# Golden Tortoise Beetle Optimizer: A Novel Nature-Inspired Meta-heuristic
Algorithm for Engineering Problems

Omid Tarkhaneh<EMAIL_ADDRESS>Neda Alipour<EMAIL_ADDRESS>Amirahmad Chapnevis<EMAIL_ADDRESS>Haifeng Shen<EMAIL_ADDRESS>Department of Computer Science, University of Tabriz, Tabriz, Iran.  Deparment
of Electrical Engineering, Shahed University, Tehran, Iran  Department of
Computer Science, Virginia Commonwealth University, Richmond, USA  Peter Faber
Business School, Australian Catholic University, Sydney, Australia

###### Abstract

This paper proposes a novel nature-inspired meta-heuristic algorithm called
the Golden Tortoise Beetle Optimizer (GTBO) to solve optimization problems. It
mimics golden tortoise beetle’s behavior of changing colors to attract
opposite sex for mating and its protective strategy that uses a kind of anal
fork to deter predators. The algorithm is modeled based on the beetle’s dual
attractiveness and survival strategy to generate new solutions for
optimization problems. To measure its performance, the proposed GTBO is
compared with five other nature-inspired evolutionary algorithms on 24 well-
known benchmark functions investigating the trade-off between exploration and
exploitation, local optima avoidance, and convergence towards the global
optima is statistically significant. We particularly applied GTBO to two well-
known engineering problems including the welded beam design problem and the
gear train design problem. The results demonstrate that the new algorithm is
more efficient than the five baseline algorithms for both problems. A
sensitivity analysis is also performed to reveal different impacts of the
algorithm’s key control parameters and operators on GTBO’s performance.

###### keywords:

Golden Tortoise Beetle; optimization; Meta-heuristic; sensitivity analysis.

##  1 Introduction

Optimization is a common problem in many fields ranging from engineering
design to holiday planning [1]. In these activities, the aim is to optimize an
objective such as profit or time, or a combination of multiple objectives. An
optimization process consists of multiple input variables, a defined objective
function, and an output which is the cost or the fitness of the objective
function [1, 2]. Different algorithms have been proposed to solve different
optimization problems and based on the nature of these algorithms, they can be
classified into three main categories: deterministic algorithms, stochastic
algorithms, and hybrid algorithms. A deterministic algorithm has no randomness
and it follows a single path to reach the goal. The hill-climbing algorithm
[1, 3] is one such example of this category. Stochastic methods are based on
the concept of randomness, which helps them try different ways of searching.
One typical example of this type is the Genetic Algorithm (GA) which utilizes
semi-randomness when searching into a problem area [1, 4]. In other words,
such an algorithm tries to look for a solution using a different path in each
separate run. A hybrid algorithm is the composition of stochastic and
deterministic methods and one basic example is the hybrid of GA operators and
the hill-climbing method. For example, the starting point in the deterministic
method can be generated by random mechanisms [1].

Meta-heuristics, as a special type of stochastic algorithms, have been widely
adopted in solving different optimization problems, including but not limited
to, image segmentation [5], data clustering [6], or feature selction [7, 8].
Simplicity, flexibility, derivation-free mechanism, and local optima avoidance
are some of the advantages of these algorithms [9, 10]. First, simplicity is
inherited from natural concepts, which assist scientists to understand meta-
heuristic algorithms and apply them to solve real problems. Second,
flexibility makes these algorithms applicable to a wide range of problems
without any change to their internal structures as they can be used as
blackboxes [9, 10]. Third, meta-heuristic algorithms have established the so-
called derivation-independent mechanism where an optimization process starts
with random solutions without calculating the derivative of search spaces to
seek the optima, making them suitable for unknown spaces. Last, these
algorithms are capable of avoiding local optima due to the random operators
that stochastic meta-heuristics such as evolutionary algorithms utilize. Most
meta-heuristic algorithms use a two-phase search process: diversification
(exploration) followed by intensification (exploitation) [11]. In the
diversification phase, an optimizer utilizes stochastic operators to randomly
and globally investigate the promising area(s) of the search space. In the
intensification, it focuses on local searching around the promising area(s)
derived from the previous phase [12, 13, 14].

Nature-inspired optimizers are a specific type of meta-heuristic algorithms
that are typically modeled after biological behaviors, physical phenomena, or
evolutionary concepts. These algorithms have a wide spectrum of applications
ranging from engineering optimization [15, 16] to deep learning [17], and to
coordinating robots [18]. Section 2 details some of the representative nature-
inspired meta-heuristic algorithms. Due to the stochastic nature of an
optimization process, finding a suitable balance between diversification and
intensification is one of the most challenging tasks in the development of a
meta-heuristic algorithm. Some nature-inspired algorithms suffer from
performance degrade when they are applied to composition engineering problems
such as the welded beam design problem [19] or the gear train design problem
[10] due to the difficulty in striking a balance between exploration and
exploitation. This motivated us to search for a better solution because
according to the No Free Lunch (NFL) theorem [20], any two optimization
algorithms perform equally when solving all problems. In other words, one
algorithm may be superior in solving one set of problems but ineffective in
solving another set.

In this paper, we present a novel nature-inspired optimization algorithm
called the Golden Tortoise Beetle Optimizer (GTBO) that is inspired by the
nature-life and the behavior of golden tortoise beetle. In this algorithm,
special operators are mathematically modeled after the mating behavior and the
survival strategy of golden tortoise beetle using heuristic techniques to
generate solutions for solving optimization problems. To assess the
performance of the proposed algorithm against baselines, we employed 24 well-
known benchmark functions including unimodal, multimodal, and composite
functions [21, 22, 23] to compare and contrast their exploration,
exploitation, local minima avoidance, and convergence rates. We also conducted
Wilcoxon rank-sum test to prove that the GTBO’s superior performance is
statistically significant [24]. As the employed benchmark functions are
typically unconstrained or with only simple constraints while real-world
problems often have specific constraints and contain unknown search areas
[25], we applied the proposed algorithm to two well-known real engineering
problems including the welded beam design problem and the gear train design
problem [10] and the results again confirmed GTBO outperformed the baseline
algorithms. We further conducted a sensitivity analysis to reveal different
impacts of the algorithm’s key control parameters and operators on its
performance for the two engineering problems.

The main contributions of the paper are summarized as follows:

  1. 1.

We propose a novel nature-inspired algorithm for solving optimization problems
that are unprecedentedly modeled after golden tortoise beetle.

  2. 2.

The proposed algorithm creates a decent search mechanism that is underpinned
by the two unique color-switching and survival operators.

  3. 3.

The proposed algorithm is efficient for real-world engineering problems and
problems with unknown search areas.

  4. 4.

The proposed algorithm has few control parameters to adjust and is easy to
implement.

The rest of the paper is structured as follows: Section 2 presents a
literature review of nature-inspired meta-heuristic algorithms. Section 3
describes the proposed GTBO algorithm. Experiments setup and results are
presented in Sections 4 and 5 respectively, followed by the applications of
GTBO in two engineering problems in Section 6. Finally, Section 7 concludes
the work and suggests some directions for future research.

##  2 Related Work

Nature-inspired meta-heuristic optimization algorithms can generally be
categorized into evolutionary, physics-based, and swarm intelligence (SI)
algorithms [26].

###  2.1 Evolutionary algorithms

Evolutionary algorithms (EAs) are typically inspired by evolutionary concepts
from nature for solving optimization problems. One of the well-known
algorithms is the Genetic Algorithm (GA). This algorithm is related to
Darwin’s theory of evolutionary concepts and was proposed by Holland [4] in
1992. This simple algorithm starts the optimization process by generating a
set of initial random solutions as candidate solutions (individuals) and the
whole set of solutions is considered as a population. Each new population is
created by the combination and mutation of individuals in the previous
generations. Individuals with the highest probability are randomly selected
for creating the new population. There are some other studies based on EA such
as Differential Evolution (DE) [27], Evolutionary Programming (EP) [28],
Evolution Strategy (ES) [29], Genetic Programming (GP) [30], and Biogeography-
Based optimizer (BBO) [31].

The proposed Golden Tortoise Beetle Optimizer (GTBO) is also an evolutionary
algorithm. To the best of our knowledge, there has been no evolutionary
technique in the literature that imitates the behavior of golden tortoise
beetle.

###  2.2 Physics-based algorithms

These algorithms are developed by mimicking some of the physical rules, such
as gravity, electromagnetic force, and weights. One representative algorithm
is the Gravitational Search Algorithm (GSA) [32], which was inspired by
Newton’s law of gravity and the law of motion. This algorithm starts the
searching process by utilizing a random collection of agents that have mass
interaction with each other. Throughout iterations, the masses are attracted
to each other according to the gravitational forces between them. The heavier
masses attract bigger force. As such, the global optimum will be found by the
heaviest mass, while other masses will be attracted according to their
distances. GSA can produce high-quality solutions, but it has some drawbacks,
such as slow convergence and the tendency to become trapped in local minima.
Some of the other well-known physics-based algorithms include Charged System
Search (CSS) [33], Central Force Optimization (CFO) [34], Artificial Chemical
Reaction Optimization Algorithm (ACROA) [35], Black Hole (BH) algorithm [6],
Ray Optimization (RO) algorithm [36], and Galaxy-based Search Algorithm (GbSA)
[37].

###  2.3 Swarm Intelligence (SI) algorithms

SI algorithms are inspired by the population-based behavior of social insects
as well as other animal species. For example, Particle Swarm Optimization
(PSO) algorithm [38] models the swarming behavior of a herd of animals and
flocks of birds. This algorithm defines individuals as particles and tries to
find the optima by utilizing information obtained from each particle. This
algorithm, similar to the GA method, starts the optimization process with a
set of initial populations, or in fact random solutions as candidate
solutions, which are distributed throughout the search space. Based on the
best solution, the position and velocity of each particle are updated over the
subsequent iterations. In addition, there is another set called velocity which
preserves the number of moving particles that have moved with the best
solution gained by the swarm. The PSO algorithm has fewer parameters for
tuning and is good for multi-objective problems. The most important
disadvantages of PSO are that they easily fall into local optimum in high-
dimensional spaces and that they have a low convergence rate in iterative
processes.

