Incremental shifting vector and mixed uncertainty analysis method for reliability-based design optimization

  • Hairui Zhang
  • Hao Wang
  • Yao Wang
  • Dongpao HongEmail author
Research Paper


Reliability-based design optimization (RBDO) is a powerful tool for addressing design problems involving variables with uncertainty characteristics. In practical engineering problems, there may not be sufficient information to build probabilistic distribution models for some crucial parameters. Evidence theory has emerged to deal with inaccuracies of parameters that lack information or knowledge. In this study, the incremental shifting vector and mixed uncertainty analysis method (ISVMUA) is proposed to improve the efficiency in dealing with RBDO problems with both aleatory and epistemic uncertainties based on probability and evidence theory. Higher efficiency is achieved in ISVMUA using the following strategies: (1) the result of the mixed uncertainty analysis is directly employed to update the deterministic optimization formulation by the incremental shifting vector; (2) a new allocation strategy is proposed to reasonably decompose the total target failure probability into all the focal elements of the epistemic uncertainties; and (3) a strategy of feasibility checking is employed to identify the inactive constraints among all the quasi-equivalent probabilistic constraints to simplify the deterministic optimization problem. Four examples are investigated to demonstrate the effectiveness and efficiency of the proposed method.


Reliability-based design optimization (RBDO) Incremental shifting vector Mixed uncertainty Evidence theory 



The authors would like to deeply thank Prof. Wen Yao of the Academy of Military Science for her useful and helpful comments in improving this paper.


  1. Andre JT, Rafael HL, Leandro FFM (2016) A general RBDO decoupling approach for different reliability analysis methods. Struct Multidisc Optim 54:317–332MathSciNetCrossRefGoogle Scholar
  2. Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14(4):227–245CrossRefGoogle Scholar
  3. Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainties in applied mechanics. Elsevier Science, AmsterdamzbMATHGoogle Scholar
  4. Chen Z, Qiu H, Gao L, Su L, Li P (2013) An adaptive decoupling approach for reliability-based design optimization. Comput Struct 117:58–66CrossRefGoogle Scholar
  5. Cho TM, Lee BC (2011) Reliability-based design optimization using convex linearization and sequential optimization and reliability assessment method. Struct Saf 33(2011):42–50CrossRefGoogle Scholar
  6. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339MathSciNetCrossRefGoogle Scholar
  7. Du XP (2006) Uncertainty analysis with probability and evidence theories. The 2006 ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, PA, Sept. 10–13Google Scholar
  8. Du XP (2008) Unified uncertainty analysis by the first order reliability method. J Mech Des 130(9) paper 091401Google Scholar
  9. Du XP, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126(2):225–233CrossRefGoogle Scholar
  10. Du XP, Sudjianto A, Huang B (2005) Reliability-based design with the mixture of random and interval variables. J Mech Des 127(6):1068–2068CrossRefGoogle Scholar
  11. Elishakoff I, Colombi P (1993) Combination of probabilistic and convex models of uncertainty when sarce knowledge is present on acoustic excitation parameters. Comput Methods Appl Mech Eng 104(2):187–209CrossRefGoogle Scholar
  12. Enevoldsen I, Sorensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15(3):169–196CrossRefGoogle Scholar
  13. Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111–121Google Scholar
  14. Hu X, Chen X, Parks T, G, Yao W (2016) Review of improved Monte Carlo methods in uncertainty-based design optimization for aerospace vehicles. Prog Aerosp Sci 86:20–27Google Scholar
  15. Huang ZL, Jiang C, Zhou YS, Luo Z, Zhang Z (2016) An incremental shifting vector approach for reliability-based design optimization. Struct Multidisc Optim 53(3):523–543MathSciNetCrossRefGoogle Scholar
  16. Huang ZL, Jiang C, Zhang Z, Fang T, Han X (2017a) A decoupling approach for evidence-theory-based reliability design optimization. Struct Multidisc Optim 56:647–661CrossRefGoogle Scholar
  17. Huang ZL, Jiang C, Zhou YS, Zheng J, Long XY (2017b) Reliability-based design optimization for problems with interval distribution parameters. Struct Multidisc Optim 55:513–528MathSciNetCrossRefGoogle Scholar
  18. Jiang C, Han X, Lu GY, Liu J, Zhang Z, Bai YC (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200(33):2528–2546CrossRefGoogle Scholar
  19. Jiang C, Zhang Z, Han X, Liu J (2013) A novel evidence-theory-based reliability analysis method for structures with epistemic uncertainty. Comput Struct 129:1–12CrossRefGoogle Scholar
  20. Lee JJ, Lee BC (2005) Efficient evaluation of probabilistic constraints using an encelope function. Eng Optim 37(2):185–200CrossRefGoogle Scholar
  21. Liang J, Mourelatos ZP, Tu J (2004) A single-loop method for reliability-based design optimization. Proceedings of ASME Design Engineering Technical Conferences. Salt Lake City, UTGoogle Scholar
  22. Nikolaidis E, Burdisso R (1988) Reliability based optimization: a safety index approach. Comput Struct 28(6):781–788CrossRefGoogle Scholar
  23. Shafer G (1976) A mathematical theory of evidence. Princeton University Press, PrincetonzbMATHGoogle Scholar
  24. Tu J, Choi KK, Park YH (1999) A new study on reliability based design optimization. J Mech Des 121(4):557–564CrossRefGoogle Scholar
  25. Wu J, Luo Z, Zhang Y, Zhang N, Chen L (2013) Interval uncertain method for multibody mechanical systems using chebyshev inclusion functions. Int J Numer Methods Eng 95(7):608–630MathSciNetCrossRefGoogle Scholar
  26. Xia B, Lu H, Yu D, Jiang C (2015) Reliability-based design optimization of structural systems under hybrid probabilistic and interval model. Comput Struct 160:126–134CrossRefGoogle Scholar
  27. Yang RJ, Gu L (2004) Experience with approximate reliability-based optimization methods. Struct Multidiscip Optim 26(2):152–159MathSciNetCrossRefGoogle Scholar
  28. Yang RJ, Chuang C, Gu L, Li G (2005) Experience with approximate reliability-based optimization methods 2:an exhaust system problem. Struct Multidisc Optim 29(6):488–497CrossRefGoogle Scholar
  29. Yao W, Chen X, Luo W, van Tooren M, Guo J (2011) Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles. Prog Aerosp Sci 47(6):450–479CrossRefGoogle Scholar
  30. Yao W, Chen X, Ouyang Q, van Tooren M (2013a) A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory. Struct Multidisc Optim 48:339–354MathSciNetCrossRefGoogle Scholar
  31. Yao W, Chen X, Huang Y (2013b) Sequential optimization and mixed uncertainty analysis method for reliability-based optimization. AIAA 51(9):2266–2277CrossRefGoogle Scholar
  32. Yi P, Zhu Z, Gong JX (2016) An approximate sequential optimization and reliability assessment method for reliability-based design optimization. Struct Multidisc Optim 54:1367–1378MathSciNetCrossRefGoogle Scholar
  33. Zadeh LA (1982) A note on prototype theory and fuzzy sets. Cognition 12(3):291–297CrossRefGoogle Scholar
  34. Zhang X, Huang HZ (2010) Sequential optimization and reliability assessment for multidisciplinary design optimization under aleatory and epistemic uncertainties. Struct Multidisc Optim 40:165–175CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.China Academy of Launch Vehicle TechnologyBeijingChina

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