Soft difference-product: A new product for soft sets with its decision-making

Year 2024, Volume: 5 Issue: 2, 114 - 137, 31.12.2024

Aslıhan Sezgin , Nazlı Helin Çam

https://doi.org/10.54559/jauist.1589242

Abstract

A thorough mathematical foundation for handling uncertainty is provided by the concept of soft sets. Soft set operations are key concepts in soft set theory since they offer novel approaches to problems requiring parametric data. The “soft difference-product” a new product operation for soft sets, is proposed in this study along with all of its algebraic properties concerning different types of soft equalities and subsets. Additionally, we explore the connections between this product and other soft set operations by investigating the distributions of soft difference-product over other soft set operations. Using the uni-int operator and the uni-int decision function for the soft-difference product, we apply the uni-int decision-making method, which selects a set of optimal elements from the alternatives by giving an example that shows how the approach may be conducted effectively in various areas. Since the theoretical underpinnings of soft computing techniques are drawn from purely mathematical concepts, this study is crucial to the literature on soft sets.

Keywords

soft set, soft difference-product, soft subset, soft equal relations

Ethical Statement

No approval from the Board of Ethics is required.

Project Number

-

References

• D. A. Molodtsov, Soft set theory–first results, Computers and Mathematics with Applications 37 (4-5) (1999) 19–31.
• L. A. Zadeh, Fuzzy sets, Information Control (8) (1965) 338–353.
• P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44 (8-9) (2002) 1077–1083.
• De-Gang Chen, E. C. C. Tsang, D. S. Yeung, Some notes on the parameterization reduction of soft sets, in: Proceedings of the 2003 International Conference on Machine Learning and Cybernetics, Xi’an, 2003, pp. 1442–1445.
• De-Gang Chen, E. C. C. Tsang, X. Wang, The parametrization reduction of soft sets and its applications Computers and Mathematics with Applications 49 (5–6) (2005) 757–763.
• Z. Xiao, L. Chen, B. Zhong, S. Ye, Recognition for soft information based on the theory of soft sets, In: J. Chen (ed.), IEEE proceedings of International Conference on Services Systems and Services Management, 2005, pp 1104–1106.
• M. M. Mushrif, S. Sengupta, A. K. Ray, Texture classification using a novel, soft-set theory based classification algorithm, In: P. J. Narayanan, S. K. Nayar, H. T. Shum (Eds.), Computer Vision – ACCV 2006, Vol 3851 of Lecture Notes in Computer Science, Springer, Berlin, Heidelberg.
• M. T. Herawan, M. M. Deris, A direct proof of every rough set is a soft set, Third Asia International Conference on Modelling & Simulation, Bundang, Indonesia, 2009, pp. 119–124.
• M. T. Herawan, M. M. Deris, Soft decision making for patients suspected influenza, In: D. Taniar, O. Gervasi, B. Murgante, E. Pardede, B. O. Apduhan (Eds.), Computational Science and Its Applications -ICCSA 2010, Vol 6018 of Lecture Notes in Computer Science, Springer, Berlin, Heidelberg.
• T. Herawan, Soft set-based decision making for patients suspected influenza-like illness, International Journal of Modern Physics: Conference Series 1 (1) (2005) 1–5.
• N. Çağman, S. Enginoğlu, Soft set theory and uni-int decision making, European Journal of Operational Research 207 (2) (2010) 848–855.
• N. Çağman, S. Enginoğlu, Soft matrix theory and its decision making, Computers and Mathematics with Applications 59 (10) (2010) 3308–3314.
• X. Gong, Z. Xiao, X. Zhang, The bijective soft set with its operations, Computers and Mathematics with Applications 60 (8) (2010) 2270–2278.
• Z. Xiao, K. Gong, S. Xia, Y. Zou, Exclusive disjunctive soft sets, Computers and Mathematics with Applications 59 (6) (2010) 2128–2137.
• F. Feng, Y. Li, N. Çağman, Generalized uni-int decision making schemes based on choice value soft sets, European Journal of Operational Research 220 (1) (2012) 162–170.
• Q. Feng, Y. Zhou, Soft discernibility matrix and its applications in decision making. Applied Soft Computing (24) (2014) 749–756.
• A. Kharal, Soft approximations and uni-int decision making, The Scientific World Journal (4) (2014) 327408.
• M. K. Dauda, M. Mamat, M. Y. Waziri, An application of soft set in decision making. Jurnal Teknologi 77 (13) (2015) 119–122.
• V. Inthumathi, V. Chitra, S. Jayasree, The role of operators on soft set in decision making problems, International Journal of Computational and Applied Mathematics 12 (3) (2017) 899–910.
