The metaheuristic algorithms are widely used in solving the parameters of the optimization problem. The marine predators algorithm (MPA) is a novel populationbased intelligent algorithm. Although MPA has shown a talented foraging strategy, it still needs a balance of exploration and exploitation. Therefore, a multistage improvement of marine predators algorithm (MSMPA) is proposed in this paper. The algorithm retains the advantage of multistage search and introduces a linear flight strategy in the middle stage to enhance the interaction between predators. Predators further away from the historical optimum are required to move, increasing the exploration capability of the algorithm. In the middle and late stages, the search mechanism of particle swarm optimization (PSO) is inserted, which enhances the exploitation capability of the algorithm. This means that the stochasticity is decreased, that is the optimal region where predators jumping out is effectively stifled. At the same time, selfadjusting weight is used to regulate the convergence speed of the algorithm, which can balance the exploration and exploitation capability of the algorithm. The algorithm is applied to different types of CEC2017 benchmark test functions and three multidimensional nonlinear structure design optimization problems, compared with other recent algorithms. The results show that the convergence speed and accuracy of MSMPA are significantly better than that of the comparison algorithms.
Metaheuristic Algorithms are studied relying on the combination of stochastic algorithms and local search algorithms. They can search globally and find similar solutions to the optimal solutions. These intelligent algorithms can be applied to solve reallife optimization problems [
The advantage of MPA over other algorithms is that the predator performs multiple motions depending on the behavior of the prey. According to the strategy of the best encounter, the predator can choose Levy motion or Brownian motion, which ensures the connection between predator and prey. Specifically, the Levy strategy is used by the predator when the number of prey in the area is low and on the contrary, the predator performs Brownian motion. The MPA is divided into three stages based on the speed score. The predator’s action pattern is regulated by using a segmented search strategy. At low velocity ratios (
Although the global search capability of MPA is stronger than that of algorithms such as particle swarm optimization, it still suffers from weak convergence and easily falls into local optimum. In order to improve the performance of MPA in the search for optimization, many scholars have conducted a series of research on it. Houssein et al. [
The core of metaheuristic algorithms is exploration and exploitation. Although MPA is a novel algorithm, it still has an imbalance between exploration and exploitation. Overemphasis on exploration can lead to species moving too fast. Obviously, this approach tends to miss optimal solutions; overemphasis on exploitation can lead to aggregation and stagnation of population, which can trap the algorithm in local optimal solutions. Balancing exploration and exploitation is the key to solving the algorithmic problems. No single algorithm can be suitable for all problems, or even for different stages of a problem. Drawing on the advantages of other algorithms and then combining MPA is also a popular research.
In this paper, we introduce the linear flight strategy and PSO search mechanism into MPA. The multistage search mechanism of MPA is retained, which is an irreplaceable advantage of other algorithms. In the first stage we keep the original Brownian motion, and the marine species are evenly and randomly spread over the search area. During the second stage, the first half of predators carry out a linear flight strategy and use it at a distance from the prey, thus increasing the chance of reemployment. The second half uses the search mechanism of PSO, which adds interaction between individuals and enhances the predator’s exploitation ability. The third stage also uses this mechanism to speed up the convergence of the algorithm. In addition, this paper also uses selfadjusting weight updates to adjust the parameters, so that the performance of the algorithm can be improved.
This paper is organized as follows.
Brownian motion is a common type of irregular motion that is manifested by cluttered and disordered particles. The particles in life are usually suspended in a liquid or gas. Mathematically, Brownian motion is a random variable that follows a normal distribution, and it is also a Markov process. Its step size is a probability function defined by a Gaussian distribution. The control density function of the point of motion
The trajectories of Brownian motion in 2D and 3D space are depicted in
Levy flight was proposed by the French mathematician Levy [
The parameter
The standard deviation corresponding to the above equation satisfies the following equation.
Levy flight is considered to be the best search pattern for the algorithm when the number of prey in the environment is less in [
At the beginning of the MPA, an Elite Matrix and a Prey Matrix are constructed. The Elite Matrix is used to store information about the best predators with the purpose of monitoring prey location information and finding other prey. The Elite Matrix is represented as follows:
The prey is defined as another Prey matrix. The position of the predator is updated according to its formulation. Initializing the initial prey, where the predator constitutes the elite. The predator matrix is expressed as follows:
The MPA relies mainly on the update of the position information of these two matrices to complete the optimization process. The movement of the predator and prey in the MPA is divided into three phases with different velocities. The specific steps are as follows:
The high speed ratio occurs mainly in the initial stage. And in order to coordinate the speed of the predator and prey, the best strategy for the predator is to be at rest in the case of a high speed ratio
With a unit speed ratio, predators and prey move at essentially the same speed. At this stage, creatures need to both maintain diversity and enhance interactivity. Both exploration and exploitation are key factors influencing predation. The prey is also a predator, so the prey is responsible for exploitation and the predator is responsible for exploration. According to the movement pattern of the species, at
When the movement of the population reaches a low velocity ratio, it is necessary to improve the proximity search capability of the algorithm. So Levy motion is performed at the final stage in MPA.
According to the movement strategy of the predator, the main idea of the algorithm is to be mainly responsible for the exploration in the early stage. The search space of the solution is expanded in the first stage. In the middle stage, exploration is gradually replaced by exploitation, where they are performed simultaneously. The later stage is mainly responsible for exploitation to avoid the predator moving too fast and so missing the global optimal solution. In this paper, we still use segmented processing to balance exploration and exploitation capabilities.
When marine species
During the first half of the second stage, in order to enhance the interaction between species, we used the strategy of linear flight to guide they to move. There are two cases of linear flight: predators that are far from some prey will swim in a line towards the prey, that is, predators that are far from the global optimal position will move in a straight line. The formula is as follows:
Another case is that if the predators move to a terminal position beyond the optimal search area, the position is represented as
The second half of the search method of the predator transitions to exploitation, and the algorithm’s position update strategy differs for different search strategies and different iteration stages of the step. If the prey only performs Levy moves, it may lead to a similar behavior of all individuals falling into local optimal. When a predator stagnates and cannot leave this space using the random wandering strategy of Levy, it means that it cannot jump out of the local minimum using the position of the neighboring individuals, and then the population diversity decreases substantially, which is not conducive to population search. Therefore, in the other half of the second stage and the third stage, we use the search mechanism of PSO, which has a strong local meritseeking ability.
The search mechanism of PSO is utilized to reduce the randomness and uncertainty of marine species movement during the exploitation phase and accelerate the convergence of the algorithm. Meanwhile, we use selfadjusting weight for P to control the position update. The exploration capability of the algorithm is increased in the early stage, and the exploitation capability of the algorithm is enhanced in the later stage.
In summary, this paper provides a phased improvement of MPA. The Brownian motion in the prephase is still adopted to provide a comfortable predatory environment for marine species. The middle phase is carried out in two stages. In order to prevent predators and prey from overgathering or overdispersing, half of the species adopt a linear flight strategy. The PSO search mechanism with a very high level of interactivity is accepted by the other half. At the later stage, the creatures gather and roam around the prey, in search of the best prey. The PSO search mechanism is still adopted in order to search in the vicinity of the best individual in a small area, which allows the algorithm to converge well. At the same time, the constant P is replaced by selfadjusting weight. Updating the P parameter according to the state of the predation process can be better integrated into the predation process.
For the way of movement of marine creatures, the influence of the surrounding environment on it also needs to be considered. In the MPA algorithm, FADs are considered as the local optimal solution, which is needed to be found out. In the algorithm FADs effect is used to prevent the organisms from stopping their movement. It allows the fish to make longer distance jumps to make they active again. So the FADs are expressed as follows:
Marine memory is used to update the elite matrix. First, the fitness is calculated for the prey matrix. If the fitness of the prey is better than the corresponding result in the elite matrix, the individual is replaced, that is, the elite matrix is updated. Then the fitness of all individuals in the elite matrix is calculated to find the best individual. If it meets the requirements, the algorithm stops, otherwise it continues to iterate.
The pseudocode for the MSMPA algorithm is given in Algorithm 1 and flowchart of MSMPA is shown in
In this paper, CEC2017 benchmark functions [
All algorithms are implemented in the MATLAB 9.10 (R2021a) programming language. To evaluate the performance of the MSMPA algorithm, we selected two MPArelated recent algorithms and three advanced others. Their corresponding parameters and literature are shown in
Algorithm  Parameter settings  Year  Refs. 

