## 3 Materials and Methods
### Uncertainty Propagation
The credibility-\(\beta\) transformation assumes information reliability and turns \(\beta\) into a risk-attitude slider; pushing \(\beta\) toward zero amplifies concern for extreme losses and triggers an earlier sale, whereas \(\beta\) near one tolerates volatility and favors staying invested. Together, \(\lambda\) and \(\beta\) convert the abandonment option from a passive safety net into an active steering mechanism that aligns divestment timing with both information quality and the firm's risk appetite. ### Practical Advantages and Competitiveness of the Hybrid Random-Fuzzy DMM

One of the key strengths of the presented hybrid DMM approach lies in its ability to differentiate between uncertainty arising from inherent randomness (aleatory) and that caused by incomplete knowledge (epistemic). This distinction enables decision makers not only to model variability in market conditions and input parameters more accurately but also to explicitly recognize the role of subjective expert judgment in scenarios where historical data are scarce or unreliable. By avoiding the premature aggregation of these two types of uncertainty during simulation, the method allows for a transparent and traceable propagation of uncertainty throughout the valuation process. In contrast to traditional real option valuation models, which often rely solely on probabilistic assumptions, the hybrid DMM introduces an additional analytical layer that reflects the decision maker's attitude toward uncertainty. Through the application of three different strategies for transforming p-box results into subjective probability distributions--pignistic, ambiguity-based, and credibility-based transformations--the method provides flexibility in capturing diverse decision-maker preferences, including optimism, pessimism, and varying degrees of confidence in expert assessments. This feature empowers practitioners to adjust the valuation according to their individual risk tolerance, epistemic beliefs, and strategic posture. Compared with classical Black-Scholes-based valuations, the proposed hybrid DMM offers three tangible benefits. First, it accommodates expert-elicited confidence intervals where historical statistics are weak--typical of long-horizon investments in metallurgy, energy, or infrastructure--thereby extending option pricing to domains that lie beyond the reach of standard BS inputs. Second, by maintaining cross-parameter interdependency and actively controlling the width of the p-box, it yields narrower and more realistic ranges for real option value (ROV), mitigating both excessive optimism and overly conservative discount \"penalties.\" Third, its p-box transformations let decision makers swiftly \"tune\" the valuation to their own risk profile--something a single-point BS estimate cannot replicate. Importantly, when epistemic uncertainty collapses to zero, the hybrid DMM convergesto the conventional Black-Scholes outcome, ensuring full backward compatibility while delivering richer, decision-relevant information whenever uncertainty runs deep. One of the key advantages of this hybrid DMM is its ability to effectively separate aleatoric and epistemic uncertainties in the decision-making process. As research shows [32], understanding the nature of uncertainty can help in designing interventions to improve judgment accuracy. Investors who see market uncertainty as stemming from a lack of knowledge or skill tend to seek expert guidance and react swiftly to new information, while those who view market uncertainty as driven by random processes are more inclined to diversify their portfolios to mitigate risk [39]. Unlike other methods, such as fuzzy real options valuation (FROV), this approach avoids aggregating these uncertainties during propagation through simulation. Additionally, it introduces an additional layer that explicitly requires decision makers to address simulation results by considering potential issues like experts' over-optimism, reliance on heuristics, or other cognitive biases.