Simultaneous topology, shape, and size optimization of trusses, taking account of uncertainties using multi-objective evolutionary algorithms

  • Teerapol Techasen
  • Kittinan Wansasueb
  • Natee Panagant
  • Nantiwat Pholdee
  • Sujin BureeratEmail author
Original Article


This paper proposes the design of trusses using simultaneous topology, shape, and size design variables and reliability optimization. Objective functions consist of structural mass and reliability, while the probability of failure is set as a design constraint. Design variables are treated to simultaneously determine structural topology, shape, and sizes. Six test problems are posed and solved by a number of multi-objective evolutionary algorithms, and it is found that Hybridized Real-Code Population-Based Incremental Learning and Differential Evolution is the best performer. This work is considered an initial study for the combination of reliability optimization and simultaneous topology, shape, and sizing optimization of trusses.


Truss optimization Multi-objective evolutionary algorithms Pareto dominance Reliability index 



This work was supported by the Graduate Engineering Camp Fund, Faculty of Engineering, Khon Kaen University, Thailand, and the Thailand Research Fund (TRF).


  1. 1.
    Karagöz S, Yıldız AR (2017) A comparison of recent metaheuristic algorithms for crashworthiness optimisation of vehicle thin-walled tubes considering sheet metal forming effects. Int J Veh Des 73(1–3):179–188CrossRefGoogle Scholar
  2. 2.
    Kiani M, Yildiz AR (2015) A comparative study of non-traditional methods for vehicle crashworthiness and NVH optimization. Arch Comput Methods Eng 23(4):723–734MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Yıldız AR, Kurtuluş E, Demirci E, Yıldız BS, Karagöz S (2016) Optimization of thin-wall structures using hybrid gravitational search and Nelder–Mead algorithm. Mater Test 58(1):75–78CrossRefGoogle Scholar
  4. 4.
    Yıldız BS, Yıldız AR (2018) Comparison of grey wolf, whale, water cycle, ant lion and sine-cosine algorithms for the optimization of a vehicle engine connecting rod. Mater Test 60(3):311–315CrossRefGoogle Scholar
  5. 5.
    Yıldız BS, Yıldız AR (2017) Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes. Mater Test 59(5):425–429CrossRefGoogle Scholar
  6. 6.
    Yıldız BS (2017) A comparative investigation of eight recent population-based optimisation algorithms for mechanical and structural design problems. Int J Veh Des 73(1–3):208–2187Google Scholar
  7. 7.
    Yıldız BS (2017) Yıldız AR, Pholdee N, Bureerat S (2017) Hybrid real-code population-based incremental learning and differential evolution for many-objective optimisation of an automotive floor-frame. Int J Veh Des 73(1–3):20–538Google Scholar
  8. 8.
    Yıldız AR, Öztürk F (2010) Hybrid Taguchi-harmony search approach for shape optimization. Recent Adv Harmon Search Algorithm 8:89–98Google Scholar
  9. 9.
    Yildiz AR (2013) Comparison of evolutionary-based optimization algorithms for structural design optimization. Eng Appl Artif Intell 26(1):327–333CrossRefGoogle Scholar
  10. 10.
    Yildiz AR, Saitou K (2011) Topology synthesis of multicomponent structural assemblies in continuum domains. J Mech Des 133(1):011008–011009CrossRefGoogle Scholar
  11. 11.
    Yildiz BS, Lekesiz H, Yildiz AR (2016) Structural design of vehicle components using gravitational search and charged system search algorithms. Mater Test 58(1):79–81CrossRefGoogle Scholar
  12. 12.
    Victoria M, Querin OM, Díaz C, Martí P (2016) liteITD a MATLAB graphical user interface (GUI) program for topology design of continuum structures. Adv Eng Softw 100:126–147CrossRefGoogle Scholar
  13. 13.
    Savsani VJ, Tejani GG, Patel VK, Savsani P (2017) Modified meta-heuristics using random mutation for truss topology optimization with static and dynamic constraints. J Comput Des Eng 4(2):106–130Google Scholar
  14. 14.
    Assimi H, Jamali A, Nariman-zadeh N (2017) Sizing and topology optimization of truss structures using genetic programming. Swarm Evolut Comput 37:90–103CrossRefGoogle Scholar
  15. 15.
    Kaveh A, Ilchi Ghazaan M (2015) Hybridized optimization algorithms for design of trusses with multiple natural frequency constraints. Adv Eng Softw 79:137–147CrossRefGoogle Scholar
  16. 16.
