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Portfolio investment based on probabilistic multi-objective optimization and uniform design for experiments with mixtures
Maosheng Zhenga, Jie Yub
a Northwest University, School of Chemical Engineering, Xi'an, People's Republic of China, e-mail: mszhengok@aliyun.com, corresponding author, ORCID iD: https://orcid.org/0000-0003-3361-4060
b Northwest University, School of Life Science, Xi'an, People's Republic of China, e-mail: yujie@nwu.edu.cn,
ORCID iD: https://orcid.org/0000-0001-6606-5462
DOI: 10.5937/vojtehg71 -41828; https://doi.org/10.5937/vojtehg71-41828
FIELD: mathematics, operation research, security ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: In this paper, a new approach to solving the portfolio investment problem is formulated to handle simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return. Probability - based multi - objective optimization is combined with uniform design for experiments with mixtures to conduct processing. Methods: Preliminarily, probability - based multi - objective optimization is employed to synthesize the bi-objective problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return into a single objective one of the total preferable probability of each alternative scenario. The total preferable probability is the product of all partial preferable probabilities of each performance utility; subsequently, the method of uniform design for experiments with mixtures is used to create a set of effective sampling points for the portfolio investment problem to provide discretization in data processing and simplifying treatment, of which the proportion x, follows the constraint condition of x+ x2 + x3... + xs = 1 with the total number of variables s for x.
Results: The new approach is used to deal with the portfolio Investment problem that is, in essence, simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return, which leads to reasonable consequences. The results are with the quality of rationality from the respect of the probability theory for simultaneous optimization of multiple objectives.
Conclusion: This method naturally reflects the essence of the portfolio investment problem and opens a new way of solving the relevant problem.
Key words: portfolio investment problem, multi-objective optimization, preferable probability, discrete sampling, probability theory.
Introduction
Portfolio investment aims to diversify investment risks in an effective way. Markowitz proposed a decision - making model of portfolio investment in 1952 which is seen as the foundation of the modern portfolio theory (Wang, 2022).
In Markowitz's treatment, the expected rate of return and the variance of the rate of return are employed to evaluate risky securities, with the latter used to reflect risk. The significant consequence of Markowitz's investigation is that investors should invest their funds in several securities instead in only one, so that investment risk could be reduced and appropriate investment returns obtained.
However, Markowitz's algorithm could only deal either with maximizing the expected rate of return and setting the variance of the rate of return as a restraint condition or with minimizing the variance of the rate of return and letting the expected rate of return be a constraint condition once at a time. In other words, such an approach could not handle simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return rationally due to the lack of appropriate methodology for dealing with multi-objective optimization (Sarmas et al, 2020; Oberoi et al, 2020; Nisani & Shelef, 2021).
Recently, Zheng et al (2022a) proposed probability - based multi -objective optimization in viewpoint of system theory, which created a brand new concept of "preferable probability"; furthermore, assessments for probability - based multi - objective optimization were put forward from the respects of the probability theory and the set theory. As a rationally novel approach concerning multiple objectives, it could be used in many fields, including energy planning, programming problems, operation research, financial affairs, management programs, material selection, mechanical design, engineering design, etc.
In this article, probability - based multi - objective optimization is combined with uniform design for experiments with mixtures to deal with the portfolio problem, so as to deal with the problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return rationally.
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Solution of the portfolio problem in the light of probability - based multi - objective optimization methodology and uniform design for experiments with mixtures
In this section, probability - based multi - objective optimization and uniform design for experiments with mixtures are organically combined, which establishes a rational method for solving the portfolio investment problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return, i.e, a bi-objective problem. The probability - based multi - objective optimization method is used to transfer a bi - objective optimization problem into a single -objective optimization problem from the perspective of the probability theory naturally; the discretization of uniform design for experiments with mixtures provides an effective discrete sampling to simplify mathematical processing.
The systematic implementation is demonstrated by sub - sections A), B), and C).
