Volume 27 • Issue 1 • PP: 43-58 • 2026
Some Einstein Operations on Rough Neutrosophic Sets with their Properties
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Algebraic operations, which include addition, subtraction, division, scalar multiplication, and exponentiation, are the fundamental mathematical operations utilised in decision-making analysis. When performing on numbers, the algebraic operations are commonly referred to as arithmetic operations. Another alternative for algebraic operations, known as Einstein operations, has gained recognition for its smooth approximation and utilisation of Archimedean norms. However, it is crucial to note that Einstein operations are not designed to effectively address issues of indeterminacy, uncertainty, and lower-upper approximation. Thus, this paper defines some rough neutrosophic-based Einstein operations known as RNS Einstein addition, RNS Einstein multiplication, RNS Einstein scalar multiplication, and RNS Einstein exponentiation. By adopting rough neutrosophic sets (RNS), which incorporate neutrosophic lower and upper approximations, the proposed RNS Einstein operations offer a practical approach for handling uncertain situations. Some examples are provided to demonstrate the applicability of the RNS Einstein operations. Several desirable properties related to the defined RNS Einstein operations are investigated. Finally, the proposed RNS Einstein operations are applied in solving multi-criteria decision-making problems within a rough neutrosophic environment.
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References
[1] L. A. Zadeh, "Fuzzy sets," Information and Control, vol. 8, no. 3, pp. 338-353, 1965.
[2] Z. Pawlak, "Rough sets and fuzzy sets," Fuzzy Sets and Systems, vol. 17, no. 1, pp. 99-102, 1985.
[3] F. Smarandache, Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis, 1998.
[4] S. Broumi, F. Smarandache, and M. Dhar, "Rough neutrosophic sets," Infinite Study, vol. 32, pp. 493-502, 2014.
[5] S. Alias, D. Mohamad, and A. Shuib, "Rough neutrosophic multisets," Neutrosophic Sets and Systems, vol. 16, pp. 80-88, 2017.
[6] Q. Jin, K. Hu, C. Bo, and L. Li, "A new single-valued neutrosophic rough sets and related topology," Journal of Mathematics, vol. 1, pp. 5522021, 2021.
[7] D. J. S. Martina and G. Deepa, "Application of multi-valued rough neutrosophic set and matrix in multi-criteria decision-making: Multi-valued neutrosophic rough set and matrix," Mathematics in Applied Sciences and Engineering, vol. 4, no. 3, pp. 227–248, 2023.
[8] C. Bo, X. Zhang, S. Shao, and F. Smarandache, "New multigranulation neutrosophic rough set with applications," Symmetry, vol. 10, no. 11, p. 578, 2018.
[9] H. Zhao and H. Y. Zhang, "On hesitant neutrosophic rough set over two universes and its application," Artificial Intelligence Review, vol. 53, pp. 4387-4406, 2020.
[10] K. Rogulj, J. Kilić Pamuković, and M. Ivić, "Hybrid MCDM based on VIKOR and cross entropy under rough neutrosophic set theory," Mathematics, vol. 9, no. 12, p. 1334, 2021.
[11] E. M. Kumari and M. Thirucheran, "Rough hesitant bipolar neutrosophic sets and its applications in game theory," Advances in Applied Mathematical Sciences, vol. 21, no. 1, pp. 493-517, 2021.
[12] S. Das, R. Das, and B. C. Tripathy, "Topology on rough pentapartitioned neutrosophic set," Iraqi Journal of Science, pp. 2630-2640, 2022.
[13] A. E. Samuel and R. Narmadhagnanam, "Pi-distance of rough neutrosophic sets for medical diagnosis," Infinite Study, 2019.
[14] J. S. M. Donbosco and D. Ganesan, "The energy of rough neutrosophic matrix and its application to MCDM problem for selecting the best building construction site," Decision Making: Applications in Management and Engineering, vol. 5, no. 2, pp. 30-45, 2022.
[15] B. Praba, S. Pooja, and N. Sivakumar, "Attribute based double bounded rough neutrosophic sets in facial expression detection," Neutrosophic Sets and Systems, vol. 49, no. 1, p. 21, 2022.
[16] M. Myvizh, "Choosing the best location in the hospital by MADM strategy using the energy of a rough neutrosophic matrix," Mathematics in Engineering, Science & Aerospace (MESA), vol. 14, no. 3, 2023.
[17] M. Shazib Hameed, S. Ali, S. Shoaib, M. Mukhtar, M. Kashif Ishaq, and U. Mukhtiar, "On characterization of χ-single valued neutrosophic subgroups," Journal of Mathematics and Computer Science, vol. 24, no. 04, pp. 358–369, 2021.
[18] M. Abdel-Basset and M. Mohamed, "The role of single valued neutrosophic sets and rough sets in smart city: Imperfect and incomplete information systems," Measurement, vol. 124, 2018.
[19] M. Khan, M. Gulistan, N. Hassan, and A. M. Nasruddin, "Air pollution model using neutrosophic cubic Einstein averaging operators," Neutrosophic Sets and Systems, vol. 32, no. 1, p. 24, 2020.
[20] J. Ye, E. Türkarslan, M. Ünver, and M. Olgun, "Algebraic and Einstein weighted operators of neutrosophic enthalpy values for multi-criteria decision making in neutrosophic multi-valued set settings," Granular Computing, vol. 7, no. 3, pp. 479-487, 2022.
[21] H. M. A. Farid and M. Riaz, "Single-valued neutrosophic Einstein interactive aggregation operators with applications for material selection in engineering design: Case study of cryogenic storage tank," Complex & Intelligent Systems, vol. 8, no. 3, pp. 2131-2149, 2022.
[22] M. Jamil, F. Afzal, A. Akgül, S. Abdullah, A. Maqbool, A. Razzaque, and J. Awrejcewicz, "Einstein aggregation operators under bipolar neutrosophic environment with applications in multi-criteria decision-making," Applied Sciences, vol. 12, no. 19, p. 10045, 2022.
[23] M. Kamran, S. Ashraf, N. Salamat, M. Naeem, and T. Botmart, "Cyber security control selection based decision support algorithm under single valued neutrosophic hesitant fuzzy Einstein aggregation information," AIMS Mathematics, vol. 8, no. 3, pp. 5551-5573, 2023.
[24] D. Ajay, G. Selvachandran, J. Aldring, P. H. Thong, L. H. Son, and B. C. Cuong, "Einstein exponential operation laws of spherical fuzzy sets and aggregation operators in decision making," Multimedia Tools and Applications, vol. 82, no. 27, pp. 41767-41790, 2023.
[25] B. Li, J. Wang, L. Yang, and X. Li, "A novel generalized simplified neutrosophic number Einstein aggregation operator," Infinite Study, 2018.
[26] A. Fahmi, M. Aslam, and R. Ahmed, "Decision-making problem based on generalized interval-valued bipolar neutrosophic Einstein fuzzy aggregation operator," Soft Computing, vol. 27, no. 20, pp. 14533–14551, 2023.
[27] S. M. Zainal, A. T. Ab Ghani, and L. Abdullah, "On some logical relation properties for rough neutrosophic set," in Journal of Physics: Conference Series, vol. 1878, p. 012048, 2021.
[28] K. Mondal, S. Pramanik, and B. C. Giri, "Rough neutrosophic aggregation operators for multi-criteria decision-making," in Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets, pp. 79-105, 2019.
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