The sparrow search algorithm (SSA) is a newly proposed meta-heuristic optimization algorithm based on the sparrow foraging principle. Similar to other meta-heuristic 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 real-world problem that requires seeking the maximum or minimum value of a given objective function and they can be classified as single-objective optimization problems and multi-objective 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 swarm-based optimization algorithm that mainly simulates the process of sparrow foraging. The sparrow foraging process is a kind of discoverer-follower 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 per-formed 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 single-objective 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 SSA-1, the second chaotic map combined algorithm is named SSA-2, 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 | SSA-1 | SSA-2 | SSA-3 | SSA-4 | SSA-5 | SSA-6 | SSA-7 | SSA-8 | SSA-9 | SSA-10 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
F1 | Mean | 1.72E-129 | 3.58E-114 | 7.16E-147 | 5.43E-156 | 5.83E-128 | 9.76E-116 | 6.93E-112 | 6.58E-124 | 2.74E-89 | 7.39E-94 | 1.39E-120 |
Std | 9.40E-129 | 1.96E-113 | 2.73E-146 | 2.45E-155 | 3.19E-111 | 5.35E-115 | 3.76E-127 | 3.60E-123 | 1.50E-88 | 4.05E-93 | 7.64E-120 | |
F2 | Mean | 1.78E-53 | 4.33E-54 | 2.22E-69 | 3.61E-75 | 1.77E-66 | 4.26E-56 | 1.21E-71 | 1.43E-62 | 5.11E-38 | 8.90E-52 | 1.02E-60 |
Std | 9.73E-66 | 1.49E-53 | 1.22E-68 | 1.98E-74 | 9.71E-53 | 2.30E-55 | 6.45E-71 | 7.68E-62 | 2.80E-37 | 3.42E-51 | 5.60E-60 | |
F3 | Mean | 1.04E-88 | 1.82E-71 | 9.32E-90 | 1.84E-116 | 4.05E-82 | 1.12E-83 | 7.00E-91 | 3.29E-80 | 4.80E-59 | 1.63E-72 | 1.32E-96 |
Std | 5.70E-88 | 9.96E-82 | 5.10E-89 | 1.00E-115 | 2.22E-70 | 6.15E-83 | 3.83E-90 | 1.80E-79 | 2.63E-58 | 8.93E-72 | 5.56E-96 | |
F4 | Mean | 9.18E-79 | 2.12E-60 | 2.77E-73 | 7.99E-97 | 5.34E-66 | 4.10E-54 | 3.08E-61 | 2.41E-61 | 5.70E-46 | 2.36E-47 | 8.31E-64 |
Std | 5.03E-78 | 1.16E-59 | 1.52E-72 | 4.02E-96 | 2.92E-65 | 2.08E-53 | 1.68E-60 | 1.32E-60 | 3.12E-45 | 1.30E-46 | 3.27E-63 | |
F5 | Mean | 1.65E-04 | 1.50E-04 | 3.15E-04 | 1.79E-04 | 1.24E-04 | 1.16E-04 | 1.20E-04 | 1.21E-04 | 1.09E-04 | 8.45E-05 | 1.11E-04 |
Std | 3.36E-04 | 5.73E-04 | 7.26E-04 | 4.46E-04 | 2.16E-04 | 1.95E-04 | 1.97E-04 | 2.51E-04 | 1.90E-04 | 1.96E-04 | 2.00E-04 | |
F6 | Mean | 5.81E-08 | 8.00E-08 | 3.81E-08 | 3.36E-08 | 4.50E-08 | 5.23E-08 | 6.19E-08 | 1.85E-07 | 2.27E-07 | 6.58E-08 | 5.57E-08 |
Std | 9.15E-08 | 1.52E-07 | 5.87E-08 | 6.62E-08 | 9.31E-08 | 8.24E-08 | 1.55E-07 | 4.43E-07 | 4.66E-07 | 1.03E-07 | 1.37E-07 | |
