An improved bisection-type algorithm for control chart calibration

Calculation of the control limits is a critical step when designing control charts in statistical process control. Traditional control chart designs require the control limits to be computed so that a characteristic of the in-control run length distribution, such as the mean or median, equals a pre-determined value. When the complexity of the in-control process distribution hinders analytical methods, Monte Carlo approaches can be used to find the appropriate control limits. Among these methods, the classical bisection searching algorithm is widely used. However, a major drawback of this method is the requirement of an initial range of values for the search. Furthermore, it is computationally very demanding when multiple control charts are used simultaneously. In this paper, we present a modified bisection searching algorithm to enhance the computational efficiency. The new method eliminates the initial specification of a range for searching. Additionally, an efficient generalization of this approach is proposed to handle the multi-chart setting. Numerical results confirm that our method offers an efficient and reliable way to compute the control limits, in comparison with the conventional bisection searching algorithm and the algorithm based on stochastic approximations. A Julia computer code implementing the proposed method is provided in the supplemental materials.