MPAmu  2022  [ 

MPA  2020  [ 

HFPSO  2018  [ 

HGWOP  2020  [ 

HPSOBOA  2020  [ 
The Wilcoxon signedrank test in statistics is used to compare the differences of the experiments and make the results more credible. It compares the performance of MSMPA with other algorithms. At the “+”, “−” and “=” significance levels, MSMPA is significantly better than, significantly lower than and consistent with the algorithms being compared. The published parameters of these algorithms are kept identical in the paper without any changes.
In this section, the number of function evaluations can be expressed as
MSMPA  MPAmu  MPA  HFPSO  HGWOP  HPSOBOA  

F1  Mean  1.09E+03=  1.51E+03+  2.77E+03+  1.38E+03+  6.03E+10+  
Std  1.53E+03  1.08E+03  2.52E+03  1.30E+03  5.86E+09  
Rank  2  4  5  3  6  
F3  Mean  1.80E−03+  3.41E−02+  7.25E−02+  1.75E+00+  9.27E+04+  
Std  1.51E−03  1.86E−02  1.14E−01  2.53E+00  5.87E+03  
Rank  2  3  4  5  6  
F4  Mean  8.29E+01  8.13E+01=  8.45E+01+  1.15E+02+  2.05E+04+  
Std  6.34E+00  8.61E+00  4.45E+00  1.52E+01  2.73E+03  
Rank  3  2  4  5  6  
F5  Mean  5.94E+01  6.31E+01+  5.68E+01=  8.86E+01+  5.03E+02+  
Std  1.04E+01  1.58E+01  2.03E+01  1.04E+01  2.69E+01  
Rank  3  4  2  5  6  
F6  Mean  4.01E−01  4.34E−01+  3.96E−01=  5.10E−01+  1.11E+02+  
Std  3.79E−01  3.32E−01  2.11E−01  5.23E−01  9.52E+00  
Rank  3  4  2  5  6  
F7  Mean  9.44E+01  9.59E+01+  7.55E+01−  1.06E+02+  8.07E+02+  
Std  1.47E+01  1.20E+01  2.99E+01  1.77E+01  4.23E+01  
Rank  3  4  2  5  6  
F8  Mean  6.50E+01  6.60E+01+  7.34E+01+  8.14E+01+  4.21E+02+  
Std  1.23E+01  1.12E+01  1.61E+01  1.29E+01  2.57E+01  
Rank  2  3  4  5  6  
F9  Mean  1.75E+01  1.95E+01=  1.21E+01−  1.33E+01−  1.91E+04+  
Std  1.20E+01  1.07E+01  1.05E+01  1.67E+01  3.82E+03  
Rank  4  5  2  3  6  
F10  Mean  2.64E+03  2.61E+03=  2.77E+03+  3.08E+03+  9.15E+03+  
Std  4.14E+02  4.99E+02  5.36E+02  4.17E+02  4.68E+02  
Rank  3  2  4  5  6  
F11  Mean  2.93E+01  2.79E+01=  1.17E+02+  7.34E+01+  7.97E+03+  
Std  1.16E+01  1.58E+01  6.07E+01  3.27E+01  1.54E+03  
Rank  3  2  5  4  6  
F12  Mean  1.17E+03  1.27E+03+  1.60E+05+  2.69E+04+  1.70E+10+  
Std  3.17E+02  3.25E+02  2.10E+05  2.23E+04  2.73E+09  
Rank  2  3  5  4  6  
F13  Mean  7.77E+01  8.87E+01+  1.86E+04+  1.42E+04+  1.66E+10+  
Std  1.36E+01  1.30E+01  2.85E+04  1.14E+04  5.38E+09  
Rank  2  3  5  4  6  
F14  Mean  3.17E+01+  4.20E+01+  7.34E+03+  4.23E+03+  6.27E+06+  
Std  4.38E+00  5.69E+00  7.49E+03  3.95E+03  3.62E+06  
Rank  2  3  5  4  6  
F15  Mean  1.95E+01+  3.44E+01+  6.56E+03+  2.76E+03+  9.04E+08+  
Std  3.78E+00  5.86E+00  7.96E+03  4.71E+03  4.43E+08  
Rank  2  3  5  4  6 
MSMPA  MPAmu  MPA  HFPSO  HGWOP  HPSOBOA  

