## 2 Methodology
### The Best-Worst Method (BWM)
#### 2.2.2 BWM Calculation Process  
1. Determination of the Best and Worst Criteria;  
Let us consider a scenario where \(n\) distinct evaluation criteria are present. These criteria collectively constitute the set of evaluation criteria, denoted as Equation (1):  
\[F=\{f_{1},f_{2},\cdots,f_{n}\} \tag{1}\]  
Subject matter experts in the relevant field assess these criteria to identify the best criterion \(F_{B}\) and the worst criterion \(F_{W}\).
2. Calculation of the Comparison Vector; \(Q_{B}\);  
Field experts employ a 1-9 scale for scoring each evaluation criterion, thereby determining the relative preference of the most favorable criterion. This is achieved by assessing how the best criterion fares in comparison to the others. Consequently, we obtain the comparison vector, as shown in Equation (2):  
\[Q_{B}=\{p_{B1},p_{B2},\cdots,p_{Bn}\} \tag{2}\]  
In this vector, a value of 1 indicates that two criteria are of equal importance, whereas a value of 9 reflects the utmost importance of the best criterion \(F_{B}\) in comparison to any other criterion. Evidently, \(p_{BB}=1\).
3. Calculation of the Comparison Vector \(Q_{W}\).  
Experts in the relevant field employ a 1-9 scale to appraise each evaluation criterion, gauging its level of preference compared to the least favorable criterion. This appraisal process leads to the formulation of the comparison vector, as shown in Equation (3):  
\[Q_{W}=\{p_{1W},p_{2W},\cdots,p_{nW}\} \tag{3}\]  
In this vector, a value of 1 indicates that two criteria are of equal importance, whereas a value of 9 reflects extreme importance of a criterion relative to the worst criterion \(F_{W}\). Evidently, \(p_{WW}=1\).
4. Calculation of the Optimal Criterion Weights  
Define the optimal set of weights as \(\{w_{1}^{*},w_{2}^{*},\cdots,w_{n}^{*}\}\). This set is determined by the model that aims to minimize the maximum absolute deviation represented by Equation (4):  
\[\big{\{}|w_{B}-a_{Bj}w_{j}|,|w_{j}-a_{jW}w_{W}|\big{\}} \tag{4}\]  
Here, \(w_{B}\) represents the weight of the best criterion, \(w_{W}\) represents the weight of the worst criterion, \(w_{j}\) represents the weight of the \(j\)-th criterion, \(a_{Bj}\) represents the degree of preference of the best criterion over the \(j\)-th criterion, and \(a_{jW}\) represents the degree of preference of the \(j\)-th criterion over the worst criterion. All criterion weights are non-negative and their sum equals 1, which can be expressed as Equation (5):  
\[\begin{array}{l}\underset{s.t.}{\text{minmax}}\big{\{}|w_{B}-a_{Bj}w_{j}|,| w_{j}-a_{jW}w_{W}|\big{\}}\\ \end{array} \tag{5}\]  
In the optimization model presented, '\(s.t.\)' introduces the constraints necessary for the solution.  
Based on the properties of the BWM, the aforementioned model can be refined and transformed into a linear programming model, as shown in Equation (6):  
\[\begin{split}\min\zeta\\ s.t.\left\{\begin{array}{c|c}\left|w_{B}-a_{B}w_{j}\right|& \leq\zeta\\ \left|w_{j}-a_{jW}w_{w}\right|&\leq\zeta\\ \sum\limits_{j=1}^{n}w_{j}=1,w_{j}\geq 0;\forall j\in N\end{array}\right. \end{split} \tag{6}\]  
In this context, \(\zeta\) represents the consistency error of the expert comparison results. A smaller value of \(\zeta\) indicates a smaller error in the weight calculation results, meaning that the calculated weights are more reliable.