Extensions of Atanassov's Intuitionistic Fuzzy Interaction Bonferroni Means and Their Application to Multiple-Attribute Decision Making

The Bonferroni mean (BM) was originally presented by Bonferroni and had been generalized by many researchers on Atanassov's intuitionistic fuzzy sets (AIFSs) for its capacity to capture the interrelationship between input arguments. Nevertheless, the forms of the combinations of the newly proposed interaction theory on AIFSs with BM are very single, and the existing BMs on AIFSs are not consistent with aggregation operations on the ordinary fuzzy sets. As complements to the existing generalizations of BM under Atanassov's intuitionistic fuzzy environment, this paper develops the extended Atanassov's intuitionistic fuzzy interaction Bonferroni mean (EIFIBM) and the extended weighted Atanassov's intuitionistic fuzzy interaction Bonferroni mean, which can evolve into a series of BMs by taking different generator functions that reflect the different preference attitudes of the decision makers. In addition, some of the EIFIBMs are consistent with aggregation operations on the ordinary fuzzy sets, and some of the EIFIBMs consider the interactions between the membership and nonmembership functions of different Atanassov's intuitionistic fuzzy sets; thus, they can be used in more decision situations. We investigate the properties of these new extensions and apply them to multiple-attribute decision-making problems with admissible orders. Finally, numerical examples show the validity and feasibility of the new approaches.