F16  Mean  4.16E+02  3.59E+02=  7.55E+02+  5.04E+02+  5.95E+03+  
Std  1.46E+02  1.43E+02  2.18E+02  1.97E+02  9.46E+02  
Rank  3  2  5  4  6  
F17  Mean  7.14E+01  7.30E+01+  1.94E+02+  2.14E+02+  4.36E+03+  
Std  2.28E+01  2.65E+01  8.01E+01  1.36E+02  2.57E+03  
Rank  2  3  4  5  6  
F18  Mean  2.81E+01  3.40E+01+  1.39E+05+  1.16E+05+  1.44E+08+  
Std  2.72E+00  3.83E+00  8.13E+04  8.24E+04  1.21E+08  
Rank  2  3  5  4  6  
F19  Mean  1.91E+01  2.14E+01+  1.23E+04+  6.43E+03+  1.28E+09+  
Std  2.88E+00  3.00E+00  1.08E+04  4.69E+03  5.36E+08  
Rank  2  3  5  4  6  
F20  Mean  1.15E+02  1.15E+02+  2.44E+02+  2.12E+02+  1.48E+03+  
Std  5.48E+01  5.31E+01  1.52E+02  5.72E+01  1.69E+02  
Rank  2  3  5  4  6  
F21  Mean  1.98E+02+  1.94E+02+  2.82E+02+  2.26E+02+  7.34E+02+  
Std  7.85E+01  7.60E+01  7.82E+01  1.76E+01  4.92E+01  
Rank  3  2  5  4  6  
F22  Mean  9.84E+02+  8.30E+03+  
Std  1.72E−02  4.45E−01  1.70E−01  1.44E+03  6.97E+02  
Rank  5  6  
F23  Mean  3.41E+02  3.40E+02=  4.87E+02+  3.75E+02+  1.46E+03+  
Std  1.13E+02  1.10E+02  1.37E+02  3.98E+01  1.59E+02  
Rank  3  2  5  4  6  
F24  Mean  4.67E+02  4.58E+02−  4.63E+02=  5.66E+02+  1.77E+03+  
Std  1.20E+01  5.04E+01  1.23E+01  6.10E+01  2.14E+02  
Rank  4  2  3  5  6  
F25  Mean  3.93E+02+  3.95E+02+  3.37E+03+  
Std  1.63E+00  1.64E+00  1.75E+01  5.49E+00  5.04E+02  
Rank  4  5  6  
F26  Mean  3.01E+02+  1.62E+03+  4.85E+02+  9.65E+03+  
Std  1.65E−02  1.90E−01  9.86E+02  5.58E+02  6.82E+02  
Rank  3  5  4  6  
F27  Mean  4.98E+02  4.97E+02=  5.28E+02+  5.71E+02+  2.31E+03+  
Std  6.78E+00  6.73E+00  1.82E+01  2.12E+01  4.66E+02  
Rank  3  2  4  5  6  
F28  Mean  3.11E+02+  3.45E+02+  4.00E+02+  3.86E+02+  5.26E+03+  
Std  2.87E+01  2.78E+01  4.08E+01  4.13E+01  6.78E+02  
Rank  2  3  5  4  6  
F29  Mean  5.41E+02  5.34E+02=  5.18E+02−  7.66E+02+  9.78E+03+  
Std  5.80E+01  7.56E+01  5.54E+01  1.81E+02  6.26E+03  
Rank  4  3  2  5  6  
F30  Mean  2.23E+03+  1.52E+04+  6.94E+03+  2.49E+09+  
Std  1.02E+02  1.23E+02  1.86E+04  1.88E+03  1.27E+09  
Rank  3  5  4  6 
The experimental results we obtained are shown in
MSMPA  MPAmu  MPA  HFPSO  HGWOP  HPSOBOA  

