Optimal one-sided shewhart control charts with variable sampling intervals

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Abstract
When a control chart is used to detect changes in a process the usual practice is to take samples from the process using a fixed sampling interval between samples. This paper considers the properties of Shewhart control charts when the sampling interval used after each sample is not tixed but instead depends on what is observed in the sample. The basic rationale is that the sampling interval should be short if there is some indication of a change in the process and long if there is no indication of a change. If the indication of a change is strong enough then the chart signals in the same way as the fixed sampling interval Shewhart chart. The result is that samples will be taken more frequently when there is a change in the process, and this process change can be detected much more quickly than when fixed sampling intervals are used. Expressions for properties such as the average time to signal and the average number of samples to signal are developed. It is shown that if the sampling interval must be chosen from a range of possible sampling intervals then the optimal control chart for detecting a specified process change uses only the shortest possible interval and the longst possible interval. The chart is optimal in the sense that it mini¬mizes the average time to signal and the average number of samples to signal when the process has changed, subject to constraints on the false alarm rate and the sampling rate when the process has not changed.

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