Deriving a priority vector from a pairwise comparison matrix (PCM) is a crucial step in the Analytical Hierarchy Process (AHP). Although there exists a priority vector that satisfies the conditions of order preservation (COP), the priority vectors obtained through existing prioritization methods frequently violate these conditions, resulting in numerous COP violations. To address this issue, this paper introduces a novel procedure to manage COP violations in AHP. Firstly, we prove that the index-exchangeability condition is both a necessary and sufficient condition for determining whether a priority vector satisfies COP. This enables the direct detection of COP violations, relying solely on the pairwise comparison preferences of decision-makers, rather than the prioritization methods utilized. Subsequently, we propose the Minimal Number of Violations and Deviations Method (MNVDM) model, which aims to derive a priority vector with the minimal number of COP violations. In particular, the MNVDM can obtain a violation-free priority vector when the PCM meets the index exchangeability conditions. Furthermore, an optimization model based on minimizing information loss is designed to ensure the COP by revising the preferences when the index-exchangeability conditions are violated. Finally, the feasibility and efficiency of the proposed models are validated through numerical examples and Monte Carlo simulation experiments. Our implementation is available at: https://github.com/Tommytutu/COP.

翻译：从成对比较矩阵中推导优先向量是层次分析法的关键步骤。尽管存在满足序保持条件的优先向量，但现有优先排序方法获得的优先向量经常违反这些条件，导致大量COP违规。为解决此问题，本文提出一种处理AHP中COP违规的新方法。首先，我们证明指标可交换条件是判断优先向量是否满足COP的充分必要条件。这使得能够直接检测COP违规，仅依赖于决策者的成对比较偏好，而非所使用的优先排序方法。随后，我们提出最小违规数与偏差法模型，旨在获得具有最少COP违规次数的优先向量。特别地，当PCM满足指标可交换条件时，MNVDM可获得零违规的优先向量。此外，当违反指标可交换条件时，我们设计了基于最小化信息损失的优化模型，通过修正偏好来确保COP。最后，通过数值算例和蒙特卡洛模拟实验验证了所提模型的可行性与有效性。我们的实现代码发布于：https://github.com/Tommytutu/COP。