Application of methods of tropical optimization for evaluating alternatives based on pairwise comparisons

Abstract

Solution to the problem of evaluating alternatives based on pairwise comparisons is considered, using methods of tropical optimization. The problem of deriving the vector of scores for alternatives reduces to the approximation of pairwise comparison matrices by consistent matrices in the sense of the log-Chebyshev metric. Then the approximation problem is formulated and solved in terms of tropical mathematics. The results obtained are represented in compact vector form, ready for further analysis and practical computations. In the case when the solution is non-unique (up to a positive factor), it is suggested that the set of solutions be characterized via two solutions that are, in some sense, the best and worst solutions. As the best solution, the vector is taken which best differentiates between the alternatives with the highest and lowest scores, and as the worst, the vector which worth differentiates these alternatives. It is shown that these vectors can be obtained using methods of tropical optimization. To illustrate the results obtained, solution examples for problems of evaluating the scores of alternatives are given. Refs 23.