The sparrow search algorithm (SSA) is a newly proposed metaheuristic optimization algorithm based on the sparrow foraging principle. Similar to other metaheuristic algorithms, SSA has problems such as slow convergence speed and difficulty in jumping out of the local optimum. In order to overcome these shortcomings, a chaotic sparrow search algorithm based on logarithmic spiral strategy and adaptive step strategy (CLSSA) is proposed in this paper. Firstly, in order to balance the exploration and exploitation ability of the algorithm, chaotic mapping is introduced to adjust the main parameters of SSA. Secondly, in order to improve the diversity of the population and enhance the search of the surrounding space, the logarithmic spiral strategy is introduced to improve the sparrow search mechanism. Finally, the adaptive step strategy is introduced to better control the process of algorithm exploitation and exploration. The best chaotic map is determined by different test functions, and the CLSSA with the best chaotic map is applied to solve 23 benchmark functions and 3 classical engineering problems. The simulation results show that the iterative map is the best chaotic map, and CLSSA is efficient and useful for engineering problems, which is better than all comparison algorithms.
The optimization problem is a common realworld problem that requires seeking the maximum or minimum value of a given objective function and they can be classified as singleobjective optimization problems and multiobjective optimization problems [
However, similar to other metaheuristic algorithms, there are also problems such as reduction of population diversity and early convergence in the late iterations when solving complex optimization problems.
Based on the discussion above, a chaos sparrow search algorithm based on logarithmic spiral search strategy and adaptive step size strategy (CLSSA) is proposed in this paper, which employs three strategies to enhance the global search ability of SSA. In CLSSA, different chaotic maps are used to change the random values of the parameters in the SSA. Logarithmic spiral search strategy is used to expand the search space and enhance population diversity. Two adaptive step size strategies are applied to adjust the development and exploration ability of the algorithm. To verify the performance of CLSSA, 23 benchmark functions and three engineering problems were used for the tests. Simulation results show that the CLSSA proposed in this paper is superior to the existing methods in terms of accuracy, convergence speed and stability.
The rest of this article is organized as follows:
SSA is a novel swarmbased optimization algorithm that mainly simulates the process of sparrow foraging. The sparrow foraging process is a kind of discovererfollower model, and the detection and early warning mechanism is also superimposed. Individuals with good fitness in sparrows are the producers, and other individuals are the followers. At the same time, a certain proportion of individuals in the population are selected for detection and early warning. If a danger is found, these individuals fly away to find new position.
There are producers, followers, and guards in SSA. The location update is performed according to their respective rules. The update rules are as follows:
In
Chaos is a random phenomenon in nonlinear dynamic systems, which is regular and random, and is sensitive to initial conditions and ergodicity. According to these characteristics, chaotic graphs represented by different equations are constructed to update the random variables in the optimization algorithm.
Through experiments, it is found that the original SSA is easy to fall into the local optimum, which leads to premature convergence. As shown in
where
ID  Mapping type  Function 

1  Chebyshev map  
2  Circle map  
3  Gauss map  
4  Iterative map  
5  Logistic map  
6  Precewise map  
