No optimization algorithm can obtain satisfactory results in all optimization tasks. Thus, it is an effective way to deal with the problem by an ensemble of multiple algorithms. This paper proposes an ensemble of population-based metaheuristics (EPM) to solve single-objective optimization problems. The design of the EPM framework includes three stages: the initial stage, the update stage, and the final stage. The framework applies the transformation of the real and virtual population to balance the problem of exploration and exploitation at the population level and uses an elite strategy to communicate among virtual populations. The experiment tested two benchmark function sets with five metaheuristic algorithms and four ensemble algorithms. The ensemble algorithms are generally superior to the original algorithms by Friedman’s average ranking and Wilcoxon signed ranking test results, demonstrating the ensemble framework’s effect. By solving the iterative curves of different test functions, we can see that the ensemble algorithms have faster iterative optimization speed and better optimization results. The ensemble algorithms cannot fall into local optimum by virtual populations distribution map of several stages. The ensemble framework performs well from the effects of solving two practical engineering problems. Some results of ensemble algorithms are superior to those of metaheuristic algorithms not included in the ensemble framework, further demonstrating the ensemble method’s potential and superiority.
Ensemblepopulation-based metaheuristicsreal and virtual populationelite strategyswarm intelligenceNational Natural Science Foundation of China62073330Introduction
Multiple optimization techniques have been proposed to address the challenges of various optimization problems, which include accurate and approximate methods [1]. Accurate methods obtain an exact solution to guarantee optimality. Approximate methods can generate a high-quality solution in a reasonable time to meet the actual requirements, but they cannot ensure a globally optimal solution. As one type of approximate method, metaheuristics are general for solving optimization problems. Metaheuristics include single-solution-based and population-based by the number of solutions processed in the optimization phase [2].
Although engineers apply meta-heuristic algorithms to various optimization problems, the solution to a specific optimization problem is only sometimes satisfactory. Therefore, researchers widely use integration strategies to design general algorithms for particular problem sets. Many practical and available algorithms have been developed for single-objective optimization [3], constrained optimization [4], multi-objective optimization [5], niche [6], and others. Metaheuristic algorithms have significantly been developed and applied to solving optimization problems in many fields [7], such as scheduling [8], data mining [9], and unmanned system [10].
When engineers integrate various optimization algorithms, they have to design appropriate integration schemes to take advantage of their respective strengths. The paper proposes an ensemble of population-based metaheuristics (EPM) to solve the single-objective problem. We consider each population-based metaheuristic algorithm as a whole in the ensemble framework without understanding its characteristics and internal mechanisms. The ensemble framework puts forward the transformation between the real and virtual populations to coordinate exploration and exploitation at the population level. It also adopts an elite strategy to realize information exchanges among virtual populations to enhance the diversity of populations. The transformation between the real and virtual populations mainly solves the problem of falling into local optimum. We also apply an elite strategy to speed up the convergence. The practical issues that the ensemble framework can solve are related to the application field of its integrated meta-heuristic algorithms. If a meta-heuristic algorithm can be applied to solve a particular practical problem, then the algorithm can still solve the optimization problem in the ensemble framework.
The rest of this paper is structured as follows. Section 2 summarizes the ensemble methods of the population-based metaheuristic algorithms. Section 3 describes the design of the EPM framework and how to use it to integrate various metaheuristic algorithms. Section 4 conducts testing and analysis, where it tests the original and ensemble algorithms using two test function sets and two engineering application problems. Finally, in Section 5, the full text is summarized, pointing to future research.
Related WorkPopulation-Based Metaheuristics
Population-based metaheuristics have the same paradigm, as shown in Fig. 1. A population represents a set of solutions, and each individual represents a feasible solution. Generation strategies are designed to generate a new population, and the original individual is replaced by the newly generated individual according to selection strategies so that the population is continuously optimized until the set stop condition is met. The various population-based metaheuristic algorithms differ in how they perform the generation and selection processes.
The typical process diagram of population-based metaheuristics
The design of population-based metaheuristics is inspired by nature and human society, including evolutionary processes, physics, biology, chemistry, mathematics, human bodies, society, and so on. Evolutionary algorithms simulate the process of biological evolution, using crossover, mutation, reproduction, and other operators to produce better solutions. Typical algorithms include differential evolution [11], genetic algorithm [12], and genetic programming [13]. Biology-based algorithms mimic the behaviors in biological groups, such as foraging, mating, and movement, including particle swarm optimization [14], artificial bee swarm optimization [15], cuckoo search [16], and sand cat swarm optimization [17]. Human-based algorithms mainly simulate human body structure, way of thinking, and social behavior, including tabu search [18], cooperation search [19], brainstorming algorithms [20], heap-based optimization [21], and war strategy optimization [22]. Physics-based algorithms draw lessons from the classical laws of physics. For example, the idea of gravity search comes from the law of gravity [23]. The big bang algorithm simulates the big bang effects and the big bang processes [24]. The electromagnetic mechanism-like algorithm simulates charged particles’ attraction and repulsion mechanisms in the electromagnetic field [25]. The algorithm based on mathematics draws lessons from mathematical formulas and mathematical theorems, such as the sine cosine algorithm based on the mathematical properties of the sine and cosine functions [26], the chaotic golden section algorithm based on chaotic motion and golden section [27] and the arithmetic optimization algorithm based on the main arithmetic operators in mathematics [28].
Engineers apply metaheuristic algorithms in many fields, such as engineering, material, robot, computer, home life, and medical treatment. Chopra et al. proposed a golden jackal optimization algorithm for the economic load dispatch problem in electrical engineering [29]. Zhang et al. designed a generalized normal distribution optimization to improve the accuracy of extracting the model parameters in the photovoltaic material industry [30]. Ibrahim et al. proposed a hybrid wind-driven-based fruit fly optimization algorithm for adjusting several model parameters in the double-diode cell material industry [31]. Tang et al. combined fuzzy C-means clustering and ant colony optimization algorithm for mission planning of unmanned aerial vehicles in emergent scenarios [32]. Tian applied a backtracking search optimization algorithm to adjust the algorithm parameters of the least square support vector machine in the computer field [33]. Jordehi designed a quadratic binary particle swarm optimization to schedule multiple interruptible and non-interruptible appliances in an intelligent home [34]. Nadimi-Shahraki et al. applied an enhanced whale optimization algorithm to detect coronavirus disease 2019 [35].
The search steps of these algorithms include exploration and exploitation phases [36]. In the exploration phase, it ensures the diversity of the population as much as possible so that the individuals in the population can fully explore each area of the feature space. If there is insufficient exploration, the solution is locally optimal. In the exploitation phase, the search should focus on areas with high-quality solutions, constantly searching the neighborhood. If underexploited, there is no way to converge to a global solution.
Ensemble Approaches of Population-Based Metaheuristics
A single algorithm cannot handle all the optimization problems [37], so the ensemble of multiple meta-heuristic algorithms can increase the effect of optimization. The ensemble technology of the meta-heuristic algorithms includes low-level and high-level ensembles depending on the objects. The low-level ensemble integrates multiple search strategies and variable parameter values of a single algorithm. The high-level ensemble integrates multiple meta-heuristic algorithms and their variants.
