MALE AND FEMALE CHROMOSOMES IN GENETIC
ALGORITHMS
Ghodrat Moghadampour
VAMK, University of Applied Sciences, Technology and Communication, Wolffintie 30, 65200 Vaasa, Finland
Keywords: Evolutionary algorithm, Genetic algorithm, Distributional genetic algorithm, Function optimization, Male
and female chromosomes, Adaptive operators.
Abstract: Evolutionary algorithms work on randomly generated populations, which are converged over runs toward
the desired optima. Randomly generated populations are of different qualities based on their average fitness
values. In many cases switching all bits of a randomly generated binary individual to their opposite values
might quickly produce a better individual. This technique increases diversity among individuals in the
population and allows exploring the search space in a more rigorous way. In this research the effect of such
operation during the initialization of the population and crossover operator has been investigated.
Experimentation with 44 test problems in 2200 runs showed that this technique can facilitate producing
better individuals on average in around 32% of cases.
1 INTRODUCTION
Evolutionary algorithms are heuristic algorithms,
which imitate the natural evolutionary process and
are mainly used to solve problems which are hard to
solve in conventional ways. In an evolutionary
algorithm 1) problems are described by a set of
parameters, 2) parameters are interpreted as a set of
artificial genes, 3) genes are considered as blueprints
of individuals and 4) evolution is applied to
individuals (Fogel, Owens & Walsh 1966; Holland
1975; Krink 2005).
Evolutionary algorithms have typically five basic
components: 1) a genetic representation of a number
of solutions to the problem, 2) a way to create an
initial population of solutions, 3) an evaluation
function for rating solutions in terms of their
“fitness”, 4) “genetic” operators that alter the genetic
composition of offspring during reproduction, 5)
values for the parameters, e.g. population size,
probabilities of applying genetic operators
(Michalewicz 1996).
2 GENETIC ALGORITHMS
Most often genetic algorithms (GAs) have at least
the following elements in common: one or more
populations of chromosomes (solution candidates,
individuals), selection according to fitness, crossover
to produce new offspring, and random mutation of
offspring. A simple GA works as follows: 1) A
population of
n l -bit strings (chromosomes) is
randomly generated, 2) the fitness
)(xf of each
chromosome
x
in the population is calculated, 3)
chromosomes are selected to go through crossover
and mutation operators with
c
p and
m
p
probabilities respectively, 4) the old population is
replace by the new one, 5) the process is continued
until the termination conditions are met.
The basic concepts of GA have evolved over
time and many heuristic techniques have been
utilized to customize GA for specific types of
problems. The studies on the usage of genetic
components have been grouped into the following
categories in (Talaslioglu 2009): 1) genetic operators
with adjustable parameters and representation of
design variables, which covers a.o. representation
techniques and different operators used in GAs, 2)
constraint handling for evaluation of fitness values,
which for instance covers the design of penalty
functions and adaptive approaches for handling the
constraints.
220
Moghadampour G. (2010).
MALE AND FEMALE CHROMOSOMES IN GENETIC ALGORITHMS.
In Proceedings of the 12th International Conference on Enterprise Information Systems - Artificial Intelligence and Decision Support Systems, pages
220-225
DOI: 10.5220/0002897702200225
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2.1 Problem Encoding
Genetic algorithms work on the genotype space
(coding space) and the phenotype space (solution
space) alternatively. The genetic operators work on
the genotype space and the evaluation and selection
operators work on the phenotype space. The
mapping from the genotype space to the phenotype
space influence the performance of genetic
algorithms significantly. Natural selection is the link
between chromosomes and the performance of
decoded solutions (Gen & Cheng 2000).
The encoding methods can be divided to the
following classes: 1) binary encoding for binary,
integer and real numbers, 2) real number encoding
for real numbers only, 3) integer or literal
permutation encoding, 4) general data structure
encoding. Binary encoding requires a decoding
function and it might cause decoding anomalies
(Krink 2005). The real number encoding is
considered the best one for solving optimizations
and constrained optimizations problems (Gen et al
2000).
2.2 Genetic Operators
For any evolutionary computation technique a
representation of object variables must be chosen
along with the appropriate evolutionary computation
operators. For each representation, several operators
might be employed (Michalewicz 2000).
