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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.1 pp.35-46
DOI : https://doi.org/10.7232/iems.2020.19.1.035

# Improving the Investment Attractiveness of Energy Facilities in the Process of Environmental Modernization

Vitaly G. Korolev*
Basic Department of FAS of Russia, Plekhanov Russian University of Economics, Moscow, Russian Federation
*Corresponding Author, E-mail: vit-korolev@murdoch.in
November 22, 2019 January 11, 2020 January 16, 2020

## ABSTRACT

Energy facilities have to meet not only the environmental requirements, but also the economic attractiveness for investors. In this regard, the issue of determining the parameters that govern the structural features of ensuring the safety of energy facilities becomes relevant. Given that the main trend is an increase in the share of alternative or environmentally friendly energy and a decrease in the share of fossil fuels, an increasing interest in the world is identified with the development of atomic energetics. The paper elaborates on the matter of standardizing the environmental safety for enterprises in the energy sector. The authors suggest using a tool for determining economic feasibility upon the formation of the alternative energy industry investment project. The paper showcases the correspondence of the suggested financial model with environmental and technical requirements, as well as presents a model that meets international requirements for the design of environmentally friendly facilities. The practical significance of the study is determined by the fact that the developed principles for the design of alternative energy systems are related primarily to the requirements of a financially sound investment model. The authors suggest expanding the established toolkit for the tasks of designing separate alternative energy facilities.

## 1. INTRODUCTION

Tapping the experience in conducting operations at facilities with radiation-nuclear technologies and on the basis of cost-benefit analysis of alternative options, ultimate design and technological solutions should be selected, providing the lowest possible collective dose (Kazantzi et al., 2013). For the selected option, optimization of radiation protection shall be performed (Freeman, 2006). At this stage, the optimal access paths are selected, the applicability of shielding, decontamination, dust suppression, and other measures aimed at reduction of the collective dose is determined (Weiss et al., 1986). Activities for the implementation of any project, including the facilitation of radiation protection, should not lead to aggravation of the conditions for the implementation of other projects (Tomabechi et al., 1990). Operations should be planned in such a way as to ensure minimal additional radioactive contamination of structures and equip-ment.

At the project implementation stage, measures should also be taken to further optimize the process of conducting operations based on the As Low As Reasonably Achievable (ALARA) principle (is a safety principle designed to minimize radiation doses and releases of radioactive materials). Thus, the current dynamics analysis of the collective dose changes will assist in planning, facilitating early detection of a dose increase in comparison with the expected values (Shigemitsu et al., 2005). This will allow to take corrective measures and reduce the exposure dose of personnel. Application of special simulators will lead to a reduction in the operation time, a reduction in individual and collective doses, as well as a reduction in the estimated cost of operations due to optimization of technological operations and increase in the level of staff training (Darvas, 1990). Upon project implementation, (Piri Damagh and Faure, 2016) it is necessary to maximize the use of existing equipment and infrastructure, consider the practices of similar operations, coordinate with other work at facilities with radiation-nuclear technologies.

In the process of designing operations at any facilities with radiation-nuclear technologies, it will be necessary to ensure strict adherence to the provisions and rules in the field of radiation safety (Radandt et al., 2008). For this, it is necessary to ensure the selection of optimal design solutions and elaborate measures for radiation safety. An integrated approach to the optimization of design solutions, the organization of radiation protection should be based on the general concept of radiation safety, which is built on the ALARA principle.

## 2. MATERIALS AND METHODS

In accordance with the recommendations in the practice of radiation safety, radiation protection of humans and the environment, the justification of the applicability of a certain measure is based on the procedure for comparing the benefits from the application of the measure with the corresponding costs associated with its application. At the same time, costs are understood as a general extent of all negative factors that arise or may arise in the process or as a result of the implementation of a measure. These may be harm to health, economic losses, psychological, political, and other consequences (Logan et al., 1990). To compare costs and benefits, they should be quantified on the same scale. The cost of each measure shall be determined by the project. The determination of the monetary equivalent of both the benefits and the costs of implementing this measure is associated with certain difficulties. Therefore, in the decision-making process, it is important to correctly juxtapose all the costs and benefits of the application of the given measure (Conn et al., 1990). The measure will be justified if the benefit from its implementation will be greater than the total costs associated with its implementation, and optimal if the net benefit from its implementation – the difference between the total benefit and total costs – will be maximum. All the main methods for assessing the justification and optimizing design decisions in intervention situations are based on a structural approach. The general scheme of this ap-proach, its main elements and sequence are displayed in Figure 1.