An algorithm may be excellent at solving one type of optimization problem and not good at solving another. The search strategy of the algorithm mainly expresses this particularity. Therefore, to enhance the effect of the algorithm on optimization problems that the algorithm is not good at, a variety of search strategies can be integrated to improve the algorithm to obtain a better solution. Qin et al. proposed a self-adaptive differential evolution algorithm that randomly assigned the experimental vector generation strategy and the control parameters to population individuals based on the characteristics of the historical solution [38]. Pan et al. designed an adaptive differential evolution algorithm based on the hybrid leapfrog strategy, which can maintain the overall diversity of the population during dynamic evolution [39]. Gong et al. applied a multi-operator search strategy based on a cheap agent model, using multiple sub-generation search operators and proxy models to select the best candidate solution [40]. Ali et al. divided the population into several subgroups, each adopting a modified mutation strategy based on differential evolution [41]. Rakhshani et al. proposed a new extension of the cuckoo algorithm, which combines the exploration capabilities of computer science with the development behavior provided by the covariance matrix adaptation evolution strategy through multiple search strategies [42]. Li et al. designed an adaptive learning framework for particle swarm optimization to select the optimal method from some strategies to deal with different search spaces [43].
When the set of optimization problems is vast, a single swarm intelligence optimization algorithm cannot handle it. So researchers ensemble multiple algorithms to solve optimization problems. Lynn et al. proposed an evolutionary particle swarm optimization algorithm that integrates different particle swarm optimization algorithms [44]. Zhang et al. proposed a multivariable coordination framework, which coordinates multiple improved differential evolution algorithms [45]. Thangavelu et al. designed an island-based dynamic data exchange algorithm mixing four classical differential evolution variants that fill each island with different dynamic data exchange variables as a possible way to implement a robust dynamic data exchange system [46]. Wu et al. proposed an algorithm based on an ensemble of multiple differential evolution variants, which divides the whole population into index and reward subpopulations and allocates the reward subpopulation dynamics to the well-performing differential evolution variant algorithm according to the performance of the index subpopulation [47]. Elsayed et al. applied a group of different particle swarm optimization variants to integrate and measure the performance of varying optimization variants according to the improvement index to adaptively determine the number of individuals allocated to the variant algorithm in each generation [48]. Vrugt et al. applied an adaptive learning strategy that integrates evolution strategy, genetic algorithm, and particle swarm optimization to dynamically adjust the influence of these three algorithms on individuals [49]. Xue et al. proposed an integrated evolutionary algorithm using an adaptive learning search technique incorporating three different algorithms [50]. Elsayed et al. designed a framework with multiple adaptive algorithms and operators, using two decisions to adaptively select the appropriate algorithm and search operator [51]. The existing ensemble frameworks are usually designed for specific algorithms and their variants, focusing on complementarity and neglecting generality.
In this paper, we design an ensemble framework based on common structural features of metaheuristic algorithms rather than a specific algorithm. The ensemble framework belongs to the high-level ensemble, which could integrate various meta-heuristic and variant algorithms. These algorithms can have multiple search strategies, including low-level ensembles.
Ensemble of Population-Based Metaheuristics
This section introduces the EPM framework in detail. Section 3.1 explains the idea and features of the framework. Section 3.2 shows the stages of using the EPM framework to integrate the metaheuristic algorithm. It presents the implementation procedure and pseudocode in Section 3.3.
Basic ConceptsAlgorithm Library
After filling in multiple meta-heuristic algorithms, the ensemble framework can only solve optimization problems. These meta-heuristic algorithms are placed together in the algorithm library. It is hard to judge which population-based metaheuristic algorithm is the best for a particular optimization problem, especially when there is not enough information about the problem. While engineers can find better algorithms through trial and error, it would be time-consuming and labor-intensive. In addition, to solve some problems, a satisfactory solution cannot be obtained through a search strategy using a single algorithm. Neither the specific transition point from one algorithm to another nor which algorithms to apply is known. Each algorithm has its characteristics: local search, global search, optimization speed, and population diversity. The role of the EPM framework is to make all kinds of algorithms give full play to their respective strengths and provide a communication bridge for these algorithms. It requires that as many algorithms as possible be included in the algorithm library.
Real and Virtual Populations
The EPM framework proposes the setting and conversion of real and virtual populations. Virtual populations represent various virtual states of the population obtained by the evolution of the real population in different directions. All virtual populations come from the real population. Each virtual population applies a population-based metaheuristic algorithm, which updates the population by selection rules to obtain individuals with better fitness values. Each virtual population can exchange information to facilitate population optimization. When certain conditions are satisfied, it retains the optimal virtual population as the optimal population, which materializes into a real population. After that, it uses the same operation to virtualize and set up the next population that meets certain conditions, constantly optimizing the population until reaching the stop condition. The adjustment of parameters for each swarm intelligence algorithm is not considered, nor is the specific update strategy within the population. The setting of the real and virtual populations is a new idea for coordinated exploration and exploitation at the population level.
Elite Strategy
The exchange of information among virtual populations enables the population to obtain satisfactory solutions more quickly in the renewal process, reducing the incidences of falling into local optima. The algorithm framework designed in this study uses an elite strategy to communicate between virtual populations. To minimize the interference in the virtual population, the individual with the smallest fitness value is an elite in each virtual population. If multiple individuals have the same fitness value, it randomly selects one as the elite. It brings these elites together to form a group and selects the best elite according to their fitness. When meeting certain conditions, the best elite individual is added to each virtual population to complete the exchange of information between populations. In this way, it reduces the interference of virtual populations and realizes population optimization with the help of the optimal individual’s influence on the population.
The Framework of EPM
The EPM framework consists of three stages: the initial stage, the update stage, and the final stage. The update stage includes two sub-stages high-level generation and selection. Fig. 2 shows the entire process diagram of EPM. These three stages are described in detail below.
The entire process diagram of EPM
The Initial Stage
The search space dimension of the optimization problem is D. Its upper boundary U is (u1,u2,…,ud,…,uD), and its lower boundary L is (l1,l2,…,ld,…,lD). The number of individuals in the population is N, and the population set X is {X1,X2,…,Xn,…,XN}. The position of the nth individual Xn is (x1n,x2n,…,xdn,…,xDn), where xdn is calculated according to Eq. (1), r is a random number in the [0,1].
It selects the required algorithm in the algorithm library, and the number of algorithms selected is M. The set of algorithms A is {A1,A2,…Am,…,AM}. R represents the real population, and XR(t) represents the population at the tth iteration. The virtual population set V is {V1,V2,…,Vm,…,VM}. XVm(t) represents the virtual population corresponding to the mth algorithm at the tth iteration. When t=0, XR(t) and XVm(t) are obtained from Eq. (2).
xdn=ud+r(ud−ld)XVm(0)=XR(0)=X(0),m=1,2,3,…M
The Update Stage
At the high-level generation process, each virtual population adopts the corresponding algorithm to update separately, where their internal update strategy set Rw is {Rw1,Rw2,…,Rwm,…,RwM}. Rwm represents the internal update strategy of the algorithm Am, and the update process of the virtual population position XVm is shown in Fig. 3.
The updating process diagram of the virtual population position
At the (t+1)th iteration, XVm′(t+1) is obtained from XVm(t) through the generation and selection strategy within the algorithm Am, as shown in Eq. (3). However, it is not the virtual population for the next update XVm(t+1), which needs to be determined according to the high-level selection process.
