1 Introduction
1.1 Meta-heuristic methods
1.2 Related work
1.3 Motivation of the work
1.4 Contribution and innovation to the work
2 The black-winged kite algorithm (BKA)
2.1 Inspiration and behavior of black-winged kites
2.2 Mathematical model and algorithm
2.2.1 Initialization phase
2.2.2 Attacking behavior
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y i, j t and y i, j t + 1 represent the position of the ith Black-winged kites in the jth dimension in the t and (t + 1)th iteration steps, respectively.
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r is a random number that ranges from 0 to 1, and p is a constant value of 0.9.
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T is the total number of iterations, and t is the number of iterations that have been completed so far.
2.2.3 Migration behavior
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L j t represents the leading scorer of the Black-winged kites in the jth dimension of the tth iteration so far.
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y i, j t and y i, j t + 1 represent the position of the ith Black-winged kites in the jth dimension in the t and (t + 1)th iteration steps, respectively.
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Fi represents the current position in the jth dimension obtained by any Black-winged kite in the t iteration.
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Fri represents the fitness value of the random position in the jth dimension obtained from any Black-winged kites in the t iteration.
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C(0,1) represents the Cauchy mutation (Jiang, et al. 2023). The definition is as follows:
2.3 The balance and diversity analyses
2.4 Computational complexity
2.4.1 Time complexity
2.4.2 Space complexity
3 Experimental results and discussion
3.1 The benchmark set and compared algorithms
3.2 Sensitivity analysis
Function | p = 0.9 | p = 0.7 | p = 0.5 | p = 0.3 |
---|---|---|---|---|
F1 | 3.27E+04 | 1.71E+08 | 5.07E+08 | 9.44E+08 |
F3 | 3.00E+02 | 3.80E+03 | 5.26E+03 | 4.76E+03 |
F4 | 4.10E+02 | 4.18E+02 | 4.52E+02 | 4.84E+02 |
F5 | 5.27E+02 | 5.39E+02 | 5.28E+02 | 5.40E+02 |
F6 | 6.17E+02 | 6.20E+02 | 6.23E+02 | 6.26E+02 |
F7 | 7.99E+02 | 7.77E+02 | 7.69E+02 | 7.88E+02 |
F8 | 8.28E+02 | 8.38E+02 | 8.45E+02 | 8.37E+02 |
F9 | 1.14E+03 | 1.22E+03 | 1.23E+03 | 1.28E+03 |
F10 | 1.80E+03 | 1.87E+03 | 1.89E+03 | 1.91E+03 |
F11 | 1.14E+03 | 1.15E+03 | 1.20E+03 | 1.18E+03 |
F12 | 2.52E+04 | 6.82E+05 | 1.14E+06 | 5.76E+05 |
F13 | 2.23E+03 | 3.91E+03 | 2.42E+03 | 4.74E+03 |
F14 | 1.48E+03 | 1.49E+03 | 1.49E+03 | 1.50E+03 |
F15 | 1.55E+03 | 1.76E+03 | 1.71E+03 | 2.21E+03 |
F16 | 1.74E+03 | 1.80E+03 | 1.79E+03 | 1.80E+03 |
F17 | 1.75E+03 | 1.76E+03 | 1.80E+03 | 1.78E+03 |
F18 | 2.29E+03 | 2.69E+03 | 9.15E+03 | 4.69E+03 |
F19 | 1.96E+03 | 1.94E+03 | 2.40E+03 | 2.00E+03 |
F20 | 2.07E+03 | 2.10E+03 | 2.08E+03 | 2.14E+03 |
F21 | 2.28E+03 | 2.28E+03 | 2.30E+03 | 2.26E+03 |
F22 | 2.35E+03 | 2.34E+03 | 2.33E+03 | 2.53E+03 |
F23 | 2.63E+03 | 2.63E+03 | 2.66E+03 | 2.65E+03 |
F24 | 2.78E+03 | 2.75E+03 | 2.78E+03 | 2.75E+03 |
F25 | 2.91E+03 | 2.93E+03 | 2.95E+03 | 2.99E+03 |
F26 | 2.99E+03 | 3.09E+03 | 3.34E+03 | 3.24E+03 |
F27 | 3.10E+03 | 3.12E+03 | 3.12E+03 | 3.13E+03 |
F28 | 3.29E+03 | 3.27E+03 | 3.33E+03 | 3.33E+03 |
F29 | 3.22E+03 | 3.26E+03 | 3.25E+03 | 3.28E+03 |
F30 | 1.12E+06 | 2.52E+05 | 1.11E+06 | 2.10E+06 |
w/t/l | 21/1/7 | 4/1/24 | 2/0/27 | 1/1/27 |
Function | p = 0.9 | p = 0.7 | p = 0.5 | p = 0.3 |
---|---|---|---|---|
F1 | 3.02E+02 | 2.57E+03 | 2.12E+03 | 2.76E+03 |
F2 | 4.03E+02 | 4.88E+02 | 4.87E+02 | 4.42E+02 |
F3 | 6.30E+02 | 6.37E+02 | 6.30E+02 | 6.29E+02 |
F4 | 8.20E+02 | 8.20E+02 | 8.22E+02 | 8.24E+02 |
F5 | 1.12E+03 | 1.15E+03 | 1.16E+03 | 1.13E+03 |
F6 | 1.94E+03 | 3.40E+03 | 2.19E+03 | 2.76E+03 |
F7 | 2.04E+03 | 2.05E+03 | 2.06E+03 | 2.05E+03 |
F8 | 2.22E+03 | 2.23E+03 | 2.23E+03 | 2.24E+03 |
F9 | 2.53E+03 | 2.58E+03 | 2.57E+03 | 2.60E+03 |
F10 | 2.67E+03 | 2.57E+03 | 2.56E+03 | 2.55E+03 |
F11 | 2.71E+03 | 2.92E+03 | 2.80E+03 | 2.70E+03 |
F12 | 2.87E+03 | 2.89E+03 | 2.87E+03 | 2.87E+03 |
w/t/l | 7/3/2 | 0/1/11 | 0/2/10 | 2/1/9 |
3.3 The results of the algorithm on different test sets
3.3.1 Evaluation of 18 functions and qualitative analysis
Name | Function | D | Range | min |
---|---|---|---|---|
Sphere |
\(F_{1} \left( x \right) = \sum\limits_{i = 1}^{D} {x_{i}^{2} }\) | 30 | [− 100, 100] | 0 |
Schwefel 2.22 |
