CC BY
7
-□ □-
Робота видноситься до областi моделю-вання системних процеыв з метою вибору найкращого варiанту для прийняття ршення. Встановлено, що для достовiрностiрезультату моделювання необхдно накладати обме-ження на параметри модельованого проце-су. Розроблено метод визначення параметрiв обмеження на область допустимихзмт пара-метрiв моделей коротких операцш при порiв-нянш операцшних процеыв на базi операцш рiзног тривалостi
Ключовi слова: дослидження операцш, моделювання операцшних процеыв, метод моделювання, верифшащя формули ефектив-ностi
□-□
Работа относится к области моделирования системных процессов с целью выбора наилучшего варианта для принятия решения. Установлено, что для достоверности результата моделирования необходимо накладывать ограничения на параметры моделируемого процесса. Разработан метод определения параметров ограничения на область допустимых изменений параметров моделей коротких операций при сравнении операционных процессов на базе операций разной продолжительности
Ключевые слова: исследование операций, моделирование операционных процессов, метод моделирования, верификация формулы
эффективности -□ □-
UDC 007.5
|DOI: 10.15587/1729-4061.2018.126446|
DEVELOPMENT OF THE METHOD FOR MODELING OPERATIONAL PROCESSES FOR TASKS RELATED TO DECISION
MAKING
I. Lutsenko
Doctor of Technical Sciences, Professor* E-mail: [email protected] I. Oksanych PhD, Associate Professor* E-mail: [email protected] I. Shevchenko Doctor of Technical Science, Professor* E-mail: [email protected] N. Karabu t Senior Lecturer Department of modeling and software Kryvyi Rih National University Vitaliya Matusevycha str., 11, Kryvyi Rih, Ukraine, 50027 E-mail: [email protected] *Department of Information and Control Systems Kremenchuk Mykhailo Ostrohradskyi National University Pershotravneva str., 20, Kremenchuk, Ukraine, 39600
1. Introduction
Investment possibilities of the society and, therefore, the pace of development of industry depend on the efficiency of utilizing available resources.
To argue that a decision that was made improves efficiency, without using thorough methods of operations research, is possible only in one case. This is the case when, as a result of a new technological solution, operational costs and operation time decrease while the quality and cost estimation of the total output product of an operation do not decrease [1, 2].
All other cases, involving decision making, require the application of special methods of operations research. Success stability in the investment activity depends on the reliability of results of such a research.
Complexity of solving this problem is well traced in the history of formation and development of operations research [3-5]. The rhetoric and research methods have not changed much since publication of the first monographs on operations research.
The issues of operations research are well known to professionals who use results of operations research in the problems on optimization of systems processes [6, 7]. "Engineers, researchers, economists and designers continuously
offer new, "universal, precise and clear" objective functions. In 1967, one author managed to collect more than 100 criteria of optimization of separation processes. Upon completion of classification it became clear that there was no universal criterion and selection of a criterion of optimization or efficiency of the process is not an easy task" [6].
Complexities, associated with the development and verification of the effectiveness formula, resulted in a rather lengthy crisis in operations research, where, up to now, it is considered preferable to explore operations by the results of modeling of operational processes [8-11].
However, similarly to the case of verification of the effectiveness formula, the key to success lies not so much in the method of research, but in the reliability of results, obtained from its use.
Since modeling of operational processes is the basic tool for direct assessment of operation effectiveness, development of a method for improving the accuracy of modeling is an important scientific task.
2. Literature review and problem statement
It is necessary to compare alternative variants of operations in order to make a well-grounded decision. In this case,
©
one must verify the formula that is supposed to be used as a criterion for comparison. However, the verification process requires the use of the method of mathematical modeling. The use of this method, in turn, requires verification of a model of the operational process.
If verification procedure did not lead to positive results, this choice is carried out using the method of mathematical modeling. However, the need for verification of the operational process remains an open question.
As it is shown by an analysis of sources [12-15], focusing on the issue of operations research, the emphasis in these papers is shifted towards development of the methods for modeling business processes. Thus, paper [12] considers the use of the methods of linear programming, discrete programming [13], non-linear programming [14], dynamic programming [15] and so on.
