A Multi-Disturbance Marine Predator Algorithm Based on Oppositional Learning and Compound Mutation

Abstract

Marine Predator Algorithm (MPA) is a meta-heuristic algorithm based on the foraging behavior of marine animals. It has the advantages of few parameters, simple setup, easy implementation, accurate calculation, and easy application. However, compared with other meta-heuristic algorithms, this algorithm has some problems, such as a lack of transition between exploitation and exploration and unsatisfactory global optimization performance. Aiming at the shortage of MPA, this paper proposes a multi-disturbance Marine Predator Algorithm based on oppositional learning and compound mutation (mMPA-OC). Firstly, the optimal value selection process is improved by using Opposition-Based Learning mechanism and enhance MPA’s exploration ability. Secondly, the combined mutation strategy was used to improve the predator position updating mechanism and improve the MPA’s global search ability. Finally, the disturbances factors are improved to multiple disturbances factors, so that the MPA could maintain the population diversity. In order to verify the performance of the mMPA-OC, experiments are conducted to compare mMPA-OC with seven meta-heuristic algorithms, including MPA on different dimensions of the CEC-2017 benchmark function, complex CEC-2019 benchmark function, and engineering optimization problems. Experiments have shown that the mMPA-OC is more efficient than other meta-heuristic algorithms.

1. Introduction

In recent years, as optimization problems in the real world have become more difficult and complex, some scientific researchers have been attracted to studying optimization problems. Ordinary optimization algorithms include the gradient descent method, Newton method, conjugate gradient method, and Lagrange multiplier method. Because of their flexibility, robustness and self-organization, meta-heuristic algorithms are more attractive than ordinary optimization methods in solving complex optimization problems, providing a powerful tool for solving them.

Meta-heuristics are methods that solve optimization problems by simulating various natural processes, biological or human thinking. In general, meta-heuristic algorithms can be divided into four categories: algorithms based on biological evolution, algorithms based on swarm intelligence, algorithms based on physical phenomena, and algorithms based on human social behavior [1,2]. The evolution process of organisms is based on biological evolution in nature; examples include Genetic Algorithm (GA) [3], Evolutionary Programming (EP) [4], Differential Evolution (DE) [5], and Biogeography-Based Optimizer (BBO) [6]. Swarm intelligence-based algorithms are inspired by the foraging behavior of social creatures, such as Particle Swarm Optimization (PSO) [7], Artificial Bee Colony (ABC) [8], Cuckoo Search (CS) [9], Bat Algorithm (BA) [10], Gray Wolf Optimizer (GWO) [11], and Ant Colony Optimization (ACO) [12]. Algorithms based on physical phenomena are inspired by objective physical regulars, such as Black Hole (BH) [13], Gravitational Search Algorithm (GSA) [14], Small World Optimization Algorithm (SWOA) [15], Ray Optimization (RO) [16], Central Force Optimization (CFO) [17], Charged System Search (CSS) [18], Simulated Annealing (SA) [19]. Algorithms based on human social behavior are inspired by the structure of human social networks, such as Group Counseling Optimization (GCO) [20] and Teaching-Learning-Based Optimization (TLBO) [21].

In recent years, many novel meta-heuristic algorithms have been proposed by scholars, such as the Whale Optimization Algorithm (WOA) [22], Moth Flame Optimization (MFO) [23], Butterfly Optimization Algorithm (BOA) [24], Salp Swarm Algorithm (SSA) [25], Galactic Swarm Optimization (GSO) [26], Sine Cosine Algorithm (SCA) [27], Harmony Search (HS) [28], Group Search Optimizer (GSO) [29], Political Optimizer (PO) [30], Majority–Minority Cellular Automata Algorithm (MMCAA) [31]. They perform well on a variety of complex optimization problems, such as multi-objective scheduling [32], image segmentation [33], fault diagnosis [34], and feature selection [35].

Marine Predators Algorithm (MPA) [36] is a novel meta-heuristic optimization algorithm proposed by Afshin Faramarzi et al. in 2020, which is inspired by the predation behavior of marine animals. Marine predators choose the best foraging strategy between Lévy flight wander and Brownian motion to achieve the purpose of predation. The algorithm has the advantages of easy implementation and higher convergence accuracy, so it has attracted the wide attention of scholars once it was proposed. As an excellent meta-heuristic algorithm, MPA has been applied to many optimization fields and it has achieved good results. For example, Ho et al. [37] combined MPA and Feedforward Neural Network (FNN) to train a neural network for structural damage detection. Dinh [38] used the MPA for parameter tuning to realize image fusion, so that the output image has good quality. Chen et al. [39] proposed that MPA was used to accurately determine the parameters of the PV model. Ridha [40] applied the MPA to the single-diode and dual-diode photovoltaic models, and accurately determined the parameters of the photovoltaic model. Xia [41] applied the MPA to the optimization problem of wind turbine blade shape, saving raw materials. Houssein [42] used the MPA to automatically search and optimize CNN parameters to minimize user variability in CNN training. Oszust [43] proposed an optimized extreme learning machine based on MPA (MPA-ELM) to predict the thermal displacement of the motorized spindle model with higher prediction accuracy.

Despite this, the algorithm still has problems, such as lack of transition between exploitation and exploration and the ease with which it becomes stuck in local optima [44,45,46,47]. Many scholars have studied these problems and proposed many improvements. For example, Elaziz et al. [44] proposed that Differential evolution (DE) can enhance the exploration phase of MPA and make the algorithm jump out of local optimum. Zhong et al. [45] combined the teaching-based optimization algorithm with the MPA and added effective mutation and crossover strategies to avoid falling into the local optimum. Wang et al. [46] added the position update strategy of PSO algorithm to MPA to enhance the global search ability of MPA. Oszust [43] used the local escape operator (LEO) to enhance the balance between exploitation and exploration of MPA, and to overcome the premature convergence. The above improvements reduce the possibility of MPA falling into local optima and optimize the performance of MPA. The above researches promote the rapid development of MPA, but the algorithm still has some shortcomings such as weak global search ability, slow convergence speed and unbalanced exploration, and exploitation when dealing with complex optimization problems. Therefore, it is still the focus of research to improve MPA and apply it to different complex optimization problems.

The innovative contributions of this paper are as follows:

• A variant of MPA, a multiple disturbance Marine Predator Algorithm based on opposition-based learning and composite mutation (mMPA–OC) is proposed.
• The Opposition–Based Learning mechanism was used to improve the optimal value selection process and MPA’s exploration ability. The composite mutation strategy was used to improve the update mechanism of the predator’s position to improve the global search ability of MPA. The disturbances factors are improved to multiple disturbances factors, so that the MPA maintains the population diversity in the iterative process.
• In order to verify the effectiveness of mMPA–OC, different CEC benchmark functions and engineering problems are used to evaluate the performance.

The paper is arranged as follows: Section 2 introduces the basic MPA. Section 3 presents the proposed mMPA-OC. Section 4 presents the results of numerical simulation experiments and analyzes. Section 5 presents and analyzes the results of mMPA-OC on engineering optimization problems. Section 6 presents the conclusion and outlook.

2. Marine Predators Algorithm

The Marine Predator Algorithm is a new meta-heuristic optimization algorithm proposed by Faramarzi et al. [36] in 2020. It simulates the behavior of marine animals such as swoosh fish, sharks, sun fish and horse fish to find food, and abstracts it as the marine predators choose different foraging strategies under different moving speed ratios [48]. The MPA has included many excellent strategies in other algorithms, such as iterative segmentation, adaptive operators, elite selection, Levy flight, which makes MPA have better optimization ability. The MPA can be divided into five phases: initialization phase, exploration phase, transition phase, exploitation phase, and disturbance phase. Among them, the exploration phase, transition phase, and exploitation phase are the main parts of the optimization process of MPA.

2.1. Initialization Phase

As an optimization algorithm based on swarm intelligence, MPA is the same as other similar methods in its initial solution distribution: it is random in the search space. MPA initializes the population by Equation (1):

$$X_i=LB+ran{d}_1\ast (UB-LB)i=1,2,\dots ,n$$

(1)

where, $UB$ and $LB$ are the upper and lower boundaries of the search space, respectively, and $rand$${}_1$ are random numbers between 0 and 1, and n is the population size. In the MPA, predators are divided into elite predators and common predators, and the two predators perform predation activities at the same time. Therefore, the elite matrix $Elite$ and the prey matrix $Prey$ are defined to store the positions of elite predators and other predators, respectively, as shown in Equations (2) and (3).

$$Elite={\left(\begin{array}{cccc}{x}_{1,1}^I& x_{1,2}^I& \cdots & x_{1,d}^I\\ x_{2,1}^I& x_{2,2}^I& \cdots & x_{2,d}^I\\ ⋮& ⋮& \ddots & ⋮\\ x_{n,1}^I& x_{n,2}^I& \cdots & x_{n,d}^I\end{array}\right)}_{n\ast d}$$

(2)

$$Prey={\left(\begin{array}{cccc}{x}_{1,1}& x_{1,2}& \cdots & x_{1,d}\\ x_{2,1}& x_{2,2}& \cdots & x_{2,d}\\ ⋮& ⋮& \ddots & ⋮\\ x_{n,1}& x_{n,2}& \cdots & x_{n,d}\end{array}\right)}_{n\ast d}$$

(3)

In Equation (2), the elite matrix is composed of the elite predator $X^I$ repeated n times, $X^I$ represents the elite predator vector, n represents the population number, and d represents the dimension of the search space. In Equation (3), the prey matrix has the same dimension as the elite matrix, and $x_{i,j}$ denotes the ith dimension of the jth predator.

2.2. Exploration Phase

At the beginning of the algorithm iteration ($iteration$ ≤ $\frac{Ma{x}_{-}iterations}{3}$), the prey moves faster than the predator ($v\ge 10$), and the algorithm enters the exploration phase. The prey searches the entire solution space with Brownian motion, and the predators remain motionless. The exploration phase is defined as follows:

$$\begin{array}{cc}\hfill & {step}_i=RB\otimes \left({Elite}_i-RB\otimes {Prey}_i\right)\hfill \\ \hfill & {Prey}_i={Prey}_i+P\times {rand}_2\otimes {\mathrm{step}}_i\hfill \end{array}$$

(4)

where $RB$ stands for Brownian motion, $P=0.5$, $rand$${}_2$ is a random number between zero and one, ⊗ represents the multiplication of elements. $Elit{e}_i$ represents the predator, $Pre{y}_i$ represents the prey ($i=1,2,\cdots ,n$), n denotes the number of populations.

2.3. Transition Phase

When $\frac{Ma{x}_{-}iterations}{3}<$ $iterations$ ≤ $\frac{2\times Ma{x}_{-}iterations}{3}$, MPA is in the transition phase from exploration to exploitation. In this phase, the predator and prey move at the same speed ($v=1$) and the population is divided into two parts: the predator for exploration and the prey for exploitation. The mathematical model for this phase is as follows:

While $\frac{Ma{x}_{-}iterations}{3}<$ $iterations$ ≤ $\frac{2\times Ma{x}_{-}iterations}{3}$

$$\begin{array}{cc}\hfill & {step}_i=RL\otimes \left({Elite}_i-RL\otimes {Prey}_i\right)\hfill \\ \hfill & {Prey}_i={Prey}_i+P\times {rand}_3\otimes {step}_i\hfill \end{array}i\le n/2$$

(5)

$$\begin{array}{cc}\hfill & {step}_i=RB\otimes \left(Elite-RB\otimes {Prey}_i\right)\hfill \\ \hfill & {Prey}_i={Prey}_i+P\times CF\otimes {step}_i\hfill \end{array}i>n/2$$

(6)

In Equation (5), $RL$ denotes the Lévy flight vector and $rand$${}_3$ is a random vector between zero and one. In Equation (6), $CF$ is a decay factor that causes the predator’s step size to decrease, and its expression is $CF={\left(1-\frac{iteration}{Ma{x}_{-}iterations}\right)}^{2\frac{iteration}{Ma{x}_{-}iterations}}$, $Ma{x}_{-}iterations$ indicates the maximum number of iterations.

