A Large-Scale Group Decision Making Model Based on Trust Relationship and Social Network Updating

1  Introduction

Due to the increasing complexity of the social and economic environment, it is increasingly difficult to rely on a single decision maker (DM) to make effective decisions. Therefore, many organizations use multiple members in the decision-making process, which is called group decision making (GDM). GDM is a participatory process, which selects the best solution from alternatives by considering the personal opinions of multiple experts [1]. However, with the rapid development of technology and society [2], the group sizes have gradually become larger and more complex [3,4]. Generally, a group with more than 20 members is defined as a large-scale group.

In the background of LGDM, decision makers (DMs) may have diverse opinions because of the different knowledge and experience. Therefore, how to help DMs reach consensus becomes a key issue. In order to deal with the complexity and uncertainty of the LGDM problem, this study starts from the following aspects: the dimensional reduction of large-scale DMs [5], the consensus measure and the improving method.

The dimensional reduction of large-scale DMs. There are two main directions for reducing the size of large-scale DMs. The one is based on the department or field in which the DM belongs. For example, Liu et al. [6] classified the DMs according to the DMs’ school, and determined the percentage distribution on evaluations of each group concerning each alternative. The characteristics of this direction are simple and convenient. But in the real process, judgment information given by the same type of DMs is not necessarily the same. Therefore, the clustering method based on the evaluation value or preference value of DM is used to reduce the dimension of large-scale DMs. For example, Wu et al. [7] used the k-means method to cluster a large amount of hesitant fuzzy preference information and improved consensus level based on three-level consensus measures and a local feedback strategy. Yang et al. [8] investigated the additive consistency of the intuitionistic fuzzy preference relations in group decision making using T-normalized intuitionistic fuzzy priority vectors. In the above studies, DMs are considered as independent individuals. However, there are social relationships among DMs, especially the trust relationship which is clearly existed and important in reality.

Social networking applications generate a huge amount of data daily. Meanwhile, social networks (SNs) have become a growing field of research due to the heterogeneity of data and structures, as well as their size and dynamics [9]. Some studies have proven the advantages of social networking, such as social network-based recommendation systems [10–12], online review websites incorporating the social-networking function [3,13], collaborative networks in the con53text of publications and citations [14,15], preventing the spread of rumors and misinformation by identifying influencers [16–19]. Information on SNs can not only enrich and improve the DMs’ preference information, promote and accelerate consensus reaching process, but more importantly, reduce the dimension of large-scale DMs. Trust is a special case in social relations, and some studies have also analyzed the impact of trust relations on clustering [20,21]. Therefore, in the framework of social network-group decision making (SN-GDM), it is novel and feasible to use trust relationship as a reliable source of member weight information. However, most existing studies only consider the trust relationship between nodes without considering the trust degree of different nodes; they also ignore the generation process of relationship network, and cannot automatically cluster in the process of dimension reduction. In addition, previous studies that applied SNA to large group networks did not consider network construction and update based on DMs feedback, except for the trust relationship among DMs.

The consensus measure and improving method. On the one hand, an interesting issue within the group decision theory is the consensus measure, and the key of the problem is how to determine the DMs’ weight and subgroup weight in the process of decision matrix aggregation. In GDM, the decision matrix is generally aggregated through subjective or objective weighting to perform consensus measures. In particular, expert weights may be adjusted during the consensus reaching process. Pang et al. [22] developed an extended TOPSIS method and aggregation-based method for multi-attribute group decision making (MAGDM) with probabilistic linguistic information in the case where the attribute weights are unknown and partially known. Wu et al. [23] used trust score values to assign importance weights to experts. For LGDM, the weight of individual DMs and the weight of each subgroup need to be determined after clustering. For example, Wu et al. [5] determined the expert weight through the centrality of the network. Wu et al. [7] determined the subgroup weight based on the number of experts in the subgroup. Shi et al. [24] used a uniform aggregation operator to update the weights in the consensus reaching process (CRP). According to the above research, group consensus can be improved quickly by adjusting the weight of DMs and subgroups in the CRP. This is also in line with reality. The influence of some DMs may increase in the process of DM interactions, resulting in the corresponding changes in weights. Although the above studies involve weight changes in the CRP, the trust relationship between DMs is rarely considered. Generally speaking, DM with strong trust relationships has a greater influence. In the process of interaction, DMs with strong trust relationships will affect the preferences of DMs with weaker trust relationships.

