Process modelling and
optimization aid
FONTEIX Christian
Professor of Chemical Engineering
Polytechnical National Institute of Lorraine
Chemical Engineering Sciences Laboratory
Process modelling and
optimization aid
MultiCriteria Decision Aid
FONTEIX Christian
Professor of Chemical Engineering
Polytechnical National Institute of Lorraine
Chemical Engineering Sciences Laboratory
Multicriteria decision aid
Generalities
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Modelling of human preferences
Opposite objectives
Subjectivity
Context influence (social, economic…)
Lobbying
Expert system structure of the model
Mathematical functions
Difficulty to determine the best structure of the model
and its characteristics (rules) or parameters
• Robustness of a decision
Multicriteria decision aid
Generalities
• Different methods
• Sequential search by single point
• The decision maker indicate at each step the search
direction. Disadvantages : no entire view of the
problem, a priori expression of the preferences
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Using a preferences modelling on an alternatives list
To list all the alternatives at first and choose a model
To determine the prefered alternatives with the model
Advantages : entire view and a posteriori expression of
the preferences
Multicriteria decision aid
Preferences modelling types
• MCDA is just an aid
• It is an information for the decision maker
• The decision maker choose the final decision
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Parameters determination of preference modelling
Preferences extraction by cognitive determination
Preferences extraction by alternatives comparizon
Preferences extraction by parametric identification
from « preferences measurement »
• Group consensus
Multicriteria decision aid
Preferences modelling types
• Parametric identification from « preferences
measurement »
• Choice of a representative set of alternatives
• The decision maker score each alternative of the set
• The decision maker rank the alternatives of the set
• 3 classifications for preferences modelling
• Classification by the result (ranking of classification of
the totality of the alternatives, or classification)
• Compensative or non compensative models
• Partial or total aggregation
Multicriteria decision aid
Preferences modelling types
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Classification by the result
Alternatives comparizon 2 by 2
Determination of a Preferences Relationship System
Detection and delating of Cycles
Determination of several classification
Choice of one classification
A
E
C
B
F
H
I
G
Multicriteria decision aid
Preferences modelling types
• Classification by the compensation type
• Compensative type : for each alternative a bad criterion
value can be compensate by a good criterion value (2 
criteria)
• Non compensative type : for each alternative a bad
criterion value cannot be compensate by a good
criterion value (2  criteria)
• Classification by aggregation type
• Total aggregation : the score depends on one alternative
• Partial aggregation : the score of one alternative
depends also on the others
Multicriteria decision aid
Preferences modelling types
Type
Total
aggregation
Compensative
MAUT (AHP)
Non
compensative
OWA
Choquet
integral
Partial
aggregation
PROMETHEE
ELECTRE
Rough Sets
Multicriteria decision aid
Preferences modelling types
• MAUT : Multi Attribute Utility Theory
• AHP : Analytical hierarchical Process (method for
MAUT parameters determination)
• OWA : Ordered Weighted Average
• Choquet integral : complex model (criteria interactions)
• ELECTRE : ELimination Et Choix Traduisant la REalité
• PROMETHEE : Preference Ranking Organisation
METHod for Enrichment Evaluations
• GAIA : Graphical Analysis for Interactive Assistance
(for PROMETHEE)
Multicriteria decision aid
Preferences modelling types
• Rough Sets : mathematical theory which extend the set
theory (as fuzzy sets)
• Authors have used the rough sets theory to develop a
simple fast multicriteria decision aid
• In rough sets theory a set is defined by a maximum set
and a minimum set
Multicriteria decision aid
MAUT
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x : alternative of which we calculate the score
fi(x) : value of the criterion i for the alternative x
ui(fi(x)) : utility value corresponding to criterion i
wi : weight associated to criterion i
s(x) : score of the alternative x
u(s(x)) : total utility of the alternative x
n : number of criteria
u(), ui() : utility functions
n
usx    w i ui  f i x  with
i1
n
w
i1
i
1
Multicriteria decision aid
MAUT
• The weights define the relative importance of criteria
• The utility functions define the intrinsic exigency of the
decision maker on each criterion
• The utility ([0, 1]) must be maximized
• For a criterion to be minimized :
Utility
1
Non exacting
Neutral
Exacting decison maker
0
Minimum
Criterion value
Maximum
Multicriteria decision aid
MAUT
• Problem on neutral case
• Example (2 criteria to be maximized) :
2  f1

