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Expert Systems With Applications
journal homepage: www.elsevier.com/locate/eswa
Highlights
Reptile Search Algorithm (RSA): A nature-inspired
meta-heuristic optimizer
Expert Systems With Applications xxx (xxxx) xxx
Laith Abualigah
∗
, Mohamed Abd Elaziz, Putra Sumari, Zong Woo Geem, Amir H. Gandomi
• Developed a novel optimization algorithm inspired by hunting behaviour of Reptiles (RSA).
• Tested RSA against classical, CEC2017, CEC2019 test functions and engineering problems.
• Compared the RSA to other well-known optimization algorithms.
• Demonstrated effectiveness and superiority of the proposed RSA.
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Contents lists available at ScienceDirect
Expert Systems With Applications
journal homepage: www.elsevier.com/locate/eswa
Reptile Search Algorithm (RSA): A nature-inspired meta-heuristic optimizer
Laith Abualigah
a,b,
∗
, Mohamed Abd Elaziz
c
, Putra Sumari
b
, Zong Woo Geem
d
,
Amir H. Gandomi
e
a
Faculty of Computer Sciences and Informatics, Amman Arab University, Amman 11953, Jordan
b
School of Computer Sciences, Universiti Sains Malaysia, Pulau Pinang 11800, Malaysia
c
Department of Mathematics, Faculty of Science, Zagazig University, Egypt
d
Department of Energy and Information Technology, Gachon University, Seongnam-si, Republic of Korea
e
Faculty of Engineering and Information Technology, University of Technology Sydney, Ultimo, NSW 2007, Australia
A R T I C L E I N F O
Keywords:
Reptile Search Algorithm (RSA)
Optimization algorithms
Meta-heuristics
Real-word problems
Optimization problems
A B S T R A C T
This paper proposes a novel nature-inspired meta-heuristic optimizer, called Reptile Search Algorithm (RSA),
motivated by the hunting behaviour of Crocodiles. Two main steps of Crocodile behaviour are implemented,
such as encircling, which is performed by high walking or belly walking, and hunting, which is performed
by hunting coordination or hunting cooperation. The mentioned search methods of the proposed RSA are
unique compared to other existing algorithms. The performance of the proposed RSA is evaluated using twenty-
three classical test functions, thirty CEC2017 test functions, ten CEC2019 test functions, and seven real-world
engineering problems. The obtained results of the proposed RSA are compared to various existing optimization
algorithms in the literature. The results of the tested three benchmark functions revealed that the proposed
RSA achieved better results than the other competitive optimization algorithms. The results of the Friedman
ranking test proved that the RSA is a significantly superior method than other comparative methods. Finally,
the results of the examined engineering problems showed that the RSA obtained better results compared to
other various methods.
1. Introduction1
There is a well-known theory said, ‘‘survival of the fittest’’. Hence,2
predators have to choose a robust approach to maximize the hunting3
the prey in natural life (Bartumeus, da Luz, Viswanathan, & Cata-4
lan, 2005). Generally, the foraging activity of various animals in na-5
ture is a dramatically random walk approach; a stochastic manner in6
which the next position is subject to the current situation/position7
and a change likelihood to the next position which can be mathemati-8
cally represented as an optimization technique (Shcherbacheva, 2019;9
Viswanathan et al., 1999). These approaches have been developed by10
the environment and typically chosen by predators to survive.11
There are two main classes for optimization techniques: (1) deter-12
ministic methods, which are divided into linear and non-linear ap-13
proaches (Horst & Tuy, 2013). These approaches are the most used14
deterministic methods, designated by utilizing the gradient erudition15
of the problem to explore the search space and find the optimal16
solution (Abualigah, 2020a; Abualigah & Diabat, 2020). In spite of17
these approaches are useful for linear search problems (Unimodal),18
∗
Corresponding author at: Faculty of Computer Sciences and Informatics, Amman Arab University, Amman 11953, Jordan.
