On the complexity and modeling of the electric vehicle sharing problem

We introduce the electric vehicle sharing problem (EVSP), a problem that arises from the planning and operation of electric car-sharing systems which allow one-way rental of vehicles. The problem aims at finding the maximum total daily rental time in which customers' demands are assigned to the existing fleet. In addition, either all of the customer's demands are completely fulfilled or the customer does not use the system at all. We show that the EVSP is NP-hard, and we provide four mixed-integer linear programming formulations based on space-time network flow models, along with some theoretical results. We perform a comprehensive computational study of the behavior of the proposed formulations using two benchmark sets, one of which is based on real-world data from an electric car-sharing system located in Fortaleza, Brazil. The results show that our best formulation is effective in solving instances where each customer has only one demand. In general, we were able to optimally solve at least 55% of the instances within the time limit of one hour.