A clustering- and maximum consensus-based model for social network large-scale group decision making with linguistic distribution
Introduction
Generally speaking, group decision making (GDM) is a kind of problem, in which more than two decision makers (DMs) are involved, and the priorities of a set of alternatives are obtained based on DMs’ preference matrices. In the GDM process, determining the weight of DMs is crucial, which will affect the result of decision making directly, and much literature have paid attention to this point [1].
In recent years, with the rapid development of the society and the expansion of technological paradigms [2], the decision environment and decision groups have changed radically. Today’s GDM problems often contain a large number of participants who are all related to the decision-making results, which calls for a paritcular resolution framework. Based on this situation, the large-scale GDM (LGDM) problem was proposed, in which a larger number of DMs take part in the decision process and their responsibility for the decision result. Wu et al. [3] classified GDM problem with more than 20 DMs as LGDM problem, in which large-scale DMs usually need to reach a consensus. Due to the new challenges caused by this situation, such as summarized in [4]: (1) Scalability, (2) Time cost, (3) Constant preference supervision, (4) Stronger disagreement positions, (5) Difficulties to understand/visualize the current state of agreement, new models for LGDM are necessary regarding classic GDM.
To improve the quality of the LGDM, many consensus-based methods have been put forward [5], [6], [7], [8], [9]. Labella et al. [5] compared the classical consensus reaching processes applied to LGDM, analyzed their performance, and summarized the main challenges. Palomares et al. [6] presented a consensus model suitable to manage large-scales of DMs, which incorporates a fuzzy clustering-based scheme to detect and manage individual and sub-group non-cooperative behaviors. Zhang et al. [7] proposed a novel consensus reaching model for the heterogeneous large-scale GDM with the individual concerns and satisfactions. Dong et al. [8] propose a novel framework based on a self-management mechanism for noncooperative behaviors in large-scale consensus reaching process. Li et al. [9] addressed LGDM problem with the personalized individual semantics. In these works, the consistency and similarity of DMs’ preferences have attracted wide attention [8], [9], the former can reflect the rationality and logicality of DMs’ preference information [10], while the latter reports the distance of DMs’ opinions. A consistency-driven optimization model is introduced in [9] to obtain the personalized individual semantics of linguistic terms with linguistic preference relation, and the Grey clustering algorithm is used in [8], which is based on the similarity measure between experts. Although numerous research methods for LGDM problems have been reported, most works generated the weight of decision-making groups only from single source, this can be irrational due to there are multiple influencing factors for DMs’ weight.
Meanwhile, in real-life LGDM problem, almost all DMs are in social network, that is, DMs are not isolated decision-making units, but have social relations with each other. We can call the LGDM problem in the context of social network as SNLGDM (social network LGDM) problem, which usually has the following remarkable features: (1) it contains a large number of DMs from different departments and professional fields; (2) there are conflicts among DMs, so a consensus process is needed to reach agreement; (3) DMs are in social network, and there are certain social relations among them.
In dealing with social relations, social network analysis (SNA) has attracted much attention. SNA is proposed by sociologists based on knowledge of graph theory and mathematical models, and has been proved to be an effective tool to describe the social relationships among experts in LGDM [11]. Meanwhile, due to the characteristics of easy evaluation and transmissibility, trust relationship is widely used to represent the social relations between DMs, and scholars have conducted extensive researches on it under SNLGDM problem. The application of SNA using trust relationship in LGDM mainly includes the following aspects: (1) completing preference information of DMs [12]; (2) ranking options on the basis of weight vector generated through SNA [13]; (3) detecting conflict in decision-making process [14]; (4) detecting DMs’ non-cooperative behavior by using SNA [15]; (5) generating comprehensive similarity matrix by considering experts’ social relationships and preference information simultaneously [16]; and (6) addressing consensus problem with social relationships among DMs [17]. However, only a few works have focused on generating the weight of decision groups based on SNA, and even in these articles, the social network is the only source to generate the weight of DMs [3], as mentioned above, this may be unreasonable. In fact, for all social network GDM (SNGDM) problems, to the best of our knowledge, only few articles represented by [18], [19] have considered the weight of DMs by multiple reliable sources, furthermore, only few works represented by [20], [21] are in the context of LGDM.
