Artificial Intelligence Review (2024) 57:59

https://doi.org/10.1007/s10462-023-10653-7

A novel metaheuristic inspired by horned lizard defense

tactics

Hernán Peraza‑Vázquez1 · Adrián Peña‑Delgado2 · Marco Merino‑Treviño1,3 ·

Ana Beatriz Morales‑Cepeda3 · Neha Sinha4

Accepted: 19 December 2023 / Published online: 16 February 2024

© The Author(s) 2024

Abstract
This paper introduces HLOA, a novel metaheuristic optimization algorithm that mathemat-
ically mimics crypsis, skin darkening or lightening, blood-squirting, and move-to-escape
defense methods. In crypsis behavior, the lizard changes its color by becoming translu-
cent to avoid detection by its predators. The horned lizard can lighten or darken its skin,
depending on whether or not it needs to decrease or increase its solar thermal gain. The
skin darkening or lightening strategy is modeled by including the stimulating hormone
melanophore rate( 𝛼-MHS) that influences these skin color changes. Further, the move-
to-evasion strategy is also mathematically described. The horned lizard’s shooting blood
defense mechanism, described as a projectile motion, is also modeled. These strategies bal-
ance exploitation and exploration mechanisms for local and global search over the solution
space. HLOA performance is benchmarked with sixty-three optimization problems from
the literature, testbench problems provided in IEEE CEC- 2017 “Constrained Real-Param-
eter Optimization”, analyzed for dimensions 10, 30, 50, and 100, as well as testbench func-
tions from IEEE CEC-06 2019 “100-Digit Challenge”. Moreover, three real-world con-
straint optimization applications from IEEE CEC2020 and two engineering problems, the
multiple gravity assist optimization and the optimal power flow problem, are also studied.
Wilcoxon and Friedman statistics tests compare the HLOA algorithm results against ten
recent bio-inspired algorithms. Wilcoxon shows that HLOA provides the optimal solution
for most testbench functions more effectively than competing algorithms. At the same time,
the Friedman statistics test ranks the HLOA first, and the n-dimensional analysis shows
that it performs better on the constrained optimization problems for dimensions 50 and
100. The source code is free and available from https://​www.​mathw​orks.​com/​matla​bcent​
ral/​filee​xchan​ge/​159658-​horned-​lizard-​optim​izati​on-​algor​ithm-​hloa.

Keywords Optimization · Bio-inspired algorithm · Metaheuristics · Horned Lizard ·

Crypsis

Extended author information available on the last page of the article

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Fig. 1  Metaheuristics classification

1 Introduction

Optimization determines the optimal values of a problem’s variables to minimize (or maxi-
mize) an objective function. It can be represented in the following way:
Given a function f to minimize, where f ∶ ℝD → ℝ , with D as the dimension of the
problem (number of variables), sought a vector xo ∈ ℝD such that f (xo ) ⩽ f (x) ∀ x ∈ ℝD ,
while xo satisfying inequality and equality constraints, gp (x0 ) ⩽ 0 and, hq (x0 ) = 0 , with
p = 1, 2, .., P and, q = 1, 2, .., Q , where P and Q, are number of inequality and equal-
ity constraints, respectively. Additionally, it must be met that xil ⩽ xi ⩽ xiu , where
i = 1, 2, .., D and i-th variable varies in the interval [xil , xiu ].
Optimization problem-solving strategies are classified as Deterministic or Stochas-
tic. Deterministic methods are grouped as gradient-based or not gradient-based, where
both classifications show good performance for linear, convex, and simple optimization
tasks. Nevertheless, these methods don’t work in certain cases that involve complex
problems, objective functions that can’t be differentiated, search spaces that aren’t lin-
ear, problems that aren’t convex, and NP-hard problems. Given that many real-world
optimization problems are NP-hard, the scientific community frequently uses stochastic
methodologies such as metaheuristics instead of deterministic methods.
Some of the metaheuristics and their classification are depicted in Fig. 1 and
described as follows:

• Evolutionary algorithm: Genetic Algorithm (GA) (Holland 2006), Differential Evo-

lution (DE) (Ahmad et al. 2022), Cultural Algorithm (Maheri et al. 2021)(CA),
Memetic Algorithm (MA) (Ahrari and Essam 2022), Evolutionary Strategy (ES)
(Beyer et al. 2002), and Gradient Evolution Algorithm (GEA) (Kim and Lee 2023).
• Swarm-based algorithm: Particle Swarm Optimization(PSO) (Kennedy and Eberhart
1995), Dingo Optimization Algorithm (DOA) (Peraza-Vázquez et al. 2021), Black

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A novel metaheuristic inspired by horned lizard defense tactics Page 3 of 65 59

Widow Optimization Algorithm (BWOA) (Peña-Delgado et al. 2020), Jumping Spi-

der optimization Algorithm (JSOA) (Peraza-Vázquez et al. 2021), Fennec Fox Algo-
rithm (FFA) (Trojovska et al. 2022), Artificial Rabbits Optimization (ARO) (Wang
et al. 2022), Tunicate Swarm Algorithm (TSA) (Kaur et al. 2020), and Mountain
Gazelle Optimizer (Abdollahzadeh et al. 2022).
• Ancient-based algorithm: Giza Pyramids Construction (GPC) (Harifi et al. 2021) and
Al-Biruni Earth Radius (BER) Metaheuristic Search Optimization Algorithm (El-
Kenawy et al. 2022).
• Physics-based algorithm: Simulated Annealing (SA) (Duan and Hou 2021), Thermal
Exchange Optimization (TEO) (Kaveh and Dadras 2017), Gravitational Search Algo-
rithm (GSA) (Mittal et al. 2021), Momentum Search Algorithm (MSA) (Dehghani
and Samet 2020), Water Cycle Algorithm (WCA) (Sadollah et al. 2016), Electro-Mag-
netism Optimization (EMO) (Abedinpourshotorban et al. 2016), Kepler optimization
algorithm (KOA) (Abdel-Basset et al. 2023), and Cyclical Parthenogenesis Algorithm
(CPA) (Kaveh and Bakhshpoori 2019)
• Chemistry-based algorithm: Chemical Reaction Optimization (CRO) (Lam and Li
2012), Crystal Structure Algorithm (CryStAl) (Talatahari et al. 2021), and Artificial
Chemical Process (ACP) (Irizarry 2004).
• Human-based algorithm: Sewing Training-Based Optimization (STBO) (Dehghani
et al. 2022), Society Civilization Algorithm (SCA) (Ray and Liew 2003), Anarchic
Society Optimization (ASO) (Ahmadi-Javid 2011), Teamwork Optimization Algorithm
(TOA) (Dehghani and Trojovský 2021), Imperialist Competitive Algorithm (ICA)
(Atashpaz-Gargari and Lucas 2007), and Coronavirus Mask Protection Algorithm
(CMPA) (Yuan et al. 2023).
• Plant-based algorithm: Invasive Weed Optimization(IWO) (Xing and Gao 2014), Arti-
ficial Plant Optimization (APO) (Zhao et al. 2011), Artificial Root Foraging Algorithm
(ARFA) (Liu et al. 2017), Rooted Tree Optimization (RTO) (Labbi et al. 2016), Artifi-
cial Algae Algorithm (AAA) (Uymaz et al. 2015), Willow Catkin Optimization Algo-
rithm (WCO) (Pan et al. 2023), Strawberry Algorithm (SBA) (Khan 2018), and Water-
wheel Plant Algorithm (Abdelhamid et al. 2023).
• Music-based / Art-based algorithm: Harmony Search Algorithm (HSA) (Kim 2016),
Chaotic harmony search algorithm (CJSA) (Alatas 2010), Musical Composition Algo-
rithm (MMC) (Mora-Gutiérrez et al. 2014), Melody Search Algorithm (MSA) (Ashrafi
and Dariane 2011,) and Stochastic Paint Optimizer (Kaveh et al. 2022).
• Sport-based algorithm: Volleyball Premier League (VPL) (Moghdani and Salimifard
2018), Puzzle Optimization Algorithm (POA) (Patil et al. 2022), Running City game
optimizer (RCGO) (Ma et al. 2023), Football Game-Based Optimization (FGBO)
(Fadakar and Ebrahimi 2016), and Alpine Skiing Optimization (ASO) (Yuan et al.
2022, 2023a, b).
• Mathematical-based algorithm: Stochastic Fractal Search (SFS) (Salimi 2015), Hyper-
Spherical Search (HSS) (Karami et al. 2014), Arithmetic Optimization Algorithm
(AOA) (Abualigah et al. 2021), and Sine-Cosine Algorithm (SCA) (Mirjalili 2016).
• Single-solution-based algorithm: Large Neighbourhood Search (LNBS) (Pisinger and
Ropke 2010), Tabu Search (TS) (Yu et al. 2023), and Variable Neighbourhood Search
(VNBS) (Hansen and Mladenovići 2018).
• Hybrid algorithm: Cuckoo optimization algorithm and SailFish optimizer (COA-SFO)
(Ikram et al. 2023), Hybrid Mutualism Mechanism-inspired Butterfly and Flower Polli-
nation Optimization Algorithm (HMMB-FPOA) (Pratha et al. 2023), Butterfly Optimi-
zation Algorithm Combined with Black Widow Optimization (BFA-BWOA) (Xu et al.

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59 Page 4 of 65 H. Peraza‑Vázquez et al.

2022), Equilibrium Whale Optimization Algorithm (EWOA) (Tan and Mohamad-Saleh

2023), Improved Dingo Optimization Algorithm (IDOA) (Almazán-Covarrubias et al.
2022), Cuckoo Search and Stochastic Paint Optimizer (CSSPO) (Ismail et al. 2023),
Elite opposition-based learning, Chaotic k-best gravitational search strategy, and Grey
wolf optimizer (EOCSGWO) (Yuan et al. 2022), Adaptive resistance and Stamina
strategy-based Dragonfly algorithm (ARSSDA) (Yuan et al. 2020), and Coulomb force
search strategy-based dragonfly algorithm (CFSSDA) (Yuan et al. 2020).

Since no algorithm can effectively and efficiently solve any optimization or instances of the
same problem (Joyce and Herrmann 2017), the scientific community keeps developing new
algorithms to solve challenging optimization problems that outperform or vie with those
already described in the literature.
A generic metaheuristic framework consists of four phases described below and
depicted in Fig. 2.
Phase 1: A set of initial population vectors are randomly generated. This population will
evolve with each iteration. Each vector represents the search agent, where the population
size can affect the algorithm’s performance.

Fig. 2  The generic framework of a Metaheuristic Algorithm

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Phase 2: Fitness is the result of evaluating the entire population of the objective func-
tion of each vector. When minimizing, the population’s lowest fitness vector is best.
Phase 3: Here, it is required to incorporate mathematical functions recombining the
vectors. These functions can model the behavior of living beings or physical/chemical phe-
nomena as bio-inspired functions.
Phase 4: The vector with the best fitness is then compared to the fitness of the previously
recombined vectors in each iteration. If the number of iterations has not been met, stop condi-
tion, it returns to phase 2. Otherwise, the best fitness value reached is reported. The number
of iterations, known as generations, is a value defined before starting the algorithm that influ-
ences the algorithm’s performance in this evolutionary process.
Finally, every metaheuristic should have a good balance between the ability to explore
(diversify) and exploit (intensify) the search solution space. In other words, it should have
global and local search strategies to improve its performance.
This paper proposes a Horned Lizard Optimization Algorithm (HLOA) as a novel swarm-
based algorithm. This algorithm was inspired by how the horned lizard reptile conceals and
defends itself from predators. The contributions of this work can be briefed as follows:

• A novel bio-inspired optimization algorithm that encompasses all aspects of the Horned
Lizard’s behavior to defend himself from their predators. Four defense strategies include
crypsis behavior, skin darkening or lightening blood-Squirting, and move-to-escape.
Also, the alpha-melanophore stimulating hormone rate, which influences their skin color
change, is considered. These strategies could inspire other researchers to explore new
directions and applications in the bio-inspired algorithms field, leading to a proliferation of
related research.
• HLOA performance is evaluated in the following set of functions: IEEE CEC 2017 ”Con-
strained Real-Parameter Optimization” for 10-dimensional, 30-dimensional, 50-dimen-
sional, and 100-dimensional benchmark problems, IEEE CEC06-2019 "100-Digit Chal-
lenge" test functions, and sixty-three testbench functions from the literature. Furthermore,
three real-world constraint optimization applications from CEC2020 and two engineering
problems, multiple gravity assist optimization and the optimal power flow problem, were
also tested.
• When comparing the HLOA algorithm to other approaches, it allows the scientific com-
munity to understand the relative strengths and weaknesses of different techniques evalu-
ated with the same test instances.
• The algorithm is validated with Friedman tests, Wilcoxon tests, and convergence analy-
ses. The results are compared to those of ten recently developed bio-inspired metaheuristic
algorithms.
• The scientific community can access the HLOA’s Matlab source code to support this
study’s findings.

The remaining sections of this paper are structured as follows: The second section illustrates
the HLOA bio-inspiration, a detailed mathematical formulation, the time complexity, and the
pseudo-code. Then, the performance of the proposed approach is benchmarked with several
testbench functions, and their comparison with ten recent bio-inspired algorithms is presented
in the third section. In the fourth section, the algorithm’s results and discussion are presented.
The fifth section describes the application of HLOA to real-world optimization problems and
the constraint-handling technique employed. Finally, the paper summarizes the conclusions
and future work.

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59 Page 6 of 65 H. Peraza‑Vázquez et al.

2 Horned lizard optimization algorithm (HLOA)

2.1 Biological fundamentals

The horned lizard is scientifically known as Phrynosoma. It is an endemic reptile from

south-central regions of the US to northeastern Mexico. They are adapted to arid or semi-
arid extreme temperature areas. Horned lizards feed on various species, including grass-
hoppers, crickets, beetles, spiders, ticks, butterflies, and moths (Leaché and McGuire
2006). Their primary passive method of defense is crypsis (Stevens and Merilaita 2011;
Ruxton et al. 2004). This method consists of the capacity to blend in with its surroundings
through color, pattern, and shape. For example, the color pattern of the Horned Lizards
changes geographically to match the terrain, and their spines cover their body outlines,
making them hard to spot (Ruxton et al. 2004). Another passive defensive strategy is move-
to-escape. Moreover, when threatened, this lizard employs aggressive tactics such as expel-
ling a short stream of blood that travels more than a meter away (Cooper and Sherbrooke
2010; Middendorf 2001). It should be noted that all reptiles, horned lizards included, resort
to thermoregulation since they cannot produce their body heat, depending on the surround-
ing temperature, to maintain their warmth (Lara-Reséndiz et al. 2015; Grigg and Buckley
2013). In addition, the horned lizard can lighten or darken its skin, depending on whether
or not it needs to decrease or increase its solar thermal gain. Thus, at high temperatures
(25◦ - 40◦ C), they acquire lighting color, whereas at low temperatures (16◦ - 17◦ C), they
acquire darkened skin. Dark skin does not reflect any color; on the contrary, it absorbs
all wavelengths of light, turning them into heat. The rapid color change of the skin of the
horned lizard is due to the effects of temperature on their alpha-melanophore stimulating
hormone (𝛼-MSH) (Sherbrooke 1997) (Fig. 3).

2.2 Mathematical model and optimization algorithm

As previously described, the lizard can defend itself by changing its colors to match its
surroundings. Additionally, it can lighten or darken its skin, depending on whether it needs
to increase or decrease its solar thermal gain. The rate of the lizard’s alpha-melanophore-
stimulating hormone (𝛼-MSH) is a factor in this rapid color change. Moreover, it can also
shoot a short stream of blood to defend against its prey. In this work, each of these lizards’

Fig. 3  Horned Lizard Phrynosoma. Photograph by Brdavids (published under a CC BY 2.0 license)

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A novel metaheuristic inspired by horned lizard defense tactics Page 7 of 65 59

defense behaviors, described before, are mathematically modeled as part of the optimiza-
tion algorithm.

2.2.1 Strategy 1: Crypsis behavior

Crypsis is the process through which an organism can blend in with its surroundings by
imitating characteristics of the environment, such as color and texture, or even by becom-
ing translucent, making it difficult for predators or prey to detect or recognize them, see
Fig. 5. It is an adaptive behavior that helps organisms avoid detection, thus increasing their
chances of survival in the wild world (Ruxton et al. 2004). As the scope of this work is
based on the horned lizard, it is to be noted that its crypsis method is mathematically repre-
sented through color theory (Westland et al. 2012; Niall 2017).
On the other hand, The International Commission on Illumination (CIE) (Niall 2017)
standardized light sources by the amount of emitted energy, throughout the visible spec-
trum (400 to 700 nm), at each wavelength. In addition, the organization defined a color
evaluation system, e.g., L*a*b system for Cartesian coordinates and L*C*h system for
polar coordinates, to compute a color in a color space.
In the L*a*b system, L* indicates the luminosity, and a* and b* are the chromatic coor-
dinates, as shown below.
{
+a, indicates Red
a∗ =
{ −a, indicates Green (1)
∗ +b, indicates Yellow
b =
−b, indicates Blue

In the L*C*h system, L* defines lightness, C* specifies color intensity, and h* indi-
cates hue angle (an angular measurement). Hue moves in a circle around the "equator"
to describe the color family (red, yellow, green, and blue) and all the colors in between.
i.e., The numbers on the hue circle range from 0 to 360, starting with red at 0 degrees,
then moving counterclockwise through yellow, green, blue, then back to red. The L axis
describes the luminous intensity of the color. Comparing the value makes it possible to
classify colors as light or dark. Both color system representations are shown in Figs. 4 and
5.
The transformation of rectangular coordinates to polar coordinates can be seen in Eq. 2.
√
c∗ = a∗2 + b∗2
� ∗�
b (2)
h = arcTg ∗
a
c* and h values correspond to chroma (or saturation) and hue, respectively. The value of h
is the hue angle and is expressed in degrees ranging from 0 ◦ to 360◦ . The inverse formulas
are as follows:
a∗ = c∗ cos (h)
b∗ = c∗ sin(h) (3)

Without loss of generality, let the ordered pair (a∗p , b∗q ) and (a∗r , b∗s ) be any two colors, with
p ≠ q ≠ r ≠ s. So, any two new colors, e.g., colorVar1 and colorVar2, can be obtained with
the following arithmetic operations shown in Eq. 4

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59 Page 8 of 65 H. Peraza‑Vázquez et al.

Fig. 4  Representation of the

color space for the CIE L*a*b
and L*C*h systems

Fig. 5  Horned Lizard Crypsis. Photograph by Paul Asman and Jill Lenoble (published under a CC BY 2.0
license)

colorVar1 = b∗p − a∗q − a∗r + b∗s

(4)
colorVar2 = b∗p − a∗q + a∗r − b∗s

These colors can be represented in a single equation, as shown below.

