Risk Averse Second Best Toll Pricing
Existing second best toll pricing (SBTP) models determine optimal tolls of a subset of links in a transportation network by minimizing certain system objective, while the traffic flow pattern is assumed to follow user equilibrium (UE). We show in this paper that such toll design approach is risk prone, which tries to optimize for the best-case scenario, if the UE problem has multiple solutions. Accordingly, we propose a risk averse SBTP approach aiming to optimize for the worst-case scenario, which can be formulated as a min-max problem. We establish a general solution existence condition for the risk averse model and discuss in detail that such a condition may not be always satisfied in reality. In case a solution does not exist, it is possible to replace the exact UE solution set by a set of approximate solutions. This replacement guarantees the solution existence of the risk averse model. We then develop a scheme such that the solution set of an affine UE can be explicitly expressed. Using this explicit representation, an improved simplex method can be adopted to solve the risk averse SBTP model.
KeywordsTransportation Hull Lution Toll
Unable to display preview. Download preview PDF.
The authors would like to thank the four anonymous referees for their insightful comments and helpful suggestions on an earlier version of this paper. The first author also appreciates the insightful discussions with Dr. Paul Tseng at the University of Washington on the fortified descent simplex method. This work is supported in part by Air Force Office of Scientific Research Grant FA9550-07-1-0389, and National Science Foundation Grants DMI- 052 1953, DMS-0427689 and IIS-0511905.
- Ban, X., Ferris, M.C. and Liu, H. (2007). An MPCC formulation and numerical studies for continuous network design with asymmetric user equilibria. Submitted for publication.Google Scholar
- Ban, X., Ferris, M.C. and Tang, L. (2009a). Risk neutral second best toll pricing. Technical Report RPI-WP-200902, http://www.rpi.edu/banx/publications/RNSBTP.pdf.
- Ban, X., Liu, H. and Ferris, M.C. (2006). A link-node based complementarity model and its solution algorithm for asymmetric user equilibria. Proceedings of the 85th Annual Meeting of Transportation Research Board (CD-ROM).Google Scholar
- Ban, X., Lu, S., Ferris, M.C. and Liu, H. (2009b). Considering risk-taking in second best toll pricing. Technical Report RPI-WP-200901, Internet Link: http://www.rpi.edu/banx/publications/RASBTP.pdf.
- Bertsekas, D.P. (1995). Nonlinear Programming. Athena Scientific.Google Scholar
- Facchinei, F. and Pang, J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems: Vol. I & II. Springer.Google Scholar
- Lawphongpanich, S., and Hearn, D. (2004). An MPEC approach to second-best toll pricing. Mathematical Programming B, 101, 33-55.Google Scholar
- Lawphongpanich, S., Hearn, D. and Smith, M.J. (2006). Mathematical And Computational Models for Congestion Charging. Springer.Google Scholar
- Luo, Z.Q., Pang, J.S. and Ralph, D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press.Google Scholar
- Nagurney, A. (1998). Network Economics: A Variational Inequality Approach (2nd Edition). Kluwer Academic Publishers.Google Scholar
- Robinson, S.M. (1981). Some continuity properties of polyhedral multifunctions. Mathematical Programming Studies, 14, 206-214.Google Scholar
- Rockafellar, R.T. and Wets, R.J.B. (1998). Variational Analysis. Number 317 in Grundlehren der Mathematischen Wissenschaften. Springer-Verlag.Google Scholar
- Sheffi, Y. (1985). Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall.Google Scholar
- Yang, H. and Huang, H.J. (2005). Mathematical and Economic Theory of Road Pricing. Elsevier.Google Scholar