Abstract
This study proposes a largescale group decisionmaking (LSGDM) consensus model considering the experts’ adjustment willingness based on the interactive weights’ determination, which can be applied to an LSGDM problem through a case of earthquake shelters. The main contributions of our research are of three aspects as follows. First, the determination method of the interactive weight, which obtains the DMs’ attitude towards the decisionmaking results, is presented. The subgroups’ weights are calculated, and the unit adjustment cost for each DM is defined. Second, we introduce an objective consensus threshold by the mean and variance of the consensus level for each subgroup. Subsequently, an identification rule is performed to determine the DM to be adjusted with the large difference and the low adjustment cost. And we developed a novel consensus adjustment method that takes the DMs’ adjustment willingness into account to retain as much original information as possible. Thirdly, in order to reduce the subjectivity of the preset consensus threshold and the maximum number of iterations, an objective consensus termination condition that combines the current group consensus level and the consensus adjustment rate is put forward. Finally, the proposed model has demonstrated its effectiveness and superiority based on the comparative and sensitive analysis through a practical example.
1. Introduction
Group decisionmaking (GDM) is a process in which, under certain constraints, some experts or decisionmakers (DMs) obtain the optimal from several feasible alternatives by expressing their opinions or preferences [1]. With the development of social, science, and technology, the complexity of decisionmaking events has becoming increasingly high [1], the ambiguity and uncertainty of the decisionmaking environment and context have become increasingly high, and the number and diversity of DMs participating in decisionmaking issues have increased rapidly. GDM has developed into largescale group decisionmaking (LSGDM) [2–6], multiattribute LSGDM [7–10] and so on. Compared with GDM, the number of experts and DMs involved in LSGDM is larger, usually more than 20 [4, 9, 11, 12]. The differences in DMs’ backgrounds and knowledge are greater, and thereby the consensus level among DMs is lower.
In recent years, LSGDM and fuzzy mathematics, game theory, computers, information technology, and other theories are being integrated and developed. The research of multiattribute LSGDM mainly focuses on the expression of DMs’ or experts’ preference information [13, 14], clustering [12], aggregation of group preference information [2, 13–16], determination of weight [3, 17, 18], and consensus reaching process (CRP) [2, 3, 9, 14, 19]. Many methods of expression of experts’ preference information have proposed, such as fuzzy preference information, linguistic preference information, and random preference information. Liu et al. [20] transformed intervalvalued intuitionistic fuzzy numbers into singlevalued numbers and then proposed a twostage regularized generalized canonical correlation analysis decisionmaking method based on multiblock analysis to address the MALSGDM problem in the intervalvalued intuitionistic fuzzy environment. Bai et al. [11] developed an LSGDM model with cooperative behavior based on social network analysis, considering the propagation of decisionmakers’ preferences by considering the propagation of DMs’ preferences. Zhen et al. [21] proposed a computational model based on the use of extended linguistic hierarchies and used multigranularity linguistic distribution to provide interpretable aggregate linguistic results to experts in order to maximize information retention.
To reduce the dimension and complexity of the decisionmaking process, many clustering methods have been proposed and applied, such as the kmeans clustering method [4, 12], fuzzy cmeans clustering method [22], vector spacebased clustering method [23], the and transitive closure clustering method [5]. Several researchers have proposed some novel clustering methods that can be used in LSGDM problems from different perspectives. For instance, Du et al. [6] developed a new clustering method considering both trust relationships and opinion similarity in a social network context. In this study, the kmeans clustering method is utilized to reduce the dimension of the LSGDM problem.
However, despite a number of LSGDM methods having been proposed, these methods are used in specific situations. For instance, in Liu et al. [24]’s research, DMs cannot give complete and accurate evaluation information at once, and it is therefore difficult for decisionmaking groups to reach a consensus at once. In addition, some proposed LSGDM models include the determination process of the DMs’ weight, but the weight of the DM is relatively simple, that is, the dynamics of the decisionmaking process are not taken into account. Next, CRP, a rather critical and essential process, reduces and even eliminates the conflicts of group and further improves the effectiveness and rationality of decision results. But in the existing literature, the threshold determination of many CRP is very subjective, which is not conducive to the objectivity of decisionmaking results. Therefore, our research intends to answer the following questions:(a)How can we not only ensure that more DMs participate in the decisionmaking process but also make the results represent the opinions and attitudes of more DMs?(b)How to obtain a more scientific and accurate consensus threshold?(c)Under what conditions can the consensus adjustment process terminate automatically?
Based on the above analysis, an LSGDM consensus model considering the experts’ adjustment willingness based on the interactive weights is proposed in this study. Its innovations and contributions are shown as follows:(1)A novel method of weight determination has been developed, which considers the DMs’ attitude towards the decisionmaking results, thereby ensuring the effective participation of DMs. Moreover, to improve the rationality of LSGDM, a harmonious degree is taken into account in the calculation of subgroups’ weight.(2)In the consensus measure process, a more reasonable consensus measure method is introduced, which considers both the differences between the DMs’ opinion and the group opinion and the harmonious degree. In the consensus feedback process, an identification process is presented, which considers the unit adjustment cost for each DM. Then, a new adjustment process is constructed, which the evaluation information is less distorted or lost by considering the experts’ adjustment willingness.(3)An objective calculation method of the consensus threshold is presented from the perspective of the mean and variance of the consensus level, and then a termination condition that considers both the current consensus level and the consensus adjustment rate is designed to objectively terminate the CRP. It not only compares the current consensus level with the consensus threshold but also compares the consensus adjustment rate. As a result, this method can address the subjectivity and unreasonableness of the preset consensus threshold and the maximum number of iterations to a certain extent.
