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

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.

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