XVm′(t+1)=Rwm(XVm(t))
The entire high-level selection process is shown in Fig. 4. The elite swarm consists of individuals corresponding to the minimum fitness value in each virtual population. The set of these fitness values E is {E1,E2,…,Em,…,EM}, where Em represents the minimum fitness value in the mth virtual population. At the (t+1)th iteration, Em(t+1) is calculated by Eq. (4), where f() represents calculating the fitness value. B is the minimum in the set E, and B(t+1) is calculated by Eq. (5). Then, the individual with the minimum fitness value of the elite swarm B is considered as the best elite. Its corresponding virtual population is called the optimal virtual population Vb and its corresponding individual position is XeVb. If there are multiple individuals corresponding to the minimum fitness value, random selection is performed.
H is the minimum fitness value of the best elite in history. When B(t+1)≥H, all virtual populations are preserved, and no information exchange is carried out among them. The virtual populations to be updated next time is shown in Eq. (6).
XVj(t+1)=XVj′(t+1),j=1,2,…M
When B(t+1)<H, if B(t+1)≪H, the optimal virtual population Vb is materialized to the real population. The real population generates new virtual populations to be updated next time, as shown in Eq. (7). In this process, the parameters in the algorithms corresponding to the original virtual populations are retained. In this paper, it can be determined that the difference is large if one of the following three conditions is met: {B(t+1)×H>0,B(t+1)>0,a≤0.1}, {B(t+1)×H>0,H<0,a≥10}, {B(t+1)×H<0,eB(t+1)<eH}, where a is the ratio of B(t+1) to H, as shown in Eq. (8). If B(t+1) is not much different from H, the best elite is copied and distributed to other virtual populations and an individual in each virtual population is randomly replaced in the process. For virtual population Vm, q is a random integer in [1,N], representing a random individual which will be replaced by the best elite. The improved population will be updated in the next time as the new virtual populations, and the process is shown in Eq. (9).
XVj(t+1)=XR(t+1)=XVb′(t+1),j=1,2,…,Na=B(t+1)H,H≠0XVj(t + 1)⟵XqVj′(t+1)=XeVb(t+1)XVj′(t+1),j=1,2,…,M
The Final Stage
The update stage repeats until reaching the maximum of iterations T, then the whole process ends. In the last iteration, the optimal virtual population Vb is materialized to obtain the real population, as shown in Eq. (10). After the last iteration, the optimal virtual population is transformed into the real population.
XR(T)=XVb(T)
General Process of EPM
The ensemble range covered by the EPM framework includes all population-based metaheuristic algorithms and their modified versions. The larger the number of algorithms in the algorithm library, the better the possibility of covering many aspects of the optimization problem. First, it selects an appropriate number of algorithms from the algorithm library for the ensemble. In this process, the number of algorithms can be set without prior knowledge and then selected randomly. In the case of specific prior knowledge, it chooses the appropriate algorithm according to the previous knowledge. The pseudocode of EPM is presented below:
Experiment Results
It applies ensemble algorithms to two test sets, 23 standard benchmark functions [52] and the 2017 congress on evolutionary computation (CEC2017) benchmark functions [53]. Section 4.1 describes these functions in detail. Section 4.2 presents the five algorithms in the algorithm library along with a comprehensive comparison between the integrated and original algorithms, introducing the parameters of each algorithm in the experiment. Section 4.3 illustrates the experimental results of the nine algorithms. It gives the iterative curves of all algorithms on some functions in Section 4.4 to directly observe the optimization effect and convergence of the algorithms. Section 4.5 shows the evolution of the real and virtual populations in the optimization process. Section 4.6 applies the ensemble framework to solve two engineering application problems.
Benchmark Functions
We select two sets of benchmark functions. One set consists of the 23 standard benchmark functions of the experiment, which are described in detail in Table 1, including function type, function name, and function definition. D denotes the number of independent variables in the function, and the initialization range means the range of variables. The optimal value corresponding to each function is the global optimum. In these functions, F1–F13 are high-dimensional with a dimension of 30; F1–F7 are unimodal; F8–F13 are multi-peak; F14–F23 are low-dimensional with only a few local minima. The other set is the CEC2017 benchmark functions set, as described in Table 2. Among these functions, C1–C2 are unimodal; C3–C9 are simple multi-peak; C10–C19 are mixed; C20–C29 are compound. The variable dimensions range from [−100, 100]. We set the dimensions uniformly to 30 and 50.
23 standard benchmark functions [<xref ref-type="bibr" rid="ref-52">52</xref>]
Function type
No.
Function name
D
Initialization range
Global optimum
F1
Sphere model
30
−100≤xi≤100
0
F2
Schwefel’s problem 2.22
30
−10≤xi≤10
0
F3
Schwefel’s problem 1.2
30
−100≤xi≤100
0
Unimodal test functions
F4
Schwefel’s problem 2.21
30
−100≤xi≤100
0
F5
Generalized Rosenbrock’s function
30
−30≤xi≤30
0
F6
Step function
30
−100≤xi≤100
0
F7
Quartic function
30
−1.28≤xi≤1.28
0
F8
Generalized Schwefel’s problem 2.26
30
−500≤xi≤500
−418.9829×d
F9
Generalized Rastrigin’s function
30
−5.12≤xi≤5.12
0
F10
Ackley’s function
30
−32≤xi≤32
0
Multimodal test functions
F11
Generalized Griewank function
30
−600≤xi≤600
0
F12
Generalized Penalized functions
30
−50≤xi≤50
0
F13
Generalized Penalized functions
30
−50≤xi≤50
0
F14
Shekel’s Foxholes function
2
−65.53≤xi≤65.53
1
F15
Kowalik’s function
4
−5≤xi≤5
0.0003
F16
Six-hump camel-back function
2
−5≤xi≤5
−1.0316
F17
Branin function
2
−5≤xi≤10,0≤x2≤15
0.398
Multimodal test functions with fix dimension
F18
Goldstein-price function
2
−5≤xi≤5
3
F19
Hartman’s family
3
0≤xi≤1
−3.86
F20
Hartman’s family
6
0≤xi≤10
−3.32
F21
Shekel’s family
4
0≤xi≤10
−10.1532
F22
Shekel’s family
4
0≤xi≤10
−10.4028
F23
Shekel’s family
4
0≤xi≤10
−10.5363
The CEC2017 benchmark functions [<xref ref-type="bibr" rid="ref-53">53</xref>]
Function type
No.
