Distance and similarity measures for Pythagorean fuzzy sets

The concept of Pythagorean fuzzy sets is very much applicable in decision science because of its unique nature of indeterminacy. The main feature of Pythagorean fuzzy sets is that it is characterized by three parameters, namely, membership degree, non-membership degree, and indeterminate degree, in such a way that the sum of the square of each of the parameters is one. In this paper, we present axiomatic definitions of distance and similarity measures for Pythagorean fuzzy sets, taking into account the three parameters that describe the sets. Some distance and similarity measures in intuitionistic fuzzy sets, viz, Hamming, Euclidean, normalized Hamming, and normalized Euclidean distances, and similarities are extended to Pythagorean fuzzy set setting. However, it is discovered that Hamming and Euclidean distances and similarities fail the metric conditions in Pythagorean fuzzy set setting whenever the elements of the two Pythagorean fuzzy sets, whose distance and similarity are to be measured, are not equal. Finally, numerical examples are provided to illustrate the validity and applicability of the measures. These measures are suggestible to be resourceful in multicriteria decision-making problems (MCDMP) and multiattribute decision-making problems (MADMP), respectively.