Mixture Designs

Presented byAhmed Hassan Sayed

The American University in Cairo

Interdisciplinary Engineering Programs

ENGR 592

Dr. Lotfi GaafarMixture DesignsSimplex Lattice Simplex

Centroid

Mixture Designs

Introduction• In many cases, products are made by blending more than one

ingredient together. • Usually the manufacturer of each of these products is

interested in one or more properties of the final product, which depends on the proportions of the ingredients used

• Examples: – Cake formulations (by blending baking powder, shortening, four,

sugar, and water), property of interest is the fluffiness of cake.– Construction Concrete (made by mixing sand, water, and

cement), property of interest is the compressive strength

Mixture DesignsMixture Designs: When to

use?• Experiments in which the response depends on

factors that represents proportions of a blend.

• The fact that the proportions must sum up to a constant (usually 1 or 100%) makes this type of experiments a class on its own.

Mixture DesignsMixture designs:

Representation• From previous discussion:

• Geometrically, it’s represented by a q-1 dimensional simplex (q is no. of components)

• All experimental points lies on, or inside the simplex region:– 2-component: a straight line.– 3-component: a triangle.– 4-component: tetrahedron.

0.1...211

q

q

ii xxxx

Mixture DesignsMixture designs:

Representation• Points on the vertices of the simplex region are called

pure or single component mixture.• Triangular coordinates (3 component system):

– Points (1,0,0), (0,1,0), and (0,0,1) corresponds to the pure blends of A, B, and C.

Mixture DesignsMixture designs:

Representation• The response can be represented by the surface above

the triangle in 3D or as a contour plot, where each contour line represents a specific response.

Mixture DesignsSteps in Planning a Mixture

Experiment1. Objective and problem statement.2. Select the mixture component, and the levels they will assume

with (proportions).3. Identify the property of interest or the response variable.4. Decide what's the appropriate model to fir your data, and choose

a design that will be enough to both fit the model and allows for statistical test of the model.

5. Perform the experiment based on the design points.6. Analyze the data (Ex: ANOVA)7. Draw conclusions from the analysis and state your

recommendations.

Mixture DesignsObjectives & Underlying

Assumptions• The objective of a mixture experiments can be

one of the following:

– Empirically predict the response of interest for certain components proportions.

– Obtain some "measure of influence" of single components and with other components as well on the response of interest.

Mixture DesignsObjectives & Underlying

Assumptions• The assumptions made for this type of

experiments:– Errors and normally independently distributed with a

mean of zero and a constant variance.– The true response surface is continuous over the

entire region.– The response is assumed to be only dependent on

the ingredients proportions and not on the amount of the mixture itself.

Mixture DesignsFit Models: Canonical

Polynomials• Consider fitting a 1st order polynomial to a 3 –

component mixture

• Due to the fact that all proportions must sum up to one, this can be rewritten as (Multiply intercept by the components sum):

• Where:

332211 xxxy o

3*32

*21

*1 xxxy

1*1 o 2

*2 o 3

*3 o

Mixture Designs

Simplex Lattice Designs• Points are evenly distributed allover the

simplex range. (a lattice)• For a model of degree m, the proportions

assumed by each components are:

• For a q component mixture, the lattice is referred to by the notation {q, m}

1,...,2,1,0mm

xi

Mixture DesignsSimplex Lattice : {3,2}

Example• For q= 3, m = 2. The proportions assumed are:

• Using all possible Combinations:

1,21,0ix

)2/1,2/1,0(),2/1,0,2/1(),0,2/1,2/1(),1,0,0(),0,1,0(),0,0,1(),,( 321 xxx

Mixture DesignsSimplex Lattice : {3,2}

Example• Points on the vertices represent pure

mixtures, points on the edges represent binary blends. )0,5.0,5.0(

)1,0,0()5.0,5.0,0(

)0,0,1(

)0,5.0,5.0(

)1,0,0(

)5.0,5.0,0(

)5.0,0,5.0(

)0,1,0(

Mixture Designs

Simplex Centroid Designs• Consists of all possible mixtures that have

equally weighted proportions of one to q components.

• q permutations of (1,0,0..,0), (q Choose 2) permutations of (1/2,1/2,0,0,…,0) till we reach the overall centroid (1/q, 1/q, …, 1/q)ز

Mixture Designs

)3/1,3/1,3/1(

)1,0,0()5.0,5.0,0(

)0,0,1(

)0,5.0,5.0(

)1,0,0(

)5.0,5.0,0(

)5.0,0,5.0(

)0,1,0(

)3/1,3/1,3/1(

Simplex Centroid : 3 component Example

• For q = 3, the simplex space will contain the following points:

(1,0,0), (0,1,0), (0,0,1), (.5,.5,0), (.5,0,.5), (0,.5,.5), (1/3, 1/3, 1/3).

Mixture Designs

Mixture Designs: ANOVA• Augmented points are usually added to give

more degrees of freedom for error lack of fit, and model significance analysis.

