Question

A process which is in a state of statistical control (within ± 3σ) has an estimate of standard deviation (σ) 2 mm. The specification limits for the corresponding product are 120 ± 8 mm. When process mean shifts from 118 mm to 122 mm with no change in process standard deviation, the difference in process capability index Cpk is ______

Answer 1

Concept:

Process capability index(\(C_{pk}\)): It is a statistical tool, to measure the ability of a process to produce output within customer's specification limits

\({{\rm{C}}_{{\rm{pk}}}}{\rm{\;}} = {\rm{min\;}}\left[ {\frac{{{\rm{USL\;}} - {\rm{\;μ }}}}{{3{\rm{σ }}}},{\rm{\;}}\frac{{{\rm{μ \;}} - {\rm{\;LSL}}}}{{3{\rm{σ }}}}} \right]\)

where, USL = upper specification limits, LSL = lower specification limit, σ = standard deviation, μ = process mean

Calculation:

Given:

USL = 120 + 8 = 128 mm, LSL = 120 - 8 = 112 mm, σ = 2 mm

(i) When μ = 118 mm

\({{\rm{C}}_{{\rm{pk}}}}{\rm{\;}} = {\rm{min\;}}\left[ {\frac{{{\rm{128\;}} - {\rm{\;118 }}}}{{3\ \times\ {\rm{2 }}}},{\rm{\;}}\frac{{{\rm{118 \;}} - {\rm{\;112}}}}{{3\ \times\ {\rm{2 }}}}} \right]\)

\(C_{pk}\) = min [1.67, 1] = 1

(ii) When μ = 122 mm

\({{\rm{C}}_{{\rm{pk}}}}{\rm{\;}} = {\rm{min\;}}\left[ {\frac{{{\rm{128\;}} - {\rm{\;122 }}}}{{3\ \times\ {\rm{2 }}}},{\rm{\;}}\frac{{{\rm{122 \;}} - {\rm{\;112}}}}{{3\ \times\ {\rm{2 }}}}} \right]\)

\(C_{pk}\) = min [1, 1.6671] = 1

∴ The difference in process capability is 1 - 1 = 0