7  Sine map  
8  Singer map  
9  Sinusoidal map  
10  Tent map 
It can be seen from the
In the SSA, two strategies are used for the location update of the guards. The Gaussian distribution is used to generate the step size for individuals with poor fitness. It can be seen from the
For the individuals with poor fitness, when the dominant population of the updated sparrow is better than the dominant population of the previous generation, the larger step size of the Cauchy distribution is used to make the poor individual approach to the dominant population quickly; while when the dominant population of the updated sparrow is weaker than the dominant population of the previous generation, indicating that the renewal effect of this generation is not good, the smaller step size of Gaussian distribution is used to strengthen the search of the space near the individual. For individuals with better fitness, the adaptive step strategy is used. As can be seen from the
The pseudo code and flow chart of CLSSA is shown in Algorithm 2 and
In
In this paper, 23 classical test functions are employed, including 7 unimodal functions, 6 multimodal functions and 10 fixed dimensional functions. The above test functions are all singleobjective functions. The unimodal function F1–F7 has only one global optimal value, which is mainly used to test the development ability of the algorithm; the multimodal function has multiple local minima, which can be used to test the exploration ability of the algorithm. The benchmark function is shown in
Test function  Name  Type  Range  Optimum  

Sphere  US  30  [−100, 100]  0  
Schwefel 2.22  UN  30  [−10, 10]  0  
Schwefel 1.2  UN  30  [−100, 100]  0  
Schwefel 2.21  US  30  [−100, 100]  0  
Rosenbrock  UN  30  [−30, 30]  0  
Step  US  30  [−100, 100]  0  
Quartic  US  30  [−1.28, 1.28]  0  
Schwefel 2.26  MS  30  [−500, 500]  −418.9829*D  
Rastrigin  MS  30  [−5.12, 5.12]  0  
Ackley  MS  30  [−32, 32]  8.8818e−16  
Griewank  MN  30  [−600, 600]  0  
Penalized  MN  30  [−50, 50]  0  
Penalized2  MN  30  [−50, 50]  0  
Foxholes  MS  2  [−65.53, 65.53]  0.998004  
Kowalik  MS  4  [−5, 5]  0.0003075  
Six Hump Camel Back  MN  2  [−5, 5]  −1.03163  
Branin  MS  2  [−5, 10]×[0, 15]  0.398  
Goldstein Price  MN  2  [−5, 5]  3  
Hartman 3  MN  3  [0, 1]  −3.8628  
Hartman 6  MN  6  [0, 1]  −3.32  
Langermann 5  MN  4  [0, 10]  −10.1532  
Langermann 7  MN  4  [0, 10]  −10.4029  
Langermann 10  MN  4  [0, 10]  −10.5364 
Ten kinds of chaotic maps are combined with SSA algorithm to form new algorithms, the first chaotic map combined algorithm is named SSA1, the second chaotic map combined algorithm is named SSA2, and so on. The ten combined algorithms are compared with SSA in the benchmark function. In order to make a fair comparison, on the same experimental platform, the number of populations is set to 50, and the maximum number of iterations is 300. Except for using chaotic sequences to replace parameter
ID  SSA  SSA1  SSA2  SSA3  SSA4  SSA5  SSA6  SSA7  SSA8  SSA9  SSA10  

F1  Mean  1.72E129  3.58E114  7.16E147  5.43E156  5.83E128  9.76E116  6.93E112  6.58E124  2.74E89  7.39E94  1.39E120 
Std  9.40E129  1.96E113  2.73E146  2.45E155  3.19E111  5.35E115  3.76E127  3.60E123  1.50E88  4.05E93  7.64E120  
F2  Mean  1.78E53  4.33E54  2.22E69  3.61E75  1.77E66  4.26E56  1.21E71  1.43E62  5.11E38  8.90E52  1.02E60 
Std  9.73E66  1.49E53  1.22E68  1.98E74  9.71E53  2.30E55  6.45E71  7.68E62  2.80E37  3.42E51  5.60E60  
F3  Mean  1.04E88  1.82E71  9.32E90  1.84E116  4.05E82  1.12E83  7.00E91  3.29E80  4.80E59  1.63E72  1.32E96 
Std  5.70E88  9.96E82  5.10E89  1.00E115  2.22E70  6.15E83  3.83E90  1.80E79  2.63E58  8.93E72  5.56E96  
F4  Mean  9.18E79  2.12E60  2.77E73  7.99E97  5.34E66  4.10E54  3.08E61  2.41E61  5.70E46  2.36E47  8.31E64 
Std  5.03E78  1.16E59  1.52E72  4.02E96  2.92E65  2.08E53  1.68E60  1.32E60  3.12E45  1.30E46  3.27E63  
