Table 3 Simulation results on UBFs
fn | Criteria | Algorithms | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Traditional algorithms | DE variants | PSO variants | Hybrid variants | Proposed algorithms | ||||||||
EO | HHO | JADE | SHADE | HEPSO | RPSOLF | PSOSCALF | FAPSO | aDE | aPSO | haDEPSO | ||
f1 | Mean | 3.32e−40 | 2.03e + 00 | 1.87e−31 | 1.42e−09 | 16.26772 | 5.065e−269 | 1.11014e−20 | 2.87e−127 | 0.00e + 000 | 0.00e + 000 | 0.00e + 000 |
|  | std | 6.78e−40 | 4.04e−01 | 6.43e−31 | 3.09e−09 | 10.01293 | 0.00e + 00 | 1.83289e−20 | 1.76e−127 | 0.00e + 000 | 0.00e + 000 | 0.00e + 000 |
| Â | Rank | 4 | 9 | 5 | 7 | 8 | 2 | 6 | 3 | 1 | 1 | 1 |
f2 | Mean | 7.12e−23 | 1.70e + 00 | 2.79e−15 | 0.0087 | 1.28424 | 1.000e−134 | 4.09460e−11 | 1.02e−17 | 0.00e + 000 | 0.00e + 000 | 0.00e + 000 |
|  | std | 6.36e−23 | 7.37e−02 | 9.51e−15 | 0.0213 | 0.41611 | 3.753e−134 | 5.68981e−11 | 1.43e−17 | 0.00e + 000 | 0.00e + 000 | 0.00e + 000 |
| Â | Rank | 3 | 9 | 4 | 7 | 8 | 2 | 6 | 5 | 1 | 1 | 1 |
f3 | Mean | 8.06e−09 | 1.17e + 02 | 1.10e−03 | 15.4352 | 7.423e + 03 | 7.791e−249 | 2.16858e−12 | 1.68e−11 | 0.00e + 000 | 0.17e−129 | 0.00e + 000 |
|  | std | 1.60e−08 | 5.28e + 00 | 5.14e−03 | 9.9489 | 7.423e + 03 | 0.00e + 00 | 1.03815e−11 | 2.49e−11 | 0.00e + 000 | 1.67e−131 | 0.00e + 000 |
| Â | Rank | 6 | 9 | 7 | 8 | 10 | 2 | 4 | 5 | 1 | 3 | 1 |
f4 | Mean | 5.39e−10 | 2.05e+00 | 1.66e−03 | 0.9796 | 23.95145 | 1.937e−157 | 8.47410e−08 | 4.09e+03 | 3.28e−101 | 8.35e−098 | 0.00e+000 |
|  | std | 1.38e−09 | 7.40e−02 | 1.98e−03 | 0.7995 | 7.71460 | 1.061e−156 | 1.23324e−07 | 6.53e+02 | 7.12e−103 | 7.08e−098 | 0.00e+000 |
| Â | Rank | 5 | 9 | 7 | 8 | 10 | 2 | 6 | 11 | 3 | 4 | 1 |
f5 | Mean | 2.53e+01 | 2.95e+00 | 1.18e+01 | 24.4743 | 2.380e+03 | 27.42672 | 21.97646 | 6.55e−11 | 1.25e−021 | 4.85e−012 | 2.21e−033 |
|  | std | 0.16e+00 | 8.36e−02 | 1.57e+01 | 11.2080 | 1.852e+03 | 0.24848 | 0.54774 | 1.99e−11 | 1.85e−023 | 3.21e−012 | 3.72e−037 |
| Â | Rank | 7 | 9 | 10 | 6 | 11 | 8 | 5 | 4 | 2 | 3 | 1 |
f6 | Mean | 8.29e−06 | 2.49e+00 | 4.59e−31 | 5.31e−10 | 21.55405 | 2.98244 | 7.13998e−12 | 2.37e−12 | 0.00e+000 | 1.75e−032 | 0.00e+000 |
|  | std | 5.02e−06 | 8.25e−02 | 1.65e−30 | 6.35e−10 | 9.33263 | 0.23250 | 3.65884e−11 | 1.84e−13 | 0.00e+000 | 9.16e−035 | 0.00e+000 |
| Â | Rank | 8 | 9 | 3 | 7 | 6 | 10 | 5 | 4 | 1 | 2 | 1 |
f7 | Mean | 1.17e−02 | 8.20e+00 | 6.49e−03 | 0.0235 | 0.12982 | 0.00104 | 0.00012 | 0.00e+00 | 2.19e−003 | 5.11e−001 | 1.07e−003 |
|  | std | 6.54e−04 | 1.69e−01 | 2.48e−03 | 0.0088 | 0.09727 | 7.644e−04 | 0.00010 | 0.00e+00 | 1.10e−004 | 1.70e−002 | 1.40e−005 |
| Â | Rank | 6 | 11 | 5 | 9 | 10 | 7 | 4 | 1 | 3 | 8 | 2 |
f8 | Mean | − 9016.34 | 4.86e+00 | − 1.24e+04 | − 11,713.1 | − 2.139e+03 | − 3.254e+03 | − 12,569.48 | 2.48e−11 | − 1.25e+004 | − 6.37e+004 | − 1.25e+004 |
