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This work considers those road networks in which there are multi-route choices for bifurcation-destination (or origin-destination) pairs, and designs a real-time variable message sign (VMS)-based routing control strategy in the model predictive control (MPC) framework. The VMS route recommendation provided by the traffic management authority is directly considered as the control variable, and the routing control model is established, in which a multi-dimensional control vector is introduced to describe the influence of route recommendations on flow distribution. In the MPC framework, a system optimum routing strategy with the constraints regarding drivers’ acceptability with recommended routes is designed, which can not only meet the traffic management authority’s control requirement but also improve drivers’ satisfaction with the route guidance system. The simulation carried out shows that the proposed routing control can effectively mitigate traffic congestion, reduces followers’ time delay, and improves drivers’ satisfaction with routing control in road networks.

The notion of dynamic user equilibrium (DUE) and dynamic user optimum (DUO) were still treated interchangeably in the dynamic traffic assignment (DTA) literature until tolerance-based DUO was recently proposed as alternative definitions of DUO. The latter distinguishes DUO from DUE, and the tolerances can be prescribed a priori (i.e. DUO with fixed tolerance) or endogenously (i.e. DUO with variable tolerance). These tolerances reflect the differences in travel costs between used paths of an origin–destination (OD) pair. Since DUE requires zero difference in travel costs between used paths of an OD pair, it can be regarded as DUO with zero tolerance. This paper first compares the three alternative definitions of DUO and discusses their relationships. Secondly, the existence conditions of these DUO solutions are discussed. A key conclusion is that DUO with variable tolerance always exists even when the other two types of DUO solutions may not. The second conclusion is that, when the tolerances are equal to zero, the solution must be a DUE solution. Furthermore, DUE is a special case of DUO with variable tolerance; since the tolerance in the concept of DUO with variable tolerance is an endogenous variable, if a DUE solution exists, an efficient method for solving for DUO with variable tolerance should directly find such a DUE solution. All these are illustrated by means of three examples. Finally, it is acknowledged that the analysis of DUO with variable tolerance in the same rigor as that of DUE is a challenging problem and worthy of further research.

Models to describe or predict of time-varying traffic flows and travel times on road networks are usually referred to as dynamic traffic assignment (DTA) models or dynamic user equilibrium (DUE) models. The most common form of algorithms for DUE consists of iterating between two components namely dynamic network loading (DNL) and path inflow reassignment or route choice. The DNL components in these algorithms have been investigated in many papers but in comparison the path inflow reassignment component has been relatively neglected. In view of that, we investigate various methods for path inflow reassignment that have been used in the literature. We compare them numerically by embedding them in a DUE algorithm and applying the algorithm to solve DUE problems for various simple network scenarios. We find that the choice of inflow reassignment method makes a huge difference to the speed of convergence of the algorithms and, in particular, find that ‘travel time responsive’ reassignment methods converge much faster than the other methods. We also investigate how speed of convergence is affected by the extent of congestion on the network, by higher demand or lower capacity. There appears to be much scope for further improving path inflow reassignment methods.

In this paper, we establish a bi-level optimization model for the equilibrium transportation problem concerning both capacity expansion and road toll pricing under the user equilibrium conditions. The bi-level optimization problem is reformulated as a mathematical programming problem with complementarity constraints (MPCC), which fails to satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ). We adopt a smoothing approach to overcome the lack of constraint qualifications in the MPCC problem. Under mild conditions, it has been proven that the sequence of the global optimal solutions generated by solving corresponding smoothing subproblems converges to one optimal solution of the original MPCC problem. Numerical experiments show that the proposed method is practical in solving user equilibrium transportation problems with capacity expansion combining road toll pricing.

In the literature, two different models have been used to compute link travel times in dynamic traffic assignment (DTA), and elsewhere we investigated how these are affected by discretising the link length. Here we consider discretising time as well as space (the link length). We vary the discretising of time with spatial discretisation held fixed, and vice versa, and also vary both together. The results show that “coordinated” discretisation is usually the most efficient in approximating the limit solution (continuous time, continuous space) and, even when it is not the most efficient, it has other advantages. The results have implications for algorithms for DTA and for the choice of discrete versus continuous time models. For example, refining the discretisation of time (without refining it for space) can make the solution less accurate, so that in the widely used whole-link models (i.e. without spatial discretisation) it is more efficient to use the largest feasible time steps, close to the link travel time.