APPLICATION OF PROBABILITY-BASED MULTI-OBJECTIVE OPTIMIZATION IN MATERIAL ENGINEERING Текст научной статьи по специальности «Медицинские технологии»

QPI/in/lHAflHI/1 HAyMHM PAflOBM OPimHAflbHblE HAyMHblE CTATbll ORIGINAL SCIENTIFIC PAPERS

APPLICATION OF PROBABILITY-BASED MULTI-OBJECTIVE OPTIMIZATION IN MATERIAL ENGINEERING

Maosheng Zheng

Northwest University, School of Chemical Engineering,

Xi'an, People's Republic of China,

e-mail: [email protected],

ORCID iD: https://orcid.org/0000-0003-3361-4060

DOI: 10.5937/vojtehg70-35366; https://doi.org/10.5937/vojtehg70-35366

FIELD: Mathematics, Materials ARTICLE TYPE: Original scientific paper

Abstract:

Introduction/purpose: Althought many methods have been proposed to deal with the problem of material selection, there are inherent defects of additive algorithms and subjective factors in such methods. Recently, a probability-based multi-objective optimization was developed to solve the inherent shortcomings of the previous methods, which introduces a novel concept of preferable probability to reflect the preference degree of the candidate in the optimization. In this paper, the new method is utilized to conduct an optimal scheme of the switching material of the RF-MEMS shunt capacitive switch, the sintering parameters of natural hydroxyapatite and the optimal design of the connecting claw jig.

Methods: All performance utility indicators of candidate materials are divided into two groups, i.e., beneficial or unbeneficial types for the selection process; each performance utility indicator contributes quantitatively to a partial preferable probability and the product of all partial preferable probabilities makes the total preferable probability of a candidate, which transfers a multi-objective optimization problem into a single-objective optimization one and represents a uniquely decisive index in the competitive selection process.

Results: Cu is the appropriate material in the material selection for RF -MEMS shunt capacitive switches; the optimal sintering parameters of natural hydroxyapatite are at 1100 C and 0 compaction pressure; and the optimal scheme is scheme No 1 for the optimal design of a connecting claw jig.

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Conclusion: The probability-based multi-objective optimization can be easily used to deal with an optimal problem objectively in material engineering.

Key words: multi-objective optimization, probability theory, preferable probability, material engineering, scheme selection.

Introduction

It has been more than 40 years (Ashby, 2000) since early works in material selection appeared; many methods have been proposed to analyze a big amount of data involved in the material selection process so as to obtain an appropriate result.

Various algorithms (techniques) have been developed, including Ashby's method (Ashby, 2000; Ashby et al, 2004), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Vlse Kriterijumska Optimizacija Kompromisno Resenje (VIkOr), Multi Attribute Decision Making (MADM), Analytical Hierarchy Process (AHP), Simple Additive Weighted (SAW) method and Multi-Objective Optimization on the basis of Ratio Analysis (MOORA), etc (Zheng et al, 2021). Ashby's method is difficult to be applied in cases which involve multiple criteria of selection (Ashby, 2000; Ashby et al, 2004; Zheng et al, 2021). Deshmukh et al employed the multi-objective optimization (MOO) techniques of TOPSIS and VIKOR to perform the material selection of the switching structure for RE-MEMS shunt capacitive switches (Deshmukh & Angira, 2019). However, there exist inherent problems of additive algorithms and subjective factors in the MADM, AHP, MOORA, TOPSIS and VIKOR due to their fatal scaling or normalization processes (Zheng et al, 2021).

Recently, a new probability-based multi-objective optimization method was developed (Zheng et al, 2021), attempting to solve the inherent problems of personal and subjective factors in the previous multi-objective optimization methods. The novel concept of preferable probability was introduced to reflect the preference degree of a candidate in the optimization where all performance utility indicators of candidates are divided into beneficial or unbeneficial types for the selection. Each performance utility indicator of a candidate contributes to a partial preferable probability quantitatively, and the total preferable probability of a candidate is the product of all partial preferable probabilities from the viewpoint of the probability theory, which is the overall and unique decisive index in the competitive selection process. The new multi-objective optimization method was also extended with the application of the multi-objective orthogonal test design method (OTDM) and the

uniform test design method (UTDM), which results in appropriate achievements (Zheng et al, 2021).

