Exact Algorithms for Electric Vehicle-Routing Problems with Time Windows

Effective route planning for battery electric commercial vehicle (ECV) fleets has to take into account their limited autonomy and the possibility of visiting recharging stations during the course of a route. In this paper, we consider four variants of the electric vehicle-routing problem with time windows: (i) at most a single recharge per route is allowed, and batteries are fully recharged on visit of a recharging station; (ii) multiple recharges per route, full recharges only; (iii) at most a single recharge per route, and partial battery recharges are possible; and (iv) multiple, partial recharges. For each variant, we present exact branch-price-and-cut algorithms that rely on customized monodirectional and bidirectional labeling algorithms for generating feasible vehicle routes. In computational studies, we find that all four variants are solvable for instances with up to 100 customers and 21 recharging stations. This success can be attributed to the tailored resource extension functions (REFs) that enable efficient labeling with constant time feasibility checking and strong dominance rules, even if these REFs are intricate and rather elaborate to derive. The studies also highlight the superiority of the bidirectional labeling algorithms compared to the monodirectional ones. Finally, we find that allowing multiple as well as partial recharges both help to reduce routing costs and the number of employed vehicles in comparison to the variants with single and with full recharges.