The nonlinear equation system approach to solving dynamic user optimal simultaneous route and departure time choice problems

Dynamic user optimal simultaneous route and departure time choice (DUO-SRDTC) problems are usually formulated as variational inequality (VI) problems whose solution algorithms generally require continuous and monotone route travel cost functions to guarantee convergence. However, the monotonicity of the route travel cost functions cannot be ensured even if the route travel time functions are monotone. In contrast to traditional formulations, this paper formulates a DUO-SRDTC problem (that can have fixed or elastic demand) as a system of nonlinear equations. The system of nonlinear equations is a function of generalized origin-destination (OD) travel costs rather than route flows and includes a dynamic user optimal (DUO) route choice subproblem with perfectly elastic demand and a quadratic programming (QP) subproblem under certain assumptions. This study also proposes a solution method based on the backtracking inexact Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, the extragradient algorithm, and the Frank-Wolfe algorithm. The BFGS method, the extragradient algorithm, and the Frank-Wolfe algorithm are used to solve the system of nonlinear equations, the DUO route choice subproblem, and the QP subproblem, respectively. The proposed formulation and solution method can avoid the requirement of monotonicity of the route travel cost functions to obtain a convergent solution and provide a new approach with which to solve DUO-SRDTC problems. Finally, numeric examples are used to demonstrate the performance of the proposed solution method.

[1]  Ziyou Gao,et al.  A Cumulative Perceived Value-Based Dynamic User Equilibrium Model Considering the Travelers’ Risk Evaluation on Arrival Time , 2012 .

[2]  T. Iryo.  Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology , 2013 .

[3]  W. Y. Szeto,et al.  A nonlinear equation system approach to the dynamic stochastic user equilibrium simultaneous route and departure time choice problem , 2015 .

[4]  Qiang Meng,et al.  A Computational Model for the Probit-Based Dynamic Stochastic User Optimal Traffic Assignment Problem , 2012 .

[5]  Hai Yang,et al.  The equilibrium-based origin-destination matrix estimation problem , 1994 .

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

[7]  Tao Yao,et al.  A partial differential equation formulation of Vickrey’s bottleneck model, part II: Numerical analysis and computation , 2013 .

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

[9]  Richard Mounce,et al.  Route swapping in dynamic traffic networks , 2011 .

[10]  Henry X. Liu,et al.  Continuous-time point-queue models in dynamic network loading , 2012 .

[11]  Kenneth A. Small,et al.  THE SCHEDULING OF CONSUMER ACTIVITIES: WORK TRIPS , 1982 .

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

[13]  Xiwen Lu,et al.  A new backtracking inexact BFGS method for symmetric nonlinear equations , 2008, Comput. Math. Appl..

[14]  Chris Hendrickson,et al.  Schedule Delay and Departure Time Decisions in a Deterministic Model , 1981 .

[15]  Barbara Panicucci,et al.  A path-based double projection method for solving the asymmetric traffic network equilibrium problem , 2007, Optim. Lett..

[16]  Georgia Perakis,et al.  An Analytical Model for Traffic Delays and the Dynamic User Equilibrium Problem , 2006, Oper. Res..

[17]  Ismail Chabini.  Analytical Dynamic Network Loading Problem: Formulation, Solution Algorithms, and Computer Implementations , 2001 .

[18]  Jin-Su Mun,et al.  A solution algorithm for a dynamic deterministic user equilibrium assignment model with departure time choice , 2011 .

[19]  W. Y. Szeto,et al.  Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance , 2014, 1402.1211.

[20]  Tao Yao,et al.  A partial differential equation formulation of Vickrey’s bottleneck model, part I: Methodology and theoretical analysis , 2013 .

[21]  E. Khobotov.  Modification of the extra-gradient method for solving variational inequalities and certain optimization problems , 1989 .

[22]  Markos Papageorgiou,et al.  Dynamic modeling, assignment, and route guidance in traffic networks , 1990 .

[23]  Che-Fu Hsueh,et al.  Discrete-Time Dynamic User-Optimal Departure Time/Route Choice Model , 1998 .

[24]  S. Travis Waller,et al.  Reliable evacuation planning via demand inflation and supply deflation , 2010 .

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

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

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

[28]  Sangjin Han,et al.  Dynamic traffic modelling and dynamic stochastic user equilibrium assignment for general road networks , 2003 .

[29]  Yu Nie,et al.  Equilibrium analysis of macroscopic traffic oscillations , 2010 .

[30]  W. Y. Szeto,et al.  A CELL-BASED SIMULTANEOUS ROUTE AND DEPARTURE TIME CHOICE MODEL WITH ELASTIC DEMAND , 2004 .