At the same time, no attention is paid to assessment of reliability of research results, obtained using the methods of mathematical modeling [16-19].
Lack of due attention to the problems of validity of results of operations identification by the effectiveness criterion contributed to creation of the "industry" of optimization criteria.
It is a common practice to try using technical indicators in problems of decision making. Thus, in paper [20], there is an attempt to establish relation of effectiveness and the amount of the "critical load". In article [21], researchers try to solve the problem of effectiveness evaluation using the "reliability" indicator, in paper [22] - with the help of "filling criterion", while "energy of sampling errors" is used in [23]. In paper [24], effectiveness is defined as the magnitude of minimal deviation of an object from the assigned trajectory.
Many papers are dedicated to the issue of energy efficiency [25]. Traditionally, there are many papers, in which costs minimum [26] or a combination of indicators by the Paretto principle [27] are used as the effectiveness criterion. Such criteria are integrated, and particular indicators in their composition pass through the scaling phase with the use of weight coefficients. In this case, weight coefficients are determined based of a subjective view of developers on the extent to which a criterion corresponds to the possibility to assess the effectiveness of an operational process.
Given this, none of the considered indicators passed the verification procedure using the methods of mathematical modeling.
Thus, there is now a scientific issue related to reliability of the decision results. The abundance of various evaluation indicators indirectly point out these problems [20-27]. The structure of these indicators is different and hence, incomparable evaluation results will be obtained under the same conditions.
The developed methods of indirect evaluation [12-19] do not fully solve the problem of estimation reliability, since verification of adequacy of models is carried out at the level of checking technical parameters.
When it comes to models, everything is fine. They can adequately display the physics of the process of transition, grinding, smelting, lump formation, etc. However, to make a decision, it is necessary to compare inputs and outputs of operational processes, taking into consideration their different duration.
In article [28], it is shown that certain constraints, associated with the loss inter-operational transitions, must be
imposed on parameters of models in order to obtain reliable data on a modeling result.
Thus, the scientific problem is to determine parameters of such comparable models of operational processes, the results of which can be reliable in problems of decision making.
3. The aim and objectives of the study
The aim of present study is to develop a modeling method by the introduction of constraints on the parameters of operations of the compared operational processes in decision making problems.
To accomplish the aim, the following tasks have been set:
- to state the rules of determining the region of permissible values of cost estimation of the output product of the operational process, consisting of short operations;
- to develop the method for determining the region of impermissible values of cost estimation of the output product of the operational process for problems of mathematical modeling with a view to decision making and verification of evaluation indicators;
- to carry out test studies of the method operation on the example of the two evaluation indicators, designed to determine effectiveness of the studied operations.
4. Determining the rules of comparison of results of operational processes
Any system operation is carried out for solving the problem of increasing the value (cost estimation) of the output operation products (PE) in relation to the value (cost estimation) of input operation products (RE) [29]. Operation representation as signals of registration of the value motion at its input and output is used in the problems of operations research. As a result, solutions of the problems of this class of operations are identified relative to the criterion of resource use efficiency [30]. The use of results of such identification enables making the best decision.
It is possible to directly perform identification of operations by estimating its parameters using the effectiveness formula. But to do it, it is necessary to verify the effectiveness formula [31-33]. As there is no standard of effectiveness, verification is based on the methods of mathematical modeling.
The second opportunity to identify an operation is to use the same method of mathematical modeling, when by final results of the modeled operational process one makes a judgment on effectiveness of an operation it is based on.
Motion of products at the input and output of an operation in the general case has the form of flows of resource consumption and resource efficiency, distributed in time. Reducing these flows to comparable magnitudes makes it possible to represent these flows in the form of the united flows of values at the input (re(t)) and at the output of an output of the operation (pe(t)) [29].
Such operation model is defined as a global operation model.
