Faculty of Engineering

Modelling the behaviour of networks and their users

We are surrounded by network structures. Everyone interacts with telecommunications, transportation and energy networks every day. These networks can be modelled mathematically and the models can be used to predict the behaviour and performance of the network or its users. For example, how will road users sharing a congested transportation network change their route choice to save travel time?


To understand the behaviour of networks and their users it is necessary to understand situations where the network users do not change behaviour as there is no incentive to do so. This is termed ‘equilibrium’. For instance, for our road users in a congested transport network, when the network is at equilibrium the journey times on all routes are equal and a single user changing route will not improve their travel time.

Mathematical modelling of networks is currently restricted to situations where there is a single measure of cost or benefit, such as travel time. This project aims to develop models for situations where there are multiple conflicting and incomparable costs or benefits. In the given example this might include travel time, toll cost or environmental considerations which are not easy to value in the same way.

The fundamentals of this research can be applied to networks arising in transportation, energy, economics, supply chains etc.


Key focus areas/issues

Much is known about single objective network equilibrium problems and a variety of algorithms have been proposed. These algorithms depend on the assumptions made on the network and the associated flow and cost functions. However, it is often necessary to consider multiple conflicting and incommensurable objectives in network equilibrium problems. There are currently no algorithms to solve these problems.

This project has four specific objectives:

  1. Multiobjective multicommodity flow problems – this work seeks to identify flows of multiple commodities through a network that satisfy demand such that multiple costs of flow are minimised.
  2. Multiobjective equilibration algorithms – using algorithms to find efficient paths for multicommodity flow between two components of the network.
  3. Theory of multiobjective equilibrium problems – studying the behaviour of the developed algorithms
  4. Experimental verification - applying the developed algorithms to particular network types such as transportation networks and supply chains.

Key people


Andrea Raith
Email: a.raith@auckland.ac.nz
Phone: +64 9 373 7599 extn 82421


Related publications

The following publications are from previous relevant research: 

M. Ehrgott. Multicriteria Optimization, 2nd edition, Springer, 2005. 

J.Y.T. Wang, A. Raith and M. Ehrgott. Tolling Analysis with Bi-objective Traffic Assignment. In M. Ehrgott, B. Naujoks, T. Stewart, J. Wallenius (eds.) Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems, Springer, 117-129, 2010.

A. Raith, and M. Ehrgott `On vector equilibria, vector optimisation and vector variational inequalities’. Journal of Multi-Criteria Decision Analysis, in print.

A. Raith and M. Ehrgott. A two-phase algorithm for the bi-objective integer minimum cost flow problem. Computers & Operations Research 36, 1945-1945, 2009.

A. Raith and M. Ehrgott, A comparison of solution strategies for biobjective shortest path problems. Computers & Operations Research 36, 1299-1331, 2009.

A. Raith; C. van Houtte, J.Y.T. Wang and M. Ehrgott. Applying bi-objective shortest path methods to model cycle route-choice. In Proceedings of 32nd Australasian Transportation Research Forum, http://www.cmsl.co.nz/assets/sm/4433/61/paper106-Raith.pdf, 2009.

J.Y.T. Wang, R. Lindsey and H. Yang. Nonlinear pricing on private roads with congestion and toll collection costs. Transportation Research Part B, 2010, doi:10.1016/j.trb.2010.05.004



This project has been made possible with the support of the Marsden Fund.