A Modified Manta Ray Foraging Optimization for Global Optimization Problems

The Manta ray foraging optimization (MRFO) is a novel swarm-based metaheuristic optimizer. It is mainly modeled by simulating three foraging behaviors of the Manta rays, which has a good performance. However, several drawbacks of MRFO have been noticed by analyzing its mathematical model. Random selection of reference points in the early iterations weakens the exploitation capability of MRFO. Chain foraging tends to lead the algorithm into local optimum. In addition, the algorithm suffers from the deficiency of decreasing population diversity in the late iteration. To address these shortcomings, a modified MRFO using three strategies, called m-MRFO, is proposed in this paper. An elite search pool (ESP) is established in this paper to enhance exploitation capability. By using adaptive control parameter strategies (ACP), we expand the range of MRFO’s exploration in the early iterations and enhance the accuracy of exploitation in the later iterations, balancing exploiting and exploring capabilities. Furthermore, we use a distribution estimation strategy (DES) to adjust the evolutionary direction using the dominant population information to promote convergence. The m-MRFO performance was verified by selecting 23 classical test functions and CEC2017 test suite. The significance of the results was also verified by Friedman test, Wilcoxon test and Iman-Davenport test. Moreover, we have confirmed the potential of m-MRFO to solve real-world problems by solving three engineering design problems. The simulation results show that the improvement strategy proposed in this paper can effectively improve the performance of MRFO. m-MRFO is highly competitive.


I. INTRODUCTION
Global optimization problems can be found in almost every field of finance, engineering, and science. With the development of science and technology, increasingly complex optimization problems are emerging. Many realistic optimization problems are accompanied by several difficulties: expensive computational costs, complex non-linear constraints, dynamic objective functions and huge search ranges [1]. In this case, it is a challenge to efficiently find a solution that satisfies the constraint. Conventional mathematical or numerical programming methods are overwhelmed when faced with multiple types of non-integrable, non-continuous The associate editor coordinating the review of this manuscript and approving it for publication was Jagdish Chand Bansal.
problems [2]. In addition, they have difficulty in balancing accuracy and time cost when solving large-scale real-world multimodal problems. Metaheuristic optimization algorithms, as a class of stochastic optimization algorithms, perform better in balancing the quality of solution and time cost. It has been widely used to solve complex optimization problems in natural and engineering fields due to its simple structure and its non-reliance on the gradient information of specific problems [3]. More and more scholars are paying attention to and working on metaheuristic optimization algorithms.
In the past decades, various algorithms have been proposed one after another. In general, meta-heuristic optimization algorithms can be divided into three groups [4]: evolution-based algorithms, physics-based algorithms, and swarm-based algorithms. Genetic algorithm (GA) [5] that simulates the survival of the fittest mechanism in nature is a widely used evolutionary algorithm. In addition, other evolutionary algorithms have been proposed, including differential evolution (DE) [6], evolutionary programming (EP) [7], and evolutionary strategies (ES) [8]. The popularity of these evolutionary algorithms has also encouraged more and more researchers to study and propose other evolutionary algorithms [9]- [11]. The physics-based algorithms construct optimization models by emulating the physical laws of the universe. Simulated annealing (SA) [12] inspired by annealing phenomena in metallurgy is one of the best-known physics-based algorithms. Apart from SA, other physics-based algorithms have been proposed, such as gravity search algorithm (GSA) [13], nuclear reaction optimizer (NRO) [14], sine cosine algorithm (SCA) [15], black hole algorithm (BOA) [16], and water cycle algorithm (WCA) [17]. The swarm-based optimization algorithm performs by simulating the social behavior of the population. Particle swarm optimization (PSO) [18] and ant colony optimization (ACO) [19] are two classical swarm-based optimization algorithms. They perform by simulating a bird colony and an ant colony cooperating in foraging, respectively. There are other swarm intelligence optimization algorithms including: grey wolf optimizer (GWO) [20], whale optimization algorithm (WOA) [21], sparrow search algorithm (SSA) [22], firefly algorithm (FA) [12], artificial bee colony algorithm (ABC) [23], and so on [24]- [27].
Recently, a swarm-based algorithm called Manta ray foraging optimization (MRFO) that emulates the foraging behavior of manta rays is proposed by Zhao in 2020 [28]. As a newly proposed algorithm, MRFO is quickly applied to solve various engineering optimization problems. Abd Elaziz et al. [29] use fractional order calculus to enhance MRFO and applies it to multilevel thresholding image segmentation. Ghosh et al. [30] propose a binary version of MRFO and solves the feature selection problem. An improved version of MRFO is proposed by Xu et al. [31] and high-temperature proton exchange membrane fuel cell is analyzed and optimized. Hassan et al. [32] propose an improved MRFO with a hybrid gradient-based optimizer and uses it to solve the economic emission dispatch problem. The global maximum power point tracker based on MRFO is proposed by Fathy et al. [33].
However, MRFO also has the shortcomings of insufficient exploitation ability, decreasing population diversity, and easy to fall into local optimum. These deficiencies are mainly caused by the imbalance in the exploitation and exploration of the search space by the algorithm. In order to enhance the algorithm performance and balance the exploitation and exploration capabilities, in this paper, a modified MRFO (m-MRFO) with three improvement strategies is proposed. An elite search pooling strategy is proposed to improve the algorithm exploitation ability for the deficiency of too slow convergence due to random selection of reference points in the early iterations. To balance the algorithm exploitation and exploration, an adaptive parameter control strategy is proposed. And a Gaussian probability model is used to describe the dominant population distribution and modify the evolutionary direction, thus improving the algorithm performance.
To fully verify the performance of m-MRFO, 51 functions and 3 engineering design problems are used. And the superiority of the algorithm is verified by numerical analysis, convergence analysis, stability analysis, Wilcoxon test and Friedman test. The main contributions of this paper are as follows: (1) An elite search pool is introduced to improve the exploitation of the algorithm. (2) To balance the exploitation and exploration of the algorithm, an adaptive parameter control strategy is proposed. (3) The evolutionary direction is modified using Gaussian probability model to improve the performance of m-MRFO. (4) The superiority of m-MRFO is tested on 51 test functions and 3 engineering design problems.
The remainder of this paper is organized as follows. A review of the basic MRFO is presented in Section II. Section III provides a detailed description of the proposed m-MRFO. In Section IV, the effectiveness of the proposed improvement strategy is verified using the classical test functions and CEC 2017 test suite. Furthermore, the m-MRFO is applied to solve three engineering design problems in Section IV. Finally, we summarize this work in Section V and offer directions for future research.

