Advertisement

On shadowed hypergraphs

  • Tamunokuro Opubo William-WestEmail author
Methodologies and Application
  • 2 Downloads

Abstract

Fuzzy hypergraphs are useful tools for representing granular structures when describing the relations between objects and set of hyperedges, at minute detail, in a specific granularity. Their merit over classical hypergraphs lies in their ability to model uncertainty that may arise with objects and/or granules incident to each other. Many previous studies on fuzzy hypergraphs seek to exploit their strong descriptive potential to analyze n-ary relations in several contexts. However, due to the very detailed numeric membership grades of their objects, fuzzy hypergraphs are sensitive to noise, which may be overwhelming in their general interpretation. To address this issue, a principle of least commitment to certain membership grades is sort by embracing a framework of shadowed sets. The specific concern of this paper is to study a noise-tolerable framework, viz. shadowed hypergraph. Our goal is to capture and quantify noisy objects in clearly marked out zones. We discuss some characteristic properties of shadowed hypergraph and describe an algorithm for transforming a given fuzzy hypergraph into its resulting shadowed hypergraph. Finally, some illustrative examples are provided to demonstrate the essence of shadowed hypergraphs.

Keywords

Fuzzy sets Shadow sets Fuzzy hypergraphs 

Notes

Acknowledgements

The author sincerely wishes to thank the Editor-in-Chief Professor Antonio Di Nola, and the anonymous reviewers for their technical comments and other valuable suggestions.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.

References

  1. Akram M, Alshehri NO (2015) Tempered interval—valued fuzzy hypergraphs. UPB Sci Bull Ser A 77:39–48MathSciNetzbMATHGoogle Scholar
  2. Akram M, Dudek WA (2013) Intuitionistic fuzzy hypergraphs with applications. Inf Sci 218:182–193MathSciNetCrossRefzbMATHGoogle Scholar
  3. Berge C (1970) Graphes et hypergraphes. Dunod, PariszbMATHGoogle Scholar
  4. Chen SM (1997) Interval-valued fuzzy hypergraph and fuzzy partition. IEEE Trans Syst Man Cybern Part B Cybern 27:725–733CrossRefGoogle Scholar
  5. Chen G, Zhong N, Yiyu Y (2008) A hypergraph model of granular computing. In: IEEE international conference on granular computing, pp 26–28Google Scholar
  6. Deng XF, Yao YY (2014) Decision-theoretic three-way approximations of fuzzy sets. Inf Sci 279:702–715MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkzbMATHGoogle Scholar
  8. Fahmi A, Abdullah S, Amin F, Siddiqui N (2017) Aggregation operators on triangular cubic fuzzy numbers and its application to multi-criteria decision making problems. J Intell Fuzzy Syst.  https://doi.org/10.3233/jifs-162007 Google Scholar
  9. Fahmi A, Abdullah S, Amin F, Sajjad MA, Khan WA (2018a) Trapezoidal cubic fuzzy number Einstein hybrid weighted averaging operators and its application to decision making. Soft Comput.  https://doi.org/10.1007/s00500-018-3242-6 Google Scholar
  10. Fahmi A, Abdullah S, Amin F, Ali A, Khan WA (2018b) Some geometric operators with triangular cubic linguistic hesitant fuzzy number and their application in group decision-making. J Intell Fuzzy Syst.  https://doi.org/10.3233/jifs-18125 Google Scholar
  11. Fahmi A, Amin F, Abdullah S, Ali A (2018c) Cubic fuzzy einstein aggregation operators and its application to decision making. Int J Syst Sci.  https://doi.org/10.1080/00207721.2018.1503356 MathSciNetGoogle Scholar
  12. Ibrahim AM, William-West TO (2019) Induction of shadowed sets from fuzzy sets. Granul Comput 4(1):27–38CrossRefGoogle Scholar
  13. Kaufmann A (1977) introduction a la thiorie des sous-ensemble flous, vol 4. Masson, ParisGoogle Scholar
  14. Lee-Kwang H, Lee KM (1995) An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy hypergraphs and fuzzy partitions. IEEE Trans Syst Man Cybern 25(1):196–201CrossRefGoogle Scholar
  15. Lee-Kwang H, Lee K (2011) Fuzzy hypergraph and fuzzy partition. IEEE Trans Syst Man Cybern 25:196–201MathSciNetCrossRefzbMATHGoogle Scholar
  16. Motro A (1988) VAGUE: a user interface to relational databases that permits vague queries. ACM Trans Inf Syst 6(3):187–214CrossRefGoogle Scholar
  17. Parvathi R, Thilagavathi S, Karunambigai MG (2009) Intuitionistic fuzzy hypergraphs. Cybern Inf Technol 9:46–53MathSciNetGoogle Scholar
  18. Pedrycz W (1998) Shadowed sets: representing and processing fuzzy sets. IEEE Trans Syst Man Cybern Part B Cybern 28:103–109CrossRefGoogle Scholar
  19. Pedrycz W (2005) Interpretation of clusters in the framework of shadowed sets. Pattern Recogn Lett 26:2439–2449CrossRefGoogle Scholar
  20. Pedrycz W, Vukovich G (2002) Granular computing with shadowed sets. Int J Intell Syst 17(2):173–197CrossRefzbMATHGoogle Scholar
  21. Stell G (2010) Relational granularity for hypergraphs. LNAI 6086:267–276Google Scholar
  22. Tahayori H, Sadeghian A, Pedrycz W (2013) Induction of shadowed sets based on the gradual grade of fuzziness. IEEE Trans Fuzzy Syst 21(5):937–949CrossRefGoogle Scholar
  23. Wald A (1945) Sequential tests of statistical hypotheses. Ann Math Stat 16:117–186MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wang Q, Gong Z (2018) An application of fuzzy hypergraphs and hypergraphs in granular computing. Inf Sci 42:296–314MathSciNetCrossRefGoogle Scholar
  25. William-West TO, Singh D (2018) Information granulation for rough fuzzy hypergraphs. Granul Comput 3:75–92CrossRefGoogle Scholar
  26. William-West TO, Ibrahim AM, Kana AFD (2019) Shadowed set approximation of fuzzy sets based on nearest quota of fuzziness. Ann Fuzzy Math Inform 4(1):27–38Google Scholar
  27. Wong MH, Leung KS (1990) A fuzzy database-query language. Inf Syst 15(5):583–590CrossRefGoogle Scholar
  28. Yager R (1977) Multi-objective decision using fuzzy sets, decision sciences. Int J Man-Mach Stud 9:375–382CrossRefzbMATHGoogle Scholar
  29. Yang T, Wu X (2014) Dimensionality reduction of hypergraph information system. In: IEEE international conference on granular computing (GrC), pp 346–351Google Scholar
  30. Yao YY, Deng XF (2011) Sequential three-way decisions with probabilistic rough sets. In: Wang Y et al (eds) ICCI-CC 2011, pp 120–125Google Scholar
  31. Yao Y, Wang S, Deng X (2017) Constructing shadowed sets and three-way approximations of fuzzy sets. Inf Sci 413(1):132–153MathSciNetCrossRefGoogle Scholar
  32. Zhang Y, Yao JT (2018) Game theoretic approach to shadowed sets: a three-way tradeoff perspective. Inf Sci.  https://doi.org/10.1016/j.ins.2018.07.058 Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Physical SciencesAhmadu Bello UniversityZariaNigeria

Personalised recommendations