Conjoint measurement and rough set approach for multicriteria sorting problems in presence of ordinal criteria

We consider a multicriteria sorting problem consisting in assignment of some actions to some predefined and preference-ordered decision classes. The actions are described by a finite set of criteria. The sorting task is usually performed using one of three preference models: discriminant function (as in scoring methods, discriminant analysis, UTADIS), outranking relation (as in ELECTRE TRI) or decision rules. A challenging problem in multicriteria sorting is the aggregation of ordinal criteria. To handle this problem some max-min aggregation operators have been considered, with the most general one - the fuzzy integral of Sugeno (1974). We show show that the decision rule model has some advantages over the integral of Sugeno. More generally, we consider the multicriteria sorting problem in terms of conjoint measurement and prove a representation theorem stating an equivalence of a very simple cancellation property, a general discriminant function and a specific outranking relation, on the one hand, and a decision rule model on the other hand. Moreover, we consider a more general decision rule model based on the rough sets theory being one of emerging methodologies for extraction of knowledge from data. The advantage of the rough sets approach in comparison to competitive methodologies is the possibility of handling inconsistent data that are often encountered in preferential information, due to hesitation of decision makers, unstable character of their preferences, imprecise or incomplete information and the like. Therefore, we propose a general model of conjoint measurement that, using the basic concepts of the rough set approach (lower and upper approximation), is able to represent these inconsistencies by a specific discriminant function. We show that these inconsistencies can also be represented in a meaningful way by "if...,then..." decision rules induced from rough approximations.