II. THE BASIC MRFO
In this section, the basic steps of MRFO are described. MRFO is performed by simulating three foraging strategies of manta rays, namely chain foraging, cyclone foraging and somersault foraging. Similar to other swarm-based metaheuristic algorithms, MRFO generates initial populations randomly in the search space. Then it is updated by the three strategies mentioned above. The mathematical models for these three foraging strategies are given respectively below.

A. CHAIN FORAGING
The manta rays form a foraging chain by linking their heads and tails in a line. MRFO considers that the best solution is a higher concentration of plankton, which is the target food for manta rays. While the first individual moves only towards food, the rest of the individuals move not only towards food but also towards individuals located in front of themselves in the foraging chain. The mathematical model of chain foraging is described as follows.
where x t i is the position of the i th individual at generation t. r i ∈ [0, 1], i = 1, 2, 3, 4 are uniformly distributed random vectors. x t best is the plankton with the highest concentration, VOLUME 9, 2021 that is, the optimal individual. NP is the number of populations. α is a weight coefficient.

B. CYCLONE FORAGING
When manta rays find plankton in deep water, they form long foraging chains and then move toward food in a spiral. This behavior is similar to WOA, but in addition to spiraling close to food, it also follows the individuals in front of it. The mathematical model of cyclone foraging can be given by the following equation.
where r i ∈ [0, 1], i = 5, 6 is uniformly distributed random vectors. β is the weight coefficient. r 7 ∈ [0, 1] is a uniformly distributed random number. iter max and iter are the maximum number of iterations and the current number of iterations, respectively. In Eq. (3), food is mainly used as a reference point for spiral foraging, which contributes to the full exploitation of the space near food. In addition, to expand the search range, a randomly generated location in the search space is used as a reference location for spiral foraging. This allows all individuals to search for areas far from their current best position. The random spiral foraging mechanism focuses mainly on exploration, allowing MRFO to perform a broad global search. The specific mathematical model is described as follows.
x rand = lb + r 8 · (ub − lb) (5) x t+1 where x rand is a random position randomly produced in the search space. r i ∈ [0, 1], i = 8, 9, 10 are uniformly distributed random vectors. ub and lb are the upper and lower bounds of the search space, respectively.

