An intensive practical course mainly for PhD-students on the use of designs of experiments (DOE) and response surface methodology (RSM) for optimization problems. The course covers relevant background, nomenclature and general theory of DOE and RSM modelling for factorial and optimisation designs in addition to practical exercises in Matlab. Due to time limitations, the course concentrates on linear and quadratic models on the k≤3 design dimension. This course is an ideal starting point for every experimental engineering wanting to work effectively, extract maximal information and predict the future behaviour of their system. Mikko Mäkelä (DSc, Tech) is a postdoctoral fellow at the Swedish University of Agricultural Sciences in Umeå, Sweden and is currently visiting the Department of Chemical Engineering at the University of Alicante. He is working in close cooperation with Paul Geladi, Professor of Chemometrics, and using DOE and RSM for process optimization mainly for the valorization of industrial wastes in laboratory and pilot scales.”

1. 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

2. 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

3. 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

4. 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

5. Session 1
Introduction
 Why experimental design
Factorial design
 Design matrix
 Model equation = coefficients
 Residual
 Response contour

6. If the current location is
known, a response surface
provides information on:
- Where to go
- How to get there
- Local maxima/minima
Response surfaces

7. Is there a difference?
vs. ?
Mäkelä et al., Appl. Energ. 131 (2014) 490.

8. Research problem
܂,۾
 A and B constant reagents
 C reaction product (response), to be maximized
 T and P reaction conditions (continuous factors), can be regulated

9. Response as a contour plot
What kind of equation could
describe C behaviour as a
function of T and P?
C = f(T,P)

10. 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

11. How can we do it?
The expert method

12. How can we do it?
The shotgun method

13. How can we do it?
The ”Soviet” method
 xk possibilities with k
factors on x levels
 2 factors on 4 levels = 16
experiments

14. How can we do it?
The classical method
P fixed
x
T fixed

15. How can we do it?
Factorial design
 ΔT, ΔP
 Factor interaction (diagonal)

16. Why experimental design?
 Reduce the number of experiments
→ Cost, time
 Extract maximal information
 Understand what happens
 Predict future behaviour

17. 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

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

19. 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!

20. 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

21. 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

22. 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?

23. 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!

24. 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

25. 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!

26. 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.

27. 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

28. 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

29. 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!

30. 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

31. Factorial design
- +
T
+
-
P
+
-
K
- +
TxP
- +
TxK
PxK
+
-
Main effects and interactions:

32. 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?

33. Factorial design
An estimate of model error needed
 Center-points
 Duplicated experiments
 Model residual
܍ ൌ ܡ െ ܆܊ ൌ ܡ െ ࢟ෝ
ݕ௜
݁௜
ݕො௜

34. Factorial design
Error estimation allows significant testing
Remove insignificant coefficients
 Leave main effects
 Important interaction, main effect
cannot be removed

35. Factorial design
Error estimation allows significant testing
Remove insignificant coefficients
 Leave main effects
 Important interaction, main effect
cannot be removed
Recalculate significance upon removal!

36. Factorial design
Model residuals
 Checking model adequacy
 Finding outliers
 Normally distributed
→ Random error
Several ways to present residuals
 Possibility for response transformation

37. Factorial design
R2 statistic
 Explained variability of
measured response
R2 = 0.9962
 99.6% explained

38. Factorial design
More things to look at
 Normal distribution of coefficients
 Residual
 Standardized residual
 Residual histogram
 Residual vs. time
 ANOVA

39. Factorial design

40. 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

41. Session 1
Introduction
 Why experimental design
Factorial design
 Design matrix
 Model equation = coefficients
 Residual
 Response contour

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

43. 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

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