2.4. Exploition Phase

When $\frac{2\times Ma{x}_{-}iterations}{3}$ $<iteration$ ≤ $Ma{x}_{-}iterations$, the MPA is in the exploition phase, the motion speed of the predator is faster than the motion speed of the prey ($v=0.1$), and the best predation mode of the predator is Lévy flight. Therefore, predators mainly exploit locally in this phase. Its mathematical model is defined as follows:

While $\frac{2\times Ma{x}_{-}iterations}{3}$ $<iteration$ ≤ $Ma{x}_{-}iterations$

$$\begin{array}{cc}\hfill & ste{p}_i=RL\otimes (RL\otimes Elit{e}_i-Pre{y}_i)\hfill \\ \hfill & Pre{y}_i=Elit{e}_i+P\times CF\otimes ste{p}_i\hfill \end{array}i=1,2,\dots ,n$$

(7)

2.5. Disturbance Mechanism

The artificial disturbance mechanism was added to the MPA, that is, the influence of eddy formation and Fish Aggregation Devices (FADs) on predators in the ocean was considered. Overall, 80% of fish prey near themselves, and only 20% make long jumps, due to vortices and fish-gathering devices. The mathematical expression is as follows:

$${Prey}_i=\left\{\begin{array}{cc}{Prey}_i+CF\left[LB+r_1\otimes (UB-LB)\right]\otimes U\hfill & if{r}_1\le FAD\mathrm{s}\hfill \\ {Prey}_i+\left[FADs\left(1-r_1\right)+r_1\right]\left({Prey}_{s1}-{Prey}_{s2}\right)\hfill & if{r}_1>FADs\hfill \end{array}\right.$$

(8)

where $FADs=0.2$, U is the binary vector of zeros and ones, r${}_1$ is a random number between 0 and 1, s${}_1$ and s${}_2$ represents any two predators in the matrix $Prey$. Marine predators have good memories of where they foraged. Implemented $FADS$ after updating, Evaluate $Pre{y}_i$ for fitness. If $Pre{y}_i$ fitness is smaller than its previous value, $Elit{e}_i$ is updated so the quality of the solution keeps improving. Pseudocode of the Marine Predator Algorithm is shown in Algorithm 1.

Algorithm 1: Marine Predator Algorithm.

Initialize populations (Prey) based on Equation (1) While termination criteria are not met Structure the matrix $Elite$ and the matrix $Prey$ based on Equations (2) and (3), calculate the fitness saving   If $iteration$ ≤ $\frac{Ma{x}_{-}iterations}{3}$     Update matrix $Prey$ based on Equation (4)   Else if $\frac{Ma{x}_{-}iterations}{3}<$ $iteration$ ≤ $\frac{2\times Ma{x}_{-}iterations}{3}$     For the first half of the populations $(i\le n/2)$     Update matrix $Prey$ based on Equation (5)     For the other half of the populations $(n/2<i\le n)$     Update matrix $Prey$ based on Equation (6)   Else if $\frac{2\times Ma{x}_{-}iterations}{3}$ $<iteration$ ≤ $Ma{x}_{-}iterations$     Update matrix $Prey$ based on Equation (7)   End if   Accomplish memory saving and $Elite$ update   Applying $FADs$ effect and update based on Equation (8) End while

3. Proposed Algorithm

In view of the lack of transition between MPA exploition and exploration, the OBL mechanism is used to generating reverse solutions, which improved the exploration ability of MPA. This paper adds an OBL mechanism to the optimal value selection process to improve the exploration ability of MPA. Aiming at the problem that MPA is prone to becoming stuck in local optima, the composite mutation strategy is used to improve the update mechanism of predator position to improve the MPA’s global search ability. Aiming at the problem that the single artificial disturbance process in MPA is obviously inconsistent with the actual predation process of predators, and it is less likely to jump out of the local optimal, the artificial disturbance factors are improved to multiple artificial disturbance factors, so that the MPA’s population diversity is maintained in the iterative process.

3.1. Opposition-Based Learning

The OBL mechanism was proposed by Tizhoosh [49] in 2005. Its principle is as follows: first, the inverse solution of the current solution is calculated, and then the optimal solution is selected from the current solution and its inverse solution to update. Previous studies have found that the reverse solution is 50% more likely to be better than the current solution [50], and it can effectively increase the diversity of the population. Therefore, the OBL mechanism has been applied to the improvement of a variety of algorithms and achieved good improvement results. Yu [50] used the OBL mechanism to reduce the probability of Grey Wolf Algorithm falling into local optimum and increase the diversity of the population. Gupta [51] used OBL to optimize HHO, which improved the search efficiency of HHO, effectively alleviating the problems of suboptimal solution stagnation and premature convergence.

The opposition-based learning mechanism is defined as follows: Let $P=\left(y_1,y_2,\cdots ,{\mathrm{y}}_d\right)$ be a point in M dimensional space, where $y_1,y_2,\cdots ,{\mathrm{y}}_d$ are all real numbers,$y_i\in \left[{\mathrm{a}}_i,{\mathrm{b}}_i\right]$. Then the opposite point $\overline{P}=\left(\overline{y_1},\overline{y_2},\cdots ,\overline{y_d}\right)$ is defined as:

$$\overline{y_i}=a_i+b_i-y_i$$

(9)

In this paper, the OBL mechanism is used to improve the MPA, increase the algorithm‘s population diversity, and enhance the exploration ability of the algorithm. As shown in Equation (10).

$${Prey}_{1i}=UB+LB-{Prey}_i$$

(10)

where ${Prey}_{1i}$ is the reverse position of ${Prey}_i$, compare the fitness of ${Prey}_{1i}$ and ${Prey}_i$. If the fitness value of ${Prey}_{1i}$ is smaller than the fitness value of ${Prey}_i$, ${Prey}_{1i}$ is updated to ${Prey}_i$, so that the quality of the solution is constantly improved. In this way, the population diversity of MPA is increased, the probability of MPA finding a better location is increased, and the exploration capability of MPA is enhanced.

3.2. Compound Mutation Strategy

The compound mutation strategy can enhance the exploration and exploitation ability of swarm intelligence algorithm. For this reason, many researchers have proposed various DE algorithms by applying multiple mutation schemes to enhance the performance of swarm intelligence algorithms. Gao [52] proposed a PSO with composite mutation strategy, which makes PSO have stronger exploration ability and faster convergence rate. Cui [53] designed a new adaptive multi-subgroup DE algorithm to improve the optimization performance. Zhang [54] used the compound mutation strategy to balance the exploitation and exploration trend of SSA, and increased the global convergence ability of SSA.

The Composite mutation strategy has been used to improve a variety of algorithms and achieved remarkable results. Therefore, this paper adopts the local mutation scheme of Composite DE (CoDE) [55], which performs well in the CEC-2017 competition, to improve the global exploration ability of MPA. The generation equations of the three mutation locations ${Prey}_{2i,j}$, ${Prey}_{3i,j}$, ${Prey}_{4i,j}$ after using the local mutation scheme is shown in Equations (11)–(13):

$${Prey}_{2i,j}=\left\{\begin{array}{cc}{Prey}_{r1,j}+F_1\times \left({Prey}_{r2,j}-{Prey}_{r3,j}\right)& ifrand\left(\right)<C_{r1}\\ {Prey}_{i,j}& otherwise\end{array}\right.$$

(11)

where $r_1$, $r_2$, $r_3$ is a random integer ranging from 1 to n, and $F_1$ represents the scale coefficient, whose value is set to 1.0. $C_{r1}$ stands for mutation rate and its value is set to 0.1.

$$Pre{y}_{3i,j}=\left\{\begin{array}{c}\begin{array}{c}Pre{y}_{r4,j}+F_2\times (Pre{y}_{r5,j}-Pre{y}_{r6,j})\hfill \\ +F_2\times (Pre{y}_{r7,j}-Pre{y}_{r8,j})ifrand\left(\right)<C_{r2}\end{array}\\ Pre{y}_{i,j}\mathrm{otherwise}\end{array}\right.$$

(12)

where $r_4$, $r_5$, $r_6$, $r_7$, $r_8$ is a random integer ranging from 1 to n, $F_2$ represents the scale coefficient, whose value is set to 0.8. $C_{r2}$ stands for mutation rate and its value is set to 0.6.

$$Pre{y}_{4i,j}=\left\{\begin{array}{c}\begin{array}{c}Pre{y}_{\mathrm{i},j}+rand\left(\right)\times (Pre{y}_{r9,j}-Pre{y}_{i,j})\hfill \\ +F_3\times (Pre{y}_{r10,j}-Pre{y}_{r11,j})ifrand\left(\right)<C_{r3}\end{array}\\ Pre{y}_{i,j}\mathrm{otherwise}\end{array}\right.$$

(13)

where $r_9$, $r_{10}$, $r_{11}$ is a random integer ranging from 1 to n, $F_3$ represents the scale coefficient, whose value is set to 0.8. $C_{r3}$ stands for mutation rate and its value is set to 0.9.

Among the three mutation positions ${Prey}_{2i,j}$, ${Prey}_{3i,j}$, ${Prey}_{4i,j}$ of the $ith$ predator, firstly, the boundary of the candidate position was corrected, and then the greedy selection mechanism was used to select the position with the lowest fitness value ${Prey}_{2i,j}$, ${Prey}_{3i,j}$, ${Prey}_{4i,j}$ and ${Prey}_i$ as the predator position.

3.3. Multiple Disturbances Strategy

The disturbance mechanism of the marine predator algorithm, which can make MPA overcome the premature convergence problem in the optimization process, simulates the influence of eddy formation and Fish Aggregation Devices (FADs) during the process of predation. However, in the actual process of predation, the marine predators will be affected by the device more than once. Therefore, it is considered to run the disturbances mechanism several times in order to simulate the actual predation process of marine predators more accurately. It operates as follows:

$For$ disturbances = 1:2

$${Prey}_i=\left\{\begin{array}{cc}{Prey}_i+CF\left[LB+r_1\otimes (UB-LB)\right]\otimes U\hfill & if{r}_1\le FAD\mathrm{s}\hfill \\ {Prey}_i+\left[FADs\left(1-r_1\right)+r_1\right]\left({Prey}_{s1}-{Prey}_{s2}\right)\hfill & if{r}_1>FADs\hfill \end{array}\right.$$

(14)

Multiple disturbances strategy can not only maintain the population‘s diversity, but also greatly increase the probability of the algorithm to jump out of local optimal.

3.4. The Proposed mMPA-OC Algorithm

Firstly, an OBL mechanism is added to the optimal value selection process to improve the exploration ability of MPA. Secondly, the composite mutation strategy was used to improve the update strategy of predator position to improve the global search ability of MPA. Finally, the interference factors are improved to multiple disturbance factors, so that MPA can maintain the population diversity in the iterative process. The pseudocode of mMPA-OC is shown in Algorithm 2.

Algorithm 2: mMPA-OC Algorithm.

Initialize populations (Prey) based on Equation (1) While termination criteria are not met Structure the matrix $Elite$ and the matrix $Prey$ based on Equations (2) and (3), calculate the fitness saving   If $iteration$ ≤ $\frac{Ma{x}_{-}iterations}{3}$     Update matrix $Prey$ based on Equation (4)   Else if $\frac{Ma{x}_{-}iterations}{3}<$ $iteration$ ≤ $\frac{2\times Ma{x}_{-}iterations}{3}$     For the first half of the populations $(i\le n/2)$      Update matrix $Prey$ based on Equation (5)     For the other half of the populations $(n/2<i\le n)$      Update matrix $Prey$ based on Equation (6)   Else if $\frac{2\times Ma{x}_{-}iterations}{3}$ $<iteration$ ≤ $Ma{x}_{-}iterations$     Update matrix $Prey$ based on Equation (7)   End (if)   Accomplish memory saving and matrix $Elite$ update   Based Equation (10) compute the reverse solution, According to fitness update prey   For disturbances = 1:2    Applying FADs effect and update based on Equation (8)   End for   Generate three mutation positions based on the Equations (11)–(13)   Update prey location based on greedy selection mechanism End while

3.5. Computational Complexity

In order to better understand mMPA-OC, the space complexity and time complexity of the proposed algorithm are briefly analyzed in this section.