In cases where distributed nature of products motion can be neglected, the global operation model is replaced with a simple global operation model. In this model, function re(t) is replaced with parameter RE and function pe(t) is replaced with parameter PE. Here
-t -t RE = J re (t)dt; PE = J pe (t)dt,
where tS is the moment of start of the model of simple operation, tF is the moment of end of the model of simple operation.
Transition from functions re(t) and pe(t) to parameters RE and PE requires explicit determining of time of a simple model of operation (TO). That is,
TO=tF-ts.
Subsequently, the research will be carried out with the use of the global model of a simple operation in the form of threesome of parameters (RE, TO, PE), where PE>RE.
It is always possible to determine coefficient of operational transformation (k). Thus,
PE=kRE.
For cost-effective operation k>1.
Graphically, the model of such operational transformation can be represented as a marked vector. The length of the vector displays duration of the operation model (Fig. 1).
re
2E
TOa
Fig. 1. Graphical model of operation in the form of a marked vector with parameters
As was noted, in order to determine which of the alternative operations (Fig. 2) is more profitable for an enterprise, it is enough to compare the values of their effectiveness indicators.
REs
S
pe.
TOs
S
ret
PEL
TOt
Fig. 2. Examples of models of compared operations: RES, TOS, PES — parameters of a short operation S; REl, TOl, PEl — parameters of a long operation L
The problem is in the existence of an infinitely large set of indicators, the structure of which enables processing parameters RE, TO and PE of compared operations.
For example, it is possible to compare the operations (Fig. 2) using two alternative indicators
V =
PE - RE RE TO
and V„ =
_ (PE - RE)2
RE ■ PE TO
Let operation S be represented by the threesome (RES=2, TOS=2, PES=2.2), and operation L - by the threesome (REl=2, TOl=4, PEL=2.4).
Then, VAS=0.05, VAL=0.0375, and VBS=0.023, VBL=0.012. In this case, values of indicators VA and VB are matched. They both indicate that operation S is better than operation L.
If operation L is represented by the threesome (REL=2, TOl=4, PEl=2.4), the situation changes: VAL=0.05, and Vdl=0.021.
That is, indicator VA points out equality of operations S and L, and indicator VB continues to distinguish operation S as more profitable.
The only way to verify evaluation adequacy is the method of mathematical modeling.
The use of the method implies implementation of a consistent operational process S and L using such unchanged parameters of initial operation S and L, as well as time and added value factor. In this case, cost estimation of the output product of every previous operation is the cost estimation of the input product of the following operation (Fig. 3).
2
S
2.2
2.2 S,
2.42
2.4
4
Fig. 3. Modeling of operational process S by creating operation S2
This approach to modeling is necessary because the resource efficiency research involves modeling the processes, within which the immediate use of the operation results in the following operations occurs.
It was established that operation effectiveness does not change unless the operation time and value-added factor change [33].
Modeling shows that operation of S type is more effective than operation of L type, because by the time of the finish of the second operation, the final result of the operational process, based on operations of S type, is higher.
There seem to be all grounds for this. The initial cost estimation of input operation products is the same and operational processes finish also at the same time. However, this statement is true only in the case if a model of an operational process is adequate.
This is due to the fact that transfer of the product from the output of operation S1 to the input of operation S2 cannot proceed without losses. That is why the model in Fig. 3 must be converted to the model in Fig. 4.
XE
2.2
2.2 S2
2.42
2.4
4
Fig. 4. Model of operational processes, taking into consideration losses of transition between short operations
It means that it is necessary to determine the value of losses of transition XE in order to verify expressions VA and VB.
S
S
2
L
2
2
l
S
2
2
L
2
2
For a particular verified indicator, the region of uncertainty of cost estimation of the output product can be determined in relation to the specific parameters of a long operation. To do this, parameters of short operation S are selected so that value PES of the last operation of the cycle at the moment of comparison should be numerically equal to value PEL.
This effect can be achieved if kS! = kL, where N is multiplicity factor of a short operation (Fig. 5).
