# An Improved Bat Algorithm Based on Lévy Flights and Adjustment Factors

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Enhanced Bat Algorithm

#### 2.1. Bat Algorithm

#### 2.2. Dynamically Decreasing Inertia Weight

#### 2.3. Lévy Flights

#### 2.4. Speed Adjustment Factor

#### 2.5. The Pseudocode of the LAFBA

1. Define objective function $f\left(x\right)$, $x={\left({x}_{1},\dots ,{x}_{d}\right)}^{T}$ |

2. Set the initial value of population size n, $\alpha ,\text{}\gamma ,$ and $N\_gen$ |

3. Initialize pulse rates ${r}_{i}$ and loudness ${A}_{i}$ |

4. Initialize the bat population (Equation (1)) |

5. Evaluate and find ${x}_{*}\text{}\mathrm{where}\text{}{x}_{*}\in \left\{1,2,\dots ,n\right\}$ |

6. while t ≤ N_gen |

7. for $i$ = 1 to n |

8. Adjust frequency (Equation (2)) |

9. Update inertia weight (Equation (9)) and $\mathrm{L}\text{\xe9}\mathrm{vy}\left(\mathrm{d}\right)$ (Equation (11)) |

10. Update the velocity (Equation (8)) and position vector (Equation (13)) of the bat |

11. if (rand > ${r}_{i}$) |

12. Select a solution among the best solutions |

13. Generate a local solution around selected best (Equation (5)) |

14. end if |

15. Evaluate objective function |

16. if (rand < ${A}_{i}$ & f(${x}_{i}$) < f(${x}_{*}$)) |

17. ${x}_{*}={x}_{i}$ |

18. f(${x}_{*}$) = f(${x}_{i}$) |

19. Increase ${r}_{i}$ (Equation (7)) |

20. Reduce ${A}_{i}$ (Equation (6)) |

21. end if |

22. if ($f({x}_{i}^{t+1})<f\left({x}_{*}\right)$) |

23. Update the best solution ${x}_{*}$ |

24. end if |

25. end for |

26. Rank the bats and find the current best ${x}_{*}$ |

27. $t=t+1$ |

28. end while |

29. Return ${x}_{*}$, postprocess results and visualization |

## 3. Numerical Simulation and Analysis

#### 3.1. Parameters Setting

#### 3.2. Standard Optimization Functions

#### 3.3. Simulation Result Comparison and Analysis

#### 3.4. Convergence Curve Analysis

## 4. LAFBA for Classical Engineering Problems

#### 4.1. Tension/Compression Spring Design

Consider | $\overrightarrow{x}=\left[{x}_{1}\text{}{x}_{2}\text{}{x}_{3}\right]=\left[d\text{}D\text{}N\right],$ |

Minimize | $f\left(\overrightarrow{x}\right)=\left({x}_{3}+2\right){x}_{2}{x}_{1}^{2},$ |

Subject to | ${g}_{1}\left(\overrightarrow{x}\right)=1-\frac{{x}_{2}^{3}{x}_{3}}{71785{x}_{1}^{4}}\le 0,$ |

${g}_{2}\left(\overrightarrow{x}\right)=\frac{4{x}_{2}^{2}-{x}_{1}{x}_{2}}{12566\left({x}_{2}{x}_{1}^{3}-{x}_{1}^{4}\right)}+\frac{1}{5108{x}_{1}^{2}}-1\le 0,$ | |

${g}_{3}\left(\overrightarrow{x}\right)=1-\frac{140.45{x}_{1}}{{x}_{2}^{2}{x}_{3}}\le 0,$ | |

${g}_{4}\left(\overrightarrow{x}\right)=\frac{{x}_{1}+{x}_{2}}{1.5}-1\le 0,$ | |

Variable range | $0.05\le {x}_{1}\le 2.00$, $0.25\le {x}_{2}\le 1.30$, $2.00\le {x}_{3}\le 15.0$. |

#### 4.2. Welded Beam Design

Consider | $\overrightarrow{x}=\left[{x}_{1}\text{}{x}_{2}\text{}{x}_{3}\text{}{x}_{4}\right]=\left[h\text{}l\text{}t\text{}b\right],$ |

Minimize | $f\left(\overrightarrow{x}\right)=1.10471{x}_{1}^{2}{x}_{2}+0.04811{x}_{3}{x}_{4}\left(14.0+{x}_{2}\right),$ |

Subject to | ${g}_{1}\left(\overrightarrow{x}\right)=\tau \left(\overrightarrow{x}\right)-{\tau}_{max}\le 0,$ |

${g}_{2}\left(\overrightarrow{x}\right)=\sigma \left(\overrightarrow{x}\right)-{\sigma}_{max}\le 0,$ | |

${g}_{3}\left(\overrightarrow{x}\right)=\delta \left(\overrightarrow{x}\right)-{\delta}_{max}\le 0,$ | |

${g}_{4}\left(\overrightarrow{x}\right)={x}_{1}-{x}_{4}\le 0,$ | |

${g}_{5}\left(\overrightarrow{x}\right)=p-{p}_{c}\left(\overrightarrow{x}\right)\le 0,$ | |

${g}_{6}\left(\overrightarrow{x}\right)=0.125-{x}_{1}\le 0,$ | |

${g}_{7}\left(\overrightarrow{x}\right)=0.10471{x}_{1}^{2}+0.04811{x}_{3}{x}_{4}\left(14.0+{x}_{2}\right)-5.0\le 0,$ | |

Variable range | $0.1\le {x}_{1}\le 2,$$0.1\le {x}_{2}\le 10,$$0.1\le {x}_{3}\le 10,$$0.1\le {x}_{4}\le 2,$ |

where | $\tau \left(\overrightarrow{x}\right)=\sqrt{{\left({\tau}^{\prime}\right)}^{2}+2{\tau}^{\prime}{\tau}^{\u2033}\frac{{x}_{2}}{2R}+{\left({\tau}^{\u2033}\right)}^{2}},$ ${\tau}^{\prime}=\frac{p}{\sqrt{2}{x}_{1}{x}_{2}},{\tau}^{\u2033}=\frac{MR}{J},$ |

