Optimal exponentially weighted moving average (X)over-bar charts with estimated parameters based on median run length and expected median run length

摘要

This paper examines the exponentially weighted moving average (EWMA) (X) over bar chart with estimated process parameters. As the run length distribution is skewed when the process is in-control or slightly out-of-control, the average run length (ARL) provides less meaningful interpretation of a chart's performance. Therefore, in this paper, the median run length (MRL) and expected MRL (EMRL) are used as alternative performance criteria. Additionally, the methodology for computing the EMRL of the EWMA (X) over bar chart with known process parameters is presented. Since the performance of the EWMA (X) over bar chart is affected by estimation error, a study on the minimum number of Phase-I samples required so that the chart with estimated parameters has a desired performance is conducted. As this study reveals that a large number of Phase-I samples are needed, optimal design procedures for minimising the out-of-control MRL and EMRL of the EWMA (X) over bar chart with estimated process parameters are suggested. By using these proposed procedures, the EWMA (X) over bar chart with estimated parameters will have a closer performance to its known parameters counterpart, even with a reasonable number of Phase-I samples. The construction of the MRL based EWMA (X) over bar chart with estimated parameters is illustrated using real data and compared with the corresponding chart based on ARL.