1. 1
A Hybrid Multi-Objective Evolutionary Algorithm
Using an Inverse Neural Network
A. Gaspar-Cunha(1), A. Vieira(2), C.M. Fonseca(3)
(1)
IPC- Institute for Polymers and Composites, Dept. of Polymer Engineering,
University of Minho, Guimarães, Portugal
(2)
ISEP and Computational Physics Centre,
Coimbra, Portugal
(3)
CSI- Centre for Intelligent Systems, Faculty of Science and Technology,
University of Algarve, Faro, Portugal
HYBRID METAHEURISTICS (HM 2004)
ECAI 2004, Valencia, Spain
August, 2004
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

2. 2
INTRODUCTION
Most real optimization problems are multiobjective
Example: Simultaneous minimization of the cost and maximization
of the performance of a specific system
Dominated solution
Cost
Single optimum
(maximal performance)
Performance
Single optimum
(minimal cost)
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Multiple optima
(both objectives optimized)
PARETO FRONTIER
(set of non-dominated solutions)
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

3. 3
INTRODUCTION
Computation time required to evaluate the solutions
Start
Engineering problems:
Initialise Population
i=0
Black Box
Numerical modelling
routines
• Finite elements
• Finite differences
• Finite volumes
• etc
Evaluation
Assign Fitness
Fi
Convergence
criterion
satisfied?
i=i+1
no
Selection
HIGH COMPUTATION TIMES
yes
Stop
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Recombination
Dept. Polymer Engineering
University of Minho

4. 4
INTRODUCTION
OBJECTIVES:
• Develop an efficient multi-objective optimization
algorithm
• Reduce the number of evaluations of objective
functions necessary
• Compare performance with existing algorithms
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

5. 5
CONTENTS
• Multi-Objective Evolutionary Algorithm (MOEA)
• Artificial Neural Networks (ANN)
• Hybrid Multi-Objective Algorithm (MOEA-IANN)
• Results and Discussion
• Conclusions
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

6. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA6
How to deal with multiple criteria (or objectives)?
Single objective
(for example, weighted sum)
 0 ≤ wj ≤ 1

 ∑ wj = 1

 0 ≤ Fj ≤ 1
0 ≤ FOi ≤ 1

q
FOi = ∑ w j F j
j =1
Decision made before the search
Pareto Frontier
Multiobjective optimization
Decision made after the search
Objective 2
200
1
190
180
2
170
5
6
3
4
160
500
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Faculty of Science and Technology
University of Algarve
1000
Objective 1
1500
Dept. Polymer Engineering
University of Minho
2000

7. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA7
Basic functions of a MOEA:
Maintaining a diverse
nondominated set
(Density estimation)
Density
C2
Archiving
Fitness
C1
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Preventing nondominated
solutions from being lost
(Elitist population - archiving)
Guiding the population
towards the Pareto set
(Fitness assignment)
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Dept. Polymer Engineering
University of Minho

8. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA8
Reduced Pareto Set G.A. with Elitism (RPSGAe)
Start
RPSGAe sorts the population individuals in a number of
pre-defined ranks using a clustering technique, in order
to reduce the number of solutions on the efficient
frontier.
Initialise Population
i=0
a) Rank the individuals using a clustering
Evaluation
algorithm;
b) Calculate
Assign Fitness
Fi
i=i+1
the
fitness
using
a
ranking
function;
c) Copy the best individuals to the external
population;
Convergence
criterion
satisfied?
no
Selection
yes
Stop
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Recombination
d) If the external population becomes full:
- Apply the clustering algorithm to the
external population;
- Copy the best individuals to the internal
population;
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Dept. Polymer Engineering
University of Minho

9. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA9
Clustering algorithm example
NR = r
N=15; Nranks=3
N ranks
r=2; NR=10
r=1; NR=5
C2
N
C2
1
1
2
1
12
1
2
1
2
1
1
2
1
C1
1
C1
Gaspar-Cunha, A., Covas, J.A. - RPSGAe - A Multiobjective Genetic Algorithm with Elitism: Application
to Polymer Extrusion, in Metaheuristics for Multiobjective Optimisation, Lecture Notes in Economics and
Mathematical Systems, Gandibleux, X.; Sevaux, M.; Sörensen, K.; T'kindt, V. (Eds.), Springer, 2004.
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Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

10. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA10
Clustering algorithm example
r=3; NR=15
Fitness - Linear ranking :
C2
1
2( SP − 1) ( N + 1 − i )
FOi = 2 − SP +
N
23
12 3
2
1 3
2
FO(1) = 2.00
FO(2) = 1.87
31
23
FO(3) = 1.73
1
C1
RPSGAe
• Number of Ranks - Nranks
Parameters:
• Limits of indifference of the clustering algorithm - limit
• N. of individuals copied to the external population - Next
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Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

11. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA11
Reduced Pareto Set G.A. with Elitism (RPSGAe)
Internal
population
Next
Internal
population
(Generation n)
External
population
External
population
(Generation n)
Generation 1
Generation 2
Generation 3
Generation 4
Next
Generation 5
Generation n
Order of the RPSGAe: O(Nranks q N2)
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

12. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA12
How the basic functions are accomplished in the RPSGAe :
1. Guiding the population towards the Pareto set
Fitness assignment: ranking function based
reduction of the Pareto Set
on the
2. Maintaining a diverse nondominated set
Density estimation: ranking function based on the reduction
of the Pareto Set
3. Preventing nondominated solutions from being lost
Elitist population: periodic copy of the best solutions (to the
main population), selected with the method of Pareto set
reduction
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Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

13. 13
ARTIFICIAL NEURAL NETWORKS – ANN
Artificial Neural Networks
•
ANN implemented by a Multilayer Preceptron is a flexible scheme capable of
approximating an arbitrary complex function;
•
The ANN builds a map between a set of inputs and the respective outputs;
•
A feed-forward neural network consists of an
array of input nodes connected to an array of
output nodes through successive intermediate
layers;
•
•
Each connection between nodes has a weight,
initially random, which is adjusted during a
training process;
The output of each node of a specific layer is a
function of the sum on the weighted signals
coming from the previous layer;
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Input
Layer
Hidden
Layer
Output
Layer
P1
C1
P2
C2
...
...
Pi
Cj
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University of Minho

14. 14
HYBRID MULTI-OBJECTIVE ALGORITHM
Two possible approachs to reduce the computation time
1. During evaluation – Some solutions can be evaluated
using an approximate function, such as Fitness Inheritance,
Artificial Neural Networks, etc (this reduce the number of
exact evaluations necessary).
2. During recombination – Some individuals can be
generated using more efficient methods (this produce a fast
approximation to the optimal Pareto frontier, thus the
number of generations is reduced).
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Faculty of Science and Technology
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Dept. Polymer Engineering
University of Minho

15. 15
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN
Use of ANN to “Evaluate” some Solutions
Start
Artificial Neural Network
Initialise Population
i=0
Parameters
to optimise
P1
Convergence
criterion
satisfied?
i=i+1
no
P2
C2
...
Pi
Assign Fitness
Fi
C1
...
Evaluation
Criteria
Cj
Selection
yes
Stop
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Engenharia do Porto
Recombination
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

16. 16
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN
Use of ANN to “Evaluate” some Solutions – Method A
Proposed by K. Deb et. al
Neural Network
learning using
some solutions
of the p
generations
Neural Network
learning using
some solutions
of the p
generations
p generations r generations
p generations r generations
RPSGA with RPSGA with
Neural
exact
Network
function
evaluation
evaluation
RPSGA with RPSGA with
Neural
exact
Network
function
evaluation
evaluation
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Engenharia do Porto
Faculty of Science and Technology
University of Algarve
...
...
p generations
RPSGA with
exact
function
evaluation
Dept. Polymer Engineering
University of Minho

17. 17
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN
Use of ANN to “Evaluate” some Solutions – Method B
Neural Network
learning using
some solutions
of the p
generations
Neural Network
learning using
some solutions
of the p
generations
eNN > allowed error
p generations r generations
RPSGA with RPSGA with:
exact
• All solutions
function
(N) evaluated
evaluation
by Neural
Network
• M evaluated
by exact
function
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M
e NN =
S
j =1
(C
NN
i, j
i =1
∑ ∑
− Ci , j
)
2
S
M
eNN > allowed error
p generations r generations
RPSGA with RPSGA with:
exact
• All solutions
function
(N) evaluated
evaluation
by Neural
Network
• M evaluated
by exact
function
Faculty of Science and Technology
University of Algarve
...
...
p generations
RPSGA with
exact
function
evaluation
Dept. Polymer Engineering
University of Minho

18. HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN
18
Use of an Inverse ANN as “Recombination” operator
Start
Recombination operators:
Initialise Population
• Crossover
i=0
• Mutation
• Inverse ANN (IANN)
Evaluation
Criteria
Variables
C1
V1
C2
V2
Selection
...
...
Recombination
Cq
VM
Assign Fitness
Fi
Convergence
criterion
satisfied?
i=i+1
no
yes
Stop
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Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

