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Evaluation of steel turning by means of probability - based multi - objective optimization with appropriate numbers of attributes
Maosheng Zhenga, Jie Yub
a Northwest University, School of Chemical Engineering, Xi'an, People's Republic of China, e-mail: [email protected], corresponding author, ORCID iD: https://orcid.org/0000-0003-3361-4060
b Northwest University, School of Life Science & Technology, Xi'an, People's Republic of China, e-mail: [email protected],
ORCID iD: https://orcid.org/0000-0001-6606-5462
DOI: 10.5937/vojtehg71 -42843; https://doi.org/10.5937/vojtehg71-42843
FIELD: mathematics, materials ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: Turning is a typical machining process. However, an appropriate solution for a concurrent optimization of minimizing surface roughness, minimizing cutting forces and vibrations, and maximizing the material removal rate in turning processes has not been found yet. This article formulates the rule of separating an independent attribute from multiple attributes by using the linear correlation coefficient in the spirit of the cluster analysis first. Moreover, the evaluation of the concurrent optimization of steel turning by means of probability - based multi -objective optimization (PMOO) is taken as an example to show the procedure including the separation of an independent attribute from multiple attributes by using PMOO.
Methods: PMOO is a promising solution for turning processes. It is necessary to have an independent attribute in the evaluation of PMOO to be analogical as an independent event in the view of the probability theory. The separation of an independent attribute from multiple attributes by using the linear correlation coefficient is conducted in the spirit of the cluster analysis. It further assumes that if the linear correlation coefficient of two attributes in the cluster analysis is higher than 0.8, i.e., in case of very strong correlation, then they could be put into one category, and only one of them could be taken as an independent attribute to join the evaluation of PMOO.
Results: The formulation reflects the essence of PMOO and its application in material machining rationally, which opens a new way for solving the
relevant problem. The example of the parameter optimization of steel turning by means of PMOO indicates the rationality of the appropriate solution.
Conclusion: This innovative study has practical significance of making the utilization of PMOO method reasonable by providing a rational rule of separating independent attributes from multiple attributes of PMOO.
Key words: multi - objectives, cluster analysis, independent attributes, linear correlation coefficient, metal turning.
Introduction
Turning is a typical machining process. The number of turning machines is about 30% of all cutting machines in a cutting workshop (Thien Van et al, 2021; Hegde et al, 2022; Yildiz et al, 2023, Nguyen & Vo Thi, 2022).
The surface roughness of machining, cutting forces, vibrations, and the material removal rate (MRR) are usually used as the assessed attributes of quality evaluation for the overall machining process. In order to ensure the minimum value of surface roughness, Taguchi design and the Response Surface Method (RSM) were frequently used to conduct the optimization of cutting parameters such as cutting velocity, feed rate, and cutting depth in the turning process for various materials with minimizing surface roughness, or minimizing cutting forces, or maximizing the MRR, solely (Thien Van et al, 2021; Hegde et al, 2022; Yildiz et al, 2023, Nguyen & Vo Thi, 2022; Irzaev et al, 2021).
However, until now, an appropriate solution for a concurrent optimization of minimizing surface roughness, minimizing cutting forces and vibrations, and maximizing the MRR in turning processes has not been achieved yet (Thien Van et al, 2021; Hegde et al, 2022; Yildiz et al, 2023, Nguyen & Vo Thi, 2022; Irzaev et al, 2021).
Actually, the concurrent optimization of minimizing surface roughness, minimizing cutting forces and vibrations, and maximizing the MRR in turning processes is a typical optimization problem with multiple objectives (attributes); it is essentially focused on the simultaneity of the optimization of multiples objectives.
Probability - based multi - objective optimization (PMOO) is a newly developed approach to conduct the concurrent optimization problem of multiple attributes (objectives). A new idea of preferable probability and its assessment have been put forward (Zheng et al, 2024).
The core content of PMOO is taking the "simultaneous optimization of multiple attributes" from the entire or systematic viewpoint of the
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system theory; therefore, a probability - based method was formulated on the basis of the probability theory and the set theory, taking each attribute as an independent event in the subsequent treatment.
