解决层次分析法中优先级排序不稳定问题，确保结果唯一可靠。
Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process
- 引入锚定正则化项，融合多种优先级计算方法作为数学约束。
- 实验表明能显著降低不同方法间的误差，且收敛到唯一解。
- 适合需要稳定决策结果的复杂评估场景，如多准则决策系统。
成对互逆矩阵是层次分析法（AHP）这一决策模型的核心。尽管直接最小二乘法（DLS）提供了一种无需复杂变换即可推导优先级向量的直观机制，但其存在多个解的问题。在高度不一致情况下，如循环矛盾时，这种非凸性会导致多个不同的全局极小值，使优先级排序极度依赖初始算法猜测。为克服这一结构性缺陷，本文提出锚定正则化直接最小二乘法（ARDLS）优化模型。该模型将归一化、特征向量法、奇异值分解、余弦最大化及加权最小二乘的闭式解（伪逆格拉姆矩阵）等已确立的优先级计算算子作为理论锚点，嵌入正则化惩罚项中。这一整合系统性地打破数学对称性，引导优化景观收敛至唯一全局极小值。大量数值实验与模拟验证了，ARDLS框架可有效降低各已有优先级算子间的均方根误差，同时保证严格的数学唯一性。所提出的ARDLS或将成为AHP在众多应用领域中的理想替代方案。
原文摘要 · Abstract (English)
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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