First of all, because the size of the matrix Elite and matrix Prey required by the algorithm is $n\times d$, therefore the maximum space required by the algorithm is $O(n\times d)$.

Second, for time complexity, initializing the population and setting parameters requires $O(n\times d)$ where n is the number of populations and d is the problem’s dimension. In the process of constructing predator matrix $Prey$ and elite matrix $Elite$, requires n; The optimization process, namely the corresponding three phase, requires $O(n\times d)$; In the multiple disturbances, it requires n; in the reverse learning phase, it requires n; in the complex mutation stage, $O(n\times d)$ is required; the outermost loop does t, where t is the number of iterations. Finally, the time complexity of mMPA-OC is $O(t\times n\times d)$.

4. Numerical Experiments and Analysis

The performance of meta-heuristics is measured by experimental results on test functions. Due to the randomness of the meta-heuristic algorithm, mMPA-OC needs sufficient testing to ensure that the performance of the algorithm is not random. Therefore, we set up different experiments to verify the effectiveness of the mMPA-OC.

The first set of experiments is to verify the performance of the mMPA-OC in different dimensions of CEC-2017 with rotation and translation characteristics [56]; the details are shown in

Table 1. Summary of the CEC-2017 function test set.

Class	No	Functions	Dim	Boundary	Optima
Unimodal Functions	1	Shifted and Rotated Bent Cigar	10, 30	[−100,100]	100
	3	Shifted and Rotated Zakharov	10, 30	[−100,100]	300
Simple Multimodal Functions	4	Shifted and Rotated Rosenbrock	10, 30	[−100,100]	400
	5	Shifted and Rotated Rastrigin	10, 30	[−100,100]	500
	6	Shifted and Rotated Expanded Scaffer’s F6	10, 30	[−100,100]	600
	7	Shifted and Rotated Lunacek Bi_Rastrigin	10, 30	[−100,100]	700
	8	Shifted and Rotated Non-Continuous Rastrigin	10, 30	[−100,100]	800
	9	Shifted and Rotated Levy	10, 30	[−100,100]	900
	10	Shifted and Rotated Schwefel	10, 30	[−100,100]	1000
Hybrid Functions	11	Hybrid 1 (N = 3)	10, 30	[−100,100]	1100
	12	Hybrid 2 (N = 3)	10, 30	[−100,100]	1200
	13	Hybrid 3 (N = 3)	10, 30	[−100,100]	1300
	14	Hybrid 4 (N = 4)	10, 30	[−100,100]	1400
	15	Hybrid 5 (N = 4)	10, 30	[−100,100]	1500
	16	Hybrid 6 (N = 4)	10, 30	[−100,100]	1600
	17	Hybrid 7 (N = 5)	10, 30	[−100,100]	1700
	18	Hybrid 8 (N = 5)	10, 30	[−100,100]	1800
	19	Hybrid 9 (N = 5)	10, 30	[−100,100]	1900
	20	Hybrid 10 (N = 6)	10, 30	[−100,100]	2000
Composition Functions	21	Hybrid 1 (N = 3)	10, 30	[−100,100]	2100
	22	Composition 2 (N = 3)	10, 30	[−100,100]	2200
	23	Composition 3 (N = 4)	10, 30	[−100,100]	2300
	24	Composition 4 (N = 4)	10, 30	[−100,100]	2400
	25	Composition 5 (N = 5)	10, 30	[−100,100]	2500
	26	Composition 6 (N = 5)	10, 30	[−100,100]	2600
	27	Composition 7 (N = 6)	10, 30	[−100,100]	2700
	28	Composition 8 (N = 6)	10, 30	[−100,100]	2800
	29	Composition 9 (N = 3)	10, 30	[−100,100]	2900
	30	Composition 10 (N = 3)	10, 30	[−100,100]	3000

in order to verify the mMPA-OC in convergence rate and accuracy, and the effectiveness of the mMPA-OC in high-dimensional problems.

The second set of experiments was conducted to verify the performance of the improved algorithm in 10 test functions of CEC-2019, which are characterized by dynamic, niche composition, and many other types of problems with higher difficulty coefficients. The details of CEC-2019 are shown in

Table 2. Summary of the CEC-2019 function test set.

No	Function	Dim	Boundary	Optima
1	Storn’s Chebyshev Polynomial Fitting Problem	9	[−8192,8192]	1
2	Inverse Hilbert Matrix Problem	16	[−16,384,16,384]	1
3	Lennard–Jones Bestimum Energy Cluster	18	[−4,4]	1
4	Rastrigin	10	[−100,100]	1
5	Griewangk	10	[−100,100]	1
6	Weierstrass	10	[−100,100]	1
7	Modified Schwefel	10	[−100,100]	1
8	Expanded Schaffer’s F6	10	[−100,100]	1
9	Happy Cat	10	[−100,100]	1
10	Ackley	10	[−100,100]	1

to verify the effectiveness of the mMPA-OC on different test problems.

At the same time, for the fairness of experiments, each group of experiments selected a representative meta-heuristic algorithm for comparison. Each group of experiments evaluated the improved algorithm from three aspects. The first aspect is to compare the mean ($Mean$), minimum ($Best$), and standard deviation ($Std$) of the test function results. The second aspect is the analysis of the convergence rate of the optimization solution process. Finally, the third aspect is a non-parametric statistical test, namely the Friedman test, the Wilcoxon rank-sum test and Bonferonni–Holm test.

4.1. The Experimental Setup

In order to compare the performance of various algorithm in complex numerical optimization problems, this paper selected the representative meta-heuristic optimization algorithm with good solving ability in recent years as the comparison. These algorithms are DE [5], PSO [7], SSA [25], GWO [11], SCA [25], MPA [32], AOA [57]. For the fairness and rationality of the experiment, the parameter Settings of the selected comparison algorithm are consistent with those of the original literature. For all algorithms, the number of populations is set to 25, the number of iterations of each function is 500, and each algorithm is run 30 times independently. The $Mean$ obtained by 30 independent calculations can greatly reduce the impact of accidental errors, so it is used as an index to evaluate the convergence accuracy of the algorithm. The closer its value to the theoretical value, the better the optimization effect of the algorithm. The $Std$ is used to evaluate the stability and robustness of the algorithm, and a smaller value indicates a more stable algorithm. The $Best$ represents the optimal solution that the algorithm can find, and the smaller the $Best$ is, the stronger the optimization ability of the algorithm.

4.2. Experiments on CEC-2017

In this section, we evaluate different dimensions of the CEC2017 function, to verify the convergence rate and accuracy of the mMPA-OC in different dimensions.

4.2.1. Experiments on CEC-2017 (10 Dimensions)

Analysis of Convergence Accuracy

This section compares the accuracy of mMPA-OC with other comparison algorithms on the CEC-2017 (10 dimensions).

Table 3. Comparison of experimental results on single-peak and multi-peak functions of CEC-2017 (10 dimensions).

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F1	Mean	100	106.8276	8,108,315.844	1,061,382,100	1843.8038	10,664,648,726	163,470,763.9	10,692.9039
	Best	100	100.1804	1,208,907.685	569,859,889.3	100.2589	4,729,457,814	16,763.8129	926.8083
	Std	0	6.729	8,657,916.616	336,289,717.6	1752.3964	4,069,722,672	636,396,645.2	12,658.4727
F3	Mean	300	300	5957.27	3430.088	437.0965	13,083.9099	4276.9467	8330.1159
	Best	300	300	582.6085	892.23	300	8066.2468	320.551	2902.241
	Std	0	0	4460.7107	1557.6146	209.7594	2622.3508	3768.2704	3138.024
F4	Mean	400.0726	401.5475	416.593	462.3447	412.0975	1242.2072	419.4951	406.5477
	Best	400	400.0007	402.5074	432.1187	400.0935	633.5613	402.5088	404.8564
	Std	0.1015	1.3163	24.1903	23.7661	20.8928	340.9392	17.4984	0.792
F5	Mean	510.0037	512.438	533.3588	557.7172	553.9596	568.0954	522.4643	517.2904
	Best	503.9798	503.9798	510.5889	537.6987	512.9344	533.9107	507.1906	510.7732
	Std	3.8263	4.816	11.3082	7.9618	20.6045	17.832	10.8424	3.9419
F6	Mean	600.0019	600.0584	620.5732	621.9675	629.6076	642.5492	601.6539	600
	Best	600.0003	600.0003	603.9594	616.4607	605.3087	633.8042	600.0523	600
	Std	0.0033	0.2152	10.5186	4.2584	12.9463	5.1598	2.1251	0
F7	Mean	720.2035	725.794	755.5331	785.0931	739.9699	802.4523	736.7592	730.2992
	Best	713.265	716.205	732.4352	758.5851	721.4695	779.06	714.7309	723.4814
	Std	4.5453	6.3526	13.5028	12.359	12.5601	14.092	13.3338	3.352
F8	Mean	808.3908	812.2049	828.6228	847.7033	828.4226	841.4078	817.6611	817.7312
	Best	803.9798	804.9748	810.8841	834.2953	813.9294	826.695	805.8513	812.0345
	Std	2.8941	3.5915	10.0128	7.182	8.7665	9.1585	6.5663	2.826
F9	Mean	900	900.1924	1023.2362	1089.7086	1135.9462	1426.7447	918.2368	900
	Best	900	900	910.6348	935.6915	905.3043	1097.6163	900.2621	900
	Std	0	0.3114	170.4269	109.6609	257.0835	211.6418	23.602	0
F10	Mean	1401.9491	1491.5483	2075.7787	2512.8428	2159.8623	2278.1824	1710.1302	1825.1921
	Best	1187.0988	1122.3681	1532.0746	1991.1334	1125.9705	1780.6166	1075.761	1492.0982
	Std	110.5593	168.8878	264.8628	228.571	405.4517	276.9056	368.5721	116.0331

shows the results of the Unimodal Function and Simple Multimodal Function,

Table 4. Comparison of experimental results on CEC-2017 (10 dimensions) Hybrid function.