[ ]
colorVar = b∗p − a∗q ± a∗r − b∗s (5)

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Eq. 5 can be represented in the inverse form as follows:

[ ]
colorVar = c1 sin(hp ) − c1 cos(hq ) ± c2 cos(hr ) − c2 sin(hs ) (6)

where the angles (hue) meets hp ≠ hq ≠ hr ≠ hs, and chroma c1 ≠ c2. Finally, c1 and c2 are
factorized as shown in Eq. 7.
[ ] [ ]
colorVar = c1 sin(hp ) − cos(hq ) ± c2 cos(hr ) − sin(hs ) (7)

An equation that contains the arithmetic operation of chromatic coordinates represented in

Eq. 7, can be seen below.
( )
→ → 𝜕⋅t
x i (t + 1) = x best (t) + 𝜕 −
Max_iter
[ ( ) ( →
)] (8)
→ → →
c1 sin( x r1 (t)) − cos( x r2 (t)) − (−1)𝜎 c2 cos(xr3 (t)) − sin( x r4 (t))

→
Where x i (t + 1) is the new search agent position (horned lizard) in the solution search
→
space for the generation t + 1, x best (t) is the best search agent for the generation t; r1, r2, r3
and, r4 are integer random numbers generated between 1 and the utmost number of search
agents, with r1 ≠ r2 ≠ r3 ≠ r4; x r1 (t), x r2 (t), x r3 (t) and, x r4 (t) are the r1, r2, r3, r4-th search
→ → → →

agent selected; Max_iter represents the utmost number of iterations (generations), 𝜎 is a

binary value obtained by Algorithm 1, 𝜕 is set to 2, and, c1, c2, with c1 ≠ c2, are random
numbers taken from Table 29 containing the normalized color palette.

Algorithm 1  𝜎 procedure

2.2.2 Strategy 2: Skin darkening or lightening

The horned lizard can lighten or darken its skin, depending on whether or not it needs
to decrease or increase its solar thermal gain (Sherbrooke and Sherbrooke 1988). Thermal
energy obeys the same conservation laws as light energy (Burtt 1981). Therefore, it is the key
to the relationship between color and temperature. Thus, colors that reflect lighter repel more
heat. In this way, dark colors absorb more heat because they absorb more light energy (Burtt
1981). The color changes in the skin of the horned lizard are represented by Eqs. 9 and 10.
Equation 9 represents the lightning-skin strategy. Whereas, Eq. 10 represents the darkening-
skin strategy.

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Table 1  The color palette used Name Color Hexadecimal Decimal Normalized
for lightening or darkening the
skin
Lightening1 E8E8E8 15263976 0.0
Lightening2 9398BF 9672895 0.4046661
Darkening1 763660 7747080 0.5440510
Darkening2 161617 1447447 1

(→ )
→ → 1 →
x worst (t) = x best (t) + Light1 sin x r1 (t) − x r2 (t)
2
(→ ) (9)
𝜎1
→
− (−1) Light2 sin x r3 (t) − x r4 (t)
2
(→ )
→ → 1 →
x worst (t) = x best (t) + Dark1 sin x r1 (t) − x r2 (t)
2
(→ ) (10)
𝜎1
→
− (−1) Dark2 sin x r3 (t) − x r4 (t)
2
Where Light1 and Light2 are random numbers generated between Lightening1 (0 value) and
Lighthening2 (0.4046661 value), using these normalized values taken from Table 1. Analo-
gously, Dark1 and Dark2 are random numbers generated between Darkening1 (0.5440510
value) and Darkening2(1 value), also using→the normalized →
values from Table 1.
In addition, for both Eqs., 9 and 10, x worst (t) and x best (t) are the worst and the best
search agent found, respectively. r1, r2 , r3 and, r4 are integer random numbers generated
≠ ≠ ≠
→ →
between
→
1 and
→
the utmost number of search agents, with r 1 r 2 r 3 r 4 ; x r1 (t) , x r2 (t),
x r3 (t) and, x r4 (t) are the r1, r2 , r3, r4-th search agent selected. Finally, 𝜎 is a binary value
obtained by algorithm 1. The skin color change strategy is shown in Algorithm 2.

Algorithm 2  Darkening or lightening of skin procedure

Notice that the worst search agent in the t iteration is replaced by the new one
obtained by the skin-darkening or skin-lightening strategy.

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2.2.3 Strategy 3: blood‑squirting

The Horned lizard fends off enemies by shooting blood from its eyes (Middendorf
2001). The shooting blood defense mechanism can be represented as a projectile
motion, depicted in Fig. 6. To obtain the equations of motion, we separate the projectile
motion into its two components, X-axis (horizontal) and Y-axis (vertical):
In the horizontal direction, the shot of blood describes a uniform line movement, so
its equation of motion will be given by:
t→

∫0
→ → → →
𝜐 = 𝜐0 + g dt = 𝜐0 + g t (11)

In the vertical direction, the shot of blood describes a uniformly accelerated rectilinear
motion, it is as follows:
t( )

∫0
→ → → → → → 1→
r = r0 + 𝜐o + g t dt = r0 + 𝜐o t + g t2 (12)
2

→
(13)
→
r0 =0

The vector equations, position, and velocity, are represented by Eqs. 14 and 15,
respectively.
→ ( )→
→ 1
𝜐 0 = 𝜐0 cos(𝛼)t j + (𝜐0 sin(𝛼))t − gt2 k (14)
2

( )→ ( )→
(15)
→ →
𝜐 = r = 𝜐0 cos(𝛼) j + 𝜐0 sin(𝛼) − gt k

Finally, the trajectory can be expressed as follows:

Fig. 6  Horned lizard shooting

blood

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59 Page 12 of 65 H. Peraza‑Vázquez et al.

[ ( ) ]
→ t →
x i (t + 1) = vo cos 𝛼 + 𝜀 x best (t)
Max_iter
[ ( ) ] (16)
𝛼t →
+ vo sin 𝛼 − − g + 𝜀 x i (t)
Max_iter
→
Where x i (t + 1) is the new search→
agent position (horned lizard) in the solution search
space for→the generation t + 1, x best (t) is the best search agent found, the current search
agent is x i (t), Max_iter represents the utmost number of iterations (generations), t is the
current iteration, v0 is set to 1 seg, 𝛼 is set to 𝜋2 , 𝜖 is set to 1E-6 and, g is the gravity of the
earth, 0.009807 km∕s2

2.2.4 Strategy 4: move‑to‑escape

In this strategy, the horned lizard performs a random fast move around the environment to
escape predators. Ruxton et al. (2004). A function that includes a local and global move-
ment is proposed for the mathematical modeling of this ( evasion
)→ strategy; it is described in
Eq. 17 and depicted in Fig. 7. In this equation walk 21 − 𝜀 x i (t) is a local motion around
→ →
x i (t), and adding x best (t) generates a displacement through the solution search space (the
global movement).
( )→
→ → 1
x i (t + 1) = x best (t) + walk − 𝜀 x i (t) (17)
2
→
Where x i (t + 1) is the new search
→
agent position (horned lizard) in the solution search
space for the generation t + 1, x best (t) is the best search agent for the generation t, walk is a
random number generated between -1 and 1, 𝜖 is a random number generated →
from a stand-
ard Cauchy distribution with the mean and 𝜎 set to 0 and 1, respectively. x i (t) is the current
i-th search agent in the t generation.

2.2.5 Strategy 5: ˛ ‑melanophore stimulating hormone (˛ ‑MSH) rate

The horned lizard can lighten or darken its skin, depending on whether or not it needs to
decrease or increase its solar thermal gain. The rapid alteration in coloration observed
on the skin of horned lizards can be attributed to the influence of temperature on the
𝛼-melanophore stimulating hormone (𝛼-MSH). Additional information regarding
the study on hormone levels in horned lizards can be seen in Sherbrooke (1997). In
this research, the horned lizards’ 𝛼-melanophore rate value is defined in the following
equation:
Fitnessmax − Fitness(i)
melanophore(i) = (18)
Fitnessmax − Fitnessmin

Where Fitnessmin and Fitnessmax are the best and the worst fitness value in the current t gen-
eration, respectively, whereas fitness(i) is the current fitness value of the i-th search agent.
The melanophore(i) value vector obtained by computing Eq. 18 is normalized in the
interval of [0, 1]. A low 𝛼-MSH rate, less than 0.3, replaces search agents in Eq. 19, as
described in Algorithm 3.

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Fig. 7  Horned lizard escaping from predators

1
x�⃗i (t) = x�⃗best (t) + [�x⃗r1 (t) − (−1)𝜎 x�⃗r2 (t)] (19)
2
→ →
Where x i (t) is the current search agent, x best (t) is the best search agent found, r1 and r2
are integer random numbers generated between 1 and the utmost number of search agents,
with r1 ≠ r2, x r1 (t) and, x r2 (t), are the r1, r2-th search agent selected and, 𝜎 is a binary value
→ →

obtained by Algorithm 1

Algorithm 3  𝛼-melanophore procedure

13
59 Page 14 of 65 H. Peraza‑Vázquez et al.

2.2.6 The HLOA’s time complexity

The HLOA’s time complexity analysis includes the analysis of the initialization of the
population, fitness evaluation, and updating search agents (lizards). The initialization of
the HLOA population is O(PopSize x D), where popSize is the number of search agents
(Lizards), and D is the dimension of the optimization problem (design variables number).
O(T) represents the time complexity of computing the fitness value, i.e. objective function
value. Thus, The amount of time required for the initial evaluation of fitness is bounded
by O(PopSize × O ( T )). Therefore, the HLOA main loop’s computational complexity is
O(MaxIteration x PopSize x (D+ O(T))), as summarized in Algorithm 4 (Fig. 8 ).

2.2.7 Pseudo code for HLOA

In Algorithm 4, the pseudo-code of the HLOA algorithm is described.

Algorithm 4  Horned Lizard Optimizer Algorithm

13
A novel metaheuristic inspired by horned lizard defense tactics Page 15 of 65 59

Fig. 8  HLOA flowchart

13
59 Page 16 of 65 H. Peraza‑Vázquez et al.

3 Experimental setup

The numerical efficiency and stability of the HLOA algorithm were evaluated by solving
63 classical benchmark optimization functions reported in the literature. Each of these
functions is described in Appendix A, Tables 30, 31, and 32, where Dim represents the
function’s dimension, Interval is the boundary of the search space for the function and, the
optimum value is fmin. The HLOA algorithm was compared with ten recent bio-inspired
algorithms as described below:

• Jumping Spider Optimization Algorithm (JSOA): The algorithm mimics the behavior
of the Arachnida Salticidade spiders in nature and mathematically models its hunting
strategies: search, persecution, and jumping skills to get the prey (Peraza-Vázquez et al.
2021).
• Black Widow Optimization Algorithm (BWOA): It is based on modeling different spi-
ders’ movement strategies for courtship-mating and the pheromone rate associated with
cannibalistic behavior in female spiders (Peña-Delgado et al. 2020).
• Coot Bird Algorithm (COOT): The Coot algorithm imitates two different modes of
movement of birds on the water surface (Naruei and Keynia 2021).
• Crystal Structure Algorithm (CSA): The algorithm is based on the principles underpin-
ning the natural occurrence of crystal structures forming from the addition of the basis
to the lattice points, which may be observed in the symmetrical arrangement of con-
stituents in crystalline minerals (Talatahari et al. 2021).
• Dingo Optimization Algorithm (DOA): The algorithm mimics the social behavior of
the Australian dingo dog. Its inspiration comes from the hunting strategies of dingoes
attacking by persecution, grouping tactics, and scavenging behavior (Peraza-Vázquez
et al. 2021).
• Enhanced Jaya Algorithm (EJAYA): The classic version of the Jaya algorithm has the
defect of easily getting trapped in local optima. This updated version uses the popula-
tion information more efficiently to improve its performance (Zhang et al. 2021).
• Rat Swarm Optimizer (RSO): The inspiration of this optimizer is the attacking and
chasing behaviors of rats in nature (Dhiman et al. 2021).
• Smell Agent Optimization (SAO): The algorithm is based on the relationships between a
smelling agent and an object evaporating a smell molecule (Salawudeen et al. 2021).
• Tunicate Swarm Algorithm (TSA): The algorithm imitates jet propulsion and swarm
behaviors of tunicates during the navigation and foraging process (Kaur et al. 2020).
• Wild Horse Optimizer (WHO): The algorithm is based on the social behavior of wild
horses, such as grazing, chasing, dominating, leading, and mating. The mathematical
model includes the representation of mares, foals, and stallions living in groups (Naruei
and Keynia 2021).

Each algorithm’s benchmark function was executed 30 times, with the population size and
number of iterations set to 30 and 200, respectively. Furthermore, the Wilcoxon signed-
rank test was used to compare their performance, and the ranking of each algorithm was
obtained by the Friedman test. It is to be noted that the three best-ranked algorithms deter-
mined by this selection are then evaluated on IEEE CEC 2017 and CEC 2019, as described
below.

13
A novel metaheuristic inspired by horned lizard defense tactics Page 17 of 65 59

Table 2  Initial values for the Algorithm Parameters Value

controlling parameters of
algorithms
For all algorithms Population size for all problems 30
Maximum iterations for Testbench 200
Functions and Real-World Problems 30
Number of replications for Testbench
Functions
HLOA Does not use additional parameters –
JSOA Does not use additional parameters –
BWOA Does not use additional parameters –
DOA Hunting probability (P) 0.5
Scavenger probability (Q) 0.7
COOT R1, R2, R3, R4 [0,1]
R [− 1,1]
CSA r1, r2, r3, r4 [0,1]
EJAYA​ Does not use additional parameters –
RSO R [1,5]
C [0,2]
SAO r0, r1, r2, r3 (0,1]
olfaction capacity (Olf) 0.75
Step (SL) 0.9
TSA c1,c2,c3, rand [0,1]
Pmin 1
Pmax 4
WHO R [− 2,2]
R1, R2, R3 [0,1]
PC 0.13
PS 0.2

Table 3  IEEE CEC-C06 2019 Benchmarks “The 100-Digit Challenge:”

No Function Dim Interval fmin

CEC-1 Storn’s Chebyshev polynomial fitting problem 9 [− 8192, 8192] 1

CEC-2 Inverse Hilbert matrix problem 16 [− 16284,16284] 1
CEC-3 Lennard–Jones minimum energy cluster 18 [− 4, 4] 1
CEC-4 Rastrigin’s function 10 [− 100,100] 1
CEC-5 Griewangk’s function 10 [− 100,100] 1
CEC-6 Weierstrass function 10 [− 100,100] 1
CEC-7 Modified Schwefel’s function 10 [− 100,100] 1
CEC-8 Expanded Schaffer’s F6 function 10 [− 100,100] 1
CEC-9 Happy Cat function 10 [− 100,100] 1
CEC-10 Ackley function 10 [− 100,100] 1

IEEE CEC 2017 Testbech Functions: Experimental studies are performed on 28

10-dimensional, 30-dimensional, 50-dimensional, and 100-dimensional benchmark prob-
lems from the IEEE CEC 2017 “Constrained Real-Parameter Optimization”, as outlined in
Table 33.

13
 Table 4  Comparison of optimization results obtained for 63 benchmark functions
Algorithms HLOA JSOA BWOA

Best Ave Std Best Ave. Std. Best Ave. Std.

13
F1 − 8.88E-16 − 8.88E-16 0.00E+00 − 8.88E-16 − 8.88E-16 0.00E+00 − 8.88E-16 − 8.88E-16 0.00E+00
59 Page 18 of 65

F2 − 2.00E+02 − 2.00E+02 0.00E+00 − 2.00E+02 − 2.00E+02 0.00E+00 − 2.00E+02 − 2.00E+02 0.00E+00

F3 − 1.86E+02 − 1.86E+02 8.54E-14 − 1.86E+02 − 1.86E+02 8.67E-14 − 1.86E+02 − 1.86E+02 4.84E-04
F4 − 4.59E+00 − 4.27E+00 4.32E-01 − 4.59E+00 − 4.56E+00 1.61E-01 − 4.59E+00 − 4.35E+00 5.98E-01
F5 − 1.08E+00 − 1.08E+00 2.70E-16 − 1.08E+00 − 1.08E+00 1.80E-16 − 1.08E+00 − 1.07E+00 1.50E-02
F6 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F7 − 2.83E+13 − 2.28E+13 8.97E+12 − 2.83E+13 − 2.33E+13 9.94E+12 − 2.83E+13 − 1.96E+13 1.21E+13
F8 1.00E+00 1.00E+00 0.00E+00 1.00E+00 1.00E+00 0.00E+00 1.00E+00 1.00E+00 0.00E+00
F9 4.47E-21 1.27E-01 2.89E-01 1.73E-21 1.02E-01 2.63E-01 3.72E-17 1.02E-01 2.63E-01
F10 − 1.07E+02 − 9.96E+01 1.30E+01 − 1.07E+02 − 1.03E+02 7.91E+00 − 1.07E+02 − 9.96E+01 1.30E+01
F11 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F12 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F13 6.36E-23 8.57E-18 2.54E-17 4.96E-22 3.76E-16 1.62E-15 2.55E-14 5.33E-11 2.69E-10
F14 1.85E-23 3.83E-15 1.91E-14 2.75E-20 1.17E-12 6.43E-12 1.75E-14 3.49E-08 1.83E-07
F15 2.59E-109 8.05E-91 4.37E-90 3.90E-75 1.98E-67 9.22E-67 1.69E-253 1.23E-194 0.00E+00
F16 2.71E-03 3.61E-02 1.64E-02 1.48E-02 4.64E-02 8.58E-03 5.65E-03 4.52E-02 1.00E-02
F17 − 2.06E+00 − 2.06E+00 1.63E-15 − 2.06E+00 − 2.06E+00 9.76E-16 − 2.06E+00 − 2.06E+00 1.03E-13
F18 − 2.48E+04 − 2.48E+04 3.24E-12 − 2.48E+04 − 2.48E+04 8.46E-12 − 2.48E+04 − 1.90E+04 1.07E+04
F19 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00
F20 − 1.00E+00 − 1.00E+00 2.93E-13 − 1.00E+00 − 1.00E+00 2.06E-17 − 1.00E+00 − 1.00E+00 3.46E-12
F21 8.63E-128 8.90E-102 4.85E-101 3.38E-77 1.83E-69 8.81E-69 2.80E-264 2.72E-193 0.00E+00
F22 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00
F23 3.00E+00 7.50E+00 1.60E+01 3.00E+00 3.00E+00 9.91E-15 3.00E+00 9.30E+00 2.09E+01
F24 − 2.87E+00 − 2.87E+00 3.95E-16 − 2.87E+00 − 2.87E+00 3.87E-16 − 2.87E+00 − 2.87E+00 1.86E-15
F25 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F26 8.65E-02 2.59E-01 1.91E-01 6.47E-02 4.13E-01 1.78E-01 1.61E-01 4.91E-01 2.10E-01
F27 4.33E-21 6.00E-17 1.65E-16 1.83E-19 4.77E-16 1.96E-15 7.69E-14 8.87E-11 2.39E-10
H. Peraza‑Vázquez et al.
 Table 4  (continued)
Algorithms HLOA JSOA BWOA

Best Ave Std Best Ave. Std. Best Ave. Std.