F7 | Mean | 3.49E-04 | 4.32E-04 | 4.46E-04 | 4.14E-04 | 3.36E-04 | 5.13E-04 | 4.15E-04 | 5.22E-04 | 4.37E-04 | 5.32E-04 | 6.03E-04 |
Std | 2.81E-04 | 4.05E-04 | 3.12E-04 | 3.93E-04 | 2.85E-04 | 4.20E-04 | 4.29E-04 | 4.80E-04 | 3.49E-04 | 3.61E-04 | 4.67E-04 | |
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 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 |
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 | 3.16E-09 | 3.05E-09 | 4.30E-09 | 1.68E-09 | 4.77E-09 | 1.78E-09 | 7.68E-10 | 4.42E-09 | 2.99E-09 | 5.57E-09 | 6.23E-09 |
Std | 5.79E-09 | 5.65E-09 | 1.41E-08 | 3.14E-09 | 9.89E-09 | 2.65E-09 | 1.53E-09 | 8.63E-09 | 5.75E-09 | 1.60E-08 | 1.29E-08 | |
F13 | Mean | 5.12E-08 | 9.81E-08 | 2.00E-08 | 6.78E-08 | 3.39E-08 | 1.36E-07 | 1.31E-08 | 3.03E-08 | 3.99E-08 | 3.50E-08 | 2.29E-08 |
Std | 1.50E-07 | 3.33E-07 | 4.66E-08 | 1.60E-07 | 5.96E-08 | 3.90E-07 | 2.01E-08 | 5.49E-08 | 6.15E-08 | 5.93E-08 | 5.03E-08 | |
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 | 3.08E-04 | 3.08E-04 | 3.08E-04 | 3.18E-04 | 3.08E-04 | 3.08E-04 | 3.08E-04 | 3.08E-04 | 3.08E-04 | 3.08E-04 | 3.08E-04 |
Std | 4.44E-08 | 9.86E-07 | 4.64E-07 | 5.56E-05 | 1.27E-06 | 5.17E-07 | 2.58E-06 | 4.41E-07 | 8.24E-07 | 3.34E-06 | 2.78E-07 | |
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 | 5.53E-16 | 6.12E-16 | 5.90E-16 | 5.76E-16 | 5.53E-16 | 5.98E-16 | 5.68E-16 | 5.61E-16 | 5.98E-16 | 6.12E-16 | 6.12E-16 | |
F17 | Mean | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 |
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 | 2.87E-15 | 2.11E-15 | 4.93E+00 | 2.11E-15 | 1.84E-15 | 8.24E+00 | 4.93E+00 | 9.34E+00 | 2.51E-15 | 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 | 2.42E-15 | 2.39E-15 | 2.39E-15 | 2.34E-15 | 2.39E-15 | 2.37E-15 | 2.36E-15 | 2.39E-15 | 2.40E-15 | 2.34E-15 | 2.48E-15 | |
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 | 5.70E-02 | 6.03E-02 | 5.99E-02 | 5.83E-02 | 6.05E-02 | 5.92E-02 | 5.99E-02 | 5.54E-02 | 5.54E-02 | 6.05E-02 | 5.83E-02 | |
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 | 9.70E-01 | 2.61E+00 | 2.65E+00 | 1.62E+00 | 1.35E+00 | 2.39E+00 | 9.70E-01 | 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 | 9.27E-04 | 1.87E+00 | 2.43E+00 | 9.87E-01 | 1.37E+00 | 1.87E+00 | 1.87E+00 | 7.86E-07 | 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 | SSA-1 | SSA-2 | SSA-3 | SSA-4 | SSA-5 | SSA-6 | SSA-7 | SSA-8 | SSA-9 | SSA-10 |
---|---|---|---|---|---|---|---|---|---|---|---|
