# Fuzzy-Based Multivariate Analysis for Input Modeling of Risk Assessment in Wind Farm Projects
## 7 Conclusions  
Input modeling is the first step in MCS-based risk assessment of construction projects. Typically derived using historical project data, the use of MCS-based risk assessment in newer projects has been limited. Because of this, the subjective knowledge of risk analysts at a detailed level has not been properly considered when developing probability distributions for Monte Carlo risk assessment in construction. Therefore, this research tried to address the limitation of input modeling for risk assessment when historical data are lacking with only detailed subjective knowledge available. In particular, this research tried to devise a method that can make used of the experts' subjective knowledge with minimized biases when deriving a probability distribution for a risk factor impact. This research has developed a method capable of capturing and modeling expert subjective knowledge by deriving a flexible generalized Beta distribution through fuzzy logic. Distribution for risk factor impacts with either one type of impact (i.e., cost or time) or both (i.e., cost and time) can be developed with the proposed method. Risk factors that have both cost and schedule impact can be modeled using bivariate distributions by considering the correlation between cost and schedule impact of a risk factor, with the fitted generalized Beta distribution representing the marginal distributions for schedule and cost-risk impact. The method was applied to assess one risk factors common to wind farm construction. Sensitivity analysis was performed, and the model was found to be sensitive to changes in the mapping values of a risk factor impact range. In contrast, changing the shape of the membership function did not affect the resulting distribution. We found that the decomposition of a risk factor into its root causes and decomposition of the risk impact into smaller ranges allows the experts to depict their subjective knowledge more accurately, comprehensively, and with lower biases. An implication of these findings is that both expert subjective knowledge and risk impact decomposition should be taken into account when deriving input distributions for MCS risk assessment. This will enable the application of quantitative methods in the early stages of a project, thereby improving decision-making processes. The proposed approach enables the effective, representative, and comprehensive elicitation of the probability distribution for input modeling in MCS-based risk assessment when only a detailed level of subjective knowledge exists.  
**Supplementary Materials:** The following are available online at [[http://www.mdpi.com/1999-4893/13/12/325/s1](http://www.mdpi.com/1999-4893/13/12/325/s1)]([http://www.mdpi.com/1999-4893/13/12/325/s1](http://www.mdpi.com/1999-4893/13/12/325/s1)), Supplement SA: Detailed methodology for deriving a marginal Beta distribution from expert opinion of risk impact on cost and schedule, Supplement SB: Derivation of the marginal distribution using _SimphonyProject.NET._  
**Author Contributions:** Conceptualization, E.M. and S.A.; methodology, E.M. and S.A.; validation, E.M. and S.A.; formal analysis, E.M.; investigation, E.M. and P.J.; writing--original draft preparation, E.M.; writing--review and editing, E.M., P.J., and S.A.; supervision, S.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This study is supported by Future Energy Systems research as part of the Canada First Research Excellent Fund (CFREF FES-T11-P01).  
**Acknowledgments:** The authors would like to acknowledge [PERSON] for her assistance with manuscript editing and composition.  
**Conflicts of Interest:** The authors declare no conflict of interest.