Research Article
Modified Atom Search Optimization Based on Immunologic Mechanism and Reinforcement Learning
Table 4
Test statistical results of functions f1∼f21.
| Functions | Algorithms | Min | Mean | Max | STD |
| f1(x) | MASO | 2.99E − 45 | 1.82E − 40 | 3.5E − 39 | 7.82E − 40 | ASO | 7.33E − 27 | 2.64E − 26 | 1.05E − 25 | 3.08E − 26 | FPA | 0.006587 | 0.013937 | 0.023855 | 0.004844 | BAT | 1.09E − 05 | 155.0339 | 891.9841 | 257.7465 | PSO | 3.869551 | 20.12067 | 77.25433 | 16.40947 | CS | 0.004142 | 173.506 | 1688.332 | 391.8062 | ACOR | 6.98E − 13 | 2.01E − 12 | 3.85E − 12 | 7.74E − 13 | CSA | 1.29E − 12 | 8.89E − 09 | 7.83E − 08 | 1.9E − 08 |
| f2(x) | MASO | 2.41E − 34 | 1.63E − 29 | 1.34E − 28 | 3.9E − 29 | ASO | 2.16E − 13 | 5.32E − 13 | 9.67E − 13 | 2.51E − 13 | FPA | 0.677651 | 1.159856 | 1.89272 | 0.340836 | BAT | 0.014345 | 42.11026 | 124.5488 | 48.56557 | PSO | 6.26162 | 9.511573 | 16.34641 | 2.636531 | CS | 0.024397 | 1.543266 | 8.754166 | 2.237715 | ACOR | 3.46E − 07 | 0.000387 | 0.006348 | 0.001411 | CSA | 9.9E − 07 | 4.2E − 05 | 0.000133 | 3.52E − 05 |
| f3(x) | MASO | 1.5E − 47 | 5.73E − 42 | 8.19E − 41 | 1.87E − 41 | ASO | 5.73E − 26 | 1.73E − 25 | 3.92E − 25 | 1.03E − 25 | FPA | 0.000957 | 0.001794 | 0.002651 | 0.000518 | BAT | 0.000105 | 0.000156 | 0.000193 | 2.85E − 05 | PSO | 1.681533 | 8.073872 | 24.85504 | 6.662011 | CS | 0.138672 | 63.08822 | 290.7739 | 80.09947 | ACOR | 7.82E − 14 | 2.16E − 13 | 4.42E − 13 | 1.09E − 13 | CSA | 1.27E − 12 | 1.63E − 09 | 1.89E − 08 | 4.14E − 09 |
| f4(x) | MASO | 0 | 0 | 0 | 0 | ASO | 0 | 0 | 0 | 0 | FPA | 0 | 0 | 0 | 0 | BAT | 4214 | 10180.7 | 16319 | 3108.576 | PSO | 57 | 191.7 | 405 | 90.82435 | CS | 9 | 1558.45 | 6328 | 1791.922 | ACOR | 0 | 0 | 0 | 0 | CSA | 0 | 0 | 0 | 0 |
| f5(x) | MASO | 2.34E − 06 | 1.47E − 05 | 3.73E − 05 | 9.24E − 06 | ASO | 0.003764 | 0.00772 | 0.011135 | 0.002189 | FPA | 0.006508 | 0.020989 | 0.029649 | 0.005437 | BAT | 0.218579 | 0.403253 | 0.531159 | 0.080489 | PSO | 0.273095 | 0.62667 | 1.206052 | 0.285222 | CS | 0.94235 | 4.144344 | 13.32052 | 2.939838 | ACOR | 0.016274 | 0.032401 | 0.053185 | 0.009913 | CSA | 3.99E − 06 | 5.29E − 05 | 0.000164 | 4.22E − 05 |