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. CLSSA-1 removes both logarithmic spiral strategy and adaptive step strategy; CLSSA-2 removes chaotic map and adaptive step strategy at the same time; CLSSA-3 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 | CLSSA-1 | CLSSA-2 | CLSSA-3 | CLSSA |
---|---|---|---|---|---|
F1 | 5.16E-109 | 0.00E+00 | 6.10E-157 | 5.56E-235 | 6.90E-201 |
F2 | 6.02E-67 | 1.26E-144 | 2.07E-70 | 1.56E-124 | 1.40E-110 |
F3 | 4.72E-95 | 2.73E-219 | 3.83E-109 | 1.15E-234 | 7.26E-159 |
F4 | 1.42E-69 | 9.39E-156 | 1.07E-79 | 5.59E-138 | 5.99E-100 |
F5 | 1.02E-04 | 3.44E-04 | 1.94E-06 | 3.06E-04 | 1.79E-05 |
F6 | 5.09E-08 | 6.23E-08 | 4.87E-10 | 9.87E-07 | 2.97E-09 |
F7 | 5.31E-04 | 2.08E-04 | 3.80E-04 | 2.79E-04 | 3.09E-04 |
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 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 |
F11 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |
F12 | 1.73E-09 | 3.65E-08 | 2.36E-11 | 4.41E-08 | 7.08E-10 |
F13 | 6.66E-08 | 7.19E-08 | 7.27E-09 | 5.72E-07 | 1.70E-09 |
F14 | 5.80E+00 | 3.01E+00 | 1.20E+00 | 2.37E+00 | 1.52E+00 |
F15 | 3.08E-04 | 3.08E-04 | 3.35E-04 | 3.08E-04 | 3.08E-04 |
F16 | −1.03E+00 | −1.03E+00 | −1.03E+00 | −1.03E+00 | −1.03E+00 |
F17 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 |
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 fixed-dimensional 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 | CLSSA-1 | CLSSA-2 | CLSSA-3 | 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 | 2.0E-42 | 6.9E+02 | 2.1E-02 | 9.9E-120 | 3.9E+01 | 4.9E-63 | 1.1E-18 | 1.1E+02 | 2.1E+00 | 1.2E+02 | 2.8E+00 | 4.4E+03 | 2.8E-204 |
Mean | 9.7E-52 | 2.8E+02 | 4.4E-04 | 0.0E+00 | 4.5E-17 | 3.8E-79 | 3.9E-20 | 2.0E+00 | 1.2E+00 | 2.8E+01 | 1.4E+00 | 2.2E+03 | 0.0E+00 | |
Std | 1.1E-41 | 4.5E+02 | 4.2E-02 | 5.4E-119 | 4.1E+01 | 2.6E-62 | 9.0E-19 | 1.4E+02 | 5.7E-01 | 7.7E+01 | 7.4E-01 | 1.5E+03 | 0.0E+00 | |
F2 | Best | 1.0E-29 | 4.9E+00 | 2.3E-01 | 2.9E-67 | 2.7E-02 | 8.6E-33 | 1.9E-11 | 1.2E-01 | 1.9E+01 | 2.1E+01 | 5.3E-01 | 5.6E+01 | 1.2E-105 |
Mean | 3.0E-34 | 4.9E-01 | 1.1E-02 | 0.0E+00 | 2.8E-08 | 3.7E-39 | 4.2E-12 | 1.6E-02 | 5.8E-01 | 2.8E+00 | 3.4E-01 | 3.3E+01 | 0.0E+00 | |
Std | 2.2E-29 | 4.1E+00 | 2.0E-01 | 1.6E-66 | 8.2E-02 | 4.4E-32 | 1.2E-11 | 1.3E-01 | 3.8E+01 | 1.7E+01 | 6.5E-02 | 1.3E+01 | 6.5E-105 | |
F3 | Best | 4.4E+04 | 5.3E+05 | 6.3E+02 | 3.2E-94 | 8.9E+02 | 3.6E-53 | 1.3E-03 | 1.1E+04 | 3.5E+02 | 2.2E+04 | 1.1E+03 | 4.5E+03 | 1.4E-144 |
Mean | 2.6E+04 | 5.4E+04 | 1.3E+02 | 0.0E+00 | 3.6E+02 | 3.5E-64 | 4.6E-06 | 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 | 1.7E-93 | 3.2E+02 | 1.9E-52 | 2.2E-03 | 7.3E+03 | 1.4E+02 | 1.1E+04 | 5.5E+02 | 1.5E+03 | 7.9E-144 | |
