An enhanced honey badger algorithm based on Lévy flight and refraction opposition-based learning for engineering design problems

Abstract

Honey badger algorithm (HBA) is a recently developed meta-heuristic algorithm, which mainly simulates the dynamic search behavior of honey badger in wild nature. Similar to other basic algorithms, HBA may suffer from the weakness of poor convergence accuracy, inadequate balance between exploration and exploitation, and ease of getting trapped into the local optima. In order to address these drawbacks, this paper proposes an enhanced honey badger algorithm (EHBA) to improve the search quality of the basic method from three aspects. First, we introduce the highly disruptive polynomial mutation to initialize the population. This is considered from increasing the population diversity. Second, Lévy flight is integrated into the position update formula to boost search efficiency and balance exploration and exploitation capabilities of the algorithm. Furthermore, the refraction opposition-based learning is applied to the current global optimum of the swarm to help the population jump out of the local optima. To validate the function optimization performance, the proposed EHBA is comprehensively analyzed on 18 standard benchmark functions and IEEE CEC2017 test suite. Compared with the basic HBA and seven state-of-the-art algorithms, the experimental results demonstrate that EHBA can outperform other competitors on most of the test functions with superior solution accuracy, local optima avoidance, and stability. Additionally, the applicability of the proposed method is further highlighted by solving four engineering design problems. The results indicate that EHBA also has competitive performance and promising prospects for real-world optimization tasks.

Keywords

Honey badger algorithm highly disruptive polynomial mutation Lévy flight refraction opposition-based learning engineering design problems

1 Introduction

Optimization is the process of finding the best solution for a given problem without violating any constraint conditions. With the development of science and technology, increasingly complex optimization problems are emerging [1 –3]. Nevertheless, the traditional mathematics-based programming techniques confront the challenges of large-scale, high-dimensional, and sub-optimal search regions, so they cannot efficiently obtain the optimal results on such problems [4 –6]. The demand for more robust optimization tools is imperious. In the past few decades, meta-heuristic algorithms (MAs) have shown excellent performance in dealing with these complex optimization problems. Compared with conventional methods, MAs are usually able to provide the possible optimal solution within a reasonable amount of time, which is attributed to their distinctive advantages of few parameters, conceptual simplicity, no gradient information required, and strong ability to bypass the local optima [7]. Therefore, MAs have witnessed a rapid rise in popularity and been widely applied to various scenarios.

MAs find the optimal solution by simulating the stochastic phenomena in nature. Based on the sources of inspiration, MAs can generally be divided into three dominant categories [8, 9]: evolution-inspired, physics-inspired, and swarm-inspired. To be more specific, evolutionary algorithms are inspired by the laws of natural evolution, such as mutation, crossover, and selection. The most representative paradigm of evolutionary algorithms is genetic algorithm (GA) [10], which imitates Darwin’s theory of evolution. Of course, differential evolution (DE) [11], evolutionary programming (EP) [12], and biogeography based optimization (BBO) [13] are also covered in this branch. Physics-inspired algorithms are derived from a variety of physical concepts in the universe, which mainly contain simulated annealing (SA) [14], gravitational search algorithm (GSA) [15], multi-verse optimizer (MVO) [16], sine cosine algorithm (SCA) [17], and arithmetic optimization algorithm (AOA) [18]. Swarm-inspired algorithms imitate the collective behavior of creatures in nature. Particle swarm optimization (PSO) [19], first proposed by Kennedy and Eberhart in 1995 on the basis of the bird flocking behavior, is the most classical example in these algorithms. In PSO, each bird is considered as a candidate solution. To gain the global optimum, each bird would move around the search area and update its current position. Some other popular swarm intelligence algorithms are ant colony optimization (ACO) [20], firefly algorithm (FA) [21], ant lion optimizer (ALO) [22], grey wolf optimizer (GWO) [23], and whale optimization algorithm (WOA) [24]. Apart from the algorithms described above, more MAs are designed to solve the optimization problems in the real world, for instance, harris hawks optimization (HHO) [25], spotted hyena optimizer (SHO) [8], salp swarm algorithm (SSA) [26], moth-flame optimization algorithm (MFO) [27], butterfly optimization algorithm (BOA) [28], emperor penguin optimizer (EPO) [29], slime mould algorithm (SMA) [30], aquila optimizer (AO) [31], and gorilla troops optimizer (GTO) [32].

For MAs, it is critical to retain an appropriate balance between global exploration and local exploitation [1, 33]. In the initial stage, the search agents explore the entire search space as far as possible to determine the potential area with the optimal solution. Then, in the later stage, an algorithm conducts more local exploitation to increase the accuracy of the attained optimal solution. Even though hundreds of MAs have already been proposed, there is a constant need for further innovations in optimization algorithms. As per the no free lunch (NFL) theorem [34], no one algorithm can be suitable for all types of optimization problems. At the same time, it has been indicated that most MAs still have the limitations of slow convergence speed, imbalance between exploitation and exploration, and falling into the local optimum [35]. In addition to designing new algorithms, this theorem also prompts many scholars to improve existing algorithms with different effective strategies. To accelerate convergence, Tang et al [36] put forward a modified variant of particle swarm optimization called TACPSO through introducing acceleration coefficients and stochastic velocity. Nadimi-Shahraki et al. [37] embedded the dimension learning-based hunting mechanism into grey wolf optimizer to help the algorithm escape from the local optima. The proposed IGWO has been successfully used to solve 30 benchmark functions and four engineering design problems. In [38], Li et al. introduced the chaotic perturbation to improve the solution accuracy and robustness of ant colony optimization in the link prediction. Hosseinzadeh et al. [39] employed Lévy flight to promote the search performance of butterfly optimization algorithm and applied it to address the data placement and scheduling problem in edge computing environments. Nguyen et al. [40] proposed an improved slime mould algorithm based on opposition-based learning, namely ISMA, and implemented it for the optimal control of cascade hydropower station. Jia et al. [41] constructed a multi-strategy emperor penguin optimizer by incorporating the highly disruptive polynomial mutation, Lévy flight, and thermal exchange operator. Experimental results showed that it is an effective multilevel thresholding segmentation algorithm for colour satellite images. Meanwhile, Goodarzian et al. [42] combined firefly algorithm with simulated annealing and social engineering optimization respectively, and presented two hybrid algorithms named HFFA-SA and HSEO-FFA to find Pareto optimal solutions for the green medicine supply chain network design problem.

Honey badger algorithm (HBA) is a novel meta-heuristic algorithm developed by Hashim et al. [43] in 2021, which emulates the intelligent foraging behavior of honey badger when digging and searching for honey. Preliminary studies suggest that HBA can provide outstanding results on benchmark test problems. However, HBA also has the drawbacks of the decrease in population diversity, poor solution accuracy, and premature convergence when solving complex optimization problems. Nowadays, researches on the HBA algorithm have been carried out successively, but the proposed algorithm lacks global improvement mechanisms. As a result, NFL theorem motivates us to enhance this latest swarm-inspired algorithm.

Given the above discussion, a new enhanced honey badger algorithm (EHBA) is proposed in this paper, which introduces three search strategies to improve the overall optimization performance of HBA. First of all, the highly disruptive polynomial mutation is used to initialize the positions of honey badger to increase the population diversity. To speed up the convergence speed and balance exploration and exploitation trends of the algorithm, Lévy flight is integrated into the position update rules. Besides, the refraction opposition-based learning helps to enlarge the search area, thus strengthening the ability of jumping out of the local optima. To fully verify the effectiveness of the proposed EHBA, 18 standard benchmark functions, IEEE CEC2017 test suite, and four engineering design problems are employed for comparison experiments. The results demonstrate that EHBA outperforms the basic HBA and several advanced meta-heuristic algorithms in terms of solution accuracy, local optima avoidance, and robustness.

The structure of this paper is organized as follows. Section 2 presents the mathematical model of the basic HBA algorithm. A detailed description of three enhancement strategies and the proposed EHBA is provided in Section 3. In Section 4, we conduct a series of simulation experiments to evaluate the performance of EHBA and discuss the obtained results. Based on this, the proposed method is applied to solve four engineering design problems in Sections 5. Finally, the conclusion of this paper and future research direction are shown in Section 6.

2 Honey badger algorithm

As a nature-inspired optimization algorithm, HBA mimics the foraging behavior of honey badger [43]. To locate food sources (i.e., honey), honey badgers commonly adopt two ways: either relying on their smelling capability or following a honey-guide bird. In the former mode, the honey badger approximates the prey position based on its sense of smell. Once reaching there, it would move around the beehive, then choose the proper spot to dig and eventually catch the prey. In the latter mode, the honey badger locates the hives directly with the help of a honey-guide bird. With the above concept, the search process of HBA is modeled into three parts: initialization phase, digging phase, and honey phase, which are briefly described below.

2.1 Initialization phase

Suppose there are N (population size) honey badgers in the D-dimensional search space. The position of the i^th honey badger individual is defined as $X_{i} = (x_{i}^{1}, x_{i}^{2}, \dots, x_{i}^{D}), i = 1, 2, \dots, N$ . Therefore, the population of all honey badgers in the initialization phase can be generated using Equation (1). $X_{i} = {lb}_{i} + r_{1} \times ({ub}_{i} - {lb}_{i})$ (1) where r₁ is a random number in the range [0, 1], X_i is the position of the i^th honey badger in a population of N, ub_iand lb_i are the upper and lower boundaries of the search range, respectively.

2.2 Digging phase

In this phase, the honey badger efficiently gets close to the prey through continuous movement and sniffing. First, the idea of the smell intensity I_i is presented, which depends on the concentration strength of the prey and the distance between it to the i^th honey badger. The higher the intensity, the faster the honey badger will move and vice versa. The value of smell intensity is determined using Equations (2)–(4). $I_{i} = r_{2} \times \frac{S}{4 π d_{i}^{2}}$ (2) $S = {(X_{i} - X_{i + 1})}^{2}$ (3) $d_{i} = X_{prey} - X_{i}$ (4) where r₂ denotes a random number between [0, 1], S denotes the concentration strength, X_i+1 denotes the position of the (i + 1) ^th honey badger, X_preyrepresents the position vector of the prey (i.e., global optimal position found so far), d_i represents the distance between the prey and the i^th honey badger.

Then, in order to ensure a stable transition between exploration and exploitation in the HBA algorithm, the density coefficient α that decreases with iteration is introduced to control the time-varying randomization as follows: $α = C \times \exp (\frac{- t}{Maxiter})$ (5) where C is a constant equal to 2 at default, t is the number of the current iteration, Maxiter denotes the maximum number of iterations.

On the basis of smell intensity I_i, distance information d_i and density coefficient α, the honey badger acts like Cardioid shape [44] to update its position during the digging activity, which can be simulated as Equation (6). $\begin{matrix} X_{new} = X_{prey} + F \times γ \times I_{i} \\ \times X_{prey} + F \times r_{3} \times α \times d_{i} \\ \times | cos (2 π r_{4}) \times [1 - cos (2 π r_{5})] | \end{matrix}$ (6) where X_new is the new position of honey badger in the next iteration, γ is a constant equal to 6 at default, r₃, r₄ and r₅ are all random numbers between [0, 1], cos(·) denotes the cosine function, whereas the parameter F represents the flag used to change search direction for providing agents with a high opportunity to thoroughly scan the search space, and it is calculated as follows: $F = {\begin{matrix} 1 & if r_{6} ⩽ 0.5 \\ - 1 & else \end{matrix}$ (7) where r₆ denotes a random number between [0, 1].

2.3 Honey phase

In the honey phase, another pattern when the honey badger follows the honey-guide bird to the hive is simulated as Equation (8). $X_{new} = X_{prey} + F \times r_{7} \times α \times d_{i}$ (8) where X_new is the new position of honey badger in the next iteration, X_preyis the prey position, r₇ is a random number in the range [0, 1], d_i, α and F are also calculated using Equations (4), (5) and (7). The pseudo-code of HBA is presented in Algorithm 1.

