Distrust Behavior in Social Network Large-Scale Group Decision Making and Its Application in Water Pollution Management

Abstract

Distrust behavior is a human behavior that has a significant impact on water pollution management, but it is neglected in existing approaches. To solve this problem, we design a large-scale group decision making in social network (LSGDM-SN) approach based on distrust behavior and apply it to water pollution management. The purpose of this paper is to develop an LSGDM-SN method to assist managers choose the optimal water pollution management plan. In the presented method, fuzzy preference relations (FPRs) are used to express experts’ assessment of alternatives. To utilize the proposed LSGDM-SN approach to solve the water pollution problem, a novel agglomerative hierarchical clustering (AHC) method is proposed by combing preference similarity and social relationships. Afterward, consensus feedback based on distrust behavior and social network analysis (SNA) is developed to encourage the subset to modify its FPR. A mechanism for the identification and management of distrust behavior is introduced. Based on the situations of distrust behaviors, two pieces of feedback advice are provided to the subset to adjust its FPR. Subsequently, a score function of the FPR is proposed to obtain the best solution for water pollution management. Finally, some comparative analyses and discussions demonstrate the effectiveness and feasibility of the proposed method.

1. Introduction

With the rapid development of information technology, the number of people participating in group decision making gradually presents a large-scale trend. Recently, the promotion of social software, such as Facebook and WeChat, provides technical support for decision makers to communicate and exchange information. Hence, the large-scale group decision making in social network (LSGDM-SN) approach has attracted lots of attention and becomes a research hotspot [1,2,3,4]. The LSGDM-SN problem refers to the process in which more than 20 experts with social relationships express their opinion toward alternatives, and work together to reach a group consensus to select the best solution [5,6,7]. Generally, in the LSGDM-SN problem, fuzzy preference relation (FPR) is utilized to express the opinions of experts toward alternatives. Through FPR, experts can use several numbers to present the uncertainty of individual evaluation. To date, many discussions about the LSGDM-SN problem have been proposed. Additionally, the approach of LSGDM-SN is widely used in the field of emergency management and resource optimization [8,9,10].

Indeed, there are large-scale experts participating in the LSGDM problem. In order to improve the efficiency of information aggregation and consensus reaching, many scholars have proposed clustering methods to decrease the dimension of experts [11,12]. Based on this idea, many clustering methods have been successively applied to the LSGDM problem to divide experts with similar opinions into several small subsets [13,14]. For instance, Wu et al. [13] proposed a time series clustering method to classify alternatives into different clusters, thereby the information entropy regarding multiple alternatives is calculated. Liu et al. [15] proposed a clustering method considering both the opinion similarity and individual concern similarity of DMs. Tang et al. [16] considered the independence of communities and used the overlapping community detection method to reduce the dimension of heterogeneous experts. However, most of the above studies ignore the social relationships among experts. Because of the existence of social relationships in LSGDM-SN, it is unreasonable to ignore the trust and distrust relationships among experts.

In order to ensure the rationality of LSGDM-SN, the consensus-reaching process (CRP) is explored to eliminate opinion conflicts among large-scale experts [17,18]. Several CRP models in the LSGDM-SN problem have been discussed from multiple aspects [19,20,21], for example, the consensus of LSGDM-SN with evaluation information [14,22]. Moreover, Wang et al. [23] used the Louvain method to classify the large-scale experts into small subsets according to the trust relationships, so as to deal with the two-stage consensus problem with different power structures. Furthermore, experts come from different knowledge fields and represent different social interests, this situation leads to conflicts and non-cooperative behaviors existing extensively in LSGDM-SN. Dong et al. [24] proposed a trust consensus feedback mechanism based on leaders’ preference adjustment and trust relationships. Based on the optimal feedback model, Wu et al. [25] proposed a new framework to prevent manipulation behavior in the process of consensus in social networks. In the CRP, the moderator faces large-scale experts and complex decision-making tasks. It is relatively hard for the moderator to grasp the information of every expert. As a result, the suggestions provided by the moderator may not meet the expectations of some experts. In addition, experts are mostly experienced decision makers who may not change their opinions based on the suggestions provided by the moderator. This situation results in the expert generating distrust behavior toward the moderator. However, in the CRP of LSGDM-SN, the existing literature rarely takes the distrust behavior of the expert into account.

In China, because of the lack of attention to pollutant emissions, water pollution constantly occurs in many provinces. Water pollution not only makes factories stop production but also creates a bad social impact and great economic losses. Water pollution threatens the sustainable development of society. In recent years, a series of water pollution events have appeared in China, for instance, cancer caused by water pollution in Shandong province in 2007, water pollution in the Songhua River in 2010, industrial wastewater discharge in Jiangxi province in 2011, and so on. For improving the efficiency of water pollution management and reducing the occurrence of water pollution events, the Chinese government has issued relevant policies and laws. In addition, there are multiple studies about water pollution management [26,27]. For instance, Zhang et al. [28] discussed the potential pollution factors in the Licun River based on Spearman’s rank correlation coefficient and the comprehensive pollution index. According to the evidence from a quasi-natural experiment in China, Pan and Fan [29] explored the benefit of environmental information disclosure in managing water pollution. Han et al. [30] discussed the challenging issues of sustainable water management in coastal areas in China. Taking Tianjin in China as an example, Wang et al. [31] presented the regional water pollution management pathways and effects under strengthened policy constraints. By reviewing the above literature, it signifies that the water pollution issue is still a hotpot in the field of environmental protection. Simultaneously, all these existing studies have provided a significant reference in publishing water pollution management strategies.

As for water pollution management, the most crucial problem is to select an effective alternative to reduce the appearance of pollution. Hence, in some situations, water pollution management can be deemed as an LSGDM-SN problem. Nevertheless, according to the above studies regarding water pollution management, we discover that from the perspective of LSGDM-SN, considering the distrust behavior of experts to explore water pollution management is still a challenge. It is necessary to formulate a selection method for water pollution management according to the LSGDM-SN theory.