On the other hand, scholars have proposed consensus improvement methods for LGDM consensus problems [7,25–28], which are mainly divided into automatic methods [29,30] and interactive methods [31,32]. The automatic feedback mechanism saves time because it does not require additional expert interaction to carry out the consensus-improving process. For example, Zhang et al. [33] proposed an automatic feedback mechanism for group decision making based on the distribution linguistic preference relations. Perez et al. [34] overcame the problem of the moderator, giving a way to use an automatic system to compute and send customized advice to the experts if there is not enough consensus. The interactive feedback mechanism requires expert interaction, which takes more time but the results obtained are more accurate. For example, for the cooperative and noncooperative behaviors of experts, Quesada et al. [35] introduced a method to deal with noncooperative behaviors, which used an informal weighted scheme to assign weights to experts. Gou et al. [36] built a consensus-reaching model of noncooperative behavior and deal with noncooperative behavior and preference information. In addition, Gou et al. [37] used multi-stage interactive consensus reaching algorithms to deal with multi-expert decision making problems with language preference ordering. For the dynamic adjustment process of consensus reaching, in [1], a non-linear programming model was constructed to dynamically adjust the experts’ weights in consensus reaching process. Wu et al. [7] proposed an LGDM consensus model which allowed clusters change. And as the clusters changed in every interactive consensus round, the consensus process evolution could be captured. Besides, some studies have also considered social networks. For example, in [5], after the feedback mechanism is executed, clustering and consensus measures are performed again, the process of consensus improvement is not involved. However, the social network was applied in [5] which applied IT2-TOPSIS to obtain the optimal solution directly. In the above research on consensus improving, social networks, trust networks, interaction rules, and the update and optimization of the entire network structure in the adjustment process are rarely considered. However, in the actual adjustment process, it is necessary to consider the acceptability of information and let DMs with higher trust interact with each other. The interaction process will inevitably lead to changes in the trust relationship among members of the large group. Therefore, it is necessary to consider the update of the trust network in the process of consensus improvement.

Trust relationships have been applied in various fields and have brought great benefits to various industries. In the business field, marketing methods such as fan economy, word-of-mouth bonus, media marketing, onlookers, and participation experience have brought unexpected dividends to enterprises in the era of mobile internet. Among them, the cultivation of trust relationships is the key, and trust is the core of this emotional marketing. Similarly, in academia, the citation relationship between different scholars also reflects the trust between each other, which may further lead to collaboration relationships. In politics, the trust relationship between voters will also greatly affect the election results. Therefore, it is necessary to consider the trust relationship and trust network in LGDM. In addition, considering trust networks based on group classification of decision makers can better reflect the trust relationship, and the trust degree and the degree of information difference can be better reflected in the group consensus. Considering trust relationships can achieve more effective interactions in the feedback adjustment process.

In view of the necessity of trust in decision making, this study will take the trust relationship through the entire decision-making process, from social network construction and clustering to weight determination and consensus measurement, as well as consensus reaching process and network updating. In comparison with the previous consensus model, the consensus model proposed in this study has some distinctive features:

First, in previous studies, the SNA was typically used to simply represent the relationship of DMs. It did not really combine the actual social activities of the DMs. Considering the social trust relationships between DMs for the LGDM, this study builds a social network between DMs and uses the Louvain method to detect community based on the trust relationship.

Second, in the process of determining the weight of individual DMs and the weight of each subgroup, the trust weight and the network weight are fully considered. Compared with weights based on network degree centrality or subgroup size, it is more convincing to assign weights based on trust relationships and to make corresponding weight changes in the feedback adjustment process. In addition, the weight for individual DMs and the weight of each subgroup will update accordingly in the consensus reaching process.

Third, the proposed model allows for changes in the trust network. Individuals are able to modify their preferences in the reaching consensus process, so the trust relationship will change, which will make the number of subgroups and members of each subgroup likely to change. In addition, the consensus rules and adjustment rules proposed in the model are simple and can be used to guide modifications.

The remainder of this study is organized as follows. Section 2 introduces related concepts, such as social relationship and social impact analysis, possibility distribution based on hesitant fuzzy elements, and probability distribution based on fuzzy preference relations. Section 3 presents the proposed SNA-based LGDM method. In Section 4, an illustrative example is provided to show the applicability of the proposed method. Section 5 compares and analyzes the similarities and differences between the proposed method and the other methods in detail, cutting from the three perspectives of trust relationship, social network and subgroup classification method. Finally, Section 6 concludes this paper with future perspectives.

2  Preliminaries

2.1 Measurement of Trust among DMs

In this study, we distinguish senders and receivers of social ties, which means the DM’s online network is directed. The in-degree of a DM refers to the social ties that the DM has received from other DMs. The out-degree indicates the social ties that the DM has sent to other DMs [38]. The definitions of in-degree centrality and out-degree centrality are described as follows.

Definition 1 [39]: Let G=(E,L) be a directed graph, E={e1,e2,…,em} be the set of nodes and L={l1,…,lr,…,lq} be the set of directed edge between pairs of nodes.

(1) The number of edges originating from node er is called the out-degree centrality index of the node er.

d+(er)=∑t=1ma(er,et)(1)

where d+(er) represents the out-degree centrality of the DM er. If there is a link from er to et, then a(er,et)=1; Otherwise a(er,et)=0.

(2) The number of edges terminating from the node er is called the in-degree centrality index of the node er.

d−(er)=∑t=1ma(et,er)(2)

where d−(er) represents the in-degree centrality of the DM er. If there is a link from et to er, then a(et,er)=1; otherwise a(et,er)=0.

The relationship strength between members shows the level of trust between those members [38]. Within online social networks, members can declare friendship with one another by establishing social. If two DMs have more common social connections in an online social network, we can conclude that they have deeper social ties. This study uses degree centrality to calculate social connection strength between er and eh [22]:

SCrh=nrhd+(er)+d+(eh)−2−nrh(3)

where nrh is the number of common social connections between er and eh, which is measured by the number of common edges of er and eh. The more common social connections of er and eh are the stronger the social tie between er and eh is.

