Neural Computing and Applications
https://doi.org/10.1007/s00521-019-04170-4
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REVIEW ARTICLE
A survey of symbiotic organisms search algorithms and applications
Mohammed Abdullahi1 • Md Asri Ngadi2 • Salihu Idi Dishing1,2 • Shafi’i Muhammad Abdulhamid3
Mohammed Joda Usman4
•
Received: 16 May 2018 / Accepted: 27 March 2019
Ó Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract
Nature-inspired algorithms take inspiration from living things and imitate their behaviours to accomplish robust systems in
engineering and computer science discipline. Symbiotic organisms search (SOS) algorithm is a recent metaheuristic
algorithm inspired by symbiotic interaction between organisms in an ecosystem. Organisms develop symbiotic relationships such as mutualism, commensalism, and parasitism for their survival in ecosystem. SOS was introduced to solve
continuous benchmark and engineering problems. The SOS has been shown to be robust and has faster convergence speed
when compared with genetic algorithm, particle swarm optimization, differential evolution, and artificial bee colony which
are the traditional metaheuristic algorithms. The interests of researchers in using SOS for handling optimization problems
are increasing day by day, due to its successful application in solving optimization problems in science and engineering
fields. Therefore, this paper presents a comprehensive survey of SOS advances and its applications, and this will be of
benefit to the researchers engaged in the study of SOS algorithm.
Keywords Symbiotic organisms search Metaheuristics algorithms Optimization Bio-inspired algorithms
Local search Global search
1 Introduction
& Mohammed Abdullahi
abdullahilwafu@abu.edu.ng
Md Asri Ngadi
dr.asri@utm.my
Salihu Idi Dishing
sidishing@abu.edu.ng
Shafi’i Muhammad Abdulhamid
shafii.abdulhamid@futminna.edu.ng
Mohammed Joda Usman
umjoda@gmail.com
1
Department of Computer Science, Ahmadu Bello University,
Zaria, Nigeria
2
Department of Computer Science, Faculty of Computing,
Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia
3
Department of Cyber Security Science, Federal University of
Technology Minna, Minna, Nigeria
4
Department of Mathematics, Bauchi State University Gadau,
PMB 068, Bauchi, Bauchi State, Nigeria
Optimization algorithms are mostly inspired by nature,
usually based on swarm intelligence. Swarm intelligence is
an area of artificial intelligence (AI) that is concerned with
a collective behaviour within distributed and self-organized
systems [1–5]. Different optimization algorithms are based
on different inspirations, and these algorithms have been
widely applied in optimization of problems in science,
technology, and engineering problems [6–10]. The traditional swarm-intelligence optimization algorithms include
evolutionary algorithms like GA [11], DE [12], and swarm
intelligence algorithms like the PSO [13], bees algorithm
(BA) [14], particle bee algorithm (PBA) [15], ant colony
optimization (ACO) [16, 17], and artificial bee colony
(ABC) [18]. Recently, the area of swarm intelligence has
witnessed development of promising optimization algorithms such as symbiotic organisms search (SOS) [19],
cuckoo search (CS) [20], bat algorithm, fire fly algorithm
(FA) [21], and cat swarm optimization (CSO) [22], while
new algorithms like Beetle Antennae Search (BAS)
[23, 24] and Eagle Perching Optimizer (EPO) [25] emerged
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Neural Computing and Applications
Fig. 1 Positioning SOS within
the classes of metaheuristics
very recently. Figure 1 shows the position of the newly
introduced SOS within the context of metaheuristics.
The traditional algorithms like PSO, GA, and ACO
have been proven to be effective and robust in solving
various classes of optimization problems. However, these
algorithms have limitations like entrapment in local
minima, high computational complexity, slow convergence rate, and unsuitability for some class of objective
functions. SOS is a stochastic king of metaheuristic
algorithm, and it searches for set of solutions by means of
randomization. SOS is a SI-based optimization techniques
introduced by Cheng and Prayogo [19], and it was
motivated by interactive behaviour by organisms for
survival. Some organisms in the ecosystem depend on
other species for their survival, and this dependency-based
survival is called symbiotic association. Mutualism,
commensalism, and parasitism are the main forms of
symbiotic association in an ecosystem. Mutualism association is when two organisms interact for mutual benefit,
that is, both benefits from the relationship. Commensalism
is when one organism develops a relationship with a pair
of specie, while one specie acquires benefit from the
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relationship and the other specie is not harmed. Parasitism
is when two species develop a relationship, and one
specie acquires benefits from the relationship, while the
other specie is harmed.
In the SOS algorithm, the mutualism and commensalism phases concentrate on generating new organisms and
thus select the best organism for survival. The operations
of these phases enable the search procedure to discover
diverse solutions in the search space, thereby improving
the exploration ability of the algorithm [26]. On the other
hand, the parasitic phase enables the search procedure to
avoid the solution being trapped in a likely local optima,
thus improving the exploitation ability of the algorithm
[26]. The standard SOS algorithm was proposed to handle
continuous benchmark and engineering problems, which
was shown to be robust and has faster convergence speed
when compared with GA [11], PSO [13], DE [12], BA
[14], and PBA [15] which are the traditional metaheuristic
algorithms. The three phases of the SOS algorithm are
simple to operate, with only simple mathematical operations to code. Further, unlike the competing algorithms,
SOS does not use tuning parameters, which enhances its
Neural Computing and Applications
performance stability. The SOS algorithm operations did
not require algorithm-specific turning parameters unlike
other metaheuristic algorithms like GA which needs to
turn crossover and mutation rates, and PSO needs to turn
cognitive factor, social factor, and inertia weight. This
feature of SOS is considered an advantage since the
improper turning of the parameters could prolong the
computation time and cause premature convergence
[19, 27].
We thus conclude that the novel SOS algorithm, while
robust and easy to implement, is able to solve various
numerical optimization problems despite using fewer
control parameters than competing algorithms. Therefore,
interests of researchers in using SOS for handling optimization problems are increasing day by day. Therefore,
this paper presents a comprehensive survey of SOS
advances and its applications, and this will be of benefit
to the researchers engaged in the study of SOS
algorithm.
The contributions of this paper are as follows:
– Comprehensive presentation of SOS algorithm and
basic steps of SOS algorithms.
– Review of advances of SOS algorithm and its
applications.
– Review of the modifications of SOS for handling
continuous, discrete, multi-objective, and large-scale
optimization problems.
– Emphasis on the future research direction of SOS.
The structure of the remaining parts of the paper is
follows: The biological foundations of SOS algorithm are
discussed in Sect. 2, and then, the features of the algorithm are explained, and thereafter, the structure of the
algorithm is presented. Section 3.1 provides the evolution
SOS algorithms under which various modified and
hybrid versions of SOS algorithms are discussed. The
extensive review of the application areas for which SOS
algorithms have been applied is presented in Sect. 4. The
pronounced application areas of SOS algorithms include
combinatorial, continuous, and multi-objective optimization. Besides optimization, application of SOS algorithm
covered engineering and other real-world problems such
as power systems, transportation, design and optimization
of engineering structures, economic dispatch problems,
wireless communication, and marching learning. Section 5 presents discussion and potential future research
works, and finally, Sect. 6 draws concluding remarks on
the paper.
