Abstract
Salp swarm algorithm (SSA) is a unique swarm intelligent algorithm widely used for various practical applications due to its simple framework and good optimization performance. However, like other swarm-based algorithms, SSA yields local optimal solutions and has a slow convergence rate and low solution accuracy when dealing with high-dimensional global optimization problems. Based on quadratic interpolation and a local escape operator (LEO), a salp swarm optimization algorithm (QSSALEO) is proposed to address these issues. Quadratic interpolation around the best search agent aids QSSALEO's exploitation ability and solution accuracy, whereas the local escaping operator employs random operators to escape local optima. These tactics complement one another to help SSA promote convergence performance. Furthermore, the algorithm strives for a balance of exploitation and exploration. The proposed QSSALEO method was tested using the CEC 2017 benchmark with 50 and 100 decision variables, as well as seven CEC2008lsgo test functions with 200, 500, and 1000 decision variables, and its performance was compared to that of other metaheuristic algorithms and advanced algorithms, including seven salp swarm variants. The experimental results reveal that QSSALEO outperforms SSA and other population-based algorithms regarding convergence rate and solution correctness. The QSSALEO was then evaluated as a feature selection algorithm on 19 datasets (including three high-dimensional datasets). Friedman and Wilcoxon rank-sum statistical tests are also used to analyze the results. According to experimental data and statistical tests, the QSSALEO algorithm is very competitive and often superior to the algorithms employed in research. Therefore, the proposed method can also be considered a specialized large-scale global optimization optimizer whose performance surpasses state-of-the-art algorithms such as CMA-ES and SHADE. The algorithm source code is available at https://github.com/MohammedQaraad/An-Innovative-Quadratic-interpolation-Salp-Swarm.
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Appendix
Appendix
1.1 Appendix 1. Comparison results of the QSSALEO on unimodal functions with traditional algorithms during 2500 iterations.
Fun | D | Criteria | CSO | SSA | PSO | WOA | BAT | HHO | SCA | MFO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|---|
F1 | 50 | Avg | 1.802E+12 | 3.972E+09 | 1.523E+11 | 1.162E+10 | 1.358E+12 | 7.099E+11 | 5.275E+11 | 4.880E+11 | 4.012E+03 |
Std | 2.192E+11 | 2.537E+09 | 4.952E+10 | 4.472E+09 | 3.237E+10 | 7.922E+10 | 5.719E+10 | 1.969E+11 | 2.537E+09 | ||
Med | 1.779E+12 | 3.430E+09 | 1.549E+11 | 1.010E+10 | 7.765E+10 | 7.318E+11 | 5.219E+11 | 5.181E+11 | 5.181E+11 | ||
100 | Avg | 4.618E+12 | 2.357E+11 | 1.125E+12 | 2.261E+11 | 3.489E+12 | 2.066E+12 | 1.871E+12 | 1.377E+12 | 7.085E+03 | |
Std | 2.631E+11 | 4.572E+10 | 1.857E+11 | 4.430E+10 | 4.693E+11 | 1.210E+11 | 1.249E+11 | 5.616E+11 | 8.378E+03 | ||
Med | 4.584E+12 | 2.345E+11 | 1.154E+12 | 2.308E+11 | 3.454E+12 | 2.080E+12 | 1.855E+12 | 1.344E+12 | 1.344E+12 | ||
F2 | 50 | Avg | 7.261E+05 | 1.122E+05 | 1.630E+05 | 1.931E+05 | 4.182E+06 | 2.035E+05 | 1.457E+05 | 3.009E+05 | 1.588E+04 |
Std | 1.585E+06 | 2.506E+04 | 2.967E+04 | 6.001E+04 | 1.505E+04 | 2.757E+04 | 2.341E+04 | 1.047E+05 | 2.341E+04 | ||
Med | 3.806E+05 | 1.095E+05 | 1.616E+05 | 1.825E+05 | 8.602E+04 | 2.003E+05 | 1.420E+05 | 2.793E+05 | 2.793E+05 | ||
100 | Avg | 9.529E+05 | 3.671E+05 | 4.689E+05 | 8.409E+05 | 5.765E+06 | 3.530E+05 | 4.085E+05 | 9.229E+05 | 1.778E+05 | |
Std | 3.647E+05 | 5.821E+04 | 5.168E+04 | 1.550E+05 | 1.475E+07 | 9.945E+03 | 4.767E+04 | 1.626E+05 | 1.639E+04 | ||
Med | 8.541E+05 | 3.633E+05 | 4.553E+05 | 8.447E+05 | 1.174E+06 | 3.567E+05 | 4.085E+05 | 9.321E+05 | 9.321E+05 | ||
Rank | 50 | W/T/L | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 2/0/0 |
100 | W/T/L | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 0/0/2 | 2/0/0 |
1.2 Appendix 2. Comparison results of the QSSALEO on multimodal functions with traditional algorithms during 2500 iterations.
Fun | D | Criteria | CSO | SSA | PSO | WOA | BAT | HHO | SCA | MFO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|---|
F3 | 50 | Avg | 6.454E+04 | 8.077E+02 | 4.753E+03 | 1.154E+03 | 5.232E+04 | 2.114E+04 | 8.630E+03 | 5.359E+03 | 5.688E+02 |
Std | 1.402E+04 | 9.771E+01 | 1.296E+03 | 1.460E+02 | 4.641E+02 | 4.230E+03 | 1.756E+03 | 3.475E+03 | 9.771E+01 | ||
Med | 6.602E+04 | 7.918E+02 | 4.930E+03 | 1.123E+03 | 1.190E+03 | 2.077E+04 | 8.607E+03 | 4.342E+03 | 4.342E+03 | ||
100 | Avg | 1.942E+05 | 3.583E+03 | 2.536E+04 | 4.469E+03 | 1.381E+05 | 6.554E+04 | 3.857E+04 | 3.535E+04 | 6.998E+02 | |
Std | 3.139E+04 | 1.059E+03 | 3.986E+03 | 9.564E+02 | 3.221E+04 | 9.588E+03 | 6.727E+03 | 1.472E+04 | 5.576E+01 | ||
Med | 1.911E+05 | 3.450E+03 | 2.566E+04 | 4.359E+03 | 1.314E+05 | 6.742E+04 | 3.837E+04 | 3.418E+04 | 3.418E+04 | ||
F4 | 50 | Avg | 1.481E+03 | 8.506E+02 | 9.740E+02 | 9.764E+02 | 1.176E+03 | 9.511E+02 | 1.092E+03 | 9.908E+02 | 8.515E+02 |
Std | 6.610E+01 | 7.137E+01 | 6.156E+01 | 7.472E+01 | 4.057E+01 | 3.603E+01 | 3.351E+01 | 9.151E+01 | 3.351E+01 | ||
Med | 1.486E+03 | 8.457E+02 | 9.729E+02 | 9.589E+02 | 7.339E+02 | 9.484E+02 | 1.095E+03 | 9.703E+02 | 9.703E+02 | ||
100 | Avg | 2.677E+03 | 1.544E+03 | 1.793E+03 | 1.732E+03 | 2.151E+03 | 1.694E+03 | 1.970E+03 | 1.873E+03 | 1.357E+03 | |
Std | 1.150E+02 | 1.023E+02 | 7.583E+01 | 1.425E+02 | 1.643E+02 | 5.982E+01 | 6.461E+01 | 1.622E+02 | 6.113E+01 | ||
Med | 2.683E+03 | 1.523E+03 | 1.801E+03 | 1.732E+03 | 2.158E+03 | 1.687E+03 | 1.985E+03 | 1.870E+03 | 1.870E+03 | ||
F5 | 50 | Avg | 7.741E+02 | 6.839E+02 | 6.848E+02 | 7.185E+02 | 7.118E+02 | 7.021E+02 | 6.998E+02 | 6.894E+02 | 6.748E+02 |
Std | 1.194E+01 | 1.113E+01 | 8.197E+00 | 1.568E+01 | 1.041E+01 | 6.522E+00 | 5.907E+00 | 1.282E+01 | 5.907E+00 | ||
Med | 7.734E+02 | 6.834E+02 | 6.856E+02 | 7.190E+02 | 6.340E+02 | 7.025E+02 | 6.981E+02 | 6.871E+02 | 6.871E+02 | ||
100 | Avg | 7.678E+02 | 6.909E+02 | 7.041E+02 | 7.110E+02 | 7.114E+02 | 6.936E+02 | 7.138E+02 | 7.023E+02 | 6.754E+02 | |
Std | 8.654E+00 | 6.891E+00 | 6.701E+00 | 1.162E+01 | 1.231E+01 | 4.320E+00 | 6.984E+00 | 1.044E+01 | 4.185E+00 | ||
Med | 7.686E+02 | 6.913E+02 | 7.036E+02 | 7.073E+02 | 7.092E+02 | 6.939E+02 | 7.141E+02 | 7.011E+02 | 7.011E+02 | ||
F6 | 50 | Avg | 4.077E+03 | 1.483E+03 | 1.515E+03 | 1.791E+03 | 3.003E+03 | 1.750E+03 | 1.707E+03 | 2.173E+03 | 1.345E+03 |
Std | 2.826E+02 | 1.626E+02 | 7.016E+01 | 9.950E+01 | 1.004E+02 | 5.799E+01 | 7.866E+01 | 5.200E+02 | 5.799E+01 | ||
Med | 4.111E+03 | 1.435E+03 | 1.496E+03 | 1.780E+03 | 1.098E+03 | 1.773E+03 | 1.714E+03 | 2.121E+03 | 2.121E+03 | ||
100 | Avg | 9.061E+03 | 3.304E+03 | 3.181E+03 | 3.568E+03 | 6.129E+03 | 3.336E+03 | 3.715E+03 | 5.403E+03 | 2.661E+03 | |
Std | 5.728E+02 | 1.799E+02 | 1.606E+02 | 1.834E+02 | 1.153E+03 | 9.997E+01 | 1.548E+02 | 9.758E+02 | 2.677E+02 | ||
Med | 9.076E+03 | 3.317E+03 | 3.210E+03 | 3.572E+03 | 5.834E+03 | 3.356E+03 | 3.698E+03 | 5.675E+03 | 5.675E+03 | ||
F7 | 50 | Avg | 1.772E+03 | 1.185E+03 | 1.250E+03 | 1.253E+03 | 1.746E+03 | 1.206E+03 | 1.406E+03 | 1.403E+03 | 1.161E+03 |
Std | 7.072E+01 | 7.845E+01 | 5.276E+01 | 5.448E+01 | 8.739E+01 | 3.710E+01 | 3.010E+01 | 7.913E+01 | 3.010E+01 | ||
Med | 1.774E+03 | 1.179E+03 | 1.246E+03 | 1.255E+03 | 1.037E+03 | 1.200E+03 | 1.411E+03 | 1.373E+03 | 1.373E+03 | ||
100 | Avg | 3.134E+03 | 1.967E+03 | 2.134E+03 | 2.094E+03 | 3.010E+03 | 2.017E+03 | 2.339E+03 | 2.570E+03 | 1.770E+03 | |
Std | 1.110E+02 | 1.107E+02 | 9.283E+01 | 1.228E+02 | 1.707E+02 | 6.178E+01 | 7.515E+01 | 1.904E+02 | 8.575E+01 | ||
Med | 3.110E+03 | 1.989E+03 | 2.095E+03 | 2.098E+03 | 2.965E+03 | 2.015E+03 | 2.342E+03 | 2.561E+03 | 2.561E+03 | ||
F8 | 50 | Avg | 7.208E+04 | 1.486E+04 | 1.990E+04 | 2.848E+04 | 1.649E+04 | 1.488E+04 | 2.573E+04 | 1.867E+04 | 1.093E+04 |
Std | 1.002E+04 | 3.043E+03 | 3.964E+03 | 8.005E+03 | 4.102E+03 | 1.171E+03 | 4.033E+03 | 5.450E+03 | 1.171E+03 | ||
Med | 7.268E+04 | 1.459E+04 | 2.026E+04 | 2.641E+04 | 1.189E+04 | 1.463E+04 | 2.542E+04 | 1.782E+04 | 1.782E+04 | ||
100 | Avg | 1.631E+05 | 3.950E+04 | 6.715E+04 | 5.839E+04 | 3.398E+04 | 3.209E+04 | 8.236E+04 | 5.090E+04 | 2.319E+04 | |
Std | 1.543E+04 | 5.132E+03 | 9.643E+03 | 1.183E+04 | 6.300E+03 | 2.894E+03 | 8.021E+03 | 7.859E+03 | 1.293E+03 | ||
Med | 1.645E+05 | 3.988E+04 | 6.611E+04 | 5.779E+04 | 3.335E+04 | 3.143E+04 | 8.189E+04 | 5.055E+04 | 5.055E+04 | ||
F9 | 50 | Avg | 1.570E+04 | 8.175E+03 | 1.457E+04 | 1.194E+04 | 1.147E+04 | 1.140E+04 | 1.502E+04 | 8.900E+03 | 8.472E+03 |
Std | 5.936E+02 | 8.483E+02 | 7.082E+02 | 1.400E+03 | 2.588E+03 | 1.303E+03 | 3.663E+02 | 1.228E+03 | 3.663E+02 | ||
Med | 1.578E+04 | 8.008E+03 | 1.469E+04 | 1.196E+04 | 7.076E+03 | 1.111E+04 | 1.506E+04 | 8.895E+03 | 8.895E+03 | ||
100 | Avg | 3.342E+04 | 2.000E+04 | 3.214E+04 | 2.595E+04 | 2.674E+04 | 2.582E+04 | 3.221E+04 | 1.880E+04 | 1.648E+04 | |
Std | 6.980E+02 | 1.647E+03 | 5.891E+02 | 2.259E+03 | 2.088E+03 | 1.789E+03 | 5.919E+02 | 2.304E+03 | 2.099E+03 | ||
Med | 3.346E+04 | 2.008E+04 | 3.213E+04 | 2.589E+04 | 2.718E+04 | 2.579E+04 | 3.238E+04 | 1.934E+04 | 1.934E+04 | ||
Rank | 50 | W/T/L | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 7/0/0 |
100 | W/T/L | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 7/0/0 |
1.3 Appendix 3. Comparison results of the QSSALEO on hybrid functions with traditional algorithms during 2500 iterations.
