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Salp swarm algorithm (SSA) is a simple and effective bioinspired algorithm that is gaining popularity in global optimization problems. In this paper, first, based on the pinhole imaging phenomenon and oppositionbased learning mechanism, a new strategy called pinholeimagingbased learning (PIBL) is proposed. Then, the PIBL strategy is combined with orthogonal experimental design (OED) to propose an OPIBL mechanism that helps the algorithm to jump out of the local optimum. Second, a novel effective adaptive conversion parameter method is designed to enhance the balance between exploration and exploitation ability. To validate the performance of OPLSSA, comparative experiments are conducted based on 23 widely used benchmark functions and 30 IEEE CEC2017 benchmark problems. Compared with some wellestablished algorithms, OPLSSA performs better in most of the benchmark problems.
In recent years, metaheuristics have received incredible attention worldwide, and their great success on global optimization tasks has established superb beliefs for researchers, motivating them to develop more algorithms with good performance. Basically, metaheuristics are classified into four categories, namely swarmbased, humanbased, evolutionbased and physicsbased approaches. Among them, swarm intelligencebased methods have attracted most enthusiastic admiration, and they usually metaphorically represent some unique swarming behavior of organisms in the nature. The most classical swarm intelligent algorithm, particle swarm optimization (PSO) algorithm (
Swarm intelligent algorithms have emerged in various scientific and engineering fields, they are based on different metaphors, and their mathematical models consequently differ, which correspond to distinctive search mechanisms. Nevertheless, the framework of these algorithms is broadly the same, all divided into two phases: exploration (cohesion) and exploitation (alignment) (
Recently, a novel natureinspired swarm intelligent technique, namely salp swarm algorithm (SSA) (
Although the SSA algorithm has shown excellent performance on global optimization problems, it still suffers from premature convergence and insufficient solution accuracy when largescale optimization tasks and complex restricted engineering design issues. To address these limitations of the standard SSA, many highperformance SSAbased algorithms have been developed. In,
Many existing SSA variants focus mainly on alleviating the shortcomings of lack of convergence accuracy and unbalanced exploitation and exploration suffered by the basic SSA. For this purpose, different strategies have been injected into SSA and achieved remarkable results. However, these two limitations have not been completely solved and there are still research gaps. Moreover, the “No Free Lunch” theorems (
The remainder of this work is structured as follows: The standard SSA, including the principle, mathematical model and shortcoming analysis, is presented in
The SSA algorithm is a wellestablished swarm intelligent approach inspired by the unexplained behavior of salp swarm that organize in chains to improve foraging efficiency in oceans. SSA, resembling other population intelligencebased methodologies, commences its search process with a suit of randomly generated search agents, each of which indicates a solution to the pending problem. SSA compartmentalizes the salp population into two groups: leaders and followers. The leader is the key member, which plays a leadership role and at the front of the chain to lead the population in search of food. The followers, on the other hand, move implicitly or outright along the trajectory of the leader.
In SSA, the leading salp changes position depending on the following formula.
The mathematical equation used to change the followers’ positions is as follows:
Algorithm 1 outlines the pseudo code of the basic SSA.
The SSA approach begins the search with a reservation number of randomly generated search agents, and subsequently continuously updates the population position according to the objective value of the optimized function. The fitness of the population is evaluated after each iteration and the best individual is assigned as the current food source, which is the desired goal pursued by the leader, and the followers intuitively or implicitly follow the leading salp, the salp chain thus continuously approaching the food source. Notably, the value of the control parameter
In this subsection, we analyze the key limitations of the basic SSA, which is the inference and motivation behind the current work. The details are as follows:
1) First of all, there is only one parameter
2) Second, the position update equation of the followers in SSA does not have any control parameters, and although this can reduce the computational cost, it will make this movement mechanism sluggish and the algorithm is thus prone to fall into local optimum.
3) In addition, adaptive is a novel and effective technique that helps the algorithm to adjust the movement pattern autonomously during the search process, however, such operator is lacking in SSA.
4) Finally, maintaining a desirable balance between exploitation and exploration is the goal pursued by all swarmbased intelligence techniques, and a lot of research has focused on enhancing the capability of SSA algorithms in this regard to strengthen its overall performance, but there is still a gap in this context.
The above analyzed weak points of SSA have promoted the authors to discern that the algorithm has some drawbacks to be rectified, and this is the motivation behind proposing a novel version of SSA. Each of the embedded adjustments will be described in detail in the next section.
As discussed previously, the swarm intelligent algorithm covers two phases, namely exploration and exploitation. Exploration maintains a superiority in global search, and the strong exploration capability is conductive to improving the convergence speed. On the other hand, the domain nature of exploitation is shown in the local search, and the powerful development capability is beneficial to boost the convergence accuracy. Maintaining a proper equilibrium between exploration and exploitation holds a key position in the performance of swarm intelligent approaches, which is also a research gap that the current community tries to bridge. In this study, two straightforward but applicable mechanisms are integrated into SSA to produce an enhanced balanced SSA variants with better performance. This section provides an indepth discussion of the designed components and the OPLSSA algorithm.
In standard SSA, according to the location changing pattern of the leading salp, a prospective candidate search agent location is gained by directing the leader to the food source. Followers chase the leader explicitly or indirectly, gathering near the perceived global optimum in the later phase of the search. As a result, the standard SSA is inclined to converge prematurely. Therefore, improving the ability of the approach to avoid local optima has been considered as the most critical and necessary research goal in SSA improvement. To enhance the global exploration ability of swarm intelligent metaheuristic techniques, the most common method used in the published literatures is oppositionbased learning (OBL). For example,
Pinhole imaging is a general physical phenomenon in which a light source passes through a small hole in a plate and an inverted real image is formed on the other side of the plate. Motivated by the discovery that there are close similarities between the pinhole imaging phenomenon and the OBL mechanism, this paper proposes a pinholeimagingbased learning (PIBL) mechanism and applies it to the current leader to augment the exploration capability of SSA for unknown areas.
The schematic diagram of PIBL on the leading salp.
In
Let
Generalizing
In this paper, we use the proposed PIBL mechanism to help the leader search for unknown regions, thus improving the global search ability of the algorithm and avoiding the premature convergence due to the lack of exploration capability. However, similar to OBL, PIBL also suffers from the problem of “dimensional degradation”, i. t., the current leader improves only in some dimensions after jumping to the PIBL leader, while some other dimensions are even farther from the global optimum. To solve this problem, we introduce orthogonal experimental design (OED) and combine it with the PIBL strategy to design the orthogonal pinholeimagingbased learning (OPIBL) mechanism.
The OED is an auxiliary tool that can find the optimal combination of experiments by a reasonable number of trials. For example, for an experiment with 3 levels and 4 factors, it would take 81 attempts using a trialanderror approach. In contrast, by adopting an OED, only 9 sets of representative combinations need to be evaluated to determine the optimal combination of the experiment.
Orthogonal array of