The remainder of this paper is organized as follows: In Section 2, we briefly summarize the recent literature review from the following four aspects: the aggregation of group preference information, the interactive process, the weight determination method, and the CRP. Section 3 introduces the multiattribute LSGDM problems, and then the kmeans clustering method is used to classify the DMs into several subgroups. The determination methods of the DMs, the attributes, the subgroups, and the unit adjustment cost are shown in Section 4. Section 5 presents the proposed consensus model, which includes the consensus measurement process and the consensus feedback process and explains how its framework is conducted. In Section 6, the proposed model is applied to a case study to illustrate its effectiveness and rationality. The comparative analysis and the sensitivity analysis are performed in Sections 7 and 8, respectively, to further validate the proposed model. We draw the conclusions of this research in Section 9.
2. Literature Review
In this study, we summarize the existing literature from the following four perspectives: the aggregation of group preference information, the interactive process, the weight determination method, and the CRP.
2.1. Research on the Aggregation of Group Preference Information
Many scholars are interested in the aggregation of group preference information, which is mainly to obtain the results of group clustering. Xu et al. [19] presented a twostage method to support the CRP. The first stage classifies and obtains several subgroups by utilizing the selforganizing maps, and then an iterative algorithm is proposed to obtain the group preference for each subgroup. The second stage treats the group preference of each subcluster as the representative preference and collapses each subcluster to form a smaller and more manageable group. Liu et al. [25, 26] utilized the idea of principal component analysis (PCA), regarded attributes and decision makers as intervalvalued intuitionistic fuzzy variables, transformed them into several independent variables, and then combined them with the traditional preference aggregation operator to obtain a decisionmaking method to solve the complex multiattribute LSGDM problems. Chen et al. [27] developed a twotier collective opinion generation framework integrating professional knowledge structure and risk preference to generate collective preference assessment, and thereby to obtain an accurate and reliable alternative ranking. In this paper, we utilize weighted arithmetic averaging (WAA) to aggregate each DM’s opinion.
2.2. Studies about the Interactive Process
In reality, DMs often cannot give complete and accurate evaluation information at once, it is difficult for decisionmaking groups to reach a consensus at once, and thereby, the evaluation information needs to be continuously supplemented and adjusted. Therefore, the interactive process is not only necessary but significant to avoid the limitations of the DMs’ opinions, improve the effectiveness of the DMs’ participation, and ensure the rationality of the decisionmaking results.
Zeng et al. [28] developed an interactive procedure for GDM based on intuitionistic fuzzy preference relations, in which the similarity measures between the collective preference relation and the intuitionistic fuzzy ideal solution are used to rank the alternatives. Liao and Xu [29] established an optimization model for determining the weight and an interactive model of a multiattribute decisionmaking problem with hesitant fuzzy information to make the decision more reasonable. Ding et al. [30] proposed an interactive method to deal with the probability hesitation fuzzy multiattribute GDM problem with incomplete attribute weight information, which can reflect the DMs’ subjective desirability and reduce the effect of unfair arguments on the decision results. Therefore, in our research, the interactive process is utilized for the experts’ weights determination to improve the rationality of the decisionmaking results.
2.3. The Aspects of the Weight Determination
The determination of weight is a hot issue in LSGDM problems, which includes mainly the weights of experts’ or DMs’, the weights of the attributes of the alternatives, and the weights of the subgroups. For the determination method of experts or DMs, Meng et al. [14] integrated objectively cooperation networks and references network of DMs to construct a directed and weighted social network, and then obtained the DMs’ relative weights. Wan et al. [31] show that DM’s weight can be obtained through a programming model by minimizing the distance between individual semantics and collective. Liu et al. [17] proposed a double weight determination method of experts by utilizing mathematical programming and information entropy for multiattribute LSGDM in an intervalvalued intuitionistic fuzzy environment. Wan et al. [32] developed a similarity determination method to calculate the weight for each DM and constructed two programming models to obtain the optimal weight for each attribute.
Related to the studies of the attributes’ weight, Zhong et al. [4] developed an approach to determine the attributes and their weights based on the social media data relevant to decisionmaking problems by using the term frequencyinverse document frequency (TFIDF) method. It considers both the experts’ opinions and the views of stakeholders. In this study, the interactive process is introduced in the determination process of the DMs’ weights. For the subgroups’ weights, Liu et al. [8] set an equal weight for different subgroups, while Xu et al.[1] determine the subgroups’ weights according to the size of the subgroup.
In this study, by considering the DMs’ attitude towards the decisionmaking results (i.e., satisfaction degree), the DMs’ weights are updated and obtained, and then the attributes’ weights are obtained within the subgroups. The weights of the subgroups are calculated and updated by considering the number within the subgroup and the level of the subgroup’s satisfaction degree (i.e., harmonious degree).