2
f2
if
8
f
u1  2
2
f1  0,2
u2  2 f 2
 1 
f1  0, 
 2 
u2  1 u1 
2
u  w1u1  w 2 u2  w1u1  1 w1 1 u1 
2
u  1 w1 u12  2  3w1 u1  1 w1 
• Parabole with one minimum
• Prefered alternatives are f1 or f2 maximum  w1

• No compromise
Multicriteria decision aid
MAUT
• A limit case is given by :
2
 1 
2  f1 

f2 
f1  0,2 f1  0, 
 2 
8
f
if u1  2 u2  1 u1 u2  2 f 2
2
u  w1u1  w 2 u2  w1u1  1 w1 1 u1 
u  1 2w1 u1  1 w1 
• If w1<0.5 the prefered alternative is f2 maximum
• If w1>0.5
the prefered alternative is f1 maximum

• If w1=0.5 all the alternatives are equivalent
Multicriteria decision aid
MAUT
• In practice :
2  f1 

f2 
8
2
u1  p1
f2
2
u2  2 p 2 2 f 2
u  w1u1  w 2 u2
w1  w 2  1
p1  1 p2  1

f1  0,2
 1 
f1  0, 
 2 
Multicriteria decision aid
OWA
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x : alternative of which we calculate the score
fi(x) : value of the criterion i for the alternative x
ui(fi(x)) : performance value corresponding to criterion i
wj : weight non associated to criterion i, associated to a
performance level j
Example : 0.3, 0.1, 0.8, 0.5 are ranked as 0.8, 0.5, 0.3, 0.1
and (1)=3, (2)=4, (3)=1, (4)=2
s(x) : score of the alternative x
u(s(x)) : total performance of the alternative x
n : number of criteria
u(), ui() : performance (utility) functions
Multicriteria decision aid
OWA
• The weights define the relative importance of
performance level
• The performance functions define the intrinsic exigency
of the decision maker on each performance level
• The performance ([0, 1]) must be maximized
n
usx    w (i)ui  f i x  with
i1
n
w
j1
j
1
Multicriteria decision aid
OWA
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Example
Weight associated to performance level 1, w1=0.5
Weight associated to performance level 2, w2=0.2
Weight associated to performance level 3, w3=0.2
Weight associated to performance level 4, w4=0.1
u1=0.3, u2=0.1, u3=0.8, u4=0.5
u1 is the performance 3, the associated weight is w3=0.2
u2 is the performance 4, the associated weight is w4=0.1
u3 is the performance 1, the associated weight is w1=0.5
u4 is the performance 2, the associated weight is w2=0.2
u=0.2*0.3+0.1*0.1+0.5*0.8+0.2*0.5=0.57
Multicriteria decision aid
PROMETHEE
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The score of one alternative depends also on the others
Analogy with football :
PROMETHEE is a championship
The alternatives are the team
The score of one alternative is the goal average
The number of goals obtained by team i on team j is ij
The number of teams is N
The score of i is :
N
N
j i
j i
si   ij   ji
Multicriteria decision aid
PROMETHEE
• There is one match between teams i and j
• The goals obtained by team i depends of its player
quality (and the player quality of team j)
• Each player of team i play against the corresponding
player of team j
• The participation of a player to the score depends on
the quality difference of the 2 corresponding players :
Cijk  gk ( ijk ) with  ijk  f k x j  f k x i 
if f k must be min imized
• fk is the value of criterion k

