Table 7 Evaluation results of well-known & top-performing algorithms for the functions F31 to F51.
From: Learning cooking algorithm for solving global optimization problems
\({\text {Functions}}\) | \({\text {PSO}}\) | \({\text {TSA}}\) | \({\text {SSA}}\) | \({\text {MVO}}\) | \({\text {GWO}}\) | \({\text {WOA}}\) | \({\text {GJO}}\) | \({\text {LSHADE}}\) | \({\text {CMAES}}\) | \({\text {LCA}}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|
\({\text {F31}}\) | Mean | –5.633E+06 | –2519.79 | –3.263E+08 | –2540 | –2540 | –2560 | –2540 | –2520 | –3.03E+301 | \(-2600\) |
Std | 4.531E+04 | 100.56 | 7.148E+06 | 94.032 | 94.032 | 82.078 | 94.032 | 100.52 | NA | 0.6545 | |
Best | –5.703E+06 | \(-2600\) | –3.379E+08 | \(-2600\) | \(-2600\) | \(-2600\) | \(-2600\) | \(-2600\) | –5.94E+302 | \(-2600\) | |
Worst | –5.539E+06 | –2398.99 | –3.172E+08 | –2400 | –2400 | –2400 | –2400 | –2400 | –1.45E+192 | \(-2600\) | |
Median | –5.629E+06 | –2599.49 | –3.281E+08 | \(-2600\) | \(-2600\) | \(-2600\) | \(-2600\) | \(-2600\) | –7.50E+255 | \(-2600\) | |
\({\text {F32}}\) | Mean | 9.808E–21 | 1.827E–84 | 5.047E–11 | 4.048E–04 | 6.133E–118 | 5.475E–158 | 4.487E–219 | 3.296E–36 | 0.136153 | 0 |
Std | 1.354E–20 | 5.552E–84 | 2.616E–11 | 2.618E–04 | 1.565E–117 | 0 | 0 | 1.160E–35 | 0.091605 | 0 | |
Best | 1.217E–22 | 1.933E–91 | 1.897E–11 | 1.019E–04 | 7.510E–121 | 1.989E–169 | 4.273E–230 | 9.272E–44 | 0.037223 | 0 | |
Worst | 4.267E–20 | 1.806E–83 | 1.034E–10 | 8.838E–04 | 5.118E–117 | 5.474E–157 | 4.484E–218 | 5.190E–35 | 0.40012 | 0 | |
Median | 1.645E–21 | 8.026E–88 | 4.956E–11 | 3.425E–04 | 3.982E–120 | 5.042E–166 | 9.533E–224 | 4.418E–38 | 0.120975 | 0 | |
\({\text {F33}}\) | Mean | 3.929E–02 | 4.603E–05 | 8.692E–01 | 3.761E–01 | 3.740E–06 | 6.371E–06 | 4.489E–09 | 4.290E–04 | 1.852E–01 | 0 |
Std | 3.131E–02 | 4.765E–05 | 8.597E–01 | 1.572E–01 | 4.180E–06 | 1.247E–05 | 1.272E–08 | 3.710E–04 | 1.210E–01 | 0 | |
Best | 1.051E–02 | 1.908E–06 | 4.569E–01 | 2.139E–01 | 8.266E–07 | 2.406E–15 | 5.559E–14 | 1.802E–05 | 5.336E–02 | 0 | |
Worst | 1.162E–01 | 1.393E–04 | 3.358E+00 | 7.637E–01 | 1.308E–05 | 3.488E–05 | 4.167E–08 | 1.428E–03 | 4.742E–01 | 0 | |
Median | 2.947E–02 | 2.109E–05 | 5.812E–01 | 3.307E–01 | 1.917E–06 | 3.571E–07 | 7.753E–12 | 2.959E–04 | 1.324E–01 | 0 | |
\({\text {F34}}\) | Mean | 1.912E–29 | 6.000E–01 | 2.838E–14 | 6.454E–06 | 1.393E–09 | 8.329E–07 | 3.718E–06 | 0 | 5.663E–05 | 0 |
