Sustainable Entrepreneurship: Interval Analysis in Risk Management and Uncertain Economies

Abstract

Sustainable management in high-tech enterprises is a key aspect of successfully operating modern companies, especially under conditions of risk and uncertainty. This study reviews the field of sustainable management and interval analysis and identifies the main trends and challenges facing high-tech enterprises in the modern world. This study emphasizes the importance of applying interval analysis in making strategic decisions and developing sustainable business models that can adapt to variable environments. This paper presents empirical data, illustrating the practical application of interval analysis tools in the management in high-tech enterprises. It analyzes the effectiveness and potential of this approach to increase the levels of sustainability and competitiveness of organizations in constantly changing business environments. In general, this article is a valuable contribution to the development of sustainable management theory and practice for high-tech enterprises, enriching the existing knowledge in this area and offering new perspectives for research and practical application. Our research has been validated and is presented in the results section. The purpose of this study is to present current developments in methodologies and tools for risk measurement within the probabilistic paradigm of uncertainty, which are supposed to be used in relation to the economic evaluation of real investment projects. The methodological directions or approaches to risk measurement formed in this context are (1) based on quantile measures, within which the quantitative aspect of risk is modeled using quantile quantiles of the distribution of a random variable describing the possible (predicted) results of economic activity; (2) the Monte Carlo method, which is a tool for evaluating the indicators of economic efficiency and risk in justifying real investments, taking into account different distribution laws and mutual relations for the financial and economic parameters of the investment project, as well as its computational and instrumental elaboration.

1. Introduction

The investment process is integral to the activities of any high-tech enterprise. Through investment implementation, a wide range of priority tasks are addressed to ensure the sustainable development of a high-tech enterprise, such as forming and maintaining (reproducing) its operating system, increasing production capacity, and expanding and growing business lines [1].

The main form of sustainable management in high-tech enterprises, namely their planning and realization of real investments, is project-based. One of the main decision-making tasks in real investments involves choosing the best (optimal) investment project from a set of alternatives. This generally involves multiple criteria. In the multi-criteria structure of this task, if we proceed from the paradigm of non-determinism (uncertainty) of the economic environment, we can distinguish two levels [2].

Firstly, the economic efficiency of a high-tech project is a complex characteristic. Analyzing and evaluating this involves using a few indicators that serve as partial criteria.

Secondly, for an interested person (subject/object of decision-making, expert) or a group of such individuals, the uncertainty of the economic environment, characterized by the lack of comprehensive and accurate information about the problem under study, becomes a source of risk. This must be considered, and the quantitative assessment involves multiple criteria [3].

A fundamental difference between making a rational (optimal) decision under conditions of uncertainty with inherent risks—where it is necessary to analyze the available alternatives and choose the best ones—and making decisions in conditions of certainty (determinism) is that, in the first case, each alternative under study is not described by the only possible consequence or result, but by the distribution of possible results, of which, only one should be realized if the corresponding alternative is chosen [4].

This particular result can be assumed, but cannot be reliably predicted. Hence, the problem cannot be formulated as it would be in a deterministic situation—where the choice is about selecting an alternative that is guaranteed to provide the optimal (maximum, or minimum) result. Instead, it is reduced to choosing an alternative with the most preferable or attractive (optimal) distribution of possible results.

Assessing the economic efficiency of an investment project consists of evaluating a set of criteria that are calculated from the cash flow data [5]. Neither the flow of investments nor the flow of current (operating) payments and receipts can be planned accurately. Regarding current investment analysis and investment management, issues related to uncertainty and risk while preparing decisions regarding real investments are central. Thus, sensitivity analysis is one of the simplest and most accessible methods in accounting for uncertainty and risk when economically justifying investment projects at high-tech enterprises. Recent research studies on investment analysis and management highlight several key themes. International trade and investment are interconnected, with export policies influencing global economic cooperation [5,6]. Sustainable investing and financing play a crucial role in achieving sustainable development goals, with studies examining investor motivations, investment performance, and policy enablers [5,7].

As a basic method, this method is usually included in the functionality of specialized software tools to support analytical and managerial activities in business planning and investment analysis [8]. As the review results show, in modern studies devoted to qualitative and quantitative risk analyses of investment projects, along with the development of the latest tools and approaches, there is still active attention placed on traditional or classical methods [9]. If we look at sensitivity analysis from this angle, we can conclude that it is of interest to further expand its application to a wider range of investment design problems than its traditional version suggests [10].

The essence of traditional sensitivity analysis in the context of real investments is a sequential-unit assessment of how changes in initial parameters or variables (risk factors) influence the economic efficiency indicator (criterion) of the investment project under consideration [11,12]. Full-scale sensitivity analysis encompasses all indicators, which are defined as criteria in particular situations of investment projection situations. A greater impact of a parameter change on the criterion indicator is one reason to consider this parameter as more “risk-forming” with respect to the criterion.

Conducting a traditional sensitivity analysis for a criterion indicator (criterion) involves the following:

• Establishing the basic (baseline) scenario. It is fixed, i.e., the most probable (expected) values of the initial parameters (variables, arguments) of the criterion under study. The baseline estimates of parameters correspond to the baseline levels of the criteria conditioned by them.
• Determining the parameters (risk factors) of the criterion indicator for which the sensitivity analysis will be performed.
• Setting variation ranges. The boundaries or variation ranges (intervals) in relation to the baseline values of the parameters selected for the study are set. At this step, within the framework of traditional sensitivity analysis, it is possible to distinguish the different variants (versions). In the first variant, specified ranges are defined exactly (one for each parameter), and are, at the same time, generally differentiated for different parameters, based on the perceptions and experiences of the person concerned (the “decision-making object, the expert”). Such a mono-interval approach may also be based on (or utilize) general recommendations (if they exist), which reflect actual investment practices within the economy or specific industry of the country concerned (for the time period relevant to the investment project under study). An example can be found in this paper. In this case, the ranges for a single parameter should generally be set individually. If the interested party does not see sufficient grounds for such differentiation, a single system of ranges for all analyzed parameters can be used. The results of poly-interval sensitivity analyses are typically visualized via graphs.
• The degree of sensitivity of the criteria under study within the selected parameters is assessed. Regarding mono-interval determination of parameter variation ranges, the degree of sensitivity can be assessed in two ways. According to the first option, the assessment is based on the value interval (variation range) of the criterion indicator, corresponding to the intervals of possible values of the parameters selected for analysis: the larger the resulting interval, the greater the sensitivity and vulnerability of the criterion. Within the framework of this approach, it is easy to see that the degree of sensitivity can also be assessed by only considering part of the interval that contains unfavorable deviations. The comparison should be made with the level of the criterion defined as the baseline. The corresponding quantitative indicator of the degree of sensitivity can be conveniently and correctly labeled as the baseline variation spread. The second option is to turn to a toolkit that is based on the use of the concept of elasticity and is considered universal [13,14,15,16].

