Process capability

Process capability
See also
Downside deviation Interval scale Evaluation criteria Quality loss function Training objective Cronbach Alpha Fitness for use Performance specification Parametric analysis

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.

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 .

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:

• C_p : $C_{p}={\frac {USL-LSL}{6\sigma }}$

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_pk : $C_{pk}=min[{\frac {USL-\mu }{3\sigma }},{\frac {\mu -LSL}{3\sigma }}]$

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_pm : $C_{pm}={\frac {C_{p}}{\sqrt {1-({\frac {\mu -T}{\sigma }})^{2}}}}$

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_pmk : $C_{pmk}={\frac {C_{pk}}{\sqrt {1-({\frac {\mu -T}{\sigma }})^{2}}}}$

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_{p lower} : $C_{pu}={\frac {\mu -LSL}{3\sigma }}$

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

• C_{p upper} : $C_{pl}={\frac {USL-\mu }{3\sigma }}$

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