## 3 Materials and Methods  
### Modeling Uncertainty by Correlated Random-Fuzzy Geometric Brownian Motion (GBM)  
The standard method for evaluating real options assumes that the value of the parameter will fluctuate according to a Geometric Brownian Motion (GBM) over a defined period. The parameter \(q\) follows GBM, if it satisfies the diffusion equation [40,41,42,4]:  
\[dq_{t}=\mu q_{t}dt+\sigma q_{t}dW_{t}, \tag{1}\]  
where \(W_{t}\sim N(0,t)\), and \(\mu\) and \(\sigma\) are drift and volatility. Applying [PERSON]'s Lemma and Euler discretization over interval \(\Delta t=[t,t-1]\), we can write the formulas for the prediction \(q_{t}\)[4,40,41,42,4]  
\[q_{t}=q_{t-1}(\mu,\sigma,Z_{t})=q_{t-1}exp((\mu-\frac{\sigma^{2}}{2})\Delta t +\sigma Z\sqrt{\Delta t}), \tag{2}\]  
where \(Z\) is an independent, normal random variate. Frequently, the available statistical data are inadequate or unsuitable for estimating parameters such as \(\mu\) and \(\sigma\). In scenarios where statistical data are insufficient, these parameters are estimated or refined by subject matter experts. Consequently, \(\mu\) and \(\sigma\) become imprecise variables, aptly represented using fuzzy numbers. In our approach, when confronted with insufficient statistical data, we address the uncertainty in drift and volatility by representing them as possibilistic parameters: \(\widetilde{\mu},\widetilde{\sigma}\), described by fuzzy numbers. Consequently, Equation (2) can be rewritten as  
\[\widetilde{q}_{t}=\widetilde{q}_{t-1}(\widetilde{\mu},\widetilde{\sigma},Z_{ t})=\widetilde{q}_{t-1}exp[(\widetilde{\mu}-\frac{\widetilde{\sigma}^{2}}{2}) \Delta t+Z_{t}\widetilde{\sigma}\sqrt{\Delta t}], \tag{3}\]  
where \(\widetilde{q}_{t}\) is random-fuzzy variable. This method allows for the differentiation between uncertainties stemming from aleatory and epistemic sources to provide more realistic representation of GBM. It is noteworthy that when predicting the fluctuation of \(q\), the literature typically assumes that all values in triple \(\left(\widetilde{\mu},\widetilde{\sigma},Z_{t}\right)\) are represented by fuzzy numbers, resulting in a fuzzy variable as the outcome of the process. However, in our scenario, we encounter a combination of possibilistic \(\widetilde{\mu},\widetilde{\sigma}\) and probabilistic \(Z_{t}\). To better understand what is meant by a possibilistic distribution, it is helpful to consider its roots in possibility theory. As described by [PERSON] and [PERSON] [43], a possibility distribution--also known as a fuzzy number or a nested interval--represents an alternative to probability for encoding expert opinions. It maps intervals of parameter values to degrees of possibility (\(\mu\)), which indicate how plausible each value is. These intervals, called \(\alpha\)-cuts, provide bounds that reflect varying levels of confidence. For instance, the support of the distribution includes all possible values (\(\mu=0\)), while the core identifies the most plausible value (\(\mu=1\)). This allows for a flexible and intuitive way to express epistemic uncertainty when precise probabilities are difficult to assign. The versatility of Equation (3) means that it can effectively represent various parameters such as unit prices, raw materials prices, or market volumes. These parameters are often correlated and deeply impact the final valuation outcome. Therefore, incorporating these interdependencies in the model is crucial for achieving precise and reliable results. To incorporate this correlation, we use the method described in [44]. Let us suppose that we analyze \(I\) parameters. Additionally, it is assumed that there may be defined subsets \(I_{k}\) of correlated parameters \(k\in K\). Each \(I_{k}\) includes a subset of correlated parameters.