Let the risk object (production program, investment project, or securities portfolio) be described by n initial parameters (variables, factors), denoted as $x_i$, $i=\overline{1,n}$, and m resulting or benchmark indicators, denoted as $y_j,$ $j=\overline{1,m}$, between which, there is a functional dependence, i.e., $y_j=f_j\left(x_1,\dots ,x_n\right),$ $j=\overline{1,m}$.

Then, the elasticity, or coefficient of elasticity, of the indicator $y_j,$ $j\in \left\{1,m\right\}$ and the parameter $x_i,$ $i\in \left\{1,n\right\}$ is defined like this.

To change a parameter with respect to its value, $x_i,$ in the interval, $∆x_i$, we have the following:

$$E_{j,i}\left(x_i,∆x_i\right)={\displaystyle \frac{∆y_j}{y_j}}:{\displaystyle \frac{∆x_i}{x_i}}$$

(1)

At point $x_i$ (provided that the function is differentiated in it $f_j$), we have the following:

$$E_{j,i}\left(x_i\right)={\displaystyle \frac{\partial y_j}{y_j}}\ast {\displaystyle \frac{\partial x_i}{x_i}}.$$

(2)

According to the given relations, elasticity (elasticity coefficient) shows the percentage change in the value of the resulting indicator (criterion) when the value of the corresponding factor indicator (parameter, argument) changes by one percent. The nature of the relationship (direct or inverse) between the analyzed indicators is taken into account. Hence, the higher modulo level of the elasticity coefficient indicates greater sensitivity of the criterion under study for the given parameter. Similar to the latter, and taking into account the necessary adjustments, the degree of sensitivity is assessed when several ranges of possible fluctuations are allowed for the parameters. In this case, for a single parameter, the elasticity should be calculated for each range formed for it. Accordingly, the degree of sensitivity of the criterion under consideration is determined by the set of elasticity coefficients found.

5.

Based on the results of the previous step, the parameters of the criterion indicator are ranked in terms of their risk-forming ability. According to the key concept of sensitivity analysis, a higher level of sensitivity of a criterion to changes in a given parameter is interpreted as an indication of its greater risk-taking ability. When using a mono-interval approach, where the range or base range of the variation of a criterion indicator is used as a measure of sensitivity, the value is a sufficient indicator of the degree of risk formation of the corresponding parameter. If the mono-interval variation of the parameter values is implemented, when sensitivity is measured using the elasticity coefficient, the sufficiency of the latter (for assessing the riskiness of the parameters under study) may be questionable. In this case, a full-fledged ranking of parameters as risk factors requires introducing some additional criterion/criteria or aspect(s). The role of the additional criterion can be fulfilled by the degree of predictability (predictability) of parameters. The formalization of the analysis of the risk-forming significance (weighting) of a parameter within the specified two criteria is carried out with the help of an appropriate matrix (i.e., sensitivity or predictability matrix) [17,18]. Obviously, in the considered approach, the degree of predictability of a parameter and the value of the interval of its possible values can be understood as mutually dependent characteristics (a larger interval indicates less predictability, and vice versa, a smaller interval of possible values indicates greater predictability).

When using multiple variation ranges for the parameters of the criterion indicator, the assessment of their risk-forming level can be carried out based on the elasticity coefficient alone as well as on a two-dimensional basis, i.e., with the addition of the elasticity analysis to the predictability analysis (if there are appropriate grounds for this). As reflected in the information above, the scope of the traditional sensitivity analysis involves the individual investment project. At the same time, in many cases, economically justifying real investments presents itself as a problem when choosing the best (optimal) investment project from a set of alternatives. This necessitates the development of the considered toolkit and modification, adapted to the comparative evaluation of alternative (competing) investment projects [11,19]. A detailed account is presented as follows: Section 2 proposes a theoretical framework for sustainable development and management, including sustainable entrepreneurship. Section 3 details the methods and models of the study. Section 4 presents the qualitative and quantitative results, including the priorities for sustainability, and Section 5 presents the conclusions and implications for upcoming research.

2. Theoretical Background

Wang (2009) and Li-xin (2006) both emphasize the importance of systematically managing risks in technological innovation, with a focus on identifying and evaluating these risks. They propose a gray hierarchy evaluation model, based on the gray system theory and AHP method, to quantitatively assess innovation risks [20,21].

Harris (2006) further underscores the need for responsive management skills in high-tech entrepreneurship, particularly in the face of uncertainty and evaluation challenges [22]. Jia (2008) extends this discussion to the realm of venture capital, highlighting the need for cautious program selection and effective innovative management to reduce risks [23,24].

Soofifard (2016) and Tabatabay (2022) both propose models for selecting risk response strategies in project management, with Soofifard focusing on project costs, schedule, and quality, and Tabatabay considering primary and secondary risks in petrochemical projects [25,26]. Both models use fuzzy multi-objective methods to address uncertainty. Mohagheghi (2016) extends this work to sustainable project portfolio selection, incorporating interval-valued fuzzy sets to evaluate financial return, risk, and non-financial criteria [27,28]. These studies collectively contribute to the development of a comprehensive model for selecting response strategies for primary and secondary project risks under interval-valued fuzzy uncertainty.

Reference [29] explores the uncertainties and risks associated with investments in green hydrogen. The authors employ a stochastic approach using Monte Carlo simulations to model the financial and operational uncertainties involved in green hydrogen projects. This approach allows for a more comprehensive understanding of the potential variability in outcomes, providing a range of possible scenarios rather than a single deterministic result. The study focuses on several key variables, including market demand, technological advancements, regulatory changes, and cost fluctuations, which are critical to the success and viability of green hydrogen investments. The research findings highlight the significant impacts of these uncertainties on the expected return on investment (ROI) and underline the importance of robust risk management strategies. The Monte Carlo simulations reveal that while green hydrogen presents substantial opportunities for sustainable energy development, it also comes with high variability in financial returns due to the nascent stage of the technology and market volatility. This paper presents a crucial and timely analysis of the green hydrogen sector, which is increasingly seen as a cornerstone of future sustainable energy systems. The use of Monte Carlo simulations to assess risk and uncertainty is particularly valuable, as it offers a more realistic and nuanced view of potential investment outcomes. Given the high stakes and significant capital required for green hydrogen projects, this research provides essential insights for investors, policymakers, and stakeholders [30]. Thus, the authors’ methodology is sound, and well thought out, and reflects the complexity of market and technological factors affecting green hydrogen investments. The results emphasize the need for careful planning and risk mitigation strategies that are often overlooked in the enthusiasm for green energy solutions [31,32].