2 Symbiotic organisms search
2.1 Foundations of SOS
Symbiotic organism search (SOS) was introduced in [19],
and it was motivated by interactive behaviour by
organisms for survival. The principal idea behind SOS is
the simulation of forms symbiotic association in an
ecosystem which comprises of three stages. In the first
stage which is called the mutualism phase, a pair of
organisms interact for mutual benefit and neither of them
is harmed from the interaction. A classic example of
mutualism association is an interaction between bees and
flowers. Bees collect nectar from flower for the production of honey, and nectar collection process by bees
enables the transfer of pollen grains which aid pollination. Therefore, the involved organisms interact for
mutual benefit from the relationship. During the second
stage which is called commensalism phase, a pair of
organisms engage in a symbiotic relation where one of
the organisms acquires benefits from the relationship and
the other neither benefits nor harmed. A relationship
between remora fish and sharks is a typical example of
commensalism association. Remora fish rides on shark
for food, and shark neither benefits nor harmed from the
relationship. During the third stage, called parasitism
phase, a pair of organisms engage in a symbiotic relation
where one of the organisms acquires benefits from the
relationship and the other is harmed. An example of
parasitic association is a relationship between anopheles
mosquito and human host. An anopheles mosquito
transmits plasmodium parasite to human host which
could cause the death of human host if his/her system
cannot fight against the parasite.
2.2 Characteristics of SOS
The SOS algorithm is population (ecosystem) based with
a pre-set size called eco_size. An initial ecosystem is
randomly generated similar to other evolutionary algorithms (EAs). An organism (individual) Xi within the
ecosystem represents a candidate solution to a given
optimization problem. Xi is a D-dimensional vector of
real values, where D is the dimension of a given optimization problem. Then, mutualism, commensalism, and
parasitism phases are used to improve certain organisms,
where an organism is replaced if its new solution is better
than the old one. The algorithm iterates until the stopping
criterion is reached.
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Neural Computing and Applications
2.2.1 Mutualism phase
Xi ¼ Xi þ rð 1; 1Þ ðXbest
During the mutualism phase, the algorithm tries to improve
an organism Xi and an arbitrary organism Xj ði 6¼ jÞ by
moving their positions towards the best organism Xbest by
taking into consideration the current mean (mutual benefit
factor)of the organisms Xi and Xj , and this represents the
qualities of the organisms Xi and Xj from the current
generation. Equations (1) and (2) modelled how organisms
improvement may be influenced by the difference between
the best organism and qualities of the mutual organisms.
The parameters ri and BF are applied to the models to
provide source of randomness; r1 ð0; 1Þ and r2 ð0; 1Þ are
uniformly generated random number in the interval 0 to 1;
and BF is the benefit factor which is either 1 or 2 which
emphasizes the importance of the benefit of mutual
relationship.
where rð 1; 1Þ is a vector of uniformly distributed random
numbers between
1 and 1. i ¼ 1; 2; 3; . . .; ecosize;
j 2 f1; 2; 3; . . .; ecosizejj 6¼ ig; ecosize is the number of
organisms in the search space.
Xi ¼Xi þ r1 ð0; 1Þ ðXbest
MV BF1 Þ
ð1Þ
Xj ¼Xj þ r2 ð0; 1Þ ðXbest
MV BF2 Þ
ð2Þ
1
MV ¼ ðXi þ Xj Þ
2
ð3Þ
where r1 ð0; 1Þ and r2 ð0; 1Þ are vectors of uniformly distributed random numbers between 0 and 1;
i ¼ 1; 2; 3; :::; ecosize; j 2 f1; 2; 3; . . .; ecosizejj 6¼ ig; ecosize is the number of organisms in the search space.
2.2.2 Commensalism phase
During the commensalism phase, an organism Xi tries to
improve its self by interacting with an arbitrary organism
Xj , where i is not equal to j. The improvement is
attempted by moving the position of Xi towards the
position of the best organism XðbestÞ by taking into
account the current position of the arbitrary organism Xj .
Equation 4 modelled how organism improvement may be
influenced by the difference between the best organism
and an arbitrary organism Xj . The parameter r is applied
to the model to provide source of randomness; rð 1; 1Þ is
a uniformly generated random number in the interval - 1
to 1.
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Xj Þ
ð4Þ
2.2.3 Parasitism phase
In parasitism phase, an artificial parasite called parasite
vector is created by cloning an ith organism Xi and
modifying it using randomly generated number. Then, Xj
is randomly selected from ecosystem, and fitness values
of parasite vector and Xj are computed. If the parasite
vector is fitted than Xj , then Xj is replaced by the parasite
vector, otherwise Xj survives to the next generation of
ecosystem and parasite vector is discarded. This phase
increases the exploitation and exploration capability of
the algorithm by randomly removing the inactive solution
and introducing the active ones. Consequently, premature
convergence could be avoided and convergence rate could
be improved.
2.3 Structure of SOS
The pseudocode of the SOS algorithm is presented in
Algorithm 1. The population of organisms is initialized,
typically using uniformly generated random number. The
SOS procedure is performed within the while loop (lines
2–32 in Algorithm 1) which comprises of the following
steps. Firstly, the mutual vector and benefit factor parameters are computed, and then, the mutualism phase modifies
the current solution and a randomly selected solution (lines
4–17 in Algorithm 1). Then, the commensalism phase is
applied to modify the current solution to introduce the
exploration into the space (lines 18–23 in Algorithm 1).
Finally, parasitism phase is applied to prevent the search
procedure from getting entrapped in the local optima (lines
23–30 in Algorithm 1). The process stops when the stopping criteria are reached.
Neural Computing and Applications
Algorithm 1 Symbiotic Organisms Search Algorithm [19]
Input: Set ecosize, create population of organisms Xi , i = 1, 2, 3, ..., ecosize, initialize Xi ,
Set stopping criteria.
Output: Optimal solution
1: Identify the best organism Xbest
2: while stopping criterion is not met do
3:
for i = 1 to ecosize do
4:
Mutualism Phase
X +X
⊲ (j =
i)
5:
MV = i 2 j
6:
BF 1 = round(1 + r1 (0, 1))
7:
BF 2 = round(1 + r2 (0, 1))
8:
Xi∗ = Xi + r1 (0, 1) ∗ (Xbest − M V ∗ BF 1 )
9:
Xj∗ = Xj + r2 (0, 1) ∗ (Xbest − M V ∗ BF 2 )
10:
Evaluate Xi∗
11:
if Xi∗ then is better than Xi
12:
Xi ← Xi∗
13:
end if
14:
Evaluate Xj∗
15:
if Xj∗ then is better than Xj
16:
Xj ← Xj∗
17:
end if
18:
Commensalism Phase
⊲ (j = i)
19:
Xi∗ = Xi + r(−1, 1) ∗ (Xbest − Xj )
20:
Evaluate Xi∗
21:
if Xi∗ then is better than Xi
22:
Xi ← Xi∗
23:
end if
24:
Parasitism Phase
25:
Create parasite vector
26:
Evaluate parasite vector
27:
if parasite vector then is better than Xj
28:
Xj ← parasite vector
29:
end if
30:
Identify the best organism Xbest
31:
end for
32: end while
3 Taxonomy of SOS advances
Figure 2 presents taxonomy of the recent advances of the
SOS algorithm. The SOS advance is classified into main SOS
algorithm, SOS evolution, and SOS applications. The SOS
evolution includes modifications and hybridization. The
SOS application includes the multi-objective optimization,
combinatorial optimization, continuous optimization, engineering applications, and other application areas.
3.1 Evolution of symbiotic organisms search
algorithms
The standard SOS was developed as global optimizer for
continuous optimization problems, which was shown to be
robust and has faster convergence speed when compared
with genetic algorithm (GA) [11], particle swarm optimization (PSO) [13], differential evolution (DE) [12], bees
algorithm (BA) [14], and particle bee algorithm (PBA) [15]
which are the traditional metaheuristic algorithms. The
Fig. 2 Taxonomy of SOS advances
difficulty arose when SOS could not obtain efficient solutions for other complex optimization problems, which is
turn with the no-free-lunch theorem [28]. To overcome the
difficulties, SOS algorithms have undergone several
hybridizations and modifications to provide optimal and
efficient solutions for various optimization problems. This
section overviews the developments of SOS algorithms in
terms of the modifications and hybridizations. Figure 3
presents the taxonomy of SOS evolution.
