Resolution IV Screening on a Wave-Solder Line
A Green Belt is screening four factors to reduce bridging defects on a wave-solder process: A preheat temperature (100/120 °C), B conveyor speed (0.9/1.2 m/min), C flux density (0.80/0.90 g/cm³), D solder pot temperature (245/260 °C). Budget allows 8 factorial runs plus 3 center points, randomized, with no replication. The team runs a 2^(4−1) design with D = ABC, which is Resolution IV. Analysis shows C and B as significant main effects. The A×C interaction term is the largest two-factor term at p = 0.02. The three center points give a significantly higher defect rate than the factorial runs (p = 0.03). The team's statistician is on vacation. The plant manager wants to lock in the settings now because the project is behind schedule, arguing: 'A×C is significant, so set A high and C low and we are done.'
1. State the alias structure of this design and assess what can and cannot be concluded about the A×C interaction.
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With D = ABC, the defining relation is I = ABCD, and the design is Resolution IV. That means:
- Main effects are clear of two-factor interactions (A = BCD, B = ACD, C = ABD, D = ABC).
- Two-factor interactions are aliased in pairs: AB = CD, AC = BD, AD = BC.
So the 'significant A×C' term is really the sum of A×C and B×D. The data cannot distinguish a preheat × flux effect from a conveyor speed × pot temperature effect. Acting on the manager's instruction - A high, C low - could be setting the machine to exploit a completely different physical mechanism, or to exploit nothing at all, and you would not know which.
Two further cautions:
- With no replication, the error estimate is pooled from the smallest effects, which is fragile in an 8-run design. A p-value of 0.02 on a pooled-error basis should be treated as a signal to investigate, not a settled fact.
- The main effects B and C are clean and can be acted on directionally; that part of the finding is trustworthy.
Do not lock settings on a confounded interaction. The correct statement to the manager is: 'We know B and C matter; we do not yet know which interaction we are seeing.'
2. What does the center-point result tell you, and how does it affect the decision to lock settings now?
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The center points sit at the middle of the factor ranges (110 °C, 1.05 m/min, 0.85 g/cm³, 252.5 °C). Their defect rate being significantly higher than the factorial runs (p = 0.03) is a curvature test, and it rejects the assumption that the response is a flat plane over this region.
Implications:
- A two-level design fits only a linear model plus interactions. If curvature is present, the model's predicted optimum may lie somewhere the surface is not flat, and the predicted response at any setting is biased.
- The curvature means the best region is probably toward a corner or edge of the current cube - or, less likely, that the center is a local worst point in a bowl. Either way, the linear model cannot locate it.
- Locking settings now would deploy a model that has already failed a goodness-of-fit check, on a process where defect reduction is the goal.
What to do: treat the center-point result as a stop sign for 'lock in now'. Either confirm the linear conclusion within a narrower region using the foldover, or, if curvature is confirmed, add axial points to fit a quadratic and find the real optimum. In either case, run confirmation trials at the predicted setting before changing the production recipe.
3. Recommend the next experimental step given a tight budget, with justification for the design choice.
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Step 1 - foldover (8 additional runs). Run the same 8 treatment combinations with all signs reversed, which corresponds to D = −ABC. Combining the two blocks gives a 16-run Resolution IV design that de-aliases all two-factor interactions: AC is now estimated separately from BD. This directly answers the open question and costs one more day of trials rather than a rework of the whole project.
Step 2 - decide based on the de-aliased result.
- If AC is real, optimize along that interaction with a small follow-up (a 2-factor design or a targeted set of runs varying A and C).
- If BD is real, the physical story changes - investigate conveyor speed vs pot temperature, which is plausible because dwell time and solder temperature jointly set the thermal profile.
Step 3 - handle curvature. Because curvature was significant, add axial points (a small central composite design, 6–8 runs) or run a narrow-region 3-level design on the two or three factors that survive. Fit a quadratic, locate the optimum, then run 3 confirmation runs at the predicted setting before the recipe is changed.
Step 4 - fix the method for next time. Add replication or at least 3–5 center points as standard, and never let a Resolution IV design be the last experiment before a production change.