Iterative methods for solving an equilibrium problem arising in transit deregulation

The authors consider the context of a deregulated transit system involving private cars, bus passengers, and bus operators where the latter freely decide which line they operate. They assume that car drivers and transit users strive to minimize individual travel times whereas bus operators maximize individual profits. Within each system under consideration-cars, passengers, buses-a state of equilibrium can be characterized as the solution of a variational inequality. In this paper the authors assume that the combined model is solved by the Gauss-Seidel approach described in Fernández and Marcotte. Special attention is paid to the solution of the bus operators equilibrium for which several solution algorithms-Jacobi-Newton, Newton-Jacobi, fixed point iterations-are proposed. Numerical results are presented for a small network and for the Santiago (Chile) transit network.