Augmented Box-Behnken Designs for Fitting Third-Order Response Surfaces

Box-Behnken designs are popular with experimenters who wish to estimate a second-order model, due to their having three levels, their simplicity and their high efficiency for the second-order model. However, there are situations in which the model is inadequate due to lack of fit caused by higher-order terms. These designs have little ability to estimate third-order terms. Using combinations of factorial points, axial points, and complementary design points, we augment these designs and develop catalogues of third-order designs for 3–12 factors. These augmented designs can be used to estimate the parameters of a third-order response surface model. Since the aim is to make the most of a situation in which the experiment was designed for an inadequate model, the designs are clearly suboptimal and not rotatable for the third-order model, but can still provide useful information.