F5  Mean  1.65E04  1.50E04  3.15E04  1.79E04  1.24E04  1.16E04  1.20E04  1.21E04  1.09E04  8.45E05  1.11E04 
Std  3.36E04  5.73E04  7.26E04  4.46E04  2.16E04  1.95E04  1.97E04  2.51E04  1.90E04  1.96E04  2.00E04  
F6  Mean  5.81E08  8.00E08  3.81E08  3.36E08  4.50E08  5.23E08  6.19E08  1.85E07  2.27E07  6.58E08  5.57E08 
Std  9.15E08  1.52E07  5.87E08  6.62E08  9.31E08  8.24E08  1.55E07  4.43E07  4.66E07  1.03E07  1.37E07  
F7  Mean  3.49E04  4.32E04  4.46E04  4.14E04  3.36E04  5.13E04  4.15E04  5.22E04  4.37E04  5.32E04  6.03E04 
Std  2.81E04  4.05E04  3.12E04  3.93E04  2.85E04  4.20E04  4.29E04  4.80E04  3.49E04  3.61E04  4.67E04  
F8  Mean  8.19E+03  7.93E+03  8.15E+03  8.16E+03  8.21E+03  8.21E+03  8.12E+03  8.03E+03  8.09E+03  8.27E+03  8.08E+03 
Std  6.62E+02  7.21E+02  6.60E+02  6.95E+02  7.76E+02  5.45E+02  5.86E+02  4.94E+02  7.59E+02  6.80E+02  6.19E+02  
F9  Mean  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00 
Std  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
F10  Mean  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16 
Std  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
F11  Mean  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00 
Std  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
F12  Mean  3.16E09  3.05E09  4.30E09  1.68E09  4.77E09  1.78E09  7.68E10  4.42E09  2.99E09  5.57E09  6.23E09 
Std  5.79E09  5.65E09  1.41E08  3.14E09  9.89E09  2.65E09  1.53E09  8.63E09  5.75E09  1.60E08  1.29E08  
F13  Mean  5.12E08  9.81E08  2.00E08  6.78E08  3.39E08  1.36E07  1.31E08  3.03E08  3.99E08  3.50E08  2.29E08 
Std  1.50E07  3.33E07  4.66E08  1.60E07  5.96E08  3.90E07  2.01E08  5.49E08  6.15E08  5.93E08  5.03E08  
F14  Mean  4.51E+00  5.55E+00  6.19E+00  4.95E+00  5.87E+00  5.93E+00  5.80E+00  3.15E+00  5.54E+00  5.32E+00  7.10E+00 
Std  5.07E+00  5.44E+00  5.79E+00  5.56E+00  5.57E+00  5.52E+00  5.61E+00  4.37E+00  5.46E+00  5.35E+00  5.47E+00  
F15  Mean  3.08E04  3.08E04  3.08E04  3.18E04  3.08E04  3.08E04  3.08E04  3.08E04  3.08E04  3.08E04  3.08E04 
Std  4.44E08  9.86E07  4.64E07  5.56E05  1.27E06  5.17E07  2.58E06  4.41E07  8.24E07  3.34E06  2.78E07  
F16  Mean  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00  1.03E+00 
Std  5.53E16  6.12E16  5.90E16  5.76E16  5.53E16  5.98E16  5.68E16  5.61E16  5.98E16  6.12E16  6.12E16  
F17  Mean  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01 
Std  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
F18  Mean  3.90E+00  3.00E+00  3.00E+00  3.90E+00  3.00E+00  3.00E+00  5.70E+00  3.90E+00  6.60E+00  3.00E+00  4.80E+00 
Std  4.93E+00  2.87E15  2.11E15  4.93E+00  2.11E15  1.84E15  8.24E+00  4.93E+00  9.34E+00  2.51E15  6.85E+00  
F19  Mean  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00  3.86E+00 
Std  2.42E15  2.39E15  2.39E15  2.34E15  2.39E15  2.37E15  2.36E15  2.39E15  2.40E15  2.34E15  2.48E15  
F20  Mean  3.24E+00  3.27E+00  3.27E+00  3.25E+00  3.26E+00  3.25E+00  3.25E+00  3.29E+00  3.29E+00  3.26E+00  3.28E+00 
Std  5.70E02  6.03E02  5.99E02  5.83E02  6.05E02  5.92E02  5.99E02  5.54E02  5.54E02  6.05E02  5.83E02  
F21  Mean  8.79E+00  9.81E+00  8.62E+00  7.94E+00  9.64E+00  9.81E+00  9.13E+00  8.79E+00  9.81E+00  9.47E+00  9.13E+00 
Std  2.29E+00  1.29E+00  2.38E+00  2.57E+00  1.56E+00  1.29E+00  2.07E+00  2.29E+00  1.29E+00  1.76E+00  2.07E+00  
F22  Mean  8.81E+00  1.02E+01  8.45E+00  8.28E+00  9.87E+00  1.00E+01  8.99E+00  1.02E+01  1.00E+01  9.87E+00  9.69E+00 
Std  2.48E+00  9.70E01  2.61E+00  2.65E+00  1.62E+00  1.35E+00  2.39E+00  9.70E01  1.35E+00  1.62E+00  1.84E+00  
F23  Mean  9.82E+00  1.05E+01  9.82E+00  9.09E+00  1.04E+01  1.02E+01  9.82E+00  9.82E+00  1.05E+01  9.82E+00  9.82E+00 
Std  1.87E+00  9.27E04  1.87E+00  2.43E+00  9.87E01  1.37E+00  1.87E+00  1.87E+00  7.86E07  1.86E+00  1.87E+00  
‘+/=/−  8/9/6  9/8/6  8/7/8  10/6/7  10/6/7  7/9/9  8/6/9  9/6/8  9/6/8 
It can be observed from
