Capstone: a DMAIC project end to end

(a) Problem: "Over the last quarter, 4.2 % of brazed cores (42 per 1,000) failed the final helium leak test, against a historical level the plant regards as normal of about 1 %, causing rework and scrap and a customer escalation." Goal: "Reduce the leak-test reject rate to at most 1.0 % (10 per 1,000) within four months, confirmed by a p chart over 30 production shifts, without increasing cycle time or rework."

(b) SIPOC. Suppliers: tube and header supplier, filler-clad sheet supplier, flux supplier. Inputs: tubes, headers, fins, baffles, flux, furnace atmosphere. Process: expand tubes into headers → assemble fins and baffles → clamp in fixture → flux → braze in furnace → cool → leak test. Outputs: leak-tight cores; leak-test records. Customers: the assembly plant; the end customer; the repair station (as an internal, unwanted customer).

(c) Primary CTQ: leak-tight joints; metric: leak-test reject rate per lot, with the leak location recorded. Secondary metric: braze-joint gap at the tube-to-header joint (the physical variable behind the CTQ, once Measure confirms it), and cycle time, which must not grow.

(d) The plant manager (cost, customer), the customer's quality engineer (evidence, timing), the furnace and fixture owners (their processes will be examined), the leak-test operators (they record the data and locate the leaks), the repair station (their workload should fall), and finance (the benefit calculation, which is theirs to make, not the Green Belt's).

(a) The part × operator interaction is not significant (F = 0.34, p = 0.99), so it is pooled into repeatability. Operators differ (F = 9.07, p = 0.0005): a small but real reproducibility component.

(b) Standard deviations: repeatability 0.00254 mm, reproducibility 0.00161 mm, gauge R&R 0.00301 mm, part-to-part 0.02475 mm, total 0.02493 mm. %R&R of study variation = 12.1 %; of tolerance = 6 × 0.00301 / 0.10 = 18.1 %; ndc = 1.41 × 0.02475 / 0.00301 = 11.6, reported as 11. Repeatability (10.2 %) dominates reproducibility (6.5 %).

(c) Acceptable with a note: under 10 % would be unconditional; 10 to 30 % is the "may be acceptable depending on the application" band of the AIAG MSA guidelines (cited in Module 4 as guidelines, confirmed through secondary sources), and 18 % of a 0.10 mm tolerance means the gauge alone uses a fifth of it.^[3] ndc of 11 is well above the guideline of 5. Proceed with the capability study, record the gauge's contribution, and note that the operator difference (about 0.0016 mm) is worth a look at how the pins are seated.

(a) 259 leakers in 6,037 cores: p̄ = 0.0429. Limits for a lot of 150: 0.0429 ± 3 × 0.0165, upper 0.0925, lower below zero; for 250: upper 0.0813, lower 0.0045. No lot is beyond its limits and no Western Electric rule fires (Nelson's test 3, six rising, fires once at lot 14, a pattern this weak is a false alarm with all tests switched on). The reject rate is stable: the process reliably makes about four leakers in a hundred. That is a common-cause problem, which is what a DMAIC project is for; a special-cause problem would need the reaction plan, not a project.

(b) p̂ = 4.29 %, Wilson 95 % interval 3.81 to 4.83 %; DPMO 42,902 (one opportunity per core, the honest count); Z_long-term = 1.72, "sigma level" 3.22 with the 1.5 shift, which nobody should put on the charter without the Module 6 caveats.

(c) N = 1.96² p̂(1 − p̂)/0.005² = 6,310 cores, about a month at the current volume. The interval on the baseline is already about ±0.5 points because the study has that many cores.

(d) The charter's 42 per 1,000 sits inside the interval; the baseline is confirmed at 43 per 1,000 over these 30 shifts.

(a, b) Tube-to-header joints account for 168 of 259 leakers, 64.9 %; with tube-to-tube joints the first two categories cover 80.7 %. The project's scope narrows to the tube-to-header joint: fix that and the reject rate falls by roughly two thirds at most, to about 4.29 × (1 − 64.9/100) ≈ 1.5 % if every header leak were eliminated, which is still above the goal. The other categories will need attention later, and the charter's goal should be read with that arithmetic in mind.