A) Fundamental spirit of probabilistic multi - objective optimization
In the spirit of probability - based multi - objective optimization, each objective can be analogically represented as a single event in a system (Zheng et al, 2022a) and the whole event of multi - objective simultaneous optimization corresponds to the product of all single objectives (events). All performance utility indexes of a candidate are preliminarily classified into two types: i.e., beneficial type and unbeneficial type, in accordance with the role and preference of a candidate in the optimization, respectively. Specifically, the assessment of the preferable probability Pij of both beneficial indicators and unbeneficial indicators can be carried out according to the evaluation procedure in Fig. 1 (Zheng et al, 2022a).
The meanings of the quantities and the factors in Fig. 1 are as follows:
Pij indicates the partial preferable probability of the j-th performance utility indicator of the i-th alternative scenario, X; n expresses the total number of the alternative scenario; m reflects the total number of the performance (objective); x represents the arithmetic value of the j-th
performance utility indicator; Xjmax and Xjmin show the maximum and minimum values of the j-th performance utility indicator, respectively; aj
and ps express the normalized factors of the j-th performance utility indicator Xj in the beneficial status and in the unbeneficial status, individually; the beneficial status or the unbeneficial status of the j-th performance utility indicator Xj is determined according to its specific role or preference in the instant problem; and Pi represents the total (overall) preferable probability of the i-th alternative scenario (Zheng et al, 2022a).
Here, as to the portfolio investment problem, a simultaneous maximization of the rate of return and minimization of the variance of the rate of return is a typical bi-objective problem which contains a beneficial indicator and an unbeneficial indicator, respectively.
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Figure 1 - Evaluation procedure of the PMOO method Рис. 1 - Процедура оценки метода многокритериальной оптимизации, основанной на вероятности Слика 1 - Поступак евалуаци^е метода вишекритериумске оптимизаци^е засноване на вероватноПи
B) Discrete sampling in the spirit of uniform design for experiments with mixtures
In this bi - objective portfolio investment problem, the total preferable probability is the decisive objective function which needs to be maximized in a high-dimensional space, so the complex data treatment might be involved; thus uniform design for experiments with mixtures (UDEM) can be employed to simplify data processing rationally.
Uniform design for experiments with mixtures (UDEM), based on the good lattice point (GLP), was proposed by Fang et al (2018). The method of UDEM can be used to create a set of effective sampling points for experimental design with the restraint of xi + X2 + X3... + Xs =1 for the proportion xi with the total number of s (Fang et al, 2018); therefore, it can be used as an efficient sampling method for the portfolio investment problem here to conduct the simplification of data processing with discretization.
In addition, Fang especially developed uniform design tables and their usage tables for proper application (Fang et al, 2018).
According to Fang et al (2018), the concrete steps of uniform design for experiments with mixtures (UDEM) are generally as follows:
I. Selection of the uniform design table
Given the number of mixtures s and the number of sampling points n, select the corresponding table U*n(nf) or Un(n') and the usage table from the uniform design table provided by Fang et al (2018), and the number of columns of the usage table is selected as s-1 at this time. Give a mark of the original elements in the uniform design table U*n(nf) or Un(n') with
{qik}.
II. Constructing a new element cki
For each i, construct its new element cki according to the following formula,
cki = (2qki - 1)/(2n). (1)
III. Constructing uniform sampling points for the mixtures, xki
xu = (1 - cJS— iU 1 CuSij x, = Us 1 CuSl
h\ h JUj=1 k, j, i =1, s-i h Hj=1 k, j, k =1, n (2)
Thus, {xk} gives the corresponding uniform design table UMn(ns) of the mixture under the conditions of s and n.
C) Portfolio investment problem
According to Markowitz's study, the return rate function f1 and the risk function f2 could be introduced, and their expressions are, respectively,
s
f = E(R) = X XU , (3)
i =1
f2 = [{x1C>1f + (x2CT2)2+ (x3&3)2+--+ (XnOn)2+ r1,2(x10-1)(x2&2) + y1,3(X1C1)(X3C3)
+ Y1A(x1C1)(x4C4) +..+rij(xiC)(xjC)+---+rs-1,s(xs-1Cs-1)(xsCs)f5. (4)
In Eq. (3), jUi is the rate of return of the i-th security and s is the total number of the securities. In Eq. (4), yj is the correlation coefficient between the i-th security and the j-th security; c is the risk of the i-th security.
