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Complex & Intelligent Systems (2025) 11:177
https://doi.org/10.1007/s40747-025-01791-2
ORIGINAL ARTICLE
Chaos-enhanced metaheuristics: classification, comparison, and
convergence analysis
Abdelhadi Limane1·Farouq Zitouni1·Saad Harous2·Rihab Lakbichi1·Aridj Ferhat1·
Abdulaziz S. Almazyad3·Pradeep Jangir4,5 ·Ali Wagdy Mohamed6,7
Received: 3 October 2024 / Accepted: 24 January 2025
© The Author(s) 2025
Abstract
Chaos theory, with its unique blend of randomness and ergodicity, has become a powerful tool for enhancing metaheuris-
tic algorithms. In recent years, there has been a growing number of chaos-enhanced metaheuristic algorithms (CMAs),
accompanied by a notable scarcity of studies that analyze and organize this field. To respond to this challenge, this paper
comprehensively analyzes recent advances in CMAs from 2013 to 2024, proposing a novel classification scheme that system-
atically organizes prevalent and practical approaches for integrating chaos theory into metaheuristic algorithms based on their
strategic roles. In addition, a list of 27 standard chaotic maps is explored, and a summary of the application domains where
CMAs have demonstrably improved performance is provided. To experimentally demonstrate the capability of chaos theory
to enhance metaheuristic algorithms that face common issues such as susceptibility to local optima, non-smooth transitions
between global and local search phases, and decreased diversity, we developed a chaotic variant of the recently proposed
RIME optimizer, which also encounters these challenges to some extent. We tested C-RIME on the CEC2022 benchmark
suite, rigorously analyzing numerical results using statistical metrics. Non-parametric statistical tests, including the Friedman
and Wilcoxon signed-rank tests, were also used to validate the findings. The results demonstrated promising performance,
with 14 out of 21 chaotic variants outperforming the non-chaotic variant, whereas the piecewise map-based variant achieved
the best results. In addition, C-RIME outperformed ten state-of-the-art metaheuristic algorithms regarding solution quality
and convergence speed.
Keywords Metaheuristic algorithms ·Chaos theory ·Chaotic maps ·Chaotic variant ·RIME optimizer
BAbdelhadi Limane
limane.abdelhadi@univ-ouargla.dz
Farouq Zitouni
zitouni.farouq@univ-ouargla.dz
Saad Harous
harous@sharjah.ac.ae
Rihab Lakbichi
lakbichi.rihab@univ-ouargla.dz
Aridj Ferhat
ferhat.aridj@univ-ouargla.dz
Abdulaziz S. Almazyad
mazyad@ksu.edu.sa
Pradeep Jangir
pkjmtech@gmail.com
Ali Wagdy Mohamed
aliwagdy@gmail.com
1Laboratory of Artificial Intelligence and Information
Technologies, Department of Computer Science and
Information Technology, Kasdi Merbah University, Ouargla,
Algeria
2Department of Computer Science, College of Computing and
Informatics, University of Sharjah, Sharjah, UAE
3Department of Computer Engineering, College of Computer
and Information Sciences, King Saud University, 11543
Riyadh, Saudi Arabia
4Department of Biosciences, Saveetha School of Engineering,
Saveetha Institute of Medical and Technical Sciences,
Chennai 602105, India
5Innovation Center for Artificial Intelligence Applications,
Yuan Ze University, Taoyüan 320315, Taiwan
6Operations Research Department, Faculty of Graduate Studies
for Statistical Research, Cairo University, Giza 12613, Egypt
7Applied Science Research Center, Applied Science Private
University, Amman 11931, Jordan
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177 Page 2 of 42 Complex & Intelligent Systems (2025) 11:177
Introduction
Background
Metaheuristic algorithms (MAs) represent a popular approach
for solving various benchmark and real-world optimization
problems across a wide range of disciplines, including the
Internet of Things [150], machine learning [119], civil engi-
neering [81], and mechanical engineering [7]. In many of
these optimization problems, finding the absolute optimal
solution (global optimum) is not always required. Finding
a solution close to the best (near-optimal) often proves
sufficient. MAs prioritize finding good-quality solutions effi-
ciently, trading off optimality for faster computation time.
Under the umbrella of the No-Free-Lunch theory [181],
which underscores the absence of a universally optimal
problem-solver, researchers have developed an increasing
number of MAs inspired by diverse natural phenomena, from
animal behaviours to celestial mechanics and in a burst of
innovation, last year (2023) witnessed the emergence of over
100 novel MAs.
As the number of MAs continues to increase, several
taxonomies have been proposed. These taxonomies offer
researchers and practitioners a range of benefits, including
improved organization of the field, and a basis for conduct-
ing fair comparisons between different algorithms. One of
the earliest known taxonomies is based on the number of
solutions used in the search process: (i) Trajectory-based
and (ii) Population-based. Trajectory-based algorithms have
seen relatively limited new developments in recent years.
The most popular taxonomy for MAs categorizes them based
on their source of inspiration. Many variants of this taxon-
omy have emerged. For instance, according to [112], MAs
can be categorized into: (i) Swarm Intelligence: this cate-
gory accounts for the vast majority of MAs, (ii) Physics
and Chemistry, (iii) Social Human behaviour,(iv)Breeding-
based Evolution,(v)Plants Based,(vi)Miscellaneous:this
category includes any algorithm not classified elsewhere.
Another taxonomy, proposed in the same study by Molina
et al. [112], focuses on the behaviour of the search opera-
tors. This taxonomy categorizes algorithms according to how
they generate new solutions. One category includes algo-
rithms that rely on mutation or shifting existing solutions
(Differential vector movement). The other category includes
algorithms that combine multiple existing solutions to cre-
ate new ones (Solution creation). Another recent study by
Rajwar et al. [136] proposed a simple taxonomy based on
the number of parameters required by the algorithm, exclud-
ing common parameters such as the population size and the
number of iterations. This ranges from zero-parameter (Free-
parameter based) algorithms to algorithms with up to five
parameters (Penta-parameter based). A final category exists
for algorithms with more than five parameters.
The development of MAs is often an iterative process.
After designing an initial algorithm, researchers can identify
areas for improvement, such as limitations in exploitation or
exploration capabilities. To address these weaknesses, new
variants of the algorithm are created through the applica-
tion of diverse techniques. These techniques include transfer
functions [98], Lévy flights, opposition-based learning, the
Nelder-Mead method, elitist strategies, and various mutation
operators [171]. Chaos theory, which is the focus of this arti-
cle, is another such technique with significant potential to
enhance MAs.
Chaos theory has its origins in the 1960s, with the term
itself coined by Li and Yorke [94] in 1975. Chaos is often
associated with dynamical systems. These systems, gov-
erned by defined rules, can exhibit surprisingly complex and
unpredictable behaviour. Even slight initial variations signif-
icantly impact long-term outcomes in these systems. Edward
Lorenz, the spiritual father of this theory, made groundbreak-
ing discoveries during his research on weather prediction. His
experiments revealed significant discrepancies in long-term
forecasts due to minor variations in initial data. This phe-
nomenon, later popularized as the butterfly effect, became a
cornerstone of chaos theory.
A unique set of properties characterizes chaos: (i) even
minor changes in the initial state of a chaotic system can lead
to dramatically different outcomes over time, (ii) while not
truly random, chaotic systems exhibit behaviour that appears
statistically similar to random processes and (iii) over time, a
chaotic system explores all possible states within its defined
boundaries, never repeating a specific trajectory, a property
called ergodicity.
Due to these properties, chaos theory is applicable in a
wide range of fields such as: control theory, cryptography,
and network science [104]. As this article will explore, chaos
theory also holds particular promise for enhancing optimiza-
tion algorithms, specifically MAs.
Chaos-enhanced metaheuristic algorithms (CMAs) pro-
pose that integrating chaos theory can improve algorithmic
performance in several ways. Firstly, by introducing a degree
of randomness, chaos can help CMAs escape local optima,
preventing them from becoming trapped in suboptimal solu-
tions [75,79,183,202]. Secondly, chaos can enhance the
diversity of solutions, thus enriching the solution space [75,
118]. Thirdly, the incorporation of chaos may lead to acceler-
ated convergence, enabling CMAs to reach optimal solutions
more efficiently [160]. Lastly, chaos theory is believed to
provide a mechanism for achieving a better balance between
exploration and exploitation within the search process [23,
120]. However, CMAs have some limitations. One key draw-
back is that researchers often lack assurance at the outset
that integrating chaos theory will definitively enhance per-
formance. In certain cases, the application of chaos may
have minimal or even negligible impact on the optimization
123
Complex & Intelligent Systems (2025) 11:177 Page 3 of 42 177
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
Years
0
50
100
150
200
250
300
Number of publications
Article Conference paper Other
Fig. 1 Number of published documents containing the term ‘Chaos’ or
‘Chaotic’ and at least one of the terms ‘Nature-Inspired Algorithm’,
‘Evolutionary Algorithm’, ‘Swarm Intelligence’, or ‘Metaheuristic
Algorithm’ in title, abstract, or keywords. Data was retrieved from Sco-
pus on June 13, 2024
results. Additionally, selecting an appropriate chaotic map
from a broad set of options can be computationally intensive
and may require extensive trial and error.
The increasing interest and development in chaotic meta-
heuristic algorithms can be observed in Fig. 1, which displays
the number of documents published each year from 2013 to
June 2024. The data is categorized by type: journal paper,
conference paper, and other publications. Notably, there has
been a significant rise in publications from 2020 onwards,
peaking in 2022 and 2023 with over 250 publications yearly.
This trend indicates a growing recognition of the poten-
tial benefits of chaos theory in metaheuristic optimization,
reflecting the expanding research efforts and applications
in this field. The research line from 2013 to 2024 can be
partitioned into three interrelated sub-periods, each mark-
ing a distinct phase in the evolution of CMAs. The first
sub-period, spanning 2013 and 2014, serves as an exten-
sion of earlier efforts, during which researchers primarily
focused on developing chaotic variants of well-established
algorithms such as genetic algorithms (GA) [50] and parti-
cle swarm optimization (PSO) [57]. The second sub-period,
beginning around 2015, reflects a notable shift in focus as
the research community started to recognize the potential
of integrating chaos theory into metaheuristic algorithms.
This phase is distinguished by foundational studies, includ-
ing surveys, reviews, and bibliographic analyses of CMAs
[55]. While these efforts provided valuable insights and laid
the groundwork for future research, the overall number of
contributions during this period remained limited. The final
sub-period, spanning from 2020 to the present, represents a
phase of rapid growth and innovation. This explosion phase is
closely tied to the widespread advancements in metaheuris-
tic algorithm development. During this period, the scope
of chaotic metaheuristics has expanded significantly, mov-
ing beyond well-established algorithms to include newly
developed ones. Remarkably, some algorithms now feature
multiple chaotic variants introduced by different researchers,
highlighting the increasing interest and diversification in this
field.
Overview of previous analyses of CMAs
While numerous studies have developed chaotic variants of
several MAs, there is a noticeable gap in the comprehensive
analysis of these studies. Only a limited number of articles
have addressed this topic so far. A comprehensive review
article focusing on the integration of chaos theory into the
firefly algorithms, a particular swarm-based metaheuristic
algorithm, is conducted in Fister Jr et al. [55]. The review
extensively categorized chaos-based firefly algorithms based
on their type (modified or hybrid), the parameters adjusted
using chaos, and their application domain (global optimiza-
tion or specific engineering problems). Furthermore, their
review classified these algorithms according to the chaos
map employed. The findings indicate that the logistic map
was the most frequently utilized. The study in Kaveh [82]
presented a concise review of chaos-embedded metaheuris-
tic algorithms. This study reviewed chaotic systems and
listed nine specific chaos maps. In addition, the work intro-
duced a novel particle swarm optimization (PSO) scheme
that integrates chaos theory. The efficacy of the proposed
scheme was evaluated using two large-scale test problems.
The same domain is briefly reviewed in ATALI et al. [28].
Their review classified the examined papers based on the
underlying inspiration for the algorithms employed. These
categories included swarm-based algorithms, nature-inspired
and evolutionary algorithms, bioinspired algorithms, and
physics-based algorithms. Recently, Pluhacek et al. [131]
performed a comprehensive bibliographic analysis focus-
ing on the integration of chaos maps with metaheuristic
algorithms. Then, they highlighted three well-established
algorithms: Genetic Algorithm (GA), differential evolution
(DE), and particle swarm optimization (PSO).
Unlike previous studies that classified algorithms based
on inspiration such as swarm-based, bioinspired, or physics-
based algorithms, our classification takes a more targeted
approach by focusing specifically on the role of chaos in
CMAs. This allows us to offer a unique perspective by catego-
rizing algorithms based on how chaos is utilized within them,
rather than relying on a general classification. Furthermore,
whereas some previous studies are restricted to bibliographic
analyses of a few well-known algorithms, such as GA, PSO,
and DE, our approach is more comprehensive. By analyzing
a wider range of CMAs, we deliver a deeper and more exten-
123
177 Page 4 of 42 Complex & Intelligent Systems (2025) 11:177
sive understanding of the diverse ways chaos can enhance
metaheuristic performance, setting our work apart in both
scope and practical relevance.
In addition, this work goes beyond analyzing existing
algorithms by contributing to developing a new chaotic vari-
ant of the RIME optimizer [158]. This effort aligns with our
objective to bridge the gap between theoretical understanding
and practical application of CMAs. While the proposed clas-
sification scheme provides a comprehensive framework for
examining how chaos theory integrates with metaheuristic
algorithms, the experiments with the chaotic RIME opti-
mizer demonstrate the practical potential of chaos theory to
improve solution quality and accelerate algorithm conver-
gence.
Contribution of the paper
This article aims to offer an in-depth analysis of recent
research on CMAs. A thorough examination of existing lit-
erature from 2013 to 2024, was conducted to achieve this.
This involved assembling and meticulously examining rele-
vant articles. The key contributions of this paper are:
•This paper presents an extensive list of 27 chaos maps,
detailing their governing equations and definition
domains. To the best of our knowledge, no existing
study has reached this number. This comprehensive list-
ing serves as a valuable resource for researchers who aim
to utilize a wide variety of chaos maps in their optimiza-
tion algorithms.
•We propose a classification scheme that categorizes the
CMAs into six distinct categories. This scheme provides
a more specific and useful framework for understanding
CMAs, unlike previous surveys which often only mention
the substitution of randomness with chaos in a general
way. This paper analyses over 70 related studies, and
categorizes them according to our proposed classification
scheme.
•We provide a detailed summary of the applications of
CMAs across different fields. This summary demon-
strates the practical utility and effectiveness of CMAs
in solving real-world problems.
•We develop a novel chaotic variant of the RIME opti-
mizer, testing 21 different chaotic maps and evaluating
its performance using the CEC2022 benchmark suite.
Organization of the paper
The remainder of this paper is organized as: section two
presents the foundation of CMAs by exploring commonly
used chaos maps. Section three categorizes and analyzes
the various approaches for integrating chaos theory into
metaheuristic algorithms. Six distinct approaches will be
identified and reviewed in dedicated subsections. Subse-
quently, section four explores the diverse applications of
CMAs across different fields. To further investigate the
potential of CMAs, section five introduces a novel variant, the
Chaotic RIME optimizer. Section six then details the exper-
iments conducted to evaluate the performance of C-RIME,
and compare it to existing algorithms. Finally, section seven
summarizes this comprehensive review.
Chaotic maps
The reviewed papers consistently demonstrate the integration
of chaos theory into metaheuristic algorithms through the
use of chaotic sequences generated by discrete-time chaotic
systems (DTCS). These DTCS, despite their inherent com-
plexity, can be surprisingly simple to model mathematically.
A general representation of a DTCS is expressed by Eq. (1).
n+1=τ(
n), n={ωi
n},1≤i≤d(1)
where:
•n+1: The state of the system at time n+1, represented
as a vector or set of dchaotic variables.
•τ: The nonlinear map or function that updates the state
of the system from time nto n+1.
•n: The state of the system at time n, represented as a
vector or set of dchaotic variables.
•ωi
n:The
i-th
component of the state vector at time n.
Most chaotic maps used in integrating chaos theory into
MAs are either one- or two-dimensional. The following sub-
sections offer a detailed overview of these commonly utilized
discrete chaotic maps from 2013 to 2024, including their
equations and domains of definition. Figure 2visualizes these
chaos maps normalized to the range (0, 1), with each iterated
over 250 iterations.
Logistic map
The logistic map, a one-dimensional polynomial map intro-
duced by May [107], demonstrated how a very simple non-
linear dynamical equation could lead to complex behaviour
and chaotic dynamics. The formula is shown in Eq. (2)[122,
141].
ωn+1=α·ωn·(1−ωn)(2)
where ωn∈(0,1),ω
0/∈1
4,1
2,3
4and αis a control param-
eter set to 4.
123
Complex & Intelligent Systems (2025) 11:177 Page 5 of 42 177
Fig. 2 Visualization of different chaos maps
123
177 Page 6 of 42 Complex & Intelligent Systems (2025) 11:177
Chebyshev map
The equation of the chebyshev map is shown in Eq. (3)[60,
122,141].
ωn+1=cos (α·arccos (ωn)) (3)
The chebyshev map iterates within the bounds of (−1,1).
Sine map
An example of unimodal chaotic maps is the sine map.Itis
defined by Eq. (4)[60,141].
ωn+1=α
4sin (π·ωn)(4)
This map is defined for ωn∈0,α
4, where 0 <α≤4
Singer map
The singer map is another example that demonstrates how
simple deterministic equations can yield complex, seemingly
random sequences that are highly sensitive to initial con-
ditions. Equation (5) formulates this one-dimensional map
[122,141,166].
