6 Chemical Process Concentration CUSUM

EXAMPLE 15-6 Chemical Process Concentration CUSUM

  A Tabular CUSUM

  L s 1i2 ⫽ max30, 1␮ 0 ⫺K 2⫺x i ⫹s L 1i ⫺ 12 4

  We will illustrate the tabular CUSUM by applying it to the chemical process concentration data in Table 15-7. The

  ⫽ max 30, 199 ⫺ 12 ⫺ x i ⫹s L 1i ⫺ 12 4

  process target is ␮ 0 ⫽ 99, and we will use K ⫽ 1 as the ref-

  ⫽ max 30, 98 ⫺ x i ⫹s L 1i ⫺ 12 4

  erence value and H ⫽ 10 as the decision interval. The rea- sons for these choices will be explained later.

  Therefore, for observation 1 the CUSUMs are

  Table 15-8 shows the tabular CUSUM scheme for the chemical process concentration data. To illustrate the calcula-

  s H 112 ⫽ max30, x 1 ⫺ 100 ⫹ s H 102 4

  tions, note that

  ⫽ max

  30, 102.0 ⫺ 100 ⫹ 04 ⫽ 2.0

  H s 1i2 ⫽ max30, x i ⫺ 1␮ 0 ⫹K 2⫹s H 1i ⫺ 12 4

  and

  ⫽ max i 30, x ⫺ 199 ⫹ 12 ⫹ s H 1i ⫺ 12 4 s L 112 ⫽ max30, 98 ⫺ x 1 ⫹s L 102 4

  ⫽ max i 30, x ⫺ 100 ⫹ s H 1i ⫺ 12 4 ⫽ max

  30, 98 ⫺ 102.0 ⫹ 04 ⫽ 0

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  CHAPTER 15 STATISTICAL QUALITY CONTROL

  Table 15-8 The Tabular CUSUM for the Chemical Process Concentration Data

  Observation

  Upper CUSUM

  Lower CUSUM

  as shown in Table 15-8. The quantities n H and n L in Table 15-8

  Next Steps: The limits for the CUSUM charts may be used

  indicate the number of periods that the CUSUM s H (i) or s L (i)

  to continue to operate the chart in order to monitor future

  have been nonzero. Notice that the CUSUMs in this example

  productions.

  never exceed the decision interval H

  10. We would there-

  fore conclude that the process is in control.

  When the tabular CUSUM indicates that the process is out of control, we should search for the assignable cause, take any corrective actions indicated, and restart the CUSUMs at zero. It may be helpful to have an estimate of the new process mean following the shift. This can be computed from

  s H 1i2

  0 K

  n ,

  if s

  H H 1i2 H

  ˆ μ

  s L 1i2

  (15-32)

  0 K

  n ,

  if s

  L 1i2 H

  L

  It is also useful to present a graphical display of the tabular CUSUMs, which are

  sometimes called CUSUM status charts. They are constructed by plotting s H (i) and s L (i) ver-

  sus the sample number. Figure 15-21 shows the CUSUM status chart for the data in Example

  15-6. Each vertical bar represents the value of s H (i) and s L (i) in period i. With the decision in-

  terval plotted on the chart, the CUSUM status chart resembles a Shewhart control chart. We have also plotted the sample statistics x i for each period on the CUSUM status chart as the

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  15-8 TIME-WEIGHTED CHARTS

  Figure 15-21 The

  CUSUM status chart

  for Example 15-6.

  Sample number

  solid dots. This frequently helps the user of the control chart to visualize the actual process performance that has led to a particular value of the CUSUM.

  The tabular CUSUM is designed by choosing values for the reference value K and the decision interval H. We recommend that these parameters be selected to provide good average run-length values. There have been many analytical studies of CUSUM ARL performance. Based on these studies, we may give some general recommendations for selecting H and K.

  Define H h X and K k X , where X is the standard deviation of the sample variable used

  in forming the CUSUM (if n

  1, ). X X Using h 4 or h 5 and k 1兾2 will gener- ally provide a CUSUM that has good ARL properties against a shift of about 1 X (or 1 X ) in

  the process mean. If much larger or smaller shifts are of interest, set k

  兾2, where is the size

  of the shift in standard deviation units.

  To illustrate how well the recommendations of h

  4 or h 5 with k 1兾2 work, con- sider these average run lengths in Table 15-9. Notice that a shift of 1 X would be detected in

  Table 15-9 Average Run Lengths for a CUSUM Control Chart

  with k = 1 兾2 Shift in Mean

  (multiple of X )

  h 4 h 5

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  CHAPTER 15 STATISTICAL QUALITY CONTROL

  either 8.38 samples (with k ⫽ 1兾2 and h ⫽ 4) or 10.4 samples (with k ⫽ 1兾2 and h ⫽ 5). By

  comparison, Table 15-6 shows that an X chart would require approximately 43.9 samples, on

  the average, to detect this shift.

  These design rules were used for the CUSUM in Example 15-6. We assumed that the process standard deviation ␴ ⫽ 2. (This is a reasonable value; see Example 15-2.) Then with k ⫽ 1兾2 and h ⫽ 5, we would use

  K

  ⫽ k␴ ⫽ 1 ⁄ 2 122 ⫽ 1 and H ⫽ h␴ ⫽ 5 122 ⫽ 10

  in the tabular CUSUM procedure.