Function name
Dim
Initialization range
Global optimum
Unimodal functions
C1
Shifted and rotated bent cigar function
30, 50
−100≤xi≤100
100
C2
Shifted and rotated Zakharov function
30, 50
−100≤xi≤100
200
C3
Shifted and rotated Rosenbrock’s function
30, 50
−100≤xi≤100
300
C4
Shifted and rotated Rastrigin’s function
30, 50
−100≤xi≤100
400
C5
Shifted and rotated expanded Scaffer’s F6 function
30, 50
−100≤xi≤100
500
Simple multimodal functions
C6
Shifted and rotated Lunacek Bi_Rastrigin function
30, 50
−100≤xi≤100
600
C7
Shifted and rotated non-continuous Rastrigin’s function
30, 50
−100≤xi≤100
700
C8
Shifted and rotated Levy function
30, 50
−100≤xi≤100
800
C9
Shifted and rotated Schwefel’s function
30, 50
−100≤xi≤100
900
C10
Hybrid function 1
30, 50
−100≤xi≤100
1000
C11
Hybrid function 2
30, 50
−100≤xi≤100
1100
C12
Hybrid function 3
30, 50
−100≤xi≤100
1200
C13
Hybrid function 4
30, 50
−100≤xi≤100
1300
C14
Hybrid function 5
30, 50
−100≤xi≤100
1400
Hybrid functions
C15
Hybrid function 6
30, 50
−100≤xi≤100
1500
C16
Hybrid function 7
30, 50
−100≤xi≤100
1600
C17
Hybrid function 8
30, 50
−100≤xi≤100
1700
C18
Hybrid function 9
30, 50
−100≤xi≤100
1800
C19
Hybrid function 10
30, 50
−100≤xi≤100
1900
Composition functions
C20
Composition function 1
30, 50
−100≤xi≤100
2000
C21
Composition function 2
30, 50
−100≤xi≤100
2100
C22
Composition function 3
30, 50
−100≤xi≤100
2200
C23
Composition function 4
30, 50
−100≤xi≤100
2300
C24
Composition function 5
30, 50
−100≤xi≤100
2400
C25
Composition function 6
30, 50
−100≤xi≤100
2500
C26
Composition function 7
30, 50
−100≤xi≤100
2600
C27
Composition function 8
30, 50
−100≤xi≤100
2700
C28
Composition function 9
30, 50
−100≤xi≤100
2800
C29
Composition function 10
30, 50
−100≤xi≤100
2900
Original Algorithms and Ensemble Algorithms
In the experiment, we select some new representative algorithms for the ensemble, which are Harris hawks optimization (HHO) [54], seagull optimization algorithm (SOA) [55], slime mold algorithm (SMA) [56], salp swarm algorithm (SSA) [57], manta ray foraging optimization (MRFO) algorithm [58]. The EPM class algorithms represent the algorithms integrated under the EPM framework. This paper names the algorithms integrated by the EPM framework as “EPM + number”, where “number” represents the number of filled algorithms in the ensemble framework. It obtains the EPM2 algorithm by integrating HHO and SOA, the EPM3 algorithm by integrating HHO, SOA, and SMA, the EPM4 algorithm by integrating HHO, SOA, SMA, and SSA, the EPM5 algorithm by integrating HHO, SOA, SMA, SSA, and MRFO. The ensemble framework does not interfere with the updated formulas of the integrated algorithms, so researchers can adjust their parameters according to the characteristics of the original algorithms. The parameters of the original algorithms are consistent with the papers that proposed these algorithms. The parameters in integrated algorithms are the same as the original algorithms’ parameters. It lists the settings of the parameters in Table 3.
The parameter settings of the algorithms
Algorithm
Parameter
Introduction and settings
HHO
r1, r2, r3, r4, q, E0, r, J, S, β
r1, r2, r3, r4, q, E0, r are all random numbers in [0, 1]; J is the following number within [0, 2]; S is a random vector of dimension D; the elements are random numbers in [0, 1]; and β is set to 1.5.
SOA
fc, rd, θ, u, v, e
fc decreases linearly from 2 to 0, rd is a random number in [0, 1]; θ is a random angle in [0, 2π]; u = 1, v = 1; e is the base of the natural logarithm.
SMA
rand, z, vb, vc, r
rand is in [0, 1], and the location update parameter z = 0.03; vb is in [−a, a]; vc linearly decreases from 1 to 0, and r represents a random value in [0, 1].
SSA
c1, c2, c3
c1 decreases from 2 to 0, and c2 and c3 are random numbers in [0, 1].
MRFO
r, r2, r3, rand
r, r1, r2, r3 and rand represent random numbers in [0, 1].
Comparison Results
The average (AVE) and standard deviation (STD) of the best solutions obtained by all test algorithms in 25 independent runs and 500 iterations in each dimension were recorded (see Tables 4–6) and compared.
The comparison results of 23 standard benchmark functions by different optimization algorithms
Function
Metric
EPM5
EPM4
EPM3
EPM2
HHO
SOA
SMA
SSA
MRFO
F1
AVE
0
0
0
0
0
0
0
3.09E−09
0
STD
0
0
0
0
0
0
0
7.33E−10
0
F2
AVE
0
0
0
0
0
0
0
0.355149
0
STD
0
0
0
0
0
0
0
0.784676
0
F3
AVE
0
0
0
0
0
2.45912
0
1.77E−08
0
STD
0
0
0
0
0
5.722051
0
4.23E−09
0
F4
AVE
0
0
0
0
0
0
0
4.48E−05
0
STD
0
0
0
0
0
0
0
2.86E−05
0
F5
AVE
4.07E−26
7.13E−10
1.62E−09
2.13E−09
1.03E−06
18.78447
0.000457
35.12469
3.85E−11
STD
1.59E−25
1.78E−09
3.82E−09
5.7E−09
1.24E−06
13.60155
0.000292
44.13715
4.72E−11
F6
AVE
4.44E−33
3.15E−11
4.46E−11
6.52E−11
1.34E−08
0.563822
1.61E−06
2.87E−09
7.4E−34
STD
1.06E−32
5.12E−11
1.02E−10
1.51E−10
1.61E−08
0.542323
5.27E−07
5.49E−10
2.56E−33
F7
AVE
1.81E−06
1.74E−06
1.77E−06
2.12E−06
2.11E−06
4.14E−06
2.7E−06
0.00271
2.−06
STD
1.44E−06
1.5E−06
1.54E−06
1.87E−06
1.47E−06
4.02E−06
3.11E−06
0.001164
1.45E−06
F8
AVE
−12569.5
−12569.5
−12569.5
−12569.5
−12569.5
−12569.5
−12569.5
−7895.55
−8805.3
STD
1.96E−12
8.4E−10
8.01E−09
7.67E−11
6.6E−05
2.81E−05
0.000112
720.5855
480.4721
F9
AVE
0
0
0
0
0
0
0
55.08084
0
STD
0
0
0
0
0
0
0
14.91848
0
F10
AVE
8.88E−16
8.88E−16
8.88E−16
8.88E−16
8.88E−16
8.88E−16
8.88E−16
1.743802
8.88E−16
STD
0
0
0
0
0
0
0
0.922858
0
F11
AVE
0
0
0
0
0
0
0
0.014162
0
STD
0
0
0
0
0
0
0
0.012837
0
F12
AVE
1.59E−32
3.87E−12
8.57E−12
1.23E−11
1.05E−09
0.01385
4.64E−07
0.355485
1.58E−32
STD
3.1E−34
4.38E−12
1.04E−11
2.02E−11
1.52E−09
0.011173
4.2E−07
0.615933
1.2E−34
F13
AVE
1.38E−32
5.84E−11
