## 3 Interval Type-2 Fuzzy Risk Analysis Model (IT2 FRAM)  
IT2 FRAM is a multi-step model employing fuzzy arithmetic to analyze risk and opportunity events to determine contingency reserve for construction projects. Figure 3 presents the five steps of IT2 FRAM and their outputs. In steps 1 and 2, the work, cost, and risk breakdown structures (WBS, CBS, and RBS) are determined. In step 3, the membership functions of the linguistic terms for risks and opportunities are determined using interval type-2 fuzzy sets and the principle of justifiable granularity as explained in Sections 2.2-2.4. Then, in step 4, the identified risks and opportunities are assessed by SMEs using linguistic terms and their related fuzzy numbers. Finally, the contingency reserve can be calculated in step 5 using fuzzy arithmetic as explained in Section 2.1. In step 1, the WBS and CBS are developed. The WBS is the foundation of IT2 FRAM assuming that each project contains up to a three-level WBS, namely work package, activity, and task. As an example, Figure 4 shows a three-level WBS of a wind farm project illustrated in FRA(c). The CBS must be developed after establishing the WBS to determine the cost of the work packages, activities, and tasks. Developing the event breakdown structure (EBS) and the identification of potential risk and opportunity events are step 2 in IT2 FRAM. Since there is no consensus on the standard categorization of risk and opportunity events [16], different combinations of risk and opportunity identification methods can be employed, ranging from information-gathering methods to analysis-based techniques. [PERSON] and [PERSON] [43] conducted a systematic review and content analysis of 130 papers from journals with high  
Figure 3: Steps of IT2 FRAM (modified from [1]). impact factors in the construction engineering and management area published between 1990 and 2017. They propose eleven categories of risk and opportunity events, which are considered as the default template of IT2 FRAM. These event categories are depicted in Figure 5: resource-related, management, technical, construction, site conditions, contractual and legal, economic, financial, environmental, social, political, and health and safety. In step 3, the linguistic terms and scales must be established and optimized to assess the probability and impact of the events. Triangular or trapezoidal fuzzy numbers represent linguistic  
Figure 4: Three-level work breakdown structure (WBS) comprising work packages, activities, and tasks in FRA®. Figure 5: Two-level event breakdown structure (EBS) in FRA®. terms. According to [PERSON] and [PERSON] [8], [PERSON] [44], and [PERSON] [45], triangular and trapezoidal shapes are the most common shapes for fuzzy numbers that have supports with the open intervals of real numbers. Triangular fuzzy numbers are a special case of trapezoidal fuzzy numbers. In IT2 FRAM and according to [PERSON] [46], the probability and impact of events are commonly determined by five linguistic terms namely, _very low, low, medium, high,_ and _very high._ A sample of triangular membership functions for risk and opportunity probability with respective linguistic terms is presented in Figure 6.  
Different membership functions for probability and impact must be formed and aggregated to benefit from the knowledge and experience of all decision makers and SMEs on a project. Type-1 fuzzy sets project only one crisp number for the membership degree of each linguistic term, while interval type-2 fuzzy sets return an interval. Therefore, interval type-2 fuzzy sets are preferable, and they provide more information than type-1 fuzzy sets. An interval type-2 fuzzy set also covers all opinions. Figure 7 shows a hypothetical case study of various membership functions of the linguistic term _very low_ for _risk probability_ as determined by seven SMEs. For instance, in Figure 7, based on the opinion of SME 1, the risks with very low probability are those risks with an occurrence probability of less than 6 percent with the full membership degree in 0.