XE
2.2
2.2 S,
2.42
2.42
4
Fig. 5. Models of operational processes with the same result at the moment of comparison
The left boundary of the region of impermissible values of PEsl is determined by value PESL=kRE. The right boundary is determined in each case separately for each verified indicator.
Thus, for indicator VA, XE is determined from equality
PES + XE - RES _ PEL - REL
REsTOs
REl TOl '
(1)
a parameter of the output operation product is carried out by results of comparison of a short and a long operation multiple to operation duration. Such comparison is performed using the verified effectiveness criterion.
In this case, operational processes, constructed with the use of compared operations, must have the same initial investments and the same results of operational processes at the moment of their comparison without taking into consideration the losses of operational transitions.
3. In the case of making a decision on operations effectiveness, it is done by comparing results of operational processes at the moment of their simultaneous completion. In this case, the region of permissible values of cost estimations of output products of a short operation is determined either experimentally or with the use of the maximum estimated magnitude of losses as a result of processing the data of a set of evaluation criteria from p. 2.
5. Development of the method for modeling operational
processes
Taking into account the above, the method of formation of models of operations in problems of decision making can be represented in the form of the following steps:
1. Parameters of a long operation (REL, TOL, PEL), where PEl>REl are determined.
2. The value of added value factor of a short operation is determined from expression
k - Nfh~ ks - i]KL-
hence
(PEr - RE, )RESTOS XE —L-Ü—S-S + RE - PE<.
RE, TO,
Having substituted the data of operations S and L (Fig. 5) in equation (1), we will obtain that XE=0.01. Then PESR=kRE+XE=2.21.
For indicator VB, value of XE is determined from equality
(PES + XE - RES )2 RES (PES + XE) TO2S
(PE, - RE, )2 RE, PEl TOl'
(2)
and can be determined with the use of the algorithmic methods.
Having substituted the data of operations S and L (Fig. 5) in equation (2), we will obtain PESR=kRE+XE= =0.000238.
Then PESR=kRE+XE=2.200238.
Thus, it is possible to state rules for comparing results of operational processes, based on the use of operations of different duration.
1. It is necessary to determine the region of values of output operation products in comparable cost magnitudes based on operations of different duration in order to have a possibility to compare results of operational processes. In this case, cost estimations of output operation must be within such range of magnitudes, in which the magnitude of losses of transition between successive short operations can be ignored.
2. When using results of modeling in problems of estimation indicator verification, the region of permissible values of
3. The left boundary of impermissible values of PESL of a short operation is determined from expression PESL=RE-kS.
4. Determining the width of the region of impermissible values of parameter PES.
4. 1. In the case of modeling of operational processes for the verification purpose, the width of the region of impermissible values of PES is determined from the condition of equality of the left and the right parts of the verified expression.
In this case, the data of parameters of a short operation with variable XE, taking into consideration uncertainty of parameter PES., are substituted into the left part of the verified expression. The data of the parameters of a long operation are substituted into the right part of the verified expression.
The width of the region of impermissible values of PES is determined by solving equality relative to XE.
4. 2. In the case of modeling operational processes with a view to decision making or optimization, it is supposed that the effectiveness formula is not verified. To determine the width of the region of impermissible values of PES, a set of unverified estimated indicators with the required formal features is determined. The indicator that requires the widest uncertainty region is selected among these indicators. Magnitudes of XE, determined with the use of this indicator are used to determine the width of the region of impermissible values of PES.
5. The right boundary of parameter PESR of a short operation is determined from expression PESR=kRE+XE.
6. Steps 3-5 are repeated for each short operation within the interval of duration of a long operation.
S
2
1
2
L
2
2
6. Solution to the problem of modeling operations using the developed method
Let us consider the problem of modeling operational processes with the aim of verification of formula
V _ (PE - RE)2
B RE ■ PE ■ TO2'
Let us assume that a long operation has the following parameters: REL=2, TOL=4, PEL=2.42.
We will determine parameters of uncertainty zone for an alternative operation:
RES=REl=2, TOs=TOl/2=2, ks _,Jk~ _ 1.1.
