## 3 LRPTW Model
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Practical Significance** \\ \hline \(n\) & Number of vehicles \\ \(m\) & Number of receiving points \\ \(a\) & Number of new warehouses \\ \(k\) & Vehicle number \(k\)\(\in\)[1, 2, \(\ldots\), n] \\ \(i,j,q\) & Receiving point number \\ \(p\) & Distribution center number \\ \(f_{1}\) & The cost of each new warehouse \\ \(f_{2}\) & Cost per vehicle per kilometer \\ \(f_{3}\) & Fixed cost per vehicle \\ \(f_{4}\) & Cooling mass produced per minute \\ \(f_{5}\) & Mass produced by fuel consumption per kilometer driven \\ \(f_{6}\) & Product loss coefficient \\ \(D_{ij}\) & The distance between the delivery point and \\ \(T_{ij}\) & The time between the delivery point and \\ \(X_{k}\) & 0.1 decision variable, indicating that the first vehicle completes the \\ \(X_{jk}\) & delivery operation at the receiving point \\ \(X_{ijk}\) & 0.1 decision variable, indicating that the first vehicle is the receiving point \\ \(X_{ipk}\) & 0.1 decision variable, indicating that the first vehicle completes the \\ \(Q_{k}\) & delivery operation for the distribution center and receiving point \\ \(Q_{l}\) & Actual load of the k-th vehicle \\ \(Q_{l}\) & Required weight at delivery point i \\ \(Q\) & Delivery vehicle maximum load \\ \(L_{k}\) & Actual distance traveled by the k-th vehicle \\ \(L\) & Maximum driving distance of delivery vehicles \\ \(e_{i}\) & Earliest delivery time at delivery point i \\ \(e_{p}\) & Delivery center p earliest delivery time \\ \(l_{i}\) & Latest delivery time at delivery point i \\ \(l_{p}\) & Latest delivery time for distribution center p \\ \(A_{ik}\) & Actual delivery time at delivery point i \\ \(A_{pk}\) & Actual delivery time of distribution center p \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical symbols. \[Cost_{2}=\sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{k=1}^{n}D_{ij}\times f_{2}\times X_{ijk} \tag{5}\]  
\[Cost_{3}=n\times f_{3} \tag{6}\]  
\[CO_{21}=\sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{k=1}^{n}T_{ij}\times f_{4}\times X_{ijk} \tag{7}\]  
\[CO_{22}=\sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{k=1}^{n}D_{ij}\times f_{5}\times X_{ijk} \tag{8}\]  
\[Loss=\sum_{k=1}^{n}Q_{k}\times f_{6} \tag{9}\]  
\(Cost_{1}\) is the cost of a new warehouse, \(Cost_{2}\) is the fuel cost, \(Cost_{3}\) is the vehicle fixed cost, \(CO_{21}\) is the refrigeration carbon emissions, \(CO_{22}\) is the fuel consumption carbon emissions, and \(Loss\) is the product loss. Subject to:  
\[X_{ijk}=\left\{ \begin{array}{l}1,\text{the delivery operation of customer i}\\ \text{and j are completed by the $k_{th}$ vehicle}\\ 0,\text{others}\end{array} \right. \tag{10}\]  
\[X_{ik}=\left\{ \begin{array}{l}1,\text{the delivery operation of customer}\\ \text{i is carried out by the $k_{th}$ vehicle}\\ 0,\text{others}\end{array} \right. \tag{11}\]  
\[\sum_{i=1}^{m}\sum_{k=1}^{n}X_{ipk}=\sum_{i=1}^{m}\sum_{k=1}^{n}X_{pik} \tag{12}\]  
\[\sum_{i=1}^{m}X_{ijk}=\sum_{i=1}^{m}X_{jik}(k\in n,\forall j\in m) \tag{13}\]  
\[Q_{k}\leq Q \tag{14}\]  
\[\sum_{k=1}^{n}X_{ik}=1(\forall i\in m) \tag{15}\]  
\[Q\geq Q_{i}(\forall i\in m) \tag{16}\]  
\[\sum_{i=1}^{m}X_{iqk}=\sum_{j=1}^{m}X_{qjk}(k\in n,\forall q\in m) \tag{17}\]  
\[e_{i}\leq A_{ik}\leq l_{i} \tag{18}\]  
\[e_{p}\leq A_{pk}\leq l_{p} \tag{19}\]  
\[L_{k}=\sum_{i=1}^{m}X_{ijk}\times D_{ij}(k\in n,\forall j\in m) \tag{20}\]  
\[L_{k}\leq L \tag{21}\]  
In this model, Equation (1) represents the cost objective function of the multi-objective Location-Routing Problem, aiming to minimize the distribution cost while satisfying the constraints. Equation (2) represents the carbon dioxide emission objective function, aiming to minimize carbon dioxide emissions during the distribution process while satisfying the constraints. Equation (3) represents the product loss objective function, aiming to minimize the loss of fresh agricultural products during the distribution process while satisfying the constraints.