## 3 LRPTW Model  
The problem model studied in this article originates from the order distribution of fresh agricultural products on an e-commerce platform. The distribution process can be abstracted into central warehouses, auxiliary warehouses, customer points, and distribution vehicles. The distribution warehouses can choose a central warehouse and an uncertain number of auxiliary warehouses. Within the distribution warehouse, multiple vehicles can be used simultaneously. Due to the construction cost of the auxiliary warehouses, time window limitations at customer points, and load and energy limitations of the distribution vehicles, the path planning problem can be abstracted as LRPTW. In LRPTW, there are distribution warehouses and receiving points. Vehicles departing from the distribution warehouses eventually return to these distribution warehouses. The vehicle parameters of the distribution warehouses are the same, but the time window at customer points varies. Reasonable assumptions can simplify the model construction process, reduce the consideration of secondary factors, and decrease the difficulty of model construction. Therefore, to control some unexplained variables and directly reflect the basic problems of the model, this article simplifies the actual situation. It proposes conditions for an appropriate environment required for model building. **Customer points:** To meet the needs of each customer point, only one delivery vehicle is allowed to serve it and visit it only once. Additionally, the demand at any receiving point must be less than the maximum load of the vehicle. **Delivery vehicles:** All delivery vehicles are of the same type with identical load and speed. **Delivery routes:** Based on demand, the distribution center has enough vehicles to deliver each route on time. Transportation is one-way from the distribution center to the receiving point. **Distribution warehouses:** The start and final destination of each delivery vehicle are the same distribution warehouse, with no midway allocation. Figure 1: A realistic model of the LRPTW problem. **Load Limit:** The total demand from all customers on the same route must not exceed the rated load of the vehicle. **Energy limitations:** Each delivery vehicle needs to return to the distribution center for energy replenishment, and the one-way distance cannot exceed the maximum driving distance. **Time window restriction:** The time window should at least meet the condition that the vehicle path contains only one demand node. Delivery vehicle arrival times and loading operations should occur within the acceptable time window at the delivery point. **Other settings:** The delivery process does not consider road congestion, adverse weather conditions, or vehicle breakdowns. The definitions of mathematical symbols used in the problem model are shown in Table 1. In the multi-objective location-route problem, the three objectives are the minimum cost objective, the minimum carbon emission objective, and the minimum product loss objective. The mixed-integer programming model of this problem is as follows:  
Objective:  
\[\text{MinCost}=\text{Cost}_{1}+\text{Cost}_{2}+\text{Cost}_{3} \tag{1}\]  
\[\text{MinCO}_{2}=\text{CO}_{21}+\text{CO}_{22} \tag{2}\]  
\[\text{MinLoss}=\text{Loss} \tag{3}\]  
W.R.T:  
\[Cost_{1}=a\times f_{1} \tag{4}\]