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Fig. 3 Taxonomy of SOS
evolution
3.1.1 Modified symbiotic organisms search algorithms
Since the introduction of SOS algorithm, it has undergone
several modifications to provide optimal and efficient
solutions for various optimization problems. Modifications
have been applied to several components of SOS such as
solution initialization [29, 30], SOS phases [29–33],
introduction of new phases [34], and fitness function
evaluation [29]. The summary of modified SOS algorithms
is presented in Table 1
Modified symbiotic organisms search (MSOS) algorithm was proposed by [31] to improve the convergence
rate and accuracy of SOS algorithm. In MOS, the ecosize is
divided into three inhabitants and the integrated inhabitant
is executed using predefined probabilities. The inhabitants
and probabilities are used to update mutualism, commensalism, and parasitism phases, respectively. The MSOS
algorithm introduces, in all of its phases, new relations to
update the solutions to improve its capacity of identifying
stable and of high-quality solutions in a reasonable time.
Furthermore, to increase the capacity of exploring the
MSOS algorithm in finding the most promising zones, it is
endowed with a chaotic component generated by the
logistic map.
New relations for updating the phases of SOS were
introduced by [29] to improve the quality of solution, and
chaotic sequence generated by logistic map was employed
to improve the exploration capability of SOS. In this paper,
this technique is referred to as MSOS1. The introduced
relations are applied to update the mutualism and commensalism phases. Modification to benefit factors of SOS
algorithms was proposed in [35], otherwise known as SOS–
AFB. The factors were adaptively determined based on the
fitness of the current organism and the best organism. The
adaptive benefit factors strengthen the exploration capability when the organisms Xi or Xj ði 6¼ jÞ are far from the
Table 1 Modified SOS algorithms
S/n
Modified technique(s)
Modification
Compared
algorithms
Result
1
MSOS [31, 32]
Changed the organism structure and
partition based eco-size called inhabitants
SOS and harmony
search (HS)
MSOS outperforms SOS and HS in terms of
speed, accuracy and convergence
2
MSOS1 [29]
SOS and other
metaheuristic
algorithms in
the literature
MSOS1 performs better than basic SOS and
recently proposed techniques for solving
large scale economic dispatch problem
3
SOS–ABF [35]
New relations for updating mutualism and
commensalism phases, elimination of
parasitism phase, and replacement of
random number components with logistic
chaotic map
Introduced adaptive benefit factors
SOS, PSO,
DPSO,CSS, and
CBO
The SOS and its variants used in this work
ranks high among the different techniques
under consideration
4
SOSCanonical , SOSBasic ,
SOSSR 1 , SOSSR 2 ,
ISOSSR 1 , and
ISOSSR 2 [34]
Introduced competition and amensalism
phases
SOS and different
variants of PSO
The computational time of ISOSSR 1 is a
little more than PSOSR 1 , but is marginally
less than PSOSR 2 ; conversely, ISOSSR 2
shows more computational time associated
with SOSSR 1 , PSOSR 1 , and PSOSR 2
5
I-SOS [30]
Replaced random number components with
random weighted reflective parameter,
and introduced predation phase
DE, PSO, and
SOS
Ecosize and number of function evaluation
are investigated by varying these two
parameters. ISOS outperformed the other
algorithm comparison schemes
6
SMSOS [33]
Introduced simplex method [36] to improve
exploration and exploitation capability
ABC, BA, CS,
FPA, and GWO
SMSOS converges faster with higher
precision and better robustness while
achieving shortest flight path and avoiding
threat areas
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Neural Computing and Applications
best organism (Xbest ), while the exploitation capability is
strengthen when the interacting organisms are closer to the
best organism. The competition and amensalism interaction
strategies were proposed by [34] in addition to mutualism,
commensalism, and parasitism phases proposed in the
standard SOS. The idea of competition phase is to generate
an organism that can compete with the current organism Xi .
An organism Xj is randomly selected from the ecosystem,
and a new organism called Xcompetitor is generated. If
f ðXcompetitor Þ is better than f ðXi Þ, then Xi is updated with
Xcompetitor . The concept of amensalism phase is similar to
that of competition phase, but amensalism organism is
unaffected by the interaction, while the other organism is
negatively affected. This phase introduces diversity into
search space, thereby avoiding premature convergence and
local optima entrapment. The commensalism organism
Xamensalis is generated by mutating the randomly selected
organism Xj , and Xj is replaced by Xamensalis .
Predation phase and a random weighted reflective
parameter are proposed by [30] to improve the performance of SOS algorithm. The new improved SOS is called
ISOS. The proposed predator phase modelled the interaction between the predator and prey in an ecosystem. The
predator feeds on the prey which always eventually leads to
the death of the prey. The proposed random weighted
parameter is used to replace the generated normally distributed random numbers in the mutualism and commensalism phase, respectively. Banerjee and Chattopadhyay
[32] presents a modified SOS (MSOS) algorithm for optimizing the performance of 3-dimensional turbo code (3DTC) in controlling error coding scheme to improve adequate redundancy in communications system. The proposed MSOS improves the performance of bit error rate
(BER) of 3D-TC as compared to 8-state Duo-Binary Turbo
Code (DB-TC) of the Digital Video Broadcasting Return
Channel via Satellite standard (DVB-RCS) and Serially
Concatenated Convolution Turbo Code (SCCTC)
structures.
Miao et al. [33] proposed a modified SOS algorithm
based on simplex method (SMSOS) in order to obtain
optimal flight route for unmanned combat aerial vehicle
(UCAV) systems under multi-constrained global optimization problem. The modified algorithm adopted simplex method that improves the population diversity and
increases exploration and exploitation. The method has
been added into the phases of the original SOS such as
mutualism, commensalism, and parasitism. Thus, prevent
the local optimal solution of the algorithm from premature
convergence. Simulation has been conducted in the
MATLAB environment, and the result obtained shows the
efficiency of the modified SOS for UCAV compared with
other state-of-the-art algorithms in finding the shortest path
to evade the randomly generated threat areas with high
optimization precision. However, modified SOS tests the
flight route path only and did not take into consideration of
other parameters used by the UCAV. A different modification of SOS that uses complex method has been developed in order to extend the CSOS diversification that can
lead to high precision in obtaining global optimum solution
[33].
3.1.2 Hybrid symbiotic organisms search algorithms
The traditional SOS algorithm suffers from local optima
entrapment like other metaheuristic algorithms which leads
to premature convergence [37]. The prominent way to
improve the search ability of metaheuristic algorithms is by
hybridization. Hybridization technique tries to integrate the
advantages of two or more techniques to substantially
reduce their disadvantages [37]. The hybrid algorithms
result to improvements in convergence speed and quality of
solutions [37]. Table 2 summarizes hybrid-based SOS
algorithms
Hybrid SOS algorithms have been proposed by incorporating local search techniques [38, 41]. Hybrid algorithms are not only able to offer faster speed of
convergence, but also produce better quality solutions. In
[38], a hybrid SOS is proposed by combining SOS algorithm with simple quadratic interpolation (SQI) to enhance
the global search of SOS algorithm. The performance of
the proposed algorithm was tested on real-world and largescale benchmark functions like CEC2005 and CEC2010.
The results obtained by the proposed algorithm were
compared with other classical metaheuristic algorithms.
The proposed algorithm outperforms other state-of-the-art
algorithms in terms of quality of solutions and convergence
rate. Saha and Mukherjee [39] hybridized SOS with a
chaotic local search (CSOS) to improve the solution
accuracy and convergence speed of the SOS algorithm, and
the parasitism phase of the standard SOS is not considered
in the proposed CSOS in order to reduce the computational
complexity. The performance evaluation of CSOS was
carried out using both twenty-six unconstrained benchmark
test functions and real-world power system problem. The
CSOS algorithm obtained superior results over the compared algorithms for both benchmark function optimization
and power engineering optimization task.
