Integrating Probability and Possibility Theory: A Novel Approach to Valuing Real Options in Uncertain Environments

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This paper introduces a novel hybrid approach to real option valuation that can be particularly useful in industries where expert judgment plays a significant role and historical data is limited or unreliable—such as the metallurgical, energy, and infrastructure sectors. The method enables decision makers to rigorously incorporate both objective data and subjective beliefs into investment analysis, providing more realistic valuation under deep uncertainty. It is especially applicable to long-term capital-intensive projects where strategic flexibility and uncertainty about future market conditions are crucial.

Abstract

The article presents a new method for evaluating investment projects in uncertain conditions, assuming that uncertainty may have two origins: aleatory (related to randomness) and epistemic (due to incomplete knowledge). Epistemic uncertainty is rarely considered in investment analysis, which can result in undervaluing the future opportunities and risks. Our contribution is built around a correlated random–fuzzy Geometric Brownian Motion, a hybrid Monte Carlo engine that propagates mixed uncertainty into a probability box, combined with three p-box-to-CDF transformations (pignistic, ambiguity-based and credibility-based) to reflect decision-maker attitudes. Our approach utilizes the Datar–Mathews method (DM method) to gather relevant information regarding the potential value of a real option. By combining probabilistic and possibilistic approaches, the proposed valuation model incorporates hybrid Monte Carlo simulation and a random–fuzzy Geometric Brownian Motion, considering the interdependence between parameters. The result of the hybrid simulation is a pair of upper and lower cumulative probability distributions, known as a p-box, which represents the uncertainty range of the Net Present Value (NPV). We propose three transformations of the p-box into a subjective probability distribution, which allow decision makers to incorporate their subjective beliefs and risk preferences when performing real option valuation. Thus, our approach allows the combination of objective available information about valuation of investment with the decision maker’s attitude in front of partial ignorance. To demonstrate the effectiveness of our approach in practical scenarios, we provide a numerical illustration that clearly showcases how our approach delivers a more precise valuation of real options.

1. Introduction

The Datar–Mathews method (DM method) is a particularly interesting but still relatively underexplored area in simulation-based real option valuation (ROV). The method proposed by Datar and Mathews in 2004 provides an easy way to determine the real option value of a project by using the average of positive outcomes [1,2]. It uses Monte Carlo simulation to propagate uncertainty and to create pay-off distribution of an investment alternative. The resulting pay-off distribution is flexible, allowing for any compounding interval and free selection of discount rates [3]. Additionally, the results of the DM method converge with the Black & Scholes result when the Monte Carlo simulation is run enough times. In this article, we propose a method for performing the DM method in a hybrid environment, which refers to a scenario where the parameters utilized for evaluating the investment cannot be precisely determined.

In real option valuation, the accuracy of input parameters greatly influences the financial terms, such as cash flow and profit. Analysts frequently estimate parameters using past data and simultaneously make subjective guesses about future economic changes [4,5,6,7]. It results in the additional type of uncertainty in describing parameters, which, apart from random uncertainty, arises from a lack of knowledge or inherent vagueness [7]. The exclusive reliance on probability theory in these situations can result in a false sense of precision and an incorrect assumption that outcomes are inherently predictable and repeating in nature [8]. There is therefore a natural need to distinguish between different types of uncertainty in ROV.

Entrepreneurial activities encompass different levels of informational reliability, giving rise to situations where risks are linked to the probabilities of expected outcomes, while uncertainty arises when these probabilities cannot be precisely determined. It follows from the fact that valuing investment projects usually involves both types of uncertainty—aleatoric and epistemic. Aleatoric uncertainty is observed variability while epistemic uncertainty arises from incomplete knowledge. To effectively capture non-probabilistic uncertainty, it becomes crucial to incorporate to the model both probabilistic and possibilistic approaches [9]. Experts’ opinions or imprecise estimates may be introduced to the model in the form of possibilistic distribution as they better express vagueness, imprecision, or ambiguity [10]. In the field of ROV, the probabilistic Black–Scholes (B-S) model provides robust framework to model aleatoric uncertainty of many variables such as market size or product prices [5,6,11,12,13,14]. Therefore, a combination of both probabilistic and possibilistic approaches is necessary for accurately modeling and evaluating the uncertainties that exist in ROV.

Several factors led us to the writing of this article. The first is that a pure probabilistic approach is difficult to use in valuing investment projects using the DM method due to the inherent vagueness that investors encounter. The most widely used probabilistic real option valuation techniques—such as the analytical Black–Scholes model [15], the Cox–Ross–Rubinstein binomial lattice [16], Monte Carlo-based extensions of the Datar–Mathews framework [1], and decision-tree analysis [17]—require the analyst to specify every source of uncertainty with a precise probability distribution derived from rich historical data. This assumption is problematic when knowledge is incomplete, and parameters are only available as expert judgements or coarse intervals (epistemic uncertainty). In such cases the purely probabilistic scheme collapses into an ‘illusion of exactness’: it treats ignorance as if it were randomness, bans interval-valued inputs, and therefore often understates risk or overstates confidence.

The second factor is the significant reliance on expert estimates in the metallurgical industry, where investment lifecycles typically span 10–12 years. Moreover, the relevance of historical data diminishes rapidly after 2–3 years, making it impractical to assert any predictive value of time series beyond 5 years [18]. The final reason is the interdependency among different products’ demands. This connection creates a challenge when trying to forecast the demand for a single product in isolation, as it depends on the demand for other products within the same product line [19].

This study introduces a novel method for valuing investment projects that accounts for both imprecise and random uncertainties, while explicitly modeling the interdependencies among parameters within the Brownian Motion framework. Conventional ROV models tend to overlook the relationships between parameters, which can result in inaccurate forecasts and less precise valuation [4,7,19,20]. While some research has examined the correlation between parameters when using a probabilistic approach [5], current methods have limitations in fully addressing the complexities of aleatory and epistemic uncertainty. By incorporating these correlations in the DM, the proposed approach can provide a more realistic and precise evaluation of investment projects, while also accounting for the vagueness and randomness of the decision-making context.

The structure of the article is as follows: Firstly, we conducted a comprehensive review of real options in a hybrid environment. Next, we introduced the method of propagating uncertainty in such an environment using Monte Carlo simulation. Subsequently, we explored strategies for selecting subjective probability distributions that consider the decision maker’s perspective when faced with incomplete knowledge. To illustrate the effectiveness of our approach, we provided a clear example demonstrating how it enhances the precision of real options valuation.

2. Fuzzy Sets in Real Options Pricing

Real options thinking begins with the realization that any irreversible investment embeds an option to defer, expand, contract, or abandon, and that the economic value of that flexibility depends entirely on how we describe the unknown future. By following the successive languages scholars have used to encode “the unknown,” this seeks to move beyond single-distribution probability.

Frank H. Knight’s classic distinction [21] between measurable risk and non-probabilistic uncertainty had already warned that statistical regularities cannot exist for truly novel ventures, and that entrepreneurial profit is therefore a reward for bearing the latter. Ronald A. Howard’s Bayesian decision-analysis calculus [22] translated Knight’s insight into an operational tool: managers could express private beliefs as subjective probabilities and utilities, mapping even sparse evidence into coherent decision trees. Dixit and Pindyck’s Investment Under Uncertainty [23] established the dominant paradigm in real asset valuation by applying the Black–Scholes framework under the assumption of Geometric Brownian Motion, known probability distributions, constant volatility, perfect liquidity, and complete markets—requiring extensive historical data and the ability to assign precise probabilities to all future states. However, practical implementation revealed significant limitations. The mathematical complexity proved inaccessible to many practitioners, while the assumption of known probability distributions became problematic for innovative projects lacking historical data.