Social interaction strength is a combination of time length, emotional intensity, intimacy (mutual confiding), and reciprocal services that characterize the ties [40]. This study measures interaction strength between er to eh by interaction frequency frh. Normalized interaction strength between er and eh is calculated by:

SIrh=frh−frminfrmax−frmin(4)

where frmax and frmin are the maximum interaction frequency and minimum interaction frequency from er  to other DM et,t∈{1,2,…,m}, respectively. Interaction frequency SIrh is assumed to be fixed during the decision-making process. Note that interaction strength SIrh may not be the same as SIhr. The reason is that SIrh refers to interaction strength between er and eh evaluated by er, while SIhr is interaction strength between eh and er evaluated by eh.

Let λ be a weight for balancing the importance of connection strength and interaction strength. For example, λ<0.5 reflects that the interaction strength is more informative than connection strength. In this paper, λ takes 0.4. The strength of social ties between er and eh (evaluated by er) is defined by aggregating the connection strength and interaction strength of er and eh:

TSrh=λSCrh+(1−λ)SIrh(5)

2.2 Possibility Distribution Based Hesitant Fuzzy Element

Let X={x1,x2,…,xn}(n≥2) be a finite set of alternatives. E={e1,…,er,…,em} is a set of DMs.

Each DM gives a judgment on the various alternatives. As we know, preference relations are a classical and powerful preference structure to represent the preferences in GDM problems. Fuzzy preference relations (FPR) have been found to be effective when dealing with uncertain information [41–45]. Therefore, this study uses FPR to represent each DM’s opinion on alternatives X.

Definition 2 [7]: A FPR on X is represented by a matrix B=(bij)n×n⊂X×X, where bij=μB(xi,xj)∈[0,1] indicates the assessment for the pair (xi,xj). In addition, the additive reciprocity holds, that is, bij+bji=1,i,j=1,2,…,n.

Definition 3 [7]: The possibility distribution-based hesitant fuzzy element (PDHFE) can be expressed as follows:

h(p)={hl(pl)|l=1,2,…,#h}(6)

where hl,l=1,2,…,#h are the membership degrees and #h is the cardinality of h. pl denotes the possibility of hl. If there is only one membership value in a given PDHFE, then the bracket can be dropped; for example, we have 0.5 = 0.5 = 0.5(1). For simplicity, h(p)=hp.

Definition 4 [7]: A PDFPR on X is given by the matrix H=(hpij)n×n⊂X×X, where hpij={hijl(pijl)|l=1,2,…,#hpij} is a PDHFE. Moreover, the hpij meets the following conditions:

hijl+hji#hpij−l+1=1,hij=0.5(7)

pijl=pji#hpij−l+1,hijl≤hijl+1,i,j=1,2,…,n(8)

where hijl is the lth possible value of hij and pijl is the probability of hijl.

Definition 5 [7]: The expected value or mean for hp can be defined as follows:

Expect(hp)=∑l=1#hphlpl(9)

Definition 6 [7]: The distance between hp(1) and hp(2) is defined as follows:

d(hp(1),hp(2))=∑l1=1#hp(1)∑l2=1#hp(2)pl1(1)×pl2(2)|hl1(1)−hl2(2)|(10)

where hp(1)={hl(1)pl(1)|l=1,2,…,#hp(1)} and hp(2)={hl(2)pl(2)|l=1,2,…,#hp(2)} are two PDHFEs. It is easy to see that 0≤d{hp(1),hp(2)}≤1.

3  Consensus Framework and Model for the LGDM

3.1 Problem Description and Consensus Framework

Suppose that there are m DMs who provide their preferences for n alternatives, and the preference relations of DMs can be expressed by FPR. Where E={e1,e2,…,em} represents m DMs, X={x1,x2,…,xn} represents n alternatives. The pairwise comparison matrix given by the decision maker is denoted as Bk=(bijk)n×n,k=1,2,…,m. This study also collects the number of likes and reviews that the DMs interact with other members in their daily lives, and counts the frequency of interaction between DMs. A trust network is constructed by calculating the degree of trust by the interaction frequency between DMs. For the three aspects mentioned in the introduction, the paper will take the following solutions, and Fig. 1 illustrates the overall solution clearly:

Figure 1: The overall solution for LGDM problems based on trust relation and changeable network

(1) Firstly, according to the social relations between DMs, a directed social relationship network can be constructed. Then the trust relationships among the members are calculated. Using the Louvain method which has been widely used in large-scale network community detection [46], the large-scale social network is divided into several subgroups. This part is detailed in Section 3.2.

(2) Secondly, the weight of the DMs in the subgroup and the weight of each subgroup based on trust relationships are calculated by combining the SNA and the Louvain method, then the consensus index is obtained. Details of trust weight determination and consensus measure are given in Section 3.3.

(3) Thirdly, if the group consensus does not reach the predefined threshold, the feedback mechanism considering the trust relationship between the members is used to adjust the consensus until the threshold is reached, and the network will change during this process. Details of the consensus reaching process are given in Section 3.4.

(4) Finally, the DMs’ preferences are aggregated according to the network relations, and the alternatives are sorted to select the best ones.

