The Neutrosophic Soft Set (NSS) is an advanced and highly effective expansion of soft sets, specifically designed to handle parameterized values of alternatives. As an enhanced version of fuzzy soft sets, it provides a novel mathematical framework that offers significant advantages in dealing with uncertain information. This model is created by merging soft sets and neutrosophic sets, providing a robust approach to uncertainty management. Various algorithms have been proposed for making neutrosophic decisions using NSSs. However, these algorithms neglect external effective that influence the Decision- Making (DM) process, focusing solely on parameters. To address this issue, the article introduces the concept of Effective Neutrosophic Soft Sets (ENSSs). Additionally, we extend and generalize the innovative concept of Effective Fuzzy Soft Sets (EFSSs) to accommodate three independent membership criteria, aiming to enhance effectiveness and realism. We also introduce operations on ENSSs, including subset, complement, union, intersection, AND, and OR, which are defined along with illustrative examples. Furthermore, we examine some of its properties. Moreover, we present applications of this concept in DM problems and Medical Diagnosis (MD).

[1] Abu Qamar M. and Hassan N., “Generalized Q- Neutrosophic Soft Expert Set for Decision under Uncertainty,” Symmetry, vol. 10, no. 11, pp. 1-16, 2018. https://www.mdpi.com/2073- 8994/10/11/621

[2] Al-Hijjawi S. and Alkhazaleh S., “Possibility Neutrosophic Hypersoft Set,” Neutrosophic Sets and Systems, vol. 53, no. 1, pp. 117-129, 2023. https://digitalrepository.unm.edu/nss_journal/vol53/iss1/7

[3] Al-Hijjawi S., Ahmad A., and Alkhazaleh S., “Time Q-Neutrosophic Soft Expert Set,” International Journal of Neutrosophic Science, vol. 19, no. 1, pp. 088-28, 2022. DOI:10.54216/IJNS.190101

[4] Alkhazaleh S., “Effective Fuzzy Soft Set Theory and its Applications,” Applied Computational Intelligence and Soft Computing, vol. 2022, pp. 1- 12, 2022. https://doi.org/10.1155/2022/6469745

[5] Alkhazaleh S. and Beshtawi E., “Effective Fuzzy Soft Expert Set Theory and its Applications,” International Journal of Fuzzy Logic and Intelligent Systems, vol. 23, no. 2, pp. 192-204, 2023. https://doi.org/10.5391/IJFIS.2023.23.2.192

[6] Alkhazaleh S. and Hazaymeh A., “N-Valued Refined Neutrosophic Soft Sets and their Applications in Decision Making Problems and Medical Diagnosis,” Journal of Artificial Intelligence and Soft Computing Research, vol. 8, no. 1, pp. 79-86, 2018. DOI:10.1515/jaiscr-2018- 0005

[7] Alkhazaleh S. and Salleh A., “Fuzzy Soft Expert Set and its Application,” Applied Mathematics, vol. 5, no. 9, pp. 1349-1368, 2014. https://doi.org/10.4236/am.2014.59127

[8] Alkhazaleh S. and Salleh A., “Soft Expert Sets,” Advances in Decision Sciences, vol. 2011, pp. 1- 12, 2011. https://doi.org/10.1155/2011/757868

[9] Alkhazaleh S., Salleh A., and Hassan N., “Possibility Fuzzy Soft Set,” Advances in Decision Sciences, vol. 2011, pp. 1-18, 2011. http://doi.org/10.1155/2011/479756

[10] Alkhazaleh S., Salleh A., and Hassan N., “Soft Multisets Theory,” Applied Mathematical Sciences, vol. 5, no. 72, pp. 3561-3573, 2011. https://www.researchgate.net/publication/235631549_Soft_multisets_theory

[11] Al-Sharqi F., Al-Qudah Y., and Alotaibi N., “Decision-Making Techniques Based on Similarity Measures of Possibility Neutrosophic Soft Expert Sets,” Neutrosophic Sets and Systems, vol. 55, no. 1, pp. 358-382, 2023. https://digitalrepository.unm.edu/nss_journal/vol 55/iss1/22

[12] Al-Sharqi F., Al-Quran A., and Romdhini M., “Decision-Making Techniques Based on Similarity Measures of Possibility Interval Fuzzy Soft Environment,” Iraqi Journal for Computer Science and Mathematics, vol. 4, no. 4, pp. 18-29, 2023. https://doi.org/10.52866/ijcsm.2023.04.04.003

[13] Broumi S. and Smarandache F., “Intuitionistic Neutrosophic Soft Set,” Journal of Information and Computing Science, vol. 8, no. 2, pp. 130-140, 2013.

