New types of aggregation functions for interval-valued fuzzy setting and preservation of pos-B and nec-B-transitivity in decision making problems

In this contribution new types of aggregation functions for interval-valued fuzzy setting are introduced. They are called necessary and possible aggregation functions, respectively. In the monotonicity conditions for these aggregation functions the classical monotonicity for intervals is replaced with the new comparability relations. These relations follow naturally from the interpretations of interval-valued fuzzy sets and together with the classically used monotonicity for intervals, form a family of all possible approaches to define relations of comparability for intervals. Moreover, in this paper dependencies between considered families of aggregation functions are presented. Furthermore, transitivity properties of interval-valued fuzzy relations, based on these new comparability relations, are studied and preservation of them by possible and necessary aggregation functions are considered.