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

Abstract

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, it is susceptible to solution non-uniqueness. Under high levels of inconsistency, such as severe cyclic contradictions, the DLS optimization landscape becomes non-convex, yielding multiple distinct global minima. Consequently, priority rankings become unstable and critically dependent on initial algorithmic guesses. Furthermore, established prioritization operators (POs), including normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), frequently generate disparate outcomes. To overcome these structural deficiencies, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model as a harmonizing framework. ARDLS integrates uniquely determined established POs as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries and tilts the optimization landscape to guarantee convergence upon a single, unique global minimum. By minimizing the root mean square variance (RMSV) of the initial baseline vectors, ARDLS effectively unifies these divergent solutions. Comprehensive numerical experiments validate that the framework successfully fine-tune the solution of established POs by reducing RMSV while ensuring strict mathematical uniqueness. The practical utility of the method is further demonstrated through a numerical case study resolving an innovation fund dilemma in FinTech project selection. The proposed ARDLS approach offers a robust alternative to classical AHP across a wide range of decision-making domains.