Fun | D | Criteria | CSO | SSA | PSO | WOA | BAT | HHO | SCA | MFO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|---|
F10 | 50 | Avg | 4.977E+04 | 2.524E+03 | 4.505E+03 | 2.783E+03 | 9.222E+04 | 1.637E+04 | 8.829E+03 | 2.228E+04 | 1.374E+03 |
Std | 1.426E+04 | 5.261E+02 | 1.191E+03 | 6.232E+02 | 1.923E+03 | 2.679E+03 | 1.956E+03 | 1.646E+04 | 5.261E+02 | ||
Med | 4.817E+04 | 2.402E+03 | 4.120E+03 | 2.677E+03 | 4.662E+03 | 1.719E+04 | 8.579E+03 | 1.725E+04 | 1.725E+04 | ||
100 | Avg | 7.992E+05 | 6.706E+04 | 1.168E+05 | 1.521E+05 | 2.214E+06 | 2.649E+05 | 1.200E+05 | 1.727E+05 | 3.281E+03 | |
Std | 1.864E+06 | 1.481E+04 | 1.748E+04 | 7.024E+04 | 6.540E+06 | 8.548E+04 | 2.047E+04 | 9.759E+04 | 4.422E+02 | ||
Med | 4.337E+05 | 6.856E+04 | 1.196E+05 | 1.350E+05 | 5.923E+05 | 2.352E+05 | 1.244E+05 | 1.781E+05 | 1.781E+05 | ||
F11 | 50 | Avg | 8.715E+11 | 2.843E+09 | 3.763E+10 | 6.611E+09 | 7.070E+11 | 3.838E+11 | 1.224E+11 | 6.277E+10 | 2.381E+08 |
Std | 2.132E+11 | 2.641E+09 | 1.675E+10 | 3.649E+09 | 1.301E+10 | 1.112E+11 | 2.931E+10 | 4.455E+10 | 2.641E+09 | ||
Med | 8.373E+11 | 1.834E+09 | 3.448E+10 | 5.755E+09 | 5.699E+09 | 3.967E+11 | 1.243E+11 | 4.834E+10 | 4.834E+10 | ||
100 | Avg | 2.331E+12 | 2.004E+10 | 2.434E+11 | 3.298E+10 | 2.046E+12 | 1.161E+12 | 6.433E+11 | 4.275E+11 | 1.163E+09 | |
Std | 3.823E+11 | 1.024E+10 | 6.763E+10 | 1.166E+10 | 3.434E+11 | 1.960E+11 | 1.067E+11 | 2.547E+11 | 4.515E+08 | ||
Med | 2.388E+12 | 1.745E+10 | 2.296E+11 | 3.018E+10 | 2.077E+12 | 1.155E+12 | 6.304E+11 | 4.361E+11 | 4.361E+11 | ||
F12 | 50 | Avg | 5.714E+11 | 1.136E+05 | 5.642E+09 | 1.795E+08 | 4.989E+11 | 1.700E+11 | 4.212E+10 | 2.332E+10 | 1.287E+05 |
Std | 2.047E+11 | 6.688E+04 | 3.459E+09 | 2.499E+08 | 1.131E+09 | 1.173E+11 | 1.563E+10 | 2.875E+10 | 6.688E+04 | ||
Med | 5.895E+11 | 9.423E+04 | 4.518E+09 | 1.125E+08 | 1.139E+09 | 1.409E+11 | 3.750E+10 | 7.490E+09 | 7.490E+09 | ||
100 | Avg | 7.046E+11 | 1.524E+05 | 4.485E+10 | 5.507E+08 | 5.809E+11 | 2.978E+11 | 1.281E+11 | 9.287E+10 | 7.008E+04 | |
Std | 1.292E+11 | 1.410E+05 | 1.536E+10 | 2.991E+08 | 1.360E+11 | 4.924E+10 | 2.388E+10 | 6.045E+10 | 2.202E+04 | ||
Med | 7.308E+11 | 9.138E+04 | 4.148E+10 | 4.556E+08 | 5.975E+11 | 2.820E+11 | 1.311E+11 | 8.116E+10 | 8.116E+10 | ||
F13 | 50 | Avg | 1.342E+08 | 7.774E+05 | 1.179E+06 | 2.292E+06 | 1.390E+08 | 3.041E+07 | 4.860E+06 | 2.360E+06 | 1.231E+05 |
Std | 9.520E+07 | 7.672E+05 | 1.201E+06 | 1.707E+06 | 8.484E+05 | 3.166E+07 | 3.373E+06 | 4.014E+06 | 7.672E+05 | ||
Med | 1.040E+08 | 5.727E+05 | 6.766E+05 | 2.127E+06 | 6.438E+05 | 1.999E+07 | 3.737E+06 | 1.033E+06 | 1.033E+06 | ||
100 | Avg | 2.689E+08 | 1.015E+07 | 1.276E+07 | 1.029E+07 | 2.436E+08 | 3.207E+07 | 3.700E+07 | 2.777E+07 | 5.794E+05 | |
Std | 1.285E+08 | 7.267E+06 | 8.654E+06 | 4.329E+06 | 1.670E+08 | 1.779E+07 | 1.534E+07 | 3.425E+07 | 2.885E+05 | ||
Med | 2.455E+08 | 9.230E+06 | 9.528E+06 | 1.029E+07 | 2.103E+08 | 2.967E+07 | 3.385E+07 | 1.473E+07 | 1.473E+07 | ||
F14 | 50 | Avg | 1.547E+11 | 6.185E+04 | 1.030E+08 | 2.669E+07 | 1.083E+11 | 1.938E+10 | 6.003E+09 | 1.531E+09 | 6.209E+04 |
Std | 6.306E+10 | 3.138E+04 | 9.247E+07 | 4.638E+07 | 1.330E+09 | 1.468E+10 | 2.984E+09 | 2.910E+09 | 3.138E+04 | ||
Med | 1.464E+11 | 5.368E+04 | 6.167E+07 | 6.974E+06 | 5.678E+06 | 1.708E+10 | 5.575E+09 | 3.652E+05 | 3.652E+05 | ||
100 | Avg | 3.141E+11 | 8.790E+04 | 2.040E+09 | 9.488E+07 | 2.981E+11 | 1.247E+11 | 4.045E+10 | 2.984E+10 | 5.427E+04 | |
Std | 7.472E+10 | 5.068E+04 | 1.144E+09 | 1.103E+08 | 7.629E+10 | 3.283E+10 | 1.013E+10 | 2.845E+10 | 1.926E+04 | ||
Med | 3.150E+11 | 7.396E+04 | 1.743E+09 | 5.970E+07 | 2.976E+11 | 1.273E+11 | 4.085E+10 | 2.127E+10 | 2.127E+10 | ||
F15 | 50 | Avg | 1.128E+04 | 4.004E+03 | 4.502E+03 | 5.516E+03 | 1.005E+04 | 7.402E+03 | 5.859E+03 | 4.471E+03 | 3.990E+03 |
Std | 1.748E+03 | 5.624E+02 | 6.527E+02 | 8.221E+02 | 5.171E+02 | 1.753E+03 | 4.389E+02 | 5.127E+02 | 4.389E+02 | ||
Med | 1.087E+04 | 3.842E+03 | 4.574E+03 | 5.421E+03 | 3.145E+03 | 7.033E+03 | 5.916E+03 | 4.468E+03 | 4.468E+03 | ||
100 | Avg | 3.115E+04 | 8.345E+03 | 1.185E+04 | 1.382E+04 | 2.548E+04 | 1.856E+04 | 1.377E+04 | 8.614E+03 | 6.761E+03 | |
Std | 5.042E+03 | 9.769E+02 | 1.048E+03 | 1.868E+03 | 4.655E+03 | 3.387E+03 | 8.264E+02 | 9.457E+02 | 8.469E+02 | ||
Med | 2.938E+04 | 8.336E+03 | 1.164E+04 | 1.338E+04 | 2.582E+04 | 1.778E+04 | 1.365E+04 | 8.634E+03 | 8.634E+03 | ||
F16 | 50 | Avg | 1.184E+05 | 3.658E+03 | 3.427E+03 | 4.193E+03 | 1.482E+05 | 5.216E+03 | 4.710E+03 | 4.535E+03 | 3.402E+03 |
Std | 1.634E+05 | 3.799E+02 | 3.704E+02 | 4.726E+02 | 2.495E+02 | 1.085E+03 | 2.911E+02 | 1.526E+03 | 2.911E+02 | ||
Med | 7.325E+04 | 3.568E+03 | 3.375E+03 | 4.113E+03 | 2.855E+03 | 4.952E+03 | 4.716E+03 | 4.182E+03 | 4.182E+03 | ||
100 | Avg | 3.547E+07 | 6.352E+03 | 7.671E+03 | 9.070E+03 | 2.913E+07 | 1.246E+06 | 3.178E+04 | 1.396E+04 | 5.870E+03 | |
Std | 3.877E+07 | 7.095E+02 | 7.814E+02 | 1.455E+03 | 2.968E+07 | 1.707E+06 | 3.442E+04 | 8.849E+03 | 7.443E+02 | ||
Med | 2.106E+07 | 6.446E+03 | 7.626E+03 | 8.740E+03 | 1.565E+07 | 7.740E+05 | 1.758E+04 | 1.028E+04 | 1.028E+04 | ||
F17 | 50 | Avg | 3.569E+08 | 5.877E+06 | 7.889E+06 | 1.633E+07 | 4.527E+08 | 7.121E+07 | 2.754E+07 | 1.294E+07 | 9.575E+05 |
Std | 2.342E+08 | 4.167E+06 | 5.373E+06 | 1.274E+07 | 1.130E+07 | 4.117E+07 | 1.397E+07 | 1.501E+07 | 4.167E+06 | ||
Med | 3.408E+08 | 4.547E+06 | 6.203E+06 | 1.167E+07 | 3.500E+06 | 5.392E+07 | 2.351E+07 | 9.159E+06 | 9.159E+06 | ||
100 | Avg | 6.341E+08 | 9.402E+06 | 1.277E+07 | 6.782E+06 | 5.800E+08 | 4.460E+07 | 6.887E+07 | 1.558E+07 | 1.043E+06 | |
Std | 2.607E+08 | 6.544E+06 | 5.676E+06 | 3.344E+06 | 3.754E+08 | 2.905E+07 | 2.736E+07 | 2.202E+07 | 4.230E+05 | ||
Med | 5.907E+08 | 6.943E+06 | 1.191E+07 | 5.771E+06 | 4.783E+08 | 3.940E+07 | 6.400E+07 | 8.656E+06 | 8.656E+06 | ||
F18 | 50 | Avg | 7.482E+10 | 1.854E+07 | 1.567E+08 | 2.907E+07 | 1.795E+04 | 7.659E+09 | 3.744E+09 | 8.190E+08 | 4.147E+06 |
Std | 2.695E+10 | 1.984E+07 | 2.194E+08 | 6.049E+07 | 8.186E+07 | 8.203E+09 | 1.666E+09 | 2.304E+09 | 3.705E+03 | ||
Med | 7.234E+10 | 8.922E+06 | 6.868E+07 | 1.058E+07 | 5.009E+06 | 6.083E+09 | 3.407E+09 | 3.988E+07 | 3.988E+07 | ||
100 | Avg | 3.435E+11 | 9.470E+07 | 6.097E+09 | 1.340E+08 | 2.906E+11 | 1.260E+11 | 3.739E+10 | 2.363E+10 | 2.529E+07 | |
Std | 9.166E+10 | 9.252E+07 | 2.139E+09 | 9.164E+07 | 8.846E+10 | 3.809E+10 | 1.217E+10 | 2.550E+10 | 1.619E+07 | ||
Med | 3.490E+11 | 5.832E+07 | 6.007E+09 | 1.107E+08 | 3.066E+11 | 1.233E+11 | 3.662E+10 | 1.716E+10 | 1.716E+10 | ||
F19 | 50 | Avg | 4.480E+03 | 3.212E+03 | 3.812E+03 | 3.765E+03 | 4.237E+03 | 3.522E+03 | 4.007E+03 | 3.843E+03 | 3.036E+03 |
Std | 2.392E+02 | 2.869E+02 | 3.241E+02 | 3.512E+02 | 4.889E+02 | 2.806E+02 | 1.611E+02 | 2.922E+02 | 1.611E+02 | ||
Med | 4.499E+03 | 3.235E+03 | 3.884E+03 | 3.813E+03 | 3.144E+03 | 3.596E+03 | 4.007E+03 | 3.797E+03 | 3.797E+03 | ||
100 | Avg | 8.150E+03 | 5.317E+03 | 7.362E+03 | 6.475E+03 | 6.527E+03 | 6.129E+03 | 7.480E+03 | 5.832E+03 | 5.325E+03 | |
Std | 3.363E+02 | 4.961E+02 | 3.151E+02 | 6.275E+02 | 6.295E+02 | 4.514E+02 | 2.955E+02 | 4.978E+02 | 5.837E+02 | ||
Med | 8.178E+03 | 5.350E+03 | 7.378E+03 | 6.457E+03 | 6.365E+03 | 6.080E+03 | 7.518E+03 | 5.983E+03 | 5.983E+03 | ||
Rank | 50 | W/T/L | 0/0/10 | 2/0/8 | 0/0/9 | 0/0/10 | 1/0/9 | 0/0/10 | 0/0/10 | 0/0/10 | 7/0/3 |
100 | W/T/L | 0/0/10 | 0/0/10 | 0/0/10 | 0/0/10 | 0/0/10 | 0/0/10 | 0/0/10 | 0/0/10 | 10/0/0 |
1.4 Appendix 4. Comparison results of the QSSALEO on composite functions with traditional algorithms during 2500 iterations.
