Abstract
The aim of the current paper is to introduce a global optimization algorithm, inspired from the survival strategies of flying foxes during a heatwave, called as Flying Foxes Optimization (FFO). The proposed method exploits a Fuzzy Logic (FL) technique to determine the parameters individually for each solution, thus resulting in a parameters-free optimization algorithm. To evaluate FFO, 56 benchmark functions, including the CEC2017 test function suite and three real-world engineering problems, are employed and its performance is compared to those of state-of-the-art metaheuristics, when it comes to global optimization. The comparison results reveal that the proposed FFO optimizer constitutes a powerful attractive alternative for global optimization.
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Funding
This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme «Human Resources Development, Education and Lifelong Learning» in the context of the project “Strengthening Human Resources Research Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation (ΙΚΥ).
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Appendices
Appendix A
This Appendix presents the comparison of FFO’s performance with that of the rest state-of-the-art optimization approaches. Bold values correspond to the best located values.
f | Statistics | FFO | FA | HS | BA | IWO | BAT | CS |
|---|---|---|---|---|---|---|---|---|
1 | Min | 6.36E−55 | 8.94E+00 | 2.73E+00 | 6.44E+02 | 6.65E−01 | 6.20E+02 | 6.68E+02 |
Average | 5.45E−54 | 1.49E+01 | 3.51E+00 | 7.52E+02 | 1.01E+00 | 7.48E+02 | 7.60E+02 | |
Median | 3.40E−54 | 1.41E+01 | 3.41E+00 | 7.55E+02 | 9.56E−01 | 7.57E+02 | 7.67E+02 | |
Max | 2.03E−53 | 2.40E+01 | 4.83E+00 | 7.98E+02 | 1.67E+00 | 7.84E+02 | 8.17E+02 | |
Std | 5.70E−54 | 3.63E+00 | 5.66E−01 | 3.92E+01 | 2.57E−01 | 3.84E+01 | 3.83E+01 | |
2 | Min | 2.99E+01 | 1.15E+03 | 1.13E+03 | 2.04E+06 | 1.88E+02 | 2.23E+06 | 1.57E+06 |
Average | 5.69E+01 | 2.95E+03 | 1.60E+03 | 2.46E+06 | 9.60E+02 | 2.64E+06 | 2.46E+06 | |
Median | 4.47E+01 | 3.02E+03 | 1.66E+03 | 2.43E+06 | 7.57E+02 | 2.67E+06 | 2.54E+06 | |
Max | 1.02E+02 | 4.82E+03 | 2.24E+03 | 2.95E+06 | 2.61E+03 | 2.97E+06 | 2.91E+06 | |
Std | 2.26E+01 | 1.16E+03 | 3.43E+02 | 2.28E+05 | 7.00E+02 | 2.18E+05 | 3.07E+05 | |
3 | Min | 1.83E−53 | 1.86E+02 | 4.11E+01 | 1.49E+04 | 1.00E+02 | 1.53E+04 | 1.43E+04 |
Average | 1.57E−52 | 3.15E+02 | 5.89E+01 | 1.75E+04 | 2.81E+02 | 1.72E+04 | 1.68E+04 | |
Median | 6.42E−53 | 3.16E+02 | 5.83E+01 | 1.76E+04 | 2.56E+02 | 1.72E+04 | 1.69E+04 | |