This simplified scheme does not reflect all pos-sible situations that may arise upon performing research. For instance, this diagram does not include feedback between stages or the ability to view certain intermediate results in the process of analysis.

At the first stage, the choice of project imple-mentation options is very important. It consists of three operations (Piri Damagh and Faure, 2016): listing of all possible options; a preliminary analysis to cut off those options that are obviously impossible; selection of options that will directly take part in the optimization procedure.

After determination of the problem, the scope of work and possible action options, a very important stage is to determine the research criteria (Goss, 2002). Failure to include criteria that may affect the effectiveness of measures, or an incorrect assessment of their role may lead to an erroneous final result (Gribust, 2019). Upon optimization, not only those criteria that (directly or indirectly) belong to the criteria for radiation protection may be considered, but also the criteria outside its scope (Gribust, 2019). From a practical standpoint, the definition of criteria for the analysis of environmental safety of options for performing or not performing operations at facilities with radiation-nuclear technologies using environmental safety indicators was carried out (Veselov, 2011). Envi-ronmental safety indicators of radiation protection are the doses, costs, risks of potential accidents, etc., which can be divided into two categories. The first category includes indicators that should always be included in the analytical optimization procedure (Devault et al., 2017). These are the cost of the measure and the value of the collective dose (Hansen, 2000). The second category includes indicators that are not always included in the analysis, but they can be included by decision makers to refine the analysis in order to obtain more accurate results. These are the individual dose distribution, the probability of events, the reliability of a constructive solution, etc. (Hohl and Tisdell, 1993). When all environmental safety indicators that are necessary to consider are defined, it may happen that some of them cannot be quantified for inclusion in the analytical procedure (Nordenstam, 1996). In this case, the decision maker has to qualitatively assess these environmental safety indicators, and the result of this assessment shall then be involved in further analysis. (Streffer et al., 2003). To facilitate the objective of finding the best option, the theory of expert systems, methods of multicriteria analysis and the theory of fuzzy models may be applied (Osipov, 2019).

In this case, the effectiveness of the criterion is quite minimal, therefore, the objective should be consid-ered as single-criterion, while the choice issue is always defined as multicriteria (Bull et al., 1998). To solve such problems, the International Commission on Radiation Protection (ICRP) developed a number of recommendations. The most important component of the ICRP optimization approach is the quantitative determination of the results of optimization studies in applicable cases.

## 3. RESULTS AND DISCUSSION

Proceeding from the analysis of provisions, rules and recommendations, decision making criteria for as-sessing the environmental safety level of facilities with radiation-nuclear technologies are defined as fol-lows:

• 1) nuclear incident prevention;

• 2) radiation protection criteria, which consist of:

• - the implementation of radiation protection of personnel, the population and the environment in accordance with the requirements of laws, provisions and rules adopted in the country;

• - the implementation of radiation protection of personnel in accordance with the ALARA prin-ciple;

• - assessment of radiation consequences for per-sonnel, which includes exposure, associated with both the activities upon the implementation of the main objective, and presence in the exclusion zone;

• - assessment of the radiological consequences of potential accidents for personnel, the population and the environment;

• 3) Radioactive Waste Management (RWM) cri-teria;

• 4) general technical criteria, which consist of: creating defenсe in depth; quality assurance; ex-perience exchange; human factor inclusion; application of proven engineering practice.

It’s should be to determine the specified criteria for the performance or non-performance of works at facilities with radiation-nuclear technologies for practical use with help of the following seven environmental safety indicators:

• - financial assessment of operations (including protection upon operations with radioactive waste (RWM));

• - received effective dose of the collective type (CED) which is inevitably obtained by a person in the implementation of measures;

• - the risk of radiological contamination among personnel (excluding CED, which is inevitably obtained by a person upon the implementation of measures);

• - collective radiological risk of the population (including potential exposure of the population in the event of possible emergencies upon perfor-mance or non-performance of scheduled opera-tions);

• - financial risk (including the potential costs of eliminating the consequences of potential emer-gencies upon performance or non-performance of scheduled operations);

• - project implementation time;

• - possible uncertainty value.

Certain indicators of environmental safety have a different nature of formation and significance, which does not allow a direct assessment of the level of environmental safety of facilities with radiation-nuclear technologies. To perform such an assessment, it is necessary to determine the adequate transformation of these environmental safety indicators into a dimensionless mathematical space to obtain an integral estimation of the level of environmental safety. For mathematical transformation in this quality, this paper suggests to use the method of multicriteria analysis of a function of sufficient significance.

Within the framework of the set objective, we have developed an algorithm for its implementation by making the optimal decision (Figure 2).