Fun | D | Criteria | CSO | SSA | PSO | WOA | BAT | HHO | SCA | MFO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|---|
F20 | 50 | Avg | 3.427E+03 | 2.634E+03 | 2.813E+03 | 2.959E+03 | 3.151E+03 | 3.035E+03 | 2.908E+03 | 2.788E+03 | 2.655E+03 |
Std | 1.339E+02 | 6.514E+01 | 4.596E+01 | 1.013E+02 | 5.125E+01 | 8.516E+01 | 4.460E+01 | 7.420E+01 | 4.460E+01 | ||
Med | 3.413E+03 | 2.646E+03 | 2.819E+03 | 2.953E+03 | 2.530E+03 | 3.014E+03 | 2.903E+03 | 2.781E+03 | 2.781E+03 | ||
100 | Avg | 5.221E+03 | 3.489E+03 | 3.889E+03 | 4.235E+03 | 4.984E+03 | 4.576E+03 | 4.048E+03 | 3.761E+03 | 3.384E+03 | |
Std | 2.371E+02 | 1.552E+02 | 1.080E+02 | 1.535E+02 | 2.372E+02 | 2.392E+02 | 9.891E+01 | 1.518E+02 | 1.706E+02 | ||
Med | 5.229E+03 | 3.477E+03 | 3.885E+03 | 4.210E+03 | 5.022E+03 | 4.536E+03 | 4.042E+03 | 3.743E+03 | 3.743E+03 | ||
F21 | 50 | Avg | 1.753E+04 | 1.046E+04 | 1.593E+04 | 1.327E+04 | 1.390E+04 | 1.340E+04 | 1.665E+04 | 1.049E+04 | 9.609E+03 |
Std | 6.786E+02 | 1.850E+03 | 1.745E+03 | 1.324E+03 | 2.528E+03 | 1.208E+03 | 4.319E+02 | 1.010E+03 | 4.319E+02 | ||
Med | 1.746E+04 | 1.031E+04 | 1.639E+04 | 1.348E+04 | 8.790E+03 | 1.333E+04 | 1.674E+04 | 1.068E+04 | 1.068E+04 | ||
100 | Avg | 3.570E+04 | 2.277E+04 | 3.444E+04 | 2.934E+04 | 2.901E+04 | 2.881E+04 | 3.462E+04 | 2.074E+04 | 1.959E+04 | |
Std | 5.994E+02 | 3.944E+03 | 9.004E+02 | 1.442E+03 | 2.140E+03 | 1.822E+03 | 4.825E+02 | 1.812E+03 | 1.632E+03 | ||
Med | 3.570E+04 | 2.305E+04 | 3.454E+04 | 2.931E+04 | 2.952E+04 | 2.889E+04 | 3.470E+04 | 2.074E+04 | 2.074E+04 | ||
F22 | 50 | Avg | 4.941E+03 | 3.165E+03 | 3.443E+03 | 3.721E+03 | 4.618E+03 | 4.234E+03 | 3.590E+03 | 3.231E+03 | 3.177E+03 |
Std | 3.582E+02 | 9.977E+01 | 8.097E+01 | 1.883E+02 | 9.548E+01 | 2.132E+02 | 7.492E+01 | 7.564E+01 | 7.492E+01 | ||
Med | 4.972E+03 | 3.147E+03 | 3.451E+03 | 3.751E+03 | 2.989E+03 | 4.207E+03 | 3.578E+03 | 3.219E+03 | 3.219E+03 | ||
100 | Avg | 7.841E+03 | 4.052E+03 | 4.883E+03 | 5.034E+03 | 6.516E+03 | 6.155E+03 | 5.047E+03 | 3.957E+03 | 3.743E+03 | |
Std | 7.697E+02 | 1.924E+02 | 1.603E+02 | 2.216E+02 | 3.222E+02 | 4.134E+02 | 1.198E+02 | 1.533E+02 | 2.105E+02 | ||
Med | 7.832E+03 | 4.022E+03 | 4.898E+03 | 5.037E+03 | 6.491E+03 | 6.006E+03 | 5.038E+03 | 3.954E+03 | 3.954E+03 | ||
F23 | 50 | Avg | 5.453E+03 | 3.294E+03 | 3.664E+03 | 3.776E+03 | 4.864E+03 | 4.496E+03 | 3.775E+03 | 3.246E+03 | 3.239E+03 |
Std | 5.384E+02 | 9.187E+01 | 7.984E+01 | 1.723E+02 | 1.376E+02 | 2.405E+02 | 6.064E+01 | 5.091E+01 | 5.091E+01 | ||
Med | 5.434E+03 | 3.270E+03 | 3.653E+03 | 3.773E+03 | 3.165E+03 | 4.486E+03 | 3.768E+03 | 3.243E+03 | 3.243E+03 | ||
100 | Avg | 1.358E+04 | 4.789E+03 | 6.537E+03 | 6.260E+03 | 1.032E+04 | 9.445E+03 | 6.884E+03 | 4.592E+03 | 4.264E+03 | |
Std | 1.151E+03 | 2.263E+02 | 4.259E+02 | 4.944E+02 | 9.426E+02 | 8.380E+02 | 2.341E+02 | 2.120E+02 | 2.201E+02 | ||
Med | 1.364E+04 | 4.819E+03 | 6.486E+03 | 6.232E+03 | 1.026E+04 | 9.267E+03 | 6.929E+03 | 4.543E+03 | 4.543E+03 | ||
F24 | 50 | Avg | 3.193E+04 | 3.312E+03 | 5.748E+03 | 3.487E+03 | 2.509E+04 | 1.017E+04 | 7.382E+03 | 6.354E+03 | 3.061E+03 |
Std | 6.108E+03 | 8.815E+01 | 6.977E+02 | 1.335E+02 | 4.286E+02 | 9.735E+02 | 7.762E+02 | 3.849E+03 | 8.815E+01 | ||
Med | 3.259E+04 | 3.290E+03 | 5.652E+03 | 3.462E+03 | 3.608E+03 | 1.011E+04 | 7.134E+03 | 4.698E+03 | 4.698E+03 | ||
100 | Avg | 7.226E+04 | 6.001E+03 | 1.180E+04 | 5.585E+03 | 4.748E+04 | 1.972E+04 | 1.795E+04 | 1.252E+04 | 3.382E+03 | |
Std | 1.133E+04 | 6.119E+02 | 1.334E+03 | 4.327E+02 | 9.318E+03 | 1.592E+03 | 2.090E+03 | 4.562E+03 | 5.809E+01 | ||
Med | 7.154E+04 | 6.000E+03 | 1.169E+04 | 5.534E+03 | 4.604E+04 | 1.981E+04 | 1.746E+04 | 1.161E+04 | 1.161E+04 | ||
F25 | 50 | Avg | 2.574E+04 | 8.186E+03 | 1.085E+04 | 1.398E+04 | 2.165E+04 | 1.492E+04 | 1.282E+04 | 9.048E+03 | 6.538E+03 |
Std | 2.820E+03 | 2.675E+03 | 7.221E+02 | 1.215E+03 | 7.765E+02 | 7.011E+02 | 4.847E+02 | 8.458E+02 | 4.847E+02 | ||
Med | 2.562E+04 | 8.475E+03 | 1.081E+04 | 1.417E+04 | 6.629E+03 | 1.479E+04 | 1.277E+04 | 8.941E+03 | 8.941E+03 | ||
100 | Avg | 7.546E+04 | 2.530E+04 | 2.923E+04 | 3.408E+04 | 6.918E+04 | 4.500E+04 | 3.764E+04 | 2.067E+04 | 2.020E+04 | |
Std | 5.348E+03 | 4.351E+03 | 2.092E+03 | 3.802E+03 | 1.012E+04 | 2.540E+03 | 2.281E+03 | 1.936E+03 | 6.508E+03 | ||
Med | 7.575E+04 | 2.565E+04 | 2.881E+04 | 3.385E+04 | 6.804E+04 | 4.505E+04 | 3.723E+04 | 2.098E+04 | 2.098E+04 | ||
F26 | 50 | Avg | 8.007E+03 | 3.878E+03 | 4.606E+03 | 4.286E+03 | 3.200E+03 | 6.351E+03 | 4.579E+03 | 3.646E+03 | 3.778E+03 |
Std | 1.113E+03 | 1.688E+02 | 1.867E+02 | 4.691E+02 | 1.018E+02 | 9.094E+02 | 1.743E+02 | 1.230E+02 | 6.401E−05 | ||
Med | 8.130E+03 | 3.864E+03 | 4.684E+03 | 4.151E+03 | 3.640E+03 | 6.339E+03 | 4.608E+03 | 3.658E+03 | 3.658E+03 | ||
100 | Avg | 1.535E+04 | 4.561E+03 | 6.511E+03 | 5.236E+03 | 3.200E+03 | 1.208E+04 | 7.791E+03 | 4.160E+03 | 3.906E+03 | |
Std | 1.432E+03 | 3.047E+02 | 5.675E+02 | 7.152E+02 | 8.459E−05 | 1.755E+03 | 4.410E+02 | 2.111E+02 | 2.323E+02 | ||
Med | 1.558E+04 | 4.497E+03 | 6.428E+03 | 5.039E+03 | 3.200E+03 | 1.225E+04 | 7.798E+03 | 4.106E+03 | 4.106E+03 | ||
F27 | 50 | Avg | 1.849E+04 | 3.831E+03 | 5.611E+03 | 4.264E+03 | 3.300E+03 | 9.891E+03 | 7.310E+03 | 8.507E+03 | 3.304E+03 |
Std | 2.195E+03 | 2.585E+02 | 5.386E+02 | 2.741E+02 | 5.211E+02 | 8.839E+02 | 6.693E+02 | 1.117E+03 | 3.661E−05 | ||
Med | 1.847E+04 | 3.751E+03 | 5.717E+03 | 4.245E+03 | 4.404E+03 | 9.849E+03 | 7.295E+03 | 8.865E+03 | 8.865E+03 | ||
100 | Avg | 5.591E+04 | 7.751E+03 | 1.334E+04 | 7.187E+03 | 3.300E+03 | 2.330E+04 | 2.254E+04 | 2.047E+04 | 3.465E+03 | |
Std | 5.302E+03 | 1.443E+03 | 1.805E+03 | 6.741E+02 | 7.431E−05 | 1.568E+03 | 1.785E+03 | 3.778E+03 | 4.584E+01 | ||
Med | 5.688E+04 | 7.911E+03 | 1.331E+04 | 7.025E+03 | 3.300E+03 | 2.346E+04 | 2.219E+04 | 2.041E+04 | 2.041E+04 | ||
F28 | 50 | Avg | 7.208E+05 | 6.262E+03 | 7.035E+03 | 8.313E+03 | 4.121E+05 | 2.819E+04 | 8.007E+03 | 5.752E+03 | 6.257E+03 |
Std | 1.062E+06 | 7.372E+02 | 8.103E+02 | 9.872E+02 | 3.547E+02 | 2.448E+04 | 8.454E+02 | 6.151E+02 | 6.151E+02 | ||
Med | 2.641E+05 | 6.215E+03 | 7.157E+03 | 8.264E+03 | 4.812E+03 | 1.969E+04 | 7.943E+03 | 5.622E+03 | 5.622E+03 | ||
100 | Avg | 3.164E+06 | 1.260E+04 | 1.537E+04 | 1.650E+04 | 3.323E+06 | 1.641E+05 | 2.357E+04 | 4.639E+04 | 1.081E+04 | |
Std | 2.341E+06 | 1.667E+03 | 2.083E+03 | 2.669E+03 | 4.187E+06 | 1.030E+05 | 6.140E+03 | 1.041E+05 | 7.850E+02 | ||
Med | 2.692E+06 | 1.269E+04 | 1.523E+04 | 1.597E+04 | 1.585E+06 | 1.320E+05 | 2.191E+04 | 1.253E+04 | 1.253E+04 | ||
F29 | 50 | Avg | 1.044E+11 | 5.302E+08 | 8.971E+08 | 5.825E+08 | 8.860E+10 | 1.247E+10 | 6.105E+09 | 1.866E+09 | 1.799E+08 |
Std | 4.397E+10 | 2.303E+08 | 5.753E+08 | 3.097E+08 | 2.399E+08 | 7.920E+09 | 1.983E+09 | 3.894E+09 | 2.303E+08 | ||
Med | 9.897E+10 | 5.117E+08 | 7.147E+08 | 5.299E+08 | 3.399E+08 | 9.554E+09 | 6.234E+09 | 1.181E+08 | 1.181E+08 | ||
100 | Avg | 5.255E+11 | 1.798E+09 | 2.074E+10 | 2.898E+09 | 4.476E+11 | 2.123E+11 | 7.870E+10 | 3.784E+10 | 4.415E+08 | |
Std | 9.543E+10 | 1.024E+09 | 7.127E+09 | 1.296E+09 | 1.014E+11 | 6.481E+10 | 1.681E+10 | 2.535E+10 | 1.901E+08 | ||
Med | 5.224E+11 | 1.593E+09 | 1.913E+10 | 2.536E+09 | 4.301E+11 | 2.088E+11 | 7.572E+10 | 3.375E+10 | 3.375E+10 | ||
Rank | 50 | W/T/L | 0/0/10 | 2/0/9 | 0/0/10 | 0/0/10 | 2/0/8 | 0/0/10 | 0/0/10 | 1/0/9 | 4/0/6 |
100 | W/T/L | 0/0/10 | 0/0/10 | 0/0/10 | 0/0/10 | 2/0/8 | 0/0/10 | 0/0/10 | 0/0/10 | 8/0/2 |
1.5 Appendix 5. Overall effectiveness OE of the QSSALEO with traditional algorithms.
Dimensions | Criteria | CSO | SSA | PSO | WOA | BAT | HHO | SCA | MFO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|
50 | W/T/L | 0/0/29 | 4/0/25 | 0/0/29 | 0/0/29 | 2/0/27 | 0/0/29 | 0/0/29 | 1/0/28 | 22/0/7 |
OE | 0% | 13.79% | 0% | 0% | 6.89% | 0% | 0% | 3.44% | 75.68% | |
100 | W/T/L | 0/0/29 | 0/0/29 | 0/0/29 | 0/0/29 | 2/0/27 | 0/0/29 | 0/0/29 | 0/0/29 | 27/0/2 |
OE | 0% | 0% | 0% | 0% | 6.89% | 0% | 0% | 0% | 93.10% |
1.6 Appendix 6. Wilcoxon rank-sum (p value) of the QSSALEO versus other traditional algorithms on CEC2017 with 50 and 100 dimensions.
Fun | D | CSO | SSA | PSO | WOA | BAT | HHO | SCA | MFO |
|---|---|---|---|---|---|---|---|---|---|
F1 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F2 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F3 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F4 | 50 | < 0.05 | 0.738113285 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F5 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | 0.7616 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.7616 | |
F6 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F7 | 50 | < 0.05 | 0.222170531 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F8 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F9 | 50 | < 0.05 | 0.129455778 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.104139005 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F10 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F11 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F12 | 50 | < 0.05 | 0.591599389 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F13 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F14 | 50 | < 0.05 | 0.591599389 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F15 | 50 | < 0.05 | 0.993795993 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F16 | 50 | < 0.05 | < 0.05 | 0.680259258 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F17 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F18 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F19 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | 0.968987463 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F20 | 50 | < 0.05 | 0.396691029 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F21 | 50 | < 0.05 | 0.205001323 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.065354024 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F22 | 50 | < 0.05 | 0.797490352 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F23 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.423194286 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F24 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F25 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.114460738 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.137504391 | |
F26 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F27 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F28 | < 0.05 | < 0.05 | 0.528807429 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F29 | 50 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.240345789 |