Another well-known SI algorithm is Ant Colony Optimization (ACO) [39] that was
inspired by the social behavior of ants in nature. The goal of this algorithm
is to find the best and shortest path to the source of food. Candidate
solutions are improved over the course of an iteration. ACO algorithm can
search among a population in parallel and also allow the rapid discovery of
good solutions. However, it has an uncertain time to convergence as well as
dependent sequences of random decisions. A further novel algorithm is Grey
Wolf Optimization (GWO) [9], which was inspired by the leadership hierarchy
and hunting mechanism of grey wolves. There are four levels of wolves called
alpha, beta, omega, and delta respectively each of which has a certain
responsibility. The hunting process can be divided into three phases,
including tracking and chasing the prey, harassing the prey, and finally
attacking the prey. However, this algorithm sometimes has a low local search
ability as well as a slow convergence. In addition, there are a lot of other
SI algorithms such as Artificial Bee Colony Algorithm (ABC) [40], Black Widow
Optimization (BWO) [41], Cuckoo Search Algorithm (CS) [42], Ant Lion Optimizer
(ALO) [10], Artificial Fish-Swarm Algorithm (AFSA) [43], Wasp Swarm Algorithm
[44], and Fruit fly Optimization Algorithm (FOA) [45].

Table 1 briefly compares some of the nature-inspired meta-heuristic
optimization algorithms including the proposed new algorithm GTBO in terms of
convergence, local minima avoidance, exploration, exploitation, and balance
between exploration and exploitation. Regarding the GA, there is a common
problem of local minima stagnation which is due to the lack of population
diversity. Another issue is the poor balance between exploration and
exploitation, which can be solved by adjusting the selection approach [46]. As
for PSO, a major shortcoming is also to do with a poor balance between
exploration and exploitation. Besides, it cannot properly maintain particle
diversity, which results in premature convergence. However, this algorithm has
a high exploitation capability and is easy to implement [47]. For the ABC
algorithm, it can get trapped into local minima especially in solving complex
multimodal problems and it has poor exploitation. However, the algorithm
benefits from good exploration [48, 49]. Similarly in algorithms such as GSA,
there is a risk of premature convergence and the balance between exploration
and exploitation is also poor. GSA uses complex operators that require high
computational time and hence has a slow searching speed especially in the last
few iterations so it cannot guarantee global optima in some cases [50]. Other
algorithms such as ALO [10] and BWO [41] have good exploitation, but their
exploration is poor especially in multimodal and composite functions.

In contrast, the proposed GTBO performs all well on convergence rate, good
balance between exploration and exploitation, and local minima avoidance. With
its unique operators, it can effectively explore and exploit the search space,
which in turn ensures a good convergence speed and maintains a good balance
between exploration and exploitation. Furthermore, random selection of some
solutions helps the algorithm increase the probability of avoiding local
minima.

Table 1: Comparison of optimization algorithms Algorithm | Convergence | Local Minima Avoidance | Exploration | Exploitation | Balance of Exploration & Exploitation
---|---|---|---|---|---
GA | Poor | Poor | Good | Average | Poor
DE | Average | Average | Good | Average | Poor
GSA | Poor | Poor | Good | Average | Poor
ACO | Average | Good | Poor | Good | Average
PSO | Poor | Poor | Poor | Good | Poor
ABC | Average | Good | Good | Poor | Poor
ALO | Average | Good | Poor | Good | Poor
BWO | Average | Good | Poor | Good | Poor
GTBO | Good | Good | Good | Good | Good

##  3 Golden Tortoise Beetle Optimizer (GTBO)

###  3.1 Lifestyle of Golden Tortoise Beetles

The golden tortoise beetle is one of three species of tortoise beetles that
can be found in Florida, eastern North America. They can likely be found
wherever all members of the morning glory family (Convolvulaceae) and also
sweet potato are found. Both larvae and adults feed on foliage [51, 52, 53].
They are known as ‘golden bugs’ and grow to around 5.0 to 7.0 mm in length.
Larvae of this type of insects have moveable anal fork, e.g. peddler, which
they hold over their backs as a shield to cover their bodies and deter
potential enemies [51]. It is very common among golden tortoise beetles to
hide and then demonstrate certain patterns, a strategy to deter their
predators [52].

More than 30 years ago, they became the first known insect species with the
ability to extremely change their colors ranging from brownish and purplish to
bright orange or gold during copulation or agitation because of optical
illusion [54]. They have two states including resting and disturbed. In the
resting state, these insects have their normal color of gold which arises from
a chirped multilayer reflector maintained in a perfect coherent state by the
presence of humidity in the porous patches within each layer. In the disturbed
state, such as holding them between fingers or applying pressure on them, they
will have a low-saturated red appearance resulting from the destruction of
this reflector by the expulsion of the liquid from the porous patches, turning
the multilayer into a translucent slab that leaves an unobstructed view of the
deeper-lying, pigmented red substrate [55].

Due to the metallic quality and glare gold color of these insects, it is
generally difficult for birds to see them. By changing from golden to orange
and spotty, they can mimic lady beetles and with the different forms of
defense, birds cannot recognize the difference [54]. Figure 1 shows the adult
golden and spotty tortoise beetles. The time of changing color from dull red
and spotty to golden is when they signal to the opposite sex to prove that
they are mature, which means beetles that are not mature enough cannot produce
golden color. Professor of biology Edward M. Barrows from Georgetown
University, realized that these insects can not only last their copulation
anywhere from 15 to 583 minutes but also alter their color in as quickly as
two minutes [54]. Another important observation is that these beetles usually
have a brilliant metallic golden color when first seen on a plant, but the
golden appearance is lost when they are dead and dry. They are naturally red-
brown with spots when they are dead, and they return to golden or red when
they are brought back to normal temperature. Elimination of moisture from the
cuticle causes them to lose their metallic color and turn to red, the same
reddish appearance when they are in a disturbed state. A further observation
is a tense and competitive relationship among tortoise beetles, which is
called ‘arms-race’.

Figure 1: (a) Adult golden tortoise beetle (b) adult spotted tortoise beetle.

The lifestyle of golden tortoise beetles has inspired us to come up with a
novel optimization method that mathematically models their color-switching
behavior for attracting the opposite sex to mating and reproduction and their
survival mechanism for protecting themselves from predators. In this research,
each beetle represents a solution or an individual. The GTBO utilizes the
reproduction concept to generate solutions and the survival mechanism to
determine which solutions will get to the next iteration. In the following
sections, we will present the two key operators, followed by the optimization
algorithm.

###  3.2 Inspired Operators

The color switching operator is modeled after a golden tortoise beetle’s
behavior of changing its color when mating and at the time of disturbance. The
color change is done by changing the liquid content of the interference layer
under the pressure of fluid injections by the golden tortoise beetle, for
example, changing the refractive index will change the dominant reflected
wavelength [56]. To formulate the changing color operation, we use the
concepts of the reflective index and the length of wavelength in thin layer
interference. Consider $x$ and $y$ as two materials that are placed on top of
each other like a stack. The optical thickness for each layer is $1/4$
wavelength. In other words, $d_{x}.n_{x}$=$d_{y}.n_{y}$ where $d_{x}$ and
$d_{y}$ denote the layer thickness and $n_{x}$ and $n_{y}$ are the reflective
indices [56]. The assumed reflected color can be described in Equation 1 when
we have stacks placed in the higher reflective index.

$h\cdot\lambda=(d_{x}\cdot n_{x}\cdot cos(\theta_{x})+d_{y}\cdot n_{y}\cdot
cos(\theta_{y}))\text{,}$ (1)

where $h$ is a constant value, $\lambda$ denotes the wavelength of the
reflected light, and $\theta_{x}$ and $\theta_{y}$ are the normal angles. In
order to determine the dominating wavelength ($\lambda$), we adopt Vigneron’s
method [57] defined for a multilayer reflector containing two alternating
different layers, as formulated by Equation 2.

$\lambda=\frac{2\alpha\sqrt{\bar{\phi}^{2}-sin^{2}(\theta_{z})}}{k}\text{,}$
(2)

where $\lambda$ denotes the wavelength, $\bar{\phi}$ is the mean of the
reflective index, $\alpha$ indicates the thickness of the mentioned layers,
$k$ is a constant integer number, and $\theta_{z}$ is a normal angle. Mature
golden tortoise beetles utilize these mechanisms to change colors for mating
and reproducing the next generations. Figure 2 depicts the visual aspect of
gold in the chromaticity coordinates reflected in the yellow hue region of the
CIE diagram. GTBO uses the color switching operator to produce new
generations.

Figure 2: Chromaticity coordinates.

The survival operator is modeled after a beetle’s protective mechanism against
predators. Female beetles usually lay eggs in clusters in the places such as
stems of the host leaves when they become yellowish or reddish larvae within
approximately ten days. The larvae collect their shed skins using a structure
called anal fork which works as a fecal shield to protect them against
predators such as ants. However, some of these larvae become mature but many
may not [54, 58]. For simplicity in generating the survived beetles, this
study utilized a new crossover operator to generate $S_{p}$ percent of these
beetles in the algorithm.

###  3.3 Generating Initial Solutions

The successfulness (efficiency) of each beetle in attracting the opposite sex
via changing the color to gold and in protecting the larvae from predators has
a great impact on its reproduction of the next generation. To solve an
optimization problem, the problem variables need to be displayed as a matrix
and in GTBO, the matrix is called a “position”. The more successful a beetle
is in mating and in protecting larvae, the more efficient a solution is. Each
solution converges to an optimal value over an evolutionary process as a
female beetle will be attracted by a golden male beetle. The more golden the
male beetle is, the more the female will change its position, and the closer
it will move towards the male. In an optimization problem with $n$ dimensions,
the problem variables can be represented as a $m\times n$ position matrix as
follows:

$P_{beetle}=\begin{pmatrix}b_{1,1}&b_{1,2}&\cdots&b_{1,n}\\\
b_{2,1}&b_{2,2}&\cdots&b_{2,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
b_{m,1}&b_{m,2}&\cdots&b_{m,n}\end{pmatrix}$

where $bi,j$ is the value of the j-th variable (dimension) of the i-th beetle,
$m$ is the number of beetles, and $n$ is the number of variables. The values
of the defined matrix are float values.