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F11	Mean	1103.4891	1104.1364	1281.9724	1259.5273	1191.3504	3354.6329	1139.664	1106.0263
	Best	1101.0112	1101.5978	1154.1689	1171.8384	1114.7385	1293.9305	1105.5102	1103.9669
	Std	1.2925	1.3009	108.1073	79.6123	49.6161	1728.1785	35.4107	1.3954
F12	Mean	1240.5562	1308.7265	3,230,858.996	21,377,525.08	2,789,474.332	432,667,659.8	1,051,695.492	1,095,867.871
	Best	1200.343	1203.5592	33,493.0474	2,841,849.046	22,150.2276	138,437.5232	18,893.007	250,517.6037
	Std	53.0655	74.4526	5,644,195.597	12,709,367.48	3,387,559.385	432,830,989.4	1,772,161.396	969,349.5173
F13	Mean	1304.4426	1308.9554	40,434.0397	82,836.5458	18,372.9432	8755.3468	12,717.5124	5458.9172
	Best	1300.1077	1301.4537	4576.6037	3679.2476	2962.0021	3641.9745	2357.5097	1839.8396
	Std	2.4144	3.288	47,557.3757	58,029.1041	13,541.0253	5331.6244	8828.8129	4421.4108
F14	Mean	1402.6926	1408.6092	2672.7999	2821.571	6039.5533	11,204.2191	3285.5448	1883.5571
	Best	1400.0023	1401.018	1492.7123	1543.5738	1483.8057	1687.888	1474.4304	1421.6967
	Std	1.2558	8.0957	2509.8955	1301.5523	4822.5189	8911.9233	2000.6228	968.4925
F15	Mean	1500.5491	1501.5445	7998.0207	4053.8962	29,219.8144	18,537.5347	6658.6737	1872.0453
	Best	1500.0444	1500.1732	2891.5448	1867.6562	1725.7407	3339.3339	1711.2978	1516.6168
	Std	0.4381	0.7921	4822.0205	2086.5598	40343.4605	6398.6788	4547.513	401.708
F16	Mean	1601.7153	1610.1882	1828.5972	1842.7075	1971.0083	2109.1143	1769.0236	1636.9271
	Best	1600.3814	1601.6186	1635.8182	1689.0373	1675.2534	1758.9776	1605.2372	1604.0572
	Std	1.0481	22.4468	124.2733	97.2775	145.2623	149.8626	134.8513	46.3899
F17	Mean	1717.2059	1721.4231	1822.6115	1797.6155	1794.8546	1881.578	1762.7652	1714.5795
	Best	1701.9063	1700.5126	1756.4502	1748.4453	1737.9088	1750.336	1727.8175	1701.8373
	Std	9.0696	10.0999	53.0339	20.7544	62.2821	105.755	20.4612	8.8204
F18	Mean	1800.8938	1809.2932	90,944.4631	362,483.3345	19,698.2444	7,995,222.952	25,226.8988	6237.7304
	Best	1800.1453	1803.761	3142.5952	39,744.1004	3881.2493	3318.0853	3300.1188	2393.893
	Std	0.7168	4.6172	164,478.7197	282,886.676	12,313.2544	30,473,369.49	14,511.9651	4248.5487
F19	Mean	1900.8767	1901.7436	7380.7955	11,372.8388	10,661.52	149,380.3247	19,619.968	3074.5644
	Best	1900.0913	1900.4799	2065.7355	2484.498	1967.2902	2368.5756	1928.3734	1905.9442
	Std	0.442	0.7419	7630.2208	8020.5709	9014.0731	96,392.4919	49,536.8252	1975.285
F20	Mean	2014.2019	2023.412	2125.1832	2131.7449	2212.7991	2159.3552	2104.6432	2000.2411
	Best	2000.9962	2000.0039	2059.1384	2070.7585	2095.2963	2047.3442	2027.4857	2000
	Std	9.7962	9.8009	47.54	39.5295	78.0492	91.0083	63.741	0.3387

shows the results of the Hybrid Functions, and

Table 5. Comparison of experimental results on CEC-2017 (10-dimensions) combined functions.

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F21	Mean	2200	2200.1556	2320.03	2310.5843	2334.4961	2339.3412	2306.4462	2303.7184
	Best	2200	2200.0002	2205.9721	2212.0808	2200	2226.5385	2201.1102	2225.1284
	Std	0	0.4811	36.6974	63.0239	52.6469	37.161	36.4069	36.2225
F22	Mean	2279.5231	2287.3144	2311.8083	2413.8976	2304.0333	3048.2127	2380.1052	2303.7112
	Best	2200.0001	2200.0026	2259.0062	2348.3776	2239.1749	2539.9307	2301.8293	2301.1204
	Std	39.517	31.4418	13.8889	37.6343	12.9538	363.6323	266.9115	2.5557
F23	Mean	2610.5461	2612.7036	2635.8853	2663.946	2666.7291	2760.5573	2625.5951	2619.4209
	Best	2605.1177	2605.1294	2617.644	2644.2775	2610.205	2677.446	2607.5145	2611.4611
	Std	4.0451	4.4424	14.1558	10.0113	42.124	49.6746	13.1716	3.7865
F24	Mean	2565.8858	2508.0594	2756.1926	2793.9359	2724.3776	2844.062	2751.498	2744.1484
	Best	2500	2500.0007	2519.9315	2773.1282	2500	2644.0742	2732.4154	2623.0139
	Std	105.0136	43.8255	60.8266	7.1696	128.7548	103.0452	14.0368	31.2292
F25	Mean	2881.005	2897.8673	2943.6753	2975.1482	2929.0971	3404.3929	2937.1138	2927.91
	Best	2600.0458	2897.7429	2899.3518	2932.9428	2897.9914	3017.2301	2903.5571	2898.8279
	Std	77.2381	0.4669	36.7961	19.3857	30.1638	253.2932	19.0204	15.6694
F26	Mean	2816.6686	2844.0255	3196.1072	3251.4646	2943.435	4055.5569	3261.3199	2989.3457
	Best	2600.0007	2600.008	2854.519	3015.1466	2600.0001	3409.7005	2807.1331	2913.7001
	Std	117.6878	93.253	325.1367	398.2489	240.3498	308.8662	440.5729	43.2016
F27	Mean	3089.3949	3089.9817	3174.0925	3106.407	3120.5145	3270.1402	3112.9911	3091.7492
	Best	3089.0055	3088.978	3074.1695	3098.8631	3091.7734	3155.1816	3090.9987	3090.2317
	Std	0.4056	1.1961	49.1313	4.4923	33.3561	67.3763	28.4897	1.4882
F28	Mean	3100.0002	3101.8567	3289.5001	3357.2857	3297.555	3819.2845	3390.0391	3345.4925
	Best	3100	3100.0006	3272.5011	3244.5531	3100	3401.0833	3179.301	3208.123
	Std	0.0002	10.1621	13.3484	92.0799	124.7313	138.3026	116.5221	78.2294
F29	Mean	3145.3909	3155.7852	3299.7075	3280.5186	3312.8269	3459.6432	3214.9734	3200.8089
	Best	3130.4101	3134.285	3165.749	3174.5907	3157.0794	3181.5614	3154.8275	3164.4616
	Std	9.756	20.2514	71.081	58.3767	85.9996	201.9068	52.6096	19.9764
F30	Mean	3471.0628	3936.7449	92,479.3729	1,749,161.064	954,889.1999	39,227,575.78	1,144,625.586	152,386.1769
	Best	3394.8434	3405.8359	3720.3605	44,102.5397	38,412.1961	322,791.3601	7034.7464	16,203.7498
	Std	230.796	1032.6363	289,931.5158	1,129,451.248	1,480,086.714	31,423,163.17	1,467,710.156	178,563.0271

shows the results of the Composition Functions. It should be noted that the numbers in bold represent the relative best value for the comparison algorithm.

It can be seen from Table 3 that mMPA-OC has the best indexes on Unimodal Functions functions. mMPA-OC performs best on Simple Multimodal Functions, F4, F5, and F9 in all indicators, and the $Best$ is very close to the theoretical optimal value. On F6, although the performance of mMPA-OC is not the best, it is close to the theoretical optimal value. On F7 and F8, the $Best$ and $Mean$ found by mMPA-OC are the best, but DE $Std$ is the best. mMPA-OC performs well on the $Mean$ of F10, but the $Best$ found by GWO performs best.

As for the Hybrid function, F11–F16, and F18–F19 are the best in all performance mMPA-OC. On F17, DE performed well in $Mean$ and $Std$, MPA performed well in the optimal solution $Best$, but mMPA-OC performed better in $Mean$ and $Std$ than MPA. DE was the best on F20, mMPA-OC is second better than MPA.

For the combined function, mMPA-OC is the best in all performance on F21 and F28–F30, the $Best$ and the $Mean$ values of mMPA-OC on F22, F23, F25 are the best of all algorithms, but the $Std$ is not the best. On F24, mMPA-OC performs best in the $Best$, but slightly worse in the $Mean$ and $Std$. On F26, mMPA-OC performs well in the $Mean$, but not the best in the $Best$ and $Std$. On F27, The $Mean$ and $Std$ obtained by mMPA-OC are good, but the $Best$ is not the best.

From the above results, we can infer that mMPA-OC can achieve results close to the theoretical optimal value to a large extent and is relatively stable compared to other algorithms.

Analysis of Convergence Rate

This section uses the convergence curve to assess the convergence rate of mMPA-OC to further verify the performance of mMPA-OC. Figure 1 and Figure 2 specifically shows the convergence characteristics of the comparison algorithm on F1, F3, F8, F10, F12, F14, F16, F18, F20, F21, F23, F25, F26, and F30. The convergence curve shows that when the iterations reaches 500, the convergence accuracy of mMPA-OC is very close to MPA on F3, F14, F18, and F21, but the convergence rate of mMPA-OC is fast. Among the convergence curves, mMPA-OC has the fastest convergence rate and the highest convergence accuracy.

In general, mMPA-OC has the best convergence performance, MPA, SSA and DE have weak convergence performance, and GWO, SCA, AOA and PSO have the weakest convergence performance. These results show that mMPA-OC can quickly find a better location and achieve efficient search.

Analysis of Statistical Results

In this section, the most commonly used test, namely the Wilcoxon rank sum test, Friedman test, and Bonferroni–Holm test, were used to verify the statistical difference between other algorithms and mMPA-OC in CEC-2017 (10 dimensions).

Compare Wilcoxon Rank Sum Test

In this paper, the significance level of 0.05 was used. The p-value of Wilcoxon rank sum test was compared with the significance level of 0.05 [58] to evaluate the performance of mMPA-OC. The p-values of the Wilcoxon rank sum test for mMPA-OC and other compared algorithms are shown in

Table 6. p-values of the Wilson rank-sum test of mMPA-OC and the comparison algorithm in CEC-2017 (10 dimensions).

	mMPA-OC & MPA	mMPA-OC & PSO	mMPA-OC & SCA	mMPA-OC & SSA	mMPA-OC & AOA	mMPA-OC & GWO	mMPA-OC & DE
F1	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F3	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F4	4.69 $\times {10}^{-8}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	8.15 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F5	2.51 $\times {10}^{-2}$	1.46 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	6.07 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.87 $\times {10}^{-7}$	9.06 $\times {10}^{-8}$
F6	4.42 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F7	5.87 $\times {10}^{-4}$	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.50 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$	1.29 $\times {10}^{-6}$	3.82 $\times {10}^{-9}$
F8	8.66 $\times {10}^{-5}$	9.92 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.31 $\times {10}^{-8}$	7.39 $\times {10}^{-11}$
F9	3.69 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.08 $\times {10}^{-5}$
F10	4.51 $\times {10}^{-2}$	6.07 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.57 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$	4.71 $\times {10}^{-4}$	6.70 $\times {10}^{-11}$
F11	1.19 $\times {10}^{-1}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.34 $\times {10}^{-11}$	1.20 $\times {10}^{-8}$
F12	2.96 $\times {10}^{-5}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F13	1.73 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F14	3.16 $\times {10}^{-5}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F15	1.86 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F16	1.87 $\times {10}^{-7}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.08 $\times {10}^{-11}$
F17	5.19 $\times {10}^{-2}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.01 $\times {10}^{-1}$
F18	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F19	1.60 $\times {10}^{-7}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F20	1.78 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.69 $\times {10}^{-11}$	3.69 $\times {10}^{-11}$
F21	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	5.57 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F22	5.75 $\times {10}^{-2}$	2.44 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$	3.16 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.98 $\times {10}^{-11}$
F23	7.01 $\times {10}^{-2}$	3.69 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.33 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	1.49 $\times {10}^{-6}$	6.52 $\times {10}^{-9}$
F24	6.10 $\times {10}^{-3}$	2.15 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.99 $\times {10}^{-4}$	1.41 $\times {10}^{-9}$	1.29 $\times {10}^{-9}$	4.20 $\times {10}^{-10}$
F25	5.56 $\times {10}^{-4}$	2.87 $\times {10}^{-10}$	3.69 $\times {10}^{-11}$	7.38 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.16 $\times {10}^{-10}$	2.67 $\times {10}^{-9}$
F26	3.99 $\times {10}^{-4}$	8.10 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	2.28 $\times {10}^{-1}$	3.02 $\times {10}^{-11}$	8.10 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$
F27	9.03 $\times {10}^{-4}$	9.51 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	6.70 $\times {10}^{-11}$
F28	4.98 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	9.51 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F29	5.55 $\times {10}^{-2}$	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	6.07 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.37 $\times {10}^{-10}$	3.34 $\times {10}^{-11}$
F30	2.49 $\times {10}^{-6}$	5.49 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$

.