|  | std | 595.1113 | 1.03e+00 | 1.27e+02 | 230.49 | 8.282e+02 | 2.860e+02 | 2.39996e−07 | 6.44e−12 | 1.07e−017 | 2.10e−001 | 0.00e+000 |
| Â | Rank | 6 | 4 | 1 | 8 | 5 | 7 | 1 | 2 | 1 | 9 | 1 |
f9 | Mean | 0.00e+00 | 3.77e+00 | 1.71e−04 | 8.5332 | 42.00118 | 0.00e+00 | 0.00e+00 | 0.00e+00 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
|  | std | 0.00e+00 | 8.87e−01 | 1.52e−04 | 2.1959 | 7.08632 | 0.00e+00 | 0.00e+00 | 0.00e+00 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
| Â | Rank | 1 | 3 | 2 | 4 | 5 | 1 | 1 | 1 | 1 | 1 | 1 |
f10 | Mean | 8.34e−14 | 3.75e+00 | 1.31e−14 | 0.3957 | 2.83842 | 4.085e−15 | 2.24609e−11 | 4.86e−15 | 1.44e−015 | 1.97e−014 | 2.88e−016 |
|  | std | 2.53e−14 | 8.75e−01 | 2.46e−14 | 0.5868 | 0.66134 | 1.084e−15 | 2.33542e−11 | 1.74e−15 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
| Â | Rank | 7 | 8 | 5 | 10 | 9 | 3 | 8 | 4 | 2 | 6 | 1 |
f11 | Mean | 0.00e+00 | 4.17e+00 | 2.87e−03 | 0.0048 | 1.16858 | 0.00e+00 | 0.00e+00 | 1.74e−16 | 0.00e+000 | 3.37e−111 | 0.00e+000 |
|  | std | 0.00e+00 | 5.56e−01 | 7.85e−03 | 0.0077 | 0.12602 | 0.00e+00 | 0.00e+00 | 3.60e−16 | 0.00e+000 | 1.11e−119 | 0.00e+000 |
| Â | Rank | 1 | 7 | 4 | 5 | 6 | 1 | 1 | 3 | 1 | 2 | 1 |
f12 | Mean | 7.97e−07 | 1.90e+01 | 1.73e−02 | 0.0346 | 0.47856 | 0.26157 | 8.46465e−14 | 1.57e−32 | 1.05e−032 | 3.34e−002 | 3.73e−033 |
|  | std | 7.69e−07 | 3.31e+00 | 7.74e−02 | 0.0875 | 0.22623 | 0.03386 | 2.79106e−13 | 0.00e+00 | 2.77e−034 | 1.02e−004 | 3.18e−034 |
| Â | Rank | 5 | 11 | 6 | 8 | 10 | 9 | 4 | 3 | 2 | 7 | 1 |
f13 | Mean | 0.029295 | 1.89e+01 | 5.45e−24 | 7.32e−04 | 1.85056 | 2.05282 | 0.00399 | 1.58e−32 | 2.09e−021 | 9.30e−004 | 1.58e−032 |
|  | std | 0.035271 | 1.56e+00 | 2.58e−23 | 0.0028 | 0.65246 | 0.16579 | 0.00928 | 0.00e+00 | 3.27e−023 | 3.71e−004 | 1.89e−043 |
| Â | Rank | 7 | 10 | 2 | 4 | 8 | 9 | 6 | 1 | 3 | 5 | 1 |
f14 | Mean | 0.99800 | 9.98e−01 | 9.98e−01 | 0.998004 | 0.99800 | 1.54064 | 1.13027 | 9.98e−001 | 9.98e−001 | 9.98e−001 | 9.98e−001 |
|  | std | 1.54e−16 | 9.23e−01 | 0.00e+00 | 5.83e−17 | 9.219e−17 | 1.84429 | 0.50338 | 1.27e−08 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
| Â | Rank | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 1 |
f15 | Mean | 0.00239 | 3.10e−04 | 3.01e−03 | 0.002374 | 6.404e−04 | 0.00171 | 3.13244e−04 | 3.95e−04 | 3.83e−004 | 3.99e−004 | 3.02e−004 |
|  | std | 0.00609 | 1.97e−04 | 6.92e−03 | 0.0061 | 2.801e−04 | 0.00508 | 2.17489e−05 | 6.02e−08 | 9.23e−012 | 2.96e−007 | 2.38e−018 |
| Â | Rank | 11 | 2 | 8 | 10 | 7 | 9 | 3 | 5 | 4 | 6 | 1 |
f16 | Mean | − 1.03161 | −1.03e+00 | − 1.03e+00 | − 1.03161 | − 1.03161 | − 1.03161 | − 1.0316 | −1.03e+00 | −1.03e+000 | −1.03e+000 | −1.03e+000 |
|  | std | 6.04e−16 | 6.78e−16 | 6.78e−16 | 6.51e−16 | 3.554e−15 | 1.650e−05 | 4.40244e−16 | 0.00e+00 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
