Table 2.
Experimental results obtained on benchmark test functions.
| Function/Dim/Measures | MHGS | HGS | CDO | SCSO | AOA | AGWO | WOA | GWO | PSO | DEAH | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| F1 | 30 | Mean | 0.00E + 00 | 3.27e-311 | 1.42E-261 | 4.50E-234 | 2.73E-108 | 3.39E-91 | 8.07E-150 | 3.51E-59 | 1.29E + 02 | 2.99e + 10 |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.49E-107 | 1.17E-90 | 4.36E-149 | 4.79E-59 | 1.34E + 01 | 6.8e + 09 | ||
| 100 | Mean | 0.00E + 00 | 4.90e-322 | 1.03E-236 | 1.36E-214 | 1.73E-02 | 1.55E-41 | 8.06E-146 | 2.65E-29 | 1.32E + 03 | 2.060E + 04 | |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 7.82E-03 | 3.62E-41 | 4.41E-145 | 3.32E-29 | 1.14E + 02 | 1.5E + 04 | ||
| 500 | Mean | 0.00E + 00 | 1.98E-306 | 2.05E-222 | 5.91E-204 | 5.95E-01 | 1.04E-18 | 4.90E-144 | 1.57E-12 | 3.30E + 04 | - | |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 4.08E-02 | 7.86E-19 | 2.59E-143 | 7.08E-13 | 2.16E + 03 | - | ||
| F2 | 30 | Mean | 0.00E + 00 | 2.33E-155 | 2.13E-133 | 3.95E-122 | 0.00E + 00 | 6.50E-55 | 1.31E-104 | 7.90E-35 | 6.54E + 01 | - |
| Std | 0.00E + 00 | 1.27E-154 | 4.39E-133 | 1.23E-121 | 0.00E + 00 | 1.22E-54 | 5.28E-104 | 7.60E-35 | 1.42E + 01 | - | ||
| 100 | Mean | 0.00E + 00 | 5.46E-162 | 2.04E-121 | 1.02E-110 | 5.61E-91 | 7.94E-27 | 2.60E-102 | 5.67E-18 | 2.05E + 21 | - | |
| Std | 0.00E + 00 | 2.99E-161 | 2.17E-121 | 5.37E-110 | 3.07E-90 | 8.05E-27 | 9.30E-102 | 2.60E-18 | 1.03E + 22 | - | ||
| 500 | Mean | 0.00E + 00 | 2.99E + 264 | 5.98E-113 | 1.90E-107 | 2.43E-04 | 1.52E-12 | 1.49E-100 | 5.78E-08 | 4.15E + 138 | - | |
| Std | 0.00E + 00 | Inf | 5.39E-113 | 9.96E-107 | 5.63E-04 | 8.25E-13 | 4.80E-100 | 1.70E-08 | 2.11E + 139 | - | ||
| F3 | 30 | Mean | 0.00E + 00 | 1.67E-170 | 1.52E-202 | 3.49E-197 | 7.84E-03 | 1.04E-24 | 2.06E + 04 | 6.81E-14 | 3.92E + 02 | 3.000E + 02 |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.49E-02 | 2.25E-24 | 1.01E + 04 | 3.31E-13 | 9.51E + 01 | 2.0E-05 | ||
| 100 | Mean | 0.00E + 00 | 3.12E-85 | 7.55E-122 | 4.78E-184 | 4.14E-01 | 1.02E + 02 | 8.39E + 05 | 1.57E + 01 | 2.40E + 04 | 3.000E + 02 | |