$M=p\left(L+\frac{{x}_{2}}{2}\right),$ $R=\sqrt{\frac{{x}_{2}^{2}}{4}+{\left(\frac{{x}_{1}+{x}_{3}}{2}\right)}^{2}},$ | |

$J=2\left\{\sqrt{2}{x}_{1}{x}_{2}\left[\frac{{x}_{2}^{2}}{12}+{\left(\frac{{x}_{1}+{x}_{3}}{2}\right)}^{2}\right]\right\},$ $\sigma \left(\overrightarrow{x}\right)=\frac{6PL}{{x}_{4}{x}_{3}^{2}},\delta \left(\overrightarrow{x}\right)=\frac{4P{L}^{3}}{E{x}_{3}^{3}{x}_{4}}$ | |

${P}_{C}\left(\overrightarrow{x}\right)=\frac{4.013E\sqrt{\frac{{x}_{3}^{2}{x}_{4}^{6}}{36}}}{{L}^{2}}\left(1-\frac{{x}_{3}}{2L}\sqrt{\frac{E}{4G}}\right),$ | |

$p=6000\text{}\mathrm{lb},\text{}L=14\text{}\mathrm{in}.,{\delta}_{max}=0.25\text{}\mathrm{in}.,$ | |

E = 30 × 10^{6} psi, G = 12 × 10^{6} psi, | |

${\tau}_{max}=13,600\text{}\mathrm{p}\mathrm{s}\mathrm{i},{\sigma}_{max}=30,000\text{}\mathrm{p}\mathrm{s}\mathrm{i}.$ |

