| In reality,the uncertainty of complex world shows in many aspects,including vagueness,fuzziness,randomness,roughness and hesitation,etc.The traditional mathematical methods have not been able to satisfy the requirements for dealing with the uncertainty.In this context,the researchers have proposed a series of new mathematical approaches to deal with vagueness and uncertainty,such as fuzzy set theory,rough set theory,soft set theory and the latest hesitant fuzzy set theory,etc.As a hot research direction in recent years,hesitant fuzzy set theory has attracted the attention of many researchers.There have been many fruitful results of hesitant fuzzy sets both in theory and applications.In this study,we concentrate on the fundamental theory of hesitant fuzzy sets and focus on the fusion of hesitant fuzzy sets and other uncertain theories like rough sets and soft sets.The main contributions of this dissertation are summarized as follows:1.We present a novel framework for the study of hesitant fuzzy rough sets by integrating rough sets with hesitant fuzzy sets.Lower and upper approximations of hesitant fuzzy sets with respect to a hesitant fuzzy approximation space are first defined,which improve the hesitant fuzzy rough set proposed by Yang et al.Properties of hesitant fuzzy approximation operators are also examined.Meanwhile,it is proved that the set of all lower approximation sets based on a hesitant fuzzy reflexive and transitive approximation space forms a hesitant fuzzy topology.And conversely,hesitant fuzzy topological space satisfying a certain condition can be associated with a hesitant fuzzy reflexive and transitive approximation space such that the induced lower hesitant fuzzy rough approximation operator is exactly the hesitant fuzzy interior operator of the given topology.That is to say,there exists a one-to-one correspondence between the set of all hesitant fuzzy reflexive and transitive approximation spaces and the set of all hesitant fuzzy rough topological spaces.2.Through a combination of hesitant fuzzy sets with multi-granulation rough sets,multi-granulation rough sets are generalized into hesitant fuzzy environment.Two types of the multi-granulation hesitant fuzzy rough set model are constructed: one is the optimistic multi-granulation hesitant fuzzy rough set;the other is the pessimistic multigranulation hesitant fuzzy rough set.The relationships among the single-granulation hesitant fuzzy rough sets,optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets are also established.In order to further measure uncertainty in multi-granulation hesitant fuzzy rough sets,the concepts of rough measure and rough measure about the parameters α and β are presented and some of their interesting properties are examined.Finally,we propose a reduction approach in multi-granulation hesitant fuzzy decision information systems based on multi-granulation hesitant fuzzy rough sets to eliminate redundant hesitant fuzzy granulations.And an example is provided to illustrate the validity of this approach.3.We generalize hesitant fuzzy rough sets into two universes environment.By a constructive approach,the hesitant fuzzy rough set model over two universes is first presented and some properties of this model are further discussed.Rough measure of hesitant fuzzy set in hesitant fuzzy approximation space over two universes is investigated.The union,the intersection and the composition of hesitant fuzzy rough sets over two universes are also discussed.Finally,we give a new approach of decision making in uncertainty environment by using the hesitant fuzzy rough set over two universes.Two practical applications are also provided to illustrate the validity of this approach.4.By combining the dual hesitant fuzzy set and soft set theory,we introduce an extension of soft set theory,which is called dual hesitant fuzzy soft sets.Further some operations on the dual hesitant fuzzy soft sets are investigated,such as complement operation,“AND” and “OR” operations,ring sum and ring product operations.Then,we present an adjustable approach to dual hesitant fuzzy soft sets based on decision making and some numerical examples are provided to illustrate the developed approach.Finally,the weighted dual hesitant fuzzy soft set is introduced and its application in decision making problem is also shown. |
