Traditional group decision making (GDM) problems typically aim to obtain a complete ranking of all considered alternatives from best to worst. However, in numerous real-life scenarios, there are instances where it is imperative to assign each alternative into one of two rank levels, creating a ranking where one subset of alternatives is prioritized above the other subset of alternatives. These scenarios are known as 2-rank GDM problems. While a range of methods exist for addressing 2-rank GDM problems, most are specifically tailored to multiattribute decision making situations, thereby limiting their applicability in scenarios involving preference relations. The linguistic preference relation (LPR) is an effective representation tool of decision makers’ (DMs’) preferences for pairwise comparisons of alternatives using linguistic terms. Since words may have different meanings for different DMs, a phenomenon known as personalized individual semantics (PISs), the modeling of linguistic PISs in 2-rank GDM problems with LPRs is worth investigating and challenging to address. Consequently, this article develops models to support consensus reaching for 2-rank linguistic GDM problems with PISs and consistency of DMs. Specifically, PIS consistency-driven models are initially employed to measure and improve the consistency of the LPRs of the individual DMs with unacceptable consistency level. Based on this foundation, the 2-rank vectors for both individuals and the group are determined. Subsequently, a 2-rank consensus measurement method is proposed on which a 2-rank consensus reaching process is designed to support DMs in improving their consensus levels. This involves the development of a PISs-based minimum adjustment consensus optimization model and a PISs-based individual consensus level maximization model. An algorithm to implement the proposed consensus reaching framework is also provided. Finally, numerical experiments and simulation results are reported to demonstrate the effectiveness of the proposed method.

In multi-criteria sorting (MCS) problems, decision makers often express indirect and potentially conflicting preferences, especially when these preferences come from multiple sources or take different forms. Effectively handling such inconsistency is essential for generating reliable sorting results in MCS problems. This paper introduces a novel approach tailored to resolving inconsistency in MCS problems with heterogeneous preferences. The proposed framework begins with a consistency checking model to detect conflicts in the provided heterogeneous preferences. If the inconsistency is detected, a two-stage adjustment model will be applied: the first stage minimizes the number of adjustments, while the second stage preserves high-confidence preferences wherever possible. Once the adjusted preferences are consistent, a sorting result determination model is used to assign alternatives to predefined categories, with an emphasis on maximizing discriminative power between categories.

Efficient and robust data clustering remains a challenging task in data analysis. Recent efforts have explored the integration of granular-ball (GB) computing with clustering algorithms to address this challenge, yielding promising results. However, existing methods for generating GBs often rely on single indicators to measure GB quality and employ threshold-based or greedy strategies, potentially leading to GBs that do not accurately capture the underlying data distribution. To address these limitations, this article leverages the principle of justifiable granularity (POJG) to measure the quality of a GB for clustering tasks and introduces a novel GB generation method, termed GB-POJG. Specifically, a comprehensive metric integrating the coverage and specificity of a GB is introduced to assess GB quality. Utilizing this quality metric, GB-POJG incorporates a strategy of maximizing overall quality and an anomaly detection method to determine the generated GBs and identify abnormal GBs, respectively. Compared to previous GB generation methods, GB-POJG maximizes the overall quality of generated GBs while ensuring alignment with the data distribution, thereby enhancing the rationality of the generated GBs. Experimental results obtained from both synthetic and publicly available datasets underscore the effectiveness of GB-POJG, showcasing improvements in clustering accuracy and normalized mutual information. All codes have been released at https://zenodo.org/records/13643332.

In the field of linguistic decision making, it is widely acknowledged that different individuals may have different understandings of the same linguistic information. Consequently, the modeling of personalized individual semantics (PISs) has emerged as a prominent research avenue within the domain of linguistic decision making. In this article, we study multicriteria decision making problems with incomplete linguistic preference relations (ILPRs), particularly in scenarios where the marginal utility function of each criterion remain elusive. We develop a new approach for modeling PISs of decision makers from the preference disaggregation perspective. The proposed method initiates by representing the marginal utility of each alternative regarding to every criterion, employing piecewise linear functions. It then customizes linguistic preference values within an ILPR by translating them into numerical preference intensities. Subsequently, this article formulates optimization models to check and rectify the inconsistencies between the multicriteria assessment information and the known elements in the ILPR. Furthermore, an optimization model is devised with the objective of minimizing the inconsistency index of the complete linguistic preference relation, thereby enabling the estimation of missing elements of the ILPR and the ranking of alternatives. Consequently, this method also facilitates the derivation of marginal utility functions and PISs for the decision maker. The proposed method is illustrated using a practical scenario involving a car purchase problem, substantiated by a series of simulation experiments and comparative analysis, ultimately providing validation for the proposed approach.

