6: II 7: III
6.1 Time Weighted Control Charts When the In-control µ and σ are known
Time Weighted Control Charts in Phase II
Shewhart control charts like the X −R charts discussed in Chapter 4 are very useful in Phase I. They can quickly detect a large shift in the mean or variance, and the patterns on the chart defined by the Western Electric rules mentioned inSection 4.2.3, are helpful in hypothesizing the cause of out-of- control conditions. If it can be demonstrated that a process is in "in-control"
with a capability index of 1.5 or higher in Phase I, then there will be no more than 1,350 ppm (or 0.135%) nonconforming output in future production (as long as the process mean does not shift by more that 1.5 standard deviations).
Therefore, to maintain a high quality level it is important that small shifts in the mean on the order of one standard deviation are quickly detected and corrected in during Phase II monitoring.
However, Shewhart charts are not sensitive to small shifts in the mean when used for Phase II process monitoring. When the only out-of-control sig- nal used on a Shewhart chart is a point beyond the control limits, the OC may be too high and the ARL too long to quickly detect a small shift in the process mean. If additional sensitizing rules (like the Western Electric rules) are used in Phase II to decrease the out-of-control ARL, they also dramatically de- crease the in-control ARL. This could cause unnecessary process adjustments or searches for causes. The solution is to use time weighted control charts.
These charts plot the cumulative or weighted sums of all past observations, and are more likely to detect a small shift in the process mean.
6.1 Time Weighted Control Charts When the In-control
The vertical dashed line in the figure shows the point where the mean increased from 50 to 56.5. However, there are no points outside the control limits. It can be observed that all the points lie above the centerline on the right side of the chart.
R># random data from normal N(50,5) followed by N(56.6, 5) R>set.seed(109)
R>x1<-rnorm(7,50,5) R>set.seed(115) R>x2<-rnorm(8,56.6,5)
R># individuals chart assuming mu=50, sigma=5 R>library(qcc)
R>qcc(x1,type="xbar.one",center=50,std.dev=5, newdata=x2)
This might suggest that the mean has increased, but, even using the West- ern Electric sensitizing rules, no out-of-control signal should be assumed until the 7th consecutive point above the center line occurs at the 14th point on the chart.
FIGURE 6.1: Individuals Chart of Random Data
To enhance the sensitivity of the control chart, without using the Western Electric rules that decrease the ARL0 and increase the chance of a false pos- itive, consider the cumulative sums of deviations from the centerline defined in Equation 6.1
Ci=
i
X
k=1
(Xk−µ), (6.1)
and shown inTable 6.1.
TABLE 6.1: Cumulative Sums of Deviations from the Mean observation value Deviation fromµ=50 Ci
1 40.208 −9.792 −9.792 2 56.211 6.211 −3.581 3 51.236 1.236 −2.345 4 60.686 10.686 8.341 5 45.230 −4.770 3.571 6 49.849 −0.151 3.420
7 52.491 2.491 5.912
8 59.762 9.762 15.674 9 59.462 9.462 25.135 10 59.302 9.302 34.437 11 55.679 5.679 40.117 12 57.155 7.155 47.272 13 60.219 10.219 57.491 14 56.770 6.771 64.261 15 55.949 5.948 70.210
Figure 6.2 is a plot of the cumulative sums Ci by observation number i. In this figure, it can be seen that the cumulative sums of deviations from µ = 50 remain close to zero for subgroups 1–7. These were generated from the normal distribution with mean 50. However, beginning at subgroup 8, where the process mean increased, the cumulative sums of deviations begin to increase noticeably.
FIGURE 6.2: Plot of Cumulative Sums of Deviations fromTable 6.1
This illustrates a time weighted weighted chart. Although the figure ap- pears to show an out-of-control signal, there are no formal control limits on the graph. Using the tabularCusum control chartis one way to obtain them.
6.1.1 Cusum Charts
Standardized tabular Cusum control charts work with individual measure- ments rather than subgroups. Two sets of cumulative sums of the standardized individual deviations from the mean are plotted on the tabular Cusum control chart. The first are the sums of the positive deviations,Ci+ from the process mean, and the second set are the sums of the negative deviations Ci−, from the process mean. The definition of these two cumulative sums are shown in Equation 6.2, and they are illustrated using the random data fromFigure 6.1 andTable 6.2.
Ci+ =max[0, yi−k+Ci−+1] (6.2) Ci− =max[0,−k−yi+Ci−−1],
where yi = (xi−µ0)/σ0, and µ0 = 50 andσ0 = 5 are the known in-control process mean and standard deviation. The constant kis generally chosen to be 1/2 the size of the mean shift in standard deviations that you would like to detect quickly. In order to detect a one standard deviation shift in the mean quickly,k= 1/2, and decision limits are placed on the chart at±h. An out-of- control signal occurs if the upper CusumCi+exceeds +hor the lower Cusum Ci+falls below−h.