Guha et al. [41] presented an hybrid SOS called quasioppositional symbiotic organism search (QOSOS) algorithm for handling load frequency control (LFC) problem
of the power system. The theory of quasi-oppositional
based learning (Q-OBL) is integrated into SOS to avoid
entrapment in local optima and improve convergence rate.
Authors considered two-area and four-area interconnected
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Neural Computing and Applications
Table 2 Hybridized SOS algorithms
S/n
Modified
technique(s)
Hybridized with
Compared algorithms
Result
1
HSOS [38]
Simple quadratic interpolation
(SQI)
SOS, PSO, GA and CLPSO
The proposed algorithm outperformed other state-ofthe-art algorithms in terms of quality of solutions and
convergence rate
2
CSOS [39]
Chaotic local search (CLS)
SOS, PSO, GA, GA–PSO,
and TLBO
3
CSOS-I [40]
Chaotic local search (CLS)
SOS and harmony search
(HS)
CSOS algorithm produces better result over the other
comparison methods in terms of convergence rate and
global search ability for both benchmark functions for
some selected real-world power problems
MSOS outperforms SOS and HS in terms of speed,
accuracy, and convergence
4
QOSOS [41]
Quasi-oppositional based
learning (Q-OBL)
SOS, PSO, GA, CLPSO,
etc
QOSOS tuned PID-controller makes the LFC system
robust and shows moderately improved and steady
results over the wide-ranging of different system
parameters
5
hSOS–SA [42]
Simulated annealing (SA)
TLBO, ABC, PSO, LUS,
and BBO
The hybridization with SA prevent the technique from
premature convergence that leads to stable, fast, and
reliable AVR response
6
SASOS-II [43]
Simulated annealing (SA)
DE and its variants
SASOS efficiently minimize the considered models of
the problem
7
SASOS-I [44]
Simulated annealing (SA)
SOS
SASOS outperforms SOS in terms of convergence
speed and solution quality
8
SOS–SA [45]
Simulated annealing (SA)
GA–PSO–ACO, MSA–
IBS, LBSA, and SOS
The SOS–SA outperforms the compared algorithms in
terms of convergence and average execution time
power system to test the effectiveness of the proposed
algorithm. The dynamic performance of the test power
systems obtained by QOSOS is better than that of SOS and
comprehensive learning PSO. QOSOS also showed better
robustness and sensitivity for the test power systems.
To address the issue of SOS been likely trapped in the
local optima, attempts have made by researchers to
hybridize simulated annealing local search technique with
SOS algorithm [43–46]. Çelik and Öztürk [46] proposed
the hybridization of SOS with simulated annealing (hSOS–
SA) for integration into the design of proportional– integral–derivative (PID) controller for automatic voltage
regulator (AVR) by taking into consideration both time and
frequency domain specifications. The SOS optimizes the
PID parameters used by the technique in order to improve
the stability of the system. Furthermore, the hybridization
with SA prevents the technique from premature convergence that leads to stable, fast and reliable AVR response.
Moreover, [44] proposed SASOS-I algorithm for task
scheduling in cloud computing environment, and the
SASOS-I avoided likely entrapment of SOS procedure in
the local optima while minimizing task makespan and
response time. Ezugwu et al. [45] presented SOS-SA
algorithm for solving travelling salesman problems, and the
proposed algorithm reduces execution time while reducing
the convergence speed of the SOS procedure. Sulaiman
et al. [43] presented symbiotic organism search-based
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simulated annealing (SASOS-II) local search technique for
solving directional overcurrent relay (DOCR) to minimize
the sum of the operating times of all primary relays considering standard IEEE3-, 4-, and 6-bus systems. The
SASOS-III algorithm efficiently minimizes the considered
models of the problem.
4 Applications of symbiotic organisms
search algorithms
The SOS algorithm and its variants have been used in
solving various optimization problems in the field of science and engineering. The taxonomy of applications of
SOS algorithm is illustrated in Fig. 4. As seen it the figure,
SOS algorithm has been applied to combinatorial, continuous, and multi-objective optimization problems. Additionally, it has been used for solving problems in
transportation, machine learning, and cloud computing.
Finally, SOS algorithms usage has been on the increase in
areas of engineering. These areas include environmental/
economic dispatch, power systems optimization, design of
engineering structures, wireless communications, construction project scheduling, electromagnetic optimization,
and reservoir optimization.
Neural Computing and Applications
Fig. 4 Taxonomy of SOS
applications
4.1 Symbiotic organisms search for classes
of optimization problems
Some efforts have been made in adapting and modifying
SOS algorithms to handle various classes of optimization
problems such as multi-objective optimization, constrained
optimization, combinatorial optimization, and continuous
optimization which are important aspect of facilitating
design and optimization of various problems in engineering
and computer science.
4.1.1 Multi-objective symbiotic organisms search
algorithms
Multi-objective optimization problems involve many conflicting objectives, thus improving one objective lead to
deterioration of other objectives [47–56]. There is no single
optimal solution that can optimize multi-objective optimization problems (MOP) with conflicting objectives,
rather a set of optimal trade-off solutions known as Pareto
optimal solutions. Many practical optimization problems
consist of multiple objectives. The objectives often conflict
with one another. Improving one objective may lead to
decline in quality of another objective. Thus, there is no
single solution which can optimize all the objectives.
However, a set of optimal trade-off solutions are significant
for decision-making. Metaheuristic algorithms have proven
to be able to provide approximate solutions to MOP.
Recently, SOS has been applied to solve MOP. Since the
original SOS cannot be adapted directly to handle multiobjective optimization problems, three issues have to be
considered when extending SOS to handle multi-objective
optimization problems. First, how to choose the global and
local best organisms to guide the search of an organism.
Second, how to maintain good solutions found so far.
Lastly, how to constraints in the case of constrained
optimization problems. Table 3 presents multi-objective
SOS algorithms.
Dosoglu et al. [57] proposed SOS for handling economic emission load dispatch (EELD) problem for thermal
generator in power systems to minimize operating cost and
emission while satisfying load demand. The multi-objectives are converted into a single objective using weighted
sum approach. The efficiency of the proposed algorithm
was tested on various standard power systems IEEE
3-machines 6-bus, IEEE 5-machines 14-bus, IEEE 6-machines 30-bus test systems for both with transmission loss
and without transmission loss. The obtained results by the
proposed algorithm are better as compared to other optimization algorithms like GA, DEA, PSO, bees algorithm
(BA), mine blast algorithm (MBA), and cuckoo search
(CS). Tran et al. [58] presents multi-objective SOS
(MSOS) for optimizing trade-off among project duration,
project cost, and the utilization of multiple work shift
schedules. The optimal trade-off while maintaining availability constraints is essential to enhance the success of
entire construction project. They employed a selection
mechanism introduced by [62] for selection of candidates
solution to facilitate the generation of good Pareto front.
The ecosize remains unchanged during the optimization
process, the best solutions are chosen from the combined
ecosystem, and a two-solution dominance approach is used.
The combination of the current population and advanced
population is larger than the ecosize. Therefore, the ecosize
solutions are selected based on the non-dominated sorting
technique [11] and crowding entropy sorting technique
[63]. A study of performance of the MSOS was carried out
using a case study of construction projects, and the
obtained results were compared with known algorithms
like non-dominated sorting genetic algorithm II (NSGAII), the multiple objective particle swarm optimization
(MOPSO), the multiple objective differential evolution
(MODE), and the multiple objective artificial bee colony
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Neural Computing and Applications
Table 3 Multi-objective SOS algorithms
S/n
Multi-objective
technique
Solution
approach
Application domain
Compared algorithms
Result
1
SOS [57]
Weighted
sum
Economic emission load dispatch
(EELD) problem for thermal
generators in power systems
NSGA-II, FCGA, and BBO
The proposed algorithm outperformed
other state-of-the-art algorithms for
EELD problems
2
MOSOS [58]
Pareto
Time–cost–labour utilization
trade-off problem
NSGA-II, MOPSO,
MODE, and MOABC
The MSOS algorithm demonstrates
improved diversity and generated
better Pareto fronts
3
MSOS [59]
Pareto
Constrained truss design problem
NSGA-II, MOPSO,
MOCBO, MGE, and
MGP
MSOS demonstrated superior
performance over the compared
algorithm
4
MSOS, IMSOS
[60]
Pareto
Constrained brushless direct
current (dc) motor design
NSGA-II
The proposed approach provided
better performance in terms of
Pareto front and normalized
Euclidean distances
5
CMSOS [61]
Pareto
Task scheduling in cloud
computing
EMS-C, ECMSMOO, and
BOGA
The CMSOS algorithm outperformed
the compared algorithms in terms of
Pareto fronts for makespan and cost
(MOABC). MSOS was found to be efficient in solving
trade-off among project duration, cost, and labour utilization and finds Pareto optimal solutions that are useful for
assisting construction project decision-makers.