F4 | Best | 5.5E+01 | 9.5E+00 | 5.3E+00 | 6.2E-65 | 7.2E+00 | 2.8E-33 | 9.8E-05 | 3.6E+01 | 2.5E+00 | 6.3E+01 | 1.6E+00 | 3.5E+01 | 5.2E-117 |
Mean | 6.1E-01 | 3.6E+00 | 2.6E+00 | 0.0E+00 | 3.9E+00 | 3.0E-39 | 1.9E-05 | 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 | 3.4E-64 | 1.6E+00 | 1.2E-32 | 6.5E-05 | 1.0E+01 | 1.2E+00 | 8.8E+00 | 2.1E-01 | 3.7E+00 | 1.8E-116 | |
F5 | Best | 2.9E+01 | 2.4E+03 | 1.1E+02 | 1.0E-04 | 1.2E+02 | 1.1E-02 | 2.7E+01 | 2.2E+05 | 4.5E+02 | 5.4E+06 | 2.3E+02 | 1.6E+06 | 6.1E-07 |
Mean | 2.8E+01 | 2.0E+02 | 2.3E+01 | 3.1E-08 | 2.6E+01 | 1.2E-03 | 2.5E+01 | 1.1E+03 | 4.5E+01 | 5.0E+03 | 5.8E+01 | 4.4E+05 | 3.2E-27 | |
Std | 3.0E-01 | 3.0E+03 | 8.0E+01 | 1.6E-04 | 7.2E+01 | 1.8E-02 | 8.2E-01 | 5.1E+05 | 6.2E+02 | 2.9E+07 | 2.1E+02 | 8.3E+05 | 1.6E-06 | |
F6 | Best | 1.8E+00 | 6.6E+02 | 2.7E-02 | 4.4E-08 | 4.6E+01 | 8.3E-05 | 5.6E-01 | 8.7E+01 | 2.2E+00 | 1.5E+03 | 2.9E+00 | 4.1E+03 | 4.6E-10 |
Mean | 8.4E-01 | 2.4E+02 | 2.3E-04 | 2.2E-11 | 4.8E-17 | 7.6E-07 | 7.0E-05 | 9.6E+00 | 9.1E-01 | 3.7E+01 | 1.7E+00 | 2.4E+03 | 0.0E+00 | |
Std | 6.4E-01 | 3.0E+02 | 3.3E-02 | 8.1E-08 | 4.3E+01 | 8.9E-05 | 3.4E-01 | 1.3E+02 | 6.2E-01 | 4.4E+03 | 7.0E-01 | 8.1E+02 | 1.7E-09 | |
F7 | Best | 4.7E-03 | 7.2E-02 | 2.9E-02 | 6.7E-04 | 4.5E-02 | 1.5E-04 | 2.2E-03 | 2.1E-01 | 4.0E-02 | 5.3E-01 | 1.3E-02 | 1.1E+00 | 3.2E-04 |
Mean | 2.1E-05 | 1.0E-02 | 1.5E-02 | 3.1E-05 | 1.1E-02 | 1.0E-05 | 3.1E-04 | 1.6E-02 | 1.2E-02 | 1.2E-01 | 4.2E-03 | 3.7E-01 | 2.0E-05 | |
Std | 6.1E-03 | 5.2E-02 | 1.1E-02 | 5.7E-04 | 2.5E-02 | 1.2E-04 | 9.5E-04 | 1.9E-01 | 1.4E-02 | 8.1E-01 | 4.4E-03 | 4.2E-01 | 2.7E-04 | |
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 | 5.1E-13 | 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 | 6.7E-15 | 4.8E+00 | 1.3E+00 | 8.9E-16 | 1.6E-03 | 8.9E-16 | 2.2E-10 | 1.4E+01 | 2.0E+00 | 1.4E+01 | 6.5E-01 | 1.3E+01 | 8.9E-16 |
Mean | 8.9E-16 | 2.1E+00 | 3.2E-02 | 8.9E-16 | 4.5E-09 | 8.9E-16 | 8.6E-11 | 2.2E-01 | 6.4E-01 | 2.7E+00 | 3.6E-01 | 7.2E+00 | 8.9E-16 | |
Std | 4.2E-15 | 2.0E+00 | 9.5E-01 | 0.0E+00 | 6.2E-03 | 0.0E+00 | 1.0E-10 | 8.0E+00 | 4.5E-01 | 7.0E+00 | 8.8E-02 | 2.2E+00 | 0.0E+00 | |
F11 | Best | 7.4E-18 | 2.2E+02 | 5.3E-02 | 0.0E+00 | 1.1E+02 | 0.0E+00 | 6.9E-03 | 1.7E+00 | 9.7E-01 | 7.9E+00 | 1.0E+00 | 3.9E+01 | 0.0E+00 |
Mean | 0.0E+00 | 1.5E+02 | 3.7E-03 | 0.0E+00 | 8.3E+01 | 0.0E+00 | 0.0E+00 | 6.6E-01 | 8.4E-01 | 1.5E+00 | 9.8E-01 | 2.4E+01 | 0.0E+00 | |
Std | 2.8E-17 | 3.0E+01 | 7.4E-02 | 0.0E+00 | 1.6E+01 | 0.0E+00 | 8.6E-03 | 1.1E+00 | 4.4E-02 | 2.3E+01 | 2.7E-02 | 8.1E+00 | 0.0E+00 | |
