# Consistency and stability in aggregation operators with data structure

Abstract

In this work, we continue with a previous work in which we analyzed and defined notions of consistency, stability and continuity of a family of aggregation operators (FAO) when the data is unstructured. Here we use these concepts to tackle with aggregation problem for those situations in which the information or data that has to be aggregated has an inherent structure. In particular, we will focus on two structures that has received an important attention during last decades due to it application is different fields as Muticriteria Decision Analysis: The linear order structures and the hierarchical structures from a prioritized point of view.

Description

In this work, we will deal with two different but related problems for a family of aggregation operators. On one hand, we will deal with a dimensional problem of this family. It is important to notice 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, and therefore a relation between {An} and {Am} does not necessarily exist in a family of aggregation operators. In this context, it seems natural to incorporate some properties to maintain the logical consistency between these families 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 remains the same throughout the possible changes in the dimension n of the data. The second concept presented in this work is related with the consistency in of the aggregation operator family when the data that have to be aggregated possess an inherent structure. In our opinion, the idea and definition of consistency of a aggregation family should also take into account the structure of the data. Data structure refers to the relative position of information units. Some preliminary ideas can be founded, in which the structure of a set of fuzzy classes is included in order to distinguish between some apparently similar situations which however present different underlying structures (as the interval-valued fuzzy sets and the intuitionistic fuzzy sets. In other areas as probability theory, it is clear that the estimation process is dependent of the structure of the data that has to be aggregated. These situations are usually represented as a lineal structure (as in a time series) or by means of graph (as in Markov chains).. In this work, we show how it is possible to extend some of these stability ideas for those situations in which the data present some structures

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