0.7)$
a B	${\tilde{\lambda }}_{\mathit{BP}}=(0.2,0.2,0.5)$	${\tilde{\lambda }}_{\mathit{BN}}=(0.2,0.3,0.6)$
a N	${\tilde{\lambda }}_{\mathit{NP}}=(0.3,0.4,0.6)$	${\tilde{\lambda }}_{\mathit{NN}}=(0.1,0.3,0.5)$

$\alpha =\frac{(l_{\mathit{PN}}-l_{\mathit{NN}})+2(m_{\mathit{PN}}-m_{\mathit{NN}})+(u_{\mathit{PN}}-u_{\mathit{NN}})}{(l_{\mathit{PN}}-l_{\mathit{NN}})+(l_{\mathit{NP}}-l_{\mathit{PP}})+2(m_{\mathit{PN}}-m_{\mathit{NN}})+2(m_{\mathit{NP}}-m_{\mathit{PP}})+(u_{\mathit{NP}}-u_{\mathit{PP}})+(u_{\mathit{PN}}-u_{\mathit{NN}})}=0.39$ ;

$\gamma =\frac{(l_{\mathit{PN}}-l_{\mathit{BN}})+2(m_{\mathit{PN}}-m_{\mathit{BN}})+(u_{\mathit{PN}}-u_{\mathit{BN}})}{(l_{\mathit{BP}}-l_{\mathit{PP}})+(l_{\mathit{PN}}-l_{\mathit{BN}})+2(m_{\mathit{BP}}-m_{\mathit{PP}})+2(m_{\mathit{PN}}-m_{\mathit{BN}})+(u_{\mathit{BP}}-u_{\mathit{PP}})+(u_{\mathit{PN}}-u_{\mathit{BN}})}=0.54$ ; $\beta =\frac{(l_{\mathit{BN}}-l_{\mathit{NN}})+2(m_{\mathit{BN}}-m_{\mathit{NN}})+(u_{\mathit{BN}}-u_{\mathit{NN}})}{(l_{\mathit{NP}}-l_{\mathit{BP}})+(l_{\mathit{BN}}-l_{\mathit{NN}})+2(m_{\mathit{NP}}-m_{\mathit{BP}})+2(m_{\mathit{BN}}-m_{\mathit{NN}})+(u_{\mathit{NP}}-u_{\mathit{BP}})+(u_{\mathit{BN}}-u_{\mathit{NN}})}=0.88$ .

7 Conclusions and remarks

In this paper, firstly, we establish a three-way conflict analysis model on the triangular fuzzy information system of [-1, 1]. Furthermore, we use the expectation of the triangular fuzzy number to reasonably describe the agent’s specific attitudes towards the issues, and we use the two attributes of the triangular fuzzy number, expectation and variance, to characterize the degree of the agent’s attitudes towards the issues and discuss the three coalitional sets of the agent’s attitudes towards a single issue. Secondly, considering that the issues do not have the same importance, we attach weights to the corresponding issues with the help of the Sugeno measure, thus discussing the agent’s attitudes toward multiple issues and obtaining three coalition sets of the agent toward multiple issues. Thirdly, we define the weighted distance function between agents by revising the distance of triangular fuzzy numbers and investigated the relationship between agents. Finally, we illustrate how the thresholds α and β have been calculated using triangular fuzzy decision-making rough set theory.

Although this paper generalizes, improves and perfects the existing conflict analysis model on the triangular fuzzy conflict information system, it does not give the causes of conflict and conflict resolution strategies, which will be the limitation of this paper. Therefore, in the future, we will work on the \hfilneg following studies: (i) Finding the causes of conflict on the triangular fuzzy conflict information system; (ii) How to construct a conflict resolution model; (iii) How to model conflict analysis and resolution in an incomplete and dynamic triangular fuzzy conflict information system.