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Holland, J.H. Erratum: Genetic Algorithms and the Optimal Allocation of Trials. Siam J. Comput.
**1974**, 3, 326. [Google Scholar] [CrossRef] - Kennedy, J.; Eberhart, R. Particle swarm optimization. IEEE Int. Conf. Neural Netw.
**1995**, 2002, 1942–1948. [Google Scholar] - Eberhart, R.; Kennedy, J. A new optimizer using particle swarm theory. In MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science; IEEE Press: Piscataway, NJ, USA, 1995; pp. 39–43. [Google Scholar]
- Dorigo, M.; Colorni, V.A. Ant system: Optimization by a colony of cooperating agents. IEEE Trans. Syst. Man Cyber. Part B Cyber.
**1996**, 26, 29–41. [Google Scholar] [CrossRef] [PubMed] - Karaboga, D. An Idea Based on Honey Bee Swarm for Numerical Optimization. TR-06; Erciyes University: Kayseri, Turkey, 2005. [Google Scholar]
- Gandomi, A.H.; Alavi, A.H. Krill herd: A new bio-inspired optimization algorithm. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 4831–4845. [Google Scholar] [CrossRef] - Yang, X.S.; Deb, S. Cuckoo Search via Lévy flights. In Proceedings of the 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India, 9–11 December 2009; pp. 210–214. [Google Scholar]
- Yang, X.S.; Deb, S. Engineering Optimisation by Cuckoo Search. Int. J. Math. Modell. Numer. Optim.
**2010**, 1, 330–343. [Google Scholar] [CrossRef] - Mirjalili, S. Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowl.-Based Syst.
**2015**, 89, 228–249. [Google Scholar] [CrossRef] - Sergeyev, Y.D.; Kvasov, D.E.; Mukhametzhanov, M.S. Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms. Math. Comput. Simul.
**2017**, 141, 96–109. [Google Scholar] [CrossRef] - Yang, X.S. A New Metaheuristic Bat-Inspired Algorithm. Comput. Knowl. Technol.
**2010**, 284, 65–74. [Google Scholar] - Ramli, M.R.; Abas, Z.A.; Desa, M.I.; Abidin, Z.Z.; Alazzam, M.B. Enhanced Convergence of Bat Algorithm Based on Dimensional and Inertia Weight Factor. J. King Saud Univ.-Comput. Inf. Sci.
**2018**. [Google Scholar] [CrossRef] - Banati, H.; Chaudhary, R. Multi-Modal Bat Algorithm with Improved Search (MMBAIS). J. Comput. Sci.
**2017**, 23, 130–144. [Google Scholar] [CrossRef] - Al-Betar, M.A.; Awadallah, M.A.; Faris, H.; Yang, X.S.; Khader, A.T.; Alomari, O.A. Bat-inspired Algorithms with Natural Selection mechanisms for Global optimization. Neurocomputing
**2018**, 273, 448–465. [Google Scholar] [CrossRef] - Li, Y.; Pei, Y.H.; Liu, J.S. Bat optimization algorithm combining uniform variation and Gaussian variation. Control Decis.
**2017**, 32, 1775–1781. [Google Scholar] - Chakri, A.; Khelif, R.; Benouaret, M.; Yang, X.S. New directional bat algorithm for continuous optimization problems. Expert Syst. Appl.
**2017**, 69, 159–175. [Google Scholar] [CrossRef] [Green Version] - Al-Betar, M.A.; Awadallah, M.A. Island Bat Algorithm for Optimization. Expert Syst. Appl.
**2018**, 107, 126–145. [Google Scholar] [CrossRef] - Laudis, L.L.; Shyam, S.; Jemila, C.; Suresh, V. MOBA: Multi Objective Bat Algorithm for Combinatorial Optimization in VLSI. Proc. Comput. Sci.
**2018**, 125, 840–846. [Google Scholar] [CrossRef] - Tawhid, M.A.; Dsouza, K.B. Hybrid Binary Bat Enhanced Particle Swarm Optimization Algorithm for solving feature selection problems. Appl. Comput. Inf.
**2018**. [Google Scholar] [CrossRef] - Osaba, E.; Yang, X.S.; Diaz, F.; Lopez-Garcia, P.; Carballedo, R. An improved discrete bat algorithm for symmetric and asymmetric Traveling Salesman Problems. Eng. Appl. Artif. Intell.
**2016**, 48, 59–71. [Google Scholar] [CrossRef] - Mohamed, T.M.; Moftah, H.M. Simultaneous Ranking and Selection of Keystroke Dynamics Features Through A Novel Multi-Objective Binary Bat Algorithm. Future Comput. Inf. J.
**2018**, 3, 29–40. [Google Scholar] [CrossRef] - Hamidzadeh, J.; Sadeghi, R.; Namaei, N. Weighted Support Vector Data Description based on Chaotic Bat Algorithm. Appl. Soft Comput.
**2017**, 60, 540–551. [Google Scholar] [CrossRef] - Qi, Y.H.; Cai, Y.G.; Cai, H. Discrete Bat Algorithm for Vehicle Routing Problem with Time Window. Chin. J. Electron.
**2018**, 46, 672–679. [Google Scholar] - Bekdaş, G.; Nigdeli, S.M.; Yang, X.S. A novel bat algorithm based optimum tuning of mass dampers for improving the seismic safety of structures. Eng. Struct.
**2018**, 159, 89–98. [Google Scholar] [CrossRef] - Ameur, M.S.B.; Sakly, A. FPGA based hardware implementation of Bat Algorithm. Appl. Soft Comput.
**2017**, 58, 378–387. [Google Scholar] [CrossRef] - Chaib, L.; Choucha, A.; Arif, S. Optimal design and tuning of novel fractional order PID power system stabilizer using a new metaheuristic Bat algorithm. Ain Shams Eng. J.
**2017**, 8, 113–125. [Google Scholar] [CrossRef] [Green Version] - Mohammad, E.; Sayed-Farhad, M.; Hojat, K. Bat algorithm for dam–reservoir operation. Environ. Earth Sci.
**2018**, 77, 510. [Google Scholar] - Liu, J.S.; Ji, H.Y.; Li, Y. Robot Path Planning Based on Improved Bat Algorithm and Cubic Spline Interpolation. Acta Autom. Sin.
**2019**. [Google Scholar] - Shi, Y.; Eberhart, R. Modified particle swarm optimizer. Proc. IEEE ICEC Conf. Anchorage
**1999**, 69–73. [Google Scholar] - Du, Y.H. Advanced Mathematics; Beijing Jiaotong University Press: Beijing, China, 2014. [Google Scholar]
- Ball, F.; Bao, Y.N. Predict Society; Contemporary China Publishing House: Beijing, China, 2007. [Google Scholar]
- Yang, X.S.; Karamanoglu, M.; He, X. Flower pollination algorithm: A novel approach for multiobjective optimization. Eng. Optim.
**2014**, 46, 1222–1237. [Google Scholar] [CrossRef] - Jamil, M.; Yang, X.S. A Literature Survey of Benchmark Functions for Global Optimization Problems. Mathematics
**2013**, 4, 150–194. [Google Scholar] - Mirjalili, S. SCA: A Sine Cosine Algorithm for solving optimization problems. Knowl.-Based Syst.
**2016**, 96, 120–133. [Google Scholar] [CrossRef] - Arora, S.; Singh, S. Butterfly optimization algorithm: A novel approach for global optimization. Soft Comput.
**2019**, 23, 715–734. [Google Scholar] [CrossRef] - Derrac, J.; García, S.; Molina, D.; Herrera, F. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut. Comput.
**2011**, 1, 3–18. [Google Scholar] [CrossRef] - Wilcoxon, F. Individual Comparisons by Ranking Methods. Biom. Bull.
**1945**, 1, 80–83. [Google Scholar] [CrossRef] - García, S.; Molina, D.; Lozano, M.; Herrera, F. A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: A case study on the CEC’2005 Special Session on Real Parameter Optimization. J. Heuristics
**2009**, 15, 617–644. [Google Scholar] - Zhao, Z.Y. Introduction to optimum design. Probabilistic Eng. Mech.
**1990**, 5, 100. [Google Scholar] - Belegundu, A.D.; Arora, J.S. A study of mathematical programming methods for structural optimization. Part I: Theory. Int. J. Numer. Methods Eng.
**2010**, 21, 1601–1623. [Google Scholar] [CrossRef] - Kaveh, A.; Talatahari, S. An improved ant colony optimization for constrained engineering design problems. Eng. Comput.
**2010**, 27, 155–182. [Google Scholar] [CrossRef] - Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. GSA: A Gravitational Search Algorithm. Inf. Sci.
**2009**, 179, 2232–2248. [Google Scholar] [CrossRef] - He, Q.; Wang, L. An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng. Appl. Artif. Intell.
**2007**, 20, 89–99. [Google Scholar] [CrossRef] - Mezura-Montes, E.; Coello, C.A.C. An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. Int. J. Gen. Syst.
**2008**, 37, 443–473. [Google Scholar] - Coello Coello, C.A. Use of a Self-Adaptive Penalty Approach for Engineering Optimization Problems. Comput. Ind.
**2000**, 41, 113–127. [Google Scholar] [CrossRef] - Mahdavi, M.; Fesanghary, M.; Damangir, E. An improved harmony search algorithm for solving optimization problems. Appl. Math. Comput.
**2007**, 188, 1567–1579. [Google Scholar] [CrossRef] - Mirjalili, S.; Lewis, A. The Whale Optimization Algorithm. Adv. Eng. Softw.
**2016**, 95, 51–67. [Google Scholar] [CrossRef] - Li, L.J.; Huang, Z.B.; Liu, F.; Wu, Q.H. A heuristic particle swarm optimizer for optimization of pin connected structures. Comput. Struct.
**2007**, 85, 340–349. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey Wolf Optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Krohling, R.A.; Coelho, L.D.S. Coevolutionary Particle Swarm Optimization Using Gaussian Distribution for Solving Constrained Optimization Problems. IEEE Trans. Cyber.
**2007**, 36, 1407–1416. [Google Scholar] [CrossRef] - Coello, C.; Carlos, A. constraint-handling using an evolutionary multi objective optimization technique. Civ. Eng. Environ. Syst.
**2000**, 17, 319–346. [Google Scholar] [CrossRef] - Deb, K. Optimal design of a welded beam via genetic algorithms. AIAA J.
**1991**, 29, 2013–2015. [Google Scholar] - Deb, K. An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng.
**2000**, 186, 311–338. [Google Scholar] [CrossRef] - Lee, K.S.; Geem, Z.W. A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice. Comput. Methods Appl. Mech. Eng.
**2005**, 194, 3902–3933. [Google Scholar] [CrossRef] - MartÍ, V.; Robledo, L.M. Multi-Verse Optimizer: A nature-inspired algorithm for global optimization. Neural Comput. Appl.
**2016**, 27, 495–513. [Google Scholar] - Askarzadeh, A. A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm. Comput. Struct.
**2016**, 169, 1–12. [Google Scholar] [CrossRef] - Ragsdell, K.M.; Phillips, D.T. Optimal Design of a Class of Welded Structures Using Geometric Programming. J. Eng. Ind.
**1976**, 98, 1021–1025. [Google Scholar] [CrossRef]

**Table 1.**Comparison results of Lévy flights and adjustment factors (LAFBA) and other five algorithms for benchmark functions with D = 10. BA, bat algorithm; PSO, particle swarm optimization; MFO, moth flame optimization; SCA, sine cosine algorithm; BOA, butterfly optimization algorithm.