Std | 3.915E–29 | 5.026E–01 | 2.892E–14 | 9.092E–06 | 8.038E–10 | 1.509E–06 | 5.835E–06 | 0 | 7.594E–05 | 0 | |
Best | 0 | 7.450E–09 | 8.906E–16 | 5.052E–08 | 3.183E–10 | 7.660E–12 | 5.679E–10 | 0 | 2.384E–06 | 0 | |
Worst | 9.555E–29 | 1.0003 | 8.228E–14 | 2.313E–05 | 2.638E–09 | 3.770E–06 | 1.469E–05 | 0 | 3.362E–04 | 0 | |
Median | 0 | 1 | 1.624E–14 | 8.876E–07 | 1.104E–09 | 7.099E–08 | 1.649E–08 | 0 | 3.073E–05 | 0 | |
\({\text {F35}}\) | Mean | 0 | 0 | 5.480E–12 | 1.714E–04 | 0 | 0 | 0 | 0 | 0 | 0 |
Std | 0 | 0 | 4.281E–12 | 1.351E–04 | 0 | 0 | 0 | 0 | 0 | 0 | |
Best | 0 | 0 | 2.957E–13 | 1.532E–05 | 0 | 0 | 0 | 0 | 0 | 0 | |
Worst | 0 | 0 | 1.223E–11 | 4.835E–04 | 0 | 0 | 0 | 0 | 0 | 0 | |
Median | 0 | 0 | 4.154E–12 | 1.517E–04 | 0 | 0 | 0 | 0 | 0 | 0 | |
\({\text {F36}}\) | Mean | 0 | 1.800E–01 | 1.504E–14 | 3.950E–07 | 8.052E–08 | 2.051E–04 | 7.572E–07 | 0 | 0 | 0 |
Std | 0 | 5.540E–01 | 1.587E–14 | 3.354E–07 | 5.999E–08 | 1.669E–04 | 8.078E–07 | 0 | 0 | 0 | |
Best | 0 | 4.319E–07 | 4.824E–16 | 6.200E–08 | 1.487E–08 | 7.203E–06 | 1.751E–08 | 0 | 0 | 0 | |
Worst | 0 | 1.800E+00 | 4.752E–14 | 1.229E–06 | 2.215E–07 | 5.154E–04 | 2.446E–06 | 0 | 0 | 0 | |
Median | 0 | 2.881E–06 | 9.595E–15 | 3.015E–07 | 7.783E–08 | 1.822E–04 | 4.921E–07 | 0 | 0 | 0 | |
\({\text {F37}}\) | Mean | \(-1.9093\) | \(-1.80126\) | \(-1.89264\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.89296\) | \(-1.8013\) |
Std | 0.09078 | 5.026E–05 | 0.09389 | 2.200E–16 | 2.200E–16 | 2.200E–16 | 2.200E–16 | 2.200E–16 | 0.060191 | 2.200E–16 | |
Best | \(-1.988\) | \(-1.8013\) | \(-1.988\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.9883\) | \(-1.8013\) | |
Worst | \(-1.8013\) | \(-1.8012\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8049\) | \(-1.8013\) | |
Median | \(-1.9679\) | \(-1.8013\) | \(-1.8846\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.8013\) | \(-1.91565\) | \(-1.8013\) | |
\({\text {F38}}\) | Mean | –4.672E+00 | –4.072E+00 | –4.016E+00 | –4.242E+00 | –4.534E+00 | –3.814E+00 | –3.897E+00 | –4.687658 | –4.50042 | –3.257E+00 |
Std | 4.752E–02 | 6.054E–01 | 6.354E–01 | 3.449E–01 | 5.962E–02 | 5.724E–01 | 2.747E–01 | 9.112E–16 | 0.179110 | 1.260E–01 | |
Best | –4.704E+00 | –4.589E+00 | –4.645E+00 | –4.687E+00 | –4.645E+00 | –4.665E+00 | –4.452E+00 | –4.687658 | –4.6877 | –3.426E+00 | |
