A Novel Crossover Operator for Genetic Algorithms: Ring Crossover

Abstract

The genetic algorithm (GA) is an optimization and search technique based on
the principles of genetics and natural selection. A GA allows a population
composed of many individuals to evolve under specified selection rules to a
state that maximizes the "fitness" function. In that process, crossover
operator plays an important role. To comprehend the GAs as a whole, it is
necessary to understand the role of a crossover operator. Today, there are a
number of different crossover operators that can be used in GAs. However, how
to decide what operator to use for solving a problem? A number of test
functions with various levels of difficulty has been selected as a test polygon
for determine the performance of crossover operators. In this paper, a novel
crossover operator called 'ring crossover' is proposed. In order to evaluate
the efficiency and feasibility of the proposed operator, a comparison between
the results of this study and results of different crossover operators used in
GAs is made through a number of test functions with various levels of
difficulty. Results of this study clearly show significant differences between
the proposed operator and the other crossover operators.

Full text

A Novel Crossover Operator for Genetic Algorithms:
Ring Crossover
Yılmaz KAYA
1
Murat UYAR
2
Ramazan TEKĐN
3
1
Siirt University, Department of Computer Engineering, Siirt, Turkey, yilmazkaya1977@gmail.com
2
Siirt University, Department of Electric and Electronics Engineering, Siirt, Turkey, muratuyar1@gmail.com
3
Batman University, Department of Computer Engineering, Batman, Turkey, ramazan.tekin@batman.edu.tr
Abstract
—
The genetic algorithm (GA) is an optimization and
search technique based on the principles of genetics and
natural selection. A GA allows a population composed of many
individuals to evolve under specified selection rules to a state
that maximizes the “fitness” function. In that process,
crossover operator plays an important role. To comprehend
the GAs as a whole, it is necessary to understand the role of a
crossover operator. Today, there are a number of different
crossover operators that can be used in GAs. However, how to
decide what operator to use for solving a problem? A number
of test functions with various levels of difficulty has been
selected as a test polygon for determine the performance of
crossover operators.
In this paper, a novel crossover operator called ‘ring
crossover’ is proposed. In order to evaluate the efficiency and
feasibility of the proposed operator, a comparison between the
results of this study and results of different crossover operators
used in GAs is made through a number of test functions with
various levels of difficulty. Results of this study clearly show
significant differences between the proposed operator and the
other crossover operators.
Keywords : Genetic algorithm, crossover operator, ring crossover
I.
I
NTRODUCTION
Genetic algorithms (GAs) represent general-purpose
search and optimization technique based on evolutionary
ideas of natural selection and genetics. They simulate
natural processes based on principles of Lamarck and
Darwin. In 1975, Holland developed this idea in his book
“Adaptation in natural and artificial systems”. He described
how to apply the principles of natural evolution to
optimization problems and built the first GAs. Holland’s
theory has been further developed and now GAs stand up as
a powerful tool for solving search and optimization
problems. GAs are based on the principle of genetics and
evolution [1] . Today, there exists many variations on GAs
and term “genetic algorithm” is used to describe concepts
sometimes very far from Holland’s original idea [2]. The
two most commonly employed genetic search operators are
crossover and mutation. Crossover produces offspring by
recombining the information from two parents. Mutation
prevents convergence of the population by flipping a small
number of randomly selected bits to continuously introduce
variation. The driving force behind GAs is the unique
cooperation between selection, crossover and mutation
operator. A genetic operator is a process used in GAs to
maintain genetic diversity. The most widely used genetic
operators are recombination, crossover and mutation.
The main goal of this paper is to introduce a new
crossover operator called ring crossover (RC) and present
the performance of this crossover operator. The rest of this
paper is organized as follow. In section 2, definitions and
concepts of the different crossover operators are introduced.
In section 3, the proposed method in this study is given. In
section 4, a number of the functions widely used in
performance evaluation of GA operators are defined. In
section 5, the optimization results and performance
comparison of proposed method are shown. Finally,
conclusions are discussed in section 6.
II. C
ROSSOVER OPERATORS
The crossover operator is a genetic operator that combines
(mates) two chromosomes (parents) to produce a new
chromosome (offspring). The idea behind crossover is that
the new chromosome may be better than both of the parents
if it takes the best characteristics from each of the parents.
Crossover occurs during evolution according to a user-
definable crossover probability. For purpose of this work,
only crossover operators that operate on two parents and
have no self-adaptation properties will be considered.
A. Single Point Crossover
When performing crossover, both parental chromosomes
are split at a randomly determined crossover point.
Subsequently, a new child genotype is created by appending
the first part of the first parent with the second part of the
second parent [3, 4]. A single crossover point on both
parents' organism strings is selected. All data beyond that
point in either organism string is swapped between the two
parent organisms. Figure 1 shows the single point crossover
(SPC) process.

