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Specifying the control limits is one of the critical decisions that must be made in designing a control chart. By moving the control limits further from the center line, we decrease the risk of a type I error that is, the risk of a point falling beyond the control limits, indicating an out-of-control condition when no assignable cause is present. However, widening the control limits will also increase the risk of a type II error that is, the risk of a point falling between the control limits when the process is really out of control. If we move the control limits closer to the center line, the opposite effect is obtained: The risk of type I error is increased, while the risk of type II error is decreased. The control limits on a Shewhart control chart are customarily located a distance of plus or minus three standard deviations of the variable plotted on the chart from the center line. That is, the constant k in equation 16-1 should be set equal to 3. These limits are called 3-sigma control limits. A way to evaluate decisions regarding sample size and sampling frequency is through the average run length (ARL) of the control chart. Essentially, the ARL is the average number of points that must be plotted before a point indicates an out-of-control condition. For any Shewhart control chart, the ARL can be calculated from the mean of a geometric random variable (Montgomery 2001). Suppose that p is the probability that any point exceeds the control limits. Then
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Thus, for an X chart with 3-sigma limits, p 0.0027 is the probability that a single point falls outside the limits when the process is in control, so ARL 1 p 1 0.0027 370
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is the average run length of the X chart when the process is in control. That is, even if the process remains in control, an out-of-control signal will be generated every 370 points, on the average. Consider the piston ring process discussed in Section 16-4.2, and suppose we are sampling every hour. Thus, we will have a false alarm about every 370 hours on the average. Suppose we are using a sample size of n 5 and that when the process goes out of control the mean shifts to 74.0135 millimeters. Then, the probability that X falls between the control limits of Fig. 16-3 is equal to P 373.9865 X 74.0135 when 73.9865 74.0135 Pc 0.0045 0.5 P 3 6 Z 04 74.01354 74.0135 74.0135 Z d 0.0045
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Therefore, p in Equation 16-28 is 0.50, and the out-of-control ARL is ARL 1 p 1 0.5 2
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Table 16-6 Average Run Length (ARL) for an X Chart with 3-Sigma Control Limits Magnitude of Process Shift 0 0.5 1.0 1.5 2.0 3.0 ARL n 1 370.4 155.2 43.9 15.0 6.3 2.0 ARL n 4 370.4 43.9 6.3 2.0 1.2 1.0
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That is, the control chart will require two samples to detect the process shift, on the average, so two hours will elapse between the shift and its detection (again on the average). Suppose this approach is unacceptable, because production of piston rings with a mean diameter of 74.0135 millimeters results in excessive scrap costs and delays nal engine assembly. How can we reduce the time needed to detect the out-of-control condition One method is to sample more frequently. For example, if we sample every half hour, only one hour will elapse (on the average) between the shift and its detection. The second possibility is to increase the sample size. For example, if we use n 10, the control limits in Fig. 16-3 narrow to 73.9905 and 74.0095. The probability of X falling between the control limits when the process mean is 74.0135 millimeters is approximately 0.1, so p 0.9, and the out-of-control ARL is ARL 1 p 1 0.9 1.11
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Thus, the larger sample size would allow the shift to be detected about twice as quickly as the old one. If it became important to detect the shift in the rst hour after it occurred, two control chart designs would work:
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