2.4. CRP Studies
CRP is a rather critical and essential segment in LSGDM problems [33, 34], which is reducing and eliminating the conflict of a group and improving the effectiveness and rationality of decision results. Zhong et al. [4] presented a multistage hybrid consensusachieving model by integrating both cardinal consensus and ordinal consensus and applied it in the scene of the selection of a hotel for the centralized isolation of entry personnel during the COVID19 epidemic. For the consensus feedback process in the CRP, experts may not tolerate their opinions being modified unrestricted during the CRP. Hence, all experts have an accepted modification for their opinions, which can be presented as the adjustment willingness. However, few studies have focused on the experts’ adjustment willingness. Zhong et al. [5] proposed a nonthreshold consensus model, which includes an objective termination condition for CRP. It can reduce the subjectivity of the predefined consensus threshold and the maximum number of iterations to a certain extent. In addition, Wan et al. [35] developed a novel twostage CRP method considering DM’s willingness to modify preference information.
Therefore, our research presents an objective calculation method of the consensus threshold from the perspective of the mean and variance of the consensus level and then develops a termination condition that considers both the current consensus level and the consensus adjustment rate to objectively terminate the CRP.
2.5. Research on the Case Application
Many scholars have proposed many LSGDM approaches from the perspective of practice and application. Xiao et al. [36] established the civil engineering construction contractor selection framework in the LSGDM environment by considering the interaction within and between the management layers of the consensus model. Chen et al. [37] determined passenger demands and evaluated their satisfaction by using a combination of online review analysis and LSGDM based on a case study of a highspeed rail system in China.
3. Preliminaries
3.1. The Multiattribute LSGDM Problems Description
LSGDM is the process of selecting the best option from the opinion of many DMs, who express their opinion based on the decisionmaking information provided for alternatives [11]. Accordingly, let X = {x1, …, xp, …, xP}(P ≥ 2) be the set of alternatives, E = {e1, …, em,…, eM}(M ≥ 20) be the set of experts and DMs, and F = {f1, …, fn, …, fN}(N ≥ 2) be the set of attributes for each alternative.
First, the DM em provides his or her evaluation information Qm = ()P × N(m = 1, …, M), where represents the evaluation value of the attribute fn on the alternative xp for the DM em. Then, the DM em provides his or her allowed modification values and , which represents, respectively, the DM em is allowed to modify the positive and negative range of the evaluation information they provide. It is noting that the allowed modification values represent the DMs’ adjustment willingness and the values of and are both positive. For instance, a DM em provides the values of , and , the modification value of can be more acceptable in the interval [max(, 0), min(, 1)]. Note that the value of should be in the range [0, 1] before and after adjustment. The greater the value of + , the lower the difficulty of adjusting the evaluation information, and the higher the concession degree of the DM em in order to reach group consensus. However, if the adjustment value exceeds the allowed modification range of the DM, it will pay an enormous adjustment cost, and the evaluation information of the DM will be forced to change, resulting in information distortion. Therefore, this situation is not considered in this paper.
In this paper, the weight vectors of attribute for each alternative are denoted as wm = [, …, , …, ], where represents a weight value of the attribute fn that the DM em provided according to his or her knowledge and experience, 0 ≤ ≤ 1, and . The set of the DMs’ weights is denoted as Wt = {m = 1, …, M}, where means the weight value of the DM em participating in the subgroup given at the tth stage. Clearly, W1 is the initial set of the DMs’ weights.
Generally, the LSGDM process usually involves the following four stages: Stage 1. Clustering. In order to reduce the complexity of the LSGDM problem and the calculation process, a clustering method is generally utilized to divide all DMs to several subgroups according to some rules. In this study, the kmeans clustering method is utilized according to the opinion similarity. The details are shown in Section 3.2. Stage 2. Aggregate the opinion. The weighted arithmetic averaging (WAA) operator is usually used to aggregate each DM’s evaluation information to a subgroup’s decision matrix and each subgroup’s decision matrix into the collective decision matrix [38]. For a LSGDM problem, suppose that is the em’s weight in the subgroup Gk, is the Gk’s weight. Then, the aggregation process can be derived as where meets 0 ≤ ≤ 1 and = 1. where meets 0 ≤ ≤ 1 and = 1. Stage 3. The CRP. Due to the large number and the complex background of DMs in the subgroup, the consensus level is lower. It aims to obtain an acceptable consensus level. If the consensus is not reached, then the consensus feedback process should be executed. The details are shown in Section 5. Stage 4. Selection process. After obtaining the collective decision matrix by equation (2), the collective opinion score s(xp) for each alternative xp is derived as where ∈ [0, 1] is the weight of attribute fn for the group, and = 1. ’s calculation equation is shown in Section 4.2 by equation (22).
3.2. The DMs Clustering Process
In this study, all DMs are classified as K subgroups by using kmeans clustering method. The algorithm is given as Algorithm 1.

In this paper, K meets the condition 2 ≤ K ≤ . The number of the DMs is denoted as in the subgroup Gk.
4. Several Important Concepts in This Study
4.1. The Determination of the DMs’ Weights and the Attributes’ Weights
There are many factors that affect the determination of the DMs’ weights, such as the subjective degree of the DMs’ opinion, DMs’ attitude towards the overall or intragroup opinion, and so on. Apparently, the higher the subjective degree of the DM’s opinion or preference, the lower the objectivity of the DM’s decision results, the smaller the influence of the DM on the final results of decisionmaking, and the DM’s weight value therefore should be decreased to some extent. Again, the higher the DM’s attitude towards the overall or intragroup opinion or preference, the more satisfied the DM is with the clustering results, the greater the DM’s recognition of the results of group decisionmaking, and the DM’s weight value therefore should be enhanced to a certain extent. In this study, the determination methods of the DMs’ weights and the attributes’ weights are developed in Algorithm 2.

