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The linguistic Pythagorean fuzzy set (LPFS) is an important implement for modeling the uncertain and imprecise information. In this paper, a novel TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method is proposed for LPFSs based on correlation coefficient and entropy measure. To this end, the correlation coefficient is proposed for the relationship measurement between LPFSs. Afterwards, two entropy measures are developed to calculate the attribute weight information. Then, a novel linguistic Pythagorean fuzzy TOPSIS (LPF-TOPSIS) method is proposed to solve multiple attribute decision-making problems. Finally, the LPF-TOPSIS method is applied to handle a case concerning the selection of firewall productions, and then, a case concerning the security evaluation of computer systems is given to conduct the comparative analysis between the proposed LPF-TOPSIS method and previous decision-making methods for validating the superiority of the proposed LPF-TOPSIS method.

Compared to the intuitionistic fuzzy sets, the Pythagorean fuzzy sets (PFSs) can provide the decision makers with more freedom to express their evaluation information. There exist some research results on the correlation coefficient between PFSs, but sometimes they fail to deal with the problems of disease diagnosis and cluster analysis. To tackle the drawbacks of the existing correlation coefficients between PFSs, some novel directional correlation coefficients are put forward to compute the relationship between two PFSs by taking four parameters of the PFSs into consideration, which are the membership degree, non-membership degree, strength of commitment, and direction of commitment. Afterwards, two practical examples are given to show the application of the proposed directional correlation coefficient in the disease diagnosis, and the application of the proposed weighted directional correlation coefficient in the cluster analysis. Finally, they are compared with the previous correlation coefficients that have been developed for PFSs.