3. Methods and Models

The following model features of the stated problem can be distinguished as follows:

• Parameter sets (variables, arguments), which are considered as risk factors in the context of individual efficiency criteria, for alternative investment projects, may or may not coincide with each other;
• When comparing the riskiness of investment alternatives (in the context of separate performance criteria), the integrated sensitivity of the latter should be assessed and compared, which takes into account the parallel (simultaneous) change in the values of all risk-forming (risk-relevant) parameters.

Regarding traditional sensitivity analysis, its desired modification can be constructed in two versions: mono- and poly-intervals. Let us attempt to formulate a variant of the solution of the problem under consideration (for the mono-interval formulation first).

Let us turn to a variant of the mono-interval version when the variation range of the criterion is used as a measure of sensitivity. A realistic and acceptable approach to forming a modified sensitivity analysis model involves finding and quantifying the interval (range) of possible values of a criterion indicator (attributable to the variation of all its parameters, which are considered risk-forming) within the specified intervals for these parameters.

The constructive embodiment of the idea outlined above and the development of the desired modification of sensitivity analysis can be carried out using the interval analysis (mathematics) methodology. Based on the mathematical apparatus of this theory, for the rational choice of the best investment alternative, the following steps can be proposed to assess the sensitivity of a specific efficiency criterion to simultaneous changes in a set of parameters within a particular investment project [33,34]:

• The baseline scenario of project implementation is identified, which is described by the baseline values of the initial parameters of the criterion under consideration and the level of the latter that corresponds to them.
• The parameters of the criterion indicator are determined, which are regarded as risk-forming (risk-relevant) and for which sensitivity analysis will be performed.
• The limits or ranges (intervals) of variation with respect to the baseline level of the parameter values selected for analysis are set.
• Based on the interval analysis (mathematics), an interval estimate of the criterion under study is calculated, corresponding to the variation ranges of the values of the parameters adopted in the previous step.

The variation spread and the baseline variation spread for the obtained interval serve as integrated indicators of the sensitivity of the criterion indicator to simultaneous fluctuations in the values of risk-forming parameters.

The methodological guidelines and the scheme of integrated sensitivity assessment presented above allow us to formulate a model for the comparative assessment of the economic efficiency of alternative investment projects using sensitivity analysis (employing a mono-interval variation of parameters):

• For each investment project from the set of investment alternatives (project variants), the basic scenario of its realization is determined. The most expected (probable) values of the initial parameters (variables, arguments) should be taken as basic data.
• A set of criterion indicators (partial criteria) is defined, where the economic efficiency of the compared investment projects should be assessed. It should be remembered that assessing the effectiveness of real investments enables the use of partial criteria that reflect their financial effects, profitability, and payback period. At the same time, the indicators concerning, respectively, the first two and the third of the named aspects have different optimization directions or ingredients. The criteria surrounding financial effects and profitability are optimized in the maximum direction (i.e., they are positive or have positive ingredients), while the payback indicators are optimized in the minimum direction (respectively, they are negative or have negative ingredients).
• For the base level of the initial parameters of investment projects, the corresponding values of the partial efficiency criteria selected in the previous step are calculated.
• For each investment project, initial parameters are determined, which are considered risk factors and for which sensitivity analysis will be performed.
• The limits or ranges (intervals) of variation with respect to the baseline values of the initial parameters selected for the sensitivity analysis are set.
• Interval estimates of the partial criteria of their economic efficiency are calculated for the investment projects under consideration with respect to the variation ranges of risk-forming parameters set in the previous step, using interval analysis or mathematics. After that, for each interval estimate, we calculate the values of unfavorable deviations of values from the base level of the corresponding partial criterion, i.e., the base variation spread, which is regarded as an absolute indicator of the sensitivity measure. In this case, the nature of deviations (favorable/unfavorable) should be determined based on the optimization direction (ingredient) of the analyzed partial criterion.
• Regarding the partial efficiency criteria of the investment projects under study, based on the values of the baseline variation spread (obtained in the previous step to ensure comparability (comparability)), the sensitivity measure is calculated in relative terms. It is proposed to use the ratio of the baseline variation spread of a partial criterion to the increment of economic efficiency, which is provided when its baseline level is reached in comparison with the threshold (limit) value. It is natural to establish the latter based on the boundaries of interval estimates of the analyzed partial criterion within the set of compared investment projects.

4. Results and Discussion

Based on the economic evaluation of real investments using sensitivity analysis, each private performance criterion that is taken to compare investment alternatives should be detailed in two aspects:

• The basic level of this partial criterion, which reflects the most expected (probable) course of the investment project realization;
• The sensitivity of this partial criterion to changes in initial parameters, which is interpreted as its risk burden (its riskiness).

For indicators reflecting the above two aspects, in accordance with their role, it is reasonable to refer to them as detailed criteria.

4.1. Formatting the Mathematical Components

According to the previous steps, as well as the remarks made, the comparative economic evaluation of alternative investment projects using sensitivity analysis can be carried out based on a generalized (integrated) indicator (criterion), which, if we limit ourselves to the additive variant of criteria convolution, can be formulated as follows:

$${SI}_j={\sum }_{i=1}^L{a}_l(b_{l1}^H{K}_{l2}^{bs}+b_{l2}^H{CBR}^m({\stackrel{̿}{K}}_{lj},K_{lj}^{bs})), j=\overline{1,m},$$

(3)

$${}_{ }{}^H{K}_{lj}^{bs}=\left\{\begin{array}{c}{\displaystyle \frac{K_{lj}^{bs}-K_{minl}}{K_{maxl}-K_{minl}}}, if K_l=K_l^{+}\\ {\displaystyle \frac{K_{maxl}-K_{lj}^{bs}}{K_{maxl}-K_{min}l}}, if K_l=K_l^{-}\end{array}\right.,$$

(4)

$${}_{ }{}^H{CBR}^m\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right)=1-{CBR}^m\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right),$$

