## 2 Related Work
### Vehicle Routing Problems  
Given a set of transport orders and a fleet of vehicles with given capacities, the VRP aims to minimize the total traveled distance; the number of tours; or--more generally--the travel costs for fulfilling all transport orders meeting all given constraints, which due to capacity constraints typically relate to the volume or weight of the given transport items. For private entities, minimizing traveled distances is critical because it directly affects load factors and costs. On the other hand, for the public actors reducing the traveled distance is paramount as it directly affects pollution and city traffic. The VRP was first introduced by [10] in the late 1950s. They formulated the mathematical programming approach and algorithm to solve the problem of delivering gasoline to gas stations. Over the years, the VRP has proven to be a valuable tool for optimizing transport operations. Minimizing the total traveled distance or optimizing the number of vehicles required can significantly improve transport costs. Descriptions of VRP methods can be found in lecture books on logistics. A comprehensive explanation of several algorithms and their function, including the enrichment towards the team orientation problem (TOP), can be found, e.g., in [11]. In this context, ref. [12] provided a comprehensive and extended overview of the VRPs for LML and its diverse applications, with a particular focus on incorporating the perspectives of local authorities and companies. Additionally, several operational tools were introduced by [13; 14; 15; 16]. These studies offer valuable insights into practical solutions and strategies to address the challenges posed by LML, thus enhancing the efficiency and sustainability of transport operations in urban areas. In the realm of alternative delivery locations, several notable studies have contributed to the field. In [17], the authors presented an early work that addresses the vehicle routing problem with roaming delivery locations. In their model, customers receive deliveries in different non-overlapping time windows at different delivery points, e.g., by getting packages delivered to their parked car throughout the day. The authors highlight the advantages of roaming delivery over traditional home delivery and emphasize its cost-effectiveness. Ref. [18] looked at a different facet of alternative delivery, considering direct-to-customer delivery or delivery through pickup points. They framed this as a multi-depot two-echelon vehicle routing problem. While their study examined various delivery options, it did not consider capacity constraints at pickup points. In this regard, ref. [19] studied a distribution system with capacitated locker stations and ignored home deliveries and customer time windows. However, they considered different sizes for the APLs. Nevertheless, the restriction to only one parcel per slot simplified the packing decisions. The work of [20] addressed the inclusion of capacity constraints at alternative delivery locations. However, they did not include time windows for customers at their home locations in the problem. In addition, ref. [21] addressed the vehicle routing problem with private and shared delivery locations, which is closely related to our problem. Their private locations represented home delivery points, while the shared locations were APLs stations. However, they did not consider the different parcel and slot sizes. Their approach uses a matheuristic based on finding large neighborhoods, with a repair step using a mixed integer programming model and integrated iterated local search to avoid getting stuck in local optima. A study by [22] explored the concept of mobile APLs with dynamically changing locations. They focused on optimally locating these mobile APLs so that customers stay within a certain range of their assigned station during the planning horizon. The goal was to minimize the locker fleet. Ref. [23] proposed a VRP with soft time windows for last-mile delivery and studied the impact of the introduction of lockers.