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Given a railway line with n stops and the number of travelers between each pair of stops, we show how to split these stops into k different fare zones in order to maximize the benefit obtained from the sale of tickets to the travelers. We present a method to obtain this solution that is based on finding the longest path in a weighted root tree. This method improves in terms of efficiency the combinatorial method, where all the possible distributions have to be considered for deciding which is the optimal one., The authors would like to thank RENFE for his collaboration and providing us several tables of data of the flow of passengers of the commuter trains of the Valencia region. J. Alberto Conejero is supported by MEC Project MTM2013-47093-P and Esther Sanabria-Codesal is supported by MEC Project MTM2012-33073.

The management of a car-rental service becomes more complex as long as one-way bookings between different depots are accepted. These bookings can increase the operational costs due to the necessity of moving vehicles from one depot to another by the company staff in order to attend previously accepted bookings. We present an iterative model based on flows on networks for the acceptance of bookings by a car-rental service that permits one-way reservations. Our model lets us also recover the movement of the fleet of vehicles between the depots over the time. In addition, it also permits including restrictions on the amount of cars managed at every single depot. These results can be of interest for an electric car-rental service that operates at different depots within a city or region., J. Alberto Conejerois is supported by MEC Project MTM2013-47093-P. The authors thank Victor Fernandez and Raul Urbano from the Mobincity Project for their helpful discussions on the topic. Esther Sanabria-Codesal is supported by MEC Project MTM2012-33073.