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# Shewhart Charts versus Manhattan Control

We appreciate that it is easy to ‘find fault’ and far more difficult to come up with a better alternative. We believe that our findings about the Shewhart chart (and many others) amount to more than ‘fault finding’. We have identified serious short comings.

The remainder of this article is based on simulations from a simulation program. The simulation program generates random numbers from a normal distribution with mean zero and standard deviation of 2. As sub-group sizes of 4 are used, the standard error is 1. The conclusions based on this data will apply to other means and standard deviations.

The range chart method is used to determine control limits for the Shewhart X-Bar chart. Conclusions obtained with this chart are also obtained with the S chart.

#### Simulation 1

This simulation simulated a process that is stable for a while and then is effected by a disturbance, which increases the process average by two standard deviations. After a while the disturbance disappears and the process reverts to normal. The top chart in Figure 1a shows how the process changed.

Figure 1(a): Simulation 1

The second chart in Figure 1a is a simple Shewhart chart. The last chart is the Manhattan chart using exactly the same data. Observe above that the Manhattan control chart almost identically follows the real process.

If we simply look at the run chart and ignore the control limits most will see that the process appears to have shifted. However, this is not how the Shewhart chart is supposed to be used. Control limits were introduced to remove the subjective basis, to avoid different interpretations by different people. If the Shewhart chart is to be used objectively then the above chart will advise the operator that a problem occurred at the 54th (above) and at the 110th point (below). These points are way of the mark and misleading. They are nowhere near the occurrence and disappearance of the problem and operators will search for assignable causes in the wrong areas of the chart.

The Manhattan control correctly indicated when the process changed and hence operators have a better chance of identifying the process.

Sometimes a run and trend test is used to increase the sensitivity of the Shewhart chart. Figure 1b is a repeat of the simulation using the runs test based on 8 points.

Figure 1(b): Simulation 1 with a run and trend test

Again the Manhattan control chart has shown the true situation. The Shewhart chart shows three problems but it again is misleading not reflecting the true process and once again operators would have difficulties identifying the assignable causes. They would search for causes at the wrong time.

#### Simulation 2

Shewhart charts, and indeed all other conventional charts test for changes relative to the center line. Significance testing should be relative on the previous change. If we do not check for changes relative to previous changes we will not be able to detect all assignable causes for serious problems. Quality improvement is about detecting and removing assignable causes. Figure 2 shows a simulation that has resulted in several step changes.

Figure 2: Simulation 2

The Manhattan control chart shows all changes. The Shewhart shows some points out of control but in the wrong areas. By using the Shewhart chart objectively there is no indication of the many step changes and since each step change is due to new disturbances there is little change of identifying the causes.

Even a subjective assessment of the run chart does not identify the step changes, showing the futility of run chart inspections. A visual inspection of the plotted points only indicates a positive trend followed by a negative trend.

#### Simulation 3 (Flat Liner)

A flat liner is a statistically in-control process. The process mean is stable. Theoretically the Shewhart control chart should not show a point out of control. The Manhattan should be a flat line. In practice there will be false alarms. The Manhattan control chart has proved to be far superior in detecting changes. However, how does it perform if there are no changes? Having too many false alarms would be undesirable.

Simulations have shown that the number of false alarms is comparable. However, when the Shewhart chart uses a run and trend test the false alarms increases for that chart, leading to credibility problems.

#### Conclusion

The Shewhart chart does not show the onset and duration of a change. It does not show relative changes. Run tests tend to confuse the picture. Manhattan control charts better show the real changes of the process and therefore will lead to real quality improvements. The Shewhart chart was only designed to detect that the process is out of control. This is not enough and most quality engineers know that their process is out of statistical control without needing a control chart.

To improve the process we need to know when a problem started and how long it lasted for. This information is also vital for product retrievals. The Manhattan chart was designed for this, the Shewhart chart was not. The simulation has demonstrated that the Shewhart chart provides the quality engineer with a misleading picture of what is happening with the process. Most quality engineers are not aware of this and assume that when a point is out of control, or a run is shown then that is where the problem has occurred. We must now ask, should a tool be used that does not provide the correct picture of what is happening to the process when a better tool exists? Should we use a tool that will prompt operators to search for assignable causes in the wrong points in time?

The reason for the problem with Shewhart charts is simple. Noise is plotted and the most elementary method is used to identify changes in the process. The method, originating from the 1920s is just too simple. The Manhattan algorithm is far more exhaustive is its approach.

Manhattan charts have far greater flexibility. The algorithm can be modified to filter certain changes, it can be modified to deal with trends and extreme non-normality. Manhattan theory can be used in stability analysis, relationship analysis and much more. Manhattan charts require far less training. Shewhart charts are expensive to implement. Training is required to explain how to set up and interpret and when to apply the many charts and supplementary rules. Yes, Manhattan charts can be used to control variability.

There are many more problems with other control charts, such as the Cusum and EWMA.

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