Max | 1.40E−51 | 4.69E+02 | 7.96E+01 | 1.90E+04 | 4.46E+02 | 2.02E+04 | 1.91E+04 | |
Std | 2.75E−52 | 8.47E+01 | 1.04E+01 | 1.08E+03 | 1.08E+02 | 1.17E+03 | 1.54E+03 | |
4 | Min | 9.58E−81 | 7.65E−10 | 8.03E−12 | 1.74E−06 | 2.13E−06 | 1.90E−06 | 2.03E−07 |
Average | 1.90E−67 | 7.53E−08 | 8.27E−11 | 6.27E−03 | 1.21E−05 | 6.34E−03 | 8.50E−03 | |
Median | 1.22E−72 | 3.36E−08 | 5.71E−11 | 1.35E−03 | 9.76E−06 | 1.11E−03 | 3.27E−03 | |
Max | 4.38E−66 | 3.24E−07 | 3.11E−10 | 3.82E−02 | 4.46E−05 | 4.05E−02 | 9.57E−02 | |
Std | 8.66E−67 | 8.89E−08 | 8.74E−11 | 1.06E−02 | 9.02E−06 | 1.10E−02 | 2.09E−02 | |
5 | Min | − 1.00E+00 | − 9.96E−01 | − 9.84E−01 | − 6.11E−01 | − 8.90E−01 | − 5.73E−01 | − 6.11E−01 |
Average | − 1.00E+00 | − 9.90E−01 | − 9.77E−01 | − 4.56E−01 | − 8.37E−01 | − 4.40E−01 | − 4.33E−01 | |
Median | − 1.00E+00 | − 9.91E−01 | − 9.76E−01 | − 4.65E−01 | − 8.39E−01 | − 4.37E−01 | − 4.23E−01 | |
Max | − 1.00E+00 | − 9.79E−01 | − 9.71E−01 | − 2.37E−01 | − 7.85E−01 | − 2.72E−01 | − 2.69E−01 | |
Std | 0.00E+00 | 4.09E−03 | 3.18E−03 | 1.09E−01 | 3.15E−02 | 8.36E−02 | 1.10E−01 | |
6 | Min | 5.63E−29 | 1.30E+02 | 4.98E+01 | 1.53E+03 | 2.23E+02 | 1.54E+03 | 1.54E+03 |
Average | 1.63E−28 | 1.69E+02 | 6.56E+01 | 1.62E+03 | 2.88E+02 | 1.64E+03 | 1.65E+03 | |
Median | 1.62E−28 | 1.73E+02 | 6.41E+01 | 1.62E+03 | 2.77E+02 | 1.64E+03 | 1.64E+03 | |
Max | 3.31E−28 | 2.06E+02 | 8.45E+01 | 1.73E+03 | 3.75E+02 | 1.72E+03 | 1.75E+03 | |
Std | 7.74E−29 | 2.10E+01 | 8.52E+00 | 5.00E+01 | 4.23E+01 | 5.04E+01 | 4.88E+01 | |
7 | Min | 5.42E+00 | 1.61E+01 | 2.10E+01 | 7.06E+01 | 3.50E+01 | 7.54E+01 | 7.42E+01 |
Average | 1.22E+01 | 2.05E+01 | 2.32E+01 | 7.89E+01 | 4.69E+01 | 7.92E+01 | 7.94E+01 | |
Median | 1.34E+01 | 2.01E+01 | 2.32E+01 | 7.91E+01 | 4.69E+01 | 7.94E+01 | 8.00E+01 | |
Max | 1.75E+01 | 2.62E+01 | 2.54E+01 | 8.28E+01 | 5.68E+01 | 8.17E+01 | 8.23E+01 | |
Std | 4.26E+00 | 2.32E+00 | 1.12E+00 | 2.73E+00 | 6.35E+00 | 1.71E+00 | 2.08E+00 | |
8 | Min | 1.17E−27 | 2.78E+02 | 1.91E+01 | 1.36E+37 | 2.09E+09 | 6.08E+16 | 5.90E+39 |
Average | 3.87E−27 | 3.35E+02 | 2.55E+01 | 3.19E+61 | 2.08E+26 | 3.21E+62 | 1.79E+61 | |
Median | 3.31E−27 | 3.45E+02 | 2.51E+01 | 1.61E+54 | 1.50E+16 | 3.62E+56 | 1.13E+54 | |
Max | 1.00E−26 | 3.96E+02 | 3.35E+01 | 4.37E+62 | 2.48E+27 | 4.16E+63 | 3.00E+62 | |
Std | 1.94E−27 | 3.02E+01 | 4.05E+00 | 1.02E+62 | 6.51E+26 | 1.02E+63 | 6.76E+61 | |
9 | Min | 1.13E+01 | 1.89E+02 | 1.85E+02 | 5.56E+02 | 5.89E+02 | 5.86E+02 | 6.19E+02 |
Average | 2.23E+01 | 2.47E+02 | 2.38E+02 | 8.18E+02 | 8.12E+02 | 8.44E+02 | 8.02E+02 | |
