EAO: Enhanced aquila optimizer for solving optimization problem

Abstract

The Aquila optimization (AO) algorithm has the drawbacks of local optimization and poor optimization accuracy when confronted with complex optimization problems. To remedy these drawbacks, this paper proposes an Enhanced aquila optimization (EAO) algorithm. To avoid elite individual from entering the local optima, the elite opposition-based learning strategy is added. To enhance the ability of balancing global exploration and local exploitation, a dynamic boundary strategy is introduced. To elevate the algorithm’s convergence rapidity and precision, an elite retention mechanism is introduced. The effectiveness of EAO is evaluated using CEC2005 benchmark functions and four benchmark images. The experimental results confirm EAO’s viability and efficacy. The statistical results of Freidman test and the Wilcoxon rank sum test are confirmed EAO’s robustness. The proposed EAO algorithm outperforms previous algorithms and can useful for threshold optimization and pressure vessel design.

Keywords

Aquila optimization algorithm optimization function kapur entropy threshold optimization pressure vessel design

1 Introduction

Optimization is the process of determining which of all the possible solutions, given certain limitations, is the optimum alternative. When a problem has multiple solutions, optimization can be applied. This problem is called the optimization problem. The objective function, decision variable, and constraint condition make up its fundamental three elements. The objective function helps the solution measure what is optimal. The decision variable can change the optimization result by adjusting its value in a controllable range. Constraints are the limits on these decision variables. Optimization problems [1] can be divided into constrained and unconstrained problems, continuous optimization problems and discrete optimization problems, stochastic optimization problems and deterministic optimization problems, linear programming problems and nonlinear programming problems, convex optimization problems and non-convex optimization problems, global optimization problems and local optimization problems. The stochastic optimization problem refers to the objective function or constraint function involving random variables with uncertainty. Deterministic optimization problems do not contain random variables, and the optimization result of the problem is fixed. The linear programming problem refers to the fact that both the objective function and the constraint function are linear functions. It is easy to solve, and the main methods are the simplex method and the interior point method. Any optimization problem that maximizes a concave function on a convex set or minimizes a convex function on a convex set is called a convex optimization problem. If the function domain is not a convex set or the objective function is not a convex function, then the corresponding optimization problem is a non-convex optimization problem.

The problem that the decision variable’s value should be taken under certain limitations in order to get the chosen objective function to attain the optimal value is referred to as the optimal method. In the mathematical sense, the goal of an optimization method is to maximize or minimize the objective function’s value while staying within the bounds of a given set of equations or inequalities. The matching algorithm type is chosen based on the optimization problem’s features in order to find the best solution. Algorithm type can be separated into two categories: precise optimization algorithms and approximate optimization algorithms. The problem scale and operation time requirements for a precision optimization algorithm are very high. An algorithm that can provide a reasonably good solution in an appropriate period and effectively balance the quality of the solution and the operation time is known as an approximate optimization method. The most popular of these is the approximate optimization algorithm’s meta-heuristic algorithm. Although finding the ideal solution is not assured, the meta-heuristic algorithm is independent of the problem. It now places greater emphasis on the quality of the objective function’s solution than it does on the problem itself. It balances exploration and exploitation capacities [2] in addition to not requiring gradient information [3]. Numerous study findings demonstrate that the meta-heuristic algorithm has the advantages of self-adaptation, parallelism, robustness, and solving ability. Moreover, when dealing with multi-variable multi-dimensional optimization problems that are not limited by dimension, the meta-heuristic algorithm does not need to know the mathematical characteristics of the optimal solution of the optimization problem, and directly finds the optimal solution of the optimization problem [4]. Although it is not guaranteed that the optimal solution can be obtained, it can obtain the solution closest to the optimal solution in the shortest time, and it can use random processes to obtain the optimal output [5]. Therefore, it has been widely used. But the meta-heuristic algorithm must balance exploration and exploitation abilities to play a better role. Excessive exploration will slow down the rate of convergence, while excessive exploitation may lead to local optimality. Because of the method’s randomness, a partial meta-heuristic algorithm may have good exploitation capabilities but not enough exploration capabilities, causing it to remain locally optimal. Alternatively, it may have excellent exploration capabilities but insufficient exploitation capabilities. Furthermore, the No Free Lunch (NFL) theory [6] asserts that meta-heuristic algorithms cannot consistently outperform on all optimization problems [7]. This means that no algorithm is capable of solving all issues optimally, while other algorithms are only suboptimal at solving specific problems optimally. To address a particular problem, it is important to create a new or enhanced high-performance algorithm. Interested readers can refer to [11 –13] for a detailed list and a detailed survey of meta-heuristic algorithms proposed over the last fifty years.

A meta-heuristic algorithm named AO algorithm was proposed by Laith Abualigah in 2021 [9]. Table 1 lists some new meta-heuristic algorithms proposed by Laith Abualigah. The algorithm simulates various hunting behaviors of Aquila in nature. The global exploration ability of the algorithm reflects the fast-moving hunting strategy, and the algorithm’s ability to exploit locally impacts the slow-moving hunting strategy [10]. Moreover, accurate results are obtained in realizing many kinds of optimization problems. Despite the author’s showing the effectiveness of AO’s search on a variety of test functions, the Aquila optimizer’s solution quality and its ability to balance exploration and exploitation could still be improved. Exploration is the process of choosing novel algorithmic solutions [8]. Stronger exploration means the algorithm is more likely to explore uncharted solution regions, increasing the likelihood that it may discover better solutions. Exploration is often frequent in the early stages of the algorithm and gradually decreases with the iteration of the algorithm to make more use of existing optima solutions. Exploitation refers to the use of known optimal solutions in an algorithm. Higher exploitation means that the algorithm is more inclined to use a known good solution in order to expect further optimization around that solution. Exploitation can hasten the convergence of algorithms to local optima, but it can also cause them to miss the global optima and fall into local optima. To balance the exploration and exploitation abilities of the algorithm, the advantages of each strategy can be combined into one algorithm, and the disadvantages of the algorithm can be eliminated while the advantages are utilized. Numerous academics have developed different strategies to remedy its flaws and enhance AO’s effectiveness. To improve the exploration and exploitation capabilities of AO, Serdar Ekinci et al. used the Nelder-Medad search method and stochastic learning mechanism, proposed the bAO (balanced Aquila optimizer), and applied it to air-fuel ratio system management [14]. By adopting opposition-based learning and the chaotic local search with shrinking strategy into the CmOBL-AO algorithm, Serdar Ekinci et al. improved the performance of the vehicle cruise control system [15]. Serdar Ekinci et al. merged the aforementioned two ideas to improve AO. They utilized the Nelder-Mead simplex search strategy, modified the opposition-based learning mechanism, and then applied it to automatic voltage regulator control [16].

