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The extension of intuitionistic fuzzy sets (IFS) to Pythagorean fuzzy sets (PFS) is a significant advancement, addressing the inherent limitations of IFS. This study introduces a novel entropy measure specifically designed for Pythagorean fuzzy sets, establishing its axiomatic definition and presenting key properties. Decision making guided by entropy is advantageous, as it effectively mitigates ambiguity with increasing entropy values. Furthermore, a numerical example is provided to facilitate a comparative assessment of our newly introduced entropy measure in contrast to existing PFS entropy measures. The validation of our findings is achieved through the application of the COPRAS method, which determines decision outcomes based on a multitude of influencing factors. Notably, the determination of weights in this method is underpinned by the utilization of our innovative entropy measure. [ABSTRACT FROM AUTHOR]

The quality and efficiency for urban water environment treatment projects plays an important role for social and economic sustainable development of urban. Considering the uncertainty and complexity and the fuzziness of human’s thinking in urban water environment treatment projects, this study established evaluation method for quality management performance under Pythagorean fuzzy environment. Firstly, the weight vector is obtained using Pythagorean fuzzy entropy. Second, to measure the evaluation result in terms of score, a transfer equation used to transform the Pythagorean fuzzy number to performance evaluation score is proposed. Third, a quality management performance evaluation method combining Pythagorean fuzzy set and TOPSIS method is established. Finally, applying the method to a real situation of quality management activities in the construction phase of urban water environment treatment project, the performance results is C, which shows the feasibility and applicability of the presented approach. [ABSTRACT FROM AUTHOR]

Entropy and cross-entropy are very vital for information discrimination under complicated Pythagorean fuzzy environment. Firstly, the novel score factors and indeterminacy factors of intuitionistic fuzzy sets (IFSs) are proposed, which are linear transformations of membership functions and non-membership functions. Based on them, the novel entropy measures and cross-entropy measures of an IFS are introduced using Jensen Shannon-divergence (J-divergence). They are more in line with actual fuzzy situations. Then the cases of Pythagorean fuzzy sets (PFSs) are extended. Pythagorean fuzzy entropy, parameterized Pythagorean fuzzy entropy, Pythagorean fuzzy cross-entropy, and weighted Pythagorean fuzzy cross-entropy measures are introduced consecutively based on the novel score factors, indeterminacy factors and J-divergence. Then some comparative experiments prove the rationality and effectiveness of the novel entropy measures and cross-entropy measures. Additionally, the Pythagorean fuzzy entropy and cross-entropy measures are designed to solve pattern recognition and multiple criteria decision making (MCDM) problems. The numerical examples, by comparing with the existing ones, demonstrate the applicability and efficiency of the newly proposed models. [ABSTRACT FROM AUTHOR]

Entropy and cross-entropy are very vital for information discrimination under complicated Pythagorean fuzzy environment. Firstly, the novel score factors and indeterminacy factors of intuitionistic fuzzy sets (IFSs) are proposed, which are linear transformations of membership functions and non-membership functions. Based on them, the novel entropy measures and cross-entropy measures of an IFS are introduced using Jensen Shannon-divergence (J-divergence). They are more in line with actual fuzzy situations. Then the cases of Pythagorean fuzzy sets (PFSs) are extended. Pythagorean fuzzy entropy, parameterized Pythagorean fuzzy entropy, Pythagorean fuzzy cross-entropy, and weighted Pythagorean fuzzy cross-entropy measures are introduced consecutively based on the novel score factors, indeterminacy factors and J-divergence. Then some comparative experiments prove the rationality and effectiveness of the novel entropy measures and cross-entropy measures. Additionally, the Pythagorean fuzzy entropy and cross-entropy measures are designed to solve pattern recognition and multiple criteria decision making (MCDM) problems. The numerical examples, by comparing with the existing ones, demonstrate the applicability and efficiency of the newly proposed models. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods.

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods. [ABSTRACT FROM AUTHOR]