# A mathematical programming algorithm for optimal bus frequencies

Sammanfattning: Assume that a bus network is given, i.e. we are given a network of streets on which certain bus lines have been set up. Let the total number of buses be given. Assume furthermore that the total demand for bus transportation is given in the form of the marginal totals of an origin-destination matrix, i.e. the total demand for travel from certain origins as well as the total demand for travel to certain destinations is given.Problem: Determine the complete travel pattern and decide which bus frequencies to use on the various lines.The problem is formulated as a non-linear programming problem The most interesting features are that the model explicitly takes into account capacity constraints on the buses, and that the distribution of trips between different zones is influenced by the frequencies on the bus lines. The model also takes into account modal split between bus riding and walking. (An extension to a model handling modal split between car and bus is formulated but not solved.)The model is intended for usage in medium to long range planning.An iterative algorithm to solve this problem is developed. The algorithm is shown to converge to stationary points. As a part of the algorithm, an algorithm for the combined distribution-assignment problem in traffic planning is developed using decomposition.The model has been used on the bus network in the town of Linköping (80.000 inhabitants). The planning model suggests certain actions which are in agreement with the actions actually taken by the bus operator.

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