Mathematical Analysis of Some Rough Hybridized Models

Abstract

The classical set theory has been extended to many valuable set theories due to the ongoing interests in knowledge representation and analysis of incompleteness and uncertainty in data analysis. Among other theories, rough set theory and fuzzy set theory have proved to be worthwhi le extensions. Fuzzy set theory was introduced by Zadeh [86] to model vague concepts. Elements of the universe are graded on the basis of their attributes; where grades are assigned from the interval [0, 1]. While as a mathematical tool, theory of rough sets was initially proposed by Pawlak [46] to handle imprecision and incompleteness in information systems. The initial approach adopted by Pawlak includes partitioning the universe set into granules (classes) of elements which are indistinguishable or indiscernible subject to the available information. Using these classes, an arbitrary subset of a universe is approximated by two definable subsets called lower and upper approximations. The concealed knowledge in the information system can thus be revealed by using the concepts of upper and lower approximations in the rough set theory. Another tool to study uncertainty is the theory of hesitant fuzzy sets presented by Torra [69]. In real life situations, it is sometimes very difficult to decide a definite membership grade for an element of a set. This type of situation occurs in multi-criteria decision making. When a group of specialists tries to decide a membership grade for an element of the set, it may not be unique. This hesitancy leads to define hesitant fuzzy sets.