3.2 Trust Weight Determination and Consensus Measure

In determining the weight of individual DM and the weight of each subgroup, compared with weighting based on network degree centrality or subgroup size, the trust weight and the network weight are fully considered. Firstly, this study uses the Louvain method to classify DMs, and obtains the comprehensive weight of each DM by the network weights and trust weights. Then, based on the reciprocal of the distance from each subgroup to the network center, the weight of each subgroup is calculated. Finally, based on the work of Herrera-Viedma et al. [47], the main steps of the consensus measure are listed. In addition, the weights for individual DM and the weight of each subgroup will change accordingly in the consensus reaching process.

3.2.1 Trust Network and Louvain Method in Community Detection

Community, also called a cluster or module, is a group of vertices which probably share common properties. Community detection refers to the recognition of modules and their boundaries based on the structural positions of vertices and the classification of vertices [48]. The community detection method is the key based on a trust network for DMs to reduce dimensionality clustering. Some methods have been developed, such as the GN method [49], the spin-glass method [50], the random walk community detection [51], the label propagation method [52] and so on. The Louvain method is a commonly used community detection method, which is based on the modularity theory proposed by Blondel et al. [46]. This method is an agglomerative clustering algorithm, which reveals the complete hierarchical community structure of the network and can cluster subgroups automatically without setting the initial number of subgroups. Therefore, this study uses the Louvain method for DMs in LGDM problems.

Assuming that a network has t nodes, the algorithm of the Louvain method can be expressed as follows [39]:

Step 1: Each node in the network is treated as a separate community. In the beginning, the number of communities is the same as the number of nodes.

Step 2: Assign each node to the community according to the node near each node, calculate the value of ΔQ before and after the module allocation, and record the maximum ΔQ value of the node. If maxΔQ>0, then the node d^α is assigned to the community where the node with the maximum ΔQ value is located, otherwise, it will remain unchanged.

The definition of ΔQ is as follows:

ΔQ=[∑in+2e^r,in2m−(∑tot+e^r2m)]−[∑in2m−(∑tot2m)2−(e^r2m)2]

where ∑in is the total degree of all edges in the community c, ∑tot is the total degree of all edge associated to the node in the community c, e^r,in is the number of edges of the node from node er(r=1,2,…,m) to all nodes in the community c, m is the number of all edges in the network.

Step 3: Repeat Step 2 until the network in the community no longer changes.

Step 4: Streamline the entire network and treat the nodes in a community as a new node. The weights between nodes within the community are converted to the weights of the new nodes. The weights between the edges of the community are converted to weights between the edges of the new nodes. The boundary value of the weight is 1.

Step 5: Repeat Steps 1–4 until the modules of the entire network no longer change.

3.2.2 The Comprehensive Weight of Each Decision Maker

(1) Network weights of each DM

Set V nodes in the network. Firstly, the degree centrality CD(er) and the eigenvector centrality CE(er) of each node er(r=1,2,…,m) are calculated by using Pajek software, and then they are standardized [5]:

CD′(er)=CD(er)∑r=1mCD(er)(11)

CE′(er)=CE(er)∑r=1mCE(er)(12)

Combining the normalized degree centrality CD′(er) with the eigenvector centrality CE′(er), calculate the mixed center degree CF(er) of the node er(r=1,2,…,m):

CF(er)=σCD′(er)+(1−σ)CE′(er)(13)

According to [49], the value of σ is generally 0.5.

Suppose that m nodes in the network are divided into p subgroups, and there are s nodes in the subgroup Gk(1≤k≤p), the network weight of the node er(er∈ck,1≤r≤s) can be calculated as:

wr1=CF(er)∑r=1sCF(er)(14)

(2) Trust weights of each DM

Assigning weight to each DM is an important part of the decision-making process and plays a key role in obtaining the final solution. For online social networks, historical interaction information can provide a reliable source for accessing DMs.

Social influence is defined as the individual’s thoughts, feelings, attitudes or behaviors can affect others when interacting with other individuals or groups. DMs with higher social impact have the ability to influence the opinions of other members [47]. Using social network analysis techniques, the social impact of each DM in the group can be obtained. The more social relations a person has with others, the more influence the decision maker will be. In the social network analysis method, the centrality of the in-degree is used to quantify the social influence of DM in a network, and the social influence can reflect the importance of DM to a certain extent. Therefore, the social impact of er,r=1,2,…,m is defined by Eq. (15), and the trust weight is determined by Eq. (16).

d′−(er)=1m−1∑h=1mTShr,∀r=1,2,…,m(15)

wr2=d′−(er)∑h=1md′−(er),∀r=1,2,…,m(16)

Then the comprehensive weight of each DM can be calculated as w=θwr2+(1−θ)wr1.

3.2.3 The Weight of Each Subgroup

The network weight between subgroups can be calculated from the reciprocal of the distance from each subgroup to the network center. The further the distance, the smaller the weight. The main steps for calculating weights are shown below [5]:

Step 1: Calculate the fusion centrality of the network.

CF=1|M|∑r∈MCF(er)(17)

where M={e1,e2,…,er,…,et} represents all nodes of the network.

Step 2: Calculate the fusion centrality of each subgroup.

CFk=1|Sk|∑r∈SkCF(er)(18)

where M={S1∪S2∪…∪Sk∪…∪Sp}, Sk={e1,e2,…,er,…,es}, |Sk| represents the number of all nodes of the subgroup ck, er is the nodes in the subgroup ck.