Over the past two decades, researchers have responded by pluralizing the very notion of uncertainty. Fuzzy real options [18] fundamentally shift from randomness to imprecision and vagueness as primary uncertainty characteristics. It replaces complex stochastic processes with expert-generated scenarios (optimistic, most likely, pessimistic) and employs possibilistic statistics rather than probabilistic moments. The Datar–Mathews [1] method calculates option value as the average of positive outcomes from a project’s NPV distribution, effectively creating a truncated pay-off distribution. This preserves familiar DCF modeling while incorporating real options thinking.

Beyond that described above, various paradigms have emerged to address specific characteristics of uncertainty. These include possibility theory, which interprets unknowns as degrees of plausibility rather than frequencies; interval and gray number analysis, which represent parameters as bounded ranges in the absence of reliable probabilities; and approaches such as info-gap theory, Dempster–Shafer evidence theory, and robust optimization, which prioritize resilience to ignorance over precise prediction. However, the mathematical complexity and interpretive challenges associated with these methods have limited their adoption in practice. As a result, probability theory and possibility theory based on fuzzy numbers remain the most popular frameworks, as they offer a more intuitive and accessible foundation for decision makers.

Fuzzy set theory has emerged as a prominent tool for addressing uncertainty in real option valuation, particularly when precise probabilistic data is unavailable. Many papers explore the possibility of using fuzzy sets or possibility theory to real options valuations. Carlsson et al. [24] developed a methodology for valuing options on R&D projects, in which future cash flows were estimated by trapezoidal fuzzy numbers. They presented a fuzzy mixed integer programming model for the R&D optimal portfolio selection problem. Carlsson and Fuller [8] use possibility theory to study fuzzy real option valuation. The present values of expected cash flows and expected costs are estimated by authors also by trapezoidal fuzzy numbers. They determined the optimal exercise time by the help of possibilistic mean value and variance of fuzzy numbers. Garcia [25] used the fuzzy real options valuation model in a real investment project from the energy sector. In Garcia’s study, the theoretical framework proposed by Carlsson and Fullér [8] is applied to analyze the timing and selection of power plants from among various investment alternatives. Further, the study conducted by Allenotor and Thulasiram [26] employs a fuzzy trinomial real options model for the valuation of grid resources, substantiating the model’s applicability through empirical analysis. Additionally, Tao [27] formulated an extensive methodology that integrates fuzzy risk assessment with a real options strategy, specifically for the evaluation of information technology investments in nuclear power stations. Zeng et al. [28] compared the application of traditional Net Present Value method with real options in investment evaluations, analyzed the uncertainty of a power grid investment project, and discussed how to make the investment decision when investment cost and cash flow are both fuzzy numbers. Kahraman and Ucal [29] used the certainty equivalent approach for real options valuation in an oil investment with fuzzified data. Ho and Liao [30] propose a fuzzy approach for investment project valuation in uncertain environments from the aspect of real options. Lee et al. [12] utilized the principles of fuzzy decision theory combined with Bayes’ theorem to assess fuzziness in option analysis practices.

However, it should be emphasized that none of the above-mentioned studies consider the problem defined in the first part of this article. In a systematic review of the interactions between fuzzy set theory and option pricing conducted in [9], the fuzzy-random approach to option pricing was represented in approximately 35% of 240 reviewed papers. However, the fuzzy real option was only represented by 13 papers. To the knowledge of the authors, there are few approaches to the analysis of real options in a hybrid environment. In [7,31], a valuation of the portfolio of real options under both types of uncertainty is presented. In [32], the generalized fuzzy–stochastic multi-mode real options model is developed. The authors of [33] introduce a fuzzy–random extension of the Ho–Lee term structure model by incorporating fuzzy volatility into a binomial lattice framework, enhancing the pricing of interest rate derivatives like caplets under uncertainty.

The equivalent of the DM method in a fuzzy environment is the “fuzzy pay-off” method proposed by Collan (a good summary can be found in [18]) (FROV). Collan in [34] compares the use of the Datar–Mathews method and the fuzzy pay-off method in the analysis of investment cases with different levels of complexity. The results of the two methods are compared through numerical illustrations, showing that FROV is sufficient for problems with low complexity, but the DM method is better suited for more complex ones. In [35], Borges states that there are situations in which the original “fuzzy pay-off” calculates the real option value to be negative, what is theoretically incorrect. He proposed the center of gravity approach to real option valuation (PROBROV). Stoklasa in [36,37] presents a possibilistic method for real ROV and compares the aforementioned methods and fully possibilistic fuzzy ROV (POSROV).

Many prevailing theories in judgment and decision making treat uncertainty as a singular concept. However, in our approach, we differentiate between two distinct types of uncertainty: epistemic (events that are potentially knowable) and aleatory (inherently random events). It is crucial to emphasize that, within our framework, this differentiation leans more towards a philosophical perspective rather than a purely pragmatic one. This approach has been gaining popularity [38,39].

All the literature analyzed in Section 2 was retrieved from Web of Science Core Collection, Scopus, and Google Scholar (search window 2000–2024).

3. Materials and Methods

3.1. 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 [4,40,41,42]:

$$d{q}_t=\mu q_{t}dt+\sigma q_{t}d{W}_t,$$

(1)

where $W_t~N(0,t)$, and μ and σ are drift and volatility. Applying Ito’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]

$$q_t=q_{t-1}({\mu ,\sigma ,Z_t)=q}_{t-1}\mathit{exp}((\mu -{\displaystyle \frac{{\sigma }^2}{2}})\Delta t+\sigma Z\sqrt{\Delta t}),$$

(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: $\tilde{\mu },\tilde{\sigma }$, described by fuzzy numbers. Consequently, Equation (2) can be rewritten as

$${\tilde{q}}_t={\tilde{q}}_{t-1}(\tilde{\mu }, \tilde{\sigma },Z_t)={\tilde{q}}_{t-1}\mathit{exp}[(\tilde{\mu }-{\displaystyle \frac{{\tilde{\sigma }}^2}{2}})\Delta t+Z_t\tilde{\sigma }\sqrt{\Delta t}],$$

(3)

where ${\tilde{q}}_t$ 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 $(\tilde{\mu }, \tilde{\sigma },Z_t)$ 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 $\tilde{\mu }, \tilde{\sigma }$ 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 Dubois and Prade [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 (μ), which indicate how plausible each value is. These intervals, called α-cuts, provide bounds that reflect varying levels of confidence. For instance, the support of the distribution includes all possible values (μ = 0), while the core identifies the most plausible value (μ = 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. Therefore, Equation (3) can be expressed as the following forecasting model for parameter q_i

$${\tilde{q}}_{i,t}={\tilde{q}}_{i,t-1}\mathit{exp}[({\tilde{\mu }}_i-{\displaystyle \frac{{{\tilde{\sigma }}_i}^2}{2}})\Delta t+{\widehat{Z}}_{i,t}{\tilde{\sigma }}_i\sqrt{\Delta t}], i\in I_k.$$

(4)

Here, it is assumed that ${\tilde{\mu }}_i$ and ${\tilde{\sigma }}_i$ are possibilistic parameters, ${\widehat{Z}}_{i,t}$ is a correlated random normal variate and ${\tilde{q}}_t$ is a random–fuzzy variable. To define ${\widehat{Z}}_{i,t}$, the Cholesky decomposition may be utilized to decompose the correlation matrix (as described in [45]). Possibilistic distribution may be viewed as a nested set of intervals (called α-cut)

$${\tilde{\mu }}_{\alpha }=\left\{x\in R|{\tilde{\mu }}_{\alpha }\ge \alpha \right\},0\le \alpha \le 1.$$

(5)