Traditional group decision-making problems focus on obtaining a complete ranking of all alternatives from best to worst. However, in many real-life scenarios, there are instances where it is necessary to assign only two rank levels to alternatives, creating a ranking where one subset of alternatives is prioritized above another subset. These scenarios are referred to as 2-rank group decision-making problems. Linguistic preference relations serve as an effective tool for expressing decision-makers preferences, as they allow comparisons between two alternatives at a time using linguistic terms. Nonetheless, in 2-rank group decision-making problems with linguistic preference relations, it is common for the same linguistic term to hold different meanings for different decision-makers, a phenomenon known as personalized individual semantics (PISs). Addressing how to model PISs in 2-rank group decision-making problems presents a significant challenge. In this paper, we develop a consensus-reaching model for 2-rank linguistic group decision-making problems, incorporating PISs and consistency control for decision-makers. Specifically, we first employ consistency-driven models to evaluate and improve the consistency of each decision-makers linguistic preference relations. Based on this foundation, we determine the 2-rank preference vectors for both individuals and the group. Subsequently, we propose a 2-rank consensus measurement method and design a 2-rank consensus-reaching process to help decision-makers enhance their consensus level. This involves the development of a PIS-based consensus level maximization model and a PIS-based minimum adjustment model. Furthermore, we introduce an algorithm to implement the consensus-reaching framework. Ultimately, numerical experiments and simulation results are provided to demonstrate the effectiveness of the proposed method.

In multicriteria group decision-making (MCGDM) problems, there exist the situations that alternatives need to be assigned to several predefined ordered categories rather than be ranked from the most preferred to the least preferred, which is called multicriteria group sorting problems. To address multicriteria group sorting problems, it is necessary to implement a consensus reaching process to fully consider the opinion of each decision maker and reduce the conflict among them. To do so, this article proposes a consensus reaching model for multicriteria group sorting problems based on the threshold-based value-driven sorting method from the perspective of minimum adjustment. Specifically, we first define the consensus measure to calculate the agreement degree among decision makers by considering the ordinal information and cardinal information at the same time. On this basis, we construct a minimum adjustment optimization model in terms of the threshold-based value-driven sorting method to assist decision makers in modifying their decision matrices and further promote consensus. Followed by this, an optimization model that aims to minimize the distance between all decision makers and the group with respect to alternatives’ comprehensive values and sorting results is developed to determine the group sorting result for alternatives. Moreover, a numerical application of urban park management, some sensitivity analysis, and simulation experiments are provided to justify the proposed method. Experimental results reveal that the proposed method is effective in promoting consensus.

In large-scale group decision-making (LSGDM), incomplete preferences often arise due to the complexity of LSGDM or the limited experience of decision-makers (DMs). When DMs provide preferences using incomplete fuzzy preference relations (IFPRs), most studies focus on cardinal information, overlooking the vital ordinal relations in IFPRs. However, a high level of cardinal consensus may not represent unanimous pairwise comparisons among DMs’ preferences. In contrast, ordinal relations are crucial for ranking results by enabling pairwise comparisons. Therefore, this paper presents an LSGDM framework mining ordinal-cardinal group consensus by prioritizing ordinal relations followed by cardinal information of IFPRs. We begin by extracting ordinal relations from IFPRs. Next, an ordinal clustering optimization model is constructed to minimize overall conflicts. Finally, a value function-based consensus model is developed, identifying a consensus ranking by considering ordinal and cardinal information within subgroups. By linking this value function with IFPRs, each subgroup achieves a complete fuzzy preference relation (FPR) that is both ordinal and cardinal consistent. The application example shows the feasibility of this approach. Numerical analyses validate the clustering optimization model's effectiveness in reducing the global average conflict degree, and simulation studies with varying levels of incompleteness in FPRs demonstrate the consensus model's robustness in achieving consistent ranking results.

Consistency and consensus are important issues for linguistic group decision making (GDM), which have been extensively studied by scholars. Nevertheless, most of previous consensus reaching models focus on adjusting decision makers’ preference relations and ignore the individual consistency, which results in that individual consistency may be destroyed by using these consensus reaching models. Moreover, it has been accepted that words mean different things for different people and thus, it is also necessary to model decision makers’ personalized individual semantics (PISs) in linguistic GDM. This work focuses on developing some PIS-based consistency control and consensus reaching models for linguistic GDM. First, we analyze the problems existing in previous PIS models and then develop a minimum adjustment-based optimization model to test and improve the individual consistency for a linguistic preference relation (LPR). Followed by this, a PIS-based individual consensus-level maximization model and a PIS-based minimum adjustment model are established for consensus reaching in linguistic GDM, in which individual consistency control is considered. Furthermore, an algorithm for consensus reaching is proposed based on these models. To justify the proposed models and algorithm, some numerical results and simulation analysis are provided eventually.

