Capability and Calibration Pressure on a Shaft Line
A CNC turning cell machines a hydraulic shaft to 40.000/39.975 mm. Over the last 200 pieces the mean is 39.993 mm with σ = 0.0030 mm, and the drawing requires Cpk ≥ 1.33. At commissioning the cell had Cp = 1.39. The tool-wear offset is set once at the start of each 8-hour shift and is not touched again; the operator says the size “climbs” by about 0.004 mm across a shift. The customer has already reported two shafts at 40.002 mm. The production manager wants to keep shipping from the finished-goods bin after “a quick 100% check with the old air gauge”, which is overdue for calibration, and asks the shift supervisor to sign the inspection report as if the gauge were current.
1. Calculate Cp and Cpk, estimate the expected fraction outside specification, and explain what the two indices say about the process.
Read the worked answer
Cp = tolerance width/(6σ) = 0.025/(6 × 0.0030) = 0.025/0.018 = 1.39. This matches the commissioning value and shows that the inherent spread is capable of meeting 1.33 if the mean is centred.
Cpk = min(USL − x̄, x̄ − LSL)/(3σ) = min(40.000 − 39.993, 39.993 − 39.975)/(0.009) = min(0.007, 0.018)/0.009 = 0.78. The process is off-centre towards the upper limit, so its actual capability is only 0.78 - well below 1.33. The expected oversize fraction is the normal tail above USL: z = 0.007/0.0030 = 2.33, giving about 1%, or roughly 50 parts per 5,000. The lower tail is negligible (z = 6).
The two indices together say: do not scrap the process, centre it. But centring alone is not sufficient because of drift. If the mean drifts 0.004 mm over a shift, the total requirement is 8σ + drift ≈ 0.028 mm, which exceeds the 0.025 mm tolerance. To hold Cpk 1.33 at both ends of a shift, σ must be reduced to about (0.025 − 0.004)/8 ≈ 0.0026 mm, or the drift must be compensated in-process so the effective drift band is about 0.001 mm.
2. How should the supervisor respond to the request to use the old gauge and sign the report?
Read the worked answer
The supervisor should refuse to sign. An inspection record signed against an out-of-calibration gauge is not evidence of conformance; it is a false record, regardless of whether the gauge happens to be reading correctly. The old air gauge may have bias or wear that is unknown, and its calibration status is precisely what the customer and the quality system rely on. Signing “as if current” would be falsification, a breach of professional ethics, and potentially a safety issue because the shaft is a hydraulic component.
The correct response is to quarantine product since the last valid calibration or last verified check, sort it with a calibrated gauge after a gauge R&R study, and document the containment. The customer should be notified of the escape and the containment, with an 8D or equivalent corrective-action report. Production pressure does not change the validity of the measurement. If a temporary check is needed to keep the line running, it can be used for screening only, clearly labelled as non-conformance screening, with final release based on calibrated inspection.
3. Outline the process changes needed to reach and sustain Cpk ≥ 1.33.
Read the worked answer
First, centre the process: with σ = 0.0030, a mean of about 39.9875 gives Cpk 1.39, so the tool offset should be set to bring the start-of-shift mean to about 39.985–39.987, allowing for the expected climb. Second, control the drift: introduce automatic or frequent in-process gauging with tool-wear compensation, or change the insert grade, coolant strategy, and spindle warm-up so that drift is reduced. The target is an effective mean band of about 0.001–0.002 mm across the shift, which with σ = 0.0030 gives 8σ + 0.002 = 0.026 mm - still marginal, so a modest reduction in σ to about 0.0028 mm is also needed.
Third, install X̄-R or X̄-s control charts with reaction rules and a capability study after 100–200 parts. Fourth, validate the measurement system: calibration, gauge R&R below 10–15% of tolerance, and a clear gauge control plan. Fifth, mistake-proof the offset entry and record every change. Only after a fresh capability study demonstrates Cpk ≥ 1.33 on consecutive shifts should the cell be released for normal production.