E-mail addresses: Aligah.2020@gmail.com (L. Abualigah), abd_el_aziz_m@yahoo.com (M.A. Elaziz), putras@usm.my (P. Sumari), geem@gachon.ac.kr
(Z.W. Geem), Amirhossein.Gandomi@uts.edu.au (A.H. Gandomi).
they are susceptible to local optima trap when implementing to non- 19
linear search problems (Multimodal), including real-world optimization 20
problems. To overcome this problem, several methods can be used 21
such as a different initial population strategy, hybridize, or modify the 22
algorithm (Luenberger, Ye, et al., 1984). (2) Stochastic methods, which 23
are another alternative that generates and utilizes random variables 24
such as the meta-heuristic optimization algorithms. These optimiza- 25
tion algorithms are utilized to globally search in the available search 26
space of the problem to obtain the near-optimal solution (Gardiner 27
et al., 1985). The advantages of these algorithms are simplicity, flex- 28
ibility, independency to the problem, easy to use, and gradient-free 29
nature (Abualigah, 2020b; Abualigah, Shehab, Alshinwan, Mirjalili, and 30
Abd Elaziz, 2020). 31
Recently, meta-heuristic algorithms have been successfully em- 32
ployed to address various complicated optimization problems (Gan- 33
domi, 2014; Gandomi & Alavi, 2012). These algorithms are exposed 34
to be more robust than the other traditional methods that are based 35
on conventional logic or mathematical programming. Exploration (di- 36
versification) and exploitation (Intensification) search strategies are 37
https://doi.org/10.1016/j.eswa.2021.116158
Received 20 May 2020; Received in revised form 7 September 2021; Accepted 24 October 2021
L. Abualigah et al.
two chief merits of the meta-heuristic algorithms. The exploration1
search strategy works to ensure that the algorithm explores the given2
search space widely and efficiently. While the exploitation search3
strategy searches around the obtained-optimal solution to find the4
best candidate solution. The main objectives of introducing advanced5
meta-heuristic algorithms are to solve various complicated optimization6
problems faster and to get more robust optimization methods (Erol7
& Eksin, 2006; Geem, Kim, & Loganathan, 2001; Kaveh & Farhoudi,8
2013).9
According to the nature of inspiration, meta-heuristic algorithms10
are classified into main four classes (Abualigah & Diabat, 2021): (1)11
evolutionary-based algorithms (EA) (Fonseca & Fleming, 1995), (2)12
swarm-based intelligence (SI) (Parpinelli & Lopes, 2011), (3) physics-13
based methods (PM) (Biswas, Mishra, Tiwari, & Misra, 2013), and14
(4) human-based methods (HM) (Kosorukoff, 2001). EAs mimic the15
behaviour of natural evolution. These algorithms utilize operators16
(i.e., crossover and mutation) inspired by biologies. The most com-17
monly utilized EA is GA. GA uses these operators to generate improved18
solutions. Other examples of the EA class are evolutionary program-19
ming (Fogel, Owens, & Walsh, 1966), differential evolution (Storn20
& Price, 1997), and evolution strategy (Hansen, Müller, & Koumout-21
sakos, 2003). SI simulates the social behaviour of animals in swarms22
(i.e., herds, flocks, or schools). Mainly, the characteristic of this class is23
the sharing of joint information of all animals through the optimization24
process. The most utilized algorithm in this class is particle swarm25
optimization (PSO), developed by Kennedy and Eberhart (Eberhart &26
Kennedy, 1995). The Fox Red Optimization Algorithm is suggested27
based on a mathematical representation of red fox rules and hunting for28
food (Połap & Woźniak, 2021). Black Widow Optimization Algorithm29
is proposed in Hayyolalam and Kazem (2020) based on the inspiration30
of mating behaviour of black widow spiders. Other examples of SI31
class are Salp Swarm Algorithm (Abualigah, Shehab, Alshinwan, &32
Alabool, 2019), Ant Colony Optimization (Dorigo, Birattari, & Stutzle,33
2006), and Dolphin Echolocation (Kaveh & Farhoudi, 2013). PMs are34
inspired by physical laws in life, and mainly defines the communication35
