## 2 Problem Modeling
### Problem Description  
A schematic diagram illustrating the scenario discussed in this paper is presented in Figure 1. The solid black lines denote the vehicle delivery routes along standard urban roads, while the dashed black lines indicate urban roads that are disrupted and out of service. The solid blue lines represent the drone delivery routes. In this context, the co-distribution center dispatches multiple delivery vehicles, some of which are equipped with drones, to fulfill customer demand points within the jurisdictional area to the greatest extent possible. We define the set of customer points \(\mathrm{C}=\{1,\ldots,n\}\), assuming that all customer demand points are distributed along the urban roads. Each customer point may be serviced by, at most, one type of delivery facility, either a vehicle or a drone. In summary, the problem articulated in this paper can be characterized as a graph (V, E), where \(\mathrm{V}=\mathrm{C}\cup\{0,n+1\}\) represents the set of all nodes (0 and \(n+1\) corresponding to the same co-distribution center), and \(\mathrm{E}=\{(i,j),i,j\in\mathrm{V},i\
eq j\}\) denotes the set of all arcs. In the area affected by road disturbances within the delivery range, a delivery vehicle is equipped with a drone to facilitate collaborative delivery. This vehicle is capable of servicing customer locations along the typically navigable roads. Concurrently, it functions as a mobile base station, providing a platform for the drone to replace its batteries and load or unload express items, while also serving as a designated take-off and landing site for  
Figure 1: Schematic representation of the delivery scenario in a road-disturbed area. the drone. To enhance the management of drones, it is essential that each drone takes off and lands on a fixed vehicle, with additional provision for take-off and landing at the co-distribution center. Consequently, the set of dynamic take-off and landing points for drones can be defined as \(\mathrm{S}=\{0,1,\ldots\ldots\mathrm{n},\mathrm{n}+1\}\). During the delivery operation, a single launch of a drone is capable of servicing multiple customer locations. The set of potential paths for the drone is defined as \(\mathrm{L}=\{1,\ldots\ldots,\mathrm{n}-1\}\), which encompasses all feasible drone trajectories. Upon reaching a customer location, the drone is programmed to automatically unload the goods and swiftly depart. After fulfilling delivery tasks for several customers, in accordance with its endurance capacity, the drone returns to a designated meeting point, where it rendezvous with the assigned vehicle to land and perform necessary actions such as battery replacement and the loading or unloading of express items. Given the endurance and load capacity constraints of drones, each drone is assigned a flight time limit of and a load limit of Q. The model presented in this paper addresses the problem of regional overall distribution. It explores various combinations of collaborative distribution involving both vehicles and drones. Let U represent the set of drones and K denote the set of vehicles. The distances associated with each arc traversed by vehicles and drones are denoted as \(\mathrm{d}^{\mathrm{k}}_{\mathrm{ij}}\) and \(\mathrm{d}^{\mathrm{u}}_{\mathrm{il}}\), respectively, while the corresponding travel times are represented as \(\mathrm{t}^{\mathrm{k}}_{\mathrm{ij}}\) and \(\mathrm{t}^{\mathrm{u}}_{\mathrm{il}}\). The collaborative distribution involving drones and vehicles converges at a designated meeting point. Given that the research scenario outlined in this paper is predicated on the variability of road network performance during emergencies, the planning of drone take-off and landing points must be dynamically adjusted based on real-time information. Consequently, there are no predetermined conditions for establishing drone take-off and landing sites. It is imperative to ensure that a drone does not arrive at the meeting point prior to the vehicle, as this poses significant safety risks.