## 2 Problem Description and Modeling
### Mathematical Model  
The goal of this paper is to optimize the delivery distance and vehicle average loading rate simultaneously. Recording a single vehicle loading only increases the algorithm time, so this paper only considers the average loading rate. The symbol definitions are shown in Table 1.  
Figure 1: The 3L-CVRP problem. The mathematical model is as follows:  
(a) Equation (1) represents the minimization of the total travel distance, as follows:  
\[\text{minf}=\sum_{i\in M}\sum_{j\in M}\sum_{k\in K}x_{ij}^{k}\cdot y_{ki}D_{ij},i\
eq j \tag{1}\]  
(b) Equation (2) represents the maximization of the average loading rate, as follows:  
\[\text{max}V=\sum_{k\in K}\frac{\sum_{i\in N}\sum_{b\in B_{i}}l_{ib}w_{ib}l_{ib} y_{ki}}{V_{k}} \tag{2}\]  
(c) Equations (3)-(12) are the constraint conditions; Equation (3) ensures that each customer point is served only once per trip, as follows:  
\[\sum_{i\in N\cup M}\sum_{k\in K}x_{ij}^{k}=1,j\in N,i\
eq j \tag{3}\]  
(d) Equation (4) specifies that the number of vehicle trips used by the distribution center cannot exceed the total number of elements in set \(K\), as follows:  
\[\sum_{k\in K}\sum_{i\in N}x_{0i}^{k}=\sum_{k\in K}\sum_{j\in N}x_{0}^{k}\leq|K| \tag{4}\]  
(e) Equation (5) represents the flow balance constraint, meaning the inflow and outflow of the same node must be equal, as follows:  
\[\sum_{i\in M}x_{ia}^{k}-\sum_{j\in M}x_{aj}^{k}=0,\ a\in M,k\in K,i\
eq j\
eq o \tag{5}\]  
(f) Equations (6) and (7) represent the uniqueness constraint for flow direction, as follows:  
\[\sum_{i\in N,i\
eq j}x_{ij}^{k}=y_{kj},\ j\in N,k\in K \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Meaning** & **Parameters** & **Meaning** \\ \hline \(i,j,o\) & Customer ID & \(b,d\) & Parcel ID \\ \hline \(B_{i}\) & Number of parcels required by the \(i\)-th customer point & \(K\) & Set of vehicle trips \\ \hline \(k\) & Vehicle trip ID & \(D_{ij}\) & Distance from customer point \(i\) to customer point \(j\) \\ \hline \(x_{ij}^{k}\) & Binary variable: equal to 1 if the \(k\)-th vehicle trip is from point \(i\) to point \(j\); otherwise, 0 & \(y_{ki}\) & Binary variable: equal to 1 if customer point \(i\) is served by the \(k\)-th vehicle trip; otherwise, 0 \\ \hline \(f\) & Total travel distance of the delivery vehicles & \(M\) & Node set (including the distribution center and \(N\) customer points) \\ \hline \(ort\) & Equal to 1 or 2, representing two placement orientations & \(N\) & Customer point set \\ \hline \(L,W,H\) & Length, width, and height of the delivery vehicle compartment & \(V\) & Average loading rate of delivery vehicles \\ \hline \(l_{ib},l_{ad}\) & Length of the \(b\)-th parcel at customer point \(i\) and the \(d\)-th parcel at customer point \(o\) & \(h_{ib},h_{ad}\) & Height of the \(b\)-th parcel at customer point \(i\) and the \(d\)-th parcel at customer point \(o\) \\ \hline \(w_{ib},w_{ad}\) & Width of the \(b\)-th parcel at customer point \(i\) and the \(d\)-th parcel at customer point \(o\) & \(X_{ibk},X^{\prime}_{iik}\), & \(X\)-axis coordinates of the bottom-left-near corner and the opposite corner of the \(b\)-th parcel at customer point \(i\) and the \(d\)-th parcel at customer point \(o\) \\ \hline \(Y_{ibk},Y^{\prime}_{iik}\), & \(Y\)-axis coordinates of the bottom-left-near corner and the opposite corner of the \(b\)-th parcel at customer point \(i\) and the \(d\)-th parcel at customer point \(o\) on the \(k\)-th trip \\ \hline \(Y_{ibk},Y^{\prime}_{iik}\), & \(Y\)-axis coordinates of the bottom-left-near corner and the opposite corner of the \(b\  
\[\sum_{j\in N,j\
eq i}x_{ij}^{k}=y_{ki},i\in N,k\in K \tag{7}\]
2. Equation (8) ensures that the total volume of parcels loaded for each trip does not exceed the volume capacity of the vehicle compartment, as follows: \[\sum_{b\in B_{i}}y_{ki}\cdot l_{ib}\cdot w_{ib}\cdot h_{ib}\leq L\cdot W\cdot H \cdot\alpha,\;i\in N,b=\{1,2,3,\ldots,B_{i}\},k\in K\] (8) 2.