| Std | 0.00E + 00 | 1.71E-84 | 4.14E-121 | 0.00E + 00 | 2.14E-01 | 3.56E + 02 | 1.69E + 05 | 2.97E + 01 | 4.89E + 03 | 3.8E-02 | ||
| 500 | Mean | 0.00E + 00 | 1.79E-29 | 1.56E-16 | 1.18E-176 | 3.49E + 01 | 7.19E + 05 | 3.01E + 07 | 1.30E + 05 | 6.88E + 05 | - | |
| Std | 0.00E + 00 | 9.78E-29 | 8.53E-16 | 0.00E + 00 | 2.02E + 01 | 2.76E + 05 | 7.93E + 06 | 4.94E + 04 | 1.67E + 05 | - | ||
| F4 | 30 | Mean | 0.00E + 00 | 6.49E-132 | 9.47E-123 | 6.08E-100 | 1.80E-02 | 1.03E-23 | 3.34E + 01 | 1.68E-14 | 4.54E + 00 | 4.384E + 02 |
| Std | 0.00E + 00 | 3.56E-131 | 2.43E-122 | 3.19E-99 | 1.97E-02 | 3.82E-23 | 3.13E + 01 | 3.00E-14 | 3.18E-01 | 3.5E + 01 | ||
| 100 | Mean | 0.00E + 00 | 2.32E-146 | 1.27E-113 | 3.78E-96 | 8.85E-02 | 7.87E + 00 | 7.75E + 01 | 4.15E-03 | 1.54E + 01 | 6.399E + 02 | |
| Std | 0.00E + 00 | 8.66E-146 | 4.52E-113 | 1.43E-95 | 1.04E-02 | 2.13E + 01 | 2.13E + 01 | 1.19E-02 | 1.59E + 00 | 4.6E + 01 | ||
| 500 | Mean | 0.00E + 00 | 2.10E-139 | 6.79E-96 | 3.33E-93 | 1.71E-01 | 9.92E + 01 | 7.78E + 01 | 5.71E + 01 | 3.22E + 01 | - | |
| Std | 0.00E + 00 | 1.13E-138 | 3.50E-95 | 8.24E-93 | 1.06E-02 | 2.53E-01 | 2.45E + 01 | 6.31E + 00 | 3.01E + 00 | - | ||
| F5 | 30 | Mean | 2.82E-03 | 1.45E + 01 | 2.73E + 01 | 2.81E + 01 | 2.83E + 01 | 2.67E + 01 | 2.73E + 01 | 2.70E + 01 | 1.38E + 05 | 6.754E + 02 |
| Std | 4.59E-03 | 1.20E + 01 | 2.56E-01 | 9.69E-01 | 3.23E-01 | 4.67E-01 | 5.52E-01 | 7.37E-01 | 3.78E + 04 | 4.2E + 01 | ||
| 100 | Mean | 5.23E-02 | 6.02E + 01 | 9.81E + 01 | 9.84E + 01 | 9.88E + 01 | 9.79E + 01 | 9.77E + 01 | 9.74E + 01 | 5.58E + 06 | 1.397E + 03 | |
| Std | 1.19E-01 | 4.66E + 01 | 3.15E-01 | 4.54E-01 | 9.29E-02 | 5.54E-01 | 3.90E-01 | 6.84E-01 | 1.34E + 06 | 1.0E + 02 | ||
| 500 | Mean | 1.52E-01 | 3.95E + 02 | 4.97E + 02 | 4.98E + 02 | 4.99E + 02 | 4.98E + 02 | 4.96E + 02 | 4.98E + 02 | 2.63E + 08 | - | |
| Std | 4.33E-01 | 2.01E + 02 | 7.38E-02 | 2.90E-01 | 6.63E-02 | 1.69E-01 | 3.91E-01 | 3.99E-01 | 5.73E + 07 | - | ||
| F6 | 30 | Mean | 5.99E-08 | 8.96E-07 | 7.50E + 00 | 1.84E + 00 | 2.81E + 00 | 1.21E + 00 | 1.20E-01 | 6.70E-01 | 1.32E + 02 | 6.334E + 02 |
| Std | 1.05E-07 | 1.37E-06 | 0.00E + 00 | 5.58E-01 | 2.51E-01 | 3.43E-01 | 1.20E-01 | 3.30E-01 | 1.53E + 01 | 6.0E + 00 | ||
