Uncertainty measurements for Pythagorean fuzzy set and their applications in multiple-criteria decision making

Pythagorean fuzzy set (PFS) is an effective approach for handling uncertainty in practical applications, whereas how to measure the uncertainty degree of PFS is still an open issue. In this paper, we propose a novel entropy measure of PFS by taking into account both Pythagorean fuzziness entropy in terms of the membership and non-membership degrees, and Pythagorean hesitation entropy in terms of hesitation degree. To be specific, we firstly propose a new cross-entropy measure of PFS to assess the difference of uncertainty between PFSs. By measuring the cross-entropy with the crisp set, we then propose a new entropy measure of PFS, which mathematically proves the satisfaction requirements of the axioms defined for the entropy of PFS. Based on that, we propose weighted cross-entropy and entropy measures which can be used for different applications. Furthermore, we illustrate the superiority of the proposed entropy measures of PFS by the comparisons with existing entropy measures of PFS. In addition, we propose an entropy-based multi-criteria decision-making (MCDM) method, which intends to minimize the total uncertainty amount of decision matrix to make the deciding result more convincing. As a result, we conduct a sensitivity analysis of the proposed weighted entropy measure in the MCDM method, and a company investment case study is attached to illustrate the effectiveness of the proposed MCDM method. The experimental result shows that the MCDM method based on the newly defined weighted entropy measure is stable, even in the setting of the variation of parameter.