It is noted that the purpose of the Step 10 is to obtain the attitude of DMs, i.e., the satisfaction degree, and then to adjust the weights of DMs according to it. The decision information is fed back to the DMs, and then their attitude towards the decision result is obtained. The step, which is necessary and significant, cannot only improve the participation of DMs in the decisionmaking process and better reflect the attitude of DMs, but make the decisionmaking results more scientific and reasonable.
4.2. The Weight Determination for Each SubGroup
There are several factors that affect the weight value of the subgroup, such as the number of the subgroup and the harmonious degree of the subgroup. If the subgroup has more DMs, the higher weight value should be given to the subgroup. Conversely, the fewer xDMs in the subgroup, the lower the weight value of the subgroup. Moreover, the greater the harmonious degree of the subgroup, the higher the support of the DMs in the subgroup for the decision results, the larger the satisfaction degree of the DMs in the subgroup with the clustering, and the weight value of the subgroup should be increased appropriately. The algorithm for the weight determination of the subgroup has been developed in Algorithm 3.

According to the weight of each subgroup and the weight of the attribute for each subgroup, the collective attribute weight vector wG = [, …, ] is derived as
4.3. The Determination Method of the Unit Adjustment Cost
In the existing studies, the unit adjustment cost for each DM sometimes is given in advance [39–41], while others is calculated according to some rules [42]. For instance, Labella et al. [42] developed an objective metric based on the cost of modifying experts’ opinions to evaluate CRPs in GDM problems, which is based on two novel minimum cost consensus (MCC) models that consider the distance of the DMs to the collective opinion and also ensure the minimum consistency among DMs.
However, in reality, the unit adjustment cost for each DM is many factors involved. DM may have different perspective for the same problem under the different context. Moreover, DM may have different expectation for the LSGDM result, and then have different attitude. Therefore, the unit adjustment cost must be related to the DMs’ individual characteristics. In this study, we determine the unit adjustment cost of each DM according to two factors: the DMs’ adjustment willingness and and the satisfaction degree . For the and , the larger the value of + , the higher the concession degree of the DM em in order to reach group consensus, the lower the difficulty of adjusting the evaluation information, the unit adjustment cost of em should be smaller. On the contrary, if the lower the value of + , the unit adjustment cost of em should be larger. For the , the larger the value of , the more satisfied the DM em is with the current decisionmaking result, the less the difficulty of adjusting the em’s evaluation information to the subgroup collective decision result, and the unit adjustment cost of em should be smaller. Conversely, if the value of is smaller, the unit adjustment cost of em should be higher.
As aforementioned, in this paper, we define the unit adjustment cost for each DM as follows.
Definition 1. Suppose that the unit adjustment cost of em is cm, the allowed modification values of em are and , and the satisfaction degree of em is .where is a scale factor and represents the proportion of the allowed modification range. Apparently, cm increases monotonically with respect to (), and decreases with respect to .
Theorem 1. The value of cm is in the interval of [0, 1].
The proof of Theorem 1 is shown in Appendix A.
5. The Proposed Consensus Model
The CRP mainly includes two parts: consensus measure and consensus feedback. The details of the method of consensus measure are shown in Section 5.1, and Section 5.2 presents the consensus feedback process.
5.1. Consensus Measure
This procedure aims to judge whether an acceptable consensus level among group is reached or not. Clearly, in reality, the collective consensus level is related not only to the differences between the DMs’ and the subgroups’ opinion but to the attitude for each DM in a subgroup, i.e. the harmonious degree of a subgroup. Therefore, we define the consensus measure method as follows.
Definition 2. Suppose that Qm = ()P × N is the evaluation information of the DM em, = ()P × N is the evaluation information of the subgroup Gk obtained by equation (1), and represents the harmonious degree for the subgroup Gk by equation (19). The differences between the DM em’s and the subgroup Gk’s opinion can be derived aswhere the greater the value of ∈ [0, 1], the larger the deviation between em and Gk. And, the differences within the subgroup Gk’s opinion can be presented aswhere the larger the value of ∈ [0, 1], the lower the opinion similarity in the subgroup Gk. The consensus level CLk in a subgroup Gk is derived asAccordingly, the group consensus level GCL can be calculated bywhere GCL meets 0 ≤ GCL ≤ 1. Obviously, the greater the value of GCL, the higher the consensus level between the group. If GCL = 1, it means that a complete consensus has been reached, however, is almost impossible. Hence, soft consensus is generally a rule for LSGDM problems [43–45].
∈ [0, 1] is usually a preset consensus threshold. If GCL ≥ , then it means that an acceptable consensus level has been reached among the group. Otherwise, consensus feedback is an imperative process to improve the consensus level, and let GCL0 = GCL. Consequently, the determination method of is important for consensus process. Generally speaking, the consensus threshold is composed of the level of each individual, which indicates that the overall threshold must reflect the willingness of the individual. Therefore, without losing generality, the value of is obtained from the perspective of mean and variance.
