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Messina, Montgomery, Keats & Runger

schemes are highly ineffective in signaling that a sustained shift has occurred in the process. Although the simulation results for these two schemes are shown in this study, the only scheme that will be employed for control actions is the combined EWMA prediction error and tracking signal scheme. Therefore, our comments will address only this scheme. In all cases, the combined EWMA prediction error and tracking signal scheme outperforms the EPC scheme alone. However, when a smali sustained shift occurs in the process, the scheme has difficulty finding it. The simulation results show a maximum failure to detect the smali shift of 0.065. The reason this occurs is that if the predicted error EWMA doesn't find the shift immediately, it will elevate to the new shift level and never detect it.

Table 2 presents the average run lengths (ARL's) observed in the simulation in Table 1. The values shown in the table are based only on the number of simulations that signaled a sustained shift . The smallest shifts are the most difficult to detect, as one would expect. One reason that the average run lengths are so long for these smali shifts is that for smali shifts active control often compensates for the shift magnitude. However, in all cases, the addition of the SPC rule reduces variability.

Example of the Effects of a Trend on Control Actions The effects of a trend on the control actions is now explored. Figurę 7 is a realization of a typical pattem that will be encountered for control actions when a trend of magnitude 1 unit per period occurs in the process at observation 251. The performance measure associated with this EPC scheme is 89.973. The magnitude of this performance measure indicates that the MMSE controller in eąuation (3) is ineffective for adjusting the process when an assignable cause in the form of a trend occurs in the process.

Figurę 8 is a plot of the first 256 residuals obtained from fitting an EWMA with 1 = 0.9 to the control actions. The reason that only 256 are shown instead of 500 is that the combined prediction error EWMA and tracking signal scheme signaled that the shift in the process was found at this observation. Figurę 9 is the prediction eiror EWMA with I = 0.1 and Figurę 10 is the smoothed error tracking signal with K3 = 0.5 and a two-in-a-row nile. This combined scheme ąuickly detects the assignable cause and the associated performance measure is 2.719. This performance measure is a weighted measure as described by eąuation (7). Figures 11 and 12 are the CUSUM control charts using values of K = Ict and K = 2a respectively. The performance measures for these two schemes are 89.973. This is the exact performance measure obtained from using only an EPC scheme. This tends to indicate that when assignable causes in the form of trends exists in the process,