5. Examples
5.2 Vertical Board Density Data (VDP)
5.2.2 Monitoring the VDP Data
We use the RMVE chart in Equation (9) to monitor the VDP profiles. The results are as follows. Both the OAAT and the Delete-All schemes remove the 4th, 9th, 17th, 20th, and 24th profiles, see Figure 39(a)(b). Figure 39(c) shows that the remaining 18 profiles are now in control.
For saving computing time, we suggest using the Delete-All scheme to the VDP data.
(a) (b)
(c)
Figure 39: The RMVE chart when monitor VDP profile. (a) The 24 boards at the first iteration, (b) the 22 boards at the 2nd iteration, and (c) the 18 boards at the final iteration.
We compute the T statistic for the VDP data, and find that02 T = 4.622674. With02 T this 02
large, we could just use the Delete-All scheme to monitor profiles. Figure 40 shows all 24 filled profiles by the model (16). We notice that the highest board (the 3rd board) and the lowest board (the 6th board) are not removed by both schemes. The same argument about the 44th profile of the response-dose data may be applied here.
Figure 40: The 3rd and 6th profiles are not removed.
5.3 The T Statistic for Examples 02
We compute the T statistic for the bioassay data by Equation (9). The02 T02 statistic is 0.992673 for does-response profile monitoring. Although 0.992673 is almost equal to the cutoff point 1.0, for being conservative, we suggest using the OAAT scheme. The T02 statistic is 0.2747621 for the variance of profiles monitoring, so we should use the OAAT scheme.
For the VDP data, the T02 statistic is 4.622674, which is so large that we could just use the Delete-All scheme. Hence, before monitoring real data we may use the statistic in Equation (8) and (9) to decide which scheme to use.
In summary, the results of these two examples agree with the decision criterion T . The 02 Bioassay data in Subsection 5.1 demonstrate a case that the OAAT scheme performs better than the Delete-All scheme while the VDP example in Subsection 5.2 demonstrates a case that we could use the Delete-All scheme to save some computing.
6. Conclusion
We use the RMVE control chart to monitor profiles, and compare the OAAT scheme with the Delete-All scheme. The study indicates that the OAAT scheme performs better than the Delete-All scheme. The OAAT scheme is to run through the whole iterative procedure by removing out-of-control points one at a time at each iteration and then perform the investigation for all alarms after all the remaining samples are all in control. This practice may save tremendous amount of time and money in bringing process to in-control state.
In general, we suggest using the OAAT scheme. This method can lower the false-alarm rate and retain almost the same detecting power when compared with the Delete-All scheme.
However, the OAAT scheme needs more iterations of control charting than the Delete-All scheme.
Thus we suggest computing the T0 or T02 statistic before applying the monitoring scheme to data.
To decide which scheme to use, if the statistic is greater than or equal 1.0 or so, then we could use the Delete-All scheme. The two examples, the bioassay data and the VDP data, successfully demonstrate the usefulness of the judging criterion of T . 02
We use the Bioassay Data to demonstrate that the OAAT scheme performs better than the Delete-All scheme and use the VDP Data to demonstrate how to use a statistic to decide which scheme to use in order to save some time.
We could give different weights to the parameters estimated from the profile data, because the extent of importance on each parameter of profiles may be different. Developing an adequate monitoring scheme for such processes is a potential future research topic.
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Appendix A
The Moment Generating Function (MGF) is
2 2 2
2 2 2 2 2
Putting the MGF to work again:
2 2 2 2 2 1 2 1
2 2
We first consider the case of the standard multivariate normal distribution. Suppose we want to truncate the distribution such that the p-dimension cube A≡ −[ a a, ]p, covers 1− of the α
2 ( ) 1Φ a − = p1−α.
.. We first transform Y into a standard multivariate normal. Variate Y* by
Appendix B
49 22.09057 32.20893
94 21.01456 29.34401
95 21.0017 29.86019
96 21.50454 30.15476
97 21.91065 30.01665
98 20.51235 27.45656
99 20.15563 29.87197
100 20.0556 28.50683