Improvement of Multi-population
Genetic Algorithms Convergence Time
Maria Angelova, Tania Pencheva
maria.angelova@clbme.bas.bg,
tania.pencheva@clbme.bas.bg
Fermentation processes
Fermentation processes (FP) are widely used in different branches of
industry – in the production of pharmaceuticals, chemicals and
enzymes, yeast, foods and beverages.


Fermentation processes are:
characterized as complex, dynamic systems with interdependent and
time-varying process variables;
described by non-linear models with a very complex structure.
An important step for adequate modeling of non-linear models of FP is
the choice of a certain optimization procedure for model parameter
identification.
Aims of the investigation





The influence of five of the main genetic algorithm
parameters to be investigated for six modifications of multipopulation
genetic
algorithms
(MpGA)
towards
convergence time:
generation gap - GGAP
crossover rate - XOVR
mutation rate - MUTR
insertion rate - INSR
migration rate - MIGR
MpGA performance to be demonstrated for parameter
identification of S. cerevisiae fed-batch cultivation.
Genetic algorithms
Genetic algorithms (GA) :
- are a direct random search technique for finding global
optimal solution in complex multidimensional search space;
- are based on mechanics of natural selection and natural
genetics;
- have advantages such as hard problems solving, noise
tolerance, easy to interface and hybridize;
- are proved to be very suitable for the optimization of highly
non-linear problems,
- are applied in the area of biotechnology, especially for
parameter identification of fermentation process models.
Multi-population genetic algorithms

Simple genetic algorithm (SGA) works with a population
of coded parameters set called “chromosomes”. Each of
these artificial chromosomes is composed of binary
strings (or genes) of certain length (number of binary
digits). Each gene contains information for the
corresponding parameter.

Multi-population genetic algorithm (MpGA) is a single
population genetic algorithm, in which many populations,
called subpopulations, evolve independently from each
other for a certain number of generations. After a certain
number of generations (isolation time), a number of
individuals are distributed between the subpopulations.
MpGA modifications






Six kinds of MpGA are investigated towards improvement of
algorithms convergence time. MpGA differ from each other in the
sequence of execution of main genetic operators’ selection,
crossover and mutation:
MpGA-SCM (coming from sequence selection, crossover,
mutation);
MpGA-CMS (crossover, mutation, selection);
MpGA-SMC (selection, mutation, crossover);
MpGA-MCS (mutation, crossover, selection);
MpGA-SC (selection, crossover);
MpGA-CS (crossover, selection) is newly developed here, provoked
by the promising results obtained when selection operator is
processed after crossover in SGA.
MpGA-CS
The main idea of this modification is that the individuals are reproduced
processing only crossover and avoiding mutation.
In the beginning, MpGA-CS generates a random population of
n chromosomes, i.e. suitable solutions for the problem. In order to prevent the
loss of reached good solution by crossover, selection has been processed after
crossover. Parents’ genes combine to form a whole new chromosome during
the crossover. After the reproduction, the MpGA-CS calculates the objective
function for the offspring and the best fitted individuals from the offspring are
selected to replace the parents, according to their objective values. When a
certain number of generations is fulfilled, the MpGA-CS is terminated.
Range of investigated genetic algorithm
parameters
Very big generation gap value does not improve performance of GA,
especially regarding how fast the solution will be found. Mutation is
randomly applied with low probability, typically in the range 0.01 and 0.1.
A higher crossover rate introduces new strings more quickly into the
population. A low crossover rate may cause stagnation due to the lower
exploration rate. Insertion rate is a general measure how many of the
individuals produced at each population are inserted into the new
generation. Migration rate characterized the number of exchanged
individuals.
GGAP XOVR MUTR INSR MIGR
0.5
0.67
0.8
0.9
-
0.65
0.75
0.85
0.95
-
0.02
0.04
0.06
0.08
0.1
0.5
0.6
0.8
0.9
1
0.2
0.4
0.6
0.8
0.1
Mathematical model
of S. cerevisiae fed-batch cultivation
dX
F
= μX - X
dt
V
dS
F
= -qS X +  Sin - S 
dt
V
dE
F
= qE X - E
dt
V

