Process performance index

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In process improvement efforts, the process performance index is an estimate of the process capability of a process during its initial set-up, before it has been brought into a state of statistical control.

Formally, if the upper and lower specifications of the process are USL and LSL, the estimated mean of the process is $\hat{\mu}$ , and the estimated variability of the process is $\hat{\sigma}$ , then the process performance index is defined as:

$\hat{P}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \times \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \times \hat{\sigma}} \Bigg]$

$\hat{\sigma}$ is estimated using the sample standard deviation. P_pk may be negative if the process mean falls outside the specification limits (because the process is producing a large proportion of defective output).

Some specifications may only be one sided (for example, strength). For specifications that only have a lower limit, $\hat{P}_{p,lower} = {\hat{\mu} - LSL \over 3 \times \hat{\sigma}}$ ; for those that only have an upper limit, $\hat{P}_{p,upper} = {USL - \hat{\mu} \over 3 \times \hat{\sigma}}$ .

Practitioners may also encounter $\hat{P}_{p} = \frac{USL - LSL} {6 \times \hat{\sigma}}$ , a metric that does not account for process performance that is not exactly centered between the specification limits, and therefore is interpreted as what the process would be capable of achieving if it could be centered and stabilized.

[edit] Interpretation

Larger values of P_pk may be interpreted to indicate that a process that is more capable of producing output within the specification limits, though this interpretation is controversial.

Strictly speaking, from a statistical standpoint, P_pk is meaningless if the process under study is not in control because one cannot reliably estimate the process underlying probability distribution, let alone parameters like $\hat{\mu}$ and $\hat{\sigma}$ . Furthermore, using this metric of past process performance to predict future performance is highly suspect.

From a management standpoint, when an organization is under pressure to set up a new process quickly and economically, P_pk is a convenient metric to gauge how set-up is progressing (increasing P_pk being interpreted as "the process capability is improving"). The risk is that P_pk is taken to mean a process is ready for production before all the kinks have been worked out of it.

Once a process is put into a state of statistical control, process capability is described using process capability indices, which are formulaically identical to P_pk (and P_p). The indices are named differently to call attention to whether the process under study is believed to be in control or not.