What is the Cpm index and how does it relate to Cpk?

As the bumper sticker says, “Shifts Happen” and so we generally want to use the slightly more liberal process capability of C_pk ≥ 1.33.

In my last column, I was asked about a capability index called C_pk. C_pk is a measure of a process’ ability to conform to specification. A C_pk = 1 means that if nothing changes, 99.73% of the process output will be within specification, as in Figure 1.

Figure 1 – A process with C_p = C_pk = C_pm = 1

But as the bumper sticker says, “Shifts Happen” and so we generally like to see a process capability of C_pk ≥ 1.33 as in Figure 2.

Figure 2 – A process with a C_p = C_pk = C_pm = 1.33

With this capability, a moderately large unexpected event (a shift of one standard deviation as in Figure 3) can happen and we have time to fix it before we start having a lot of parts fall out of specification.

Figure 3 – The Figure 2 process shifted by one standard deviation: C_p = 1.33 C_pk = 1 C_pm = 0.940

But I left you hanging when I mentioned that conformance to specification is not the whole picture. Depending on your industry, you may have heard of another capability index called C_pm . And to understand C_pm, you need to know how to define quality.

In the really old days (1950-1970), we talked about quality in terms of acceptable levels of defects: in spec was “good quality” and out of spec, “bad quality.” MIL-STD-105 told us how many defective parts we could knowingly ship to the military, and since the standard was adopted widely in industry, it told you how many defective parts you could knowingly ship to pretty much everyone! Defective parts were considered part of the process.

Until the day when it became painfully obvious to industry that defective product was not in fact inevitable. Suddenly our customers started demanding that ALL their parts be within specification.

At about the same time, we began to understand that the costs associated with product or service variation within the specification was not equal. Parts that are closer to the upper spec limit or lower spec limit cost us, and our customers, more money to manufacture and use. Parts which are closer to the customer’s desired target or nominal cost less to manufacture and our customers less to use.

Since we make the same revenue on parts that are barely in spec as we do parts that are on target, parts that are near target are more profitable for us and our customer. Therefore we changed the definition of quality to “good quality is conformance to target or nominal in the absence of defects.” Now a part that was not in specification did not possess any type of quality – it was simply unacceptable. A part that was barely in spec had poor quality and a part that was near the target had high quality.

A good estimate of the costs wasted by the process due to deviation from target is a parabola, as proposed by Genichi Taguchi. The costs have a minimum at the target, are pretty small around the target, and start increasing rapidly as we approach the specification limits.

Figure 4 – The Taguchi Loss Function

This was a huge change in thinking. Since C_pk measured conformance to specification, it no longer described all that we wanted from our processes. For example, we might have a process that produces very consistent output, but is off target, as in Figure 5.

Figure 5 – A low-variability, off-target process with a C_p = 3 C_pk = 1.33 C_pm = 0.588

If high quality is defined as on or near target, the chances of the customer getting high quality is essentially zero, even with a C_pk = 1.33. At least with Figure 3, some customers get what they want, even though the variation and probability of going out of spec are higher.

So another capability index was proposed based on the Taguchi Loss Function, one that penalized you like C_pk if your process went out of spec, but one that also penalized you if your output was off target. This index is called C_pm and is calculated this way:

Where µ is the process average and σ is the standard deviation. You can see how if you are on target, C_pm gives you the same number as C_p and C_pk . But as your process average moves off of the target, you start incurring a greater and greater penalty in C_pm.

Now go back and take a look at the effect changes in the process have on C_pm . We still want a C_p = C_pk = C_pm ≥ 1.33, so you can see which process meets that criterion. Overall, C_pm is the best indicator of how your customers experience the quality of your product or service, though you get valuable process information from all these indices.

Don’t forget – capability indices like these are only meaningful if the process is in control and the specification limits and target are based on the customer needs.