Table 4 The statistical results of unimodal and multimodal test functions.
From: A novel meta-heuristic algorithm based on candidate cooperation and competition
Function | Statistic | CCCA | PSO | FA | CSA | HMS | ICA | TLBO | BSO |
|---|---|---|---|---|---|---|---|---|---|
Best | 1.17E− 06 | 1.19E+02 | 5.58E+04 | 2.65E+02 | 1.37E−05 | 6.39E+03 | 1.67E+04 | 3.43E+03 | |
Ave | 2.59E− 06 | 2.98E+02 | 6.76E+04 | 4.26E+02 | 1.16E+00 | 7.98E+03 | 1.90E+04 | 5.70E+03 | |
Std | 1.04E− 06 | 1.12E+02 | 5.16E+03 | 1.16E+02 | 3.10E+00 | 1.68E+03 | 1.43E+03 | 1.42E+03 | |
Least | 4.66E− 06 | 4.60E+02 | 7.47E+04 | 6.57E+02 | 1.05E+01 | 1.20E+04 | 2.10E+04 | 8.28E+03 | |
\({{{F}}_{1}}(x)\) | Time (s) | 1.54 | 0.08 | 30.84 | 0.22 | 28.57 | 4.64 | 0.34 | 2.13 |
Best | 1.01E+00 | 1.41E+00 | 2.74E+03 | 5.86E+00 | 3.24E+00 | 3.11E+01 | 5.78E+01 | 2.58E+01 | |
Ave | 1.83E+00 | 2.06E+00 | 1.43E+07 | 8.45E+00 | 1.18E+01 | 4.99E+01 | 1.02E+02 | 3.28E+01 | |
Std | 3.63E− 01 | 6.14E−01 | 1.94E+07 | 1.32E+00 | 3.56E+00 | 9.93E+00 | 7.31E+01 | 4.80E+00 | |
Least | 2.33E+00 | 3.66E+00 | 6.08E+07 | 9.88E+00 | 1.60E+01 | 6.64E+01 | 3.13E+02 | 4.09E+01 | |
\({{{F}}_{2}}(x)\) | Time (s) | 1.52 | 0.09 | 31.75 | 0.22 | 29.11 | 4.71 | 0.39 | 2.31 |
Best | 1.59E+01 | 1.18E+05 | 3.16E+07 | 1.27E+05 | 3.15E−02 | 3.49E+06 | 8.46E+06 | 2.01E+06 | |
Ave | 3.24E+02 | 2.86E+05 | 3.85E+07 | 3.00E+05 | 1.43E+03 | 4.95E+06 | 1.07E+07 | 2.91E+06 | |
Std | 2.79E+02 | 1.04E+05 | 4.23E+06 | 1.33E+05 | 2.92E+03 | 9.27E+05 | 1.40E+06 | 6.21E+05 | |
Least | 8.84E+02 | 4.76E+05 | 4.49E+07 | 5.38E+05 | 9.85E+03 | 6.85E+06 | 1.28E+07 | 3.88E+06 | |
\({{{F}}_{3}}(x)\) | Time (s) | 1.62 | 0.09 | 31.17 | 0.22 | 29.35 | 4.67 | 0.33 | 2.14 |
Best | 7.82E+00 | 4.95E+00 | 8.10E+01 | 8.67E+00 | 1.27E− 120 | 2.95E+01 | 4.02E+01 | 2.17E+01 | |
Ave | 1.10E+01 | 7.68E+00 | 8.67E+01 | 1.10E+01 | 4.09E− 01 | 3.85E+01 | 4.32E+01 | 2.71E+01 | |
Std | 1.75E+00 | 1.25E+00 | 2.74E+00 | 1.47E+00 | 1.21E+00 | 5.79E+00 | 2.47E+00 | 4.13E+00 | |
Least | 1.38E+01 | 9.44E+00 | 9.05E+01 | 1.41E+01 | 4.05E+00 | 4.55E+01 | 4.82E+01 | 3.66E+01 | |
\({{{F}}_{4}}(x)\) | Time (s) | 1.65 | 0.09 | 31.16 | 0.23 | 28.64 | 4.85 | 0.37 | 2.23 |
