The use of electric vehicles (EVs) in freight and passenger transportation gives birth to a new family of vehicle routing problems (VRPs), the so-called electric VRPs (e-VRPs). As their name suggests, e-VRPs extend classical VRPs to account (mainly) for two constraining EV features: the short driving range and the long battery charging time. As a matter of fact, routes performed by EVs usually need to include time-consuming detours to charging stations. Most of the existing literature on e-VRPs relies on one of the following assumptions: i) vehicles recharge to their battery to its maximum level every time they reach a charging station or ii) the amount of battery charge is a linear function of the charging time. In practical situations, however, the amount of charge (and thus the time spent at each charging point) is a decision variable and battery charge levels are a concave function of the charging times. In this research we introduce the electric vehicle routing problem with non-linear charging functions (e-VRP-NLCF). We propose a mixed-integer linear programming (MILP) formulation that, running on a commercial solver, is able to solve small instances of the problem. To tackle large-scale instances we propose a metaheuristic that uses a MILP formulation to find the optimal charging policy. We report on extensive computational experiments evaluating the performance of the proposed methods and analyzing the impact on the solutions of different charging policy assumptions.
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