*2.1. Pay-Off Method*

The fuzzy pay-off method [21,34] approaches building the distribution through setting scenarios. Usually, three scenarios are defined, pessimistic, realistic, and optimistic, although more can be created if reasonable. First, managers are asked to provide estimates of input values, like costs, prices, production volumes, etc., for every scenario. The idea is to generate the estimates for the worst possible scenario (pessimistic) such that nothing worse can happen, for the best possible scenario (optimistic) such that nothing "better" can be expected to happen, and the one with the most realistic estimates (realistic or best estimate). Second, net present value is calculated for each scenario. Third, the three NPVs are used to form a triangular pay-off distribution for the project NPVs are mapped on the value (x) axis, while the *y*-axis depicts the membership degree within the set of possible outcomes. Full membership (equal to 1) is assigned to the "realistic" scenario value, and limit to zero membership to the pessimistic and the optimistic scenario NPVs, implying that anything worse or better correspondingly is not expected to take place. The relationship between the positive and the negative and the realistic scenario value is assumed to be linear. Thus, in

the final stage, a triangle is formed that represents the possibilistic range of the project's NPVs and that is treated as a triangular fuzzy number, for details see [13,26]. company the material provided for decision-making. For example, the mean value of the distribution and the variance can be calculated. Furthermore, the real option value can be

Descriptive statistics can be calculated directly from the pay-off distribution and ac-

mapped on the value (x) axis, while the *y*-axis depicts the membership degree within the set of possible outcomes. Full membership (equal to 1) is assigned to the "realistic" scenario value, and limit to zero membership to the pessimistic and the optimistic scenario NPVs, implying that anything worse or better correspondingly is not expected to take place. The relationship between the positive and the negative and the realistic scenario value is assumed to be linear. Thus, in the final stage, a triangle is formed that represents the possibilistic range of the project's NPVs and that is treated as a triangular fuzzy num-

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Descriptive statistics can be calculated directly from the pay-off distribution and accompany the material provided for decision-making. For example, the mean value of the distribution and the variance can be calculated. Furthermore, the real option value can be computed based on the expected mean of the positive part of the distribution [34,35]. The main steps of the method are visualized in Figure 1. computed based on the expected mean of the positive part of the distribution [34,35]. The main steps of the method are visualized in Figure 1. The fuzzy pay-off method has been used in many application areas, including energy and oil investments [26,36–38], screening and selection of research and development projects [39,40], and management of a patent portfolios [41].

**Figure 1.** The fuzzy pay-off method in four steps. **Figure 1.** The fuzzy pay-off method in four steps.

ber, for details see [13,26].

*2.2. Simulation-Based Profitability Analysis and Simulation Decomposition*  Simulation-based analysis is based on two parts, one part is a (computer) model that contains stylized (often much-simplified) structure of the studied system that nevertheless The fuzzy pay-off method has been used in many application areas, including energy and oil investments [26,36–38], screening and selection of research and development projects [39,40], and management of a patent portfolios [41].

#### carries a strong resemblance with reality. The system model includes a number of inputs *2.2. Simulation-Based Profitability Analysis and Simulation Decomposition*

and outputs to and from the system that can be studied to understand what the system "does". The best system models have a high requisite variety (requisite complexity) [42] and thus offer relatively high credibility by way of fidelity with the real world. Systemmodels may also be dynamic and change as a function of (simulation) time. The second part is simulation, which is typically arranged by means of automated software inputting a large number of input variable-combinations (vectors, input scenarios) into the system and collecting the corresponding output values. The input value-combinations are selected from input-value distributions that are pre-determined for each input (and may also be single values, crisp). The output-values are typically presented as histograms or frequency distributions and it is common to assume that the distribution is a probabilistic representation of the occurrence frequency of the outputs from the system. A Monte Carlo simulation is a simulation, where the input value selection is made randomly by the simulation software from the input-value distributions for a typically pre-set number of times [43,44]. In the context of ex-ante profitability analysis or policy-effect analysis the system underlying the simulation analysis is the profitability analysis cash-flow model of the invest-Simulation-based analysis is based on two parts, one part is a (computer) model that contains stylized (often much-simplified) structure of the studied system that nevertheless carries a strong resemblance with reality. The system model includes a number of inputs and outputs to and from the system that can be studied to understand what the system "does". The best system models have a high requisite variety (requisite complexity) [42] and thus offer relatively high credibility by way of fidelity with the real world. System-models may also be dynamic and change as a function of (simulation) time. The second part is simulation, which is typically arranged by means of automated software inputting a large number of input variable-combinations (vectors, input scenarios) into the system and collecting the corresponding output values. The input value-combinations are selected from input-value distributions that are pre-determined for each input (and may also be single values, crisp). The output-values are typically presented as histograms or frequency distributions and it is common to assume that the distribution is a probabilistic representation of the occurrence frequency of the outputs from the system. A Monte Carlo simulation is a simulation, where the input value selection is made randomly by the simulation software from the input-value distributions for a typically pre-set number of times [43,44].

ment that is facing the policy, and the cash-flows that are received by the investment are regulated by the policy as a function of the environment that the investment is facing, described in terms of the input-variable value-combinations. This means that the system used includes both the profitability analysis model and the policy-model. The simulation software is then used to reveal the outputs from the system under various (randomly drawn) real-world scenarios [45,46]. In the context of ex-ante profitability analysis or policy-effect analysis the system underlying the simulation analysis is the profitability analysis cash-flow model of the investment that is facing the policy, and the cash-flows that are received by the investment are regulated by the policy as a function of the environment that the investment is facing, described in terms of the input-variable value-combinations. This means that the system used includes both the profitability analysis model and the policy-model. The simulation software is then used to reveal the outputs from the system under various (randomly drawn) real-world scenarios [45,46].

