Dynamic user equilibrium with side constraints for a traffic network : theoretical development and numerical solution algorithm

This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler's discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.

[1]  Tao Yao,et al.  Securitizing congestion: The congestion call option , 2008 .

[2]  W. Y. Szeto,et al.  DYNAMIC TRAFFIC ASSIGNMENT: PROPERTIES AND EXTENSIONS , 2006 .

[3]  Hai Yang,et al.  TRIAL-AND-ERROR IMPLEMENTATION OF MARGINAL-COST PRICING ON NETWORKS IN THE ABSENCE OF DEMAND FUNCTIONS , 2004 .

[4]  Wang,et al.  Review of road traffic control strategies , 2003, Proceedings of the IEEE.

[5]  Hani S. Mahmassani,et al.  Equivalent gap function-based reformulation and solution algorithm for the dynamic user equilibrium problem , 2009 .

[6]  Markos Papageorgiou,et al.  Store-and-forward based methods for the signal control problem in large-scale congested urban road networks , 2009 .

[7]  Andy H.F. Chow,et al.  Dynamic system optimal traffic assignment – a state-dependent control theoretic approach , 2009 .

[8]  Yu Nie.  A Cell-based Merchant-Nemhauser Model for the System Optimum Dynamic Traffic Assignment Problem , 2010 .

[9]  H. M. Zhang,et al.  Delay-Function-Based Link Models: Their Properties and Computational Issues , 2005 .

[10]  Haijun Huang,et al.  Mathematical and Economic Theory of Road Pricing , 2005 .

[11]  Terry L. Friesz,et al.  Dynamic Network User Equilibrium with State-Dependent Time Lags , 2001 .

[12]  Terry L. Friesz,et al.  Approximate network loading and dual-time-scale dynamic user equilibrium , 2011 .

[13]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[14]  G. Perakis,et al.  A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders , 2007 .

[15]  Malachy Carey,et al.  A framework for user equilibrium dynamic traffic assignment , 2009, J. Oper. Res. Soc..

[16]  T. Noiri.  On weakly continuous mappings , 1974 .

[17]  M. Patriksson,et al.  An augmented lagrangean dual algorithm for link capacity side constrained traffic assignment problems , 1995 .

[18]  Patrice Marcotte,et al.  On the Existence of Solutions to the Dynamic User Equilibrium Problem , 2000, Transp. Sci..

[19]  Terry L. Friesz,et al.  Solving the Dynamic Network User Equilibrium Problem with State-Dependent Time Shifts , 2006 .

[20]  David K. Smith,et al.  Mathematical Programming: Theory and Algorithms , 1986 .

[21]  Vittorio Astarita,et al.  A CONTINUOUS TIME LINK MODEL FOR DYNAMIC NETWORK LOADING BASED ON TRAVEL TIME FUNCTION , 1996 .

[22]  Hai-Jun Huang,et al.  Modeling and solving the dynamic user equilibrium route and departure time choice problem in network with queues , 2002 .

[23]  Terry L. Friesz,et al.  A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem , 1993, Oper. Res..

[24]  Jin‐Su Mun,et al.  Traffic Performance Models for Dynamic Traffic Assignment: An Assessment of Existing Models , 2007 .

[25]  Malachy Carey,et al.  Behaviour of a whole-link travel time model used in dynamic traffic assignment , 2002 .

[26]  Shing Chung Josh Wong,et al.  Heuristic algorithms for simulation-based dynamic traffic assignment , 2010 .

[27]  Robin Lindsey.  Existence, Uniqueness, and Trip Cost Function Properties of User Equilibrium in the Bottleneck Model with Multiple User Classes , 2004, Transp. Sci..

[28]  H. M. Zhang,et al.  Solving the Dynamic User Optimal Assignment Problem Considering Queue Spillback , 2010 .

[29]  Y. W. Xu,et al.  Advances in the Continuous Dynamic Network Loading Problem , 1996, Transp. Sci..

[30]  H. Maurer,et al.  SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control , 2000 .

[31]  Torbjörn Larsson,et al.  Side constrained traffic equilibrium models: analysis, computation and applications , 1999 .

[32]  Bin Ran,et al.  for dynamic user equilibria with exact flow propagations , 2008 .

[33]  Roger L. Tobin,et al.  Uniqueness and computation of an arc-based dynamic network user equilibrium formulation , 2002 .

[34]  Torbjörn Larsson,et al.  A column generation procedure for the side constrained traffic equilibrium problem , 2004 .

[35]  G. Leoni.  A First Course in Sobolev Spaces , 2009 .

[36]  T. Friesz.  Dynamic Optimization and Differential Games , 2010 .

[37]  H. M. Zhang,et al.  A Comparative Study of Some Macroscopic Link Models Used in Dynamic Traffic Assignment , 2005 .

[38]  Hai Yang,et al.  Departure time, route choice and congestion toll in a queuing network with elastic demand , 1998 .

[39]  W. Y. Szeto,et al.  A cell-based variational inequality formulation of the dynamic user optimal assignment problem , 2002 .

[40]  Lorenzo Meschini,et al.  Macroscopic arc performance models with capacity constraints for within-day dynamic traffic assignment , 2005 .

[41]  Hai Yang,et al.  Modeling the capacity and level of service of urban transportation networks , 2000 .

[42]  Malachy Carey,et al.  Comparing whole-link travel time models , 2003 .

[43]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.