ωn+1=α·(7.86 ·ωn−23.31 ·ω2
n
+28.75 ·ω3
n−13.302875 ·ω4
n)(5)
Here, αis a control parameter that can take any value in the
range [0.9,1.08]and ωnis restricted to the interval (0,1).
Sinusoidal map
The sinusoidal map is one of the simplest chaotic maps. The
map equation is expressed in Eq. (6)[122,141].
ωn+1=a·ω2
n·sin (π·ωn)(6)
A simplified form of this equation is obtained when a=2.3
and ω0=0.7, resulting in:
ωn+1=sin (π·ωn)(7)
The sinusoidal map operates on values strictly between
zero and one.
Circle map
The circle map is another commonly used one-dimensional
chaotic map, described by Eq. (8)[102,122,141].
ωn+1=ωn+β−α
2·πsin (2·π·ωn)(
mod 1)(8)
It exhibits chaotic behaviour when α=0.5 and β=0.2.
The chaotic sequence generated by this map falls within the
interval (0,1).
The circle map can also be represented as shown in Eq.
(9)[55]:
ωn+1=ωn+β−(α−2·π)sin (2·π·ωn)(
mod 1)
(9)
ICMIC map
The iterative chaotic map with infinite collapses (ICMIC) is
represented by Eq. (10)[60,67].
ωn+1=sin α
ωn(10)
Here, α∈(0,∞)is a control parameter that influences the
behaviour of the map. Unlike many of the reviewed chaotic
maps in this paper, the ICMIC map exhibits ergodicity over
the interval (−1,1).
Another form of the iterative chaotic map is given by Eq.
(11)[141].
ωn+1=sin α·π
ωn(11)
In this form, the range of parameter αis reduced to values in
(0,1)rather than (0,∞).
Cubic map
The cubic map is a one-dimensional map that generates
chaotic sequences in (0,1). Its equation can be defined as
shown in Eq. (12)[22,60].
ωn+1=ρ·ωn·1−ω2
n(12)
Typically, when ρ=2.59, the cubic map exhibits chaotic
dynamics.
Sawtooth map
The sawtooth map is a specialized form of the Bernoulli shift
map when the parameter pis set to 0.5. The map acts on the
unit interval (0,1)and it is defined by Eq. (13)[67,168].
ωn+1=2·ωn(mod 1)(13)
123
Complex & Intelligent Systems (2025) 11:177 Page 7 of 42 177
Neuron map
The neuron map can be represented as shown in Eq. (14)[60,
137].
ωn+1=α−2·tanh (β)·e−3·ω2
n(14)
Here, αand βare two equation parameters, when α=0.5
and β=5. This map generates chaotic sequence in −3
2,1
2.
Tent m a p
The tent map is a one-dimensional piecewise linear map,
which shows linearity within its segments but not as a whole.
It combines features from both the bit shift map and the logis-
tic map with α=4. The tent map is defined by Eq. (15)[124].
ωn+1=μ·ωn,if ωn<0.5
μ·(1−ωn), if ωn≥0.5(15)
The tent map is defined for ωn∈(0,1)and it exhibits chaotic
behaviour within specific ranges of parameters, such as when
μ=2.
Gauss map
The Gauss map, alternatively referred to as the mouse map in
certain contexts, represents a fundamental one-dimensional
and nonlinear piecewise-defined dynamical system capable
of generating chaotic sequences. Its formulation is captured
by Eq. (16)[122,141] for values of ωnvarying between 0
and 1.
ωn+1=0ωn=0
1
ωn(mod1)Otherwise (16)
Intermittency map
The intermittency map is defined by Eq. (17)[67].
ωn+1=+ωn+σ·ωκ
n0<ω
n≤p
ωn−p
1−pp<ω
n<1(17)
where p∈(0,1),σ =1−−p
pκ, p,κ =2, and ωn∈
(0,1).
Bernoulli shift map
The Bernoulli shift map is a one-dimensional map character-
ized by its simplicity and it consists of two linear segments.
This map has only one parameter, denoted as p, which is
defined within the interval (0,1). The map is modelled by
Eq. (18)[184].
ωn+1=⎧
⎨
⎩
ωn
1−p0<ω
n≤1−p
ωn−(1−p)
p1−p<ω
n<1(18)
The parameter pinfluences the behaviour of the map, dic-
tating the transition points between the two linear segments.
The Bernoulli shift map is defined on the interval (0,1).
Kent map
The Kent map is a one-dimensional discrete dynamical sys-
tem defined within the unit interval (0,1). Its mathematical
formulation is represented by the Eq. (19)[168,199].
ωn+1=⎧
⎨
⎩
ωn
p0<ω
n≤p
1−ωn
1−pp<ω
n<1(19)
where p∈(0,1)is a control parameter that governs the
map’s behaviour.
PLC map
The piecewise linear chaotic map is a one-dimensional
chaotic system characterized by two linear piecewise seg-
ments. It is defined by Eq. (20)[151].
ωn=ωn
p0<ω
n<p
(1−ωn)·(1−p)p≤ωn<1(20)
Here, ωnand pare bounded within the interval (0,1).
Liebovitch map
The Liebovitch map, proposed by Liebovitch and Toth, is a
three-piecewise linear segment map defined by Eq. (21)[67]:
ωn+1=⎧
⎪
⎨
⎪
⎩
α·ωn0<ω
n≤p1
p2−ωn
p2−p1p1<ω
n≤p2
1−β·(1−ωn)p2<ω
n<1
(21)
Here, p1and p2are parameters in the interval (0,1)with
p1<p2,αand βare calculated as shown in Eqs. (22) and
(23).
α=p2
p1
(1−(p2−p1)) (22)
β=1
p2−1(( p2−1)−p1(p2−p1)) (23)
The map operates within the interval (0,1).
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177 Page 8 of 42 Complex & Intelligent Systems (2025) 11:177
Piecewise map
The piecewise map is defined by Eq. (24)[67,168]:
ωn+l=⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
ωn
p0≤ωn<p
ωn−p
0.5−pp≤ωn<1
2
1−p−ωn
0.5−p
1
2≤ωn<1−p
1−ωn
p1−p≤ωn<1
(24)
Here, p∈0,1
2, and the generated chaotic sequence falls
within the interval (0,1).
Zaslavskii map
In 1978, George M. Zaslavsky defined one of the funda-
mental two-dimensional dynamical systems, known as the
Zaslavskii map [192]. This map is formulated as shown in
Eq. (25)[168].
ωn+1=ωn+v·(1+μ·φn)
+·v·μ·cos (2·π·ωn)(
mod 1)
φn+1=e−r·φn+cos ·2·π·ω1
n(mod 1)
(25)
Here, r,v and represent three parameters of the equa-
tion. The map demonstrates chaotic behaviour under specific
conditions, namely when r=3,v =400
3, and =0.3. Addi-
tionally, μ=1−e−r
r.
Arnold’s cat map
The Arnold’s cat map, named after the renowned mathe-
matician Vladimir Arnold, is a classic example of a two-
dimensional map used in the study of dynamical systems
and chaos theory. The map is defined by Eq. (26)[129,130,
155].
ωn+1=ωn+φn(mod 1)
φn+1=ωn+αφn(mod 1)(26)
where the parameter αis set to 0.1.
Sinai map
The sinai map is formulated by Eq. (27)[129,130,168].
ωn+1=ωn+φn+α·cos (2·π·φn)(
mod 1)
φn+1=ωn+2·φn(mod 1)(27)
Here, αis a parameter set to 0.1.
Hénon map
Hénon [69] introduced the two-dimensional Hénon map as
a reductionist model with essential properties similar to the
Lorenz system. This map operates by iteratively mapping
points within a plane back onto itself. The Hénon map is
defined by Eq. (28)[168].
ωn+1=1−α·ω2
n+φn
φn+1=β·ωn
(28)
The Hénon map can also be represented as a two-step recur-
rence relation:
ωn+1=1−α·ω2
n+β·ωn−1(29)
Typically, suggested values for the parameters αand βare 1.4
and 0.3, respectively. These values are often used to demon-
strate the chaotic behaviour of the Hénon map.
Lozi map
Lozi [103] introduced a two-dimensional chaotic map named
after himself, known as the Lozi map.TheLozi map is a
variation of the Hénon map, where the quadratic term (ω2
n)
is replaced by the absolute value (|ωn|). Mathematically, the
Lozi map can be expressed by Eq. (30)[129,130,172].
ωn+1=1−α·|ωn|+φn
φn+1=β·ωn
(30)
Here, aand bare control parameters. Typical values used to
demonstrate chaotic behaviour include α=1.7 and β=0.5.
Ikeda map
The Ikeda map, discovered by Ikeda [77], is a discrete-time
dynamical system. The 2D form of the Ikeda map is detailed
in Dressler’s work [48] and is formulated by Eq. (31)[168].
ωn+1=1+α·(ωn·cos (t)−φn·sin (t))
φn+1=α·(ωn·sin (t)+φn·cos (t)) (31)
where t=0.4−6
(1+ω2
n+φ2
n)and α=0.7.
Tinkerbell map
The tinkerbell map is a two-dimensional discrete dynamical
system defined by Eq. (32)[129,130,172].
ωn+1=ω2
n−φ2
n+α·ωn+β·φn
φn+1=2·ωn·φn+δ·ωn+σ·φn
(32)
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Complex & Intelligent Systems (2025) 11:177 Page 9 of 42 177
where α,β,δand σare the system parameters. Common
values found in the literature include α=0.9, β=−0.6013,
δ=2, and σ=0.5.
Dissipative standard map
The dissipative standard map is encapsulated in Eq. (33)
[129,130,172].
ωn+1=ωn+φn(mod 2π)
φn+1=α·φn+β·sin (ωn)(
mod 2π)(33)
The dissipative standard map demonstrates chaotic dynam-
ics for parameter values α=0.6 and β=8.8.
Burgers map
The Burgers’ map is a well-known system of two coupled,
non-linear differential equations used to represent a variety
of dynamical phenomena. It is defined by Eq. (34)forthe
state variables ωnand φnat the (n+1)-th time step [129,
130,172].
ωn+1=αωn−φ2
n
φn+1=βφn+ωnφn
(34)
Here, αand βare system parameters that influence the map’s
behaviour. A common set of parameter values found in the
literature is α=0.75 and β=1.75.
Common issues in chaos maps
Chaos maps, despite their fascinating properties, can present
some challenges and require consideration when studying
them. Here are some common issues to keep in mind:
•The first issue: while chaos maps are inherently sensitive
to initial conditions, certain maps can exhibit periodic
behaviour under specific conditions, rendering the gen-
erated sequence non-chaotic. To address this and ensure
the desired chaotic behaviour, one common approach
involves injecting a small amount of noise into the sys-
tem. This can be achieved by adding a term proportional
to a random number between 0 and 1, scaled by a small
positive constant, as shown in Eq. (35).
ωn=ωn+r. (35)
Here, ωrepresents the chaotic variable, ris a random
number between 0 and 1. Finally, is a small positive
number. This injects a slight randomness into the system,
helping to nudge it away from falling into periodic orbits
and promoting more chaotic behaviour.
•The second issue: another crucial aspect to consider is
normalization. Since different chaotic maps can have
distinct output ranges (e.g., (−1,1),(−1.5,0.5)), nor-
malization helps to ensure that the generated chaotic
sequences align with the desired range. This is often
necessary for further analysis or integration with other
systems that operate within specific value boundaries.
Min-max scaling to a specific range (e.g., 0–1) is a com-
mon normalization technique.
In addition, it is crucial to note that the majority of chaos
maps, used in CMAs, have a discrete nature. A limited num-
ber of continuous chaos maps has been used by researchers
to enhance the performance of metaheuristic algorithms.
Among these systems, the Lorenz system [50]isthemost
prominent.
Approaches of integrating chaos theory into
metaheuristic algorithms
This section presents a novel classification of prevalent and
effective approaches for integrating chaos theory into meta-
heuristic algorithms. The classification delineates six distinct
categories: (i) Chaotic population initialization strategies;
(ii) Chaos-based boundary handling technique; (iii) Chaos-
enhanced search operators; (iv) Chaos-driven high-level
strategies; (v) Chaos local search strategies and (vi) Chaos
as a source of inspiration.
Chaotic population initialization strategies
The essence of most metaheuristic algorithms is to design
robust search operators. The performance of the devel-
oped algorithm depends not only on these operators but
also on the initialization phase. Efficient initialization is
often instrumental in achieving fast convergence and opti-
mal performance, whereas poor initialization can lead to
slow convergence and unsatisfactory results. Numerous tech-
niques have been proposed in the literature to address this
critical phase. Among these,
random initialization
stands out
as the simplest and most widely utilized approach. In ran-
dom initialization, the initial population can be generated
using Eq. (36).
xn,0
d=x−
d+rn
d·(x+
d−x−
d), 1≤d≤D,1≤n≤N
(36)
Here, xn,0
drepresents the value of the
d-th
variable for n-th
individual in the initial population, with x+
dand x−
ddenoting
the upper and lower bounds of the d-th variable, respec-
tively. The term rn
dcorresponds to a randomly generated
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177 Page 10 of 42 Complex & Intelligent Systems (2025) 11:177
value uniformly distributed between 0 and 1, Nrepresents the
population size, and Dis the total dimensionality (number
of
decision variables
) in the optimization problem.
Chaotic initialization introduces a key modification to the
standard initialization process. The random variable (rn
d)in
Eq. (36) is replaced with a chaotic variable (cn
d)generated by
a pre-selected chaotic map. All other elements of the equation
(upper and lower bounds, dimensionality, population size)
remain the same. The new formulation is given by Eq. (37).
xn,0
d=x−
d+cn
d·(x+
d−x−
d), 1≤d≤D,1≤n≤N
(37)
In simpler terms, chaotic initialization involves two key
steps:
•Generating chaotic sequences: This step involves the
creation of sequences containing chaotic numbers. The
initial value for each sequence is chosen randomly
between 0 and 1.
•Mapping the chaotic sequences: These sequences are
then mapped to the search space using Eq. (37). The
upper and lower bounds (x+
dand x−
d) of each variable
define the valid range for the mapping.
Importantly, tackling the small periodic cycles is crucial to
ensure the effectiveness of this initialization approach. The
detailed steps for chaos-based initialization are outlined in
Algorithm 1.
Algorithm 1: Chaos-based initialization
1Set parameters Nand D;
2for n←0to Ndo
3for d←1to Ddo
4if n=0then
// Generate the heads
5c0
d←rand(0,1);
6else
// Prevent the periodic cycles
7if c is plunged into a periodic cycle then
8perform small random perturbation;
9end
// Generate the chaotic number
10 cn
d←ChaosMapcn−1
d;
// Perform mapping into the search
space
11 xn,0
d←x−
d+cn
d·(x+
d−x−
d);
12 end
13 end
14 end
Indeed, chaos maps create sequences that are pseudo-
random yet highly diverse, yielding a structured spread of
initial solutions. This quality enhances the initial search
diversity, helping the algorithm to avoid premature con-
vergence and achieve faster, more reliable convergence.
Numerous studies support that chaotic sequences cover the
search space more uniformly than using the traditional ran-
dom initialization, providing a more advantageous starting
point for optimization. This attribute makes chaotic initializa-
tion particularly effective for high-dimensional and complex
problems where robust exploration is critical [143]. This
approach, which introduces chaos into the initialization phase
is exemplified in the work conducted by Ebrahimzadeh and
Jampour [50]. In their study, the genetic algorithm, a classi-
cal evolutionary metaheuristic algorithm, was initialized with
chaotic sequences generated by the Lorenz chaotic system.
Additionally, the inherent randomness in the key operators,
crossover and mutation, was further enriched by incorporat-
ing outputs from the Lorenz chaotic system. Experimental
results showcased an accelerated convergence rate for the
chaotic GA compared to the traditional GA. The chaotic dif-
ferential bee colony algorithm, a hybrid algorithm, utilized
the logistic map to generate initial candidate solutions, as
demonstrated by Lu et al. [106]. This utilization aimed to
enhance solution diversity and accelerate convergence speed.
Huang et al. [75] incorporated chaos theory into the cuckoo
search algorithm (CS) across three stages, including the ini-
tialization phase. In this phase, they employed five different
chaotic maps (logistic, tent, Gauss, sinusoidal, and circle)
to initialize the positions of host nests. In similar work [24],
ten chaos maps have been employed to generate initial butter-
flies of the butterfly optimization algorithm (Chaotic BOA1).
Suresh and Lal [160] investigated chaotic initialization for
DPSO, evaluating 10 chaotic maps and selecting the logis-
tic map for its effectiveness. Their proposed chaotic DPSO
outperforms five existing chaotic variants in optimization
algorithms. Tian and Shi [164] proposed a modification to
the PSO algorithm to enhance its stability. Their approach
involved utilizing the logistic chaos map for population ini-
tialization. The suboptimal convergence rate and heightened
likelihood of becoming trapped in local optima, which are
characteristic of the whale optimization algorithm (WOA),
prompted Chen et al. [41] to address these issues by inte-
grating quasi-oppositional based learning into the solution
update mechanism and employing chaos theory to initialize
the population.
In addition to the effectiveness demonstrated in men-
tioned earlier algorithms, chaotic initialization has been
successfully applied to a wide range of other optimization
algorithms, including artificial bee colony algorithm [93],
bacterial foraging optimization [197], ant lion optimizer
algorithm [202], moth-flame optimization algorithm [70],
particle swarm optimization [165], multi-verse optimiza-
tion algorithm [159], modified whale optimization algorithm
[96], grasshopper optimization algorithm [49], multi-swarm
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Complex & Intelligent Systems (2025) 11:177 Page 11 of 42 177
whale optimizer [177], whale optimization algorithm [46],
adaptive chimp optimization algorithm [179], archimede
optimization algorithm [35].