According to the objective evaluation in probability - based multi -objective optimization methodology (Zheng et al, 2022a), f1 exhibits the bigger the better, which therefore belongs to the beneficial objective, and f2 manifests the smaller the better, which thus belongs to the unbeneficial objective, respectively.
Therefore, the answer to the portfolio problem is the optimization of a bi-objective problem. Furthermore, probability - based multi - objective optimization methodology can be used to assess it reasonably. Of course, all the evaluations in probability - based multi - objective optimization methodology can be applied rationally.
Applications
In this section, an example illustrates the use of the above steps for solving the portfolio investment problem.
Take a combination of four securities, i.e., securities A, B, C, and D as a typical example. The specific optimization process is explained in detail.
Let the expected rate of return of Security A be j1 = 11.29%, and let its standard deviation of return be c = 24.53%; let the expected rate of return of Security B be u = 18.10%, and let its standard deviation of return be C2 = 19.94%; let the expected rate of return of Security C be j3 = 8.29%, and let its standard deviation of return be C3 = 11.80%; and let the expected rate of return of Security D be u = 1152%, with its standard deviation of return of C4 = 12.75%. Furthermore, it is assumed
that the following correlation coefficients between two of the above securities are /12 = 0, y1,3 = 0.68, y1,4 = 0, /2,3 = 0, /2,4 = 0, /3,4 = 0.
Now we need to make a decision of simultaneous optimization of both the maximization of the rate of return and minimization of the variance of the rate of return on this portfolio investment.
Solution
In this section, the problem of "portfolio" is analyzed based on probability - based multi - objective optimization methodology. This is a typical bi-objective optimization problem.
Let x1, x2, x3 and x4 be the investment percentages of four securities, A, B, C and D, respectively. There is actually a constraint condition for this problem, i.e., x1 + x2 + x3 + x4= 1; therefore, it has actually three independent variables, namely x1, x2 and x3.
Since the sampling points of this bi-objective optimization problem are positioned in the 4 - dimensional space, it is necessary to include at least 23 sampling points with the characteristics of the "good lattice point" in the effective region for the discretization of data processing (Yu et al, 2022; Zheng et al, 2022a; Zheng et al, 2022b).
According to Fang et al (2018), this is a "uniform design for experiments with mixtures" problem due to the constraint condition of the four variables. Let us take the uniform table U*23(237) as the initial table to construct a uniform test design table UM23(234) with mixtures, as shown in Table 1.
The uniform test table UM23(234) with the mixtures of Table 1 is based on the uniform design table U*23(237). Because here the number of variables s equals to 4, and n equals to 23, from the above rules, xk1 = 1 - ck13, xk2 = cM1/3{1 - ck12), xk3 = ck11/3ck21/2-(1-ck3), xk4 = ck13-ck21/2-ck3 (Fang et al, 2018).
Furthermore, we can get the values of the rate of return function f1 and the risk function f2, the distribution of their partial preferable probability and their total preferable probability, as well as the ranking at the sampling points (alternative scenario), which are shown in Table 2.
Fig. 2 shows the variation of the return rate with respect to risk at the discrete sampling points. The results reflect that the 2nd discrete sampling point gives the maximum total preferable probability closely followed by the 6th sampling point; therefore, they could be taken as the optimal solution to this portfolio problem.
Regarding the 2nd sampling point, the corresponding investment ratio is X1" = 0.0222, X2" = 0.3494, X3" = 0.2596, X4" = 0.3688, which leads to the rate of return of 12.98% and the risk of 9.09%.
As for the 6th sampling point, its investment ratio is at x/ = 0.0871, X2' = 0.4665, X3' = 0.1068, X4 = 0.3397, and its obtained rate of return is 14.23% with the risk of 10.73%.
