Reliability-Based Robust Optimization Applied to Engineering System Design

  • Fran Sérgio LobatoEmail author
  • Márcio Aurelio da Silva
  • Aldemir Aparecido CavaliniJr.
  • Valder SteffenJr.


Traditionally, concerning the engineering system design, the vector of design variables is considered as being formed by deterministic quantities (free of errors, i.e., without the influence of uncertainties). However, small variations in these quantities can affect the considered objective functions and, consequently, the resulting design of the system. Probabilistic methods have been proposed to deal with this kind of problem. These methodologies can be classified according to two main categories: robust design (RD) and reliability-based design (RBO). In this contribution, reliability and robustness are used simultaneously in engineering system design. The proposed approach is based on the Differential Evolution (DE) algorithm in association with the following strategies: (1) Mean Effective Concept (MEC) (robustness); and (2) Inverse Reliability Analysis (IRA) (reliability-based design). Mathematical and engineering test cases are studied to evaluate the proposed methodology. The obtained results demonstrate that the proposed technique represents an interesting alternative to reliability-based robust design of engineering systems.



The authors acknowledge the Brazilian research agencies CNPq, FAPEMIG (APQ-02284-15), and CAPES for the financial support of this research work through the National Institute of Science and Technology on Smart Structures in Engineering (INCT-EIE).


