## 2 Fuzzy Sets in Real Options Pricing  
Real options thinking begins with the realization that any irreversible investment embeds an option to defer, expand, contract, or abandon, and that the economic value of that flexibility depends entirely on how we describe the unknown future. By following the successive languages scholars have used to encode \"the unknown,\" this seeks to move beyond single-distribution probability. [PERSON]'s classic distinction [21] between measurable risk and non-probabilistic uncertainty had already warned that statistical regularities cannot exist for truly novel ventures, and that entrepreneurial profit is therefore a reward for bearing the latter. [PERSON]'s Bayesian decision-analysis calculus [22] translated [PERSON]'s insight into an operational tool: managers could express private beliefs as subjective probabilities and utilities, mapping even sparse evidence into coherent decision trees. [PERSON] and [PERSON]'s Investment Under Uncertainty [23] established the dominant paradigm in real asset valuation by applying the Black-Scholes framework under the assumption of Geometric Brownian Motion, known probability distributions, constant volatility, perfect liquidity, and complete markets--requiring extensive historical data and the ability to assign precise probabilities to all future states. However, practical implementation revealed significant limitations. The mathematical complexity proved inaccessible to many practitioners, while the assumption of known probability distributions became problematic for innovative projects lacking historical data. Over the past two decades, researchers have responded by pluralizing the very notion of uncertainty. Fuzzy real options [18] fundamentally shift from randomness to imprecision and vagueness as primary uncertainty characteristics. It replaces complex stochastic processes with expert-generated scenarios (optimistic, most likely, pessimistic) and employs possibilistic statistics rather than probabilistic moments. The Datar-Mathews [1] method calculates option value as the average of positive outcomes from a project's NPV distribution, effectively creating a truncated pay-off distribution. This preserves familiar DCF modeling while incorporating real options thinking. Beyond that described above, various paradigms have emerged to address specific characteristics of uncertainty. These include possibility theory, which interprets unknowns as degrees of plausibility rather than frequencies; interval and gray number analysis, which represent parameters as bounded ranges in the absence of reliable probabilities; and approaches such as info-gap theory, Dempster-Shafer evidence theory, and robust optimization, which prioritize resilience to ignorance over precise prediction. However, the mathematical complexity and interpretive challenges associated with these methods have limited their adoption in practice. As a result, probability theory and possibility theory based on fuzzy numbers remain the most popular frameworks, as they offer a more intuitive and accessible foundation for decision makers. Fuzzy set theory has emerged as a prominent tool for addressing uncertainty in real option valuation, particularly when precise probabilistic data is unavailable. Many papers explore the possibility of using fuzzy sets or possibility theory to real options valuations. [PERSON] et al. [24] developed a methodology for valuing options on R&D projects, in which future cash flows were estimated by trapezoidal fuzzy numbers. They presented a fuzzy mixed integer programming model for the R&D optimal portfolio selection problem. [PERSON] and [PERSON] [8] use possibility theory to study fuzzy real option valuation. The present values of expected cash flows and expected costs are estimated by authors also by trapezoidal fuzzy numbers. They determined the optimal exercise time by the help of possibilistic mean value and variance of fuzzy numbers. [PERSON] [25] used the fuzzy real options valuation model in a real investment project from the energy sector.