Order based hierarchies on hesitant fuzzy approximation space
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ORIGINAL ARTICLE
Order based hierarchies on hesitant fuzzy approximation space Eric C. C. Tsang1 · Jingjing Song1 · Degang Chen2 · Xibei Yang3 Received: 7 July 2017 / Accepted: 27 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract 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. Keywords Granular computing · Hesitant fuzzy approximation space · Hesitant fuzzy relation · Hesitant fuzzy set · Hierarchy
1 Introduction Granular computing was first proposed by Zadeh, which focused on a general theory and methodology for problem solving and information process by considering multiple levels of granularity [8, 50, 53]. There are three basic issues in granular computing, i.e., information granulation, organization and causation. Zadeh pointed out that the information granulation involves decomposition of whole into parts, the * Jingjing Song [email protected] 1
Faculty of Information Technology, Macau University of Science and Technology, Macau, People’s Republic of China
2
Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People’s Republic of China
3
School of Computer Science and Technology, Jiangsu University of Science and Technology, Zhenjiang 212003, Jiangsu, People’s Republic of China
organization involves integration of parts into whole, and the causation involves association of
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