Definition 3. Suppose that and are the reasonable thresholds, the calculation process is shown aswhere = CLk and = represent the mean and standard deviation of the consensus level CLk(k = 1, …, K), respectively. We define that the consensus threshold is the average value of and , namely,The determination method of the consensus threshold is to make the setting of the consensus threshold more objective from the perspective of the current consensus level. In reality, the consensus level is related to the experts’ evaluation information, which means the consensus level is uncertainty in the group. Therefore, to reduce the chance, the calculation of the consensus threshold is by weighting two reasonable thresholds in this study, which can reduce the subjectivity of the predefined it to a certain extent.
5.2. Consensus Feedback
Consensus feedback aims to obtain a highconsensus level in the group. Generally, the feedback process includes two parts: identification and adjustment.
5.2.1. Identification Process
It aims to determine the subgroup and the DM to be adjusted, which have the maximum differences. First, the determination of the subgroup to be adjusted is necessary. Due to the consensus level involves the difference between opinions and harmonious degree and harmonious degree does not change, the determination method of the adjusted subgroup focus on the difference between opinions, and the equation is = max{}(k = 1, …, K). Then, the DM em to be modified is obtained in the subgroup Gk by the cm = min{em ∈ Gk}. Therefore, the DM em of the subgroup Gk should be modified in the adjustment process. It should be noted that a DM can be adjusted at most once in the feedback process.
5.2.2. Adjustment Process
This process goals to modified the DMs’ evaluation information according to some advices. Most of existing studies generally are given on adjustment strategies based on mathematical analysis. In this study, the DMs’ evaluation information is modified according to both the subgroup’s opinion and the DM’s adjustment willingness. Detailed procedure is given in Algorithm 4.

The CRP is an iterative process that should be terminated by a condition. For the existing consensus model, a consensus threshold and the maximum iterative number generally should be set subjectively in advance [2, 4, 9, 39–42, 44–46]. In this study, we refer to the concept of consensus improvement rate proposed by Zhong et al. [5], the consensus adjustment rate is defined to judge whether the adjustment process is terminated. The detailed procedure is performed as follows.
We suppose that in the tth stage, the consensus adjustment cost ACt is defined as
Then, the total adjustment cost TACt before the tth stage can be derived as
Definition 4. (see [5]).Suppose that consensus adjustment rate is denoted as CARt before the tth stage, which considers both the adjustment cost and the consensus improvement rate. The equation can be given as follows:where GCLt represents the group consensus level after adjusting the tth round. If CARt ≤ CARt + 1, it means the consensus adjustment rate is well, and then the CRP can be proceed; otherwise, the CRP should be terminated, and the (t + 1)th adjustment should be restored. Thus, the tth adjustment is the last adjustment. To compare the termination condition, i.e., consensus adjustment rate CARt, the consensus feedback process for at least one round.
5.3. The Framework of the Proposed Consensus Model
As the above aforementioned, the procedure of the proposed consensus model can be summarized as follows, and the framework can be shown in Figure 1 (Algorithm 5).
6. Case Study
6.1. Case Background
Earthquake is a natural phenomenon. According to statistics, the number of earthquakes in China from 2010 to 2020 was 6029, of which 48 were above magnitude 6.0. The earthquake will not only cause huge losses to the regional economy, but also cause huge casualties. Therefore, it is very critical to prepare before the earthquake to reduce casualties.
Tangshan, in Hebei Province of China, is an earthquake prone area. A county in this city plans to build several earthquake shelters. However, due to limited conditions, only one earthquake shelter can be built according to the existing resources. There are four feasible alternatives to choose from: (1) x1: near the station; (2) x2: near a school; (3) x3: near the residential area; (4) x4: near the hospital. The 20 experts participated in the building of the project, including government officials, construction builders, emergency management experts and many other fields. These experts need to evaluate the feasible alternatives from the following five factors: (1) f1: cost; (2) f2: capacity; (3) f3: construction difficulty; (4) f4: individual preference; (5) f5: time constraint. For example, when building a large capacity shelter, its manufacturing cost, construction difficulty and construction time will usually increase, so everyone has different attitude on these factors. Due to each alternative has its own advantages and disadvantages, it is necessary to select a satisfactory alternative through group decisionmaking. The information provided by experts is reported in Appendix B.