\(\begin{gathered} F_{11} \left( x \right) = \sum\limits_{i = 1}^{\dim } {\left[ {y_{i}^{2} - 10\cos \left( {2\pi y_{i} } \right) + 10} \right]} , \hfill \\ y_{i} = \left\{ {x_{i} ,\left| {x_{i} < 0.5} \right|} \right. \hfill \\ \end{gathered}\) | 30 | [− 10, 10] | 0 |
Schwefel 1.2 |
\(F_{3} \left( x \right) = \sum\limits_{i = 1}^{D} {\left( {\sum\limits_{j = 1}^{i} {x_{j} } } \right)^{2} }\) | 30 | [− 100, 100] | 0 |
Schwefel 2.21 |
\(F_{4} \left( x \right) = \max_{i} \left\{ {\left| {x_{i} } \right|,1 \le x_{i} \le D} \right\}\) | 30 | [− 10, 10] | 0 |
Quartic |
\(F_{5} \left( x \right) = \sum\limits_{i = 1}^{Dim} {Dim \times x_{i}^{2} } + rand\left( {0,1} \right)\) | 30 | [− 1.28, 1.28] | 0 |
Sum power |
\(F_{6} = \sum\limits_{i = 1}^{Dim} {\left| {x_{i} } \right|}^{{\left( {i + 1} \right)}}\) | 30 | [− 1, 1] | 0 |
Sum squares |
\(F_{7} \left( x \right) = \sum\limits_{{i = 1}}^{{Dim}} {Dim \times x_{i}^{2} }\) | 30 | [− 10, 10] | 0 |
Zakharov |
\(F_{8} \left( x \right) = \sum\limits_{i = 1}^{dim} {x_{i}^{2} } + \left( {\sum\limits_{i = 1}^{\dim } {0.5ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{\dim } {0.5ix_{i} } } \right)^{4}\) | 30 | [− 5, 10] | 0 |
Noise |
\(F_{9} \left( x \right) = \sum\limits_{i = 1}^{D} {ix_{i}^{4} }\) | 30 | [− 1.28,1.28] | 0 |
Name | Function | D | Range | min |
---|---|---|---|---|
Rastrigin |
\(F_{10} \left( x \right) = \sum\limits_{i = 1}^{\dim } {\left[ {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right]}\) | 30 | [− 5.12, 5.12] | 0 |
NCRastrigin |
\(\begin{array}{*{20}c} {F_{{11}} \left( x \right) = \sum\limits_{{i = 1}}^{{\dim }} {\left[ {y_{i}^{2} - 10\cos \left( {2\pi y_{i} } \right) + 10} \right]} ,} \\ {y_{i} = \left\{ {\begin{array}{*{20}c} {x_{i} ,\left| {x_{i} } \right| < 0.5} \\ {round\left( {2x_{i} } \right)/2,\left| {x_{i} } \right| > 0.5} \\ \end{array} } \right.} \\ \end{array}\) | 30 | [− 5.12, 5.12] | 0 |
Ackley |
\(\begin{gathered} f_{12} \left( x \right) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{\dim }\sum\limits_{i = 1}^{\dim } {x_{i}^{2} } } } \right) + \hfill \\ \exp \left( {\frac{1}{\dim }\sum\limits_{i = 1}^{\dim } {\cos \left( {2\pi x_{i} } \right)} } \right) + 20\_\exp \left( 1 \right) \hfill \\ \end{gathered}\) | 30 | [− 50, 50] | 0 |
Griewank |
\(F_{{13}} \left( x \right) = \frac{1}{{4000}}\sum\limits_{{i = 1}}^{{\dim }} {x_{i}^{2} } - \prod\limits_{{i = 1}}^{{\dim }} {\cos \left( {\frac{{x_{i} }}{{\sqrt i }}} \right)} + 1\) | 30 | [− 600, 600] | 0 |
Alpine |
\(f_{14} \left( x \right) = \sum\limits_{i = 1}^{\dim } {\left| {x_{i} \times \sin \left( {x_{i} } \right) + 0.1x_{i} } \right|}\) | 30 | [− 10, 10] | 0 |
Weierstrass |
\(\begin{gathered} f_{15} \left( x \right) = \sum\limits_{i = 1}^{\dim } {\left( {\sum\limits_{k = 0}^{{k_{\max } }} {\left[ {a^{k} \cos \left( {2\pi b^{k} \left( {x_{i + 0.5} } \right)} \right)} \right]} } \right)} - \hfill \\ \dim \cdot \sum\limits_{k = 0}^{{k_{\max } }} {\left[ {a^{k} \cos \ge \left( {2\pi b^{k} \cdot 0.5} \right)} \right]} ,a = 0.5,b = 3,k_{\max } = 20 \hfill \\ \end{gathered}\) | 30 | [− 1, 1] | 0 |
Solomon |
\(f_{16} \left( x \right) = 1 - \cos \left( {2\pi \sqrt {\sum\limits_{i = 1}^{{Dimx_{i}^{2} }} {} } } \right) + 0.1\sqrt {\sum\limits_{i = 1}^{Dim} {x_{i}^{2} } }\) | 30 | [− 100, 100] | 0 |
Bohachevsky |
\(f_{17} \left( x \right) = \sum\limits_{i = 1}^{D - 1} {\left[ \begin{gathered} x_{i}^{2} + 2x_{i + 1}^{2} - 0.3\cos \left( {3\pi x_{i} } \right) \hfill \\ - 0.4\cos \left( {4\pi x_{i + 1} } \right) + 0.7 \hfill \\ \end{gathered} \right]}\) | 30 | [− 10, 10] | 0 |
Generalized schaffer |
\(\begin{gathered} f_{18} = 0.5 + \pi \left( {\left( {{\text{Sin}} \left( {\sum\limits_{i = 1}^{D} {x_{i}^{2} } } \right)} \right)^{2} - 0.5} \right) \times \hfill \\ \left( {1 + 0.001\left( {\sum\limits_{i = 1}^{D} {x_{i}^{2} } } \right)} \right)^{ - 2} \hfill \\ \end{gathered}\) | 30 | [− 100, 100] | 0 |
function | Index | BKA | MVO | SCA | GWO | MPA | RIME | ALO | WOA | STOA | DO |
---|---|---|---|---|---|---|---|---|---|---|---|
F1 | Avg | 9.68E−81 | 1.52E−02 | 5.25E−10 | 4.94E−56 | 2.73E−30 | 1.68E−02 | 7.19E−09 | 1.56E−74 | 1.40E−16 | 1.04E−11 |
Std | 3.06E−80 | 5.71E−03 | 1.66E−09 | 1.53E−55 | 4.88E−30 | 1.90E−02 | 2.43E−09 | 4.16E−74 | 2.31E−16 | 1.09E−11 | |
Rank | 1 | 9 | 7 | 3 | 4 | 10 | 8 | 2 | 5 | 6 | |
F2 | Avg | 2.47E−49 | 4.95E−02 | 1.93E−09 | 3.36E−33 | 4.39E−17 | 3.10E−02 | 5.01E−01 | 4.58E−51 | 1.35E−09 | 9.53E−07 |
Std | 5.52E−49 | 3.22E−02 | 2.51E−09 | 3.23E−33 | 4.32E−17 | 9.48E−03 | 7.06E−01 | 9.00E−51 | 2.38E−09 | 7.13E−07 | |
Rank | 2 | 9 | 6 | 3 | 4 | 8 | 10 | 1 | 5 | 7 | |