For LGDM problems, it’s critical to reduce the complexity by lessening the large-scale DMs’ dimension. Analysis techniques [22], [23], [24] are commonly used to deal with the complexity of LGDM problem among existing researches. Liu et al. [22] proposed an interval-valued intuitionistic fuzzy principal component analysis model for the complex multi-attribute LGDM problem. Zhu et al. [23] developed a new clustering procedure combining three-dimensional gray relational analysis and the concept of hierarchical clustering. Wu et al. [24] employed the interval type-2 fuzzy equivalence clustering analysis to classify DMs to reduce the dimension of the large-scale DMs. Based on the clustering methods mentioned above and DMs’ preference information, large-scale DMs can be partitioned into several sub-groups, in which members have similar preference. However, sometimes it’s unreliable to use preference information of DMs [7], especially for heterogeneous network. Based on trust relationship under social network context, a network partition method is designed by Dong et al. [25] to deal with the dimension problem. Based on Dong et al.’s research [25], Wu et al. [13] proposed a two-stage trust network partition model. However, both Dong et al.’s method [25] and Wu et al.’s method [13] will produce isolated nodes in the process of network partition. And isolated nodes are often at a disadvantage in the weight distribution, which may be detrimental to the subsequent consensus process.
For linguistic LGDM problems, representing the group’s linguistic preference is another crucial step. To solve this problem, methods of resolution have been given. By directly extending the mathematical model, which is developed by traditional linguistic GDM researches, to linguistic LGDM problems [26], Wan et al. [27] devised a probabilistic linguistic preference relations-based method, while He et al. [28] proposed an extended TODIM method based on shadowed sets to model linguistic variables. Similarly, based on linguistic term with hesitant fuzzy information, models are advanced in [29]. In fact, if the linguistic term set is adopted in the context of LGDM problems, then group’s preference tends to be reflected in the distribution of these terms. DMs’ evaluation of alternatives can be retained by this representation method. For conventional linguistic GDM problems, the introduction of linguistic models and computational processes to deal with linguistic information may mean that the derived information is a gross over simplification of the facts. To handle this problem, Zhang et al. [30] adopted the linguistic distribution (LD) assessments, which can retain the initial decision information of DMs in the group to the greatest extent. For this reason, LD is more practical for LGDM problems with linguistic term.
Given all the above analysis, in this study, we are committed to putting forward a clustering- and maximum consensus-based resolution framework with LD for LGDM problem in social network environment. SNA will play a role in multiple stages of the model, and multiple reliable indexes are comprehensively considered when determining weight. Meanwhile, the optimization model based on maximum consensus is used to consider the effect and efficiency of decision-making.
In the clustering- and maximum consensus-based consensus method, independent sub-groups are obtained from the division of large-scale DMs through proposed SAN-based trust network clustering model according to trust relationship, and there are no isolated nodes. Then LD assessment, which minimizes the loss of decision information, is used to represent the preference relation of each sub-group. Following this, the consistency and similarity indexes are obtained from subgroups’ preference analysis, while the in-centrality degree indexes are gained by social trust network analysis. Considering these three dependable sources, the comprehensive weight of sub-groups is determined by devised maximum consensus-based model, whose objective is maximizing the level of consensus between sub-groups and collective matrix. Meanwhile, collective ranking of alternatives can be obtained through the collective preference relation.
It’s worth mentioning that, clustering and computing weights are based on different criteria, SNA-based clustering method can reasonably complete the task of dimensionality reduction and improve the degree of harmony, while comprehensive weight method can make the decision result more reasonable and improve the efficiency of decision-making. Combining the two can play a better effect than a single use.
In this study, we try to achieve the following contributions:
- (1)
Establish a solution framework for SNLGDM problems based on multi-stage utilization of SNA. SNA technology has been proved to be an effective tool to describe the social relationship between experts, but in the current LGDM research, the development and utilization of SNA is not enough. Our solution framework uses SNA technology in multiple stages to make it play an innovative role in building social trust network, dividing large-scale DMs and generating sub-group weights.
- (2)
Develop a SNA-based network clustering model to handle large-scale DMs’ dimension problem according to trust relationship. This can avoid the problem that DMs’ preference information is sometimes unreliable, especially for heterogeneous network. At the same time, there is no isolated node in this clustering process, which is more beneficial to the subsequent consensus process, because the preference information of isolated nodes is often ignored due to their disadvantage in weight distribution.
