Table 5.
Results of single-peak and multi-peak benchmark functions.
| Function | Criteria | IWHO | WHO | MA | PSO | NGO | SSA | GWO | COOT |
|---|---|---|---|---|---|---|---|---|---|
| F1 | avg | 0 | 6.629 × 10−91 | 4.714 × 10−12 | 2.737 × 10−10 | 3.979 × 10−211 | 1.122 × 10−43 | 4.529 × 10−59 | 1.321 × 10−39 |
| std | 0 | 3.567 × 10−90 | 2.421 × 10−11 | 1.086 × 10−9 | 0 | 4.747 × 10−43 | 9.959 × 10−12 | 7.114 × 10−39 | |
| F2 | avg | 1.025 × 10−291 | 2.506 × 10−50 | 9.438 × 10−8 | 4.021 × 10−4 | 4.77 × 10−106 | 1.461 × 10−26 | 1.109 × 10−34 | 3.414 × 10−26 |
| std | 0 | 1.344 × 10−49 | 1.563 × 10−7 | 3.329 × 10−3 | 2.553 × 10−105 | 6.542 × 10−26 | 1.475 × 10−34 | 1.834 × 10−25 | |
| F3 | avg | 0 | 1.572 × 10−56 | 1.859 × 103 | 1.085 | 7.325 × 10−207 | 7.957 × 10−38 | 3.953 × 10−15 | 5.870 × 10−44 |
| std | 0 | 7.437 × 10−56 | 9.828 × 102 | 3.62 × 10−1 | 0 | 4.285 × 10−37 | 1.486 × 10−14 | 3.109 × 10−43 | |
| F4 | avg | 2.494 × 10−82 | 6.154 × 10−37 | 3.457 × 101 | 1.102 × 101 | 4.512 × 10−103 | 1.243 × 10−37 | 1.642 × 10−14 | 3.053 × 10−7 |
| std | 0 | 1.232 × 10−36 | 8.807 | 3.057 × 102 | 2.311 × 10−102 | 6.698 × 10−37 | 2.238 × 10−14 | 1.644 × 10−6 | |
| F5 | avg | 2.485 × 10−1 | 5.292 × 101 | 5.338 × 101 | 2.987 × 101 | 2.810 × 101 | 8.877 × 10−3 | 2.686 × 101 | 2.971 × 101 |
| std | 2.283 × 10−1 | 10.059 × 102 | 4.787 × 101 | 1.551 × 101 | 9.521 × 10−1 | 7.998 × 10−3 | 9.602 × 10−1 | 7.874 | |
| F6 | avg | 4.562 × 10−10 | 1.595 × 10−8 | 6.424 × 10−14 | 8.325 × 10−10 | 2.553 | 2.641 × 10−5 | 6.713 × 10−1 | 8.996 × 10−3 |
| std | 1.356 × 10−9 | 4.095 × 10−8 | 1.867 × 10−13 | 1.826 × 10−9 | 5.301 × 10−1 | 1.352 × 10−5 | 3.432 × 10−1 | 5.114 × 10−3 | |
| F7 | avg | 3.703 × 10−5 | 3.498 × 10−4 | 1.234 × 10−2 | 1.982 × 10−2 | 1.079 × 10−4 | 1.077 × 10−3 | 9.155 × 10−4 | 1.098 × 10−3 |
| std | 3.150 × 10−5 | 1.879 × 10−4 | 2.861 × 10−3 | 3.175 × 10−3 | 2.009 × 10−5 | 1.331 × 10−4 | 1.368 × 10−4 | 9.571 × 10−4 | |
| F8 | avg | −1.232 × 104 | −9.025 × 103 | −1.053 × 104 | −2.752 × 103 | −7.968 × 103 | −8.195 × 103 | −6.279 × 103 | −7.543 × 103 |
| std | 4.721 × 102 | 7.426 × 102 | 3.316 × 102 | 3.766 × 102 | 5.581 × 102 | 2.483 × 103 | 7.877 × 102 | 6.622 × 102 | |
| F9 | avg | 0 | 0 | 7.617 | 4.905 × 101 | 0 | 0 | 1.518 × 10−1 | 4.661 × 10−13 |
| std | 0 | 0 | 4.252 | 1.301 × 101 | 0 | 0 | 8.176 × 10−1 | 2.334 × 10−12 | |
| F10 | avg | 8.881 × 10−16 | 3.611 × 10−15 | 3.965 × 10−1 | 5.111 × 10−6 | 3.73 × 10−15 | 8.881 × 10−16 | 1.581 × 10−14 | 2.836 × 10−11 |
| std | 0 | 1.760 × 10−15 | 5.369 × 10−1 | 7.288 × 10−6 | 1.421 × 10−15 | 0 | 2.495 × 10−15 | 1.525 × 10−10 | |
| F11 | avg | 0 | 0 | 1.709 × 10−2 | 1.397 × 101 | 0 | 0 | 3.712 × 10−3 | 3.33 × 10−17 |
