Comparison of Genetic Algorithm (GA) and Particle Swarm Optimization Algorithm (PSO) for Discrete and Continuous Size Optimization of 2D Truss Structures

Journal of Soft Computing in Civil Engineering 3-2 (2019) 76-97
How to cite this article: Akbari M, Henteh M. Comparison of genetic algorithm (GA) and particle swarm optimization algorithm
(PSO) for discrete and continuous size optimization of 2D Truss Structures. J Soft Comput Civ Eng 2019;3(2):76–97.
https://doi.org/10.22115/scce.2019.195713.1117.
2588-2872/ © 2019 The Authors. Published by Pouyan Press.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Contents lists available at SCCE
Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Comparison of Genetic Algorithm (GA) and Particle Swarm
Optimization Algorithm (PSO) for Discrete and Continuous Size
Optimization of 2D Truss Structures
M. Akbari1*
, M. Henteh2
1. Assistant Professor, Civil Engineering Department, University of Kashan, Kashan, Iran
2. Ph.D. Candidate, Structure Engineering, Faculty of Civil Engineering, Semnan University, Semnan, Iran
Corresponding author: makbari@kashanu.ac.ir
https://doi.org/10.22115/SCCE.2019.195713.1117
ARTICLE INFO ABSTRACT
Article history:
Received: 24 July 2019
Revised: 02 November 2019
Accepted: 03 November 2019
Optimization of truss structures including topology, shape
and size optimization were investigated by different
researchers in the previous years. The aim of this study is
discrete and continuous size optimization of two-dimensional
truss structures with the fixed topology and the shape. For
this purpose, the section area of the members are considered
as the decision variables and the weight minimization as the
objective function. The constraints are the member stresses
and the node displacements which should be limited at the
allowable ranges for each case. In this study, Genetic
Algorithm and Particle Swarm Optimization algorithm are
used for truss optimization. To analyse and determine the
stresses and displacements, OpenSees software is used and
linked with the codes of Genetic Algorithm and Particle
Swarm Optimization algorithm provided in the MATLAB
software environment. In this study, the optimization of four
two-dimensional trusses including the Six-node, 10-member
truss, the Eight-node, 15-member truss, the Nine-node, 17-
member truss and the Twenty-node, 45-member truss under
different loadings derived from the literature are done by the
Genetic Algorithm and Particle Swarm Optimization
algorithm and the results are compared with those of the
other researchers. The comparisons show the outputs of the
Genetic Algorithm are the most generally economical among
the different studies for the discrete size cases while for the
continuous size cases, the outputs of the Particle Swarm
Optimization algorithm are the most economical.
Keywords:
Particle swarm optimization
algorithm;
Genetic algorithm;
2D truss structures;
Discrete and continuous sizes;
OpenSees.

M. Akbari, M. Henteh/ Journal of Soft Computing in Civil Engineering 3-2 (2019) 76-97 77
1. Introduction
Trusses, as simple structure and rapid analysis, are often used to examine and compare different
optimization algorithms. Therefore, optimal design of truss structures is an active branch of
optimization research. Truss structures are widely used for cost-effectiveness, ease of
implementation, the need for specialized equipment for execution, and the need for today's
human to be used for structures with large openings without central columns. The optimization
of a design is the main purpose of any designer who tries to choose a combination of different
factors or to make a decision or to produce a device in such a way as to meet a set of
requirements and criteria. Generally, truss structures are optimized in three ways:1-Optimizing
the size or optimizing the cross-section, in which case the cross-section of the members is
selected as the design variable and the coordinates of the nodes and the topology of the structure
are fixed [1]. 2-Optimization of the shape in which the coordinates of nodes are considered as
design variables [2]. 3-Optimization of the topology, in which case how the nodes connect
together by members is examined [3].
Natural-based methods try to regulate the random search process using the rules governing
nature. One of the most prominent methods is genetic algorithm and particle swarm optimization
algorithm. The idea of an evolutionary algorithm was first raised by Rechenberg in 1960. His
research was about evolutionary strategies. Later, his theory was examined by many scholars to
lead to the design of a genetic algorithm(GA). Genetic algorithm is, in fact, a computer search
method based on optimization algorithms based on the structure of genes and chromosomes that
was introduced by John Holland in 1975 at the University of Michigan [4], and developed by a
group of his students such as Goldberg and Ann Arbor. Particle Swarm Optimization(PSO)
algorithm is an optimization technique based on a population of initial responses. This technique
was first designed by Kennedy and Eberhart in 1995 based on the social behavior of bird and fish
species [5,6].
The PSO demonstrates its proper functioning in many areas, such as finding optimal functions
for functions, training neural networks, controlling fuzzy systems, and other issues where genetic
algorithms can be applied to them. Many scientists and engineers are currently developing and
improving these algorithms at universities and research centers around the world. There has been
a lot of research on optimization of truss structures.
In 1992, Rajeev and Krishnamoorthy, based on Goldberg's research, used a simple genetic
algorithm to optimize trusses [7]. In 1995, Hajela and Lee provided a two-step method for
optimizing trusses [8]. In 1998, the Camp and other colleagues presented the optimal design of
two-dimensional structures using the genetic algorithm [9].
In 2002, Fourie and Groenwold optimized the size and shape of truss structures by particle
swarm algorithm [10]. In the same year, Li and colleagues, by means of the particle swarm
algorithm, in various ways, optimized truss structures [11].
Kaveh and Talatahari using the algorithm called Big Crunch Algorithm optimize the size of truss
structures in 2009 [12]. Kaveh and Malakouti Rad introduced a hybrid genetic algorithm and

