Virtual Logic-Based Fuzzy Parameterized Fuzzy Hypersoft Sets: An Algorithmic Framework for Improved Decision-Making

Abstract

The primary aim of this study is to address potential inaccuracies in membership values expressed by decision-makers when dealing with uncertainty in complex environments. To achieve this, the concept of virtual logic is incorporated as a refinement mechanism. Building on this foundation, we construct the notion of a virtual fuzzy parameterized fuzzy hypersoft set (vfp-fhs-set) by integrating virtual logic with hypersoft sets, which are designed to more effectively represent the components of parameter sets. Through this integration, a novel hybrid mathematical model is introduced, enhancing parameterization with fuzzy membership values and providing a more robust framework for decision-making under uncertainty. Several fundamental set operations associated with this structure are formally examined to establish its theoretical soundness. Furthermore, a decision-making algorithm based on vfp-fhs-sets is proposed and illustrated with an application, demonstrating the practical utility of the model. Comparative analysis with existing approaches highlights the advantages of incorporating virtual logic, particularly in improving precision and reliability in the ranking of alternatives.

Keywords

• fp-fhs-set
• vfp-fhs-set
• Algorithm
• Decision-making

Sanal Mantık Tabanlı Bulanık Parametreli Bulanık Hiper Yumuşak Kümeler: Gelişmiş Karar Verme İçin Algoritmik Bir Çerçeve

Abstract

Bu çalışmanın temel amacı, karmaşık ortamlarda belirsizlikle başa çıkarken karar vericiler tarafından ifade edilen üyelik değerlerindeki olası yanlışlıkları ele almaktır. Bunu başarmak için, sanal mantık kavramı bir iyileştirme mekanizması olarak dahil edilmiştir. Bu temel üzerine inşa ederek, sanal mantığı parametre kümelerinin bileşenlerini daha etkili bir şekilde temsil etmek üzere tasarlanmış hiperyumuşak kümelerle entegre ederek sanal bulanık parametreli bulanık hiperyumuşak küme (vfp-fhs-kümesi) kavramını inşa ediyoruz. Bu entegrasyon yoluyla, bulanık üyelik değerleriyle parametrelendirmeyi geliştiren ve belirsizlik altında karar alma için daha sağlam bir çerçeve sağlayan yeni bir hibrit matematiksel model tanıtılmıştır. Bu yapıyla ilişkili birkaç temel küme işlemi, teorik sağlamlığını belirlemek için resmi olarak incelenmiştir. Ayrıca, vfp-fhs-kümeleri tabanlı bir karar alma algoritması önerilmiş ve bir uygulama ile gösterilerek modelin pratik faydası gösterilmiştir. Mevcut yaklaşımlarla karşılaştırmalı analiz, özellikle alternatiflerin sıralanmasında hassasiyeti ve güvenilirliği artırmada sanal mantığın dahil edilmesinin avantajlarını vurgulamaktadır.

Keywords

• fp-fhs-küme
• vfp-fhs-küme
• Algoritma
• Karar verme

References

1. [1] Zadeh L.A., Fuzzy Sets, Information and Control, 1965, 8, pp. 338–353.
2. [2] Molodtsov D., Soft Set Theory-First Results, Computers and Mathematics with Applications, 1999, 37, pp. 19–31.
3. [3] Palanikumar M. and Iampan A., A Novel Approach to Decision Making Based on Type-II Generalized Fermatean Bipolar Fuzzy Soft Sets, International Journal of Innovative Computing, Information and Control, 2022, 18(3), pp. 769–783.
4. [4] Voskoglou M.G., A Hybrid Model for Decision Making Utilizing TFNs and Soft Sets as Tools, Equations, 2022, 2, pp. 65–69.
5. [5] Dalkılıç O., Relations on Neutrosophic Soft Set and Their Application in Decision Making, Journal of Applied Mathematics and Computing, 2021, 67, pp. 257–273.
6. [6] Dalkılıç O., On Topological Structures of Virtual Fuzzy Parametrized Fuzzy Soft Sets, Complex and Intelligent Systems, 2022, 8, pp. 337–348.
7. [7] Ali G. and Ansari M.N., Multiattribute Decision-Making under Fermatean Fuzzy Bipolar Soft Framework, Granular Computing, 2022, 7(2), pp. 337– 352.
8. [8] Çetkin V., Bornological Spaces in the Context of Fuzzy Soft Sets, Filomat, 2022, 36(4), pp. 1341–1350.