In this paper, the new probability-based multi-objective optimization method is used to perform the optimal scheme in material engineering, which includes the selection of switching material of the RF-MEMS shunt capacitive switch, the optimization of the sintering parameters of natural hydroxyapatite and the optimal design of a connecting claw jig.

Brief introduction to the new multi-objective optimization method

In the new probability-based multi-objective optimization method (Zheng et al, 2021), a beneficial utility index of material performance indicator contributes to a partial preferable probability in a positively linear manner, i.e.,

Pij = OijXij, i = 1, 2,...,n; j = 1, 2,..m. (1)

In Eq. (1), Xij is the jh beneficial utility index of the material performance indicator of the i1h candidate material; Pij represents the partial preferable probability of the beneficial utility index X-,; n is the total number of candidate materials in the material group involved; m is the total number of the utility indices of each candidate material in the group; a; is the normalized factor of the ;1h utility index of the material performance indicator, a = 1/(nX.), and X,. is the arithmetic mean

U ^ J ^ J

value of the utility index of the material performance indicator in the material group involved.

Equivalently, the unbeneficial utility index of the material performance indicator contributes to a partial preferable probability in a negatively linear manner, i.e.,

Pij = /3ij(Xjmax + Xjmin -Xi), i = 1, 2,...,n; j = 1, 2,..m. (2)

In Eq. (3), Xjmax and Xjmin represent the maximum and minimum values of the utility indices Xj of the material performance indicator in the material group, respectively, and ft is the normalized factor of the j1h utility indices of the material performance indicator, ft = 1/[n(Xjmin + Xjmax) - nX} ].

Moreover, the total / comprehensive preferable probability of the /1h candidate material is the product of its partial preferable probability P,j of each utility index of the material performance indicator in the overall

selection due to the "simultaneous optimization" of the multi-objects in the viewpoint of probability theory (Zheng et al, 2021), i.e.,

m

Pi = Pi * Pi2 ••• Pim =n Pj . (3)

j=1

The total preferable probability of a candidate is the uniquely decisive index in the overall selection process competitively, which transfers a multi-objective optimization problem (MOOP) into a single -objective optimization one. The main characteristic of the new probability-based multi-objective optimization is that the treatment for both beneficial utility index and unbeneficial utility index is equivalent and conformable, which is without any artificial or subjective scaling factors involved in the process.

Application of probability-based multi-objective optimization in material engineering

1) Multi-objective optimization in the material selection of RF-MEMS shunt capacitive switches

Radio Frequency Micro Electro Mechanical Systems (RF-MEMS) is a promising technology for implementing passive devices in future wireless communication systems (Deshmukh & Angira, 2019). Switches have drawn more attention due to their frequent use in many cases in wireless communication systems. An RF-MEMS technology-based switch has low insertion loss, high isolation, high linearity and less power consumption (Deshmukh & Angira, 2019). Its shunt capacitive switch has two stable states i.e., up-state and down-state (Deshmukh & Angira, 2019). Power can flow from the input port to the output port in the switch upstate, while it stands at the off-state in its down-state (Deshmukh & Angira, 2019; Angira & Rangra, 2015a; Angira & Rangra, 2015b).

The optimization of the performance of the switching structure involves many parameters (criteria), such as pull-in voltage, RF response (insertion loss and isolation), maximum displacement, thermal conductivity, etc (Deshmukh & Angira, 2019; Angira & Rangra, 2015a; Angira & Rangra, 2015b). Since many parameters are involved, it can be seen as a MOOP in the performance optimization of the switching material selection. Therefore, a MOOP can be used to decrease human effort since a large number of materials are available in practice, forming a material bank together with many manufacturing processes and selection attributes (Zheng et al, 2021).

Yang et al pointed out that if different normalization methods are applied, significant different results may be produced (Yang et al, 2021). Podviezko et al also stressed that different normalization of data applying to popular MCDM methods such as SAW or TOPSIS could lead to significant differences in the assessment (Podviezko & Podvezko, 2015). As a consequence, many researchers paid a lot of attention to the choice of the normalization type. However, it is still puzzling which normalization method is better and how to determine final results of material selection from different normalization algorithms.

