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Manhattan or Shewhart chart?

Dr Juergen Ude Dr Juergen Ude

Introduction

There are many schools of thoughts on the best application of control charts. The general opinion is that the control chart is a tool for continuous improvement, whereby assignable causes are identified and removed. A less preferred application is to use the control chart as a basis of making a compensating process adjustment.

Grant and Leavenworth (1988) note that “One important tool in statistical quality control is the Shewhart control chart.. The power of the Shewhart technique lies in its ability to separate out these assignable causes of quality variation. This makes possible the diagnosis and correction of many production troubles and often brings substantial improvements in product quality and reduction of spoilage and rework.”

Donald J. Wheeler (ASQC STNL Vol 15, No4) notes “ Control Charts... are primarily intended for a different role – the detection of assignable causes in a timely manner so that the organisation may act to remove those causes and thereby to improve the process output. The uses of control charts as a process adjustment mechanism is one of the less sophisticated uses of the charts.

Douglas C. Montgomery (1985) states “The most important use of a control chart is to improve process...... the routine and attentive use of control charts will identify assignable causes. If these causes can be eliminated from the process, variability will be reduced and the process will be improved.

The Shewhart Control chart is not the effective tool for continuous improvement that we are led to believe. There are many who share this belief and there are of course many who object to it. In the spirit of continuous improvement of the tools the quality profession uses, the following will try and explain the problems with the Shewhart Chart and then discuss an alternative method.

So what can possibly be wrong with the Shewhart Chart?

Quite a few things. Some are well known problems, and some are fundamental, but overlooked or not publicised. Publicised problems include low sensitivity and control limits that are often too tight and sometimes too wide. Training costs are high. Shewhart Charts are not simple. Users need to learn to distinguish between X-bar, Individuals, S, Range, N,P,NP and U charts, and understand the many supplementary tests.

As these problems are well documented I will not discuss these here.

The Shewhart Chart was not designed to show the onset and duration of a change

This is a fundamental problem. To identify an assignable cause we need to know when a problem started and when it finished. We can then relate the start and finish time to changes in other variables.

The Shewhart Chart will only advise that there is a problem with the process. The objective however is not to be told that there is a problem. The objective is to identify the assignable cause and remove it.

Unfortunately many SPC courses and text books imply that when a point falls outside a control limit then this is where the problem occurred and we need to simply establish the assignable cause. Many examples will show an out-of-control point with an annotation such as ‘batch problem’, falsely implying that when a point falls inside a control limit everything is OK, and if a point falls outside everything is not ok. The reality is that it takes time for a point to fall outside a control limit, or a run to form, usually long after the assignable causes manifests itself. Operators may thus falsely believe that the process is in control when a points falls with limits, or search for causes in the wrong areas when a point falls outside control limits (or when a statistical run in obtained)

The Shewhart Chart does not test for relative changes.

This is another fundamental problem. Significance testing for out-of-control points and run tests are relative to the reference line. This means that if there is a new change no test is performed to identify the new change. This is not conducive to improving the process. We need to be informed of all introductions of assignable causes if we wish to improve the process.

It is ironical that the design of the Shewhart control chart is more appropriate for adjusting the process when it drifts from target. Most experts do not recommend the control chart to be used for this. Instead, the commonly recommended application is to identify and remove assignable causes.

Demonstrating the problems through simulation

A very effective way of demonstrating the problems is through simulation. Before I do this I must explain the fundamentals.

Let us assume that we adjust a process to fill the net weight of a consumer product to 200g. It is an established fact that not every packet will weigh 200g. The process will produce packets that weigh more or less than 200g, and if we averaged every packet produced, and a sufficiently large number of packets was produced, then the average will be 200g. If we were to take a sample of 5 packets and average this, then this average may or may not be equal to 200g. We must distinguish between the process mean and the sample mean. The sample mean is an estimator of the process mean and in this case the average of the five sample averages. The process mean is the true process average, in this case 200g. The simulations will generate sample means about the process mean.

After simulating the data, the control chart is then applied to the generated random numbers to see how well it can reflect the process disturbances. Normally the probability distribution is chosen to match the probability distribution of the chosen process. For our simulation we will use the Normal distribution.

When performing simulations, one must run many simulations. A single simulation is misleading and can lead to bias. Thousands of simulations have been run, from many perspectives, for many different control charts, using a wide range of decision rules. For practical reasons only limited outcomes are to be shown below. However they are representative of all simulations. The objective here is to give you a feel for the problem.

EXAMPLE ONE

Our first simulation example is based on the generation of two disturbances causing a change of 2 standard errors lasting 5 points each

Change Analysis chart 1 - Simulating two disturbances Figure 1: Simulating two disturbances

In most instances the Shewhart Chart, with or without a run test failed to even show the presence of the ‘assignable causes’.

Change Analysis chart 1 - Simulation which compares Manhattan with Shewhart Figure 2: A typical result showing how the shewhart chart failed

It is the course understandable why the run test did not work either. Each change was only 5 points in duration. The duration was not large enough for the run test, based on 8 points, to indicate out-of-control.

EXAMPLE TWO

For disturbances of longer duration similar results are obtained. For example another simulation looked at the effectiveness of identifying the onset and duration of a two standard error change that manifested itself for a longer duration (10).

Rarely did points fall outside control limits. Sometimes a run would be appear in the correct place, sometimes in an incorrect place. Shewhart Charts with run tests often provided confusing results. For example Figure 4 shows runs before and after the change.

Change Analysis chart 1 - Simulation which compares Manhattan with Shewhart Figure 3: A single 2 standard error disturbance to the process
Change Analysis chart 1 - Simulation which compares Manhattan with Shewhart Figure 4: One long change of 2 standard error change

Based on up to two simple process disturbances of a relatively large change, the simulations have shown that the Shewhart chart failed virtually at all the times to indicate where the process first changed and for how long. Using a run test tended to result in more false alarms and very often in signalling out of control states in the ‘wrong’ areas. The latter leads to confusion, and waste of effort.

EXAMPLE THREE

In practice many process disturbances may occur over the period of sampling. The following example (Figure 5) shows a simulation model where several changes were simulated. Such a dirty process is not unusual in practice. Figure 6, shows an example of a typical output. Whether a run test was used or not, the resolution in detecting the various changes was extremely poor. It is impossible to detect where the changes occurred.

There just appears to be one big upward trend followed by a downward trend. This makes it hard to pin point the cause.

Change Analysis chart 1 - Simulation which compares Manhattan with Shewhart Figure 5: Simulation model where we have simulated several step changes, of large magnitude (2 standard errors, and long duration, (10) points each.
Change Analysis chart 1 - Simulation which compares Manhattan with Shewhart Figure 6: Output for multiple change simulation

In Figure 6, we find only two out-of-control points. This shows how ineffective control limits really are. If we use the run test, almost every point is out of control. Is this a serious problem? Consider the actions of an operator trained to react only when a point falls outside control limits. It is not often that an operator reacts on basis of two points. Consider an operator taught to react to a run test. He would throw his hands in the air. Where will he look for an assignable cause?. Consider an operator taught to take pattern and trends into account (back to subjective decision making). She too would throw her hands in the air.

Conclusion

A Shewhart chart is only able to advise whether a process is in statistical control or not. (This in itself is a vague statement. What is statistical control?) A tool that can only advise that a problem exist is not an effective improvement tool. We know life is full of problems. What we need are tools that provide solutions, not tell us what we know already.

According to surveys, over 90 percent of processes are ‘out-of-control’, even when diligent process control is practised. The Shewhart chart therefore only tells us what we know with high probability anyway. An effective tool is one that will aid in identifying the cause.

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