An algorithm inspired by bee colonies coupled to an adaptive penalty method for truss structural optimization problems

  • Afonso Celso de Castro LemongeEmail author
  • Grasiele Regina Duarte
  • Leonardo Goliatt da Fonseca
Technical Paper


The constrained optimization problems are very common in the engineering field. For instance, in civil, aeronautical, mechanical engineering and so on, this type of problem is largely used to find the best designs of structures leading to a structural optimization problem to be solved. Commonly, these problems consist in to find structures with the minimum weight, subject to a set of constraints such as allowable stress, displacements, natural frequencies of vibration and stability criteria. Besides the traditional optimization methods, consolidated through the decades, the evolutionary algorithms, in general inspired by natural phenomenona, have been playing an important role showing robustness to solve this kind of problem. In 2005, the artificial bee colony algorithm (ABC), inspired by the foraging of bee colonies, was proposed to solve multimodal and multidimensional optimization problems. This paper proposes, analyzes and discusses the coupling of ABC to variants of an adaptive penalty method, handling the constraints, to solve traditional problems of truss structural optimization. The results obtained are compared with the literature showing that the proposed strategy can be efficient and competitive.


Constrained optimization Truss structural optimization Artificial bee colony algorithm Adaptive penalty method 



The authors thank the Graduate Program in Computational Modeling (UFJF) and Brazilian Agencies CNPq (Grants 306186/2017-9 and 429639/2016-3), FAPEMIG (Grants TEC PPM 174/18 and TEC APQ 01606/15) and CAPES for the financial support.


