Pairwise Comparison Matrix

• Pairwise comparison matrices are structured mathematical tools used to encode, analyze, and derive priority judgments based on multiplicative ratio assessments.
• They incorporate methods such as the eigenvector, geometric mean, and tropical optimization approaches to extract weight vectors and assess consistency.
• Applications span decision analysis, group aggregation, and clustering, leveraging advanced concepts in linear algebra, optimization, and combinatorial theory.

A pairwise comparison matrix (PCM) is a structured mathematical object used to encode, analyze, and derive priorities among alternatives or criteria based on multiplicative ratio judgments. PCMs are central to multi-criteria decision making frameworks such as the Analytic Hierarchy Process (AHP), and they underpin advanced research directions in matrix theory, optimization, and data aggregation. Their study involves diverse mathematical disciplines, including linear algebra, combinatorics, optimization, Lie theory, and tropical mathematics.

1. Definition, Reciprocity, and Consistency

A pairwise comparison matrix of order $n$, $A = [a_{ij}]$, is a square matrix with entries $a_{ij} > 0$ for all $i, j$. The following properties are standard:

• Reciprocity: $a_{ij} a_{ji} = 1$ for all $i, j$, with diagonal $a_{ii} = 1$.
• Consistency: $a_{ij} a_{jk} = a_{ik}$ for all $i, j, k$ (multiplicative transitivity).

A matrix $A$ is consistent if there exists a positive weight vector $s = (s_1, \ldots, s_n)$ such that $a_{ij} = s_i / s_j$. If the entries deviate from this relation, $A$ is inconsistent.

Consistency can also be characterized in log-space: if $x_{ij} = \log a_{ij}$, then $A$ is consistent if $x_{ij} + x_{jk} = x_{ik}$ and $x_{ji} = -x_{ij}$ for all $i, j, k$ (1311.0748).

2. Weight Extraction and Priority Derivation Methods

Eigenvector Method (EV/Saaty)

The principal right eigenvector $w^{EV}$ of $A$, normalized so $\sum_k w_k=1$, is found by solving:

$$A\,w^{EV} = \lambda_{\max}\, w^{EV}, \quad w^{EV} > 0$$

For consistent matrices, $\lambda_{\max} = n$ and $w^{EV}_i/w^{EV}_j = a_{ij}$. For inconsistent matrices, the ratios approximate the judgments (Herman et al., 2015).

Geometric Mean (GM, Logarithmic Least Squares, LLSM)

The GM method computes for each row:

$$w_i^{GM} = \left( \prod_{j=1}^n a_{ij} \right)^{1/n}, \quad w_i = \frac{w_i^{GM}}{\sum_{k=1}^n w_k^{GM}}$$

This yields the unique minimizer of the sum of squared logarithmic errors. The GM weights coincide with the solution obtained by projecting the log-matrix onto the consistent subspace with respect to the Frobenius inner product (Koczkodaj et al., 2015, Koczkodaj et al., 2020).

Heuristic Rating Estimation (HRE)

When some weights are known (references), HRE computes unknown weights as the fixed point of an averaging process, leading to a linear system $A w_U = b$, where $w_U$ are unknown weights and $b$ involves the known references. Existence and uniqueness under bounded inconsistency is characterized via M-matrix theory (Kułakowski, 2013, Kułakowski, 2014).

Tropical and Orthogonal Projection Approaches

Tropical optimization minimizes the maximum log-error (log-Chebyshev sense), yielding solutions via the Kleene star and tropical spectral radius (Krivulin, 2015). Orthogonal projections in the log-space under various inner products (including weighted Frobenius) produce consistent approximations that minimize reconstruction error in the chosen norm (Benitez et al., 18 Mar 2024, Koczkodaj et al., 2020, Koczkodaj et al., 2021).

3. Inconsistency Indices and Monitoring

Several indices measure the degree of inconsistency:

• Saaty’s Consistency Index (CI):

$CI(A) = \frac{\lambda_{\max}(A) - n}{n-1}$

Frequently scaled by a random index $RI_n$ to form the Consistency Ratio $CR = CI/RI_n$ (1311.0748, 1311.0748).

• Koczkodaj’s Index (CM/KI):

For all triads $(i, j, k)$:

$$KI(A) = \max_{i < j < k} \min \left\{ |1 - a_{ik}/(a_{ij} a_{jk})|,\, |1 - (a_{ij} a_{jk})/a_{ik}| \right\}$$

Interpreted as the largest minimal violation of transitivity in any triad (1311.0748, Herman et al., 2015).

• Peláez–Lamata Determinant-based Index (CI):

Uses the average of determinants of triads (1311.0748).

• Distance-based Inconsistency:

Minimizes the maximal distortion over all triads. Efficiently monitored using LP in incomplete matrices to provide real-time feedback (Bozoki et al., 2015).

Thresholds (e.g., $CR < 0.10$) are commonly used for acceptability, though their theoretical justification remains debated. Advanced inconsistency reduction is formulated as mixed-integer convex programming, enabling targeted repair of PCMs with minimal edits (1311.0748).

