【免费】蜣螂优化算法（DBO）源代码+DBO原始Paper+23个经典测试函数

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The Journal of Supercomputing

https://doi.org/10.1007/s11227-022-04959-6

1 3

Dung beetle optimizer: anew meta‑heuristic algorithm

forglobal optimization

JiankaiXue

1,2

· BoShen

1,2

Accepted: 16 November 2022

2022

Abstract

In this paper, a novel population-based technique called dung beetle optimizer

(DBO) algorithm is presented, which is inspired by the ball-rolling, dancing, forag-

ing, stealing, and reproduction behaviors of dung beetles. The newly proposed DBO

algorithm takes into account both the global exploration and the local exploitation,

thereby having the characteristics of the fast convergence rate and the satisfactory

solution accuracy. A series of well-known mathematical test functions (including

both 23 benchmark functions and 29 CEC-BC-2017 test functions) are employed to

evaluate the search capability of the DBO algorithm. From the simulation results, it

is observed that the DBO algorithm presents substantially competitive performance

with the state-of-the-art optimization approaches in terms of the convergence rate,

solution accuracy, and stability. In addition, the Wilcoxon signed-rank test and the

Friedman test are used to evaluate the experimental results of the algorithms, which

proves the superiority of the DBO algorithm against other currently popular optimi-

zation techniques. In order to further illustrate the practical application potential, the

DBO algorithm is successfully applied in three engineering design problems. The

experimental results demonstrate that the proposed DBO algorithm can eﬀectively

deal with real-world application problems.

Keywords Convergence rate· Dung beetle optimizer (DBO)· Engineering

applications· Particle swarm optimization (PSO)· Swarm intelligence

* Bo Shen

bo.shen@dhu.edu.cn

College ofInformation Science andTechnology, Donghua University, Shanghai, China

Engineering Research Center ofDigitalized Textile andFashion Technology, Ministry

ofEducation, Shanghai, China

J.Xue, B.Shen

1 3

1 Introduction

The optimization problems have been the focus of research for a long time and

has existed in a variety of real-world systems, including fault diagnosis systems

[1, 2], energy management systems [3, 4], forecasting systems [5, 6], and so on

[7, 8]. It should be noticed that a large number of complex optimization problems

(e.g., the NP-complete problems) are particularly diﬃcult to be settled using the

conventional mathematical programming techniques such as the conjugate gra-

dient and quasi-Newton methods [9]. In this regard, a great number of swarm

intelligence (SI) optimization algorithms have been introduced with the merits

of the easy implementation, self-learning ability, and simple framework. Speciﬁ-

cally, the SI system can be viewed as a swarm where each individual denotes

a candidate solution in the entire search space. In addition, the characteristic of

the SI system is that the individual interactions promote the appearance of the

intelligence behavior. It is worth pointing out that the realization of the optimiza-

tion process mainly includes the following two steps: 1) creating a group of ran-

dom individuals within the scope of the search space and 2) combining, moving,

or evolving these random individuals during the iteration process. This is also

the main framework of almost all SI-based techniques in dealing with optimiza-

tion problems. Note that the diﬀerence for each optimization algorithm is how

to design new strategies (especially combining, moving, or evolving) during the

optimization process.

For example, a well-known population-based technique, namely the particle

swarm optimization (PSO) technique, has gained much research attention with

the advantages of the fast convergence rate, few parameters, and satisfactory solu-

tion accuracy [10, 11]. To be speciﬁc, in a standard PSO algorithm, all parti-

cles thoroughly explore and exploit the search space of the optimization problem

according to their velocity and position. Then, the movement of the whole group

evolves from the disorder to the order, and eventually all particles are clustered at

the best position. It should be mentioned that the position of each individual can

gradually converge to the global optimal solution during the evolution process,

which mainly depends on following two important positions: 1) one is the per-

sonal best position of each individual and 2) the other is the global best position

of the entire population. In addition, the ant colony optimization (ACO) algo-

rithm has become as a famous SI-based technique as the PSO approach as well

[12, 13]. More speciﬁcally, in a ACO algorithm, the path of each ant indicates

a feasible solution and the paths of all ants constitute the solution space. Note

that the selection of the pheromone is of crucial importance for the ant colony

to search the globally optimal path in the whole optimization problem space. In

such a case, the shortest path is obtained in the direction of the stronger level of

pheromone.

