 5. Process Improvement
5.5.4. What is a mixture design?

## Simplex-lattice designs

Definition of simplex-
lattice points
A {q, m} simplex-lattice 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,
xi = 0, 1/m, 2/m, ... , 1 for i = 1, 2, ... , q
and all possible combinations (mixtures) of the proportions from this equation are used.
Except for the center, all design points are on the simplex boundaries Note that the standard Simplex-Lattice and the Simplex-Centroid designs (described later) are boundary-point 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 simplex-type designs with interior design points.
Example of a three-
component simplex lattice design
Consider a three-component mixture for which the number of equally spaced levels for each component is four (i.e., xi = 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} simplex-lattice 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} simplex-lattice design.
Design table
TABLE 5.3  Simplex Lattice Design
X1 X2 X3

0 0 1
0 0.333 0.667
0 0.667 0.333
0 1 0
0.333 0 0.667
0.333 0.333 0.333
0.333 0.6667 0
0.667 0 0.333
0.667 0.333 0
1 0 0
Diagram showing configuration of design runs FIGURE 5.9  Configuration of Design Runs for a {3,3} Simplex-Lattice Design

The number of design points in the simplex-lattice is (q+m-1)!/(m!(q-1)!).

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 x1 + x2 + ... + xq = 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} simplex-lattice design and substituting into this function the dependence relationship among the xi terms. The number of terms in the {q, m} polynomial is (q+m-1)!/(m!(q-1)!), as stated previously. This is equal to the number of points that make up the associated {q, m} simplex-lattice 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} simplex-lattice design is
$$E(Y) = \beta_{0} + \beta_{1}x_{1} + \cdots + \beta_{q}x_{q}$$
Multiplying β0 by (x1 + x2 + ... + xq = 1), the resulting equation is
$$E(Y) = \beta_{1}^{\star}x_{1} + \cdots + \beta_{q}^{\star}x_{q}$$
with $$\small \beta_{i}^{\star} = \beta_{0} + \beta_{i}$$ for all i = 1, ..., q.

First-
order canonical form
This is called the canonical form of the first-order 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 $E(Y) = \sum_{i=1}^{q}{\beta_{i}x_{i}}$ Quadratic $E(Y) = \sum_{i=1}^{q}{\beta_{i}x_{i}} + \sum_{i=1}^{q}{\sum_{i < j}^{q}{\beta_{ij}x_{i}x_{j}}}$ Cubic $\begin{array}{lcl} E(Y) & = & \sum_{i=1}^{q}{\beta_{i}x_{i}} + \sum_{i=1}^{q}{\sum_{i < j}^{q}{\beta_{ij}x_{i}x_{j}}} + \\ & & \sum_{i=1}^{q}{\sum_{i < j}^{q} {\delta{ij}x_{i}x_{j}(x_{i} - x_{j})}} + \\ & & \sum_{k=1}^{q}{\sum_{j < k}^{q}{\sum_{i < j}^{q} {\beta_{ijk}x_{i}x_{j}x_{k}}}} \end{array}$ Special Cubic $\begin{array}{lcl} E(Y) & = & \sum_{i=1}^{q}{\beta_{i}x_{i}} + \sum_{i=1}^{q}{\sum_{i < j}^{q}{\beta_{ij}x_{i}x_{j}}} + \\ & & \sum_{k=1}^{q}{\sum_{j < k}^{q}{\sum_{i < j}^{q} {\beta_{ijk}x_{i}x_{j}x_{k}}}} \end{array}$
Linear blending portion The terms in the canonical mixture polynomials have simple interpretations. Geometrically, the parameter βi in the above equations represents the expected response to the pure mixture xi=1, xj=0, i ≠ j, and is the height of the mixture surface at the vertex xi=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 three-component mixture problem. The three components are Polyethylene (X1), polystyrene (X2), and polypropylene (X3), 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} simplex-lattice 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} Simplex-Lattice Yarn Elongation Problem
Table showing the simplex-
lattice design and observed responses
TABLE 5.4  Simplex-Lattice Design for Yarn Elongation Problem
X1 X2 X3 Observed
Elongation Values

0.0 0.0 1.0 16.8, 16.0
0.0 0.5 0.5 10.0, 9.7, 11.8
0.0 1.0 0.0 8.8, 10.0
0.5 0.0 0.5 17.7, 16.4, 16.6
0.5 0.5 0.0 15.0, 14.8, 16.1
1.0 0.0 0.0 11.0, 12.4
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
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  1.55e-13
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 t-tests for each of the parameters in the model. The three cross product terms are significant (X1*X2, X3*X1, X3*X2), indicating a significant quadratic fit.
$$\small \hat{y} = 11.7 x_{1} + 9.4 x_{2} + 16.4 x_{3} + 19.0 x_{1} x_{2} + 11.4 x_{1} x_{3} - 9.6 x_{2} x_{3}$$  