Abstract
An interval-valued Pythagorean hesitant fuzzy set (IVPHFS) not only can be regarded as the union of some interval-valued Pythagorean fuzzy sets but also represent the Pythagorean hesitant fuzzy elements in the form of interval values. So IVPHFSs are extensions of Pythagorean hesitant fuzzy sets (PHFSs) and interval-valued Pythagorean fuzzy sets (IVPFSs), which are powerful tools to represent more complicated, uncertain and vague information. This paper focuses on the four kinds of correlation coefficients for PHFSs, and extends them to the correlation coefficients and the weighted correlation coefficients for IVPHFSs. In the processing, we develop the least common multiple expansion (LCME) method to solve the problem that the cardinalities of Pythagorean hesitant fuzzy elements (PHFEs) (or interval-valued Pythagorean hesitant fuzzy elements (IVPHFEs)) are different. In addition, we propose score functions and accuracy functions of Pythagorean fuzzy elements (PFEs) (or interval-valued Pythagorean fuzzy elements (IVPFEs)) to rank all the PFEs (or IVPFEs) in a PHFE (or an IVPHFE). Especially, score functions and accuracy functions of IVPFEs are both presented as interval numbers. Then use the comparison method of interval numbers to compare two revised IVPHFEs in order to keep the original fuzzy information as far as possible. What’s more, we define the local correlations and local informational energies which can depict the similarity between two IVPHFEs more meticulously and completely. At last the numerical examples to show the feasibility and applicability of the proposed methods in multiple criteria decision making (MCDM) and clustering analysis.
Highlights
Correlation plays an important role in mathematics, statistics and engineering sciences
We mainly focus on the correlation measures for Pythagorean hesitant fuzzy sets (PHFSs) and interval-valued Pythagorean hesitant fuzzy set (IVPHFS)
What’s more, we introduce the concepts of the local correlations and the local informational energies, and we deduce the four formulas of correlation coefficients for PHFSs and IVPHFSs and the four weighted correlation coefficients for IVPHFSs
Summary
Correlation plays an important role in mathematics, statistics and engineering sciences. We mainly focus on the correlation measures for PHFSs and IVPHFSs. Torra and Narukawa [35] and Torra [36] introduced the concept of hesitant fuzzy sets (HFSs) which permit the membership degree of an object to a set of several possible values. Meng and Chen [24] proposed the correlation coefficients of HFSs based on fuzzy measures and they [25] extended the method to study the case of interval-valued hesitant fuzzy sets. Some scholars combined PFSs with HFSs and introduced Pythagorean hesitant fuzzy sets (PHFSs), but their definitions are little different.
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