, Volume 45, Issue 5, pp 693–704 | Cite as

On the Weibull cost estimation of building frames designed by simulated annealing

  • Ignacio Paya-ZafortezaEmail author
  • Víctor Yepes
  • Fernando González-Vidosa
  • Antonio Hospitaler
Simulation, Optimization & Identification


This paper proposes a general methodology to determine the number of numerical tests required to provide a solution for a heuristic optimization problem with a user-defined accuracy as compared to a global optimal solution. The methodology is based on the extreme value theory and is explained through a problem of cost minimization for reinforced concrete building frames. Specifically, 1000 numerical experiments were performed for the cost minimization of a two-bay and four-floor frame using the Simulated Annealing (SA) algorithm. Analysis of the results indicates that (a) a three-parameter Weibull distribution function fits the results well, (b) an objective and general procedure can be established to determine the number of experiments necessary to solve an optimization problem with a heuristic which generates independent random solutions, and (c) a small number of experiments is enough to obtain good results for the structural engineer.


Optimization Reinforced concrete Weibull distribution Extreme value theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Grierson DE (1994) Practical optimization of structural steel frameworks. In: Adeli H (ed) Advances in design optimization. Taylor & Francis, London Google Scholar
  2. 2.
    Hernández S, Fontan A (2002) Practical applications of design optimization. WIT Press, Southampton Google Scholar
  3. 3.
    Dreo J, Petrowsky A, Siarry P, Taillard E, Chatterjee A (2006) Metaheuristics for hard optimization. Methods and case studies. Springer, Berlin zbMATHGoogle Scholar
  4. 4.
    van Laarhoven PJM, Aarts EHL (1987) Simulated annealing: theory and applications. Kluwer Academic, Dordrecht zbMATHGoogle Scholar
  5. 5.
    Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, New York zbMATHGoogle Scholar
  6. 6.
    Adeli H, Sarma KC (2006) Cost optimization of structures. Fuzzy logic, genetic algorithms and parallel computing. Wiley, Chichester CrossRefGoogle Scholar
  7. 7.
    Goldberg DE, Samtani MP (1986) Engineering optimization via genetic algorithms. In: ASCE proceedings of the ninth conference on electronic computation, New York, pp 471–482 Google Scholar
  8. 8.
    Coello CA, Christiansen AD, Santos F (1997) A simple genetic algorithm for the design of reinforced concrete beams. Eng Comput 13(4):185–196 CrossRefGoogle Scholar
  9. 9.
    Panigrahi SK, Chakraverty S, Mishra BK (2009) Vibration based damage detection in a uniform strength beam using genetic algorithm. Meccanica. doi: 10.1007/s11012-009-9207-1 MathSciNetGoogle Scholar
  10. 10.
    Bassir DH, Zapico JL, González MP, Alonso R (2007) Identification of a spatial linear model based on earthquake-induced data and genetic algorithm with parallel selection. Int J Simul Multidiscip Des Optim 1(1):39–48 CrossRefGoogle Scholar
  11. 11.
    Sid B, Domaszewski M, Peyraut F (2007) An adjacency representation for structural topology optimization using genetic algorithm. Int J Simul Multidiscip Des Optim 1(1):49–54 CrossRefGoogle Scholar
  12. 12.
    Balling RJ, Yao X (1997) Optimization of reinforced concrete frames. ASCE J Struct Eng 123(2):193–202 CrossRefGoogle Scholar
  13. 13.
    Ceranic B, Fryer C, Bines RW (2001) An application of simulated annealing to the optimum design of reinforced concrete retaining structures. Comput Struct 79(17):1569–1581 CrossRefGoogle Scholar
  14. 14.
    González-Vidosa F, Yepes V, Alcalá J, Carrera M, Perea C, Payá-Zaforteza I (2008) Optimization of reinforced concrete structures by simulated annealing. In: Tan CM (ed) Simulated annealing. I-Tech Education and Publishing, Vienna, pp 307–320. Available in Accessed 6 Feb 2009 Google Scholar
  15. 15.
    Paya-Zaforteza I, Yepes V, Hospitaler A, González-Vidosa F (2009) CO2-optimization of reinforced concrete frames by simulated annealing. Eng Struct 31(7):1501–1508 CrossRefGoogle Scholar
  16. 16.
    Yepes V, Alcalá J, Perea C, González-Vidosa F (2008) A parametric study of optimum earth-retaining walls by simulated annealing. Eng Struct 30(3):821–830 CrossRefGoogle Scholar
  17. 17.
    Perea C, Alcalá J, Yepes V, González-Vidosa F, Hospitaler A (2008) Design of reinforced concrete bridge frames by heuristic optimization. Adv Eng Softw 39(3):676–688 CrossRefGoogle Scholar
  18. 18.
