# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 3 Materials and Methods
### Data Processing  
#### 3.2.1 Marginal Distributions  
Fuzzy set theory is then used to scientifically quantify the combined influence of the root causes to derive the marginal probability distribution functions that will be input into the MCS models. The workflow for deriving the marginal distributions is illustrated in Figure 4, and a step-by-step procedure of the proposed method is detailed in Supplement SA. This study has chosen a Beta distribution to represent the marginal distribution of schedule or cost impact. A Beta distribution was selected because (1) its specialized form, the PERT distribution, is commonly applied in risk assessment studies [54], (2) it is a bounded distribution with finite limits, making it intuitively plausible to many decision-makers in risk analysis [54], (3) it is flexible, taking any shape according to its shape parameters (i.e., \(\beta\) and \(\alpha\)) [55; 56], and (4) it is frequently used to describe variability or uncertainty over a fixed (i.e., bounded) range [18]. The steps for integrating subjective knowledge into the Beta distributions are detailed as follows. Figure 3: Research methodology. A membership function is usually used in fuzzy sets to represent the relationship between a range of possible values and a linguistic term [57]. The membership function assigns a membership degree, in which the relation of these values to the linguistic term is defined within the interval [0, 1], representing no and full membership, respectively. In general, the membership of a fuzzy set, \(A\), in the case of a discrete universe of discourse, \(X\), is usually expressed as follows:  
\[A\ =\ \sum\
olimits^{n}_{1=1}\frac{\mu_{i}}{x_{i}}=\ \frac{\mu_{1}}{x_{1}}+\ \frac{\mu_{2}}{x_{2}}+\ldots\ldots+\frac{\mu_{n}}{x_{n}}=\ \sum\mu_{A}(x)/x, \tag{2}\]  
where \(\mu_{i}=\mu_{i}(x_{i})\) is the degree of belonging of element \(x_{i}\) to set \(A\), and \(n=\) the number of elements in set \(A\). In construction applications, triangular and trapezoidal fuzzy numbers are usually used to represent the membership function. The development of the membership function, including the selection of linguistic terms and the range of values that a linguistic term represent, is determined by the analyst. Once root causes are identified and assessed linguistically, the quantification analysis can be conducted. The fuzzy logic quantification algorithm combines the adverse consequence (C) (i.e., the contribution of the root cause to the overall cost or schedule-risk impact as a percentage) and the frequency of occurrence (\(F\)) for each root cause scenario. This is accomplished by calculating a fuzzy relation matrix between (\(F\)) and (\(C\)), resulting in \(R\) (\(F\), \(C\)), which is the Cartesian product \(F\times C\). The elements of \(R\) (\(F\), \(C\)) are computed as follows:  
\[\mu_{R}(x_{i},y_{i})=min[\mu_{F}(x_{i}),\mu_{C}(y_{i})], \tag{3}\]  
where \(x_{i}\) = an element of universe \(X\); \(y_{i}\)= an element of universe \(Y\); \(\mu_{R}(x_{i},y_{i})\) = the membership value of element \((x_{i},y_{i})\) in the fuzzy relation \(R\); \(min\) = the minimum values of both elements \(x_{i}\) and \(y_{i}\); \(\mu_{F}(x_{i})\) = the membership value of element \(x_{i}\) in fuzzy set F; and \(\mu_{C}(y_{i})\) = the membership value of element \(y_{i}\) in fuzzy set \(C\). Once all of the fuzzy relation matrices have been calculated, the fuzzy logic quantification algorithm calculates the union matrix of all the fuzzy relation matrices, thereby representing the combined effect of all root cause scenarios. The union of two fuzzy relations, for example \(S\) and \(Z\), is denoted by \(S\bigcup Z\), and the membership function is calculated as follows:  
\[\mu_{S\cup Z}(x_{i},y_{i})=max[\mu_{s}(x_{i},y_{i}),\mu_{z}(x_{i},y_{i})], \tag{4}\]  
Figure 4: Quantification steps for marginal distribution. where \(max\) = the maximum value of both relations \(s\) and \(z\).