1. Process modelling and optimization aid
FONTEIX Christian
Professor of Chemical Engineering
Polytechnical National Institute of Lorraine
Chemical Engineering Sciences Laboratory

2. Process modelling and optimization aid Evolutionary algorithm
FONTEIX Christian
Professor of Chemical Engineering
Polytechnical National Institute of Lorraine
Chemical Engineering Sciences Laboratory

3. Evolutionary algorithm Generalities
Objective : to minimize F(x)
F is a scalar
x is a vector of n elements
Some elements of x are reals
Some elements of x are integers
F can be non continuous and non derivable
F have several optima
There is no constraint (if not, the best way is multicriteria optimization)
For each element :

4. Evolutionary algorithm Generalities
Genetic algorithm : coding x in binary string
Difficulty : to choose the best coding (classic number decomposition on base 2 leads to many new optima which return convergence difficulties)
Evolutionary algorithm : coding x in real numbers
An integer is the integer part of a real number
Vector x corresponds to the true values (comprehensive)
Diploïd algorithm exists, simpler is haploïd algorithm

5. Evolutionary algorithm Generalities
Optimization by solutions population
Using of DARWIN evolution concepts
Genetic operators : death, birth, crossover, mutation
Birth from 2 parents (the survivals)
Death of the most misfit individuals (elitist selection)
The population bordline corresponds to a level line
Interest : population is a picture of F surface
Evolutionary algorithm is fast, simple and easy to use

6. Evolutionary algorithm Word list
Population : set of individuals
Generation : way from the last population to the new
Chromosome : vector x
Genotype : x for haploïd algorithm
Phenotype : set of genotype and corresponding F value (define the fitness of the individuals)
Child viability : better of equal to the last survival
(a child less good that the last survival is delated)
An EA don’t search the best solution but delate the worst

7. Evolutionary algorithm Standard parameters
Vector x size : n
Population size : N=(20 to 40)*n
Number of survivals : S=N/2
Number of mutants : M=N/10
End test of EA : biodiversity lost (BL), convergence stagnation (CS) and time limitation (TL)
BL : F(worst) - F(best) < threshold
CS : best individual don’t change during 5 generations
TL : maximum number of generation

8. Evolutionary algorithm WorkingF
Last survival
Population=8
Death=4
Population 1
Population 2
Population 3
Dead
1st selection
2nd selection x
Child

9. Evolutionary algorithm Process scheme
Initialization
Population ranking
End test
End
Death
New population
Birth

10. Evolutionary algorithm Initializationx1 x2 x3 xN
Random of

11. Evolutionary algorithm Population ranking
From the best individual to the worst
After ranking x1 is the best individual
F(x1) is the smallest value of F in the population
After ranking xN is the worst individual
F(xN) is the highest value of F in the population

12. Evolutionary algorithm End test
Lost of biodiversity :
Security
number of new generation  maximum number

13. Evolutionary algorithm Deathx1 x2 x3 xS
xS+1 xN
Survivals
Delated

14. Evolutionary algorithm Mutants
Random of M individuals
These individuals survive if
Alone M* mutants survive (maybe M*=0)
The M* are situated in the new population from xS+1 to xS+M* (survivals from old population : from x1 to xS)

15. Evolutionary algorithm Birth by crossover
Birth of children from 2 parents (new parents for each)
These individuals survive if
The viability of each child is verified at each birth
The birth process stop if N-S-M* viable children are born (needed children for population restoration)
The N-S-M* are situated in the new population from xS+M*+1 to xN

16. Evolutionary algorithm Birth by crossover
Random choice of 2 parents from survivals
Determination of the best parent P1
Random choice of i for each vector element
Caculation of
Calculation of corresponding F value and viability validation

17. Evolutionary algorithm Birth by crossoverx1
Domain of child birth
Lowest
probability P1 P2
Highest
probability x2

18. Evolutionary algorithm Birth by crossover
1.2 1 i
-0.2

19. Evolutionary algorithm Birth by crossover
Possibility of slipping
of the population
Optimum
Constraints
or bordlines
Population
domain

20. Evolutionary algorithm Convergence example
Minimization of

21. Evolutionary algorithm Confidence domain determination
The end test must be modified :
Fisher Snedecor test for confidence domain : F(x)  B
The biodiversity lost test become : F(xN)  B
No security test
The selection test must be modified at last generation :
S=N/2 for all generation excepted the last
Last generation : B replace F(xS) if F(xN/2) < B
At last generation the viability test is F(xC)  B
No other modification