A Generalization of the Concave Integral in Terms of Decomposition of the Integrated Function for Bipolar Scales

Abstract

In the context of the multiple-criteria decision aid (MCDA), several fuzzy integrals concerning capacities (non-additive measures) have been introduced by various researchers in the last sixty years. Recently, Lehrer has proposed a new integral for capacities known as concave integral. The concave integral is based on the decomposition of random variables into simple ingredients. The concave integral concerning capacity is defined as the maximum value obtained among all its decompositions. The paper aims to model a new integration based on the decomposition of random variables into simple ingredients for multi-criteria decision making support when underlying scales are bipolar. This paper proposes a generalization of the concave integral in terms of decompositions of the integrated function to be suitable for bipolar scales. We show that the random variable is analyzed as a combination of indicators, where each allowed decomposition has a value determined by the bi-capacity. Lastly, we illustrate our framework by a practical example.