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Application of tropical optimization for solving multicriteria problems of pairwise comparisons using log-Chebyshev approximation

Nikolai Krivulin

TL;DR

Methods and results of tropical optimization are applied to develop a new approach for handling the multiobjective optimization problem according to various principles of optimality, which is given in a compact vector form ready for formal analysis and efficient computation.

Abstract

We consider a decision-making problem to find absolute ratings of alternatives that are compared in pairs under multiple criteria, subject to constraints in the form of two-sided bounds on ratios between the ratings. Given matrices of pairwise comparisons made according to the criteria, the problem is formulated as the log-Chebyshev approximation of these matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank) to minimize the approximation errors for all matrices simultaneously. We rearrange the approximation problem as a constrained multiobjective optimization problem of finding a vector that determines the approximating consistent matrix. The problem is then represented in the framework of tropical algebra, which deals with the theory and applications of idempotent semirings and provides a formal basis for fuzzy and interval arithmetic. We apply methods and results of tropical optimization to develop a new approach for handling the multiobjective optimization problem according to various principles of optimality. New complete solutions in the sense of the max-ordering, lexicographic ordering and lexicographic max-ordering optimality are obtained, which are given in a compact vector form ready for formal analysis and efficient computation. We present numerical examples of solving multicriteria problems of rating four alternatives from pairwise comparisons to illustrate the technique and compare it with others.

Application of tropical optimization for solving multicriteria problems of pairwise comparisons using log-Chebyshev approximation

TL;DR

Methods and results of tropical optimization are applied to develop a new approach for handling the multiobjective optimization problem according to various principles of optimality, which is given in a compact vector form ready for formal analysis and efficient computation.

Abstract

We consider a decision-making problem to find absolute ratings of alternatives that are compared in pairs under multiple criteria, subject to constraints in the form of two-sided bounds on ratios between the ratings. Given matrices of pairwise comparisons made according to the criteria, the problem is formulated as the log-Chebyshev approximation of these matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank) to minimize the approximation errors for all matrices simultaneously. We rearrange the approximation problem as a constrained multiobjective optimization problem of finding a vector that determines the approximating consistent matrix. The problem is then represented in the framework of tropical algebra, which deals with the theory and applications of idempotent semirings and provides a formal basis for fuzzy and interval arithmetic. We apply methods and results of tropical optimization to develop a new approach for handling the multiobjective optimization problem according to various principles of optimality. New complete solutions in the sense of the max-ordering, lexicographic ordering and lexicographic max-ordering optimality are obtained, which are given in a compact vector form ready for formal analysis and efficient computation. We present numerical examples of solving multicriteria problems of rating four alternatives from pairwise comparisons to illustrate the technique and compare it with others.
Paper Structure (31 sections, 11 theorems, 139 equations)

This paper contains 31 sections, 11 theorems, 139 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 Matrix
  5. Best and Worst Differentiating Solutions
  6. Multicriteria Pairwise Comparison Problems
  7. Max-Ordering Solution
  8. Lexicographic Ordering Solution
  9. Lexicographic Max-Ordering Solution
  10. Preliminary Algebraic Definitions, Notation and Results
  11. Idempotent Semifield
  12. Matrix and Vector Algebra
  13. Vector Inequalities
  14. Tropical Optimization Problems
  15. Conjugate Quadratic Optimization Problem
  16. ...and 16 more sections

Key Result

Lemma 1

For any column-regular matrix $\bm{A}$ and regular vector $\bm{d}$, all solutions of inequality I-Axleqd are given by the inequality $\bm{x}\leq(\bm{d}^{-}\bm{A})^{-}$.

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • proof
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 4 more