Qualitative Integrals and Desintegrals: How to Handle Positive and Negative Scales in Evaluation

Abstract

Integrals are currently used in multiple criteria analysis for synthesizing into a global evaluation the advantages possessed by a potential choice. As such, integrals are operators that increase with the criteria evaluations. However, an item may be also evaluated in terms of its defects. Then the more and the greater the defects, the smaller the evaluation should be. An operator that can provide a synthesis of the defects of an item in this sense is called a desintegral. Desintegrals are maximal when no defects at all are present, while integrals are maximal when all advantages are sufficiently present. So, the greater the value of an integral, or a desintegral, the better the corresponding item since advantages are greater, or defects are smaller respectively. Desintegrals implicitly refer to a negative scale, since an order-reversing mapping of the scale used for evaluating each criterion transforms the degree to which the value is advantageous into a degree to which it is disadvantageous, and conversely. In this paper, we provide an organised description of counterparts to Sugeno integrals that synthesize positive or negative evaluations in the qualitative framework of a totally ordered residuated lattice equipped with an involutive negation. We exploit three kinds of criteria weighting schemes that are allowed by this algebraic structure.