Short consideration for application eamples of health-care system integration using hyperfuzzy sets and hyperrough sets

Authors

  • Takaaki Fujita * Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan.

https://doi.org/10.22105/thi.v2i1.30

Abstract

To model diverse real-world phenomena, a range of uncertainty-handling concepts has been actively studied, including Fuzzy Sets [1, 2], Rough Sets [3], Intuitionistic Fuzzy Sets [4], Paraconsistent Sets [5], Neutrosophic Sets [6, 7], Hyperneutrosophic Sets [8], Plithogenic Sets [9], and others. Among these extensions of fuzzy sets, Hyperfuzzy Sets are of particular significance. A hyperfuzzy set extends the notion of fuzzy sets to a hierarchical structure, enabling a more refined and flexible representation of uncertainty.
In addition, the notion of a HyperRough Set generalizes the classical setting to multi-attribute data by assigning, to each attribute profile, a subset of the universe and then taking rough approximations with respect to a fixed indiscernibility relation. However, research on real-life applications of Hyperfuzzy Sets and HyperRough Sets remains limited. This paper explores application examples drawn from real-world scenarios by examining system integration using the Hyperfuzzy Set and HyperRough Set frameworks. Note that Health-Care system integration is the process of connecting distinct subsystems or components into a unified, functional, and efficient whole. 

Keywords:

Hyperfuzzy set, Fuzzy set, System integration, Set theory, Rough set, Hyperrough set

