## 4 Case Study
### Decision Problem  
Designing city logistics involves developing a comprehensive strategy to efficiently and sustainably manage the movement of goods within urban areas. The goal is to minimizecosts, reduce congestion and environmental impacts, and ensure timely deliveries [32]. The case study is related to the design of an innovative last-mile distribution system based on a combination of traditional vans and _Autonomous Delivery Robots (ADRs)_[33]. ADRs are self-driving vehicles intended to transport goods from one location to another without human intervention. These robots utilize advanced technologies such as artificial vision, sensors, and GPS to navigate and perform their delivery tasks efficiently. Different companies have carried out pilot initiatives for testing ADRs: for example, _Starship Technologies_ is running a self-driving robot service in the city of London [34], as well as _e-Novia_ and _Twinwheel_ in different cities [35]. Recently, _Aramer_ launched a self-driving delivery robot [36], while _FedEx_ experimented its six-wheeled autonomous delivery robot, called _Roxo SameDay Bot_[37]. In our application setting, the decision makers want to design a last-mile distribution system that utilizes a fleet of vans, each equipped with a sub-fleet of ADRs, to drop off small parcels at customer locations. Deliveries can be performed by directly visiting customers with the van or by parking the van at designated locations (_parking slots_, PSs), from which the final leg of the delivery is completed either on foot by the driver or by ADRs. ADRs can autonomously navigate sidewalks at pedestrian speed to deliver parcels to customers' doors. Formally, the problem is defined on a directed multigraph \(G=(V,A)\) where \(V\) is the vertex set and \(A\) is the arc set. The vertex set is partitioned into three subsets: \(V=\{0\}\cup P\cup C\), with \(0\) representing the van depot, \(P\) denoting the set of available parking slots, and \(C\) comprising the set of customers to be serviced within a planning horizon of length \(T\). The depot has a time window \([0,T]\), and each parking slot \(p\in P\) has a number of availability windows of the form \([e_{p},l_{p}]\) during which parking reservations can be made. Each customer \(c\in C\) has a demand \(q\) expressed as the number of parcels to be delivered. For each pair of vertices \(i,j\in V\), set \(A\) contains two arcs: \((i,j)_{d}\) for driving and \((i,j)_{w}\) for walking (traversed by either a driver or an ADR). The travel time for walking arcs is calculated based on constant speeds \(v_{dri}\) and \(v_{adv}\) for drivers and ADRs, respectively. The driving time \(\tau_{ij}(t)\) varies according to the time of day and depends on departure time \(t\) from node \(i\)[38]. Customer requests must be satisfied by a fleet of homogeneous vans, each with a capacity \(Q_{van}\) and equipped with \(m\) ADRs. ADRs have a capacity \(Q_{adv}\) and a maximum travel distance \(L_{adv}\) per full charge. Drivers, who can also perform deliveries on foot, have a capacity \(Q_{dri}\) per walking trip and a maximum daily walking distance \(L_{dri}\). The cost structure includes daily fixed costs for vans, drivers, and ADRs, as well as variable costs per kilometer for vans. Each customer must be visited once, vans must return to the depot before the end of the day, and capacity and distance constraints must be satisfied. The decision-making process is divided into two main components:  
* _Parking Slot Selection_ (PSS), a semi-structured decision, which consists in locating \(k\) out \(|P|\) parking slots (see [39] for a review on location models). This decision involves human judgment to incorporate implicit preferences and contextual factors. LLMs can assist in generating insights to inform the decision-making process and suggest alternative locations for parking slots based on stakeholders' preferences and feedback.