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


Upper and/or lower bound constraints may be present  In mixture designs when there are constraints on the component proportions, these are often upper and/or lower bound constraints of the form L_{i} ≤ x_{i} ≤ U_{i}, i = 1, 2,..., q, where L_{i} is the lower bound for the ith component and U_{i} the upper bound for the ith component. The general form of the constrained mixture problem is  
Typical additional constraints 
L_{i} x_{i} U_{i}, for i = 1, 2,..., q 

Example using only lower bounds 
Consider the following case in which only the lower bounds in the
above equation are imposed, so that the constrained mixture problem
becomes
L_{i} ≤ x_{i} ≤ 1, for i = 1, 2,..., q


Feasible mixture region  The feasible mixture space is shown in the figure below. Note that the existence of lower bounds does not affect the shape of the mixture region, it is still a simplex region. In general, this will always be the case if only lower bounds are imposed on any of the component proportions.  
Diagram showing the feasible mixture space 
FIGURE 5.12: The Feasible Mixture Space (Shaded Region) for Three Components with Lower Bounds 

A simple transformation helps in design construction and analysis  Since the new region of the experiment is still a simplex, it is possible to define a new set of components that take on the values from 0 to 1 over the feasible region. This will make the design construction and the model fitting easier over the constrained region of interest. These new components ( \( x_{i}^{\star} \) ) are called pseudo components and are defined using the following formula  
Formula for pseudo components 
\[ x_{i}^{\star} = \frac{x_{i}  L_{i}} {1  L} \] with \[ L = \sum_{i=1}^{q}{L_{i}} < 1 \] denoting the sum of all the lower bounds. 

Computation of the pseudo components for the example  In the three component example above, the pseudo components are  


Constructing the design in the pseudo components 
Constructing a design in the pseudo components is accomplished by
specifying the design points in terms of the
and then converting them to the original component settings using


Select appropriate design  In terms of the pseudo components, the experimenter has the choice of selecting a SimplexLattice or a SimplexCentroid design, depending on the objectives of the experiment.  
Simplexcentroid design example (after transformation)  Suppose, we decided to use a Simplexcentroid design for the threecomponent experiment. The table below shows the design points in the pseudo components, along with the corresponding setting for the original components.  
Table showing the design points in both the pseudo components and the original components 


Use of pseudo components (after transformation) is recommended  It is recommended that the pseudo components be used to fit the mixture model. This is due to the fact that the constrained design space will usually have relatively high levels of multicollinearity among the predictors. Once the final predictive model for the pseudo components has been determined, the equation in terms of the original components can be determined by substituting the relationship between x_{i} and \( x_{i}^{\star} \).  
Doptimal designs can also be used  Computeraided designs (Doptimal, for example) can be used to select points for a mixture design in a constrained region. See Myers and Montgomery (1995) for more details on using Doptimal designs in mixture experiments.  
Extreme vertice designs are another option  Note: There are other mixture designs that cover only a subportion or smaller space within the simplex. These types of mixture designs (not covered here) are referred to as extreme vertices designs. (See chapter 11 of Myers and Montgomery (1995) or Cornell (1990). 