# A conceptual comparison of several metaheuristic algorithms on continuous optimisation problems

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## Abstract

The field of continuous optimisation has witnessed an explosion of the so-called new or novel metaheuristic algorithms. Though not all of these algorithms are efficient as proclaimed by their inventors, a few of them have proved to be very efficient and thus have become popular tools for solving complex optimisation problems. Therefore, there is a need for a systematic analysis approach to fairly evaluate and compare the results of some of these optimisation algorithms. In this paper, a set of well-known mathematical benchmark functions are compiled to provide an easily accessible collection of standard benchmark test problems for continuous global optimisation. This set of test problems are used to investigate the computational capabilities and the microscopic behaviour of twelve different metaheuristic algorithms. The required number of function evaluations for reaching the best solution and the run-time complexity of the algorithms are compared. Furthermore, statistical tests are conducted to validate the concluding remarks.

## Keywords

Metaheuristics Population-based metaheuristics Swarm intelligence Continuous domain optimisation## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

## References

- 1.Zhou JX, Zou W (2008) Meshless approximation combined with implicit topology description for optimisation of continua. Struct Multidiscip Optim 36(4):347–353CrossRefGoogle Scholar
- 2.Luo Z, Zhang N, Wang Y, Gao W (2013) Topology optimisation of structures using meshless density variable approximants. Int J Numer Methods Eng 93(4):443–464zbMATHCrossRefGoogle Scholar
- 3.Lin J, Chen CS, Liu CS, Lu J (2016) Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions. Comput Math Appl 72(3):555–567MathSciNetzbMATHCrossRefGoogle Scholar
- 4.Lin J, Reutskiy SY, Lu J (2018) A novel meshless method for fully nonlinear advection–diffusion–reaction problems to model transfer in anisotropic media. Appl Math Comput 339:459–476MathSciNetCrossRefGoogle Scholar
- 5.Fu ZJ, Xi Q, Chen W, Cheng AHD (2018) A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations. Comput Math Appl 76:760–773MathSciNetCrossRefGoogle Scholar
- 6.Fister I, JrFister I, Yang X-S, Brest J (2013) A comprehensive review of firefly algorithms. Swarm Evolut Comput 13:34–46CrossRefGoogle Scholar
- 7.Jamil M, Yang XS (2013) A literature survey of benchmark functions for global optimisation problems. Int J Math Model Numer Optim 4(2):150–194zbMATHGoogle Scholar
- 8.Dieterich JM, Hartke B (2012) Empirical review of standard benchmark functions using evolutionary global optimisation. Appl Math 3:1552–1564CrossRefGoogle Scholar
- 9.Holland JH (1992) Genetic algorithms. Sci Am 267(1):66–73CrossRefGoogle Scholar
- 10.Goldbarg EF, Goldbarg MC, de Souza GR (2008) Particle swarm optimisation algorithm for the traveling salesman problem. In: Traveling salesman problem, ed: InTechGoogle Scholar
- 11.Dorigo M, Birattari M, Stutzle T (2006) Ant colony optimisation. IEEE Comput Intell Mag 1(4):28–39CrossRefGoogle Scholar
- 12.Cheng M-Y, Prayogo D (2014) Symbiotic organisms search: a new metaheuristic optimisation algorithm. Comput Struct 139:98–112CrossRefGoogle Scholar
- 13.Yang X-S, Deb S (2009) Cuckoo search via Lévy flights. In: 2009. NaBIC 2009. World Congress on nature & biologically inspired computing, pp 210–214Google Scholar