With the increase of technological and societal demands, more and more decision makers are involved in the process of group decision making, which is called large-scale group decision making (LGDM). For an LGDM problem with linguistic information, it is common that different decision makers tend to provide linguistic assessments defined on multigranular linguistic term sets due to the difference in knowledge and culture background, and that hesitant fuzzy linguistic term sets (HFLTSs) are used by decision makers to model the hesitancy of their assessments. This article proposes first an algorithm to represent a linguistic distribution assessment (LDA) using a hesitant linguistic distribution (HLD). Two other algorithms are then proposed to transform an unbalanced HFLTS into a balanced LDA and to transform a balanced LDA into an unbalanced LDA, respectively. An approach is then proposed to deal with multiattribute LGDM problems with multigranular unbalanced hesitant fuzzy linguistic information based on these algorithms. In the proposed approach, all unbalanced hesitant fuzzy linguistic information is transformed into LDAs defined on a balanced linguistic term set, and then an LDA-based clustering algorithm is devised to cluster decision makers. Based on the clustering result, decision makers' linguistic distribution decision matrices are further fused to obtain collective assessments of alternatives. In order to provide easy-to-understand linguistic results for decision makers, all LDAs of alternatives are represented by HLDs defined on each decision maker's initial linguistic term set. Finally, an example for the selection of subway lines is used to demonstrate the proposed approach.

Large-scale group decision making problems exist widely in human being's daily life. In this paper, a new approach to large-scale multi-attribute group decision making with multi-granular unbalanced linguistic information is developed. First, an algorithm is proposed to represent the initial multi-granular unbalanced linguistic information of decision makers with the use of unbalanced linguistic distribution assessments. Based on the gain and loss of an unbalanced linguistic distribution assessment over another, the classical TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) method is then extended to derive a raking of alternatives for large-scale multi-attribute group decision making problems. Finally, an example for talent selection is used to demonstrate the feasibility of the proposed approach.

Linguistic large-scale group decision making (LGDM) problems are more and more common nowadays. In such problems a large group of decision makers are involved in the decision process and elicit linguistic information that are usually assessed in different linguistic scales with diverse granularity because of decision makers' distinct knowledge and background. To keep maximum information in initial stages of the linguistic LGDM problems, the use of multigranular linguistic distribution assessments seems a suitable choice, however, to manage such multigranular linguistic distribution assessments, it is necessary the development of a new linguistic computational approach. In this paper, it is proposed a novel computational model based on the use of extended linguistic hierarchies, which not only can be used to operate with multigranular linguistic distribution assessments but also can provide interpretable linguistic results to decision makers. Based on this new linguistic computational model, an approach to linguistic large-scale multiattribute group decision making is proposed and applied to a talent selection process in universities.

The hesitant fuzzy linguistic term set (HFLTS) is a useful tool for a decision maker to elicit hesitant fuzzy linguistic information in decision making problems. In this paper, it is proposed the concept of unbalanced HFLTS based on the unbalanced linguistic term set. To fuse unbalanced HFLTSs in decision making, the unbalanced hesitant fuzzy linguistic weighted averaging operator and the unbalanced hesitant fuzzy linguistic ordered weighted averaging operator are developed, which follow the paradigm of computing with words and aggregate a collection of unbalanced HFLTSs into an unbalanced linguistic distribution assessment to provide interpretable results for decision makers. The proposed aggregation operators are further extended to fuse multi-granular unbalanced HFLTSs and applied to deal with multi-attribute group decision making problems. Eventually, an example for investment alternative selection is provided to demonstrate the proposed aggregation operators and the decision making approach.

Consistency and consensus play an important role in group decision making based on pairwise comparison matrices. In this paper, both the consistency and consensus issues for group decision making with interval pairwise comparison matrices are investigated. First, two minimum adjustment-based consistency improving models are proposed to simplify and improve the consistency improving model developed in a recent paper. Subsequently, to help decision makers reach consensus in group decision making, the group consensus degree is defined and then two minimum adjustment-based consensus reaching models are also developed, in which both the consistency and consensus issues are considered. It is also pointed out that the proposed consensus reaching model can generalize the consistency improving model. Through the proposed models, the consistency and consensus level for the interval pairwise comparison matrices can be improved, which will be helpful for deriving satisfactory and reasonable decision results in group decision making problems. Eventually, two numerical examples are provided to demonstrate the effectiveness of the proposed models.

The hesitant fuzzy linguistic term set is a useful tool for decision makers to express their linguistic assessments over alternatives. In this paper, some new operations of hesitant fuzzy linguistic term sets are proposed based on 2-tuple linguistic aggregation operators and distribution linguistic aggregation operators, which can avoid the loss of information and make the aggregation results interpretable. Based on the proposed aggregation operators, an approach to multi-attribute group decision making with hesitant fuzzy linguistic term sets is developed. Finally, an example is used to demonstrate the feasibility and effectiveness of the proposed approach.