# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 2 Literature Review  
### MCS for Risk Assessment and Input Modeling  
Risk assessment is conducted by evaluating the probability of occurrence and the impact of risk factors to determine their severity on project outcomes. Mathematically, this is accomplished by multiplying probability (\(P\)) by impact (\(I\)) [14; 15] and summing the results to obtain an overall effect of risk factor, \(n\), on project cost and time, as per Equation (1):  
\[\text{S}=\sum_{i=1}^{n}P_{i}*I_{i} \tag{1}\]  
MCS is a probabilistic technique for the quantitative analysis of risks in the construction industry [16]. MCS makes use of probability distributions rather than deterministic values to model the uncertainty associated with a particular input [9]. In a MCS risk analysis experiment, the term \(I_{i}\) in Equation (1) is replaced with a probability distribution function representing the impact on cost or schedule. A MCS experiment for assessing project risks is performed as follows. First, the baseline cost and schedule of the individual activities are prepared [9]. Then, risk factors affecting the project are identified, and the cost impact, schedule impact, probability of occurrence, and affected work-packages are determined for each individual risk factor [9]. If a risk occurs while running the simulation experiment, the cost and schedule impacts are added to the affected activities. An example project, consisting of three activities (A, B, C) and one risk factor affecting Activity A, is presented in Figure 2.  
A key limitation for the practical application of MCS in risk assessment is the development of risk impact probability distributions that are not readily available because of insufficient historical data. These distributions, therefore, must be derived from other existing information and knowledge [17; 18]. The development of such distributions, known as input modeling, is widely discussed in literature due to its impact on simulation outputs and, consequently, on the quality of decisions made based on the simulation results [19]. Input modeling in the case of absent data is known as an elicitation process. Here, expert judgment, recognized as a type of scientific data [20], is elicited and used to construct probability distributions [21]. Elicitation of expert judgment can take three forms, where experts are asked to specify (1) the cumulative distribution function, (2) the density distribution function, or (3) to provide partial information about the distribution such as mean, standard deviation, or several quintiles of the distribution [22]. The elicitation process typically involves the elicitation of the most likely, maximum, and minimum values, or estimating the mean and variance from experts [23]. Several studies have provided a review and guidelines of the statistical methods used to elicit probability distributions [22; 24; 25]. For example, [PERSON] [26] concluded that multiple experts should be asked to provide upper, lower, and most likely values for an uncertain quantity, which can then be fitted to a triangular distribution. In contrast, [PERSON] et al. [21] developed a web-based tool that has five optional methods for the elicitation process, including a roulette method, quartile method, tertile method, probability method, and hybrid method. Another challenge of elicitation for risk assessment is the introduction of biases arising from the inherent subjectivity of risk evaluation. Although experts are considered to be knowledgeable and experienced, their judgment may, as a result of biases, be inaccurate--especially when judging probability [27]. There are two major types of biases: cognitive and motivational. Cognitive biases are defined as \"systematic deviations [of expert evaluation] from logic, probability, or rational choice theory\" [28] and are often associated with heuristic judgment processes [29].