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In numerous practical vehicle-routing applications, larger vehicles are employed as mobile depots to support a fleet of smaller vehicles that perform certain tasks. The mobile depots offer the possibility of keeping the task vehicles operational by supplying them en route with certain resources. For example, in two-echelon distribution systems, small task vehicles are used to navigate narrow streets and to deliver/collect goods or to collect waste, and larger vehicles serve as mobile depots to replenish the goods to be delivered or to receive collected goods or waste at the outskirts of the urban area. Accessibility constraints may also be imposed by regulations on emissions, which make some areas only accessible for environmentally friendly vehicles such as, for example, battery-powered electric vehicles. Especially if the respective refueling infrastructure is sparse, mobile refueling stations seem to be an interesting alternative. In this paper, we introduce the vehicle-routing problem with time windows and mobile depots (VRPTWMD) to capture the routing decisions of the described applications in a generalized fashion. The VRPTWMD is characterized by fleets of task vehicles (TVs) and support vehicles (SVs). The SVs may serve as mobile depots to restore either the load or the fuel capacity of the TVs that are used to fulfill the customer requests. We present a mixed-integer program for the VRPTWMD with which small instances can be solved using a commercial solver. Moreover, we develop a high-quality hybrid heuristic composed of an adaptive large neighborhood search and a path relinking approach to provide solutions on larger problem instances. We use a newly generated set of large VRPTWMD instances to analyze the effect of different problem characteristics on the structure of the identified solutions. In addition, our approach shows very convincing performance on benchmark instances for the related two-echelon multiple-trip VRP with satellite synchronization, which can be viewed as a special case of the VRPTWMD. Our heuristic is able to significantly improve a large part of the previous best-known solutions while spending notably less computation time than the comparison algorithm from the literature.

We study a class of vehicle‐routing problems with simultaneous pickup and delivery (VRPSPD). In VRPSPDs, each customer may require a certain quantity of goods delivered from the depot and a quantity of goods to be picked up and returned to the depot. Besides the standard VRPSPD, we address (1) the VRPSPD with time limit (VRPSPDTL), which imposes a time limit on the routes of the transportation vehicles, (2) the VRPSPD with time windows (VRPSPDTW), which takes customer time windows into account, (3) the VRP with divisible deliveries and pickups (VRPDDP), which allows for fulfilling the delivery and pickup requests of each customer in two separate visits, (4) the previously unstudied VRP with restricted mixing of divisible deliveries and pickups (VRPRMDDP), which accounts for the difficulty of rearranging the vehicle load by additionally requiring that a certain percentage of the vehicle capacity must remain unoccupied when both types of demand are simultaneously loaded, and (5) the previously unstudied VRPDDP with time windows (VRPDDPTW). We develop a hybrid heuristic solution method which combines an adaptive large neighborhood search algorithm with a path relinking approach, called ALNS‐PR, and we demonstrate the competitiveness of our algorithm on benchmark instances proposed in the literature. Especially on VRPSPDTL, VRPSPDTW, and VRPDDP instances, our ALNS‐PR proves to be superior to the majority of comparison algorithms and is able to obtain numerous new best solutions.