Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process

Kevin Kam Fung Yuen

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Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process

Kevin Kam Fung Yuen

arXiv:2608.21187v1Aug 21, 2026math.OCcs.AImath.NA

TL;DR

Inconsistent pairwise reciprocal matrices can make DLS non-convex, yielding multiple solutions and unstable rankings. ARDLS regularizes DLS toward uniquely determined prioritization operators, and the paper reports unique convergence with reduced RMSE across settings. Its regularization depends on λ, while incomplete comparisons and broader practical validation remain future work.

Problem
DLS priority extraction from highly inconsistent pairwise reciprocal matrices can produce multiple global minima, making priority rankings sensitive to initial algorithmic guesses.
Method
ARDLS adds a regularization penalty anchored by uniquely determined established prioritization operators to restore strict convexity and select a unique solution.
Results
ARDLS eliminates initial-value sensitivity and guarantees convergence to a unique global priority vector while reducing RMSE across low- and higher-dimensional inconsistent matrices.
Takeaways & Limitations
ARDLS offers a unified approach for deriving stable priority vectors by bridging competing prioritization operators within a mathematically sound framework.
Takeaways & Limitations
The regularization outcome depends heavily on selecting an appropriate λ, and extension to incomplete pairwise comparisons remains future work.

Abstract

from arXiv · show

Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP), a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, the DLS provides multiple solutions. Under high levels of inconsistency, such as cyclic contradictions, this non-convexity yields multiple distinct global minima, resulting in unstable priority rankings that critically depend on initial algorithmic guesses. To overcome this structural deficiency, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model. ARDLS integrates uniquely determined established prioritization operators, such as normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries, tilting the optimization landscape to guarantee convergence upon a single, unique global minimum. Comprehensive numerical experiments and simulations validate that the ARDLS framework successfully reduces root mean square error among established priority operators, while guaranteeing strict mathematical uniqueness. The proposed ARDLS may be the ideal alternative for the AHP applied to many application domains.

1. Introduction

AHP prioritization from inconsistent pairwise reciprocal matrices remains difficult because established operators lack consistent superiority and DLS can have multiple optima. ARDLS addresses this non-uniqueness by anchoring DLS with established prioritization operators and evaluates the resulting framework numerically.

Broad evaluations find that no individual prioritization operator consistently outperforms the others under judgmental inconsistencies.
DLS is arithmetically appealing but lacks a closed-form analytical solution and is fundamentally non-convex.
DLS non-convexity can produce distinct priority vectors with the same minimized objective value, making standard numerical optimization unstable and computationally challenging.
ARDLS incorporates robust, uniquely determined prioritization operators as theoretical anchors within a regularization penalty to address DLS multiple minima.
The study evaluates ARDLS through RMSE analysis and comprehensive numerical experiments across inconsistency thresholds to assess uniqueness, usability, feasibility, and stability.

2 Prioritization Operators

This section reviews established prioritization operators and describes normalization-based procedures for deriving priority vectors from pairwise comparison matrices.

The section reviews established prioritization operators, including four foundational normalization techniques whose names follow their operational calculation steps.
NRS sums each row’s elements and normalizes by the grand total, producing priority-vector components that sum to unity.
NRCS sums each column, takes the reciprocal of each column sum, and normalizes the resulting values to sum to unity.
AMNC divides each matrix element by its column sum and computes each priority weight as the arithmetic mean across its normalized row elements.

2.4 Normalization of the Geometric Means of Rows (NGMR) / Logarithmic Least Squares (LLS)

NGMR derives normalized priorities from row-wise geometric means, while the geometric mean vector is also the exact closed-form solution to the LLS optimization problem.

2.4 NGMR: NGMR calculates each row’s geometric mean as the n-th root of its element product, then scales the resulting vector so its components sum to unity.
2.4 LLS: The geometric mean vector is the exact closed-form solution to the Logarithmic Least Squares optimization problem.
2.4 NGMR: A subsequent scaling step converts the geometric-mean vector into the final normalized priority vector.

