# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 6 Discussion  
Difficulty selecting an input distribution that comprehensively represents risk impact [67] has limited the use of probabilistic simulation-based risk assessment in practice--particularly in the planning and construction phase of wind farm projects that are characterized by a lack of historical data. While various methods and recommendations to overcome this challenge have been suggested in literature [67], existing methods are not capable of incorporating a large amount of subjective knowledge in a time-sensitive, practical manner. Indeed, a flexible method that allows the expert to represent their subjective knowledge as a probability distribution has not been described in literature. The proposed approach addresses this limitation by allowing a risk analyst to (1) reliably assess the risk impact based on subjective knowledge and expertise, (2) consider the root causes of a risk factor when calculating its impact, (3) model the dependence between the cost and schedule impacts of a risk factor that has both cost and schedule impact, (4) reduce biases in expert evaluation through the decomposition of a risk factor, and (5) overcome the limitation for using MCS in practice (i.e., the need for historical data) [68].  
This research study proposed a fuzzy-based multivariate analysis approach to address limitations regarding data availability for construction risk assessment of onshore wind projects. The proposed approach was used to successfully solve an illustrative case study of one risk factor common to wind projects. Benefits of proposed method, which included the ability to incorporate detailed subjective knowledge of an expert through the consideration of additional risk factor details, were demonstrated. Furthermore, mapping values provided by experts within the impact range were found to have a considerable impact on the density function of resulting distribution (Figure 11). This was in contrast to minimal impact observed following modifications to the fuzzy membership function of adverse consequence (\(C\)) and frequency of occurrence (\(F\)) for evaluating root causes (Figures 9 and 10). Although the proposed approach shares similarities with previously developed methods (i.e., determination of minimum and maximum impact values), here, the probability density function of the resulting distribution is enhanced by two notable contributions, namely (1) calculating the most probable value from the combination of the root causes and their assessment and (2) mapping values to a risk impact range. Notably, this method can be applied to any type of project characterized by access to limited data and is, therefore, not limited to wind farm construction.  
While the proposed method showed an improvement over previously developed approaches, the findings of this study should be interpreted in consideration of the following limitations. First, the membership function used in this study is a linear triangular and trapezoidal membership function. Other non-linear shapes for membership functions were not investigated and may affect the results. Nevertheless, most membership functions can be accurately represented by either triangular or trapezoidal functions [69]. Second, the risk impact range was divided into three subsets; the examination of more subsets may help experts to better express their belief and confidence about a risk impact. Third, the multivariate approach proposed here will require additional data pooling efforts to ensure that data for all impact dimensions are collected. Models must be built on information that is available and may need to be adjusted in instances where data are lacking, incomplete, or in an inappropriate format for analysis. In this research, a normal copula was used based on the recommendations of previous studies. Future work will include testing multiple copulas, comparing their performance, and developing a complete MCS model for risk assessment considering the developed probability distributions by the proposed method.