Kybernetika 49 no. 3, 465-486, 2013

Measuring consistency and inconsistency of pair comparison systems

Jaroslav Ramík and Milan Vlach


In this paper we deal with mathematical modeling of real processes that are based on preference relations in the sense that, for every pair of distinct alternatives, the processes are linked to a value of preference degree of one alternative over the other one. The use of preference relations is usual in decision making, psychology, economics, knowledge acquisition techniques for knowledge-based systems, social choice and many other social sciences. For designing useful mathematical models of such processes, it is very important to adequately represent properties of preference relations. We are mainly interested in the properties of such representations which are usually called reciprocity, consistency and transitivity. In decision making processes, the lack of reciprocity, consistency or transitivity may result in wrong conclusions. That is why it is so important to study the conditions under which these properties are satisfied. However, the perfect consistency or transitivity is difficult to obtain in practice, particularly when evaluating preferences on a set with a large number of alternatives. Under different preference representation structures, the multiplicative and additive preference representations are incorporated in the decision problem by means of a transformation function between multiplicative and additive representations. Some theoretical results on relationships between multiplicative and additive representations of preferences on finite sets are presented and some possibilities of measuring their consistency or transitivity are proposed and discussed. Illustrative numerical examples are provided.


multi-criteria optimization, pair-wise comparison matrix, AHP


90B50, 90C29, 91B08


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