key: cord-0995376-pjlwleyv
authors: Lu, Yanling; Xu, Yejun; Herrera-Viedma, Enrique; Han, Yefan
title: Consensus of large-scale group decision making in social network: the minimum cost model based on robust optimization
date: 2020-08-29
journal: Inf Sci (N Y)
DOI: 10.1016/j.ins.2020.08.022
sha: 05437565bbee6636d76cf9b9f46f219d678bcce7
doc_id: 995376
cord_uid: pjlwleyv

Recently, large-scale group decision making (LSGDM) in social network comes into being. In the practical consensus of LSGDM, the unit adjustment cost of experts is difficult to obtain and may be uncertain. Therefore, the purpose of this paper is to propose a consensus model based on robust optimization. This paper focuses on LSGDM, considering the social relationship between experts. In the presented model, an expert clustering method, combining trust degree and relationship strength, is used to classify experts with similar opinions into subgroups. A consensus index, reflecting the harmony degree between experts, is devised to measure the consensus level among experts. Then, a minimum cost model based on robust optimization is proposed to solve the robust optimization consensus problem. Subsequently, a detailed consensus feedback adjustment is presented. Finally, a case study and comparative analysis are provided to verify the validity and advantage of the proposed method.

With the development of information technology, such as e-democracy and peer to peer working, a large number of decision makers (experts) are involved in decision making problems. This has led to the large-scale group decision making (LSGDM) to become a hotspot [40, 24, 25] . Generally, LSGDM is a special 5 type of decision making in which no fewer than 20 experts collaboratively choose the best alternative [9] . A number of practical applications [8] , such as porous media [41] and petroleum engineering [27] , involve LSGDM. In actual LSGDM, the experts may not be independent individuals, they may have specific relationships or be stakeholders. Consequently, their relationships may accelerate 10 the exchange of internal information, thereby affecting the final decision results.

LSGDM in social network has received increasing attention recently [30, 23] .

As a new type of decision case, LSGDM in social network refers to the process in which experts rely on social relations and opinion support from close friends to make decisions. In this decision process, the trust relationship between experts 15 can potentially affect the process of clustering, consensus and opinion gathering.

Several studies have demonstrated that social network has a positive impact on missing value estimation [30] and achieving LSGDM [26] . For example, based on trust relationships among experts, Tian et al. [30] constructed a multi-objective programming model to estimate incomplete interval type-2 fuzzy information. 20 Considering social trust behavior, Wu et al. [38] proposed a two-stage trust network partition algorithm to reduce the complexity of the LSGDM problem. In addition, many studies have developed various methods based on trust relations in LSGDM, such as trust-based consensus [19] , trust-based conflict detection and elimination [23] and non-cooperative behavior management based on trust 25 relations [42] .

To eliminate conflicts between experts, a consensus reaching process (CRP) is often used, which aims to achieve a collective solution as close to a unanimous agreement as possible [19, 30, 23] . To date, several types of CRPs in group decision making (GDM) have been developed from the following five aspects: 30 (1) CRPs with preferred representative structure [16, 11, 43, 18] , investigating various CRPs based on preference order, utility function and preference relations, (2) individual consistency in the consensus problem [39, 44, 21] , exploring how to guarantee individual consistency in the CRPs, (3) dynamic consensus realization process [1, 7, 20] , seeking solutions for cases where the decision el-35 ements are dynamically changed, (4) CRPs in social network [32, 36, 35, 48] , considering the social relationship between experts in CRPs, and (5) minimum cost consensus models [4, 12, 17] . It can be observed that all the five aspects have the same goal: to eliminate conflicts among experts and to improve the quality of decision result. 40 In LSGDM, however, experts come from different fields and represent different interests. This causes the LSGDM consensus to take longer and be costlier than GDM. Therefore, LSGDM consensus requires a capable moderator to devote time and resources to convincing experts to change their opinions into an acceptable level. To model this kind of consensus, the minimum cost consensus 45 model is selected from the five models introduced. The minimum cost consensus model was first constructed by Ben-Arieh and Easton [4] using a distance measured method. On the basis of Ben-Arieh and Easton's work, Gong et al. [11] further developed the minimum cost consensus model with interval preference opinions. Recently, Labella et al. [17] developed an objective metric based on a 50 comprehensive minimum cost model to evaluate the performance of CRPs.

