Design of Experiments

Box-Behnken with Categorical Factors (Cartesian × Continuous BB)

Extends the Box-Behnken design to mixed factors — continuous get the classical 3-level RSM plan, categorical are Cartesian-crossed alongside.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Response Surface > Create Response Surface Design (Box-Behnken with categorical option)

Description

Box-Behnken has the advantage that no corner points of the design cube are embedded — all runs sit on the midpoints of cube edges. This is useful when extreme factor combinations are physically impossible or unsafe. With categorical factors, the classical Box-Behnken would try to place a category at a ‘midpoint‘ position (e.g. between Steel and Carbide), which is meaningless. The mixed variant cuts the problem analogously to the CCD case: Box-Behnken over the continuous factors only, then Cartesian-cross with the categorical level combinations. Requires at least 3 continuous factors (Box-Behnken is undefined for k < 3).

Formulas

Run count

N = N_{BB} (k_{cont}) \cdot j \in cat \prod ∣ L_{j} ∣

Run count = (Box-Behnken over continuous) × (product of all categorical level counts)

Continuous BB structure

BB (k) : k_{cont} \geq 3, Eckpunkte des Designs = 0 (keine W \overset{u}{¨} rfel-Ecken)

Box-Behnken uses only edge midpoints of the k-dimensional cube

Assumptions

At least 3 continuous factors (Box-Behnken prerequisite)
Categorical factors have finite level sets

Limitations

Box-Behnken not available for k_cont = 2 — use CCD for two continuous factors
Like CCD: estimates categorical main effects and categorical × continuous interactions, not Cat × Cat

References

Box, G.E.P. & Behnken, D.W. (1960). Some new three level designs for the study of quantitative variables, Technometrics 2:455-475
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Ed., Wiley — Chapter 7.7