Methods of Tropical Optimization in Rating Alternatives Based on Pairwise Comparisons

We apply methods of tropical optimization to handle problems of rating alternatives on the basis of the log-Chebyshev approximation of pairwise comparison matrices. We derive a direct solution in a closed form, and investigate the obtained solution when it is not unique. Provided the approximation problem yields a set of score vectors, rather than a unique (up to a constant factor) one, the set is reduced to those vectors, which least and most differentiate between the alternatives with the highest and lowest scores, and thus can be representative of the entire solution. We offer a technique to find these vectors, and give an illustrative numerical example. Key-Words: idempotent semifield, tropical optimization, matrix approximation, pairwise comparison, consistent matrix. MSC (2010): 65K10, 15A80, 65K05, 41A50, 90B50