F3 | Avg | 4.20E−92 | 1.54E−03 | 1.29E−05 | 1.12E−26 | 1.54E−16 | 4.21E−03 | 2.44E−04 | 3.91E+00 | 1.56E−12 | 8.79E−08 |
Std | 1.33E−91 | 1.30E−03 | 2.34E−05 | 2.98E−26 | 2.47E−16 | 2.91E−03 | 5.83E−04 | 5.58E+00 | 3.12E−12 | 9.93E−08 | |
Rank | 1 | 8 | 6 | 2 | 3 | 9 | 7 | 10 | 4 | 5 | |
F4 | Avg | 1.81E−38 | 9.71E−03 | 1.67E−04 | 2.08E−19 | 1.27E−13 | 1.55E−02 | 1.74E−04 | 3.71E−01 | 1.41E−07 | 9.24E−06 |
Std | 5.71E−38 | 3.48E−03 | 2.42E−04 | 2.71E−19 | 9.35E−14 | 5.31E−03 | 1.19E−04 | 6.62E−01 | 1.20E−07 | 9.14E−06 | |
Rank | 1 | 8 | 6 | 2 | 3 | 9 | 7 | 10 | 4 | 5 | |
F5 | Avg | 3.04E−04 | 2.56E−03 | 1.54E−03 | 4.89E−04 | 7.71E−04 | 3.43E−03 | 3.64E−02 | 2.16E−03 | 2.92E−03 | 2.53E−03 |
Std | 1.83E−04 | 1.79E−03 | 8.77E−04 | 2.77E−04 | 8.52E−04 | 1.07E−03 | 2.26E−02 | 2.52E−03 | 2.50E−03 | 1.57E−03 | |
Rank | 1 | 7 | 4 | 2 | 3 | 9 | 10 | 5 | 8 | 6 | |
F6 | Avg | 4.69E−118 | 3.60E−08 | 1.12E−29 | 5.30E−121 | 2.14E−62 | 4.97E−11 | 2.85E−07 | 1.42E−109 | 2.21E−39 | 8.00E−15 |
Std | 1.48E−117 | 3.23E−08 | 3.35E−29 | 1.67E−120 | 3.96E−62 | 9.49E−11 | 1.38E−07 | 4.50E−109 | 6.99E−39 | 1.03E−14 | |
Rank | 2 | 9 | 6 | 1 | 4 | 8 | 10 | 3 | 5 | 7 | |
F7 | Avg | 5.40E−79 | 1.44E−03 | 4.22E−14 | 1.53E−58 | 1.18E−31 | 1.27E−03 | 2.04E−07 | 5.27E−82 | 1.59E−18 | 1.58E−12 |
Std | 1.70E−78 | 1.45E−03 | 8.55E−14 | 4.49E−58 | 1.58E−31 | 9.73E−04 | 2.30E−07 | 1.04E−81 | 2.61E−18 | 1.28E−12 | |
Rank | 2 | 10 | 6 | 3 | 4 | 9 | 8 | 1 | 5 | 7 | |
F8 | Avg | 1.78E−70 | 5.92E−04 | 6.09E−15 | 7.25E−57 | 3.30E−31 | 1.45E−03 | 1.82E−10 | 3.74E−80 | 2.50E−18 | 1.98E−12 |
Std | 5.61E−70 | 2.25E−04 | 1.25E−14 | 2.09E−56 | 4.05E−31 | 5.36E−04 | 1.01E−10 | 1.18E−79 | 3.91E−18 | 3.31E−12 | |
Rank | 2 | 9 | 6 | 3 | 4 | 10 | 8 | 1 | 5 | 7 | |
F9 | Avg | 3.89E−04 | 8.34E−02 | 2.22E−03 | 5.99E−04 | 4.62E−04 | 5.66E−02 | 2.06E−02 | 2.33E−03 | 2.34E−03 | 8.16E−03 |
Std | 2.54E−04 | 4.01E−02 | 2.43E−03 | 3.25E−04 | 1.66E−04 | 3.68E−02 | 1.13E−02 | 2.93E−03 | 2.19E−03 | 4.82E−03 | |
Rank | 1 | 10 | 4 | 3 | 2 | 9 | 8 | 5 | 6 | 7 | |
F10 | Avg | 0.00E+00 | 1.47E+01 | 5.31E−03 | 1.42E−15 | 0.00E+00 | 3.89E+00 | 2.33E+01 | 2.84E−15 | 2.36E+00 | 3.38E+00 |
Std | 0.00E+00 | 7.96E+00 | 1.14E−02 | 4.49E−15 | 0.00E+00 | 1.78E+00 | 8.66E+00 | 8.99E−15 | 4.22E+00 | 2.35E+00 | |
Rank | 1 | 8 | 4 | 2 | 1 | 7 | 9 | 3 | 5 | 6 | |
F11 | Avg | 0.00E+00 | 1.36E+01 | 3.97E+00 | 1.73E+00 | 2.01E−01 | 2.30E+00 | 1.93E+01 | 1.78E−16 | 3.10E+00 | 1.40E+00 |
Std | 0.00E+00 | 3.78E+00 | 5.61E+00 | 1.99E+00 | 6.32E−01 | 9.47E−01 | 6.52E+00 | 5.62E−16 | 2.33E+00 | 1.26E+00 | |
Rank | 1 | 9 | 8 | 5 | 3 | 6 | 10 | 2 | 7 | 4 | |
F12 | Avg | 4.44E−16 | 6.16E+00 | 2.00E+01 | 7.19E−15 | 5.16E−14 | 2.09E+00 | 2.31E−01 | 4.71E−15 | 2.00E+01 | 9.99E+00 |
Std | 0.00E+00 | 9.54E+00 | 1.19E−03 | 1.12E−15 | 1.11E−13 | 6.28E+00 | 4.87E−01 | 1.50E−15 | 1.89E−04 | 1.05E+01 | |
Rank | 1 | 7 | 10 | 3 | 4 | 6 | 5 | 2 | 9 | 8 | |
F13 | Avg | 0.00E+00 | 3.37E−01 | 1.11E−01 | 1.80E−02 | 0.00E+00 | 1.70E−01 | 2.47E−01 | 3.67E−02 | 7.23E−02 | 8.59E−02 |
Std | 0.00E+00 | 1.09E−01 | 2.08E−01 | 1.34E−02 | 0.00E+00 | 7.36E−02 | 9.20E−02 | 8.15E−02 | 1.03E−01 | 6.98E−02 | |
Rank | 1 | 9 | 6 | 2 | 1 | 7 | 8 | 3 | 4 | 5 | |
F14 | Avg | 2.89E−50 | 2.05E−01 | 2.23E−03 | 1.17E−04 | 4.46E−16 | 1.56E−02 | 4.80E−01 | 4.08E−01 | 4.42E−02 | 4.52E−02 |
Std | 4.25E−50 | 1.40E−01 | 6.86E−03 | 2.11E−04 | 9.52E−16 | 1.64E−02 | 6.62E−01 | 1.29E+00 | 1.40E−01 | 1.02E−01 | |
Rank | 1 | 8 | 4 | 3 | 2 | 5 | 10 | 9 | 6 | 7 | |
F15 | Avg | 0.00E+00 | 1.14E+00 | 0.00E+00 | 1.67E+00 | 0.00E+00 | 1.83E−01 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 2.58E−01 |
Std | 0.00E+00 | 4.22E−01 | 0.00E+00 | 1.60E+00 | 0.00E+00 | 1.52E−01 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 2.14E−01 | |
Rank | 1 | 3 | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | |
F16 | Avg | 1.21E−45 | 1.29E−01 | 9.95E−02 | 9.95E−02 | 9.95E−02 | 1.29E−01 | 6.96E−01 | 1.29E−01 | 9.95E−02 | 1.29E−01 |
Std | 3.82E−45 | 9.44E−02 | 2.04E−06 | 4.94E−10 | 5.19E−17 | 9.44E−02 | 2.57E−01 | 9.44E−02 | 4.43E−08 | 9.44E−02 | |
Rank | 1 | 3 | 2 | 2 | 2 | 3 | 4 | 3 | 2 | 3 | |
F17 | Avg | 0.00E+00 | 1.79E−01 | 3.86E−14 | 0.00E+00 | 0.00E+00 | 4.59E−04 | 1.19E+00 | 0.00E+00 | 0.00E+00 | 4.65E−13 |
Std | 0.00E+00 | 4.25E−01 | 1.17E−13 | 0.00E+00 | 0.00E+00 | 2.54E−04 | 1.07E+00 | 0.00E+00 | 0.00E+00 | 7.95E−13 | |
Rank | 1 | 5 | 2 | 1 | 1 | 4 | 1548nudb246789 | 1 | 1 | 3 | |
F18 | Avg | 3.98E−01 | 3.98E−01 | 4.00E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 |
Std | 0.00E+00 | 9.55E−07 | 1.89E−03 | 2.95E−05 | 5.17E−15 | 9.99E−07 | 6.58E−14 | 8.18E−06 | 3.43E−04 | 5.09E−11 | |
