DOE in DMAIC: 2^k Factorial Design for Process Optimization

A designed experiment changes several inputs (Xs) at once, on purpose and in a planned pattern, so the team can measure how each one moves the output (Y) and whether the inputs interact. In DMAIC it is one of the strongest ways to turn suspected root causes into confirmed ones and to choose the settings the new process will run at.

What the exam expects

Design of experiments sits in Improve (VII.A) of the 2022 CSSBB Body of Knowledge. Expect to define the vocabulary (factor, level, response, treatment combination, replication, randomization, blocking, aliasing, resolution), compute main and interaction effects from a two-level table, read main-effects and interaction plots, and choose between a full and a fractional factorial.

Key concepts

• 2^k full factorial: k factors, each at a low (−1) and high (+1) level, every combination run once per replicate. Three factors give 2^3 = 8 runs.
• Main effect: average Y at the factor's high level minus average Y at its low level. With N runs, effect = contrast ÷ (N/2).
• Interaction: the effect of one factor depends on the level of another. The AB column is the row-by-row product of the A and B signs. Non-parallel lines on an interaction plot signal it.
• Coefficient vs effect: in coded units the regression coefficient is half the effect, because the factor moves two units (−1 to +1).
• Randomization protects against lurking variables such as tool wear or ambient drift. Replication gives an estimate of pure error. Blocking removes a known nuisance source, such as raw-material lot or shift.
• Fractional factorial 2^(k−p): fewer runs, paid for with aliasing. Resolution III: main effects aliased with two-factor interactions. Resolution IV: main effects clear of two-factor interactions, but two-factor interactions aliased with each other. Resolution V: two-factor interactions clear of each other.
• Unreplicated designs have no pure error, so judge effects with a normal or half-normal probability plot, Lenth's method, or by pooling negligible higher-order terms.

Worked example: a 2^3 on peel strength

A team studies peel strength (N, larger is better) of a bonded joint. A = cure temperature (150 / 170 °C), B = clamp pressure (2 / 4 bar), C = dwell time (20 / 30 s). One replicate, run in random order, listed here in standard order.

Reading it: temperature dominates. Pressure looks negligible on its own (2.0 N), but the AB interaction (8.0 N) shows pressure helps only at high temperature, so B stays in the model (hierarchy) and the main-effects plot alone would mislead. Dwell time is small, so the team can set it at the cheaper 20 s, after a half-normal plot confirms that C, AC, BC and ABC fall on the noise line. Recommended settings: 170 °C, 4 bar, 20 s. The reduced model (A, B, AB) predicts 83.0 N there; run 4, at exactly those settings, gave 82 N. A confirmation run at those settings comes before the pilot.

Common exam traps

• Dividing the contrast by N instead of N/2 for an effect, or reporting the effect when the question asks for the coefficient (half the effect).
• Dropping a small main effect that is part of a large interaction.
• Interpreting main effects alone when the interaction plot shows crossing or diverging lines.
• Claiming statistical significance from a single unreplicated run set with no error estimate.
• Choosing a resolution III fraction when two-factor interactions must be estimated.
• Running in standard order, which lets time trends masquerade as factor effects.
• Skipping the confirmation run and going straight to rollout.

Check yourself

1. A 2^4 full factorial has how many runs per replicate, and what divisor turns a contrast into an effect? (16 runs; divide by 8.)
2. An effect of 12 in coded units corresponds to what regression coefficient? (6.)
3. In a resolution IV design, what are two-factor interactions aliased with? (Other two-factor interactions.)
4. Why does B stay in the model above although its main effect is only 2.0? (It is part of the AB interaction; hierarchy.)
5. What does randomizing the run order protect against? (Unknown time-related or lurking variables biasing the effects.)

Related: Full vs fractional factorial · Interaction plots · Blocking & randomization · Improve tollgate

More: DMAIC overview · Project charter · VOC & CTQ · SIPOC · MSA in DMAIC · Process capability · Data collection planning · Graphical analysis · Root cause analysis · Multi-vari analysis · Hypothesis testing · Regression · DOE · FMEA · Define tollgate · Change management · Financial benefits · Team dynamics · Improve tollgate · Piloting & scaling · Poka-yoke (BB) · Solution selection · Control plan · SPC in DMAIC · Sustaining gains

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