Figure 1. Single point crossover
B. Two Point Crossover
Apart from SPC, many different crossover algorithms
have been devised, often involving more than one cut point.
It should be noted that adding further crossover points
reduces the performance of the GA. The problem with
adding additional crossover points is that building blocks are
more likely to be disrupted. However, an advantage of
having more crossover points is that the problem space may
be searched more thoroughly. In two-point crossover (TPC),
two crossover points are chosen and the contents between
these points are exchanged between two mated parents [5, 6]
Figure 2. Two point crossover
In figure 2, the arrows indicate the crossover points.
Thus, the contents between these points are exchanged
between the parents to produce new children for mating in
the next generation.
C. Intermediate Crossover
Intermediate creates offsprings by a weighted average of
the parents. Intermediate crossover (IC) is controlled by a
single parameter Ratio:
offspring = parent1+rand*Ratio*(parent2 - parent1)
If Ratio is in the range [0,1] then the offsprings produced
are within the hypercube defined by the parents locations at
opposite vertices. Ratio can be a scalar or a vector of length
number of variables. If Ratio is a scalar, then all of the
offsprings will lie on the line between the parents. If Ratio is
a vector then children can be any point within the hypercube
[7].
D. Heuristic Crossover
In heuristic crossover (HC), heuristic returns an offspring
that lies on the line containing the two parents, a small
distance away from the parent with the better fitness value
in the direction away from the parent with the worse fitness
value. The default value of Ratio is 1.2. If parent1 and
parent2 are the parents, and parent1 has the better fitness
value, the function returns the child [7],
offspring = parent2 + Ratio * (parent1 - parent2)
E. Arithmetic Crossover
In arithmetic crossover (AC), arithmetic creates children
that are the weighted arithmetic mean of two parents.
Children are feasible with respect to linear constraints and
bounds. Alpha is random value between [0,1]. If parent1 and
parent2 are the parents, and parent1 has the better fitness
value, the function returns the child [7],
offspring =alpha*parent1 + (1-alpha)*parent2
III. P
ROPOSED CROSSOVER OPERATOR
:
RING CROSSOVER
The operator called ring crossover is consisted of four
steps. The steps of the proposed operator in this paper are
shown in figure 3. All of the steps in the algorithm are
discussed one by one.
Step-1: In this step, two parents such as parent1 and parent2
are considered for the crossover process, as shown in fig.
3(a).
Step-2: The chromosomes of parents are firstly combined
with a form of ring, as shown in fig. 3(b). Later, a random
cutting point is decided in any point of ring.
Step-3: The children are created with a random number
generated in any point of ring according to the length of the
combined two parental chromosomes. With reference to the
cutting point in step 2, while one of the children is created in
the clockwise direction, the other one is created in direction
of the anti-clockwise, as shown in fig. 3(c).
Step-4: In this step, swapping and reversing process is
performed in the RC operator, as shown in fig. 3(d). In
swapping process, a number of genes are swapped in
crossed parents. In reversion process, the remaining genes
are reversed in crossed parents. As the length of ring is
equal to the total length of both of parents and the children
are created according to a random point of ring, more
variety can be provided in possible number of children by
RC operator according to SPC and TPC operators.