Table 1
New meta-heuristic algorithms proposed by Laith Abualigah

Algorithm Inspiration Cite

Reptile Search Algorithm The hunting behavior of crocodiles [17]

The Arithmetic Optimization Algorithm Four operations in arithmetic [18]

Multi-verse optimizer algorithm The principle that matter in the universe is transferred from white holes to black holes through wormholes [19]

Salp swarm algorithm Group behavior of salp chains [20]

Prairie dog optimization algorithm Foraging and burrowing behavior of prairie dogs in their natural habitat [21]

Dwarf mongoose optimization algorithm Foraging behavior of dwarf mongoose populations [22]

Gazelle Optimization Algorithm The behavior of a gazelle to escape from predators [23]

Harris hawks optimization Harris hawks predation behavior [24]

Ebola Optimization Search Algorithm How Ebola virus disease spreads [25]

Algorithm	Inspiration	Cite
Reptile Search Algorithm	The hunting behavior of crocodiles	[17]
The Arithmetic Optimization Algorithm	Four operations in arithmetic	[18]
Multi-verse optimizer algorithm	The principle that matter in the universe is transferred from white holes to black holes through wormholes	[19]
Salp swarm algorithm	Group behavior of salp chains	[20]
Prairie dog optimization algorithm	Foraging and burrowing behavior of prairie dogs in their natural habitat	[21]
Dwarf mongoose optimization algorithm	Foraging behavior of dwarf mongoose populations	[22]
Gazelle Optimization Algorithm	The behavior of a gazelle to escape from predators	[23]
Harris hawks optimization	Harris hawks predation behavior	[24]
Ebola Optimization Search Algorithm	How Ebola virus disease spreads	[25]

The aforementioned papers use different strategies to improve AO. In this paper, to heighten the optimization performance of AO, the EAO algorithm is proposed. Although the number of iterations in the exploration phase is 2/3 of the total number of iterations in the AO algorithm, the exploitation phase is usually not enough. The elite retention strategy with strong exploitation ability is used to grant exploitation ability greater significance. The optimum value is the primary driver of the AO position update mode; if the optimal value is local optimal, the entire algorithm will switch to local search. The elite opposition-based learning strategy is used to prevent the optimum solution from slipping into the local optimal and to enhance the global aspect of the optimal solution. This increases the population’s diversity and broadens the algorithm’s exploration stage. It also helps the algorithm fairly balance its exploration and exploitation capabilities, thereby raising the likelihood that the global solution will be discovered sooner. After several iterations of solving the function, the population will be approaching the optimal solution. The fixed upper and lower bounds cannot play a good constraint role in the late iteration. So, the dynamic boundary strategy is introduced to balance exploration and exploitation abilities. It also increases convergence speed. The CEC2005 test function optimization is handled by EAO to validate its effectiveness. EAO is also used to solve threshold optimization problems and pressure vessel design problems. Compared with several meta-heuristic algorithms, the superiority and practicability of EAO are fully verified.

The significant ideas are briefly outlined here.

The elite opposition-based learning strategy, dynamic boundary strategy, and elite retention mechanism are used to improve the optimization performance of AO.

On the CEC2005 benchmark function, EAO’s performance is validated by comparison with other optimization algorithms.

Kapur entropy is used as the objective function by five meta-heuristic algorithms, including EAO. Then the threshold is selected according to the image histogram. The results attest to the efficacy of EAO in practical application.

The efficiency of EAO algorithm is examined for solving the engineering design problems.

AO algorithm’s mathematical model is introduced in Sect. 2. Section 3 describes the EAO algorithm. The concepts of Kapur entropy and pressure vessel are described in Sect. 4. The EAO’s superiority is proven in Sect. 5. The summary and prospect are described in Sect. 6.

2 Aquila optimization (AO) algorithm

The AO algorithm, which imitates the seeking and catching prey behavior of the Aquila, is divided into four stages: broad exploration, limited exploration, broad exploitation, and narrowed exploitation.

2.1 Expanded exploration (X1)

The Aquila performs a high soar with a straight stoop in this step. Equation (1) expresses this strategy. $\begin{matrix} po s_{1} (t + 1) = po s_{best} (t) \times (1 - \frac{t}{T}) + \\ (Po s_{M} (t) - Po s_{best} (t) \times rand) \end{matrix}$ (1)

Where Pos₁(t + 1) generated by the first method on t + 1 iteration, Pos_best(t) is the optimal solution obtained by the t iterations, Pos_M(t) is the mean position of the population at the T iterative. N and Dim stand in for the array numbers and array dimensions of the arrow, respectively. $Po s_{M} (t) = \frac{1}{N} \sum_{i = 1}^{N} Po s_{i} (t), \forall j = 1, 2, \dots, Dim$ (2)

where Dim is the dimension size of the problem and N is the number of candidate solution.

2.2 Narrowed exploration (X2)

The Aquila catches the prey in this step by contouring the flight with a short slide strike. Equation (3) expresses this strategy. $P o s_{2} (t + 1) = P o s_{b e s t} (t) \times L e v y (D) + P o s_{R} (t) + (y - x) \times r a n d$ (3)

Where, Pos₂ (t + 1) generated by the second method on t + 1 iteration; Levy (Dim) is the levy distribution function. Pos_R(t) is a random solution in the t iteration; y and x represent the spiral shape in the search; rand is a random value of (0,1). $Levy (Dim) = s \times \frac{u \times σ}{| v |^{\frac{1}{β}}}$ (4)

where s is a constant value fixed to 0.01, u and v are random numbers between 0 and 1. σ is calculated using Equation (5). $σ = (\frac{Γ (1 + β) \times sin e (\frac{π β}{2})}{Γ (\frac{1 + β}{2}) \times β \times 2^{(\frac{β - 1}{2})}})$ (5)

where β is a constant value fixed to 1.5. y and x are used to present the spiral shape in the search, which are calculated as follows. $y = r \times cos (θ)$ (6) $x = r \times sin (θ)$ (7)

where $r = r_{1} + U \times Di m_{1}$ (8) $θ = - ω \times Di m_{1} + θ_{1}$ (9) $θ_{1} = \frac{3 \times π}{2}$ (10)

Where r₁ takes a value between 1 and 20 for fixed the number of search cycles, and U is a small value fixed to 0.00565. Dim₁ is integer numbers from 1 to the length of the search space, and ω is a small value fixed to 0.005.