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

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

[33]  Andy H.F. Chow.  Properties of system optimal traffic assignment with departure time choice and its solution method , 2009 .

[34]  Jong-Shi Pang,et al.  Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach , 2012 .

[35]  M. Fukushima.  On the dual approach to the traffic assignment problem , 1984 .

[36]  W. Y. Szeto,et al.  A Cell‐Based Model for Multi‐class Doubly Stochastic Dynamic Traffic Assignment , 2011, Comput. Aided Civ. Infrastructure Eng..

[37]  Torbjörn Larsson,et al.  A dual scheme for traffic assignment problems , 1997 .

[38]  S. Travis Waller,et al.  A Dynamic Route Choice Model Considering Uncertain Capacities , 2012, Comput. Aided Civ. Infrastructure Eng..

[39]  W. Y. Szeto,et al.  A cell-based dynamic traffic assignment model: Formulation and properties , 2002 .

[40]  Deepak K. Merchant,et al.  A Model and an Algorithm for the Dynamic Traffic Assignment Problems , 1978 .

[41]  Terry L. Friesz,et al.  A differential variational inequality formulation of dynamic network user equilibrium with elastic demand , 2014 .

[42]  W. Y. Szeto,et al.  An intersection-movement-based stochastic dynamic user optimal route choice model for assessing network performance , 2015 .

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

[44]  Malachy Carey,et al.  An approach to modelling time-varying flows on congested networks , 2000 .

[45]  B. Ran,et al.  A discrete time dynamic flow model and a formulation and solution method for dynamic route choice , 2005 .

[46]  Satish V. Ukkusuri,et al.  On the holding-back problem in the cell transmission based dynamic traffic assignment models , 2012 .

[47]  Terry L. Friesz,et al.  A Discrete Time, Nested Cost Operator Approach to the Dynamic Network User Equilibrium Problem , 1995, Transp. Sci..

[48]  W. Y. Szeto,et al.  Dynamic traffic assignment: model classifications and recent advances in travel choice principles , 2012 .

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

[50]  George L. Nemhauser,et al.  Optimality Conditions for a Dynamic Traffic Assignment Model , 1978 .

[51]  W. Vickrey.  Congestion Theory and Transport Investment , 1969 .

[52]  C. Y. Wang,et al.  Dynamic capacitated user-optimal departure time/route choice problem with time-window , 2001, Eur. J. Oper. Res..

[53]  Hillel Bar-Gera,et al.  The effect of signalised intersections on dynamic traffic assignment solution stability , 2014 .

[54]  Carlos F. Daganzo,et al.  Queue Spillovers in Transportation Networks with a Route Choice , 1998, Transp. Sci..

[55]  W. Y. Szeto,et al.  An Intersection-Movement-Based Dynamic User Optimal Route Choice Problem , 2013, Oper. Res..

[56]  David Bernstein,et al.  An alternative formulation of the simultaneous route and departure-time choice equilibrium problem , 1996 .

[57]  Erik T. Verhoef,et al.  Step tolling with bottleneck queuing congestion , 2012 .

[58]  M O Ghali,et al.  TRAFFIC ASSIGNMENT, TRAFFIC CONTROL AND ROAD PRICING. , 1993 .

[59]  Malachy Carey,et al.  Dynamic traffic assignment approximating the kinematic wave model: System optimum, marginal costs, externalities and tolls , 2012 .

[60]  G. F. Newell.  A simplified theory of kinematic waves in highway traffic, part I: General theory , 1993 .

[61]  Shing Chung Josh Wong,et al.  A predictive dynamic traffic assignment model in congested capacity-constrained road networks , 2000 .

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

[63]  Tao Yao,et al.  Existence of simultaneous route and departure choice dynamic user equilibrium , 2012, 1211.0898.

[64]  Isaak Yperman,et al.  The Link Transmission Model for dynamic network loading , 2007 .

[65]  Richard S. Higgins,et al.  Raising Rivals’ Costs , 2014 .

[66]  Athanasios K. Ziliaskopoulos,et al.  A Linear Programming Model for the Single Destination System Optimum Dynamic Traffic Assignment Problem , 2000, Transp. Sci..

[67]  Malachy Carey,et al.  Optimal Time-Varying Flows on Congested Networks , 1987, Oper. Res..

[68]  W. Y. Szeto,et al.  Discretised link travel time models based on cumulative flows: Formulations and properties , 2011 .

[69]  Yun-Bin Zhao,et al.  Global bounds for the distance to solutions of co-coercive variational inequalities , 2007, Oper. Res. Lett..

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

[71]  Robin Lindsey,et al.  DEPARTURE TIME AND ROUTE CHOICE FOR THE MORNING COMMUTE , 1990 .