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.