Solving 2nd Best Toll Pricing Problems

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Solving 2nd Best Toll Pricing Problems

• Donald W. HearnSiriphong LawphongpanichCenter
  for Applied OptimizationIndustrial and Systems
  EngineeringUniversity of Florida

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Outline

• Introduction
• 2nd Best Toll Pricing Problem as MPEC
• FD-VI
• ED-VI
• Equivalent Formulations
• ED-KKT
• Properties of 2nd Best Tolls
• ED-EX
• Cutting Constraint Algorithm
• Numerical Examples
• Conclusions

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Traffic Congestion
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Electronic Toll Collection Facilities
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Congestion Charging in London
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Publications in Congestion Toll Pricing

• Hearn, D. W., Bounding Flows in Traffic
  Assignment Models,'' Research Report 80-4, Dept.
  of Industrial and Systems Engineering, University
  of Florida, Gainesville, FL, 1980.
• Bergendorff, P., The Bounded Flow Approach to
  Congestion Pricing, Master's Thesis, Center for
  Applied Optimization, University of Florida, and
  Department of Mathematics, Royal Institute of
  Technology, Stockholm, 1995.
• Bergendorff, P., D. W. Hearn, and M. V. Ramana,
  Congestion Toll Pricing of Traffic Networks,
  Network Optimization, P. M. Pardalos, D. W. Hearn
  and W. W. Hager (Eds.), Springer-Verlag Series,
  Lecture Notes in Economics and Mathematical
  Systems, 1997, pp. 51-71.
• Hearn, D. W. and Ramana, M. V., Solving
  Congestion Toll Pricing Models, in Equilibrium
  and Advanced Transportation Modeling, P. Marcotte
  and S. Nguyen (Eds.), Kluwer Academic Publishers,
  1998, pp. 109-124.
• Hearn, D. W. and Yildirim, M. B., A Toll
  Pricing Framework for Traffic Assignment Problems
  with Elastic Demand, Current Trends in
  Transportation and Network Analysis papers in
  honor of Michael Florian, Kluwer Academic
  Publishers, 2001.

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Publications in Congestion Toll Pricing (cont.)

• Hearn, D. W., Yildirim, M. B., Ramana, M. V. and
  Bai, L. H., Computational Methods for
  Congestion Toll Pricing Models, Proceedings of
  The 4th International IEEE Conference on
  Intelligent Transportation Systems, 2001.
• Yildirim, M. B., Congestion Toll Pricing Models
  and Methods for Variable Demand Networks, PhD
  Dissertation, Department of Industrial Systems
  Engineering, University of Florida, Gainesville,
  FL, 2001
• Yildirim, M. B. and Hearn, D. W., A First Best
  Toll Pricing Framework for Variable Demand
  Traffic Assignment Problems, submitted for
  publication.
• Lawphongpanich, S. and Hearn, D.W., On the
  Second-Best Toll Pricing Problem, submitted for
  publication.
• Bai, L., Hearn, D.W., and Lawphongpanich, S.,
  Heuristics for the Minimum Toll Booth Problem,
  submitted for publication.
• Bai. L, Hearn, D.W., Lawphongpanich, S.,
  Decomposition Techniques for the Minimum Toll
  Revenue Problem, submitted for publication.

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Introduction

• The 2nd Best tolling pricing problem assumes that
  some arcs are not tollable.
• Our research goals
• Study the problem as a mathematical program with
  equilibrium constraints (MPEC)
• Examine the relationship between 2nd best tolls
  and marginal social cost pricing
• Develop a solution procedure based on existing
  nonlinear programming algorithms

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2nd Best Toll Pricing Problem Notation

• Indices
• a arcs or links in the network
• k origin-destination (or OD) pairs
• Problem data or parameters
• Ek OD vector for the kth OD pair, i.e., Ek
  ep eq
• bk demand for the kth OD pair (when the demand
  is fixed)
• A node-arc incidence matrix of the traffic
  network
• Y set of non-tollable arcs
• s(v) travel time or cost vector whose element,
  sa(v) denotes the travel time for link a.
• w(t) inverse demand vector whose element, wk(tk)
  can be interpreted as the benefit gained from
  making tk trips from the origin to the
  destination of OD pair k.
• Variables
• ? toll vector
• v,u total flow vector
• t,d demand vector
• xk flow vector for OD pair k

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2nd Best Toll Pricing Problem Fixed Demand

• where

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2nd Best Toll Pricing Problem Elastic Demand

• where

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Elastic Demand Equivalent Formulation 1

• Because it satisfies SBQC, ED-VI is equivalent to

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Properties of ED-KKT

• The last constraint in ED-KKT implies that the
  total toll revenue is constant with respect to an
  associated toll set.

where
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Properties of ED-KKT Example.