9. [9] Dalkılıç O., Generalization of Neutrosophic Parametrized Soft Set Theory and Its Applications, Journal of Polytechnic, 2022, 25(2), pp. 675–684.
10. [10] Liu J.B., Ali S., Mahmood M.K. and Mateen M.H., On m-Polar Diophantine Fuzzy N-Soft Set with Applications, Combinatorial Chemistry and High Throughput Screening, 2022, 25(3), pp. 536–546.
11. [11] Dalkılıç O., Determining the Membership Degrees in the Range (0, 1) for Hypersoft Sets Independently of the Decision-Maker, International Journal of Systems Science, 2022, 53(8), pp. 1733–1743.
12. [12] Smarandache F., Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set, Neutrosophic Sets and Systems, 2018, 22, pp. 168–170.
13. [13] Smarandache F., Soft Set Product Extended to Hypersoft Set and Indetermsoft Set Cartesian Product Extended to Indetermhypersoft Set, Journal of Fuzzy Extension and Applications, 2022, 3(4), pp. 313–316.
14. [14] Jafar M.N., Saeed M., Khan K.M., Alamri F.S. and Khalifa H.A.E.W., Distance and Similarity Measures Using Max-Min Operators of Neutrosophic Hypersoft Sets with Application in Site Selection for Solid Waste Management Systems, IEEE Access, 2022, 10, pp. 11220–11235.
15. [15] Saeed M., Rahman A.U., Ahsan M. and Smarandache F., Theory of Hypersoft Sets: Axiomatic Properties, Aggregation Operations, Relations, Functions and Matrices, Neutrosophic Sets and Systems, 2022, 51, pp. 744–765.
16. [16] Ihsan M., Saeed M. and Rahman A.U., Neutrosophic Hypersoft Expert Set: Theory and Applications, Neutrosophic Sets and Systems, 2022, 50, pp. 431–458.
17. [17] Rahman A.U., Saeed M., Khalifa H.A.E.W. and Afifi W.A., Decision Making Algorithmic Techniques Based on Aggregation Operations and Similarity Measures of Possibility Intuitionistic Fuzzy Hypersoft Sets, AIMS Mathematics, 2022, 7(3), pp. 3866–3895.
18. [18] Çağman N., Enginoğlu S. and Çıtak F., Fuzzy Soft Set Theory and Its Applications, Iranian Journal of Fuzzy Systems, 2011, 8(3), pp. 137–147.
19. [19] Yolcu A. and Öztürk T.Y., Fuzzy Hypersoft Sets and Its Application to Decision-Making, Theory and Application of Hypersoft Set, 2021, pp. 50–64.
20. [20] Yolcu A., Smarandache F. and Öztürk T.Y., Intuitionistic Fuzzy Hypersoft Sets, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2021, 70(1), pp. 443–455.
21. [21] Çağman N., Çıtak F. and Enginoğlu S., FP-Soft Set Theory and Its Applications, Annals of Fuzzy Mathematics and Informatics, 2021, 2(2), pp. 219– 226.
22. [22] Dalkılıç O. and Demirtaş N., VFP-Soft Sets and Its Application on Decision Making Problems, Journal of Polytechnic, 2021, 24(4), pp. 1391–1399.
23. [23] Çağman N., Çıtak F. and Enginoğlu S., Fuzzy Parameterized Fuzzy Soft Set Theory and Its Applications, Turkish Journal of Fuzzy Systems, 2010, 1(1), pp. 21–35.
24. [24] Dalkılıç O., An Application of VFPFSS’s in Decision Making Problems, Journal of Polytechnic, 2022, 25(2), pp. 491–501.
25. [25] Rahman A.U., Saeed M. and Khalifa H.A.W., Decision Making Application Based on Parameterization of Fuzzy Hypersoft Set with Fuzzy Setting, Italian Journal of Pure and Applied Mathematics, 2022, 48, pp. 1033–1048.
26. [26] Rahman A.U., Saeed M., Mohammed M.A., Jaber M.M. and Garcia-Zapirain B., A Novel Fuzzy Parameterized Fuzzy Hypersoft Set and Riesz Summability Approach Based Decision Support System for Diagnosis of Heart Diseases, Diagnostics, 2022, 12(7), pp. 1546.
27. [27] Rahman A.U., Smarandache F., Saeed M. and Khan K.A., Fuzzy Parameterized Extensions of Hypersoft Set Embedded with Possibility-Degree Settings: An Overview, Neutrosophic Sets and Systems, 2025, 87, pp. 713–741.
28. [28] Rahman A.U., An Intuitionistic Fuzzy Parameterized Decision-Making Approach for Evaluating Site Suitability for Educational Institutions Using Hypersoft Settings, International Journal of Mathematics, Statistics, and Computer Science, 2025, 3, pp. 442–456.
29. [29] Ihsan M., Saeed M.H., Alburaikan A. and Khalifa H.A.W., Product Evaluation through Multi-Criteria Decision Making Based on Fuzzy Parameterized Pythagorean Fuzzy Hypersoft Expert Set, AIMS Mathematics, 2022, 7(6), pp. 11024–11052.
30. [30] Saeed M.H., Razaq F., Saeed M., Habib M.S. and Hwang S.J., A Decision-Theoretic Framework for Wastewater Treatment Performance Assessment Based on a Fuzzy Parameterized Fuzzy Hypersoft Set Approach, Scientific Reports, 2025, 15(1), pp. 20706.