Optimization and Engineering

, Volume 15, Issue 1, pp 137–165 | Cite as

A surrogate-based multiobjective metaheuristic and network degradation simulation model for robust toll pricing



Robust transportation network design problems generally rely on systems engineering methods that share common research gaps. First, problem sizes are constrained due to the use of multi-objective solution algorithms that are notoriously inefficient due to computationally expensive function evaluations. Second, link disruptions at a network level are difficult to model realistically. In this paper, a stochastic search metaheuristic based on radial basis functions is proposed for constrained multiobjective problems. It is proven to converge, and compared with conventional metaheuristics for four representative test problems. A scenario simulation method based on multivariate Bernoulli random variables that accounts for correlations between link failures is proposed to sample scenarios for a mean-variance toll pricing problem. Four tests are conducted on the classical Sioux Falls network to gain some insights into the algorithm, the simulation model, and to the robust toll pricing problem. The first test empirically measures the efficiency of the simulation algorithm and approximate Pareto set by obtaining a standard error in the ε-indicator measure for a given number of scenarios and iterations. The second test compares the dominance of the proposed heuristic’s solutions with a conventional multiobjective genetic algorithm by comparing the average ε-indicator. The third test quantifies the gap due to falsely assuming that link failures are independent of each other when they are not. The last test quantifies the value of having the flexibility to adapt a Pareto set of toll pricing solutions to changing probability regimes such as peak and off-peak hurricane or snow seasons.


Multiobjective Surrogate model Radial basis function Robust optimization Multivariate Bernoulli Toll pricing problem Network design 



One of the authors, Joseph Chow, was financially supported while he was a Ph.D. student by the FHWA Office of Professional and Corporate Development under the Dwight D. Eisenhower Transportation Graduate Fellowship. The two authors were also partly supported by the University of California Transportation Center under the research project entitled “Large Scale Real Option Models for Network Investment Planning and Operational Risk Hedging”. The support is gratefully acknowledged. The algorithm presented here for the MO-RBF supercedes the one presented in Joseph Chow’s dissertation.