From the p-value test results given in Table 6, it can be seen that there is no significant difference between the mMPA-OC and the other seven algorithms only in a few functions. For example, in F5, F11, F17, F22, F23, and F29, there is no significant difference between the mMPA-OC and MPA, and on F26 there is no significant difference between the mMPA-OC and SSA, which is reflected in the performance of mMPA-OC not being very good. In short, mMPA-OC differs greatly from other algorithms. Therefore, comparison of Wilcoxon rank sum test results shows that mMPA-OC has superior performance.

Compare Friedman Test

In order to further illustrate the comprehensive performance of mMPA-OC and other algorithms, the Friedman test was introduced to test the data. The Friedman tests were performed with a significance level of 0.05 to obtain the average ranking of each algorithm. The results are shown in

Table 7. Average ranking of the Friedman test on CEC-2017 (10 dimensions).

Algorithm	mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
Mean rank	1.1379	2.1034	5.4138	6.2414	5.3793	7.7241	4.7931	3.2069
Final rank	1	2	6	7	5	8	4	3

. Furthermore, mMPA-OC has the best comprehensive performance in CEC-2017 test set (10 dimensions), followed by MPA, DE, GWO, SSA, PSO, SCA, and AOA.

Bonferroni–Holm Test

In order to further verify performance differences between the proposed algorithm mMPA-OC and its peers. Bonferroni–Holm tests were used to test the p-values obtained by Wilcoxon rank sum test. When the significance level is 0.025, the Logistic values of Bonferroni–Holm tests for mMPA-OC compared with other algorithms in CEC-2017 (10 dimensions) are obtained as shown in the

Table 8. Logistic values of Bonferroni–Holm tests for mMPA-OC compared with other algorithms in CEC-2017 (10 dimensions).

	mMPA-OC & MPA	mMPA-OC & PSO	mMPA-OC & SCA	mMPA-OC & SSA	mMPA-OC & AOA	mMPA-OC & GWO	mMPA-OC & DE
F1	1	1	1	1	1	1	1
F3	1	1	1	1	1	1	1
F4	0	1	1	1	1	1	1
F5	1	1	1	1	1	1	1
F6	1	1	1	1	1	1	1
F7	1	1	1	1	1	1	1
F8	1	1	1	1	1	1	1
F9	0	1	1	1	1	1	1
F10	0	1	1	1	1	1	1
F11	1	1	1	1	1	1	1
F12	1	1	1	1	1	1	1
F13	1	1	1	1	1	1	1
F14	1	1	1	1	1	1	1
F15	1	1	1	1	1	1	1
F16	1	1	1	1	1	1	1
F17	0	1	1	1	1	1	0
F18	1	1	1	1	1	1	1
F19	1	1	1	1	1	1	1
F20	0	1	1	1	1	1	1
F21	1	1	1	1	1	1	1
F22	0	1	1	1	1	1	1
F23	0	1	1	1	1	1	1
F24	0	1	1	1	1	1	1
F25	1	1	1	1	1	1	1
F26	1	1	1	1	1	1	1
F27	1	1	1	1	1	1	1
F28	1	1	1	1	1	1	1
F29	0	1	1	1	1	1	1
F30	1	1	1	1	1	1	1

. The results show that the difference between mMPA-OC and its peers is statistically significant.

Compared with Other Improved Algorithms

In this section, we conduct the comparison experiment of mMPA-OC and two latest modified MPAs on CEC2017 (10 dims). while TLMPA [45] was proposed in 2020, MSMPA [47] was proposed in 2022. Comparing the test results, it can be seen that the proposed algorithm has 16 optimal results and 9 suboptimal results on 29 test functions. It can be concluded that the proposed algorithm is competitive with other improvements for MPA. The comparison of mMPA-OC and other improved MPAs is shown in

Table 9. Comparison of mMPA-OC with other improved algorithms in CEC-2017 (10 dimensions).

	mMPA-OC		TLMPA [45]		MSMPA [47]
	Mean	Std	Mean	Std	Mean	Std
F1	100	0	100	0	107	7.31
F3	300	0	300	0	300	0
F4	400.0726	0.1015	400.03	0.06	401	0.912
F5	510.0037	3.8263	510.58	2058	509	0.039
F6	600.0019	0.0033	600	0	600	0.0155
F7	720.2035	4.5453	723.63	3.08	724	4.31
F8	808.3908	2.8941	811.56	3.72	810	3.05
F9	900	0	900	0	900	0.175
F10	1401.9491	110.5593	1613.14	113.19	1460	9.41
F11	1103.4891	1.2925	1102.51	1.08	1100	1.88
F12	1240.5562	53.0655	1310.64	95.32	1290	60
F13	1304.4426	2.4144	1309.72	2.36	1310	3.22
F14	1402.6926	1.2558	1402.71	1.88	1410	4.66
F15	1500.5491	0.4381	1500.76	0.36	1500	0.789
F16	1601.7153	1.0481	1604.05	4.61	1600	1.22
F17	1717.2059	9.0696	1706.29	4.82	1720	9.07
F18	1800.8938	0.7168	1801.47	1.65	1810	3.53
F19	1900.8767	0.442	1900.45	0.47	1900	0.682
F20	2014.2019	9.7962	2000.72	3.61	2020	10.5
F21	2200	0	2214.99	37.45	2210	38.6
F22	2279.5231	39.517	2288.34	29.04	2290	26.1
F23	2610.5461	4.0451	2611.55	4.04	2610	3.45
F24	2565.8858	105.0136	2587.59	112.7	2550	97.9
F25	2881.005	77.2381	2905.56	17.29	2900	0.0093
F26	2816.6686	117.6878	2879.99	76.11	2760	10.6
F27	3089.3949	0.4056	3089.36	0.58	3090	0.419
F28	3100.0002	0.0002	3108.92	95.29	3110	51.6
F29	3145.3909	9.756	3165.07	11.13	3150	12.1
F30	3471.0628	230.796	4133.44	3178.85	3470	69.5

.

4.2.2. Experiments on CEC-2017 (30 Dimensions)

We conducted experiments based on CEC-2017 (30-dimensions) to verify the effectiveness of the mMPA-OC for solving higher-dimensional problems.

Analysis of Convergence Accuracy

In this section, the convergence accuracy of mMPA-OC is compared with other algorithms on the CEC-2017(30 dimensions). The results for Unimodal function and Simple Multimodal function are presented in

Table 10. Comparison of experimental results on single-peak and multi-peak functions of CEC-2017 (30 dimensions).

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F1	Mean	42,182.50063	1,294,122.701	12,922,443,488	20,666,583,679	122,494.247	55,220,766,498	3,886,271,590	990,394.399
	Best	5560.88699	66,870.19604	5,429,136,519	13,038,925,561	8616.060888	32,234,584,745	828,838,627.5	120,694.7251
	Std	35,535.61668	1,349,814.717	6,301,414,674	4,261,179,855	172,517.2824	9,973,432,269	2,532,897,207	790,074.4228
F3	Mean	1996.326789	9141.842858	145,908.3146	92,702.37928	75,813.35145	83,952.74582	70,405.81736	169,970.3748
	Best	364.9041071	2067.04235	59,466.21112	46,576.17093	25,686.07082	59,268.54975	44,922.37915	118,659.4953
	Std	1298.035752	4136.318306	49,126.76055	20,057.50471	26,863.06493	8984.218676	15,513.64254	32,593.307
F4	Mean	499.7804649	506.0088473	2536.364476	2922.009032	545.4611684	14,441.64737	686.3956453	522.0442862
	Best	467.8360878	476.5938901	827.0856318	1477.230752	475.9992276	7706.914711	567.819881	495.1244623
	Std	22.8973307	17.14127549	1144.518336	957.523058	46.64038666	5627.891781	82.65292593	20.88511283
F5	Mean	579.3948743	617.9093447	789.872256	830.268896	726.9839312	896.8573171	643.2966936	694.0952688
	Best	552.8156458	558.8030149	696.8631336	753.4065639	626.3734703	819.2773451	590.8492131	660.0310588
	Std	14.86724156	23.74085381	54.18131288	39.38164433	59.69027928	38.4052921	41.67733932	13.29234634
F6	Mean	603.2557297	610.6667441	669.3466865	665.7943635	661.2356692	683.8376316	613.9177854	600.1691854
	Best	600.9790419	604.0560878	638.4494488	654.1626578	638.8042274	660.5653423	606.0947963	600.0927924
	Std	1.403005234	4.448962132	12.29793956	6.763730468	10.40866695	7.593890492	4.970431006	0.045275737
F7	Mean	845.7248071	864.8199904	1300.223373	1259.05305	969.8810264	1409.510269	932.944045	931.499091
	Best	782.8268592	810.3196962	1153.192758	1135.5894	861.3813241	1316.214984	823.0886023	893.1720415
	Std	31.50585511	29.83219259	87.8592846	70.22390698	66.03176677	56.51161031	55.26901543	13.87776347
F8	Mean	874.2971663	907.8467772	1046.90385	1104.863143	958.0147696	1121.452392	913.1056072	997.8145444
	Best	849.7488228	870.2565545	984.8416277	1029.22054	885.5973031	1030.011903	862.9636365	979.6881102
	Std	16.55124425	17.52388178	32.71540352	27.26076277	28.60640877	36.12472219	28.89665738	11.09386178
F9	Mean	1100.928478	1617.244645	9858.646183	8627.201757	5899.232799	8074.554642	2420.937586	2118.072252
	Best	940.8448295	1018.890211	5351.806258	5152.005263	3715.768964	6243.809679	1221.031563	1404.080004
	Std	217.1184266	378.6530134	2418.424546	1890.531562	1037.79086	1233.264503	1105.724071	467.7150445
F10	Mean	4099.667046	4508.391413	7585.100057	8870.758277	5196.392039	8026.575198	5159.132986	7700.291644
	Best	2740.474353	3087.469784	5854.172578	7736.241937	3744.512352	6624.089711	3352.352877	6976.181046
	Std	574.7750946	528.1053996	767.3836531	452.065567	633.5282205	617.8704318	1407.922492	308.9169598

, the results for Hybrid function are presented in

Table 11. Comparison of experimental results on CEC-2017 (30-dimensional) mixed function.