9.1E−11
9.06E−11
6.77E−09
0.132341
8.44E−07
0.002637
1.355246
STD
1.04E−33
8.07E−11
1.42E−10
1.91E−10
1.19E−08
0.12463
5.17E−07
0.004789
1.412608
F14
AVE
0.998004
0.998004
0.998004
1.037765
1.037765
4.687508
0.998004
0.998004
0.998004
STD
0
7.02E−15
5.48E−14
0.198805
0.198805
4.688223
3.01E−15
1.83E−16
0
F15
AVE
0.000307
0.000307
0.000308
0.000308
0.000308
0.001109
0.000368
0.000815
0.000385
STD
2.06E−19
1.21E−11
6.38E−08
4.96E−08
4.58E−08
0.00072
0.000192
0.000363
0.000253
F16
AVE
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
STD
6.8E−16
1.33E−15
2.2E−14
5.87E−14
9.27E−15
3.21E−07
1.25E−12
1.38E−14
6.8E−16
F17
AVE
0.397887
0.397887
0.397887
0.397887
0.397887
0.398059
0.397887
0.397887
0.397887
STD
0
2.35E−14
3.56E−09
1.75E−08
1.14E−08
0.000401
1.96E−09
1.32E−14
0
F18
AVE
3
3
3
4.08
5.16
12.20428
3
3
3
STD
1.26E−15
2.91E−13
5.97E−12
5.400001
7.475962
13.71568
3.01E−14
2.46E−13
5.94E−16
F19
AVE
−3.86278
−3.86278
−3.86278
−3.86278
−3.86278
−3.74259
−3.86278
−3.86278
−3.86278
STD
2.27E−15
1.81E−14
5.47E−08
1.39E−06
6.15E−07
0.170625
4.29E−09
5.74E−15
2.27E−15
F20
AVE
−3.29346
−3.26493
−3.25541
−3.25539
−3.25538
−2.96169
−3.21261
−3.21261
−3.26493
STD
0.051824
0.060624
0.060235
0.060236
0.060235
0.170635
0.03292
0.03292
0.060624
F21
AVE
−10.1532
−10.1532
−10.1532
−6.07478
−5.05519
−4.44062
−10.1532
−9.74901
−9.94928
STD
5.44E−15
1.94E−05
1.81E−05
2.081226
5.5E−06
1.754009
7.69E−06
1.398954
1.0196
F22
AVE
−10.4029
−10.4029
−10.4029
−6.57593
−5.08767
−5.83812
−10.4029
−9.58101
−10.1903
STD
1.92E−15
5.69E−06
2.45E−05
2.435741
6.22E−06
2.508465
1.02E−05
2.305573
1.063054
F23
AVE
−10.5364
−10.5364
−10.5364
−6.21004
−5.5611
−4.8448
−10.5364
−10.322
−10.5364
STD
1.92E−15
1.48E−05
2.86E−05
2.207733
1.497374
2.028815
7.76E−06
1.072153
1.81E−15
The comparison results of the CEC2017 benchmark functions by different optimization algorithms (D = 30)
EPM5
EPM4
EPM3
EPM2
HHO
SOA
SMA
SSA
MRFO
C1
AVE
902.7646
3290.374
2685.852
489759.6
5825.588
4.71E+10
7489.476
3354.55
3953.039
STD
1118.24
4044.062
4243.236
2418277
3309.647
9.59E+09
7115.915
3783.203
4491.838
C2
AVE
200
200.0002
200.0012
8.88E+33
4.77E+31
2.57E+49
200.0002
200.0001
200
STD
1.76E−05
6.6E−05
0.00085
4.28E+34
1.64E+32
1.28E+50
0.000123
3.04E−05
9.63E−06
C3
AVE
300
300
300.847
28223.06
32767.91
137237.7
300.001
300
300
STD
0
7.04E−09
0.5768
5322.439
4405.763
45948.26
0.000584
3.77E−09
0
C4
AVE
404.4552
436.6484
428.9964
662.9686
634.4399
11593.27
488.2914
479.8565
409.3811
STD
11.44006
25.74595
23.56127
218.7391
108.0743
3366.472
5.52105
25.40835
21.0248
C5
AVE
536.893
531.9988
559.8022
730.6306
746.4483
880.6916
575.2127
609.7334
649.9196
STD
9.786207
12.33574
16.24438
33.09082
29.05142
43.61556
18.01149
29.2981
40.06929
C6
AVE
600.8313
600.5959
601.5444
653.6156
657.7857
678.0107
600.3176
625.4053
619.4164
STD
0.593362
0.375077
0.882561
7.245739
4.69095
9.483725
0.192387
11.44769
12.61308
C7
AVE
766.9689
761.2482
792.5594
1199.308
1256.393
1418.424
811.1121
859.4338
989.8419
STD
8.979743
10.43814
15.01733
63.29684
66.28971
58.77982
20.65983
35.38709
86.96931
C8
AVE
836.8598
833.5277
850.9737
960.6261
968.3868
1106.081
877.8322
894.8789
932.9259
STD
6.6184
9.776067
14.33767
23.16343
25.06896
28.21193
13.90581
31.50098
28.2599
C9
AVE
900.3583
900.6909
900.5986
4955.688
5063.042
9446.444
1506.398
2220.573
3669.055
STD
0.374084
0.724316
0.718643
698.6464
609.5586
1539.526
1027.192
879.4835
1097.606
C10
AVE
2628.831
2540.615
2678.661
6063.822
5253.509
9541.206
3559.888
4575.969
4808.949
STD
308.2059
345.6339
473.6041
994.8354
885.5511
529.2566
475.7757
705.6062
715.4212
C11
AVE
1126.983
1132.647
1139.906
1276.364
1287.052
10240.64
1194.034
1256.286
1184.318
STD
8.540598
10.10047
14.1604
75.00597
64.74324
3050.719
41.25983
44.91092
31.17383
C12
AVE
15027.23
88204.73
100155.4
45934047
67881154
8.45E+09
238634.7
771305.1
19146.42
STD
9080.098
56539.9
60453.42
72150740
1.1E+08
3.71E+09
145317.3
689276.8
12496.58
C13
AVE
5477.185
27176.19
25679.45
173224
19554468
2.3E+09
26860.77
95078.71
16508.89
STD
4442.871
20246.44
25013.57
194785
97111555
3.12E+09
25417.04
62176.79
12679.22
C14
AVE
1567.886
3082.811
4749.069
339770.8
280555
1441032
9348.684
3512.879
1604.296
STD
45.90245
1584.016
3683.581
329934.4
287152.8
1445004
6823.33
2262.68
71.7545
C15
AVE
3623.123
8074.151
10605.32
33162.86
37733.69
4.18E+08
30796.05
50766.18
5584.899
STD
2810.202
6215.589
8942.196
29081.38
32078.31
5.53E+08
11458.03
40469.1
6645.642
C16
AVE
1801.661
1949.337
2125.943
3153.129
3258.431
4787.399
2400.258
2407.28
2631.215
STD
130.349
205.3241
214.2251
495.0602
448.0817
824.6578
303.5794
228.4946
279.2315
C17
AVE
1832.242
1820.747
1955.324
2518.757
2620.966
3751.296
2063.12
1993.909
2129.348
STD
57.19301
63.95016
131.0033
317.6166
327.4079
1679.326
186.789
164.0414
215.483
C18
AVE
22870.17
60371.9
65056.1
1053220
924321.3
53180666
166444.2
86677.09
29835.96
STD
19990.05
38430.41
48955.33
1342197
827943.6
75027162
76311.93
48923.04
17856.1
C19
AVE
3824.338
13040.8
10533.51
50554.58
42888.47
4.7E+08
41470.4
108043.6
10004.24
STD
3315.617
12306.69
10690.86
37080.57
56123.46
6.29E+08
17380.8
49528.17
10141.75
C20
AVE
2119.088
2148.06
2169.274
2616.076
2650.025
2957.38
2313.325
2351.115
2380.781
STD
65.23756
90.19009
116.4725
153.5199
168.6338
200.5934
129.0021
146.3238
209.8808
C21
AVE
2338.545
2338.708
2354.376
2506.347
2548.73
2681.681
2388.157
2405.765
2412.039
STD
9.689742
8.679737
14.91288
40.69084
46.96825
61.8029
23.58705
33.65642
27.48967
C22
AVE
2300.121
2300.223
2301.24
6730.48
6983.713
9654.64
5344.452
3911.237
2607.776
STD
0.57528
0.597588
1.233973
1600.559
678.1923
1098.703
530.8082
2053.508