• A. O. Atagün, H. Kamacı, O. Oktay, Reduced soft matrices and generalized products with applications in decision making, Neural Computing and Applications (29) (2018) 445–456.
• H. Kamacı, K. Saltık, H. F. Akız, A. O. Atagün, Cardinality inverse soft matrix theory and its applications in multicriteria group decision making, Journal of Intelligent & Fuzzy Systems 34 (3) (2018) 2031–2049.
• J. L. Yang, Y.Y. Yao, Semantics of soft sets and three-way decision with soft sets, Knowledge-Based Systems 194 (2020) 105538.
• S. Petchimuthu, H. Garg, H. Kamacı, A. O. Atagün, The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM, Computational and Applied Mathematics, 39 (2) (2020) 1–32.
• İ. Zorlutuna, Soft set-valued mappings and their application in decision making problems, Filomat 35 (5) (2021) 1725–1733.
• P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Computers and Mathematics with Applications, 45 (1) (2003) 555–562.
• D. Pei, D. Miao, From soft sets to information systems, in: X. Hu, Q. Liu, A. Skowron, T. Y. Lin, R. R. Yager, B. Zhang (Eds.), IEEE International Conference of Granular Computing, Beijing, 2005, pp. 617–621.
• M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications, 57 (9) (2009) 1547–1553.
• C. F. Yang, A note on: “Soft set theory” [Computers & Mathematics with Applications 45 (2003), 4-5, 555–562], Computers and Mathematics with Applications. 56 (7) (2008) 1899–1900.
• F. Feng, Y. M. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing 14 (2010) 899–911.
• Y. Jiang, Y. Tang, Q. Chen, J. Wang, S. Tang, Extending soft sets with description logics, Computers and Mathematics with Applications 59 (6) (2010) 2087–2096.
• M. I. Ali, M. Shabir, M. Naz, Algebraic structures of soft sets associated with new operations, Computers and Mathematics with Applications. 61 (9) (2011) 2647–2654.
• C. F. Yang, A note on soft set theory, Computers and mathematics with applications, 56 (7) (2008) 1899–1900.
• I. J. Neog, D. K. Sut, A new approach to the theory of soft set, International Journal of Computer Applications 32 (2) (2011) 1–6.
• L. Fu, Notes on soft set operations, ARPN Journal of Systems and Softwares 1 (6) (2011) 205–208.
• X. Ge, S. Yang, Investigations on some operations of soft sets, World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences, 5 (3) (2011) 370–373.
• D. Singh, I. A. Onyeozili, Some conceptual misunderstanding of the fundamentals of soft set theory, ARPN Journal of Systems and Softwares, 2 (9) (2012) 251–254.
• D. Singh, I. A. Onyeozili, Some results on distributive and absorption properties on soft operations, IOSR Journal of Mathematics, 4 (2) (2012) 18–30.
• D. Singh, I. A. Onyeozili, On some new properties on soft set operations, International Journal of Computer Applications. 59 (4) (2012) 39–44.
• D. Singh, I. A. Onyeozili, Notes on soft matrices operations, ARPN Journal of Science and Technology 2 (9) (2012) 861–869.
• P. Zhu, Q. Wen, Operations on soft sets revisited, Journal of Applied Mathematics (2013) (2013) 1–7.
• J. Sen, On algebraic structure of soft sets, Annals of Fuzzy Mathematics and Informatics, 7 (6) (2014) 1013–1020.
• Ö. F. Eren, On operations of soft sets, Master’s Thesis Ondokuz Mayıs University (2019) Samsun.
• N. S. Stojanovic, A new operation on soft sets: extended symmetric difference of soft sets, Military Technical Courier 69 (4) (2021) 779–791.
• A. Sezgin, E. Yavuz, A new soft set operation: Soft binary piecewise symmetric difference operation, Necmettin Erbakan University Journal of Science and Engineering 5 (2) (2023) 189–208.
• A. Sezgin, M. Sarıalioğlu, A new soft set operation: Complementary soft binary piecewise theta operation, Journal of Kadirli Faculty of Applied Sciences 4 (2) 325–357.
• A. Sezgin, N. Çağman, A new soft set operation: Complementary soft binary piecewise difference operation, Osmaniye Korkut Ata University Journal of the Institute of Science and Technology 7 (1) (2024) 58–94.
• A. Sezgin, F. N. Aybek, N. B. Güngör, A new soft set operation: Complementary soft binary piecewise union operation, Acta Informatica Malaysia (7) 1 (2023) 38–53.
• A. Sezgin, F. N. Aybek, A. O. Atagün, A new soft set operation: Complementary soft binary piecewise intersection operation, Black Sea Journal of Engineering and Science 6 (4) (2023) 330–346.
• A. Sezgin, A. M. Demirci, New soft set operation: Complementary soft binary piecewise star operation, Ikonion Journal of Mathematics 5 (2) (2023) 24–52.
• K. Y. Qin, Z. Y. Hong, On soft equality, Journal of Computational and Applied Mathematics, 234 (5) (2010) 1347–1355.
