Journal of Global Optimization

, Volume 7, Issue 4, pp 381–405 | Cite as

Bilevel programming in traffic planning: Models, methods and challenge

  • Athanasios Migdalas


Well-founded traffic models recognize the individual network user's right to the decision as to when, where and how to travel. On the other hand, the decisions concerning management, control, design and improvement investments are made by the public sector in the interest of the society as a whole. Hence, transportation planning is a characteristic example of a hierarchical process, in which the public sector at one level makes decisions seeking to improve the performance of the network, while at another level the network users make choices with regard to route, travel mode, origin and destination of their travel. Our objective is to provide a review on the current state of research and development in bilevel programming problems that arize in this context, and attract the attention of the global optimization community to this problem class of imense practical importance.


Bilevel programming network problems traffic planning 


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  1. 1.
    E. Aarts and J. Korst (1990)Simulated Annealing and Boltzmann Machines — A Stochastic Approach to Combinatorial Optimization and Neural Computing, John Wiley & Sons, ChichesterGoogle Scholar
  2. 2.
    H. Aashtiani and T. Magnanti (1981) Equilibria on a congested transportation network,SIAM Journal on Algebraic and Discrete Methods, vol. 2, pp. 213–226Google Scholar
  3. 3.
    M.S. Abdulaal and L.J. LeBlanc (1979) Continuous Equilibrium Network Design Models,Transportation Research, vol. 13B, pp. 19–32Google Scholar
  4. 4.
    F.A. Al-Khayyal, R. Horst and P.M. Pardalos (1992) Global Optimization of Concave Functions subject to Quadratic Constraints: An Application in Nonlinear Bilevel Programming, In: [5]., pp. 125–147Google Scholar
  5. 5.
    G. Anandalingam and T.L. Friesz (1992) (Ed.s)Hierarchical Optimization, Annals of Operations Research, vol. 34, J.C. Baltzer AG, Basel.Google Scholar
  6. 6.
    J.P. Aubin (1979)Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam.Google Scholar
  7. 7.
    J.F. Bard (1983) An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem,Operations Research, vol. 31, pp. 670–684Google Scholar
  8. 8.
    J.F. Bard (1988) Convex Two-Level Optimization,Mathematical Programming, vol. 40, pp. 15–27Google Scholar
  9. 9.
    J.F. Bard (1991) Some Properties of the Bilevel Programming Problem,Journal of Optimization Theory and Applications, vol. 68, pp. 371–378Google Scholar
  10. 10.
    J.F. Bard and J.E. Falk (1982) An Explicit Solution to the Multilevel Programming Problem,Computers and Operations Research, vol. 9, pp. 77–100Google Scholar
  11. 11.
    T. Başar and G.J. Olsder (1982)Dynamic Noncooperative Game Theory, Academic Press, London.Google Scholar
  12. 12.
    M. Beckmann, C.B. McGuire and C.B. Winsten (1956)Studies in Economics of Transportation, Yale University PressGoogle Scholar
  13. 13.
    O. Ben-Ayed, C.E. Blair (1990) Computational Difficulties of Bilevel Linear Programming,Operations Research, vol. 38, pp. 556–560Google Scholar
  14. 14.
    O. Ben-Ayed, D. E. Boyce and C.E. Blair (1988) A General Bilevel Linear Programming Formulation of the Network Design Problem,Transportation Research, vol. 22B, pp. 311–318Google Scholar
  15. 15.
    O. Ben-Ayed, C.E. Blair, D.E. Boyce and L.J. LeBlanc (1992) Construction of a Real-World Bilevel Linear Programming Model of the Highway Network Design Problem, In: [5]., pp. 219–254Google Scholar
  16. 16.
    J. Berechman (1984) Highway-Capacity Utilization and Investment in Transportation Corridors,Environment and Planning, vol. 16A, pp. 1475–1488Google Scholar
  17. 17.
    M. Bierlaire and P.L. Toint (1995) MEUSE: An Origin-Destination Matrix Estimator that Exploits Structure,Transportation Research, vol. 29B, pp. 47–60Google Scholar
  18. 18.
    D.E. Boyce (1984) Urban Transportation Network-Equilibrium and Design Models: Recent Achievements and Future Prospects,Environment and Planning, vol. 16A, pp. 1445–1474Google Scholar
  19. 19.
    G.E. Cantarella and A. Sforza (1987) Methods for Equilibrium Network Traffic Signal Setting, In: [75]., pp. 69–89Google Scholar
  20. 20.