A) Utility indices of the material performance indicators in the material selection of RF - MEMS shunt capacitive switches

In the study of Deshmukh & Angira (2019), the optimal objectives for this purpose are low pull-in voltage, low RF loss, high thermal conductivity and maximum displacement of the beam structure. As a result, the square root of Young's modulus of the material E05, the electrical resistive coefficient pe, the thermal conductivity of the material A, the ratio of the fracture strength at to Young's modulus E of the material, ot/E, are taken as the optimal utility indices of the material attribute indicators (Deshmukh & Angira, 2019).

B) Divisions of the utility indices in the material selection of RF - MEMS shunt capacitive switches

From analyzing the requirements of the optimizations of the bridge of RF-MEMS shunt capacitive switches, i.e., higher pull-in voltage (Vp), lower RF loss, higher thermal conductivity and the higher maximum displacement of the switch beam (Deshmukh & Angira, 2019), the utility indices of the square root of Young's modulus of the material, E05, the thermal conductivity of the material, A, the ratio of the fracture strength at to Young's modulus E of the material, at/E, belong to the beneficial type of the material performance index, while the electrical resistive coefficient, pe, belongs to the unbeneficial type of the material performance index in the assessment.

C) Assessment results

The values of the conventional material performance indicators for various materials are given in Table 1 (Deshmukh & Angira, 2019).

The partial preferable probabilities of the utility indices of E05, A and pe and at/E and the total preferable probabilities are assessed according to Equations (1) through (5), respectively, shown in Table 2. In addition,

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the ranking here by using the new probability-based multi-objective optimization method is given in Table 2 together with those of Vikor and Topsis from Ref. (Deshmukh & Angira, 2019) for comparison.

Table 1 - Conventional material performance indicators for various materials (Deshmukh

& Angira, 2019)

Таблица 1 - Стандартные показатели эффективности материалов для различных материалов (Deshmukh & Angira, 2019) Табела 1 - Индикатори уобича]ених перформанси материала за различите материале (Deshmukh & Angira, 2019)

Mat. Young's modulus E (GPa) Electrical resistive coefficient p (0 m) 10-8 Thermal conductivity 1 (W/mK) Fracture strength crf (MPa) (OT/E) x103

Ni 193 6.99 90 345 1.7876

Au 70 2.44 315 220 3.1429

Al 70 2.82 204 47 0.6714

Ag 83 1.59 407 110 1.3253

Pt 168 10.5 73 125 0.7440

Cu 117 1.68 386 314 2.6838

Cr 279 12.9 90 370 1.3262

W 411 5.28 163 1725 4.1971

Co 209 6.24 69 675 3.2297

Fe 211 9.61 73 540 2.5592

Table 2 - Partial preferable probabilities and total preferable probabilities for various materials for shunt capacitive switch optimization Таблица 2 - Частичные предпочтительные вероятности и общие предпочтительные вероятности для различных материалов при оптимизации

емкостного шунтирующего переключателя Табела 2 - Делимичне поже^не вероватноПе и укупне поже^не вероватноПе за различите материале у оптимизации капацитивног прекидача шанта

Mat. PeW.5 Ppe Pi Pof/E Ptxlû4 Rank here Rank Vikor Rank Topsis

Ni 0.1073 0.0884 0.0481 0.0825 0.3766 6 6 6

Au 0.0646 0.1420 0.1684 0.1451 2.2423 3 1 1

Al 0.0646 0.1375 0.1091 0.0310 0.3005 7 4 4

Ag 0.0704 0.1520 0.2176 0.0612 1.4242 4 3 3

Pt 0.1001 0.0470 0.0390 0.0343 0.0631 10 8 9

Cu 0.0835 0.1510 0.2064 0.1239 3.2248 1 2 2

Cr 0.1290 0.0187 0.0481 0.0612 0.0712 9 10 10

W 0.1566 0.1085 0.0872 0.1937 2.8697 2 9 7

Co 0.1117 0.0972 0.0369 0.1490 0.5971 5 5 5

Fe 0.1122 0.0575 0.0390 0.1181 0.2975 8 7 8

It can be seen from Table 2 that the appropriate material from the new multi-objective optimization method is Cu, which is different from those of Vikor and Topsis from (Deshmukh & Angira, 2019); this is because of the inherent defects of personal and subjective factors in Vikor and Topsis (Deshmukh & Angira, 2019).