Therefore, the purpose of this paper is to develop an LSGDM-SN method to assist managers choose the optimal water pollution management plan. The contribution of this paper can be elaborated from several aspects. Firstly, considering the social relationships and FPRs of experts, an AHC method is designed to classify many experts into small subsets. Then, a consensus measure is presented to check whether a large group of experts have reached an acceptable degree of consensus. Furthermore, distrust behavior identification and management are shown in the consensus feedback process. Finally, a score function of the FPR is proposed to get the optimal solution for water pollution management.

The main innovations are as follows:

(1)

The AHC method is utilized to decrease the complexity of LSGDM-SN with the FPR, in which both the trust relationships and distrust relationships among experts are incorporated into the clustering method. Then, the concept of preference similarity, trust similarity, and distrust similarity are proposed to compute the degree of overall similarity among experts. Meanwhile, the algorithm for the AHC of LSGDM-SN is presented.

(2)

Consensus feedback based on distrust behavior and social network analysis (SNA) is presented to encourage the subset to modify its FPR based on different distrust types. In the identification process, both the distrust score and the degree of difference are incorporated to measure the distrust degree of the subset. Based on the cases of distrust behaviors, two pieces of feedback advice are provided to the subset to adjust its FPR.

(3)

A score function of FPR is designed to choose the best alternative for water pollution management. By calculating the final score of the collective FPR, we rank all alternatives. Then, the one with the highest score is selected as the best solution for water pollution management. Finally, a framework for the proposed LSGDM-SN considering distrust behavior is depicted to visualize the decision process.

The effectiveness of the proposed approach is demonstrated by a case study of water pollution management. Moreover, some comparative analysis and discussion are offered to present the validity of the proposed LSGDM-SN with distrust behavior. From the results, it illustrates that distrust behavior and weight determination of the expert have a significant influence on the alternative ranking of water pollution management.

The rest of this paper is organized as follows. In Section 2, some preliminaries regarding the FPR and SNA are reviewed. In Section 3, large-scale expert clustering based on the AHC approach is proposed. Section 4 presents the consensus of LSGDM-SN based on distrust behavior and SNA, then a selection method is also provided. Subsequently, in Section 5, a case study is shown to illustrate the application of the proposed LSGDM-SN in water pollution management. Section 6 conducts a comparative analysis and discussion. Finally, conclusions are given in Section 7.

2. Preliminaries

In this section, some related preliminaries about the FPR are introduced. Then, SNA and other concepts are illustrated.

2.1. Fuzzy Preference Relation

Let X = {x₁, x₂, …, x_n} be the set of alternatives, where N = {1, 2, …, n}. E = {e₁, e₂, …, e_m} denotes the set of experts in the LSGDM-SN problem, they utilize FPRs to express their evaluations toward alternatives.

Definition 1

([32]). The FPR R of all alternatives X = {x₁, x₂, …, x_n} is a fuzzy relation on the product set X × X with membership function μ_R: X × X → [0, 1], μ_R (x_i, x_j) = r_ij, satisfying

$$r_{ij}+r_{ji}=1,r_{ii}=0.5,i,j\in N$$

(1)

Usually, an n × n matrix R = (r_ij)_n×n is used to represent the FPR, where r_ij is the preference degree of x_i over x_j. Particularly, 0 ≤ r_ij < 0.5 signifies that the expert prefers x_j to x_i. If 0.5 < r_ij < 1, it means x_i is preferred to x_j by experts. In particular, r_ij = 0.5 means that the expert has no preference for x_i and x_j, and r_ij = 1 indicates that x_i is definitely preferred to x_j.

Definition 2

([33]). An FPR R = (r_ij)_n×n has additive consistency if r_ij + r_jl + r_li = 1.5, for ∀ i, l, j ∈ N.

Generally, in the discussion about FPRs, the additive consistency of the FPR is deemed an important research hotpot. Ma et al. [34] proposed the definition of additive consistency in FPRs.

Definition 3

([34]). Let H = (h_ij)_n×n be an FPR, it is an additive consistency FPR if it satisfies

$$h_{ij}=\frac{1}{n}{\displaystyle \sum _{l=1}^n(r_{il}+r_{lj})-0.5}$$

(2)

where all i, l, j = 1, 2, …, n with i < l < j and r_ij + r_ji = 1.

The additive consistency of an FPR can ensure the rationality of individual preferences. Therefore, the consistency index (CI) of the FPR based on additive consistency is proposed by [35]:

$$CI(R)=\sqrt{\frac{2}{n(n-1)}{\displaystyle \sum _{j=i+1}^n{\displaystyle \sum _{i=1}^{n-1}{(r_{ij}-h_{ij})}^2}}}$$

(3)

Obviously, 0 ≤ CI(R) ≤ 1. In practice, we usually set a consistency threshold ($\overline{\mathit{CI}}$) to weigh whether the consistency level of the FPR is acceptable. If CI(R) <$\overline{\mathit{CI}}$, we use the algorithm in [35] to repair the consistency of the FPR. Because the problem of consistency is not our research problem, this paper assumes that the FPRs expressed by experts have acceptable consistency.

2.2. SNA

As a useful social analysis tool, SNA generally is used to investigate the relationships among social members. In the field of GDM, SNA is frequently used in the opinion dynamic and consensus process.

There are three main parts in SNA, including experts, the relationships among experts, and the socio-matrix, which are presented in

Table 1. Three main parts in SNA.

Socio-Matrix	Graph	Algebraic
G = $\left(\begin{array}{ccccc}0& 0& 1& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 1& 0\end{array}\right)$		e1Be3 e2Be1 e3Be2, e3Be4 e1Be5 e5Be4, e5Be1

.

(1)

Socio-matrix: The matrix G = (g_kq)_m×m is utilized to present the relationships data among experts. If expert e_k has no relationship with e_q, the value of g_kq is equal to 0. Otherwise, the value of g_kq is 1 if there is an existing relationship between e_k and e_q.

(2)

Graph: Social networks can also be represented by a graph, where the points are connected by straight lines. In the network diagram, e_k→e_q indicates that there is a direct relationship from e_k to e_q.

(3)

Algebraic: The algebraic can denote the social relationships among experts and the relationship combinations.

In the socio-matrix of Table 1, the relationship information among experts is not quantified. To solve this problem, Wu et al. [36] defined the concept of trust function, it is shown as follows.

Definition 4

([36]). Let t_kq and d_kq be the trust degree and distrust degree from e_k to e_q, respectively. The set of trust functions is denoted by tf_kq = (t_kq, d_kq), where t_kq, d_kq ∈ [0, 1].

Example 1.

According to Definition 4, we assume that the social relationships among five experts are shown in Table 1, then the social assessment matrix TF = (tf_kq)_m×m between experts can be expressed by:

$$TF=\left(\begin{array}{ccccc}0& 0& (0.2,0.6)& 0& 0\\ (0.7,0.5)& 0& 0& 0& 0\\ 0& (0.9,0.8)& 0& (0.3,0.1)& 0\\ 0& 0& 0& 0& (0.4,0.2)\\ (0.8,0.6)& 0& 0& (0.2,0.5)& 0\end{array}\right)$$

In this social assessment matrix, tf₁₃ = (0.2, 0.6) signifies that the trust value from e_k to e_q is 0.2, while the distrust value from e_k to e_q is 0.6.

Usually, the relationship information in the social network can be propagated. For example, if expert e_k trusts e_q, and e_q trusts e_y, then e_k may trust e_y to some extent. That is, the expert e_q acts as an intermediary to achieve trust propagation from e_k to e_y. In a social network, trust can be propagated by one or more intermediaries. However, if there is more than one middleman to propagate information, the information may be distorted. Therefore, in order to ensure the accuracy of information propagation, this paper assumes that information can only be transmitted by one intermediary. Taking Figure 1 as an example, there is no direct trust and distrust from e_k to e_y, but trust and distrust can be propagated through e_k→e_q→e_y.

Wu et al. [36] proposed the uninorm trust propagation operator based on the t-norm to quantify the process of information propagation in social networks. As shown in Figure 1, e_k→e_q→e_y is the information propagation path from e_k to e_y, and the indirect trust function value from e_k to e_y can be obtained using the uninorm trust propagation operator

$$(t_{ky},d_{ky})=\left\{\begin{array}{ll}(t_{qy},d_{qy}),& \mathrm{if}(t_{kq},d_{kq})=(1,0)\\ (t_{kq},d_{kq}),& \mathrm{if}(t_{qy},d_{qy})=(1,0)\\ (0,1),& \mathrm{if}(t_{kq},d_{kq})\vee (t_{qy},d_{qy})=(0,1)\\ \left(\frac{t_{kq}{t}_{qy}}{t_{kq}{t}_{qy}+(1-t_{kq})(1-t_{qy})},\frac{t_{kq}{d}_{qy}}{t_{kq}{d}_{qy}+(1-t_{kq})(1-d_{qy})}\right),& \mathrm{otherwise}\end{array}\right.$$

(4)

where (t_kq, d_kq) is the trust function from e_k to e_q, and (t_qy, d_qy) is the trust function from e_q to e_y.

In many cases, there may be multiple trust function propagation paths from e_k to e_y, as shown in Figure 2. There are two trust function propagation paths from e_k to e_y: e_k→e_q1→e_y and e_k→e_q2→e_y. We propose the trust aggregation operator in the process of trust propagation according to the weighted average operator.

Definition 5.

Let ${\mathit{tf}}_{\mathit{ky}}^{\mathit{\gamma }}{=(t}_{\mathit{ky}}^{\mathit{\gamma }}{,d}_{\mathit{ky}}^{\mathit{\gamma }})$ (γ = 1,…,$\overline{\gamma }$) be the trust function from e_k to e_y in path γ, the indirect trust function value from e_k to e_y is defined as

$$t{f}_{ky}=\frac{1}{\overline{\gamma }}{\displaystyle \sum _{\gamma =1}^{\overline{\gamma }}t{f}_{ky}^{\gamma }}$$

(5)

Definition 6.

Let tf_kq = (t_kq, d_kq) (k, q ∊ E) be the trust function from e_k to e_q, and tso_q and dso_q are the trust score and distrust score from e_k to e_q, respectively. The values of tso_q and dso_q are calculated by

$$ts{o}_q=\frac{1}{m-1}{\displaystyle \sum _{k=1,k\ne q}^m{t}_{kq}}$$

(6)

$$ds{o}_q=\frac{1}{m-1}{\displaystyle \sum _{k=1,k\ne q}^m{d}_{kq}}$$

(7)

Clearly, the values of tso_q and dso_q are within the interval [0, 1].

3. Large-Scale Expert Clustering Based on AHC Approach

In the LSGDM-SN problem, clustering analysis aims to divide experts into several small subsets and improves the efficiency of decision making. The traditional clustering methods include fuzzy clustering [37], the Louvain method [23], and the AHC method [38]. If fuzzy clustering is utilized in LSGDM-SN, an expert may belong to multiple subsets at the same time, which makes the clustering results meaningless. Regarding the Louvain method, it conducts expert clustering only based on the social network relations among experts, and it ignores the subjective preferences of experts. However, the AHC method can overcome the above two defects and makes the clustering result more reasonable. Specifically, the AHC method classifies large-scale experts based on preference similarity and social similarity among experts, and each expert only belongs to one subset. In addition, AHC method has higher robustness and efficiency in dealing with LSGDM-SN problems. Based on these advantages, this paper adopts AHC method to carry out expert clustering.

AHC is carried out according to the similarity of two experts. Generally, the similarity between two experts is determined by aggregating the preference similarity and the social similarity. The higher the similarity between the two experts, the more likely they are to be clustered into the same subset.

Definition 7.

Let $R^k{=(r}_{\mathit{ij}}^k{)}_{n\times n}$ and $R^q{=(r}_{\mathit{ij}}^q{)}_{n\times n}$ be two FPRs provided by experts e_k and e_q, respectively. Let PS_kq = (ps_kq)_n×n be the preference similarity matrix among experts e_k and e_q. Based on the Manhattan distance, the value of ps_kq is calculated by

$$p{s}_{kq}=1-\frac{2}{n(n-1)}{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^n|r_{ij}^k-r_{ij}^q|}}$$

(8)

Obviously, the value of ps_kq is within the interval [0, 1], and ps_kq = ps_qk.

Definition 8.