78 M. Akbari, M. Henteh/ Journal of Soft Computing in Civil Engineering 3-2 (2019) 76-97
particle swarm in 2010 for power analysis and design [13]. In the same year, Kaveh and
Talatahari designed the optimal design of skeletal structures through an algorithm called the
Charged System Search Algorithm and the optimal design of skeletal structures using the
Imperialist Competitive Algorithm [14,15].
Kaveh and Abbasgholiha, using an algorithm called the Big Crunch Algorithm in 2011,
optimized steel frames [16]. Martini, in 2011, optimized the size, shape, and topology of truss
structures using the Harmony Search Method [17]. Hajirasouliha and colleagues, in 2011,
optimized the topology for seismic design of truss structures [18].
Richardson and colleagues Contributed to the optimization of the multi-objective topology of
truss structures with the kinematic stability repair in 2012 [19]. Miguel, in 2012, optimized the
shape and size of truss structures with dynamic limitations using a modern initiative algorithm
[20]. Makiabadi and colleagues in 2013 designed the optimal design of truss bridges using an
optimization algorithm based on training techniques [21]. Leandro and colleagues Assisted in
optimizing the size, shape and topology of truss structures in 2013 [22].
In 2014, Gandomi offered a new approach to optimization using an algorithm called the Interior
Search Algorithm [23]. Kazemzadeh Azad and Hasancebi in 2014 presented a method for
optimizing the size of truss structures based on An Elitist Self-Adaptive Step-Size Search
Algorithm [24]. In 2014, Kaveh and Mahdavai presented a new, highly innovative method of
Colliding Bodies Optimization for the optimal design of truss structures of continuous size [25].
Kaveh and colleagues in 2015, used an algorithm called An improved magnetic charged system
search for optimization of truss structures with continuous and discrete variables [26].
The purpose of this study is to optimize the discrete and continuous size of two-dimensional
trusses with topology and fixed shape. For this purpose, two genetic algorithms and a particle
swarm algorithm are used and the efficiency of each of the algorithms is investigated in a few
case studies of discrete and continuous truss size. In the following, a general overview of the
genetic algorithm and particle swarm algorithm is presented. Then the truss design optimization
model formulation is expressed and by introducing the trusses, the results of the two algorithms
are presented and compared with the results of previous researchers.
2. Genetic algorithm (GA)
In this algorithm, each point (solution) of the decision-making space is replicated to a
chromosome, so that each chromosome is composed of decision-making variables sub-strands
that contain multiple genes. The general trend of the genetic algorithm is that initially a primary
population of chromosomes is generated randomly. Then, the Crossover and Mutation operators
affect the primary population and create a chosen population. Then, the selection operator selects
among the selected population, according to the merits of a new population called children. This
population replaces the primary population and forms the population of the next generation. This
process is repeated so that the conditions for reaching the final answer are provided. The
convergence condition can be repeated up to a certain generation or non-response to several
generations.