In the above minimum cost consensus models, the unit adjustment cost of experts is usually considered to be fixed. In practice, however, it is always uncertain and reflected by interval values [20] , a distribution uncertainty set [13] and a box set [14] . Robust optimization is a new method of investigating the 55 uncertain optimization problem [31, 10, 28] . It originates in robust control theory and is a supplement to stochastic optimization. Different from stochastic optimization, robust optimization does not assume the distribution of uncertain parameters. The purpose of robust optimization is to obtain a solution that satisfies all constraints and optimizes the value of the objective function in the 60 worst case. Many theories and methods have been developed based on robust optimization. For example, Bandi and Bertsimas [2] suggested that the option pricing problem was a robust optimization problem, and reduced the worst-case replication error to a minimum. Ji et al. [15] proposed a fuzzy robust weighted method for a bilevel game model. Yanıkoglu et al. [45] introduced adjustable 65 robust optimization. A guide on how to apply an adjustable robust optimization method is given in the work of Yanıkoglu et al. [45] .

The CRP in social network and the consensus models mentioned above are very insightful, but they still need to be improved to address the actual CRP problem:

(1) The traditional consensus in social network only considers the trust and distrustful relationship between experts. Such a relationship is regarded as the input information in a CRP, without considering the internal diversity of social relationships. For LSGDM problems, if experts develop closer relationships, they are more likely to form similar opinions regarding an alternative. 75 In contrast, if their relationships are weak, their opinions may display obvious differences. Therefore, when considering the relationship information among experts, how can an LSGDM consensus be achieved?

(2) The existing minimum cost models mostly assume that the unit adjustment cost given by the moderator is definite and known beforehand. This 80 certain value is used directly in the feedback process without considering the uncertain situation. However, it is hardly a guarantee that the unit adjustment cost is certain and known in the actual LSGDM problem. The reason for this is that because the number of experts that take part in the LSGDM is more than or equal to 20 [9] , it is difficult for the moderator to master the compensation 85 information of each expert. Once an uncertain situation is encountered, the feasibility and validity of the consensus process will be largely hampered for LSGDM problems.

(3) To achieve consensus, the previous minimum cost models only focus on minimizing the total compensation cost provided by the moderator. However, 90 when the unit adjustment cost is uncertain, each expert usually expects the unit adjustment cost paid them to be maximized. Thus, this case will lead the total compensation cost paid by the moderator to reach a maximum, which is the worst-case consensus problem. Therefore, facing this case, we should consider how to minimize the total compensation cost in the worst-case consensus 95 problem.

Motivated by the limitations summarized above, we solve the LSGDM consensus in social network using the minimum cost model based on robust optimization. In the proposed LSGDM consensus, considering the relationship information among experts, a novel expert clustering is given to improve the 100 efficiency of the LSGDM consensus. Considering the relationship strength between experts, a new consensus measure method to manage opinion difference between experts is proposed. According to the harmony degree between experts, a consensus index (CI) is developed to measure the consensus level of LSGDM in social network. To address the uncertain situation of the unit adjustment cost, 105 an ellipsoidal set restricting the unit adjustment cost is proposed. Then, in the feedback adjustment process, a consensus model based on robust optimization is constructed to optimize the total compensation cost in the worst case. To the best of our knowledge, this is the first consensus of LSGDM in social network based on robust optimization.

The remainder of this paper is organized as follows. Section 2 reviews some preliminaries on the minimum cost consensus model, social network analysis and trust propagation in social network. In Section 3, the CRP of LSGDM in social network based on robust optimization is shown. In Section 4, a case study is discussed, which illustrates the application of the proposed consensus 115 model in LSGDM in social network. Following this, a comparative analysis and discussion are presented in Section 5. Finally, conclusions are provided in Section 6.

In this section, we first introduce the LSGDM problem. Then, the minimum 120 cost consensus model is presented. Finally, the social network analysis is briefly described, along with some related definitions. To understand LSGDM better, we describe the difference between GDM and LSGDM based on three points: the characteristics of experts, the clustering process and the consensus measure.

(1) The characteristics of experts 130 In GDM, only a few experts are usually considered. However, in LSGDM, the number of group members is greater than or equal to 20. In addition, experts in GDM are generally described as independent individuals, neglecting the objective trust relationships between them. However, because LSGDM involves so many experts, this premise rarely holds. Therefore, social relationships can 135 be widely observed among experts in LSGDM.

