Cpk, Ppk and Tolcap: Why Tolcap predicts Cpk

18th May 2017

Tolcap enables Designers to predict the process capability of parts even before prototype samples are produced. Tolcap predicts Cpk - but what is Cpk, and is it the most appropriate process capability index to predict?

The simplest process capability index: Cp.

The simplest process capability index, cp, illustration

Recall Control Charts. To set up a control chart for a dimension on a part the dimension is measured on a sample batch of the parts. The mean (average) and standard deviation, or “sigma” of the measurements are calculated. The action limits on the Control Chart are then conventionally set at plus and minus three sigma from the mean.

To measure process capability, an obvious approach was to compare the tolerance spread with the control chart action limits. Cp does just that:
Cp = (Upper Tolerance Limit - Lower Tolerance Limit) / (Upper Action Limit - Lower Action Limit)
 = (Upper Tolerance Limit - Lower Tolerance Limit) / 6σ

Process Shift and Cpk

One problem with Cp is that it assumes the mean of the distribution of the parts is at the centre of the tolerance band. Real life processes don’t behave that way. Consider a couple of examples:
1) Turning & Boring. For each batch or production run, a setter installs and adjusts the cutting tools to produce the required dimension. There is a limit to the precision the setter can achieve in centring the distribution of the parts around the nominal dimension between the limits.
2) Plastic Injection Moulding. As the tool is used it will wear. The toolmaker will make the tool with this in mind: splines on the tool that produce grooves in the part will wear away and the grooves in the part will get smaller as the tool wears - and vice versa. So the toolmaker makes the splines on the tool oversize to start, to maximise tool life.

For these reasons, process capability is measured using the index Cpk:

Process capability index, cpk, illustration

Cpk is calculated by working out which is the nearer limit to the mean of the measured sample. If the mean is nearer the lower limit (as in the diagram), then
Cpk = (Mean - LowerTolerance Limit) / 3σ
But if the mean is nearer the Upper Tolerance Limit, then
Cpk = (UpperTolerance Limit - Mean ) / 3σ
Of course if the mean actually does coincide with the centre of the distribution, then Cpk = Cp, but in general it is smaller.

Cpk or Ppk?

In Process Capability (Cp, Cpk) and Process Performance (Pp, Ppk). The formulae for Cpk and Ppk are exactly the same, but the difference is how sigma is calculated. For Ppk, data is collected for the whole history of the part, whereas for Cpk provides a “snapshot” using data from one batch or run of the process.

So as a designer, which would you like to know in advance: Cpk or Ppk? Well obviously both of them! OK, which is more useful? Design for Six Sigma recognises process shift - and also tells us that variation (sigma) is what we have to ensure we have in control, because there is usually something we can do about process shift. Going back to the Turning and Boring example, if the average dimension is too small, we call the setter back. In the case of Injection Moulding, we call in the toolmaker to adjust or refurbish the die - or eventually we find and accept it’s time to replace the tooling. That 1.5σ process shift by the way comes from the idea of maybe having to keep a control chart to monitor process shift: The action limits on the mean on an Xbar-R chart with four samples come out at +/- 1.5σ. So on this basis, Tolcap predicts Cpk rather than Ppk.

Ppk for Ford PPAP

Tolcap acknowledges the Process design of the Ford PPAP (Pre-production Part Approval) process. Ford require suppliers to ensure their parts will prove process capable in series production at Cpk=1.33. That is why Tolcap’s default acceptable Cpk target is 1.33. In support of this requirement, Ford requires suppliers to demonstrate Ppk=1.67 on the final pre-production parts manufactured on the production line process. Ppk? It is perhaps easy to dodge an explanation by saying that here “P” stands for “Preliminary”, but is it actually Process Performance they mean?

If the pre-production parts are all from one batch, it makes no difference: the batch is the entire history anyway. But there may be different batches. In this case it doesn’t seem unreasonable that the supplier demonstrates the process will consistently be well within limits, even if the mean has to be re-adjusted, and there is a margin to allow for variation of materials and conditions once full scale production is under way.

Written by:

Richard Batchelor MA, MBA, CEng, FIEE

Richard Batchelor MA, MBA, CEng, FIEE

Richard is a founding member of the Capra Technology team.