C. SOMERSAULT FORAGING
In this phase, the food location is considered as a pivot point. Each individual flip around the pivot and thus searches for a new location. The mathematical model of this phase is represented as follows.
x t+1 where S is the somersault factor that decides the somersault range of manta rays and S = 2. r 11 and r 12 are two random numbers in [0, 1].
MRFO regulates the exploration and exploitation behavior by controlling the change of (iter/iter max ). When (iter/iter max ) < rand, the exploration behavior is mainly performed, and food sources are randomly generated as reference points in the search space. When (iter/iter max ) ≥ rand, the optimal individual is used as a reference point, which facilitates the exploitation of the algorithm. In addition, a random number is used to select chain foraging or spiral foraging. After that, Somersault foraging is performed.

III. THE MODIFIED MRFO
To overcome the shortcomings of MRFO, we use an elite search pool instead of randomly generated individuals as reference points to improve the algorithm exploitation performance. In addition, we make a good transition from exploration to exploitation with an adaptive parameter control strategy. And a balance between them is achieved. To modify the evolutionary direction, we use a distribution estimation strategy. By sampling the dominant population information, we enhance the population diversity and improve the algorithm performance. The mathematical model of m-MRFO is described in detail as follows.

A. ELITE SEARCH POOL STRATEGY (ESP)
Analysis of Eq. (5) and Eq. (6) shows that the reference location of cyclone foraging is randomly generated in the search space in the early iterative stage. While this facilitates the algorithm to search more space, the large range of random positions weakens the algorithm exploitation ability and slows down the convergence speed. In order to enhance the exploitation capability while still retaining its ability to search large spaces, an elite search pool strategy is proposed in this paper. We put the current best three individuals into a set. As shown in Eq. (8).
where X esp1 , X esp2 and X esp3 are the best three individuals. The reference point is chosen randomly from these three individuals each time. By using the ESP strategy, the position of the reference point is changed from randomly generated to one of the best three individuals. This greatly enhances the algorithm exploitation capability. Meanwhile, the three individuals are chosen randomly, which to some extent avoids the prematureness of the algorithm caused by the optimal individual falling into the local optimum. To balance the algorithm exploitation and exploration, we also add a new individual to the ESP. This new individual is randomly composed of the best three individuals. This retains the possibility to select dominant individuals and also provides the option to select positions at a longer range. Thus, the final mathematical model of the ESP strategy is described as follows: where r i ∈ [0, 1], i = 13, 14, 15 are uniformly distributed random vectors.

B. ADAPTIVE CONTROL PARAMETER STRATEGY (ACP)
The original MRFO balances the search behavior by controlling the value of (iter/iter max ). (iter/iter max ) is a linearly increasing variable, which does not accurately reflect and accommodate the complex nonlinear search process. The nonlinear parameter control strategy is an effective measure to prevent premature. Several researchers have proposed various nonlinear parametric control strategies for balancing exploitation and exploration [34]- [36].In this paper, we propose an adaptive control parameter strategy with a mixture of sine and cosine functions. The specific mathematical model of parameter Coef is as follows.
As shown in Figure 1, the new strategy focuses more on exploration in the early stage to avoid the algorithm from falling into local optimum. In the later stage, it keeps the exploitation of large probability, which helps the algorithm to accelerate the convergence. Furthermore, we note that the parameter S is constant when the original MRFO performs somersault foraging, which is not beneficial for the algorithm to perform effectively. In the early stages of optimization, the algorithm performs more exploratory behavior, so S needs to be large enough to search more space. In the late iteration, the algorithm needs to be more precise for exploitation. At this point, too large S will lead to a weakening of the algorithm's exploitation ability. Therefore, we need a smaller value of S. Therefore, we propose a linear decreasing strategy with parameter S. The mathematical model is as follows.
where Smin and Smax are the maximum and minimum values of the parameter S, respectively.

