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. 2022 Oct 17;12:17387. doi: 10.1038/s41598-022-22458-9

Table 5.

Evaluation results on fixed-dimensional multimodal functions.

GA PSO GSA TLBO MVO GWO WOA TSA MPA RSA AVOA STBO
F14 Mean 0.998102 3.212919 3.564456 0.998007 0.998004 4.221422 3.88582 6.56528 0.998004 4.469522 1.14691 0.998004
Best 0.998004 0.998004 0.998004 0.998004 0.998004 0.998004 0.998004 0.998004 0.998004 1.002309 0.998004 0.998004
Worst 0.998721 10.76318 8.840836 0.998034 0.998004 12.67051 10.76318 12.67051 0.998004 10.76318 2.982105 0.998004
Std 0.000213 2.88548 2.189673 6.96E−06 7.30E−12 4.652866 4.146898 4.89171 1.82E−10 3.084637 0.485651 0
Median 0.998004 2.487068 2.806896 0.998004 0.998004 1.990054 1.495017 4.948548 0.998004 2.982105 0.998004 0.998004
Rank 5 7 8 4 2 10 9 12 3 11 6 1
F15 Mean 0.01273 0.001638 0.002156 0.003375 0.007576 0.001357 0.000693 0.008509 0.000307 0.001305 0.000429 0.000307
Best 0.000767 0.000307 0.000923 0.000309 0.000336 0.000307 0.000308 0.000308 0.000307 0.000727 0.000308 0.000307
Worst 0.026092 0.020363 0.00352 0.020364 0.020363 0.020363 0.002252 0.020942 0.000307 0.002601 0.001223 0.000307
Std 0.010579 0.004442 0.000483 0.007325 0.009629 0.004478 0.000523 0.010056 2.92E−19 0.000536 0.000231 2.99E−19
Med 0.012594 0.000307 0.002076 0.00032 0.000755 0.000308 0.000475 0.000779 0.000307 0.001149 0.000322 0.000307
Rank 12 7 8 9 10 6 4 11 2 5 3 1
F16 Mean − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.02846 − 1.03163 − 1.02844 − 1.03163 − 1.03163
Best − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03162 − 1.03163 − 1.03163
Worst − 1.03161 − 1.03163 − 1.03163 − 1.03162 − 1.03163 − 1.03163 − 1.03163 − 1 − 1.03163 − 1 − 1.03163 − 1.03163
Std 4.37E−06 1.14E−16 1.25E−16 2.49E−06 4.97E−08 3.27E−09 5.93E−11 0.009735 2.10E−10 0.007256 7.20E−15 7.75E−16
Median − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03163 − 1.03067 − 1.03163 − 1.03163
Rank 6 1 1 7 5 4 2 8 3 9 1 1
F17 Mean 0.524411 0.539013 0.397887 0.422207 0.397887 0.397888 0.397888 0.397906 0.397887 0.638307 0.397887 0.397887
Best 0.397887 0.397887 0.397887 0.39789 0.397887 0.397887 0.397887 0.397888 0.397887 0.397901 0.397887 0.397887
Worst 2.791186 2.791184 0.397887 0.882291 0.397888 0.397894 0.397891 0.397971 0.397887 5.040108 0.397887 0.397887
Std 0.534343 0.538701 0 0.108293 6.52E−08 1.63E−06 8.40E−07 2.48E−05 2.71E−09 1.036187 0 0
Med 0.397892 0.397887 0.397887 0.397972 0.397887 0.397888 0.397887 0.397894 0.397887 0.402551 0.397887 0.397887
Rank 8 9 1 7 3 5 4 6 2 10 1 1
F18 Mean 5.729191 3 3 3.000001 3 3.000011 3.000005 15.15005 3 6.169642 3.000002 3
Best 3.000044 3 3 3 3 3 3 3 3 3 3 3
Worst 30.53809 3 3 3.000003 3.000001 3.000033 3.000032 84.00069 3 39.23578 3.000012 3
Std 8.39291 2.76E−15 2.92E−15 8.96E−07 2.54E−07 9.56E−06 8.89E−06 29.67426 5.39E−14 9.865771 3.08E−06 1.81E−16
Median 3.001628 3 3 3 3 3.000009 3.000001 3.000008 3 3.000086 3.000001 3
Rank 10 2 3 6 5 9 8 12 4 11 7 1
F19 Mean − 3.86228 − 3.86278 − 3.86278 − 3.86203 − 3.86278 − 3.86096 − 3.8602 − 3.86273 − 3.86278 − 3.80846 − 3.86278 − 3.86278
Best − 3.86278 − 3.86278 − 3.86278 − 3.8627 − 3.86278 − 3.86278 − 3.86278 − 3.86278 − 3.86278 − 3.85487 − 3.86278 − 3.86278
Worst − 3.85745 − 3.86278 − 3.86278 − 3.8548 − 3.86278 − 3.8549 − 3.85378 − 3.86256 − 3.86278 − 3.68429 − 3.86278 − 3.86278
Std 0.001431 2.09E−15 1.87E−15 0.001716 7.41E−08 0.002948 0.002953 5.09E−05 2.28E−15 0.04789 3.22E−13 2.78E−15
Med − 3.86277 − 3.86278 − 3.86278 − 3.86249 − 3.86278 − 3.86246 − 3.86139 − 3.86275 − 3.86278 − 3.82667 − 3.86278 − 3.86278
Rank 5 1 1 6 3 7 8 4 1 9 2 1
F20 Mean − 3.19552 − 3.29822 − 3.322 − 3.23822 − 3.2446 − 3.23965 − 3.2753 − 3.23237 − 3.322 − 2.63831 − 3.28617 − 3.322
Best − 3.3214 − 3.322 − 3.322 − 3.31043 − 3.32199 − 3.32199 − 3.32198 − 3.32137 − 3.322 − 3.15625 − 3.322 − 3.322
Worst − 2.99692 − 3.2031 − 3.322 − 3.08169 − 3.20259 − 3.02064 − 3.10782 − 2.84 − 3.322 − 1.30322 − 3.19994 − 3.322
Std 0.093531 0.048793 3.95E−16 0.065246 0.058264 0.095719 0.074262 0.146905 4.20E−16 0.417228 0.056151 2.49E−16
Median − 3.18946 − 3.322 − 3.322 − 3.1998 − 3.20302 − 3.26252 − 3.32127 − 3.31998 − 3.322 − 2.73165 − 3.322 − 3.322
Rank 9 2 1 7 5 6 4 8 1 10 3 1
F21 Mean − 5.89083 − 5.77879 − 6.17737 − 5.84595 − 8.51163 − 9.64743 − 8.36636 − 6.52198 − 10.1532 − 5.0552 − 10.1532 − 10.1532
Best − 9.0381 − 10.1532 − 10.1532 − 9.44872 − 10.1532 − 10.1531 − 10.1529 − 10.138 − 10.1532 − 5.0552 − 10.1532 − 10.1532
Worst − 2.34247 − 2.63047 − 2.63047 − 3.80037 − 2.63047 − 5.10034 − 5.05374 − 2.60298 − 10.1532 − 5.0552 − 10.1532 − 10.1532
Std 2.512564 3.703566 3.699126 1.769566 2.623726 1.55506 2.493026 3.337297 1.95E−15 2.48E−07 6.57E−15 3.65E−15
Med − 6.83679 − 2.68286 − 3.51696 − 5.02319 − 10.1531 − 10.1527 − 10.1469 − 5.04462 − 10.1532 − 5.0552 − 10.1532 − 10.1532
Rank 8 10 7 9 4 3 5 6 1 11 2 1
F22 Mean − 7.21825 − 6.31807 − 10.4029 − 8.09591 − 9.6056 − 10.4024 − 8.0395 − 7.53629 − 10.4029 − 5.08767 − 10.4029 − 10.4029
Best − 10.1952 − 10.4029 − 10.4029 − 9.92173 − 10.4029 − 10.4028 − 10.4028 − 10.3998 − 10.4029 − 5.08767 − 10.4029 − 10.4029
Worst − 2.62184 − 1.83759 − 10.4029 − 4.21215 − 5.08765 − 10.4018 − 2.76572 − 1.82822 − 10.4029 − 5.08767 − 10.4029 − 10.4029
Std 2.472441 3.837031 2.97E−15 1.699295 1.947221 0.00028 3.034341 3.483042 3.65E−15 5.80E−07 2.41E−14 2.88E−15
Median − 7.89012 − 4.40599 − 10.4029 − 8.81648 − 10.4029 − 10.4025 − 10.3974 − 10.1566 − 10.4029 − 5.08767 − 10.4029 − 10.4029
Rank 9 10 2 6 5 4 7 8 1 11 3 1
F23 Mean − 5.78525 − 5.62285 − 10.5364 − 8.30576 − 9.99556 − 9.72457 − 8.77684 − 5.46265 − 10.5364 − 5.12847 − 10.5364 − 10.5364
Best − 10.417 − 10.5364 − 10.5364 − 9.98248 − 10.5364 − 10.5364 − 10.5364 − 10.4691 − 10.5364 − 5.12848 − 10.5364 − 10.5364
Worst − 2.38428 − 2.42173 − 10.5364 − 4.06348 − 5.12846 − 2.42172 − 2.42169 − 1.67573 − 10.5364 − 5.12847 − 10.5364 − 10.5364
Std 2.966829 3.755817 1.73E−15 1.491856 1.664511 2.497522 3.196353 3.753624 2.51E−15 1.29E−06 7.08E−15 6.93E−16
Med − 6.05259 − 3.35328 − 10.5364 − 8.76487 − 10.5364 − 10.536 − 10.5338 − 2.84687 − 10.5364 − 5.12847 − 10.5364 − 10.5364
Rank 8 9 1 7 4 5 6 10 3 11 2 1
Sum rank 80 58 33 68 46 59 57 85 21 98 30 10
Mean rank 8 5.8 3.3 6.8 4.6 5.9 5.7 8.5 2.1 9.8 3 1
Total rank 10 7 4 9 5 8 6 11 2 12 3 1