On Sugeno integral as an aggregation function

Aggregation refers to the process of combining numerical values x1, . . . , xm into a single one M (x1, . . . , xm), so that the final result of aggregation takes into account all the individual values. In decision making, values to be aggregated are typically preference or satisfaction degrees and thus belong to the unit interval [0, 1]. This paper aims at investigating the Sugeno integral which can be regarded as an aggregation function. In particular, we show that any Sugeno integral is a weighted max-min function, that is, setting X = {1, . . . , m}, a function of the form M (x1, . . . , xm) = ∨ T⊆X [aT ∧ ( ∧ i∈T xi)], aT ∈ [0, 1], where a is a set function satisfying a∅ = 0 and ∨ T⊆X aT = 1. Such functions are investigated in this paper. We also show that those functions can also be written as M (x1, . . . , xm) = ∧ T⊆X [bT ∨ ( ∨ i∈T xi)], bT ∈ [0, 1], (weighted min-max functions) where b is a set function satisfying b∅ = 1 and ∧ T⊆X bT = 0. The correspondance formulae b = b(a) and a = a(b) are given as well. For instance, we have (.1 ∧ x1) ∨ (.3 ∧ x2) ∨ (x2 ∧ x3) = (.1 ∨ x2) ∧ (.3 ∨ x3) ∧ (x1 ∨ x2). We also propose an axiomatic characterization of this class of functions based on some aggregation properties: the increasingness and the stability for minimum and maximum with the same unit. Most of these results are applied to the Sugeno integral. In particular, we can derive equivalent expressions and characterize the family of all the Sugeno integrals. We also consider particular weighted max-min functions: Boolean max-min functions, weighted maximum and minimum functions, ordered weighted maximum and minimum functions, partial maximum and minimum functions, order statistics and associative medians. Of course, all these functions are Sugeno integrals.