Exact D-optimal designs for a linear log contrast model with mixture experiment for three and four ingredients

This study investigates the exact D-optimal designs of the linear log contrast model using the mixture experiment suggested by Aitchison and Bacon-Shone (1984) and the design space restricted by Lim (1987) and Chan (1988). Results show that for three ingredients, there are six extreme points that can be divided into two non-intersect sets S1 and S2. An exact N-point D  -optimal design for N=3p+q,p≥1,1≤q≤2N=3p+q,p≥1,1≤q≤2 arranges equal weight n/N,0≤n≤pn/N,0≤n≤p at the points of S1 (S2) and puts the remaining weight (N−3n)/N(N−3n)/N on the points of S2 (S1) as evenly as possible. For four ingredients and N=6p+q,p≥1,1≤q≤5N=6p+q,p≥1,1≤q≤5, an exact N-point design that distributes the weights as evenly as possible among the six supports of the approximate D-optimal design is exact D-optimal.

► Exact D-optimal design puts supports on those of the approximate D-optimal design.
► Suitable Lagrange interpolation polynomial is used for the information matrices.
► The geometric-arithmetic means inequality for matrices is used to show the optimal results.
► The final design structure has to do with the minimum run size of the approximate optimal designs.