Figure 1.
Ref. [
30] (
a). Sobol sequential random number distribution. (
b). Pseudo-random number sequence distribution.
Figure 1.
Ref. [
30] (
a). Sobol sequential random number distribution. (
b). Pseudo-random number sequence distribution.
Figure 2.
Comparison between the original Q and the improved Q.
Figure 2.
Comparison between the original Q and the improved Q.
Figure 3.
Learning strategy simulation.
Figure 3.
Learning strategy simulation.
Figure 4.
The flowchart of the SNDSO.
Figure 4.
The flowchart of the SNDSO.
Figure 5.
The mean value-based rank-filled of different parameter combinations.
Figure 5.
The mean value-based rank-filled of different parameter combinations.
Figure 6.
Convergence plot of the different population size on the CEC2015.
Figure 6.
Convergence plot of the different population size on the CEC2015.
Figure 7.
The population diversity of SO and the SNDSO.
Figure 7.
The population diversity of SO and the SNDSO.
Figure 8.
The exploration and exploitation of the SNDSO.
Figure 8.
The exploration and exploitation of the SNDSO.
Figure 9.
Mean ranking of algorithms on different test dimensions.
Figure 9.
Mean ranking of algorithms on different test dimensions.
Figure 10.
Convergence plot of different algorithms on the CEC2017 test function set (Dim = 30).
Figure 10.
Convergence plot of different algorithms on the CEC2017 test function set (Dim = 30).
Figure 11.
Boxplot of different algorithms on the CEC2017 test function set (Dim = 30).
Figure 11.
Boxplot of different algorithms on the CEC2017 test function set (Dim = 30).
Figure 12.
Box plots of algorithms dealing with FS problems.
Figure 12.
Box plots of algorithms dealing with FS problems.
Figure 13.
Convergence plot of the algorithm dealing with the FS problems.
Figure 13.
Convergence plot of the algorithm dealing with the FS problems.
Figure 14.
Ranking of all metrics for algorithms dealing with FS problems.
Figure 14.
Ranking of all metrics for algorithms dealing with FS problems.
Figure 15.
3-bar truss design problem.
Figure 15.
3-bar truss design problem.
Figure 16.
10-bar truss design problem.
Figure 16.
10-bar truss design problem.
Figure 17.
Tension/compression spring design problem.
Figure 17.
Tension/compression spring design problem.
Figure 18.
Construction of speed reducer.
Figure 18.
Construction of speed reducer.
Figure 19.
Welded beam design problem.
Figure 19.
Welded beam design problem.
Table 1.
Parameter information for 16 advanced algorithms.
Table 1.
Parameter information for 16 advanced algorithms.
Algorithms | Time | Parameters Settings |
---|
SO | 2022 | |
DBO | 2022 | |
Salp Swarm Algorithm (SSA) [34] | 2017 | |
TLBO | 2011 | |
Fuzzy Self-Tuning Differential Evolution (FSTDE) [35] | 2020 | |
Artificial Hummingbird Algorithm (AHA) [36] | 2021 | |
Whale Optimization Algorithm (WOA) [37] | 2016 | |
HHO | 2019 | |
Butterfly Optimization Algorithm (BOA) [38] | 2019 | |
Artificial Butterfly Optimization (ABO) [39] | 2017 | |
Particle Swarm Optimization (PSO) [40] | 1995 | |
GWO | 2014 | |
Bezier Search Differential Evolution (BESD) [41] | 2020 | |
Moth-Flame Optimization (MFO) [42] | 2015 | |
Equilibrium Optimizer (EO) [43] | 2020 | |
SNDSO | NA | |
Table 2.
The CEC2015 test function set.
Table 2.
The CEC2015 test function set.
Problem | Types | Name | Optimum |
---|
CEC2015_F1 | Unimodal | Rotated high conditioned elliptic function | 100 |
CEC2015_F2 | | Rotated cigar function | 200 |
CEC2015_F3 | Simple Multimodal | Shifted and rotated Ackley’s function | 300 |
CEC2015_F4 | | Shifted and rotated Rastrigin’s function | 400 |
CEC2015_F5 | | Shifted and rotated Schwefel’s function | 500 |
CEC2015_F6 | Hybrid | Hybrid function 1 (N = 3) | 600 |
CEC2015_F7 | | Hybrid function 2 (N = 4) | 700 |
CEC2015_F8 | | Hybrid function 3 (N = 5) | 800 |
CEC2015_F9 | Composition | Composition function 1 (N = 3) | 900 |
CEC2015_F10 | | Composition function 2 (N = 3) | 1000 |
CEC2015_F11 | | Composition function 3 (N = 5) | 1100 |
CEC2015_F12 | | Composition function 4 (N = 5) | 1200 |
CEC2015_F13 | | Composition function 5 (N = 5) | 1300 |
CEC2015_F14 | | Composition function 6 (N = 7) | 1400 |
CEC2015_F15 | | Composition function 7 (N = 10) | 1500 |
Search range: [−100, 100] | | | |
Table 3.
Results of the different population size on the CEC2015.
Table 3.
Results of the different population size on the CEC2015.
Problem | Metrics | = 10 | = 20 | = 60 | = 120 | = 30 |
---|
CEC2015_F1 | Mean | 1.1095 × 106 | 8.0399 × 105 | 8.2751 × 105 | 1.5211 × 106 | 6.2595 × 105 |
| Rank | 4 | 2 | 3 | 5 | 1 |
| Wilcoxon | = | = | = | - | NA |
CEC2015_F2 | Mean | 2.7250 × 103 | 2.3829 × 103 | 3.6684 × 103 | 1.7561 × 103 | 7.6041 × 102 |
| Rank | 4 | 3 | 5 | 2 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F3 | Mean | 3.2103 × 102 | 3.2103 × 102 | 3.2096 × 102 | 3.2097 × 102 | 3.2093 × 102 |
| Rank | 4 | 4 | 2 | 3 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F4 | Mean | 5.5304 × 102 | 5.0796 × 102 | 5.4226 × 102 | 4.9738 × 102 | 4.6580 × 102 |
| Rank | 5 | 3 | 4 | 2 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F5 | Mean | 4.7199 × 103 | 5.7387 × 103 | 6.4086 × 103 | 6.7890 × 103 | 5.8907 × 103 |
| Rank | 1 | 2 | 4 | 5 | 3 |
| Wilcoxon | + | = | - | - | NA |
CEC2015_F6 | Mean | 1.5306 × 105 | 1.4430 × 105 | 1.9620 × 105 | 1.3288 × 105 | 5.5288 × 104 |
| Rank | 4 | 3 | 5 | 2 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F7 | Mean | 7.1553 × 102 | 7.1206 × 102 | 7.1266 × 102 | 7.1128 × 102 | 7.1069 × 102 |
| Rank | 5 | 3 | 4 | 2 | 1 |
| Wilcoxon | - | = | - | = | NA |
CEC2015_F8 | Mean | 8.3496 × 104 | 4.5179 × 104 | 4.1471 × 104 | 6.2157 × 104 | 2.3666 × 104 |
| Rank | 5 | 3 | 2 | 4 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F9 | Mean | 1.0807 × 103 | 1.0160 × 103 | 1.0034 × 103 | 1.0034 × 103 | 1.0033 × 103 |
| Rank | 5 | 4 | 2 | 2 | 1 |
| Wilcoxon | - | - | - | = | NA |
CEC2015_F10 | Mean | 1.0534 × 105 | 5.9160 × 104 | 1.0865 × 105 | 1.3714 × 105 | 5.7913 × 104 |
| Rank | 3 | 2 | 4 | 5 | 1 |
| Wilcoxon | - | = | - | - | NA |
CEC2015_F11 | Mean | 2.1071 × 103 | 1.8789 × 103 | 1.7977 × 103 | 1.7727 × 103 | 1.7314 × 103 |
| Rank | 5 | 4 | 3 | 2 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F12 | Mean | 1.3511 × 103 | 1.3543 × 103 | 1.4004 × 103 | 1.4004 × 103 | 1.3350 × 103 |
| Rank | 2 | 3 | 4 | 4 | 1 |
| Wilcoxon | - | - | - | - | NA |
CEC2015_F13 | Mean | 1.3007 × 103 | 1.3000 × 103 | 1.3000 × 103 | 1.3000 × 103 | 1.3000 × 103 |
| Rank | 5 | 1 | 1 | 1 | 1 |
| Wilcoxon | - | - | + | + | NA |
CEC2015_F14 | Mean | 3.7289 × 104 | 3.6974 × 104 | 3.5288 × 104 | 3.5486 × 104 | 3.4832 × 104 |
| Rank | 5 | 4 | 2 | 3 | 1 |
| Wilcoxon | - | - | = | - | NA |
CEC2015_F15 | Mean | 1.6000 × 103 | 1.6000 × 103 | 1.6000 × 103 | 1.6000 × 103 | 1.6000 × 103 |
| Rank | 1 | 1 | 1 | 1 | 1 |
| Wilcoxon | - | = | - | - | NA |
Mean Rank | | 3.87 | 2.80 | 3.07 | 2.87 | 1.13 |
+/-/= | | 1/13/1 | 0/10/5 | 1/12/2 | 1/12/2 | NA |
Table 4.
Friedman mean rank test for CEC2015.
Table 4.
Friedman mean rank test for CEC2015.
Problem | SO | SSO | NSO | DSO | SNDSO |
---|
CEC2015_F1 | 5 | 4 | 3 | 2 | 1 |
CEC2015_F2 | 5 | 4 | 3 | 2 | 1 |
CEC2015_F3 | 4 | 3 | 2 | 5 | 1 |
CEC2015_F4 | 4 | 3 | 5 | 2 | 1 |
CEC2015_F5 | 1 | 2 | 3 | 4 | 5 |
CEC2015_F6 | 5 | 4 | 3 | 2 | 1 |
CEC2015_F7 | 4 | 4 | 3 | 2 | 1 |
CEC2015_F8 | 5 | 3 | 4 | 2 | 1 |
CEC2015_F9 | 5 | 2 | 4 | 1 | 3 |
CEC2015_F10 | 5 | 3 | 4 | 2 | 1 |
CEC2015_F11 | 5 | 3 | 4 | 2 | 1 |
CEC2015_F12 | 3 | 4 | 1 | 2 | 5 |
CEC2015_F13 | 5 | 4 | 3 | 1 | 2 |
CEC2015_F14 | 5 | 3 | 1 | 4 | 2 |
CEC2015_F15 | 4 | 4 | 2 | 3 | 1 |
Mean Rank | 4.33 | 3.33 | 3.00 | 2.40 | 1.80 |
Final Rank | 5 | 4 | 3 | 2 | 1 |
Table 5.
The CEC2017 test function set.
Table 5.
The CEC2017 test function set.
Problem | Types | Name | Optimum |
---|
CEC2017_F1 | Unimodal | Shifted and Rotated Bent Cigar Function | 100 |
CEC2017_F3 | | Shifted and Rotated Zakharov Function | 300 |
CEC2017_F4 | Multimodal | Shifted and Rotated Rosenbrock’s Function | 400 |
CEC2017_F5 | | Shifted and Rotated Rastrigin’s Function | 500 |
CEC2017_F6 | | Shifted and Rotated Expanded Scaffer’s F6 Function | 600 |
CEC2017_F7 | | Shifted and Rotated Lunacek Bi-Rastrigin Function | 700 |
CEC2017_F8 | | Shifted and Rotated Non-Continuous Rastrigin’s Function | 800 |
CEC2017_F9 | | Shifted and Rotated Lévy Function | 900 |
CEC2017_F10 | | Shifted and Rotated Schwefel’s Function | 1000 |
CEC2017_F11 | Hybrid | Hybrid function 1 (N = 3) | 1100 |
CEC2017_F12 | | Hybrid function 1 (N = 3) | 1200 |
CEC2017_F13 | | Hybrid function 3 (N = 3) | 1300 |
CEC2017_F14 | | Hybrid function 4 (N = 4) | 1400 |
CEC2017_F15 | | Hybrid function 5 (N = 4) | 1500 |
CEC2017_F16 | | Hybrid function 6 (N = 4) | 1600 |
CEC2017_F17 | | Hybrid function 6 (N = 5) | 1700 |
CEC2017_F18 | | Hybrid function 6 (N = 5) | 1800 |
CEC2017_F19 | | Hybrid function 6 (N = 5) | 1900 |
CEC2017_F20 | | Hybrid function 6 (N = 6) | 2000 |
CEC2017_F21 | Composition | Composition function 1 (N = 3) | 2100 |
CEC2017_F22 | | Composition function 2 (N = 3) | 2200 |
CEC2017_F23 | | Composition function 3 (N = 4) | 2300 |
CEC2017_F24 | | Composition function 4 (N = 4) | 2400 |
CEC2017_F25 | | Composition function 5 (N = 5) | 2500 |
CEC2017_F26 | | Composition function 6 (N = 5) | 2600 |
CEC2017_F27 | | Composition function 7 (N = 6) | 2700 |
CEC2017_F28 | | Composition function 8 (N = 6) | 2800 |
CEC2017_F29 | | Composition function 9 (N = 3) | 2900 |
CEC2017_F30 | | Composition function 10 (N = 3) | 3000 |
Search range: [−100, 100] |
Table 6.
Numerical results of the different algorithms on solving the CEC2017 test function set (Dim = 30).
Table 6.
Numerical results of the different algorithms on solving the CEC2017 test function set (Dim = 30).
