## 3 Methodology
### Selection of Replacement Plan Using Hybrid Chaotic Whale Optimization Algorithm
#### 3.2.1 Analytic Hierarchy Process  
Under the suggestions of experts selected from relevant research field via the Delphi method [37], the hierarchy structure model of performance-based optimal replacement plan selection is constructed and illustrated in Figure 4, which takes 12 kinds of design, construction, and management requirements into consideration.  
In accordance with the 1-9 evaluation scale method [38], pairwise comparison matrices will be determined via the Delphi method. To put it in practical terms, the comparison matrix \(X\) is exhibited as an example:  
\[X=\left(x_{ij}\right)_{3\times 3}=\left(\begin{array}{ccc}1&Y_{1}/Y_{2}&Y_{1}/Y_{3 }\\ Y_{2}/Y_{1}&1&Y_{2}/Y_{3}\\ Y_{3}/Y_{1}&Y_{3}/Y_{2}&1\end{array}\right) \tag{5}\]  
where \(Y_{1}/Y_{2}\) is the relative priority of criterion \(Y_{1}\) with respect to criterion \(Y_{2}\), whose specific value is determined by the results of the above-mentioned 1-9 evaluation scale method, and the specific evaluation regulations are demonstrated in Table 12.  
After the construction of comparison matrices, the asymptotic normalization coefficient method is utilized to realize the process of level simple sequence (LLS), where the eigenvector of comparison matrix \(w_{i}\) is defined as the hierarchical weight vector and calculated as follows [3]:  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Evaluation Scale** & **Comparison of the Priority** \\ \hline
1 & \(Y_{1}\) and \(Y_{2}\) are equally important \\
3 & \(Y_{1}\) is slightly important compared with \(Y_{2}\) \\
5 & \(Y_{1}\) is obviously important compared with \(Y_{2}\) \\
7 & \(Y_{1}\) is highly important compared with \(Y_{2}\) \\
9 & \(Y_{1}\) is extremely important compared with \(Y_{2}\) \\
2, 4, 6, 8 & Intermediate values used to represent compromise \\ \hline \hline \end{tabular}
\end{table}
Table 12: Evaluation scale classification.  
Figure 4: Hierarchy structure model of the performance-based replacement method selection.  
\[M_{i}=\prod\limits_{j=1}^{n}x_{ij},\ \ \overline{w_{i}}=\sqrt[n]{M_{i}},\ \ w_{i}= \frac{\overline{w_{i}}}{\sum\limits_{j=1}^{n}\overline{w_{j}}}\ (i=1,\ 2\ldots n) \tag{6}\]  
where \(M_{i}\) is the chain-multiplication result of \(x_{ij}\), and \(n\) is the order of comparison matrix.  
Simultaneously, the related maximum eigenvalue is \(\lambda_{\max}=\sum\limits_{i=1}^{n}\frac{\left(xw\right)_{i}}{mw_{i}}\), where \(\left(Xw\right)_{i}\) is the part \(i\) of \(Xw\).  
In accordance with obtained \(\lambda_{max}\), the consistency ratio (CR) and consistency index (CI) are introduced to perform a consistency test, which demonstrates the rationality of the constructed comparison matrix and can be calculated as follows:  
\[CR=\frac{CI}{RI} \tag{7}\] \[CI=\frac{\lambda_{max}-n}{n-1}\]  
where \(RI\) is the random consistency index that corresponds to the order of comparison matrix and can be defined as Table 13 illustrates.  
If \(CR<0.10\), the matrix consistency is acceptable. Otherwise, the matrix is inconsistent and is required to be modified. In particular, the matrix is defined as the complete consistency matrix when \(CR\) is equal to 0.