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)

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

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.