Adaptive penalty function and distance measure were
proposed by [59] to handle both equality and inequality
constraints associated with multi-objective optimization
problems. The values of the penalty function and distance
measure are changed according to the fitness and the
average constraint violation of individual to modify the
objective function. The modified objective function is used
at non-dominated sorting stage to obtain the optimal
solution in feasible and infeasible region. The performance
of the proposed algorithms was evaluated using eighteen
benchmark multi-objective functions. The results of the
simulation indicate superior performance of the proposed
algorithm over other multi-objective optimization algorithms like multi-objective particle swarm optimization
(MOPSO), multi-objective colliding bodies optimization
(MOCBO), non-dominated sorting genetic algorithm II
(NSGA-II), and two gradient-based multi-objective algorithms such as Multi-Gradient Explorer (MGE) and MultiGradient Pathfinder (MGP). Ayala et al. [60] proposed
multi-objective SOS (MSOS) algorithm for solving electromagnetic optimization problems, and the MSOS algorithm is based on non-dominance sorting and crowding
distance criterion. Furthermore, an improved MSOS
(IMSOS) was proposed by replacing the random components of SOS with Gaussian probability distribution function. The results of the proposed algorithms shown
interesting performance over NSGA-II.
Abdullahi et al. [61] proposed a chaotic symbiotic
organisms search (CMSOS) algorithm for solving multiobjective large-scale task scheduling optimization problem
123
on IaaS cloud computing environment. The chaotic
sequence technique is employed to ensure diversity among
organisms for global convergence. The CMSOS algorithm
ensured optimal trade-offs between execution time and
cost.
4.1.2 Combinatorial optimization
SOS algorithm and its variants have been utilized for
solving combinatorial optimization problems. Table 4
presents the summary of combinatorial SOS-based algorithms. Cheng et al. [27] proposed discrete version of SOS
for solving multiple resources leveling in the multiple
projects scheduling problem (MRLMP) to reduce fluctuation in resource utilization during the span of project
implementation. The proposed optimization model transforms continuous solutions to discrete solutions to handle
MRLMP, because SOS was designed originally for continuous optimization space and MRLMP is a discrete
optimization problem. The authors compared the performance of the SOS with GA, PSO, and DE on discrete
optimization space using accuracy, solution stability, and
satisfaction as performance measure. The SOS was found
to be more reliable and efficient as indicated experimental
results and statistical tests. Verma et al. [64] proposed a
discrete version of SOS for congestion management (CM)
problem in deregulated electricity market to reduce load
rescheduling cost. The proposed algorithm considered line
loading and load bus voltage constraints while handling
CM problem to minimize rescheduling cost of generators.
The effectiveness of the proposed method in various test
cases, and the results obtained by SOS were compared to
those of simulated annealing (SA), random search method
(RSM), and PSO. The proposed method proved to be
Neural Computing and Applications
Table 4 Combinatorial optimization-based SOS algorithms
S/n
Technique
Application domain
Compared
algorithms
Result
1
DSOS–MRLMP [27]
Resource leveling in project
scheduling
GA, PSO, and
DE
The proposed algorithm significantly
performs better than the compared
algorithms in terms of the start time
2
SOS [58]
Transmission congestion
management in deregulated power
system
PSO, RSM,
RCGA, and
SA
The SOS algorithm efficiently minimizes
congestion management for modified IEEE
30- and 57-bus systems
3
SOS [65]
Capacitated vehicle routing problem
No comparison
The SOS algorithm showed reasonable
computational time
4
DSOS [66]
Task scheduling in cloud computing
PSO, SA_PSO
DSOS performs better than compared
algorithms for large number of tasks
5
SASOS [44]
Task scheduling in cloud computing
SOS
SASOS converges faster than SOS algorithm
6
SOSCanonical , SOSBasic , SOSSR 1 ,
SOSSR 2 , ISOSSR 1 , and
ISOSSR 2 [34]
Capacitated routing problem
SOS
SASOS converges faster than SOS algorithm
7
SOS [67]
Capacitated routing problem
DE, GA, and
PSO
SASOS converges faster than SOS algorithm
efficient on modified IEEE 30- and 57-bus test power
system for CM problems. Eki et al. [65] proposed a discrete version of SOS for handling capacitated vehicle
routing problem (CVRP), a decoding method for dealing
with discrete problem setting of CVRP. The proposed
algorithm was tested on a set of classical benchmark
problems, and the obtained results showed the superior
performance of SOS over the best known results. Authors
applied swap reverse local search technique after the execution of the three phases of SOS to improve the quality of
solution.
Abdullahi et al. [66] presents discrete version of SOS
(DSOS) for solving task scheduling problem in cloud
computing environment. It transforms continuous solutions
into discrete solutions to fit the task scheduling problem in
cloud computing environment. The proposed method was
compared with PSO and its variants, and DSOS proved to
be more effective and efficient than PSO and its variants.
DSOS outperformed PSO and its variants in terms of
convergence rate, quality of solution, and scalability.
Abdullahi and Ngadi [44] presents hybridized SA and SOS
(SASOS) for solving task scheduling problem in cloud
computing environment. The SA algorithm was used to
improve the local search ability of SOS. The proposed
algorithm was used to improve the task execution makespan by taking into consideration the utilization of computing resources. The performance of SASOS was
compared with that basic SOS, and SASOS performs better
in terms of convergence rate, quality of solution, and
scalability. Vincent et al. [34] proposed SOS for solving
capacitated vehicle routing problem (CVRP), and the aim
of CVRP is to minimize the total route cost when deciding
the routes for a set of vehicles. The authors proposed two
solution representations for transforming the solution from
continuous to discrete search space. They further apply
local search strategy and introduced two new interaction
called competition and amensalism. Performance of the
proposed SOS was evaluated using two set of known
benchmark problems. The results obtained by the proposed
SOS are better than that of basic SOS and PSO. Zhang
et al. [67] developed machine learning framework based on
regularized extreme learning machine (ELM) and SOS.
SOS is employed for optimizing input weights, bias and
regularized factor parameters, while ELM computes the
output weights to save computing time. The proposed
algorithm was evaluated using data from University of
California, Irvine (UCI) dataset repository, and it was
found to perform better than the classical classification
algorithms like scalable vector machine (SVM), leastsquares support-vector machine (LS-SVM), and backpropagation (BP).
