The Difficulty in Designing Shewhart X-bar and X Control Charts with Estimated Parameters

• Autores: Nesma A. Saleh, Mohamed A. Mahmoud, Matthew J. Keefe, William H. Woodall
• Localización: Journal of quality technology: A quarterly journal of methods applications and related topics, ISSN 0022-4065, Nº. 2, 47, 2015, págs. 127-138
• Idioma: inglés
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• Resumen

  • The performance of the Shewhart X-bar control chart with estimated in-control parameters has been discussed a number of times in the literature. Previous studies showed that at least 400/(n - 1) phase I samples, where n being greater than 1 is the sample size, are required so that the chart performs on average as if the in-control process parameter values were known. This recommendation was based on the in-control expected average run length (ARL) performance. The reliance on the expected ARL metric, however, averages across the practitioner-to-practitioner variability. This variability occurs due to the different historical data sets that practitioners use, which results in varying parameter estimates, control limits, and in-control ARL values. This article shows that taking this type of variability into consideration leads to far larger amounts of phase I data than what was previously recommended. This is to ensure low levels of variation in the in-control ARL values among practitioners. The standard deviation of the ARL (SDARL) metric is used to evaluate performance for various amounts of phase I data. The authors show that no realistic phase I sample size is sufficient to have confidence that the attained in-control ARL is close to the desired value. The authors additionally investigate the effect of different process standard deviation estimators on the X-bar chart performance, showing that it is best to use a biased estimator. They also study the design of the X-chart for the case n = 1, drawing similar conclusions regarding the amount of phase I data. An alternative approach to designing control charts is recommended.