CMM - Centro de Modelamiento Matemático

Modelamiento de Sistemas de Transporte

Estado: finalizado

Investigadores CMM
Marcos Kiwi, Raúl Manasevich, Martín Matamala, Jean-Bernard Baillon.

Otros Investigadores
Francisco Martínez (U. de Chile), Michael Florian (Montreal), Cristián Cortés (U. de Chile), Michel Gendreau (Montreal).

Estudiantes
Pedro Jara, José Vaisman.

Our interest is the analysis of transportation networks within a large city, in order to provide a stronger scientific ground to assist the decisions made by the competent authorities. In addition, several problems in transportation science are still on the way of attaining full maturity concerning its correct mathematical formulation and solution.

Equilibrium and Congestion in Public Transportation Networks

The goal here is to model the equilibrium state on a static transit network, incorporating the congestion effects upon the users decisions. In contrast with private transportation networks, the effects of congestion upon public transportation systems have until very recently been essentially neglected or introduced in a very unsatisfactory and mathematically incorrect way. This raises substantial doubts concerning the validity of using such models to describe the situation of highly congested networks as in most large cities. We fill this gap by focusing on a hyper-path formulation of the equilibrium, where we fully incorporate congestion by means of different approximations based on queuing theory. We have proved existence of equilibrium, and derived different characterizations. The latter has enabled us to devise an algorithm for computing an equilibrium, which has been tested on real-world networks. At the same time we have studied the congestion processes occurring at bus stops, using both simulation and queuing theory, in order to provide a more realistic treatment of congestion.

Algorithms for Computing User Equilibrium in Networks

User equilibrium models are used to describe the equilibrium state on a multiple origin-destination network in which travel time along each arc is flow-dependent. The standard approach to solve this problem is to directly apply Wardrop’s equilibrium conditions and to transform them into an equivalent convex minimization problem. In this area we have studied an alternative approach based on a combinatorial decomposition of the network into regions to which Wardrop’s conditions are applied by switching flows. This local approach iteratively solves the network user equilibrium and has been shown to be convergent in the case of cyclic networks. From a computational point of view, the method has been tested on real-world instances comparing favorably with respect to the more classical methods.

Urban Land-use Equilibrium Models

The spatial distribution of activities in the urban context is the result of a large number of individual location decisions taken by agents (households and firms). Each individual makes an optimal choice considering the available location set, his/her preferences and the constraints (budget and time). Location choices are different because the built environment is heterogeneous, including here the distribution of activities itself plus the accessibility provided by the available transport system, but also because of the diversity of agents preferences and buildings available at each location. Urban economics assumes that location equilibrium requires that each location is assigned to that agent which submits the highest bid. Suppliers decide which new land and the type of building to develop as to maximize profit given the maximum bidder rule. All agents take decisions simultaneously with complete information about the decisions been made by others. The existence and sensitivity analysis of an equilibrium for this market is a complex problem because: The equilibrium is achieved by the adjustment of rents for each location; demand is the result of a bid-auction process for each building-location option; supply follows a profit maximizing behavior; equilibrium is attained by the simultaneous satisfaction of the constraints.

Routing on Networks with Server Latency

This subject is concerned with routing on a communication/transportation network on which links may be traversed in either direction but there is a latency period each time one switches direction. Assuming that requests arrive to both link ends under a known pattern, the question is to devise a routing strategy that optimizes a given efficiency measure such as total or maximum queue length, average or maximum waiting time, throughput, etc. This occurs for instance in the case of a one-lane road that must be shared for traffic in both directions. Our study has focused on analyzing a feasibility question, namely, does there exist a control policy capable of routing the flow and keeping the waiting queues at nodes of bounded length? Different situations have been analyzed, depending on the amount of information that the controllers placed on each node may exchange.

Dynamic and Continuous Models for Traffic Flow

Since it is fairly obvious that flow and congestion is an eminently dynamic process, we started a research activity in the form of an internal seminar to explore alternative ways of incorporating a dynamic dimension to the network equilibrium models. Several models have been proposed for the dynamics of traffic flow along a route, in terms of PDE's describing the space/time evolution of the state variables (velocity, density and flow). The goal is to understand the evolution of shock waves within the traffic. Because of their complexity, the use of these models has been essentially restricted to modeling an isolated route segment, while the effect of the rest of the system is included through the boundary conditions.

Subir