2.3 Expanded exploitation (X3)

The Aquila executes a low fly with a gradual decline in attack in this step. Equation (11) expresses this strategy. $\begin{matrix} Po s_{3} (t + 1) = (Po s_{best} (t) - Po s_{M} (t)) \times α - \\ rand + ((Ub - Lb) \times rand + Lb) \times δ \end{matrix}$ (11)

Where Pos₃ (t + 1) generated by the second method on t + 1 iteration; The upper and lower bounds of population location is represented Ub and Lb, respectively. α and δ are fixed value of 0.1.

2.4 Narrowed exploitation (X4)

This strategy is known as moving and catching prey. Hunting huge prey use this strategy. Equation (12) is a mathematical expression of this method. $P o s_{4} (t + 1) = QF \times P o s_{b e s t} (t) - (G_{1} \times P o s (t) \times r a n d) - G_{2} \times L e v y (D) + r a n d \times G_{1}$ (12)

Where, Pos₄ (t + 1) generated by the second method on t + 1 iteration; QF represents the quality function used to balance the search strategy. G₁ represents the various movements that a predator uses to track its prey during hunting. G₂ is a decreasing value from 2 to 0 that represents the flight gradient of the AO as it follows the prey from the first to the last position. t and T represent the current and maximum iterations, respectively. $QF (t) = t^{\frac{2 \times rand - 1}{{(1 - T)}^{2}}}$ (13) $G_{1} = 2 \times rand - 1$ (14) $G_{2} = 2 \times (1 - \frac{t}{T})$ (15)

3 Enhanced aquila optimization (EAO) algorithm

The Aquila individuals are represented by the blue particles in Fig. 1, while the prey, or the optimal position, is represented by the red. Figure 1(a) shows that the Aquila individuals are more decentralized in the search space at the beginning of the iteration. Currently, population diversity is better, and the population has powerful global search capabilities.

Fig. 1

The location distribution of the population in different periods.

Figure 1 (b)–(d) shows Aquila individuals influenced by the preceding generation’s position constantly approach optimal as iterative searches carry out. This behavior reduces the diversity of populations. If the optimal particle at this time is a local optimum, AO will easily fall into the local optimum. Therefore, an elite opposition-based learning strategy is introduced to prevent the best solution from falling into the local optimum.

Seeking a balance between exploitation and exploratory abilities is the core of meta-heuristic algorithms. The quality of the optimal solution will be further enhanced if both of abilities are balanced effectively [26, 27]. To better balance the two abilities, a dynamic boundary strategy is presented. Worst Individual fitness will have an impact on the next generation position update and decrease the optimization accuracy during the algorithm iteration period. Therefore, the elite retention strategy is designed to improve the optimization accuracy and global convergence ability. Algorithm 1 shows the pseudocode of EAO. Figure 2 shows the detailed flow chart of EAO.

Fig. 2

Flow chart of EAO.

Algorithm 1 The pseudocode of EAO

Initialize the population Xand the parameters.

While (t < = T)

Calculate the fitness value of the function.

Calculate the best solution Pos_best

Update the best solution Pos_best according to Algorithm 2.

Carry out Dynamic Boundary strategy according to Algorithm 3.

for (i = 1,2 . . . , N) do

Calculate the mean solution X_M(t) for the current iteration.

Update the x, y, G₁, G₂, Levy(D), etc.

ift < = (2/3) *T then

if rand < = 0.5 then

Update the current solution using Equation (1).

Evaluate the fitness value

else

Update the current solution using Equation (3).

Evaluate the fitness value.

End if

else

if rand < = 0.5 then

Update the current solution using Equation (12).

Evaluate the fitness value.

Else

Update the current solution using Equation (13).

Evaluate the fitness value

end if

Carry out Elite Retention mechanism according to Algorithm 4.

end for

end while

return The best solution Pos_best.

3.1 Elite opposition-based learning

Tizhoosh proposed the Opposition-Based Learning (OBL) strategy in 2005. The study found that the reverse solution has a 50% better chance of achieving the global ideal than the current solution [28]. Because elite individuals possess more serviceable information than other popular individuals, the reverse solution generated by elite individuals through opposite learning is more likely to be close to the global optimal solution. This method is called Elite Opposition-Based Learning (EOBL). It not only increases the quality of the solution and the diversity of the population but also prevents the algorithm from entering the precocious.

For the optimal solution Pos_best in the solution space, there is always the corresponding reverse solution Pos_best^*, which is calculated by Eq. (16). If the fitness value of Pos_best^* is better than Pos_best’s fitness value, Pos_best^* is updated to the optimal solution. The pseudocode for this procedure is shown in Algorithm 2. $Po {s_{best}}^{*} = Ub + Lb - Po s_{best}$ (16)

Algorithm 2 The pseudocode of EOBL

Input the current best position Pos_best

Use Eq. (16) to generate the opposite solution Pos_best^* of Pos_best

if (F_obj(Pos_best)>F_obj(Pos_best)) then

Pos_best = Pos_best*

end if

Output Pos_best

3.2 Dynamic boundary strategy

In 2022, Jiankai Xue et al. proposed a dung beetle optimization algorithm by simulations the foraging behavior of dung beetles in nature [29]. To provide a safe environment for their larvae, female dung beetles must carefully select where to lay their eggs. Inspired by this, the author presented a dynamic boundary strategy. Equation (17) shows its mathematical expression, and Fig. 3 shows its conceptual model. The black circle symbolizes the population individuals, the red circle represents the upper and lower boundaries of the search range, and the brown circle represents the current local optimal location Pos_best. The R-value has a significant role in determining the dynamic boundary range. Dynamic boundary can optimize the algorithm’s convergence accuracy and convergence speed. Algorithm 3 shows its pseudocode.

Fig. 3

Conceptual model of dynamic boundary strategy.

$\begin{matrix} L b^{*} = max (Po s_{best} \times (1 - R), Lb), \\ U b^{*} = max (Po s_{best} \times (1 + R), Ub) \end{matrix}$ (17)

Where, Lb^* and Ub^* respectively represent the upper and lower bounds of the optimal hunting region, R = (1-t)/T.