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F11	Mean	1198.111328	1213.511928	4625.59875	4139.229599	1460.554197	9516.813398	2522.682524	2128.050893
	Best	1127.143647	1148.450132	2102.448569	2342.5548	1272.662329	4312.21918	1372.813369	1444.757575
	Std	34.34277308	29.56551235	2205.870563	1131.405696	159.0913465	3281.199419	1321.685389	813.0570137
F12	Mean	630,554.2723	1,618,609.873	26,229,8740	2,662,909,795	39,488,993.65	14,685,258,660	138,514,181.6	30,813,285.77
	Best	18,155.62017	114,474.7007	80,085,784.45	888,221,541.3	4,352,912.417	8,528,127,904	5,415,646.89	9,509,025.965
	Std	682,378.395	1,361,850.423	149,814,158.9	956,093,526	41,786,086.75	3,638,880,699	148,843,762.5	15,592,738.45
F13	Mean	1621.57658	7335.847578	6,254,323.877	1,246,434,428	89,376.63849	12,720,791,543	19,810,084.23	4,111,696.121
	Best	1448.98089	3499.37311	930,936.1215	425,636,626.8	20,257.741	5,181,599,004	40,816.37699	423,533.289
	Std	170.09789	2333.571437	4,609,517.375	513,557,603.9	49,459.28151	5,343,770,111	63,340,877.02	3,302,270.622
F14	Mean	1441.893542	1486.11475	190,070.8079	819,455.9544	441,998.8711	4,204,442.154	682,355.4525	460,770.9919
	Best	1431.044694	1448.229096	9313.379368	141,111.8668	5963.904711	101,715.112	7440.456946	28,903.51463
	Std	6.435523748	17.52021901	224,746.7845	451,333.8154	386,426.0348	3,141,820.846	909,216.3139	432,112.5578
F15	Mean	1589.169659	1779.883698	2,554,385.835	69,626,410.92	34,485.30008	409,626,559.6	621,614.2374	944,879.4466
	Best	1530.706886	1631.204295	70,477.69153	3,837,870.616	6888.63852	17,884.97979	14,612.27273	65,236.70263
	Std	28.1226353	79.54382121	11,596,949.62	69,156,949.82	26,283.73154	529,660,184.5	1,001,420.41	868,593.7792
F16	Mean	2234.17381	2448.000328	3459.314781	4109.093128	3235.361513	5623.880422	2773.739087	3072.995614
	Best	1657.814557	2012.940408	2690.595523	3577.317593	2571.767158	3787.674141	2083.904676	2727.350064
	Std	218.1261359	215.4089878	388.7831725	311.8323531	406.1625848	1225.195907	387.3942008	187.4076652
F17	Mean	1857.691056	1915.22387	2590.795767	2833.332164	2436.907645	6051.009549	2141.28347	2211.508781
	Best	1761.689584	1787.410485	1890.151411	2256.476859	2042.331902	2654.573856	1785.108415	1938.868372
	Std	74.65630401	105.2666646	263.6206959	217.2106479	271.6981257	3018.525755	212.5609811	121.3702865
F18	Mean	2019.519302	3079.167694	3,564,313.301	16,475,978.49	2,486,312.371	32,361,179.98	3,691,766.078	3,191,591.88
	Best	1842.557775	2128.692528	211,460.2577	3,943,823.946	294,731.7335	2,435,144.759	70,417.05849	1,257,845.528
	Std	171.9084059	1075.026009	3,782,269.33	8,896,736.987	2,962,883.884	24,063,290.29	5,615,533.333	1,605,533.552
F19	Mean	1927.440769	1989.482726	10,933,812.04	111,670,459.1	8,162,084.747	753,180,793	2,532,275.903	843,861.0162
	Best	1918.080605	1942.21941	26,6189.1843	21,916,322.28	70,016.91147	2,076,596.539	6158.737264	122,729.6263
	Std	6.046823475	29.31001575	13,675,902.93	83,264,644.91	4,918,593.502	585,886,052.1	7,808,509.054	822,262.9829
F20	Mean	2225.333015	2327.802362	2813.63039	2908.671988	2769.288474	2879.294661	2508.509997	2512.480112
	Best	2055.829959	2124.204998	2480.689911	2564.178303	2316.741249	2392.034584	2180.837385	2240.343975
	Std	114.8740254	101.3253257	173.2153674	163.0862191	301.9609215	227.7024178	225.6131741	138.4528088

, and the results for Composition function are presented in

Table 12. Comparison of experimental results on CEC-2017 (30-dimensional) combinatorial functions.

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F21	Mean	2366.16272	2379.396604	2585.619153	2604.445594	2514.221649	2702.343449	2415.610694	2492.451478
	Best	2332.283277	2229.107462	2499.043923	2560.516016	2446.685652	2616.305719	2364.196848	2463.934067
	Std	15.52000315	35.29535399	44.92389714	20.32751014	50.66219474	52.37238577	27.94015691	11.89111618
F22	Mean	2775.71317	2314.222463	7871.594172	9984.73825	4676.083823	8914.777019	5216.19368	7128.466697
	Best	2300.733746	2304.95843	3174.372024	5455.495034	2300.816597	5950.678444	2399.384781	3022.438351
	Std	1237.475233	6.339510027	1805.632724	992.3580935	2613.23147	1118.267786	1727.634709	1955.515728
F23	Mean	2707.264603	2734.285614	3105.755218	3083.031912	3025.543682	3628.660005	2809.751615	2841.193252
	Best	2669.099101	2671.322996	2936.56325	3009.553509	2684.345746	3309.137517	2740.838308	2817.92824
	Std	15.27517354	29.7640222	86.53248724	42.62796051	123.3648787	155.7881019	49.91415177	15.06073136
F24	Mean	2882.356892	2897.827657	3295.1231	3264.798671	3101.767247	3901.528453	2956.505115	3038.737564
	Best	2848.613601	2854.90465	3068.154665	3163.769519	2994.213827	3523.054912	2873.823404	3013.088665
	Std	17.63301225	23.21260688	117.9616688	42.37870369	69.0709779	207.216424	49.16516788	14.48515637
F25	Mean	2897.536719	2914.229262	3350.222575	3662.701095	2975.933394	5755.805367	3057.794292	2898.606938
	Best	2883.76333	2884.801208	3042.010482	3341.7416	2912.613618	4388.145604	2957.697808	2890.451576
	Std	16.64408793	20.33170492	214.187784	219.2677532	48.54949565	774.624494	76.13993148	6.663182868
F26	Mean	3488.134472	3387.128079	8072.958222	8049.309668	5218.498654	10771.38502	5114.73175	5565.532713
	Best	2900.336077	2911.299347	4538.206895	7261.454867	2801.865057	9146.924852	4195.066038	5248.104381
	Std	714.4476505	612.3547519	1305.878392	447.7536611	2115.923606	861.4001182	492.8098306	132.3567493
F27	Mean	3214.765437	3227.458308	3200.007143	3574.772451	3354.138091	4727.05599	3287.22779	3230.328489
	Best	3191.47946	3205.908144	3200.006775	3398.5637	3238.66275	3872.489746	3232.618421	3216.658404
	Std	11.20406522	17.12003434	0.000140891	80.94057892	95.30269168	394.8392583	33.28001186	6.354347982
F28	Mean	3090.000002	3100.001324	3291.752602	3308.718235	3294.342966	3828.816377	3368.39253	3324.526818
	Best	2800.000057	3100.000616	3272.501081	3238.565507	3100.000013	3449.360214	3167.718878	3176.822487
	Std	54.77224538	0.000386339	12.81682933	79.17289652	126.5297553	141.0515553	87.23018358	89.86123431
F29	Mean	3567.404824	3762.747664	4931.134752	5333.574202	4603.525496	8284.230744	3975.010859	4184.722548
	Best	3402.458241	3445.51866	4028.017024	4954.967139	4008.619252	5138.157565	3581.611646	3861.138053
	Std	106.4221859	156.3873518	519.3277082	270.2138512	310.4802353	6414.773921	232.6293185	153.6286832
F30	Mean	7386.445363	27,380.52712	27,479,160.45	200,765,178.6	10,251,055.41	2,420,933,115	13,899,516.33	672,438.0501
	Best	5323.048733	10,343.75986	3,949,404.871	34,199,558.31	474,611.0326	354,733,990	2,225,792.497	169,746.2971
	Std	1252.386349	20,194.76458	22,842,729.5	93,741,282.02	9,391,200.27	1,147,208,084	11,553,559.77	552,842.0405

. Again, the numbers in bold represent the best value for the comparison algorithm. It can be seen from Table 8 that mMPA -OC algorithm has the best performance in all Unimodal function F1 and F3.

In Table 10, mMPA-OC is the best in Simple Multimodal function, F9 is the best in all performance, on F4, F5, F7, F8, F10, the $Best$ and the $Mean$ is the best, but $Std$ is not the best among all comparison algorithms. DE has the best performance on the function F6, but the $Best$ found by mMPA-OC is also close to the theoretical optimal value, and better than MPA.

As for the Hybrid function in Table 11, mMPA-OC of F12–F15 and F17–F19 is the best in all performance, mMPA-OC is the $Best$ and the $Mean$ of F11, F16, and F20 is the best. However, the $Std$ are not the best performing among the compared algorithms.

As for the composition functions in Table 12, F29 and F30 are the best in all performance mMPA-OC. On F22 and F27, the $Best$ found by mMPA-OC is the best, but $Std$ and $Mean$ are not the best among all comparison algorithms. MPA performs well on F22, while PSO performs well on F27. On F23–F25 and F28, mMPA-OC performed well in $Best$ and the $Mean$, but $Std$ was not the best among all comparison algorithms. On F21, the $Mean$ found by mMPA-OC is the best among all algorithms, but the $Std$ of DE is the best, and the $Best$ of MPA is best. The performance of mMPA-OC was not the best on F26.

From the above results, we can infer that the convergence performance of mMPA-OC is also improved in different dimensions of the same test function.

Analysis of Convergence Rate

In order to further evaluate the performance of mMPA-OC, the convergence curve is used in this section to evaluate the convergence rate of mMPA-OC in high dimensions. Figure 3 and Figure 4 specifically show the convergence performance of the mMPA-OC and other algorithms on F1, F3, F5, F7, F9, F12, F14, F16, F18, F20, F21, F23, F25, F27, and F30 of CEC-2017 (30-dimensional). Although the convergence rate on F1 is good in the early phase of SSA, the convergence rate of mMPA-OC is faster than SSA in the period phase of searching, PSO converges faster than it on F27. However, the mMPA-OC curve is always at the bottom, indicating that the mMPA-OC has fast convergence rate and convergence accuracy. These results indicate that mMPA-OC in CEC-2017 (30 dimensions) can approach the optimal solution faster than other comparison algorithms.

Analysis of Statistical Results

In order to verify the statistical difference between other algorithms and mMPA-OC in CEC-2017 (30 dimensions), and further illustrate the comprehensive performance of mMPA-OC and its comparison algorithm, the most commonly used Wilcoxon rank sum test, Friedman test, and Bonferroni–Holm test were used to evaluate the algorithm.

Compare Wilcoxon Rank Sum Test

According to the p-values of the Wilcoxon rank sum test in

Table 13. p-values of the Wilcoxon rank sum test of mMPA-OC and the comparison algorithm in CEC-2017 (30 dimensions).

	mMPA-OC & MPA	mMPA-OC & PSO	mMPA-OC & SCA	mMPA-OC & SSA	mMPA-OC & AOA	mMPA-OC & GWO	mMPA-OC & DE
F1	8.15 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.51 $\times {10}^{-2}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.50 $\times {10}^{-11}$
F3	2.03 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F4	1.45 $\times {10}^{-1}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.02 $\times {10}^{-5}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.56 $\times {10}^{-4}$
F5	2.39 $\times {10}^{-8}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.46 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$
F6	8.15 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F7	2.24 $\times {10}^{-2}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.20 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	1.07 $\times {10}^{-7}$	1.21 $\times {10}^{-10}$
F8	5.53 $\times {10}^{-8}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	7.39 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.38 $\times {10}^{-7}$	3.02 $\times {10}^{-11}$
F9	1.01 $\times {10}^{-8}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.46 $\times {10}^{-10}$	1.33 $\times {10}^{-10}$
F10	3.50 $\times {10}^{-3}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.36 $\times {10}^{-7}$	3.02 $\times {10}^{-11}$	8.66 $\times {10}^{-5}$	3.02 $\times {10}^{-11}$
F11	1.09 $\times {10}^{-1}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F12	7.30 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F13	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F14	6.07 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F15	4.08 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F16	9.03 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.20 $\times {10}^{-7}$	3.02 $\times {10}^{-11}$
F17	2.61 $\times {10}^{-2}$	8.99 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.25 $\times {10}^{-7}$	6.70 $\times {10}^{-11}$
F18	2.67 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F19	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F20	7.70 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	9.92 $\times {10}^{-11}$	4.08 $\times {10}^{-11}$	5.19 $\times {10}^{-7}$	2.92 $\times {10}^{-9}$
F21	2.05 $\times {10}^{-3}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.82 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$
F22	5.86 $\times {10}^{-6}$	1.78 $\times {10}^{-10}$	3.69 $\times {10}^{-11}$	1.86 $\times {10}^{-3}$	4.08 $\times {10}^{-11}$	1.01 $\times {10}^{-8}$	8.89 $\times {10}^{-10}$
F23	1.68 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.20 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F24	2.24 $\times {10}^{-2}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.18 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$
F25	2.53 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.46 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.99 $\times {10}^{-2}$
F26	9.93 $\times {10}^{-2}$	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	7.96 $\times {10}^{-3}$	3.02 $\times {10}^{-11}$	1.21 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$
F27	1.68 $\times {10}^{-3}$	8.48 $\times {10}^{-9}$	3.02 $\times {10}^{-11}$	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.98 $\times {10}^{-11}$	9.06 $\times {10}^{-8}$
F28	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.95 $\times {10}^{-11}$
F29	7.22 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.82 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$
F30	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$

, it shows that mMPA-OC is not significantly different from the other seven comparison algorithms only in a few functions. For example, on F4, F7, and F26, there is no significant difference between the mMPA-OC and MPA, and most of these cases occur when the convergence accuracy of the mMPA-OC is not as good as that of the MPA. Therefore, a comparison of Wilcoxon rank sum test results showed that mMPA-OC still had superior performance in CEC-2017 (30 dimensions).

