## 1 Introduction
Moreover, FRAM addresses the measurement imprecision and the subjective uncertainty of experts' opinions when assessing the probability and impact of risks and opportunities. Finally, FRAM enables risk analysts to estimate contingency at different levels of confidence. FRAM has limitation despite all the mentioned advantages. To implement FRAM in practice, it is necessary to determine the membership functions of linguistic terms pertaining to risk probability, risk impact, opportunity probability, and opportunity impact. FRAM does not propose a systematic method for determining the membership functions of linguistic terms for probability and impact, which are the foundations of its risk analysis process. Moreover, FRAM fails to aggregate the opinions of different subject matter experts (SMEs) about the membership functions of the aforementioned linguistic terms. The membership functions of linguistic terms can vary depending on how the characteristics of each project affect experts' judgements based on their risk attitude, knowledge, experience, and so on. In general, the two main categories of estimating membership functions are expert-driven and data-driven approaches [19]. In expert-driven approaches, the elicitation of membership functions is considered as a process of knowledge acquisition via eligible experts. The most common method in expert-driven approaches is the AHP [20], which enables experts to perform pairwise evaluations of alternatives in order to determine their membership function. Membership functions in data-driven approaches, however, are elicited based on the organization (structuring) of data, such as in fuzzy clustering [21]. There are some limitations to eliciting membership functions through the aforementioned approaches. For example, AHP, as the most common expert-driven method, is not applicable in forming the membership functions of risk analysis linguistic terms in FRAM. To employ AHP, all risks and opportunities must be considered as alternatives for pairwise comparison, which can be impossible or very time-consuming. Moreover, the aggregation of different opinions of SMEs is impossible through AHP. Besides, according to [PERSON] and [PERSON] [19], there are no explicit performance indexes invoked by the AHP approach. However, since industries suffer from accessing qualified data about risk management, data-driven approaches are not applicable in most cases. Moreover, they may cause fuzzy sets that are not semantically meaningful, which means that fuzzy clustering could result in some \"crowded\" fuzzy sets with unclear meaning and they would need to be optimized [19]. These further adjustments during the optimization process could hinder the interpretability aspect. Various optimization methods are employed to adjust fuzzy sets including the simulated annealing algorithm [22], genetic algorithm [23; 24], and tabu search [25]. To address these gaps, the objective of this paper is to propose an interval type-2 fuzzy risk analysis model (IT2 FRAM) that extends FRAM [1] for determining contingency reserve. The proposed method employs interval type-2 fuzzy sets (introduced by [PERSON] [14]) in order to provide a broaderknowledge representation and approximate reasoning for computing with words. Because \"words mean different things to different people\" [26; 27], wider knowledge representation in terms of a spread in membership values through type-2 fuzzy sets is more useful as compared to the standard fuzzy sets [27; 28; 29; 30; 31]. IT2 FRAM aggregates the opinion of SMEs using optimized interval type-2 fuzzy sets. The principle of justifiable granularity [11] is employed for determining the optimized interval type-2 membership functions of risk analysis concepts (i.e., linguistic variables including probability and impact). This principle provides an alternative to clustering methods in constructing information granules based on the criteria of coverage and specificity of data [32].