Cusum Control Charts

17.6 Cusum Control Charts

  The disadvantage of the Shewhart-type control charts, developed and illustrated in the preceding sections, lies in their inability to detect small changes in the mean. A quality control mechanism that has received considerable attention in the statistics literature and usage in industry is the cumulative sum (cusum) chart. The method for the cusum chart is simple and its appeal is intuitive. It should become obvious to the reader why it is more responsive to small changes in the mean. Consider a control chart for the mean with a reference level established at value

  W . Consider particular observations X 1 ,X 2 ,...,X r . The first r cusums are

  S 1 =X 1 −W S 2 =S 1 + (X 2 −W) S 3 =S 2 + (X 3 −W)

  .. . S r =S r −1 + (X r − W ).

  It becomes clear that the cusum is merely the accumulation of differences from the reference level. That is,

  The cusum chart is, then, a plot of S k against time.

  Suppose that we consider the reference level W to be an acceptable value of the mean μ. Clearly, if there is no shift in μ, the cusum chart should be approximately horizontal, with some minor fluctuations balanced around zero. Now, if there is only a moderate change in the mean, a relatively large change in the slope of the cusum chart should result, since each new observation has a chance of contributing

  a shift and the measure being plotted is accumulating these shifts. Of course, the signal that the mean has shifted lies in the nature of the slope of the cusum chart. The purpose of the chart is to detect changes that are moving away from the reference level. A nonzero slope (in either direction) represents a change away from the reference level. A positive slope indicates an increase in the mean above the reference level, while a negative slope signals a decrease.

  Cusum charts are often devised with a defined acceptable quality level (AQL) and rejectable quality level (RQL) preestablished by the user. Both represent values of the mean. These may be viewed as playing roles somewhat similar to those of the null and alternative mean of hypothesis testing. Consider a situation where the analyst hopes to detect an increase in the value of the process mean. We shall

  use the notation μ 0 for AQL and μ 1 for RQL and let μ 1 >μ 0 . The reference level

  The values of S r (r = 1, 2, . . . .) will have a negative slope if the process mean is at

  μ 0 and a positive slope if the process mean is at μ 1 .

  Chapter 17 Statistical Quality Control

  Decision Rule for Cusum Charts

  As indicated earlier, the slope of the cusum chart provides the signal for action by the quality control analyst. The decision rule calls for action if, at the rth sampling period,

  d r > h,

  where h is a prespecified value called the length of the decision interval and

  d r =S r − min S .

  1 ≤i≤r−1 i

  In other words, action is taken if the data reveal that the current cusum value exceeds by a specified amount the previous smallest cusum value.

  A modification in the mechanics described above makes employing the method easier. We have described a procedure that plots the cusums and computes differ- ences. A simple modification involves plotting the differences directly and allows for checking against the decision interval. The general expression for d r is quite simple. For the cusum procedure where we are detecting increases in the mean,

  d r = max[0, d r −1 + (X r − W )].

  The choice of the value of h is, of course, very important. We do not choose in this book to provide the many details in the literature dealing with this choice. The reader is referred to Ewan and Kemp, 1960, and Montgomery, 2000b (see the Bibliography) for a thorough discussion. One important consideration is the expected run length . Ideally, the expected run length is quite large under μ = μ 0

  and quite small when μ = μ 1 .

  Review Exercises

  17.1 Consider X 1 ,X 2 ,...,X n independent Poisson

  random variables with parameters μ 1 ,μ 2 ,...,μ n . Use

  the properties of moment-generating functions to show

  n

  that the random variable X is a Poisson random

  variable with mean μ i and variance 13 2.4009 μ 0.0077 i . 14 2.3992 0.0107

  17.2 Consider the following data taken on subgroups

  of size 5. The data contain 20 averages and ranges on

  the diameter (in millimeters) of an important compo-

  nent part of an engine. Display ¯

  X- and R-charts. Does

  the process appear to be in control?

  17.3 Suppose for Review Exercise 17.2 that the buyer

  has set specifications for the part. The specifications

  require that the diameter fall in the range covered by

  2.40000 ± 0.0100 mm. What proportion of units pro-

  duced by this process will not conform to specifica-

  17.4 For the situation of Review Exercise 17.2, give

  numerical estimates of the mean and standard devia-

  Review Exercises

  tion of the diameter for the part being manufactured process producing a certain type of item that is consid- in the process.

  ered either defective or not defective. Twenty samples are taken.

  17.5 Consider the data of Table 17.1. Suppose that (a) Construct a control chart for control of proportion additional samples of size 5 are taken and tensile

  defective.

  strength recorded. The sampling produces the follow- ing results (in pounds per square inch).

  (b) Does the process appear to be in control? Explain.

  Number of

  Sample Items

  (a) Plot the data, using the ¯

  X- and R-charts for the

  preliminary data of Table 17.1.

  (b) Does the process appear to be in control? If not, 17.9 For the situation of Review Exercise 17.8, sup-

  explain why.

  pose that additional data are collected as follows:

  Sample

  Number of Defective Items

  17.6 Consider an in-control process with mean μ = 25

  and σ = 1.0. Suppose that subgroups of size 5 are √

  used with control limits μ ± 3σ n, and centerline at

  μ. Suppose that a shift occurs in the mean, and the

  new mean is μ = 26.5.

  (a) What is the average number of samples required

  (following the shift) to detect the out-of-control sit-

  (b) What is the standard deviation of the number of

  runs required?

  10 7 Does the process appear to be in control? Explain.

  17.7 Consider the situation of Example 17.2. The fol-

  lowing data are taken on additional samples of size 5. 17.10 A quality control effort is being undertaken for

  Plot the ¯

  X- and S-values on the ¯

  X- and S-charts that

  a process where large steel plates are manufactured and

  were produced with the data in the preliminary sam- surface defects are of concern. The goal is to set up ple. Does the process appear to be in control? Explain

  a quality control chart for the number of defects per

  why or why not.

  plate. The data are given below. Set up the appropri- ate control chart, using this sample information. Does

  the process appear to be in control?

  Number of

  Sample Defects

  17.8 Samples of size 50 are taken every hour from a

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  Chapter 18

  Bayesian Statistics