[14] Deli I. and Broumi S., “Neutrosophic Soft Matrices and NSM-Decision Making,” Journal of Intelligent and Fuzzy Systems, vol. 28, no. 5, pp. 2233-2241, 2015. https://doi.org/10.3233/IFS- 141505

[15] Hassan N., Uluçay V., and Şahin M., “Q- Neutrosophic Soft Expert Set and its Application in Decision Making,” International Journal of Fuzzy System Applications, vol. 7, no. 4, pp. 37- 61, 2018. https://doi.org/10.4018/IJFSA.2018100103

[16] Maji P., “Neutrosophic Soft Set,” Annals of Fuzzy Mathematics and Informatics, vol. 5, no. 1, pp. 157-168, 2013. https://fs.unm.edu/Maji- NeutrosophicSoftSet.pdf

[17] Maji P., Biswas R., and Roy A., “Soft Set Theory,” Computers Mathematical with  Applications, vol. 45, no. 4, pp. 555-562, 2003. https://doi.org/10.1016/S0898-1221(03)00016-6

[18] Maji P., Roy A., and Biswas R., “An Application of Soft Sets in a Decision Making Problem,” Computers Mathematical with Applications, vol. 44, no. 8, pp. 1077-1083, 2002. https://doi.org/10.1016/S0898-1221(02)00216-X

[19] Majumdar P. and Samanta S., “Generalized Fuzzy Soft Sets,” Computers and Mathematics with Applications, vol. 59, no. 4, pp. 1425-1432, 2010. https://doi.org/10.1016/j.camwa.2009.12.006

[20] Molodtsov D., “Soft Set Theory-first Results,” Computers and Mathematics with Applications, From Neutrosophic Soft Set to Effective Neutrosophic Soft Set Generalizations ... 519 vol. 37, no. 2, pp. 19-31, 1999. https://doi.org/10.1016/S0898-1221(99)00056-5

[21] Roy A. and Maji P., “A Fuzzy Soft Set Theoretic Approach to Decision Making Problems,” Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412-418, 2007. https://doi.org/10.1016/j.cam.2006.04.008

[22] Saeed M., Ahsan M., Siddique M., and Ahmad M., “A Study of the Fundamentals of Hypersoft Set Theory,” International Journal of Scientific and Engineering Research, vol. 11, no. 1, pp. 220-239, 2020. https://www.ijser.org/researchpaper/A-Study-of-The-Fundamentals-of-Hypersoft-Set-Theory.pdf

[23] Sahin M., Alkhazaleh S., and Ulucay V., “Neutrosophic Soft Expert Sets,” Applied Mathematics, vol. 6, no. 1, pp. 116-127, 2015. https://doi.org/10.4236/am.2015.61012

[24] Saqlain M., Moin S., Jafar M., Saeed M., and Smarandache F., “Aggregate Operators of Neutrosophic Hypersoft Set,” Neutrosophic Sets and Systems, vol. 32, no. 1, pp. 294-306, 2020. https://digitalrepository.unm.edu/nss_journal/vol 32/iss1/18

[25] Saqlain M., Jafar N., Moin S., Saeed M., and Broumi S., “Single and Multi-Valued Neutrosophic Hypersoft Set and Tangent Similarity Measure of Single Valued Neutrosophic Hypersoft Sets,” Neutrosophic Sets and Systems, vol. 32, no. 1, pp. 317-329, 2020. https://api.semanticscholar.org/CorpusID:216486 181

[26] Saqlain M. and Xin X., “Interval Valued, m-Polar and m-Polar Interval Valued Neutrosophic Hypersoft Sets,” Neutrosophic Sets and Systems, vol. 36, pp. 389-399, 2020. https://api.semanticscholar.org/CorpusID:226439 299

[27] Smarandache F., A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, InfoLearnQuest, 2006. https://zenodo.org/records/49174 (fifth edition)

[28] Smarandache F., “Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set,” Neutrosophic Sets and Systems, vol. 22, no. 1, pp. 168-170, 2018. file:///C:/Users/user/Downloads/ExtensionOfSoft SetToHypersoft.pdf

[29] Uluay V., Ahin M., and Hassan N., “Generalized Neutrosophic Soft Expert Set for Multiple-Criteria Decision-Making,” Symmetry, vol. 10, no. 10, pp. 1-17, 2018. https://doi.org/10.3390/sym10100437

[30] Yan L., “Modeling Fuzzy Data with Fuzzy Data Types in Fuzzy Database and XML Models,” The International Arab Journal of Information Technology, vol. 10, no. 6, pp. 43-48, 2013. shttps://www.ccis2k.org/iajit/PDF/vol.10,no.6/48 47.pdf

[31] Zadeh L., “Fuzzy Sets,” Information and Control,” vol. 8, no. 3, pp. 338-253, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X

[32] Zulqarnain R., Xin X., Saqlain M., and Smarandache F., “Generalized Aggregate Operators on Neutrosophic Hypersoft Set,” Neutrosophic Sets and Systems, vol. 36, no. 1, pp. 271-281, 2020. https://digitalrepository.unm.edu/nss_journal/vol 36/iss1/20