R-optimal designs for second-order Scheffé model with qualitative factors

Considering a mixture model with qualitative factors, the $R$-optimal design problem is investigated when the levels of the qualitative factor interact with the mixture factors. In this paper, the conditions for $R$-optimality of designs with mixture and qualitative factors are presented. General analytical expressions are also derived for the decision function under the $R$-optimal designs, in order to verify that the resulting designs satisfy the general equivalence theorem. In addition, the relative efficiency of the $R$-optimal design is discussed.