Table 1 - Uniform test table UM23(234) with the mixtures based on the uniform design
table U*23(237)
Таблица 1 - Таблица унифицированного теста UM23(234) с вариантами, основанными на таблице унифицированной модели U*23(237) Табела 1 -Табела униформног теста UM23(234) са вари]аци]ама заснованим на табели униформног дизайна U*23(237)
No. qio q2o q3o Ci C2 C3 xi X2 X3 X4
1 11 17 19 0.4565 0.7174 0.8044 0.2300 0.1178 0.1276 0.5246
2 22 10 14 0.9348 0.4130 0.5870 0.0222 0.3494 0.2596 0.3688
3 9 3 9 0.3696 0.1087 0.3696 0.2824 0.4810 0.1492 0.0874
4 20 20 4 0.8478 0.8478 0.1522 0.0535 0.0750 0.7389 0.1326
5 7 13 23 0.2826 0.5435 0.9783 0.3438 0.1725 0.0105 0.4733
6 18 6 18 0.7609 0.2391 0.7609 0.0871 0.4665 0.1068 0.3397
7 5 23 13 0.1957 0.9783 0.5435 0.4195 0.0063 0.2621 0.3121
8 16 16 8 0.6739 0.6739 0.3261 0.1233 0.1570 0.4850 0.2347
9 3 9 3 0.1087 0.3696 0.1087 0.5228 0.1871 0.2586 0.0315
10 14 2 22 0.5870 0.0652 0.9348 0.1627 0.6235 0.0139 0.1999
11 1 19 17 0.0217 0.8043 0.7174 0.7209 0.0288 0.0707 0.1796
12 12 12 12 0.5 0.5 0.5 0.2063 0.2325 0.2806 0.2806
13 23 5 7 0.9783 0.1957 0.2826 0.0073 0.5536 0.3150 0.1241
14 10 22 2 0.4130 0.9348 0.0652 0.2553 0.0247 0.6731 0.0470
15 21 15 21 0.8913 0.6304 0.8913 0.0376 0.1982 0.0831 0.6811
16 8 8 16 0.3261 0.3261 0.6739 0.3117 0.2953 0.1282 0.2649
17 19 1 11 0.8043 0.0217 0.4565 0.0700 0.7929 0.0745 0.0626
18 6 18 6 0.2391 0.7609 0.2391 0.3793 0.0793 0.4119 0.1295
19 17 11 1 0.7174 0.4565 0.0217 0.1048 0.2903 0.5917 0.0131
20 4 4 20 0.1522 0.1522 0.8478 0.4661 0.3256 0.0317 0.1766
21 15 21 15 0.6304 0.8913 0.6304 0.1425 0.0479 0.2992 0.5104
22 2 14 10 0.0652 0.5870 0.4130 0.5975 0.0941 0.1810 0.1274
23 13 7 5 0.5435 0.2826 0.1957 0.1839 0.3822 0.3490 0.0849
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Table 2 - Evaluation results of the rate of return f1, the risk f2, the preferable probability and the ranking at the sampling points
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Таблица 2 - Результаты оценки ставки доходности ^, риска f2, предпочтительной вероятности и ранжирования по точкам выборки
Табела 2 - Резултати процене стопе приноса Ъ, ризика f2, поже^не вероватноПе и рангираъе по тачкама узорковаъа
No. Values of fi and f2 Partial preferable probability Overall preferable probability Rank
fi f2 Pfi Pf2 PtxIO3
1 0.1183 0.0979 0.0417 0.0510 2.1241 9
2 0.1298 0.0909 0.0457 0.0533 2.4346 1
3 0.1414 0.1268 0.0498 0.0416 2.0703 11
4 0.0961 0.0992 0.0337 0.0506 1.7125 17
5 0.1254 0.1099 0.0442 0.0471 2.0800 10
6 0.1423 0.1073 0.0501 0.0479 2.4017 2
7 0.1062 0.1321 0.0374 0.0397 1.4907 18
8 0.1096 0.0917 0.0386 0.0530 2.0452 12
9 0.1180 0.1552 0.0415 0.03234 1.3434 20
10 0.1554 0.1334 0.0547 0.0395 2.1591 6
11 0.1132 0.1841 0.0399 0.0229 0.9142 23
12 0.1210 0.0968 0.0426 0.0514 2.1880 5
13 0.1414 0.1179 0.0498 0.0445 2.2152 4
14 0.0945 0.1306 0.0333 0.0404 1.3433 21
15 0.1255 0.0970 0.0442 0.0513 2.2663 3
16 0.1298 0.1107 0.0457 0.0468 2.1404 7