Chaos-based boundary handling technique
Respecting boundary constraints is crucial in metaheuris-
tic algorithm design. Each variable xdtypically has upper
and lower bounds x+
dand x−
d, respectively, such that x−
d≤
xd≤x+
d. The initial population of solutions is often gen-
erated using Eq. (36), which inherently respects this type
of constraint. However, issues may arise when updating
the population using certain operators, leading to exceed-
ing the boundaries. Several strategies exist to handle this
constraint. A common approach is random reinitialization,
where solutions exceeding the boundaries are replaced by
others randomly generated within the feasible space.
Chaos has been successfully integrated into the mecha-
nisms of boundary handling in various research studies by
using Eq. (37) to generate solutions in the search space chaot-
ically instead of the solutions that violate the boundary limits.
For instance, Huang et al. [75] introduced a chaos-adjusted
boundary handling approach for the cuckoo search optimiza-
tion algorithm. In their work, they effectively substituted
values that violate boundary constraints with new ones gen-
erated chaotically using Eq. (37). This methodology has also
been applied to moth-flame optimization, as demonstrated by
Hongwei et al. [70], to manage the positions of moths flying
out of the boundaries.
Chaos-enhanced search operators
The structure of metaheuristic algorithms primarily com-
prises a series of swarm/evolutionary operators that guide
candidate solutions during their exploration and exploitation
of the search space. These operators are applied in a spe-
cific order and under particular conditions unique to each
addressed algorithm. Essentially, a swarm/evolutionary oper-
ator is directly responsible for transitioning a population X
from state X(s)to the next stage X(s+1)by making specific
modifications to the candidate solutions within the popula-
tion. Mathematically, this can be encapsulated by Eq. (38).
X(s+1)=(X(s), P,) (38)
Here, X(S)is the current population, X(s+1)repre-
sents the next stage of the population after applying the
swarm/evolutionary operator, denotes a nonlinear map-
ping function that orchestrates the transition from X(s)to
X(s+1),P={p1,..., pl}forms a subset of non-random
parameters which can either be constant values or prede-
fined parameters and ={ψ1,...,ψ
m}represents a subset
of random variables injecting stochasticity into the search
operator.
Integrating chaos theory into the search operators can
influence both and P. This entails either substituting ran-
dom parameters within or fine-tuning other parameters
within P.
Random parameter substitution
An effective strategy to enhance metaheuristic algorithms
with chaos theory involves substituting either partially or
entirely the set of random parameters in Eq. (38) with
sequences generated from chaotic systems. This substitution
results in Eq. (39).
X(s+1)=(X(s), P,,ζ) (39)
where ={ψ1,...,ψ
m1}represents the set of random
parameters, while ζ={z1,...,zm2}is a set of chaotic
parameters. The total number of parameters remains con-
stant, i.e., m1+m2=m. When the random parameters are
entirely replaced by the chaotic sequences (m2=m), the set
becomes empty (=∅). The remaining components in
the equation, including X(s),X(s+1),and Phave been
defined earlier in the preceding context.
Drawing on an extensive range of sources, researchers
have utilized chaos theory to replace certain random param-
eters of displacement operators with chaotically generated
variables.
Saremi et al. [142] tested three chaos maps—circle map,
sine map, and tent map—as replacements for the random
parameter controlling the Cbest parameters in the krill herd
algorithm (KHA) for optimizing four benchmark functions.
Also, during that year Ghasemi et al. [64] introduced the
CIWO algorithm, an enhanced version of the invasive weed
optimization, which utilizes various chaos maps instead of a
normal distribution. This algorithm was applied to different
optimal power flow problems.
In their study, Pluhacek et al. [129] introduced the Chaotic
particle swarm optimization. They employ an ensemble of
six two-dimensional, discrete chaotic maps as generators of
pseudo-random numbers. They applied this approach to opti-
mize 28 benchmark functions.
In the work by Miti´cetal.[110], several one-dimensional
chaos maps are employed with the grey wolf optimizer,
where the Gaussian distribution in the exploration parame-
ter Cequation is substituted by chaotic sequences generated
by these maps. The proposed algorithm showed competitive
results for some optimization problems in the robotic field.
Hongwei et al. [70] presented the moth-flame optimiza-
tion algorithm combined with an ensemble of ten chaos maps
and applied it to 18 benchmark functions and two engineering
design problems. They applied chaos in three ways: initializa-
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177 Page 12 of 42 Complex & Intelligent Systems (2025) 11:177
tion boundary handling, and replacing the random distance
parameter.
Tharwat et al. [163] integrated eight chaotic maps into
PSO and introduced two variants, each designed to replace
one of the two random parameters with a chaotic sequence
while preserving the randomness nature of the other param-
eter. The suggested approach was applied for optimization
purposes in the context of path planning.
Bingol and Alatas [38] introduced three enhanced variants
of the optics inspired optimization (OIO) algorithm, which
utilized chaos maps such as Gauss map, circle map, logistic
map, sinusoidal map, and tent map as generators of pseudo-
random numbers. These chaotic OIO variants were tested on
six benchmark functions and two engineering design prob-
lems.
Ewees and Abd Elaziz [52] introduced a hybrid algo-
rithm designed to enhance the multi-verse optimizer, named
CMVHHO. This algorithm integrates specific operators from
the harris hawk optimizer to enhance its local search capa-
bilities. Additionally, ten chaos maps were employed to
substitute the four random parameters, with one parame-
ter replaced while the remaining three were retained. The
efficacy of CMVHHO was rigorously evaluated through
experimentation on a set of 15 benchmark functions and four
real-world engineering design problems.
Further efforts to improve MVO’s capabilities include the
recent work by Amezquita et al. [19], who proposed a series
of novel chaotic variants of MVO. These variants incorporate
fuzzy logic principles and leverage ten different chaotic maps
to enhance their optimization capabilities. The efficacy of the
proposed algorithms was assessed through their application
to the optimization of 13 benchmark functions.
Anand and Arora [20] proposed a chaotic variant of the
selfish herd optimizer for global optimization across thirteen
high-dimensional benchmark problems. Additionally, they
introduced a binary (and chaotic) variant for feature selection,
which was tested on 21 different datasets
Varol Altay and Alatas [169] introduced multiple chaotic
bird swarm algorithms designed for various benchmark func-
tions and real-world engineering optimization problems.
Chaotic maps were utilized to replace the Gaussian distri-
bution in the displacement operators. The aim is to improve
the global convergence capabilities of the original BSA.
Aydilek et al. [30] introduced CHFPSO, a hybrid meta-
heuristic algorithm, which combines the firefly algorithm
and particle swarm optimization. Chaotic numbers replaced
random numbers within the operators, which enhance its per-
formance on CEC2015 benchmark problems.
In the same year, Lekouaghet et al. [91] designed and
employed a chaotic variant of the Rao-1 algorithm to esti-
mate optimal values for solar photovoltaic (PV) cell/module
models. This modified algorithm employs the logistic map
to replace a random parameter in the search operator.
Also, during that year Dehkordi et al. [45] introduced
an improved version of the Harris Hawk optimizer called
NCHHO, which utilizes nonlinear control parameters to
balance exploration and exploitation phases. In addition,
chaos maps were employed to enhance the optimization per-
formance of the original HHO. Their primary application
focused on solving vehicle routing optimization problems.
In a related line of research, Onay and Aydemir [123]pro-
posed and investigated three chaotic variants of the hunger
games search (HGS) algorithm. These adaptations employ
ten distinct chaotic maps. The performance of these chaotic
HGS variants was evaluated using a diverse set of 23 bench-
mark problems, CEC2017, and three engineering design
optimization problems.
In their work, Altay [17] proposed a novel chaotic
metaheuristic algorithm termed the chaotic slime mould opti-
mization algorithm (CSMA). This algorithm employs ten
distinct chaotic maps to achieve the best performance. The
efficacy of CSMA was comprehensively evaluated through
experimentation on a set of 62 benchmark functions and three
real-world engineering design problems.
Building upon the dwarf mongoose optimization algo-
rithm (DMO) introduced less than two years ago, the authors
of [4] proposed a chaotic variant termed CDMO for feature
selection.
In addition, the chaotic Archimedes optimization algo-
rithm (CAO) was developed in Bencherqui et al. [35]. This
algorithm leverages ten well-known chaotic maps individu-
ally to replace the random parameters within the standard AO
algorithm. This substitution aims to enhance the algorithm’s
ability to escape from local optima, leading to improved
optimization performance. The efficacy of CAO was compre-
hensively evaluated through testing on a set of 23 benchmark
functions, three real-world engineering design problems, and
its application to signal feature extraction tasks.
The recently developed reptile search algorithm was
enhanced by integrating an iterative map in the work of
Elashry et al. [51]. Four random parameters were replaced
with chaotic numbers generated using the iterative map, with
the aim to enhance the diversity capability of the predecessor
algorithm and prevent it from becoming stuck in local optima.
This chaotic variant was applied to optimization problems in
wireless sensor networks.
Non-random parameter adjusting
Chaos, not only serves as a substitute for random parame-
ters but also exhibits the capability to control other essential
parameters, denoted as Pin Eq. (38), employed by the search
operators in meta-heuristic algorithms. This leads to the
introduction of a modified formulation of the search oper-
ator as shown in Eq. (40).
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Complex & Intelligent Systems (2025) 11:177 Page 13 of 42 177
X(s+1)=(X(s), P,,) (40)
where P={p1,..., pl1}represents the set of non-random
parameters, ={g1,...,gl2}is the set of chaotically
controlled non-random parameters and l1+l2=l.The
remaining symbols are defined as previously described.
Gandomi et al. [57] introduced a chaotic accelerated PSO,
wherein they chaotically adjusted the parameter β, which was
traditionally set as a constant or linearly varied between two
predefined values. The variant based on the sinusoidal map
proved to be the most promising.
In the same year, Gandomi et al. [56] introduced chaos into
the firefly algorithm by leveraging twelve distinct chaotic
maps. These chaotic maps were employed to adjust cru-
cial parameters, specifically the light absorption coefficient,
attractiveness coefficient, and enhancing the algorithm’s per-
formance.
To enhance the krill herd algorithm [174], the attraction
parameter βhas been regulated by applying chaotic maps and
APSO has been used as a mutation to introduce a new hybrid
algorithm called chaotic particle-swarm krill herd algorithm.
Similarly, the authors of [175] used twelve different chaotic
maps for tuning the inertia weights of the KHA.
In the regular cuckoo search algorithm, a fixed step size α
is typically used. However, a newer approach has been pro-
posed that utilizes chaotic sequences instead of a fixed value
in the cuckoo search algorithm. Huang et al. [75] demon-
strated the effectiveness of dynamically adjusting the step
size between 0.1 and 0.3 for different tests. By incorporating
five different chaos maps, this method significantly enhances
the algorithm’s exploration ability and prevents premature
convergence to local solutions, thereby making the CCS opti-
mization algorithm more effective overall. A related study by
Wang et al. [176] involved adjusting the step size parameter
using chaotic sequences normalized between 0 and 2 and
exploring twelve distinct chaos maps.
In the original water cycle algorithm, the C value is a pre-
determined parameter. Heidari et al. [67] introduced CWCA
I, where the parameter C is adjusted chaotically using a lin-
early increasing operator. In their study, C was defined to
vary between 1 and 2.
In addition, the grey wolf optimizer employs chaos to
adjust its control parameters Aand C[105,120]. Similarly,
the Grasshopper Optimization Algorithm utilizes chaos to
dynamically adjust parameters c1and c2[23], while the
arithmetic optimization algorithm incorporates chaos for
adjusting
MOA
and
MOP
[95]. Moreover, the social group
optimization algorithm benefits from chaos in adjusting its
self-introspection parameter C[116].
Recently,an improved version of the whale optimization
algorithm is introduced in Zitouni and Harous [205]. In
this approach, the search space is restricted by utilizing
the nelder-mead algorithm and applied a chaotic control
to the vector a. This control employed a chaotic logistic
map, which, unlike the original algorithm where the val-
ues of this vector decreased linearly from 2 to 0, provided
enhanced optimization capabilities. Sarangi and Mohapa-
tra [141] directly replaced the coefficient vector (Cofi)
of the mountain gazelle optimizer with values generated
chaotically. The Cofiplays a crucial role in balancing the
intensification and diversification processes and is integrated
into three equations of the algorithm.
The primary advantage of this approach is the reduction
in the number of user-defined parameters, as determining the
optimal value for a parameter can be challenging, particularly
when dealing with various optimization problems.
Chaos-driven high-level strategies
In the abstract structure of metaheuristic algorithms, there
are various components, including a set of phases, each with
a distinct behaviour. Some phases are dedicated to explo-
ration, others to exploitation, and some serve both roles.
Many modern algorithms incorporate more than one phase
for exploration and one for exploitation. Moreover, the appli-
cation of these phases is not necessarily sequential. At this
level, high-level strategies come into play, controlling the
entire search process. These strategies determine, at each
iteration, which search operator should be applied, which
solutions should be updated, and how the balance between
exploration and exploitation is managed.
In the reviewed papers, many researchers have focused
on integrating chaos maps into the high-level strategies of
metaheuristic algorithms. These chaos maps contribute to
enhance the overall control and decision-making processes,
determining which search operators to apply and when, and
optimizing the balance between exploration and exploitation.
Jordehi [79] employed 11 different chaotic maps to adjust
the pulse emission rate parameter rin bat swarm optimiza-
tion (Bat II). This parameter determines the balance between
generating solutions randomly and using a specific equation.
Instead of using a fixed value for pto balance the explo-
ration and exploitation capabilities of the butterfly algorithm,
Arora and Singh [24] utilized a sequence of chaotic numbers
generated using 10 chaos maps (BOA II). A chaotic-driven
high-level mechanism can be seen also in the chaotic water
cycle algorithm (CWCA II) [67]. The whale optimization
algorithm incorporates different distinct strategies to update
the positions of whales during each iteration, including the
spiral model and the shrinking encircling mechanism. The
trade-off between these two strategies is determined by a
parameter denoted as pr ob. Oliva et al. [118] utilized the
Singer map to model the value of this parameter. The mod-
elling of foraging and vigilance behaviours of birds in the
bird swarm algorithm, and it involves probability parameter
P. In the eighth variant (CMBSA8) proposed by Varol Altay
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177 Page 14 of 42 Complex & Intelligent Systems (2025) 11:177
and Alatas [169], several chaos maps have been employed
in this probabilistic decision process. Seven novel scenarios
for integrating chaos maps into the arithmetic optimization
algorithm (CAOA) are proposed in Aydemir [29]. These
scenarios involve replacing the three random parameters
responsible for operator selection within each iteration of
the algorithm with chaotic sequences generated from differ-
ent chaotic maps. The work in Zermani et al. [194] adjusted
the same three random parameters using a logistic map, and
employed orthogonal-based learning for further enhance-
ment.
Chaos local search strategies
Metaheuristic algorithms must achieve a suitable compro-
mise between the exploration of new regions and the exploita-
tion of promising areas in the search space for optimal
performance. By integrating chaotic local search strate-
gies, metaheuristic algorithms can enhance their exploitation
capabilities in promising regions of the search space, while
still maintaining a sufficient degree of exploration. This
hybridization aims to improve the overall optimization per-
formance of the algorithms and their ability to converge to
global optimal solutions.
To address the problem of premature convergence in the
artificial bee colony algorithm, Liao et al. [97] employed
a chaotic local search strategy. The proposed algorithm
successfully solved the short-term hydrothermal scheduling
problem in the field of power systems. The chaotic gravi-
tational search algorithm (CGSA-II) introduced by Gao et
al. [58] utilizes a chaotic local search strategy. This strategy
intensifies the exploitation of the most promising regions sur-
rounding the current optimal solution by conducting a local
search based on the logistic map. A similar approach is pre-
sented in [151], which utilizes various chaotic maps for the
local search operator.
One of the enhancement, introduced in the chaotic dif-
ferential bee colony optimization algorithm [106]isthe
integration of a chaotic local search mechanism, unlike the
original version of the algorithm, which employs a random
search strategy for the scout bees.
To improve the search performance of the grey wolf opti-
mizer,Yuetal.[186] presented 12 chaotic variants. These
variants are integrated into a chaotic local search strategy,
each with a different chaotic map. These variants were
applied to 29 benchmark functions. In a related study, Xu
et al. [184] introduced two chaotic variants of the grey wolf
optimizer. The first variant incorporates a single chaotic map
for local search operations and the second variant employs a
selective hybridization strategy that utilizes multiple chaotic
maps.
In addition to the standard selection, crossover, and muta-
tion operations of genetic algorithms, Zhang et al. [196]
proposed the integration of a chaotic local search operator.
This chaotic local search component enhances the algo-
rithm’s exploitation capabilities. The resulting genetic chaos
optimization algorithm is particularly well-suited for optimal
time trajectory planning problems, where its improved local
search performance demonstrates significant advantages.
Zhang et al. [197] proposed the chaotic BFO algorithm,
which incorporates a chaotic local search mechanism into
the bacterial foraging optimization algorithm. The objective
of this hybridization was to improve the algorithm’s perfor-
mance in optimizing a diverse set of 23 benchmark functions,
as well as to address real-world engineering problems from
the IEEE CEC 2011 benchmark suite.
Also, Liu et al. [100] introduced a hybrid approach that
integrated a chaos search operator into the PSO algorithm.
This novel technique, known as CPSO, aimed to enhance the
performance of the PSO algorithm by leveraging the ergodic
and unpredictable nature of chaos. The proposed CPSO algo-
rithm was applied to the parameter estimation problem for a
nonlinear sun shadow model, demonstrating its effectiveness
in achieving accurate parameter estimates compared to the
standard PSO and DE algorithms.
