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Modelling the effects of road pricing

Stewart, Kathryn


Kathryn Stewart


The classical road tolling problem is to toll network links such that under the principles of Wardropian User Equilibrium (UE) assignment, a System Optimising (SO) flow pattern is obtained. Such toll sets are however non-unique and further optimisation is possible: for example, minimal revenue tolls create the desired SO flow pattern at minimal additional cost to the user. In the case of deterministic assignment, the minimal revenue toll problem is capable of solution by various methods, such as linear programming (Bergendorff et al, 1997) and by reduction to a multi-commodity max- flow problem (Dial, 2000). However, these methods are restricted in their application to small networks or problems of special structure.

However, it is generally accepted that deterministic models are less realistic than stochastic, and thus it is of interest to investigate the principles of tolling under stochastic modelling conditions. This thesis develops methodologies to examine the minimal revenue toll problem in the case of Stochastic User Equilibrium. In examining the case of Stochastic User Equilibrium the ‘desired flow pattern’ to be created must first be determined. The classical economics solution of replacing cost flow functions with marginal cost flow functions does not generally result in the total network cost being minimised in the stochastic case. Thus tolls which are analogous to Marginal Social Cost Pricing (MSCP) in the deterministic case do not give the System Optimal solution but do result in drivers being charged for their externalities. If the true system optimal flow pattern is desired it may be possible to derive tolls which are unrelated to MSCP. This thesis examines tolling methodologies to achieve or to closely approach the deterministic SO solution under SUE and examines both path-based exact methods and link-based heuristic methods.

It may however be considered to be more desirable in the stochastic case to produce instead a ‘Stochastic System Optimum’ (SSO) where the perceived total network cost is minimised. The SSO solution may be achieved by applying MSCP-tolling under SUE (Maher et al, 2005). This thesis examines tolling solutions for inducing the Stochastic System Optimal (SSO) under SUE and compares such toll sets with those derived to induce SO flows.

Initial work was based on the assumption of a fixed demand stochastic equilibrium model. It is clear that imposing tolls on a network, will directly affect demand as well as being able to influence route choice and this paper investigates tolling under Stochastic User Equilibrium with elastic demand (SUEED). Elastic demand may be readily included in stochastic equilibrium models (Maher et al, 1999) and MSCP tolls may be easily derived by using marginal cost functions in an SUEED algorithm. In the case of deterministic assignment with elastic demand a Social Optimum with Elastic demand (SOED) has been formulated where economic benefit is to be maximised; under this formulation it has been shown that all toll-sets associated with the flow/demand pattern which minimises the SOED objective function generate the same revenue and that marginal social cost price tolls are such a toll set (Hearn and Yildirim, 2002; Larsson and Patriksson; 1998).

This thesis extends the SOED formulation to SSOED (Stochastic Social Optimum with Elastic Demand) and discusses the extension of the fixed revenue result to tolling to achieve SSOED under SUEED. It further examines the case where the economic benefit is not to be maximised and tolling may be used to achieve an SSO flow pattern for a particular OD matrix where minimal revenue tolls may be desired. The earlier heuristic used to create toll sets to induce the 'true SO' flow pattern under stochastic assignment methods presupposes that the desired flow pattern is fixed and may be determined. In the case of elastic demand, further iteration is required to account for the change in the 'desired flow pattern' as each link toll is increased and the fixed demand heuristic is extended to include this. This heuristic may also be utilised with a minor modification to produce toll sets to induce SSO solutions under SUEED and these are compared to toll sets derived to induce SO flow patterns under SUEED.

Numerical results are given for small toy networks and logit-based SUE is used primarily for reasons of mathematical tractability although the heuristics derived are equally valid for use with any stochastic assignment method.


Stewart, K. Modelling the effects of road pricing. (Thesis). Napier University. Retrieved from

Thesis Type Thesis
Deposit Date Oct 25, 2019
Publicly Available Date Oct 25, 2019
Keywords road tolling; minimal revenue tolls; Stochastic User Equilibrium; flow patterns
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