A Novel Generalized Interval-Valued Neutrosophic Rough Soft Set Framework for Enhanced Decision-Making: Application in Water Quality Assessment

Anjan  Mukherjee ¹ ( Department of Mathematics, Tripura University Agartala, Agartala, India. )
Ajoy Kanti Das ² ( Department of Mathematics, Tripura University Agartala, Agartala, India. )
Nandini  Gupta ³ ( Department of Mathematics, Bir Bikram Memorial College, Agartala, India )
Suman  Patra ⁴ ( Department of Mathematics, Tripura University Agartala, Agartala, India. )
Tahir  Mahmood ⁵ ( Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan. )
Suman  Das ⁶ ( Department of Education (ITEP), NIT, Agartala, Tripura, India. )
Rakhal  Das ⁷ ( Department of Mathematics, The ICFAI University Tripura, Agartala, India. )

Submited date : 2024-12-20 Accepted date : 2025-01-29

Keywords: Fuzzy Set, Neutrosophic Set, Soft Set, Rough Set, Generalized Neutrosophic Set, Interval-Valued Neutrosophic Set.,

Abstract :

This study introduces a novel framework, Generalized Interval-Valued Neutrosophic Rough Soft Sets (GIVNRS sets), designed to improve handling uncertainty, imprecision, and vagueness in complex decision-making scenarios. By integrating soft, rough, and generalized interval-valued neutrosophic set theories, the framework offers a robust methodology for addressing indeterminacy and incomplete data. The theoretical foundation of GIVNRS sets is built upon fundamental operations, including intersection, union, complement, and novel aggregation union operators tailored for multi-criteria decision-making (MCDM) applications. The practical applicability of the framework is demonstrated through a water quality assessment, where it successfully classifies river segments based on key water quality parameters such as pH, Dissolved Oxygen (DO), and Biochemical Oxygen Demand (BOD). The case study results show that the pollution scores for the river segments were computed, classifying the segments such as “Good,” “Moderate,” and “Poor,” with corresponding pollution levels. These findings highlight the framework’s ability to manage incomplete and inconsistent data, providing a reliable and comprehensive water quality evaluation. Compared to traditional models, the GIVNRS set approach offers enhanced flexibility, stability, and adaptability. This study not only contributes to the theoretical development of neutrosophic, soft, and rough set theories but also establishes GIVNRS sets as a powerful tool for water quality decision-making. Future research will explore further advancements in the application and computational efficiency of this framework.

References:

[1] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8(3): 338-353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
[2] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20(1): 87-96. DOI: https://doi.org/10.1016/S0165-0114(86)80034-3
[3] Pawlak Z. Rough sets. International Journal of Computer and Information Sciences. 1982; 11(5): 341- 356. DOI: https://doi.org/10.1007/BF01001956
[4] Molodtsov D. Soft set theory: First results. Computers and Mathematics with Applications. 1999; 37(4-5): 1931. DOI:
https://doi.org/10.1016/S0898-1221(99)00056-5
[5] Smarandache F. A unifying field in logics: Neutrosophy: Neutrosophic probability, set, and logic. Rehoboth, USA: American Research Press; 1998.
[6] Smarandache F. Neutrosophic set: A generalization of the intuitionistic fuzzy sets. International Journal of Pure and Applied Mathematics. 2005; 24(3): 287-297.
[7] Broumi S, Smarandache F. Interval-valued neutrosophic rough sets. International Journal of Computational Mathematics. 2015; 232919, 13 pages. DOI: https://doi.org/10.1155/2015/232919
[8] Mukherjee A, Das R. Generalized interval-valued neutrosophic set and its application. Transactions of Fuzzy Sets and Systems. 2025; 4(2): 1-13. DOI:https://doi.org/10.71602/tfss.2025.1130042
[9] WHO. Guidelines for drinking water quality. 4th ed. Geneva, Switzerland: World Health Organization; 2011.
[10] CPCB. Guidelines for Water Quality Management. Central Pollution Control Board, Delhi, India: East Arjun Nagar; 2008.
[11] Salama AA, Alblowi SA. Generalized neutrosophic set and generalized neutrosophic topological spaces. Computer Science and Engineering. 2012; 2(7): 129-132. DOI: https://doi.org/10.5923/j.computer.20120207.01
[12] Broumi S, Smarandache F, Dhar M. Rough neutrosophic sets. Italian Journal of Pure and Applied Mathematics. 2014; 32(32): 493-502. DOI: https://doi.org/10.5281/zenodo.30310
[13] Broumi S, Smarandache F. Interval-valued neutrosophic soft rough sets. International Journal of Computational Mathematics. 2015; 2: 1-13. DOI: https://doi.org/10.1155/2015/232919
[14] Yang H-L, Bao Y-L, Guo Z-L. Generalized interval neutrosophic rough sets and its application in multiattribute decision making. Filomat. 2018; 32(1): 11-33. DOI: https://doi.org/10.2298/FIL1801011Y
[15] Saha A, Broumi S. New operators on interval valued neutrosophic sets. Neutrosophic Sets and Systems. 2019; 28(10): 128-137.
[16] Sahoo B, Debnath BK. A novel hybrid spherical fuzzy multi-criteria decision-making approach to select the best hydroelectric power plant source in India. Renewable Energy Focus. 2024; 51(16): 100650. DOI: https://doi.org/10.1016/j.ref.2024.100650
[17] Saha A, Majumder P, Dutta D, Debnath BK. Multi-attribute decision making using q-rung orthopair fuzzy weighted fairly aggregation operators. Journal of Ambient Intelligence and Humanized Computing. 2021; 12: 81498171. DOI: https://doi.org/10.1007/s12652-020-02551-5
[18] Das AK, Gupta N, Mahmood T, Tripathy BC, Das R, Das S. An innovative fuzzy multi-criteria decisionmaking model for analyzing anthropogenic influences on urban river water quality. Iran Journal of Computer Science. 2025; 8: 103-124. DOI: https://doi.org/10.1007/s42044-024-00211-x
[19] Patel A, Chitnis K. Application of fuzzy logic in river water quality modeling for analysis of industrialization and climate change impact on Sabarmati river. Water Supply. 2022; 22(1): 238250. DOI: https://doi.org/10.2166/ws.2021.275
[20] Das AK, Gupta N, Mahmood T, Tripathy BC, Das R, Das S. (1st ed.). Boca Raton, USA: CRC Press; 2025.
[21] Das AK, Granados C, An advanced approach to fuzzy soft group decision-making using weighted average ratings. SN Computer Science. 2021; 2(6): 471. DOI: https://doi.org/10.1007/s42979-021-00873-5
[22] Das AK, On partially included intuitionistic fuzzy rough relations. Afrika Matematika. 2016; 27: 993- 1001. DOI:
https://doi.org/10.1007/s13370-016-0395-2
[23] Das AK, Granados C. Preference intuitionistic fuzzy rough relation and its theoretical approach. Journal of the Indian Mathematical Society. 2023; 90(3-4): 199-212. DOI: https://doi.org/10.18311/jims/2023/27812
[24] Das AK, Granados C, Bhattacharya J. Some new operations on fuzzy soft sets and their applications in decision-making. Songklanakarin Journal of Science and Technology. 2022; 44(2): 440-449. DOI: https://doi.org/10.14456/sjst-psu.2022.61
[25] Granados C, Das AK, Das B. Some continuous neutrosophic distributions with neutrosophic parameters based on neutrosophic random variables. Advances in the Theory of Nonlinear Analysis and its Applications. 2022; 6(3): 380-389. DOI: https://doi.org/10.31197/atnaa.1056480
[26] Granados C, Das AK, Osu BO. Weighted neutrosophic soft multiset and its application to decision making. Yugoslav journal of operations research. 2023; 33(2): 293-308. DOI: https://doi.org/10.2298/YJOR220915034G
[27] Mukherjee A, Das AK. Topological structure formed by intuitionistic fuzzy rough relations. Journal of the Indian Mathematical Society. 2016; 83(1-2): 135-144.