The advantages of PMOO are its probabilistic foundation in view of the system theory, rationality and certainty of its solution without any artificial factors, and a simple and convenient algorithm in mathematical treatment, which are obviously superior to other methods of multi-objective optimization such as the Analytic Hierarchy Process (AHP), the Vlsekriterijumsko KOmpromisno Rangiranje (VIKOR), the Technique of Ranking Preferences by Similarity to the Ideal Solution (TOPSIS), Multi-Objective Optimization (MOO) on the basis of the Ratio Analysis (MOORA) , the Pareto solution, the Grey Relational Analysis (GRA), etc. (Zheng et al, 2024; Salomon, 2019). Besides, this approach is superior regarding simplicity in data processing to other metaheuristics.
The new approach could be employed in many fields involving multiple attributes, including energy planning, economic affairs, operation research, programming problems, material selection, mechanical design, etc. Therefore, PMOO is a promising solution for a concurrent optimization of minimizing surface roughness, minimizing cutting forces and vibrations, and maximizing the MRR of turning processes in view of the system theory (Zheng et al, 2024).
Moreover, from the perspective of the probability theory and the set theory, the intersection of independent events and the joint probability of independent events could be used to characterize the concurrent occurrence of multiple independent events as the concurrent optimization of multiple attributes. In this way, when it allocates each attribute to an independent event, the problem of the simultaneous optimization of multiple attributes becomes "rule - based". However, the allocation of each attribute to an independent event naturally relies on the separation of independent events from multiple attributes such that the PMOO method could be used rationally.
Thus, separating an independent event from multiple attributes is of considerable significance to ensure the appropriate application of the PMOO method in material selection.
While, the cluster analysis could fortunately be employed to conduct the separation of an independent attribute from multiple attributes. By classifying things rationally, problems in the material world could be clarified and understood gradually (Backhaus et al, 2021; Scitovski et al, 2021). In the process of the cluster analysis, the class is often not given in advance, but it needs to be determined according to the characteristics of the observed data, and there is no need to make any assumptions
about the number and structure of classes. In the clustering results, attributes belonging to the same class tend to be similar to each other in a sense, while attributes belonging to different classes tend to be dissimilar. The purpose of the cluster analysis is to classify attributes into several classes according to certain rules. The cluster analysis can be divided into the Q - type cluster analysis and the R - type cluster analysis in accordance with different classification objectives. The Q -type clustering analysis is for samples and the R - type clustering is for performances (Backhaus et al, 2021; Scitovski et al, 2021).
Generally speaking, according to the degree of similarity, attributes (or samples) are classified one by one; closely related classes are clustered into a small taxon, and then gradually expanded, so that the alienated ones are clustered into a large taxon, until all samples (or performances) are clustered, forming a cluster diagram that represents the affinity. Samples (or performances) are classified in accordance with some requirements in turn (Backhaus et al, 2021; Scitovski et al, 2021).
The general viewpoint of classification is that the closer the similarity of attributes is, the closer their similarity coefficient is to 1 or -l, while the similarity coefficient of unrelated attributes is closer to 0.
Those with higher similarity are classified into one category, and those with higher dissimilarity are classified into different categories. The distance in the variable "space" is the characteristic between the "points". Each sample is regarded as a point in the P-dimensional space, and the distance between the points is measured by some kind of measurement. The points that are closer to each other belong to one category while the points that are farther away belong to different categories.
This paper mainly focuses on separating independent attributes from multiple attributes of PMOO in respect of the R - type cluster analysis rationally, so as to guarantee the appropriate application of the PMOO method in material selection first. The evaluation of steel turning by means of PMOO is presented as an example of the process of separating independent attributes from multiple attributes for subsequent evaluation.
Procedure of separating an independent attribute from multiple attributes in PMOO for material machining by means of the cluster analysis
The formulation of separating an independent attribute from multiple attributes in PMOO for material machining by means of the cluster analysis is as follows:
1. Representative of similarity
As a representative of similarity, the linear correlation coefficient is frequently employed as a branch to identify similarity (Backhaus et al, 2021; Scitovski et al, 2021).
The linear correlation coefficient is defined by
rjk=Z(* -uj ) ■ -uk) / [Z(yj -uJ )2 ■ Z(y-k -uk)2]05
(i)
In equation (1), rjk is the linear correlation coefficient which is employed to identify the degree of linear correlation between two attributes yij and yik; uj is the average value of the j-th attribute and uk is the average value of the k-th attribute.
