introduced the Genetic Algorithm (GA) as a method that was going to be
efficient, easy to use, and applicable to a wide range of optimization problems. The performance
of GA applied to a
given optimization problem is affected by a number of factors. One of the
most important factors is the parameters manipulation strategy such as crossover, mutation, and
population size. The parameters values are problem dependent, therefore, the optimal s
these parameters must be chosen carefully for a given problem. Finding of parameter settings that
work well in one problem is not a trivial task and it is time consuming.
Two methods are used to find the best parameters values: parameter tuning
control. Parameter tuning involves finding a good value for the parameters before the GA run and
fixing it for all generations and all chromosomes in the population. Dejong
static values for the parameters, which were good for the classes of test functions he used
(the test functions are a numeric problems). The values he found were: population size equal to
50, probability of crossover equal to 0.6, and probability of mutati
on equal to 0.001. Grensfelt
used GA to find a set of static parameter values on the same test functions used by DeJong
. The values he obtained were: populati
on size equal to 30, crossover probability equal to
0.95, and mutation probability equal to 0.01. These parameter values give reasonable
performance for the studied class of test function. The second method of finding a set of
parameter values suitable for
a given optimization problem is parameter control. In this method
one starts with certain initial parameters values. Then these values are changed either in an
adaptive way by using feedback information during the GA run or using a preset formula.
er control was early realized by Rechenberg
in his 1/5 successful rule. According to
this rule the ratio of successful mutations to all mutation should be 1/5
If the ratio is greater
than 1/5 the mutation step size should be decreased, if it is less than 1/5 the mutation step size
should be increased
Eiben, Hinterding, and Michalewicz
classified parameter control into three classes:
deterministic, adaptive and self
adaptive which are described in more details below.
In the deterministic approach the parameters are changed according to a heuristic formula
ally depends on time schedule, and uses no feedback from the GA run. Fogarty,
changed the mutation probability in GA by decreasing it exponentially over
generations. They started with an initial muta
tion probability and then halved it in the next
generation adding to it a base line value. The base line value is employed to keep the mutation
probability from becoming too small later stages of generation.
where t is the generatio
n number and 1/240 is the base line value used.
Hesser, and Manner
derived a general formula to calculate the probability of
mutation on the basis of the generation number and certain constant parameters di
fferent for each
problem. Unfortunately, these constants are not easy to calculate for some optimization problems.
are the constant parameter.
ion size, string length of the
chromosome, and the generation number respectively.
, and Back
, experimentally, found the optimal mutation rate is 1/
is the string length of the chromosome) on (1+1) algorithm. (1+1) algorithm single parent
produce one offspring by means of mutation, then the best of them will survive to the next
then, suggested a time dependent mutation, where the mutation
probability is decreased over generation on the basis of the generation number. The dynamics of
mutation probability is controlled by the maximum num
ber of generations where it is set before
the GA run.
are the string length of the chromosome, the maximum number of generations,
and the number of current generation.
ported excellent experimental results were obtained for hard combinatorial
, and Back
investigated the mutation
probability control when no crossover was used. Gabriela
stated that fixed and small
mutation probability if crossover used w
ill give better performance. He, experimentally, proved
that using a time dependent variant of the mutation might improve the GA performance if no
crossover is used which agrees with the conclusions of
In adaptive parameter control feedback information is extracted on how well the search is
going and used to control the values and direction of the paramete
rs values. Adaptive control
parameter were first used by Rechenberg
in his 1/5 successful rule of controlling the
mutation step size in evolution strategy. Bryant, and Julstrom
used the contribution of the
parameter value in making new chromosomes with a fitness better than the median of the
population at that time. This contribution is recorded as a credit for that parameter value. The
credit assigned for the
parameter value controls its decrease or increase. Schlierkamp, and
adapted the population sizes using competing subpopulations. In this method,
multiple populations with different sizes are run at the same time.
After each generation the
subpopulation which has the best fitness is stored in a quality record. After a number of
generations the population with the highest quality record is increased. All others are decreased
according to their quality record. In a s
imilar fashion, Hinterding
ran three populations
simultaneously. These populations had an initial size ratio of 1:2:4. After a certain time interval
the size of each population was halved, doubled, or mainta
ined depending on its best fitness.