Using thecusumfunction in theqccpackage, a Cusum chart of the random data is illustrated in the R code below and shown inFigure 6.3. The argument decision.interval=5 specifies h to be 5 and makes the ARL0 = 465; the argumentse.shift=1sets the shift in the process mean to be detected as 1, measured in standard deviations, (i.e. k = 1/2); and the arguments center and std.dev specify the known process mean and standard deviation. The function plotsCi+ on the upper half of the graph and−Ci− on the lower side.
R>#standardized Cusum chart assuming mu=50, sigma=5 R>library(qcc)
R>y1<-(x1-50)/5 R>y2<-(x2-50)/5
R>cusum(y1,center=0,std.dev=1,se.shift=1, decision.interval=5,newdata=y2)
The Cusum chart shown in Figure 6.3 (and the values of Ci+ and −Ci− inTable 6.2) shows an out-of-control signal on the 12th individual value and indicates the process mean has increased.
If the Phase I OCAP indicates that the out-of-control signal can be cor- rected by a change in the level of a manipulable variable, then automated manufacturing processes often use the tabular Cusums in Equation 6.2 in a feedback loop. In this loop, the manipulable variable is automatically adjusted whenever an out-of-control signal is detected.
FIGURE 6.3: Standardized Cusum Chart of the Random Data in Table 6.1 withk= 1/2 andh= 5
TABLE 6.2: Standardized Tabular Cusums
Individual value xi yi Ci+ Ci− 1 40.208 −1.958 0.000 −1.458 2 56.211 1.242 0.742 0.000 3 51.236 0.247 0.489 0.000 4 60.686 2.137 2.126 0.000 5 45.230 −0.954 0.673 −0.454 6 49.849 −0.030 0.142 0.000 7 52.491 0.498 0.141 0.000 8 59.762 1.952 1.593 0.000 9 59.462 1.892 2.985 0.000 10 59.302 1.860 4.346 0.000 11 55.679 1.136 4.982 0.000 12 57.155 1.431 5.913 0.000 13 60.219 2.043 7.457 0.000 14 56.770 1.354 8.311 0.000 15 55.949 1.190 9.000 0.000
When Ci+ exceeds h, to give an out-of-control signal, an estimate of the current value of the process mean is given by Equation 6.3 .
ˆ
µc=µ+σk+σCi+
N+ , (6.3)
where ˆµcis the estimate of the current mean,N+is the consecutive number of nonzero values ofCi+ up to and including the point where it exceededh, and
µ, and σare the known in-control values of the process mean and standard deviation.
For example, the Cusum chart in Figure 6.3 first exceeds h = 5 when Ci+= 5.913, and at that point there areN+ = 11 consecutive positive values forCi+. So the estimate of the process mean at that point is:
ˆ
µc = 50 + 5(.5) +(5)(5.913)
11 = 55.19.
Therefore, in an automated process, the manipulable variable would be changed to reduce the process mean by 55.19−50 = 5.19 to get it back to the known in-control level.
When−Ci− is less than −h, to give an out-of-control signal, an estimate of the current mean of the process is given by:
ˆ
µc=µ−σk−σCi−
N− , (6.4)
and the manipulable would be changed to increase the process mean back to the known in-control level.
6.1.1.1 Headstart Feature
If an out-of-control signal is detected in Phase II monitoring with a process that is not automated, or does not contain a manipulable variable with a known relationship to the process mean, then the process is usually stopped until a change or adjustment can be made. In this situation, it might be desirable to use a more sensitive control chart immediately after restarting the process, just in case the adjustment was not effective.
Lucas and Crosier[69] suggested a simple adjustment to the Cusum chart called the fast initial response (FIR) or headstart feature to make it more sensitive. Instead of starting the Cusum with C0+ = 0.0 and C0− = 0.0 as illustrated inTable 6.2, they suggested setting C0+ =h/2, andC0− =−h/2.
This is illustrated with inTable 6.3using the first four data points in Table 6.2, as if the process were just restarted. In this caseh= 5 soC0+= 2.5 and C0−=−2.5.
TABLE 6.3: Standardized Taular Cusums with FIR of h/2
Individual value xi yi Ci+ Ci− 1 40.208 −1.958 0.042 -3.958 2 56.211 1.242 0.784 -2.216 3 51.236 0.247 0.531 -1.469 4 60.686 2.137 2.168 0.000
By using a headstart of h/2 and h = 5 on a Cusum chart, Lucas and
Crosier Lucas and Crosier[69] showed that the average run length (ARL) for detecting a one standard deviation change in the process mean was reduced from 10.4 to 6.35 (a 39% decrease). The ARL0, or average run length for detecting a false out-of-control signal when the mean has not changed from the known in-control value, is only decreased by 7.5% (from 465 to 430).
