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. 2025 Feb 17;15:5730. doi: 10.1038/s41598-025-88883-8

Table 3.

Statistical results for Spacing on ZDT, DTLZ, WFG, UF and CF problems.

Function Index Algorithms
MOMVO MSSA MOEAD MOEBS
ZTD1 Ave 0.026459 0.011731 0.013979 0.006526
Std 0.083189 0.003861 0.004078 0.000589
ZTD2 Ave 0.049249 0.014362 0.011457 0.005759
Std 0.117064 0.006560 0.003621 0.002442
ZTD3 Ave 0.108414 0.012892 0.014752 0.007169
Std 0.193046 0.004540 0.007811 0.000661
ZTD4 Ave 1.048411 0.012032 0.105549 0.006058
Std 2.843320 0.003810 0.429861 0.003506
ZTD6 Ave 0.044969 0.013930 0.010304 0.251907
Std 0.041661 0.004551 0.004606 0.260172
DTLZ1 Ave 2.726574 0.591521 1.045438 4.199137
Std 0.456379 0.807114 0.539217 1.343409
DTLZ2 Ave 0.068741 0.012420 0.067143 0.076190
Std 0.012837 0.009472 0.007988 0.007696
DTLZ3 Ave 23.178768 4.356206 6.195430 28.465642
Std 3.647969 6.338016 2.380846 8.496647
DTLZ4 Ave 0.063679 0.023347 0.048773 0.117965
Std 0.010617 0.017199 0.027854 0.034067
DTLZ5 Ave 0.020626 0.014997 0.014925 0.036219
Std 0.013399 0.007854 0.003137 0.014184
DTLZ6 Ave 0.011793 0.041339 0.015963 0.008058
Std 0.012261 0.037144 0.004450 0.000883
DTLZ7 Ave 0.064497 0.046943 0.006793 0.056736
Std 0.008312 0.026421 0.020207 0.035528
WFG1 Ave 0.138023 0.103604 0.041242 0.135123
Std 0.018245 0.005639 0.020383 0.026284
WFG2 Ave 0.013139 0.027049 0.050835 0.121488
Std 0.001234 0.011530 0.026267 0.047342
WFG3 Ave 0.015524 0.034422 0.045398 0.040556
Std 0.001380 0.011515 0.017320 0.007244
WFG4 Ave 0.018550 0.032411 0.034418 0.021353
Std 0.001265 0.017734 0.008099 0.002749
WFG5 Ave 0.018758 0.028893 0.040085 0.020727
Std 0.001819 0.014732 0.009440 0.003985
WFG6 Ave 0.019684 0.038801 0.042859 0.035936
Std 0.001994 0.014583 0.010125 0.011437
WFG7 Ave 0.019608 0.035109 0.041497 0.049148
Std 0.002232 0.019626 0.012306 0.013961
WFG8 Ave 0.031513 0.035221 0.042582 0.104775
Std 0.026971 0.015369 0.009984 0.044722
WFG9 Ave 0.017388 0.038483 0.039209 0.018282
Std 0.001930 0.021641 0.009116 0.004334
WFG10 Ave 0.024650 0.037757 0.059506 0.038616
Std 0.008543 0.014741 0.033912 0.005922
UF1 Ave 0.007880 0.004532 0.001702 0.044787
Std 0.005658 0.007875 0.002221 0.046068
UF2 Ave 0.010643 0.014314 0.010836 0.086456
Std 0.009050 0.012210 0.004727 0.066344
UF3 Ave 0.029637 0.002103 0.001735 0.029597
Std 0.010066 0.005219 0.003132 0.060371
UF4 Ave 0.010837 0.009884 0.012177 0.007954
Std 0.004303 0.010191 0.005080 0.002267
UF5 Ave 0.034639 0.002677 0.002387 0.366548
Std 0.037212 0.010935 0.004784 0.296823
UF6 Ave 0.040039 0.000334 0.002625 0.265813
Std 0.000000 0.001168 0.007402 0.169029
UF7 Ave 0.007608 0.010135 0.005436 0.051190
Std 0.006473 0.007378 0.007234 0.030607
UF8 Ave 0.260435 0.150251 0.051048 1.044351
Std 0.152174 0.236871 0.023417 0.475604
UF9 Ave 0.213146 0.087874 0.043289 0.899606
Std 0.146353 0.090730 0.028973 0.477132
UF10 Ave 0.627649 0.282037 0.041046 2.768668
Std 0.385007 0.355436 0.053757 2.233361
CF1 Ave 0.029862 0.011287 0.016632 0.004730
Std 0.035021 0.004153 0.009126 0.000427
CF2 Ave 0.018650 0.005358 0.002823 0.064963
Std 0.045332 0.008679 0.002844 0.110851
CF3 Ave 0.199220 0.000242 0.002150 0.622956
Std 0.000000 0.000744 0.004646 0.569364
CF4 Ave 0.012645 0.003991 0.003735 0.437485
Std 0.016190 0.006786 0.009463 0.494674
CF5 Ave - 0.002202 0.003613 1.441032
Std - 0.004987 0.010211 1.551803
CF6 Ave 0.019671 0.009398 0.007727 0.389627
Std 0.013569 0.005310 0.011447 0.351685
CF7 Ave 0.284863 0.006070 0.004083 1.164972
Std 0.307611 0.013523 0.009015 1.374945
CF8 Ave 0.295503 0.184859 0.057122 2.864021
Std 0.241285 0.129685 0.060418 1.250903
CF9 Ave 0.229793 0.158875 0.051472 1.047098
Std 0.211594 0.212562 0.031089 0.642722
CF10 Ave 0.695424 0.495872 0.042921 2.720145
Std 0.484528 0.629724 0.053518 1.791807