On endogenously distinguishing inactive paths in stochastic user equilibrium : a convex programming approach with a truncated path choice model

Abstract

This paper develops a convex programming approach with a truncated path choice model to resolve a fundamental drawback of conventional stochastic user equilibrium (SUE) models; that is, assign strictly positive flow to a path, irrespective of the length of its travel time. The centerpiece of the truncated path choice model is to truncate the path choice probability to zero when the travel time exceeds travelers’ maximum acceptable travel time, while the choice probabilities of other paths follow the utility maximization principle. Although the truncated path choice model has a non-smooth expression, the truncated SUE condition can be equivalently formulated as a twice-differentiable convex mathematical programming (MP), which has a simple structure comparable to that of Fisk’s MP formulation of the multinomial logit SUE model. Moreover, the origin–destination pair-specific parameter of the maximum acceptable travel time, which is explicitly expressed in the truncated path choice model, is endogenized and implicit in the devised MP formulation. This substantially decreases the computational effort required for parameter calibration. The desirable MP formulation enables us to establish the existence and uniqueness of equilibrium path flow under mild assumptions and to develop convergent and efficient solution algorithms. Specifically, we develop a path-based gradient projection algorithm incorporating an Armijo-type Barzilai-Borwein step size scheme for solving the truncated SUE model. Numerical results demonstrate the validity of the truncated SUE model and the efficiency and robustness of the devised algorithm.