Up to now, many novel SI-based techniques have been proposed with the pur-

pose of obtaining more optimizers with good search ability that can solve various

practical optimization problems more eﬃciently, and their performance has been

veriﬁed by a large number of experiments in the existing literature. Similarly,

1 3

Dung beetle optimizer: anew meta‑heuristic algorithm for…

these new SI-based algorithms also mainly mimic the social behaviors of living

creatures (e.g., ﬁsh, insects and birds) in nature [14]. For example, the grey wolf

optimizer (GWO) algorithm has been presented in [15], which simulates the lead-

ership hierarchy (including alpha, beta, delta, and omega) and hunting behavior

of grey wolves. Note that the hunting mechanism of the GWO algorithm mainly

consists of searching, encircling, and attacking prey. Recently, the whale optimi-

zation algorithm (WOA) has been put forward in [16] where the bubble-net hunt-

ing strategy has been ﬁrst proposed that mimics the social behavior of humpback

whales. More recently, a new SI-based technique, the Harris hawks optimizer

(HHO) algorithm, has been proposed in [17], which simulates the living behavior

of Harris’ hawks. The HHO algorithm has attracted the attention of researchers

from both industrial and academic societies due to its strong capability of search-

ing around the global optimum [18–20]. Moreover, there are several promising

SI-based algorithms as well, see [21–25] for more discussions.

It is well known that a great number of SI-based algorithms have played a vitally

important role in many practical applications. For example, it has been shown in

[26, 27] that the ACO algorithm has the satisfactory search capability in dealing

with the traveling salesman problems. Similarly, during the past few decades, the

PSO algorithm has been widely implemented in solving the complex optimization

problems such as large-scaled complex networks [28], fuzzy systems [29], compli-

cated multimodal optimization systems [30] and other ﬁelds [31]. Furthermore, in

the past few years, the newly developed SI-based approaches have also been suc-

cessfully applied to various applications with the purpose of providing more suitable

solutions for diﬀerent optimization problems. For instance, in [32], by introducing

the two new strategies such as the reverse learning and the Levy ﬂight disturbance,

an improved the WOA (IWOA) has been proposed and then is used to identify

the unknown parameters with regard to a static var compensator (SVC) model in

order to verify the optimization performance of the IWOA. In [33], a hybrid HHO

algorithm, namely comprehensive learning Harris hawks-equilibrium optimization

(CLHHEO) algorithm, has been designed by utilizing the comprehensive learning

strategy, the operator of equilibrium optimizer, and the terminal replacement mecha-

nism. The CLHHEO algorithm is successfully used in the optimization of several

engineering design problems (e.g., the car side impact design and the multiple disc

clutch brake design). The sparrow search algorithm (SSA) has been designed in [23]

and a random walk SSA (RWSSA) has been proposed in [34] to optimize the dis-

tribution and signal coverage of 5G networks in open-pit mines. In summary, the

emerging SI-based techniques have received much research attention in dealing with

various practical applications.

On the other hand, according to No Free Lunch (NFL) theorem, we know that

no single algorithm can deal with all optimization problems. In other words, the

optimization performance of an algorithm may perform well in one set of problems

and poorly in another. Therefore, the NFL theorem encourages ﬁnding and develop-

ing more optimizers with satisfactory performance. Motivated by the above discus-

sions, a new SI-based technique called dung beetle optimizer (DBO) is developed

with the aim of providing a more eﬃcient optimizer to solve complex optimization

problems. Correspondingly, the main contributions of this paper can be highlighted

J.Xue, B.Shen

1 3

into the following three aspects: 1) the DBO algorithm is put forward where ﬁve

diﬀerent updating rules that are inspired by the ball-rolling, dancing, foraging,

stealing, and reproduction behaviors of dung beetles are designed to help ﬁnd high-

quality solutions; 2) the DBO algorithm is comprehensively evaluated through a

series of mathematical test functions (including both 23 benchmark functions and

29 CEC-BC-2017 test functions) and the experimental results indicate that the DBO

algorithm more competitive performance compared with the state-of-the-art opti-

mization techniques; and 3) the DBO algorithm is successfully employed in several

practical engineering design problems, which illustrates that this algorithm is prom-

ising for solving real-world application problems.