  1. Abdulaal M, LeBlanc LJ (1979) Continuous equilibrium network design models. Transp Res, Part B, Methodol 13:19–32 CrossRefGoogle Scholar
  2. Abramson MA, Asaki TJ, Dennis JE, Magallanez R, Sottile MJ (2011) An efficient class of direct search surrogate methods for solving expensive optimization problems with CPU-time-related functions. Struct Multidiscip Optim 45(1):53–64 CrossRefGoogle Scholar
  3. Bakkaloglu M, Wylie JJ, Wang C, Ganger GR (2002) Modeling correlated failures in survivable storage systems. In: Fast abstract at international conference on dependendable systems and networks Google Scholar
  4. Ban X, Ferris MC, Tang L, Lu S (2013) Risk neutral second best toll pricing. Transp Res, Part B, Methodol 48:67–87 CrossRefGoogle Scholar
  5. Björkman M, Holmström K (2000) Global optimization of costly nonconvex functions using radial basis functions. Optim Eng 1(4):373–397 MATHMathSciNetCrossRefGoogle Scholar
  6. Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805 MATHMathSciNetCrossRefGoogle Scholar
  7. Boyles S, Kockelman KM, Waller TS (2010) Congestion pricing under operational, supply-side uncertainty. Transp Res, Part C, Emerg Technol 18(4):519–535 CrossRefGoogle Scholar
  8. Budiman M (2004) Matlab utility: Latin hypercube sampling. budiman@acss.usyd.edu.au Google Scholar
  9. Campigotto P, Passerini A, Battiti R (2013) Active learning of Pareto fronts. Departmental technical report, University of Trento, DISI-13-001 Google Scholar
  10. Chafekar D, Shi L, Rasheed K, Xuan J (2005) Multiobjective GA optimization using reduced models. IEEE Trans Syst Man Cybern, Part C, Appl Rev 35(2):261–265 CrossRefGoogle Scholar
  11. Chang CH, Tung YK, Yang JC (1994) Monte Carlo simulation for correlated variables with marginal distributions. J Hydraul Eng 120(3):313–331 CrossRefGoogle Scholar
  12. Chen A, Yang H, Lo HK, Tang WH (2002) Capacity reliability of a road network: an assessment methodology and numerical results. Transp Res, Part B, Methodol 36(3):225–252 CrossRefGoogle Scholar
  13. Chen A, Subprasom K, Ji Z (2006) A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem. Optim Eng 7(3):225–247 MATHMathSciNetCrossRefGoogle Scholar
  14. Chow JYJ (2010) Flexible management of transportation networks under uncertainty. PhD Dissertation, University of California, Irvine, 256 pp Google Scholar
  15. Chow JYJ, Regan AC, Arkhipov DI (2010) Fast converging global heuristic for continuous network design problem using radial basis functions. Transp Res Rec 2196:102–110 CrossRefGoogle Scholar
  16. Curtis W, Zikan K, Sowizral H (2006) Random sampling for multivariate Bernoulli variables. US Patent 7,006,954 B1, 28 Feb 2006 Google Scholar
  17. Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Multidiscip Optim 14(1):63–69 CrossRefGoogle Scholar
  18. Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New York MATHGoogle Scholar
  19. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197 CrossRefGoogle Scholar
  20. Du ZP, Nicholson A (1997) Degradable transportation systems: sensitivity and reliability analysis. Transp Res, Part B, Methodol 31(3):225–237 CrossRefGoogle Scholar
  21. Ekström J, Engelson L, Rydergren C (2009) Heuristic algorithms for a second-best congestion pricing problem. NETNOMICS 10(1):85–102 CrossRefGoogle Scholar
  22. Eldred MS, Dunlavy DM (2006) Formulations for surrogate-based optimization with data fit, multifidelity, and reduced-order models. In: Proceedings of the 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, VA Google Scholar
  23. Evans GW, Stuckman B, Mollaghasemi M (1991) Multicriteria optimization of simulation models. In: Nelson BL, Kelton WD, Clark GM (eds) Proceedings of the winter simulation conference, pp 894–900 Google Scholar
  24. Fang H, Horstemeyer MF (2006) Global response approximation with radial basis functions. Eng Optim 38(4):407–424 MathSciNetCrossRefGoogle Scholar
  25. Farina M, Amato P (2005) Linked interpolation-optimization strategies for multicriteria optimization problems. Soft Comput 9(1):54–65 CrossRefGoogle Scholar
  26. Frostig E (2001) Comparison of portfolios which depend on multivariate Bernoulli random variables with fixed marginals. Insur Math Econ 29(3):319–331 MATHMathSciNetCrossRefGoogle Scholar
  27. Gardner LM, Boyles SD, Waller ST (2011) Quantifying the benefit of responsive pricing and travel information in the stochastic congestion pricing problem. Transp Res, Part A, Policy Pract 45(3):204–218 CrossRefGoogle Scholar
  28. Gutmann HM (2001) A radial basis function method for global optimization. J Glob Optim 19(3):201–227 MATHMathSciNetCrossRefGoogle Scholar
  29. Karoonsoontawong A, Waller ST (2007) Robust dynamic continuous network design problem. Transp Res Rec 2029:58–71 CrossRefGoogle Scholar
  30. Le MN, Ong YS, Menzel S, Jin Y, Sendhoff B (2012) Evolution by adapting surrogates. Evol Comput. doi: 10.1162/EVCO_a_00079 Google Scholar
  31. LeBlanc LJ, Morlok EK, Pierskalla WP (1974) An accurate and efficient approach to equilibrium traffic assignment on congested networks. Transp Res Rec 491:12–23 Google Scholar
  32. Li H, Bliemer MCJ, Bovy PHL (2007) Optimal toll design from reliability perspective. In: Proc., 6th triennnial conference on transportation analysis, Phuket, Thailand Google Scholar
  33. Li X, Ouyang Y (2010) A continuum approximation approach to reliable facility location design under correlated probabilistic disruptions. Transp Res, Part B, Methodol 44(4):535–548 CrossRefGoogle Scholar
  34. Li ZC, Lam WHK, Wong SC, Sumalee A (2012) Environmentally sustainable toll design for congested road networks with uncertain demand. Int J Sustain Transp 6(3):127–155 CrossRefGoogle Scholar
  35. Lo HK, Tung YK (2003) Network with degradable links: capacity analysis and design. Transp Res, Part B, Methodol 37(4):345–363 CrossRefGoogle Scholar