Algorithm 3 The pseudocode of dynamic boundary

Input the maximum iteration number T, the current iteration number t.

R = (1-t)/T

update upper and lower boundary using Eq. (17).

3.3 Elite retention mechanism

The worst individual will impact the algorithm’s ability to optimize, thus the worst individual in the current iteration will be replaced by the best individual. The elite retention mechanism can prevent the best individual from being lost or destroyed. It improves the convergence rate and ability of global convergence of algorithm. And the pseudocode of elite retention mechanism is shown in Algorithm 4. $Po s_{worst} = Po s_{best}$ (18)

Where Pos_worst is the worst individual.

Algorithm 4 The pseudocode of elite retention

Input the best solution Pos_best

Evaluate the worst solution Pos_worst

Relace Pos_best with Pos_worst using Equation (18).

4 Optimization problems

4.1 Threshold optimization based on EAO

The threshold selection method [30] based on the image histogram has outstanding stability and easiest implementation. It is a commonly used parallel region technology. Single threshold selection is to traverse the 0-L gray value of the image, then obtain the result table and sort it after L times of calculation. The gray value corresponding to the optimal result in the result table is the threshold to be selected. Finally, the target and background regions are separated using the best threshold. Twice threshold selected have L*L combinations, so need L² calculations to obtain the optimal threshold. In this manner, L^K times must be calculated for K threshold selection. Calculation time of multi-threshold selection increases exponentially as the number of thresholds increases. The exhaustive search for the optimal threshold is an NP-hard combinatorial optimization problem with a relatively low accuracy [31]. The challenge of choosing the appropriate threshold can be designed as a restricted optimization problem to reduce calculating time. The EAO algorithm optimizes the threshold selected approach as the objective function. Kapur entropy is the most commonly used threshold selection method [32]. Therefore, EAO combined with Kapur entropy is used to select the threshold.

4.1.1 Kapur entropy

In information theory, entropy is a physical variable used to measure the uniformity of distribution. The greater the entropy, the more uniform the distribution [33]. In image, entropy represents tightness and separateness between image categories [34]. Kapur entropy finds threshold pixel by the probability distribution of the gray histogram, so that the distribution of information between the target region and the background region in the image is the largest [35].

For the multilevel threshold selected problem, let the threshold be M = (m₁, m₂, . . . , m_K|0≤m₁ _< m₂ _< ... < m_k≤L), where K is the threshold number. Each threshold corresponds to an entropy value. The entropy values corresponding to the K thresholds are H₀, H₁,... H_K, Kapur entropy expression between different classes is as follows Equation (19): ${\begin{matrix} H_{0} = - \sum_{i = 0}^{m_{1} - 1} \frac{P_{i}}{w_{0}} ln \frac{P_{i}}{w_{0}}, w_{0} = \sum_{i = 0}^{m_{1} - 1} P_{i} \\ H_{1} = - \sum_{i = m_{1}}^{m_{2} - 1} \frac{P_{i}}{w_{1}} ln \frac{P_{i}}{w_{1}}, w_{1} = \sum_{i = m_{1}}^{m_{2} - 1} P_{i} \\ ⋮ \\ H_{K} = - \sum_{i = m_{K}}^{L - 1} \frac{P_{i}}{w_{K}} ln \frac{P_{i}}{w_{K}}, w_{K} = \sum_{i = m_{K}}^{L - 1} P_{i} \end{matrix}$ (19)

The set of thresholds with the maximum entropy of formula Equation (20) is the optimal threshold: ${\vec{m}}^{*} = \underset{0 ⩽ m_{1} ⩽ m_{2} ⩽ \dots ⩽ m_{K} ⩽ L}{arg max} (\sum_{i = 0}^{K} H_{i})$ (20)

4.1.2 Kapur entropy combined with EAO

In this paper, Kapur entropy is regarded as the fitness function of EAO. EAO can reduce the time and space complexity of Kapur entropy finding thresholds. The position of the Aquila is the threshold. The gray level 0 is the lower bounds and gray level 255 is upper bounds in threshold optimization problem. The algorithm flow chart is shown in Fig. 4.

Fig. 4

Threshold optimization flow chart based on EAO.

4.2 Pressure Vessel Design Problem based on EAO

Pressure vessel design is a typical constrained optimization problem. It is challenging to resolve because it is an entirely nonlinear and integer programming problem. The goal of this problem is to minimize the total capacity cost of the container design on the premise of pressure resistance. The pressure vessel diagram is shown in Fig. 5. L is the length of the section, R is the radius of the inner wall, and T_s and T_h are the wall thickness of the cylinder and the wall thickness of the head, respectively. L, R, T_s, and T_h are the four optimization variables of the pressure vessel design problem.

Fig. 5

Pressure vessel diagram.

The objective function of the problem is expressed as follows Equation (22).

$x = [x_{1}, x_{2}, x_{3}, x_{4}] = [T_{s}, T_{h}, R, L]$ (21) $\begin{matrix} min f (x) = 0.622 d_{1} rL + 1.7781 d_{2} r^{2} \\ + 3.1661 {d_{1}}^{2} L + 19.84 {d_{1}}^{2} r \end{matrix}$ (22)

The target constraints are as follows. $g_{1} (x) = - x_{1} + 0.0193 x_{3} ⩽ 0$ (23) $g_{2} (x) = - x_{2} + 0.00954 x_{3} ⩽ 0$ (24) $g_{3} (x) = - π {x_{3}}^{2} x_{4} - \frac{4 π}{3} {x_{3}}^{3} + 1296000 ⩽ 0$ (25) $g_{4} (x) = x_{4} - 240 ⩽ 0$ (26) $0 ⩽ x_{1}, x_{2} ⩽ 100, 10 ⩽ x_{3}, x_{4} ⩽ 100$ (27)

5 Experiment and analysis

When an algorithm is used to solve the test function, it is actually solving the unconstrained optimization problem [36]. According to the search pattern, the smallest value of function corresponds to the location where the eagle is searching for its prey. To verify the feasibility and effectiveness of the proposed EAO algorithm, EAO is compared with other algorithms on sixteen CEC2005 benchmark functions. These comparison algorithms fall into two groups: EAO variants and some classical metaheuristic algorithms. To reduce the error caused by randomness, the mean, standard deviation, and ideal value of each algorithm run separately 30 times are recorded. And the best value of each function is highlighted in bold. To ascertain whether there were statistically significant differences between the data, the Friedman test and Wilcoxon rank sum test were utilized. For a random block design, the Friedman test is used to determine whether there is a difference between various population distributions from various associated samples [37]. It works well with non-parametric, irregularly distributed data. The Wilcoxon rank sum test is based on the rank sum of sample data and is a common nonparametric statistical method used to compare the equality of two related or independent samples [38]. Because it does not make special assumptions about data distribution, it can be applied to complex data distribution. In all test experiments, the maximum number of iterations was set to 500 and the population size to 30. All simulation experiments are based on Windows10 64-bit operating system, 2.40GHz CPU, 8GB memory. And Matlab R2017b software is used to compile and run.

