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


Definition of simplex lattice points 
A {q, m} simplexlattice design for q components consists
of points defined by the following coordinate settings: the proportions
assumed by each component take the m+1 equally spaced values
from 0 to 1,


Except for the center, all design points are on the simplex boundaries  Note that the standard SimplexLattice and the SimplexCentroid designs (described later) are boundarypoint designs; that is, with the exception of the overall centroid, all the design points are on the boundaries of the simplex. When one is interested in prediction in the interior, it is highly desirable to augment the simplextype designs with interior design points.  
Example of a three component simplex lattice design 
Consider a threecomponent mixture for which the number of equally spaced levels for each component is four (i.e., x_{i} = 0, 0.333, 0.667, 1). In this example q = 3 and m = 3. If one uses all possible blends of the three components with these proportions, the {3, 3} simplexlattice then contains the 10 blending coordinates listed in the table below. The experimental region and the distribution of design runs over the simplex region are shown in the figure below. There are 10 design runs for the {3, 3} simplexlattice design.  
Design table 


Diagram showing configuration of design runs 
FIGURE 5.9 Configuration of Design Runs for a {3,3} SimplexLattice Design The number of design points in the simplexlattice is (q+m1)!/(m!(q1)!). 

Definition of canonical polynomial model used in mixture experiments  Now consider the form of the polynomial model that one might fit to the data from a mixture experiment. Due to the restriction x_{1} + x_{2} + ... + x_{q} = 1, the form of the regression function that is fit to the data from a mixture experiment is somewhat different from the traditional polynomial fit and is often referred to as the canonical polynomial. Its form is derived using the general form of the regression function that can be fit to data collected at the points of a {q, m} simplexlattice design and substituting into this function the dependence relationship among the x_{i} terms. The number of terms in the {q, m} polynomial is (q+m1)!/(m!(q1)!), as stated previously. This is equal to the number of points that make up the associated {q, m} simplexlattice design.  
Example for a {q, m=1} simplex lattice design 
For example, the equation that can be fit to the points from a
{q, m=1} simplexlattice design is


First order canonical form 
This is called the canonical form of the firstorder mixture model. In general, the canonical forms of the mixture models (with the asterisks removed from the parameters) are as follows:  
Summary of canonical mixture models 


Linear blending portion  The terms in the canonical mixture polynomials have simple interpretations. Geometrically, the parameter β_{i} in the above equations represents the expected response to the pure mixture x_{i}=1, x_{j}=0, i ≠ j, and is the height of the mixture surface at the vertex x_{i}=1. The portion of each of the above polynomials given by \[ \sum_{i=1}^{q}{\beta_{i}x_{i}} \] is called the linear blending portion. When blending is strictly additive, then the linear model form above is an appropriate model.  
Three component mixture example 
The following example is from Cornell (1990) and consists of a threecomponent mixture problem. The three components are Polyethylene (X_{1}), polystyrene (X_{2}), and polypropylene (X_{3}), which are blended together to form fiber that will be spun into yarn. The product developers are only interested in the pure and binary blends of these three materials. The response variable of interest is yarn elongation in kilograms of force applied. A {3,2} simplexlattice design is used to study the blending process. The simplex region and the six design runs are shown in the figure below. The design and the observed responses are listed in Table 5.4. There were two replicate observations run at each of the pure blends. There were three replicate observations run at the binary blends. There are 15 observations with six unique design runs.  
Diagram showing the designs runs for this example 
FIGURE 5.10 Design Runs for the {3,2} SimplexLattice Yarn Elongation Problem 

Table showing the simplex lattice design and observed responses 


Fit a quadratic mixture model 
The design runs listed in the above table are in standard order. The
actual order of the 15 treatment runs was completely randomized.
Since there are three
levels of each of the three mixture components, a quadratic mixture
model can be fit to the data. The results of the model fit are shown
below. Note that there was no intercept in the model.
Summary of Fit RSquare 0.951356 RSquare Adj 0.924331 Root Mean Square Error 0.85375 Mean of Response 13.54 Observations (or Sum Wgts) 15 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Prob > F Model 5 2878.27 479.7117 658.141 1.55e13 Error 9 6.56 0.7289 C Total 14 2884.83 Parameter Estimates Term Estimate Std Error t Ratio Prob>t X1 11.7 0.603692 19.38 <.0001 X2 9.4 0.603692 15.57 <.0001 X3 16.4 0.603692 27.17 <.0001 X2*X1 19 2.608249 7.28 <.0001 X3*X1 11.4 2.608249 4.37 0.0018 X3*X2 9.6 2.608249 3.68 0.0051 

Interpretation of results  Under the parameter estimates section of the output are the individual ttests for each of the parameters in the model. The three cross product terms are significant (X_{1}*X_{2}, X_{3}*X_{1}, X_{3}*X_{2}), indicating a significant quadratic fit.  
The fitted quadratic model 
The fitted quadratic mixture model is
\( \small \hat{y} = 11.7 x_{1} + 9.4 x_{2} + 16.4 x_{3} + 19.0 x_{1} x_{2} + 11.4 x_{1} x_{3}  9.6 x_{2} x_{3} \) 

Conclusions from the fitted quadratic model  Since b_{3} > b_{1} > b_{2}, one can conclude that component 3 (polypropylene) produces yarn with the highest elongation. Additionally, since b_{12} and b_{13} are positive, blending components 1 and 2 or components 1 and 3 produces higher elongation values than would be expected just by averaging the elongations of the pure blends. This is an example of 'synergistic' blending effects. Components 2 and 3 have antagonistic blending effects because b_{23} is negative.  
Contour plot of the predicted elongation values 
The figure below is the contour plot of the elongation values. From
the plot it can be seen that if maximum elongation is desired, a blend
of components 1 and 3 should be chosen consisting of about 75%  80%
component 3 and 20%  25% component 1.
FIGURE 5.11 Contour Plot of Predicted Elongation Values from {3,2} SimplexLattice Design The analyses in this page can be obtained using R code. 