Table 13 Wilcoxon rank–sum test and t-test results for LCA versus well–known and top-performing algorithms for 51 benchmark functions.
From: Learning cooking algorithm for solving global optimization problems
Functions | LCA vs | LCA vs | LCA vs | LCA vs | LCA vs | LCA vs | LCA vs | LCA vs | LCA vs | |
|---|---|---|---|---|---|---|---|---|---|---|
PSO | TSA | SSA | MVO | GWO | WOA | GJO | LSHADE | CMAES | ||
\({\text {F1}}\) | \(p-value\) | 9.119E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.190E–05 | 9.120E–05 |
\(t-test\) | –3.935468 | –2.185655 | –20.00023 | –15.98234 | –2.6901698 | 0 | –1.805788 | –3.915415 | –4.914035 | |
\({\text {F2}}\) | \(p-value\) | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.140E–05 | 9.130E–05 |
\(t-test\) | –8.304950 | –2.3503565 | –3.656791 | –13.62703 | –5.797903 | –1.462105 | –3.059250 | –2.431003 | –11.34986 | |
\({\text {F3}}\) | \(p-value\) | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.160E–05 | 9.090E–05 |
\(t-test\) | –9.509E+00 | –2.192E+00 | –5.670E+00 | –1.043E+01 | –1.481E+00 | –8.789E+00 | –1.501E+00 | –5.716330 | –6.599026 | |
\({\text {F4}}\) | \(p-value\) | 9.110E–05 | 9.101E–05 | 9.101E–05 | 9.100E–05 | 9.110E–05 | 9.101E–05 | 9.110E–05 | 9.160E–05 | 9.110E–05 |
\(t-test\) | –1.979E+01 | –2.090E+00 | –9.698E+00 | –1.069E+01 | –6.560E+00 | –5.699E+00 | –1.646E+00 | –17.18328 | –15.38482 | |
\({\text {F5}}\) | \(p-value\) | 9.110E–05 | 9.110E–05 | 9.120E–05 | 9.100E–05 | 9.101E–05 | 9.110E–05 | 9.120E–05 | 9.120E–05 | 9.140E–05 |
\(t-test\) | –5.222E+00 | –1.546E+02 | –2.351E+00 | –3.373E+00 | –1.991E+02 | –1.634E+02 | –1.628E+02 | –7.078189 | –19.44548 | |
\({\text {F6}}\) | \(p-value\) | 9.110E–05 | 9.110E–05 | 9.100E–05 | 9.110E–05 | 9.120E–05 | 9.110E–05 | 9.110E–05 | 9.170E–05 | 9.130E–05 |
\(t-test\) | –7.445E+00 | –3.806E+01 | –1.059E+01 | –1.637E+01 | –1.144E+01 | –3.256E+00 | –2.339E+01 | –1.511763 | –5.993604 | |
\({\text {F7}}\) | \(p-value\) | 9.110E–05 | 9.110E–05 | 9.120E–05 | 9.110E–05 | 9.110E–05 | 9.110E–05 | 9.120E–05 | 9.210E–05 | 9.210E–05 |
\(t-test\) | –2.011E+01 | –1.487E+01 | –1.073E+01 | –1.666E+01 | –9.593E+00 | –4.683E+00 | –9.720E+00 | –10.48742 | –9.983163 | |
\({\text {F8}}\) | \(p-value\) | 9.10E–05 | 9.10E–05 | 9.10E–05 | 9.10E–05 | 9.10E–05 | 9.10E–05 | 9.10E–05 | 9.63E–05 | 9.12E–05 |
\(t-test\) | –2.752E+01 | –3.992E+01 | 1.833E+01 | –6.688E+01 | –5.972E+01 | –3.539E+00 | –3.386E+01 | –7.709736 | 0 | |