6.1.1.2 ARL of Shewhart and Cusum Control Charts for Phase II Monitoring
If individual measurements are monitored rather than subgroups of data in Phase II monitoring, and a Shewhart chart is used, individual measurements would be plotted with the center lineµand the control limits µ±3σ, where µ and σ are the known values of the in-control process mean and standard deviation.
The OC or operating characteristic for the Shewhart chart is the probabil- ity of falling within the control limits, and it is a function of how far the mean has shifted from the known in-control mean (as discussed in Section 4.5.1).
The average run length (ARL) is the expected value of a Geometric random with ARL= 1/(1−OC). The OC for a Shewhart when the mean has shifted k standard deviations to the left or right can be calculated with the pnorm function in R aspnorm(3-k)-pnorm(-3-k)
Vance[97] published a computer program for finding the ARLs for Cusum control charts. The xcusum.arl function in the R package spc calculates ARLs for Cusum charts with or without the headstart feature by numerically approximating the integral equation used to find the ex- pected or average run lengths. An example call of the xcusum.arl function isxcusum.arl(k=.5, h=5, mu = 1.0, hs=2.5, sided="two"). k=.5 spec- ifies k = 1/2.h=5 specifies the decision interval h= 5. mu = 1.0 indicates that the ARL for detecting a one σ shift in the mean should be calculated.
hs=2.5 specifies the headstart feature to be half the decision interval, and sided="two"indicates that the ARL for a two sided Cusum chart should be calculated.
The R code below was used to calculate the ARL for the Shewhart individ- uals chart for each of the mean shift values in the vectormuusing thepnorm function; the ARL for the Cusum chart; and the ARL for the Cusum chart with theh/2 headstart feature is calculated using thexcusum.arl function.
R>#ARL individuals chart with 3 sigma limits R>mu<-c(0.0,.5,1.0,2.0,3.0,4.0,5.0)
R>OC3=c(pnorm(3-mu[1])-pnorm(-3-mu[1]), pnorm(3-mu[2])-pnorm(-3-mu[2]), pnorm(3-mu[3])-pnorm(-3-mu[3]),
pnorm(3-mu[4])-pnorm(-3-mu[4]), pnorm(3-mu[5])-pnorm(-3-mu[5]), pnorm(3-mu[6])-pnorm(-3-mu[6]), pnorm(3-mu[7])-pnorm(-3-mu[7])) R>ARL3=1/(1-OC3)
R>#
R>#ARL for Cusum with and without the headstart feature R>library(spc)
R>ARLC<-sapply(mu,k=.5,h=5,sided="two",xcusum.arl)
R>ARLChs<-sapply(mu,k=.5,h=5,hs=2.5,sided="two",xcusum.arl) R>round(cbind(mu,ARL3,ARLC,ARLChs),digits=2)
To get a vector of Cusum ARLs corresponding to a whole vector of possible mean shift values, the Rsapply function was used in the code above. This function calculates the ARL for each value of the mean shift by repeating the call toxcusum.arlfor each value in the vectormu. The calculated ARLs form the body ofTable 6.4.
TABLE 6.4: ARL for Shewhart and Cusum Control Charts
Cusum with Shift in Mean Shewhart Cusum Chart h/2 headstart Multiples ofσ Individuals Chart k= 1/2, h= 5 k= 1/2, h= 5
0.0 370.40 465.44 430.39
0.5 155.22 38.00 28.67
1.0 43.89 10.38 6.35
2.0 6.30 4.01 2.36
3.0 2.00 2.57 1.54
4.0 1.19 2.01 1.16
5.0 1.02 1.69 1.02
In this table we can see that the ARL for a Shewhart chart when the mean has not shifted from the known in-control mean is 370 (rounded to the nearest integer), while the ARLs for Cusum charts with or without the headstart feature are longer (i.e. 430–465). Therefore, false positive signals are reduced when using the Cusum chart.
The ARL for detecting a small shift in the mean of the order of 1σis over 40 for a Shewhart chart plotting individual measurements. Recall from Section 4.5.1, that the ARL for detecting a 1σ shift in the mean using a Shewhart X chart with subgroups of size n= 5 was approximately 5 (exactly 4.495).
That impliesn×ARL≈5×4.49 = 22.45 total measurements. Therefore, the ShewhartXchart can detect small shifts in the mean with fewer measurements than a Shewhart chart plotting individual measurements.