# A Study on the Vehicle Routing Planning Method for Fresh Food Distribution
## 2 Problem Description and Model Construction
### Problem Description
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbols** & \multicolumn{1}{c}{**Description of Symbol**} \\ \hline \(E\) & The set of large warehouses in a city center, \(\{e|e\in E\}\) \\ \(H\) & The set of candidate points for the front warehouse, \(\{h|h\in H\}\) \\ \(I\) & The set of customer points for distribution, \(\{i|i\in I\}\) \\ \(W=I\cup H\) & The set of the customer point \(i\) and the front warehouse \(h\) \\ \(K\) & The set of delivery vehicles, \(\{k|k\in K\}\) \\ \((i,j)\) & The point from the distribution center to the front warehouse \(i,j\in H,i\
eq j\) \\ \(p_{i}\) & The demand at the customer point (kg) \\ \(\mathcal{O}_{h}\) & The capacity of the front warehouse (kg) \\ \(\mathcal{C}_{h}\) & The unit fixed cost of the front warehouse \(h\) (CNY) \\ \(T_{h}\) & The operating cost of the front warehouse \(h\) (CNY) \\ \(C_{ch}\) & The unit distance transportation cost of trunk distribution vehicles from the central warehouse to the front warehouse (CNY/km) \\ \(d_{ch}\) & The distance from the central warehouse to the front warehouse of the main delivery vehicle \\ \(d_{hi}\) & The distance between the front warehouse and the customer point \\ \(d_{himax}\) & The farthest distance between customer points that regional vehicles can deliver to and the front warehouse \\ \(t_{ch}\) & The time from the central warehouse to the front warehouse for the main delivery vehicle \\ \(t_{k}^{k}\) & The service time of the vehicle \(k\) at the customer point \(i\) \\ \(t_{i}/t_{j}\) & The time when the vehicle arrives at the customer point \(i\) or \(j\) \\ \(t_{di}\) & The waiting time for the vehicle at the customer point \(i\) \\ \(C_{k1}\) & The fixed cost per unit distance of mainline vehicles (CNY/km) \\ \(C_{p}\) & The transportation cost per unit distance of branch line vehicles (CNY/km) \\ \(C_{k2}\) & The fixed cost per unit distance of branch line vehicles (CNY/km) \\ \(C_{rk1}\) & The unit time cooling cost of mainline vehicles (CNY/h) \\ \(\theta_{1}\) & The unit loss cost from the large warehouse in the city center to the front warehouse \\ \(\theta_{2}\) & The unit loss cost from the front warehouse to the customer point \(i\) \\ \(F_{k}\) & The vehicle capacity \\ \(\omega_{t}\) & The cargo damage amount at time \(t\) (kg) \\ \(\delta\) & The corruption rate of the goods \\ \(g_{ch}\) & The transportation volume from the large warehouse in the city center to the front warehouse \(h\) (kg) \\ \(d_{ij}\) & The distance from the customer point \(i\) to the customer point \(j\) (km) \\ \(R_{hi}\) & The distance between the front warehouse and the customer point \\ \(\Delta t_{i}\) & The service time of the vehicle at the customer point \(i\) \\ \(ET_{i}\) & The earliest expected arrival time at the customer point \(i\) \\ \(LT_{i}\) & The latest expected arrival time at the customer point \(i\) \\ \(\beta_{1}\) & The penalty coefficient per unit time that the vehicle performs the service earlier than the earliest expected service start time of the customer point \(i\) \\ \(\beta_{2}\) & The penalty coefficient per unit time that the vehicle performs the service later than the latest expected service start time of the customer point \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptions of the symbols. Constraints s.t. :  
\[1\leq\sum\limits_{h\in H}Y_{h} \tag{9}\]  
\[\sum\limits_{h\in H}R_{hi}Y_{h}\geq 1\forall_{i}\in I \tag{10}\]  
\[Y_{h}=\{0,1\}\forall h\in H \tag{11}\]  
Equation (9) indicates that the demand point is serviced by the front warehouse at least once. Equation (10) represents the construction of at least one front warehouse to cover demand points. Equation (11) represents the value constraint of the decision variable \(Y_{h}\). Lower layer planning. Based on the location scheme of the initial front warehouse and the customer point division of the upper layer model, decisions are made on the terminal distribution path for each customer point in the region.