Rough sets methodology for sorting problems in presence of multiple attributes and criteria Salvatore Greco a, * , Benedetto Matarazzo a , Roman Slowinski b a Faculty of Economics, University of Catania, Corso Italia, 55, 95129 Catania, Italy b Institute of Computing Science, Poznan University of Technology, 60-965 Poznan, Poland Abstract We consider a sorting (classification) problem in the presence of multiple attributes and criteria, called the MA&C sorting problem. It consists in assignment of some actions to some pre-defined and preference-ordered decision classes. The actions are described by a finite set of attributes and criteria. Both attributes and criteria take values from their domains; however, the domains of attributes are not preference-ordered, while the domains of criteria (scales) are totally ordered by preference relations. Among the attributes we distinguish between qualitative attributes and quantitative attributes. In order to construct a comprehensive preference model that could be used to support the sorting task, we consider preferential information of the decision maker (DM) in the form of assignment examples, i.e. exemplary assignments of some reference actions to the decision classes. The preference model inferred from these examples is a set of ‘‘if ... , then ... ’’ decision rules. The rules are derived from rough approximations of decision classes made up of reference actions. They satisfy conditions of completeness and dominance, and manage with possible ambiguity (inconsistencies) in the set of examples. Our idea of rough approximations involves three relations together: indiscernibility, similarity and dominance defined on qualitative and quantitative attributes, and on criteria, respec- tively. The usefulness of this approach is illustrated by an example. Ó 2002 Published by Elsevier Science B.V. Keywords: Rough sets; Sorting; Classification; Multiple criteria decision analysis; Decision rules 1. Introduction The main difficulty with application of many existing multiple-criteria decision aiding (MCDA) methods lies in acquisition of the decision maker’s (DM’s) preferential information. Very often, this information has to be given in terms of preference model parameters, like importance weights, sub- stitution rates and various thresholds. It is gener- ally acknowledged, however, that people prefer to make exemplary decisions than to explain them in terms of the preference model adopted by the an- alyst. For this reason, the idea of inferring pref- erence models from exemplary decisions provided by the DM is very attractive. The exemplary de- cisions may, however, be inconsistent because of European Journal of Operational Research 138 (2002) 247–259 www.elsevier.com/locate/dsw * Corresponding author. E-mail addresses: salgreco@mbox.unict.it (S. Greco), matarazz@mbox.unict.it (B. Matarazzo), slowinsk@sol.put.poznan.pl (R. Slowinski). 0377-2217/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII:S0377-2217(01)00244-2