In fact, the evaluation result of the new probability-based method for multi-objective optimization in material selection is no need to equal to those of other previous approaches exactly due to their involvements of personal or other subjective coefficients.

2) Optimization of sintering parameters of natural hydroxyapatite

Abifarin conducted the optimization of hydroxyapatite (HAp) mechanical characteristics using Taguchi grey relational analysis design which includes hardness and compressive strength (Abifarin, 2021). Three levels of sintering temperature and two levels of compaction pressure are employed during sintering (Abifarin, 2021). The design and the results are shown in Table 3. Again, the probability-based multi-objective optimization is used to conduct the assessment with hardness and compressive strength as the beneficial type index. The evaluation results are shown in Table 4.

Table 3 - Design and the results of HAp Таблица 3 - Разработка и результаты гидроксиапатита Табела 3 - Про]ектова^е и резултати хидроксиапатита

No Pressure Temperature °C Hardness Compressive Strength

1 0 900 0.54 0.39

2 0 1000 0.838 0.58

3 0 1100 0.940 0.84

4 5 900 0.656 0.34

5 5 1000 0.929 0.5

6 5 1100 1.103 0.69

Table 4 - Evaluation results of HAp Таблица 4 - Результаты оценки гидроксиапатита Табела 4 - Резултати оцене хидроксиапатита

Partial preferable probability Total

No Hardness Strength Pt*102 Rank

1 0.1079 0.1168 1.2596 6

2 0.1674 0.1737 2.9069 3

3 0.1878 0.2515 4.7225 1

4 0.1311 0.1018 1.3340 5

5 0.1856 0.1497 2.7781 4

6 0.2203 0.2066 4.5518 2

CD

Table 4 indicates that the optimal sintering parameters of natural hydroxyapatite are at 1100°C and 0 compaction pressure.

3) Optimal design of a connecting claw jig

Yan et al conducted the multi-objective optimal design of a connecting claw jig with ANSYS Workbench finite element analysis software (Yan et al, 2021). The maximum equivalent stress (MPa) Yi, the weight (kg) Y2, the minimum safety factor Y3 and the maximum deformation (mm) Y4 of the claw jig are taken as the optimization objectives, while the thickness of the substrate FD1 (mm) xi, the angle of the connecting claw A1 (°) X2, the thickness of the connecting claw FD2 (mm) X3, and the outside diameter of the jig base R1 (mm) X4 are taken as the input variables.

After the simulation and the analysis, three candidate schemes with good objective functions are selected by the system, as shown in Table 5. The object Y3 is a beneficial type index, while Yi, Y2 and Y4 are all unbeneficial type indexes. The evaluation results are shown in Table 6.

Table 6 shows that the optimal scheme is scheme No 1.

Table 5 - Three candidate schemes of the connecting claw jig Таблица 5 - Три возможные схемы кулачковой муфты Табела 5 - Шеме три кандидата за кануасту споjницу

Original scheme X1 (mm) X2 (°) X3 (mm) X4 (mm) Y1 (MPa) Y2 (kg) Y3 Y4 (mm)

56 79 38 32.5 151.22 6.176 1.6532 1.171

1 54.125 73.13 35.414 31.051 128.42 5.615 1.9467 0.923

2 48.125 73.77 37.982 32.395 143.31 5.577 1.7444 1.043

3 46.625 76.38 39.908 30.715 161.48 5.620 1.5482 1.375

Table 6 - Evaluation results of the connecting claw jig Таблица 6 - Результаты оценки кулачковой муфты Табела 6 - Резултати оцена кануасте cnojHUце

No. Partial preferable probability Total

Y1 Y2 Y3 Y4 Pt*100 Rank

1 0.3700 0.3327 0.3716 0.3870 1.7697 1

2 0.3358 0.3349 0.3329 0.3532 1.3229 2

3 0.2942 0.3324 0.2955 0.2598 0.7507 3

CD

Conclusion

The application of the new probability-based multi-objective optimization method in dealing with three optimal problems of material engineering has shown that: the appropriate material (Cu) is successfully selected, which meets the requirements of the optimizations of the bridge of RF - MEMS shunt capacitive switches; the optimal sintering parameters of natural hydroxyapatite are at 1100°C and 0 compaction pressure; and the optimal scheme of the connecting claw jig is scheme No 1. The main feature of the new probability-based multi-objective optimization method is that the treatment is equivalent and conformable for both the beneficial utility index and the unbeneficial utility index, without any artificial or subjective scaling factors involved in the process.