4.1.3 Continuous optimization
The original SOS was applied to continuous optimization
problems taken into account the benchmarks of the standard mathematical optimization functions. Summary of the
continuous optimization-based SOS algorithms is presented in Table 5. SOS was originally proposed in [19] for
solving unconstrained mathematical and engineering
design problems. The evaluation of SOS was carried out on
standard numerical benchmark test functions and compared
with GA, PSO, PBA, BA, and DE. The results obtained by
SOS indicate its superior performance for handling
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Neural Computing and Applications
Table 5 Continuous optimization-based SOS algorithms
S/n
Technique
Application domain
Compared algorithms
Result
1
SOS [19]
Mathematical benchmark functions, and
engineering design optimization
PSO, PSOPC, HPSO,
DHPSACO, and MBA
The proposed algorithm significantly
performs better than the compared
algorithms
2
SOS [68]
Energy optimization (Thomson’s
problem)
EA_SG2
The proposed algorithm provides better
optimal values of energy
3
HSOS
[38]
Real parameter optimization problem
using benchmark functions (CEC2005
and CE2010), Frequency modulation
sounds parameter identification problem
and spread spectrum radar polyphase
code design problem
Standard algorithms in the
literature
The HSOS algorithm outperforms compared
algorithms in terms of numerical results for
tested multimodal functions and considered
optimization problems
complex numerical optimization problems. SOS was also
found to achieve better results when tested with some
practical structural design problems.
Kanimozhi et al. [68] proposed SOS for solving
Thomson’s problem by determining optimal configuration
of N identical point charges on a unit sphere. The performance of the proposed algorithm was carried on values of
N ranging from 3 to 300. The results of SOS algorithm
provide better optimum energy values than that of evolutionary algorithm and spectral gradient for unconstrained
condition (EA_SG2) method. Nama et al. [38] presents
hybrid SOS by combining SOS algorithm with simple
quadratic interpolation (SQI) to enhance the global search
of SOS algorithm. The performance of the proposed
algorithm was tested on real-world and large-scale benchmark functions like CEC2005 and CEC2010. The results
obtained by the proposed algorithm were compared with
other classical metaheuristic algorithms. The proposed
algorithm outperforms other state-of-the-art algorithms in
terms of quality of solutions and convergence rate.
4.2 SOS algorithms for engineering applications
The growing popularity of SOS as a robust and efficient
metaheuristic algorithm has attracted attention of
researchers in using SOS algorithm to solve optimization
problems arising from various disciplines of science and
Table 6 SOS algorithms for
economic dispatch
123
S/n
engineering. While significant success has been achieved in
this areas in recent times, optimization of problems in these
areas still remains active research issue. As a result, SOS
algorithm has been applied to solve optimization problems
in the class like economic dispatch [29, 57, 69], power
optimization [31, 32], construction project scheduling [58],
design
optimization
of
engineering
structures
[30, 35, 59, 70], transportation [34, 65], energy optimization [68], wireless communication [71, 72], and machine
learning [73, 74]. With the trend of application of SOS to
optimization problems, SOS has shown to provide allpurpose principles that can easily be adapted to solve wide
range of optimization problems in various domains.
4.2.1 Economic dispatch
Economic dispatch problem has attracted several research
attentions because of the high concerns about environmental pollution. SOS algorithm and its variants have been
applied in solving economic dispatch problems by optimising pollution emission and cost of generation. Summary
of SOS application to economic dispatch problems is presented in Table 6. Rajathy et al. [75] present a novel
method of using symbiotic organism search algorithm in
solving security-constrained economic dispatch proposed
SOS for solving security-constrained Economic Load
Dispatch Problem. The 6-bus system was used to test the
Application
References
1
Economic/emission dispatch problem
[57]
2
Economic/emission dispatch problem
[29]
3
Large-scale economic dispatch with valve-point effects
[69]
4
Dynamic economic dispatch with valve-point effects
[77]
5
Bid-based economic load dispatch
[76]
6
Security-constrained economic dispatch
[75]
Neural Computing and Applications
Table 7 SOS algorithms for
power system optimization
S/n
Application
References
1
Power optimization of three-dimensional turbo code
[31, 32]
2
Optimal power flow based on valve-point effect and prohibited zones
[78]
3
Optimal static VAr compensator (SVC) installation problem
[82]
4
Optimal capacitor placement
[83]
5
Optimal power flow of power system with FACTS
[80]
6
Minimization of network power loss while satisfying the power demand
[84]
7
Minimization of network power loss while satisfying the power demand
[84]
8
Load frequency control
[41, 85]
9
Short-term hydrothermal scheduling
[86]
10
Optimal coordination of directional over-current relays
[81]
11
Power transmission congestion management in deregulated environment
[64]
12
Real power loss minimization
[79]
efficiency of the proposed algorithm. The presented simulation results for Economic Load Dispatch with and
without transmission constraints showed better convergent
rate for the proposed algorithm. Tiwari and Pandit [76]
present SOS for handling bid-based economic dispatch
problem for deregulated electricity market. SOS algorithm
tries to minimize generator cost while satisfying load
demands. The efficiency of the proposed algorithm was
tested on IEEE-30 bus system with six generators, two
customers, and two dispatch periods under low, medium,
and high bidding strategies. The results obtained by SOS
were compared with other algorithms such as DE and PSO.
SOS was found to produce better result than DE and PSO.
Sonmez et al. [77] proposed SOS for handling dynamic
economic dispatch problem in modern power system. They
considered constraints like ramp rate limits, prohibiting
operating zones and valve-point effects. The proposed
algorithm was evaluated using 5 units, 10 units, and 13
units systems as test cases. The results of SOS were found
to be more robust and converge faster than other metaheuristic algorithms like PSO, GA, SA, and DE. Guvenc
et al. [69] proposed SOS for solving both classical and
non-convex economic load dispatch (ELD) problems.
Authors used three different test cases like 3-unit, 15-unit,
and 38-unit power systems to show the efficiency and
reliability of the proposed algorithm. SoS outperformed
other metaheuristic algorithm in terms of solution quality
and convergence rate for both classical and non-convex
problems. Secui [29] proposed a modified SOS for solving
economic dispatch problem by taking into consideration
various constraints valve-point effects, the prohibited
operating zones (POZ), the transmission line losses, multifuel sources, as well as other operating constraints of the
generating units and power system. New relations for
updating the phases of SOS were introduced, and chaotic
sequence generated by logistic map was employed to
improve the exploration capability of SOS. The
performance of the proposed algorithm was tested for
various systems like 13 units, 40 units, 80 units, 160 units,
and 320 units. The proposed algorithm showed better
performance compared to other optimization techniques for
solving economic dispatch problems.
4.2.2 Power systems optimization
Several related optimization problems have been solved
using SOS algorithm, and these problems include optimal
power flow, efficient allocation of compensators and
capacitors in power systems, optimization of power loss,
and power transmission congestion management. Summary
of SOS application to power systems optimization problems is presented in Table 7. Duman [78] proposed SOS
for solving optimal power flow (OPF) problem with valvepoint effect and prohibited zones in modern power systems.
The efficiency of the proposed algorithm is tested on
modified IEEE 30-bus test system using various cases like
without valve-point effect and prohibited zones, with prohibited zones and with valve-point effect, with valve-point
effect, and prohibited zones. The obtained results for all the
test scenarious by the proposed method were compared
with other metaheuristic algorithms in the literature, and
SOS proved to be more effective and robust for all the test
cases. Banerjee and Chattopadhyay [31] proposed a modified SOS (MSOS) for synthesis of 3-dimensional turbo
code for optimization of bit error rate (BER) performance
in communication systems. Authors grouped the ecosystem
into three inhabitants each with it associated probability.
The simulation result indicates that MSOS is performed
better in terms of accuracy, speed, and convergence as
compared with SOS and harmony search (HS) algorithm.
Guha et al. [41] proposed hybrid SOS called quasi-oppositional symbiotic organism search (QOSOS) algorithm
for handling load frequency control (LFC) problem of the
power system. The theory of quasi-oppositional based
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Neural Computing and Applications
Table 8 SOS algorithms for
design optimization of
engineering structures
S/n
Application
References
1
Structural design optimization
[30, 35, 59, 70]
2
Optimum design of frame and grillage systems
[87]
3
Gas transmission compressor design problem
[88]
4
Constrained truss design problems
[30]
learning (Q-OBL) is integrated into SOS to avoid entrapment in local optima and improve convergence rate.
