Integrating short turning and deadheading in the optimization of transit services

Urban transit demand exhibits peaks in time and space, which can be efficiently served by means of different fleets, increasing frequencies in those groups of stops with larger passenger inflow. In this paper we develop a model that combines short turning and deadheading in an integrated strategy for a single transit line, where the optimization variables are both of a continuous and discrete nature: frequencies within and outside the high demand zone, vehicle capacities, and those stations where the strategy begins and ends. We show that closed solutions can be obtained for frequencies in some cases, which resembles the classical "square root rule". Unlike the existing literature that compares different strategies with a given normal operation (no strategy - single frequency), we use an optimized base case, in order to assess the potential benefits of the integrated strategy on a fair basis. We found that the integrated strategy can be justified in many cases with mixed load patterns, where unbalances within and between directions are observed. In general, the short turning strategy may yield large benefits in terms of total cost reductions, while low benefits are associated with deadheading, due to the extra cost of running empty vehicles in some sections.