Optimal Three-Level Designs for Response Surfaces in Spherical Experimental Regions

Since 1960, Box–Behnken designs have been very popular with experimenters wishing to estimate a second-order model in three or four factors. This popularity is due to these three-level designs' simplicity and high efficiency. However, as the number of factors increases, the run size of Box–Behnken designs increases rapidly, making them less attractive. The purpose of this article is to recommend the use of D-optimal and I-optimal three-level designs for spherical design regions involving three or more factors. Using an optimal design algorithm offers flexibility of run size. In addition, optimal-design criteria permit construction of second-order designs for more complex response surface applications involving mixture or qualitative factors or requiring split-plot randomization. The restriction to three-level designs provides great practical convenience and often little loss in efficiency, provided the design is not nearly saturated.

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