Interdependence between ${\tilde{\mu }}_i$ and ${\tilde{\mu }}_j$,$i,j\in I_k$ can be modeled by interval regression [40]. Below, we present the equations that define the relationships between these variables, using the notation of α-levels [38]

$${\tilde{\mu }}_{i,\alpha }\ge \mathrm{i}\mathrm{n}\mathrm{f}\left({\mathrm{a}}_1^{\mathrm{i},\mathrm{j}}\right)\times {\tilde{\mu }}_{j,\alpha }+\mathrm{i}\mathrm{n}\mathrm{f}\left({\mathrm{a}}_2^{\mathrm{i},\mathrm{j}}\right), i,j\in I_k,$$

(6)

$${\tilde{\mu }}_{i,\alpha }\le \mathrm{s}\mathrm{u}\mathrm{p}\left({\mathrm{a}}_1^{\mathrm{i},\mathrm{j}}\right)\times {\tilde{\mu }}_{j,\alpha }+\mathrm{s}\mathrm{u}\mathrm{p}\left({\mathrm{a}}_2^{\mathrm{i},\mathrm{j}}\right), i,j\in I_k,$$

(7)

where $a_1^{i,j}$, $a_2^{i,j}$ represent the coefficients of the interval regression equations, which are pivotal in establishing the dependencies between parameters ${\tilde{\mu }}_i$ and ${\tilde{\mu }}_j$. These coefficients $a_1^{i,j}$ and $a_2^{ij}$ can be ascertained using the method proposed by Hladik and Černy (crisp input–crisp output variant) [46].

Interval regression, applied here to quantify the interdependence among imprecise drift parameters, yields coefficient bands rather than point slopes (Equations (6) and (7), and results in Tables 3 and 4). The lower edge of each band reflects the most conservative transmission of an economic shock—e.g., a sudden rise in scrap-steel prices—into the revenue stream, while the upper edge captures the most favorable response. Interpreted as interval elasticities, these bands can be propagated directly through the hybrid simulation. Moreover, because the bands are fitted simultaneously for mutually dependent parameters, the method curbs the arbitrariness of expert elicitation: permissible fuzzy drifts are no longer assigned in isolation but must satisfy data-driven dependence constraints, ensuring internal consistency across correlated inputs and sharpening subsequent valuation results.

3.2. 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 Baudrit [50].

Let ${ICF}_t\left(q_i\right),$ 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 α-cuts with step $∆\alpha$.

Using the α-levels notation, the constraints on the possibilistic parameters ${\tilde{\mu }}_i$ and ${\tilde{\sigma }}_i$ can be defined as follows:

$$\mathrm{i}\mathrm{n}\mathrm{f}\left({\tilde{\mu }}_{i,\alpha }\right)\le {\tilde{\mu }}_i\le \mathrm{s}\mathrm{u}\mathrm{p}\left({\tilde{\mu }}_{i,\alpha }\right),0\le \alpha \le 1,i\in I,$$

(8)

$$\mathrm{i}\mathrm{n}\mathrm{f}\left({\tilde{\sigma }}_{i,\alpha }\right){\le \sigma }_i\le \mathrm{s}\mathrm{u}\mathrm{p}\left({\tilde{\sigma }}_{i,\alpha }\right),0\le \alpha \le 1,i\in I.$$

(9)

Consequently, Equations (8) and (9) constitute additional constraints in ICF calculation. The realization of ${\widehat{Z}}_{i,t}$ 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.

Algorithm 1 Calculation of pay-off distribution in hybrid environment

Require: J, $∆\alpha$, T begin  Set j = 0  while $j\le J$   $\alpha =0$   while $\alpha \le 1$    calculate sup() and inf() of ${\tilde{q}}_{i,0,\alpha }$(${\tilde{\sigma }}_{i,\alpha }$,${\tilde{\mu }}_{i,\alpha }$,0) (Equation (4))     subject to      α-level constraints ${\tilde{\sigma }}_{i,\alpha },{\tilde{\mu }}_{i,\alpha }$ (Equations (8) and (9))      interdependence between ${\tilde{\mu }}_{i,\alpha }$ (Equations (6) and (7))    for t = 0 to T−1     Generate vector of ${\widehat{Z}}_{i,t+1}$ (MC Sampling with Cholesky decomposition)     Forecast vector of ${\tilde{q}}_{i,t+1,\alpha }({\tilde{\sigma }}_{i,\alpha },{\tilde{\mu }}_{i,\alpha }, {\widehat{Z}}_{i+1,t})$ (Equation (4))     Solve ${\overline{ICF}}_{t,\propto ,j}\left[{\tilde{q}}_{i,t,\alpha }\right({\tilde{\sigma }}_{i,\alpha },{\tilde{\mu }}_{i,\alpha }, {\widehat{Z}}_{i,t}\left)\right]\to \mathrm{m}\mathrm{a}\mathrm{x}$      subject to       financial constraints     Solve ${\underset{\_}{ICF}}_{t,\propto ,j}\left[{\tilde{q}}_{i,t,\alpha }\right({\tilde{\sigma }}_{i,\alpha },{\tilde{\mu }}_{i,\alpha }, {\widehat{Z}}_{i,t}\left)\right]\to \mathrm{m}\mathrm{i}\mathrm{n}$      subject to       financial constraints    Compute $f_{\propto ,j}=[{\underset{\_}{npv}}_{\propto ,j}\underset{\_}{,} {\overline{npv}}_{\propto ,j}]$    $\alpha =\alpha +∆\alpha$   j = j + 1 end

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)

$${Result}_{symhyb}=\left\{f_{\alpha ,j}\subseteq \Theta |f_{\propto ,j}=[{\underset{\_}{npv}}_{\alpha ,j}\underset{\_}{,} {\overline{npv}}_{\propto ,j}],0\le \alpha \le 1,j\in J,npv\in R\right\}.$$

(10)

Let m be a function assigning a mass probability to each $f_{\propto ,j}$, such as $m\left(f_{\propto ,j}\right)={(A\ast J)}^{-1}$, where A is the number of $\alpha$-levels used in the simulation. The belief regarding the occurrence of a given NPV can be described by a pair of probabilities (belief):

$$\underset{\_}{P}({npv=npv}_0)=\sum _{npv\subseteq f_{\propto ,j}}m(f_{\propto ,j})$$

(11)

and (plausibility)

$$\overline{P}(npv={npv}_0)=\sum _{npv\cap f_{\propto ,j}\ne \varnothing }m(f_{\propto ,j}).$$

(12)

This pair defines an interval commonly interpreted as the upper and lower probability bounds of $P\left({npv=npv}_0\right)$.

Let $\underset{\_}{\Phi },\overline{\Phi }:R\to \left[\mathrm{0,1}\right]$ be non-decreasing functions such as ${\forall }_x\overline{\Phi }\left(x\right)\ge \underset{\_}{\Phi }\left(x\right)$. An interval

$$[\underset{\_}{\Phi },\overline{\Phi }]=\{\Phi :R\to [\mathrm{0,1}], \mathrm{non}-\mathrm{decreasing}|{\forall }_{x\in R}\underset{\_}{\Phi }\left(npv\right)\le \Phi \left(npv\right)\le \overline{\Phi }\left(npv\right)\}$$

(13)

is called probability box (p-box), and functions $\overline{\Phi }$ i $\underset{\_}{\Phi }$ are called upper and lower cumulative pay-off distribution. The exact form of the cumulative distribution function ${\Phi }^{\ast }\left(npv\right)$ is unknown, but it is known to lie between these bounds. It can be shown [46] that if the $npv\subseteq R$, then $\underset{\_}{\Phi }\left(x\right)=\underset{\_}{P}\left(npv\in \left(-\infty ,x\right]\right), \overline{\Phi }\left(x\right)=\overline{P}(npv\cap (-\mathrm{\infty },\mathrm{x}]\ne \varnothing ))$. 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.