The value of parameter PES of the left boundary of uncertainty zone PESL=kSRES=2.2.
Determining the right boundary of the uncertainty zone from equality of the ratio
(PES + XE - RES )2 _ (PEl - REl )2
RES •(PES + XE)■ TO] ~ REL PEL TO2L '
Having substituted values of operation SL and L, we will obtain
(2.2 + XE - 2)2 _ (2.42 - 2)2
2■ (2.2 + XE)■ 22 _ 2^2.42^42 ,
hence, XE=0.000238.
Thus, PESR=2.2+0.000238=2.200238.
We will construct three models of short operations, in which two values of cost estimation of the output product are at the edges of uncertainty region and one value gets into this region. Let it be values PESL-=2.19, PESR+=2.21, PEsn=2.2002 (Fig. 6).
Value added factor
1.21 | 1.095 | 1.1001 1.105
Operational processes
¿1 S2 53
Time 0 2 2 2 2
1
2 2.19 2.2002 2.21
3
4 2.42 2.39805 2.42044 2.442
VB 0.227789 0.20605 0.22771 0.2494
Estimation indicator
Fig. 6. Comparison of effectiveness of operations by results of modeling and with the use of estimation indicator VB
One can see that the resulting value of operational process S2 at the moment of comparison (2.42044) is higher than the value of operation L, obtained at the same moment (2.42). Therefore, modeling operation with parameter k=1.001 is impermissible.
Similar results were obtained for indicator VA (Fig. 7).
The final value of operational process S2 is higher than the final value of the operational process L. In this case
Value added factor
1.21 | 1.095 | 1.105 | 1.11
Operational processes
u ¿1 ¿2 ¿3
Time 0 2 2 2 2
1
2 2.19 2.21 2.22
3
4 2.42 2.39805 2.44205 2.4642
VA 0.0525 0.0475 0.0525 0.055
Estimation indicator
Fig. 7. Comparison of effectiveness of operations by results of modeling and with the help of estimation indicator VA
Therefore, modeling of operations with parameter k= =1.105 is impermissible
7. Discussion of results of research related to the development of the method for modeling operational processes
The method for determining optimal control using the methods of mathematical modeling is quite popular. In this case, much attention is paid to accuracy of construction of models of operational processes and enhancing the accuracy of calculation methods. At the same time, little importance is attached to determining constraints that should be imposed on modeled objects.
Comparison of operational processes with similar initial investment is considered. This is a necessary condition, which allows making a judgment about comparability of final results of the operational process, of course, only if the operations of compared processes finish simultaneously.
The multiplicity condition for short operations of one process in relation to long operations of the second process is very strict and is hardly possible to be met while solving most practical tasks.
In this sense, this research rather indicates the existence of a problem in this regard than offers a ready practical solution. Obviously, the studies, connected with the necessity of taking into consideration inter-operation losses, are still to be carried out.
In addition, in the case of modeling cumulative operational processes, it is required to keep in mind that these losses increase with an increase in value added factor and the absolute magnitude of cost estimation of a transferred product (Fig. 8).
\2_2.662
0.0008413
I2— 2.30738 -► 2.30738 2.662 -►
|2 '-► 2.2 --► 2.42 2.662
Fig. 8. Change in inter-operational losses at an increase in magnitude of value added factor and cost estimation of output operation product
On the other hand, in the vast majority of cases, there is no need to use the methods of mathematical modeling to obtain prognostic estimates of effectiveness. In the case if there is an optimization process, it is much easier to compare separate operations rather than operational processes, generated by them. In this case, the proposed method of modeling allows avoiding errors in selection of evaluation expressions, which are planned to be used as an effectiveness criterion, in verification problems.
8. Conclusions
1. The rules for determining the region of permissible values for cost estimation of output products of a short operation in problems of mathematical modeling with a view to making a decision and verification of estimation indicators were stated.
The set of rules is reduced to the following points:
- input and output quantitative parameters of operations must be reduced to comparable cost magnitudes;
- compared estimates of output products of a short operation must have the same initial investment and start simultaneously;
- cost estimates of output products of a short operation must be in such a range of magnitudes, within which losses
of transition between successive short operations can be ignored;
- comparison is carried out at the time of simultaneous completion of compared operational processes.