  1. 1.
    Akay B, Karaboga D (2012) Artificial bee colony algorithm for large-scale problems and engineering design optimization. J Intell Manuf 23(4):1001–1014Google Scholar
  2. 2.
    Aragon V, Victoria S, Esquivel S, Coello C (2010) A modified version of a T-cell algorithm for constrained optimization problems. Int J Numer Methods Eng 84(3):351–378zbMATHGoogle Scholar
  3. 3.
    Bacanin N, Tuba M (2012) Artificial bee colony (ABC) algorithm for constrained optimization improved with genetic operators. Stud Inform Control 21(2):137–146Google Scholar
  4. 4.
    Barbosa H, Bernardino H, Barreto A (2010) Using performance profiles to analyze the results of the 2006 CEC constrained optimization competition. In: IEEE congress on evolutionary computation (CEC), pp 1–8Google Scholar
  5. 5.
    Barbosa HJ, Lemonge AC (2002) An adaptive penalty scheme in genetic algorithms for constrained optimization problems. In: In Proceedings of the genetic and evolutionary computation conference, vol 2, pp 287–294Google Scholar
  6. 6.
    Barbosa HJ, Lemonge AC (2008) An adaptive penalty method for genetic algorithms in constrained optimization problems. In: Frontiers in evolutionary robotics. InTechGoogle Scholar
  7. 7.
    Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming—theory and algorithms, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  8. 8.
    Capriles PVSZ, Fonseca LG, Barbosa HJC, Lemonge ACC (2007) Rank-based ant colony algorithms for truss weight minimization with discrete variables. Commun Numer Methods Eng 23(6):553–575. MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cheng MY, Prayogo D (2014) Symbiotic organisms search: a new metaheuristic optimization algorithm. Comput Struct 139:98–112.
  10. 10.
    Cheng MY, Prayogo D, Wu YW, Lukito MM (2016) A hybrid harmony search algorithm for discrete sizing optimization of truss structure. Autom Constr 69(Supplement C):21–33.
  11. 11.
    Datta R, Deb K (2015) Evolutionary constrained optimization. Springer, BerlinzbMATHGoogle Scholar
  12. 12.
    Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(24):311–338zbMATHGoogle Scholar
  13. 13.
    Dede T, Bekiroglu S, Ayvaz Y (2011) Weight minimization of trusses with genetic algorithm. Appl Soft Comput 11(2):2565–2575. Google Scholar
  14. 14.
    Degertekin S, Hayalioglu M (2013) Sizing truss structures using teaching-learning-based optimization. Comput Struct 119:177–188.
  15. 15.
    Ding M, Chen H, Lin N, Jing S, Liu F, Liang X, Liu W (2017) Dynamic population artificial bee colony algorithm for multi-objective optimal power flow. Saudi J Biol Sci 24(3):703–710. Computational Intelligence Research & Approaches in Bioinformatics and Biocomputing
  16. 16.
    Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Progr 91(2):201–213MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dolan ED, More JJ (2002) Benchmarking optimization software with performance profiles. Math Progr 91(2):201–213MathSciNetzbMATHGoogle Scholar
  18. 18.
    Duarte GR, de Castro Lemonge AC, da Fonseca LG (2017) An algorithm inspired by social spiders for truss optimisation problems. Eng Comput 34(8):2767–2792. Google Scholar
  19. 19.
    Farshi B, Alinia-ziazi A (2010) Sizing optimization of truss structures by method of centers and force formulation. Int J Solids Struct 47(1819):2508–2524zbMATHGoogle Scholar
  20. 20.
    Fonseca L, Barbosa H, Lemonge A (2010) On similarity-based surrogate models for expensive single-and multi-objective evolutionary optimization. In: Tenne Y, Goh CK (eds) Computational intelligence in expensive optimization problems. Springer, Berlin, pp 219–248Google Scholar
  21. 21.
    Gao W, Liu S, Huang L (2012) A global best artificial bee colony algorithm for global optimization. J Comput Appl Math 236(11):2741–2753MathSciNetzbMATHGoogle Scholar
  22. 22.
    Garcia RP, Monta BG, Carvalho ECR, Barbosa HJC, Bernardino HS, Lemonge ACC (2013) Novas variantes para o método de penalização adaptativo (apm) para problemas de otimização com restrições. In: XLV Simpósio Brasileiro de Pesquisa Operacional - SBPO, pp 2017–2028Google Scholar
  23. 23.
    Ghasemi M, Hinton E, Wood R (1999) Optimization of trusses using genetic algorithms for discrete and continuous variables. Eng Comput 16(3):272–303zbMATHGoogle Scholar
  24. 24.
    Groenwold AA, Stander N, Snyman JA (1999) A regional genetic algorithm for the discrete optimal design of truss structures. Int J Numer Methods Eng 44(6):749–766.<749::AID-NME523>3.0.CO;2-FGoogle Scholar
  25. 25.
    Hadidi A, Azad SK, Azad SK (2010) Structural optimization using artificial bee colony algorithm. In: 2nd international conference on engineering optimizationGoogle Scholar
  26. 26.
    Haftka RT, Gurdal Z (1992) Elements of structural optimization, 3rd edn. Springer, BerlinzbMATHGoogle Scholar
  27. 27.
    Harrell LJ, Ranjithan SR (1999) Evaluation of alternative penalty function implementations in a watershed management design problem. In: Banzhaf W, Daida J, Eiben AE, Garzon MH, Honavar V, Jakiela M, Smith RE (eds.) Proceedings of the genetic and evolutionary computation conference, vol 2, pp 1551–1558. Morgan Kaufmann, Orlando, Florida, USAGoogle Scholar
