Process capability index


In process improvement efforts, the process capability index or process capability ratio is a statistical measure of process capability: the ability of a process to produce output within specification limits. The concept of process capability only holds meaning for processes that are in a state of statistical control. Process capability indices measure how much "natural variation" a process experiences relative to its specification limits and allows different processes to be compared with respect to how well an organization controls them.
If the upper and lower specification limits of the process are USL and LSL, the target process mean is T, the estimated mean of the process is \hat and the estimated variability of the process is \hat, then commonly accepted process capability indices include:
IndexDescription
\hat_p = \frac Estimates what the process is capable of producing if the process mean were to be centered between the specification limits. Assumes process output is approximately normally distributed.
\hat_ = Estimates process capability for specifications that consist of a lower limit only. Assumes process output is approximately normally distributed.
\hat_ = Estimates process capability for specifications that consist of an upper limit only. Assumes process output is approximately normally distributed.
\hat_ = \min \BiggEstimates what the process is capable of producing, considering that the process mean may not be centered between the specification limits. \hat_ < 0 if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed.
\hat_ = \frac Estimates process capability around a target, T. \hat_ is always greater than zero. Assumes process output is approximately normally distributed. \hat_ is also known as the Taguchi capability index.
\hat_ = \frac Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed.

\hat is estimated using the sample standard deviation.

Recommended values

Process capability indices are constructed to express more desirable capability with increasingly higher values. Values near or below zero indicate processes operating off target or with high variation.
Fixing values for minimum "acceptable" process capability targets is a matter of personal opinion, and what consensus exists varies by industry, facility, and the process under consideration. For example, in the automotive industry, the Automotive Industry Action Group sets forth guidelines in the Production Part Approval Process, 4th edition for recommended Cpk minimum values for critical-to-quality process characteristics. However, these criteria are debatable and several processes may not be evaluated for capability just because they have not properly been assessed.
Since the process capability is a function of the specification, the Process Capability Index is only as good as the specification. For instance, if the specification came from an engineering guideline without considering the function and criticality of the part, a discussion around process capability is useless, and would have more benefits if focused on what are the real risks of having a part borderline out of specification. The loss function of Taguchi better illustrates this concept.
At least one academic expert recommends the following:
SituationRecommended minimum process capability for two-sided specificationsRecommended minimum process capability for one-sided specification
Existing process1.331.25
New process1.501.45
Safety or critical parameter for existing process1.501.45
Safety or critical parameter for new process1.671.60
Six Sigma quality process2.002.00

However where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive.

Relationship to measures of process fallout

The mapping from process capability indices, such as Cpk, to measures of process fallout is straightforward. Process fallout quantifies how many defects a process produces and is measured by DPMO or PPM. Process yield is the complement of process fallout and is approximately equal to the area under the probability density function \Phi = \frac \int_^\sigma e^ \, dt if the process output is approximately normally distributed.
In the short term, the relationships are:
CpkSigma level Area under the
probability density function		Process yieldProcess fallout
0.3310.682689492168.27%317311
0.6720.954499736195.45%45500
1.0030.997300203999.73%2700
1.3340.999936657599.99%63
1.6750.999999426799.9999%1
2.0060.999999998099.9999998%0.002

In the long term, processes can shift or drift significantly. If there was a 1.5 sigma shift 1.5σ off of target in the processes, it would then produce these relationships:
CpkAdjusted
Sigma level 		Area under the
probability density function		Process yieldProcess fallout
0.3310.308537538730.85%691462
0.6720.691462461369.15%308538
1.0030.933192798793.32%66807
1.3340.993790334799.38%6209
1.6750.999767370999.9767%232.6
2.0060.999996602399.99966%3.40

Because processes can shift or drift significantly long term, each process would have a unique sigma shift value, thus process capability indices are less applicable as they require statistical control.

Example

Consider a quality characteristic with target of 100.00 μm and upper and lower specification limits of 106.00 μm and 94.00 μm respectively. If, after carefully monitoring the process for a while, it appears that the process is in control and producing output predictably, we can meaningfully estimate its mean and standard deviation.
If \hat and \hat are estimated to be 98.94 μm and 1.03 μm, respectively, then
Index
\hat_p = \frac = \frac = 1.94
\hat_ = \min \Bigg = \min \Bigg = 1.60
\hat_ = \frac = \frac = 1.35
\hat_ = \frac = \frac = 1.11

The fact that the process is running off-center is reflected in the markedly different values for Cp, Cpk, Cpm, and Cpkm.