Algorithm 1

pseudo-code of the basic HBA
1. Initialize the population size N and maximum number of iterations Maxiter
2. Initialize the random population positions X_i (i = 1, 2, …, N)
3. Calculate the fitness values of each honey badger individual X_i and assign to f_i (i = 1, 2, …, N)
4. Save the best position as X_prey and assign fitness to f_prey
5. Whilet ⩽ Maxiterdo
6. Calculate the density coefficient α using Equation (5)
7. For each honey badger do
8. Calculate the smell intensity I_i using Equation (2)
9. Ifrand < 0.5 thenrand is a random number in the interval [0, 1]
10. Update the honey badger position X_new using Equation (6)
11. E lse
12. Update the honey badger position X_new using Equation (8)
13. End If
14. Calculate the fitness of the new position X_new and assign to f_new
15. Iff_new ⩽ f_ithen
16. Let X_i = X_new and f_i = f_new
17. End If
18. Iff_new ⩽ f_preythen
19. Let X_prey = X_new and f_prey = f_new
20. End If
21. End For
22. t = t + 1
23. End While
24. ReturnX_prey

3 The proposed EHBA algorithm

To overcome the drawbacks in the basic HBA, an enhanced version named EHBA is proposed in this section. First, the highly disruptive polynomial mutation is introduced to generate mutation candidate solutions, which increases the population diversity. Besides, Lévy flight is utilized to modify the position update formula to facilitate a better balance between the algorithm exploration and exploitation. In order to explore more unknown search areas and avoid falling into the local optima, we use the refraction opposition-based learning to evaluate the opposite solution of the global optimum. The detailed process is described below.

3.1 Highly disruptive polynomial mutation

The mutation operator is an exploratory method to generate new candidate solutions and provide additional diversity when solving a specific optimization problem [45, 46]. In fact, the highly disruptive polynomial mutation (HDPM) is an improved version of the polynomial mutation [47] to solve its problem that the mutation has no effect when a variable reaches the boundaries. Compared with the other mutation mechanisms, HDPM has been proved to be more accurate and consistent on most benchmark functions. The formula of HDPM is presented as follows: $X_{i}^{'} = X_{i} + δ_{k} \cdot (u b_{i} - l b_{i})$ (9) where $X_{i}^{'}$ denotes the offspring position, X_i denotes the parent position, ub_iand lb_i are the upper and lower boundaries of the search range, respectively. The coefficient δ_k is calculated according to Equations (10)–(12). $δ_{1} = \frac{X_{i} - {lb}_{i}}{{ub}_{i} - {lb}_{i}}$ (10) $δ_{2} = \frac{{ub}_{i} - X_{i}}{{ub}_{i} - {lb}_{i}}$ (11)

$δ_{k} = {\begin{matrix} {[2 r_{8} + (1 - 2 r_{8}) \cdot {(1 - δ_{1})}^{η_{m} + 1}]}^{\frac{1}{η_{m} + 1} - 1}, if r_{8} ⩽ 0.5 \\ 1 - {[2 (1 - r_{8}) + 2 (r_{8} - 0.5) \cdot {(1 - δ_{2})}^{η_{m} + 1}]}^{\frac{1}{η_{m} + 1}}, otherwise \end{matrix}$ (12) where r₈ is a random number in the range [0, 1], η_m is the distribution index. As can be noticed in Equation (12), HDPM is able to continue to explore the whole target area even if the variable is near one of its boundaries. By adding HDPM into the initialization phase of the basic algorithm, it is beneficial to enrich the population diversity and lay a good foundation for exploration.

3.2 Lévy flight

It has been indicated that the flying behavior of many animals and insects in nature shows the typical characteristics of Lévy flight [48, 49]. Broadly speaking, Lévy flight is a class of random walk. In this methodology, the jump size alternates between short-distance and occasional long-distance steps drawn from the Lévy probability distribution. Due to its robust search efficiency, Lévy flight has been successfully designed in many optimizes, such as harris hawks optimization (HHO) [25] and aquila optimizer (AO) [31]. The formula of Lévy flight is expressed as follows: $L & eacute; vy = s \times \frac{u \times σ}{| v |^{1 / β}}$ (13) where s is a constant equal to 0.01, u and v are random numbers between [0, 1]. The value of σ is computed using Equation (14). $σ = {\frac{Γ (1 + β) \times \sin (π \times \frac{β}{2})}{Γ [\frac{(1 + β)}{2}] \times β \times 2^{\frac{(β - 1)}{2}}}}^{\frac{1}{β}}$ (14) where Γ (·) denotes the standard gamma function, β is a constant equal to 1.5.

Figure 1 illustrates the distribution and 2D trajectory of Lévy flight within 500 steps. It can be observed from the figure that an emblematical motion mode of Lévy flight works like this: the particle starts with a local movement, performing a number of small steps, and then it carries out a large step followed by another local movement. This factor contributes to undertaking both global and local search simultaneously. In the proposed EHBA, we try to integrate Lévy flight into the position update formula of search agents to boost the search efficiency and enhance the algorithm’s exploration and exploitation trends to a certain extent. After the enhancement, the position of honey badger can be updated as shown in Equations (16).

Fig. 1

Distribution and 2D trajectory of Lévy flight.

$\begin{matrix} X_{new} = X_{prey} + F \times γ \times I_{i} \\ \times X_{prey} + F \times r_{3} \times α \times d_{i} \\ \times | cos (2 π r_{4}) \times [1 - cos (2 π r_{5})] | \times Lévy \end{matrix}$

$X_{new} = X_{prey} + (F \times r_{7} \times α \times d_{i}) \times Lévy$ (15)

3.3 Refraction opposition-based learning

Refraction opposition-based learning (ROBL), incorporating the traditional opposition-based learning [50] and refraction principle in physics, is a powerful technique to improve meta-heuristic algorithms [51, 52]. The ideology of ROBL is to evaluate the fitness values of the current solution and its opposite solution simultaneously, and then choose the better one to proceed with the next iteration. In the basic HBA, according to the position update rules, the new candidate position of honey badger is obtained by guiding the current individual to the best position of the swarm (i.e., X_prey). In the later stage of search, all the remaining honey badgers in the population prefer to cluster around the global optimum, thus leading to a decrease in the population diversity and even premature convergence. Hence, ROBL is introduced in this paper to help the global optimum move to a potential superior area and minimize the likelihood of falling into the local optima. Figure 2 illustrates the process of ROBL for the global optimum (X_prey) on one-dimensional space.

Fig. 2

ROBL for the global optimum on one-dimensional space.

In Fig. 2, the cardinal point O denotes the midpoint of the x-axis search interval [a, b], the y-axis is the normal, X and X^* are called the incidence point and refraction point, respectively, φ₁ and φ₂ are the angle of incidence and refraction, respectively, and the length of incidence light OX and refraction light OX^* are h and h′, respectively. Let the projection of incidence point X on the coordinate axis be X_prey, and its opposite position based on ROBL is defined as $X_{prey}^{*}$ . The geometric relationships in this figure can be expressed as follows: $sin φ_{1} = ((a + b) / 2 - X_{prey}) / h$ (16) $sin φ_{2} = (X_{prey}^{*} - (a + b) / 2) / h^{'}$ (17)

According to Equations (18), the formula for the refraction index η is given as follows: $η = \frac{sin φ_{1}}{sin φ_{2}} = \frac{h^{'} ((a + b) / 2 - X_{prey})}{h (X_{prey}^{*} - (a + b) / 2)}$ (18)

Here, let k = h′/h, the opposite solution $X_{prey}^{*}$ can be calculated by modifying Equation (19): $X_{prey}^{*} = \frac{a + b}{2} + \frac{a + b}{2 k η} - \frac{X_{prey}}{k η}$ (19)

Generally, Equation (20) could also be extended to handle D-dimensional decision variables as follows: $X_{prey}^{j *} = \frac{a_{j} + b_{j}}{2} + \frac{a_{j} + b_{j}}{2 k η} - \frac{X_{prey}^{j}}{k η}$ (20) where a_j and b_j are the upper and lower limits of the j^th dimension (j = 1, 2, ⋯ , D), respectively, $X_{prey}^{j *}$ denotes the inverse solution of $X_{prey}^{j}$ in the j^th dimension.

3.4 Algorithm flowchart

Based on the enhanced strategies stated earlier in subsections 3.1 to 3.3, the flowchart and pseudo-code of the proposed EHBA algorithm in this paper are depicted in Fig. 3 and Algorithm 2, respectively.

Fig. 3

Flow chart of the proposed EHBA algorithm.

Algorithm 2

Pseudo-code of EHBA
1. Initialize the population size N and the maximum number of iterations t ⩽ Maxiter
2. Initialize the random positions of all honey badgers X_i (i = 1, 2, …, N)
3. Generate the mutant positions of honey badgers using Equation (9)
4. Calculate the fitness values of each honey badger individual X_i and assign to f_i (i = 1, 2, …, N)
5. Save the best position as X_prey and assign fitness to f_prey
6. Whilet ⩽ Maxiterdo
7. Calculate the density coefficient α using Equation (5)
8. For each honey badger do
9. Calculate the smell intensity I_i using Equation (2)
10. Ifrand < 0.5 thenrand a random number in the interval [0, 1]
11. Update the honey badger position X_new using Equation (15)
12. E lse
13. Update the honey badger position X_new using Equation (16)
14. End If
15. Calculate the fitness of the new position X_new and assign to f_new
16. Iff_new ⩽ f_ithen
17. Let X_I = X_new and f_i = f_new
18. End If
19. Iff_new ⩽ f_preythen
20. Let X_prey = X_new and f_prey = f_new
21. End If
22. Perform ROBL to evaluate the opposite position of X_prey using Equation (21) and select the better one as the next optimal position
23. End For
24. t = t + 1
25. End While
26. Return X_prey

4 Experimental results and discussions

In this section, the effectiveness and feasibility of the proposed EHBA are comprehensively validated on two sets of optimization problems. The first group consists of 18 standard benchmark functions selected from the literature [43], which are used to evaluate the performance of the algorithm in solving simple numerical optimization problems. The second group consists of 29 CEC2017 functions, and they are used to evaluate the performance of the algorithm with regard to addressing complicated numerical optimization problems. All the experiments are implemented in MATLAB R2017a software (version 9.2.0) with Microsoft Windows 10 system, and the hardware platform of the computer is configured as Intel (R) Core (TM) i5-6300HQ CPU @ 2.30GHz and 16GB RAM.

4.1 Experiment on standard benchmark functions

In this subsection, a total of 18 standard benchmark functions are employed for comparative experiments to investigate the performance of the proposed algorithm. First, the definitions of these functions and parameter settings are described. Then, EHBA, the basic HBA and some other state-of-the-art meta-heuristic algorithms are tested on these functions concurrently. The obtained quantitative results, boxplot, convergence behavior, and average computation time are analyzed. Final, we assess the scalability of EHBA.

4.1.1 Benchmark functions

The 18 benchmark functions can be divided into three types: unimodal (UM), multimodal (MM), and fix-dimension multimodal (FM). The unimodal benchmark functions (F₁-F₆) have only one global optimal solution, which are frequently used to detect the exploitation competence and convergence rate of the algorithm. The second category, namely the multimodal benchmark functions (F₇-F₁₁), on the other hand, consist of multiple local minima and one global optimum. This type of functions can be employed to examine the algorithm’s capability to explore and escape from local optima. Moreover, the fix-dimension multimodal benchmark functions (F₁₂-F₁₈) are a combination of the previous two types of functions but with lower dimensions, and they are designed to understand the stability of the algorithm between exploration and exploitation. The expression, name, spatial dimension (Dim), search range, and theoretical minimum of these functions are outlined in Table 1. Figure 4 visualizes the search space of 18 benchmark functions.

