The Promotion And Application Of The Fuzzy Rough Sets

Fuzzy sets and rough sets theories are two mathematical tools to solve the knowledge of ambiguity and uncertainty. During the past decades, Scholars have presented some extended forms of Fuzzy sets and rough sets in succession. At the same time, the integration of the two theories has also gained a rapid development, fuzzy rough sets and the various extensive forms of the fuzzy rough sets are proposed. On the basis of existing achievements, the further expansion studies of fuzzy rough sets are given in this thesis. Firstly, focus on researching hesitation fuzzy sets, the related properties of the dual hesitant fuzzy rough sets are discussed, and the hesitant fuzzy rough sets and the dual hesitant fuzzy rough sets models based on inclusion degree are established. Secondly, according to discussing the relationship of the various extended fuzzy sets, fuzzy rough sets are further extended to type-2 intuitionistic fuzzy rough sets. The main work is as follows:(1)By using the score function of the hesitant fuzzy sets, the inclusion degree of fuzzy sets is extended to the hesitant fuzzy set. The axiomatic definition and some calculation formulas of hesitant fuzzy inclusion degree are given. Secondly, based on the division of hesitation fuzzy, the conception of hesitant fuzzy inclusion approximation space has been proposed, the model based on the hesitant fuzzy rough set has been established and some basic properties of approximate operator has been discussed. then, an attribute reduction method is given based on inclusion degree in the hesitation fuzzy approximation space, which not only gains the maximum reduction sets, but also obtains corresponding decision rules accordingly, and the effectiveness and feasibility of the proposed approach is proved by a practical example. Finally, the above theory and method will be further expanded, the dual hesitant fuzzy set model based on inclusion degree is built, the related properties are discussed, and the method of attribute reduction is given by a practical example.(2)According to investigating the basic concepts of the rough hesitant fuzzy sets, the rough membership function of hesitant fuzzy sets is developed and its basic properties are researched, then the uncertainty measurement of rough hesitant fuzzy set is given by using the hesitant fuzzy entropy of rough membership function. In addition, some of the relevant properties of entropy are discussed. At the same time, based on the concepts and operational properties of dual hesitant fuzzy sets, the definition of dual hesitant fuzzy relation is given at first. Then, we will integrate the dual hesitant fuzzy sets with the rough sets, the dual hesitant fuzzy rough sets model is constructed in the dual hesitant fuzzy approximate space, and some basic properties of the model are discussed. Afterwards, the definitions of reflexivity, symmetry, transmission and equivalent dual hesitant fuzzy relation are given, and the characteristic relationship between the dual hesitant fuzzy relation and the upper and lower approximation operators is discussed.(3)The relationship between type-2 fuzzy sets and interval-value fuzzy sets, hesitant fuzzy sets, intuitionistic fuzzy sets and other sets is discussed, the relationship between the extended type-2 fuzzy sets, that is type-2 intuitionistic fuzzy sets, and interval-value fuzzy sets, hesitant fuzzy sets, intuitionistic fuzzy sets and other sets is further discussed. In addition, the basic operations between type-2 fuzzy sets and hesitant fuzzy sets and other sets are studied, and an example is provided to validate the effectiveness and feasibility. On the basis of the above discuss, a pair of type-2 intuitionistic fuzzy upper and lower approximation operators are defined in the type-2 intuitionistic fuzzy approximation space. The specific change forms of upper and lower approximation operators are discussed when the type-2 intuitionistic fuzzy relations degenerate into the common type-2 fuzzy relations and general equivalence relations. Then, the definition of type-2 intuitionistic fuzzy sets inclusion relation, and some properties of type-2 intuitionistic fuzzy upper and lower approximation operators are studied. The type-2 intuitionistic fuzzy relations under three characteristics are also defined. Finally, the relations between these three special type-2 intuitionistic fuzzy relations and the characteristics of approximation operators are discussed.