6.2. DecisionMaking Process
The proposed consensus model is applied to increase the consensus level and to obtain the optimal alternative, the steps of which are shown as follows. Noting that = = = γ = 0.2, = 0.5, and a = 0.8 in this case: Step 1. The 20 DMs are divided into 5 subgroups by using the Algorithm 1 as shown in Table 1. Step 2. Calculate the subgroups’ original evaluation matrices through equation (1), that is, Then, calculate the DMs’ weights and the attributes’ weights in Gk(k = 1,2, …, 5) by using Algorithm 2. And compute the subgroups’ weights and the collective attributes’ weights through Algorithm 3. The calculation results are shown in Table 2. Noting that in this step the computation results of the harmonious degree of the subgroups are shown in Table 3. Step 3. Based on equation (22), the unit adjustment cost for each DM can be calculated and then obtained, and the results are reported in Table 4. Step 4. The CRP. First, the computation results of the current consensus level and other information and the consensus threshold are recorded in Tables 5 and 6, respectively. Apparently, GCL0 = GCL = 0.8302 < δ = 0.8590, the consensus feedback then should be executed. The opinion of the DM e8 in G5 is first modified through adjustment process, and then obtain the adjustment result, i.e. GCL1 = 0.8362 < δ = 0.8590. The consensus adjustment process therefore should be executed again. The DM e19 in G1 is identified and adjusted, and then obtain the adjustment result, i.e. GCL2 = 0.8410 < δ = 0.8590. In this time, CAR1 = 2.4717 < CAR2 = 2.6229, so the CRP should not be terminated. Subsequently, the evaluation information of the DM e20 in G5 need to be modified, then the adjustment result is GCL3 = 0.8446 < δ = 0.8590, and CAR2 = 2.6229 < CAR3 = 3.0673. The consensus feedback should be executed again. The opinion of the DM e4 in G4 is modified. The adjustment result is GCL4 = 0.8489 < δ = 0.8590, and CAR3 = 3.0673 > CAR4 = 2.7436. The consensus feedback process should be terminated and T = 3. In addition, the fourth iteration result should be restored, and the final group consensus level is GCLT = 0.8446. The adjustment process of the consensus level and the change process of CARt are shown in Tables 7 and 8, respectively. The expert adjusted evaluation matrices are obtained and reported in Appendix B. And the subgroups’ adjusted evaluation matrices can be derived by equation (1), that is, Step 5. Based on the above calculation results, the group’s evaluation information is obtained, that is, and the score value for each alternative can be derived by equation (3), that is s(x1) = 0.4739, s(x2) = 0.5413, s(x3) = 6383, and s(x4) = 0.4901. Furthermore, the ranking of the alternative is x3 x2 x4 x1, and the optimal alternative is x3.

6.3. Discussion
In this case, the consensus level is changed from 0.8302 to 0.8446. Although the consensus threshold δ = 0.8590 has not been reached, and the consensus level has also been greatly improved. Also, the original information has been retained as much as possible in the consensus feedback process. Therefore, the proposed consensus model considering the interactive weights and the experts’ adjustment willingness is verified to be effective.
7. Comparative Analysis
In this section, to demonstrate that the characteristics of the proposed consensus method is imperative, comparative analyses that without the interactive and without the experts’ adjustment willingness, respectively, are implemented in Section 7.1 and Section 7.2.
7.1. The Proposed Model without considering the Interactive Weights
Continuing the case that given in Section 5, the proposed consensus model is executed without considering the determination of the interactive weight. Therefore, from Step 10 to Step 12 of Algorithm 2 and from Step 2 to Step 4 of Algorithm 3 are missing. The analyses process is performed as follows: Step 1A. The content of this part is the same as the Step 1 of Section 6.2. Step 2A. The subgroups’ original evaluation matrices do not change. But the weight determination process and results are changed and shown in Table 9. Step 3A. Due to the interactive process is missing, the unit adjustment cost is not affected by harmonious degree. Thus, the determination method of the unit adjustment cost is simplified, that is, The unit adjustment cost without considering the interactive weights can be derived and shown in Table 10. Step 4A. The computation results of the current consensus level and other information and the consensus threshold are reported in Tables 11 and 12, respectively. Apparently, GCL’0 = GCL’ = 0.8976 < δ = 0.9068, the consensus feedback then should be executed. The adjustment process of the consensus level is shown in Table 13. It should be noted that in the identification process the DM who have a higher weight is adjusted if there are several DMs who have the same unit adjustment. Step 5A. Based on the above calculation results, the group’s evaluation information is obtained, that is, and the score value for each alternative can be derived by equation (3), that is s(x1) = 0.4674, s(x2) = 0.5284, s(x3) = 0.6243, and s(x4) = 0.4808. Furthermore, the ranking of the alternative is x3 x2 x4 x1, and the optimal alternative is x3.
7.2. The Consensus Feedback without considering the Adjustment Willingness
Similar to Section 7.1, the proposed consensus model is executed without considering the adjustment willingness. Therefore, Step 8 and Step 9 of Algorithm 2 are missing, and Algorithm 4 is no longer available. The analyses process is shown as follows: Step 1B. The content of this part is the same as the Step 1 of Section 6.2. Step 2B. The subgroups’ original evaluation matrices do not change. But the weight determination process and results are changed and shown in Table 14. Step 3B. Due to the adjustment willingness of the DM is missing, the unit adjustment cost is not affected by the allowed modification range. Thus, the determination method of the unit adjustment cost is simplified, that is, The unit adjustment cost without considering the adjustment willingness can be derived and shown in Table 15. Step 4B. The computation results of the current consensus level and other information and the consensus threshold are reported in Tables 16 and 17, respectively. Apparently, GCL’0 = GCL’ = 0.8306 < δ = 0.8592, the consensus feedback then should be executed. In this subsection, a novel adjustment rule is used, and the details are carried out as follows. Suppose that the evaluation matrices of the DM and the subgroup to be adjusted are and , then the adjusted evaluation matrix of the DM is = 1/2 + . The adjustment process of the consensus level is shown in Tables 18 and 19, respectively. In this consensus feedback process, the maximum adjustment round T is 2. Step 5B. Based on the above calculation results, the group’s evaluation information is obtained, that is, and the score value for each alternative can be derived by equation (3), that is s(x1) = 0.4713, s(x2) = 0.5263, s(x3) = 0.6223, and s(x4) = 0.4843. Furthermore, the ranking of the alternative is x3 x2 x4 x1, and the optimal alternative is x3.