(2) The clustering process Because LSGDM is the more complicate process, it is difficult to solve LS-GDM with traditional GDM methods. Therefore, expert clustering is often used to classify experts with similar opinions into one subgroup. This process can 140 effectively reduce the necessary number of experts in LSGDM and improve the efficiency of consensus.

(3) The consensus measure In GDM, the consensus measure is conducted orderly on the expert and group level. However, in order to accelerate the speed of the consensus mea-145 sure, the consensus measure is usually implemented at the subgroup level in LSGDM. This step avoids massive calculations on the large-scale expert level, which further improves the efficiency of consensus in LSGDM.

Based on CRP and by considering the aggregation process of expert opinions, Zhang et al. [46] constructed the minimum cost consensus model by employing an aggregation operator: is the only independent variable in the nonlinear optimization model (1) . In order to solve o ′ i , a transformation process occurs in [46] , which ensures that the solution process is more operational.

Theorem 1 [46] By setting

Theorem 1 indicates that the optimal consensus opinion o ′ i can be derived 165 by solving the linear programming model (2) .

In minimum cost consensus models (1) and (2), the unit adjustment cost c i is deemed to be certain and already known. However, due to the diversity of experts and the complexity of LSGDM, the unit adjustment cost c i is difficult to derive, and it is usually uncertain. Therefore, in this paper, we consider the 170 value of c i to be unknown.

Social network analysis investigates the relationships between social members, such as group members, enterprises or countries [26, 8, 29] Social network analysis consists of three elements: the set of experts, the relationship between them, and the sociometric data. Detailed information is depicted in Table 1 .

(1) Sociometric data: The relationship data between experts is presented in a matrix, called the sociometric matrix. In the sociometric matrix, t ij = 1 denotes that expert e i has a direct trust relationship with e j . Otherwise, the value of t ij is equal to 0 if expert e i does not have a direct trust relationship with e j .

(2) Graph: A social network is represented by a graph in which points are connected by straight lines. In the graph, e i → e j signifies a direct trust relationship between e i and e j .

(3) Algebraic representation: This representation distinguishes between several different relationships and represents a combination of relationships.

However, the above sociometric only describes the relationship between experts, it does not accurately quantify the evaluation opinion and relationship strength between experts. Therefore, we adopt trust information with relationship strength to express the relationship information between experts in social network. 200 Liu et al. [23] introduced the basic concept of relationship strength between experts, which is provided in Definition 1.

Definition 1 [23] Trust information with relationship strength is defined in the form of a 2-tuple, and it is expressed as

where the first component t ij denotes the trust degree from experts e i to e j , satisfying the condition 0 ≤ t ij ≤ 1. If t ij is equal to 0, it implies that expert e i fully distrusts e j . In contrast, expert e i will absolutely trust e j if the value of t ij 205 is 1. The second component s ij is the relationship strength between experts e i and e j . Relationship strength represents the contact frequency between experts.

If the value of s ij is high, the relationship between experts e i and e j is closer. For convenience, T S = (ts ij ) m×m is referred to as a trust matrix with relationship strength. 

In social network, information is transitive. Thus, trust can be spread by one or more mediators. Trust information may be distorted when information is propagated through multiple mediators. So, in this paper, we only consider the case in which trust is transmitted by one mediator. Taking Fig.1 as an example, 215 there is no direct relationship between e i and e j , but trust propagation can be achieved through mediator e f . For example, through a trust propagation path e i → e f → e j , the trust evaluation from e i to e j can be reached.

To achieve trust propagation in social network, we firstpresent the concept of uninorms. Then the uninorm propagation operator and other definitions are 220 explained.

Definition 3 [36] Assume that U is a uninorm trust propagation operator defined as a mapping. Corresponding two 2-tuples (t if , s if ) and (t f j , s f j ), we have

In the process of trust propagation, an expert is more likely to share in-225 formation with an expert with whom he shares a close relationship. In other words, the relationship strength s ij as a key factor affects the trust propagation efficiency. According to Liu et al. [23] , trust propagation efficiency can be deemed a function of relationship strength.

Definition 4 [23] Assume that the relationship information of expert e i toward expert e j is ts ij = (t ij , s ij ), then the propagation efficiency from e i to e j is

where p(s) belongs to the interval [0, 1]. The greater the value of s ij , the greater 230 the value of p(s). When the contact frequency between e i and e j is 0, there is no trust propagation between them. However, trust propagation efficiency between experts e i and e j will be greatest if their relationship strength is 1.