- (3)
Present a maximum consensus-based method to generate the sub-groups’ comprehensive weights by considering multiple dependable indexes. In real decision-making problems, DMs’ weights are influenced by multiple factors, therefore, this weight determination method is more reasonable. Meanwhile, under the emergent decision environment where DMs need to draw a consensus expeditiously, this method can maximize the degree of consensus and make the decision result accepted by more DMs, both the efficiency and the effect of the consensus process are considered. Specifically, in the situation that DMs provide sufficient decision-making information, that is, their preference matrix and social trust relationships are available, it’s more applicable to take above three reliable factors into account.
The remainder of paper is organized in the following order: Section 2 reviews some knowledge related to linguistic distribution, social network analysis and network partition. The SNLGDM problem and a resolution framework for it are presented in Section 3. Then, Section 4 proposes a social trust network clustering model to reduce the large-scale DMs into several independent sub-networks. In Section 5, using constructed maximum consensus-based model, sub-groups’ comprehensive weight and collective preference matrix are obtained. Following this, in order to present the validity of this research, a numerical example and the coefficient analysis is carried out in Section 6. Section 7 presents some comparisons with related GDM methods. Conclusively, conclusion and future research interests are seen in Section 8.
Section snippets
Preliminaries
In this section, we briefly review the basic concepts and knowledge related to linguistic distribution, social network analysis and network partition.
SNLGDM problem and a resolution framework for it
The SNLGDM problem is formally presented, meanwhile we design a consensus framework for it, which regarding on social trust network clustering and maximum consensus degree.
Social trust network clustering model
In the network partition obtained by Dong et al [25], followers may have multiple leaders, that is, a follower may belong to multiple sub-groups. Meanwhile, if a DM does not have any trust relation with other DMs, he cannot be divided into a sub-group.
To solve above problems and get independent sub-groups according to large-scale DMs’ trust relationship, Wu et al. [13] devised a network partition method with two stages. However, Wu et al.’s model [13] only considered the isolated nodes that
Maximum consensus-based optimization model
At the beginning of this part, the vectors of consistency, similarity and in-centrality degree indexes are gained through preference matrix analysis and social trust network analysis respectively. Then sub-groups’ comprehensive weight vector can be obtained by the linear combination of the above three vectors. In order to determine the combination coefficients, a maximum consensus-based method is proposed, which can maximize the level of consensus between sub-groups and collective preference
Example of the SNLGDM problem
Simulating the SNLGDM problem in real world, a numerical example is advanced to illustrate the effectiveness and practicability of the proposed consensus method. The main existing studies defined the GDM problem, which number of involved DMs is not less than 20, as LGDM problem [3], [31]. In our illustrative example, a set of 20 DMs and a set of four alternatives are involved. The linguistic term set that the DMs used is defined as:
Comparison analysis
In this section, we divide the proposed method into two stages: clustering and consensus, and compare with the related methods respectively to examine the superiority of the proposed method.
Following this, comparisons between this paper and some related GDM methods are reported. We sort out the main features of 10 related models for GDM problem and compare them with devised clustering- and maximum consensus-based method, which can elucidate the essential characteristics of our proposal.
It’s
Conclusion
In this study, we investigated the SNALGM problem and proposed a clustering- and maximum consensus-based consensus method with linguistic distribution. Compared with the existing consensus method in the LGDM, the proposed consensus reaching model incorporates two new practical decision features:
(1) Handling the dimension problem for LGDM according to the trust relationship. The method in this paper mainly depends on the social network of DMs for clustering. It only needs DMs to give the trust
Compliance with Ethical Standards
(1) Disclosure of potential conflicts of interest.
We declare that we do have no commercial or associative interests that represent a conflict of interests in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work.
(2) Research involving human participants and/or animals.
This article does not contain any studies with human participants or animals performed by any of the authors.
Credit author
Peide Liu, the first
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This paper is supported by the National Natural Science Foundation of China (No. 71771140), Project of cultural masters and “the four kinds of a batch” talents, the Taishan Scholars Project of Shandong Province, Shandong Provincial Key Research and Development Program(Major Scientific and Technological Innovation Project) (Nos. 2021SFGC0102, 2020CXGC010110).
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