Lower and upper bounds for scheduling multiple balancing vehicles in bicycle-sharing systems

Public bicycle-sharing systems have been implemented in many big cities around the world to face many public transport problems. The exploitation and the management of such transportation systems imply crucial operational challenges. The balancing of stations is the most crucial question for their operational efficiency and economic viability. In this paper, we study the balancing problem of stations with multiple vehicles by considering the static case. A mathematical formulation of the problem is proposed, and two lower bounds based on Eastman's bound and SPT rule are developed. Moreover, we proposed four upper bounds based on a genetic algorithm, a greedy search algorithm and two hybrid methods that integrate a genetic algorithm, a local search method and a branch-and-bound algorithm. The developed lower and upper bounds are tested and compared on a large set of instances.