Bipolar decomposition integrals

Abstract

The idea of decomposition integral, inspired by the concept of Lebesgue integral, is a common framework for unifying many nonlinear integrals, such as the Choquet, the Shilkret, the PAN, and the concave integrals. This framework concerns aggregation on a unipolar scale, and depends on the distinguished decomposition system under some constraints on the sets being considered for each related integral. The aim of this paper is to provide a general framework to deal with integrals concerning aggregation on unipolar and bipolar scales. To achieve this aim, we propose in this paper an extension of the idea of decomposition integral of the integrated function to be suitable for bipolar scales depending on the distinguished bipolar decomposition system under some constraints on the bipolar collections being considered for each related bipolar fuzzy integral. Then, we introduce some properties of bipolar decomposition integrals, including those establishing that our approach covers the Cumulative Prospect Theory (CPT) model and the integrals with respect to bipolar capacities. Finally, we conclude with certain directions on some additional findings related to the research.