## 5 Methodology
### Real-Time Hybrid Compaction Quality Evaluation Using \(i\)-AHP Method
#### 5.2.2 \(i\)-AHP Model  
Based on the interval model, the \(i\)-AHP model is established by extending the deterministic numbers to uncertain-but-bounded intervals, and the process for the real-time hybrid evaluation is as follows:  
1. Construct the \(i\)-AHP system and the rating matrix. The \(i\)-AHP system is established as shown in Figure 2. Thereafter, a scoring criterion is necessary by using expert grading method. Then the rating matrix of each element can be obtained as: \[\mathbf{s}=[s_{1},s_{2},\cdots,s_{n}],\] (13) where \(n\) denotes the total number of elements in the alternative level.
2. Construct the interval comparison matrices through pairwise comparison. As shown in Figure 2, the comparison matrices include the criterion level to objective level (\(\mathbf{P}^{I}\)) and the alternative level to criterion level (\(\mathbf{A}^{I}\)). \[\mathbf{P}^{I}=\begin{array}{c}P_{1}\\ P_{2}\\ \vdots\\ P_{m}\end{array}\left[\begin{array}{c}P_{1}\\ P_{12}\\ P_{21}^{I}\end{array}\cdots\begin{array}{c}P_{m}\\ P_{1m}^{I}\\ \vdots\\ P_{m2}^{I}\end{array}\cdots\begin{array}{c}P_{m}\\ P_{1m}^{I}\\ \cdots\\ \end{array}\right],\] (14) \[\mathbf{A}^{I}=\begin{array}{c}A_{1}\\ A_{2}\\ A_{2}\\ \vdots\\ A_{n}\end{array}\left[\begin{array}{c}A_{1}\\ A_{12}^{I}\end{array}\cdots\begin{array}{c}A_{n}^{I}\\ A_{2n}^{I}\end{array}\right],\] (15) where \(m\) is the total number of elements in criteria level. \(x_{ij}^{I}=\left[x_{ij}^{-},x_{ij}^{+}\right]\)(\(x=p,a\)) denote the interval model for comparing the importance of the \(i\)th and the \(j\)th element, and \(a_{ij}^{I}=0(i\
eq j)\) when \(A_{i}\) and \(A_{j}\) are in different criteria. The elements in the comparison matrices satisfy the reciprocal principle, namely \(x_{ij}^{\pm}=1/x_{ji}^{\mp}(x=p,a)\), and the importance degree is listed in Table 2.  
Calculate the interval weight vectors of the interval comparison matrices, including the criterion level to objective level (\(\overline{\textbf{p}}^{I}\)) and the alternative level to criterion level (\(\overline{\textbf{a}}^{I}\)). The expression of \(\overline{\textbf{p}}^{I}\) is: \[\left\{\begin{array}{l}\overline{\textbf{p}}^{I}=\left[\overline{\textbf{p} }^{-},\overline{\textbf{p}}^{+}\right]=\left[\varepsilon_{1}\textbf{x}^{-}, \varepsilon_{2}\textbf{x}^{+}\right]\\ \varepsilon_{1}=\sqrt{\sum_{j=1}^{m}\frac{1}{\sum_{i=1}^{n}p_{ij}^{+}}}\\ \varepsilon_{2}=\sqrt{\sum_{j=1}^{m}\frac{1}{\sum_{i=1}^{n}p_{ij}^{+}}}\end{array} \right.,\] (16) where \(\textbf{x}^{-}\) and \(\textbf{x}^{+}\) are the normalized eigenvectors of \(\textbf{P}^{-}\) and \(\textbf{P}^{+}\).
4. Check the consistency of the interval comparison matrices. Firstly, the fuzzy degree of interval number model \(x^{I}\) is defined as: \[\delta\Big{(}x^{I}\Big{)}=\frac{2x^{r}}{x^{c}}.\] (17) Then the fuzzy degree of \(\textbf{P}^{I}\) can be expressed as: \[\delta\Big{(}\textbf{P}^{I}\Big{)}=\frac{2x^{r}}{m(m-1)}\sum\
olimits_{i=1} ^{m}\sum\
olimits_{j=1,i\
eq 1}^{m}\delta\Big{(}p_{ij}^{I}\Big{)}.\] (18) The criterion of consistency check is: \[\left\{\begin{array}{l}C.R.=\frac{\eta-m}{(m-1)R.I.}\leq 0.1\times\Big{(}1+ \delta\Big{(}\textbf{P}^{I}\Big{)}\Big{)}\\ \eta=\lambda^{-}+\Big{(}\frac{\varepsilon}{1+c}\Big{)}(\lambda^{+}-\lambda^{- })\\ c=\frac{1-\sum_{j=1}^{m}w_{j}^{+}}{\sum_{j=1}^{m}w_{j}^{+}-1}\end{array}\right.,\] (19) where \(C.R.\) denotes the consistency ratio. \(R.I.\) denotes the average random consistency index, which can be queried in Reference [20]. The error of the interval comparison matrices will be too large when Equation (19) can't be satisfied, which requires a new judgement.
5. Define and calculate the \(HCI.\) The interval total weight vectors can be obtained based on the above process, which is expressed as: \[\textbf{v}^{I}=\overline{\textbf{p}}^{I}\overline{\textbf{a}}^{I},\] (20) then the final hybrid evaluation index \(HCI\) can be defined as Equation (21), here \(HCI^{I}\) denotes the interval \(HCI\), and the higher index shows the better compaction quality. \[HCI^{I}=\textbf{sv}^{I}.\] (21)