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Shewhart vs Manhattan Control

Dr Juergen Ude Dr Juergen Ude

Shewhart charts

These were first introduced by Dr. Walter Shewhart early this century. Their logic is extremely simple. There is a natural amount of variation in measurements, which we should not react to other than through major process changes. Limits are placed at positions within which we expect the natural variation of the process to fall in. The charts are easy to use and ideal for an environment without computers.

Decisions are based on single points only. They are simply a plot of sampling noise. Shewhart Charts are insensitive to small changes, they do not estimate the onset, duration and magnitude of a process change. They are not appropriate to perform run chart correlations. They assume a stable process mean, an assumption that often does not apply in practice. When this assumption is violated, control limits become too tight and sometimes too wide.

Shewhart charts are often supplemented by other decision rules, to increase their sensitivity, but these increase the false alarm rate.

Cusum charts

Sums of differences between actual data and a reference point are accumulated and plotted. A change in the process mean is shown as a change in the slope. A V-mask is often used to detect a significant change. Most computer software make use of Johnson’s method to determine the V-Mask. (Closely related to the V-Mask is the Decision Interval method).

The variability of Cusums is dependent on the variability of “Total” and, hence, a parabolic shape seems more appropriate. Some authors have reported using parabolic shape masks. Using a V-Mask helps to explain why these charts are less sensitive to large changes.

There is a fundamental flaw in the theory of Johnson’s method. The flaw means that the actual run length performance of Cusums based on Johnson’s method is not as expected.

Cusums are not popular because some have difficulty interpreting the Cusum plot.

Exponential and moving averages charts

These use forecasting smoothing techniques. Real-time process control has certain problems which make the application of Shewhart and Cusums difficult. The objective is for operators to take corrective action whenever there are changes in the process. However, this is easier said than done. One major problem is that changes are often not sustained. Reacting to these changes can increase variability. (After corrective action was taken the forces that would have corrected the process in its own can still act and thus increase variation.) In these situations it seems logical for operators to concern themselves only with overall long-term trends, ignoring the short term perturbations of the process. Exponential and moving averages charts serve this purpose well.

These charts are not the most suitable for off-line control. Here we are interested in identifying assignable causes of variation. Due to smoothing, there is a sluggishness in the response to changes and hence onsets and durations of changes cannot be easily estimated, making it difficult to identify causes.

Manhattan Control

These are based on comprehensive significance testing that involves many points. Changes in the process are shown in the form of plateaus, as shown in Figure 1. The advantages over other charts are:

  • They estimate the onset and duration of a change. This information is important to decide how much product to impound. It also makes detecting assignable causes of variation easier.
  • They estimate the magnitude of the change. This is required to place perspective on results.
  • They are highly sensitive. It seems unlikely that their sensitivity can be bettered.
  • They are easy to understand. Operators can make more sense out of charts that plot the signal instead of the noise.
  • Applications are many including analysis of service performance.
  • They are ideal for run chart correlations. Finger print comparisons can be performed on several variables to establish correlations, not possible through ordinary regression analysis.
  • They seem to be very robust to non-normality, for changes of duration greater than five points and, hence, can often be applied to attribute data. A transformation may also be used to normalize the data.
An example of a change analysis chart (Manhattan control chart)

The main disadvantage is that they require a computer. The usual problems of significance testing also apply; e.g. changes depend on the amount of data and magnitude of change. This, however, applies to all control charts. Another disadvantage is that trends will be seen as step changes, but we have never found this to be a problem. Standard Shewhart and Cusum charts also do not cater for trends.

Average Run Length Performance

Average run length is the average number of points that are plotted before the system detects a change. For a basic Shewhart chart it takes, on average, 44 points before a standard error change is detected.

Unfortunately, much of the theory and literature on run lengths make the assumption that the change is relative to a known level of a parameter and, hence, do not indicate the true performance. In practice we usually do not know what the true parameter is. Shewhart charts, for example, are typically applied by computing the average of, say 25 to 50 points and then placing control limits about this average. If during this change a shift has occurred this will affect the average and, hence the position of control limits and sensitivity.

Manhattan charts are based on significance testing of relative changes. They make no assumption on the current parameter. The average run length depends on the level of significance, the duration of the plateau prior to the change and the magnitude of the change. Our research has shown for a very large initial plateau the average run length to detect a one standard error change is nine with the same type I error as a Shwehart chart. This is better than most, if not all, other charts and can probably not be improved upon.

In practice the duration of plateaus prior to a change will not be as large. Also the run length distribution is variable. Taking these factors into account we recommend a Type I error of .0001(c.f.0027 Shewhart Type I error). At this level one can expect an ARL of around 15 for a one standard error change.

Conclusion

A decision on which is the most appropriate tool will ultimately be based on practice, not theory. (Too much theory is based on assumptions that do not hold in practice.)

Shewhart charts are unlikely ever to be completely replaced, because they serve a useful purpose when there is no computing power. For on-line SPC where the objective is to make adjustments to the process as changes are detected, exponential charts are probably the most appropriate. For off-line SPC, as part of the continuous improvement process, Manhattan control is, in my opinion, the step forward.

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