C. DISTRIBUTION ESTIMATION STRATEGY (DES)
The original MRFO's chain foraging strategy uses the optimal individual and neighboring individuals for position updating.
This leads to a premature convergence of the algorithm if the optimal individual has fallen into a local optimum, then the chain rule leads all subsequent individuals to approach the locally optimal individual. In order to enhance the algorithm performance, a distribution estimation strategy is proposed in this paper with the following mathematical model.
where   (14) and equation (15) If rand < 0.5 If Coef > rand Update Manta ray based on equation (3)  Else Update Manta ray based on equation (6)  Greedy strategy is adopted to select the offspring Calculate the S based on equation (12) Update Manta ray based on equation (7) Boundary control; calculate fitness of each agent Greedy strategy is adopted to select the offspring End while Output the best solution VOLUME 9, 2021

D. TIME COMPLEXITY OF THE MRFO
The time complexity of mrfo can be seen from the literature [28] as follows.
In this paper, three improvement strategies are proposed, ESP and ACP do not change the time complexity. The time complexity of the covariance matrix of DES is O(T (NP/2·D 2 )). Therefore, the time complexity of m-MRFO is shown below.
where T is the maximum value of the iteration, NP is the number of individuals, and D is the number of variables.

IV. EXPERIMENTS RESULTS
To verify the performance of m-MRFO, two different sets of benchmark functions are used for testing. The first group includes 23 classical test functions with the information shown in Table 1. The second group is the CEC2017 test suite, which has the detailed information shown in Table 2.
There are 51 test functions in total in the two test suites, which can be divided into unimodal test functions, multimodal test functions, low-dimensional test functions, hybrid functions and composite functions. The unimodal test functions have only one global optimal solution, so they are often applied to evaluate the exploitation capability of an algorithm. The multimodal test functions, on the other hand, have multiple global optimal solutions and are therefore used to test the exploration capability of the algorithms. The low-dimensional test functions check the algorithm's ability to explore in low dimensions. Hybrid and composite functions are more complex and can be used to verify the overall performance of an algorithm.
There are four parts of the experiment on m-MRFO. In the first part of the experiment, we need to determine the values of Smax and Smin. The values of Smax and Smin affect the development and exploration ability of the algorithm, so we adopt 23 classical test functions to get their optimal values. Secondly, to verify the effectiveness of the three improvement strategies proposed in this paper, we employ the CEC2017 test suite for testing. In the third and fourth parts of the experiment, m-MRFO is compared with other algorithms using the classical test functions and the CEC2017 test suite, respectively. In the last part, three engineering optimization problems are used to test the proposed m-MRFO performance.
To ensure fair comparison, for the classical test functions, all algorithms adopt the same dimensions, the maximum number of iterations is set to 300, the number of populations is set to 50, and all test functions are run 30 times independently. For CEC2017, Dim, iter max , and NP are set uniformly to 30, 600, and 500, respectively. all test functions are run independently 51 times. The experiments in this paper were conducted on a computer with an AMD R7 4800U processor and 16 GB RAM. Programming was performed using MATLAB R2016b.

A. ANALYSIS OF Smax AND Smin PARAMETER SETTINGS
The algorithm parameter settings have a great impact on its performance. In the original MRFO, S is the key parameter for somersault foraging. We control the variation of S by the parameters Smax and Smin to balance the algorithm exploitation and exploration. In this section, we need to identify the VOLUME 9, 2021  In this part, 23 classical test functions are used to compare the optimization results of m-MRFO for the above 30 (5 × 6) different parameter settings. Each test function was run 30 times independently, and a totally of 20,700 data were obtained. Due to the large amount of data, the experimental results are not compared specifically, but the differences are reflected by ranking the simulation results under different parameter settings by Friedman test.
Based on the experimental data, the Friedman test results are given for unimodal test functions and multimodal test functions, respectively. As shown in Figure 3, for the unimodal test functions, the algorithm mostly performs poorly when Smax = {2.4, 2.2}. And when Smax = {1.8, 1.6}, the algorithm is generally better. On the other hand, the algorithm performs poorly when Smin has a value that is too large or too small. The algorithms are generally better when Smin = 1.For the multimodal test functions, the influence of the parameter settings is exactly the opposite of the case for the unimodal test functions. In addition, to identify the optimal values of Smax and Smin, we present the results of the Friedman test considering all the tested functions. The results show that the algorithm performs best when Smax = 2.4, Smin = 1.4.