\({\text {F9}}\) | \(p-value\) | 9.120E–05 | 9.110E–05 | 9.130E–05 | 9.110E–05 | NA | NA | NA | 9.160E–05 | 9.130E–05 |
\(t-test\) | –24.34744 | –20.97211 | –22.28598 | –13.95689 | –1.452966 | 0 | 0 | –1.001548 | –96.23406 | |
\({\text {F10}}\) | \(p-value\) | 9.110E–05 | 8.701E–05 | 9.120E–05 | 9.110E–05 | 3.101E–05 | 1.220E–04 | 8.110E–06 | 9.170E–05 | 9.140E–05 |
\(t-test\) | –1.566E+00 | –5.276E+00 | –7.926E+00 | –8.549E+00 | –2.397E+01 | –5.811E+00 | –6.520E+15 | –20.25979 | –7.952658 | |
\({\text {F11}}\) | \(p-value\) | 9.10E–05 | NA | 9.10E–05 | 9.10E–05 | NA | NA | NA | 9.18E–05 | NA |
\(t-test\) | –7.880752 | –4.248139 | –6.080502 | –25.43402 | –1.452966 | 0 | 0 | –3.348001 | 0 | |
\({\text {F12}}\) | \(p-value\) | 9.110E–05 | 9.120E–05 | 9.120E–05 | 9.130E–05 | 9.110E–05 | 9.120E–05 | 9.110E–05 | 9.160E–05 | 9.131E–05 |
\(t-test\) | –6.544E+00 | –1.282E+01 | –8.636E+00 | –7.885E+00 | –8.596E+00 | –5.554E+00 | –1.318E+01 | –1.836609 | –1.907916 | |
\({\text {F13}}\) | \(p-value\) | 9.112E–05 | 9.111E–05 | 9.101E–05 | 9.103E–05 | 9.110E–05 | 9.100E–05 | 9.104E–05 | 9.170E–05 | 9.160E–05 |
\(t-test\) | –2.184E+00 | –1.924E+01 | –2.716E+00 | –8.278E+00 | –1.117E+01 | –7.612E+00 | –4.083E+01 | –1.270109 | –3.599126 | |
\({\text {F14}}\) | \(p-value\) | 0.000185 | 1.935E–04 | NA | NA | NA | NA | 3.836E–04 | 8.140E–06 | 9.140E–05 |
\(t-test\) | –4.885E+00 | –5.617E+00 | –1.452E+00 | 0 | –3.078E+00 | –4.358E+00 | –6.120E+00 | –5889555 | –5.871148 | |
\({\text {F15}}\) | \(p-value\) | 9.210E–05 | 1.761E–03 | 9.210E–05 | 9.110E–05 | 3.176E–01 | 2.616E–04 | 8.753E–02 | 1.340E–02 | 9.190E–05 |
\(t-test\) | –8.627E+00 | –2.371E+00 | –1.197E+01 | –2.452E+00 | –2.301E+00 | –4.317E+00 | –1.490E+00 | –1.043669 | –6.670370 | |
\({\text {F16}}\) | \(p-value\) | NA | NA | NA | NA | NA | NA | NA | NA | NA |
\(t-test\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
\({\text {F17}}\) | \(p-value\) | 0.000122 | 0.68782 | 0.000122 | 0.000122 | 0.1189 | 0.000122 | 0.039403 | 8.57E–05 | 0.000122 |
\(t-test\) | 5.097E+00 | –9.910E–01 | 5.097E+00 | 5.097E+00 | –5.554E–01 | 5.097E+00 | –6.535E–01 | 5.432E+00 | 5.097E+00 | |
\({\text {F18}}\) | \(p-value\) | NA | NA | NA | NA | NA | NA | NA | NA | NA |
\(t-test\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | \(-2.664508\) | |
\({\text {F19}}\) | \(p-value\) | 6.629E–05 | 2.823E–03 | 6.629E–05 | 6.629E–05 | 4.464E–03 | 5.813E–01 | 5.802E–01 | 4.978350 | 6.629E–05 |
