Axiomatization of utility, outranking and decision-rule preference models for multiple-criteria classification problems under partial inconsistency with the dominance principle

Multiple-criteria classification (sorting) problem concerns assignment of actions (objects) to some pre-defined and preference-ordered decision classes. The actions are described by a finite set of criteria, i.e. attributes, with preference-ordered scales. To perform the classification, criteria have to be aggregated into a preference model which can be: utility (discriminant) function, or outranking relation, or "if..., then..." decision rules. Decision rules involve partial profiles on subsets of criteria and dominance relation on these profiles. A challenging problem in multiple-criteria decision making is the aggregation of criteria with ordinal scales. We show that the decision rule model we propose has advantages over a general utility function, over the integral of Sugeno, conceived for ordinal criteria, and over an outranking relation. This is shown by basic axioms characterizing these models. Moreover, we consider a more general decision rule model based on the rough set theory. The advantage of the rough set approach compared to competitive methodologies is the possibility of handling partially inconsistent data that are often encountered in preferential information, due to hesitation of decision makers, unstable character of their preferences, imprecise or incomplete knowledge and the like. We show that these inconsistencies can be represented in a meaningful way by "if..., then..." decision rules induced from rough approximations. The theoretical results reported in this paper show that the decision rule model is the most general aggregation model among all the considered models.