The dynamic system of the traffic assignment problem: Part I. Theory

In this paper, we first define so-called J-functions of First-In-First-Out violation, from which we define J-equilibria and a corresponding dynamic model of changing path flows, called J-system. Then we demonstrate that for fixed demand the objective function in the BMW’s mathematical programming formulation (Beckmann, McGuire, and Winsten, 1956) can be interpreted as the total energy of J-system. This leads to a new definition of user equilibria as the only globally stable steady states of J-system. In this sense, J-system is the dynamic system of the traffic assignment problem. We further show the existence of J-system as the dynamic system of the traffic assignment problem with variable demand. From this study, we can see that the new formulation and the BMW formulation are internally related, and the former can have as wide applications as the latter. Then in a following paper, we develop computational methods intrinsic to this simple and intuitive formulation for solving traffic assignment problems. In yet another following paper, we propose J-system formulation of dynamic traffic assignment problem.