References

  1. [1] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338-353.https://doi.org/10.1016/S0019-9958(65)90241-X
  2. [2] Mordeson, J. N., & Nair, P. S. (2000). Fuzzy graphs and fuzzy hypergraphs (Vol. 46). Phys-
  3. [3] ica.https://www.amazon.nl/-/en/John-N-Mordeson/dp/3790812862
  4. [4] Akram, M. (2011). Bipolar fuzzy graphs. Information sciences, 181(24), 5548-5564. https://doi.org/10.1016/j.ins.2011.07.037
  5. [5] Wen-Ran Zhang. Bipolar fuzzy sets. 1997.
  6. [6] Atanassov, K. T., & Gargov, G. (2017). Intuitionistic fuzzy logics. Berlin: Springer International Publishing.https://doi.org/10.1007/978-3-319-48953-7
  7. [7] Atanassov, K. T. (2012). On intuitionistic fuzzy sets theory (Vol. 283). Springer.https://doi.org/10.1007/978-3-642-29127-2
  8. [8] Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Infinite study.https://books.google.nl/books/about/Single_Valued_Neutrosophic_Sets.html?id=RFbVDwAAQBAJ&redir_esc=y
  9. [9] Broumi, S., Bakali, A., & Bahnasse, A. (2018). Neutrosophic sets: An overview. Infinite Study, 410 (1).https://books.google.nl/books/about/Neutrosophic_Sets_An_Overview.html?id=pftuDwAAQBAJ&redir_esc=y
  10. [10] Broumi, S., Talea, M., Bakali, A., Smarandache, F., & Kumar, P. K. (2017, May). Shortest path problem on single valued neu-
  11. [11] trosophic graphs. In 2017 international symposium on networks, computers and communications (ISNCC) (pp. 1-6). IEEE.https://doi.org/10.1109/ISNCC.2017.8071993
  12. [12] Torra, V., & Narukawa, Y. (2009, August). On hesitant fuzzy sets and decision. In 2009 IEEE international conference on
  13. [13] fuzzy systems (pp. 1378-1382). IEEE. https://doi.org/10.1109/FUZZY.2009.5276884
  14. [14] Xu, Z. (2014). Hesitant fuzzy sets theory (Vol. 314). Switzerland: Springer International Publishing.
  15. [15] https://doi.org/10.1007/978-3-319-04711-9
  16. [16] Smarandache, F. (2018). Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic Sets and
  17. [17] Systems, 22, 168.https://digitalrepository.unm.edu/nss_journal/vol22/iss1/13/
  18. [18] Smarandache, F. (2018). Plithogenic set, an extension of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets-revisited. In-
  19. [19] finite study.https://philarchive.org/rec/SMAPSA-2
  20. [20] Sultana, F., Gulistan, M., Ali, M., Yaqoob, N., Khan, M., Rashid, T., & Ahmed, T. (2023). A study of plithogenic graphs:
  21. [21] applications in spreading coronavirus disease (covid-19) globally. Journal of ambient intelligence and humanized computing, 14(10), 13139-13159. https://doi.org/10.1007/s12652-022-03772-6
  22. [22] Das, S., Ghorai, G., & Pal, M. (2022). Picture fuzzy tolerance graphs with application. Complex & Intelligent Systems, 8(1),
  23. [23] -554. https://doi.org/10.1007/s40747-021-00540-5
  24. [24] Khan, W. A., Arif, W., Nguyen, Q. H., Le, T. T., & Van Pham, H. (2024). Picture fuzzy directed hypergraphs with applica-
  25. [25] tions toward decision-making and managing hazardous chemicals. IEEE Access, 12, 87816-87827.Bhttps://doi.org/10.1109/ACCESS.2024.3415120
  26. [26] Cuong, B. C., & Kreinovich, V. (2013, December). Picture fuzzy sets-a new concept for computational intelligence problems.
  27. [27] In 2013 third world congress on information and communication technologies (WICT 2013) (pp. 1-6). IEEE. https://doi.org/10.1109/WICT.2013.7113099
  28. [28] Song, S. Z., Kim, S. J., & Jun, Y. B. (2017). Hyperfuzzy ideals in bck/bci-algebras. Mathematics, 5(4), 81.
  29. [29] https://doi.org/10.3390/math5040081
  30. [30] Ghosh, J., & Samanta, T. K. (2012). Hyperfuzzy set and hyperfuzzy group. International Journal of Advanced Science and Tech-
  31. [31] nology, 41, 27-38. https://www.earticle.net/Article/A206708
  32. [32] Fujita, T., Mehmood, A., & Ghaib, A. A. (2025). Hyperfuzzy control system and superhyperfuzzy control system. Smart Multi-
  33. [33] Criteria Analytics and Reasoning Technologies, 1(1), 1-21. https://doi.org/10.65069/smart1120252