| 100 | Mean | 5.56E-05 | 8.16E-05 | 2.50E + 01 | 1.34E + 01 | 1.74E + 01 | 1.40E + 01 | 1.84E + 00 | 9.39E + 00 | 1.32E + 03 | 6.556E + 02 | |
| Std | 1.32E-04 | 1.56E-04 | 0.00E + 00 | 1.33E + 00 | 6.20E-01 | 5.55E-01 | 6.45E-01 | 8.92E-01 | 1.16E + 02 | 4.5E + 00 | ||
| 500 | Mean | 2.56E-04 | 3.68E-04 | 1.25E + 02 | 1.02E + 02 | 1.15E + 02 | 1.09E + 02 | 2.03E + 01 | 9.34E + 01 | 3.28E + 04 | - | |
| Std | 2.78E-04 | 5.95E-04 | 0.00E + 00 | 4.10E + 00 | 1.05E + 00 | 9.04E-01 | 6.64E + 00 | 2.23E + 00 | 2.70E + 03 | - | ||
| F7 | 30 | Mean | 3.31E-04 | 3.89E-04 | 4.47E-05 | 7.80E-05 | 2.59E-05 | 6.23E-04 | 1.82E-03 | 8.70E-04 | 9.66E + 01 | 1.113E + 03 |
| Std | 3.23E-04 | 6.14E-04 | 3.77E-05 | 1.06E-04 | 3.48E-05 | 6.18E-04 | 1.88E-03 | 4.62E-04 | 2.83E + 01 | 1.2E + 02 | ||
| 100 | Mean | 3.78E-04 | 5.02E-04 | 6.01E-05 | 8.83E-05 | 2.91E-05 | 2.00E-03 | 2.62E-03 | 2.78E-03 | 1.92E + 03 | 3.604E + 03 | |
| Std | 2.91E-04 | 7.45E-04 | 4.40E-05 | 1.11E-04 | 2.24E-05 | 1.15E-03 | 2.97E-03 | 1.27E-03 | 1.26E + 02 | 3.4E + 02 | ||
| 500 | Mean | 5.19E-04 | 6.69E-04 | 9.04E-05 | 1.35E-04 | 6.01E-05 | 9.65E-03 | 1.45E-03 | 1.08E-02 | 5.80E + 04 | - | |
| Std | 4.38E-04 | 9.01E-04 | 6.97E-05 | 2.11E-04 | 4.73E-05 | 5.43E-03 | 1.69E-03 | 2.86E-03 | 2.09E + 03 | - | ||
| F8 | 30 | Mean | −1.26E + 04 | −1.26E + 04 | −4.16E + 03 | −6.58E + 03 | −5.74E + 03 | −3.74E + 03 | −1.14E + 04 | −6.36E + 03 | −6.85E + 03 | 9.541E + 02 |
| Std | 5.17E-03 | 1.32E-01 | 3.20E + 02 | 9.41E + 02 | 4.21E + 02 | 3.08E + 02 | 1.49E + 03 | 8.08E + 02 | 7.60E + 02 | 4.1E + 01 | ||
| 100 | Mean | −4.19E + 04 | −4.19E + 04 | −6.74E + 03 | −1.99E + 04 | −1.10E + 04 | −6.70E + 03 | −3.72E + 04 | −1.66E + 04 | −2.22E + 04 | 1.772E + 03 | |
| Std | 4.98E + 00 | 9.86E + 00 | 6.02E + 02 | 2.16E + 03 | 7.93E + 02 | 4.18E + 02 | 5.27E + 03 | 1.31E + 03 | 1.87E + 03 | 1.6E + 02 | ||
| 500 | Mean | −2.09E + 05 | −2.09E + 05 | −1.37E + 04 | −6.96E + 04 | −2.45E + 04 | −1.50E + 04 | −1.88E + 05 | −5.81E + 04 | −9.68E + 04 | - | |
| Std | 1.95E + 01 | 5.90E + 01 | 1.18E + 03 | 5.61E + 03 | 1.36E + 03 | 1.10E + 03 | 2.55E + 04 | 1.33E + 04 | 1.79E + 04 | - | ||
