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

TL;DR

This paper presents an algebraic method using tropical optimization to solve a complex bi-criteria decision-making problem involving pairwise comparisons and box constraints, providing an analytical solution and Pareto front characterization.

Contribution

It introduces a novel algebraic approach leveraging tropical optimization to analytically solve a constrained bi-criteria rating problem based on pairwise comparisons.

Findings

Derived an analytical solution for the bi-criteria problem

Reduced the problem to solving parameterized inequalities

Identified the Pareto front for the decision problem

Abstract

We consider a decision-making problem to evaluate absolute ratings of alternatives that are compared in pairs according to two criteria, subject to box constraints on the ratings. The problem is formulated as the log-Chebyshev approximation of two pairwise comparison matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank), to minimize the approximation errors for both matrices simultaneously. We rearrange the approximation problem as a constrained bi-objective optimization problem of finding a vector that determines the approximating consistent matrix, and then represent the problem in terms of tropical algebra. We apply methods and results of tropical optimization to derive an analytical solution of the constrained problem. The solution consists in introducing two new variables that describe the values of the objective functions and allow reducing the problem to the solution of a system of parameterized inequalities constructed for the unknown vector, where the new variables play the role of parameters. We exploit the existence condition for solutions of the system to derive those values of the parameters that belong to the Pareto front inherent to the problem. Then, we solve the system for the unknown vector and take all solutions that correspond to the Pareto front, as a complete solution of the bi-objective problem. We apply the result obtained to the bi-criteria decision problem under consideration and present illustrative examples.