Median | 2.11E+01 | 2.48E+02 | 2.40E+02 | 8.35E+02 | 7.87E+02 | 8.69E+02 | 8.00E+02 | |
Max | 3.59E+01 | 2.99E+02 | 3.12E+02 | 1.02E+03 | 1.19E+03 | 9.69E+02 | 9.65E+02 | |
Std | 7.52E+00 | 2.94E+01 | 2.97E+01 | 1.17E+02 | 1.58E+02 | 1.03E+02 | 9.82E+01 | |
10 | Min | 1.69E+01 | 1.24E+02 | 1.70E+01 | 2.78E+02 | 1.68E+02 | 2.99E+02 | 2.99E+02 |
Average | 4.37E+01 | 1.79E+02 | 2.17E+01 | 3.27E+02 | 2.29E+02 | 3.32E+02 | 3.36E+02 | |
Median | 4.28E+01 | 1.71E+02 | 2.17E+01 | 3.26E+02 | 2.21E+02 | 3.30E+02 | 3.43E+02 | |
Max | 7.86E+01 | 2.72E+02 | 2.74E+01 | 3.97E+02 | 3.24E+02 | 3.75E+02 | 3.61E+02 | |
Std | 1.40E+01 | 4.54E+01 | 3.00E+00 | 2.87E+01 | 3.88E+01 | 1.91E+01 | 1.79E+01 | |
11 | Min | 0.00E+00 | 0.00E+00 | 9.01E−09 | 1.19E−08 | 3.08E−06 | 9.33E−07 | 2.37E−06 |
Average | 0.00E+00 | 5.33E−16 | 2.18E−07 | 1.66E−02 | 3.65E−05 | 1.16E−02 | 2.61E−02 | |
Median | 0.00E+00 | 0.00E+00 | 2.07E−07 | 2.78E−03 | 3.17E−05 | 2.20E−03 | 7.51E−03 | |
Max | 0.00E+00 | 3.55E−15 | 5.40E−07 | 5.72E−02 | 1.00E−04 | 5.93E−02 | 9.28E−02 | |
Std | 0.00E+00 | 1.30E−15 | 1.27E−07 | 2.30E−02 | 2.54E−05 | 1.74E−02 | 3.25E−02 | |
12 | Min | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |
Average | 0.00E+00 | 0.00E+00 | 9.86E−04 | 3.40E−05 | 2.15E−08 | 5.46E−05 | 8.01E−05 | |
Median | 0.00E+00 | 0.00E+00 | 2.22E−16 | 8.38E−07 | 7.27E−15 | 1.36E−08 | 1.37E−06 | |
Max | 0.00E+00 | 0.00E+00 | 9.86E−03 | 2.20E−04 | 3.32E−07 | 3.73E−04 | 4.43E−04 | |
Std | 0.00E+00 | 0.00E+00 | 3.04E−03 | 6.16E−05 | 7.59E−08 | 1.08E−04 | 1.41E−04 | |
13 | Min | 0.00E+00 | 4.42E+01 | 5.17E−01 | 2.48E+01 | 1.12E+01 | 1.93E+01 | 2.36E+01 |
Average | 5.08E−14 | 5.18E+01 | 7.40E−01 | 3.24E+01 | 1.69E+01 | 3.24E+01 | 3.20E+01 | |
Median | 5.41E−14 | 5.20E+01 | 6.87E−01 | 3.19E+01 | 1.63E+01 | 3.26E+01 | 3.20E+01 | |
Max | 6.59E−14 | 6.24E+01 | 1.09E+00 | 3.99E+01 | 2.20E+01 | 4.33E+01 | 4.02E+01 | |
Std | 1.25E−14 | 5.42E+00 | 1.58E−01 | 4.09E+00 | 2.93E+00 | 5.93E+00 | 4.81E+00 | |
14 | Min | 4.00E−01 | 4.00E+00 | 3.20E+00 | 2.82E+01 | 7.11E+00 | 2.78E+01 | 2.67E+01 |
Average | 4.72E−01 | 4.95E+00 | 3.73E+00 | 2.94E+01 | 7.96E+00 | 2.97E+01 | 2.94E+01 | |
Median | 5.00E−01 | 4.95E+00 | 3.73E+00 | 2.95E+01 | 7.91E+00 | 2.98E+01 | 2.96E+01 | |
Max | 6.00E−01 | 5.80E+00 | 4.60E+00 | 3.03E+01 | 8.82E+00 | 3.09E+01 | 3.09E+01 | |
Std | 5.34E−02 | 5.75E−01 | 4.19E−01 | 6.57E−01 | 3.96E−01 | 8.34E−01 | 1.11E+00 | |
15 | Min | 5.98E−23 | 3.48E+07 | 2.51E+07 | 1.77E+11 | 3.89E+07 | 1.58E+11 | 1.52E+11 |
Average | 9.00E−07 | 1.10E+08 | 3.92E+07 | 2.00E+11 | 9.02E+07 | 1.98E+11 | 2.01E+11 | |