F12 | Best | 9.3E-02 | 1.8E+00 | 9.3E-01 | 2.8E-09 | 1.8E+00 | 6.3E-06 | 3.8E-02 | 3.5E+05 | 2.5E+00 | 8.5E+06 | 1.1E-02 | 6.7E+04 | 4.9E-11 |
Mean | 3.2E-02 | 3.5E-01 | 3.8E-03 | 1.2E-11 | 3.7E-01 | 5.6E-08 | 7.0E-03 | 1.2E+00 | 4.3E-01 | 6.6E+00 | 4.0E-03 | 2.5E+02 | 2.4E-28 | |
Std | 9.0E-02 | 1.4E+00 | 8.1E-01 | 7.4E-09 | 1.0E+00 | 7.1E-06 | 2.1E-02 | 9.1E+05 | 1.6E+00 | 4.7E+07 | 1.9E-02 | 9.2E+04 | 1.5E-10 | |
F13 | Best | 1.3E+00 | 2.5E+01 | 6.4E-01 | 2.4E-08 | 1.5E+01 | 5.7E-05 | 4.7E-01 | 1.6E+06 | 2.1E-01 | 3.9E+03 | 1.4E-01 | 1.7E+06 | 3.6E-09 |
Mean | 7.7E-01 | 5.4E+00 | 1.3E-03 | 1.9E-11 | 6.3E-02 | 7.1E-08 | 2.5E-04 | 1.1E+01 | 9.4E-02 | 3.5E+01 | 7.8E-02 | 2.5E+05 | 2.9E-22 | |
Std | 4.4E-01 | 1.4E+01 | 8.6E-01 | 5.6E-08 | 7.1E+00 | 7.2E-05 | 2.3E-01 | 3.5E+06 | 9.1E-02 | 5.6E+03 | 3.7E-02 | 1.4E+06 | 1.7E-08 | |
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 | 1.8E-16 | 5.8E-17 | 5.3E+00 | 4.2E+00 | 9.5E-01 | 3.8E+00 | 8.9E-01 | 6.7E-11 | 1.2E+00 | 3.9E+00 | 1.8E-03 | 2.5E+00 | |
F15 | Best | 1.1E-03 | 1.4E-03 | 5.6E-04 | 3.1E-04 | 6.0E-03 | 4.1E-04 | 4.4E-03 | 1.1E-03 | 5.4E-03 | 1.0E-03 | 2.4E-03 | 7.9E-04 | 3.1E-04 |
Mean | 3.1E-04 | 3.1E-04 | 3.1E-04 | 3.1E-04 | 1.8E-03 | 3.1E-04 | 3.1E-04 | 4.8E-04 | 4.9E-04 | 4.9E-04 | 3.8E-04 | 5.4E-04 | 3.1E-04 | |
Std | 6.3E-04 | 3.6E-03 | 3.8E-04 | 1.7E-06 | 4.1E-03 | 2.4E-04 | 8.1E-03 | 3.6E-04 | 8.4E-03 | 3.5E-04 | 4.9E-03 | 1.4E-04 | 2.4E-07 | |
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 | 1.8E-09 | 6.0E-16 | 6.3E-16 | 5.5E-16 | 5.0E-16 | 3.0E-10 | 3.4E-08 | 3.6E-05 | 7.3E-07 | 6.8E-16 | 4.1E-12 | 1.8E-07 | 5.5E-16 | |
F17 | Best | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 |
Mean | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | 4.0E-01 | |
Std | 2.6E-05 | 0.0E+00 | 0.0E+00 | 0.0E+00 | 0.0E+00 | 2.5E-05 | 1.3E-06 | 1.4E-03 | 1.7E-07 | 0.0E+00 | 2.1E-11 | 1.9E-09 | 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 | 1.5E-15 | 1.7E-15 | 4.9E+00 | 4.0E-15 | 3.4E-07 | 8.2E-05 | 4.4E-05 | 5.8E-06 | 1.4E-15 | 4.9E+00 | 8.5E-07 | 4.9E-15 | |
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 | 3.8E-03 | 3.7E-03 | 2.7E-15 | 2.3E-15 | 2.5E-03 | 3.7E-03 | 1.9E-03 | 1.6E-03 | 2.3E-06 | 2.7E-15 | 5.7E-14 | 1.2E-06 | 2.4E-15 | |
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 | 7.1E-02 | 3.6E-01 | 6.0E-02 | 6.0E-02 | 1.4E-15 | 1.1E-01 | 6.9E-02 | 2.5E-01 | 6.0E-02 | 6.2E-02 | 6.0E-02 | 1.5E-02 | 5.9E-02 | |
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 | 5.8E-15 | 3.3E+00 | 1.8E+00 | 3.6E+00 | 4.1E-03 | 1.3E+00 | 2.1E+00 | 3.1E+00 | 3.7E+00 | 3.6E+00 | 1.3E-01 | 5.4E-15 | |
