Abstract
In this paper, we propose a new type of fuzzy covering-based rough set model over two different universes by using Zadeh’s extension principle. We mainly address the following issues in this paper. First, we present the definition of fuzzy \(\beta\)-neighborhood, which can be seen as a fuzzy mapping from a universe to the set of fuzzy sets on another universe and study its properties. Then we define a new type of fuzzy covering-based rough set model on two different universes and investigate the properties of this model. Meanwhile, we give a necessary and sufficient condition under which two fuzzy \(\beta\)-coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation. Moreover, matrix representations of the fuzzy covering lower and fuzzy covering upper approximation operators are investigated. Finally, we propose a new approach to a kind of multiple criteria decision making problem based on fuzzy covering-based rough set model over two universes. The proposed models not only enrich the theory of fuzzy covering-based rough set but also provide a new perspective for multiple criteria decision making with uncertainty.
Highlights
This paper firstly studies fuzzy covering-based rough set on two different universes and gives an illustrated example of multiple criteria decision making problem, which is discussed by using fuzzy covering-based rough set
We present the basic description of a multiple criteria decision making problem under the framework of fuzzy β-neighborhood over two universes, and give a general decision making methodology for multiple criteria decision making problem by using the fuzzy covering-based rough set theory over two universes
Fuzzy β-covering is a new notion defined by Ma in [32], which can build a bridge between covering-based rough set theory and fuzzy set theory
Summary
This paper firstly studies fuzzy covering-based rough set on two different universes and gives an illustrated example of multiple criteria decision making problem, which is discussed by using fuzzy covering-based rough set. Rough set theory (RST) was originally proposed by Pawlak [40, 41] in 1982 as a useful mathematical tool for dealing with the vagueness and granularity in information systems and data analysis. This theory can approximately characterize an arbitrary subset of a universe by using two definable subsets called lower and upper approximations [8]. Yeung et al [76] have proposed some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and studied the relationship among them from the viewpoint of constructive approach.
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