Algorithm | Function | Best | Worst | Average | SD | Function | Best | Worst | Average | SD |
---|---|---|---|---|---|---|---|---|---|---|

LAFBA | F1 | 0 | 0 | 0 | 0 | F6 | 0 | 7.89 × 10^{−16} | 1.05 × 10^{−16} | 1.98 × 10^{−16} |

BA | 1.27 × 10^{1} | 1.46 × 10^{2} | 7.99 × 10^{1} | 3.52 × 10^{1} | 7.48 × 10^{−2} | 8.76 × 10^{1} | 3.03 × 10^{1} | 2.63 × 10^{1} | ||

PSO | 2.17 × 10^{−1} | 2.45 | 9.51 × 10^{−1} | 4.52 × 10^{−1} | 5.19 × 10^{−6} | 6.80 × 10^{−3} | 8.89 × 10^{−4} | 1.54 × 10^{−3} | ||

MFO | 7.92 × 10^{−11} | 7.58 × 10^{−1} | 1.51 × 10^{−1} | 1.44 × 10^{−1} | 9.89 × 10^{−17} | 1.48 × 10^{−13} | 1.10 × 10^{−14} | 2.82 × 10^{−14} | ||

SCA | 1.34 × 10^{−14} | 8.98 × 10^{−1} | 1.56 × 10^{−1} | 2.16 × 10^{−1} | 4.64 × 10^{−18} | 1.69 × 10^{−10} | 7.33 × 10^{−12} | 3.16 × 10^{−11} | ||

BOA | 3.11 × 10^{−14} | 1.45 × 10^{−12} | 2.95 × 10^{−13} | 3.26 × 10^{−13} | 5.84 × 10^{−12} | 1.03 × 10^{−11} | 8.24 × 10^{−12} | 1.16 × 10^{−12} | ||

LAFBA | F2 | 0 | 3.08 × 10^{−31} | 1.63 × 10^{−32} | 5.77 × 10^{−32} | F7 | 0 | 1.53 × 10^{−15} | 1.67 × 10^{−16} | 3.98 × 10^{−16} |

BA | 6.74 × 10^{−14} | 7.29 × 10^{−13} | 2.85 × 10^{−13} | 1.51 × 10^{−13} | 4.63 × 10^{−6} | 4.76 × 10^{1} | 7.93 | 9.83 | ||

PSO | 2.62 × 10^{−11} | 8.69 × 10^{−7} | 7.48 × 10^{−8} | 1.91 × 10^{−7} | 5.19 × 10^{−6} | 6.80 × 10^{−3} | 8.89 × 10^{−4} | 1.54 × 10^{−3} | ||

MFO | 2.11 × 10^{−29} | 3.17 × 10^{−21} | 1.11 × 10^{−22} | 5.78 × 10^{−22} | 1.17 × 10^{−16} | 5.02 × 10^{−13} | 5.50 × 10^{−14} | 1.16 × 10^{−13} | ||

SCA | 1.15 × 10^{−30} | 7.54 × 10^{−20} | 3.46 × 10^{−21} | 1.42 × 10^{−20} | 9.03 × 10^{−18} | 4.87 × 10^{−12} | 3.70 × 10^{−13} | 9.31 × 10^{−13} | ||

BOA | 4.12 × 10^{−15} | 1.17 × 10^{−14} | 7.28 × 10^{−15} | 1.84 × 10^{−15} | 6.40 × 10^{−12} | 1.26 × 10^{−11} | 9.46 × 10^{−12} | 1.46 × 10^{−12} | ||

LAFBA | F3 | 0 | 2.74 × 10^{−8} | 7.01 × 10^{−9} | 8.65 × 10^{−9} | F8 | 0 | 3.26 × 10^{−15} | 4.89 × 10^{−16} | 9.14 × 10^{−16} |

BA | 1.21 × 10^{1} | 1.94 × 10^{1} | 1.74 × 10^{1} | 1.51 | 1.31 | 8.01 × 10^{1} | 2.58 × 10^{1} | 1.92 × 10^{1} | ||

PSO | 2.94 × 10^{−3} | 1.17 | 3.73 × 10^{−1} | 5.25 × 10^{−1} | 1.05 × 10^{−3} | 2.96 × 10^{−1} | 5.96 × 10^{−2} | 7.38 × 10^{−2} | ||

MFO | 2.99 × 10^{−8} | 5.09 | 4.07 × 10^{−1} | 1.05 | 2.45 × 10^{−6} | 2.51 × 10^{2} | 2.57 × 10^{1} | 5.80 × 10^{1} | ||

SCA | 7.17 × 10^{−10} | 1.69 × 10^{−4} | 7.00 × 10^{−6} | 3.07 × 10^{−5} | 1.81 × 10^{−9} | 9.82 × 10^{−3} | 9.40 × 10^{−4} | 2.48 × 10^{−3} | ||

BOA | 1.65 × 10^{−9} | 6.14 × 10^{−9} | 3.49 × 10^{−9} | 1.20 × 10^{−9} | 7.47 × 10^{−12} | 1.14 × 10^{−11} | 9.61 × 10^{−12} | 1.09 × 10^{−12} | ||

LAFBA | F4 | 0 | 9.59 × 10^{−14} | 1.50 × 10^{−14} | 2.48 × 10^{−14} | F9 | 0 | 1.23 × 10^{−8} | 2.51 × 10^{−9} | 4.09 × 10^{−9} |

