A Study on the Power Functions of the Shewhart $\bar{X}$ Chart via Monte Carlo Simulation

Abstract

The Shewhart $\bar{X}$ control chart is used to monitor shifts in the process mean. However, it is less sensitive to small shifts. The Shewhart $\bar{X}$ chart’s sensitivity can be enhanced by reducing the width of the control limits, increasing the subgroup size and using detection rules to enhance the chart’s sensitivity. However, these actions will influence the power functions of the Shewhart $\bar{X}$ chart. A probability table providing the probabilities of detecting shifts in the mean, calculated using the formulae is recommended. However, the main setback is that the calculations of the probabilities using the formulae are complicated, laborious and time consuming. In this paper, a Monte Carlo simulation using the Statistical Analysis System (SAS) is conducted to compute these probabilities. The probabilities computed via Monte Carlo simulation are closed to that obtained using formulae. Therefore, the Monte Carlo simulation method is recommended as it provides savings, in terms of time and cost. In addition, the Monte Carlo simulation method is also more flexible in calculating the probabilities, for different combinations of the detection rules. The results obtained will enable practitioners to design and implement the Shewhart $\bar{X}$ chart more effectively.