Process performance index: Difference between revisions

Revision as of 20:22, 1 April 2010

Revision as of 07:48, 4 April 2010

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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]].<REFref>{{Citation | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]] | dateyear = 2005 | ___location = [[Hoboken, New Jersey]] | pages = 348 – 349 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 9780471656319 | oclc = 56729567}}</REFref>

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:

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<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 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.

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==Interpretation==

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.{{cnCitation 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>.<REFref>{{Citation | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]] | dateyear = 2005 | ___location = [[Hoboken, New Jersey]] | page = 349 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 9780471656319 | 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.}}</REFref> Furthermore, using this metric of past process performance to predict future performance is highly suspect.<REFref>{{Citation | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]] | dateyear = 2005 | ___location = [[Hoboken, New Jersey]] | page = 349 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 9780471656319 | 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.}}</REFref>

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.