## 1 Introduction
On the one hand, deterministic approaches, which are based on the intuition and experience of experts, have difficulty calculating the exposure of risk events and determining the appropriate contingency applying to a single crisp value [10]. Moreover, deterministic techniques fail to consider opportunities. On the other hand, in the probabilistic approaches, the value of contingency reserve can be affected by the lack of quality and quantity in historical data, since these techniques significantly rely on historical data [5]. Additionally, probabilistic techniques assume that cost variations are inherently random. Many find it difficult to perform an accurate and precise risk assessment, since the data are either scarce or of low quality [11]. Fuzzy logic, which is based on the fuzzy set theory developed by [PERSON] [12,13], fills the gap for classic techniques as it handles such uncertainties. Applying fuzzy logic, experts are able to assess the probability and impact of events with linguistic terms such as _very low_, _medium_, and _high_, which can be represented by fuzzy numbers [14]. Fuzzy numbers are a special type of fuzzy sets employed to represent the values of real-world parameters when the exact amounts cannot be measured due to a lack of information or knowledge [15]. Reviewing the literature shows that fuzzy logic alone or as integrated with other techniques can be employed to address the limitations associated with classic contingency reserve determination tools and techniques. A hybrid method that integrates the fuzzy set theory with the Monte Carlo simulation, proposed by [PERSON] et al. [10], can handle both random and subjective uncertainties. However, this suggested method fails to determine the individual effect of each risk event, and instead calculates the range estimation of the combined effect of risk events. Another method proposed by [PERSON] and [PERSON] [16] combines the analytic hierarchy process (AHP) with fuzzy set theory to prioritize different risk factors in a building project. However, this proposed fuzzy AHP method [16] fails to deal with definite scales and has a high potential of encountering inconsistencies during pairwise comparison. In another study [17], fault tree analysis (FTA) and fuzzy set theory are integrated for the quantitative assessment of risk events; however,this hybrid approach is unable to handle the drawbacks of the FTA method, which does not model large systems and is inflexible for incorporating later changes. Failure mode and effect analysis (FMEA), AHP, and fuzzy set theory are combined by [PERSON] et al. [15] to assess risks and determine contingency; however, establishing clearly defined terms for its input and output variables requires a significant effort. To capture the interdependencies among different risk events and variables, a fuzzy system dynamic model [18] has been proposed; however, it has difficulty establishing the feedback loops and the mathematical equations. [PERSON] et al. [1] proposed using fuzzy arithmetic-based risk analysis method (FRAM) to fill the gap by addressing the imprecision in measurement and the subjective uncertainty inherent in experts' estimations. FRAM applies a fuzzy arithmetic procedure that solves the problem of substantial reliance on historical data in probabilistic methods. The fuzzy arithmetic procedure employs expert judgment, linguistic scales, and fuzzy numbers resulting in the flexibility of FRAM. Moreover, experts are able to customize linguistic scales and fuzzy numbers for different types of projects and phases. FRAM also considers risk attitude in terms of its contingency calculation and output determination methods. Compared to fuzzy FMEA [15], FRAM does not rely on complicated failure cause-and-effect scenarios in its computation procedures. Moreover, FRAM does not depend on feedback loops with complex mathematical equations when several variables are considered in the fuzzy system dynamics model [18].