The first step of this algorithm is to analyse the impact of the designed operations. Proceeding from the analysis, a list of possible options is developed (step 2) and, using a qualitative analysis, the most promising ones are selected for further study (step 3). After that, the selection criteria for the option are specified (step 5) and numerical indicators of environmental safety are calculated with specification of limits of the parameter changes (step 6). For quality indicators of environmental safety, a group of experts evaluates options that are analyzed with specification of limits of the parameter changes (step 7) and determine the weight of each of the criteria (step 8). Next, for each option, a generalized performance indicator $W i$ is calculated (step 9), among which the maximum value $W m a x$ is determined and the nearest largest value of a generalized performance indicator $W m a x − 1$ (the maxi-mum of the remaining values) (step 10). Since some of the environmental safety indicators are evaluative in nature, it is necessary to analyze the sensitivity of the pa-rameter variation obtained from the choice within the limits of the changes (step 11), that is, to determine whether the choice of the optimal option will change upon different values of the calculation parameters outside the changes. Next, the minimum value of the generalized efficiency indicator $m i n ( W m a x )$ is determined for the best option upon varying the parameters and the maximum value of the generalized performance indicator $m a x ( W m i n − 1 )$ for the closest option upon variation of parameters. If $m i n ( W m a x ) > m a x ( W m a x − 1 )$ (step 12), then the resulting option will be optimal (step 13). If $m i n ( W m a x )$ $≤ m a x ( W m a x − 1 )$ (step 12), then either all options are equivalent to each other, or it is necessary to readjust the list of criteria, clarify the limits of the parameter changes, evaluate the quality indicators of environmental safety and determine the weight of the criteria.

The next step of the algorithm is to determine the criteria for making the optimal decision using envi-ronmental safety indicators. Considering the requirements for the performance of operations on the economic evaluation of the object of study, the following indicators of environmental safety and optimal decision making are suggested:

• 1) $C$ : the cost of operations (including the costs of radiation protection and RWM);

• 2) $D$ : collective effective dose (CED) upon the implementation of measures;

• 3) $R p$ : collective radiological risk of personnel (excluding CED upon the implementing the measure);

• 4) $R N$ : collective radiological risk of the popu-lation;

• 5) $F$ : financial risk;

• 6) $T$ : project implementation time;

• 7) $Q$ : the magnitude of the risk of deterioration of the conditions for routine maintenance, extraction of FCM, etc. (the possible uncertainty value).

Selected criteria are often predominantly quantitative. The values of environmental safety $C$ (in thousands, dollars), $D$ (in Sieverts) and $T$ (in years) indicators are determined in the process of the project implementation development.

Personnel (population) collective radiological risk $R p$ $( R N )$ can be represented as follows:

$R p , N = r E p , N ∑ i ∈ Θ ( p i ∑ j ∈ Θ N j D j )$
(1)

where $r E p = 5.6 × 10 − 2 , 3 в − 1 ( r E N = 7.3 × 10 − 2 , 3 в − 1 )$ – risk factor per unit of dose, years;

• $p i$ : probability of the event leading to the exposure of personnel (population) established by the regulatory document, year;

• $N J$ : number of personnel (population),

• $D j$ : dose at a certain distance and at a certain concentration in a certain place, direction from the scene of the accident, Sv;

• $Ω$ : multitude of places of localization of per-sonnel (population) in the sanitary protection zone of an object with radiation-nuclear tech-nologies.

Financial risk $F$ may be represented as fol-lows:

$F = ∑ i ∈ Ω p i C i p$
(2)

where

• $p i$ : the probability of an event leading to exposure of personnel or population;

• $C i p$ : accident liquidation cost of with probability $p i$ , in thousands dollars; $Ω$ – the multitude of accidents, the radiological risk of which for the personnel (population) exceeds the value of the minimum environmental risk.

This indicator displays that it is determined mainly by quality. This qualitative type assessment can be performed in three different types. And with that, expert techniques may be applied depending on the choice of methodology.

To calculate the possible uncertainty value (risk of deterioration of the conditions for routine maintenance, extraction of FCM, etc.) $Q$ , it’s should be use the direct assessment method based on a universal 9-point scale. Having analyzed the possible qualitative indicators of environmental safety, the rating scale will look as follows: 1 – risk that can be neglected; 3 – low risk; 5 – constant risk; 7 – high risk; 9 – unacceptable risk; 2, 4, 6, 8 – intermediate values between adjacent scale values.

By a limitation that can be considered a discrepancy in the opinions of experts, the authors determine the size of the discrepancies between the individual structures by 15-20%. Moreover, such a discrepancy is permitted only under the condition that there are no other methods of verification of the received data and their interpretation.

