Table 5 Statistical results of eight CEAHA versions with eight different chaotic maps on 23 test functions.
From: Enhanced artificial hummingbird algorithm with chaotic traversal flight
Fun | Index | AHA-Cubic | AHA | AHA-Tent | AHA-Bernoulli | AHA-PLCM | AHA-Logistic | AHA-Chebyshev | AHA-2D-NHM | AHA-Circle |
|---|---|---|---|---|---|---|---|---|---|---|
F1 | Mean | 4.66E−321 | 1.50E−138+ | 4.41E−232+ | 7.35E−227+ | 2.91E−232+ | 5.43E−322\(\approx\) | 2.77E−321\(\approx\) | 6.26E−207+ | 2.17E−203+ |
Std | 0.00E+00 | 8.22E−138 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F2 | Mean | 5.83E−163 | 1.20E−76+ | 9.94E−119+ | 2.93E−119+ | 7.77E−119+ | 1.31E−161\(\approx\) | 5.55E−165\(\approx\) | 6.76E−105+ | 9.58E−104+ |
Std | 3.14E−162 | 3.22E−76 | 5.23E−118 | 1.53E−118 | 3.69E−118 | 7.20E−161 | 0.00E+00 | 3.69E−104 | 4.44E−103 | |
F3 | Mean | 1.69E−303 | 4.83E−131+ | 4.28E−221+ | 1.16E−215+ | 5.65E−214+ | 1.55E−293\(\approx\) | 1.38E−305\(\approx\) | 4.73E−191+ | 9.14E−185+ |
Std | 0.00E+00 | 2.63E−130 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F4 | Mean | 7.57E−158 | 1.03E−66+ | 1.25E−111+ | 1.44E−113+ | 5.03E−114+ | 9.63E−154\(\approx\) | 7.94E−161\(\approx\) | 3.81E−100+ | 3.75E−97+ |
Std | 4.14E−157 | 3.41E−66 | 6.85E−111 | 4.44E−113 | 1.96E−113 | 5.27E−153 | 4.29E−160 | 1.31E−99 | 1.93E−96 | |
F5 | Mean | 1.18E+01 | 2.63E+01+ | 8.93E+00\(\approx\) | 1.10E+01\(\approx\) | 6.43E+00\(\approx\) | 8.75E+00\(\approx\) | 1.04E+01\(\approx\) | 9.43E+00- | 4.78E+00- |
Std | 1.24E+01 | 4.17E−01 | 1.19E+01 | 1.23E+01 | 1.07E+01 | 1.21E+01 | 1.20E+01 | 1.22E+01 | 9.48E+00 | |
F6 | Mean | 0.00E+00 | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) |
Std | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F7 | Mean | 4.11E−05 | 1.63E−04+ | 6.91E−05\(\approx\) | 6.68E−05+ | 6.88E−05+ | 4.50E−05\(\approx\) | 4.62E−05\(\approx\) | 6.48E−05+ | 8.61E−05+ |
Std | 3.97E−05 | 1.27E−04 | 6.61E−05 | 4.78E−05 | 4.69E−05 | 3.09E−05 | 3.65E−05 | 4.60E−05 | 6.85E−05 | |
F8 | Mean | − 1.26E+04 | -1.18E+04+ | − 1.26E+04- | − 1.26E+04- | − 1.26E+04\(\approx\) | − 1.26E+04\(\approx\) | − 1.26E+04\(\approx\) | − 1.26E+04- | − 1.26E+04- |
Std | 2.63E−01 | 3.69E+02 | 1.60E−01 | 1.07E−01 | 2.17E−01 | 1.71E−01 | 1.56E−01 | 5.85E−02 | 1.38E−01 | |
F9 | Mean | 0.00E+00 | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) |
Std | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F10 | Mean | 8.88E−16 | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) | 8.88E−16\(\approx\) |
Std | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F11 | Mean | 0.00E+00 | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) | 0.00E+00\(\approx\) |
Std | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F12 | Mean | 7.05E−07 | 3.46E−05+ | 4.47E−07 | 6.26E−07\(\approx\) | 3.76E−07- | 9.59E−07\(\approx\) | 9.28E−07\(\approx\) | 1.74E−07 | 2.85E−07 |
Std | 5.24E−07 | 3.51E−05 | 4.86E−07 | 5.15E−07 | 2.75E−07 | 6.24E−07 | 7.53E−07 | 2.09E−07 | 1.91E−07 | |