1065.147
C23
AVE
2685.296
2683.946
2703.451
3166.514
3206.271
3328.737
2731.09
2739.832
2790.808
STD
9.284617
11.54514
13.65706
135.4268
150.4809
139.2706
19.91425
25.89812
44.39223
C24
AVE
2856.568
2857.054
2888.855
3335.164
3359.277
3583.298
2929.555
2898.986
2979.212
STD
12.73916
11.06584
14.77498
141.8446
141.0508
195.7917
21.97067
24.94498
66.27963
C25
AVE
2885.649
2886.709
2886.745
2958.363
2952.888
4572.914
2886.848
2893.028
2895.376
STD
1.601232
0.681923
0.685797
32.52224
39.78769
460.3713
1.030813
14.86022
16.38967
C26
AVE
3725.473
3934.553
4036.657
7607.326
8304.568
10358.2
4472.035
4464.339
4870.579
STD
464.9816
165.8759
275.9082
1334.879
1258.937
1119.672
235.3289
831.3642
1641.959
C27
AVE
3166.58
3168.203
3168.97
3611.143
3741.738
3976.634
3206.848
3224.959
3267.911
STD
10.76602
8.738246
8.745193
148.5094
197.485
322.848
12.5343
15.85913
24.35195
C28
AVE
3100
3100.003
3100.089
3453.845
3394.916
6406.415
3224.79
3132.602
3126.623
STD
0
0.015433
0.026477
160.969
169.3918
710.1894
66.59074
47.08401
55.40508
C29
AVE
3390.706
3437.384
3431.83
5012.228
5167.606
6355.715
3697.814
3789.834
3942.537
STD
79.95842
94.77418
108.0931
434.1358
712.0787
824.497
174.1942
173.2771
271.9866
The comparison results of the CEC2017 benchmark functions by different optimization algorithms (D = 50)
EPM5
EPM4
EPM3
EPM2
HHO
SOA
SMA
SSA
MRFO
C1
AVE
1564.423
3363.094
5868.441
15223.61
14185.43
1.01E+11
15409.67
4773.522
4792.013
STD
2074.851
3416.559
6299.051
5317.337
4192.902
8.13E+09
11336.18
6920.711
6722.137
C2
AVE
200
202.2101
340.291
1.86E+59
5.31E+59
6.26E+84
201.7396
200.8243
200
STD
9.59E−06
7.80E+00
90.16485
8.38E+59
2.66E+60
3.12E+85
7.430594
4.12E+00
1.48E−05
C3
AVE
300
300
303.7261
44216.94
47427.25
252790.3
300.0082
300
300
STD
4.84E−13
2.31E−08
1.75844
7642.29
9387.97
90954.65
0.002057
9.08E−09
2.39E−13
C4
AVE
417.1868
433.3292
430.2082
933.5133
945.979
30758.01
504.3023
544.9028
422.8925
STD
14.18587
20.07223
1.988095
599.3669
483.4245
6286.819
59.34425
47.54378
28.53133
C5
AVE
586.8017
570.0516
615.8567
852.495
849.6292
1150.826
687.0763
742.8122
815.8375
STD
11.33574
19.6727
18.62161
33.7776
26.29409
36.83114
41.034
59.28636
51.54474
C6
AVE
604.7959
603.0001
606.5931
661.39
663.1714
694.9141
601.0333
638.0651
640.2838
STD
1.728911
1.287632
2.364182
5.401158
4.73241
7.585304
0.506592
9.609981
11.76814
C7
AVE
826.695
834.6281
879.2088
1648.979
1693.569
2042.556
926.7816
1009.936
1367.78
STD
15.39112
13.70104
28.97706
80.90631
53.91651
46.16475
45.60423
72.26719
178.9801
C8
AVE
885.8617
870.7809
915.449
1144.098
1163.041
1441.307
963.9156
1038.168
1119.021
STD
16.68351
17.65313
18.10443
28.4163
31.71651
33.01192
24.77768
57.34869
54.97388
C9
AVE
937.6997
920.8308
1088.04
12716.74
12612.68
34561.62
5948.958
8208.887
9954.461
STD
61.96252
26.5344
363.9745
1621.532
985.5352
5698.828
3876.126
2373.46
2379.059
C10
AVE
3694.749
3761.504
3700.033
7626.868
8230.649
13892.13
6280.105
7617.053
7348.594
STD
558.9742
532.997
582.0079
935.1439
1204.979
758.1973
626.1485
783.3506
939.2402
C11
AVE
1177.74
1166.028
1180.263
1396.777
1383.864
22591.94
1303.236
1322.368
1261.253
STD
15.6531
9.219968
16.98053
71.8303
87.64393
4489.043
63.97801
37.66977
30.2221
C12
AVE
37036.41
893674.8
991485.5
1.37E+08
1.35E+08
5.26E+10
1598430
4238014
83609.84
STD
22014.37
627633
857632.9
2.04E+08
2.99E+08
1.81E+10
1074183
2520333
41626.2
C13
AVE
4075.313
16996.81
14110.42
212873.9
125767.5
1.82E+10
29634.34
130309.7
4124.072
STD
3396.271
12763.81
11479.36
507175.8
94114.22
9.25E+09
10151.04
75453.32
4256.024
C14
AVE
2426.744
7447.321
15906.74
1680775
1705491
31389494
52348.57
15112.99
3072.003
STD
658.6872
6164.037
11337.99
3798183
3885700
32006311
25969.55
12145.55
1094.581
C15
AVE
5616.482
12308.82
13023.14
4694751
4537062
6.16E+09
29350.23
39421.45
9413.125
STD
4499.291
7888.652
8786.184
23254860
22398016
3.85E+09
6731.168
22304.52
6287.582
C16
AVE
2237.506
2332.344
2553.069
4028.273
4048.173
8094.206
3164.62
3063.408
3297.917
STD
247.3496
249.5483
240.8204
513.0978
698.4317
1485.932
368.3294
326.5094
521.3009
C17
AVE
2257.742
2274.431
2474.799
3639.855
3672.466
7247.96
2871.545
2960.49
3205.788
STD
155.2337
193.8793
194.0557
452.6595
473.4265
3760.402
242.0486
268.2297
326.7872
C18
AVE
19549.7
80794.79
90703.99
4242293
2935428
1.02E+08
207761.6
107742.9
33126
STD
12822.92
27734.12
50939.99
6473234
3355806
80050046
126316.6
35916.97
22584.63
C19
AVE
4774.262
10989.99
9610.29
62330.2
52244.67
2.15E+09
14225.18
201180.9
14846.17
STD
3910.518
12388.52
8085.775
45900.33
27950.68
1.68E+09
17052.33
54726.35
11388.86
C20
AVE
2287.938
2319.214
2533.119
3162.177
3257.845
4062.8
2818.477
2957.316
3179.556
STD
108.3397
139.2058
213.9936
311.8407
258.3304
404.9723
288.1831
253.0242
349.4163
C21
AVE
2388.669
2375.095
2420.159
2785.253
2812.891
3138.545
2492.834
2505.974
2563.45
STD
14.39998
19.76868
21.43885
72.10099
82.51934
97.2633
36.10054
45.65499
65.47573
C22
AVE
2300.425
2300.273
2302.295
10608.09
10422.03
16728.95
8257.222
9165.022
8619.539
STD
0.791301
0.679075
1.070631
1533.443
1014.829
853.0564
988.791
952.0107
1623.496
C23
AVE
2804.425
2796.67
2831.246
3801.936
3871.871
4141.234
2917.741
2938.457
3118.995
STD
14.84545
21.28376
27.53321
184.6423
223.2331
194.9199
44.99394
40.59267
101.1374
C24
AVE
2978.396
2970.417
3030.666
4096.05
4039.954
4620.13
3095.88
3078.31
3285.498
STD
14.84988
23.40616
28.51908
215.5657
174.8984
271.4433
36.24266
40.7089
104.6463
C25
AVE
2974.773
2971.509
2975.22
3249.563
3258.095
14143.84
3017.684
3023.769
3052.767
STD
8.554417
17.51097
12.64854
105.7085
93.53723
1790.841
38.19031
45.47396
35.97161
C26
AVE
3945.906