avg.Rank  2.172414  2.206897  2.37931  4.655172  3.206897  6 
totalRank  1  2  3  5  4  6 
+/=/−  11/16/2  15/7/7  27/0/2  20/1/8  29/0/0 
In hybrid functions, MSMPA is particularly effective in finding the best results for F14 and F15. The second ranking among the ten functions reaches five times. In the
In the composition functions, the optimal average of functions F21, F22, F25, F26, F28 and F30 is represented by MSMPA, which obtains the most optimal values among all algorithms. In the computation of these functions, MPA performs better than MPAmu. This indicates that the multistage improvement mechanism of MSMPA as well as the selfadjusting weight work well for them. The connection between the prestage organisms is strengthened by the linear flight strategy, which increases the information transfer. The movement patterns of the later stage organisms are exploited by the PSO search mechanism, which is used to find the optimal prey.
In fact, MPArelated algorithms all have similar convergence curves. How to find a better convergence curve in the similarity is the direction of improvement in this paper. As can be seen from the table, MSMPA has significantly better computational accuracy for the three metaheuristics HFPSO, HGWOP and HPSOBOA. Likewise, it is excellent in the comparison with the two MPAs. It indicates that the population diversity is improved after using the linear flight strategy and selfadjusting weight. However, in the convergence graph, the convergence speed of MSMPA is not significant and only slightly faster than that of MPAs. This means that MSMPA has improved the convergence speed and population diversity compared with the comparison algorithms, but its convergence could still be improved in the future.
In this section, the algorithm in this paper focuses on the analysis of three engineering optimization problems. Of course problems have been solved by other algorithms as well, so we can compare and analyze the practical applications of MSMPA. To eliminate randomness and variability, we choose the conventional
The design problem of pressure vessel was first introduced by Kannan et al. [
There are also many algorithms that solve the problem, such as [
Methods  Best  Worst  Median  Mean  Std 

Sandgren [ 
8129.1036  NA  NA  NA  NA 
Gandomi et al. [ 
6059.714  6495.3470  NA  6447.7360  502.693 
Coello Coello et al. [ 
6059.7208  7544.4925  6257.5943  6440.3786  448.4711 
He et al. [ 
6059.7143  NA  NA  6289.92881  305.78 
Akay et al. [ 
6059.714339  NA  NA  6245.308144  205 
Hassanien et al. [ 
6059.606944  6061.034418  6059.606944  6059.844857  0.0582763 
MSMPA  5930.69992  6166.632401  5981.6469  5988.2459  58.424 
Method  

Sandgren [ 
1,125000  0.625000  47.700000  117.701000  8129.1036 
Gandomi et al. [ 
0.812500  0.437500  42.0984456  176.6365958  6059.7143348 
Coello et al. [ 
0.812500  0.437500  42.098400  176.6372000  6059.7208 
He et al. [ 
0.812500  0.437500  42.098445  176.6365950  6059.7143 
Akay et al. [ 
0.812500  0.437500  42.098446  176.636596  6059.714339 
Hassanien et al. [ 
0.812500  0.437500  42.100204  176.614800  6059.606944 
MSMPA  0.8066  0.3971  41.7911  180.4828  5930.69992 
The welded beam design optimization problem is a commonly used engineering optimization problem with the structure shown in
The calculation results of different algorithms for this problem are shown in
Methods  Best  Worst  Median  Mean  Std 

Coello Coello [ 
1.748309  1.785835  NA  1.771973  0.01122 
Dimopoulos [ 
1.731186  NA  NA  NA  NA 
He et al. [ 
1.72802  1.782143  NA  1.748831  0.012926 
MezuraMontes et al. [ 
1.724852  NA  NA  1.725  1.00E−15 
Gandomi et al. [ 
1.7312065  2.3455793  NA  1.878656  0.2677989 
Mehta et al. [ 
1.724855  1.72489  1.724861  1.724865  NA 
Akay et al. [ 
1.724852  NA  NA  1.741913  0.031 
Feng et al. [ 
1.695505466  1.87392175  1.700841829  1.724051281  0.047618 
MSMPA  1.69546  NA  1.7032  NA  NA 
Method  