Figure 3. Ring crossover
IV. T
EST FUNCTIONS
The proposed method must be tested by a number of the
functions widely used in performance evaluation of GA
operators such as crossover. Test functions used in this paper
have two important features: modality and separability.
Unimodal function is a function with only one global
optimum. Function is multimodal if it has two or more local
optima. Multimodal functions are more difficult to optimize
compared to unimodal functions [8].
A. Sphere Function
Sphere function is a test function proposed by De Jong. It
has been widely used in evaluation of genetic algorithms and
development of the theory of evolutionary strategies. Sphere
function is a simple, continuous and strongly convex function.
Sphere function is unimodal and additively separable.
Boundaries are set at [-5.12; 5.12]. Sphere function’s global
minimum is in point x=0 with value f(x)=0 [8,9]. The simplest
test function is De Jong’s function 1. It is continues, convex
and unimodal. This function is defined as shown below.
2
1
( )
D
i
i
f x x
=
=
∑
(1)
This function is defined as F1 in paper.
B. Axis Parallel Hyper-Ellipsoid Function
This function is similar to Sphere function. It is also known
as weighted sphere model. It is also unimodal and additively
separable. Boundaries are set at [-5.12; 5.12]. Function’s
global minimum is in point x=0 with value f(x)=0 [8,9]. The
axis parallel hyper-ellipsoid is similar to De Jong's function 1.
It is also known as the weighted sphere model. Again, it is
continues, convex and unimodal. It is defined as shown below.
2
1
( )
D
i
i
f x ix
=
=
∑
(2)
This function is defined as F2 in paper.
C. Rotated Hyper-Ellipsoid Function
This function represents an extension of the axis parallel
hyper-ellipsoid function. With respect to the coordinate axes,
this function produces rotated hyperellipsoids. It is continues,
convex and unimodal. Boundaries are set at [-65.536; 65.536].
Function’s global minimum is in point x=0 with value f(x)=0
[9]. An extension of the axis parallel hyper-ellipsoid is
Schwefel's function 1.2. With respect to the coordinate axes,
this function produces rotated hyper-ellipsoids. It is continues,
convex and unimodal. This function is defined as shown
below.
2
1 1
( )
D i
j
i j
f x x
= =
 
=
 
 
∑ ∑
(3)
This function is defined as F3 in paper.
D. Normalized Schwefel Function
The surface of Schwefel function is composed of a great
number of peaks and valleys. The function has a second best
minimum far from the global minimum where many search
algorithms are trapped. Moreover, the global minimum is near
the bounds of the domain. Schwefel’s function is deceptive in
that the global minimum is geometrically distant, over the
parameter space, from the next best local minimum. Schwefel
function is multimodal and additively separable. Boundaries
are set at [-500; 500]. Function’s global minimum is in point
x=420.968 with value f(x)=-418.9829 [8] . Schwefel's function
[Sch81] is deceptive in that the global minimum is
geometrically distant, over the parameter space, from the next
best local minima. Therefore, the search algorithms are
potentially prone to convergence in the wrong direction. It is
defined as shown below.
1
( ) sin( | |
D
i i
i
f x x x
=
= −
∑
(4)
This function is defined as F4 in paper.
E. Generalized Rastrigin Function
Rastrigin function was constructed from Sphere adding a
cosine modular term. Its contour is made up of a large number
of local minima whose value increases with the distance to the
global minimum. Thus, the test function is highly multimodal.
However, the location of the local minima’s are regularly
distributed. Rastrigin function is additively separable.
Boundaries are set at [-5.12; 5.12]. Function’s global
minimum is found in point x=0 with value f(x)=0 [8,10].

