Process performance index: Difference between revisions

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{{Disputed|date=February 2021}}In [[process improvement]] efforts, the '''process performance index''' is an estimate of the [[process capability]] of a [[Process (engineering)|process]] during its initial set-up, ''before'' it has been brought into a state of [[statistical control]].<ref>{{Citation | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]] | year = 2005 | ___location = [[Hoboken, New Jersey]] | pages = 348–349 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567 | url-status = dead | archiveurl = https://web.archive.org/web/20080620095346/http://www.eas.asu.edu/~masmlab/montgomery/ | archivedate = 2008-06-20 }}</ref>

Formally, if the upper and lower [[Specification (technical standard)|specifications]] of the process are USL and LSL, the estimated mean of the process is <MATH>\hat{\mu}</MATH>, and the estimated variability of the process (expressed as a [[standard deviation]]) is <MATH>\hat{\sigma}</MATH>, then the process performance index is defined as:

:<MATH>\hat{P}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \times \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \times \hat{\sigma}} \Bigg]</MATH>

<MATH>\hat{\sigma}</MATH> is estimated using the [[Unbiased estimation of standard deviation|sample standard deviation]]. P<SUB>pk</SUB> 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, <MATH>\hat{P}_{p,lower} = {\hat{\mu} - LSL \over 3 \times \hat{\sigma}}</MATH>; for those that only have an upper limit, <MATH>\hat{P}_{p,upper} = {USL - \hat{\mu} \over 3 \times \hat{\sigma}}</MATH>.

Practitioners may also encounter <MATH>\hat{P}_{p} = \frac{USL - LSL} {6 \times \hat{\sigma}}</MATH>, a metric that failsdoes tonot account for process performance that is not exactly centered between the specification limits, and therefore is limitedinterpreted inas itswhat the process would be capable of achieving if it could be centered and utilitystabilized.

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

==Interpretation==

Larger values of P<SUB>pk</SUB> may be interpreted to indicate that a process is more capable of producing output within the specification limits, though this interpretation is controversial.{{Citation needed|date=April 2010}} Strictly speaking, from a statistical standpoint, P<SUB>pk</SUB> 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 <MATH>\hat{\mu}</MATH> and <MATH>\hat{\sigma}</MATH>.<ref>{{Citation | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]] | year = 2005 | ___location = [[Hoboken, New Jersey]] | page = 349 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567 | quote = However, please note that if the process is '''not''' in control, the indices P<SUB>p</SUB> and P<SUB>pk</SUB> have no meaningful interpretation relative to process capability, because they cannot predict process performance. | url-status = dead | archiveurl = https://web.archive.org/web/20080620095346/http://www.eas.asu.edu/~masmlab/montgomery/ | archivedate = 2008-06-20 }}</ref>{{Disputed inline|date=February 2021}} Furthermore, using this metric of past process performance to predict future performance is highly suspect.<ref>{{Citation | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]] | year = 2005 | ___location = [[Hoboken, New Jersey]] | page = 349 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567 | quote = Unless the process is stable (in control), no index is going to carry useful predictive information about process capability or convey any information about future performance. | url-status = dead | archiveurl = https://web.archive.org/web/20080620095346/http://www.eas.asu.edu/~masmlab/montgomery/ | archivedate = 2008-06-20 }}</ref>{{Disputed inline|date=February 2021}}

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

Practitioners may also encounter <MATH>\hat{P}_{p} = \frac{USL - LSL} {6 \times \hat{\sigma}}</MATH>, a metric that fails to account for process performance that is not exactly centered between the specification limits, and therefore is limited in its utility.

Once a process is put into a state of statistical control, process capability is described using [[Process capability index|process capability indices]], which are formulaically identical to P<SUB>pk</SUB> (and P<SUB>p</SUB>).{{Disputed inline|date=February 2021}} The indices are named differently in order to call attention to whether the process under study is believed to be in control or not.

==Interpretation==

==Example==

Larger values of P<SUB>pk</SUB> may be interpreted to indicate that a process that is more capable of producing output within the specification limits, though this interpretation is controversial.

Consider a quality characteristic with a target of 100.00&nbsp;[[Micrometre|μm]] and upper and lower specification limits of 106.00&nbsp;μm and 94.00&nbsp;μm, respectively. If, after carefully monitoring the process for a while, it appears that the process is out of control and producing output unpredictably (as depicted in the [[run chart]] below), one can't meaningfully estimate its mean and standard deviation. In the example below, the process mean appears to drift upward, settle for a while, and then drift downward.

[[File:ProcessPerformanceExample.svg]]

Strictly speaking, from a statistical standpoint, P<SUB>pk</SUB> 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 <MATH>\hat{\mu}</MATH> and <MATH>\hat{\sigma}</MATH>. 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<SUB>pk</SUB> is a convenient metric to gauge how set-up is progressing (increasing P<SUB>pk</SUB> being interpreted as "the process capability is improving"). The risk is that P<SUB>pk</SUB> is taken to mean a process is ready for production before all the kinks have been worked out of it.

If <MATH>\hat{\mu}</MATH> and <MATH>\hat{\sigma}</MATH> are estimated to be 99.61&nbsp;μm and 1.84&nbsp;μm, respectively, then

Once a process is put into a state of statistical control, process capability is described using [[Process capability index|process capability indices]], which are formulaically identical to P<SUB>pk</SUB> (and P<SUB>p</SUB>). The indices are named differently to call attention to whether the process under study is believed to be in control or not.

{| class="wikitable"

! Index

|-

| <MATH>\hat{P}_p = \frac{USL - LSL} {6 \hat{\sigma}} = \frac{106.00 - 94.00} {6 \times 1.84} = 1.09</MATH>

|-

| <MATH>\hat{P}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \hat{\sigma}} \Bigg] = \min \Bigg[ {106.00 - 99.61 \over 3 \times 1.84}, { 99.61 - 94 \over 3 \times 1.84} \Bigg] = 1.02</MATH>

|}

That the process mean appears to be unstable is reflected in the relatively low values for P<SUB>p</SUB> and P<SUB>pk</SUB>. The process is producing a significant number of defectives, and, until the [[Common-cause and special-cause|cause]] of the unstable process mean is identified and eliminated, one really can't meaningfully quantify how this process will perform.

==See also==

*[[Process (engineering)]]

*[[Process capability]]

*[[Process capability index]]

==References==

{{Reflist}}

{{DEFAULTSORT:Process Performance Index}}

[[Category:Index numbers]]

[[Category:ProcessStatistical managementprocess control]]

[[Category:Quality control]]