Using a slightly different approach, Lobo, and Harik
ran a population with a small size and
doubling it at genotype convergence (all chromosomes in the population have the
genotype). In order to accelerate the some time tedious genotype convergence he ran several
populations simultaneously. According to
Annunziato, and Pizzuti
the environment contains
some useful informat
ion that could be used to stabiles or control the parameter strategy. The
parameter values should be changed according to how a chromosome in the population interacts
with the others. The rate of change of the parameter values are controlled by the current
adaptive control, is the third approach used described in the literature. It was first
used by Schwefl
in the evolution strategy where he tried to control the mutation step size.
omosome in the population combined with its own mutation variance as part of the
chromosome structure, and this mutation variance is subjected to mutation, recombination, and
selection as well as the solution parameters. Back
work into GA. He
added extra bits at the end of each chromosome in the population to control the mutation, and
crossover probability. At first the mutation, and crossover
probability bits were purely random as
the parameters solution. These bits were subjected to mutation and crossover as well, better
values of parameter gave better chromosomes and then at the end better values dominated the
population. Another way to self
adapt the parameters values, described by Srivniras, and Patniak
is by assigning mutation and crossover probability for each chromosome on the basis of its
fitness and the environments property at that time
. Arabas, Michalewicz, and Mulawka
the age of the chromosomes to control the population size. Each created chromosome was
assigned a lifetime to control how many generations would survive before being de
lifetime of chromosome was determined by its fitness and the current state of the search.
approach for deterministic and self
ntrol, as well as the
methodologies of Lobo, and Harik
and of Annunziato, and Pizzuti
for adaptive control
were used by us and are described in details further
2. Test Genetic Algorithms
2.1. Traditional steady state genetic algorithm
Traditional steady state genetic algorithm is used to serve as a benchmark for other
variants. Gray code is used, a mutation rate P
=1/l, crossover rate P
= 0.9, populatio
N=60, uniform crossover and bit flip mutation. In TGA the worst chromosome is deleted to make
a room for children. The chromosome length will be 200 bits for all test functions except f
be 60 bits, which consists of 10x20 =200 bits (10x6=60 fo
) for ten function variables
(dimension n=10) and at the end of chromosome 2x10=20 bits are added for two self
. The GA terminates when the optimum is found or the maximum number of
fitness evaluations is reached. The maxim
um number of evaluations is 500000 for f
, and f
and 200000 for f
adaptive crossover, mutation probability, and population size
of genetic algorithm
mutation rate is encoded in extra bits at the end of each
chromosome in the population within a range between 0.001 and 0.25, and the same for crossover
within a range between 0 and 1. Mutation then takes places in tw
o steps. First mutate the
bits that encode the mutation rate only and immediately decoded to establish a new mutation rate.
Second, the new mutation rate is used to mutate the main bits (those encode the solution). For
reproduction two chromosomes are sele
cted by Tournament selection. The bits that encode the
crossover rate in the selected chromosome are decoded to establish crossover rate P
. A random
below 1 is compared with P
of the selected chromosome, if
is lower chromosome P
the member i
s ready to mate. If both selected chromosomes are ready to mate two children are
created by uniform crossover, then mutated and inserted in the population. While if both selected
chromosomes reject to mate two children created by mutation only. If one of b
chromosomes willing to mate and the other doesn’t. One child created by mutating the
chromosome who doesn’t like to mate. The willing chromosome is on hold and the next parent
selection round only picks one other parent.
The number of generati
ons the chromosome stays alive proposed by Arabas
is used to
adapt the population size. Every new chromosome created assigned age “Remaining Life
Time (RLT)” according to its fitness by a bi
are minimum and maximum remaining life time, and the set to 1, and 11
are the worst, and best chromosome fitness in the
is the average fitness of the population.
is the i
At each cycle the RLT of each chromosome in the population is decremented by one. If the RLT
of a chromosome reaches zero i
t is removed from the population.