of the candidate solutions based on controlling rules of the physical36
methods. One of the most commonly utilized algorithms in PM class is37
Simulated Annealing, which utilizes thermodynamics laws to heating38
and the next handler cooling of an element to grow the volume39
of its crystals. Other examples of PM class are Gravitational Search40
Algorithm (Rashedi, Nezamabadi-Pour, & Saryazdi, 2009), Henry Gas41
Solubility Optimization (Hashim, Houssein, Mabrouk, Al-Atabany, &42
Mirjalili, 2019), and Charged System Search (Kaveh & Talatahari,43
2010b). Finally, HMs are motivated by human communications and44
behaviour in communities. This class is consequently emphasized to45
generate better solutions until met the termination criteria. Examples46
of HMs are the Imperialist Competitive Algorithm (ICA) (Atashpaz-47
Gargari & Lucas, 2007), and Teaching-Learning-Based Optimization48
(TLBO) (Rao, Savsani, & Vakharia, 2011). Moreover, these various49
methods have been widely used to solve different problems such50
as image segmentation (Abuowaida, Chan, Alshdaifat, & Abualigah,51
2021), microscopy image analysis (Altabeeb, Mohsen, Abualigah, &52
Ghallab, 2021; Połap, 2020), task scheduling (Abd Elaziz, Abualigah, &53
Attiya, 2021), economic emission dispatch (Hassan, Kamel, Abualigah,54
& Eid, 2021), optimal allocation of power resources (Eid, Kamel,55
& Abualigah, 2021), feature selection (Jiang, Luo, Wei, Abualigah,56
et al., 2021), vulnerability detection (Şahin & Abualigah, 2021), images57
classification (Yousri et al., 2021), intrusion detection system (Safaldin,58
Otair, & Abualigah, 2021), identifying photovoltaic models (Yousri59
et al., 2020), and others.60
Genetic Algorithms (GA) (Holland et al., 1992), Harmony Search61
(HS) Algorithm (Geem et al., 2001), Cuckoo Search Optimization62
(CS) (Yang & Deb, 2009), Krill Herd Algorithm (KHA) (Gandomi &63
Alavi, 2012), Gray Wolf Optimizer (GWO) (Mirjalili, Mirjalili, & Lewis,64
2014), Artificial Bee Colony (ABC) (Karaboga & Akay, 2009), Aquila65
Optimizer (AO) (Abualigah et al., 2021), and Arithmetic Optimiza- 66
tion Algorithm (AOA) (Abualigah, Diabat, Mirjalili, Abd Elaziz and 67
Gandomi, 2021) are some of the common traditional meta-heuristics 68
optimization algorithms. Despite the success of traditional and recent 69
optimization algorithms, no algorithm can guarantee to achieve the 70
best global optimum solutions for various optimization problems. This 71
has been proven by the theorem of the No-Free-Lunch in search and 72
optimization (Abualigah, Diabat and Geem, 2020; Wolpert & Macready, 73
1997). This theory motivated us to introduce a new optimization 74
algorithm and solve various optimization problems more efficiently. 75
We intend to propose a more dynamic and effective algorithm; 76
this paper introduces a new natural-inspired based meta-heuristic op- 77
timizer, called Reptile Search Algorithm (RSA). This algorithm is stim- 78
ulated by the encircling and hunting behaviours of Crocodiles in the 79
real-life. The main difference between the proposed RSA and others 80
is that RSA has a unique procedure modelled to update the solutions’ 81
positions using four novel mechanisms. For example, encircling is 82
performed by high walking or belly walking, and hunting is conducted 83
by hunting coordination or hunting cooperation. The main motivation 84
behind RSA is to find powerful search methods that can produce better 85
quality solutions for the complicated problems and get new best results 86
that can help solve complex real-world problems. A set of twenty-three 87
classical test functions, thirty CEC2017 test functions, and ten CEC2019 88
test functions is used to verify the robustness and effectiveness of 89
the proposed RSA rigorously. Moreover, eight real-world engineering 90
problems are used to investigate the effectiveness of the proposed RSA 91
further. 92
The remainder of the paper is organized as: Section 2 describes the 93
Reptile Search Algorithm developed in this paper. Section 3 presents 94
