POO-LPSP: Parallel Osprey Optimized Least Penalty-Squared Prioritization Methods for Priority Derivation in the Analytic Hierarchy Process

In the Analytic Hierarchy Process (AHP), deriving priority vectors from pairwise reciprocal matrices (PRMs) is a critical step. While traditional methods like the Eigenvector method are standard, they often struggle with inconsistent judgments and rank reversal. This paper addresses the computational difficulty of solving advanced, non-linear optimization models—specifically the Least Penalty-Squared Prioritization (LPSP) models—designed to improve prioritization reliability by minimizing penalty-weighted variance.

The author develops two refined LPSP models: the Least Product of Penalty and Direct Squares (LPPDS) and the Least Product of Penalty and Weighted Squares (LPPWS). These models incorporate a new metric, the Root Mean Penalty-Squared Variance (RMPSV), which integrates structural penalty logic with variance data to handle consistency violations more effectively. Because these models are non-convex and non-differentiable, the author introduces the Parallel Osprey Optimization Algorithm (POOA). This metaheuristic is enhanced with parallel computing, constraint-handling mechanisms, and an adaptive early-stopping criterion to efficiently navigate the search space and avoid local minima.

The POO-LPSP method provides a robust framework for priority derivation that outperforms traditional methods by explicitly penalizing inconsistent judgments. By utilizing parallel processing, the POOA significantly reduces the computational bottleneck associated with evaluating complex penalty matrices across large populations. The practical utility of this approach was validated through a Generative AI (GAI) vendor selection case study, demonstrating that POO-LPSP serves as a reliable alternative to Saaty's Eigen system method, particularly in scenarios where decision-makers require higher consistency and mathematical rigor.