Inclusion measure for typical hesitant fuzzy sets,the relative similarity measure and fuzzy entropy

Typical hesitant fuzzy sets (THFSs), possessing a finite-set-valued fuzzy membership degrees called typical hesitant fuzzy elements (THFEs), is a special kind of hesitant fuzzy sets. Fuzzy inclusion relationship, as the order structure in fuzzy mathematics, plays an elementary role in the theoretical research and practical applications of fuzzy sets. In this paper, a new partial order for THFEs is defined via the disjunctive semantic meaning of a set, based on which fuzzy inclusion relationship is defined for THFSs. Furthermore, inclusion measures are defined to present the quantitative ranking of every two THFEs and THFSs and different inclusion measures are constructed. The related similarity measure, distance and fuzzy entropy of THFSs are presented and their relationship with inclusion measures are investigated. Finally, an example is given to show that the inclusion measure can be applied effectively in hesitant fuzzy multi-attribute decision making.