A generalization of stability for families of aggregation operators

In this paper we extend the notion of stability of each aggregation function as presented in a previous paper of the authors, where an aggregation function was denominated Family of Aggregation Operators (FAO) in order to stress that aggregation operators within an aggregation function should be consistent. We will show that the previous definition presents certain problems when dealing with a FAO that is not idempotent or continuous. Meanwhile the previous definition was based on the study of fixed points, without taking into account the environment of each point, as many applications demand, the new approach we propose now is more flexible, and at the end allows a more appropriate definition of the consistency of a FAO. It will be shown that under certain conditions our new proposal is equivalent to the previous definition of strict stability, but the new definition covers families of aggregation operators that are not strictly stable, but intuitively stable, and vice versa. In short, we will see that our new proposal fits much better the intuition of stability of a FAO. This reformulation of stability is based upon the concept of penalty function as a measure of proximity between a value and an array of values.