## 3 Problem Description and Model Formulation
### Problem Statement  
Consider a region with \(S\) construction sites where the demolition of existing buildings takes place (indexed by \(s\), \(s=1,\ldots,S\)), resulting in the generation of construction waste. Generally, waste generated will be disposed of through landfilling, wherein it is collected and transported to designated landfill sites. Given that many resourceful materials can be recycled from waste and reused in new projects, the regional government is now planning to select from \(R\) candidate locations to establish waste recycling centers (indexed by \(r\)). The maximum area that can be used to establish recycling centers at location \(r\) is known as \(a_{r}\). The constructed area of the recycling center at location \(r\), denoted by \(\alpha_{r}\) (to be determined), should be no greater than \(a_{r}\). The capacity of the recycling center at location \(r\) is \(e^{1}\alpha_{r}\) ton, where \(e^{1}\) is a coefficient representing the waste handling capacity per unit area of recycling centers (ton/m\({}^{2}\)). The construction waste can be delivered either to landfill sites or to recycling centers (if established) for disposal. If the waste is delivered to a landfill site, it will be directly landfilled. However, if the waste is delivered to recycling centers, it will be sorted and reused. Denote \(e^{2}\) as a conversion rate of waste to materials, i.e., \(e^{2}\) tons of building materials can be recycled from one ton of construction waste. Subsequently, the recycled materials will then be sold to \(S^{\prime}\) construction sites of new projects for reutilization to foster a sustainable cycle. Additionally, \(s\) is used to index the construction site requiring recycled materials, where \(s=S+1,\ldots,S+S^{\prime}\). The objective of the government is to maximize waste recycling, i.e., the weight of recycled building materials delivered to construction sites for reusing, subject to a predetermined cost threshold \(C^{max}\). Specifically, the unit cost of establishing a recycling center at location \(r\) per square meter (including land acquisition and construction costs) is denoted as \(C^{1}_{r}\). The transportation cost between the construction site \(s\) and recycling center \(r\) is denoted by \(C^{2}_{s,r}\). If \(s=1,\ldots,S\), it indicates the cost of delivering one ton of construction waste from a construction site \(s\) to a recycling center \(r\); if \(s=S+1,\ldots,S+S^{\prime}\), it indicates the cost of delivering one ton of recycled materials from a recycling center \(r\) to a construction site \(s\). Therefore, the total costs used to establish the recycling centers and transport waste and materials should be no greater than \(C^{max}\). As site selection for recycling centers is a long-term decision, we shall comprehensively account for the stochastic amount of construction waste generated from demolition activities and the stochastic amount of building materials necessitated by future projects. Consider \(\Omega\) scenarios (indexed by \(\omega\)), each with a possibility of \(p_{\omega}\), and \(\sum_{\omega=1}^{\Omega}p_{\omega}=1\). The stochastic parameter \(V_{s,\omega}\) is used to indicate 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 each scenario \(\omega\). The uncertainty of parameter \(V_{s,\omega}\) leads to the formulation of the investigated problem as a two-stage stochastic program. In the first stage, the government decides the establishment of recycling centers (i.e., variable \(\alpha_{r}\)).