box-behnken design
**Box-Behnken design** is a **response surface methodology (RSM)** experimental design that efficiently fits **second-order (quadratic) models** without requiring experiments at extreme corner conditions (all factors simultaneously at their highest or lowest levels). It is an alternative to the Central Composite Design (CCD).
**Design Structure**
- Box-Behnken designs combine **two-level factorial designs** for pairs of factors with **center points**.
- Each factor appears at only **three levels**: −1, 0, and +1.
- Critically, the design **never includes corner points** where all factors are simultaneously at extreme levels — all runs have at least one factor at its center value.
**Example: 3-Factor Box-Behnken (15 runs)**
| Run | A | B | C |
|-----|---|---|---|
| 1–4 | ±1 | ±1 | 0 |
| 5–8 | ±1 | 0 | ±1 |
| 9–12 | 0 | ±1 | ±1 |
| 13–15 | 0 | 0 | 0 |
Pairs of factors are varied in a 2² factorial pattern while the remaining factor is at center (0). Plus 3 center point replicates.
**Advantages Over CCD**
- **No Extreme Corners**: Avoids conditions where all factors are at their extreme levels simultaneously — these conditions may be physically impractical, dangerous, or outside equipment capability.
- **Fewer Runs**: For 3 factors, Box-Behnken uses **15 runs** vs. CCD's **20 runs** (with 6 axial + 6 center). For 4 factors: 27 vs. 30.
- **Spherical Design**: All design points are approximately the same distance from the center — providing more uniform prediction quality.
- **Three Levels Only**: No axial (star) points extending beyond the factorial range — stays within the original factor ranges.
**Disadvantages**
- **No Corner Coverage**: Cannot evaluate the response at extreme combinations — the model may be less accurate at corners.
- **Not Sequential**: Unlike CCD, which can be built up from a factorial design by adding axial/center points, Box-Behnken requires running all points together.
- **Limited Blocking**: More difficult to split into blocks compared to CCD.
**Semiconductor Applications**
- **Safe Operating Conditions**: When running at all extreme conditions simultaneously risks wafer damage, equipment limits, or safety hazards.
- **Narrow Process Windows**: When the design space is tightly constrained and extending beyond it (as CCD axial points require) is not possible.
- **Efficient Optimization**: When the primary goal is finding an optimum within the current operating range with minimal runs.
**Choosing Between CCD and Box-Behnken**
| Criterion | CCD | Box-Behnken |
|-----------|-----|------------|
| **Extreme conditions OK?** | Yes (needed for axial points) | No (avoids extremes) |
| **Sequential from factorial?** | Yes (add axial/center points) | No (new design) |
| **Prediction at corners?** | Better | Worse |
| **Number of runs** | Slightly more | Slightly fewer |
Box-Behnken designs are the **preferred RSM design** when operating at extreme factor combinations is impractical — they provide efficient quadratic modeling while keeping all experiments within safe, achievable processing conditions.