\(t-test\) | 4.997E+00 | 3.549E+00 | 4.997E+00 | 4.997E+00 | 2.593E+00 | 1.025E+00 | –7.341E–01 | 4.978E+00 | 4.997E+00 | |
\({\text {F20}}\) | \(p-value\) | 0.000085 | 9.10E–05 | 9.10E–05 | 8.673E–05 | 8.673E–05 | 1.435E–04 | 9.10E–05 | 7.191E–05 | 8.131E–05 |
\(t-test\) | 1.125E+01 | 1.001E+01 | 1.059E+01 | 1.124E+01 | 1.016E+01 | 5.524E+00 | 9.920E+00 | 1.332E+01 | 1.208E+01 | |
\({\text {F21}}\) | \(p-value\) | 1.220E–04 | 9.10E–05 | 3.125E–02 | 1.651E–04 | 8.773E–05 | 4.515E–04 | 9.10E–05 | 8.139E–06 | NA |
\(t-test\) | –6.304E+00 | –6.656E+00 | –2.797E+00 | –3.559E+00 | –1.454E+00 | –1.474E+00 | –3.565E+00 | –7.881E+08 | 0 | |
\({\text {F22}}\) | \(p-value\) | 1.953E–03 | 9.10E–05 | 3.125E–02 | 3.908E–04 | 8.673E–05 | 9.10E–05 | 8.673E–05 | 8.140E–06 | NA |
\(t-test\) | –4.214E+00 | –3.772E+00 | –2.787E+00 | –4.358E+00 | –7.756E+00 | –3.059E+00 | –1.463E+00 | 2.761E+10 | 0 | |
\({\text {F23}}\) | \(p-value\) | 3.125E–02 | 9.10E–05 | NA | 1.854E–04 | 8.283E–05 | 8.975E–05 | 9.10E–05 | 8.140E–06 | NA |
\(t-test\) | –2.853E+00 | –5.448E+00 | –2.138E+00 | –4.359E+00 | –1.453E+00 | –4.389E+00 | –1.461E+00 | 5.677E+09 | 0 | |
\({\text {F24}}\) | \(p-value\) | 9.10E–05 | 1.651E–04 | 9.10E–05 | NA | NA | NA | NA | NA | NA |
\(t-test\) | 4.325E+02 | –9.124E+00 | 2.746E+02 | 0 | 0 | 0 | 0 | 0 | 0 | |
\({\text {F25}}\) | \(p-value\) | 9.121E–05 | 9.113E–05 | 9.098E–05 | 9.090E–05 | 9.110E–05 | 9.130E–05 | 9.099E–05 | 9.1013E–05 | 9.121E–05 |
\(t-test\) | –2.044E+00 | –2.818E+00 | –4.536E+00 | –4.385E+00 | –3.459E+00 | –1.48E+00 | –1.945E+00 | –3.536E+00 | –5.204E+00 | |
\({\text {F26}}\) | \(p-value\) | 4.569E–05 | 1.007E+00 | 2.505E–01 | 1.402E–02 | 1.402E–02 | 9.10E–05 | 9.10E–05 | 4.569E–05 | 1.070E–04 |
\(t-test\) | 1.388E+01 | –2.840E+00 | –2.174E+00 | –1.448E+00 | –1.448E+00 | 1.388E+01 | 1.386E+01 | 1.388E+01 | 9.901E+00 | |
\({\text {F27}}\) | \(p-value\) | NA | 0.00006103 | NA | 0.0009765 | NA | NA | NA | NA | NA |
\(t-test\) | 0 | –2.516 | 0 | –1.448 | 0 | 0 | 0 | 0 | 0 | |
\({\text {F28}}\) | \(p-value\) | 9.077E–05 | 9.077E–05 | 9.077E–05 | 9.077E–05 | 9.077E–05 | NA | NA | 9.207E–05 | 9.116E–05 |
\(t-test\) | –2.26 | 0 | –4.789 | –7.09 | 0 | 0 | 0 | –1.001894843 | 0 | |
\({\text {F29}}\) | \(p-value\) | 9.110E–05 | 9.120E–05 | 9.100E–05 | 9.130E–05 | 9.140E–05 | 9.110E–05 | 9.110E–05 | NA | 9.160E–05 |
\(t-test\) | –6.454E+00 | –3.562E+00 | –1.536E+00 | –3.318E+00 | –6.512E+00 | –3.927E+00 | –7.644E+00 | 0 | –1.701E+00 | |