Best | 2.18E+01 | 1.31E+04 | 1.11E+10 | 4.10E+05 | 2.90E+01 | 2.42E+08 | 1.22E+09 | 5.66E+07 | |
Ave | 3.53E+03 | 5.15E+04 | 2.16E+10 | 1.39E+06 | 9.40E+03 | 8.09E+08 | 2.17E+09 | 2.04E+08 | |
Std | 6.17E+03 | 4.72E+04 | 5.16E+09 | 5.78E+05 | 1.99E+04 | 3.69E+08 | 6.13E+08 | 1.06E+08 | |
Least | 1.97E+04 | 1.81E+05 | 2.84E+10 | 2.48E+06 | 6.77E+04 | 1.32E+09 | 3.24E+09 | 4.19E+08 | |
\({{{F}}_{5}}(x)\) | Time (s) | 1.68 | 0.09 | 30.84 | 0.24 | 28.26 | 4.73 | 0.32 | 2.17 |
Best | 1.18E− 06 | 8.18E+01 | 5.49E+04 | 2.94E+02 | 7.27E+00 | 5.91E+03 | 1.30E+04 | 3.89E+03 | |
Ave | 1.97E− 06 | 4.00E+02 | 6.23E+04 | 4.93E+02 | 1.43E+01 | 8.45E+03 | 1.89E+04 | 5.49E+03 | |
Std | 4.82E− 07 | 2.56E+02 | 5.57E+03 | 1.78E+02 | 1.68E+01 | 1.40E+03 | 2.35E+03 | 1.16E+03 | |
Least | 2.68E− 06 | 9.20E+02 | 7.18E+04 | 8.38E+02 | 6.43E+01 | 1.03E+04 | 2.14E+04 | 7.92E+03 | |
\({{{F}}_{6}}(x)\) | Time (s) | 1.49 | 0.09 | 32.72 | 0.21 | 28.44 | 4.73 | 0.32 | 2.14 |
Best | 1.32E− 03 | 4.59E−02 | 5.72E+01 | 1.27E−01 | 9.47E−03 | 1.27E+00 | 1.29E+01 | 8.18E−01 | |
Ave | 3.22E− 03 | 1.55E−01 | 7.76E+01 | 2.11E−01 | 8.51E−02 | 3.90E+00 | 2.01E+01 | 3.05E+00 | |
Std | 1.44E− 03 | 7.45E−02 | 1.16E+01 | 7.29E−02 | 5.01E−02 | 2.45E+00 | 3.89E+00 | 1.76E+00 | |
Least | 5.92E− 03 | 2.90E−01 | 9.49E+01 | 3.81E−01 | 1.77E−01 | 8.37E+00 | 2.42E+01 | 7.11E+00 | |
\({{{F}}_{7}}(x)\) | Time (s) | 2.65 | 0.26 | 39.34 | 0.82 | 28.69 | 5.24 | 0.75 | 2.44 |
Best | −1.12E+04 | −3.30E+03 | −3.22E+03 | −7.95E+03 | −4.46E+03 | −8.05E+03 | −5.97E+03 | −3.13E+03 | |
Ave | −9.87E+03 | −2.78E+03 | −2.58E+03 | −6.52E+03 | −4.24E+03 | −6.68E+03 | −4.93E+03 | −2.27E+03 | |
Std | 6.81E+02 | 2.89E+02 | 2.91E+02 | 7.42E+02 | 1.85E+02 | 8.16E+02 | 7.56E+02 | 4.52E+02 | |
Least | −8.66E+03 | −2.40E+03 | −2.24E+03 | −5.58E+03 | −3.89E+03 | −5.28E+03 | −3.14E+03 | −1.64E+03 | |
\({{{F}}_{8}}(x)\) | Time (s) | 2.11 | 0.12 | 33.88 | 0.38 | 28.51 | 5.01 | 0.43 | 2.32 |
Best | 5.00E+00 | 1.60E+01 | 3.01E+02 | 7.60E+01 | 0.00E+00 | 1.26E+02 | 2.41E+02 | 1.88E+02 | |
Ave | 1.05E+01 | 3.16E+01 | 3.67E+02 | 9.05E+01 | 2.00E− 01 | 2.00E+02 | 2.52E+02 | 2.35E+02 | |
