S1 - Process product optimization using design experiments and response surface methodolgy

Process/product optimization
using design of experiments and
response surface methodology
M. Mäkelä
Sveriges landbruksuniversitet
Swedish University of Agricultural Sciences
Department of Forest Biomaterials and Technology
Division of Biomass Technology and Chemistry
Umeå, Sweden

DOE and RSM
You
DOE RSM
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 1
Introduction
 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
x
T fixed

How can we do it?
Factorial design
 ΔT, ΔP
 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
2
3

Factorial design
T
3
P
N:o T P
1 80 2
2 120 2
3 80 3
4 120 3
2
80 120

Factorial design
T
1
P
-1 1
-1
In coded levels
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
The smallest possible full factorial design!

Factorial design
45 75
T
1
P
25 35
-1 1
-1
Design matrix:
N:o T P C
1 -1 -1 25
2 1 -1 35
3 -1 1 45
4 1 1 75

Factorial design
45 75
T
1
P
25 35
-1 1
-1
Average T effect:
T = ଻ହାଷହ
ଶ െ ସହାଶହ
ଶ ൌ 20
Average P effect:
P = ଻ହାସହ
ଶ െ ଷହାଶହ
ଶ ൌ 30
Interaction (TxP) effect:
TxP = ଻ହାଶହ
ଶ െ ଷହାସହ
ଶ ൌ 10

Research problem
܂,۾,۹
 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:
ݕࢉ ൌ ݂ ܂, ۾, ۹ ൅ ߝ
ݕ ൌ ߚ଴ ൅ ߚଵݔଵ ൅ ߚଶݔଶ ൅ ⋯ ൅ ߚ௞ݔ௞ ൅ ߝ
In matrix notation:
ܡ ൌ ܆܊ ൅ ܍ →
yଵ
yଶ
⋮
y୬
ൌ
1 ݔଵଵ ݔଶଵ ⋯ ݔଵ௞
1 ݔଵଶ ݔଶଶ ⋯ ݔଶ௞
1 ⋮ ⋮ ⋱ ⋮
1 ݔଵ௡ ݔଶ௡ ⋯ ݔ௡௞
b଴
bଵ
⋮
b୩
൅
eଵ
eଶ
⋮
e୬
Measure Choose
Unknown!

Factorial design
First step
 Selection and coding of factor levels
→ Design matrix
T = [80, 120]
P = [2, 3]
K = [0.5, 1]
0.5
3
1
P
2
80 120
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:
܆ ൌ
െ1 െ1
1 െ1
െ1 1
1 1
, ܆ᇱ ൌ െ1 1 െ1 1
െ1 െ1 1 1
2x4
܆ᇱ܆ ൌ െ1 1 െ1 1
െ1 െ1 1 1
4x2
െ1 െ1
1 െ1
െ1 1
1 1
2x2
ൌ 4 0
0 4
If two columns are orthogonal, corresponding variables are linearly independent,
i.e., assessed independent of each other.

Factorial design
Design matrix:
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
-1
1
1
45 80
54 68
52 83
60 72
-1
-1 1
T
P
K

Factorial design
Model equation, main terms:
ݕ ൌ ߚ଴ ൅ ߚଵݔଵ ൅ ߚଶݔଶ ൅ ߚଷݔଷ ൅ ߝ
where
ݕ denotes response
ݔ௜ factor (T, P or K)
ߚ௜ coefficient
ߝ residual
ߚ଴ 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
܊ ൌ
b଴
bଵ
bଶ
bଷ
ൌ
64.2
11.5
െ2.5
0.8
 bo average value (mean term)
 Large coefficient → important factor
 Interactions usually present
Due to coding, the coefficients are comparable!

Factorial design
Model equation with interactions:
ݕ ൌ ߚ଴ ൅ ߚଵݔଵ ൅ ߚଶݔଶ ൅ ߚଷݔଷ ൅ ߚଵଶݔଵݔଶ ൅ ߚଵଷݔଵݔଷ ൅ ߚଶଷݔଶݔଷ ൅ ߚଵଶଷݔଵݔଶݔଷ ൅ ߝ
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 design
Equation = coefficients
܊ ൌ
b଴
bଵ
bଶ
bଷ
bଵଶ
bଵଷ
bଶଷ
bଵଶଷ
ൌ
64.2
11.5
െ2.5
0.8
0.8
5.0
0
0.3
 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
܍ ൌ ܡ െ ܆܊ ൌ ܡ െ ࢟ෝ
ݕ௜
݁௜
ݕො௜

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
Prediction:
T = 110
K = 0.9
P = 2 (min. level)
Coded location:
ܠܕ ൌ 1 0.5 െ1 0.6 0.3
Predicted response:
ݕො௠ ൌ 74.5 േ 2.4

Nomenclature
Factorial design
Screening
Design matrix
Model equation
Response
Effect (main/interaction)
Coefficient
Significance
Contour
Residual

Thank you for listening!
 Please send me an email that you are attending the course
mikko.makela@slu.se

S1 - Process product optimization using design experiments and response surface methodolgy

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