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.
The full consistency of Saaty's matrix of preference intensities used in AHP is practically unachievable for a large number of objects being compared. There are many procedures and methods published in the literature that describe how to assess whether Saaty's matrix is "consistent enough". Consistency is in these cases measured for an already defined matrix (i.e. ex-post). In this paper we present a procedure that guarantees that an acceptable level of consistency of expert information concerning preferences will be achieved. The proposed method is based on dividing the process of inputting Saaty's matrix into two steps. First, the ordering of the compared objects with respect to their significance is determined using the pairwise comparison method. Second, the intensities of preferences are defined for the objects numbered in accordance with their ordering (resulting from the first step). In this paper the weak consistency of Saaty's matrix is defined, which is easy to check during the process of inputting the preference intensities. Several propositions concerning the properties of weakly consistent Saaty's matrices are proven in the paper. We show on an example that the weak consistency, which represents a very natural requirement on Saaty's matrix of preference intensities, is not achieved for some matrices, which are considered "consistent enough" according to the criteria published in the literature. The proposed method of setting Saaty's matrix of preference intensities was used in the model for determining scores for particular categories of artistic production, which is an integral part of the Registry of Artistic Results (RUV) currently being developed in the Czech Republic. The Registry contains data on works of art originating from creative activities of Czech art colleges and faculties. Based on the total scores achieved by these institutions, a part of the state budget subsidy is being allocated among them.