Problem | Metric | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA | SNDSO |
---|
CEC2017_F1 | Mean | 7.6968 × 103 | 1.7314 × 106 | 4.7140 × 103 | 3.9611 × 103 | 8.2179 × 103 | 5.6786 × 103 | 9.7321 × 107 | 5.1689 × 102 |
| Std | 6.9515 × 103 | 2.9496 × 106 | 5.6379 × 103 | 4.3817 × 103 | 5.6047 × 103 | 5.9722 × 103 | 5.6634 × 107 | 3.0473 × 102 |
| FEV | 1.5298 × 102 | 5.8481 × 102 | 5.2639 × 101 | 3.7545 × 10−1 | 2.0682 × 103 | 3.4245 × 102 | 1.7628 × 107 | 3.0000 × 10−1 |
CEC2017_F3 | Mean | 2.7926 × 104 | 3.9541 × 104 | 4.2415 × 102 | 4.1228 × 103 | 9.9256 × 104 | 3.1930 × 103 | 2.0375 × 105 | 1.8652 × 103 |
| Std | 8.3168 × 103 | 1.2925 × 104 | 1.6859 × 102 | 2.5277 × 103 | 1.8787 × 104 | 1.8332 × 103 | 6.1654 × 104 | 4.6252 × 1013 |
| FEV | 1.1898 × 104 | 2.0305 × 104 | 1.0637 × 10−2 | 4.0060 × 102 | 6.8241 × 104 | 5.9081 × 102 | 1.0931 × 105 | 1.5652 × 103 |
CEC2017_F4 | Mean | 5.0176 × 102 | 5.1928 × 102 | 4.9866 × 102 | 4.9621 × 102 | 4.9478 × 102 | 4.9794 × 102 | 5.8535 × 102 | 4.0437 × 102 |
| Std | 2.9394 × 101 | 3.2965 × 101 | 1.8208 × 101 | 2.7301 × 101 | 2.8417 × 100 | 2.5441 × 101 | 4.6642 × 101 | 0.0000 × 100 |
| FEV | 7.1148 × 101 | 5.4783 × 101 | 7.3208 × 101 | 4.3529 × 101 | 9.0041 × 101 | 6.4576 × 101 | 1.0389 × 102 | 4.3721 × 100 |
CEC2017_F5 | Mean | 6.1882 × 102 | 7.3835 × 102 | 6.4518 × 102 | 6.1376 × 102 | 5.9197 × 102 | 6.4690 × 102 | 8.0220 × 102 | 5.3363 × 102 |
| Std | 3.6416 × 101 | 5.2716 × 101 | 3.3602 × 101 | 2.6023 × 101 | 9.4389 × 100 | 3.3996 × 101 | 6.2190 × 101 | 4.3588 × 100 |
| FEV | 6.4804 × 101 | 9.2447 × 101 | 9.4521 × 101 | 5.9699 × 101 | 6.8614 × 101 | 8.2581 × 101 | 2.0024 × 102 | 3.2838 × 101 |
CEC2017_F6 | Mean | 6.0921 × 102 | 6.4427 × 102 | 6.4042 × 102 | 6.1550 × 102 | 6.0000 × 102 | 6.0229 × 102 | 6.7349 × 102 | 6.0010 × 102 |
| Std | 6.3719 × 100 | 1.0389 × 101 | 1.0706 × 101 | 7.2252 × 100 | 9.7157 × 10−3 | 2.9894 × 100 | 1.2214 × 101 | 7.5138 × 10−2 |
| FEV | 1.5847 × 100 | 2.3438 × 101 | 2.1631 × 101 | 4.7650 × 100 | 0.0000 × 100 | 6.1249 × 10−2 | 4.8646 × 101 | 0.0000 × 100 |
CEC2017_F7 | Mean | 8.3350 × 102 | 9.9429 × 102 | 8.6975 × 102 | 9.0679 × 102 | 8.1462 × 102 | 9.9040 × 102 | 1.2773 × 103 | 8.9628 × 102 |
| Std | 3.9154 × 101 | 9.0041 × 101 | 3.6262 × 101 | 5.4023 × 101 | 7.8481 × 100 | 8.0980 × 101 | 7.9178 × 101 | 2.8136 × 101 |
| FEV | 8.3719 × 101 | 1.5847 × 102 | 1.0696 × 102 | 1.1479 × 102 | 9.6365 × 101 | 1.4059 × 102 | 3.5467 × 102 | 1.4163 × 102 |
CEC2017_F8 | Mean | 9.1339 × 102 | 1.0168 × 103 | 9.2138 × 102 | 8.8497 × 102 | 8.9763 × 102 | 9.2023 × 102 | 1.0050 × 103 | 8.3930 × 102 |
| Std | 2.9846 × 101 | 5.3370 × 101 | 4.0855 × 101 | 2.2114 × 101 | 1.2430 × 101 | 2.1571 × 101 | 5.6911 × 101 | 4.5539 × 100 |
| FEV | 5.0871 × 101 | 1.1342 × 102 | 5.0743 × 101 | 5.3728 × 101 | 7.4855 × 101 | 8.8552 × 101 | 1.3946 × 102 | 3.4824 × 101 |
CEC2017_F9 | Mean | 1.5316 × 103 | 4.7118 × 103 | 3.7953 × 103 | 1.9050 × 103 | 1.2497 × 103 | 3.4366 × 103 | 1.0079 × 104 | 9.0056 × 102 |
| Std | 4.2244 × 102 | 1.7258 × 103 | 1.4379 × 103 | 6.7509 × 102 | 2.7801 × 102 | 1.0277 × 103 | 4.9202 × 103 | 3.1873 × 10−1 |
| FEV | 1.6041 × 102 | 1.5195 × 103 | 4.3583 × 102 | 2.0039 × 102 | 7.3508 × 101 | 1.0426 × 103 | 3.4726 × 103 | 2.2055 × 1011 |
CEC2017_F10 | Mean | 3.8240 × 103 | 6.0211 × 103 | 4.8265 × 103 | 7.7712 × 103 | 4.7344 × 103 | 4.3533 × 103 | 6.5874 × 103 | 6.6151 × 103 |
| Std | 6.4217 × 102 | 9.2260 × 102 | 6.1594 × 102 | 6.6894 × 102 | 2.6319 × 102 | 6.7587 × 102 | 7.7786 × 102 | 4.8435 × 102 |
| FEV | 1.5094 × 103 | 3.1788 × 103 | 2.7820 × 103 | 3.7052 × 103 | 3.0434 × 103 | 2.2264 × 103 | 3.8403 × 103 | 4.6036 × 103 |
CEC2017_F11 | Mean | 1.2546 × 103 | 1.4409 × 103 | 1.3000 × 103 | 1.2379 × 103 | 1.7796 × 103 | 1.1910 × 103 | 2.9283 × 103 | 1.1454 × 103 |
| Std | 5.8729 × 101 | 1.0587 × 102 | 4.6247 × 101 | 4.3927 × 101 | 4.7503 × 102 | 3.4546 × 101 | 1.3831 × 103 | 8.5309 × 10−1 |
| FEV | 3.9386 × 101 | 1.8816 × 102 | 1.0572 × 102 | 4.6258 × 101 | 1.7625 × 102 | 3.9416 × 101 | 4.7664 × 102 | 3.6000 × 101 |
CEC2017_F12 | Mean | 1.5552 × 107 | 2.9351 × 107 | 6.8690 × 106 | 8.1761 × 104 | 5.2830 × 106 | 1.0133 × 106 | 1.1565 × 108 | 6.4213 × 104 |
| Std | 1.6079 × 107 | 6.5031 × 107 | 4.0910 × 106 | 9.1597 × 104 | 2.3165 × 106 | 6.4323 × 105 | 6.4869 × 107 | 4.4952 × 104 |
| FEV | 3.3228 × 104 | 2.9947 × 105 | 1.3244 × 106 | 1.4293 × 104 | 1.9861 × 106 | 2.3776 × 105 | 7.7703 × 106 | 1.3400 × 104 |
CEC2017_F13 | Mean | 5.8820 × 105 | 1.5221 × 106 | 8.8660 × 104 | 1.7018 × 104 | 4.3172 × 105 | 1.9841 × 104 | 2.4293 × 105 | 2.9001 × 104 |
| Std | 7.9095 × 105 | 1.8575 × 106 | 4.4136 × 104 | 1.6763 × 104 | 3.8698 × 105 | 1.7751 × 104 | 3.6378 × 105 | 1.9471 × 104 |
| FEV | 3.2675 × 103 | 2.2544 × 104 | 1.6615 × 104 | 4.5050 × 102 | 6.6059 × 103 | 3.7402 × 102 | 2.9488 × 104 | 1.2620 × 103 |
CEC2017_F14 | Mean | 2.0610 × 104 | 5.6819 × 104 | 1.4945 × 104 | 1.1986 × 104 | 3.0733 × 105 | 2.8129 × 104 | 1.8931 × 106 | 1.7391 × 103 |
| Std | 2.5389 × 104 | 5.3657 × 104 | 1.1383 × 104 | 1.2171 × 104 | 1.8644 × 105 | 2.7480 × 104 | 1.9436 × 106 | 1.8178 × 101 |
| FEV | 7.2600 × 102 | 2.4485 × 103 | 2.2189 × 103 | 8.9801 × 102 | 3.7286 × 104 | 6.7992 × 102 | 4.9313 × 104 | 3.3579 × 102 |
CEC2017_F15 | Mean | 1.1524 × 105 | 9.4785 × 104 | 7.3017 × 104 | 4.7432 × 103 | 1.0602 × 105 | 4.2974 × 103 | 1.4447 × 105 | 1.5873 × 103 |
| Std | 1.5235 × 105 | 1.1166 × 105 | 6.7115 × 104 | 4.0869 × 103 | 8.9260 × 104 | 3.3613 × 103 | 1.4741 × 105 | 5.1624 × 101 |
| FEV | 7.7956 × 102 | 1.4919 × 103 | 2.3426 × 104 | 2.2252 × 102 | 2.0646 × 104 | 6.9976 × 101 | 2.6461 × 104 | 7.7890 × 101 |
CEC2017_F16 | Mean | 2.7576 × 103 | 3.2647 × 103 | 2.5474 × 103 | 2.4505 × 103 | 2.5264 × 103 | 2.8006 × 103 | 3.8146 × 103 | 1.8520 × 103 |
| Std | 3.9252 × 102 | 4.4304 × 102 | 3.0154 × 102 | 3.3139 × 102 | 1.5372 × 102 | 2.6103 × 102 | 5.3913 × 102 | 1.3876 × 1012 |
| FEV | 5.5222 × 102 | 8.4239 × 102 | 4.8915 × 102 | 1.8320 × 102 | 6.2821 × 102 | 5.3819 × 102 | 1.1894 × 103 | 2.5201 × 102 |
CEC2017_F17 | Mean | 2.2504 × 103 | 2.4617 × 103 | 2.1528 × 103 | 1.9619 × 103 | 2.0370 × 103 | 2.2582 × 103 | 2.5932 × 103 | 1.7633 × 103 |
| Std | 1.9061 × 102 | 2.5665 × 102 | 1.5217 × 102 | 1.3291 × 102 | 1.0840 × 102 | 1.9367 × 102 | 2.8520 × 102 | 1.6006 × 101 |
| FEV | 1.4067 × 102 | 1.4015 × 102 | 2.1115 × 102 | 6.5383 × 101 | 1.5264 × 102 | 2.8154 × 102 | 2.9875 × 102 | 3.7609 × 101 |
CEC2017_F18 | Mean | 8.8315 × 105 | 1.7061 × 106 | 2.0075 × 105 | 3.2319 × 105 | 9.1437 × 105 | 1.5366 × 105 | 5.2409 × 106 | 1.3902 × 105 |
| Std | 8.4346 × 105 | 2.5172 × 106 | 1.1560 × 105 | 2.5306 × 105 | 5.9038 × 105 | 1.1665 × 105 | 5.0927 × 106 | 9.7676 × 104 |
| FEV | 7.6403 × 104 | 4.6738 × 104 | 3.2624 × 104 | 7.4136 × 104 | 1.4604 × 105 | 2.7265 × 104 | 9.7192 × 104 | 6.0568 × 103 |
CEC2017_F19 | Mean | 7.5455 × 105 | 4.5696 × 105 | 7.0117 × 105 | 7.1248 × 103 | 6.4803 × 104 | 9.6477 × 103 | 6.0112 × 106 | 1.9815 × 103 |
| Std | 1.1022 × 106 | 6.9521 × 105 | 4.3723 × 105 | 5.3588 × 103 | 3.2324 × 104 | 1.0424 × 104 | 6.1383 × 106 | 9.2504 × 1013 |
| FEV | 4.8615 × 102 | 8.3936 × 102 | 1.1094 × 104 | 7.6690 × 101 | 3.0306 × 103 | 2.5298 × 102 | 2.5268 × 104 | 8.1462 × 101 |
CEC2017_F20 | Mean | 2.5598 × 103 | 2.5509 × 103 | 2.5201 × 103 | 2.3121 × 103 | 2.3024 × 103 | 2.5035 × 103 | 2.7778 × 103 | 2.1702 × 103 |
| Std | 2.0656 × 102 | 2.1082 × 102 | 1.6065 × 102 | 1.2509 × 102 | 1.0797 × 102 | 1.6943 × 102 | 2.5342 × 102 | 3.2149 × 100 |
| FEV | 9.8836 × 101 | 1.8688 × 102 | 2.0014 × 102 | 1.0238 × 102 | 5.2641 × 101 | 1.5974 × 102 | 3.2116 × 102 | 1.6652 × 102 |
CEC2017_F21 | Mean | 2.4163 × 103 | 2.5492 × 103 | 2.4291 × 103 | 2.3890 × 103 | 2.3918 × 103 | 2.4273 × 103 | 2.5706 × 103 | 2.4378 × 103 |
| Std | 2.9850 × 101 | 6.2977 × 101 | 3.8786 × 101 | 2.6984 × 101 | 2.5175 × 101 | 3.2751 × 101 | 5.6185 × 101 | 2.6464 × 101 |
| FEV | 2.5595 × 102 | 3.2172 × 102 | 2.6957 × 102 | 2.3983 × 102 | 1.6685 × 102 | 2.6328 × 102 | 3.6268 × 102 | 1.6500 × 102 |
CEC2017_F22 | Mean | 4.5892 × 103 | 4.2780 × 103 | 5.1976 × 103 | 2.7059 × 103 | 4.4198 × 103 | 2.3013 × 103 | 6.5914 × 103 | 2.3007 × 103 |
| Std | 1.7805 × 103 | 2.0568 × 103 | 2.1077 × 103 | 1.5310 × 103 | 1.8632 × 103 | 1.9340 × 100 | 2.1882 × 103 | 1.2356 × 100 |
| FEV | 1.0013 × 102 | 1.0060 × 102 | 1.0000 × 102 | 1.0000 × 102 | 1.0000 × 102 | 1.0003 × 102 | 2.2039 × 102 | 1.0000 × 102 |
CEC2017_F23 | Mean | 2.8423 × 103 | 2.9816 × 103 | 2.7687 × 103 | 2.7882 × 103 | 2.7651 × 103 | 2.7923 × 103 | 3.0865 × 103 | 2.7175 × 103 |
| Std | 5.0849 × 101 | 9.0469 × 101 | 3.6346 × 101 | 3.7569 × 101 | 1.5082 × 101 | 3.0857 × 101 | 1.0749 × 102 | 6.4086 × 100 |
| FEV | 4.3670 × 102 | 4.7053 × 102 | 4.0921 × 102 | 4.1249 × 102 | 4.2546 × 102 | 4.4529 × 102 | 5.6973 × 102 | 4.1630 × 102 |
CEC2017_F24 | Mean | 2.9720 × 103 | 3.1301 × 103 | 2.9304 × 103 | 2.9448 × 103 | 2.9926 × 103 | 3.0378 × 103 | 3.1978 × 103 | 2.9477 × 103 |
| Std | 5.4106 × 101 | 8.7737 × 101 | 3.0603 × 101 | 3.3581 × 101 | 1.7276 × 101 | 5.8788 × 101 | 9.9094 × 101 | 2.2396 × 101 |
| FEV | 5.0120 × 102 | 5.9157 × 102 | 4.7639 × 102 | 4.9562 × 102 | 5.5719 × 102 | 5.4004 × 102 | 5.9375 × 102 | 4.9662 × 102 |
CEC2017_F25 | Mean | 2.8880 × 103 | 2.9424 × 103 | 2.9017 × 103 | 2.9065 × 103 | 2.8907 × 103 | 2.9018 × 103 | 3.0148 × 103 | 2.8896 × 103 |
| Std | 4.0438 × 100 | 4.7463 × 101 | 2.0943 × 101 | 1.9977 × 101 | 1.9521 × 100 | 2.0768 × 101 | 3.8347 × 101 | 6.0523 × 100 |
| FEV | 3.8363 × 102 | 3.8411 × 102 | 3.8691 × 102 | 3.8475 × 102 | 3.8807 × 102 | 3.8380 × 102 | 4.3809 × 102 | 3.8300 × 102 |
CEC2017_F26 | Mean | 6.1434 × 103 | 6.2145 × 103 | 4.4528 × 103 | 4.7555 × 103 | 4.5716 × 103 | 4.0319 × 103 | 7.8932 × 103 | 4.8406 × 103 |
| Std | 6.9520 × 102 | 9.5024 × 102 | 1.2151 × 103 | 1.1539 × 103 | 3.4823 × 102 | 1.7759 × 103 | 1.1293 × 103 | 4.0412 × 102 |
| FEV | 1.9104 × 103 | 7.7789 × 102 | 2.0000 × 102 | 2.0000 × 102 | 9.2522 × 102 | 2.0008 × 102 | 3.4186 × 103 | 1.4534 × 103 |
CEC2017_F27 | Mean | 3.2844 × 103 | 3.3124 × 103 | 3.2456 × 103 | 3.2620 × 103 | 3.2136 × 103 | 3.2670 × 103 | 3.4302 × 103 | 3.2306 × 103 |
| Std | 4.1969 × 101 | 7.1139 × 101 | 2.6213 × 101 | 3.8189 × 101 | 4.6635 × 100 | 2.3967 × 101 | 1.0217 × 102 | 1.4670 × 101 |
| FEV | 5.2317 × 102 | 5.2360 × 102 | 5.0525 × 102 | 5.2010 × 102 | 5.0656 × 102 | 5.2238 × 102 | 5.5078 × 102 | 4.9882 × 102 |
CEC2017_F28 | Mean | 3.2402 × 103 | 3.4542 × 103 | 3.2208 × 103 | 3.2205 × 103 | 3.2877 × 103 | 3.2135 × 103 | 3.3762 × 103 | 3.2129 × 103 |
| Std | 4.5262 × 101 | 4.1670 × 102 | 2.5381 × 101 | 2.0421 × 101 | 2.0831 × 101 | 1.4817 × 101 | 3.9377 × 101 | 1.7401 × 101 |
| FEV | 4.0248 × 102 | 4.2633 × 102 | 3.7469 × 102 | 3.9105 × 102 | 4.4680 × 102 | 3.8273 × 102 | 5.0563 × 102 | 3.7427 × 102 |
CEC2017_F29 | Mean | 4.1437 × 103 | 4.3638 × 103 | 4.0972 × 103 | 3.8030 × 103 | 3.6447 × 103 | 3.8094 × 103 | 4.9466 × 103 | 3.6595 × 103 |
| Std | 3.1847 × 102 | 3.8875 × 102 | 2.4977 × 102 | 2.1535 × 102 | 1.0393 × 102 | 1.9355 × 102 | 4.8903 × 102 | 1.8744 × 102 |
| FEV | 7.4529 × 102 | 6.9844 × 102 | 7.7266 × 102 | 5.2767 × 102 | 5.1652 × 102 | 5.5320 × 102 | 8.6103 × 102 | 5.0545 × 102 |
CEC2017_F30 | Mean | 6.8100 × 105 | 2.3906 × 106 | 2.5414 × 106 | 8.1994 × 103 | 5.2204 × 104 | 9.5798 × 103 | 1.9892 × 107 | 6.9760 × 103 |
| Std | 1.3978 × 106 | 2.6885 × 106 | 1.7516 × 106 | 2.7841 × 103 | 3.0443 × 104 | 2.9485 × 103 | 1.2561 × 107 | 1.1063 × 103 |
| FEV | 3.2696 × 103 | 2.2775 × 104 | 2.8069 × 105 | 2.2858 × 103 | 1.1279 × 104 | 3.4988 × 103 | 3.4805 × 106 | 2.2809 × 103 |
Rank First | | 2 | 0 | 2 | 2 | 4 | 1 | 0 | 18 |
Mean Rank | 4.7931 | 6.6207 | 4.2414 | 3.1724 | 3.6897 | 3.7931 | 7.7586 | 1.9310 |
Final Rank | 6 | 7 | 5 | 2 | 3 | 4 | 8 | 1 |
Table 7.
Numerical results of the different algorithms on solving the CEC2017 test function set (Dim = 50).