  36. Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26(6):369–395 MATHMathSciNetCrossRefGoogle Scholar
  37. Mulvey JM, Vanderbei RJ, Zenio SA (1995) Robust optimization of large-scale systems. Oper Res 43(2):264–281 MATHMathSciNetCrossRefGoogle Scholar
  38. Ng MW, Lo HK (2013) Regional air quality conformity in transportation networks with stochastic dependencies: a theoretical copula-based model. In: Networks and spatial economics. doi: 10.1007/s11067-013-9185-7 Google Scholar
  39. Pilát M, Neruda R (2012) Aggregate meta-models for evolutionary multiobjective and many-objective optimization. Neurocomputing. doi: 10.1016/j.neucom.2012.06.043 Google Scholar
  40. Regis RG, Shoemaker CA (2007) A stochastic radial basis function method for the global optimization of expensive functions. INFORMS J Comput 19(4):497–509 MATHMathSciNetCrossRefGoogle Scholar
  41. Rodríguez JE, Medaglia AL, Coello Coello CA (2009) Design of a motorcycle frame using neuroacceleration strategies in MOEAs. J Heuristics 15(2):177–196 MATHCrossRefGoogle Scholar
  42. Schütze O, Laumanns M, Coello CA, Dellnitz M, Talbi EG (2008) Convergence of stochastic search algorithms to finite size Pareto set approximations. J Glob Optim 41(4):559–577 MATHCrossRefGoogle Scholar
  43. Sharma S, Ukkusuri SV, Mathew TV (2009) A Pareto optimal multi-objective optimization for the robust transportation network design problem. Transp Res Rec 2090:95–104 CrossRefGoogle Scholar
  44. Shan S, Wang GG (2005) An efficient Pareto set identification approach for multiobjective optimization on black-box functions. J Mech Des 127(5):866–874 MathSciNetCrossRefGoogle Scholar
  45. Simpson TW, Peplinski JD, Koch PN, Allen JK (2001) Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 17(2):129–150 MATHCrossRefGoogle Scholar
  46. Siu BWY, Lo HK (2008) Doubly uncertain transportation network: degradable capacity and stochastic demand. Eur J Oper Res 191(1):166–181 MATHMathSciNetCrossRefGoogle Scholar
  47. Small KA (1992) Using the revenues from congestion pricing. Transportation 19:359–381 CrossRefGoogle Scholar
  48. Smilowitz KR, Daganzo CF (2007) Continuum approximation techniques for the design of integrated package distribution systems. Networks 50(3):183–196 MATHMathSciNetCrossRefGoogle Scholar
  49. Sumalee A, Watling DP (2003) Travel time reliability in a network with dependent link modes and partial driver response. J East Asia Soc Transp Stud 5:1687–1701 Google Scholar
  50. Sumalee A, Watling DP (2008) Partition-based approach for estimating travel time reliability with dependent failure probability. J Adv Transp 42(3):213–238 MathSciNetCrossRefGoogle Scholar
  51. Sumalee A, Xu W (2011) First-best marginal cost toll for a traffic network with stochastic demand. Transp Res, Part B, Methodol 45(1):41–59 CrossRefGoogle Scholar
  52. Teugels JL (1990) Some representations of the multivariate Bernoulli and binomial distributions. J Multivar Anal 32(2):256–268 MATHMathSciNetCrossRefGoogle Scholar
  53. Verhoef ET (2002a) Second-best pricing in general static transportation networks with elastic demands. Reg Sci Urban Econ 32(3):281–310 CrossRefGoogle Scholar
  54. Verhoef ET (2002b) Second-best congestion pricing in general networks. Heuristic algorithms for finding second-best optimal toll levels and toll points. Transp Res, Part B, Methodol 36(8):707–729 CrossRefGoogle Scholar
  55. Voutchkov I, Keane A (2010) Multi-objective optimization using surrogates. In: Tenne Y, Goh CK (eds) Computational intelligence in optimization: adaptation, learning, and optimization, vol 7, pp 155–175 CrossRefGoogle Scholar
  56. Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129(4):370–380 MathSciNetCrossRefGoogle Scholar
  57. Wang H, Mao W, Shao H (2013) Stochastic congestion pricing among multiple regions: competition and cooperation. J Appl Math 2013:1–12 MathSciNetGoogle Scholar
  58. Wilson B, Cappelleri D, Simpson TW, Frecker M (2001) Efficient Pareto frontier exploration using surrogate approximations. Optim Eng 2(1):31–50 MATHMathSciNetCrossRefGoogle Scholar
  59. Yang H, Bell MGH (1997) Traffic restraint, road pricing and network equilibrium. Transp Res, Part B, Methodol 31(4):303–314 CrossRefGoogle Scholar
  60. Yang H, Huang HJ (1998) Principle of marginal-cost pricing: how does it work in a general road network? Transp Res, Part A, Policy Pract 32(1):45–54 CrossRefGoogle Scholar
  61. Yang H, Zhang X (2003) Optimal toll design in second-best link-based congestion pricing. Transp Res Rec 1857:85–92 CrossRefGoogle Scholar
  62. Yang BS, Yeun YS, Ruy WS (2002) Managing approximation models in multiobjective optimization. Struct Multidiscip Optim 24(2):141–156 CrossRefGoogle Scholar
  63. Yao T, Wei MM, Zhang B, Friesz T (2012) Congestion derivatives for a traffic bottleneck with heterogeneous commuters. Transp Res, Part B, Methodol 46(10):1454–1473 CrossRefGoogle Scholar
  64. Yin Y (2008) Robust optimal traffic signal timing. Transp Res, Part B, Methodol 42(10):911–924 CrossRefGoogle Scholar
  65. Yin Y, Madanat SM, Lu X-Y (2009) Robust improvement schemes for road networks under demand uncertainty. Cent Eur J Oper Res 198:470–479 MATHMathSciNetCrossRefGoogle Scholar
  66. Zhang HM, Ge YE (2004) Modeling variable demand equilibrium under second-best road pricing. Transp Res, Part B, Methodol 38(8):733–749 CrossRefGoogle Scholar
  67. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195 CrossRefGoogle Scholar
  68. Zitzler E, Thiele L, Laumanns M, Fonseca CM, da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Civil EngineeringRyerson UniversityTorontoCanada
  2. 2.Department of Computer Science, Institute of Transportation StudiesUniversity of California, IrvineIrvineUSA

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