Choquet integral on Liu´s generalized intuitionistic fuzzy numbers

How to find a good aggregation operation and an operation-invariant partial order for all operations are the most important key issues in intuitionistic fuzzy multi-criteria fuzzy decision making problems. In this paper, a Choquet integral aggregation operation on Liu´s generalized intuitionistic fuzzy numbers is proposed, a generalized condition of an operation-invariant partial order for Choquet integral aggregation operation of any given addition and scalar multiplication operations is also proposed, and a very useful theorem is further proposed. According to this useful theorem, we know that if a totally partial order is operation-invariant for Atanassov´s addition operation and scalar multiplication on intuitionistic fuzzy numbers, then it is also operation-invariant for Choquet integral operation on intuitionistic fuzzy numbers. Moreover, if a totally partial order is operation-invariant for Liu´s generalized addition operation and scalar multiplication on Liu´s generalized intuitionistic fuzzy numbers, then it is also operation-invariant for Choquet integral operation on Liu´s generalized intuitionistic fuzzy numbers. Most of all, it is pointed out that although Xu´s order is a totally partial order but not a operation-invariant partial order for Choquet integral operation on intuitionistic fuzzy numbers. Up to now, only Liu´s order which proposed by author´s previous work is an operation-invariant partial order for Choquet integral operation not only on intuitionistic fuzzy numbers but also on Liu´s generalized intuitionistic fuzzy numbers, which can be used to handle intuitionistic fuzzy multi-criteria fuzzy decision making problems and Liu´s generalized intuitionistic fuzzy multi-criteria fuzzy decision making problems.