Gao et al. [59] investigated different approaches to
hybridize the JADE algorithm, a prominent DE variant, with
chaotic local search mechanisms. The authors conducted a
comprehensive evaluation of four hybrid JADE-based algo-
rithms on a set of 53 benchmark functions from the CEC 2005
and CEC 2013 benchmark suites. Furthermore, the proposed
hybrid algorithms were applied to real-world engineering
optimization problems
Xu et al. [183] proposed a novel hybrid optimization
algorithm called CLSGMFO, which combines the moth-
flame optimization algorithm with chaotic local search and
Gaussian mutation operators. The developed CLSGMFO
algorithm was applied to the parameter tuning of the ker-
nel extreme learning machine model, a machine learning
technique used for supervised classification tasks. The per-
formance of the optimized KELM model utilizing the
CLSGMFO algorithm was compared against various state-
of-the-art classifiers.
Recently, Gharehchopogh et al. [63] introduced a novel
variant of the farmland fertility algorithm known as the
CQFFA. This variant incorporates both the quasi-
oppositional-based learning strategy and chaotic local search
techniques. The efficacy of the CQFFA was evaluated using
a diverse set of benchmark functions and engineering prob-
lems, showcasing its potential in optimization tasks.
Compared to the other approaches, chaotic local search
strategies do not adjust the parameter values of the enhanced
algorithm. Instead, they incorporate chaotic systems as a
local search mechanism, improving the algorithm’s ability
to explore and exploit the search space effectively.
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Complex & Intelligent Systems (2025) 11:177 Page 15 of 42 177
Chaos as a source of inspiration
Most metaheuristic algorithms can be categorized as
metaphor-based, drawing inspiration from various sources
such as human and animal behaviours [207], plant growth,
physical phenomena [206], and evolutionary theory. These
inspirations often lead to the development of innovative opti-
mization algorithms. In line with this trend, Talatahari and
Azizi [162] have introduced the chaos game optimization
algorithm, which is inspired by certain dynamics found in
chaos theory. Notably, chaos serves as a source of inspira-
tion in this context rather than being integrated as a tool to
directly enhance the mathematical model.
Applications of CMAs
The integration of chaos theory into the development of
metaheuristic algorithms has gained significant attention in
recent years. This approach is being increasingly explored to
tackle a diverse array of real-world problems, particularly
those characterized by non-linearity, multiple non-linear
constraints, and high dimensionality. Such problems often
present substantial challenges for exact methods, which can
struggle to find solutions within reasonable time frames, even
for standard metaheuristic algorithms. The inherent unpre-
dictability and ergodicity of chaotic systems make them
well-suited for enhancing the exploration and exploitation
capabilities of metaheuristic algorithms, leading to more
effective and efficient problem-solving strategies.
CMAs have found applications in various machine learn-
ing domains. For instance, in feature selection [2], chaotic
algorithms help identify the most relevant features for pre-
dictive modelling, thereby improving the performance and
interpretability of machine learning models. Similarly, in
classification tasks [99], these algorithms enhance the abil-
ity to find optimal decision boundaries, leading to improved
classification performance.
Image processing is another domain where CMAs have
made significant contributions. By optimizing parameters
and improving segmentation techniques, these algorithms
enhance the quality and efficiency of image analysis tasks.
For example [54], chaotic algorithm has been used to opti-
mize edge detection methods, resulting in more accurate and
faster image segmentation.
In the field of parameter estimation, CMAs have proven
effective for various models. For example, they have been
employed to accurately estimate parameters for photovoltaic
models [61], which are critical to optimize the performance of
solar energy systems. Similarly, in infinite impulse response
models [46], chaotic algorithms help to find the optimal filter
coefficients, leading to better signal processing outcome.
Medical diagnosis is another area where CMAs have
shown great potential. For example, in the diagnosis of
breast cancer and Parkinson’s disease [178], these algorithms
improve the accuracy of diagnostic models by optimizing
feature selection and classification parameters. In diabetes
and erythematous-squamous diagnosis [177], chaotic algo-
rithms optimize the generalization capability of SVM by
effectively handling the non-linear and high-dimensional
nature of medical data.
CMAs are also widely used in power systems. They have
been applied to optimize reactive power dispatch [1], eco-
nomic dispatch [153], electrical power distribution systems
[33], and optimal power flow [39]. By incorporating chaotic
maps, these algorithms can more effectively explore and
exploit the complex, high-dimensional search spaces typical
in power system problems, leading to improved operational
efficiency and stability. In robotics, these algorithms con-
tribute to path planning [101], and optimization of controller
parameters [120], leading to more efficient and intelligent
robotic systems.
Engineering design problems [35], which often involve
complex optimization tasks, also benefit from the application
of CMAs. These algorithms aid to find the optimal design
parameters, improve performance, and reduce the cost of
engineering solutions.
In general, the application of CMAs is approximately the
same as for standard metaheuristic algorithms; however, the
latter often provide more promising results. Table 1pro-
vides a comprehensive overview of various applications of
CMAs, and summarizes the contributions of approximately
130 research papers in different domains.
Chaotic RIME optimizer (C-RIME)
The RIME optimizer, introduced by Su et al. [158], represents
a novel, physics-inspired, stochastic, and population-based
metaheuristic algorithm. This optimizer employs two prin-
cipal search operators: the soft-rime search operator and the
hard-rime search operator. These operators are designed to
promote the essential components of any metaheuristic algo-
rithm: exploration, exploitation, and avoidanceof entrapment
in local optima. In fact, the simplicity of the RIME optimizer,
coupled with its strong performance, suggests that integrating
additional techniques could lead to even better results. How-
ever, the original RIME faces some common issues, such as
getting trapped in local optima, insufficiently smooth transi-
tions from global to local search, and decreased diversity in
search solutions. Introducing chaotic theory offers a promis-
ing approach to address these issues, as noted in the literature.
In this section, we first provide the mathematical equa-
tions of the original RIME optimizer. Then, we explain the
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177 Page 16 of 42 Complex & Intelligent Systems (2025) 11:177
Table 1 Summary of applications of CMAs
Application Abbr Predecessor algorithm Year References
Pow e r s y stem s
Reactive power dispatch CKHA Krill Herd Algorithm 2015 [115]
CBA Bat Algorithm 2020 [113]
CTFWO Turbulent Flow of Water-based Optimization 2022 [1]
Economic dispatch CDBCO Differential Bee Colony Optimization 2014 [106]
CGBABCA Global Best Artificial Bee Colony Algorithm 2015 [148]
CSADHS Self-Adaptive Differential Harmony Search 2015 [135]
CBA Bat Algorithm 2016 [8]
CKH Krill Herd Algorithm 2017 [32]
MP-CJAYA JAYA Algorithm 2018 [188]
CIHSA Improved Harmony Search Algorithm 2019 [138]
MOMVO Multi-Verse Optimization algorithm 2020 [159]
IABC Artificial Bee Colony 2021 [15]
CSGO Social Group Optimization 2021 [149]
CSMA Slime Mould Algorithm 2022 [153]
CSHO Spotted Hyena Optimization 2023 [109]
Electrical power distribution system OCDE Differential Evolution 2017 [86]
QOCHHO Harris Hawk’s Optimizer 2022 [33]
Optimal power flow CSADHS Self-adaptive Differential Harmony Search Algorithm 2013 [26]
CIOA Invasive Weed Optimization 2014 [64]
CKHA Krill Herd Algorithm 2015 [114]
CSS Cuckoo Search 2017 [31]
ICEFO Electromagnetic Field Optimization 2020 [39]
CAVOA African Vultures Optimization Algorithm 2024 [111]
Short-term hydrothermal scheduling ACABC Artificial Bee Colony 2013 [97]
Hybrid electric powertrain intelligent sizing CAPSO Particle Swarm Optimization 2017 [204]
Economic energy scheduling CGTO Gorilla Troop’s Optimizer 2023 [37]
Load margin assessment CCSA Crow Search Algorithm 2023 [36]
Hybrid renewable energy system ICGO Grasshopper Optimizer 2024 [195]
Power losses in power distribution networks CGFA Golden Flower Algorithm 2023 [161]
123
Complex & Intelligent Systems (2025) 11:177 Page 17 of 42 177
Table 1 continued
Application Abbr Predecessor algorithm Year References
Machine learning techniques
Feature selection CBPSO Particle Swarm Optimization 2016 [78]
CAO Antlion Optimization 2016 [193]
CCSO Chicken Swarm Algorithm 2017 [10]
CWOA Whale Optimization Algorithm 2018 [144]
CSSA Salp Swarm Algorithm 2018 [145]
CCSA Crow Search Algorithm 2019 [146]
CMVO Multi-Verse Optimizer 2019 [53]
CDA Dragonfly Algorithm 2019 [147]
CBBHA Black Hole Algorithm 2020 [133]
CFCSA Crow Search Optimization Algorithm 2020 [21]
CSHO Selfish Herd Optimizer 2020 [20]
CISA Chaotic Interior Search Algorithm 2020 [25]
OCBSOD Brainstorm optimization 2020 [117]
CGSK Gaining Sharing Knowledge-based optimization 2021 [9]
CGSA Group Search Optimizer 2022 [5]
ISCA Sine Cosine Algorithm 2022 [166]
CGBO chaotic gradient-based optimizer 2022 [2]
BCHOAFS Horse Herd Optimization Algorithm 2023 [191]
CTSO Transient Search Algorithm 2023 [18]
CFA Firefly Algorithm 2023 [65]
CRSA Reptile Search Algorithm 2023 [6]
DC-GPO Gold-Panning Optimizer 2023 [180]
CDMO Dwarf Mongoose Optimization 2024 [4]
Image processing CDPSO Darwinian Particle Swarm Optimization 2017 [160]
CEFO Electromagnetic Field Optimization 2019 [156]
CSOS Symbiotic Organisms Search 2021 [40]
CER Rao algorithm 2023 [121]
Coronavirus Optimization Algorithm 2023 [71]
DC-GPO Gold-Panning Optimizer 2023 [180]
Classification MCPS Particle Swarm Optimization 2015 [27]
CPSO Particle Swarm Optimization 2015 [99]
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177 Page 18 of 42 Complex & Intelligent Systems (2025) 11:177
Table 1 continued
Application Abbr Predecessor algorithm Year References
Parameters estimation
Photovoltaic cells models MPCOA Mutative-scale Parallel Chaos Optimization Algorithm 2014 [189]
CWOA Whale Optimization Algorithm 2017 [118]
ILCOA Chaotic Optimization Algorithm 2019 [132]
LCROA Rao-1 Optimization Algorithm 2021 [91]
ACGWO Grey Wolf Optimization 2021 [127]
FCHHHO Harris Hawks Optimization 2023 [61]
Fuel cells models CEPSO Particle Swarm Optimization 2021 [126]
CGOW Grey Wolf Optimization 2022 [173]
CBSSO Shark Smell Optimization 2023 [66]
Adaptive IIR model COWOA Whale Optimization Algorithm 2023 [46]
Semi-empirical mathematical models – Aquila Optimization Algorithm 2024 [168]
Robotics
Path planning – Genetic Chaos Optimization Algorithm 2017 [196]
CGWO Grey Wolf Optimizer 2018 [110]
SATC-ALO Antlion optimizer algorithm 2019 [202]
CPSO Particle Swarm Optimization 2019 [163]
CPSO-ACO Particle Swarm Optimization + Ant Colony Optimization 2019 [42]
CAOSA Aquila Optimizer + Simulated Annealing 2022 [11]
CMPA Marine Predators Algorithm 2023 [134]
ACEO Adaptive Equilibrium Optimizer 2023 [101]
parameter optimization for HOSM controllers GWO Grey Wolf Optimizer 2017 [120]
Robot localization CPSO Particle Swarm Optimization 2019 [190]
ANCOA Adaptive niching Chaos Optimization Algorithm 2022 [139]
Offline robotic manipulator controller tuning CDE Differential Evolution 2024 [128]
Medical diagnosis
Parkinson’s disease CMFO Moth-Flame Optimization 2017 [178]
Breast cancer CMFO Moth-Flame Optimization 2017 [178]
CMWOA Whale Optimization Algorithm 2020 [177]
Diabetes/erythemato-squamous CMWOA Whale Optimization Algorithm 2020 [177]
COVID-19 BIAS Interactive Autodidactic School Algorithm 2023 [62]
The electroencephalographic signal decomposition-based optimization problems CGRGLS Golden Ration Guided Local Search 2023 [85]
123
Complex & Intelligent Systems (2025) 11:177 Page 19 of 42 177
Table 1 continued
Application Abbr Predecessor algorithm Year References
Wireless sensor networks
ICAEA Immune Chaotic Adaptive Evolutionary Algorithm 2022 [198]
CGTOA Gorilla Troops Optimization Algorithm 2023 [154]
IBWOA Black Widow Optimization Algorithm 2023 [182]
PSO-VDCOA Particle Swarm Optimization 2022 [200]
CRO Rider Optimization 2023 [140]
CRSA Reptile Search Algorithm 2024 [51]
Engineering design problems
CAPSO Particle Swarm Optimization 2013 [57]
AFA Adaptive Firefly Algorithm 2015 [34]
CWCA Water Cycle Algorithm 2017 [67]
CMFO Moth-Flame Optimization 2019 [70]
CMVHHO Multi-Verse Optimizer + Harris Hawks Optimizer 2020 [52]
COA Chaotic Optimization Algorithm 2020 [84]
CMBSA Bird Swarm Algorithm 2020 [169]
OBCWOA Whale Optimization Algorithm 2020 [41]
CEBA Bat algorithm 2020 [187]
OCSSA Salp Swarm Algorithm 2020 [201]
CRUN Runge Kutta Optimization 2022 [185]
CAOA Arithmetic Optimization Algorithm 2022 [95]
cFTL Follow The Leader algorithm 2022 [152]
CHGS Hunger Games Search 2022 [123]
CSMA Slime Mould Optimization Algorithm 2022 [17]
CEFO Electromagnetic Field Optimization 2023 [76]
CSGO Social Group Optimization 2023 [116]
CMPA Marine Predators Algorithm 2023 [87]
CSGO Social Group Optimization 2023 [116]
CSHO SeaHorse Optimization 2023 [125]
CHHO Harris Hawks Optimizer 2023 [47]
CMPA Marine Predators Algorithm 2023 [88]
CSHO Spotted Hyena Optimizer 2023 [89]
CTSO Transient Search Algorithm 2023 [18]
CGRAD Gradient Based Optimizer 2023 [167]
CQFFA Farmland Fertility Algorithm 2023 [63]
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177 Page 20 of 42 Complex & Intelligent Systems (2025) 11:177
Table 1 continued
Application Abbr Predecessor algorithm Year References
CSFLA Shuffled Frog-Leaping Algorithm 2023 [83]
CABC Artificial Bee Colony 2023 [83]
CICA Imperialist Competitive Algorithm 2023 [83]
CAO Archimede Optimization Algorithm 2024 [35]
CMGO Mountain Gazelle Optimizer 2024 [141]
Other applications
Manufacturer scheduling problem/Cloud manufacturing COA Chaos Optimization Algorithm 2019 [74]
QoS-aware web service composition CHHO Harris Hawks Optimization 2020 [92]
Cryptology GFA Firefly Algorithm 2020 [13]
XGWO Crossover Grey Wolf Optimizer 2023 [90]
Internet of vehicles NCHHO Harris Hawks Optimization 2021 [45]
Random forest hyperparameters optimization COSMA Slime Mould Algorithm 2022 [203]
Optimizing bridge maintenance plans ECDE Differential Evolution 2022 [3]
Crude oil time series prediction CHGSO Henry Gas Solubility Optimization 2022 [80]
CCO-AVOA Vulture Optimization Algorithm 2023 [12]
Electromagnetic optimization problems CCPA Colony Predation Algorithm 2023 [73]
Workflow scheduling in cloud CO-DWO Dwarf Mongoose Optimization
Task scheduling in resource-limited cyber-physical systems CMPO Marine Predators Optimization 2023 [16]
Test case prioritization CFFO Flower Fruit-fly Optimization 2023 [170]
Smart farming CJO Jaya Optimization algorithm 2023 [14]
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Complex & Intelligent Systems (2025) 11:177 Page 21 of 42 177
modifications we have introduced to enhance the overall per-
formance of the RIME optimizer.
Soft-rime search operator
The soft-rime search operator is designed to achieve a balance
between exploration and exploitation in a step-wise manner.
It updates the position of each rime-particle within the rime-
agents population using Eq. (41).
xn,New
d=xbest
d+Rime Factor ·U(x−
d,x+
d)(41)
In Eq. (41), xbest
drepresents the
d-th
dimension of the
best rime-agent found so far. The term U(x−
d,x+
d)denotes
a uniformly distributed random number between the lower
bound x−
dand the upper bound x+
dof the
d-th
dimension,
and the
RimeFactor
is computed using Eq. (42).
Rime Factor =r1·cos(θ) ·β(42)
The rime factor is controlled by a random number r1
between −1 and 1, an angle θ, and an environmental fac-
tor β.Bothαand βare calculated using Eqs. (43) and (44),
respectively.
θ=0.1·π·t
T(43)
β=1−round(w.t/T)
w(44)
The application probability of the soft-rime search oper-
ator is determined by √t/T, where t represents the current
iteration and Tis the maximum number of iterations.
Hard-rime search operator
To avoid premature convergence and becoming trapped in
local optima, the hard-rime search operator plays a criti-
cal role. The application probability of the hard-rime search
operator for each rime-particle is related to the normalized fit-
ness value of its rime-agent, ensuring that rime-agents with
poorer fitness are more likely to be updated. The operator
follows the mechanism given in Eq. (45).
xn,New
d=xbest
d(45)
Enhancements to the RIME optimizer
We propose several enhancements to improve the perfor-
mance of the RIME optimizer.