(5)

$${CBR}^m\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right)=\left\{\begin{array}{c}{\displaystyle \frac{BR\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right)}{\left({\stackrel{̿}{K}}_{lj}-K_{minl}\right)}}, if K_l=K_l^{+}\\ {\displaystyle \frac{BR\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right)}{K_{max}l-K_{lj}^{bs}}}, if K_l-K_l^{-}\end{array},\right.$$

(6)

$$BR\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right)=\left\{\begin{array}{c}{K}_{lj}^{bs}-{\underset{\_}{K}}_{lj}, if K_l=K_l^{+} \\ {\overline{K}}_{lj}-K_{lj}^{bs}, if K_l-K_l^{-}\end{array},\right.$$

(7)

$$l=\overline{1,L}, j=\overline{1,m} ,$$

$$K_{minl}=min\left\{{\underset{\_}{K}}_{lj}|j=\overline{1,m}\right\},l=\overline{1,L},$$

(8)

$$K_{maxl}=max\left\{{\overline{K}}_{lj}|j=\overline{1,m}\right\},l=\overline{1,L},$$

(9)

$$0<b_{l1}<1,$$

(10)

$$0<b_{l2}<1,$$

(11)

$$b_{l1}+b_{l2}=1,$$

(12)

$$0<a_l<1,$$

(13)

$${\sum }_{l=1}^L{a}_l=1,$$

(14)

where ${SI}_j$—generalized (integrated) criterion of economic efficiency of the j-th investment project;

L—number of private criteria used to evaluate the real investment efficiency, by means of which the choice of the optimal investment project is made;

m—number of investment projects in the aggregate, from which the best project is selected;

$K_l$—l-th partial criterion of the efficiency of real investments, from their set, by means of which the choice of the best investment project is made;

$K_{lj}^{bs}$—value of the l-th partial efficiency criterion, corresponding to the baseline scenario of realization of the j-th investment project;

${}_{ }{}^H{K}_{lj}^{bs}$—normalized value of the indicator $K_{lj}^{bs};$

${CBR}^m({\stackrel{̿}{K}}_{lj},K_{lj}^{bs})$—sensitivity coefficient based on the baseline variation spread of values for the l-th partial efficiency criterion of the j-th investment project;

${}_{ }{}^H{CBR}^m\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right)$—normalized value of the indicator ${CBR}^m({\stackrel{̿}{K}}_{lj},K_{lj}^{bs})$;

$K_{minl},K_{maxl}$—respectively, the minimum and maximum values of the relevant (i.e., the one that is taken into account) variation range (interval) of values of the l-th partial performance criterion;

$K_l=K_l^{+},K_l-K_l^{-}$—fixing, respectively, the positive and negative ingredients for the l-th partial performance criterion;

${\stackrel{̿}{K}}_{lj}$—interval estimation of the l-th partial efficiency criterion of the j-th investment project, corresponding to the ranges (intervals) of possible (probable) fluctuations in the initial parameters established by the interested party (decision-maker, expert);

${\underset{\_}{K}}_{lj},{\overline{K}}_{lj}$—respectively, the lower and upper limits of the interval estimation of the l-th partial efficiency criterion of the j-th investment project;

$BR({\stackrel{̿}{K}}_{lj},K_{lj}^{bs})$—is the baseline variation range of values of the l-th partial efficiency criterion for the j-th investment project;

$b_{l1},b_{l2}$—weighting coefficients for the indicators, respectively, $K_{lj}^{bs}$ and ${CBR}^m\left({\stackrel{̿}{K}}_{lj},K_{lj}^{bs}\right);$$a_l$—weight coefficient for the l-th partial efficiency criterion.

Among the compared variants of real investments, the best project should be recognized as the one for which the generalized criterion of economic efficiency acquires the highest value, in this case, ${SI}_j\in \left[\mathrm{0,1}\right], j=\overline{1,m}.$ The risk component in the proposed approach can be accounted for in a slightly different way if, instead of sensitivity coefficients based on the baseline variation spread as indicators of the risk measure in relation (3), we use the worst values within the interval estimates of partial performance criteria. Taking into account this substitution, the corresponding model, if we refer to additive convolution, takes the following form:

$${SI}_j={\sum }_{l=1}^L{a}_l(b_{l1}{}_{ }{}^H{K}_{lj}^{bs}+b_{l2}{}_{ }{}^{H}B({\stackrel{̿}{K}}_{lj})), j=\overline{1,m},_{ }^{ }_{ }^{ }$$

(15)

$${}_{ }{}^H{K}_{lj}^{bs}=\left\{\begin{array}{c}{\displaystyle \frac{K_{lj}^{bs}-K_{minl}}{K_{maxl}-K_{minl}}}, if K_l=K_l^{+}\\ {\displaystyle \frac{K_{maxl}-K_{lj}^{bs}}{K_{maxl}-K_{min}l}}, if K_l=K_l^{-}\end{array},\right.$$

(16)

$${}_{ }{}^{H}B({\stackrel{̿}{K}}_{lj})=\left\{\begin{array}{c}{\displaystyle \frac{B\left({\stackrel{̿}{K}}_{lj}\right)-K_{minl}}{K_{maxl}-K_{minl}}}, if K_l=K_l^{+} \\ {\displaystyle \frac{K_{maxl}-B\left({\stackrel{̿}{K}}_{lj}\right)}{K_{maxl}-K_{minl}}}, if K_l=K_l^{-} \end{array},\right.$$

(17)

$$B\left({\stackrel{̿}{K}}_{lj}\right)=\left\{\begin{array}{c}{\underset{\_}{K}}_{lj}, if K_l=K_l^{+} \\ {\overline{K}}_{lj}, if K_l=K_l^{-}\end{array},\right.$$

(18)

$$l=\overline{1,L}, j=\overline{1,m},$$

$$K_{minl}=min\left\{{\underset{\_}{K}}_{lj}|j=\overline{1,m}\right\}, l=\overline{1,L},$$

(19)

$$K_{maxl}=max\left\{{\overline{K}}_{lj}|j=\overline{1,m}\right\}, l=\overline{1,L},$$

(20)

$$0<b_{l1}<1,$$

(21)

$$0<b_{l2}<1,$$

(22)

$$b_{l1}+b_{l2}=1,$$

(23)

$$0<a_l<1,$$

(24)

$${\sum }_{l=1}^L{a}_l=1,$$

(25)

where $B\left({\stackrel{̿}{K}}_{lj}\right)$—the worst value of the l-th partial efficiency criterion of the j-th investment project within the interval assessment, corresponding to the ranges (intervals) of possible (probable) fluctuations established by the interested party of initial parameters;

${}_{ }{}^{H}B({\stackrel{̿}{K}}_{lj})$—normalized value of the indicator $B\left({\stackrel{̿}{K}}_{lj}\right);$

$b_{l1}, b_{l2}$—weighting coefficients for the indicators, respectively $K_{lj}^{bs}$ and $B\left({\stackrel{̿}{K}}_{lj}\right)$.

