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  • 1.
    Ekström, Joakim
    et al.
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, The Institute of Technology.
    Sumalee, Agachai
    Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong, China.
    Lo, Hong K.
    Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
    Optimizing Toll Locations and Levels Using a Mixed Integer Linear Approximation Approach2012In: Transportation Research Part B: Methodological, ISSN 0191-2615, E-ISSN 1879-2367, Vol. 46, no 7, p. 834-854Article in journal (Refereed)
    Abstract [en]

    This paper addresses the toll design problem of finding the toll locations and levels in a congestion pricing scheme, which minimize the total travel time and the toll-point cost (set-up and operational costs of the toll collecting facilities). Road users in the network are assumed to be distributed according to the principle of user equilibrium, with the demand assumed to be fixed and given a priori. The toll design problem is commonly formulated as a nonlinear program, which in general is non-convex and non-smooth, and thus difficult to solve for a global optimum. In this paper, the toll design problem is approximated by a mixed integer linear program (MILP), which can be solved to its globally optimal solution. The MILP also gives a lower bound estimation of the original non-linear problem, and the accuracy of the approximation is improved by iteratively updating the MILP. To demonstrate the approach, we apply the algorithm to two networks: a smaller network with 18 links and 4 OD-pairs to illustrate the properties of the approach, and the Sioux Falls network with 87 links and 30 OD-pairs to demonstrate the applicability of the approach.

  • 2.
    Lundgren, Jan
    et al.
    Linköping University, Department of Science and Technology. Linköping University, The Institute of Technology.
    Peterson, Anders
    Linköping University, Department of Science and Technology. Linköping University, The Institute of Technology.
    A Heuristic for the Bilevel Origin–Destination Matrix Estimation Problem2008In: Transportation Research Part B: Methodological, ISSN 0191-2615, E-ISSN 1879-2367, Vol. 42, no 4, p. 339-354Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the estimation of an origin–destination (OD) matrix, given a target OD-matrix and traffic counts on a subset of the links in the network. We use a general nonlinear bilevel minimization formulation of the problem, where the lower level problem is to assign a given OD-matrix onto the network according to the user equilibrium principle. After reformulating the problem to a single level problem, the objective function includes implicitly given link flow variables, corresponding to the given OD-matrix. We propose a descent heuristic to solve the problem, which is an adaptation of the wellknown projected gradient method. In order to compute a search direction we have to approximate the Jacobian matrix representing the derivatives of the link flows with respect to a change in the OD-flows, and we propose to do this by solving a set of quadratic programs with linear constraints only. If the objective function is differentiable at the current point, the Jacobian is exact and we obtain a gradient. Numerical experiments are presented which indicate that the solution approach can be applied in practice to medium to large size networks.

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