the results, discussion, and evaluation of RSA on various optimization 95
problems. Section 4 presents the conclusion of the current work and 96
recommends future directions. Thus, two mathematical models were 97
introduced to update the positions of candidate solutions; one for 98
diverse search and another toward the optimal search region. 99
2. The Reptile Search Algorithm (RSA) 100
In this section, the exploration (global search) and exploitation 101
(local search) phases of the proposed Reptile Search Algorithm (RSA) 102
are presented, which is inspired by the encircling mechanisms, hunt- 103
ing mechanisms, and the social behaviour of Crocodiles in nature. 104
Crocodiles behaviours consist of encircling and hunting the prey. These 105
mechanisms are mathematically modelled to present the proposed RSA 106
and perform the optimization processes. RSA is a population-based 107
and gradient-free method, so it can be used to address complicated or 108
straightforward optimization problems subject to specific constraints. 109
Cohesive groups are helpful for active co-operation between Crocodiles 110
and also maximize their robustness. 111
2.1. Biology and behaviour of Crocodiles 112
Crocodiles (subfamily includes the true crocodiles) are colossal 113
semi-aquatic semiaquatic creepers that live everywhere in tropics such 114
as Australia, Africa, Asia, and the Americas. The word crocodile refers 115
to only the kinds within the subfamily of ‘‘Crocodylinae’’. Generally, a 116
crocodile’s physical characteristics support them to be a strong preda- 117
tor (Dinets, Brueggen, & Brueggen, 2015; Kushlan & Mazzotti, 1989). 118
Their external shape is a sign of its water and predatory lifestyle. 119
Crocodiles have very little resistance to a flow of air and water (stream- 120
lined body); this shape increases its speed. It makes their movement 121
easier, which enables them to move quickly. As well, Crocodiles also 122
raise their feet to the side during walking, which runs faster. Crocodiles’ 123
webbed feet allow them to walk turns and sudden movements quickly 124
in the swimming. These feet are a distinctive feature, where the animals 125
usually move from one place to another by walking (Dinets, 2015; Platt 126
et al., 2006). The main characterizations of the Crocodile behaviour are 127
given as follows. 128
L. Abualigah et al.
Vision: Crocodiles have perfect night eyesight and are mainly nightly1
hunters. They utilize the weakness (i.e., miserable night eye-2
sight) of prey animals to their support.3
Hunting and diet: Crocodiles are snare predators, looking for nearby4
fish or land animals, and then running out to attack. Crocodiles5
hunt fish, reptiles, crustaceans, amphibians, mollusks, mammals,6
and birds, and sometimes they eat smaller crocodiles. Crocodiles7
often hunt small fish and invertebrates, gradually moving on8
to larger prey. Crocodiles are predators (in cold-blooded); they9
have a prolonged metabolism so that they can survive for a10
long time without food. Notwithstanding their appearance of11
moving slow, crocodiles have a swift beating. They are top and12
powerful predators in their habitat, and several classes have13
been recognized, attacking and killing other predators such as14
sharks and deer. As well, when given a chance, they would15
prey on young or dying elephants and other different animals.16
Evidence implies that crocodiles also feed on several kinds of17
fruits.18
Locomotion: Crocodiles can run very fast over small distances, even19
out of water. When a crocodile moves quickly, it keeps its legs in20
a straighter and more straightforward situation under its body21
(called the high walk). This kind of walk allows crocodiles a high22
speed.23
Cognition: Crocodiles own some advanced cognitive skills. They can24
recognize and utilize patterns of prey behaviour, such as while25
prey closes to the river to drink frequently.26
Hunting: Crocodiles are advanced hunters: cooperate as a team to27
hunt their prey. Hunting is carried out by crocodiles cooper-28
