## 4 Implementation of IT2 FRAM in FRA(r) and Discussion
To find the best fit linear equation, we solve linear equations between (0,1) and the mean point of  
Figure 11: Converting the optimized interval type-2 fuzzy set membership function of _very low_ to a type-1 fuzzy set. Figure 10: Optimized interval type-2 fuzzy set membership function of _very low_ after optimization. each interval set that provides an intercept value at the \(x\) axis. Solving the linear equation between points (0,1) and (1,0.75) yields the intercept on the \(x\) axis at \(x=4\). We draw other lines by solving the line equations between (0,1) and the other mean points. The line between (0,1) and (7,0.05) gives the intercept value on the \(x\)-axis at \(x=7.37\). Thus, all the calculated intercepts on the \(x\)-axis are at \(x=\{4,5,5.45,\ 6.67,\ 6.25,\ 6.31,\ 7.37\}\). This represents a region of uncertainty between \(x=4\) and \(x=7.37\), which is a direct consequence of the differences in opinion of the SMEs. This region of uncertainty forms the FOU of the interval type-2 fuzzy set with the triangular membership function. The interval type-2 set can be modeled as described in Section 2.3. Statistically, the word being modeled should be contained within \(\left[x_{m}^{l},x_{m}^{r}\right]\). Here, \(x_{m}^{l}\) is the mean of the left end points of the interval type-2 fuzzy set and \(x_{m}^{r}\) is the mean of the right end points. Assuming that the end point uncertainties disappear, then the above interval type-2 fuzzy set reduces to a type-1 fuzzy set with \(a=b=x_{m}^{r}=x_{m}^{l}\). The mean of these points is at \(x_{m}^{r}=5.86\) with the standard deviation \(s=1.046\). The resulting type-1 fuzzy set is highlighted with red color. Similarly, the optimized membership functions are obtained for other linguistic terms. Figure 12 illustrates the optimized membership functions of all the linguistic terms for risk probability. In step 4, as illustrated in Figure 13, the identified local and global risk and opportunity events are assigned to work packages, activities, and tasks, and the probability and impact of these events are assessed on the basis of linguistic terms (type-1 fuzzy sets determined in step 3). Figure 12: Optimized membership functions of the linguistic terms for risk probability. Figure 13: Assigning and assessing the local events. Finally, fuzzy arithmetic is employed to calculate the work package and project contingency reserve using FRA(r). IT2 FRAM provides the user with the choice of standard fuzzy arithmetic or extended fuzzy arithmetic, the latter of which uses four different t-norms. The resulting fuzzy value of contingency reserve can be presented both as an interval value using the confidence level and as a crisp value based on the selection of a single value (defuzzification) method (Figure 14). The defuzzified single value of the total project contingency reserve based on the COA is CAD 7,307,032, and at an \(\alpha\)-cut level of 0.50 there is a confidence level (possibility degree) of 0.5 that the project contingency will be between CAD 932,573 and CAD 9,890,760. Figure 15 is the project summary graph report created by FRA(r) that provides a visualization of the defuzzified contingency values. This report is only available when a single defuzzified value is being used as the output in FRA(r). Figure 14: Assigning and assessing the local events. Figure 15: Local, global, and total contingency values. In Table 3, several contingency reserve determination methods are summarized and compared based on ten criteria. IT2 FRAM provides a unique structured way to develop, optimize, and aggregate the linguistic terms. IT2 FRAM addresses the limitations of the other methods of contingency reserve determination. The interval type-2 fuzzy sets in IT2 FRAM capture more uncertainties, provide better knowledge representation, and consider several experts' opinions. The principle of justifiable granularity optimizes these interval type-2 fuzzy sets by maximizing the performance index of two criteria--coverage and specificity--which helps minimize the effects of outlier opinions of SMEs.