Selection Methods for Genetic Algorithms

Abstract

Based on a study of six well known selection methods often used in
genetic algorithms, this paper presents a technique that benefits their
advantages in terms of the quality of solutions and the genetic diversity. The
numerical results show the extent to which the quality of solution depends on
the choice of the selection method. The proposed technique, that can help
reduce this dependence, is presented and its efficiency is numerically
illustrated.

Full text

Int. J. Emerg. Sci., 3(4), 333-344, December 2013
ISSN: 2222-4254
© IJES
333
Selection Methods for Genetic Algorithms
Khalid Jebari, Mohammed Madiafi
Laboratory, Faculty of Sciences, Mohammed V–Agdal University, UM5A, Rabat, Morocco,
Ben M’sik Faculty of Sciences, Hassan II Mohammedia University, UH2M, Casablanca, Morocco,
khalid.jebari@gmail.com, madiafi.med@gmail.com
Abstract. Based on a study of six well known selection methods often used in
genetic algorithms, this paper presents a technique that benefits their
advantages in terms of the quality of solutions and the genetic diversity. The
numerical results show the extent to which the quality of solution depends on
the choice of the selection method. The proposed technique, that can help
reduce this dependence, is presented and its efficiency is numerically
illustrated.
Keywords: Evolutionary Computation, Genetic Algorithms, Genetic
Operator, Selection Pressure, Genetic Diversity.
1 INTRODUCTION
The theory of genetic algorithms (GAs) was originally developed by John Holland
in 1960 and was fully developed in his book " Adaptation in Natural and Artificial
Systems ", published in 1975 [1]. His initial goal was not to develop an optimization
algorithm , but rather to model the adaptation process, and show how this process
could be useful in a computing system. The GAs are stochastic search methods
using the concepts of Mendelian genetics and Darwinian evolution [2,3]. Such
heuristics have been proved effective in solving a variety of hard real-world
problems in many application domains including economics, engineering,
manufacturing, bioinformatics, medicine, computational science, etc [4].
In principle, a population of individuals selected from the search space , often in a
random manner, serves as candidate solutions to optimize the problem [3]. The
individuals in this population are evaluated through ( "fitness" ) adaptation function.
A selection mechanism is then used to select individuals to be used as parents to
those of the next generation. These individuals will then be crossed and mutated to
form the new offspring. The next generation is finally formed by an alternative
mechanism between parents and their offspring [4]. This process is repeated until a
certain satisfaction condition.
More formally, a standard GA can be described by the following pseudo-code:

Khalid Jebari, and Mohammed Madiafi
334
SGA_pseudo-code
{
 Choose an initial population of individuals:
(0)P
;
 Evaluate the fitness of all individuals of
(0)P
;
 Choose a maximum number of generations:
max
t
;
 While (not satisfied and
max
tt
) do {
-
1tt
;
- Select parents for offspring production;
- Apply reproduction and mutation operators;
- Create a new population of survivors:
()Pt
;
- Evaluate
()Pt
;
}
 Return the best individual of
()Pt
;
}
As we can see from the pseudo-code above, a GA is a parametrical algorithm
whose application to a given problem requires setting parameters and making
decisions about:
 the way parents are selected for offspring production;
 selected parents are recombined;
 population size is adjusted;
 individuals are mutated, or crossed;
 probability of crossover and probability of mutation that explore other areas
of research space are chosen [5].
The ultimate goal is the right choice of these parameters [5]. Research has
therefore looked to find techniques to dynamically adjust and improve the quality of
the solution. We find in the literature a lot of attention to the genetic operators
exploration, and much more to population size [5]. By cons less attention is paid to
the selection operator. Without this operator, genetic algorithms are only simple
random methods give different values each time [3]. Individuals who have a higher
fitness value have a high probability of being selected for the next generation by this
operator. Selection method focuses research in promising areas of the search space.
The balance between exploitation and exploration is essential for the behaviour of
genetic algorithms. It can be adjusted by the selection pressure of the selection
operator and the probability of crossover and mutation. In the case of selection, a
strong selection pressure may cause the algorithm converges to a local optimum,
while a low selection pressure may cause the AG to random results that differ from
one run to another [6]. The selection operator is aimed at exploiting the best
characteristics of good candidate solutions in order to improve these solutions
throughout generations, which, in principle, should guide the GA to converge to an
acceptable and satisfactory solution of the optimisation problem at hand [2].
Selection operator is the most important parameter that may influence the
performance of a GA [7]. It is aimed at exploiting the best characteristics of good

