Multi-Granularity Probabilistic Rough Fuzzy Sets for Interval-Valued Fuzzy Decision Systems

The probabilistic rough set (PRS) model, through the incorporation of error levels, represents a quantitative extension of the classical rough set model. It serves as a fundamental expansion that enables robust fault tolerance capabilities by employing relative quantitative description. However, when confronted with interval-valued fuzzy data, the PRS model is rendered ineffective. The primary reason for this lies in the absence of a unique equivalence relation in interval-valued decision systems. This paper presents a novel approach to address this limitation. In this paper, we first propose a fuzzy similarity relation based on diversity function, which establishes a viable foundation for constricting models of probabilistic rough fuzzy set and multi-granularity probabilistic rough set models for interval-valued fuzzy decision systems. Then the decision rules are derived from the presented three kinds of multi-granularity probabilistic rough fuzzy sets, respectively. In order to elucidate the concepts of interval-valued probabilistic rough fuzzy sets and multi-granularity probabilistic rough fuzzy sets, a case study is considered, which is helpful for applying these theories to deal with practical issues.