Obviously, the linear correlation coefficient is just the right coefficient to reflect the linear proportional relationship between two attribute indexes y-j and y^, it is more reasonable to reflect the similarity between samples or attributes; in addition, the linear correlation coefficient also has the invariance of normalization similar to the equation (Backhaus et al, 2021; Scitovski et al, 2021).
2. Rules of separating an independent attribute from multiple attributes
As mentioned previously, in the PMOO method for material selection, allocating each attribute to an independent event depends on differentiating an independent event from multiple attributes through the cluster analysis. This section gives the formulation of separation of an independent attribute from multiple attributes.
In the light of the general rule of the R - type clustering analysis for performance classification and the advantage of the linear correlation coefficient in the cluster analysis, the linear correlation coefficient is employed to formulate the separation of an independent attribute from multiple attributes. The appropriate rules are given in the following steps:
a) Evaluations of the similarity of attributes and classification
The linear correlation coefficient in the cluster analysis is used to characterize the similarity of attribute indexes in the performance classification first.
b) Identification of the attribute category
As for the attribute classification, it further assumes that if the linear correlation coefficient of two attributes in the cluster analysis is higher than 0.8, i.e., in case of very strong correlation, they can be put into one category, and only one of them can be used as an independent attribute to join the evaluation of PMOO while the attributes with the linear correlation coefficient lower than 0.8 in the cluster analysis are considered to be in different categories.
c) Evaluation of an independent attribute in PMOO for material machining
Take each independent attribute to join the evaluation of PMOO for material machining only. Especially, if more attributes than the independent attribute are used to join the evaluation and the analysis of the multi - objective optimization problem, it is equivalent to the increase of the weighting factors of the corresponding attributes.
Application in the optimization of the steel turning parameters
Thien Van et al. reported the results of the multi - objective optimization problem of turning EN 10503 steel by using the VIKOR method (Thien Van et al, 2021). However, the shortcomings of the "closeness" to the "virtual ideal solution" and the additional weighting factor of the VIKOR method remained.
In this article, the multi - objective optimization problem of the turning of EN 10503 steel is re-analyzed by means of PMOO with the cluster analysis rationally. In Thien Van's research, the cutting velocity n, the feed rate f, the depth of cut t, and the insert nose radius r were chosen as the input parameters with three levels for each parameter. Taguchi's orthogonal array Lg(34) was used to conduct the design and experiments, as shown in Table 1. The surface roughness Ra, the cutting force components Fx, Fy, and Fz (in the x, y, and z directions), the vibration component amplitudes Ax, Ay and Az (in the x, y, and z directions), and the material removal rate (the MRR) were taken as their evaluated attributes (objectives). Their results are given in Table 2.
Table 1 - Experiment design with L9(34) Таблица 1 - Планирование эксперимента с L9(34) Табела 1 - Диза]н експеримента са L9(34)
No. Coded value Actual value
n f t r n (rev/min) f (mm/rev) t (mm) r (mm)
1 1 1 1 1 460 0.08 0.20 0.4
2 1 2 2 2 460 0.194 0.35 0.6
3 1 3 3 3 460 0.302 0.50 1.2
4 2 1 2 3 650 0.08 0.35 1.2
5 2 2 3 1 650 0.194 0.50 0.4
6 2 3 1 2 650 0.302 0.20 0.6
7 3 1 3 2 910 0.08 0.50 0.6
8 3 2 1 3 910 0.194 0.20 1.2
9 3 3 2 1 910 0.302 0.35 0.4
Table 2 - Experimental results with the L9(34) design Таблица 2 - Результаты эксперимента разработки L9(34) Табела 2 - Резултати експеримента са дизайном L9(34)
No. Ra (Mm) Fx (N) Fy (N) Fz (N) Ax (Mm) Ay (Mm) Az (Mm) MRR (mm3/s)
1 0.840 85.2740 24.9800 107.4400 2.385 5.3594 5.5826 7.948
2 0.605 166.2340 47.5420 230.3210 3.9816 8.5019 9.0195 54.471
3 0.644 563.7300 153.285 965.2270 5.9601 12.1603 16.2276 178.071
4 1.122 219.2030 64.0220 335.7370 5.9392 8.8440 13.9882 57.823
5 0.669 152.2660 38.5830 191.5410 4.3123 7.6545 9.3600 42.398
6 0.643 175.3230 44.1470 211.6830 5.0853 9.9639 12.5087 31.447
7 0.621 191.0840 51.7270 300.1620 4.4647 7.4923 10.1177 60.009
8 0.729 212.9260 59.1170 307.8790 5.8284 8.4602 14.1956 33.694
9 0.675 124.9690 40.5450 164.2060 6.2633 10.1637 15.2682 38.130
Let us study the similarity of attributes (objectives) first.