Ellsberg’s experiments [53] show that choices in risky situations may differ from those in situations where information about chances are imprecise. Ellsberg 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 $\Phi \left(npv\right)$. 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.

3.3. Transformation of p-Box into Subjective Pay-Off Distribution

Different approaches can be applied to choose the subjective cumulative distribution function $\Phi \left(npv\right)$ depending on the decision maker’s preferences and the decision context.

The pignistic transformation approach was proposed by Smets [56,57], and is grounded in Laplace’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 $f_{\propto ,j}$ from the p-box result set as a set of pay-offs with a unit mass probability $m\left(f_{\propto ,j}\right)$. The expected value of each interval is derived by assuming a uniform distribution across the interval $[{\underset{\_}{npv}}_{\propto ,j}, {\overline{npv}}_{\propto ,j}]$. The value of ${\Phi }_{bet}\left(npv\right)$ is calculated as [56]

$${\Phi }_{bet}(npv)={\sum }_{f_{\propto ,j}\subseteq \{npv\le {\overline{npv}}_{\propto ,j}\}}m(f_{\propto ,j}).$$

(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, Jeffrey 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:

$${\Phi }_{amb}\left(\kappa ,\lambda \right)=\lambda \cdot {\overline{\Phi }}^{-1}\left(\kappa \right)+\left(1-\lambda \right)\cdot {\underset{\_}{\Phi }}^{-1}\left(\kappa \right),\kappa \in \left[\mathrm{0,1}\right].$$

(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 Liu [55]. Dubois [58] extends the measure proposed by Liu with the coefficient of risk aversion β

$${\Phi }_{cred}(npv,\beta )=\beta \cdot \overline{\Phi }\left(\right(-\infty ,npv\left]\right)+(1-\beta )\cdot \underset{\_}{\Phi }(-\infty ,npv]).$$

(16)

For β = 0.5, ${\Phi }_{cred}(npv,0.5)$ is equal to the Liu’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 Hurwitz criterion and Liu’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.

3.4. 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-λ (Hurwicz) rule introduces the parameter λ as a measure of confidence in the p-box bounds themselves. Practically, λ quantifies the decision maker’s belief in the epistemic reliability of expert data: values near 1 signal full trust that experts have captured the phenomenon accurately (the lower the value, the more the optimistic bound dominates), whereas values near 0 correspond to extreme skepticism, privileging the upper, more pessimistic CDF. Thus, the Hurwicz rule acts as a rapid “sensitivity gauge” to the quality of expert assessments—ideal for “what-if analysts are off by 30%?” scenarios.

The credibility-β (Dubois–Liu) transformation assumes the p-box itself is already accepted and lets the decision maker embed a personal attitude toward risk inside those bounds. Here, β is a left-tail slider: low values (β → 0) emphasize concern for extreme losses, while β → 1 reflects greater tolerance of risk. This approach is suited to questions such as “how worried am I about the left tail of NPV?” or whenever an individual risk profile must be imposed after the p-box has been deemed trustworthy.

In short: λ encodes trust in expert data, β encodes risk aversion. Choose λ when exploring information quality; choose β when the information is stable and the key issue is how the decision maker values risky tails.

3.5. Datar–Mathews Method in Random–Fuzzy Environment

In its classical form, the DM method relies on Monte Carlo simulation to model the uncertainty associated with investment project. The analysis begins by identifying the parameters that drive profitability and describing their uncertainty in terms of costs and revenues. A Monte Carlo run then generates the probability distribution of the project’s Net Present Value (NPV). In the case of complex computational models, it may be necessary to implement parallel computing to speed up the Monte Carlo simulation [60]. The project is subsequently interpreted as a call option: negative NPVs are set to zero, reflecting the real-world practice of abandoning unprofitable ventures.

The ROV is calculated as

$${ROV}_{hyb}=U_r\ast E\left(npv>0\right)={\displaystyle \frac{{\int }_0^{\infty }\phi \left(npv\right)dnpv}{{\int }_{-\infty }^{\infty }\phi \left(npv\right)dnpv}}{\int }_0^{\infty }npv \phi (npv)dnpv,$$

(17)

where U_r is the probability that the investment will be profitable (success ratio) and $E(npv>0)$ denotes the expected value of the positive part of the NPV distribution. Figure 1 illustrates how the Datar–Mathews method can be applied in a hybrid (probabilistic–possibilistic) environment.

3.6. Comparison with Existing Approaches

Before we proceed to the analysis of the numerical case, we wanted to describe the similarities and differences between our approach and the fuzzy pay off method for real option valuation (FROV) [18]. Both approaches are based on the Datar–Matthews method but differ in their assumptions. Although both approaches have their roots in possibility theory [58], we employ a fuzzy–random approach, whereas FPOV utilizes pure fuzzy.

In both approaches, the value of a real option is calculated by multiplying the “centroid” of the positive side of the NPV distribution by the proportion of the area of the NPV distribution that represents positive outcomes. The distinction lies in the fact that in our approach, we determine this value based on the subjective probability distribution of NPV derived from the p-box, while FROV utilizes fuzzy NPV for this purpose. Additionally, we use the expected value as the centroid, while variants of FPOV utilize possibilistic mean, center of gravity, or other interpretations. In essence, we have control over the shape of the NPV distribution, whereas FPOV interprets its expected value based on the distribution.

It should be emphasized that using the method proposed by us, ROV, is possible with both methods—ours and Collan’s. This is made possible by relying on hybrid simulation, which yields a p-box. It can be demonstrated that this structure can be transformed into both a probability distribution and a fuzzy random variable, thereby resulting in a fuzzy NPV [61,62]. Further research is needed to determine the relationship between our approach and the approach proposed in Collan’s work.

4. Results

This example considers a steel manufacturer that is weighing the construction of a new production line for organic-coated sheets. The firm currently produces conventional sheets but now has an opportunity to adopt organic-coating technology. Management must determine whether building the new facility is economically justified. Before construction can begin, the company must prepare project documentation and complete environmental assessments to obtain the required permits; we assume these preparatory activities will cost USD 20,000.

To assist in the decision-making process, the company conducts Net Present Value (NPV) and real option analyses. These analyses involve assessing the future cash flows with the help of experts. The key question is whether the company should invest the funds to obtain a return on investment.

Model for Estimating the Value of Project in Hybrid Environment

In this section, we introduce the NPV model that underpins the real option valuation of a hypothetical production system (see Figure 2). The model extends the framework described in [58].

The project under consideration is the construction of a new plant for manufacturing an organic-coated (OC) steel sheet. The OC sheet is produced from hot-dip-galvanized (HDG) sheet and enjoys both a broader range of applications and a higher unit value. The main input to the process is cold-rolled (CR) sheet, which is first converted to HDG sheet; part of that output is sold directly, while the rest is further processed into OC sheet and sold. Both the HDG and OC stages generate steel-scrap by-products that are likewise sold.