B. ANALYSIS OF M-MRFO IMPROVEMENT STRATEGIES
The improvement proposed in this paper for the original MRFO consists of three parts: search pool strategy, adaptive control parameter strategy and distribution estimation strategy. To evaluate the effectiveness of different improvement strategies, we propose three m-MRFO derivation algorithms with different improvement strategies, as shown in Table 3. m-MRFO-1 utilizes ESP to enhance the algorithm performance. m-MRFO-2 is used for evaluating the effectiveness of the ACP. The DES is fused into m-MRFO-3. The performance of the five algorithms is compared using the CEC2017 test suite. The five algorithms run under the same experimental parameters. Each function is run independently 51 times. The mean error results for each algorithm are listed in Table 4, and the Friedman test results for the five algorithms are given in the last row.
From the statistical results, it is clear that the m-MRFO with the complete improvement strategies performs best with a Friedman test ranking value of 1.25. The ranking values of  the three derived algorithms with one improved strategy are also better than the original MRFO. The ranking values for the three derived algorithms are 3.39, 3.57 and 2.25, respectively. Thus, the impact of these three improvement strategies on the performance of the algorithm can be obtained in descending order: EDS > ESP > ACP. m-MRFO-3 performs the best among the three derived algorithms, proving that the utilization of DES can effectively enhance the performance of the algorithm. It generates offspring using the overall distribution information of the dominant population, which effectively avoids the deficiency of falling into local optimum caused by the population following only the VOLUME 9, 2021 optimal individual. The performance of m-MRFO-1 in solving the multimodal test functions is similar to that of m-MRFO-3 with better results. This is due to the introduction of ESP, which effectively expands the search range by randomly selecting an individual in the ESP as a reference point, thus enhancing the algorithm's ability to solve multimodal problems. Figure 4 presents the ranking radar diagram of the five algorithms. It can be observed that the area enclosed by m-MRFO is the smallest, which visually indicates that m-MRFO has the best performance.

C. ANALYSIS OF CLASSICAL TEST FUNCTIONS TEST
In this part, the performance of m-MRFO on classical test functions will be verified. A comparison with m-MRFO is performed using other advanced algorithms. The parameter settings of each algorithm are given in Table 5. The experimental parameters and environment are consistent with Section IV A. The algorithms employed for comparison include flower pollination algorithm (FPA) [37], biogeography-based optimization (BBO) [9], moth-flame optimization (MFO) [38], multi-verse optimizer (MVO) [39], sine cosine algorithm (SCA) [15], sparrow search algorithm (SSA) [22], particle swarm optimization (PSO) [18], whale optimization algorithm (WOA) [21] and gravitational search algorithm (GSA) [13].
As shown in Table 6, m-MRFO performs best in solving five of the seven unimodal test functions (F1-F4, F7). The m-MRFO provides the second-best solution in solving F5 and F6. It is due to the replacement of randomly generated reference points using ESP, which further enhances the  (F9-F11). However, m-MFRO gives unsatisfactory results in solving F8 and F13. The analysis of the results of the low-dimensional test functions shows that m-MFRO achieves the best solution on 9 of the 10 test functions (4 first and 5 second). PSO, SSA and MFO are in the second, third and fourth positions. In addition, we evaluate the performance of the algorithms using the Friedman test. All algorithms are ranked according to the mean value. The results show that m-MRFO ranks first with a ranking value of 2.28. The statistical results show that m-MRFO has the best performance compared to the nine algorithms mentioned above for solving the classical test functions.