F28 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15
F29 − 6.74E-01 − 6.74E-01 1.32E-16 − 6.74E-01 − 6.74E-01 3.57E-17 − 6.74E-01 − 6.74E-01 1.16E-09
F30 3.81E-20 5.28E-02 1.54E-01 4.99E-18 6.92E-03 3.79E-02 9.02E-12 7.52E-01 3.70E-01
F31 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00
F32 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00
F33 3.22E+01 3.46E+01 1.80E+00 3.20E+01 3.20E+01 1.08E-14 3.20E+01 3.20E+01 1.08E-14
F34 9.00E-01 9.00E-01 4.52E-16 9.00E-01 9.00E-01 4.52E-16 9.00E-01 9.00E-01 4.52E-16
F35 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F36 6.98E+02 1.29E+03 3.35E+02 1.66E+03 2.14E+03 1.48E+02 2.94E+03 4.35E+03 6.48E+02
F37 2.36E-05 8.12E-04 8.88E-04 1.08E-05 2.95E-04 3.31E-04 2.16E-05 4.44E-04 3.99E-04
F38 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F39 − 5.00E+00 − 1.95E+00 2.43E+00 0.00E+00 1.82E-38 6.19E-38 − 5.00E+00 − 1.67E-01 9.13E-01
F40 2.24E-02 2.68E+01 7.13E+00 1.07E-16 4.08E-01 1.11E+00 2.89E+01 2.90E+01 2.93E-02
A novel metaheuristic inspired by horned lizard defense tactics

F41 2.77E-53 1.43E-45 7.41E-45 0.00E+00 1.10E-37 5.37E-37 1.95E-117 4.36E-78 2.39E-77
F42 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F43 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F44 1.57E-03 1.57E-03 1.10E-06 1.57E-03 1.57E-03 6.92E-07 1.57E-03 1.57E-03 1.24E-06
F45 2.93E-01 2.93E-01 5.50E-10 2.93E-01 2.93E-01 3.93E-06 2.93E-01 2.93E-01 6.01E-06
F46 2.31E-51 5.23E-47 2.16E-46 0.00E+00 2.70E-39 4.38E-39 2.60E-136 4.26E-97 2.30E-96
F47 7.73E-57 2.02E-48 3.97E-48 0.00E+00 7.56E-40 2.61E-39 8.32E-125 9.58E-95 5.23E-94
F48 2.00E-58 2.80E-46 1.38E-45 0.00E+00 2.89E-39 1.11E-38 6.01E-126 2.20E-91 1.20E-90
F49 0.00E+00 0.00E+00 0.00E+00 0.00E+00 7.48E-297 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F50 3.90E+03 5.84E+03 1.02E+03 3.82E-04 3.04E+03 1.48E+03 7.10E+03 8.41E+03 5.64E+02
F51 − 4.45E+02 − 3.30E+02 1.07E+02 − 4.45E+02 − 4.41E+02 9.00E+00 − 1.96E+02 − 1.46E+02 9.58E+00
F52 − 3.86E+02 − 3.21E+02 4.09E+01 − 3.86E+02 − 3.66E+02 4.27E+01 − 2.22E+02 − 1.57E+02 1.81E+01
F53 − 3.19E+32 − 1.85E+31 6.25E+31 − 8.75E+33 − 4.23E+32 1.68E+33 − 5.94E+25 − 3.05E+24 1.10E+25
Page 19 of 65 59

13
 Table 4  (continued)
Algorithms HLOA JSOA BWOA

Best Ave Std Best Ave. Std. Best Ave. Std.

13
F54 5.61E-104 1.97E-89 1.03E-88 0.00E+00 1.67E-67 4.30E-67 1.59E-242 2.98E-199 0.00E+00
59 Page 20 of 65

F55 − 1.03E+03 − 9.87E+02 1.94E+01 − 1.17E+03 − 8.23E+02 2.37E+02 − 7.46E+02 − 6.42E+02 5.21E+01
F56 1.55E-101 3.73E-88 2.04E-87 0.00E+00 2.23E-65 1.19E-64 4.78E-258 8.80E-197 0.00E+00
F57 4.20E-119 6.21E-106 3.37E-105 0.00E+00 4.17E-70 2.19E-69 4.29E-257 9.11E-192 0.00E+00
F58 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F59 4.29E-43 3.70E-07 1.88E-06 2.18E-34 9.39E-30 1.84E-29 5.39E-82 3.13E-18 1.72E-17
F60 3.51E-12 5.52E-12 3.61E-12 0.00E+00 3.51E-13 1.07E-12 3.51E-12 6.67E-08 2.38E-07
F61 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00
F62 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 0.00E+00
F63 9.30E-106 4.87E-90 2.66E-89 3.29E-72 1.26E-63 4.29E-63 7.42E-263 6.29E-194 0.00E+00
Algorithms

DOA COOT CSA

Best Ave. Std. Best Ave. Std. Best Ave. Std.

F1 − 8.88E-16 − 8.88E-16 0.00E+00 6.16E-10 3.08E-06 8.60E-06 3.37E-08 2.55E-04 4.37E-04

F2 − 2.00E+02 − 2.00E+02 0.00E+00 − 2.00E+02 − 2.00E+02 4.57E− 10 − 2.00E+02 − 2.00E+02 3.08E-11
F3 − 1.86E+02 − 1.86E+02 1.78E-06 − 1.86E+02 − 1.86E+02 2.37E-07 − 1.86E+02 − 1.86E+02 4.30E-08
F4 − 4.59E+00 − 4.59E+00 2.02E-12 − 4.59E+00 − 4.59E+00 1.07E-05 − 4.59E+00 − 4.59E+00 1.55E-04
F5 − 1.08E+00 − 1.08E+00 2.06E-16 − 1.08E+00 − 1.08E+00 2.18E-16 − 1.08E+00 − 1.08E+00 3.10E-09
F6 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F7 − 2.83E+13 − 1.20E+13 1.33E+13 − 2.43E+11 − 6.88E+10 6.87E+10 − 2.83E+13 − 2.76E+13 9.37E+11
F8 1.00E+00 1.00E+00 0.00E+00 1.00E+00 1.00E+00 4.39E-11 1.00E+00 1.00E+00 3.84E-14
F9 0.00E+00 2.54E-02 1.39E-01 9.34E-17 2.13E-09 5.94E-09 3.77E-07 2.04E-05 2.14E-05
F10 − 1.07E+02 − 1.06E+02 3.55E+00 − 1.07E+02 − 1.07E+02 1.81E-02 − 1.07E+02 − 1.07E+02 6.07E-03
F11 0.00E+00 0.00E+00 0.00E+00 0.00E+00 7.25E-16 2.69E-15 0.00E+00 0.00E+00 0.00E+00
H. Peraza‑Vázquez et al.
 Table 4  (continued)
Algorithms

DOA COOT CSA

Best Ave. Std. Best Ave. Std. Best Ave. Std.

F12 0.00E+00 0.00E+00 0.00E+00 0.00E+00 9.95E-14 5.17E-13 0.00E+00 0.00E+00 0.00E+00
F13 0.00E+00 2.50E-10 1.36E-09 2.29E-19 1.71E-08 8.51E-08 1.41E-06 1.98E-04 1.74E-04
F14 1.38E-87 1.57E-04 8.61E-04 4.17E-17 7.37E-12 2.40E-11 2.19E-05 5.39E-04 3.98E-04
F15 1.94E-174 3.79E-30 2.07E-29 2.78E-25 6.11E-09 3.32E-08 8.10E-18 1.20E-09 2.28E-09
F16 1.70E-03 9.71E-02 3.25E-01 1.13E-02 2.69E-01 6.96E-01 1.56E-01 4.35E-01 2.17E-01
F17 − 2.06E+00 − 2.06E+00 1.06E-08 − 2.06E+00 − 2.06E+00 4.74E-07 − 2.06E+00 − 2.06E+00 8.34E-07
F18 − 2.48E+04 − 2.48E+04 7.28E-03 − 2.48E+04 − 2.48E+04 1.49E-05 − 2.48E+04 − 2.48E+04 7.55E-01
F19 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 9.98E-01 1.16E-02 − 1.00E+00 − 1.00E+00 1.02E-12
F20 − 1.00E+00 − 1.00E+00 1.36E-08 − 1.00E+00 − 1.00E+00 8.48E-09 − 1.00E+00 − 1.00E+00 2.19E-04
F21 0.00E+00 1.44E-42 7.86E-42 1.69E-36 4.14E-23 2.18E-22 7.59E-25 2.51E-20 5.38E-20
F22 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 1.00E+00 2.31E-12 − 1.00E+00 − 1.00E+00 3.20E-10
F23 3.00E+00 3.00E+00 1.76E-07 3.00E+00 3.00E+00 6.38E-09 3.00E+00 3.00E+00 8.27E-04
A novel metaheuristic inspired by horned lizard defense tactics

F24 − 2.87E+00 − 2.87E+00 6.85E-16 − 2.87E+00 − 2.87E+00 3.67E-12 − 2.87E+00 − 2.87E+00 3.66E-07
F25 0.00E+00 0.00E+00 0.00E+00 0.00E+00 3.29E-08 1.61E-07 0.00E+00 2.26E-06 1.13E-05
F26 7.67E-02 4.20E-01 2.11E-01 5.57E-02 1.68E-01 7.15E-02 6.80E-02 1.50E-01 4.57E-02
F27 0.00E+00 6.39E-08 3.50E-07 4.22E-15 4.08E-05 1.85E-04 6.40E-05 1.55E-03 1.51E-03
F28 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15
F29 − 6.74E-01 − 6.74E-01 1.62E-06 − 6.74E-01 − 6.74E-01 3.29E-11 − 6.74E-01 − 6.74E-01 7.67E-08
F30 0.00E+00 2.47E-01 2.56E-01 2.47E-11 7.54E-03 3.88E-02 1.72E-05 3.30E-04 3.14E-04
F31 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00
F32 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00
F33 3.20E+01 3.20E+01 1.08E-14 3.25E+01 3.57E+01 1.62E+00 3.20E+01 3.20E+01 1.08E-14
F34 9.00E-01 9.00E-01 4.61E-16 9.00E-01 1.41E+00 1.16E+00 9.00E-01 9.00E-01 5.45E-06
F35 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Page 21 of 65 59

13
 Table 4  (continued)
Algorithms

DOA COOT CSA

13
Best Ave. Std. Best Ave. Std. Best Ave. Std.
59 Page 22 of 65

F36 2.00E+03 2.98E+03 6.44E+02 3.93E+02 1.63E+03 5.49E+02 1.48E+03 2.20E+03 3.49E+02
F37 1.09E-04 6.61E-04 6.20E-04 8.20E-04 8.89E-03 6.02E-03 2.61E-06 2.62E-03 1.63E-03
F38 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.78E-02 9.74E-02 1.25E-12 6.34E-06 1.86E-05
F39 − 5.00E+00 − 7.97E-01 1.82E+00 − 4.69E+00 − 4.52E+00 1.06E-01 − 2.26E+00 − 7.24E-01 6.54E-01
F40 2.88E+01 2.89E+01 3.79E-02 2.83E+01 3.32E+01 1.03E+01 2.88E+01 2.88E+01 3.02E-02
F41 0.00E+00 3.33E-03 1.82E-02 4.12E-11 1.12E-01 1.14E-01 2.83E-06 3.50E-02 4.68E-02
F42 0.00E+00 0.00E+00 0.00E+00 0.00E+00 2.22E-17 8.94E-17 0.00E+00 0.00E+00 0.00E+00
F43 0.00E+00 0.00E+00 0.00E+00 0.00E+00 2.09E-14 1.14E-13 0.00E+00 0.00E+00 0.00E+00
F44 1.57E-03 1.59E-03 5.54E-05 1.57E-03 1.78E-03 5.08E-04 1.57E-03 1.57E-03 2.49E-06
F45 2.93E-01 2.93E-01 1.62E-05 2.93E-01 2.93E-01 2.17E-05 2.93E-01 2.93E-01 1.88E-06
F46 2.02E-93 1.07E-15 5.84E-15 1.90E-10 9.38E-05 2.13E-04 3.24E-07 4.24E-03 1.33E-02
F47 2.59E-126 1.18E-14 6.04E-14 2.05E-10 3.10E-05 8.85E-05 3.75E-08 2.45E-04 4.04E-04
F48 8.96E-86 1.13E-18 6.19E-18 1.63E-10 4.54E+20 2.48E+21 3.44E-07 2.83E-03 4.76E-03
F49 0.00E+00 1.86E-57 1.02E-56 2.55E-116 7.01E-64 2.67E-63 1.82E-110 1.53E-38 8.18E-38
F50 5.81E+03 7.78E+03 8.56E+02 4.25E+03 6.10E+03 9.03E+02 6.40E+03 7.71E+03 3.87E+02
F51 − 1.46E+02 − 1.15E+02 2.40E+01 − 2.49E+02 − 1.54E+02 3.78E+01 − 1.81E+02 − 1.48E+02 1.22E+01
F52 − 2.42E+02 − 1.60E+02 3.85E+01 − 1.93E+02 − 1.40E+02 2.12E+01 − 2.19E+02 − 1.65E+02 1.86E+01
F53 − 2.74E+26 − 9.29E+24 4.99E+25 − 3.45E+24 − 1.76E+23 6.41E+23 − 1.33E+25 − 6.57E+23 2.44E+24
F54 3.86E-153 4.26E-19 2.33E-18 5.89E-22 1.70E-11 8.58E-11 5.17E-17 9.36E-09 3.08E-08
F55 − 7.88E+02 − 7.22E+02 5.16E+01 − 1.07E+03 − 9.89E+02 5.48E+01 − 8.57E+02 − 8.10E+02 2.57E+01
F56 5.94E-226 1.83E-24 1.00E-23 5.36E-20 1.90E-09 6.99E-09 9.94E-13 2.99E-07 1.15E-06
F57 1.95E-272 4.08E-48 1.96E-47 3.86E-40 2.91E-23 1.24E-22 4.01E-28 3.52E-23 1.32E-22
F58 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
F59 1.67E-107 1.63E-11 7.33E-11 5.31E-13 5.21E-05 1.53E-04 9.80E-11 1.30E-07 3.84E-07
H. Peraza‑Vázquez et al.
 Table 4  (continued)
Algorithms

DOA COOT CSA

Best Ave. Std. Best Ave. Std. Best Ave. Std.

F60 0.00E+00 8.27E-09 2.44E-08 3.51E-12 7.50E-12 5.83E-12 3.51E-12 3.63E-12 2.23E-13
F61 − 1.00E+00 − 8.00E-01 6.09E-01 − 1.00E+00 − 3.05E-01 9.49E-01 − 1.00E+00 − 1.00E+00 1.13E-09
F62 − 1.00E+00 − 1.00E+00 5.96E-08 − 1.00E+00 − 6.43E-01 4.33E-01 − 1.00E+00 − 9.79E-01 5.12E-02
F63 1.58E-219 1.02E-23 5.57E-23 1.01E-26 4.42E-04 2.42E-03 5.58E-14 8.93E-09 2.43E-08
Algorithms

EJAYA​ RSO SAO TSA WHO

Best Ave Std Best Ave. Std. Best Ave. Std. Best Ave. Std. Best Ave. Std.

F1 3.46E+00 6.33E+00 1.73E+00 − 8.88E-16 − 8.88E-16 0.00E+00 4.34E-02 5.97E-01 1.33E+00 7.59E-05 2.13E+00 1.60E+00 8.18E-11 1.19E-08 2.21E-08
F2 − 2.00E+02 − 2.00E+02 1.09E-10 − 2.00E+02 − 2.00E+02 0.00E+00 − 2.00E+02 − 1.98E+02 3.26E+00 − 2.00E+02 − 2.00E+02 2.99E-14 − 2.00E+02 − 2.00E+02 0.00E+00
F3 − 1.86E+02 − 1.86E+02 8.67E-14 − 1.86E+02 − 1.86E+02 7.88E-09 − 1.86E+02 − 1.75E+02 2.76E+01 − 1.86E+02 − 1.86E+02 4.79E-04 − 1.86E+02 − 1.86E+02 8.67E-14
A novel metaheuristic inspired by horned lizard defense tactics

F4 − 4.59E+00 − 4.59E+00 2.86E-08 − 4.59E+00 − 4.59E+00 5.39E-04 − 4.59E+00 − 3.16E+00 1.45E+00 − 4.59E+00 − 4.27E+00 4.32E-01 − 4.59E+00 − 4.56E+00 1.61E-01
F5 − 1.08E+00 − 1.08E+00 4.12E-17 − 1.08E+00 − 1.08E+00 9.61E-05 − 1.08E+00 − 1.04E+00 3.75E-02 − 1.08E+00 − 1.08E+00 2.68E-10 − 1.08E+00 − 1.08E+00 0.00E+00
F6 0.00E+00 7.47E+00 4.83E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.08E+00 7.77E-01 1.28E+01 3.06E+01 7.49E+00 0.00E+00 0.00E+00 0.00E+00
F7 − 4.70E+09 − 9.99E+08 1.30E+09 − 1.13E+05 − 1.01E+04 2.37E+04 − 2.83E+13 − 1.40E+13 1.13E+13 − 1.35E+10 − 5.93E+08 2.46E+09 − 4.53E+11 − 6.24E+10 9.25E+10
F8 1.00E+00 1.00E+00 1.06E-12 1.00E+00 1.00E+00 0.00E+00 1.01E+00 5.54E+02 1.40E+03 1.00E+00 1.00E+00 0.00E+00 1.00E+00 1.00E+00 0.00E+00
F9 1.07E-20 1.86E-14 3.60E-14 1.65E-03 6.75E-02 1.90E-01 1.61E-06 4.09E-03 5.01E-03 1.98E-09 1.49E-01 3.03E-01 0.00E+00 2.54E-02 1.39E-01
F10 − 1.07E+02 − 1.07E+02 5.59E-06 − 1.07E+02 − 1.07E+02 5.46E-01 − 1.07E+02 − 1.06E+02 2.23E+00 − 1.07E+02 − 1.04E+02 7.37E+00 − 1.07E+02 − 1.07E+02 4.71E-14
F11 0.00E+00 2.96E-16 1.26E-15 0.00E+00 0.00E+00 0.00E+00 1.04E-03 1.62E+01 4.48E+01 0.00E+00 1.71E-04 9.39E-04 0.00E+00 0.00E+00 0.00E+00
F12 0.00E+00 4.63E-17 1.84E-16 0.00E+00 0.00E+00 0.00E+00 1.56E-03 3.05E+00 7.98E+00 0.00E+00 9.46E-02 1.10E-01 0.00E+00 0.00E+00 0.00E+00
F13 1.10E-23 6.45E-21 1.50E-20 1.13E-03 2.96E-02 2.69E-02 2.47E-06 8.70E-02 4.74E-01 2.37E-06 6.00E-01 2.96E+00 0.00E+00 2.10E-31 1.15E-30
F14 5.07E-25 5.37E-22 1.68E-21 4.44E-07 1.20E-03 1.46E-03 9.88E-06 1.13E-02 3.08E-02 1.74E-05 2.70E-04 2.72E-04 1.38E-87 1.05E-31 5.76E-31
F15 3.78E-01 1.66E+00 1.17E+00 0.00E+00 5.99E-115 1.11E-114 2.68E-04 1.21E-02 1.09E-02 1.97E-10 1.00E-08 1.90E-08 1.18E-21 2.20E-18 3.56E-18
F16 2.15E-02 9.85E-02 6.33E-02 2.08E-01 1.52E+00 7.87E-01 1.60E-01 1.24E+00 7.71E-01 5.42E-02 4.60E-01 2.21E-01 1.54E-04 1.24E-02 9.51E-03
F17 − 2.06E+00 − 2.06E+00 1.15E-08 − 2.06E+00 − 2.06E+00 4.97E-05 − 2.06E+00 − 2.06E+00 8.61E-04 − 2.06E+00 − 2.06E+00 6.07E-07 − 2.06E+00 − 2.06E+00 1.63E-14
Page 23 of 65 59

13
 Table 4  (continued)
Algorithms

EJAYA​ RSO SAO TSA WHO

13
Best Ave Std Best Ave. Std. Best Ave. Std. Best Ave. Std. Best Ave. Std.
59 Page 24 of 65

F18 − 2.48E+04 − 2.48E+04 6.43E-05 − 2.48E+04 − 2.48E+04 1.12E+00 − 2.48E+04 − 2.33E+04 2.24E+03 − 2.48E+04 − 2.31E+04 6.29E+03 − 2.48E+04 − 2.48E+04 9.38E-05
F19 − 1.00E+00 − 9.99E-01 2.79E-03 − 1.00E+00 − 1.00E+00 0.00E+00 − 9.98E-01 − 9.48E-01 3.99E-02 − 1.00E+00 − 9.43E-01 1.95E-02 − 1.00E+00 − 9.98E-01 1.16E-02
F20 − 1.00E+00 − 1.00E+00 7.55E-08 − 9.98E-01 − 9.23E-01 1.01E-01 − 1.00E+00 − 5.27E-01 4.93E-01 − 1.00E+00 − 6.00E-01 4.98E-01 − 1.00E+00 − 1.00E+00 0.00E+00
F21 4.95E-23 6.33E-15 3.21E-14 0.00E+00 1.42E-112 7.75E-112 5.65E-04 1.61E-01 2.23E-01 4.30E-57 2.86E-21 1.57E-20 1.68E-80 8.27E-67 2.50E-66
F22 − 1.00E+00 − 9.95E-01 4.29E-03 − 1.00E+00 − 1.00E+00 0.00E+00 − 1.00E+00 − 9.96E-01 4.30E-03 − 1.00E+00 − 1.00E+00 4.26E-10 − 1.00E+00 − 1.00E+00 0.00E+00
F23 3.00E+00 3.00E+00 5.45E-15 3.00E+00 3.00E+00 1.14E-03 3.00E+00 3.79E+00 1.68E+00 3.00E+00 1.11E+01 1.76E+01 3.00E+00 3.00E+00 2.30E-15
F24 − 2.87E+00 − 2.87E+00 5.89E-16 − 2.87E+00 − 2.87E+00 2.02E-04 − 2.87E+00 − 2.87E+00 3.61E-05 − 2.87E+00 − 2.84E+00 2.12E-01 − 2.87E+00 − 2.87E+00 4.74E-16
F25 1.24E+00 1.92E+00 5.09E-01 0.00E+00 0.00E+00 0.00E+00 7.07E-05 2.81E+00 5.39E+00 5.65E-07 1.91E-02 2.07E-02 0.00E+00 9.88E-14 2.76E-13
F26 8.42E-02 1.69E-01 6.14E-02 1.31E-01 2.99E-01 5.73E-02 1.25E-02 7.30E-02 4.34E-02 1.50E-01 3.44E-01 8.40E-02 3.87E-02 9.96E-02 5.56E-02
F27 9.93E-12 2.67E-06 7.05E-06 1.18E-04 9.37E-02 1.39E-01 6.90E-07 1.69E-02 3.03E-02 3.55E-05 9.06E-04 8.31E-04 0.00E+00 8.95E-15 2.92E-14
F28 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15 − 1.51E+01 − 1.51E+01 5.42E-15
F29 − 6.74E-01 − 6.74E-01 1.65E-14 − 6.74E-01 − 5.76E-01 1.46E-01 − 6.74E-01 − 5.99E-01 1.45E-01 − 6.74E-01 − 6.74E-01 3.40E-06 − 6.74E-01 − 6.74E-01 8.50E-17
F30 2.84E-15 3.80E-10 8.43E-10 3.30E-01 8.77E-01 1.92E-01 8.38E-07 9.56E-02 1.61E-01 2.65E-06 6.31E-05 6.95E-05 3.65E-30 1.13E-01 2.29E-01
F31 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00 2.02E+02 2.02E+02 0.00E+00
F32 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00 1.00E+02 1.00E+02 0.00E+00
F33 3.22E+01 3.49E+01 1.48E+00 4.41E+01 4.41E+01 7.23E-15 3.20E+01 3.20E+01 1.08E-14 3.20E+01 3.20E+01 1.08E-14 3.21E+01 3.43E+01 1.59E+00
F34 1.38E+00 2.66E+00 5.77E-01 9.00E-01 2.69E+00 1.68E+00 1.05E+00 6.05E+00 3.02E+00 3.74E+00 4.78E+00 7.23E-01 9.00E-01 1.61E+00 6.48E-01
F35 5.78E-22 1.19E-07 4.00E-07 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.82E-06 4.45E-06 1.74E-40 6.78E-29 2.31E-28 0.00E+00 0.00E+00 0.00E+00
F36 1.72E+05 3.47E+06 3.64E+06 3.03E+03 4.86E+03 8.15E+02 1.56E+03 3.29E+07 1.08E+08 2.76E+02 9.70E+02 3.83E+02 2.61E+02 9.15E+02 4.46E+02
F37 5.63E-02 1.78E-01 7.91E-02 7.75E-07 1.19E-03 1.50E-03 3.89E-03 5.64E-02 5.54E-02 5.58E-03 3.47E-02 2.15E-02 1.55E-04 3.30E-03 2.85E-03
F38 1.13E+02 1.55E+02 2.01E+01 0.00E+00 0.00E+00 0.00E+00 1.40E-01 7.88E+01 8.70E+01 1.53E+02 2.08E+02 3.49E+01 0.00E+00 1.65E-03 5.07E-03
F39 − 4.58E+00 − 4.24E+00 2.21E-01 − 5.00E+00 − 4.83E+00 9.12E-01 9.73E-03 1.12E-01 1.57E-01 − 5.00E+00 − 5.00E+00 2.64E-04 − 4.86E+00 − 4.52E+00 3.46E-01
F40 4.03E+01 9.37E+01 3.94E+01 2.88E+01 2.89E+01 8.82E-02 1.10E-01 5.76E+00 9.96E+00 2.71E+01 2.85E+01 6.19E-01 2.64E+01 6.06E+01 5.75E+01
F41 2.38E+00 3.43E+00 6.93E-01 0.00E+00 3.66E-02 4.90E-02 9.99E-02 1.44E+00 1.60E+00 3.00E-01 5.30E-01 1.18E-01 4.72E-05 1.02E-01 6.70E-02
F42 4.66E-15 1.69E-09 8.94E-09 0.00E+00 0.00E+00 0.00E+00 4.88E-07 3.79E-02 5.30E-02 0.00E+00 7.07E-04 8.81E-04 0.00E+00 0.00E+00 0.00E+00
F43 8.26E-14 1.05E-09 3.19E-09 0.00E+00 0.00E+00 0.00E+00 2.04E-07 4.80E-02 5.64E-02 0.00E+00 1.04E-04 5.72E-04 0.00E+00 0.00E+00 0.00E+00
H. Peraza‑Vázquez et al.
 Table 4  (continued)
Algorithms

EJAYA​ RSO SAO TSA WHO

Best Ave Std Best Ave. Std. Best Ave. Std. Best Ave. Std. Best Ave. Std.

F44 1.57E-03 1.64E-03 1.41E-04 1.57E-03 1.57E-03 5.58E-06 1.58E-03 2.53E-02 3.67E-02 1.57E-03 1.57E-03 4.19E-06 1.57E-03 1.57E-03 4.33E-06
F45 2.93E-01 2.93E-01 3.37E-05 2.93E-01 2.93E-01 1.30E-05 2.93E-01 3.07E-01 2.21E-02 2.93E-01 2.93E-01 3.39E-06 2.93E-01 2.93E-01 6.15E-10
F46 1.98E+01 4.26E+01 1.56E+01 0.00E+00 9.23E-51 3.70E-50 2.87E-01 3.73E+01 7.69E+01 1.42E-04 1.03E-03 8.30E-04 1.16E-10 2.24E-08 2.79E-08
F47 9.69E+00 1.27E+01 1.96E+00 0.00E+00 4.82E-09 2.64E-08 1.80E-02 1.29E+00 3.04E+00 1.10E+00 6.49E+00 3.42E+00 1.35E-08 4.24E-06 7.24E-06
F48 2.11E+02 1.16E+15 6.32E+15 0.00E+00 1.20E-53 4.71E-53 4.20E-01 9.72E+18 3.41E+19 1.53E-04 1.17E-03 1.28E-03 4.46E-08 2.09E+02 2.96E+02
F49 7.83E-04 6.24E+00 1.73E+01 0.00E+00 0.00E+00 0.00E+00 6.16E-21 9.53E-11 3.49E-10 2.13E-28 3.36E-19 1.20E-18 4.37E-91 4.05E-62 2.22E-61
F50 6.75E+03 7.77E+03 4.93E+02 5.06E+03 7.34E+03 1.50E+03 2.04E+02 7.55E+03 3.21E+03 5.78E+03 6.85E+03 6.30E+02 2.87E+03 4.47E+03 6.56E+02
F51 − 1.49E+02 − 1.18E+02 1.27E+01 − 2.12E+02 − 1.63E+02 2.08E+01 − 4.45E+02 − 1.56E+02 8.59E+01 − 1.58E+02 − 1.33E+02 1.45E+01 − 3.09E+02 − 2.27E+02 2.59E+01
F52 − 1.55E+02 − 1.23E+02 1.50E+01 − 2.15E+02 − 1.72E+02 3.43E+01 − 3.81E+02 − 1.51E+02 7.23E+01 − 1.74E+02 − 1.38E+02 1.91E+01 − 2.31E+02 − 2.06E+02 1.73E+01
F53 − 4.55E+23 − 1.93E+22 8.32E+22 − 7.51E+27 − 6.48E+26 1.56E+27 − 2.36E+30 − 1.99E+29 5.72E+29 − 7.53E+24 − 7.21E+23 1.99E+24 − 1.23E+28 − 5.64E+26 2.27E+27
F54 5.31E-02 3.16E-01 3.43E-01 0.00E+00 4.27E-101 2.28E-100 1.25E-04 1.89E-02 3.05E-02 6.92E-11 9.60E-09 1.35E-08 6.56E-23 1.56E-17 7.72E-17
F55 − 1.05E+03 − 8.80E+02 9.92E+01 − 8.75E+02 − 6.58E+02 8.45E+01 − 1.17E+03 − 1.17E+03 5.52E-01 − 9.66E+02 − 8.51E+02 6.99E+01 − 1.08E+03 − 1.02E+03 3.34E+01
F56 3.42E+00 1.39E+01 7.97E+00 0.00E+00 1.95E-98 8.44E-98 1.35E-02 5.33E-01 5.58E-01 1.77E-08 5.70E-07 9.45E-07 4.29E-20 1.77E-16 4.32E-16
A novel metaheuristic inspired by horned lizard defense tactics

F57 7.83E-25 1.58E-21 3.32E-21 0.00E+00 6.44E-116 2.94E-115 1.04E-04 9.31E-03 1.85E-02 2.84E-55 5.97E-02 1.22E-01 1.74E-80 6.23E-67 3.00E-66
F58 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 8.08E-02 1.51E-01 0.00E+00 4.62E-22 1.92E-21 0.00E+00 0.00E+00 0.00E+00
F59 1.21E-03 3.54E-01 9.14E-01 0.00E+00 6.38E-18 2.85E-17 6.49E-05 7.06E-03 1.24E-02 8.12E-03 2.11E+02 9.93E+02 4.91E-21 1.02E-11 5.03E-11
F60 7.39E-12 1.06E-11 1.66E-12 0.00E+00 1.93E-10 4.12E-10 3.55E-12 4.30E-08 1.25E-07 6.47E-09 2.74E-07 6.78E-07 1.48E-11 2.18E-11 3.32E-12
F61 9.95E-01 9.95E-01 3.39E-16 − 1.00E+00 4.63E-01 8.97E-01 9.95E-01 9.96E-01 4.76E-04 9.97E-01 9.98E-01 2.33E-04 − 1.00E+00 − 8.67E-01 5.06E-01
F62 5.62E-13 3.27E-12 2.48E-12 − 1.00E+00 − 2.00E-01 4.07E-01 1.43E-16 2.15E-14 5.00E-14 4.30E-13 8.97E-13 6.89E-13 − 9.97E-01 − 4.19E-01 3.99E-01
F63 3.53E+01 8.73E+01 3.98E+01 0.00E+00 1.18E-100 6.46E-100 1.02E+00 6.25E+01 7.92E+01 1.02E-03 2.10E-02 3.43E-02 1.73E-11 1.38E-06 3.46E-06
Page 25 of 65 59

13
59 Page 26 of 65 H. Peraza‑Vázquez et al.

IEEE CEC-06 2019 Testbech Functions: The “100-Digit Challenge” testbench functions
from IEEE CEC-06 2019 are described in Table 3.
The entire parameter settings for each algorithm are shown in Table 2. All experiments
were conducted on a standard desktop computer with the following specifications: Intel
core i7-10750 H CPU, 2.60 GHz, 32 GB RAM, Linux Kubuntu 20.04 LTS operating sys-
tem, and implemented in MATLAB R2021b.

4 Results and discussion

This section presents the computational results of HLOA on benchmark optimization prob-
lems. The comparison results are shown in Table 4, displaying the best, mean, and stand-
ard deviation values. Moreover, Figs. 9, 10, and 11 summarize the convergence graphs of
all functions versus all algorithms chosen for this investigation. To analyze the significant
differences between the results of the proposed HLOA and the other algorithms, a non-
parametric Wilcoxon Signed-rank test with a significance level of %5 was conducted. This
trial determines the significance level of two algorithms. An algorithm is statistically sig-
nificant if the calculated p-value is less than 0.05. Table 5 summarizes the result of this
test. Furthermore, the eleven algorithms were ranked by computing the Friedman test for
63 benchmark functions. Friedman test ranks the algorithms according to their average
performance and generates a ranking score, where a lower value indicates a better per-
formance. The results are shown in Table 6. According to the statistical data presented
in Table 4, the Horned Lizard Optimization Algorithm (HLOA) can achieve exceptional
outcomes. Among the 63 benchmark functions, there are unimodal and multimodal func-
tions to test the algorithm’s capabilities related to the exploitation and exploration in the
solution space. A more reliable way to compare the performance of algorithms when
solving benchmark functions is through statistical tests. The results of the Wilcoxon rank-
sum test, in Table 5, indicate that HLOA outperformed the following algorithms: Black

Table 5  Statistical results of

Wilcoxon signed-rank test for HLOA vs JSOA HLOA vs BWOA
HLOA versus other algorithms (+/=/-) p-value (+/=/-) p-value
for 63 functions, with a 22/26/15 7.92E-01 28/19/16 3.40E-04
significance level of %5
HLOA vs DOA HLOA vs COOT
(+/=/-) p-value (+/=/-) p-value
36/16/11 6.54E-03 48/5/10 1.99E-04
HLOA vs CSA HLOA vs EJAYA​
(+/=/-) p-value (+/=/-) p-value
43/9/11 5.79E-03 47/3/13 1.40E-05
HLOA vs RSO HLOA vs SAO
(+/=/-) p-value (+/=/-) p-value
30/16/17 1.67E-03 54/2/7 7.60E-07
HLOA vs TSA HLOA vs WHO
(+/=/-) p-value (+/=/-) p-value
52/4/7 2.00E-06 31/13/19 4.31E-01

The bold numbers in the table indicate a significant difference between

the two related Algorithms where HLOA was outstanding

13
 Table 6  Friedman test of all compared algorithms for 63 functions
HLOA JSOA BWOA DOA COOT CSA EJAYA​ RSO SAO TSA WHO

Sum of ranks 248.22 250.11 313.74 340.2 411.39 381.78 475.65 364.14 560.07 529.2 291.06
A novel metaheuristic inspired by horned lizard defense tactics

Mean of ranks 3.94 3.97 4.98 5.4 6.53 6.06 7.55 5.78 8.89 8.4 4.62
Overall ranks 1 2 4 5 8 7 9 6 11 10 3

The bold number in a table indicates the order of the ranked algorithms by the Freidman test
Page 27 of 65 59

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 Table 7  IEEE CEC 2017 Benchmarks “Constrained Real-Parameter Optimization” results for Dimension 10, from the best ranked algorithms
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

13
59 Page 28 of 65

C01 7.72e+0 9.46e+1 9.06e+1 1.20e+3 3.73e+3 2.27e+3 7.11e+2 3.28e+3 1.94e+3 9.05e−2 6.15e+0 6.87e+0
C02 9.38e+0 9.02e+1 7.10e+1 1.04e+3 4.49e+3 2.89e+3 1.04e+3 3.88e+3 3.02e+3 1.26e−2 5.28e+0 8.46e+0
C03 1.91e+3 1.11e+4 7.00e+3 2.02e+3 7.21e+3 3.00e+3 1.36e+3 8.91e+3 4.68e+3 6.06e+2 4.44e+3 2.35e+3
C04 6.49e+1 9.24e+1 2.15e+1 7.71e+1 1.12e+2 1.78e+1 8.69e+1 1.19e+2 2.04e+1 1.90e+1 4.32e+1 1.14e+1
C05 1.65e+1 9.47e+4 2.45e+5 9.70e+2 6.52e+6 2.04e+7 7.08e+2 8.07e+6 3.23e+7 5.56e−1 9.79e+0 1.26e+1
C06 2.86e+2 1.01e+4 1.32e+4 1.06e+3 4.56e+4 4.41e+4 7.28e+2 3.05e+4 3.47e+4 1.22e+2 1.82e+3 1.52e+3
C07 -8.01e+1 2.36e+6 6.34e+6 -6.63e+1 9.84e+7 8.12e+7 -1.45e+2 5.70e+7 5.96e+7 -1.62e+2 -4.72e+1 4.71e+1
C08 1.55e+3 7.87e+5 1.16e+6 1.66e+6 1.90e+7 1.22e+7 3.98e+6 4.09e+7 3.26e+7 7.51e−2 1.88e+3 5.76e+3
C09 1.30e+0 4.92e+2 1.23e+3 4.09e+0 1.31e+6 2.96e+6 1.28e+1 6.08e+5 1.62e+6 -1.82e−2 7.54e+1 3.36e+2
C10 2.91e+3 1.52e+6 4.25e+6 1.82e+10 1.33e+11 1.08e+11 4.62e+9 1.84e+11 1.87e+11 9.63e−4 5.44e+0 2.52e+1
C11 4.92e+2 9.56e+5 2.09e+6 1.09e+7 7.54e+7 3.81e+7 1.98e+6 4.91e+7 3.37e+7 2.96e+1 2.81e+6 1.08e+7
C12 8.24e+0 3.38e+1 2.03e+1 1.34e+9 6.64e+9 5.11e+9 4.68e+8 4.47e+9 4.35e+9 3.99e+0 5.24e+0 4.68e+0
C13 9.80e+3 2.46e+7 4.79e+7 1.18e+8 8.49e+9 5.87e+9 2.03e+8 4.32e+9 3.01e+9 3.89e−3 5.60e+1 1.25e+2
C14 3.32e+0 2.67e+3 7.22e+3 3.81e+8 1.10e+10 8.91e+9 1.49e+8 7.78e+9 8.73e+9 2.38e+0 2.90e+0 3.02e−1
C15 1.18e+1 1.70e+1 3.12e+0 3.27e+8 3.71e+9 3.49e+9 1.49e+1 2.26e+9 3.37e+9 5.50e+0 1.06e+1 2.80e+0
C16 5.03e+1 6.60e+1 1.04e+1 5.65e+1 2.71e+9 3.65e+9 5.03e+1 1.64e+9 3.71e+9 1.26e+1 2.28e+1 6.82e+0
C17 7.99e+10 2.89e+11 1.29e+11 3.82e+10 3.82e+10 2.33e−5 3.82e+10 3.82e+10 2.33e−5 8.79e+10 2.75e+11 1.34e+11
C18 5.74e+6 5.98e+11 2.26e+12 3.35e+16 2.11e+19 2.90e+19 2.86e+13 1.03e+19 2.16e+19 3.13e+1 1.69e+7 6.51e+7
C19 1.77e+11 1.78e+11 2.34e+8 1.78e+11 1.78e+11 1.56e+8 1.77e+11 1.78e+11 1.81e+8 1.76e+11 1.77e+11 3.78e+8
C20 5.48e−1 1.83e+0 4.86e−1 8.71e−1 1.98e+0 4.23e−1 8.17e−1 1.72e+0 4.17e−1 5.86e−1 1.15e+0 2.51e−1
C21 2.40e+1 1.63e+6 7.08e+6 3.56e+9 6.16e+10 4.01e+10 2.17e+9 4.63e+10 4.63e+10 3.99e+0 9.13e+0 8.03e+0
C22 1.22e+5 1.09e+8 1.51e+8 3.95e+9 6.19e+10 4.57e+10 9.30e+7 3.65e+10 3.08e+10 9.50e+0 3.38e+4 8.24e+4
C23 3.60e+0 5.25e+5 1.24e+6 4.36e+9 9.42e+10 8.44e+10 1.65e+9 7.94e+10 8.32e+10 2.73e+0 3.25e+0 2.52e−1
C24 1.18e+1 1.67e+1 2.43e+0 5.76e+8 3.24e+10 2.53e+10 9.72e+8 3.84e+10 4.03e+10 8.64e+0 1.17e+1 2.26e+0
C25 4.40e+1 7.04e+1 1.08e+1 4.53e+8 4.13e+10 3.46e+10 4.57e+7 3.19e+10 3.51e+10 1.88e+1 3.80e+1 1.26e+1
H. Peraza‑Vázquez et al.
 Table 7  (continued)
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

C26 1.21e+5 2.04e+5 2.28e+5 7.62e+9 7.21e+10 3.92e+10 3.40e+8 2.78e+10 2.23e+10 1.21e+5 1.21e+5 4.08e−2
C27 1.67e+7 1.76e+14 3.33e+14 2.98e+17 1.89e+21 3.17e+21 1.54e+18 2.21e+21 3.95e+21 9.80e+1 3.04e+14 9.02e+14
C28 1.78e+11 1.78e+11 1.50e+8 1.78e+11 1.78e+11 1.54e+8 1.77e+11 1.78e+11 1.96e+8 1.77e+11 1.78e+11 2.41e+8
A novel metaheuristic inspired by horned lizard defense tactics
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 Table 8  IEEE CEC 2017 Benchmarks “Constrained Real-Parameter Optimization” results for Dimension 30, from the best ranked algorithms
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

13
59 Page 30 of 65

C01 3.72e+3 6.69e+3 1.75e+3 1.18e+4 2.69e+4 1.19e+4 1.41e+4 2.85e+4 1.03e+4 2.27e+3 6.03e+3 3.18e+3
C02 3.17e+3 5.74e+3 1.73e+3 1.22e+4 2.72e+4 1.04e+4 1.43e+4 5.06e+4 4.59e+4 1.21e+3 4.50e+3 1.81e+3
C03 5.44e+4 2.37e+5 1.10e+5 5.52e+4 2.54e+5 1.22e+5 8.97e+4 2.17e+5 1.10e+5 2.92e+4 6.22e+4 2.73e+4
C04 3.23e+2 3.95e+2 3.58e+1 3.86e+2 4.28e+2 2.36e+1 3.73e+2 4.31e+2 3.05e+1 1.48e+2 2.22e+2 4.94e+1
C05 3.26e+4 4.85e+5 1.14e+6 2.67e+5 2.10e+6 5.12e+6 9.99e+4 9.44e+6 2.58e+7 1.14e+2 2.16e+4 1.14e+5
C06 1.75e+3 2.73e+4 3.74e+4 2.11e+3 1.16e+5 1.34e+5 2.45e+3 7.87e+4 9.28e+4 3.02e+3 8.70e+3 3.60e+3
C07 -7.52e+1 6.85e+8 6.36e+8 8.47e+8 3.81e+9 1.49e+9 1.96e+9 4.42e+9 1.21e+9 -3.11e+2 2.86e+7 7.00e+7
C08 3.44e+7 9.76e+7 4.17e+7 1.20e+8 1.07e+9 8.94e+8 6.21e+8 6.86e+9 8.43e+9 7.91e+7 2.24e+9 3.69e+9
C09 1.55e+6 7.20e+8 1.69e+9 8.73e+9 1.20e+11 8.65e+10 8.11e+8 4.02e+10 5.34e+10 3.06e+0 1.05e+6 5.73e+6
C10 2.35e+9 8.71e+10 7.74e+10 3.27e+12 6.31e+12 3.34e+12 2.92e+12 2.17e+13 1.21e+13 2.39e+9 6.02e+10 7.11e+10
C11 1.47e+7 1.96e+8 1.44e+8 1.42e+9 2.91e+9 7.20e+8 1.00e+9 2.36e+9 8.47e+8 2.01e+6 1.81e+8 2.96e+8
C12 2.78e+6 4.83e+7 6.62e+7 9.08e+10 2.16e+11 6.04e+10 9.00e+10 2.10e+11 5.66e+10 5.29e+4 4.50e+7 1.03e+8
C13 3.84e+8 2.71e+9 3.01e+9 1.05e+11 2.15e+11 5.43e+10 9.15e+10 2.19e+11 5.87e+10 1.88e+7 2.91e+8 7.16e+8
C14 8.75e+6 1.37e+8 1.32e+8 1.06e+11 3.83e+11 1.14e+11 1.90e+11 4.24e+11 1.04e+11 2.40e+5 3.21e+7 5.75e+7
C15 1.81e+1 2.30e+1 3.84e+0 4.63e+10 1.33e+11 4.38e+10 1.34e+10 1.24e+11 5.00e+10 1.49e+1 2.57e+1 6.99e+0
C16 1.95e+2 2.26e+2 1.52e+1 4.76e+10 1.35e+11 4.23e+10 6.17e+10 1.40e+11 4.34e+10 1.07e+2 1.56e+2 2.03e+1
C17 2.41e+12 5.62e+12 1.71e+12 3.54e+11 3.54e+11 0.00e+0 3.54e+11 3.54e+11 0.00e+0 2.28e+12 5.76e+12 1.60e+12
C18 2.05e+13 1.59e+16 4.11e+16 5.42e+20 1.62e+21 6.13e+20 5.65e+20 1.46e+21 5.59e+20 3.31e+12 1.76e+16 6.51e+16
C19 1.85e+12 1.85e+12 8.56e+8 1.85e+12 1.85e+12 5.41e+8 1.85e+12 1.85e+12 7.97e+8 1.84e+12 1.84e+12 1.87e+9
C20 6.37e+0 8.49e+0 9.15e−1 7.11e+0 9.44e+0 8.31e−1 7.26e+0 8.83e+0 8.06e−1 5.81e+0 7.22e+0 6.22e−1
C21 1.97e+8 3.89e+9 3.61e+9 1.84e+12 4.84e+12 1.59e+12 1.99e+12 3.69e+12 1.29e+12 1.07e+8 4.57e+9 6.01e+9
C22 2.81e+9 1.59e+10 9.78e+9 1.77e+12 4.45e+12 1.57e+12 9.45e+11 4.03e+12 1.95e+12 3.36e+8 9.56e+9 9.97e+9
C23 3.49e+8 7.73e+9 5.94e+9 2.16e+12 8.54e+12 3.37e+12 2.78e+12 6.70e+12 2.64e+12 1.89e+8 4.01e+9 5.20e+9
C24 1.81e+1 8.73e+7 3.38e+8 1.44e+12 3.72e+12 1.21e+12 1.21e+12 3.80e+12 1.38e+12 1.81e+1 1.91e+8 8.86e+8
C25 1.95e+2 1.13e+8 4.24e+8 4.73e+11 4.04e+12 1.79e+12 1.30e+12 3.23e+12 1.31e+12 1.76e+2 2.23e+9 8.59e+9
H. Peraza‑Vázquez et al.
 Table 8  (continued)
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

C26 1.15e+8 3.26e+9 2.24e+9 1.14e+12 3.95e+12 1.75e+12 1.76e+12 4.11e+12 1.58e+12 8.23e+6 3.64e+9 5.13e+9
C27 6.24e+16 2.01e+19 3.14e+19 1.50e+23 1.88e+24 1.47e+24 6.83e+22 9.30e+23 8.05e+23 5.38e+15 4.86e+19 1.29e+20
C28 1.85e+12 1.85e+12 7.62e+8 1.85e+12 1.85e+12 6.30e+8 1.85e+12 1.85e+12 7.19e+8 1.85e+12 1.85e+12 8.13e+8
A novel metaheuristic inspired by horned lizard defense tactics
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 Table 9  IEEE CEC 2017 Benchmarks “Constrained Real-Parameter Optimization” results for Dimension 50, from the best ranked algorithms
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

13
59 Page 32 of 65

C01 1.40e+4 2.09e+4 3.73e+3 3.90e+4 1.10e+5 4.31e+4 3.64e+4 1.14e+5 6.50e+4 1.41e+4 2.63e+4 8.64e+3
C02 1.23e+4 2.61e+4 3.04e+4 4.39e+4 1.53e+5 8.01e+4 4.02e+4 1.27e+5 8.43e+4 9.65e+3 1.77e+4 4.40e+3
C03 1.77e+5 9.63e+5 5.38e+5 2.45e+5 9.23e+5 6.32e+5 1.69e+5 6.87e+5 4.39e+5 7.11e+4 1.58e+5 5.95e+4
C04 6.61e+2 7.28e+2 4.51e+1 7.39e+2 7.81e+2 2.56e+1 7.09e+2 7.85e+2 3.76e+1 3.44e+2 5.33e+2 9.98e+1
C05 2.65e+5 6.17e+6 1.98e+7 4.61e+5 4.54e+7 7.29e+7 3.27e+5 5.36e+7 8.84e+7 3.74e+3 2.18e+5 5.12e+5
C06 2.57e+3 2.65e+4 4.24e+4 6.73e+3 1.57e+5 1.88e+5 3.20e+3 1.12e+5 1.28e+5 5.89e+3 1.46e+4 4.98e+3
C07 1.11e+9 4.43e+9 1.79e+9 2.45e+9 1.15e+10 5.72e+9 9.45e+9 1.71e+10 3.77e+9 -2.28e+2 9.22e+8 7.92e+8
C08 2.20e+8 7.06e+8 2.98e+8 8.94e+8 6.45e+9 7.02e+9 1.74e+9 3.14e+10 3.68e+10 5.71e+8 5.77e+10 7.21e+10
C09 2.61e+8 7.34e+10 1.18e+11 2.28e+11 9.82e+11 3.79e+11 1.45e+11 8.18e+11 4.61e+11 7.83e+2 1.29e+7 4.93e+7
C10 8.61e+11 2.45e+12 1.09e+12 2.00e+13 7.84e+13 5.55e+13 1.64e+13 8.19e+13 5.33e+13 7.12e+11 3.27e+12 2.49e+12
C11 3.50e+8 1.72e+9 8.56e+8 6.23e+9 9.23e+9 1.42e+9 4.67e+9 7.96e+9 1.68e+9 7.67e+7 2.89e+9 4.87e+9
C12 4.30e+8 1.67e+9 1.07e+9 5.71e+11 8.04e+11 9.41e+10 5.11e+11 7.70e+11 1.10e+11 2.73e+8 5.91e+9 6.48e+9
C13 4.70e+9 3.77e+10 3.81e+10 4.83e+11 8.56e+11 1.37e+11 4.52e+11 7.88e+11 1.47e+11 2.65e+9 1.46e+10 1.12e+10
C14 9.97e+8 4.73e+9 4.28e+9 8.43e+11 1.57e+12 2.51e+11 9.22e+11 1.53e+12 2.62e+11 5.33e+8 1.90e+10 2.13e+10
C15 2.12e+1 2.58e+1 4.57e+0 3.08e+11 5.45e+11 1.19e+11 2.72e+11 5.16e+11 1.09e+11 2.12e+1 2.37e+8 8.65e+8
C16 3.39e+2 3.88e+2 2.29e+1 3.17e+11 5.44e+11 9.80e+10 3.34e+11 5.33e+11 9.96e+10 2.40e+2 4.57e+8 2.50e+9
C17 8.83e+12 2.02e+13 3.86e+12 1.04e+12 1.04e+12 7.45e−4 1.04e+12 1.04e+12 7.45e−4 1.21e+13 1.94e+13 4.24e+12
C18 1.70e+16 1.83e+18 1.91e+18 3.96e+21 7.29e+21 1.43e+21 3.60e+21 6.47e+21 1.61e+21 3.91e+17 4.77e+19 7.98e+19
C19 5.28e+12 5.28e+12 2.20e+9 5.28e+12 5.28e+12 9.79e+8 5.28e+12 5.28e+12 1.28e+9 5.26e+12 5.27e+12 4.34e+9
C20 1.40e+1 1.66e+1 1.37e+0 1.61e+1 1.78e+1 8.23e−1 1.50e+1 1.73e+1 1.08e+0 1.29e+1 1.45e+1 9.36e−1
C21 5.04e+9 6.52e+10 4.37e+10 6.93e+12 1.04e+13 1.75e+12 5.03e+12 9.95e+12 1.88e+12 9.92e+9 1.58e+11 1.31e+11
C22 2.59e+10 1.82e+11 1.42e+11 5.31e+12 1.08e+13 1.94e+12 6.11e+12 9.88e+12 1.75e+12 1.75e+10 3.65e+11 6.20e+11
C23 2.04e+10 1.05e+11 6.55e+10 1.26e+13 2.04e+13 4.39e+12 1.27e+13 2.06e+13 3.42e+12 4.62e+10 3.50e+11 3.20e+11
C24 1.81e+1 8.70e+9 1.70e+10 4.46e+12 9.09e+12 1.86e+12 4.93e+12 8.89e+12 2.24e+12 2.75e+1 7.44e+10 1.05e+11
C25 3.77e+2 7.08e+9 1.26e+10 5.98e+12 9.49e+12 1.71e+12 5.36e+12 8.83e+12 2.02e+12 4.96e+8 8.49e+10 1.20e+11
H. Peraza‑Vázquez et al.
 Table 9  (continued)
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

C26 6.33e+9 4.62e+10 3.65e+10 5.91e+12 9.97e+12 2.03e+12 5.93e+12 1.01e+13 1.96e+12 9.95e+9 1.60e+11 1.53e+11
C27 2.63e+19 5.81e+20 5.24e+20 7.36e+23 3.46e+24 1.57e+24 1.30e+24 3.53e+24 1.57e+24 5.14e+18 4.43e+21 8.06e+21
C28 5.28e+12 5.28e+12 1.69e+9 5.28e+12 5.28e+12 1.07e+9 5.28e+12 5.28e+12 9.14e+8 5.28e+12 5.28e+12 1.75e+9
A novel metaheuristic inspired by horned lizard defense tactics
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 Table 10  IEEE CEC 2017 Benchmarks “Constrained Real-Parameter Optimization” results for Dimension 100, from the best ranked algorithms
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

