# Box-Behnken Response Surface Methodology

Video 7. Box Behnken Response Surface Methodology RSM Design and Analysis Example using Minitab & MS Excel

Box-Behnken Design, BBD for the response surface methodology, RSM, is specially designed to fit a second-order model, which is the primary interest in most RSM studies. To fit a second-order regression model (quadratic model), the BBD only needs three levels for each factor (Figure 15), rather than five levels in CCD (Figure 14). The BBD set a mid-level between the original low- and high-level of the factors, avoiding the extreme axial (star) points as in the CCD. Moreover, the BBD uses face points, often more practical, rather than the corner points in CCD. The addition of the mid-level point allows the efficient estimation of the coefficients of a second-order model (Box et al., 2005). The BBD is almost rotatable as the CCD. Moreover, often, the BBD requires a smaller number of experimental runs.

Figure 14. Central Composite Design, CCD for Rotatability (left) and Face Center Design (right). Note. The central composite design, CCD with the axial points at the face is known as the face-centered central composite design or the face-centered cube if the axial points are placed at the face.

Figure 15. Two Representation of the Box-Behnken Design, BBD for RSM

# How to Design the Box-Behnken RSM

Video 8 demonstrates an overview of the design, analysis, and explanation of the results for the Box-Behnken Design, BBD.

The BBD uses the 22 full factorial design to generate for the higher number of factors by systematically adding a mid-level between the low and the high levels of the factors. Table 5 and Table 6 provide the Box-Behnken designs for three, four, and five factors, respectively. Many designs can be found in any standard statistical package such as Minitab, Design Experts, JMP and SAS. To design the BBD, simply the 22 full factorial is used as the base design and then orthogonal blocks are created using the mid-levels for the other factors. Therefore, it becomes a three-level factorial design, for which a full quadratic (second order) model can be fitted for the response surface.

Table 5

Box-Behnken Design for Three and Four Factors

Table 6

Box-Behnken Design for Five Factors

# How to Analyze and Explain the Results

Video 8 demonstrates an overview of the design, analysis, and explanation of the results for the Box-Behnken Design, BBD. The analysis and the explanation for the results are exactly the same as the central composite design, CCD. Let’s look at the human comfort study with the lighting factor, X3 added to it. The X1 and X2 are the temperature (degree Fahrenheit) and the humidity (%), respectively. Data is collected from randomly selected 15 individuals with the specified room temperature, humidity, and lighting level conditions (Table 7).

Table 7

Human Comfort vs Temperature, Humidity, and Lighting

The design can be either created in MS Excel or Minitab, or using any software demonstrated in Video 8. Although the CCD and the BBD are different, their analyses are exactly the same and so the interpretations of the results. For example, the overall model and the quadratic square terms are significant (ANOVA table in Figure 16). The variations in the response are also explained well by the model terms, which is about 99% (the model summary table in Figure 16). The Pareto analysis shows that most effects are coming from the square terms of both the humidity and temperature only.

Figure 16. Box-Behnken Example Analysis Results

Figure 17. Box-Behnken Pareto Analysis of the Effects

The response surfaces and the contour plots for all combinations for variables are provided in Figure 18. The response surface between the humidity and the temperature shows the curvature, the effects from both square terms (top-left graph in Figure 18). The Oval dark reason in the contour plot between the humidity and temperature shows the maximum comfort (top-right graph in Figure 18). As the lighting effect was observed to be insignificant, the contour plots show almost vertical parallel lines when plotted against either of the significant variables of humidity or temperature (bottom four graphs of Figure 18). These vertical parallel lines indicate that as we proceed along the X3 variable (lighting factor), no changes in comfort are observed. However, as we proceed along either the X1 or X2, there is a significant change observed in the comfort values as it can be seen through the color gradients. The response surfaces show a significant curvature (effect of the square terms) for both humidity and temperature while plotted against the lighting factor.

Figure 18. Box-Behnken Response Surfaces and Contour Plots

The optimization results in Figure 19 indicate that both the middle level of the humidity and the temperature are observed to be the best to achieve the maximum human comfort. The selected lighting conditions do not affect human comfort in this study.

Figure 19. Box-Behnken Response Surfaces Optimization Output

# Is Box-Behnken a Better Design than the Central Composite Design?

Video 9 shows a comparison analysis between the central composite design, CCD, and the Box-Behnken Design, BBD.

Video 9. Is Box Behnken Better than the Central Composite Design in the Response Surface Methodology

All models are wrong (Box et al., 2005)! Therefore, developing a less inaccurate model would be preferable. Any model within a smaller reason is more accurate than a wider range, the Box-Behnken will arguably provide a better estimation of the parameters, as the levels are not too extreme as the central composite designs. For example, in the study of human comfort by the temperature and the humidity, the low level of the temperature of 65-degree Fahrenheit is already low (Table 8). If the axial points are placed even further lower than this level, this could be a wastage of resources because we already know that this temperature will produce a lower comfort rating. It does not make much sense to study something that we already know. Moreover, in Box-Behnken, most treatment combinations use the mid-levels of the other factors. Therefore, the study is conducted around the expected optimum reasons in the most experimental runs. For example, there is a total of 150 mid-levels as compared to only 40 low or high levels in five-factor Box-Behnken Design in Table 6. This experiment will be more practical to conduct than a central composite design with five factors as the central composite will contain many extreme points.

Table 8

Comparison between the Central Composite and the Box-Behnken Designs

Nevertheless, the central composite design is the traditional fractional factorial design of experiments. Therefore, it has all the advantages of the fractional factorial design. Moreover, the CCD is rotatable, while the Box-Behnken is nearly rotatable or rotatable for some specific designs.

As the central composite design consists of five levels for each factor, it will be possible to test up to a fourth-order model. However, the Box-Behnken design consists of only three levels for each factor. Therefore, only a second-order model is possible for the Box-Behnken design.

Generally, for more well-informed processes, Box-Behnken could be more useful, while the central composite could be more useful in relatively unknown processes. This could be the reason why the central composite design is used more than the Box-Behnken design because most studies are conducted on to find something new. Nevertheless, for more refinement and optimization, Box-Behnken will provide more precision.

In summary, both designs have their advantages and disadvantages. The designers can choose any of these two depending on the optimization goals.