References

Abifarin, J.K. 2021 Taguchi grey relational analysis on the mechanical properties of natural hydroxyapatite: effect of sintering parameters. The International Journal of Advanced Manufacturing Technology, 117, pp.49-57. Available at: https://doi.org/10.1007/s00170-021-07288-9.

Angira, M. & Rangra, K. 2015a. Design and investigation of a low insertion loss, broadband, enhanced self and hold down power RF-MEMS switch. Microsystem Technologies, 21(6), pp.1173-1178. Available at: https://doi.org/10.1007/s00542-014-2188-6.

Angira, M. & Rangra, K. 2015b. Performance improvement of RF-MEMS capacitive switch via asymmetric structure design. Microsystem Technologies, 21(7), pp.1447-1452. Available at: https://doi.org/10.1007/s00542-014-2222-8.

Ashby, M.F. 2000. Multi-Objective optimization in material design and selection. Acta Materialia, 48(1), pp.359-369. Available at: https://doi.org/10.1016/S1359-6454(99)00304-3.

Ashby, M.F., Bréchet, Y.J.M., Cebon, D. & Salvo, L. 2004. Selection strategies for materials and processes. Materials & Design, 25(1), pp.51-67. Available at: https://doi.org/10.1016/S0261-3069(03)00159-6.

Deshmukh, D. & Angira, M. 2019. Investigation on Switching Structure Material Selection for RF-MEMS Shunt Capacitive Switches Using Ashby, TOPSIS and VIKOR. Transactions on Electrical and Electronic Materials, 20, pp.181-188. Available at: https://doi.org/10.1007/s42341-018-00094-3.

Podviezko, A. & Podvezko, V. 2015. Influence of data transformation on multicriteria evaluation result. Procedia Engineering, 122, pp.151-157. Available at: https://doi.org/10.1016/j.proeng.2015.10.019.

Yan, Y., Fu, N., Zhang, X., Wang, C., Sun, J. & Lu, L. 2021. Research on the Multi-Objective Optimization Design of Connecting Claw Jig. International Journal of Steel Structures, 21(6), pp.1911-1920. Available at: https://doi.org/10.1007/s13296-021-00542-6.

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Yang, W-C., Chon, S-K., Choe, C-M. & Yang, J-Y. 2021. Materials selection method using TOPSIS with some popular normalization methods. Engineering Research Express, 3(1), art.number: 015020. Available at: https://doi.org/10.1088/2631-8695/abd5a7.

Zheng, M., Wang, Y. & Teng, H. 2021. A New "Intersection" Method for Multi-Objective Optimization in Material Selection. Tehnicki glasnik, 15(4), pp.562-568. Available at: https://doi.org/10.31803/tg-20210901142449.

ПРИМЕНЕНИЕ МНОГОЦЕЛЕВОЙ ОПТИМИЗАЦИИ, ОСНОВАННОЙ НА ВЕРОЯТНОСТИ В МАТЕРИАЛОВЕДЕНИИ

Маошенг Чжэн

Северо-западный политехнический университет, факультет химической инженерии, г. Сиань, Народная Республика Китай

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА:

27.47.00 Математическая кибернетика; 27.47.19 Исследование операций, 45.00.00 ЭЛЕКТРОТЕХНИКА: 45.09.00 Электротехнические материалы ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: Несмотря на то, что существует множество методов решения проблемы выбора материала на основе аддитивных алгоритмов, такие алгоритмы по своей сути содержат недостатки и субъективные факторы. С целью преодоления недостатков этих методов, недавно была разработана многокритериальная вероятностная

оптимизация, которая вводит новую концепцию предпочтительной вероятности, показывающую степень предпочтительности кандидата в процессе оптимизации. В данной статье используется новый метод в разработке оптимальной схемы материала для емкостного шунтирующего переключателя в радиочастотных микроэлектромеханических системах (РЧ МЭМС), параметрах спекания природного гидроксиапатита, а также для оптимальной разработки кулачковой муфты.