Let TF = (tf_kq)_m×m be the complete matrix of the trust function, where tf_kq = (t_kq, d_kq) is the trust function value given by e_k to e_q. The trust similarity ts_kq and distrust similarity ds_kq between experts e_k and e_q are defined as follows

$$t{s}_{kq}=1-\left|ts{o}_k-ts{o}_q\right|$$

(9)

$$d{s}_{kq}=1-\left|ds{o}_k-ds{o}_q\right|$$

(10)

where tso_k is the trust score of expert e_k, and dso_k denotes the distrust score of expert e_k.

Definition 9.

Let ps_kq, ts_kq, and ds_kq be the preference similarity, trust similarity, and distrust similarity between experts e_k and e_q, respectively. The overall similarity os_kq between experts e_k and e_q is defined as follows

$$o{s}_{kq}=\lambda p{s}_{kq}+(1-\lambda )(t{s}_{kq}+d{s}_{kq})$$

(11)

Clearly, the greater the value of os_kq, the higher the similarity between experts e_k and e_q.

After calculating the overall similarity, the AHC method is initiated to divide the large-scale experts into several subsets. For similarity, the subset is denoted as su^p (p = 1,…, υ), and experts in subset su^p are denoted as $e_k^p$ (k = 1,…,|su^p|).

The detailed AHC method for LSGDM-SN is presented in Algorithm 1.

Algorithm 1. The detailed AHC method.

Input$:\mathrm{the}\mathrm{FPRs}\mathrm{of}m\mathrm{experts},\mathrm{complete}\mathrm{matrix}\mathrm{of}\mathrm{trust}\mathrm{function}\mathit{TF}={\left({\mathit{tf}}_{\mathit{kq}}\right)}_{m\times m},\mathrm{the}\mathrm{parameter}\mathit{\lambda },\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{subsets} \mathit{\upsilon }$. Output: subsets su1, …, sup, …, suυ. Step 1: Regard each expert ek (k =1, …, m) as one initial subset. Step 2: Calculating the values of pskq, tskq, dskq and oskq for each initial subset based on Equations (8)–(11). Step 3: Selecting the maximum oskq, then classify initial subsets eq and ek into one new subset. Step 4: Deleting subsets eq and ek in Step 3, the overall similarity is recalculated based on Definition 9. Step 5: Steps 3 and 4 are repeated constantly. If all subsets are merged into one subset, the process is ended. Step 6: Setting the number of subsets$\upsilon$, output the clustering result su1, …, sup, …, suυ.

Remark 1.

Initially, the data of opinion similarity and trust relationship are inputted into the AHC method. Through the AHC method, large-scale experts can be classified into several small subsets, in which the dimension of LSGDM-SN is reduced. Next, both the consensus with distrust behavior and the selection process are conducted on the basis of the subset. It can minimize the computational complexity of the LSGDM-SN problem. Therefore, expert clustering based on the AHC method largely improves the efficiency of the LSGDM-SN problem.

4. Consensus of LSGDM-SN Based on Distrust Behavior and SNA

In this section, a consensus measure method is proposed first. Then, the consensus-reaching process based on distrust behavior and SNA is presented. Furthermore, a selection process is given. Finally, a framework of the detailed LSGDM-SN process is depicted.

4.1. Consensus Measure

After the AHC process ends, the consensus measure for each subset is proceeded. Thus, we can further determine whether consensus feedback is necessary according to the results of the consensus measure.

Definition 10.

Let $R^{p,1}{=(r}_{\mathit{ij}}^{p,1}{)}_{n\times n}$, $R^{p,2}{=(r}_{\mathit{ij}}^{p,2}{)}_{n\times n}$,…, $R^{p,\left|{\mathit{su}}^p\right|}{=(r}_{\mathit{ij}}^{p,\left|{\mathit{su}}^p\right|}{)}_{n\times n}$ be the FPRs of experts in subset su^p, ${\omega }^{p,1}$, ${\omega }^{p,2}$,…, ${\omega }^{p,\left|{\mathit{su}}^p\right|}$ be the set of weights of experts in subset su^p. The FPR $R^p{=(r}_{\mathit{ij}}^p{)}_{n\times n}$ of subset su^p is computed by

$$r_{ij}^p={\displaystyle \sum _{k=1}^{\left|s{u}^p\right|}{\omega }^{p,k}{r}_{ij}^{p,k}}$$

(12)

where |su^p| is the number of experts in subset su^p. The weight of expert ω^p,k can be obtained by ω^p,k = 1/|su^p|.

Next, the collective FPR $\overline{R}{=\left({\overline{r}}_{\mathit{ij}}\right)}_{n\times n}$ can be derived through

$${\overline{r}}_{ij}={\displaystyle \sum _{p=1}^{\upsilon }{\phi }^p{r}_{ij}^p}$$

(13)

where φ^p represents the weight of subset su^p and it is obtained by φ^p = 1/υ. Clearly, the value of φ^p is within [0, 1] and ${{\displaystyle \sum }}_{p=1}^{\upsilon }{\phi }^p$= 1.

Definition 11.

The consensus level of subset CLS^p and collective CLC can be determined by

$$CL{S}^p=1-\frac{2}{n(n-1)}{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^n|r_{ij}^p-{\overline{r}}_{ij}|}}$$

(14)

$$CLC={\displaystyle \sum _{p=1}^{\upsilon }{\phi }^{p}CL{S}^p}$$

(15)

It can be observed that CLC ∊ [0, 1]. Usually, a consensus threshold θ is proposed to judge whether consensus feedback is required. The selection process is activated if CLC ≥ θ. Otherwise, the feedback adjustment is entered until CLC exceeds the predefined θ.

4.2. Consensus Feedback Based on Distrust Behavior and SNA

The LSGDM-SN problem aims to derive the most optimal alternative efficiently and quickly. If CLC < θ, the feedback adjustment is continued and a moderator provides modification advice to the subset with the minimum consensus level. However, in the LSGDM-SN problem, the experts in the selected subset have rich educational and social experiences, and they are usually professional decision makers. As a result, selected subsets may exhibit distrust behavior toward the suggestion provided by the moderator. Generally, the distrust behavior means that the selected subset does not trust the moderator if the distrust degree of the selected subset is more than the given threshold. The rationality and validity of LSGDM-SN will hardly be guaranteed if the distrust behavior of the selected subset is ignored. Hence, the identification and management of distrust behavior should be considered in the consensus feedback process.