Rank | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
Average Ranking | 1.22 | 7.33 | 5.00 | 2.50 | 2.61 | 6.78 | 7.29 | 3.50 | 4.61 | 5.33 | |
Final Ranking | 1 | 10 | 6 | 2 | 3 | 8 | 9 | 4 | 5 | 7 |
3.3.2 Evaluation of the CEC-2017 suite test
Index | BKA | GJO | PSO | AVOA | SHO | SCSO | SSA | AO | COA | |
---|---|---|---|---|---|---|---|---|---|---|
F1 | Avg | 3.27E+04 | 3.16E+08 | 8.47E+05 | 3.46E+03 | 9.75E+08 | 1.15E+08 | 2.19E+10 | 7.55E+05 | 8.44E+09 |
Std | 1.13E+04 | 1.95E+08 | 3.01E+05 | 4.39E+03 | 1.10E+09 | 3.06E+08 | 8.97E+07 | 5.27E+05 | 3.49E+09 | |
F3 | Avg | 3.00E+02 | 7.93E+03 | 3.03E+02 | 3.00E+02 | 9.35E+03 | 5.10E+03 | 3.40E+04 | 4.08E+03 | 2.16E+04 |
Std | 1.87E−01 | 4.42E+03 | 5.91E−01 | 4.10E−06 | 4.39E+03 | 5.56E+03 | 3.35E+02 | 2.55E+03 | 4.28E+03 | |
F4 | Avg | 4.10E+02 | 4.32E+02 | 4.10E+02 | 4.18E+02 | 4.38E+02 | 4.34E+02 | 4.81E+03 | 4.33E+02 | 1.34E+03 |
Std | 1.82E+01 | 1.07E+01 | 6.77E+00 | 1.48E+01 | 1.22E+01 | 1.43E+01 | 1.03E+02 | 2.75E+01 | 2.98E+02 | |
F5 | Avg | 5.27E+02 | 5.33E+02 | 5.38E+02 | 5.35E+02 | 5.38E+02 | 5.35E+02 | 6.23E+02 | 5.28E+02 | 5.72E+02 |
Std | 1.58E+01 | 4.25E+00 | 1.35E+01 | 1.13E+01 | 8.46E+00 | 1.33E+01 | 4.11E+00 | 1.32E+01 | 1.03E+01 | |
F6 | Avg | 6.17E+02 | 6.04E+02 | 6.14E+02 | 6.11E+02 | 6.12E+02 | 6.19E+02 | 6.48E+02 | 6.13E+02 | 6.33E+02 |
Std | 5.58E+00 | 2.13E+00 | 6.23E+00 | 7.66E+00 | 5.62E+00 | 1.07E+01 | 1.12E+00 | 5.75E+00 | 4.45E+00 | |
F7 | Avg | 7.99E+02 | 7.54E+02 | 7.89E+02 | 7.68E+02 | 7.90E+02 | 7.60E+02 | 1.00E+03 | 7.62E+02 | 8.54E+02 |
Std | 3.77E+01 | 2.69E+01 | 3.13E+01 | 2.36E+01 | 1.95E+01 | 1.82E+01 | 8.43E+00 | 2.08E+01 | 3.06E+01 | |
F8 | Avg | 8.28E+02 | 8.31E+02 | 8.46E+02 | 8.57E+02 | 8.30E+02 | 8.31E+02 | 9.54E+02 | 8.31E+02 | 8.76E+02 |
Std | 5.64E+00 | 1.11E+01 | 2.12E+01 | 1.62E+01 | 1.00E+01 | 1.08E+01 | 1.01E+00 | 8.97E+00 | 1.19E+01 | |
F9 | Avg | 1.14E+03 | 9.34E+02 | 1.25E+03 | 1.16E+03 | 1.10E+03 | 1.09E+03 | 3.43E+03 | 1.00E+03 | 1.68E+03 |
Std | 1.16E+02 | 5.55E+01 | 1.62E+02 | 2.97E+02 | 1.93E+02 | 1.65E+02 | 3.67E+02 | 5.73E+01 | 2.58E+02 | |
F10 | Avg | 1.80E+03 | 1.62E+03 | 1.99E+03 | 1.98E+03 | 1.53E+03 | 1.85E+03 | 4.56E+03 | 1.79E+03 | 2.62E+03 |
Std | 3.17E+02 | 3.12E+02 | 4.36E+02 | 2.79E+02 | 2.35E+02 | 2.91E+02 | 5.66E+01 | 2.70E+02 | 1.94E+02 | |
F11 | Avg | 1.14E+03 | 1.18E+03 | 1.14E+03 | 1.16E+03 | 1.14E+03 | 1.16E+03 | 6.49E+04 | 1.19E+03 | 1.83E+03 |
Std | 3.06E+01 | 5.63E+01 | 1.16E+01 | 6.96E+01 | 1.01E+01 | 5.09E+01 | 9.19E+02 | 6.53E+01 | 6.11E+02 | |
F12 | Avg | 2.52E+04 | 5.53E+05 | 4.63E+05 | 1.20E+06 | 7.19E+05 | 2.53E+05 | 3.12E+09 | 4.96E+06 | 2.22E+08 |
Std | 3.85E+04 | 7.60E+05 | 4.57E+05 | 1.53E+06 | 5.95E+05 | 2.98E+05 | 3.78E+07 | 6.30E+06 | 2.00E+08 | |
F13 | Avg | 2.23E+03 | 1.49E+04 | 1.67E+04 | 7.15E+03 | 9.06E+03 | 1.04E+04 | 1.73E+09 | 1.90E+04 | 5.13E+05 |
Std | 9.92E+02 | 8.74E+03 | 1.14E+04 | 6.76E+03 | 4.88E+03 | 7.76E+03 | 1.72E+07 | 9.31E+03 | 5.43E+05 | |
F14 | Avg | 1.48E+03 | 2.95E+03 | 2.26E+03 | 2.21E+03 | 3.98E+03 | 3.65E+03 | 1.69E+03 | 2.64E+03 | 1.57E+03 |
Std | 3.17E+01 | 1.87E+03 | 1.17E+03 | 1.06E+03 | 1.47E+03 | 1.89E+03 | 8.11E+00 | 1.28E+03 | 7.75E+01 | |
F15 | Avg | 1.55E+03 | 4.00E+03 | 4.62E+03 | 3.37E+03 | 2.68E+03 | 4.12E+03 | 1.23E+04 | 6.24E+03 | 8.05E+03 |
Std | 1.89E+01 | 1.07E+03 | 2.29E+03 | 1.53E+03 | 1.17E+03 | 1.74E+03 | 1.81E+02 | 1.32E+03 | 3.63E+03 | |
F16 | Avg | 1.74E+03 | 1.81E+03 | 1.84E+03 | 1.83E+03 | 1.80E+03 | 1.85E+03 | 3.13E+03 | 1.77E+03 | 2.11E+03 |
Std | 1.17E+02 | 1.64E+02 | 1.11E+02 | 1.46E+02 | 9.80E+01 | 1.50E+02 | 3.95E+01 | 1.27E+02 | 9.87E+01 | |
F17 | Avg | 1.75E+03 | 1.77E+03 | 1.77E+03 | 1.76E+03 | 1.75E+03 | 1.76E+03 | 2.95E+03 | 1.77E+03 | 1.82E+03 |
Std | 1.75E+01 | 2.24E+01 | 4.05E+01 | 1.52E+01 | 1.29E+01 | 1.64E+01 | 1.24E+02 | 1.95E+01 | 3.01E+01 | |
F18 | Avg | 2.29E+03 | 3.23E+04 | 1.94E+04 | 9.90E+03 | 1.89E+04 | 2.12E+04 | 9.67E+09 | 2.80E+04 | 2.83E+06 |
Std | 4.63E+02 | 6.42E+03 | 1.63E+04 | 7.69E+03 | 8.35E+03 | 1.58E+04 | 5.30E+07 | 1.39E+04 | 8.73E+06 | |
F19 | Avg | 1.96E+03 | 3.44E+04 | 4.28E+03 | 7.01E+03 | 8.67E+03 | 3.40E+04 | 3.96E+08 | 9.45E+03 | 1.54E+04 |
Std | 4.36E+01 | 8.16E+04 | 1.99E+03 | 7.67E+03 | 5.03E+03 | 8.03E+04 | 1.20E+07 | 9.58E+03 | 2.68E+04 | |
F20 | Avg | 2.07E+03 | 2.13E+03 | 2.17E+03 | 2.12E+03 | 2.07E+03 | 2.12E+03 | 2.99E+03 | 2.10E+03 | 2.23E+03 |
Std | 2.33E+01 | 5.90E+01 | 6.28E+01 | 7.40E+01 | 5.95E+01 | 4.57E+01 | 1.52E+02 | 5.81E+01 | 7.81E+01 | |