atively based on coordination and collaboration. Coordinated29
hunting is a sophisticated form of collaborative hunting, in30
which particular predators associate in each other’s movements31
adjusted on the target-prey during organized hunting. Coordi-32
nated hunting is a sophisticated form of cooperative hunting,33
in which specific predators associate in each other’s actions,34
which is considered to be uncommon for the animal lacking a35
backbone. In some instances, individual Crocodile runs on the36
same role (being the ambusher or the driver) during various37
hunts, as seen in lions.38
Coordination and collaboration of crocodiles: Crocodiles hunt in a39
team, as the modern study has shown. This distinguished them,40
members, as one of the best sophisticated and intelligent teams41
that can make cooperation between different individuals with42
different roles. Seeing crocodiles hunting operation is very com-43
plicated. They hunt by ambush; they occasionally eat due to44
their metabolism is slow, and almost all hunting happens at45
night and maybe in shallow waters. For example, the crocodiles46
moved together to force a group of fishes to gather in a small47
group (dense group). They then took turns hunting fish from this48
group (bait ball). Next, the crocodiles will take turns cutting49
over the circle centroid, attack the fishes. Often, Crocodiles of50
various sizes follow various roles. More giant Crocodiles lead51
fishes from the deeper area of a lagoon into the shallows, where52
smaller, more intelligent Crocodiles prevents its escape. Other53
cases included a crocodile scared a pig or a zebra (wildebeests),54
making it move quickly into a lagoon where other crocs were55
disappearing and waiting for the attack. Hunting behaviour of56
Crocodiles is relatively constant; when a Crocodile caught a57
prey, it would move out of the hunting space and join the58
Crocodiles again after eating its victim. It seems that the animals59
after hunting take some resting and wait their turn to join the60
active hunting team.61
A conclusion is that crocodiles are one of the most intelligent 62
and expert hunters and possibly next behind humans. We modelled 63
Crocodile behaviours as a mathematical optimization, and it is deter- 64
mining the best solution subjected to specific constraints. Optimization 65
problems occur in various quantitative disciplines, from engineering, 66
economics, and computer sciences to operations research and industry, 67
and improvements in searching techniques have been attracting interest 68
in several domains of sciences. The main inspiration of the proposed 69
algorithm (RSA) derived from the encircling and hunting the prey. In 70
the following subsections, the descriptions of these processes in the RSA 71
are discussed. RSA is then proposed based on the mathematical model. 72
2.2. Initialization phase 73
In RSA, the optimization process starts with a set of candidate 74
solutions (𝑋) as shown in Eq. (1), which is generated stochastically, 75
and the best-obtained solution is considered as the nearly the optimum 76
in each iteration. 77
𝑋 =
𝑥
1,1
⋯ 𝑥
1,𝑗
𝑥
1,𝑛−1
𝑥
1,𝑛
𝑥
2,1
⋯ 𝑥
2,𝑗
⋯ 𝑥
2,𝑛
⋯ ⋯ 𝑥
𝑖,𝑗
⋯ ⋯
⋮ ⋮ ⋮ ⋮ ⋮
𝑥
𝑁−1,1
⋯ 𝑥
𝑁−1,𝑗
⋯ 𝑥
𝑁−1,𝑛
𝑥
𝑁,1
⋯ 𝑥
𝑁,𝑗
𝑥
𝑁,𝑛−1
𝑥
𝑁,𝑛
(1) 78
where 𝑋 is a set of the candidate solutions that are generated randomly 79
by using Eq. (2), 𝑥
𝑖,𝑗
denotes to the 𝑗
𝑡ℎ
position of the 𝑖
𝑡ℎ
solution, 𝑁 80
is the number of candidate solutions, and 𝑛 denotes to the dimension 81
size of the given problem,. 82
𝑥
𝑖𝑗
= 𝑟𝑎𝑛𝑑 × (𝑈𝐵 − 𝐿𝐵) + 𝐿𝐵, 𝑗 = 1, 2, … , 𝑛 (2) 83
where 𝑟𝑎𝑛𝑑 is a random value, 𝐿𝐵 and 𝑈𝐵 denote to the lower and 84
upper bound of the given problem, respectively. 85
2.3. Encircling phase (exploration) 86
In this section, the exploratory behaviour (encircling) of RSA is 87
introduced. According to the encircling behaviour, Crocodiles have two 88
movements during the encircling are high walking and belly walk. 89
These movements refer to different reigns, which commitment to the 90