Step 3: Calculate the relative distance λk′ of CF(ck) and CF in community ck.

λk′=1|CFk−CF|(19)

Step 4: Standardize the weight λk′.

λk′=λk′∑k=1pλk′(20)

The trust relation between subgroups can be reconstructed by treating all nodes in a subgroup as a new node, and then calculating the trust weights between subgroups. Finally, the network weights and trust weights of the subgroups are integrated, and the comprehensive weights of the subgroups are obtained.

Next, the consensus measure will be computed based on Ht,t=1,2,…,K. Based on the work of Herrera-Viedma et al. [47], the main steps of the consensus measure are as follows:

Step 1: Calculate the similarity matrix. Let Hk=(hpij,k)n×n and Ht=(hpij,t)n×n correspond to the probability distribution of subgroup Gk and Gt based on the fuzzy preference relationship. Where hpij,k={hij,kl(pij,kl)|l=1,2,…,#hpij,k} or hpij,t={hij,tl(pij,tl)|l=1,2,…,#hpij,t} is a probability distribution based on hesitant fuzzy elements. For each pair of subgroups Gk and Gt(k<t,t∈{1,2,…,K}), the similarity matrix SMkt=(smijkt)n×n is calculated as follows:

SMkt=(−⋯sm1nkt⋮⋱⋮smn1kt⋯−)(21)

where smijkt is the similarity of the subgroups Gk and Gt with respect to the alternatives (xi,xj),i,j=1,2,…,n. According to definition 6, smijkt can be expressed as follows:

smijkt=1−d(hpij,k,hpij,t)=∑l1=1#hpij,k∑l2=1#hpij,tpl1ij,k×pl2ij,t|hl1ij,k−hl2ij,t|(22)

Step 2: Calculate the consensus matrix. By aggregating the similarity matrix of each pair of subgroups (Gk,Gt), a consensus matrix CM=(cmij)n×n can be obtained. Then this study normalizes the weights of the subgroup pairs (Gk,Gt), ωkt is defined as follows:

ωkt=wGkwGt∑k=1K−1∑t=k+1KwGkwGt(23)

cmij can be expressed as follows:

cmij=∑k=1K−1∑t=k+1Kωktsmijkt(24)

Step 3: Calculate the large-scale group consensus index (LGCI).

(1) Calculate the consensus degree of the pair of alternatives. The consensus degree cpij of the alternatives pair (xi,xj) can be obtained from the consensus matrix CM=(cmij)n×n. cpij can be expressed as follows:

cpij=cmij,i,j=1,2,…,n,i≠j(25)

(2) Calculate the consensus degree of the alternatives. The consensus degree cai, i=1,2,…,n of each alternative xi can be calculated as follows:

cai=∑j=1,j≠incpijn−1(26)

(3) Calculate the consensus degree of the preference relationship. This study refers to the consensus degree of the large group preference relationship as the consensus index LGCI, and the formula is as follows:

LGCI=∑i=1ncain(27)

3.3 Consensus Reaching Process Based on Social Network Updating

Based on the above discussion, this study can get consensus at different levels. Assume LGCI¯ is a predefined threshold. If LGCI≥LGCI¯, all DMs reach a higher level of consensus and the consensus adjustment process ends. Otherwise, this study will adjust the preference values of DMs with low consensus. In this section, this study proposes a feedback mechanism to help DMs in each subgroup change their preferences based on [37]. The mechanism consists of four identification rules and two direction rules to detect which subgroups need to change their preferences, and the DMs’ preference relations will change during identification, clustering, and location.

Moreover, the strength of ties between members shows the trust degree between those members. DMs with strong trust relationships have a greater impact. In the process of interaction, DMs with strong trust relationships will affect the preferences of DMs with weaker trust relationships. At the same time, the trust relation between DMs will also change. Therefore, this study will reflect the results of the feedback in the social network and change the connection between DMs to achieve a higher consensus faster according to the trust relationship between DMs.

3.3.1 Identification Rules

The identification rules can obtain alternative comparison pairs, subgroups, and DMs continuously and accurately. Therefore, DMs can change their preferences in a precise way.

Identification Rule 1: Identify alternatives whose preferences need to be changed. The identified alternatives can be expressed as follows:

ALT={xi|cai<LGCI¯,i=1,2,…,n}(28)

Identification Rule 2: Identify the locations that need to be changed. For any xi∈ALT, the identified set of locations can be expressed as follows:

Posi={(i,j)|xi∈ALT∧cpij<LGCI¯,i<j}(29)

Identification Rule 3: Determine the ideal and non-ideal sets for each identified part. For (i,j)∈Posi, ideal set CLUij+ and non-ideal set CLUij− can be expressed as follows:

CLUij+={Gk+(xi,xj)|k+=kargmax{∑t=1,t≠kKsmijtk}}(30)

CLUij−={G1,G2,…,Gk}∖CLUij+(31)

Identification Rule 4: Identify the DMs who need to change their preferences. For the identified non-ideal set Gk−(xi,xj), this study can use a deeper consensus measure to determine the DMs with only the lowest level of consensus.