Authors considered two-area and four-area interconnected
power system to test the effectiveness of the proposed
algorithm. The dynamic performance of the test power
systems obtained by QOSOS is better than that of SOS and
comprehensive learning PSO. QOSOS also showed better
robustness and sensitivity for the test power systems.
Balachennaiah and Suryakalavathi [79] presents SOS for
solving optimal power flow (OPF) problem, and they
attempt to optimize real power loss (RPL) of a transmission
system while considering certain constraints. The efficiency of the proposed method is experimented on New
England 39-bus system, and the results obtained are compared with those of interior point successive linear programming (IPSLP) and bacteria foraging algorithm (BFA).
Authors observed superior performance of the proposed
algorithm over IPSL and BFA. Prasad and Mukherjee [80]
proposed SOS for optimal power flow (OPF) problem of
power system equipped with flexible AC transmission
systems. The authors used modified IEEE-30 and IEEE-57
bus test systems to test the efficiency of the proposed
algorithm. The OPF was formulated with objective functions such as fuel cost minimization, transmission active
power loss minimization, emission reduction, and minimization of combined economic and environmental cost.
The simulation results are obtained by the proposed algorithm more effective as compared with hybrid tabu search
and simulated annealing (TS/SA) and differential evolution
(DE).
Saha et al. [81] proposed SOS algorithm for solving
directional over-current relays (DORs) coordination optimization problem in power systems. Authors validated the
computational capability of the proposed using IEEE 6-bus
and WSCC 9-bus test systems. The obtained results
showed significant reduction in operating time of relays
while maintaining reliable coordination margin for primary/backup pair as compared to particle swarm optimization (PSO) and teaching learning-based optimization
(TLBO). Guha et al. [41] proposed SOS algorithm to
solving load frequency control problem (LFC) for design
and analysis of interconnected two-area reheat thermal
power plant equipped with proportional–integral–derivative (PID) controller. The proposed algorithms enhanced
the stability of the power system as compared to DE and
123
PSO. Saha and Mukherjee [39] hybridized SOS with a
chaotic local search (CSOS) to improve the solution
accuracy and convergence speed of the SOS algorithm, and
the parasitism phase of the standard SOS is not considered
in the proposed CSOS in order to reduce the computational
complexity. The performance evaluation of CSOS was
carried out using both twenty-six unconstrained benchmark
test functions and real-world power system problem. The
CSOS algorithm obtained superior results over the compared algorithms for both benchmark function optimization
and power engineering optimization task.
4.2.3 Engineering structures
The need for energy-efficient and eco-friendly buildings
has been a major concern worldwide. Modern building
designs are tailored towards low-carbon emission and
energy efficiency. Therefore, optimal design of residential
buildings needs to consider conflicting objectives like low
cost, energy efficiency, and minimal environmental impact.
SOS algorithms have been employed for solving optimization problems arising from such design specification,
and the summary of the application is presented in Table 8.
Talatahari [87] presents discrete version of SOS for
solving structural optimization problem. The 3-bay
24-story frame problem was used to evaluate the efficiency
of the proposed algorithm. The proposed algorithm performs better than other metaheuristic algorithms like ACO,
HS, and imperialist competitive algorithm (ICA). Tejani
et al. [35] introduced adaptive benefit factors in basic SOS
to keep the balance between exploration and exploitation of
search space. The efficiency of the proposed algorithm was
tested on sets of engineering structure optimization problems. Six planar and space trusses subject to multiple
natural frequency constraints were used to study the
effectiveness of the proposed algorithm. The performance
of the proposed algorithms was compared with other
metaheuristic algorithms like NHGA, NHPGA, CSS,
enhanced CSS, HS, FA, CSS–BBBC, OC, GA, hybrid OC–
GA, CBO, 2D–CBO, PSO, and DPSO. Prayogo et al. [70]
presents SOS algorithm for solving civil engineering
problem with many design variables and constraints. The
performance of SOS was evaluated using benchmark
problems and three civil engineering problems, and the
results of simulation indicated that SOS is more effective
Neural Computing and Applications
Table 9 Other engineering applications
Area
Construction project scheduling
Energy optimization (Thomson’s
problem)
Wireless communications
Application
Reference
Time, cost, and labour utilization trade-off
[58]
Optimizing multiple resources leveling
[27]
Minimizing energy of point charges on a sphere
[68]
Synthesis of antenna arrays
[71]
Design of linear antenna arrays with low sidelobes level
[72]
Nonlinear optimization
Frequency modulation sounds parameter identification and spread spectrum radar polyphase
code design
[38]
Electromagnetic optimization
Multi-objective electromagnetic optimization
[60]
Magnetic levitation system
Controlling instability and high input nonlinearity of magnetic levitation system
[91]
Water optimization
Reservoir optimization
[82]
and efficient than the compared algorithms. The proposed
model is a promising tool for assisting civil engineers to
make decisions to minimize the expenditure of material
and financial resources.
to 8-state Duo-Binary Turbo Code (DB-TC) of the Digital
Video Broadcasting Return Channel via Satellite standard
(DVB-RCS) and Serially Concatenated Convolution Turbo
Code (SCCTC) structures.
4.2.4 Wireless communication
4.2.5 Other engineering applications
Improved SOS algorithms have been to found optimal
solutions in terms of covered and efficient energy consumption in wireless communications systems than the
original SOS and their counterpart swarm intelligence
algorithms. Dib [71] proposed SOS for handling design of
linear antenna arrays with low sidelobes level. The proposed algorithm produces a radiation pattern with minimum sidelobe level as compared with other metaheuristics
algorithms like PSO, biography-based optimization (BBO),
and Taguchi. Das et al. [89] presents SOS for determining
optimal size and location of distributed generation (DG) in
radial distributed network (RDN) for the reduction in network loss taking into consideration deterministic load
demand. The performance of the proposed algorithm was
tested using different RDNs like 33-bus and 69-bus distribution networks. The results obtained by the proposed
algorithm are compared with other metaheuristic algorithms such as particle swarm optimization (PSO), teaching-learning-based optimization (TLO), cuckoo search
(CS), artificial bee colony (ABC), gravitational search
algorithm (GSA), and stochastic fractal search (SFS). The
proposed algorithm offers better solution in terms of minimum loss and convergence mobility.
Banerjee and Chattopadhyay [32] presents a modified
SOS (MSOS) algorithm for optimizing the performance of
3-dimensional turbo code (3D-TC) in controlling error
coding scheme to improve adequate redundancy in communications system. The proposed MSOS improves the
performance of bit error rate (BER) of 3D-TC as compared
Application of SOS algorithms to other engineering areas
is presented in Table 9. Bozorg-Haddad et al. [90] applied
SOS algorithm to optimize single objective and multi-objective reservoir optimization problems. The proposed
algorithm outperforms GA and water cycle algorithm
(WCA) for solving single objective and multi-objective
reservoir formulations. Ayala et al. [60] proposed multiobjective SOS (MSOS) algorithm for solving electromagnetic optimization problems, and the MSOS algorithm is
based on non-dominance sorting and crowding distance
criterion. Furthermore, an improved MSOS (IMSOS) was
proposed by replacing the random components of SOS with
Gaussian probability distribution function. The results of
the proposed algorithms shown interesting performance
over NSGA-II. Nama et al. [30] presents improved version
of SOS (I-SOS) for solving unconstrained global optimization problems. The authors proposed an additional
phase named predation phase to improve the performance
of the algorithm, and they also introduce random weighted
reflection vector to improve the search capability of SOS.
The performance of the proposed algorithm was tested on a
set of benchmark functions and compared with other stateof-the-art metaheuristics algorithms like PSO, DE, and
basic SOS. The performance results indicate that I-SOS
outperformed PSO, DE, and SOS.
