## 2 Theoretical Background
### Analytic Hierarchy Process (AHP)  
The analytic hierarchy process is a decision method proposed by [PERSON] (1980). It mainly applies to uncertain situations and multicriteria decision-making problems [34; 35]. The AHP systemizes complicated multicriteria problems with a concise hierarchical framework. A decision maker carries out pairwise comparison between two criteria on their relative importance within a level. After that, a pairwise comparison matrix is constructed in order to determine the relative importance between the criteria. The overall prioritized vector of the entire level can be calculated by making the levels serially connected, leading to the weight of each evaluation criterion. The quantized result assists decision-makers in making comprehensive evaluations of alternative schemes in order to determine their priority and reduce the risk of making a wrong decision. [PERSON] et al. proposed a new approach of calculating weights to replace the pairwise comparison of the analytic hierarchy process (AHP), called the voting analytic hierarchy process. This approach is simpler than AHP and can calculate weights systematically. It can also determine the priority of various suppliers by determining each one's score.  
The procedure of the analytic hierarchy process includes five steps:  
Step 1: Delimit the decision-making problem.  
Step 2: Build the hierarchical framework.  
Step 3: Construct the pairwise comparison matrix, as shown in Table 1, which is the assessment scale and relative comparison of the hierarchy analysis.  
Step 4: Calculate the eigenvalues.  
Step 5: Examine the consistency.  
When calculating the eigenvectors in Step 4, the following algorithms can be used:  
1. Theoretical analysis of eigenvalues and eigenvectors.
2. Four approximate solutions that were proposed by [PERSON]: Normalization of the means of row vectors, also called normalization of the row average; normalization of the means of column vectors, also called the means of normalized columns; normalization of the geometric means of  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Scale** & **Definition** \\ \hline
1 & Equally important \\
3 & Slightly important \\
5 & Important \\
7 & Very important \\
9 & Absolutely important \\
2, 4, 6, 8 & Intermediate values \\ \hline \hline \end{tabular}
\end{table}
Table 1: Assessment scale and relative definitions of hierarchy analysis.  
row vectors, also called normalization of the geometric mean of the rows; and the average of the inverse of the normalized rows.  
The consistency examination in Step 5 determines whether the evaluation results obtained from the pairwise comparison are consistent. In other words, it determines whether the experts' preference satisfies the transitivity. [PERSON] proposed the use of the consistency index (Consistency Index, _C.I._) and consistency ratio (Consistency Ratio, _C.R._) to carry out the examination. If both _C.I._ and _C.R._ are less than 0.1, it indicates that the pairwise matrix has consistency. The equations of _C.I._ and _C.R._ are as follows:  
\[\text{C.I.}=\frac{\lambda_{\text{max}}-n}{n-1} \tag{16}\]  
where \(\lambda_{\text{max}}\) is the maximum eigenvalue of the matrix, and \(n\) is the order of the matrix (number of parameters); and  
\[\begin{array}{ll}&\text{C.R.}<0.1\Rightarrow\text{OK}\\ &\text{C.R.}=\frac{C.I.}{R.I.}\text{ }\text{C.R.}=\text{ }\text{Consistencyratio}\\ &\text{R.I.}=\text{ }\text{Consistencyindex}\end{array} \tag{17}\]  
where n is the number of evaluation criteria, and _R.I._ is the random index, and its value increases with the number of criteria, as shown in Table 2.