Std | 3.50E+00 | 1.03E+01 | 3.49E+01 | 9.60E+00 | 4.00E− 01 | 2.89E+01 | 1.08E+01 | 2.51E+01 | |
Least | 1.80E+01 | 4.70E+01 | 4.11E+02 | 1.07E+02 | 1.00E+00 | 2.28E+02 | 2.70E+02 | 2.70E+02 | |
\({{{F}}_{9}}(x)\) | Time (s) | 1.96 | 0.11 | 33.41 | 0.35 | 28.63 | 4.74 | 0.42 | 2.65 |
Best | 3.36E+00 | 2.26E+00 | 2.00E+01 | 4.68E+00 | 8.88E− 16 | 1.37E+01 | 1.84E+01 | 1.17E+01 | |
Ave | 4.08E+00 | 2.70E+00 | 2.00E+01 | 6.06E+00 | 5.49E− 04 | 1.52E+01 | 1.93E+01 | 1.31E+01 | |
Std | 6.37E−01 | 4.03E−01 | 1.23E−03 | 6.62E−01 | 1.15E− 03 | 8.99E−01 | 3.61E−01 | 9.73E−01 | |
Least | 5.23E+00 | 3.39E+00 | 2.00E+01 | 6.90E+00 | 3.83E− 03 | 1.69E+01 | 1.96E+01 | 1.47E+01 | |
\({{{F}}_{10}}(x)\) | Time (s) | 2.00 | 0.11 | 33.46 | 0.37 | 28.53 | 4.79 | 0.43 | 2.24 |
Best | 1.20E−03 | 5.41E+00 | 4.61E+02 | 2.41E+00 | 1.20E− 39 | 3.41E+01 | 1.61E+02 | 2.94E+01 | |
Ave | 5.45E− 03 | 9.38E+00 | 6.03E+02 | 4.19E+00 | 1.25E−02 | 8.02E+01 | 1.78E+02 | 5.30E+01 | |
Std | 2.70E− 03 | 2.59E+00 | 6.91E+01 | 1.57E+00 | 2.04E−02 | 2.55E+01 | 1.49E+01 | 1.77E+01 | |
Least | 9.75E− 03 | 1.37E+01 | 6.78E+02 | 7.06E+00 | 5.55E−02 | 1.17E+02 | 2.13E+02 | 8.56E+01 | |
\({{{F}}_{11}}(x)\) | Time (s) | 1.85 | 0.12 | 34.59 | 0.44 | 29.05 | 4.78 | 0.47 | 2.21 |
Best | 6.71E+01 | 8.18E+00 | 2.14E+08 | 6.39E+01 | 3.14E+00 | 1.28E+05 | 9.94E+06 | 6.31E+02 | |
Ave | 1.16E+02 | 3.67E+01 | 3.94E+08 | 9.17E+01 | 5.26E+00 | 1.85E+06 | 1.87E+07 | 6.10E+04 | |
Std | 3.09E+01 | 2.56E+01 | 1.33E+08 | 2.14E+01 | 2.27E+00 | 2.06E+06 | 5.86E+06 | 1.70E+05 | |
Least | 1.61E+02 | 9.54E+01 | 6.30E+08 | 1.26E+02 | 9.63E+00 | 7.73E+06 | 3.33E+07 | 5.72E+05 | |
\({{{F}}_{12}}(x)\) | Time (s) | 2.17 | 0.12 | 40.83 | 0.48 | 31.52 | 4.96 | 0.78 | 2.29 |
Best | 1.10E+00 | 3.85E− 01 | 7.48E+08 | 1.34E+01 | 4.54E+00 | 5.11E+06 | 2.55E+07 | 2.76E+05 | |
Ave | 7.47E+00 | 4.78E+00 | 9.70E+08 | 3.75E+01 | 4.86E+00 | 1.02E+07 | 6.86E+07 | 2.13E+06 | |
Std | 8.16E+00 | 2.75E+00 | 1.31E+08 | 1.75E+01 | 2.21E− 01 | 2.71E+06 | 2.65E+07 | 1.74E+06 | |
Least | 2.81E+01 | 1.04E+01 | 1.20E+09 | 7.07E+01 | 5.24E+00 | 1.42E+07 | 1.25E+08 | 6.36E+06 | |
\({{{F}}_{13}}(x)\) | Time (s) | 2.34 | 0.13 | 41.98 | 0.63 | 28.89 | 5.08 | 0.77 | 2.34 |