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59 Page 34 of 65

C01 6.61e+4 1.04e+5 1.60e+4 1.93e+5 4.70e+5 1.49e+5 2.08e+5 4.13e+5 1.60e+5 6.03e+4 1.29e+5 4.16e+4
C02 6.58e+4 1.38e+5 7.54e+4 2.12e+5 5.57e+5 1.69e+5 1.96e+5 5.16e+5 2.16e+5 5.52e+4 1.06e+5 4.06e+4
C03 7.89e+5 1.85e+6 8.11e+5 6.25e+5 2.03e+6 9.76e+5 4.19e+5 2.11e+6 1.04e+6 2.32e+5 5.86e+5 2.05e+5
C04 1.45e+3 1.60e+3 6.44e+1 1.56e+3 1.65e+3 4.91e+1 1.62e+3 1.69e+3 4.14e+1 1.05e+3 1.35e+3 2.08e+2
C05 1.28e+6 3.63e+6 1.12e+7 1.27e+6 7.04e+6 1.94e+7 1.14e+6 2.38e+7 4.91e+7 1.85e+5 2.25e+6 2.29e+6
C06 4.67e+3 4.08e+4 4.67e+4 9.12e+3 3.65e+5 2.35e+5 2.40e+4 3.23e+5 2.88e+5 8.15e+3 2.24e+4 7.67e+3
C07 2.58e+10 4.54e+10 8.84e+9 2.20e+10 3.89e+10 1.76e+10 6.88e+10 9.78e+10 1.11e+10 6.51e+9 2.30e+10 9.30e+9
C08 6.52e+9 1.35e+10 4.33e+9 1.63e+10 7.68e+10 6.90e+10 4.76e+10 3.52e+11 3.43e+11 3.44e+10 1.77e+12 1.85e+12
C09 3.41e+11 1.49e+12 8.02e+11 2.66e+12 4.91e+12 1.13e+12 1.66e+12 4.70e+12 1.22e+12 1.05e+8 9.62e+9 2.19e+10
C10 3.03e+13 4.85e+13 1.27e+13 1.27e+14 5.33e+14 5.85e+14 1.43e+14 5.94e+14 4.02e+14 4.39e+13 9.76e+13 3.81e+13
C11 8.92e+9 2.21e+10 6.07e+9 3.67e+10 4.55e+10 3.95e+9 2.67e+10 4.23e+10 5.26e+9 5.56e+9 1.77e+11 5.01e+11
C12 5.89e+10 1.55e+11 5.12e+10 4.86e+12 5.73e+12 3.66e+11 4.37e+12 5.59e+12 5.33e+11 2.11e+11 9.13e+11 4.67e+11
C13 1.17e+11 6.00e+11 3.10e+11 5.21e+12 5.95e+12 3.13e+11 5.07e+12 5.95e+12 4.08e+11 3.64e+11 9.67e+11 7.79e+11
C14 1.20e+11 3.52e+11 1.30e+11 9.59e+12 1.13e+13 7.21e+11 8.25e+12 1.13e+13 9.15e+11 3.50e+11 1.57e+12 8.26e+11
C15 2.43e+1 2.31e+10 4.93e+10 3.35e+12 4.24e+12 4.15e+11 3.42e+12 4.27e+12 3.89e+11 2.45e+10 4.56e+11 4.14e+11
C16 7.23e+2 9.18e+9 2.08e+10 3.46e+12 4.37e+12 3.33e+11 3.07e+12 4.20e+12 4.43e+11 1.45e+10 3.82e+11 3.03e+11
C17 8.19e+13 1.12e+14 1.25e+13 6.47e+12 6.47e+12 4.97e−3 6.47e+12 6.47e+12 4.97e−3 7.32e+13 1.05e+14 1.44e+13
C18 1.48e+20 1.09e+21 1.06e+21 9.07e+22 1.13e+23 9.26e+21 7.68e+22 1.07e+23 1.33e+22 1.46e+21 3.04e+22 4.68e+22
C19 2.16e+13 2.16e+13 5.19e+9 2.16e+13 2.16e+13 2.71e+9 2.16e+13 2.16e+13 2.09e+9 2.15e+13 2.15e+13 1.54e+10
C20 3.40e+1 3.80e+1 1.87e+0 3.59e+1 3.96e+1 1.66e+0 3.56e+1 3.92e+1 1.50e+0 3.31e+1 3.51e+1 1.01e+0
C21 8.33e+11 2.65e+12 7.64e+11 5.08e+13 6.53e+13 6.33e+12 4.35e+13 6.16e+13 7.05e+12 2.77e+12 1.00e+13 4.51e+12
C22 1.71e+12 3.86e+12 1.35e+12 4.48e+13 6.36e+13 8.99e+12 4.82e+13 6.49e+13 6.47e+12 3.71e+12 1.11e+13 1.05e+13
C23 1.60e+12 5.50e+12 1.58e+12 9.85e+13 1.29e+14 1.27e+13 7.72e+13 1.23e+14 1.52e+13 5.15e+12 2.08e+13 1.53e+13
C24 8.69e+11 2.04e+12 8.04e+11 3.94e+13 5.91e+13 7.27e+12 3.48e+13 5.58e+13 8.28e+12 2.27e+12 9.30e+12 7.58e+12
C25 7.96e+11 1.90e+12 8.51e+11 4.17e+13 5.79e+13 6.56e+12 4.60e+13 5.98e+13 5.55e+12 2.57e+12 8.76e+12 4.06e+12
H. Peraza‑Vázquez et al.
 Table 10  (continued)
Function HLOA JSOA BWOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

C26 1.19e+12 2.73e+12 1.13e+12 4.67e+13 6.31e+13 6.59e+12 4.55e+13 6.12e+13 6.98e+12 3.49e+12 9.78e+12 5.82e+12
C27 3.99e+22 1.25e+23 6.64e+22 1.66e+25 3.05e+25 5.20e+24 9.92e+24 2.57e+25 7.40e+24 8.53e+22 1.33e+24 1.36e+24
C28 2.16e+13 2.16e+13 2.68e+9 2.16e+13 2.16e+13 2.10e+9 2.16e+13 2.16e+13 2.21e+9 2.16e+13 2.16e+13 2.52e+9
A novel metaheuristic inspired by horned lizard defense tactics
Page 35 of 65 59

13
59 Page 36 of 65 H. Peraza‑Vázquez et al.

Table 11  Statistical results of Wilcoxon signed-rank test for IEEE CEC 2017 benchmark functions at
Dimension 10, with a significance level of %5
HLOA vs JSOA HLOA vs BWOA HLOA vs WHO

(+/=/-) p-value (+/=/-) p-value (+/=/-) p-value

25/0/3 2.25E-4 23/0/5 5.85E-4 2/0/26 4.16E-4

The bold numbers in the table indicate a significant difference between the two related Algorithms where
HLOA was outstanding

Table 12  Friedman test for IEEE HLOA JSOA BWOA WHO

CEC 2017 benchmark functions
at Dimension 10
Mean of ranks 2.21 3.59 3.05 1.14
Overall ranks 2 4 3 1

The bold number in a table indicates the order of the ranked algo-
rithms by the Freidman test

Table 13  Statistical results of Wilcoxon signed-rank test for IEEE CEC 2017 benchmark functions at
Dimension 30, with a significance level of %5
HLOA vs JSOA HLOA vs BWOA HLOA vs WHO

(+/=/-) p-value (+/=/-) p-value (+/=/-) p-value

27/0/1 4.60E-05 24/0/4 2.94E-04 9/0/19 3.74E-01

The bold numbers in the table indicate a significant difference between the two related Algorithms where
HLOA was outstanding

Widow Optimization Algorithm (BWOA), Dingo Optimization Algorithm (DOA), Coot

Bird Algorithm (COOT), Crystal Structure Algorithm (CSA), Enhanced Jaya Algorithm
(EJAYA), Rat Swarm Optimizer (RSO), Smell Agent Optimization (SAO) and, Tunicate
Swarm Algorithm (TSA). Meanwhile, there are no significant differences between HLOA
versus the Jumping Spider Optimization Algorithm (JSOA) and Wild Horse Optimizer
(WHO). Moreover, in the Friedman test analysis, the HLOA algorithm is ranked first,
whereas the JSOA, WHO, and BWOA are ranked second, third, and fourth, respectively.
Based on this analysis, in the rest of the paper, these algorithms are only used in IEEE CEC
2017 ”Constrained Real-Parameter Optimization”, IEEE CEC-06 2019 “100-Digit Chal-
lenge”, and real-world applications.
On the other hand, the findings obtained by computational analysis of HLOA on bench-
mark functions from IEEE CEC 2017 “Constrained Real-Parameter Optimization” prob-
lems for dimensions 10, 30, 50, and 100 are shown in Tables 7, 8, 9, and 10. Say tables dis-
play the best, mean, and standard deviation computed. To examine the disparities among
the algorithms, a Wilcoxon signed-rank test was used with a significance level of 5%. The
ranking between the algorithms HLOA, BWOA, JSOA, and WHO was determined using
the Friedman test.
Table 11 summarizes the Wilcoxon rank-sum test for 10-dimensional problems and
indicates that HLOA outperformed all the algorithms. Nevertheless, in Table 12 "Friedman
test", it is ranked in second place.

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A novel metaheuristic inspired by horned lizard defense tactics Page 37 of 65 59

Table 14  Friedman test for IEEE HLOA JSOA BWOA WHO

CEC 2017 benchmark functions
at Dimension 30
Mean of ranks 1.86 3.45 3.27 1.43
Overall ranks 2 4 3 1

The bold number in a table indicates the order of the ranked algo-
rithms by the Freidman test

Table 15  Statistical results of Wilcoxon signed-rank test for IEEE CEC 2017 benchmark functions at
Dimension 50, with a significance level of %5
HLOA vs JSOA HLOA vs BWOA HLOA vs WHO

(+/=/-) p-value (+/=/-) p-value (+/=/-) p-value

26/0/2 6.10E-5 26/0/2 6.10E-5 16/0/12 6.51E-02

The bold numbers in the table indicate a significant difference between the two related Algorithms where
HLOA was outstanding

For 30-dimensional problems, the Wilcoxon rank-sum test in Table 13 shows that
HLOA outperformed the JSOA and BWOA algorithms. Meanwhile, there are no signifi-
cant differences with WHO. Whereas, in Table 14 "Friedman test" ranks HLOA in second
place.
According to the results presented in Table 15, it can be observed that in the case
of 50-dimensional problems, the HLOA algorithm demonstrated superior performance
compared to the JSOA and BWOA algorithms, as indicated by the Wilcoxon rank-sum
test. In contrast, there are no substantial disparities with the WHO. In comparison, the
Friedman test assigns the highest rank to HLOA. See Table 16.
In the context of 100-dimensional problems, the Wilcoxon rank-sum test in Table 17
shows that HLOA outperformed all algorithms. Meanwhile, the Friedman test has deter-
mined that HLOA holds the highest rank, as seen in Table 18.
Additionally, the computational results of HLOA on benchmark functions from CEC-
06 2019 “The 100-Digit Challenge” problems are shown in Table 19, displaying the best,
mean, and standard deviation values. Furthermore, Fig. 12 summarizes the convergence
graphs of all functions versus the best-ranked algorithms. To analyze the significant differ-
ences between the results, Wilcoxon Signed-rank test with a significance level of %5 was
carried out. Table 20 summarizes the result of this test and indicates that HLOA outper-
formed Black Widow Optimization Algorithm (BWOA). Meanwhile, there are no signifi-
cant differences between HLOA with the Jumping Spider Optimization Algorithm (JSOA)
and Wild Horse Optimizer (WHO).

Table 16  Friedman test for IEEE HLOA JSOA BWOA WHO

CEC 2017 benchmark functions
at Dimension 50
Mean of ranks 1.57 3.52 3.20 1.71
Overall ranks 1 4 3 2

The bold number in a table indicates the order of the ranked algo-
rithms by the Freidman test

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59 Page 38 of 65 H. Peraza‑Vázquez et al.

Table 17  Statistical results of Wilcoxon signed-rank test for IEEE CEC 2017 benchmark functions at
Dimension 100, with a significance level of %5
HLOA vs JSOA HLOA vs BWOA HLOA vs WHO

(+/=/-) p-value (+/=/-) p-value (+/=/-) p-value

26/0/2 1.19E-04 26/0/2 9.90E-05 17/0/11 9.43E-03

The bold numbers in the table indicate a significant difference between the two related Algorithms where
HLOA was outstanding

Table 18  Friedman test for IEEE HLOA JSOA BWOA WHO

CEC 2017 benchmark functions
at Dimension 100
Mean of ranks 1.54 3.48 3.16 1.82
Overall ranks 1 4 3 2

The bold number in a table indicates the order of the ranked algo-
rithms by the Freidman test

5 Real‑world applications

In this section, the capabilities of the HLOA were tested by solving five optimization prob-
lems, which are three Real-World Single Objective Bound Constrained Numerical Optimi-
zation problems taken from the CEC 2020 special session (Kumar et al. 2020), the Multi-
ple Gravity Assist (MGA) problems provided by the European Space Agency (ESA) [89]
and the Optimal Power Flow Problem. For all engineering problems solved, HLOA was
compared against the three best-ranked algorithms calculated from the Friedman test, see
Table 6.

5.1 Constraint handling

The Penalization of Constraints method was used for constraint handling. The mathemati-
cal formulation of this method is described in Eq. 20 and taken from Peraza-Vázquez et al.
(2021).

if MCV( x ) ≤ 0
{ → →
→ f ( x ),
F( x ) = → (20)
fmax + MCV( x ), otherwise.
→
Where f ( x ) is the fitness function value of a feasible solution (a solution that does not
violate constraints), whereas
→
fmax is the fitness function value of the worst solution in the
population, and MCV( x ) is the Mean Constraint Violation (Peraza-Vázquez et al. 2021)
represented in Eq. 21.
p m
∑ ∑
Gi (x→ ) + Hj (x→ )
i=1 j=1 (21)
MCV(x→ ) =
p+m

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 Table 19  IEEE CEC-C06 2019 Benchmarks ‘‘The 100-Digit Challenge:” results
Algorithms
Function HLOA BWOA JSOA WHO
Best Ave Std Best Ave Std Best Ave Std Best Ave Std

CEC-1 -4.441E-16 -4.441E-16 0.000E+00 4.238E+04 1.704E+05 2.671E+05 -4.441E-16 -4.441E-16 0.000E+00 2.593E-10 1.720E-08 4.305E-08
CEC-2 -2.000E+02 -2.000E+02 0.000E+00 1.735E+01 1.787E+01 4.941E-01 -2.000E+02 -2.000E+02 0.000E+00 -2.000E+02 -2.000E+02 0.000E+00
CEC-3 -1.864E+02 -1.864E+02 8.672E-14 1.270E+01 1.270E+01 1.218E-03 -1.864E+02 -1.864E+02 8.672E-14 -1.864E+02 -1.864E+02 8.672E-14
CEC-4 -4.590E+00 -4.502E+00 2.687E-01 2.377E+03 8.875E+03 5.894E+03 -4.590E+00 -4.561E+00 1.608E-01 -4.590E+00 -4.561E+00 1.608E-01
A novel metaheuristic inspired by horned lizard defense tactics

CEC-5 -1.077E+00 -1.077E+00 2.220E-16 1.764E+00 3.122E+00 8.127E-01 -1.077E+00 -1.077E+00 1.797E-16 -1.077E+00 -1.077E+00 0.000E+00
CEC-6 0.000E+00 0.000E+00 0.000E+00 8.312E+00 1.120E+01 1.216E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
CEC-7 -2.835E+13 -1.755E+13 1.196E+13 4.019E+02 9.562E+02 3.151E+02 -2.835E+13 -2.592E+13 7.524E+12 -4.755E+11 -7.761E+10 1.036E+11
CEC-8 1.000E+00 1.000E+00 0.000E+00 5.289E+00 6.361E+00 4.656E-01 1.000E+00 1.000E+00 0.000E+00 1.000E+00 1.000E+00 0.000E+00
CEC-9 1.553E-21 7.621E-02 2.325E-01 2.285E+02 1.207E+03 8.101E+02 2.301E-20 3.273E-17 1.072E-16 0.000E+00 9.244E-34 5.063E-33
CEC-10 -1.068E+02 -1.035E+02 7.374E+00 2.021E+01 2.049E+01 1.340E-01 -1.068E+02 -1.022E+02 8.369E+00 -1.068E+02 -1.061E+02 3.552E+00
Page 39 of 65 59

13
59 Page 40 of 65 H. Peraza‑Vázquez et al.

Fig. 9  Convergence curves of the best-ranked algorithms by Friendam test. From F1 to F20, functions
shown in Appendix A

Fig. 10  Convergence curves of the best-ranked algorithms by Friendam test. From F21 to F42, functions
shown in Appendix A

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Fig. 11  Convergence curves of the best-ranked algorithms by Friendam test. From F43 to F63, functions
shown in Appendix A

Table 20  Statistical results of HLOA vs JSOA HLOA vs BWOA HLOA vs WHO

Wilcoxon signed-rank test for
CEC 2019, with a significance
(+/=/-) p-value (+/=/-) p-value (+/=/-) p-value
level of %5
1/6/3 0.465 10/0/0 0.005 2/5/3 0.686

The bold numbers in the table indicate a significant difference between

the two related Algorithms where HLOA was outstanding

Fig. 12  Convergence Curves of CEC 2019 functions

13
59 Page 42 of 65 H. Peraza‑Vázquez et al.

Table 21  Comparison Results Algorithms Optimal values for variables

of the Process Flow Sheeting
problem x1 x2 x3 fmin

HLOA 0.94194 −2.1 1 1.076554818

JSOA 0.94194 −2.1 1 1.076554818
BWOA 0.94194 −2.1 1 1.076554818
WHO 0.94194 −2.1 1 1.076554818

→ →
Here, the MCV( x ) is the mean sum of the inequalities (Gi ( x )) and the equalities ( Hj ( x ) )
→

constraints, depicted by Eq. 22 and 23, respectively. Notice that the inequality gi ( x ) and
→

→
equality hj ( x ) constraints only have a value, the punishment if the constraint is violated.

0, if gi ( x ) ≤ 0
{ →
→
Gi ( ) =
x → (22)
gi ( x ), otherwise.

if |hj ( x )| − 𝛿 ≤ 0
{ →
→ 0,
Hj ( x ) = → (23)
|hj ( x )|, otherwise

5.2 Process flow sheeting problem

This non-convex constrained optimization problem has three decision variables with
three inequality constraints (Kumar et→al. 2020) as described in Eq. 24. The best-known
feasible objective function value is f ( x ) = 1.0765430833.
→
Minimize f ( x ) = −0.7x3 + 5(0.5 − x1 )2 + 0.8
g1 ( x ) = −e(x1 −0.2) − x2 ≤ 0
→
Subject to
g2 ( x ) = x2 + 1.1x3 ≤ −1.0 (24)
→

g3 ( x ) = x1 − x3 ≤ 0.2
→

with bounds ∶ 0.2 ≤ x1 ≤ 1, −2.22554 ≤ x2 ≤ 1, x3 ∈ {0, 1}

In Table 21, the comparison results show that HLOA, JSOA, BWOA, and WHO
reported feasible and competitive solutions, as seen in the convergence graph in Fig. 13.
The HLOA difference with the best-known feasible objective function value for all algo-
rithms is 1.17347E-05.

