# Location Selection Methods for Urban Terminal Co-Distribution Centers with Air-Land Collaboration
## 3 Modeling
### Model Construction  
Considering the service capacity and coverage of the different types of co-distribution centers and with the goal of minimizing the total cost of network operations and the overall distribution time, a multi-objective planning and layout model is constructed. \[\text{min}Z=a\sum Z_{time}+g\sum Z_{cost} \tag{1}\]  
\[\sum Z_{time}=\sum_{u=1}^{a}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{u}}{v_{u}} \cdot y_{ij}^{u}+\sum_{k=1}^{b}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{k}}{v_ {k}}\cdot y_{ij}^{k} \tag{2}\]  
\[\sum Z_{cost}=\sum_{s=1}^{2}\sum_{i=1}^{n}C_{s}\cdot x_{is}+\sum_{s=1}^{2}\sum _{i=1}^{n}\sum_{j=1}^{m}\left(c_{v}^{u}y_{ij}^{u}+c_{v}^{k}y_{ij}^{k}\right) \cdot x_{is} \tag{3}\]  
\[\sum_{i=1}^{n}\sum_{s=1}^{2}x_{is}\leq n \tag{4}\]  
\[\sum_{s=1}^{2}x_{is}\leq 1 \tag{5}\]  
\[\sum_{i}^{n}y_{ij}=1,\forall j\in J \tag{6}\]  
\[\sum_{i=1}^{n}\sum_{j=1}^{m}\left(y_{ij}^{u}+y_{ij}^{k}\right)h_{j}\leq\sum_{ s=1}^{2}x_{is}E_{s},\forall i\in I \tag{7}\]  
\[A_{ij}-y_{ij}^{u}\geq 0\;\forall i\in I,\forall j\in J \tag{8}\]  
\[A_{ij}y_{ij}^{u}d_{ij}^{u}\leq\frac{D_{u}}{2}\;\forall i\in I,\forall j\in J, \forall u\in U \tag{9}\]  
\[q_{ij}^{u}\leq Q_{u}\;\forall i\in I,\forall j\in J \tag{10}\]  
\[\sum Z_{time}=\sum_{u=1}^{a}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{u}}{v_{u}} \cdot y_{ij}^{u}+\sum_{k=1}^{b}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{k}}{v_ {k}}\cdot y_{ij}^{k} \tag{2}\]0  
\[\sum Z_{time}=\sum_{u=1}^{a}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{u}}{v_{u}} \cdot y_{ij}^{u}+\sum_{k=1}^{b}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{k}}{v_ {k}}\cdot y_{ij}^{k} \tag{2}\]1  
\[\sum Z_{time}=\sum_{u=1}^{a}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{u}}{v_{u}} \cdot y_{ij}^{u}+\sum_{k=1}^{b}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{k}}{v_ {k}}\cdot y_{ij}^{k} \tag{2}\]2  
\[\sum Z_{time}=\sum_{u=1}^{a}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{u}}{v_{u}} \cdot y_{ij}^{u}+\sum_{k=1}^{b}\sum_{i=1}^{n}\sum_{j=1}^{m}\frac{d_{ij}^{k}}{v_ {k}}\cdot y_{ij}^{k} \tag{2}\]3  
In the model, Equation (1) is the overall objective function of the mixed-integer planning. Equation (2) is the distribution time function, including the total time for UAVs and vehicles to complete the distribution task, and Equation (3) is the cost function, which includes the construction cost of warehouses, facility renovation, and platform construction, as well as distribution costs, such as the cost of the purchase of facilities and equipment, energy consumption, human resources, and equipment maintenance. Equation (4) imposes \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol Type** & **Notation Expression** & **Meaning** \\ \hline \multirow{5}{*}{Variants} & \multirow{2}{*}{\(x_{is}\)} & Whether the alternative landing site i is to be constructed as an s-type co-distribution center, with a value of 1 for yes and 0 for no \\ \cline{2-3}  & & Whether customer demand point j is delivered by UAV co-distribution point i, with a value of 1 for yes and 0 for no \\ \cline{2-3}  & & Whether customer demand point j is delivered by vehicle co-distribution point i, with a value of 1 for yes and 0 for no \\ \cline{2-3}  & & Whether the distribution path between UAV co-distribution point i and customer demand point j crosses the no-fly zone, with a value of 1 for no and 0 for yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._a constraint on the number of landing points, i.e., the total number of co-distribution centers cannot exceed the number of alternative co-distribution centers. Equation (5) states that any alternative co-distribution center must be either an air or a ground center. Equation (6) is a constraint on the matching of supply and demand points (i.e., for any demand point, there is only one co-distribution center responsible for the distribution) and prevents the existence of demand points that cannot be distributed or distribution locations with more than one point of departure and landing.