The electric vehicle sharing problem (EVSP) arises from the planning and operation of one-way electric car-sharing systems. It aims to maximize the total rental time of a fleet of electric vehicles while ensuring that all the demands of the customer are fulfilled. In this paper, we expand the knowledge on the complexity of the EVSP by showing that it is NP-hard to approximate it to within a factor of $n^{1-\epsilon}$ in polynomial time, for any $\epsilon > 0$, where $n$ denotes the number of customers, unless P = NP. In addition, we also show that the problem does not have a monotone structure, which can be detrimental to the development of heuristics employing constructive strategies. Moreover, we propose a novel approach for the modeling of the EVSP based on energy flows in the network. Based on the new model, we propose a relax-and-fix strategy and an exact algorithm that uses a warm-start solution obtained from our heuristic approach. We report computational results comparing our formulation with the best-performing formulation in the literature. The results show that our formulation outperforms the previous one concerning the number of optimal solutions obtained, optimality gaps, and computational times. Previously, $32.7\%$ of the instances remained unsolved (within a time limit of one hour) by the best-performing formulation in the literature, while our formulation obtained optimal solutions for all instances. To stress our approaches, two more challenging new sets of instances were generated, for which we were able to solve $49.5\%$ of the instances, with an average optimality gap of $2.91\%$ for those not solved optimally.

翻译：电动汽车共享问题（EVSP）源于单向电动汽车共享系统的规划与运营，旨在确保满足所有客户需求的同时，最大化电动汽车车队的总租赁时间。本文拓展了对EVSP复杂性的认知，证明除非P=NP，否则对于任意ε>0，该问题在多项式时间内无法近似到n^(1-ε)因子以内（其中n表示客户数量）。此外，我们还证明该问题不具有单调结构，这对采用构造性策略的启发式算法开发可能产生不利影响。同时，我们提出了一种基于网络能量流的EVSP建模新方法。基于新模型，我们提出了一种松弛-固定策略和一种精确算法，该算法使用从启发式方法获得的暖启动解。我们报告了将所提公式与文献中性能最优公式进行比较的计算结果。结果表明，在获得最优解的数量、最优性差距和计算时间方面，我们的公式均优于先前公式。此前，文献中性能最优的公式仍有32.7%的算例未求解（在1小时时间限制内），而我们的公式为所有算例都获得了最优解。为验证方法的有效性，我们生成了两组更具挑战性的新算例，其中49.5%的算例成功求解，未最优求解的算例平均最优性差距为2.91%。