Worst | –4.537E+00 | –2.747E+00 | –2.600E+00 | –3.686E+00 | –4.495E+00 | –2.885E+00 | –3.618E+00 | –4.687658 | –4.136 | –3.056E+00 | |
Median | –4.686E+00 | –4.407E+00 | –4.101E+00 | –4.331E+00 | –4.494E+00 | –3.615E+00 | –3.844E+00 | –4.687658 | –4.56845 | –3.227E+00 | |
\({\text {F39}}\) | Mean | –8.869E+00 | –6.493E+00 | –6.496E+00 | –7.708E+00 | –7.910E+00 | –5.962E+00 | –6.184E+00 | –9.660151 | –6.92552 | –5.661E+00 |
Std | 5.371E–01 | 5.907E–01 | 7.129E–01 | 7.124E–01 | 1.216E+00 | 6.052E–01 | 1.035E+00 | 3.645E–15 | 0.631307 | 4.317E–01 | |
Best | –9.518E+00 | –7.531E+00 | –8.136E+00 | –8.865E+00 | –9.503E+00 | –7.046E+00 | –7.723E+00 | –9.660151 | –8.6191 | –6.336E+00 | |
Worst | –7.990E+00 | –5.855E+00 | –5.517E+00 | –6.589E+00 | –5.489E+00 | –5.014E+00 | –4.310E+00 | –9.660151 | –5.9688 | –5.227E+00 | |
Median | –8.796E+00 | –6.261E+00 | –6.361E+00 | –7.905E+00 | –8.085E+00 | –5.854E+00 | –6.285E+00 | –9.660151 | –6.8364 | –5.449E+00 | |
\({\text {F40}}\) | Mean | 0 | 0 | 2.664E–16 | 8.298E–09 | 0 | 0 | 0 | 0 | 1.703E–05 | 0 |
Std | 0 | 0 | 3.022E–16 | 9.054E–09 | 0 | 0 | 0 | 0 | 1.401E–05 | 0 | |
Best | 0 | 0 | 0 | 1.338E–09 | 0 | 0 | 0 | 0 | 2.181E–06 | 0 | |
Worst | 0 | 0 | 8.881E–16 | 2.991E–08 | 0 | 0 | 0 | 0 | 5.771E–05 | 0 | |
Median | 0 | 0 | 2.220E–16 | 3.508E–09 | 0 | 0 | 0 | 0 | 1.385E–05 | 0 | |
\({\text {F41}}\) | Mean | 0 | 0 | 1.007E–11 | 1.187E–04 | 0 | 0 | 0 | 0 | 5.112E–03 | 0 |
Std | 0 | 0 | 1.197E–11 | 1.229E–04 | 0 | 0 | 0 | 0 | 5.660E–03 | 0 | |
Best | 0 | 0 | 6.614E–13 | 4.069E–06 | 0 | 0 | 0 | 0 | 2.104E–04 | 0 | |
Worst | 0 | 0 | 4.002E–11 | 3.553E–04 | 0 | 0 | 0 | 0 | 2.137E–02 | 0 | |
Median | 0 | 0 | 5.375E–12 | 8.998E–05 | 0 | 0 | 0 | 0 | 3.596E–03 | 0 | |
\({\text {F42}}\) | Mean | 0 | 4.525E–02 | 4.931E–12 | 3.013E–05 | 0 | 2.275E–16 | 0 | 0 | 1.285E–03 | 0 |
Std | 0 | 9.286E–02 | 4.915E–12 | 2.564E–05 | 0 | 1.913E–16 | 0 | 0 | 8.107E–04 | 0 | |
Best | 0 | 0 | 2.765E–13 | 1.852E–07 | 0 | 0 | 0 | 0 | 5.142E–05 | 0 | |
Worst | 0 | 2.262E–01 | 1.326E–11 | 8.532E–05 | 0 | 4.996E–16 | 0 | 0 | 2.587E–03 | 0 | |
Median | 0 | 0 | 2.097E–12 | 2.857E–05 | 0 | 2.220E–16 | 0 | 0 | 1.346E–03 | 0 | |
\({\text {F43}}\) | Mean | \(-25\) | \(-24.99872\) | \(-25\) | \(-25\) | \(-24.9985\) | \(-25\) | \(-24.99759\) | \(-24.9999\) | \(-24.9996\) | \(-25\) |
Std | 0 | 0.000996 | 0 | 0 | 0.002237 | 0 | 0.003963 | 7.290E–15 | 0.000256 | 0 | |