Based on trust propagation and propagation efficiency, according to [23] , the trust value of experts e i to e j in Fig.1 is calculated as Definition 5 [23] When there are multiple trust propagation paths, the trust propagation value from e i to e j is defined as

where p pathp ij is the overall propagation efficiency for path p, and t pathp ij represents the fully propagated trust value of e i to e j .

Example 1 In Fig.3 , two paths e 1 → e 2 → e 3 and e 1 → e 5 → e 3 can simultaneously propagate trust information from e 1 to e 3 . In this situation, the trust value from e 1 to e 3 is as follows:

In this section, we present the proposed consensus problem with a strength 240 relationship and uncertain unit adjustment cost. In Section 3.1, a detailed framework regarding the CRP of LSGDM with an uncertain cost is proposed.

In Section 3.2, a new K-means algorithm is presented to classify experts into different subgroups. A consensus measure considering the harmony degree is given in Section 3.3. Finally, a minimum cost consensus model based on robust 245 optimization is provided in Section 3.4.

To discuss the consensus decision problem of LSGDM in the social network environment, this paper uses the trust evaluation value and relationship strength (1) Expert clustering: Implement Algorithm 1 to classify the experts into several subgroups based on the comprehensive evaluation among them.

(2) Consensus measure: Use Eqs. (14) and (15) to calculate the consensus index of the subgroup (CIC) and the CI of the collective, respectively.

(3) Feedback adjustment process: Use the minimum cost model based on 260 robust optimization to achieve LSGDM consensus in social network. For simplicity, the notation for this paper is as follows: 

Considering the sociality of experts, expert clustering is to classify experts with similar opinions into one subgroup, which can improve the efficiency of consensus in LSGDM when the unit adjustment of an expert is unknown. There are many different clustering methods, including fuzzy clustering methods [3] and the K-means algorithm [40] . In traditional clustering methods, such as 275 fuzzy clustering methods [3] , each expert may be assigned to many subgroups simultaneously. This does not effectively provide a convincing clustering result for LSGDM, which leaves the clustering process meaningless. However, the Kmeans algorithm can overcome the defect of some fuzzy clustering methods, which makes the clustering more reasonable as every expert is assigned to one 280 and only one subgroup. Moreover, the K-means clustering method has enough robustness to determine a clustering result [40] . It has also demonstrated high efficiency in dealing with LSGDM. Motivated by these advantages, the K-means algorithm is used to implement expert clustering in this paper.

The K-means algorithm is based on the distance difference between expert 285 and subgroup center to conduct clustering: the smaller the opinion distance between expert and subgroup center, the greater their similarity, and the more likely the expert belongs to the subgroup. Here are the basic principles of the K-means algorithm:

(1) The subgroup center cl k is initialized, then the distance value d k i calcu-290 lated.

(2) The distance between each e i and each cl k is compared in turn, and the e i is assigned to the subgroup for which d k i has the minimum value. Relationship strength, which indicates the intimacy or contact frequency between experts in social network, is one of the most important factors affecting 295 human relationships. If the relationship between e i and e j is strong, e i will have a greater comprehensive evaluation toward e j . However, if the relationship between them is weak, e i may have a lower comprehensive evaluation of e j .

Thus, relationship strength can reflect the intimacy between experts and can impact the final comprehensive evaluation among experts. Therefore, in order to 300 consider relationship strength in the clustering process effectively, we propose an expert clustering method that combines trust degree and relationship strength.

Let ts ij = (t ij , s ij ) be the relationship information given by e i to e j , then the comprehensive evaluation from e i toward e j is defined as: Generally, we select one of the CE i as the subgroup center. The similarity degree between e i and subgroup center cl k is expressed by Euclidean distance. It 310 is shown as Definition 7.

The Euclidean distance between expert e i and subgroup center cl k on trust degree and relationship strength is defined as

where ce ij denotes expert e i 's comprehensive evaluation of expert e j , 1 ≤ j ≤ m, j ̸ = i, and ce kj denotes the comprehensive evaluation of expert e j by subgroup center cl k .

Remark 3 For Definition 7, the smaller the value d k i , the larger the similarity 315 degree between expert e i and subgroup cl k . When the similarity degree is larger, expert e i is more likely to be classified into subgroup cl k .