F21 | Avg | 2.28E+03 | 2.32E+03 | 2.34E+03 | 2.28E+03 | 2.32E+03 | 2.28E+03 | 2.79E+03 | 2.30E+03 | 2.37E+03 |
Std | 6.29E+01 | 3.88E+01 | 5.37E+01 | 7.68E+01 | 4.15E+01 | 6.29E+01 | 1.49E+01 | 4.64E+01 | 3.99E+01 | |
F22 | Avg | 2.35E+03 | 2.34E+03 | 2.55E+03 | 2.31E+03 | 2.41E+03 | 2.31E+03 | 5.14E+03 | 2.31E+03 | 3.07E+03 |
Std | 1.45E+02 | 4.13E+01 | 5.69E+02 | 5.30E+00 | 1.26E+02 | 9.35E+00 | 2.11E+01 | 3.02E+00 | 3.10E+02 | |
F23 | Avg | 2.63E+03 | 2.63E+03 | 2.70E+03 | 2.64E+03 | 2.66E+03 | 2.64E+03 | 4.01E+03 | 2.63E+03 | 2.71E+03 |
Std | 1.98E+01 | 1.26E+01 | 5.30E+01 | 2.13E+01 | 1.82E+01 | 1.07E+01 | 3.71E+01 | 1.22E+01 | 3.19E+01 | |
F24 | Avg | 2.78E+03 | 2.77E+03 | 2.88E+03 | 2.72E+03 | 2.73E+03 | 2.75E+03 | 3.38E+03 | 2.74E+03 | 2.85E+03 |
Std | 2.43E+01 | 1.04E+01 | 3.34E+01 | 1.19E+02 | 1.17E+02 | 6.91E+01 | 1.90E+00 | 8.63E+01 | 5.74E+01 | |
F25 | Avg | 2.91E+03 | 2.96E+03 | 2.93E+03 | 2.94E+03 | 2.94E+03 | 2.94E+03 | 4.73E+03 | 2.93E+03 | 3.52E+03 |
Std | 1.87E+01 | 4.36E+01 | 2.35E+01 | 2.15E+01 | 2.99E+01 | 1.76E+01 | 5.09E+00 | 2.47E+01 | 2.05E+02 | |
F26 | Avg | 2.99E+03 | 3.04E+03 | 3.63E+03 | 3.22E+03 | 3.44E+03 | 3.06E+03 | 5.54E+03 | 3.01E+03 | 4.00E+03 |
Std | 2.12E+02 | 9.99E+01 | 6.53E+02 | 4.93E+02 | 4.41E+02 | 1.23E+02 | 1.50E+02 | 1.80E+02 | 2.57E+02 | |
F27 | Avg | 3.10E+03 | 3.11E+03 | 3.21E+03 | 3.10E+03 | 3.14E+03 | 3.12E+03 | 4.82E+03 | 3.10E+03 | 3.20E+03 |
Std | 1.07E+01 | 2.90E+01 | 5.54E+01 | 1.65E+01 | 2.93E+01 | 3.88E+01 | 3.03E+01 | 5.57E+00 | 2.81E+01 | |
F28 | Avg | 3.29E+03 | 3.36E+03 | 3.29E+03 | 3.38E+03 | 3.47E+03 | 3.34E+03 | 3.94E+03 | 3.40E+03 | 3.77E+03 |
Std | 1.14E+02 | 9.94E+01 | 1.26E+02 | 9.86E+01 | 1.71E+02 | 1.29E+02 | 3.74E+00 | 8.24E+01 | 1.13E+02 | |
F29 | Avg | 3.22E+03 | 3.22E+03 | 3.34E+03 | 3.29E+03 | 3.24E+03 | 3.24E+03 | 4.19E+03 | 3.24E+03 | 3.42E+03 |
Std | 5.33E+01 | 4.70E+01 | 1.46E+02 | 5.01E+01 | 4.78E+01 | 4.34E+01 | 4.83E+01 | 5.19E+01 | 1.13E+02 | |
F30 | Avg | 1.12E+06 | 7.62E+05 | 3.88E+05 | 2.96E+05 | 1.45E+06 | 2.03E+06 | 1.94E+08 | 2.93E+05 | 5.41E+06 |
Std | 1.44E+06 | 8.59E+05 | 4.59E+05 | 3.63E+05 | 1.77E+06 | 3.48E+06 | 7.35E+06 | 5.11E+05 | 6.00E+06 |
3.3.3 Evaluation of the CEC-2022 objective functions
Index | BKA | GJO | PSO | AVOA | SHO | SCSO | SSA | AO | COA | |
---|---|---|---|---|---|---|---|---|---|---|
F1 | Avg | 3.02E+02 | 4.27E+03 | 3.02E+02 | 3.07E+02 | 3.86E+03 | 2.25E+03 | 1.07E+04 | 1.92E+03 | 8.05E+03 |
Std | 5.22E+01 | 1.83E+03 | 5.71E-01 | 1.58E+01 | 2.16E+03 | 2.11E+03 | 9.23E+01 | 1.19E+03 | 1.98E+03 | |
F2 | Avg | 4.03E+02 | 4.41E+02 | 4.03E+02 | 4.35E+02 | 4.44E+02 | 4.28E+02 | 7.61E+03 | 4.79E+02 | 1.51E+03 |
Std | 1.17E+02 | 2.39E+01 | 3.63E+00 | 3.59E+01 | 2.82E+01 | 3.43E+01 | 5.95E+01 | 1.16E+02 | 6.83E+02 | |
F3 | Avg | 6.30E+02 | 6.11E+02 | 6.29E+02 | 6.21E+02 | 6.12E+02 | 6.19E+02 | 7.00E+02 | 6.13E+02 | 6.50E+02 |
Std | 9.79E+01 | 7.89E+00 | 1.52E+01 | 1.38E+01 | 5.45E+00 | 1.56E+01 | 4.78E+00 | 6.33E+00 | 9.71E+00 | |
F4 | Avg | 8.20E+02 | 8.30E+02 | 8.25E+02 | 8.30E+02 | 8.24E+02 | 8.28E+02 | 9.02E+02 | 8.22E+02 | 8.50E+02 |
Std | 1.65E+02 | 1.04E+01 | 1.30E+01 | 7.41E+00 | 4.74E+00 | 6.34E+00 | 1.38E+00 | 4.76E+00 | 7.95E+00 | |
F5 | Avg | 1.12E+03 | 1.02E+03 | 1.16E+03 | 1.27E+03 | 1.15E+03 | 1.18E+03 | 2.69E+03 | 1.09E+03 | 1.37E+03 |
Std | 4.31E+02 | 1.06E+02 | 1.51E+02 | 1.57E+02 | 1.31E+02 | 2.15E+02 | 2.57E+02 | 1.71E+02 | 1.35E+02 | |
F6 | Avg | 1.94E+03 | 8.83E+03 | 7.84E+03 | 2.97E+03 | 4.27E+03 | 5.15E+03 | 2.18E+08 | 1.53E+04 | 1.44E+07 |
Std | 6.64E+01 | 2.26E+03 | 2.79E+03 | 1.18E+03 | 1.51E+03 | 2.20E+03 | 1.77E+07 | 1.09E+04 | 1.52E+07 | |
F7 | Avg | 2.04E+03 | 2.04E+03 | 2.06E+03 | 2.04E+03 | 2.04E+03 | 2.06E+03 | 2.72E+03 | 2.04E+03 | 2.10E+03 |
Std | 9.28E+01 | 1.95E+01 | 2.71E+01 | 7.57E+00 | 2.36E+01 | 2.28E+01 | 2.39E+02 | 1.02E+01 | 1.45E+01 | |
F8 | Avg | 2.22E+03 | 2.23E+03 | 2.23E+03 | 2.23E+03 | 2.22E+03 | 2.23E+03 | 2.99E+03 | 2.23E+03 | 2.24E+03 |
Std | 1.57E+02 | 4.10E+00 | 2.51E+00 | 2.60E+00 | 3.10E+00 | 3.30E+00 | 5.80E+01 | 7.31E+00 | 1.72E+01 | |
F9 | Avg | 2.53E+03 | 2.60E+03 | 2.53E+03 | 2.56E+03 | 2.60E+03 | 2.56E+03 | 3.06E+03 | 2.57E+03 | 2.75E+03 |
Std | 1.92E+02 | 4.83E+01 | 1.80E-03 | 6.20E+01 | 1.58E+01 | 3.11E+01 | 6.07E+00 | 2.95E+01 | 4.37E+01 | |
F10 | Avg | 2.67E+03 | 2.60E+03 | 2.56E+03 | 2.57E+03 | 2.55E+03 | 2.54E+03 | 5.18E+03 | 2.56E+03 | 2.72E+03 |
Std | 2.36E+02 | 5.31E+01 | 7.29E+01 | 6.74E+01 | 6.03E+01 | 6.33E+01 | 5.02E+01 | 6.56E+01 | 1.99E+02 | |