2.1. Coding and how to form chromosomes
Encoding is largely dependent on the problem, and for each particular issue, its aspects must be
measured and an efficient way used. In this study, chromosome genes can include integers or real
numbers according to the size of the design (discrete or continuous). For example, to design a
15-member (Eight-node) truss loop of 16 sizes available, each gene can be one of the numbers
from 1 to 16 and with continuous size, the real numbers between the minimum and the maximum
available size.
2.2. Crossover operator
One of the main operators of the genetic algorithm is the crossover operator. The crossover
process is the ability to change the feature of the scheme among the members of the population
in order to improve the suitability of the next generation of designs. This is similar to the transfer
of genetic traits to the processes of the birth of living beings, which are formed by RNA and
DNA. The crossover operator leads searching in the space you decide. In this process, locations
are randomly determined along the chromosome, and the genes of these chromosomes are
replaced with two new chromosomes.
The proportion of the population of the children due to the Crossover is called the percentage of
Crossover, which is considered to be 80% in this study. In the event of high cross-link rates, most
chromosomes participate in the next generation. On the one hand, with a decrease in the rate of
Crossover, a relatively large number of chromosomes appear to be present in the next generation.
2.3. Mutation operator
The goal of the mutation is to create more dispersion within the exploration space of the design.
A mutation operator randomly changes one or more genes of a chromosome. This allows you to
check other spatial searches. In the event of a small mutation rate, the practical purpose of the
mutation is violated and it does not have the desired effect. If the mutation rate is large, the
genetic algorithm is led to a disorder and its convergence will be considerably reduced.
In the mutation operator, the gene is randomly determined from the length of the chromosome,
and then the amount is randomly changed. The ratio of the mutated population to the population
is called the percentage of Mutation, which is considered to be 30% in this study. In addition, the
ratio of the number of selected chromosome genes to the mutation that changes in this process is
the total number of chromosome genes known as mutation rates and according to the surveys is
0.3 in this study.
2.4. Generation of second-generation population
There are several methods to produce next generation population. In this paper, the population of
the current generation, the population of the children caused by the Crossover, together with the
mutated populations, forms a population that needs to be selected from the population, from the
point of view of fitness and population size, as chromosomes of the next generation.

2.5. Selection operator
For the processes of Crossover and Mutation, it is necessary to select a number of chromosomes
from the population of the present generation of chromosomes. For this purpose, there are
different methods such as Roulettle wheel method, Tournoment method, random method, etc. In
this study, the Roulettle wheel selection method is used. In this method, the probability of
selecting a chromosome i (Pi) is proportional to the fitness of that chromosome (Fi) which can be
expressed as follows:
p
i
i n
i
i 1
F
P
F



(1)
Which np is population size, and in this study the size of the population is 100. The fitness level
of each chromosome is determined further. On the Roulettle wheel, each section that has more
area, has a greater chance to choose.
3. Algorithm of particle swarm optimization (PSO)
Like all other evolutionary algorithms, the PSO optimization algorithm also begins by creating a
random population of solutions, which is herein referred to as a group of particles. The
specification of each particle in a group is determined by a set of parameters whose optimal
values should be determined. In this method, each particle represents a point of the issue space.
Each of the particles has memory, which remembers the best position in the search space.
Therefore, the motion of each particle occurs in two directions:
1-To the best position that the particle has been chosen so far.
2- To the best position that all particles have taken.
In this way, changing the position of each particle in the search space is influenced by the
experience and knowledge of itself and its neighbors. Suppose that in a particular problem that,
D-dimensional space, and the i particle of the group can be represented by a vector of velocity
and a position vector. Changing the position of each particle is possible by changing the structure
of the position and the previous velocity. Each particle contains information that includes the
best value so far(Personal best) and has the position of Xi
j,t. This information is the result of a
comparison of the efforts that each particle makes to find the best answer. Each particle also
finds the best answer so far received in the whole group, comparing the optimal values of
different particles(Global best).
Each particle tries to change its position using the following information to achieve the best
answer:
Current Location (Xi
j,t)
Current Velocity (Vi
j,t)
The distance between the current and the optimal Personal situation (Xpbestij)

The distance between the current position and the Global optimal (Xgbestj)
Thus, the velocity of each particle and, consequently, its new position change as follows:
   
i i i i i
j, t 1 j, t 1 1 j j, t 2 2 j j, t
V w V c r Xpbest X c r Xgbest X
           (2)
i i i
j, t 1 j, t j, t 1
X X V
 
