Enhanced quadratic approximation integrated with butterfly optimization: a new search algorithm tested on structural and mathematical problems


  • Ali Mortazavi Graduate School of Natural and Applied Sciences, Ege University, Izmir, 35100 (Turkey)
  • Soner Seker Civil Engineering Department, Uşak University, Uşak, 64000 (Turkey)




quadratic approximation, butterfly optimization algorithm, hybrid methods


The Butterfly Optimization Algorithm (BOA) is a swarm based technique, inspired from mating and food searching process of butterflies, developed in last year. Experiments indicate that BOA provides substantial exploration capability on conventional non-constrained benchmark problems, however for the cases with more complex and noisy domains the algorithm can easily be trapped into local minima due to its restricted exploitation behavior. To tackle this issue, current study deals with introducing an alternative search strategy to explore the region of the search domain with high certainty. Such that, firstly a weighted agent is defined and then a quadratic search is performed in the vicinity of this pre-defined agent. This alternative search strategy is named as Enhanced Quadratic Approximation (EQA) and it is combined with BOA method to improve its exploitation behavior and provide an efficient search algorithm. Thus, obtained new method is named as Enhanced Quadratic Approximation Integrated with Butterfly Optimization (EQB) algorithm. Different properties of proposed EQB are tested on mathematical and structural benchmark problems. Acquired results show that the introduced algorithm, in comparison with its parent method and some other well-stablished reported algorithms in the literature, provides a competitive performance in terms of stability, accuracy and convergence rate.


Arora, S., & Singh, S. (2019). Butterfly optimization algorithm: a novel approach for global optimization. Soft Computing, 23(3), 715–734. https://doi.org/10.1007/s00500-018-3102-4

Cheng, M.-Y., & Prayogo, D. (2014). Symbiotic Organisms Search: A new metaheuristic optimization algorithm. Computers & Structures, 139, 98–112. https://doi.org/https://doi.org/10.1016/j.compstruc.2014.03.007

Das, K. N., & Singh, T. K. (2014). Drosophila Food-Search Optimization. Applied Mathematics and Computation, 231, 566–580. https://doi.org/https://doi.org/10.1016/j.amc.2014.01.040

Dede, T., & Ayvaz, Y. (2015). Combined size and shape optimization of structures with a new meta-heuristic algorithm. Applied Soft Computing, 28, 250–258. https://doi.org/https://doi.org/10.1016/j.asoc.2014.12.007

Deep, K., & Das, K. (2009). Performance improvement of real coded genetic algorithm with Quadratic Approximation based hybridisation. International Journal of Intelligent Defence Support Systems, 2(4), 319-334.

Degertekin, S. O. (2012). Improved harmony search algorithms for sizing optimization of truss structures. Computers & Structures, 92, 229–241. https://doi.org/https://doi.org/10.1016/j.compstruc.2011.10.022

Ding, Z. H., Huang, M., & Lu, Z. R. (2016). Structural damage detection using artificial bee colony algorithm with hybrid search strategy. Swarm and Evolutionary Computation, 28, 1–13. https://doi.org/https://doi.org/10.1016/j.swevo.2015.10.010

Dorigo, M., & Blum, C. (2005). Ant colony optimization theory: A survey. Theoretical Computer Science, 344(2), 243–278. https://doi.org/https://doi.org/10.1016/j.tcs.2005.05.020

Finotto, V. C., da Silva, W. R. L., Valášek, M., & Štemberk, P. (2013). Hybrid fuzzy-genetic system for optimising cabled-truss structures. Advances in Engineering Software, 62, 85–96. https://doi.org/https://doi.org/10.1016/j.advengsoft.2013.04.012

Gonçalves, M. S., Lopez, R. H., & Miguel, L. F. F. (2015). Search group algorithm: A new metaheuristic method for the optimization of truss structures. Computers & Structures, 153, 165–184. https://doi.org/https://doi.org/10.1016/j.compstruc.2015.03.003