Методы: Все показатели полезности материалов-кандидатов делятся на две группы, полезные или невыгодные для процесса отбора; каждый показатель полезности вносит количественный вклад в частичную предпочтительную вероятность, а произведение всех частичных предпочтительных вероятностей составляет общую

ПРИМЕНА ВИШЕКРИТЕРШУМСКЕ ОПТИМИЗАЦШЕ

предпочтительную вероятность кандидата, что переводит проблему многокритериальной оптимизации в проблему оптимизации с одним критерием и представляет собой уникальный индекс в процессе конкурсного отбора.

Результаты: Медь оказалась подходящим материалом при 5 выборе материалов для емкостных шунтирующих переключателей в радиочастотных микроэлектромеханических ф системах (РЧ МЭМС); оптимальные параметры спекания го природного гидроксиапатита - 1100° С при нулевом давлении щ сжатия, а оптимальной схемой проектирования кулачковой § муфты является схема №1.

Выводы: Многокритериальная оптимизация на основе вероятностей может широко применяться при принятии § объективных решений оптимальных проблем в матереаловедении.

Ключевые слова: многокритериальная оптимизация, теория вероятностей, предпочтительная вероятность,

материаловедение, выбор схемы. -§

тз ф

(Л

ЗАСНОВАНА НА ВЕРОВАТНОЪИ У ТЕХНОЛОГИИ МАТЕРИАЛА -2

Маошенг Ценг

Универзитет Северозапад, Факултет хеми]ског инженерства, Си]ан, Народна Република Кина £5

.о го .о

ОБЛАСТ: математика, матери]али ВРСТА ЧЛАНКА: оригинални научни рад

Сажетак:

Увод/цил>: Иако посто}и много метода за решаваъе проблема < селекци'е материала заснованих на адитивним алгоритмима, такви алгоритми инхерентно садрже недостатке и суб}ективне с, факторе. Ради превазилажеъа слабости поменутих метода, недавно }е разви]ена вишекритери}умска оптимизаци}а заснована на вероватноЬи ко}а уводи нови концепт пожеъне вероватноЬе ко]и показу}е степен пожел>ности кандидата при оптимизации. У овом раду користи се нов метод за изво^еъе оптималне шеме за материал за капацитивну склопку шанта у радиофреквенци]ским микроелектромеханичким системима (РФ МЕМС), за параметре синтероваъа природног хидроксиапатита, као и за оптимално про]ектовак>е кануасте спо]нице.

Методе: Сви показатели перформанси корисности материала -кандидата деле се на корисне и некорисне за селекциу. Сваки показател перформанси корисности квантитативно доприноси делимичноj пожелноj вероватноЬи, док производ свих делимичних пожелних вероватноЬа чини укупну пожелну вероватноПу кандидата, чиме се проблем вишекритери]умске оптимизаци]е преводи у проблем ]еднокритериумске оптимизаци]е и представла jединствени одлучу]уПи индекс у компетитивном процесу селекци'е.

Резултати: Бакар се показао као одеовара}уЬи материал при селекции материала за капацитивне склопке шанта у радиофреквенци]ским микроелектромеханичким системима (РФ МЕМС). Оптимални параметри синтероваша природное хидроксиапатита су 1100 °C и нулти притисак саби}аша, а оптимална шема за про}ектоваше кануасте спо}нице }есте шема бро] 1.

Заклучак: Вишекритери]умска оптимизаци}а на бази вероватноЬе може се jедноставно применити за об]ективно решаваше оптималное проблема у технологии материала.

Клучне речи: вишекритериумска оптимизаци]а, теори]а вероватноЬе, пожелна вероватноЬа, технолоеи}а материала, селекци}а шеме.

Paper received on / Дата получения работы / Датум приема чланка: 12.12.2021. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 29.12.2021.

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 31.12.2021.

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