  (3)
In this case ,Vi
j,t+1, Vi
j,t, Xi
j,t, Xi
j,t+1, Xpbesti
j, Xgbestj, are respectively the variation velocity of
the j-th decision of the i-th particle in the new repetition, the velocity of the j-th decision of the i-
th particle in the current repetition, the variable value of the j-th decision, the i-th particle in the
new repetition, the best value of the j-th decision variable of i-th particle it has chosen and the
best value of the j-th decision variable of the best particle (the best position ever taken by all the
particles). Figure (1) shows the motion of a particle and updates its velocity.
Fig. 1. Move a particle and update the Velocity.
r1 and r2 are random numbers between zero and one that are used to preserve diversity and
variety of the group.
c1, c2 are cognitive and social parameters, respectively. Choosing the appropriate value for these
parameters will accelerate the convergence of the algorithm and prevent early convergence in
local optimizations.
The parameter w is the weighted inertia used to ensure convergence in PSO. Weighted inertia is
used to control the effect of previous Velocity records on current Velocities. In this study,
according to the studies, the values of the parameters c1, c2 and w are considered to be 2, 2, 1,
respectively.
4. Formation of optimization model
4.1. The objective function
In an optimization problem, design variables are obtained in such a way that, while satisfying all
design constraints, the objective function has the lowest possible value. The objective function in
this paper is the weight of the truss structure and is defined as follows:

J
j j
j 1
W ρ L A

  
 (4)
Where j is the number of members of the truss, ρ the density of the members of the trusses, Lj the
length of the member of j-th of truss and the Aj is the cross section of the j-th member of the
truss.
4.2. Terms and constraints of truss structures design
4.2.1. Criteria and constraints related to axial stresses of truss members
Because the structure is truss so, only the force in the structure is axial force. As a result, the
Axial stress must be less than the permitted stress and:
a a
f F
 (5)
In which the fa is existing stress (force on the cross-section of the element) and the Fa is
permitted axial stresses.
4.2.2. Criteria and constraints related to displacement truss nodes
In truss structures, the displacement of truss nodes is important, which is usually limited. can be
stated as:
x x
u U
 (6)
y y
u U
 (7)
Where ux, uy, Ux and Uy are respectively the existing displacement of i-th node in the x direction,
the existing displacement of i-th node in the y direction, allowed displacement of i-th node in the
x direction, and allowed displacement of the i-th node in the y direction.
4.3. Penalty function
The calculation of the genetic algorithm and the particle swarm algorithm are set for unbound
functions. Therefore, in order to apply this method, the objective function of the target function
set and the governing constraints must be converted to an equivalent free function. The most
common method for forming an equivalent free function is the penalty function. In this paper, the
definition of the penalty function is as follows:
K
k
k 1
Penalty λ V

  (8)
In these relations, λ, the coefficient of the penalty function, Vk indicates the degree of violation
of the constraint of k-th and k represents the number of constraints of the optimization model.

4.4. Penalized objective function
To penalty-tune the objective function, the method of sum has been used. This method is such
that the target function is combined with the penalty function of the violation of the constraint
and if the violation function is zero (that is, all constraints are satisfied), the result is acceptable
and otherwise the result is not acceptable. Given the above, the value of the penalized objective
function, according to the following equation:
P
W W Penalty
  (9)
4.5. Fitness function
In the literature of the genetic algorithm they use a function called fitness, which acts in the
minimization problem unlike the objective function. In this study, the fitness function for each
chromosome of a population is determined according to the following exponential relation:
P
P max
W
F exp
W 
 
 
 
 
(10)
In the above relation, WP-max is the maximum value of the objective function of the fined
chromosomes of the current generation.
5. Convergence condition
In this paper, the condition for convergence is the passage of a certain number of repetitions in
the Particle Swarm Optimization and Genetic Algorithm. If this number is reached, the best
answer of this repetition is given as an optimal response. Determining the number of repetitions
to terminate the optional algorithm can only be determined based on the experience of the
program implementation.
6. Design examples
In order to verify the validity of the results and demonstrate the effectiveness of the proposed
algorithms, several examples of scientific papers have been evaluated. In this study, OpenSees
software [27] was used to analyze the structure and obtain the forces of members and
displacement of the nodes, and the code for optimizing the genetic algorithm and the particle
swarm algorithm was written in the Matlab software [28] programming environment.
6.1. Six-node two dimensional truss (10 members)
In the first example, the Six-node two dimensional truss was evaluated. Figure (2) shows the
geometric properties, loading, and support conditions for this truss. The materials used in this
truss have an elastic modulus(E) of 10000000 lb/in2
and density(ρ) of 0/1 lb/in3
. Maximum
allowed stress (Fall) ±25000 lb/in2
, Maximum allowed node displacement (Umax) in both vertical
and horizontal directions is ±2 in.