BA | 1.79 × 10^{1} | 8.76 × 10^{1} | 4.78 × 10^{1} | 1.95 × 10^{1} | 1.65 | 5.08 | 3.46 | 9.33 | ||

PSO | 1.31 | 3.02 × 10^{1} | 9.41 | 6.93 | 2.14 × 10^{−2} | 2.94 × 10^{−1} | 8.88 × 10^{−2} | 5.66 × 10^{−2} | ||

MFO | 5.97 | 6.28 × 10^{1} | 2.70 × 10^{1} | 1.39 × 10^{1} | 9.20 × 10^{−2} | 4.80 | 1.37 | 1.16 | ||

SCA | 2.56 × 10^{−12} | 2.41 × 10^{1} | 2.44 | 6.64 | 1.09 × 10^{−7} | 3.34 × 10^{−3} | 3.59 × 10^{−4} | 6.97 × 10^{−4} | ||

BOA | 4.26 × 10^{−14} | 5.67 × 10^{1} | 3.10 × 10^{1} | 2.18 × 10^{1} | 3.77 × 10^{−9} | 5.34 × 10^{−9} | 4.50 × 10^{−9} | 4.30 × 10^{−10} | ||

LAFBA | F5 | 0 | 6.11 × 10^{−16} | 8.23 × 10^{−17} | 1.52 × 10^{−16} | F10 | 0 | 9.09 × 10^{−16} | 9.00 × 10^{−17} | 2.36 × 10^{−16} |

BA | 9.72 × 10^{−3} | 2.28 × 10^{−1} | 1.31 × 10^{−1} | 5.67 × 10^{−2} | 6.91 | 1.58 × 10^{2} | 6.14 × 10^{1} | 3.83 × 10^{1} | ||

PSO | 9.72 × 10^{−3} | 7.82 × 10^{−2} | 2.67 × 10^{−2} | 1.68 × 10^{−2} | 7.82 × 10^{−4} | 9.71 × 10^{−2} | 2.89 × 10^{−2} | 3.03 × 10^{−2} | ||

MFO | 3.72 × 10^{−2} | 2.28 × 10^{−1} | 1.28 × 10^{−1} | 4.60 × 10^{−2} | 1.59 × 10^{−5} | 1.75 × 10^{1} | 3.35 | 6.25 | ||

SCA | 9.72 × 10^{−3} | 3.72 × 10^{−2} | 1.06 × 10^{−2} | 5.02 × 10^{−3} | 8.55 × 10^{−10} | 0.02047 | 9.25 × 10^{−4} | 0.003736 | ||

BOA | 3.72 × 10^{−2} | 8.08 × 10^{−2} | 7.18 × 10^{−2} | 1.47 × 10^{−2} | 5.97 × 10^{−12} | 1.11 × 10^{−11} | 9.02 × 10^{−12} | 1.38 × 10^{−12} |

Algorithm | Function | Best | Worst | Average | SD | Function | Best | Worst | Average | SD |
---|---|---|---|---|---|---|---|---|---|---|

LAFBA | F1 | 0 | 1.33 × 10^{−15} | 1.42 × 10^{−16} | 3.01 × 10^{−16} | F6 | 0 | 1.50 × 10^{−14} | 3.34 × 10^{−15} | 4.40 × 10^{−15} |

BA | 9.85 × 10^{1} | 5.36 × 10^{2} | 3.23 × 10^{2} | 1.08 × 10^{2} | 1.85 | 3.35 × 10^{2} | 1.86 × 10^{2} | 8.73 × 10^{1} | ||

PSO | 5.80 × 10^{−2} | 3.46 × 10^{−1} | 1.66 × 10^{−1} | 7.21 × 10^{−2} | 1.49 × 10^{−1} | 9.03 × 10^{−1} | 3.74 × 10^{−1} | 1.56 × 10^{−1} | ||

MFO | 9.48 × 10^{−1} | 2.71 × 10^{2} | 2.22 × 10^{1} | 6.11 × 10^{1} | 6.42 × 10^{−3} | 2.62 × 10^{1} | 3.55 | 9.05 | ||

SCA | 5.39 × 10^{−1} | 7.04 | 1.49 | 1.21 | 8.99 × 10^{−5} | 1.55 | 9.02 × 10^{−2} | 2.81 × 10^{−1} | ||

BOA | 7.17 × 10^{−13} | 1.73 × 10^{−11} | 6.82 × 10^{−12} | 4.58 × 10^{−12} | 9.88 × 10^{−12} | 1.20 × 10^{−11} | 1.10 × 10^{−11} | 5.75 × 10^{−13} | ||

LAFBA | F2 | 0 | 1.14 × 10^{−28} | 6.72 × 10^{−30} | 2.13 × 10^{−29} | F7 | 0 | 1.75 × 10^{−13} | 3.06 × 10^{−14} | 5.03 × 10^{−14} |

BA | 2.18 × 10^{−11} | 6.16 × 10^{−8} | 2.18 × 10^{−9} | 1.14 × 10^{−8} | 4.00 × 10^{1} | 1.31 × 10^{3} | 5.37 × 10^{2} | 2.94 × 10^{2} | ||

PSO | 4.78 × 10^{−4} | 2.69 | 1.16 × 10^{−1} | 4.96 × 10^{−1} | 2.22 | 3.37 × 10^{1} | 8.28 | 8.07 | ||

MFO | 3.59 × 10^{−6} | 2.86 × 10^{−3} | 2.43 × 10^{−4} | 5.34 × 10^{−4} | 3.87 × 10^{−2} | 7.87 × 10^{2} | 2.01 × 10^{2} | 2.24 × 10^{2} | ||

SCA | 5.29 × 10^{−7} | 2.01 × 10^{−1} | 1.03 × 10^{−2} | 3.66 × 10^{−2} | 1.44 × 10^{−3} | 6.09 | 4.90 × 10^{−1} | 1.12 | ||

BOA | 8.92 × 10^{−15} | 1.56 × 10^{−14} | 1.15 × 10^{−14} | 1.35 × 10^{−15} | 1.10 × 10^{−11} | 1.37 × 10^{−11} | 1.23 × 10^{−11} | 7.91 × 10^{−13} | ||

