Stochastic Equilibrium Assignment with Variable Demand: Theoretical and Implementation Issues

Stochastic Equilibrium Assignment with Variable Demand: Theoretical and Implementation Issues


G Cantarella, A Cartenì, S de Luca, University of Salerno, IT


A fixed-point approach is presented for assignment with variable demand. It allows to specify a wide range of models and solution algorithms w.r.t. different sets of variables among demand or path or arc flows and/or satisfaction, path or arc flows.


Models for traffic assignment to transportation networks simulate how demand and supply interact each other in transportation systems. These models allow the calculation of performance measures and user flows for each supply element (network arc), resulting from origin-destination demand flows, path choice behaviour, and the interactions between supply and demand. Assignment models play a central role in developing a complete model for a transportation system; their results, in turn, are the inputs for the design and/or evaluation of transportation projects.
In such a context, assignment with variable demand seems relevant for urban planning over a medium-long term horizon. Recently, it has been pointed out that transport models should reflect all significant traveller reactions (recently Smith, 2009; see also Commission for Integrated Transport, 2004; COMSIS, 1996; DfT, 2005). In particular, trip generation, trip distribution, modal split and route choice should be modelled in a consistent process based on the equilibrium between transport supply and travel demand. Such an issue is all the more significant when congestion of the transport network is not negligible and when several alternative modes/routes are available.
Since Wardrop (1952), several papers focused on equilibrium assignment with variable demand may be found in literature. Among them, a consistent modelling approach is based on fixed-point models which allows us to analyse existence and uniqueness of solution as well to specify several solution algorithms. This approach was introduced by Daganzo (1983), and further developed by Cantarella (1997).
In this paper a general fixed-point modelling framework is presented that allows dealing with stochastic user equilibrium assignment with variable demand. According to the so-called ?internal? approach, the demand model is embedded within the network loading map to be combined with the arc cost vector function. The proposed modelling framework is general enough to accommodate most existing demand models and path choice models as well as class-specific arc cost functions (provided that they are specified through a linear transformation of a commonly used arc cost function).

A wide range of models and solution algorithms are described w.r.t. different sets of variables among demand or path or arc flows as well as satisfaction, path or arc flows. The adopted fixed-point framework allows to easily defining conditions for solution existence and uniqueness, as well as for algorithm convergence.
Described models and algorithms are also compared with those resulting from the so-called ?external? approach. According to the latter approach an equilibrium is searched between the demand flows resulting from the demand model, and the costs resulting from an assignment with constant demand. This approach is very common in practice, presumably since it can easily be implemented; still it may be lead to much less effective methods w.r.t. the above described ?internal? ones.
Results of applications to a real-scale urban network, regarding the city of Salerno (population 150 000) in Italy, are discussed to support comparison and to address main implementation issues. Theoretical results also apply to extra-urban / regional / national networks, but results of comparison analysis may not, due to the different structure of available path choice sets and the larger scale.

Keywords: stochastic assignment; equilibrium; variable demand.


Association for European Transport