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Mean-based incomplete pairwise comparisons method with the reference values

Konrad Kułakowski, Anna Kędzior, Jacek Szybowski, Jiri Mazurek

TL;DR

It is proved that the newly introduced geometric HRE method for incomplete data is optimal and the existence of a feasible solution for the second arithmetic variant of the HRE method is provided.

Abstract

In this article, we propose two quantitative methods for calculating weight vectors for incomplete pairwise comparison matrices using reference values. Both procedures are extensions of arithmetic and geometric heuristic estimation (HRE) methods. The proposed solutions allow flexible selection of the number of reference alternatives and the range of comparisons, from the acceptable minimum to a complete set. In this paper, we prove that the newly introduced geometric HRE method for incomplete data is optimal. For this method, we also prove the existence of a feasible solution. In the paper, we also provide sufficient conditions for the existence of a solution for the second arithmetic variant of the HRE method. We illustrate the presented methods with numerical examples.

Mean-based incomplete pairwise comparisons method with the reference values

TL;DR

It is proved that the newly introduced geometric HRE method for incomplete data is optimal and the existence of a feasible solution for the second arithmetic variant of the HRE method is provided.

Abstract

In this article, we propose two quantitative methods for calculating weight vectors for incomplete pairwise comparison matrices using reference values. Both procedures are extensions of arithmetic and geometric heuristic estimation (HRE) methods. The proposed solutions allow flexible selection of the number of reference alternatives and the range of comparisons, from the acceptable minimum to a complete set. In this paper, we prove that the newly introduced geometric HRE method for incomplete data is optimal. For this method, we also prove the existence of a feasible solution. In the paper, we also provide sufficient conditions for the existence of a solution for the second arithmetic variant of the HRE method. We illustrate the presented methods with numerical examples.
Paper Structure (13 sections, 14 theorems, 97 equations)

This paper contains 13 sections, 14 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pairwise comparisons
  4. Matrices and functions
  5. Heuristic Rating Estimation Method
  6. Arithmetic Incomplete Heuristic Rating Estimation
  7. Geometric Incomplete Heuristic Rating Estimation
  8. Solution of Incomplete Heuristic Rating Estimation
  9. Solution of arithmetic approach
  10. The existence of a solution in the geometric approach
  11. Optimality of the geometric approach
  12. Discussion
  13. Summary

Key Result

Theorem 5

Varga2000mia An $n\times n$ matrix $C$ is irreducible if and only if its directed graph $G(C)$ is strongly connected.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Theorem 8
  • Theorem 9
  • Corollary 10
  • ...and 18 more