## 3 Materials and Methods
### Uncertainty Propagation
Together, the first two metrics reveal where epistemic uncertainty is most acute, while the latter two quantify how much ignorance remains overall:

\[\Delta_{max}=\underset{x\in\mathbb{R}}{sup}\big{[}\overline{\Phi}(npv)- \underline{\Phi}(npv)\big{]}, \tag{33}\]

\[npv^{*}=\underset{npv\in\mathbb{R}}{arg\max}\big{[}\overline{\[PERSON]}(npv)- \underline{\[PERSON]}(npv)\big{]}, \tag{34}\]

Figure 6.0

Figure 6.1

The quantitative indicators in Table 5 reveal two distinct pictures of epistemic uncertainty, depending on whether statistical dependence among the inputs is acknowledged. In the interdependent scenario, the p-box exhibits its largest vertical separation at an NPV of about \(-\)29,600 EUR; there, the upper and lower cumulative-distribution functions diverge by \(\Delta_{\max}\approx\) 0.726. Although this represents a pronounced local \"spike\" of ignorance, the uncertainty is highly concentrated: the integral of the gap over the whole NPV range--our area metric A--amounts to only 1.8 \(\times\) 10\({}^{5}\), and the average width of the p-box is 0.225. When dependence is ignored, the pointwise gap never exceeds 0.2, and its peak shifts deep into the loss tail at roughly \(-\)133,000 EUR. Yet, the epistemic envelope flares out over a far broader span of outcomes, so the accumulated area under the gap almost doubles to 3.9 \(\times\) 10\({}^{5}\), while the mean width falls to 0.133 because much of the separation occurs where both CDFs are already close to 0 or 1. For decision makers, this contrast is critical. The interdependent model delivers a leaner, but more informative uncertainty statement: it pinpoints a single _NPV_ zone--near the break-even region--where an additional measurement or expert elicitations would have maximum value, while confirming that elsewhere the ignorance is comparatively small. By spreading uncertainty across a wider domain, the _independent_ model inflates the total epistemic mass and may drive overly conservative risk assessments, yet provides little guidance on where further information would be most effective. In short, modeling the true dependence structure reduces the overall epistemic burden by roughly one half and focuses attention on the one part of the outcome space where knowledge is genuinely lacking. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Assumption** & \(\Delta_{\max}\) & \(npv^{\star}\) & **Area A** & **Average Width \(\overline{\Delta}\)** \\ \hline Interdependent (a) & 0.73 & \(-\)29,642 & 1.79 \(\times\) 10\({}^{5}\) & 0.22 \\ Independent (b) & 0.52 & \(-\)133,037 & 3.87 \(\times\) 10\({}^{5}\) & 0.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Quantitative indicators of epistemic uncertainty for independent and interdepended assumption. Aleatoric variability is embedded in the shape of each cumulative-distribution curve inside a probability box. To give it a numerical description, we employ two distribution-agnostic indices: mean slope and inter-quartile range. The mean slope Smean is the average gradient of the \(\Phi_{crd}(npv,0.5)\) over the 10-90% probability range; the larger this value, the steeper the distribution function and the more tightly clustered (less random) the outcomes. The inter-quantile range IQR = npv0.75-npv0.25 measures the width of the central 50% of the distribution and grows in direct proportion to the aleatoric variability of the quantity under study. The results of the analysis are presented in Table 6.

Preserving dependency contracts the output distribution (Table 6): in the dependent-input case Smean increases by roughly 2.5 times, while the central spread IQR drops by more than 50%. Conversely, treating inputs as independent exaggerates aleatoric variability, yielding a flatter p-box and an inflated IQR that can mislead analysts about the system's true randomness. ### Hybrid Real Option Valuation Results

The results presented in Table 7 provide a comprehensive comparison of the various approaches considered.