Optimal constrained design of control charts using stochastic approximations

In statistical process monitoring, control charts typically depend on a set of tuning parameters besides its control limit(s). Proper selection of these tuning parameters is crucial to their performance. In a specific application, a control chart is often designed for detecting a target process distributional shift. In such cases, the tuning parameters should be chosen such that some characteristic of the out-of-control (OC) run length of the chart, such as its average, is minimized for detecting the target shift, while the control limit is set to maintain a desired in-control (IC) performance. However, explicit solutions for such a design are unavailable for most control charts, and thus numerical optimization methods are needed. In such cases, Monte Carlo-based methods are often a viable alternative for finding suitable design constants. The computational cost associated with such scenarios is often substantial, and thus computational efficiency is a key requirement. To address this problem, a two-step design based on stochastic approximations is presented in this paper, which is shown to be much more computationally efficient than some representative existing methods. A detailed discussion about the new algorithm’s implementation along with some examples are provided to demonstrate the broad applicability of the proposed methodology for the optimal design of univariate and multivariate control charts. Computer codes in the Julia programming language are also provided in the supplemental material.

在统计过程监控领域，控制图通常除控制限外，还依赖一组调优参数。合理选取这些调优参数对控制图的性能至关重要。在特定应用场景中，控制图往往被设计用于检测目标过程的分布偏移，此时需对调优参数进行选择，使得控制图的失控（out-of-control, OC）运行长度的某项特征（如其平均值）在检测目标偏移时达到最小，同时通过设置控制限以维持期望的在控（in-control, IC）性能。然而，对于多数控制图而言，此类设计并无显式解，因此需要采用数值优化方法。在此类场景中，基于蒙特卡洛的方法常是获取合适设计常数的可行替代方案，但该类方法对应的计算成本往往较高，故计算效率成为核心需求。为解决该问题，本文提出了一种基于随机逼近的两步设计方法，经验证其计算效率远优于若干代表性现有方法。本文还详细讨论了新算法的实现细节，并辅以若干示例，以展示所提方法在单变量与多变量控制图优化设计中的广泛适用性。补充材料中还提供了基于Julia编程语言的计算机代码。