Based on the SNA and preference deviation, the distrust degree of the selected subset is defined as follows.

Definition 12.

Let dso^p,1, …, dso^p,k,…, ${\mathit{dso}}^{{p,|\mathit{su}}^p|}$ be the distrust scores of experts in subset su^p, and $R^p{=(r}_{\mathit{ij}}^p{)}_{n\times n}$ be the FPR of subset su^p, $\overline{R}{=\left({\overline{r}}_{\mathit{ij}}\right)}_{n\times n}$ be the collective FPR, the distrust degree dd^p of subset su^p toward moderator is defined by

$$d{d}^p=\alpha \times {\displaystyle \sum _{k=1}^{\left|s{u}^p\right|}{\omega }^{p,k}ds{o}^{p,k}}+(1-\alpha )\times \frac{2}{n(n-1)}{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^n|r_{ij}^p-{\overline{r}}_{ij}|}}$$

(16)

The first part ${\displaystyle {\displaystyle \sum }_{k=1}^{{|\mathit{su}}^p|}}{\omega }^{p,k}{\mathit{dso}}^{p,k}$ denotes the distrust score of the subset, and the second part $\frac{2}{n(n-1)}{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{\displaystyle {\displaystyle \sum }_{j=i+1}^n}{|r}_{\mathit{ij}}^p-{\overline{r}}_{\mathit{ij}}|$ signifies the degree of difference between the subset’s FPR and the overall FPR. Where α is a parameter that weighs the distrust score and the degree of difference, its value is within [0, 1]. In this paper, we consider these two parts equally important, so α = 0.5.

When the subset su^p with the minimum consensus level is found, the moderator works to provide modification advice for the subset. Given the threshold of distrust degree β, the corresponding modification strategy for the subset su^p with minimum consensus level is summarized in the following two cases.

Case 1.

dd^b ≤ β, the subset trusts in the moderator to some extent. That is, the subset is willing to adopt the suggestions from the moderator. Therefore, the FPR of the subset is changed as follows.

$$r_{ij}^{p,(\tau +1)}=\sigma \times r_{ij}^{p,(\tau )}+(1-\sigma )\times {\overline{r}}_{ij}$$

(17)

Case 2.

dd^b > β, the subset absolutely distrusts the suggestion of the moderator in the feedback process. In this case, we assume that the subset can be influenced by the subset with the maximum consensus level. Hence, the FPR of the subset is modified by

$$r_{ij}^{p,(\tau +1)}=\sigma \times r_{ij}^{p,(\tau )}+(1-\sigma )\times \max \left\{CL{S}^{p,(\tau )}\right\}$$

(18)

where the value of σ is within [0, 1]. max {CLS^p,(τ)} means the subset su^p with maximum consensus level in iteration τ.

Based on the two cases, the subset with distrust behavior is identified and managed until the CLC reaches the predefined threshold.

Remark 2.

In distrust behavior management, to motivate the subset to participate in the consensus, two different cases are designed to modify the FPR. Indeed, this gives the subset more freedom to modify its preference. Traditional consensus feedback neglects the distrust behavior of the subset, their corresponding suggestions are conducted without requiring the participation of subsets. Nevertheless, the proposed CRP fully considers the distrust behavior of the subset, which is obviously different from the automatic feedback mechanisms. The proposed method can solve the situation that subset distrusts the advice of the moderator. Simultaneously, it effectively addresses the distrust behavior of the subset in the CRP of the LSGDM-SN problem.

4.3. Selection Process

The selection process aims to calculate the final score of each alternative and select the most optimal alternative for LSGDM-SN. Usually, based on the weighted average operator, the final score of the alternative x_i is

$$fs(x_i)=\frac{1}{n}{\displaystyle \sum _{j=1}^n{\overline{r}}_{ij}}$$

(19)

Finally, the one with the maximum score is selected as the best alternative.

4.4. The Proposed Framework of LSGDM-SN

This section exhibits a framework for the proposed LSGDM-SN to make the proposed method more clearly. The detailed LSGDM-SN process is depicted in Figure 3.

5. Case Study

In this section, a practical application of the proposed LSGDM-SN approach considering distrust behavior in water pollution management is presented.

With the progress of industry and social development, water pollution is becoming increasingly serious and has become a worldwide environmental problem. Water pollution affects industrial production and product quality, and even makes production impossible. In addition, water pollution affects people’s lives and directly endangers people’s health. In order to reduce the probability of water pollution, and improve the efficiency of water pollution management, City A decides to beforehand develop a water pollution control plan.

Combined with the objectives of the local environmental protection department, the government manager provides four emergency plans for water pollution: x₁, x₂, x₃, and x₄. Subsequently, 20 experts with social relationships from different sectors of society are invited to participate in the LSGDM-SN [2,4,5,6]. The detailed evaluation of 20 experts on the four alternatives is shown in

Table 2. The detailed evaluation of 20 experts on the four alternatives.