100 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
1.7 Appendix 7. Summary of Freidman test results on CEC2017’s test functions with dimensions 50 and 100
Algorithm | Dimension | Average rank | Overall rank |
|---|---|---|---|
CSO | 50 | 8.69 | 9 |
100 | 8.74 | 9 | |
SSA | 50 | 2.43 | 2 |
100 | 2.63 | 2 | |
PSO | 50 | 4.35 | 4 |
100 | 4.71 | 5 | |
WOA | 50 | 4.57 | 5 |
100 | 4.21 | 3 | |
BAT | 50 | 7.04 | 8 |
100 | 7.10 | 8 | |
HHO | 50 | 6.21 | 7 |
100 | 5.89 | 6 | |
SCA | 50 | 5.90 | 6 |
100 | 6.10 | 7 | |
MFO | 50 | 4.13 | 3 |
100 | 4.35 | 4 | |
QSSALEO | 50 | 1.68 | 1 |
100 | 1.27 | 1 |
1.8 Appendix 8. Comparison results of the QSSALEO with some recent algorithms during 2500 iterations.
F | Cr | RW-GWO | HI-WOA | LNIMRA | PPSO-W | PPSO | LJA | CPSO | WFOA | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|
F1 | Avg | 1.283E+10 | 1.639E+12 | 2.382E+12 | 1.236E+09 | 2.568E+09 | 3.316E+12 | 3.974E+12 | 2.408E+12 | 7.085E+03 |
Std | 2.567E+09 | 1.217E+11 | 1.466E+11 | 3.238E+09 | 4.371E+09 | 6.474E+11 | 6.825E+11 | 1.396E+11 | 8.378E+03 | |
Med | 1.297E+10 | 1.647E+12 | 2.413E+12 | 2.363E+08 | 9.568E+08 | 3.319E+12 | 4.158E+12 | 2.387E+12 | 2.387E+12 | |
F2 | Avg | 8.560E+05 | 3.571E+05 | 2.902E+05 | 1.844E+05 | 1.638E+05 | 1.387E+06 | 7.775E+05 | 2.662E+12 | 1.778E+05 |
Std | 3.065E+05 | 5.043E+03 | 2.219E+04 | 9.234E+04 | 2.757E+04 | 4.888E+05 | 1.057E+05 | 9.576E+12 | 1.639E+04 | |
Med | 8.032E+05 | 3.587E+05 | 2.909E+05 | 1.545E+05 | 1.588E+05 | 1.283E+06 | 7.583E+05 | 1.020E+11 | 1.020E+11 | |
F3 | Avg | 1.354E+03 | 4.421E+04 | 6.852E+04 | 9.558E+02 | 9.868E+02 | 1.203E+05 | 1.322E+05 | 9.222E+04 | 6.998E+02 |
Std | 1.436E+02 | 7.350E+03 | 1.232E+04 | 9.083E+01 | 8.877E+01 | 3.887E+04 | 3.573E+04 | 1.338E+04 | 5.576E+01 | |
Med | 1.321E+03 | 4.412E+04 | 7.115E+04 | 9.397E+02 | 9.712E+02 | 1.097E+05 | 1.387E+05 | 8.903E+04 | 8.903E+04 | |
F4 | Avg | 1.522E+03 | 1.890E+03 | 1.902E+03 | 1.411E+03 | 1.393E+03 | 2.291E+03 | 2.390E+03 | 2.142E+03 | 1.357E+03 |
Std | 8.857E+01 | 9.198E+01 | 4.734E+01 | 7.814E+01 | 6.824E+01 | 1.354E+02 | 1.340E+02 | 5.432E+01 | 6.113E+01 | |
Med | 1.523E+03 | 1.859E+03 | 1.917E+03 | 1.398E+03 | 1.393E+03 | 2.264E+03 | 2.393E+03 | 2.146E+03 | 2.146E+03 | |
F5 | Avg | 6.941E+02 | 7.119E+02 | 7.103E+02 | 6.825E+02 | 6.840E+02 | 7.367E+02 | 7.481E+02 | 7.265E+02 | 6.754E+02 |
Std | 5.657E+00 | 6.237E+00 | 3.656E+00 | 4.936E+00 | 5.342E+00 | 7.214E+00 | 1.782E+01 | 4.667E+00 | 4.185E+00 | |
Med | 6.950E+02 | 7.121E+02 | 7.106E+02 | 6.818E+02 | 6.858E+02 | 7.369E+02 | 7.504E+02 | 7.266E+02 | 7.266E+02 | |
F6 | Avg | 3.038E+03 | 3.836E+03 | 3.721E+03 | 3.219E+03 | 3.242E+03 | 5.400E+03 | 8.624E+03 | 4.114E+03 | 2.661E+03 |
Std | 2.029E+02 | 7.444E+01 | 1.022E+02 | 1.687E+02 | 1.469E+02 | 1.192E+03 | 9.155E+02 | 7.442E+01 | 2.677E+02 | |
Med | 3.008E+03 | 3.832E+03 | 3.737E+03 | 3.248E+03 | 3.256E+03 | 5.191E+03 | 8.695E+03 | 4.100E+03 | 4.100E+03 | |
F7 | Avg | 1.903E+03 | 2.222E+03 | 1.993E+03 | 2.104E+03 | 2.436E+03 | 2.643E+03 | 2.822E+03 | 2.695E+03 | 1.770E+03 |
Std | 9.018E+01 | 1.216E+02 | 8.927E+01 | 1.048E+02 | 1.172E+02 | 1.653E+02 | 1.769E+02 | 6.481E+01 | 8.575E+01 | |
Med | 1.901E+03 | 2.177E+03 | 1.967E+03 | 2.093E+03 | 2.422E+03 | 2.645E+03 | 2.796E+03 | 2.684E+03 | 2.684E+03 | |
F8 | Avg | 5.044E+04 | 6.222E+04 | 6.140E+04 | 2.768E+04 | 2.870E+04 | 1.059E+05 | 1.242E+05 | 7.671E+04 | 2.319E+04 |
Std | 5.120E+03 | 5.638E+03 | 2.454E+03 | 3.207E+03 | 3.363E+03 | 1.863E+04 | 2.142E+04 | 3.589E+03 | 1.293E+03 | |
Med | 4.941E+04 | 6.210E+04 | 6.122E+04 | 2.717E+04 | 2.812E+04 | 1.012E+05 | 1.258E+05 | 7.665E+04 | 7.665E+04 | |
F9 | Avg | 2.005E+04 | 2.969E+04 | 2.855E+04 | 1.779E+04 | 1.780E+04 | 3.234E+04 | 3.373E+04 | 3.220E+04 | 1.648E+04 |
Std | 1.505E+03 | 2.138E+03 | 1.025E+03 | 1.947E+03 | 1.594E+03 | 1.126E+03 | 1.384E+03 | 1.126E+03 | 2.099E+03 | |
Med | 2.029E+04 | 3.008E+04 | 2.835E+04 | 1.798E+04 | 1.782E+04 | 3.224E+04 | 3.374E+04 | 3.230E+04 | 3.230E+04 | |
F10 | Avg | 4.820E+04 | 1.864E+05 | 1.136E+05 | 9.601E+03 | 5.673E+03 | 5.257E+05 | 4.565E+05 | 2.820E+05 | 3.281E+03 |
Std | 1.852E+04 | 2.764E+04 | 2.004E+04 | 1.270E+04 | 4.478E+03 | 2.514E+05 | 1.533E+05 | 9.170E+04 | 4.422E+02 | |
Med | 4.330E+04 | 1.875E+05 | 1.155E+05 | 3.971E+03 | 4.538E+03 | 5.017E+05 | 4.482E+05 | 2.680E+05 | 2.680E+05 | |
F11 | Avg | 3.343E+09 | 5.684E+11 | 1.140E+12 | 3.762E+09 | 2.639E+09 | 1.591E+12 | 1.868E+12 | 1.543E+12 | 1.163E+09 |
Std | 1.775E+09 | 1.086E+11 | 2.230E+11 | 5.973E+09 | 8.317E+09 | 3.155E+11 | 4.165E+11 | 1.272E+11 | 4.515E+08 | |
Med | 2.939E+09 | 5.577E+11 | 1.199E+12 | 1.423E+09 | 1.034E+09 | 1.596E+12 | 1.760E+12 | 1.545E+12 | 1.545E+12 | |
F12 | Avg | 8.381E+07 | 1.272E+11 | 2.903E+11 | 5.017E+08 | 2.944E+07 | 4.449E+11 | 4.576E+11 | 3.755E+11 | 7.008E+04 |
Std | 1.357E+08 | 2.707E+10 | 7.765E+10 | 1.497E+09 | 1.610E+08 | 1.185E+11 | 1.512E+11 | 5.350E+10 | 2.202E+04 | |
Med | 2.125E+07 | 1.278E+11 | 2.891E+11 | 6.610E+04 | 5.066E+04 | 4.502E+11 | 4.610E+11 | 3.790E+11 | 3.790E+11 | |
F13 | Avg | 6.062E+06 | 8.733E+06 | 3.207E+06 | 4.401E+05 | 5.473E+05 | 2.060E+08 | 2.364E+08 | 6.334E+08 | 5.794E+05 |
Std | 2.507E+06 | 2.985E+06 | 2.467E+06 | 3.231E+05 | 2.890E+05 | 1.672E+08 | 2.002E+08 | 4.781E+08 | 2.885E+05 | |
Med | 5.172E+06 | 8.186E+06 | 2.193E+06 | 3.451E+05 | 5.048E+05 | 1.667E+08 | 1.699E+08 | 5.706E+08 | 5.706E+08 | |
F14 | Avg | 8.554E+07 | 5.048E+10 | 9.808E+10 | 3.906E+04 | 2.934E+04 | 2.055E+11 | 2.372E+11 | 3.011E+11 | 5.427E+04 |
Std | 2.975E+08 | 1.294E+10 | 4.058E+10 | 1.568E+04 | 1.323E+04 | 7.409E+10 | 1.095E+11 | 3.875E+10 | 1.926E+04 | |
Med | 6.004E+06 | 5.081E+10 | 9.008E+10 | 3.610E+04 | 2.859E+04 | 1.953E+11 | 2.149E+11 | 2.916E+11 | 2.916E+11 | |
F15 | Avg | 7.203E+03 | 1.711E+04 | 1.459E+04 | 6.889E+03 | 6.924E+03 | 2.066E+04 | 5.598E+03 | 2.851E+04 | 6.761E+03 |
Std | 7.317E+02 | 1.579E+03 | 2.421E+03 | 8.083E+02 | 8.771E+02 | 3.459E+03 | 7.717E+02 | 2.822E+03 | 8.469E+02 | |
Med | 7.198E+03 | 1.677E+04 | 1.393E+04 | 6.829E+03 | 7.097E+03 | 2.014E+04 | 5.638E+03 | 2.858E+04 | 2.858E+04 | |
F16 | Avg | 7.852E+03 | 3.433E+04 | 2.651E+05 | 6.391E+03 | 6.316E+03 | 9.392E+06 | 5.020E+03 | 4.567E+07 | 5.870E+03 |
Std | 1.869E+03 | 3.763E+04 | 4.935E+05 | 7.105E+02 | 7.677E+02 | 2.165E+07 | 6.005E+02 | 4.060E+07 | 7.443E+02 | |
Med | 7.110E+03 | 2.093E+04 | 6.223E+04 | 6.289E+03 | 6.370E+03 | 2.214E+06 | 4.911E+03 | 2.441E+07 | 2.441E+07 | |
F17 | Avg | 7.616E+06 | 1.365E+07 | 4.071E+06 | 7.758E+05 | 8.007E+05 | 3.662E+08 | 3.915E+08 | 2.829E+08 | 1.043E+06 |
Std | 3.285E+06 | 4.283E+06 | 4.848E+06 | 1.124E+06 | 3.557E+05 | 2.281E+08 | 2.456E+08 | 2.489E+08 | 4.230E+05 | |
Med | 6.903E+06 | 1.312E+07 | 2.351E+06 | 4.301E+05 | 7.988E+05 | 3.167E+08 | 2.872E+08 | 2.039E+08 | 2.039E+08 | |
F18 | Avg | 4.021E+07 | 4.837E+10 | 8.973E+10 | 1.315E+06 | 8.850E+05 | 1.819E+11 | 2.377E+11 | 2.910E+11 | 2.529E+07 |
Std | 2.936E+07 | 1.427E+10 | 4.176E+10 | 1.786E+06 | 1.366E+06 | 8.458E+10 | 7.455E+10 | 5.335E+10 | 1.619E+07 | |
Med | 3.400E+07 | 4.548E+10 | 9.880E+10 | 5.597E+05 | 3.519E+05 | 1.716E+11 | 2.271E+11 | 2.765E+11 | 2.765E+11 | |
F19 | Avg | 5.483E+03 | 6.429E+03 | 6.190E+03 | 5.702E+03 | 5.566E+03 | 8.098E+03 | 6.329E+03 | 7.569E+03 | 5.325E+03 |
Std | 4.786E+02 | 6.004E+02 | 3.441E+02 | 6.004E+02 | 6.466E+02 | 7.048E+02 | 1.097E+03 | 5.511E+02 | 5.837E+02 | |
Med | 5.354E+03 | 6.308E+03 | 6.271E+03 | 5.649E+03 | 5.501E+03 | 7.957E+03 | 6.862E+03 | 7.536E+03 | 7.536E+03 | |
F20 | Avg | 3.554E+03 | 4.236E+03 | 4.285E+03 | 3.847E+03 | 3.812E+03 | 4.602E+03 | 4.566E+03 | 5.040E+03 | 3.384E+03 |
Std | 1.614E+02 | 1.385E+02 | 1.973E+02 | 1.663E+02 | 2.221E+02 | 2.155E+02 | 2.077E+02 | 1.822E+02 | 1.706E+02 | |
Med | 3.503E+03 | 4.254E+03 | 4.291E+03 | 3.780E+03 | 3.799E+03 | 4.616E+03 | 4.526E+03 | 5.015E+03 | 5.015E+03 | |
F21 | Avg | 2.255E+04 | 3.142E+04 | 3.098E+04 | 2.063E+04 | 2.209E+04 | 3.426E+04 | 3.562E+04 | 3.433E+04 | 1.959E+04 |
Std | 1.358E+03 | 2.372E+03 | 7.127E+02 | 1.764E+03 | 2.274E+03 | 1.347E+03 | 1.392E+03 | 1.105E+03 | 1.632E+03 | |
Med | 2.254E+04 | 3.085E+04 | 3.089E+04 | 2.021E+04 | 2.182E+04 | 3.408E+04 | 3.583E+04 | 3.462E+04 | 3.462E+04 | |
F22 | Avg | 4.301E+03 | 5.082E+03 | 5.434E+03 | 5.213E+03 | 5.149E+03 | 5.456E+03 | 7.267E+03 | 7.447E+03 | 3.743E+03 |
Std | 1.539E+02 | 3.135E+02 | 2.832E+02 | 4.471E+02 | 5.131E+02 | 2.914E+02 | 6.630E+02 | 2.726E+02 | 2.105E+02 | |
Med | 4.277E+03 | 5.018E+03 | 5.430E+03 | 5.204E+03 | 5.065E+03 | 5.405E+03 | 7.331E+03 | 7.430E+03 | 7.430E+03 | |
F23 | Avg | 5.079E+03 | 6.639E+03 | 7.237E+03 | 8.407E+03 | 8.211E+03 | 7.857E+03 | 1.265E+04 | 1.422E+04 | 4.264E+03 |
Std | 2.737E+02 | 5.539E+02 | 4.792E+02 | 1.462E+03 | 1.423E+03 | 7.851E+02 | 1.404E+03 | 1.273E+03 | 2.201E+02 | |
Med | 5.062E+03 | 6.609E+03 | 7.059E+03 | 8.624E+03 | 8.200E+03 | 7.861E+03 | 1.259E+04 | 1.385E+04 | 1.385E+04 | |
F24 | Avg | 3.927E+03 | 1.550E+04 | 2.463E+04 | 3.544E+03 | 3.641E+03 | 4.387E+04 | 6.311E+04 | 2.752E+04 | 3.382E+03 |
Std | 1.307E+02 | 1.354E+03 | 2.346E+03 | 9.074E+01 | 7.417E+01 | 1.419E+04 | 1.609E+04 | 3.029E+03 | 5.809E+01 | |
Med | 3.920E+03 | 1.548E+04 | 2.473E+04 | 3.536E+03 | 3.651E+03 | 3.709E+04 | 6.004E+04 | 2.701E+04 | 2.701E+04 | |
F25 | Avg | 2.194E+04 | 3.931E+04 | 4.799E+04 | 2.674E+04 | 2.865E+04 | 5.010E+04 | 7.247E+04 | 6.360E+04 | 2.020E+04 |
Std | 2.951E+03 | 2.136E+03 | 3.497E+03 | 7.179E+03 | 5.077E+03 | 6.769E+03 | 1.132E+04 | 2.987E+03 | 6.508E+03 | |
Med | 2.242E+04 | 3.892E+04 | 4.877E+04 | 2.751E+04 | 2.888E+04 | 5.076E+04 | 7.219E+04 | 6.425E+04 | 6.425E+04 | |
F26 | Avg | 3.200E+03 | 7.644E+03 | 6.086E+03 | 4.683E+03 | 4.499E+03 | 1.004E+04 | 1.338E+04 | 1.824E+04 | 3.906E+03 |
Std | 4.356E−04 | 7.748E+02 | 7.188E+02 | 6.605E+02 | 5.638E+02 | 1.782E+03 | 1.748E+03 | 1.885E+03 | 2.323E+02 | |
Med | 3.200E+03 | 7.804E+03 | 6.040E+03 | 4.544E+03 | 4.372E+03 | 9.783E+03 | 1.311E+04 | 1.819E+04 | 1.819E+04 | |
F27 | Avg | 3.300E+03 | 1.610E+04 | 3.076E+04 | 3.679E+03 | 3.710E+03 | 4.096E+04 | 5.288E+04 | 1.610E+04 | 3.465E+03 |
Std | 4.884E−04 | 1.157E+03 | 2.365E+03 | 2.510E+02 | 7.173E+01 | 9.127E+03 | 8.733E+03 | 1.157E+03 | 4.584E+01 | |
Med | 3.300E+03 | 1.615E+04 | 3.069E+04 | 3.612E+03 | 3.699E+03 | 3.956E+04 | 5.061E+04 | 1.615E+04 | 1.615E+04 | |
F28 | Avg | 8.085E+03 | 3.380E+04 | 3.973E+04 | 1.065E+04 | 1.010E+04 | 9.277E+05 | 1.065E+04 | 3.380E+04 | 1.081E+04 |
Std | 1.143E+03 | 1.639E+04 | 2.742E+04 | 1.343E+03 | 9.387E+02 | 1.546E+06 | 1.343E+03 | 1.639E+04 | 7.850E+02 | |
Med | 7.847E+03 | 3.000E+04 | 2.843E+04 | 1.055E+04 | 1.004E+04 | 3.787E+05 | 1.055E+04 | 3.000E+04 | 3.000E+04 | |
F29 | Avg | 8.839E+07 | 9.920E+10 | 1.752E+11 | 7.590E+07 | 2.406E+07 | 2.806E+11 | 3.113E+11 | 1.752E+11 | 4.415E+08 |
Std | 3.467E+07 | 2.465E+10 | 7.835E+10 | 2.758E+08 | 1.755E+07 | 1.041E+11 | 1.184E+11 | 7.835E+10 | 1.901E+08 | |
Med | 8.247E+07 | 1.019E+11 | 1.675E+11 | 1.748E+07 | 1.797E+07 | 2.512E+11 | 2.854E+11 | 1.675E+11 | 1.675E+11 | |
Rank W/T/L | 3/0/25 | 0/0/29 | 0/0/29 | 2/0/27 | 3/0/26 | 0/0/29 | 1/0/28 | 0/0/29 | 20/0/9 | |