7.3. Discussion
Based on the above introduction, the differences between the proposed model and without considering the interactive weights, the proposed model and without the experts’ adjustment willingness are shown as follows.
7.3.1. The Subgroups’ Consensus Level
The proposed model, without considering the interactive weights, and without the experts’ adjustment willingness of the subgroups’ consensus levels are shown in Figure 2. Specifically, the ranking of the initial consensus level is CL_{3} > CL_{4} > CL_{2} > CL_{1} > CL_{5} in the models M1 and M3 while model M2 is CL_{3} > CL_{2} > CL_{4} > CL_{1} > CL_{5}. The consensus levels of the subgroups G3 and G5 of the M1 and M3 are 0.9600 and 0.7707, respectively, while the M2 are 1.0000 and 0.8584.
For the model M2, the stage showing the decision results to experts is omitted in the process of weight determination. Hence, the decision information is not fed back to DM in time, and the decisionmaker’s attitude towards the decision information is not obtained. Although the consensus level of the M2 is higher than the M1 (i.e. the proposed model), the calculation of consensus level is inaccurate and partial. Therefore, the decisionmaking result cannot fully represent the wishes and attitudes of the DMs.
For the model M3, the stage reflecting the DMs’ adjustment willingness information is removed. Thus, the adjustment attitude of DMs to their own evaluation information cannot be known, and the lack and distortion of the DMs’ evaluation information may lead. Although the consensus level for each subgroup of the M3 is very similar with the M1, the DMs’ adjustment willingness (i.e. the stubborn degree to their own opinion) is omitted, and the acquisition of consensus level is not scientific and reasonable. Therefore, it may lead to the unreasonable decisionmaking result due to the DMs’ opinion is distorted or ignored.
7.3.2. The Changes of the Group Consensus Level GCL_{t} and the Adjustment Cost
The proposed model, without considering the interactive weights, and without the experts’ adjustment willingness of the group consensus level GCL_{t} and the adjustment cost are record in Table 20.
For the model M2, after 2 rounds of the consensus feedback process, the consensus threshold and an acceptable consensus level has been reached. Due to the lack of timely attitude of DMs towards decisionmaking information, the satisfaction degree for each DM is not obtained. Also, the consensus level of the subgroup does not involve the harmonious degree, namely the recognition of the subgroup is not presented. Therefore, the consensus level of the M2 improves faster than the M1 because it does not consider the harmonious degree of the subgroup. Meanwhile, the calculation method of the unit adjustment cost is partial, and the adjustment cost of the M2 increases faster than the M1 and then is inaccurate and unscientific. Therefore, the decisionmaking result of the M1 is more reasonable than M2.
For the model M3, after 2 rounds of the consensus feedback process, the termination condition is satisfied and the final consensus level can be obtained. Due to the lack of the DMs’ adjustment willingness information, the distortion of the DMs’ opinion may lead. And, the consensus level of the M3 increases slower than the M1. For instance, after 2 rounds of the iteration process the group consensus level of the M3 is 0.8365 while the M1 is 0.8410. And t after one round the group consensus level of the M1 is 0.8362. It is obvious that the result of the M3 adjustment twice is similar to that of the M1 adjustment once. Meanwhile, the computation process of the unit adjustment cost is also partial, and the adjustment cost of the M3 is lower than the M1 and then is inaccurate and unreasonable. From the perspective of the experts’ adjustment willingness, the decisionmaking result of the M1 is more scientific than M3.
In summary, it is apparent that both considering the interactive weights and the experts’ adjustment willingness for the CRP is not only very reasonable but imperative.
8. Sensitivity Analysis
In this section, sensitivity analyses that the unit adjustment cost and the harmonious degree in the CRP, respectively, are presented in Section 8.1 and Section 8.2.
8.1. The Effect of the Unit Adjustment Cost cm in Consensus Feedback Process
In Section 6, the unit adjustment cost cm for each DM is obtain based on equation (22) and the value of is 0.5. In this subsection, a discussion based on the different unit adjustment cost is introduced. To further prove the rationality of the proposed consensus model, a sensitivity analysis is conducted with different values of = {0.1, 0.3, 0.5, 0.7, 0.9}. The unit adjustment cost for each DM, the group consensus levels, and the score of the alternatives with different are recorded in Figure 3, Tables 21, and 22, respectively.
According to Figure 3, it is apparent that the unit adjustment cost cm increases with respect to the value of for each DM. For the different value of , the unit adjustment cost between two decision makers may be different. For example, the unit adjustment costs of e14 and e15 are 0.5000 and 0.6775 when = 0.5, respectively, while they are 0.8608 and 0.7827 when = 0.9. In other words, c14 < c15 when = 0.5 while c14 > c15 when = 0.9. Therefore, we set a moderate value (i.e. = 0.5) in the case study due to we cannot accurately obtain the importance of the experts’ adjustment willingness or the satisfaction degree. For Table 21, the termination round is the first stage when = 0.1 and 0.3, while the termination round is the third stage when = 0.5, 0.7, and 0.9. This is because with the increase of adjustment cost, there are more opportunities, however, if the adjustment cost is too high, it is unfavorable for the consensus feedback process. Therefore, setting = 0.5 is reasonable in the case study.
