5.
Process Improvement
5.5. Advanced topics 5.5.4. What is a mixture design?


Definition of simplex centroid designs 
A second type of mixture design is the simplexcentroid design. In the qcomponent simplexcentroid design, the number of distinct points is 2^{q}  1. These points correspond to q permutations of (1, 0, 0, ..., 0) or q single component blends, the \( \small \left( \begin{array}{c} q \\ 2 \end{array} \right) \) permutations of (.5, .5, 0, ..., 0) or all binary mixtures, the \( \small \left( \begin{array}{c} q \\ 3 \end{array} \right) \) permutations of (1/3, 1/3, 1/3, 0, ..., 0), ..., and so on, with finally the overall centroid point (1/q, 1/q, ..., 1/q) or qnary mixture.  
Model supported by simplex centroid designs 
The design points in the SimplexCentroid design will support
the polynomial
\[ \begin{array}{lcl} E(Y) & = & \sum_{i=1}^{q}{\beta_{i}x_{i}} + \sum_{i=1}^{q}{\sum_{i < j}^{q}{\beta_{ij}x_{i}x_{j}}} + \\ & & \sum_{k=1}^{q}{\sum_{j < k}^{q}{\sum_{i < j}^{q} {\beta_{ijk}x_{i}x_{j}x_{k}}}} + \cdots + \\ & & \beta_{12 \dots q}x_{i}x_{j} \dots x_{q} \end{array} \] which is the qthorder mixture polynomial. For q = 2, this is the quadratic model. For q = 3, this is the special cubic model. 

Example of runs for three and four components 
For example, the fifteen runs for a four component (q = 4)
simplexcentroid design are:
