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Tolled multi-class traffic equilibria and toll sensitivities

  • Research Paper
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EURO Journal on Transportation and Logistics

Abstract

We review properties of tolled equilibria in road networks, with users differing in their time values, and study corresponding sensitivities of equilibrium link flows w.r.t. tolls. Possible applications include modeling of individual travellers that have different trip purposes (e.g. work, business, leisure, etc.) and therefore perceive the relation between travel time and monetary cost in dissimilar ways. The typical objective is to reduce the total value of travel time (TVT) over all users. For first best congestion pricing, where all links in the network can be tolled, the solution can be internalized through marginal social cost (MSC) pricing. The MSC equilibrium typically has to be implemented through fixed tolls. The MSC as well as the fixed-toll equilibrium problems can be stated as optimization problems, which in general are convex in the fixed-toll case and non-convex in the MSC case. Thus, there may be several MSC equilibria. Second-best congestion pricing, where one only tolls a subset of the links, is much more complex, and equilibrium flows, times and TVT are not in general differentiable w.r.t. tolls in sub-routes used by several classes. For generic tolls, where the sets of shortest paths are stable, we show how to compute Jacobians (w.r.t positive tolls) of link flows and times as well as of the TVT. This can be used in descent schemes to find tolls that minimize the TVT at least locally. We further show that a condition of independent equilibrium cycles, together with a natural extension of the single class regularity condition of strict complementarity, leads to genericity, and hence existence of said Jacobians.

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Notes

  1. This assumes that flows of all classes have the same influence on the link time. This can usually be achieved by measuring the class flows in passenger car units (PCU:s). In this case, the toll levied per vehicle may be \( p_{a} \) or some class dependent positive multiple of \( p_{a} \). The analysis of the current paper can straightforwardly be performed also for these situations. But to retain notational simplicity we do the analysis for the case that all PCU:s equal 1.

  2. The construction of the sets \( {\mathcal{C}}_{k} \) and \( {\bar{\mathcal{C}}}_{0} \) can be performed in following way. First, for each origin and class, find the equilibrium cycles in the corresponding reduced SP-graph, and select a subset as a basis. Then for each class, select a basis \( {\bar{\mathcal{C}}}_{k} \) as a subset of the basis cycles for the different origins. Put in \( \overline{\overline{{\mathcal{C}}}}_{0} \) any untolled cycle in \( {\bar{\mathcal{C}}}_{k} \) that is a class k equilibrium cycle for some other class k’. Finally, define \( {\mathcal{C}}_{k} = {\bar{\mathcal{C}}}_{k} { \setminus }\overline{\overline{{\mathcal{C}}}}_{0} \) and \( {\bar{\mathcal{C}}}_{0} \) as a basis of \( \overline{\overline{{\mathcal{C}}}}_{0} . \)

References

  • Bazaraa MS, Jarvis J, Sherali HD (2009) Linear Programming and Network Flows. Wiley, NJ

    Book  Google Scholar 

  • Beckmann M, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven

    Google Scholar 

  • Clarke F (1983) Optimization and nonsmooth analysis. Wiley Interscience, New York

    Google Scholar 

  • Clarke F (1989) Methods of dynamic and nonsmooth optimization, regional conference series in applied mathematics, vol 57. SIAM, Philadephia

  • Dafermos SC (1973) Toll patterns for multiclass-user transportation networks. Transp Sci 7:211–223

    Article  Google Scholar 

  • Dial RB (1999a) Network-optimized road pricing: part I: a parable and a model. Oper Res 47:54–64

    Article  Google Scholar 

  • Dial RB (1999b) Network-optimized road pricing: Part II: algorithms and examples. Oper Res 47:327–336

    Article  Google Scholar 

  • Ekström J, Engelson L, Rydergren C (2009) Heuristic algorithms for a second-best congestion pricing problem NETNOMICS 10:85–102

    Google Scholar 

  • Eliasson, J. (2000), The use of average values of time in road pricing. A note on a common misconception. In: Eliasson J (ed) Transport and location analysis. Dissertation, Dept. of Infrastructure and Planning, Royal Institute of Technology, Stockholm

  • Engelson L, Lindberg PO, Daneva M (2003) Multi-class user equilibria under social marginal cost pricing. In: Leopold-Wildburger U, Rendl F, Wäscher G (eds) Operations research proceedings 2002. Springer, Berlin, pp 174–179

  • Engelson L, Lindberg PO (2006) Congestion pricing of road networks with users having different time values. In: Lawphongpanich S, Hearn D, Smith M (eds) Mathematical and computational models for congestion charging, 81–104, applied optimization, 101. Springer, New York

    Google Scholar 

  • Hare W, Sagastizabal C (2010) A redistributed proximal bundle method for nonconvex optimization. SIAM J Optim 20(5):2442–2473

    Article  Google Scholar 

  • Hearn DW, Ramana MV (1998) Solving congestion toll pricing models. In: Marcotte P, Nguyen S (eds) Equilibrium and advanced transportation modeling. Kluwer Academic Publishers, New York, pp 109–124

  • Hearn DW, Yildirim MB (2002) A toll pricing framework for traffic assignment problems with elastic demand. In: Gendreau M, Marcotte P (eds) Transportation and network analysis: current trends. Kluwer Academic Publishers, New York

  • Inregia (2001) Case study: Österleden. A basis for planning of transport systems in cities (in Swedish). Inregia, Stockholm

  • Leurent F (1998) Sensitivity and error analysis of the dual criteria traffic assignment model. Transp Res 32B:189–204

    Article  Google Scholar 

  • Lindberg PO (2010) A note on two papers by dial. In: Lindberg PO. Aspects of static multi-class traffic equilibria under congestion pricing. Licentiate thesis, KTH. TRITA-TEC-LIC, ISSN 1653-445X. http://kth.diva-portal.org/smash/record.jsf?searchId=2&pid=diva2:319920

  • Lindberg PO (2013) Sensitivity analysis of traffic equilibria with applications to OD-estimation, talk at the 2013 hEART conference in Stockholm

  • Lindberg PO, Engelson L (2004) Convexification of the traffic equilibrium problem with social marginal cost tolls, operations research proceedings 2003. Springer, Berlin, pp 141–148

    Google Scholar 

  • Lu S (2008) Sensitivity of static traffic user equilibria with perturbations in arc cost function and travel demand. Transp Sci. 42:105–123

    Article  Google Scholar 

  • Patriksson M (1994) The traffic assignment problem: models and methods. VSP, Utrecht

    Google Scholar 

  • Patriksson M (2004) Sensitivity analysis of traffic equilibria. Transp Sci. 38:258–281

    Article  Google Scholar 

  • Patriksson M, Rockafellar RT (2003) Sensitivity analysis of aggregated variational inequality problems, with application to traffic equilibria. Transp Sci. 37:56–68

    Article  Google Scholar 

  • Sandholm WH (2002) Evolutionary implementation and congestion pricing. Rev Ecol Stud 68:667–689

    Article  Google Scholar 

  • Tobin RL, Friesz TL (1988) Sensitivity analysis for equilibrium network flow. Transp Sci. 22:242–250

    Article  Google Scholar 

  • Verhoef ET (2002) Second-best congestion pricing in general networks: heuristic algorithms for finding second-best optimal toll levels and toll points. Transp Res 36B:707–729

    Article  Google Scholar 

  • Verhoef ET, Nijkamp P, Rietveld P (1995) Second-best regulation of road transport externalities. J Transp Econ Policy 29(4):147–167

    Google Scholar 

  • Yang H (1997) Sensitivity analysis for the elastic demand network equilibrium problem with applications. Transp Res 31B:55–70

    Article  Google Scholar 

  • Yang H, Huang H-J (2005) Mathematical and economic theory of road pricing. Emerald Group Publishing, West Yorkshire

  • Yang H, Bell MGH (2007) Sensitivity analysis of network traffic equilibria revisited: the corrected approach. In: Heydecker B (ed) Mathematics in transport. Elsevier, Amsterdam, pp 373–411

  • Yang H, Huang H-J (2004) The multi-class, multi-criteria traffic network equilibrium and systems optimum problem. Transp Res 38B:1–15

    Article  Google Scholar 

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Acknowledgments

The main parts of this paper were delivered at the TSL workshop Congestion Management of Transportation Systems on the Ground and in the Air, at Asilomar 2011. This research was partially supported by the Centre for Transport studies, KTH, Sweden.

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Authors and Affiliations

  1. Centre for Traffic Studies, and Department of Transport Science, KTH Royal Institute of Technology, 10044, Stockholm, Sweden

    P. O. Lindberg

  2. Department of Transport Science, KTH Royal Institute of Technology, Stockholm, 10044, Sweden

    Leonid Engelson

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  1. P. O. Lindberg
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  2. Leonid Engelson
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Correspondence to P. O. Lindberg.

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Lindberg, P.O., Engelson, L. Tolled multi-class traffic equilibria and toll sensitivities. EURO J Transp Logist 4, 197–222 (2015). https://doi.org/10.1007/s13676-014-0058-0

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