OE | 13.79% | 0% | 0% | 6.89% | 13.79% | 0% | 3.44% | 0% | 68.96% |
1.9 Appendix 9. Wilcoxon rank-sum of the QSSALEO versus other advanced algorithms on CEC2017.
Fun | RW-GWO | HI-WOA | LNIMRA | PPSO-W | PPSO | LJA | CPSO | WFOA |
|---|---|---|---|---|---|---|---|---|
1 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
2 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
3 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
4 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.07246672 | < 0.05 | < 0.05 | < 0.05 |
5 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
6 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
7 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
8 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
9 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
10 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
11 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.55977641 | < 0.05 | < 0.05 | < 0.05 |
12 | < 0.05 | < 0.05 | < 0.05 | 0.55977641 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
13 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.57029143 | < 0.05 | < 0.05 | < 0.05 |
14 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
15 | < 0.05 | < 0.05 | < 0.05 | 0.69169337 | 0.65761242 | < 0.05 | < 0.05 | < 0.05 |
16 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.12945578 | < 0.05 | < 0.05 | < 0.05 |
17 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
18 | 0.05882719 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
19 | 0.25303015 | < 0.05 | < 0.05 | < 0.05 | 0.16873193 | < 0.05 | < 0.05 | < 0.05 |
20 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
21 | < 0.05 | < 0.05 | < 0.05 | 0.12945578 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
22 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
23 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
24 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
25 | 0.57029143 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
26 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
27 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
28 | < 0.05 | < 0.05 | < 0.05 | 0.3629539 | < 0.05 | < 0.05 | 0.3629539 | < 0.05 |
29 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
1.10 Appendix 10. Friedman test result of the QSSALEO versus other advanced algorithms on CEC2017.
Fun | RW-GWO | HI-WOA | LNIMRA | PPSO-W | PPSO | LJA | CPSO | WFOA | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|
1 | 3.931 | 5.000 | 6.586 | 2.138 | 2.931 | 8.207 | 6.552 | 8.655 | 1.000 |
2 | 6.759 | 4.931 | 3.897 | 1.897 | 1.828 | 7.793 | 8.828 | 6.621 | 2.448 |
3 | 3.966 | 5.069 | 6.069 | 2.414 | 2.621 | 8.172 | 7.241 | 8.448 | 1.000 |
4 | 3.724 | 5.345 | 5.655 | 2.310 | 2.345 | 8.138 | 7.103 | 8.724 | 1.655 |
5 | 3.897 | 5.724 | 5.345 | 2.379 | 2.517 | 8.172 | 7.138 | 8.621 | 1.207 |
6 | 2.310 | 5.862 | 5.172 | 3.172 | 3.345 | 7.897 | 7.138 | 8.931 | 1.172 |
7 | 2.207 | 4.828 | 3.034 | 4.034 | 6.034 | 7.448 | 7.862 | 8.517 | 1.034 |
8 | 4.103 | 5.414 | 5.483 | 2.310 | 2.552 | 8.207 | 7.000 | 8.793 | 1.138 |
9 | 3.517 | 6.034 | 5.276 | 2.414 | 2.517 | 7.621 | 7.517 | 8.552 | 1.552 |
10 | 4.000 | 6.069 | 4.966 | 2.310 | 2.552 | 8.345 | 7.172 | 8.414 | 1.172 |
11 | 3.586 | 5.034 | 6.069 | 2.621 | 1.931 | 7.897 | 7.655 | 8.345 | 1.862 |
12 | 3.828 | 5.034 | 6.310 | 2.310 | 1.690 | 8.069 | 7.586 | 8.000 | 2.172 |
13 | 5.034 | 5.690 | 4.138 | 1.724 | 2.138 | 7.759 | 8.379 | 7.862 | 2.276 |
14 | 4.000 | 5.138 | 5.966 | 2.000 | 1.552 | 7.448 | 8.690 | 7.759 | 2.448 |
15 | 3.793 | 7.069 | 6.172 | 3.310 | 3.310 | 7.793 | 8.966 | 1.483 | 3.103 |
16 | 4.414 | 6.207 | 6.759 | 3.379 | 3.310 | 8.069 | 8.897 | 1.414 | 2.552 |
17 | 4.897 | 5.828 | 3.966 | 1.586 | 2.103 | 8.172 | 7.621 | 8.207 | 2.621 |
18 | 3.621 | 5.207 | 6.000 | 1.621 | 1.448 | 7.207 | 8.552 | 8.034 | 3.310 |
19 | 2.931 | 5.621 | 4.897 | 3.759 | 2.966 | 8.655 | 8.069 | 5.552 | 2.552 |
20 | 2.103 | 5.310 | 5.793 | 3.517 | 3.345 | 7.241 | 8.931 | 7.414 | 1.345 |
21 | 3.379 | 5.862 | 5.552 | 2.103 | 2.897 | 7.345 | 7.655 | 8.586 | 1.621 |
22 | 2.034 | 4.207 | 5.828 | 4.724 | 4.414 | 5.793 | 8.586 | 8.414 | 1.000 |
23 | 2.000 | 3.586 | 4.448 | 5.897 | 5.724 | 5.379 | 8.724 | 8.241 | 1.000 |
24 | 3.966 | 5.000 | 6.241 | 2.172 | 2.793 | 8.207 | 6.793 | 8.759 | 1.069 |
25 | 1.759 | 4.966 | 6.276 | 3.172 | 3.483 | 6.724 | 8.241 | 8.690 | 1.690 |
26 | 1.000 | 6.069 | 4.966 | 3.552 | 3.414 | 6.966 | 8.897 | 8.000 | 2.138 |
27 | 1.000 | 5.500 | 7.172 | 3.207 | 3.793 | 7.966 | 5.500 | 8.862 | 2.000 |
28 | 1.138 | 6.948 | 7.103 | 3.534 | 3.103 | 9.000 | 6.948 | 3.534 | 3.690 |
29 | 2.828 | 5.310 | 6.672 | 1.690 | 1.552 | 8.103 | 6.672 | 8.241 | 3.931 |
Avg | 3.301 | 5.444 | 5.580 | 2.802 | 2.904 | 7.717 | 7.756 | 7.575 | 1.923 |
Rank | 4.00 | 5.00 | 6.00 | 2.00 | 3.00 | 8.00 | 9.00 | 7.00 | 1.00 |
1.11 Appendix 11. Wilcoxon rank-sum test result of the QSSALEO versus other traditional algorithms on CEC2008lsgo.
Fun | D | CSO | SSA | PSO | WOA | GWO |
|---|---|---|---|---|---|---|
F1 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F2 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F3 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F4 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F5 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F6 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.056776 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F7 | 200 | < 0.05 | < 0.05 | < 0.05 | 0.7498769 | < 0.05 |
500 | 0.141670 | < 0.05 | < 0.05 | 0.3880841 | 0.228116250 | |
1000 | < 0.05 | < 0.05 | < 0.05 | 0.8702912 | < 0.05 |
1.12 Appendix 12. Comparison results of the QSSALEO with some improved SSA’s algorithms during 2500 iterations.
F | Criteria | ESSA | HSSASCA | ISSA | ISSA_OBL | IWSSA | STS-SSA | TVSSA | SSA-FGWO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|
F1 | Avg | 8.827E+10 | 2.361E+12 | 1.284E+04 | 1.569E+11 | 2.240E+12 | 2.667E+12 | 3.761E+05 | 2.098E+04 | 7.085E+03 |
Std | 1.928E+10 | 1.549E+11 | 1.693E+04 | 3.985E+10 | 1.038E+11 | 9.447E+10 | 1.996E+06 | 2.227E+04 | 8.378E+03 | |
Med | 8.571E+10 | 2.358E+12 | 6.436E+03 | 1.510E+11 | 2.221E+12 | 2.688E+12 | 6.629E+03 | 1.470E+04 | 2.853E+03 | |
F2 | Avg | 3.006E+05 | 1.038E+06 | 2.797E+05 | 2.543E+05 | 3.449E+05 | 3.511E+05 | 3.111E+05 | 2.941E+05 | 1.778E+05 |
Std | 1.521E+04 | 3.015E+05 | 1.443E+04 | 2.031E+04 | 1.299E+04 | 1.212E+04 | 5.292E+04 | 7.616E+04 | 1.639E+04 | |
Med | 3.013E+05 | 9.958E+05 | 2.807E+05 | 2.530E+05 | 3.466E+05 | 3.509E+05 | 3.113E+05 | 2.841E+05 | 1.801E+05 | |
F3 | Avg | 1.931E+03 | 5.647E+04 | 7.467E+02 | 2.365E+03 | 6.599E+04 | 1.030E+05 | 7.668E+02 | 7.250E+02 | 6.998E+02 |
Std | 2.859E+02 | 1.236E+04 | 4.313E+01 | 5.839E+02 | 5.487E+03 | 1.078E+04 | 5.436E+01 | 5.165E+01 | 5.576E+01 | |
Med | 1.909E+03 | 5.392E+04 | 7.520E+02 | 2.198E+03 | 6.721E+04 | 1.010E+05 | 7.629E+02 | 7.237E+02 | 7.031E+02 | |
F4 | Avg | 1.470E+03 | 1.989E+03 | 1.257E+03 | 1.416E+03 | 2.097E+03 | 2.134E+03 | 1.333E+03 | 1.257E+03 | 1.357E+03 |
Std | 7.087E+01 | 6.050E+01 | 6.995E+01 | 7.334E+01 | 3.611E+01 | 2.856E+01 | 1.044E+02 | 9.174E+01 | 6.113E+01 | |
Med | 1.459E+03 | 1.989E+03 | 1.249E+03 | 1.404E+03 | 2.098E+03 | 2.138E+03 | 1.335E+03 | 1.252E+03 | 1.359E+03 | |
F5 | Avg | 6.888E+02 | 7.213E+02 | 6.925E+02 | 6.865E+02 | 7.267E+02 | 7.286E+02 | 6.768E+02 | 6.766E+02 | 6.754E+02 |
Std | 6.711E+00 | 7.390E+00 | 7.820E+00 | 5.085E+00 | 2.327E+00 | 2.471E+00 | 5.606E+00 | 4.649E+00 | 4.185E+00 | |
Med | 6.885E+02 | 7.206E+02 | 6.721E+02 | 6.872E+02 | 7.270E+02 | 7.285E+02 | 6.764E+02 | 6.755E+02 | 6.761E+02 | |
F6 | Avg | 2.385E+03 | 3.915E+03 | 2.489E+03 | 3.464E+03 | 3.834E+03 | 4.026E+03 | 2.127E+03 | 1.970E+03 | 2.661E+03 |
Std | 1.343E+02 | 1.008E+02 | 8.945E+02 | 2.202E+02 | 8.159E+01 | 4.667E+01 | 1.902E+02 | 2.335E+02 | 2.677E+02 | |
Med | 2.396E+03 | 3.924E+03 | 2.070E+03 | 3.489E+03 | 3.838E+03 | 4.040E+03 | 2.095E+03 | 1.961E+03 | 2.654E+03 | |
F7 | Avg | 1.855E+03 | 2.434E+03 | 1.868E+03 | 1.902E+03 | 2.539E+03 | 2.616E+03 | 1.895E+03 | 1.867E+03 | 1.770E+03 |
Std | 6.579E+01 | 8.771E+01 | 3.664E+02 | 8.392E+01 | 4.861E+01 | 3.571E+01 | 3.833E+02 | 1.436E+02 | 8.575E+01 | |
Med | 1.748E+03 | 2.438E+03 | 1.759E+03 | 1.888E+03 | 2.537E+03 | 2.620E+03 | 1.730E+03 | 1.970E+03 | 1.790E+03 | |
F8 | Avg | 3.912E+04 | 8.851E+04 | 2.351E+04 | 3.211E+04 | 8.260E+04 | 8.299E+04 | 2.601E+04 | 2.554E+04 | 2.319E+04 |
Std | 4.089E+03 | 1.730E+04 | 2.539E+03 | 3.799E+03 | 3.463E+03 | 3.647E+03 | 3.284E+03 | 2.686E+03 | 1.293E+03 | |
Med | 3.892E+04 | 8.634E+04 | 2.437E+04 | 3.228E+04 | 8.368E+04 | 8.315E+04 | 2.694E+04 | 2.516E+04 | 2.323E+04 | |
F9 | Avg | 1.940E+04 | 3.266E+04 | 1.563E+04 | 1.874E+04 | 3.235E+04 | 3.242E+04 | 1.557E+04 | 1.616E+04 | 1.648E+04 |
Std | 1.008E+03 | 1.401E+03 | 1.493E+03 | 1.431E+03 | 6.314E+02 | 5.480E+02 | 1.380E+03 | 1.552E+03 | 2.099E+03 | |
Med | 1.918E+04 | 3.256E+04 | 1.521E+04 | 1.913E+04 | 3.251E+04 | 3.241E+04 | 1.562E+04 | 1.589E+04 | 1.607E+04 | |
F10 | Avg | 7.488E+04 | 2.044E+05 | 1.002E+04 | 3.585E+04 | 1.850E+05 | 2.504E+05 | 5.350E+03 | 8.209E+03 | 3.281E+03 |
Std | 2.707E+04 | 6.640E+04 | 4.192E+03 | 8.530E+03 | 3.161E+04 | 6.234E+04 | 1.122E+03 | 2.351E+03 | 4.422E+02 | |
Med | 6.867E+04 | 1.825E+05 | 8.850E+03 | 3.680E+04 | 1.823E+05 | 2.411E+05 | 5.401E+03 | 7.583E+03 | 3.195E+03 | |
F11 | Avg | 8.628E+09 | 1.199E+12 | 2.872E+09 | 9.239E+09 | 1.059E+12 | 1.765E+12 | 2.436E+09 | 2.860E+09 | 1.163E+09 |
Std | 3.763E+09 | 2.191E+11 | 1.098E+09 | 3.024E+09 | 1.188E+11 | 1.167E+11 | 1.195E+09 | 1.430E+09 | 4.515E+08 | |
Med | 7.629E+09 | 1.189E+12 | 2.690E+09 | 9.394E+09 | 1.082E+12 | 1.781E+12 | 2.282E+09 | 2.585E+09 | 1.081E+09 | |
F12 | Avg | 5.127E+08 | 3.417E+11 | 9.544E+04 | 3.850E+10 | 2.623E+11 | 4.558E+11 | 7.639E+04 | 1.097E+05 | 7.008E+04 |
Std | 2.717E+08 | 8.214E+10 | 3.190E+04 | 1.175E+11 | 3.301E+10 | 5.437E+10 | 2.333E+04 | 4.930E+04 | 2.202E+04 | |