\({\text {F30}}\) | \(p-value\) | 9.103E–05 | 1.005E–02 | 9.103E–05 | 6.370E–05 | 6.177E–02 | 2.455E–01 | 2.455E–01 | 6.370E–05 | 9.16E–05 |
\(t-test\) | 7.862E+02 | –5.338E+00 | 1.338E+02 | 0 | –4.358E+00 | –2.179E+00 | –2.179E+00 | 0 | 2.39E+307 | |
\({\text {F31}}\) | \(p-value\) | 9.120E–05 | 8.670E–02 | 9.110E–05 | 1.007E+00 | 1.007E+00 | 2.310E–01 | 1.007E+00 | 3.071E–01 | 9.180E–05 |
\(t-test\) | 5.556E+02 | –3.547E+00 | 2.041E+02 | –2.833E+00 | –2.833E+00 | –2.156E+00 | –2.833E+00 | –3.539E+00 | 2.08E+302 | |
\({\text {F32}}\) | \(p-value\) | 9.120E–05 | 9.110E–05 | 9.120E–05 | 9.100E–05 | 9.130E–05 | 9.110E–05 | 9.150E–05 | 9.140E–05 | 9.091E–05 |
\(t-test\) | –3.23822 | –1.47163 | –8.62344 | –6.91458 | –1.75204 | 0 | 0 | –1.27038 | –6.64696 | |
\({\text {F33}}\) | \(p-value\) | 9.120E–05 | 9.098E–05 | 9.100E–05 | 9.110E–05 | 9.130E–05 | 9.120E–05 | 9.100E–05 | 9.190E–05 | 9.140E–05 |
\(t-test\) | –5.611E+00 | –4.319E+00 | –4.521E+00 | –1.069E+01 | –4.00E+00 | –2.283E+00 | –1.577E+00 | –5.179E+00 | –6.842E+00 | |
\({\text {F34}}\) | \(p-value\) | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | NA | 9.120E–05 |
\(t-test\) | –2.185E+00 | –5.339E+00 | –4.389E+00 | –3.175E+00 | –7.751E+00 | –2.467E+00 | –2.850E+00 | 0 | –3.335E+00 | |
\({\text {F35}}\) | \(p-value\) | NA | NA | 9.120E–05 | 9.130E–05 | NA | NA | NA | NA | NA |
\(t-test\) | 0 | 0 | –5.725E+00 | –5.671E+00 | 0 | 0 | 0 | 0 | 0 | |
\({\text {F36}}\) | \(p-value\) | NA | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | 9.120E–05 | NA | NA |
\(t-test\) | 0 | –1.453E+00 | –4.238E+00 | –5.266E+00 | –6.003E+00 | –5.496E+00 | –4.192E+00 | 0 | 0 | |
\({\text {F37}}\) | \(p-value\) | 4.882E–04 | 7.813E–03 | 1.953E–03 | NA | NA | NA | NA | NA | 9.130E–05 |
\(t-test\) | 5.320E+00 | \(-3.559E+00\) | 4.350E+00 | 0 | 0 | 0 | 0 | 0 | 6.810E+00 | |
\({\text {F38}}\) | \(p-value\) | 9.10E–05 | 4.600E–04 | 4.600E–04 | 9.10E–05 | 9.10E–05 | 2.254E–03 | 9.10E–05 | 8.67E–05 | 9.19E–05 |
\(t-test\) | 4.698E+01 | 5.897E+00 | 5.241E+00 | 1.199E+01 | 4.094E+01 | 4.252E+00 | 9.474E+00 | 5.074E+01 | 2.537E+01 | |
\({\text {F39}}\) | \(p-value\) | 9.120E–05 | 9.110E–05 | 9.130E–05 | 9.101E–05 | 9.110E–05 | 2.814E–02 | 3.724E–02 | 8.975E–05 | 1.250E–04 |
\(t-test\) | 2.081E+01 | 5.084E+00 | 4.484E+00 | 1.099E+01 | 7.793E+00 | 1.814E+00 | 2.086E+00 | 4.142E+01 | 7.393E+00 | |
\({\text {F40}}\) | \(p-value\) | NA | NA | NA | 9.130E–05 | NA | NA | NA | NA | 9.210E–05 |