• Y. B. Jun, X. Yang, A note on the paper “Combination of interval-valued fuzzy set and soft set” [Comput. Math. Appl. 58 (2009) 521–527], Computers and Mathematics with Applications 61 (5) (2011) 1468–1470.
• X. Y. Liu, F. F. Feng, Y. B. Jun, A note on generalized soft equal relations, Computers and Mathematics with Applications 64 (4) (2012) 572–578.
• F. Feng, L. Yongming, Soft subsets and soft product operations, Information Sciences (232) (2013) 44–57.
• M. Abbas, B. Ali, S. Romaguer, On generalized soft equality and soft lattice structure, Filomat 28 (6) (2014) 1191–1203.
• M. Abbas, M. I. Ali, S. Romaguera, Generalized operations in soft set theory via relaxed conditions on parameters, Filomat 31 (19) (2017) 5955–5964.
• T. Alshami, Investigation and corrigendum to some results related to g-soft equality and g f-soft equality relations, Filomat 33 (11) (2019) 3375–3383.
• T. Alshami, M. El-Shafei, T-soft equality relation, Turkish Journal of Mathematics 44 (4) (2020) 1427–1441.
• B. Ali, N. Saleem, N. Sundus, S. Khaleeq, M. Saeed, R. A. George, Contribution to the theory of soft sets via generalized relaxed operations, Mathematics 10 (15) (2022) 26–36.
• A. Sezgin, A. O. Atagün, N. Çağman, A complete study on and-product of soft sets, Sigma Journal of Engineering and Natural Sciences (In Press)
• A. S. Sezer, Certain characterizations of LA-semigroups by soft sets, Journal of Intelligent and Fuzzy Systems 27 (2) (2014) 1035–1046.
• A. S. Sezer, A new approach to LA-semigroup theory via the soft sets, Journal of Intelligent and Fuzzy System 26 (5) (2014) 2483–2495.
• A. S. Sezer, N. Çağman, A. O. Atagün, Soft intersection interior ideals, quasi-ideals and generalized bi-ideals: a new approach to semigroup theory II, Journal of Multiple-valued Logic and Soft Computing 23 (1-2) (2014) 161–207.
• A. Sezgin, A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals, Algebra Letters 2016 (2016) 3 1–46.
• M. Tunçay, A. Sezgin, Soft union ring and its applications to ring theory, International Journal of Computer Applications 151 (9) (2016) 7–13.
• E. Muştuoğlu, A. Sezgin, Z. K. Türk, Some characterizations on soft uni-groups and normal soft uni-groups, International Journal of Computer Applications 155 (10) (2016) 1–8.
• A. Khan, M. Izhar, A. Sezgin, Characterizations of Abel Grassmann’s groupoids by the properties of double-framed soft ideals, International Journal of Analysis and Applications 15 (1) (2017) 62–74.
• A. Sezgin, N. Çağman, A. O. Atagün, A completely new view to soft intersection rings via soft uni-int product, Applied Soft Computing 54 (2017) 366–392.
• A. Sezgin, A new view on AG-groupoid theory via soft sets for uncertainty modeling, Filomat 32 (8) (2018) 2995–3030.
• A. O. Atagün, A. Sezgin, Soft subnear-rings, soft ideals and soft N-subgroups of near-rings, Mathematical Sciences Letters 7 (1) (2018) 37–42.
• M. Gulistan, F. Feng, M. Khan, A. Sezgin, Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets, Mathematics (6) (2018) 293.
• T. Mahmood, Z. U. Rehman, A. Sezgin, Lattice ordered soft near rings, Korean Journal of Mathematics 26 (3) (2018) 503–517.
• C. Jana, M. Pal, F. Karaaslan, A. Sezgi̇n, (α, β)-Soft intersectional rings and ideals with their applications. New Mathematics and Natural Computation 15 (02) (2019) 333–350.
• A. O. Atagün, H. Kamacı, İ. Taştekin, A. Sezgin, P-properties in near-rings, Journal of Mathematical and Fundamental Sciences 51 (2) (2019) 152–167.
• Ş. Özlü, A. Sezgin, Soft covered ideals in semigroups, Acta Universitatis Sapientiae, Mathematica 2 (2) (2020) 317–346.
• A. Sezgin, A. O. Atagün, N. Çağman, H. Demir, On near-rings with soft union ideals and applications, New Mathematics and Natural Computation 18 (2) (2022) 495–511.
• T. Manikantan, P. Ramasany, A. Sezgin, Soft quasi-ideals of soft near-rings, Sigma Journal of Engineering and Natural Science 41 (3) (2023) 565–574.
• K. Naeem, Soft set theory & soft sigma algebras, LAP LAMBERT Academic Publishing, 2017.
• M. Riaz, K. Naeem, Novel concepts of soft sets with applications, Annals of Fuzzy Mathematics & Informatics 13 (2) (2017) 239–251.
• M. Riaz, K. Naeem, Measurable soft mappings, Punjab University Journal of Mathematics 48 (2) (2016) 19–34.
• S. Memiş, Another view on picture fuzzy soft sets and their product operations with soft decision-making, Journal of New Theory (38) (2022) 1–13.
• K. Naeem, S. Memiş, Picture fuzzy soft σ-algebra and picture fuzzy soft measure and their applications to multi-criteria decision-making, Granular Computing 8 (2) (2023) 397–410.