# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 4 Application and Results
### Data Processing  
#### 4.2.1 Marginal Distributions  
The assessment (Table 1) includes the frequency of occurrence (\(F\)) and adverse consequence (\(C\)) of the scenario/root cause in linguistic terms. The linguistic terms were then represented using a membership function chosen by the analyst. In this example, the following membership functions, adopted from [58], were used to represent (\(F\)) and (\(C\)), as shown in Tables 2 and 3, respectively: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Frequency of**} \\  & **Root Cause/Scenario** & **Occurrence** & **Adverse Consequence** \\  & & **(F)** & **(C)** \\ \hline
1 & Construction noise is low & Likely & Very small \\
2 & Construction noise is medium & Likely & Large \\
3 & Construction noise is high & Unlikely & Large \\
4 & Harm to activities is low & Unlikely & Small \\
5 & Harm to activities is medium & Somewhat likely & Large \\
6 & Harm to activities is high & Unlikely & Very large \\
7 & Traffic disturbance is low & Very likely & Very small \\
8 & Traffic disturbance is medium & Somewhat likely & Large \\
9 & Traffic disturbance is high & Unlikely & Very large \\
10 & Poor communication & Unlikely & Medium \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Root causes of risk factors with accompanying scenarios and assessment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Element of**} & \multicolumn{5}{c}{**Frequency of Occurrence (F)**} \\ \cline{2-5}  & \multirow{2}{*}{**Very Unlikely**} & \multirow{2}{*}{**Unlikely**} & **Somewhat** & **Likely** & **Very Likely** \\  & & & **Likely** & **(C)** \\ \hline
0 & 1 & 0 & 0 & 0 & 0 \\
0.1 & 0.8 & 0.8 & 0 & 0 & 0 \\
0.2 & 0.2 & 1.0 & 0 & 0 & 0 \\
0.3 & 0 & 0.8 & 0.5 & 0 & 0 \\
0.4 & 0 & 0 & 0.8 & 0 & 0 \\
0.5 & 0 & 0 & 1 & 0.5 & 0 \\
0.6 & 0 & 0 & 0.8 & 0.8 & 0 \\
0.7 & 0 & 0 & 0.5 & 1.0 & 0.5 \\
0.8 & 0 & 0 & 0 & 0.8 & 0.8 \\
0.9 & 0 & 0 & 0 & 0.6 & 0.9 \\
1.0 & 0 & 0 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Membership function for frequency of occurrence (\(F\)). The adverse consequence of root causes was related to the overall risk impact subjectively (i.e., if the consequence is very small, the impact will be small) by the expert, as shown in Table 6.  
Once assessed, each relationship was represented using a fuzzy relation matrix as per Equation (3). An example of fuzzy relation for row No. 3, Q (\(C_{medium}\), \(I_{medium}\)), is provided Table 7.  
After establishing all relations, a fuzzy union matrix, \(V\), between all relations was established, as detailed in Table 8, using Equations (4) and (5). Fuzzy composition matrix \(U\circ V(x_{i},z_{k})\) of the union matrices \(U\) and \(V\) (Tables 5 and 8, respectively) was calculated using Equation (6). The resulting matrix is provided as Table 9. The multiplication of \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Adverse Consequence (C)** & **Impact (I)** \\ \hline
1 & Very small & Small \\
2 & Small & Small \\
3 & Medium & Medium \\
4 & Large & Large \\
5 & Very large & Large \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relationships between adverse consequence (C) and impact (I).