## 3 Problem Description and Model Formulation
### Model Formulation  
According to the analysis of the investigated problem, it can be formulated into a two-stage stochastic programming model. This study focuses on two underlying assumptions: (1) this paper only considers recyclable construction waste, while non-recyclable waste will still be disposed of via landfilling; and (2) there is no upper limit on the capacity of a landfill site because all construction waste is delivered to the landfill site prior to the establishment of recycling centers. The notation used in this paper is listed as follows:  
**Indices:**  
\(r\): index for the location of the recycling center  
\(s\): index for the construction site  
\(\omega\): index for the scenario  
**Parameters:**  
\(R\): the number of candidate locations for establishing recycling centers  
\(S\): the number of construction sites that generate construction waste (\(s=1,\ldots,S\))  
\(S^{\prime}\): the number of construction sites that require recycled building materials (\(s=S+1,\ldots,S+S^{\prime}\))  
\(\Omega\): the number of scenarios  
\(a_{r}\): the maximum area of the recycling center at location \(r\)  
\(e^{1}\): the waste handling capacity per unit area of recycling centers  
\(e^{2}\): the weight of building materials recycled per ton of construction waste  
\(C_{r}^{1}\): the unit cost of establishing a recycling center at a location \(r\)  
\(C_{s,r}^{2}\): the unit cost of delivering one ton of construction waste from the construction site \(s=1,\ldots,S\) to the recycling center \(r\) or the unit cost of delivering one ton of recycled materials from the recycling center \(r\) to the construction site \(s=S+1,\ldots,S+S^{\prime}\) (CNY/ton)  
\(C_{s}^{3}\): the unit cost of delivering one ton of construction waste from the construction site \(s=1,\ldots,S\) to the landfill site (CNY/ton)  
\(C^{max}\): the maximum total costs  
\(p_{\omega}\): the possibility of scenario \(\omega\)  
\(V_{s,\omega}\): the weight of construction waste generated by the construction site \(s=1,\ldots,S\) or the weight of recycled building materials required by the construction site \(s=S+1,\ldots,S+S^{\prime}\) in scenario \(\omega\)Variables: \(\alpha_{r}\): continuous, the constructed area of the recycling center at location \(r=1,\ldots,R\)  
\(\beta_{s,r,\omega}\): continuous, the weight of construction waste delivered from construction site \(s=1,\ldots,S\) to recycling center \(r\) or the weight of recycled building materials from recycling center \(r\) to construction site \(s=S+1,\ldots,S+S^{\prime}\) in scenario \(\omega\)  
\(\gamma_{s,\omega}\): continuous, the weight of construction waste delivered from the construction site \(s=1,\ldots,S\) to the landfill site in the scenario \(\omega\)  
**Mathematical model:**  
Based on the above definition of indices, parameters, and variables, a two-stage stochastic programming model is formulated as follows. \[ \begin{array}{ll}\textbf{[M1]}\text{Maximize}\ \sum_{\omega=1}^{\Omega}p_{\omega} \!\!\sum_{r=1}^{R}\sum_{s=S+1}^{S+S^{\prime}}\beta_{s,r,\omega}&\text{(1)}\\ \text{s.t.}&\gamma_{s,\omega}+\sum_{r=1}^{R}\beta_{s,r,\omega}=V_{s,\omega}& \forall s=1,\ldots,S,\ \omega=1,\ldots,\Omega&\text{(2)}\\ \sum_{s=1}^{S}\beta_{s,r,\omega}\leq e^{\frac{1}{4}}\alpha&\forall r=1,\ldots,R,\omega=1,\ldots,\Omega&\text{(3)}\\ \sum_{r=1}^{S}\beta_{s,r,\omega}\geq\sum_{s=S+1}^{S+S^{\prime}}\beta_{s,r, \omega}&\forall r=1,\ldots,R,\omega=1,\ldots,\Omega&\text{(4)}\\ \sum_{r=1}^{R}\beta_{s,r,\omega}\leq V_{s,\omega}&\forall s=S+1,\ldots,S+S^{ \prime},\omega=1,\ldots,\Omega&\text{(5)}\\ \sum_{r=1}^{R}\left(C_{r}^{1}\alpha_{r}+\sum_{s=1}^{S+S^{\prime}}C_{s,r}^{2} \beta_{s,r,\omega}\right)+\sum_{s=1}^{S}C_{s}^{3}\gamma_{s,\omega}\leq C^{max}& \forall\omega=1,\ldots,\Omega&\text{(6)}\\ 0\leq\alpha_{r}\leq a_{r}&\forall r=1,\ldots,R&\\ \beta_{s,r,\omega}\geq 0&\forall s=1,\ldots,S+S^{\prime},r=1,\ldots,R,\omega=1, \ldots,\Omega&\text{(8)}\\ \gamma_{s,\omega}\geq 0&\forall s=1,\ldots,S,\ \omega=1,\ldots,\Omega&\text{(9)} \end{array}