Order based hierarchies on hesitant fuzzy approximation space

Granular computing which focuses on everyday and commonly used concepts and notions is a new field of multi-disciplinary study in dealing with theories, methodologies and techniques. As an important role in granular computing, hierarchy has attracted considerable attention. We are usually hesitant and irresolute for one thing when making decisions, which leads to a set of possible membership degrees. However, the existing hierarchies focus on crisp environment or fuzzy environment in which each element of the set has only one membership degree. To fill this gap, we research the hierarchies on hesitant fuzzy information granulations whose information granule has at least one membership degree of one object to the reference set. Firstly, we put forward new orders on hesitant fuzzy sets to characterize the hierarchies on hesitant fuzzy sets, the relationships of these orders are also researched. Moreover, we characterize the hierarchies on hesitant fuzzy information granulations from the viewpoint of granular computing. And then, new orders are presented to characterize the hierarchies on hesitant fuzzy information granulations. The order based hierarchies on hesitant fuzzy approximation space provide us with a more comprehensible perspective for the study of granular computing. Finally, two examples are given. One example is employed to compare the differences among the proposed orders on hesitant fuzzy sets, the other example is illustrated to show the orders on hesitant fuzzy sets that can be applied to hesitant fuzzy multi-attribute decision making. The results show that the orders proposed in this paper are effective to characterize the hierarchies on hesitant fuzzy approximation space.