| Â | Rank | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
f17 | Mean | 0.397887 | 3.98e−01 | 3.98e−01 | 0.397887 | 0.39788 | 0.39837 | 0.39787 | 3.98e−001 | 3.98e−001 | 3.98e−001 | 3.98e−001 |
|  | std | 0.00e+00 | 2.54e−06 | 0.00e+00 | 3.24e−16 | 6.594e−13 | 5.267e−04 | 3.66527e−15 | 0.00e+000 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
| Â | Rank | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 |
f18 | Mean | 3.00e+00 | 3.00e+00 | 3.00e+00 | 3.00e+00 | 0.65246 | 3.00002 | 3. 00e+00 | 3.00e+000 | 3.00e+000 | 3.00e+000 | 3.00e+000 |
|  | std | 1.56e−15 | 0.00e+00 | 1.82e−15 | 1.87e−15 | 5.146e−11 | 1.658e−05 | 5.96540e−13 | 5.31e−016 | 0.00e+000 | 1.33e−018 | 0.00e+000 |
| Â | Rank | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
f19 | Mean | − 3.86278 | −3.86e+00 | − 3.86e+00 | − 3.86278 | − 3.86278 | − 3.85923 | − 3.86278 | −3.87e+000 | −3.86e+000 | −3.86e+000 | −3.86e+000 |
|  | std | 2.59e−15 | 2.44e−03 | 2.71e−15 | 2.69e−15 | 1.008e−13 | 0.00283 | 8.31755e−15 | 2.11e−004 | 0.00e+000 | 3.36e−021 | 0.00e+000 |
| Â | Rank | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 1 | 1 | 1 |
f20 | Mean | − 3.2687 | −3.322 | − 3.29e+00 | − 3.27047 | − 3.31803 | − 3.10441 | − 3.27168 | − 3.29e+000 | − 3.32e+000 | − 3.27e+000 | − 3.32e+000 |
|  | std | 0.05701 | 0.137406 | 5.11e−02 | 0.0599 | 0.02170 | 0.15760 | 0.06371 | 0.00e+000 | 0.00e+000 | 0.00e+000 | 0.00e+000 |
| Â | Rank | 6 | 1 | 4 | 5 | 2 | 7 | 5 | 3 | 1 | 5 | 1 |
f21 | Mean | − 8.55481 | −10. 1531 | − 9.14e+00 | − 9.2343 | − 10.15319 | − 4.76171 | − 10.15319 | − 7.51e+00 | − 1.01e+001 | − 9.87e+001 | − 1.01e+001 |
|  | std | 2.76377 | 0.885673 | 2.06e+00 | 1.3969 | 2.680e−05 | 0.73723 | 4.46227e−15 | 1.21e−01 | 0.00e+000 | 1.07e−011 | 0.00e+000 |
| Â | Rank | 5 | 1 | 4 | 3 | 1 | 6 | 1 | 3 | 1 | 2 | 1 |
f22 | Mean | − 9.3353 | −10.4015 | − 9.88e+00 | − 10.2809 | − 10. 4015 | − 4.81927 | − 10. 40,156 | − 6.04e+01 | − 1.04e+001 | − 9.87e+000 | − 1.04e+001 |
|  | std | 2.43834 | 1.352375 | 1.61e+00 | 1.3995 | 0.01728 | 0.75699 | 1.80672e−15 | 2.14e−01 | 2.91e−012 | 2.08e−002 | 1.10e−016 |
| Â | Rank | 5 | 1 | 3 | 2 | 1 | 7 | 1 | 6 | 1 | 4 | 1 |
f23 | Mean | − 9.63655 | −10.5364 | − 1.03e+01 | 63.333 | − 10.53640 | − 5.06376 | − 10.53640 | − 5.64e+00 | − 1.05e+001 | − 9.13e+000 | − 1.05e+001 |
|  | std | 2.38811 | 0.927655 | 1.40e+00 | 80.872 | 5.845e−07 | 0.82968 | 4.84794e−15 | 5.70e−02 | 1.07e−021 | 2.05e−003 | 0.00e+000 |
| Â | Rank | 3 | 1 | 2 | 7 | 1 | 6 | 1 | 5 | 1 | 4 | 1 |
Sum of rank | Â | 102 | 119 | 87 | 124 | 125 | 107 | 75 | 75 | 35 | 78 | 24 |
Average | Â | 4.43 | 5.17 | 3.78 | 5.39 | 5.43 | 4.65 | 3.26 | 3.26 | 1.52 | 3.39 | 1.04 |
overall rank | Â | 6 | 8 | 5 | 9 | 10 | 7 | 3 | 3 | 2 | 4 | 1 |