19. 19
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN
Set of Solutions Generated with the IANN
Selection of n+q solutions from the
• 3.q extreme solutions
• n interior solutions
For j = 1, ..., q
(where, q is the number of criteria) :
∆C2
c
Criterion 2
present population to generate:
b
e1
C j = C 'j + ∆C j
Points 1, 2, …, n:
a
1
2
3
a
4
e2
b
c
Criterion 1 ∆C1
Point ej to a: C j = C j + ∆C j
'
Point ej to b: C j ( j =i ) = C 'j
∧ C j ( j ≠i ) = C 'j + ∆C j
Point ej to c: C j ( j =i ) = C j − ∆C j
'
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∧ C j ( j ≠i ) = C 'j + ∆C j
Faculty of Science and Technology
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Dept. Polymer Engineering
University of Minho

20. HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN
Set of Solutions Generated with the IANN
Use of IANN to generate
new solutions
c
e1
a
1
2
3
a
4
e2
Parameter 2
Criterion 2
∆C2
b
b
c
Criterion 1 ∆C1
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University of Algarve
2
1
b
a
c
4
e1
a
3
e2
b
c
Parameter 1
Dept. Polymer Engineering
University of Minho
20

21. HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN
21
MOEA-IANN Algorithm Parameters
 Number of Ranks - Nranks
 N. of individuals copied to the external population - Next
 Limits of indifference of the clustering algorithm – limit
 Criteria variation at beginning - ∆Cinit
 Criteria variation at end - ∆Cf
 N. of generations which individuals are used to train the IANN – Ngen
 Rate of individuals generated with the IANN – IR
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Dept. Polymer Engineering
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22. 22
RESULTS AND DISCUSSION – Test problems
K. Deb et. al - Test Problem Generator
Minimize f1 ( x1 ) ,
Minimize f 2 ( x2 ) ,
 
Minimize
f q −1 ( xq −1 ),
f q ( x ) = g ( xq ) h( f1 ( x1 ) , f 2 ( x2 ) ,  , f q −1 ( xq −1 ), g ( xq ) ),
Minimize
Subject to
x
xi ∈ ℜ i , for i = 1, 2,  , q.
2 Criteria
2C-ZDT1 (Convex): M = 30; xi ∈ [0, 1]
1.00
f1 ( x1 ) = x1

= g × 1 −


where, g ( x 2 , , x M
Instituto Superior de
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0.60
f1 

g

) = 1+ 9 ∑
M
i =2
f2
f 2 ( x 2 , , x M )
0.80
0.40
0.20
xi
M −1
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0.00
0
0.2
0.4
0.6
0.8
f1
Dept. Polymer Engineering
University of Minho
1

23. 23
RESULTS AND DISCUSSION – Test problems
2 Criteria
2C-ZDT2 (Non-convex): M = 30; xi ∈ [0, 1]
1.00
f1 ( x1 ) = x1
  f1  2 
= g × 1 − 
  g 
 


where, g ( x 2 , , x M ) = 1 + 9
∑
M
i=2
0.60
f2
f 2 ( x 2 , , x M )
0.80
0.40
0.20
xi
0.00
M −1
0
0.4
0.6
1
1.00
f1 ( x1 ) = x1
0.60
f1

−  f 1  sin (10 π f1 ) 

g  g



) = 1+ 9 ∑
M
0.20
f2

f 2 ( x 2 ,  , x M ) = g × 1 −


-0.200
xi
0.4
0.6
-0.60
M −1
0.2
-1.00
i=2
f1
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0.8
f1
2C-ZDT3 (Discrete): M = 30; xi ∈ [0, 1]
where, g ( x 2 , , x M
0.2
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0.8
1

24. 24
RESULTS AND DISCUSSION – Test problems
2 Criteria
2C-ZDT4 (Multimodal): M = 10; x1 ∈ [0, 1]; xi ∈ [-5, 5]
1.40
1.20
f1 ( x1 ) = x1

= g × 1 −


f1


g

f2
f 2 ( x 2 , , x M )
1.00
0.80
0.60
(
where, g ( x 2 , , x M ) = 1 + 10 ( M − 1) + ∑i = 2 xi2 − 10 cos( 4 π xi )
M
0.40
)
0.20
0.00
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
f1
2C-ZDT6 (Non-uniform): M = 10; xi ∈ [0, 1]
1.00
f1 ( x1 ) = 1 − exp(−4 x1 ) sin 6 (6 π x1 )
  f1  2 
= g × 1 − 
  g 
 


where, g ( x 2 , , x M )
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 ∑M xi
= 1 + 9 i = 2
 M −1