  34. [34] Fujita, T. (2027). Hyperfuzzy and superhyperfuzzy group decision-making. Spectrum of Decision Making and Applications, 1-18.
  35. [35] https://doi.org/10.31181/sdmap4158
  36. [36] Fujita, T., Mehmood, A., & Ghaib, A. A. (2025). Hyperfuzzy offgraphs: A unified graph-based theoretical decision framework for
  37. [37] hierarchical under off-uncertainty. Applied Decision Analytics, 1(1), 1-14. https://ada-journal.org/index.php/ada/article/view/1 [23] Fujita, T. (2025). Weak hyperfuzzy set and weak superhyperfuzzy set. Computational Methods, 2(1), 11-
  38. [38] https://www.jpub.org/journal-admin/uploads/articles/cm212.pdf
  39. [39] Fujita, T. Foundations of (m, n; L)-SuperHyperFuzzy Sets.https://doi.org/10.31224/5070
  40. [40] Fujita, T. Foundations of (m, n)-superhyperfuzzy, superhyperneutrosophic, and superhyperplithogenic
  41. [41] sets.https://doi.org/10.31224/4913
  42. [42] Fujita, T. (2025). HyperFuzzy MultiSet and (ᶃ, ᶄ)-Superhyperfuzzy Multiset with some applications. Authorea Preprints.
  43. [43] https://www.techrxiv.org/doi/full/10.36227/techrxiv.175554727.79729950
  44. [44] Pawlak, Z. (1982). Rough sets. International journal of computer & information sciences, 11(5), 341-
  45. [45] https://doi.org/10.1007/BF01001956
  46. [46] Broumi, S., Smarandache, F., & Dhar, M. (2014). Rough neutrosophic sets. Neutrosophic sets and systems, 32, 493-502.
  47. [47] https://philarchive.org/rec/BRORNS-2
  48. [48] Morsi, N. N., & Yakout, M. M. (1998). Axiomatics for fuzzy rough sets. Fuzzy sets and Systems, 100(1-3), 327-
  49. [49] https://doi.org/10.1016/S0165-0114(97)00104-8
  50. [50] Lenz, O. U., Cornelis, C., & Peralta, D. (2022, July). Fuzzy-rough-learn 0.2: a python library for fuzzy rough set algorithms and
  51. [51] one-class classification. In 2022 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 1-8). IEEE. https://doi.org/10.1109/FUZZ-IEEE55066.2022.9882778
  52. [52] Atagün, A. O., & Kamacı, H. (2023). Strait fuzzy sets, strait fuzzy rough sets and their similarity measures-based decision making
  53. [53] systems. International Journal of Systems Science, 54(12), 2519-2535. https://doi.org/10.1080/00207721.2023.2233971
  54. [54] Zhan, J., Ali, M. I., & Mehmood, N. (2017). On a novel uncertain soft set model: Z-soft fuzzy rough set model and corre-
  55. [55] sponding decision making methods. Applied Soft Computing, 56, 446-457. https://doi.org/10.1016/j.asoc.2017.03.038
  56. [56] Zhao, H., & Zhang, H. Y. (2020). On hesitant neutrosophic rough set over two universes and its application. Artificial Intelligence
  57. [57] Review, 53(6), 4387-4406. https://doi.org/10.1007/s10462-019-09795-4
  58. [58] Bo, C., Zhang, X., Shao, S., & Smarandache, F. (2018). Multi-granulation neutrosophic rough sets on a single domain and
  59. [59] dual domains with applications. Symmetry, 10(7), 296. https://doi.org/10.3390/sym10070296
  60. [60] Bo, C., Zhang, X., Shao, S., & Smarandache, F. (2018). New multigranulation neutrosophic rough set with applications. Sym-
  61. [61] metry, 10(11), 578. https://doi.org/10.3390/sym10110578
  62. [62] Ma, T., & Tang, M. (2006, October). Weighted rough set model. In Sixth International Conference on Intelligent Systems Design
  63. [63] and Applications (Vol. 1, pp. 481-485). IEEE. https://doi.org/10.1109/ISDA.2006.280
  64. [64] Sateesh, N., Srinivasa Rao, P., & Rajya Lakshmi, D. (2023). Optimized ensemble learning‐based student's performance pre-
  65. [65] diction with weighted rough set theory enabled feature mining. Concurrency and Computation: Practice and Experience, 35(7), e7601. https://doi.org/10.1002/cpe.7601
  66. [66] Aitha, N., & Srinadas, R. (2009). A strategy to reduce the control packet load of aodv using weighted rough set model for manet.
  67. [67] The International Arab Journal of Information Technology, 8(1), 108-117. https://iajit.org/PDF/vol.8,no.1/17.pdf
  68. [68] Fujita, T. (2025). Hyperrough topsis method and superhyperrough topsis method. Uncertainty Discourse and Applications, 2
