where p is the probability that a sample exceeds the control limits, R is the run length and E[-} denotes the expected value. For an x chart with 3a limits, the probability that a point will be outside the control limits even though the process is in control is p = 0.0027. Consequently, the ARL(0) is ARL = l/p = 1/0.0027 = 370. For other types of charts such as CUSUM, it is difficult or impossible to derive ARL(0) values based on theoretical arguments. Instead, the magnitude of the level change to be detected is selected and Monte Carlo simulations are run to compute the run lengths, their averages and variances.
6.1.2 Cumulative Sum (CUSUM) Charts
The cumulative sum (CUSUM) chart incorporates all the information in a data sequence to highlight changes in the process average level. The values to be plotted on the chart are computed by subtracting the overall mean t±Q from the data and then accumulating the differences. For a sample size n > 1, denote the average of the jth sample x}. The quantity i
j=i is plotted against the sample number i. CUSUM charts are more effective than Shewhart charts in detecting small process shifts, since they combine
information from several samples. CUSUM charts are effective with samples of size 1. The CUSUM values can be computed recursively
If the process is in-control at the target value /iq ■ the CUSUM S't should meander randomly in the vicinity of 0. If the process mean is shifted, an upward or downward trend will develop in the plot. Visual inspection of changes of slope indicates the sample number (and consequently the time) of the process shift. Even when the mean is on target, the CUSUM Si may wander far from the zero line and give the appearance of a signal of change in the mean. Control limits in the form of a V-mask were employed when CUSUM charts were first proposed in order to decide that a statistically significant change in slope has occurred and the trend of the CUSUM plot is different than that of a random walk. CUSUM plots generated by a computer became more popular in recent years and the V-mask has been replaced by upper and lower confidence limits of one-sided CUSUM charts.
One-Sided CUSUM charts are developed by plotting i
j=l where K is the reference value to detect an increase in the mean level. If Si becomes negative for /i, > it is reset to zero. When S, exceeds the decision interval H, a statistically significant increase in the mean level is declared. Values for K and H can be computed from the relations:
Given the a and (3 probabilities, the size of the shift in the mean to be detected (A), and the standard deviation of the average value of the variable x (<ts), the parameters in Equation 6.24 are:
A two-sided CUSUM can be generated by running two one-sided CUSUM charts simultaneously with the upper and lower reference values. The recursive formulae for high and low side shifts that include resetting to zero are
respectively. The starting values are usually set to zero, Sh(0) = Sl(0) = 0. When Sn(i) or Si(i) exceeds the decision interval H, the process is out-of-control. Average Run Length (ARL) based methods are usually utilized to find the chart parameter values H and K. The rule of thumb for ARL (A) for detecting a shift of magnitude A in the mean when A ^ 0 and A > K is
Two-sided CUSUM sometimes is called as the tabular CUSUM. Whereas the monitoring results are usually given as tabulated form, it is useful to present graphical display for tabular CUSUM. These charts are generally called as CUSUM status charts .
Example Develop the CUSUM chart to detect a shift in the mean of magnitude S = A/a$ =2, with a = 0.01 and ¡3 = 0.05. Using Eq. 6.25, d = 2.28. H and K are computed from Eq. 6.24 as H = 2.62 and K = 1.15, and the one-sided CUSUM charts are based on Eq. 6.26. The resulting charts where the first twenty samples are used to develop the charts are shown in Figure 6.6. Since the observation 23 exceeds the decision interval H at which Sh > H = 2.62, we would conclude that the process is out of control at that point.
6.1.3 Moving Average Control Charts for Individual Measurement s
Individual data (sample size n=l) are common in many process industries. Continuous streams of data are more common for continuous processes. MA charts may be used for monitoring successive batches by using end of batch quality measurements. MA charts may also be used for data with small variation, collected during a batch run. Instantaneous variations in a batch run may be small, but the process varies over time. Selecting a group of successive measurements close together in time will include mostly variation due to measurement and sampling error. In such situations, statistical monitoring of the process can be achieved by using moving-average (MA) charts. In MA charts, averages of the consecutive data groups of size a are plotted. The control limit computations are based on averages and standard deviation values computed from moving ranges. Since each MA point has (a — 1) common data points, the successive MAs are highly autocorrelated. This autocorrelation is ignored in the usual construction of these charts. The MA control charts should not be used with strongly autocorrelated data. The MA charts detect small drifts efficiently (better than x chart) and they can be used when the original data do not have Normal distribution.
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