The fitness value of each golden tortoise beetle is obtained by evaluating the
cost function. Thus, the profit function of the proposed algorithm can be
defined as follows:

$F_{beetle}=\begin{pmatrix}f([b_{1,1}&b_{1,2}&\cdots&b_{1,n}])\\\
f([b_{2,1}&b_{2,2}&\cdots&b_{2,n})]\\\ \vdots&\vdots&\ddots&\vdots\\\
f([b_{m,1}&b_{m,2}&\cdots&b_{m,n})]\end{pmatrix}$

where $F_{beetle}$ shows the fitness value for each beetle and $f$ denotes the
objective function. This profit function can be defined both as a minimization
or maximization function.

The algorithm utilizes a random operator to initialize the first beetle. Real-
valued amounts are uniformly initialized randomly between the lower and upper
bounds of the dimensions $[X_{min},X_{max}]$ where
$X_{min}={{X^{1}}_{min},\cdots,X^{D}_{min}}$ and
$X_{max}={X^{1}_{max},\cdots,X^{D}_{max}}$.

Equation 3 defines the initial value of the k-th parameter in the i-th beetle
at the generation $G=1$ for a problem with $D$ dimensions:

$x^{k}_{i,1}=x^{k}_{min}+rand(0,1)\cdot(x^{k}_{max}-x^{j}_{min}),k=1,2,\cdots,D.$
(3)

This type of initialization has also been used in different evolutionary
algorithms such as differential evolution (DE) algorithm [5].

###  3.4 Switching Color Operator

The solutions for the number of mature beetles are generated using the
following equations (Equations 4,$\cdots$, 6).

$V^{G}_{i}=X^{G}_{i}+S_{color}\cdot(X_{r_{1}}^{G}-X^{G}_{best})\text{,}$ (4)

where $X^{G}_{i}$ is the position of current female beetle in generation $G$,
which moves towards the golden male beetle $X_{r_{1}}$ in generation $G$ that
has a golden color with a value determined by the color switching operator
$S_{color}$. In particular, $r_{1}$ is a randomly generated integer in
$[1,NP]$ excluding $i$, where $NP$ is the number of beetle populations, while
$X^{G}_{best}$ is the solution with the best fitness at the generation $G$.

The female beetle changes its position in order to mate with the golden beetle
showing an attractive golden color and reproduce the next generation. The
best-achieved solution in each generation is preserved as it is subtracted
from other parts of the generated solutions. The value of $S_{color}$ is
mathematically modeled using Equations 5 and 6:

$S_{color}=(d_{x}\cdot n_{x}\cdot cos(\theta_{x})+d_{y}\cdot n_{y}\cdot
cos(\theta_{y}))+(h\cdot\lambda)\text{,}$ (5)

where:

$\begin{cases}d_{x},n_{x}=&Randn()\\\ \\\ d_{y},n_{y}=&Randn()\cdot\beta\\\
\\\ \bar{\phi}=&Cauchy(\sigma,\mu)\\\ \\\ \theta_{x},\theta_{y}=&\beta\\\ \\\
\alpha,k,h=&rand()\\\ \\\ \theta_{z}=&2\cdot\pi\cdot rand()\\\ \end{cases}$

(6)

In particular, $Randn()$ is a normal random function which generates numbers
in $[1,n]$, $\beta$ is a uniform random function which generates numbers in
the range of $[0.1,0.9]$, $rand()$ is a random number generator between 0 and
1, and Cauchy is the standard Cauchy distribution where $\sigma$ is the
location parameter and $\mu$ is a scaling parameter, which are initialized to
0.5 and 0.2 respectively. The values generated by the Cauchy distribution is
initialized to 0.5 when it is outside the lower bound of zero and upper bound
of 1 in the algorithm. Furthermore, variables $h$, $\lambda$, $\bar{\phi}$,
$\alpha$, $k$, and $\theta_{z}$ are defined in Equations 1 and 2.

###  3.5 Survival Operator

According to the above-mentioned details about the eggs laid by tortoise
beetles, some beetles’ eggs will survive due to the efficiency of their
protective strategies for deterring predators. In GTBO, for the sake of
simplicity yet without losing generality, a crossover operator is considered
for producing $Sp$ percent of the survived beetles, which will change to the
larvae in the future and then become mature beetles. The survival operator is
defined using Equations 7 and 8.

$\begin{cases}Beetle_{1}=&\alpha\cdot
x_{r_{1}}+(1-\alpha)\cdot(x_{r_{2}}-\sigma_{1})\\\ \\\ Beetle_{2}=&\alpha\cdot
x_{r_{2}}+(1-\alpha)\cdot(x_{r_{1}}-\sigma_{2})\\\ \end{cases}$

(7)

where $x_{r_{1}}$ and $x_{r_{2}}$ are two randomly chosen solutions in the
range of $[1,NP]$ where $NP$ is the total number of populations, and $\alpha$
is an uniform random number between [0, 1]. Variables $\sigma_{1}$ and
$\sigma_{2}$ are defined in Equation 8:

$\begin{cases}\sigma_{1}=&(1-\epsilon)\cdot(x_{best}-x_{r_{1}})\\\ \\\
\sigma_{2}=&(1-\epsilon)\cdot(x_{best}-x_{r_{2}})\\\ \\\
\epsilon=&\frac{\alpha\cdot\eta}{|p|^{\beta}}\par\end{cases}$

(8)

where $x_{best}$ is the best achieved solution so far, $\eta$ and $p$ are
normal numbers associated with the size of solution $x_{r_{1}}$, $\beta$ is a
continuous uniform random number generated between [0.1, 0.5], and $\epsilon$
is a variable which is determined based on the values of $\alpha$, $\beta$,
$\eta$, and $p$. Although there are some other selection operator strategies
such as Roulette wheel selection method [59], the proposed crossover operator
used a random selection strategy for choosing two beetles for generating new
solutions.

###  3.6 The GTBO Algorithm

Data: $X$,$M_{r}(MatureRate)$, $S_{r}(SurvivalRate)$

Result: $X$ // return new generated beetle population

1

$X\leftarrow\\{x_{1},x_{2}...x_{n}\\}$ // initialize solutions randomly and
determine fitness values using Eq. 3

$x^{*}\leftarrow best(X)$ // assign current best

$it\leftarrow 0$ // initial iteration variable

$NP\leftarrow$k // total number of beetle population

$NFE\leftarrow 0$ // curent value of function evaluation

$MP\leftarrow M_{r}*NP$ // number of matured beetle population

$SP\leftarrow S_{r}*NP$ // number of Survival beetle population

2 begin

3

4 while  _Stop criterion not satisfied ( $NFE<=TotalNFE$)_ do

5

6 for  _$i\leftarrow 1$ to $MP$ _ do

7 for  _each beetle $X_{i}$ _ do

Generate solution $V_{i}^{G}$ by Eq. 4 // Switching Color Operator

8 Evaluate the fitness value of generated solution

9 Store the generated solution in Mature Population.

10

11

12 for  _$i\leftarrow 1$ to $SP$ _ do

13 for  _each beetle $X_{i}$ _ do

Select two solution randomly. Generate solutions using Eq. 7 // Reproduction
operator

14 Evaluate the fitness value of generated solution

15 Store the generated solution in survival population.

16

17

18 start again from line 8 if the stopping criterion not satisfied

19 Merge the beetle population

20 Sort the population based on the fitness value and select the best NP ones.

21 Output the result.

22

Algorithm 1 GTBO

According to the pseudocode in Algorithm 1, the GTBO starts with the
initialization phase where the key control parameters such as the $M_{r}$
(mature rate) and $S_{r}$ (survival rate) are defined. The values of these
control parameters should be defined in the range of (0, 1]. The number of
mature beetles is determined based on the whole number of the beetle
population $NP$ through the multiplication of the mature rate by $NP$.
Similarly, the number of survived beetles are determined according to the
survival rate and the number of beetle’s population. After the initialization,
the algorithm starts the main loop where it continues until the stopping
criterion is satisfied.

In the main loop, first, the switching color mechanism is used to generate
solutions for the number of mature beetles. Every mature beetle can change its
color to golden in order to attract and mate with the opposite sex to improve
its chance for reproduction. All the generated solutions are stored in the
repository called the matured population. After this stage, the next operator
starts and generates solutions using the number of survival beetles, which
then change to matured beetles. The generated solutions at this stage are also
stored in a repository called survived population. Finally, the algorithm
iterates the main loop again and checks the stopping condition, and when it is
satisfied it merges the population of matured beetles with the survived
beetles and selects the best-generated solution and outputs the result. GTBO’s
main characteristics can be summarized as follows:

1) The color switching operator and the survival operator help the algorithm
guarantee balance between exploration and exploitation due to the random
selection mechanism.
2) Due to the random selection of some solutions in the population-based
algorithm, there is a high probability of local optima avoidance.
3) The algorithm has a good population diversity as matured and survived
beetle population is merged and then sorted.
4) In every iteration, the best solution will preserve, update, and compare
with generated solutions (elite).
5) The algorithm has few parameters to adjust.

