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16.0 Process Capability Analysis

In some applications, there are specific requirements associated with the performance measurement of interest that must be met for the process to be operating in an acceptable manner. These requirements are usually described in terms specification limits or standards.

The following three process capability parameters are required to assess process capability with respect to the specification limits.

In non-technical terms a process is capable if it consistently produces the performance measurement on target within the specification limits.

16.1 Capability Indices

The following three “capability indices” are measures of the conformance of the performance measurement to the target value and specification limits.

Pp=(USL−LSL)6σ(16-1)P_{p} = \frac{(USL - LSL)}{6\sigma} \tag{16-1}

Equation 16-1. The PP Capability Index

Ppl=(μ−LSL)3σ(16-2)P_{pl} = \frac{(\mu - LSL)}{3\sigma} \tag{16-2}

Equation 16-2. The Ppl Capability Index

Ppu=(USL−μ)3σ(16-3)P_{pu} = \frac{(USL - \mu)}{3\sigma} \tag{16-3}

Equation 16-3. The Ppu Capability Index

Values for these indices below 1 indicate that the process is not capable of producing the performance measurement within the specification limits. Typically values greater than 1.33 indicate that the process is currently capable of meeting the specification limits.

16.2 Estimation of the Capability Indices

Recall that if {Y1Y_{1}, Y2Y_{2}, … ,YNY_{N}} is a random sample drawn from a N(μ, σ2)N(\mu,\ \sigma^{2}) distribution, then the expected value for the sample standard deviation is

μS=αNσ\mu_{S} = \alpha_{N}\sigma

So an unbiased estimate for σ is given by

σ^=SαN∙\widehat{\sigma} = \frac{S}{\alpha_{N}} \bullet

The process capability indices are estimated as follows where Y‾\overline{Y} is the overall average for the performance measurement.

P^p=(USL−LSL)6σ^(16-4){\widehat{P}}_{p} = \frac{(USL - LSL)}{6\widehat{\sigma}} \tag{16-4}

Equation 16-4. Estimated Capability Index

P^pl=(Y‾‾..−LSL)3σ^(16-5){\widehat{P}}_{pl} = \frac{\left( {\overline{\overline{Y}}}_{..} - LSL \right)}{3\widehat{\sigma}} \tag{16-5}

Equation 16-5. Estimated Capability Index

P^pu=(USL−Y..‾‾)3σ^(16-6){\widehat{P}}_{pu} = \frac{\left( USL - \overline{\overline{Y_{..}}} \right)}{3\widehat{\sigma}} \tag{16-6}

Equation 16-6. Estimated Capability Index

16.3 The Percent of Product Outside of the Specification Limits

The percent of process output that is outside of the specification limits is calculated directly using the following formulas.

Percent Below the LSL=(Number of Units in the Sample Below the LSLTotal Sample Size−N)x100(16-7)Percent\ Below\ the\ LSL = \left( \frac{Number\ of\ Units\ in\ the\ Sample\ Below\ the\ LSL}{Total\ Sample\ Size - N} \right)x100 \tag{16-7}

Equation 16-7. Percent of Process Output Below the Lower Specification Limit

Percent Above the USL=(Number of Units in the Sample Above the USLTotal Sample Size−N)x100(16-8)Percent\ Above\ the\ USL = \left( \frac{Number\ of\ Units\ in\ the\ Sample\ Above\ the\ USL}{Total\ Sample\ Size - N} \right)x100 \tag{16-8}

Equation 16-8. Percent of Process Output Above the Upper Specification Limit

Percent  Outside the Specification LimitsPercent\ \ Outside\ the\ Specification\ Limits
=(Number of Units in the Sample Outside of the Specification LimitsTotal Sample Size−N)x100(16-9)= \left( \frac{Number\ of\ Units\ in\ the\ Sample\ Outside\ of\ the\ Specification\ Limits}{Total\ Sample\ Size - N} \right)x100 \tag{16-9}

Equation 16-9. Percent of Process Output Outside of the Specification Limits

The results of the process capability analyses are presented in Figure 16-1.

