Reformulating the traffic equilibrium problem via a smooth gap function

This paper proposes an alternate formulation of the traffic assignment problem using route flows and the shortest Origin-Destination (OD) travel times as the decision variables. This is accomplished through defining a gap function to convert the Nonlinear Complementarity Problem (NCP) formulation to an equivalent Mathematical Program (MP). This formulation has two advantages: 1.(i) it can model assignment problems with general route costs which cannot be accomplished with existing formulations that use link-flow variables 2.(ii) the objective function is smooth, convex, and bounded, which permits efficient MP algorithms for its solution. Two solution approaches are developed to solve the proposed formulation. The first is based on a set of working routes, which are modeled as ''known a priori'' based on travelers' preferences or interviews. The second approach uses a column generation procedure to generate a new route in each iteration on a need basis. For each approach, we use a Successive Quadratic Programming (SQP) algorithm to solve for the solutions. To show that the formulation is correct, we solve a small example with a general route cost and compare it to the classic traffic equilibrium problem which assumes an additive route cost function. Finally, numerical results for a medium-sized network are provided to demonstrate the feasibility of the solution approach.

[1]  F. Giannessi,et al.  Variational inequalities and network equilibrium problems , 1995 .

[2]  Shing Chung Josh Wong,et al.  A path-based traffic assignment algorithm based on the TRANSYT traffic model , 2001 .

[3]  W. Andrew,et al.  LO, and A. , 1988 .

[4]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[5]  Mike Smith,et al.  An algorithm for solving asymmetric equilibrium problems with a continuous cost-flow function , 1983 .

[6]  E. Cascetta,et al.  STOCHASTIC USER EQUILIBRIUM ASSIGNMENT WITH EXPLICIT PATH ENUMERATION: COMPARISON OF MODELS AND ALGORITHMS , 1997 .

[7]  Larry J. LeBlanc,et al.  AN EFFICIENT APPROACH TO SOLVING THE ROAD NETWORK EQUILIBRIUM TRAFFIC ASSIGNMENT PROBLEM. IN: THE AUTOMOBILE , 1975 .

[8]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[9]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[10]  Yosef Sheffi,et al.  Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods , 1985 .

[11]  David Bernstein,et al.  The Traffic Equilibrium Problem with Nonadditive Path Costs , 1995, Transp. Sci..

[12]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[13]  Hong Kam Lo,et al.  Traffic equilibrium problem with route-specific costs: formulation and algorithms , 2000 .

[14]  R. Asmuth.  Traffic network equilibria , 1978 .

[15]  R. Jayakrishnan,et al.  A FASTER PATH-BASED ALGORITHM FOR TRAFFIC ASSIGNMENT , 1994 .

[16]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

[17]  Clermont Dupuis,et al.  An Efficient Method for Computing Traffic Equilibria in Networks with Asymmetric Transportation Costs , 1984, Transp. Sci..

[18]  A. Nagurney.  Network Economics: A Variational Inequality Approach , 1992 .

[19]  Mike Smith,et al.  A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks , 1993 .

[20]  H. Z. Aashtiani.  The multi-modal traffic assignment problem. , 1979 .

[21]  Bin Ran,et al.  MODELING DYNAMIC TRANSPORTATION NETWORKS , 1996 .

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

[23]  H. Lo.  A DYNAMIC TRAFFIC ASSIGNMENT FORMULATION THAT ENCAPSULATES THE CELL-TRANSMISSION MODEL , 1999 .

[24]  Mike J. Smith.  The existence and calculation of traffic equilibria , 1983 .

[25]  M G Bell,et al.  Path flow estimation and traffic system management , 1997 .

[26]  João Soares,et al.  Testing a New Class of Algorithms for Nonlinear Complementarity Problems , 1995 .

[27]  R Jayakrishnan,et al.  A PATH-BASED GRADIENT PROJECTION ALGORITHM: EFFECTS OF EQUILIBRATION WITH A RESTRICTED PATH SET UNDER TWO FLOW UPDATE POLICIES , 1998 .

[28]  Donald W. Hearn,et al.  The gap function of a convex program , 1982, Operations Research Letters.