Some new types induced complex intuitionistic fuzzy Einstein geometric aggregation operators and their application to decision-making problem

The objective of this research is to develop some novel operational laws based of T-norm and T-conorm and then using these operational laws to develop several Einstein operators for aggregating the different complex intuitionistic fuzzy numbers (CIFNs) by considering the dependency between the pairs of its membership degrees. In the existing studies of fuzzy and its extensions, the uncertainties present in the data are handled with the help of degrees of membership that are the subset of real numbers, which may also loss some valuable data and hence consequently affect the decision results. A modification to these, complex intuitionistic fuzzy set handles the uncertainties with the degree whose ranges are extended from real subset to the complex subset with unit disk and hence handle the two-dimensional information in a single set. Thus, motivated by this and this paper we present some novel methods such as complex intuitionistic fuzzy Einstein weighted geometric aggregation (CIFEWGA) operator, complex intuitionistic fuzzy Einstein ordered weighted geometric aggregation (CIFEOWGA) operator, complex intuitionistic fuzzy Einstein hybrid geometric aggregation (CIFEHGA) operator, induced complex intuitionistic fuzzy Einstein ordered weighted geometric aggregation (I-CIFEOWGA) operator and induced complex intuitionistic fuzzy Einstein hybrid geometric aggregation (I-CIFEHGA) operator. We present some of their desirable properties such as idempotency, boundedness and monotonicity. Furthermore, based on these methods a multi-attribute group decision-making problem developed under complex intuitionistic fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to show the effectiveness, importance and efficiency of the novel approach.