A modified gradient projection algorithm for solving the elastic demand traffic assignment problem Seungkyu Ryu a , Anthony Chen a,n , Keechoo Choi b a Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110, USA b Department of Transportation Engineering, Ajou University, Suwon, 442-749, Reppublic of Korea article info Available online 11 February 2014 Keywords: Elastic demand User equilibrium Gradient projection Traffic assignment Path-based algorithm abstract This paper develops a path-based traffic assignment algorithm for solving the elastic demand traffic assignment problem (EDTAP). A modified path-based gradient projection (GP) method combined with a column generation is suggested for solving the equivalent excess-demand reformulation of the problem in which the elastic demand problem is reformulated as a fixed demand problem through an appropriate modification of network representation. Numerical results using a set of real transportation networks are provided to demonstrate the efficiency of the modified GP algorithm for solving the excess-demand formulation of the EDTAP. In addition, a sensitivity analysis is conducted to examine the effects of the scaling parameter used in the elastic demand function on the estimated total demand, number of generated paths, number of used paths, and computational efforts of the modified GP algorithm. Published by Elsevier Ltd. 1. Introduction An elastic-demand traffic assignment problem (EDTAP) is one that establishes the equilibrium between supply functions and demand functions in a transportation network. In the traffic assignment problem, the supply functions are determined by the link travel time characteristics on the network, and the demand functions are determined by the user benefits derived from travel [15]. At equilibrium, the link flows, link travel times, path flows, path travel times, origin–destination (O–D) travel demands, and O–D travel times are consistent and satisfy the user equilibrium (UE) conditions [30,28]. That is, the travel times on all used paths between any O–D pair are equal, and are also equal to or less than the travel times on any unused paths. In addition, the O–D travel demands should satisfy the demand functions. Beckmann et al. [3] provided the first convex programming formulation for the user equilibrium (UE) traffic assignment problem with endogenously determined travel demands. Based on this seminal work, many researchers have considered different formulation approaches and solution algorithms to enhance the modeling realism and applica- tions of the UE model with elastic demand. In terms of formulation approaches, Aashtiani [1] gave the first nonlinear complementarity problem (NCP) formulation for modeling the interactions in a multi- modal network. Dafermos [13] offered a variational inequality (VI) formulation for the multimodal traffic equilibrium model with elastic demand, where the link travel costs depend on the entire link flow vector and the travel demands depend on the entire mode- specificO–D cost vector. Fisk and Boyce [14] provided alternative VI formulations for the network equilibrium travel choice pro- blem, which does not required invertibility of the travel demand function. Cantarella [7] provided a fixed point (FP) formulation for the multi-mode multi-user equilibrium assignment with elastic demand, where users have different behavioral characteristics as well as different choice sets. As for solution algorithms, the convex combinations method (or the Frank–Wolfe algorithm) used for solving the UE traffic assignment problem with fixed demand is perhaps the most commonly used approach as it can be readily adapted to solve the elastic demand version with minor additional computational effort needed to compare the current shortest path cost with the current value of the inverse demand function [28]. Gartner [17,18] summarized three approaches for modeling the generalized traffic equilibrium problem as an equivalent network in which the elastic demand functions are represented by appropriate generating links: (1) minimum-cost circulation, (2) zero-cost overflow, and (3) excess demand. Fukushima [16] explored the possibility of solving the EDTAP via its dual problem as a nonsmooth convex programming formulation, while Nagurney [25] extended the concept of equilibration operator introduced by Dafermos and Sparrow [12] for solving the excess demand reformulation of the EDTAP. Babonneau and Vial [2] proposed a variant of the analytic center cutting plane method for solving the EDTAP with emphasis on Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research http://dx.doi.org/10.1016/j.cor.2014.01.012 0305-0548 Published by Elsevier Ltd. n Corresponding author. Tel.: þ1 435 797 7109; fax: þ1 435 797 1185. E-mail address: anthony.chen@usu.edu (A. Chen). Computers & Operations Research 47 (2014) 61–71