Table 1
Descriptions of 18 benchmark functions

Function Name Dim Range F _min Type

$F_{1} (x) = \sum_{i = 1}^{D} x_{i}^{2}$ Sphere 30 [–100,100] 0 UM

$F_{2} (x) = \sum_{i = 1}^{D} | x_{i} | + \prod_{i = 1}^{D} | x_{i} |$ Schwefel 2.22 30 [–10,10] 0 UM

$F_{3} (x) = \sum_{i = 1}^{D} {(\sum_{j = 1}^{D} x_{j})}^{2}$ Schwefel 1.2 30 [–100,100] 0 UM

$F_{4} (x) = \max_{i} {| x_{i} |, 1 ⩽ i ⩽ D}$ Schwefel 2.21 30 [–100,100] 0 UM

$F_{5} (x) = \sum_{i = 1}^{D} {(| x_{i} + 0.5 |)}^{2}$ Step 30 [–100,100] 0 UM

$F_{6} (x) = \sum_{i = 1}^{D} {ix}_{i}^{4} + random [0, 1]$ Quartic 30 [–1.28,1.28] 0 UM

$F_{7} (x) = \sum_{i = 1}^{D} [x_{i}^{2} - 10 cos (2 π x_{i}) + 10]$ Rastrigin 30 [–5.12,5.12] 0 MM

$F_{8} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{D} \sum_{i = 1}^{D} x_{i}^{2}}) - exp (\frac{1}{D} \sum_{i = 1}^{D} cos (2 π x_{i})) + 20 + e$ Ackley 30 [–32,32] 0 MM

$F_{9} (x) = \frac{1}{4000} \sum_{i = 1}^{D} x_{i}^{2} - \prod_{i = 1}^{D} cos (\frac{x_{i}}{\sqrt{i}}) + 1$ Griewank 30 [–600,600] 0 MM

$F_{10} (x) = \frac{π}{D} {10 sin (π y_{1}) + \sum_{i = 1}^{D - 1} {(y_{i} - 1)}^{2} [1 + 10 {sin}^{2} (π y_{i + 1})] + {(y_{D} - 1)}^{2}} + \sum_{i = 1}^{D} u (x_{i}, 10, 100, 4)$ $y_{i} = 1 + \frac{x_{i} + 1}{4}, u (x_{i}, a, k, m) = {\begin{matrix} k {(x_{i} - a)}^{m} x_{i} > a \\ 0 - a < x_{i} < a \\ k {(- x_{i} - a)}^{m} x_{i} < - a \end{matrix}$ Penalized 30 [–50,50] 0 MM

$F_{11} (x) = 0.1 {\sin^{2} (3 π x_{i}) + \sum_{i = 1}^{D} {(x_{i} - 1)}^{2} [1 + \sin^{2} (3 π x_{i} + 1)] + {(x_{D} - 1)}^{2} [1 + \sin^{2} (2 π x_{n})]} + \sum_{i = 1}^{D} u (x_{i}, 5, 100, 4)$ Penalized 2 30 [–50,50] 0 MM

$F_{12} (x) = 4 x_{1}^{2} - 2.1 x_{1}^{4} + \frac{1}{3} x_{1}^{6} + x_{1} x_{2} - 4 x_{2}^{2} + 4 x_{2}^{4}$ Six-hump Camel Back 2 [–5,5] –1.0316 FM

$F_{13} (x) = {(x_{2} - \frac{5.1}{4 π^{2}} x_{1}^{2} + \frac{5}{π} x_{1} - 6)}^{2} + 10 (1 - \frac{1}{8 π}) \cos x_{1} + 10$ Branin 2 [–5,5] 0.398 FM

$F_{14} (x) = [1 + {(x_{1} + x_{2} + 1)}^{2} (19 - 14 x_{1} + 3 x_{1}^{2} - 14 x_{2} + 6 x_{1} x_{2} + 3 x_{2}^{2})] \times [30 + 30 + {(2 x_{1} - 3 x_{2})}^{2} \times (18 - 32 x_{1} + 12 x_{1}^{2} + 48 x_{2} - 36 x_{1} x_{2} + 27 x_{2}^{2})]$ Goldstein Price 2 [–2,2] 3 FM

$F_{15} (x) = - \sum_{i = 1}^{4} c_{i} exp (- \sum_{j = 1}^{3} a_{ij} (x_{j} - p_{ij})^{2})$ Hartman 3 3 [–1,2] –3.8628 FM

$F_{16} (x) = - \sum_{i = 1}^{4} c_{i} exp (- \sum_{j = 1}^{6} a_{ij} (x_{j} - p_{ij})^{2})$ Hartman 6 6 [0,1] –3.32 FM

$F_{17} (x) = - \sum_{i = 1}^{5} [(X - a_{i}) (X - a_{i})^{T} + c_{i}]^{- 1}$ Langermann 5 4 [0,10] –10.1532 FM

$F_{18} (x) = - \sum_{i = 1}^{7} [(X - a_{i}) (X - a_{i})^{T} + c_{i}]^{- 1}$ Langermann 7 4 [0,10] –10.4028 FM

Function	Name	Dim	Range	F _min	Type
$F_{1} (x) = \sum_{i = 1}^{D} x_{i}^{2}$	Sphere	30	[–100,100]	0	UM
$F_{2} (x) = \sum_{i = 1}^{D} \| x_{i} \| + \prod_{i = 1}^{D} \| x_{i} \|$	Schwefel 2.22	30	[–10,10]	0	UM
$F_{3} (x) = \sum_{i = 1}^{D} {(\sum_{j = 1}^{D} x_{j})}^{2}$	Schwefel 1.2	30	[–100,100]	0	UM
$F_{4} (x) = \max_{i} {\| x_{i} \|, 1 ⩽ i ⩽ D}$	Schwefel 2.21	30	[–100,100]	0	UM
$F_{5} (x) = \sum_{i = 1}^{D} {(\| x_{i} + 0.5 \|)}^{2}$	Step	30	[–100,100]	0	UM
$F_{6} (x) = \sum_{i = 1}^{D} {ix}_{i}^{4} + random [0, 1]$	Quartic	30	[–1.28,1.28]	0	UM
$F_{7} (x) = \sum_{i = 1}^{D} [x_{i}^{2} - 10 cos (2 π x_{i}) + 10]$	Rastrigin	30	[–5.12,5.12]	0	MM
$F_{8} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{D} \sum_{i = 1}^{D} x_{i}^{2}}) - exp (\frac{1}{D} \sum_{i = 1}^{D} cos (2 π x_{i})) + 20 + e$	Ackley	30	[–32,32]	0	MM
$F_{9} (x) = \frac{1}{4000} \sum_{i = 1}^{D} x_{i}^{2} - \prod_{i = 1}^{D} cos (\frac{x_{i}}{\sqrt{i}}) + 1$	Griewank	30	[–600,600]	0	MM
$F_{10} (x) = \frac{π}{D} {10 sin (π y_{1}) + \sum_{i = 1}^{D - 1} {(y_{i} - 1)}^{2} [1 + 10 {sin}^{2} (π y_{i + 1})] + {(y_{D} - 1)}^{2}} + \sum_{i = 1}^{D} u (x_{i}, 10, 100, 4)$ $y_{i} = 1 + \frac{x_{i} + 1}{4}, u (x_{i}, a, k, m) = {\begin{matrix} k {(x_{i} - a)}^{m} x_{i} > a \\ 0 - a < x_{i} < a \\ k {(- x_{i} - a)}^{m} x_{i} < - a \end{matrix}$	Penalized	30	[–50,50]	0	MM
$F_{11} (x) = 0.1 {\sin^{2} (3 π x_{i}) + \sum_{i = 1}^{D} {(x_{i} - 1)}^{2} [1 + \sin^{2} (3 π x_{i} + 1)] + {(x_{D} - 1)}^{2} [1 + \sin^{2} (2 π x_{n})]} + \sum_{i = 1}^{D} u (x_{i}, 5, 100, 4)$	Penalized 2	30	[–50,50]	0	MM
$F_{12} (x) = 4 x_{1}^{2} - 2.1 x_{1}^{4} + \frac{1}{3} x_{1}^{6} + x_{1} x_{2} - 4 x_{2}^{2} + 4 x_{2}^{4}$	Six-hump Camel Back	2	[–5,5]	–1.0316	FM
$F_{13} (x) = {(x_{2} - \frac{5.1}{4 π^{2}} x_{1}^{2} + \frac{5}{π} x_{1} - 6)}^{2} + 10 (1 - \frac{1}{8 π}) \cos x_{1} + 10$	Branin	2	[–5,5]	0.398	FM
$F_{14} (x) = [1 + {(x_{1} + x_{2} + 1)}^{2} (19 - 14 x_{1} + 3 x_{1}^{2} - 14 x_{2} + 6 x_{1} x_{2} + 3 x_{2}^{2})] \times [30 + 30 + {(2 x_{1} - 3 x_{2})}^{2} \times (18 - 32 x_{1} + 12 x_{1}^{2} + 48 x_{2} - 36 x_{1} x_{2} + 27 x_{2}^{2})]$	Goldstein Price	2	[–2,2]	3	FM
$F_{15} (x) = - \sum_{i = 1}^{4} c_{i} exp (- \sum_{j = 1}^{3} a_{ij} (x_{j} - p_{ij})^{2})$	Hartman 3	3	[–1,2]	–3.8628	FM
$F_{16} (x) = - \sum_{i = 1}^{4} c_{i} exp (- \sum_{j = 1}^{6} a_{ij} (x_{j} - p_{ij})^{2})$	Hartman 6	6	[0,1]	–3.32	FM
$F_{17} (x) = - \sum_{i = 1}^{5} [(X - a_{i}) (X - a_{i})^{T} + c_{i}]^{- 1}$	Langermann 5	4	[0,10]	–10.1532	FM
$F_{18} (x) = - \sum_{i = 1}^{7} [(X - a_{i}) (X - a_{i})^{T} + c_{i}]^{- 1}$	Langermann 7	4	[0,10]	–10.4028	FM

Fig. 4

3D view of the search space for 18 benchmark functions.

4.1.2 Parameter settings

In order to demonstrate improvements and superiority of the proposed algorithm, the experimental results of EHBA are compared with those of the basic HBA [43], ALO [22], HHO [25], WOA [24], MVO [16], MFO [27], TACPSO [36], and IGWO [37]. For a fair comparison, the maximum iteration and population size of all mentioned algorithms are set as 500 and 30, respectively. As per the literatures [51, 52], in the proposed EHBA, we set the refraction index η = 100 and k = 1000. Besides, the other specific parameters of comparison algorithms are set as recommended in the original literatures, as shown in Table 2. These parameter settings guarantee the fairness of the experiment and allow each algorithm to take advantage of its optimization characteristic. To decrease random errors, all algorithms run independently 30 times on each benchmark function.

Table 2
Parameter settings for EHBA and other comparison algorithms

Reference Algorithm Parameter setting

[22] ALO k = 500

[25] HHO E₀ ∈ [- 1, 1] ; E₁ = [2, 0]

[24] WOA a₁ = [2, 0] ; a₁ = [- 1, - 2] ; b = 1

[16] MVO existenceprobability ∈ [0.2, 1] ; travellingdistancerate ∈ [0.6, 1]

[27] MFO b = 1 ; t = [- 1, 1] ; a ∈ [- 1, - 2]

[43] HBA γ = 6 ; C = 2

[36] TACPSO ω_max = 0.9 ; ω_min = 0.4 ; C_max = 2.5 ; C_min = 0.5

[37] IGWO a = [2, 0]

— EHBA γ = 6 ; C = 2 ; η = 100 ; k = 1000

Reference	Algorithm	Parameter setting
[22]	ALO	k = 500
[25]	HHO	E₀ ∈ [- 1, 1] ; E₁ = [2, 0]
[24]	WOA	a₁ = [2, 0] ; a₁ = [- 1, - 2] ; b = 1
[16]	MVO	existenceprobability ∈ [0.2, 1] ; travellingdistancerate ∈ [0.6, 1]
[27]	MFO	b = 1 ; t = [- 1, 1] ; a ∈ [- 1, - 2]
[43]	HBA	γ = 6 ; C = 2
[36]	TACPSO	ω_max = 0.9 ; ω_min = 0.4 ; C_max = 2.5 ; C_min = 0.5
[37]	IGWO	a = [2, 0]
—	EHBA	γ = 6 ; C = 2 ; η = 100 ; k = 1000

4.1.3 Comparison of EHBA with other meta-heuristic algorithms

In this subsection, we test EHBA, HBA, and seven meta-heuristic algorithms on 18 different types of benchmark functions. To analyze the experimental results more clearly and intuitively, we adopt the average fitness (Avg) and standard deviation (Std) as two measures. The average fitness represents the convergence accuracy of the algorithm, which is calculate as follows: $Avg = \frac{1}{n} \sum_{i = 1}^{n} R_{i}$ (21) where n denotes the number of times that an algorithm has executed, R_i is the result of each optimization. The closer the average value to the theoretical minimum, the higher convergence accuracy of the algorithm. While the standard deviation characterizes the deviation degree between the experimental results and the average values. The formula for the standard deviation is shown as follows: $Std = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(R_{i} - Avg)}^{2}}$ (22)

In the experiment, the smaller the value of the standard deviation, the better robustness of the algorithm. After 30 independent runs, the average fitness and standard deviation results obtained by the nine optimizers are reported in Table 3. It should be noted that in all tables below, the best values have been highlighted in bold. From the data in this table, it is apparent that the proposed EHBA provides better performance on most of the test functions.

