## 3 Methods and Data Sources
### MSW Management Statement and Model Formulation  
This study models the municipal waste collection and transfer system of Beijing as a multi-objective site allocation model with stochastic demand, aiming to optimize the waste collection and transfer processes. We focus on three primary objectives: enhancing residents' quality of life, achieving economic efficiency, and promoting environmental protection. The model identifies waste generation nodes within various communities and potential future waste treatment centers, while discussing how to optimize among these three objectives to ensure both the realism and feasibility of the model. To this end, several necessary assumptions are made in this paper:  
* The demand of each garbage generating node is an independent random variable that undergoes certain fluctuations and the distribution obeys a normal distribution \(N(\mu,\sigma^{2})\), where \(\mu\) is the average value of the garbage generation obtained from the prediction and \(\sigma\) is set artificially. * The demand at a single waste generation node must not exceed the maximum capacity of the vehicle to ensure that an empty vehicle can deliver all the waste from a single node. * A garbage generation node can only be served by one vehicle but can be performed by splitting into two services. If the vehicle is full, the vehicle returns to the waste disposal center to be emptied and then returns to the waste generation node for service. * All vehicles are identical and have a limited maximum distance they can travel. * The time required to dispose of the waste at the waste disposal center is the same and constant. * The model also needs to consider the cost of returning the vehicle to the waste disposal center and the cost of the truck driver's wages for the time spent at work. For the purpose of mathematical description, the symbols in the equations are described in Symbol Definitions. The decision variables for this problem contain the following: \(X_{ijk}\), if the vehicle \(k\) travels from node \(i\) to node \(j\), then \(X_{ijk}=1\); otherwise \(X_{ijk}=0\); \(y_{ik}\), \(1\) if the node \(i\) is assigned to the vehicle \(k\) for service, \(0\) otherwise; \(Z_{i}\), assigned \(1\) if the waste disposal center \(i\) is selected for selection, \(0\) otherwise. \[D(car_{i})=dis_{car_{i},center_{1}}+\sum_{i=1}^{k}dis_{center_{i},center_{i+ 1}}+uIDis(car) \tag{5}\]  
\[T(cari)=time_{cari,center_{i}}+\sum_{i=1}^{k}time_{center_{i},center_{i+1}}+ \sum_{i=1}^{k}Serv_{i}+uLTime(car) \tag{6}\]  
\[uLDis(car)=\sum_{i=1}^{uL}{(dis_{center}+dis_{center})} \tag{7}\]  
\[uLTime(car)=N(uL)\times uLTm_{i}+\sum_{i=1}^{uL}{(time_{center}+time_{center})} \tag{8}\]  
\[Sa(car_{i})=\left\{ \begin{aligned} T(car_{i})Saly& T(car_{i})\leq 8\\ 8 Saly+T(car_{i})Saly& T(car_{i})>8\end{aligned} \right. \tag{9}\]  
To facilitate the expression of the objective function, some variables appearing in the objective function are defined in the above equation. Equation (5) represents the sum of distances that the vehicle \(i\) needs to travel while performing the service, including the distance from the waste disposal center to the first waste generation node and the distance between different waste generation nodes for waste collection. In addition, if the vehicle finds that the remaining capacity does not meet the requirement after reaching the waste generation node, it needs to return to the waste disposal center for unloading and then return for further loading. Equation (6) is the time consumed by each vehicle, including the vehicle operation time from the waste disposal center to the first waste generation node, the vehicle operation time for collecting waste between different waste generation nodes and the service time of each waste generation node, and the corresponding time needs to be considered after considering the problem of returning to the waste disposal center midway.