Process/product optimization using design of experiments and response surface methodology

Mikko Mäkelä

Sveriges landbruksuniversitetSwedish University of Agricultural Sciences

Department of Forest Biomaterials and TechnologyDivision of Biomass Technology and ChemistryUmeå, Sweden

Contents

Practical course, arranged in 4 individual sessions:

Session 1 – Introduction, factorial design, first order models

Session 2 – Matlab exercise: factorial design

Session 3 – Central composite designs, second order models, ANOVA,

blocking, qualitative factors

Session 4 – Matlab exercise: practical optimization example on given data

Session 1

Introduction

Why experimental design

Factorial design

Design matrix

Model equation = coefficients

Residual

Response contour

Session 2

Factorial design

Research problem

Design matrix

Model equation = coefficients

Degrees of freedom

Predicted response

Residual

ANOVA

R2

Response contour

Session 3

Central composite designs

Design variance

Common designs

Second order models

Stationary points

ANOVA

Blocking

Confounding

Qualitative factors

Central composite designs

f(x) f(x)

x1 x2 x1 x2x3

First order f(x) Second order f(x)

Central composite designs

Second order models through

Center-points

Axial points

αnc

Central composite designs

Center-points (nc)

Pure error (lack of fit)

Curvature

Axial points (α)

Quadratic terms

Spherical designα > 1

Cuboidal designα = 1

Central composite designs

Design characteristics nc and α

Pure error (lack of fit)

Estimated error distribution

Area of operability

Control over factor levels

Central composite designs

Practical design optimality

Model parameters (βi)

Prediction ( ) quality

Prediction ( ) quality emphasized

Design rotatability

r

[0, 0]

SPV = f(r)

Scaled prediction variance (SPV):

SPVNVar x

σ

Central composite designs

CCD, k 2, 2, 5

CCD, k 2, 2, 1

Scaled prediction variance

Central composite designs

Common designs

Central composite α > 1

Central composite designs

Common designs

Central composite α = 1

Central composite designs

Common designs

Box-Behnken

Second order models

First order models

Main effects

Main effects + interactions

Second order models

Main effects + interactions + quadratic terms

⋯

Second order models

N:o xi xj xij xii xjj

1 -1 -1 1 12 1 -1 -1 13 -1 1 -1 14 1 1 1 15 -α 0 0 06 α 0 0 07 0 -α 0 α2

8 0 α 0 α2

9 0 0 0 010 0 0 0 011 0 0 0 0

Design matrix, k = 2

Factorial

Axial

Center-points

Research problem

A central composite design was

performed for a tire tread compound

Two factors x1 and x2

Axial distance α = 1.633

N:o of center-point nc = 4

Measured response, y

Tire abrasion index

Factor Factor levelsx1 -1.633 -1 0 1 1.633x2 -1.633 -1 0 1 1.633

Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 275.

Research problemN:o x1 x2 x12 x11 x22 y

1 -1 -1 1 1 1 2702 1 -1 -1 1 1 2703 -1 1 -1 1 1 3104 1 1 1 1 1 2405 -1.633 0 0 2.667 0 5506 1.633 0 0 2.667 0 2607 0 -1.633 0 0 2.667 5208 0 1.633 0 0 2.667 3809 0 0 0 0 0 52010 0 0 0 0 0 29011 0 0 0 0 0 58012 0 0 0 0 0 590

Factorial

Axial

Center-points

Research problem

Unrefined coefficients

Contour

Second order models

Second order models can include stationary points:

Saddle point Maximum/minimum

Second order models

Stationary point character can be described

Fitted second order model (k = 2)

Derivation 0 results in

2 0

2 0

Second order models

For analysing a stationary point

′ where

⋯ , ⋮ and

/2 ⋯ /2⋯ /2⋱ ⋮

sym.

→ location and character

Second order models

Stationary point location

From the previous example

0.5 0.2

485.8

Second order models

Stationary point character

/2 ⋯ /2⋯ /2⋱ ⋮

sym.

Eigenvalues

, , ⋯ , all < 0 → Maximum

, , ⋯ , all > 0 → Minimum

, , ⋯ , mixed in sign → Saddle point..

Coefficients

Response dependent of a coefficient

H0: ⋯ 0

H1: 0 for at least one j

Lack of fit

Corrected cp residuals vs. others

→ Sufficiently fitted model?

ANOVA

ANOVA based on the F test

Tests if two sample populations

have equal variances (H0)

Ratio of variances and respective dfs

Distribution for every combination of dfs

One- or two-tailed

Alternative hypothesis (H1)

upper one-tailed (reject H0 if F F ∝, df , df )

ANOVA

ANOVA

Parameter df Sum of squares (SS)

Meansquare (MS) F-value p-value

Total corrected n-1 SStot MStot

Regression k SSmod MSmod MSmod/MSres

<0.05>0.05

Residual n-p SSres MSres

Lack of fitn-p-

(nc-1) SSlof MSlof MSlof/MSpe

<0.05>0.05

Pure error nc-1 SSpe MSpe

p = k + 1

MS = SS / df

Research problem

An extraction process (x1,x2,x3) was studied using a cuboidal central

composite design (α = 1, nc = 3) for maximizing yield

Statistically significant coefficients x1, x2, x3 and x12

Responses (in order): 56.6, 58.5, 48.9, 55.2, 61.8, 63.3, 61.5, 64, 61.3,

65.5, 64.6, 65.9, 63.6, 65.0, 62.9, 63.8, 63.5

Present a full ANOVA table

Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 266.

Research problem

Sum of squares for pure error

SS of center-points corrected for the (center-point) mean

Parameter df Sum of squares (SS)

Meansquare (MS) F-value p-value

Total corrected

Regression

Residual

Lack of fit

Pure error

ANOVA

Response transformations or modification of model terms might alleviate

lack of fit

Blocking

Blocking/confounding can be used to separate nuisance effects Different batches of raw materials

Varying conditions on different days

Blocking

Replicated designs arranged in different blocks

Confounding

A single design divided into different blocks

→ 2k design in 2p blocks where p < k

In a 23 design with 2 blocks, confound nuisance to x123

N:o x1 x2 x3 x123 y1 - - - - 902 + - - + 643 - + - + 814 + + - - 635 - - + + 776 + - + - 617 - + + - 888 + + + + 53

Blocking

E.g. 2 blocks based on the x123 interaction (randomized within blocks)

Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 126.

Blocking

b(2:8) bs(2:8)

11.90.92.41.40.91.63.4

11.93.42.41.40.90.91.6

Qualitative factors

Design factors can be

Quantitative (continuous)

Qualitative (discrete)

→ Use of switch variables for discrete factors

E.g. effect of temperature and solvent (A, B or C) on extraction

where

1 if A isdiscrete level and 1 ifB isthediscrete level0 otherwise 0 otherwise

Qualitative factors

Session 3

Central composite designs

Design variance

Common designs

Second order models

Stationary points

ANOVA

Blocking

Confounding

Qualitative factors

Nomenclature

Center-point

Axial point

Lack of fit

Prediction

Rotatability

Stationary point

Saddle point

Minimum

Maximum

Analysis of variance (ANOVA)

Response transformation

Blocking

Confounding

Qualitative factors

Contents

Practical course, arranged in 4 individual sessions:

Session 1 – Introduction, factorial design, first order models

Session 2 – Matlab exercise: factorial design

Session 3 – Central composite designs, second order models, ANOVA,

blocking, qualitative factors

Session 4 – Matlab exercise: practical optimization example on given

data

Thank you for listening!

Please send me an email that you are attending the course

mikko.makela@slu.se