1  2  3  4  
1  1  1  1  
2  1  2  2  2 
3  1  3  3  3 
4  2  1  2  3 
5  2  2  3  1 
6  2  3  1  2 
7  3  1  3  2 
8  3  2  1  3 
9  3  3  2  1 
In each iteration, the OPIBL mechanism is used for the leader, the dimension of the problem to be solved is considered as the factor of the OED, and the leader and the PIBL leader are regarded as the two levels of the OED. The designed OPIBL mechanism considers the information of the current leader and the PIBL leader, and retains the respective dominant dimensions to combine as a promising partial PIBL individual, called OPIBL leader. In this way, the OPIBL mechanism can effectively avoid the “dimensional degradation” problem caused by PIBL and significantly help the leader to quickly approach the global optimal solution. To visualize the process of the leader jumps to OPIBL leader, we consider a 7dimensional problem and draw the schematic diagram of the leader jumps to the OPIBL leader according to
Construct OPIBL leader.
In order not to increase the computational complexity of the algorithm, only the leader executes the OPIBL operation. Then, evaluate both the current leader and the OPIBL leader, and reserve the highquality search agent.
For natureinspired swarm intelligent algorithms, strong exploration capability is beneficial to improve the convergence speed, while powerful exploitation ability is conductive to refine the convergence accuracy. Maintaining a proper balance between exploration and exploitation can effectively boost the overall performance of the algorithm, which is a research difficulty that the metaheuristic community has been trying to conquer with great effort. In the basic SSA, the follower updates its position according to
Based on the above analysis, this paper proposes an adaptive position update mechanism for follower salps to replace the original formula, namely
In PSO, the inertia weight coefficient factor changes dynamically during the search process to help the algorithm switch between exploitation and exploration operations. With the progress of the research on metaheuristic techniques, the inertia weight has been introduced into many swarm intelligencebased approaches to improve their performance. For example,
The inertia weight
In summary,
The flow chart of OPLSSA.
In this section, 23 classical benchmark test functions are solved using the OPLSSA algorithm to synthetically verify its effectiveness and applicability. The results obtained by the proposed approach on the test cases are recorded and compared with some wellestablished metaheuristic techniques, including the basic SSA, the forefront swarm intelligent algorithms, and the popular SSA variants. All experiments were implemented under MATLAB 2016b software, operating system used is Microsoft WINDOWS 10 64bit Home, and simulations supported by Intel (R) Core (TM) i77700 CPU at 3.60 GHz with 8.00 GB RAM.
In this subsection, the selected 23 benchmark functions that include both multimodal and unimodal functions are reported, as shown in
The characteristics of the classical benchmark functions.
Function type  Function formulation  Search range 


Unimodal 

[−100,100]  0 

[−10,10]  0  

[−100,100]  0  

[−100,100]  0  

[−100,100]  0  

[−1.28,1.28]  0  

[−1.28,1.28]  0  

[−1,1]  0  

[−100,100]  0  

[−100,100]  0  

[−1,1]  0  
Multimodal 

[−5.12,5.12]  0 

[−32,32]  0  

[−600,600]  0  

[−10,10]  0  

[−10,10]  0  

[−1,1]  0  

[−5,10]  0  

[−10,10]  0  

[−100,100]  0  

[−15,15]  0  

[−10,10]  0  

[−1,1]  0 
To test the performance of the advocated OPLSSA algorithm, 23 classical benchmark functions reported in
(OBSSA) (
Comparisons of nine algorithms on 23 test functions with 100 dimensions.
Function  Results  SSA  LSSA  ASSA  GSSA  OBSSA  ASSO  RDSSA  IWOSSA  OPLSSA 

