As the bumper sticker says, “Shifts Happen” and so we generally want to use the slightly more liberal process capability of Cpk ≥ 1.33.
In my last column, I was asked about a capability index called Cpk. Cpk is a measure of a process’ ability to conform to specification. A Cpk = 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 Cp = Cpk = Cpm = 1
But as the bumper sticker says, “Shifts Happen” and so we generally like to see a process capability of Cpk ≥ 1.33 as in Figure 2.
Figure 2 – A process with a Cp = Cpk = Cpm = 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: Cp = 1.33 Cpk = 1 Cpm = 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 Cpm . And to understand Cpm, 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 Cpk 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 Cp = 3 Cpk = 1.33 Cpm = 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 Cpk = 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 Cpk if your process went out of spec, but one that also penalized you if your output was off target. This index is called Cpm 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, Cpm gives you the same number as Cp and Cpk . But as your process average moves off of the target, you start incurring a greater and greater penalty in Cpm.
Now go back and take a look at the effect changes in the process have on Cpm . We still want a Cp = Cpk = Cpm ≥ 1.33, so you can see which process meets that criterion. Overall, Cpm 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.
Click here to learn how to customize WinSPC to display the Cpm index.
How to Display the Cpm Index in WinSPC
The Cpm capability index can be viewed in a few different places within WinSPC. The Plant Monitor and Capability Report automatically display the value. In the Data Collection and Query Tool screens, Cpm is viewed by changing the status bar properties.
The status bar is the located at the very bottom of the WinSPC screen.
- You may change the status bar properties by right-clicking on the status bar and selecting Properties. This opens the Status Bar Properties dialog box.
- Under Statistics Based On Samples, check the Cpm checkbox and click OK.
Note: You may view the formula used to derive Cpm, or any other statistic, by rolling your mouse over the item.
The Cpm statistic is now displayed in the status bar.
The Cpm capability index can also be viewed in the Plant Monitor and Capability Report.
To review statistics in the Plant Monitor, simply roll your mouse over a colored cell and the information is displayed.
In the Capability Report, Cpm and all other statistics are displayed when the report is run…