International Journal of Emerging Sciences 3(4), 333-344, December 2013
335
candidate solutions in order to improve these solutions throughout generations,
which, in principle, should guide the GA to converge to an acceptable and
satisfactory solution of the optimisation problem at hand [6].
In the literature there are several selection methods: Roulette Wheel Selection
[8], Stochastic Universal Sampling, the tournament selection and the selection of
Boltzmann and others [9].
However, despite decades of research, there are no general guidelines or
theoretical support concerning the way of selecting a good selection method for
each problem [10]. This can be a serious problem because, as we will see through
numerical results, a non-suitable selection operator can lead to poor performance of
the GA in terms of both rapidity and reliability. To illustrate this problem we will
consider a set of six popular selection methods that we apply to the optimization
problem of four benchmark functions. In the following sections, we first provide a
brief description of each studied selection method. In section 3, we outline the
presentation of a technique that we propose in order to help reducing the influence
of selection operator on the global performance of a GA. Numerical results are
presented and discussed in section 4. Section 5 is devoted to our concluding remarks
and future works.
2 SELECTION METHODS FOR GAS
As mentioned before, six different selection methods are considered in this work,
namely: the roulette wheel selection (RWS), the stochastic universal sampling
(SUS), the linear rank selection (LRS), the exponential rank selection (ERS), the
tournament selection (TOS), and the truncation selection (TRS). In this section, we
provide a brief description of each studied selection method.
2.1 Roulette Wheel Selection (RWS)
The conspicuous characteristic of this selection method is the fact that it gives to
each individual i of the current population a probability
()pi
of being selected [10],
proportional to its fitness
()fi
1
()
() ()
n
j
fi
pi fj


(1)
Where n denotes the population size in terms of the number of individuals. The
RWS can be implemented according to the following pseudo-code
RWS_ pseudo-code
{
 Calculate the sum
1()
n
i
S f i


;

Khalid Jebari, and Mohammed Madiafi
336
 For each individual
1in
do {
- Generate a random number
 
0,S


;
-
0iSum 
;
0j
;
- Do {
-
()iSum iSum f j
;
-
1jj
;
} while(
iSum


and
jn
)
- Select the individual
j
;}
}
Note that a well-known drawback of this technique is the risk of premature
convergence of the GA to a local optimum, due to the possible presence of a
dominant individual that always wins the competition and is selected as a parent.
2.2 Stochastic Universal Sampling (SUS)
The SUS [8] is a variant of RWS aimed at reducing the risk of premature
convergence. It can be implemented according to the following pseudo-code:
SUS_ pseudo-code
{
 Calculate the mean
1
1 ( )
n
i
f n f i


;
 Generate a random number
 
0,1


;

(1)Sum f
;
delta f


;
0j
;
 Do {
- If (
delta Sum
) {
- select the jth individual;
-
delta delta Sum
;
}
else {
-
1jj
;
-
()Sum Sum f j
;
}
} while (
jn
)
}

International Journal of Emerging Sciences 3(4), 333-344, December 2013
337
2.3 Linear Rank Selection (LRS)
LRS [9] is also a variant of RWS that tries to overcome the drawback of premature
convergence of the GA to a local optimum. It is based on the rank of individuals
rather than on their fitness. The rank n is accorded to the best individual whilst the
worst individual gets the rank 1. Thus, based on its rank, each individual i has the
probability of being selected given by the expression
()
() ( 1)
rank i
pi nn

(2)
Once all individuals of the current population are ranked, the LRS procedure can
be implemented according to the following pseudo-code:
LRS_ pseudo-code
{
 Calculate the sum
1
2.001
vn

;
 For each individual
1in
do {
- Generate a random number
 
0,v


;
- For each
1jn
do {
- If (
()pj


) {
- Select the jth individual;
- Break;
}
}
}
}
2.4 Exponential Rank Selection (ERS)
The ERS [10] is based on the same principle as LRS, but it differs from LRS by the
probability of selecting each individual. For ERS, this probability is given by the expression:
 
( ) 1.0 exp rang i
pi c


 

(3.a)
with
 
 
 
 
21
61
nn
cnn
  
  
(3.b)
Once the n probabilities are computed, the rest of the method can be described by the
following pseudo-code:
ERS_ pseudo-code

Khalid Jebari, and Mohammed Madiafi
338
{
 For each individual
1in
do {
- Generate a random number
12
,
9cc




;
- For each
1jn
do {
- If (
()pj


) {
- Select the jth individual;
- Break;
} // end if
} // end for j
}// end for i
}
2.5 Tournament Selection (TOS)
Tournament selection [11] is a variant of rank-based selection methods. Its principle
consists in randomly selecting a set of k individuals. These individuals are then
ranked according to their relative fitness and the fittest individual is selected for
reproduction. The whole process is repeated
n
times for the entire population.
Hence, the probability of each individual to be selected is given by the expression:
 
 
1
1 if 1, 1
()
0 if ,
k
nk
n
Ci n k
C
pi
i n k n


  




(4)
Technically speaking, the implementation of TOS can be performed according to
the pseudo-code:
TOS_ pseudo-code
{
 Create a table
t
where the
n
individuals are placed in
a randomly chosen order
 For
1 to in
do {
- for
1 to jn
do {
-
1()i t j
;
- For
1 to mn
do {
-
2()i t j m
;
- If (
12
( ) ( )f i f i
) the select
1
i
else select
2
i
;
}// end for m
-
j j k
;
} // end for j

International Journal of Emerging Sciences 3(4), 333-344, December 2013
339
}// end for i
}
2.6 Truncation Selection (ERS)
The truncation selection [10] is a very simple technique that orders the candidate
solutions of each population according to their fitness. Then, only a certain portion
p
of the fittest individuals are selected and reproduced
1p
times. It is less used in
practice than other techniques, except for very large population. The pseudo-code of
the technique is as follows:
TRS_ pseudo-code
{
 Order the
n
individuals of
()Pt
according to their
fitness;
 Set the portion
p
of individuals to select
(e.g.
10% 50%p
);

int( )sp n p
// selection pressure;
 Select the first
sp
individuals;
}
3 DESCRIPTION OF THE PROPOSED TECHNIQUE
After implementing the six selection methods described in the previous section and
tested them on the optimization problem of a variety of test functions we found that
results differ significantly from one selection method to another. This poses the
problem of selecting the adequate method for real-world problems for which no
posterior verification of results is possible.
To help mitigating this non-trivial problem we present in this section the outlines
of a new selection procedure that we propose as an alternative, which can be useful
when no single other technique can be used with enough confidence. Our technique
is a dynamic one in the sense that the selection protocol can vary from one
generation to another. The underling idea consists in finding a good compromise
between proportional methods, which decrease the effect of selection pressure and
assure some genetic diversity within the population, but may increase the
convergence time; and elitist methods that reduce the convergence time but may
increase the effect of selection pressure and, therefore, the risk of converging to
local minima.
To achieve this goal, more than one selection method are applied at each generation,
but in a competitive way meaning that only results provided by the selection
operator with the best performance are actually taken into account. To assess and

Khalid Jebari, and Mohammed Madiafi
340
compare the performance of candidate selection methods two objective criteria are
employed. The first criterion is the quality of solution; it can easily be measured as a
function of the fitness
*
f
of the best individual. The second criterion is the genetic
diversity, which is less evident to quantify than the first one. In this work, as a
measure of the genetic diversity within the population
()Pt
we propose the mean
population diversity at generation t over 30 runs is calculated according to the
formula:
 
   








30
1 1
),(
)1ln()max( 1
30
1
)(
k
n
i
n
ij ij
ij tkd
nd
tDiv
(5)
Where Di j (k, t) is the Euclidean distance between the i-th and j-th individuals at
generation t of the k-th run, n is the population size, max(dij)is the maximum distance
supposed between individuals of the population and t the number of generations or
iterations. As a measure of the quality of the solution at each generation we used
the criterion
2
min
2
max
*
ff
f
Quality 

(6)
Where
max
f
and
min
f
denote respectively the maximum and the minimum values of
the fitness at generation
t
, and
*max
ff
or
*min
ff
depending on the nature of the
problem, which can be either a maximization or a minimization problem. Finally, in
order to combine the two criteria in a unique one we used the relation
Quality
t
t
Div
t
C11 

(7)
4 EXPERIMENTAL STUDY
In this section, we present some examples of numerical results obtained by applying
the six selection methods, as well as the proposed Combined Selection (CS)
procedure, described in the previous sections, to the problem of genetic
optimization of a set of four well-known benchmark functions [12]. The first
function is a classical multi modal and high dimension function defined on the
interval [-30,30] by the expression:

















 



n
ii
n
iix
n
x
neexf 1
1
2)2cos(
1
1
2.0
20))(min(

(8)
The second test function is also a mono-dimensional example but it is a deceptive
one in the sense that it possesses, as depicted by Fig. 1, two local maxima in which
optimization algorithms may be trapped.