##  4 Experiments

###  4.1 Benchmark functions and baseline algorithms

In this section, several benchmark functions are employed to evaluate the
performance of the proposed algorithm by using different criteria. In this
regard, three classes of benchmark test functions are employed to prove the
efficiency of the proposed algorithm [21, 22, 60]. The first class is unimodal
functions described in Table 2. In this class, there is one single optimum
where the efficiency of the algorithm can be assessed through the exploitation
and the convergence speed. Another class is to do with multimodal functions
where it consists of more than one optimum. This class is more complicated
compared to the unimodal with which the proposed algorithm can be evaluated in
terms of both exploration and avoidance of local minima [10]. This type of
class is listed in Table 3 [21, 22, 60]. The third class of test functions is
hybrid composite functions described in CEC 2005 special session [23]. One
algorithm should properly make a trade-off between exploration and
exploitation to achieve efficient results in composite functions [10]. Thus,
this class of functions, which are listed in Table 4, tests the ability of the
proposed algorithm in terms of the balance between both exploration and
exploitation. The 3-dimensional shapes of the unimodal and multimodal
functions are presented in Figures 3 and 4. Regarding the baseline algorithms,
this study employed standard GA [4], ABC [49], PSO [61], Black Widow
Optimization (BWO) algorithm [41], and Ant Lion Optimizer (ALO) algorithm
[10].

###  4.2 Evaluation criteria and parameter settings

The number of population ($NP$) for all the algorithms is initialized to 100.
As a fair stopping condition, this research utilizes the total number of
function evaluations (TotalNFE) initialized to 100,000 for all the algorithms.
All algorithms performed 30 independent runs using Matlab 2016 on a computer
with Intel Core i3, 2.5 GHz, 4 GB RAM running Windows 7. Table
LABEL:tab:parameters shows the parameter settings in detail. The parameter
settings of the ALO algorithm are determined based on its original paper. Some
criteria such as the best, mean fitness value, and standard deviation (std)
are employed to show the results of all algorithms. However, in order to show
the superiority of the proposed algorithm, this research conducted the
Wilcoxon signed rank-sum test to show that the proposed algorithm is
statistically significant with respect to the baseline algorithms [62]. The
details of this test will be discussed in the later sections.

##  5 Results and Discussions

###  5.1 Results on unimodal functions

Tables 6, 5.1, and 8 reveal the results related to the unimodal test
functions. In these tables, best denotes the best-achieved fitness value over
30 independent runs, mean shows the average value of fitness value, and STD
indicates the standard deviation of obtained results. The superior results of
the GTBO are shown in boldface.

Table 2:  Unimodal benchmark functions. (Dim: Dimension, range: Lower and upper bound of problem, $f_{min}$: global minimum value of function ). Function | Dim | Range | $f_{min}$
---|---|---|---
$F_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}$ | 30, 200 | [-100, 100] | 0
$F_{2}(x)=\sum_{i=1}^{n}|x_{i}|+\displaystyle\prod_{i=1}^{n}|x_{i}|$ | 30, 200 | [-10, 10] | 0
$F_{3}(x)=\sum_{i=1}^{n}(\sum_{j-1}^{i}x_{j})^{2}$ | 30, 200 | [-100, 100] | 0
$F_{4}(x)=\displaystyle max_{i}\\{|x_{i}|,1\leq i\leq n\\}$ | 30, 200 | [-100, 100] | 0
$F_{5}(x)=\sum_{i=1}^{n-1}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]$ | 30, 200 | [-30, 30] | 0
$F_{6}(x)=\sum_{i=1}^{n}([x_{i}+0.5])^{2}$ | 30, 200 | [-100, 100] | 0
$F_{7}(x)=\sum_{i=1}^{n}ix_{i}^{4}+random[0,1)$ | 30, 200 | [-1.28, 1.28] | 0
Table 3:  Multimodal benchmark functions. (Dim: Dimension, range: Lower and upper bound of problem, $f_{min}$: global minimum value of function). Function | Dim | Range | $f_{min}$
---|---|---|---
$F_{8}(x)=\sum_{i=1}^{n}-x_{i}sin(\sqrt{|x_{i}|})$ | 30, 200 | [-500, 500] | $-418.9829\times Dim$
$F_{9}(x)=\sum_{i=1}^{n}[x_{i}^{2}-10cos(2\pi x_{i})+10]$ | 30, 200 | [-5.12, 5.12] | 0
$F_{10}(x)=-20exp(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}})-exp(\frac{1}{n}\sum_{i=1}^{n}cos(2\pi x_{i}))+20+e$ | 30, 200 | [-32, 32] | 0
$F_{11}(x)=\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}cos(\frac{x_{i}}{\sqrt{i}})+1$ | 30, 200 | [-600, 600] | 0
$F_{12}(x)=\frac{\pi}{n}\\{10sin(\pi y_{1})+\sum_{i=1}^{n-1}(y_{i}-1)^{2}[1+10sin^{2}(\pi y_{i+1})]+(y_{n}-1)^{2}\\}+\sum_{i=1}^{n}u(x_{i},10,100,4)$ | 30, 200 | [-50, 50] | 0
$F_{13}(x)=0.1\\{sin^{2}(3\pi x_{1})+\sum_{i=1}^{n}(x_{i}-1)^{2}[1+sin^{2}(3\pi x_{i}+1)]+(x_{n}-1)^{2}[1+sin^{2}(2\pi x_{n})]\\}+\sum_{i=1}^{n}u(x_{i},5,100)$ | 30, 200 | [-50, 50] | 0

Figure 3: F1-F7 represnts the 3-D versions of unimodal functions.

Figure 4: F8-F13 represnts the 3-D versions of multimodal functions.  Table 4:  Hybrid composition benchmark functions (CEC2005) used for experimental study (Dim: Dimension, Min, Max: Limits of search space, H: Hybrid, C: Composition ). Function No | Name | Type | Min | Max | Dim
---|---|---|---|---|---
F15 | hybrid comp. Fn 1 | HC | -5 | 5 | 30
F16 | Rotated hybrid comp. Fn 1 | HC | -5 | 5 | 30
F17 | Rotated hybrid comp. Fn 1 with noise | HC | -5 | 5 | 30
F18 | Rotated hybrid comp. Fn 2 | HC | -5 | 5 | 30
F19 | Rotated hybrid comp. Fn 2 with narrow global optimal | HC | -5 | 5 | 30
F20 | Rotated hybrid comp. Fn 2 with the global optimum | HC | -5 | 5 | 30
F21 | Rotated hybrid comp. Fn 3 | HC | -5 | 5 | 30
F22 | Rotated hybrid comp. Fn 3 with high condition number matrix | HC | -5 | 5 | 30
F23 | Non-continuous rotated hybrid comp. Fn 3 | HC | -5 | 5 | 30
F24 | Rotated hybrid comp. Fn 4 | HC | -5 | 5 | 30
F25 | Rotated hybrid comp. Fn 4 without Bounds | HC | -2 | 5 | 30
Table 5: Parameter settings of the employed algorithms

Algorithm Parameter Description GA $pc=0.67$ Crossover Rate $Mc=0.33$ Mutation
Rate PSO $w=2$ Inertia Weight $bg=2.2$ Best global experience $wdamp=0.98$
$bp=2.4$ Best personal experience BWO $PP=0.6$ Procreate Rate $CR=0.44$
Cannibalism Rate ABC $No.\ \ of\ \ food\ \ sources=NP/2$ $Limit=15$ GTBOA
$M_{r}\ \ (Mature\ \ Rate)=0.4$ Rate. of Matured Beetles $S_{r}\ \ (Survival\
\ Rate)=0.2$ Rate. of Survived Beetles $TotalNFE=100,000$ No. of Function
Evaluations

Table 6:  Unimodal benchmark functions results of all employed algorithms for 30 dimensions. Function No | Criteria | GA | PSO | ABC | ALO | BWO | GTBOA
---|---|---|---|---|---|---|---
F1 | Best | 1.2131e-06 | 108.9183 | 3.0858 | 1.7124 | 0.0011 | 2.2912e-19
| Mean | 1.6048 | 375.9294 | 14.3653 | 1.8804 | 0.0029 | 3.4636e-18
| STD | 3.1829 | 240.5136 | 10.8668 | 0.2171 | 0.0019 | 4.2131e-18
F2 | Best | 7.5194e-05 | 2.1456 | 1.5127 | 1.1277 | 0.0031 | 5.9375e-12
| Mean | 0.1006 | 10.6079 | 2.4668 | 41.7537 | 0.0072 | 1.7371e-11
| STD | 0.1373 | 7.5088 | 0.7499 | 51.9005 | 0.0066 | 1.2793e-11
F3 | Best | 445.2838 | 1.4940e+04 | 2.3119e+04 | 36.6311 | 141.9181 | 4.1325
| Mean | 689.4357 | 2.4175e+04 | 2.8024e+04 | 44.9914 | 230.9100 | 14.7848
| STD | 223.5618 | 4.1746e+03 | 2.3374e+03 | 7.2157 | 84.8917 | 12.5700
F4 | Best | 4.6389e-04 | 32.0965 | 56.8363 | 0.9720 | 0.6664 | 0.3167
| Mean | 0.5985 | 36.4333 | 68.2251 | 2.7303 | 1.3689 | 1.3985
| STD | 0.8674 | 3.1428 | 4.5274 | 1.6668 | 0.4055 | 1.0517
F5 | Best | 162.9445 | 4.1259e+04 | 564.6546 | 37.5728 | 100.2362 | 24.3109
| Mean | 1.3213e+03 | 9.8076e+04 | 905.6610 | 303.8040 | 216.1671 | 45.2228
| STD | 1.2222e+03 | 7.2228e+04 | 478.0250 | 499.1320 | 76.1047 | 28.4930
F6 | Best | 3.3299e-04 | 66.9998 | 2.1334 | 1.6228 | 4.3832e-04 | 7.7615e-20
| Mean | 0.6190 | 279.7871 | 9.5629 | 1.9982 | 0.0019 | 3.3433e-18
| STD | 0.8442 | 164.8274 | 9.4036 | 0.2518 | 0.0013 | 5.3386e-18
F7 | Best | 0.0027 | 0.2594 | 0.5100 | 0.0057 | 0.0014 | 0.0054
| Mean | 0.0068 | 0.9902 | 1.2994 | 0.0093 | 0.0040 | 0.0132
| STD | 0.0027 | 1.0903 | 0.5285 | 0.0038 | 0.0020 | 0.0060
Table 7:  Statistical analysis (Wilcoxon rank-sum test) on unimodal test
functions with 30 dimensions