Each of the two proposed models is adaptable to the poly-interval variation of values of the initial parameters of the criterion indicators. From a practical point of view, it seems reasonable that the poly-interval variation of parameters be carried out in a single scale of gradation (successive levels). In this case, it is possible to recommend scales with a small number of gradations—three or five—given in qualitative (verbal, linguistic) form. Regarding a three-level scale, its individual gradations can be qualitatively labeled as follows: small variation, medium variation, significant (substantial) variation. If a five-level scale is used, its individual gradations can be labeled as sufficiently small variation, small variation, medium variation, significant variation, or sufficiently large variation. Taking into account the remarks made, the desired adaptations of the models under consideration can be constructed as follows, when taking into account the risk aspect by means of sensitivity coefficients based on the base spread of variation:

$${SI}_j={\sum }_{l=1}^L{a}_l\left[b_{l1}{}_{ }{}^H{K}_{lj}^{bs}+b_{l2}\left[{\sum }_{i=1}^n{c}_i{}_{ }{}^H{CBR}^m({\stackrel{̿}{K}}_{lj},K_{lj}^{bs})\right]\right], j=\overline{1,m},$$

(26)

$${}_{ }{}^H{K}_{lj}^{bs}=\left\{\begin{array}{c}{\displaystyle \frac{K_{lj}^{bs}-K_{minl}}{K_{maxl}-K_{minl}}}, if K_l=K_l^{+}\\ {\displaystyle \frac{K_{maxl}-K_{lj}^{bs}}{K_{maxl}-K_{min}l}}, if K_l=K_l^{-}\end{array}, \right. l=\overline{1,L}, j=\overline{1,m},$$

(27)

$$K_{minl}=min\left\{{\underset{\_}{K}}_{lj}^{(i)}|j=\overline{1,m}, i=\overline{1,n}\right\}=min\left\{{\underset{\_}{K}}_{lj}^{(n)}|j=\overline{1,m}\right\},l=\overline{1,L},$$

(28)

$$K_{maxl}=max\left\{{\overline{K}}_{lj}^{(i)}|j=\overline{1,m}, i=\overline{1,n}\right\}=max\left\{{\overline{K}}_{lj}^{(n)}|j=\overline{1,m}\right\},l=\overline{1,L},$$

(29)

$${}_{ }{}^H{CBR}^m({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs})=1-{CBR}^m\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right),$$

(30)

$${CBR}^m\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right)=\left\{\begin{array}{c}{\displaystyle \frac{BR({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs})}{K_{lj}^{bs}-K_{minli}}}, if K_l=K_l^{+} \\ {\displaystyle \frac{BR({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs})}{K_{maxli}-K_{lj}^{bs}}}, if K_l=K_l^{-} \end{array},\right.$$

(31)

$$BR\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right)=\left\{\begin{array}{c}{K}_{lj}^{bs}-{\underset{\_}{K}}_{lj}^{(i)}, if K_l=K_l^{+} \\ {\overline{K}}_{lj}^{(i)}-K_{lj}^{bs}, if K_l=K_l^{-}\end{array},\right.$$

(32)

$$l=\overline{1,L}, j=\overline{1,m}, i=\overline{1,n},$$

$$K_{minli}=min\left\{{\underset{\_}{K}}_{lj}^{(i)}|j=\overline{1,m}\right\}, l=\overline{1,L}, i=\overline{1,n},$$

(33)

$$K_{maxli}=max\left\{{\overline{K}}_{lj}^{(i)}|j=\overline{1,m}\right\}, l=\overline{1,L}, i=\overline{1,n},$$

(34)

$$0<c_i<1,$$

(35)

$${\sum }_{i=1}^n{c}_i=1,$$

(36)

$$0<b_{l1}<1,$$

(37)

$$0<b_{l2}<1,$$

(38)

$$b_{l1}+b_{l2}=1,$$

(39)

$$0<a_l<1,$$

(40)

$${\sum }_{l=1}^L{a}_l=1,$$

(41)

where ${CBR}^m\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right)$—sensitivity coefficient based on the indicator of the baseline variation spread of values for the l-th partial efficiency criterion of the j-th investment project, within the i-th gradation of the degree of variation of parameter values;

${}_{ }{}^H{CBR}^m({\stackrel{̿}{K}}_{lj}^{(i)},_{ }^{ }{K}_{lj}^{bs})$—normalized value of the indicator ${CBR}^m\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right);$

$K_{minl},K_{maxl}$—respectively, the minimum and maximum values of the relevant range (interval) of variation of values of the l-th partial efficiency criterion;

${\stackrel{̿}{K}}_{lj}^{\left(i\right)}$—interval assessment within the i-th gradation of the l-th partial efficiency criterion of the j-th investment project, assuming that ${\stackrel{̿}{K}}_{lj}^{\left(i\right)}\subset {\stackrel{̿}{K}}_{lj}^{\left(i+1\right)}, i=\overline{1,n-1};$

${\underset{\_}{K}}_{lj}^{(i)},{\overline{K}}_{lj}^{(i)}$—respectively, the lower and upper boundaries of interval estimation of the l-th partial efficiency criterion of the j-th investment project within the i-th gradation of the degree of variation of parameter values;

$BR\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right)$—is the baseline variation range of values of the l-th partial efficiency criterion for the j-th investment project within the i-th gradation of the degree of variation of parameter values;

$K_{minli},K_{maxli}$—respectively, the minimum and maximum values of the relevant range (interval) of variation of values of the l-th partial efficiency criterion within the i-th gradation of the degree of variation of parameter values; and within the i-th gradation of the degree of variation of parameter values;

$c_i$—weighting factor for the indicator ${CBR}^m\left({\stackrel{̿}{K}}_{lj}^{(i)},K_{lj}^{bs}\right);$

$b_{l1}$—weighting factor for the indicator $K_{lj}^{bs}$;

$b_{l2}$—weight coefficient of the unit based on a set of sensitivity coefficients within the accepted system of gradations of the degrees of variation of the parameter values for the l-th partial efficiency criterion.