The similarity analysis of these data shows that there is a strong linear correlation among the cutting force components of Fx, Fy and Fz, and among the vibration component amplitudes of Ax, Ay and Az of the turning process.
The linear correlation coefficients of Fx vs Fy and Fx vs Fz are rFxFy = 99.72 % and rFxFz = 99.64%, respectively, see Figure 1. The linear correlation coefficients of Ax vs Ay and Ax vs Az are rAxAy = 97.77 % and rAxAz = 80.39%, respectively, see Figure 2.
Figure 1 - Linear correlations of Fx vs Fy and Fx vs Fz Рис. 1 - Линейные корреляции между Fx и Fy, а также между Fx и Fz Слика 1 - Линеарне корелаци^е измену Fx и Fy, као и измену Fx и Fz
Figure 2 - Linear correlations of Ax vs Ay and Ax vs Az Рис. 2 - Линейные корреляции между Ax и A y, а также между Ax и Az Слика 2 - Линеарне корелаци^е измену Ax и A y, као и измену Ax и Az
As stated in the previous section, since there is a strong linear relationship among Fx, Fy and Fz, only one component of them can be employed as the independent attribute to join the evaluation of PMOO. The same applies for Ax, Ay and Az. Therefore, Fx and Ax are taken as the independent attributes to join the evaluation of PMOO.
Finally, the surface roughness Ra, the cutting force components Fx, the vibration component amplitudes Ax, and the material removal rate MRR were taken as the actual evaluated independent multiple attributes.
Furthermore, in accordance with PMOO, the MRR belongs to the beneficial performance index to join the evaluation of partial preferable probability while Ra, Fx, and Ax, belong to the unbeneficial performance index to join the evaluation of their partial preferable probabilities.
The assessed consequences are shown in Table 3 which indicates that experiment No. 2 has the highest total preferable probability Pt at the first glance, followed by experiment No. 7.
However, the optimal configuation of Thien Van et al. is just experiment No. 7 which obviously exibits poorer responses than experiment No. 2 integrally (see the following detail for comparison).
As for experiment No. 2, the responses of the surface roughness, the cutting force and the vibration component amplitudes (in the X, Y, and Z directions), and the material removal rate (MRR) of experiment No. 2 are 0.605 |jm, 166.2340 N, 47.5420 N, 230.3210 N, 3.9816 |jm, 8.5019 jm, 9.0195 jm, and 54.471 mm3/s, respectively, while in experiment No. 7, the responses of the surface roughness, the cutting force and the vibration component amplitudes (in the X, Y, and Z directions), and the material removal rate (MRR) are 0.621 jm, 191.084 N, 51.727 N, 300.162 N, 4.465 jm, 7.492 jm, 10.118 jm, and 60.009 mm3/s, respectively.
Furthermore, the range analysis can be conducted for the total preferable probability Pt to perform successive optimization, as shown in Table 4. It indicates that the order of impact of the input variables is r > t >f > n, and the subsequent optimal configuration will be r2t3f2n3.
Table 3 - Assessed results of preferable probability and ranking
Таблица 3 - Полученные результаты предпочтительной вероятности и
ранжирования
Табела 3 - Добц'ени резултати поже^не вероватноПе и рангираъе
No. PRa PFx Pax Pmrr Pt*104 Rank
1 0.0986 0.1427 0.1863 0.0158 0.4135 9
2 0.1247 0.1222 0.1388 0.1081 2.2875 1
3 0.1204 0.0216 0.0800 0.3533 0.7344 6
4 0.0673 0.1088 0.0806 0.1147 0.6767 8
5 0.1176 0.1258 0.1290 0.0841 1.6051 3
6 0.1205 0.1199 0.1060 0.0624 0.9558 4
7 0.1230 0.1159 0.1245 0.1191 2.1123 2
8 0.1110 0.1104 0.08309 0.0669 0.6870 7
9 0.1170 0.1327 0.0710 0.0757 0.8329 5
Table 4 - Results of the range analysis Таблица 4 - Результаты анализа ранжирования Табела 4 - Резултати анализе рангираша
Level n f t r
1 1.1451 1.0675 0.6854 0.9505
2 1.0792 1.5265 1.2657 1.7852
3 1.2107 0.8410 1.4839 0.6994
Range 0.1315 0.6855 0.7985 1.0858
Order 4 3 2 1
Optimum пз f2 t3 Г2
Conclusion
By using the linear correlation coefficient as similarity of the cluster analysis to conduct the classification of attributes, the separation of an independent attribute from multiple attributes could be performed rationally for the assessment of PMOO for material machining. In the evaluation, only each independent attribute could join the evaluation of PMOO. If more attributes than an independent attribute are used to join the analysis and the evaluation of multi - objective optimization problem, it is equivalent to the increase of the weighting factors of the corresponding attributes. The example of parameter optimization of steel turning by means of PMOO indicates the rationality of the appropriate solution.
References
Backhaus, K., Erichson, B., Gensler, S., Weiber, R. & Weiber, T. 2021. Multivariate Analysis: An Application-Oriented Introduction. Wiesbaden: Springer Fachmedien. Available at: https://doi.org/10.1007/978-3-658-32589-3.
Hegde, A., Hindi, J., Gurumurthy, B.M., Sharma, S. & Ki, A. 2022. Machinability study and optimization of tool life and surface roughness of ferrite: Bainite dual phase steel. Journal of Applied Engineering Science, 20(2), pp.358364. Available at: https://doi.org/10.5937/jaes0-32927.
Irzaev, G., Kanaev, M. & Isalova, M. 2021. Selection of the preferred design for manufacturability by constructing the Pareto tuple. Journal of Applied Engineering Science, 19(2), pp.275-281. Available at: https://doi.org/10.5937/jaes0-26922.
Nguyen, H.S., & Vo Thi, N.U. 2022. Multi-Objective Optimization in Turning Process Using RIM Method. Applied Engineering Letters: Journal of Engineering and Applied Sciences, 7(4), pp.143-153. Available at: https://doi.org/10.18485/aeletters.2022.7.4.2.
Salomon, S. 2019. Active Robust Optimization: Optimizing for Robustness of Changeable Products. Cham: Springer. Available at: https://doi.org/10.1007/978-3-030-15050-1.
Scitovski, R., Sabo, K., Martinez-Alvarez, F. & Ungar, S. 2021. Cluster Analysis and Applications. Cham: Springer. Available at: https://doi.org/10.1007/978-3-030-74552-3.
Thien Van, N., Tien Hoang, D., Trung Duc, D. & Nguyen, N.-T. 2021. Multi-objective optimization of turning process using a combination of Taguchi and VIKOR methods. Journal of Applied Engineering Science, 19(4), pp.868-873. Available at: https://doi.org/10.5937/jaes0-29654.
Yildiz, A., Ugur, L. & Parlak, I.E. 2023. Optimization of the Cutting Parameters Affecting the Turning of AISI 52100 Bearing Steel Using the Box-Behnken Experimental Design Method. Applied Sciences, 13(1), art.number:3. Available at: https://doi.org/10.3390/app13010003.
Zheng, M., Yu, J., Teng, H., Cui, Y. & Wang, Y. 2024. Probability-Based Multi-objective Optimization for Material Selection, 2nd Edition. Singapore: Springer. Available at: https://doi.org/10.1007/978-981-99-3939-8.
Оценка точения стали, основанная на вероятности многоцелевой оптимизации с соответствующим количеством атрибутов
Маошенг Чжэна, Джи Йюб
Северо-западный политехнический университет, г. Сиань, Народная Республика Китай а факультет химической инженерии, корресподент
б факультет естественных наук и технологий
РУБРИКА ГРНТИ: 27.47.00 Математическая кибернетика,
81.09.00 Материаловедение ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: Точение - это типичный процесс механической обработки металла. Однако подходящее решение по одновременной оптимизации минимизации шероховатости поверхности, минимизации силы резания и вибраций, максимального увеличения скорости удаления стружки в течение точения пока не найдено. В данной статье на основании кластерного анализа сформулировано правило выделения независимого атрибута из множества атрибутов с использованием коэффициента линейной корреляции. Помимо того, на примере оценки одновременной оптимизации токарной обработки стали с помощью многокритериальной вероятностной оптимизации (PMOO) продемонстрирована процедура выделения независимого атрибута из множества атрибутов с помощью PMOO.