| f6(x) | MASO | 2.41E − 60 | 6.82E − 51 | 1.26E − 49 | 2.82E − 50 | ASO | 2.19E − 52 | 1.46E − 51 | 5.24E − 51 | 1.51E − 51 | FPA | 3.28E − 11 | 3.39E − 10 | 1.43E − 09 | 3.33E − 10 | BAT | 6.29E − 11 | 1.02E − 10 | 1.45E − 10 | 2.17E − 11 | PSO | 0.001601 | 0.010088 | 0.038612 | 0.010173 | CS | 0.000168 | 0.34018 | 2.726353 | 0.655415 | ACOR | 2.22E − 14 | 2.49E − 13 | 1.82E − 12 | 3.95E − 13 | CSA | 3.56E − 31 | 1.06E − 23 | 1.02E − 22 | 2.42E − 23 |
| f7(x) | MASO | 2.22E − 14 | 3.83E − 13 | 5.36E − 12 | 1.18E − 12 | ASO | 4.35E − 14 | 7.41E − 14 | 1.47E − 13 | 2.97E − 14 | FPA | 0.17994 | 0.618946 | 1.286978 | 0.260178 | BAT | 10.99074 | 13.41295 | 19.95889 | 1.88331 | PSO | 3.213753 | 5.385464 | 7.818361 | 1.282747 | CS | 8.631765 | 13.19331 | 16.18969 | 1.882678 | ACOR | 2.3E − 07 | 4.2E − 07 | 5.56E − 07 | 1.07E − 07 | CSA | 2.84E − 07 | 1.95E − 05 | 4.98E − 05 | 1.54E − 05 |
| f8(x) | MASO | 1.11E − 16 | 1.65E − 15 | 5.66E − 15 | 1.65E − 15 | ASO | 0 | 0 | 0 | 0 | FPA | 0.086888 | 0.136963 | 0.230776 | 0.044916 | BAT | 19.19178 | 53.02061 | 94.33961 | 21.77808 | PSO | 0.939324 | 1.178857 | 1.343306 | 0.102008 | CS | 0.610131 | 4.621765 | 33.27042 | 8.201352 | ACOR | 4.88E − 12 | 0.022291 | 0.247193 | 0.069061 | CSA | 3.57E − 10 | 4.25E − 08 | 4.24E − 07 | 9.87E − 08 |
| f9(x) | MASO | 0 | 4.97E − 13 | 3.18E − 12 | 9.06E − 13 | ASO | 9.949591 | 14.5264 | 20.89413 | 3.356089 | FPA | 53.45329 | 72.74809 | 84.86339 | 8.314181 | BAT | 38.80531 | 79.29986 | 169.1439 | 35.18297 | PSO | 75.51275 | 110.7482 | 169.454 | 25.42386 | CS | 30.82795 | 51.57561 | 84.51111 | 14.55878 | ACOR | 183.2672 | 214.6805 | 237.272 | 12.35372 | CSA | 1.76E − 12 | 2.82E − 09 | 7.92E − 09 | 2.7E − 09 |
| f10(x) | MASO | 0.000382 | 0.000382 | 0.000382 | 1.35E − 11 | ASO | 4106.132 | 4650.915 | 5626.077 | 521.8557 | FPA | 3244.947 | 3698.782 | 4219.299 | 237.132 | BAT | 5972.237 | 7099.381 | 8123.466 | 576.3557 | PSO | 3897.536 | 5375.416 | 6743.524 | 814.7786 | CS | 1848.715 | 2793.954 | 4026.17 | 610.2232 | ACOR | 1478.361 | 5880.244 | 7607.56 | 360.132 | CSA | 0.000382 | 0.000382 | 0.000382 | 7E − 08 |
| f11(x) | MASO | 3.44E − 33 | 1.83E − 05 | 9.49E − 05 | 3.11E − 05 | ASO | 4.23E − 14 | 8.76E − 14 | 1.97E − 13 | 4.72E − 14 | FPA | 3.227243 | 6.669646 | 9.146383 | 1.526006 | BAT | 0.834363 | 3.926399 | 9.459001 | 2.576305 | PSO | 2.748819 | 6.186047 | 10.35792 | 2.432746 | CS | 0.039447 | 0.854302 | 2.652989 | 0.811748 | ACOR | 0.034574 | 0.302129 | 5.21918 | 1.157372 | CSA | 7.64E − 08 | 5.4E − 06 | 2.31E − 05 | 5.86E − 06 |
