# 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
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Adverse**} & \multicolumn{8}{c}{**Impact \(\times 10^{3}\)**} \\ \cline{2-13}  & **10** & **12.5** & **15** & **17.5** & **20** & **22.5** & **25** & **27.5** & **30** & **32.5** & **35** & **37.5** & **40** \\ \hline
0 & 1.0 & 0.9 & 0.8 & 0.7 & 0.6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0.1 & 0.9 & 0.9 & 0.8 & 0.7 & 0.6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0.2 & 0.5 & 0.5 & 0.5 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0.3 & 0 & 0 & 0 & 0 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0 & 0 & 0 \\
0.4 & 0 & 0 & 0 & 0 & 0.7 & 0.8 & 0.8 & 0.8 & 0.7 & 0 & 0 & 0 \\
0.5 & 0 & 0 & 0 & 0 & 0.7 & 0.85 & 1.0 & 0.85 & 0.7 & 0 & 0 & 0 & 0 \\
0.6 & 0 & 0 & 0 & 0 & 0.7 & 0.8 & 0.8 & 0.8 & 0.7 & 0 & 0 & 0 & 0 \\
0.7 & 0 & 0 & 0 & 0 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0 & 0 & 0 \\
1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0.7 & 0.8 & 0.9 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Fuzzy union matrix \(V\) of all relationships between adverse consequence (C) and impact (I). Figure 6: Ranges of risk impact along with values and their membership degree. the rows' summation and frequency of occurrence is provided. Row No. 9 (Table 9, grey) was selected, as it provided the maximum value for the product. After selecting the subset, row No. 9, that represents the cost-risk impact, the probability of each element in the cost-risk impact subset was calculated, according to Equation (7), as follows:  
\(P(R_{I}=10)=P(R_{I}=12.5)=P(R_{I}=15)=P(R_{I}=35)=P(R_{I}=37.5)=P(R_{I}=40) \frac{0.8}{7.3}=0.1095\),  
\(P(R_{I}=17.5)=P(R_{I}=32.5)=\frac{0.7}{7.3}=0.0958\)  
\(P(R_{I}=22.5)=P(R_{I}=25)=P(R_{I}=27.5)=\frac{0}{7.3}=0\),  
\(P(R_{I}=30)=\frac{0.5}{7.3}=0.0685\),  
\(P(R_{I}=20)=\frac{0.6}{7.3}=0.082\)  
The mean and the variance were calculated, according to Equations (8) and (9), as follows:  
\(\mu_{I}=[(10*0.1095) +(12.5*0.1095)+(15*0.1095)+(17.5*0.0958)\) \(+(20*0.082)+(22.5*0)+(25*0)+(27.5*0)+(30*0.0685)\) \(+(32.5*0.0958)+(35*0.1095)+(37.5*0.1095)\) \(+(40*0.1095)]*10^{3}=24910\) $  
\(\sigma^{2}_{I}=\left[(10^{2}*0.1095) +(12.5^{2}*0.1095)+(15^{2}*0.1095)+(17.5^{2}*0.0958)\right.\)  
\(+(20^{2}*0.082)+(22.5^{2}*0)+(25^{2}*0)+(27.5^{2}*0)+\ (30^{2}*0.0685)\)  
\(+(32.5^{2}*0.0958)+\ (35^{2}*0.1095)+(37.5^{2}*0.1095)\)  
\(+(40^{2}*0.1095)\big{]}*10^{6}-(24910)^{2}=120488150\)  
Finally, the shape parameters of the risk impact Beta distribution were calculated, based on the minimum, maximum, mean, the variance, according to Equations (10) and (11), as follows:  
\(\alpha=\frac{24910-10000}{40000-10000}\bigg{[}\frac{(24910-10000)(40000-2491 0)}{120488150}-1\bigg{]}=0.43\)  
\(\mathrm{s}=0.43\left[\frac{40000-24910}{24910-10000}\right]=0.43\)  
The resulting probability density function of the Beta distribution is presented in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{**Impact \(\times 10^{3}\)**} & \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Method**} \\ \cline{2-2} \cline{10-14}  & & & & & & & & & & & & & & & & \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0.1 & 0.8 & 0.8 & 0.7 & 0.7 & 0.8 & 0.8 & 0.8 & 0.7 & 0.7 & 0.8 & 0.8 & 0.8 & 10.0 & 1.00 \\
0.2 & 1.0 & 0.9 & 0.8 & 0.7 & 0.7 & 0.85 & 1 & 0.85 & 0.7 & 0.7 & 0.8 & 0.9 & 1.0 & 10.9 & 2.18 \\
0.3 &The marginal distribution of the schedule-risk impact was constructed in a manner similar to the marginal distribution of the cost-risk impact. The same root causes were used together with their evaluation in terms of frequency of occurrence (\(F\)) and adverse consequence (C). The mapping values were also the same as the cost-risk impact. Only the lower and upper limits of the distribution were changed. The lower limit was set to 1 day and the upper limit to 10 days.