Table 3

Comparison results of EHBA and other algorithms on 18 benchmark functions

Function	Criteria	ALO	HHO	WOA	MVO	MFO	HBA	TACPSO	IGWO	EHBA
F ₁	Avg	1.40E-03	3.62E-93	1.15E-70	1.21E+00	2.34E+03	8.40E-136	2.64E-01	1.40E-28	0.00E+00
	Std	1.29E-03	1.98E-92	5.94E-70	3.17E-01	5.04E+03	2.19E-135	7.74E-01	2.00E-28	0.00E+00
F ₂	Avg	5.35E+01	3.72E-49	1.29E-50	1.45E+00	3.02E+01	1.46E-72	1.10E+00	7.22E-18	0.00E+00
	Std	5.14E+01	2.00E-48	5.52E-50	2.44E+00	2.03E+01	3.06E-72	2.52E+00	8.85E-18	0.00E+00
F ₃	Avg	4.15E+03	5.08E-77	4.79E+04	2.30E+02	2.06E+04	7.35E-94	8.32E+02	1.04E-03	0.00E+00
	Std	2.05E+03	2.42E-76	1.31E+04	1.00E+02	1.24E+04	3.88E-93	1.00E+03	1.63E-03	0.00E+00
F ₄	Avg	1.64E+01	2.64E-48	4.94E+01	2.12E+00	6.80E+01	1.22E-57	1.03E+01	1.42E-05	0.00E+00
	Std	4.46E+00	1.38E-47	2.83E+01	9.44E-01	7.61E+00	2.13E-57	3.62E+00	1.02E-05	0.00E+00
F ₅	Avg	1.28E-03	1.47E-03	4.48E-01	1.24E+00	2.01E+03	1.70E-02	5.83E-01	3.22E-02	3.78E-05
	Std	1.24E-03	2.59E-03	2.52E-01	4.50E-01	4.08E+03	6.30E-02	2.09E+00	8.34E-02	4.38E-05
F ₆	Avg	2.34E-01	1.06E-04	3.61E-03	3.53E-02	3.69E+00	4.13E-03	1.72E-01	2.45E-03	7.89E-05
	Std	8.38E-02	1.11E-04	4.53E-03	1.01E-02	6.86E+00	3.85E-03	5.03E-01	1.15E-03	6.28E-05
F ₇	Avg	7.69E+01	0.00E+00	3.79E-15	1.25E+02	1.55E+02	0.00E+00	7.11E+01	1.86E+01	0.00E+00
	Std	2.30E+01	0.00E+00	2.08E-14	2.63E+01	3.84E+01	0.00E+00	2.24E+01	6.18E+00	0.00E+00
F ₈	Avg	4.78E+00	8.88E-16	4.68E-15	2.32E+00	1.52E+01	8.88E-16	2.23E+00	6.08E-14	8.88E-16
	Std	2.94E+00	0.00E+00	2.27E-15	3.30E+00	6.91E+00	0.00E+00	8.17E-01	8.89E-15	0.00E+00
F ₉	Avg	5.74E-02	0.00E+00	1.60E-02	8.69E-01	2.79E+01	0.00E+00	1.35E-01	4.88E-03	0.00E+00
	Std	2.62E-02	0.00E+00	4.99E-02	7.54E-02	4.19E+01	0.00E+00	1.35E-01	6.94E-03	0.00E+00
F ₁₀	Avg	1.24E+01	5.91E-04	2.48E-02	1.65E+00	8.53E+06	8.67E-02	1.92E+00	2.16E-04	2.74E-06
	Std	4.28E+00	7.46E-04	1.59E-02	1.02E+00	4.67E+07	6.02E-02	1.18E+00	1.14E-03	9.02E-07
F ₁₁	Avg	2.89E+01	5.25E-03	4.88E-01	1.73E-01	1.37E+07	5.19E-01	5.02E+00	1.02E-01	3.19E-04
	Std	1.68E+01	3.46E-03	2.27E-01	8.28E-02	7.49E+07	3.11E-01	4.87E+00	1.18E-01	2.67E-04
F ₁₂	Avg	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316	-1.0316
	Std	1.09E-13	2.47E-09	5.31E-09	4.42E-07	6.78E-16	6.05E-12	7.25E-10	8.32E-16	4.65E-16
F ₁₃	Avg	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01
	Std	1.10E-13	8.46E-06	7.65E-06	6.20E-07	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
F ₁₄	Avg	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00
	Std	9.30E-13	2.84E-06	2.32E-04	3.18E-06	1.69E-15	1.48E-07	2.49E-15	6.58E-15	1.73E-15
F ₁₅	Avg	-3.8625	-3.8604	-3.8467	-3.8626	-3.8625	-3.8619	-3.8627	-3.8627	-3.8628
	Std	4.62E-13	3.53E-03	2.69E-02	3.04E-06	2.71E-15	2.40E-03	2.60E-15	1.16E-12	2.46E-15
F ₁₆	Avg	-3.2620	-3.1010	-3.2449	-3.2614	-3.2068	-3.2719	-3.2564	-3.2598	-3.3141
	Std	6.10E-02	1.06E-01	1.19E-01	6.17E-02	5.91E-02	8.21E-02	6.35E-02	3.02E-02	1.03E-02
F ₁₇	Avg	-6.7869	-5.2137	-8.3461	-6.7911	-6.3044	-9.9024	-6.3950	-10.0198	-10.1532
	Std	2.90E+00	8.91E-01	2.58E+00	2.90E+00	3.33E+00	1.37E+00	3.46E+00	7.31E-01	5.41E-15
F ₁₈	Avg	-5.6831	-5.4194	-7.1176	-8.8359	-7.6630	-8.5261	-7.3275	-10.1803	-10.4029
	Std	3.01E+00	1.28E+00	3.02E+00	2.70E+00	3.45E+00	3.17E+00	3.64E+00	1.22E+00	1.85E-03

Specifically, when solving the unimodal benchmark functions F₁-F₆, only EHBA is able to accurately converge to the theoretical global optimum (0) on F₁-F₄. And the solution accuracy of EHBA has a dramatic improvement over the basic HBA. For F₅ and F₆, although EHBA cannot obtain the optimal value, it still marginally outperforms all other methods and shows superior results. In terms of the standard deviation, EHBA also performs best on all test functions. In light of the characteristics of the unimodal functions, it hopefully proves that the proposed EHBA has competitive local exploitation potential, which mainly owes to Lévy flight.

When solving the multimodal benchmark functions F₇-F₁₁, the average fitness and standard deviation of EHBA are obviously better than the remaining eight algorithms on F₁₀ and F₁₁. For F₇-F₉, EHBA shares the same performance as HHO and HBA. Meanwhile, it is notable that the proposed algorithm can stably achieve the global optimum on F₇ and F₉. The multimodal functions contain many local optima in the search space. Hence, these results prove EHBA can effectively get rid of local optima to find high-quality solutions, which is attributed to its excellent exploration trend enhanced by HDPM and ROBL strategies.

The fix-dimension multimodal benchmark functions F₁₂-F₁₈ are designed to evaluate the algorithm’s stability in the transition between exploration and exploitation stages. As we can see, the proposed EHBA obtains the best results on F₁₅-F₁₈. For F₁₂ and F₁₄, all algorithms provide the consistent average fitness, but the standard deviation of EHBA is the smallest among them. This implies that EHBA has better robustness. For F₁₃, EHBA performs equally to MFO, HBA, TACPSO, and IGWO, but much better than the others. These results show that the proposed algorithm in this paper can well balance the behaviors of exploration and exploitation in the search space.

In order to better understand the distribution properties of the enhanced algorithm, the corresponding boxplots of 12 typical benchmark functions are drawn in Fig. 5. It can be clearly seen that the proposed EHBA exhibits good consistency on these functions and generates no outliers during the iteration. The obtained median, maximum and minimum values of EHBA are more concentrated than the comparison algorithms. The above validates the strong stability of EHBA.

Fig. 5

Boxplot analysis of EHBA and other algorithms on 12 benchmark functions.

To illustrate the convergence performance of EHBA, Fig. 6 records the convergence curves of each algorithm in solving 12 selected functions. From Fig. 6, we notice that the proposed EHBA begins to converge from the initial stage of iterations on F₁-F₄, and its convergence curves have a greater decay rate than all other comparison algorithms. For F₆, EHBA has a similar tendency to HHO and HBA, but it eventually converges with higher accuracy. For functions F₇-F₉, EHBA can efficiently reach the global optimum with the least number of iterations. For F₁₀, EHBA performs slightly worse than HHO in the initial stage of iterations, however, it shows superiority in the latter stage and obtains a better result. The convergence curves of nine algorithms are quite close on F₁₂ and F₁₄, but EHBA still has a bit advantage in convergence speed. For F₁₈, EHBA also presents a faster convergence speed and higher solution accuracy. These analyses reveal that the proposed EHBA has better convergence behavior with the help of Lévy flight and ROBL.

Fig. 6

Convergence curves of EHBA and other algorithms on 12 benchmark functions.

Table 4 reports the average computation time spent on F₁-F₁₈. The total runtime of each algorithm is calculated and sorted as follows: ALO (123.18s)>IGWO (12.44s)>HHO (5.46s)>EHBA (4.70s)>MVO (4.03s)>HBA (3.68s)>MFO (2.37s)>WOA (2.19s)>TACPSO (2.16s). As it presents, the process consumption of TACPSO is the least, whereas EHBA requires more runtime than HBA, which ranks fourth to last. The main reason for this is that three enhancement strategies increase the steps of HBA and extra time. On the whole, our proposed algorithm is acceptable in view of its remarkable search performance.

Table 4

Comparison results of the average computation time for different algorithms (unit: s)

Function	ALO	HHO	WOA	MVO	MFO	HBA	TACPSO	IGWO	EHBA
F ₁	9.98E+00	1.61E-01	8.93E-02	2.27E-01	9.38E-02	1.83E-01	7.44E-02	6.76E-01	2.12E-01
F ₂	1.01E+01	1.53E-01	9.64E-02	1.97E-01	9.76E-02	1.68E-01	8.01E-02	6.27E-01	2.26E-01
F ₃	1.04E+01	7.79E-01	3.42E-01	4.47E-01	3.51E-01	4.30E-01	3.34E-01	1.21E+00	5.31E-01
F ₄	9.81E+00	1.68E-01	7.62E-02	2.03E-01	9.59E-02	1.76E-01	7.30E-02	6.09E-01	2.17E-01
F ₅	9.94E+00	2.07E-01	9.22E-02	2.05E-01	9.65E-02	1.59E-01	9.93E-02	6.02E-01	2.11E-01
F ₆	1.03E+01	2.84E-01	1.34E-01	2.42E-01	1.42E-01	2.22E-01	1.24E-01	7.19E-01	2.69E-01
F ₇	1.05E+01	2.16E-01	7.46E-02	2.08E-01	9.63E-02	1.65E-01	8.13E-02	6.39E-01	2.13E-01
F ₈	1.01E+01	2.33E-01	8.27E-02	2.15E-01	1.07E-01	1.71E-01	1.06E-01	6.36E-01	2.24E-01
F ₉	1.01E+01	2.59E-01	1.03E-01	2.23E-01	1.17E-01	1.88E-01	9.81E-02	6.61E-01	2.41E-01
F ₁₀	1.03E+01	5.85E-01	2.31E-01	3.69E-01	2.62E-01	3.36E-01	2.45E-01	9.62E-01	4.00E-01
F ₁₁	1.15E+01	6.94E-01	2.61E-01	4.44E-01	2.95E-01	3.68E-01	2.68E-01	1.08E+00	4.50E-01
F ₁₂	9.01E-01	1.82E-01	4.76E-02	1.21E-01	5.41E-02	1.20E-01	4.89E-02	4.88E-01	1.73E-01
F ₁₃	8.40E-01	1.38E-01	4.29E-02	1.17E-01	4.68E-02	1.24E-01	4.28E-02	5.02E-01	1.75E-01
F ₁₄	8.67E-01	1.44E-01	4.14E-02	1.13E-01	4.68E-02	1.14E-01	4.56E-02	4.92E-01	1.84E-01
F ₁₅	1.27E+00	2.33E-01	7.82E-02	1.41E-01	8.33E-02	1.66E-01	7.91E-02	5.80E-01	2.13E-01
F ₁₆	2.28E+00	2.48E-01	8.16E-02	1.42E-01	8.67E-02	1.56E-01	8.04E-02	5.86E-01	2.08E-01
F ₁₇	1.62E+00	3.42E-01	1.27E-01	1.88E-01	1.31E-01	1.97E-01	1.23E-01	6.48E-01	2.55E-01
F ₁₈	1.76E+00	4.36E-01	1.86E-01	2.30E-01	1.62E-01	2.34E-01	1.57E-01	7.34E-01	2.97E-01

Furthermore, the Wilcoxon rank-sum test [53], a nonparametric statistical method is utilized to examine whether there are significant differences between EHBA and other comparison algorithms in a statistical sense. For Wilcoxon rank-sum test, the significance level is fixed to 0.05 and the obtained p-values are listed in Table 5. In this table, the sign “+/=/-” denotes that EHBA performs better, identically to, or poorer than the comparison algorithm respectively, and the last line counts the total number of all signs. It can be observed from Table 5 that the proposed EHBA outperforms ALO on 18 functions, HHO on 15 functions, WOA on 17 functions, MVO on 18 functions, MFO on 17 functions, HBA on 14 functions, TACPSO on 15 functions, and IGWO on 15 functions, respectively. Therefore, on basis of the statistical theory, EHBA is significantly different from the other eight algorithms, and it is considered as the best optimizer among them.

