Table 4.
Comparisons of VGHHO and six approaches for 18 classical benchmark problems with 1000 dimensions in Table 1
| Function | Index | BOA | SOA | HHO | AGDE | EEGWO | ISCA | VGHHO |
|---|---|---|---|---|---|---|---|---|
| f1 | Mean | 3.15E-11 | 8.02E-01 | 7.12E-95 | 5.17E + 05 | 0 | 0 | 0 |
| Std | 3.20E-12 | 2.80E-01 | 6.82E-95 | 4.78E + 04 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f2 | Mean | NA | 1.62E-02 | 2.12E-47 | NA | 0 | 1.26E-207 | 0 |
| Std | NA | 4.43E-03 | 4.73E-47 | NA | 0 | 0 | 0 | |
| Ranking | 6 | 5 | 4 | 6 | 1 | 3 | 1 | |
| f3 | Mean | 1.50E-08 | 9.97E + 01 | 8.92E-48 | 9.51E + 01 | 2.55E-214 | 1.46E-203 | 0 |
| Std | 8.56E-10 | 1.00E-01 | 6.99E-48 | 1.97E-01 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 4 | 6 | 2 | 3 | 1 | |
| f4 | Mean | 9.99E + 02 | 1.21E + 04 | 8.34E + 00 | 8.03E + 08 | 9.99E + 02 | 9.99E + 02 | 9.39E-01 |
| Std | 2.87E-02 | 1.04E + 04 | 7.02E + 00 | 3.51E + 07 | 4.99E-02 | 1.66E-02 | 6.05E-01 | |
| Ranking | 3 | 6 | 2 | 7 | 3 | 3 | 1 | |
| f5 | Mean | 2.33E-03 | 6.72E-01 | 1.23E-03 | 1.13E + 04 | 9.13E-05 | 9.99E-05 | 4.80E-05 |
| Std | 2.44E-04 | 2.45E-01 | 1.58E-03 | 2.39E + 03 | 6.66E-05 | 1.91E-04 | 3.96E-05 | |
| ranking | 5 | 6 | 4 | 7 | 2 | 3 | 1 | |
| f6 | Mean | 3.22E-11 | 5.23E + 00 | 2.56E-93 | 2.42E + 06 | 0 | 0 | 0 |
| Std | 3.07E-12 | 1.29E + 00 | 4.37E-93 | 2.30E + 05 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f7 | Mean | 7.50E-13 | 3.24E + 00 | 4.85E-124 | 1.60E-06 | 0 | 0 | 0 |
| Std | 6.08E-13 | 6.38E-01 | 5.33E-124 | 5.71E-07 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 4 | 6 | 1 | 1 | 1 | |
| f8 | Mean | 3.56E-11 | 4.27E + 04 | 5.97E-90 | 7.75E + 09 | 0 | 0 | 0 |
| Std | 2.44E-12 | 4.30E + 04 | 6.41E-90 | 7.38E + 08 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f9 | Mean | 1.89E + 03 | 5.04E + 01 | 0 | 1.15E + 04 | 0 | 0 | 0 |
| Std | 4.21E + 03 | 7.96E + 01 | 0 | 5.16E + 02 | 0 | 0 | 0 | |
| Ranking | 6 | 5 | 1 | 7 | 1 | 1 | 1 | |
| f10 | Mean | 1.28E-08 | 2.00E + 01 | 8.88E-16 | 1.72E + 01 | 8.88E-16 | 8.88E-16 | 8.88E-16 |
| Std | 7.25E-10 | 4.46E-05 | 0 | 2.81E-01 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 1 | 6 | 1 | 1 | 1 | |
| f11 | Mean | 3.12E-11 | 2.01E-01 | 0 | 4.22E + 00 | 0 | 0 | 0 |
| Std | 4.10E-12 | 1.63E-01 | 0 | 1.19E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 1 | 7 | 1 | 1 | 1 | |
| f12 | Mean | 4.90E-09 | 1.27E-01 | 2.89E-48 | 1.34E + 03 | 2.61E-218 | 8.06E-208 | 0 |
| Std | 2.17E-09 | 2.19E-01 | 2.37E-48 | 4.61E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 2 | 3 | 1 | |
| f13 | Mean | 3.59E-11 | 9.96E + 04 | 5.73E-94 | 4.57E + 04 | 0 | 0 | 0 |
| Std | 5.94E-12 | 6.95E + 01 | 4.60E-94 | 7.07E + 03 | 0 | 0 | 0 | |
| ranking | 5 | 7 | 4 | 6 | 1 | 1 | 1 | |
| f14 | Mean | 3.60E-01 | 8.80E-01 | 3.53E-46 | 8.43E + 01 | 3.15E-47 | 1.22E-58 | 0 |
| Std | 4.30E-02 | 1.30E-01 | 3.27E-46 | 2.92E + 00 | 6.52E-47 | 2.46E-58 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 3 | 2 | 1 | |
| f15 | Mean | 1.97E-11 | 2.18E-04 | 2.24E-97 | 2.24E + 02 | 0 | 0 | 0 |
| Std | 1.31E-11 | 2.49E-04 | 1.96E-97 | 2.48E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f16 | Mean | 2.88E-11 | 9.94E-03 | 7.85E-96 | 9.80E + 02 | 0 | 0 | 0 |
| Std | 2.25E-12 | 5.93E-03 | 6.53E-96 | 7.76E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f17 | Mean | 5.28E-06 | 3.80E + 03 | 3.72E-24 | 2.59E + 03 | 0 | 0 | 0 |
| Std | 6.03E-07 | 1.37E + 00 | 4.10E-24 | 7.81E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 4 | 6 | 1 | 1 | 1 | |
| f18 | Mean | 6.87E-14 | 4.80E-03 | 3.89E-291 | 5.49E + 00 | 0 | 0 | 0 |
| Std | 3.02E-14 | 5.59E-03 | 0 | 1.05E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| Average ranking | 5.00 | 6.17 | 3.39 | 6.67 | 1.39 | 1.61 | 1.00 | |
| Total ranking | 5 | 6 | 4 | 7 | 2 | 3 | 1 | |
NA represents no available solution
The best value of each function is highlighted in bold in the table