5.1 Ablation analysis: Strategy effectiveness

As described in Sect. 3, EAO improves the performance of AO by introducing three strategies: elite opposition-based learning strategy, dynamic boundary strategy and elite retention mechanism. To verify the effectiveness of the three strategies is the goal of this selection. To do this, this paper carry out three accessional experiments. EAO only adopted the elite opposition-based learning strategy in the first experiment, and named this algorithm as EOBL-AO (Elite Opposition-Based Learning AO). In the second experiment, EAO only adopted the dynamic boundary strategy and named this algorithm DB-AO (Dynamic Boundary AO). In the third experiment, only the elite retention mechanism was used, and the algorithm was named ER-AO (Elite Retention mechanism AO). The parameters of these three algorithms are the same as those of AO. Table 2 lists the results of the five algorithms on 16 benchmark functions and the ranking of the Friedman test. Figure 6 shows the convergence curves of the five algorithms on partial reference functions.

Fig. 6

Convergence comparison diagram of algorithms using different strategies.

Table 2

Experimental results among different algorithm

FUNC	Results	AO	ER-AO	EOBL -AO	DB-AO	EAO
F1	Mean	3.06E-106	6.35E-113	8.68E-112	6.40E-115	2.96E-116
	Std	1.68E-105	3.47E-112	4.75E-111	3.51E-114	1.62E-115
	Best	1.01E-156	5.69E-207	1.12E-156	4.09E-165	7.44E-208
F2	Mean	5.37E-53	2.76E-58	2.96E-54	7.20E-54	5.62E-62
	Std	2.94E-52	1.31E-57	1.62E-53	3.94E-53	2.88E-61
	Best	1.29E-82	6.63E-108	2.81E-80	7.82E-80	2.28E-118
F3	Mean	6.00E-102	4.35E-116	5.71E-102	1.90E-100	1.11E-117
	Std	3.28E-101	2.06E-115	3.13E-101	1.01E-99	5.97E-117
	Best	1.76E-154	1.57E-212	4.70E-160	3.79E-164	1.53E-206
F4	Mean	1.50E-54	1.62E-58	9.55E-55	5.39E-55	5.42E-62
	Std	8.22E-54	8.28E-58	5.19E-54	2.95E-54	2.96E-61
	Best	3.52E-81	3.11E-104	4.74E-84	6.93E-81	5.08E-102
F5	Mean	0.0019	0.0018	8.19E-03	0.0261	0.0298
	Std	0.0027	0.0041	0.0017	0.0397	0.0707
	Best	2.70E-06	5.52E-07	8.00E-06	2.66E-05	3.53E-06
F6	Mean	5.63E-05	2.87E-05	3.30E-05	7.33E-07	6.86E-07
	Std	1.16E-03	4.81E-05	6.29E-05	1.18E-06	1.79E-06
	Best	5.33E-09	4.05E-08	1.87E-07	4.20E-09	2.51E-09
F7	Mean	1.60E-03	9.32E-05	9.69E-05	1.47E-03	9.24E-05
	Std	1.71E-03	9.42E-05	1.21E-03	1.23E-03	1.01E-03
	Best	6.39E-06	6.95E-06	3.14E-06	1.22E-06	1.17E-06
F9	Mean	0	0	0	0	0
	Std	0	0	0	0	0
	Best	0	0	0	0	0
F10	Mean	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16
	Std	0	0	0	0	0
	Best	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16
F11	Mean	0	0	0	0	0
	Std	0	0	0	0	0
	Best	0	0	0	0	0
F12	Mean	3.87E-06	4.31E-06	5.31E-06	7.78E-08	1.49E-07
	Std	5.70E-06	6.89E-06	7.43E-06	1.08E-07	1.87E-07
	Best	4.66E-08	1.48E-09	4.08E-08	1.14E-09	4.01E-10
F13	Mean	1.52E-05	2.40E-05	1.89E-05	4.83E-05	7.49E-05
	Std	3.38E-05	3.46E-05	2.26E-05	8.75E-05	1.07E-03
	Best	1.12E-08	5.50E-08	1.78E-09	7.41E-09	5.65E-07
F14	Mean	3.0201	3.0934	1.3944	3.4478	1.5604
	Std	3.7554	3.2112	0.9227	3.7557	0.8108
	Best	0.9980	0.9980	0.9980	0.9980	0.9980
F15	Mean	5.45E-03	5.51e-03	6.23E-03	5.02E-03	4.83E-03
	Std	1.62E-03	1.65E-03	4.12E-03	1.36E-03	1.31E-03
	Best	3.17E-03	3.28E-03	3.34E-03	3.42E-03	3.22E-03
F16	Mean	–1.0311	–1.0312	–1.0312	–1.0312	–1.0313
	Std	5.69E-03	3.97E-03	4.59E-03	3.93E-03	1.63E-03
	Best	–1.0316	–1.0316	–1.0316	–1.0316	–1.0316
F17	Mean	0.3982	0.3981	0.3982	0.3981	0.3980
	Std	5.43E-03	6.06E-03	3.34E-03	3.13E-03	1.72E-03
	Best	0.3979	0.3979	0.3979	0.3979	0.3979
Friedman		3.5729	2.8021	3.3021	3.1979	2.1250

As can be seen from Table 2, EAO obtains six optimal means, seven optimal standard deviations and six optimal values, and the Friedman test result is the first, indicating that the combination of three strategies can make AO achieve better results. For the unimodal function F1-F2, the elite retention mechanism plays a more promoting role than the other two strategies. For multi-modal functions F12-F14, the elite opposition-based learning strategy provides great help to EAO optimization. For fixed multi-peak function F15-F17, the dynamic boundary strategy improves the convergence accuracy of EAO. From the Friedman test ranking, it can be seen that the algorithms using the three strategies are ranked superior to AO, indicating that the three strategies play a role in most functions, and the elite retention mechanism serving a more promotional effect than the other two strategies.