  1. 1.
    Agarwal, H.: Reliability based design optimization: formulations and methodologies. PhD. thesis, University of Notre Dame (2004)Google Scholar
  2. 2.
    Aoues, Y., Chateauneuf, A.: Benchmark study of numerical methods for reliability-based design. Struct. Multidiscip. Optim. 41, 277–294 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carter, A.D.S.: Mechanical Reliability and Design. Wiley, New York (1997)CrossRefGoogle Scholar
  4. 4.
    Castro, R.E.: Optimization of structures with multi-objective through genetic algorithms. D.Sc. thesis (in Portuguese), COPPE/UFRJ, Rio de Janeiro (2001)Google Scholar
  5. 5.
    Deb, K., Gupta, H.: Introducing robustness in multiobjective optimization. Evol. Comput. 14(4), 463–494 (2006)CrossRefGoogle Scholar
  6. 6.
    Deb, K., Padmanabhan, D., Gupta, S., Mall, A.K.: Handling uncertainties through reliability-based optimization using evolutionary algorithms. IEEE Trans. Evol. Comput. 13(5), 1054–1074 (2009)CrossRefGoogle Scholar
  7. 7.
    Der-Kiureghian, A., De Stefano, M.: Efficient algorithm for second-order reliability analysis. J. Mech. Eng. ASCE1 17(12), 2904–2923 (1991)CrossRefGoogle Scholar
  8. 8.
    Du, X.: Probabilistic engineering design – first order and second reliability methods. University of Missouri, Rolla (2005)Google Scholar
  9. 9.
    Du, X., Chen, W.: Sequential optimization and reliability assessment method for efficient probabilistic design. J. Mech. Des. 126, 225–233 (2004)CrossRefGoogle Scholar
  10. 10.
    Fiessler, B., Neumann, H.-J., Rackwitz, R.: Quadratic limit states in structural reliability. J. Mech. Eng. ASCE 105(4), 661–676 (1979)Google Scholar
  11. 11.
    Gholaminezhad, I., Jamali, A., Assimi, H.: Multi-objective reliability-based robust design optimization of robot gripper mechanism with probabilistically uncertain parameters. Neural Comput. Appl. 1, 1–12 (2016)Google Scholar
  12. 12.
    Jeong, S.B., Park, G.J.: Reliability-based robust design optimization using the probabilistic robustness index and the enhanced single loop single vector approach. In: 10th World Congress on Structural and Multidisciplinary Optimization, Orlando (2013)Google Scholar
  13. 13.
    Keshtegar, B., Chakraborty, S.: An efficient-robust structural reliability method by adaptive finite-step length based on Armijo line search. Reliab. Eng. Syst. Saf. 172, 195–206 (2018)CrossRefGoogle Scholar
  14. 14.
    Lagaros, N.D., Plevris, V., Papadrakakis, M.: Reliability based robust design optimization of steel structures. Int. J. Simul. Multidiscip. Des. Optim. 1, 19–29 (2007)CrossRefGoogle Scholar
  15. 15.
    Lee, M.C.W., Mikulik, Z., Kelly, D.W., Thomson, R.S., Degenhardt, R.: Robust design – a concept for imperfection insensitive composite structures. Compos. Struct. 92(6), 1469–1477 (2010)CrossRefGoogle Scholar
  16. 16.
    Leidemer, M.N.: Proposal of evolutionary robust optimization methodology using the unscented transform applicable to circuits for RF circuits/microwave. MSc. Thesis (in Portuguese), University of Brasilia, Brasilia (2009)Google Scholar
  17. 17.
    Liao, K.W., Ha, C.: Application of reliability-based optimization to earth-moving machine: hydraulic cylinder components design process. Struct. Multidiscip. Optim. 36(5), 523–536 (2008)CrossRefGoogle Scholar
  18. 18.
    Liao, K.W., Ivan, G.: A single loop reliability-based design optimization using EPM and MPP-based PSO. Lat. Am. J. Solids Struct. 11, 826–847 (2014)CrossRefGoogle Scholar
  19. 19.
    Melchers, R.E.: Structural Reliability Analysis and Prediction. Wiley, Chichester (1999)Google Scholar
  20. 20.
    Moreira, F.R., Lobato, F.S., Cavalini, A.A. Jr., Steffen, V. Jr.: Robust multi-objective optimization applied to engineering systems design. Lat. Am. J. Solids Struct. 13, 1802–1822 (2016)CrossRefGoogle Scholar
  21. 21.
    Paenk, I., Branke, J., Jin, Y.: Efficient search for robust solutions by means of evolutionary algorithms and fitness approximation. IEEE Trans. Evol. Comput. 10, 405–420 (2006)CrossRefGoogle Scholar
  22. 22.
    Phadke, M.S.: Quality Engineering Using Robust Design. Prentice Hall, Englewood Cliffs (1989)Google Scholar
  23. 23.
    Qu, X., Haftka, R.T.: Reliability-based design optimization using probabilistic sufficiency factor. Struct. Multidiscip. Optim. 27(5), 314–325 (2004)CrossRefGoogle Scholar
  24. 24.
    Ramu, P., Qu, X., Youn, B.D., Haftka, R.T., Choi, K.K.: Safety factor and inverse reliability measures. In: Proceeding of 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs (2004)Google Scholar
  25. 25.
    Ravichandran, G.: Integrated reliable and robust design. Missouri University of Science and Technology (2011)Google Scholar
  26. 26.
    Ritto, T.G., Sampaio, R., Cataldo, E.: Timoshenko beam with uncertainty on the boundary conditions. J. Braz. Soc. Mech. Sci. Eng. 30(4), 295–303 (2008)CrossRefGoogle Scholar
  27. 27.
    Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23, 470–472 (1952)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sampaio, R., Soize, C.: On measures of nonlinearity effects for uncertain dynamical systems application to a Vibro-impact system. J. Sound Vib. 303, 659–674 (2007)CrossRefGoogle Scholar
  29. 29.
    Shahraki, A.F., Noorossana, R.: A combined algorithm for solving reliability-based robust design optimization problems. J. Math. Comput. Sci. 7, 54–62 (2013)CrossRefGoogle Scholar
  30. 30.
    Soize, C.: A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. J. Sound Vib. 288(3), 623–652 (2005)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Souza, D.L., Lobato, F.S., Gedraite, R.: Robust multiobjective optimization applied to optimal control problems using differential evolution. Chem. Eng. Technol. 1, 1–8 (2015)Google Scholar
  32. 32.
    Storn, R., Price, K.: Differential evolution – a simple and efficient adaptive scheme for global optimization over continuous spaces. Int. Comput. Sci. Inst. 12, 1–16 (1995)Google Scholar
  33. 33.
    Taguchi, G.: Taguchi on Robust Technology Development – Bringing Quality Engineering Upstream. ASME Press, New York (1993)CrossRefGoogle Scholar
  34. 34.
    Thanedar, P.B., Kodiyalam, S.: Structural optimization using probabilistic constraints. Struct. Multidiscip. Optim. 4, 236–240 (1992)CrossRefGoogle Scholar
  35. 35.
    Tichy, M.: First-order third-moment reliability method. Struct. Saf. 16(2), 189–200 (1994)CrossRefGoogle Scholar
  36. 36.
    Wang, S., Li, Q., Savage, G.J.: Reliability-based robust design optimization of structures considering uncertainty in design variables. Math. Probl. Eng. 2015, 1–8 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Wang, Y., Tuo, Y., Yang, S.X., Biglarbegian, M., Fu, M.: Reliability-based robust dynamic positioning for a turret-moored floating production storage and offloading vessel with unknown time-varying disturbances and input saturation (2018). CrossRefGoogle Scholar
  38. 38.
    Wu, Y.T., Shin, Y., Sues, R., Cesare, M.: Safety-factor based approach for probability-based design optimization. In: Proceedings of the 42rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2001)Google Scholar
  39. 39.
    Zhao, Y.G., Ono, T.: A general procedure for first/second-order reliability method (FORM/SORM). Struct. Saf. 21(1), 95–112 (1999)CrossRefGoogle Scholar
  40. 40.
    Zhao, Y.G., Ono, T.: New approximations for SORM: part 2. J. Mech. Eng. ASCE1 25(1), 86–93 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Fran Sérgio Lobato
    • 1
    Email author
  • Márcio Aurelio da Silva
    • 2
  • Aldemir Aparecido CavaliniJr.
    • 2
  • Valder SteffenJr.
    • 2
  1. 1.NUCOP-Laboratory of Modeling, Simulation, Control and Optimization, School of Chemical EngineeringFederal University of UberlândiaUberlândiaBrazil
  2. 2.LMEst-Structural Mechanics Laboratory, School of Mechanical EngineeringFederal University of UberlândiaUberlândiaBrazil

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