•Utilization of the pbest technique: the RIME optimizer
has a structure similar to the DE algorithm. In the RIME
optimizer, the soft-rime search operator corresponds to
the mutation operator in DE, while the hard-rime search
operator corresponds to the crossover operator. DE fea-
tures various mutation strategies, some guided by the
best-so-far individual and others using a random indi-
vidual to direct the search process. An advanced DE
variant, JADE, introduced the current-to-pbest strategy,
which involves selecting a random individual from the
top p% of the population.
Given the structural similarities between RIME and DE,
and the demonstrated performance of the pbest tech-
nique, we decided to incorporate the pbest concept into
the RIME optimizer instead of using the current best-
so-far individual. Typically, pis set to a constant value
(e.g. 10%). However, in recent studies [157], pvalue has
been set to change dynamically. In our case, the value
of pvaries linearly from 30% to 20% throughout the
iterations.
•Probability adjustment of the hard-rime search operator:
The probability of applying the hard-rime search operator
is set to a constant value of 0.1, rather than being based
on the normalized fitness value.
•Chaotic adjustment of the rime factor: the RimeFactor
in the soft-rime search operator (i.e. Eq. (42)) is adjusted
using chaotic maps, enhancing the balance between
exploration and exploitation. The modified RimeFactor
is given by Eq. (46).
Rime Factor =c·cos(θ) ·β(46)
Here, cis a chaotic number generated from a pre-selected
chaotic map.
Figure 3provides a graphical representation of chaotic
RimeFator
, where the chaotic parameter cin Eq. (46) is gen-
erated by different chaos maps (i.e. the first 21 chaos maps
listed in “Chaotic maps” section) during 500 iterations.
These modifications are aimed at enhancing the overall
optimization performance. The first modification increases
the diversity of solutions, thereby improving the algorithm’s
ability to locate the global optimum. The second modification
maintains the simplicity of implementation, ensuring that the
enhancements do not introduce undue complexity. The final
modification addresses a critical parameter, the RimeFactor,
by adjusting it chaotically. This adjustment achieves a more
balanced trade-off between exploration and exploitation, and
it helps prevent the algorithm from becoming trapped in local
optima. Algorithm 2outlines the main steps of the C-Rime
optimizer.
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177 Page 22 of 42 Complex & Intelligent Systems (2025) 11:177
Fig. 3 The behaviour of chaos-based RIME Factor parameter during 500 iterations for different chaos maps
123
Complex & Intelligent Systems (2025) 11:177 Page 23 of 42 177
Algorithm 2: Pseudo code of C-RIME optimizer
Input: Population size N, Maximum iterations T, Bounds
[X−,X+]
Output: Best solution found Xbest
1Initialize population X(0)randomly within bounds [X−,X+];
2Evaluate the fitness of each individual in X;
3Set Xbest as the best solution found so far;
4Initialize t←0;
5while t<Tdo
6Select Xpbest from top p% of the population;
7for each individual X nin X(t−1)do
8Calculate θand βusing Eq. (43)andEq.(44);
9Generate chaotic number cusing a preselected chaos map;
10 Calculate Rime Factor using (46);
// soft rime search operator
11 for each dimension j do
12 if rand <√t/Tthen
13 xn,New
d=xpbest
d+Rime Factor ·U(x−
d,x+
d);
14 end
15 end
// hard rime search operator
16 for each dimension j do
17 if rand <0.1then
18 xn,New
d=xpbest
d;
19 end
20 end
21 Perform boundary handling technique;
22 Evaluate the fitness of Xn,New;
23 perform positive greedy selection strategy;
24 end
25 t←t+1;
26 end
27 return Xbest ;
Experiments and results
In this section, we present a comprehensive evaluation of the
proposed chaotic RIME optimizer through extensive exper-
iments.
Experimental configuration
The performance of the chaotic RIME algorithm was evalu-
ated using twelve benchmark functions from the CEC2022
benchmark suite, categorized into four groups: one uni-
modal function (F1), four basic functions (F2−F5), three
hybrid functions (F6−F8), and four composition functions
(F9−F12).
The chaotic RIME optimizer was benchmarked against
several state-of-the-art metaheuristic algorithms, including:
grey wolf optimizer (GWO) [108], harris hawks optimiza-
tion (HHO) [68], lévy flight distribution (LFD) [72], optical
microscope algorithm (OMA) [43], coati optimization algo-
rithm (COA) [44], chaotic arithmetic optimization algorithm
(C-AOA) [95], chaotic slime mould optimization algorithm
(C-SMA) [17], chaotic hunger games search optimization (C-
HGS) [123], chaotic dwarf mongoose optimization algorithm
(C-DMOA) [4], and chaotic aquila optimization algorithm
(C-AO) [168]. The selection of these algorithms was based
on their established effectiveness and relevance within the
optimization literature. Specifically, GWO, HHO, and LFD
are well-recognized for their robust performance across var-
ious optimization tasks. OMA and COA represent recently
developed and highly cited metaheuristics, showcasing inno-
vative approaches in the field. Additionally, the inclusion
of C-AOA, C-SMA, C-HGS, C-DMOA, and C-AO ensures
a comprehensive comparison with other chaos-based meta-
heuristic algorithms pertinent to our study. Of the optimizers
selected, only OMA and COA are based on free parame-
ters. The remaining algorithms are parameterized according
to the recommendations of the original authors, with spe-
cific parameter values detailed in Table 2. Our proposed
algorithm, C-RIME, also includes three principal parameters
that were carefully tuned to enhance performance. The first
parameter, an inherent parameter w, comes from the origi-
nal RIME algorithm and is set to 5, as recommended by the
original authors. The other two parameters, pandr, are intro-
duced in our work and were fine-tuned through an empirical
trial-and-error approach. For parameter p, which controls
the proportion of the population used in the pbest strategy,
we tested several configurations, varying pin intervals of
[40–30%],[30–20%],[20–10%],[40–20%], and [30–10%]
of the total population. For r, which determines the proba-
bility of applying the hard-rime search operator, we tested
the values of 0.1, 0.2, and 0.3. This systematic exploration
led us to evaluate 15 configurations in total (5 values for p
and 3 for r), from which we identified the optimal settings
as pdecreasing linearly from 30 to 20% and r=0.1.
All reported results indicate the function error values, cal-
culated as specified in Eq. (47).
erri=Fi(x)−Fi(x∗), 1≤i≤numberO f Runs (47)
Here, xrepresents the obtained solution in the i-th run,
x∗is the target fitness value.
Each algorithm was run for 30 independent trials to
account for stochastic variability. The performance metrics
used are minimum value (MIN), maximum value (MAX),
average value (AVG), and standard deviation (STD), as
shown in Eqs. (48), (49), (50), and (51).
MIN =min
i∈{1,...,n}erri(48)
MAX =max
i∈{1,...,n}erri(49)
AVG =1
n
n
i=1
erri(50)
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177 Page 24 of 42 Complex & Intelligent Systems (2025) 11:177
Table 2 The parameter settings
Optimizer Parameter Value
GWO aDecreases linearly from 2 to 0
HHO β1.5
LFD CSV 0.5
Threshold 2
β1.5
α110
α20.00005
α30.005
∂10.9
∂20.1
OMA – –
COA – –
C-AOA α5
μ0.499
MOP
min 0.2
MOP
max 1
C-SMA z0.03
C-HGS l0.03
LH 100
C-DMOA – –
C-AO α0.1
δ0.1
STD =
1
n
n
i=1
(erri−err ∗)2(51)
In these equations, nrepresents the number of runs, erri
denotes the obtained function error value, and err ∗is equal
to AV G as calculated by Eq. (50).
The population size and the number of function eval-
uations were kept consistent across all algorithms, set to
30 and 200,000, respectively. The experiments were carried
out using the MATLAB software (r2023a) in an environ-
ment with the following specifications: Processor: Intel(R)
Core(TM) i7-10510U CPU @ 1.80 GHz 2.30 GHz, RAM:
16 GB, OS: Windows 11.
Performance comparison of chaotic RIME optimizer
variants
This section presents an analysis of the performance of
various chaotic maps when integrated with the RIME opti-
mizer. We aim to identify the most effective configuration to
enhance the optimizer performance in the CEC2022 bench-
mark suite (dimension = 10).
For nomenclature, each variant is designated as CXX-
RIME, where XX represents the identifier of the chaos map
used to generate the chaotic sequence in Eq. (46). For exam-
ple, C01-RIME refers to the variant that incorporates all
modifications described in “Chaotic RIME optimizer (C-
RIME)” section, and utilizes the chaos map number 01
(logistic map) as listed in “Chaotic maps” section. Similarly,
C02-RIME employs map number 02 (Chebyshev map),…,
and C21-RIME uses map number 21 (sinai map). The origi-
nal RIME optimizer and a variant using random distribution
(RND-RIME) are also included for comparison.
The performance results of the chaotic RIME optimizers
are summarized in Table 3. To determine the overall effec-
tiveness of each variant, the Friedman test was employed.
The results are presented in Table 4.
The analysis of these results reveals significant variabil-
ity in the performance of different chaotic variants. Some
of them, such as C10-RIME and C16-RIME, consistently
demonstrate superior optimization capabilities across mul-
tiple functions. For example, C10-RIME excels particularly
for F1,F3, and F5, achieving the lowest values of all per-
formance metrics (MIN, MAX, AVG, and STD). Similarly,
C16-RIME shows outstanding performance for functions F7
and F8, indicating its robustness and effectiveness in these
specific optimization scenarios.
By contrast, other chaotic variants do not perform as
well. For example, C08-RIME underperforms by compar-
ison to the random distribution variant (RND-RIME) in
certain cases. This suggests that not all chaotic maps enhance
the optimizer’s effectiveness equally, and some may even
degrade the performance. It highlights the importance of
selecting an appropriate chaotic map tailored to the specific
characteristics of the optimization problem at hand.
Delving into function-specific results, the superior perfor-
mance of certain chaotic variants becomes more pronounced.
For instance, for function F1, C10-RIME stands out with the
lowest values across all metrics, showcasing its exceptional
optimization capability. Conversely, C16-RIME shows the
highest values for this function, highlighting its ineffective-
ness in this particular scenario. For function F2, C14-RIME
achieves the best mean performance, surpassing most other
variants and both baseline optimizers. For function F3, C10-
RIME once again excels, followed closely by C11-RIME,
C15-RIME, and C19-RIME, with ten chaotic variants out-
performing RND-RIME and fourteen surpassing the original
RIME.
The function F4presents a different landscape, with
C06-RIME and C17-RIME showing excellent performance
through the smallest gap error, best AVG, and MAX values,
respectively. In addition, this function shows the original
RIME, with several surpassing the random variant. For
function F5, C10-RIME continues to dominate, followed
by C17-RIME, C12-RIME, and C21-RIME, although C04-
RIME and C08-RIME lag behind RND-RIME.
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Complex & Intelligent Systems (2025) 11:177 Page 25 of 42 177
Table 3 Results of RIME, RND-RIME and all chaotic variants of RIME optimizer
Algorithm MIN MAX AVG STD Rank MIN MAX AVG STD Rank
F1F2
C01-RIME 4.71E−04 6.71E−03 1.87E−03 1.16E−03 22 4.86E+00 8.92E+00 7.94E+00 1.40E+00 21
C02-RIME 4.49E−04 4.21E−03 1.68E−03 9.16E−04 19 3.14E−01 8.92E+00 6.60E+00 2.88E+00 10
C03-RIME 2.92E−04 3.43E−03 1.71E−03 8.02E−04 20 4.44E−01 8.92E+00 7.64E+00 1.94E+00 19
C04-RIME 2.79E−04 3.98E−03 1.34E−03 9.65E−04 16 2.31E−04 8.92E+00 6.45E+00 3.34E+00 8
C05-RIME 7.30E−04 4.30E−03 1.79E−03 7.28E−04 21 4.55E−01 8.92E+00 7.41E+00 2.11E+00 17
C06-RIME 3.06E−04 2.35E−03 7.89E−04 5.10E−04 13 2.94E−04 8.92E+00 6.22E+00 3.30E+00 5
C07-RIME 3.42E−04 5.29E−03 1.60E−03 9.61E−04 18 9.58E−02 8.92E+00 7.06E+00 2.64E+00 12
C08-RIME 4.16E−04 3.05E−03 1.35E−03 5.84E−04 17 1.03E−03 8.92E+00 7.08E+00 2.72E+00 14
C09-RIME 3.00E−04 2.17E−03 8.21E−04 4.57E−04 14 5.47E−02 8.92E+00 6.72E+00 2.77E+00 11
C10-RIME 2.73E−05 2.46E−04 9.33E−05 5.47E−05 1 3.35E−02 8.92E+00 6.31E+00 3.32E+00 6
C11-RIME 2.31E−04 2.15E−03 9.22E−04 5.14E−04 15 2.62E−02 8.92E+00 5.86E+00 3.35E+00 2
C12-RIME 1.87E−04 1.62E−03 6.92E−04 3.57E−04 9 3.70E−01 8.92E+00 6.48E+00 2.46E+00 9
