#### The Electric Two-echelon Vehicle Routing Problem

##### Ulrich Breunig, Roberto Baldacci, Richard F. Hartl, Thibaut Vidal

Two-echelon distribution systems are attractive from an economical standpoint and help to keep large vehicles out of city centers. Large trucks can be used to deliver goods to intermediate facilities in accessible locations, whereas smaller vehicles allow to reach the final customers. Due to their reduced size and emissions, companies consider using an electric fleet of terrestrian or aerial vehicles for last mile deliveries. Route planning in multi-tier logistics leads to notoriously difficult problems. This difficulty is accrued in the presence of an electric fleet, since each vehicle operates on a smaller range, and may require visits to charging stations. To study these challenges, we introduce the Electric Two-echelon Vehicle Routing Problem as a prototypical problem. We propose a large neighbourhood search metaheuristic as well as an exact mathematical programming algorithm, which uses decomposition techniques to enumerate promising first-level solutions, in conjunction with bounding functions and route enumeration for the second-level routes. These algorithms produce optimal or near-optimal solutions for the problem, and allow us to evaluate the impact of several defining features of optimized battery-powered distribution networks. We created representative E2EVRP benchmark instances to simulate realistic metropolitan areas. In particular, we observe that the detour miles due to recharging decrease proportionally to $1/\rho^x$ with $x \approx 5/4$ as a function of the charging stations density $\rho$; e.g., in a scenario where the density of charging stations is doubled, recharging detours are reduced by 58\%. Finally, we evaluate the trade-off between battery capacity and detour miles. This estimate is critical for strategic fleet-acquisition decisions, in a context where large batteries are generally more costly and less environment-friendly.

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