  69. [69] (1), 61-75. https://doi.org/10.48313/uda.vi.59
  70. [70] Fujita, T., & Mehmood, A. (20252). Graded HyperRough Set and Linguistic HyperRough Set. Journal of Mathematical Struc-
  71. [71] tures and Applications (GJMSA). 12(1), 1–23, 2025. https://americaspg.com/article/pdf/4086
  72. [72] Fujita, T., & Smarandache, F. (2025). A concise introduction to hyperfuzzy, hyperneutrosophic, hyperplithogenic, hypersoft, and hy-
  73. [73] perrough sets with practical examples. Neutrosophic Sets and Systems, 80, 609-
  74. [74] file:///C:/Users/NoteBook/OneDrive/Desktop/37HyperFuzzy.pdf
  75. [75] Fujita, T. (2025). Neighborhood hyperrough set and neighborhood superhyperrough set. Pure Mathematics for Theoretical
  76. [76] Computer Science, 5(1), 34-47.https://B2n.ir/wu5764
  77. [77] Takaaki, F., & Smarandache, F. (2025). Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncer-
  78. [78] tainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Fifth volume: Various SuperHyperConcepts (Collected Papers). https://philpapers.org/rec/TAKAUC-2
  79. [79] Nacaroglu, Y., Akgunes, N., Pak, S., & Cangul, I. N. (2021). Some graph parameters of power set graphs. Advances & Applica-
  80. [80] tions in Discrete Mathematics, 26(2), 211-219. file:///C:/Users/NoteBook/OneDrive/Desktop/basilan%20(1).pdf
  81. [81] Shalu, M. A., & Yamini, S. D. (2014). Counting maximal independent sets in power set graphs. Indian Institute of Information
  82. [82] Technology Design & Manufacturing (IIITD&M) Kancheepuram, India. https://B2n.ir/hd4340
  83. [83] Jech, T. (2003). Set theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics. Springer
  84. [84] Monographs in Mathematics.
  85. [85] Jun, Y. B., Hur, K., & Lee, K. J. (2017). Hyperfuzzy subalgebras of bck/bci-algebras. Annals of Fuzzy Mathematics and
  86. [86] Informatics.
  87. [87] Nazari, Z., & Mosapour, B. (2018). The entropy of hyperfuzzy sets. Journal of Dynamical Systems and Geometric Theories, 16(2),
  88. [88] -185. https://doi.org/10.1080/1726037X.2018.1436270
  89. [89] Smarandache, F. (2024). Foundation of superhyperstructure & neutrosophic superhyperstructure. Neutrosophic Sets and Systems,
  90. [90] (2024), 367-381. https://fs.unm.edu/NSS/SuperHyperStructure.pdf
  91. [91] Khali, H. E., GÜNGÖR, G. D., & Zaina, M. A. N. (2022). Neutrosophic SuperHyper Bi-Topological Spaces: Original Notions
  92. [92] and New Insights. Neutrosophic Sets and Systems, 51(1), 3. https://digitalrepository.unm.edu/cgi/viewcontent.cgi?article=2116&context=nss_journal
  93. [93] Das, A. K., Das, R., Das, S., Debnath, B. K., Granados, C., Shil, B., & Das, R. (2025). A comprehensive study of neutrosophic super-
  94. [94] hyper bci-semigroups and their algebraic significance. Transactions on Fuzzy Sets and Systems, 8(2), 80. https://doi.org/10.71602/tfss.2025.1198050
  95. [95] Al-Odhari, A. (2025). Neutrosophic Power-Set and Neutrosophic Hyper-Structure of Neutrosophic Set of Three Types. Annals of
  96. [96] Pure and Applied Mathematics, 31(2), 125-146. http://dx.doi.org/10.22457/apam.v31n2a05964
  97. [97] Smarandache, F. S. (2023). Neutrosophic SuperHyperStructure: Current understanding and future directions, Neutrosophic
  98. [98] Systems With Applications, 2023, 12, 68-76. https://philarchive.org/archive/SMASSN
  99. [99] Smarandache, F. (2017). Hyperuncertain, superuncertain, and superhyperuncertain sets/logics/probabilities/statistics. Infinite
  100. [100] Study. https://fs.unm.edu/CR/HyperUncertain-SuperUncertain-SuperHyperUncertain.pdf
  101. [101] Pawlak, Z. (1998). Rough set theory and its applications to data analysis. Cybernetics & Systems, 29(7), 661-688.
  102. [102] https://doi.org/10.1080/019697298125470
  103. [103] Pawlak, Z. (1991). Rough sets: Theoretical aspects of reasoning about data. Kluwer Academic Publishers.
  104. [104] https://cir.nii.ac.jp/crid/1971430859860150820
  105. [105] Zimmerman, N. A. (1969, August). System integration as a programming function. In Proceedings of the 1969 24th national con-