Table 5

Comparison results of Wilcoxon rank-sum test between EHBA and other algorithms on 18 benchmark functions

Function	EHBA vs ALO		EHBA vs HHO		EHBA vs WOA		EHBA vs MVO		EHBA vs MFO		EHBA vs HBA		EHBA vs TACPSO		EHBA vs IGWO
	p-value	win	p-value	win	p-value	win	p-value	win	p-value	win	p-value	win	p-value	win	p-value	win
F ₁	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+
F ₂	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+
F ₃	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+
F ₄	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+	1.21E-12	+
F ₅	1.61E-06	+	3.02E-11	+	6.70E-11	+	3.02E-11	+	3.02E-11	+	5.49E-08	+	1.41E-09	+	1.17E-02	+
F ₆	1.83E-08	+	1.83E-08	+	3.85E-06	+	1.83E-08	+	1.83E-08	+	1.83E-08	+	1.83E-08	+	5.57E-05	+
F ₇	1.21E-12	+	NaN	=	3.34E-05	+	1.21E-12	+	1.21E-12	+	NaN	=	1.21E-12	+	1.21E-12	+
F ₈	1.21E-12	+	NaN	=	3.04E-10	+	1.21E-12	+	1.21E-12	+	NaN	=	1.21E-12	+	8.09E-13	+
F ₉	1.21E-12	+	NaN	=	8.15E-02	-	1.21E-12	+	1.21E-12	+	NaN	=	1.21E-12	+	6.61E-05	+
F ₁₀	3.02E-11	+	2.90E-04	+	3.02E-11	+	3.02E-11	+	3.02E-11	+	3.02E-11	+	3.02E-11	+	7.24E-02	-
F ₁₁	7.09E-08	+	3.02E-11	+	9.26E-09	+	6.70E-11	+	3.02E-11	+	1.55E-09	+	8.88E-06	+	4.08E-11	+
F ₁₂	8.87E-12	+	8.87E-12	+	8.87E-12	+	8.87E-12	+	8.87E-12	+	8.87E-12	+	2.85E-10	+	4.14E-06	+
F ₁₃	1.20E-12	+	4.57E-12	+	1.21E-12	+	1.21E-12	+	NaN	=	NaN	=	NaN	=	NaN	=
F ₁₄	2.64E-11	+	2.64E-11	+	2.64E-11	+	2.64E-11	+	2.64E-11	+	2.64E-11	+	2.16E-01	-	2.64E-11	+
F ₁₅	1.41E-11	+	1.41E-11	+	1.41E-11	+	1.41E-11	+	1.41E-11	+	4.20E-08	+	9.33E-03	+	1.69E-01	-
F ₁₆	1.95E-07	+	5.48E-11	+	1.17E-09	+	1.28E-09	+	4.77E-05	+	6.13E-03	+	1.00E+00	-	4.58E-04	+
F ₁₇	1.41E-11	+	1.41E-11	+	1.41E-11	+	1.41E-11	+	4.51E-05	+	1.57E-11	+	2.26E-08	+	9.93E-05	+
F ₁₈	7.27E-11	+	2.89E-10	+	1.77E-10	+	2.38E-10	+	8.26E-09	+	4.32E-08	+	1.19E-10	+	2.89E-10	+
+/=/-	18/0/0		15/3/0		17/0/1		18/0/0		17/1/0		14/4/0		15/1/2		15/1/2

4.1.4 Scalability analysis

Scalability reflects the execution efficiency of an algorithm in different dimensions. As the dimensional size of the optimization problem increases, many algorithms are highly prone to lose their effectiveness. To assess the scalability’s impact on the performance of EHBA, the proposed algorithm is used to solve 11 benchmark functions F₁-F₁₁ in Table 1 with higher dimensions. We expand the dimension from 30 to 50, 100 and 500, respectively. The average fitness results of the basic HBA and EHBA are shown in Table 6.

Table 6
Average fitness values of HBA and EHBA in different dimensions on 11 benchmark functions

Function 50 100 500

HBA EHBA HBA EHBA HBA EHBA

F ₁ 1.73E-128 0.00E+00 7.28E-121 0.00E+00 3.78E-113 0.00E+00

F ₂ 2.60E-68 0.00E+00 1.38E-65 0.00E+00 2.20E-60 0.00E+00

F ₃ 1.55E-86 0.00E+00 2.09E-77 0.00E+00 2.77E-64 0.00E+00

F ₄ 5.78E-50 0.00E+00 6.89E-40 0.00E+00 3.13E-28 0.00E+00

F ₅ 1.52E+00 6.01E-03 1.02E+01 2.71E+00 9.84E+01 1.84E+01

F ₆ 2.49E-01 3.40E-04 2.56E-01 4.64E-04 3.14E-01 5.81E-04

F ₇ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

F ₈ 1.99E+00 8.88E-16 2.13E+00 8.88E-16 2.66E+00 8.88E-16

F ₉ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

F ₁₀ 4.78E-02 1.04E-03 2.41E-01 8.80E-02 1.48E+00 6.55E-01

F ₁₁ 2.50E+00 2.95E-02 8.27E+00 8.12E-01 1.95E+01 3.81E+00

Function	50	100	500
F ₁	1.73E-128	0.00E+00	7.28E-121	0.00E+00	3.78E-113	0.00E+00
F ₂	2.60E-68	0.00E+00	1.38E-65	0.00E+00	2.20E-60	0.00E+00
F ₃	1.55E-86	0.00E+00	2.09E-77	0.00E+00	2.77E-64	0.00E+00
F ₄	5.78E-50	0.00E+00	6.89E-40	0.00E+00	3.13E-28	0.00E+00
F ₅	1.52E+00	6.01E-03	1.02E+01	2.71E+00	9.84E+01	1.84E+01
F ₆	2.49E-01	3.40E-04	2.56E-01	4.64E-04	3.14E-01	5.81E-04
F ₇	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
F ₈	1.99E+00	8.88E-16	2.13E+00	8.88E-16	2.66E+00	8.88E-16
F ₉	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
F ₁₀	4.78E-02	1.04E-03	2.41E-01	8.80E-02	1.48E+00	6.55E-01
F ₁₁	2.50E+00	2.95E-02	8.27E+00	8.12E-01	1.95E+01	3.81E+00

Table 7

Descriptions of IEEE CEC2017 test suite

Function	Name	Dim	Range	F _min
Unimodal functions
CEC-1	Shifted and Rotated Bent Cigar Function	10	[-100,100]	100
CEC-3	Shifted and Rotated Zakharov Function	10	[-100,100]	300
Multimodal Functions
CEC-4	Shifted and Rotated Rosenbrock’s Function	10	[-100,100]	400
CEC-5	Shifted and Rotated Rastrigin’s Function	10	[-100,100]	500
CEC-6	Shifted and Rotated Expanded Scaffer’s F6 Function	10	[-100,100]	600
CEC-7	Shifted and Rotated Lunacek Bi_Rastrigin Function	10	[-100,100]	700
CEC-8	Shifted and Rotated Non-Continuous Rastrigin’s Function	10	[-100,100]	800
CEC-9	Shifted and Rotated Levy Function	10	[-100,100]	900
CEC-10	Shifted and Rotated Schwefel’s Function	10	[-100,100]	1000
Hybrid Functions
CEC-11	Hybrid Function 1 (N = 3)	10	[-100,100]	1100
CEC-12	Hybrid Function 2 (N = 3)	10	[-100,100]	1200
CEC-13	Hybrid Function 3 (N = 3)	10	[-100,100]	1300
CEC-14	Hybrid Function 4 (N = 4)	10	[-100,100]	1400
CEC-15	Hybrid Function 5 (N = 4)	10	[-100,100]	1500
CEC-16	Hybrid Function 6 (N = 4)	10	[-100,100]	1600
CEC-17	Hybrid Function 6 (N = 5)	10	[-100,100]	1700
CEC-18	Hybrid Function 6 (N = 5)	10	[-100,100]	1800
CEC-19	Hybrid Function 6 (N = 5)	10	[-100,100]	1900
CEC-20	Hybrid Function 6 (N = 6)	10	[-100,100]	2000
Composition Functions
CEC-21	Composition Function 1 (N = 3)	10	[-100,100]	2100
CEC-22	Composition Function 2 (N = 3)	10	[-100,100]	2200
CEC-23	Composition Function 3 (N = 4)	10	[-100,100]	2300
CEC-24	Composition Function 4 (N = 4)	10	[-100,100]	2400
CEC-25	Composition Function 5 (N = 5)	10	[-100,100]	2500
CEC-26	Composition Function 6 (N = 5)	10	[-100,100]	2600
CEC-27	Composition Function 7 (N = 6)	10	[-100,100]	2700
CEC-28	Composition Function 8 (N = 6)	10	[-100,100]	2800
CEC-29	Composition Function 9 (N = 3)	10	[-100,100]	2900
CEC-30	Composition Function 10 (N = 3)	10	[-100,100]	3000

From Table 6, it is clear that the optimization accuracy of both algorithms gradually decreases with the increase in the dimension. This is because the larger the dimensions, the more complex the search domain. However, the performance of EHBA does not deteriorate significantly. Especially for functions F₁-F₆, F₈, F₉, and F₁₁, EHBA can always find more promising solutions than the basic HBA. The overall results provide strong evidence that EHBA can not only solve low-dimensional problems at ease but also maintain good search ability when tackling high-dimensional problems.

4.2 Experiment on IEEE CEC2017 test suite

The standard benchmark function experiment has demonstrated the excellent performance of HBA in solving simple problems. Nevertheless, with the development of metaheuristics, more and more algorithms are able to take good results in the standard test functions. To further verify the superiority of the enhanced algorithm in this paper, we use the IEEE CEC2017 test suite [54] to evaluate the performance of EHBA on challenging problems. This test suite is composed of 29 functions with over half of complex hybrid and composition functions. The function name, dimension size (Dim), search range, and optimal objective value of these functions are outlined in Table 7. As described in the previous subsection, the proposed EHBA is compared with HBA and the other seven optimizers. The parameter settings for all involved algorithms are consistent with those in Table 2. Meanwhile, the maximum iteration and population size are set as 500 and 30, respectively. After 30 independent runs, the average value and standard deviation results are shown in Table 8.