Table 7.
Numerical results of the different algorithms on solving the CEC2017 test function set (Dim = 50).
Problem | Metric | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA | SNDSO |
---|
CEC2017_F1 | Mean | 6.9187 × 106 | 7.4556 × 107 | 8.0054 × 103 | 1.4179 × 106 | 2.9963 × 106 | 1.2886 × 105 | 6.6916 × 108 | 3.5994 × 102 |
| Std | 2.1993 × 107 | 6.7658 × 107 | 9.6800 × 103 | 4.8212 × 106 | 1.0251 × 107 | 1.0840 × 105 | 4.5185 × 108 | 2.1919 × 102 |
| FEV | 4.6129 × 104 | 7.0033 × 101 | 1.0451 × 100 | 1.2442 × 103 | 9.8313 × 103 | 1.2159 × 104 | 2.5159 × 108 | 7.4084 × 101 |
CEC2017_F3 | Mean | 1.2267 × 105 | 1.5214 × 105 | 3.4862 × 104 | 8.1620 × 104 | 2.6856 × 105 | 3.2442 × 104 | 1.7291 × 105 | 7.7868 × 104 |
| Std | 1.4828 × 104 | 2.1483 × 104 | 1.1200 × 104 | 1.8059 × 104 | 3.3418 × 104 | 8.7966 × 103 | 5.0815 × 104 | 1.4224 × 104 |
| FEV | 9.0859 × 104 | 7.7104 × 104 | 1.3097 × 104 | 4.1987 × 104 | 2.1575 × 105 | 1.7594 × 104 | 1.0912 × 105 | 1.2900 × 104 |
CEC2017_F4 | Mean | 6.0746 × 102 | 8.7549 × 102 | 5.8011 × 102 | 5.8780 × 102 | 6.4978 × 102 | 5.5063 × 102 | 1.0277 × 103 | 4.7390 × 102 |
| Std | 6.9045 × 101 | 6.5721 × 102 | 4.4475 × 101 | 5.3160 × 101 | 2.1518 × 101 | 4.6665 × 101 | 1.5149 × 102 | 1.9662 × 101 |
| FEV | 7.4978 × 101 | 1.9135 × 102 | 7.5973 × 101 | 5.8647 × 101 | 2.0447 × 102 | 7.9359 × 101 | 3.9024 × 102 | 4.1900 × 101 |
CEC2017_F5 | Mean | 7.5092 × 102 | 9.7259 × 102 | 8.2475 × 102 | 7.4058 × 102 | 7.9370 × 102 | 8.2279 × 102 | 9.5229 × 102 | 6.2193 × 102 |
| Std | 5.5773 × 101 | 9.3113 × 101 | 7.8566 × 101 | 2.9907 × 101 | 1.6782 × 101 | 4.0968 × 101 | 5.6026 × 101 | 1.7461 × 101 |
| FEV | 1.7688 × 102 | 2.6013 × 102 | 1.8116 × 102 | 1.6417 × 102 | 2.6179 × 102 | 2.3185 × 102 | 3.6051 × 102 | 9.7653 × 101 |
CEC2017_F6 | Mean | 6.2409 × 102 | 6.6478 × 102 | 6.5591 × 102 | 6.3224 × 102 | 6.0011 × 102 | 6.0872 × 102 | 6.8597 × 102 | 6.0086 × 102 |
| Std | 7.4489 × 100 | 9.9216 × 100 | 8.7479 × 100 | 6.3110 × 100 | 1.7096 × 10−1 | 9.2411 × 100 | 1.0239 × 101 | 5.2345 × 10−1 |
| FEV | 9.2075 × 100 | 4.8632 × 101 | 4.0624 × 101 | 2.2247 × 101 | 1.7858 × 10−4 | 1.1440 × 100 | 6.2466 × 101 | 3.5901 × 10−8 |
CEC2017_F7 | Mean | 9.2640 × 102 | 1.3833 × 103 | 1.1295 × 103 | 1.2087 × 103 | 1.0142 × 103 | 1.3585 × 103 | 1.7766 × 103 | 1.1296 × 103 |
| Std | 3.7379 × 101 | 1.3256 × 102 | 8.6389 × 101 | 8.6085 × 101 | 1.3079 × 101 | 1.6460 × 102 | 1.0199 × 102 | 4.4823 × 101 |
| FEV | 1.7303 × 102 | 4.3420 × 102 | 2.6969 × 102 | 3.9655 × 102 | 2.8788 × 102 | 3.9277 × 102 | 8.6269 × 102 | 3.4330 × 102 |
CEC2017_F8 | Mean | 1.0545 × 103 | 1.2686 × 103 | 1.0908 × 103 | 1.0483 × 103 | 1.0923 × 103 | 1.1190 × 103 | 1.2704 × 103 | 1.0200 × 103 |
| Std | 5.5812 × 101 | 8.1043 × 101 | 6.6961 × 101 | 2.9157 × 101 | 1.6373 × 101 | 4.0922 × 101 | 6.8501 × 101 | 6.9378 × 1013 |
| FEV | 1.3085 × 102 | 3.2021 × 102 | 1.6218 × 102 | 1.9803 × 102 | 2.5740 × 102 | 2.5571 × 102 | 3.5310 × 102 | 2.1996 × 102 |
CEC2017_F9 | Mean | 3.8962 × 103 | 1.5487 × 104 | 1.2444 × 104 | 1.0414 × 104 | 6.8864 × 103 | 1.0718 × 104 | 2.8280 × 104 | 9.7836 × 102 |
| Std | 1.4863 × 103 | 6.2322 × 103 | 3.4933 × 103 | 4.5990 × 103 | 1.1708 × 103 | 2.3244 × 103 | 9.1805 × 103 | 2.0102 × 101 |
| FEV | 1.1897 × 103 | 4.1300 × 103 | 4.7410 × 103 | 1.5465 × 103 | 3.9690 × 103 | 5.5871 × 103 | 1.5885 × 104 | 5.1329 × 101 |
CEC2017_F10 | Mean | 6.2709 × 103 | 1.0253 × 104 | 7.8386 × 103 | 1.3325 × 104 | 9.4782 × 103 | 6.8469 × 103 | 1.0782 × 104 | 1.1965 × 104 |
| Std | 1.2427 × 103 | 2.0070 × 103 | 1.0851 × 103 | 1.7078 × 103 | 4.3288 × 102 | 8.3414 × 102 | 1.3334 × 103 | 5.6182 × 102 |
| FEV | 3.0879 × 103 | 6.5402 × 103 | 4.0211 × 103 | 7.8028 × 103 | 7.3751 × 103 | 4.2084 × 103 | 7.1276 × 103 | 9.7405 × 103 |
CEC2017_F11 | Mean | 1.4158 × 103 | 1.7496 × 103 | 1.4536 × 103 | 1.3465 × 103 | 4.5340 × 103 | 1.3105 × 103 | 2.4296 × 103 | 1.2678 × 103 |
| Std | 6.0419 × 101 | 3.5635 × 102 | 9.0311 × 101 | 5.2032 × 101 | 1.3635 × 103 | 1.3283 × 102 | 4.6763 × 102 | 2.7636 × 101 |
| FEV | 1.7758 × 102 | 1.7585 × 102 | 2.2206 × 102 | 1.5636 × 102 | 9.5835 × 102 | 8.7702 × 101 | 7.0643 × 102 | 1.1480 × 102 |
CEC2017_F12 | Mean | 4.8320 × 107 | 2.1037 × 108 | 3.5248 × 107 | 1.8175 × 106 | 9.5338 × 107 | 6.3373 × 106 | 5.3407 × 108 | 1.1190 × 106 |
| Std | 5.1193 × 107 | 1.4152 × 108 | 2.3263 × 107 | 1.5567 × 106 | 4.1975 × 107 | 4.8968 × 106 | 2.8960 × 108 | 6.1463 × 105 |
| FEV | 3.1048 × 106 | 1.9643 × 107 | 5.9824 × 106 | 2.4000 × 105 | 2.1259 × 107 | 9.8789 × 105 | 1.0417 × 108 | 2.3581 × 105 |
CEC2017_F13 | Mean | 2.0862 × 106 | 6.8353 × 106 | 1.5866 × 105 | 1.1587 × 104 | 2.9434 × 106 | 1.9045 × 104 | 8.7713 × 106 | 5.4735 × 103 |
| Std | 3.8336 × 106 | 9.5547 × 106 | 1.3681 × 105 | 8.4234 × 103 | 3.1917 × 106 | 1.3722 × 104 | 1.0043 × 107 | 5.5104 × 103 |
| FEV | 6.5636 × 103 | 5.3423 × 104 | 3.6804 × 104 | 1.1738 × 103 | 5.2960 × 104 | 2.2740 × 103 | 9.6757 × 105 | 4.0670 × 102 |
CEC2017_F14 | Mean | 3.2331 × 105 | 1.7161 × 106 | 1.1267 × 105 | 7.0472 × 104 | 1.6706 × 106 | 1.9638 × 105 | 1.9814 × 106 | 6.2322 × 104 |
| Std | 3.9031 × 105 | 2.0354 × 106 | 9.6405 × 104 | 6.9838 × 104 | 7.6149 × 105 | 1.6106 × 105 | 1.3820 × 106 | 5.6623 × 104 |
| FEV | 2.1032 × 104 | 7.0635 × 103 | 1.8319 × 104 | 2.8652 × 103 | 3.6429 × 105 | 7.1815 × 103 | 3.8966 × 105 | 2.8123 × 103 |
CEC2017_F15 | Mean | 5.1368 × 105 | 9.9105 × 106 | 7.3417 × 104 | 1.0053 × 104 | 3.2126 × 105 | 1.0031 × 104 | 8.9389 × 105 | 5.2488 × 103 |
| Std | 9.5256 × 105 | 4.1805 × 107 | 5.8702 × 104 | 6.3377 × 103 | 5.4247 × 105 | 7.0448 × 103 | 1.2026 × 106 | 2.6713 × 103 |
| FEV | 1.7157 × 103 | 9.8815 × 103 | 1.7637 × 104 | 6.5671 × 102 | 1.8698 × 104 | 2.9670 × 102 | 4.8331 × 104 | 2.3731 × 102 |
CEC2017_F16 | Mean | 3.7962 × 103 | 4.6345 × 103 | 3.4240 × 103 | 2.8993 × 103 | 3.4476 × 103 | 3.3279 × 103 | 5.4054 × 103 | 2.9814 × 103 |
| Std | 5.0250 × 102 | 5.5184 × 102 | 4.2740 × 102 | 4.1400 × 102 | 3.1194 × 102 | 4.5405 × 102 | 9.2631 × 102 | 4.6252 × 1013 |
| FEV | 1.0685 × 103 | 1.8452 × 103 | 1.1548 × 103 | 6.3845 × 102 | 1.2548 × 103 | 9.3065 × 102 | 2.3727 × 103 | 1.3814 × 103 |
CEC2017_F17 | Mean | 3.4495 × 103 | 3.9403 × 103 | 3.3833 × 103 | 2.9285 × 103 | 3.2845 × 103 | 3.3346 × 103 | 4.1581 × 103 | 2.7380 × 103 |
| Std | 3.8626 × 102 | 4.1380 × 102 | 2.6674 × 102 | 2.3148 × 102 | 2.5534 × 102 | 3.3678 × 102 | 5.5396 × 102 | 9.1598 × 101 |
| FEV | 8.6558 × 102 | 1.0934 × 103 | 1.2407 × 103 | 7.8211 × 102 | 6.5466 × 102 | 1.0820 × 103 | 1.5645 × 103 | 9.7232 × 102 |
CEC2017_F18 | Mean | 4.1225 × 106 | 5.6941 × 106 | 1.4912 × 106 | 1.3823 × 106 | 6.3645 × 106 | 1.4399 × 106 | 1.4816 × 107 | 7.5714 × 105 |
| Std | 4.1214 × 106 | 5.5995 × 106 | 1.2275 × 106 | 1.5643 × 106 | 2.8237 × 106 | 9.7924 × 105 | 1.4508 × 107 | 8.5371 × 105 |
| FEV | 1.8350 × 105 | 2.3902 × 105 | 1.4596 × 105 | 1.5753 × 105 | 1.1319 × 106 | 1.2854 × 105 | 3.0932 × 106 | 4.9942 × 104 |
CEC2017_F19 | Mean | 7.4091 × 105 | 1.7095 × 106 | 2.2760 × 106 | 1.6291 × 104 | 9.4985 × 104 | 2.0754 × 104 | 4.1310 × 106 | 1.5621 × 104 |
| Std | 1.5208 × 106 | 2.1557 × 106 | 1.1176 × 106 | 8.8087 × 103 | 7.2886 × 104 | 1.0205 × 104 | 4.5821 × 106 | 1.0684 × 104 |
| FEV | 8.0289 × 102 | 2.2697 × 104 | 3.0789 × 104 | 2.6735 × 102 | 1.6106 × 104 | 2.1731 × 103 | 3.7980 × 104 | 2.4295 × 102 |
CEC2017_F20 | Mean | 3.1515 × 103 | 3.5412 × 103 | 3.1829 × 103 | 3.2374 × 103 | 3.1152 × 103 | 3.3052 × 103 | 3.7748 × 103 | 2.8105 × 103 |
| Std | 2.6731 × 102 | 3.0993 × 102 | 3.3733 × 102 | 3.8443 × 102 | 1.3312 × 102 | 3.1463 × 102 | 4.1470 × 102 | 4.6252 × 1013 |
| FEV | 5.9017 × 102 | 7.2049 × 102 | 4.0829 × 102 | 4.5669 × 102 | 9.1138 × 102 | 6.9915 × 102 | 8.1249 × 102 | 8.1047 × 102 |
CEC2017_F21 | Mean | 2.5461 × 103 | 2.8119 × 103 | 2.5766 × 103 | 2.5385 × 103 | 2.6145 × 103 | 2.5708 × 103 | 2.9476 × 103 | 2.5539 × 103 |
| Std | 5.6417 × 101 | 1.1103 × 102 | 5.9013 × 101 | 4.0144 × 101 | 1.7521 × 101 | 4.5508 × 101 | 1.1614 × 102 | 3.4725 × 100 |
| FEV | 3.4883 × 102 | 4.0307 × 102 | 3.5726 × 102 | 3.8067 × 102 | 4.7680 × 102 | 3.7283 × 102 | 6.2497 × 102 | 3.4814 × 102 |
CEC2017_F22 | Mean | 8.4564 × 103 | 1.1198 × 104 | 9.6777 × 103 | 1.2086 × 104 | 1.1192 × 104 | 9.4874 × 103 | 1.3106 × 104 | 1.3641 × 104 |
| Std | 1.1669 × 103 | 1.4956 × 103 | 9.5175 × 102 | 5.2042 × 103 | 4.0537 × 102 | 9.6525 × 102 | 1.2338 × 103 | 4.5931 × 102 |
| Fev | 3.6249 × 103 | 6.1686 × 103 | 6.0139 × 103 | 1.1986 × 102 | 8.0207 × 103 | 5.8052 × 103 | 8.5548 × 103 | 1.0371 × 104 |
CEC2017_F23 | Mean | 3.1523 × 103 | 3.3931 × 103 | 2.9982 × 103 | 3.0791 × 103 | 3.0451 × 103 | 3.0896 × 103 | 3.7187 × 103 | 2.9921 × 103 |
| Std | 1.0811 × 102 | 1.7441 × 102 | 6.3050 × 101 | 9.7515 × 101 | 1.9855 × 101 | 7.2203 × 101 | 1.9317 × 102 | 1.3876 × 10−12 |
| FEV | 6.6113 × 102 | 7.9823 × 102 | 6.0478 × 102 | 5.9582 × 102 | 7.1553 × 102 | 6.5093 × 102 | 1.1280 × 103 | 6.9206 × 102 |
CEC2017_F24 | Mean | 3.2843 × 103 | 3.6589 × 103 | 3.1353 × 103 | 3.2079 × 103 | 3.3189 × 103 | 3.4237 × 103 | 3.7508 × 103 | 3.2258 × 103 |
| Std | 1.0628 × 102 | 1.8788 × 102 | 5.4742 × 101 | 7.5968 × 101 | 2.7426 × 101 | 8.6706 × 101 | 1.4682 × 102 | 4.5243 × 101 |
| FEV | 7.0727 × 102 | 9.0542 × 102 | 6.3868 × 102 | 6.8689 × 102 | 8.6670 × 102 | 8.4307 × 102 | 1.0048 × 103 | 7.4271 × 102 |
CEC2017_F25 | Mean | 3.0912 × 103 | 3.1443 × 103 | 3.0614 × 103 | 3.1051 × 103 | 3.0832 × 103 | 3.0959 × 103 | 3.3785 × 103 | 3.0403 × 103 |
| Std | 3.8128 × 101 | 5.8831 × 101 | 2.7413 × 101 | 2.8687 × 101 | 2.5435 × 101 | 2.1759 × 101 | 1.2905 × 102 | 6.5164 × 100 |
| FEV | 5.1172 × 102 | 5.5999 × 102 | 5.1491 × 102 | 5.5081 × 102 | 5.4066 × 102 | 5.4804 × 102 | 6.2332 × 102 | 5.1054 × 102 |
CEC2017_F26 | Mean | 8.5278 × 103 | 8.9769 × 103 | 5.5778 × 103 | 8.8695 × 103 | 6.8610 × 103 | 5.5005 × 103 | 1.3040 × 104 | 4.9156 × 103 |
| Std | 1.0729 × 103 | 2.0750 × 103 | 2.1942 × 103 | 1.8372 × 103 | 2.8307 × 102 | 3.0981 × 103 | 1.5662 × 103 | 1.2641 × 102 |
| FEV | 3.8610 × 103 | 1.1398 × 103 | 3.0000 × 102 | 1.6879 × 103 | 3.3537 × 103 | 3.0798 × 102 | 6.8374 × 103 | 2.2925 × 103 |
CEC2017_F27 | Mean | 3.7602 × 103 | 3.9652 × 103 | 3.5008 × 103 | 3.7192 × 103 | 3.3612 × 103 | 3.6790 × 103 | 4.3759 × 103 | 3.3521 × 103 |
| Std | 1.2393 × 102 | 1.6747 × 102 | 1.0366 × 102 | 1.5364 × 102 | 3.2691 × 101 | 1.6853 × 102 | 4.6830 × 102 | 6.1003 × 10−1 |
| FEV | 8.4008 × 102 | 9.7285 × 102 | 6.5867 × 102 | 7.2196 × 102 | 5.8008 × 102 | 7.4193 × 102 | 1.0044 × 103 | 5.7849 × 102 |
CEC2017_F28 | Mean | 3.3211 × 103 | 4.7840 × 103 | 3.3189 × 103 | 3.3785 × 103 | 3.7817 × 103 | 3.3655 × 103 | 4.0351 × 103 | 3.3181 × 103 |
| Std | 2.8688 × 101 | 1.8489 × 103 | 3.4704 × 101 | 4.3927 × 101 | 8.2740 × 102 | 3.5919 × 101 | 2.3307 × 102 | 6.6993 × 10−1 |
| FEV | 4.6491 × 102 | 5.4597 × 102 | 4.5936 × 102 | 4.9127 × 102 | 5.5535 × 102 | 4.9703 × 102 | 9.0185 × 102 | 4.5730 × 102 |
CEC2017_F29 | Mean | 5.1569 × 103 | 6.1511 × 103 | 5.0660 × 103 | 4.8640 × 103 | 4.2899 × 103 | 4.3632 × 103 | 7.8499 × 103 | 4.1893 × 103 |
| Std | 5.4474 × 102 | 1.1348 × 103 | 3.9181 × 102 | 4.8829 × 102 | 1.9336 × 102 | 2.8144 × 102 | 1.1466 × 103 | 4.4961 × 102 |
| FEV | 1.3573 × 103 | 1.7673 × 103 | 1.3787 × 103 | 1.2392 × 103 | 8.9380 × 102 | 8.7807 × 102 | 3.3468 × 103 | 6.3319 × 102 |
CEC2017_F30 | Mean | 8.7878 × 106 | 3.1713 × 107 | 6.3653 × 107 | 1.0419 × 106 | 2.3058 × 106 | 1.0052 × 106 | 1.7282 × 108 | 1.1149 × 106 |
| Std | 1.9232 × 107 | 2.7160 × 107 | 1.2583 × 107 | 1.9536 × 105 | 5.3038 × 105 | 2.9745 × 105 | 5.6393 × 107 | 2.5320 × 105 |
| FEV | 8.2780 × 105 | 5.1846 × 106 | 3.0958 × 107 | 7.5182 × 105 | 1.2682 × 106 | 7.2445 × 105 | 7.8861 × 107 | 7.0535 × 105 |
Rank First | | 3 | 0 | 1 | 2 | 1 | 2 | 0 | 20 |
Mean Rank | 4.2069 | 6.7931 | 3.8621 | 3.5517 | 4.4483 | 3.5172 | 7.7241 | 1.8966 |
Final Rank | 5 | 7 | 4 | 3 | 6 | 2 | 8 | 1 |
Table 8.
