# Process capability

Process capability | |
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**Process capability** is combined by two words. Process is "one or more actions or operations planned, executed and evaluated to achieve a stated goal" and capability is "the degree to which the process achieves the stated goal". So process capability is statistical measurable property of a process to the given specification^{[1]}.

## Measuring Process Capability

**Voice of the process (VOP)** is random behavior of process. **Voice of the customer (VOC)** is what customers need and what they expectate for the result of the process. VOP and VOC speak in other languages. VOP knows what it produces and this is random variation. VOC consists of histograms, data and probability models which predict future behavior of process. Process capability tools interpret VOP and VOC. Process capability study estimates variation of a process and decide if it is capable of meet the targets specifed by customer. To set compartment we use LSL and USL. **LSL** is Lower Specification Limit. **USL** is Upper Specification Limit. When random variation of process is contained in this compartment we can say process is capable ^{[2]}.

## Process Capability Indices

Measures of process capability is Process Capability Indices (**PCIs**). Now popular is **capability ratio (CR)**. To calculate CR we just need to calculate C_{p}. Estimate C_{p} to show where process be centered between specification. We can also calculate other indicators. For example, estimate process capability with the lower limit only or upper limit only, process capability around a target (T). C_{pk} is centralized indicator, but both of them shows how close you are to your average performance. Process capability is strong related with Process Performance^{[3]}:

**C**\[C_p=\frac{USL-LSL}{6\sigma}\]_{p}

Generally accepted minimum value for C_{p} is 1, but expected is 1,33. Larger value is better. Then we can say process is capable. At the value 1 process yield is 99,7%, but **at the value 1,33 process yield is 99,99%**. For example, at the value 0,5 process yield is only 86,8%. We must remember that C_{p} can be not centered between users requirements.

**C**\[C_{pk}=min[\frac{USL-\mu}{3\sigma},\frac{\mu-LSL}{3\sigma}]\]_{pk}

Interpretion for C_{pk} is such as for C_{p}, but C_{pk} is centered indicator. When C_{pk} is 1,33 or more is capable and meets specification limits.

**C**\[C_{pm}=\frac{C_p}{\sqrt{1-(\frac{\mu-T}{\sigma})^2}}\]_{pm}

C_{pm} at least 1 is good and 1,33 expected. Is related to C_{p}. It shows potential gain to be obtained by move mean closer to target. Target can be not centered between users specifications.

**C**\[C_{pmk}=\frac{C_{pk}}{\sqrt{1-(\frac{\mu-T}{\sigma})^2}}\]_{pmk}

C_{pmk} at least 1 is good and 1,33 expected. Is related to C_{pk}. Interpretation is such as for C_{pm}, but with the difference C_{pmk} is centered between users requirements.

**C**\[C_{pu}=\frac{\mu-LSL}{3\sigma}\]_{p lower}

C_{p lower} etimates capability for requirements that consist with lower limit only. With C_{p upper} is a component of C_{pk}.

**C**\[C_{pl}=\frac{USL-\mu}{3\sigma}\]_{p upper}

C_{p upper} etimates capability for requirements that consist with upper limit only. With C_{p lower} is a component of C_{pk}.

Where:

- LSL - Lower Specification Limit
- USL - Upper Specification Limit
- \(\sigma\) - standard deviation
- \(\mu\) - mean

Assuming that the distribution is approximately normally distributed.

## Footnotes

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## References

- Hawley G. O., (1962).
*Reliability Through Process Capability Studies*, Sandia Corporation, p. 1-5 - Kortz S. and others, (1993).
*Process Capability Indices*, CRC Press - Pearn W. L. and others, (2006).
*Encyclopedia And Handbook Of Process Capability Indices: A Comprehensive Exposition Of Quality Control Measures*, World Scientific - Polhemus N. W., (2017).
*Process Capability Analysis: Estimating Quality*, CRC Press - Relyea D. B., (2011).
*The Practical Application of the Process Capability Study: Evolving From Product Control to Process Control*, CRC Press - Sleeper A., (2005).
*Design for Six Sigma Statistics, Chapter 6 - Measuring Process Capability*, McGraw Hill Professional, p. 319-325.

**Author:** Adam Widła