A copula-based approach to aggregation operators

Aggregation operators transform a finite number of inputs, called arguments, into a single output. They are applied in many theoretical and practical domains and in particular aggregation operators play important role in different approaches to decision making, where values to be aggregated are typically preference or satisfaction degrees. Many operators of different type have been considered in connection with different situations. This note develops a new unified approach to copula-based modeling and characterizations of aggregation operators. This approach was originally proposed in [6] for a particular class of aggregation operators. The concept of copula was introduced long ago (Sklar 1959), but only recently its potential for applications has become clear. Copulae permit to represent joint distribution functions by splitting the marginal behavior, embedded in the marginal distributions, from the dependence captured by the copula itself. So the copula approach is particularly useful when we investigate the interaction between different arguments of aggregation operators. For example, in multi-criteria decision making problems the decision criteria present some interaction, whose nature varies from one situation to another. The problem of modeling interaction remains a difficult question in the theory of aggregation operators. In this context we also focus on the study of the class of Archimedean copulae, that have proven useful for modeling dependence in a variety of settings and that form a dense subclass of the class of associative copulae.