Soft Set And Hesitant Fuzzy Set With Their Application In Decision Making

There are a variety of uncertain problems in real world. Among of them can not be dealt effectively with using the existing tool, such as fuzzy set, interval mathematics, rough set and so on. For solving these problems, Molodsov and Torra proposed the soft set and hesitant fuzzy set, respectively. These are the new mathematic tools for modeling uncertainties. A soft set is a pair consisting of a parameter set and a set-valued mapping from the parameter set into the power set of the universe of discourse. Fuzzy sets can be considered as a special case of soft sets. In the research on the theory and application of soft sets, the combination of soft sets and other uncertain method is an important research direction. And a lot of research results have been obtained, such as fuzzy soft set, soft rough set, type2fuzzy soft set, interval-valued fuzzy soft set and so on. Hesitant fuzzy set permits the membership degree of an element to a set to be represented as several possible values in [0,1], which is a new generalization of fuzzy set. The basic unit of hesitant fuzzy set is the hesitant fuzzy element, which can well describe the state of hesitation when it is difficult to make decisions. The study on hesitant fuzzy set includes mainly distance, similarity measure, aggregation operator and their application in multiple attribute decision making.Although the study of soft set and hesitant fuzzy set has made many achieve-ments, but the research is still in the primary stage and still need further to be developed and improved in many aspects. Based on the previous results, we con-tinue to investigate deeply the theory and application of soft set and hesitant fuzzy set. The detailed arrangement of this paper stands out as follows:(1) Study on several hybrid models of soft set and their application in decision making. By combining soft set and hesitant fuzzy set, generalized fuzzy soft set and interval-valued fuzzy set and intuitionistic fuzzy set, we propose hesitant fuzzy soft set, generalized interval-valued fuzzy soft set and generalized intuitionistic fuzzy soft set, respectively. Some basic operations on the proposed soft set are defined. For hesitant fuzzy soft set, the extended-strict intersection, extended-strict union, strict-strict intersection and strict-strict union are also defined. Some properties of those operators are studied. Furthermore, we discuss the lattice structures of intuitionistic fuzzy soft set and hesitant fuzzy soft set. Based on generalized interval-valued fuzzy soft set and hesitant fuzzy soft set, we propose an approach for multiple attribute decision making, respectively. Meanwhile a similarity measure of intuitionistic fuzzy soft set is presented and an application in medical diagnosis is given.(2) Study on similarity measures of hesitant fuzzy set and their application in decision making. We adjust the axiomatic definitions of distance and similarity measure between hesitant fuzzy sets. On the basis of the new definitions, we propose a series of distance measures on hesitant fuzzy sets, and get the similarity measures corresponding those distances. Based on the set-theoretic approach and matching function, we also develop several similarity measure on hesitant fuzzy set. Moreover, according to the definition of positive and negative ideal hesitant fuzzy set, we propose a method for multiple attribute decision making based on the proposed similarity measures.(3) Study on some hesitant fuzzy Einstein aggregation operators and their ap-plication in multiple attribute decision making. To distinguish two hesitant fuzzy elements, we define an accuracy function of hesitant fuzzy elements and propose a new method to compare two hesitant fuzzy elements. We also give some Ein-stein operations on hesitant fuzzy element and discuss some of their properties. Based on those operations, we propose the hesitant fuzzy Einstein weighted av-eraging (HFEWAε) and the hesitant fuzzy Einstein ordered weighted averaging (HFEOWAε) operators. Moreover, by combining generalized mean and prior-itized averaging operators, we develop the generalized hesitant fuzzy prioritized Einstein weighted averaging (GHFPEWAε) and the generalized hesitant fuzzy prioritized Einstein weighted geometric (GHFPEWGε) operators. At the same time, some desirable properties of those operators and the relationships between the proposed operators and the existing operators are discussed. Base on the hesitant fuzzy Einstein aggregation operators and the generalized hesitant fuzzy prioritized Einstein aggregation operators, we develop a new approach for multiple attribute decision making and multiple attribute group decision making, respectively.