C13-RIME 1.66E−04 2.01E−03 5.45E−04 3.99E−04 3 3.79E−03 8.92E+00 7.06E+00 2.47E+00 13
C14-RIME 1.24E−04 2.41E−03 6.17E−04 4.99E−04 6 1.88E−02 8.92E+00 5.82E+00 3.37E+00 1
C15-RIME 1.40E−04 2.00E−03 6.59E−04 4.08E−04 8 2.06E−03 8.92E+00 6.19E+00 3.56E+00 4
C16-RIME 1.02E−03 6.17E−03 2.58E−03 1.23E−03 23 1.05E−03 8.92E+00 7.68E+00 2.05E+00 20
C17-RIME 1.56E−04 1.41E−03 7.41E−04 3.07E−04 12 4.04E−02 8.92E+00 7.26E+00 2.51E+00 16
C18-RIME 9.88E−05 1.40E−03 7.22E−04 3.64E−04 11 4.15E−03 8.92E+00 6.19E+00 2.90E+00 3
C19-RIME 3.10E−05 1.44E−03 5.91E−04 3.28E−04 5 1.98E−02 8.92E+00 7.09E+00 2.55E+00 15
C20-RIME 2.27E−04 2.08E−03 7.01E−04 3.77E−04 10 2.73E−02 8.92E+00 6.44E+00 2.97E+00 7
C21-RIME 1.78E−04 1.54E−03 6.50E−04 3.19E−04 7 3.48E−03 7.08E+01 8.88E+00 1.19E+01 23
RND-RIME 7.93E−05 1.43E−03 5.71E−04 3.12E−04 4 4.01E+00 8.92E+00 7.51E+00 1.76E+00 18
RIME 7.34E−05 1.20E−03 4.03E−04 2.26E−04 2 5.15E−05 7.08E+01 8.41E+00 1.70E+01 22
F3F4
C01-RIME 2.76E−03 2.21E−02 8.54E−03 4.08E−03 17 2.98E+00 1.89E+01 9.25E+00 3.84E+00 19
C02-RIME 4.47E−03 1.50E−02 8.80E−03 2.69E−03 20 3.98E+00 1.39E+01 8.29E+00 2.60E+00 8
C03-RIME 3.78E−03 2.89E−02 9.88E−03 5.32E−03 22 2.98E+00 1.39E+01 8.03E+00 2.73E+00 6
C04-RIME 2.25E−03 1.83E−02 7.14E−03 3.22E−03 14 3.98E+00 1.59E+01 9.29E+00 3.29E+00 20
C05-RIME 3.28E−03 2.66E−02 9.02E−03 5.51E−03 21 1.99E+00 1.69E+01 8.19E+00 3.89E+00 7
C06-RIME 2.51E−03 1.68E−02 6.19E−03 3.00E−03 10 9.95E−01 1.39E+01 7.69E+00 2.79E+00 2
C07-RIME 2.80E−03 2.12E−02 8.74E−03 3.64E−03 19 3.98E+00 1.69E+01 8.92E+00 3.09E+00 15
C08-RIME 3.48E−03 1.48E−02 8.68E−03 2.79E−03 18 3.98E+00 1.65E+01 9.17E+00 3.06E+00 18
C09-RIME 2.47E−03 2.02E−02 7.12E−03 3.85E−03 13 3.98E+00 1.89E+01 8.86E+00 3.62E+00 13
C10-RIME 8.38E−04 4.36E−03 2.15E−03 8.84E−04 1 4.97E+00 2.29E+01 1.02E+01 3.61E+00 22
C11-RIME 2.06E−03 9.27E−03 5.19E−03 2.00E−03 2 2.98E+00 1.49E+01 8.42E+00 2.76E+00 9
C12-RIME 2.56E−03 2.15E−02 6.10E−03 3.20E−03 9 1.99E+00 1.39E+01 8.46E+00 3.31E+00 10
C13-RIME 1.19E−03 1.43E−02 5.77E−03 3.02E−03 6 2.98E+00 1.69E+01 9.09E+00 3.70E+00 16
C14-RIME 1.63E−03 1.95E−02 6.33E−03 3.33E−03 12 1.99E+00 1.59E+01 7.86E+00 3.38E+00 4
C15-RIME 1.94E−03 1.24E−02 5.54E−03 2.42E−03 3 4.97E+00 1.49E+01 9.65E+00 2.83E+00 21
C16-RIME 3.38E−03 2.24E−02 1.01E−02 4.53E−03 23 4.97E+00 1.59E+01 9.09E+00 2.69E+00 17
C17-RIME 1.35E−03 1.94E−02 7.19E−03 4.36E−03 15 2.98E+00 1.29E+01 7.40E+00 2.68E+00 1
C18-RIME 2.01E−03 1.69E−02 5.71E−03 2.73E−03 5 3.98E+00 1.69E+01 8.47E+00 3.10E+00 11
C19-RIME 2.18E−03 1.20E−02 5.62E−03 2.49E−03 4 1.99E+00 1.39E+01 7.83E+00 3.13E+00 3
C20-RIME 2.07E−03 1.07E−02 5.95E−03 2.05E−03 7 2.98E+00 1.49E+01 8.82E+00 3.39E+00 12
123
177 Page 26 of 42 Complex & Intelligent Systems (2025) 11:177
Table 3 continued
F3F4
C21-RIME 1.90E−03 1.31E−02 5.96E−03 2.91E−03 8 2.98E+00 1.79E+01 8.92E+00 3.63E+00 14
RND-RIME 2.16E−03 2.60E−02 6.27E−03 4.43E−03 11 2.98E+00 1.49E+01 7.99E+00 2.93E+00 5
RIME 1.48E−03 2.76E−02 8.08E−03 5.66E−03 16 2.98E+00 4.88E+01 2.18E+01 9.39E+00 23
F5F6
C01-RIME 1.88E−05 8.98E−02 3.09E−03 1.61E−02 9 7.04E+00 6.20E+03 1.29E+03 1.71E+03 1
C02-RIME 1.65E−05 1.79E−01 6.11E−03 3.21E−02 16 2.87E+01 6.21E+03 1.96E+03 1.83E+03 15
C03-RIME 1.44E−05 8.97E−02 3.11E−03 1.61E−02 10 1.19E+01 6.24E+03 1.95E+03 1.92E+03 14
C04-RIME 1.19E−05 8.97E−02 9.09E−03 2.68E−02 21 4.51E+00 6.22E+03 1.98E+03 1.89E+03 16
C05-RIME 1.81E−05 8.98E−02 3.13E−03 1.61E−02 11 2.63E+01 6.23E+03 1.41E+03 1.33E+03 5
C06-RIME 1.27E−05 8.96E−02 9.02E−03 2.69E−02 19 8.06E+00 6.23E+03 1.79E+03 1.97E+03 11
C07-RIME 1.49E−05 8.97E−02 6.10E−03 2.23E−02 15 8.53E+01 6.22E+03 2.10E+03 1.63E+03 17
C08-RIME 2.41E−05 4.54E−01 1.82E−02 8.26E−02 22 7.07E+01 6.21E+03 1.95E+03 1.95E+03 13
C09-RIME 6.86E−06 8.96E−02 3.06E−03 1.61E−02 7 3.01E+01 6.21E+03 2.74E+03 2.19E+03 23
C10-RIME 8.28E−07 4.62E−05 1.53E−05 1.24E−05 1 4.41E+00 6.20E+03 2.21E+03 2.30E+03 19
C11-RIME 1.73E−05 8.97E−02 6.24E−03 2.23E−02 17 1.27E+02 6.25E+03 2.44E+03 2.11E+03 21
C12-RIME 1.34E−05 2.14E−04 6.07E−05 5.21E−05 3 2.08E+01 6.21E+03 1.67E+03 1.82E+03 10
C13-RIME 7.75E−06 5.97E−04 7.49E−05 1.14E−04 5 1.17E+01 6.23E+03 2.14E+03 2.11E+03 18
C14-RIME 9.19E−06 3.74E−04 8.05E−05 8.47E−05 6 7.91E+01 6.20E+03 1.80E+03 1.84E+03 12
C15-RIME 1.16E−05 8.96E−02 6.04E−03 2.23E−02 14 7.14E+00 6.25E+03 1.39E+03 1.77E+03 4
C16-RIME 2.86E−05 8.97E−02 3.14E−03 1.61E−02 12 4.98E+00 4.75E+03 1.31E+03 1.34E+03 2
C17-RIME 6.51E−06 1.69E−04 5.25E−05 4.43E−05 2 9.37E+00 6.23E+03 2.39E+03 2.25E+03 20
C18-RIME 7.11E−06 8.96E−02 3.06E−03 1.61E−02 8 3.79E+01 3.90E+03 1.57E+03 1.27E+03 7
C19-RIME 7.00E−06 8.96E−02 9.01E−03 2.69E−02 18 1.58E+01 6.22E+03 1.62E+03 1.82E+03 9
C20-RIME 7.52E−06 8.96E−02 6.01E−03 2.23E−02 13 1.28E+01 4.55E+03 1.46E+03 1.45E+03 6
C21-RIME 1.28E−05 2.12E−04 6.35E−05 5.18E−05 4 3.44E+01 6.22E+03 1.39E+03 1.67E+03 3
RND-RIME 1.06E−05 8.96E−02 9.03E−03 2.68E−02 20 9.89E+00 6.22E+03 1.59E+03 1.87E+03 8
RIME 1.11E−05 9.10E−01 1.49E−01 2.26E−01 23 1.87E+01 6.26E+03 2.58E+03 2.16E+03 22
F7F8
C01-RIME 8.64E−03 2.00E+01 1.02E+01 9.83E+00 18 8.74E−02 2.03E+01 8.38E+00 9.64E+00 3
C02-RIME 1.06E−02 2.01E+01 9.60E+00 9.78E+00 16 1.28E−01 2.06E+01 9.76E+00 9.77E+00 7
C03-RIME 8.91E−03 2.01E+01 1.03E+01 9.77E+00 20 1.62E−01 2.09E+01 1.24E+01 9.64E+00 16
C04-RIME 8.58E−03 2.00E+01 6.06E+00 9.13E+00 5 7.63E−02 2.08E+01 1.03E+01 9.94E+00 8
C05-RIME 1.45E−02 2.00E+01 6.87E+00 9.30E+00 8 9.99E−02 2.02E+01 9.68E+00 9.79E+00 6
C06-RIME 1.07E−02 2.00E+01 7.58E+00 9.46E+00 9 1.07E−01 2.04E+01 8.35E+00 9.66E+00 2
C07-RIME 1.29E−02 2.00E+01 8.94E+00 9.69E+00 12 1.24E−01 2.08E+01 1.36E+01 9.29E+00 20
C08-RIME 1.78E−02 2.00E+01 9.17E+00 9.53E+00 13 1.19E−01 2.05E+01 1.09E+01 9.87E+00 10
C09-RIME 1.33E−02 2.01E+01 9.52E+00 9.82E+00 14 5.55E−02 2.06E+01 1.03E+01 9.88E+00 9
C10-RIME 6.20E−03 2.00E+01 9.57E+00 9.76E+00 15 6.85E−02 2.03E+01 1.29E+01 9.53E+00 17
C11-RIME 1.28E−02 2.00E+01 6.26E+00 9.01E+00 6 1.89E−01 2.09E+01 1.24E+01 9.61E+00 15
C12-RIME 6.93E−03 2.01E+01 4.24E+00 7.90E+00 2 9.17E−02 2.07E+01 9.63E+00 9.88E+00 5
C13-RIME 8.55E−03 2.01E+01 1.08E+01 9.90E+00 21 8.30E−02 2.07E+01 1.36E+01 9.23E+00 21
C14-RIME 6.63E−03 2.01E+01 1.01E+01 9.91E+00 17 1.17E−01 2.09E+01 1.12E+01 9.77E+00 12
C15-RIME 7.04E−03 2.01E+01 6.87E+00 9.30E+00 7 9.22E−02 2.07E+01 1.49E+01 8.74E+00 22
C16-RIME 1.08E−02 2.01E+01 3.60E+00 7.35E+00 1 1.30E−01 2.10E+01 7.79E+00 9.52E+00 1
123
Complex & Intelligent Systems (2025) 11:177 Page 27 of 42 177
Table 3 continued
F7F8
C17-RIME 9.87E−03 2.01E+01 7.72E+00 9.38E+00 10 5.80E−02 2.08E+01 1.11E+01 9.73E+00 11
C18-RIME 5.72E−03 2.00E+01 4.95E+00 8.32E+00 4 6.04E−02 2.03E+01 1.29E+01 9.50E+00 18
C19-RIME 6.97E−03 2.00E+01 4.89E+00 8.35E+00 3 8.42E−02 2.03E+01 1.23E+01 9.65E+00 14
C20-RIME 9.51E−03 2.06E+01 1.02E+01 9.85E+00 19 8.99E−02 2.04E+01 9.09E+00 9.68E+00 4
C21-RIME 6.89E−03 2.01E+01 8.92E+00 9.70E+00 11 1.08E−01 2.08E+01 1.36E+01 9.31E+00 19
RND-RIME 1.57E−02 2.00E+01 1.09E+01 9.80E+00 22 1.28E−01 2.04E+01 1.17E+01 9.66E+00 13
RIME 7.52E−03 2.01E+01 1.22E+01 9.59E+00 23 5.16E−02 2.08E+01 1.57E+01 8.36E+00 23
F9F10
C01-RIME 2.29E+02 2.29E+02 2.29E+02 7.81E−06 23 1.00E+02 2.13E+02 1.18E+02 4.06E+01 21
C02-RIME 2.29E+02 2.29E+02 2.29E+02 8.07E−06 20 1.25E−01 2.12E+02 1.02E+02 5.41E+01 9
C03-RIME 2.29E+02 2.29E+02 2.29E+02 8.11E−06 21 3.48E+00 2.13E+02 9.52E+01 4.39E+01 5
C04-RIME 2.29E+02 2.29E+02 2.29E+02 3.54E−06 10 6.31E−02 2.13E+02 9.20E+01 4.69E+01 3
C05-RIME 2.29E+02 2.29E+02 2.29E+02 6.56E−06 16 1.88E−01 2.10E+02 1.02E+02 6.04E+01 8
C06-RIME 2.29E+02 2.29E+02 2.29E+02 5.85E−06 17 6.27E−02 2.15E+02 1.05E+02 5.11E+01 12
C07-RIME 2.29E+02 2.29E+02 2.29E+02 4.39E−06 14 1.00E+02 2.19E+02 1.26E+02 4.70E+01 22
C08-RIME 2.29E+02 2.29E+02 2.29E+02 9.09E−06 18 1.88E−01 1.00E+02 9.37E+01 2.45E+01 4
C09-RIME 2.29E+02 2.29E+02 2.29E+02 4.62E−06 15 2.50E−01 2.08E+02 1.05E+02 4.10E+01 10
C10-RIME 2.29E+02 2.29E+02 2.29E+02 3.86E−07 1 1.87E−01 2.13E+02 9.54E+01 5.23E+01 6
C11-RIME 2.29E+02 2.29E+02 2.29E+02 6.01E−06 19 2.50E−01 2.14E+02 1.05E+02 4.25E+01 14
C12-RIME 2.29E+02 2.29E+02 2.29E+02 3.37E−06 7 1.88E−01 2.10E+02 1.08E+02 4.65E+01 15
C13-RIME 2.29E+02 2.29E+02 2.29E+02 3.51E−06 3 6.25E−02 2.09E+02 9.79E+01 3.24E+01 7
C14-RIME 2.29E+02 2.29E+02 2.29E+02 4.18E−06 6 2.50E−01 2.14E+02 1.11E+02 5.87E+01 16
C15-RIME 2.29E+02 2.29E+02 2.29E+02 3.76E−06 11 1.88E−01 2.13E+02 1.14E+02 6.79E+01 19
C16-RIME 2.29E+02 2.29E+02 2.29E+02 6.19E−06 22 1.88E−01 2.10E+02 1.05E+02 4.97E+01 13
C17-RIME 2.29E+02 2.29E+02 2.29E+02 3.48E−06 8 6.27E−02 2.11E+02 9.08E+01 4.01E+01 2
C18-RIME 2.29E+02 2.29E+02 2.29E+02 2.52E−06 4 1.25E−01 2.07E+02 8.40E+01 4.55E+01 1
C19-RIME 2.29E+02 2.29E+02 2.29E+02 3.90E−06 9 6.26E−02 2.12E+02 1.12E+02 5.00E+01 17
C20-RIME 2.29E+02 2.29E+02 2.29E+02 2.31E−06 5 1.25E−01 2.13E+02 1.16E+02 5.89E+01 20
C21-RIME 2.29E+02 2.29E+02 2.29E+02 4.17E−06 13 2.50E−01 2.08E+02 1.05E+02 4.19E+01 11
RND-RIME 2.29E+02 2.29E+02 2.29E+02 4.39E−06 12 1.88E−01 2.15E+02 1.12E+02 5.03E+01 18
RIME 2.29E+02 2.29E+02 2.29E+02 1.04E−06 2 6.77E+00 2.23E+02 1.40E+02 6.06E+01 23
F11 F12
C01-RIME 5.09E−02 3.00E+02 1.62E+01 5.93E+01 9 1.59E+02 1.66E+02 1.61E+02 1.78E+00 5
C02-RIME 5.09E−02 3.00E+02 2.05E+01 6.40E+01 12 1.59E+02 1.64E+02 1.61E+02 1.66E+00 9
C03-RIME 3.68E−02 3.00E+02 2.01E+01 6.41E+01 10 1.59E+02 1.64E+02 1.61E+02 1.57E+00 1
C04-RIME 4.50E−02 3.00E+02 6.01E+01 1.07E+02 23 1.59E+02 1.66E+02 1.61E+02 2.04E+00 13
C05-RIME 4.57E−02 3.00E+02 1.51E+01 5.94E+01 7 1.59E+02 1.63E+02 1.61E+02 1.56E+00 3
C06-RIME 3.35E−02 8.19E−02 5.85E−02 1.44E−02 1 1.59E+02 1.66E+02 1.61E+02 1.91E+00 12
C07-RIME 4.98E−02 3.00E+02 2.51E+01 6.81E+01 15 1.59E+02 1.66E+02 1.61E+02 1.88E+00 2
C08-RIME 3.75E−02 1.50E+02 1.01E+01 3.75E+01 6 1.59E+02 1.63E+02 1.61E+02 1.72E+00 7
C09-RIME 4.00E−02 3.00E+02 3.01E+01 7.15E+01 17 1.59E+02 1.67E+02 1.62E+02 2.06E+00 15
C10-RIME 1.33E−02 3.00E+02 4.01E+01 7.69E+01 20 1.59E+02 1.67E+02 1.62E+02 2.33E+00 20
C11-RIME 4.06E−02 3.00E+02 2.03E+01 6.40E+01 11 1.59E+02 1.64E+02 1.61E+02 1.92E+00 11
C12-RIME 2.67E−02 1.50E+02 3.01E+01 6.01E+01 18 1.59E+02 1.67E+02 1.62E+02 1.97E+00 22
C13-RIME 2.98E−02 1.50E+02 1.53E+01 4.51E+01 8 1.59E+02 1.67E+02 1.62E+02 2.41E+00 18
123
177 Page 28 of 42 Complex & Intelligent Systems (2025) 11:177
Table 3 continued
F11 F12
C14-RIME 3.80E−02 3.00E+02 2.51E+01 6.81E+01 14 1.59E+02 1.67E+02 1.62E+02 2.38E+00 21
C15-RIME 3.78E−02 4.00E+02 3.85E+01 8.73E+01 19 1.59E+02 1.65E+02 1.62E+02 1.67E+00 16
C16-RIME 5.38E−02 3.49E−01 1.25E−01 5.23E−02 2 1.59E+02 1.64E+02 1.61E+02 1.80E+00 6
C17-RIME 3.43E−02 3.00E+02 4.01E+01 8.61E+01 21 1.59E+02 1.65E+02 1.61E+02 2.03E+00 4
C18-RIME 2.61E−02 1.50E+02 5.32E+00 2.70E+01 4 1.59E+02 1.66E+02 1.62E+02 2.09E+00 19
C19-RIME 3.51E−02 3.00E+02 2.51E+01 7.83E+01 13 1.59E+02 1.64E+02 1.62E+02 1.79E+00 14
C20-RIME 4.13E−02 1.50E+02 5.07E+00 2.70E+01 3 1.59E+02 1.66E+02 1.61E+02 2.10E+00 10
C21-RIME 3.35E−02 3.00E+02 3.01E+01 8.13E+01 16 1.59E+02 1.64E+02 1.61E+02 1.77E+00 8
RND-RIME 3.34E−02 1.50E+02 1.01E+01 3.75E+01 5 1.59E+02 1.67E+02 1.62E+02 2.46E+00 17
RIME 1.85E−02 3.00E+02 4.51E+01 9.61E+01 22 1.59E+02 1.69E+02 1.63E+02 2.74E+00 23
The bold values indicate the best-performing results for each criterion
Table 4 Friedman test results for RIME, RND-RIME and all Chaotic variants
Algorithm C01-RIME C02-RIME C03-RIME C04-RIME C05-RIME C06-RIME C07-RIME C08-RIME
Mean ranks 14.00 13.42 13.67 13.08 10.83 9.42 15.08 13.33
Overall ranking 21 18 20 16 9 2 22 17
Algorithm C09-RIME C10-RIME C11-RIME C12-RIME C13-RIME C14-RIME C15-RIME C16-RIME
Mean ranks 13.42 10.75 11.83 9.92 11.58 10.58 12.33 11.83
Overall ranking 19 8 12 4 11 7 14 13
Algorithm C17-RIME C18-RIME C19-RIME C20-RIME C21-RIME RND-RIME RIME
Mean ranks 10.17 7.92 10.33 9.67 11.42 12.75 18.67
Overall ranking 5 163101523
The bold values indicate the best-performing results for each criterion
For function F6, C10-RIME and C18-RIME exhibit out-
standing performance with the lowest error and standard
deviation values, while C01-RIME ranks the highest overall,
and the original RIME is last. The functions F7and F8are
notable for C16-RIME’s top-ranking performance in both,
demonstrating their specific strength in these optimization
tasks.
The remaining functions, F9through F12, also reflect
similar trends. For functions F9and F10, variants such as
C10-RIME and C18-RIME stand out, showcasing their abil-
ity to achieve low error rates and high consistency. Function
F11 shows that C06-RIME and C16-RIME perform excep-
tionally well, while function F12 highlights the effectiveness
of C03-RIME and C07-RIME.