f1  Mean  1.29E+03  2.50E03  5.11E02  1.23E15  3.92E32  2.02E26  3.93E65  1.75E06  0 
Std  3.17E+02  2.20E03  3.97E02  4.22E15  4.65E32  3.11E27  1.75E64  1.61E06  0  
frank  9  7  8  5  3  4  2  6  1  
f2  Mean  8.71E+02  2.00E01  2.01E02  7.11E16  3.00E32  9.87E27  8.16E39  4.69E07  0 
Std  2.42E+02  1.74E01  8.20E03  2.86E15  3.50E32  1.48E27  4.47E38  4.08E07  0  
frank  9  8  7  5  3  4  2  6  1  
f3  Mean  5.02E+04  8.39E+03  1.79E+04  1.43E+04  4.77E30  1.92E25  5.28E34  1.01E+05  0 
Std  2.29E+04  5.89E+03  9.86E+03  8.51E+03  8.26E30  7.04E26  2.89E33  3.21E+04  0  
frank  8  5  7  6  3  4  2  9  1  
f4  Mean  2.73E+01  2.25E+01  9.3313  2.39E+01  5.52E17  3.45E14  5.53E30  4.47E+01  0 
Std  2.8509  5.2878  2.6632  3.2136  3.56E17  3.08E15  2.40E29  7.0144  0  
frank  8  6  5  7  3  4  2  9  1  
f5  Mean  2.83E+03  8.1000  1.46E+01  0  0  0  0  1.0667  0 
Std  8.76E+02  5.0402  7.0392  0  0  0  0  1.7991  0  
frank  9  7  8  1  1  1  1  6  1  
f6  Mean  2.97E01  1.17E02  1.06E07  1.86E02  3.97E70  1.44E59  1.95E95  3.67E12  0 
Std  1.91E01  1.70E02  1.14E07  2.55E02  7.31E70  4.82E60  1.07E94  8.94E12  0  
frank  9  7  6  8  3  4  2  5  1  
f7  Mean  2.6326  6.81E01  8.53E02  1.13E01  7.85E05  1.14E04  7.41E04  8.06E02  8.96E05 
Std  4.78E01  2.71E01  2.09E02  1.45E01  9.21E05  1.03E04  5.22E04  4.73E02  9.53E05  
frank  9  8  6  7  1  3  4  5  2  
f8  Mean  3.36E06  2.59E10  1.98E26  7.00E50  9.10E39  1.17E35  2.74E74  1.33E17  0 
Std  2.60E06  1.21E09  7.55E26  3.83E49  2.83E38  2.33E35  1.50E73  4.51E17  0  
frank  9  8  6  3  4  5  2  7  1  
f9  Mean  7.14E+07  1.2397  1.26E+02  1.20E14  2.88E27  1.36E21  6.63E33  9.80E03  0 
Std  2.67E+07  9.42E01  5.88E+01  4.08E14  4.05E27  5.33E22  3.63E32  6.50E03  0  
frank  9  7  8  5  3  4  2  6  1  
f10  Mean  6.72E+13  1.36E+09  9.78E+05  1.53E+13  5.34E35  5.37E32  2.55E82  1.69E+05  0 
Std  4.72E+13  3.58E+09  1.12E+06  1.27E+13  1.55E34  7.22E32  1.39E81  5.07E+05  0  
frank  9  7  6  8  3  4  2  5  1  
f11  Mean  1.7951  1.47E04  2.78E10  5.71E17  7.60E33  5.19E30  5.47E76  5.77E09  0 
Std  1.5770  3.60E04  7.29E10  2.49E16  2.92E32  1.15E29  2.99E75  2.62E08  0  
frank  9  8  6  5  3  4  2  7  1  
f12  Mean  2.36E+02  1.95E+02  2.38E+02  2.88E11  0  0  0  3.22E+02  0 
Std  3.42E+01  1.13E+02  1.31E+02  1.27E10  0  0  0  1.14E+02  0  
frank  7  6  8  5  1  1  1  9  1  
f13  Mean  1.03E+01  8.47E02  2.93E02  1.32E09  8.88E16  1.96E14  4.20E15  6.63E01  8.88E16 
Std  1.4847  3.05E01  1.03E02  1.80E09  0  2.27E15  9.01E16  3.6301  0  
frank  9  7  6  5  1  4  3  8  1  
f14  Mean  1.35E+01  2.74E02  5.33E02  1.29E14  0  0  0  3.60E03  0 
Std  4.7887  4.76E02  3.61E02  6.18E14  0  0  0  8.60E03  0  
frank  9  7  8  5  1  1  1  6  1  
f15  Mean  2.86E+01  1.35E+01  1.04E+01  4.42E14  1.49E17  1.14E14  1.65E26  1.88E+01  0 
Std  6.1418  1.29E+01  6.2120  1.40E13  5.88E18  7.75E16  6.69E26  1.56E+01  0  
frank  9  7  6  5  3  4  2  8  1  
f16  Mean  3.51E+02  1.77E+02  4.31E+01  4.73E08  1.99E32  7.97E27  6.69E47  8.3098  0 
Std  6.68E+02  1.38E+02  2.49E+01  1.69E07  1.96E32  1.22E27  3.63E46  2.21E+01  0  
frank  9  8  7  5  3  4  2  6  1  
f17  Mean  4.9384  1.3822  1.36E04  0  0  0  0  1.99E09  0 
Std  6.59E01  7.74E01  1.69E04  0  0  0  0  9.33E10  0  
frank  9  8  7  1  1  1  1  6  1  
f18  Mean  7.16E+01  1.78E02  8.50E03  9.29E09  2.09E+02  1.75E28  4.35E47  2.16E02  2.92E44 
Std  1.95E+01  1.08E02  4.90E03  2.12E08  3.03E+01  3.42E29  2.38E46  2.77E02  8.67E45  
frank  8  6  5  4  9  3  1  7  2  
f19  Mean  1.89E+01  3.40E03  1.48E04  4.36E17  8.03E35  4.02E29  1.48E43  2.55E09  0 
Std  5.7632  4.70E03  7.83E05  1.00E16  9.15E35  5.04E30  8.13E43  1.80E09  0  
frank  9  8  7  5  3  4  2  6  1  
f20  Mean  5.0612  2.83E01  1.8212  5.69E10  4.79E09  1.28E07  6.21E15  7.26E02  0 
Std  2.18E01  238E01  5.49E01  5.11E10  1.36E09  4.05E09  2.58E14  2.26E01  0  
frank  9  7  8  5  4  3  2  6  1  
f21  Mean  1.90E+02  1.34E+01  3.94E01  1.22E16  0  0  0  1.34E06  0 
Std  2.88E+01  6.4554  5.12E01  3.94E16  0  0  0  7.37E07  0  
frank  9  8  7  5  1  1  1  6  1  
f22  Mean  4.1930  4.1724  4.1941  9.39E01  4.65E09  1.19E07  1.86E02  3.5709  0 
Std  1.88E01  3.04E01  1.61E01  4.69E01  1.36E09  4.29E09  2.18E02  1.1627  0  
frank  8  7  9  5  2  3  4  6  1  
f23  Mean  2.59E04  1.16E05  5.96E11  8.13E05  5.24E108  1.64E92  6.43E209  2.79E10  0 
Std  1.65E04  9.04E06  9.92E11  1.08E04  1.72E107  8.86E93  0  7.19E10  0  
frank  9  7  5  8  3  4  2  6  1  
Average frank  8.7391  7.1304  6.7826  5.1739  2.6956  3.2174  1.9565  6.5652  1.0869  
Overall frank  9  8  7  5  3  4  2  6  1 
Statistical conclusions based on Wilcoxon signedrank test on 100dimensional benchmark problems.
Function  SSA 
LSSA 
ASSA 
GSSA 
OBSSA 
ASSO 
RDSSA 
IWOSSA 