D. ANALYSIS OF THE CEC 2017 TEST
The classical test functions can verify the performance of algorithms to a certain extent. However, with the development of intelligent optimization algorithms, more and more algorithms have better performance in the classical test functions. In order to further verify the superiority of the m-MFRO proposed in this paper, more complex test functions are needed for testing. Therefore, we use the IEEE CEC2017 test suite to further validate the performance of the improved algorithms. This test set consists of more complex and difficult test functions. It has been widely used to evaluate the performance of various newly proposed and improved algorithms. In this part, eight recently proposed algorithms are evaluated for comparison with m-MRFO. These state-of-the-art algorithms include artificial ecosystem-based VOLUME 9, 2021 optimization (AEO) [40], harris hawks optimization (HHO) [41], virus colony search (VCS) [42], arithmetic optimization algorithm (AOA) [43], slime mould algorithm (SMA) [44], jellyfish search (JS) [45], pathfinder algorithm (PFA) [46] and tunicate swarm algorithm (TSA) [47]. All algorithm parameters are set the same as the original literature, as shown in Table 7. In this paper, the performance of m-MRFO is comprehensively evaluated by numerical analysis, convergence analysis, stability analysis, Wilcoxon test, Friedman test, and Iman-Davenport test. Table 8 lists the numerical statistics for each algorithm independently solving the CEC2017 test suite 51 times. Analysis of Table 8 shows that although m-MRFO does not achieve the optimal value of 0 for the unimodal test function F24, it provides the best solution of the nine algorithms. This demonstrates the superiority of m-MRFO in solving pathological functions and once again validates that improved strategies can effectively improve the exploitation capability. In the multimodal test functions F25-F31, all algorithms perform differently. The m-MRFO performs best on F25 and F30. SMA provides the best solutions for F26, F28 and F31. JS achieves the best answers for F27 and F29. The average rank of m-MRFO in solving the multimodal test functions is better than all the comparison algorithms, which indicates the competitive exploration capability of m-MRFO. In the hybrid functions and composite functions, m-MFRO outperforms the comparison algorithm overall. Specifically, m-MFRO achieves the best results on eight of the nine hybrid functions. m-MFRO ranks in at least the top three of all 11 composite functions. The results for the hybrid and composite functions illustrate that m-MRFO achieves a good balance between exploitation and exploration behaviors and has a strong potential for solving complex real-world optimization problems.
To analyze the distribution characteristics of the solutions solved by the improved algorithms, the box diagrams are shown in Figure 5 based on the results of each algorithm solving the test function 51 times independently. For each algorithm, the center mark of each box indicates the median of the results of 51 solving functions, the bottom and top edges of the box indicate first and third-degree points, and the symbol ''+'' indicates bad values that are not inside the box. As can be seen from Figure 5, for F24, F25, F30, F35, F37 and F42, there are no bad values for m-MRFO, which indicates that the distribution of solutions obtained from m-MRFO is more concentrated and m-MRFO has better stability. For other test functions with some bad values, the distribution of the m-MRFO solutions is also more concentrated compared to the comparison algorithm. In conclusion, the variance of the m-MRFO solving test functions is much lower and the stability is better than that of the comparison algorithm.
Convergence speed and convergence accuracy are important indicators of algorithm performance. Figure 6 shows the mean error convergence curves for each algorithm solving the test functions. It can be seen that m-MFRO has a faster convergence speed and better convergence accuracy. In the convergence curve of the unimodal test function F24, m-MFRO has the fastest convergence speed and the highest accuracy. This indicates that m-MFRO has better exploitation capability. The convergence curves of the multimodal functions show that m-MFRO can explore well and thus avoid the local optimum. In most hybrid and composite functions, m-MRFO can achieve a better result quickly. This demonstrates that m-MFRO transitions well from exploration to exploitation, balancing the behaviors of exploration and exploitation in the search space.
The literature [48], [49] show that analyzing algorithm performance based on mean values alone is not sufficient. To avoid coincidence in the test, we adopt a variety of statistical analyses to verify algorithm performance.
In this paper, the Wilcoxon signed-rank test is first used to verify whether m-MRFO is significantly different from the comparison algorithms in statistical sense. Table 8 presents the results of the Wilcoxon signed rank test for each algorithm and m-MRFO at the significance level α = 0.05. In the Table 9, the symbol ''+'' indicates that m-MRFO outperforms the comparison algorithm. The symbol ''-'' indicates that m-MRFO underperforms the comparison algorithm. The symbol ''='' indicates that m-MRFO performs similarly to the comparison algorithm. The symbol ''R+'' is a positive rank value indicating the extent to which m-MRFO is better than the comparison algorithm, and ''R-'' indicates the opposite result. Counting the number of ''+/=/-'' for each algorithm, it can be seen that m-MRFO has the best performance among the algorithms involved in the test. m-MRFO outperforms all the comparison algorithms on at least 15 functions, which shows that m-MFRO is statistically significantly different from the other algorithms.
In addition, to check the differences and rankings between several algorithms, another non-parametric multiple comparison method is used in this paper: the Friedman test. A lower ranking in the Friedman test means a better performance, and the Friedman test compares three aspects: mean, standard deviation and time. As shown in Table 9, the probability of significance for the three aspects of the Friedman  test is significantly less than 0.05. Therefore, the hypothesis is rejected and the performance of the seven compared algorithms is significantly different. In terms of mean and time, m-MRFO performed best. As for standard deviation, m-MRFO ranks behind JS, but still outperforms the rest of the algorithms. To further analyze the differences between the algorithms, a post-hoc Iman-Davenport's test was employed. The Iman-Davenport test is based on the F-distribution with (k − 1) and (k − 1)(N − 1) degrees of freedom. It can be seen from  16, 9.4705e-12 and 0, respectively. To find differences in all algorithms, critical difference (CD) based on the Nemenyi test was used. The critical value q α is 2.3053, so the CD is 1.6873. A post-hoc test concludes that if the difference in Friedman ranking values between the two algorithms is less than the CD value, there is no significant difference between the two algorithms; conversely, there is a significant difference between the algorithms.

