Three-way decisions with reflexive probabilistic rough fuzzy sets

The theory of three-way decisions is used to classify a set of objects into the three disjoint parts of the positive, negative and boundary regions. This paper mainly studies three-way decisions with reflexive probabilistic rough fuzzy sets. We first discuss reflexive probabilistic rough sets in a reflexive probabilistic approximation space. For any fuzzy set and a level value, reflexive probabilistic rough fuzzy sets are introduced based on a pair of thresholds and a reflexive binary relation. Related properties of them are investigated. The lower and upper reflexive probabilistic rough fuzzy approximations are monotonic when the two thresholds and the level value increase or decrease. To give the physical interpretation of the required thresholds in reflexive probabilistic rough fuzzy sets, a set of states is constructed based on any fuzzy set and a level value. Three-way classifications in reflexive probabilistic rough fuzzy sets are then discussed. This method gives the values of the pair of thresholds using the loss functions. Meanwhile, using the minimum-risk decision rules, the lower and upper reflexive probabilistic rough fuzzy sets are constructed, which are consistent with the meaning of the lower and upper approximations in rough set theory. Related algorithm and example are also shown to explain the proposed method.