Methods of tropical optimization in multicriteria problems of raiting alternatives from pairwise comparisons

The paper deals with the application of methods and results of tropical mathematics, which focuses on the theory and applications of algebraic systems with idempotent operations, to the development of a multicriteria decision-making procedure. A problem is considered to evaluate ratings of alternatives from pairwise comparisons of the alternatives under several criteria, and from pairwise comparisons of the criteria. To solve the problem, a decision-making procedure is proposed based on the Chebyshev approximation, in logarithmic scale, of pairwise comparison matrices by reciprocally symmetrical matrices of unit rank (consistent matrices), which determine the elements in the vectors of weights of criteria and ratings of alternatives. First, the approximation problem for the matrix of pairwise comparison of criteria is solved to find the weights of criteria. Then, the weighted pairwise comparison matrices of alternatives are approximated by a common consistent matrix, which gives the required vector of ratings of alternatives. If the result is not unique (up to a positive factor), an additional problem of analyzing the solutions is solved to find vectors that can be considered, in a sense, as the worst and best solutions. In the framework of the proposed procedure, the problems of approximation and analysis of solutions are formulated as tropical optimization problems, which have direct analytical solutions in a compact vector form. An example of the application of the procedure to solve the known problem by T. Saaty on selecting a school is given.