## 3 The UDVRP Model
### Model Formulation  
The meanings of parameters and variables are as Table 3.  
The objective of this approach is to minimize the aggregate costs, which encompass transportation expenses, energy consumption costs, and penalty costs. The objective function is defined by Equation (3).  
\[\begin{split} minf&=\sum_{k\in k}\sum_{i\in N}\sum_{ j\in N}(wd_{ij}+c)x_{ij}^{k}+h\sum_{k\in K}\sum_{i\in N}\sum_{j\in N}ed_{ij}(p+\\ q_{i}^{k})x_{ij}^{k}&+\sum_{i=1}^{n}P_{i}\end{split} \tag{3}\]  
The constraints calculated by Equation (3) are as follows:  
1. Route optimization constraints: \[\sum_{i=1}^{n}\sum_{j=1}^{n}d_{ij}x_{ij}^{k}\leq D,i,j=1,2,...,n\] (4) \[\sum_{k\in K}y_{i}^{k}=1,i=1,2,...,n\] (5) \[\sum_{i=1}^{n}\sum_{k=1}^{K}x_{ij}^{k}=1,j=1,2,...,n\] (6) \[\sum_{j=1}^{n}\sum_{k=1}^{K}x_{ij}^{k}=1,i=1,2,...,n\] (7) \[\sum_{j\in N}x_{ij}^{k}-\sum_{j\in N}x_{ij}^{k}=0,k=1,2,\cdots,\] (8) \[\sum_{i=1}^{n}x_{is}^{k}-\sum_{j=1}^{n}x_{sj}^{k}=0\] (9) where \(y_{i}^{k}=1\) means served, \(y_{i}^{k}=0\) means not served. If vehicle \(k\) travels directly from distribution point \(i\) to \(j\), \(x_{ij}^{k}=1\), otherwise \(x_{ij}^{k}=0\). The Equation (4) signifies that the driving distance of each vehicle must not surpass the maximum distance driven. The constraint Equations (5)-(7) mean that each distribution point is only served by one vehicle; Equation (8) means that the number of unmanned delivery vehicles arriving and leaving any distribution node is the same; Equation (9) ensures that the driving trajectory of each vehicle must be It is a closed loop.
2. Time optimization constraints: \[t_{i}+T_{i}+t_{ij}\leq M\big{(}1-x_{ij}\big{)}+t_{j},\forall i,j\in N,\forall k\in K\] (10)If vehicle \(k\) selects the route from node \(i\) to \(j\), the total time at node \(i\) (arrival time plus service time) cannot exceed \(M\); otherwise, if the vehicle \(k\) does not choose this path, it considers only the arrival time at node \(j\).  
\[t_{j}=\sum_{i=1}^{n}x_{ij}\big{(}t_{i}+T_{i}+t_{ij}\big{)},\forall i,j\in N, \forall k\in K \tag{11}\]  
Equation (11) sums up the contributions of arrival time, service time, and travel time from all nodes \(i\) to \(j\) where \(x_{ij}=1\) (indicating the route is selected). The chosen paths have a direct impact on the arrival time to node \(j\).
3. Load optimization constraints:  
\[\sum_{i=1}^{n}r_{i}y_{i}^{k}\leq Q,i=1,2,...,n \tag{12}\]  
\[q_{s}^{k}=\sum_{i\in N}\sum_{j\in N,j\
eq i}n_{i}x_{ij}^{k},\forall k\in K \tag{13}\]  
\[q_{j}^{k}\geq q_{i}^{k}+m_{j}-n_{j}-M\big{(}1-x_{ij}^{k}\big{)},\forall i,j\in N,i\
eq j,\forall k\in K \tag{14}\]  
\[0\leq q_{i}^{k}\leq Q,\forall i\in N,k\in K \tag{15}\]  
Equation (12) ensures that the load capacity of each vehicle cannot surpass the maximum load; Equation (13) demonstrates that the initial load capacity of the vehicle when leaving the distribution center is equal to the sum of the cargo volume at the service distribution point; Equation (14) indicates the sequence of vehicles The cargo capacity of serving two distribution points changes; Equation (15) indicates that the vehicle's cargo capacity is always between 0 and Q.
4. penalty cost optimization:  
\[P(i)=\alpha\Bigg{(}1-\beta\tanh\Bigg{(}\frac{1}{n}\sum_{i=1}^{n}S(i)\Bigg{)} \Bigg{)} \tag{16}\]  
Equation (16) represents the penalty cost generated by customer satisfaction.