Med | 4.999E+08 | 3.285E+11 | 9.235E+04 | 4.304E+04 | 2.621E+11 | 4.725E+11 | 7.550E+04 | 1.047E+05 | 6.482E+04 | |
F13 | Avg | 1.299E+07 | 5.252E+07 | 2.267E+06 | 5.671E+06 | 5.765E+07 | 1.054E+08 | 1.562E+06 | 1.685E+06 | 5.794E+05 |
Std | 4.386E+06 | 3.417E+07 | 1.249E+06 | 1.483E+06 | 1.446E+07 | 3.844E+07 | 6.772E+05 | 9.122E+05 | 2.885E+05 | |
Med | 1.244E+07 | 3.866E+07 | 2.042E+06 | 5.326E+06 | 5.692E+07 | 9.779E+07 | 1.449E+06 | 1.560E+06 | 4.948E+05 | |
F14 | Avg | 1.044E+08 | 1.601E+11 | 1.028E+05 | 2.006E+05 | 1.114E+11 | 2.451E+11 | 6.783E+04 | 1.013E+05 | 5.427E+04 |
Std | 7.338E+07 | 6.096E+10 | 4.868E+04 | 7.184E+03 | 2.196E+10 | 2.861E+10 | 2.217E+04 | 3.125E+04 | 1.926E+04 | |
Med | 7.972E+07 | 1.471E+11 | 9.821E+04 | 1.845E+04 | 1.123E+11 | 2.436E+11 | 6.826E+04 | 1.035E+05 | 5.752E+04 | |
F15 | Avg | 6.989E+03 | 1.624E+04 | 6.246E+03 | 9.486E+03 | 1.798E+04 | 2.471E+04 | 6.527E+03 | 6.310E+03 | 6.761E+03 |
Std | 8.326E+02 | 2.083E+03 | 8.271E+02 | 1.054E+03 | 1.330E+03 | 1.981E+03 | 8.193E+02 | 9.522E+02 | 8.469E+02 | |
Med | 7.184E+03 | 1.628E+04 | 6.270E+03 | 9.482E+03 | 1.807E+04 | 2.501E+04 | 6.567E+03 | 6.074E+03 | 6.486E+03 | |
F16 | Avg | 6.087E+03 | 1.734E+06 | 5.443E+03 | 6.124E+03 | 3.968E+05 | 9.088E+06 | 5.775E+03 | 5.388E+03 | 5.870E+03 |
Std | 5.924E+02 | 2.399E+06 | 5.564E+02 | 6.603E+02 | 2.406E+05 | 4.607E+06 | 7.074E+02 | 5.274E+02 | 7.443E+02 | |
Med | 6.093E+03 | 7.681E+05 | 5.419E+03 | 6.166E+03 | 3.014E+05 | 9.545E+06 | 5.691E+03 | 5.448E+03 | 6.009E+03 | |
F17 | Avg | 1.018E+07 | 5.592E+07 | 2.972E+06 | 3.501E+06 | 1.038E+08 | 2.143E+08 | 2.571E+06 | 3.876E+06 | 1.043E+06 |
Std | 4.361E+06 | 3.486E+07 | 2.045E+06 | 9.376E+05 | 2.544E+07 | 7.324E+07 | 1.272E+06 | 1.806E+06 | 4.230E+05 | |
Med | 9.139E+06 | 4.680E+07 | 2.578E+06 | 3.291E+06 | 1.020E+08 | 2.069E+08 | 2.543E+06 | 3.831E+06 | 1.037E+06 | |
F18 | Avg | 9.677E+07 | 1.573E+11 | 5.613E+07 | 2.236E+06 | 1.055E+11 | 2.451E+11 | 4.125E+07 | 6.703E+07 | 2.529E+07 |
Std | 1.301E+08 | 4.598E+10 | 3.655E+07 | 1.968E+06 | 2.091E+10 | 2.280E+10 | 3.175E+07 | 4.418E+07 | 1.619E+07 | |
Med | 6.192E+07 | 1.585E+11 | 6.106E+07 | 1.818E+06 | 1.088E+11 | 2.441E+11 | 3.872E+07 | 6.114E+07 | 2.253E+07 | |
F19 | Avg | 5.500E+03 | 7.713E+03 | 5.055E+03 | 5.206E+03 | 7.574E+03 | 7.484E+03 | 5.042E+03 | 5.434E+03 | 5.325E+03 |
Std | 5.766E+02 | 6.073E+02 | 5.568E+02 | 5.069E+02 | 2.204E+02 | 2.689E+02 | 5.060E+02 | 4.888E+02 | 5.837E+02 | |
Med | 5.441E+03 | 7.705E+03 | 4.968E+03 | 5.197E+03 | 7.624E+03 | 7.550E+03 | 4.960E+03 | 5.460E+03 | 5.310E+03 | |
F20 | Avg | 3.263E+03 | 4.325E+03 | 3.037E+03 | 3.663E+03 | 4.334E+03 | 5.370E+03 | 3.229E+03 | 2.995E+03 | 3.384E+03 |
Std | 1.092E+02 | 1.732E+02 | 1.163E+02 | 1.688E+02 | 1.219E+02 | 1.684E+02 | 1.512E+02 | 1.058E+02 | 1.706E+02 | |
Med | 3.266E+03 | 4.323E+03 | 3.023E+03 | 3.682E+03 | 4.353E+03 | 5.390E+03 | 3.184E+03 | 3.003E+03 | 3.428E+03 | |
F21 | Avg | 2.305E+04 | 3.464E+04 | 2.580E+04 | 2.337E+04 | 3.475E+04 | 3.490E+04 | 2.617E+04 | 2.065E+04 | 1.959E+04 |
Std | 1.600E+03 | 1.075E+03 | 8.138E+03 | 1.696E+03 | 4.958E+02 | 5.301E+02 | 7.886E+03 | 3.732E+02 | 1.632E+03 | |
Med | 2.337E+04 | 3.452E+04 | 2.019E+04 | 2.325E+04 | 3.473E+04 | 3.503E+04 | 2.178E+04 | 2.061E+04 | 1.979E+04 | |
F22 | Avg | 3.543E+03 | 5.479E+03 | 3.519E+03 | 4.675E+03 | 5.270E+03 | 6.831E+03 | 3.674E+03 | 3.521E+03 | 3.743E+03 |
Std | 5.769E+01 | 2.779E+02 | 1.613E+02 | 2.828E+02 | 1.105E+02 | 2.694E+02 | 1.384E+02 | 1.225E+02 | 2.105E+02 | |
Med | 3.537E+03 | 5.476E+03 | 3.502E+03 | 4.715E+03 | 5.265E+03 | 6.800E+03 | 3.669E+03 | 3.500E+03 | 3.750E+03 | |
F23 | Avg | 4.320E+03 | 7.673E+03 | 4.077E+03 | 5.637E+03 | 7.500E+03 | 1.173E+04 | 4.181E+03 | 4.048E+03 | 4.264E+03 |
Std | 1.265E+02 | 6.571E+02 | 1.401E+02 | 4.850E+02 | 3.237E+02 | 6.061E+02 | 1.671E+02 | 1.718E+02 | 2.201E+02 | |
Med | 4.314E+03 | 7.568E+03 | 4.043E+03 | 5.546E+03 | 7.543E+03 | 1.171E+04 | 4.139E+03 | 4.019E+03 | 4.219E+03 | |
F24 | Avg | 4.823E+03 | 2.431E+04 | 3.423E+03 | 4.879E+03 | 2.294E+04 | 2.929E+04 | 3.461E+03 | 3.529E+03 | 3.382E+03 |
Std | 2.937E+02 | 3.381E+03 | 7.816E+01 | 2.865E+02 | 1.347E+03 | 1.416E+03 | 6.456E+01 | 9.853E+01 | 5.809E+01 | |
Med | 4.796E+03 | 2.383E+04 | 3.413E+03 | 4.878E+03 | 2.287E+04 | 2.937E+04 | 3.470E+03 | 3.587E+03 | 3.387E+03 | |
F25 | Avg | 1.333E+04 | 4.238E+04 | 1.358E+04 | 2.462E+04 | 4.741E+04 | 5.693E+04 | 1.416E+04 | 1.312E+04 | 2.020E+04 |
Std | 5.431E+03 | 3.190E+03 | 1.356E+03 | 4.702E+03 | 1.781E+03 | 1.753E+03 | 5.007E+03 | 1.391E+03 | 6.508E+03 | |
Med | 1.101E+04 | 4.210E+04 | 1.352E+04 | 2.571E+04 | 4.735E+04 | 5.708E+04 | 1.531E+04 | 1.303E+04 | 2.249E+04 | |
F26 | Avg | 3.737E+03 | 8.623E+03 | 3.642E+03 | 4.967E+03 | 9.318E+03 | 1.435E+04 | 3.845E+03 | 3.719E+03 | 3.906E+03 |
Std | 1.709E+02 | 1.212E+03 | 8.559E+01 | 4.812E+02 | 8.162E+02 | 9.449E+02 | 1.409E+02 | 1.263E+02 | 2.323E+02 | |
Med | 3.709E+03 | 8.248E+03 | 3.633E+03 | 4.873E+03 | 9.440E+03 | 1.428E+04 | 3.838E+03 | 3.691E+03 | 3.894E+03 | |
F27 | Avg | 5.579E+03 | 2.747E+04 | 3.484E+03 | 6.000E+03 | 2.862E+04 | 3.662E+04 | 3.524E+03 | 3.491E+03 | 3.465E+03 |
Std | 6.237E+02 | 3.858E+03 | 4.124E+01 | 5.755E+02 | 1.617E+03 | 1.259E+03 | 4.471E+01 | 3.640E+01 | 4.584E+01 | |
Med | 5.550E+03 | 2.883E+04 | 3.499E+03 | 5.979E+03 | 2.881E+04 | 3.666E+04 | 3.524E+03 | 3.488E+03 | 3.455E+03 | |
F28 | Avg | 7.541E+03 | 2.294E+05 | 8.711E+03 | 1.186E+04 | 9.514E+04 | 8.299E+05 | 9.918E+03 | 8.898E+03 | 1.081E+04 |
Std | 7.472E+02 | 2.451E+05 | 8.153E+02 | 1.422E+03 | 3.896E+04 | 3.386E+05 | 1.114E+03 | 6.358E+02 | 7.850E+02 | |
Med | 7.555E+03 | 1.436E+05 | 8.720E+03 | 1.160E+04 | 9.301E+04 | 7.748E+05 | 9.688E+03 | 8.795E+03 | 1.093E+04 | |
F29 | Avg | 7.237E+08 | 2.808E+11 | 7.536E+08 | 1.268E+09 | 1.938E+11 | 3.954E+11 | 8.287E+08 | 7.890E+08 | 4.415E+08 |
Std | 2.831E+08 | 7.012E+10 | 2.985E+08 | 5.204E+08 | 4.062E+10 | 3.636E+10 | 4.456E+08 | 3.957E+08 | 1.901E+08 | |
Med | 8.532E+08 | 2.851E+11 | 8.011E+08 | 1.200E+09 | 1.922E+11 | 4.024E+11 | 6.697E+08 | 7.474E+08 | 3.888E+08 | |
Rank W/T/L | 1/0/28 | 0/0/29 | 3/0/21 | 1/0/28 | 0/0/29 | 0/0/29 | 2/0/27 | 6/0/23 | 16/0/13 | |
OE | 3.44% | 0.00% | 10.34% | 3.44% | 0.00% | 0.00% | 6.89% | 20.68% | 55.17% |
1.13 Appendix 13. Wilcoxon rank-sum of the QSSALEO vs. some improved SSA’s algorithms on CEC2017.
Fun | ESSA | HSSASCA | ISSA | ISSA_OBL | IWSSA | STS-SSA | TVSSA | SSA-FGWO |
|---|---|---|---|---|---|---|---|---|
1 | < 0.05 | < 0.05 | 0.199498378 | < 0.05 | < 0.05 | < 0.05 | 0.234174458 | < 0.05 |
2 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
3 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.163970523 |
4 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.396691029 | < 0.05 |
5 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.646404177 | 0.405412107 |
6 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
7 | 0.104139005 | < 0.05 | 0.821595142 | < 0.05 | < 0.05 | < 0.05 | 0.646404177 | < 0.05 |
8 | < 0.05 | < 0.05 | 0.259544222 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
9 | < 0.05 | < 0.05 | 0.188817733 | < 0.05 | < 0.05 | < 0.05 | 0.234174458 | 0.882550076 |
10 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
11 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
12 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.10749411 | < 0.05 |
13 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
14 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
15 | 0.129455778 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.498735035 | 0.070028526 |
16 | 0.518681718 | < 0.05 | < 0.05 | 0.286756545 | < 0.05 | < 0.05 | 0.199498378 | < 0.05 |
17 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
18 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.159311066 | < 0.05 |
19 | 0.234174458 | < 0.05 | 0.110934253 | 0.47920362 | < 0.05 | < 0.05 | 0.088592954 | 0.570291431 |
20 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
21 | < 0.05 | < 0.05 | 0.074973975 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
22 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.301060782 | < 0.05 |
23 | 0.104139005 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.118074856 | < 0.05 |
24 | < 0.05 | < 0.05 | 0.065354024 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
25 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
26 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.396691029 | < 0.05 |
27 | < 0.05 | < 0.05 | 0.114460738 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
28 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
29 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
1.14 Appendix 14. Wilcoxon rank-sum test result of the QSSALEO versus other advanced algorithms on CEC2008lsgo.
Fun | D | PPSO | PPSO_W | DESAP-abs | SHADE | CMA-ES | Large-scale LM-CMA | Large-scale QIWOA | Large-scale DSCA | Large-scale SSA-FGWO |
|---|---|---|---|---|---|---|---|---|---|---|
F1 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F2 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.25303 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F3 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F4 | 200 | < 0.05 | < 0.05 | 0.993796 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | 0.396691 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F5 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F6 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
F7 | 200 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
500 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | |
1000 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 | < 0.05 |
1.15 Appendix 15. Friedman test result of the QSSALEO versus some improved SSA’s algorithms on CEC2017.
Fun | ESSA | HSSASCA | ISSA | ISSA_OBL | IWSSA | STS-SSA | TVSSA | SSA-FGWO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|
