## 4 Computational Experiments
### Experimental Results
#### 4.2.2 Comparative Results  
Next, a comparative analysis is conducted between the optimal results by solving [M1], denoted as \(\alpha_{r}^{*}\) and [M2], denoted as \(\alpha_{r}^{**}\). This comparative analysis is narrated through a meticulously designed process, articulated in the following steps:  
1. [M1] is solved to obtain the optimal solution, \(\alpha_{r}^{*}\), alongside the corresponding optimal objective value, \(\mathbb{Z}^{*}\).
2. Subsequently, [M2] is resolved to determine the solution, \(\alpha_{r}^{**}\). For each scenario \(\omega=1,\ldots,\Omega\), the comparison model [M3] is then solved to obtain the objective value  
Figure 2: An optimal recycling plan solved by [M1].  
specific to each scenario. Following the acquisition of objective values across all scenarios, we compute the mean objective value, \(\overline{z}^{**}\), serving as a critical measure for assessing model performance under varied conditions.
3. The final stage of our analysis involves a comprehensive comparison between the objective values \(\overline{z}^{*}\) and \(\overline{z}^{**}\). The mathematical model [M3] is presented as follows:  
**Newly defined parameters:**  
\(\alpha_{r}^{act}\): the optimal value of \(\alpha_{r}\) by solving the model [M2] (i.e., \(\alpha_{r}^{**}\))  
\(V_{s}^{act}\): the weight of construction waste generated by the construction site \(s=1,\dots,S\) or the weight of recycled building materials required by the construction site \(s=S+1,\dots,S+S^{\prime}\)  
**Newly defined variables:**  
\(\beta_{s,r}^{act}\): continuous, the weight of construction waste delivered from the construction site \(s=1,\dots,S\) to the recycling center \(r\) or the weight of recycled building materials from the recycling center \(r\) to the construction site \(s=S+1,\dots,S+S^{\prime}\)  
\(\gamma_{s}^{act}\): continuous, the weight of construction waste delivered from the construction site \(s=1,\dots,S\) to the landfill site  
**Mathematical model:**  
\(\gamma_{s}^{act}+\sum_{r=1}^{R}\beta_{s,r}^{act}=V_{s}^{act}\) (18) \(\forall s=1,\dots,S\) (19) \(\forall r=1,\dots,R\) (20) \(e^{2}\sum_{s=1}^{S}\beta_{s,r}^{act}=\sum_{s=S+1}^{S^{\prime}}\beta_{s,r}^{act}\) \(\forall r=1,\dots,R\) (21) \(\forall s=S+1,\dots,S+S^{\prime}\) (22) \(\forall s=1,\dots,R\) (23) \(\beta_{s,r}^{act}\geq 0\) \(\forall s=1,\dots,S+S^{\prime},r=1,\dots,R\) (24) \(\gamma_{s}^{act}\geq 0\) \(\forall s=1,\dots,S\) (25)  
The algorithmic framework for comparing the solutions derived by deterministic programming is presented in Algorithm 1.  
By solving [M2], we obtain the optimal results about \(\alpha_{r}^{**}\), as outlined in Table 8. Additionally, we engage the same scenario in Section 4.2 and solve [M3] for optimal illustration. Figure 3 illustrates the recycling plan formulated by model [M3], which does not account for uncertainties in construction waste management. The red font indicates that less construction waste is recycled at a construction site compared to model [M1], while the green font signifies more recycling. Specifically, the red label \"BY (less 64,430 tons)\" denotes that model [M3] recycles 64,430 tons less than model [M1] at the BY construction site. A notable improvement is observed when compared to the results solved by [M1]. In the plan derived by [M3], approximately 2,241,227 tons of construction waste are recycled. This figure is about 7% lower than the results achieved by [M1]. Figure 4 compares the relative values of recycled construction waste managed by [M1] and [M3]. We assume that all amounts of recycled construction waste solved by [M3] are standardized to 0.5, with the maximum relative value set to 1. To substantiate the superiority of [M1], we conduct tests using 100 sets of stochastic numbers ranging from 80% to 120%, reflecting variations in the volume of construction waste and the demand for recycled materials. The results consistently show that the recycling rates achieved by [M1] surpass those of [M3].