, it is shown as follows. We suppose the number of τ is equal to 2.

(1) According to Definition 6, the comprehensive evaluation matrix is calculated as: (3) Finally, for expert e 2 , the Euclidean distances of d 1 2 and d 4 2 are calculated, respectively, giving us 4 2 , expert e 2 is classified into subgroup 4. For expert e 3 , similarly, the Euclidean distances of d 1 3 and d 4 3 are also calculated, from which we obtain d 1 3 = 0.546 and d 4 3 = 0.806. Thus expert e 3 is classified into subgroup 1 because the value d 1 3 is lower than d 4 3 .

(4) Finally, we obtain the clustering result. The first subgroup is {e 1 , e 3 } and the second is

The detailed expert clustering process for LSGDM in social network is expressed in Algorithm 1.

The expert clustering process for LSGDM in social network.

There are m experts involved in the LSGDM problem. Each expert has certain social relationship with other experts, and he gives relationship information about the other experts ts ij = (t ij , s ij ).

The relationship information matrix T S = (ts ij ) m×m , and the number of subgroup τ .

Step 1. Apply Eq. (8) to calculate the value of ce ij .

Step 2. Initial subgroup center cl k is selected from comprehensive evaluation matrix

Step 3. Utilize Eq.(9) to compute the Euclidean distance between expert e i and subgroup center cl k , and compare the value of

Step 4. The sample mean of each subgroup is used as the new subgroup centering.

Step 5. Repeat Step 2 and Step 3 until the subgroup center no longer changes. 

Using Algorithm 1, large-scale experts in social network can be classified into τ subgroups. To facilitate the subsequent consensus process, we assume that the subgroup with a large number of experts should be assigned large weight.

The weight of subgroup k can be calculated as follows:

Then, taking relationship strength into account, a consensus measure is designed to calculate the consensus degree among experts.

Let o k i , o k j be the opinion of expert e i and expert e j in subgroup k, respectively. o k i and o k j are real numbers. Let SM k = (sm k ij ) m×m be the similarity matrix of subgroup k, where sm k ij represents the similarity degree between experts e i and e j in subgroup k, and sm k ij is defined as

In addition, in social network, the relationship between experts may increase 335 or decrease the harmony degree between them. In other words, when expert e i has a strong relationship with expert e j , there is a higher harmony degree between them. However, if expert e i has a weak relationship with expert e j , the harmony degree between them will be very low, even though there is a high similarity degree. Thus, the harmony degree between experts e i and e j in 340 subgroup k is defined as

The harmony degree between experts e i and e j in subgroup k is defined as:

The larger the value of h k ij , the higher the harmony degree between e i and e j in subgroup k. Obviously, H k is a symmetric matrix where the elements on the main diagonal are 0.

Let HC = (hc ij ) m×m be the harmony degree matrix of the collective. Using the weighted averaging(WA) operator, it can be computed as

When the number of experts in subgroup k is m − n, but the number of 

In the aggregation process, we suppose that ω 1 = 0.3, ω 2 = 0.4 and ω 3 = 0.3.

Because the three matrices have different dimensions, the dimensions of the matrix (h 1 ij ) 3×3 and (h 2 ij ) 4×4 are extended to 5 × 5. Therefore, the harmony degree matrix of the collective can be calculated as 

The CIC of subgroup k is defined as:

Similarly, the CI of the collective can be calculated as: 

In this section, we propose a feedback adjustment process that consists of 

When the CI among experts is unacceptable, the opinion of each expert 365 needs to be further adjusted according to the minimum cost model. In the exiting minimum cost consensus models, it is assumed that the precise unit adjustment cost is known. In reality, however, the unit adjustment cost is normally difficult to obtain and is uncertain. When the unit adjustment cost is uncertain, experts always want to get the highest unit adjustment cost from 370 the moderator. So, the total compensation cost paid by the moderator may be very high, which is the worst-case consensus problem. In order to deal with this problem, we propose to minimize the total compensation cost in the worst case, which is the consensus problem based on robust optimization.