To apply the Datar–Mathews (DM) method we must derive φ (npv), the NPV density obtained from a Monte Carlo simulation of the new OC plant. This calls for incremental cash-flow modeling: cash flows in the investment scenario (which includes the OC facility) are compared with those in the baseline scenario (the status quo without the new plant). The project NPV can then be computed with Equations (18)–(32).

$$NPV={\displaystyle \frac{{\sum }_{t=1}^{T}IC{F}_t}{{\left(1+WACC\right)}^t}}-{CF}_0, t=1,\dots ,T$$

(18)

$$\begin{gathered} \mathrm{IC}{\mathrm{F}}_{\mathrm{t}}=\left[{\left(PA{T}_t+D{A}_{\mathit{OC},t}+D{A}_{\mathit{HDG},t}+\Delta W{C}_t\right)}_{\mathit{inv}}-{\left(\mathit{PA}{T}_t+D{A}_{\mathit{HDG},t}+\Delta W{C}_t\right)}_{\mathit{base}}\right] \\ +1_{t=T}\left(R{V}_{T,\mathit{inv}}-R{V}_{T,\mathit{base}}\right) \end{gathered}$$

(19)

$$\begin{gathered} {PAT}_t^{scen}={Rev}_t^{scen}-{TC}_t^{scen}-\max \left({Rev}_t^{scen}-{TC}_t^{scen},0\right)\ast TA, \\ scen\in \{base,inv\} \end{gathered}$$

(20)

$${Rev}_t^{inv}={P_t^{OC}SV}_t^{OC}+{P_t^{HDG}SV}_t^{HDG,inv}$$

(21)

$${Rev}_t^{base}={SV}_t^{HDG,base}{P}_t^{HDG}$$

(22)

$${SV}_t^{OC,inv}=\min \left({SF}_t^{OC},{CAP}^{OC}\right)$$

(23)

$${SV}_t^{HDG,base}=\min \left({SF}_t^{HDG},{CAP}^{HDG}\right)$$

(24)

$${SF}_t^{prod}={ACF}_t^{prod}{MS}^{prod},prod\in \{OC,HDG\}$$

(25)

$${TC}_t^{base}={SV}_t^{HDG,base}{(VC\_RM}_t^{HDG}+{VC\_O}^{HDG})+{DA}^{HDG}+{FC}^{HDG}$$

(26)

$$\begin{gathered} {TC}_t^{inv}={SV}_t^{OC,inv}\left({VC\_RM}_t^{OC}+{VC\_O}^{OC}\right)+{SV}_t^{HDG,inv}{(VC\_RM}_t^{HDG}+{VC\_O}^{HDG}) \\ +{DA}^{HDG}+{FC}^{HDG}+{DA}^{OC}+{FC}^{OC} \end{gathered}$$

(27)

$${VC\_RM}_t^{HDG}={PUC}^{HDG}{P}_t^{CR}+\left({1-PU}^{HDG}\right)P_t^{scrap}$$

(28)

$${VC\_RM}_t^{OC}={PUC}^{OC}{P}_t^{HDG}-\left({1-PUC}^{OC}\right)P_t^{scrap}$$

(29)

$${∆NWC}_t^{scen}={NWC}_t^{scen}-{NWC}_{t-1}^{scen},t=1,\dots ,T$$

(30)

$$\begin{gathered} {NWC}_t^{scen}={\displaystyle \frac{{Rev}_t^{scen}}{CTR}}+{\displaystyle \frac{{Rev}_t^{scen}}{DTR}}+{\displaystyle \frac{{TC}_t^{scen}-\sum _{prod}{DA}_t^{prod}}{ITO}} \\ -{\displaystyle \frac{{TC}_t^{scen}-\sum _{prod}{DA}_t^{prod}}{DTP}},scen\in \{base,inv\} \end{gathered}$$

(31)

$$\begin{gathered} {RV}_T^{base}={\displaystyle \frac{{Rev}_T^{scen}}{CTR}}+0.7{\displaystyle \frac{{Rev}_T^{scen}}{DTR}}+0.7{\displaystyle \frac{{TC}_T^{scen}-\sum _{prod}{DA}_T^{prod}}{ITO}} \\ -{\displaystyle \frac{{TC}_T^{scen}-\sum _{prod}{DA}_T^{prod}}{DTP}},scen\in \left\{base,inv\right\}, \end{gathered}$$

(32)

The notation used in model are presented in

Table 1. Notation for parameters and variables used in NPV calculation.

Indices

t—investment period: {1, …, T}; scen—scenario: base—baseline, inv—investment; prod—product: OC—OC sheet, HDG—HDG sheet, CR—CR sheet, scrap—scrap;

Input parameters

TA—tax, ${\mathit{C}\mathit{A}\mathit{P}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—capacity of prod plan; ${\mathit{M}\mathit{S}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—market share of prod product; DTR—receivable turnover ratio; CTR—cash turnover ratio; ITO—inventory turnover ratio; DTP—payable turnover ratio; PUCprod—per-unit consumption of sheet required to produce prod sheet; ${\mathit{F}\mathit{C}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—fixed cost of prod plan; ${\mathit{V}\mathit{C}\_\mathit{O}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—other annual variable production costs per ton for prod sheet; Auxiliary variables ${\mathit{P}\mathit{A}\mathit{T}}_{\mathit{t}}^{\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—profit after tax in scen scenario in t; ${\mathit{D}\mathit{A}}_{\mathit{t}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—annual depreciation of prod plan in t; ${∆\mathit{W}\mathit{C}}_{\mathit{t}}^{\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—the change in net working capital in scen scenario in t; ${\mathit{R}\mathit{V}}_{\mathit{T}}^{\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—residual value in scen scenario in T; ${\mathit{R}\mathit{e}\mathit{v}}_{\mathit{t}}^{\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—revenue in scen scenario in t; ${\mathit{T}\mathit{C}}_{\mathit{t}}^{\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—total cost in scen scenario in t; ${\mathit{S}\mathit{V}}_{\mathit{t}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d},\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—sales volume of prod product in scen scenario in t; ${\mathit{N}\mathit{W}\mathit{C}}_{\mathit{t}}^{\mathit{s}\mathit{c}\mathit{e}\mathit{n}}$—net working capital in year t in scenario scen; ${\mathit{V}\mathit{C}\_\mathit{R}\mathit{M}}_{\mathit{t}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—cost of CR sheet per ton of prod sheet in year t; ${\mathit{S}\mathit{F}}_{\mathit{t}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—sales forecast for prod product;

Uncertain variables

${\mathit{P}}_{\mathit{t}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—price of prod per ton in year t; ${\mathit{A}\mathit{C}\mathit{F}}_{\mathit{t}}^{\mathit{p}\mathit{r}\mathit{o}\mathit{d}}$—apparent consumption forecast for prod product in t.

.

The Net Present Value of the proposed OC-sheet line is obtained with Equation (18), in which the incremental cash flows are defined by Equation (19) and decomposed by Equations (20)–(29) into operating profit after tax, depreciation, variable input costs and fixed overheads. The annual variation in net working capital is captured by Equation (30), while the stock of working capital itself is evaluated with the revised Equation (31), which relates cash, inventories, receivables and payables to four turnover ratios (CTR, ITO, DTR and DTP). At the end of the planning horizon, the model releases tied-up capital according to the Wilcox convention [5] implemented in the revised Equation (32), i.e., it recovers 100% of cash, 70% of the book value of inventories and receivables, and the full 100% of outstanding payables.

Prices and apparent consumption for the analyzed products are inter-related rather than independent, displaying complex patterns of correlation and mutual dependence. Two relationship types can be distinguished: correlations and reciprocal dependencies. The pair-wise price correlations are summarized in

Table 2. Correlation matrix for product prices.

	Scrap	CR Sheet	HDG Sheet	OC Sheet
Scrap	1.000	0.955	0.892	0.853
CR sheet	0.955	1.000	0.961	0.921
HDG sheet	0.892	0.961	1.000	0.911
OC sheet	0.853	0.921	0.911	1.000

, and the correlation coefficient between the apparent consumption of HDG sheet and OC sheet is 0.532.

Interval-regression coefficients that characterize the interdependencies among the drift parameters (μ_i) of the individual price series are provided in

Table 3. Interval-regression coefficient capturing the interdependence between drift parameters of the individual product-price series. Each bracket shows [lower; upper] the bounds derived from Equations (6) and (7), respectively; a₁ is the slope, a₂ the intercept.