  28. 28.
    Juang DS, Chang WT (2006) A revised discrete lagrangian-based search algorithm for the optimal design of skeletal structures using available sections. Struct Multidiscip Optim 31(3):201–210. Google Scholar
  29. 29.
    Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. Tech. Rep. TR06, Erciyes University, Engineering Faculty, Kayseri, TurkiyeGoogle Scholar
  30. 30.
    Karaboga D, Akay B (2011) A modified artificial bee colony (ABC) algorithm for constrained optimization problems. Appl Soft Comput 11(3):3021–3031Google Scholar
  31. 31.
    Karaboga D, Basturk B (2007) Artificial bee colony (abc) optimization algorithm for solving constrained optimization problems. In: Foundations of fuzzy logic and soft computing, lecture notes in computer science, vol 4529, pp 789–798 Springer, BerlinGoogle Scholar
  32. 32.
    Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Glob Optim 39(3):459–471MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kaveh A, Farhoudi N (2013) A new optimization method: dolphin echolocation. Adv Eng Softw 59:53–70Google Scholar
  34. 34.
    Kaveh A, Ghazaan MI, Bakhshpoori T (2013) An improved ray optimization algorithm for design of truss structures. Period Polytech Civ Eng 57(2):97Google Scholar
  35. 35.
    Kaveh A, Khayatazad M (2012) A new meta-heuristic method: ray optimization. Comput Struct 112:283–294Google Scholar
  36. 36.
    Kaveh A, Talatahari S (2009) A particle swarm ant colony optimization for truss structures with discrete variables. J Constr Steel Res 65(8):1558–1568.
  37. 37.
    Krempser E, Bernardino HS, Barbosa HJ, Lemonge AC (2017) Performance evaluation of local surrogate models in differential evolution-based optimum design of truss structures. Eng Comput 34(2):499–547. Google Scholar
  38. 38.
    Lamberti L (2008) An efficient simulated annealing algorithm for design optimization of truss structures. Comput Struct 86(1920):1936–1953Google Scholar
  39. 39.
    Lee KS, Geem ZW (2004) A new structural optimization method based on the harmony search algorithm. Comput Struct 82(910):781–798Google Scholar
  40. 40.
    Lee KS, Geem ZW, ho Lee S, woong Bae K (2005) The harmony search heuristic algorithm for discrete structural optimization. Eng Optim 37(7):663–684. MathSciNetGoogle Scholar
  41. 41.
    Lemonge AC, Barbosa HJ, Bernardino HS (2015) Variants of an adaptive penalty scheme for steady-state genetic algorithms in engineering optimization. Eng Comput 32(8):2182–2215Google Scholar
  42. 42.
    Lemonge ACC, Barbosa HJC (2004) An adaptive penalty scheme for genetic algorithms in structural optimization. Int J Numer Methods Eng 59(5):703–736zbMATHGoogle Scholar
  43. 43.
    Li L, Huang Z, Liu F (2009) A heuristic particle swarm optimization method for truss structures with discrete variables. Comput Struct 87(7):435–443. Google Scholar
  44. 44.
    Mernik M, Liu SH, Karaboga D, Crepinsek M (2015) On clarifying misconceptions when comparing variants of the artificial bee colony algorithm by offering a new implementation. Inform Sci 291:115–127MathSciNetzbMATHGoogle Scholar
  45. 45.
    Pan QK, Tasgetiren MF, Suganthan P, Chua T (2011) A discrete artificial bee colony algorithm for the lot-streaming flow shop scheduling problem. Inform Sci 181(12):2455–2468MathSciNetGoogle Scholar
  46. 46.
    Perez R, Behdinan K (2007) Particle swarm approach for structural design optimization. Comput Struct 85(1920):1579–1588Google Scholar
  47. 47.
    Quan H, Shi X (2008) On the analysis of performance of the improved artificial-bee-colony algorithm. In: Fourth international conference on natural computation, 2008. ICNC 08, vol 7, pp 654–658Google Scholar
  48. 48.
    Rajeev S, Krishnamoorthy C (1992) Discrete optimization of structures using genetic algorithms. J Struct Eng 118(5):1233–1250Google Scholar
  49. 49.
    Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2012) Mine blast algorithm for optimization of truss structures with discrete variables. Comput Struct 102–103:49–63.
  50. 50.
    Sonmez M (2011) Artificial bee colony algorithm for optimization of truss structures. Appl Soft Comput 11(2):2406–2418MathSciNetGoogle Scholar
  51. 51.
    Sonmez M (2011) Discrete optimum design of truss structures using artificial bee colony algorithm. Struct Multidiscip Optim 43(1):85–97. Google Scholar
  52. 52.
    Storn R, Price K (1997) Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359MathSciNetzbMATHGoogle Scholar
  53. 53.
    Talatahari S, Nouri M, Tadbiri F (2012) Optimization of skeletal structural using artificial bee colony algorithm. Int J Optim Civ Eng 2(4).
  54. 54.
    Wu SJ, Chow PT (1995) Steady-state genetic algorithms for discrete optimization of trusses. Comput Struct 56(6):979–991.

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Afonso Celso de Castro Lemonge
    • 1
    Email author
  • Grasiele Regina Duarte
    • 2
  • Leonardo Goliatt da Fonseca
    • 1
  1. 1.Department of Applied and Computational MechanicsFederal University of Juiz de ForaJuiz de ForaBrazil
  2. 2.Graduate Program of Computational ModelingFederal University of Juiz de ForaJuiz de ForaBrazil

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