Fig. 2. Six-node two dimensional truss.
This truss is divided into three different sizes in discrete or continuous size and loaded as
follows:
1- Mode 1
Loads of p1 and p2 are 100000 and zero and the sections used to design this truss from the set of
section lists (A) selected as follows:
A={1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.63, 2.88, 2.93, 3.09, 3.13, 3.38, 4.47, 3.55, 3.63, 3.84,
3.87, 3.88, 4.18, 4.22, 4.49, 4.59, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.50, 13.50, 13.90, 14.20,
15.50, 16.00, 16.90, 18.80, 19.90, 22.00, 22.90, 26.50, 30, 33.50} (in2
)
Table 1
Comparison of optimal Six-node two dimensional Truss-Mode 1.
This study
Kazemzadeh
Azad et al.
[29]
Li et al.
[11]
Li et al.
[11]
Li et al.
[11]
Nana korn et
al. [30]
Camp et al.
[9]
Coello et al.
[31]
Rajeev et al.
[7]
Design
variables
(in2
)
PSO
GA
GSS
HPSO
PSOPC
PSO
Mathematical
algorithm
Mathematical
algorithm
Mathematical
algorithm
Mathematical
algorithm
33.5
33.5
30
30
30
30
33.5
30
30
33.5
A1
1.62
1.62
1.62
1.62
1.8
1.62
1.62
1.62
1.62
1.62
A2
22.9
22.9
22.9
22.9
26.5
30
22.9
26.5
22.9
22
A3
15.5
14.2
13.9
13.5
15.5
13.5
15.5
13.5
13.5
15.5
A4
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
A5
1.62
1.62
1.62
1.62
1.62
1.8
1.62
1.62
1.62
1.62
A6
7.22
7.97
11.5
7.97
11.5
11.5
7.22
7.22
13.9
14.2
A7
22.9
22.9
22.9
26.5
18.8
18.8
22.9
22.9
22
19.9
A8
22
22
22
22
22
22
22
22
22
19.9
A9
1.62
1.62
1.62
1.8
3.09
1.8
1.62
1.62
1.62
2.62
A10
5499.3
5490.74
5533.66
5531.98
5593.44
5581.76
5499.3
5556.9
5586.59
5613.84
Weight
(lb)
In Table 1, a summary of the best designs presented so far is presented with the results of this
research. As shown in Table 1, the amount of weight obtained from the genetic algorithm is

much less than the particle swarm algorithm and other algorithms used by other researchers. In
addition, the path of the convergence of the genetic algorithm and the particle swarm algorithm
for the Six-node two dimensional truss-mode 1 are shown in Figures (3) and (4).
In these forms, the structure weight reduction process with the Number of Function Evaluation
(NFE) is observed during the execution of the algorithm.
Fig. 3. Convergence pattern of the Genetic Algorithm for Six-Node two dimensional truss-Mode 1.
Fig. 4. Convergence pattern of the particle swarm algorithm for Six-node two dimensional truss-Mode 1.
2- Mode 2
Loads of p1 and p2 are 100000 (lb) and zero and The members of the structure are selected from
the interval of the series (A) as follows:
0.1≤ A≤ 35 (in2
)
research.

As shown in Table 2, the amount of weight obtained from the particle swarm algorithm is far less
than the genetic algorithm and other algorithms used by other researchers.
In addition, the path of the convergence of the genetic algorithm and the particle swarm
algorithm for the six-node two dimensional truss mode- 2 are shown in Figures (5) and (6). In
these forms, the structure weight reduction process is observed with the Number of Function
Evaluation.
Table 2
Comparison of optimal Six-node two dimensional truss-Mode 2.
This study
Eskandar
et al.
[32]
Kaveh&
Malakouti
rad [13]
Kaveh&
Kalatjari
[33]
Kaveh
and
Rahami
[34]
Hadidi
et al.
[35]
Hadidi
et al.
[35]
Li et al.
[36]
Koohestani
and
Kazemzadeh
Azad[37]
Design
variables
(in2
)
PSO
GA
WCA
HGAPSO
FMGA
FMGA
MABC
ABC
HPSO
ARCGA
30.598
30.3653
30.53
30.63
29.50
30.67
30.6573
34.3057
30.704
30.5984
A1
0.1
0.100066
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1002
A2
23.171
23.3523
23.05
23.06
23.50
22.87
23.0429
20.6728
23.167
23.1714
A3
15.1958
15.075
15.03
15.01
15.50
15.34
15.2821
14.5074
15.183
15.1958
A4
0.1
0.100009
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
A5
0.54084
0.560738
0.56
0.59
0.5
0.46
0.5626
0.6609
0.551
0.5409
A6
7.46250
7.44587
7.48
7.49
7.50
7.48
7.4721
7.8696
7.46
7.4625
A7
21.0346
21.0846
21.12
21.10
21.50
20.96
21.0084
20.3461
20.978
21.0346
A8
21.5182
21.5954
21.63
21.56
21.50
21.70
21.5094
22.0232
21.508
21.5182
A9
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
A10
5060.86
5061.0067
5061.02
5061.40
5067.30
5061.90
5060.97
5095.33
5060.92
5060.90
Weight
(lb)
Fig. 5. Convergence pattern of the Genetic Algorithm for the Six- node two dimensional Truss-Mode 2.