When modeling the risk component using marginal adverse values, we have the following:

$${SI}_j={\sum }_{l=1}^L{a}_l\left(b_{l1}{}_{ }{}^H{K}_{lj}^{bs}+b_{l2}\left[\sum _{i=1}^n{c}_i{}_{ }{}^{H}B({\stackrel{̿}{K}}_{lj}^{(i)})\right]\right),j=\overline{1,m},$$

(42)

$${}_{ }{}^H{K}_{lj}^{bs}=\left\{\begin{array}{c}{\displaystyle \frac{K_{lj}^{bs}-K_{minl}}{K_{maxl}-K_{minl}}}, if K_l=K_l^{+}\\ {\displaystyle \frac{K_{maxl}-K_{lj}^{bs}}{K_{maxl}-K_{min}l}}, if K_l=K_l^{-}\end{array}, \right. l=\overline{1,L}, j=\overline{1,m},$$

(43)

$${}_{ }{}^{H}B({\stackrel{̿}{K}}_{lj}^{(i)})_{ }^{ }=\left\{\begin{array}{c}{\displaystyle \frac{B({\stackrel{̿}{K}}_{lj}^{(i)})-K_{minl}}{K_{maxl}-K_{minl}}}, if K_l=K_l^{+} \\ {\displaystyle \frac{K_{maxl}-B({\stackrel{̿}{K}}_{lj}^{\left(i\right)})}{K_{maxl}-K_{minl}}}, if K_l=K_l^{-} \end{array},\right.$$

(44)

$$B\left({\stackrel{̿}{K}}_{lj}^{(i)}\right)=\left\{\begin{array}{c}{\underset{\_}{K}}_{lj}^{(i)}, if K_l=K_l^{+}\\ {\overline{K}}_{lj}^{(i)}, if K_l=K_l^{-} \end{array},\right.$$

(45)

$$l=\overline{1,L}, j=\overline{1,m}, i=\overline{1,n},$$

$$K_{minl}=min\left\{{\underset{\_}{K}}_{lj}^{(i)}|j=\overline{1,m}, i=\overline{1,n}\right\}=min\left\{{\underset{\_}{K}}_{lj}^{(n)}|=\overline{1,m}\right\}, l=\overline{1,L},$$

(46)

$$K_{maxl}=max\left\{{\overline{K}}_{lj}^{(i)}|j=\overline{1,m}, i=\overline{1,n}\right\}=max\left\{{\overline{K}}_{lj}^{(n)}|j=\overline{1,m}\right\}, l=\overline{1,L},$$

(47)

$$0<c_i<1,$$

(48)

$${\sum }_{i=1}^n{c}_i=1,$$

(49)

$$0<b_{l1}<1,$$

(50)

$$0<b_{l2}<1,$$

(51)

$$b_{l1}+b_{l2}=1,$$

(52)

$$0<a_l<1,$$

(53)

$${\sum }_{l=1}^L{a}_l=1,$$

(54)

where $B({\stackrel{̿}{K}}_{lj}^{\left(i\right)})$—the worst value of the l-th partial efficiency criterion of the j-th investment project within the interval assessment, corresponding to the i-th gradation of the degree of variation of parameter values;

${}_{ }{}^{H}B({\stackrel{̿}{K}}_{lj}^{(i)})$—normalized value of the indicator $B({\stackrel{̿}{K}}_{lj}^{\left(i\right)})$;

$c_i$—weighting factor for the indicator $B({\stackrel{̿}{K}}_{lj}^{\left(i\right)})$;

$b_{l1}$—weighting factor for the indicator $K_{lj}^{bs}$;

$b_{l2}$—weight coefficient of the unit based on the set of extremely unfavorable values within the accepted system of gradations of the degree of variation of parameter values for the l-th partial efficiency criterion.

Along with the approach given by relations (46) and (47), parameters $K_{minl}$, $K_{maxl}$ can be found in a slightly different way, based on arithmetic weighted averages of the lower and upper bounds of the interval estimates, respectively ${\stackrel{̿}{K}}_{lj}^{\left(i\right)}, i=\overline{1,n}:$

$$K_{minl}=min\left\{\sum _{i=1}^n{c}_i{\underset{\_}{K}}_{lj}^{(i)}|j=\overline{1,m}\right\}, l=\overline{1,L},$$

(55)

$$K_{maxl}=max\left\{{\sum }_{i=1}^n{c}_i{\overline{K}}_{lj}^{(i)}|j=\overline{1,m}\right\}, l=\overline{1,L}.$$

(56)

Thus, relevant tools have been formulated to enable sustainable economic development in risk management and uncertain economy. In particular, a model was proposed in which three local criteria, one for grouping values of the nondeterministic evaluation of the efficiency criterion and two indicators of the risk degree, are combined using additive-multiplicative convolution. The initial methodological approach of this model is an attempt to take into account the substantive and formal features of the local criteria selected for the formation of an optimal investment decision to a greater extent than for standardized (additive, multiplicative) convolutions.

4.2. Practical Application: Estimation of Economic Efficiency of Project Measures on Real Investments under Conditions of Uncertainty

The introduction of tools for assessing the economic efficiency of investment projects in conditions of uncertainty, or taking into account uncertainty in the practical activity of enterprises, enables the formation of an appropriate organizational and economic mechanism.

From the perspective of the nature of localization in relation to the enterprise, the prerequisites (determinants) of the productive use of tools for the economic justification of real investment projects, taking into account uncertainty, can be divided into two groups: external and internal. The logic for assessing the feasibility of implementation, using the systematic (systematized) analytical toolkit developed in this work, is structured around these two groups of prerequisites. If we limit ourselves to a three-level evaluation scale (low level (L), medium level (M), and high level (H)), it is expressed by the matrix presented below (

Table 1. Matrix for determining the level of expediency of the complex implementation of tools to assess the economic efficiency of investment projects at the enterprise, taking into account uncertainty.

Qualitative assessment of the level of internal environment prerequisites	Qualitative Assessment of the Level of Prerequisites of the External Environment
		L	M	H
	Low level (L)	Low	Low	Low
	Medium level (M)	Low	Medium	High
	High level (H)	Low	High	High

Source: compiled by the author.

).

A set of partial financial ratios is formed $X_i$, $i=\overline{1,n},$ reflecting various aspects of the financial conditions and business activities of the enterprise in a sufficiently complete and non-excessive manner (i.e., without duplication, if possible).