1 | 5.03 | 7.76 | 2.38 | 5.97 | 7.24 | 9.00 | 2.48 | 2.93 | 2.21 |
2 | 5.00 | 9.00 | 3.79 | 2.52 | 7.03 | 7.45 | 5.45 | 3.76 | 1.00 |
3 | 5.14 | 7.14 | 2.83 | 5.86 | 7.86 | 9.00 | 3.21 | 2.07 | 1.90 |
4 | 5.38 | 7.03 | 2.03 | 4.59 | 8.21 | 8.76 | 3.24 | 2.03 | 3.72 |
5 | 5.31 | 7.34 | 2.17 | 5.10 | 8.10 | 8.55 | 2.83 | 3.00 | 2.59 |
6 | 3.62 | 7.62 | 3.67 | 5.86 | 6.97 | 8.74 | 2.41 | 1.69 | 4.41 |
7 | 2.69 | 6.72 | 4.17 | 4.90 | 7.59 | 8.86 | 3.52 | 3.48 | 3.07 |
8 | 6.00 | 8.21 | 2.14 | 4.69 | 7.83 | 7.97 | 3.28 | 3.00 | 1.90 |
9 | 5.41 | 8.07 | 2.52 | 4.86 | 7.86 | 8.07 | 2.28 | 2.97 | 2.97 |
10 | 5.93 | 7.72 | 3.48 | 5.07 | 7.66 | 8.62 | 2.17 | 3.34 | 1.00 |
11 | 5.34 | 7.79 | 3.07 | 5.62 | 7.21 | 9.00 | 2.52 | 3.17 | 1.28 |
12 | 5.90 | 7.86 | 3.52 | 2.24 | 7.00 | 8.97 | 2.97 | 4.00 | 2.55 |
13 | 5.97 | 7.55 | 4.10 | 5.03 | 7.72 | 8.69 | 2.83 | 2.76 | 1.28 |
14 | 6.00 | 7.90 | 3.93 | 1.07 | 7.24 | 8.86 | 3.21 | 3.93 | 2.69 |
15 | 3.76 | 7.17 | 2.48 | 5.97 | 7.83 | 9.00 | 2.93 | 2.66 | 3.21 |
16 | 4.31 | 7.83 | 2.45 | 4.38 | 7.24 | 8.93 | 3.28 | 2.52 | 4.07 |
17 | 5.90 | 7.14 | 3.17 | 3.66 | 7.97 | 8.90 | 2.93 | 3.86 | 1.48 |
18 | 4.69 | 7.93 | 4.17 | 1.14 | 7.14 | 8.93 | 3.48 | 4.69 | 2.83 |
19 | 4.10 | 8.28 | 3.00 | 3.24 | 7.93 | 7.79 | 3.14 | 4.07 | 3.45 |
20 | 3.72 | 7.59 | 1.97 | 5.69 | 7.41 | 9.00 | 3.41 | 1.59 | 4.62 |
21 | 4.31 | 7.28 | 4.17 | 4.59 | 7.38 | 7.97 | 4.52 | 2.69 | 2.10 |
22 | 2.59 | 7.76 | 4.00 | 6.00 | 7.21 | 9.00 | 3.90 | 2.21 | 2.34 |
23 | 4.28 | 7.59 | 2.21 | 6.00 | 7.41 | 9.00 | 2.93 | 1.93 | 3.66 |
24 | 5.45 | 7.69 | 2.14 | 5.55 | 7.41 | 8.90 | 2.66 | 3.41 | 1.79 |
25 | 2.55 | 7.10 | 2.62 | 5.45 | 7.90 | 9.00 | 3.14 | 2.45 | 4.79 |
26 | 2.69 | 7.41 | 2.03 | 5.97 | 7.59 | 9.00 | 3.83 | 2.52 | 3.97 |
27 | 5.28 | 7.55 | 2.38 | 5.72 | 7.45 | 9.00 | 3.28 | 2.66 | 1.69 |
28 | 1.34 | 7.69 | 2.52 | 5.62 | 7.38 | 8.93 | 3.93 | 2.69 | 4.90 |
29 | 3.50 | 7.97 | 3.57 | 5.17 | 7.10 | 8.93 | 3.45 | 3.38 | 1.93 |
Avg | 4.52 | 7.64 | 2.99 | 4.74 | 7.51 | 8.72 | 3.21 | 2.95 | 2.74 |
Rank | 5.000 | 8.000 | 3.000 | 6.000 | 7.000 | 9.000 | 4.000 | 2.000 | 1.000 |
1.16 Appendix 16. Comparison results of the QSSALEO on CEC2008lsgo with traditional algorithms during 2500 iterations.
Fun | D | Criteria | CSO | SSA | PSO | WOA | GWO | QSSALEO |
|---|---|---|---|---|---|---|---|---|
F1 | 200 | Avg | 9.414E+05 | 1.865E+05 | 4.844E+05 | 1.207E+05 | 1.885E+05 | -4.500E+02 |
Std | 4.476E+04 | 2.474E+04 | 3.117E+04 | 1.536E+04 | 2.114E+04 | 2.016E−07 | ||
Med | 9.436E+05 | 1.840E+05 | 4.797E+05 | 1.218E+05 | 1.836E+05 | − 4.500E+02 | ||
500 | Avg | 6.635E+05 | 1.197E+06 | 1.498E+06 | 8.075E+05 | 8.264E+05 | − 4.361E+02 | |
Std | 4.865E+04 | 5.376E+04 | 5.437E+04 | 3.411E+04 | 4.870E+04 | 5.266E+00 | ||
Med | 6.587E+05 | 1.198E+06 | 1.492E+06 | 8.076E+05 | 8.235E+05 | − 4.356E+02 | ||
1000 | Avg | 2.415E+06 | 2.932E+06 | 3.155E+06 | 2.251E+06 | 2.183E+06 | 3.906E+04 | |
Std | 1.112E+05 | 6.835E+04 | 4.440E+04 | 4.829E+04 | 5.550E+04 | 3.675E+03 | ||
Med | 2.397E+06 | 2.925E+06 | 3.161E+06 | 2.257E+06 | 2.180E+06 | 3.953E+04 | ||
F2 | 200 | Avg | − 2.952E+02 | − 3.493E+02 | − 3.962E+02 | − 3.756E+02 | − 3.683E+02 | − 4.064E+02 |
Std | 4.684E+00 | 3.595E+00 | 1.951E+00 | 8.571E+00 | 1.508E+00 | 1.436E+00 | ||
Med | − 2.950E+02 | − 3.495E+02 | − 3.964E+02 | − 3.743E+02 | − 3.678E+02 | − 4.065E+02 | ||
500 | Avg | − 3.152E+02 | − 3.381E+02 | − 3.904E+02 | − 3.204E+02 | − 3.556E+02 | − 4.023E+02 | |
Std | 4.750E+00 | 2.713E+00 | 2.131E+00 | 5.832E+00 | 5.713E−01 | 5.857E−01 | ||
Med | − 3.154E+02 | − 3.386E+02 | − 3.905E+02 | − 3.201E+02 | − 3.558E+02 | − 4.024E+02 | ||
1000 | Avg | − 3.059E+02 | − 3.339E+02 | − 3.872E+02 | − 3.698E+02 | − 3.524E+02 | − 4.007E+02 | |
Std | 3.235E+00 | 2.615E+00 | 1.284E+00 | 9.165E+00 | 8.673E−01 | 9.060E−01 | ||
Med | − 3.060E+02 | − 3.342E+02 | − 3.872E+02 | − 3.680E+02 | − 3.527E+02 | − 4.009E+02 | ||
F3 | 200 | Avg | 7.157E+11 | 4.582E+10 | 1.180E+11 | 1.596E+10 | 3.902E+10 | 2.423E+03 |
Std | 6.829E+10 | 8.966E+09 | 1.843E+10 | 3.022E+09 | 8.717E+09 | 3.579E+03 | ||
Med | 7.260E+11 | 4.725E+10 | 1.209E+11 | 1.606E+10 | 3.801E+10 | 1.071E+03 | ||
500 | Avg | 2.748E+11 | 3.881E+11 | 5.042E+11 | 1.856E+11 | 2.677E+11 | 2.544E+05 | |
Std | 4.380E+10 | 2.918E+10 | 2.555E+10 | 1.468E+10 | 1.539E+10 | 2.130E+05 | ||
Med | 2.751E+11 | 3.894E+11 | 5.037E+11 | 1.857E+11 | 2.645E+11 | 1.868E+05 | ||
1000 | Avg | 1.243E+12 | 1.152E+12 | 1.185E+12 | 6.799E+11 | 7.717E+11 | 1.693E+09 | |
Std | 9.166E+10 | 3.963E+10 | 3.062E+10 | 2.158E+10 | 2.659E+10 | 3.882E+08 | ||
Med | 1.235E+12 | 1.151E+12 | 1.185E+12 | 6.759E+11 | 7.711E+11 | 1.667E+09 | ||
F4 | 200 | Avg | 3.917E+03 | 2.192E+03 | 2.720E+03 | 2.196E+03 | 1.532E+03 | 1.453E+03 |
Std | 1.394E+02 | 1.041E+02 | 1.291E+02 | 1.483E+02 | 1.249E+02 | 4.870E+01 | ||
Med | 3.930E+03 | 2.178E+03 | 2.709E+03 | 2.199E+03 | 1.539E+03 | 1.459E+03 | ||
500 | Avg | 4.894E+03 | 7.025E+03 | 7.945E+03 | 6.867E+03 | 5.705E+03 | 3.882E+03 | |
Std | 2.470E+02 | 1.739E+02 | 1.722E+02 | 3.169E+02 | 1.513E+02 | 9.036E+01 | ||
Med | 4.856E+03 | 7.016E+03 | 7.944E+03 | 6.882E+03 | 5.708E+03 | 3.864E+03 | ||
1000 | Avg | 1.349E+04 | 1.551E+04 | 1.691E+04 | 1.476E+04 | 1.342E+04 | 8.669E+03 | |
Std | 3.167E+02 | 2.567E+02 | 2.096E+02 | 3.311E+02 | 2.296E+02 | 1.593E+02 | ||
Med | 1.350E+04 | 1.541E+04 | 1.691E+04 | 1.475E+04 | 1.341E+04 | 8.668E+03 | ||
F5 | 200 | Avg | 7.577E+03 | 1.387E+03 | 3.431E+03 | 8.054E+02 | 1.241E+03 | − 1.800E+02 |
Std | 4.056E+02 | 2.208E+02 | 1.921E+02 | 1.026E+02 | 1.670E+02 | 4.542E−03 | ||
Med | 7.538E+03 | 1.403E+03 | 3.394E+03 | 7.957E+02 | 1.227E+03 | − 1.800E+02 | ||
500 | Avg | 5.310E+03 | 9.373E+03 | 1.165E+04 | 6.303E+03 | 6.456E+03 | − 1.790E+02 | |
Std | 4.377E+02 | 4.877E+02 | 3.203E+02 | 2.709E+02 | 3.699E+02 | 1.958E−01 | ||
Med | 5.311E+03 | 9.391E+03 | 1.166E+04 | 6.274E+03 | 6.393E+03 | − 1.790E+02 | ||
1000 | Avg | 2.130E+04 | 2.549E+04 | 2.759E+04 | 1.978E+04 | 1.906E+04 | 1.717E+02 | |
Std | 7.809E+02 | 8.309E+02 | 4.443E+02 | 5.076E+02 | 4.410E+02 | 4.258E+01 | ||
Med | 2.130E+04 | 2.552E+04 | 2.760E+04 | 1.993E+04 | 1.912E+04 | 1.729E+02 | ||
F6 | 200 | Avg | − 1.187E+02 | − 1.195E+02 | − 1.193E+02 | − 1.207E+02 | − 1.210E+02 | − 1.213E+02 |
Std | 2.695E−02 | 4.275E−02 | 9.039E−02 | 1.486E−02 | 3.999E−01 | 7.468E−01 | ||
Med | − 1.187E+02 | − 1.195E+02 | − 1.193E+02 | − 1.207E+02 | − 1.210E+02 | − 1.211E+02 | ||
500 | Avg | − 1.201E+02 | − 1.192E+02 | − 1.191E+02 | − 1.198E+02 | − 1.204E+02 | − 1.213E+02 | |
Std | 1.429E−01 | 2.770E−02 | 5.016E−02 | 2.308E−02 | 1.746E−01 | 4.594E−01 | ||
Med | − 1.201E+02 | − 1.192E+02 | − 1.191E+02 | − 1.198E+02 | − 1.205E+02 | − 1.212E+02 | ||
1000 | Avg | − 1.194E+02 | − 1.191E+02 | − 1.189E+02 | − 1.195E+02 | − 1.204E+02 | − 1.209E+02 | |
Std | 4.118E−02 | 1.408E−02 | 2.362E−02 | 1.541E−02 | 3.326E−01 | 2.755E−01 | ||
Med | − 1.194E+02 | − 1.191E+02 | − 1.189E+02 | − 1.195E+02 | − 1.205E+02 | − 1.208E+02 | ||
F7 | 200 | Avg | − 2.311E+05 | − 3.628E+05 | − 1.644E+05 | − 4.169E+05 | − 5.394E+05 | − 4.280E+05 |
Std | 1.330E+04 | 3.966E+04 | 2.087E+02 | 5.313E+04 | 4.466E+04 | 6.214E+04 | ||
Med | − 2.278E+05 | − 3.628E+05 | − 1.644E+05 | − 4.133E+05 | − 5.424E+05 | − 4.261E+05 | ||
500 | Avg | − 7.841E+05 | − 6.430E+05 | − 4.309E+05 | − 9.152E+05 | − 9.259E+05 | − 8.895E+05 | |
Std | 7.940E+04 | 4.889E+04 | 1.682E+04 | 9.648E+04 | 5.440E+04 | 1.880E+05 | ||
Med | − 7.902E+05 | − 6.420E+05 | − 4.340E+05 | − 9.207E+05 | − 9.192E+05 | − 8.831E+05 | ||
1000 | Avg | − 1.248E+06 | − 1.066E+06 | − 7.257E+05 | − 1.961E+06 | − 1.454E+06 | − 1.957E+06 | |
Std | 8.321E+04 | 5.821E+04 | 6.946E+03 | 2.470E+05 | 7.890E+04 | 2.752E+05 | ||
Med | − 1.250E+06 | − 1.058E+06 | − 7.269E+05 | − 1.967E+06 | − 1.446E+06 | − 2.027E+06 | ||
Rank | 200 | W/T/L | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 1/0/6 | 6/0/1 |
500 | W/T/L | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 1/0/6 | 6/0/1 | |