17 0.1648 0.1601 0.0580 0.0308 1.7848 16
18 0.1062 0.1330 0.0374 0.0396 1.4806 19
19 0.1150 0.1064 0.0405 0.0482 1.9519 14
20 0.1345 0.1356 0.0474 0.0387 1.8348 15
21 0.1084 0.0921 0.0382 0.0529 2.0187 13
22 0.1142 0.1637 0.0402 0.0296 1.1893 22
23 0.1287 0.1104 0.0453 0.0469 2.1267 8
Figure 2 - Variation of the return rate with respect to the risk at the discrete sampling
points
Рис. 2 - Варианты ставки доходности по сравнению с риском в дискретных
точках выборки
Слика 2 - Варц'ацц'а стопе приноса у односу на ризик у тачкама дискретное
узорковаъа
Conclusion
In this paper, the probability-based multi-objective optimization method is combined with uniform design for experiments with mixtures to study the portfolio investment problem, which aims to give a rational approach to handling the problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return; the analysis shows that probability - based multi - objective optimization methodology could provide the optimal solution with the characteristics of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return; uniform design for experiments with mixtures could be used to properly conduct discretization for data processing and simplification.
References
Fang, K.-T., Liu, M.-Q., Qin, H. & Zhou, Y.-D. 2018. Theory and Application of Uniform Experimental Designs. Singapore: Springer. Available at: https://doi.org/10.1007/978-981-13-2041-5.
Nisani, D. & Shelef, A. 2021. A statistical analysis of investor preferences for portfolio selection. Empirical Economics, 61, pp.1883-1915. Available at: https://doi.org/10.1007/s00181-020-01947-8.
Oberoi, S., Girach, M.B. & Chakrabarty, S.P. 2020. Can robust optimization offer improved portfolio performance? An empirical study of Indian market. Journal of Quantitative Economics, 18, pp.611-630. Available at: https://doi.org/10.1007/s40953-020-00205-z.
Sarmas, E., Xidonas, P. & Doukas, H. 2020. Multicriteria Portfolio Construction with Python. Cham, Switzerland: Springer. Available at: https://doi.org/10.1007/978-3-030-53743-2.
Wang, S. 2022. Securities Investment: Theory and Practice. Beijing, China: Science Press (in Chinese). ISBN: 9787030630469.
Yu, J., Zheng, M., Wang, Y. & Teng, H. 2022. An efficient approach for calculating a definite integral with about a dozen of sampling points. Vojnotehnicki glasnik/Military Technical Courier, 70(2), pp.340-356. Available at: https://doi.org/10.5937/vojtehg70-36029.
Zheng, M., Teng, H., Yu, J., Cui, Y. & Wang Y. 2022a. Probability-Based Multi-objective Optimization for Material Selection. Singapore: Springer. Available at: https://doi.org/10.1007/978-981-19-3351-6.
Zheng, M., Teng, H., Wang, Y. & Yu, J. 2022b. Appropriate Algorithm for Assessment of Numerical Integration. In: 2022 International Joint Conference on Information and Communication Engineering (JCICE), Seoul, Republic of Korea, pp.18-22, May 20-22. Available at:
https://doi.org/10.1109/JCICE56791.2022.00015.
