## 3 Materials and Methods
### Uncertainty Propagation
This transformation represents the investor's mental attitude towards the success of the investment or his epistemic knowledge about the effects of the investment. Based on the pignistic transformation, the real option's value is lower than the acquisition cost, implying a negative investment decision. However, when the \(\text{ROV}_{\text{amb}}\) and \(\text{ROV}_{\text{cred}}\) transformations are considered, the real option value exceeds the acquisition cost. Therefore, these transformations support the initiation of preparatory steps for building the organic coating plant, as this would create an opportunity to capitalize on future investments in the production of the replacement product. ### Benchmarking Against the Fuzzy Pay-Off Method

To validate our hybrid approach, we additionally computed the fuzzy real options value (FROV) following [PERSON]'s methodology [18]. Given the p-box structure of our results, we implemented two variants of FROV calculation. The classical approach computes ROV as \(\text{A}^{+}\times\text{E}(\text{A}^{+})\), where \(\text{A}^{+}\) represents the success probability and \(\text{E}(\text{A}^{+})\) denotes the expected positive NPV [66]. However, since the p-box provides an interval [lower CDF, upper CDF] at the zero-crossing point, we must choose how to aggregate this uncertainty. We adopted the arithmetic mean of the bounds, yielding \(\text{A}^{+}=(0.3079+0.9549)/2=0.6314\). The alternative approach directly integrates the positive NPVs weighted by their probability differentials, calculated as \(\text{ROV}=\int(\text{NPV}>0)\text{NPV}\times\text{dP}\), where P represents the average of lower and upper CDFs (credibility (\(\beta=0.5\)) or credibility distribution by Liu [59]. The FROV calculations yielded two distinct values: the classical method produced \(\text{ROV}=44.25\) thousand USD, while the integration approach resulted in \(\text{ROV}=26.66\) thousand USD. Remarkably, the integration-based FROV exactly matches our credibility transformation (\(\beta=0.5\)), confirming that both methods represent a balanced risk aversion. This convergence demonstrates that our hybrid methods provide a more general framework, where classical FROV represents a specific case that has been extended in subsequent research. Importantly, Collan's original FROV assumes direct epistemic determination of fuzzy numbers (possibility distributions) without simulation, making it particularly suitable for cases where expert judgment directly defines the uncertainty structure. In contrast, our method combines Monte Carlo simulation with possibilistic parameters, enabling a systematic exploration of both aleatory and epistemic uncertainties. This hybrid framework not only encompasses FROV as a special case but also provides decision makers with multiple transformation approaches (pignistic, ambiguity, credibility) that reflect different attitudes toward uncertainty, thus offering a more comprehensive toolkit for real option valuation under conditions where both types of uncertainty are present. ## 6 Conclusions

This study introduces a novel hybrid approach to real option valuation that integrates probabilistic and possibilistic frameworks within the Dantz-Mathews method (DMM). To the best of our knowledge, this is the first adaptation of the DMM to explicitly accommodate both aleatory and epistemic uncertainties throughout the entire valuation process. By combining random and fuzzy representations of uncertainty, the proposed method provides a more comprehensive and realistic assessment of investment projects operating under deep uncertainty and limited data availability. ### Managerial Insights

By integrating epistemic and aleatory uncertainties, the proposed framework extends the classical option chain and equips decision makers with tools to manage the breadth of uncertainty in line with stakeholder preferences.