Planejamento e Otimização de

Experimentos

Mixture Experiments

Prof. Dr. Anselmo E de Oliveira

anselmo.quimica.ufg.br

anselmo.disciplinas@gmail.com

• In mixture experiments, the factors are the

components of ingredients of a mixture, and

consequently their levels are not independent

0 ≤ 𝑥𝑖 ≤ 1 𝑖 = 1,2, … , 𝑝 components

and

𝑥1 + 𝑥2 +⋯+ 𝑥𝑝 = 1

𝒙𝟏 + 𝒙𝟐 = 𝟏

x2

x1 0 1

1

𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 = 𝟏

Trilinear coordinate system

Simplex Designs

• Simplex Designs are used to study the effects of

the mixture components on the response variable

• 𝒑,𝒎 Simplex lattice design

𝑝 components

𝑚+ 1 equally spaced values from 0 to 1

𝑥𝑖 = 0,1

𝑚,2

𝑚,… , 1

All possible combinations (mixtures) of the proportions

are used

Ex: 𝒑 = 𝟑 and 𝒎 = 𝟐 3,2 simplex lattice design

𝑥𝑖 = 0,1

2, 1 𝑖 = 1,2,3

Six runs:

X1, X2, X3 = 1,0,0 , 0,1,0 , 0,0,1 ,1

2,1

2, 0 ,

1

2, 0,1

2, 0,1

2,1

2

The number of points in

a {𝑝,𝑚} simplex lattice

design is

𝑁 =𝑝 +𝑚 − 1 !

𝑚! 𝑝 − 1 !

• An alternative to simplex lattice design

is the simplex centroid design

2𝑝 − 1 points corresponding to 1

𝑝,1

𝑝, … ,1

𝑝

3,2 simplex lattice design: 6 runs

1,0,0 , 0,1,0 , 0,0,1 ,1

2,1

2, 0 ,

1

2, 0,1

2, 0,1

2,1

2

3,2 simplex centroid design: 23 − 1 = 7 runs

1,0,0 , 0,1,0 , 0,0,1 ,1

2,1

2, 0 ,1

2, 0,1

2, 0,1

2,1

2 𝟏

𝟑,𝟏

𝟑,𝟏

𝟑

{3,2}

{4,3}

• It is usually desirable to augment the simplex

lattice or simplex centroid

Mixture Models

• Linear

𝑦 = 𝑏𝑖𝑥𝑖

𝑝

𝑖=1

• Quadractic

𝑦 = 𝑏𝑖𝑥𝑖

𝑝

𝑖=1

+ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑝

𝑖<𝑗

• Full Cubic

𝑦 = 𝑏𝑖𝑥𝑖

𝑝

𝑖=1

+ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑝

𝑖<𝑗

+ 𝛿𝑖𝑗𝑥𝑖𝑥𝑗

𝑝

𝑖<𝑗

𝑥𝑖 − 𝑥𝑗 + 𝑏𝑖𝑗𝑘𝑥𝑖𝑥𝑗𝑥𝑘𝑖<𝑗<𝑘

• Special Cubic

𝑦 = 𝑏𝑖𝑥𝑖

𝑝

𝑖=1

+ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑝

𝑖<𝑗

+ 𝑏𝑖𝑗𝑘𝑥𝑖𝑥𝑗𝑥𝑘𝑖<𝑗<𝑘

– Linear blending portion

𝑏𝑖𝑥𝑖

𝑝

𝑖=1

– When curvature arises from nonlinear blending

between component pairs, the parameters 𝑏𝑖𝑗

represent either synergistic or antagonistic blending

– Higher order terms are frequently necessary in

mixture models

Plots

Mixture Designs with

• Example

Three components – polyethylene(𝑥1), polystyrene(𝑥2), and polypropylene(𝑥3) – were

blended to form fiber that will be spun into yarn for

draperies. The response variable of interest is yarn

elongation in kilograms of force applied. A {3,2}

simplex lattice design is used to study the product.

Design Point

Component Proportions Observed Elongation values

Average Elongation value (𝒚 ) 𝑥1 𝑥2 𝑥3

1 1 0 0 11.0,12.4 11.7

2 ½ ½ 0 15.0,14.8,16.1 15.3

3 0 1 0 8.8,10.0 9.4

4 ½ 0 ½ 17.7,16.4,16.6 16.9

5 0 ½ ½ 10.0,9.7,11.8 10.5

6 0 0 1 16.8,16.0 16.4

install.packages("mixexp") library(mixexp) dat <- SLD(3,2) #{3,2} simplex lattice design DesignPoints(dat) #plot mixvars <-c("x1","x2","x3") y <- c(11.7,15.3,9.4,16.9,10.5,16.4) MixModel(dat,"y",mixvars,2) #second-order mixture polinomial coefficients Std.err t.value Prob x1 11.7 NaN NaN NaN x2 9.4 NaN NaN NaN x3 16.4 NaN NaN NaN x2:x1 19.0 NaN NaN NaN x3:x1 11.4 NaN NaN NaN x2:x3 -9.6 NaN NaN NaN Residual standard error: NaN on 0 degrees of freedom Corrected Multiple R-squared: 1 Call: lm(formula = mixmodnI, data = frame) Coefficients: x1 x2 x3 x1:x2 x1:x3 x2:x3 11.7 9.4 16.4 19.0 11.4 -9.6 𝑦 = 11.7𝑥1 + 9.4𝑥2 + 16.4𝑥3 + 19.0𝑥1𝑥2 + 11.4𝑥1𝑥3 − 9.6𝑥2𝑥3

because 𝑏3 > 𝑏1 > 𝑏2 we would conclude that component3 (polypropylene) produces yarn with the highest elongation

because 𝑏12 and 𝑏13 are positive, blending components 1 and 2 or 1 and 3 produces higher elongation values than would be expected just by averaging the elongations of the pure blends (synergistic blending effects)

components 2 and 3 have antagonistic blending effects

MixturePlot(dat$x3,dat$x2,dat$x1,y, x3lab="Fraction x3", x2lab="Fraction x2", x1lab="Fraction x1", corner.labs=c("x3","x2","x1"), constrts=FALSE,contrs=TRUE,cols=TRUE, mod=2,n.breaks=9)

If the maximum elongation is desired, a blend of components 1 and 3 should be chosen consisting of about 80% component 3 and 20% component 1

• In some mixture problems, constraints on the individual components arise – Lower bound constraint

𝑙𝑖 ≤ 𝑥𝑖 ≤ 1 When only lower bound constraints are present, the feasible design region is still a simplex

– Pseudocomponents

𝑥𝑖′ =𝑥𝑖 − 𝑙𝑖

1 − 𝑙𝑖𝑝𝑗=1

with 𝑙𝑖𝑝𝑗=1 < 1 and 𝑥1

′ + 𝑥2′ +⋯+ 𝑥𝑝

′ = 1

𝑥𝑖 = 𝑙𝑖 + 1 − 𝑙𝑖

𝑝

𝑗=1

𝑥𝑖′

– If the components have both upper and lower bound constraints, the feasible region is no longer a simplex: irregular polytope • Computer-generated optimal designs

feasible design region

Simplex centroid design