Experts	FPR	Experts	FPR
e1	$\left(\begin{array}{cccc}0.5& 0.4& 0.5& 0.9\\ 0.6& 0.5& 0.4& 0.2\\ 0.5& 0.6& 0.5& 0.2\\ 0.1& 0.8& 0.8& 0.5\end{array}\right)$	e2	$\left(\begin{array}{cccc}0.5& 0.7& 0.1& 0.1\\ 0.3& 0.5& 0.4& 0.2\\ 0.9& 0.6& 0.5& 0.7\\ 0.9& 0.8& 0.3& 0.5\end{array}\right)$
e3	$\left(\begin{array}{cccc}0.5& 0.7& 0.8& 0.9\\ 0.3& 0.5& 0.8& 1.0\\ 0.2& 0.2& 0.5& 0.2\\ 0.1& 0& 0.8& 0.5\end{array}\right)$	e4	$\left(\begin{array}{cccc}0.5& 0& 0.1& 0.6\\ 1.0& 0.5& 0.2& 0.2\\ 0.9& 0.8& 0.5& 0.2\\ 0.4& 0.8& 0.8& 0.5\end{array}\right)$
e5	$\left(\begin{array}{cccc}0.5& 0.1& 0.9& 0.3\\ 0.9& 0.5& 0.5& 0.6\\ 0.1& 0.5& 0.5& 0.1\\ 0.7& 0.4& 0.9& 0.5\end{array}\right)$	e6	$\left(\begin{array}{cccc}0.5& 0.3& 0.5& 0.1\\ 0.7& 0.5& 0.3& 0.5\\ 0.5& 0.7& 0.5& 0.3\\ 0.9& 0.5& 0.7& 0.5\end{array}\right)$
e7	$\left(\begin{array}{cccc}0.5& 0.9& 0.7& 1.0\\ 0.1& 0.5& 0.4& 0.1\\ 0.3& 0.6& 0.5& 0.3\\ 0& 0.9& 0.7& 0.5\end{array}\right)$	e8	$\left(\begin{array}{cccc}0.5& 0.7& 0.4& 0\\ 0.3& 0.5& 0.7& 0.5\\ 0.6& 0.3& 0.5& 0.9\\ 1.0& 0.5& 0.1& 0.5\end{array}\right)$
e9	$\left(\begin{array}{cccc}0.5& 0.9& 0.8& 0.2\\ 0.1& 0.5& 0.7& 0.7\\ 0.2& 0.3& 0.5& 0.8\\ 0.8& 0.3& 0.2& 0.5\end{array}\right)$	e10	$\left(\begin{array}{cccc}0.5& 0.7& 0.9& 0.5\\ 0.3& 0.5& 0.9& 0.6\\ 0.1& 0.1& 0.5& 0.5\\ 0.5& 0.4& 0.5& 0.5\end{array}\right)$
e11	$\left(\begin{array}{cccc}0.5& 0.5& 1.0& 0.2\\ 0.5& 0.5& 0.2& 0.9\\ 0& 0.8& 0.5& 0.7\\ 0.8& 0.1& 0.3& 0.5\end{array}\right)$	e12	$\left(\begin{array}{cccc}0.5& 0.6& 0.5& 0.6\\ 0.4& 0.5& 0.5& 0.8\\ 0.5& 0.5& 0.5& 0.5\\ 0.4& 0.2& 0.5& 0.5\end{array}\right)$
e13	$\left(\begin{array}{cccc}0.5& 0.2& 1.0& 0\\ 0.8& 0.5& 0.2& 0.7\\ 0& 0.8& 0.5& 0.4\\ 1.0& 0.3& 0.6& 0.5\end{array}\right)$	e14	$\left(\begin{array}{cccc}0.5& 0.2& 0.3& 0.9\\ 0.8& 0.5& 0.4& 0.4\\ 0.7& 0.6& 0.5& 0.3\\ 0.1& 0.6& 0.7& 0.5\end{array}\right)$
e15	$\left(\begin{array}{cccc}0.5& 0.6& 0.8& 0.2\\ 0.4& 0.5& 0.7& 0.3\\ 0.2& 0.3& 0.5& 0.9\\ 0.8& 0.7& 0.1& 0.5\end{array}\right)$	e16	$\left(\begin{array}{cccc}0.5& 0.4& 0.4& 0.8\\ 0.6& 0.5& 0.7& 0.7\\ 0.6& 0.3& 0.5& 0.5\\ 0.2& 0.3& 0.5& 0.5\end{array}\right)$
e17	$\left(\begin{array}{cccc}0.5& 0.8& 0.4& 0.8\\ 0.2& 0.5& 0.7& 0.1\\ 0.6& 0.3& 0.5& 0.1\\ 0.2& 0.9& 0.9& 0.5\end{array}\right)$	e18	$\left(\begin{array}{cccc}0.5& 0.6& 0.7& 0.1\\ 0.4& 0.5& 0.5& 0.8\\ 0.3& 0.5& 0.5& 0.1\\ 0.9& 0.2& 0.9& 0.5\end{array}\right)$
e19	$\left(\begin{array}{cccc}0.5& 0.2& 0.3& 0.2\\ 0.8& 0.5& 0.7& 0.9\\ 0.7& 0.3& 0.5& 0.3\\ 0.8& 0.1& 0.7& 0.5\end{array}\right)$	e20	$\left(\begin{array}{cccc}0.5& 0.4& 0.8& 0.2\\ 0.6& 0.5& 0.9& 0.2\\ 0.2& 0.1& 0.5& 0.2\\ 0.8& 0.8& 0.8& 0.5\end{array}\right)$

. Moreover, the social assessments between the 20 experts are depicted in

Table 3. The social assessments between 20 experts.

	e1	e2	e3	…	e18	e19	e20
e1	(1, 0)	(0.9, 0.3)	(0.8, 0.7)		(0.6, 1.0)	(-, -)	(-, -)
e2	(-, -)	(1, 0)	(0.8, 0.3)		(-, -)	(0.7, 0.9)	(-, -)
e3	(0.3, 0.3)	(0.1, 0.6)	(1, 0)		(0.8, 0.4)	(-, -)	(-, -)
…
e18	(-, -)	(-, -)	(-, -)		(1, 0)	(0.1, 0.1)	(0.8, 0.5)
e19	(-, -)	(-, -)	(0.7, 1.0)		(0.4, 0.9)	(1, 0)	(-, -)
e20	(-, -)	(0.7, 0.2)	(0.1, 0.7)		(0.3, 0.4)	(-, -)	(1, 0)

. Additionally, some related parameters are predefined as follows.

In the expert clustering stage, the parameter λ equals 1/3, and the number of subsets$\upsilon$ is set to 7.

In the consensus measure process, the consensus threshold θ is 0.95.

In the consensus feedback process, the parameter σ is 0.5, and the threshold of distrust degree β is 0.3.

Based on the FPRs of 20 experts and social assessments, using Equations (8)–(11), the overall similarity between any two experts can be obtained. Then, applying the AHC algorithm, 20 experts can be classified into seven subsets. The clustering result is su¹ = {e₇, e₁₀, e₁₃}, su² = {e₂, e₃, e₄, e₉, e₁₁, e₁₇, e₁₉}, su³ = {e₆, e₁₅}, su⁴ = {e₁, e₅, e₈}, su⁵ = {e₁₄, e₁₈}, su⁶ = {e₁₆, e₂₀}, su⁷ = {e₁₂}. This is depicted in Figure 4.

