Algebraic solution to constrained bi-criteria decision problem of rating alternatives through pairwise comparisons

Nikolai Krivulin

Published 2019-11-21Version 1

We consider a decision-making problem to evaluate absolute ratings (priorities, scores) of alternatives (choices, decisions) from the results of their pairwise comparisons according to two equally weighted criteria, subject to constraints on the rates. First, we formulate the problem as a bi-objective optimization problem that consists in the simultaneous constrained approximation, in the Chebyshev sense in logarithmic scale, of pairwise comparison matrices for each criteria by a common consistent matrix of unit rank, which determines the vector of ratings. Furthermore, we represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution approach involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the optimization problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the optimization problem.