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.

In this manuscript, the notions of q-rung orthopair fuzzy sets (q-ROFSs) and complex fuzzy sets (CFSs) are combined is to propose the complex q-rung orthopair fuzzy sets (Cq-ROFSs) and their fundamental laws. The Cq-ROFSs are an important way to express uncertain information, and they are superior to the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the real part (similarly for imaginary part) of complex-valued membership degree and the qth power of the real part (similarly for imaginary part) of complex-valued non‐membership degree is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we develop the score function, accuracy function and comparison method for two Cq-ROFNs. Based on Cq-ROFSs, some new aggregation operators are called complex q-rung orthopair fuzzy weighted averaging (Cq-ROFWA) and complex q-rung orthopair fuzzy weighted geometric (Cq-ROFWG) operators are investigated, and their properties are described. Further, based on proposed operators, we present a new method to deal with the multi‐attribute group decision making (MAGDM) problems under the environment of fuzzy set theory. Finally, we use some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.

The q-rung orthopair fuzzy set (q-ROFS) as a generalization of fuzzy set (FS), is characterized by membership and non-membership, and the sum of their q-powers is restricted to [0,1]. In this manuscript, a new Complex q-rung orthoapir fuzzy set (Cq-ROFS) is proposed by combining q-ROFS and complex fuzzy set (CFS). Cq-ROFS is a better way to process uncertain and imprecise information in decision making, which is characterized by complex-valued membership and complex-valued non-membership. First, some fundamental operational laws, score function and accuracy function, and comparison method are proposed. Further, because the vector similarity measures (VSMs) play a key role in statistics, physics and engineering sciences, some VSMs called Jaccard similarity measures (JSMs), dice similarity measures (DSMs) and cosine similarity measures (CSMs) for Cq-ROFSs and interval-valued complex q-rung orthopair fuzzy sets (IVCq-ROFSs) are investigated. Moreover, the hybrid vector similarity measures (HVSMs) called variation co-efficient similarity measures (VCSMs) for Cq-ROFSs and IVCq-ROFSs are also proposed and their properties are discussed. Finally, in order to demonstrate the feasibility of the investigated HVSMs, the existing similarity measures about complex Pythagorean fuzzy sets (CPFSs) and complex intuitionistic fuzzy sets (CIFSs) are compared with the proposed methods by numerical examples of medical diagnosis and pattern recognition.