As shown in Fig. 6, ER-AO converges faster on unimodal functions. In other words, the convergence performance of EAO mainly depends on the elite retention mechanism. On multi-modal functions, the EOBL-AO algorithm and DB-AO algorithm greatly promote the convergence speed of EAO, indicating that the elite opposition-based learning strategy and dynamic boundary mechanism can make EAO have better optimization effect. The convergence graphs of F6 and F12 functions clearly show the ability of the algorithm to jump out of local optimal in the later iteration, indicating that the three strategies can help EAO to get rid of local optimal stagnation. On the fixed multi-peak function, there is almost no difference between the convergence curves of the algorithms, which shows that the three strategies can not only ensure the efficiency of convergence speed, but also improve the convergence accuracy of the algorithms.

5.2 Compare with other meta-heuristic algorithms

To further verify the accuracy and reliability of EAO algorithm, the performances of EAO, AO, Firefly algorithm (FA) [39], Fruit Fly Optimization Algorithm (FOA) [40] and Tunicate Swarm Algorithm (TSA) [41] are compared on 16 benchmark functions. The parameter settings of each algorithm are shown in Table 3. FOA has no extra parameters to set. Table 4 shows the statistical results of the five algorithms and the Friedman rank test results. The convergence curves of the five algorithms are shown in Fig. 7.

Fig. 7

Convergence diagram of five algorithms.

Table 3

Parameter values of each algorithm

Algorithms	Parameters	Value
AO	α	0.1
	δ	0.1
	U	0.0265
	r1	10
	ω	0.005
EAO	α	0.1
	δ	0.1
	U	0.0265
	r1	10
	ω	0.005
FA	Beta	2
	Gamma	0.2
	Alpha_damp	098
TSA	P_min,P_max	1,4
	c₁, c₂, c₃	rand

Table 4

Comparison between EAO and other algorithms on the benchmark function

FUNC	Results	AO	FA	FOA	TSA	EAO
F1	Mean	9.79e-103	0.24035	1284.18	1.38e-39	3.61e-114
	Std	4.99e-102	0.0929	231.94	6.19e-39	1.79e-113
	Best	8.53e-162	0.0799	844.55	2.18e-44	1.32e-222
F2	Mean	1.43e-54	1.0631	15.2217	3.71e-24	4.17e-62
	Std	6.78e-54	0.2062	3.0647	9.36e-24	2.28e-61
	Best	4.03e-79	0.7437	10.3195	1.93e-27	1.91e-109
F3	Mean	4.40e-103	0.3520	4035.59	8.52e-24	4.02e-113
	Std	2.41e-102	0.2089	1480.46	3.68e-23	2.20e-112
	Best	1.17e-156	0.0754	1094.73	2.10e-33	5.96e-211
F4	Mean	5.64e-55	1.2334	20.5134	1.57e-10	9.21e-58
	Std	2.14e-54	1.4074	2.4887	3.14e-10	5.02e-57
	Best	6.44e-80	0.1611	12.3404	1.44e-12	7.47e-104
F5	Mean	0.0024	25.8432	268879.06	8.3712	0.0251
	Std	0.0039	30.9461	110202.70	0.8493	0.0468
	Best	4.91e-07	2.1671	96755.53	5.6547	8.35e-05
F6	Mean	2.82e-05	0.2401	1250.18	1.1218	3.22e-07
	Std	4.26e-05	0.1110	288.20	0.3245	4.56e-07
	Best	1.57e-08	0.1075	630.29	0.5103	1.55e-10
F7	Mean	1.03 e-03	0.0901	0.1957	0.0028	9.61e-05
	Std	6.90e-05	0.0601	0.0626	0.0019	7.81e-05
	Best	1.86e-05	0.0178	0.06047	8.34e-03	1.69e-06
F9	Mean	0	23.4267	61.685	26.8685	0
	Std	0	4.93288	8.3387	11.4177	0
	Best	0	13.5037	45.7997	7.6297	0
F10	Mean	8.88e-16	1.3532	11.7234	2.3467	8.88e-16
	Std	0	0.3525	0.8063	1.5715	0
	Best	8.88e-16	0.5815	10.1667	7.99e-15	8.88e-16
F11	Mean	0	0.1901	12.6609	0.4137	0
	Std	0	0.0795	2.3806	0.2455	0
	Best	0	0.0848	7.2163	0.0089	0
F12	Mean	8.00e-06	0.0794	53.7923	3.4279	1.25e-07
	Std	1.27e-05	0.2117	55.0051	3.6120	1.59e-07
	Best	7.51e-08	0.0027	9.8802	0.0638	2.80e-10
F13	Mean	1.43e-05	0.0510	121133.10	0.7350	8.43e-05
	Std	1.57e-05	0.0339	90106.24	0.2951	1.54e-03
	Best	6.57e-09	0.0156	6512.18	0.2006	1.25e-08
F14	Mean	3.003	1.3279	0.9980	8.6071	1.8909
	Std	3.3726	0.9817	1.50e-06	4.5516	0.9869
	Best	0.9980	0.9980	0.9980	0.9980	0.9980
F15	Mean	4.88e-03	9.28e-03	0.0026	0.0115	4.81e-03
	Std	1.27e-03	2.65e-03	0.0028	0.0199	1.06e-03
	Best	3.24e-03	6.08e-03	0.0011	3.07 e-03	3.29e-03
F16	Mean	–1.0312	–1.0315	–1.0305	–1.0274	–1.0313
	Std	4.09e-03	2.45e-03	0.0011	0.0109	4.55e-03
	Best	–1.0316	–1.0316	–1.0316	–1.0316	–1.0316
F17	Mean	0.3980	0.3979	0.3981	0.3979	0.3980
	Std	2.85e-03	2.71e-05	2.14e-03	8.01e-05	2.06 e-03
	Best	0.3980	0.3979	0.3979	0.3979	0.3980
Friedman		2.0625	3.03125	4.78125	3.5625	1.5625

As can be seen from Table 4, on unimodal function, EAO is optimal compared with other algorithms except for function F5. On multimodal functions, EAO reaches theoretical optimality for both F9 and F11 functions. However, on functions F13 and F14, EAO is inferior to AO and FA, respectively. On the fixed multimodal function F16, EAO is less than an order of magnitude worse than FA. As can be seen from the Friedman test ranking, the average ranking of EAO is the first. In most functions, EAO has better optimization ability than other algorithms.