The most commonly used genetic operators are
crossover and mutation. However, the basic ideas of
these operators have been adjusted and implemented
in many different problem specific manners by many
researchers. Several genetic operators have been
presented by Moghadampour (Moghadampour
2006); random building block operator, integer and
decimal mutation operators, variable crossover
operator and variable replacement operator. In
(Cervantes & Stephens 2008) it is argued that
applying genetic operators with probabilities
dependant on the fitness rank of a genotype or
phenotype offers a robust alternative to the simple
GA and avoids some questions of parameter tuning
without having to introduce an explicit encoded self-
adaptation mechanism.
2.2.1 Crossover
Crossover is the main distinguishing feature of a
GA. In single-point crossover a single crossover
position is chosen randomly and the parts of the two
parents divided by the crossover position are
exchanged to form two new individuals (offspring).
It recombines building blocks (schemas) on different
strings, but, it is “positional biased”: the location of
the bits in the chromosome determines the schemas
that can be created or destroyed by crossover
(Eshelman, Caruana & Schaffer 1989; Mitchell
1998).
In two-point crossover, two positions are chosen
at random and the segments between them are
exchanged. Two-point crossover reduces positional
bias and endpoint effect, it is less likely to disrupt
schemas with large defining lengths, and it can
combine more schemas than single-point crossover
(Mitchell 1998). Two-point crossover cannot
combine all schemas.
Multipoint-crossover has also been implemented,
e.g. in one method, the number of crossover points
for each parent is chosen from a Poisson distribution
whose mean is a function of the length of the
chromosome.
Parameterized uniform crossover (Spears & De
Jong 1991; Mitchell 1998) is also a multipoint-
crossover in which each bit is exchanged with
probability
p ( 8.05.0
≤
≤
p ) and any schemas
contained at different positions in the parents can
potentially be recombined in the offspring. Hence,
there is no positional bias. This implies that uniform
crossover can be highly disruptive of any schema
and may prevent co-adapted alleles from ever
forming in the population. (Mitchell 1998).
2.2.2 Mutation
The common mutation operator used in canonical
genetic algorithms to manipulate binary strings
A
A
}1,0{),...(
1
=∈= Iaaa of fixed length A was
originally introduced by Holland (1975) for general
finite individual spaces
A
AAI ...
1
×= , where
},...,{
1
i
k
iii
A
α
α
=
. By this definition, the mutation
operator proceeds by:
i. determining the position
}),...,1{(,...,
1
liii
jh
∈
to undergo mutation by a uniform random choice,
where each position has the same small
probability
m
p
of undergoing mutation,
independently of what happens at other positions.
ii. forming the new vector
),...,,,...,,,,...,(
111
11
1
1
1 A
aaaaaaaaa
h
i
h
i
h
iiiii +−+−
′
′
=
′
,
where
ii
Aa
∈
′
is drawn uniformly at random from
the set of admissible values at position
i .
MALE AND FEMALE CHROMOSOMES IN GENETIC ALGORITHMS
221

The original value
i
a at a position undergoing
mutation is not excluded from the random choice of
ii
Aa ∈
′
. This implies that although the position is
chosen for mutation, the corresponding value might
not change at all (Bäck, Fogel, Whitley & Angeline
2000).
Crossover is commonly viewed as the major
instrument of variation and innovation in GAs, with
mutation, playing a background role, insuring the
population against permanent fixation at any
particular locus (Mitchell 1998; Bäck, Fogel,
Whitley & Angeline 2000). Mutation and crossover
have the same ability for “disruption” of existing
schemas, but crossover is a more robust constructor
of new schemas (Spears 1993; Mitchell 1998).
While recombination involves more than one
parent, mutation generally refers to the creation of a
new solution form one and only one parent. Most
mutation operators for permutations are related to
operators, which have also been used in
neighbourhood local search strategies. (Whitley
2000).
2.2.3 Other Operators and Mating
Strategies
Examples of other operators used in Gas are:
inversion, gene doubling and several operators for
preserving diversity in the population. For instance,
a “crowding” operator has been used in (De Jong
1975; Mitchell 1998) to prevent too many similar
individuals (“crowds”) from being in the population
at the same time. This operator replaces an existing
individual by a newly formed and most similar
offspring. In (Mengshoel & Goldberg 2008) a
probabilistic crowding niching algorithm, in which
subpopulations are maintained reliably, is presented.
It is argued that like the closely related deterministic
crowding approach, probabilistic crowding is fast,
simple, and requires no parameters beyond those of
classical genetic algorithms.