In order to further analyze the optimization ability of the eleven algorithms, the results of these algorithms in each test function are compared and sorted according to the mean value of
ID  SSA  SSA1  SSA2  SSA3  SSA4  SSA5  SSA6  SSA7  SSA8  SSA9  SSA10 

F1  3  8  2  1  4  7  9  5  11  10  6 
F2  9  8  3  1  4  7  2  5  11  10  6 
F3  5  10  4  1  7  6  3  8  11  9  2 
F4  2  8  3  1  4  9  7  6  11  10  5 
F5  9  8  11  10  7  4  5  6  2  1  3 
F6  6  9  2  1  3  4  7  10  11  8  5 
F7  2  5  7  3  1  8  4  9  6  10  11 
F8  4  11  6  5  3  2  7  10  8  1  9 
F9  1  1  1  1  1  1  1  1  1  1  1 
F10  1  1  1  1  1  1  1  1  1  1  1 
F11  1  1  1  1  1  1  1  1  1  1  1 
F12  6  5  7  2  9  3  1  8  4  10  11 
F13  8  10  2  9  5  11  1  4  7  6  3 
F14  2  6  10  3  8  9  7  1  5  4  11 
F15  1  8  5  11  7  3  9  4  6  10  2 
F16  1  1  1  1  1  1  1  1  1  1  1 
F17  1  1  1  1  1  1  1  1  1  1  1 
F18  7  5  1  7  3  3  10  6  11  2  9 
F19  1  1  1  1  1  1  1  1  1  1  1 
F20  11  5  4  10  6  9  8  1  2  7  3 
F21  8  2  10  11  4  1  7  9  3  5  6 
F22  9  1  10  11  6  3  8  2  4  5  7 
F23  10  2  9  11  3  4  8  7  1  5  6 
Mean ranks  4.69  5.08  4.43  4.52  3.91  4.30  4.73  4.65  5.21  5.17  4.82 
In order to further prove the effectiveness of using chaotic sequences to replace SSA algorithm parameter,
Combined with the above analysis, the chaotic mapping sequence can promote the improvement of SSA performance, and iterative mapping has the best effect on improving the performance of the SSA. Therefore, in the next part of the CLSSA performance test, the iterative mapping sequence is used to replace the random value parameter
As mentioned above, this paper mainly uses three strategies to improve SSA, so three different derivative algorithms are designed to evaluate the impact of these three strategies on the algorithm. These three derivation algorithms are obtained by removing the corresponding improvement strategy from CLSSA. CLSSA1 removes both logarithmic spiral strategy and adaptive step strategy; CLSSA2 removes chaotic map and adaptive step strategy at the same time; CLSSA3 removes chaotic map strategy and logarithmic spiral strategy at the same time. 23 benchmark functions are used to compare the performance of the three derived algorithms with SSA and CLSSA. Each algorithm runs 30 times independently on each test function, and the statistical average results are shown in
Function  SSA  CLSSA1  CLSSA2  CLSSA3  CLSSA 

F1  5.16E109  0.00E+00  6.10E157  5.56E235  6.90E201 
F2  6.02E67  1.26E144  2.07E70  1.56E124  1.40E110 
F3  4.72E95  2.73E219  3.83E109  1.15E234  7.26E159 
F4  1.42E69  9.39E156  1.07E79  5.59E138  5.99E100 
F5  1.02E04  3.44E04  1.94E06  3.06E04  1.79E05 
F6  5.09E08  6.23E08  4.87E10  9.87E07  2.97E09 
F7  5.31E04  2.08E04  3.80E04  2.79E04  3.09E04 
F8  −8.13E+03  −7.75E+03  −7.78E+03  −8.65E+03  −8.48E+03 
F9  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00 
F10  8.88E16  8.88E16  8.88E16  8.88E16  8.88E16 
F11  0.00E+00  0.00E+00  0.00E+00  0.00E+00  0.00E+00 
F12  1.73E09  3.65E08  2.36E11  4.41E08  7.08E10 
F13  6.66E08  7.19E08  7.27E09  5.72E07  1.70E09 
F14  5.80E+00  3.01E+00  1.20E+00  2.37E+00  1.52E+00 
F15  3.08E04  3.08E04  3.35E04  3.08E04  3.08E04 
F16  −1.03E+00  −1.03E+00  −1.03E+00  −1.03E+00  −1.03E+00 
F17  3.98E01  3.98E01  3.98E01  3.98E01  3.98E01 
F18  3.00E+00  3.90E+00  4.80E+00  6.60E+00  3.00E+00 
F19  −3.86E+00  −3.86E+00  −3.86E+00  −3.86E+00  −3.86E+00 
F20  −3.27E+00  −3.26E+00  −3.27E+00  −3.28E+00  −3.27E+00 
F21  −8.96E+00  −8.80E+00  −1.02E+01  −9.81E+00  −1.02E+01 
F22  −9.34E+00  −8.28E+00  −1.04E+01  −1.00E+01  −1.04E+01 
F23  −9.63E+00  −8.91E+00  −1.04E+01  −1.05E+01  −1.05E+01 
In order to verify the performance of CLSSA, the proposed CLSSA is compared with the SSA, WOA, BSO [
As shown in