    Rahami H, Kaveh A, Gholipour Y (2008) Sizing, geometry and topology optimization of trusses via force method and genetic algorithm. Eng Struct 30(9):2360–2369CrossRefGoogle Scholar
  17. 17.
    Assimi H, Jamali A (2018) A hybrid algorithm coupling genetic programming and Nelder–Mead for topology and size optimization of trusses with static and dynamic constraints. Expert Syst Appl 95:127–141CrossRefGoogle Scholar
  18. 18.
    Tejani GG, Savsani VJ, Bureerat S, Patel VK (2018) Topology and size optimization of trusses with static and dynamic bounds by modified symbiotic organisms search. J Comput Civil Eng 32(2):04017085–04017011CrossRefGoogle Scholar
  19. 19.
    Tejani GG, Savsani VJ, Bureerat S, Patel VK, Savsani P (2018) Topology optimization of truss subjected to static and dynamic constraints by integrating simulated annealing into passing vehicle search algorithms. Eng Comput. Google Scholar
  20. 20.
    Kaveh A, Zolghadr A (2011) Shape and size optimization of truss structures with frequency constraints using enhanced charged system search algorithm. Asian J Civil Eng 12:487–509Google Scholar
  21. 21.
    Šilih S, Kravanja S, Premrov M (2010) Shape and discrete sizing optimization of timber trusses by considering joint flexibility. Adv Eng Softw 41(2):286–294CrossRefzbMATHGoogle Scholar
  22. 22.
    Lieu Q, Do DTT, Lee J (2018) An adaptive hybrid evolutionary firefly algorithm for shape and size optimization of truss structures with frequency constraints. Comput Struct 195:99–112CrossRefGoogle Scholar
  23. 23.
    Tejani GG, Savsani VJ, Patel VK (2016) Adaptive symbiotic organisms search (SOS) algorithm for structural design optimization. J Comput Des Eng 3(3):226–249Google Scholar
  24. 24.
    Panagant N, Bureerat S (2018) Truss topology, shape and sizing optimization by fully stressed design based on hybrid grey wolf optimization and adaptive differential evolution. Eng Optim 47:1–17CrossRefGoogle Scholar
  25. 25.
    Noilublao N, Bureerat S (2011) Simultaneous topology, shape and sizing optimisation of a three-dimensional slender truss tower using multiobjective evolutionary algorithms. Comput Struct 89(23–24):2531–2538CrossRefGoogle Scholar
  26. 26.
    Mortazavi A, Toğan V (2016) Simultaneous size, shape, and topology optimization of truss structures using integrated particle swarm optimizer. Struct Multidiscip Optim 54(4):715–736MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tomšič P, Duhovnik J (2014) Simultaneous topology and size optimization of 2D and 3D trusses using evolutionary structural optimization with regard to commonly used topologies. Adv Mech Eng 6:864–867Google Scholar
  28. 28.
    Zhou M, Pagaldipti N, Thomas HL, Shyy YK (2004) An integrated approach to topology, sizing, and shape optimization. Struct Multidiscip Optim 26(5):308–317CrossRefGoogle Scholar
  29. 29.
    Tejani GG, Savsani VJ, Patel VK, Mirjalili S (2018) An improved heat transfer search algorithm for unconstrained optimization problems. J Comput Des EngGoogle Scholar
  30. 30.
    Tejani GG, Savsani VJ, Patel VK, Savsani PV (2018) Size, shape, and topology optimization of planar and space trusses using mutation-based improved metaheuristics. J Comput Des Eng 5(2):198–214Google Scholar
  31. 31.
    Tejani GG, Savsani VJ, Patel VK, Bureerat S (2017) Topology, shape, and size optimization of truss structures using modified teaching-learning based optimization. Adv Comput Des 2:313–331Google Scholar
  32. 32.
    Gomes HM (2011) Truss optimization with dynamic constraints using a particle swarm algorithm. Expert Syst Appl 38(1):957–968CrossRefGoogle Scholar
  33. 33.
    Kaveh A, Zolghadr A (2014) Democratic PSO for truss layout and size optimization with frequency constraints. Comput Struct 130:10–21CrossRefGoogle Scholar
  34. 34.
    Yang IT, Hsieh YH (2012) Reliability-based design optimization with cooperation between support vector machine and particle swarm optimization. Eng Comput 29(2):151–163CrossRefGoogle Scholar
  35. 35.
    Kaveh A, Talatahari S (2010) A charged system search with a fly to boundary method for discrete optimum design of truss structures. Asian J Civ Eng 11:277–293Google Scholar
  36. 36.