2.5 Eigenvector (EV)

The Eigenvector operator derives priorities from the principal right eigenvector associated with the matrix’s maximum eigenvalue, which also supports consistency-ratio evaluation.

The Eigenvector operator solves the characteristic eigensystem to obtain the non-normalized principal right eigenvector associated with the maximum eigenvalue of matrix A.
The principal eigenvalue λmax is computed as part of the Eigenvector method.
λmax is used to form the Consistency Ratio for evaluating the consistency of a pairwise comparison matrix.

2.6 Singular Value Decomposition (SVD)

The SVD approach factorizes a pairwise reciprocal matrix into three components and derives priority weights from its principal singular vectors, followed by normalization.

2.6 Singular Value Decomposition (SVD): SVD is presented as a framework for priority elicitation in the Analytic Hierarchy Process.Its role is to derive priorities from the matrix’s singular-vector structure.
2.6 Singular Value Decomposition (SVD): SVD factorizes the pairwise reciprocal matrix as A = UΣV^T.The factorization provides the three-component structure used for priority elicitation.
2.6 Singular Value Decomposition (SVD): The priority weights w_i combine elements from the principal left singular vector and reciprocal elements of the principal right singular vector.The resulting weights are then normalized.

2.7 Cosine maximization (CosMax)

CosMax derives priorities by maximizing directional alignment between the weight vector and matrix columns, and this study further provides a direct closed-form solution.

2.7 Cosine maximization (CosMax): CosMax maximizes directional alignment between the priority weight vector and the matrix’s column profiles.The method formulates priority elicitation as an alignment objective.
2.7 Cosine maximization (CosMax): Its objective can be written as the sum of cosines between the weight vector and the matrix columns.The angles θ_j measure alignment with each column vector a_·j.
2.7 Cosine maximization (CosMax): This study derives a direct, single-stage algebraic closed-form solution for the CosMax model.The derivation replaces the previously outlined five-step optimization procedure.

2.8 Least Squares operators

The least-squares operators derive priorities by minimizing squared errors, with WLS addressing DLS’s structural deficiencies and PIGM supplying an explicit solution.

2.8 Least Squares operators: DLS directly minimizes the sum of squared errors to achieve the least RMSV.Its appeal is the direct arithmetic form of the optimization objective.
2.8 Least Squares operators: WLS was introduced to address severe structural deficiencies in the DLS optimization problem.The passage presents WLS as an alternative least-squares formulation.
2.8 Least Squares operators: PIGM offers an explicit solution within the Inverse Gram Matrix family after WLS lacked a closed-form analytic solution.Its role is to provide a closed-form weighted-least-squares solution.
2.8 Least Squares operators: The vector r is nonzero, and e is a compatible column vector of ones for the matrix G^-1.These conditions define the notation used in the PIGM formulation.

3. Prioritization result variance measures

RMSE is introduced as a performance metric for comparing prioritization operators; although DLS attains the lowest RMSE, it can produce multiple solutions, while ARDLS targets uniqueness and RMSE improvement.

3. Prioritization result variance measures: RMSE provides a rigorous metric for quantifying native errors when evaluating prioritization operators.It advances beyond Total Deviation and Euclidean Distance as comparison measures.
3. Prioritization result variance measures: The RMSE expression can be represented compactly through element-wise matrix operations.This form supports efficient matrix-centric computation.
3. Prioritization result variance measures: DLS yields the lowest RMSE but often produces multiple solutions, whereas ARDLS is proposed to provide uniqueness while outperforming other POs in RMSE.The result passage presents the trade-off between DLS’s error performance and solution multiplicity.

4. Anchored Regularized Direct Least Squares

ARDLS combines Direct Squares Error with an anchor regularization penalty to address DLS non-uniqueness. A sufficiently large regularizer reshapes the objective into a strictly convex landscape with a unique global minimum.

