发表机构
Monash University Malaysia（马来西亚莫纳什大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对层次分析法中直接最小二乘法存在多解的问题，提出整合既定优先级算子的锚定正则化直接最小二乘模型，可保证收敛到唯一全局最小值，降低均方根误差，为层次分析法提供理想替代方案。
AI 中文摘要
成对互反矩阵是层次分析法（AHP）的基础，层次分析法是一种决策模型。直接最小二乘（DLS）方法提供了一种直观的机制来推导优先级向量，无需复杂变换，但该方法会产生多个解。在高不一致性（如循环矛盾）下，这种非凸性会产生多个不同的全局最小值，导致不稳定的优先级排序，严重依赖算法的初始猜测。为克服这一结构缺陷，本文提出了锚定正则化直接最小二乘（ARDLS）优化模型。ARDLS 将唯一确定的既定优先级算子（如归一化技术、特征向量法、奇异值分解、余弦最大化及伪逆 Gram 矩阵（加权最小二乘的闭式解））作为理论锚点整合到正则化罚项中。这种整合系统性地打破了数学对称性，调整了优化空间，以保证收敛到单一、唯一的全局最小值。综合数值实验与模拟验证，ARDLS 框架成功降低了既定优先级算子间的均方根误差，同时保证了严格的数学唯一性。所提出的 ARDLS 或为应用于诸多领域的 AHP 的理想替代方案。
英文摘要
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.
Comments21 pages, 11 tables, 5 figures; Interactive demonstrations can be accessed at https://kkfyuen.github.io/ardlsDemos/ and are archived at https://doi.org/10.5281/zenodo.22343042