    E. Cascetta and S. Nguyen (1988) A Unified Framework for Estimating or Updating Origin-Destination Matrices from Traffic Counts,Transportation Research, vol. 22B, pp. 437–455Google Scholar
  21. 21.
    Y. Chen and M. Florian (1991) The Nonlinear Bilevel Programming Problem — A General Formulation and Optimality Conditions, Centre de recherche sur les transports, Publication No.794, Université de Montréal, C.P.6128, Montréal, Québec, H3C3J7, CanadaGoogle Scholar
  22. 22.
    S. Dafermos (1980) Traffic Equilibrium and Variational Inequalitie,Transportation Science, vol. 14, pp. 42–54Google Scholar
  23. 23.
    S. Dafermos (1982) The General Multimodal Network Equilibrium Problem with Elastic Demand,Networks, vol. 12, pp. 57–72Google Scholar
  24. 24.
    S. Dafermos and F.T. Sparrow (1969) The Traffic Assignment Problem for a General Network,Journal of Research of the National Bureau of Standards, vol. 73B, pp. 91–118Google Scholar
  25. 25.
    C. Daganzo (1982) Unconstrained Extremal Formulations of Some Transportation Equilibrium Problems,Transportation Science, vol 16, pp. 332–361Google Scholar
  26. 26.
    O. Damberg and A. Migdalas (1995) Efficient Solution of Traffic Assignment Problems with a Distributed Simplicial Decomposition Algorithm, (To appear)Google Scholar
  27. 27.
    G.A. Davis (1994) Exact Local Solution of the Continuous Network Design Problem via Stochastic User Equilibrium Assignment,Transportation Research, vol. 28B, pp. 61–75Google Scholar
  28. 28.
    R.S. Dembo and U. Tulowitzki (1988) Computing Equilibria on Large Multicommodity Networks: An Application of Truncated Quadratic Programming Algorithms,Networks, vol. 18, pp. 273–284Google Scholar
  29. 29.
    A.M. deSilva and G.P. McCormick (1992) Implicit Defined Optimization Problems, In: [5], pp. 107–124Google Scholar
  30. 30.
    A.V. Fiacco (1976) Sensitivity Analysis for Nonlinear Programming Using Penalty Methods,Mathematical Programming, vol. 10, pp. 287–311Google Scholar
  31. 31.
    A.V. Fiacco (1983)Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New YorkGoogle Scholar
  32. 32.
    C.S. Fisk (1984) Game Theory and Transportation Systems Modelling,Transportation Research, vol. 18B, pp. 301–313Google Scholar
  33. 33.
    C.S. Fisk (1984) Optimal Signal Controls on Congested Networks,Ninth International Symposium on Transportation and Traffic Theory, VNU Science Press, pp. 197–216Google Scholar
  34. 34.
    C.S. Fisk (1986) A Conceptual Framework for Optimal Transportatiion Systems Planning with Integrated Supply and Demand Models,Transportation Science, vol. 20, pp. 37–47Google Scholar
  35. 35.
    C.S. Fisk (1988) On Combining Maximum Entropy Trip Matrix Estimation with User Assignment,Transportation Research, vol. 22B, pp. 69–73Google Scholar
  36. 36.
    C.S. Fisk (1989) Trip Matrix Estimation from Link Traffic Counts: The Congested Network Case,Transportation Research, vol. 23B, pp. 331–336Google Scholar
  37. 37.
    C.S. Fisk and D. Boyce (1983) A Note on Trip Matrix Estimation from Link Traffic Count Data,Transportation Research, vol. 17B, pp. 245–250Google Scholar
  38. 38.
    C.S. Fisk and S. Nguyen (1982) Solution Algorithms for Network Equilibrium Models with Assymetric User Costs,Transportation Science, vol. 16, pp. 361–381Google Scholar
  39. 39.
    M. Florian and Y. Chen (1991) A Bilevel Programming Approach to Estimating O-D Matrix by Traffic Counts, Centre de recherche sur les transports, Publication No.750, Université de Montréal, C.P.6128, Montréal, Québec, H3C3J7, CanadaGoogle Scholar
  40. 40.
    M. Florian and Y. Chen (1992) A Successive Linear Approximation Method for the Bilevel O-D Matrix Adjustment Problem, Centre de recherche sur les transports, Publication No.807, Université de Montréal, C.P.6128, Montréal, Québec, H3C3J7, CanadaGoogle Scholar
  41. 41.
    M. Florian and Y. Chen (1993) A Coordinate Descent Method for the Bilevel O-D Matrix Adjustment Problem, Centre de recherche sur les transports, Publication No.807, Université de Montréal, C.P.6128, Montréal, Québec, H3C3J7, Canada. (Paper presented atthe IFORS Conference in Lisbon, Portugal, July 1993)Google Scholar
  42. 42.
