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

M. Mäkelä

Sveriges landbruksuniversitetSwedish University of Agricultural Sciences

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

DOE and RSM

DOE RSM

You

Design of experiments (DOE)

Planning experiments

→ Maximum information from

minimized number of experiments

Response Surface Methodology (RSM)

Identifying and fitting an appropriate

response surface model

→ Statistics, regression modelling &

optimization

What to expect?

Background and philosophy

Theory

Nomenclature

Practical demonstrations and exercises (Matlab)

What not?

Matrix algebra

Detailed equation studies

Statistical basics

Detailed listing of possible designs

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 1Introduction

Why experimental design

Factorial design

Design matrix

Model equation = coefficients

Residual

Response contour

If the current location is

known, a response surface

provides information on:

- Where to go

- How to get there

- Local maxima/minima

Response surfaces

Is there a difference?

vs. ?

Mäkelä et al., Appl. Energ. 131 (2014) 490.

Research problem

,

A and B constant reagents

C reaction product (response), to be maximized

T and P reaction conditions (continuous factors), can be regulated

Response as a contour plot

What kind of equation could

describe C behaviour as a

function of T and P?

C = f(T,P)

What else do we want to know?

Which factors and interactions are important

Positions of local optima (if they exist)

Surface and surface function around an

optimum

Direction towards an optimum

Statistical significance

How can we do it?

The expert method

How can we do it?

The shotgun method

How can we do it?

The ”Soviet” method

xk possibilities with k

factors on x levels

2 factors on 4 levels = 16

experiments

How can we do it?

The classical method

P fixed

T fixed

x

How can we do it?

Factorial design

∆T, ∆P

Factor interaction (diagonal)

Why experimental design?

Reduce the number of experiments

→ Cost, time

Extract maximal information

Understand what happens

Predict future behaviour

Challenges

Multiple factors on multiple levels

6 factors on 3 levels, 36 experiments

Reduce number of factors

Only 2 levels

→ Discard factors

= SCREENING

1

23

1

2

Factorial design

T

P

N:o T P

1 80 2

2 120 2

3 80 3

4 120 32

3

12080

Factorial design

T

P

1-1-1

1In coded levels

The smallest possible full factorial design!

N:o T T coded

P P coded

1 80 -1 2 -1

2 120 1 2 -1

3 80 -1 3 1

4 120 1 3 1

Factorial design

T

P

25 35

45 75

1-1-1

1Design matrix:

N:o T P C

1 -1 -1 25

2 1 -1 35

3 -1 1 45

4 1 1 75

Factorial design

T

P

25 35

45 75

1-1-1

1Average T effect:

T = 20

Average P effect:

P = 30

Interaction (TxP) effect:

TxP = 10

Research problem

, ,

A and B constant reagents

C reaction product (response), to be maximized

T, P and K reaction conditions (continuous factors) at two different levels

Number of experiments 23 = 9 ([levels][factors])

How to select proper factor levels?

Research problem

Empirical model:

, ,

⋯

In matrix notation:

→

yy⋮y

1 ⋯1 ⋯1 ⋮ ⋮ ⋱ ⋮1 ⋯

bb⋮b

ee⋮e

Measure ChooseUnknown!

Factorial design

First step

Selection and coding of factor levels

→ Design matrix

T = [80, 120]

P = [2, 3]

K = [0.5, 1]

0.5

280 120

3

1

P

T

K

Factorial design

Factorial design matrix

Notice symmetry in diffent colums

Inner product of two colums is zero

E.g. T’P = 0

This property is called orthogonality

N:o Order T P K

1 -1 -1 -1

2 1 -1 -1

3 -1 1 -1

4 1 1 -1

5 -1 -1 1

6 1 -1 1

7 -1 1 1

8 1 1 1

Randomize!

Orthogonality

For a first-order orthogonal design, X’X is a diagonal matrix:

If two columns are orthogonal, corresponding variables are linearly independent, i.e., assessed independent of each other.

1 11 11 11 1

, 1 1 1 11 1 1 1

1 1 1 11 1 1 1

1 11 11 11 1

4 00 4

2x4

4x22x2

Factorial design

N:o T P K Resp. (C)

1 -1 -1 -1 60

2 1 -1 -1 72

3 -1 1 -1 54

4 1 1 -1 68

5 -1 -1 1 52

6 1 -1 1 83

7 -1 1 1 45

8 1 1 1 80

Design matrix:

-1

-1-1 1

1

1

60 72

52 83

6854

45 80

T

PK

Factorial design

Model equation, main terms:

where

denotes response

factor (T, P or K)

coefficient

residual

mean term (average level)

N:o T P K Resp. (C)

1 -1 -1 -1 60

2 1 -1 -1 72

3 -1 1 -1 54

4 1 1 -1 68

5 -1 -1 1 52

6 1 -1 1 83

7 -1 1 1 45

8 1 1 1 80

Factorial design

Equation = coefficients

bbbb

64.211.52.50.8

bo average value (mean term) Large coefficient → important factor

Interactions usually present

Due to coding, the coefficients are comparable!

Factorial designModel equation with interactions:

N:o T P K TxP TxK PxK TxPxK Resp. (C)

1 -1 -1 -1 1 60

2 1 -1 -1 -1 72

3 -1 1 -1 1 54

4 1 1 -1 -1 68

5 -1 -1 1 -1 52

6 1 -1 1 1 83

7 -1 1 1 -1 45

8 1 1 1 1 80

Factorial design

+-

T

+

-P

+-

K

TxP

- +

TxK

+-

PxK

+

-

Main effects and interactions:

Factorial designEquation = coefficients

bbbbbbbb

64.211.52.50.80.85.000.3

Large interaction b13 (TxK)

Important interaction, main effects cannot be removed

→ Which coefficients to include?

Factorial design

An estimate of model error needed

Center-points

Duplicated experiments

Model residual

Factorial design

Error estimation allows significant testing

Remove insignificant coefficients

Leave main effects

Important interaction, main effect

cannot be removed

Factorial design

Error estimation allows significant testing

Remove insignificant coefficients

Leave main effects

Important interaction, main effect

cannot be removed

Recalculate significance upon removal!

Factorial design

Model residuals

Checking model adequacy

Finding outliers

Normally distributed

→ Random error

Several ways to present residuals

Possibility for response transformation

Factorial design

R2 statistic

Explained variability of

measured response

R2 = 0.9962

99.6% explained

Factorial design

More things to look at

Normal distribution of coefficients

Residual Standardized residual

Residual histogram

Residual vs. time

ANOVA

Factorial design

Factorial design

Prediction:

T = 110

K = 0.9

P = 2 (min. level)

Coded location:

1 0.5 1 0.6 0.3

Predicted response:

74.5 2.4

Session 1Introduction

Why experimental design

Factorial design

Design matrix

Model equation = coefficients

Residual

Response contour

Nomenclature

Factorial design

Screening

Design matrix

Model equation

Response

Effect (main/interaction)

Coefficient

Significance

Contour

Residual

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