Toll Pricing and Heterogeneous Users: Approximation Algorithms for Finding Bicriterion Time-Dependent Efficient Paths in Large-Scale Traffic Networks

Hani Mahmassani, Xuesong Zhou, Chung-Cheng Lu
2005 Transportation Research Record  
This paper presents both exact and approximation algorithms for finding extreme efficient time-dependent shortest paths for use with dynamic traffic assignment applications to networks with variable toll pricing and heterogeneous users (with different value of time preferences). A parametric least-generalized cost path algorithm is presented to determine a complete set of extreme efficient time-dependent paths that simultaneously consider travel time and cost criteria. However, exact procedures
more » ... may not be practical for large networks. For this reason, approximation schemes are devised and tested. Based on the concept of ⑀-efficiency in multiobjective shortest path problems, a binary search framework is developed to find a set of extreme efficient paths that minimize expected approximation error, with the use of the underlying value of time distribution. Both exact and approximation schemes (along with variants) are tested on three actual traffic networks. The experimental results indicate that the computation time and the size of the solution set are jointly determined by several key parameters such as the number of time intervals and the number of nodes in the network. The results also suggest that the proposed approximation scheme is computationally efficient for large-scale bi-objective time-dependent shortest path applications while maintaining satisfactory solution quality. Road pricing is increasingly considered as an effective demand management strategy to reduce traffic congestion and improve system performance during peak periods in many metropolitan areas. In capacity-limited transportation networks, the planning and operation of various road pricing strategies, such as road tolls, cordon (area) tolls, and high-occupancy toll lanes, call for path choice models that take into account two essential decision attributes: travel time and out-of-pocket cost. In a utility maximization framework, each trip maker can be assumed to select a path that minimizes a generalized cost function where travel time is weighted by the trip maker's value of time (VOT). The VOT relative to each trip represents how much money the trip maker is willing to trade off for unit time saving. Various empirical studies (1, 2) have suggested that the VOT varies significantly across individuals because of different socioeconomic characteristics, trip purposes, attitudes, and inherent preferences. To capture the heterogeneous preferences underlying trip makers' route choice behavior, several bi-objective traffic assignment models (3, 4) (S. C. Dafermos, unpublished manuscript, 1981) have been proposed to generalize the classic single-objective traffic assignment by relaxing the VOT from a constant to a continuously distributed random variable. Solution procedures for the bi-objective traffic assignment problem typically involve a direction-finding step, which needs to determine bi-objective minimum paths that simultaneously seek to minimize conflicting path attributes. In addition, many applications in transportation planning and operations require efficient solution algorithms for optimum path problems with multiple objectives, such as travel time, reliability, and population exposure in the problem of shipping hazardous materials (5). All these applications underscore the need for computationally efficient and tractable solution procedures for multiobjective shortest path problems (MOSPs) in transportation networks. Because no unique optimal path usually exists in terms of all the objectives in a general network, the MOSP aims to find a set of nondominated (i.e., Pareto optimal or efficient) paths. The weighting method combines different attributes into a single utility function and systematically varies the weights (e.g., VOT) to generate nondominated solution sets. This method has been widely applied to solve the MOSP, because it can utilize efficient solution algorithms for the resulting single-objective shortest path problem. In the recursive weighting algorithm presented by Dial (6), a weighting parameter is iteratively generated based on the slope of the line connecting the two extremes (i.e., the trade-off between two solutions), and the search process is repeated recursively until all (or a given number of) the efficient paths are found. Another common weighting method is to calculate shortest paths with randomly generated VOT from a given distribution function (7) . Henig (8) introduced the concept of extreme (or supported) efficient paths, which correspond to extreme points in the boundary of the convex hull containing all the efficient points in the criterion space. He further pointed out that the weighting method can enumerate only the extreme efficient paths because nonextreme efficient solutions are dominated by a convex combination of extreme efficient solutions. The parametric shortest path method was also applied by several researchers, such as Henig (8) , Mote et al. (9), and Dial (10), to efficiently identify extreme efficient shortest paths. By computing the sensitivity range of the weighting parameter, the parametric weighting method is able to move directly from one extreme efficient solution to the next one without redundant calculations. As a result, the feasible range of VOT can be partitioned into a number of intervals, each corresponding to a shortest path tree. Another method to identify the entire nondominated solution set is the multilabeling approach pro-
doi:10.3141/1923-04 fatcat:ktguow3mt5ganpv6gdmesw624e