Exact D-optimal designs for a second-order response surface model on a circle with qualitative factors

In this article, exact D-optimal designs for a second-order response surface model on a circular design region with qualitative factors are investigated. Based on this design region, an exact D-optimal design with regular polygon structure is made up according to the remainder terms of the numbers of experimental trials at each qualitative levels divided by 6. The complete proofs of exact D-optimality for models including two quantitative factors and one 2-level qualitative factor are presented as well as those for a model with only quantitative factors. When the qualitative factor has more than 2 levels, a method is proposed for constructing exact designs with high efficiency. Exact D-optimal designs with minimal supports are also proposed for practical consideration.