## 3 Methodology
### Calculation of Evaluation Criteria Weights Based on Rough BWM and Entropy Weight Method
#### 3.2.2 Calculation of Objective Weights of Evaluation Criteria Using the Entropy Weight Method  
The entropy weight method describes the uncertainty of evaluation information in objectively analyzing the importance of the evaluation criteria [38; 48]. To address biases in the subjective weighting solutions, an initial evaluation matrix for solution values is first constructed using the evaluation criteria. The EWM is then adopted to calculate the objective weight of each evaluation criterion. Step 1. The CV evaluation criteria are first divided into quantitative and qualitative criteria. Then, the DM \(k\) evaluates the solution, and the evaluations of all DMs are gathered to construct the initial evaluation matrix \(\text{IEM}=(x_{\text{j}})_{K\times n}\), where \(1\leq i\leq K\). In particular, qualitative criteria are evaluated using a 9-point scale. For quantitative criteria, each DM suggests an appropriate value according to his own perception and experience. For example, based on the criterion E1 in our case, the value judgments of CS\({}_{1}\) are assigned by four DMs, which are 1.75 (t), 1.80 (t), 1.50 (t), and 1.75 (t), respectively. Step 2. Equations (1) and (2) are used to convert the evaluation data of the DMs into rough numbers and construct the rough value matrix (RVM) of the solutions, \(\text{RVM}=(x^{\prime}_{\text{j}})_{K\times n}\). This matrix is adopted to compute objective evaluation data for the criteria weights. Step 3. The RVM of the solutions in terms of CV criteria is first constructed, and then normalized using Equation (8). Subsequently, the rough values of the solutions in terms of cost criteria are converted into utility values to produce the normalized rough value matrix \(\text{NRM}=(f^{\text{L}}_{ij})_{K\times n}\), which is adopted in subsequent dimensionless entropy weight and criteria effect calculations, as expressed in Equation (9). \[r_{ij}=[r^{\text{L}}_{ij^{\text{L}}},r^{\text{L}}_{ij}]=\left\{ \begin{array}[ ]{ll}\left[f^{\text{L}}_{ij}\Big{/}\sqrt{\sum\limits_{i}{(f^{\text{L}}_{ij}) ^{2}}},h^{\text{L}}_{ij}\Big{/}\sqrt{\sum\limits_{i}{(f^{\text{L}}_{ij})^{2}} }\right],\ j\in\text{benefit criterion}\\ \left[\left(1/f^{\text{L}}_{ij}\right)\Big{/}\sqrt{\sum\limits_{i}{(1/f^{ \text{L}}_{ij})^{2}}},\left(1/f^{\text{L}}_{ij}\right)\Big{/}\sqrt{\sum \limits_{i}{(1/f^{\text{L}}_{ij})^{2}}}\right],\ j\in\text{cost criterion} \end{array} \right. \tag{9}\]  
where \(f_{ij}^{\text{L}}\) and \(f_{ij}^{\text{L}}\) denote the lower and upper bounds of the rough number of criteria j in solution i, respectively. Step 4. The information entropy ej of each evaluation criterion is calculated based on the normalized RVM. \(e_{j}\) represents a measure of the uncertainty of the solution values for each criterion \(j\); the higher the value of \(e_{j}\), the more consistent the data for an evaluation criterion, and the lower its weight. The formulas for calculating \(e_{j}\) are expressed in Equations (10) and (11):\[e_{j}=\lambda(-\frac{1}{\ln\kappa}\sum\limits_{i=1}^{K}{\rm{g}}_{ij}\ln{g}_{ij})+(1 -\lambda)(-\frac{1}{\ln\kappa}\sum\limits_{i=1}^{K}{z}_{ij}\ln{z}_{ij}) \tag{10}\]  
\[{\rm{g}}_{ij}=\frac{\left(r_{ij}^{L}+r_{ij}^{L}\right)/2}{\sum\limits_{i=1}^{K }{\left(r_{ij}^{L}+r_{ij}^{L}\right)/2}},z_{ij}=\frac{1-\left(r_{ij}^{L}-r_{ ij}^{L}\right)}{K-\sum\limits_{i=1}^{K}{\left(r_{ij}^{L}-r_{ij}^{L}\right)}} \tag{11}\]  
where \(\lambda\) (\(0<\lambda<1\)) is the balancing coefficient for the median of the rough number and uncertainty of DM evaluations. In this study, \(\lambda\) was set to 0.5 in all \(e_{j}\) calculations. Finally, Equation (12) is adopted to compute the entropy weight of each evaluation criterion using their information entropy, thus generating their objective weights.