International Journal of Emerging Sciences 3(4), 333-344, December 2013
341
Figure 1. The deceptive test function f2(x).
2()fx
is defined on the interval
 
0,6
by the expression
  
 
 
2
1 for 0,1
0.3 1 for 1,3
() 0.3 5 for 3,5
2 10 for 5,6
xx
xx
fx xx
xx

  

   

  

(9)
The third example is the bi-variable Shubert function [3] defined, for
1010 1 x
and
1010 2 x
, by
55
3 1 2 1 2
11
( , ) ( cos(( 1) )( cos(( 1) ))
ii
f x x i i x i i i x i

    

(10)
And the last example is the bi-variable instance of the more general Rastrigin
function [4] defined by:
 n
iin xxnxxxf 12
214 ))2cos(10(10),...,,(

(11)
for
nixi,...,2,1 ,12.512.5 
Results are summarized in Tables 1 to 3. Table 1 shows for each selection and
each test function, the number of generations the GA needed to converge to an
acceptable solution.
Analysis of Table I shows significant differences in convergence speed of the GA
for the six studied selection methods, particularly in the case of the deceptive
example,
2
f
.
Table 1. Number of Generations needed for convergence
Test
Functions
Selection Method
RWS
SUS
LRS
ERS
TOS
TRS
CS
f1
40
34
48
16
7
5
9
f2
229
173
290
290
14
18
27

Khalid Jebari, and Mohammed Madiafi
342
f3
43
16
18
18
10
8
14
f4
27
9
14
14
3
3
5
Table 2. Percentage of Selection Pressure
Test
Functions
Selection Method
RWS
SUS
LRS
ERS
TOS
TRS
CS
f1
12.3
12.2
22.8
22.9
80
70
84
f2
2.3
2.3
1.3
5.4
75
33
70
f3
10
4.8
4.8
4.8
80
65
80
f4
25.1
19
4.8
4.8
65
50
70
Table 2 shows the percentage of selection pressure for each studied selection
method and each function. The selection pressure,
sp
, of a given selection method
is defined as the number of generations after which the best individual dominates
the population. As to the percentage of selection pressure, it is defined by
min
sp sp
where
min
sp
denotes the minimal selection pressure observed among all the studied
parent selection methods. We can remark that proportional methods maintain the
genetic diversity more than elitist ones.
Table 3 provides sample results related to another aspect of this study. It is the
aspect of quality assessment of the optimum provided by the GA for each test
function and for each selection method, including the combined selection method
we proposed in this work. To measure this quality, we used the relative error
f
ff
ff

*
(12)
where
*
f
is the optimum provided by the algorithm and
f
the actual optimum,
which is a priori known.
Table 3. Relative errors in percentage of the optima
Test
Functions
Selection Method
RWS
SUS
LRS
ERS
TOS
TRS
CS
f1
18
7
7
7
5
7
4
f2
5
7
0
5
0
0
0
f3
34
34
34
34
34
34
0
f4
98
98
96
96
95
96
0
By analysing this table we can see clearly that the proposed method of selection
performs better than the six studied selection methods.

International Journal of Emerging Sciences 3(4), 333-344, December 2013
343
5 CONCLUSION
In this paper, six well-known selection methods for GAs are studied, implemented
and their relative performance analysed and compared using a set of four
benchmark functions. These methods can be categorised into two categories:
proportional and elitist. The first category uses a probability of selection
proportional to the fitness of each individual. Selection methods of this category
allow maintaining a genetic diversity within the population of candidate solutions
throughout generations, which is a good property that prevents the GA from
converging to local optima. But, on the other hand, these methods tend to increase
the time of convergence.
By contrast, selection methods of the second category select only the best
individuals, which increases the speed of convergence but at the risk of converging
to local optima due to the loss of genetic diversity within the population of candidate
solutions.
Starting from these observations, we have conducted a preliminary study aimed at
combining the advantages of the two categories. This study resulted in a new
combined selection procedure whose outlines are presented in this paper. The main
idea behind this procedure is the use of more than one selection method in a
competitive way together with an objective criterion which allows choosing the best
selection method to adopt at each generation.
The proposed technique was successfully applied to the optimisation problem of a
set of well-known benchmark functions, which encourages farther developments of
this idea.
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