Compare Friedman Test

The Friedman test was introduced to test the data, to illustrate the performance of mMPA-OC and its comparison algorithm in the CEC-2017 test set (30 dimensions). The Friedman tests were performed with a significance level of 0.05 to obtain the average ranking of each algorithm, and the results are shown in

Table 14. Average ranking of the Friedman test on CEC-2017 (30 dimensions).

Algorithm	mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
Mean rank	1.1379	2.1034	5.8258	6.4793	4.2414	7.7241	4.1379	4.0345
Final rank	1	2	6	7	5	8	4	3

. It shows that the mMPA-OC has the best comprehensive performance in the CEC-2017 (30 dimensions), followed by MPA, DE, GWO, SSA, PSO, SCA and AOA.

Bonferroni–Holm Test

In order to further investigate performance differences between proposed algorithm mMPA-OC and its peers. Bonferroni–Holm tests were used to test the p-values obtained by Wilcoxon rank sum test. When the significance level is 0.025, the Logistic values of Bonferroni–Holm tests for mMPA-OC compared with other algorithms in CEC-2017 (30 dimensions) are obtained as shown in the

Table 15. Logistic values of Bonferroni–Holm tests for mMPA-OC compared with other algorithms in CEC-2017 (30 dimensions).

	mMPA-OC & MPA	mMPA-OC & PSO	mMPA-OC & SCA	mMPA-OC & SSA	mMPA-OC & AOA	mMPA-OC & GWO	mMPA-OC & DE
F1	1	1	1	0	1	1	1
F3	1	1	1	1	1	1	1
F4	0	1	1	1	1	1	1
F5	1	1	1	1	1	1	1
F6	1	1	1	1	1	1	1
F7	0	1	1	1	1	1	1
F8	1	1	1	1	1	1	1
F9	1	1	1	1	1	1	1
F10	0	1	1	1	1	1	1
F11	0	1	1	1	1	1	1
F12	1	1	1	1	1	1	1
F13	1	1	1	1	1	1	1
F14	1	1	1	1	1	1	1
F15	1	1	1	1	1	1	1
F16	1	1	1	1	1	1	1
F17	0	1	1	1	1	1	0
F18	1	1	1	1	1	1	1
F19	1	1	1	1	1	1	1
F20	1	1	1	1	1	1	1
F21	1	1	1	1	1	1	1
F22	1	1	1	1	1	1	1
F23	1	1	1	1	1	1	1
F24	0	1	1	1	1	1	1
F25	1	1	1	1	1	1	0
F26	0	1	1	0	1	1	1
F27	1	1	1	1	1	1	1
F28	1	1	1	1	1	1	1
F29	1	1	1	1	1	1	1
F30	1	1	1	1	1	1	1

. The results show that the difference between mMPA-OC and its peers is statistically significant.

4.3. Experiments on CEC-2019

In order to verify the performance of the mMPA-OC in different test functions, we conducted experiments on CEC-2019, which is characterized by dynamic, niche composition, computationally expensive and many other types of problems with a higher degree of difficulty.

4.3.1. Analysis of Convergence Accuracy

In this section, the convergence accuracy performance of mMPA-OC and other comparison algorithms on the CEC-2019 is introduced. The summary results are shown in

Table 16. Comparison of experimental results in CEC-2019.

		mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
F1	mean	1	1.0796	8,044,536.8	6,771,320.343	1,511,907.772	7,077,275.736	74,023.9946	7132100.557
	Best	1	1	610,350.7776	35.5759	81,411.9328	1.0556	1	1,317,465.473
	Std	0	0.1881	9,818,726.755	10,233,145.43	1,182,947.654	12,124,639.33	141,898.6632	3,705,531.336
F2	mean	3.0346	10.7719	3745.4234	4632.6977	1142.6944	12,917.2973	471.8143	4382.4856
	Best	2.1037	3.4058	1057.3624	1308.569	261.5518	7091.803	14.9859	2698.3515
	Std	0.2922	6.2957	2017.6746	2137.3641	842.6056	3155.681	312.5551	796.7693
F3	mean	1.4512	1.4957	8.8932	10.2006	4.021	10.449	2.8815	7.9373
	Best	1	1	4.4189	7.4601	1.5378	7.9291	1.0031	6.1993
	Std	0.3178	0.3554	1.906	0.9706	1.5499	1.0707	1.954	0.5986
F4	mean	11.1839	13.8684	34.7305	53.2534	55.8883	63.27	21.1792	17.0837
	Best	5.9748	6.9698	16.2928	36.3291	23.884	29.0831	9.097	12.854
	Std	2.7611	4.1886	16.6509	8.3496	21.203	11.8383	7.777	2.1346
F5	mean	1.0524	1.0868	2.1297	11.7114	1.2412	88.5376	1.9451	1.1955
	Best	1.0152	1.0123	1.7845	4.0034	1.0615	20.6859	1.3075	1.0946
	Std	0.0253	0.0486	0.2097	4.9863	0.1246	31.062	0.8014	0.0633
F6	mean	1.1633	1.4802	6.7484	8.3122	6.46	11.0913	3.6432	1.9132
	Best	1.0035	1.0084	4.0718	5.0231	1.9906	7.353	1.4174	1.056
	Std	0.2091	0.3593	1.6778	1.297	1.973	1.2387	1.2052	0.7157
F7	mean	523.6464	584.2882	1177.606	1627.6451	1287.966	1424.4459	761.5338	824.5373
	Best	249.7686	241.8689	536.8633	1264.7746	638.5018	930.3358	169.6064	535.2754
	Std	143.0002	191.1648	318.3026	202.6899	353.3968	249.2289	321.0158	155.7938
F8	mean	3.4409	3.6856	4.5487	4.5705	4.63	4.7204	4.0514	4.0483
	Best	2.6203	2.6824	4.0178	3.8977	3.2856	3.8677	2.6286	3.1103
	Std	0.3671	0.3539	0.2993	0.2239	0.5171	0.346	0.5323	0.3036
F9	mean	1.0907	1.1004	1.463	1.7417	1.3678	3.1136	1.2128	1.2744
	Best	1.0366	1.0165	1.2311	1.4192	1.1026	1.2737	1.1007	1.1769
	Std	0.0265	0.0391	0.1685	0.3398	0.1909	0.7894	0.076	0.0486
F10	mean	19.0806	20.3429	21.4577	21.5027	20.4499	21.1423	21.227	21.1906
	Best	1.0005	1.0249	21.1993	21.3283	4.5742	20.9669	14.0142	21.118
	Std	5.8784	3.6486	0.0921	0.0749	2.9984	0.0704	1.3664	0.0456

. The optimal value will be bolded. It can be seen from Table 16 that mMPA-OC has the best performance on F1–F3 and F6, and is close to the theoretical global optimal value in F1 and F6. On F4, F8, and F10, the $Best$ and $Mean$ are the best, but the $Std$ of F4 and F10 is slightly worse than that of DE, and the $Std$ of F8 is slightly worse than that of SSA. On F6, the $Best$ found by mMPA-OC is the best. On F5, F7, and F9, the $Mean$ and $Std$ obtained by mMPA-OC are the best, but the $Best$ obtained by mMPA-OC is the best in MPA. In conclusion, compared with these seven meta-heuristic optimization algorithms, the global optimal value searched by MPA-OC is closest to the theoretical global optimal value. This indicates that mMPA-OC search ability is stronger.

4.3.2. Analysis of Convergence Rate

In this section, the convergence curve in CEC-2019 is used to evaluate the convergence rate of mMPA-OC and verify the performance of mMPA-OC. Figure 5 shows the convergence characteristics of the algorithm on F1, F4–F10 of CEC-2019. The curve of mMPA-OC is always under the iteration curve, indicating that mMPA-OC algorithm has a fast convergence rate. On F5, although the early convergence rate of search is not as good as that of DE, it does not fall into local optimum. Therefore, it can be shown that compared with other comparison algorithms, mMPA-OC in CEC-2019 can approach the optimal solution and search the optimal solution more quickly.

4.3.3. Analysis of Statistical Results

In order to verify the statistics difference between mMPA-OC and other algorithms in CEC-2019, the Wilcoxon rank-sum test, Friedman test, and Bonferroni–Holm tests are used in this section to evaluate their statistical differences.

Compare Wilcoxon Rank-Sum Test

Table 17. p-values of the Wilson rank-sum test of mMPA-OC and the comparison algorithm in CEC-2019.

	mMPA-OC & MPA	mMPA-OC & PSO	mMPA-OC & SCA	mMPA-OC & SSA	mMPA-OC & AOA	mMPA-OC & GWO	mMPA-OC & DE
F1	4.06 $\times {10}^{-6}$	2.98 $\times {10}^{-11}$	2.98 $\times {10}^{-11}$	2.98 $\times {10}^{-11}$	2.98 $\times {10}^{-11}$	3.44 $\times {10}^{-10}$	2.98 $\times {10}^{-11}$
F2	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$
F3	2.42 $\times {10}^{-2}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	1.33 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	1.86 $\times {10}^{-6}$	3.02 $\times {10}^{-11}$
F4	3.34 $\times {10}^{-3}$	3.69 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.83 $\times {10}^{-8}$	1.41 $\times {10}^{-9}$
F5	4.71 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.61 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.50 $\times {10}^{-11}$
F6	1.25 $\times {10}^{-4}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	6.07 $\times {10}^{-11}$	3.26 $\times {10}^{-7}$
F7	1.86 $\times {10}^{-1}$	2.61 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	1.61 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	0.001442328	2.39 $\times {10}^{-8}$
F8	0.015014133	4.08 $\times {10}^{-11}$	4.08 $\times {10}^{-11}$	1.41 $\times {10}^{-9}$	4.08 $\times {10}^{-11}$	3.09 $\times {10}^{-6}$	2.83 $\times {10}^{-8}$
F9	0.157975689	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.87 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	3.16 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$
F10	0.171450047	3.02 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	2.00 $\times {10}^{-5}$	6.72 $\times {10}^{-10}$	4.20 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$

shows the p-value test results of Wilcoxon rank-sum test. It can be seen that there is no significant difference between the mMPA-OC and the other algorithms only in a few functions. For example, on F7, there is no significant difference between mMPA-OC and MPA. Therefore, a comparison of Wilcoxon rank-sum test results shows that mMPA-OC still has superior performance in CEC-2019.

Compare Friedman Test

In order to further illustrate the performance of mMPA-OC, the Friedman test was introduced for the data. At the significance level of 0.05, the Friedman test was conducted to obtain the average ranking of each algorithm. The results are listed in

Table 18. Average ranking of the Friedman test on CEC-2019.

Algorithm	mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
Mean rank	1	2	5.9	6.8	4.9	7.3	3.8	4.3
Final rank	1	2	6	7	5	8	3	4

, and mMPA-OC has the best comprehensive performance in the CEC-2019, followed by MPA, GWO, DE, SSA, PSO, SCA, and AOA.

Bonferroni–Holm Test

In order to further investigate performance differences between proposed algorithm mMPA-OC and other algorithms. Bonferroni–Holm tests were used to test the p-values obtained by Wilson’s rank-sum test. When the significance level is 0.025, the Logistic values of Bonferroni–Holm tests for mMPA-OC compared with other algorithms in CEC-2019 are obtained as shown in the

Table 19. Logistic values of Bonferroni–Holm tests for mMPA-OC compared with other algorithms in CEC-2019.

	mMPA-OC & MPA	mMPA-OC & PSO	mMPA-OC & SCA	mMPA-OC & SSA	mMPA-OC & AOA	mMPA-OC & GWO	mMPA-OC & DE
F1	1	1	1	1	1	1	1
F2	1	1	1	1	1	1	1
F3	0	1	1	1	1	1	1
F4	1	1	1	1	1	1	1
F5	1	1	1	1	1	1	1
F6	1	1	1	1	1	1	1
F7	0	1	1	1	1	1	1
F8	0	1	1	1	1	1	1
F9	0	1	1	1	1	1	1
F10	0	1	1	1	1	1	1

. The results show that the difference between mMPA-OC and its peers is statistically significant.

