Research On The Uncertainty Measures Of Pythagorean Fuzzy Sets And Their Applications

As generalizations of fuzzy sets,Pythagorean fuzzy sets and interval-valued Pythagorean hesitant fuzzy sets allow more flexibility in handling uncertain information.And their theories are widely used in pattern recognition,cluster analysis,multi-attribute decision making and other fields.Pythagorean fuzzy entropy can reflect the uncertainty of Pythagorean fuzzy information.Pythagorean fuzzy cross-entropy can measure the difference between two Pythagorean fuzzy sets.The similarity measure and distance measure of interval-valued Pythagorean hesitant fuzzy sets can measure the closeness between two interval-valued Pythagorean hesitant fuzzy sets.This paper discusses the uncertainty measurement of Pythagorean fuzzy sets and interval-valued Pythagorean hesitant fuzzy sets.The main contents are as follows:This paper first proposes new score function and uncertainty function of Pythagorean fuzzy sets.Then a new Pythagorean fuzzy entropy formula based on the two functions and Shannon entropy is introduced.This formula not only overcomes former defects of fuzzy entropy,but also satisfies the basic properties of fuzzy entropy.Furthermore,considering the different importance of the two functions,the parameterized Pythagorean fuzzy entropy is raised.After that,based on score function,uncertainty function and -divergence,we introduce a novel Pythagorean fuzzy cross-entropy and prove that it satisfies the basic cross-entropy properties.The Pythagorean fuzzy entropy and cross-entropy are also combined to solve pattern recognition and multi-attribute decision making problems under Pythagorean fuzzy environment.Then,according to the fuzzy closeness degree,the similarity measure of intervalvalued Pythagorean hesitant fuzzy sets is introduced,which is proved to satisfy the basic properties of the similarity measure.In addition,it is applied to the clustering algorithm to solve the clustering problem under interval-valued Pythagorean hesitant fuzzy environment.Finally,the distance measures is proposed to solve multi-attribute decision making problems under interval-valued Pythagorean hesitant fuzzy environment.