F13 | Mean | 1.15E−05 | 1.74E+00+ | 7.97E−06\(\approx\) | 7.10E−06- | 7.47E−06\(\approx\) | 1.03E−05\(\approx\) | 9.54E−06\(\approx\) | 1.98E−06- | 4.37E−06- |
Std | 9.41E−06 | 5.65E−01 | 7.06E−06 | 6.19E−06 | 6.93E−06 | 8.41E−06 | 5.86E−06 | 1.86E−06 | 3.54E−06 | |
F14 | Mean | 9.98E−01 | 1.06E+00\(\approx\) | 9.98E−01\(\approx\) | 9.98E−01\(\approx\) | 9.98E−01\(\approx\) | 9.98E−01\(\approx\) | 9.98E−01\(\approx\) | 9.98E−01\(\approx\) | 9.98E−01\(\approx\) |
Std | 0.00E+00 | 3.62E−01 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F15 | Mean | 3.07E−04 | 3.07E−04+ | 3.07E−04\(\approx\) | 3.07E−04\(\approx\) | 3.07E−04\(\approx\) | 3.07E−04\(\approx\) | 3.07E−04\(\approx\) | 3.07E−04\(\approx\) | 3.07E−04\(\approx\) |
Std | 5.77E−16 | 3.58E−15 | 9.75E−17 | 6.12E−16 | 7.79E−18 | 6.46E−18 | 3.37E−18 | 4.88E−18 | 1.83E−17 | |
F16 | Mean | -1.03E+00 | -1.03E+00\(\approx\) | -1.03E+00\(\approx\) | -1.03E+00\(\approx\) | -1.03E+00\(\approx\) | -1.03E+00\(\approx\) | -1.03E+00\(\approx\) | -1.03E+00- | -1.03E+00\(\approx\) |
Std | 6.12E−16 | 6.05E−16 | 5.83E−16 | 6.05E−16 | 5.83E−16 | 5.76E−16 | 5.90E−16 | 5.53E−16 | 5.76E−16 | |
F17 | Mean | 3.98E−01 | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) | 3.98E−01\(\approx\) |
Std | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.59E−13 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
F18 | Mean | 3.00E+00 | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) | 3.00E+00\(\approx\) |
Std | 1.27E−15 | 1.34E−15 | 1.25E−15 | 1.26E−15 | 1.25E−15 | 2.16E−15 | 1.28E−15 | 1.29E−15 | 1.33E−15 | |
F19 | Mean | -3.86E+00 | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) | -3.86E+00\(\approx\) |
Std | 2.71E−15 | 2.71E−15 | 2.71E−15 | 2.71E−15 | 2.71E−15 | 2.71E−15 | 2.71E−15 | 2.71E−15 | 2.71E−15 | |
F20 | Mean | -3.32E+00 | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) | -3.32E+00\(\approx\) |
Std | 1.36E− 15 | 2.17E− 02 | 1.36E− 15 | 1.36E− 15 | 1.34E− 15 | 1.36E− 15 | 1.34E− 15 | 1.36E− 15 | 1.34E− 15 | |
F21 | Mean | -1.02E+01 | − 1.02E+01\(\approx\) | -1.02E+01\(\approx\) | -1.02E+01\(\approx\) | -1.02E+01\(\approx\) | − 1.02E+01\(\approx\) | − 1.02E+01\(\approx\) | -1.02E+01\(\approx\) | -1.02E+01\(\approx\) |
Std | 6.33E−15 | 1.35E−05 | 6.51E−15 | 6.14E−15 | 6.51E−15 | 6.39E−15 | 6.33E−15 | 6.68E−15 | 6.51E−15 | |
F22 | Mean | − 1.04E+01 | -1.04E+01\(\approx\) | -1.04E+01\(\approx\) | − 1.04E+01\(\approx\) | − 1.04E+01 | − 1.04E+01\(\approx\) | − 1.04E+01\(\approx\) | − 1.04E+01\(\approx\) | − 1.04E+01\(\approx\) |
Std | 0.00E+00 | 2.70E−11 | 3.30E−16 | 4.66E−16 | 8.73E−16 | 3.30E−16 | 5.71E−16 | 4.66E−16 | 4.66E−16 | |
F23 | Mean | − 1.05E+01 | − 1.05E+01\(\approx\) | − 1.05E+01\(\approx\) | − 1.05E+01\(\approx\) | − 1.05E+01\(\approx\) | − 1.05E+01\(\approx\) | − 1.05E+01\(\approx\) | -1.05E+01\(\approx\) | -1.05E+01\(\approx\) |
Std | 2.62E−15 | 3.92E−15 | 1.23E−15 | 2.49E−15 | 9.33E−16 | 1.04E−15 | 1.14E−15 | 2.62E−15 | 1.14E−15 | |
Cubic wins | + | 10 | 4 | 5 | 5 | 0 | 0 | 5 | 5 | |
Similar | \(\approx\) | 13 | 17 | 16 | 16 | 23 | 23 | 13 | 14 | |
Other wins | – | 0 | 2 | 2 | 2 | 0 | 0 | 5 | 4 | |
