Balancing the stations of a self service "bike hire" system

This paper is motivated by operating self service transport systems that flourish nowadays. In cities where such systems have been set up with bikes, trucks travel to maintain a suitable number of bikes per station. It is natural to study a version of the C-delivery TSP defined by Chalasani and Motwani in which, unlike their definition, C is part of the input: each vertex v of a graph G = (V,E) has a certain amount xv of a commodity and wishes to have an amount equal to yv (we assume that ∑ v∈V xv = ∑ v∈V yv and all quantities are assumed to be integers); given a vehicle of capacity C, find a minimal route that balances all vertices, that is, that allows to have an amount yv of the commodity on each vertex v. This paper presents among other things complexity results, lower bounds, approximation algorithms, and a polynomial algorithm when G is a tree. Mathematics Subject Classification. ???, ??? 1. Self service “bike hire” systems In 2007, the bike world experienced a great event: the beginning of the Vélib’ system in Paris. Vélib’ is a self service bike hire system in which bikes are left at the free disposal of users throughout the whole city. Users can take a bike at any time and at any station and replace it at any time and at any station they want, under the conditions that there is bike at the starting station and a free place at the final station. Such systems already existed before Vélib’ but what made Vélib’ special is its size, which is largely bigger than previous systems of this kind : 1500 stations, almost 20000 bikes, more than 70000 travels each day. Very quickly, operating such a system has proved difficult. In addition to the servicing, which has to face a quite important number of intentional damages, the stations have to keep a good ratio between the total number of places and the number of bikes in each station (this target ratio is called in Paris “taux de November 15, 2010. 1 Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France. [email protected] 2 Ecole Polytechnique, 91128 Palaiseau Cedex, France. [email protected] 3 Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France. [email protected] 4 Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France. [email protected] 5 Ecole Polytechnique, 91128 Palaiseau Cedex, France. [email protected] 6 ∗Corresponding author. Université Paris Est, LVMT, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France. [email protected] 7 Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France. [email protected] c © EDP Sciences 2001 ha l-0 06 01 44 3, v er si on 1 22 J un 2 01 1 Author manuscript, published in "RAIRO-Operations Research (2011) DOI: 10.1051/ro/2011102" DOI : 10.1051/ro/2011102