Simulation decomposition is based on the Monte Carlo simulation framework and thus, in contrast with the pay-off method, belongs to the probabilistic framework. Simulation Decomposition decomposes the results of the simulated output probability distribution into sub-distributions that are matched with the input variable value range combinations from which they result. The input range combinations can be understood as scenarios. This input-output matching reveals important information about cause and effect and allows decision-makers to better understand what effect the various scenarios will have

on the output. The procedure is based on (i) identifying the relevant variables that can be affected by the project owner, their relevant "states", and boundaries for each state; (ii) forming "groups" or scenarios by combining the states; (iii) running the simulation, while keeping track on the input-output "inference"; (iv) visualizing the results such that the outcome resulting from each input group (scenario) is separately visualized and allows better understanding of "what leads to what". The procedure is depicted in Figure 2. The detailed description of the procedure, how the results from it are visualized, and available implementation tools can be found in [22,47]. and allows decision-makers to better understand what effect the various scenarios will have on the output. The procedure is based on (i) identifying the relevant variables that can be affected by the project owner, their relevant "states", and boundaries for each state; (ii) forming "groups" or scenarios by combining the states; (iii) running the simulation, while keeping track on the input-output "inference"; (iv) visualizing the results such that the outcome resulting from each input group (scenario) is separately visualized and allows better understanding of "what leads to what". The procedure is depicted in Figure 2. The detailed description of the procedure, how the results from it are visualized, and available implementation tools can be found in [22,47].

Simulation decomposition is based on the Monte Carlo simulation framework and thus, in contrast with the pay-off method, belongs to the probabilistic framework. Simulation Decomposition decomposes the results of the simulated output probability distribution into sub-distributions that are matched with the input variable value range combinations from which they result. The input range combinations can be understood as scenarios. This input-output matching reveals important information about cause and effect

**Figure 2.** Schematic visualization of the simulation decomposition procedure. **Figure 2.** Schematic visualization of the simulation decomposition procedure.

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If simulation decomposition is performed by using a single variable, one can see the influence of this variable on the outcome in the presence of other uncertainties. The two extremes would be zero (low) influence, if all the scenarios are "lying" on top of each other (share same output values on the *x*-axis), and strong influence, if the scenarios are vertically separated from each other (do not share same output values on the *x*-axis). If the decomposition is performed by using two or more variables, one can observe the interplay of variables and possible synergies, if any are hidden in the system. The more nonlinearities and various what-if rules the system has, the more valuable the decomposition potentially becomes. If simulation decomposition is performed by using a single variable, one can see the influence of this variable on the outcome in the presence of other uncertainties. The two extremes would be zero (low) influence, if all the scenarios are "lying" on top of each other (share same output values on the *x*-axis), and strong influence, if the scenarios are vertically separated from each other (do not share same output values on the *x*-axis). If the decomposition is performed by using two or more variables, one can observe the interplay of variables and possible synergies, if any are hidden in the system. The more nonlinearities and various what-if rules the system has, the more valuable the decomposition potentially becomes.

Simulation decomposition has demonstrated its value in renewable energy policy analysis [22,48], in other environmental policy issues [25,49], and can be generally applied to any problem modeled with Monte Carlo simulation independent of the context [47]. Simulation decomposition has demonstrated its value in renewable energy policy analysis [22,48], in other environmental policy issues [25,49], and can be generally applied to any problem modeled with Monte Carlo simulation independent of the context [47].

A similar scenario decomposition can be made within the possibilistic framework, by framing an input-output system by using a fuzzy inference system (FIS), see [24]. This approach has benefits and drawbacks. Using FIS avoids simulation and thus requires less computational time, however, the necessity of manual construction of the many scenarios typically overrides the time savings. In the simulation decomposition method, scenarios are created and valued automatically, based on the user-specified partitions of the input variables. A similar scenario decomposition can be made within the possibilistic framework, by framing an input-output system by using a fuzzy inference system (FIS), see [24]. This approach has benefits and drawbacks. Using FIS avoids simulation and thus requires less computational time, however, the necessity of manual construction of the many scenarios typically overrides the time savings. In the simulation decomposition method, scenarios are created and valued automatically, based on the user-specified partitions of the input variables.

#### *2.3. Numerical Assumptions 2.3. Numerical Assumptions*

This study makes numerical assumptions based on publicly available literature and following the practice presented in [50]. The economic life of a biorefinery plant typically varies between 20–25 years, and in this study, the lifetime of 20 years is used. The corporate tax-rate is assumed to be 20% and the discount is set at 10%. The numerical assumptions about the biofuel production-plant investment are estimates taken from [51]. These estimates include the investment cost of a 500 million liters per year of renewable diesel production 430 M€ and operating cost of 0.86 EUR/liter. The assumptions related to This study makes numerical assumptions based on publicly available literature and following the practice presented in [50]. The economic life of a biorefinery plant typically varies between 20–25 years, and in this study, the lifetime of 20 years is used. The corporate tax-rate is assumed to be 20% and the discount is set at 10%. The numerical assumptions about the biofuel production-plant investment are estimates taken from [51]. These estimates include the investment cost of a 500 million liters per year of renewable diesel production 430 M€ and operating cost of 0.86 EUR/liter. The assumptions related to policies supporting the use of biofuels are related to the Finnish biofuel policy. Tax-rates used in this study are retrieved from the Finnish Tax Administration (2021), and the amount of penalty for not achieving the required share of biofuels is retrieved from the decisions of the Finnish Parliament (2018). All numerical assumptions made in this study are listed in Table 1.


**Table 1.** Numerical assumptions for the studied system.
