Table 7 The comparison of obtained solutions for fixed-dimensional multimodal functions.
F | D | EI | PSO | SCA | WOA | HRO | GWO | MSGWO1 | I-GWO | MSGWO2 | HRO-GWO |
|---|---|---|---|---|---|---|---|---|---|---|---|
F7 | 2 | Best | 9.980e−01 | 9.980e−01 | 9.980e−01 | 9.980e−01 | 9.980e−01 | 9.980e−01 | 9.980e−01 | 9.980e−01 | 9.980e−01 |
Avg | 9.980e−01 | 1.064e+00 | 1.886e+00 | 9.980e−01 | 4.518e+00 | 1.296e+00 | 1.097e+00 | 3.844e+00 | 9.980e−01 | ||
Wor. | 9.980e−01 | 2.982e+00 | 1.096e+01 | 9.980e−01 | 1.267e+01 | 2.982e+00 | 2.982e+00 | 1.076e+01 | 9.980e−01 | ||
Std | 2.629e−07 | 3.562e−01 | 1.864e+00 | 1.900e−09 | 4.457e+00 | 4.324e−01 | 4.324e−01 | 3.570e+00 | 0.000e+00 | ||
F8 | 2 | Best | 3.000000 | 3.000000 | 3.000000 | 2.999999 | 3.000000 | 3.000000 | 3.000000 | 3.000000 | 2.999999 |
Avg | 3.055987 | 3.000007 | 3.000004 | 2.999999 | 3.000004 | 3.000006 | 3.000000 | 3.000001 | 2.999999 | ||
Wor. | 3.341624 | 3.000030 | 3.000035 | 2.999999 | 3.000017 | 3.000031 | 3.000000 | 3.000004 | 2.999999 | ||
Std | 8.164e−02 | 8.581e−06 | 7.592e−06 | 5.790e−16 | 3.672e−06 | 7.628e−06 | 2.329e−07 | 9.602e−07 | 3.765e−16 | ||
F9 | 3 | Best | − 3.862782 | − 3.862025 | − 3.862781 | − 3.862782 | − 3.862782 | − 3.862782 | − 3.862782 | − 3.862782 | − 3.862782 |
Avg | − 3.862781 | − 3.855774 | − 3.861263 | − 3.862782 | − 3.862186 | − 3.862183 | − 3.862037 | − 3.859235 | − 3.862782 | ||
Wor. | − 3.862778 | − 3.852876 | − 3.854901 | − 3.862782 | − 3.854901 | − 3.854901 | − 3.854930 | − 3.854900 | − 3.862782 | ||
Std | 2.559e−05 | 2.928e−03 | 2.450e−03 | 8.882e−16 | 1.941e−03 | 1.732e−03 | 2.193e−03 | 3.920e−03 | 4.200e−15 | ||
F10 | 6 | Best | − 3.315432 | − 3.170276 | − 3.322003 | − 3.322021 | − 3.322020 | − 3.322019 | − 3.322020 | − 3.322020 | − 3.322021 |
Avg | − 3.244581 | − 2.976402 | − 3.260889 | − 3.285593 | − 3.240311 | − 3.183589 | − 3.222477 | − 3.214057 | − 3.262561 | ||
Wor. | − 3.075045 | − 1.921499 | − 3.077863 | − 3.198163 | − 3.086691 | − 3.015834 | 3.133509 | − 3.086690 | − 3.203102 | ||
Std | 7.368e−02 | 2.404e−01 | 7.802e−02 | 5.473e−02 | 8.585e−02 | 8.967e−02 | 5.123e−02 | 8.442e−02 | 5.946e−02 | ||
F11 | 4 | Best | − 9.942076 | − 4.981549 | − 10.153152 | − 10.153200 | − 10.153185 | − 10.153112 | − 10.153138 | − 5.055197 | − 10.153200 |
Avg | − 7.672811 | − 2.985946 | − 9.642503 | − 7.162445 | −9.645591 | − 7.863472 | − 8.883117 | − 5.055195 | − 9.689436 | ||
Wor. | − 3.254147 | − 0.879667 | − 5.055189 | − 2.630472 | − 5.055196 | − 5.055190 | − 5.055197 | − 5.055193 | − 5.055198 | ||
Std | 1.845e+00 | 1.795e+00 | 1.529e+00 | 3.663e+00 | 1.523e+00 | 2.531e+00 | 2.200e+00 | 1.098e−06 | 1.814e+00 | ||
F12 | 4 | Best | − 10.536325 | − 8.444841 | − 10.536371 | − 10.536410 | − 10.536400 | − 10.536372 | − 10.536399 | − 10.536310 | − 10.536410 |
Avg | − 10.535102 | − 4.984155 | − 10.155362 | − 10.130943 | − 10.536323 | − 10.265787 | − 10.265851 | − 5.398869 | − 10.536328 | ||
Wor. | − 10.532792 | − 0.945504 | − 5.128472 | − 2.427335 | − 10.536210 | − 5.128479 | − 5.128478 | − 5.128474 | − 10.464117 | ||
Std | 1.024e−03 | 1.761e+00 | 1.345e+00 | 1.767e+00 | 4.951e−05 | 1.179e+00 | 1.179e+00 | 1.179e+00 | 1.564e−02 | ||
Rank | – | w/t/l | 0/4/14 | 0/1/17 | 0/1/17 | 1/12/5 | 1/2/15 | 0/1/17 | 0/2/16 | 0/2/16 | 4/12/2 |
