R-optimal designs for mixture models in two types of regions

This article presents a study of R-optimality for mixture models in two types of regions. One is the direct sum experimental region for additive mixture models. Sufficient conditions are given so that R-optimal design for additive mixture models can be constructed from the R-optimal designs for homogeneous models in sub-mixture systems. On the other hand, we explore the R-optimality of product-type design in cuboidal regions, and the optimal design can be constructed from transformed models. Two examples are given to show the methods of constructing optimal designs.