## 3 Materials and Methods
### Uncertainty Propagation
The value of \(\Phi_{bet}(npv)\) is calculated as [56]

\[\Phi_{bet}(npv)=\sum_{f_{\infty,j}\subseteq\{npv\subseteq\overline{npv}_{ \infty,j}\}}m(f_{\infty,j}). \tag{14}\]

This distribution reflects a neutral attitude, where the decision-maker \"bets\" equally on all possible outcomes within the specified intervals. It serves as the least biased representation of the decision maker's state of knowledge, aligning with the observed outcome of the simulation [56]. In this transformation, the decision maker does not introduce any additional information into the model. The Hurwicz criterion provides a way to interpolate between the best case (upper bound) and the worst case (lower bound) represented by the p-box [58]. By using this criterion, [PERSON] introduced the concept of ambiguity, which represents a state between complete ignorance and perfect knowledge of probability. The ambiguity distribution may be defined as follows:

\[\Phi_{amb}(\kappa,\lambda)=\lambda\cdot\overline{\Phi}^{-1}(\kappa)+(1- \lambda)\cdot\underline{\Phi}^{-1}(\kappa),\kappa\in[0,1]. \tag{15}\]

where \(\lambda\) takes values from 0 to 1 and is called the indicator of the pessimism (optimism) of a decision maker. The optimism index may be treated as a simple measure of belief in expert's opinion. The third approach is a generalization of the credibility theory proposed by [PERSON] [55]. [PERSON] [58] extends the measure proposed by [PERSON] with the coefficient of risk aversion \(\beta\)

\[\Phi_{cred}(npv,\beta)=\beta\cdot\overline{\Phi}((-\infty,npv])+(1-\beta)\cdot \underline{\Phi}(-\infty,npv]). \tag{16}\]

For \(\beta=0.5\), \(\Phi_{cred}(npv,0.5)\) is equal to the [PERSON]'s credibility distribution [59]. Credibility distribution may be treated as decision-maker attitude in the face of uncertainty. Polarization and vagueness represent systematic biases in decision making when confronted with uncertainty. Polarization occurs when probabilities significantly deviate from the average, reflecting overconfidence, while vagueness involves adjusting probabilities closer to the average, often as an overcorrection to polarization or due to manipulation for favorable average outcomes. The [PERSON] criterion and [PERSON]'s credibility distribution are complementary to each other, but they should be used in different scenarios. This approach is similar to those proposed in [18; 35; 36]. The choice of an approach to transform the p-box into a cumulative distribution function depends on the analyst and scenario. Each of the approaches is valuable in different situations (which will be discussed in the case study and in Section 3.4). It is usually necessary to compare all strategies simultaneously. After choosing an approach, the cumulative distribution function \(\Phi_{\{pig,amb,cred\}}\) is transformed into a probability density function \(j\) or pay-off distribution This allows for the specification of a pay-off distribution that enables the determination of the ROV. ### Decision-Theoretic Scope and Limits

The three p-box transformations therefore differ not in the data they require--each always starts from the lower and upper cumulative-distribution bounds--but in the subjective element the decision maker chooses to inject. The pignistic transformation of Smets demands nothing from the decision maker: it functions as a purely neutral reference, consistent with the principle of insufficient reason. Operationally, it is equivalent to a classical, fully stochastic Monte Carlo simulation. One \"average\" (pignistic) CDF is treated as an ordinary probability distribution, from which thousands of NPV paths are sampled. Risk analysis then proceeds exactly as in standard financial studies--without extra subjective parameters and without distinguishing epistemic from aleatory uncertainty, relying solely on conventional sampling from a single common distribution. The ambiguity-\(\lambda\) (Hurwicz) rule introduces the parameter \(\lambda\) as a measure of confidence in the p-box bounds themselves.