Enhanced Designs for the Multi-Region Run Sum x̄ Control Chart Based on the Median Run Length Metric

Extensive research has focused on the multi-region run sum x̄ control chart, a sophisticated procedure for monitoring process mean shifts, particularly utilising the average run length (ARL) metric. Nevertheless, the skewed nature of the run-length distribution can render the ARL metric misleading, and occasionally ineffective, when assessing a control chart's performance. The employment of the ARL metric in designing the run sum x̄ chart can erode practitioners’ confidence. This might be due to its complexities and difficulties in interpreting the complicated statistical concepts involved. To address this issue, the median run length (MRL) metric is proposed in this paper, which has been endorsed by various researchers due to its robustness towards skewness in the run-length distribution. We design two novel optimal run sum x̄ charts utilising MRL and expected MRL (EMRL) metrics, under the assumptions of deterministic and unknown shit-size scenarios, respectively. Specifically, the performances of the 4-region and 7-region run sum x̄ charts in both the zero-state and steady-state scenarios are developed using the Markov chain approach. Our comparative studies reveal that the proposed MRL- and EMRL-optimal run sum x̄ charts surpass the optimal Shewhart x̄ and exponentially weighted moving average (EWMA) x̄ charts in terms of their average detection speeds, especially for moderate to large levels of process mean shifts. Practical examples from different industries are demonstrated using the proposed optimal run sum x̄ chart.