Abstract
This study proposes a modified Elephant Herding Optimization algorithm to enhance the capability of a classical algorithm for convalescent convergence rate and precision to solve global optimization problems. The proposed Improved Elephant Herding Optimization (IEHO) uses an opposition learning-based initialization to get a better initial population. A sine cosine-based clan updating operator updates the clan individuals towards or outwards their clan leaders. Levy flight distribution with step size controller is applied to perform a local and global search on newly updated positions. The separating operator is modified to maintain a balance between exploration and exploitation of the algorithm. In addition, an elitism strategy is introduced to retain the fittest individual in the consequent iterations. The effectiveness of IEHO is validated on 97 benchmark functions which include unimodal, multimodal, and CEC-BC-2017 functions. The performance of IEHO is compared to fourteen state-of-the-art algorithms along with the winner algorithm of CEC-BC-2017. Friedman's mean rank test shows the dominance of the proposed algorithm for unimodal and multimodal functions. The proposed IEHO algorithm secures the best rank for all 97 benchmark functions. Finally, the applicability of IEHO is shown on five real-world engineering design problems. Results have proven that IEHO performed superior or equivalent to the algorithms reported in the literature and evaluated in this work.
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Appendix
Appendix
In the following table, f. no. represents the function number, function name defines the name of the function, dim represents the number of dimensions (design variables) of the function, range defines the lower and upper bound of search space for the function, global value defines the global optimum value of the function.
F. no. | Function name | Dim | Range | Global value |
|---|---|---|---|---|
Unimodal functions with fixed dimension | ||||
F1 | Beale | 2 | [− 4.5, 4.5] | 0 |
F2 | Booth | 2 | [− 10, 10] | 0 |
F3 | Brent | 2 | [− 10, 10] | 0 |
F4 | Matyas | 2 | [− 10, 10] | 0 |
F5 | Schaffer N. 4 | 2 | [− 100, 100] | 0.292579 |
F6 | Wayburn Seader 3 | 2 | [− 500, 500] | 19.10588 |
F7 | Leon | 2 | [− 1.2, 1.2] | 0 |
F8 | Cube | 2 | [− 10, 10] | 0 |
F9 | Zettl | 2 | [− 5, 10] | − 0.00379 |
Unimodal functions with variable dimensions | ||||
F10 | Sphere | 30 | [− 100, 100] | 0 |
F11 | Power Sum | 30 | [− 1, 1] | 0 |
F12 | Schwefel’s 2.20 | 30 | [− 100, 100] | 0 |
F13 | Schwefel’s 2.21 | 30 | [− 100, 100] | 0 |
F14 | Step | 30 | [− 100, 100] | 0 |
F15 | Stepint | 30 | [− 5.12, 5.12] | − 155 |
F16 | Schwefel’s 2.22 | 30 | [− 100, 100] | 0 |
F17 | Schwefel’s 2.23 | 30 | [− 10, 10] | 0 |
F18 | Rosenbrock | 30 | [− 30, 30] | 0 |
F19 | Brown | 30 | [− 1, 4] | 0 |
F20 | Dixon and Price | 30 | [− 10, 10] | 0 |
F21 | Powell singular | 30 | [− 4, 5] | 0 |
F22 | Xin− She Yang | 30 | [− 20, 20] | − 1 |
F23 | Perm 0, D, beta | 5 | [− Dim, Dim] | 0 |
F24 | Sum squares | 30 | [− 10, 10] | 0 |
Multimodal functions with fixed− dimension | ||||
F25 | Egg Crate | 2 | [− 5, 5] | 0 |
F26 | Ackley N.3 | 2 | [− 32, 32] | − 195.629 |
F27 | Adjiman | 2 | [− 1, 2] | − 2.02181 |
F28 | Bird | 2 | [− 2 \(\pi\), 2 \(\pi\)] | − 106.765 |
F29 | Camel 6 Hump | 2 | [− 5, 5] | − 1.0316 |
F30 | Branin RCOS | 2 | [− 5, 5] | 0.397887 |
F31 | Goldstien Price | 2 | [− 2, 2] | 3 |
F32 | Hartman 3 | 3 | [0, 1] | − 3.86278 |
F33 | Hartman 6 | 6 | [0, 1] | − 3.32236 |
F34 | Cross-in-tray | 2 | [− 10, 10] | − 2.06261 |
F35 | Bartels Conn | 2 | [− 500, 500] | 1 |
F36 | Bukin 6 | 2 | [(− 15, − 5), (− 5, − 3)] | 180.3276 |
F37 | Carrom table | 2 | [− 10, 10] | − 24.1568 |
F38 | Chichinadze | 2 | [− 30, 30] | − 43.3159 |
