Process Capability Dispute on a Ø25 mm Shaft Line
A Tier-2 auto-component plant machines a critical shaft to Ø25.000 ± 0.020 mm on a CNC lathe. A 50-piece capability study, taken after a tool change and with the machine warm, gives a stable X̄–R chart with mean 25.006 mm and within-subgroup standard deviation 0.004 mm. The customer's drawing requires Cpk ≥ 1.33; the plant's internal target is Cp ≥ 1.67. The production head wants to place an order for a new precision grinder costing ₹85 lakh, arguing that the lathe cannot hold the tolerance. The quality head wants to recentre the tool offset and continue. The measurement system has a repeatability standard deviation of 0.0025 mm and has not been formally studied for reproducibility. The next dispatch is 10 days away, and 4,000 pieces are in stock, all made to the current setting. You are asked to chair the review.
1. Calculate Cp and Cpk from the study and interpret what they say about the process.
Read the worked answer
Tolerance width = USL − LSL = 25.020 − 24.980 = 0.040 mm.
Cp = 0.040/(6 × 0.004) = 1.667. This meets the internal target of 1.67 and shows that the process spread is small enough for the tolerance.
Cpk = min[(25.020 − 25.006), (25.006 − 24.980)]/(3 × 0.004) = min(0.014, 0.026)/0.012 = 1.167.
Cpk is below the customer requirement of 1.33. The gap is not caused by excessive variation but by the mean being 0.006 mm above the nominal 25.000 mm. The process is off-centre toward the upper limit. Buying a grinder to reduce spread would not address the dominant problem, which is centring.
2. Decide the immediate action: buy the grinder, recentre the tool offset, or do both. Include the expected Cpk after recentring and the disposition of the 4,000 pieces in stock.
Read the worked answer
Recentre first. Shifting the mean down by 0.006 mm to 25.000 mm gives Cpk = 0.020/(3 × 0.004) = 1.667, which meets both the customer requirement and the internal target, using the existing lathe. The grinder is not justified by this data alone.
The stock needs sorting because it was made at mean 25.006 mm. Using the observed σ = 0.004 mm, P(X > 25.020) = P(Z > 3.5) ≈ 0.00023, or about 0.9 piece in 4,000. The lower limit is far away (Z ≈ −6.5). So the expected reject count is small, but 100% inspection or guard-banded sorting near the upper limit is prudent. Note also that the gauge repeatability is 0.0025 mm, giving 6σ_gauge = 0.015 mm, or 37.5% of the tolerance; the observed σ therefore contains substantial measurement variation. After correcting for gauge variance, the true process σ is about 0.0031 mm, so the recentred process is even more capable than the observed figures suggest. The high GRR still means the measurement system must be improved before capability is finally signed off.
3. What medium-term actions and documentation would you require before accepting the process, and why is a single capability study insufficient?
Read the worked answer
A single 50-piece study after a tool change does not capture tool wear, thermal drift, operator variation, or lot-to-lot material effects. Require a tool-wear study over a full tool life, an X̄–R or X̄–s control chart with defined reaction plan, and a gauge R&R study with reproducibility included. Target GRR below 10% for capability decisions; 37.5% is too high for accepting Cpk at the customer boundary.
Document the recentring offset, first-piece approval, revised control plan, calibration status, and operator training. Keep the 4,000-piece sorting record and quarantine any parts above the guard band. If tool wear shows a consistent upward drift, add automatic tool-offset compensation or a mid-batch offset check. Recalculate capability after the offset change over at least 25 subgroups. Only if the recentred process fails to hold Cpk ≥ 1.33 over tool life should capital purchase of a grinder be reopened.