Process Capability Index (Cpk)

Cpk fixes what Cp ignores by measuring only to the nearer specification limit: $\min[(\mathit{USL}-\mu)/3\sigma, (\mu-\mathit{LSL})/3\sigma]$. Take the process from the Cp example, tolerance 9.500 to 10.500 with $\sigma = 0.125$, and run it at 10.100 instead of 10.000. The upper side gives $0.400/0.375 = 1.07$ and the lower gives $0.600/0.375 = 1.60$, so Cpk is 1.07 against a Cp of 1.33. That quarter of an index point is pure off-centring, and no amount of variation reduction recovers it.

Cpk can never exceed Cp, and the two are equal only when the mean sits dead centre. Reading them together is the whole diagnosis. Cp high and Cpk low means the process is capable but mis-centred, which is usually a machine adjustment and a good afternoon's work. Cp and Cpk both low means the process is too variable for the tolerance, which is a much longer project. Reporting Cpk alone throws away the half of the message that tells you which situation you are in.

The solver offers the inverses with a warning attached, because the minimum is a many-to-one function. Two different means give the same Cpk, one on each side of centre, so solving for $\mu$ returns the upper branch and you mirror it about the middle of the band for the other. Solving for a specification limit has the same problem: Cpk pins down only the nearer limit and says nothing about the far one. Solving for $\sigma$ is the one clean inverse, because sigma scales both sides equally and never changes which one is binding.