5. Engineering Optimization Problem

In order to verify the effectiveness of the mMPA-OC in practical engineering applications, the design problems of the welded beam, chemical vessel, and gear train are used to evaluate the performance of the algorithm. The proposed mMPA-OC is compared with PSO, DE, SSA, SCA, GWO, AOA, and MPA. The experiment set the population size to 50 and the maximum number of iterations to 100. Moreover, the experiment results of each algorithm are recorded after 30 independent runs.

5.1. Design of Welding Beam

The main purpose of the welding beam design problem is to minimize the manufacturing cost, which was proposed by Coello [59]. Its four design variables are welding thickness $t=\left(q_1\right)$, height $h=\left(q_2\right)$, length $l=\left(q_3\right)$, and strip thickness $s=\left(q_4\right)$. The objective function of this problem can be formulated as follows:

minimize:

$$f\left(X\right)=1.10471q_1^2{q}_2+0.04811q_3{q}_4\left(14.0+q_2\right)$$

(15)

subject to:

$$\begin{array}{c}{G}_1\left(X\right)=\tau \left(X\right)-{\tau }_{max}\le 0,G_2\left(X\right)=\sigma \left(X\right)-{\sigma }_{max}\le 0,\hfill \\ G_3\left(X\right)=\delta \left(X\right)-{\delta }_{max}\le 0,G_4\left(X\right)=q_1-q_4\le 0,\hfill \\ G_5\left(X\right)=P-P_c\left(X\right)\le 0,G_6\left(X\right)=0.125-q_1\le 0,\hfill \\ G_7\left(X\right)=1.10471q_1^2+0.04811q_3{q}_4\left(14.0+q_2\right)-5.0\le 0,\hfill \end{array}$$

(16)

$$\begin{array}{c}\varphi \left(X\right)=\sqrt{{\left({\varphi }^{\prime }\right)}^2+2{\varphi }^{\prime }{\varphi }^{″}\frac{q_2}{2R}+{\left({\varphi }^{″}\right)}^2},\hfill \\ {\varphi }^{\prime }=\frac{P}{\sqrt{2}{q}_1{q}_2},{\varphi }^{″}=\frac{MR}{J},\hfill \\ M=P\left(L+\frac{q_2}{2}\right),R=\sqrt{\frac{q_2^2}{4}+{\left(\frac{q_1+q_3}{2}\right)}^2},\hfill \\ J=2\left\{\sqrt{2}{q}_1{q}_2\left[\frac{q_2^2}{4}+{\left(\frac{q_1+x_3}{2}\right)}^2\right]\right\},\hfill \\ \sigma (\overrightarrow{X})=\frac{6PL}{q_4{q}_3^2},\delta (\overrightarrow{X})=\frac{6P{L}^3}{E{q}_3^2{x}_4},\hfill \\ P_c(\overrightarrow{X})=\frac{4.013E\sqrt{q_3^2{q}_4^6/36}}{L^2}\left(1-\frac{q_3}{2L}\sqrt{\frac{E}{4G}}\right),\hfill \\ P=6000lb,L=14in,{\delta }_{max}=0.25\mathrm{in},\hfill \\ E=3\times {10}^7\mathrm{psi},G=1.2\times {10}^7\mathrm{psi},\hfill \\ {\tau }_{max}=1.36\times {10}^4\mathrm{psi},{\sigma }_{max}=3\times {10}^4\mathrm{psi},\hfill \end{array}$$

(17)

variable range: $0.1\le q_1,q_4\le 2,0.1\le q_2,q_3\le 10$. The experimental results of various metaheuristic algorithms when dealing with the welded beam design problem are shown in

Table 20. Comparison results between mMPA-OC and the other seven meta-heuristics algorithms on the welded beam design problem.

Algorithm	mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
$Mean$	$\mathbf{1.69}\times {\mathbf{10}}^{\mathbf{1}}$	1.70 $\times {10}^1$	2.59 $\times {10}^1$	1.71 $\times {10}^1$	1.95 $\times {10}^1$	1.95 $\times {10}^1$	1.09 $\times {10}^{14}$	1.09 $\times {10}^{14}$
$Min$	$\mathbf{1.70}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1.70}\times {\mathbf{10}}^{\mathbf{1}}$	1.94 $\times {10}^1$	1.70 $\times {10}^1$	1.83 $\times {10}^1$	1.72 $\times {10}^1$	1.09 $\times {10}^{14}$	1.09 $\times {10}^{14}$
$SD$	3.25 $\times {10}^{-4}$	1.10 $\times {10}^{-2}$	3.76 $\times {10}^{-1}$	7.14 $\times {10}^{-3}$	9.49 $\times {10}^{-2}$	2.69 $\times {10}^{-1}$	0	4.90 $\times {10}^{-1}$

, and the results show that mMPA-OC performs the best. Therefore, the mMPA-OC can effectively solve the welding beam design problem.

5.2. Pressure Vessel Design

The problem of pressure vessel design [60] is to minimize its production cost $f\left(q\right)$ while meeting production requirements. Its four design variables are: the thickness of the shell $t_S\left(=q_1\right)$, the thickness of the head $t_H\left(=q_2\right)$, the inner radius $t_R\left(=q_3\right)$, and the length of the cylindrical part of the container $t_L\left(=q_4\right)$. Moreover, $q_1$ and $q_2$ are integer multiples of 0.625, and are continuous variables. The objective function of this problem is as follows.

$$minf\left(x\right)=0.6224q_1{q}_3{q}_4+1.7781q_2{q}_3^2+3.1661q_1^2{q}_4+19.84q_1^2{q}_3$$

(18)

The constraints are as follows:

$$\begin{array}{c}{G}_1\left(x\right)=-q_1+0.0193q_3\le 0\hfill \\ G_2\left(x\right)=-q_2+0.00954q_3\le 0\hfill \\ G_3\left(x\right)=-\pi q_3^2{q}_4-\frac{4}{3}\pi q_3^3+1296000\le 0,\hfill \\ G_4\left(x\right)=q_4-240\le 0\hfill \end{array}$$

(19)

The range of the variable-is:

$$minf\left(q\right)=0.6224q_1{q}_3{q}_4+1.7781q_2{q}_3^2+3.1661q_1^2{q}_4+19.84q_1^2{q}_3,$$

(20)

subject to:

$$\begin{array}{c}{G}_1\left(q\right)=-q_1+0.0193q_3\le 0\hfill \\ G_2\left(q\right)=-q_2+0.00954q_3\le 0\hfill \\ G_3\left(q\right)=-\pi q_3^2{q}_4-\frac{4}{3}\pi q_3^3+1296000\le 0\hfill \\ G_4\left(q\right)=q_4-240\le 0\hfill \end{array}$$

(21)

variable range:

$$\begin{array}{cc}\hfill q_1,q_2& \in W\{1\times 0.0625,2\times 0.0625,3\times 0.0625,\dots ,1600\times 0.0625,\},\hfill \end{array}$$

$$10\le q_3,q_4\le 200.$$

Table 21. Comparison results between mMPA-OC and the other seven meta-heuristics algorithms on the pressure vessel design problem.

Algorithm	mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
$Mean$	2302.55	2302.59	5778.64	5227.90	3507.21	204,449.50	204,449.50	204,584.87
$Min$	2302.55	2302.55	3797.11	2322.54	2335.69	204,345.84	204,345.84	204,323.08
$SD$	0.00	0.17	1148.71	1582.19	345.47	97.97	97.97	367.49

records the results of the comparison between mMPA-OC and seven meta-heuristics on the pressure vessel design problem, and the results shows that mMPA-OC performs the best. Therefore, the mMPA-OC can effectively solve the pressure vessel design problem.

5.3. Gear Train Design Problem

The gear train design problem is an unconstrained discrete design problem in mechanical engineering [61]. The objective of this problem is to minimize the ratio of the angular velocity of the output axis to the angular velocity of the input axis. Its four design variables are: $N_A\left(=q_1\right)$, $N_B\left(=q_2\right)$, $N_C\left(=q_3\right)$, and $N_D\left(=q_4\right)$. The objective function of this problem can be formulated as follows:

minimize:

$$f\left(X\right)={\left(\frac{1}{6.931}-\frac{q_3{q}_2}{q_1{q}_4}\right)}^2$$

(22)

variable range:

$$q_1,q_2,q_3,q_4\in \{12,13,14,\dots ,60\}$$

Table 22. Comparison results between mMPA-OC and the other seven meta-heuristics on the gear train design problem.

Algorithm	mMPA-OC	MPA	PSO	SCA	SSA	AOA	GWO	DE
$Mean$	2.00 $\times {10}^{-9}$	2.16 $\times {10}^{-9}$	9.45 $\times {10}^{-9}$	1.01 $\times {10}^{-8}$	1.14 $\times {10}^{-9}$	8.53 $\times {10}^{-8}$	6.83 $\times {10}^{-10}$	3.13 $\times {10}^{-10}$
$Min$	2.70 $\times {10}^{-12}$	2.70 $\times {10}^{-12}$	9.94 $\times {10}^{-11}$	2.31 $\times {10}^{-11}$	2.31 $\times {10}^{-11}$	3.30 $\times {10}^{-9}$	2.70 $\times {10}^{-12}$	2.70 $\times {10}^{-12}$
$SD$	1.32 $\times {10}^{-9}$	2.34 $\times {10}^{-9}$	1.10 $\times {10}^{-8}$	1.63 $\times {10}^{-8}$	8.28 $\times {10}^{-10}$	1.13 $\times {10}^{-7}$	5.68 $\times {10}^{-10}$	3.98 $\times {10}^{-10}$

records the comparison results between mMPA-OC and seven meta-heuristics on the gear train design problem, and the results shows that mMPA-OC performs the best. Therefore, the mMPA-OC can effectively solve the problem of gear train design.

6. Conclusions

In this paper, we propose a multi-disturbance Marine predator algorithm based on opposition-based learning and compound mutation. MPA lacks the transition between exploitation and exploration, and the algorithm easily becomes stuck in a local optimum. Therefore, we propose three improvements to overcome these shortcomings. Firstly, the opposition-based learning mechanism was added to the optimal value selection process to improve the MPA’s exploration ability. Secondly, the composite mutation strategy was used to improve the predator position update mechanism to improve the MPA’s global search ability. Finally, the disturbance factors are improved to multiple disturbance factors, so that MPA can maintain the population diversity in the iterative process. In order to evaluate the performance of mMPA-OC, experiments are designed to compare with seven algorithms MPA, PSO, DE, GWO, SSA, SCA and AOA on different dimensions of CEC-2017 and on the more complex CEC-2019 function test set. For the experimental results, we evaluate the algorithms in terms of convergence accuracy, stability, convergence rate, Friedman test, Wilcoxon test, and Bonferonni–Holm test. The experimental results show that the mMPA-OC can not only maintain the diversity of the population and greatly increase the ability to jump out of local optimum, but also can well balance the exploration phase and the exploitation phase. In order to evaluate the effect of mMPA-OC on real engineering optimization problems, mMPA-OC is compared with seven meta-heuristic algorithms. The experimental results show that the mMPA-OC has better optimization effect than other meta-heuristic algorithms. In conclusion, mMPA-OC is an effective improvement of MPA, which has certain development prospects and is a feasible method to solve complex optimization problems, which is worth our research. For example, whether other strategies can be introduced on the basis of mMPA-OC algorithm to improve its performance ability, whether it can be improved into an optimizer to solve multi-objective optimization problems, whether it can be improved into an optimizer to solve 0-1 knapsack problem, and whether it can be better applied to practical problems.