Model formulation: ARDLS minimizes the sum of Direct Squares Error and an Anchor Regularization Penalty based on a pre-established priority vector.The anchor penalty discourages estimated weights from deviating excessively from weights produced by another prioritization operator.
Parameter behavior: The regularization parameter interpolates between DLS at λ→0 and the anchor vector at λ→∞.The appropriate setting of λ determines the optimal balance between data-driven and prior weights.
DLS convexity: DLS is unique for consistent or slightly inconsistent matrices but may have multiple local or global minima under high inconsistency.A positive local convexity bound guarantees uniqueness only within a coordinate basin and does not exclude distinct global solutions elsewhere.
ARDLS convexity: The anchor penalty adds a strongly convex quadratic component that can force the ARDLS objective into strict convexity when λ is sufficiently large.The resulting landscape has a single, unique global minimum.
Regularization threshold: If Δ_DLS≤0, ARDLS guarantees a unique solution when λ exceeds the minimum regularizer bound λ̃.The bound is derived from the worst-case minimum curvature across identified DLS global solutions, with a safety margin α>1 defining a safe regularizer.
Practical configuration: ARDLS can estimate the DLS curvature from anchor weights before optimization and initialize the solver at those weights for stable, reproducible convergence.This heuristic helps assess likely non-convexity and guides the solver into the same local valley across runs.

5. Graphical and Numerical analysis

Numerical and graphical analyses show that DLS is unique under consistent or mild inconsistency but can become multi-modal under severe cyclic inconsistency. ARDLS regularization removes competing minima, stabilizes convergence, and improves error metrics across 3 × 3 and 6 × 6 cases.

Case 2: Mild inconsistency: Under slight inconsistency, DLS retains a strictly unimodal convex landscape and converges to a unique global minimum despite an elevated error floor.For CR = 0.0156, DLS reaches objective value 0.3491 at w = (0.148,0.622,0.229).
Case 2: Mild inconsistency: ARDLS consistently converges to w* ≈ (0.1485,0.6221,0.2295) and reduces baseline DSE values of 0.3719–0.7041 to 0.3491.The anchor regularization penalty remains between 0.0001 and 0.0010, while RMSE(w*) = 0.1970 outperforms every individual base operator.
Case 3: Severe cyclic inconsistency: Under severe cyclic inconsistency with CR = 1.94, unregularized DLS has three distinct global solutions with minimum objective value 28.445 and strong sensitivity to initialization.The symmetric initialization reaches a saddle region at objective value 28.688 with ΔDLS = −9, whereas perturbations reach asymmetric basins with ΔDLS = 29.941.
Case 3: Severe cyclic inconsistency: Safe regularization with λ* = 36 transforms the multi-modal DLS landscape into a strictly unimodal basin and guarantees convergence to one global minimum independently of initial values.The regularizer eliminates competing global basins while retaining standard anchor vectors.
Case 4: Higher-dimensional PRM: In the highly inconsistent 6 × 6 case, ARDLS stabilizes results across variable prioritization operators, yielding DSE(w*) = 60.015 and RMSV(w*) = 1.2912.Unregularized baseline DSE ranges from 73.16 to 138.22 and baseline RMSV from 1.42 to 1.95.

6. Conclusion and future study

ARDLS addresses severe non-convexity in DLS by selecting a curvature-based regularization penalty that restores strict global convexity. Numerical evaluations show reduced initial-value sensitivity across matrix sizes, while future work targets incomplete comparisons and practical software validation.

ARDLS dynamically evaluates local curvature bounds to determine a safe regularization penalty λ∗ that restores strict global convexity.
Numerical evaluations show that ARDLS eliminates initial-value sensitivity in both 3 × 3 cyclic and 6 × 6 highly inconsistent matrices.
ARDLS guarantees convergence to a unique robust global priority vector while reducing Root Mean Square Error across diverse prioritization operators.
Future work should adapt the regularizer to incomplete pairwise comparisons and alternative non-linear objectives such as Weighted Least Squares.
Integrating ARDLS into practical software and testing it in real-world domains would further validate user satisfaction and decision stability.