Statement of the rules allows using the capabilities of the verified expression itself to determine parameters of modeled operational processes.
2. The method for determining the region of permissible values for parameters of short operations of compared processes was developed. This makes it possible to formalize the solution of the problem of determining constraints on generated model data.
The method is based on determining:
- parameters of an operation of longer duration;
- parameters of a short operation based on parameters of a long operation;
- boundaries of the region of impermissible values of output parameters of short operations.
3. Verification of the developed method of modeling on the example of comparing results of two operational processes was performed. Comparison of the results of operational processes with the results of estimation of separate operations of there processes was obtained. It was shown that in the region, excluded from consideration, with the use of the proposed method, there is a mismatch of modeling results with the results of direct estimation of operations using the verified indicators.
References
1. The technology of production of a copper - aluminum - copper composite to produce current lead buses of the high - voltage plants / Dragobetskii V., Shapoval A., Naumova E., Shlyk S., Mospan D., Sikulskiy V. // 2017 International Conference on Modern Electrical and Energy Systems (MEES). 2017. P. 400-403. doi: 10.1109/mees.2017.8248944
2. High-energy ultrasound to improve the quality of purifying the particles of iron ore in the process of its enrichment / Morkun V., Gubin G., Oliinyk T., Lotous V., Ravinskaia V., Tron V. et. al. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 6, Issue 12 (90). P. 41-51. doi: 10.15587/1729-4061.2017.118448
3. Bellman R. The theory of dynamic programming // Bull. Amer. Math. Soc. 1954. Issue 6. P. 503-515.
4. Akof R., Sasieni M. Osnovy issledovaniya operaciy. Moscow: Mir, 1971. 534 p.
5. Taha A. H. Operations research: an introduction. Pearson, 1995. 813 p.
6. Barskiy L. A., Kozin V. Z. Sistemniy analiz v obogashchenii poleznyh iskopaemyh. Moscow: Nedra, 1978. 486 p.
7. Rais A., Viana A. Operations Research in Healthcare: a survey // International Transactions in Operational Research. 2010 Vol. 18, Issue 1. P. 1-31. doi: 10.1111/j.1475-3995.2010.00767.x
8. Zaky M. A., Machado J. A. T. On the formulation and numerical simulation of distributed-order fractional optimal control problems // Communications in Nonlinear Science and Numerical Simulation. 2017. Vol. 52. P. 177-189. doi: 10.1016/j.cn-sns.2017.04.026
9. Siegfried R. Special issue: Leveraging modeling and simulation as a service for the future modeling and simulation ecosystem // The Journal of Defense Modeling and Simulation: Applications, Methodology, Technology. 2017. Vol. 14, Issue 2. P. 113-113. doi: 10.1177/1548512917691652
10. Hu X., Li S. E., Kum D. Modeling and Control Problems in Sustainable Transportation and Power Systems // Mathematical Problems in Engineering. 2016. Vol. 2016. P. 1-3. doi: 10.1155/2016/1691250
11. Morkun V., Savytskyi O., Tymoshenko M. Optimization of the second and third stages of grinding based on fuzzy control algorithms // Metallurgical and Mining Industry. 2015. Issue 8. P. 22-25.
