## 3 Materials and Methods
### Uncertainty Propagation
Practically, \(\lambda\) quantifies the decision maker's belief in the epistemic reliability of expert data: values near 1 signal full trust that experts have captured the phenomenon accurately (the lower the value, the more the optimistic bound dominates), whereas values near 0 correspond to extreme skepticism, privileging the upper, more pessimistic CDF. Thus, the Hurwicz rule acts as a rapid \"sensitivity gauge\" to the quality of expert assessments--ideal for \"what-if analysts are off by 30%?\" scenarios. The credibility-\(\beta\) (Dubois-Liu) transformation assumes the p-box itself is already accepted and lets the decision maker embed a personal attitude toward risk inside those bounds. Here, \(\beta\) is a left-tail slider: low values (\(\beta\to 0\)) emphasize concern for extreme losses, while \(\beta\to 1\) reflects greater tolerance of risk. This approach is suited to questions such as \"how worried am I about the left tail of NPV?\" or whenever an individual risk profile must be imposed after the p-box has been deemed trustworthy. In short: \(\lambda\) encodes trust in expert data, \(\beta\) encodes risk aversion. Choose \(\lambda\) when exploring information quality; choose \(\beta\) when the information is stable and the key issue is how the decision maker values risky tails. ### Data-Mathews Method in Random-Fuzzy Environment

In its classical form, the DM method relies on Monte Carlo simulation to model the uncertainty associated with investment project. The analysis begins by identifying the parameters that drive profitability and describing their uncertainty in terms of costs and revenues. A Monte Carlo run then generates the probability distribution of the project's Net Present Value (NPV). In the case of complex computational models, it may be necessary to implement parallel computing to speed up the Monte Carlo simulation [60]. The project is subsequently interpreted as a call option: negative NPVs are set to zero, reflecting the real-world practice of abandoning unprofitable ventures. The ROV is calculated as

\[ROV_{hyb}=U_{r}*E(npv>0)=\frac{\int_{0}^{\infty}\varphi(npv)dnpv}{\int_{-\infty }^{\infty}\varphi(npv)dnpv}\int_{0}^{\infty}npv\ \ \varphi(npv)dnpv, \tag{17}\]

where U\({}_{r}\) is the probability that the investment will be profitable (_success ratio_) and \(E(npv>0)\) denotes the expected value of the positive part of the NPV distribution. Figure 1 illustrates how the Datar-Mathews method can be applied in a hybrid (probabilistic-possibilistic) environment. ### Comparison with Existing Approaches

Before we proceed to the analysis of the numerical case, we wanted to describe the similarities and differences between our approach and the fuzzy pay off method for real option valuation (FROV) [18]. Both approaches are based on the Datar-Matthews method but differ in their assumptions. Although both approaches have their roots in possibility theory [58], we employ a fuzzy-random approach, whereas FPOV utilizes pure fuzzy. In both approaches, the value of a real option is calculated by multiplying the \"centroid\" of the positive side of the NPV distribution by the proportion of the area of the NPV distribution that represents positive outcomes. The distinction lies in the fact that in our approach, we determine this value based on the subjective probability distribution of NPV derived from the p-box, while FROV utilizes fuzzy NPV for this purpose. Additionally, we use the expected value as the centroid, while variants of FPOV utilize possibilistic mean, center of gravity, or other interpretations. In essence, we have control over the shape of the NPV distribution, whereas FPOV interprets its expected value based on the distribution. It should be emphasized that using the method proposed by us, ROV, is possible with both methods--ours and [PERSON]'s. This is made possible by relying on hybrid simulation, which yields a p-box. It can be demonstrated that this structure can be transformed into both a probability distribution and a fuzzy random variable, thereby resulting in a fuzzy NPV [61; 62].