## 5 Practical Problem Solving and Analysis
### Establishment of VRPTW Model  
The vehicle routing problem with the time window constraints (VRPTW) optimization model is as follows:  
\[\min Z=\left(c_{0}\sum_{k\in K}y_{k}+c_{1}\sum_{k\in K}\sum_{i\in V}\sum_{j\in V }d_{ij}x_{ijk}\right)+epu\sum_{i\in N}\max\left(e_{i}-t_{i}^{1},0\right)+lpu\sum_ {i\in N}\max\left(t_{i}^{1}-l_{i},0\right) \tag{20}\]  
The objective function indicates that the total delivery cost was the lowest. The total cost consists of the vehicle fixed cost, travel distance cost, and time window penalty cost. The model constraints are as follows. \[\sum_{j\in N,i\
eq j}x_{ijk}=1,\forall i\in N,k\in K \tag{21}\]  
\[\sum_{j\in V,i\
eq j}x_{0 jk}\leq 1,k\in K \tag{22}\]  
\[\sum_{k\in K}y_{k}\leq|K| \tag{23}\]  
\[\sum_{i\in V,i\
eq j}x_{ijk}-\sum_{i\in V,j\
eq i}x_{ijk}=0,\forall j\in V,k\in K \tag{24}\]  
\[0\leq h_{jk}^{1}\leq h_{ik}^{1}-p_{i}x_{ijk}+C\left(1-x_{ijk}\right),\forall i \in V,\forall j\in V,i\
eq j,k\in K \tag{25}\]  
\[0\leq h_{0k}\leq C,k\in K \tag{26}\]  
\[t\omega_{i}=\max\left[0,\left(e_{i}-t_{i}^{1}\right)\right],i\in N \tag{27}\]  
\[t_{i}^{2}=t_{i}^{1}+tf_{i}+t\omega_{i}+tc_{i},i\in N\cup F \tag{28}\]  
\[t_{ij}=\frac{d_{ij}}{S},\forall i,j\in V \tag{29}\]  
\[\sum_{j\in N,i\
eq j}x_{ijk}=1,\forall i\in N,k\in K \tag{21}\]0 \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Implication** \\ \hline
0 & Distribution center \\ \(N\) & Collection of customer points, \(N=\{1,\,2,\,3,\ldots,n\}\) \\ \(K\) & Using the set of vehicles, \(K=\{1,\,2,\,3,\ldots,k\}\) \\ \(V\) & The set of all nodes, \(V=0\cup N\) \\ \(V_{k}\) & Node set of vehicles \(k\) driving path \\ \(d_{ij}\) & Distance from node \(i\) to node \(j\) \\ \(p_{i}\) & Demand of customer point \(i\) \\ C & Maximum loading capacity of vehicle \\ \(c_{ijk}\) & Load of vehicle \(k\) from node \(i\) to node \(j\) \\ \(e_{i}\) & The earliest service time of customer node \(i,\,i\in N\) \\ \(l_{i}\) & The latest service time of customer node \(i,\,i\in N\) \\ \(t_{i}^{1}\) & The time when the vehicle reaches customer node \(i,\,i\in N\) \\ \(t_{i}^{2}\) & The time when the vehicle leaves customer node \(i,\,i\in N\) \\ \(t_{ui}^{2}\) & The waiting time of vehicle at customer node \(i,\,i\in N\) \\ \(tf_{i}\) & Vehicle service time at node \(i,\,i\in N\) \\ \(t_{ij}\) & Travel time of vehicle from node \(i\) to node \(j,\,i,j\in V\) \\ \(v\) & Vehicle speed \\ \(epu\) & Unit penalty cost for vehicles arriving earlier than left time window \\ \(lpu\) & Unit penalty cost for vehicle arriving later than right time window \\ \(\alpha_{0}\) & Fixed cost per unit vehicle \\ \(c_{1}\) & Unit distance cost \\ \(S\) & Vehicle driving distance \\ \(h_{k}^{1}\) & The load of vehicle \(k\) arriving customer node \(i,\,i\in V_{k}\cup N\) \\ \(h_{k}^{2}\) & The load of vehicle \(k\) leaving customer node \(i,\,i\in V_{k}\cup N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameter list. Equation (21) indicates that the demand of each customer point is satisfied only once by one vehicle. Equation (22) indicates that each vehicle only serves customer points on one path. Equation (23) indicates that the number of vehicles put into use does not exceed the total number of vehicles in the distribution center. Equation (24) indicates that the traffic flow remains balanced at each customer point. Equation (25) represents the residual cargo weight of the vehicle at customer point \(j\). Equation (26) indicates that the load of the vehicle does not exceed the maximum load of the vehicle in the process of driving. Equation (27) indicates the waiting time \(t\omega_{i}(=e_{i}-t_{i}^{1})\) for the vehicle when it arrives at the customer point early. If the vehicle does not arrive in advance, \(t\omega_{i}=0\). Equation (28) represents the time when the vehicle departs from customer point \(i\) to the next customer point. Equation (29) represents the driving time of the vehicle from customer point \(i\) to customer point \(j\). Equation (30) represents the time when the vehicle arrives at customer point \(j\).