Design of experiments (DOE) for engineers: 2-level factorial designs explained

Theory: why change everything at once?

The traditional way to optimise a process is to change one factor at a time while holding the others constant. It is slow, and it cannot show interactions: situations where the effect of one factor depends on the setting of another, for example a higher holding pressure that helps only when the melt is cold. In practice many process problems are exactly such interactions.

A 2-level full factorial design (2^k) tests every combination of k factors, each at a low (−) and a high (+) setting. Three factors need 2³ = 8 runs. Every run contributes to the estimate of every effect, so the experiment is far more efficient than one factor at a time.

The effect of a factor is the average response at its high setting minus the average at its low setting. An interaction effect AB is calculated the same way, using the product of the coded settings of A and B as the “setting”.

Whether an effect is real or only noise is judged in one of two ways. With replicates (every combination run more than once) the scatter between repeats gives the pure error and each effect gets a t-test and a p-value. Without replicates, Lenth’s method treats the many small effects as noise and flags the effects that stand out from them; it works well from 3 factors up.

Step by step

1. Define the response and the goal. One measurable result (shrinkage, force, diameter, cycle time) and whether it should be maximised, minimised or brought to a target.
2. Check the measurement system. If the gauge cannot see the differences you expect, the experiment will show nothing. Run a Gage R&R first if in doubt.
3. Choose the factors and levels. Use the team’s process knowledge, a fishbone diagram, existing data (regression, multi-vari). Write down how every other setting is held constant.
4. Decide on replicates. Two replicates give real p-values and protect against a single bad run. One replicate is acceptable for screening 4 or 5 factors.
5. Randomise the run order and follow it, so that drift (tool wear, warm-up, material batch, operator) is spread over all settings instead of looking like a factor effect.
6. Run the trials, let the process stabilise after each change, and record every result and anything unusual.
7. Analyse in the DOE tool: Pareto of effects, main effects, interactions and best settings.
8. Confirm the best setting with a few extra runs, check the capability there and update the control plan and the PFMEA.

Case study: shrinkage of an injection moulded part

A moulded housing shrinks too much and the mounting holes drift towards the tolerance limit. The team suspects three settings and plans a 2³ design with 2 replicates (16 runs, random order). This is the Load sample data of the DOE tool.

The analysis (8 degrees of freedom for the pure error, R² 99.7 %) gives these effects on shrinkage in %:

Melt temperature and holding pressure both reduce shrinkage strongly. The positive AB interaction means the two do not simply add up: the interaction plot shows that a higher pressure reduces shrinkage a lot when the melt is at 210 °C, but much less at 240 °C. Cooling time is statistically significant, but its effect (0.06 %) is small in practice, so the team keeps the shorter, cheaper 10 s cycle.

For the lowest shrinkage the tool recommends 240 °C, 80 bar, 10 s with a predicted shrinkage of 1.28 %, against a grand mean of 1.59 % over the trial. Three confirmation shots at this setting measure 1.27 to 1.30 %. The team updates the process sheet and the control plan and checks the hole position capability in the next run.

Common mistakes

• Levels too close together. The effect disappears in the noise. Move the settings as far apart as the process safely allows.
• Not randomising. Running all low settings first and all high settings later turns warm-up or tool wear into a fake factor effect.
• Ignoring interactions. Reading only the main effects can recommend a setting that is worse in combination. Always look at the interaction plots for significant interactions.
• Confusing statistical and practical significance. With good replicates even a tiny effect becomes significant. Ask whether its size matters for the product and the cost.
• Skipping the confirmation runs. The prediction is a model. A few runs at the chosen setting prove it before the control plan changes.
• Uncontrolled other settings. Anything that changes during the trial and is not a factor (material lot, operator, ambient temperature) must be held constant or recorded.

Frequently asked questions