Line Optimization in Public Transport Systems
Line planning is one of the strategic tasks a transport company is faced with. The aim is to create a line plan with line routes and service frequencies. Line optimization means to determine a line plan that is optimal regarding to a defined objective lik
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1 Introduction Line planning is one of the strategic tasks a transport company is faced with. The aim is to create a line plan with line routes and service frequencies. Line optimization means to determine a line plan that is optimal regarding to a defined objective like the number of direct travelers [3], the total ride time, number of changes [5], the total cost [2] or the total traveling time. The literature offers approaches with choosing lines from a given set as well as construct line routes from the scratch [1], [4]. All of these approaches presume a given origin-destination-matrix. At least for urban areas this is not realistic. The most important questions of a traffic planner of a transportation company are: ”How much does the new line plan cost?” and ”How many passengers will go by public transport under the new circumstances?”. Obviously it is necessary to consider the movement in demand for public transport within line optimization. In this paper we include frequency depending changing times. In urban public transport systems often more than one line connects two points in a direct way. The expected traveling time is therefore lower than riding time plus half of the frequency time of the used line(s). The waiting times will decrease if there are e.g. two lines that connect two points by parallel line routes. In practice, transport companies take advantage of lines that are parallel in the city and separate in the periphery to give a good service in the area with a great demand and connect the suburbs more efficient with the city. By experience (i.e. tested with data of Dresden)
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Michael J. Klier and Knut Haase
minimizing traveling times without regarding parallel line routes yields unrealistic results for the waiting times.
2 Model In this section we present a model that can cope with (partially) parallel lines and traveling time dependent passenger demand. 2.1 Assumptions Let G [V, A] be a directed graph with a set of nodes V and a set of node connecting arcs A. The nodes represent stops for public transport. The arcs symbolize connections between nodes that can be passed by public transport vehicles. For each arc a ride time tij is defined. Furthermore, we know a set of line routes L. The arcs (i, j) which are part of the route of line l are given by set Aˆlij . F is a set of possible frequencies a line can be operated with. Every node pair that is connected by at least one potential line route yields for each combination of potential line routes l and frequencies f one arc. The larger the pool of lines L is, the more such arcs are required. For practical reasons we generate combinations with no more than five parallel line routes. Thus for each combination the expected traveling time can be estimated. Furthermore, we are able to calculate the proportion of passengers for each line frequency combination within a subset of lines. We assume that a path p ∈ P is a connection between one pair of nodes u, v i.e. possibility for passengers to get from node u to node v. While the arcs are direct connections
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