Interval-valued intuitionistic fuzzy hybrid weighted averaging operator based on Einstein operation and its application to decision making

The notion of interval-valued intuitionistic fuzzy set (IVIFS) is a generalization of that of Atanassov's intuitionistic fuzzy set (AIFS). The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. In this paper, we define some Einstein operations on IVIFS and develop three arithmetic averaging operators, such as the interval-valued intuitionistic fuzzy Einstein weighted averaging (IVIFWA(epsilon)) operator, interval-valued intuitionistic fuzzy Einstein ordered weighted averaging (IVIFOWA(epsilon)) operator, and interval-valued intuitionistic fuzzy Einstein hybrid weighted averaging (IVIFHWA(epsilon)) operator, for aggregating interval-valued intuitionistic fuzzy information. The IVIFHWA(epsilon) operator generalizes both the IVIFWA(epsilon) and IVIFOWA(epsilon) operators. Moreover, we establish various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. Finally, we apply the IVIFHWA(epsilon) operator to multiple attribute decision making with interval-valued intuitionistic fuzzy information.