Fig. 6. Convergence pattern of the particle swarm algorithm for the Six-node two dimensional Truss-
Mode 2.
3- Mode 3
Loads of p1 and p2 are 150000 (lb) and 50000 (lb) and the members of the structure are selected
from the interval of the series (A) as follows:
0.1≤ A≤ 35 (in2
)
In Table 3, a summary of the best projects presented so far is presented along with the results of
this research. As shown in Table 3, the amount of weight obtained from the particle swarm
algorithm is far less than the genetic algorithm and other algorithms used by other researchers. In
for the six- two dimensional truss-mode 3 are shown in Figures (7) and (8). In these forms, the
structure weight loss process is observed with the Number of Function Evaluation.
Table 3
Comparison of optimal Six-node two dimensional truss-Mode 3.
This study
Hadidi et
al.[35]
Hadidi et
al. [35]
Koohestani and
Kazemzadeh Azad
[37]
Li et al.
[36]
Khan et al. [38]
Design
variables
(in2
)
PSO
GA
MABC
ABC
ARCGA
HPSO
Mathematical
algorithm
23.6383
23.8709
23.6383
24.8143
23.5986
23.353
24.72
A1
0.1
0.1
0.1
0.1
0.1009
0.1
0.1
A2
25.3230
25.0552
26.3237
26.0480
25.1175
25.502
26.54
A3
14.41
14.6923
14.4108
14.8772
14.5383
14.25
13.22
A4
0.1
0.100008
0.1001
0.1
0.1001
0.1
0.108
A5
1.970
1.96982
1.9707
2.0055
1.9713
1.972
4.835
A6
12.3780
12.443
12.3781
12.4467
12.3923
12.363
12.66
A7
12.7738
12.78170
12.7739
12.6835
12.7439
12.894
13.78
A8
20.2678
20.0339
20.2678
18.8669
20.3697
20.356
18.44
A9
0.1
0.1
0.1
0.1
0.1
0.101
0.1
A10
4676.96
4677.655
4677.06
4691.07
4677.24
4677.29
4792.52
Weight (lb)

Fig. 7. Convergence pattern of the Genetic Algorithm for the Six-node two dimensional truss-Mode 3.
Fig. 8. Convergence pattern of particle swarm algorithm for Six-node two dimensional truss-Mode 3.
6.2. Eight-node two dimensional truss (15 members)
In the second example, the Eight-node two dimensional truss is evaluated. Figure (9) shows the
geometric properties, loading, and support conditions for this truss. The materials used in this
truss have an elastic modulus(E) of 200 GPa and density(ρ) 7800 kg/m3
. Maximum allowed
stress (Fall) ±120 MPa, Maximum allowed node displacement(Umax) in both vertical and
horizontal directions ±10mm and load P is 35 KN.
Fig. 9. Eight-node two dimensional truss.

