New paper accepted in Fuzzy Sets and Systems. The inception of this paper came over a year ago with a few emerging papers based on our generalized Bonferroni means paper that extended the function to Atanassov and interval-valued fuzzy sets. The main thing that wasn’t really dealt with well by these emerging papers was the weighting convention.
Title: On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts
Authors: Beliakov, G. and James, S.
Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous ex- tensions of the weighted arithmetic mean and ordered weighted aver- aging operator also have the absorbent element ⟨1, 0⟩, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the oper- ations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit anal- ogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions
with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.