## 2 Preliminaries
### Analytic Hierarchy Process  
Analytic hierarchy process (AHP), introduced by [PERSON] (1980) [59], is an effective tool for dealing with complex decision making problem [60; 61; 62]. The weight determination process quantifies the subjective assessment of experts, and can check the consistency of decision-makers' evaluation. Generally, the process of applying AHP can be divided into three steps.  
Step 1: Constructing the pair-wise comparison judgement matrix.  
Assume that \(n\) pieces of decision elements are presented as \((F_{1},F_{2},F_{3},\cdots,F_{n})\). In order to compute the weight of decision elements, a comparison judgement matrix represented as \(M_{n*n}=[m_{ij}]\) is created:  
\[M_{n*n}=\left[\begin{array}{cccc}1&m_{12}&\cdots&m_{1n}\\ m_{21}&1&\cdots&m_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ m_{n1}&m_{n2}&\ldots&1\end{array}\right],\]  
which satisfies:  
\[m_{ij}=\frac{1}{m_{ji}}. \tag{5}\]  
Each entry \(m_{ij}\) represents the importance of the \(i\)th element on the \(j\)th element. If \(m_{ij}>1\), then the \(i\)th element is more important than the \(j\)th element, while if \(m_{ij}<1\), then the \(i\)th element is less important than the \(j\)th element. If two elements have the same importance, then the entry \(m_{ij}\) is 1. The relative importance between two decision elements is measured according to a numerical scale from 1 to 9, as shown in Table 1.  
Step 2: Calculating the weight of decision elements.  
The eigenvector of \(M_{n*n}\) can be denoted as \(\overline{\omega}\) = (\(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\),... \(\omega_{n})^{T}\), which is calculated using:  
\[M\overline{w}=\lambda_{max}\overline{w}, \tag{6}\]  
where \(\lambda_{max}\) is the largest eigenvalue of matrix \(M_{n*n}\). The eigenvector corresponding to the largest eigenvalue can be viewed as the final criterion for ranking goals.  
Step 3: Checking the consistency index.  
A consistency index (CI) is used to measure the consistency within each pair-wise comparison judgement matrix, which is defined as:  
\[CI=\frac{\lambda_{max}-n}{n-1}. \tag{7}\]  
Accordingly, the consistency ratio (CR) can be calculated as follows:  
\[CR=\frac{CI}{RI}, \tag{8}\]  
where RI is the random index. RI and CR are related to the dimension of the matrix, which is listed in Table 2.  
Generally, if CR \(>\) 0.1, the consistency of the pair-wise comparison matrix \(M\) is unacceptable and the elements in the matrix should be revised. Otherwise, \(M\) is considered acceptable and the eigenvector \(\omega\) is treated as the weighing vector after normalization.  
There are three main advantages in AHP: simplicity, practicability, and systematicness. Simplicity refers to the fact that the computation to determine the weight of criteria is concise and the result is clear in order to make decision-making convenient. Practicability means that AHP can deal with a wide range of problems compared to traditional optimization methods with the combination of qualitative and quantitative analysis. Systematicness can be understood to describe how AHP comprehensively regards the object as a system and then makes decisions according to the decomposition, comparison, and judgement.  
\begin{table}
\begin{tabular}{c c} \hline
**Value of \(m_{ij}\)** & **Interpretation** \\ \hline
1 & \(i\) and \(j\) are equally important \\
3 & \(i\) is slightly more important than \(j\) \\
5 & \(i\) is more important than \(j\) \\
7 & \(i\) is strongly more important than \(j\) \\
9 & \(i\) is absolutely more important than \(j\) \\
2, 4, 6, 8 & intermediate values between the two adjacent judgements \\ \hline \end{tabular}
\end{table}
Table 1: Pair-wise comparison scale for analytical hierarchy process (AHP) preference.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Dimension** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline RI & 0.00 & 0.58 & 0.90 & 1.12 & 1.24 & 1.32 & 1.41 & 1.45 & 1.51 \\ \hline \end{tabular}
\end{table}
Table 2: Values of the random index (RI).