As shown in Fig. 7, EAO converges quickly before 50 iterations, indicating that EAO is highly sensitive to the optimal value and can quickly find the optimal value region. For F15 function, the early convergence rate of EAO is lower than FA, but the convergence curve of EAO has many turning points, indicating that EAO has the ability to jump out of the local optima. After 400 iterations, the convergence speed and convergence accuracy of EAO are gradually better than that of FA. Every turn of the convergence curve of functions F6, F9, F12 reflects the ability of the algorithm to jump out of the optimal. Compared with the four algorithms, EAO has advantages in convergence speed and convergence accuracy, but it needs to be further enhanced in F5, F13 functions optimization problems.

5.3 Statistical test

To determine whether the EAO results are statistically significantly different from the best results of other algorithms, the Wilcoxon rank sum test [42] was used in this paper, and was conducted at the significance level of 0.05. Table 5 is plays the Wilcoxon rank sum test P-values for EAO and other methods on 16 benchmark functions. “R” is the result of significance judgment. The performance of EAO is better than, worse than or equal to other algorithms is respectively symbolized as “+”, “–” and “=”. NAN indicates that the optimization indexes of EAO algorithm and comparison algorithm are equal, and significance judgment cannot be made. p < 0.05 indicates that there are significant differences between the two comparison algorithms.

Table 5
Wilcoxon rank sum test results of EAO and other algorithms

FUNC AO FA FOA TSA

P R P R P R P R

F1 0.0083 + 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F2 0.0026 + 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F3 0.0053 + 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F4 0.0072 + 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F5 2.1765e-05 – 3.3384e-11 + 3.0199e-11 + 3.0199e-11 +

F6 3.9648e-08 + 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F7 0.4553 = 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F9 NAN = 1.2118e-12 + 1.2118e-12 + 1.2118e-12 +

F10 NAN = 1.2118e-12 + 1.2118e-12 + 1.2009e-12 +

F11 NAN = 1.2118e-12 + 1.2118e-12 + 1.2118e-12 +

F12 5.5727e-10 + 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F13 0.0615 = 3.0199e-11 + 3.0199e-11 + 3.0199e-11 +

F14 0.0487 + 7.9820e-08 – 0.5895 = 7.0852e-08 +

F15 0.9234 = 1.9568e-10 + 3.0199e-11 + 0.2519 =

F16 0.2643 = 2.6320e-04 – 3.7704e-04 + 5.5386e-07 –

F17 0.4733 = 4.1460e-06 – 0.0575 = 0.0701 =

FUNC	AO	FA	FOA	TSA
F1	0.0083	+	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F2	0.0026	+	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F3	0.0053	+	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F4	0.0072	+	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F5	2.1765e-05	–	3.3384e-11	+	3.0199e-11	+	3.0199e-11	+
F6	3.9648e-08	+	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F7	0.4553	=	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F9	NAN	=	1.2118e-12	+	1.2118e-12	+	1.2118e-12	+
F10	NAN	=	1.2118e-12	+	1.2118e-12	+	1.2009e-12	+
F11	NAN	=	1.2118e-12	+	1.2118e-12	+	1.2118e-12	+
F12	5.5727e-10	+	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F13	0.0615	=	3.0199e-11	+	3.0199e-11	+	3.0199e-11	+
F14	0.0487	+	7.9820e-08	–	0.5895	=	7.0852e-08	+
F15	0.9234	=	1.9568e-10	+	3.0199e-11	+	0.2519	=
F16	0.2643	=	2.6320e-04	–	3.7704e-04	+	5.5386e-07	–
F17	0.4733	=	4.1460e-06	–	0.0575	=	0.0701	=

As can be seen from Table 5, EAO performs much better than other algorithms on unimodal functions. This demonstrates the reliability of EAO’s test results and demonstrates EAO’s ability to exploit. On the F9-F11 multimodal function, there is not a significant distinction between EAO and AO. On the fixed multimodal function F15-F17, the R value of EAO versus AO is “=”. The optimization performance of EAO is better than that of AO in terms of mean value, variance, and ideal value, but there is no discernible difference between them because the improvement space is not very wide. Compared with FA, the R value of EAO on F16-F17 is “–”, which indicates that EAO has weak exploration ability on fixed multimodal functions.

5.4 Time complexity analysis

The efficiency of an algorithm can be measured by the time complexity and the space complexity. In most cases, the time complexity of the algorithm is compared. As can be seen from the flow chart in Fig. 2, EAO mainly consists of the following parts.

Population initialization, time complexity is O(N).

Calculate the fitness function and the global optimal solution, and the time complexity is O(f(Dim)).

The time complexity of updating population location is O(T*N)+O(T*N*Dim); Both the reverse elite solution and the elite retention time complexity are O (1).

To determine whether the algorithm stops iteration, the time complexity is O (1).

Therefore, the time complexity of EAO algorithm is O(N*(T*Dim+1)), which is the same as the time complexity of AO algorithm [14], which means that EAO does not sacrifice operation efficiency in exchange for performance improvement.

5.5 Application for threshold optimization

To further verify the effectiveness of EAO, four groups of test images [43] as shown in Fig. 8 were selected for experiments.

Fig. 8

The selected images and its corresponding histogram.

Three standard metrics, Peak Signal to Noise Ratio (PSNR) [44], Structural Similarity index (SSIM) [45], and Feature Similarity index (FSIM) [46], are chosen to quantitatively evaluate the selected threshold performance of each algorithm, as shown in Tables 6–8. The larger the three indicators, the better the selected threshold.

Table 6–8 shows that the Kapur entropy based EAO algorithm achieves greater FSIM and SSIM values than other algorithms for K = 8,10. It shows that EAO is more effective in high-dimensional threshold. The PSNR value obtained cannot clearly show the advantage of EAO. Therefore, to reflect the differences among algorithms, Friedman test was performed on the data in Table 6–8. The test results are shown in Fig. 9. Figure 9 shows that the ranking of the three indicators is highest when the threshold value of EAO is 8, 10, demonstrating that EAO is more favorable and stable in the selection of high-dimensional threshold.