F11 | Avg | 2.71E+03 | 2.89E+03 | 2.69E+03 | 2.76E+03 | 2.96E+03 | 2.98E+03 | 5.13E+03 | 2.67E+03 | 3.89E+03 |
Std | 1.75E+02 | 2.60E+02 | 1.28E+02 | 1.59E+02 | 2.52E+02 | 2.22E+02 | 7.62E+00 | 1.04E+02 | 3.92E+02 | |
F12 | Avg | 2.87E+03 | 2.87E+03 | 2.94E+03 | 2.87E+03 | 2.90E+03 | 2.87E+03 | 4.73E+03 | 2.87E+03 | 2.98E+03 |
Std | 5.49E+00 | 7.26E+00 | 6.36E+01 | 2.66E+00 | 3.64E+01 | 9.90E+00 | 2.68E+01 | 1.90E+00 | 9.10E+01 |
3.4 Nonparametric statistical analysis
Function | BKA vs GJO | BKA vs PSO | BKA vs AVOA | BKA vs SHO | BKA vs SCSO | BKA vs SSA | BKA vs AO | BKA vs COA |
F1 | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(−) | 1.83E−04(+) | 5.39E−02(=) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F3 | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(−) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F4 | 2.83E−03(+) | 1.04E−01(=) | 1.76E−01(=) | 2.20E−03(+) | 2.20E−03(+) | 1.83E−04(+) | 3.61E−03(+) | 1.83E−04(+) |
F5 | 1.86E−01(=) | 1.62E−01(=) | 3.07E−01(=) | 1.21E−01(=) | 1.86E−01(=) | 1.83E−04(+) | 8.50E−01(=) | 1.83E−04(+) |
F6 | 7.69E−04(−) | 2.73E−01(=) | 6.40E−02(=) | 5.39E−02(=) | 9.10E−01(=) | 1.83E−04(+) | 1.62E−01(=) | 1.83E−04(+) |
F7 | 5.80E−03(−) | 4.73E−01(=) | 1.04E−01(=) | 9.10E−01(=) | 9.11E−03(−) | 1.83E−04(+) | 1.73E−02(−) | 5.80E−03(+) |
F8 | 5.21E−01(=) | 3.12E−02(+) | 3.30E−04(+) | 6.23E−01(=) | 3.07E−01(=) | 1.83E−04(+) | 3.85E−01(=) | 1.83E−04(+) |
F9 | 4.40E−04(−) | 1.04E−01(=) | 4.27E−01(=) | 3.07E−01(=) | 2.41E−01(=) | 1.83E−04(+) | 3.61E−03(−) | 2.46E−04(+) |
F10 | 2.73E−01(=) | 3.85E−01(=) | 2.73E−01(=) | 3.76E−02(−) | 5.21E−01(=) | 1.83E−04(+) | 8.50E−01(=) | 4.40E−04(+) |
F11 | 7.57E−02(=) | 8.50E−01(=) | 9.70E−01(=) | 7.34E−01(=) | 6.78E−01(=) | 1.83E−04(+) | 7.57E−02(=) | 1.83E−04(+) |
F12 | 1.31E−03(+) | 5.83E−04(+) | 2.83E−03(+) | 1.01E−03(+) | 1.31E−03(+) | 1.83E−04(+) | 3.30E−04(+) | 1.83E−04(+) |
F13 | 1.83E−04(+) | 1.31E−03(+) | 7.28E−03(+) | 2.46E−04(+) | 5.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F14 | 2.57E−02(+) | 3.12E−02(+) | 2.20E−03(+) | 7.69E−04(+) | 5.80E−03(+) | 1.83E−04(+) | 1.83E−04(+) | 7.28E−03(+) |
F15 | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 5.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F16 | 3.07E−01(=) | 1.62E−01(=) | 2.73E−01(=) | 2.41E−01(=) | 8.90E−02(=) | 1.83E−04(+) | 5.71E−01(=) | 1.83E−04(+) |
F17 | 2.57E−02(+) | 1.62E−01(=) | 1.40E−01(=) | 9.70E−01(=) | 8.90E−02(=) | 1.83E−04(+) | 1.73E−02(+) | 2.46E−04(+) |
F18 | 1.83E−04(+) | 3.30E−04(+) | 2.46E−04(+) | 2.46E−04(+) | 1.31E−03(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F19 | 4.59E−03(+) | 1.83E−04(+) | 1.83E−04(+) | 2.83E−03(+) | 2.83E−03(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F20 | 1.73E−02(+) | 5.83E−04(+) | 1.86E−01(=) | 1.04E−01(=) | 2.11E−02(+) | 1.83E−04(+) | 2.41E−01(=) | 5.83E−04(+) |
F21 | 4.73E−01(=) | 3.76E−02(+) | 7.91E−01(=) | 1.40E−01(=) | 1.00E+00(=) | 1.83E−04(+) | 1.00E+00(=) | 5.80E−03(+) |
F22 | 3.76E−02(+) | 1.86E−01(=) | 7.34E−01(=) | 1.13E−02(+) | 4.73E−01(=) | 1.83E−04(+) | 2.73E−01(=) | 3.30E−04(+) |
F23 | 4.73E−01(=) | 5.83E−04(+) | 2.41E−01(=) | 2.83E−03(+) | 7.57E−02(=) | 1.83E−04(+) | 1.86E−01(=) | 3.30E−04(+) |
F24 | 6.23E−01(=) | 1.83E−04(+) | 4.27E−01(=) | 1.00E+00(=) | 2.41E−01(=) | 1.83E−04(+) | 2.12E−01(=) | 1.31E−03(+) |
F25 | 2.83E−03(+) | 6.40E−02(=) | 4.59E−03(+) | 5.80E−03(+) | 2.20E−03(+) | 1.83E−04(+) | 1.40E−02(+) | 1.83E−04(+) |
F26 | 2.12E−01(=) | 1.40E−02(+) | 2.41E−01(=) | 1.13E−02(+) | 1.86E−01(=) | 1.83E−04(+) | 6.23E−01(=) | 1.83E−04(+) |
F27 | 2.41E−01(=) | 3.30E−04(+) | 1.00E+00(=) | 4.40E−04(+) | 9.70E−01(=) | 1.83E−04(+) | 1.86E−01(=) | 1.83E−04(+) |
F28 | 3.12E−02(+) | 4.73E−01(=) | 1.72E−02(+) | 1.73E−02(+) | 1.86E−01(=) | 1.83E−04(+) | 1.71E−03(+) | 1.83E−04(+) |
F29 | 9.70E−01(=) | 4.52E−02(+) | 1.13E−02(+) | 2.41E−01(=) | 3.85E−01(=) | 1.83E−04(+) | 5.21E−01(=) | 1.83E−04(+) |
F30 | 6.23E−01(=) | 7.33E−01(=) | 5.20E−01(=) | 6.77E−01(=) | 4.27E−01(=) | 1.79E−04(+) | 6.77E−01(=) | 1.12E−02(+) |
+/=/− | 14/12/3 | 16/13/0 | 10/17/2 | 15/13/1 | 10/18/1 | 29/0/0 | 12/15/2 | 29/0/0 |
Function | BKA vs GJO | BKA vs PSO | BKA vs AVOA | BKA vs SHO | BKA vs SCSO | BKA vs SSA | BKA vs AO | BKA vs COA |
---|---|---|---|---|---|---|---|---|
F1 | 1.83E−04(+) | 2.83E−03(+) | 4.52E−02(−) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F2 | 5.83E−04(+) | 5.71E−01(=) | 9.08E−03(+) | 4.59E−03(+) | 1.73E−02(+) | 1.83E−04(+) | 3.30E−04(+) | 1.83E−04(+) |
F3 | 7.69E−04(−) | 5.71E−01(=) | 8.90E−02(=) | 7.69E−04(+) | 6.40E−02(=) | 1.83E−04(+) | 7.69E−04(−) | 1.01E−03(+) |
