Abstract: The arithmetic operations on intuitionistic fuzzy sets defined by Atanassov are the most popular in the intuitionistic fuzzy set theory. Based on these operations, many commonly used methods for solving the decision-making (DM) problems under intuitionistic fuzzy environment have been developed and various aggregation operators have been proposed. However, there have been revealed some undesirable properties of these operators, such as the inconsistency with the operations on the ordinary fuzzy sets (FSs), the non-monotonicity of the addition and multiplication operations or non-monotonicity under multiplication by a scalar. We show in this paper that these drawbacks pertain to also the recently proposed combined aggregations operators such as intuitionistic fuzzy Heronian mean (IFHM), intuitionistic fuzzy interaction partitioned Bonferroni mean (IFIPBM), intuitionistic fuzzy Dombi Bonferroni mean (IFDBM), intuitionistic fuzzy Maclaurin symmetric mean (IFMSM), Pythagorean fuzzy Maclaurin symmetric mean (PFMSM), q-rung orthopair fuzzy power Maclaurin symmetric mean (q-ROFPMSM), Muirhead mean (IFWMM) and intuitionistic fuzzy hybrid weighted arithmetic and geometric aggregation operators (IFHWAGA).This paper proposes some new arithmetic operations on Atanassov’s intuitionistic fuzzy sets that have good algebraic properties, such as idempotency, commutativity, monotonicity and monotonicity under multiplication by a scalar. Based on the proposed operations, the intuitionistic fuzzy weighted arithmetic mean and intuitionistic fuzzy weighted geometric mean operators with the acceptable properties are developed. Some illustrative examples are performed to demonstrate effectiveness and reliability of our method. Finally, in order to verify the validity of the proposed method in solving real-life DM problems, an application example is conducted with a comparative analysis with other existing methods.
Keywords: Intuitionistic fuzzy sets, operations, aggregation operator, decision making
