## 3 Methodology
### Calculation of Evaluation Criteria Weights Based on Rough BWM and Entropy Weight Method
#### 3.2.1 Calculation of Subjective Weights of Evaluation Criteria Using Rough BWM
 \tag{3}\]  
Subsequently, the averages of the rough series can be expressed as rough numbers of the comparison vectors. The rough numbers, \(\overline{RN}(h_{Bj})\) and \(\overline{RN}(h_{jW})\), defined by Equation (4), can be transformed as follows: \(\overline{RN}(h_{Bj})=\{[\ \underline{h_{B1}},\overline{h_{B1}}],[\ \underline{h_{B2}},\overline{h_{B2}}],\ \ldots,\ [\underline{h_{Bn}}, \overline{h_{Bn}}]\}\), \(\overline{RN}(h_{jW})=[\ \underline{h_{1W}},\ \overline{h_{1W}}],[\ \underline{h_{2W}},\ \overline{h_{2W}}],\ \ldots,\ [ \underline{h_{nW}},\ \overline{h_{nW}}]\}\). \[ \begin{array}{c}\overline{RN}(h_{Bj})=[\underline{h_{Bj}},\overline{h_{Bj}} ]=\frac{1}{m}\sum\limits_{k=1}^{m}\ [h_{Bj}^{k},\overline{h_{Bj}^{k}}]\\ \overline{RN}(h_{jW})=[\underline{h_{jW}},\overline{h_{jW}}]=\frac{1}{m}\sum \limits_{k=1}^{m}\ \underline{[h_{jW}^{k}},\overline{h_{jW}^{k}}]\end{array}  \tag{4}\]  
where \(\underline{h_{Bj}^{k}}\) and \(\overline{h_{Bj}^{k}}\) represent the upper and lower boundaries of the rough number \(\overline{RN}(h_{Bj})\), respectively, while \(\underline{h_{jW}^{k}}\) and \(\overline{h_{jW}^{k}}\) denote the upper and lower bounds of the rough number \(\overline{RN}(h_{jW})\), respectively. Step 4. A nonlinear min-max mathematical programming problem is constructed, as expressed in Equation (5). This is used to evaluate the rough weight of each criterion, \(RN(w_{j})\). In addition, \(d([x],[y])\) represents the degree of separation between the \(RN(x)\) and \(RN(y)\) rough sets [46], while \([w_{B}]\) and \([w_{W}]\) denote the weights of the best and worst criteria, respectively, as expressed in Equation (6). \[ \begin{array}{c}\underset{j}{\text{minmax}}\Big{\{}d(\frac{[w_{Bj}]}{[w_{j} ]},\ [h_{Bj}]),d(\frac{[w_{j}]}{[w_{j}]},\ [h_{jW}])\Big{\}};\\ \text{s.t.}\quad\sum\limits_{j=1}^{n}\frac{w_{j}}{\underline{v}_{j}}\leq 1, \text{ for all }j.\\ \quad\sum\limits_{j=1}^{n}\overline{w_{j}}\geq 1, \text{ for all }j.\\ \quad\overline{w_{j}}\geq\underline{w_{j}}\geq 0,\text{ for all }j.\end{array}  \tag{5}\]  
\[ \begin{array}{c}d([x],[y])=\sqrt{(\underline{x}-\underline{y})^{2}+( \overline{x}-\overline{y})^{2}}\end{array}  \tag{6}\]  
Based on the rules of arithmetic operations with rough numbers and the characteristics of min-max programming problems [13], to obtain the optimal weights \(w_{j}=[w_{1}^{*},w_{2}^{*},\ldots,w_{n}^{*}]\) and goals constraint \(\xi\), an improved mathematical model was proposed by [PERSON] [47], and the objective function was customized as follows:\[ \begin{array}{ll}\min&\zeta;\\ \text{s.t.}&\left|\underline{w_{\text{B}}-\overline{h_{\text{B}}y}\overline{w_{ \text{j}}}}\right|\leq\zeta,\ \left|\overline{w_{\text{B}}}-\underline{h_{\text{B}}y}w_{\text{j}}\right|\leq \zeta,\ \text{for all }j.\\ &\left|\underline{w_{\text{j}}-\overline{h_{\text{J}}y}\overline{w_{\text{W}} }}\right|\leq\zeta,\ \left|\overline{w_{\text{j}}}-\underline{h_{\text{J}}w} \underline{w_{\text{W}}}\right|\leq\zeta,\ \text{for all }j.\\ &\sum\limits_{j=1}^{n}\underline{w_{\text{j}}}\leq 1,\ \sum\limits_{j=1}^{n} \overline{w_{\text{j}}}\geq 1,\ \text{for all }j\\ &\overline{w_{\text{j}}}\geq\underline{w_{\text{j}}}\geq 0,\ \text{for all }j.\end{array}  \tag{7}\]  
where \(\xi\) denotes the goals constraint, and \(\{\left[\underline{w_{\text{1}}},\overline{w_{\text{1}}}\right]\), \(\left[\underline{w_{\text{2}}},\overline{w_{\text{2}}}\right]\),..., \(\left[\underline{w_{n}},\overline{w_{n}}\right]\}\) represent the optimal rough weights of criteria. The Lingo18 software is used to solve the aforementioned linear programming problem, while Equation (8) is adopted to normalize the rough weights of the evaluation criteria and calculate their crisp weights (\(w_{\text{j}}\), where \(w_{\text{j}}\in\) [0,1). \[w_{\text{j}}=(1-\alpha)\underline{w_{\text{j}}}+\alpha\overline{w_{\text{j}}} \tag{8}\]