Set function representations of evaluation values, weighted evaluation values, fuzzy measures, and Choquet integrals and an alternative representation of Choquet integrals

Abstract

A set function representation preserves rank information among the evaluation values of a sample, and an aggregated set function of samples also preserves rank information. We apply this set function representation to the evaluation values, the weighted evaluation values, the OWA operator, and the Choquet integral. Since the Choquet integral can be represented as the product of two set function representations, a fuzzy measure and a set function representation of the evaluation values, a set function representation of the Choquet integral is also possible. The Choquet integral is the product of two set function functions, but it can be expressed as the product of three or more set function representations. Therefore, we represent the Choquet integral as the product of three set function representations: importance to the criteria, importance to the rank, and the evaluation values. Finally, we examine the properties of these three product expressions.