For the design of the eight-node two dimensional truss, members of the structure are selected
from the set of section lists (A) as follows:
A={113.2, 143.2, 145.9, 174.9, 185.9, 235.9, 265.9, 297.1, 308.6, 334.3, 338.2, 497.8, 507.6,
736.7, 791.2, 1063.7} (mm2
)
research. As shown in Table 4, the amount of weight obtained from the genetic algorithm is
much lower than the particle swarm algorithm and other algorithms used by other researchers. In
are shown for the eight-node two dimensional truss in forms (10) and (11). In these forms, the
structure weight loss process is observed with the Number of Function Evaluation.
Table 4
Comparison of optimal results of Eight-node two dimensional truss.
This study
Li et al. [11]
Sabour et al. [39]
Sadollah et al. [40]
Eskandar et al. [32]
Li et al.
[11]
Li et al.
[11]
HGA[41]
Design variables
(mm2
)
PSO
GA
HPSO
ICA
ICACO
MBA
WCA
PSOPC
PSO
Mathematical
algorithm
185.9
113.2
113.2
113.2
185.9
308.6
A1
113.2
113.2
113.2
113.2
113.2
174.9
A2
143.2
113.2
113.2
113.2
143.2
338.2
A3
113.2
113.2
113.2
113.2
113.2
143.2
A4
736.7
736.7
736.7
736.7
736.7
736.7
A5
143.2
113.2
113.2
113.2
143.2
185.9
A6
113.2
113.2
113.2
113.2
113.2
265.9
A7
736.7
736.7
736.7
736.7
736.7
507.6
A8
113.2
113.2
113.2
113.2
113.2
143.2
A9
113.2
113.2
113.2
113.2
113.2
507.6
A10
113.2
113.2
113.2
113.2
113.2
297.1
A11
113.2
113.2
113.2
113.2
113.2
174.9
A12
113.2
113.2
113.2
185.9
113.2
297.1
A13
334.3
113.2
334.3
334.3
334.3
235.9
A14
334.3
113.2
334.3
334.3
334.3
265.9
A15
108.84
95.9401
105.735
108.96
108.84
142.117
Weight (kg)

Fig. 10. Convergence pattern of the Genetic Algorithm for the Eight-node two dimensional truss.
Fig. 11. Convergence pattern of the particle swarm algorithm for the Eight-node two dimensional truss.
6.3. Nine-node two dimensional truss (17 members)
In the third example, the Nine-node two dimensional truss has been evaluated. Figure (12) shows
the geometric features, loading, and supporting conditions for this truss. The materials used in
this truss have an elastic modulus(E) of 30000000 lb/in2
and density(ρ) 0.268 lb/in3
. Maximum
, maximum allowed node displacement (Umax) in both vertical
and horizontal directions ±2 in and load P is 100000 lb.

Fig. 12. Nine-node two-dimensional truss.
For the design of the nine-node two dimensional truss, members of the structure are selected
0.1≤A (in2
)
research. As shown in Table 5, the amount of weight obtained from the particle swarm algorithm
is far less than the genetic algorithm and other algorithms used by other researchers. In addition,
the path of the convergence of the genetic algorithm and the particle swarm algorithm for nine
nodes two dimensional truss are shown in forms (13) and (14). In these forms, the structure
weight loss process is observed with the Number of Function Evaluation.
Table 5
Comparison of Optimal Nine-node two-dimensional Trusses.
This study
Kazemzadeh Azad
and Hasancebi[24]
Hadidi et
al. [35]
Hadidi et
al.[35]
Koohestani and
Kazemzadeh Azad
[37]
Design
variables (in2
)
PSO
GA
ESASS
MABC
ABC
ARCGA
15.93
16
15.9324
15.6762
12.9587
15.891
A1
0.1
0.1
0.1
0.1
0.1
0.105
A2
12.070
12.299
12.0193
12.0491
11.5965
12.101
A3
0.1
0.1
0.1
0.1
0.1
0.1
A4
8.066
8
8.1001
8.1312
6.3320
8.075
A5
5.562
5.52090
5.53
5.62020
6.5356
5.541
A6
11.933
11.903
11.9209
11.8822
12.4792
11.97
A7
0.1
0.1
0.1
0.1
0.1
0.1
A8
7.945
7.91538
8.0128
8.0517
9.0901
7.955
A9
0.1
0.1
0.1
0.1
0.1
0.1
A10
4.0545
4.0522
4.0715
4.0912
5.1578
4.07
A11
0.1
0.1
0.1
0.1
0.1
0.1
A12
5.657
5.66065
5.6726
5.6746
6.4197
5.705
A13
4
3.96854
4.0154
3.9864
4.0553
3.975
A14
5.558
5.56092
5.5286
5.6792
5.7984
5.516
A15
0.1
0.1
0.1
0.1
0.1
0.1
A16
5.579
5.51966
5.5739
5.4907
6.8470
5.563
A17
2581.88
2582.0672
2581.93
2582.27
2642.45
2581.95
Weight(lb)

Fig. 13. Convergence pattern of the Genetic Algorithm for the Nine-node two dimensional truss.
Fig. 14. Convergence pattern of the particle swarm algorithm for the Nine-node two dimensional truss.
6.4. Twenty-node two dimensional truss (45 members)
In the last example, the Twenty-node two dimensional truss was evaluated. Figure (15) shows the
geometric properties, loading, and support conditions of this truss. The materials used in this
truss have an elastic modulus (E) of 30000000 lb/in2
and density (ρ) 0.283 lb/in3
. Maximum
, Maximum allowed node displacement (Umax) in both vertical
and horizontal directions ±2 in and load P is 100000 lb.
Fig. 15. Twenty-node two dimensional truss.