| f12(x) | MASO | 1.15E − 31 | 6.29E − 28 | 1.18E − 26 | 2.63E − 27 | ASO | 2.78E − 27 | 6.39E − 27 | 2.1E − 26 | 5.59E − 27 | FPA | 1.055809 | 2.341216 | 3.399438 | 0.776936 | BAT | 3.166887 | 8.757063 | 15.72329 | 3.538368 | PSO | 1.089431 | 7.019646 | 16.18996 | 3.659522 | CS | 9.73E − 09 | 0.005653 | 0.104223 | 0.02323 | ACOR | 1.69E − 11 | 9.04E − 11 | 4.43E − 10 | 9.82E − 11 | CSA | 1.24E − 13 | 5.21E − 11 | 4.53E − 10 | 1.13E − 10 |
| f13(x) | MASO | 2.22E − 16 | 9.1E − 16 | 5.33E − 15 | 1.14E − 15 | ASO | 0.111929 | 0.171566 | 0.247934 | 0.04255 | FPA | 0.164183 | 0.23541 | 0.288289 | 0.035616 | BAT | 0.493711 | 0.498347 | 0.499585 | 0.001342 | PSO | 0.213232 | 0.336072 | 0.442089 | 0.05312 | CS | 0.470226 | 0.493116 | 0.498482 | 0.006452 | ACOR | 0.018867 | 0.036172 | 0.046726 | 0.006468 | CSA | 1.33E − 15 | 4.72E − 12 | 2.33E − 11 | 6.56E − 12 |
| f14(x) | MASO | 1.78E − 15 | 1.44E − 14 | 4.09E − 14 | 1.04E − 14 | ASO | 12 | 18.70114 | 35.0005 | 6.254622 | FPA | 58.21314 | 75.72416 | 92.27162 | 10.98064 | BAT | 142.0001 | 225.425 | 320.25 | 43.94807 | PSO | 51.61432 | 97.72042 | 154.1477 | 27.68505 | CS | 22.03245 | 65.54915 | 124.1758 | 23.27628 | ACOR | 159.1282 | 193.0334 | 215.8543 | 15.42857 | CSA | 5.74E − 11 | 5.9E − 09 | 2.73E − 08 | 8.01E − 09 |
| f15(x) | MASO | 9.9E − 13 | 7.77E − 09 | 8.32E − 08 | 2E − 08 | ASO | 13.22012 | 14.72793 | 15.51416 | 0.63515 | FPA | 23.19234 | 26.9907 | 29.18676 | 1.618782 | BAT | 0.095172 | 18.58866 | 73.64575 | 30.2613 | PSO | 305.1689 | 574.6557 | 1477.057 | 316.5058 | CS | 92.87089 | 4700.287 | 44298.9 | 10461.76 | ACOR | 24.81962 | 25.10222 | 25.36799 | 0.158517 | CSA | 2.24E − 11 | 1.35E − 09 | 5.65E − 09 | 1.97E − 09 |
| f16(x) | MASO | 3.12E − 32 | 1.12E − 29 | 2.02E − 28 | 4.51E − 29 | ASO | 3.53E − 29 | 8.02E − 29 | 1.31E − 28 | 3.4E − 29 | FPA | 0.795144 | 1.715761 | 2.908752 | 0.550818 | BAT | 0.519122 | 11.97827 | 28.55629 | 9.510974 | PSO | 3.03539 | 7.040329 | 11.93878 | 2.719544 | CS | 7.55E − 08 | 0.019757 | 0.366427 | 0.081707 | ACOR | 9.06E − 09 | 2.61E − 07 | 2.1E − 06 | 4.61E − 07 | CSA | 2.13E − 14 | 3.58E − 11 | 1.54E − 10 | 4.63E − 11 |
