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Decision-making problem is a fundamental activity in human life every day. Regarding group decision-making, individuals have insufficient comprehension of the decision objects owing to the limited experience and knowledge and drawback the evaluation process. Therefore, the evaluation information is missing and the group cannot get the precise information for decision objects resulting in the generation of incomplete information systems. It is meaningful for individuals finding an approach to Leverage and tap into existing information. On the basis, the work presented in this paper focuses on group decision-making problems (GDM) with incomplete information. Under the framework of group decision-making, a GDM approach with an incomplete information system is presented combining the formal concept analysis and rough set theory. Firstly, we propose a concept called formal decision context and discuss the properties of the formal decision context, which could divide the incomplete information system into many cognitive units, and the definitions of cognitive units will be given. Then, the property and the relations of various cognitive units will be discussed. Additionally, the upper and lower approximation space under the formal decision context is founded considering the risk attitudes integrating fuzzy rough set and formal concept context. Considering the order relation of various cognitive units and decision objects, we introduce the dominance relation and the importance degree of each cognitive unit under the framework of formal decision context. Furthermore, a theoretical framework of the multi-criteria decision-making problem is presented and two models for the multi-criteria group decision-making problem are discussed combined with fuzzy rough set and formal decision context from two perspectives, including qualitative analysis and quantitative analysis. Finally, an illustrated example is applied to verify the validity and effectiveness of the proposed models. The contributions of this paper include two aspects. On the one hand, this paper provides a new perspective and paradigm considering MCGDM problems with the incomplete information system from the perspective of group consensus. On the other hand, this paper gives two approaches for group decision-making problems from sorting and set theory by the perspective of multi-granular rough set and group consensus. • A model of formal decision context with incomplete information to describe the cognitive situation. • An approach to MCDM problem combined rough set with formal decision context with incomplete information. • Two approaches to MCGDM problem from qualitative analysis and quantitative analysis. [ABSTRACT FROM AUTHOR]

This paper proposes a rectified complex spherical fuzzy set (rCTSFS) model that enables the phase term of a complex number to function truthfully to its inherent meaning of representing directions, phases, or color hues. In addition, this paper proposes the score and accuracy functions for rectified complex spherical fuzzy numbers (rCTSFn), which allows the three types of membership degrees of an rCTSFn to fulfill human judgment/intuition. The proposed rCTSFS model proves to be a productive extension of the complex spherical fuzzy set (CSFS), complex fuzzy set (CFS), and spherical fuzzy set (SFS) models. On the other hand, Dombi t -norms prove more flexible and comprehensive than some of the other families of triangular norms, such as the algebraic t -norms and the Einstein t -norms, due to the presence of a parameter γ. The parameter γ determines the amount of aggressiveness at estimating the maximum and the minimum of a population based on a sample obtained. Therefore, this paper proposes two Dombi-Choquet integral operators, namely, the rectified complex t -spherical fuzzy arithmetic Dombi-Choquet integral (r C T S F A γ , w λ ) operator and the rectified complex t -spherical fuzzy geometric Dombi-Choquet integral (r C T S F G γ , w λ ) operator. A multi-criteria decision making (abbr. MCDM) algorithm utilizing the two Dombi-Choquet integral operators is innovated. The proposed Dombi-Choquet MCDM algorithm for the rCTSFS model is applied to an MCDM problem related to diabetic retinopathy detection on five real-life fundus images of different severity levels taken from the Messidor2 dataset. Our newly proposed algorithm proves to be the only algorithm that yields the correct diagnostic results that match the hard truth. On the other hand, none of the 50 algorithms observed among recent works in literature can produce the correct diagnostic results, even after lending the fuzzification procedure innovated in this work to them. [ABSTRACT FROM AUTHOR]

• The centroid distance of PFNs is proposed and its properties are discussed. • Propose a score function based on the centroid distance and discuss its properties. • Applications of Pythagorean fuzzy clustering, ranking and MCDM are developed. • Some advantages of the proposed methods are showed through comparative analysis. By using Pythagorean fuzzy set (PFS), people can quantitatively describe and process information with fuzziness, and then make reasonable and effective judgments or decisions. In this process, distance measure often serves as a useful and important tool. The purpose of this paper is to explore some information processing methods and techniques in Pythagorean fuzzy environment. Firstly, the research progress of PFS is reviewed, and the main research directions and some hot issues in relevant research fields are discussed. Secondly, by analyzing the information of hesitation degree of Pythagorean fuzzy number (PFN), definitions of hesitation factor and centroid distance measure based on the representation of centroid coordinates of the hesitation regions are proposed, and properties of hesitation factor and centroid distance measure are discussed. Finally, the novel clustering algorithm, ranking method and multi criteria decision making (MCDM) method are developed in Pythagorean fuzzy environment, which are based on the new centroid distance measure. The effectiveness of the proposed algorithm and methods are verified through numerical examples and comparative analysis. The research results indicate that the centroid distance measure proposed in this paper can effectively characterize the distance between PFNs, thus making the clustering algorithm of PFNs, the ranking method of PFNs, and the Pythagorean fuzzy MCDM method have obvious superiorities and advantages. [ABSTRACT FROM AUTHOR]

Given that the majority of existing approaches for multi-criteria group decision making (MCGDM) rely solely on the preferences of decision makers (DMs) and fail to consider the various relationships between alternatives, this paper attempts to model the relevant relational structures using graphs and introduce the concept of graph neural networks (GNNs) in the context of group decision-making. By leveraging the powerful expressive capabilities of GNNs, the aim is to mine additional information pertinent to the decision-making process and screen out alternatives for the final decision. To begin, we provide a mapping of MCGDM to the graph domain and construct a corresponding relation graph among alternatives. Additionally, to deal with uncertain or vague information, we transform the group decision-making problem into a Pythagorean fuzzy environment and define a novel measure of entropy specifically designed for Pythagorean fuzzy sets (PFSs) in the entropy weight model to determine the weights of criteria. Simultaneously, we propose a new distance measure for PFSs, which is then applied to the extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method to rank alternatives. Furthermore, we develop a GNNs-based Pythagorean fuzzy MCGDM approach that incorporates the aforementioned techniques for group decision-making. Finally, to validate the effectiveness and superiority of this approach, we employ it to address a supplier selection issue. Compared with baseline group decision-making approaches, our approach can indeed capture the relationships among alternatives in complex group decision-making scenarios and outperforms the best-performing baseline by nearly 2.8% in terms of ranking accuracy. • A mapping from multi-criteria group decision making to the graph domain is provided. • A normalized entropy measure is defined in Pythagorean fuzzy sets. • A modified Pythagorean fuzzy distance measure is proposed. • A Pythagorean fuzzy group decision-making approach based on graph neural networks is developed. [ABSTRACT FROM AUTHOR]