Reduction of variables and constraints in fitting antibuoyant fuzzy measures to data using linear programming

The discrete Choquet integral with respect to various types of fuzzy measures serves as an important aggregation function which accounts for mutual dependencies between the inputs. The Choquet integral can be used as an objective (or constraint) in optimisation problems, and the type of fuzzy measure used determines its complexity. This paper examines the class of antibuoyant fuzzy measures, which restrict the supermodular (convex) measures and satisfy the Pigou–Dalton progressive transfers principle. We determine subsets of extreme points of the set of antibuoyant fuzzy measures, whose convex combinations form a basis of three proposed algorithms for random generation of fuzzy measures from that class, and also for fitting fuzzy measures to empirical data or solving best approximation problems. Potential applications of the proposed methods are envisaged in social welfare, ecology, and optimisation.