LAFBA | F3 | 0 | 1.04 × 10^{−7} | 2.42 × 10^{−8} | 3.32 × 10^{−8} | F8 | 0 | 3.70 × 10^{−14} | 7.39 × 10^{−15} | 1.15 × 10^{−14} |

BA | 1.36 × 10^{1} | 1.90 × 10^{1} | 1.75 × 10^{1} | 1.15 | 1.21 × 10^{1} | 2.27 × 10^{3} | 2.42 × 10^{2} | 4.05 × 10^{2} | ||

PSO | 1.52 | 4.28 | 2.90 | 5.77 × 10^{−1} | 5.01 × 10^{1} | 4.20 × 10^{2} | 1.65 × 10^{2} | 8.14 × 10^{1} | ||

MFO | 1.25 | 1.98 × 10^{1} | 1.51 × 10^{1} | 5.34 | 2.06 × 10^{2} | 9.81 × 10^{2} | 5.09 × 10^{2} | 1.97 × 10^{2} | ||

SCA | 3.78 × 10^{−2} | 2.03 × 10^{1} | 7.69 | 8.97 | 4.31 × 10^{1} | 2.05 × 10^{2} | 1.26 × 10^{2} | 4.20 × 10^{1} | ||

BOA | 5.53 × 10^{−9} | 7.04 × 10^{−9} | 6.24 × 10^{−9} | 3.84 × 10^{−10} | 8.71 × 10^{−12} | 1.18 × 10^{−11} | 1.05 × 10^{−11} | 7.86 × 10^{−13} | ||

LAFBA | F4 | 0 | 2.49 × 10^{−12} | 2.57 × 10^{−13} | 6.51 × 10^{−13} | F9 | 0 | 6.42 × 10^{−8} | 1.62 × 10^{−8} | 2.38 × 10^{−8} |

BA | 5.97 × 10^{1} | 2.77 × 10^{2} | 1.43 × 10^{2} | 5.73 × 10^{1} | 3.80 | 8.41 | 6.17 | 1.05 | ||

PSO | 6.26E × 10^{1} | 1.39 × 10^{2} | 9.11 × 10^{1} | 2.09 × 10^{1} | 4.02 × 10^{−1} | 1.49 | 7.33 × 10^{−1} | 2.34 × 10^{−1} | ||

MFO | 1.24 × 10^{2} | 2.84 × 10^{2} | 1.75 × 10^{2} | 3.39 × 10^{1} | 5.80 | 8.48 | 7.31 | 6.34 × 10^{−1} | ||

SCA | 1.476745263 | 1.48 × 10^{2} | 4.68 × 10^{1} | 3.24 × 10^{1} | 1.11 | 6.68 | 3.98 | 1.26 | ||

BOA | 0 | 2.19 × 10^{2} | 3.93 × 10^{1} | 8.01 × 10^{1} | 4.30 × 10^{−9} | 5.59 × 10^{−9} | 5.13 × 10^{−9} | 2.75 × 10^{−10} | ||

LAFBA | F5 | 0 | 1.64 × 10^{−14} | 2.62 × 10^{−15} | 4.82 × 10^{−15} | F10 | 0 | 6.76 × 10^{−14} | 1.10 × 10^{−14} | 1.82 × 10^{−14} |

BA | 1.78 × 10^{−1} | 3.73 × 10^{−1} | 3.06 × 10^{−1} | 6.35 × 10^{−2} | 1.57 × 10^{2} | 2.68 × 10^{3} | 7.54 × 10^{2} | 4.79 × 10^{2} | ||

PSO | 3.72 × 10^{−2} | 2.28 × 10^{−1} | 9.21 × 10^{−2} | 3.74 × 10^{−2} | 3.53 | 3.39 × 10^{1} | 1.32 × 10^{1} | 7.06 | ||

MFO | 3.12 × 10^{−1} | 3.73 × 10^{−1} | 3.42 × 10^{−1} | 1.72 × 10^{−2} | 7.39 | 1.65 × 10^{2} | 6.46 × 10^{1} | 3.86 × 10^{1} | ||

SCA | 3.72 × 10^{−2} | 1.27 × 10^{−1} | 4.87 × 10^{−2} | 2.21 × 10^{−2} | 3.02 | 7.53 × 10^{1} | 3.54 × 10^{1} | 1.91 × 10^{1} | ||

BOA | 7.85 × 10^{−2} | 1.27 × 10^{−1} | 1.17 × 10^{−1} | 1.87 × 10^{−2} | 9.36 × 10^{−12} | 1.23 × 10^{−11} | 1.11 × 10^{−11} | 6.22 × 10^{−14} |

**Table 3.**Comparison results of LAFBA and other five algorithms for benchmark functions with D = 100.

Algorithm | Function | Best | Worst | Average | SD | Function | Best | Worst | Average | SD |
---|---|---|---|---|---|---|---|---|---|---|

LAFBA | F1 | 0 | 1.67 × 10^{−15} | 2.87 × 10^{−16} | 5.31 × 10^{−16} | F6 | 0 | 1.81 × 10^{−13} | 5.31 × 10^{−14} | 6.00 × 10^{−14} |

BA | 5.33 × 10^{2} | 2.05 × 10^{3} | 1.31 × 10^{3} | 3.86 × 10^{2} | 4.33 × 10^{2} | 2.06 × 10^{3} | 1.15 × 10^{3} | 4.26 × 10^{2} | ||

PSO | 1.11 | 1.03 × 10^{1} | 4.16 | 2.19 | 1.56 | 6.14 × 10^{1} | 1.35 × 10^{1} | 1.38 × 10^{1} | ||

MFO | 4.57 × 10^{2} | 1.03 × 10^{3} | 6.68 × 10^{2} | 1.29 × 10^{2} | 1.27 × 10^{2} | 2.41 × 10^{2} | 1.90 × 10^{2} | 3.25 × 10^{1} | ||

SCA | 1.40 × 10^{1} | 2.31 × 10^{2} | 1.01 × 10^{2} | 6.37 × 10^{1} | 3.02 | 9.46 × 10^{1} | 3.31 × 10^{1} | 2.13 × 10^{1} | ||