Based on Table 22, we found that the scores for each alternative are very similar. Due to the different unit adjustment cost, the different value of may lead to different DM to be adjusted in the same stage. Therefore, it will be a small difference in the score of the alternative with the different . However, the ranking of the alternatives is the same with the different . Therefore, it means that setting = 0.5 is reasonable in the case study, and the proposed consensus model considering both the interactive weights and the experts’ adjustment willingness is stability and rationality.
8.2. The Effect of Harmonious Degree in the Consensus Level Calculation
In Section 6, the consensus level for each subgroup is computed based on equation (14) and the value of a is 0.8. To further prove the rationality and stability of the proposed consensus model, a sensitivity analysis is introduced with different values of a = {0.6, 0.7, 0.8, 0.9, 1.0}. The consensus thresholds, the group consensus levels and the adjustment costs, and the score of the alternatives with different values of a are reported in Tables 23–25, respectively.
For Tables 23 and 24, it is apparent that the consensus thresholds and the initial group consensus levels increases with respect to a. Although the increase of the importance of the subgroups’ harmonious degree will lead to the decrease of the subgroups’ and the collective consensus level, it is very critical for the decisionmaking results. The consensus level of the subgroup not only represents the consistency of the opinion within subgroup, but also represents the DMs’ attitude towards the decisionmaking results and the stability of the sub group’s opinion. Therefore, it is reasonable and imperative that the consensus level considers both the opinion’s consistency and the DMs’ attitude. Meanwhile, the final group consensus levels satisfies the equation CAR_{T} ≤ CAR_{T+1} when a = 0.6, 0.7 and 0.8, while that meets the equation GCL_{T} ≥ δ when a = 0.9 and 1.0. This is because with the increase of a, the differences between the consensus threshold δ and the initial consensus level GCL0 decreases, the consensus termination conditions are easy to meet than before. For example, the difference between the consensus threshold δ and the initial consensus level GCL0 is 0.0468 when a = 0.6 while 0.0097 when a = 1.0. Hence, with the increase of a, the difference between the current GCLt and δ is less and less. Also, the constraint condition CAR_{T} ≤ CAR_{T+1} can obtain the GCLT faster. Therefore, it is reasonable that the importance of the harmonious degree is 0.2 (i.e. a = 0.8) in the case study, and the termination condition is necessary and rationality.
According to Table 25, we found that the scores of alternatives are the same when a is 0.6 and 1.0, 0.7 and 0.8, respectively. This is because the iteration round is the same. Interestingly, the ranking of alternatives is the same when a is different. It shows that the proposed consensus model is more stable and the decisionmaking result is more scientific.
9. Conclusion
In this study, we propose a consensus model of LSGDM considering the interactive weights’ determination and the experts’ adjustment willingness, and apply it to select the building of an earthquake shelter. The main contributions and innovations of this research are shown as follows:(1)We develop a novel method of weight determination, which considers the DMs’ attitude towards the decisionmaking results, thereby ensuring the effective participation of DMs. Moreover, to improve the rationality of LSGDM, the harmonious degree is conducted in the calculation of subgroups’ weight. It is of significance for the DMs more involved in the decisionmaking process and the decisionmaking result more reflect the willingness of DMs.(2)By considering the experts’ adjustment willingness, it is ensured that the evaluation information is less distorted or lost. Moreover, the unit adjustment cost is designed. Subsequently, to improve the efficiency of CRP, an identification rule combines the unit adjustment cost and the consensus level is presented to retain as much original information as possible in consensus feedback process, which can easily reach an acceptable level of consensus.(3)An objective calculation method of the consensus threshold is conducted, and then a termination condition that considers both the current consensus level and the consensus adjustment rate is designed to objectively terminate the CRP. It not only compares the current consensus level with consensus threshold but compares the consensus adjustment rate. As a result, this method can address the subjectivity and unreasonableness of the preset consensus threshold and the maximum number of iterations to a certain extent.
Despite several valuable findings obtained by our research, there remain some limitations that should be further dealt with in the future. In this study, in addition to the interactive weights of the proposed model, the professional knowledge background and decisionmaking experience and other individual attributes of DMs should be considered. Moreover, with the development of social networks, the relationship between DMs that become complex should be considered in LSGDM. Also, the behavioral factors (i.e. noncooperation) [47] and psychological factor (i.e. selfconfident) [48] of DMs should also be conducted, it will be a meaningful research for the LSGDM problems.
Appendix
A: The Proof of Theorem 1
Proof. First, the values of and are both positive and in the interval [0, 1], so we have 0 ≤ + ≤ 2 and 0 ≤ ≤ . For the function y = cos(x), y is monotone decreasing when x is in the interval [0, ]. Thus, we can obtain that the value of is in the interval of [0, 1]. Then, the value of is positive and in the interval [0, 1]. The function z = sin() is monotone increasing when x is in the interval [0, 1], so the value of z is in the interval [0, 1]. Therefore, the value of cm is in the interval of [0, 1].
B: The Information Provided by Experts
The information provided by experts are reported in Tables 26–28. In this paper, the value range of is from 0 to 1 (Tables 29 and 30).
Data Availability
The data used to support the findings of this paper are included within the article (Case Study section and Appendix).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors appreciate the financial support of the National Key Research and Development Program of China (Grant no. 2018YFB1402500) and the Natural Science Foundation of Heilongjiang Province (Grant no. JJ2021LH1530). The Reform and Develop HighLevel Talent Projects in Local Universities Supported by the Central Government (2020GSP13)