4. Structure, Generator Sets, and Reconstruction

In a consistent $n \times n$ PCM, all entries can be generated from any spanning tree of $n-1$ multiplicative entries—referred to as generators. Reconstruction proceeds by solving a sparse linear system in log-space:

$y_{i_k} - y_{j_k} = \log(v_k),\ k=1,\dots,n-1$

where $y_i = \log s_i$ encode the weights. The full matrix is then $a_{ij} = \exp(y_i - y_j)$ (Koczkodaj et al., 2013). The minimal generator property highlights foundational combinatorial connections (Cayley’s formula: $n^{n-2}$ generator sets).

This formalism clarifies sensitivity to error propagation and the critical role of spanning-tree selection in minimizing distortion.

5. Orthogonalization and Lie-Algebraic Decomposition

Orthogonal projections of log-transformed PCMs onto the consistent subspace (of dimension $n-1$) yield the closest consistent matrix in a given inner product, typically the Frobenius or a weighted variant. The orthogonal complement quantifies pure inconsistency.

• Generalized Frobenius Orthogonalization: An explicit $W$-orthogonal basis can be constructed with Gram–Schmidt, costing $O(n^3)$. The projection recovers the consistent approximation $A_{l}$ and the inconsistency component $A_{h}$ (Benitez et al., 18 Mar 2024, Koczkodaj et al., 2020, Koczkodaj et al., 2015).
• Lie-Theoretic Decomposition: PCMs are considered as elements of a matrix Lie group under Hadamard multiplication, with the log-domain mapping their geometry into skew-symmetric matrices. Decomposition yields $A = A_{\rm approx} \circ A_{\rm ortho}$, separating the nearest consistent matrix from the inconsistency residual (Koczkodaj et al., 2021).

For non-abelian generalizations (e.g., group-valued PCMs), consistency is expressed as the existence of $\{w_i \in G\}$ such that $g_{ij} = w_i^{-1} w_j$, and the projection problem becomes a Riemannian optimization in the group manifold (Koczkodaj et al., 2016).

6. Efficiency, Pareto Optimality, and Weight Vector Properties

A weight vector $w$ is efficient (Pareto-optimal) if no other $w'$ approximates all matrix elements at least as well, and some strictly better, i.e.,

$$|\!| a_{ij} - w'_i/w'_j |\!| \leq |\!| a_{ij} - w_i/w_j |\!| \;\forall (i, j)$$

and strict inequality for at least one $(i, j)$. The principal eigenvector is always weakly efficient but can be strictly inefficient (Bozóki et al., 2016). Linear programming formulations allow testing for efficiency and constructing dominating efficient vectors. For PCMs differing from consistency in a single reciprocal pair (simple perturbed), the principal eigenvector is efficient (Ábele-Nagy et al., 2015).

Monte Carlo studies show that for modest levels of inconsistency, GM and EV methods produce virtually indistinguishable weights (maximal observed norm differences below 0.1%) (Herman et al., 2015, Kułakowski et al., 2020). The discrepancy between methods grows with inconsistency but rarely exceeds empirically established error bounds for $KI < 0.2$.

7. Applications, Aggregation, and Clustering

PCMs are extensively used in group decision contexts. When aggregating multiple PCMs, clustering approaches (e.g., $k$-medoids) are applied directly to PCMs in log-Euclidean or compatibility-based dissimilarity metrics. This enables detection of outlier decision-makers, quantification of reliability (variance of inconsistency within clusters), and robust aggregation by choosing representative medoids rather than averaging PCMs (Ágoston et al., 8 Feb 2024). Constraints on inconsistencies of cluster centers are naturally incorporated as linear constraints in the clustering LP.

Table: Methods for Deriving Weights from PCMs

Method	Solution Principle	Distance/Optimality Criteria
Eigenvector (EV)	Principal r.e. eigenvector of $A$	Nonlinear eigenproblem, L2 global fit
Geometric Mean (GM)	Row-wise geometric mean, normalized	Log-LS, Frobenius projection
HRE	Linear system using references and averaging heuristics	Diagonal-dominant Jacobi, M-matrix
Tropical Optimization	Minimax (max-log-error) tropical opt.	Log-Chebyshev (∞-norm)
Orthogonal Projection	Euclidean or weighted log-space projection	Closest in norm induced by inner prod.

Each method offers distinct computational, interpretive, and robustness properties, especially for inconsistent or incomplete data. For "not-so-inconsistent" matrices, practical differences are negligible for most applications (Herman et al., 2015).

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The theory and methodology of pairwise comparison matrices integrate linear and nonlinear optimization, matrix factorization, combinatorial enumeration, and algebraic geometry, establishing PCMs as a central object in mathematical decision modeling, with advanced tools for efficiency testing, inconsistency monitoring, and structured aggregation. The technical literature provides comprehensive frameworks for projection, perturbation, and dynamic analysis of PCMs, underlined by precise bounds, empirical validation, and efficient algorithms (Koczkodaj et al., 2015, 1311.0748, Bozóki et al., 2016, Benitez et al., 18 Mar 2024, Kułakowski et al., 2020).