F4 | 4.52E−02(+) | 3.07E−01(=) | 2.11E−02(+) | 2.12E−01(=) | 2.57E−02(+) | 1.83E−04(+) | 1.86E−01(=) | 1.83E−04(+) |
F5 | 2.11E−02(−) | 7.91E−01(=) | 3.12E−02(+) | 6.78E−01(=) | 9.70E−01(=) | 1.83E−04(+) | 3.07E−01(=) | 7.69E−04(+) |
F6 | 1.83E−04(+) | 1.83E−04(+) | 1.40E−02(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F7 | 7.34E−01(=) | 1.04E−01(=) | 2.41E−01(=) | 4.73E−01(=) | 2.12E−01(=) | 1.83E−04(+) | 3.85E−01(=) | 1.83E−04(+) |
F8 | 4.52E−02(+) | 1.13E−02(+) | 4.27E−01(=) | 9.10E−01(=) | 1.86E−01(=) | 1.83E−04(+) | 4.52E−02(+) | 5.83E−04(+) |
F9 | 1.83E−04(+) | 1.01E−03(+) | 1.04E−01(=) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) | 1.83E−04(+) |
F10 | 9.10E−01(=) | 1.04E−01(=) | 9.10E−01(=) | 5.21E−01(=) | 3.07E−01(=) | 1.83E−04(+) | 8.50E−01(=) | 1.04E−01(=) |
F11 | 1.13E−02(+) | 2.11E−02(+) | 2.73E−01(=) | 7.28E−03(+) | 2.11E−02(+) | 1.83E−04(+) | 2.57E−02(+) | 1.83E−04(+) |
F12 | 6.78E−01(=) | 2.46E−04(+) | 7.34E−01(=) | 4.40E−04(+) | 2.73E−01(=) | 1.83E−04(+) | 4.73E−01(=) | 1.83E−04(+) |
+/=/− | 7/3/2 | 6/6/0 | 4/7/1 | 5/7/0 | 6/6/0 | 12/0/0 | 6/5/1 | 11/1/0 |
Function | BKA | GJO | PSO | AVOA | SHO | SCSO | SSA | AO | COA |
---|---|---|---|---|---|---|---|---|---|
F1 | 2.20 | 6.00 | 4.40 | 1.00 | 6.60 | 3.70 | 9.00 | 4.10 | 8.00 |
F3 | 2.00 | 6.30 | 3.00 | 1.00 | 6.20 | 4.60 | 9.00 | 4.90 | 8.00 |
F4 | 2.00 | 5.30 | 2.20 | 2.70 | 5.60 | 5.50 | 9.00 | 4.70 | 8.00 |
F5 | 2.90 | 3.70 | 4.40 | 4.20 | 4.70 | 4.70 | 9.00 | 3.40 | 8.00 |
F6 | 5.20 | 1.70 | 4.30 | 3.20 | 4.20 | 5.40 | 9.00 | 4.10 | 7.90 |
F7 | 5.30 | 2.40 | 5.00 | 3.20 | 5.60 | 3.40 | 9.00 | 3.30 | 7.80 |
F8 | 2.80 | 3.60 | 5.00 | 6.60 | 3.60 | 3.30 | 9.00 | 3.60 | 7.50 |
F9 | 5.00 | 1.80 | 5.60 | 4.00 | 4.40 | 4.40 | 9.00 | 3.20 | 7.60 |
F10 | 4.20 | 3.10 | 4.70 | 5.40 | 2.20 | 4.60 | 9.00 | 3.90 | 7.90 |
F11 | 3.60 | 5.20 | 3.20 | 3.50 | 3.40 | 3.90 | 9.00 | 5.20 | 8.00 |
F12 | 1.60 | 3.90 | 4.60 | 4.30 | 4.90 | 3.30 | 9.00 | 5.40 | 8.00 |
F13 | 1.10 | 5.40 | 5.40 | 3.10 | 4.10 | 4.00 | 9.00 | 5.70 | 7.20 |
F14 | 1.60 | 5.00 | 4.30 | 4.70 | 7.00 | 6.50 | 5.40 | 6.60 | 3.90 |
F15 | 1.10 | 4.90 | 5.00 | 3.60 | 3.10 | 4.70 | 9.00 | 6.70 | 6.90 |
F16 | 3.20 | 4.30 | 4.60 | 4.20 | 4.30 | 4.60 | 9.00 | 3.10 | 7.70 |
F17 | 2.90 | 5.40 | 4.30 | 3.80 | 2.80 | 4.30 | 9.00 | 4.80 | 7.70 |
F18 | 1.10 | 6.50 | 4.10 | 3.10 | 4.40 | 4.40 | 9.00 | 6.00 | 6.40 |
F19 | 1.30 | 5.00 | 3.90 | 4.50 | 5.30 | 5.70 | 9.00 | 5.40 | 4.90 |
F20 | 2.70 | 4.90 | 5.50 | 4.50 | 2.50 | 4.30 | 9.00 | 4.30 | 7.30 |
F21 | 3.50 | 4.50 | 5.50 | 4.00 | 4.60 | 4.10 | 9.00 | 3.20 | 6.60 |
F22 | 2.90 | 5.40 | 3.60 | 2.90 | 6.30 | 3.50 | 9.00 | 3.50 | 7.90 |
F23 | 2.40 | 2.60 | 7.30 | 3.80 | 5.50 | 3.70 | 9.00 | 3.20 | 7.50 |
F24 | 4.00 | 3.40 | 7.60 | 3.70 | 4.30 | 2.90 | 9.00 | 3.20 | 6.90 |
F25 | 1.60 | 4.70 | 3.30 | 4.90 | 4.20 | 4.90 | 9.00 | 4.40 | 8.00 |
F26 | 2.80 | 3.50 | 5.90 | 4.30 | 5.40 | 3.70 | 9.00 | 3.10 | 7.30 |
F27 | 2.70 | 2.70 | 7.00 | 3.20 | 5.70 | 3.50 | 9.00 | 3.90 | 7.30 |
F28 | 2.50 | 4.00 | 2.80 | 3.90 | 5.50 | 4.20 | 9.00 | 5.20 | 7.90 |
F29 | 3.20 | 2.60 | 5.60 | 5.40 | 4.10 | 3.90 | 9.00 | 3.80 | 7.40 |
F30 | 4.00 | 4.50 | 3.60 | 3.00 | 5.50 | 5.00 | 9.00 | 3.00 | 7.40 |
Average | 2.81 | 4.22 | 4.68 | 3.78 | 4.69 | 4.30 | 8.88 | 4.31 | 7.34 |
Ranking | 1 | 4 | 6 | 2 | 7 | 3 | 9 | 5 | 8 |
Function | BKA | GJO | PSO | AVOA | SHO | SCSO | SSA | AO | COA |
---|---|---|---|---|---|---|---|---|---|
F1 | 1.90 | 6.40 | 2.70 | 1.40 | 5.70 | 5.00 | 8.90 | 5.00 | 8.00 |
F2 | 2.10 | 5.40 | 1.90 | 4.50 | 4.90 | 4.30 | 9.00 | 4.90 | 8.00 |
F3 | 5.70 | 2.40 | 5.90 | 4.60 | 2.80 | 4.00 | 9.00 | 2.90 | 7.70 |
F4 | 2.70 | 5.40 | 4.10 | 5.10 | 3.70 | 4.70 | 9.00 | 2.60 | 7.70 |
F5 | 4.10 | 2.60 | 4.40 | 6.30 | 4.20 | 4.00 | 9.00 | 3.30 | 7.10 |
F6 | 1.20 | 6.00 | 5.40 | 2.50 | 3.10 | 3.80 | 9.00 | 6.00 | 8.00 |
F7 | 4.00 | 3.80 | 5.80 | 3.10 | 3.40 | 4.80 | 9.00 | 3.50 | 7.60 |
F8 | 2.90 | 4.20 | 5.60 | 3.50 | 2.80 | 4.10 | 9.00 | 5.40 | 7.50 |
F9 | 1.90 | 5.60 | 2.70 | 2.20 | 6.20 | 4.40 | 9.00 | 5.00 | 8.00 |
F10 | 4.60 | 5.30 | 3.20 | 4.60 | 3.70 | 3.40 | 9.00 | 4.70 | 6.50 |
F11 | 2.30 | 4.70 | 3.60 | 3.60 | 5.60 | 5.30 | 9.00 | 2.90 | 8.00 |
F12 | 3.00 | 2.90 | 7.10 | 3.20 | 6.00 | 3.30 | 9.00 | 2.90 | 7.60 |
Average | 3.03 | 4.56 | 4.37 | 3.72 | 4.34 | 4.26 | 8.99 | 4.09 | 7.64 |
Ranking | 1 | 7 | 6 | 2 | 5 | 4 | 9 | 3 | 8 |
3.5 Effectiveness analysis
BKA (w/t/l) | GJO (w/t/l) | PSO (w/t/l) | AVOA (w/t/l) | SHO (w/t/l) | SCSO (w/t/l) | SSA (w/t/l) | AO (w/t/l) | COA (w/t/l) | |
