While performing a web search, I remembered how difficult the concept of “process stability” can be. How do you know when a process is stable?
D. C. Montgomery, one of the recognized authorities on the subject of statistical process control, seems to give conflicting advice on this. For instance, he’s careful to point out the assumptions underlying all of the measures that one would use on a process, and unstable processes invalidate most or all of these assumptions. How do you know if a process is stable if none of your analyses are applicable?
Process stability needs an operational definition. Luckily, there are at least two:
1) No signals on the appropriate process behavior chart (a.k.a. control chart);
2) Cpk / Ppk == 1 and Cp / Pp == 1
Signals on a process behavior chart do not necessarily mean that a process is out of control (i.e. false signals are possible, and expected at certain mathematically determinable rates), but we can be sure of process stability if there are no signals.
Likewise, we can take issue with using the process capability indices Pp, Ppk, Cp and Cpk in this manner. All assume a normal distribution, which you only get with a stable process, so you shouldn’t trust them as measures of process capability. In this case, that’s fine: don’t report the actual values; just report the ratio of Cp to Pp or Cpk to Ppk. When the ratio is 1, the process is stable; the larger the ratio, the worse the process. Donald Wheeler discusses this use of Ppk and Cpk, and the measures’ relation to production costs, in his latest column for Quality Digest.
Whether or not the process is economical (i.e. Cpk and Ppk are high enough) is a question completely separate from stability.
I was discussing this with a friend who, for various reasons, needs to allow for some process drift. In other words, a Ppk less than Cpk is expected and acceptable, but only up to a certain point. The nice thing about the Cpk/Ppk ratio is that it’s simple: a ratio of 1 means the process is stable; a ratio greater than 1 means the process is not stable; a ratio of less than 1 means someone has made a mistake or is lying. If we need to allow for some process drift, we lose this simplicity.
So suppose that we have a Cpk of 1.66. There are then five standard deviations between the process mean and the nearest specification limit. Assuming a process drift of 1.5 Sigmas, our Ppk is 1.16, giving us a ratio Cpk/Ppk of 1.43. If, however, our Cpk is 1.00, then a process drift of 1.5 Sigmas gives us a Cpk/Ppk ratio of 2.00.
With an allowed process drift of a fixed number of Sigma, it’s no longer so simple to determine, from the Cpk/Ppk ratio, whether or not a process is “stable” within the limits set by management.
A slightly more sophisticated calculation is needed, then. What we can calculate is the ratio
(Short Term Sigma – Long Term Sigma) / Allowed Process Drift
If the result is less than or equal to 1, then the process is “good enough” (i.e. within our allowed drift). If the ratio is greater than 1, then the process is considered out of control and action needs to be taken to eliminate sources of variation. If the ratio is less than 0, then someone made a mistake or is lying (i.e. long-term Sigma can never be less than short-term Sigma).