Constrained Pythagorean Fuzzy Sets and Its Similarity Measure

Pythagorean fuzzy set (PFS) is an extension of the intuitionistic fuzzy set. It has a wider space of membership degrees. Thus, it is more capable in expressing and handling the fuzzy information in engineering practice and scientific research. However, PFSs lack a mathematical tool to express stochastic or probability information, rendering it unsuitable for application in many scenarios. In this article, an ordered number pair is used to describe fuzzy information and stochastic information under uncertain environments, namely constrained Pythagorean fuzzy set (CPFS). The CPFS has two components, $\text{CPFS}=(A,P)$, where $A$ is the classical PFS, while $P$ is a measurement of reliability for $A$. For PFS, CPFS is the first unified description of fuzzy information and probabilistic information, which is a more flexible way to describe knowledge or thinking. Furthermore, the similarity measure of CPFSs is presented, which meets the similarity measure theorems and can better indicate the flexibility of CPFSs. Numerical examples are used to demonstrate that the CPFSs similarity measure is reasonable and effective. The method of similarity measure can be degenerated to the similarity measure of PFSs under specific case and can avoid generating counter-intuitive results. In addition, similarity measure of CPFSs is applied to medical diagnosis and target classification of Iris. These experimental results have proven the practicability and effectiveness of our model.