Analytic hierarchy process: an approach based on the theory of latent variables

The paper proposes a mathematical model for evaluating the attractiveness of alternatives by a qualitative criterion, which is based on the method of evaluating latent variables according to the Rasch model. This model is intended for assessing the attractiveness of alternatives in expert evaluation using the hierarchy analysis method based on a paired comparison of alternatives. The proposed method will make it possible to obtain objective estimates of alternatives on a linear scale, which do not depend on the composition of the alternatives being evaluated. Aim. The purpose of the study is to describe a new mathematical apparatus that allows one to obtain estimates of the attractiveness of alternatives and has a number of advantages over traditional methods of evaluation. The proposed model is based on the theory of latent variables, and more precisely, on the Rasch model for estimating latent variables. Materials and methods. The basis of the model for evaluating alternatives described in the paper is the method of paired comparisons. However, unlike the traditional mathematical apparatus of the hierarchy analysis method, the method for identifying the preference of one alternative over another for each of their pairs is based on a probabilistic approach, so preference estimates are less abstract and more objective. The paper gives a mathematical substantiation of the proposed model, compares the results of estimation by the proposed model with the results obtained by traditional methods. Computational experiments are also described that substantiate the adequacy of the obtained estimates. Results. A mathematical model has been developed, which is an alternative to the hierarchy analysis method (T. Saaty) for assessing the attractiveness of alternatives using the method of paired comparisons. Unlike the traditional evaluation method, the model allows you to obtain independent estimates of alternatives on a linear scale, and the criteria for comparing alternatives are probabilistic in nature, which allows for an objective assessment. Computational experiments have shown the adequacy and good stability of the obtained estimates to changes in the initial data. Conclusion. The proposed method for processing expert information obtained from paired comparisons of alternatives will make it possible to obtain the most objective assessments of them and can be used as a mathematical apparatus for decision support systems in many areas of scientific and practical activity.