Weighted defect rates across machining cells
A pump plant's monthly output of 20,000 units comes from three machining cells. Cell M1 makes 50% of the units with a defect rate of 1.2%; M2 makes 30% with 2.0%; M3 makes 20% with 5.0%. The quality manager's dashboard reports an overall defect rate of 2.2% and states that 'M3 is responsible for almost half of all defects, so halving M3's defect rate will halve the plant's defect rate.' A customer receives a 5,000-unit monthly shipment and rejects 96 units. The plant also plans to introduce a fourth cell M4 with 10% of output and a 3.0% defect rate, taking share from M1 (down to 40%), while M2 and M3 keep their 30% and 20% shares. The sales team argues that adding M4 must improve quality because its defect rate is below the plant average. The production head wants a numerical check before approving the plan.
1. Verify the overall defect rate and M3's share of all defects. Evaluate the quality manager's claim.
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Overall defect rate = 0.50×1.2 + 0.30×2.0 + 0.20×5.0 = 0.60 + 0.60 + 1.00 = 2.2%. This matches the dashboard.
Defect contribution: M1 = 0.60 percentage points, M2 = 0.60, M3 = 1.00. M3's share = 1.00/2.20 = 45.5%, so 'almost half' is fair.
The second claim is false. Halving M3's defect rate to 2.5% changes its contribution from 0.20×5.0 = 1.00 to 0.20×2.5 = 0.50, a fall of 0.50 percentage point. Overall defects become 0.60 + 0.60 + 0.50 = 1.70%, a reduction of 0.50/2.20 = 22.7%, not 50%. Even eliminating M3 entirely gives 1.20%, a 45.5% reduction. The manager has confused share of defects with leverage on the total.
2. The customer rejects 96 units out of 5,000. Is this consistent with the stated 2.2% defect rate?
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Expected rejects = 5,000×0.022 = 110 units. The observed 96 is 14 units lower, or 12.7% below expectation. That is not a dramatic departure: for a binomial count the standard deviation is approximately √(5,000×0.022×0.978) = √107.6 = 10.4 units, so 96 is about 1.35 standard deviations below 110. Such a result can easily arise from normal shipment-to-shipment variation or from a slightly different product mix.
The correct judgement is that 96 rejects is mildly reassuring but not proof that the process has improved. One shipment cannot justify changing the quality plan; a control chart or several consecutive shipments would be needed.
3. Compute the new overall defect rate after M4 is introduced. Is the sales team's claim correct?
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New mix: M1 = 40%, M2 = 30%, M3 = 20%, M4 = 10%. New overall rate = 0.40×1.2 + 0.30×2.0 + 0.20×5.0 + 0.10×3.0 = 0.48 + 0.60 + 1.00 + 0.30 = 2.38%.
The rate rises from 2.2% to 2.38%, so the sales claim is wrong. M4's own defect rate of 3.0% is below the old plant average of 2.2%? No: 3.0% is actually above 2.2%. Even if it were below average, the mix effect matters. M4 takes 10 points of share from M1, the cleanest cell at 1.2%, so it replaces low-defect volume with higher-defect volume. The weighted average is driven by both the cell rate and the share it displaces.