Interval-valued inclusion–exclusion integral to aggregate interval-valued data based on multiple admissible orders

Abstract

As a nonlinear fuzzy aggregation function, the interval-valued Choquet integral is widely used in decision analysis, rule-based classification, and information fusion. However, its linear order between intervals is usually provided via human intervention, and different settings significantly affect calculation results. As an extended form of the Choquet integral, the Inclusion–Exclusion (IE) integral does not require sorting input variables, a property that demonstrates remarkable potential in information fusion. Nevertheless, this concept has not been extended to the interval-valued domain. Therefore, an interesting challenge arises: how to utilize the IE integral’s feature of not requiring input sorting to address the sensitivity of interval-valued Choquet integrals to order relation selection. Aiming at this sensitivity, this study proposes an interval-valued IE integral and constructs a novel aggregation model to mitigate order dependency. The research includes: (1) defining the interval-valued IE integral based on IE integral theory and analyzing its properties; (2) constructing an aggregation model by integrating interval-valued IE and Choquet integrals; (3) deriving gradient formulas for model parameters and providing the model’s computational algorithm; (4) verifying the effectiveness of the proposed method through numerical experiments and benchmark public datasets. The results provide a new methodology for addressing the order relation sensitivity of fuzzy integrals and expand the theoretical application scope of IE integrals.