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Figure 16-1. The Current Process Capability Analysis

16.4 The Potential Process Capability Analysis

If the analysis indicates that the variation in the performance measurement is significantly affected by assignable causes associated with the process design conditions or production time, then the process capability could be improved by removing those causes of variation from the process. In that case, the distribution of the performance measurement would be the same as the R2 residuals centered at the target value, and this would be the “potential capability” of the process.

The analysis of the potential process capability is useful because it provides information on the “return on investment” made to remove assignable causes discovered in the analyses. If the “potential capability” is significantly better than the current capability, then it may be economically reasonable to invest in actions to remove the assignable causes.[48]

The potential capability analyses are conducted as described above, but replacing the original performance measurement data with the R2 residuals centered at the target value.

16.4.1 The Potential Capability Indices

The standard deviation for the R2 residuals is denoted by σR2\sigma_{R2}, and this standard deviation is estimated by

σ^R2=SR2αN(16-10){\widehat{\sigma}}_{R2} = \frac{S_{R2}}{\alpha_{N}} \tag{16-10}

Equation 16-10. Potential Process Standard Deviation

where SR2S_{R2} is the sample standard deviation for the R2 residuals and N is the total samples size for the R2 residuals. The potential process capability indices are estimated as follows.

C^p=(USL−LSL)6σ^R2(16-11){\widehat{C}}_{p} = \frac{(USL - LSL)}{6{\widehat{\sigma}}_{R2}} \tag{16-11}

Equation 16-11. Estimated Potential Process Capability Index Cp

C^pl=(Y‾‾..−LSL)3σ^R2(16-12){\widehat{C}}_{pl} = \frac{\left( {\overline{\overline{Y}}}_{..} - LSL \right)}{3{\widehat{\sigma}}_{R2}} \tag{16-12}

Equation 16-12. Estimated Potential Process Capability Index Cpl

C^pu=(USL−Y‾‾..)3σ^R2(16-13){\widehat{C}}_{pu} = \frac{\left( USL - {\overline{\overline{Y}}}_{..} \right)}{3{\widehat{\sigma}}_{R2}} \tag{16-13}

Equation 16-13. Estimated Potential Process Capability Index Cpu

16.5 The Potential Percent of Product Outside of the Specification Limits

The potential percent of process output that is outside of the specification limits is calculated directly using the following formulas.

Potential Percent Below the LSL=(Number of Centered R2 Residual Below the LSLTotal Sample Size−N)x100(16-14)Potential\ Percent\ Below\ the\ LSL = \left( \frac{Number\ of\ Centered\ R2\ Residual\ Below\ the\ LSL}{Total\ Sample\ Size - N} \right)x100 \tag{16-14}

Equation 16-14. Potential Percent of Process Output Below the Lower Specification Limit

Potential Percent Above the USL=(Number of Centered R2 Residuals Above the USLTotal Sample Size−N)x100(16-15)Potential\ Percent\ Above\ the\ USL = \left( \frac{Number\ of\ Centered\ R2\ Residuals\ Above\ the\ USL}{Total\ Sample\ Size - N} \right)x100 \tag{16-15}

Equation 16-15. Potential Percent of Process Output Above the Upper Specification Limit

Potential Percent  Outside the Specification LimitsPotential\ Percent\ \ Outside\ the\ Specification\ Limits
=(Number of Centered R2 Residuals Outside of the Specification LimitsTotal Sample Size−N)x100(16-16)= \left( \frac{Number\ of\ Centered\ R2\ Residuals\ Outside\ of\ the\ Specification\ Limits}{Total\ Sample\ Size - N} \right)x100 \tag{16-16}

Equation 16-16. Potential Percent of Process Output Outside of the Specification Limits

The results of the potential process capability analyses are presented in Figure 16-2.

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Figure 16-2. The Potential Process Capability Analysis

Footnotes

  1. The economic analysis is part of the problem formulation.

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Part VII. The VAS Analytic System
15.0 Taguchi Loss Function Analysis
Part VII. The VAS Analytic System
17.0 The Test for ‘Maximum Information”