Fuzzy rough set on probabilistic approximation space over two universes and its application to emergency decision-making

Probabilistic approaches to rough sets are still an important issue in rough set theory. Although many studies have been written on this topic, they focus on approximating a crisp concept in the universe of discourse, with less effort on approximating a fuzzy concept in the universe of discourse. This article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space over two universes. We first present the definition of a lower and upper approximation of a fuzzy set with respect to a probabilistic approximation space over two universes by defining the conditional probability of a fuzzy event. That is, we define the rough fuzzy set on a probabilistic approximation space over two universes. We then define the fuzzy probabilistic approximation over two universes by introducing a probability measure to the approximation space over two universes. Then, we establish the fuzzy rough set model on the probabilistic approximation space over two universes. Meanwhile, we study some properties of both rough fuzzy sets and fuzzy rough sets on the probabilistic approximation space over two universes. Also, we compare the proposed model with the existing models to show the superiority of the model given in this paper. Furthermore, we apply the fuzzy rough set on the probabilistic approximation over two universes to emergency decision-making in unconventional emergency management. We establish an approach to online emergency decision-making by using the fuzzy rough set model on the probabilistic approximation over two universes. Finally, we apply our approach to a numerical example of emergency decision-making in order to illustrate the validity of the proposed method.