| F9 | 30 | Mean | 0.00E + 00 | 0.00E + 00 | 1.37E + 02 | 0.00E + 00 | 0.00E + 00 | 1.89E-15 | 0.00E + 00 | 4.25E-01 | 3.43E + 02 | 4.413E + 03 |
| Std | 0.00E + 00 | 0.00E + 00 | 9.90E + 01 | 0.00E + 00 | 0.00E + 00 | 1.04E-14 | 0.00E + 00 | 1.32E + 00 | 2.37E + 01 | 1.3E + 03 | ||
| 100 | Mean | 0.00E + 00 | 0.00E + 00 | 6.50E + 00 | 0.00E + 00 | 0.00E + 00 | 2.65E-14 | 7.58E-15 | 1.31E + 00 | 1.40E + 03 | 2.822E + 04 | |
| Std | 0.00E + 00 | 0.00E + 00 | 6.12E + 00 | 0.00E + 00 | 0.00E + 00 | 7.12E-14 | 4.15E-14 | 2.81E + 00 | 5.89E + 01 | 4.5E + 03 | ||
| 500 | Mean | 0.00E + 00 | 0.00E + 00 | 5.41E + 00 | 0.00E + 00 | 7.55E-07 | 3.77E-06 | 0.00E + 00 | 6.09E + 00 | 7.47E + 03 | - | |
| Std | 0.00E + 00 | 0.00E + 00 | 4.56E + 00 | 0.00E + 00 | 2.32E-06 | 2.06E-05 | 0.00E + 00 | 5.85E + 00 | 1.47E + 02 | - | ||
| F10 | 30 | Mean | 8.88E-16 | 8.88E-16 | 4.44E-15 | 8.88E-16 | 8.88E-16 | 7.28E-15 | 4.56E-15 | 1.63E-14 | 8.53E + 00 | 4.712E + 03 |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.45E-15 | 1.74E-15 | 3.00E-15 | 3.11E-01 | 7.6E + 02 | ||
| 100 | Mean | 8.88E-16 | 8.88E-16 | 4.44E-15 | 8.88E-16 | 7.34E-05 | 2.01E-14 | 4.09E-15 | 1.09E-13 | 1.20E + 01 | 1.603E + 04 | |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 4.02E-04 | 4.44E-15 | 2.85E-15 | 9.22E-15 | 3.13E-01 | 1.4E + 03 | ||
| 500 | Mean | 8.88E-16 | 8.88E-16 | 4.44E-15 | 8.88E-16 | 7.65E-03 | 5.96E-11 | 4.20E-15 | 5.06E-08 | 1.64E + 01 | - | |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 5.55E-04 | 3.43E-11 | 2.46E-15 | 1.03E-08 | 4.56E-01 | - | ||
| F11 | 30 | Mean | 0.00E + 00 | 0.00E + 00 | 2.59E-03 | 0.00E + 00 | 1.02E-01 | 0.00E + 00 | 0.00E + 00 | 2.60E-03 | 1.03E + 00 | 1.279E + 03 |
| Std | 0.00E + 00 | 0.00E + 00 | 4.81E-03 | 0.00E + 00 | 6.24E-02 | 0.00E + 00 | 0.00E + 00 | 5.72E-03 | 8.75E-03 | 6.4E + 01 | ||
| 100 | Mean | 0.00E + 00 | 0.00E + 00 | 2.37E-03 | 0.00E + 00 | 4.47E + 02 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.34E + 00 | 2.503E + 03 | |
| Std | 0.00E + 00 | 0.00E + 00 | 4.73E-03 | 0.00E + 00 | 1.54E + 02 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 2.62E-02 | 3.5E + 02 | ||