Table 8
Comparison results of EHBA and other algorithms on IEEE CEC2017 test suite

F_n Criteria ALO HHO WOA MVO MFO HBA TACPSO IGWO EHBA

CEC-1 Avg 3.07E+03 4.32E+06 8.31E+07 1.75E+04 9.94E+07 4.45E+03 4.71E+04 6.59E+04 2.74E+03

Std 3.91E+03 1.53E+07 1.17E+08 7.99E+03 3.50E+08 4.16E+03 4.95E+03 3.30E+04 3.02E+03

CEC-3 Avg 5.01E+02 7.16E+02 6.54E+03 3.00E+02 1.15E+04 3.00E+02 3.00E+02 3.02E+02 3.00E+02

Std 6.51E+02 4.09E+02 5.84E+03 1.71E-01 1.04E+04 4.56E-01 3.10E-02 2.25E+00 6.29E-06

CEC-4 Avg 4.07E+02 4.35E+02 4.59E+02 4.08E+02 4.24E+02 4.12E+02 4.10E+02 4.08E+02 4.04E+02

Std 1.15E+01 3.80E+01 6.10E+01 1.27E+01 3.13E+01 1.24E+01 1.75E+01 1.73E+01 7.47E-01

CEC-5 Avg 5.26E+02 5.60E+02 5.57E+02 5.20E+02 5.35E+02 5.22E+02 5.24E+02 5.13E+02 5.15E+02

Std 1.20E+01 1.70E+01 1.89E+01 8.27E+00 1.04E+01 1.04E+01 1.21E+01 6.16E+00 9.24E+00

CEC-6 Avg 6.16E+02 6.39E+02 6.37E+02 6.04E+02 6.03E+02 6.02E+02 6.01E+02 6.01E+02 6.00E+02

Std 9.12E+00 9.55E+00 8.93E+00 8.98E+00 3.59E+00 3.91E+00 1.90E-01 7.53E-01 3.68E-02

CEC-7 Avg 7.49E+02 7.93E+02 7.85E+02 7.39E+02 7.42E+02 7.39E+02 7.29E+02 7.31E+02 7.34E+02

Std 1.35E+01 2.35E+01 2.16E+01 1.36E+01 1.55E+01 1.00E+01 8.36E+00 1.14E+01 1.21E+01

CEC-8 Avg 8.24E+02 8.32E+02 8.47E+02 8.22E+02 8.35E+02 8.18E+02 8.16E+02 8.20E+02 8.07E+02

Std 9.57E+00 1.04E+01 1.73E+01 9.52E+00 1.32E+01 6.20E+00 7.20E+00 6.20E+00 4.19E+00

CEC-9 Avg 1.15E+03 1.52E+03 1.66E+03 9.00E+02 1.06E+03 9.29E+02 9.11E+02 9.01E+02 9.00E+02

Std 2.25E+02 1.97E+02 5.39E+02 7.44E-01 3.56E+02 1.07E+02 2.28E+01 1.11E-02 2.95E+00

CEC-10 Avg 2.05E+03 2.13E+03 2.18E+03 1.91E+03 1.86E+03 1.84E+03 1.74E+03 1.87E+03 1.67E+03

Std 3.88E+02 2.96E+02 3.51E+02 3.13E+02 3.22E+02 3.72E+02 3.49E+02 4.86E+02 2.73E+02

CEC-11 Avg 1.19E+03 1.18E+03 1.23E+03 1.13E+03 1.21E+03 1.13E+03 1.12E+03 1.11E+03 1.12E+03

Std 6.18E+01 3.98E+01 7.52E+01 1.99E+01 1.30E+02 3.85E+01 5.02E+01 2.48E+00 1.80E+01

CEC-12 Avg 2.30E+06 3.77E+06 3.54E+06 1.75E+06 1.76E+06 1.92E+04 2.27E+04 3.89E+04 1.31E+04

Std 2.74E+06 3.39E+06 5.85E+06 1.92E+06 3.54E+06 1.60E+04 1.92E+04 4.26E+04 9.16E+03

CEC-13 Avg 1.31E+04 1.47E+04 1.49E+04 1.04E+04 1.53E+04 1.17E+04 1.06E+04 1.99E+04 1.03E+04

Std 1.02E+04 9.57E+03 1.05E+04 1.03E+04 1.28E+04 1.13E+04 9.84E+03 1.48E+04 9.55E+03

CEC-14 Avg 2.51E+03 2.11E+03 2.77E+03 2.56E+03 5.00E+03 2.04E+03 2.46E+03 1.67E+03 1.56E+03

Std 2.13E+03 7.46E+02 1.40E+03 1.61E+03 6.77E+03 1.46E+03 4.97E+03 1.57E+01 1.34E+02

CEC-15 Avg 1.64E+04 6.97E+03 9.31E+03 3.64E+03 9.45E+03 2.21E+03 2.49E+03 2.38E+03 2.04E+03

Std 1.12E+04 3.30E+03 8.92E+03 2.96E+03 1.03E+04 1.86E+02 1.31E+03 2.19E+03 1.01E+02

CEC-16 Avg 1.86E+03 1.90E+03 2.00E+03 1.84E+03 1.77E+03 1.73E+03 1.72E+03 1.61E+03 1.69E+03

Std 1.61E+02 1.11E+02 1.62E+02 1.68E+02 1.14E+02 1.00E+02 1.17E+02 3.21E+00 9.62E+01

CEC-17 Avg 1.79E+03 1.79E+03 1.83E+03 1.78E+03 1.79E+03 1.76E+03 1.75E+03 1.75E+03 1.74E+03

Std 5.69E+01 6.21E+01 9.40E+01 5.38E+01 5.58E+01 3.25E+01 3.15E+01 2.99E+01 1.34E+01

CEC-18 Avg 1.79E+04 2.03E+04 1.79E+04 1.60E+04 2.44E+04 1.14E+03 1.75E+04 1.21E+04 1.14E+03

Std 1.32E+04 1.25E+04 1.29E+04 1.41E+04 1.39E+04 7.19E+03 1.10E+04 6.45E+03 9.08E+03

CEC-19 Avg 1.59E+04 1.49E+04 1.15E+05 4.49E+03 1.36E+04 4.07E+03 5.55E+03 3.96E+03 3.88E+03

Std 1.09E+04 1.21E+04 3.06E+05 3.88E+03 1.94E+04 5.98E+03 5.28E+03 4.21E+03 3.29E+03

CEC-20 Avg 2.15E+03 2.21E+03 2.22E+03 2.11E+03 2.10E+03 2.08E+03 2.05E+03 2.07E+03 2.03E+03

Std 8.95E+01 6.18E+01 8.66E+01 7.96E+01 7.96E+01 6.91E+01 1.71E+01 5.22E+01 1.10E+01

CEC-21 Avg 2.31E+03 2.34E+03 2.33E+03 2.30E+03 2.33E+03 2.31E+03 2.27E+03 2.38E+03 2.29E+03

Std 4.21E+01 5.32E+01 5.90E+01 4.71E+01 3.60E+01 3.88E+01 6.01E+01 5.50E+01 4.87E+01

CEC-22 Avg 2.35E+03 2.37E+03 2.57E+03 2.42E+03 2.31E+03 2.30E+03 2.30E+03 2.30E+03 2.30E+03

Std 2.54E+02 3.10E+02 5.33E+02 3.07E+02 1.81E+01 2.69E+01 9.18E+00 3.41E+00 1.02E+01

CEC-23 Avg 2.63E+03 2.68E+03 2.66E+03 2.63E+03 2.65E+03 2.63E+03 2.63E+03 2.62E+03 2.62E+03

Std 1.13E+01 3.39E+01 2.47E+01 9.85E+00 1.19E+01 1.17E+01 2.09E+01 8.55E+00 9.56E+00

CEC-24 Avg 2.73E+03 2.83E+03 2.78E+03 2.75E+03 2.76E+03 2.75E+03 2.72E+03 2.76E+03 2.71E+03

Std 7.78E+01 4.95E+01 4.14E+01 8.00E+01 5.04E+01 7.41E+01 6.36E+01 9.90E+01 3.64E+01

CEC-25 Avg 2.93E+03 2.93E+03 2.98E+03 2.93E+03 2.95E+03 2.93E+03 2.92E+03 2.94E+03 2.92E+03

Std 2.35E+01 2.19E+01 5.03E+01 2.18E+01 2.99E+01 6.32E+01 2.42E+01 1.97E+01 2.22E+01

CEC-26 Avg 3.20E+03 3.61E+03 3.55E+03 3.01E+03 3.04E+03 3.11E+03 2.95E+03 2.92E+03 2.94E+03

Std 4.15E+02 5.45E+02 5.34E+02 3.23E+02 1.04E+02 4.00E+02 1.26E+02 1.69E+02 2.98E+02

CEC-27 Avg 3.11E+03 3.16E+03 3.14E+03 3.09E+03 3.10E+03 3.12E+03 3.10E+03 3.12E+03 3.09E+03

Std 2.38E+01 5.40E+01 4.07E+01 1.10E+01 3.16E+00 4.08E+01 1.05E+01 3.32E+01 4.63E-01

CEC-28 Avg 3.36E+03 3.45E+03 3.55E+03 3.31E+03 3.34E+03 3.43E+03 3.30E+03 3.35E+03 3.30E+03

Std 1.57E+02 1.66E+02 1.73E+02 1.19E+02 9.24E+01 3.16E+02 2.24E+02 1.44E+02 1.04E+02

CEC-29 Avg 3.27E+03 3.36E+03 3.40E+03 3.24E+03 3.24E+03 3.27E+03 3.26E+03 3.21E+03 3.16E+03

Std 7.61E+01 1.93E+01 1.12E+02 6.74E+01 6.55E+01 7.39E+01 7.78E+01 1.08E+02 5.25E+01

CEC-30 Avg 5.37E+05 1.95E+06 1.78E+06 6.86E+05 7.74E+05 9.35E+05 2.53E+05 1.35E+06 7.43E+05

Std 1.10E+06 2.45E+06 2.08E+06 6.63E+05 7.24E+05 9.11E+05 4.94E+05 2.49E+06 8.15E+05

F_n	Criteria	ALO	HHO	WOA	MVO	MFO	HBA	TACPSO	IGWO	EHBA
CEC-1	Avg	3.07E+03	4.32E+06	8.31E+07	1.75E+04	9.94E+07	4.45E+03	4.71E+04	6.59E+04	2.74E+03
	Std	3.91E+03	1.53E+07	1.17E+08	7.99E+03	3.50E+08	4.16E+03	4.95E+03	3.30E+04	3.02E+03
CEC-3	Avg	5.01E+02	7.16E+02	6.54E+03	3.00E+02	1.15E+04	3.00E+02	3.00E+02	3.02E+02	3.00E+02
	Std	6.51E+02	4.09E+02	5.84E+03	1.71E-01	1.04E+04	4.56E-01	3.10E-02	2.25E+00	6.29E-06
CEC-4	Avg	4.07E+02	4.35E+02	4.59E+02	4.08E+02	4.24E+02	4.12E+02	4.10E+02	4.08E+02	4.04E+02
	Std	1.15E+01	3.80E+01	6.10E+01	1.27E+01	3.13E+01	1.24E+01	1.75E+01	1.73E+01	7.47E-01
CEC-5	Avg	5.26E+02	5.60E+02	5.57E+02	5.20E+02	5.35E+02	5.22E+02	5.24E+02	5.13E+02	5.15E+02
	Std	1.20E+01	1.70E+01	1.89E+01	8.27E+00	1.04E+01	1.04E+01	1.21E+01	6.16E+00	9.24E+00
CEC-6	Avg	6.16E+02	6.39E+02	6.37E+02	6.04E+02	6.03E+02	6.02E+02	6.01E+02	6.01E+02	6.00E+02
	Std	9.12E+00	9.55E+00	8.93E+00	8.98E+00	3.59E+00	3.91E+00	1.90E-01	7.53E-01	3.68E-02
CEC-7	Avg	7.49E+02	7.93E+02	7.85E+02	7.39E+02	7.42E+02	7.39E+02	7.29E+02	7.31E+02	7.34E+02
	Std	1.35E+01	2.35E+01	2.16E+01	1.36E+01	1.55E+01	1.00E+01	8.36E+00	1.14E+01	1.21E+01
CEC-8	Avg	8.24E+02	8.32E+02	8.47E+02	8.22E+02	8.35E+02	8.18E+02	8.16E+02	8.20E+02	8.07E+02
	Std	9.57E+00	1.04E+01	1.73E+01	9.52E+00	1.32E+01	6.20E+00	7.20E+00	6.20E+00	4.19E+00
CEC-9	Avg	1.15E+03	1.52E+03	1.66E+03	9.00E+02	1.06E+03	9.29E+02	9.11E+02	9.01E+02	9.00E+02
	Std	2.25E+02	1.97E+02	5.39E+02	7.44E-01	3.56E+02	1.07E+02	2.28E+01	1.11E-02	2.95E+00
CEC-10	Avg	2.05E+03	2.13E+03	2.18E+03	1.91E+03	1.86E+03	1.84E+03	1.74E+03	1.87E+03	1.67E+03
	Std	3.88E+02	2.96E+02	3.51E+02	3.13E+02	3.22E+02	3.72E+02	3.49E+02	4.86E+02	2.73E+02
CEC-11	Avg	1.19E+03	1.18E+03	1.23E+03	1.13E+03	1.21E+03	1.13E+03	1.12E+03	1.11E+03	1.12E+03
	Std	6.18E+01	3.98E+01	7.52E+01	1.99E+01	1.30E+02	3.85E+01	5.02E+01	2.48E+00	1.80E+01
CEC-12	Avg	2.30E+06	3.77E+06	3.54E+06	1.75E+06	1.76E+06	1.92E+04	2.27E+04	3.89E+04	1.31E+04
	Std	2.74E+06	3.39E+06	5.85E+06	1.92E+06	3.54E+06	1.60E+04	1.92E+04	4.26E+04	9.16E+03
CEC-13	Avg	1.31E+04	1.47E+04	1.49E+04	1.04E+04	1.53E+04	1.17E+04	1.06E+04	1.99E+04	1.03E+04
	Std	1.02E+04	9.57E+03	1.05E+04	1.03E+04	1.28E+04	1.13E+04	9.84E+03	1.48E+04	9.55E+03
