## 3 Materials and Methods
### Uncertainty Propagation  
The uncertainty propagation is achieved through hybrid simulation, which combines the Monte Carlo technique with the extension principle of fuzzy set theory. The method has been applied across various fields, such as LCA [47], environmental science [48], and construction [49]. It is based on assumptions of IRS (Independent Random Set) simulation proposed by [PERSON] [50]. Let \(ICF_{t}(q_{i})\), which represents incremental cashflow of investment in year \(t\in T\) (\(T\) being the economic life-cycle of project). To perform hybrid simulation, all fuzzy variables are partitioned into \(\alpha\)-cuts with step \(\Delta\alpha\). Using the \(\alpha\)-levels notation, the constraints on the possibilistic parameters \(\widetilde{\mu}_{i}\) and \(\widetilde{\sigma}_{i}\) can be defined as follows:  
\[\inf\!\left(\widetilde{\mu}_{i,\Delta}\right)\leq\widetilde{\mu}_{i}\leq\sup \!\left(\widetilde{\mu}_{i,\Delta}\right)\!,0\leq\alpha\leq 1,i\in I, \tag{8}\]  
\[\inf\!\left(\widetilde{\sigma}_{i,\Delta}\right)\leq\sigma_{i}\leq\sup\!\left( \widetilde{\sigma}_{i,\Delta}\right)\!,0\leq\alpha\leq 1,i\in I. \tag{9}\]  
Consequently, Equations (8) and (9) constitute additional constraints in \(ICF\) calculation. The realization of \(\hat{Z}_{i,I}\) is sampled from the probability distributions with Cholesky decomposition [51]. Formally, the hybrid simulation is presented in Algorithm 1. The prediction of future parameter values in the simulation loop is based on Equation (4), which captures the hybrid uncertainty propagation through random-fuzzy Geometric Brownian Motion with parameter interdependencies. ```
0:\(J\), \(\Delta\alpha\), \(T\) begin  Set \(j=0\) while\(j\leq J\) \(\alpha=0\) while\(\alpha\leq 1\)  calculate \(\sup\!\left(\right)\) and \(\inf\!\left(\right)\) of \(\widetilde{q}_{i,0,\Delta}\!\left(\widetilde{\sigma}_{i,\Delta}\widetilde{ \mu}_{i,\Delta},0\right)\) (Equation (4))  subject to \(\alpha\)-level constraints \(\widetilde{\sigma}_{i,\Delta},\widetilde{\mu}_{i,\Delta}\) (Equations (8) and (9))  interdependence between \(\widetilde{\mu}_{i,\Delta}\) (Equations (6) and (7)) for\(t=0\) to \(T-1\)  Generate vector of \(\hat{Z}_{i,I+1}\) (MC Sampling with Cholesky decomposition)  Forecast vector of \(\widetilde{q}_{i,I+1,\Delta}\!\left(\widetilde{\sigma}_{i,\Delta},\widetilde{ \mu}_{i,\Delta}^{\top},\hat{Z}_{i+1,I}\right)\) (Equation (4))  Solve \(\widetilde{ICF}_{t,\kappa,j}\!\left[\widetilde{q}_{i,I,\Delta}\!\left( \widetilde{\sigma}_{i,\Delta},\widetilde{\mu}_{i,\Delta}^{\top},\hat{Z}_{i,I} \right)\right]\rightarrow\max\)  subject to  financial constraints  Solve \(\widetilde{ICE}_{t,\kappa,j}\!\left[\widetilde{q}_{i,I,\Delta}\!\left( \widetilde{\sigma}_{i,\Delta},\widetilde{\mu}_{i,\Delta}^{\top},\hat{Z}_{i,I} \right)\right]\rightarrow\min\)  subject to  financial constraints  Compute \(f_{\kappa,j}=\left[\underline{n\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\



The output of hybrid simulation is a set of random intervals \(Result_{symhyb}\). To obtain a pay-off distribution from these results, the Dempster-Shafer theory is applied [52]. Let \(\Theta\) be a non-empty set called a Frame of Discernment, that describes all possible interval values of the NPV (Net Present Value) of the evaluated investment. The simulation results yield a collection of premises (called _focal elements_)

\[Result_{symhyb}=\Big{\{}f_{\alpha,j}\subseteq\Theta\Big{|}f_{\alpha,j}=[ \underline{npv}_{\alpha,j},\overline{npv}_{\alpha,j}],0\leq\alpha\leq 1,j\in J, npv\in R\Big{\}}. \tag{10}\]

Let \(m\) be a function assigning a mass probability to each \(f_{\alpha,j}\), such as \(m\big{(}f_{\alpha,j}\big{)}=(A*J)^{-1}\), where \(A\) is the number of \(\alpha\)-levels used in the simulation.