# 11.3.1 - Two Major Types of Mixture Designs

11.3.1 - Two Major Types of Mixture Designs

## Simplex Lattice Design

A {p,m} simplex lattice design for p factors (components) is defined as all possible combination of factor levels defined as

$$x_i=0,\frac{1}{m},\frac{2}{m},\cdots,1 \qquad i=1,2,\ldots,p$$

As an example, the simplex lattice design factor levels for the case of {3,2} will be

$$x_i=0,\frac{1}{2},1 \qquad i=1,2,3$$

Which results in the following design points:

$$(x_1,x_2,x_3)=\{(1,0,0),(0,1,0),(0,0,1),(\frac{1}{2},\frac{1}{2},0),(\frac{1}{2},0,\frac{1}{2}),(0,\frac{1}{2},\frac{1}{2})\}$$

## Simplex Centroid Design

This design which has $$2^{p}-1$$ design points consist of p permutations of (1,0,0,…,0), permutations of $$(1,0,0,\ldots,0),\displaystyle{p\choose 2}$$, permutations of $$(\dfrac{1}{2},\dfrac{1}{2},0,\ldots,0),\displaystyle{p\choose 3}$$, and the overall centroid $$\displaystyle(\dfrac{1}{p},\dfrac{1}{p},\cdots,\dfrac{1}{p})$$. Some simplex centroid designs for the case of p = 3 and p = 4 can be find in Figure 11.41.

Minitab handles mixture experiments which can be accessed through Stat > DOE > Mixture. It allows for building and analysis of Simplex Lattice and Simplex Centroid designs. Furthermore, it covers a third design which is named, Extreme Vertex Design. Application of Extreme Vertex designs are for cases where we have upper and lower constraints on some or all of the components making the design space smaller than the original region.

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