CEC-14	Avg	2.51E+03	2.11E+03	2.77E+03	2.56E+03	5.00E+03	2.04E+03	2.46E+03	1.67E+03	1.56E+03
	Std	2.13E+03	7.46E+02	1.40E+03	1.61E+03	6.77E+03	1.46E+03	4.97E+03	1.57E+01	1.34E+02
CEC-15	Avg	1.64E+04	6.97E+03	9.31E+03	3.64E+03	9.45E+03	2.21E+03	2.49E+03	2.38E+03	2.04E+03
	Std	1.12E+04	3.30E+03	8.92E+03	2.96E+03	1.03E+04	1.86E+02	1.31E+03	2.19E+03	1.01E+02
CEC-16	Avg	1.86E+03	1.90E+03	2.00E+03	1.84E+03	1.77E+03	1.73E+03	1.72E+03	1.61E+03	1.69E+03
	Std	1.61E+02	1.11E+02	1.62E+02	1.68E+02	1.14E+02	1.00E+02	1.17E+02	3.21E+00	9.62E+01
CEC-17	Avg	1.79E+03	1.79E+03	1.83E+03	1.78E+03	1.79E+03	1.76E+03	1.75E+03	1.75E+03	1.74E+03
	Std	5.69E+01	6.21E+01	9.40E+01	5.38E+01	5.58E+01	3.25E+01	3.15E+01	2.99E+01	1.34E+01
CEC-18	Avg	1.79E+04	2.03E+04	1.79E+04	1.60E+04	2.44E+04	1.14E+03	1.75E+04	1.21E+04	1.14E+03
	Std	1.32E+04	1.25E+04	1.29E+04	1.41E+04	1.39E+04	7.19E+03	1.10E+04	6.45E+03	9.08E+03
CEC-19	Avg	1.59E+04	1.49E+04	1.15E+05	4.49E+03	1.36E+04	4.07E+03	5.55E+03	3.96E+03	3.88E+03
	Std	1.09E+04	1.21E+04	3.06E+05	3.88E+03	1.94E+04	5.98E+03	5.28E+03	4.21E+03	3.29E+03
CEC-20	Avg	2.15E+03	2.21E+03	2.22E+03	2.11E+03	2.10E+03	2.08E+03	2.05E+03	2.07E+03	2.03E+03
	Std	8.95E+01	6.18E+01	8.66E+01	7.96E+01	7.96E+01	6.91E+01	1.71E+01	5.22E+01	1.10E+01
CEC-21	Avg	2.31E+03	2.34E+03	2.33E+03	2.30E+03	2.33E+03	2.31E+03	2.27E+03	2.38E+03	2.29E+03
	Std	4.21E+01	5.32E+01	5.90E+01	4.71E+01	3.60E+01	3.88E+01	6.01E+01	5.50E+01	4.87E+01
CEC-22	Avg	2.35E+03	2.37E+03	2.57E+03	2.42E+03	2.31E+03	2.30E+03	2.30E+03	2.30E+03	2.30E+03
	Std	2.54E+02	3.10E+02	5.33E+02	3.07E+02	1.81E+01	2.69E+01	9.18E+00	3.41E+00	1.02E+01
CEC-23	Avg	2.63E+03	2.68E+03	2.66E+03	2.63E+03	2.65E+03	2.63E+03	2.63E+03	2.62E+03	2.62E+03
	Std	1.13E+01	3.39E+01	2.47E+01	9.85E+00	1.19E+01	1.17E+01	2.09E+01	8.55E+00	9.56E+00
CEC-24	Avg	2.73E+03	2.83E+03	2.78E+03	2.75E+03	2.76E+03	2.75E+03	2.72E+03	2.76E+03	2.71E+03
	Std	7.78E+01	4.95E+01	4.14E+01	8.00E+01	5.04E+01	7.41E+01	6.36E+01	9.90E+01	3.64E+01
CEC-25	Avg	2.93E+03	2.93E+03	2.98E+03	2.93E+03	2.95E+03	2.93E+03	2.92E+03	2.94E+03	2.92E+03
	Std	2.35E+01	2.19E+01	5.03E+01	2.18E+01	2.99E+01	6.32E+01	2.42E+01	1.97E+01	2.22E+01
CEC-26	Avg	3.20E+03	3.61E+03	3.55E+03	3.01E+03	3.04E+03	3.11E+03	2.95E+03	2.92E+03	2.94E+03
	Std	4.15E+02	5.45E+02	5.34E+02	3.23E+02	1.04E+02	4.00E+02	1.26E+02	1.69E+02	2.98E+02
CEC-27	Avg	3.11E+03	3.16E+03	3.14E+03	3.09E+03	3.10E+03	3.12E+03	3.10E+03	3.12E+03	3.09E+03
	Std	2.38E+01	5.40E+01	4.07E+01	1.10E+01	3.16E+00	4.08E+01	1.05E+01	3.32E+01	4.63E-01
CEC-28	Avg	3.36E+03	3.45E+03	3.55E+03	3.31E+03	3.34E+03	3.43E+03	3.30E+03	3.35E+03	3.30E+03
	Std	1.57E+02	1.66E+02	1.73E+02	1.19E+02	9.24E+01	3.16E+02	2.24E+02	1.44E+02	1.04E+02
CEC-29	Avg	3.27E+03	3.36E+03	3.40E+03	3.24E+03	3.24E+03	3.27E+03	3.26E+03	3.21E+03	3.16E+03
	Std	7.61E+01	1.93E+01	1.12E+02	6.74E+01	6.55E+01	7.39E+01	7.78E+01	1.08E+02	5.25E+01
CEC-30	Avg	5.37E+05	1.95E+06	1.78E+06	6.86E+05	7.74E+05	9.35E+05	2.53E+05	1.35E+06	7.43E+05
	Std	1.10E+06	2.45E+06	2.08E+06	6.63E+05	7.24E+05	9.11E+05	4.94E+05	2.49E+06	8.15E+05

Fig. 7

Ranking of EHBA and other algorithms on IEEE CEC2017 test suite.

As can be seen from Table 8, EHBA provides the best results on all unimodal functions CEC-1 and CEC-3. Although HBA and TACPSO obtain the same average fitness as EHBA on CEC-3, the standard deviation of EHBA is smaller. For multimodal functions, EHBA performs best on CEC-4, CEC-6, CEC-8, CEC-9 and CEC-10. IGWO and TACPSO have satisfactory solutions on CEC-5 and CEC-7, respectively. For hybrid functions and composite functions, EHBA outperforms the other eight algorithms on 12 out of 20 test functions. These results reveal that EHBA can solve a wide variety of complex optimization problems as well.

Figure 7 shows the radar ranking diagram of the nine algorithms on this test suite. The smaller the region surrounded by each curve, the better performance of the algorithm. It is obvious that EHBA has the smallest enclosed region, thereby visually proving that the proposed multi-strategy combination EHBA has superior performance in comparison.

5 EHBA for solving engineering design problems

In this section, the proposed EHBA is used to solve four constrained real-world engineering problems: three-bar truss design problem, pressure vessel design problem, tension/compression spring design problem, and speed reducer design problem, to highlight its performance at the practical application level. For the sake of convenience, the death penalty function [55] is adopted here to deal with the infeasible solutions subjected to these constraints during optimization. Likewise, EHBA runs independently 30 times for each project with the population size and maximum iteration are set to 30 and 500, respectively. The obtained results are compared against those of different well-known meta-heuristic algorithms released in the literature.

5.1 Three-bar truss design problem

The three-bar truss design problem is one of the most representative optimization applications in the field of civil engineering. As illustrated in Fig. 8, the main objective of this issue is to find three structural decision variables, namely the cross-sectional area of component 1 (A₁), cross-sectional area of component 2 (A₂), and cross-sectional area of component 3 (A₃), where A₁ = A₃, to minimize the cost weight of a three-bar truss. The problem is defined mathematically as follow.

Fig. 8

Schematic view of three-bar truss design problem.

Consider $\vec{x} = [x_{1}, x_{2}] =] A_{1}, A_{2}]$

Minimize $f (\vec{x}) = (2 \sqrt{2} x_{1} + x_{2}) \times l$

Subject to $g_{1} (\vec{x}) = \frac{\sqrt{2} x_{1} + x_{2}}{\sqrt{2} x_{1}^{2} + 2 x_{1} x_{2}} P - σ ⩽ 0$ $g_{2} (\vec{x}) = \frac{x_{2}}{\sqrt{2} x_{1}^{2} + 2 x_{1} x_{2}} P - σ ⩽ 0$ $g_{3} (\vec{x}) = \frac{1}{\sqrt{2} x_{2} + x_{1}} P - σ ⩽ 0$

Variable range $0 ⩽ x_{1}, x_{2} ⩽ 1$ where $l = 100 cm, P = 2 {KN / cm}^{2}, σ = {2 KN / cm}^{2} .$

In Table 9, the experimental results of EHBA and other famous algorithms are listed. As it presents, EHBA provides better outcomes than majority of other algorithms. Additionally, the results of EHBA are very competitive compared to SSA, GEO, AHA, and L-SHADE, which indicates the merits of EHBA in realizing the minimum cost of this problem.

Table 9

Comparison results for three-bar truss design problem

Algorithm	Optimum variables		Optimum
	A₁ (x₁)	A₂ (x₂)	cost
MFO [27]	0.78824	0.40947	263.89598
MVO [16]	0.78860	0.40845	263.89585
SSA [26]	0.78867	0.40828	263.89584
MBA [57]	0.78857	0.40856	263.89585
GEO [58]	0.78867	0.40826	263.89584
GOA [59]	0.78890	0.40762	263.89588
AHA [60]	0.78868	0.40822	263.89584
AOA [18]	0.79369	0.39426	263.91540
L-SHADE	0.78895	0.40747	263.89594
IPOP-CMA-ES	0.78290	0.42484	263.92090
EHBA	0.78859	0.40849	263.89584

5.2 Pressure vessel design problem

The pressure vessel design problem first proposed by Kannan and Kramer [56] aims to reduce the overall fabrication cost of a pressure vessel as much as possible. Figure 9 illustrates the structure of the pressure vessel and associated parameters distribution. In this design, four decision variables are considered, including the thickness of the shell (T_s), thickness of the head (T_h), inner radius (R), and length of the cylindrical portion (L). The mathematical model of this optimization can be formulated as follows.

Consider $\vec{x} = [x_{1} x_{2} x_{3} x_{4}] = [T_{s} T_{h} RL]$

Fig. 9

Schematic view of pressure vessel design problem.

