The 2-additive decomposition integrals and their applications

Abstract

Decomposition integral with respect to a capacity (monotone measure) is a common framework for unifying some nonlinear integrals, such as the Choquet, the Shilkret, the PAN, the PC, and the concave integrals. The formula of decomposition integral concerning capacities depends on the distinguished decomposition system under some constraints on the sets being considered for each related integral. The Möbius representation of a monotone set function is a fundamental concept permitting the derivation of simple expressions of nonlinear integrals. In this paper, we propose Möbius representation of some decomposition integrals based on Lovász idea of an extension of pseudo-boolean functions for the decomposition systems to get different types of decomposition integrals. Then, we introduce a new type of integral to unify the Choquet, Shilkret, PAN, and PC integrals in terms of the Möbius transform. Consequently, we then propose the expressions of computing decomposition integrals concerning a 2-additive capacity. Finally, an illustrative example is used to demonstrate the applicability of the proposed results and simplicity of computing the 2-additive decomposition integrals.