Line Planning for Different Demand Periods

Line planning as one of the first planning stages in public transport is a well-researched topic. Nearly all models start with the assumption that the demand for public transport is known and fixed. For real-world applications this assumption is not practicable, since there are different demands depending on the period of the day and the day of the week, e.g., the high demand in morning traffic differs from the demand during a week-day, or from the low demand on Sunday's afternoons, or at night. Planning lines for different demand periods comes with two conflicting goals: On the one hand, the line concept should be adapted as good as possible to the respective demand. On the other hand, the lines should be as similar as possible for different demand periods, e.g., the line plan for Sunday afternoon should be related to the one on Monday morning. In this paper, we show that line planning for different demand periods can be modeled and solved: We introduce the multi-period line planning problem which is to find optimized line concepts for each demand period which are similar (enough) to each other. To this end, we discuss three different approaches to define the (dis)similarity between line concepts. These are frequency-based concepts, and concepts taking the number of different lines and the shape of the lines into account. For the latter, we use Wasserstein distances for modeling the similarity between two line concepts. We show that for all these similarity measures the line planning problem can be formulated as an integer linear program and solved efficiently. Our experiments furthermore show the differences of the resulting line concepts, and that the similarity of line concepts between different demand periods and the quality of the line concept are conflicting goals.