Numerical results of the different algorithms on solving the CEC2017 test function set for (Dim = 100).
Table 8.
Numerical results of the different algorithms on solving the CEC2017 test function set for (Dim = 100).
Problem | Metric | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA | SNDSO |
---|
CEC2017_F1 | Mean | 1.4868 × 107 | 1.2012 × 1010 | 1.7449 × 104 | 4.7467 × 109 | 1.4211 × 109 | 1.6540 × 107 | 1.3190 × 1010 | 4.1728 × 106 |
| Std | 9.8621 × 106 | 3.0708 × 1010 | 1.8279 × 104 | 3.2922 × 109 | 1.0071 × 109 | 8.0912 × 106 | 2.9395 × 109 | 4.3502 × 106 |
| FEV | 4.7215 × 106 | 9.3130 × 108 | 1.4521 × 102 | 5.2721 × 108 | 1.1191 × 108 | 6.2539 × 106 | 8.1442 × 109 | 1.0330 × 106 |
CEC2017_F3 | Mean | 3.1826 × 105 | 3.5045 × 105 | 3.3968 × 105 | 3.5863 × 105 | 7.5227 × 105 | 1.7160 × 105 | 8.8699 × 105 | 2.8479 × 105 |
| Std | 1.6276 × 104 | 2.1552 × 104 | 5.6005 × 104 | 3.1149 × 104 | 6.8323 × 104 | 1.8385 × 104 | 1.4860 × 105 | 3.0252 × 104 |
| FEV | 2.7353 × 105 | 3.1973 × 105 | 2.1703 × 105 | 3.1219 × 105 | 6.0402 × 105 | 1.3234 × 105 | 6.0987 × 105 | 1.1546 × 105 |
CEC2017_F4 | Mean | 8.0983 × 102 | 7.1986 × 103 | 7.4117 × 102 | 1.5056 × 103 | 1.4770 × 103 | 8.8926 × 102 | 3.4724 × 103 | 7.1327 × 102 |
| Std | 7.7472 × 101 | 1.2426 × 104 | 7.0513 × 101 | 6.0066 × 102 | 2.7408 × 102 | 6.1654 × 101 | 8.1815 × 102 | 5.7007 × 100 |
| FEV | 2.7359 × 102 | 6.8363 × 102 | 2.1066 × 102 | 6.3863 × 102 | 7.4973 × 102 | 3.1354 × 102 | 1.8707 × 103 | 2.1023 × 102 |
CEC2017_F5 | Mean | 1.1981 × 103 | 1.4930 × 103 | 1.3424 × 103 | 1.1735 × 103 | 1.5732 × 103 | 1.3275 × 103 | 1.6677 × 103 | 1.0779 × 103 |
| Std | 1.3490 × 102 | 2.3215 × 102 | 9.8293 × 101 | 7.7532 × 101 | 3.4885 × 101 | 8.0051 × 101 | 1.0943 × 102 | 2.9857 × 100 |
| FEV | 4.2868 × 102 | 6.7197 × 102 | 6.4373 × 102 | 5.5087 × 102 | 9.7439 × 102 | 5.8016 × 102 | 9.3461 × 102 | 4.2197 × 102 |
CEC2017_F6 | Mean | 6.3946 × 102 | 6.6900 × 102 | 6.6458 × 102 | 6.4883 × 102 | 6.1423 × 102 | 6.2281 × 102 | 6.9402 × 102 | 6.1089 × 102 |
| Std | 6.4676 × 100 | 1.3498 × 101 | 6.7739 × 100 | 3.5262 × 100 | 3.1049 × 100 | 1.1149 × 101 | 9.6397 × 100 | 3.9044 × 100 |
| FEV | 2.7805 × 101 | 5.1826 × 101 | 5.2668 × 101 | 4.1098 × 101 | 7.5212 × 100 | 5.7378 × 100 | 7.9053 × 101 | 4.0310 × 100 |
CEC2017_F7 | Mean | 1.3392 × 103 | 2.7088 × 103 | 2.0119 × 103 | 2.3081 × 103 | 1.8157 × 103 | 2.7683 × 103 | 3.5326 × 103 | 2.0046 × 103 |
| Std | 6.6652 × 101 | 2.3312 × 102 | 2.1511 × 102 | 2.2947 × 102 | 4.0286 × 101 | 2.2595 × 102 | 2.0664 × 102 | 9.0887 × 101 |
| FEV | 4.9533 × 102 | 1.5441 × 103 | 9.5752 × 102 | 1.0520 × 103 | 1.0377 × 103 | 1.5885 × 103 | 2.4294 × 103 | 1.1673 × 103 |
CEC2017_F8 | Mean | 1.5331 × 103 | 1.8716 × 103 | 1.6860 × 103 | 1.5646 × 103 | 1.8610 × 103 | 1.7292 × 103 | 2.1121 × 103 | 1.7468 × 103 |
| Std | 1.1504 × 102 | 2.2745 × 102 | 1.2936 × 102 | 6.2945 × 101 | 3.3031 × 101 | 1.0320 × 102 | 1.1743 × 102 | 6.3554 × 101 |
| FEV | 5.0844 × 102 | 7.9423 × 102 | 6.0075 × 102 | 5.8318 × 102 | 9.8365 × 102 | 7.3487 × 102 | 1.1584 × 103 | 8.2957 × 102 |
CEC2017_F9 | Mean | 1.3834 × 104 | 4.8142 × 104 | 2.8214 × 104 | 5.0604 × 104 | 6.0725 × 104 | 2.4006 × 104 | 5.4802 × 104 | 7.5261 × 103 |
| Std | 4.6874 × 103 | 1.9125 × 104 | 4.0159 × 103 | 7.3175 × 103 | 7.3588 × 103 | 1.1907 × 103 | 1.4342 × 104 | 4.3529 × 102 |
| FEV | 7.0570 × 103 | 2.0040 × 104 | 1.9304 × 104 | 3.2202 × 104 | 4.4448 × 104 | 1.8221 × 104 | 3.9125 × 104 | 5.8194 × 103 |
CEC2017_F10 | Mean | 1.7802 × 104 | 2.1714 × 104 | 1.5705 × 104 | 3.0993 × 104 | 2.5432 × 104 | 1.4925 × 104 | 2.4489 × 104 | 1.3257 × 104 |
| Std | 6.5174 × 103 | 5.0846 × 103 | 2.0305 × 103 | 1.3443 × 103 | 5.5155 × 102 | 1.2892 × 103 | 2.1063 × 103 | 3.0233 × 102 |
| FEV | 9.5836 × 103 | 1.2903 × 104 | 8.9378 × 103 | 2.4359 × 104 | 2.3096 × 104 | 1.1515 × 104 | 1.9736 × 104 | 8.8399 × 103 |
CEC2017_F11 | Mean | 1.9876 × 104 | 9.4062 × 104 | 4.0115 × 103 | 5.2462 × 103 | 7.9567 × 104 | 2.5881 × 104 | 1.1377 × 105 | 3.6060 × 103 |
| Std | 5.3712 × 103 | 4.3169 × 104 | 4.9910 × 102 | 1.5042 × 103 | 1.5685 × 104 | 8.2292 × 103 | 4.3960 × 104 | 8.1875 × 101 |
| FEV | 8.9681 × 103 | 2.4329 × 104 | 1.8129 × 103 | 2.1088 × 103 | 4.6559 × 104 | 1.2416 × 104 | 5.8445 × 104 | 2.4255 × 103 |
CEC2017_F12 | Mean | 2.9106 × 108 | 1.1369 × 109 | 3.5080 × 108 | 2.0413 × 108 | 1.2527 × 109 | 5.2596 × 107 | 2.5511 × 109 | 1.9809 × 107 |
| Std | 2.4654 × 108 | 6.7029 × 108 | 1.4736 × 108 | 4.2336 × 108 | 4.9437 × 108 | 2.0206 × 107 | 8.1236 × 108 | 1.0721 × 107 |
| FEV | 2.9009 × 107 | 2.1694 × 108 | 1.0297 × 108 | 2.4188 × 107 | 4.7365 × 108 | 1.9081 × 107 | 8.4941 × 108 | 7.3683 × 106 |
CEC2017_F13 | Mean | 5.9408 × 106 | 7.8630 × 107 | 7.9810 × 104 | 1.8059 × 104 | 1.4897 × 107 | 7.1682 × 104 | 2.1065 × 107 | 9.1063 × 103 |
| Std | 9.2130 × 106 | 1.1189 × 108 | 2.6687 × 104 | 9.1815 × 103 | 3.7752 × 107 | 1.7234 × 105 | 1.2487 × 107 | 6.0468 × 103 |
| FEV | 1.3747 × 104 | 1.1936 × 105 | 3.9130 × 104 | 5.3202 × 103 | 4.5138 × 105 | 9.1499 × 103 | 5.9844 × 106 | 2.1061 × 103 |
CEC2017_F14 | Mean | 3.9787 × 106 | 8.2981 × 106 | 1.0560 × 106 | 9.2195 × 105 | 1.8961 × 107 | 1.8705 × 106 | 8.2176 × 106 | 8.6645 × 105 |
| Std | 3.4041 × 106 | 7.4345 × 106 | 5.7090 × 105 | 6.9576 × 105 | 6.3840 × 106 | 7.9368 × 105 | 3.2630 × 106 | 6.5123 × 105 |
| FEV | 3.2703 × 105 | 1.2417 × 106 | 2.9645 × 105 | 1.0003 × 105 | 3.7998 × 106 | 8.2419 × 105 | 1.6813 × 106 | 1.8694 × 105 |
CEC2017_F15 | Mean | 2.2059 × 106 | 7.7609 × 106 | 6.5415 × 104 | 4.9141 × 103 | 1.6483 × 106 | 1.5691 × 104 | 6.4756 × 106 | 4.7736 × 103 |
| Std | 3.7094 × 106 | 2.0048 × 107 | 2.3604 × 104 | 3.1590 × 103 | 9.4550 × 105 | 1.6832 × 104 | 8.2966 × 106 | 3.0661 × 103 |
| FEV | 3.9203 × 103 | 8.2628 × 104 | 2.4897 × 104 | 6.9523 × 102 | 1.8201 × 105 | 7.3170 × 102 | 6.5846 × 105 | 6.5191 × 102 |
CEC2017_F16 | Mean | 6.2231 × 103 | 8.2772 × 103 | 6.6311 × 103 | 5.5385 × 103 | 8.0065 × 103 | 6.0381 × 103 | 1.2883 × 104 | 4.3093 × 103 |
| Std | 1.0154 × 103 | 1.5317 × 103 | 7.9175 × 102 | 6.8389 × 102 | 3.6608 × 102 | 8.1836 × 102 | 2.0352 × 103 | 2.7394 × 102 |
| FEV | 3.1390 × 103 | 3.8411 × 103 | 3.7423 × 103 | 2.4399 × 103 | 5.6271 × 103 | 2.5774 × 103 | 8.1093 × 103 | 2.4399 × 103 |
CEC2017_F17 | Mean | 5.9066 × 103 | 7.4802 × 103 | 5.4078 × 103 | 5.5310 × 103 | 7.0543 × 103 | 5.2076 × 103 | 8.5219 × 103 | 4.7031 × 103 |
| Std | 6.1978 × 102 | 9.7309 × 102 | 4.8077 × 102 | 6.7704 × 102 | 3.0935 × 102 | 4.7908 × 102 | 1.4958 × 103 | 6.8721 × 101 |
| FEV | 2.9537 × 103 | 3.9465 × 103 | 2.7332 × 103 | 2.4419 × 103 | 4.7217 × 103 | 2.6648 × 103 | 4.7590 × 103 | 2.9166 × 103 |
CEC2017_F18 | Mean | 6.9131 × 106 | 1.0726 × 107 | 2.3631 × 106 | 1.6111 × 106 | 2.3376 × 107 | 2.7110 × 106 | 6.7446 × 106 | 1.1198 × 106 |
| Std | 5.3414 × 106 | 8.7218 × 106 | 1.2875 × 106 | 8.5861 × 105 | 7.8307 × 106 | 9.1411 × 105 | 2.4345 × 106 | 2.5593 × 105 |
| FEV | 1.1602 × 106 | 8.4044 × 105 | 7.3946 × 105 | 3.3682 × 105 | 1.0564 × 107 | 1.2178 × 106 | 2.0269 × 106 | 3.2756 × 105 |
CEC2017_F19 | Mean | 7.6261 × 106 | 1.2565 × 107 | 8.7799 × 106 | 6.5990 × 103 | 4.7735 × 106 | 9.7817 × 103 | 3.4037 × 107 | 3.4885 × 103 |
| Std | 1.1230 × 107 | 1.2509 × 107 | 4.3892 × 106 | 1.0065 × 104 | 2.4319 × 106 | 8.8265 × 103 | 2.3245 × 107 | 1.4784 × 103 |
| FEV | 1.0252 × 103 | 6.4900 × 105 | 1.0673 × 106 | 5.5047 × 102 | 1.8684 × 106 | 4.9506 × 102 | 7.8735 × 106 | 3.7586 × 102 |
CEC2017_F20 | Mean | 5.5846 × 103 | 6.3989 × 103 | 5.3158 × 103 | 6.6005 × 103 | 5.6453 × 103 | 5.2582 × 103 | 6.5832 × 103 | 4.8005 × 103 |
| Std | 1.0286 × 103 | 5.3245 × 102 | 6.3847 × 102 | 9.7729 × 102 | 2.8089 × 102 | 5.6959 × 102 | 7.2198 × 102 | 4.5339 × 102 |
| FEV | 2.2168 × 103 | 3.4322 × 103 | 1.9833 × 103 | 1.7423 × 103 | 2.9388 × 103 | 2.2310 × 103 | 3.2995 × 103 | 2.2949 × 103 |
CEC2017_F21 | Mean | 3.0570 × 103 | 3.8122 × 103 | 3.1587 × 103 | 3.1650 × 103 | 3.4220 × 103 | 3.0887 × 103 | 4.0694 × 103 | 3.0026 × 103 |
| Std | 1.3963 × 102 | 2.0786 × 102 | 1.4560 × 102 | 1.3860 × 102 | 3.7380 × 101 | 9.5127 × 101 | 1.6996 × 102 | 3.6710 × 101 |