exploration search (globally). Crocodile movements (high and belly 91
walking) cannot allow them to approach the target prey due to their 92
disturbance easily, unlike another search phase (hunting phase). Hence, 93
the exploration search discovers a wide search space; it can find the 94
density area maybe after several endeavours. In addition, the explo- 95
ration mechanisms (high and belly walking) are operated at this stage 96
of optimization to support the other phase (hunting/exploration) in the 97
search process through extensive and spread research. 98
The RSA can transfer between encircling (exploration) and hunting 99
(exploitation) search phases, this change between various behaviours 100
is done based on four conditions; divide the total number of iterations 101
into four parts. The exploration mechanisms of RSA explore the search 102
regions and approach to find a better solution based on two main search 103
strategies (high walking strategy and belly walking strategy). 104
This phase of searching is conditioned on two conditions. The high 105
walking movement strategy is conditioned by 𝑡 ≤
𝑇
4
, and the belly 106
walking movement strategy is conditioned by 𝑡 ≤ 2
𝑇
4
and 𝑡 >
𝑇
4
. 107
This means that this condition will be satisfied for almost the half 108
number of exploration iterations (High walking) and another half for 109
the Belly walking. These are two exploration search methods. Note, a 110
stochastic scaling coefficient is examined for the element to generate 111
more diverse-solutions and explore diverse-regions. We employed the 112
most straightforward rule, which can mimic the encircling behaviour of 113
Crocodiles. In this paper, the position updating equations are proposed 114
for the exploration phase as in Eq. (3). 115
𝑥
(𝑖,𝑗)
(𝑡+1) =
𝐵𝑒𝑠𝑡
𝑗
(𝑡) × −𝜂
(𝑖,𝑗)
(𝑡) × 𝛽 − 𝑅
(𝑖,𝑗)
(𝑡) × 𝑟𝑎𝑛𝑑, 𝑡 ≤
𝑇
4
𝐵𝑒𝑠𝑡
𝑗
(𝑡) × 𝑥
(𝑟
1
,𝑗)
× 𝐸𝑆(𝑡) × 𝑟𝑎𝑛𝑑, 𝑡 ≤ 2
𝑇
4
𝑎𝑛𝑑 𝑡 >
𝑇
4
(3) 116
L. Abualigah et al.
Fig. 1. Encircling the prey, when (𝑡 ≤
𝑇
2
).
where 𝐵𝑒𝑠𝑡
𝑗
(𝑡) is the 𝑗
𝑡ℎ
position in the best-obtained solution so far,1
𝑟𝑎𝑛𝑑 denotes to a random number between 0 and 1, 𝑡 is the number2
of the current iteration, and 𝑇 is the maximum number of iterations.3
𝜂
(𝑖,𝑗)
denotes to the hunting operator for the 𝑗
𝑡ℎ
position in the 𝑖
𝑡ℎ
4
solution, which is calculated using Eq. (4). 𝛽 is a sensitive parameter,5
controls the exploration accuracy (i.e., High walking) for encircling6
phase over the course of iterations, which is fixed equal to 0.1. Reduce7
function (𝑅
(𝑖,𝑗)
) is a value used to reduce the search area, which is8
calculated using Eq. (5). 𝑟
1
is a random number between [1 𝑁] and9
𝑥
(𝑟
1
,𝑗)
denotes to a random position of the 𝑖
𝑡ℎ
solution. 𝑁 is the number10
of the candidate solutions. Evolutionary Sense (𝐸𝑆(𝑡)) is a probability11
ratio takes randomly decreasing values between 2 and −2 throughout12
the number of iterations, which is calculated using Eq. (6).13
𝜂
(𝑖,𝑗)
= 𝐵𝑒𝑠𝑡
𝑗
(𝑡) × 𝑃
(𝑖,𝑗)
, (4)14
𝑅
(𝑖,𝑗)
=
𝐵𝑒𝑠𝑡
𝑗
(𝑡) − 𝑥
(𝑟
2
,𝑗)
𝐵𝑒𝑠𝑡
𝑗
(𝑡) + 𝜖
, (5)15
𝐸𝑆(𝑡) = 2 × 𝑟
3
×
1 −
1
𝑇
, (6)16
where, 𝜖 a small value and 𝑟
2
is a random number between [1 𝑁].17
In Eq. (6), 2 is used as a correlation value to give values between 2 and18
0, 𝑟
3
denotes to a random integer number between −1 and 1. 𝑃
(𝑖,𝑗)
is19
the percentage difference between the 𝑗
𝑡ℎ
position of the best-obtained20
solution and the 𝑗
𝑡ℎ
position of the current solution, which is calculated21
using Eq. (7).22
𝑃
(𝑖,𝑗)
= 𝛼 +
𝑥
(𝑖,𝑗)
− 𝑀(𝑥
𝑖
)
𝐵𝑒𝑠𝑡
𝑗
(𝑡) × (𝑈𝐵
(𝑗)
− 𝐿𝐵
(𝑗)
) + 𝜖
, (7)23
where 𝑀(𝑥
𝑖
), as in Eq. (7), is the average positions of the 𝑖
𝑡ℎ
solution,24
which is calculated using Eq. (8). 𝑈 𝐵
(𝑗)
and 𝐿𝐵
(𝑗)
are the upper and25
lower boundaries of the 𝑗
𝑡ℎ
position, respectively. 𝛼 is a sensitive pa-26
rameter, controls also the exploration accuracy (the difference between27
candidate solutions) for the hunting cooperation over the course of28
iterations, which is fixed equal to 0.1 in this paper.29
𝑀(𝑥
𝑖
) =
1
𝑛
𝑛
𝑗=1
𝑥
(𝑖,𝑗)
, (8)30
Fig. 2. Attacking the prey, when (𝑡 >
𝑇
2
).