When solving the multimodal test functions F8–F13, GSA, PSO, BBO and MFO outperform CLSSA in solving F8. For F9–F11, CLSAA, SSA, HHO can all stably converge to the optimal value. WOA can obtain the optimal value, but it is not stable. The CLSSA has the highest accuracy for F12–F13, with optimal values improved by 17 and 11 orders of magnitude compared to SSA. When solving the fixeddimensional multimodal functions F14–F23, the CLSSA performs poorly for F14, outperforming only SSA, WOA, GSA, GWO and BBO. For the F15, optimal values can be obtained for CLSSA and SSA, but CLSSA is more stable than SSA. All algorithms have similar performance at F16, and all can obtain optimal values. GSA is the most stable and CLSAA is the second most stable. The CLSSA outperforms WOA, HHO, GSA, CSA, MVO, BBO and FPA for F17, with performance comparable to other algorithms. As for F18, the stability of CLSSA is only weaker than BSO, PSO, and GSA. The CLSSA outperforms all comparison algorithms for F19, F21 and F23. The GSA performs best for F22, with the CLSSA second best. In all multimodal test functions, CLSSA performs better than SSA, which shows that the logarithmic spiral strategy proposed in this paper can significantly improve the performance of algorithm exploration.
Function  SSA  CLSSA1  CLSSA2  CLSSA3  CLSSA 

F1  5  1  4  2  3 
F2  5  1  4  2  3 
F3  5  2  4  1  3 
F4  5  1  4  2  3 
F5  3  5  1  4  2 
F6  3  4  1  5  2 
F7  5  1  4  2  3 
F8  3  5  4  1  2 
F9  1  1  1  1  1 
F10  1  1  1  1  1 
F11  1  1  1  1  1 
F12  3  4  1  5  2 
F13  3  4  2  5  1 
F14  5  4  1  3  2 
F15  2  4  5  1  3 
F16  1  1  1  1  1 
F17  1  1  1  1  1 
F18  2  3  4  5  1 
F19  1  1  1  1  1 
F20  4  5  2  1  3 
F21  4  5  1  3  1 
F22  4  5  2  3  1 
F23  4  5  3  2  1 
Mean ranks  3.086957  2.826087  2.304348  2.304348  1.826087 
ID  Index  WOA  BSO  PSO  SSA  GSA  HHO  GWO  SCA  MVO  MFO  BBO  PFA  CLASSA 

F1  Best  2.0E42  6.9E+02  2.1E02  9.9E120  3.9E+01  4.9E63  1.1E18  1.1E+02  2.1E+00  1.2E+02  2.8E+00  4.4E+03  2.8E204 
Mean  9.7E52  2.8E+02  4.4E04  0.0E+00  4.5E17  3.8E79  3.9E20  2.0E+00  1.2E+00  2.8E+01  1.4E+00  2.2E+03  0.0E+00  
Std  1.1E41  4.5E+02  4.2E02  5.4E119  4.1E+01  2.6E62  9.0E19  1.4E+02  5.7E01  7.7E+01  7.4E01  1.5E+03  0.0E+00  
F2  Best  1.0E29  4.9E+00  2.3E01  2.9E67  2.7E02  8.6E33  1.9E11  1.2E01  1.9E+01  2.1E+01  5.3E01  5.6E+01  1.2E105 
Mean  3.0E34  4.9E01  1.1E02  0.0E+00  2.8E08  3.7E39  4.2E12  1.6E02  5.8E01  2.8E+00  3.4E01  3.3E+01  0.0E+00  
Std  2.2E29  4.1E+00  2.0E01  1.6E66  8.2E02  4.4E32  1.2E11  1.3E01  3.8E+01  1.7E+01  6.5E02  1.3E+01  6.5E105  
F3  Best  4.4E+04  5.3E+05  6.3E+02  3.2E94  8.9E+02  3.6E53  1.3E03  1.1E+04  3.5E+02  2.2E+04  1.1E+03  4.5E+03  1.4E144 
Mean  2.6E+04  5.4E+04  1.3E+02  0.0E+00  3.6E+02  3.5E64  4.6E06  1.4E+03  9.1E+01  5.4E+03  4.9E+02  1.6E+03  0.0E+00  
Std  8.7E+03  6.4E+05  3.6E+02  1.7E93  3.2E+02  1.9E52  2.2E03  7.3E+03  1.4E+02  1.1E+04  5.5E+02  1.5E+03  7.9E144  
F4  Best  5.5E+01  9.5E+00  5.3E+00  6.2E65  7.2E+00  2.8E33  9.8E05  3.6E+01  2.5E+00  6.3E+01  1.6E+00  3.5E+01  5.2E117 
Mean  6.1E01  3.6E+00  2.6E+00  0.0E+00  3.9E+00  3.0E39  1.9E05  1.7E+01  1.0E+00  3.9E+01  1.1E+00  2.5E+01  0.0E+00  
Std  2.7E+01  3.3E+00  1.8E+00  3.4E64  1.6E+00  1.2E32  6.5E05  1.0E+01  1.2E+00  8.8E+00  2.1E01  3.7E+00  1.8E116  
F5  Best  2.9E+01  2.4E+03  1.1E+02  1.0E04  1.2E+02  1.1E02  2.7E+01  2.2E+05  4.5E+02  5.4E+06  2.3E+02  1.6E+06  6.1E07 
Mean  2.8E+01  2.0E+02  2.3E+01  3.1E08  2.6E+01  1.2E03  2.5E+01  1.1E+03  4.5E+01  5.0E+03  5.8E+01  4.4E+05  3.2E27  
Std  3.0E01  3.0E+03  8.0E+01  1.6E04  7.2E+01  1.8E02  8.2E01  5.1E+05  6.2E+02  2.9E+07  2.1E+02  8.3E+05  1.6E06  
F6  Best  1.8E+00  6.6E+02  2.7E02  4.4E08  4.6E+01  8.3E05  5.6E01  8.7E+01  2.2E+00  1.5E+03  2.9E+00  4.1E+03  4.6E10 
Mean  8.4E01  2.4E+02  2.3E04  2.2E11  4.8E17  7.6E07  7.0E05  9.6E+00  9.1E01  3.7E+01  1.7E+00  2.4E+03  0.0E+00  
Std  6.4E01  3.0E+02  3.3E02  8.1E08  4.3E+01  8.9E05  3.4E01  1.3E+02  6.2E01  4.4E+03  7.0E01  8.1E+02  1.7E09  