Median | 1.71E−13 | 1.08E+08 | 3.94E+07 | 1.98E+11 | 8.62E+07 | 1.99E+11 | 2.02E+11 | |
Max | 2.03E−05 | 2.58E+08 | 5.53E+07 | 2.24E+11 | 1.62E+08 | 2.24E+11 | 2.28E+11 | |
Std | 4.00E−06 | 5.40E+07 | 8.46E+06 | 1.49E+10 | 3.03E+07 | 1.61E+10 | 1.86E+10 | |
16 | Min | − 1.96E+03 | 1.99E+07 | 1.29E+07 | 8.93E+10 | 2.00E+07 | 7.77E+10 | 7.54E+10 |
Average | − 1.93E+03 | 6.14E+07 | 2.22E+07 | 1.04E+11 | 3.15E+07 | 1.01E+11 | 1.03E+11 | |
Median | − 1.94E+03 | 5.21E+07 | 2.01E+07 | 1.05E+11 | 3.02E+07 | 1.03E+11 | 1.06E+11 | |
Max | − 1.86E+03 | 1.45E+08 | 3.52E+07 | 1.18E+11 | 4.37E+07 | 1.15E+11 | 1.18E+11 | |
Std | 2.94E+01 | 3.73E+07 | 6.58E+06 | 8.17E+09 | 6.65E+06 | 9.45E+09 | 1.15E+10 | |
17 | Min | 1.97E−11 | 4.26E−02 | 3.21E−02 | 3.51E+15 | 6.06E+03 | 1.17E+15 | 1.45E+15 |
Average | 7.20E−08 | 6.77E+00 | 1.34E−01 | 1.44E+17 | 1.51E+10 | 1.24E+17 | 3.22E+17 | |
Median | 3.50E−09 | 4.25E−01 | 1.17E−01 | 8.21E+16 | 3.15E+07 | 3.05E+16 | 7.71E+16 | |
Max | 1.33E−06 | 1.17E+02 | 3.57E−01 | 7.65E+17 | 2.99E+11 | 1.17E+18 | 2.11E+18 | |
Std | 2.63E−07 | 2.61E+01 | 8.34E−02 | 1.84E+17 | 6.67E+10 | 2.72E+17 | 6.05E+17 | |
18 | Min | 7.88E−03 | 6.43E−02 | 1.88E−01 | 8.56E+00 | 1.09E+00 | 9.84E+00 | 7.49E+00 |
Average | 1.44E−02 | 1.55E−01 | 2.41E−01 | 2.79E+01 | 1.95E+00 | 2.44E+01 | 2.54E+01 | |
Median | 1.37E−02 | 1.39E−01 | 2.31E−01 | 1.97E+01 | 1.85E+00 | 2.03E+01 | 1.99E+01 | |
Max | 2.44E−02 | 3.06E−01 | 3.04E−01 | 9.58E+01 | 3.36E+00 | 6.69E+01 | 5.99E+01 | |
Std | 3.69E−03 | 5.88E−02 | 3.25E−02 | 2.19E+01 | 6.32E−01 | 1.36E+01 | 1.46E+01 | |
19 | Min | 0.00E+00 | 1.32E−56 | 2.79E−15 | 8.92E−71 | 7.39E−11 | 2.12E−70 | 4.31E−70 |
Average | 0.00E+00 | 3.52E−55 | 3.04E−13 | 5.20E−69 | 3.42E−09 | 9.46E−69 | 8.69E−69 | |
Median | 0.00E+00 | 2.91E−55 | 1.33E−13 | 1.82E−69 | 2.87E−09 | 5.90E−69 | 4.19E−69 | |
Max | 0.00E+00 | 8.92E−55 | 1.44E−12 | 2.70E−68 | 1.22E−08 | 4.12E−68 | 4.24E−68 | |
Std | 0.00E+00 | 2.62E−55 | 4.08E−13 | 7.18E−69 | 3.10E−09 | 1.11E−68 | 1.07E−68 | |
20 | Min | 0.00E+00 | 0.00E+00 | 6.90E−14 | 5.75E−09 | 8.47E−12 | 4.16E−10 | 1.63E−09 |
Average | 0.00E+00 | 0.00E+00 | 5.35E−12 | 7.23E−06 | 6.56E−10 | 4.12E−06 | 3.77E−06 | |
Median | 0.00E+00 | 0.00E+00 | 6.09E−12 | 7.17E−07 | 4.39E−10 | 2.91E−07 | 1.67E−06 | |
Max | 0.00E+00 | 0.00E+00 | 1.48E−11 | 5.08E−05 | 3.45E−09 | 3.29E−05 | 2.78E−05 | |
Std | 0.00E+00 | 0.00E+00 | 4.14E−12 | 1.56E−05 | 8.42E−10 | 8.39E−06 | 6.28E−06 | |
21 | Min | 0.00E+00 | 0.00E+00 | 4.63E−13 | 1.57E−05 | 2.68E−10 | 1.98E−05 | 1.97E−05 |