F39 | Cross function | 2 | [− 10, 10] | 0 |
F40 | Cross leg table | 2 | [− 10, 10] | − 1 |
F41 | Crowned cross | 2 | [− 10, 10] | 0.0001 |
F42 | Easom | 2 | [− 100, 100] | − 1 |
F43 | Giunta | 2 | [− 1, 1] | 0.060447 |
F44 | Helical valley | 3 | [− 10, 10] | 0 |
F45 | Himmelblau | 2 | [− 5, 5] | 0 |
F46 | Holder | 2 | [− 10, 10] | − 19.2085 |
F47 | Pen holder | 2 | [− 11, 11] | − 0.96354 |
F48 | Test tube holder | 2 | [− 10, 10] | − 10.8723 |
F49 | Shubert | 2 | [− 10, 10] | − 186.731 |
F50 | Shekel | 4 | [0, 10] | − 10.5364 |
F51 | Three-Hump Camel | 2 | [− 5, 5] | 0 |
Multimodal function with variable dimension | ||||
F52 | Schwefel’s 2.26 | 30 | [− 500, 500] | − 418.983 |
F53 | Rastrigin | 30 | [− 5.12, 5.12] | 0 |
F54 | Periodic | 30 | [− 10, 10] | 0.9 |
F55 | Qing | 30 | [− 500, 500] | 0 |
F56 | Alpine N. 1 | 30 | [− 10, 10] | 0 |
F57 | Xin-She Yang | 30 | [− 5, 5] | 0 |
F58 | Ackley | 30 | [− 32, 32] | 0 |
F59 | Trignometric 2 | 30 | [− 500, 500] | 0 |
F60 | Salomon | 30 | [− 100, 100] | 0 |
F61 | Styblinski-Tang | 30 | [− 5, 5] | − 1174.98 |
F62 | Griewank | 30 | [− 100, 100] | 0 |
F63 | Xin-She Yang N. 4 | 30 | [− 10, 10] | − 1 |
F64 | Xin-She Yang N. 2 | 30 | [− 2 \(\pi\), 2 \(\pi\)] | 0 |
F65 | Gen. penalized | 30 | [− 50, 50] | 0 |
F66 | Penalized | 30 | [− 50, 50] | 0 |
F67 | Michalewics | 30 | [0, \(\pi\)] | − 29.6309 |
F68 | Quartic noise | 30 | [− 1.28, 1.28] | 0 |
CEC-BC-2017 Functions | ||||
F69 | Shifted and rotated bent cigar function | 10 | [− 100, 100] | 100 |
F70 | Shifted and rotated rosenbrock function | 10 | [− 100, 100] | 300 |
F71 | Shifted and rotated rastrigin function | 10 | [− 100, 100] | 400 |
F72 | Shifted and rotated expanded Scaffer’s F6 function | 10 | [− 100, 100] | 500 |
F73 | Shifted and rotated lunacek bi Rastrigin function | 10 | [− 100, 100] | 600 |
F74 | Shifted and rotated non-continuous Rastrigin’s function | 10 | [− 100, 100] | 700 |
F75 | Shifted and rotated Levy function | 10 | [− 100, 100] | 800 |
F76 | Shifted and rotated Schwefel’s function | 10 | [− 100, 100] | 900 |
F77 | Hybrid Function 1 (N = 3) | 10 | [− 100, 100] | 1000 |
F78 | Hybrid Function 2 (N = 3) | 10 | [− 100, 100] | 1100 |
F79 | Hybrid Function 3 (N = 3) | 10 | [− 100, 100] | 1200 |
F80 | Hybrid Function 4 (N = 4) | 10 | [− 100, 100] | 1300 |
F81 | Hybrid Function 5 (N = 4) | 10 | [− 100, 100] | 1400 |
F82 | Hybrid Function 6 (N = 4) | 10 | [− 100, 100] | 1500 |
F83 | Hybrid Function 6 (N = 5) | 10 | [− 100, 100] | 1600 |
F84 | Hybrid Function 6 (N = 5) | 10 | [− 100, 100] | 1700 |
F85 | Hybrid Function 6 (N = 5) | 10 | [− 100, 100] | 1800 |
F86 | Hybrid Function 6 (N = 6) | 10 | [− 100, 100] | 1900 |
F87 | Composite Function 1 (N = 3) | 10 | [− 100, 100] | 2000 |
F88 | Composite Function 2 (N = 3) | 10 | [− 100, 100] | 2100 |
F89 | Composite Function 3 (N = 4) | 10 | [− 100, 100] | 2200 |
F90 | Composite Function 4 (N = 4) | 10 | [− 100, 100] | 2300 |
F91 | Composite Function 5 (N = 5) | 10 | [− 100, 100] | 2400 |
F92 | Composite Function 6 (N = 5) | 10 | [− 100, 100] | 2500 |
F93 | Composite Function 7 (N = 6) | 10 | [− 100, 100] | 2600 |
F94 | Composite Function 8 (N = 6) | 10 | [− 100, 100] | 2700 |
F95 | Composite Function 9 (N = 6) | 10 | [− 100, 100] | 2800 |
F96 | Composite Function 10 (N = 3) | 10 | [− 100, 100] | 2900 |
F97 | Composite Function 11 (N = 3) | 10 | [− 100, 100] | 3000 |
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Singh, H., Singh, B. & Kaur, M. An improved elephant herding optimization for global optimization problems. Engineering with Computers 38 (Suppl 4), 3489–3521 (2022). https://doi.org/10.1007/s00366-021-01471-y
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DOI: https://doi.org/10.1007/s00366-021-01471-y