12. Juraj S. Introduction to Operations Research. Columbia University, 2014. 117 p.
13. Sottinen T. Operations Research with GNU Linear Programming. Kit ORMS, 2009. 200 p.
14. Hillier F. S., Lieberman G. J. Introduction to operations research. McGraw-Hil, 2001. 1214 p.
15. Liberti L. Problems and exercises in Operations Research. Ecole Polytechnique, 2006. 128 p.
16. Sharma J. K. Operations Research. Theory and Application. Macmillan, 2009. 958 p.
17. Larnder H. OR Forum - The Origin of Operational Research // Operations Research. 1984. Vol. 32, Issue 2. P. 465-476. doi: 10.1287/opre.32.2.465
18. Hoeve W. J. Operations Research Techniques in Constraint Programming. ILLC Dissertation Series, 2005. 154 p.
19. Murthy P. R. Operation Research. New age international publishers, 2007. 705 p.
20. Wu Y., Cheng T. C. E. Henig efficiency of a multi-criterion supply-demand network equilibrium model // Journal of Industrial and Management Optimization. 2006. Vol. 2, Issue 3. P. 269-286. doi: 10.3934/jimo.2006.2.269
21. Optimization of the structure of multifunctional information systems according to the criterion of a required value of the efficiency ratio / Anishchenka U. V., Kryuchkov A. N., Kul'bak L. I., Martinovich T. S. // Automatic Control and Computer Sciences. 2008. Vol. 42, Issue 4. P. 203-209. doi: 10.3103/s0146411608040068
22. Miskowicz M. Efficiency of Event-Based Sampling According to Error Energy Criterion // Sensors. 2010. Vol. 10, Issue 3. P. 2242-2261. doi: 10.3390/s100302242
23. Xu Q., Wehrle E., Baier H. Adaptive surrogate-based design optimization with expected improvement used as infill criterion // Optimization. 2012. Vol. 61, Issue 6. P. 661-684. doi: 10.1080/02331934.2011.644286
24. Xia L. Optimization of Markov decision processes under the variance criterion // Automatica. 2016. Vol. 73. P. 269-278. doi: 10.1016/j.automatica.2016.06.018
25. Vasilyev E. S. Optimization of the architecture of a charge pump device on the basis of the energy efficiency criterion // Journal of Communications Technology and Electronics. 2013. Vol. 58, Issue 1. P. 95-99. doi: 10.1134/s1064226913010099
26. Shorikov A. F., Rassadina E. S. Multi-criterion optimization of production range generation by an enterprise // Economy of Region. 2010. Issue 2. P. 189-196. doi: 10.17059/2010-2-18
27. Integrating Hierarchical Clustering and Pareto-Efficiency to Preventive Controls Selection in Voltage Stability Assessment / Mansour M. R., Delbem A. C. B., Alberto L. F. C., Ramos R. A. // Lecture Notes in Computer Science. 2015. P. 487-497. doi: 10.1007/978-3-319-15892-1_33
28. Development of a verification method of estimated indicators for their use as an optimization criterion / Lutsenko I., Fomovskaya E., Oksanych I., Koval S., Serdiuk O. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 2, Issue 4 (86). P. 17-23. doi: 10.15587/1729-4061.2017.95914
29. Lutsenko I. Optimal control of systems engineering. Development of a general structure of the technological conversion subsystem (Part 2) // Eastern-European Journal of Enterprise Technologies. 2015. Vol. 1, Issue 2 (73). P. 43-50. doi: 10.15587/17294061.2015.36246
30. Lutsenko I. Definition of efficiency indicator and study of its main function as an optimization criterion // Eastern-European Journal of Enterprise Technologies. 2016. Vol. 6, Issue 2 (87). P. 24-32. doi: 10.15587/1729-4061.2016.85453
31. Development of a method for the accelerated two-stage search for an optimal control trajectory in periodical processes / Lutsen-ko I., Fomovskaya O., Konokh I., Oksanych I. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 3, Issue 2 (87). P. 47-55. doi: 10.15587/1729-4061.2017.103731
32. Formal signs determination of efficiency assessment indicators for the operation with the distributed parameters / Lutsenko I., Fomovskaya E., Oksanych I., Vikhrova E., Serdiuk O. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 1, Issue 4 (85). P. 24-30. doi: 10.15587/1729-4061.2017.91025
33. Development of the method for testing of efficiency criterion of models of simple target operations / Lutsenko I., Vihrova E., Fomovskaya E., Serdiuk O. // Eastern-European Journal of Enterprise Technologies. 2016. Vol. 2, Issue 4 (80). P. 42-50. doi: 10.15587/1729-4061.2016.66307