Function No.  GTBOA vs GA GTBOA vs PSO GTBOA vs ABC pval T+ T- winner pval T+
T- winner pval T+ T- winner F1 0.0118$(<0.05^{+})$ 751 239 +
1.0202e-187$(<0.05^{+})$ 957 33 + 1.0029e-123$(<0.05^{+})$ 693 297 + F2
0.3843$(>0.05^{+})$ 565 425 - 4.4929e-161$(<0.05^{+})$ 971 19 +
9.4246e-152$(<0.05^{+})$ 852 138 + F3 5.5314e-05$(<0.05^{+})$ 535 455 +
6.9057e-236$(<0.05^{+})$ 936 54 + 1.6768e-283$(<0.05^{+})$ 982 8 + F4
0.4210$(>0.05^{+})$ 0 990 - 1.0075e-240$(<0.05^{+})$ 932 58 +
3.9268e-320$(<0.05^{+})$ 990 0 + F5 1.6447e-78$(<0.05^{+})$ 723 267 +
1.7986e-210$(<0.05^{+})$ 980 10 + 1.3470e-133$(<0.05^{+})$ 872 118 + F6
2.3143e-31$(<0.05^{+})$ 611 379 + 4.1734e-186$(<0.05^{+})$ 972 18 +
2.7740e-135$(<0.05^{+})$ 884 106 + F7 1.0620e-170$(<0.05^{+})$ 710 280 +
9.8038e-235$(<0.05^{+})$ 979 11 + 1.9982e-261$(<0.05^{+})$ 990 0 + GTBOA vs
ALO GTBOA vs BWO pval T+ T- winner pval T+ T- winner F1
1.2234e-140$(<0.05^{+})$ 843 147 + 8.5090e-19$(<0.05^{+})$ 589 401 + F2
1.5410e-181$(<0.05^{+})$ 971 19 + 5.0797e-17$(<0.05^{+})$ 582 408 + F3
3.3480e-06$(<0.05^{+})$ 691 299 + 2.0306e-11$(<0.05^{+})$ 610 380 + F4
3.4211e-46$(<0.05^{+})$ 900 90 + 5.9719e-27$(<0.05^{+})$ 655 335 + F5
7.2719e-101$(<0.05^{+})$ 939 51 + 4.7413e-69$(<0.05^{+})$ 706 284 + F6
1.0515e-154$(<0.05^{+})$ 936 54 + 3.5595e-26$(<0.05^{+})$ 611 379 + F7
0.0820$(>0.05^{+})$ 345 645 - 0.3214$(<0.05^{+})$ 102 888 +

Table 8:  Unimodal benchmark functions results of all employed algorithms for 50 dimensions. Function No | Criteria | GA | PSO | ABC | ALO | BWO | GTBOA
---|---|---|---|---|---|---|---
F1 | Best | 1.0000e-03 | 2.1415e+03 | 104.5097 | 6.1849 | 3.1374 | 6.8041e-09
| Mean | 0.2401 | 3.9719e+03 | 262.8478 | 7.5865 | 9.6358 | 9.1314e-08
| STD | 0.3944 | 1.0605e+03 | 185.3239 | 0.9823 | 4.0663 | 8.2333e-08
F2 | Best | 4.0000e-04 | 30.8469 | 4.1226 | 3.2234 | 0.5971 | 1.6904e-06
| Mean | 0.3842 | 79.0294 | 5.8825 | 82.7113 | 1.0424 | 5.3055e-06
| STD | 0.6819 | 33.8814 | 1.0043 | 95.6385 | 0.3665 | 2.9952e-06
F3 | Best | 1.2141e+03 | 6.6226e+04 | 6.7060e+04 | 886.5140 | 567.1294 | 1.2137e+03
| Mean | 1.9825e+03 | 8.1653e+04 | 8.2528e+04 | 1.0102e+03 | 870.9053 | 3.0783e+03
| STD | 669.1000 | 1.0852e+04 | 8.4185e+03 | 91.0529 | 231.4373 | 1.2043e+03
F4 | Best | 0.0397 | 51.3212 | 77.3875 | 8.2060 | 5.6129 | 16.0928
| Mean | 1.8005 | 56.5572 | 80.5567 | 12.1953 | 8.4819 | 21.4955
| STD | 3.1948 | 2.8711 | 1.9138 | 4.1685 | 2.4756 | 3.5943
F5 | Best | 270.7661 | 1.2681e+06 | 2.6798e+03 | 100.1233 | 563.2197 | 73.7331
| Mean | 9.6687e+03 | 3.0598e+06 | 6.1883e+03 | 191.6021 | 1.9829e+03 | 149.3488
| STD | 1.2861e+04 | 1.8600e+06 | 2.7620e+03 | 98.9993 | 1.4415e+03 | 81.9379
F6 | Best | 1.0000e-04 | 1.9402e+03 | 40.8407 | 5.5331 | 3.9743 | 5.6347e-09
| Mean | 1.6303 | 4.1254e+03 | 210.3022 | 7.9905 | 7.7142 | 5.4944e-08
| STD | 2.6221 | 1.5306e+03 | 106.4296 | 1.7843 | 2.9658 | 6.0208e-08
F7 | Best | 0.0153 | 1.8539 | 4.1916 | 0.0185 | 0.0120 | 0.0283
| Mean | 0.0364 | 7.4114 | 7.4776 | 0.0297 | 0.0291 | 0.0556
| STD | 0.0236 | 9.6894 | 2.5425 | 0.0089 | 0.0134 | 0.0259

Table 9:  Multimodal benchmark functions results of all employed algorithms for 30 dimensions. Function No | Criteria | GA | PSO | ABC | ALO | BWO | GTBOA
---|---|---|---|---|---|---|---
F8 | Best | -4.1661e+03 | -3833.3258 | -4.0026e+03 | -5.7714e+03 | -4.1663e+03 | -1.0570e+04
| Mean | -3.9527e+03 | -3.3568e+03 | -3.8028e+03 | -5.5843e+03 | -3.9670e+03 | -9.3533e+03
| STD | 352.6358 | 269.1801 | 121.9016 | 134.6784 | 151.6641 | 773.4089
F9 | Best | 2.5011e-12 | 66.7150 | 59.1072 | 32.9624 | 0.0094 | 22.8840
| Mean | 0.0267 | 173.4088 | 66.0430 | 51.4169 | 1.6281 | 35.4868
| STD | 0.0552 | 49.5622 | 5.7181 | 18.9709 | 2.2689 | 12.0436
F10 | Best | 4.8116e-04 | 4.0877 | 4.9133 | 0.7608 | 0.0055 | 1.1199e-10
| Mean | 0.0869 | 6.1922 | 7.3804 | 1.8218 | 0.0125 | 4.2979e-10
| STD | 0.1442 | 1.2298 | 1.4983 | 0.9408 | 0.0089 | 2.4641e-10
F11 | Best | 1.9055e-11 | 1.5373 | 1.0836 | 0.9664 | 0.0036 | 0.0000
| Mean | 0.4100 | 5.2037 | 1.2179 | 1.0016 | 0.0586 | 0.0031
| STD | 0.4439 | 2.1055 | 0.1136 | 0.0202 | 0.0688 | 0.0047
F12 | Best | 5.8804e-06 | 20.3593 | 0.0348 | 3.1373 | 0.0012 | 4.4676e-09
| Mean | 7.6819e-04 | 1.0549e+04 | 0.2412 | 5.7674 | 0.0411 | 2.0542e-04
| STD | 0.0018 | 2.5201e+04 | 0.1640 | 2.7118 | 0.0588 | 4.9198e-04
F13 | Best | 4.9435e-05 | 226.5227 | 0.3491 | 0.4885 | 0.0573 | 4.3196e-08
| Mean | 0.4128 | 3.2245e+05 | 0.8159 | 0.6337 | 0.3013 | 0.0015
| STD | 0.6804 | 5.7834e+05 | 0.4147 | 0.1448 | 0.2701 | 0.0038

Table 10:  Statistical analysis (Wilcoxon rank-sum test) on multimodal test
functions with 30 dimensions

Function No.  GTBOA vs GA GTBOA vs PSO GTBOA vs ABC pval T+ T- winner pval T+
T- winner pval T+ T- winner F8 1.3227e-263$(<0.05^{+})$ 990 0 +
2.6385e-309$(<0.05^{+})$ 990 0 + 5.9288e-323$(<0.05^{+})$ 990 0 + F9
0.2412$(>0.05^{+})$ 0 990 - 1.1810e-97$(<0.05^{+})$ 932 27 +
1.2306e-75$(<0.05^{+})$ 766 224 + F10 5.3312e-09$(<0.05^{+})$ 513 477 +
3.8201e-193$(<0.05^{+})$ 972 18 + 3.1481e-214$(<0.05^{+})$ 986 4 + F11
4.0220e-56$(<0.05^{+})$ 679 311 + 6.2604e-192$(<0.05^{+})$ 969 21 +
5.4183e-163$(<0.05^{+})$ 896 94 + F12 0.0925$(>0.05^{+})$ 115 384 -
9.0311e-95$(<0.05^{+})$ 482 17 + 1.1016e-18$(<0.05^{+})$ 369 130 + F13
1.0087e-26$(<0.05^{+})$ 401 98 + 5.9091e-75$(<0.05^{+})$ 473 26 +
3.3476e-10$(<0.05^{+})$ 342 157 + GTBOA vs ALO GTBOA vs BWO pval T+ T- winner
pval T+ T- winner F8 3.9598e-190$(<0.05^{+})$ 873 113 +
9.8074e-247$(<0.05^{+})$ 950 40 + F9 2.0175e-56$(<0.05^{+})$ 884 106 +
0.0725$(>0.05^{+})$ 0 990 - F10 5.4181e-159$(<0.05^{+})$ 936 54 +
5.3513e-25$(<0.05^{+})$ 607 383 + F11 1.3326e-160$(<0.05^{+})$ 932 58 +
8.3918e-36$(<0.05^{+})$ 635 355 + F12 9.4212e-65$(<0.05^{+})$ 434 65 +
0.0052$(<0.05^{+})$ 272 227 + F13 2.5806e-33$(<0.05^{+})$ 439 60 +
0.0322$(<0.05^{+})$ 276 223 +

