Average Run Length (ARL) On A Control Chart

Dr Juergen Ude Dr Juergen Ude

This article will discuss the concept of average run lengths as applies to Statistical Process Control Charts.

For effective process control we need a control chart which is sufficiently sensitive to rapidly detect a change in the process average, whilst rarely giving false alarm signals.

A measure of sensitivity is the average run length (ARL). The average run length is simply the average number of inspections required before an out-of-control point is reached. Consistent with the above, the ARL for an in-control process should be large and for an out-of-control process small.

SPC as conventionally taught often does not take ARLs into account. Normally a sub-group of say five is chosen arbitrarily and control limits set at the average or minus three times the standard deviation divided by the square root of the sample size.

This may however not provide a sufficiently sensitive control chart. To detect a one standard error range in the average could take on average 44 inspections to detect. In this particular instance this is equivalent to .45 standard deviation change. .45 may not seem much, but if the standard deviation is large then this can prove costly in some applications. A classical example is weight control in the confectionery industry. A .5 standard deviation could represent several percent of a chocolate bar and if a process shift of this magnitude is undetected then several hundred thousand dollars of confectionery may be given away.

A two standard error change will on average take six inspections to detect. This is equivalent to approximately a one standard deviation change if the subgroup size is equal to five.

Of more concern is that in the real world the process average is often not stable i.e. there is inherent time to time variation. Adjusting sample size may then have little effect. It may therefore, on average, take up to 44 inspections to detect a one standard deviation change. Alternatively, it may take approximately 6 inspections to detect a two standard deviation change. Actual run lengths as large as 100 and 20, respectively are also likely to occur from time to time.

Considering that actual inspection frequencies may be as low as one/hour (even though usually specified at one/half hour) considerable off-centre product may be produced before being detected. For example, a whole days production may pass before a two standard deviation change is detected.

It is therefore unwise to assume that just because SPC has been applied, product received or manufactured will have been tightly controlled. If tight control is important then more sensitive control charts may required. The required sensitivity is an economic decision and in fact there are algorithms available that can for some applications calculate “optimum” parameters for the control chart, including inspection frequency.

Alternatively, if the within sample variation is negligible then the sample size and control limits can be sortably adjusted to achieve the required degree of sensitivity by use of a suitable ARL table.

If there is considerable between sample variation or if individual charts are used, then a more sensitive control chart will be required. An example is the Cusum chart, but users are warned that there may be an ambiguity in the theory. My own personal favorite is the exponential control chart, whose sensitivity is comparable to the Cusum chart.

Sensitively of control charts is often enhanced by using warning limits and supplementary rules. These are typically applied to conventional three sigma charts without account of the effect on the average run length on the in-control process.

An alternative to using more sensitive control charts is to simply increase the inspection frequency.

To conclude this introduction to average run lengths, setting up SPC should not simply involve taking an arbitrary sub-group of five and then using three sigma control limits. The required sensitivity must be taken into account. Setting up a control chart should therefore, amongst other considerations, involve a sensitivity analysis to determine the effect of a departure from the target value and economic calculations to determine the best control chart parameters.