Small Box–Behnken design

Box-Behnken design has been popularly used for the second-order response surface model. It is formed by combining two-level factorial designs with incomplete block designs in a special manner—the treatments in each block are replaced by an identical design. In this paper, we construct small Box–Behnken design. These designs can fit the second-order response surface model with reasonably high efficiencies but with only a much smaller run size. The newly constructed designs make use of balanced incomplete block design (BIBD) or partial BIBD, and replace treatments partly by 2III3−1 designs and partly by full factorial designs. It is shown that the orthogonality properties in the original Box and Behnken designs will be kept in the new designs. Furthermore, we classify the parameters into groups and introduce Group Moment Matrix (GMM) to estimate all the parameters in each group. This allows us to significantly reduce the amount of computational costs in the construction of the designs.