| 500 | Mean | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 7.96E + 03 | 6.84E-04 | 0.00E + 00 | 1.79E-13 | 1.09E + 01 | - | |
| Std | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.81E + 03 | 3.75E-03 | 0.00E + 00 | 8.40E-14 | 7.29E-01 | - | ||
| F12 | 30 | Mean | 6.74E-10 | 8.53E-09 | 1.29E + 00 | 6.72E-02 | 3.75E-01 | 8.97E-02 | 6.62E-03 | 3.74E-02 | 4.88E + 00 | 1.926E + 04 |
| Std | 1.21E-09 | 1.14E-08 | 2.70E-01 | 2.64E-02 | 4.68E-02 | 2.06E-02 | 6.68E-03 | 2.41E-02 | 6.89E-01 | 1.4E + 04 | ||
| 100 | Mean | 1.32E-06 | 1.41E-06 | 1.27E + 00 | 3.14E-01 | 8.61E-01 | 4.50E-01 | 1.53E-02 | 2.42E-01 | 1.13E + 03 | 1.343E + 06 | |
| Std | 2.68E-06 | 3.20E-06 | 8.10E-02 | 7.71E-02 | 1.95E-02 | 5.92E-02 | 5.79E-03 | 5.28E-02 | 2.05E + 03 | 5.1E + 05 | ||
| 500 | Mean | 1.20E-06 | 1.99E-02 | 1.21E + 00 | 7.22E-01 | 1.07E + 00 | 9.31E-01 | 3.98E-02 | 7.45E-01 | 1.67E + 07 | - | |
| Std | 2.83E-06 | 1.09E-01 | 8.57E-03 | 6.27E-02 | 1.32E-02 | 1.79E-02 | 1.37E-02 | 2.77E-02 | 5.87E + 06 | - | ||
| F13 | 30 | Mean | 4.10E-08 | 3.25E-03 | 3.92E-01 | 2.33E + 00 | 2.80E + 00 | 1.05E + 00 | 2.59E-01 | 5.53E-01 | 2.33E + 01 | - |
| Std | 6.31E-08 | 1.78E-02 | 5.87E-02 | 3.73E-01 | 1.06E-01 | 2.33E-01 | 2.13E-01 | 2.39E-01 | 3.84E + 00 | - | ||
| 100 | Mean | 3.60E-05 | 3.55E-01 | 4.62E + 00 | 9.67E + 00 | 9.94E + 00 | 7.88E + 00 | 1.76E + 00 | 6.40E + 00 | 2.01E + 05 | - | |
| Std | 8.62E-05 | 1.35E + 00 | 3.34E-01 | 1.54E-01 | 5.89E-02 | 2.09E-01 | 7.74E-01 | 4.54E-01 | 1.05E + 05 | - | ||
| 500 | Mean | 9.98E-05 | 4.66E + 00 | 4.36E + 01 | 4.98E + 01 | 5.02E + 01 | 4.85E + 01 | 1.14E + 01 | 4.61E + 01 | 1.09E + 08 | - | |
| Std | 2.08E-04 | 1.42E + 01 | 9.81E-01 | 7.09E-02 | 2.36E-02 | 2.52E-01 | 4.28E + 00 | 5.95E-01 | 3.43E + 07 | - | ||
| F14 | 2 | Mean | 9.98E-01 | 2.95E + 00 | 1.27E + 01 | 4.72E + 00 | 1.00E + 01 | 2.57E + 00 | 1.69E + 00 | 3.41E + 00 | 3.16E + 00 | - |
| Std | 0.00E + 00 | 3.97E + 00 | 2.33E + 00 | 4.10E + 00 | 3.56E + 00 | 2.43E + 00 | 1.86E + 00 | 3.82E + 00 | 3.19E + 00 | - | ||
| F15 | 4 | Mean | 3.08E-04 | 5.84E-04 | 3.26E-04 | 3.60E-04 | 1.36E-02 | 2.41E-03 | 5.79E-04 | 2.98E-03 | 5.50E-03 | - |