The function of acceptable significance for our research is determined by the fact that it is first necessary to determine which indicators are significant and which are not. If quantitative indicators still prevail, then they should be considered as priority ones. Since the indicator of possible uncertainty, denoted as $D$ prevails, then parameters of a qualitative type are considered primary, and quantitative are secondary. If the indicators stably remain constant and do not have a clearly defined priority, then we should talk about such calculation parameters as $Q$ – collective effective dose (CED), $R p$ – collective radiological risk of personnel and $R N$ – collective radiological risk of the population not risk averse or indifferent, and indicators $C$ – cost of operations and $F$ – financial risk, may be exposed or indifferent to risk. For indicators $T$ – project implementation time and $Q$ – possible uncertainty value, the authors assume that they are indifferent to risk.

The next stage of the analysis is that the authors calculate the limiting values for quantitative indicators, which are characterized by the level of environmental safety. If all parameters are considered, it’s obtain the following equation:

$u ( x m i n ) = u ( 0 ) = 1$
(3)
$u ( x m a x ) = 0$
(4)

If the upper limit remains sufficiently critically determined, then the economic types of investment attractiveness for objects may be considered on the basis of calculating the approximation to the axis of values. The authors define these functions at points $x 0 , 5 = ( x m a x − x m i n ) / 2 , x 0 , 25 = ( x m a x − x m i n ) / 4$ and $x 0 , 75 = 3 × ( x m a x − x m i n ) / 4 : U ( x 0 , 5 ) , U ( x 0 , 25 ) , U ( x 0 , 75 )$ . Permissible significance in this case can be expressed graphically.

After studying the experience of using the method of multicriteria analysis of a function of sufficient significance and studying possible types of functions of sufficient significance for all quantitative indicators, the following types of functions of sufficient significance should be used.

For indicators that have an upper limit, monotonically decrease and are indifferent to risk, simple linear functions of the form (the general form of such functions was described in the works of American scientists, R. Keeney and H. Raiffa (Keeney and Raiffa, 1993)) are used:

$u i ( x ) = 1 − x x m a x$
(5)

where $i = 1 , … , 7 ;$ $x$ – indicator value; $x m a x$ – maxi-mum indicator value.

A graphic display of the form of the above functions of sufficient significance is provided in Figure 3.

If the investment limits, which should be mini-mally designated for the project, are calculated with a significant problem or significant resources are spent on it, then it is worthwhile to say that this is reflected due to the implementation of the values of certain functions. Thus, for example, if a small value of environmental risk $u j ( x )$ has high values, then it should be insensitive to the values of the argument. If the regions of large values are in the region of the function $u j ( x )$ , then the value of the minimum investment is impossible in principle. And with that, the function $u j ( x )$ X will remain stable.

Having researched the possible options for the form of a function of sufficient significance with the above properties, the function $u ( x ) = e − a x + b e − c x$ was selected for the values $x , { a , b , c } > 0$ . This function is one of the group of functions that were proposed by the American scientists, R. Keeney and H. Raiffa, to con-struct functions of sufficient significance that reflect a downward risk aversion. In the selected function, it was first accepted that $b = 0$ and the behaviour of the graph of the function $u ( x ) = e − a x$ was analysed. After the analysis, it was decided to continue to consider functions of the form $u ( x ) = E X P ( − a x n )$ . Having researched the dependence of the graph of this function on the value of the degree of the variable $x$ , it was decided that $n = 4$ corresponds to the set objectives. Then our function will have the form $u ( x ) = E X P ( − a x 4 )$ . A graphic image of the group of the above functions depending on the selected constant with an argument is displayed in Fig. 4. This figure shows that at $α = 0 , 7$ the value of the function $u ( 0 , 5 ) = 0 , 5$ , so finally the function had the form $u ( x ) = E X P 0 , 7 x 4$ . In the process of adapting the specified function to the listed conditions, a function of sufficient significance of a special kind was found that meets the set requirements and has the following form:

$u i ( x ) = E X P ( − 0 , 7 ( x / A ) 4 )$
(6)

where i =1,..,7; x – criterion value; F – constant, defined as the average value of the i-th environmental safety indicator of all options considered;

A graphical representation of the above func-tions of sufficient significance is displayed in Fig-ure 4.

