Bi-Objective Network Equilibrium, Traffic Assignment and Road Pricing

Multi-objective equilibrium models of traffic assignment state that users of road networks travel on routes that are efficient with respect to several objectives, such as travel time and toll. This concept provides a general framework for modelling traffic flow in tolled road networks. We present the concept of time surplus maximisation as a way of handling user preferences. Given a toll, users have a maximum time they are willing to spend for a trip. Time surplus is this maximum time minus actual travel time. A rational user can be assumed to maximise time surplus, leading to the definition of time surplus maximisation bi-objective user equilibrium. We propose to use such models on the lower level of bi-level models for pricing in road networks under multiple upper level objectives such as minimising total travel time and emissions. In such a model a multi-objective optimisation problem at the upper level is combined with a multi-objective equilibrium problem at the lower level. 1 Traffic Assignment and User Equilibrium Traffic assignment models the route choice of users of a road network. Given a set of origin-destination (OD) pairs and demand for travel between these OD pairs, it determines how many users choose each of the available routes, and thereby the amount of traffic on each section of the road network. Conventional traffic assignment is based on the assumption that all users want to minimise their travel time, or more generally, a generalised cost function c(xp) = m(xp)+αt(xp), (1) where xp represents traffic flow on route p, t is travel time, which is dependent on flow, and m is a monetary cost comprising of tolls, vehicle operating cost etc. Judith Y.T. Wang · Matthias Ehrgott Department of Engineering Science, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand, e-mail: {m.ehrgott,j.wang}@auckland.ac.nz 1 2 Judith Y.T. Wang and Matthias Ehrgott that may also depend on flow and α is value of time. A user will choose the route between their origin and destination that has the least value of c(xp). The traffic assignment problem is based on Wardrop’s principle of user equilibrium [9], which can be stated as follows: Under user equilibrium conditions traffic arranges itself in such a way that no individual trip maker can improve their generalised cost by unilaterally switching routes. In other words, at equilibrium, the generalised cost of any used route between an OD pair must be equal and less than that of any unused route. It is important to note that (1) is the linear combination of two components, time and monetary cost. In fact these are two different objective functions. Several authors have recognised this and suggested bi-objective traffic assignment models, see the references in [8]. However, these models are restrictive, by keeping the assumption of the existence of an additive generalised cost (or sometimes generalised time) function (1). Moreover, there is evidence, that users in reality do not behave according to this assumption, see references in [8]. In [8] we have suggested a more general bi-objective user equilibrium condition, that assumes that all users have the two objectives of minimising travel time and minimising toll cost. Under bi-objective user equilibrium (BUE) conditions traffic arranges itself in such a way that no individual trip maker can improve either his/her toll or travel time or both without worsening the other objective by unilaterally switching routes. We have shown that, even if considering all possible values of time, i.e. α ∈ [0,∞), in (1), bi-objective models based on generalised cost provide only a subset of all possible solutions to traffic assignment that satisfy the BUE condition. Hence the definition of BUE provides an appropriate general framework for the study of traffic assignment in tolled road networks. Furthermore, in [7] we have suggested the time surplus maximisation concept as a new route choice model that addresses the stochastic nature of route choice behaviour and the variability among users on their willingness to pay. It is based on the idea of time surplus. We assume that a user has in his mind a maximum time he is willing to spend in traffic, given any level of toll. If τ p is the toll on route p for OD pair k and the travel time is t(xp) then the time surplus on route p for individual i is t ip = t max i ( τ p )