| f17(x) | MASO | 1.26E − 32 | 2.26E − 31 | 1.86E − 30 | 4.61E − 31 | ASO | 6.55E − 28 | 2.14E − 27 | 5.54E − 27 | 1.53E − 27 | FPA | 0.075634 | 0.159202 | 0.383942 | 0.076028 | BAT | 28.02352 | 59.78051 | 75.72239 | 10.54314 | PSO | 3.843779 | 71.81837 | 922.9239 | 200.6675 | CS | 3.36E − 22 | 8.9E − 22 | 1.77E − 21 | 4.31E − 22 | ACOR | 3.17E − 08 | 8.81E − 07 | 6.2E − 06 | 1.69E − 06 | CSA | 9.23E − 13 | 2.67E − 10 | 8.43E − 10 | 2.34E − 10 |
| f18(x) | MASO | 2.22E − 14 | 1.37E − 13 | 8.96E − 13 | 2.02E − 13 | ASO | 4.71E − 14 | 9.18E − 14 | 1.75E − 13 | 3.96E − 14 | FPA | 1.169421 | 1.510639 | 2.147189 | 0.28967 | BAT | 12.84255 | 14.03792 | 16.05812 | 0.884501 | PSO | 3.803351 | 5.659908 | 7.923137 | 0.986783 | CS | 14.20253 | 16.00982 | 17.58788 | 1.084051 | ACOR | 3.56E − 07 | 8.04E − 07 | 1.54E − 06 | 3.01E − 07 | CSA | 6.75E − 07 | 2.1E − 05 | 5.99E − 05 | 1.93E − 05 |
| f19(x) | MASO | 0 | 5E − 13 | 4.66E − 12 | 1.05E − 12 | ASO | 8.954632 | 16.01884 | 19.89918 | 3.492605 | FPA | 59.36055 | 76.74372 | 100.1136 | 10.82332 | BAT | 31.84028 | 84.16432 | 198.9923 | 34.74113 | PSO | 68.84252 | 109.8572 | 159.7136 | 22.65396 | CS | 65.6753 | 109.6626 | 154.4972 | 25.50229 | ACOR | 202.1726 | 219.1054 | 230.5434 | 8.423366 | CSA | 6.31E − 12 | 5.12E − 09 | 3.79E − 08 | 1.01E − 08 |
| f20(x) | MASO | 0 | 1.46E − 15 | 8.99E − 15 | 1.99E − 15 | ASO | 0 | 0 | 0 | 0 | FPA | 0.053602 | 0.127282 | 0.226275 | 0.05216 | BAT | 30.91076 | 58.58257 | 119.8216 | 20.73163 | PSO | 0.744758 | 1.095692 | 1.310903 | 0.116757 | CS | 0.868257 | 3.707799 | 19.43772 | 4.708682 | ACOR | 8.4E − 10 | 0.00311 | 0.062197 | 0.013908 | CSA | 2.6E − 13 | 3.35E − 08 | 1.31E − 07 | 4.47E − 08 |
| f21(x) | MASO | 1140.688 | 2142.009 | 2596.788 | 363.9108 | ASO | 4046.881 | 4920.749 | 5321.667 | 391.9594 | FPA | 2480.206 | 2843.365 | 3076.285 | 173.4922 | BAT | 3444.441 | 5378.728 | 7371.65 | 998.61 | PSO | 1906.464 | 4896.28 | 6944.173 | 1242.374 | CS | 3032.338 | 4125.281 | 5258.021 | 555.419 | ACOR | 2691.551 | 3918.742 | 5687.103 | 463.1189 | CSA | 7233.943 | 8295.476 | 9227.791 | 487.6725 |
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The best results are shown in bold.
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