2.3. Adaptive Genetic Algorithm by Reproduction and Competition
Pizzuti and Annunziato
introduced a dynamic environment determined by the
reproduction and competition rules a
mong chromosomes. The adaptation mechanisms of the
parameters probability are controlled by the environment. The environment is constrained by the
maximum population size which it is set before the GA run. The environment adaptive rules
control the paramet
ers probability called Population density or meeting probability. Meeting
probability defined as:
are the current population size and the maximum population size respectively.
o chromosomes meet then they can create children: by crossover (bisexual) or by
fighting (competition) for natural resources (the stronger kill the weaker), otherwise the current
chromosome can differentiate (mono
sexual or mutation) to create child. The c
rossover rate and
competition rate are defined as:
is the crossover rate and
is the meeting probability.
is the competition rate and
is the crossover rate .
Initially a random number, limited by the maximum population size is generated for the
initial population size. Then at each iteration we select the i
chromosome of the population, for i
from 1 to population si
ze. Then randomly select another chromosome from the population. A
below 1 compared with the meeting probability P
happened. If interact happened, another random
below 1 compared to P
ren are created by uniform crossover and immediately inserted in the population. If
, competition will take place then the weaker chromosome is removed from the
population. If meeting not happened, one child is created by mutation and immedia
tely inserted in
2.4. Adaptive population size of genetic algorithm
To overcome the population size problem and drift velocity, Lobo
establishing a race among populations of various
sizes. Multiple populations with different sizes
are running simultaneously. Lobo method gives priority to the smaller population size by giving
them more function evaluations. Initially runs population 1 for 4 generations, and then runs
population 2 for 1
generation, then population 1for 3 more generation, then population 2 for 1
generation, and so on. At any time if the average fitness of the smaller population is less than the
larger one, then the smaller population will be destroyed. The crossover rate
and mutation rate
for each population are the same and they are set before the run of GA. P
=0.9, and P
the chromosome length.
2.5. Deterministic mutation rate of genetic algorithm
In this al
gorithm we used a formula depends on time
“generation counter” and
to change the mutation rate Back
. Generation counter constrained by
ximum number of generation
. The formula of mutation defined as:
are the chromosome’s length, current generations, and maximum number
At each iteration one chromosome is selected to be parent by tournament selection. Then
one child is created by mutation and the better of both parent and child survives for the next
generation. No crossover used to create chi
3. Test Functions
To evaluate the performance of the parameter
less GA algorithm, we used the same test
functions used by Back
. For selecting the test function Back followed the guidelines reported
, and Back
. The test functions should:
Contain problems resistant to Hill
Contain nonlinear, non separable problems,
Contain scalable funct
Have a canonical form,
Include a few unimodal functions for comparison of efficiency (convergence velocity),
Include a few multimodal functions of different complexity with a large number of local
Include multimodal functions with irregular a
rrangement of local optima,
dimensional functions, because these are better representatives to real
Ten variables are used (dimension n=10) for each test function and constrained in the interval
. Tow dimensional domain are used to draw the surface
of each function except function five. To draw the surface of function five we implement a program
generate a binary string randomly several times, then counter the number of 1’s in the string.
er of 1’s in the string determines which part of the function five must be use. At the end we
will have a matrix of values that represent the surface amplitude.
The test functions suite the rules above:
a sphere model after De Jong
. It is continuous, convex and unimodal.
The function has global optimum at point zero,
is the generalized Rosenbrock
function. The function has global optimum inside a
long, narrow, parabolic shaped flat valley. To find the valley is trivial, however convergence to
the global optimum is difficult,
is the generalized Ackley
function. It is a variant multimodal function
with global minimum located at the origin with function value of zero
is the generalized Rastrigin
function. It is a non
multimodal function. This function is a fairly diffi
cult problem due to its large search space and
its large number of local minima,
is the fully deceptive six
bit function Deb
. All functions have dimension n=1
use 20 bits/variable except f
which uses 6 bits/variable.
Ackley, D.H., A connectionist machine for genetic hill climbing,
Kluwer, Boston, 1987.
