An assignment problem in a network with asymmetric travel costs

In this work we consider the assignment problem of the traffic in a network whose links present cost functions with asymmetric costs. We suppose that the network presents a constant flow, and that always all the flow can be assigned in a way optimum to the different roads; that is, minimizing the global utilization cost of the network, according to the Wardrop's principle. The iterative methods based on the algorithm of Frank–Wolfe are difficult to apply given the impossibility of the objective function as established by Beckman, McGuire and Winston, compelling to the employment of the algorithms of diagonalization. The present work accomplishes an implementation of an optimum assignment process based on the active set method using the approximation outlined by Gill, Murray and Wright, considering a quasi-Newton line search. Our algorithm shows better results than the typical algorithms of diagonalization, with greater precision and a smaller number of iterations, as well as a smaller operational cost.