M¨ obius representation of the bipolar decomposition integrals and its applications

Document Type : Research Paper

Authors

1 Department of Applied Sciences, University of Technology, Al Sina´ a Street, P. O. Box (19006), Baghdad, Iraq

2 Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´ eho 11, 810 05 Bratislava, Slovakia

3 Department of Algebra and Geometry, Palack´y University of Olomouc, 17. listopadu 12, 779 00 Olomouc, Czech Republic

Abstract

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.

Keywords

References

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Iranian Journal of Fuzzy Systems
Volume 22, Issue 4
July and August 2025
Pages 1-15

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