Supplemental Material for Chapter 5
S5.1. A Simple Alternative to Runs Rules on the Chart
It is well-known that while Shewhart control charts detect large shifts quickly, they are relative insensitive to small or moderately-sized process shifts. Various sensitizing rules (sometimes called runs rules) have been proposed to enhance the effectiveness of the chart to detect small shifts. Of these rules, the Western Electric rules are among the most popular. The western Electric rules are of the r out of m form; that is, if r out of the last m consecutive points exceed some limit, an out of control signal is generated.
In a very fundamental paper, Champ and Woodall (1987) point out that the use of these sensitizing rules does indeed increase chart sensitivity, but at the expense of (sometimes greatly) increasing the rate of false alarms, hence decreasing the in-control ARL. Generally, I do not think that the sensitizing rules should be used routinely on a control chart, particularly once the process has been brought into a state of control. They do have some application in the establishment of control limits (Phase 1 of control chart usage) and in trying to bring an unruly process into control, but even then they need to be used carefully to avoid false alarms.
Obviously, Cusum and EWMA control charts provide an effective alternative to Shewhart control charts for the problem of small shifts. However, Klein (2000) has proposed another solution. His solution is simple but elegant: use an r out of m consecutive point rule, but apply the rule to a single control limit rather than to a set of interior “warning” type limits. He analyzes the following two rules:
- If two consecutive points exceed a control limit, the process is out of control. The width of the control limits should be 1.78.
- If two out of three consecutive points exceed a control limit, the process is out of control. The width of the control limits should be 1.93.
These rules would be applied to one side of the chart at a time, just as we do with the Western Electric rules.
Klein (2000) presents the ARL performance of these rules for the chart, using actual control limit widths of , as these choices make the in-control ARL exactly equal to 370, the values associated with the usual three-sigma limits on the Shewhart chart. The table shown below is adapted from his results. Notice that Professor Klein’s procedure greatly improves the ability of the Shewhart chart to detect small shifts. The improvement is not as much as can be obtained with an EWMA or a Cusum, but it is substantial, and considering the simplicity of Klein’s procedure, it should be more widely used in practice.
Shift in process mean, in standard deviation units / ARL for the Shewhart chart with three-sigma control limits / ARL for the Shewhart chart with control limits / ARL for the Shewhart chart with control limits0 / 370 / 350 / 370
0.2 / 308 / 277 / 271
0.4 / 200 / 150 / 142
0.6 / 120 / 79 / 73
0.8 / 72 / 44 / 40
1 / 44 / 26 / 23
2 / 6.3 / 4.6 / 4.3
3 / 2 / 2.4 / 2.4
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