Implication Relations, Equivalence Relations and Hierarchical Structure of Attributes in Multiple Criteria Decision Making

Abstract

This paper presents an efficient technique for computing implication structures, especially for Multiple Attribute Decision Making (MADM) problems. The method based on the generalized modus ponens. Implication means that there is a membership of the binary relation that a good performance level of the attribute A_i implies a good performance level of an other attribute A_j for each combination. Obviously, if such implications are found one can get some positive effects:

• 1 reduction of the number of relevant attributes,

• 2 computing a number measuring the inner contradictions between the attributes of the decision making problem,

• 3 obtaining a hierarchy of the attributes. Equivalence classes arise during the computation. Basic information is given by the decision matrix. The method needs upper and lower satisfaction levels for each single criterion. Using a so-called implication threshold, given by the decision maker, one can get a crisp implication structure of crisp equivalence classes. A measuring number of the inner contradiction leads to the degree of carefulness, needed in the following MADM steps. Some methods in MADM (like, for instance, Saaty’s hierarchical method) are based on a hierarchical structure of attributes. The method, presented in this paper, helps in this case, if such a hierarchy cannot be given by the decision maker.