Table 4delineates the mean ranks derived from the Fried-
man test and the overall rankings of all chaotic variants of
the RIME optimizer. The mean rank serves as a consoli-
dated metric of each variant’s performance across the twelve
benchmark functions (F1−F12). A lower mean rank signi-
fies superior overall performance. Based on the data shown
in Table 4C18-RIME emerges as the top performer, securing
the lowest mean ranks and the highest overall rankings.
Notably, all variants outperform the original optimizer,
which ranked last, thereby demonstrating the effectiveness
of the non-chaotic modifications implemented on this opti-
mizer. The efficacy of the chaotic improvement is evident as
14 out of 21 variants outperformed the RND-RIME. These
include: C06-RIME, C20-RIME, C12-RIME, C17-RIME,
C19-RIME, C14-RIME, C10-RIME, C5-RIME, C21-RIME,
C13-RIME, C11-RIME, C16-RIME, and C15-RIME, in
descending order of performance.
Based on these findings, the variant with the most robust
performance, C18-RIME, is selected for subsequent exper-
iments and will henceforth be referred to as C-RIME.
C18-RIME is selected because of its robust performance
across a variety of optimization scenarios, thereby ensuring
its effectiveness in further testing.
Comparison between C-RIME and state-of-the-art
The purpose of this experiment is to benchmark the per-
formance of the chaotic RIME optimizer against several
state-of-the-art metaheuristic algorithms. Table 5summa-
123
Complex & Intelligent Systems (2025) 11:177 Page 29 of 42 177
Table 5 Results of C-RIME and competitive algorithms (dimension = 10)
Algorithm MIN MAX AVG STD Rank MIN MAX AVG STD Rank
F1F2
GWO 5.80E−02 6.03E+03 6.86E+02 1.37E+03 9 2.94E−01 7.12E+01 2.28E+01 2.10E+01 8
HHO 2.41E−01 7.33E−01 4.41E−01 1.05E−01 6 3.16E−02 8.67E+01 2.04E+01 2.86E+01 7
LFD 2.57E−09 2.20E+02 1.81E+01 5.38E+01 8 5.83E−02 8.68E+01 3.80E+01 3.44E+01 9
OMA 1.71E−13 2.22E−12 5.55E−13 4.68E−13 2 1.02E−03 7.15E+01 1.12E+01 2.34E+01 4
COA 4.61E+03 1.03E+04 7.66E+03 1.43E+03 11 1.65E+02 2.90E+03 1.20E+03 6.84E+02 11
C-AOA 8.50E+00 4.05E+03 1.71E+03 1.15E+03 10 1.32E−03 2.70E+02 9.45E+01 5.89E+01 10
C-HGS 2.07E−07 1.50E−03 2.53E−04 3.76E−04 4 4.37E+00 8.01E+01 1.37E+01 2.13E+01 6
C-SMA 8.10E−06 1.50E−04 4.40E−05 3.22E−05 3 4.70E+00 8.92E+00 7.32E+00 1.85E+00 3
C-DMOA 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1 1.30E−01 4.58E+00 1.86E+00 1.10E+00 1
C-AO 1.17E−01 1.06E+01 1.79E+00 2.07E+00 7 5.71E−01 9.09E+01 1.12E+01 1.50E+01 5
C-RIME 9.88E−05 1.40E−03 7.22E−04 3.64E−04 5 4.15E−03 8.92E+00 6.19E+00 2.90E+00 2
F3F4
GWO 6.42E−03 2.94E+00 4.91E−01 7.68E−01 6 4.98E+00 2.68E+01 1.40E+01 5.14E+00 3
HHO 1.36E+00 4.38E+01 2.16E+01 1.20E+01 9 1.00E+01 4.08E+01 2.77E+01 7.37E+00 7
LFD 2.69E+00 3.97E+01 1.72E+01 7.23E+00 8 1.29E+01 5.77E+01 2.99E+01 1.30E+01 10
OMA 1.48E−06 1.11E+00 6.54E−02 2.06E−01 4 6.40E+00 2.15E+01 1.37E+01 4.50E+00 2
COA 2.51E+01 6.86E+01 4.66E+01 9.40E+00 11 2.42E+01 6.40E+01 5.01E+01 1.05E+01 11
C-AOA 1.59E+01 5.36E+01 3.03E+01 8.23E+00 10 1.99E+01 4.78E+01 2.94E+01 7.23E+00 9
C-HGS 4.50E−05 2.50E+00 4.15E−01 5.19E−01 5 8.95E+00 5.27E+01 2.87E+01 1.03E+01 8
C-SMA 1.24E−02 7.63E−02 2.38E−02 1.17E−02 3 4.97E+00 4.08E+01 2.14E+01 8.53E+00 5
C-DMOA 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1 1.00E+01 3.15E+01 2.26E+01 4.77E+00 6
C-AO 1.41E+00 2.65E+01 7.39E+00 5.10E+00 7 5.99E+00 4.08E+01 2.01E+01 8.90E+00 4
C-RIME 2.01E−03 1.69E−02 5.71E−03 2.73E−03 2 3.98E+00 1.69E+01 8.47E+00 3.10E+00 1
F5F6
GWO 2.52E−02 5.30E+01 8.04E+00 1.35E+01 5 5.53E+02 6.39E+03 3.35E+03 2.13E+03 7
HHO 5.43E+01 7.52E+02 3.94E+02 1.76E+02 8 9.54E+01 6.32E+03 1.11E+03 1.41E+03 2
LFD 1.63E+02 7.69E+02 4.43E+02 1.66E+02 10 1.09E+02 5.02E+03 1.77E+03 1.42E+03 5
OMA 4.25E−10 1.77E+01 1.55E+00 3.33E+00 4 7.12E+01 2.83E+03 5.42E+02 6.60E+02 1
COA 9.45E+01 9.19E+02 5.62E+02 1.73E+02 11 2.27E+05 1.21E+08 1.65E+07 2.48E+07 11
C-AOA 1.23E+02 7.27E+02 4.13E+02 1.53E+02 9 2.23E+01 5.01E+03 1.72E+03 1.22E+03 4
C-HGS 8.95E−02 5.68E+02 5.26E+01 1.17E+02 7 7.69E+01 6.31E+03 3.29E+03 2.11E+03 6
C-SMA 1.82E−05 1.73E+00 2.24E−01 3.94E−01 3 5.31E+02 6.29E+03 3.98E+03 1.72E+03 8
C-DMOA 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1 4.69E+02 7.18E+04 1.92E+04 1.93E+04 10
C-AO 7.87E+00 1.68E+02 5.04E+01 4.14E+01 6 1.01E+03 1.44E+04 5.62E+03 3.35E+03 9
C-RIME 7.11E−06 8.96E−02 3.06E−03 1.61E−02 2 3.79E+01 3.90E+03 1.57E+03 1.27E+03 3
F7F8
GWO 5.12E−02 5.72E+01 2.41E+01 1.28E+01 6 4.23E+00 2.90E+01 2.16E+01 5.93E+00 4
HHO 2.89E+00 1.44E+02 3.86E+01 2.57E+01 8 6.02E+00 6.00E+01 2.79E+01 9.97E+00 9
LFD 5.20E+00 6.48E+01 3.86E+01 1.52E+01 9 2.16E+01 3.92E+01 2.75E+01 4.57E+00 8
OMA 2.33E+00 2.90E+01 1.57E+01 9.14E+00 2 7.69E+00 2.76E+01 2.27E+01 5.14E+00 5
COA 4.74E+01 1.26E+02 8.47E+01 1.63E+01 11 2.84E+01 9.69E+01 4.07E+01 1.53E+01 10
C-AOA 4.83E+01 1.19E+02 8.39E+01 2.26E+01 10 1.91E+01 1.18E+02 4.68E+01 2.33E+01 11
123
177 Page 30 of 42 Complex & Intelligent Systems (2025) 11:177
Table 5 continued
F7F8
C-HGS 1.46E−05 2.01E+01 1.67E+01 7.45E+00 3 8.21E−02 3.02E+01 1.78E+01 8.26E+00 2
C-SMA 1.25E−02 2.16E+01 1.86E+01 5.58E+00 4 2.79E−01 2.16E+01 2.00E+01 3.70E+00 3
C-DMOA 1.71E+01 2.50E+01 2.24E+01 1.59E+00 5 4.05E+00 3.02E+01 2.60E+01 4.78E+00 7
C-AO 9.62E+00 5.49E+01 2.99E+01 8.87E+00 7 2.16E+01 3.11E+01 2.55E+01 2.32E+00 6
C-RIME 5.72E−03 2.00E+01 4.95E+00 8.32E+00 1 6.04E−02 2.03E+01 1.29E+01 9.50E+00 1
F9F10
GWO 2.29E+02 3.52E+02 2.60E+02 3.18E+01 9 1.00E+02 3.43E+02 1.90E+02 5.89E+01 9
HHO 2.29E+02 3.76E+02 2.39E+02 3.66E+01 7 1.00E+02 2.67E+02 1.75E+02 6.85E+01 8
LFD 1.97E+02 3.11E+02 2.53E+02 3.25E+01 8 1.01E+02 1.02E+02 1.01E+02 2.32E−01 2
OMA 2.29E+02 2.29E+02 2.29E+02 2.68E−12 3 1.00E+02 2.11E+02 1.06E+02 2.28E+01 4
COA 3.55E+02 5.56E+02 4.46E+02 3.89E+01 11 1.17E+02 6.14E+02 2.67E+02 1.36E+02 11
C-AOA 2.71E+02 3.61E+02 3.24E+02 2.47E+01 10 6.90E+00 4.99E+02 2.01E+02 1.13E+02 10
C-HGS 2.29E+02 2.29E+02 2.29E+02 8.53E−14 2 6.89E+00 2.32E+02 1.19E+02 5.64E+01 6
C-SMA 2.29E+02 2.29E+02 2.29E+02 2.43E−05 5 1.00E+02 2.23E+02 1.08E+02 3.00E+01 5
C-DMOA 1.86E+02 1.86E+02 1.86E+02 5.68E−14 1 1.00E+02 1.61E+02 1.03E+02 1.10E+01 3
C-AO 2.29E+02 2.81E+02 2.37E+02 1.30E+01 6 1.01E+02 2.33E+02 1.64E+02 5.95E+01 7
C-RIME 2.29E+02 2.29E+02 2.29E+02 2.52E−06 4 1.25E−01 2.07E+02 8.40E+01 4.55E+01 1
F11 F12
GWO 1.25E−01 6.49E+02 2.09E+02 1.82E+02 9 1.60E+02 1.93E+02 1.66E+02 8.62E+00 5
HHO 3.12E+00 4.01E+02 1.86E+02 1.42E+02 7 1.64E+02 3.51E+02 1.89E+02 3.76E+01 8
LFD 4.47E−05 3.12E+02 7.06E+01 8.56E+01 5 1.59E+02 2.00E+02 1.88E+02 1.48E+01 7
OMA 3.18E−12 4.00E+02 1.42E+02 1.26E+02 6 1.65E+02 1.72E+02 1.69E+02 1.58E+00 6
COA 2.36E+02 1.92E+03 9.41E+02 3.88E+02 11 1.75E+02 3.57E+02 2.53E+02 4.51E+01 11
C-AOA 1.45E−05 6.62E+02 2.76E+02 1.54E+02 10 1.79E+02 4.04E+02 2.42E+02 4.59E+01 10
C-HGS 1.95E−06 5.84E+02 2.01E+02 1.84E+02 8 1.59E+02 1.66E+02 1.63E+02 1.49E+00 3
C-SMA 2.36E−03 4.00E+02 6.34E+01 1.37E+02 3 1.59E+02 1.63E+02 1.61E+02 1.54E+00 1
C-DMOA 0.00E+00 1.77E+02 5.11E+01 7.24E+01 2 1.67E+02 2.00E+02 1.96E+02 8.86E+00 9
C-AO 2.31E+00 3.06E+02 6.76E+01 8.31E+01 4 1.61E+02 1.72E+02 1.65E+02 2.13E+00 4
C-RIME 2.61E−02 1.50E+02 5.32E+00 2.70E+01 1 1.59E+02 1.66E+02 1.62E+02 2.09E+00 2
Bold values indicate the best-performing results for each benchmark
rizes the experimental results for a dimension of 10, and the
results for a dimension of 20 are presented in Table 6.
The detailed analysis of these results reveals the high per-
formance of the enhanced optimizer. For instance, for F4,
C-RIME shows the lowest MIN value, indicating its supe-
rior capability to find the optimal solution. The AVG and
STD values for C-RIME in F4 are also notably low, suggest-
ing high accuracy and stability. This trend is evident in most
functions, such as F7,F8,F10 and F11, where C-RIME out-
performs the other algorithms by a significant margin. For
functions like F2,F3,F5and F12, while some competitive
algorithms like C-DMOA show competitive performance,
C-RIME still maintains comparable results, highlighting its
versatility.
Table 6extends the comparison to a higher dimensional
space (dimension = 20), providing insights into the scalabil-
ity of the algorithms. Higher dimensional spaces pose more
complex challenges for optimization algorithms.
For this more challenging scenario, C-RIME continues to
demonstrate its superior performance. For example, for F4,
F6,F7,F8, and F10, C-RIME achieves the first rank.
However, the increased dimensionality highlights the
strengths of some other algorithms for specific functions. For
example, for F6and F10, while C-RIME performs efficiently,
LFD and OMA algorithms show competitive or marginally
superior performance in certain metrics. This indicates that
while C-RIME is highly effective, other algorithms can also
be strong contenders depending on the problem’s character-
istics.