2.4. Hunting phase (exploitation) 31
In this section, the exploitative behaviour (hunting) of RSA is in- 32
troduced. According to the hunting behaviour, Crocodiles have two 33
strategies during the hunting are hunting coordination and cooperation. 34
These strategies refer to different intensify techniques, which commit- 35
ment to the exploitation search (locally). Crocodile strategies (hunting 36
coordination and cooperation) allow them to approach the target prey 37
easily due to their intensification, unlike encircling mechanisms. Hence, 38
the exploitation search discovers the near-optimal solution, maybe after 39
several endeavours. Besides, the exploitation mechanisms are operated 40
at this stage of optimization to conduct an intensification search near 41
the optimal solution and emphasized communication between them. 42
The exploitation mechanisms of RSA exploit the search space and 43
approach to find the optimal solution based on using two main search 44
strategies (i.e., (1) hunting coordination and (2) hunting coopera- 45
tion), which is modelled as in Eq. (9). The searching in this phase 46
is conditioned as the hunting coordination strategy is conditioned by 47
𝑡 ≤ 3
𝑇
4
and 𝑡 > 2
𝑇
4
, otherwise, the hunting cooperation strategy is 48
performed, when 𝑡 ≤ 𝑇 and 𝑡 > 3
𝑇
4
. Note, stochastic coefficients are 49
considered to generate more dense-solutions and exploit the promising 50
regions (locally). We employed the most straightforward rule, which 51
can mimic the hunting behaviour of Crocodiles. In this paper, the 52
following position updating equations are proposed for the exploitation 53
phase (Eq. (9)): 54
𝑥
(𝑖,𝑗)
(𝑡 + 1) =
𝐵𝑒𝑠𝑡
𝑗
(𝑡) × 𝑃
(𝑖,𝑗)
(𝑡) × 𝑟𝑎𝑛𝑑, 𝑡 ≤ 3
𝑇
4
𝑎𝑛𝑑 𝑡 > 2
𝑇
4
𝐵𝑒𝑠𝑡
𝑗
(𝑡) − 𝜂
(𝑖,𝑗)
(𝑡) × 𝜖 − 𝑅
(𝑖,𝑗)
(𝑡) × 𝑟𝑎𝑛𝑑, 𝑡 ≤ 𝑇 𝑎𝑛𝑑 𝑡 > 3
𝑇
4
(9) 55
where 𝐵𝑒𝑠𝑡
𝑗
(𝑡) is the 𝑗
𝑡ℎ
position in the best-obtained solution so far, 56
𝜂
(𝑖,𝑗)
denotes to the hunting operator for the 𝑗
𝑡ℎ
position in the 𝑖
𝑡ℎ
57
solution, which is calculated using Eq. (4). 𝑃
(𝑖,𝑗)
is the percentage 58
difference between the 𝑗
𝑡ℎ
position of the best-obtained solution and the 59
𝑗
𝑡ℎ
position of the current solution, which is calculated using Eq. (7). 60
𝜂
(𝑖,𝑗)
denotes to the hunting operator for the 𝑗
𝑡ℎ
position in the 𝑖
𝑡ℎ
61
solution, which is calculated using Eq. (4). 𝜖 a small value. 𝑅
(𝑖,𝑗)
is a 62
value used to reduce the search area, which is calculated using Eq. (5). 63
In this respect, Figs. 1 and 2 show that when 𝑡 ≤
𝑇
2
, the encircling 64
phase (exploration) happens, otherwise; when 𝑡 >
𝑇
2
, the hunting phase 65
(exploitation) occurs to be close enough to prey when attacking. 66
