New measures of statistical performance of process control charts

Traditional measures of statistical performance of SPC control charts are widely used to evaluate the performance of a control chart. Pretty often, even if the control chart is designed from an economic point of view, statistical properties are set as necessary conditions (constraints), to control performance. The usual statistical measure used to evaluate the performance of a chart is either the false alarm rate, α, or the equivalent average run length, ARL0. Usually, the condition is that α should not exceed 0.27% (probability of false alarm for a Shewhart X‾$\bar{X}$ chart with k = 3), or, that the ARL0 (=1/α), should be no less than 370. This paper revisits the issue of statistical properties of a chart, shows that no α constraint suffices to rationalize the performance of the control procedure, in terms of limiting the false alarms to an acceptable level, and proposes new measures of statistical performance that can be used for most commonly used control charts. In addition, a by-product of the paper is a generic model development that can be used for the computation of statistical (and economic) measures of performance for control charts, even for the more advanced, adaptive control charts.