    Kaveh A, Zakian P (2014) Enhanced bat algorithm for optimal design of skeletal structures. Asian J Civil Eng 15:179–212Google Scholar
  37. 37.
    Kaveh A, Jafari L, Farhoudi (2015) Truss optimization with natural frequency constraints using a dolphin echolocation algorithm. Asian J Civil Eng 16:29–46Google Scholar
  38. 38.
    Mandhyan A, Srivastava G, Krishnamoorthi S (2016) A novel method for prediction of truss geometry from topology optimization. Eng Comput 33(1):95–106CrossRefGoogle Scholar
  39. 39.
    Cheng MY, Prayogo D (2016) A novel fuzzy adaptive teaching–learning-based optimization (FATLBO) for solving structural optimization problems. Eng Comput 33(1):55–69CrossRefGoogle Scholar
  40. 40.
    Kaintura A, Spina D, Couckuyt I, Knockaert L, Bogaerts W, Dhaene T (2017) A kriging and stochastic collocation ensemble for uncertainty quantification in engineering applications. Eng Comput 33(4):935–949CrossRefGoogle Scholar
  41. 41.
    Shobeir V (2016) The optimal design of structures using ACO and EFG. Eng Comput 32(4):645–653CrossRefGoogle Scholar
  42. 42.
    Shi J, Cao J, Cai K, Wang Z, Qin QH (2016) Layout optimization for multi-bi-modulus materials system under multiple load cases. Eng Comput 32(4):745–753CrossRefGoogle Scholar
  43. 43.
    Ho-Huu V, Nguyen-Thoi T, Vo-Duy T, Nguyen-Trang T (2016) An adaptive elitist differential evolution for optimization of truss structures with discrete design variables. Comput Struct 165:59–75CrossRefGoogle Scholar
  44. 44.
    Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37(5):447–465CrossRefzbMATHGoogle Scholar
  45. 45.
    Kaveh A, Laknejadi K (2012) A hybrid evolutionary graph-based multi-objective algorithm for layout optimization of truss structures. Acta Mech 224(2):343–364MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Veldhuizen DAV, Lamont G (2000) Multiobjective evolutionary algorithms: analyzing the state-of-the-art. Evol Comput 8(2):125–147CrossRefGoogle Scholar
  47. 47.
    Zhou A, Qu BY, Li H, Zhao SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evolut Comput 1(1):32–49CrossRefGoogle Scholar
  48. 48.
    Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3:257–271CrossRefGoogle Scholar
  49. 49.
    Srisomporn S, Bureerat S (2008) Geometrical design of plate-fin heat sinks using hybridization of MOEA and RSM. IEEE Trans Compon Packag Technol 31(2):351–360CrossRefGoogle Scholar
  50. 50.
    Pholdee N, Bureerat S (2013) Hybridisation of real-code population-based incremental learning and differential evolution for multiobjective design of trusses. Inf Sci 223:136–152MathSciNetCrossRefGoogle Scholar
  51. 51.
    Santawy MFE, Ahmed AN (2012) A multi-objective chaotic harmony search technique for structural optimization. Comput Sci 1:9–12Google Scholar
  52. 52.
    Gholizadeh S, Baghchevan A (2017) Multi-objective seismic design optimization of steel frames by a chaotic meta-heuristic algorithm. Eng Comput 33(4):1045–1060CrossRefGoogle Scholar
  53. 53.
    Nasrollahi A (2017) Optimum shape of large-span trusses according to AISC-LRFD using ranked particles optimization. J Constr Steel Res 134:92–101CrossRefGoogle Scholar
  54. 54.
    Lim J, Lee B (2015) A semi-single-loop method using approximation of most probable point for reliability-based design optimization. Struct Multidiscip Optim 53:745–757MathSciNetCrossRefGoogle Scholar
  55. 55.
    Coletti G, Petturiti D, Vantaggi B (2017) Fuzzy memberships as likelihood functions in a possibilistic framework. Int J Approx Reas 88:547–566MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Komal (2018) Fuzzy reliability analysis of DFSMC system in LNG carriers for components with different membership function. Ocean Eng 155:278–294CrossRefGoogle Scholar
  57. 57.
    Kharmanda G, Sharabatey S, Ibrahim H, Makhloufi A, Elhami A (2009) Reliability-based design optimization using semi-numerical strategies for structural engineering applications. Int J CAD/CAM 9:1–16Google Scholar
  58. 58.