(c) Why the header joints leak. A Pareto locates, it does not explain. It also depends on the operators' classification, which is an attribute measurement system in its own right: an attribute agreement analysis (Module 4) on a sample of leakers would confirm that "tube-to-header" means the same thing to all three operators before the project bets on it.

(a) x̄̄ = 0.1124 mm, R̄ = 0.0344 mm; UCL_x̄ = 0.1124 + 0.577 × 0.0344 = 0.1124 + 0.0198 = 0.1323, LCL_x̄ = 0.0926; UCL_R = 2.114 × 0.0344 = 0.0727. No point beyond a limit on either chart, no Western Electric rule: stable.

(b) Anderson-Darling on the 125 values: p = 0.91. No evidence against normality, and a gap of this kind (a difference of two machined dimensions) has no physical reason to be skewed.

(c) σ̂_within = 0.0344 / 2.326 = 0.01479 mm; overall s = 0.01414 mm; ratio 0.96. Mean 0.1124 mm, 0.0124 mm above the 0.10 mm target: the upper side governs. Overall (2026 Cp/Cpk, since stable): Cp = 0.10 / 0.08485 = 1.18; Cpu = 0.0376 / 0.04242 = 0.89; Cpl = 1.47; Cpk = 0.89, 95 % interval 0.76 to 1.01. Within (Cwk, legacy Cpk): Cwu = 0.0376 / 0.04437 = 0.85, Cwk = 0.85. k = 0.25: a quarter of the half-tolerance is lost to centring.

(d) Predicted 3,938 PPM beyond the limits, almost all above 0.15 mm; observed 0 of 125, consistent with 0.4 %.

(e) A Cpk of 0.89 predicts 0.4 % of cores with a mean gap beyond 0.15 mm, not 4 %. The reconciliation is physical, not statistical: a joint leaks when its own gap exceeds what the filler can bridge, which the braze engineers put at about 0.14 mm, and the plotted value is the mean of six joints. The fraction of cores whose mean gap exceeds 0.14 mm under the fitted normal is 2.6 %, and individual joints scatter further than the core mean. So the gap distribution is entirely consistent with a few per cent of header leaks, and the specification limit of 0.15 mm is too generous for the process's purpose, a Module 15 conversation for later. The capability study says: stable, centred high, too wide for a joint that must not exceed 0.14 mm.

(a) x̄ = 599.83 °C, MR̄ = 2.110 °C, σ̂ = 1.871 °C; limits 594.22 to 605.44 °C, MR limit 6.89 °C. Stable; no rule fires.

(b) The furnace peak temperature is stable to about ±5.6 °C around 600 °C. It cannot be the source of a reject rate that is itself stable at 4 % unless the whole band is wrong, which the braze engineers can answer from the filler's melting range, and the DOE below will test. It is not ruled out as a factor; it is ruled out as a special cause, which is a different thing. Everyone stops blaming the furnace, for now.

(a) Machine: tube expansion mandrel wear, fixture clamp force, furnace peak temperature and dwell. Material: tube outside diameter and header hole diameter (the gap is their difference), clad thickness, flux quantity. Method: expansion depth, fixture loading sequence, flux application. Measurement: pin-gauge use (now characterised), leak-location classification. Man: fixture loading by shift. Environment: furnace atmosphere dew point.

(b) Three lines, ratings on the 1 to 10 scales of the AIAG-VDA FMEA handbook and its action-priority logic, which replaces the older RPN product:^[4] (1) Failure mode: joint gap above bridging limit; effect: leak at test, severity 8 (escape to the customer if the test misses); cause: insufficient tube expansion; current control: none on the gap, leak test at the end; occurrence 6, detection 4; action priority high. (2) Gap above limit; cause: fixture clamp force low or uneven; control: monthly fixture check; occurrence 4, detection 5; high. (3) Incomplete filler flow; cause: furnace peak temperature low; control: continuous thermocouple log with alarm; occurrence 2, detection 2; low. The ratings are judgements and are recorded as such; the value of the exercise is that it puts expansion and clamping ahead of the furnace, which is where the data from Step 5 already pointed.