Independent Variable		Dependent Variable
		Scrap	CR Sheet	HDG Sheet	OC Sheet
Scrap	a1		[−0.060; 1.943]	[0.369; 1.378]	[0.589; 1.554]
	a2		[−0.018; 0.001]	[−0.006; −0.002]	[0.003; 0.0098]
CR sheet	a1	[−0.897; 2.914]		[−1.611; 3.287]	[−1.689; 3.951]
	a2	[−0.013; 0.041]		[−0.015; 0.039]	[−0.031; 0.069]
HDG sheet	a1	[0.721; 1.489]	[0.399; 1.598]		[0.531; 1.811]
	a2	[0.003; 0.006]	[−0.011; −0.003]		[0.004; 0.016]
OC sheet	a1	[0.063; 1.689	[−2.160; 3.866]	[−0.599; 2.112]
	a2	[−0.012; 0.003]	[−0.062; 0.040]	[−0.022; 0.007]

; the corresponding coefficients for apparent consumption appear in

Table 4. The interval-regression coefficient capturing the interdependence between drift parameters of apparent consumption for the HDG sheet and OC sheet. Each bracket shows [lower; upper] the bounds derived from Equations (6) and (7), respectively; a₁ is the slope, a₂ the intercept.

Independent Variable		Dependent Variable
		HDG Sheet	OC Sheet
HDG sheet	a1		[−0.239; 0.800]
	a2		[−0.050; 0.167]
OC sheet	a1	[−1.589; 3.461]
	a2	[−0.040; 0.088]

. Interval-regression coefficients (a1, a2) were computed by solving the pair of inequalities given in Equations (6) and (7), following Hladík & Černý’s method.

For a steel plant operating with the leverage typical of the industry, we assume a real weighted-average cost of capital (WACC) of 10% p.a. and a real risk-free rate of 5% p.a. [4].

Polish historical datasets (1996–2022, constant prices-2022 USD) show that, in real terms, OC-sheet prices increased at an average annual rate of 0.89%. Over the same period, average annual growth rates were 1.46% for HDG sheet, 1.73% for CR sheet, and 1.63% for scrap. To generalize these trends we represent the common drift term across all products by the trapezoidal fuzzy number (0.85%, 1.10%, 1.30%, 1.50%), reflecting the structural shifts observed in the market—particularly during the 2006–2007 economic boom.

A review of apparent-consumption data for 1996–2022 shows that demand for HDG sheet and OC sheet grew on average by 7.55% and 8.65% per year, respectively. To generalize these dynamics, we adopt a common drift term for all products, expressed as the trapezoidal fuzzy number (6.0%, 6.5%, 7.0%, 7.5%), which captures the structural changes the sector underwent—particularly in the boom years of 2006–2007.

Volatility is estimated as the standard deviation of log-returns over 2000–2022. 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:

• Prices
  • Scrap: (13.0%, 14.0%, 15.0%, 16.0%);
  • CR sheet and HDG sheet: (15.0%, 17.0%, 18.0%, 20.0%);
  • OC sheet: (10.0%, 11.0%, 12.0%, 13.0%).
• Apparent consumption
  • HDG sheet: (8.0%, 9.0%, 10.0%, 11.0%);
  • OC sheet: (12.0%, 13.0%, 14.0%, 15.0%).

The trapezoidal fuzzy numbers (Figure 3) reported above were derived by the following algorithm. First, an annual time series of prices or demand is compiled in the original currency (PLN), converted to constant-2022 USD (using the average PLN/USD exchange rate for each year and deflating with the U.S. CPI, 2022 = 100), and cleansed of obvious outliers. Second, the series’ annual log-return statistics are computed: the mean serves as a point estimate of drift, while the standard deviation represents volatility. Third, structural breaks identified by domain experts are flagged so the algorithm can temper over-optimistic means or under-estimated risk. Fourth, the core interval [c,d] of the trapezoid is centered on the point estimate and symmetrically widened by a small margin that reflects “normal-year” variability. Fifth, the support bounds [a,b] are set by extending the core with a larger safety margin that captures plausible extremes stemming from shocks or expert judgment. Sixth, if several products share the same underlying driver (e.g., a common boom), the algorithm either pools them into a single trapezoid or retains product-specific ones, depending on a similarity test for their empirical moments. Seventh, each draft trapezoid is validated by superimposing its membership function on the empirical histogram: at least 90% of observations must lie inside [a,b] and roughly half inside [c,d]; otherwise, the margins are adaptively rescaled, and the previous steps repeated.

Other cost assumptions are as follows. Variable production costs (energy, labor, maintenance) amount to roughly USD 116 per ton for HDG sheet and USD 182 per ton for OC sheet. Fixed operating costs are estimated at USD 31.28 million per year for the existing HDG line and an incremental USD 13.44 million per year for the proposed OC line. The OC-line capital outlay is about USD 42 million. Market-share targets are 30% for HDG sheet and 25% for OC sheet, reflecting their expected competitive positions.

Under these assumptions, the NPV of building OC-sheet plant was built.

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 (β = 0.5), ambiguity (λ = 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.

5.1. Decomposition of Epistemic and Aleatory Uncertainty

Figure 4 reports the resulting probability box, $[\underset{\_}{\Phi },\overline{\Phi }]$, 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.

To translate this graphical separation into decision-relevant, quantitative indicators of the epistemic component, we introduce four complementary metrics, reported in

Table 5. Quantitative indicators of epistemic uncertainty for independent and interdepended assumption.

Assumption	Δmax	npv*	Area A	$\mathbf{Average} \mathbf{Width} \overline{\mathsf{\Delta }}$
Interdependent (a)	0.73	−29,642	1.79 × 105	0.22
Independent (b)	0.52	−133,037	3.87 × 105	0.13

. 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].

Together, the first two metrics reveal where epistemic uncertainty is most acute, while the latter two quantify how much ignorance remains overall:

$${\Delta }_{max}=\underset{x\in R}{\mathit{sup}}\left[\overline{\Phi }\left(npv\right)-\underset{\_}{\Phi }\left(npv\right)\right],$$

(33)

$${npv}^{\ast }=\mathit{arg}\underset{\mathit{npv}\in R}{\mathit{max}}\left[\overline{\Phi }\left(npv\right)-\underset{\_}{\Phi }\left(npv\right)\right],$$

(34)

$$A={\int }_{{npv}_{min}}^{{npv}_{max}}\left[\overline{\Phi }\left(npv\right)-\underset{\_}{\Phi }\left(npv\right)\right]dnpv,$$

(35)

$$\overline{\Delta }={\displaystyle \frac{A}{{npv}_{max}-{npv}_{min}}}.$$

(36)

The quantitative indicators in Table 5 reveal two distinct pictures of epistemic uncertainty, depending on whether statistical dependence among the inputs is acknowledged. In the interdependent scenario, the p-box exhibits its largest vertical separation at an NPV of about −29,600 EUR; there, the upper and lower cumulative-distribution functions diverge by Δ_max ≈ 0.726. Although this represents a pronounced local “spike” of ignorance, the uncertainty is highly concentrated: the integral of the gap over the whole NPV range—our area metric A—amounts to only 1.8 × 10⁵, and the average width of the p-box is 0.225.

When dependence is ignored, the pointwise gap never exceeds 0.2, and its peak shifts deep into the loss tail at roughly −133,000 EUR. Yet, the epistemic envelope flares out over a far broader span of outcomes, so the accumulated area under the gap almost doubles to 3.9 × 10⁵, while the mean width falls to 0.133 because much of the separation occurs where both CDFs are already close to 0 or 1.