Hasançebi, O., Çarbaş, S., Doğan, E., Erdal, F., & Saka, M. P. (2010). Comparison of non-deterministic search techniques in the optimum design of real size steel frames. Computers & Structures, 88(17), 1033–1048. https://doi.org/https://doi.org/10.1016/j.compstruc.2010.06.006

Javidy, B., Hatamlou, A., & Mirjalili, S. (2015). Ions motion algorithm for solving optimization problems. Applied Soft Computing, 32, 72–79. https://doi.org/https://doi.org/10.1016/j.asoc.2015.03.035

Liang, Y.-C., & Cuevas Juarez, J. R. (2016). A novel metaheuristic for continuous optimization problems: Virus optimization algorithm. Engineering Optimization, 48(1), 73–93. https://doi.org/10.1080/0305215X.2014.994868

Moloodpoor, M., Mortazavi, A., & Ozbalta, N. (2019). Thermal analysis of parabolic trough collectors via a swarm intelligence optimizer. Solar Energy, 181, 264–275. https://doi.org/https://doi.org/10.1016/j.solener.2019.02.008

Montes, E., Coello, C., & Velazquez-Reyes, J. (2014). Proceedings of Fourth International Conference on Soft Computing for Problem Solving. India: Springer.

Mortazavi, A. (2019a). Comparative assessment of five metaheuristic methods on distinct problems. Dicle University Journal of Engineering, 10(3), 879–898.

Mortazavi, A. (2019b). Interactive fuzzy search algorithm: A new self-adaptive hybrid optimization algorithm. Engineering Applications of Artificial Intelligence, 81, 270–282. https://doi.org/https://doi.org/10.1016/j.engappai.2019.03.005

Mortazavi, A. (2019c). The Performance Comparison of Three Metaheuristic Algorithms On the Size, Layout and Topology Optimization of Truss Structures. Mugla Journal of Science and Technology, 5(2), 28–41.

Mortazavi, A. (2020). A new fuzzy strategy for size and topology optimization of truss structures. Applied Soft Computing, 93, 106412. https://doi.org/https://doi.org/10.1016/j.asoc.2020.106412

Mortazavi, A. (2021a). Bayesian Interactive Search Algorithm: A New Probabilistic Swarm Intelligence Tested on Mathematical and Structural Optimization Problems. Advances in Engineering Software, 155, 102994. https://doi.org/https://doi.org/10.1016/j.advengsoft.2021.102994

Mortazavi, A. (2021b). Size and layout optimization of truss structures with dynamic constraints using the interactive fuzzy search algorithm. Engineering Optimization, 53(3), 369–391. https://doi.org/10.1080/0305215X.2020.1726341

Mortazavi, A., & Togan, V. (2017a). Sizing and layout design of truss structures under dynamic and static constraints with an integrated particle swarm optimization algorithm, 51, 239–252. https://doi.org/10.1016/j.asoc.2016.11.032

Mortazavi, A., & Togan, V. (2017b). Triangular units based method for simultaneous optimizations of planar trusses. Advances in Computational Design, 2(3), 195–210.

Mortazavi, A., & Toğan, V. (2016). Simultaneous size, shape, and topology optimization of truss structures using integrated particle swarm optimizer. Structural and Multidisciplinary Optimization, 54(4). https://doi.org/10.1007/s00158-016-1449-7

Mortazavi, A., Toğan, V., Daloğlu, A., & Nuhoglu, A. (2018). A. Comparison of Two Metaheuristic Algorithms on Sizing and Topology Optimization of Trusses and Mathematical Functions. Gazi University Journal of Science, 31(2), 416–435.

Mortazavi, A., Toğan, V., & Moloodpoor, M. (2019). Solution of structural and mathematical optimization problems using a new hybrid swarm intelligence optimization algorithm. Advances in Engineering Software, 127, 106-123. https://doi.org/10.1016/j.advengsoft.2018.11.004

Mortazavi, A., Toǧan, V., & Nuhoglu, A. (2018). Comparison of Two Metaheuristic Algorithms on Sizing and Topology Optimization of Trusses and Mathematical Functions. Gazi University Journal of Science, 31(2), 416–435.