The same result can be accomplished by using an
explicit “fitness sharing” function (Goldberg &
Smith 1987; Mitchell 1998) whose idea is to
decrease each individual’s fitness by an explicit
increasing function of the presence of other similar
population members. In some cases, this operator
induces appropriate “speciation”, allowing the
population members to converge on several peaks in
the fitness landscape (Goldberg et al. 1987; Mitchell
1998). However, the same effect could be obtained
without the presence of an explicit sharing function
(Smith, Forrest & Perelson 1993; Mitchell 1998).
Diversity in the population can also be promoted
by putting restrictions on mating. For instance,
distinct “species” tend to be formed if only
sufficiently similar individuals are allowed to mate
(Deb & Goldberg 1989; Mitchell 1998). Another
attempt to keep the entire population as diverse as
possible is disallowing mating between too similar
individuals, “incest” (Eshelman 1991; Eshelman &
Schaffer 1991; Mitchell 1998). Another solution is
to use a “sexual selection” procedure; allowing
mating only between individuals having the same
“mating tags” (parts of the chromosome that identify
prospective mates to one another). These tags, in
principle, would also evolve to implement
appropriate restrictions on new prospective mates
(Holland 1975; Mitchell 1998).
Another solution is to restrict mating spatially.
The population evolves on a spatial lattice, and
individuals are likely to mate only with individuals
in their spatial neighbourhoods. Such a scheme
would help preserve diversity by maintaining
spatially isolated species, with innovations largely
occurring at the boundaries between species (Hillis
1992; Mitchell 1998).
3 APPLYING MALE-FEMALE
PATTERN
In this research bit string
2
c is considered to be the
female of bit string
1
c if the value of each locus in
2
c has the opposite value of the equivalent locus in
1
c . The same idea can be expressed in the following
way:
212211
^1: llclcl =∈∀∧∈∀
(1)
The male-female concepts were applied during
population initialization at the beginning of the
algorithm and during each crossover operator. In the
following we go through each procedure in detail.
3.1 Initialization of the Population
Binary initialization of individuals was performed to
assure the real random initialization of the
population. This was simply done by randomly
initializing each locus of gene in each chromosome
with 0 or 1. During initialization, for each individual
a mate of the opposite sex was created. This was
done simply by inverting each gene in the individual
to its opposite value.
The motivation for this operation was the
observation that flipping all bits in a string could
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lead to rapid fitness improvement. Moreover,
crossover with two bit strings of opposite values
increases the chance of producing better offspring.
For each individual, both male and female
chromosomes were evaluated. Of these
chromosomes, the fitter one was set to be the male
chromosome and the other one was set to be the
female chromosome. It’s clear that the order of male
and female chromosomes could have been well vice
versa.
This process was repeated until all members of
the population were created. Therefore, each
individual was actually presented by two
chromosomes: a male chromosome and a female
chromosome. However, operations were aimed at
the male chromosomes by default.
After each evaluation of the population,
individuals were sorted in ascending order according
to their fitness values. This helped dividing the
population to three separate parts: 25th percentile
(lower quartile), 75th percentile (higher quartile) and
interquartile range (above the lower quartile and
below the higher quartile). This division was
necessary in order to recognize the most critical
areas in the distribution of the population and focus
genetic operators on most promising individuals.
This also helped implementing genetic operators
more precisely and avoiding precious processing
time.
It is assumed that as a result, genetic operators
will be more efficient and improve the population
more rapidly. This division will also help
maintaining diversity in the population while for
instance individuals in the higher quartile will go
through continuous evolution process and improve
more. Furthermore, the division of the population
helps focusing evolutionary operators intentionally
on certain individuals instead of hoping that a
random process would take care of the process and
select fitter individuals for different operators.
3.2 The Higher Quartile Crossover
Operator
The higher quartile crossover (HQC) operator
implements the idea of the well-known one-point
crossover. However, parents are selected for
breeding in a new way. For this operator, two
different parents,
1
p and
2
p and a crossover point
cp were randomly selected. Then, the male
chromosome of parent
1
p was crossed over with the
male and female chromosomes of parent
2
p . As a
result, four new offspring were created. These new
offspring then went through the survivor selection
procedure for possible replacement.
The new idea here is that the crossover operator
is repeated with the same parent indexes and on the
same crossover point as long as a better offspring is
created. It is important to notice that the same parent
indexes do not necessarily mean the same parents
since a better offspring might have replaced the
parent during the previous survivor selection.
