Box-Behnken Design — Three-Level Response Surface Methodology
Box-Behnken Response Surface Design · Also known as: BBD, Box-Behnken, Box-Behnken RSM design, three-level incomplete factorial design
The Box-Behnken design (BBD) is an efficient response surface methodology design that fits a full second-order polynomial model using three levels of each factor. Introduced by Box and Behnken in 1960, it places experimental points at the midpoints of the edges of a hypercube and at the center, avoiding the corner points where all factors are simultaneously at their extreme levels. This structure makes BBD particularly attractive when extreme-level combinations are physically impossible, costly, or unsafe to test.
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When to use it
Use Box-Behnken design when you need to fit a full second-order (quadratic) response surface model with 3 to 7 continuous factors and the corner points of the experimental space are unsafe, infeasible, or prohibitively expensive to test — for example, when extreme settings of multiple process variables simultaneously could damage equipment or pose safety hazards. BBD is appropriate when the number of experimental runs must be kept moderate and all factors can each individually reach their high and low settings. Do NOT use BBD if you need a design that supports higher-order terms beyond quadratic, if you have categorical factors, if extrapolation to extreme vertex points is the goal, or if only two factors are studied (a central composite design is more efficient for k = 2). Also avoid BBD when a corner-point run is actually needed to meet a downstream specification requirement.
Strengths & limitations
- Requires fewer experimental runs than a three-level full factorial, making it cost- and time-efficient for fitting second-order models.
- No corner point runs — particularly valuable when simultaneously extreme settings of all factors are physically hazardous, prohibitively expensive, or process-unfeasible.
- All factor-level combinations in the design matrix are near-center or on mid-edge positions, which are often the safest and most representative operating conditions.
- Near-rotatability: prediction variance is approximately constant at equal distances from the design center, giving uniform estimation accuracy across the experimental region.
- Straightforward to augment with center-point replicates that provide a pure estimate of experimental error and support a lack-of-fit test.
- Not suitable for sequential experimentation: unlike central composite designs, the BBD cannot be built from an earlier two-level factorial screening experiment.
- Does not estimate interactions at the extreme factor combinations, since corner points are absent; if behavior at the corners is critical, a central composite or full factorial design is needed.
- Coverage of the experimental region is not uniform — the corners of the hypercube are entirely unsampled, which can introduce prediction bias if the true optimum lies near a corner.
- For five or more factors the number of runs can grow substantially, reducing its efficiency advantage over central composite designs.
Frequently asked
How does Box-Behnken design differ from central composite design?
Both designs fit a second-order polynomial model for response surface analysis, but they differ in structure. The central composite design (CCD) includes a two-level factorial (corner) block plus axial (star) points that extend beyond the factor range, and can be built sequentially from a prior screening experiment. The BBD has no corner points and no points outside the factor range — it uses only three levels per factor. Choose BBD when corner runs are infeasible; choose CCD when sequential building or properties at the extreme corners are important.
How many runs does a Box-Behnken design require?
The number of runs depends on the number of factors k. For k = 3: 15 runs (12 edge midpoints + 3 center points); for k = 4: 27 runs; for k = 5: 46 runs. These counts include a standard number of center-point replicates. Compare this to a three-level full factorial, which would require 27, 81, and 243 runs for k = 3, 4, and 5 respectively — BBD offers dramatic savings.
Can I use Box-Behnken design with more than seven factors?
BBD is generally used for 3 to 7 factors. For more than seven factors the run count grows substantially and alternative approaches such as definitive screening designs or optimal computer-generated designs are usually more efficient. The original Box and Behnken (1960) paper described designs up to seven factors.
What software can run a Box-Behnken analysis?
BBD is supported in most major statistical packages: JMP (Design of Experiments module), Minitab (Response Surface section of DOE), SAS (PROC OPTEX or PROC RSREG), and R (the rsm package with the bbd() function). Python users can use the pyDOE2 library. All of these generate the design matrix, fit the second-order model, and provide ANOVA, response surface plots, and optimization output.
Is Box-Behnken design rotatable?
BBD is approximately rotatable, meaning the prediction variance is nearly constant at equal distances from the design center, but strict rotatability depends on the number of factors. For k = 3, the standard BBD is exactly rotatable with the commonly used center-point count. For other values of k, rotatability is approximate. In practice this near-rotatability is sufficient for most engineering and scientific applications.
Sources
- Box, G. E. P., & Behnken, D. W. (1960). Some new three level designs for the study of quantitative variables. Technometrics, 2(4), 455–475. DOI: 10.1080/00401706.1960.10489912 ↗
- Myers, R. H., Montgomery, D. C., & Anderson-Cook, C. M. (2016). Response Surface Methodology: Process and Product Optimization Using Designed Experiments (4th ed.). Wiley. ISBN: 978-1118916025
How to cite this page
ScholarGate. (2026, June 3). Box-Behnken Response Surface Design. ScholarGate. https://scholargate.app/en/experimental-design/box-behnken-design
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