Research On Algebraic Structures Of Hesitant Fuzzy Sets

In most decision problems,decision makers often hesitate between several possible values,when they are determining the membership degree of an element.And the number of possible values offered by different decision makers is usually different.In order to solve such problems,Torra defined hesitant fuzzy sets,which is a new extension of fuzzy sets.Hesitant fuzzy sets extend the membership degree of an element from a number of interval[0,1]to a subset of[0,1].Therefore,hesitant fuzzy sets can depict the hesitation of experts in decision making effectively.This thesis discusses the lattice structure of hesitant fuzzy sets,and extends the partial order of hesitant fuzzy sets to the set of nonempty finite vectors V([0,1]),then the lattice structure of V([0,1])is discussed.Also,the thesis defines some operations about q-rung hesitant fuzzy sets,and some of their properties are studied.The main results are summarized as follows:1.The thesis discusses the relationship between partial orders ?1,?2 and ?*on hesitant fuzzy sets,and proves that the partial order ?*is contained in the partial orders?1,?2.Based on the order?*,the thesis obtains that the set of nonempty finite subsets in[0,1]neither form a lattice nor form a multilattice.Then the order ?*is extended to the order ?V*on the set of nonempty finite vectors V([0,1]),and the order ?V*is turned out be a preorder.At the same time,an equivalence relaton ? is defined on V([0,1]).And the preorder ?V*is extended to the order(?)on the set of all equivalence classes V([0,1])/?.It proves that(?)is a partial order on V([0,1])/?.Then,it is proved that V([0,1])/? can form a multilattice based on the partial order(?).2.Some basic operations of q-rung fuzzy sets are defined and their properties are discussed.By combining q-rung fuzzy sets and hesitant fuzzy sets,the concept of q-rung hesitant fuzzy sets is proposed.Then,some basic operations of q-rung hesitant fuzzy sets are defined such as geometry,arithmetic,multiplication,and some properties of these operations are researched.Based on the operations of q-rung hesitant fuzzy sets,the multicriteria decision making model is constructed,and the feasibility and validity of the proposed model are verified by examples.