For decision makers, this contrast is critical. The interdependent model delivers a leaner, but more informative uncertainty statement: it pinpoints a single NPV zone—near the break-even region—where an additional measurement or expert elicitations would have maximum value, while confirming that elsewhere the ignorance is comparatively small. By spreading uncertainty across a wider domain, the independent model inflates the total epistemic mass and may drive overly conservative risk assessments, yet provides little guidance on where further information would be most effective. In short, modeling the true dependence structure reduces the overall epistemic burden by roughly one half and focuses attention on the one part of the outcome space where knowledge is genuinely lacking.

Aleatoric variability is embedded in the shape of each cumulative-distribution curve inside a probability box. To give it a numerical description, we employ two distribution-agnostic indices: mean slope and inter-quartile range. The mean slope S_mean is the average gradient of the ${\Phi }_{cred}(npv,0.5)$ over the 10–90% probability range; the larger this value, the steeper the distribution function and the more tightly clustered (less random) the outcomes. The inter-quantile range IQR = npv_0.75-npv_0.25 measures the width of the central 50% of the distribution and grows in direct proportion to the aleatoric variability of the quantity under study. The results of the analysis are presented in

Table 6. Simple aleatoric-uncertainty metrics for considered assumptions.

Assumption	Smean	IQR
Interdependent (a)	2.51 × 10−6	180
Independent (b)	1.00 × 10−6	434

.

Preserving dependency contracts the output distribution (Table 6): in the dependent-input case S_mean increases by roughly 2.5 times, while the central spread IQR drops by more than 50%. Conversely, treating inputs as independent exaggerates aleatoric variability, yielding a flatter p-box and an inflated IQR that can mislead analysts about the system’s true randomness.

5.2. Hybrid Real Option Valuation Results

The results presented in

Table 7. Comparison of simulation results and transformations (thousand USD).

	$\underset{\_}{\mathit{\Phi }}$	$\overline{\mathit{\Phi }}$	${\mathit{\Phi }}_{\mathit{b}\mathit{e}\mathit{t}}\left(\mathit{n}\mathit{p}\mathit{v}\right)$	${\mathit{\Phi }}_{\mathit{a}\mathit{m}\mathit{b}}\left(\mathit{\kappa },0.5\right)$	${\mathit{\Phi }}_{\mathit{c}\mathit{r}\mathit{e}\mathit{d}}(\mathit{n}\mathit{p}\mathit{v},0.5)$
E(npv)	−3.42 USD	1.05 USD	−2.48 USD	−492 USD	−1.18 USD
Std dev	4.13 USD	3.16 USD	3.45 USD	2603 USD	3.06 USD
Success ratio	8%	80%	15%	54%	55%
ROV	2.41 USD	50.91 USD	4.60 USD	23.72 USD	26.66 USD

provide a comprehensive comparison of the various approaches considered. Utilizing the data from Table 7 allows for conducting valuation analyses using both the Net Present Value (NPV) and real option valuation (ROV) methods. The classical NPV analysis, irrespective of the transformation employed, indicates a negative decision regarding the initiation of the investment Figure 6. Essentially, initiating a project that does not add value to the company would not be rational. Although the standard deviation remains consistent across scenarios, the substantial percentage difference between the optimistic and pessimistic probability bounds highlights the critical importance of considering the option value in investment decisions. Figure 5 and Figure 6 reveal that the project’s Net Present Value is not characterized by a single figure but by a band of possible outcomes whose width is driven by epistemic + aleatory uncertainty.

Conversely, adopting a real option analysis perspective provides an alternative viewpoint. Real option valuation offers insights into the project’s potential upside value. If the potential benefits from the project exceed the cost of obtaining the real option, which enables capturing upside potential in response to changes, then acquiring the option becomes a sensible investment. In this analysis, three distinct outcomes are derived due to the application of three different ROV methodologies.

Using pure results from the hybrid simulation (see Figure 6 and the numerical summary in Table 7), the probability of success exhibits a broad range from 8% to 80%, and the corresponding ROV spans from 2 to over 26 thousand USD. The considerable disparity between the optimistic and pessimistic bounds emphasizes the importance of distinguishing between data accuracy and expert estimations in the valuation process. Decision makers must evaluate whether to rely on this information for investment decisions. If they find the range excessively broad, they may opt for additional research to reduce subjective uncertainty represented by fuzzy numbers for GBM parameters. However, further analysis may not always be feasible due to various constraints. In such instances, decision makers might opt for one of the available transformations.

The pignistic probability transformation can serve as a baseline reference. The unit mass probability represents the subjective probability for each interval $f_{\propto ,j}$. Here, “subjective” implies that $m(f_{\propto ,j})$ is proportional to its appearance in simulation outcomes. This interpretation suggests the probability of epistemic states represented by $f_{\propto ,j}$. When each state within intervals is equally probable, the pignistic transformation aligns closely with pure Monte Carlo simulation. In our example, ${\Phi }_{bet}\left(npv\right)$ closely mirrors $\underset{\_}{\Phi }$ (with a success probability of 15%). Thus, the probability of success is influenced by the decision-maker’s stance toward uncertainty and expert assessments.

The selection of other transformations depends on the perceived nature of epistemic uncertainty. The ambiguity transformation, as illustrated in this study using a weighting of 0.5, is particularly suitable for sensitivity analyses. This measure reflects the decision maker’s confidence level in expert opinions, influencing the weighting of optimistic versus pessimistic probability bounds.

Finally, the third transformation reflects the decision maker’s attitude towards uncertainty, indicating their preference for optimism or pessimism. This approach resembles the possibilistic fuzzy pay-off method for real option valuation described in [56]. It should be emphasized, however, that this transformation does not take into account investors’ risk aversion. This transformation represents the investor’s mental attitude towards the success of the investment or his epistemic knowledge about the effects of the investment.

Based on the pignistic transformation, the real option’s value is lower than the acquisition cost, implying a negative investment decision. However, when the ROV_amb and ROV_cred transformations are considered, the real option value exceeds the acquisition cost. Therefore, these transformations support the initiation of preparatory steps for building the organic coating plant, as this would create an opportunity to capitalize on future investments in the production of the replacement product.

5.3. Benchmarking Against the Fuzzy Pay-Off Method

To validate our hybrid approach, we additionally computed the fuzzy real options value (FROV) following Collan’s methodology [18]. Given the p-box structure of our results, we implemented two variants of FROV calculation. The classical approach computes ROV as A⁺ × E(A⁺), where A⁺ represents the success probability and E(A⁺) denotes the expected positive NPV [66]. However, since the p-box provides an interval [lower CDF, upper CDF] at the zero-crossing point, we must choose how to aggregate this uncertainty. We adopted the arithmetic mean of the bounds, yielding A⁺ = (0.3079 + 0.9549)/2 = 0.6314. The alternative approach directly integrates the positive NPVs weighted by their probability differentials, calculated as ROV = ∫(NPV > 0) NPV × dP, where P represents the average of lower and upper CDFs (credibility (β = 0.5) or credibility distribution by Liu [59].

The FROV calculations yielded two distinct values: the classical method produced ROV = 44.25 thousand USD, while the integration approach resulted in ROV = 26.66 thousand USD. Remarkably, the integration-based FROV exactly matches our credibility transformation (β = 0.5), confirming that both methods represent a balanced risk aversion.

This convergence demonstrates that our hybrid methods provide a more general framework, where classical FROV represents a specific case that has been extended in subsequent research. Importantly, Collan’s original FROV assumes direct epistemic determination of fuzzy numbers (possibility distributions) without simulation, making it particularly suitable for cases where expert judgment directly defines the uncertainty structure. In contrast, our method combines Monte Carlo simulation with possibilistic parameters, enabling a systematic exploration of both aleatory and epistemic uncertainties. This hybrid framework not only encompasses FROV as a special case but also provides decision makers with multiple transformation approaches (pignistic, ambiguity, credibility) that reflect different attitudes toward uncertainty, thus offering a more comprehensive toolkit for real option valuation under conditions where both types of uncertainty are present.

