# Improvements of Design for Safety in Construction through Multi-Participants Perception Analysis
## 3 Materials and Methods
### Calculation and Analysis of the Weighting Factors  
#### 3.3.1 Explanation of F-AHP  
AHP was developed by [PERSON] in 1980 [53]. It is a methodology that stratifies decision-making problems (e.g., goal-criteria alternatives) and systematically evaluates the relative importance of various alternatives through a pairwise comparison of the elements of a hierarchy. In addition, it can measure the consistency of results on importance through what is termed the consistency ratio (CR). Here, the AHP analysis proceeds as follows: (i) the decision-making problems are stratified into goals, criteria, and alternatives; (ii) a pairwise comparison of the decision-making elements is conducted at a given level of the hierarchy, with the pairwise comparison matrix calculated using Equation (1); (iii) the pairwise comparison matrix is converted into a normalized matrix calculated using Equation (2), and the average value of each row of the normalized matrix is calculated to derive a weighting factor; and (iv) to prove the effectiveness of the weighting factor, the CR calculated using Equation (4) is presented using the consistency index (CI) calculated using Equation (3) and the random index (RI) presented in Table 5 [53; 54; 55]. \[A=\left( \begin{array}{ccc}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{n1}&\cdots&a_{nn}\end{array} \right) \tag{1}\]  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Criteria**} & \multicolumn{6}{c}{**Levene’s Test for the Equality of Variances**} & \multicolumn{2}{c}{**Welch’s Test**} \\ \cline{2-7}  & & \multicolumn{6}{c}{**Independent Variable**} \\ \cline{2-7}  & F & df & Sig. & Statistic & df & Sig. \\ \hline Eight & & & & & & \\ DfS & 2.87 & 7 & 0.006 & 3.79 & 7 & 0.001 \\ alternatives & & & & & & \\ \hline Five & & & & & & \\ construction & & & & & & \\ project & & & & & & \\ participants & & & & & & \\ \hline \multirow{3}{*}{Two  career levels} & \multicolumn{6}{c}{**Levene’s test for the equality of variances**} & \multicolumn{6}{c}{**T-test for Equality of Means**} \\ \cline{2-7}  & & & & & & & \\ \cline{1-1}  & & & & & & &  
Here, matrix A is the comparison matrix, \(a_{ij}\) is the \(j\)-th alternative at the \(i\)-th alternative, and \(n\) is the number of alternatives. \[P=\left( \begin{array}{ccc}p_{11}&\cdots&p_{1n}\\ \vdots&\ddots&\vdots\\ p_{n1}&\cdots&p_{nn}\end{array} \right) \tag{2}\]  
In this equation, matrix \(P\) is the normalized matrix and \(P_{ij}\) is \(a_{ij}\) divided by the sum of the \(a_{ij}\) columns. \[CI=\left(\lambda_{max}-n\right)/(n-1) \tag{3}\]  
Here, \(\lambda_{\max}\) is the maximum Eigen value of the comparison matrix,  
\[CR=CI/RI, \tag{4}\]  
where \(CR\) is the consistency ratio of the respondents and \(RI\) is the random index. AHP can analyze differences in opinions between the respondent groups because it can establish a plan by reflecting the perceptions of all construction participants. However, the traditional AHP cannot easily reflect the opinions of respondents precisely because with uncertainty of information, it is difficult to provide an exact numerical point for a perception [56; 57; 58]. Accordingly, to overcome this problem, fuzzy AHP (F-AHP) is utilized in this study. The triangular fuzzy value for a pair-wise comparison can take the features of the interval perception and the most likely values. F-AHP utilizes the following equations. First, the judgement matrix \(\text{F}_{n\times n}\), which is the ratio of the interval values, is calculated by Equation (5). Second, the value of the fuzzy synthetic extent about the \(i\)th object (Pi) is calculated by Equation (6). Third, the possibility degree \(\mu(d)\) can be expressed by Equation (7). Forth, the weighting vector of F-AHP can be expressed by Equation (8) [56]. \[F=\left( \begin{array}{ccc}1&\cdots&P_{1n}\\ \vdots&\ddots&\vdots\\ P_{n1}&\cdots&1\end{array} \right) \tag{5}\]  
Here, \(P_{1n}\) is the ratio of the interval value for the first factor and the interval value for the \(n\)-th factor, and \(P_{n1}\) is the inverse of \(P_{1n}\).