Abstract
As an extension of fuzzy sets, hesitant bipolar-valued fuzzy set is a new mathematical tool for dealing with fuzzy problems, but it still has the problem with the inadequacy of the parametric tools. In order to further improve the accuracy of decision making, a new mixed mathematical model, named hesitant bipolar-valued fuzzy soft set, is constructed by combining hesitant bipolar-valued fuzzy sets with soft sets. Firstly, some related theories of hesitant bipolar-valued fuzzy sets are discussed. Secondly, the concept of hesitant bipolar-valued fuzzy soft set is given, and the algorithms of complement, union, intersection, “AND,” and “OR” are defined. Based on the above algorithms, the corresponding results of operation are analyzed and the relevant properties are discussed. Finally, a multiattribute decision-making method of hesitant bipolar-valued fuzzy soft sets is proposed by using the idea of score function and level soft sets. The effectiveness of the proposed method is illustrated by an example.
Highlights
E rest of this paper is arranged as follows
A multiattribute decision-making approach for hesitant bipolar-valued fuzzy soft sets is given. is approach demonstrates that hesitant bipolar-valued fuzzy soft sets can be converted into simpler fuzzy soft sets by considering the score matrix
The theory of hesitant bipolar-valued fuzzy soft sets is firstly proposed based on soft sets and hesitant bipolar-valued fuzzy sets. en, some basic operations and corresponding properties are defined
Summary
Throughout the paper, let U be an initial limited universe set, E be the set of all possible parameters with respect to U, and D ⊆ E. Let P(U) denote the power set of U. A hesitant bipolar-valued fuzzy set H on U is defined as. Let Z (ZP, ZN) be a HBVFE and the score function S(Z) is defined as. (1) If S(ZH1(u)) ≤ S(ZH2(u)), H1 is called as a hesitant bipolar-valued fuzzy quasisubset of H2, denoted by H1⋐H2 or H2⋑H1. (2) If S(ZH1(u)) S(ZH2(u)), H1 and H2 are called hesitant bipolar-valued fuzzy quasiequal, denoted by H1 ≈ H2. Let H 1, H 2 ∈ HBVFS(U); (1) Complement: Hc1 〈u, ZHc1 (u)〉 | u ∈ U. Let H1, H2, H3 ∈ HBVFS(U); we have some operational laws, such as (1) (Hc1)c H1. It can be obtained from eorem 1 and Definition 7. Let H1, H2 ∈ HBVFS(U); we have (1) H1⋒H2⋐ H1⋐H1⋓H2. (2) It is similar to the proof of conclusion 1
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