Best | \(-25\) | \(-24.9998\) | \(-25\) | \(-25\) | \(-25\) | \(-25\) | \(-25\) | \(-24.9999\) | \(-24.9999\) | \(-25\) | |
Worst | \(-25\) | \(-24.9968\) | \(-25\) | \(-25\) | \(-24.9925\) | \(-25\) | \(-24.9879\) | \(-24.9999\) | \(-24.9992\) | \(-25\) | |
Median | \(-25\) | \(-24.999\) | \(-25\) | \(-25\) | \(-24.9995\) | \(-25\) | \(-24.9993\) | \(-24.9999\) | \(-24.9997\) | \(-25\) | |
\({\text {F44}}\) | Mean | 0 | 0 | 1.337E–14 | 0 | 0 | 0 | 0 | 0 | 4.501E–06 | 0 |
Std | 0 | 0 | 1.698E–14 | 0 | 0 | 0 | 0 | 0 | 1.199E–05 | 0 | |
Best | 0 | 0 | 7.658E–17 | 0 | 0 | 0 | 0 | 0 | 1.001E–09 | 0 | |
Worst | 0 | 0 | 6.020E–14 | 0 | 0 | 0 | 0 | 0 | 3.954E–05 | 0 | |
Median | 0 | 0 | 8.522E–15 | 0 | 0 | 0 | 0 | 0 | 1.237E–07 | 0 | |
\({\text {F45}}\) | Mean | 0 | 3.409E–06 | 1.262E–14 | 1.590E–07 | 1.824E–06 | 4.402E–12 | 2.724E–06 | 0 | 1.252E–07 | 0 |
Std | 0 | 6.038E–06 | 2.422E–14 | 1.688E–07 | 1.695E–06 | 1.029E–11 | 4.129E–06 | 0 | 2.670E–07 | 0 | |
Best | 0 | 3.326E–09 | 1.579E–18 | 5.559E–09 | 2.502E–08 | 1.034E–15 | 5.573E–08 | 0 | 1.678E–11 | 0 | |
Worst | 0 | 2.024E–05 | 8.119E–14 | 4.924E–07 | 4.873E–06 | 3.413E–11 | 1.142E–05 | 0 | 9.250E–07 | 0 | |
Median | 0 | 5.062E–07 | 3.225E–15 | 6.255E–08 | 1.515E–06 | 7.322E–14 | 5.841E–07 | 0 | 2.886E–08 | 0 | |
\({\text {F46}}\) | Mean | \(-1.0341\) | \(-0.985586\) | \(-1.0341\) | \(-1.0341\) | \(-1.0341\) | \(-1.0341\) | \(-0.959692\) | \(-1.0341\) | \(-1.0341\) | \(-1.0341\) |
Std | 0 | 0.099343 | 0 | 0 | 0 | 0 | 0.152476 | 0 | 0 | 0 | |
Best | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | |
Worst | \(-1.034\) | –0.79193 | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | |
Median | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | \(-1.034\) | |
\({\text {F47}}\) | Mean | –4.825E+00 | –4.011E+00 | –4.053E+00 | –4.535E+00 | –4.439E+00 | –4.103E+00 | –4.299E+00 | –4.825E+00 | –4.824E+00 | –4.62031 |
Std | 9.112E–16 | 7.259E–01 | 5.938E–01 | 4.537E–01 | 4.850E–01 | 5.964E–01 | 6.390E–01 | 9.112E–16 | 1.822E–15 | 0.2888552 | |
Best | –4.825E+00 | –4.822E+00 | –4.825E+00 | –4.825E+00 | –4.825E+00 | –4.825E+00 | –4.825E+00 | –4.825E+00 | –4.824E+00 | –4.6849 | |
Worst | –4.825E+00 | –2.895E+00 | –2.895E+00 | –3.860E+00 | –3.860E+00 | –2.895E+00 | –2.916E+00 | –4.825E+00 | –4.824E+00 | –3.3931 | |
Median | –4.825E+00 | –3.858E+00 | –3.860E+00 | –4.825E+00 | –4.825E+00 | –3.863E+00 | –4.599E+00 | –4.825E+00 | –4.824E+00 | –4.6849 | |