6. Conclusions

This study introduces a novel hybrid approach to real option valuation that integrates probabilistic and possibilistic frameworks within the Datar–Mathews method (DMM). To the best of our knowledge, this is the first adaptation of the DMM to explicitly accommodate both aleatory and epistemic uncertainties throughout the entire valuation process. By combining random and fuzzy representations of uncertainty, the proposed method provides a more comprehensive and realistic assessment of investment projects operating under deep uncertainty and limited data availability.

6.1. Managerial Insigts

By integrating epistemic and aleatory uncertainties, the proposed framework extends the classical option chain and equips decision makers with tools to manage the breadth of uncertainty in line with stakeholder preferences. The method preserves the formal rigor of real option theory while providing high process transparency, which renders it particularly valuable in technology-intensive industries that operate under ambiguous market data.

Our option-based framework aligns with the strategic-management view of the firm as a portfolio of interrelated real options. The option-chain concept advanced by Bowman and Hurry [67] conceptualizes strategic decision making as a sequence of mutually dependent real options, each of which endogenously generates a further set of development paths. Within this framework, three pivotal stages can be distinguished: (i) the potential option, an opportunity space that arises from entrepreneurial alertness; (ii) the real option, i.e., a variant that, having passed an initial screen, is admitted to the portfolio of active projects; and (iii) the managerial decision to execute, continue, or abandon the project in the light of current market signals.

Our study transfers this logic to a single, capital-intensive investment and fuses the option-chain theory with a hybrid p-box-based valuation approach. In the epistemic phase, domain experts elicit the full catalog of project development scenarios (potential options) and identify functional dependencies among economic, technological, and regulatory parameters; the richer the organizational knowledge, the narrower the epistemic bounds become. The width of those bounds also proxies’ industry turbulence, thereby linking sectoral innovativeness directly to the number and breadth of potential options.

The first filter in the chain is a hybrid conversion algorithm that operationalizes hypothesis H2 in Burger-Helmchen’s structural-equation model [68]. It removes configurations that violate corporate financial and risk constraints and collapses the multidimensional set of potential options into a single real option evaluated with the p-box technique. At this juncture, we introduce an ambiguity transformation that calibrates top management’s trust in expert knowledge; if the epistemic intervals remain excessively wide, management can revert to the identification stage or reject the project altogether.

The next stage comprises a sequential information-update loop implemented through the hybrid DMM. Every new market signal triggers a re-sampling of the p-box, thereby formalizing the canonical defer–expand–abandon cycle that characterizes an option chain. The parameter λ, interpreted as endogenous risk aversion, is adjusted dynamically in response to competitive threats and operational capabilities.

This framework can also be used as an extension of the approach proposed in Čirjevskis—abandonment option [69]. By embedding a salvage-value process SVal_t into the hybrid DM simulation and redefining the pay-off as max(npv_t, SVal_t), every Monte Carlo path automatically replaces catastrophic outcomes with a liquidation floor. In p-box terms the left tail is clipped, the vertical gap between the upper and lower CDFs shrinks, and quantitative indicators such as the area metric A and average width contract, signaling a tangible reduction in epistemic ignorance and therefore of the risk-premium managers must charge. The abandonment option becomes a built-in stop loss: even projects NPV is negative can remain defensible because value destruction is capped at the salvage price.

Once this floor is in place, the three p-box transformations act as managerial levers that decide when (and whether) to exercise it. The ambiguity-λ rule lets executives express how much they trust the market estimates behind SVal_t: a low λ inflates the p-box and encourages delaying CAPEX until better M&A data arrive, while a high λ narrows the band and highlights a favorable exit window. The credibility-β transformation assumes information reliability and turns β into a risk-attitude slider; pushing β toward zero amplifies concern for extreme losses and triggers an earlier sale, whereas β near one tolerates volatility and favors staying invested. Together, λ and β convert the abandonment option from a passive safety net into an active steering mechanism that aligns divestment timing with both information quality and the firm’s risk appetite.

6.2. Practical Advantages and Competitiveness of the Hybrid Random–Fuzzy DMM

One of the key strengths of the presented hybrid DMM approach lies in its ability to differentiate between uncertainty arising from inherent randomness (aleatory) and that caused by incomplete knowledge (epistemic). This distinction enables decision makers not only to model variability in market conditions and input parameters more accurately but also to explicitly recognize the role of subjective expert judgment in scenarios where historical data are scarce or unreliable. By avoiding the premature aggregation of these two types of uncertainty during simulation, the method allows for a transparent and traceable propagation of uncertainty throughout the valuation process.

In contrast to traditional real option valuation models, which often rely solely on probabilistic assumptions, the hybrid DMM introduces an additional analytical layer that reflects the decision maker’s attitude toward uncertainty. Through the application of three different strategies for transforming p-box results into subjective probability distributions—pignistic, ambiguity-based, and credibility-based transformations—the method provides flexibility in capturing diverse decision-maker preferences, including optimism, pessimism, and varying degrees of confidence in expert assessments. This feature empowers practitioners to adjust the valuation according to their individual risk tolerance, epistemic beliefs, and strategic posture.

Compared with classical Black–Scholes-based valuations, the proposed hybrid DMM offers three tangible benefits. First, it accommodates expert-elicited confidence intervals where historical statistics are weak—typical of long-horizon investments in metallurgy, energy, or infrastructure—thereby extending option pricing to domains that lie beyond the reach of standard BS inputs. Second, by maintaining cross-parameter interdependency and actively controlling the width of the p-box, it yields narrower and more realistic ranges for real option value (ROV), mitigating both excessive optimism and overly conservative discount “penalties.” Third, its p-box transformations let decision makers swiftly “tune” the valuation to their own risk profile—something a single-point BS estimate cannot replicate. Importantly, when epistemic uncertainty collapses to zero, the hybrid DMM converges to the conventional Black–Scholes outcome, ensuring full backward compatibility while delivering richer, decision-relevant information whenever uncertainty runs deep.

One of the key advantages of this hybrid DMM is its ability to effectively separate aleatoric and epistemic uncertainties in the decision-making process. As research shows [32], understanding the nature of uncertainty can help in designing interventions to improve judgment accuracy. Investors who see market uncertainty as stemming from a lack of knowledge or skill tend to seek expert guidance and react swiftly to new information, while those who view market uncertainty as driven by random processes are more inclined to diversify their portfolios to mitigate risk [39]. Unlike other methods, such as fuzzy real options valuation (FROV), this approach avoids aggregating these uncertainties during propagation through simulation. Additionally, it introduces an additional layer that explicitly requires decision makers to address simulation results by considering potential issues like experts’ over-optimism, reliance on heuristics, or other cognitive biases. Consequently, the proposed approach allows decision makers to assess the influence of such factors on the final decision, while also providing quantitative descriptions of their approach to uncertainty.

Overall, this paper introduces a comprehensive and robust hybrid DMM that accounts for various sources of uncertainty and provides valuable insights for decision makers grappling with real options under high uncertainty. By incorporating both aleatoric and epistemic uncertainties, the method enables a thorough analysis of investment projects and facilitates the integration of expert knowledge, ultimately bridging the gap between real-world decision making and real option valuation. Because the algorithm treats aleatory and epistemic components separately and only then reunifies them in the pay-off stage, it is immediately applicable to any sector—energy, infrastructure, life-sciences R&D—where data scarcity co-exists with expert insight. Only the cash-flow model needs adaptation; the uncertainty propagation and valuation core remain unchanged.