- 14.Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimisation: artificial bee colony (ABC) algorithm. J Glob Optim 39(3):459–471zbMATHCrossRefGoogle Scholar
- 15.Yang X-S (2010) A new metaheuristic bat-inspired algorithm. In: González JR, Pelta DA, Cruz C, Terrazas G, Krasnogor N (eds) Nature inspired cooperative strategies for optimisation (NICSO 2010). Springer, Berlin, pp 65–74CrossRefGoogle Scholar
- 16.Ab Wahab MN, Nefti-Meziani S, Atyabi A (2015) A comprehensive review of swarm optimisation algorithms. PLoS ONE 10(5):e0122827CrossRefGoogle Scholar
- 17.Yang X-S (2012) Flower pollination algorithm for global optimisation. In: International conference on unconventional computing and natural computation, pp 240–249Google Scholar
- 18.Mehrabian AR, Lucas C (2006) A novel numerical optimisation algorithm inspired from weed colonization. Ecol Inf 1(4):355–366CrossRefGoogle Scholar
- 19.Pham DT, Castellani M (2009) The bees algorithm: modelling foraging behaviour to solve continuous optimisation problems. Proc Inst Mech Eng Part C J Mech Eng Sci 223(12):2919–2938CrossRefGoogle Scholar
- 20.Dolan ED, Moré JJ (2002) Benchmarking optimisation software with performance profiles. Math Program 91(2):201–213MathSciNetzbMATHCrossRefGoogle Scholar
- 21.Ma H, Simon D, Fei M, Chen Z (2013) On the equivalences and differences of evolutionary algorithms. Eng Appl Artif Intell 26(10):2397–2407CrossRefGoogle Scholar
- 22.Ma H, Ye S, Simon D, Fei M (2017) Conceptual and numerical comparisons of swarm intelligence optimisation algorithms. Soft Comput 21(11):3081–3100CrossRefGoogle Scholar
- 23.Civicioglu P, Besdok EA (2013) A conceptual comparison of the Cuckoo-search, particle swarm optimisation, differential evolution and artificial bee colony algorithms. Artif Intell Rev 39(4):315–346CrossRefGoogle Scholar
- 24.Soler-Dominguez A, Juan AA, Kizys R (2017) A survey on financial applications of metaheuristics. ACM Comput Surv (CSUR) 50(1):15CrossRefGoogle Scholar
- 25.Yang XS (2010) Engineering optimisation: an introduction with metaheuristic applications. Wiley, New YorkCrossRefGoogle Scholar
- 26.Ezugwu AE, Prayogo D (2019) Symbiotic organisms search algorithm: theory, recent advances and applications. Expert Syst Appl 119(1):184–209CrossRefGoogle Scholar
- 27.Ali MM, Khompatraporn C, Zabinsky ZB (2005) A numerical evaluation of several stochastic algorithms on selected continuous global optimisation test problems. J Glob Optim 31(4):635–672zbMATHCrossRefGoogle Scholar
- 28.Yang XS (2010) Firefly algorithm, Lévy flights and global optimization. In: Bramer M, Ellis R, Petridis M (eds) Research and development in intelligent systems, vol XXVI. Springer, London, pp 209–218CrossRefGoogle Scholar
- 29.Yang XS, Karamanoglu M, He X (2014) Flower pollination algorithm: a novel approach for multiobjective optimisation. Eng Optim 46(9):1222–1237MathSciNetCrossRefGoogle Scholar
- 30.Gendreau M, Potvin JY (2010) Handbook of metaheuristics. Springer, New YorkzbMATHCrossRefGoogle Scholar
- 31.Amodeo L, Talbi EG, Yalaoui F (eds) (2018) Recent developments in metaheuristics. Springer, BerlinzbMATHGoogle Scholar
- 32.Zhang L, Liu L, Yang X-S, Dai Y (2016) A novel hybrid firefly algorithm for global optimisation. PLoS ONE 11(9):e0163230CrossRefGoogle Scholar
- 33.Ross K (2017) ISM206: metaheuristics. https://classes.soe.ucsc.edu/ism206/Fall05/Lecture12.pdf. Accessed 23 Jan 2018