f1  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f2  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f3  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f4  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f5  1.2118E12  1.1808E12  1.1941E12  N/A  N/A  N/A  N/A  1.4306E04 
f6  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f7  3.0199E11  3.0199E11  3.0199E11  3.0199E11  0.6952  0.2062  1.2870E09  3.0199E11 
f8  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  4.5736E12  1.2118E12 
f9  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f10  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f11  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f12  1.2118E12  1.2118E12  1.2118E12  1.4545E04  N/A  N/A  N/A  1.2118E12 
f13  1.2118E12  1.2118E12  1.2118E12  1.2118E12  N/A  5.6687E13  7.1518E13  1.2118E12 
f14  1.2118E12  1.2118E12  1.2118E12  6.6067E05  N/A  N/A  N/A  1.2118E12 
f15  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f16  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f17  1.2118E12  1.2118E12  1.2118E12  N/A  N/A  N/A  N/A  1.2118E12 
f18  3.0199E11  3.0199E11  3.0199E11  3.0199E11  3.0199E11  3.0199E11  3.0199E11  3.0199E11 
f19  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f20  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f21  1.2118E12  1.2118E12  1.2118E12  8.1523E02  N/A  N/A  N/A  1.2118E12 
f22  8.6253E13  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2078E12  1.2098E12 
f23  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
+/ = /  23/0/0  23/0/0  23/0/0  21/2/0  17/6/0  17/6/0  17/5/1  23/0/0 
From
Radar plot for consolidated ranks of 23 benchmarkproblems with the SSA variants.
In this subsection, the developed OPLSSA algorithm was compared with marine predators algorithm (MPA) (
Comparisons of nine algorithms on 23 test functions with 100 dimensions.
Function  Results  TSA  MPA  HGS  AOA  IGWO  WEMFO  DMMFO  OGWO  OPLSSA 

