Mixture designs for constrained components in orthogonal blocks

It is often the case in mixture experiments that some of the ingredients, such as additives or flavourings, are included with proportions constrained to lie in a restricted interval, while the majority of the mixture is made up of a particular ingredient used as a filler. The experimental region in such cases is restricted to a parallelepiped in or near one corner of the full simplex region. In this paper, orthogonally blocked designs with two experimental blends on each edge of the constrained region are considered for mixture experiments with three and four ingredients. The optimal symmetric orthogonally blocked designs within this class are determined and it is shown that even better designs are obtained for the asymmetric situation, in which some experimental blends are taken at the vertices of the experimental region. Some examples are given to show how these ideas may be extended to identify good designs in three and four blocks. Finally, an example is included to illustrate how to overcome the problems of collinearity that sometimes occur when fitting quadratic models to experimental data from mixture experiments in which some of the ingredient proportions are restricted to small values.