\(t-test\) | 0 | 0 | –3.942E+00 | –4.098E+00 | 0 | 0 | 0 | 0 | –5.435E+00 | |
\({\text {F41}}\) | \(p-value\) | NA | NA | 9.120E–05 | 9.130E–05 | NA | NA | NA | NA | 9.206E–05 |
\(t-test\) | 0 | 0 | –3.761E+00 | –4.321E+00 | 0 | 0 | 0 | 0 | –4.039E+00 | |
\({\text {F42}}\) | \(p-value\) | NA | NA | 9.130E–05 | 9.110E–05 | NA | NA | NA | NA | 9.170E–05 |
\(t-test\) | 0 | –2.179E+00 | –4.487E+00 | –5.256E+00 | 0 | –5.318E+00 | 0 | 0 | –7.089E+00 | |
\({\text {F43}}\) | \(p-value\) | NA | 9.10E–05 | NA | NA | 4.436E–04 | NA | 1.990E–04 | 8.139E–06 | 8.641E–05 |
\(t-test\) | 0 | –5.743E+00 | 0 | 0 | –2.997E+00 | 0 | –2.719E+00 | –2.748E+08 | –6.281E+00 | |
\({\text {F44}}\) | \(p-value\) | NA | NA | 9.120E–05 | NA | NA | NA | NA | NA | 9.160E–05 |
\(t-test\) | 0 | 0 | –3.521E+00 | 0 | 0 | 0 | 0 | 0 | –1.679E+00 | |
\({\text {F45}}\) | \(p-value\) | NA | 9.120E–05 | 9.130E–05 | 9.110E–05 | 9.100E–05 | 9.120E–05 | 9.160E–05 | NA | 9.130E–05 |
\(t-test\) | 0 | –2.524E+00 | –2.330E+00 | –4.212E+00 | –4.812E+00 | –1.912E+00 | –2.950E+00 | 0 | –2.096E+00 | |
\({\text {F46}}\) | \(p-value\) | NA | NA | NA | NA | NA | NA | NA | NA | NA |
\(t-test\) | 0 | –2.179E+00 | 0 | 0 | 0 | 0 | –2.179E+00 | 0 | 0 | |
\({\text {F47}}\) | \(p-value\) | 9.150E–05 | 9.120E–05 | 9.130E–05 | 9.110E–05 | 9.160E–05 | 1.435E–04 | 1.435E–04 | 9.180E–05 | 9.083E–05 |
\(t-test\) | 3.169E+00 | –3.485E+00 | –3.838E+00 | –7.052E–01 | –1.436E+00 | –3.487E+00 | –2.047E+00 | 3.169E+00 | 3.164E+00 | |
\({\text {F48}}\) | \(p-value\) | 9.130E–05 | 9.150E–05 | 3.256E–03 | 9.160E–05 | 9.140E–05 | 9.120E–05 | 6.966E–04 | 9.140E–05 | 9.210E–05 |
\(t-test\) | 5.362E+01 | 5.186E+00 | –1.993E+00 | 3.729E+00 | 9.452E+00 | –2.954E+00 | –1.258E+00 | 5.362E+01 | 2.851E+01 | |
\({\text {F49}}\) | \(p-value\) | NA | 9.110E–05 | 9.130E–05 | 9.160E–05 | 9.120E–05 | 9.110E–05 | 9.150E–05 | NA | 9.140E–05 |
\(t-test\) | –2.171E+00 | –2.849E+00 | –3.201E+00 | –3.418E+00 | –3.390E+00 | –1.959E+00 | –2.656E+00 | 0 | –2.087E+00 | |
\({\text {F50}}\) | \(p-value\) | NA | 9.120E–05 | 9.130E–05 | 9.140E–05 | 9.160E–05 | 9.130E–05 | 9.110E–05 | NA | 9.120E–05 |
\(t-test\) | 0 | –2.298E+00 | –3.114E+00 | –1.540E+00 | –1.710E+00 | –1.599E+00 | –3.317E+00 | 0 | –1.712E+00 | |
\({\text {F51}}\) | \(p-value\) | NA | 9.140E–05 | 9.160E–05 | 9.120E–05 | 9.110E–05 | 9.180E–05 | 9.120E–05 | NA | 9.140E–05 |
\(t-test\) | –1.452E+00 | –3.039E+00 | –4.451E+00 | –3.149E+00 | –2.739E+00 | –3.139E+00 | –4.335E+00 | 0 | –1.559E+00 |