In the robust optimization consensus problem, a method of dealing with this uncertain cost is proposed. Assuming that we know the nominal value of an uncertain unit cost, it is denoted as c * i . Rather than assuming that c i is in an interval [20] , we suppose that c i belongs to an ellipsoidal set. Every point of the ellipsoid is a possible value of unit cost c i . The ellipsoidal set is shown as:

where δ i is a parameter that represents the deviation between the uncertain 375 unit adjustment cost and the nominal value.

In the minimum cost model (2), the unit adjustment cost c i is certain. We mainly focus on minimizing the total compensation cost. However, in the robust optimization consensus problem, we minimize the total compensation cost based on robust optimization. Hence, the model of the robust optimization consensus 380 problem is defined as It can be observed that the proposed model is more complex than the minimum cost consensus models (2) . Firstly, we consider the uncertainty of the unit adjustment cost in the proposed model. The unit adjustment cost is assumed in an ellipsoidal set. Moreover, our model is based on robust optimization where, 400 rather than optimizing the minimum total cost, one optimizes the worst-case total cost with respect to an ellipsoidal set. Finally, unlike the linear model (2), the robust optimization model (16) is a difficult problem to solve.

After building the model of the robust optimization consensus problem, we next discuss how to solve this model when both unit adjustment cost and consensus opinion are unknown. To deal with this problem, a robust counterpart is proposed to determine the total compensation cost and the optimal consensus opinion of every expert. 

Proof 1 Due to the unit adjustment cost, c i belongs to ellipsoidal set Γ, that is c i = c * i ± δ i . The objective function of the robust optimization consensus model consists of two parts: a certain part and an uncertain part. Thus, the

where the variance of random part ξ is

Consequently, the robust optimization value of M inmax

The idea of dealing with uncertainty is as follows. In Eq. (19) , the greater the variance, the larger the deviation 415 degree between the uncertain unit cost and the nominal value. It is the worstcase uncertainty. Hence, we take the worst situation where the value of ξ is equal to V ar 1/2 (t) in Eq. (19) . With this approach, the objective function of the robust optimization consensus problem is

the robust counterpart of the robust optimization consensus is shown as model 420 (17) .

The robust counterpart of the robust optimization consensus can then be solved using MATLAB. Solving model (17) 

In the following, we discuss how to estimate the unit adjustment cost of expert e i . 

Considering the first principle, the unit adjustment cost is limited to an ellip-soidal set, it has been bound. Therefore, we have the following result

In addition, the detailed CRP of LSGDM in social network is shown in Step 1. Using Eq. (7) to calculate the missing value of T S matrix, then we obtain the complete T S matrix.

Step 2. Using Algorithm 1 to classify experts who have similar comprehensive evaluation into one subgroup. Then, large-scale experts can be classified into τ subgroups.

Step 3. Set χ = 0, utilizing Eq.(14) to calculate the consensus index of subgroup CIC k χ .

Step 4. Utilize Eq.(15) to compute the consensus index CIχ. If CIχ ≥ σ, go to step 5.

Otherwise, go to the next step.

Step 5. Select the subgroup with lowest CIC k χ to adjust. Using model (17) to determine the optimal consensus opinion o ′ i,χ and total compensation cost Bχ, then estimate the unit adjustment cost c i,χ . In the next iteration, let o i,χ+1 = o ′ i,χ , χ = χ + 1 and B = B χ−1 + Bχ, then return to Step 2.

Step 6. When the acceptable consensus index is reached, output the consensus index of subgroup CIC k χ , consensus index CIχ, optimal consensus opinion o i,χ and total compensation cost Bχ.

In January 2020, COVID-19 broke out in Wuhan, China. The virus is highly infectious, and the number of infected people in China is soaring. According In a compensation negotiation, the market manager is the moderator while tenants play the role of DMs. Naturally, the market manager needs to negotiate with the tenants about the compensation price. In the negotiation process, the 460 market manager pays a unit negotiation cost to the tenants so as to change their initial compensation prices. However, due to the COVID-19 outbreak, the market manager lacks experience in determining the unit negotiation cost. The unit negotiation cost is difficult to obtain and is unknown. In this situation, each tenant seeks to receive the highest unit negotiation cost from the market 465 manager. So, the total negotiation cost paid the market manager may be very high, which is the worst-case consensus problem. Therefore, we intend to minimize the total negotiation cost in the worst case, which is the consensus problem Table 2 . Then, some relevant parameters and assumptions are set as

(1) It is assumed that the propagation path greater than or equal to two mediators is invalid. This is because the relationship strength value ranges from 0 to 1. The more mediators involved, the lower the propagation efficiency.

