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

Omid Tarkhaneh tarkhanehomid@gmail.com Neda Alipour nedalipur@gmail.com Amirahmad Chapnevis chapnevisa@vcu.edu Haifeng Shen haifeng.shen@acu.edu.au 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.

We propose a novel nature-inspired algorithm for solving optimization problems that are unprecedentedly modeled after golden tortoise beetle.
2.

The proposed algorithm creates a decent search mechanism that is underpinned by the two unique color-switching and survival operators.
3.

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

Refer to caption — 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.

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

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

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

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.

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.

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.

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
$F_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}$	30, 200	[-100, 100]
$F_{2}(x)=\sum_{i=1}^{n}\|x_{i}\|+\displaystyle\prod_{i=1}^{n}\|x_{i}\|$	30, 200	[-10, 10]
$F_{3}(x)=\sum_{i=1}^{n}(\sum_{j-1}^{i}x_{j})^{2}$	30, 200	[-100, 100]
$F_{4}(x)=\displaystyle max_{i}\{\|x_{i}\|,1\leq i\leq n\}$	30, 200	[-100, 100]
$F_{5}(x)=\sum_{i=1}^{n-1}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]$	30, 200	[-30, 30]
$F_{6}(x)=\sum_{i=1}^{n}([x_{i}+0.5])^{2}$	30, 200	[-100, 100]
$F_{7}(x)=\sum_{i=1}^{n}ix_{i}^{4}+random[0,1)$	30, 200	[-1.28, 1.28]

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

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

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

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

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

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