5.3 Process synthesis problem

This problem has seven decision variables and nine inequality constraints with non-linear-
ities in real and binary variables (Kumar et al. 2020). The mathematical→
representation is
shown in Eq. 25. The best-known feasible objective function value is f ( x ) = 2.9248305537
.

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A novel metaheuristic inspired by horned lizard defense tactics Page 43 of 65 59

Fig. 13  Convergence graph of the Process Flow Sheeting Problem

→
Minimize f ( x ) = (1 − x1 )2 + (2 − x2 )2 + (3 − x3 )2 + (1 − x4 )2 +
(1 − x5 )2 + (1 − x6 )2 − ln(1 + x7 )
g 1 ( x ) = x1 + x2 + x3 + x 4 + x 5 + x6 ≤ 5
→
Subject to
g2 ( x ) = x1 2 + x2 2 + x3 2 + x6 2 ≤ 5.5
→

g3 ( x ) = x1 + x4 ≤ 1.2
→

g4 ( x ) = x2 + x5 ≤ 1.8
→

g5 ( x ) = x3 + x6 ≤ 2.5 (25)
→

g6 ( x ) = x1 + x7 ≤ 1.2
→

g7 ( x ) = x22 + x52 ≤ 1.64

→

g8 ( x ) = x32 + x62 ≤ 4.25

→

g9 ( x ) = x32 + x52 ≤ 4.64

→

with bounds 0 ≤ x1 , x2 , x3 ≤ 1,
x4 , x5 , x6 , x7 ∈ {0, 1}

In Table 22, the comparison results show that all algorithms reported feasible and com-
petitive solutions, as seen in the convergence graph in Fig. 14. Note that the HLOA and
BWOA have the most competitive values, whilst HLOA difference to the best-know feasi-
ble objective function value is 1.11E-04.

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59 Page 44 of 65 H. Peraza‑Vázquez et al.

Table 22  Comparison results of the Process Synthesis problem

Optimal values Algorithms
for variables
HLOA JSOA BWOA WHO

x1 0.19849 0.19468 0.19754 0.19551

x2 1.2806 1.2806 1.2805 1.2806
x3 1.9546 1.955 1.9548 1.9549
x4 1 1 1 1
x5 0 0 0 0
x6 0 0 0 0
x7 1 1 1 1
fmin 2.92494158706362 2.92495120667677 2.92487042815638 2.92496872672763

Fig. 14  Convergence graph of the Process Synthesis Problem

5.4 Optimal design of an industrial refrigeration system

This problem stated in Eq. 26, have fourteen decision variables and fifteen inequal-
ity constraints formulated and a non-linear inequality-constrained optimization problem
→
(Kumar et al. 2020). Where the best-known feasible objective function value is f ( x ) =
3.22130008E-02.

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A novel metaheuristic inspired by horned lizard defense tactics Page 45 of 65 59

→
f ( x ) = 63098.88x2 x4 x12 + 5441.5x22 x12 + 115055.5x21.664 x6 + 6172.27x22 x6
+63098.88x1 x3 x11 + 5441.5x12 x11 + 115055.5x11.664 x5 + 6172.27x12 x5 +
Minimize 140.53x1 x11 + 281.29x3 x11 + 70.26x12 + 281.29x32 +
−1 2
14437x81.8812 x12
0.3424
x10 x14 x1 x7 x9−1 +
20470.2x72.893 x11
0.316 2
x1
g1 ( x ) = 1.524x7−1 ≤ 1
→
Subject to
g2 ( x ) = 1.524x8−1 ≤ 1
→

= 0.07789x1 − 2x7−1 x9 ≤ 1
→
g3 ( x )
≤1
→
g4 ( x ) = 7.05305x9−1 x12 x10 x8−1 x2−1 x14
−1

x14 ≤ 1
→
−1
g5 ( x ) = 0.0833x13
x12 x8 x10 ≤ 1
→
−1 2.1195 2.1195 −1 0.2 −1
g6 ( x ) = 47.136x20.333 x10 x12 − 1.333x8 x13 + 62.08x13
≤1
→
g7 ( x ) = 0.04771x10 x81.8812 x12
0.3424

≤1
→
g8 ( x ) = 0.0488x9 x71.893 x11
0.316

g9 ( x ) = 0.0099x1 x3−1 ≤ 1
→

g10 ( x ) = 0.0193x2 x4−1 ≤ 1

→

g11 ( x ) = 0.0298x1 x5−1 ≤ 1

→

g12 ( x ) = 0.056x2 x6−1 ≤ 1

→

g13 ( x ) = 2x9−1 ≤ 1
→

≤1
→
−1
g14 ( x ) = 2x10
≤1
→
−1

0.001 ≤ xi ≤ 5, i = 1, 2, .., 14
g1 ( x ) = x12 x11
with bounds
(26)

In Table 23, the comparison results show that HLOA and JSOA algorithms reported
feasible solutions, whereas BWOA and WHO results are infeasible, as seen in the con-
vergence graph, see Fig. 15. Note that the HLOA algorithm is ranked as the first-best
obtained solution, and the difference with the best-known feasible objective function value
is improved by -9.93E-04.

5.5 Multiple gravity assist (MGA) optimization problem: cassini spacecraft

trajectory design

The Multiple Gravity Assist (MGA) problem is a straightforward benchmark for evaluat-
ing global optimization techniques in Space Mission Design-related challenges. The math-
ematical representation is a finite-dimension global optimization problem with nonlinear
constraints. For an interplanetary probe powered by a chemical propulsion engine to travel
from the Earth to another planet or asteroid, the best potential trajectory must be found.
The MGA mathematical approach can be found in Zuo et al. (2016); Wagner and Wie
(2015). Where the European Space Agency (ESA) raises an MGA issue with the Cassini
spacecraft trajectory design problem [89]. The objective of this mission is to reach Saturn
and get captured by its gravity into an orbit having pericenter radius rp set to 108950 km,
and eccentricity fixed to 0.98. The lower and upper variable bounds are shown in Table 24.

13
59 Page 46 of 65 H. Peraza‑Vázquez et al.

Table 23  Comparison results of the Optimal Design of an Industrial Refrigeration System

Optimal values Algorithms
for variables
HLOA JSOA BWOA WHO

x1 0.001 0.001 0.001 0.001

x2 0.001 0.0010003 0.001 0.001
x3 0.001 0.001 0.001 0.001
x4 0.001 0.001 0.001 0.001
x5 0.001 0.001 0.001 0.0010002
x6 0.001 0.001 0.001 0.001
x7 1.5284 1.58 1.5239 1.5082
x8 1.524 1.5241 1.3249 1.524
x9 4.8055 4.9599 2.895 4.9999
x10 2.0036 2.1421 2.0499 2.0189
x11 0.001 0.001 0.0074473 0.015076
x12 0.001 0.001 0.001 0.015076
x13 0.0072995 0.007508 0.0074762 0.025947
x14 0.087629 0.089784 0.079109 0.025947
fmin 0.033206223934 0.034988 0.052832403932352∗ 0.044098911284881∗

* The solution does not satisfy one or more constraints

Fig. 15  Convergence graph of the Optimal Design of an Industrial Refrigeration System. The dotted lines
represent an infeasible solution shown by an algorithm

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A novel metaheuristic inspired by horned lizard defense tactics Page 47 of 65 59

Table 24  Lower and Upper State Variable Lower Bounds Upper Bounds Units
Bounds of variables
x(1) t0 -1000 0 MJD2000∗
x(2) T1 30 400 days
x(3) T2 100 470 days
x(4) T3 30 400 days
x(5) T4 400 2000 days
x(6) T5 1000 6000 days

* Modified Julian Date 2000

Table 25  Constraints on the Fy-by pericenters km

various fly-by pericenters
rp1 6351.8
rp2 6351.8
rp3 6778.1
rp4 671492

The Constraints of the various fly-by pericenters are shown in Table 25. The planetary
fly-by sequence and→more details can be found in [89]. The best-known feasible objective
function value is f ( x ) = 4.9307.
The comparison results in Table 26 show that HLOA, JSOA, BWOA, and WHO
reported feasible solutions. The HLOA and WHO showed competitive results, whereas
BWOA was outstanding, as seen in the convergence graph in Fig. 16. The HLOA differ-
ence with the best-known feasible objective function value is 4.26E-01.

5.6 Optimal power flow

The optimal power flow (OPF) is a non-linear optimization problem that combines an opti-
mization function with the power flow problem to calculate the operating conditions of a
power system network subjected to practical and physical constraints (Nucci et al. 2021;
Huneault and Galiana 1991), OPF mathematical formulation can be described as follows:
Minimize J(x, u),
Subjected to g(x, u) = 0, (27)
and h(x, u) ≤ 0
Where J(x, u) is the objective function, g(x, u) is the set of equality constraints, while
h(x, u) are the inequality constraints. The set of control variables u, defined in Eq. 28,
is PG , active power generation at the PV buses; VG , voltage magnitude; QC , shunt Volt-
Amperes Reactive (VAR) compensator; T transformer tap settings. Subindices NG, NC,
and NT are the number of generators, the number of regulating transformers, and the VAR
compensators, respectively.

13
59 Page 48 of 65 H. Peraza‑Vázquez et al.

Fig. 16  MGA Optimization: Cassini Spacecraft Trajectory Design Problem

[ ]
uT = PG2 ⋯ PGNG , VG1 ⋯ VGNG , QC1 ⋯ QCNC , T1 ⋯ TNT (28)

The set of state variables xT , stated in Eq. 29, are:

PG1: Active power generation at slack bus

VL : Voltage at PQ buses
QG : Generators reactive power output
Sl : Line flow, transmission line loadings
[ ]
xT = PG1 , VL1 ⋯ VLNL , QG1 ⋯ QGNC , Sl1 ⋯ Slnl (29)

Where subindices NL and nl are the numbers of load buses and transmission lines,
respectively.
The real and reactive power equality constraints taken from the power flow equations are
defined in Eq. 30 and 31.
NB
∑ [ ]
PGi − PDi − Vi Vj Gij cos(𝜃ij ) + Bij sin(𝜃ij ) = 0 (30)
j=1

NB
∑ [ ]
QGi − QDi − Vi Vj Gij sin(𝜃ij ) + Bij cos(𝜃ij ) = 0 (31)
j=1

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A novel metaheuristic inspired by horned lizard defense tactics Page 49 of 65 59

Table 26  Comparison results for the Cassini Spacecracft Trajectory Design

Optimal values for Algorithms
variables
HLOA JSOA BWOA WHO

x1 −767.4051 −803.93006 −789.577110052954 −766.16821

x2 173.92445 161.05094 158.658439545012 170.39407
x3 414.38194 449.38348 449.385957886722 416.44468
x4 52.698608 60.027747 54.3533764089 52.857507
x5 1041.7422 795.06518 1028.07660808395 1040.5838
x6 4696.4909 3837.03 4557.84620923363 4575.138
fmin 5.3580 5.7519 4.93248908962165 5.3302

Table 27  Generator, transformer Min Max Initial Min Max Initial

and shunt var compensator
inequality constraints
PG1 50 200 99.23 T11 0.9 1.1 1.078
PG2 20 80 80 T12 0.9 1.1 1.069
PG5 15 50 50 T15 0.9 1.1 1.032
PG8 10 35 20 T36 0.9 1.1 1.068
PG11 10 30 20 QC10 0 5 0
PG13 12 40 20 QC12 0 5 0
VG1 0.95 1.1 1.05 QC15 0 5 0
VG2 0.95 1.1 1.04 QC17 0 5 0
VG5 0.95 1.1 1.01 QC20 0 5 0
VG8 0.95 1.1 1.01 QC21 0 5 0
VG11 0.95 1.1 1.05 QC23 0 5 0
VG13 0.95 1.1 1.05 QC24 0 5 0
QC29 0 5 0

Where PG and QG are the active and reactive power generation, whereas PD and QD are the
active and reactive load demand, NB is the number of buses, Gij and Bij are the conductance
and susceptance between bus i and j of the admittance matrix Yij = Gij + jBij . The inequal-
ity constraints of the OPF formulation [?], summarized in Table 27, are defined in Eq. 32
to 35. Where Eq. 32 represent the generator constraints; Eq. 33 represent the transformer
constraints; Eq. 34 define the shunt VAR compensator constraints; and Eq. 35 represent the
security constraints.

VGmin ≤ VGi ≤ VGmax , i = 1, ⋯ , NG

≤ PGi ≤ Pmax
i i

Pmin , i = 1, ⋯ , NG (32)
≤ QGi ≤
G i G i

Qmin
Gi Qmax
Gi , i = 1, ⋯ , NG

Timin ≤ Ti ≤ Timax , i = 1, ⋯ , NT (33)

13
59 Page 50 of 65 H. Peraza‑Vázquez et al.

Ci ≤ QGCi ≤ QCi , i = 1, ⋯ , NG
Qmin (34)
max

VLmin ≤ VLi ≤ VLmax , i = 1, ⋯ , NL

Sli ≤ Slmax , i = 1, ⋯ , nl
(35)
i i

The Black Widow Optimization (BWOA), Jumping Spider Optimization (JSOA), and
Wild Horse Optimizer (WHO) algorithms previously ranked by the Friedman test, see
Table 6, are contrasted against the Horned Lizard optimization algorithm (HLOA) to
solve the Optimal Power Flow problem for the IEEE-30 bus test system. The IEEE test
system, depicted in Fig. 17, consists of six generators placed at nodes 1, 2, 5, 8, 11, and

Fig. 17  IEEE 30-bus system (Nusair and Alasali 2020)

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A novel metaheuristic inspired by horned lizard defense tactics Page 51 of 65 59

13, highlighted in red color, four transformers located in lines 11, 12, 15, and 36, high-
lighted in blue color and nine reactive compensators at nodes 17, 20, 21, 23, 24 and 29
highlighted in yellow. Line and node numbering is depicted in green and black color,
respectively. Three cases of studies are conducted, the minimization of generation fuel
cost and the minimization of active and reactive power transmission losses. In the first
case, the objective function represents the total fuel of the six generator units, and it is
defined as follows:
NG
∑
J= fi ($∕h) (36)
i=1

Where fi is a quadratic function as described in Ela et al. (2010).

The objective function for the minimization of active power transmission losses is defined
in Eq. 37, whilst the reactive power minimization function is stated in Eq. 38 (Fig. 17).
NB NB NB
∑ ∑ ∑
J= Pi = PGi − PDi (37)
i=1 i=1 i=1

NB NB NB
∑ ∑ ∑
J= Qi = PQi − QDi (38)
i=1 i=1 i=1

From the convergence graph analysis, it is confirmed that HLOA outperforms

BWOA, WHO, and JSOA algorithms for the first and third case of study, see Fig. 18
and 20 respectively. However, for the second case of study, minimization of active
power transmission losses, the HLOA algorithm achieves very competitive results
as illustrated in Fig. 19. All tests were run for a population size of 30 and 500 itera-
tions. Table 28 summarizes the algorithms’ minimum obtained values for each case
of study.

Fig. 18  First study case, Minimi-

zation of fuel cost

13
59 Page 52 of 65 H. Peraza‑Vázquez et al.

Fig. 19  Second study case,

minimization of active power
transmission losses

Fig. 20  Third study case,

minimization of reactive power
transmission losses

Table 28  Obtained results Minimization studies BWOA WHO JSOA HLOA

Fuel cost 819.191 811.648 811.165 810.464

Active Power 3.367 3.391 3.408 3.471
Reactive Power −19.943 −20.404 −19.698 −21.682

6 Conclusion

This paper presents a novel metaheuristic optimization algorithm inspired by the

horned lizard’s defense behavior, named the Horned Lizard Optimization Algorithm
(HLOA). The defense behavior included the following four methods: crypsis (the abil-
ity of an organism to conceal itself by having a color, pattern, and shape that allows it
to blend into the surrounding environment), skin darkening or lightening, bloodstream
shooting, and move-to-escape. Furthermore, the rapid skin color change is influenced

13
A novel metaheuristic inspired by horned lizard defense tactics Page 53 of 65 59

by the 𝛼-melanophore stimulating hormone (alpha-MHS) rate, is also modeled. These

models progressively refine the search vectors by evolving them (recombining vectors)
at each iteration. They provide a suitable balance between exploration and exploitation
in the solution search space. The algorithm’s performance was evaluated on Sixty-three
well-known testbench functions, twenty-eight functions for 10-dimensional, 30-dimen-
sional, 50-dimensional, and 100-dimensional from IEEE CEC-2027 “Constrained
Real-Parameter Optimization”, and ten functions from IEEE CEC06-2019 “100-Digit
Challenge”. Furthermore, five real-world optimization problems, the Multiple Gravity
Assist Optimization Problem, the Optimal Power Flow problem, and the rest of the
problems are taken from CEC2020. Moreover, the HLOA performance is compared
to ten of the most recent algorithms published in the scientific literature. The statisti-
cal results show that the HLOA algorithm outperforms BWOA, DOA, COOT, CSA,
EJAYA, RSO, SAO, and TSA. At the same time, it has competitive results with JSOA
and WHO algorithms.
The following are the findings and conclusions of this study:

• The algorithm has no parameters to configure in modeling the Horned Lizard

defense tactics. The only parameters of the algorithm are those common to all bio-
inspired, that is, the size of the population and the number of iterations.
• The Wilcoxon rank-sum test demonstrates that, in many situations, HLOA is signif-
icantly superior to alternative bio-inspired algorithms. The HLOA also ranks high-
est compared to other algorithms by the Friedman statistics test.
• For 10-dimensional problems, the Wilcoxon rank-sum test shows that HLOA out-
performed all the algorithms. Nevertheless, it is ranked in second place by the
Friedman test. Furthermore, the Wilcoxon rank-sum demonstrates that HLOA out-
performs JSOA and BWOA algorithms for 30-dimensional problems. Meanwhile,
there are no significant differences with WHO. In addition, HLOA ranks second
in Friedman’s test. In the case of 50-dimensional problems, the HLOA algorithm
demonstrated superior performance compared to the JSOA and BWOA algorithms,
as indicated by the Wilcoxon rank-sum test. In contrast, there are no substantial
disparities with the WHO. In comparison, the Friedman test assigns the highest
rank to HLOA. Finally, in the context of 100-dimensional problems, the Wilcoxon
rank-sum test shows that HLOA outperformed all algorithms. Meanwhile, the
Friedman test has determined that HLOA holds the highest rank. This n-dimen-
sional analysis shows that the HLOA algorithm performs better for dimensions 50
and 100.
• HLOA shows competitive results on the Process Flow Sheeting Problem, Process Syn-
thesis Problem, and Multiple Gravity Assist Optimization Problem. While showing the
best performance on the Optimal Design of an Industrial Refrigeration System
• In the optimal power flow problem, HLOA better minimizes fuel cost and reactive
power than BWOA, WHO, and JSOA algorithms.
• HLOA can solve real-world problems with unknown search spaces with outstanding
results.

