When is it convenient to split public transport lines?

Abstract

To serve a public transport demand on a corridor, it is possible to have just one line serving all of it, or two of them each serving one segment - which we refer to as one divided line. In this paper, we investigate theoretically under which conditions a divided line is better than a single line, minimizing the sum of users’ and operators’ costs. Both solutions are often encountered in real-life networks, with the division typically occurring on CBDs or subcenters. Understanding which is the optimal alternative is particularly relevant in the context of public transport network design, where most heuristics do not consider splitting lines as a potential step to improve their solutions. We formulate the problem in a generalized linear city, and optimize the frequency and vehicle size of each of the lines involved, to then compare the resulting costs of the full line versus the line divided at a given node. We prove that there are three conditions that favor the partition of a line, namely i) inducing few transfers, ii) a large difference in the maximum flows between the two segments, and iii) the segment with the lowest maximum flow being long. These conditions are synthesized into a divisibility index that can be calculated for each node, which is used to propose two algorithms to rapidly find where to split the line if convenient. The approach is tested numerically in a linear version of the Parametric City Model, where we compare the optimal set of lines with the ones resulting after dividing the full line following our algorithms with very good results.