Supplemental Material for Chapter 10
S10.1. Difference Control Charts
The difference control chart is briefly mentioned in Chapter 10, and a reference is given to a paper by Grubbs (1946). There are actually two types of difference control charts in the literature. Grubbs compared samples from a current production process to a reference sample. His application was in the context of testing ordinance. The plotted quantity was the difference in the current sample average and the reference sample average. This quantity would be plotted on a control chart with center line at zero and control limits at , where are the average ranges for the reference samples (1) and the current production samples (2) used to establish the control limits.
The second type of difference control chart was suggested by Ott (1947), who considered the situation where differences are observed between paired measurements within each subgroup (much as in a paired t-test), and the average difference for each subgroup is plotted on the chart. The center line for this chart is zero, and the control limits are at is the average of the ranges of the differences. This chart would be useful in instrument calibration, where one measurement on each unit is from a standard instrument (say in a laboratory) and the other is from an instrument used in different conditions (such as in production).
S10.1. Control Charts for Contrasts
There are many manufacturing processes where process monitoring is important but traditional statistical control charts cannot be effectively used because of rational subgrouping considerations. Examples occur frequently in the chemical and processing industries, stamping, casting and molding operations, and electronics and semiconductor manufacturing.
As an illustration, consider a furnace used to create an oxide layer on silicon wafers. In each run of the furnace a set of m wafers will be processed, and at the completion of the run a single measurement of oxide thickness will be taken at each of n sites or locations on each wafer. These mn thickness measurements will be evaluated to ensure the stability of the process, check for the possible presence of assignable causes, and to determine any necessary modifications to the furnace operating conditions (or the recipe) before any subsequent runs are initiated. Figure S10,1, adapted from Runger and Fowler (1999) and Czitrom and Reece (1997), shows a typical oxidation furnace with m = 4 wafers and n = 9 sites on each wafer. In Chapter 6 of the textbook there is an example illustrating an aerospace casting where vane height and inner diameter are the characteristics of interest. Each casting has five vanes that are measured to monitor the height characteristic and the diameter of a casting is measured at 24 locations using a coordinate measuring machine.
In these applications it would be inappropriate to monitor the process with traditional charts. For example in the oxidation furnace, assuming a rational subgroup of either n = 9 or n = 36 is not correct because all sites experience the processing activities during each furnace run simultaneously. That is, there is much less variability between the observations at the 9 sites than would be anticipated in observations collected from a process where all measurements reflect the processing activity(s) each unit experiences independently. What usually occurs when this misapplication of the standard charts is implemented is that the control limits on the chart will be too narrow. Then if the process experiences moderate run-to-run variability, there will be many out-of-control points on the chart that engineers and process operating personnel cannot associate with specific upsets or assignable causes.
Furnace 6 2 9
Location1 3 1 5
7 4 8
Furnace · · ·
Location2
Furnace
Location3
Furnace
Location4
Figure S10.1. Diagram of a Furnace where four wafers are simultaneously processed and nine quality measurements are performed on each wafer.
The most widely used approach to monitoring these processes is to first consider the average of all mn observations from a run as a single observation and to use a Shewhart control chart for individuals to monitor the overall process mean. The control limits for this chart are usually found by applying a moving range to the sequence of averages. Thus, the control limits for the individuals chart reflect run-to-run variability, not variability within a run. The variability within a run is monitored by applying a control chart for s (the standard deviation) or to all mn observations from each run. It is interesting to note that this approach is so widely used that at least one popular statistical software package (Minitab) includes it as a standard control charting option (called the “between – within” procedure in Minitab). This procedure was illustrated in Example 5-11.
Runger and Fowler (1999) show how the structure of the data obtained on these processes can be represented by an analysis of variance model, and how control charts based on contrasts can be designed to detect specific assignable causes of potential interest. Below we briefly review their results and relate them to some other methods. Then we analyze the average run performance of the contrast charts and show that the use of specifically designed contrast charts can greatly enhance the ability of the monitoring scheme to detect assignable causes. We confine our analysis to Shewhart charts, but both Cusum and EWMA control charts would be very effective alternatives, because they are more effective in detecting small process shifts, which are likely to be of interest in many of these applications
Contrast Control Charts
We consider the oxidation process in Figure S10.1, but allow m wafers in each run with n measurements or sites per wafer. The appropriate model for oxide thickness is
(S10.1)
where yij is the oxide thickness measurement from run i and site j, ri is the run effect, sj is the site effect, and is a random error component. We assume that the site effects are fixed effects, since the measurements are generally taken at the same locations on all wafers. The run effect is a random factor and we assume it is distributed as . We assume that the error term is distributed as . Notice that equation (S10.1) is essentially an analysis of variance model.
Let yt be a vector of all measurements from the process at the end of run t. It is customary in most applications to update the control charts at the completion of every run. A contrast is a linear combination of the elements of the observation vector yt , say
where the elements of the vector c sum to zero and, for convenience, we assume that the
contrast vector has unit length. That is,
Any contrast vector is orthogonal to the vector that generates the mean, since the mean can be written as
Thus, a contrast generates information that is different from the information produced by the overall mean from the current run. Based on the particular problem, the control chart analyst can choose the elements of the contrast vector c to provide information of interest to that specific process.
For example, suppose that we were interested in detecting process shifts that could cause a difference in mean thickness between the top and bottom of the furnace. The engineering cause of such a difference could be a temperature gradient along the furnace from top to bottom. To detect this disturbance, we would want the contrast to compare the average oxide thickness of the top wafer in the furnace to the average thickness of the bottom wafer. Thus, if m = 4, the vector c has mn = 36 components, the first 9 of which are +1, the last 9 of which are –1, and the middle 18 elements are zero. To normalize the contrast to unit length we would actually use
One could also divide the elements of c by nine to compute the averages of the top and bottom wafers, but this is not really necessary.