| FEV | 6.8050 × 102 | 1.2228 × 103 | 7.6721 × 102 | 7.9855 × 102 | 1.2575 × 103 | 7.7488 × 102 | 1.6789 × 103 | 6.7067 × 102 |
CEC2017_F22 | Mean | 1.8219 × 104 | 2.4529 × 104 | 1.8144 × 104 | 3.3452 × 104 | 2.7358 × 104 | 1.9550 × 104 | 2.8377 × 104 | 1.4152 × 104 |
| Std | 5.9709 × 103 | 4.1212 × 103 | 1.2555 × 103 | 1.3092 × 103 | 6.5514 × 102 | 1.5628 × 103 | 1.8139 × 103 | 4.8339 × 102 |
| FEV | 9.8554 × 103 | 1.5735 × 104 | 1.4081 × 104 | 2.7075 × 104 | 2.3742 × 104 | 1.4398 × 104 | 2.3383 × 104 | 1.1282 × 104 |
CEC2017_F23 | Mean | 3.7221 × 103 | 4.5920 × 103 | 3.5889 × 103 | 3.8700 × 103 | 3.6461 × 103 | 3.4056 × 103 | 4.8941 × 103 | 3.3200 × 103 |
| Std | 1.8177 × 102 | 2.6556 × 102 | 1.2188 × 102 | 1.8632 × 102 | 3.3515 × 101 | 9.4038 × 101 | 2.5252 × 102 | 1.0746 × 102 |
| FEV | 1.1322 × 103 | 1.6966 × 103 | 1.0322 × 103 | 1.2778 × 103 | 1.2450 × 103 | 9.1690 × 102 | 2.1345 × 103 | 8.4364 × 102 |
CEC2017_F24 | Mean | 4.7531 × 103 | 5.6227 × 103 | 4.1241 × 103 | 4.7745 × 103 | 4.2536 × 103 | 4.3269 × 103 | 6.1524 × 103 | 4.1195 × 103 |
| Std | 2.7775 × 102 | 5.3680 × 102 | 1.5204 × 102 | 2.7944 × 102 | 4.5410 × 101 | 1.4636 × 102 | 3.7959 × 102 | 5.9064 × 100 |
| FEV | 1.8741 × 103 | 2.3076 × 103 | 1.4539 × 103 | 1.8498 × 103 | 1.7744 × 103 | 1.6858 × 103 | 3.1289 × 103 | 1.3184 × 103 |
CEC2017_F25 | Mean | 3.5054 × 103 | 4.1471 × 103 | 3.4398 × 103 | 3.9761 × 103 | 4.2853 × 103 | 3.5404 × 103 | 4.9228 × 103 | 3.4132 × 103 |
| Std | 9.1402 × 101 | 1.9847 × 103 | 6.3834 × 101 | 2.4419 × 102 | 2.0112 × 102 | 9.5560 × 101 | 2.9775 × 102 | 1.6637 × 101 |
| FEV | 8.4263 × 102 | 9.4549 × 102 | 8.3100 × 102 | 1.0874 × 103 | 1.4336 × 103 | 8.4758 × 102 | 1.9113 × 103 | 9.0037 × 102 |
CEC2017_F26 | Mean | 1.9256 × 104 | 2.0619 × 104 | 1.4997 × 104 | 2.7735 × 104 | 1.6571 × 104 | 1.9742 × 104 | 3.4396 × 104 | 1.2943 × 104 |
| Std | 2.2483 × 103 | 3.3522 × 103 | 2.9101 × 103 | 1.9883 × 103 | 4.2314 × 102 | 7.2992 × 103 | 3.3903 × 103 | 1.0609 × 103 |
| FEV | 1.3375 × 104 | 1.2864 × 104 | 5.1434 × 102 | 2.1666 × 104 | 1.3152 × 104 | 1.4614 × 103 | 2.5222 × 104 | 9.0576 × 103 |
CEC2017_F27 | Mean | 4.0880 × 103 | 4.5178 × 103 | 3.7788 × 103 | 4.4496 × 103 | 3.6224 × 103 | 3.8888 × 103 | 5.4459 × 103 | 3.6176 × 103 |
| Std | 1.9366 × 102 | 4.7320 × 102 | 1.3347 × 102 | 3.6220 × 102 | 4.2204 × 101 | 1.4642 × 102 | 6.8519 × 102 | 8.1102 × 101 |
| FEV | 1.0197 × 103 | 1.2296 × 103 | 8.5464 × 102 | 1.0258 × 103 | 7.8690 × 102 | 9.3425 × 102 | 1.6822 × 103 | 7.8548 × 102 |
CEC2017_F28 | Mean | 3.5962 × 103 | 1.3182 × 104 | 3.5170 × 103 | 4.4893 × 103 | 1.5828 × 104 | 3.7064 × 103 | 6.1924 × 103 | 3.6687 × 103 |
| Std | 5.7846 × 101 | 7.8642 × 103 | 4.3478 × 101 | 4.5355 × 102 | 8.4106 × 101 | 9.2488 × 101 | 5.7373 × 102 | 6.1108 × 101 |
| FEV | 6.8568 × 102 | 1.1881 × 103 | 6.2523 × 102 | 9.8239 × 102 | 1.2906 × 104 | 7.5927 × 102 | 2.4250 × 103 | 6.2190 × 102 |
CEC2017_F29 | Mean | 8.6005 × 103 | 1.0030 × 104 | 9.3258 × 103 | 8.0419 × 103 | 7.9940 × 103 | 6.9411 × 103 | 1.5746 × 104 | 6.4913 × 103 |
| Std | 1.0500 × 103 | 1.4044 × 103 | 6.8766 × 102 | 6.6941 × 102 | 3.1488 × 102 | 5.2665 × 102 | 2.0600 × 103 | 1.8257 × 102 |
| FEV | 4.0623 × 103 | 4.9307 × 103 | 5.0186 × 103 | 3.9790 × 103 | 4.2054 × 103 | 2.9818 × 103 | 9.3382 × 103 | 3.3646 × 103 |
CEC2017_F30 | Mean | 4.9725 × 105 | 4.7897 × 107 | 7.2061 × 107 | 1.2749 × 105 | 9.4935 × 105 | 3.2649 × 105 | 7.3937 × 108 | 5.9693 × 104 |
| Std | 3.2208 × 105 | 3.8380 × 107 | 3.2944 × 107 | 1.1631 × 105 | 4.0375 × 105 | 2.3659 × 105 | 3.3343 × 108 | 5.0741 × 104 |
| FEV | 1.1235 × 105 | 7.2025 × 106 | 2.4246 × 107 | 1.3064 × 104 | 2.1795 × 105 | 6.7028 × 104 | 2.1584 × 108 | 1.3063 × 104 |
Rank First | | 2 | 0 | 2 | 0 | 0 | 1 | 0 | 24 |
Mean Rank | 3.7931 | 6.5862 | 3.5172 | 4.4828 | 5.4138 | 3.3793 | 7.4828 | 1.3448 |
Final Rank | 4 | 7 | 3 | 5 | 6 | 2 | 8 | 1 |
Table 9.
p-values of CEC2017 test functions (Dim = 30).
Table 9.
p-values of CEC2017 test functions (Dim = 30).
Problem | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA |
---|
CEC2017_F1 | 3.3068 × 10−8/- | 9.7373 × 10−11/- | 2.6520 × 10−6/- | 2.7506 × 10−5/- | 2.8900 × 10−11/- | 5.4788 × 10−10/- | 2.9617 × 10−11/- |
CEC2017_F3 | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/+ | 2.0511 × 10−5/- | 1.2118 × 10−12/- | 1.5363 × 10−4/- | 1.2118 × 10−12/- |
CEC2017_F4 | 3.3593 × 10−11/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- |
CEC2017_F5 | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- |
CEC2017_F6 | 3.0180 × 10−11/- | 3.0180 × 10−11/- | 3.0180 × 10−11/- | 3.0180 × 10−11/- | 2.1183 × 10−11/+ | 1.1731 × 10−9/- | 3.0180 × 10−11/- |
CEC2017_F7 | 1.5964 × 10−7/+ | 1.1674 × 10−5/- | 2.7548 × 10−3/+ | 4.2039 × 10−1/= | 3.0199 × 10−11/+ | 3.8349 × 10−6/- | 3.0199 × 10−11/- |
CEC2017_F8 | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- |
CEC2017_F9 | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- |
CEC2017_F10 | 3.0199 × 10−11/+ | 6.6689 × 10−3/+ | 1.7769 × 10−10/+ | 2.4386 × 10−9/- | 3.0199 × 10−11/+ | 3.0199 × 10−11/+ | 4.6427 × 10−1/= |
CEC2017_F11 | 1.2354 × 10−9/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 4.9074 × 10−9/- | 1.4634 × 10−11/- |
CEC2017_F12 | 2.6695 × 10−9/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 8.4180 × 10−1/= | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- |
CEC2017_F13 | 1.3345 × 10−1/= | 5.5727 × 10−10/- | 4.3106 × 10−8/- | 6.0971 × 10−3/+ | 1.7294 × 10−7/- | 4.0595 × 10−2/+ | 9.9186 × 10−11/- |
CEC2017_F14 | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- |
CEC2017_F15 | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 3.0159 × 10−12/- | 1.7203 × 10−12/- | 7.7129 × 10−11/- | 1.7203 × 10−12/- |
CEC2017_F16 | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 3.3593 × 10−11/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- |
CEC2017_F17 | 2.8628 × 10−11/- | 2.8628 × 10−11/- | 2.8628 × 10−11/- | 9.4287 × 10−11/- | 2.8628 × 10−11/- | 2.8628 × 10−11/- | 2.8628 × 10−11/- |
CEC2017_F18 | 6.0104 × 10−8/- | 2.0283 × 10−7/- | 2.9205 × 10−2/- | 2.6806 × 10−4/- | 1.2057 × 10−10/- | 6.6273 × 10−1/= | 6.7220 × 10−10/- |
CEC2017_F19 | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 3.3593 × 10−11/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- |
CEC2017_F20 | 4.7536 × 10−10/- | 2.5174 × 10−11/- | 2.5174 × 10−11/- | 1.6724 × 10−7/- | 7.3944 × 10−9/- | 4.7536 × 10−10/- | 2.5174 × 10−11/- |
CEC2017_F21 | 3.8481 × 10−3/+ | 3.8249 × 10−9/- | 2.0621 × 10−1/= | 7.0881 × 10−8/+ | 4.1825 × 10−9/+ | 2.4581 × 10−1/= | 5.4941 × 10−11/- |
CEC2017_F22 | 3.8116 × 10−9/- | 6.6672 × 10−11/- | 7.0948 × 10−9/- | 3.3622 × 10−5/- | 2.0265 × 10−9/- | 1.1652 × 10−5/- | 3.0066 × 10−11/- |
CEC2017_F23 | 2.1549 × 10−12/- | 1.7203 × 10−12/- | 2.6318 × 10−8/- | 6.2561 × 10−11/- | 3.7688 × 10−12/- | 2.1549 × 10−12/- | 1.7203 × 10−12/- |
CEC2017_F24 | 6.3533 × 10−2/= | 3.0199 × 10−11/- | 2.6077 × 10−2/+ | 3.7904 × 10−1/= | 1.1737 × 10−9/- | 1.4294 × 10−8/- | 3.0199 × 10−11/- |
CEC2017_F25 | 5.5611 × 10−4/+ | 9.2603 × 10−9/- | 1.8577 × 10−1/= | 8.2919 × 10−6/- | 2.1540 × 10−6/- | 3.5012 × 10−3/- | 3.0199 × 10−11/- |
CEC2017_F26 | 3.8249 × 10−9/- | 1.4294 × 10−8/- | 8.7663 × 10−1/= | 3.4783 × 10−1/= | 6.7869 × 10−2/= | 7.9590 × 10−3/+ | 3.0199 × 10−11/- |
CEC2017_F27 | 1.4733 × 10−7/- | 4.1825 × 10−9/- | 6.3772 × 10−3/- | 5.8587 × 10−6/- | 9.8329 × 10−8/+ | 7.0881 × 10−8/- | 3.6897 × 10−11/- |
CEC2017_F28 | 1.1738 × 10−3/- | 6.0658 × 10−11/- | 2.5188 × 10−1/= | 1.4945 × 10−1/= | 4.0772 × 10−11/- | 5.3951 × 10−1/= | 3.0199 × 10−11/- |
CEC2017_F29 | 1.1023 × 10−8/- | 1.5465 × 10−9/- | 6.5183 × 10−9/- | 3.6709 × 10−3/- | 4.5530 × 10−1/= | 3.5638 × 10−4/- | 6.6955 × 10−11/- |
CEC2017_F30 | 2.3738 × 10−8/- | 2.9916 × 10−11/- | 2.9916 × 10−11/- | 8.2293 × 10−2/= | 2.9916 × 10−11/- | 4.7229 × 10−6/- | 2.9916 × 10−11/- |
+/-/= | 4/23/2 | 1/28/0 | 4/21/4 | 2/21/6 | 5/22/2 | 3/23/3 | 0/28/1 |
Table 10.
p-values of CEC2017 test functions (Dim = 50).
Table 10.
p-values of CEC2017 test functions (Dim = 50).
