Picture Hesitant Fuzzy Set and Its Application to Multiple Criteria Decision-Making

Abstract

Abstract:To address the complex multiple criteria decision-making (MCDM) problems in practice, this article proposes the picture hesitant fuzzy set (PHFS) theory based on the picture fuzzy set and the hesitant fuzzy set. First, the concept of PHFS is put forward, and its operations are presented, simultaneously. Second, the generalized picture hesitant fuzzy weighted aggregation operators are developed, and some theorems and reduced operators of them are discussed. Third, the generalized picture hesitant fuzzy prioritized weighted aggregation operators are put forward to solve the MCDM problems that the related criteria are at different priorities. Fourth, two novel MCDM methods combined with the proposed operators are constructed to determine the best alternative in real life. Finally, two numerical examples and an application of web service selection are investigated to illustrate the effectiveness of the proposed methods. The sensitivity analysis shows that the different values of the parameter λ affect the ranking of alternatives, and the proposed operators are compared with several existing MCDM methods to illustrate their advantages.Keywords: multiple criteria decision-making; picture hesitant fuzzy set; generalized picture hesitant fuzzy weighted aggregation operator; generalized picture hesitant fuzzy prioritized weighted aggregation operator

Multi-Granulation Picture Hesitant Fuzzy Rough Sets

“…The notion that arises is given in the next definition: Definition 6. [33] Let X be a non-empty set. A picture hesitant fuzzy set (abbreviated as PHFS) on the set X is…”

“…In a different line of thought, situations where the decision makers hesitate among various possible values appeal to hesitant fuzzy sets [32] and the hybrid models. For example, Wang and Li [33] have recently introduced picture hesitant fuzzy sets (PHFS).…”

“…As single granulation has limitations, as already explained, we adopt a structure based on multi-granulation, which is more suited for multi-source information systems. We point out that multi-granulation rough sets have been used in various fields, including decision making [34], feature selection [35], etc.…”

Linguistic Z-number weighted averaging operators and their application to portfolio selection problem

“…The fuzzy set theory, which was introduced by Zhdeh [1], has created massive progress in the representation of uncertain and ambiguous data and has successfully been applied in the different area. Following Zadeh [1], some authors developed the fuzzy set theory and presented its extended forms such as the interval-valued fuzzy sets [2], the type-2 fuzzy sets [3], the fuzzy multi-sets [4], the intuitionistic fuzzy sets [5], the hesitant fuzzy sets [6], the interval-valued hesitant fuzzy sets [7], picture fuzzy sets [8], Picture Hesitant Fuzzy Set [9], Proportional hesitant fuzzy linguistic term set [10] and proportional interval type-2 hesitant fuzzy set [11]. However, the fuzzy set theory and its extended forms are inefficient at representing the reliability of the information.…”

Hesitant Probabilistic Fuzzy Information Aggregation Using Einstein Operations

“…This method has been worked out to some extent. Wang and Li [30] proposed the picture hesitant fuzzy set to express the uncertainty and complexity of experts' opinions and applied them to solve diverse situations during MCDM processes. Interval-valued HFSs have been used in the applications of group decision making in [31].…”