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With the generalisation of intuitionistic fuzzy sets (IFSs), picture fuzzy sets (PFSs) have been developed based on membership, neutral membership, and non-membership degrees. Compared with IFSs, PFSs can more accurately represent uncertainty in real-world decision-making problems. Recently, the research on decision-making theories and methods under picture fuzzy environments has been rapidly developing. Therefore, this manuscript presents a systematic review of picture fuzzy decision-making theories and methods, including the context in which this field was developed, the current status and advancements of this field, and the main research results obtained with the picture fuzzy information. First, this review introduces the development process of PFSs and the current status of corresponding theories. Next, the basic theories based on PFSs, including operations rules, measures, and aggregation operators are introduced. Afterward, this review summarises the current research on multi-criteria decision-making (MCDM), multi-criteria group decision-making (MCGDM), and large-scale group decision-making (LSGDM) methods with picture fuzzy information. Finally, future research directions of picture fuzzy decision-making theories and methodologies are discussed. [ABSTRACT FROM AUTHOR]

Large-scale group decision-making (LSGDM) problems involve two processes: a clustering process is first implemented to break down a larger group into several smaller subgroups for simplification purposes, after which a consensus reaching process (CRP) is utilized to eliminate information conflicts among large-scale decision-makers (DMs). Opinion conflicts are inevitable in large DM groups due to self-interest biases and information blind spots. To address these limitations, this study proposes a twofold feedback mechanism to address both non-cooperative behaviors and information hesitation. Firstly, based on detailed review of existing similarity measures, we defined a novel picture fuzzy Chi-square similarity measure (NPFCS) to enhance accuracy. To distinguish subgroups, the second procedure was developed to establish a weight assignment model following the implementation of the developed similarity-based fuzzy clustering algorithm. In the third step, decision opinions are examined from two distinct perspectives: the hesitancy degrees and non-cooperative behaviors, to ensure favorable decision outcomes. Alongside the definition of a hierarchical mechanism for detecting DMs with information blind spots, the identified DMs were mandated to resubmit individual matrices. This measure aims to prevent the aggregation of negative impacts on information credibility caused by the major principle. Furthermore, non-cooperative behaviors were identified at the element level of multi-criteria matrices with corresponding adjustments performed automatically. Next, a real-world site selection problem involving thirty DMs was solved using the proposed method to illustrate its application. Moreover, the results of sensitivity analysis demonstrate that our proposal is robust to changes in model parameters, and several comparisons were conducted from two perspectives, thus confirming the superiority of the proposed approach. • Thispaperdefinesa novelChi-squarepicture fuzzy similaritymeasure. • The weightingassignmentconsidersboth size and cohesion of subgroups. • A consensus methodology respondsto hesitancy degrees and non-cooperative behaviors. • The developed twofold feedback mechanism promotes information modificationprecisely [ABSTRACT FROM AUTHOR]

In the multi‐criteria group decision‐making (MCGDM) problems with great uncertainty, making full use of participants' evaluation information could help improve the accuracy and reliability of decision results. Probabilistic linguistic term set (PLTS) is an effective tool to represent qualitative data and can fully express the hesitation and preference of decision makers. Therefore, this paper aims to propose an MCGDM method based on PLTSs. In the proposed method, the projection of PLTSs is explored to measure the distance and angle differences between two objects, and Bayesian best–worst method (Bayesian BWM) is used to determine the aggregated final weights of criteria. Besides, the elimination and choice translating reality III (ELECTRE III) method combined with distillation algorithm deals with the projection of PLTSs to obtain the alternatives' ranking of each decision maker. Then, the weighted convex median voting rule is developed to integrate the rankings results regarding all decision makers, which can solve the conflict of ranking results among experts and ensure that the comprehensive ranking results are reasonable and practical. Finally, a case study of health‐care waste management is designed and comparative analyses are implemented to show the effectiveness and advantages of the proposed method. [ABSTRACT FROM AUTHOR]

Interval rough numbers play an important role in dealing with complex fuzzy relationships. In this paper, a group decision-making (GDM) model based on interval rough multiplicative reciprocal (IRMR) matrix is proposed. Firstly, the inconsistency, satisfactory consistency and complete consistency of the IRMR matrix are defined from the perspective of set theory. Secondly, an improved method for the inconsistent IRMR matrix is introduced to address the inconsistent preference matrix in GDM. We define the uniform approximation matrix of the IRMR matrix, prove its existence, and provide a new calculation method for the sorting vector of IRMR matrix. Finally, the multiplicative reciprocal matrix obtained with a weighted geometric operator assembly is still the IRMR matrix. A GDM algorithm of the IRMR matrix is presented. The proposed algorithm is demonstrated using an illustrative example, and its feasibility and effectiveness are verified through comparison with other existing methods. [ABSTRACT FROM AUTHOR]

As a variation of fuzzy sets and intuitionistic fuzzy sets, neutrosophic sets have been developed to represent uncertain, imprecise, incomplete and inconsistent information that exists in the real world. Simplified neutrosophic sets (SNSs) have been proposed for the main purpose of addressing issues with a set of specific numbers. However, there are certain problems regarding the existing operations of SNSs, as well as their aggregation operators and the comparison methods. Therefore, this paper defines the novel operations of simplified neutrosophic numbers (SNNs) and develops a comparison method based on the related research of intuitionistic fuzzy numbers. On the basis of these operations and the comparison method, some SNN aggregation operators are proposed. Additionally, an approach for multi-criteria group decision-making (MCGDM) problems is explored by applying these aggregation operators. Finally, an example to illustrate the applicability of the proposed method is provided and a comparison with some other methods is made. [ABSTRACT FROM AUTHOR]