Инвестиционный портфель, основанный на вероятностной многоцелевой оптимизации и унифицированной модели с вариационными экспериментами
Маошенг Чжэна, Джи Йюб
Северо-западный политехнический университет, г. Сиань, Народная Республика Китай а факультет химической инженерии, корресподент
б факультет естественных наук
РУБРИКА ГРНТИ: 27.47.00 Математическая кибернетика,
27.47.19 Исследование операций ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: В данной статье представлен новый подход к решению задач инвестиционного портфеля путем одновременной оптимизации максимизации ставки доходности и минимизации дисперсии ставки доходности. Обработка выполняется путем сочетания вероятностной многокритериальной оптимизации с унифицированной моделью вариационных экспериментов. Методы: Вероятностная многокритериальная оптимизация в основном используется при сочетании двухкритериальной задачи одновременной оптимизации максимизации ставки доходности и
минимизации дисперсии ставки доходности в однокритериальной задаче общей предпочтительной вероятности по каждому альтернативному сценарию. Общая предпочтительная вероятность является произведением всех частных вероятностей предпочтительной полезности
производительности, поэтому метод унифицированной модели вариационных экспериментов используется при создании множества эффективных точек выборки для решения задачи инвестиционного портфеля с целью достижения дискретизации при обработке данных и упрощения процедуры, в которой пропорция хI подчиняется условию ограничения хI + Х2 + хз. . . + Хз = 1 с общим числом переменных s для х..
Результаты: В решении задач инвестиционного портфеля используется новый подход, который по сути представляет собой одновременную оптимизацию максимизации ставки доходности и минимизации дисперсии ставки доходности, что приводит к разумным результатам. Результаты обладают качеством рациональности с точки зрения теории вероятностей для одновременной оптимизации нескольких целей. Выводы: Настоящий метод естественным образом отражает суть проблемы инвестиционного портфеля и дает новый способ решения важнейшей задачи.
Ключевые слова: задача инвестиционного портфеля, многокритериальная оптимизация, предпочтительная вероятность, дискретная выборка, теория вероятностей.
Инвестициони портфолио заснован на пробабилистичщ вишекритер^умсщ оптимизации и униформном диза]ну за експерименте са вари]аци]ама
Маошенг Ценг8, Ъаи иуб
Универзитет Северозапад, Си]ан, Народна Република Кина а Факултет хеми]ског инженерства, аутор за преписку
б Факултет природних наука
ОБЛАСТ: математика, рачунарске науке КАТЕГОРИИ (ТИП) ЧЛАНКА: оригинални научни рад
Сажетак:
Увод/циъ: Формулисан jе нови приступ решаваъу проблема инвестиционог портфол^а помогу истовременог оптимизоваъа максимизац^е стопе приноса и минимизац^е вар^ансе стопе приноса. Процесираъе се изводи комбинац^ом вишекритер^умске оптимизац^е засноване на вероватноЬи са униформним дизаjном за експерименте са вар^ац^ама.
Методе: Вишекритер^умска оптимизац^а заснована на вероватноЬи превасходно се користи да се двокритерц'умски проблем истовременог оптимизована максимизац^е стопе приноса и минимизац^е вар^ансе стопе приноса синтетише у jеднокритериjумски проблем укупне пожелне вероватноЬе сваког алтернативног сценар^а. Укупна пожелна вероватноЬа jе производ свих парц^алних пожелних вероватноЬа корисности сваке перформансе. Дакле, метод униформног дизаjна за експерименте са вар^ац^ама користи се за креиране скупа ефективних тачака узоркована за проблем инвестиционог портфол^а како би се постигла дискретизац^а у процесирану података и поjедноставио поступак у щем пропорц^а х, следи услов ограничена xi + Х2 + хз. . . + xs = 1 са укупним броjем вар^абли s за х..
Резултати: Нови приступ користи се за решаване проблема инвестиционог портфол^а щи, у суштини, представла истовремено оптимизоване максимизац^е стопе приноса и минимизац^е вар^ансе стопе приноса, што води до разумливих последица. Резултати имаjу квалитет рационалности са становишта теорбе вероватноЬе за истовремену оптимизац^у вишеструких цилева.
Заклучак: Оваj метод природно одсликава суштину проблема инвестиционог портфол^а и пружа нов начин за негово решаване.
Клучне речи: проблем инвестиционог портфол^а, вишекритер^умска оптимизац^а, пожелна вероватноЬа, дискретно узорковане, теорба вероватноЬе.
Paper received on / Дата получения работы / Датум приема чланка: 18.12.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 05.06.2023.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 07.06.2023.
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© 2023 Авторы. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «Creative Commons» (http://creativecommons.org/licenses/by/3.0/rs/).
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