BOA | 4.79 × 10^{−12} | 1.99 × 10^{−11} | 1.29 × 10^{−11} | 4.35 × 10^{−12} | 1.09 × 10^{−11} | 1.34 × 10^{−11} | 1.19 × 10^{−11} | 5.62 × 10^{−13} | ||

LAFBA | F2 | 0 | 5.18 × 10^{−27} | 5.64 × 10^{−28} | 1.11 × 10^{−27} | F7 | 0 | 5.14 × 10^{−12} | 1.14 × 10^{−12} | 1.90 × 10^{−12} |

BA | 1.58 × 10^{−7} | 1.51 | 2.02 × 10^{−1} | 4.22 × 10^{−1} | 2.37 × 10^{3} | 2.11 × 10^{4} | 9.63 × 10^{3} | 4.51 × 10^{3} | ||

PSO | 1.28 | 1.31 × 10^{1} | 4.69 | 2.88 | 2.12 × 10^{2} | 4.43 × 10^{3} | 6.72 × 10^{2} | 9.23 × 10^{2} | ||

MFO | 2.76 | 1.33 × 10^{1} | 7.31 | 2.53 | 5.75 × 10^{3} | 1.40 × 10^{4} | 9.27 × 10^{3} | 2.35 × 10^{3} | ||

SCA | 1.61 | 9.52 | 4.29 | 1.89 | 2.39 × 10^{2} | 3.80 × 10^{3} | 1.17 × 10^{3} | 7.31 × 10^{2} | ||

BOA | 1.10 × 10^{−14} | 1.58 × 10^{−14} | 1.29 × 10^{−14} | 1.02 × 10^{−15} | 1.19 × 10^{−11} | 1.53 × 10^{−11} | 1.35 × 10^{−11} | 8.61 × 10^{−13} | ||

LAFBA | F3 | 0 | 1.53 × 10^{−7} | 3.86 × 10^{−8} | 5.92 × 10^{−8} | F8 | 0 | 1.57 × 10^{−12} | 1.68 × 10^{−13} | 3.74 × 10^{−13} |

BA | 1.51 × 10^{1} | 1.92 × 10^{1} | 1.78 × 10^{1} | 8.52 × 10^{−1} | 3.44 × 10^{2} | 1.74 × 10^{3} | 8.28 × 10^{2} | 2.86 × 10^{2} | ||

PSO | 4.56 | 8.12 | 6.12 | 9.17 × 10^{−1} | 7.99 × 10^{2} | 3.06 × 10^{3} | 1.52 × 10^{3} | 5.23 × 10^{2} | ||

MFO | 1.93 × 10^{1} | 1.99 × 10^{1} | 1.97 × 10^{1} | 1.62 × 10^{−1} | 2.39 × 10^{3} | 5.00 × 10^{3} | 3.99 × 10^{3} | 6.68 × 10^{2} | ||

SCA | 8.28 | 2.06 × 10^{1} | 1.68 × 10^{1} | 4.74 | 9.96 × 10^{2} | 2.00 × 10^{3} | 1.44 × 10^{3} | 2.26 × 10^{2} | ||

BOA | 5.14 × 10^{−9} | 6.81 × 10^{−9} | 5.85 × 10^{−9} | 3.48 × 10^{−10} | 8.18 × 10^{−12} | 1.19 × 10^{−11} | 1.04 × 10^{−11} | 8.34 × 10^{−13} | ||

LAFBA | F4 | 0 | 3.41 × 10^{−11} | 1.03 × 10^{−11} | 1.11 × 10^{−11} | F9 | 0 | 1.80 × 10^{−7} | 4.13 × 10^{−8} | 6.89 × 10^{−8} |

BA | 2.02 × 10^{2} | 8.14 × 10^{2} | 4.60 × 10^{2} | 1.49 × 10^{2} | 5.36 | 9.19 | 7.04 | 1.06 | ||

PSO | 4.28 × 10^{2} | 7.24 × 10^{2} | 5.65 × 10^{2} | 6.46 × 10^{1} | 1.19 | 2.84 | 1.77 | 4.13 × 10^{−1} | ||

MFO | 8.15E × 10^{2} | 1.10 × 10^{3} | 9.19 × 10^{2} | 7.34 × 10^{−3} | 8.95 | 9.69 | 9.37 | 2.02 × 10^{−1} | ||

SCA | 3.24 × 10^{1} | 6.68 × 10^{2} | 2.59 × 10^{2} | 1.43 × 10^{2} | 8.50 | 9.47 | 9.15 | 2.18 × 10^{−1} | ||

BOA | 0 | 3.51 × 10^{−1} | 1.17 × 10^{−2} | 6.40 × 10^{−2} | 4.66 × 10^{−9} | 5.89 × 10^{−9} | 5.28 × 10^{−9} | 2.71 × 10^{−10} | ||

LAFBA | F5 | 0 | 2.66 × 10^{−13} | 5.69 × 10^{−14} | 7.18 × 10^{−14} | F10 | 0 | 1.96 × 10^{−12} | 5.81 × 10^{−13} | 7.43 × 10^{−13} |

BA | 3.73 × 10^{−1} | 4.72 × 10^{−1} | 4.44 × 10^{−1} | 2.52 × 10^{−2} | 2.47 × 10^{3} | 1.37 × 10^{4} | 6.31 × 10^{3} | 2.72 × 10^{3} | ||

PSO | 7.82 × 10^{−2} | 3.12 × 10^{−1} | 1.92 × 10^{−1} | 5.56 × 10^{−2} | 1.29 × 10^{2} | 4.26 × 10^{2} | 2.55 × 10^{2} | 7.16 × 10^{1} | ||

MFO | 4.60 × 10^{−1} | 4.76 × 10^{−1} | 4.70 × 10^{−1} | 3.45 × 10^{−3} | 4.13 × 10^{2} | 9.28 × 10^{2} | 6.59 × 10^{2} | 1.39 × 10^{2} | ||

SCA | 1.78 × 10^{−1} | 3.47 × 10^{−1} | 2.83 × 10^{−1} | 4.28 × 10^{−2} | 4.48 × 10^{2} | 1.38 × 10^{3} | 7.26 × 10^{2} | 1.94 × 10^{2} | ||