---|---|---|---|---|---|---|---|---|---|
CEC-2017 | 11/10/8 | 3/2/24 | 0/3/26 | 3/3/23 | 1/3/25 | 0/1/28 | 0/0/29 | 0/2/27 | 0/0/29 |
CEC-2022 | 2/6/4 | 2/2/8 | 0/3/9 | 0/2/10 | 0/2/10 | 1/1/10 | 0/0/12 | 1/2/9 | 0/0/12 |
Total | 13/16/12 | 5/4/32 | 0/6/35 | 3/5/33 | 1/5/35 | 1/2/38 | 0/0/41 | 1/4/36 | 0/0/41 |
OE (%) | 70.7% | 22.0% | 14.6% | 19.5% | 14.6% | 7.3% | 0% | 12.2% | 0% |
3.6 Limitation analysis
4 BKA for solving engineering problems
4.1 Pressure vessel design
Algorithm | Optimal values for variables | Optimal cost | |||
---|---|---|---|---|---|
Ts | Th | R | L | ||
BKA | 0.778433 | 0.384690 | 40.319619 | 200 | 5887.364927 |
GJO | 0.778523 | 0.403115 | 40.332023 | 200 | 5908.557674 |
PSO | 0.958559 | 0.510067 | 49.120228 | 105.66659 | 6289.337794 |
AVOA | 1.244248 | 0.615032 | 64.468805 | 13.297390 | 7254.449800 |
SHO | 0.779035 | 0.384660 | 40.327793 | 199.650290 | 5889.368900 |
SCSO | 0.987235 | 0.488050 | 51.151977 | 89.460729 | 6347.596799 |
SSA | 56.132650 | 55.950231 | 56.158215 | 55.928870 | 4.49E + 06 |
AO | 1.037249 | 0.514761 | 53.556647 | 74.087978 | 6582.536753 |
COA | 1.134869 | 1.674960 | 56.788030 | 52.204932 | 13362.472751 |
4.2 Design issue with tension/compression springs
Algorithm | Optimal values for variables | Optimal cost | ||
---|---|---|---|---|
d | D | N | ||
BKA | 0.051173 | 0.344426 | 12.047782 | 0.01267027 |
GJO | 0.050468 | 0.3276388 | 13.255784 | 0.01268300 |
PSO | 0.05 | 0.3170802 | 14.076339 | 0.01274300 |
AVOA | 0.05 | 0.317425 | 14.02777 | 0.012719054 |
SHO | 0.0508 | 0.334800 | 11.702 | 0.012681 |
SCSO | 0.051592 | 0.354395 | 11.426859 | 0.01271702 |
SSA | 0.077347 | 1.34212 | 1.960143 | 0.03179716 |
AO | 0.061211 | 0.624604 | 4.419888 | 0.015023988 |
COA | 0.05571 | 0.4614 | 7.11158 | 0.013048048 |
4.3 Welded beam design
Algorithm | Optimal values for variables | Optimal cost | |||
---|---|---|---|---|---|
h | l | t | b | ||
BKA | 0.205730 | 3.470488 | 9.036622 | 0.205730 | 1.724853 |
GJO | 0.205803 | 3.468938 | 9.036642 | 0.205837 | 1.725582 |
PSO | 0.209488 | 3.45523 | 8.927898 | 0.215478 | 1.783039 |
AVOA | 0.20592 | 3.468021 | 9.032414 | 0.205921 | 1.725545 |
SHO | 0.20585 | 3.46946 | 9.03276 | 0.20591 | 1.7259 |
SCSO | 0.205709 | 3.471169 | 9.036666 | 0.205731 | 1.724928 |
SSA | 2.330378 | 2.682921 | 2.616964 | 2.453113 | 5.20E + 14 |
AO | 0.200517 | 3.654022 | 9.057898 | 0.206271 | 1.749188 |
COA | 0.25499 | 2.998089 | 7.969305 | 0.330805 | 2.371241 |
4.4 Speed reducer design problem
Algorithm | Optimal values for variables | Optimal cost | ||||||
---|---|---|---|---|---|---|---|---|
b | m | p | l1 | l2 | d1 | d2 | ||
BKA | 3.5 | 0.7 | 17 | 7.3 | 7.71532 | 3.350215 | 5.286654 | 2994.47107 |
GJO | 3.5 | 0.7 | 17 | 7.32 | 7.72122 | 3.35025 | 5.28665 | 2994.80495 |
PSO | 3.5 | 0.7 | 17 | 8.3 | 7.8 | 3.352412 | 5.286715 | 3005.763 |
AVOA | 3.5 | 0.7 | 17 | 7.3 | 7.71532 | 3.350215 | 5.286654 | 2994.47109 |
SHO | 3.5 | 0.7 | 17 | 7.3 | 7.7163 | 3.3502 | 5.2867 | 2994.504 |
SCSO | 3.5 | 0.7 | 17 | 7.32 | 8.029658 | 3.350294 | 5.286794 | 3001.69686 |
SSA | 7.62 | 8.33 | 8.27 | 8.04 | 8.038538 | 7.823429 | 7.862038 | 2.11E + 16 |
AO | 3.58 | 0.7 | 17 | 7.41 | 7.843836 | 3.363452 | 5.319559 | 3052.2253 |
COA | 3.5 | 0.7 | 25.72 | 7.33 | 7.974521 | 3.569454 | 5.339685 | 4940.841 |
4.5 Three-bar truss design problem
Algorithm | Optimal values for variables | Optimal weight | |
---|---|---|---|
x1 | x2 | ||
BKA | 0.788675 | 0.408248 | 263.895843 |
GJO | 0.788657 | 0.408299 | 263.895844 |
PSO | 0.788919 | 0.404741 | 263.896200 |
AVOA | 0.788983 | 0.407378 | 263.895913 |
SHO | 0.788898 | 0.40762 | 263.895881 |
SCSO | 0.788334 | 0.409214 | 263.895959 |
SSA | 0.707614 | 0.704996 | 270.642993 |
AO | 0.788981 | 0.407368 | 263.895929 |
COA | 0.788496 | 0.408285 | 263.895844 |
4.6 Analysis of the results of engineering design problems
5 Conclusion and future works
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Evaluate the performance of BKA using the CEC-2017 test set, CEC-2022 test set, and 18 complex functions, demonstrating superior results across various characteristics and complexities.
-
Statistical validation using the Friedman and Wilson sign rank tests, with BKA securing first place, confirming its effectiveness and scientific reliability.
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Practical application of BKA in five engineering cases involving challenging conditions and constrained search spaces, where it shows significant superiority by quickly converging to high-quality solutions and exhibiting excellent performance.