Consistency and stability in aggregation operators: An application to missing data problems

An aggregation operator [1, 5, 7, 8, 9, 12] is usually defined as a real function An such that, from n data items x1, . . . ,xn in [0,1], produces an aggregated value An(x1, . . . ,xn) in [0,1] [4]. This definition can be extended to consider the whole family of operators for any n instead of a single operator for an specific n. This has led to the current standard definition [4, 15] of a family of aggregation operators (FAO) as a set {An : [0,1]n → [0,1],n ∈ N}, providing instructions on how to aggregate collections of items of any dimension n. This sequence of aggregation functions {An}n∈N is also called extended aggregation functions (EAF) by other authors [15, 5]. In this work, we will deal with two different but related problems for extended aggregation functions or family of aggregation operators On one hand, let us remark that in practice, it is frequent that some information can get lost, be deleted or added, and each time a cardinality change occurs a new aggregation operator Am has to be used to aggregate the new collection of m elements. However, it is important to remark that a relation between {An} and {Am} does not necessarily exist in a family of aggregation operators as defined in [4]. In this context, it seems natural to incorporate some properties to maintain the logical consistency between operators in a FAO when changes on the cardinality of the data occur, for which we need to be able to build up a definition of family of aggregation operators in terms of its logical consistency, and solve each problem of aggregation without knowing apriori the cardinality of the data. This is, the operators that compose a FAO have to be somehow related, so the aggregation process