Building and Using Control Charts - Six Sigma

Six Sigma DMAIC projects start with an established process, so the first thing to do is to establish the current level of performance and stability. The only effective way is to use control charts, an approach pioneered by Dr. Shewhart in the late 1920s and, although understanding has developed a little since then, the basic approach has remained intact. This section of the notes explains the appropriate approaches to generating process learning from Shewhart’s approach to charting.

Run Charts: The First Ste

The first step in putting data into context is to see it as part of the history of the process. This is best achieved by the use of run charts. Such diagrams (see below) allow judgments to be made about process trends or shits. They often also compare the current status of the process to the target or budget associated with that process.

Whilst it can easily be seen that this is a significant improvement on making judgments based on the comparison of two adjacent points it is still not particularly scientific. Questions arising from such charts might include: when is a trend significant? How much of a shit has to occur before we act? How does the target relate to the actual performance of the process?

Run chart

Shewhart Charts: Application of Economic and Scientific Principles

The lack of convincing answers to these questions shows the vulnerability of this approach. Shewhart uses the empirical rule for homogenous data to set up rules by which we can make consistent judgments about changes in the process.

A Control chart

The concept of natural limits for a process means that we can distinguish significant changes from insignificant ones: Special Causes from Common Causes of variation. Since the decision rules are based upon characteristics of all homogenous data sets rather than the specific attributes of one particular distribution this is a very robust model.

Note that texts which claim that control charts are based upon the normal distribution and the central limit theorem are moving away from the original work conducted by Shewhart and are, in fact, not following consistent logic. For example, whilst the central limit theorem works for the average chart it does not apply to the range charts for the subgroup sizes typically used, nor can it apply to the individuals chart where there are no subgroup averages for the theorem to apply to.

Shewhart general approach to process control is to take a subgroup of the data and extrapolate from the results of this subgroup to make predictions for the population. The two elements of the subgroup to which control are applied are the average and the range. It is appropriate at this point to discuss the relative roles of these two elements.

Role of the Average Chart

The average chart is concerned with variation between subgroups. The control limits are based upon 3 sigma for the subgroup average distribution. They are essentially testing if individual subgroup averages vary more than could be expected given the variability within individual subgroups. To this end the control limits are calculated using the average range of subgroup data as an estimate of this short - term variability.

Role of the Range Chart

The range chart is concerned with variation within subgroups. The control limits are based upon 3 sigma for the subgroup range distribution. They are essentially testing if the variation within each subgroup is similar to the variation within the other subgroups. To this end the control limits are calculated using the average range of subgroup data as an estimate of this within subgroup variability.

Rational Sub grouping

There is a requirement which underpins the application of the average and range chart. The requirement is known as ‘rational sub grouping’. Since the control limits of the average chart are calculated using subgroup range data we are assuming that the range of a subgroup is a reliable estimate of short term variability. If the subgroup range is regularly distorted by special causes then the control limits will be distorted leading to incorrect decisions.

We need to select subgroups in such a way as to minimize this possibility and ensure homogeneity within the subgroup. The best way of achieving this is to select them so that they are produced at approximately the same time – usually consecutively within the process. However, rational sub grouping is also about thinking about the context for the data. What are the sources of variation present? What questions are the charts addressing? Specifically, any natural subgroups which occur within the data need to be considered. If you ignore a natural subgroup and force the data into another pattern you will be creating irrational subgroups which will distort the process control.

Inappropriate sub grouping is a particular issue with data which naturally occurs in a subgroup size of one. Examples of this might include monthly values (e.g. sales figures), periodic measurements from a continuous process or final test values for a series of complex products. If we accept the statistical wisdom that control charts only work because of the central limit theorem we would group the data, but if we group together, for example, five consecutive months of sales data because there would be a virtual certainty that a special cause would intervene within the subgroup (promotions, product launches, etc.). This would distort the calculated control limits and lead to poor decision making.