4502.135
4913.446
12205.31
12226.77
17256.95
5719.404
4752.921
6541.465
STD
887.4889
197.8206
249.1071
904.1735
1051.926
960.7725
373.1578
2102.883
3501.497
C27
AVE
3170.97
3170.389
3170.612
4830.036
4949.775
6216.204
3323.926
3368.092
3592.878
STD
9.292116
9.521278
8.661707
446.8078
543.9687
712.5707
65.79624
77.34347
145.1447
C28
AVE
3258.189
3259.178
3259.365
3757.093
3737.579
11912.49
3284.561
3301.465
3290.119
STD
1.823178
2.67E−01
0.280697
311.0017
217.6791
1688.529
22.72083
16.07822
28.54801
C29
AVE
3477.968
3495.964
3509.711
6852.671
6928.21
26533.06
4176.37
4571.589
4437.38
STD
159.98
140.3959
225.5474
1164.126
1186.805
17826.29
325.5151
294.3504
444.2482
The population size is 70, and the number of iterations is 500 × D. D is the dimension of the test function. According to the characteristics of the EPM framework, the more algorithms are integrated into the framework, the more times the function will be evaluated. To compare the results of the integration algorithm and the original algorithm more fairly, we design a perfect ensemble (PE) that takes the optimal value from the effects of multiple original algorithms. In each independent run, it records the best result of several original algorithms as the result of this run, and after 25 runs, all the results are averaged. This paper names the results obtained by the PE framework as “PE + number”, where “number” represents the number of integrated results in the framework. It expresses the results obtained by averaging the optimal values of HHO and SOA as PE2, the results obtained by averaging the running results of HHO, SOA, and SMA as PE3, the results obtained by averaging the optimal values of HHO, SOA, SMA, and SSA as PE4, the results obtained by averaging the optimal values of HHO, SOA, SMA, SSA, and MRFO as PE5.
Table 7 displays the results of Friedman’s mean rank test for the benchmark functions. On the 23 standard benchmark functions, the mean rankings of EPM5, EPM3, and EPM2 are higher than those of PE5, PE3, and PE2, respectively; the mean ranking of EPM4 is lower than that of PE4. On the CEC2017 benchmark functions with 30 dimensions, the mean rankings of EPM5, EPM4, EPM3, and EPM2 are higher than that of PE5, PE4, PE3, and PE2, respectively. On the CEC2017 benchmark functions with 50 dimensions, the mean rankings of EPM5, EPM4, EPM3, and EPM2 are higher than that of PE5, PE4, PE3, and PE2, respectively. In most cases, the algorithms in the ensemble framework get a better ranking. Increasing the number of integrated metaheuristic algorithms enhances the average optimization ability of the overall ensemble framework.
Friedman’s mean ranking test results of benchmark functions
Benchmark functions
Comparison
EPM
PE
23 standard benchmark functions
EPM2 vs. PE2
1.3478
1.6522
EPM3 vs. PE3
1.5435
1.4565
EPM4 vs. PE4
1.4348
1.5652
EPM5 vs. PE5
1.4348
1.5652
CEC2017 benchmark functions (D = 30)
EPM2 vs. PE2
1.3103
1.6897
EPM3 vs. PE3
1.1034
1.8966
EPM4 vs. PE4
1.2069
1.7931
EPM5 vs. PE5
1.0862
1.9138
CEC2017 benchmark functions (D = 50)
EPM2 vs. PE2
1.4138
1.5862
EPM3 vs. PE3
1.1034
1.8966
EPM4 vs. PE4
1.1379
1.8621
EPM5 vs. PE5
1.1207
1.8793
Table 8 displays the p-value test results on benchmark functions. On the 23 standard benchmark functions, the p-value resulting from the comparison between EPM2 and PE2 is less than 0.1 and more significant than 0.05, indicating that the results of the algorithms are different at the significance level of 0.1. The p-values resulting from the comparisons between EPM3 and PE3, between EPM4 and PE4, and between EPM5 and PE5 are greater than 0.1, indicating no significant difference between the results. On the CEC2017 benchmark functions with 30 dimensions, the p-value resulting from the comparison between EPM2 and PE2 is greater than 0.1, indicating no significant difference between the results. The p-values for the comparison between EPM3 and PE3, between EPM4 and PE4, and between EPM5 and PE5 are less than 0.01, indicating that the results of these algorithms are significantly different at the significance level of 0.01. On the CEC2017 benchmark functions with 50 dimensions, the p-value resulting from the comparison between EPM2 and PE2 is greater than 0.1, indicating no significant difference between the results. The p-values for the comparison between EPM3 and PE3, between EPM4 and PE4, and between EPM5 and PE5 are less than 0.01, indicating that the results of these algorithms are significantly different at the significance level of 0.01.
Wilcoxon signed ranking test results of benchmark functions
Benchmark functions
Comparison
R+
R-
p-value
α
23 standard benchmark functions
EPM2 vs. PE2
195
81
9.95E−02
0.1
EPM3 vs. PE3
131
145
1.00E+00
>0.1
EPM4 vs. PE4
175
101
1.56E−01
>0.1
EPM5 vs. PE5
171.5
104.5
1.39E−01
>0.1
CEC2017 benchmark functions (D = 30)
EPM2 vs. PE2
265
170
3.04E−01
>0.1
EPM3 vs. PE3
427
8
5.90E−06
0.01
EPM4 vs. PE4
378
57
5.19E−04
0.01
EPM5 vs. PE5
414.5
20.5
2.52E−05
0.01
CEC2017 benchmark functions (D = 50)
EPM2 vs. PE2
222
213
9.22E−01
>0.1
EPM3 vs. PE3
419
16
1.32E−05
0.01
EPM4 vs. PE4
413
22
2.36E−05
0.01
EPM5 vs. PE5
398.5
36.5
1.08E−04
0.01
Iterative Curves
We select some test functions, draw the iterative curve of the experimental algorithm, and visually observe the convergence of the algorithms. The iterative curves in Fig. 5 correspond to functions F3, F15, F20, C5, C16, and C20, representing the performance of EPM5 under different types of functions. It is observed that the EPM5 algorithm has a faster iterative optimization speed and achieves better results through the iterative curves.