In practice, a set of k contrasts, say
can be used to define control charts to monitor a process to detect k assignable causes of interest. These simultaneous control charts have overall false alarm rate a, where
(S10.2)
and ai is the false alarm rate for the ith contrast. If the contrasts are orthogonal, then Equation (S9-2) holds exactly, while if the contrasts are not orthogonal then the Bonferroni inequality applies and the a in Equation (S10.2) is a lower bound on the false alarm rate.
Related Procedures
Several authors have suggested related approaches for process monitoring when non-standard conditionss relative to rational subgrouping apply. Yashchin (1994), Czitrom and Reese (1997), and Hurwicz and Spagon (1997) all present control charts or other similar techniques based on variance components. The major difference in this approach in comparison to these authors is the use of an analysis-of-variance type partitioning based on contrasts instead of variance components as the basis of the monitoring scheme. Roes and Does (1995) do discuss the use of contrasts, and Hurwicz and Spagon discuss contrasts to estimate the variance contributed by sites within a wafer. However, the Runger and Fowler model is the most widely applicable of all the techniques we have encountered.
Even though the methodology used to monitor specific differences in processing conditions has been studied by all these authors, the statistical performance of these charts has not been demonstrated. We now present some performance results for Shewhart control charts.
Average Run Length Performance of Shewhart Charts
In this section we assume that the process shown in Figure S10.1 is of interest. The following scenarios are considered:
· A change in the mean of the top versus the bottom wafer.
· Changes on the left versus the right side of all wafers.
· Significant changes between the outside and the inside of each wafer.
· Four wafers are selected from the tube.
The contrasts for these charts are:
A comparison of the ARL values obtained using these contrasts and the traditional approach – an individuals control chart for the mean of all 36 observations- is presented in Tables S10.1, S10.2, and S10.3. From inspection of these tables, we see that the charts for the orthogonal contrasts, originally with the same in-control ARL as the traditional chart, are more sensitive to changes at specific locations, thus improving the chances of early detection of an assignable cause. Notice that the improvement is dramatic for small shifts, say on the order of 1.5 standard deviations or less.
A similar analysis was performed for a modified version of the process shown in figure S10.1. In this example, there are seven measurements per wafer for a total of 28 measurements in a run. There are still three measurements at the center of the wafer, but now there are only measurements at the perimeter; one in each “corner”. The same types of contrasts used in the previous example (top versus bottom, left versus right and edge versus center) were analyzed and the ARL results are presented in Tables S10.4, S10.6, and S10.6.
Table S10.1. Average Run Length Performance of Traditional and Orthogonal Contrast Charts for a shift in the Edges of all Wafers. In this chart m = 4 and n = 9.
Size of Shift
In Multiples of s
/Edge versus Center Contrast
/Traditional Chart
0.51
1.5
2
2.5
3 / 11.7
1.9
1.1
1
1
1 / 13.6
2.2
1.1
1
1
1
Table S10.2. Average Run Length Performance of Traditional and Orthogonal Contrast Charts for a shift in the Top Wafer. In this chart m = 4 and n = 9.
Size of Shift
In Multiples of s /Bottom Top versus Contrast
/Traditional Chart
0.51
1.5
2
2.5
3 / 23.4
3.9
1.5
1.1
1
1 / 47
10
3.4
1.7
1.2
1
Table S10.3. Average Run Length Performance of Traditional and Orthogonal Contrast Charts for a shift in the Left side of all Wafers. In this chart m = 4 and n = 9.
Size of Shift
In Multiples of s /Left versus Right Contrast
/Traditional Chart
0.51
1.5
2
2.5
3 / 26.7
4.6
1.7
1.1
1
1 / 57.2
13.6
4.6
2.2
1.4
1.1
Table S10.4. Average Run Length Comparison between Traditional and Orthogonal Contrast Charts for a shift in the Edge of all Wafers. In this chart m = 4 and n = 7.
Size of Shift
In Multiples of s /Edge versus Center Contrast
/Traditional Chart
0.51
1.5
2
2.5
3 / 26.7
4.6
1.7
1.1
1
1 / 46.4
9.8
3.3
1.7
1.2
1
Table S10.5. Average Run Length Comparison between Traditional and Orthogonal Contrast Charts for a change in the Top Wafer. In this chart m = 4 and n = 7.
Size of Shift
In Multiples of s /Top versus Bottom Contrast
/Traditional Chart
0.51
1.5
2
2.5
3 / 30.8
5.5
2
1.2
1
1 / 57.9
13.8
4.7
2.2
1.4
1.1
Table S10.6. Average Run Length Performance of Traditional and Orthogonal Contrast Charts for a shift in the left side of all Wafers. In this chart m = 4 and n = 7.
Size of Shift
In Multiples of s /Left versus Right Contrast
/Traditional Chart
0.51
1.5
2
2.5
3 / 26.7
4.6
1.7
1.1
1
1 / 46.4
9.8
3.3
1.7
1.2
1
Decreasing the number of measurements per wafer has increased the relative importance of the changes in the mean of a subset of the observations and the traditional control charts signal the shift faster than in the previous example. Still, note that the control charts based on orthogonal contrasts represent a considerable improvement over the traditional approach.
S10.3. Run Sum and Zone Control Charts
The run sum control chart was introduced by Roberts (1966), and has been studied further by Reynolds (1971) and Champ and Rigdon (1997). For a run chart for the sample mean, the procedure divides the possible values of into regions on either side of the center line of the control chart. If is the center line and is the process standard deviation, then the regions above the center line, say, are defined as