In particular, the following coefficients may be used for this purpose:

$X_1$—autonomy ratio (ratio of equity to the balance sheet currency);

$X_2$—current assets equity ratio (ratio of own current assets, which can be defined, in particular, as the difference between current assets and current assets) and short-term (current) liabilities to current assets);

$X_3$—intermediate liquidity ratio (ratio of cash, current financial investments, and current accounts receivable to short-term liabilities);

$X_4$—absolute liquidity ratio (ratio of cash and current financial investments to short-term liabilities);

$X_5$—turnover of all assets (ratio of proceeds from sales to the average value of assets for the period );

$X_6$—return on total assets (ratio of net profit to average assets for the period).

We partition the complete set $E=\left[\mathrm{0,1}\right]$ of financial and economic states of the enterprise into five subsets (intervals) $E_i,$ $i=\overline{\mathrm{1,5}}$, which define five qualitative levels (gradations) of financial and economic well-being. The classifier constructed in this way is presented in

Table 2. Classification of levels of the financial and economic well-being of the enterprise.

Interval (Gradation) of Values E	Name (Qualitative Identification) of the Interval (Gradation)
$0\le e\le 0.2$	$E_1$—marginal disadvantage
$0.2\le e\le 0.4$	$E_2$—disadvantage
$0.4\le e\le 0.6$	$E_3$—satisfactory well-being
$0.6\le e\le 0.8$	$E_4$—relative well-being
$0.8\le e\le 1$	$E_5$—comprehensive well-being

Source: compiled by the author.

.

For each financial ratio $X_i$, $i\in \left\{1,\dots ,n\right\},$ the complete set of values $B_i$, $i\in \left\{1,\dots ,n\right\}$ is divided into five subsets (intervals) $B_{ij}$, $i\in \left\{1,\dots ,n\right\}, j=\overline{\mathrm{1,5}}$, corresponding to five qualitative levels (gradations).

Within the above set of financial ratios, the growth of each ratio is as follows: $X_i$, $i\in \left\{1,\dots ,n\right\}$ predetermines an increase in the level of the financial and economic well-being of the enterprise.

If for a coefficient, which is included in the model, there is an opposite dependence, then its value should be replaced by the opposite one. The fixed assumption allows us to establish a correspondence where with interval $B_{ij}$, $i\in \left\{1,\dots ,n\right\}, j=\left\{1,\dots ,5\right\},$ there is a division of the coefficient values $X_i$, $i\in \left\{1,\dots ,n\right\}$, determining the financial and economic well-being (stability) of the enterprise at the level of the interval $X_j$, $j\in \left\{1,\dots ,5\right\}$. If we use the set of financial ratios given above, then we can construct a classifier, which is reflected in

Table 3. Classification of levels of financial ratios selected for use.

Indicator Designation	${\mathit{B}}_{\mathit{i}1}$$, \mathit{i}=\overline{\mathrm{1,6}}-$$\mathbf{Very} \mathbf{Low} \mathbf{Level} \mathbf{of} \mathbf{the} \mathbf{Indicator} {\mathit{X}}_{\mathit{i}}$	${\mathit{B}}_{\mathit{i}2}$$, \mathit{i}=\overline{\mathrm{1,6}}-$$\mathbf{Low} \mathbf{Level} \mathbf{of} \mathbf{the} \mathbf{Indicator} {\mathit{X}}_{\mathit{i}}$	${\mathit{B}}_{\mathit{i}3}$$, \mathit{i}=\overline{\mathrm{1,6}}-$$\mathbf{Medium} \mathbf{Level} \mathbf{of} \mathbf{the} \mathbf{Indicator} {\mathit{X}}_{\mathit{i}}$	${\mathit{B}}_{\mathit{i}4}$$, \mathit{i}=\overline{\mathrm{1,6}}-$$\mathbf{High} \mathbf{Level} \mathbf{of} \mathbf{the} \mathbf{Indicator} {\mathit{X}}_{\mathit{i}}$	${\mathit{B}}_{\mathit{i}4}$$, \mathit{i}=\overline{\mathrm{1,6}}-$$\mathbf{Very} \mathbf{High} \mathbf{Level} \mathbf{of} \mathbf{the} \mathbf{Indicator} {\mathit{X}}_{\mathit{i}}$
$X_1$	$0-0.15$	$0.15-0.25$	$0.25-0.45$	$0.45-0.65$	$0.65-1$
$X_2$	$-1-0$	$0-0.09$	$0.09-0.3$	$0.3-0.45$	$0.45-1$
$X_3$	$0-0.55$	$0.55-0.75$	$0.75-0.95$	$0.95-1.4$	$1.4-\infty$
$X_4$	$0-0.025$	$0.025-0.09$	$0.09-0.3$	$0.3-0.55$	$0.55-\infty$
$X_5$	$0-0.1$	$0.1-0.2$	$0.2-0.35$	$0.35-0.65$	$0.65-\infty$
$X_6$	$-\infty -0$	$0-0.01$	$0.01-0.08$	$0.08-0.3$	$0.3-\infty$

Source: compiled by the author.

.

In general, each financial ratio, $X_i$, $i=\overline{1,n}$, has its significance or weight in the integral assessment of the financial and economic well-being of the enterprise. By weighting factors ${(r}_i$, $i=\overline{1,n}),$ it is reasonable to determine with the help of Fishburne’s rule. Its essence is to establish weights based on a system of preferences between the indicators to be weighted. Regarding a mixed system of preferences, when there are equally important indicators together with the indicators that have different importance, the weighting coefficients in the Fishburne approach should be calculated according to the following formula:

$$r_i={\displaystyle \frac{k_i}{\sum _i^n{k}_i}},i=\overline{1,n}$$

(57)

$$\mathrm{Here} \mathrm{with}, k_n=1,$$

(58)

$$k_{i-1}=\left\{\begin{array}{c}{k}_i,X_{i-1}~X_i\\ k_i+1,X_{i-1}\succ X_i\end{array}\right.,i=\overline{n,2}.$$

(59)

Based on the financial statements data, the values of the ratios selected for use are calculated as $X_i$, $i=\overline{1,n}$.

The level of correspondence of the values of financial ratios obtained at the previous step to the levels of the used interval classifier is recognized (Table 3). The degree of correspondence of the value of the i-th coefficient to the j-th qualitative level (partitioning interval) of the classifier (${\lambda }_{ij}$) is equal to 1 if the value of the coefficient falls into this interval, and equal to 0 otherwise.

