## 4 Implementation of IT2 FRAM in FRA(r) and Discussion  
In this section, a hypothetical case study is presented as an illustration of how to implement IT2 FRAM in practice. FRA(r) is employed to implement the fuzzy arithmetic procedures of IT2 FRAM. The three-level WBS of a hypothetical onshore wind farm project includes six work packages, 11 activities, and 42 tasks. The budget is CAD 554,628,000, and the work packages and their respective costs are presented in Table 1. The default two-level EBS in FRA(r) (see Figure 5) was modified resulting in new EBS with 26 risk and opportunity events, six of which were global and 20 were local. For step 3, linguistic terms and their scales and respective fuzzy sets are established in order to evaluate the probability and impact of the risk and opportunity events. The opinions of different SMEs must be collected for the linguistic terms of probability and impact of events. In this hypothetical situation, it is assumed that there are seven SMEs whose opinions are essential for analyzing the risk and opportunity events. Table 2 summarizes their opinions about the membership function of the linguistic term _very low_ for _risk probability_. Based on the opinion of SME 1, the linguistic term _very low_ for _risk probability_ ranges from 0 to 6 percent with the membership value of 1 in 0 percent. However, for SME 4 this value is different and ranges between 0 and 8 percent. The interval type-2 fuzzy sets are used to consider all the membership functions suggested by different SMEs. As illustrated in Figure 8, an interval type-2 fuzzy set is formed by taking the minimum and maximum of each column in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Work Package Name** & **Total Cost (CAD)** \\ \hline Conceptional phase & $6,000,000 \\ Design & $3,000,000 \\ Contracting/procurement & $399,764,000 \\ Construction & $135,664,000 \\ Handover checklists & $5,000,000 \\ Operation/maintenance & $5,200,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cost of work packages. Figure 8: Membership function of interval type-2 fuzzy set of _very low_ before optimization. Then, by applying the principle of justifiable granularity and taking it as a multiplicative optimization problem, the tradeoff between specificity and coverage are performed. The lower and upper bounds of the interval type-2 fuzzy set membership function of all the intervals are calculated by maximizing the coverage and the specificity of the interval simultaneously (see Section 2.4). Figure 9 shows the tradeoff results for horizontal intervals from 1 to 4 in Figure 8.  
Figure 9: Optimizing the lower and upper bounds of interval type-2 fuzzy sets. Figure 10 shows the optimized interval type-2 fuzzy set membership function of the linguistic term _very low_ for _risk probability_. Based on the theories and concepts associated with interval type-2 fuzzy sets [26; 38; 39], it is evident that interval type-2 fuzzy sets capture more uncertainty than their type-1 counterparts. Thus, interval type-2 membership functions are used to aggregate the opinions of all the SMEs. However, to minimize the effect of outlier opinions, the principle of justifiable granularity is used. These optimized membership functions are then type-reduced to standard membership functions for crisp output calculation. Figure 11 illustrates the process of converting interval type-2 membership function to type-1 membership function based on Sections 2.2 and 2.3. The aim is to find the best fit line passing through these interval fuzzy values. Statistically, the interval fuzzy values are represented by the mean and spread. The interval sets are represented by their corresponding mean points in the \(x\)-\(y\) space, which are (0,1), (1,0.75), (2,0.6), (3,0.45), (4,0.4), (5,0.2), (6,0.05) and (7,0.05). The mean values of all the interval fuzzy sets might not lie on a straight line.