## 3 Materials and Methods
### Uncertainty Propagation
For prices, the resulting annual volatilities are 15.02% (scrap), 19.04% (CR sheet), 18.67% (HDG sheet) and 12.86% (OC sheet); for apparent consumption they are 10.43% for HDG sheet and 15.03% for OC sheet. Consistent with the treatment of drift, we represent each volatility by a trapezoidal fuzzy number:

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## 5 Discussion

Figure 4 displays the p-box of the project's Net Present Value (NPV) calculated for two dependency assumptions: (a) interdependencies between the input parameters retained and (b) without. Figure 5 shows the corresponding NPV pay-off density functions obtained after three subjective transformations of the p-box--credibility (\(\beta\) = 0.5), ambiguity (\(\lambda\) = 0.5) and pignistic. The cumulative versions of those three distributions are plotted in Figure 6. Using each transformed distribution, the real option value (ROV) is computed with Equation (17); the resulting NPV and ROV statistics are summarized in Table 7.

Figure 4: The Net Present Value (NPV) p-box of the investment project for the construction of a new organic-coated sheet plant considering interdependencies and independence between parameters. Figure 3: Trapezoidal fuzzy numbers representing annual drift and volatility for key input parameters (a) prices (b) apparent consumption. ### Decomposition of Epistemic and Aleatory Uncertainty

Figure 4 reports the resulting probability box, \(\left[\underline{\Phi},\overline{\Phi}\right]\), for two cases: (a) when interdependencies among input parameters are included and (b) when they are ignored. P-boxes let us disentangle the effects of the two main kinds of uncertainty. Relaxing the dependency assumptions (b) widens the gap between the lower and upper cumulative curves, signaling greater epistemic uncertainty. The vertical distance between these curves therefore measures epistemic uncertainty, whereas the shape of each curve captures the aleatory--or purely random--component. Recognizing the difference helps decision makers see whether risk stems mainly from limited knowledge and expert-judgment bias (epistemic) or from intrinsic variability in historical data (aleatory). The wider this band, the weaker the hard-data support and the higher the risk that the ex-post outcome will deviate from expectations. For managers, this translates into a higher risk premium (in practice: a higher hurdle IRR or a lower valuation multiple), and a higher value of managerial flexibility--a wide p-box signals that the real option to defer, stage, or abandon the investment may dominate the classical NPV. Figure 5: Comparison of NPV pay-off functions for credibility (\(\beta\) = 0.5), ambiguity (\(\lambda\) = 0.5), and pignistic transformations. Figure 6: The transformation of the NPV p-box into subjective cumulative pay-off distributions: a comparison of pignistic, ambiguity (\(\lambda\) = 0.5), and credibility (\(\beta\) = 0.5) approaches. To translate this graphical separation into decision-relevant, quantitative indicators of the epistemic component, we introduce four complementary metrics, reported in Table 5. Maximum-vertical-gap Equation (33) captures the worst-case epistemic uncertainty by taking the single largest difference between the two CDFs [63]. The location of the maximum-gap Equation (34) records the value on the horizontal axis at which that largest difference occurs, thereby pinpointing where knowledge is weakest. Area-metric Equation (35) measures the total mass of epistemic uncertainty by integrating the vertical gap across the entire domain [64]. Average-width Equation (36) normalizes the area by the length of the domain, yielding the typical breadth of the p-box and allowing scenarios defined on different ranges to be compared directly [65].