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. 2025 Apr 12;15:12690. doi: 10.1038/s41598-025-96901-y

Table 6.

Results of HV metric for WFG problems.

Problem M D MaOALO MaOPSO MaOSOS MaOMVO NSGA-III MaOMPA
WFG1 5 14 0.52627 ± 0.0780 0.50258 ± 0.0659 0.52774 ± 0.0410 0.61054 ± 0.0492 0.50660 ± 0.0347 0.72182 ± 0.0269
7 16 0.48627 ± 0.0469 0.41244 ± 0.0872 0.51360 ± 0.0296 0.44390 ± 0.0226 0.47052 ± 0.0524 0.65822 ± 0.0433
WFG2 5 14 0.94283 ± 0.00944 0.89988 ± 0.0137 0.96760 ± 0.00931 0.96031 ± 0.00862 0.95891 ± 0.0114 0.98384 ± 0.00303
7 16 0.94875 ± 0.0113 0.90821 ± 0.0248 0.94967 ± 0.0120 0.92736 ± 0.0768 0.95593 ± 0.00720 0.98495 ± 0.00469
WFG3 5 14 0.071907 ± 0.0264 0.0093326 ± 0.00848 0.036545 ± 0.0263 0.065779 ± 0.0277 0.052587 ± 0.0129 0.079647 ± 0.0178
7 16 0.019762 ± 0.0197 0.00000 ± 0.0000 0.00000 ± 0.0000 0.00000 ± 0.0000 0.0011052 ± 0.00247 0.0094679 ± 0.0143
WFG4 5 14 0.67292 ± 0.00741 0.64072 ± 0.00959 0.69270 ± 0.00413 0.64616 ± 0.00935 0.70083 ± 0.00632 0.72634 ± 0.00336
7 16 0.75539 ± 0.0108 0.66824 ± 0.0193 0.71171 ± 0.140 0.66662 ± 0.0250 0.77400 ± 0.00693 0.80128 ± 0.0117
WFG5 5 14 0.64589 ± 0.00757 0.61640 ± 0.0115 0.67809 ± 0.00224 0.62291 ± 0.00359 0.68380 ± 0.00761 0.68990 ± 0.00199
7 16 0.71775 ± 0.00916 0.61391 ± 0.0178 0.74898 ± 0.0123 0.60511 ± 0.0243 0.75675 ± 0.00909 0.76552 ± 0.00319
WFG6 5 14 0.60728 ± 0.0207 0.56411 ± 0.0114 0.64800 ± 0.0131 0.61151 ± 0.0268 0.65092 ± 0.0204 0.66104 ± 0.0297
7 16 0.70430 ± 0.0158 0.52828 ± 0.0448 0.72979 ± 0.0295 0.59003 ± 0.0338 0.72395 ± 0.00583 0.74918 ± 0.0131
WFG7 5 14 0.67088 ± 0.0262 0.62534 ± 0.0174 0.67936 ± 0.00499 0.57733 ± 0.00990 0.69261 ± 0.0126 0.74121 ± 0.0100
7 16 0.76664 ± 0.00736 0.61460 ± 0.0485 0.75424 ± 0.0342 0.54361 ± 0.0449 0.78344 ± 0.0125 0.83868 ± 0.00433
WFG8 5 14 0.54151 ± 0.0189 0.50728 ± 0.0166 0.58411 ± 0.00682 0.53178 ± 0.0133 0.56184 ± 0.0106 0.59172 ± 0.0149
7 16 0.57349 ± 0.0307 0.45802 ± 0.0597 0.65561 ± 0.00743 0.56745 ± 0.0158 0.59398 ± 0.0230 0.63311 ± 0.0148
WFG9 5 14 0.57269 ± 0.0447 0.55781 ± 0.0390 0.59047 ± 0.0153 0.53652 ± 0.0202 0.63289 ± 0.0309 0.68275 ± 0.0109
7 16 0.66804 ± 0.0302 0.54137 ± 0.0181 0.62607 ± 0.0362 0.47543 ± 0.0617 0.69360 ± 0.0230 0.66805 ± 0.0503

Significant values are in bold.