Three-way conflict analysis resolves real-world conflicts in two main steps: identifying the causes of conflict and designing feasible strategies to mitigate disputes. Existing approaches predominantly rely on linear conflict functions, which are insufficient for capturing nuanced variations in inter-agent relationships. Moreover, systematic development of feasible strategies has been limited. To address these limitations, this paper proposes a three-way conflict resolution model incorporating nonlinear conflict functions. Firstly, we introduce nonlinear conflict functions that accurately model the subtle fluctuations in conflict degrees. These functions are used to trisect both the set of agent pairs and the set of issues. We further propose a novel method to determine optimal trisections along with the corresponding optimal threshold pairs. Once the trisection phase is completed, we precisely identify the critical conflict issues that require resolution, where the parameter of conflict tolerance level is induced from the optimal issue trisection. Subsequently, conflict resolution of critical issues is formulated as an optimization problem, and two adjustment mechanisms are developed to derive distinct adjustment strategies. Finally, the applicability and superior performance of the proposed model, in terms of accurately identifying conflict issues and generating effective resolution strategies, are demonstrated through a commonly-used case study, a sensitivity evaluation, and a comparative analysis. The results confirm that our approach offers a flexible, efficient, and practical framework for three-way conflict analysis in complex decision-making contexts.

Anchor-based strategies have been widely used to accelerate spectral clustering, yet their effectiveness is directly affected by the quality of the selected anchors. Random sampling has become one of the most important anchor determination methods due to its efficiency. However, the anchors obtained by a single random sampling often fail to adequately capture the topological structure of the original data, making it difficult for the constructed anchor graph to achieve satisfactory clustering performance. To solve this problem, we propose a novel spectral embedding representation model based on random anchor graph aggregation (RAGA), in which an aggregated anchor graph can be produced to obtain enhanced sample representation capability. Specifically, we perform multiple random samplings to make the distribution of the selected anchors approximate the original data within a reasonable sampling time. Subsequently, adaptive weighted learning is performed on the contribution of the constructed multiple anchor graphs, and then an aggregated anchor graph can be formed, which can portray the topological structure of the original samples more precisely. In addition, spectral embedding and spectral rotation are integrated into a joint learning framework to reduce the model learning error accumulation caused by the traditional two-stage framework. Notably, we propose a rigorous theorem for analyzing the approximation of samples by the selected anchors in multiple random samplings. Our proposed RAGA maintains the speed advantage of random sampling while obtaining a high-quality aggregated anchor graph, enabling it to handle large-scale data scenarios. Experimental results on several benchmark datasets show that the RAGA model outperforms other state-of-the-art (SOTA) anchor graph-based clustering methods.

The twin support vector machine (TWSVM) classifier and its fuzzy variant fuzzy twin support vector machine (FTSVM) have received considerable attention due to their low computational complexity. However, their performance often deteriorates when the input data is affected by noise. To overcome this limitation, this study leverages the robustness of granular-ball computing (GBC) against noise to develop more effective classification models by integrating GBC with TWSVM and FTSVM. First, we introduce the granular-ball TWSVM (GBTWSVM) classifier, which incorporates GBC with the TWSVM framework. By replacing traditional point-wise inputs with granular-ball representations, we derive a pair of nonparallel hyperplanes for the GBTWSVM classifier by solving a quadratic programming problem. Afterwards, we develop the granular-ball FTSVM (GBFTSVM) classifier, where the membership and nonmembership functions of granular-balls are defined using Pythagorean fuzzy sets, enabling a more nuanced differentiation of the contributions of granular-balls from distinct regions within the input space. By incorporating these functions into the FTSVM framework, we derive a pair of nonparallel hyperplanes for the GBFTSVM classifier through the solution of a quadratic programming problem. Finally, we present algorithms for the GBTWSVM and GBFTSVM classifiers and evaluate their performance on 21 benchmark datasets. Experimental results demonstrate the superior scalability, computational efficiency, and robustness of the proposed classifiers in pattern recognition, highlighting their potential as advanced tools for noise-tolerant classification.

In some real-world situations, Pythagorean fuzzy sets are more powerful and effective than intuitionistic fuzzy sets to describe vague and uncertain information, and there are many Pythagorean fuzzy information systems for conflicts in which attitudes of agents on issues are depicted by Pythagorean fuzzy numbers. In this paper, we first provide the concepts of positive, neutral, and negative alliances with two thresholds and employ examples to illustrate how to compute positive, neutral, and negative alliances in Pythagorean fuzzy information systems for conflicts. Then, we focus on three-way conflict analysis based on the Bayesian minimum risk theory and explore examples to show how to compute the positive, neutral, and negative alliances with a Pythagorean fuzzy loss function given by an expert. Finally, we study how to calculate positive, neutral, and negative alliances with group decision theory and take examples to demonstrate how to construct the positive, neutral, and negative alliances with a group of Pythagorean fuzzy loss functions given by more experts.

Granular structure is a mathematical expression of knowledge in granular computing and a direct determinant of the data processing efficiency. To improve the efficiency of data processing, many scholars have studied the reduction of granular structure. The attribute reduction and the granular reduction are two types of reduction on different layers of a granular structure, with the latter being both an essential step for granular structure reduction and the foundation of the attribute reduction. Yet compared with the attribute reduction, the granular reduction has received less attention from scholars. Therefore, a fuzzy granular reduction theory and a granular matrix based on the fuzzy β-coverings is proposed in this article. The insufficiency of the existing granular reduction theory for fuzzy β-coverings is pointed out, and proper sufficient and necessary conditions for two fuzzy β-coverings generating the same upper and lower approximations are also given in this article. In addition, to reduce and evaluate a fuzzy β-covering, a novel reduction algorithm based on a granular matrix is proposed for the first time. Also, since fuzzy covering reduction is NP-hard, a heuristic greedy algorithm is designed to obtain a reduct. Numerical experiments show that the redundancy rates of neighborhood granule sets induced by some big-scale data sets exceed 99%, which indicates that the existing neighborhood granulation methods need to be urgently improved. Based on this, concise granular structures and much more efficient feature selection algorithms can be proposed in the future.

The introduction of information system homomorphisms has made a substantial contribution to attribute reduction. However, the efforts made on homomorphisms are far from sufficient. This paper further investigates homomorphisms between covering approximation spaces. First, we introduce the concepts of upper and lower homomorphisms as well as homomorphisms in order to study the relationship between covering approximation spaces. Then we present the notions of covering approximation subspaces and product spaces. We also compress covering approximation spaces and covering information systems with the aim of attribute reduction. Afterwards, by utilizing the compressions of the original spaces and systems we compress the dynamic covering approximation spaces and dynamic covering information systems. Several illustrative examples are employed to demonstrate that the homomorphisms provide an effective approach for compressing covering approximation spaces and covering information systems.

In practical situations, interval-valued fuzzy sets are of interest because fuzzy sets of this kind are frequently encountered. In this paper, motivated by the needs for solving imprecise problems, we generalize the concept of shadowed sets for understanding interval-valued fuzzy sets and provide a solution to compute a pair of thresholds by searching for a balance of uncertainty. Then we present three-way approximations of interval-valued fuzzy sets and a formulation for calculating the pair of thresholds using single-valued loss functions. We also compute three-way approximations of interval-valued fuzzy sets using interval-valued loss functions. Afterwards, we employ several examples to illustrate that how to take an action for an object with an interval-valued membership grade using an interval-valued loss function.