Parameterized fuzzy β-covering relations-based fuzzy rough set models and their applications to three-way decision and attribute reduction

In the existing research on fuzzy β-covering rough set (Fβ-CRS) models, it is often required that the parameter β of each covering be equal, which seriously restricts the flexibility and practical efficiency of the model in applications. Additionally, the existing fuzzy β-neighborhood operators (Fβ-NOs) have limited expressive power, as most of them fail to satisfy reflexivity or symmetry. Therefore, it is necessary to study a more flexible and general model. To address this issue, this paper constructs some novel Fβ-NOs and Fβ-CRS models within a new variable-scale fuzzy β-covering approximation space (VSFβ-CAS), and explores their properties. First, based on overlap functions and grouping functions, four types of Fβ-NOs satisfying reflexivity and symmetry are proposed in VSFβ-CAS, which include parameterized fuzzy β-neighborhoods and parameterized fuzzy complementary β-neighborhoods. Their properties are demonstrated and it is proven that all conform to the structural requirements of fuzzy β-covering relations. On this theoretical basis, four Fβ-CRS models with upper approximation-inclusion lower approximation relations are further constructed. The basic properties of the models and the interrelationships among different models are discussed. Finally, based on the four constructed models, we designed experiments for both samples and attributes, i.e., three-way decision and attribute reduction. First, we applied the four proposed models to three-way decision tasks: we defined both fuzzy and crisp three-way regions, and performed comparative experiments on public datasets using six measurements. Meanwhile, we introduced fuzzy dependency functions and fuzzy composite functions, and developed some variable-scale attribute reduction algorithms. We then conducted in-depth analyses on how different β values and function configurations affect experimental results, and carried out comparative experiments with existing models on public datasets. The experimental results indicate that our models not only demonstrate advantages and rationality in addressing both sample and attribute-related issues, but also exhibit higher flexibility—further validating the strong generalization ability of the proposed models.