To compare qualitative indicators of environ-mental safety, a system of expert assessments will be used by means of direct assessment method based on a universal 9-point scale. Then a function of sufficient significance for qualitative indicators can be determined as follows (i=1, ..., 7):

$u i ( 9 ) = 0 u i ( 8 ) = 0 , 125 u i ( 7 ) = 0 , 25 u i ( 6 ) = 0 , 375 u i ( 5 ) = 0 , 5 u i ( 4 ) = 0 , 625 u i ( 3 ) = 0 , 75 u i ( 2 ) = 0 , 875 u i ( 1 ) = 1$
(7)

The graph $Q$ of a function of sufficient significance for qualitative indicators of environmental safety is discrete in nature, since the function is defined only at 9 points (Figure 5).

Summing up the procedure for constructing functions of sufficient significance, the following conclusion should be made:

• 1) for a qualitative indicator of environmental safety , a function of sufficient significance has the form displayed in Figure 5;

• 2) for all quantitative indicators of environmental safety in the presence of an upper limit of the criterion value, one can use the function of sufficient significance, presented in Figure 3;

• 3) for all quantitative indicators of environmental safety in the absence of an upper limit of the indicator value, one can use the function of sufficient significance, presented in Figure 4.

Weighting factors are always determined on the basis of allocation of specialized indicators. For each -th environmental safety indicator, let denote the worst value and $b j$ – the best value. Then for positively oriented scales, it should be $w j ≤ x j ≤ b j$ . Let $I$ be the set of all numbers of indicators of environmental safety, $T$ – the subset of the set $I$ , and $T$ – its complement to $I$ , $T ¯ = I − T$ . Using $X T$ it’s necessary to denote the profile, wherein the components $x j$ of the level $b j$ for all $j ∈ T$ and $w j$ for $j ∈ T ¯$ . As $u j ( w j = 0 )$ and $u j ( b j ) = 0$, $u ( X T ) = ∑ i ∈ T k i$ then . It’s should be also define $K ( T ) = ∑ i ∈ T k i$ . Furthermore, when $T$ is a singleton set ${ J }$ , there is $u ( X T ) = k j = K ( { j } )$ . Function $K$ , defined on subsets of the set $I$ has the usual properties of a probability measure: , if S and T do not meet.

The search for the function $K$ is related to the objective of establishing the corresponding probability distribution on a finite sample space. Very often, upon determining the weight degree $K$ , as well as upon finding the numerical values of the probability measure, it is impractical to start by determining the numerical parameter of an atomic character, characterized by additional calcucalculations, such as numbers $k 1 , k 2 , …$ Instead, it may be more convenient to first set the numerical values for the subsets (that is, determine $K ( T )$ or special subsets), and then define conditional measures. In addition, to find the function $K ( T )$ , one can apply a system of expert estimates. With such determination of weighting factors, all further calculations are based on expert estimates, and the final result will depend on the quality of the input data. In order to obtain the most reliable input data, it is necessary to create an algorithm or an automated system for obtaining data by expert estimates. One way to achieve this goal is to create a problem-oriented system for determining the probabilities of critical events, in which experts and decision makers operate in an organized environment using computer simulation. The starting point for this is modern ideas in the field of technology development, expert systems, decision making theory.

The weighting factors can often be determined on the basis of scaling. Thus, having considered the possibility of using expert assessment methods, it’s shows that it is advisable to scale the assessment by 9 positions. Weighting factors can be obtained by the standard formula:

$k l = M l ∑ i = 1 n M i$
(8)

where $M l$ – median expert assessment for determining the weight of environmental safety indicator $l$ based on a universal 9-point scale.

As a basic assumption, the authors assume that all indicators are equivalent to each other: $k 1 = k 2 = k 3 = k 4 = k = k 6 = k 7 = 1 7$ . If there is any data or the relationship between the selected indicators or the decision maker has their own idea of the significance of environmental safety indicators, then it is necessary, upon using the formula (8) with help of the direct assessment method based on a 9-point scale, to determine the weighting factors according to the priorities of the indicators. As a measure of the con-sistency of expert opinions, one can accept a discrepancy in the estimates of not more than two points, which corresponds to about 20% of the error of the whole scale. There is no point in getting more accurate values, since quantitative criteria cannot be calculated with greater accuracy (instrument and measurement errors, designing work, source data, etc.).

