5. Process Improvement 5.5. Advanced topics 5.5.4. What is a mixture design? 5.5.4.3. Simplex-centroid designs
Definition of  simplex-centroid designs	A second type of mixture design is the simplex-centroid design. In the q-component simplex-centroid design, the number of distinct points is 2q-1. These points correspond to q permutations of (1, 0, 0,…,0) or q single component blends, the (q choose 2) permutations of (.5, .5, 0, …,0) or all binary mixtures, the (q choose 3) 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 q-nary mixture.
	The design points in the Simplex-Centroid design will support the polynomial
Model supported by simplex-centroid designs
	which is the q-th-order mixture polynomial. For q = 2, this is the quadratic model. For q = 3, this is the special cubic model.
	For example, the fifteen runs for a four component (q = 4) simplex-centroid design are:  (1,0,0,0), (0,1,0, 0), (0,0,1,0), (0,0,0,1), (.5,.5, 0,0), (.5,0,.5, 0) ..., (0,0,.5,.5), (1/3,1/3,1/3,0), ...,(0,1/3,1/3,1/3), (1/4,1/4,1/4,1/4). The runs for a three component simplex-centroid design of degree 2 are (1,0,0), (0,1,0), (0,0,1), (.5, .5, 0), (.5, 0, .5), (0, .5, .5), (1/3, 1/3, 1/3). However, in order to fit a first order model, only the five terms with a "1" or all "1/4's" would be needed. To fit a second order model, add the six terms with a ".5" (this also fits a saturated third order model, with no degrees of freedom left for error).