Using tropical optimization techniques in bi-criteria decision problems

Nikolai Krivulin

Published 2018-10-19Version 1

We consider decision problems of rating alternatives based on their pairwise comparisons according to two criteria. Given pairwise comparison matrices for each criterion, the problem is to find the overall priorities of each alternative. We offer a solution that involve the minimax approximation of the comparison matrices by a common (consistent) matrix of unit rank in terms of the Chebyshev metric in logarithmic scale. The approximation problem reduces to a bi-objective optimization problem to minimize the approximation errors simultaneously for both comparison matrices. We formulate the problem in terms of tropical (idempotent) mathematics, which focuses on the theory and applications of algebraic systems with idempotent addition. To solve the optimization problem obtained, we apply methods and results of tropical optimization to derive a complete Pareto-optimal solution in a direct explicit form ready for further analysis and straightforward computation, and then apply this solution to the decision problem of interest. As an illustration of the approach, we present examples of the solution of two-dimensional problems in general form, and of a decision problem with four alternatives in numerical form.