## Applicability of Sensitivity Analysis of the Traffic Assignment Problem

### Authors

BENNETT D, University of Nottingham and SHEER A F, University of London, UK

### Description

Although the basic equations for sensitivity analysis of the traffic assignment problem have been known for some years, little use has been made of them. This paper shows that the solutions for the quantities of practical interest have a simple structure

### Abstract

Although the basic equations for sensitivity analysis of the traffic assignment problem have been known for some years, little use has been made of them. This paper shows that the solutions for the quantities of practical interest have a simple structure and a form that makes calculation very easy, even for realistic networks. These quantities may be the effects of possible changes in traffic demand or changes in the cost of travel (travel time) on individual links. The basic equations of the sensitivity analysis have been discussed before, without any derivation of explicit solutions. Electrical circuit analysis has been adapted to show that graph-theoretic techniques could be used to find explicit solutions in certain cases. We have extended and generalised that approach: our aim is to show how the solutions depend explicitly on the structure of, and the levels of congestion on, the network.

We give explicit formulae for various quantities (such as changes in link-flows and journey times for given changes in demand), in a form which shows the separate dependence on the geometry and congestion levels on the network. These formulae are applicable to both user and system equilibrium criteria, and to multiple user classes, and can readily be extended to eater for asymmetric link cost (travel time) functions. It is not necessary to solve a traffic assignment problem at any stage. A procedure for evaluating these formulae quickly for a realistic network is also given.

The parameters appearing in the formulae are the partial derivatives of the link travel times with respect to the link-flows; these cannot be measured directly, though they will be zero for uncongested links. Where link-specific data is available from a SCOOT (Split, Cycle and Offset Optimisation Technique) urban traffic signal control system, this also cannot be used directly. Therefore an on-street registration plate survey was used to obtain real flow and travel time data for calibration. The region surveyed was a small network in Leicester, for which SCOOT data was available from the Instrumented City facility, and the survey was undertaken during an evening peak period in November 1995. By comparing the parameter values derived from the survey data with the SCOOT data for the links, a simple formula connecting the required parameters with SCOOT data was obtained. This procedure is not without problems. Care is needed in specifying the graph-theoretic representation of the network so that SCOOT data can be used. and the way it is collected means that some of it is unreliable. The survey produced travel times for journeys across the region, and a weighted least squares method was used to find the link travel times.

The method was applied to a larger region in Leicester, to find the predicted sensitivities to changes in demand. These were plausible; however, since the regions are physically small, the predicted changes in average journey times were less than the typical variations due to signal timings, and so a small survey of a small region does not provide a good test of the accuracy of the theory. An estimate of the effects of changes in the extrinsic features of a network is often needed: what happens if demand changes, or a link is altered by, say, traffic calming ? The model that we have developed can be used to answer such questions, and calculations are fast enough for it to be used in real time, or in an interactive planning model.

#### Publisher

Association for European Transport