0.60
f2
f 2 ( x 2 , , x M )
0.80
0.40




0.25
Faculty of Science and Technology
University of Algarve
0.20
0.00
0
0.2
0.4
f1
Dept. Polymer Engineering
University of Minho

25. 25
RESULTS AND DISCUSSION – Test problems
3 Criteria
3C-ZDT1 (Convex): M = 30; xi ∈ [0, 1]
f1 ( x1 ) = x1
1.0
f 3 ( x 3 , , x M )

= g × 1 −


where, g ( x3 , , x M
f1 f 2
g
) = 1+ 9 ∑




M
i =3
f3
f 2 ( x2 ) = x2
0.5
0.0
0.2
0.4
xi
0.6
f2
M −1
0.4
0.6
0.8
0.8
1.0
0.2
0.0
0.0
f1
1.0
3C-ZDT2 (Non-convex): M = 30; xi ∈ [0, 1]
f1 ( x1 ) = x1
1.0
f 3 ( x3 ,  , x M )
  f f 2 
= g × 1 −  1 2  
  g  
 
 
where, g ( x3 ,  , xM
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) =1+ 9 ∑
M
i =3
xi
M −1
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f3
f 2 ( x2 ) = x 2
0.5
0.0
0.2
0.4
0.6
f2
0.4
0.6
0.8
0.8
1.0
1.0
Dept. Polymer Engineering
University of Minho
f1
0.2
0.0
0.0

26. 26
RESULTS AND DISCUSSION – Test problems
3 Criteria
3C-ZDT3 (Discrete): M = 30; xi ∈ [0, 1]
f1 ( x1 ) = x1
1.00
f 2 ( x2 ) = x2
0.75
0.50

f1 f 2  f1 f 2 

 sin (10 π f1 f 2 ) 
−


g
 g 


0.00
0.0
0.2
0.4
0.6
∑i = 3 x i
M
f2
where, g ( x3 , , x M ) = 1 + 9
0.25
f3

f 3 ( x 3 ,  , x M ) = g × 1 −

M −1
-0.25
0.8
1.0
1.00
1.0
0.8
0.6
0.4
0.2
f1
1.0
0.75
1.000
0.8
0.8125
0.50
0.6250
0.25
0.4375
f3
0.6
0.2500
f2
0.00
0.06250
0.4
-0.1250
-0.25
-0.50
0.0
0.2
0.4
f1 0.6
0.6
0.4
0.8
0.2
1.0
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f2
0.8
1.0
-0.3125
0.2
0.0
0.0
-0.5000
0.2
0.4
0.6
0.8
1.0
f1
0.0
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Dept. Polymer Engineering
University of Minho
0.0
-0.50

27. 27
RESULTS AND DISCUSSION – Test problems
3 Criteria
3C-ZDT4 (Multimodal): M = 10; x1,2 ∈ [0, 1]; xi ∈ [-5, 5]
18
f1 ( x1 ) = x1
16
14
f 2 ( x2 ) = x2
f 3 ( x 3 , , x M )
12

= g × 1 −


f1 f 2
g
f3
10




8
6
4
(
where, g ( x3 , , x M ) = 1 + 10 ( M − 1) + ∑i =3 xi2 − 10 cos( 4 π xi )
M
)
0.0
0.2
0.4
0.6
f2
3C-ZDT6 (Non-uniform): M = 10; xi ∈ [0, 1]
0.4
0.6
0.8
0.8
1.0
0.2
f1
1.0
f 1 ( x1 ) = 1 − exp(−4 x1 ) sin 6 (6 π x1 )
1.0
f 2 ( x 2 ) = 1 − exp(−4 x 2 ) sin 6 (6 π x 2 )
  f f 2 
= g × 1 −  1 2  
  g  
 
 
where, g ( x3 , , x M )
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0.8
 ∑ xi
= 1 + 9 i =3
 M −1

M
0.6
f3
f 3 ( x 3 , , x M )
2
0.0
0
0.4




0.25
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0.2
0.0
0.2
0.4
0.6
f2
0.4
0.6
0.8
0.8
1.0
1.0
Dept. Polymer Engineering
University of Minho
f1
0.2
0.0
0.0