Therefore, in this case, there should be only one mediator involved in trust propagation.

(2) The acceptable consensus threshold σ is 0.88, and the maximum iteration number χ * is set to 7.

In the robust optimization consensus problem, 20 tenants in social network are classified into four subgroups. The detailed expert clustering result is shown in Table 4 . From Table 4 , we can see that tenant e 11 is classified into a separate 485 group. He does not have any neighbors who share similar opinions with him.

Hence, we consider tenant e 11 to be a non-cooperative expert. He was not invited to participate in this compensation negotiation.

By aggregating the harmony degree of cl 1 , cl 3 and cl 4 , the harmony degree 

of the collective HC 0 , we get 

The consensus index of subgroup CIC k 0 can then be calculated using Eq. Finally, consensus process is terminated after three iterations. The detailed 515 iteration process is described in Table 5 .

In this section, some comparisons with existing methods are provided to examine the superiority of the proposed method. A discussion of the comparison results is given. Finally, cost comparisons of three cases are analyzed to 520 investigate the advantages of the robust optimization method.

In this subsection, to demonstrate the effectiveness of the proposed robust consensus problem, we compare the result with existing methods and algorithms.

The comparative analysis is conducted in the context of the same case study. (1) M1: this method is proposed by Wu and Chiclana [32] . When a consensus is achieved in social network, the trust information between experts is considered. A trust consensus is proposed by Wu and Chiclana [32] in which a consensus is reached in an incomplete social network. However, this only considers the trust degree in social network; relationship strength is not taken 535 into account. In the proposed robust consensus model based on trust propagation, the initial incomplete trust network can be extended to a complete trust network. Nevertheless, in the method proposed by Wu and Chiclana [32] , both trust propagation and relationship strength are ignored. By applying this method to the case, a detailed consensus iteration result can be obtained (see 540   Table 6 ).

(2) M2: this model, proposed by Wu et al. [33] , explores social network GDM with trust propagation. It builds trust relationships between experts and assumes that trust can be propagated. However, it does not consider relationship strength in trust propagation and consensus processes. By applying Wu et 545 al.'s [33] method to the case, we can derive a detailed consensus iteration result (see Table 7 ).

(3) M3: this grey clustering algorithm method developed by Liu et al. [24] classifies experts with similar similarity degrees into a subgroup. In this algorithm, similarity degree is based only on fuzzy preference values and self-550 confidence levels; it does not consider the harmony degree between experts. We can take the harmony degree into account in the grey clustering algorithm [24] and implement it in the case. The expert clustering result is: Table 8 . Table 9 .

(5) M5: this method, a product line design proposed by Bertsimas and Misić [5] , uses the nominal model and robust model to discuss the expected revenue. The worst-case revenue also is explored in this method. When the worst case occurs, the robust optimization method is used to minimize the total negotiation cost. The main difference between our method and M5 is 570 the utilization of the worst case in the goal function. In this paper, we adjust the opinions of experts based on robust optimization without implementing a worst-case method into the process.

To carry out the worst-case method proposed by Bertsimas and Misić [5] in this case, the goal function in model (16) is changed to

where c * i is the nominal value of an uncertain unit cost, δ i is the ellipsoidal parameter. Using the worst-case method to adjust the opinions of tenants, the 575 detailed consensus iteration result are depicted in Table 10 . 

where E is the expectation and α represents the probability. B is the fixed budget limitation predefined by the moderator. By solving model (22) to adjust the opinions of the tenants, the detailed consensus iteration result are shown in 585 Table 11 .

To verify the effectiveness and necessity of the proposed method further, we obtained the consensus iteration and cost comparison for different methods. {6.14, 6.14, 6.14, 6.14, 6.14, 6.15, 6.15} 0.874 43.00 3 {11. 25, 11.25, 11.25, 11.25, 11.25, 11.25} 0.884 67.50 In Fig.6 , it is clear that the consensus-reaching speed for the proposed method is dramatically faster than methods (1)- (6) . Moreover, the proposed method needs only three iterations to achieve consensus, while the number of iterations for methods (1)-(4) are always greater than three. This indicates that 595 in real LSGDM, considering trust propagation and relationship information is more favorable for achieving consensus. That is to say, using the social network of experts to reach consensus is suitable for real LSGDM cases.