f1  Mean  3.01E10  1.80E19  2.23E152  9.49E79  2.21E12  3.59E22  3.29E+04  4.59E15  0 
Std  3.25E10  1.27E19  1.22E151  5.09E78  1.32E12  1.93E21  7.03E+03  5.59E15  0  
frank  8  5  2  3  7  4  9  6  1  
f2  Mean  1.41E10  7.00E20  8.26E149  8.91E78  1.09E12  1.68E22  1.48E+04  9.72E16  0 
Std  1.57E10  6.87E20  4.52E148  4.59E77  1.41E12  8.42E22  2.65E+03  1.36E15  0  
frank  8  5  2  3  7  4  9  6  1  
f3  Mean  1.28E+04  9.6728  3.73E18  1.14E62  5.69E+03  4.63E07  2.37E+05  8.28E+02  0 
Std  7.98E+03  1.22E+01  2.04E17  5.34E62  3.15E+03  2.53E06  4.31E+04  1.01E+03  0  
frank  8  5  3  2  7  4  9  6  1  
f4  Mean  5.59E+01  1.87E07  4.79E62  3.47E38  5.0967  2.04E10  8.89E+01  1.8357  0 
Std  1.34E+01  8.76E08  2.62E61  1.09E37  3.2803  4.26E10  2.6732  1.9532  0  
frank  7  5  2  3  8  4  9  6  1  
f5  Mean  1.34E+01  0  0  0  0  0  3.53E+04  0  0 
Std  9.6264  0  0  0  0  0  6.17E+03  0  0  
frank  8  1  1  1  1  1  9  1  1  
f6  Mean  4.99E18  8.77E41  8.25E261  4.89E164  1.72E22  7.39E46  1.02E+20  3.51E28  0 
Std  2.62E17  1.44E40  0  0  2.97E22  4.03E45  2.86E+01  7.99E28  0  
frank  8  5  2  3  7  4  9  6  1  
f7  Mean  5.22E02  2.00E03  1.90E03  7.68E04  1.44E02  1.60E03  9.24E+01  2.10E03  4.87E05 
Std  1.70E02  1.00E03  2.50E03  5.45E04  5.40E03  1.10E03  2.96E+01  1.90E03  5.19E05  
frank  8  5  4  2  7  3  9  6  1  
f8  Mean  1.83E42  4.25E60  1.52E77  7.57E187  2.57E57  1.66E82  1.40E03  1.26E61  0 
Std  9.92E42  1.12E59  8.29E77  0  1.31E56  6.92E82  1.60E03  6.84E51  0  
frank  8  5  4  2  6  3  9  7  1  
f9  Mean  7.98E07  9.23E16  3.30E160  4.07E74  3.60E09  6.95E19  2.07E+08  4.56E12  0 
Std  1.30E06  9.36E16  1.81E159  1.87E73  2.68E09  2.24E18  9.88E+07  4.89E12  0  
frank  8  5  2  3  7  4  9  6  1  
f10  Mean  4.28E02  8.66E39  1.95E54  5.97E170  4.60E10  2.88E48  2.86E+17  5.41E21  0 
Std  1.11E01  1.67E38  1.03E53  0  1.69E09  8.27E48  1.19E+17  1.21E20  0  
frank  8  5  3  2  7  4  9  6  1  
f11  Mean  7.21E20  3.89E53  1.62E43  4.26E183  6.04E28  8.17E63  9.86E01  2.85E38  0 
Std  2.95E19  7.53E53  8.76E43  0  1.43E27  4.09E62  3.91E01  9.84E38  0  
frank  8  4  5  2  7  3  6  9  1  
f12  Mean  9.86E+02  0  0  0  1.46E+02  2.85E+02  8.14E+02  1.1484  0 
Std  1.04E+02  0  0  0  4.72E+01  3.14E+02  6.94E+01  2.7139  0  
frank  9  1  1  1  6  7  8  5  1  
f13  Mean  1.25E05  5.03E11  8.88E16  1.86E+01  1.53E07  6.67E01  1.97E+01  4.91E09  8.88E16 
Std  2.86E05  3.14E11  0  5.0657  6.66E08  3.6445  3.34E01  1.57E09  0  
frank  6  3  1  8  5  7  9  4  1  
f14  Mean  1.63E02  0  0  0  4.70E03  0  3.02E+02  1.80E03  0 
Std  1.80E02  0  0  0  7.10E03  0  5.85E+01  7.00E03  0  
frank  8  1  1  1  7  1  9  6  1  
f15  Mean  1.51E+02  3.83E12  9.59E72  6.71E42  3.70E03  5.58E+01  5.79E+01  2.05E04  0 
Std  2.16E+01  3.34E12  5.25E71  2.71E41  1.90E03  2.67E+01  1.01E+01  4.99E04  0  
frank  9  4  2  3  6  7  8  5  1  
f16  Mean  1.38E+02  1.83E16  5.31E123  3.44E61  8.2595  5.44E21  1.56E+03  2.97E23  0 
Std  9.02E+01  4.55E16  2.91E122  1.88E60  4.4488  2.95E20  2.13E+02  1.37E22  0  
frank  8  6  2  3  7  5  9  4  1  
f17  Mean  3.2947  0  0  0  2.63E14  0  1.26E+01  5.39E15  0 
Std  2.8236  0  0  0  8.78E15  0  1.6209  2.58E15  0  
frank  8  1  1  1  7  1  9  6  1  
f18  Mean  9.89E12  2.99E19  1.49E109  2.08E69  1.72E12  7.12E23  5.07E+02  6.62E16  2.38E44 
Std  1.45E11  3.67E19  8.17E109  1.14E68  1.89E12  3.85E22  7.89E+01  5.22E16  4.48E15  
frank  8  5  1  2  7  4  9  6  3  
f19  Mean  1.09E+01  7.27E22  4.13E136  7.49E81  1.61E14  4.56E27  1.62E+02  5.65E27  0 
Std  5.97E+01  1.00E21  2.26E135  3.45E80  1.43E14  1.28E26  3.67E+01  2.26E26  0  
frank  8  6  2  3  7  4  9  5  1  
f20  Mean  2.8523  4.58E07  2.19E39  8.39E22  1.02E02  2.04E08  8.4517  7.17E05  0 
Std  1.4825  4.09E07  8.26E39  2.07E21  3.10E03  4.15E08  3.23E01  4.19E05  0  
frank  8  5  2  3  7  4  9  6  1  
f21  Mean  4.3276  0  0  0  2.16E12  0  2.19E+03  2.95E15  0 
Std  1.21E+01  0  0  0  1.55E12  0  4.97E+02  3.28E15  0  
frank  8  1  1  1  7  1  9  6  1  
f22  Mean  7.2165  6.13E01  3.63E38  1.71E02  3.0308  4.60E03  7.4324  8.31E01  0 
Std  5.16E01  7.47E02  1.99E37  1.78E02  4.72E01  6.80E03  2.66E01  2.58E01  0  
frank  8  5  2  4  7  3  9  6  1  
f23  Mean  8.46E14  3.86E55  6.3951  6.12E235  6.93E25  1.58E70  6.23E01  3.89E64  0 
Std  2.86E13  1.06E54  8.6601  0  2.46E24  8.54E70  2.75E01  2.12E63  0  
frank  7  5  9  2  6  3  8  4  1  
Average frank  7.9130  4.0435  2.3913  2.5217  6.5217  3.6956  8.7391  5.5612  1.0869  
Overall frank  8  5  2  3  7  4  9  6  1 
Statistical conclusions based on Wilcoxon signedrank test on 100dimensional benchmark problems.
Function  TSA 
MPA 
HGS 
AOA 
IGWO 
WEMFO 
DMMFO 
OGWO 

