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Table 3 Simulation results on UBFs

From: An advanced hybrid meta-heuristic algorithm for solving small- and large-scale engineering design optimization problems

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