Arabas, J., Michalewicz, Z., & Mulawka, J., GAVaPS
A genetic algorithm with varying
Proceeding of the 1
IEEE Conference on Evolutionary Computation,
IEEE Press, 1994.
Annunziato, M., & Pizzuti, S., Adaptive parameterization of evolutionary algorithms driven
by reproduction and competition,
Proceeding of ESIT2000, PP 246
Back, T., Self
Adaptation in genetic algorithms,
In F. J. Varela and P. Bourgine, editor,
Proceeding of the First European Conference on Artificial Life, PP 263
271, The MIT
Press, Cambridge, MA, 1992.
Back, T., The interaction of mutation ra
te, selection, and self
adaptation within a genetic
In R. Manner and B. Manderick, editors, Parallel Problem solving from Nature,
94, Elsevier Amsterdam, 1992.
Back, T., Optimal mutation rates in genetic search.
In Forrest, S. (Ed), Procee
ding the Fifth
International Conference on Genetic Algorithms PP 2
8, San Mateo, Ca: Morgan
Back, T., & Schwefel, H.
P., An overview of evolutionary algorithms for parameter
Evolutionary Computation, Vol. 1, No. 1, PP 1
Back, T., & Schwefel, H.
P., Evolution strategies I: Variants and their computational
implementation. In winter,
G., Perisux, J., Galan, M., & Cuesta, P. (Eds), Genetic
Algorithms in Engineering and Computer Science (Chapter 6, PP 11
John Wiley and Sons. 1995.
Back, T., & Schutz M., Intelligent mutation rate control in canonical genetic algorithm,
Proceeding of the International Symposium on Methodologies for Intelligent Systems, PP
Back, T., Evolutionary Algorithms in t
heory and practice,
Oxford University Press, 1996.
Back, T., & Michalewiccz, Z., Test landscapes
, In Back, T., Fogel, D.B., & Michalewicz,
Z., (Ed): Handbook of Evolutionary Computation, Chapter B2.7, PP 14
20, Institute of
Physics Publishing and Oxford Un
iversity Press, New York, 1997.
Back, T., Eiben, A.E., & Van der Vaart, N.A., An empirical study on Gas without
In Schenauer, M., Deb, K., Rudolph, G., Yao, X., Lutton, E.,Merelo, J. J., and
P. (Ed): Parallel Problem Solving from Na
ture PPSN V, Lecture Notes in
Computer Science Vol. 1917, PP 315
Bryant, A., & Julstrom, What have you done for me lately? Adapting operator probabilities
in a steady
state genetic algorithm,
Proceeding of the Sixth International Conference on
enetic Algorithms, PP 81
7, Morgan Kufmann, 1995.
Davis, L., Adapting operator probabilities in genetic algorithms.
In Schaffer, J. D. (Ed),
Proceeding of the Third International Conference on Genetic Algorithms PP 16
Mateo, Ca: Morgan Kaufman,. 19
Deb, K., Deceptive Landscape,
In Back, T., Fogel, D.B, & Michalewicz, Z. (editors):
Handbook of Evolutionary Computation, Institute of Physics Publishing & Oxford
University Press, New York, 1997.
De Jong, K. A., An analysis of the behavior of a class
of genetic adaptive systems,
Doctoral dissertation, University of Michigan, Ann Arbor, University Microfilms No 76
Eiben, A.E., Hinterding, R., & Michalewicz, Z., Parameter control in evolutionary
IEEE Transactions on Evolutionary C
omputation, Vol. 3, No. 2, PP 124
Fogarty, T., & Terence, C., Varying the probability of mutation in the genetic algorithm,
Proceeding of the Third International Conference on Genetic algorithms, PP 104
Morgan Kufmann, 1989.
Gabriela, O., In
man, H., & Hilary, B. On recombination and optimal mutation rates,
Proceedings of Genetic and Evolutionary Computation Conference
Morgan Kaufmann, San Francisco, CA, 1999.
Grefenstette, J. J., Optimization of control parameters for
In Sage, A. P.
