A novel bidirectional projection measures of circular intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems

Atanassov recently proposed a new circular intuitionistic fuzzy set (CIFS) as an extension of intuitionistic fuzzy sets to express uncertain information by a circle with centered membership, non-membership, and radius r. Circular intuitionistic fuzzy sets can express uncertain information more flexibly than the intuitionistic fuzzy set. In this paper, we first propose a new method for calculating the radius r of CIFSs by ordinary least squares (OLS). We introduce some notions, such as modules of the circular intuitionistic fuzzy set and the cosine of the included angle between membership and non-membership vectors of the circular intuitionistic fuzzy set. Then, we define a new bidirectional projection measure of circular intuitionistic fuzzy sets, which takes into account the difference between different CIFSs in terms of membership degree and non-membership degree and radius r. The proposed bidirectional projection measures show superiority compared with some recent research works through numerical examples. Finally, the method is applied to a multi-attribute decision-making problem with group expert decision-making to prove the effectiveness and accuracy of the method.