The DMs in the non-ideal set Gk−(xi,xj) can be expressed as {ek1,ek2,…,ekG(k−)}. G(k−)=#Gk−(xi,xj) is the number of DMs in Gk−(xi,xj). Note that the FPR of eky is Bky=(bij,ky)n×n. Similarly, the DMs in Gk+(xi,xj) can be represented as {et1,et2,…,etG(k+)}. G(k+)=#Gk+(xi,xj) is the number of DMs in Gk+(xi,xj). The preference of Gk+(xi,xj) is a PDHFE, which can be expressed by hpij,k+={hij,kl+(pij,k+l)|l=1,2,…,#hpij,k+}. The average preference for hpij,k+ can be calculated as follows:

Expect(hpij,k+)=∑l=1#hpij,k+hij,kl+pij,k+l(32)

Set thresholds β. The DMs who need to change their preferences (xi,xj) in Gk−(xi,xj) can calculate as follows:

EXPSij−={eky|bij,ky−Expect(hpij,k+)|>β,ky∈{k1,k2,…,kG(k−)}}(33)

3.3.2 Directions Rules and Network Adjustment

For each eky∈EXPSij−, there are two directions rules as follows [10]:

Directions Rule 1: If bij,ky<Expect(hpij,k+), then eky needs to increase the preference value of (xi,xj).

Directions Rule 2: If bij,ky>Expect(hpij,k+), then eky needs to decrease the preference value of (xi,xj).

One way to achieve the direction rule is to provide a set of values for eky∈EXPSij−. Since each element in the initial eky belongs to S[0.1,0.9], it is better to set the range of values of all recommended values in this interval.

Let Rndij,k+=Round(Expect(hpij,k+)), where Round is a method to find the nearest discrete membership in the range of S[0.1,0.9]. Then the recommended set eky can be expressed as follows:

RSeky(xi,xj)={bij,ky′|bij,ky′≠bij,ky∧bij,ky′∈(S[0.1,0.9]∧[min{bij,ky,Rndij,k+},max{bij,ky,Rndij,k+}])} By the opposite nature of each other, the preference eky and (xi,xj) can be automatically adjusted and updated.

After completing the above adjustment process, the final step is to modify the original social network.

Step 1: Let (i,j) be the position that needs to be modified, E′={e′1,e′2,…,e′v} is a set of DMs whose opinions need to be modified.

Step 2: Calculate the TSrh value according to Eq. (5), and use it as an index to evaluate the trust degree between members. Find the most trusted member E′={e′1,e′2,…,e′v} in the ideal set CLUij+, denoted as F′={f′1,f′2,…,f′v}.

Step 3: Modify the original social network. According to E′={e′1,e′2,…,e′v} and F′={f′1,f′2,…,f′v}, connect ei′ and fi′, where i=1,2,…,v.

3.4 The Procedure of the Proposed Method

The main steps of the LGDM consensus problem are summarized below. Fig. 2 shows the framework of the proposed method.

Figure 2: The framework of the proposed method

Part 1: Determine the network structure of the LGDM problem.

(1) The set of DMs and alternatives are represented by {e1Gt,e2Gt,…,emtGt} and X={x1,x2,…xn}, respectively.

(2) For each DM, a fuzzy preference matrix Bk=(bijk)n×n,k=1,2,…,m is established. DMs tend to use fuzzy preference relationships to express their opinion. Next, the consensus threshold LGCI¯, parameter β and the maximum number of iterations Maxround are set.

(3) The Louvain method is used to determine subgroups of large-scale networks. Suppose this study gets Kr subgroups, denoted by {G1,G2,…,GKr}.

Part 2: Calculate the trust weight and consensus measure.

(1) Calculate the weights of the tth (1≤i≤m) DM and the tth (1≤t≤Kr) subgroups.

(2) Calculate the PDFRP for each subgroup.

(3) Calculate the consensus of each subgroup.

Subgroup consensus can be obtained according to Section 3.3. Set the consensus is LGCIr in the current round. If LGCIr≥LGCI¯ or r>Maxround, go to Part 4; otherwise, proceed to Part 3.

Part 3: Consensus reaching process.

(1) Determine the identified DM as the corresponding new collection.

According to the four identification rules proposed in Section 3.3.1, the set of non-ideal DM can be expressed as EXPSij,r−.

(2) Direct the DMs in EXPSij,r− to modify their preferences.

Modify the FPR of the DM in EXPSij,r− based on the two direction rules in Section 3.3.2. Next, let r=r+1, the modified FPR is still represented byBk,r,k=1,2,…,m.

(3) Modify the social network.

Part 4: Select the best alternative(s).

(1) Based on the final adjusted network, the weight of each DM within the subgroup and subgroup weights are recalculated, and then the final pairwise comparison matrix is obtained by aggregating the preference matrix of all subgroups for each pair of alternatives according to Eqs. (21)–(24).

(2) After obtaining the final pairwise comparison matrix, the final ranking result is obtained by subtracting the sum of each column value from the sum of each row value of each alternative [53].

(3) Rank all alternatives in descending order by the gap between the column value and row value and choose the alternative with the smallest gap as the best alternative.