Table 11:  Multimodal benchmark functions results of all employed algorithms for 50 dimensions. Function No | Criteria | GA | PSO | ABC | ALO | BWO | GTBOA
---|---|---|---|---|---|---|---
F8 | Best | -1.8123e+04 | -1.4103e+04 | -1.4177e+04 | -9.2607e+03 | -1.2220e+04 | -1.6429e+04
| Mean | -1.5776e+04 | -1.2733e+04 | -1.3103e+04 | -9.1438e+03 | -1.0800e+04 | -1.5095e+04
| STD | 1.7020e+03 | 804.3379 | 572.4003 | 113.0223 | 1.0467e+03 | 514.2831
F9 | Best | 8.0000e-04 | 291.5818 | 142.8249 | 96.1233 | 9.4104 | 41.7882
| Mean | 1.6294 | 407.2185 | 169.8958 | 108.0825 | 21.9785 | 71.1063
| STD | 1.7271 | 82.0520 | 22.1329 | 15.0202 | 7.6228 | 21.7004
F10 | Best | 8.0000e-04 | 9.6225 | 7.5975 | 2.4662 | 0.5517 | 2.3944e-05
| Mean | 0.0839 | 11.2567 | 10.6152 | 3.0566 | 1.0042 | 0.0016
| STD | 0.0701 | 0.7568 | 1.8320 | 0.5687 | 0.3508 | 5.6849e-04
F11 | Best | 6.0000e-04 | 20.7541 | 1.7991 | 1.0630 | 1.0212 | 6.8045e-09
| Mean | 0.3076 | 38.2575 | 3.4333 | 1.0789 | 1.1043 | 0.0099
| STD | 0.4770 | 12.6355 | 0.8731 | 0.0119 | 0.0734 | 0.0254
F12 | Best | 1.6813 | 3.3053e+04 | 0.4340 | 14.5439 | 1.4612 | 0.1550
| Mean | 4.6357 | 2.2985e+06 | 1.0839 | 19.8078 | 4.9364 | 2.9127
| STD | 1.9286 | 3.0329e+06 | 0.5251 | 6.2547 | 0.2615 | 1.9893
F13 | Best | 1.0000e-04 | 1.5536e+06 | 1.4779 | 4.0195 | 6.4525 | 0.9129
| Mean | 11.1122 | 6.9882e+06 | 5.5231 | 90.7775 | 14.1978 | 9.9584
| STD | 0.1380 | 8.2902e+06 | 5.3147 | 51.2084 | 7.5786 | 6.5993

Table 12:  Hybrid composite benchmark functions results of all employed algorithms for 30 dimensions. Function No | Criteria | GA | PSO | ABC | ALO | BWO | GTBOA
---|---|---|---|---|---|---|---
F15 | Best | 261.2875 | 413.9231 | 310.3578 | 502.5654 | 520.1649 | 200.0063
| Mean | 587.2951 | 586.7310 | 481.2109 | 560.0458 | 580.2370 | 404.3135
| STD | 138.5073 | 71.9346 | 68.0997 | 53.8564 | 36.3407 | 184.1164
F16 | Best | 294.4417 | 304.6725 | 397.8783 | 118.7518 | 291.1114 | 54.3660
| Mean | 503.2913 | 424.3700 | 443.5813 | 266.0464 | 394.3834 | 110.9351
| STD | 124.2395 | 102.7154 | 19.7516 | 175.5074 | 101.6714 | 37.3781
F17 | Best | 289.1165 | 292.1645 | 636.3725 | 230.6587 | 330.7373 | 73.2936
| Mean | 487.4050 | 405.3080 | 766.5341 | 320.7117 | 500.0729 | 148.9198
| STD | 101.2404 | 110.4219 | 61.6579 | 104.6620 | 147.9874 | 108.5539
F18 | Best | 1.1030e+03 | 916.7264 | 955.7344 | 916.7859 | 1.0990e+03 | 885.7073
| Mean | 1.1517e+03 | 959.8715 | 994.9993 | 925.5520 | 1.1258e+03 | 899.2759
| STD | 29.2261 | 35.5771 | 18.7432 | 8.1187 | 15.6452 | 6.2678
F19 | Best | 1.1085e+03 | 933.1690 | 954.5782 | 931.8614 | 1.0861e+03 | 898.0658
| Mean | 1.1527e+03 | 984.4148 | 989.0768 | 974.8522 | 1.1234e+03 | 902.7453
| STD | 25.9372 | 41.1798 | 22.6374 | 47.3148 | 15.6621 | 3.3756
F20 | Best | 1.0677e+03 | 923.3271 | 967.4342 | 800.1327 | 1.0976e+03 | 890.1426
| Mean | 1.1463e+03 | 956.9104 | 997.2200 | 877.0547 | 1.1253e+03 | 900.2646
| STD | 38.6435 | 31.7415 | 15.8666 | 70.3336 | 18.0401 | 5.7584
F21 | Best | 1.2389e+03 | 1.0822e+03 | 1.0319e+03 | 500.3422 | 1.2062e+03 | 500.0000
| Mean | 1.2549e+03 | 1.1096e+03 | 1.1535e+03 | 833.3210 | 1.2177e+03 | 711.0300
| STD | 11.0959 | 12.0662 | 58.2933 | 337.3735 | 7.2659 | 275.1757
F22 | Best | 1.1649e+03 | 899.7449 | 1.0984e+03 | 1.0423e+03 | 1.0823e+03 | 581.3024
| Mean | 1.2124e+03 | 948.8016 | 1.1922e+03 | 1.0992e+03 | 1.1153e+03 | 804.6198
| STD | 29.1641 | 27.8554 | 61.1247 | 45.6596 | 27.1676 | 94.9291
F23 | Best | 1.2196e+03 | 1.1003e+03 | 1.1685e+03 | 550.6398 | 1.2005e+03 | 534.1640
| Mean | 1.2527e+03 | 1.1165e+03 | 1.2037e+03 | 1.0637e+03 | 1.2174e+03 | 683.7794
| STD | 19.0620 | 10.8319 | 18.6563 | 286.9826 | 15.2362 | 243.7339
F24 | Best | 1.2495e+03 | 937.2971 | 1.3316e+03 | 200.4646 | 1.2482e+03 | 323.1548
| Mean | 1.2795e+03 | 985.7198 | 1.4003e+03 | 638.5843 | 1.4003e+03 | 607.3364
| STD | 15.3617 | 91.8088 | 33.5210 | 600.2639 | 33.5210 | 196.7787
F25 | Best | 1.7385e+03 | 1.9230e+03 | 1.7753e+03 | 1.6282e+03 | 1.7753e+03 | 1.3589e+03
| Mean | 1.7848e+03 | 1.9684e+03 | 1.8208e+03 | 1.6326e+03 | 1.8208e+03 | 1.5204e+03
| STD | 27.3936 | 22.3652 | 27.7505 | 2.6468 | 27.7505 | 82.2932

Table 13:  Statistical analysis (Wilcoxon rank-sum test) on hybrid composite
test functions with 30 dimensions

Function No.  GTBOA vs GA GTBOA vs PSO GTBOA vs ABC pval T+ T- winner pval T+
T- winner pval T+ T- winner F15 7.8604e-70$(<0.05^{+})$ 725 265 +
4.0660e-149$(<0.05^{+})$ 863 127 + 4.2081e-131$(<0.05^{+})$ 870 120 + F16
3.0334e-246$(<0.05^{+})$ 939 51 + 1.0285e-248$(<0.05^{+})$ 939 50 +
1.7892e-310$(<0.05^{+})$ 990 0 + F17 1.3046e-142$(<0.05^{+})$ 836 154 +
1.7612e-198$(<0.05^{+})$ 936 54 + 3.1198e-314$(<0.05^{+})$ 990 0 + F18
1.2556e-312$(<0.05^{+})$ 988 2 + 1.5981e-203$(<0.05^{+})$ 888 102 +
5.3276e-284$(<0.05^{+})$ 990 0 + F19 2.8290e-319$(<0.05^{+})$ 987 3 +
2.0457e-223$(<0.05^{+})$ 900 90 + 1.5210e-316$(<0.05^{+})$ 987 3 + F20
2.1312e-286$(<0.05^{+})$ 973 17 + 1.3642e-176$(<0.05^{+})$ 850 140 +
2.6859e-270$(<0.05^{+})$ 990 0 + F21 2.8565e-288$(<0.05^{+})$ 959 31 +
1.4215e-246$(<0.05^{+})$ 925 65 + 8.5974e-264$(<0.05^{+})$ 989 1 + F22
3.7825e-319$(<0.05^{+})$ 984 6 + 3.5128e-47$(<0.05^{+})$ 691 299 +
1.0398e-201$(<0.05^{+})$ 964 26 + F23 1.7376e-289$(<0.05^{+})$ 952 38 +
1.1670e-165$(<0.05^{+})$ 844 146 + 9.2559e-276$(<0.05^{+})$ 987 3 + F24
2.1571e-291$(<0.05^{+})$ 967 23 + 1.1220e-168$(<0.05^{+})$ 901 89 +
1.0542e-311$(<0.05^{+})$ 990 0 + F25 4.3952e-320$(<0.05^{+})$ 990 0 +
2.7120e-320$(<0.05^{+})$ 990 0 + 5.3673e-320$(<0.05^{+})$ 990 0 + GTBOA vs ALO
GTBOA vs BWO pval T+ T- winner pval T+ T- winner F15 1.3780e-166$(<0.05^{+})$
879 111 + 1.7310e-176$(<0.05^{+})$ 839 151 + F16 0.0218$(<0.05^{+})$ 533 457 +
1.7374e-264$(<0.05^{+})$ 919 71 + F17 4.5041e-89$(<0.05^{+})$ 976 14 +
5.9849e-249$(<0.05^{+})$ 908 82 + F18 1.2325e-219$(<0.05^{+})$ 987 3 +
1.4120e-312$(<0.05^{+})$ 988 2 + F19 5.8540e-237$(<0.05^{+})$ 988 2 +
3.4180e-231$(<0.05^{+})$ 986 4 + F20 0.1240$(>0.05^{+})$ 252 738 -
2.2213e-302$(<0.05^{+})$ 974 16 + F21 1.4823e-81$(<0.05^{+})$ 959 31 +
1.6797e-305$(<0.05^{+})$ 953 37 + F22 5.1015e-172$(<0.05^{+})$ 942 48 +
9.5479e-183$(<0.05^{+})$ 886 104 + F23 6.6032e-50$(<0.05^{+})$ 789 201 +
9.4658e-293$(<0.05^{+})$ 943 47 + F24 0.2153$(>0.05^{+})$ 168 822 -
5.6338e-314$(<0.05^{+})$ 965 25 + F25 5.7644e-171$(<0.05^{+})$ 860 130 +
5.8710e-311$(<0.05^{+})$ 987 3 +

Figure 5: Convergence curves for all the employed algorithms.