F7  Best  4.7E03  7.2E02  2.9E02  6.7E04  4.5E02  1.5E04  2.2E03  2.1E01  4.0E02  5.3E01  1.3E02  1.1E+00  3.2E04 
Mean  2.1E05  1.0E02  1.5E02  3.1E05  1.1E02  1.0E05  3.1E04  1.6E02  1.2E02  1.2E01  4.2E03  3.7E01  2.0E05  
Std  6.1E03  5.2E02  1.1E02  5.7E04  2.5E02  1.2E04  9.5E04  1.9E01  1.4E02  8.1E01  4.4E03  4.2E01  2.7E04  
F8  Best  1.1E+82  4.9E+03  9.7E+03  8.1E+03  2.7E+03  1.2E+04  6.3E+03  3.8E+03  7.7E+03  9.1E+03  8.6E+03  6.6E+03  8.0E+03 
Mean  1.4E+81  6.2E+03  1.2E+04  9.6E+03  3.4E+03  1.3E+04  7.2E+03  4.6E+03  9.2E+03  1.1E+04  9.7E+03  7.1E+03  9.6E+03  
Std  3.9E+82  3.1E+02  1.6E+03  6.1E+02  4.0E+02  2.9E+02  9.1E+02  3.4E+02  6.8E+02  8.9E+02  6.8E+02  2.5E+02  8.6E+02  
F9  Best  0.0E+00  2.5E+01  5.7E+01  0.0E+00  1.7E+01  0.0E+00  4.2E+00  6.3E+01  1.2E+02  1.6E+02  3.8E+01  1.9E+02  0.0E+00 
Mean  0.0E+00  2.3E+00  4.1E+01  0.0E+00  1.1E+01  0.0E+00  5.1E13  6.8E+00  6.7E+01  6.8E+01  2.3E+01  1.6E+02  0.0E+00  
Std  0.0E+00  1.3E+01  1.5E+01  0.0E+00  4.7E+00  0.0E+00  4.3E+00  3.8E+01  3.1E+01  4.6E+01  1.1E+01  1.6E+01  0.0E+00  
F10  Best  6.7E15  4.8E+00  1.3E+00  8.9E16  1.6E03  8.9E16  2.2E10  1.4E+01  2.0E+00  1.4E+01  6.5E01  1.3E+01  8.9E16 
Mean  8.9E16  2.1E+00  3.2E02  8.9E16  4.5E09  8.9E16  8.6E11  2.2E01  6.4E01  2.7E+00  3.6E01  7.2E+00  8.9E16  
Std  4.2E15  2.0E+00  9.5E01  0.0E+00  6.2E03  0.0E+00  1.0E10  8.0E+00  4.5E01  7.0E+00  8.8E02  2.2E+00  0.0E+00  
F11  Best  7.4E18  2.2E+02  5.3E02  0.0E+00  1.1E+02  0.0E+00  6.9E03  1.7E+00  9.7E01  7.9E+00  1.0E+00  3.9E+01  0.0E+00 
Mean  0.0E+00  1.5E+02  3.7E03  0.0E+00  8.3E+01  0.0E+00  0.0E+00  6.6E01  8.4E01  1.5E+00  9.8E01  2.4E+01  0.0E+00  
Std  2.8E17  3.0E+01  7.4E02  0.0E+00  1.6E+01  0.0E+00  8.6E03  1.1E+00  4.4E02  2.3E+01  2.7E02  8.1E+00  0.0E+00  
F12  Best  9.3E02  1.8E+00  9.3E01  2.8E09  1.8E+00  6.3E06  3.8E02  3.5E+05  2.5E+00  8.5E+06  1.1E02  6.7E+04  4.9E11 
Mean  3.2E02  3.5E01  3.8E03  1.2E11  3.7E01  5.6E08  7.0E03  1.2E+00  4.3E01  6.6E+00  4.0E03  2.5E+02  2.4E28  
Std  9.0E02  1.4E+00  8.1E01  7.4E09  1.0E+00  7.1E06  2.1E02  9.1E+05  1.6E+00  4.7E+07  1.9E02  9.2E+04  1.5E10  
F13  Best  1.3E+00  2.5E+01  6.4E01  2.4E08  1.5E+01  5.7E05  4.7E01  1.6E+06  2.1E01  3.9E+03  1.4E01  1.7E+06  3.6E09 
Mean  7.7E01  5.4E+00  1.3E03  1.9E11  6.3E02  7.1E08  2.5E04  1.1E+01  9.4E02  3.5E+01  7.8E02  2.5E+05  2.9E22  
Std  4.4E01  1.4E+01  8.6E01  5.6E08  7.1E+00  7.2E05  2.3E01  3.5E+06  9.1E02  5.6E+03  3.7E02  1.4E+06  1.7E08  
F14  Best  2.3E+00  1.0E+00  1.0E+00  5.3E+00  7.0E+00  1.3E+00  3.9E+00  1.5E+00  1.0E+00  1.6E+00  4.6E+00  1.0E+00  1.7E+00 
Mean  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.1E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  
Std  2.6E+00  1.8E16  5.8E17  5.3E+00  4.2E+00  9.5E01  3.8E+00  8.9E01  6.7E11  1.2E+00  3.9E+00  1.8E03  2.5E+00  
F15  Best  1.1E03  1.4E03  5.6E04  3.1E04  6.0E03  4.1E04  4.4E03  1.1E03  5.4E03  1.0E03  2.4E03  7.9E04  3.1E04 
Mean  3.1E04  3.1E04  3.1E04  3.1E04  1.8E03  3.1E04  3.1E04  4.8E04  4.9E04  4.9E04  3.8E04  5.4E04  3.1E04  
Std  6.3E04  3.6E03  3.8E04  1.7E06  4.1E03  2.4E04  8.1E03  3.6E04  8.4E03  3.5E04  4.9E03  1.4E04  2.4E07  
F16  Best  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00 
Mean  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  1.0E+00  
Std  1.8E09  6.0E16  6.3E16  5.5E16  5.0E16  3.0E10  3.4E08  3.6E05  7.3E07  6.8E16  4.1E12  1.8E07  5.5E16  
F17  Best  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01 
Mean  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  4.0E01  
Std  2.6E05  0.0E+00  0.0E+00  0.0E+00  0.0E+00  2.5E05  1.3E06  1.4E03  1.7E07  0.0E+00  2.1E11  1.9E09  0.0E+00  
F18  Best  3.9E+00  3.0E+00  3.0E+00  3.9E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.9E+00  3.0E+00  3.0E+00 
Mean  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  3.0E+00  