| Std | 8.76E-08 | 2.61E-04 | 1.05E-05 | 1.72E-04 | 2.82E-02 | 6.09E-03 | 2.49E-04 | 6.93E-03 | 7.77E-03 | - | ||
| F16 | 2 | Mean | −1.03E + 00 | −1.03E + 00 | −1.00E + 00 | −1.03E + 00 | −1.03E + 00 | −1.03E + 00 | −1.03E + 00 | −1.03E + 00 | −1.03E + 00 | - |
| Std | 6.78E-16 | 6.78E-16 | 7.36E-03 | 1.40E-10 | 8.93E-08 | 2.55E-06 | 2.05E-10 | 5.64E-09 | 3.63E-03 | - | ||
| F17 | 2 | Mean | 3.98E-01 | 3.98E-01 | 4.01E-01 | 3.98E-01 | 4.05E-01 | 3.98E-01 | 3.98E-01 | 3.98E-01 | 3.99E-01 | - |
| Std | 0.00E + 00 | 0.00E + 00 | 1.43E-02 | 3.09E-08 | 5.58E-03 | 3.97E-04 | 1.14E-06 | 4.87E-07 | 6.67E-04 | - | ||
| F18 | 2 | Mean | 3.00E + 00 | 3.00E + 00 | 4.00E + 01 | 3.00E + 00 | 7.50E + 00 | 3.00E + 00 | 3.00E + 00 | 3.00E + 00 | 3.05E + 00 | - |
| Std | 1.57E-15 | 1.86E-15 | 3.67E + 01 | 9.29E-07 | 1.02E + 01 | 8.67E-06 | 1.38E-05 | 1.04E-05 | 4.74E-02 | - | ||
| F19 | 3 | Mean | −3.86E + 00 | −3.86E + 00 | −3.86E + 00 | −3.86E + 00 | −3.85E + 00 | −3.86E + 00 | −3.86E + 00 | −3.86E + 00 | −3.86E + 00 | - |
| Std | 2.71E-15 | 2.71E-15 | 1.36E-03 | 2.96E-03 | 2.83E-03 | 3.69E-03 | 3.26E-03 | 2.59E-03 | 2.89E-03 | - | ||
| F20 | 6 | Mean | −3.32E + 00 | −3.27E + 00 | −3.26E + 00 | −3.21E + 00 | −3.12E + 00 | −3.17E + 00 | −3.26E + 00 | −3.26E + 00 | −3.06E + 00 | - |
| Std | 1.36E-15 | 5.92E-02 | 4.28E-02 | 1.65E-01 | 7.13E-02 | 1.02E-01 | 8.16E-02 | 1.01E-01 | 1.12E-01 | - | ||
| F21 | 4 | Mean | −1.02E + 01 | −9.98E + 00 | −7.06E + 00 | −5.41E + 00 | −3.82E + 00 | −7.24E + 00 | −8.18E + 00 | −9.06E + 00 | −5.63E + 00 | - |
| Std | 6.39E-15 | 9.31E-01 | 1.23E + 00 | 2.52E + 00 | 1.17E + 00 | 1.56E + 00 | 2.67E + 00 | 2.26E + 00 | 1.56E + 00 | - | ||
| F22 | 4 | Mean | −1.04E + 01 | −1.04E + 01 | −6.98E + 00 | −5.89E + 00 | −4.08E + 00 | −8.44E + 00 | −8.07E + 00 | −1.02E + 01 | −5.80E + 00 | - |
| Std | 8.73E-16 | 9.90E-16 | 8.29E-01 | 2.08E + 00 | 1.67E + 00 | 1.16E + 00 | 2.95E + 00 | 9.70E-01 | 2.03E + 00 | - | ||
| F23 | 4 | Mean | −1.05E + 01 | −1.04E + 01 | −7.37E + 00 | −6.75E + 00 | −4.07E + 00 | −8.20E + 00 | −7.38E + 00 | −1.03E + 01 | −5.76E + 00 | - |
| Std | 8.73E-16 | 9.87E-01 | 1.20E + 00 | 2.52E + 00 | 9.86E-01 | 1.53E + 00 | 3.34E + 00 | 1.48E + 00 | 1.86E + 00 | - | ||