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

Nikolai Krivulin

TL;DR

This paper addresses multicriteria pairwise comparison problems where absolute ratings of alternatives are inferred from pairwise judgments across multiple criteria. It develops a log-Chebyshev approximation framework on the logarithmic scale and solves the resulting multiobjective problem through tropical algebra, yielding analytical, closed-form representations of solution sets. The approach introduces tropical optimization tools, including generating matrices and atomic solutions, to obtain best and worst differentiating rating vectors, and to aggregate across criteria via a common consistent matrix of unit rank. A numerical school-selection example demonstrates how the log-Chebyshev method compares with analytic hierarchy process and weighted geometric means, and highlights potential ranking differences and the usefulness of multiple perspectives in decision support.

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.

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

TL;DR

This paper addresses multicriteria pairwise comparison problems where absolute ratings of alternatives are inferred from pairwise judgments across multiple criteria. It develops a log-Chebyshev approximation framework on the logarithmic scale and solves the resulting multiobjective problem through tropical algebra, yielding analytical, closed-form representations of solution sets. The approach introduces tropical optimization tools, including generating matrices and atomic solutions, to obtain best and worst differentiating rating vectors, and to aggregate across criteria via a common consistent matrix of unit rank. A numerical school-selection example demonstrates how the log-Chebyshev method compares with analytic hierarchy process and weighted geometric means, and highlights potential ranking differences and the usefulness of multiple perspectives in decision support.

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.
Paper Structure (14 sections, 3 theorems, 46 equations)

This paper contains 14 sections, 3 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Single Criterion Pairwise Comparison Problem
  3. Solution Approaches to Pairwise Comparison Problem
  4. Log-Chebyshev Approximation of Pairwise Comparison Matrices
  5. Best and Worst Differentiating Solutions
  6. Multicriteria Pairwise Comparison Problem
  7. Log-Chebyshev Approximation Using Tropical Algebra
  8. Elements of Tropical Algebra
  9. Tropical Representation of Log-Chebyshev Approximation
  10. Solution of Multicriteria Pairwise Comparison Problem
  11. School Selection Example
  12. Solution by Analytic Hierarchy Process
  13. Solution by Weighted Geometric Means
  14. Solution by Log-Chebyshev Approximation

Key Result

Lemma 1

Let $\bm{A}$ be a $(n\times n)$-matrix with spectral radius $\lambda>0$. Then, the minimum in problem P-minxxAx is equal to $\lambda$, and all positive solutions are given in the parametric form

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3