Average | 4.49E−27 | 2.84E−02 | 1.37E−10 | 2.11E−03 | 1.05E−08 | 2.62E−03 | 1.50E−03 | |
Median | 0.00E+00 | 8.40E−03 | 9.98E−11 | 1.29E−03 | 6.08E−09 | 8.65E−04 | 3.67E−04 | |
Max | 1.49E−25 | 1.34E−01 | 5.87E−10 | 1.31E−02 | 7.05E−08 | 1.16E−02 | 9.15E−03 | |
Std | 2.32E−26 | 4.30E−02 | 1.49E−10 | 2.87E−03 | 1.57E−08 | 3.55E−03 | 2.21E−03 | |
22 | Min | 0.00E+00 | 0.00E+00 | 1.72E−15 | 0.00E+00 | 1.81E−10 | 0.00E+00 | 0.00E+00 |
Average | 0.00E+00 | 0.00E+00 | 1.91E−13 | 3.27E−02 | 3.42E−09 | 3.27E−02 | 6.55E−02 | |
Median | 0.00E+00 | 0.00E+00 | 8.36E−14 | 0.00E+00 | 1.58E−09 | 0.00E+00 | 0.00E+00 | |
Max | 0.00E+00 | 0.00E+00 | 1.41E−12 | 2.18E−01 | 2.73E−08 | 2.18E−01 | 2.18E−01 | |
Std | 0.00E+00 | 0.00E+00 | 3.30E−13 | 8.00E−02 | 5.95E−09 | 8.00E−02 | 1.03E−01 | |
23 | Min | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 |
Average | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 8.68E−01 | − 1.00E+00 | − 8.44E−01 | − 9.25E−01 | |
Median | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 9.28E−01 | − 1.00E+00 | − 1.00E+00 | − 9.96E−01 | |
Max | − 1.00E+00 | − 1.00E+00 | − 1.00E+00 | − 3.86E−01 | − 1.00E+00 | − 2.39E−01 | − 3.64E−01 | |
Std | 0.00E+00 | 1.52E−14 | 1.50E−14 | 1.75E−01 | 2.49E−10 | 2.64E−01 | 1.54E−01 | |
24 | Min | 0.00E+00 | 2.28E−57 | 4.77E−16 | 3.84E−71 | 2.59E−12 | 1.08E−72 | 1.68E−71 |
Average | 0.00E+00 | 2.83E−56 | 2.85E−14 | 8.03E−70 | 1.53E−10 | 7.89E−70 | 6.76E−70 | |
Median | 0.00E+00 | 2.24E−56 | 1.40E−14 | 4.32E−70 | 1.05E−10 | 4.08E−70 | 2.47E−70 | |
Max | 0.00E+00 | 7.12E−56 | 2.04E−13 | 3.99E−69 | 7.06E−10 | 3.52E−69 | 3.12E−69 | |
Std | 0.00E+00 | 2.14E−56 | 4.79E−14 | 1.02E−69 | 1.65E−10 | 1.00E−69 | 9.02E−70 | |
25 | Min | 4.44E−07 | 6.17E−02 | 1.06E−06 | 5.43E−03 | 1.27E−06 | 1.27E−02 | 7.83E−04 |
Average | 2.29E−03 | 1.39E+00 | 2.31E−02 | 1.92E+00 | 2.01E−02 | 8.24E−01 | 1.66E+00 | |
Median | 8.54E−05 | 8.35E−01 | 2.30E−02 | 7.11E−01 | 4.92E−03 | 4.27E−01 | 7.83E−01 | |
Max | 3.41E−02 | 7.87E+00 | 6.88E−02 | 6.46E+00 | 7.77E−02 | 3.98E+00 | 5.92E+00 | |
Std | 6.36E−03 | 1.93E+00 | 2.14E−02 | 2.12E+00 | 2.59E−02 | 1.10E+00 | 1.94E+00 |
Appendix Β
The aim of this Appendix is to provide a simplified Matlab code of the proposed FFO.
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Zervoudakis, K., Tsafarakis, S. A global optimizer inspired from the survival strategies of flying foxes. Engineering with Computers 39, 1583–1616 (2023). https://doi.org/10.1007/s00366-021-01554-w
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DOI: https://doi.org/10.1007/s00366-021-01554-w