A Column-Generation Approach to Line Planning in Public Transport

Abstract

T he line-planning problem is one of the fundamental problems in strategic planning of public and rail transport. It involves finding lines and corresponding frequencies in a transport network such that a given travel demand can be satisfied. There are (at least) two objectives: the transport company wishes to minimize operating costs, and the passengers want to minimize traveling times. We propose a new multicommodity flow model for line planning. Its main features, in comparison to existing models, are that the passenger paths can be freely routed and lines are generated dynamically. We discuss properties of this model, investigate its complexity, and present a column-generation algorithm for its solution. Computational results with data for the city of Potsdam, Germany, are reported.

Shuttle Planning for Link Closures in Urban Public Transport Networks

“…The proposed model formulation combines the ideas of Borndörfer et al (2007) of a path formulation with the idea of Schöbel and Scholl (2006) to include transfers, and the idea of Claessens et al (1998) to combine the capacity assignment and the opening of lines into one decision variable. We extend these ideas with the introduction of additional vehicle decision variables that relax the assignment of capacity to a line, while still including integer decision variables of which lines to operate.…”

“…In contrast to Claessens et al (1998), both Schöbel and Scholl (2006) and Borndörfer et al (2007) include continuous frequency variables, and include the dynamic routing of passengers. Both suggest a column generation approach to solve the model, replacing the multi-commodity flow model for the routing of passengers by a path formulation.…”

“…The proposed formulation combines the path-formulation of Schöbel and Scholl (2006) and Borndörfer et al (2007) with the capacity formulation of Claessens et al (1998), and adds a flexible capacity assignment. The formulation allows specifying a frequency-dependent path cost, thus including frequency-dependent passenger waiting time and transfer times.…”

“…This comes at the cost of slightly overestimating passenger transfer time and boarding time to lines where more vehicles are assigned than the minimum number required, as an assignment of more vehicles leads to higher frequency and thus lower PI, which is not included in the pre-computed path costs. By defining F based on small increments in headways, this di↵erence could be kept small, and results in a more accurate estimation of PI than in the previous models of Borndörfer et al (2007) and Schöbel and Scholl (2006) that do not include frequency-dependent path costs.…”

Planning Problems in Public Transit

“…For both problems, research codes exist. In line planning, one can deal with medium sized instances, see, e.g., [7]. Somewhat more advanced is periodic timetable optimization, which was used, e.g., to compute the timetable of the Berlin subway [25].…”