f1  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f2  1.2118E12  1.2118E12  6.6167E04  4.5736E12  1.2118E12  1.2118E12  1.2118E12  1.9346E10 
f3  1.2118E12  1.2118E12  8.8658E07  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f4  1.2118E12  1.2118E12  3.4526E07  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f5  4.5342E12  N/A  N/A  N/A  N/A  N/A  1.2118E12  N/A 
f6  3.0199E11  3.0199E11  1.0702E09  3.0199E11  3.0199E11  3.0199E11  3.0199E11  2.3715E10 
f7  3.0199E11  3.3384E11  2.1947E08  2.3715E10  3.0199E11  4.9752E11  3.0199E11  1.7769E10 
f8  1.2118E12  1.2118E12  8.87E07  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f9  1.2118E12  1.2118E12  6.6167E04  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f10  1.2118E12  1.2118E12  5.3750E06  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f11  1.2118E12  1.2118E12  5.3750E06  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f12  1.2118E12  N/A  N/A  N/A  1.2118E12  2.2130E06  1.2118E12  1.2108E12 
f13  1.2118E12  1.2118E12  N/A  1.6572E11  1.2118E12  4.5736E12  1.2118E12  1.2118E12 
f14  1.2118E12  N/A  N/A  N/A  1.2118E12  N/A  1.2118E12  1.2118E12 
f15  1.2118E12  1.2118E12  3.3149E04  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f16  1.2118E12  1.2118E12  5.3750E06  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f17  1.2118E12  N/A  N/A  N/A  1.1010E12  N/A  1.2118E12  4.4162E11 
f18  3.0199E11  3.0199E11  2.3982E11  3.0199E11  3.0199E11  3.0199E11  3.0199E11  3.0199E11 
f19  1.2118E12  1.2118E12  1.2717E05  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f20  1.2118E12  1.2118E12  5.5843E03  1.2118E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
f21  1.2118E12  N/A  N/A  N/A  1.2118E12  N/A  1.2118E12  1.2029E12 
f22  1.2118E12  1.1037E12  1.4552E04  1.2118E12  1.2118E12  1.2000E12  6.4999E13  1.2118E12 
f23  1.2118E12  1.2118E12  1.2118E12  4.5736E12  1.2118E12  1.2118E12  1.2118E12  1.2118E12 
+/ = /  23/0/0  18/5/0  16/6/1  18/5/0  22/1/0  19/4/0  23/0/0  22/1/0 
King, OPLSSA gets the highest rank, followed by HGS, AOA, WEMFO, MPA, OGWO, IGWO, TSA, DMMFO, which further indicates that the performance of OPLSSA is better than its competitors.
From
Finally, from the results generated by the Wilcoxon signed ranks test, the
Radar plot for consolidated ranks of 23 benchmark problems with OPLSSA and the Frontier algorithms.
The performance of wellestablished metaheuristic algorithms will not deteriorate drastically as the dimensionality of the tobesolved problem increases. The proposed OPLSSA algorithm aims to improve the overall performance of the basic SSA, and scalability is a key point that must be considered. In this experiment, OPLSSA was applied to address 23 benchmark functions with large scales (i.e. 10000 dimensions) in
If the result obtained by the algorithm on the benchmark function satisfies
Results obtained by OPLSSA on 10000dimensional functions.
Function  OPLSSA  