    Martínez F, Yepes V, Hospitaler A, González-Vidosa F (2007) Ant colony optimization of reinforced concrete bridge piers of rectangular hollow section. In: Proceedings of the ninth international conference on the application of artificial intelligence to civil, structural and environmental engineering (AICIVIL-COMP2007), St. Julians, Malta, paper 38 Google Scholar
  19. 19.
    Payá I, Yepes V, González-Vidosa F, Hospitaler A (2008) Multiobjective optimization of reinforced concrete building frames by simulated annealing. Comput-Aided Civ Infrastruct Eng 23:596–610 CrossRefGoogle Scholar
  20. 20.
    Kicinger R, Arciszewski T, De Jong K (2005) Evolutionary computation and structural design: a survey of the state of the art. Comput Struct 83:1943–1978 CrossRefGoogle Scholar
  21. 21.
    Nieto F, Hernández S, Jurado JA (2009) Optimum design of long-span suspension bridges considering aeroelastic and kinematic constraints. Struct Multidiscip Optim 39(2):133–151 CrossRefGoogle Scholar
  22. 22.
    Marannano G, Mariotti GV (2008) Structural optimization and experimental analysis of composite material panels for naval use. Meccanica 43(2):251–262 zbMATHCrossRefGoogle Scholar
  23. 23.
    Callegari M, Palpacelli MC (2008) Prototype design of a translating parallel robot. Meccanica 43(2):133–151 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Guadagni L (2008) Development of a collaborative optimization tool for the sizing design of aerospace structures. Int J Simul Multidiscip Des Optim 2(3):187–192 CrossRefGoogle Scholar
  25. 25.
    McRoberts K (1971) A search model for evaluating combinatorially explosive problems. Oper Res 19:1331–1349 zbMATHCrossRefGoogle Scholar
  26. 26.
    Golden BL, Alt FB (1979) Interval estimation of a global optimum for large combinatorial problems. Nav Res Logist Q 26(1):69–77 zbMATHCrossRefGoogle Scholar
  27. 27.
    Bettinger P, Boston K, Kim YH, Zhu J (2007) Landscape-level optimization using tabu search and stand density-related forest management prescriptions. Eur J Oper Res 176:1265–1282 zbMATHCrossRefGoogle Scholar
  28. 28.
    Payá-Zaforteza I (2007) Optimización heurística de pórticos de edificación de hormigón armado (Heuristic optimization of reinforced concrete building frames). PhD thesis, Departamento de Ingeniería de la Construcción, Universidad Politécnica de Valencia, Valencia (in Spanish) Google Scholar
  29. 29.
    Ministerio de Fomento (1998) EHE code of structural concrete. Ministerio de Fomento, Madrid (in Spanish) Google Scholar
  30. 30.
    Ministerio de Fomento (1988) NBE AE-88. Code about the actions to be considered in buildings. Ministerio de Fomento, Madrid (in Spanish) Google Scholar
  31. 31.
    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680 CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Cerny V (1985) Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J Opt Theory Appl 45(1):41–51 zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Medina JR (2001) Estimation of incident and reflected waves using simulated annealing. ASCE J Waterw Port Coast Ocean Eng 127(4):213–221 CrossRefGoogle Scholar
  34. 34.
    Weibull W (1951) A statistical distribution function of wide applicability. ASME J Appl Mech Trans 18(3):293–297 zbMATHGoogle Scholar
  35. 35.
    Fisher R, Tippett L (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc Camb Philos Soc 24:180–190 zbMATHCrossRefGoogle Scholar
  36. 36.
    Conover WJ (1971) Practical nonparametric statistics. Wiley, New York Google Scholar
  37. 37.
    Dannenbring DG (1977) Procedures for estimating optimal solution values for large combinatorial problems. Manag Sci 23(12):1273–1283 zbMATHCrossRefGoogle Scholar
  38. 38.
    Vasko FJ, Wilson JR (1984) An efficient heuristic for large set covering problems. Nav Res Logist Q 31:163–171 zbMATHCrossRefGoogle Scholar
  39. 39.
    ReliaSoft (2007) Weibull++7 user’s guide. Reliasoft, Tucson Google Scholar
  40. 40.
    Efron B (1979) Bootstrap methods. Ann Stat 7:1–26 zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Zoubir AM, Boashash B (1998) The bootstrap and its application in signal processing. IEEE Signal Process Mag 15(1):56–67 CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Ignacio Paya-Zaforteza
    • 1
    Email author
  • Víctor Yepes
    • 1
  • Fernando González-Vidosa
    • 1
  • Antonio Hospitaler
    • 1
  1. 1.ICITECH, Departamento de Ingeniería de la Construcción y Proyectos de Ingeniería CivilUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations