Selection of optimal target reliability in RBDO through reliability-based design for market systems (RBDMS) and application to electric vehicle design

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Reliability-based design optimization (RBDO) allows decision-makers to achieve target reliability in product performance under engineering uncertainties. However, existing RBDO studies assume the target reliability as a given parameter and do not explain how to determine the optimal target reliability. From the perspective of the market, designing a product with high target reliability can satisfy many customers and increase market demand, but it can generate a large cost leading to profit reduction of the company. Therefore, the target reliability should be a decision variable which needs to be found to maximize the company profit. This paper proposes a reliability-based design for market systems (RBDMS) framework by integrating RBDO and design for market system (DMS) approaches to find the optimal target reliability. The proposed RBDMS framework is applied to electric vehicle (EV) design problems to validate effect of the target reliability on company profit—or market share—and engineering performances of EV. Several observations about the optimal target reliability are presented from the case study with various scenarios. From the EV design case study, it is verified that the proposed RBDMS framework is an effective way of finding the optimal target reliability that maximizes the company profit, and the optimal target reliability varies depending on the situation of market and competitors.

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Fig. 4
Fig. 5
Fig. 6


SoC :

State of charge of battery

DoD :

Depth of discharge of battery

D :


F :

Additional fraction of nominal capacity

P :

Penalty factor for deeper DoD

A :

Capacity loss factor

σ :

1 − A

Π :


D :

Market demands

MC :

Manufacturing cost

C :

Compensation costs

X :

Deterministic decision variable vector

X power :

Powertrain design variable vector

R :

Target reliability

W :

Warranted battery lifetime

Price :


\( {P}_F^{\mathrm{Target}} \) :

Target probability of failure for probabilistic constraints

g :

Inequality constraint functions

G :

Probabilistic constraint functions

N S :

Number of battery cells in series

N P :

Number of battery cells in parallel

FR :

Final gear ratio

RP e :

Random parameter vector of engineering model

P :

Matrix of probabilistic performances

P MPGe :

Vector of probabilistic MPGe

P range :

Vector of probabilistic driving range

P speed :

Vector of probabilistic top speed

P accel :

Vector of probabilistic acceleration

P Batt :

Vector of probabilistic battery lifetime

PR :

Vector of engineering performances that satisfy the target reliability

A :

Advertised attribute vector

A eng :

Vector of advertised attributes determined from engineering model

f engineering :

Engineering model

f attribute :

Attribute model

f marketing :

Marketing model

f X(x):

Joint probability density function

ΩF :

Failure set


  1. Allen M, Maute K (2002) Reliability-based design optimization of aeroelastic structures. 9th AIAA/ISSMO Symposium and Exhibit on Multidisciplinary Analysis and Optimization, Atlanta

  2. Amazon (2017) Amazon Mechanical Turk. Accessed 31 March 2017

  3. AMESim (2016) AMESim. Siemens Product Lifecycle Management Software, Inc, Munich, Germany. Accessed 1 Dec 2016

  4. Berry IM (2010) The effects of driving style and vehicle performance on the real-world fuel consumption of U.S. light-duty vehicles. MS thesis, Massachusetts Institute of Technology

  5. Dong J, Choi KK, Vlahopoulos N, Wang A, Zhang W (2007) Design sensitivity analysis and optimization of high frequency radiation problems using energy finite element and energy boundary element methods. AIAA J 45(6):1187–1198

  6. Dubarry M, Vuillaume N, Liaw BY (2010) Origins and accommodation of cell variations in Li-ion battery pack modeling. Int J Energy Res 34(2):216–231

  7. Ellingwood B, Galambos TV (1982) Probability-based criteria for structural design. Struct Saf 1:15–26

  8. Energy Efficiency & Renewable Energy (2011a) Advanced vehicle testing activity: 2011 Nissan Leaf, baseline testing results, Technical Report

  9. Energy Efficiency & Renewable Energy (2011b) Advanced vehicle testing activity: 2011 Nissan Leaf, beginning-of-test battery testing results, Technical Report

  10. Environmental Protection Agency (2017) Vehicle and Fuel Emissions Testing: Dynamometer Drive Schedules. Accessed 3 March 2017

  11. Frangopol DM, Maute K (2003) Life-cycle reliability-based optimization of civil and aerospace structures. Comput Struct 81(7):397–410

  12. Frischknecht BD, Whitefoot K, Papalambros PY (2010) On the suitability of econometric demand models in design for market systems. J Mech Des 132(12):121007

  13. Gomadam PM, Weidner JW, Dougal RA, White RE (2002) Mathematical modeling of lithium-ion and nickel battery systems. J Power Sources 110(2):267–284

  14. Green PE, Krieger AM (1996) Individualized hybrid models for conjoint analysis. Manag Sci 42(6):850–867

  15. Hadigol M, Maute K, Doostan A (2015) On uncertainty quantification of lithium-ion batteries: application to an LiC6/LiCoO2 cell. J Power Sources 300:507–524

  16. Helveston JP, Liu Y, Feit EM, Fuchs E, Klampfl E, Michalek JJ (2015) Will subsidies drive electric vehicle adoption? Measuring consumer preferences in the US and China. Transport Res Part A 73:96–112

  17. Huang HZ, Liu ZJ, Murthy DNP (2007) Optimal reliability, warranty and price for new products. IIE Trans 39(8):819–827

  18. J.D. Power (2017) Predicted Reliability. Accessed 28 March 2017

  19. Jing R, Xi Z, Yang XG, Decker E (2014) A systematic framework for battery performance estimation considering model and parameter uncertainties. Int J Prognostics and Health Management 5, 10(2)

  20. Kamble SH, Mathew TV, Sharma GL (2009) Development of real-world driving cycle: case study of Pune India. Transp Res Part D 14(2):132–140

  21. Kang N (2014) Multidomain demand modeling in design for market systems. PhD thesis, University of Michigan

  22. Kang N, Feinberg FM, Papalambros PY (2013) A framework for enterprise-driven product service systems design. Proceedings of the 19th International Conference on Engineering Design, Seoul, Korea, August 4–August 7, ISBN:978–1- 904670-47-6

  23. Kang N, Feinberg FM, Papalambros PY (2015) Integrated decision making in electric vehicle and charging station location network design. J Mech Des 137(6):061402

  24. Kang N, Ren Y, Feinberg FM, Papalambros PY (2016) Public investment and electric vehicle design: a model-based market analysis framework with application to a USA-China comparison study. Des Sci 2:e6

  25. Kang N, Feinberg FM, Papalambros PY (2017) Autonomous electric vehicle sharing system design. J Mech Des 139(1):011402

  26. Kang N, Bayrak A, Papalambros PY (2018) Robustness and real options for vehicle design and investment decisions under gas Price and regulatory uncertainties. J Mech Des 140(10):101404

  27. Lawder MT, Northrop PWC, Subramanian VR (2014) Model-based SEI layer growth and capacity fade analysis for EV and PHEV batteries and drive cycles. J Electrochem Soc 161(14):A2099–A2108

  28. Lee T, Jung J (2008) A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: constraint boundary sampling. Comput Struct 86(13–14):1463–1476

  29. Lee J, Kang HY, Kwon JH, Kwak BM (2009) Reliability of aerodynamic analysis using a moment method. Int J Comput Fluid Dyn 23(6):495–502

  30. Lee I, Choi KK, Gorsich D (2010) Sensitivity analysis of FORM-based and DRM-based performance measure approach for reliability-based design optimization. Int J Numer Methods Eng 82(1):26–46

  31. Lee I, Choi KK, Zhao L (2011) Sampling-based RBDO using the stochastic sensitivity analysis and dynamic kriging method. Struct Multidiscip Optim 44(3):299–317

  32. Lee I, Shin J, Choi KK (2013) Equivalent target probability of failure to convert high-reliability model to low-reliability model for efficiency of sampling-based RBDO. Struct Multidiscip Optim 48(2):235–248

  33. Levin M, Kalal TT (2003) Improving product reliability: strategies and implementation. Wiley, New York

  34. Lewis KE, Chen W, Schmidt LC, Press A (2006) Decision making in engineering design. ASME Press, New York

  35. Lim J, Lee B, Lee I (2015) Sequential optimization and reliability assessment based on dimension reduction method for accurate and efficient reliability-based design optimization. J Mech Sci Technol 29(4):1349–1354

  36. Millner A (2010) Modeling lithium ion battery degradation in electric vehicles. Proc IEEE Conf Innovative Technol Efficient Reliable Elect Supply, Waltham, MA, pp 349–356

  37. Missoum S, Dribusch C, Beran P (2010) Reliability-based design optimization of nonlinear aeroelasticity problems. J Aircr 47(3):992–998

  38. Ning G, Haran B, Popov BN (2003) Capacity fade study of lithium-ion batteries cycled at high discharge rates. J Power Sources 117(1–2):160–169

  39. Noh Y, Choi KK, Lee I (2009) Reduction of ordering effect in reliability-based design optimization using dimension reduction method. AIAA J 47(4):994–1004

  40. Nowak AS (1995) Calibration of LRFD bridge code. Aust J Struct Eng 121(8):1245–1251

  41. Orme B (2009) The CBC/HB System for Hierarchical Bayes Estimation Version 5.0 Technical Paper. Technical Paper Series, Sawtooth Software, Sequim, WA

  42. Park DH, Lee J, Han I (2007) The effect of on-line consumer reviews on consumer purchasing intention: the moderating role of involvement. Int J Electron Commer 11(4):125–148

  43. Peterson SB, Apt J, Whitacre JF (2010) Lithium-ion battery cell degradation resulting from realistic vehicle and vehicle-to-grid utilization. J Power Sources 195(8):2385–2392

  44. Pettit CL (2004) Uncertainty quantification in aeroelasticity: recent results and research challenges. J Aircr 41(5):1217–1229

  45. Plötz P, Jakobsson N, Sprei F, Karlsson S (2017) On the distribution of individual daily vehicle driving distances. Transp Res Part B Methodol 101:213–227

  46. Qu X, Venkataraman S, Haftka RT, Johnson TF (2003) Deterministic and reliability based optimization of composite laminates for cryogenic environments. AIAA J 41(10):2029–2036

  47. Rossi P, Allenby G, McCulloch R (2005) Bayesian statistics and marketing. Wiley, Hoboken, NJ

  48. Santhanagopalan S, White RE (2012) Quantifying cell-to-cell variations in lithium ion batteries. Int J Electrochem 2012:1–10

  49. Shin J, Lee I (2014) Reliability-based vehicle safety assessment and design optimization of roadway radius and speed limit in windy environments. J Mech Des 136(8):1006–1019

  50. Shin J, Lee I (2015) Reliability analysis and reliability-based design optimization of roadway horizontal curves using a first-order reliability method. Eng Optim 47(5):622–641

  51. Thaller LH (1983) Expected cycle life vs. depth of discharge relationships of well-behaved single cells and cell strings. J Power Sources 130(5):986–990

  52. Tong W, Koh WQ, Birgersson E, Mujumdar AS, Yap C (2015) Correlating uncertainties of a lithium-ion battery: a Monte Carlo simulation. Int J Energy Res 39(6):778–788

  53. Train K (2001) A comparison of hierarchical Bayes and maximum simulated likelihood for mixed logit. Paper Presented in University of California, Berkeley, pp 1–13

  54. U.S. Federal Highways Administration (2009) U.S. Federal Highway Administration, 2009. National Household Travel Survey (NHTS). Version 2.0 datasets. November 2010 ed. FHWA, Washington, D.C

  55. Yoo D, Lee I (2014) Sampling-based approach for design optimization in the presence of interval variables. Struct Multidiscip Optim 49(2):253–266

  56. Youn BD, Choi KK, Yang RJ, Gu L (2004) Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multidiscip Optim 26(3–4):272–283

  57. Youn BD, Choi KK, Tang J (2005) Structural durability design optimization and its reliability assessment. Int J Prod Dev 1(3/4):383–401

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The authors would like to thank Alparslan Emrah Bayrak of the University of Michigan for his help in building the engineering model.


This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (No. 2017R1C1B2005266) and the development of thermoelectric power generation system and business model utilizing non-use heat of industry funded by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade Industry & Energy (MOTIE) of the Republic of Korea (No. 20172010000830).

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Correspondence to Namwoo Kang or Ikjin Lee.

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Lee, U., Kang, N. & Lee, I. Selection of optimal target reliability in RBDO through reliability-based design for market systems (RBDMS) and application to electric vehicle design. Struct Multidisc Optim 60, 949–963 (2019).

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  • Reliability-based design optimization (RBDO)
  • Design for market systems (DMS)
  • Electric vehicles
  • Target reliability
  • Uncertainty