Let $C 1 , C 2 , … , C M$ be the cost of the proposed M options, $D 1 , D 2 , … , D M$ – the CED, $R p 1 , R p 2 , … , R p M$ – the collective radiological risk of personnel; $R N 1 , R N 2 , … , R N M$ – the collective radiological risk of population; $F 1 , F 2 , … F M$ – financial risk; $T 1 , T 2 , … , T M$ – project implementation time; $Q 1 , Q 2 , … , Q M$ – the values of the risk of worsening of working conditions for all proposed M options. Then to compare options, it’s should be use the following formula:

$W i = ∑ l = 1 5 k l E X P ( − 0 , 7 ( M U i l / ( ∑ l = 1 M U j l ) ) 4 + k 6 u 6 ( T i ) + k 7 u 7 ( Q i ) )$
(9)

In the event that environmental safety indicators have the same ranks, formula (9) can be written as follows:

$W i = ( ∑ l = 1 5 E X P ( − 0.7 ( M U i l / ( ∑ j = 1 M U j l ) ) 4 + u 6 ( T i ) + u 7 ( Q i ) ) ) / 7$
(10)

The optimal option is the one that has the highest value of a generalized performance indicator $W m a x = m a x ( W i , i = 1 , … , M )$ .

To highlight the most balanced forms of imple-menting environmental safety parameters, the authors consider such a parameter as the optimal value of the calculated number of indicators. It’s should be to derive a generalized indicator of effectiveness $W i$ , which, in its turn, defines the $W m a x$ and forms the closest value of effectiveness $W m a x − 1$ (maximum value of the remainder). Next, the authors consider the minimum generalized value $m i n ( W m a x )$ and its maximum value with current parameters $m a x ( W m a x − 1 )$ . If $m i n ( W m a x ) > m a x ( W m a x − 1 )$ , then the optimal type option has gotten. However, if $m i n ( W m a x ) ≤ m a x ( W m a x − 1 )$ , then the authors consider these options as equivalent and thus it becomes possible to clarify the parameters of the investment project.

Using formula (10), it should be written down $W m a x$ and $W m a x − 1$ as follows:

$W m a x = ( E X P ( − 0.7 ( M C m a x / ( ∑ j = 1 M C j ) ) 4 ) + + E X P ( − 0.7 ( M D m a x / ( ∑ j = 1 M D j ) ) 4 ) + + E X P ( − 0.7 ( M R p m a x ) / ( ∑ j = 1 M R p j ) ) 4 ) + E X P ( − 0.7 ( M R N m a x / ( ∑ j = 1 M R N j ) ) 4 ) + E X P ( − 0.7 ( M F m a x / ( ∑ j = 1 M F j ) ) 4 ) + u 6 ( T m a x ) + u 7 ( Q m a x ) / 7$
(11)
$W m a x − 1 = ( E X P ( − 0 , 7 ( M C m a x − 1 / ( ∑ j = 1 M C j ) ) 4 ) + + E X P ( − 0.7 ( V D m a x − 1 / ( ∑ j = 1 M D j ) ) 4 ) + + E X P ( − 0.7 ( M R p m a x − 1 / ( ∑ j = 1 M R N j ) ) 4 ) + + E X P ( − 0.7 ( M F m a x − 1 ) / ( ∑ j = 1 M F j ) ) 4 ) + u 6 ( T m a x − 1 ) + u 7 ( Q m a x − 1 ) / 7$
(12)

Let $C i , Δ C$ be option implementation cost and its relative error $D i$ , $Δ D$ – CED and its relative error, $R p i$ , $Δ R p$ – the collective radiological risk of personnel and its relative error; $R N i , Δ R N$ – the collective radiolog-ical risk of the population and its relative error; $F i , Δ F$ – the financial risk and its relative error; $T i , Δ T$ – project realization time and its relative error; $Q i , Δ Q$ – the values of the risk of worsening of working conditions and its relative error for all proposed M options $( i = 1 , … , M )$ . Then, in consideration of the errors, the values of all environmental safety indicators will have the following form:

• $( 1 ± Δ c ) C i$ : the cost of implementing the option, with consideration of the error of deter-mination;

• $( 1 ± Δ R p ) R p i$ : CED, with consideration of the error of determination;

• $( 1 ± Δ R N ) R N i$ : collective radiological risk of per-sonnel with consideration of the error of determination;

• : collective radiological risk of popu-lation with consideration of the error of determination;

• $( 1 ± Δ F ) F i$ : financial risk, with consideration of the error of determination;

• $( 1 ± Δ T ) T i$ : project realization time, with consid-eration of the error of determination;

• $( 1 ± Δ Q ) Q i$ : the values of the risk of worsening of working conditions, with consideration of the error of determination.