123
Complex & Intelligent Systems (2025) 11:177 Page 31 of 42 177
Table 6 Results of C-RIME and competitive algorithms (dimension = 20)
Algorithm MIN MAX AVG STD Rank MIN MAX AVG STD Rank
F1F2
GWO 1.45E+03 1.78E+04 7.58E+03 4.40E+03 6 4.70E+01 2.31E+02 9.67E+01 4.51E+01 8
HHO 3.43E+00 1.49E+01 8.77E+00 3.10E+00 4 7.21E+00 7.60E+01 5.85E+01 1.48E+01 6
LFD 5.64E+03 2.26E+04 1.28E+04 4.65E+03 7 1.33E+01 2.56E+02 1.08E+02 5.37E+01 9
OMA 1.62E−02 3.96E+00 7.89E−01 1.04E+00 3 4.94E+00 7.44E+01 5.66E+01 1.54E+01 5
COA 2.32E+04 7.29E+04 4.46E+04 1.20E+04 10 1.35E+03 3.93E+03 2.65E+03 7.38E+02 11
C-AOA 1.49E+04 4.35E+04 2.63E+04 7.66E+03 9 4.04E+02 1.69E+03 7.90E+02 2.73E+02 10
C-HGS 8.67E−01 1.28E+04 8.85E+02 2.49E+03 5 3.81E+00 7.34E+01 4.30E+01 2.24E+01 2
C-SMA 4.40E−04 4.30E−03 1.76E−03 9.11E−04 1 6.27E+00 7.72E+01 4.43E+01 1.65E+01 3
C-DMOA 2.33E+04 8.31E+04 5.29E+04 1.45E+04 11 1.41E+01 7.28E+01 1.70E+01 1.04E+01 1
C-AO 3.78E+03 3.13E+04 1.77E+04 6.64E+03 8 4.81E+01 1.38E+02 7.46E+01 2.40E+01 7
C-RIME 3.16E−02 2.64E−01 1.09E−01 4.57E−02 2 4.49E+01 7.27E+01 5.12E+01 7.02E+00 4
F3F4
GWO 1.64E−01 1.00E+01 4.14E+00 3.05E+00 5 1.64E+01 9.11E+01 4.25E+01 1.35E+01 2
HHO 3.20E+01 6.81E+01 5.08E+01 9.52E+00 10 5.74E+01 1.31E+02 8.75E+01 1.87E+01 7
LFD 3.28E+01 6.53E+01 4.70E+01 9.58E+00 9 4.38E+01 1.60E+02 9.24E+01 2.35E+01 9
OMA 5.79E−01 2.75E+01 8.03E+00 6.48E+00 6 2.59E+01 9.64E+01 5.91E+01 1.76E+01 3
COA 5.35E+01 9.22E+01 8.08E+01 7.75E+00 11 1.47E+02 2.06E+02 1.76E+02 1.44E+01 11
C-AOA 2.27E+01 4.90E+01 3.79E+01 6.41E+00 8 4.18E+01 1.40E+02 7.81E+01 1.92E+01 6
C-HGS 5.77E−02 1.40E+01 2.43E+00 2.62E+00 4 5.68E+01 1.31E+02 8.97E+01 1.85E+01 8
C-SMA 9.17E−02 5.99E−01 2.20E−01 1.18E−01 3 3.28E+01 1.27E+02 6.79E+01 2.35E+01 5
C-DMOA 1.14E−13 1.14E−13 1.14E−13 1.26E−29 1 6.87E+01 1.22E+02 1.04E+02 9.36E+00 10
C-AO 2.01E+01 4.89E+01 2.95E+01 5.68E+00 7 3.13E+01 8.98E+01 6.35E+01 1.58E+01 4
C-RIME 1.05E−02 6.18E−02 2.92E−02 1.15E−02 2 9.95E+00 4.88E+01 2.72E+01 9.31E+00 1
F5F6
GWO 1.30E+01 7.85E+02 1.62E+02 1.65E+02 3 2.05E+02 2.02E+07 1.44E+06 4.26E+06 9
HHO 1.04E+03 2.27E+03 1.69E+03 2.86E+02 10 3.42E+03 9.63E+04 4.56E+04 2.41E+04 7
LFD 1.03E+03 3.30E+03 1.64E+03 4.68E+02 9 2.18E+02 1.84E+04 3.55E+03 3.60E+03 3
OMA 1.01E+01 7.29E+02 2.44E+02 1.66E+02 4 1.04E+02 1.22E+04 2.99E+03 3.03E+03 2
COA 2.05E+03 3.19E+03 2.69E+03 3.17E+02 11 6.38E+08 4.76E+09 2.57E+09 9.24E+08 11
C-AOA 6.66E+02 3.60E+03 1.37E+03 5.05E+02 8 7.44E+01 2.13E+07 7.12E+05 3.82E+06 8
C-HGS 1.88E+02 2.12E+03 9.47E+02 4.42E+02 6 9.67E+01 2.33E+04 8.09E+03 8.52E+03 4
C-SMA 9.99E−01 1.44E+03 3.14E+02 4.48E+02 5 1.69E+02 2.35E+04 1.93E+04 6.06E+03 5
C-DMOA 0.00E+00 8.95E−02 2.98E−03 1.61E−02 1 7.41E+04 1.28E+07 3.44E+06 3.32E+06 10
C-AO 3.38E+02 1.86E+03 1.03E+03 3.50E+02 7 6.85E+03 1.07E+05 4.08E+04 2.12E+04 6
C-RIME 2.69E−04 6.34E−01 7.63E−02 1.39E−01 2 1.33E+02 1.47E+04 2.95E+03 3.54E+03 1
F7F8
GWO 2.64E+01 1.68E+02 5.89E+01 3.07E+01 5 2.39E+01 2.66E+02 5.09E+01 5.65E+01 7
HHO 7.40E+01 3.27E+02 1.51E+02 5.04E+01 10 2.87E+01 5.83E+01 4.04E+01 9.36E+00 6
LFD 4.89E+01 1.73E+02 1.04E+02 3.41E+01 8 2.59E+01 1.74E+02 7.24E+01 5.27E+01 9
OMA 4.03E+01 9.41E+01 6.22E+01 1.49E+01 6 2.50E+01 3.91E+01 3.16E+01 2.61E+00 4
123
177 Page 32 of 42 Complex & Intelligent Systems (2025) 11:177
Table 6 continued
F7F8
COA 1.62E+02 3.46E+02 2.27E+02 4.05E+01 11 5.24E+01 5.74E+02 3.07E+02 1.34E+02 11
C-AOA 7.49E+01 2.72E+02 1.38E+02 3.90E+01 9 3.33E+01 3.61E+02 1.69E+02 9.52E+01 10
C-HGS 2.55E+01 9.78E+01 5.15E+01 1.72E+01 3 2.18E+01 3.83E+01 2.48E+01 3.78E+00 2
C-SMA 2.34E+01 8.74E+01 4.07E+01 1.80E+01 2 2.06E+01 1.64E+02 2.77E+01 2.57E+01 3
C-DMOA 2.99E+01 8.86E+01 5.31E+01 1.07E+01 4 4.18E+01 8.90E+01 6.36E+01 1.27E+01 8
C-AO 4.38E+01 1.45E+02 7.77E+01 2.31E+01 7 2.81E+01 4.90E+01 3.50E+01 4.63E+00 5
C-RIME 4.53E+00 5.13E+01 2.82E+01 9.98E+00 1 1.78E+01 2.40E+01 2.22E+01 1.04E+00 1
F9F10
GWO 1.81E+02 2.58E+02 2.08E+02 2.03E+01 8 1.00E+02 1.89E+03 8.99E+02 6.19E+02 9
HHO 1.81E+02 1.88E+02 1.84E+02 1.69E+00 6 8.01E+01 1.88E+03 6.55E+02 4.34E+02 8
LFD 1.95E+02 4.53E+02 2.67E+02 6.50E+01 9 1.01E+02 1.01E+02 1.01E+02 2.10E−01 2
OMA 1.81E+02 1.81E+02 1.81E+02 6.55E−04 2 1.01E+02 2.84E+02 1.13E+02 4.40E+01 3
COA 5.27E+02 1.89E+03 1.15E+03 3.54E+02 11 6.00E+02 5.07E+03 3.77E+03 1.23E+03 11
C-AOA 3.00E+02 1.23E+03 5.12E+02 1.89E+02 10 4.16E+00 8.10E+02 1.48E+02 2.32E+02 4
C-HGS 1.81E+02 1.92E+02 1.83E+02 2.93E+00 5 1.01E+02 9.91E+02 3.46E+02 2.38E+02 5
C-SMA 1.81E+02 1.81E+02 1.81E+02 2.18E−02 4 1.00E+02 7.67E+02 3.70E+02 2.19E+02 6
C-DMOA 1.65E+02 1.65E+02 1.65E+02 2.44E−03 11.02E+02 4.24E+03 2.17E+03 1.27E+03 10
C-AO 1.82E+02 2.16E+02 1.92E+02 9.98E+00 7 1.01E+02 2.25E+03 4.53E+02 6.34E+02 7
C-RIME 1.81E+02 1.81E+02 1.81E+02 7.39E−03 3 2.07E+00 2.54E+02 8.64E+01 6.42E+01 1
F11 F12
GWO 3.14E+02 1.70E+03 8.72E+02 3.23E+02 9 2.41E+02 3.48E+02 2.66E+02 2.29E+01 6
HHO 3.36E+02 7.57E+02 3.68E+02 7.59E+01 7 2.55E+02 6.85E+02 3.54E+02 9.57E+01 9
LFD 1.32E−03 3.59E+03 4.63E+02 6.08E+02 8 2.00E+02 4.49E+02 2.19E+02 5.75E+01 2
OMA 3.00E+02 4.00E+02 3.17E+02 3.73E+01 5 2.58E+02 3.91E+02 2.94E+02 2.68E+01 8
COA 4.45E+03 6.85E+03 6.05E+03 5.23E+02 11 4.61E+02 1.57E+03 8.86E+02 2.53E+02 11
C-AOA 2.37E+03 5.91E+03 4.43E+03 8.98E+02 10 3.66E+02 1.03E+03 6.23E+02 1.74E+02 10
C-HGS 2.17E+00 7.60E+02 3.47E+02 1.31E+02 6 2.36E+02 4.86E+02 2.59E+02 4.32E+01 5
C-SMA 4.88E−02 4.00E+02 3.00E+02 1.10E+02 3 2.35E+02 2.52E+02 2.44E+02 4.26E+00 4
C-DMOA 3.00E+02 3.00E+02 3.00E+02 8.29E−06 22.00E+02 2.00E+02 2.00E+02 1.02E−04 1
C-AO 1.82E+01 7.82E+02 2.95E+02 1.75E+02 12.50E+02 3.38E+02 2.90E+02 2.06E+01 7
C-RIME 3.00E+02 4.00E+02 3.11E+02 2.98E+01 4 2.32E+02 2.60E+02 2.43E+02 6.91E+00 3
Bold values indicate the best-performing results for each benchmark
Overall, the comparative analysis in Tables 5and 6high-
lights the robustness and scalability of C-RIME. Its superior
performance in both 10-dimensional and 20-dimensional
spaces, across a diverse set of benchmark functions, under-
scores its potential as a versatile and powerful optimization
tool. The integration of chaotic maps into the RIME optimizer
appears to significantly enhance its performance, providing
a balanced approach to exploration and exploitation.
To ensure the observed performance differences are sta-
tistically significant, we conducted Friedman and Wilcoxon
signed-rank tests. Figure 5presents the Friedman test results,
which rank the performance of the algorithms used across
multiple functions, showing that C-RIME achieves the high-
est rank, indicating its superior overall performance. Table 7
displays the Wilcoxon test results. A plus (+) symbol indi-
cates a statistically significant difference (pvalue <0.05)
between C-RIME and the compared algorithm, while an
equality (=) symbol denotes no statistically significant dif-
ference. Together, the Friedman and Wilcoxon tests confirm
that C-RIME significantly outperforms most of the compared
algorithms, further validating its effectiveness.
The convergence properties of the chaotic RIME opti-
mizer were evaluated by plotting the average error of fitness
values over iterations for selected benchmark functions. As
shown in Fig. 4, the results highlight the excellent conver-
gence rate of C-RIME, which achieved the first rank for F4,
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Complex & Intelligent Systems (2025) 11:177 Page 33 of 42 177
Table 7 Wilcoxon signed-rank test results for C-RIME versus competitive algorithms
Dim Func GWO HHO LFD OMA COA C-AOA C-HGS C-SMA C-DMOA C-AO
D10 F11.73E−06 + 1.73E−06 + 9.75E−01 = 1.73E−06 + 1.73E−06 + 1.73E−06 + 3.88E−04 + 1.73E−06 + 1.73E−06 + 1.73E−06 +
F22.05E−04 + 2.89E−01 + 1.11E−02 + 7.86E−02 = 1.73E−06 + 2.35E−06 + 6.44E−01 = 1.92E−01 = 6.34E−06 + 2.16E−05 +
F33.52E−06 + 1.73E−06 + 1.73E−06 + 8.13E−01 = 1.73E−06 + 1.73E−06 + 8.19E−05 + 1.73E−06 + 1.73E−06 + 1.73E−06 +
F42.05E−04 + 1.73E−06 + 1.73E−06 + 7.51E−05 + 1.73E−06 + 1.73E−06 + 2.13E−06 + 6.98E−06 + 2.13E−06 + 8.47E−06 +
F51.73E−06 + 1.73E−06 + 1.73E−06 + 7.69E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 5.29E−04 + 1.73E−06 + 1.73E−06 +
F66.42E−03 + 1.06E−01 + 7.19E−01 = 2.11E−03 + 1.73E−06 + 7.81E−01 = 8.31E−04 + 3.11E−05 + 6.34E−06 + 1.73E−06 +
F77.69E−06 + 1.73E−06 + 1.73E−06 + 3.59E−04 + 1.73E−06 + 1.73E−06 + 1.11E−03 + 1.13E−05 + 1.92E−06 + 1.73E−06 +
F82.41E−04 + 4.45E−05 + 1.73E−06 + 4.53E−04 + 1.73E−06 + 1.92E−06 + 1.59E−03 + 6.34E−06 + 1.24E−05 + 1.73E−06 +
F91.73E−06 + 1.73E−06 + 1.11E−03 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 +
F10 1.73E−06 + 6.98E−06 + 3.11E−05 + 2.84E−05 + 1.73E−06 + 5.79E−05 + 8.92E−05 + 1.60E−04 + 3.41E−05 + 6.98E−06 +
F11 9.32E−06 + 3.88E−06 + 2.06E−01 = 7.71E−04 + 1.73E−06 + 1.92E−06 + 6.16E−04 + 1.53E−01 = 7.34E−01 = 1.02E−05 +
F12 3.85E−03 + 1.92E−06 + 2.35E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 6.56E−02 = 7.19E−02 = 1.73E−06 + 1.80E−05 +
D20 F11.73E−06 + 1.73E−06 + 1.73E−06 + 2.41E−04 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 +
F25.75E−06 + 1.59E−03 + 1.36E−05 + 3.16E−03 + 1.73E−06 + 1.73E−06 + 2.29E−01 = 1.71E−03 + 1.92E−06 + 2.16E−05 +
F31.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 +
F48.19E−05 + 1.73E−06 + 1.73E−06 + 7.69E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 2.35E−06 +
F51.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 9.32E−06 + 1.73E−06 +
F67.73E−03 + 1.73E−06 + 2.29E−01 = 7.19E−01 = 1.73E−06 + 1.85E−01 = 4.39E−03 + 2.88E−06 + 1.73E−06 + 1.73E−06 +
F74.07E−05 + 1.73E−06 + 1.73E−06 + 2.13E−06 + 1.73E−06 + 1.73E−06 + 2.37E−05 + 7.71E−04 + 1.73E−06 + 1.73E−06 +
F81.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 1.13E−05 + 6.58E−01 + 1.73E−06 + 1.73E−06 +
F91.73E−06 + 1.73E−06 + 1.73E−06 + 6.98E−06 + 1.73E−06 + 1.73E−06 + 2.88E−06 + 4.07E−02 + 1.73E−06 + 1.73E−06 +
F10 3.88E−06 + 2.35E−06 + 4.28E−02 + 2.70E−02 + 1.73E−06 + 7.34E−01 = 5.75E−06 + 4.29E−06 + 2.35E−06 + 4.07E−05 +
F11 1.73E−06 + 7.71E−04 + 5.44E−01 = 2.70E−02 + 1.73E−06 + 1.73E−06 + 7.73E−03 + 2.13E−01 = 1.73E−06 + 7.19E−01 =
F12 2.37E−05 + 1.73E−06 + 2.77E−03 + 1.73E−06 + 1.73E−06 + 1.73E−06 + 4.90E−04 + 4.41E−01 = 1.73E−06 + 1.73E−06 +
123
177 Page 34 of 42 Complex & Intelligent Systems (2025) 11:177
Fig. 4 Convergence curves of C-RIME and competitive algorithms
123
Complex & Intelligent Systems (2025) 11:177 Page 35 of 42 177
Fig. 5 Friedman test results for C-RIME and competitive algorithms
F6,F7,F8, and F10. This superior performance is evident
as C-RIME rapidly approaches the optimal solution in these
functions. For F2, C-RIME showed similar behaviour to the
original RIME across all iterations, ultimately ranking after
C-DMOA, C-HGS, and C-SMA. It achived the second rank
for F3,F5, and F9. Additionally, C-RIME ranked third for
F12. The only cases where C-RIME performed slightly worse
than the original RIME are for F1and F11. Overall, these
convergence curves confirm C-RIME’s robust and efficient
performance.
Conclusion
This study provides a comprehensive analysis of chaotic-
based metaheuristic algorithms developed between 2013 and
2024. We identified 27 discrete chaotic maps frequently
employed to integrate chaos theory into metaheuristic algo-
rithms, detailing their equations and domains of definition.
A novel classification scheme was established, categorizing
CMAs into different categories: chaos-based initialization,
the substitution of random variables in the search opera-
tors, fine-tuning of constant/user-defined parameters, driving
high-level strategies, enhancing local search mechanisms,
and inspiring new metaheuristic algorithms.
CMAs have demonstrated remarkable efficacy in tack-
ling a diverse array of real-world problems, particularly those
characterized by non-linearity, multiple constraints, and high
dimensionality. Chaos-based metaheuristic algorithms have
found applications in various domains such as machine
learning, image processing, parameter estimation, medical
diagnosis, power systems, robotics, and engineering design.
These applications highlight the enhanced exploration and
exploitation capabilities provided by chaotic systems, lead-
ing to more efficient problem-solving strategies.
In this context, we introduced an improved variant of
the RIME optimizer, termed Chaotic RIME (C-RIME).
This variant employs 21 different chaotic maps to adjust
the RimeFactor parameter, with the piecewise map demon-
strating superior performance. The C-RIME optimizer was
benchmarked against ten diverse algorithms, including both
chaotic variants and non-chaotic counterparts. The results
demonstrably validated C-RIME’s promising performance
and accelerated convergence characteristics.
The findings of this research highlight the potential of
chaos theory to significantly enhance the performance of
metaheuristic algorithms. The field has reached a point
where chaos theory is a viable method for enhancing meta-
heuristic algorithms, comparable to other techniques such
as levy-flights and opposition-based learning. However, fur-
ther refinement is necessary, particularly in addressing the
question of whether chaos outperforms randomness and in
developing methods for directly selecting suitable chaotic
maps without extensive experimentation. A more practical,
adaptive approach to chaos map selection would be especially
valuable, as it could streamline applications and improve
the effectiveness of CMAs across varied optimization tasks.
Finally, exploring the potential of uninvestigated chaotic
maps within the CMAs holds promise for further perfor-
mance improvements.
Acknowledgements The authors present their appreciation to King
Saud University for funding the publication of this research through
the Researchers Supporting Program Number (RSPD2024R809), King
Saud University, Riyadh, Saudi Arabia.
Author Contributions Abdelhadi Limane: Conceptualization, Method-
ology, Validation, Formal analysis, Data Curation, Software, Writing—
Original Draft, Writing—Review & Editing. Farouq Zitouni: Concep-
tualization, Methodology, Validation, Formal analysis, Data Curation,
Software, Supervision, Writing—Original Draft, Writing—Review
& Editing. Saad Harous: Conceptualization, Methodology, Valida-
tion, Formal analysis, Data Curation, Supervision, Writing—Original
Draft, Writing—Review & Editing. Rihab Lakbichi: Conceptual-
ization, Methodology, Validation, Formal analysis, Data Curation,
Writing—Original Draft, Writing—Review & Editing. Aridj Fer-
hat: Conceptualization, Methodology, Validation, Formal analysis,
Data Curation, Writing—Original Draft, Writing—Review & Edit-
ing. Abdulaziz S. Almazyad: Conceptualization, Writing—Original
Draft, Writing—Review & Editing. Pradeep Jangir: Conceptualiza-
tion, Writing—Original Draft, Writing—Review & Editing. Ali Wagdy
Mohamed: Conceptualization, Writing—Original Draft, Writing—
Review & Editing.
Funding The research is funded by Researchers Supporting Program
Number (RSPD2024R809), King Saud University, Riyadh, Saudi Ara-
bia.
Data availability No data was used for the research described in the
article.
123
177 Page 36 of 42 Complex & Intelligent Systems (2025) 11:177
Code availability The Matlab code of the developed C-RIME optimizer
is publicly available at https://www.mathworks.com/matlabcentral/
fileexchange/178594-chaotic- rime-optimizer.
Declarations
Conflict of interest The authors declare that they have no known com-
peting financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Open Access This article is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License,
which permits any non-commercial use, sharing, distribution and repro-
duction in any medium or format, as long as you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons licence, and indicate if you modified the licensed mate-
rial. You do not have permission under this licence to share adapted
material derived from this article or parts of it. The images or other
third party material in this article are included in the article’s Creative
Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons
licence and your intended use is not permitted by statutory regula-
tion or exceeds the permitted use, you will need to obtain permission
directly from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by-nc-nd/4.0/.
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