2
1
( ) 10 ( 10cos(2 ))
D
i i
i
f x n x x
π
=
= − −
∑
(5)
This function is defined as F5 in paper.
F. Rosenbrock's Valley Function
Rosenbrock’s valley function (banana function) is a classic
optimization problem. The global optimum is inside a long,
narrow, parabolic shaped flat valley. To find the valley is
trivial, however convergence to the global optimum is difficult
and hence this problem has been repeatedly used in assess the
performance of optimization algorithms. Banana function is
additively separable. Boundaries are set at [-2.048; 2.048].
Function’s global minimum is found in point x=1 with value
f(x)=0 [10]. This function is defined as shown below.
1
2 2 2
1
1
( ) 100 ( ) (1 )
D
i i i
i
f x x x x
−
+
=
= − − + −
∑
(6)
This function is defined as F6 in paper.
V. E
XPERIMENTAL RESULTS
In all experiments, stochastic uniform selection was used.
Parameters of GA for experiments were as following:
Gaussian mutation with p
m
mutation coefficient of 0.01 and
crossover rate p
c
of 0.8 was used, number of independent runs
for each experiment was 30, initial population N of size 20
was randomly created and used in experiments.
Dimensionality of the search space D for all test function was
set to 30. Number of overall evaluations was set to 10000. For
all test functions, finding global minimum is the objective. All
of the experiment is realized for six different types of test
functions. A comparison between the proposed crossover
method (RC) and other crossover methods are made and the
results are comparatively presented in table 1.
Table 1. Performance comparison for the different types of test functions
Function Results SPC TPC IC HC AC
RC
F1 Best
Worst
Average
5.732
7.246
5.737
3.416
6.511
3.417
6.207
6.246
6.208
0.011
8.099
2.81
5.589
6.389
5.589
0.0027
6.163
0.3299
F2 Best
Worst
Average
70.52
105.7
70.52
68.63
94.04
68.64
64.04
80.4
64.04
0.024
87.41
5.706
73.71
89.18
73.72
0.1023
106.8
11.73
F3 Best
Worst
Average
20.79
261.7
37.02
15.06
204.8
16.22
22.86
59.24
22.87
2.36
381
17.58
24.37
47.94
24.37
4.577
108.2
18.97
F4 Best
Worst
Average
-115.7
-29.46
-115.6
-115.8
-26.85
-115.4
-114.1
-27.91
-114
-117.7
-26.1
-117.1
-113.2
-27.72
-113.1
-117.8
-27.75
-117.7
F5 Best
Worst
Average
94.69
241.3
111.3
50.84
257.7
52.15
122.6
256.6
187.3
12.68
173.1
31.98
154
251.4
154.1
2.669
232.5
3.691
F6 Best
Worst
Average
73.07
269.3
73.08
70.07
390.3
78.39
34.71
349.2
34.74
29.35
369.1
117.5
27.08
260.3
27.12
28.59
316.1
32.69
VI. C
ONCLUSION
In this paper, a new crossover operator called RC is
proposed and experiments are conducted. The proposed
operator is tested by a number of test functions with various
levels of difficulty. A comparison between the results of this
method and the results of other crossover operators are made.
RC operator gives better results according to other crossover
operators. Although the most of crossover operators showed
similar results, RC operator had slightly better results than the
other crossover for F1, F2, F4, F5 functions. For F3 function,
HC operator has slightly better result than RC operator.
However, RC operator produces better result than SPC, TPC,
IC and AC. For F6 function, the results of this study are very
close to those of AC, but in generally RC operator performed
the best results than other crossover operators.
The most important advantage of the proposed method is
that more variety is presented in possible number of children
according to SPC and TPC operators. The experiments and the
results presented in the paper clearly reveal the potential
capability of the proposed method in optimization processing
based on GA. Moreover, it has the great potential to improve
the performance of GA applications in different area of
engineering.
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