Multicriteria decision aid
PROMETHEE
• The number of goals obtained by team i on team j is :
n
 ij   w k Cijk
k1
 K

si   w k Cijk  C jik 

j1 k1
N
• n is the number of criteria
 of 2 thresholds :
• Cijk depends
The indiference threshold qk
The preference threshold pk
Multicriteria decision aid
PROMETHEE
Cijk
1
0
qk
pk
ijk
Multicriteria decision aid
Example
Emulsion polymerization : latex production
Pareto domain
Pareto frontier
Multicriteria decision aid
Example
Emulsion polymerization : latex production
5
4
5
4
3
2
3
1
1
1
2
1
Pareto domain
Pareto frontier
Multicriteria decision aid
Example
Emulsion polymerization : latex production
17
3x10
7
5
6
4
-1
4
3
2
3
5
17
|Np(tf)-Npd|
5
S0 (g .l )
a
1
2
2x10
4
3
17
1x10
2
1
0
0 ,0 0 0
1
0 ,0 0 1
0 ,0 0 2
0 ,0 0 3
-1
[A ] 0 (m o l.l )
Pareto domain
0 ,0 0 4
0
0,80
0,85
0,90
X(tf)
Pareto frontier
0,95
1,00
Multicriteria decision aid
Example
Production of cow food by extrusion
Barrel temperature
70
T (°C)
60
50
Prefered zone
40
2
3
4
D (cm)
Die
5
6
Multicriteria decision aid
Example
Production of cow food by extrusion
Best decisionnal
robustness
75
55
75
45
65
35
2
3
4
D (cm)
5
6
T (°C)
T (°C)
65
55
45
Best technical
robustness
35
2
3
4
D (cm)
5
6
Optimisation multicritère
d’un procédé de mise en pâte
à haut rendement
J. Thibault (Université d’Ottawa)
R. Lanouette (Université du Québec à TR)
C. Fonteix (LSGC, Nancy)
L.N. Kiss (Université Laval)
Outline
 Introduction
 Modelling of the fermentation process
 Multicriteria Optimisation
 Pareto domain
 Ranking algorithm : Net Flow Method
 Results
 Conclusion
Introduction
 In complex processes with a large number of
input and output variables, the determination
of a suitable set of input variables that would
provide an optimal set of outputs is a major
chalenge.
 Usually dealing with numerous conflicting
objectives.
 There is a need to develop new optimisation
methods to capture the preferences of the
decision-maker to lead to an acceptable
compromise solution.
Introduction
 New techniques in multicriteria optimisation
are now emerging in the field of engineering
to resolve problems of conflicting criteria.
 Two of these techniques have been used by
the authors:
 Net Flow Method
 Rough Sets
 They are very useful because they allow one
to rank the full domain of non-dominated
solutions (Pareto domain).
Schematic of the Process
WASHED
CHIPS
PRESTEAMING
PLUG SCREW
FEEDER
CYCLONE
WASHING
PRIMARY REFINER
BLEACHING
TOWER
SECONDARY
REFINER
WASHING
PRESS
TO CLEANING
TRANSFER
CHEST
LATENCY
CHEST
Pulp and Paper Process
Model Inputs
First stage:
Temperature
Plate gap
Interstage
Treatment:
Second stage:
Consistency
H2O2 charge
Temperature
Retention time
NaOH (Y or N)
For each output, the model is a stacked
neural network (10 levels) having 7 inputs
Pulp and Paper Process
Model Outputs
Brightness Index
maximize
Specific Refining Energy
minimize
Extractive Content
minimize
Breaking Length
maximize
Optimisation Process
Experimental Design
Experimental Data
Modelling
It is not a
one-pass
process
Optimisation
We are concerned only with the
optimisation in this presentation
Optimisation
 Approximation of the Pareto domain (with or
without NaOH leading to two separate models)
Input 1
Input 2
Input 3
Input 4
Input 5
Input 6
Simulation
One solution
Output 1
Output 2
Output 3
Output 4
Dominance and Pareto Domain
 Procedure to approximate the Pareto domain:
 Randomly select the six input variables and
calculate the four criteria for a total of 6000
points.
 Compare each point and discard those points that
are dominated for all three criteria by at least one
other point.
Point 1
66.71
7.599
0.1251
4.229
Point 2
65.76
8.603
0.1745
3.837
Dominance and Pareto Domain
Point 1
66.71
7.599
0.1251
4.229
Point 3
65.76
8.603
0.1745
4.784
A large number of points are generated and compared
until 6000 non dominated points are identified.
Optimisation
 Procedure to approximate the Pareto domain:
 Randomly select the four input variables and
calculate the three criteria for a total of 6000
points.
 Compare each point and discard those points
that are dominated for all three criteria by at
least one other point.
 Keep all non-dominated solutions and 30% of
dominated points that are dominated fewest
number of times.
 Generate new points (up to 6000 points) by
random interpolation between two points.
 Redo the procedure until 6000 non-dominated
points are obtained.
Optimisation
 The Net Flow Method requires a human expert
to give some appreciation as to the nature of
each criterion. Four pieces of information
must be provided:
 The relative importance of each criterion
(a relative weighting WK);
 The indifference threshold (QK);
 The preference threshold (PK);
 The veto threshold (VK).
0 ≤ Qk ≤ Pk ≤ Vk
Optimisation
Concordance index:
 i  [1, M ]

