Solving network equilibrium problems on multimodal urban transportation networks with multiple user classes

Abstract

A modelling approach for solving quite general network equilibrium problems (with fixed trip productions and attractions) intrinsic to the urban transport planning process is presented. The framework can consider a variety of demand models and route choice behaviours within the same implementation, including multiple user classes and combined travel modes that interact on the same physical network. The demand choices are assumed to have a hierarchical structure. When trip distribution is variable, a doubly constrained entropy‐maximizing model is considered at the first level of choice and a hierarchical logit model is used for the remaining demand choices (time of departure, travel mode, transfer point for combined modes, etc.). If trip distribution is considered to be exogenous, the demand choices are modelled as a hierarchical logit. One of the main features of the model is that it considers the effects of congestion on the road network as well as congestion and capacity constraints effects in each public transport service network. The problem is formulated mathematically as a variational inequality, with asymmetric cost functions, and solved following the diagonalization procedure. Each iteration of the aforementioned procedure solves an optimization problem using the Evans algorithm. Sufficient conditions for the existence and uniqueness of the solution to the diagonalized problem are obtained. The main results of a simple example (solved with an academic version of the proposed algorithm) are presented to show the consistency of the equilibrium flows and levels of services obtained using the model. Finally, a real scale implementation of the model is briefly described to show the feasibility of its application.

Acknowledgements

Work was funded by the Pontificia Universidad Católica de Chile. The authors are grateful for the detailed revisions and important comments and contributions given by the reviewers of the paper.

Notes

In the context of performance–demand equilibrium in transportation networks, the term ‘simultaneous’ implies consistency among the levels of service in the system and flow values (link flows and origin–destination trips) for each stage considered in a particular transportation problem (also known as combined models). On the other hand, in the context of demand models, ‘sequential’ stands for decisions made at different hierarchical levels, while ‘simultaneous’ refers to decisions made at the same level in a hierarchical structure (a decision tree).

Performance (level of service) is used instead of supply, as defined by Manheim (1979). Manheim introduced the concept of level of service (S) and demand (D) leading to an equilibrium (network equilibrium). Only when Manheim considers the notion of an operator who can adjust the level of supply or price does he uses the term ‘supply’. Sheffi (1985) also distinguishes between the notion of supply function and performance function. In any case, a performance–demand equilibrium correspond to a short‐term supply–demand equilibrium, where some variables (link capacities, transit line frequencies, etc.) are fixed.