M¨ obius representation of the bipolar decomposition integrals and its applications
نویسندگان
1 Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´ eho 11, 810 05 Bratislava, Slovakia
2 Department of Algebra and Geometry, Palack´y University of Olomouc, 17. listopadu 12, 779 00 Olomouc, Czech Republic
3 Department of Applied Sciences, University of Technology, Al Sina´ a Street, P. O. Box (19006), Baghdad, Iraq
doi
10.22111/ijfs.2025.9334چکیده
The bipolar decomposition integral, recently introduced in [5], is a general framework for handling integrals related to aggregation on unipolar and bipolar scales. The M¨obius representation is related to the notion of k-additivity of a monotone set function and allows to derive simple expressions of some nonlinear integrals. In this paper, we propose M¨obius representation for the bipolar decomposition integral, which includes the M¨obius representation of each of the bipolar Choquet integral, bipolar Shilkret integral, and bipolar Pan integral. Then, we introduce the expressions for computing bipolar decomposition integrals concerning a 2-additive bi-capacity. Lastly, a practical numerical example is provided to illustrate the applicability of the proposed results in dealing with aggregation on bipolar scales, and simplicity of calculating the 2-additive bipolar decomposition integrals.