Problem | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA |
---|
CEC2017_F1 | 2.9119 × 10−11/- | 5.3982 × 10−10/- | 2.7222 × 10−7/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- | 2.9119 × 10−11/- |
CEC2017_F3 | 1.4643 × 10−10/- | 9.9186 × 10−11/- | 6.0658 × 10−11/+ | 3.6322 × 10−1/= | 3.0199 × 10−11/- | 3.6897 × 10−11/+ | 3.3384 × 10−11/- |
CEC2017_F4 | 2.0107 × 10−10/- | 2.8003 × 10−11/- | 2.0107 × 10−10/- | 2.0107 × 10−10/- | 2.8003 × 10−11/- | 2.0946 × 10−9/- | 2.8003 × 10−11/- |
CEC2017_F5 | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- |
CEC2017_F6 | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 4.1997 × 10−10/+ | 1.2057 × 10−10/- | 3.0199 × 10−11/- |
CEC2017_F7 | 3.0199 × 10−11/+ | 2.8716 × 10−10/- | 9.7052 × 10−1/= | 2.6806 × 10−4/- | 3.3384 × 10−11/+ | 4.5726 × 10−9/- | 3.0199 × 10−11/- |
CEC2017_F8 | 4.5559 × 10−3/- | 1.2118 × 10−12/- | 2.2081 × 10−6/- | 1.9139 × 10−7/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- | 1.2118 × 10−12/- |
CEC2017_F9 | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- |
CEC2017_F10 | 3.0199 × 10−11/+ | 4.6390 × 10−5/+ | 3.0199 × 10−11/+ | 3.3681 × 10−5/- | 3.0199 × 10−11/+ | 3.0199 × 10−11/+ | 9.2113 × 10−5/+ |
CEC2017_F11 | 9.6911 × 10−11/- | 9.6911 × 10−11/- | 2.9468 × 10−11/- | 1.9801 × 10−8/- | 2.9468 × 10−11/- | 6.4124 × 10−1/= | 2.9468 × 10−11/- |
CEC2017_F12 | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 4.0595 × 10−2/- | 3.0199 × 10−11/- | 2.2273 × 10−9/- | 3.0199 × 10−11/- |
CEC2017_F13 | 6.1210 × 10−10/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 1.9963 × 10−5/- | 3.0199 × 10−11/- | 3.0811 × 10−8/- | 3.0199 × 10−11/- |
CEC2017_F14 | 7.7387 × 10−6/- | 2.0338 × 10−9/- | 3.3874 × 10−2/- | 6.1001 × 10−1/= | 3.0199 × 10−11/- | 5.4620 × 10−6/- | 3.0199 × 10−11/- |
CEC2017_F15 | 1.9916 × 10−7/- | 2.9321 × 10−11/- | 2.9321 × 10−11/- | 3.1632 × 10−3/- | 2.9321 × 10−11/- | 4.6104 × 10−3/- | 2.9321 × 10−11/- |
CEC2017_F16 | 7.4716 × 10−10/- | 1.2118 × 10−12/- | 1.5363 × 10−4/- | 1.5722 × 10−1/= | 7.4716 × 10−10/- | 9.2965 × 10−4/- | 1.2118 × 10−12/- |
CEC2017_F17 | 8.7020 × 10−8/- | 6.0335 × 10−11/- | 2.2048 × 10−11/- | 6.3408 × 10−4/- | 1.0856 × 10−9/- | 1.6157 × 10−10/- | 2.2048 × 10−11/- |
CEC2017_F18 | 1.1747 × 10−4/- | 3.0103 × 10−7/- | 2.1566 × 10−3/- | 1.1228 × 10−2/- | 8.1527 × 10−11/- | 3.1573 × 10−5/- | 4.5043 × 10−11/- |
CEC2017_F19 | 3.5012 × 10−3/- | 6.0658 × 10−11/- | 4.0772 × 10−11/- | 5.7929 × 10−1/= | 4.1997 × 10−10/- | 4.0595 × 10−2/- | 3.6897 × 10−11/- |
CEC2017_F20 | 7.4716 × 10−10/- | 3.3593 × 10−11/- | 1.3341 × 10−8/- | 2.2081 × 10−6/- | 1.2118 × 10−12/- | 7.4716 × 10−10/- | 1.2118 × 10−12/- |
CEC2017_F21 | 2.8839 × 10−1/= | 4.6957 × 10−10/- | 8.3657 × 10−2/= | 3.3109 × 10−2/+ | 2.4823 × 10−11/- | 7.5997 × 10−2/= | 2.4823 × 10−11/- |
CEC2017_F22 | 3.0199 × 10−11/+ | 2.3897 × 10−8/+ | 3.0199 × 10−11/+ | 3.3285 × 10−1/= | 3.0199 × 10−11/+ | 3.0199 × 10−11/+ | 5.3685 × 10−2/= |
CEC2017_F23 | 3.3593 × 10−11/- | 1.2118 × 10−12/- | 1.0000 × 100/= | 1.5363 × 10−4/- | 1.2118 × 10−12/- | 1.3341 × 10−8/- | 1.2118 × 10−12/- |
CEC2017_F24 | 7.2884 × 10−3/- | 3.3384 × 10−11/- | 4.3106 × 10−8/+ | 2.3399 × 10−1/= | 8.1014 × 10−10/- | 1.2057 × 10−10/- | 3.0199 × 10−11/- |
CEC2017_F25 | 4.3222 × 10−7/- | 2.2048 × 10−11/- | 2.9781 × 10−3/- | 2.2048 × 10−11/- | 4.2336 × 10−10/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- |
CEC2017_F26 | 1.7203 × 10−12/- | 9.7354 × 10−10/- | 6.0696 × 10−2/= | 5.0695 × 10−11/- | 1.7203 × 10−12/- | 6.0696 × 10−2/= | 1.7203 × 10−12/- |
CEC2017_F27 | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 7.2672 × 10−2/= | 1.4634 × 10−11/- | 1.4634 × 10−11/- |
CEC2017_F28 | 8.8753 × 10−1/= | 2.2048 × 10−11/- | 1.8273 × 10−1/= | 6.6892 × 10−9/- | 2.2048 × 10−11/- | 4.2336 × 10−10/- | 2.2048 × 10−11/- |
CEC2017_F29 | 1.8500 × 10−8/- | 9.9186 × 10−11/- | 2.6015 × 10−8/- | 1.8608 × 10−6/- | 5.0120 × 10−2/= | 2.7086 × 10−2/- | 3.0199 × 10−11/- |
CEC2017_F30 | 5.5329 × 10−8/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 4.1191 × 10−1/= | 9.9186 × 10−11/- | 1.3272 × 10−2/+ | 3.0199 × 10−11/- |
+/-/= | 3/24/2 | 2/27/0 | 4/20/5 | 1/21/7 | 4/23/2 | 4/22/3 | 1/27/1 |
Table 11.
p-values of CEC2017 test functions (Dim = 100).
Table 11.
p-values of CEC2017 test functions (Dim = 100).
Problem | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA |
---|
CEC2017_F1 | 1.1023 × 10−8/- | 3.0199 × 10−11/- | 3.0199 × 10−11/+ | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 1.8567 × 10−9/- | 3.0199 × 10−11/- |
CEC2017_F3 | 4.7445 × 10−6/- | 1.9568 × 10−10/- | 5.2650 × 10−5/- | 2.8716 × 10−10/- | 3.0199 × 10−11/- | 3.0199 × 10−11/+ | 3.0199 × 10−11/- |
CEC2017_F4 | 1.5876 × 10−9/- | 1.7203 × 10−12/- | 2.6350 × 10−2/- | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 1.9256 × 10−12/- | 1.7203 × 10−12/- |
CEC2017_F5 | 1.6513 × 10−3/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.6572 × 10−8/- | 1.4634 × 10−11/- | 6.7467 × 10−11/- | 1.4634 × 10−11/- |
CEC2017_F6 | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 9.0307 × 10−4/- | 5.0912 × 10−6/- | 3.0199 × 10−11/- |
CEC2017_F7 | 3.0199 × 10−11/+ | 3.0199 × 10−11/- | 8.5338 × 10−1/= | 3.5201 × 10−7/- | 1.4643 × 10−10/+ | 3.0199 × 10−11/- | 3.0199 × 10−11/- |
CEC2017_F8 | 2.2273 × 10−9/+ | 1.2235 × 10−1/= | 6.3533 × 10−2/= | 9.9186 × 10−11/+ | 1.0702 × 10−9/- | 5.5923 × 10−1/= | 3.0199 × 10−11/- |
CEC2017_F9 | 4.9347 × 10−11/- | 2.7047 × 10−11/- | 2.7047 × 10−11/- | 2.7047 × 10−11/- | 2.7047 × 10−11/- | 2.7047 × 10−11/- | 2.7047 × 10−11/- |
CEC2017_F10 | 1.8155 × 10−3/- | 2.2048 × 10−11/- | 1.2245 × 10−7/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 1.9603 × 10−7/- | 2.2048 × 10−11/- |
CEC2017_F11 | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.8032 × 10−5/- | 6.6759 × 10−8/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- |
CEC2017_F12 | 8.9934 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 9.7555 × 10−10/- | 3.0199 × 10−11/- | 1.3111 × 10−8/- | 3.0199 × 10−11/- |
CEC2017_F13 | 1.3289 × 10−10/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 8.1975 × 10−7/- | 3.0199 × 10−11/- | 7.3803 × 10−10/- | 3.0199 × 10−11/- |
CEC2017_F14 | 7.5991 × 10−7/- | 3.1589 × 10−10/- | 5.7460 × 10−2/= | 8.6499 × 10−1/= | 3.0199 × 10−11/- | 1.2860 × 10−6/- | 4.9752 × 10−11/- |
CEC2017_F15 | 6.1210 × 10−10/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 9.0000 × 10−1/= | 3.0199 × 10−11/- | 9.5139 × 10−6/- | 3.0199 × 10−11/- |
CEC2017_F16 | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 1.4634 × 10−11/- | 2.5707 × 10−10/- | 1.4634 × 10−11/- | 2.9596 × 10−10/- | 1.4634 × 10−11/- |
CEC2017_F17 | 1.6157 × 10−10/- | 2.2048 × 10−11/- | 3.7806 × 10−8/- | 1.9703 × 10−6/- | 2.2048 × 10−11/- | 2.6932 × 10−6/- | 2.2048 × 10−11/- |
CEC2017_F18 | 4.5411 × 10−10/- | 4.5411 × 10−10/- | 3.2721 × 10−5/- | 1.8350 × 10−1/= | 2.3890 × 10−11/- | 1.0672 × 10−10/- | 2.3890 × 10−11/- |
CEC2017_F19 | 1.9568 × 10−10/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 6.9724 × 10−3/- | 3.0199 × 10−11/- | 3.0939 × 10−6/- | 3.0199 × 10−11/- |
CEC2017_F20 | 1.4342 × 10−3/- | 2.3890 × 10−11/- | 1.4342 × 10−3/- | 2.6148 × 10−8/- | 1.1607 × 10−9/- | 2.9720 × 10−3/- | 6.5151 × 10−11/- |
CEC2017_F21 | 3.3582 × 10−1/= | 2.2048 × 10−11/- | 9.3276 × 10−7/- | 4.3222 × 10−7/- | 2.2048 × 10−11/- | 1.1190 × 10−4/- | 2.2048 × 10−11/- |
CEC2017_F22 | 4.6669 × 10−4/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.2048 × 10−11/- |
CEC2017_F23 | 9.9186 × 10−11/- | 3.0199 × 10−11/- | 2.6695 × 10−9/- | 3.0199 × 10−11/- | 3.3384 × 10−11/- | 2.3800 × 10−3/- | 3.0199 × 10−11/- |
CEC2017_F24 | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 6.4334 × 10−1/= | 1.7203 × 10−12/- | 1.7203 × 10−12/- | 5.6323 × 10−11/- | 1.7203 × 10−12/- |
CEC2017_F25 | 1.6338 × 10−5/- | 2.2048 × 10−11/- | 5.3869 × 10−2/= | 2.2048 × 10−11/- | 2.2048 × 10−11/- | 2.7240 × 10−9/- | 2.2048 × 10−11/- |
CEC2017_F26 | 2.3890 × 10−11/- | 2.3890 × 10−11/- | 4.5278 × 10−7/- | 2.3890 × 10−11/- | 2.3890 × 10−11/- | 8.5522 × 10−6/- | 2.3890 × 10−11/- |
CEC2017_F27 | 6.6955 × 10−11/- | 3.0199 × 10−11/- | 2.6784 × 10−6/- | 4.5043 × 10−11/- | 6.4142 × 10−1/= | 7.3803 × 10−10/- | 3.0199 × 10−11/- |
CEC2017_F28 | 5.2650 × 10−5/+ | 3.0199 × 10−11/- | 7.3891 × 10−11/+ | 3.6897 × 10−11/- | 3.0199 × 10−11/- | 1.3732 × 10−1/= | 3.0199 × 10−11/- |
CEC2017_F29 | 2.5872 × 10−11/- | 2.5872 × 10−11/- | 2.5872 × 10−11/- | 2.5872 × 10−11/- | 2.5872 × 10−11/- | 2.1447 × 10−4/- | 2.5872 × 10−11/- |
CEC2017_F30 | 6.0658 × 10−11/- | 3.0199 × 10−11/- | 3.0199 × 10−11/- | 5.0842 × 10−3/- | 3.3384 × 10−11/- | 8.8910 × 10−10/- | 3.0199 × 10−11/- |
+/-/= | 3/25/1 | 0/28/1 | 2/22/5 | 1/25/3 | 1/27/1 | 1/26/2 | 0/29/0 |
Table 12.
Friedman mean rank test for CEC2017.
Table 12.
Friedman mean rank test for CEC2017.
Dimensions | 30 | | 50 | | 100 | |
---|
Algorithms | Mean Rank | Final Rank | Mean Rank | Final Rank | Mean Rank | Final Rank |
---|
SO | 4.3414 | 6 | 3.9678 | 5 | 3.7161 | 4 |
DBO | 6.2241 | 7 | 6.4046 | 7 | 6.2011 | 7 |
SSA | 4.3264 | 5 | 3.9460 | 4 | 3.6270 | 3 |
TLBO | 3.4701 | 2 | 3.6563 | 2 | 4.4057 | 5 |
FSTDE | 4.1690 | 4 | 4.6552 | 6 | 5.5471 | 6 |
AHA | 3.8701 | 3 | 3.6839 | 3 | 3.3833 | 2 |
WOA | 7.5115 | 8 | 7.4287 | 8 | 7.3701 | 8 |
SNDSO | 2.0874 | 1 | 2.2575 | 1 | 1.7494 | 1 |
Table 13.
Running time of the algorithm for CEC2017 (Dim = 30).
Table 13.
Running time of the algorithm for CEC2017 (Dim = 30).
Problem | SO | DBO | SSA | TLBO | FSTDE | AHA | WOA | SNDSO |
---|
CEC2017_F1 | 6.20 × 100 | 5.24 × 100 | 4.22 × 100 | 4.82 × 100 | 1.51 × 101 | 4.11 × 100 | 3.95 × 100 | 5.20 × 100 |
CEC2017_F3 | 4.56 × 100 | 5.26 × 100 | 4.08 × 100 | 4.07 × 100 | 1.50 × 101 | 4.38 × 100 | 3.68 × 100 | 4.69 × 100 |
CEC2017_F4 | 4.78 × 100 | 5.46 × 100 | 3.90 × 100 | 4.53 × 100 | 1.50 × 101 | 4.76 × 100 | 3.99 × 100 | 4.68 × 100 |
CEC2017_F5 | 4.71 × 100 | 5.47 × 100 | 4.33 × 100 | 4.65 × 100 | 1.55 × 101 | 4.42 × 100 | 3.76 × 100 | 4.23 × 100 |
CEC2017_F6 | 5.17 × 100 | 5.86 × 100 | 4.55 × 100 | 4.95 × 100 | 1.56 × 101 | 4.90 × 100 | 4.27 × 100 | 6.62 × 100 |
CEC2017_F7 | 6.23 × 100 | 6.39 × 100 | 4.24 × 100 | 4.50 × 100 | 1.55 × 101 | 4.58 × 100 | 3.89 × 100 | 6.22 × 100 |
CEC2017_F8 | 5.52 × 100 | 5.25 × 100 | 4.13 × 100 | 4.36 × 100 | 1.52 × 101 | 4.53 × 100 | 4.02 × 100 | 5.68 × 100 |
CEC2017_F9 | 6.45 × 100 | 6.75 × 100 | 4.22 × 100 | 4.27 × 100 | 1.53 × 101 | 4.67 × 100 | 3.86 × 100 | 6.45 × 100 |
CEC2017_F10 | 6.32 × 100 | 7.01 × 100 | 4.64 × 100 | 4.32 × 100 | 1.54 × 101 | 4.73 × 100 | 3.97 × 100 | 6.10 × 100 |
CEC2017_F11 | 6.91 × 100 | 6.81 × 100 | 4.38 × 100 | 4.54 × 100 | 1.54 × 101 | 4.62 × 100 | 3.90 × 100 | 8.06 × 100 |
CEC2017_F12 | 7.48 × 100 | 7.25 × 100 | 4.69 × 100 | 4.56 × 100 | 1.54 × 101 | 4.76 × 100 | 4.07 × 100 | 7.03 × 100 |
CEC2017_F13 | 7.33 × 100 | 7.17 × 100 | 4.63 × 100 | 4.70 × 100 | 1.56 × 101 | 4.76 × 100 | 4.04 × 100 | 6.89 × 100 |
CEC2017_F14 | 6.66 × 100 | 8.31 × 100 | 4.70 × 100 | 4.32 × 100 | 1.57 × 101 | 5.03 × 100 | 4.07 × 100 | 6.00 × 100 |
CEC2017_F15 | 7.35 × 100 | 1.02 × 101 | 4.52 × 100 | 4.33 × 100 | 1.57 × 101 | 4.79 × 100 | 3.86 × 100 | 6.02 × 100 |
CEC2017_F16 | 6.95 × 100 | 7.65 × 100 | 4.67 × 100 | 4.38 × 100 | 1.58 × 101 | 4.68 × 100 | 3.94 × 100 | 6.26 × 100 |
CEC2017_F17 | 8.09 × 100 | 7.57 × 100 | 5.05 × 100 | 5.07 × 100 | 1.59 × 101 | 5.04 × 100 | 4.15 × 100 | 8.06 × 100 |
CEC2017_F18 | 7.88 × 100 | 7.95 × 100 | 5.41 × 100 | 5.32 × 100 | 1.65 × 101 | 5.02 × 100 | 4.29 × 100 | 8.18 × 100 |
CEC2017_F19 | 1.03 × 101 | 9.79 × 100 | 7.48 × 100 | 7.39 × 100 | 1.82 × 101 | 6.96 × 100 | 6.24 × 100 | 9.96 × 100 |
CEC2017_F20 | 9.70 × 100 | 8.11 × 100 | 5.80 × 100 | 5.63 × 100 | 1.63 × 101 | 5.54 × 100 | 4.66 × 100 | 8.33 × 100 |
CEC2017_F21 | 9.60 × 100 | 8.25 × 100 | 6.03 × 100 | 6.17 × 100 | 1.65 × 101 | 5.70 × 100 | 5.04 × 100 | 9.13 × 100 |
CEC2017_F22 | 1.07 × 101 | 8.72 × 100 | 6.60 × 100 | 6.85 × 100 | 1.75 × 101 | 6.31 × 100 | 5.58 × 100 | 9.67 × 100 |
CEC2017_F23 | 1.17 × 101 | 8.83 × 100 | 6.66 × 100 | 7.28 × 100 | 1.73 × 101 | 6.51 × 100 | 5.88 × 100 | 9.92 × 100 |
CEC2017_F24 | 1.18 × 101 | 8.96 × 100 | 7.05 × 100 | 6.80 × 100 | 1.73 × 101 | 6.67 × 100 | 6.08 × 100 | 9.94 × 100 |
CEC2017_F25 | 1.15 × 101 | 8.84 × 100 | 6.64 × 100 | 6.93 × 100 | 1.74 × 101 | 6.49 × 100 | 5.72 × 100 | 9.87 × 100 |
CEC2017_F26 | 1.17 × 101 | 9.20 × 100 | 9.83 × 100 | 7.21 × 100 | 1.74 × 101 | 6.87 × 100 | 6.08 × 100 | 1.02 × 101 |
CEC2017_F27 | 1.21 × 101 | 9.46 × 100 | 1.00 × 101 | 7.47 × 100 | 1.76 × 101 | 7.09 × 100 | 6.44 × 100 | 1.06 × 101 |
CEC2017_F28 | 1.20 × 101 | 9.24 × 100 | 1.07 × 101 | 7.31 × 100 | 1.75 × 101 | 7.81 × 100 | 6.37 × 100 | 1.05 × 101 |
CEC2017_F29 | 1.22 × 101 | 8.90 × 100 | 1.04 × 101 | 6.83 × 100 | 1.72 × 101 | 6.48 × 100 | 5.90 × 100 | 1.01 × 101 |
CEC2017_F30 | 1.34 × 101 | 1.07 × 101 | 1.20 × 101 | 8.43 × 100 | 1.89 × 101 | 7.98 × 100 | 7.25 × 100 | 1.17 × 101 |
Mean Rank | 6.45 | 5.62 | 3.31 | 3.07 | 8.00 | 3.00 | 1.03 | 5.52 |
Final Rank | 7 | 6 | 4 | 3 | 8 | 2 | 1 | 5 |
Table 14.