(c) Expansion and clamping can be tested by comparing the gap on leaking and passing cores (data the repair station can collect from the cores it sections) and by a designed experiment. Furnace temperature is already charted. Material diameters need incoming-inspection data the project does not yet have.

(a) Welch's two-sample t-test (Module 11). Means: leakers 0.1364 mm, passers 0.1109 mm; standard deviations 0.0109 and 0.0142; Levene p = 0.34.

(b) Difference 0.0255 mm (25.5 µm); standard error 0.00401; t = 6.37 on ν = 35.6; p < 0.0001; 95 % interval 0.0174 to 0.0336 mm; Cohen's d = 2.01. "Leaking joints had gaps 26 µm larger on average (95 % interval 17 to 34 µm); a difference this large would arise in far fewer than one study in ten thousand if leaking and passing joints had the same gap distribution."

(c) It establishes association in the direction the physics predicts, with an effect of two standard deviations, which is about as strong as observational evidence gets. It does not by itself show that reducing the gap will reduce leaks: a third factor (a tube batch with a smaller diameter, say) could produce both. The designed experiment is what turns "leaking joints have bigger gaps" into "making the gap smaller stops the leaks".

(a) A = −0.0126, B = −0.0239, C = 0.0029, AB = 0.0071, AC = −0.0001, BC = 0.0011, ABC = −0.0014 mm. Check B by hand: mean of the eight runs at increased expansion minus the eight at standard = −0.0239.

(b) Residual MS = 0.0000292 on 8 degrees of freedom (residual sd 0.0054 mm), standard error of an effect 0.0027 mm. F against a 5 % critical value of 5.32: B 78.1 (p = 0.00002), A 21.8 (p = 0.0016), AB 6.96 (p = 0.030); C 1.13 (p = 0.32) and the remaining interactions are noise. R² = 0.931. Expansion is the dominant factor, clamp force matters, the two interact, and the furnace does nothing over ±5 °C.

(c) Averaged over C: standard expansion gives 0.1343 mm at low clamp and 0.1145 mm at high clamp (the production condition); increased expansion gives 0.1033 mm at low clamp and 0.0978 mm at high. The effect of clamp force is −0.0198 mm with standard expansion and only −0.0055 mm with increased expansion: once the tube fills the hole, the clamp has little left to do. Note the production cell: 0.1145 mm, which is the baseline mean of Step 4 within noise, a consistency check that the experiment reproduced the process it was meant to study.

(d) Increased expansion with the clamp kept at 4 kN (changing one thing on the line is cheaper than two, and the interaction says the clamp can stay). Prediction from the fitted model: ŷ = 0.1124 + −0.00631 + (−0.01194) + 0.00356 = 0.0978 mm, on target, with the same spread as the experiment's replicates (about 0.0054 mm per core, which includes the gauge). It assumes the effect holds over a production run of hundreds of cores rather than eight, and that the new mandrel does not change something the experiment did not measure (tube wall thinning, fin contact). The pilot tests both.

(e) Nothing, over the range tested: C's effect (0.0029 mm) is within noise, and the furnace stays at 600 °C. The maintenance department is pleased.

(a) X̄-R: x̄̄ = 0.1003 mm, R̄ = 0.0225 mm, limits 0.0873 to 0.1133, UCL_R = 0.0476; stable, no rule fires. Normality p = 0.87. σ̂_within = 0.00968, s = 0.00928 mm (ratio 0.96), the spread reduced by a factor of 1.52 as well as the mean moved. Cp = 0.10 / 0.05569 = 1.80; Cpu = 0.0497 / 0.02784 = 1.79; Cpl = 1.81; Cpk = 1.79, 95 % interval 1.56 to 2.02; Cwk = 1.71; k = 0.005, centred. Predicted PPM beyond the specification 0.07; more usefully, the fraction of core means above the 0.14 mm bridging limit is now negligible. Observed 0 of 125 out.