Best  Worst  Mean  Std  SR%  
f1  0  0  0  0  100 
f2  0  0  0  0  100 
f3  0  0  0  0  100 
f4  0  0  0  0  100 
f5  0  0  0  0  100 
f6  0  0  0  0  100 
f7  4.63E07  2.95E04  8.56E05  8.48E05  70 
f8  0  0  0  0  100 
f9  0  0  0  0  100 
f10  0  0  0  0  100 
f11  0  0  0  0  100 
f12  0  0  0  0  100 
f13  8.88E16  8.88E16  8.88E16  0  100 
f14  0  0  0  0  100 
f15  0  0  0  0  100 
f16  0  0  0  0  100 
f17  0  0  0  0  100 
f18  3.49E43  6.13E43  4.71E43  7.40E44  100 
f19  0  0  0  0  100 
f20  0  0  0  0  100 
f21  0  0  0  0  100 
f22  0  0  0  0  100 
f23  0  0  0  0  100 
Me representative test functions were plotted. In experiment 2, the convergence graphs of the OPLSSA algorithm and eight cuttingedge approaches on some representative benchmark functions were illustrated.
From
A wellestablished swarm intelligent algorithm moves in large steps in the search space in the early iterations to locate the rough position of the global optimal solution. After the lapse of few iterations, the step size is shortened to precisely search the already explored region, thus improving the convergence accuracy. Rapid convergence rate often leads to premature convergence of the algorithm, making the solution accuracy insufficient. Improving the solution precision of the method requires performing more iterations, which will degrade the convergence speed of the algorithm. The unbalanced convergence speed and convergence precision is a weak open point that destroys the performance of the algorithm. One of the main goals of the improvements to SSA in this work is to enhance the abovementioned balance of the basic algorithm. To investigate the performance of the OPLSSA algorithm in this regard, two additional sets of experiments were performed. For experimental purpose, some of the representative benchmarks with 100dimensional in
By observing
Convergence curves of OPLSSA and other SSAbased algorithms on 15 representative benchmarks.
Convergence curves of OPLSSA and other Frontier algorithms on 15 representative benchmarks.
In this subsection, the effectiveness of OPLSSA algorithm is tested on the IEEE CEC 2017 benchmark functions. This benchmark suite contains 30 test problems. For each function, the search region in each dimension is defined as [100, 100]. All selected functions are initiated in [100, 100]^{
D
}, where
Summary of the 30 CEC 2017 benchmark problems.
Class  No.  Description  Search Range  Optimal 

Unimodal  1  Shifted and Rotated Bent Cigar Function  [−100, 100]  100 
2  Shifted and Rotated Sum of Different Power Function  [−100, 100]  200  
3  Shifted and Rotated Zakharov Function  [−100, 100]  300  
Multimodal  4  Shifted and Rotated Rosenbrock’s Function  [−100, 100]  400 
5  Shifted and Rotated Rastrigin’s Function  [−100, 100]  500  
6  Shifted and Rotated Expanded Scaffer’s Function  [−100, 100]  600  
7  Shifted and Rotated Lunacek BiRastrigin Function  [−100, 100]  700  
8  Shifted and Rotated NonContinuous Rastrigin’s Function  [−100, 100]  800  
9  Shifted and Rotated Levy Function  [−100, 100]  900  
10  Shifted and Rotated Schwefel’s Function  [−100, 100]  1000  
Hybrid  11  Hybrid Function 1 (N = 3)  [−100, 100]  1100 
12  Hybrid Function 2 (N = 3)  [−100, 100]  1200  
13  Hybrid Function 3 (N = 3)  [−100, 100]  1300  
14  Hybrid Function 4 (N = 4)  [−100, 100]  1400  
15  Hybrid Function 5 (N = 4)  [−100, 100]  1500  
16  Hybrid Function 6 (N = 4)  [−100, 100]  1600  
17  Hybrid Function 6 (N = 5)  [−100, 100]  1700  
18  Hybrid Function 6 (N = 5)  [−100, 100]  1800  
19  Hybrid Function 6 (N = 5)  [−100, 100]  1900  
20  Hybrid Function 6 (N = 6)  [−100, 100]  2000  
Composition  21  Composition Function 1 (N = 3)  [−100, 100]  2100 
22  Composition Function 2 (N = 3)  [−100, 100]  2200  
23  Composition Function 3 (N = 4)  [−100, 100]  2300  
24  Composition Function 4 (N = 4)  [−100, 100]  2400  
25  Composition Function 5 (N = 5)  [−100, 100]  2500  
26  Composition Function 6 (N = 5)  [−100, 100]  2600  
27  Composition Function 7 (N = 6)  [−100, 100]  2700  
28  Composition Function 8 (N = 6)  [−100, 100]  2800  
29  Composition Function 9 (N = 3)  [−100, 100]  2900  
30  Composition Function 10 (N = 3)  [−100, 100]  3000 
Results of CEC 2017 at 30dimensional achieved by the developed algorithm and its competitors.
Function  DA  GOA  TSA  SOA  EGWO  PBO  SSA  OPLSSA 