The remainder of this paper is structured as follows. The main inspiration and

biological foundations of this paper and the proposed DBO algorithm are introduced

in Sect. 2. Benchmark test functions, parameter settings, optimization results and

discussions are shown in Sect.3. Section4 introduces the applications of the DBO

algorithm with regard to three engineering design problems in detail. Finally, the

conclusions of on the paper are given in Sect.5.

2 Dung beetle optimizer

In this section, a novel SI-based optimization technique called the DBO algorithm is

discussed in detail, including the following two aspects: 1) the inspiration and 2) the

mathematical model.

2.1 Inspiration

There are various species for the dung beetles, such as Copris ochus Motschulsky,

Onthophagus gibbulus, Caccobius jessoensis Harold and so on. It is well known that,

the dung beetle, as a common insect in nature, feeds on the dung of animals. Note that

dung beetles are found in most parts of the world and act as decomposers in nature,

which means that they are of vital importance in the ecosystem. Research has shown

that dung beetles have an interesting habit of making dung into a ball and then rolling it

out, as captured in Fig.1. It is worth mentioning that the purpose of dung beetles is able

Fig. 1 The rolling dung ball behavior of the dung beetle

1 3

Dung beetle optimizer: anew meta‑heuristic algorithm for…

to move their dung ball as quickly and eﬃciently as possible, which can prevent them

from being competed by other dung beetles [35].

As shown in Fig.1, a dung beetle is rolling backward a dung ball that is larger than

itself. On the other hand, a fascinating behavior of dung beetle can use celestial cues

(especially the sun, the moon and polarized light) to navigate and make the dung ball

roll along a straight line [36, 37]. However, if there is no light source at all (that is,

in total darkness), the dung beetle’s path is no longer straight, but curved and some-

times even slightly round [35]. Note that a number of natural factors (such as wind and

uneven ground) can cause dung beetles to deviate from their original direction. In addi-

tion, the dung beetle is likely to encounter obstacles and be unable to move forward in

the rolling process. In this regard, dung beetles usually climb on top of the dung ball to

dance (including a series of rotations and pauses), which determines their direction of

the movement [38].

Another interesting behavior observed from the dung beetle lifestyle is that dung

balls obtained have the following two main purposes : 1) some dung balls are used to

lay eggs and raise the next generation, and 2) the rest are used as food. Speciﬁcally,

dung beetles bury the dung balls and the females lay their eggs in these dung balls. It

should be noted that the dung ball not only serves as a developmental site for the larva,

but also provides the larva with food that is essential for life. Therefore, dung balls play

an irreplaceable role in the survival of dung beetles.

A new SI optimization algorithm, the DBO technique, is mainly inspired by the ball-

rolling, dancing, foraging, stealing, and reproduction behaviors of dung beetles. In the

next subsection, the behavior of the dung beetle will be modeled mathematically with

the hope of developing a optimizer with the satisfactory search performance.

2.2 The structure andthealgorithm

According to the above discussion, we know that dung beetles need to navigate through

the celestial cues in the rolling process to keep the dung ball rolling in a straight path.

To simulate the ball rolling behavior, dung beetles are required to move in a given

direction throughout the entire search space. The trajectory of a dung beetle is given in

Fig.2. In this ﬁgure, we can see that a dung beetle use the sun to navigate, where the

red arrow indicates the rolling direction. In this paper, we assume that the intensity of

the light source also aﬀects the dung beetle’s path. During the rolling process, the posi-

tion of the ball-rolling dung beetle is updated and can be expressed as

where t represents the current iteration number,

(t)

denotes the position informa-

tion of the ith dung beetle at the tth iteration,

k ∈(0, 0.2]

denotes a constant value

which indicates the deﬂection coeﬃcient, b indicates a constant value belonging to

(0, 1),

𝛼

is a natural coeﬃcient which is assigned -1 or 1,

indicates the global

worst position,

Δx

is used to simulate changes of light intensity.

(1)

(t + 1)=x

(t)+

𝛼

× k × x

(t − 1)+b ×Δx,

Δx =∣ x

(t)−X

∣

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