Mortazavi, A., Toǧan, V., & Nuhoǧlu, A. (2017). An integrated particle swarm optimizer for optimization of truss structures with discrete variables. Structural Engineering and Mechanics, 61(3). https://doi.org/10.12989/sem.2017.61.3.359

Nobile, M. S., Cazzaniga, P., Besozzi, D., Colombo, R., Mauri, G., & Pasi, G. (2018). Fuzzy Self-Tuning PSO: A settings-free algorithm for global optimization. Swarm and Evolutionary Computation, 39, 70–85. https://doi.org/https://doi.org/10.1016/j.swevo.2017.09.001

Patel, V. K., & Savsani, V. J. (2015). Heat transfer search (HTS): a novel optimization algorithm. Information Sciences, 324, 217–246. https://doi.org/https://doi.org/10.1016/j.ins.2015.06.044

Pavithr, R. S., & Gursaran. (2016). Quantum Inspired Social Evolution (QSE) algorithm for 0-1 knapsack problem. Swarm and Evolutionary Computation, 29, 33–46. https://doi.org/https://doi.org/10.1016/j.swevo.2016.02.006

Pence, I., Cesmeli, M. S., Senel, F. A., & Cetisli, B. (2016). A new unconstrained global optimization method based on clustering and parabolic approximation. Expert Systems with Applications, 55, 493–507. https://doi.org/https://doi.org/10.1016/j.eswa.2016.02.036

Quagliaroli, M., Malerba, P. G., Albertin, A., & Pollini, N. (2015). The role of prestress and its optimization in cable domes design. Computers & Structures, 161, 17–30. https://doi.org/https://doi.org/10.1016/j.compstruc.2015.08.017

Sadollah, A., Bahreininejad, A., Eskandar, H., & Hamdi, M. (2012). Mine blast algorithm for optimization of truss structures with discrete variables. Computers & Structures, 102, 49–63. https://doi.org/https://doi.org/10.1016/j.compstruc.2012.03.013

Shi, Y., & Eberhart, R. (1998). A modified particle swarm optimizer. In 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360) (pp. 69–73). https://doi.org/10.1109/ICEC.1998.699146

Souza, R. R. de, Fadel Miguel, L. F., Lopez, R. H., Miguel, L. F. F., & Torii, A. J. (2016). A procedure for the size, shape and topology optimization of transmission line tower structures. Engineering Structures, 111, 162–184. https://doi.org/https://doi.org/10.1016/j.engstruct.2015.12.005

Suganthan, P., Hansen, N., Liang, J., Deb, K., Chen, Y., Auger, A., & Tiwari, S. (2005). Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), 341–357.

Tang, K., Li, Z., Luo, L., & Liu, B. (2015). Multi-strategy adaptive particle swarm optimization for numerical optimization. Engineering applications of artificial Intelligence, 37, 9–19. https://doi.org/https://doi.org/10.1016/j.engappai.2014.08.002

Yang XS. (2009). Firefly Algorithms for Multimodal Optimization. In O. Watanabe & T. Zeugmann (Eds.), Stochastic algorithms: foundations and applications. Berlin, Heidelberg: Springer. https://doi.org/https://doi.org/10.1007/978-3-642-04944-6_14

Zheng, Y.-J. (2015). Water wave optimization: A new nature-inspired metaheuristic. Computers & Operations Research, 55, 1–11. https://doi.org/https://doi.org/10.1016/j.cor.2014.10.008




How to Cite

Mortazavi, A., & Seker, S. (2021). Enhanced quadratic approximation integrated with butterfly optimization: a new search algorithm tested on structural and mathematical problems. Revista De La Construcción. Journal of Construction, 20(2), 215–235. https://doi.org/10.7764/RDLC.20.2.215

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