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 | 1.4E-15 | 1.6E+00 | 1.9E-03 | 1.7E+00 | 3.2E+00 | 3.3E+00 | 3.6E+00 | 3.3E-01 | 1.1E-07 | |
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 | 8.9E-01 | 2.8E+00 | 1.5E+00 | 3.3E+00 | 3.7E+00 | 3.4E+00 | 3.4E-01 | 5.6E-09 |
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 low-dimensional functions. However, the optimization algorithm is prone to fail in solving high-dimensional complex function problems. Real-world optimization problems are mostly large-scale complex optimization problems. Therefore, to verify the performance of CLSSA in high-dimensional 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 | 7.29E-41 | 8.74E+03 | 1.94E+03 | 3.16E-113 | 4.31E+03 | 6.49E-62 | 2.34E-07 | 1.65E+04 | 2.62E+02 | 9.33E+04 | 2.17E+02 | 2.35E+04 | 2.04E-201 |
F2 | 4.08E-28 | 7.01E+01 | 5.46E+01 | 3.81E-69 | 1.42E+01 | 1.55E-33 | 4.77E-05 | 1.46E+01 | 2.96E+28 | 3.06E+02 | 9.41E+00 | 1.37E+06 | 4.62E-106 |
F3 | 8.91E+05 | 8.62E+07 | 6.17E+04 | 3.53E-88 | 1.62E+04 | 1.25E-40 | 1.93E+03 | 2.48E+05 | 7.26E+04 | 2.61E+05 | 6.26E+04 | 5.38E+04 | 8.17E-115 |
F4 | 8.26E+01 | 4.01E+01 | 4.11E+01 | 1.41E-71 | 1.54E+01 | 1.80E-32 | 2.84E+00 | 9.05E+01 | 5.62E+01 | 9.16E+01 | 1.95E+01 | 4.85E+01 | 6.51E-113 |
F5 | 9.84E+01 | 2.40E+05 | 4.86E+05 | 6.60E-04 | 9.81E+04 | 7.12E-02 | 9.81E+01 | 1.40E+08 | 1.76E+04 | 2.58E+08 | 5.12E+03 | 1.60E+07 | 2.77E-04 |
F6 | 1.16E+01 | 7.93E+03 | 2.03E+03 | 2.24E-06 | 4.26E+03 | 5.35E-04 | 9.91E+00 | 1.58E+04 | 2.64E+02 | 9.32E+04 | 2.11E+02 | 2.59E+04 | 1.29E-07 |
F7 | 3.74E-03 | 1.53E+00 | 2.32E+00 | 4.68E-04 | 1.62E+00 | 2.04E-04 | 1.04E-02 | 2.18E+02 | 6.50E-01 | 3.92E+02 | 1.15E-01 | 2.34E+01 | 2.84E-04 |
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 | 7.76E-15 | 9.03E+00 | 7.75E+00 | 8.88E-16 | 4.92E+00 | 8.88E-16 | 4.90E-05 | 1.87E+01 | 6.95E+00 | 1.99E+01 | 3.30E+00 | 1.26E+01 | 8.88E-16 |
F11 | 1.57E-02 | 1.00E+03 | 2.02E+01 | 0.00E+00 | 1.21E+03 | 0.00E+00 | 1.13E-02 | 1.54E+02 | 3.55E+00 | 7.75E+02 | 2.87E+00 | 2.38E+02 | 0.00E+00 |
F12 | 2.39E-01 | 7.44E+00 | 5.88E+02 | 1.85E-08 | 7.39E+00 | 2.79E-06 | 2.94E-01 | 4.34E+08 | 2.48E+01 | 4.14E+08 | 2.73E+00 | 4.66E+06 | 8.72E-09 |
F13 | 5.18E+00 | 4.84E+03 | 1.24E+05 | 1.70E-06 | 4.73E+02 | 2.45E-04 | 6.97E+00 | 7.62E+08 | 1.82E+02 | 9.06E+08 | 1.07E+01 | 3.34E+07 | 1.90E-07 |
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 | hHHO-SCA [ |
|
---|---|---|---|---|---|---|---|
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