\({\text {F48}}\) | Mean | –8.760E+00 | –6.560E+00 | –5.349E+00 | –6.312E+00 | –7.449E+00 | –5.369E+00 | –5.602E+00 | –8.760E+00 | –8.444E+00 | –5.838515 |
Std | 0 | 5.729E–01 | 1.068E+00 | 5.138E–01 | 7.223E–01 | 6.663E–01 | 8.037E–01 | 0 | 3.282E–01 | 0.2436643 | |
Best | –8.760E+00 | –7.320E+00 | –6.666E+00 | –7.027E+00 | –8.760E+00 | –6.591E+00 | –7.007E+00 | –8.760E+00 | –8.753E+00 | –5.893 | |
Worst | –8.760E+00 | –5.316E+00 | –3.504E+00 | –5.724E+00 | –7.007E+00 | –4.518E+00 | –4.385E+00 | –8.760E+00 | –7.691E+00 | –4.8033 | |
Median | –8.760E+00 | –6.571E+00 | –5.662E+00 | –6.153E+00 | –7.008E+00 | –5.202E+00 | –5.705E+00 | –8.760E+00 | –8.563E+00 | –5.893 | |
\({\text {F49}}\) | Mean | 6.563E–29 | 1.570E–07 | 5.499E–17 | 3.152E–09 | 7.079E–08 | 3.616E–14 | 5.792E–07 | 0 | 1.696E–07 | 0 |
Std | 1.351E–28 | 2.465E–07 | 7.682E–17 | 4.124E–09 | 9.338E–08 | 8.253E–14 | 9.751E–07 | 0 | 3.635E–07 | 0 | |
Best | 0 | 9.363E–10 | 7.884E–19 | 1.828E–11 | 1.257E–11 | 3.156E–17 | 2.556E–09 | 0 | 8.054E–13 | 0 | |
Worst | 4.543E–28 | 8.583E–07 | 2.502E–16 | 1.135E–08 | 2.849E–07 | 2.731E–13 | 3.256E–06 | 0 | 1.195E–06 | 0 | |
Median | 2.524E–29 | 9.174E–08 | 1.496E–17 | 4.614E–10 | 3.056E–08 | 3.527E–15 | 2.282E–07 | 0 | 2.618E–08 | 0 | |
\({\text {F50}}\) | Mean | 0 | 2.785E–07 | 1.900E–16 | 1.062E–07 | 3.451E–06 | 7.273E–14 | 1.530E–06 | 0 | 5.528E–07 | 0 |
Std | 0 | 5.418E–07 | 2.729E–16 | 3.083E–07 | 9.023E–06 | 2.034E–13 | 2.063E–06 | 0 | 1.443E–06 | 0 | |
Best | 0 | 1.219E–10 | 8.448E–20 | 3.060E–10 | 1.050E–12 | 1.059E–16 | 1.250E–10 | 0 | 3.299E–09 | 0 | |
Worst | 0 | 1.845E–06 | 7.341E–16 | 1.007E–06 | 2.950E–05 | 6.670E–13 | 6.349E–06 | 0 | 4.759E–06 | 0 | |
Median | 0 | 1.315E–07 | 5.807E–17 | 6.862E–09 | 3.732E–08 | 3.449E–15 | 8.437E–07 | 0 | 1.768E–08 | 0 | |
\({\text {F51}}\) | Mean | 4.543E–29 | 1.253E–07 | 5.579E–16 | 7.685E–08 | 7.820E–08 | 1.758E–14 | 1.908E–07 | 0 | 8.752E–07 | 0 |
Std | 1.398E–28 | 1.843E–07 | 5.605E–16 | 1.091E–07 | 1.276E–07 | 2.505E–14 | 1.968E–07 | 0 | 2.515E–06 | 0 | |
Best | 0 | 1.080E–10 | 1.936E–17 | 6.619E–10 | 7.637E–13 | 1.720E–17 | 1.068E–10 | 0 | 9.635E–11 | 0 | |
Worst | 0 | 4.938E–07 | 1.836E–15 | 3.719E–07 | 4.139E–07 | 8.365E–14 | 5.622E–07 | 0 | 1.120E–05 | 0 | |
Median | 0 | 1.539E–08 | 3.969E–16 | 3.808E–08 | 1.646E–08 | 1.046E–14 | 1.049E–07 | 0 | 2.605E–07 | 0 |
