## 3 Materials and Methods
### Uncertainty Propagation
The belief regarding the occurrence of a given NPV can be described by a pair of probabilities (belief):

\[\underline{P}(npv=npv_{0})=\sum_{npv\subseteq f_{\alpha,j}}m(f_{\alpha,j}) \tag{11}\]

and (plausibility)

\[\overline{P}(npv=npv_{0})=\sum_{npv\cap f_{\alpha,j}\
eq\varnothing}m(f_{ \alpha,j}). \tag{12}\]

This pair defines an interval commonly interpreted as the upper and lower probability bounds of \(P(npv=npv_{0})\). Let \(\underline{\Phi},\overline{\Phi}:R\rightarrow[0,1]\) be non-decreasing functions such as \(\forall_{x}\overline{\Phi}(x)\geq\underline{\Phi}(x)\). An interval

\[\big{[}\underline{\Phi},\overline{\Phi}\big{]}=\big{\{}\Phi:R\rightarrow[0,1 ],\text{ non}-\text{decreasing}\big{|}\forall_{x\in R}\underline{\Phi}(npv) \leq\Phi(npv)\leq\overline{\Phi}(npv)\big{\}} \tag{13}\]

is called probability box (\(p\)-_box_), and functions \(\overline{\Phi}\,\mathbf{i}\,\underline{\Phi}\) are called upper and lower cumulative pay-off distribution. The exact form of the cumulative distribution function \(\Phi^{*}(npv)\) is unknown, but it is known to lie between these bounds. It can be shown [46] that if the \(npv\subseteq R\), then \(\underline{\Phi}(x)=\underline{P}(npv\in(-\infty,x])\), \(\overline{\Phi}(x)=\overline{P}(npv\cap(-\infty,x]\
eq\varnothing))\). Thus, the set of premises \(Result_{symhyb}\) can be transformed into a \(p\)-_box_ representing the upper and lower bounds of the cumulative distribution for NPV outcomes. The p-box can be interpreted in two ways: as bounds on the cumulative probability associated with a specific NPV, or as the tightest possible bounds on the distribution function given the available information. It is important to note that the p-boxes obtained from hybrid simulation are non-parametric--they are constructed without any assumptions about the shape of the underlying distribution of the variable of interest. [PERSON]'s experiments [53] show that choices in risky situations may differ from those in situations where information about chances are imprecise. [PERSON] introduced the concept of ambiguity, which is a state between complete ignorance and full knowledge of probability. Ambiguity arises from low information reliability or information inconsistencies [54]. Mathematically, ambiguity is reflected in situations where a decision maker deals with risk described using upper and lower distribution functions [55]. The decision maker aims to reduce ambiguity by finding a transformation that allows them to convert the p-box into a subjective cumulative pay-off distribution function \(\Phi(npv)\). The construction of a subjective probability distribution is, in a sense, imposed on the decision maker, who, upon encountering the ambiguity of simulation outcomes, is compelled to express their belief in the form of a probability distribution, even if his or her actual knowledge may not sufficiently justify the selection of one distribution over another. ### Transformation of p-Box into Subjective Pay-Off Distribution

Different approaches can be applied to choose the subjective cumulative distribution function \(\Phi(npv)\) depending on the decision maker's preferences and the decision context. The pignistic transformation approach was proposed by [PERSON] [56; 57], and is grounded in [PERSON]'s principle of insufficient reason. It is identical to the so-called Shapley value used in cooperative game theory as a fairness principle for sharing benefits across members of coalitions. It treats each focal element \(f_{\infty,j}\) from the p-box result set as a set of pay-offs with a unit mass probability \(m\big{(}f_{\infty,j}\big{)}\). The expected value of each interval is derived by assuming a uniform distribution across the interval \([\underline{npv}_{\infty,j},\ \overline{npv}_{\infty,j}]\).