4  Case Study

4.1 Case Description

Twenty travel enthusiasts with certain social connections are denoted as E={e1,e2,…,e20}. They are going to choose the best destination for vacation travel. After pre-evaluation, the following four alternatives are selected, denoted as X={x1,x2,x3,x4}, where x1 = Gulangyu Islet; x2 = Suzhou Gardens; x3 = Lijiang Ancient City; x4 = Dujiang Dam. Each member evaluates and judges the four alternative tourist destinations by the way of pairwise comparison. This study obtains the pairwise judgment matrix Bk,k=1,2,…,20 as shown below [7]:

B1=(0.50.90.90.80.10.50.70.80.10.30.50.40.20.20.60.5), B2=(0.50.30.70.80.70.50.30.60.30.70.50.30.20.40.70.5),…, B20=(0.50.60.40.10.40.50.30.40.60.70.50.70.90.60.30.5)

The social network relations between the 20 travel enthusiasts as shown in Fig. 3.

Figure 3: DMs’ social network

In Fig. 3, each travel enthusiast is represented in the network as a node with different colors and used to distinguish the number of ingress connections of the DMs. For example, the DM e18 has the largest indegree in the group, so the color of node 18 is dark blue. In contrast, node 13 is colored yellowish because it has only one ingress connection from e2.

4.2 Method Application

4.2.1 Classification of Network Using Lauvain Method

A network corresponding to Fig. 3 is constructed in the complex network analysis tool Pajek, as shown in Fig. 4, where nodes v1−v20 correspond to nodes 1–20 in Fig. 3, respectively.

Figure 4: Network constructed in Pajek software

Using the Lauvain package included in the Pajek software, automatic classification is performed to form the classification results as shown in Fig. 5. The subgroup is represented by the set {G1,G2,…,GKr}, which is divided into three subgroups. The subgroup consisting of yellow nodes is G1, the subgroup consisting of green nodes is G2, and the subgroup consisting of red nodes is G3, i.e., G1={v1,v3,v5,v6,v9,v10,v11,v18}, G2={v2,v12,v14,v15,v16,v19}, G1={v4,v7,v8,v13,v17,v20}.

Figure 5: Social network clustering analysis results

4.2.2 Calculate the Trust Weight and Consensus Measure

For the subgroup G1, the network is shown in Fig. 6.

Figure 6: Network of subgroup G1

According to the original data required by the software and the original data required for the centrality of the feature vector, the network weight of each node in the subgroup G1 can be obtained according to the Eq. (14), as shown in Table 1.

According to the network weights and trust weights, the weight of each DM is aggregated. And this study takes θ=0.2, then, the comprehensive weights of the nodes in the subgroup G1 are shown in Table 1.

According to the calculation results by Pajek software and Section 3.2, the network weight of each subgroup is {0.05,0.04,0.19}.

Considering each subgroup as a node, Fig. 3 can be abstracted into the structure shown in Fig. 7, where the direction of the edge represents the relationship between nodes, and the thickness of the edge represents the connection strength.

Figure 7: Directed social relationship network between subgroups

The trust weights between subgroups are {0.40,0.26,0.34}.

Similarly, by aggregating the network weights and trust weights of each subgroup, the comprehensive weight of each subgroup can be obtained {0.33,0.22,0.45}.

According to their preference for alternatives provided by the 20 DMs through FPR, the preference relationship of each subgroup also forms a probability distribution based on fuzzy preference relations [51]. According to Section 3.4, the corresponding Gk,k=1,2,3 has three PDFPRs, which can be expressed as Hk=(hpij,k)n×n,k=1,2,3.

For the subgroup G1, the PDFPR can be expressed as follows:

hp12,1={0.1(0.14),0.3(0.10),0.4(0.25),0.6(0.25),0.7(0.13),0.9(0.13)}

hp13,1={0.4(0.38),0.6(0.49),0.9(0.13)},

hp14,1={0.2(0.13),0.4(0.10),0.6(0.29),0.7(0.22),0.8(0.26)},

hp23,1={0.1(0.10),0.3(0.28),0.6(0.27),0.7(0.13),0.8(0.10),0.9(0,12)}

hp24,1={0.4(0.29),0.5(0.10),0.7(0.26),0.8(0.23),0.9(0.12)},

hp34,1={0.3(0.14),0.4(0.23),0.6(0.13),0.7(0.15),0.9(0.35)}.

For the subgroup G2, the PDFPR can be expressed as follows:

hp12,2={0.3(0.29),0.4(0.15),0.6(0.21),0.7(0.15),0.9(0.20)},

hp13,2={0.2(0.21),0.4(0.30),0.6(0.18),0.7(0.31)},

hp14,2={0.2(0.15),0.3(0.21),0.4(0.18),0.5(0.15),0.8(0.31)},

hp23,2={0.1(0.30),0.3(0.11),0.4(0.21),0.6(0.18),0.8(0.20)},

hp24,2={0.2(0.30),0.3(0.21),0.6(0.29),0.7(0.20)},

hp34,2={0.1(0.20),0.3(0.11),0.4(0.51),0.6(0.18)}.

For the subgroup G3, the PDFPR can be expressed as follows:

hp12,3={0.1(0.17),0.2(0.17),0.4(0.38),0.6(0.27)},

hp13,3={0.2(0.08),0.4(0.35),0.6(0.17),0.7(0.22),0.8(0.17)},

hp14,3={0.1(0.35),0.3(0.08),0.4(0.17),0.6(0.39)},

hp23,3={0.3(0.19),0.4(0.30),0.5(0.16),0.6(0.17),0.8(0.17)},

hp24,3={0.3(0.08),0.4(0.35),0.6(0.17),0.8(0.39)},

hp34,3={0.4(0.08),0.6(0.17),0.7(0.57),0.8(0.17)}.

