5.
Process Improvement
5.5. Advanced topics 5.5.4. What is a mixture design?
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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,
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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 |
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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)!). |
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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
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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 |
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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 |
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Table showing the simplex- lattice design and observed responses |
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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 1.55e-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 |
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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. | ||||||||||||||||||||||||||||||||||||
The fitted quadratic model |
The fitted quadratic mixture model is
\( \small \hat{y} = 11.7 x_{1} + 9.4 x_{2} + 16.4 x_{3} + 19.0 x_{1} x_{2} + 11.4 x_{1} x_{3} - 9.6 x_{2} x_{3} \) |
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Conclusions from the fitted quadratic model | Since b3 > b1 > b2, one can conclude that component 3 (polypropylene) produces yarn with the highest elongation. Additionally, since b12 and b13 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 b23 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} Simplex-Lattice Design The analyses in this page can be obtained using R code. |