Table 3.
Comparisons of VGHHO and six algorithms on 18 classical benchmark problems with 100 dimensions in Table 1
| Function | Index | BOA | SOA | HHO | AGDE | EEGWO | ISCA | VGHHO |
|---|---|---|---|---|---|---|---|---|
| f1 | Mean | 2.81E-11 | 2.88E-05 | 1.05E-96 | 3.67E + 02 | 0 | 0 | 0 |
| Std | 8.26E-13 | 1.61E-05 | 1.91E-96 | 7.32E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f2 | Mean | 1.16E + 47 | 7.31E-05 | 2.47E-48 | 1.58E + 01 | 1.95E-227 | 2.68E-208 | 0 |
| Std | 1.85E + 47 | 5.90E-05 | 3.76E-48 | 1.17E + 00 | 0 | 0 | 0 | |
| Ranking | 7 | 5 | 4 | 6 | 2 | 3 | 1 | |
| f3 | Mean | 1.46E-08 | 6.88E + 01 | 8.44E-48 | 5.88E + 01 | 4.36E-220 | 2.13E-204 | 0 |
| Std | 1.40E-09 | 2.19E + 01 | 6.62E-48 | 1.45E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 4 | 6 | 2 | 3 | 1 | |
| f4 | Mean | 9.89E + 01 | 9.81E + 01 | 3.15E-01 | 7.65E + 04 | 9.89E + 01 | 9.90E + 01 | 5.39E-02 |
| Std | 2.95E-02 | 4.10E-01 | 4.41E-01 | 1.99E + 04 | 2.78E-02 | 1.19E-02 | 8.93E-02 | |
| Ranking | 4 | 3 | 2 | 7 | 4 | 6 | 1 | |
| f5 | Mean | 1.82E-03 | 1.35E-02 | 7.20E-04 | 9.83E-01 | 3.72E-05 | 7.46E-05 | 2.00E-05 |
| Std | 1.88E-03 | 1.30E-02 | 5.92E-04 | 3.01E-01 | 3.26E-05 | 1.59E-04 | 2.86E-05 | |
| Ranking | 5 | 6 | 4 | 7 | 2 | 3 | 1 | |
| f6 | Mean | 2.78E-11 | 2.24E-05 | 2.24E-100 | 1.60E + 02 | 0 | 0 | 0 |
| Std | 3.99E-12 | 1.73E-05 | 5.13E-100 | 3.62E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f7 | Mean | 3.91E-13 | 4.14E-06 | 2.02E-124 | 2.40E-12 | 0 | 0 | 0 |
| Std | 4.73E-13 | 5.90E-06 | 3.25E-124 | 9.92E-12 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 4 | 6 | 1 | 1 | 1 | |
| f8 | Mean | 3.08E-11 | 1.69E-02 | 3.91E-93 | 6.59E + 05 | 0 | 0 | 0 |
| Std | 2.80E-12 | 3.79E-02 | 3.10E-93 | 1.21E + 05 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f9 | Mean | 1.68E + 02 | 5.08E + 00 | 0 | 5.94E + 02 | 0 | 0 | 0 |
| Std | 3.75E + 02 | 5.53E + 00 | 0 | 1.18E + 01 | 0 | 0 | 0 | |
| Ranking | 6 | 5 | 1 | 7 | 1 | 1 | 1 | |
| f10 | Mean | 1.27E-08 | 2.00E + 01 | 8.88E-16 | 4.29E + 00 | 8.88E-16 | 8.88E-16 | 8.88E-16 |
| Std | 1.68E-09 | 4.72E-04 | 0 | 1.71E-01 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 1 | 6 | 1 | 1 | 1 | |
| f11 | Mean | 2.95E-11 | 2.53E-02 | 0 | 4.22E + 00 | 0 | 0 | 0 |
| Std | 1.65E-11 | 2.84E-02 | 0 | 1.19E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 1 | 7 | 1 | 1 | 1 | |
| f12 | Mean | 2.44E-09 | 1.11E-03 | 2.52E-49 | 3.71E + 01 | 1.28E-228 | 2.14E-208 | 0 |
| Std | 2.06E-09 | 8.77E-05 | 2.06E-49 | 5.48E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 2 | 3 | 1 | |
| f13 | Mean | 3.14E-11 | 8.95E + 03 | 1.34E-94 | 2.53E + 02 | 0 | 0 | 0 |
| Std | 1.50E-11 | 1.32E + 02 | 2.22E-94 | 6.51E + 01 | 0 | 0 | 0 | |
| Ranking | 5 | 7 | 4 | 6 | 1 | 1 | 1 | |
| f14 | Mean | 3.34E-01 | 3.40E-01 | 1.52E-47 | 8.37E + 00 | 8.15E-52 | 0 | 0 |
| Std | 4.55E-02 | 5.48E-02 | 1.06E-47 | 1.31E + 00 | 1.62E-51 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 3 | 1 | 1 | |
| f15 | Mean | 7.01E-12 | 9.76E-09 | 5.50E-99 | 4.16E-01 | 0 | 0 | 0 |
| Std | 5.36E-12 | 8.24E-09 | 7.71E-99 | 5.14E-02 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f16 | Mean | 2.39E-11 | 5.85E-08 | 1.23E-98 | 2.38E + 00 | 0 | 0 | 0 |
| Std | 1.99E-12 | 5.72E-08 | 1.17E-98 | 1.11E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f17 | Mean | 3.14E-06 | 4.19E-01 | 2.11E-25 | 1.17E + 02 | 0 | 0 | 0 |
| Std | 1.50E-06 | 2.27E-01 | 1.32E-25 | 4.02E + 00 | 0 | 0 | 0 | |
| Ranking | 5 | 6 | 4 | 7 | 1 | 1 | 1 | |
| f18 | Mean | 1.60E-10 | 5.20E-20 | 0 | 2.27E-04 | 0 | 0 | 0 |
| Std | 1.63E-10 | 3.79E-20 | 0 | 2.67E-04 | 0 | 0 | 0 | |
| Ranking | 6 | 5 | 1 | 7 | 1 | 1 | 1 | |
| Average ranking | 5.17 | 5.89 | 3.22 | 6.72 | 1.50 | 1.72 | 1.00 | |
| Total ranking | 5 | 6 | 4 | 7 | 2 | 3 | 1 | |
The best value of each function is highlighted in bold in the table