(Ed), IEEE Transactions on Systems, Man, and Cybernetics, Volume SMC
1, pp 122
128, New York: IEEE, 1986.
Goldberg, D.E., Genetic algorithms in search, optimization, and machine learning,
Wesley Publishing Com
pany, Inc, 1989.
Goldberg, D.E., Sizing populations for serial and parallel genetic algorithms,
the third international Conference on Genetic Algorithms and Their applications, PP 70
79, Morgan Kaufmann, 1989.
Harik, G. R., & Lobo, F. G., A
less genetic algorithm,
Banzhaf, W., Daida, J.,
Eiben, A. E., Garzon, M. H., Honavar, V., Jakiela, M., & Smith, R. E. (Eds.) .GECCO
Proceedings of the Genetic and Evolutionary Computation Conference, PP 258
Francisco, CA: Morgan Kauf
Hesser, J. & Manner, R., Towards an optimal mutation probability in genetic algorithms,
Proceeding of the 1
Parallel Problem Solving from Nature, PP 23
32, Springer 1991.
Hinterding, R., Gaussian mutation and self
adaptation for numeric geneti
Proceeding of IEEE International Conference on Evolutionary Computation, PP 384
Hinterding, R., Michalewicz, Z., & Peachy, T. C., Self
Adaptive genetic algorithm for
Proceeding of the Fourth International Conf
erence on Parallel Problem
Solving from Nature, PP 420
429, in Lecture Notes from Computer Science, Springer
Hoffmeister, F., & Back. T., Genetic algorithms and Evolution strategies: Similarities and
In Shwefel, H
P., Manner. R.,
Parallel Problem Solving from Nature
(Lecture Note in Computer Science; Vol. 496), Springer Verlag, Berlin, 1991.
Holland, J. H., Adaptation in natural and artificial systems,
Ann Arbor, MI: University of
Michigan Press, 1975.
Muhlenbein, H., How
genetic algorithms really work: I. Mutation and Hill climbing,
Parallel Problem Solving from Nature
PPSN II, 15
Rechenberg, I., Evolutions strategie: Optimierung technischer systeme nach prinzipien der
nbrock, H.H., An Automatic method for finding the greatest or least value of a
The Computer Journal, Vol. 3, No.3, PP 175
Schaffer, J.D., Caruana, R.A., Eshelman, L.J., & Das, R., A study of control parameters
affecting online performa
nce of genetic algorithms for function optimization,
the Third International Conference on Genetic Algorithms and Their Applications, PP 51
60, Morgan Kaufmann, 1989.
Voosen, D., & Muhlenbein, H. Adaptation of population sizes by
Proceeding of International Conference on Evolutionary Computation
(ICEC’96), Negoya, Japan, PP 330
P., Numerische optimierung von computer
modellen mittels der
Volume 26 of Interdiscipl
inary systems research. Birkhauser, Basel,
P., Collective phenomena in evolutionary system,
In Preprints of the 31
Annual Meeting of the International Society for General System Research, Budapest, Vol.
2, PP 1025
& Fogarty, T., Self
Adaptation of mutation rates in a steady state genetic
Proceeding of the third IEEE Conference on Evolutionary Computation, IEEE
Smith, R., Adaptively resizing populations: An algorithm and analysis,
Fifth International Conference on Genetic Algorithms, P 653 Morgan Kaufmann, 1989.
Srinivas, M., & Patniak, L. M., Adaptive Probabilities of crossover and mutation in genetic
IEEE Transactions on Systems, Man and Cybernetics, Vol. 24, No
. 4, PP 17
Torn, A., & Zilinskas, A., Global Optimization,
Lecture Note in Computer Science; Vol.
350, Springer Verlarg, Barlin, 1989.
Van der Vaart, N.A.L., Towards Totally self
adjusting genetic algorithms: Let nature sort
Master Thesis, Le
iden University, 1999.
Whitley, L. D., Mathias, K.E., Rana, S., & Dzubera, J., Building better test functions,
Eshelman, L.J. (Ed), Proceeding of the Sixth International Conference on Genetic
Algorithms, PP 239
246, Morgan Kaufmann, San Francisco, Cali