Application of log-Chebyshev approximation and tropical algebra to multicriteria problems of pairwise comparisons

Abstract

We consider multicriteria problems of evaluating absolute ratings (scores, priorities, weights) of given alternatives for making decisions, which are compared in pairs under several criteria. Given matrices of pairwise comparisons of alternatives for each criterion and a matrix of pairwise comparisons of the criteria, the aim is to calculate a vector of individual ratings of alternatives. We formulate the problem as the Chebyshev approximation of matrices on the logarithmic scale by a common consistent matrix (a symmetrically reciprocal matrix of unit rank). We rearrange the approximation problem as a multi-objective optimization problem of finding a vector that determines the consistent matrix and hence yields a vector of ratings in question. The problem is then transformed into a series of optimization problems in the framework of tropical algebra, which focuses on the theory and application of algebraic systems with idempotent operations. To solve the optimization problems, we apply methods and results of tropical optimization, which yield analytical solutions in a form ready for further analysis and straightforward computation. To illustrate the technique implemented, we give a numerical example of solving a known problem, and compare the obtained solution with results provided by classical methods of analytic hierarchy process and weighted geometrical means.