## 3 Model Establishment  
For merchant self-operated takeaway delivery, the merchant receives the order information from the online platform. Merchants are responsible for arranging a special delivery team to complete the work. In the merchant self-operated takeaway delivery model, each rider will take goods of multiple orders within a certain period from a similar area and deliver the goods to each consumer following the route optimization result. After all delivery tasks are completed, the rider will return to the merchant's address and wait for the next round of delivery tasks. The mathematical model for the merchant self-operated takeaway delivery is established, aiming to reasonably assign consumer orders to the rider and find the shortest delivery route.  
Assuming that a rider wants to deliver goods to n customers within a specified time, the model is established as follows:  
1. The decision variable \(x_{ij}\) is expressed in Equation (1). When \(x_{ij}\) is assigned as 1, it means that the rider has reached consumer node \(j\) from consumer node \(i\). When \(x_{ij}\) is assigned as 0, it means the delivery person has not traveled from consumer node \(i\) to consumer node \(j\). \[x_{ij}=\left\{\begin{array}{ll}1,&\text{the rider travels from node $i$ to node $j$}\\ 0,&\text{otherwise}\end{array}\right.\] (1)2. The objective function is shown in Equation (2). The shorter the total distance of the rider to complete the delivery task, the higher the efficiency that can be obtained. \[\text{min}L=\sum_{i\
eq j}d_{ij}x_{ij}\] (2) where \(d_{ij}\) is the distance from node \(i\) to \(j\), and \(L\) is the total distance to complete the delivery task. (3) The constraints are shown in Formulas (3)-(5). \[\sum_{j=1}^{n}x_{ij}=1,\quad i=1,2,\ldots,n\] (3) \[\sum_{i=1}^{n}x_{ij}=1,\quad j=1,2,\ldots,n\] (4) \[x_{ij}\in\{0,1\},\quad i,j=1,2,\ldots,n,i\
eq j\] (5) Equation (3) indicates that the rider walks out from node \(i\) once, and \(i=1\) means the merchant node. Equation (4) implies that the rider walks into node \(j\) once. Equation (5) represents the value choice of the variable \(x_{ij}\). With the constraint of Equations (3) and (4), each good can be delivered to its corresponding customer. (4) Considering the consumer's location would be presented based on its latitude and longitude, the real distance between different locations was calculated according to Equation (6). The Arccos equation is used for short-distance scenarios (<5 km) due to its lower computational cost (requiring fewer trigonometric operations) and negligible precision loss (<0.01% error at 5 km) compared to the Haversine equation (requiring square roots and additional trigonometric functions), sacrificing only minor spherical curvature accuracy for significant speed gains in routing systems. \[\text{D}=\arccos[\sin(A_{Lat})\times\sin(B_{Lat})+\cos(A_{Lat})\times\cos(B_ {Lat})\times\cos(\big{|}A_{Long}-B_{Long}\big{|})]\times\text{R}\] (6) where \(A_{Lat}\) and \(A_{long}\) is the latitude and longitude of location A, \(B_{lat}\) and \(B_{Long}\) is the latitude and longitude of location B. R is the Earth's radius (6371.004 km), a standard value in geodesy adopted from the International Union of Geodesy and Geophysics (IUGG) recommendations.