Abstract.
This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints. Several equivalent nonlinear programming formulations for the two problems are discussed. One formulation leads to properties that are of interest to transportation economists. Another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems. Numerical results using transportation networks from the literature are also presented.
Similar content being viewed by others
References
Arnott, R., Small, K.: The economics of traffic congestion. Am. Sci. 20 (2), 123–127 (1994)
Auchmuty, G.: Variational principles for variational inequalities. Numer. Func. Anal. Optim. 10, 863–874 (1989)
Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Dordrect, The Netherlands 1998
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Second Edition, John Wiley & Sons, New York, New York, 1993
Brooke, A., Kendirck, D., Meeraus, A.: GAMS: A User’s Guide. The Scientific Press, South San Francisco, California, 1992
Brotcorne, L., Labbé, M., Marcotte, P., Savard, G.: A bilevel model for toll optimization on a multicommodity transportation network. Trans. Sci. 35 (4), 345–358 (2001)
Dial, R.: Minimum revenue congestion pricing Part I: A fast algorithm for the single-origin case. Trans. Res. B 33 (3), 189–202 (1999)
Dial, R.: Minimum revenue congestion pricing Part II: A fast algorithm for the general case. Trans. Res. B 34 (8), 645–665 (2000)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I and II. Springer, New York, 2003
Ferrari, P.: Road network toll pricing and social welfare. Trans. Res. B 36 (5), 471–483 (2002)
Fletcher, R., Leyffer, S.: Numerical experience with solving MPECs as NLPs. Numerical Analysis Report NA/210, Department of Mathematics, University of Dundee, 2002
Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. Numerical Analysis Report NA/209, Department of Mathematics, University of Dundee, 2002
Florian, M., Hearn, D.W.: Network equilibrium models and algorithms. Chapter 6 of Handbooks in Operations Research and Management Science. In: Network Routing, M.O. Ball, T.L. Magnanti, C.L. Monma, G.L. Nemhauser (eds.), Volume 8, North-Holland, New York, 1995
Florian, M., Guélat, J., Spiess, H.: An efficient implementation of the PARTAN variant of the linear approximation method for the network equilibrium problem. Networks 17, 319–339 (1987)
Fisk, C.S., Boyce, D.E.: Alternative variational inequality formulations of the network equilibrium-travel choice problem. Trans. Sci. 17 (4), 454–463 (1983)
Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)
Fukushima, M.: Merit functions for variational inequality and complementarity problems. In: Nonlinear Optimization and Applications, G. Di Pillo, F. Giannessi (eds.), Plenum Publishing Corporation, New York, 1996
Gartner, N.H.: Optimal traffic assignment with elastic demands: A Review, Part I. Analysis Framework. Trans. Sci. 14 (2), 174–191 (1980)
Gauvin, J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12, 136–138 (1977)
Hearn, D.W., Ramana, M.V.: Solving congestion toll pricing models. In: Equilibrium and Advanced Transportation Modeling, P. Marcotte, S. Nguyen (eds.), Kluwer Academic Publishers, Boston, 1998, pp. 109–124
Hearn, D.W., Yildirim, M.B.: A toll pricing framework for traffic assignment problems with elastic demands. In: Current Trends in Transportation and Network Analysis: Miscellanea in Honor of Michael Florian, M. Gendreau, P. Marcotte (eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001
Hearn, D.W., Yildirim, M.B.: A first-best toll pricing framework for variable demand traffic assignment problems. Trans. Res. B. To appear 2004
Johansson-Stenman, O., Sterner, T.: What is the scope for environmental road pricing? In: Road pricing, Traffic Congestion and Environment, K.J. Button, E.T. Verhoef (eds.), Edward Elgar Publishing Limited, London, England, 1998
Labbé, M., Marcotte, P., Savard, G.: A bilevel model of taxation and its application to optimal highway pricing. Manage. Sci. 44 (12), 1608–1622 (1998)
Larsson, T., Patriksson, M.: Side constrained traffic equilibrium models–traffic management through link tolls. In: Equilibrium and Advanced Transportation Modelling, P. Marcotte, S. Nguyen (eds.), Kluwer Academic Publishers, New York, 1998, pp. 125–151
LeBlanc, L.J., Morlok, E.K., Pierskalla, W.P.: An efficient approach to solving the road network equilibrium traffic assignment problem. Trans. Res. 9, 309–318 (1975)
Lim, A.: Transportation network design problems: An MPEC approach. Ph.D. Dissertation, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland, 2002
Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints, Cambridge University Press, New York, New York, 1996
Mangasarian, O.L., Fromovitz, S.: The Fritz John optimal necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)
Marcotte, P.: Network optimization with continuous control parameters. Trans. Sci. 17 (2), 181–197 (1983)
Marcotte, P., Zhu, D.L.: Exact and inexact penalty methods for the generalized bilevel programming problem. Math. Program. A 74 (2), 141–157 (1996)
McDonald, J.F.: Urban highway congestion: An analysis of second-best tolls. Trans. 22, 353–369 (1995)
Migdalas, A.: Bilevel programming in traffic planning: models, methods and challenge. J. Global Optim. 7, 381–405 (1995)
Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998
Patriksson, M., Rockafellar, R.T.: A mathematical model and descent algorithm for bilevel traffic management. Trans. Sci. 36 (3), 271–291 (2002)
Scheel, H., Scholtes, S.: Mathematical programs with complimentarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (1), 1–22 (2000)
Shimizu, K., Ishizuka, Y., Bard, J.F.: Nondifferentiable and Two-Level Mathematical Programming. Kluwer Academic Publishers, Boston, 1997
Verhoef, E.T.: Second-best congestion pricing in general static transportation networks with elastic demands. Region. Sci. Urban Econ. 32 (3), 281–310 (2002)
Verhoef, E.T.: Second-best congestion pricing in general networks: Algorithms for finding second-best optimal toll levels and toll points. Trans. Res. B 36 (8), 707–729 (2002)
Yang, H., Bell, M.G.H.: Traffic restraint, road pricing and network equilibrium. Trans. Res. B 33 (4), 303–314 (1997)
Yang, H., Lam, W.H.K.: Optimal road tolls under conditions of queuing and congestion. Trans. Res. A 30 (5), 319–332 (1996)
Yildirim, M.B.: Congestion toll pricing models and methods for variable demand networks. Dissertation, Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, 2001
Zhang, H.M., Ge, Y.E.: Modeling variable demand equilibrium under second-best road pricing. Working Paper, Institute of Transportation Studies, University of California at Davis, 2002
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by NSF grants DMI-9978642 and DMI-0300316.
Rights and permissions
About this article
Cite this article
Lawphongpanich, S., Hearn, D. An MPEC approach to second-best toll pricing. Math. Program., Ser. A 101, 33–55 (2004). https://doi.org/10.1007/s10107-004-0536-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-004-0536-5