Fig. 9

Friedman test ranking of five algorithms on FSIM, SSIM, PSNR.

Table 6

FSIM values of image segmentation by different algorithms

Images	k	EAO	AO	FA	FOA	TSA
Camera man	6	0.90332	0.905	0.8963	0.896	0.8812
	8	0.9312	0.919	0.9213	0.9154	0.9081
	10	0.9412	0.9389	0.9247	0.918	0.9139
Starfish	6	0.8494	0.8492	0.839	0.8217	0.8363
	8	0.8882	0.8843	0.8667	0.8684	0.8742
	10	0.9092	0.9097	0.8865	0.8641	0.8846
Lena	6	0.8541	0.8545	0.8591	0.8578	0.8515
	8	0.9149	0.9115	0.8913	0.9059	0.9034
	10	0.9217	0.9214	0.9365	0.9069	0.9165
Barbara	6	0.9049	0.8991	0.905	0.9032	0.898
	8	0.93	0.9024	0.9279	0.9265	0.9079
	10	0.9395	0.9213	0.9389	0.9173	0.926

Table 7

SSIM values of image segmentation by different algorithms

Images	k	EAO	AO	FA	FOA	TSA
Camera man	6	0.7287	0.7175	0.7334	0.7249	0.7261
	8	0.7663	0.7659	0.7455	0.7426	0.7542
	10	0.8427	0.796	0.7784	0.7582	0.8146
Starfish	6	0.7706	0.7218	0.7715	0.7769	0.775
	8	0.8136	0.787	0.815	0.7926	0.8136
	10	0.8459	0.828	0.8402	0.8388	0.8363
Lena	6	0.6449	0.6585	0.6524	0.6501	0.65
	8	0.7878	0.769	0.7715	0.7699	0.7258
	10	0.8308	0.7536	0.8044	0.8104	0.7481
Barbara	6	0.7578	0.7605	0.748	0.7643	0.7343
	8	0.8002	0.7573	0.7869	0.7838	0.7544
	10	0.8375	0.7763	0.8271	0.7999	0.8228

Table 8

PSNR values of image segmentation by different algorithms

Images	k	EAO	AO	FA	FOA	TSA
Camera man	6	24.4094	22.7624	24.1316	23.8859	23.7325
	8	24.8922	24.5101	25.8176	25.5739	25.3855
	10	27.0408	25.4812	27.694	25.0875	25.4891
Starfish	6	21.3892	21.5219	19.9297	21.3032	21.5001
	8	23.5823	23.4316	22.4047	22.1578	23.2416
	10	24.8432	24.9214	22.1333	23.2267	23.8527
Lena	6	18.7545	18.8541	18.8699	18.8515	18.8648
	8	23.3915	22.285	23.3599	22.686	21.0186
	10	25.9145	24.5479	21.3003	24.2848	21.4907
Barbara	6	22.0839	21.9585	21.9171	21.9329	21.567
	8	24.0095	22.1828	23.6478	24.4523	22.6023
	10	25.5066	23.28	25.2622	23.9509	24.277

5.6 Application for pressure vessel design problem

Set the population dimension to 4, which is the fourth parameter of x. The value ranges of x_1, x_2, x₃ and x₄ are set to the upper boundary [100,100,100,100] and lower boundary [0,0,10,10] of the population. If x_1, x_2, x₃ and x₄ satisfy the target constraint, the position of the individual in the population is the value of x_1, x_2, x₃ and x₄.

Through 30 independent experiments, the mean value and average running time of each algorithm were calculated. The Table 9 shows that the fitness value of EAO is the smallest compared with other algorithms; this means that using EAO to find a set of pressure vessel parameters that adhere to the constraints will reduce the cost of production. TSA and FOA take less time than EAO in terms of time. It shows that EAO sacrifices convergence speed in exchange for higher convergence accuracy. The convergence speed of EAO can be effectively improved by improving population initialization, optimizing individual iteration updates by strategy, or combining the advantages of the two algorithms.

Table 9
The results of different algorithms in pressure vessel design

Algorithm x ₁ x ₂ x ₃ x ₄ Mean Time

AO 1.332 0.654 67.776 10.432 8376.993 0.1213

FA 1.317 0.649 67.868 10.080 8271.656 1.2219

FOA 1.341 1.211 67.390 13.740 13038.902 0.0769

TSA 1.315 0.654 67.618 10.647 8291.342 0.0349

EAO 1.311 0.651 67.515 10.227 8201.342 0.1157

Algorithm	x ₁	x ₂	x ₃	x ₄	Mean	Time
AO	1.332	0.654	67.776	10.432	8376.993	0.1213
FA	1.317	0.649	67.868	10.080	8271.656	1.2219
FOA	1.341	1.211	67.390	13.740	13038.902	0.0769
TSA	1.315	0.654	67.618	10.647	8291.342	0.0349
EAO	1.311	0.651	67.515	10.227	8201.342	0.1157

6 Conclusion

In this paper, by adding elite opposition-based learning, boundary selection, and elite retention mechanisms to the AO algorithm, a new algorithm, EAO, is proposed. The algorithm avoids the population falling into the local optima, guarantees population diversity, and has the advantages of fast convergence speed and high convergence accuracy. According to the experimental results on the CEC2005 test set, the convergence speed and accuracy of EAO are better than those of AO and other meta-heuristic algorithms. To further verify its performance, EAO is used to deal with threshold optimization problems and pressure vessel design problems. The PSNR, SSIM, and FSIM values and Friedman test rankings obtained by EAO at high-dimensional thresholds are the highest. In the design of pressure vessels, EAO can achieve lower design costs. At the same time, EAO has some shortcomings. On most functions, EAO does not reach the theoretical optimal value. Although the mean and variance of EAO were better than AO on F15–F17, there was no statistically significant difference on the Wilcoxon rank sum test. EAO has the problem of low precision in the selection of low-dimensional thresholds. Compared with other algorithms, it takes too long to solve the pressure vessel problem. By getting the algorithm multi-targeted and incorporating various local search and global exploitation methods into EAO, it hopes that future studies will increase the convergence accuracy, optimization ability, and robustness of the algorithm and then apply it to new application domains.

Footnotes

Acknowledgments

This study is supported by the Natural Science Foundation of China under Grant No.62273290.

Compliance with ethical standards

The authors confirm that no conflict of interest has existed.

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