Fuzzy and interval-valued fuzzy decision-theoretic rough set approaches based on fuzzy probability measure

This paper investigates decision-theoretic rough set (DTRS) approach in the frameworks of fuzzy and interval-valued fuzzy (IVF) probabilistic approximation spaces, respectively. It takes fuzzy probability and IVF probability into consideration. Bayesian decision procedure is a basis of DTRS approach. By integrating fuzzy probability measure and IVF probability measure into Bayesian decision procedure, there come fuzzy decision-theoretic rough set (FDTRS) approach and interval-valued fuzzy decision-theoretic rough set (IVF-DTRS) approach. The new approaches have the ability to directly deal with real-valued and interval-valued data. This makes FDTRS and IVF-DTRS more applicable than DTRS. Two methods are presented to compare intervals while constructing the IVF-DTRS approach: one is compatible with DTRS and FDTRS approaches; the other is a total order based on which the decision procedure is much easier to operate. Cases of two different universes of discourse for FDTRS and IVF-DTRS are also taken into account.