Information on 12 FS datasets.
Table 14.
Information on 12 FS datasets.
Datasets | Number of Features | Number of Classifications | Dataset Size |
---|
IonosphereEW | 34 | 2 | 351 |
Breastcancer | 9 | 2 | 699 |
BreastEW | 30 | 2 | 569 |
Congress | 16 | 2 | 435 |
Wine | 13 | 3 | 178 |
Vote | 16 | 2 | 435 |
Vehicle | 18 | 4 | 846 |
Exactly | 13 | 2 | 1000 |
Glass | 9 | 7 | 214 |
HeartEW | 13 | 2 | 270 |
Zoo | 16 | 7 | 101 |
SonarEW | 60 | 2 | 208 |
Table 15.
Fitness values for algorithms dealing with FS problems.
Table 15.
Fitness values for algorithms dealing with FS problems.
Datasets | Metric | HHO | BOA | WOA | ABO | PSO | GWO | SO | SNDSO |
---|
IonosphereEW | Best | 0.040420 | 0.063193 | 0.044454 | 0.053277 | 0.056218 | 0.037479 | 0.034538 | 0.017647 |
| Mean | 0.086894 | 0.083966 | 0.072720 | 0.074230 | 0.115406 | 0.072258 | 0.049922 | 0.042078 |
| Worst | 0.146975 | 0.103613 | 0.098824 | 0.101765 | 0.179664 | 0.106555 | 0.073866 | 0.063950 |
| Rank | 4/7/7 | 8/6/5 | 5/4/3 | 6/5/4 | 7/8/8 | 3/3/6 | 2/2/2 | 1/1/1 |
Breastcancer | Best | 0.054596 | 0.046283 | 0.041647 | 0.046283 | 0.039808 | 0.063869 | 0.052758 | 0.035172 |
| Mean | 0.068263 | 0.052510 | 0.056853 | 0.050013 | 0.042366 | 0.066699 | 0.056025 | 0.040357 |
| Worst | 0.089768 | 0.059233 | 0.065707 | 0.059233 | 0.052758 | 0.072182 | 0.061071 | 0.046283 |
| Rank | 7/8/8 | 4/4/3 | 3/6/6 | 4/3/3 | 2/2/2 | 8/7/7 | 6/5/5 | 1/1/1 |
BreastEW | Best | 0.047227 | 0.042596 | 0.049263 | 0.064454 | 0.047227 | 0.039263 | 0.050560 | 0.027965 |
| Mean | 0.074485 | 0.069774 | 0.070779 | 0.079813 | 0.058561 | 0.050864 | 0.064625 | 0.038916 |
| Worst | 0.097050 | 0.083717 | 0.103156 | 0.093717 | 0.073156 | 0.063156 | 0.077050 | 0.051298 |
| Rank | 4/7/7 | 3/5/5 | 6/6/8 | 8/8/6 | 4/3/3 | 2/2/2 | 7/4/4 | 1/1/1 |
Congress | Best | 0.047629 | 0.049784 | 0.045690 | 0.057974 | 0.049784 | 0.047629 | 0.035345 | 0.026940 |
| Mean | 0.056379 | 0.069698 | 0.060841 | 0.057974 | 0.061667 | 0.047780 | 0.043434 | 0.027356 |
| Worst | 0.093319 | 0.085129 | 0.099569 | 0.057974 | 0.078879 | 0.049784 | 0.047629 | 0.033190 |
| Rank | 4/4/7 | 7/8/6 | 3/6/8 | 8/5/4 | 6/7/5 | 5/3/3 | 2/2/2 | 1/1/1 |
Wine | Best | 0.037284 | 0.056034 | 0.057974 | 0.026940 | 0.023077 | 0.047629 | 0.039440 | 0.023077 |
| Mean | 0.045057 | 0.068499 | 0.058671 | 0.026940 | 0.032147 | 0.048118 | 0.045790 | 0.023077 |
| Worst | 0.078879 | 0.089224 | 0.064224 | 0.026940 | 0.056484 | 0.056034 | 0.049784 | 0.023077 |
| Rank | 4/4/7 | 7/8/8 | 8/7/6 | 3/2/2 | 1/3/5 | 6/6/4 | 5/5/3 | 1/1/1 |
Vote | Best | 0.039440 | 0.066379 | 0.037284 | 0.026940 | 0.037284 | 0.025000 | 0.037284 | 0.022845 |
| Mean | 0.072198 | 0.080747 | 0.038118 | 0.027565 | 0.045014 | 0.029188 | 0.038118 | 0.025194 |
| Worst | 0.107974 | 0.101724 | 0.049784 | 0.039440 | 0.066379 | 0.047629 | 0.049784 | 0.039440 |
| Rank | 7/7/8 | 8/8/7 | 4/4/4 | 3/2/1 | 4/6/6 | 2/3/3 | 4/4/4 | 1/1/2 |
Vehicle | Best | 0.288955 | 0.256542 | 0.240565 | 0.257002 | 0.261867 | 0.241026 | 0.257002 | 0.203517 |
| Mean | 0.318876 | 0.297731 | 0.307540 | 0.286113 | 0.284400 | 0.257372 | 0.280079 | 0.247686 |
| Worst | 0.358876 | 0.331788 | 0.353090 | 0.316042 | 0.332479 | 0.272978 | 0.315812 | 0.268113 |
| Rank | 8/8/8 | 4/6/5 | 2/7/7 | 5/5/4 | 7/4/6 | 3/2/2 | 5/3/3 | 1/1/1 |
Exactly | Best | 0.046154 | 0.267231 | 0.046154 | 0.046154 | 0.046154 | 0.046154 | 0.046154 | 0.046154 |
| Mean | 0.252886 | 0.285922 | 0.252954 | 0.178659 | 0.161626 | 0.162836 | 0.137708 | 0.134306 |
| Worst | 0.309038 | 0.296846 | 0.291192 | 0.286692 | 0.327038 | 0.291192 | 0.303962 | 0.291192 |
| Rank | 1/6/7 | 8/8/5 | 1/7/2 | 1/5/1 | 1/3/8 | 1/4/2 | 1/2/6 | 7/1/4 |
Glass | Best | 0.279365 | 0.280159 | 0.237302 | 0.311905 | 0.279365 | 0.269048 | 0.236508 | 0.140476 |
| Mean | 0.307804 | 0.313148 | 0.258757 | 0.316905 | 0.290397 | 0.277275 | 0.242619 | 0.161217 |
| Worst | 0.355556 | 0.365079 | 0.291270 | 0.345238 | 0.300794 | 0.322222 | 0.322222 | 0.204762 |
| Rank | 5/6/7 | 7/7/8 | 3/3/2 | 8/8/6 | 5/5/3 | 4/4/5 | 2/2/4 | 1/1/1 |
HeartEW | Best | 0.137179 | 0.164103 | 0.105128 | 0.088462 | 0.123077 | 0.089744 | 0.105128 | 0.071795 |
| Mean | 0.188462 | 0.200812 | 0.176197 | 0.151923 | 0.154915 | 0.102137 | 0.120812 | 0.093803 |
| Worst | 0.230769 | 0.230769 | 0.262821 | 0.205128 | 0.197436 | 0.146154 | 0.215385 | 0.124359 |
| Rank | 7/7/6 | 8/8/6 | 4/6/8 | 2/4/4 | 6/5/3 | 3/2/2 | 4/3/5 | 1/1/1 |
Zoo | Best | 0.062500 | 0.043750 | 0.088750 | 0.037500 | 0.031250 | 0.037500 | 0.037500 | 0.025000 |
| Mean | 0.106833 | 0.103792 | 0.142292 | 0.055833 | 0.038208 | 0.057833 | 0.063208 | 0.052500 |
| Worst | 0.166250 | 0.172500 | 0.185000 | 0.108750 | 0.076250 | 0.088750 | 0.076250 | 0.082500 |
| Rank | 7/7/6 | 6/6/7 | 8/8/8 | 3/3/5 | 2/1/1 | 3/4/4 | 3/5/1 | 1/2/3 |
SonarEW | Best | 0.057236 | 0.089187 | 0.060569 | 0.075854 | 0.055285 | 0.021667 | 0.040285 | 0.015000 |
| Mean | 0.120984 | 0.144083 | 0.113477 | 0.121844 | 0.097881 | 0.053760 | 0.074385 | 0.024732 |
| Worst | 0.174756 | 0.181992 | 0.159472 | 0.171992 | 0.143089 | 0.089187 | 0.121138 | 0.035285 |
| Rank | 5/6/7 | 8/8/8 | 6/5/5 | 7/7/6 | 4/4/4 | 2/2/2 | 3/3/3 | 1/1/1 |
Mean Rank | Best | 5.250000 | 6.500000 | 4.416667 | 4.833333 | 4.083333 | 3.500000 | 3.666667 | 1.500000 |
| Mean | 6.416667 | 6.833333 | 5.750000 | 4.750000 | 4.250000 | 3.500000 | 3.333333 | 1.083333 |
| Worst | 7.083333 | 6.083333 | 5.583333 | 3.833333 | 4.500000 | 3.500000 | 3.500000 | 1.500000 |
Final Rank | Best | 7 | 8 | 5 | 6 | 4 | 2 | 3 | 1 |
| Mean | 7 | 8 | 6 | 5 | 4 | 3 | 2 | 1 |
| Worst | 8 | 7 | 6 | 4 | 5 | 2 | 2 | 1 |
Table 16.
Wilcoxon rank sum test results for algorithms dealing with FS problems.
Table 16.
Wilcoxon rank sum test results for algorithms dealing with FS problems.
Datasets | HHO | BOA | WOA | ABO | PSO | GWO | SO |
---|
IonosphereEW | 1.0555 × 10−9/- | 3.2682 × 10−11/- | 5.2685 × 10−9/- | 8.6348 × 10−11/- | 4.4491 × 10−11/- | 1.4666 × 10−8/- | 2.5330 × 10−2/- |
Breastcancer | 1.3459 × 10−11/- | 2.9249 × 10−10/- | 2.1090 × 10−10/- | 4.1276 × 10−8/- | 3.7672 × 10−2/- | 1.0952 × 10−11/- | 8.0691 × 10−12/- |
BreastEW | 3.9725 × 10−11/- | 6.4721 × 10−11/- | 3.2519 × 10−11/- | 2.9339 × 10−11/- | 5.2284 × 10−11/- | 1.9376 × 10−8/- | 3.2175 × 10−11/- |
Congress | 1.6140 × 10−12/- | 2.2370 × 10−12/- | 9.4316 × 10−13/- | 4.1574 × 10−14/- | 2.1000 × 10−12/- | 1.4043 × 10−13/- | 1.8547 × 10−12/- |
Wine | 5.3349 × 10−13/- | 1.1537 × 10−12/- | 8.6958 × 10−14/- | 1.6853 × 10−14/- | 3.9882 × 10−8/- | 4.1617 × 10−14/- | 2.4891 × 10−13/- |
Vote | 7.0983 × 10−12/- | 6.2782 × 10−12/- | 2.0056 × 10−10/- | 8.2478 × 10−6/- | 5.6734 × 10−11/- | 1.1501 × 10−6/- | 1.9095 × 10−10/- |
Vehicle | 2.9229 × 10−11/- | 9.5705 × 10−11/- | 2.5323 × 10−10/- | 1.0592 × 10−10/- | 7.8783 × 10−11/- | 7.1944 × 10−3/- | 1.0349 × 10−10/- |
Exactly | 2.9875 × 10−5/- | 5.5831 × 10−8/- | 1.3384 × 10−4/- | 8.1567 × 10−1/= | 4.7391 × 10−1/= | 1.6698 × 10−1/= | 4.1160 × 10−1/= |
Glass | 2.2623 × 10−11/- | 2.2871 × 10−11/- | 2.1779 × 10−11/- | 1.6678 × 10−11/- | 4.7919 × 10−12/- | 4.7849 × 10−12/- | 2.3095 × 10−12/- |
HeartEW | 2.8252 × 10−11/- | 2.7827 × 10−11/- | 2.4068 × 10−10/- | 1.3806 × 10−7/- | 2.8433 × 10−10/- | 7.4760 × 10−4/- | 5.2863 × 10−5/- |
Zoo | 2.2076 × 10−10/- | 1.3317 × 10−9/- | 2.2290 × 10−11/- | 7.7129 × 10−2/= | 1.6582 × 10−1/= | 1.0894 × 10−1/= | 1.8346 × 10−3/- |
SonarEW | 2.9321 × 10−11/- | 2.9321 × 10−11/- | 2.9358 × 10−11/- | 2.9321 × 10−11/- | 2.9137 × 10−11/- | 1.6445 × 10−9/- | 2.9155 × 10−11/- |
+/-/= | 0/12/0 | 0/12/0 | 0/12/0 | 0/10/2 | 0/10/2 | 0/10/2 | 0/11/1 |
Table 17.
Average classification accuracy of algorithms dealing with FS problems.
Table 17.
Average classification accuracy of algorithms dealing with FS problems.
Datasets | HHO | BOA | WOA | ABO | PSO | GWO | SO | SNDSO |
---|
IonosphereEW | 92.523810 | 92.761905 | 93.238095 | 93.571429 | 90.238095 | 93.714286 | 96.000000 | 97.666667 |
| 7 | 6 | 5 | 4 | 8 | 3 | 2 | 1 |
Breastcancer | 95.707434 | 97.745803 | 96.810552 | 98.105516 | 98.872902 | 96.786571 | 97.314149 | 98.561151 |
| 8 | 4 | 6 | 3 | 1 | 7 | 5 | 2 |
BreastEW | 94.896755 | 95.457227 | 96.135693 | 94.218289 | 96.814159 | 96.607670 | 95.634218 | 98.466077 |
| 7 | 6 | 4 | 8 | 2 | 3 | 5 | 1 |
Congress | 95.402299 | 95.172414 | 94.559387 | 94.252874 | 95.555556 | 95.593870 | 97.164751 | 97.701149 |
| 5 | 6 | 7 | 8 | 4 | 3 | 2 | 1 |
Wine | 96.475096 | 94.865900 | 94.291188 | 97.701149 | 99.619048 | 95.440613 | 95.977011 | 100.000000 |
| 4 | 7 | 8 | 3 | 2 | 6 | 5 | 1 |
Vote | 94.061303 | 94.176245 | 96.551724 | 97.701149 | 96.896552 | 99.233716 | 96.551724 | 98.659004 |
| 8 | 7 | 5 | 3 | 4 | 1 | 5 | 2 |
Vehicle | 69.013807 | 70.355030 | 69.388560 | 72.386588 | 72.741617 | 75.147929 | 72.583826 | 76.429980 |
| 8 | 7 | 5 | 3 | 4 | 1 | 5 | 2 |
Exactly | 74.950000 | 69.883333 | 74.800000 | 83.966667 | 86.600000 | 86.066667 | 89.400000 | 89.550000 |
| 6 | 8 | 7 | 5 | 3 | 4 | 2 | 1 |
Glass | 70.079365 | 69.444444 | 76.269841 | 69.603175 | 71.190476 | 72.936508 | 75.634921 | 85.873016 |
| 6 | 8 | 2 | 7 | 5 | 4 | 3 | 1 |
HeartEW | 82.222222 | 80.679012 | 83.641975 | 87.222222 | 86.604938 | 91.358025 | 90.308642 | 93.024691 |
| 7 | 8 | 6 | 4 | 5 | 2 | 3 | 1 |
Zoo | 93.500000 | 93.166667 | 88.333333 | 98.333333 | 99.666667 | 98.666667 | 96.333333 | 97.500000 |
| 6 | 7 | 8 | 3 | 1 | 2 | 5 | 4 |
SonarEW | 89.186992 | 86.910569 | 91.626016 | 89.430894 | 92.926829 | 96.341463 | 94.796748 | 99.918699 |
| 7 | 8 | 5 | 6 | 4 | 2 | 3 | 1 |
Mean Rank | 6.583333 | 6.833333 | 5.666667 | 4.750000 | 3.583333 | 3.166667 | 3.750000 | 1.500000 |
Final Rank | 7 | 8 | 6 | 5 | 3 | 2 | 4 | 1 |
Table 18.
