Does the Theoretical Development in Assignment Procedures Matter in Practice?

Does the Theoretical Development in Assignment Procedures Matter in Practice?


O A Nielsen, Danish Technical University, DK


The paper compares different solution algorithms for traffic assignment on a large scale network, showing that heuristic approaches do not converge to equilibrium, and that theoretical advances over the latest years do improve results in practice.



The theoretical foundation and formulation of solution algorithms for traffic assignment have been addressed thoroughly in the literature. One of the milestones was the book "Urban Transport Networks" by Sheffi (1985) who provided an overview of the formulation of assignment models as mathematical programmes, and who suggested practically operational solution algorithms. The book provided accordingly an overview of results priory only reported in research papers.

Although the book is twenty years old, many practitioners, software packages and even university-based models still use more simple approaches, e.g. iterative or incremental loading procedures. The argument is that they converge to 'almost the same solution' in large scale cases, and that more rigorous approaches are merely of academic interest. While examples on small test networks clearly can illustrate large differences between solution algorithms and deviances in the heuristic ones, less evidence on this seem to have been reported based on larger cases.

Within static assignment procedures, Stochastic User Equilibrium (SUE) provides from a theoretical point of view a more realistic description of behaviour than deterministic models. In research and some applied models, SUE has been extended to more complex descriptions of behaviour. Extensions are e.g. random coefficients (mixed Probit), multi-class assignment where each class has its own utility function, and more refined error terms. However, little empirical research has compared the results with more simple approaches, and hereby illustrated the benefits.


The purpose of the paper is to investigate and compare the behaviour of various formulation of the assignment model on a large-scale network (618-zones, 30,000 links) covering Copenhagen. DTU developed the software, why we were not dependent on limitation of specific commercial packages. It was therefore fairly easy to test different model assumptions and solution algorithms on the same case.

Investigations and results

The first set of experiments compares the theoretical correct solution algorithm of User Equilibrium with incremental loading and iterative assignment. It is shown, that the results - as expected - indeed are different, and especially that the iterative assignment is an improper solution method.

Following this, different formulations of the error term in SUE are investigated. It is shown, that there is a difference in the results. The core conclusion is that a truncated normal distribution does not approximate a Probit model, and that the Gamma distribution, which is additive in mean and variance and non-negative, is a better choice. It is also shown that triangular and rectangular distributions are a poor choice, although the applied literature often uses these distributions, without a thorough investigation of the implications of this.

It is evident that the User Equilibrium solution is worse in terms of system costs, than the System Optimal Solution (SO) - or equal in very simple cases. However, it is also often claimed that the Stochastic User Equilibrium solution is in practice worse than the User Equilibrium. This is however not necessarily the case, since the error term can make the solution either further away or closer to SO than the UE solution. The paper provides an example on this on a small-scale network, and show whether the same may matters on the larger scale as well. The interesting implication of this is that providing information - e.g. by traffic informatics - may not improve the overall system performance. In some cases information will make thing worse!

A special case of this investigation is calculation of impacts of a new road project in Copenhagen, where Braess paradox appear. The paradox states that new infrastructure projects sometimes make the overall network performance worse, due to the difference between SO and UE.

The paper then compares the solutions of the more advanced approaches with random coefficients and multiple classes, with the simple single-class assignment procedure used in most applied models. The conclusion on this section of the paper is, that the more advanced models lead to a dramatically change and improvement of the modelling results. This was the case describing the existing system, but more important when impacts of new projects and policy initiatives are modelled. An example is that multiple classes with different value of times, as well as distribution of these, will result in a quite different response to e.g. road pricing, than if all travellers are assumed to have the same willingness to pay.


The conclusion of the paper is that the theoretical development does matter in practice. Non-rigorous solution algorithm may not converge to the User Equilibrium or Stochastic User Equilibrium solution. The formulation and calculation of the error terms, e.g. truncation procedures, does influence the result. SUE is different and more realistic than UE. Braess paradox may exist in practice - it is not merely a theoretical construct. The more advanced models with multiple classes and random coefficients change and improve the results to a large degree. Therefore, it is recommended that assignment procedures should have more attention in practice since the specification has a significant influence on the result of project evaluations in practice.


Association for European Transport