In addition, Fig.6 also reflects how, compared with the grey clustering algorithm [24] and ARC method [25] , the new K-means clustering algorithm is more 600 efficient at classifying experts and improves the consensus negotiation process.

Regarding methods (5) and (6), although the number of iterations is the same as in the proposed method, the moderator will pay more total negotiation costs to the experts. This means that, when facing the same consensus task, the proposed method can greatly decrease the cost expenditure for moderator.

As can be seen in Fig.7 , the total negotiation cost of the proposed method is lower than in methods (1)- (6) . To achieve consensus using methods (1)-(6), the resource will occur huge consumption. This signifies that considering the so- conservatism of robust optimization [13] , the moderator will pay more in total negotiation costs than he would with robust optimization if it is used to deal with LSGDM consensus in social network.

Therefore, these advantages can mainly be contributed to the unique characteristics of the proposed method.

(1) In the expert clustering process, we propose a definition for comprehensive evaluation to express opinion information and a Euclidean distance to measure the similarity degree between experts. emergency decision making, an optimal decision usually require a relatively high consensus threshold [42] . Thus, how to use the proposed method to achieve consensus in large-group emergency decision making is a problem worth discussing further.

In this paper, to realize LSGDM consensus in social network, we describe three consensus cases. The first is the nominal case. It is assumed that the unit adjustment cost is the nominal value c * i . The second is the worst case, which deems that when the unit adjustment cost is uncertain, and each expert intends to get the maximal unit adjustment cost, the total compensation cost paid by 670 the moderator will reach the maximum. The third is the robust optimization case. Its goal is to minimize the total compensation cost in the worst case.

Thus, the total compensation costs of the three cases are calculated as

In order to discuss the advantage of robust optimization in dealing with uncertain cost, we propose two indexes to quantify the benefit of robustness.

The first is the worst-case consensus loss (WCCL). The WCCL is defined by the nominal case and the worst case as

where B n is the total compensation cost in the nominal case. Additionally, B w denotes the total compensation cost paid by the moderator when the worst-case 675 consensus happens. The WCCL measures how the moderator's total compensation cost changes when the consensus transfers from the nominal case to the worst case.

The second index we discuss is the relative improvement (RI). The RI is defined by the worst case and the robust optimization case as

where B r is the total compensation cost in the robust optimization case. Compared with the WCCL, RI measures the cost improvement of the robust opti-680 mization case.

When the ellipsoidal parameter δ is set as different values, we compute the cost of each iteration under three cases. In addition, for each δ, we compute the WCCL of the nominal case and the RI of the robust optimization case. The results are presented in Fig.8 and Table 12 .

As can be seen in Fig.8 , when the consensus cases are fixed in the same iteration, the consensus cost increases as the value of the ellipsoidal parameter δ increases. This finding implies that resource compensation will increase if the uncertain cost is in a bigger ellipsoidal set. Moreover, in each iteration, the nominal case has the lowest compensation cost. When the ellipsoidal parameter 690 δ is the same, the total compensation cost in the worst case is higher than the robust optimization case. This shows that the robust optimization method can effectively reduce the loss of total compensation cost when the unit adjustment cost is uncertain. Table 12 shows how WCCL and RI vary with the value of δ. We can see 695 from the table that for the small uncertainty (δ < 10), the loss generated in the worst case is relatively lower (e.g., for δ = 7, the WCCL is 11.67%.), and the robust optimization method provides a slight advantage in reaching the consensus process. However, for the larger value of the ellipsoidal parameter δ (δ ≥ 15), the WCCL increases distinctly, and the robust optimization method has huge 700 a advantage in reducing the total compensation cost. Therefore, when the unit adjustment cost is uncertain, the robust optimization method can minimize the total compensation cost of the worst case.

In this paper, we researched LSGDM in social network and proposed a novel (2) Considering the relationship strength between experts, a harmony degree measure method considering both opinion value and relationship strength is presented. A CI reflecting the harmony degree between experts is devised to further.

In addition, in some real LSGDM problems, experts may exhibit the game behavior for pursuing their personal interest. This paper only discusses the influence of social relationship on the CRP of LSGDM. However, the game behavior of experts is ignored in the CRP. Therefore, in future work, another 740 interesting issue to discuss would be the game behavior of experts and how that influences CRPs.

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This work was supported by the National Natural Science Foundation of Chi-