## 4 Evaluation Procedure
### Determine the Trapezoidal Fuzzy Evaluation Matrix  
The concept of trapezoidal fuzzy numbers was proposed by [PERSON] in 1965. It utilizes trapezoidal membership functions to address problems in uncertain environments. 1. Constructing a fuzzy judgment matrix  
Assuming \(U=(u_{1},u_{2},u_{3},u_{4})\), where \(u_{1},u_{2},u_{3},u_{4}\) are real numbers in the set \(U\), and \(0<u_{1}<u_{2}<u_{3}<u_{4}\). Here, \(u_{1}\) and \(u_{4}\) represent the lower and upper bounds of the trapezoidal fuzzy number \(U\), respectively, while \(u_{2}\) and \(u_{3}\) are the median values of the trapezoidal fuzzy number. The formula for the trapezoidal membership function is given as (4) [27]. \[\mu(x)=\left\{ \begin{array}{c}0,(x\leq u_{1})\\ \frac{x-u_{1}}{u_{2}-u_{1}},(u_{1}<x<u_{2})\\ 1,(u_{2}\leq x\leq u_{3})\\ \frac{u_{4}-u_{3}}{u_{4}-u_{3}},(u_{3}<x<u_{4})\\ 0,(x\geq u_{4})\end{array} \right. \tag{4}\]  
Considering the subjectivity, ambiguity, and uncertainty in the scoring provided by experts in the direction of honing, the trapezoidal fuzzy numbers are calculated using the trapezoidal centroid formula. This process transforms the general judgment matrix into a fuzzy judgment matrix. In order to better reflect the experts' thought processes, as shown in Table 3, an improved valuation method is applied to the traditional \"1-9 point scale\" method [28]. \[U=\frac{\int_{u}\mu(x)\cdot xdx}{\int_{u}\mu(x)dx}\,\frac{(u\,{}^{2}_{3}+u_{3} \cdot u_{4}+u_{4}^{2})-(u\,{}^{2}_{1}+u_{1}\cdot u_{2}+u_{2}^{2})}{3\cdot(u_{4 }+u_{3}-u_{1}-u_{2})} \tag{5}\]  
In order to facilitate subsequent calculations, a brief introduction to the arithmetic rules of trapezoidal fuzzy numbers is provided. Let two trapezoidal fuzzy numbers be \(U=(u_{1},u_{2},u_{3},u_{4})\) and \(M=(m_{1},m_{2},m_{3},m_{4})\), and let l be any constant. The arithmetic rules are as follows in Table 4:  
By using the data from Table 4 to replace the elements in the simplified matrix with trapezoidal fuzzy numbers, ensuring consistency, the fuzzy judgment matrix U expressed in terms of trapezoidal fuzzy numbers can be obtained. The simplified matrix must satisfy the consistency requirement, and the improved trapezoidal fuzzy number judgment matrix also meets the consistency requirement [28]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Operational Formula** \\ \hline addition & \(U+M=(u_{1}+m_{1},u_{2}+m_{2},u_{3}+m_{3},u_{4}+m_{4})\) \\ subtraction & \(U-M=(u_{1}-m_{4},u_{2}-m_{3},u_{3}-m_{2},u_{4}-m_{1})\) \\ multiplication & \(U\times M=(u_{1}m_{1},u_{2}m_{2},u_{3}m_{3},u_{4}m_{4})\) \\ division method & \(\frac{U}{M}=\left(\frac{u_{1}}{u_{1}},\frac{u_{2}}{u_{3}},\frac{u_{3}}{u_{2}}, \frac{u_{3}}{u_{1}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Trapezoidal Fuzzy Number Arithmetic Rules. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Comparison of the** & **Traditional** & **Improving** & **Improved aij** \\
**Importance of Element aij** & **aij Assignment** & **aij Assignment** & **Trapezoidal Fuzzy Assignment** \\ \hline equality & 1 & 5/5 & (1, 1, 1, 1) \\ A little important & 3 & 6/4 & (1, 11/9, 13/7, 7/3) \\ Obviously important & 5 & 7/3 & (3/2, 13/7, 3, 4) \\ Strongly important & 7 & 8/2 & (7/3, 3, 17/3, 9) \\ Extremely important & 9 & 9/1 & (4, 17/3, 9, 9) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: '1–9 point scale’ method and improved trapezoidal fuzzy number. Assuming that \(n\) (\(n>1\)) experts with equal status are invited to score at the same time, the reduced matrix after the consistency test is performed, and trapezoidal fuzzy processing is performed. The matrix \(U^{k}\) after fuzzification of the nth expert data is recorded as:  
\[U^{k}=\left(u_{ij}^{k}\right)_{n\times n} \tag{6}\]  
In Formula (6), \(u_{ij}^{k}=\left(u_{1 ij}^{k},u_{2 ij}^{k},u_{3 ij}^{k},u_{4 ij}^{k}\right)\), K = 1, 2, \(\cdots,n\).