Ann Oper Res (2010) 176: 41–75 DOI 10.1007/s10479-009-0566-8 Dominance-based Rough Set Approach to decision under uncertainty and time preference Salvatore Greco · Benedetto Matarazzo · Roman Slowi ´ nski Published online: 29 May 2009 © Springer Science+Business Media, LLC 2009 Abstract We consider a problem of decision under uncertainty with outcomes distributed over time. We propose a rough set model based on a combination of time dominance and stochastic dominance. For the sake of simplicity we consider the case of traditional additive probability distribution over the set of states of the world, however, we show that the model is rich enough to handle non-additive probability distributions, and even qualitative ordi- nal distributions. The rough set approach gives a representation of decision maker’s time- dependent preferences under uncertainty in terms of “if ..., then. . . ” decision rules induced from rough approximations of sets of exemplary decisions. Keywords Decision under uncertainty · Time preference · Dominance-based Rough Set Approach · Decision rules · Preference learning 1 Introduction Decision under uncertainty has been intensively investigated by many researchers (for a comprehensive review see, e.g., Fishburn 1988). In this field, the basic model is the ex- pected utility, which has been axiomatized by von Neumann and Morgenstern (1947) in case of objective probability, and by Savage (1954) in case of subjective probability. Much S. Greco · B. Matarazzo Faculty of Economics, University of Catania, 95129 Catania, Italy S. Greco e-mail: salgreco@unict.it B. Matarazzo e-mail: matarazz@unict.it R. Slowi´ nski ( ) Institute of Computing Science, Pozna´ n University of Technology, 60-965 Pozna´ n, Poland e-mail: roman.slowinski@cs.put.poznan.pl R. Slowi´ nski Systems Research Institute, Polish Academy of Sciences, 00-441 Warsaw, Poland