On the other hand, currently under development for future work is an improved version
of the HLOA algorithm to solve for multi-objective and many-objective optimization.
Moreover, efforts are focused on hyper-parameter optimization of convolutional neural
networks for medical applications.

13
59 Page 54 of 65 H. Peraza‑Vázquez et al.

Appendix A Additional Tables

Additional tables are shown in this section (See Tables 29, 30, 31, 32 and 33 ).

Table 29  Color palette in its hexadecimal, decimal, and normalized form

# Color name Hexadecimal number Decimal number Normalized number

1 White #FFFFFF 16777215 0

2 Yellow #FFFF00 16776960 0.00001519920000000
3 Orange #FF6600 16737792 0.00157159600000000
4 light coral #FF8080 16744576 0.00194543600000000
5 vivid orange #FF9900 16750848 0.00234979400000000
6 Fuchsia #FF00FF 16711935 0.00355333640000000
7 Red #FF0000 16711680 0.00390619100000000
8 pale cyan #CCFFFF 13434879 0.19921876199999999
9 lavender blue #CCCCFF 13421823 0.19999696000000000
10 Silver #C0C0C0 12632256 0.24705882400000001
11 Pale blue #9999FF 12632256 0.39999392000000000
12 vivid purple #993399 10040217 0.40155639700000001
13 dark Rose #993366 10040166 0.40155943599999999
14 Gray #808080 8421504 0.49803921600000001
15 olive green #808000 8421376 0.49804684500000002
16 purple #800080 8388736 0.49999234100000001
17 Brown #800000 8388608 0.49999999699999997
18 Dark orange #664221 6701601 0.60055342899999997
19 Hot Pink #660066 6684774 0.60155639699999996
20 Dark gray #333333 3355443 0.80000000000000000
21 Pale yellow #FFFFCC 16777164 0.90000044700000004
22 aqua #00FFFF 65535 0.99609380900000000
23 Green lime #00FF00 65280 0.99610900899999999
24 Sky Blue #00CCFF 52479 0.99687200799999998
25 bluish green #008080 32896 0.99803924499999996
26 green #008000 32768 0.99804687500000000
27 vivid blue #0066CC 26316 0.99843144399999995
28 Navy blue #0000FF 255 0.99998480099999998
29 Navy #000080 128 0.99999237100000005
30 Black #000000 0 1

13
 Table 30  Description of the Testbench Functions. From F1 to F21
ID Function Dim Interval fmin

n
� ∑ ∑n
F1 f (x) = −a.exp(−b n1 i=1 xi2 ) − exp( d1 i=1 cos(c ⋅ xi )) + a + exp(1) 30 [−32, 32] 0
� √ �
F2 f (x, y) = −200e −0.2 x2 + y2 2 [−32, 32] −200
� √ �
F3 f (x, y) = −200e −0.2 x2 + y2 + 5e(cos (3x) + sin (3y)) 2 [−32, 32] −195.629028238419
∑d−1 � � �
F4 2
f (x) = i=1 (e−0.2 xi2 +xi+1 + 3 cos(2xi ) + sin(2xi+1 ) ) 2 [−35, 35] −4.590101633799122
x
F5 f (x, y) = cos(x) sin(x) − 2 x ∈ [−1, 1] −2.02181
x2 +1
y ∈ [−1, 2]
∑n
F6 f (x) = i=1 xi sin(xi + 0.1xi ) 30 [0, 10] 0
∏n � � √
F7 f (x) = i=1 sin xi ⋅ xi 30 [0, 10] 2.808n
F8 f (x, y) = x2 + y2 + xy + sin(x) + cos(y) 2 [−500, 500] 1
F9 f (x, y) = (1.5 − x + xy)2 + (2.25 − x + xy2 )2 + (2.625 − x + xy3 )2 2 [−4.5, 4.5] 0
F10 f (x, y) = sin (x)e(1 − cos(y))2 )
+ cos(y)e((1 − sin(x))2 )
+ (x − y)2 2 [−6.28, 6.28] −106.764537
( ) ( )
[−100, 100]
A novel metaheuristic inspired by horned lizard defense tactics

F11 f (x) = x12 + 2x22 − 0.3 cos 3𝜋x1 − 0.4 cos 4𝜋x2 + 0.7 2 0
( )[ ( )]
F12 f (x) = x12 + 2x22 − 0.3 cos 3𝜋x1 cos 4𝜋x2 + 0.3 2 [−100, 100] 0
F13 f (x, y) = (x + 2y − 7)2 + (2x + y − 5)2 2 [−10, 10] 0
2 2
F14 f (x, y) = (x + 10)2 + (y + 10)2 + e−x −y 2 [−20, 0] e−200
� 2 � � �
F15 ∑n−1 � � xi+1 + 1 � � x2 + 1
i
30 [−1, 4] 0
2
f (x) = i=1 xi2 + xi+1
√
F16 f (x) = 100 |x2 − 0.01x12 | +0.01 x1 + 10 2 x1 ∈ [−15, 5] 0
x2 ∈ [−5, 3]
√
F17 x2 +y2 2 [−10, 10] −2.06261218
f(x, y) = −0.0001( sin(x) sin(y) exp ( 100 − 𝜋 ) + 1 )0.1
F18 f (x, y) = 105 x2 + y2 − (x2 + y2 )2 + 10−5 (x2 + y2 )4 2 [−20, 20] −24771.09375
√
F19 1+cos(12 x2 +y2 ) 2 [−5.2, 5.2] −1
f (x, y) = − (0.5 x2 +y2 +2)
( )
Page 55 of 65 59

13
 Table 30  (continued)
ID Function Dim Interval fmin

f (x, y) = − cos(x) cos(y)exp(−(x − 𝜋)2 − (y − 𝜋)2 ) [−100, 100] −1

13
F20 2
F21 f (x, y) = x2 + y2 + 25(sin2 (x) + sin2 (y)) 2 [−5, 5] 0
59 Page 56 of 65
H. Peraza‑Vázquez et al.
 Table 31  Description of the Testbench Functions. From F22 to F42
ID Function Dim Interval fmin
∑n 2
F22 f (x) = −exp(−0.5 i=1 xi )
30 [−1, 1] −1
( )2 ( ) ( )2 ( )
F23 f (x) = [1 + x1 + x2 + 1 19 − 14x1 + 3x12 − 14x2 + 6x1 x2 + 3x22 ][30 + 2x1 − 3x2 18 − 32x1 + 12x12 + 48x2 − 36x1 x2 + 27x22 ] 2 [−2, 2] 3
F24 sin(10𝜋x) 1 [−0.5, 2.5] −0.8690
f (x) = 2x
+ (x − 1)4
F25 ∑n xi2 ∏n x 30 [−600, 600] 0
f (x) = 1 + i=1 4000 − i=1 cos( √i )
i
�� �2 �𝛼 � ∑d �
F26 1 1 1 10 [−2, 2] 0
f (x) = �x�2 − d + d 2
�x�2 + i=1 xi + 2

F27 f (x, y) = (x2 + y − 11)2 + (x + y2 − 7)2 2 [−6, 6] 0

√
F28 x2 +y2 2 [−10, 10] −19.208
f (x, y) = −� sin(x) cos(y)exp(�1 − 𝜋
�)�
2
F29 sin2 (x+y) 2 [0, 10] 0.673667521146855
f (x, y) = √
− sin (x−y)
x2 +y2
F30 f (x, y) = 100(y − x3 )2 + (1 − x)2 2 [0, 10] 0
2 [−10, 10]
F31 f (x, y) = sin (3𝜋x) + (x − 1)2 (1 + sin2 (3𝜋y)) + (y − 1)2 (1 + sin2 (2𝜋y)) 2 0
[−10, 10]
A novel metaheuristic inspired by horned lizard defense tactics

F32 f (x, y) = 0.26(x2 + y2 ) − 0.48xy 2 0

F33 f (x, y) = sin(x + y) + (x − y)2 − 1.5x + 2.5y + 1 2 x ∈ [−1.5, 4] −1.9133
y ∈ [3, 4]
∑n ∑n 2
F34 f (x) = 1 + i=1 sin2 (xi ) − 0.1e( i=1 xi ) 30 [−10, 10] 0
∑n
F35 f (x) = i=1 �xi �i+1 30 [−1, 0] 0
∑n
F36 f (x) = i=1 (x2 − i)2 30 [−500, 500] 0
∑n
F37 f (x) = i=1 ixi4 + random[0, 1) 30 [−1.28, 1.28] 0
∑n
F38 f (x, y) = 10n + i=1 (xi2 − 10cos(2𝜋xi )) 30 [−5.12, 5.12] 0
�∑n 2 �𝛼
F39 f (x) = x1 + 𝛽 i=2 xi 30 [−5, 5] −𝛾
∑n
F40 f (x, y) = i=1 [b(xi+1 − xi2 )2 + (a − xi )2 ] 30 [−5, 5] 0
� �
F41 ∑D 2 ∑D 2 30 [−100, 100] 0
f (x) = 1 − cos(2𝜋 i=1 xi ) + 0.1 i=1 xi
F42 sin2 (x2 +y2 )2 −0.5 2 [−100, 100] 0
f (x, y) = 0.5 + 1+0.001(x2 +y2 )2
Page 57 of 65 59

13
59 Page 58 of 65 H. Peraza‑Vázquez et al.

Table 32  Description of the Testbench Functions. From F43 to F63

ID Function Dim Interval fmin

F43 f (x, y) = 0.5 + sin2 (x2 −y2 )−0.5 2 [−100, 100] 0

(1+0.001(x2 +y2 ))2
F44 f (x, y) = 0.5 + sin(cos(|x2 −y2 |))−0.5 2 [−100, 100] 0.00156685
(1+0.001(x2 +y2 ))2
F45 f (x, y) = 0.5 + cos(sin(|x2 −y2 |))−0.5 2 [−100, 100] 0.292579
(1+0.001(x2 +y2 ))2
F46 ∑d 30 [−100, 100] 0
f (x) = i=1 ��xi ��
F47 f (x) = maxi𝜖|1,d| ||xi || 30 [−100, 100] 0
F48 ∑d ∏d 30 [−100, 100] 0
f (x) = i=1 �xi � + i=1 �xi �
F49 ∑d 10 30 [−10, 10] 0
f (x) = i=1 xi
∑n √
F50 f (x) = 418.9829d − i=1 xi sin( �xi �) 30 [−500, 500] 0
F51 ∑d ∑5 30 [−10, 10] −29.6733337
f (x) i=1 j=1 j sin((j + 1)xi + j)
F52 ∑d ∑5 30 [−10, 10] −25.740858
f (x) = i=1 j=1 j cos((j + 1)xi + j)
F53 ∏d �∑5 �
30 [−10, 10] −186.7309
f (x) = i=1 j=1 cos((j + 1)xi + j)
F54 ∑d 30 [−5.12, 5.12] 0
f (x) = i=1 xi2
F55 ∑ d 30
f (x) 21 i=1 (xi4 − 16xi2 + 5xi ) [−5, 5] −39.16599
F56 ∑d 30 [−10, 10] 0
f (x) = i=1 ixi2
F57 f (x, y) = 2x2 − 1.05x4 + x6
+ xy + y2 2 [−5, 5] 0
6
F58 f (x, y, z) = 4 2
(x + y2 − xy)0.75 +z 3 [0, 2] 0
3
F59 n � �2m n 30 [−5, 5] 0
⎡ � xi � � �2 ⎤
⎢ − − xi − c ⎥
� �
f (x) = ⎢e i=1
𝛽
− 2e i=1 ⎥ ⋅ ∏n cos2 x
i=1 i
⎢ ⎥
⎢ ⎥
⎣ ⎦
∑n ∑n
F60 2
f (x) = ( i=1 �xi �)exp(− i=1 sin(xi )) 30 [−2𝜋, 2𝜋] 0
� � � n � n
F61 ∑n � �2
∑ 2 ∏ 30 [−2𝜋, 2𝜋] −1
xi
f (x) = exp − 𝛽
− 2 exp − xi cos2 (xi)
i=1 i=1 i=1
� � √
F62 30
n n
n ∑ ∑ 2 [−10, 10] −1
∑ − xi 2 − sin �xi �
f (x) = sin2 (xi ) − e i=1 e i=1

i=1
∑n ∑n ∑n
F63 f (x) = 2
i=1 xi + ( i=1 0.5ixi )2 + ( i=1 0.5ixi )4 30 [−5, 10] 0

13
A novel metaheuristic inspired by horned lizard defense tactics Page 59 of 65 59

Table 33  CEC 2017 Constrained Real-Parameter Optimization.

Problem/search range Type of objective Number of constraints
E I

C01 Non Separable 0 1

[ ]D Separable
− 100, 100
CO2 Non Separable 0 1
[ ]D Rotated Non Separable, Rotated
− 100, 100
C03 Non Separable 1 1
[ ]D Separable Separable
− 100, 100
C04 Separable 0 2
[ ]D Separable
− 10, 10
C05 Non Separable 0 2
[ ]D Non Separable,Rotated
− 10, 10
C06 Separable 6 0
[ ]D Separable
− 20, 20
C07 Separable 2 0
[ ]D Separable
− 50, 50
C08 Separable 2 0
[ ]D Non Separable
− 100, 100
C09 Separable 2 0
[ ]D Non Separable
− 10, 10
C10 Separable 2 0
[ ]D Non Separable
− 100, 100
C11 Separable 1 1
[ ]D Non Separable Non Separable
− 100, 100
C12 Separable 0 2
[ ]D Separable
− 100, 100
C13 Non Separable 0 3
[ ]D Separable
− 100, 100
C14 Non Separable 1 1
[ ]D Separable Separable
− 100, 100
C15 Separable 1 1
[ ]D
− 100, 100
C16 Separable 1 1
[ ]D Non Separable Separable
− 100, 100
C17 Non Separable 1 1
[ ]D Non Separable Separable
− 100, 100
C18 Separable 1 2
[ ]D Non Separable
− 100, 100
C19 Separable 0 2
[ ]D Non Separable
− 50, 50

13
59 Page 60 of 65 H. Peraza‑Vázquez et al.

Table 33  (continued)
Problem/search range Type of objective Number of constraints
E I

C20 Non Separable 0 2

[ ]D
− 100, 100
C21 Rotated 0 2
[ ]D Rotated
− 100, 100
C22 Rotated 0 3
[ ]D Rotated
− 100, 100
C23 Rotated 1 1
[ ]D Rotated Rotated
− 100, 100
C24 Rotated 1 1
[ ]D Rotated Rotated
− 100, 100
C25 Rotated 1 1
[ ]D Rotated Rotated
-100, 100
C26 Rotated 1 1
[ ]D Rotated Rotated
− 100, 100
C27 Rotated 1 2
[ ]D Rotated Rotated
− 100, 100
C28 Rotated 0 2
[ ]D Rotated
− 50, 50

Details of 28 test problems. D is the number of decision variables, I is the number of inequality constraints,
and E is the number of equality constraints

Acknowledgements This project was supported by Instituto Politécnico Nacional (IPN) through grant SIP−
no. 20221568 and SIP−no. 20231424. Also, by CONAHCyT (Mexican Council of Humanities, Science,
and Technology) through grant no. CF-2023-I-342

Author Contributions Conceptualization, H.P.-V.; methodology, H.P.-V.; software, H.P.-V.; validation, N.S.,
M.M-T. and A.B.M.-C.; formal analysis, M.M-T. and A.P.-D.; investigation, N.S. and A.B.M.-C.; resources,
H.P.-V.; data curation, M.M-T. and N.S.; writing-original draft preparation, H.P.-V.; writing-review and edit-
ing, H.P.-V., A.P.-D. and M-M-T.; visualization, N.S. and A.B.M.-C.; supervision, H.P.-V. and A.P.-D.; pro-
ject administration, H.P.-V.; funding acquisition, H.P.-V. and A.B.M-C.

Funding This project was supported by Instituto Politécnico Nacional (IPN) through grant SIP−no.
20221568 and SIP−no. 20231424. Also, by CONAHCyT (Mexican Council of Humanities, Science, and
Technology) through grant no. CF-2023-I-342

Code availability The source code used to support the findings of this study has been deposited in the Math-
Works repository at ((link provided if the manuscript is accepted)).

Declarations
Conflict of interest The authors declare that they have no conflict of interest.

Consent to participate All authors accept their participation and collaboration

Consent for publication All authors have read and agreed to the published version of the manuscript.

13
A novel metaheuristic inspired by horned lizard defense tactics Page 61 of 65 59

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction 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 Com-
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Authors and Affiliations

Hernán Peraza‑Vázquez1 · Adrián Peña‑Delgado2 · Marco Merino‑Treviño1,3 ·

Ana Beatriz Morales‑Cepeda3 · Neha Sinha4

* Hernán Peraza‑Vázquez
hperaza@ipn.mx
* Adrián Peña‑Delgado
apea@utaltamira.edu.mx
* Marco Merino‑Treviño
mmerino@ipn.mx
Ana Beatriz Morales‑Cepeda
ana.mc@cdmadero.tecnm.mx
Neha Sinha
neha.cse.2203005@iiitbh.ac.in
1
Instituto Politécnico Nacional, CICATA Altamira, km.14.5 Carretera Tampico ‑Puerto Industrial
Altamira, 89120 Mexico City, Tamaulipas, Mexico
2
Universidad Tecnológica de Altamira, Boulevard de los Ríos Km. 3 + 100, Puerto Industrial
Altamira, 89601 Mexico City, Tamaulipas, Mexico
3
TecNM/Instituto Tecnológico de Ciudad Madero, Juventino Rosas y Jesús Urueta s/n, Col. Los
Mangos, 89318 Ciudad Madero, Tamaulipas, Mexico
4
Department of Computer Science and Engineering, Indian Institute of Information Technology
Bhagalpur, Bhagalpur, Bihar 813210, India

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