Calculating Control Limits

In order to establish whether the process is in control we need to apply a statistical test. In the case of control charts this is the control limits. Shewhart has set down methods of calculation for the control limits for each of the charts. Those are based on the assumption of 3 sigma limits for both average and range charts.

It is worth noting that the choice of 3 Sigma is an economic rather than a statistical one (Shewhart, 1980). At this level the considers that it would be economic to find and fix the causes of any point outside the limits but uneconomic to do the same for points inside the limits.

Calculations for the Average Chart

Average chart

Calculations for the Range Chart

Tthe Control limits for the range chart are calculated as below:

Out of Control Conditions

The purpose of calculating the control limits is to support the identification of out of control conditions and subsequent process learning. We require rules to indicate when a process is out of control. The control limits provide one indication of significant shocks to the system but further rules are required in order to provide more confidence in the ability of the charts to detect smaller shits or drifts in the process. There are several approaches to this but we are going to concentrate on the most common set of rules which are known as ‘the Western Electric Detection Rules. Those are below:

Rule 1: A single point falls outside the 3 sigma control limits

Rule 2: At least 2 out of 3 consecutive values fall on the same side of, and more than 2 sigma units away from, the central line

Rule 3: At least 4 out of 5 consecutive values fall on the same side of, and more than 1 sigma unit away from, the central line

Rule 4: At least 8 consecutive values fall on the same side of the central line

It is suggested that for simplicity it may be easier to introduce the charts with only rules 1 and 4 being considered as they require no extra calculations. Whilst it is possible that this would lead to some out of control points not being spotted, it may be sensible to keep matters as simple as possible early in the introduction. When the organisation is comfortable with the application of these two simple rules then the more complex rules 2 and 3 can be introduced for more sensitivity and quicker response. The key thing to remember is that it is more important to inculcate the approach to process improvement which underpins SPC than to spot every special cause in the initial phases.

It is also worth remembering that these are generic rules which work well for a wide variety of processes. They are clearly not comprehensive but reflect a good compromise between sensitivity to special causes and usability in real - life situations. In companies where their use of data is more sophisticated and experience of using SPC is greater it is possible to observe the customisation of out of control rules for different processes. For example the following pattern might be observed in a sheet extrusion process, where sheet thickness is the quality characteristic being measured.

Non - random pattern: Cycling

There is clearly a cyclical pattern emerging. Is this, however, truly a special cause? To answer this question accurately we shall need to carefully link the observed pattern to process knowledge. In this case relevant information is that the machine operates by having an automatic sensor periodically measuring the sheet thickness and using this data to provide feedback to a controller which adjusts the speed of the extrusion screw in response to the readings. The pattern could represent the tendency to over - adjust for common cause variation.

An appropriate confirmation strategy might be either to turn the controller of and observe the result or reduce the gain on the controller to increase the variation in sheet thickness required to initiate a response.

Other potential out of control indicators might be ‘hugging’ the central line (remember we expect only 60% to 75% of the plot points to fall within +/- 1 sigma unit) which could indicate poor control limit calculations or lack of rational sub grouping so that special causes within the subgroup range have had the effect of inflating the control limits. In short, any unusual patterns might be worth investigating for correlation with features of the process.

We do, however, need to take care to avoid the phenomena of operators responding to points or combinations of points because they ‘do not like them’ rather than because they indicate the presence of a special cause within the system.

Remember; never respond to an unusual pattern unless you can link it to a process cause. The other thing to remember is that the zeal with which you investigate patterns will be limited by resources. In the case of limited resources, or early in the application where failure to find a special cause may lead to reduced credibility you may wish to stick to the Western Electric Rules.

Tampering

Tampering is a phenomenon which is all too common in manufacturing processes. It is the act of responding to special causes as if they were common causes. A typical example of this is when an operator takes samples from the process and measures them to ensure that the process is operating satisfactorily. He has a process tolerance which the has to maintain, understanding that there is a little variability in the process the knocks 10% of each tolerance limits and adjusts the process if the sample the takes is outside of this ‘preferred’ limits.

