Multi-granulation fuzzy probabilistic rough sets and their corresponding three-way decisions over two universes

This article introduces a general framework of multi-granulation fuzzy probabilistic rough sets (MG-FPRSs) models in multi-granulation fuzzy probabilistic approximation space over two universes. Four types of MG-FPRSs are established, by the four different conditional probabilities of fuzzy event. For different constraints on parameters, we obtain four kinds of each type MG-FPRSs over two universes. To find a suitable way of explaining and determining these parameters in each kind of each type MG-FPRS, three-way decisions (3WDs) are studied based on Bayesian minimum-risk procedure, i.e., the decision-theoretic rough set (DTRS) approach. The main contribution of this paper is twofold. One is to extend the fuzzy probabilistic rough set (FPRS) to MG-FPRS model over two universes. Another is to present an approach to select parameters in MG-FPRS modeling by using the process of decision-making under conditions of risk.