- 34.Silberholz J, Golden B (2010) Comparison of metaheuristics. In: Gendreau M, Potvin JY (eds) Handbook of metaheuristics. International series in operations research & management science, vol 146. Springer, BostonzbMATHGoogle Scholar
- 35.Lobo FJ, Lima CF, Michalewicz Z (eds) (2007) Parameter setting in evolutionary algorithms, 54th edn. Springer, BerlinzbMATHGoogle Scholar
- 36.Shi Y, Eberhart RC (1998) Parameter selection in particle swarm optimization. In: Porto VW, Saravanan N, Waagen D, Eiben AE (eds) Evolutionary programming VII. EP 1998. Lecture notes in computer science, vol 1447. Springer, Berlin, HeidelbergGoogle Scholar
- 37.Rajabioun R (2011) Cuckoo optimisation algorithm. Appl Soft Comput 11(8):5508–5518CrossRefGoogle Scholar
- 38.Yang XS, Deb S (2014) Cuckoo search: recent advances and applications. Neural Comput Appl 24(1):169–174CrossRefGoogle Scholar
- 39.Yang X-S (2010) Nature-inspired metaheuristic algorithms. Luniver Press, FromeGoogle Scholar
- 40.Dorigo M, Birattari M (2017) Ant colony optimization. In: Sammut C, Webb GI (eds) Encyclopedia of machine learning and data mining. Springer, BostonGoogle Scholar
- 41.Dorigo M, Di Caro G (1999) Ant colony optimisation: a new meta-heuristic. In: Evolutionary computation, 1999. CEC 99. Proceedings of the 1999 Congress on, vol 2. IEEE, pp 1470–1477Google Scholar
- 42.Darquennes D (2005) Implementation and applications of ant colony algorithms. Masters, Faculty of Computer Science, University of Namur, BelgiumGoogle Scholar
- 43.Yang X-S, He X (2013) Bat algorithm: literature review and applications. Int J Bio-Inspired Comput 5(3):141–149CrossRefGoogle Scholar
- 44.Yang XS, Gandomi AH (2012) Bat algorithm: a novel approach for global engineering optimisation. Eng Comput 29(5):464–483CrossRefGoogle Scholar
- 45.Chittka L, Thomson JD, Waser NM (1999) Flower constancy, insect psychology, and plant evolution. Naturwissenschaften 86(8):361–377CrossRefGoogle Scholar
- 46.Fister JI, Yang XS, Fister I, Brest J, Fister D (2013) A brief review of nature-inspired algorithms for optimisation. arXiv preprint arXiv:1307.4186
- 47.Ezugwu AE, Adewumi OA (2017) Discrete symbiotic organisms search algorithm for travelling salesman problem. Expert Syst Appl 30(87):70–78CrossRefGoogle Scholar
- 48.Valle YD, Venayagamoorthy GK, Mohagheghi S, Hernandez JC, Harley RG (2008) Particle swarm optimisation: basic concepts, variants and applications in power systems. IEEE Trans Evol Comput 12(2):171–195CrossRefGoogle Scholar
- 49.Cui Z (2009) Alignment particle swarm optimisation. In: 2009 8th IEEE international conference on cognitive informatics, pp 497–501Google Scholar
- 50.Kennedy J, Eberhart R (1995) Particle swarm optimisation. In: Neural networks, 1995. Proceedings, IEEE international conference on, vol 4, pp 1942–1948Google Scholar
- 51.Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimisation over continuous spaces. J Glob Optim 11(4):341–359zbMATHCrossRefGoogle Scholar
- 52.Das S, Suganthan PN (2011) Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31CrossRefGoogle Scholar
- 53.Goldberg D (1989) Genetic algorithms in optimisation, search and machine learning. Addison-Wesley, ReadingzbMATHGoogle Scholar
- 54.Tereshko V, Loengarov A (2005) Collective decision making in honey-bee foraging dynamics. Comput Inf Syst 9(3):1Google Scholar