Minimize $\begin{matrix} f (\vec{x}) = & 0.6224 x_{1} x_{3} x_{4} + 1.7781 x_{2} x_{3}^{2} \\ + 3.1661 x_{1}^{2} x_{4} + 19.84 x_{1}^{2} x_{3} \end{matrix}$

Subject to $g_{1} (\vec{x}) = - x_{1} + 0.0193 x_{3} ⩽ 0$ $g_{2} (\vec{x}) = - x_{3} + 0.00954 x_{3} ⩽ 0$ $g_{3} (\vec{x}) = - π x_{3}^{2} x_{4} - \frac{4}{3} π x_{3}^{3} + 1296000 ⩽ 0$ $g_{4} (\vec{x}) = x_{4} - 240 ⩽ 0$

Variable range $0 ⩽ x_{1}, x_{2} ⩽ 99, 10 ⩽ x_{3}, x_{4} ⩽ 200$

The comparison results of the optimum variables and cost for this problem are shown in Table 10. Compared with MFO, MVO, WOA, GWO, HHO, SMA, IGWO, IGTO, L-SHADE, and IPOP-CMA-ES, it is evident that the proposed EHBA can effectively reveal the lowest cost design $f (\vec{x}) = 5885.7039$ with corresponding optimal solution equal to $\vec{x} = [0.77820.384740.3196200.0000]$ . Therefore, it is reasonable to believe that EHBA is more suitable to deal with such a problem.

Table 10

Comparison results for pressure vessel design problem

Algorithm	Optimum variables				Optimum cost
	T_s (x₁)	T_h (x₂)	R (x₃)	L (x₄)
MFO [27]	0.8125	0.4375	42.0984	176.6366	6059.7143
MVO [16]	0.8125	0.4375	42.0907	176.7387	6060.8066
WOA [24]	0.8125	0.4375	42.0982	176.6389	6059.7410
GWO [23]	0.8125	0.4345	42.0892	176.7587	6051.5639
HHO [25]	0.8176	0.4073	42.0917	176.7196	6000.4626
SMA [30]	0.7931	0.3932	40.6711	196.2178	5994.1857
IGWO [37]	0.7790	0.3855	40.3631	199.4017	5888.3400
IGTO [61]	0.7889	0.3900	40.8764	192.4031	5904.2189
L-SHADE	0.8264	0.4077	42.9045	166.9050	5959.9314
IPOP-CMA-ES	0.7814	0.3862	40.4867	197.6869	5890.8681
EHBA	0.7782	0.3847	40.3196	200.0000	5885.7039

5.3 Tension/compression spring design problem

Just like its name implies, the final purpose of the tension/compression spring design problem is to find three structural parameters, such as wire diameter (d), mean coil diameter (D), and the number of active coils (N), to minimize the total weight of the tension/compression spring, as illustrated in Fig. 10. Besides, the constraints of shear stress, surge frequency and minimum deflection are supposed to be satisfied during the optimization process. The mathematical formulation of this problem is given as follows.

Consider $\vec{x} = [x_{1} x_{2} x_{3}] = [dDN]$

Fig. 10

Schematic view of tension/compression spring design problem.

Minimize $f (\vec{x}) = (x_{3} + 2) x_{2} x_{1}^{2}$

Subject to $g_{1} (\vec{x}) = 1 - \frac{x_{2}^{3} x_{3}}{71785 x_{1}^{4}} ⩽ 0$ $g_{2} (\vec{x}) = \frac{4 x_{2}^{2} - x_{1} x_{2}}{12566 (x_{2} x_{1}^{3} - x_{1}^{4})} + \frac{1}{5108 x_{1}^{2}} ⩽ 0$ $g_{3} (\vec{x}) = 1 - \frac{140.45 x_{1}}{x_{2}^{2} x_{3}} ⩽ 0$ $g_{4} (\vec{x}) = \frac{x_{1} + x_{2}}{1.5} - 1 ⩽ 0$

Variable range $0.05 ⩽ x_{1} ⩽ 2, 0.25 ⩽ x_{2} ⩽ 1.30, 2.00 ⩽ x_{3} ⩽ 15$

The comparison results of the optimum variables and weight obtained by different optimization techniques are recorded in Table 11. It can be seen from this table that EHBA significantly outperforms all other competitors and achieves the best weight of 0.011841, which indicates that EHBA has superior potential in solving this design problem within confined space.

Table 11

Comparison results for tension/compression spring design problem

Algorithm	Optimum variables			Optimum weight
	d (x₁)	D (x₂)	N (x₃)
MFO [27]	0.051995	0.364109	10.868422	0.012667
MVO [16]	0.052510	0.376020	10.335130	0.012790
WOA [24]	0.051207	0.345215	12.004032	0.012676
GWO [23]	0.051690	0.356737	11.288850	0.012666
HHO [25]	0.051796	0.359305	11.138859	0.012665
SSA [26]	0.051207	0.345215	12.004032	0.012676
HBA [43]	0.050600	0.355200	11.373000	0.012070
MTDE [62]	0.051689	0.356718	11.288965	0.012665
L-SHADE	0.052300	0.372000	10.447200	0.012690
IPOP-CMA-ES	0.051648	0.355719	11.348019	0.012666
EHBA	0.050592	0.330898	11.980276	0.011841

5.4 Speed reducer design problem

The speed reducer design problem is also a popular research case for engineering optimization in real life. Its goal is to reduce the speed reducer’s weight as much as possible. As illustrated in Fig. 11, there are seven decision variables involved in this optimum design that need to be optimized, namely the face width (x₁), module of teeth (x₂), number of teeth in the pinion (x₃), distance of the shafts between bearings (x₄, x₅), and diameter of the shafts (x₆, x₇). Meanwhile, the minimization process is subject to four constraints: the bending stress of the gear teeth, covering stress, stress in the shafts and transverse deflections of the shafts. The mathematical model of this problem is expressed as follows.

Fig. 11

Schematic view of speed reducer design problem.

Minimize

$\begin{matrix} f (\vec{x}) = 0.7854 x_{1} x_{2}^{2} (3.3333 x_{3}^{2} + 14.9334 x_{3} - 43.0934) \\ - 1.508 x_{1} (x_{6}^{2} + x_{7}^{2}) + 7.4777 (x_{6}^{3} + x_{7}^{3}) \end{matrix}$

Subject to $g_{1} (\vec{x}) = \frac{27}{x_{1} x_{2}^{2} x_{3}} - 1 ⩽ 0$ $g_{2} (\vec{x}) = \frac{397.5}{x_{1} x_{2}^{2} x_{3}^{2}} - 1 ⩽ 0$ $g_{3} (\vec{x}) = \frac{1.93 x_{4}^{3}}{x_{2} x_{3} x_{6}^{4}} - 1 ⩽ 0$ $g_{4} (\vec{x}) = \frac{1.93 x_{5}^{3}}{x_{2} x_{3} x_{7}^{4}} - 1 ⩽ 0$ $g_{5} (\vec{x}) = \frac{\sqrt{{(\frac{745_{x_{4}}}{x_{2} x_{3}})}^{2} + 16.9 \times 10^{6}}}{110.0 x_{6}^{3}} - 1 ⩽ 0$ $g_{6} (\vec{x}) = \frac{\sqrt{{(\frac{745 x_{4}}{x_{2} x_{3}})}^{2} + 157.5 \times 10^{6}}}{85.0 x_{6}^{3}} - 1 ⩽ 0$ $g_{7} (\vec{x}) = \frac{x_{2} x_{3}}{40} - 1 ⩽ 0$ $g_{8} (\vec{x}) = \frac{5 x_{2}}{x_{1}} - 1 ⩽ 0$ $g_{9} (\vec{x}) = \frac{x_{1}}{12 x_{2}} - 1 ⩽ 0$ $g_{10} (\vec{x}) = \frac{1.5 x_{6} + 1.9}{x_{4}} - 1 ⩽ 0$ $g_{11} (\vec{x}) = \frac{1.1 x_{7} + 1.9}{x_{5}} - 1 ⩽ 0$

Variable range $\begin{matrix} 2.6 ⩽ x_{1} ⩽ 3.6, 0.7 ⩽ x_{2} ⩽ 0.8, 17 ⩽ x_{3} \\ ⩽ 287.3 ⩽ x_{4} ⩽ 8.3, 7.8 ⩽ x_{5} \\ ⩽ 8.3, 2.9 ⩽ x_{6} ⩽ 3.9, 5.0 ⩽ x_{7} ⩽ 5.5 \end{matrix}$

Table 12 reports the comparison results of EHBA with AO, AOA, GSA, SCA, SHO, STOA, SC-GWO, L-SHADE, and IPOP-CMA-ES. Based on the data in this table, it can be noticed that the optimum weight $f (\vec{x}) = 2994.4245$ is obtained by EHBA at $\vec{x} = [3.50.7177.37.715313.350545.28665]$ . This example once again highlights the remarkable search capability of the proposed algorithm in this paper.

Table 12

Comparison results for speed reducer design problem

Algorithm	Optimum variables							Optimum weight
	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇
AO [31]	3.50210	0.7	17	7.3099	7.74760	3.35785	5.29940	3007.7328
AOA [18]	3.50384	0.7	17	7.3	7.72933	3.35649	5.28670	2997.9157
GSA [15]	3.6	0.7	17	8.3	7.8	3.36966	5.28922	3051.1200
SCA [17]	3.50875	0.7	17	7.3	7.8	3.46102	5.28921	3030.5630
SHO [8]	3.50159	0.7	17	7.3	7.8	3.35127	5.28874	2998.5507
STOA [63]	3.50124	0.7	17	7.3	7.8	3.33425	5.26538	2995.9578
SC-GWO [64]	3.50064	0.7	17	7.30643	7.80617	3.35034	5.28694	2996.9859
L-SHADE	3.50251	0.7	17	7.3	8.26350	3.35648	5.28969	3011.0315
IPOP-CMA-ES	3.49761	0.7	17	7.8041	7.8	3.39680	5.28610	2997.0380
EHBA	3.5	0.7	17	7.3	7.71531	3.35054	5.28665	2994.4245

As a summary, these observed results demonstrate that EHBA is equally practical and competitive in engineering applications. Benefiting from the introduced HDPM, Lévy flight, and ROBL strategies, EHBA accomplishes excellent optimization performance, which is highly recommended to handle more real-world tasks.

6 Conclusion and future work

In this paper, an enhanced variant of honey badger algorithm, namely EHBA, was proposed based on three search strategies. First, HDPM was introduced to enrich the diversity of the population. Then, Lévy flight was incorporated into the core position update formula to improve the search efficiency and maintain a better balance between exploration and exploitation phases. Final, ROBL was utilized to evaluate the opposition solution of the global optimum, which contributes to reduce the likelihood of falling into the local optima. The feasibility and effectiveness of the proposed EHBA is comprehensively validated on 18 standard benchmark functions and IEEE CEC2017 test suite. And the non-parametric Wilcoxon’s rank-sum test was used to investigate the significance of the results. Numerical and statistical results demonstrate that three introduced strategies greatly enhance the performance of the basic algorithm. Compared with HBA and seven state-of-the-art meta-heuristic algorithms, EHBA achieves better local optima avoidance, convergence accuracy, and stability. In addition, EHBA was applied to solve four engineering design problems to verify its applicability. The results indicate that EHBA also has competitive performance for real-world optimization applications.

However, it can be seen that the proposed EHBA still suffers from high computation cost. We believe this limitation can be mitigated by introducing some parallel mechanisms such as master-slave model, cell model, and island model. In the future work, other new optimization operators and search strategies will be used to improve the optimization skills of EHBA further. Besides, we hope to integrate HDPM, Lévy flight, and ROBL into more meta-heuristic algorithms, and investigate their performance in the field of feature selection. It is also meaningful to apply EHBA to resolve more real-life problems, including the parameter self-tuning of speed proportional integral differential (PID) controller for brushless direct current motors, the global path planning for autonomous underwater vehicles in a complex environment, and the maximum power point tracking of solar photovoltaic systems.

Footnotes

Acknowledgments

The authors are grateful to the editor and reviewers for their constructive comments and suggestions, which have improved the presentation. And this work is financially supported by the National Natural Science Foundation of China under Grant 52075090 and the Fundamental Research Funds for Central Universities under Grant 2572014BB06.

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