    Lombardi M, Haftka RT (1998) Anti-optimization technique for structural design under load uncertainties. Comput Methods Appl Mech Eng 157(1–2):19–31CrossRefzbMATHGoogle Scholar
  59. 59.
    Pholdee N, Bureerat S (2012) Performance enhancement of multiobjective evolutionary optimisers for truss design using an approximate gradient. Comput Struct 106–107:115–124CrossRefGoogle Scholar
  60. 60.
    Noilublao N, Bureerat S (2013) Simultaneous topology, shape, and sizing optimisation of plane trusses with adaptive ground finite elements using MOEAs. Math Probl Eng 2013:1–9CrossRefGoogle Scholar
  61. 61.
    Park S, Choi S, Sikorsky C, Stubbs N (2004) Efficient method for calculation of system reliability of a complex structure. Int J Solids Struct 41(18–19):5035–5050CrossRefzbMATHGoogle Scholar
  62. 62.
    Greiner D, Hajela P (2011) Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations. Struct Multid Optim 45(4):589–613MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Ho-Huu V, Nguyen-Thoi T, Le-Anh L, Nguyen-Trang T (2016) An effective reliability-based improved constrained differential evolution for reliability-based design optimization of truss structures. Adv Eng Softw 92:48–56CrossRefGoogle Scholar
  64. 64.
    Kroetz HM, Tessari RK, Beck AT (2017) Performance of global metamodeling techniques in solution of structural reliability problems. Adv Eng Softw 114:394–404CrossRefGoogle Scholar
  65. 65.
    Zhao Y, Zhang X, Lu Z (2018) Complete monotonic expression of the fourth-moment normal transformation for structural reliability. Comput Struct 196:186–199CrossRefGoogle Scholar
  66. 66.
    Ma X, Liu F, Qi Y, Li L, Jiao L, Liu M, Wu J (2014) MOEA/D with Baldwinian learning inspired by the regularity property of continuous multiobjective problem. Neurocomputing 145:336–352CrossRefGoogle Scholar
  67. 67.
    Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731CrossRefGoogle Scholar
  68. 68.
    Sivasubramani KS, Swarup S (2011) Environmental/economic dispatch using multi-objective harmony search algorithm. Electr Power Syst Res 81:1778–1785CrossRefGoogle Scholar
  69. 69.
    Aittokoski T, Miettinen K (2010) Efficient evolutionary approach to approximate the Pareto-optimal set in multiobjective optimization, UPS-EMOA. Optim Methods Softw 25(6):841–858MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Bureerat S, Sriworamas K (2013) Simultaneous topology and sizing optimization of a water distribution network using a hybrid multiobjective evolutionary algorithm. Appl Soft Comput 13(8):3693–3702CrossRefGoogle Scholar
  71. 71.
    Das S, Suganthan PN (2011) Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31CrossRefGoogle Scholar
  72. 72.
    Mlakar M, Petelin D, Tušar T, Filipič B (2015) GP-DEMO: differential evolution for multiobjective optimization based on Gaussian process models. Eur J Oper Res 243(2):347–361MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Robič T, Filipič B (2005) DEMO: differential evolution for multiobjective optimization. Lect Notes Comput Sci 3410:520–533CrossRefzbMATHGoogle Scholar
  74. 74.
    Coello Coello CA, Reyes-Sierra M (2006) Multi-objective particle swarm optimizers: a survey of the state-of the-art. Int J Comput Intell Res 2(3):287–308MathSciNetCrossRefGoogle Scholar
  75. 75.
    Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: MHS’95 proceedings of the sixth international symposium on micro machine and human science, pp 39–43Google Scholar
  76. 76.
    Ahrari A, Deb K (2016) An improved fully stressed design evolution strategy for layout optimization of truss structures. Comput Struct 164:127–144CrossRefGoogle Scholar
  77. 77.
    Techasen T, Wansasueb K, Panagant N, Pholdee N, Bureerat S (2018) Multiobjective simultaneous topology, shape and sizing optimization of trusses using evolutionary optimizers. IOP Conf Ser Mater Sci Eng 370:012029CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Teerapol Techasen
    • 1
  • Kittinan Wansasueb
    • 1
  • Natee Panagant
    • 1
  • Nantiwat Pholdee
    • 1
  • Sujin Bureerat
    • 1
    Email author
  1. 1.Sustainable and Infrastructure Research and Development Center, Department of Mechanical Engineering, Faculty of EngineeringKhon Kaen UniversityKhon KaenThailand

Personalised recommendations