Correlation Coefficients of Interval-Valued Pythagorean Hesitant Fuzzy Sets and Their Applications

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.