Average number of features for algorithms dealing with FS problems.
Table 18.
Average number of features for algorithms dealing with FS problems.
Datasets | HHO | BOA | WOA | ABO | PSO | GWO | SO | SNDSO |
---|
IonosphereEW | 6.666667 | 6.400000 | 4.033333 | 5.566667 | 9.366667 | 5.333333 | 4.733333 | 7.166667 |
| 6 | 5 | 1 | 4 | 8 | 3 | 2 | 7 |
Breastcancer | 2.666667 | 2.900000 | 2.533333 | 2.966667 | 2.900000 | 3.400000 | 2.866667 | 2.466667 |
| 3 | 5 | 2 | 7 | 5 | 8 | 4 | 1 |
BreastEW | 8.566667 | 8.666667 | 10.800000 | 8.333333 | 8.966667 | 6.100000 | 7.600000 | 7.533333 |
| 5 | 6 | 8 | 4 | 7 | 1 | 3 | 2 |
Congress | 2.400000 | 4.200000 | 1.900000 | 1.000000 | 3.466667 | 1.300000 | 2.866667 | 1.066667 |
| 5 | 8 | 4 | 1 | 7 | 3 | 6 | 2 |
Wine | 2.133333 | 3.566667 | 1.166667 | 1.000000 | 3.733333 | 1.133333 | 1.533333 | 3.000000 |
| 5 | 7 | 3 | 1 | 8 | 2 | 4 | 6 |
Vote | 3.000000 | 4.533333 | 1.133333 | 1.100000 | 2.733333 | 3.566667 | 1.133333 | 2.100000 |
| 6 | 8 | 2 | 1 | 5 | 7 | 2 | 4 |
Vehicle | 7.200000 | 5.566667 | 5.766667 | 6.766667 | 7.033333 | 6.066667 | 6.000000 | 6.400000 |
| 8 | 1 | 2 | 6 | 7 | 4 | 3 | 5 |
Exactly | 3.566667 | 1.933333 | 3.400000 | 4.466667 | 5.333333 | 4.866667 | 5.500000 | 5.233333 |
| 3 | 1 | 2 | 4 | 7 | 5 | 8 | 6 |
Glass | 3.466667 | 3.433333 | 4.066667 | 3.900000 | 2.800000 | 3.033333 | 2.100000 | 3.066667 |
| 6 | 5 | 8 | 7 | 2 | 3 | 1 | 4 |
HeartEW | 3.700000 | 3.500000 | 3.766667 | 4.800000 | 4.466667 | 3.166667 | 4.366667 | 4.033333 |
| 3 | 2 | 4 | 8 | 7 | 1 | 6 | 5 |
Zoo | 7.733333 | 6.766667 | 5.966667 | 6.533333 | 5.633333 | 7.333333 | 4.833333 | 4.800000 |
| 8 | 6 | 4 | 5 | 3 | 7 | 2 | 1 |
SonarEW | 14.200000 | 15.766667 | 22.866667 | 16.033333 | 20.533333 | 12.500000 | 16.533333 | 14.400000 |
| 2 | 4 | 8 | 5 | 7 | 1 | 6 | 3 |
Mean Rank | 5.000000 | 4.833333 | 4.000000 | 4.416667 | 6.083333 | 3.750000 | 3.916667 | 3.833333 |
Final Rank | 7 | 6 | 4 | 5 | 8 | 1 | 3 | 2 |
Table 19.
Average running time of the algorithm for the FS problem.
Table 19.
Average running time of the algorithm for the FS problem.
Datasets | HHO | BOA | WOA | ABO | PSO | GWO | SO | SNDSO |
---|
IonosphereEW | 4.803066 | 3.186414 | 2.911380 | 1.998343 | 3.126366 | 3.216816 | 3.010714 | 3.078643 |
| 8 | 6 | 2 | 1 | 5 | 7 | 3 | 4 |
Breastcancer | 5.135761 | 3.348222 | 3.066325 | 2.287341 | 3.405459 | 3.416410 | 3.295708 | 3.331846 |
| 8 | 5 | 2 | 1 | 6 | 7 | 3 | 4 |
BreastEW | 5.254400 | 3.366437 | 3.162485 | 2.208694 | 3.348828 | 3.526335 | 3.293445 | 3.321811 |
| 8 | 6 | 2 | 1 | 5 | 7 | 3 | 4 |
Congress | 4.509980 | 3.298403 | 2.732547 | 2.033697 | 3.360761 | 3.349987 | 2.994612 | 3.072036 |
| 8 | 5 | 2 | 1 | 7 | 6 | 3 | 4 |
Wine | 4.369598 | 3.298717 | 2.593746 | 2.072551 | 3.179409 | 3.554128 | 2.840827 | 3.127230 |
| 8 | 6 | 2 | 1 | 5 | 7 | 3 | 4 |
Vote | 4.595980 | 3.393188 | 2.886006 | 2.160125 | 3.421998 | 3.384964 | 2.692785 | 3.178043 |
| 8 | 6 | 3 | 1 | 7 | 5 | 2 | 4 |
Vehicle | 5.814580 | 3.750190 | 3.386057 | 2.532804 | 3.804094 | 3.702382 | 3.631501 | 3.663357 |
| 8 | 6 | 3 | 1 | 7 | 5 | 2 | 4 |
Exactly | 5.328074 | 3.797175 | 3.304813 | 2.341518 | 3.893987 | 3.748275 | 3.660842 | 3.631808 |
| 8 | 6 | 2 | 1 | 7 | 5 | 4 | 3 |
Glass | 5.061052 | 3.168219 | 3.075275 | 2.226809 | 3.274313 | 3.334704 | 2.963263 | 3.125194 |
| 8 | 5 | 3 | 1 | 6 | 7 | 2 | 4 |
HeartEW | 4.760409 | 3.242640 | 3.037463 | 2.255885 | 3.366795 | 3.359168 | 3.294718 | 3.134978 |
| 8 | 4 | 2 | 1 | 7 | 6 | 5 | 3 |
Zoo | 4.995152 | 3.205886 | 3.017472 | 2.326458 | 3.311088 | 3.318231 | 3.231036 | 3.138697 |
| 8 | 4 | 2 | 1 | 6 | 7 | 5 | 3 |
SonarEW | 4.805915 | 3.065338 | 2.900962 | 2.097988 | 3.089534 | 3.081503 | 3.073096 | 3.052341 |
| 8 | 4 | 2 | 1 | 7 | 6 | 5 | 3 |
Mean Rank | 8.000000 | 5.250000 | 2.250000 | 1.000000 | 6.250000 | 6.250000 | 3.333333 | 3.666667 |
Final Rank | 8 | 5 | 2 | 1 | 6 | 6 | 3 | 4 |
Table 20.
Optimal solution of algorithms in three-bar truss design problem.
Table 20.
Optimal solution of algorithms in three-bar truss design problem.
Algorithms | | Optimal |
---|
| | | |
---|
SO | 0.788661 | 0.408288 | 263.895844 |
BESD | 0.788661 | 0.408289 | 263.895845 |
DBO | 0.788796 | 0.407905 | 263.895854 |
MFO | 0.789257 | 0.406605 | 263.896092 |
SSA | 0.788704 | 0.408167 | 263.895845 |
PSO | 0.788637 | 0.408357 | 263.895844 |
EO | 0.788755 | 0.408023 | 263.895848 |
SNDSO | 0.788680 | 0.408233 | 263.895843 |
Table 21.
Statistical results of algorithms for the three-bar truss design problem.
Table 21.
Statistical results of algorithms for the three-bar truss design problem.
Algorithms | Best | Worst | Mean | Std |
---|
SO | 263.895844 | 263.896673 | 263.896048 | 2.428072 × 10−4 |
BESD | 263.895845 | 263.895882 | 263.895856 | 8.098410 × 10−6 |
DBO | 263.895854 | 263.908123 | 263.897815 | 2.689815 × 10−3 |
MFO | 263.896092 | 264.261487 | 263.957287 | 1.003014 × 10−1 |
SSA | 263.895845 | 263.908875 | 263.897076 | 2.415208 × 10−3 |
PSO | 263.895844 | 263.899023 | 263.896315 | 7.695109 × 10−4 |
EO | 263.895848 | 263.896450 | 263.895959 | 1.444199 × 10−4 |
SNDSO | 263.895843 | 263.896624 | 263.895952 | 1.703197 × 10−4 |
Table 22.
Optimal solution of algorithms in 10-bar truss design problem.
Table 22.
Optimal solution of algorithms in 10-bar truss design problem.
Algorithms | | Optimal |
---|
| | | | | | | | | | | |
---|
SO | 0.003616 | 0.001465 | 0.003382 | 0.001563 | 0.000066 | 0.000452 | 0.002236 | 0.002430 | 0.001296 | 0.001237 | 524.985183 |
BESD | 0.003461 | 0.001457 | 0.003600 | 0.001554 | 0.000079 | 0.000464 | 0.002304 | 0.002392 | 0.001235 | 0.001252 | 526.144065 |
DBO | 0.003562 | 0.001508 | 0.003504 | 0.001549 | 0.000065 | 0.000450 | 0.002286 | 0.002364 | 0.001228 | 0.001255 | 524.924768 |
MFO | 0.003756 | 0.001493 | 0.003227 | 0.001385 | 0.000065 | 0.000461 | 0.002517 | 0.002264 | 0.001257 | 0.001303 | 526.035144 |
SSA | 0.003600 | 0.001535 | 0.003388 | 0.001395 | 0.000065 | 0.000458 | 0.002533 | 0.002214 | 0.001192 | 0.001340 | 525.175775 |
PSO | 0.003516 | 0.001531 | 0.003523 | 0.001481 | 0.000065 | 0.000453 | 0.002394 | 0.002287 | 0.001236 | 0.001256 | 524.587317 |
EO | 0.003453 | 0.001442 | 0.003537 | 0.001484 | 0.000065 | 0.000457 | 0.002329 | 0.002423 | 0.001262 | 0.001250 | 524.551304 |
SNDSO | 0.003475 | 0.001514 | 0.003552 | 0.001428 | 0.000065 | 0.000456 | 0.002461 | 0.002281 | 0.001221 | 0.001264 | 524.582840 |
Table 23.
Statistical results of algorithms for the 10-bar truss design problem.
Table 23.
Statistical results of algorithms for the 10-bar truss design problem.
Algorithms | Best | Worst | Mean | Std |
---|
SO | 524.985183 | 538.641398 | 530.132242 | 3.842013 × 100 |
BESD | 526.144065 | 532.596966 | 529.856669 | 1.327543 × 100 |
DBO | 524.924768 | 578.222671 | 539.127969 | 1.458350 × 101 |
MFO | 526.035144 | 601.429160 | 539.770295 | 1.661953 × 101 |
SSA | 525.175775 | 540.098010 | 531.300210 | 4.484961 × 100 |
PSO | 524.587317 | 533.323805 | 528.304913 | 3.072231 × 100 |
EO | 524.551304 | 533.391502 | 528.730640 | 3.060234 × 100 |
SNDSO | 524.582840 | 532.572344 | 526.910544 | 2.741038 × 100 |
Table 24.
Optimal solution of algorithms in tension/compression spring design problem.
Table 24.
Optimal solution of algorithms in tension/compression spring design problem.
Algorithms | | Optimal |
---|
| | | | |
---|
SO | 0.051950 | 0.363029 | 10.928340 | 0.012666 |
BESD | 0.051193 | 0.344878 | 12.042946 | 0.012692 |
DBO | 0.051779 | 0.358879 | 11.163347 | 0.012665 |
MFO | 0.051973 | 0.363587 | 10.897328 | 0.012667 |
SSA | 0.050000 | 0.316801 | 14.110950 | 0.012760 |
PSO | 0.052354 | 0.372922 | 10.398588 | 0.012673 |
EO | 0.051632 | 0.355337 | 11.370360 | 0.012665 |
SNDSO | 0.051700 | 0.356984 | 11.273355 | 0.012665 |
Table 25.
Statistical results of algorithms for the tension/compression spring design problem.
Table 25.
Statistical results of algorithms for the tension/compression spring design problem.
Algorithms | Best | Worst | Mean | Std |
---|
SO | 0.012666 | 0.013782 | 0.012948 | 3.152255 × 10−4 |
BESD | 0.012692 | 0.012831 | 0.012745 | 3.829559 × 10−5 |
DBO | 0.012665 | 0.017786 | 0.013765 | 1.833997 × 10−3 |
MFO | 0.012667 | 0.017773 | 0.013680 | 1.577297 × 10−3 |
SSA | 0.012684 | 0.018161 | 0.013190 | 1.024217 × 10−3 |
PSO | 0.012673 | 0.014946 | 0.013269 | 5.539001 × 10−4 |
EO | 0.012665 | 0.013788 | 0.012994 | 3.146420 × 10−4 |
SNDSO | 0.012665 | 0.012924 | 0.012717 | 5.474825 × 10−5 |
Table 26.
Optimal solution of algorithms in speed reducer design problem.
Table 26.
Optimal solution of algorithms in speed reducer design problem.
Algorithms | | Optimal |
---|
| | | | | | | | |
---|
SO | 3.500000 | 0.700000 | 17.000000 | 7.300000 | 7.715320 | 3.350541 | 5.286654 | 2994.424466 |
BESD | 3.501806 | 0.700020 | 17.000444 | 7.338166 | 7.769756 | 3.356070 | 5.290624 | 3000.768049 |
DBO | 3.500000 | 0.700000 | 17.000000 | 7.300000 | 7.715320 | 3.350541 | 5.286654 | 2994.424466 |
MFO | 3.500000 | 0.700000 | 17.000000 | 7.300000 | 7.715320 | 3.350541 | 5.286654 | 2994.424466 |
SSA | 3.510262 | 0.700000 | 17.000000 | 7.814109 | 7.774259 | 3.358179 | 5.286675 | 3006.265463 |
PSO | 3.500000 | 0.700000 | 17.000000 | 7.300000 | 7.715320 | 3.350541 | 5.286654 | 2994.424466 |
EO | 3.500000 | 0.700000 | 17.000000 | 7.300000 | 7.715320 | 3.350541 | 5.286654 | 2994.424466 |
SNDSO | 3.500000 | 0.700000 | 17.000000 | 7.300000 | 7.715320 | 3.350541 | 5.286654 | 2994.424466 |
Table 27.
Statistical results of algorithms for the speed reducer design problem.
Table 27.
Statistical results of algorithms for the speed reducer design problem.
Algorithms | Best | Worst | Mean | Std |
---|
SO | 2994.424466 | 2996.912387 | 2994.632981 | 5.146991 × 10−1 |
BESD | 3000.768049 | 3007.986907 | 3003.622557 | 1.802085 × 100 |
DBO | 2994.424466 | 3310.500380 | 3056.462973 | 9.058321 × 101 |
MFO | 2994.424466 | 3043.034936 | 3002.578313 | 1.532264 × 101 |
SSA | 3006.265463 | 3113.751959 | 3032.921212 | 2.870079 × 101 |
PSO | 2994.424466 | 2994.424466 | 2994.424466 | 1.818989 × 10−12 |
EO | 2994.424466 | 2994.424466 | 2994.424466 | 6.498050 × 10−8 |
SNDSO | 2994.424466 | 2994.424472 | 2994.424466 | 1.145042 × 10−6 |
Table 28.
Optimal solution of algorithms in welded beam design problem.
Table 28.
Optimal solution of algorithms in welded beam design problem.
Algorithms | | Optimal |
---|
| | | | | |
---|
SO | 0.198832 | 3.337369 | 9.192024 | 0.198832 | 1.670218 |
BESD | 0.197629 | 3.373626 | 9.205042 | 0.198893 | 1.675844 |
DBO | 0.198832 | 3.337364 | 9.192022 | 0.198832 | 1.670218 |
MFO | 0.198831 | 3.337384 | 9.192024 | 0.198832 | 1.670219 |
SSA | 0.190472 | 3.477217 | 9.282487 | 0.198413 | 1.687969 |
PSO | 0.198832 | 3.337365 | 9.192024 | 0.198832 | 1.670218 |
EO | 0.198832 | 3.337365 | 9.192024 | 0.198832 | 1.670218 |
SNDSO | 0.198832 | 3.337365 | 9.192024 | 0.198832 | 1.670218 |
Table 29.
Statistical results of algorithms for the welded beam design problem.
Table 29.
Statistical results of algorithms for the welded beam design problem.
Algorithms | Best | Worst | Mean | Std |
---|
SO | 1.670218 | 1.727258 | 1.674821 | 1.130506 × 10−2 |
BESD | 1.675844 | 1.692552 | 1.684660 | 4.536354 × 10−3 |
DBO | 1.670218 | 1.824621 | 1.741124 | 5.629337 × 10−2 |
MFO | 1.670219 | 2.103518 | 1.751748 | 1.114074 × 10−1 |
SSA | 1.687969 | 1.973487 | 1.750297 | 6.508382 × 10−2 |
PSO | 1.670218 | 2.164451 | 1.714953 | 1.016073 × 10−1 |
EO | 1.670218 | 1.675576 | 1.670651 | 1.266918 × 10−3 |
SNDSO | 1.670218 | 1.670283 | 1.670224 | 1.478651 × 10−5 |