Методы: РМОО является перспективным решением в токарной обработке. В оценке РМОО необходимо присутствие независимого атрибута, аналогичного независимому событию в теории вероятностей. Выделение независимого атрибута из множества атрибутов с помощью коэффициента линейной корреляции осуществляется на основании кластерного анализа. Далее предполагается, что если коэффициент линейной корреляции двух признаков при кластерном анализе превышает 0,8, т.е. в случае очень высокой корреляции, то их можно отнести к одной категории и только один из них может рассматриваться как независимый атрибут в оценке РМОО.
Результаты: Формулировка отражает суть РМОО и ее применения в механической обработке материалов, что открывает новые возможности для решения важной задачи. Пример оптимизации параметров точения стали с помощью РМОО свидетельствует о рациональности соответствующего решения.
Выводы: Это инновационное исследование имеет практическое значение, поскольку оно подчеркивает удобство использования методов РМОО, предоставляя рациональное правило для выделения независимых атрибутов из множества атрибутов РМОО.
Ключевые слова: многоцелевой подход, кластерный анализ, независимые атрибуты, коэффициент линейной корреляции, токарная обработка металла.
Евалуаци]а окрета^а челика помойу вишекритери]умске оптимизаци]е на бази вероватнойе са одговара]уйим бро]ем атрибута
Маошенг Ценг8, Ъаи иуб
Универзитет Северозапад, Си]ан, Народна Република Кина а Факултет хеми]ског инженерства, аутор за преписку
б Факултет природних наука и технологи]а
ОБЛАСТ: математика, матери]али
КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад
Сажетак:
Увод/циъ: Окретаъе jе типичан процес машинске обраде. Ме)утим, jош н^е прона)ено адекватно решете за истовремену оптимизац^у сво)ек>а храпавости, сила резака и вибрац^а на наjмаfoу могуЯу меру уз наjвеhу брзину уклаъаъа материала при процесу окретаъа. У овом раду формулише се правило за одваjа^е независног атрибута од вишеструких атрибута коришЯеъем
^ коефиц^ента линеарне корелац^е, наjпре на основу кластер
Ф анализе. Штавише, на примеру евалуац^е истовремене
оптимизац^е окреташа челика помогу вишекритер^умске оптимизац^е на бази вероватноЯе (РМОО) приказан jе поступак одва]'ак>а независног атрибута од вишеструких атрибута шеним коришЯешем.
Методи: У прецесима окреташа РМОО би могла бити добро решете. Неопходно jе да постоjи независни атрибут у евалуац^и РМОО, слично независном дога^у у теории вероватноЯе. Е Одваjаше независног атрибута од вишеструких атрибута помоЯу
о коефиц^ента линеарне корелацц'е врши се у складу са кластер
° анализом. Претпоставъа се да ако jе коефиц^ент линеарне
с корелац^е два атрибута у кластер анализи веЯи од 0,8, односно
ако jе корелац^а веома висока, тада они могу бити ставъени у g jедну категорщу, а само jедан од ших може, као независни
ш атрибут, да се придружи евалуац^и РМОО.
> Резултати: Формулац^а одсликава суштину РМОО и шену
< примену у машинсщ обради материала на рационалан начин, што
отвара нове могуЯности за решаваше битног проблема. Пример оптимизац^е параметара окреташа челика помоЯу РМОО указу/е на рационалност одговараjуЯег решеша. ^ Закъучак: Ова иновативна студ^а има практичан значаj, jер
fi истиче погодност коришЯеша метода РМОО, представъаjуЯи
^ рационално правило за одваjаше независних атрибута од
^ вишеструких атрибута РМОО.
Къучне речи: вишекритер^умски, кластер анализа, независни ш атрибути, коефиц^ент линеарне корелац^е, окреташе метала.
о
Paper received on / Дата получения работы / Датум приема чланка: 14.02.2023. О Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 24.11.2023.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 25.11.2023.
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