dO2
= -qO2 X + k LO2 a O2* - O2
dt
dV
=F
dt

where X, S, E, O2 and O2* are concentrations of biomass, substrate (glucose), ethanol,
[g.l-1], oxygen and dissolved oxygen saturation, [%]; F – feeding rate, [l.h-1]; V – volume
of bioreactor, [l]; kLO a – volumetric oxygen transfer coefficient,[h-1]; Sin – glucose
concentration in the feeding solution, [g.l-1]; , qS, qE and qO are respectively specific
rates of growth, substrate utilization, ethanol production and dissolved oxygen
consumption, [h-1].
2
2
Specific rates
  2 S
qs 
S
E
 2 E
S  kS
E  kE
2S S
YSX S  kS
qe  
2E
YEX
E
E  kE
qo2 =qE YOE  qSYOS
where 2S , 2E – maximum growth rates of substrate and ethanol, [h-1]; kS, kE – saturation
constants of substrate and ethanol, [g.l-1]; Yij – yield coefficients, [g.g-1].
Optimization criterion:
J Y =  Y - Y *   min
2
where Y is the experimental data, Y* – model predicted data, Y = [X, S, E, O2].
Influence of GGAP in MpGA with three
genetic operators
Influence of GGAP has been investigated towards model accuracy and
convergence time.
GGAP
0.5
0.67
0.8
0.9
MpGA-SCM
J
t, s
0.0220 100.8910
0.0221 112.1720
0.0221 155.4680
0.0220 170.2660
MpGA-SMC
J
t, s
0.0220 111.7810
0.0220 141.0940
0.0220 178.9680
0.0220 340.6720
MpGA-CMS
J
t, s
0.0221 273.9060
0.0221 325.5780
0.0221 321.0160
0.0221 343.6870
MpGA-MCS
J
t, s
0.0220 307.8440
0.0220 332.0620
0.0221 373.1560
0.0221 349.7500
Influence of GGAP in MpGA with two
genetic operators
Influence of GGAP has been again investigated towards model accuracy and
convergence time.
GGAP
0.5
0.67
0.8
0.9
MpGA-CS
J
t, s
0.0223 267.9220
0.0222 331.9690
0.0223 333.6250
0.0221 357.0160
MpGA-SC
J
t, s
0.0222 111.5310
0.0224 119.7340
0.0221 153.3900
0.0220 168.2190
Comparison of MpGA results

The optimization criterion values obtained with six kinds of MpGA are
very similar - there is no loss of accuracy. The obtained results can be
grouped: MpGA-SCM with MpGA-SMC and MpGA-CMS with
MpGA-MCS, but the convergence time in second group is much bigger
than the first group.

Two algorithms without mutation execution, MpGA-SC and
MpGA-CS, can be grouped together too. In cases when algorithms are
implemented only with two operators the calculation time is much less
but for the expenses of model accuracy.

Proceeding selection operator before crossover and mutation (no matter
their order) needs much less computational time at GGAP, XOVR,
MUTR, MIGR and INSR.
Results concerning considered
GA parameters

The GGAP is the most sensitive from five investigated parameters
concerning the convergence time. Up to 40% (in case of MpGASCM,) can be saved using GGAP = 0.5 instead of 0.9 without loss of
accuracy.

Exploring different values of crossover rate no such time saving is
realized but it should be pointed that values of 0.85 for XOVR can
be assumed as more appropriate.

Exploring MUTR values of 0.02 can be assumed as more
appropriate.

In INSR and MIGR no tendency of influence can be drawn.
Optimal GA parameter values
GGAP = 0.5, XOVR = 0.85, MUTR = 0.02, INSR = 0.9 and MIGR = 0.1.
Because of the similarity of the results obtained with all six kinds of
algorithms the results obtained by the developed here MpGA-CS, are
presented.
As a result of parameter identification, the values of model parameters are
respectively: S = 0.98 [h-1], E = 0.13 [h-1], kS = 0.13 [g·l-1],
kE = 0.84 [g·l-1], YSX = 0.42 [g·g-1], YEX = 1.67 [g·g-1],
kLO a = 96.2329 [h-1], YOS = 766.7862 [g·g-1], YOE = 125.5165 [g·g-1],
while CPU time was 288.6720 s and J = 0.0221.
2
Presented results from MpGA-CS application for parameter identification
of S. cerevisiae fed-batch cultivation show the effectiveness of GA for
solving complex nonlinear problems.
Experimental and model data for biomas
and substrate concentration
Fed-batch cultivation of S. cerevisiae
Fed-batch cultivation of S. cerevisiae
30
0.2
data
data
0.18
model
model
0.16
Substrate concentration, [g/l]
Biomass concentration, [g/l]
25
20
15
10
0.14
0.12
0.1
0.08
0.06
0.04
5
0.02
0
0
5
10
Time, [h]
15
0
0
5
10
Time, [h]
15
Experimental and model data for ethanol
and dissolved oxygen concentration
Fed-batch cultivation of S. cerevisiae
Fed-batch cultivation of S. cerevisiae
1
110
data
0.9
data
100
model
90
Dissolved oxygen concentration, [%]
Ethanol concentration, [g/l]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
model
80
70
60
50
40
30
20
0
5
10
Time, [h]
15
10
0
5
10
Time, [h]
15
Analysis and conclusions
Altogether six kinds of multi-population genetic algorithms have been
examined:
- Four of them are with exchanged operators’ sequence of selection,
crossover and mutation operators;
- Two modifications are without performing of mutation operator.
The influence of some of genetic algorithm parameters, namely GGAP,
XOVR, MUTR, INSR and MIGR, has been examined for all six kinds of
genetic algorithms and the most sensitive - GGAP has been distinguished
aiming to improve the convergence time.
As “favorite” among the considered here algorithms MpGA-SCM has been
marked as the fastest one. Up to almost 40% from calculation time can be
saved in the case of MpGA-SCM application using GGAP = 0.5 instead of
0.9 without loss of model accuracy.
All modifications of MpGA show the effectiveness of genetic algorithms
for solving complex nonlinear problems.
IMACS’11
Improvement of Multi-population Genetic Algorithms Convergence Time
ACKNOWLEDGEMENTS
This work is partially supported by the European Social
Fund and Bulgarian Ministry of Education, Youth and
Science under Operative Program “Human Resources
Development”, grant BG051PO001-3.3.04/40 and National
Science Fund of Bulgaria, grant DID 02-29 “Modeling
Processes with Fixed Development Rules”.
Thank you for your attention!