A new method for conversion between pythagorean fuzzy sets and intuitionistic fuzzy sets

Abstract

Modeling with inconsistent fuzzy information is not possible for some problem types. For such cases, pythagorean fuzzy sets (PFSs) cannot be used in problem formulations and a conversion to another fuzzy set extension is needed. As a new conversion between PFSs and intuitionistic fuzzy sets (IFSs), the projective relation was proposed in the literature and its results were compared with the normalization that is the conversion method used by all. However, projective relation conversion is not valid. This conversion is based on the approach of subtraction of the part causing the inconsistency from the membership, non-membership and indeterminacy grades equally. This is not a proper approach because a negative grade is obtained when one of the membership and non-membership grades of PFS is smalller than the equally substracted part. In this study, the error in the proof of the projective relation has been discussed by presenting a counterexample. A new conversion namely “square-scaled normalization” (SSNORM) which converts PFSs to IFSs by rescaling the grades depending on the relative greatness of their squares has been offered and its score and accuracy functions have been formulated. SSNORM method has been examined on a numerical example from the manufacturing industry and the obtained results have been compared with the normalization. Although both methods obtained results close to each other, SSNORM yielded more cautious results. It reached a bigger score function value but a smaller accuracy function value compared to the normalization. SSNORM method can be preferrable alternative of the normalization if the approximation errors caused by the linear rescaling is high.