Time and toll trade-off with heterogeneous users: A continuous time surplus maximization bi-objective user equilibrium model

This paper presents a continuous time surplus maximization bi-objective user equilibrium (C-TSmaxBUE) model, in which the users’ variability toward the time and toll trade-off in a tolled road network is explicitly considered. Different users are assigned with different ratios of the time saved per unit of money (RTSMs), the values of which represent the different expectations to trade more money for less travel time. Infinite indifference curves are then generated by considering continuously distributed RTSMs in the population. Accordingly, we extend the definition of the path-based time surplus for the C-TSmaxBUE model, which can be formulated as a Beckmann-type mathematical program operating in the space of RTSM boundaries. To solve the problem, we develop a path-based single-boundary adjustment (SBA) algorithm that fits the non-additive structure of the model. The algorithm involves a Newton-type flow equilibration procedure that simultaneously adjusts RTSM boundaries and path flows, as well as a column generation scheme that firstly enumerates all efficient paths and then identifies the paths that have the potential to reduce generalized travel times. Numerical results on a small network show the feature of the equilibrium flow pattern, whereas instances of practical size confirm that SBA is an efficient tool to obtain high-quality equilibrium solutions. Compared to the original TSmaxBUE problem, we found that a larger set of efficient paths will be used when the users’ heterogeneity is considered.