Some Hamacher Aggregation Operators Based on the Interval-Valued Intuitionistic Fuzzy Numbers and Their Application to Group Decision Making

With respect to multiple attribute group decision-making problems in which attribute values take the form of the interval-valued intuitionistic fuzzy numbers, the group decision-making methods based on some Hamacher aggregation operators, which extended the algebraic aggregation operators and Einstein aggregation operators, are developed. First, an interval-valued intuitionistic fuzzy Hamacher weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher-ordered weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher hybrid weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher geometric weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher geometric-ordered weighted averaging operator, and interval-valued intuitionistic fuzzy Hamacher geometric hybrid weighted averaging operator are proposed. Some desirable properties of these operators, such as commutativity, idempotency, monotonicity, and boundedness, are studied, and some special cases in these operators are analyzed. Furthermore, two methods to multicriteria decision group making based on these operators are developed. Finally, an illustrative example is given to verify the proposed methods and to demonstrate their practicality and effectiveness.