Sadek et al. [91] proposed SOS algorithm for controlling instability and high input nonlinearity of the magnetic
levitation system, and SOS algorithm was used to provide
initial adaptive and control parameters. Adaptive fuzzy
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Neural Computing and Applications
Table 10 Other application areas of SOS algorithms
Area
Application
References
Capacitated vehicle routing
[34, 65]
Traveling salesman problem
[45, 94]
Cloud computing
Task scheduling optimization
[40, 44, 66, 92, 93]
Machine learning
Input parameter optimization in data classification
[67]
Transportation
Training of weights for wavelet neural network (WNN) for equalizer design
[73]
Optimization of weights and biases for feedforward neural networks (FNNs)
[74]
backstepping control strategy was used to approximate the
uncertain parameters and structures to minimize tracking
error and improve stability of the system. Ljapunov stability theorem was employed to update the parameters of
the improved fuzzy backstepping control during each
sampling period. The simulation results of the proposed
system were verified by the experimental results which
support theoretical background of the proposed control
method.
4.3 Other application areas
SOS algorithms have also been applied in solving problems
in the area of machine of learning [73, 74], transportation
[34, 65], and cloud computing [40, 44, 61, 66, 92, 93].
4.3.1 Machine learning
In recent years, machine learning techniques have been
widely used in scientific and industrial applications.
However, the main drawback of these techniques is how to
determine the optimal value of the parameters to be tuned.
To address this issue, metaheuristic algorithms have been
employed for parameter optimization purposes. SOS
algorithms have been proved to be efficient in tuning
parameters for machine learning techniques.
Wu et al. [74] proposed SOS algorithm for efficient
training of feedforward neural networks (FNNs) by optimizing the weights and biases, and performance of the
proposed algorithm was investigated using eight different
datasets selected from the UCI machine learning repository. The obtained results of eight datasets with different
characteristic show that the proposed approach is efficient
to train FNNs compared to other training methods that have
been used in the literature: cuckoo search (CS), PSO, GA,
multiverse optimizer (MVO), gravitational search algorithm (GSA), and biogeography-based optimizer (BBO).
Nanda and Jonwal [73] proposed SOS algorithm for
training the weights of wavelet neural network (WNN) for
equalizer design in order to prevent inter-symbol interference in communication channels. The performance of
123
SOS-based WNN trained equalizer is compared with WNN
trained by cat swarm optimization (CSO) and clonal
selection algorithm (CLONAL), particle swarm optimization(PSO), and least mean square algorithm (LMS). Furthermore, the performance of the WNN structure-based
equalizer was compared with other equalizers with structure based on functional link artificial neural network
(trigonometric FLANN), radial basis function network
(RBF), and finite impulse response filter (FIR). The results
of SOS showed robust performance in handling the burst
error conditions.
4.3.2 Other applications
Other related applications of SOS algorithms go to other
areas like transportation [34, 65] and cloud computing
[40, 44, 61, 66, 92, 93]. Summary of such applications is
presented in Table 10.
5 Discussions and future works
In this section, the features of SOS algorithm are discussed.
SOS algorithms are suitable for solving both unimodal and
multi-modal optimization problems, and they have global
and fast convergence, thus obtaining better results on both
continuous and discrete optimization problems. The fast
convergence of SOS is attributed to deterioration of
diversity among the organisms as the search procedure
progresses. For SOS algorithm to be appropriate for largescale optimization problems, there is need to establish a
balance between local and global search [95, 96].
The results of SOS algorithms are influenced by the best
solution found within the ecosystem; therefore, improving
the best solution can improve the efficiency of search
mechanism of the ecosystem. Experimental results indicate
that SOS algorithms could be successfully applied to
complex optimization problems. In some situations, the
convergence rate of SOS algorithm could be improved by
increasing diversity of solutions within the search space
using chaotic maps. Adaptive turning of SOS control
Neural Computing and Applications
parameters (benefit factors) could be suitable for improving
the global convergence. Incorporation of chaotic maps and
weighted parameters into SOS move function could
improve the convergence speed and quality of solutions.
However, there are several rooms for further modifications
and developments, for instance parallelization using multipopulation (multi-ecosystem) and co-evolutionary
scheme [97, 98].
Most of the analyzed papers hybridized SOS algorithm
with local search techniques such as simulated annealing
(SA), chaotic local search (CLS), simple quadratic interpolation (SQI), and quasi-oppositional based learning (QOBL). The SOS algorithm serves as a global optimizer,
while the local search techniques further improve the global solutions. However, SOS algorithm can be hybridized
with other robust local search and global search optimizers
such as PSO, DE, and GA. Also, hybridizations that support integration of genetic operators as well as PSO and
with SOS algorithms will be an future development
[99, 100]. Other possibles future developments could be the
hybridization of SOS algorithms with machine learning
techniques where SOS acts as metaheuristic for providing
optimal parameter settings for the machine learning techniques [101–105].
Various combinatorial optimization problems have been
solved with SOS algorithm, the algorithm found better
solution in some cases, while the solutions found in other
cases are not acceptable. The main issue here is the
imbalance between local search and global search during
the SOS procedure. The imbalance problem between local
search and global search could be addressed hybridizing
SOS algorithm with local search optimizers like variable
neighbourhood search [106], hill-climb local search [107].
Another ways of controlling balance between local and
global search are through diversity maintenance and control [108]. Maintaining proper balance between local
search and global search indicates new directions for future
development of SOS algorithm. Moreover, using SOS
algorithm for combinatorial optimization problems
requires transforming continuous variables in the search
space to discrete variables in the problem space. The SOS
algorithms for solving the constrained optimization problems regard the constraints as a penalty function which
punishes the solutions that violates the constraints known
as infeasible solutions. The value of penalty function corresponds to the extent of violation of the constraints, which
affects the fitness function. However, there is need for
more effective and efficient methods for penalizing infeasible solutions when using SOS algorithms for solving
constrained optimization problems.
In spite of the recorded success of SOS algorithm, there
has not been any attempt to present theoretical analysis of
the algorithm. Experimental results showed that SOS
algorithms often converge faster to global solutions; however, there is no theoretical analysis to show how fast the
algorithm converges. The theoretical developments of SOS
algorithms need to be explored in order to gain a thorough
understanding of how SOS algorithm works. The theories
of complexity and Markov chains can be employed for
convergence analysis and stability of main variants of SOS
algorithms []. In fact, theoretical analysis is an open
problems regarding all metaheuristic algorithms [109].
Parameter tuning is another area for all metaheuristic
algorithms. The only control parameter of SOS is the
benefit factors which control the extent of exploitation and
exploration ability of the organisms. The values for these
parameters are suggested to be statically assigned in the
original proposal of SOS algorithm. Therefore, there is
need for studies in determining optimal settings for algorithm control parameters for wider application to solve
range of optimization problems with little tuning of control
parameter values [63]. However, design of automatic
schemes for tuning algorithm control parameters in an
intelligent and adaptive manner is still an open optimization problem.
SOS algorithms have been applied to various optimization problems as evidenced from this view, and the
applications can be extended to the areas like data mining,
feature selection, bio-informatics, scheduling, and realworld large-scale optimization [110]. It is indisputable that
more applications of SOS algorithms will surface in the
near future.
6 Conclusions
SOS algorithms have been applied to optimization problems in various domains since its introduction in 2014.
SOS has proven to be efficient for optimizing complex
multidimensional search space while handling multi-objective and constrained optimization problems. Active
researches on SOS since its introduction include
hybridization, discrete optimization problems, constrained
and multi-objective optimization. Hybridization intends to
combine the strengths of SOS like global search ability and
rapid optimization, with other related techniques to address
some of the issues with SOS performance, like entrapment
in local optima. SOS algorithm has been to be very
effective and easily adaptable to various application
requirements, with potentials for hybridization and modifications though SOS still faces challenges like local
optima entrapment, imbalance between local search and
global search, constraint handling, large-scale optimization, and multi-objective optimization, and these are still
important research focus as evident from the literature.
Further understanding and refinements of SOS algorithm
123
Neural Computing and Applications
and challenges of using it to solve large-scale optimization
problems are required.
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