Mixture Designs Simplex Lattice Simplex Centroid The American University in Cairo Interdisciplinary Engineering Programs ENGR 592 Dr. Lotfi Gaafar Mixture Designs Simplex Lattice Simplex Centroid Presented by Ahmed Hassan Sayed

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 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 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.

Mixture 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 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.

Steps in Planning a Mixture Experiment Objective and problem statement. Select the mixture component, and the levels they will assume with (proportions). Identify the property of interest or the response variable. 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. Perform the experiment based on the design points. Analyze the data (Ex: ANOVA) Draw conclusions from the analysis and state your recommendations.

Objectives & 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.

Objectives & 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.

Fit 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:

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}

Simplex Lattice : {3,2} Example For q= 3, m = 2. The proportions assumed are: Using all possible Combinations:

Simplex Lattice : {3,2} Example Points on the vertices represent pure mixtures, points on the edges represent binary blends.

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

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: 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: ANOVA Augmented points as it appears on both the simplex lattice and the simplex centroid designs. ) 6 / 1 , 3 2 ( ( 1 / 3 , 1 / 3 , 1 / 3 )

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: 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: 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 R-sqr MS df SS Adj. p F Error Effect Model 0.3954 0.4884 0.0251 5.2516 4.2611 11 46.872 22.378 2 44.755 Linear 0.7107 0.822 0.0307 4.9949 2.0393 8 16.314 10.186 3 30.558 Quadratic 0.6839 0.8298 0.5878 0.3225 2.2279 7 15.596 0.719 1 Special Cubic 0.7387 0.9196 0.3452 1.4893 1.8417 4 7.367 2.743 8.229 Cubic 7.048 13 91.627 Total Adjusted

Simplex 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

Simplex Lattice & Simplex Centroid: Comparison {3,3} simplex lattice (left), 3 component augmented simplex centroid (right)

Simplex 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.

Simplex 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

Simplex 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.

Simplex 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.

Important Remarks Lower Bounds: New space is scaled. Use of pseudo-components.

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.

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.

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

Thank You! Questions ?