E. ANALYSIS OF ENGINEERING DESIGN PROBLEMS TEST
The engineering design problem is a nonlinear optimization problem with complex geometry, many design variables, and many real engineering constraints. The performance of m-MRFO is evaluated by solving real-world engineering problems. Considering that these engineering design problems are constrained optimization problems involving inequality and equality, we use penalty functions to transform the constrained optimization problems into unconstrained optimization problems. These engineering design problems are completely described in the following sections.
The pressure vessel design problem shown in Figure 8, presented by [50], is a typical hybrid optimization problem where the objective is to reduce the total cost including forming cost, material cost and welding cost. There are four different variables: vessel thickness Ts (x 1 ), head thickness Th (x 2 ), inner diameter R (x 3 ) and vessel cylindrical section length L (x 4 ). The problem can be described as Eq. (22). The comparison results are shown in Table 11 and Table 12.
The tension/compression spring design problem is a mechanical engineering design optimization problem [65], which can be used to evaluate the superiority of the algorithm. As shown in Figure 9, the goal of this problem is to reduce the weight of the spring. It includes four nonlinear inequalities and three continuous variables: wire diameter w (x 1 ), coil average diameter d (x 2 ), coil length or number L (x 3 ). The comparison results are shown in Table 13 and  Table 14. The mathematical model of this problem can be described as Eq. (23).
As shown in Figure 10, the main purpose of the welded beam design problem is to reduce the manufacturing cost of the welded beam, which mainly involves four variables:  the width h (x 1 ) and length l (x 2 ) of the weld zone, the depth t (x 3 ) and the thickness b (x 4 ), and subject to the constraints of bending stress, shear stress, maximum end deflection and load conditions. The comparison results are shown in Table 15 and Table 16. the mathematical model of the problem is described as Eq. (24).
minf (x 1 , x 2 , x 3 , x 4 ) = 1.10471x 2 1 x 2 + 0.04811x 3 x 4 (14.0 + x 2 ) subject to g 1 (X ) = τ d − τ (X ) ≥ 0    g 2 (X ) = σ d − σ (X ) ≥ 0 g 3 (X ) = x 4 − x 1 ≥ 0 g 4 (X ) = P c (X ) − P ≥ 0 where τ (X ) = (τ (X )) 2 + (τ (X )) 2 + x 2 τ (X )τ (X )/ 0.25(x 2 2 + (x 1 + x 3 ) 2 ) σ (X ) = 50400/x 2 3 x 4 P c (X ) = 64746.002(1 − 0.0282346x 3  First, ESP enhances MRFO exploitation during the cyclone foraging phase by using the three optimal individuals and their random synthetic individuals as reference points. Second, the ACP strategy balances exploitation and exploration capabilities by controlling the key parameters of MRFO. Finally, the DES strategy effectively utilizes the dominant population information to guide the evolutionary direction of the population and improve MRFO performance. Classical test functions and the CEC2017 test suite are used to verify the effectiveness of the improvement strategies and the superiority of m-MRFO. Simulation results show that the ESP strategy can effectively improve the exploitation capability. ACP achieves the balance of exploitation and exploration. DES improves the convergence speed and convergence accuracy of MRFO. To indicate whether the performance differences between the algorithms are statistically significant, the CEC2017 test results are analyzed using the Wilcoxon sign rank test, Friedman test, and post hoc Iman-Davenport test. The statistical results show that the proposed m-MRFO significantly outperforms the other algorithms. To demonstrate the performance of the proposed m-MRFO on real optimization problems three engineering design problems were employed. The results demonstrate m-MRFO can effectively solve real-world optimization problems.
In the subsequent study, we can address the following questions. First, the DES strategy increases the time complexity of MRFO. How to reduce the complexity with guaranteed performance is what we need to study further. The number and composition of individuals stored in the ESP strategy can be further investigated. Moreover, m-MRFO can be extended to solve multi-objective optimization problems. For real-world optimization problems, we plan to use m-MRFO for solving the multi-UAV cooperative path planning problem and the multi-UAV cooperative target allocation problem.