However, if the natural spread of the process is exactly equal to the process tolerances then samples can breach these natural limits without a special cause being present. If the process set precisely on nominal then the process is in its best possible condition. Consider the case when a point falls outside the ‘preferred’ limits with the process in its optimum condition.

Tampering

The resulting mismatch of requirements will lead to more and opposing adjustments, introducing more and more variation into an orginally stable situation

The rule is that adjustments and changes should not be made without knowledge. True process knowledge (‘profound knowledge’ as Deming refers to it) can only be obtained by the consistent and assiduous application of control charting principles.

It is also worthy of note that tampering is rife in non - manufacturing and boardroom areas too. How often do boards of directors think that it might just be common causes of variation when sales drop, absenteeism rises or the accounts slip into the red? We still respond as if something has changed and set up teams to put things right when nothing has actually changed in the first place.

A final point to note is that we don’t always require people to tamper; the sheet thickness example is a perfect example of where an automated feedback control system creates more problems than it solves in terms of process variability.

Selecting Subgroup Size

A compromise between time / cost to measure and sensitivity of control is the key element of this decision. The most common compromise is 5. Smaller subgroups are acceptable providing that the level of sensitivity is not compromised to too great an extent. If the control chart is sensitive enough to pick up most signals then there is no need to increase the subgroup size. See also comments on rational sub grouping which underpin any comments made there.

Selecting Sampling Regimes

This is very process dependent. You should take into account the rate of change of the process (is it stable like press tools or fast-changing like some machining processes). The faster the process changes the more frequent should be the sampling, this must be balanced against the additional effort required to take samples. Another factor is the number / value of items produced between samples as this is the quantity at risk (and which needs to be inspected if an ‘out of control’ signal is given). A common compromise is one sample per hour.

Always err on the side of too frequent sampling in the early stages and relax this as control is demonstrated by long periods of stability. An important point to note there is that in SPC we deal with random rational subgroups this means that subgroups must be randomly selected from the population and the samples forming the subgroups must be consecutively produced. If our sampling pattern is too regular we run the risk of adversely affecting the randomness of our samples (by aligning with an unknown cyclical factor such as tea breaks etc.).

Always ensure that sampling plans and data collection plans in general are properly documented so that they can be repeated consistently if required.

When to Calculate Control Limits

Control limits are calculated using subgroup data and it is conventional to wait until 20 subgroups have been generated before performing the calculation. Tthis can be done as soon as only 10 subgroups into the chart but the limits are somewhat questionable and should, in any case, be recalculated once 20 subgroups have been produced.

It is also necessary to recalculate limits once a significant positive change in the process has been identified and cemented in by cause analysis or direct action. Do not recalculate limits as a result of negative changes to the process; find out why they happened and remove the cause to restore the process to its original equilibrium position. Whilst it is customary to redraw control limits once a chart has been physically completed this is not necessary and can be counter - productive in masking slow process change over time.

Individual and Moving Range Charts

A wide range of alternative charts are available for a number of different situations. Keller (2001) for example, has a comprehensive list. However, we shall only consider one, the ix / mR chart which, along with standard average and range charts will suice for most situations.

In some circumstances a natural subgroup size of 1 suggests itself. Examples of this are monthly values (e.g. sales figures), periodic measurements from a continuous process or final test values for a series of complex products. In addition to this there may be cost implications to taking larger subgroup sizes, for example in the case of destructive testing. In such cases an individual and moving range chart is used.

Within this chart we can see that the individual measurement corresponds to the average and the moving average to the range. The short term variability is estimated by the moving range which is the positive difference between the current individual plot point and the previous one. Exactly the same logic is applied to these charts as to the average and range, estimates of 3 sigma limits are applied to both charts. the calculations for IX / MR are shown below.


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