For the design of the twenty-node two dimensional truss, members of the structure are selected
0.1≤A (in2
)
research. As shown in Table 6, the amount of weight obtained from the particle swarm algorithm
is far less than the genetic algorithm and other algorithms used by other researchers. In addition,
the path of the convergence of the genetic algorithm and the particle swarm algorithm for the
twenty node two dimensional truss are displayed in shapes (16) and (17). In these forms, the
structure weight reduction process is observed with the Number of Function Evaluation.
Table 6
Comparison of optimal results of Twenty nodes two dimensional truss.
This study
Kazemzadeh Azad and
Hasancebi [24]
Hadidi et al.
[35]
Hadidi et al.
[35]
members
(in2
)
Design
variables
PSO
GA
ESASS
MABC
ABC
4.605
4.6996
4.6052
4.5996
5.4746
1,44
A1
3.7082
3.80
3.7083
3.7966
4.5989
2,45
A2
3.1919
3.05
3.1919
3.0497
4.1703
3,43
A3
3.27558
3.28
3.2756
3.2841
3.7872
4,39
A4
0.1
0.104
0.1
0.1069
0.1
5,41
A5
3.9896
3.93
3.9896
3.9279
4.1735
6,40
A6
0.8916
0.96
0.8916
0.9649
0.9497
7,42
A7
1.2170
1.21
1.2170
1.2133
1.5902
8,38
A8
7.7323
7.65
7.7323
7.6553
6.2656
9,34
A9
2.2227
2.198
2.2227
2.1993
2.2039
10,36
A10
1.1803
1.19
1.1803
1.1929
1.3925
11,35
A11
0.1
0.1
0.1
0.1001
0.1
12,37
A12
0.1
0.1
0.1
0.1008
0.1
13,33
A13
9.3901
9.50
9.3901
9.5360
9.0689
14,29
A14
1.214895
1.21
1.2149
1.2173
1.5310
15,31
A15
1.332196
1.41
1.3322
1.4190
1.6245
16,30
A16
2.605595
2.55
2.6056
2.5513
2.9146
17,32
A17
0.1
0.1
0.1
0.1
0.1
18,28
A18
11.62655
11.50
11.6266
11.5439
9.0685
19,24
A19
1.240596
1.28
1.2406
1.2807
1.6352
20,26
A20
0.1
0.1
0.1
0.101
0.1
21,25
A21
3.792295
3.75
3.7923
3.7598
4.4798
22,27
A22
0.1
0.1
0.1
0.1017
0.1
23
A23
7967.89
7969.20
7967.98
7968.95
8267.21
Weight(lb)

Fig. 16. Convergence pattern of the Genetic Algorithm for a Twenty-node two dimensional truss.
Fig. 17. Convergence pattern of the particle swarm algorithm for Twenty-node two dimensional truss.
7. Conclusion
In this study, the discrete and continuous size optimization of two-dimensional trust was
investigated using genetic algorithm and particle swarm algorithm. To illustrate the effectiveness
of these two algorithms, the results were compared with the design results of other researchers.
In this study, four types of two-dimensional truss (Six-node, Eight-node, Nine nodes and Twenty
nodes) were investigated under varying degrees of discrete or continuous size and loading, and
the results obtained in this study are presented as follows:
1-The solved examples show that the genetic algorithm and the particle swarm algorithm have
the potential and the ability to solve the optimized optimization problems and also have good
convergence velocity.

2-Comparison results show that in discrete sizes, the designs derived from the genetic algorithm
are far more economical than other designs, and vice versa in continuous measurements, the
designs derived from the particle swarm algorithm are far more economical than other designs.
3-Comparison of weight loss diagrams in terms of number of simulations (NFE) shows that the
convergence of the genetic algorithm to the optimum solution of the particle swarm algorithm is
faster.
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