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Computationally efficient orthogonalization for pairwise comparisons method

Julio Benitez, Waldemar W. Koczkodaj, Adam Kowalczyk

TL;DR

This work addresses the challenge of approximating inconsistent pairwise comparison (PC) matrices with consistent ones by introducing a generalized Frobenius inner product to decompose the skew-symmetric PC space into a sum of additively consistent and completely inconsistent components. It develops a $W$-weighted framework, constructs a $W$-orthogonal basis for the additively consistent subspace $l_n$, and derives a projection-based method to obtain the closest consistent PC matrix, $A^* = \varphi(B_h) \cdot \varphi(B_l)$, where $B_h\in h_{n,W}$ and $B_l\in l_n$. The authors also connect the orthogonal decomposition to a graph-theoretic cycle basis for $h_n$ and provide algorithms to compute a basis for $h_n$, enabling practical and scalable implementations (including a geometric view for $n=3$). These contributions offer a rigorous, implementable pathway to robust PC analysis with tunable weighting and clear geometric interpretation. $A^*$ then yields normalized priority vectors suitable for decision-making.

Abstract

Orthogonalization is one of few mathematical methods conforming to mathematical standards for approximation. Finding a consistent PC matrix of a given an inconsistent PC matrix is the main goal of a pairwise comparisons method. We introduce an orthogonalization for pairwise comparisons matrix based on a generalized Frobenius inner matrix product. The proposed theory is supported by numerous examples and visualizations.

Computationally efficient orthogonalization for pairwise comparisons method

TL;DR

This work addresses the challenge of approximating inconsistent pairwise comparison (PC) matrices with consistent ones by introducing a generalized Frobenius inner product to decompose the skew-symmetric PC space into a sum of additively consistent and completely inconsistent components. It develops a -weighted framework, constructs a -orthogonal basis for the additively consistent subspace , and derives a projection-based method to obtain the closest consistent PC matrix, , where and . The authors also connect the orthogonal decomposition to a graph-theoretic cycle basis for and provide algorithms to compute a basis for , enabling practical and scalable implementations (including a geometric view for ). These contributions offer a rigorous, implementable pathway to robust PC analysis with tunable weighting and clear geometric interpretation. then yields normalized priority vectors suitable for decision-making.

Abstract

Orthogonalization is one of few mathematical methods conforming to mathematical standards for approximation. Finding a consistent PC matrix of a given an inconsistent PC matrix is the main goal of a pairwise comparisons method. We introduce an orthogonalization for pairwise comparisons matrix based on a generalized Frobenius inner matrix product. The proposed theory is supported by numerous examples and visualizations.
Paper Structure (8 sections, 6 theorems, 54 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theoretical foundations for decomposition of PC matrices
  3. $W$-orthogonal basis of $l_n$
  4. Properties of $h_{n,W}$
  5. Geometric interpretation of the orthogonal projection to $h_{3}$ for PC matrices
  6. Graph theory and the elements in $h_n$
  7. A basis of $h_n$
  8. Conclusions

Key Result

Lemma 2.1

Figures (2)

  • Figure 1: The tangent space $h_{3}$ and the normal space $l_{3}$. The matrix $N$ spans $h_3$ and the matrices $E_1$, $E_2$ span $l_3$. Any skew-symmetric matrix $B$ can be uniquely decomposed as $B = B_h+B_l$, where $B_h \in h_3$ and $B_l \in l_3$.
  • Figure 2: Left: graph considered in Example \ref{['examplegraph']}. Right: the reduced graph is formed by deleting node 1 and any edge containing the node 1. The order of the edges $e_1, \ldots, e_6$ is lexicographical.

Theorems & Definitions (12)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • ...and 2 more