An integrated (integral) assessment of the level of financial and economic well-being of the enterprise is determined:

$$e={\sum }_{j=1}^5{e}_j{\sum }_{i=1}^n{r}_i\ast {\lambda }_{ij},$$

(60)

the following is assumed:

$$e_j=0.1+0.2\left(j-1\right),$$

(61)

that is $e_j$ is the average value for the corresponding partition interval of the set E (Table 2).

Appendix A presents the results of assessment of indicators characterizing specific aspects of investment activity, as well as the general financial and economic well-being of a group of domestic industrial enterprises. Based on these results, the corresponding generalized (integrated) indicators can be obtained, which are reflected in

Table 4. Rating estimates of investment activities for the group of companies in the industrial sector for 2021–2023.

№	Enterprise	Rating Assessment of Investment Activity
		2021	2022	2023
1	OJSC Volgogradneftemash	0.545	0.499	0.438
2	PJSC Novatek	0.551	0.541	0.554
3	LIMITED LIABILITY COMPANY “NORTHSTAL AVIATION COMPANY”	0.213	0.167	0.278
4	Ural Automobile Plant	0.365	0.343	0.262
5	PJSC SIBUR Holding	0.319	0.273	0.330
6	FGUP NAMI	0.370	0.315	0.224
7	PEC LLC	0.248	0.280	-
8	PJSC RusHydro	0.368	0.255	-
9	PJSC Surgutneftegas	0.377	0.386	-

Source: compiled by the author.

and

Table 5. The values of the integrated indicators of financial and economic well-being for enterprises in the industry sector for 2021–2023.

№	Enterprise	Rating Assessment of Investment Activity
		2021	2022	2023
1	OJSC Volgogradneftemash	0.523	0.523	0.523
2	PJSC Novatek	0.802	0.802	0.869
3	LIMITED LIABILITY COMPANY “NORTHSTAL AVIATION COMPANY”	0.687	0.720	0.767
4	Ural Automobile Plant	0.600	0.647	0.697
5	PJSC SIBUR Holding	0.623	0.666	0.600
6	FGUP NAMI	0.833	0.800	0.867
7	PEC LLC	0.869	0.869	-
8	PJSC RusHydro	0.837	0.837	-
9	PJSC Surgutneftegas	0.538	0.500	-

Source: compiled by the author.

.

In this case, the rating principle, combined with the use of the additive convolution tool, was used as the basis for the construction of the generalized assessment of investment activity. As for the integrated indicator of financial and economic well-being, it was found using the Nedosekin-Maksimov model detailed above.

Given the characteristics of the external environments of the enterprises under study (high levels of complexity and dynamism, and, accordingly, uncertainty), and taking into account the results of investment activity assessments and general financial and economic well-being obtained for them, there is every reason to conclude that—for a typical industrial enterprise—the systematic use of modern tools to support the adoption of project investment decisions, taking into account the uncertainty factor, is relevant.

5. Conclusions

The modern methodology for probabilistic modeling of indicators of economic efficiency of investment projects consists of two approaches or methods: analysis of probability distributions and Monte Carlo simulation modeling. In terms of analytical potential, the Monte Carlo method is the main method, while the analysis of probability distributions plays a secondary role.

The advantages of the Monte Carlo method as a tool for assessing the indicators of economic efficiency and risk in economically justifying real investments include a high degree of universality, in terms of taking into account different distribution laws and mutual relation “links” for financial and economic parameters of the investment project, as well as a high degree of computational and instrumental elaboration. The limitation of this method, which has fundamental character, is that its correct application is possible only in the presence of representative statistics, or reliable awareness, on the part of the interested party, regarding the distribution laws for random values of the initial parameters of the investment project under study.

The current development of risk measurement methodology and tools is determined by the following trends:

• Systematization of available risk indicators, including through the construction of generalized mathematical constructs (models);
• Axiomatic characterization of risk measures;
• Identification and analysis of the properties of individual risk indicators and the interaction between different indicators;
• Further elaboration, improvement, and development of ways (methods) of evaluation of those or other indicators of risk degree;
• Design and development along with the probabilistic-statistical methodology of approaches to risk formalization, the field of use of which is non-stochastic (non-probabilistic) uncertainty.

Four methodological directions or approaches to risk measurement, formed within the probabilistic paradigm of uncertainty and applicable to the economic evaluation of real investment projects, are based on the following:

• Assessment of the probability of unsatisfactory results of economic activity, their non-compliance with the target (planned, desired) or maximum permissible (threshold, critical) level;
• Assessment of losses or losses in case of unfavorable (undesirable) scenarios of economic activities materialize;
• Estimation of the variability of possible (predicted) results of economic activity;
• Quantile measures, in the framework of which the quantitative aspect of risk is modeled with the help of quantiles of the distribution of a random variable describing possible (predicted) results of economic activity.

The above methodological directions or approaches to measuring risk embody the following specific indicators of risk:

• For the first approach: an indicator of the probability of unsatisfactory results of economic activity, their non-compliance with the target (planned, desired) or maximum permissible (threshold, critical) level;
• For the second approach: the expected value of undesirable consequences (losses, losses); the coefficient of expected losses or losses;
• For the third approach (when measuring risk in absolute terms): variation spread, mean absolute deviation, dispersion or variation, standard deviation, semi-variance (semi-dispersion), seven-square deviation;
• For the third approach (when assessing the degree of risk in relative terms in relative terms): coefficient of variation, coefficient of semi-variation;
• For the fourth approach: the indicator of value (capital) at risk; its adaptation in the plane of real investment problems is the indicator of the worst possible value of the investigated criterion, which—with probability—will not be exceeded (such a limit value in the unfavorable direction is effective).

As part of developing the theoretical and probabilistic methodological apparatus of risk measurement, the section under discussion proposes, among other things, the indicator of target variation spread, the coefficient of target semi-variation, and generalized versions of the modified coefficients of variation and semi-variation, which, unlike their original versions, assume any, not only positive, ingredient (direction of optimization) of the analyzed economic indicator (efficiency criterion).

For investment decision support tasks, models with different accounting schemes are proposed for use, within which, various compromise schemes for sustainable development can be implemented. Hence, there is a need for balanced development across different methodological directions (approaches) and methodological means (tools). Among other things, further development of the methodological apparatus for the rationalization of investment decisions under uncertainty (based on flexible consideration of priorities, where the principle of optimality is implemented by means of one or other variants of criteria convolution, i.e., a scalar function composed of local criterion evaluations and weighting coefficients for sustainable economic development) remains relevant. At the same time, the comprehensive consideration of the risk aspect is of particular interest.