## 4 Discussion
### Analysis Obtained Using the AHP  
In this study, the construction auditing checklist of the PCMIS was divided into five risk dimensions. The owner and supervisory unit dimension is coded as O and contains 13 risk factors; the contractor and labor dimension is coded as C and contains 10 risk factors; the material and equipment dimension is coded as M and contains seven risk factors; the construction defects dimension is coded as D and contains nine risk factors; and the relevant risks dimension is coded as R and contains seven risk factors. A total of 19 major risks were identified for Taiwanese construction projects in this study (Figure 4).  
The aforementioned five risk dimensions were used in the AHP framework. Pairwise comparison was performed to determine the relative importance of the risk factors in terms of their mean Likert scale scores, and the maximum eigenvalues of the five risk dimensions were obtained. These eigenvalues represent the weights of the 19 major risk factors. After using Equation (10) to standardize matrix A into A', the row mean of the standardize matrix was calculated to obtain the factor weights (W = [\(w_{i}\)]T), as presented in Equation (11). The eigenvectors of O, C, M, D, and R were [0.1930, 0.2076, 0.1998, 0.2032, 0.1964]T. By using Equations (5) and (6), \(\lambda_{\max}\) was obtained to be 5.0, and the CI was determined to be \(0.507\times 10^{-9}\) (<0.1); thus, the consistency requirement was met.  
The weights (scores) of the risk factors were directly obtained through the Likert scale in the questionnaire. However, the relative effects between multiple factors in a given dimension were not considered. The AHP analyzed problems through a hierarchical structure and correlations, estimated the relative importance of the factors in a given layer (dimension), and conducted evaluations with a pairwise comparison matrix. In other words, two given factors in a layer were evaluated by using the factors of the previous layer as the evaluation standard for calculating the relative importance or contribution of the two factors. Because the factor weights obtained through the AHP were relative weights that ranged between 0 and 1 and add up to a sum of 1, the convergence speed and accuracy of the model could be increased when the weights were input into the ANN.  
\[\text{A}^{\prime}\ =\ \frac{a_{ij}}{\sum\limits_{i=1}^{n}a_{ij}}\, \tag{10}\]  
\[wi=\frac{1}{n}\sum\limits_{j=1}^{n}\frac{a_{ij}}{\sum\limits_{i=1}^{n}\frac{a_{ ij}}{a_{ij}}}\ \ \ \ \forall i=1,\ 2,\ \cdots,n, \tag{11}\]  
Figure 3: Histogram of 46 risk factors.  
\[\text{A}=\begin{bmatrix}\text{O}&\text{C}&\text{M}&\text{D}&\text{R}\\ \text{O}&\text{C}&\left[\begin{array}{cccc}1&0.93&0.97&0.95&0.98\\ 1.08&1&1.04&1.02&1.06\\ \text{M}&1.04&0.96&1&0.98&1.02\\ \text{D}&1.05&0.98&1.02&1&1.03\\ \text{R}&1.02&0.95&0.98&0.97&1\end{array}\right]\]  
Row factors and standardization \(\rightarrow\)  
\[\text{A}^{\prime}=\begin{bmatrix}0.193011&0.193012&0.193013&0.193006&0.193013 \\ 0.207606&0.207660&0.207616&0.207595&0.207608\\ 0.199763&0.199756&0.199765&0.199772&0.199771\\ 0.203191&0.203197&0.203179&0.203186&0.203177\\ 0.196429&0.196429&0.196426&0.196440&0.196431\end{bmatrix}\]  
\[\text{Weight }(W)\rightarrow\begin{bmatrix}w_{1}\\ w_{2}\\ w_{3}\\ w_{4}\\ w_{5}\end{bmatrix}=\frac{1}{5}\text{A}^{\prime}=\begin{bmatrix}0.1930\\ 0.2076\\ 0.1998\\ 0.2032\\ 0.1964\end{bmatrix}\]  
Figure 4: Major risk factors identified in this study.