1000 | W/T/L | 0/0/7 | 0/0/7 | 0/0/7 | 1/0/6 | 0/0/7 | 6/0/1 | |
Overall OE | 0.00% | 0.00% | 0.00% | 4.76% | 9.52% | 85.71% |
1.17 Appendix 17. Friedman test result of the QSSALEO versus other traditional algorithms on CEC2008lsgo.
Algorithm | Dimension | Average rank | Overall rank |
|---|---|---|---|
CSO | 200 | 5.857 | 6 |
500 | 3.113 | 3 | |
1000 | 4.261 | 4 | |
SSA | 200 | 3.901 | 4 |
500 | 4.773 | 5 | |
1000 | 4.916 | 5 | |
PSO | 200 | 4.700 | 5 |
500 | 5.424 | 6 | |
1000 | 5.296 | 6 | |
WOA | 200 | 2.591 | 2 |
500 | 3.527 | 4 | |
1000 | 2.754 | 3 | |
GWO | 200 | 2.650 | 3 |
500 | 2.936 | 2 | |
1000 | 2.685 | 2 | |
QSSALEO | 200 | 1.300 | 1 |
500 | 1.227 | 1 | |
1000 | 1.089 | 1 |
1.18 Appendix 18. Friedman test result of the QSSALEO versus other advanced algorithms on CEC2008lsgo.
Algorithm | Dimension | Average rank | Overall rank |
|---|---|---|---|
PPSO | 200 | 5.901 | 6 |
500 | 5.603 | 6 | |
1000 | 5.333 | 5 | |
PPSO_W | 200 | 5.842 | 5 |
500 | 4.352 | 4 | |
1000 | 3.756 | 3 | |
DESAP-abs | 200 | 4.005 | 4 |
500 | 4.414 | 5 | |
1000 | 4.744 | 4 | |
SHADE | 200 | 3.862 | 3 |
500 | 4.118 | 3 | |
1000 | 5.406 | 6 | |
CMA-ES | 200 | 6.241 | 7 |
500 | 9.842 | 10 | |
1000 | 9.643 | 10 | |
Large-scale LM-CMA | 200 | 8.714 | 9 |
500 | 7.823 | 8 | |
1000 | 7.961 | 8 | |
Large-scale QIWOA | 200 | 8.768 | 10 |
500 | 8.300 | 9 | |
1000 | 7.961 | 8 | |
Large-scale DSCA | 200 | 6.759 | 8 |
500 | 6.113 | 7 | |
1000 | 5.833 | 7 | |
Large-scale SSA-FGWO | 200 | 3.000 | 2 |
500 | 2.704 | 2 | |
1000 | 3.192 | 2 | |
QSSALEO | 200 | 1.906 | 1 |
500 | 1.729 | 1 | |
1000 | 1.049 | 1 |
1.19 Appendix 19. Comparison results of the QSSALEO on CEC2008lsgos with advanced algorithms during 2500 iterations.
F | D | Cri | PPSO | PPSO_W | DESAP-abs | SHADE | CMA-ES | Large-scale LM-CMA | Large-scale QIWOA | Large-scale DSCA | Large-scale SSA-FGWO | QSSALEO |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
F1 | 200 | Avg | 2.677E+03 | 5.401E+03 | − 2.751E+02 | 2.271E+02 | − 4.397E+02 | 7.096E+05 | 1.164E+06 | 6.503E+05 | − 4.499E+02 | − 4.500E+02 |
Std | 8.561E+02 | 3.450E+03 | 2.694E+02 | 1.130E+03 | 1.222E+01 | 3.316E+03 | 5.888E+04 | 1.142E+04 | 4.290E−02 | 2.016E−07 | ||
Med | 2.452E+03 | 4.783E+03 | − 3.694E+02 | − 3.229E+02 | − 4.445E+02 | 7.098E+05 | 1.175E+06 | 6.501E+05 | − 4.499E+02 | − 4.500E+02 | ||
500 | Avg | 1.331E+05 | 7.549E+04 | 1.211E+05 | 1.337E+05 | 3.929E+06 | 1.750E+06 | 3.031E+06 | 1.686E+06 | 4.626E+04 | − 4.361E+02 | |
Std | 1.262E+04 | 7.553E+03 | 2.336E+04 | 2.072E+04 | 1.563E+06 | 1.183E+03 | 9.584E+04 | 1.106E+04 | 7.730E+03 | 5.266E+00 | ||
Med | 1.310E+05 | 7.352E+04 | 1.185E+05 | 1.332E+05 | 3.713E+06 | 1.750E+06 | 3.041E+06 | 1.689E+06 | 4.461E+04 | − 4.356E+02 | ||
1000 | Avg | 1.001E+06 | 7.601E+05 | 9.248E+05 | 9.088E+05 | 4.233E+07 | 3.389E+06 | 6.182E+06 | 3.334E+06 | 9.381E+05 | 3.906E+04 | |
Std | 3.644E+04 | 5.236E+04 | 6.476E+04 | 6.204E+04 | 8.435E+05 | 1.385E+03 | 1.754E+05 | 1.230E+04 | 5.635E+04 | 3.675E+03 | ||
Med | 9.982E+05 | 7.559E+05 | 9.225E+05 | 9.047E+05 | 4.227E+07 | 3.389E+06 | 6.206E+06 | 3.335E+06 | 9.379E+05 | 3.953E+04 | ||
F2 | 200 | Avg | − 3.605E+02 | − 3.625E+02 | − 3.571E+02 | − 3.578E+02 | − 3.323E+02 | − 3.527E+02 | − 3.200E+02 | − 3.537E+02 | − 3.827E+02 | − 4.064E+02 |
Std | 2.134E+00 | 2.307E+00 | 3.866E+00 | 2.912E+00 | 4.900E+01 | 3.176E−01 | 2.650E+01 | 2.343E−01 | 3.316E+01 | 1.436E+00 | ||
Med | − 3.604E+02 | − 3.622E+02 | − 3.571E+02 | − 3.581E+02 | − 3.383E+02 | − 3.526E+02 | − 3.117E+02 | − 3.536E+02 | − 4.058E+02 | − 4.065E+02 | ||
500 | Avg | − 3.538E+02 | − 3.543E+02 | − 3.432E+02 | − 3.428E+02 | 1.205E+02 | − 3.508E+02 | − 3.043E+02 | − 3.514E+02 | − 4.031E+02 | − 4.023E+02 | |
Std | 8.059E−01 | 6.353E−01 | 2.503E+00 | 2.689E+00 | 2.074E+01 | 1.233E−01 | 2.455E+01 | 2.822E−01 | 3.841E−01 | 5.857E−01 | ||
Med | − 3.535E+02 | − 3.542E+02 | − 3.431E+02 | − 3.434E+02 | 1.206E+02 | − 3.508E+02 | − 3.027E+02 | − 3.512E+02 | − 4.031E+02 | − 4.024E+02 | ||
1000 | Avg | − 3.518E+02 | − 3.521E+02 | − 3.359E+02 | − 3.346E+02 | 1.695E+02 | − 3.502E+02 | − 2.879E+02 | − 3.504E+02 | − 4.016E+02 | − 4.007E+02 | |
Std | 3.892E−01 | 4.110E−01 | 2.265E+00 | 2.456E+00 | 1.895E+01 | 1.041E−01 | 2.645E+01 | 5.451E−02 | 2.468E−01 | 9.060E−01 | ||
Med | − 3.518E+02 | − 3.521E+02 | − 3.359E+02 | − 3.343E+02 | 1.692E+02 | − 3.502E+02 | − 2.789E+02 | − 3.505E+02 | − 4.017E+02 | − 4.009E+02 | ||
F3 | 200 | Avg | 9.383E+07 | 1.774E+08 | 4.157E+06 | 5.379E+06 | 1.270E+06 | 2.287E+11 | 1.150E+12 | 2.039E+11 | 1.607E+04 | 2.423E+03 |
Std | 1.413E+08 | 3.310E+08 | 5.773E+06 | 1.004E+07 | 2.825E+06 | 2.702E+09 | 1.031E+11 | 7.805E+09 | 2.114E+04 | 3.579E+03 | ||
Med | 5.975E+07 | 5.399E+07 | 2.384E+06 | 1.895E+06 | 2.840E+05 | 2.290E+11 | 1.163E+12 | 2.032E+11 | 6.429E+03 | 1.071E+03 | ||
500 | Avg | 1.850E+10 | 7.007E+09 | 2.204E+10 | 2.361E+10 | 7.826E+13 | 6.299E+11 | 3.424E+12 | 5.985E+11 | 7.537E+09 | 2.544E+05 | |
Std | 2.928E+09 | 1.271E+09 | 5.939E+09 | 9.831E+09 | 6.881E+13 | 9.202E+08 | 2.523E+11 | 5.807E+09 | 1.176E+09 | 2.130E+05 | ||
Med | 1.784E+10 | 6.702E+09 | 2.250E+10 | 2.173E+10 | 4.647E+13 | 6.300E+11 | 3.441E+12 | 5.995E+11 | 7.569E+09 | 1.868E+05 | ||
1000 | Avg | 2.357E+11 | 1.449E+11 | 2.660E+11 | 2.730E+11 | 5.031E+14 | 1.280E+12 | 7.355E+12 | 1.250E+12 | 2.130E+11 | 1.693E+09 | |
Std | 2.179E+10 | 1.362E+10 | 2.858E+10 | 2.376E+10 | 2.720E+13 | 1.047E+09 | 4.765E+11 | 6.934E+09 | 1.541E+10 | 3.882E+08 | ||
Med | 2.370E+11 | 1.464E+11 | 2.615E+11 | 2.713E+11 | 5.076E+14 | 1.280E+12 | 7.522E+12 | 1.251E+12 | 2.113E+11 | 1.667E+09 | ||
F4 | 200 | Avg | 1.494E+03 | 1.454E+03 | 9.432E+01 | 9.113E+01 | 1.768E+03 | 2.846E+03 | 4.407E+03 | 3.204E+03 | 1.439E+03 | 1.453E+03 |
Std | 6.821E+01 | 6.121E+01 | 2.927E+01 | 3.461E+01 | 4.774E+01 | 6.961E+01 | 1.390E+02 | 3.904E+01 | 6.600E+01 | 4.870E+01 | ||
Med | 1.492E+03 | 1.445E+03 | 9.212E+01 | 8.918E+01 | 1.783E+03 | 2.847E+03 | 4.400E+03 | 3.198E+03 | 1.426E+03 | 1.459E+03 | ||
500 | Avg | 5.170E+03 | 4.872E+03 | 3.383E+03 | 3.449E+03 | 1.464E+04 | 7.934E+03 | 1.203E+04 | 8.503E+03 | 4.785E+03 | 3.882E+03 | |
Std | 1.107E+02 | 1.126E+02 | 1.322E+02 | 1.236E+02 | 3.601E+03 | 9.052E+01 | 3.604E+02 | 6.773E+01 | 1.883E+02 | 9.036E+01 | ||
Med | 5.168E+03 | 4.847E+03 | 3.391E+03 | 3.482E+03 | 1.526E+04 | 7.943E+03 | 1.205E+04 | 8.520E+03 | 4.833E+03 | 3.864E+03 | ||
1000 | Avg | 1.244E+04 | 1.186E+04 | 9.615E+03 | 9.685E+03 | 1.157E+05 | 1.678E+04 | 2.467E+04 | 1.751E+04 | 1.148E+04 | 8.669E+03 | |
Std | 2.044E+02 | 1.935E+02 | 8.596E+02 | 8.878E+02 | 2.275E+03 | 1.101E+02 | 5.767E+02 | 7.364E+01 | 1.574E+02 | 1.593E+02 | ||
Med | 1.247E+04 | 1.187E+04 | 9.477E+03 | 9.646E+03 | 1.161E+05 | 1.678E+04 | 2.475E+04 | 1.751E+04 | 1.147E+04 | 8.668E+03 | ||
F5 | 200 | Avg | − 1.517E+02 | − 1.417E+02 | − 1.776E+02 | − 1.768E+02 | − 1.789E+02 | 5.474E+03 | 9.412E+03 | 4.906E+03 | − 1.799E+02 | − 1.800E+02 |
Std | 7.751E+00 | 2.500E+01 | 2.116E+00 | 4.187E+00 | 3.266E−01 | 4.896E+00 | 6.238E+02 | 9.111E+01 | 2.603E−02 | 4.542E−03 | ||
Med | − 1.526E+02 | − 1.514E+02 | − 1.784E+02 | − 1.781E+02 | − 1.790E+02 | 5.475E+03 | 9.402E+03 | 4.924E+03 | − 1.799E+02 | − 1.800E+02 | ||
500 | Avg | 1.022E+03 | 4.837E+02 | 8.512E+02 | 8.499E+02 | 3.207E+04 | 1.381E+04 | 2.539E+04 | 1.324E+04 | 2.338E+02 | − 1.790E+02 | |
Std | 1.233E+02 | 8.017E+01 | 2.065E+02 | 2.002E+02 | 1.482E+04 | 1.696E+00 | 8.693E+02 | 1.139E+02 | 5.132E+01 | 1.958E−01 | ||
Med | 1.026E+03 | 4.450E+02 | 8.324E+02 | 8.096E+02 | 3.015E+04 | 1.380E+04 | 2.566E+04 | 1.325E+04 | 2.216E+02 | − 1.790E+02 | ||
1000 | Avg | 8.603E+03 | 5.863E+03 | 7.974E+03 | 7.737E+03 | 3.806E+05 | 2.991E+04 | 5.469E+04 | 2.934E+04 | 8.082E+03 | 1.717E+02 | |
Std | 4.987E+02 | 3.026E+02 | 5.839E+02 | 4.839E+02 | 9.471E+03 | 1.770E+00 | 1.518E+03 | 9.773E+01 | 3.466E+02 | 4.258E+01 | ||
Med | 8.539E+03 | 5.878E+03 | 7.907E+03 | 7.914E+03 | 3.803E+05 | 2.991E+04 | 5.494E+04 | 2.934E+04 | 8.149E+03 | 1.729E+02 | ||
F6 | 200 | Avg | − 1.202E+02 | − 1.202E+02 | − 1.295E+02 | − 1.291E+02 | − 1.185E+02 | − 1.193E+02 | − 1.197E+02 | − 1.207E+02 | − 1.207E+02 | − 1.213E+02 |
Std | 3.049E−01 | 2.917E−01 | 1.508E+00 | 1.606E+00 | 1.893E−02 | 4.372E−02 | 4.887E−01 | 1.521E−02 | 2.633E−07 | 7.468E−01 | ||
Med | − 1.200E+02 | − 1.200E+02 | − 1.293E+02 | − 1.292E+02 | − 1.185E+02 | − 1.193E+02 | − 1.200E+02 | − 1.207E+02 | − 1.207E+02 | − 1.211E+02 | ||
500 | Avg | − 1.201E+02 | − 1.201E+02 | − 1.258E+02 | − 1.257E+02 | − 1.184E+02 | − 1.192E+02 | − 1.197E+02 | − 1.206E+02 | − 1.207E+02 | − 1.213E+02 | |
Std | 2.511E−01 | 2.502E−01 | 5.597E−01 | 7.006E−01 | 8.270E−03 | 1.815E−02 | 4.719E−E−01 | 4.290E−01 | 1.027E−04 | 4.594E−01 | ||
Med | − 1.200E+02 | − 1.200E+02 | − 1.259E+02 | − 1.257E+02 | − 1.184E+02 | − 1.192E+02 | − 1.200E+02 | − 1.207E+02 | − 1.207E+02 | − 1.212E+02 | ||
1000 | Avg | − 1.201E+02 | − 1.202E+02 | − 1.198E+02 | − 1.198E+02 | − 1.184E+02 | − 1.191E+02 | − 1.196E+02 | − 1.205E+02 | − 1.206E+02 | − 1.209E+02 | |
Std | 2.141E−01 | 2.605E−01 | 7.681E−02 | 7.358E−02 | 6.083E−03 | 1.939E−02 | 5.538E−01 | 4.273E−01 | 1.407E−06 | 2.755E−01 | ||
Med | − 1.200E+02 | − 1.200E+02 | − 1.198E+02 | − 1.198E+02 | − 1.184E+02 | − 1.191E+02 | − 1.200E+02 | − 1.206E+02 | − 1.206E+02 | − 1.208E+02 | ||
F7 | 200 | Avg | − 1.620E+05 | − 1.640E+05 | − 1.500E+05 | − 1.508E+05 | − 7.386E+04 | − 1.133E+05 | − 1.938E+05 | − 2.130E+05 | − 3.782E+05 | − 4.280E+05 |
Std | 8.341E+04 | 9.396E+04 | 1.767E+04 | 1.184E−10 | 3.874E+03 | 2.779E+02 | 1.726E+04 | 2.078E+04 | 7.402E+04 | 6.214E+04 | ||
Med | − 1.468E+05 | − 1.468E+05 | − 1.468E+05 | − 1.508E+05 | − 7.315E+04 | − 1.133E+05 | − 1.907E+05 | − 2.073E+05 | − 3.933E+05 | − 4.261E+05 | ||
500 | Avg | − 2.476E+05 | − 2.792E+05 | − 2.569E+05 | − 4.443E+05 | − 1.241E+05 | − 2.279E+05 | − 4.176E+05 | − 4.178E+05 | − 6.908E+05 | − 8.895E+05 | |
Std | 8.880E−11 | 1.730E+05 | 6.342E+03 | 1.561E+03 | 1.264E+03 | 3.004E+01 | 5.868E+04 | 1.734E+04 | 9.964E+04 | 1.880E+05 | ||
Med | − 2.476E+05 | − 2.476E+05 | − 2.558E+05 | − 4.440E+05 | − 1.244E+05 | − 2.279E+05 | − 4.121E+05 | − 4.190E+05 | − 7.232E+05 | − 8.831E+05 | ||
1000 | Avg | − 4.567E+05 | − 4.349E+05 | − 9.331E+05 | − 3.893E+05 | − 3.893E+05 | − 3.751E+05 | − 7.322E+05 | − 6.978E+05 | − 1.099E+06 | − 1.957E+06 | |
Std | 3.918E+05 | 2.729E+05 | 7.571E+04 | 2.284E+04 | 2.284E+04 | 3.681E+03 | 8.449E+04 | 3.852E+04 | 7.964E+04 | 2.752E+05 | ||
Med | − 3.851E+05 | − 3.851E+05 | − 9.579E+05 | − 3.851E+05 | − 3.851E+05 | − 3.744E+05 | − 7.135E+05 | − 6.894E+05 | − 1.112E+06 | − 2.027E+06 | ||
Rank | 200 | W/T/L | 0/0/7 | 0/0/7 | 1/0/6 | 1/0/6 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 5/0/2 |
500 | W/T/L | 0/0/7 | 0/0/7 | 2/0/5 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/1/6 | 5/0/2 | |
1000 | W/T/L | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/0/7 | 0/1/6 | 7/0/0 | |
Overall OE | 0.0% | 0.0% | 14.28% | 4.76% | 0.0% | 0.0% | 0.0% | 0.0% | 4.76% | 80.95% |
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Qaraad, M., Amjad, S., Hussein, N.K. et al. An innovative quadratic interpolation salp swarm-based local escape operator for large-scale global optimization problems and feature selection. Neural Comput & Applic 34, 17663–17721 (2022). https://doi.org/10.1007/s00521-022-07391-2
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DOI: https://doi.org/10.1007/s00521-022-07391-2