28. 28
RESULTS AND DISCUSSION – Metrics
Hypervolume Metric (Zitzler and Thiele - 1998)
This metric calculates the dominated space volume,
enclosed by the nondominated points and the origin.
S metric:
Volume of the space dominated by
the set of objective vectors
C2
Hypervolume
C1
Criteria C1 and C2 to maximize
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However, is not possible to say
that one set is better than other
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Dept. Polymer Engineering
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29. 29
RESULTS AND DISCUSSION – Algorithm Parameters
Influence of algorithm parameters on performance
Parameter
Tested values(*)
Best results
Influence
Selected
limit
0.01; 0.05; 0.1; 0.2
[0.01; 0.2]
Small
0.01
∆ Cinit
0.3; 0.4; 0.5; 0.6
[0.3; 0.5]
Small
0.5
∆ Cf
0.0; 0.1; 0.2; 0.3
[0.0; 0.3]
Small
0.2
Ngen
5; 10; 15; 20
[5; 10]
Small
5
IR
0.35; 0.50; 0.65; 0.80
[0.35; 0.8]
Small
0.8
(*) 5 runs for each tested parameter value
• The influence of the algorithm parameters on its
performance is very small.
• Each optimisation run was carried out 21 times
using the algorithm parameters selected and
different seed values.
Algorithm Parameters:
- N = 100
- Ne = 100
- Nranks = 30
- Next = 3N/Nranks = 10
- cR = 0.8
- mR = 0.05
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

30. 30
RESULTS AND DISCUSSION – Method B
Use of ANN to “Evaluate” some Solutions – Method B
S metric, 22000 evaluations
Number of evaluations
Test
problem
Method B
RPSGAe
Decrease (%)
Method B
RPSGAe
Decrease (%)
ZDT1
0.851
0.849
0.24
10000
19000
47.4
ZDT2
0.786
0.773
1.68
15300
22000
30.5
ZDT3
2.736
2.554
7.13
18000
22000
18.2
ZDT4
0.1116
0.0807
38.29
5000
22000
77.3
ZDT6
0.599
0.571
4.90
12500
22000
43.2
• The S metric after 22000 evaluations decrease when Method B is
used
• The number of evaluations necessary to attain identical level of the
S metric decreases considerably when Method B is used
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho

31. RESULTS AND DISCUSSION – 2 Criteria Test Problems
MOEA - Inverse ANN
2C-ZDT1
1
S metric
0.8
0.6
0.4
IANN
RPSGAe
0.2
0
0
50
100
150
Generations
200
250
300
• The Inverse ANN approach has the largest improvement during the
first generations, i.e., when the solution is far from the optimum;
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho
31

32. 32
RESULTS AND DISCUSSION – 2 Criteria Test Problems
MOEA - Inverse ANN
2C-ZDT2
2C-ZDT3
2.5
S metric
3
0.8
S metric
1
0.6
0.4
IANN
RPSGAe
0.2
2
1.5
1
0
0
0
100
Generations
200
300
0
2C-ZDT4
0.15
100 Generations 200
300
2C-ZDT6
0.8
0.6
0.1
S metric
S metric
IANN
RPSGAe
0.5
0.05
IANN
RPSGAe
0
0.4
0.2
IANN
RPSGAe
0
0
Instituto Superior de
Engenharia do Porto
100Generations 200
300
Faculty of Science and Technology
University of Algarve
0
100
Generations
200
Dept. Polymer Engineering
University of Minho
300

33. RESULTS AND DISCUSSION – 3 Criteria Test Problems
MOEA - Inverse ANN
3C-ZDT1
0.8
S metric
0.6
0.4
IANN
0.2
RPSGAe
0
0
50
100
150
200
250
300
Generations
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho
33

34. 34
RESULTS AND DISCUSSION – 3 Criteria Test Problems
MOEA - Inverse ANN
3C-ZDT2
0.8
1.5
S metric
S metric
0.6
0.4
0.2
IANN
RPSGAe
1.2
0.9
0.6
IANN
RPSGAe
0.3
0
0
0
100 Generations 200
300
0
3C-ZDT4
0.06
100 Generations 200
300
3C-ZDT6
0.4
0.3
0.04
S metric
S metric
3C-ZDT3
1.8
0.02
0.2
0.1
IANN
RPSGAe
0
IANN
RPSGAe
0
0
Instituto Superior de
Engenharia do Porto
100Generations 200
300
Faculty of Science and Technology
University of Algarve
0
100 Generations 200
Dept. Polymer Engineering
University of Minho
300

35. 35
CONCLUSIONS
• Algorithm parameters have a limited influence on its
performance
• Good performance of the proposed algorithm
• The number of generations needed to reach identical level
of performance is reduced thus, the computation time is
reduced by more than 50%.
• Most improvements of the IANN approach
accomplished during the first generations
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho
are

36. 36
ANY QUESTION!?
Instituto Superior de
Engenharia do Porto
Faculty of Science and Technology
University of Algarve
Dept. Polymer Engineering
University of Minho