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. 2024 Oct 22;14:24928. doi: 10.1038/s41598-024-76010-y

Table 2.

The best fitness value, mean, and standard deviation of the 15 benchmark functions of CEC-2005 across the 6 algorithms.

Fun Algorithm Best fitness Mean Std Fun Algorithm Best fitness Mean Std
F1 PSO5 3900.756 1.998 11.226 F9 PSO5 100.272 0.298 0.990
ACO9 2138.513 7.920e−13 2.688e−12 ACO9 99.489 − 9.769e−12 6.422e−10
FA11 2153.832 7.920e−13 2.688e−12 FA11 98.003 − 9.769e−12 6.422e−10
BOA12 581.290 0.581 23.898 BOA12 49.309 0.001 0.511
DAIW-PSO34 2138.650 1.891 8.229 DAIW-PSO34 100.262 0.298 0.990
HS-PSO 1.854e−22 2.227e−13 2.476e−12 HS-PSO 0.00e+00 9.077e−12 5.246e−10
F2 PSO5 11.7842 0.014 0.580 F10 PSO5 511.864 43.507 257.337
ACO9 11.7841 2.567e−15 2.828e−15 ACO9 381.367 2.293e−09 4.322e−08
FA11 4.905e+19 − 0.235 6.054 FA11 386.955 2.293e−09 4.322e−08
BOA12 76001877.206 0.440 16.403 BOA12 118.920 11.025 684.131
DAIW-PSO34 11.784 0.014 0.580 DAIW-PSO34 380.645 43.370 220.768
HS-PSO 1.114e−14 − 2.322e−16 3.567e−16 HS-PSO 0.00E+00 1.667e−09 8.642e−09
F3 PSO5 33779.437 − 1.295 35.923 F11 PSO5 3.075 − 0.938 4.339
ACO9 21422.058 2.286e−08 7.893e−07 ACO9 0.078 0.078 0.356
FA11 21363.828 2.286e−08 7.893e−07 FA11 1050073320.173 6.057 25.655
BOA12 55131.719 1.826 185.528 BOA12 5.087e+08 2.744 117.038
DAIW-PSO34 21461.451 − 0.054 30.561 DAIW-PSO34 3.075 − 0.937 4.340
HS-PSO 7.88e-15 − 2.015e−10 2.569e−08 HS-PSO − 1.022 0.078 0.356
F4 PSO5 21.258 − 0.384 12.412 F12 PSO5 48.609 0.539 3.493
ACO9 16.712 3.303e−07 3.334e−06 ACO9 47.483 1.013 0.030
FA11 98.569 10.388 51.512 FA11 49.184 1.013 0.030
BOA12 87.680 10.393 287.585 BOA12 1.360e+09 1.008 143.213
DAIW-PSO34 16.713 0.388 9.536 DAIW-PSO34 48.342 0.539 3.495
HS-PSO 1.153e−07 1.123e−08 5.682e−08 HS-PSO − 0.096 1.006 0.030
F5 PSO5 3730.416 0.314 0.984 F13 PSO5 15.512 0.145 1.621
ACO9 4728.786 0.044 0.130 ACO9 15.511 − 1.466e−13 1.273e−13
FA11 4589.090 0.044 0.130 FA11 72.774 − 0.126 4.851
BOA12 3082143.498 1.610 24.552 BOA12 3.707e−06 5.253e−07 7.838e−06
DAIW-PSO34 3719.717 0.308 0.980 DAIW-PSO34 15.511 0.145 1.621
HS-PSO 27.057 0.046 0.136 HS-PSO 1.147e−15 − 3.824e−16 2.044e−16
F6 PSO5 2143.357 − 0.346 8.451 F14 PSO5 − 1.032 0.311 0.401
ACO9 1102.599 − 0.485 0.021 ACO9 − 1.032 − 0.285 0.415
FA11 1113.206 − 0.485 0.021 FA11 69.492 − 0.486 2.232
BOA12 75814.068 − 13.174 266.449 BOA12 − 0.845 0.240 0.763
DAIW-PSO34 1103.467 − 1.597 5.965 DAIW-PSO34 − 1.032 0.311 0.401
HS-PSO 2.260e−08 − 0.500 1.865e−05 HS-PSO − 1.032 − 0.311 0.401
F7 PSO5 13.699 0.008 0.257 F15 PSO5 14.889 5.319 4.680
ACO9 40.118 0.025 0.112 ACO9 14.889 64.228 61.085
FA11 38.703 0.025 0.112 FA11 8088.353 64.171 61.029
BOA12 12.015 0.022 0.816 BOA12 15.073 5.304 6.640
DAIW-PSO34 13.388 0.008 0.257 DAIW-PSO34 14.889 5.320 4.680
HS-PSO 10.539 0.014 0.096 HS-PSO 3.927 1.715 0.943
F8 PSO5 − 2366.845 − 77.871 253.180 – – – – –
ACO9 77.858 156.620 281.447
FA11 − 1398.593 − 9.178 430.151
BOA12 − 2407.616 − 134.117 2420.034
DAIW-PSO34 − 2366.845 − 77.871 253.180
HS-PSO − 10792.911 59.223 361.747

The bold values represent the optimal value for each benchmark function.