In bike sharing systems the quality of the service to the users strongly depends on the strategy adopted to reposition the bikes. The bike repositioning problem is in general very complex as it involves different interrelated decisions: the routing of the repositioning vehicles, the scheduling of their visits to the stations, the number of bikes to load or unload for each station and for each vehicle that visits the station. In this paper we study the problem of optimally loading/unloading vehicles that visit the same station at given time instants of a finite time horizon. The goal is to minimize the total lost demand of bikes and free stands in the station. We model the problem as a mixed integer linear programming problem and present an optimal algorithm that runs in linear time in the size of the time horizon.

Abstract We study a dynamic multiperiod routing problem where, at the beginning of each time period, a set of orders arrive that have to be fulfilled either that time period or the next. Thus, in each time period there are customers that have to be served and customers whose service may be postponed. Once it has been decided which customers to serve, an optimal route is constructed and executed. The objective of the problem is to minimize the total distance traveled during the planning horizon. Deciding which customers to serve in a time period is done on the basis of incomplete information, analyzing simultaneously customers in two consecutive periods. No knowledge is available about customers requiring service in future time periods. We introduce simple algorithms, ones which naturally arise in practice, and analyze these algorithms by studying their competitive ratio. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(4), 308–317 2007