    T. Friesz (1985) Transportation Network Equilibrium, Design and Aggregation: Key Development and Research Opportunities,Transportation Research, vol. 19A, pp. 413–427Google Scholar
  43. 43.
    T.L. Friesz and P.T. Harker (1985) Properties of the Iterative Optimization-Equilibrium Algorithm,Civil Engineering System, vol. 2, pp. 142–154Google Scholar
  44. 44.
    T.L. Friesz, R.L. Tobin and T. Miller (1988) Algorithms for Spatially Competitive Network Facility-Location,Environment and Planning, vol. 15B, pp. 191–203Google Scholar
  45. 45.
    T.L. Friesz, R.L. Tobin, H. Cho and N.J. Mehta (1990) Sensitivity Analysis Based Heuristic Algorithms for the Mathematics Programs with Variational Inequality Constraints,Mathematical Programming, vol. 48, pp. 265–284Google Scholar
  46. 46.
    T.L. Friesz, H.-J. Cho, N.J. Mehta, R.L. Tobin and G. Anandalingam (1992) A Simulated Annealing Approach to the Network Design Problem with Variational Inequality Constraints,Transportation Science, vol. 26, pp. 18–26Google Scholar
  47. 47.
    N.H. Gartner, S.B. Gershwin, J.D.C. Little and P. Ross (1980), Pilot Study of Computer-Based Urban Traffic Management,Transportation Research, vol. 14B, pp. 203–217Google Scholar
  48. 48.
    P. Hansen, B. Jaumard and G. Savard (1992) New Branch-and-Bound Rules for Linear Bilevel Programming,SIAM Journal on Scientific and Statistical Computing, vol. 13, pp. 1194–1217Google Scholar
  49. 49.
    P.T. Harker and T.L. Friesz (1984) Bounding the Solution of the Continuous Equilibrium Network Design Problem, In: Ninth International Symposium on Transportation and Traffic Theory, VNU Science Press, pp. 233–252Google Scholar
  50. 50.
    A. Haurie and P. Marcotte (1986) A Game Theoretic Approach to Network Equilibrium,Mathematical Programming Study, vol. 26, pp. 252–255Google Scholar
  51. 51.
    D.W. Hearn, S. Lawphongpanich and J.A. Ventura (1985) Finiteness in Restricted Simplicial Decomposition,Operations Research Letters, vol. 4, pp.125–130Google Scholar
  52. 52.
    G. Improta (1987) Mathematical Programming Methods for Urban Network Control, In: [75]., pp. 35–68Google Scholar
  53. 53.
    R. Jeroslow (1985) The Polynomial Hierarchy and a Simple Model for Competitive Analysis,Mathematical Programming, vol. 32, pp. 146–164Google Scholar
  54. 54.
    W. Ködel (1969)Graphentheoretische Methoden und ihre Anwendungen, Springer-Verlag, Berlin, pp. 56–59Google Scholar
  55. 55.
    C.D. Kolstad (1985) A Review of the Literature on Bilevel Mathematical Programming. Los Alamos National Laboratory, Report LA-10284-MS, Los Alamos, NM.Google Scholar
  56. 56.
    C.D. Kolstad and L.S. Lasdon (1990) Derivative Evaluation and Computational Experience with Large Bilevel Mathematical Programming,Journal of Optimization and Applications, vol. 65, pp. 485–499Google Scholar
  57. 57.
    T. Larsson and M. Patriksson (1992) Simplicial Decomposition with Disaggregated Representation for the Traffic Assignment Problem,Transportation Science, vol. 26, pp. 4–17Google Scholar
  58. 58.
    T. Larsson, A. Migdalas and M. Patriksson (1993) The Application of Partial Linearization Algorithm to the Traffic Assignment Problem,Optimization, vol. 28, pp. 47–61Google Scholar
  59. 59.
    S. Lawphongpanich and D.W. Hearn (1984) Simplicial Decomposition of the Asymmetric Traffic Assignment Problem,Transportation Research, vol. 18B, pp. 123–133Google Scholar
  60. 60.
    L.J. LeBlanc (1975) An Algorithm for the Discrete Network Design Problem,Transportation Science, vol. 9, pp. 183–199Google Scholar
  61. 61.
    L.J. LeBlanc and M. Abdulaal (1984) A Comparison of User-Optimum Versus System-Optimum Traffic Assignment in Transportation Network Design,Transportation Research, vol. 18B, pp. 115–121Google Scholar
  62. 62.
