## 2 Mathematical Model of the BE-LRP
### Model Building  
The BE-LRP is constructed as a dual objective optimization model, with two optimization objectives of minimizing the number of delivery vehicles and delivery distance. The expressions of the two objective functions are as follows:  
\[F_{1}=\min\sum_{k\in K}K_{k}(\min\{\sum_{i\in I}\sum_{j\in I}\sum_{n\in R_{k}}X_{ ijkn},1\}) \tag{1}\]  
\[F_{2}=\min\sum_{i\in I}\sum_{j\in I}\sum_{n\in R_{k}}X_{ijkn}\times D_{ij} \times\xi_{ij} \tag{2}\]  
The constraint conditions are as follows:  
\[\sum_{i\in I\cup I}\sum_{k\in K}\sum_{n\in R_{k}}X_{ijkn}=1,j\in J \tag{3}\]  
\[\sum_{i\in I\cup J}X_{ijkn}-\sum_{l\in I\cup J}X_{jlkn}=0,j\in J,k\in K,n\in R_ {k} \tag{4}\]  
Constraint (3) means that each compartment is delivered only once in the whole delivery process. Constraint (4) is the conservation of material circulation in the cabin material distribution. \[\sum_{k\in K}K_{k}(\min\{\sum_{i\in I}\sum_{j\in I}\sum_{n\in R_{k}}X_{ijkn},1\} )=|U_{k}| \tag{5}\]  
\[\sum_{i\in I}\sum_{j\in I}\sum_{k\in K}X_{ijk}=|R_{k}|,k\in K \tag{6}\]  
\[\sum_{i\in I}\sum_{j\in I}X_{ijkn}\geq\sum_{i\in I}\sum_{j\in J}X_{ijk(n+1)}k \in K,n\in R_{k},n\
eq|R_{K}| \tag{7}\]  
Constraints (5) and (6) are used to reflect the number of distribution vehicles and the number of distribution routes. Constraint (7) ensures the order of vehicle delivery service routes. \[\sum_{j\in J}Q_{j}\times\sum_{i\in I\cup J}X_{ijkn}\leq M_{k},k\in K,n\in R_{k} \tag{8}\] \begin{table}
\begin{tabular}{c c} \hline Variable & Definition \\ \hline \(D_{ij}\) & The distance between distribution center \(i\) or cabin \(c\) and \(d\), \(i,j\in I\cup J\) \\ \(\xi_{ij}\) & Obstacle blocking influence coefficient, \(i,j\in I\cup J\) \\ \(\left[\alpha_{j},\beta_{j}\right]\) & Time window for cabin \(j,j\in J\) \\ \(DT_{ikn}\) & The time when the vehicle \(k\) leaves the distribution center \(i\) on the \(n\)th route, \(i\in I,k\in K,n\in R_{k}\) \\ \(AT_{jkkn}\) & The time when the vehicle \(k\) arrives the distribution center \(i\) or cabin \(j\) on the \(n\)th route, \(j\in I\cup I,k\in K,n\in R_{k}\) \\ \hline Decision Variable & Definition \\ \hline \(X_{ijkn}\) & If the vehicle \(k\) is on the nth route from the distribution center or cabin \(i\) to \(j\), \(X_{ijkn}=1\), otherwise \(X_{ijkn}=0\), \(i,j\in I\cup J\), \(k\in K,n\in R_{k}\) \\ \(X_{ijk}\) & If the vehicle \(k\) goes from the distribution center \(i\) to the cabin \(j\), \(X_{ijk}=1\), otherwise \(X_{ijk}=0\), \(i\in I,j\in J\), \(k\in K\) \\ \(K_{k}\) & If the vehicle has a service cabin, \(K_{k}=1\), otherwise \(K_{k}=0\), \(k\in K\) \\ \(Y_{i}\) & If distribution center \(i\) is selected as cabin service, \(Y_{i}=1\), otherwise \(Y_{i}=0\), \(i\in I\) \\ \(Y_{ij}\) & If distribution center \(i\) is selected as cabin \(j\) service, \(Y_{ij}=1\), otherwise \(Y_{ij}=0\), \(i\in I\), \(j\in J\) \\ \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)\[\sum_{j\in J}Q_{j}\times Y_{lj}\leq M_{i},i\in I \tag{9}\]  
Constraints (8) and (9) are capacity constraints. Constraint (8) ensures that the total demand for cabins on each route is less than the capacity of the delivery vehicles. Constraint (9) ensures that the capacity of the distribution center is greater than the total material demand on each distribution line. \[\alpha_{i}\underset{j\in J}{\sum}x_{ijkn}\leq DT_{ikn}\leq\beta_{i}\underset{j \in J}{\sum}x_{ijkn},i\in I,k\in K,n\in R_{k} \tag{10}\]  
\[\alpha_{i}\underset{j\in J}{\sum}x_{jikn}\leq AT_{ikn}\leq\beta_{i}\underset{j \in J}{\sum}x_{jikn},i\in I,k\in K,n\in R_{k} \tag{11}\]  
\[\sum_{i\in I}AT_{ikn}\leq\sum_{i\in I}DT_{ik(n+1)},k\in K,n\in R_{k},n\
eq|R_{v}| \tag{12}\]  
Constraints (10)-(12) are time window constraints to ensure that delivery occurs within the time window of the cabin construction.