- 55.Von Frisch K (2014) Bees: their vision, chemical senses, and language. Cornell University Press, IthacaGoogle Scholar
- 56.Bozorg-Haddad O, Solgi M, Lo HA (2017) Meta-heuristic and evolutionary algorithms for engineering optimisation, vol 294. Wiley, New YorkCrossRefGoogle Scholar
- 57.Fister I Jr, Fister D, Fister I (2013) A comprehensive review of cuckoo search: variants and hybrids. Int J Math Model Numer Optim 4(4):387–409zbMATHGoogle Scholar
- 58.Qu C, He W (2015) A double mutation Cuckoo Search algorithm for solving systems of nonlinear equations. Int J Hybrid Inf Technol 8(12):433–448CrossRefGoogle Scholar
- 59.Wu Y-C, Lee W-P, Chien C-W (2011) Modified the performance of differential evolution algorithm with dual evolution strategy. In: International conference on machine learning and computing, pp 57–63Google Scholar
- 60.Carr J (2014) An introduction to genetic algorithms. Sr Proj 1:40Google Scholar
- 61.Bai Q (2010) Analysis of particle swarm optimisation algorithm. Comput Inf Sci 3(1):180Google Scholar
- 62.Ezugwu AE (2019) Enhanced symbiotic organisms search algorithm for unrelated parallel machines manufacturing scheduling with setup times. Knowl-Based Syst. https://doi.org/10.1016/j.knosys.2019.02.005
- 63.Ali N, Othman MA, Husain MN, Misran MH (2014) A review of firefly algorithm. ARPN J Eng Appl Sci 9(10):1732–1736Google Scholar
- 64.Selvi V, Umarani DR (2010) Comparative analysis of ant colony and particle swarm optimisation techniques. Int J Comput Appl (0975–8887) 5(4):1–6Google Scholar
- 65.Abreu N, Ajmal M, Kokkinogenis Z, Bozorg B (2011) Ant colony optimisation, 26. https://paginas.fe.up.pt/~mac/ensino/docs/DS20102011/Presentations/PopulationalMetaheuristics/ACO_Nuno_Muhammad_Zafeiris_Behdad.pdf. Accessed 18 Feb 2018
- 66.Mohan N, Sivaraj R, Priya RD (2016) A comprehensive review of bat algorithm and its applications to various optimisation problems. Asian J Res Soc Sci Humanit 6(11):676–690Google Scholar
- 67.Xiao-hua S, Chun-ming Y (2013) Application of bat algorithm to permutation flow-shop scheduling problem. Ind Eng J 1:022Google Scholar
- 68.Yuce B, Packianather MS, Mastrocinque E, Pham DT, Lambiase A (2013) Honey bees inspired optimisation method: the bees algorithm. Insects 4(4):646–662CrossRefGoogle Scholar
- 69.Balasubramani K, Marcus K (2014) A study on flower pollination algorithm and its applications. Int J Appl Innov Eng Manag 3(11):230–235Google Scholar
- 70.Wang Y, Li D, Lu Y, Cheng Z, Gao Y (2017) Improved flower pollination algorithm based on mutation strategy. In: Intelligent human–machine systems and cybernetics (IHMSC), 2017 9th international conference on, 2017, pp 337–342Google Scholar
- 71.Yan G, Li C (2011) An effective refinement artificial bee colony optimisation algorithm based on chaotic search and application for PID control tuning. J Comput Inf Syst 7(9):3309–3316Google Scholar
- 72.Ahmadi M, Mojallali H (2012) Chaotic invasive weed optimisation algorithm with application to parameter estimation of chaotic systems. Chaos Solitons Fractals 45(9–10):1108–1120MathSciNetCrossRefGoogle Scholar
- 73.Toksari MD (2016) A hybrid algorithm of Ant Colony Optimisation (ACO) and Iterated Local Search (ILS) for estimating electricity domestic consumption: case of Turkey. Int J Electr Power Energy Syst 78:776–782CrossRefGoogle Scholar
- 74.Abrandao.com (2017) Genetic algorithms in PHP code. http://www.abrandao.com/2015/01/simple-php-genetic-algorithm/. Accessed 12 Dec 2017