    L.J. LeBlanc, R.V. Helgason and D.E. Boyce (1985) Improved Efficiency of the Frank-Wolfe Algorithm for Convex Network Programs,Transportation Science, vol. 19, pp. 445–462Google Scholar
  63. 63.
    L.J. LeBlanc and D.E. Boyce (1986) A Bilevel Programming Algorithm for Exact Solution of the Network Design Problem with User Optimal Flows,Transportation Research, vol. 20B, pp. 259–265Google Scholar
  64. 64.
    P. Marcotte (1983) Network Optimization with Continuous Control Parameters,Transportation Science, vol. 17, pp.181–197Google Scholar
  65. 65.
    P. Marcotte (1986) Network Design Problem with Congestion Effects: A Case of Bilevel Programming,Mathematical Programming, vol. 34, pp. 142–162Google Scholar
  66. 66.
    P. Marcotte (1988) A Note on a Bilevel Programming Algorithm by LeBlanc and Boyce,Transportation Research, vol. 22B, pp. 233–237Google Scholar
  67. 67.
    P. Marcotte and G. Marquis (1992) Efficient Implementation of Heuristics for the Continuous Network Design Problem, In: [5]., pp. 163–176Google Scholar
  68. 68.
    A. Migdalas (1994) A Regularization of the Frank-Wolfe Method and Unification of Certain Nonlinear Programming Methods,Mathematical Programming, vol. 56, 331–345Google Scholar
  69. 69.
    A. Migdalas (1995) When is Stackelberg Equilibrium Pareto Optimum?, In: P. Pardalos, et al (ed.s)Advances in Multicriteria Analysis, Kluwer AcademicGoogle Scholar
  70. 70.
    A. Migdalas and H. Tuy (1995) On the Bilevel Min Norm Problem, (To appear)Google Scholar
  71. 71.
    M. Minoux (1986)Mathematical Programming — Theory and Algorithms, Translated from the Frence by S. Vajda, John Wiley & Sons, ChichesterGoogle Scholar
  72. 72.
    S. Nguyen (1977) Estimation of an O-D Matrix from Network Data — A Network Equilibrium Approach, Centre de recherche sur les transports, Publication No.60, Université de Montréal, C.P.6128, Montréal, Québec, H3C3J7, CanadaGoogle Scholar
  73. 73.
    S. Nguyen (1983) Inferring Origin-Destination Demands from Network Data, Associaziione Italiana di Recerca Operativa, Atti delle Giomate di Lavoro 1983, Napoli — Castel dell' Oro, 26–28 Settembre 1983,(see also [74])Google Scholar
  74. 74.
    S. Nguyen (1984) Estimating Origin-Destination Matrices from Observed Flows, InTransportation Planning Models, M. Florian (Ed.), pp. 363–380Google Scholar
  75. 75.
    A.R. Odoni, L. Bianco and G. Szegö (1987) (Ed.s)Flow Control of Congested Networks, NATO ASI Series, Series F: Computer and Systems Sciences, vol. 38, Springer-Verlag, Berlin.Google Scholar
  76. 76.
    M. Patriksson (1994)The Traffic Assignment Problem — Models and Methods, VSP, Utrecht, The NetherlandsGoogle Scholar
  77. 77.
    H. Poorzahedy and M.A. Turnquist (1982) Approximate Algorithms for the Discrete Network Design,Transportation Research, vol. 16B, pp. 45–55Google Scholar
  78. 78.
    M. Simaan and J.B. Cruz, Jr (1973) On the Stackelberg Strategy in Nonzero-Sum Games,Journal of Optimization Theory and Applications, vol 11, pp. 533–555Google Scholar
  79. 79.
    Y. Sheffi (1985) Urban Transportation Networks, Prentice Hall, Englewood Cliffs, NJGoogle Scholar
  80. 80.
    M.J. Smith (1979) Existence, Uniqueness and Stability of Traffic Equilibria,Transportation Research, vol. 1B, pp. 295–304Google Scholar
  81. 81.
    H. Spiess (1990) A Gradient Approach for the O-D Matrix Adjustment Problem, Centre de recherche sur les transports, Publication No.693, Université de Montréal, C.P.6128, Montréal, Québec, H3C3J7, CanadaGoogle Scholar
  82. 82.
    H. von Stackelberg (1952)The Theory of the Market Economy, Oxford University Press.Google Scholar
  83. 83.
    P.A. Steenbrink (1974) Transport Network Optimization in the Dutch Integral Transportation Study,Transportation Research, vol. 8, pp. 11–27Google Scholar
  84. 84.
    A. Steenbrink (1974)Optimization of Transport Net-works, John Wiley & Sons, LondonGoogle Scholar
  85. 85.
