Discrete Decision Problems, Multiple Criteria Optimization Classes and Lexicographic Max-ordering

The topic of this paper are discrete decision problems with multiple criteria. We rst deene discrete multiple criteria decision problems and introduce a classiication scheme for multiple criteria optimization problems. To do so we use multiple criteria optimization classes. The main result is a characterization of the class of lexicographic max-ordering problems by two very useful properties, reduction and regularity. Subsequently we discuss the assumptions under which the application of this speciic MCO class is justi-ed. Finally we provide (simple) solution methods to nd optimal decisions in the case of discrete multiple criteria optimization problems. A discrete decision problem consists of selecting from a nite set of alternatives A = fa 1 ; : : :; a m g anòptimal' one. In this paper we consider the context of multiple criteria decision making (MCDM). The solution of discrete decision problems with multiple criteria is often called multiple attribute decision making, see HY81]. Every alternative is evaluated with respect to a certain number of criteria, or objective functions. We assume that f q : A ! IR; q = 1; : : :; Q are Q real valued functions representing the different criteria. Therefore we can calculate the value of each alternative with respect to each criterion as given by v qj = f q (a j): Since the number of alternatives as well as the number of criteria is nite it is convenient to write all the values as a Q m matrix: V = (v qj) = (f q (a j)) :