Qualitative integrals and desintegrals

Sugeno integrals and their particular cases such as weighted minimum and maximum have been used in multiple-criteria aggregation when the evaluation scale is qualitative. This paper proposes two new variants of weighted minimum and maximum, where the criteria weights have a role of tolerance threshold. These variants require the use of a residuated structure, equipped with an involutive negation. We propose residuated counterpart of Sugeno integrals, where the weights bear on subsets of criteria, and we study their properties, showing they are analogous to Sugeno integrals to a large extent. Finally we propose dual aggregation operations, we call desintegral, where an item is evaluated in terms of its local defects. Desintegrals are maximal when no defects at all are present, while integrals are maximal when all advantages are sufficiently present. Qualitative integrals and desintegral are ope possible approach to bipolar evaluation processes.