Application of Tropical Optimization for Solving Multicriteria Problems of Pairwise Comparisons Using Log-Chebyshev Approximation

We consider a decision-making problem to find absolute ratings ofalternatives that are compared in pairs under multiple criteria, subject toconstraints in the form of two-sided bounds on ratios between the ratings.Given matrices of pairwise comparisons made according to the criteria, theproblem is formulated as the log-Chebyshev approximation of these matrices by acommon consistent matrix (a symmetrically reciprocal matrix of unit rank) tominimize the approximation errors for all matrices simultaneously. We rearrangethe approximation problem as a constrained multiobjective optimization problemof finding a vector that determines the approximating consistent matrix. Theproblem is then represented in the framework of tropical algebra, which dealswith the theory and applications of idempotent semirings and provides a formalbasis for fuzzy and interval arithmetic. We apply methods and results oftropical optimization to develop a new approach for handling the multiobjectiveoptimization problem according to various principles of optimality. Newcomplete solutions in the sense of the max-ordering, lexicographic ordering andlexicographic max-ordering optimality are obtained, which are given in acompact vector form ready for formal analysis and efficient computation. Wepresent numerical examples of solving multicriteria problems of rating fouralternatives from pairwise comparisons to illustrate the technique and compareit with others.