Using Equations (12)–(14), the consensus level of subset is computed as CLS^1,(0) = 0.94, CLS^2,(0) = 0.95, CLS^3,(0) = 0.92, CLS^4,(0) = 0.94, CLS^5,(0) = 0.91, CLS^6,(0) = 0.92, and CLS^7,(0) = 0.88. Through Equation (15), we have CLC⁽⁰⁾ = 0.92. Because CLC⁽⁰⁾ < $\overline{\mathit{CLC}}$, we continue the feedback process.

Due to CLS^7,(0) = min{CLS^p,(τ)|p = 1,2,…,7} = 0.88, the consensus based on distrust behavior and SNA is conducted on subset su^7,(0). By Equation (16), we have dd^7,(0) = 0.33. Since dd^7,(0) > β = 0.3, the subset su^7,(0) absolutely distrusts the suggestion of the moderator. Then, the FPR of subset su^7,(0) is modified according to Equation (18). The new FPR R^7,(0) can be found in

Table 4. The detailed consensus based on distrust behavior and SNA.

τ	CLC (τ)	sup,(τ)	sup,(τ) < β	FPR
0	0.92	su7,(0)	No	$\left(\begin{array}{cccc}0.50& 0.58& 0.48& 0.52\\ 0.42& 0.50& 0.50& 0.67\\ 0.52& 0.50& 0.50& 0.46\\ 0.48& 0.33& 0.54& 0.50\end{array}\right)$
1	0.93	su5,(0)	Yes	$\left(\begin{array}{cccc}0.50& 0.42& 0.53& 0.50\\ 0.58& 0.50& 0.51& 0.57\\ 0.47& 0.49& 0.50& 0.29\\ 0.50& 0.43& 0.71& 0.50\end{array}\right)$
2	0.94	su6,(0)	Yes	$\left(\begin{array}{cccc}0.50& 0.44& 0.56& 0.52\\ 0.56& 0.50& 0.68& 0.50\\ 0.44& 0.32& 0.50& 0.38\\ 0.48& 0.50& 0.62& 0.50\end{array}\right)$
3	0.94	su3,(0)	No	$\left(\begin{array}{cccc}0.50& 0.50& 0.57& 0.50\\ 0.50& 0.50& 0.53& 0.50\\ 0.43& 0.47& 0.50& 0.49\\ 0.50& 0.50& 0.51& 0.50\end{array}\right)$
4	0.95

. After four iterations, the consensus feedback is ended. The detailed consensus based on distrust behavior and SNA is expressed in Table 4.

By Equation (19), the final score of alternatives are fs(x₁) = 0.50, fs(x₂) = 0.51, fs(x₃) = 0.46, and fs(x₄) = 0.52. Therefore, the ranking of the four alternatives is x₄ > ≻x₂ > ≻x₁ > ≻x₃. Consequently, the best solution is x₄.

6. Comparative Analysis and Discussion

To further demonstrate the validity of the proposed LSGDM-SN with distrust behavior, this section presents some comparisons and a discussion. In Section 6.1, the analysis of the influence of distrust behavior on alternative ranking is proposed. Furthermore, the impact of weight determination on decision results is provided in Section 6.2. In addition, a sensitivity analysis of the consensus threshold is developed in Section 6.3. Section 6.4 exhibits the comparison with other related studies.

6.1. The Impact of Distrust Behavior on Alternative Ranking

In the proposed approach, the identification and management of distrust behavior are considered in the consensus feedback process. Two cases are effectively proposed to provide modification advice for subsets. However, in [39], the linear combination of the subset’s FPR and collective FPR was considered as a modified FPR of the subset. This consensus feedback in [39] neglects the check process of the subset’s distrust behavior, which defaults the subset to fully trust the moderator’s advice.

In the following, the identification of distrust behavior is omitted and the modification strategy for the subset su^b is only based on Equation (17). Then, by Equation (19), the final score of each alternative can be derived. The detailed results without the identification and management of distrust behavior are given in

Table 5. The detailed results without the identification and management of distrust behavior.

Collective FPR	fs(xi)	Ranking of Alternatives
$\left(\begin{array}{cccc}0.50& 0.48& 0.55& 0.49\\ 0.52& 0.50& 0.54& 0.53\\ 0.45& 0.46& 0.50& 0.40\\ 0.51& 0.47& 0.60& 0.50\end{array}\right)$	fs(x1) = 0.505 fs(x2) = 0.523 fs(x3) = 0.453 fs(x4) = 0.520	x2 > ≻x4 > ≻x1 > ≻x3

.

It can be observed that the result of ranking in Table 5 is different from the result shown in Section 5. Hence, it verifies that the identification and management of distrust behavior have a significant impact on the final result of the LSGDM-SN problem.

6.2. The Impact of Weight Determination on Decision Result

In our paper, the weight of the expert is determined based on the number of experts in the subset. If the number of experts in the subset is large, then the weight of each expert is relatively small. In another study [3], the weight of the expert is calculated by the SNA method. If the SNA method is used in this paper, the weight of the expert is computed by

$${\omega }^q=\frac{ts{o}_q+ds{o}_q}{{\displaystyle \sum _{q=1}^{m}ts{o}_q+ds{o}_q}}$$

In each subset, the weight of the expert is normalized and obtained by

$${\omega }^q=\frac{{\omega }^q}{{\displaystyle \sum _{q=1}^{\left|s{u}^p\right|}{\omega }^q}}$$

In the following, the weight determined by the SNA method is utilized in this paper. Then, the consensus iteration and alternative ranking can be further derived. Its detailed results are provided in

Table 6. The detailed results through considering the SNA method determine expert weight.

Consensus Iteration	fs(xi)	Ranking of Alternatives
5	fs(x1) = 0.511 fs(x2) = 0.519 fs(x3) = 0.451 fs(x4) = 0.518	x2 > ≻x4 > ≻x1 > ≻x3

.

Through comparison, it can be found that the number of iterations is more than the result obtained in Section 5. In addition, the alternative ranking is also different from the ranking result in the case study. Therefore, it can be summarized that using different methods to determine experts’ weights has some effect on consensus results and alternative rankings.

