The concept of complex fuzzy set (CFS) and complex intuitionistic fuzzy set (CIFS) is two recent developments in the field of fuzzy set (FS) theory. The significance of these concepts lies in the fact that these concepts assigned membership grades from unit circle in plane, i.e., in the form of a complex number instead from [0, 1] interval. CFS cannot deal with information of yes and no type, while CIFS works only for a limited range of values. To deal with these kinds of problems, in this article, the concept of complex Pythagorean fuzzy set (CPFS) is developed. The novelty of CPFS lies in its larger range comparative to CFS and CIFS which is demonstrated numerically. It is discussed how a CFS and CIFS could be CPFS but not conversely. We investigated the very basic concepts of CPFSs and studied their properties. Furthermore, some distance measures for CPFSs are developed and their characteristics are studied. The viability of the proposed new distance measures in a building material recognition problem is also discussed. Finally, a comparative study of the proposed new work is established with pre-existing study and some advantages of CPFS are discussed over CFS and CIFS.

The key objective of the present proposed work in this paper is introduced a new version of picture fuzzy set so called spherical fuzzy sets (SFS). spherical fuzzy set is a new extension of picture fuzzy sets and Pythagorean fuzzy sets. In spherical

The notion of bipolar soft sets has already been defined, but in this article, the notion of bipolar soft sets has been redefined, called T-bipolar soft sets. It is shown that the new approach is more close to the concept of bipolarity as compared to the previous ones, and further it is discussed that so far in the study of soft sets and their generalizations, the concept introduced in this manuscript has never been discussed earlier. We have also discussed the operational laws of T-bipolar soft sets and their basic properties. In the end, we have deliberated the algebraic structures associated with T-bipolar soft sets and the applications of T-bipolar soft sets in decision-making problems.

In this manuscript, two generalizations of fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets, known as spherical fuzzy sets and T-spherical fuzzy sets, are discussed and a numerical and geometrical comparison among them is established. A T-spherical fuzzy set can model phenomena like voting using four characteristic functions denoting the degree of vote in favor, abstinence, vote in opposition, and refusal with an infinite domain, whereas an intuitionistic fuzzy set can model only phenomena of yes or no types. First, in this manuscript, some similarity measures in the frameworks of intuitionistic fuzzy sets and picture fuzzy sets are discussed. With the help of some numerical results, it is discussed that existing similarity measures have some limitations and could not be applied to problems where information is provided in T-spherical fuzzy environment. Therefore, some new similarity measures in the framework of spherical fuzzy sets and T-spherical fuzzy sets are proposed including cosine similarity measures, grey similarity measures, and set theoretic similarity measures. With the help of some results, it was proved that the proposed similarity measures are a generalization of existing similarity measures. The newly-defined similarity measures were subjected to a well-known problem of building material recognition and the results are discussed. A comparative study of new and existing similarity measures was established and some advantages of the proposed work are discussed.