• To motivate another property, consider the
  following two-arc problem where Arc 1 is tollable
  and Arc 2 is not.

where s1(v1) v1, s2(v2) v2 2, and w(t) 9
t/2
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Properties of ED-KKT Example (cont.)

• In the literature (see, e.g., McDonald, 1995, and
  Verhoef, 2000), the optimal toll for Arc 1 is

• In this expression, the optimal toll includes a
  portion of MSCP from the non-tollable arc.
• Are there similar formulas for general networks?

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Properties of ED-KKT

• Results related to the previous question
• When certain regularity conditions hold, the
  second-best tolls can always be written as an
  expression involving marginal social cost pricing
  (MSCP) terms.
• The KKT conditions associated with ED-KKT yields
  the following expression of an optimal toll
  vector.

• An interpretation
• An optimal 2nd best toll on a link involves its
  own MSCP as well as those from non-tollable arcs
  via the KKT multipliers.

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Properties of ED-KKT Example (cont.)
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Properties of ED-KKT Example (cont.)

• An optimal solution

• Using the expression,

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Properties of ED-KKT Technical Issues

• The above result assumes that the multipliers
  exist.
• This assumption is difficult to prove.
• Scheel and Scholtes 2000 show that MFCQ is
  violated at every feasible solution of ED-KKT.
• A similar expression for the tolls can be
  obtained using the tightened NLP associated
  with ED-KKT. The multipliers for this problem
  exist when, e.g.,
• sa(v) are either linear or concave
• wk(tk) are either linear or convex.

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Elastic Demand Equivalent Formulation 2

• The set VED can be expressed as a convex
  combination of its extreme points, (ui,di), i
  1,..., n.

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Cutting Constraint Algorithm for ED-EX

• Let (u1,d1) be a system optimal solution. Set r
  1.
• Solve the following master problem

• Solve the subproblem

Otherwise, set r r 1 and go to 1.
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Numerical Results Fixed Demand

• Use GAMS
• CPLEX to solve the subproblem in Step 3.
• MINOS to solve the master problem in Step 2.
• Two networks from the literature
• Sioux Falls 76 links, 24 nodes, 528 OD pairs.
• Hull 798 links, 501 nodes, 158 OD pairs.
• Tollable arc selection
• An arc is tollable if its user equilibrium flow
  exceeds its system optimum flow by a given
  percentage (excess percentage).

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Numerical Results Fixed Demand

• Sioux Falls

• Total delay at SOPT 71.9426
• Total delay at UOPT 74.8023
• CPU times are from a 300 MHz IBM SP2 computer
  with 512 MB of RAM

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Numerical Results Fixed Demand

• Hull

• Total delay at SOPT 179063
• Total delay at UOPT 186720
• CPU times are from a 300 MHz IBM SP2 computer
  with 512 MB of RAM

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Numerical Results Elastic Demand

• Network

Inverse Demand Function wk(t) ak bkt
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Numerical Results Elastic Demand
Travel Cost function sa(v) Ta(10.15(va/Ca))
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Numerical Results Elastic Demand

• Tollable arcs

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Toll Pricing Framework

• Let (v,t, ?) be an optimal solution to ED-KKT.
  Then, ? (and an associated ? ) is one of
  possibly many solutions to the following system
  of equations

• One possibility is to choose a solution that
  optimize an objective.

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Toll Pricing Framework.

• Solve ED-VI or one of the equivalent problems to
  obtain (v,t, ?) .
• Solve the following toll selection problem

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Numerical Results Elastic Demand

• Tollable arcs

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Conclusions

• Two equivalent formulations for the 2nd best toll
  pricing problem
• Properties of the 2nd best tolls
• Under some regularity conditions, the 2nd Best
  toll vector can be written as an expression
  involving MSCP.
• Toll revenue is constant.
• Cutting constraint algorithm for ED-EX
• Converges finitely
• Solves realistic problems