Table 6 shows the results of unimodal test functions using 30 dimensions. The
GTBO reveals decent results on F1, F2, F3, F5, and F6. In these test
functions, GTBO results are superior in terms of best, mean, and standard
deviation. Other algorithms such as GA and BWO are in the next ranks.
Regarding F4 and F7, GA achieved the best results. As mentioned earlier,
unimodal test functions evaluate the algorithms in terms of exploitation and
convergence rate. In this regard, as the algorithm demonstrates superior
results in terms of unimodal test functions, it can be concluded that the GTBO
has a reasonable efficiency in terms of exploitation and convergence speed.
Figures 5 (a) and (b) show the convergence curve for function F3 and F5 which
prove the efficiency of the proposed approach in converging to the global
minimum. The algorithm’s superiority is due to its employed operators that
underpin a strong search mechanism resulting in optimal results.

It is not enough to prove the efficiency of the proposed algorithm just by
showing the achieved results of the fitness function. The results should be
tested statistically compared to the baseline algorithms (GA, PSO, ABC, ALO,
and BWO) to demonstrate that it has not achieved the results by chance. In
this regard, the Wilcoxon signed rank-sum test is employed to test the GTBO
statistically at $5\%$ significance level [24]. Table 5.1 shows the results of
GTBO as compared to the baseline algorithms where ‘$T+$’ shows the sum of
ranks for the GTBO, ‘$T-$’ shows the sum of the rank for the baseline
algorithms and $pval$ denotes the P-value achieved for each algorithm. There
are two important concepts in the Wilcoxon test denoted as null and
alternative hypothesis where the former shows that there is no significant
difference between two existing algorithms while the latter denotes the
existence of a significant difference between two algorithms. In Table 5.1,
when the winner is shown by ‘+’, it means that the GTBO is superior compared
to its counterpart as the null hypothesis is rejected at $5\%$ significance
level, however, when the winner is shown by ‘-’ sign, it means that the null
hypothesis is not rejected and there is no significant difference. According
to Table 5.1, in most cases the GTBO is statistically superior against
baseline algorithms.

The proposed approach has been applied to unimodal benchmark functions with
higher dimensions too. Table 8 shows the results of GTBO when the dimension
number is 50. In this regard, the GTBO yields superior results in 4 (F1, F2,
F5, and F6) out of the 7 test functions. The achieved results show that the
proposed algorithm can still produce superior results when the number of
dimensions increases significantly.

###  5.2 Results on multimodal test functions

The proposed algorithm is applied to some multimodal benchmark test functions
and the results are shown in Tables 9, 5.1, and 11. According to Table 9, the
GTBO shows superior results with respect to baseline algorithms when the
dimension is 30. The proposed algorithm is superior in the majority of
multimodal functions including F8, F10, F11, F12, and F13. The only exception
is F9 where the GA algorithm has achieved better results than other
algorithms. In terms of best-achieved results, the GA also shows a decent
efficiency in F13 while the GTBO is in the second rank. A multimodal test
function consists of more than one optimum and assesses the efficiency of the
algorithm with regard to exploration and the local minimum avoidance. Due to
the fact that the GTBO has revealed promising results in this type of test
functions, it can be concluded that the proposed algorithm is efficient in
terms of both exploration and local minima avoidance.

Similar to the unimodal test functions, the efficiency of the GTBO is also
evaluated by employing the Wilcoxon signed rank-sum test to show that the
proposed algorithm is statistically superior. In this regard, Table 5.1 shows
that the proposed algorithm is statistically superior compared to the baseline
methods. According to Table 5.1, the GTBO is the winner in the majority of the
cases and the only exceptions are F9 and F12.

The results of the GTBO for multimodal functions are also tested on higher
dimensions where Table 11 shows that the proposed algorithm yields reasonable
results when the dimension increases significantly. The proposed algorithm
outperforms the baseline algorithms on majority of functions including F10,
F11, F12, and F13. The poor performance of some baseline algorithms in
multimodal functions reveals the reality that some of these algorithms are not
efficient in terms of avoiding local minima to reach the global minimum.
Another important factor is the exploration of these algorithms as the
algorithms with poor exploration cannot explore the search space effectively.

###  5.3 Results on hybrid composite test functions

The proposed algorithm is applied to some composite functions and the results
are shown in Table 12. These functions are challenging as they need to
evaluate the performance of the algorithms in terms of the trade-off between
exploration and exploitation. According to Table 12, the GTBO yields superior
results in 10 out of 11 functions including F15, F16, F17, F18, F19, F21, F22,
F23, F24, and F25. The proposed approach reveals a good efficiency in terms of
convergence rate too. According to Figure 5, the proposed algorithm converges
faster to the global optima in most cases compared to the baseline algorithms.

Like other classes of test functions, the efficiency of the proposed algorithm
is assessed statistically using Wilcoxon rank-sum test and the results are
shown in Table 5.1. It is clear that the proposed algorithm demonstrates
outstanding results in 9 out of 11 test functions, proving GTBO is
statistically superior.

In this section, GTBO’s performance is evaluated using different benchmark
functions against a variety of baseline algorithms and the results reveal that
the proposed algorithm is superior against baseline algorithms for most of the
benchmark functions in terms of exploration, exploitation, avoidance of local
minima, and balance between exploration and exploitation. In the GTBO, the
employed color switching and survival operators play pivotal roles in creating
good balance between exploration and exploitation resulting in decent
performance. The color switching operator assists the algorithm in achieving
global optima, while the survival operator helps the algorithm perform a
better local search. The stopping criterion in the study is the number of
function evaluations (NFE) which to the best of our knowledge is a fair
stopping criterion. The poor performance of some algorithms lies in the
concept of trapping into local minima where they cannot explore and exploit
the search space effectively. In other words, they cannot make decent balance
between exploration and exploitation and hence they cannot find the global
minima.

##  6 Application of GTBO to Engineering Problems

This section shows the performance of applying the GTBO to two real problems
in engineering. Both are real classical engineering design problems. The first
one is called the welded beam design problem and the second one is called gear
train design problem [19]. For the problems that have constraints, the
simplest constraint handler (death penalty) is employed. Other constraint
handlers such as special operators, repair algorithms, and hybrid methods are
also used in the literature [63, 64, 65]. In the case of minimization problems
when the agents violate the constraints, the penalty function is used to
penalize the objective function resulting in larger fitness values. In this
section, the only optimum values of the algorithm through 10 independent runs
are presented and the statistical test is not performed as the statistical
significance of the proposed algorithm against the same set of baseline
algorithms was proved in the earlier experiments.

###  6.1 Welded beam design problem

The welded beam design problem is a minimization problem consisting of four
variables namely, the length of bar attached to the weld (l), the bar height
(t), the thickness of the weld (h), and the bar thickness (b) [66, 19, 67].
This problem is illustrated graphically by Figure 6 where it also shows the
structural parameters.

Figure 6: Welded beam design parameters.

In addition, this problem has some defined constraints such as the beam stress
($\alpha$), the deflection of the beam ($\beta$), the bucking load on the bar
(BL), the beam end deflection ($\gamma$), and the side constraints. The
problem is defined using Equations 9 and 10.

$\begin{cases}Minimize\ \
f(\overrightarrow{z})=1.10471z_{1}^{2}+0.04811z_{3}z_{4}(14.0+z_{2}),\\\
\overrightarrow{z}=[z_{1}z_{2}z_{3}]=[hltb],\\\ \\\ Subject\ \ to\ \ \\\
h_{1}(\overrightarrow{z})=\gamma(\overrightarrow{z})-\gamma_{max}\leq 0,\\\
h_{2}(\overrightarrow{z})=\alpha(\overrightarrow{z})-\alpha_{max}\leq 0,\\\
h_{3}(\overrightarrow{z})=z_{1}-z_{4}\leq 0,\\\
h_{4}(\overrightarrow{z})=0.10471z_{1}^{2}+0.04811z_{3}z_{4}(14.0+z_{2})-5.0\leq
0,\\\ h_{5}(\overrightarrow{z})=0.125-z_{1}\leq 0,\\\
h_{6}(\overrightarrow{z})=\gamma(\overrightarrow{z})-\gamma_{max}\leq 0,\\\
h_{7}(\overrightarrow{z})=B-BL(\overrightarrow{z})\leq 0,\\\ \\\ where\\\
0.1\leq z_{1}\leq 2,\\\ 0.1\leq z_{2}\leq 10,\\\ 0.1\leq z_{3}\leq 10,\\\
0.1\leq z_{4}\leq 10,\\\ \end{cases}$

(9)

Different values of parameters defined in Equation 9 are declared in Equation
10 as follows:

$\begin{cases}\gamma(\overrightarrow{z})=\sqrt{(\gamma^{\prime})^{2}+2\gamma^{\prime}\gamma^{n}\frac{z_{2}}{2R}+\gamma^{n2}},\\\
\gamma^{\prime}=\frac{B}{\sqrt{2}z_{1}z_{2}}\\\ \gamma^{n}=\frac{MR}{J}\\\
M=B(L+\frac{z_{2}}{2})\\\
R=\sqrt{\frac{z_{2}^{2}}{4}+(\frac{z_{1}+z_{3}}{2})^{2}}\\\
J=2\bigg{(}\sqrt{2}z_{1}z_{2}\bigg{[}{\frac{z_{2}^{2}}{4}\bigg{(}\frac{z_{1}+z_{3}}{2}\bigg{)}^{2}}\bigg{]}\bigg{)}\\\
\alpha(\overrightarrow{z})=\frac{6PL}{Z_{4}Z_{3}^{2}},\beta({\overrightarrow{z}})=\frac{4PL^{3}}{Ez_{3}^{2}z+z_{4}}\\\
BL(\overrightarrow{z})=\frac{4.013E\sqrt{\frac{z_{3}^{2}z_{4}^{6}}{36}}}{L^{2}}\bigg{(}1-\frac{z_{3}}{2L}\sqrt{\frac{E}{4G}}\bigg{)}\\\
P=6000lb,L=14in.,\beta_{max}=0.25in.,E=30\times 10^{6}psi,\\\ G=12\times
10^{6}psi,\gamma_{max}=13600psi,\alpha_{max}=30000psi.\par\end{cases}$

(10)

The goal of this problem is to minimize the fabrication cost in the welded
beam design consisting of seven constraints. It has a lot of applications in
the design of machine elements and mechanical design problems [67, 19, 68].
According to Table 6.1, the GTBO outperforms the baseline algorithms in terms
of achieving the optimal cost as well as the set of optimal values for the
four parameters, confirming that the algorithm can be used to minimize the
fabrication cost of a reasonable welded beam design.

Table 14: Results of welded beam design problem

Algorithms  Welded beam optimal values for variables h l t b Optimal Value GA
0.3979 2.2478 6.0387 0.4652 2.5890 PSO 0.2637 4.6808 7.2402 0.3497 2.6349 ABC
0.1666 4.2434 9.1902 0.2055 1.7880 ALO 0.2049 3.2735 9.0382 0.2057 1.6971 BWO
0.1865 4.0128 8.3204 0.2427 1.9039 GTBO 0.2057 3.2530 9.0366 0.2057 1.6952

###  6.2 Gear train design problem

The gear train design problem is defined using Equation 11 as follows:

$\begin{cases}Let^{\prime}s\ \
consider:(\overrightarrow{n})=[n_{1},n_{2},n_{3},n_{4}],\\\ Minimize:\ \
(\overrightarrow{n})=\bigg{(}\frac{1}{6.931}-\frac{n_{3}n_{2}}{n_{1}n_{4}}\bigg{)}^{2},\\\
\\\ Subject\ \ to:\ \ \\\ 12\leq n_{1},n_{2},n_{3},n_{4}\leq 60\\\
\end{cases}$

(11)

Figure 7: Gear train design problem parameters [10].

The gear train design problem is a mechanical engineering problem aiming to
minimize the gear ratio using four gears of a train [69, 70]. The number of
teeth of the gears is considered as the parameters of the problem. This
problem has no constraints, although the ranges of the variables can be
considered as the constraints of the problem. In Figure 7, different
parameters of this problem are depicted. According to the results in Table
6.2, the GTBO again outperforms the baseline algorithms in terms of obtaining
a superior gear ratio and optimal values for the four parameters. It is worth
noting that as the gear train design problem is a discrete problem, this
experiment also confirms that GTBO is efficient in solving discrete problems.

Table 15: Results of gear train design problem

Algorithms  Gear train optimal values for variables $n_{A}$ $n_{B}$ $n_{C}$
$n_{D}$ Optimal Value GA 50 20 17 47 1.6160e-12 PSO 28 12 16 49 7.2211e-06 ABC
43 13 16 35 1.1096e-17 ALO 58 16 29 59 2.1771e-15 BWO 50 18 17 46 7.5421e-17
GTBO 51 32 12 52 2.0211e-20

###  6.3 Sensitivity analysis on key parameters

This section discusses the sensitivity analysis of the key parameters in the
GTBO in order to understand the impacts of different values of $M_{r}$ (mature
rate) and $S_{r}$ (survival rate) on the overall algorithm performance. In
addition, the two operators that generate solutions in the algorithm are
analyzed to find out their performance and impact on the algorithm’s
efficiency.

Table 16: Sensitivity analysis on mature rate ($M_{r}$)

Function No.  GTBO optimal values for different mature rates $0.2$ $0.4$ $0.6$
$0.8$ 0.9 F2 2.4604e-14 1.8303e-11 4.4458e-09 1.4349e-07 8.0541e-07 F5 19.7865
64.3749 34.1285 31.2356 35.8112 F10 4.7373e-12 4.9744e-10 2.3079e-08
5.6245e-07 2.0360e-06 F12 0.5022 6.8447e-08 2.9187e-06 2.0542e-04 4.0006e-04
F15 421.0066 304.7715 400.0000 278.0010 285.8998 F21 511.0000 500.0000
500.0000 500.0000 500.000

Regarding the key parameters, $M_{r}$ is tested using different values of
(0.2, 0.4, 0.6, 0.8, 0.9) and $S_{r}$ is tested on the values of (0.2, 0.4,
0.6, 0.8). In every class of test function, two functions are selected
randomly. The algorithm performed 30 independent runs and used 30 dimensions
for unimodal and multimodal, as well as 10 dimensions for the composition
functions. Table 6.3 shows the sensitivity analysis results on the mature rate
($M_{r}$) control parameter. It is clear that the GTBO has a better
performance on unimodal and multimodal problems when $M_{r}$ is between [0.2,
0.4], while it has superior performance in composite problems when it has
larger values such as 0.8 and 0.9.

Table 17: Sensitivity analysis on survival rate ($S_{r}$)

Function No.  GTBO optimal values for different survival rates $0.2$ $0.4$
$0.6$ $0.8$ F2 1.4349e-07 1.5478e-07 4.4734e-07 2.1932e-06 F5 31.2356 61.6426
57.6874 90.6470 F10 5.6245e-07 1.4420e-06 2.7178e-06 1.5281e-05 F12 2.0542e-04
0.0524 0.2185 0.1459 F15 252.8611 235.5687 301.2868 261.9206 F21 500.0000
500.0000 586.9541 500.0000

According to Table 6.3, the GTBO has superior results when the value of
$S_{r}$ is initialized to 0.2. This study utilized this value in order to
obtain the best results. In this test, the value of $M_{r}$ is initialized to
0.4 for all the cases.

Based on the results in Tables 6.3 and 6.3, the optimal values of $M_{r}$ and
$S_{r}$ are 0.4 and 0.2 respectively.

Table 18:  Sensitivity analysis on operators defined in GTBO

Function No. Sensitivity analysis on GTBO operators Criteria Color Switching
Operator Survival Operator Composition of Both Operators F2 Best 1.0000e-04
0.1779 6.4949e-08 Mean 0.8523 2.3908 1.4349e-07 STD 2.5567 2.1834 6.4830e-08
F5 Best 22.7657 2.5408e+04 24.3678 Mean 43.5455 4.2193e+05 31.2356 STD 29.3606
3.2102e+05 17.3351 F10 Best 2.0000e-04 5.4435 1.9400e-07 Mean 5.4000e-04
7.2504 5.6245e-07 STD 2.5473e-04 1.3102 3.6438e-07 F12 Best 0.0065 1.9051
2.4174e-06 Mean 0.1993 88.0779 2.0542e-04 STD 0.2587 189.7275 4.9198e-04

Table 18 shows the results of sensitivity analysis on different employed
operators (color-switching operator, survival operator, and a composition of
both) in GTBO. It has been done just for some randomly chosen unimodal and
multimodal test functions over 30 independent runs. As it is shown in Table
18, the results using color-switching operator are more robust than the cases
when the algorithm just uses survival operator. Furthermore, when both
operators are used, the results are much more superior than only using one
operator individually. It further confirms that while the color switching
operator is superior in terms of global search and the survival operator
excels in local search, the composition of the two operators not only improves
the population diversity but also increases the search capability, resulting
in superior balance between exploration and the exploitation.

##  7 Conclusion and Future Works

The main contribution of this work is a novel optimization algorithm inspired
by the natural life of the golden tortoise beetle. The golden tortoise beetle
uses a color-switching mechanism in order to mate and enhance its chance for
reproduction. In addition, this beetle uses a kind of shield to protect its
larvae against predators. These notions are mathematically modeled to design a
novel optimization algorithm. The algorithm has been tested on 24 benchmark
functions using different classes of unimodal, multimodal, and hybrid
composite functions against some well-known baseline algorithms including GA,
PSO, ABC, ALO, and BWO. The superiority of the algorithm on unimodal functions
proves its efficiency in exploitation and convergence rate. In terms of
multimodal, the algorithm also outperforms other algorithms, which shows the
potential of the GTBO in exploration and the local minima avoidance. It can be
concluded that the algorithm can make a trade-off between exploration and
exploitation as it has achieved remarkable results in terms of hybrid
composite functions. The algorithm is also tested statistically using Wilcoxon
signed rank-sum test whose results have revealed that the algorithm is
statistically superior with respect to the baseline algorithms.

The algorithm performance has also been tested on two well-known engineering
problems and the results conclude that GTBO is efficient in solving real
problems with constraints and unknown search spaces. A sensitivity analysis
has also been performed to provide some evidence for choosing different values
of the key control parameters and for combining the two operators to achieve a
suitable trade-off between exploration and exploitation. Because of the random
selection of some solutions in the algorithm, the GTBO has a high probability
of escaping from the local minima. Moreover, population diversity plays a
pivotal role in achieving the global minimum.

In future work, the GTBO will be applied to different optimization problems.
The algorithm’s key control parameters such as mature rate and the survival
rate can be designed with an adaptive approach. Finally, a hybrid approach
combining the GTBO with other optimizers can be considered for further
performance improvement.

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