BOA | 1.27 × 10^{−1} | 1.54 × 10^{−1} | 1.30 × 10^{−1} | 5.35 × 10^{−3} | 9.76 × 10^{−12} | 1.43 × 10^{−11} | 1.21 × 10^{−11} | 1.06 × 10^{−12} |

**Table 4.**Results of Wilcoxon rank-sum test for LAFBA and other algorithms on 10 test functions with D = 30.

F | LAFBA vs. BA | LAFBA vs. PSO | LAFBA vs. MFO | LAFBA vs. SCA | LAFBA vs. BOA | |||||
---|---|---|---|---|---|---|---|---|---|---|

p_Value | h | p_Value | h | p_Value | h | p_Value | h | p_Value | h | |

F1 | 9.78 × 10^{−12} | 1 | 9.78 × 10^{−12} | 1 | 9.78 × 10^{−12} | 1 | 9.78 × 10^{−12} | 1 | 9.78 × 10^{−12} | 1 |

F2 | 6.51 × 10^{−11} | 1 | 6.50 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.48 × 10^{−11} | 1 |

F3 | 3.71 × 10^{−11} | 1 | 3.71 × 10^{−11} | 1 | 3.71 × 10^{−11} | 1 | 3.71 × 10^{−11} | 1 | 0.111655 | 0 |

F4 | 1.24 × 10^{−11} | 1 | 1.24 × 10^{−11} | 1 | 1.24 × 10^{−11} | 1 | 1.24 × 10^{−11} | 1 | 1.14 × 10^{−06} | 1 |

F5 | 2.23 × 10^{−11} | 1 | 1.68 × 10^{−11} | 1 | 9.12 × 10^{−12} | 1 | 2.52 × 10^{−11} | 1 | 2.55 × 10^{−11} | 1 |

F6 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.45 × 10^{−11} | 1 |

F7 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.46 × 10^{−11} | 1 |

F8 | 6.50 × 10^{−11} | 1 | 6.50 × 10^{−11} | 1 | 6.50 × 10^{−11} | 1 | 6.50 × 10^{−11} | 1 | 6.46 × 10^{−11} | 1 |

F9 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 0.043201 | 1 |

F10 | 6.50 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.51 × 10^{−11} | 1 | 6.46 × 10^{−11} | 1 |

Algorithms | Optimal Values for Variables | Optimal Cost | ||
---|---|---|---|---|

d | D | N | ||

GSA [42] | 0.050276 | 0.323680 | 13.525410 | 0.0127022 |

PSO (Ha and Wang) [43] | 0.051728 | 0.357644 | 11.244543 | 0.0126747 |

ES (Coello and Montes) [44] | 0.051989 | 0.363965 | 10.890522 | 0.0126810 |

GA(Coello) [45] | 0.051480 | 0.351661 | 11.632201 | 0.0127048 |

Improved HS (Mmahdavi et al.) [46] | 0.051154 | 0.349871 | 12.076432 | 0.0126706 |

MFO [9] | 0.051994 | 0.364109 | 10.868422 | 0.0126669 |

WOA [47] | 0.051207 | 0.345215 | 12.004032 | 0.0126763 |

Montes and Coello [48] | 0.051643 | 0.355360 | 11.397926 | 0.0126980 |

Constraint correction (Arora) [41] | 0.050000 | 0.315900 | 14.250000 | 0.0128334 |

Mathematical optimization (Belegundu) [40] | 0.053396 | 0.399180 | 9.1854000 | 0.0127303 |

LAFBA | 0.051663 | 0.356074 | 11.333400 | 0.0126720 |

Algorithms | Optimal Values for Variables | Optimal Cost | |||
---|---|---|---|---|---|

h | l | t | b | ||

GWO [49] | 0.205676 | 3.478377 | 9.03681 | 0.205778 | 1.72624 |

GSA [42] | 0.182129 | 3.856979 | 10.0000 | 0.202376 | 1.87995 |

CPSO [50] | 0.202369 | 3.544214 | 9.048210 | 0.205723 | 1.72802 |

GA(Coello) [51] | N/A | N/A | N/A | N/A | 1.8245 |

GA(Deb) [52] | N/A | N/A | N/A | N/A | 2.3800 |

GA(Deb) [53] | 0.2489 | 6.1730 | 8.1789 | 0.2533 | 2.4331 |

HS (Lee and Geem) [54] | 0.2442 | 6.2331 | 8.2915 | 0.2443 | 2.3807 |

MVO [55] | 0.2054 | 3.47319 | 9.044502 | 0.20569 | 1.72645 |

GSA [56] | 0.2057 | 3.4704 | 9.0366 | 0.2057 | 1.7248 |

MFO [9] | 0.2057 | 3.4703 | 9.0364 | 0.2057 | 1.72452 |

WOA [47] | 0.205396 | 3.484293 | 9.037426 | 0.206276 | 1.730499 |

Random [57] | 0.4575 | 4.7313 | 5.0853 | 0.6600 | 4.1185 |

Simplex [57] | 0.2792 | 5.6256 | 7.7512 | 0.2796 | 2.5307 |

David [57] | 0.2434 | 6.2552 | 8.2915 | 0.2444 | 2.3841 |

Approx [57] | 0.2444 | 6.2189 | 8.2915 | 0.2444 | 2.3815 |

LAFBA | 0.184706185 | 3.642655691 | 9.134897358 | 0.205254053 | 1.7287 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Y.; Li, X.; Liu, J.; Ruan, X.
An Improved Bat Algorithm Based on Lévy Flights and Adjustment Factors. *Symmetry* **2019**, *11*, 925.
https://doi.org/10.3390/sym11070925

**AMA Style**

Li Y, Li X, Liu J, Ruan X.
An Improved Bat Algorithm Based on Lévy Flights and Adjustment Factors. *Symmetry*. 2019; 11(7):925.
https://doi.org/10.3390/sym11070925

**Chicago/Turabian Style**

Li, Yu, Xiaoting Li, Jingsen Liu, and Ximing Ruan.
2019. "An Improved Bat Algorithm Based on Lévy Flights and Adjustment Factors" *Symmetry* 11, no. 7: 925.
https://doi.org/10.3390/sym11070925