Optimal design of time-between-event control charts with parameter estimation

The tr chart is a Shewhart-type control chart used to monitor the time between events, especially in high-quality processes. It has been shown to be more efficient than classical attribute control charts based on count data. In practical applications, the in-control process parameters are often unknown and need to be estimated from a Phase I reference sample. When the available Phase I data are small and the chart parameters have to be estimated, a popular approach is to adjust the control chart limits from a conditional perspective to avoid frequent false alarms using the exceedance probability criterion. However, this approach ignores the practitioner-to-practitioner (p-to-p) variation caused by the random Phase I reference samples, which results in getting different control limits and chart performance for each practitioner Large p-to-p variation makes practitioners to have a limited confidence on using their own estimated charts. Hence, in this article, we propose to optimize the tr chart via an exact method so that it has a minimum p-to-p variation. Comparisons between the optimal and conventionally adjusted charts are made in both the in- and out-of-control cases. The most important results are that the optimal chart has a far smaller p-to-p variation and its unconditional average run length values are closer to the desired ones compared to the conventional approach regardless of the in- or out-of-control cases. Finally, two real examples are presented to illustrate the implementation of the proposed chart.