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^ε) operator, interval-valued intuitionistic fuzzy Einstein ordered weighted averaging (IVIFOWA^ε) operator, and interval-valued intuitionistic fuzzy Einstein hybrid weighted averaging (IVIFHWA^ε) operator, for aggregating interval-valued intuitionistic fuzzy information. The IVIFHWA^ε operator generalizes both the IVIFWA^ε and IVIFOWA^ε 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^ε operator to multiple attribute decision making with interval-valued intuitionistic fuzzy information.