Iterative curves of partial test functions
Evolution Process of Real Population and Virtual Population
We analyzed the process of solving function F3 by the EPM5 algorithm to observe the relationship between the real and virtual populations and follow the evolution process more intuitively during the operation of the ensemble algorithm. Fig. 6 shows the position changes of the virtual populations in the x1 and x2 dimensions, before and after 1, 2, 5, 6, 20, 21, 100, 101, 1000, 1001, 14999, and 15000 iterations, where ‘a’ represents the population before the iteration and ‘b’ represents the population after the iteration. It lists the historical optimal solutions corresponding to the number of partial iterations in Table 9.
The virtual population distribution diagram of the EPM5 algorithm under some iterations in the process of solving the function F3
Number of partial iterations on F3 function and historical optimal solution
t
H
t
H
t
H
t
H
t
H
t
H
0
102389.5
4
0.178303
19
5.78E−16
99
2.72E−97
999
0
14998
0
1
374.6011
5
0.103464
20
1.78E−17
100
2.09E−99
1000
0
14999
0
2
314.3702
6
0.103464
21
1.48E−17
101
1.32E−99
1001
0
15000
0
In Iteration 1.a, it is observed that the positions of the five virtual populations come from the same real population before the start of the first iteration. After the high-level generation sub-stage, the five virtual populations update their locations independently, and the positions of the populations are different, as shown in Iteration 1.b. Because the fitness value of the best elite is smaller than the historical optimum, it materializes the population where the optimal elite resides, and the population distribution after materialization is shown by Iteration 2.a. It transforms the five virtual populations into the real population. In Iteration 5.b and Iteration 6.a, the best elite of the virtual populations after the fifth iteration is not less than an order of magnitude smaller than the historically best fitness value. Thus, the virtual population continues to exist. In Iteration 20.a, the entity population has converged at this time. However, after the high-level generation sub-stage, the virtual population can still be distributed in every corner of the map, as shown in Iteration 20.b. After the high-level selection sub-stage, it obtains the population distributed in iteration 21.a, and the virtual populations are transformed into the real population again. It can be seen from this figure that the premature convergence of single or multiple populations will not lead to the convergence of all populations, and other populations still have good exploratory properties. In Iteration 100.b and Iteration 101.b, it is observed that the red population has converged. But in Iteration 1000.a, the red population is dispersed again, indicating that the converged population has a chance to reexplore. In Iteration 14999.a and Iteration 15000.b, all populations have converged. The red population has several individual sporadic distributions related to the corresponding algorithm characteristics of the population.
Engineering Application Problems
The pressure vessel design optimization problem optimizes the cost of welding, material, and forming of the pressure vessel by adjusting the shell thickness, head thickness, inner diameter, and length of the vessel under four constraints [59]. The three-bar truss design optimization problem is based on the stress constraint of each bar to optimize the total weight of the bar structures [60]. Table 10 gives the mathematical description of these problems. It abbreviates the first engineering problem as E1 and the second engineering problem as E2.
The average and standard deviation of the best solutions obtained by all test algorithms in 25 independent runs and 500 iterations in each dimension were recorded and compared (see Table 11). For the optimization results of the pressure vessel design problem, EPM3, EPM4, and EPM5 are better than their integrated original algorithms. EPM2 is better than SOA but worse than HHO. For the optimization results of the three-bar truss design problem, both EPM4 and EPM5 are better than their integrated original algorithm. EPM3 is better than HHO and SOA but worse than SMA. EPM2 is better than SOA but worse than HHO. Two practical engineering problems obtain better optimization results from the average optimization results by adopting the ensemble framework.
The results of engineering optimization problems by different optimization algorithms
EPM5
EPM4
EPM3
EPM2
HHO
SOA
SMA
SSA
MRFO
E1
AVE
6066.6913
6085.2628
6098.4798
8074.5080
7964.7286
13856.8187
6216.5912
6353.9354
6271.0216
STD
12.5129
20.6386
33.2818
1752.8158
1414.2115
6088.7058
219.9751
246.7167
228.2141
E2
AVE
263.895861
263.896028
263.896757
263.917593
263.914440
266.142034
263.896307
263.896183
263.895862
STD
1.7451E−05
2.6238E−04
1.1884E−03
3.8590E−02
2.5215E−02
2.0515E+00
8.2330E−04
5.1781E−04
1.8544E−05
Conclusion
We propose a new population-based metaheuristic ensemble framework in this paper. The main innovation is to use the common structural characteristics of the population-based meta-heuristic algorithms to design a framework, which is a bridge for cooperative optimization of multiple algorithms through the transformation of virtual population and entity population and elite strategy. The framework leverages the differentiation and unification of real and virtual populations to coordinate exploration and exploitation at the population level. Meanwhile, an elitist strategy is adopted to communicate among virtual populations to guarantee diversity and reduce the internal influence on each metaheuristic algorithm. In the experiment, five algorithms, SOA, HHO, SMA, SSA, and MRFO, are integrated using the ensemble framework to obtain EPM algorithms. The EPM algorithms perform superior on 23 standard benchmark functions and the CEC2017 benchmark functions. According to Friedman’s average ranking, the results of algorithms integrated with the EPM framework, in most cases, outperformed the results of the same algorithms integrated with the PE framework, demonstrating the superiority of the ensemble framework. The ensemble framework also demonstrated excellent results in solving two practical engineering problems and obtained better solutions.
With the increase in the number of optimization algorithms integrated into the ensemble framework, the information exchange between various algorithms will increase, and these algorithms will be more likely to jump out of the local optimization. However, we cannot distinguish the role of each algorithm in optimization and the influence of algorithm parameters on the optimization results. Meanwhile, each additional algorithm in the ensemble framework will increase the calculation of the virtual population corresponding to the algorithm, the storage cost of the calculation results, and the cost of information exchange, reducing the optimization speed. In future research, we will study two aspects: improving the optimization ability and speed. We will analyze the contribution of a single algorithm in the integration framework and adjust the algorithm parameters to improve the optimization ability. We will study the factors that affect the optimization speed and reduce unnecessary computing and storage costs to improve the optimization speed according to the role of different algorithms in the different stages.
The authors would like to acknowledge the editor-in-chief, associate editors, and reviewers for their contributions to the improvement of this paper.
Funding Statement
This work was supported by National Natural Science Foundation of China under Grant 62073330. The auther J. T. received the grant.
Author Contributions
Study conception and design: H. Li, J. Tang, S. Lao; data collection: Q. Pan, J. Zhan; analysis and interpretation of results: H. Li, Q. Pan; draft manuscript preparation: H. Li, J. Zhan. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials
Data available on request from the authors. The data that support the findings of this study are available from the corresponding author, (J. T.), upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
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