• Simplex Lattice: – Overall simplex center, if not present in the design.– A 50-50 combination between the simplex center and

its vertices.• Simplex Centroid:

– The 50-50 combination between the simplex center and its vertices.

Mixture Designs

Mixture Designs: ANOVA• Augmented points as it appears on both the

simplex lattice and the simplex centroid designs.)3/1,3/1,3/1(

)1,0,0()5.0,5.0,0(

)0,0,1(

)0,5.0,5.0(

)1,0,0(

)5.0,5.0,0(

)5.0,0,5.0(

)0,1,0(

)3/2,6/1,6/1()6/1,3/2,6/1(

(6/1,6/1,3/2)

(3/1,3/1,3/1)

Mixture Designs

Mixture Designs: ANOVA• Pure Error and Lack of Fit Test:

– Replicated Runs should exist to enable it– If the residuals variability >> the pure error

variability, then it can be concluded that there are differences in the blends that your model cannot explain, and hence there's lack of fit.

Mixture Designs

Mixture Designs: ANOVA• Sequential Build up of Model:

– If one is not sure about the degree of the model, start with simple linear model and go up.

– Model significance is statistically tested each time a higher order is fitted.

– test continues until there's no significant improvement in the model fit.

Mixture Designs

Mixture Designs: ANOVA• Sequential Build up of Model: Example

ANOVA; Var.:DV (mixt4.sta)

 3 Factor mixture design; Mixture total=1., 14 Runs

Sequential fit of models of increasing complexity

 SSdfMSSSdfMS   R-sqr

ModelEffectEffectEffectErrorErrorErrorFpR-sqrAdj.

Linear44.755222.37846.872114.26115.25160.02510.48840.3954

Quadratic30.558310.18616.31482.03934.99490.03070.8220.7107

Special Cubic0.71910.71915.59672.22790.32250.58780.82980.6839

Cubic8.22932.7437.36741.84171.48930.34520.91960.7387

Total Adjusted91.627137.048       

Mixture DesignsSimplex Lattice & Simplex

Centroid: Comparison• Based on a 3 component, 10 runs

experiment.

– {3,3} simplex lattice design.– 3 component simplex centroid design with 3

augmented points

Mixture DesignsSimplex Lattice & Simplex

Centroid: Comparison• {3,3} simplex lattice (left), 3 component augmented

simplex centroid (right))0,0,1(

)3/2,0,3/1(

)3/1,0,3/2()0,3/1,3/2(

)0,3/2,3/1(

)3/2,3/1,0()3/1,3/2,0( )1,0,0()0,1,0(

)3/1,3/1,3/1(

)3/1,3/1,3/1(

)1,0,0()5.0,5.0,0(

)5.0,0,5.0(

)0,5.0,5.0(

)1,0,0(

)5.0,5.0,0(

)5.0,0,5.0(

)0,1,0(

)3/1,3/1,3/1(

)6/1,6/1,3/2(

)3/2,6/1,6/1()6/1,3/2,6/1(

Mixture DesignsSimplex Lattice & Simplex

Centroid: Comparison• Fitted Model:

– Simplex lattice:• Supports a cubic model fit:

• The last three terms allows studying changes in response shape of orders higher than quadratic for binary blends.

)()()( 323223313113212112

221123322331132112332211

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Mixture DesignsSimplex Lattice & Simplex

Centroid: Comparison• Fitted Model:

– Simplex Centroid:• Supports fitting a special quadratic model:

• This model helps to detect curvature of the response surface in the interior of the triangle, which cannot be done by the cubic model of the simplex lattice design

232112333

221122332

211123

322331132112332211

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Mixture DesignsSimplex Lattice & Simplex

Centroid: Comparison• Information Distribution throughout the

experimental Region:– Simplex Centroid: More uniform distribution in

the interior of the triangle.– Simplex Lattice: More information about

response surface behavior for binary blends.

Mixture DesignsSimplex Lattice & Simplex

Centroid: Comparison• Model Sequential Buildup:

– Augment Simplex Centroid is more powerful, where power refers to the rejection of zero lack of fit when the true surface is more complicated than it can be described by terms in the fitted model.

Mixture Designs

Important Remarks• Lower Bounds:

– New space is scaled.– Use of pseudo-components.

Mixture Designs

Important Remarks• Upper and Lower Bounds:

– Simplex space is no longer a triangle.– Cannot use standard simplex lattice or

centroid designs to predict the response.

Mixture Designs

Important Remarks• Mixture-amount Experiments

– If the third assumption of mixture experiments is violated, that is, the mixture depends on both the components proportions and the mixture amount, the experiment becomes Mixture-amount experiments.

Mixture Designs

References

Cornell, J.A., Experiments With Mixtures. John Wiley & Sons Inc. 1990Experimental Designs by statease.pdf (Statease software help file)http://www.statsoft.com/textbook/stexdes.htmlhttp://www.6sigma.us/handbook/pri/section5/pri54.htm

Mixture Designs

Thank You!

Questions?