(b) 49 leakers in 6,037 cores: p̂ = 0.81 %, Wilson interval 0.61 to 1.07 %; DPMO 8,117; Z 2.40. The p chart is stable by rule 1; rule 2 fires once at lot 10 (two of three beyond 2σ), which was investigated without finding a cause and stays in the data.

(c) The 2 × 2 table (before: 259 leak, 5,778 pass; after: 49 leak, 5,988 pass) gives χ² = 146.9 on 1 degree of freedom, p far below 0.001 (the equivalent two-proportion z is √χ² = 12.1). The reduction is 3.48 percentage points, 95 % interval 2.92 to 4.04 points, a factor of 5.3 or 81 %: about 35 fewer leakers per 1,000 cores.

(d) The goal was at most 1.0 %. The point estimate (0.81 %) meets it; the upper end of the interval (1.07 %) does not quite. Honest wording: "the reject rate after the change is 0.81 % (95 % interval 0.61 to 1.07 %); the goal of 1.0 % is met at the point estimate and the data are compatible with a true rate between about 0.6 and 1.1 %." Another 1,933 cores (10 lots) would pin the rate to ±0.4 points. The control phase's ongoing p chart will supply them.

(e) That the remaining leakers are not header leaks: the Pareto must be repeated on the after data. That the improvement survives a mandrel change, a tube supplier change and six months of production, which is what the control plan is for. And the financial figure: the Green Belt reports leakers per 1,000 and the repair station's hours; finance turns that into money.

(b) Reaction plan for the gap chart. Any point beyond a limit or a rule 2 to 4 signal on the X̄ chart: (1) re-measure the subgroup; (2) if confirmed, hold cores made since the previous subgroup and note tool, lot and operator on the chart; (3) check mandrel wear and clamp force setting; (4) if the cause is found and corrected, release the held cores after a leak test with location recording and a new subgroup inside the limits; (5) if not found within the shift, call engineering; (6) record every signal and its outcome. A point beyond the R limit: check the fixture loading and the gauge before anything else. Never offset the mandrel or the clamp toward a target on the strength of one subgroup.

(c) Ongoing monitoring. The gap and leak-rate charts run with the frozen limits above. Monthly, the quality engineer reports Cpk of the gap from the month's subgroups with its 95 % interval (from 125 values it is about ±0.46 wide at this level), and the leak rate with its interval; a change is real only when the intervals say so and the charts show it. Limits are recomputed only after a deliberate process change (a new mandrel size, a fixture rebuild) has been verified by a fresh study, never because "the process has drifted". This is what an IATF 16949 process-monitoring clause and a PPAP re-submission will ask to see: the studies, the charts, the reactions, and the record of significant events.^[5][6]

(d) Closure. The report states the baseline (4.29 %, 3.81 to 4.83 %), the cause (tube-to-header gap running high, established by the sectioning comparison and the experiment), the change (mandrel 0.05 mm larger; clamp unchanged; furnace unchanged), the evidence (Cpk 0.89 to 1.79; leak rate to 0.81 %, interval 0.61 to 1.07 %; χ² = 147), what remains (the other 81 % of the Pareto's tail, the 0.15 mm specification limit that is looser than the joint needs, the operator effect in the gauge), the control plan, and the owner of each chart. Finance attaches the benefit. The Green Belt signs, the process owner signs, and the charts belong to production from that day.

1. b. Stable, common cause.
2. 18 %.
3. 0.90.
4. c. Wrong limit for the physics.
5. a. Association; the DOE establishes causation.
6. d. Clamp can stay.
7. b. Two-proportion test with interval and charts.
8. c. Met at the point estimate; interval reported.
9. a. After a verified deliberate change.
10. d. Finance.