Using all of the above and the fact that all functions of sufficient significance are defined on the interval [0, 1], now calculate $m i n ( W m a x )$ and $m a x ( W m a x − 1 )$ :

$min ( W m a x ) = ( min ( E X P ( − 0.7 ( M ( 1 ± Δ C ) C m a x / ( ∑ j = 1 M ( 1 ± Δ C ) C j ) ) 4 ) ) + min ( E X P ( − 0.7 ( M ( 1 ± Δ D ) D m a x / ( ∑ j = 1 M ( 1 ± Δ D ) D j ) ) 4 ) ) + min ( E X P ( − 0.7 ( M ( 1 ± Δ R p ) R p m a x / ( ∑ j = 1 M ( 1 ± Δ R p ) R p j ) ) 4 ) ) + min ( E X P ( − 0.7 ( M ( 1 ± Δ R N ) R N m a x / ( ∑ j = 1 M ( 1 ± Δ R N ) R N j ) ) 4 ) ) + min ( E X P ( − 0.7 ( M ( 1 ± Δ F ) F m a x / ( ∑ j = 1 M ( 1 ± Δ F ) F j ) ) 4 ) ) + min ( ( 1 ± Δ T ) u 6 ( T m a x ) ) + min ( ( 1 ± Δ Q ) u 7 ( Q m a x ) ) / 7$
(13)
$max ( W m a x − 1 ) = ( max ( E X P ( − 0.7 ( M ( 1 ± Δ C ) C m a x − 1 / ( ∑ j = 1 M ( 1 ± Δ C ) C j ) ) 4 ) ) + max ( E X P ( − 0.7 ( M ( 1 ± Δ D ) D m a x − 1 / ( ∑ j = 1 M ( 1 ± Δ D ) D j ) ) 4 ) ) + max ( E X P ( − 0.7 ( M ( 1 ± Δ R p ) R p m a x − 1 / ( ∑ j = 1 M ( 1 ± Δ R p ) R p j ) ) 4 ) ) + max ( E X P ( − 0.7 ( M ( 1 ± Δ R N ) R N m a x − 1 / ( ∑ j = 1 M ( 1 ± Δ R N ) R N j ) ) 4 ) ) + max ( E X P ( − 0.7 ( M ( 1 ± Δ F ) F m a x − 1 / ( ∑ j = 1 M ( 1 ± Δ F ) F j ) ) 4 ) ) + max ( ( 1 ± Δ T ) u 6 ( T m a x ) ) + max ( ( 1 ± Δ Q ) u 7 ( Q m a x − 1 ) ) / 7$
(14)

Since all functions of sufficient significance are descending, (15) and (16) can be written in the following form:

$m a x ( W m a x ) = ( E X P ( − 0.7 ( M ( 1 + Δ C ) C m a x / ( ∑ j = 1 M ( 1 − Δ C ) C j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 + Δ D ) D m a x / ( ∑ j = 1 M ( 1 − Δ D ) D j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 + Δ R p ) ( R p m a x ) / ( ∑ j = 1 M ( 1 − Δ R p ) R p j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 + Δ R N ) ( R N m a x ) / ( ∑ j = 1 M ( 1 − Δ R N ) R N j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 + Δ F ) F m a x / ( ∑ j = 1 M ( 1 − Δ F ) ) F j ) ) 4 ) + ( 1 − Δ T ) u 6 ( T m a x ) + ( 1 − Δ Q ) u 7 ( Q m a x ) ) / 7$
(15)
$m a x ( W m a x − 1 ) = ( E X P ( − 0.7 ( M ( 1 − Δ C ) C m a x − 1 / ( ∑ j = 1 M ( 1 + Δ C ) C j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 − Δ D ) D m a x − 1$
(16)
$/ ( ∑ j = 1 M ( 1 + Δ D ) D j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 − Δ R p ) ( R p ( m a x − 1 ) ) / ( ∑ j = 1 M ( 1 + Δ R p ) R p j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 − Δ R N ) ( R N m a x − 1 ) / ( ∑ j = 1 M ( 1 + Δ R N ) R N j ) ) 4 ) + E X P ( − 0.7 ( M ( 1 − Δ F ) F m a x − 1 / ( ∑ j = 1 M ( 1 + Δ F ) ) F j ) ) 4 ) + ( 1 + Δ T ) u 6 ( T m a x − 1 ) + ( 1 + Δ Q ) u 7 ( Q m a x − 1 ) ) / 7$

For the final approval of the selection, the au-thors require that $m i n ( W m a x ) > m a x ( W m a x − 1 )$ or, as a con-sequence, $m i n ( W m a x ) − m a x ( W m a x − 1 ) > 0$ . Then the ob-tained option will be optimal, with consideration of the errors in determining the values of environmental safety indicators. Using (15) and (16) the authors obtain:

(17)

The condition (17) is a necessary and sufficient condition for the correct selection of the optimal solution. For a rough estimate of the error in determining the values of environmental safety indicators, it is necessary to consider each of the seven differences separately with the condition of their positivity. Then the sum of seven positive values will be a positive number and condition (17) will be satisfied. Consider the first difference:

$E X P ( − 0.7 ( M ( 1 + Δ C ) C m a x / ( ∑ j = 1 M ( 1 − Δ C ) C j ) ) 4 ) − − E X P ( − 0.7 ( M ( 1 − Δ C ) C m a x − 1 / ( ∑ j = 1 M ( 1 + Δ C ) C j ) ) 4 ) > 0$
(18)

Having made a series of transformations, the authors obtain:

(20)
(21)

Similarly, the authors obtain the following re-sults for CED errors, collective radiological risk of per-sonnel, collective radiological risk of the population, financial risk:

$Δ D < D m a x − 1 5 − D m a x 5 D + D m a x − 1 5$
(22)
$Δ R p < R p m a x − 1 5 − R N m a x 5 R p m a x − 1 5 − R p m a x − 1$
(23)
$Δ R N < R N m a x − 1 5 − R N m a x 5 R N m a x 5 − R N m a x − 1$
(24)
$Δ F < F m a x − 1 5 − F m a x 5 F m a x 5 − F m a x − 1 5$
(25)

For environmental safety indicators, the project implementation time and the values of the risk of worsening of working conditions, the conditions for the positive difference in formula (17) will be as follows:

$Δ T < u 6 ( T m a x ) − u 6 ( T m a x − 1 ) u 6 ( T m a x ) + u 6 ( T m a x − 1 )$
(26)
$Δ Q < u 7 ( Q m a x ) − u 7 ( Q m a x − 1 ) u 7 ( Q m a x ) + u 7 ( Q m a x − 1 )$
(27)

Conditions (20) – (27) are sufficient for a final conclusion. However, these conditions dictate the assertion that all values of environmental safety indicators for option should be less than the corresponding values for option . In this case, the authors can conclude about selection of the best option without calculating functions of sufficient significance. Therefore, the simultaneous performance of conditions (20) – (27) is optional. Mandatory is condition (17), the performance of which guarantees the correctness of the selection of the best option.

## 4. CONCLUSIONS

For the first time, the paper defines the indica-tors of environmental safety of decision making for different, in terms of physical content, environmental safety indicators of facilities with environmental energy technologies: costs of operations, collective effective dose, collective radiological risk of personnel, collective radiological risk of population, financial risk, performance of operations, possible uncertainty value.

A research of the type of possible functions of sufficient significance was performed, on the basis of which for the first time three functions of sufficient im-portance of a special type were proposed, which can be applied to almost all indicators of assessing the level of environmental safety and selecting the optimal option. Two functions are continuous and are used for quantitative indicators. And the third function is defined only at nine points and is used for qualitative indicators of assessment. In addition, proceeding from the analysis of the possible type of functions of sufficient significance that meet the requirements of the multicriteria analysis of environmental safety of objects, the authors discovered the functions of sufficient significance of a special type for all quantitative criteria, which is constantly decreasing and displays each value from the range of values of certain environmental safety indicators in one value from the dimensionless range of values [0, 1], so that in the space of small values of cost and dose, the function has high values and is insensitive to changes in the argument; in the space of large values of cost and dose, the function has low values and is insensitive to changes in the argument in the range of average values of the argument, which are proximal to each other, and the function has a high resolution.

A research of the construction of weighting fac-tors was performed and, for their determination, it was proposed to use the method of expert estimates based on direct assessment on the basis of a 9-point scale. For this, additional researches of the works of highly qualified specialists in the field of radiation and environmental safety were performed. Subsequent to the performed analysis, the value of the weighting factors was obtained. Using the proposed functions of sufficient significance and certain weighting factors, a generalized indicator of the effectiveness of the decision was further found. Since some of the indicators are evaluative in nature, the sensitivity analysis of the choice made from the variation of the input parameters within the limits of the changes is performed, that is, it was determined whether the choice of the optimal option will change at different values of the calculation parameters beyond the change limits.

Using the results of solving the scientific issues posed by the well-known method of multicriteria analysis, the method of multicriteria analysis of environmental safety in the design, construction and operation of clean energetics facilities was improved.

## Figure

The structure of actions during the design solutions optimization procedure in the context of decision making.

The scheme for selecting the optimal solution.

The form of a function of sufficient significance u(x) monotonously falling and indifferent to the risk of environmental safety indicators.

Type of function of sufficient significance u(x) for various values of the constant for environmental safety indicators that do not have an upper limit of values.

Type of function of sufficient significance u(x) for qualitative indicators of environmental safety.

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