## 2 Problem Modeling
### Model Construction  
The variables utilized in the planning and layout model of the air-ground cooperative co-distribution center are delineated in Table 1. The primary objective of the model is to develop a viable vehicle-drone air-ground collaborative delivery scheme. This approach aims to address the challenges associated with the delivery service to certain nodes, which may be hindered by disruptions in the road network. Additionally, the model seeks to minimize both the overall delivery costs and the collaborative waiting times. Three optimization objectives are established for this purpose. (1) Objective of Optimization for Demand Coverage  
The vulnerability of the road network is directly correlated with the likelihood of failure during disturbances, which subsequently increases the delivery failure rate at customer locations along the route. The implementation of drones in conjunction with vehicle deliveries aims to effectively address customer demand at points surrounding the compromised road nodes, thereby maximizing the coverage rate of regional delivery needs. The objective function for this component can be articulated as follows:  
\[\text{maxf}_{1}=\sum\limits_{r\in\text{R}}\lambda_{r}\frac{\sum\limits_{i\in \text{C}}\sum\limits_{\text{I}\in\text{L}}\mu_{1}^{r}\cdot(z_{\text{i}}^{k}+z_{ \text{i}\text{I}}^{u})}{\sum\limits_{i\in\text{C}}\mu_{1}^{r}} \tag{1}\]  
In the formula, the variable \(\lambda\) represents the vulnerability value of the road section. This paper integrates complex network theory with pertinent research findings, taking into account the characteristics of logistics distribution supply and demand. It emphasizes the significance of the road section \(\text{I}_{\text{r}}\) and the extent of the impact that road section failure has on the overall road network under sudden conditions. These factors are collectively utilized to assess the vulnerability of the road section, as follows:  
\[\lambda_{r}=\text{I}_{r}\cdot\Delta\text{E}(r) \tag{2}\]  
The significance of a particular road selection \(\text{I}_{\text{r}}\) is assessed through the weighted edge betweenness \(\text{B}_{\text{r}}\) associated with the logistics distribution demand along that segment. Additionally, the extent to which the failure of this road segment affects the overall network is evaluated by analyzing the change rate index \(\Delta\text{E}(r)\) of network efficiency before and after the disturbance of the road network. In accordance with complex network theory, edge betweenness is defined as the ratio of the number of shortest paths that traverse a specific edge to the total number of shortest paths within the network, as follows:  
\[\text{B}_{\text{r}}=\sum\limits_{(i^{\prime},j^{\prime})\in\text{P}}\text{N} _{i^{\prime},j^{\prime}}^{r}(r) \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Node Number** & **Node Type** & **Demand** & **Distribution Center** \\ \hline
1 & Demand Point & 42 & 5 \\
2 & Demand Point & 88 & 11 \\
3 & Demand Point & 23 & 11 \\
4 & Demand Point & 67 & 5 \\
5 & Distribution Center & 55 & 11 \\
6 & Demand Point & 30 & 5 \\
7 & Demand Point & 95 & 11 \\
8 & Demand Point & 71 & 11 \\
9 & Demand Point & 145 & 5 \\
10 & Demand Point & 82 & 5 \\
11 & Distribution Center & 38 & 11 \\
12 & Demand Point & 63 & 5 \\
13 & Demand Point & 150 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution relationships within the current road network. Among the factors considered is the betweenness of the road section, denoted as \(\rm B_{r}\). This is represented by \(\rm N_{i^{\prime}_{\rm f}j^{\prime}}\), which indicates the number of shortest paths connecting node \(\rm i^{\prime}\) and node \(\rm j^{\prime}\) within the road network. Additionally, \(\rm N_{i^{\prime}_{\rm f}j^{\prime}}(r)\) represents the number of shortest paths connecting node \(\rm i^{\prime}\) and node \(\rm j^{\prime}\) that pass through the road section \(\rm r\).