  Finally, we should note that supplemental procedures such as the Western Electric rules

  cannot be safely applied to the CUSUM, because successive values of S H (i) and S L (i) are not

  independent. In fact, the CUSUM can be thought of as a weighted average, where the weights are stochastic or random. In effect, all the CUSUM values are highly correlated, thereby causing the Western Electric rules to give too many false alarms.

  15-8.2 Exponentially Weighted Moving Average Control Chart

  Data collected in time order is often averaged over several time periods. For example, eco- nomic data is often presented as an average over the last four quarters. That is, at time t the average of the last four measurements can be written as

  x t 142 ⫽ x t ⫹ x t ⫺1 ⫹ x t

  x t

  4 4 4 ⫺2 ⫹ 4 ⫺3

  This average places weight of 1 兾4 on each of the most recent observations, and zero weight on older observations. It is called a moving average and in this case a window of size 4 is used. An average of the recent data is used to smooth the noise in the data to generate a better estimate of the process mean than only the most recent observation. However, in a dynamic environment where the process mean may change, the number of observations used to construct the average is kept to a modest size so that the estimate can adjust to any change in the process mean. Therefore, the window size is a compromise between a better statistical estimate from an average and a response to a mean change. If a window of size 10 were used in a moving average, the sta- tistic x t 1102 would have lower variability, but it would not adjust as well to a change in mean.

  For statistical process control, rather than use a fixed window size it is useful to place the greatest weight on the most recent observation or subgroup average, and then gradually decrease the weight on older observations. One average of this type can be constructed by a

  multiplicative decrease in the weights. Let ␭ⱕ1 denote a constant and ␮ 0 denote the process

  target or historical mean. Suppose that samples of size n ⱖ1 are collected and is the average x t of the sample at time t. The exponentially weighted moving average (EWMA) is

  z

  t ⫽ ␭x t ⫹␭ 11 ⫺ ␭2x t ⫺1 ⫹␭ 11 ⫺ ␭2 x t ⫺2 ⫹p⫹␭

  2 t t

  11 ⫺ ␭2 ⫺1 x 1 ⫹

  11 ⫺ ␭2 ␮ 0

  t

  ⫽a k ␭ 11 ⫺ ␭2 x t t ⫺k ⫹ 11 ⫺ ␭2 ␮ 0

  k ⫽0

  Each older observation has its weight decreased by the factor

  11 ⫺ ␭2. The weight on the starting value ␮ 0 is selected so that the weights sum to one. Here z t is also sometimes called a

  geometric average.

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  15-8 TIME-WEIGHTED CHARTS

  Figure 15-22

  EWMAs with

  and

  0.2 show a 4

  compromise between a 2 smooth curve and a

  response to a shift. 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

  The value of determines the compromise between noise reduction and response to a

  mean change. For example, the series of weights when

  0.8 are

  0.8, 0.16, 0.032, 0.0064, 0.00128, . . . and when ␭ 0.2 the weights are

  When

  0.8 the weights decrease rapidly. Most of the weight is placed on the most recent

  observation, with modest contributions to the EWMA from older measurements. In this case, the EWMA does not average noise much, but it responds quickly to a mean change. However, when

  the weights decrease much more slowly and the EWMA has substantial contri-

  butions from the more recent observations. In this case, the EWMA averages noise more, but it responds more slowly to a change in the mean. Fig. 15-22 displays a series of observations with a mean shift in the middle on the series. Notice that the EWMA with

  0.2 smooths

  the data more, but that the EWMA with

  0.8 adjusts the estimate to the mean shift more

  quickly.

  It appears that it is difficult to calculate an EWMA because at every time t a new weighted average of all previous data is required. However, there is an easy method to calculate an

  EWMA based on a simple recursive equation. Let z t z 0 0 . Then it can be shown that

  EWMA Update

  Equation

  z t

  x t 11 2z t 1 (15-33)

  Consequently, only a brief computation is needed at each time t.

  To develop a control chart from an EWMA, control limits are needed for Z t . The control limits are defined in a straightforward manner. They are placed at three standard deviations around the mean of the plotted statistic Z t . This follows the general approach for a control chart in Equation 15-1. An EWMA control chart may be applied to individual measurements as an extension to an X chart or to subgroup averages. Formulas here are developed for the

  more general case with an average from a subgroup of size n. For individual measurements n 1.

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  CHAPTER 15 STATISTICAL QUALITY CONTROL

  Because Z t is a linear function of the independent observations X 1 ,X 2 ,...,X t (and 0 ),

  the results from Chapter 5 can be used to show that

  V 1Z 2t t 2 0 1Z t 2 n

  E and

  2 冤1

  11 2 where n is the subgroup size. Therefore an EWMA control chart uses estimates of 0 and in

  冥

  the following formulas:

  EWMA Control Chart

  LCL

  2t

  0 3 1n A 2 31 11 2 4

  CL

  0 (15-34)

  UCL

  2t

  1n A 2 31 11 2 4

  Note that the control limits are not of equal width about the centerline. The control limits are calculated from the variance of Z t and that changes with time. However, for large t the variance

  of converges Z t to

  2 lim V 1Z t

  tS 2 n a 2 b

  so that the control limits tend to be parallel lines about the centerline as t increases.

  The parameters 0 and are estimated by the same statistics used in or X charts. That

  is, for subgroups

  ˆ 0 X and

  ˆ R d 2 or

  ˆ S c 4

  and for n 1

  ˆ 0 X and

  ˆ MR 1.128