  106. [106] ference (pp. 459-467). https://doi.org/10.1145/800195.805951
  107. [107] Hasselbring, W. (2000). Information system integration. Communications of the ACM, 43(6), 32-38.
  108. [108] https://doi.org/10.1145/336460.336472
  109. [109] Madni, A. M., & Sievers, M. (2014). System of systems integration: Key considerations and challenges. Systems Engineering, 17
  110. [110] (3), 330-347. https://doi.org/10.1002/sys.21272
  111. [111] Wu, T. Y., Majeed, A., & Kuo, K. N. (2010). An overview of the healthcare system in Taiwan. London journal of primary care, 3(2),
  112. [112] -119. https://doi.org/10.1080/17571472.2010.11493315
  113. [113] Budrionis, A., & Bellika, J. G. (2016). The learning healthcare system: where are we now? A systematic review. Journal of
  114. [114] biomedical informatics, 64, 87-92. https://doi.org/10.1016/j.jbi.2016.09.018
  115. [115] Wendt, C., Frisina, L., & Rothgang, H. (2009). Healthcare system types: a conceptual framework for comparison. Social Policy
  116. [116] & Administration, 43(1), 70-90. https://doi.org/10.1111/j.1467-9515.2008.00647.x
  117. [117] Touati, F., & Tabish, R. (2013). U-healthcare system: State-of-the-art review and challenges. Journal of medical systems, 37
  118. [118] (3), 9949. https://doi.org/10.1007/s10916-013-9949-0
  119. [119] Smarandache, F. (2016). Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic
  120. [120] Over-/Under-/Off-Logic, Probability, and Statistics. Infinite Study. https://digitalrepository.unm.edu/math_fsp/26?utm_-source=digitalrepository.unm.edu%2Fmath_fsp%2F26&utm_medium=PDF&utm_campaign=PDFCoverPages
  121. [121] Broumi, S., Talea, M., Bakali, A., & Smarandache, F. (2016). Single valued neutrosophic graphs. Journal of New theory, (10),
  122. [122] -101. https://dergipark.org.tr/en/pub/jnt/issue/34504/381241
  123. [123] Smarandache, F. (1999). A unifying field in Logics: Neutrosophic Logic. In Philosophy (pp. 1-141). American Research Press.
  124. [124] https://web-archive.southampton.ac.uk/cogprints.org/1919/3/eBook-Neutrosophics2.pdf
  125. [125] Smarandache, F. (2023). New types of soft sets “hypersoft set, indetermsoft set, indetermhypersoft set, and treesoft set”: an
  126. [126] improved version. Infinite Study. https://B2n.ir/wt3618
  127. [127] Ihsan, M., Rahman, A. U., & Saeed, M. (2021). Hypersoft expert set with application in decision making for recruitment process.
  128. [128] Neutrosophic Sets and Systems, 42, 191-207. file:///C:/Users/NoteBook/OneDrive/Desktop/HypersoftExpertSetApplication12.pdf [69] Diestel, R. (2025). Graph theory. Springer Nature. https://B2n.ir/pk2631
  129. [129] Bretto, A. (2013). Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 1, 209-216.
  130. [130] https://doi.org/10.1007/978-3-319-00080-0
  131. [131] Feng, Y., You, H., Zhang, Z., Ji, R., & Gao, Y. (2019). Hypergraph neural networks. In Proceedings of the AAAI conference on
  132. [132] artificial intelligence (Vol. 33, No. 01, pp. 3558-3565). https://doi.org/10.1609/aaai.v33i01.33013558
  133. [133] Smarandache, F. (2020). Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of
  134. [134] HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra. Infinite Study. https://B2n.ir/gy5830
  135. [135] Hamidi, M., Smarandache, F., & Davneshvar, E. (2022). Spectrum of superhypergraphs via flows. Journal of Mathematics, 2022
  136. [136] (1), 9158912. https://doi.org/10.1155/2022/9158912
  137. [137] Hamidi, M., Smarandache, F., & Taghinezhad, M. (2023). Decision Making Based on Valued Fuzzy Superhypergraphs. Infinite
  138. [138] Study. https://doi.org/10.32604/cmes.2023.030284

Downloads

Published

2025-06-23

Issue

Vol. 2 No. 2 (2025): Trends in Health Informatics

Section

Articles

How to Cite

Fujita, T. (2025). Short consideration for application eamples of health-care system integration using hyperfuzzy sets and hyperrough sets. Trends in Health Informatics, 2(2), 68-81. https://doi.org/10.22105/thi.v2i1.30