Optimal Design

Optimal design (also called computer-generated design or algorithmic design) is a DOE approach where a computer algorithm selects the specific experimental runs that maximize statistical efficiency for a given model, constraints, and number of runs — rather than using a pre-defined template like factorial, CCD, or Box-Behnken designs.

Why Optimal Design?

• Classical designs (factorial, CCD, Box-Behnken) work well when:
• All factors have the same number of levels.
• The design space is regular (no constraints).
• Standard models (linear or quadratic) are sufficient.
• But real semiconductor experiments often involve:
• Mixed factor types: Some continuous (temperature), some categorical (gas type, chamber identity).
• Irregular regions: Certain factor combinations are physically impossible or dangerous.
• Constrained runs: Budget limits the number of wafers available.
• Complex models: Need to estimate specific terms, not the full factorial model.
• Optimal designs handle all these situations by tailoring the run selection to the specific problem.

Types of Optimal Designs

• D-Optimal: Maximizes the determinant of the information matrix — minimizes the overall variance of parameter estimates. The most commonly used criterion.
• I-Optimal (IV-Optimal): Minimizes the average prediction variance across the design space — best for response surface prediction.
• A-Optimal: Minimizes the trace (sum of variances) of the parameter estimates.
• G-Optimal: Minimizes the maximum prediction variance — best worst-case prediction.

How It Works

• Specify the Model: Define which terms to estimate (main effects, interactions, quadratic terms).
• Define the Candidate Set: List all possible experimental runs (combinations of factor levels and constraints).
• Select Criterion: Choose D-optimal, I-optimal, etc.
• Algorithm Selects Runs: The computer uses exchange algorithms (coordinate exchange, point exchange) to find the subset of candidate runs that optimizes the chosen criterion.
• Result: A custom design that is tailored to your specific model, constraints, and budget.

Semiconductor Applications

• Mixed Factor Experiments: Optimizing etch with continuous factors (power, pressure) and categorical factors (gas chemistry type, chamber ID).
• Constrained Regions: When certain power-pressure combinations are physically unsafe or outside equipment limits.
• Augmenting Existing Data: Adding runs to an existing dataset to improve model estimation.
• Resource-Limited: When only 12 wafers are available but 6 factors need screening.

Advantages and Cautions

• Advantages: Maximum flexibility, statistical efficiency, handles any constraint or factor type.
• Cautions: The design depends on the assumed model — if the model is wrong, the design may miss important effects. Also, different software may generate different designs for the same problem.

Optimal designs are the most flexible DOE approach — they solve problems that classical designs cannot, making them essential for complex semiconductor experiments with real-world constraints.