Std  4.9E+00  1.5E15  1.7E15  4.9E+00  4.0E15  3.4E07  8.2E05  4.4E05  5.8E06  1.4E15  4.9E+00  8.5E07  4.9E15  
F19  Best  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00 
Mean  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  3.9E+00  
Std  3.8E03  3.7E03  2.7E15  2.3E15  2.5E03  3.7E03  1.9E03  1.6E03  2.3E06  2.7E15  5.7E14  1.2E06  2.4E15  
F20  Best  3.2E+00  3.1E+00  3.3E+00  3.3E+00  3.3E+00  3.1E+00  3.3E+00  2.9E+00  3.2E+00  3.2E+00  3.3E+00  3.3E+00  3.3E+00 
Mean  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  3.3E+00  
Std  7.1E02  3.6E01  6.0E02  6.0E02  1.4E15  1.1E01  6.9E02  2.5E01  6.0E02  6.2E02  6.0E02  1.5E02  5.9E02  
F21  Best  8.5E+00  1.0E+01  6.8E+00  9.5E+00  7.0E+00  5.1E+00  9.8E+00  3.1E+00  7.4E+00  6.6E+00  6.0E+00  1.0E+01  1.0E+01 
Mean  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  5.1E+00  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  
Std  3.0E+00  5.8E15  3.3E+00  1.8E+00  3.6E+00  4.1E03  1.3E+00  2.1E+00  3.1E+00  3.7E+00  3.6E+00  1.3E01  5.4E15  
F22  Best  6.5E+00  1.0E+01  8.1E+00  1.0E+01  1.0E+01  5.6E+00  1.0E+01  3.9E+00  8.5E+00  8.3E+00  6.4E+00  1.0E+01  1.0E+01 
Mean  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  6.0E+00  1.0E+01  1.0E+01  1.0E+01  1.0E+01  1.0E+01  
Std  3.4E+00  1.3E+00  3.4E+00  1.3E+00  1.4E15  1.6E+00  1.9E03  1.7E+00  3.2E+00  3.3E+00  3.6E+00  3.3E01  1.1E07  
F23  Best  6.7E+00  9.7E+00  8.1E+00  1.0E+01  1.0E+01  4.8E+00  9.3E+00  4.0E+00  8.2E+00  7.8E+00  8.6E+00  1.0E+01  1.1E+01 
Mean  1.1E+01  1.1E+01  1.1E+01  1.1E+01  1.1E+01  8.5E+00  1.1E+01  7.6E+00  1.1E+01  1.1E+01  1.1E+01  1.1E+01  1.1E+01  
Std  3.7E+00  2.1E+00  3.6E+00  1.7E+00  1.5E+00  8.9E01  2.8E+00  1.5E+00  3.3E+00  3.7E+00  3.4E+00  3.4E01  5.6E09 
Combined with the above analysis, the CLSSA proposed in this paper is better than all the comparison algorithms in 12 of the 23 benchmark functions, 11 comparison algorithms in 6 test functions, 9 comparison algorithms in 3 test functions, and CLSSA is better than SSA, in all test functions, which proves that our proposed CLSSA has obvious advantages in optimization accuracy.
In order to directly show the performance differences of each algorithm in solving the test function, the algorithms are sorted according to the mean fitness of
ID  WOA  BSO  PSO  SSA  GSA  HHO  GWO  SCA  MVO  MFO  BBO  FPA  CLSSA 

F1  4  12  6  2  9  3  5  10  7  11  8  13  1 
F2  4  10  3  2  6  3  5  7  11  12  9  13  1 
F3  12  13  6  2  7  3  4  10  5  11  8  9  1 
F4  12  9  7  2  8  3  4  11  6  13  5  10  1 
F5  5  10  6  2  8  3  7  11  9  13  8  12  1 
F6  6  11  4  2  9  3  5  10  7  12  8  13  1 
F7  5  10  7  3  9  1  4  11  8  12  6  13  2 
F8  8  11  2  6  13  1  10  12  7  3  4  9  5 
F9  1  7  9  1  6  1  5  10  11  12  8  13  1 
F10  4  10  8  1  6  1  5  13  9  12  7  11  1 
F11  4  13  6  1  12  1  5  9  7  10  8  11  1 
F12  6  8  7  2  9  3  5  10  10  13  4  11  1 
F13  8  10  7  2  9  3  6  12  5  11  4  13  1 
F14  9  1  1  12  13  5  10  6  1  7  11  4  8 
F15  7  9  4  1  13  3  11  7  12  6  10  5  1 
F16  9  4  5  3  1  8  10  13  12  6  7  11  2 
F17  13  1  1  1  1  12  7  8  9  1  11  10  1 
F18  13  2  3  12  4  6  10  9  8  1  11  7  5 
F19  10  11  3  2  8  12  9  13  7  4  5  6  1 
F20  8  11  6  5  1  12  4  13  9  10  7  2  3 
F21  6  1  9  5  8  12  4  13  7  10  11  3  1 
F22  10  4  9  5  1  12  3  13  7  8  11  6  2 
F23  11  5  9  4  2  12  6  13  8  10  7  3  1 
Mean ranks  7.60  7.95  5.56  3.39  7.08  5.34  6.26  10.6  7.91  9.04  7.73  9.04  1.86 
The black bold line is the sorting result curve of CLSSA, and it can be seen intuitively that the performance of CLSSA is in the middle level on F8 and F14, and performs better in other test functions, and its surrounding area is the smallest, indicating that CLSSA has the best optimization performance as a whole.
