## 2 Materials and Methods
### Analytic Hierarchy Process  
The AHP technique, utilized in multi-criteria decision-making problems, was pioneered by [PERSON] in 1977 [14]. In this technique, characterized by its hierarchical structure, criteria are first established in alignment with the problem's objectives. Subsequently, alternatives are evaluated based on these criteria. The steps outlined in Figure 2 are followed throughout the evaluation process. Define problem: For effective decision-making, it is essential to clearly and comprehensively define the problem and reveal the purpose. Identify criteria: Criteria suitable for the defined purpose should be identified for assessing the alternatives. Identify alternatives: Once the criteria are determined, the decision alternatives to be evaluated should be identified. Perform pairwise comparisons: Comparison matrices should be constructed by establishing the importance values of the determined criteria relative to each other. The importance scale defined by [PERSON] is utilized to ascertain the degree of importance for each criterion. The importance scale is shown in Table 1.  
Figure 1: The flowchart of study. The matrix formed by comparing the criteria in pairs is referred to as matrix A, where each element signifies their importance values relative to each other. Matrix A is a square matrix of size \(n\times n\), where n represents the number of criteria used in the evaluation. Its diagonal elements are all equal to 1. The matrix A is shown in Equation (1). \[\text{A}=a_{ij}= \begin{bmatrix}a_{11}&a_{12}&\dots&a_{1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\.&.&\\ a_{n1}&a_{n2}&\dots&a_{nn}\end{bmatrix}  \tag{1}\]  
Each element of matrix A, denoted as \(\text{a}_{\text{ij}}\), represents the importance value of criterion i relative to criterion j. In pairwise comparisons, \(\text{a}_{\text{ji}}\) is calculated as specified in Equation (2). \[a_{ji}=\frac{1}{a_{ij}} \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline
**Intensity of Importance** & **Description** \\ \hline
1 & Equal Importance \\
3 & Moderate Importance \\
5 & Strong Importance \\
7 & Very Strong Importance \\
9 & Extreme Importance \\
2-4-6-8 & Average (Intermediate) Values \\ \hline \end{tabular}
\end{table} 
Table 1: Importance scale used in criteria comparisons. Figure 2: AHP process. Once the importance values of each criterion relative to each other are determined, the weights of the criteria among all criteria are calculated. The formula in Equation (3) is used for this calculation. \[c_{ij}=\frac{a_{ij}}{\sum_{i=1}^{n}a_{ij}} \tag{3}\]  
The resulting matrix, known as the normalized matrix, is denoted as matrix C. Subsequently, the average of each row in the normalized matrix C is calculated, leading to the determination of the importance weights of the criteria (\(W_{i}\)). The formula in Equation (4) is used at this stage. \[W_{i}=\frac{\sum_{i=1}^{n}c_{ij}}{n} \tag{4}\]  
Calculation of consistency ratios: Once the importance weights of each criterion are determined, consistency ratios are calculated to assess the consistency of the decision makers' personal evaluations. For this calculation, the largest eigenvalue (\(\lambda_{max}\)) should be determined first. The formula in Equation (5) is used to determine this value. \[\lambda_{max}=\frac{\sum_{j=1}^{n}a_{ij}\cdot w_{j}}{w_{i}}\text{i}=1,2,\ldots,\text{n} \tag{5}\]  
After determining the largest eigenvalue, the \"_Consistency Index (CI)_\" and \"_Consistency Ratio (CR)_\" are then calculated. The Consistency Index is calculated by the formula in Equation (6) and the Consistency Ratio is calculated by the formula in Equation (7). \[\text{CI}=\frac{\lambda_{max}-n}{n-1} \tag{6}\]  
\[\text{CR}=\frac{CI}{RI} \tag{7}\]  
The RI value in Equation (7) represents the random index value. These values, set by [PERSON] as fixed numbers, vary depending on the value of n. The index values corresponding to different n values are presented in Table 2[15]. If the Consistency Ratio (CR) is less than 0.10, the pairwise comparison matrix is considered consistent.