Formulation of Simultaneous Car and Public Transport Network Equilibrium in the Form of Mixed Complementarity Problem in the Context of Bi-level Programming
O Ivanova, Transport and Mobility Leuven, BE
This paper presents a new formulation of simultaneous network equilibrium for car and for public transport in the form of an MCP Problem. This formulation makes it possible to incorporate network equilibrium into the bi-level programming framework.
Safety and efficiency of a transport network may be significantly improved by the correct allocation of infrastructure investments. In most cases infrastructure investment projects for a city are interrelated and their benefits cannot be estimated independently. Hence, in order to allocate investments in the best way inside a city, one should consider all possible combinations of proposed infrastructure projects and choose the most optimal one. Transport infrastructure is a public good, provided by a transport ministry. Hence, the most natural criterion for choice of the best combination of investment projects is the maximization of social welfare. In order to be able to calculate the social welfare for any given combination of infrastructure projects, a transport ministry should be able to foresee the changes in travel behavior of citizens, which are the result of new investments.
Travel behavior of citizens is summarized in the form of network equilibrium model that describes behavior of both car users and public transport users. A transport system is in equilibrium, when each network user (citizen) chooses route with the smallest transport costs, given route choices of all other users. None of the network users finds it worthwhile to deviate from his equilibrium route, if the others do not deviate. Network equilibrium depends upon a network structure and hence is influenced by infrastructural improvements.
By solving network equilibrium model for each combination of infrastructure projects, a transport ministry is able to get the forecast of travel behavior of citizens and hence calculate the social welfare measure. In order for a transport ministry to be able to consider significant number of infrastructure projects and hence increase the probability that chosen combination is close to the ultimate optimum, solution time of the network equilibrium model should be as small as possible. The optimality of chosen combination of investment projects also depends upon the functional form of social welfare measure as well as upon the effective investment budget restriction.
The present paper has the following related aims. The first one is to formulate the simultaneous network equilibrium for both car and public transport in the form of Mixed Complementarity Problem and incorporate it into bi-level programming framework. The second aim is to investigate the role, which functional form of social welfare measure plays in optimal infrastructure investment choice using Oslo case-studio.
The model presented in the paper has a structure of bi-level programming or leader-follower game (Macrotte, 1986), where a transport ministry is the leader and citizens are the followers. The lower level of the bi-level programming represents changes in traveling behavior of citizens for a given allocation of investments. The upper level represents the choice of investments allocation by a transport ministry in order to maximize social welfare.
This formulation makes it possible to implement network equilibrium without use of special transport packages as well as to incorporate it into the bi-level programming framework, with the help of which one may evaluate welfare benefits of large number of infrastructure project combinations during reasonable amounts of time and allocate network investments in the most efficient way. The paper also traces relationship between functional form of social welfare measure and ordering of infrastructure project combinations using Oslo/Akershus case-studio.
Association for European Transport