To further illustrate the convergence performance of CLSSA,
To analyze the distribution characteristics of each algorithm in the test function,
The above analysis shows that CLSSA shows strong optimization ability on lowdimensional functions. However, the optimization algorithm is prone to fail in solving highdimensional complex function problems. Realworld optimization problems are mostly largescale complex optimization problems. Therefore, to verify the performance of CLSSA in highdimensional problems, 13 algorithms were compared on the 100D test functions, and the experimental results are shown in
ID  WOA  BSO  PSO  SSA  GSA  HHO  GWO  SCA  MVO  MFO  BBO  FPA  CLSSA 

F1  7.29E41  8.74E+03  1.94E+03  3.16E113  4.31E+03  6.49E62  2.34E07  1.65E+04  2.62E+02  9.33E+04  2.17E+02  2.35E+04  2.04E201 
F2  4.08E28  7.01E+01  5.46E+01  3.81E69  1.42E+01  1.55E33  4.77E05  1.46E+01  2.96E+28  3.06E+02  9.41E+00  1.37E+06  4.62E106 
F3  8.91E+05  8.62E+07  6.17E+04  3.53E88  1.62E+04  1.25E40  1.93E+03  2.48E+05  7.26E+04  2.61E+05  6.26E+04  5.38E+04  8.17E115 
F4  8.26E+01  4.01E+01  4.11E+01  1.41E71  1.54E+01  1.80E32  2.84E+00  9.05E+01  5.62E+01  9.16E+01  1.95E+01  4.85E+01  6.51E113 
F5  9.84E+01  2.40E+05  4.86E+05  6.60E04  9.81E+04  7.12E02  9.81E+01  1.40E+08  1.76E+04  2.58E+08  5.12E+03  1.60E+07  2.77E04 
F6  1.16E+01  7.93E+03  2.03E+03  2.24E06  4.26E+03  5.35E04  9.91E+00  1.58E+04  2.64E+02  9.32E+04  2.11E+02  2.59E+04  1.29E07 
F7  3.74E03  1.53E+00  2.32E+00  4.68E04  1.62E+00  2.04E04  1.04E02  2.18E+02  6.50E01  3.92E+02  1.15E01  2.34E+01  2.84E04 
F8  2.21E+04  1.37E+04  2.77E+04  2.25E+04  5.15E+03  4.18E+04  1.66E+04  6.88E+03  2.26E+04  2.22E+04  2.35E+04  1.35E+04  2.32E+04 
F9  0.00E+00  1.32E+02  4.10E+02  0.00E+00  1.44E+02  0.00E+00  2.23E+01  3.42E+02  7.59E+02  9.12E+02  2.50E+02  9.30E+02  0.00E+00 
F10  7.76E15  9.03E+00  7.75E+00  8.88E16  4.92E+00  8.88E16  4.90E05  1.87E+01  6.95E+00  1.99E+01  3.30E+00  1.26E+01  8.88E16 
F11  1.57E02  1.00E+03  2.02E+01  0.00E+00  1.21E+03  0.00E+00  1.13E02  1.54E+02  3.55E+00  7.75E+02  2.87E+00  2.38E+02  0.00E+00 
F12  2.39E01  7.44E+00  5.88E+02  1.85E08  7.39E+00  2.79E06  2.94E01  4.34E+08  2.48E+01  4.14E+08  2.73E+00  4.66E+06  8.72E09 
F13  5.18E+00  4.84E+03  1.24E+05  1.70E06  4.73E+02  2.45E04  6.97E+00  7.62E+08  1.82E+02  9.06E+08  1.07E+01  3.34E+07  1.90E07 
In summary, compared with other algorithms, the CLSSA proposed in this paper is competitive, and the proposed improvement strategy can handle the relationship between exploitation and exploration well.
Engineering design problem is a nonlinear optimization problem with complex geometric shapes, various design variables and many practical engineering constraints. The performance of the proposed algorithm is evaluated by solving practical engineering problems. In the simulation, the population size is set to 50, and the maximum iterations is 500. The results of 30 independent runs of CLSSA are compared with those in other literatures.
The pressure vessel design optimization problem shown in
Algorithm  CLSSA  CSDE [ 
HPSO [ 
GA [ 
MBA [ 
BBBO [ 


Optimum value  Th  0.7782  0.8125  0.8125  0.9375  0.7802  1.1250 
Ts  0.3847  0.4375  0.4375  0.5000  0.3856  0.6250  
R  40.3209  42.1000  42.0984  48.3290  40.4292  58.1967  
L  199.9822  176.6000  176.6366  112.6790  198.4694  44.2721  
Optimum cost  5885.7092  6059.7100  6059.7143  6410.3811  5889.3216  7206.6400 
The tension/compression spring design problem is a mechanical engineering design optimization problem, which can be used to evaluate the superiority of the algorithm. As shown in
Algorithm  CLSSA  ALO [ 
GWO  MFO  MVO  GSA  

Optimum value  w  0.0518  0.0517  0.0508  0.0521  0.0500  0.0571 
d  0.3592  0.3569  0.3357  0.3661  0.3159  0.4843  
L  11.1441  11.2793  12.6457  10.7587  14.2583  7.6234  
Optimum cost  0.0127  0.0127  0.0127  0.0127  0.0128  0.0152 
As shown in
Algorithm  CLSSA  CDE [ 
HGA [ 
TEO [ 
HHO  hHHOSCA [ 


Optimum value  h  0.2057  0.2031  0.2057  0.2057  0.2040  0.1900 
l  3.4722  3.5430  3.4709  3.4731  3.5311  3.6965  
t  9.0362  9.0335  9.0396  9.0351  9.0275  9.3863  
b  0.2058  0.2062  0.2057  0.2058  0.2061  0.2041  
Optimum cost  1.7251  1.7335  1.7252  1.7253  1.7320  1.7790 
In this paper, we use three strategies combining chaos theory, logarithmic spiral search and adaptive steps to modify the basic sparrow search algorithm. First, the chaotic mapping is used to generate the values of the parameter