## 3 Interval Type-2 Fuzzy Risk Analysis Model (IT2 FRAM)
Lower and upper limits of the intervals can be determined by the lowest and the highest height of the triangular membership functions built based on the opinions of different SMEs. Figure 6: Membership functions of the linguistic terms for risk probability. Figure 7: Suggested membership functions of the linguistic term _very low_ for _risk probability_ by seven subject matter experts (SMEs). Then, the optimal lower and upper limits of the interval type-2 fuzzy set for each linguistic term are determined by maximizing the specificity and the coverage of each horizontal interval and simultaneously applying the principle of justifiable granularity (see Section 2.4). The intrinsic contradiction between the maximization of the coverage and the maximization of the specificity results in an optimization problem with a multiplicative form of the objective function. Having these two criteria in mind, a numeric representative, a robust estimator of the sample such as median _med_ (\(D\)), must be selected for each horizontal interval (horizontal information granule). The determination of the upper and lower bound must be realized independently but in the same way. The optimal upper bound must be obtained by maximizing the value of \(V(b)\). in Equation (17). In the same way, the lower bound must be realized based on Equation (18). This process must be repeated for all intervals for each linguistic term. Then, the optimized horizontal intervals of each linguistic term are converted into interval type-2 fuzzy sets. Such a constructed optimized interval type-2 fuzzy set represents the aggregated opinions of all SMEs without the effect of outlier opinions. A statistically representative embedded set of the constructed optimized interval type-2 fuzzy set is used in the next steps of IT2 FRAM. In step 4, the probability and impact of events are assessed. Because of the neutral wording of events, each event can be evaluated simultaneously as a risk and an opportunity. An event allocation matrix (EAM) is employed to determine the relationships among the events and the project's work packages, activities, and tasks on the basis of expert judgment and project context. Events are categorized as local and global. The global events impact several work packages, activities, and tasks and are evaluated for the assigned group. On the contrary, local events can only be assessed individually for each work package, activity, or task and so are assigned individually to individual work packages, activities, and tasks. IT2 FRAM considers two capabilities to improve the accuracy of the result: (1) determining the percentage value (between 0 and 100 percent)f each work package, activity, or task impacted by each local or global event, and (2) determining the portion of the estimated cost of the work package, activity, or task (in terms of a percentage or dollar value) affected by each local or global event. Finally, the contingency of a work package, activity, or task is calculated applying the following fuzzy arithmetic procedure with respect to local events. (1) First, the probability and impact of the risk and opportunity events are evaluated by decision makers or SMEs in terms of the optimized linguistic scales which were established in step 3. Due to the neutral wording, the local events are assessed two times, both as risk and opportunity. (2) Risk and opportunity severities are calculated as a percentage by the multiplication of probability and impact fuzzy numbers. (3) The net severity percentage of each local event is calculated by a summation of risk severity and opportunity severity. (4) The fuzzy number of net severity dollar value is calculated for each local event by the multiplication of its net severity percentage by the affected cost of the work package, activity, or task. (5) The total local contingency in dollars of the work package, activity, or task is calculated by the summation of the net severity (in dollars) of all local events affecting it.