    S. Suh and T.J. Kim (1992) Solving Nonlinear Programming Models of the Equilibrium Network Design Problem: A Comparative Review, In: [5]., pp. 203–218Google Scholar
  86. 86.
    C. Suwansirikul and T.L. Friesz (1987) Equilibrium Decomposed Optimization: A Heuristic for the Continuous Equilibrium Network Design Problem,Transportation Science, vol. 21, pp. 254–263Google Scholar
  87. 87.
    H. Tan, S.B. Gerashwin and M. Athans (1979) Hybrid Optimization in Urban Traffic Networks, Report DOT-TSC-RSP-79-7, MIT, Cambridge, Mass.Google Scholar
  88. 88.
    A.N. Tikhonov and V.Y. Arsenin (1977)Solution of Ill-Posed Problems, Translated from the Russian by F. John, John Wiley & Sons, New York.Google Scholar
  89. 89.
    R.L. Tobin (1986) Sensitivity Analysis for Variational Inequalities,Journal of Optimization Theory and Applications, vol. 48, pp. 191–204Google Scholar
  90. 90.
    R.L. Tobin and T.L. Friesz (1988) Sensitivity Analysis for Equilibrium Network Flows,Transportation Science, vol. 22, pp. 242–250Google Scholar
  91. 91.
    H. Tuy, A. Migdalas and P. Värbrand (1993) A Global Optimization Approach for the Linear Two-level Program,Journal of Global Optimization, vol. 3, pp. 1–23Google Scholar
  92. 92.
    H. Tuy, A. Migdalas and P. Varbrand (1994) A Quasiconcave Minimization Method for Solving Linear Two-Level Programs,Journal of Global Optimization, vol. 4, pp. 243–263Google Scholar
  93. 93.
    G. Ünlu (1987) A Linear Bilevel Programming Algorithm Based on Bicriteria Programming,Computers and Operations Research, vol. 14, pp. 173–179Google Scholar
  94. 94.
    J.H. Van Zuylen and L.G. Willumsen (1980) The Most Likely Trip Matrix Estimation from Traffic Counts,Transportation Research, vol. 14B, pp. 281–293Google Scholar
  95. 95.
    L.N. Vicente and P.H. Calamai (1994) Bilevel and Multilevel Programming: A Bibliographic Review,Journal of Global Optimization, vol. 5, pp. 291–306Google Scholar
  96. 96.
    L. Vicente, G. Savard and J. Júdice (1994) Descent Approaches for Quadratic Bilevel Programming,Journal of Optimization Theory and Applications, vol. 81, pp. 379–399Google Scholar
  97. 97.
    J.G. Wardrop (1952) Some Theoretical Aspects of Road Traffic Research,Proceedings of the Institute of Civil Engineering, Part II, pp. 325–378Google Scholar
  98. 98.
    U.-P. Wen and S.-T. Hsu (1989) A Note on a Linear Bilevel Programming Algorithm Based on Bicreteria Programming,Computers and Operations Research, vol. 16, pp. 79–83Google Scholar
  99. 99.
    H. Yang and S. Yagar (1994) Traffic Assignment and Traffic Control in General Freeway-Arterial Corridor Systems,Transportation Research, vol. 28B, pp. 463–486Google Scholar
  100. 100.
    H. Yang and S. Yagar (1995) Traffic Assignment and Signal Control in Saturated Road Networks,Transportation Research, vol. 29A, pp. 125–139Google Scholar
  101. 101.
    H. Yang, T. Sasaki, Y. Iida and Y. Asakura (1992) Estimation of Origin-Destination Matrices from Link Traffic Counts on Congested Networks,Transportation Research, vol. 26B, pp. 417–434Google Scholar
  102. 102.
    H. Yang, Y. Iida and T. Sasaki (1984) The Equilibrium-Based Origin-Destination Matrix Estimation Problem,Transportation Research, vol. 28B, pp. 23–33Google Scholar
  103. 103.
    H. Yang, S. Yagar, Y. Iida and Y. Asakura (1994) An Algorithm for the Inflow Control Problem on Urban Freeway Networks with User-Optimal Flows,Transportation Research, vol. 28B, pp. 123–139Google Scholar
  104. 104.
    S. Zukhovitshki, R. Polyak and M. Primak (1971) Concave n-person Games (Numerical Methods),Ekonom. i Mat. Metody, vol. 7, pp. 888–900Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Athanasios Migdalas
    • 1
  1. 1.Division of OptimizationDepartment of Mathematics Linköping Institute of Technology83 LinköpingSweden

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