Phase I analysis for manufacturing process control A combination of T2 and CUSUM method

Authors Sandeep NemmaniVanshaj Handoo

Analysis approach

Preliminary Analysis

Identify the type of data discrete/continuous

Trend analysis on the data

Identify the type of manufacturing process

Principal Component

Analysis

Use covariance matrix to convert original data into a new system of linearly independent principal components

Identify principal components contributing to about 80% of total variance using description length, Scree plot and Pareto plot

Control Charting

Perform iterations of Hotelling T2 to remove spike type of change and m-CUSUM to remove a sustained mean shift of statistical distance 3

Find the in-control parameters - 0 and 0

ChartPerformance

Use Monte Carlo simulation method to determine the Run Length distribution for both the T2 and the CUSUM charts

Preliminary Analysis

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Mean of 209 Variables A plot of the average value of the 209 variablesis shown alongside

A plot of 5 observations of the data is shown below The data intuitively appears to be a profile from a

continuous, cyclic manufacturing process Safe to assume Normal distribution of individual

variables

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1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951 1001

Cyclic Process

Cycle 1 Cycle 2

Principal Component Analysis

After finding the principal components, data dimension was reduced from 209 to 4 using a combination of Minimum Description Length, Scree plot and Pareto plot

MDL gives 36 principal components to be considered. This is further reduced to 9 by the Scree plot since the variance of components after the 9th one is insignificant

Principal Component Analysis

If we check the Pareto plot for the first 9 PCs (result from Scree plot), we see that 80% variance is contributed by only the first 4 PCs

Since 80% of variance contribution is a reasonably good level, we can go ahead and create control charts using the first 4 PCs

Control Charting: Hotelling T2 Chart The 1st iteration of the T2 chart is show below 11 points out-of-control

On the 4th iteration, we were able to get all data in-control. However, we still need to detect and eliminate any points that have a small magnitude mean shift.

Since n=1 here, we are using Case III part (a) formula of 2

approximation of the Beta function with = 0.0027

Control Charting: m-CUSUM Chart After eliminating all spikes, we use the CUSUM method to eliminate sustained mean shift

After 6 iterations of the CUSUM chart, we were ale to eliminate all points having a mean shift of statistical distance 3

We tried to detect a mean shift of statistical distance 3

In the adjacent figure, we see a large number of points going out-of-control indicating that a sustained mean shift did exist and remained undetected by the T2 chart

UCL is determined by interpolation of data in literature

We cannot, however, stop at this point an state the remaining data points are all in control

To make such a statement we need both the, T2 and the CUSUM charts to be in control at the same time

Hence, the need to check T2

again

Control Charting: Round 2 We checked T2 chart and found more points out-of control. 3 iteration later all the spikes

were eliminated

Since all spikes are now eliminated, we need to double check if the CUSUM chart is in-control

The adjacent figure shows that the CUSUM chart is indeed in control

The data is now said to be in-control for : = 0.0027 - if using a T2 chart Sustained mean shift of

statistical distance 3 if using CUSUM

Chart performance for future data It is imperative to know the performance of a control chart in terms of run length distribution We used Monte Carlo simulation method to determine ARL0 and ARL1 for T

2 and CUSUM. The results of these simulation are tabulated below

Chart Mean Shift ARL

T2

0.00 371

0.92 115.4

1.66 27.44

2.99 2.67

4.35 0.36

mCUSUM

0 200*

568 5.72

18 8.45

5 9.8

2 11.2

The average run lengths for T2 is significantly better than CUSUM

We were not able to conclusive determine the reason for this. One possible explanation is that for mCUSUM the offset range is given by k*ni which varies (increases) as niincreases on accumulation. This prevents a point getting accumulated and mCUSUMstatistic going out-of-control thus increasing the ARL.

*obtained by interpolation of data in class literature

Key learning

Owing to large dimensionality of data, it is difficult to select a particular approach for setting up a process control detection method. Using multiple univariate charts for each variable is out of question. The question then is to either use a multivariate chart like Hotelling T2 or CUSUM chart or individual Shewhart charts for un-correlated principal components. In our approach we have not selected the individual charts for principal components because a process may be out of control even when the individual charts are in control. A T2 or CUSUM chart is thus more definitive in terms of its response.

Even after determining the charts that we use it is important to determine the order of use since that can reduce the number iterations to get the in-control data. Doing CUSUM before T2 would have saved some iterations.

The Average run length distributions for T2 are better than m-CUSUM. We were not able to conclusive determine the reason for this. One possible explanation is that for m-CUSUM the offset range is given by k*ni which varies (increases) as ni increases on accumulation. This prevents a point getting accumulated and m-CUSUM statistic going out-of-control thus increasing the ARL.