6.3. Sensitivity Analysis of Consensus Threshold

In this section, we develop the sensitivity analysis to investigate the influence of consensus threshold θ on the consensus iteration and ranking of alternatives. The consensus threshold θ is set in the range 0.92 to 1, the detailed results are given in

Table 7. The consensus iteration and ranking of alternatives with different $\overline{\mathit{CLC}}$.

θ	τ	fs(x1)	fs(x2)	fs(x3)	fs(x4)	Ranking of Alternatives
0.92	0	0.509	0.527	0.453	0.512	x2 > ≻x4 > ≻x1 > ≻x3
0.93	1	0.506	0.526	0.448	0.520	x2 > ≻x4 > ≻x1 > ≻x3
0.94	3	0.509	0.519	0.455	0.517	x2 > ≻x4 > ≻x1 > ≻x3
0.95	4	0.504	0.518	0.457	0.521	x4 > ≻x2 >≻x1 > ≻x3
0.96	7	0.509	0.518	0.453	0.520	x4 > ≻x2 > ≻x1 > ≻x3
0.97	10	0.508	0.520	0.454	0.518	x2 > ≻x4 > ≻x1 > ≻x3
0.98	14	0.508	0.519	0.453	0.520	x4 > ≻x2 > ≻x1 > ≻x3
0.99	22	0.507	0.520	0.453	0.519	x2 > ≻x4 > ≻x1 > ≻x3
1	344	0.508	0.519	0.453	0.520	x4 > x2 > x1 > x3

.

As can be seen from Table 7, different values θ have an impact on the ranking of alternatives and the number of consensus iterations. In addition, Figure 5 shows how the ranking of alternatives changes with different values of θ. Obviously, similar conclusions can be obtained.

6.4. The Comparison with Other Related Studies

In this subsection, the proposed LSGDM-SN approach considering social relationships and distrust behavior is compared with other existing methods. This is presented in

Table 8. Comparison with other existing methods.

Method	Clustering Method	Social Relationship	FPR	Distrust Behavior	Water Pollution Problem
Lu et al. [40]	K-means clustering	√	×	×	×
Liu et al. [15]	DM clustering	×	×	×	×
Wang et al. [23]	Louvain algorithm	√	×	×	×
Meng et al. [4]	Trust-based density peaks clustering	√	×	×	×
Our method	AHC method	√	√	√	√

Note: The symbol “√” represents that the corresponding item is considered. However, the symbol “×” indicates that the corresponding item is neglected.

. The detailed comparative analysis and discussion are presented as follows.

Lu et al. [40] illustrated the LSGDM-SN problem by the K-means clustering method and solved the uncertainty cost based on robust optimization. The K-means clustering method in [40] neglected the situation where experts provided fuzzy assessments of alternatives. Moreover, only the uncertain unit adjustment cost was considered in [40]. The distrust behavior of the expert toward the moderator was not explored. In this paper, these two problems have been well resolved.

Liu et al. [15] designed a democratic CRP for multi-criteria LSGDM based on the DM clustering method. The clustering method only considered the opinion similarity and individual concern similarity of DM; the FPR and the social relationships among experts were ignored. In this paper, both the FPR and the social relationships are considered.

Wang et al. [23] developed a two-stage feedback mechanism with different power structures for the consensus of LSGDM. Only considering trust relationships, the Louvain algorithm was applied to achieve community clustering. They did not incorporate the preference similarity among experts, which made clustering results more objective. However, the proposed method not only considers the social relationships among experts, but the preference similarity is also discussed in this paper.

Meng et al. [4] constructed a trust-based LSGDM consensus framework and utilized it for intercity railway public–private partnership model selection. Nevertheless, they neglected the case that there were some distrust behaviors among experts. Meanwhile, their research only discussed the intercity railway public–private partnership model selection, whether the water pollution problem can be addressed is a question. However, our method considers the distrust behavior and presents a framework to select optimal alternatives for water pollution management.

Through comparative analysis and discussion, we further summarize the advantages of the proposed LSGDM-SN method. Firstly, the AHC approach considering preference similarity and social similarity is designed to achieve expert clustering. Moreover, an identification process for distrust behavior is proposed. Then, two cases managing distrust behavior are provided for the subset to modify its FPR. Finally, a score function of the FPR is given to calculate the alternatives ranking of water pollution management. The proposed method can provide a reference for managers to choose the optimal water pollution scheme.

7. Conclusions

To improve the efficiency of water pollution management, this paper focuses on the LSGDM-SN problem considering distrust behavior and its application in water pollution management. The main contributions are summarized as follows:

(1)

A novel AHC method considering preference similarity and social similarity is proposed to decrease the complexity of LSGDM-SN with FPRs. Several definitions, including preference similarity, trust similarity, and distrust similarity, are proposed to compute the degree of overall similarity among experts. Subsequently, the AHC algorithm dealing with the LSGDM-SN problem is designed.

(2)

The consensus feedback for detecting and managing distrust behavior is presented, which encourages the subset to modify its FPR based on different distrust types. To identify the distrust behavior, both the distrust score and the degree of difference are incorporated to measure the distrust degree of the subset. Based on the value of distrust degree, two pieces of modification advice are provided to subset to modify it FPR.

(3)

A score function is defined to derive the alternatives ranking in water pollution management. After computing the values of scores for all alternatives, we rank them. The optimal alternative is obtained based on the maximum value score. Finally, the LSGDM-SN framework considering distrust behavior is described to visualize the decision process.

This paper has enriched the research results of LSGDM-SN, but several aspects still need to be further analyzed and improved. In some LSGDM-SN problems, different experts may adopt different preferences to express their evaluation of alternatives. However, this paper does not consider the heterogeneous preferences of experts in LSGDM-SN. Thus, in the future, the LSGDM-SN with heterogeneous preferences can be discussed. Furthermore, the proposed operators can be extended to Fermatean fuzzy sets [41,42]; the analysis of LSGDM-SN with Fermatean fuzzy sets is also worthy of work. Additionally, in the practical LSGDM-SN problem, the efficiency of the proposed approach may be impacted by the uncertain social network. Nevertheless, the uncertain social network is not considered in the clustering and consensus process. Hence, the explosion of LSGDM-SN under uncertainty is bound to be the mainstream direction [43,44]. Finally, when large-scale experts exhibit heterogeneous behaviors, how to achieve consensus in LSGDM-SN remains to be further studied.