Coello Coello [ 
0.2088  3.4205  8.9975  0.21  1.748309 
He et al. [ 
0.202369  3.544214  9.04821  0.205723  1.728024 
MezuraMontes et al. [ 
0.20573  3.470489  9.036624  0.20573  1.724852 
Gandomi et al. [ 
0.2015  3.562  9.0414  0.2057  1.73121 
Mehta et al. [ 
0.20572885  3.47050567  9.03662392  0.20572964  1.724855 
Akay et al. [ 
0.20573  3.470489  9.036624  0.20573  1.724852 
Kai Feng et al. [ 
0.205692017  3.254453177  9.036360313  0.205753289  1.695505 
MSMPA  0.20531  3.26054  9.03670  0.20573  1.69546 
The gear train design problem was first presented by Sandgren [
The solutions of the problem by MSMPA algorithm and other algorithms are shown in
Methods  Best  Worst  Median  Mean  Std 

Sandgren [ 
5.712e−06  NA  NA  NA  NA 
Kannan et al. [ 
2.146e−08  NA  NA  NA  NA 
Deb et al. [ 
2.701e−12  NA  NA  NA  NA 
Gandomi et al. [ 
2.701e−12  2.3576e−09  NA  1.9841e−09  3.5546e−09 
Garg [ 
2.7006e12  3.2999e−09  9.9215e−10  1.2149e−09  8.7787e−10 
MSMPA  6.4396e−15  9.6490e−13  5.0074e−14  4.56886e−14  1.794e−13 
Method  Gear ratio  

Sandgren [ 
18  22  45  60  0.146667  5.712e−06 
Kannan et al. [ 
13  15  33  41  0.144124  2.146e−08 
Deb et al. [ 
19  16  49  43  0.144281  2.701e−12 
Gandomi et al. [ 
19  16  43  49  0.144281  2.701e−12 
Garg [ 
19  16  43  49  0.14428096  2.70085e−12 
MSMPA  12.7940  19.1888  60  28.3595  0.1443  6.4396e−15 
In summary, MSMPA has been effective in improving the exploration and exploitation capabilities of MPAs. The experiments for engineering optimization problem are also fruitful. Its efficiency in finding the optimal solutions has been greatly improved.
Marine predators algorithm is a new algorithm proposed in 2020 and applied to many fields. Since MPA, like other intelligent algorithms, suffers from unbalanced exploration and exploitation, we improve it. In this paper, we proposed a new multistage improvement of marine predators algorithm. Firstly, the original Brownian motion is still used in the first stage, and the predation space is well expanded up. In the middle stage, the linear motion, which is more conducive to communication between creatures, is used to facilitate the fast movement of creatures far away from the prey. In the middle and late stages, the search mechanism of PSO is adopted, which effectively increases the exploitation ability of the creatures. At the same time, the constant P was changed to selfadjusting weight, thereby adapting more to the whole algorithmic process. The proposed MSMPA has been fully evaluated on CEC2017 functions and compared with established related optimization The experimental results show that MSMPA achieves competitive or even better performance on most functions, especially on composition. Also, in the engineering optimization problem, MSMPA has better objective function values compared to other algorithms and the resulting optimal solutions do not exceed the constraints. This indicates that the MSMPA algorithm can be well applied to engineering optimization problems.
In the future, we will continue to improve the algorithm and increase its performance. As the analysis of the experiment, we found that the convergence speed of MSMPA can still be improved and we can continue to experiment in this area at a later time. In addition, the predators model is a general optimization framework that can be applied to other metaheuristic algorithms, such as differential evolution (DE), genetic algorithm (GA), and artificial bee colony algorithm (ABC).
This work was supported in part by
This paper does not contain any studies with human participants or animals performed by any of the authors.
The authors declare that they have no conflicts of interest to report regarding the present study.