 k [i, j ]  Fk (i) - Fk ( j )  j  [1, M ] j  i
 k  [1, n]

Fk being minimized
ck[i, j] 1
Δk [i, j]
0
Qk
Pk
Individual


P ck [i, j ]   k
 Pk

1
if  k [i, j ]  Qk
 k [i , j ]
if Qk   k [i, j ]  Pk
- Qk
0
if  k [i, j ]  Pk
Global
 i  [1, M ]
C[i, j ]   Wk ck [i, j ] 
k=1
 j  [1, M ]
n
Optimisation
 Discordance index:
Fk being minimized
Dk[i, j] 1
0 Qk Pk


  k [i,
Dk [i, j ]  
 Vk

0
j ] - Pk
- Pk
1
if  k [i, j ]  Pk
if Pk   k [i, j ]  Vk
if  k [i, j ]  Vk
 n
3   i  [i, M ]
 [i, j ]  C[i, j ]   1- ( Dk [i, j ])  
Δk [i, j]
k=1
  j  [i, M ]
Vk
i 
M
M
j=1
j=1
 [i, j] -  [ j, i]
Results
9.0
8.0
Y2
With NaOH
7.0
No NaOH
Y1 - ISO Brightness
Y2 - Specific refining energy
Best points are in red color
6.0
64.0
66.0
68.0
Y1
70.0
Results
4.8
With NaOH
Y3 - Extractives
Y4 - Rupture length
Best points are in red color
Y4
4.4
4.0
No NaOH
3.6
3.2
0.08
0.12
0.16
0.20
Y3
0.24
0.28
Results
1.0
With NaOH
X2
0.8
X1 - Temperature of first stage
X2 - Plate spacing
200 best points are in dark
0.6
No NaOH
0.4
0.2
0.0
0.1
0.2
0.3
X1
0.4
0.5
Results
1.0
No NaOH
X4
0.9
With NaOH
0.8
X3 - Consistency
X4 - H2O2 Charge
200 best points are in dark
0.7
0.6
0.0
0.2
0.4
0.6
X3
0.8
1.0
Results
1.0
0.8
With NaOH
X6
0.6
X5 - Temperature of second stage
X6 - Residence time
200 best points are in dark
No NaOH
0.4
0.2
0.0
0.0
0.2
0.4
0.6
X5
0.8
1.0
Conclusion
 A multicriteria optimisation routine has been
applied to optimise a pulping process.
 The ranked points of the Pareto domain allow
to visualize the zone of optimal operation and
help in designing a control strategy.