Crossover points were selected so that they were at
least two loci far from the end points of the binary
representations of the chromosomes. This was to
make sure that the operator was really crossover and
not mutation.
4 EXPERIMENTATION
The male and female chromosome pattern was
applied as part of a genetic algorithm to solve the
following minimization problems: 1) the Ackley
function, 2) the Colville function, 3) the De Jong
function F1, 4) the De Jong function F2, 5) the De
Jong function F3, 6) the De Jong function F4, 7) the
De Jong function F5, 8) the Griewank function F1,
9) the Rastrigin function, 10) the Rosenbrock
function, 11) the Schaffer function F6, and 12) the
Schaffer function F7.
During test runs the population size was set to
12, but no other fixed parameter value was used.
During each phase operators were repeated as long
as they managed to produce better offspring.
For multidimensional problems with optional
number of dimensions (
n ), the algorithm was tested
for
=
n 1, 2, 3, 4, 5, 10, 20, 50 and 100. These
problems formed 44 cases and each one was tested
in 50 runs. The efficiency of utilizing so called
female chromosome in generating better individuals
and improving the population fitness values was
studied during the test runs. Table 1 summarizes the
survival statistics of male and female chromosomes
for the test functions.
The statistics indicate that using the female
chromosome has resulted in generating better
offspring in many cases. On average the female
chromosomes resulted in generating better offspring
in around 32% of test cases. Therefore it can easily
be concluded that utilizing the female chromosome
could help genetic algorithm in producing better
individuals and accelerate the process of finding the
optimal solution.
MALE AND FEMALE CHROMOSOMES IN GENETIC ALGORITHMS
223

Table 1: Comparison of survival of male and female chromosomes in test runs. To save space, for multidimensional
problems only cases with the number of variables equals to 2 and 100 are reported.
Function Variables
Survival Rate in
Selection (%)
Survival Rate in Crossover
(%)
Worst Run Best Run
Male Female Male
Female Male Female
Ackley
2 50 50 60 39 72 27
100
50 50 52 47 100 0
Griewank
2 50 49 92 7 100 0
100
0 100 81 18 0 100
Rosenbrock F1
2 50 49 100 0 100 0
100
50 50 100 0 100 0
Rastrigin F1
2 77 22 80 20 92 8
100
50 50 57 42 66 33
Colville 4 50 49 75 25 33 66
De Jong F1 3 50 50 - - - -
De Jong F2 2 52 47 - - - -
De Jong F3 5 50 49 - - - -
De Jong F4 30 68 31 33 66 46 53
De Jong F5 2 49 50 78 21 33 66
Schaffer F6 2 50 50 77 22 66 33
Schaffer F7 2 50 50 44 55 33 66
5 CONCLUSIONS
In this paper male and female chromosome concepts
were presented and techniques for generating and
applying these patterns in practice were proposed. In
addition, individuals were organized and selected for
further processing in a new way. The population was
ordered based on individuals’ fitness values and the
ones in the higher quartile of the population
distribution were selected to go through the
crossover operator.
Experimentation showed that the male female
pattern can be useful in many cases and result in
generating better individuals during the evolutionary
process of genetic algorithm. However,
experimentation suggests that the output of applying
this technique to some extent depends on the nature
of the search space and the function to be optimized.
For symmetrical search spaces, i.e. the ones, which
divide equally on the both sides of zero (like
4.54.5 ≤≤− x ) creating so called female
chromosome out of a male chromosome will result
in creating the additive inverse of the equivalent
floating-point value of the male chromosome.
This obviously will then provide an easy and fast
way to explore the opposite areas of the search
space. For asymmetrical search spaces creating
female chromosomes will in many cases result in
creating floating-point values of different
magnitude, which also help exploring the search
space more efficiently. Another observation with
using female chromosomes was that in many cases
even though the female chromosome itself didn’t
yield a better fitness value than its male mate, they
produced better offspring than their male mates after
going through genetic operators.
5.1 Future Research
The proposed pattern can be applied to new
problems and its efficiency in helping the search
process can be further evaluated. The pattern can
also be refined and adjusted to match other types of
problems.
ACKNOWLEDGEMENTS
This research is part of PhD dissertation
(Moghadampour 2006), which was publicly
defended on the 12th of May, 2006 at the University
of Vaasa, Vaasa, Finland.
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