According to Section 3.4, the consensus degree of the three subgroups can be calculated. First, the similarity matrixes between subgroups are calculated, and the results are as follows:

SM12=(−0.73950.79900.73840.7395−0.69920.72040.79900.6992−0.69350.73840.72040.6935−),

SM13=(−0.74580.81150.70320.7458−0.75940.79110.81150.7594−0.78980.70320.79110.7898−),

SM23=(−0.72740.78740.72240.7274−0.74120.73850.78740.7412−0.68660.72240.73850.6866−).

According to Eq. (13), the normalized weights between subgroups are {0.22,0.47,0.31}.

Second, the consensus of the group is calculated, and the consensus matrix is as follows:

CMround1=(−0.73870.80130.71690.7387−0.74040.75920.80130.7404−0.73660.71690.75920.7366−)

The consensus degree between alternative pairs is cpij=cmij,i,j=1,2,3,4.

Third, the consensus degree at the level of the alternative is ca1=7523, ca2=7461, ca3=7594, ca4=7376.

Finally, the consensus degree of the group is LGCI=0.7488.

4.2.3 Consensus Reaching Process

Set LGCI¯=0.8. Because LGCI<LGCI¯, the feedback adjustment mechanism can be activated. According to Rule 1 and Rule 2 in Section 3.3.1, the preference relationship on (x2,x3) needs to be modified. According to Rule 3, ideal subgroups and non-ideal subgroups can be obtained:

CLU23+={G3}, CLU23−={G1,G2}.

Let the parameter β=0.3 and calculate Expect(hp23,3)=0.495, then this study can identify the DMs that need to readjust the preference value in subgroup G1 and subgroup G2.

DMs in subgroup G1:e5,e9,

DMs in subgroup G2:e15,e16.

Find the most trusted person for DMs e5,e9,e15,e16 in subgroup G3. The most trusted persons for e5,e9,e15,e16 are e8,e17,e8,e8, respectively. The new preference value of the DM is changed to the preference value of the most trusted person [52].

First round of adjustment:

According to Section 3.2.1, this study can get a new social network based on the Louvain method and a new subgroup classified as shown in Figs. 8 and 9.

Figure 8: Network in the first round of adjustment

Figure 9: Classification result

As can be seen from the Fig. 9, three new subgroups can be obtained in this adjustment: G1={v1,v3,v5,v8,v18}, G2={v2,v6,v12,v14,v15,v16,v19}, G3={v4,v7,v9,v10,v11,v13,v17,v20}. Then the trust weight and the network weight are integrated to determine the relative weight between the subgroups as shown in Table 2. And the weight of each subgroup is {0.21,0.25,0.54}.

For the subgroup G1, the PDFPR can be expressed as follows:

hp12,1={0.1(0.16),0.4(0.42),0.6(0.25),0.9(0.17)},

hp13,1={0.4(0.43),0.6(0.16),0.7(0.24),0.9(0.17)},

hp14,1={0.2(0.25),0.4(0.18),0.6(0.40),0.8(0.17)},

hp23,1={0.3(0.25),0.4(0.42),0.6(0.16),0.7(0.17)},

hp24,1={0.4(0.16),0.5(0.18),0.7(0.25),0.8(0.41)},

hp34,1={0.3(0.16),0.4(0.35),0.6(0.25),0.7(0.24)}.

For the subgroup G2, the PDFPR can be expressed as follows:

hp12,2={0.3(0.38),0.4(0.12),0.6(0.19),0.7(0.13),0.9(0.17)},

hp13,2={0.2(0.19),0.4(0.25),0.6(0.29),0.7(0.26)},

hp14,2={0.2(0.12),0.3(0.19),0.4(0.15),0.5(0.13),0.7(0.14),0.8(0.26)},

hp23,2={0.3(0.09),0.4(0.44),0.6(0.15),0.8(0.31)},

hp24,2={0.2(0.25),0.3(0.19),0.6(0.24),0.7(0.17),0.8(0.14)},

hp34,2={0.1(0.17),0.3(0.09),0.4(0.44),0.6(0.15),0.9(0.14)}.

For the subgroup G3, the PDFPR can be expressed as follows:

hp12,3={0.1(0.12),0.2(0.13),0.4(0.29),0.6(0.33),0.7(0.14)},

hp13,3={0.2(0.07),0.4(0.45),0.6(0.37),0.8(0.12)},

hp14,3={0.1(0.30),0.3(0.07),0.4(0.12),0.6(0.28),0.7(0.10),0.8(0.14)},

hp23,3={0.3(0.31),0.4(0.07),0.5(0.24),0.6(0.26),0.8(0.13)},

hp24,3={0.3(0.07),0.4(0.45),0.6(0.12),0.7(0.14),0.8(0.13),0.9(0.10)},

hp34,3={0.4(0.07),0.6(0.13),0.7(0.45),0.8(0.12),0.9(0.24)}.

The new similarity matrixes between subgroups are:

SM12=(−0.73510.78560.75190.7351−0.80580.73900.78560.8058−0.77530.75190.73900.7753−),