| std | 0 | 0 | 1.993 × 10−2 | 3.651 | 0 | 0 | 8.541 × 10−3 | 9.132 × 10−17 | |
| F12 | avg | 2.310 × 10−11 | 1.727 × 10−2 | 2.839 × 10−2 | 2.901 × 10−1 | 1.524 × 10−1 | 3.643 × 10−6 | 3.721 × 10−2 | 2.676 × 10−2 |
| std | 6.685 × 10−11 | 3.863 × 10−2 | 5.284 × 10−2 | 3.890 × 10−1 | 6.048 × 10−2 | 1.858 × 10−6 | 1.853 × 10−2 | 7.688 × 10−2 | |
| F13 | avg | 2.080 × 10−2 | 6.233 × 10−2 | 1.054 × 10−2 | 1.831 × 10−3 | 2.579 | 5.259 × 10−3 | 6.249 × 10−1 | 5.709 × 10−2 |
| std | 3.187 × 10−2 | 3.187 × 10−2 | 2.092 × 10−2 | 4.094 × 10−3 | 4.391 × 10−1 | 6.224 × 10−3 | 2.481 × 10−1 | 4.912 × 10−2 | |
| F14 | avg | 9.98 × 10−1 | 9.98 × 10−1 | 9.98 × 10−1 | 9.98 × 10−1 | 9.98 × 10−1 | 1.171 × 101 | 1.99 | 9.98 × 10−1 |
| std | 0 | 0 | 0 | 3.047 × 10−12 | 0 | 9.536 × 10−1 | 9.920 × 10−1 | 0 | |
| F15 | avg | 3.075 × 10−4 | 7.653 × 10−4 | 3.075 × 10−4 | 6.441 × 10−4 | 3.075 × 10−4 | 3.163 × 10−4 | 2.036 × 10−2 | 7.271 × 10−4 |
| std | 1.038 × 10−8 | 2.375 × 10−4 | 3.833 × 10−4 | 3.369 × 10−4 | 4.578 × 10−4 | 6.587 × 10−4 | 2.491 × 10−8 | 5.714 × 10−5 | |
| F16 | avg | −1.031 | −1.031 | −1.031 | −1.031 | −1.031 | −1.031 | −1.031 | −1.031 |
| std | 0 | 1.570 × 10−16 | 0 | 0 | 1.57 × 10−16 | 1.087 × 10−9 | 4.158 × 10−10 | 1.776 × 10−15 | |
| F17 | avg | 3.978 × 10−1 | 3.978 × 10−1 | 3.978 × 10−1 | 3.978 × 10−1 | 3.978 × 10−1 | 3.978 × 10−1 | 3.978 × 10−1 | 3.978 × 10−1 |
| std | 0 | 0 | 1.153 × 10−7 | 0 | 0 | 1.180 × 10−9 | 1.498 × 10−7 | 1.667 × 10−10 | |
| F18 | avg | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| std | 0 | 0 | 3.140 × 10−16 | 7.021 × 10−16 | 0 | 5.518 × 10−8 | 1.017 × 10−5 | 8.038 × 10−14 | |
| F19 | avg | −3.862 | −3.862 | −3.862 | −3.862 | −3.862 | −3.862 | −3.862 | −3.862 |
| std | 0 | 0 | 0 | 0 | 4.440 × 10−16 | 3.859 × 10−7 | 3.810 × 10−3 | 4.440 × 10−16 | |
| F20 | avg | −3.322 | −3.626 | −3.322 | −3.203 | −3.322 | −3.322 | −3.261 | −3.322 |
| std | 0 | 5.944 × 10−2 | 0 | 0 | 5.916 × 10−7 | 1.097 × 10−6 | 6.038 × 10−2 | 1.416 × 10−10 | |
| F21 | avg | −1.015 × 101 | 6.418 | −6.391 | −1.015 × 101 | −1.015 × 101 | −1.015 × 101 | −7.604 | −1.015 × 101 |
| std | 0 | 3.735 | 3.761 | 0 | 6.21 × 10−5 | 6.568 × 10−6 | 2.548 | 1.433 × 10−10 | |
| F22 | avg | −1.040 × 101 | −7.765 | −1.040 × 101 | −7.063 | −1.040 × 101 | −1.040 × 101 | −1.040 × 101 | −1.040 × 101 |
| std | 0 | 2.637 | 0 | 3.339 | 2.660 × 10−5 | 1.666 × 10−5 | 3.949 × 10−4 | 1.810 × 10−11 | |
| F23 | avg | −1.053 × 101 | −1.053 × 101 | −1.053 × 101 | −6.671 | −1.053 × 101 | −1.053 × 101 | −1.053 × 101 | −1.053 × 101 |
| std | 0 | 0 | 0 | 3.864 | 2.882 × 10−8 | 6.094 × 10−6 | 1.722 × 10−5 | 6.436 × 10−12 |