Structural and Multidisciplinary Optimization

, Volume 55, Issue 1, pp 77–89 | Cite as

Variations in the application of a budget of uncertainty optimization approach

  • Gregory D. JosephEmail author


It is possible to consider uncertainty simultaneously with the design process. In fact, a Budget of Uncertainty(BoU) can be determined alongside the design solution, allowing the determination of uncertainty intervals for selected design variables and problem parameters. This paper presents a new strategy for optimization under uncertainty which provides for this simultaneous design and uncertainty determination. To test the theory, a simple Taylor series expansion strategy is used to propagate uncertainty in a design problem’s objectives and constraints and a new BoU design algorithm is formulated. Due to the need for competing objectives, nominal performance and robust design, the new formulation is a multiobjective problem with primary and secondary weights to allow for lexicographic weights of uncertain parameters and variation between optimal and robust solutions. This paper compares and contrasts three different Goal Programming techniques as solutions to the multiobjective problem. Within the paper, the term Budget of Uncertainty (BoU) is used to describe the fundamental idea of uncertainty allocation across design variables and problem parameters as well as for a shorthand to describe the presented formulation. An engineering design problem, that of a helical spring, is presented to further illustrate the new method, and an uncertainty budget is considered which trades uncertainty in coil diameter against uncertainty in wire diameter.


Robust optimization Robust design Robustness Lexicographic weighting Optimality versus robustness Goal programming 



The author wishes to thank the journal peer review team whose detailed comments and suggestions greatly enhanced the final quality of this work.


  1. Agarwal H (2004) Reliability based design optimization: formulations and methodologies, PhD Thesis, University of Notre DameGoogle Scholar
  2. Agarwal H, Renaud JE, Preston EL, Padmanabhan D (2004) Uncertainty quantification using evidence theory in multidisciplinary design optimization. Reliab Eng Syst Saf 85(1):281– 294CrossRefGoogle Scholar
  3. Athan T, Papalambros PY (1996) A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim 27:155–176CrossRefGoogle Scholar
  4. Azarm S (1984) Local monotonicity in optimal design. PhD thesis, University of MichiganGoogle Scholar
  5. Azarm S, Li WC (1989) Multi-level design optimization using global monotonicity analysis. J Mech Transm Autom Des 111(2):259–263CrossRefGoogle Scholar
  6. Azarm S, Boyars A, Li M (2006) A new deterministic approach using sensitivity region measures for multi-objective robust and feasibility robust design optimization. Trans ASME 128:874–883CrossRefGoogle Scholar
  7. Bascaran E, Bannerot RB, Mistree F (1989) Hierarchical selection decision support problems in conceptual design. Eng Optim 14(3):207–238CrossRefGoogle Scholar
  8. Ben-Tal A, Boyd S, Nemirovski A (2006) Extending scope of robust optimization: Comprehensive robust counterparts of uncertain problems. Math Program 107(1–2):63–89MathSciNetCrossRefzbMATHGoogle Scholar
  9. Ben Tal A, Den Hertog D, De Waegenaere AM, Melenberg B, Rennen G (2011) Robust solutions of optimization problems affected by uncertain probabilities. center working paper series no. 2011-061. Available at SSRN. or doi: 10.2139/ssrn.1853428
  10. Ben-Tal A, Hertog D, Vial JP (2012) Deriving robust counterparts of nonlinear uncertain inequalities. CentER Discussion Paper Series No. 2012-053. Available at
  11. Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88.3:411–424MathSciNetCrossRefzbMATHGoogle Scholar
  13. Ben-Tal A, Nemirovski A (2002) On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM J Optim 12.3:811–833MathSciNetCrossRefzbMATHGoogle Scholar
  14. Ben-Tal A, Nemirovski A (2003) On approximate robust counterparts of uncertain semidefinite and conic quadratic programs. Syst Model Optim XX:1–22MathSciNetzbMATHGoogle Scholar
  15. Ben-Tal A, Nemirovski A, Roos C (2002) Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM J Optim 13(2):535–560MathSciNetCrossRefzbMATHGoogle Scholar
  16. Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53:464–501MathSciNetCrossRefzbMATHGoogle Scholar
  17. Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52.1:35–53MathSciNetCrossRefzbMATHGoogle Scholar
  18. Bras B, Mistree F (1993) Robust design using compromise decision support problems. Eng Optim 21 (3):213–239CrossRefGoogle Scholar
  19. Bras B, Mistree F (1995) A compromise decision support problem for axiomatic and robust design. J Mech Des 117(1):10–19CrossRefGoogle Scholar
  20. Bridgman P (1922) Dimensional analysis. Yale University Press, New HavenzbMATHGoogle Scholar
  21. Hale E, Zhang Y (2007a) Case studies for a first-order robust nonlinear programming formulation. J Optim Theory Appl 134:27–45MathSciNetCrossRefzbMATHGoogle Scholar
  22. Hamel JM, Azarm S (2011) Reducible uncertain interval design by kriging metamodel assisted multi-objective optimization. J Mech Des 133(3)Google Scholar
  23. Kokkolaras M, Mourelatos ZP, Papalambros PY (2006) Impact of uncertainty quantification on design decisions for a hydraulic-hybrid powertrain engine. In: 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials ConferenceGoogle Scholar
  24. Li M, Azarm S (2008) Multiobjective collaborative robust optimization with interval uncertainty and interdisciplinary uncertainty propagation. J Mech Des 130.8:11Google Scholar
  25. Li M, Williams N, Azarm S (2009) Interval uncertainty reduction and single-disciplinary sensitivity analysis with multi-objective optimization. J Mech Des 131(3)Google Scholar
  26. Marler R, Arora J (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26:369–395MathSciNetCrossRefzbMATHGoogle Scholar
  27. Mistree F, Hughes OF, Bras B (1993) Compromise decision support problem and the adaptive linear programming algorithm. Prog Astronaut Aeronaut 150:251–251Google Scholar
  28. Rooney WC, Biegler LT (1999) Incorporating joint confidence regions into design under uncertainty. Comput Chem Eng 23(10):1563–1575CrossRefGoogle Scholar
  29. Rooney WC, Biegler LT (2001) Design for model parameter uncertainty using nonlinear confidence regions. AIChE J 47(8):1794–1804CrossRefGoogle Scholar
  30. Shupe J, Mistree F, Sobieszanski-Sobieski J (1987) Compromise: An effective approach for the hierarchical design of structural systems. Comput Struct 26(6):1027–1037CrossRefzbMATHGoogle Scholar
  31. Zhang Y (2007b) General robust-optimization formulation for nonlinear. J Optim Theory Appl 132:111–124MathSciNetCrossRefzbMATHGoogle Scholar
  32. Vadde S, Allen J, Mistree F (1994) Compromise decision support problems for hierarchical design involving uncertainty. Comput Struct 52(4):645–658CrossRefzbMATHGoogle Scholar
  33. Zadeh L, IEEE Trans. (1963) Optimality and non-scalar-valued performance criteria. IEEE Trans Autom Control 8:59–60CrossRefGoogle Scholar
  34. Zionts S (1988) Multiple criteria mathematical programming: An updated overview and several approaches. In: Mitra G (ed) Mathematical Models for Decision Support, pp 135–167Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of HoustonHoustonUSA
  2. 2.Technical SpecialistEngineering Research and Consulting (ERC) Inc.HuntsvilleUSA

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