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 mono-directional and bi-directional 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 bi-directional labeling algorithms compared to the mono-directional ones. Finally, we find that allowing multiple as well 1 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.
This paper addresses the split-delivery vehicle routing problem with time windows (SDVRPTW) that consists of determining least-cost vehicle routes to service a set of customer demands while respecting vehicle capacity and customer time windows. The demand of each customer can be fulfilled by several vehicles. For solving this problem, we propose a new exact branch-and-price-and-cut method, where the column generation subproblem is a resource-constrained elementary shortest-path problem combined with the linear relaxation of a bounded knapsack problem. Each generated column is associated with a feasible route and a compatible delivery pattern. As opposed to existing branch-and-price methods for the SDVRPTW or its variant without time windows, integrality requirements in the integer master problem are not imposed on the variables generated dynamically, but rather on additional variables. An ad hoc label-setting algorithm is developed for solving the subproblem. Computational results show the effectiveness of the proposed method.
The vehicle routing problem with time windows consists of delivering goods at minimum cost to a set of customers using an unlimited number of capacitated vehicles assigned to a single depot. Each customer must be visited within a prescribed time window. The most recent successful solution methods for this problem are branch-and-price-and-cut algorithms where the column generation subproblem is an elementary shortest-path problem with resource constraints (ESPPRC). In this paper, we propose new ideas having the potential to improve such a methodology. First, we develop a tabu search heuristic for the ESPPRC that allows, in most iterations, the generation of negative reduced cost columns in a short computation time. Second, to further accelerate the subproblem solution process, we propose to relax the elementarity requirements for a subset of the nodes. This relaxation, however, yields weaker lower bounds. Third, we introduce a generalization of the k-path inequalities and highlight that these generalized inequalities can, in theory, be stronger than the traditional ones. Finally, combining these ideas with the most recent advances published in the literature, we present a wide variety of computational results on the Solomon's 100-customer benchmark instances. In particular, we report solving five previously unsolved instances.
This paper presents a survey of the research on the Vehicle Routing Problem with Time Windows (VRPTW), an extension of the Capacitated Vehicle Routing Problem. In the VRPTW, the service at each customer must start within an associated time window and the vehicle must remain at the customer location during service. Soft time windows can be violated at a cost while hard time windows do not allow for a vehicle to arrive at a customer after the latest time to begin service. We first present a multi-commodity network flow formulation with time and capacity constraints for the VRPTW. Approximation methods proposed in the literature to derive upper bounds are then reviewed. Then we explain how lower bounds can be obtained using optimal approaches, namely, Lagrangean relaxation and column generation. Next, we provide branching and cutting strategies that can be embedded within these optimal approaches to produce integer solutions. Special cases and extensions to the VRPTW follow as well as our conclusions. Résumé Cet article synthèse porte sur les récents développements concernant le problème du routage de véhicules sous des contraintes de fenêtres de temps. Dans ce problème, le serviceà un client doit débuterà l'intérieur d'un intervalle de temps. Celui-ci peutêtre, soit relaché au prix d'une certaine pénalité, soit rigide, auquel cas, il n'est pas permis de dépasser la limite supérieure. Nous présentons un modèle de réseau multi-flots avec des contraintes de temps et de capacité. Les méthodes heuristiques permettant de calculer des bornes supérieures sont d'abord présentées. Suivent les modèles d'optimisation basés sur la relaxation lagrangienne et la génération de colonnes pourévaluer des bornes inférieures. Enfin, on présente les stratégies de coupes et de branchements liéesà ces méthodes afin de déterminer des solutions entières. L'article se termine par l'étude de cas particuliers et d'extensions ainsi que nos conclusions.
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