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. 2024 Feb 29;14:5032. doi: 10.1038/s41598-024-54910-3

Table 4.

Evaluation outcomes for the objectives specified in the F14-F23.

F M Optimization algorithms
HO WOA GWO SSA PSO SCA FA GOA TLBO CMA-ES MFO AOA IWO
F14 Mean 0.998 3.0928 6.9207 1.1305 5.5195 2.2512 9.502 12.782 0.998 4.7816 2.51 8.0876 11.358
Best 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 1.992 0.998 0.998 0.998
Worst 0.998 10.763 12.671 1.992 12.671 10.763 21.988 23.809 0.998 11.721 10.763 12.671 23.809
Std 0 3.2723 4.9187 0.34368 3.0682 1.8878 6.2553 6.298 3.86-16 2.4391 2.3156 4.7721 7.3331
Median 0.998 1.495 7.3657 0.998 5.9288 2.0092 8.3574 12.671 0.998 3.9742 0.998 10.763 10.763
Rank 1 6 9 3 8 4 11 13 2 7 5 10 12
F15 Mean 0.00030836 0.00088694 0.0083127 0.0027545 0.0024923 0.0010454 0.0058617 0.010286 0.00036839 0.0019 0.0013293 0.0071685 0.0027859
Best 0.00030749 0.00030755 0.00030749 0.00033385 0.00030749 0.00057375 0.00030749 0.00030749 0.00030749 0.0011 0.00074582 0.00034241 0.00058505
Worst 0.00031288 0.0035616 0.056621 0.02043 0.020363 0.0016389 0.020363 0.056543 0.0012232 0.0035 0.0083337 0.069975 0.020363
Std 1.31-06 0.00071966 0.012961 0.0059824 0.0060732 0.00035949 0.0089054 0.012603 0.00018223 7.25-06 0.0013978 0.014639 0.0059637
Median 0.00030779 0.0005872 0.00045554 0.00077266 0.00030782 0.00087851 0.00030749 0.0029582 0.00030749 0.0016 0.00080207 0.00047392 0.00074539
Rank 1 3 12 8 7 4 10 13 2 6 5 11 9
F16 Mean − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 0.8956 − 0.79107 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316
Best − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316
Worst − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0314 − 0.21546 2.1043 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316
Std 5.96-16 3.09-09 9.67-09 2.66-14 6.71-16 5.70-05 0.30937 0.62776 6.65-16 6.78-16 6.78-16 1.22-07 1.44-08
Median − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316 − 1.0316
Rank 1 1 1 1 1 1 2 3 1 1 1 1 1
F17 Mean 0.39789 0.3979 0.39789 0.39789 0.39789 0.40008 0.39789 0.39789 0.39789 0.39789 0.39789 0.41197 0.39789
Best 0.39789 0.39789 0.39789 0.39789 0.39789 0.3979 0.39789 0.39789 0.39789 0.39789 0.39789 0.39813 0.39789
Worst 0.39789 0.39794 0.39789 0.39789 0.39789 0.40747 0.39789 0.39789 0.39789 0.39789 0.39789 0.4415 0.39789
Std 0 1.42-05 1.41-06 4.70-14 7.23-16 0.0023394 3.86-11 0 0 0 0 0.01286 5.57-09
Median 0.39789 0.39789 0.39789 0.39789 0.39789 0.39939 0.39789 0.39789 0.39789 0.39789 0.39789 0.40712 0.39789
Rank 1 2 1 1 1 3 1 1 1 1 1 4 1
F18 Mean 3 3.0001 3 3 3.9 3.0001 3.9 9.3 3 3 3 7.7674 3
Best 3 3 3 3 3 3 3 3 3 3 3 3 3
Worst 3 3.0006 3.0001 3 30 3.0004 30 84 3 3 3 37.986 3
Std 1.27-15 0.00013473 1.79-05 2.77-13 4.9295 0.00010081 4.9295 16.904 5.53-16 1.35-15 1.62-15 10.92 8.47-07
Median 3 3 3 3 3 3 3 3 3 3 3 3 3
Rank 1 2 1 1 3 2 3 5 1 1 1 4 1
F M Optimization algorithms
HO WOA GWO SSA PSO SCA FA GOA TLBO CMA-ES MFO AOA IWO
F19 Mean − 3.8628 − 3.8335 − 3.8595 − 3.8628 − 3.8628 − 3.8542 − 3.8628 − 3.8212 − 3.8628 − 3.8628 − 3.8628 − 3.8512 -3.8628