F1  2.04E+08  1.03E+09  1.60E+10  7.37E+09  5.85E+09  1.36E+04  6.34E+02  1.06E+02 
F2  NA  NA  NA  NA  NA  NA  NA  NA 
F3  5.43E+04  3.09E+04  4.92E+04  2.98E+04  5.28E+04  1.08E+04  1.55E+04  3.00E+02 
F4  7.76E+02  4.89E+02  3.36E+03  7.15E+02  1.33E+04  5.46E+02  5.15E+02  4.69E+02 
F5  7.67E+02  7.17E+02  8.14E+02  6.90E+02  9.43E+02  6.80E+02  6.15E+02  5.66E+02 
F6  6.65E+02  6.32E+02  6.63E+02  6.35E+02  7.04E+02  6.63E+02  6.24E+02  6.03E+02 
F7  1.01E+03  8.20E+02  1.23E+03  1.08E+03  1.43E+03  1.79E+03  8.37E+02  8.05E+02 
F8  1.04E+03  1.04E+03  1.05E+03  9.48E+02  1.20E+03  9.48E+02  9.14E+02  8.47E+02 
F9  9.31E+03  4.44E+03  8.69E+03  3.63E+03  1.58E+04  3.92E+03  3.00E+03  9.19E+02 
F10  6.54E+03  5.07E+03  6.64E+03  6.44E+03  9.83E+03  4.21E+03  3.83E+03  2.98E+03 
F11  4.34E+03  9.21E+03  4.35E+03  1.88E+03  9.16E+03  1.26E+03  1.21E+03  1.14E+03 
F12  8.31E+07  4.87E+06  2.10E+09  2.20E+08  1.67E+09  6.52E+06  1.44E+06  3.85E+05 
F13  3.73E+05  2.82E+04  1.13E+09  1.48E+08  3.65E+09  1.80E+05  5.05E+04  1.30E+04 
F14  1.46E+05  8.32E+03  1.45E+06  1.17E+05  4.16E+05  1.91E+04  3.92E+03  1.92E+03 
F15  4.98E+04  1.86E+05  2.52E+07  1.51E+05  8.07E+07  3.45E+04  1.01E+04  8.98E+03 
F16  3.39E+03  3.42E+03  2.94E+03  2.65E+03  7.26E+03  2.72E+03  2.46E+03  1.96E+03 
F17  2.80E+03  2.13E+03  2.42E+03  2.06E+03  5.42E+03  2.26E+03  1.97E+03  1.80E+03 
F18  2.81E+06  8.82E+04  1.34E+06  4.42E+05  1.19E+07  1.52E+05  1.91E+05  2.95E+04 
F19  8.46E+06  4.35E+06  7.80E+06  5.04E+06  3.09E+08  4.82E+05  1.27E+05  9.51E+03 
F20  3.02E+03  2.91E+03  2.61E+03  2.63E+03  3.55E+03  2.32E+03  2.32E+03  2.11E+03 
F21  2.60E+03  2.51E+03  2.59E+03  2.45E+03  2.77E+03  2.59E+03  2.39E+03  2.36E+03 
F22  3.13E+03  2.49E+03  7.68E+03  2.66E+03  7.66E+03  6.24E+03  2.30E+03  2.30E+03 
F23  3.37E+03  2.90E+03  3.16E+03  2.82E+03  3.82E+03  3.31E+03  2.74E+03  2.69E+03 
F24  3.56E+03  3.01E+03  3.31E+03  2.93E+03  4.09E+03  3.42E+03  2.89E+03  2.90E+03 
F25  2.98E+03  2.89E+03  3.42E+03  3.23E+03  5.56E+03  2.88E+03  2.89E+03  2.88E+03 
F26  8.75E+03  4.21E+03  7.70E+03  5.09E+03  1.22E+04  7.88E+03  5.42E+03  4.11E+03 
F27  3.32E+03  3.29E+02  3.43E+03  3.25E+03  6.03E+03  3.20E+03  3.24E+03  3.19E+03 
F28  3.53E+03  4.43E+03  4.49E+03  5.29E+03  7.35E+03  3.30E+03  3.22E+03  3.10E+03 
F29  4.85E+03  4.22E+03  4.73E+03  4.01E+03  7.75E+03  4.81E+03  3.83E+03  3.48E+03 
F30  1.63E+07  1.93E+06  1.89E+07  1.09E+07  2.48E+09  1.28E+06  3.24E+06  3.02E+05 
By observing
This paper proposes an extended version of salp swarm algorithm termed as OPLSSA. Two modifications to SSA have been introduced which make it competitive with other wellestablished swarm intelligent algorithms: First, the algorithm applied the PIBL mechanism to help the leading salp to jump out of the local optimal. Second, the algorithm uses the concept of adaptivebased mechanism to generate diversity among the followers. Both these modifications helping in boosting the balance between exploration and exploitation. The performance of the proposed algorithm has been tested on 23 classical benchmark functions and 30 IEEE CEC 2017 benchmark suite and compared with several metaheuristic techniques, including SSAbased algorithms and stateoftheart swarm intelligent algorithms. The experimental results show that OPLSSA performs better or at least comparable to the competitor methods. Therefore, the developed OPLSSA algorithm can be regarded as a promising method for global optimization problems.
In the future works, we have planned to further extend the research on this paper on the following points: for one direction, the two proposed mechanisms will be combined with other swarm intelligence based algorithms with the hope of improving their performance; for another, the proposed OPLSSA algorithm will be employed to resolve realworld problems such as feature selection, PV parameter extraction, mobile robot path planning, multithreshold image segmentation, and video coding optimization.
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
ZW and HD conceived and designed the review. ZW wrote the manuscript. PH, GD, JW, ZY, and AL all participated in the data search and analysis. ZW and JY participated in the revision of the manuscript. All authors contributed to the article and approved the submitted version.
This work was supported by the National Nature Science Foundation of China (grant no. 61461053, 61461054, and 61072079), Yunnan Provincial Education Department Scientific Research Fund Project (grant no. 2022Y008), the Yunnan University’s Research Innovation Fund for Graduate Students, China (grant no. KC22222706) and the project of fund YNWRQNBJ2018310, Yunnan Province.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.