Best − 3.8628 − 3.8628 − 3.8628 − 3.8628 − 3.8628 − 3.8621 − 3.8628 − 3.8628 − 3.8628 − 3.8628 − 3.8628 − 3.8605 -3.8628
Worst − 3.8628 − 3.0885 − 3.8549 − 3.8628 − 3.8628 − 3.8443 − 3.8628 − 3.0898 − 3.8628 − 3.8628 − 3.8628 − 3.8408 -3.8628
Std 2.70-15 0.14079 0.0036479 1.51-11 6.42-07 0.0032649 5.72-11 0.14271 2.71-15 2.71-15 2.71-15 0.0038724 6.32-07
Median − 3.8628 − 3.8611 − 3.8624 − 3.8628 − 3.8628 − 3.8542 − 3.8628 − 3.8627 − 3.8628 − 3.8628 − 3.8628 − 3.8516 − 3.8628
Rank 1 5 2 1 1 3 1 6 1 1 1 4 1
F20 Mean − 3.322 − 3.2406 − 3.1218 − 3.2275 − 3.2784 − 2.8823 − 3.2586 − 2.7966 − 3.2811 − 3.2919 − 3.2422 − 3.0755 − 3.203
Best − 3.322 − 3.3219 − 3.322 − 3.322 − 3.322 − 3.1473 − 3.322 − 3.3138 − 3.3206 − 3.322 − 3.322 − 3.1844 − 3.2031
Worst − 3.322 − 2.4308 − 2.2671 − 3.1469 − 3.2031 − 1.9133 − 3.2031 − 1.9056 − 3.1059 − 3.2031 − 3.1376 − 2.9507 − 3.2026
Std 9.78-12 0.17522 0.26562 0.058906 0.058273 0.32598 0.060328 0.36014 0.066162 0.051459 0.063837 0.063759 0.00012908
Median − 3.322 − 3.3191 − 3.1676 − 3.1988 − 3.322 − 2.9914 − 3.2031 − 2.9387 − 3.3109 − 3.322 − 3.2031 − 3.0905 − 3.2031
Rank 1 7 10 8 4 12 5 13 3 2 6 11 9
F21 Mean − 10.153 − 7.9465 − 8.6325 − 7.5665 − 6.3165 − 2.1516 − 5.6345 − 5.2171 − 9.778 − 7.9121 − 6.3132 − 3.4351 − 6.6391
Best − 10.153 − 10.153 − 10.153 − 10.153 − 10.153 − 6.0051 − 10.153 − 10.153 − 10.153 − 10.153 − 10.153 − 5.5613 − 10.153
Worst − 10.153 − 2.6294 − 5.0552 − 2.6305 − 2.6305 − 0.35136 − 2.6305 − 2.6305 − 3.9961 − 2.6829 − 2.6305 − 1.9507 − 2.6305
Std 4.74-06 2.7979 2.3619 3.5062 3.6985 1.872 3.1766 2.9687 1.4345 3.4819 3.5169 0.97125 3.4561
Median − 10.153 − 10.137 − 10.152 − 10.153 − 5.078 − 0.88031 − 5.0552 − 5.0552 − 10.153 − 10.153 − 5.078 − 3.2531 − 5.1008
Rank 1 4 3 6 8 13 10 11 2 5 9 12 6
F22 Mean − 10.403 − 6.573 − 9.5722 − 8.8874 − 6.7572 − 2.7098 − 5.3848 − 5.5286 − 9.7141 − 10.403 − 8.2382 − 3.7002 − 7.4415
Best − 10.403 − 10.401 − 10.403 − 10.403 − 10.403 − 6.3217 − 10.403 − 10.403 − 10.403 − 10.403 − 10.403 − 6.8593 − 10.403
Worst − 10.403 − 1.8372 − 2.7659 − 2.7519 − 1.8376 − 0.52104 − 1.8376 − 1.8376 − 5.0265 − 10.403 − 2.7519 − 1.2708 − 1.8376
Std 6.16-05 3.2924 2.1838 2.8452 3.7466 1.9244 3.194 3.3448 1.7896 1.65-15 3.3738 1.2624 3.7449
Median − 10.403 − 5.0875 − 10.402 − 10.403 − 7.7659 − 2.6079 − 3.7243 − 3.7243 − 10.403 − 10.403 − 10.403 − 3.6181 − 10.403
Rank 1 9 3 5 8 13 11 10 2 1 6 12 7
F23 Mean − 10.536 − 6.8188 − 9.2737 − 8.2567 − 6.0645 − 4.1564 − 4.7569 − 5.3577 − 10.123 − 10.536 − 7.9819 − 4.6738 − 8.3548
Best − 10.536 − 10.534 − 10.536 − 10.536 − 10.536 − 8.3393 − 10.536 − 10.536 − 10.536 − 10.536 − 10.536 − 8.6767 − 10.536
Worst − 10.536 − 1.8588 − 2.4217 − 2.4217 − 1.8595 − 0.94428 − 1.6766 − 1.8595 − 3.8354 − 10.536 − 2.4217 − 1.8573 − 2.4217
Std 2.99-05 3.4247 2.9055 3.3562 3.7424 1.5765 3.0762 3.5273 1.5801 1.78-15 3.6868 1.5405 3.437
Median − 10.536 − 5.1284 − 10.536 − 10.536 − 3.8354 − 4.6344 − 3.8354 − 3.8354 − 10.536 − 10.536 − 10.536 − 4.8892 − 10.536
Rank 1 8 4 6 9 13 11 10 2 1 7 12 5
Sum rank 10 47 46 40 50 68 65 85 17 26 42 81 52
Mean rank 1 4.7000 4.6000 4 5 6.8000 6.5000 8.5000 1.7000 2.6000 4.2000 8.1000 5.2000
Total rank 1 7 6 4 8 11 10 13 2 3 5 12 9