Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 91–127 | Cite as

Rough fuzzy digraphs with application

  • Muhammad AkramEmail author
  • Fariha Zafar
Original Research


Rough set theory is a mathematical tool to deal with incomplete and vague information. Fuzzy set theory deals the problem of how to understand and manipulate imperfect knowledge. The aim of this research is to construct a framework for handling vague information by applying some new concept of rough fuzzy digraphs. In this research study, we present certain new aspects of rough fuzzy digraphs (RFDs) based on rough fuzzy set model. We discuss complement and \(\mu \)-complement of RFDs. We discuss the concept of isomorphisms between RFDs and the irregularity of RFDs in detail. We consider an application of our proposed hybrid decision-making method: RFDs. We also describe our hybrid decision-making method as an algorithm.


Rough fuzzy relation Irregular rough fuzzy digraphs Hybrid decision-making method Algorithm 

Mathematics Subject Classification

03E72 68R10 68R05 



The authors are very thankful to the Editor and referees for their valuable comments and suggestions for improving the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of the research article.


  1. 1.
    Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181, 5548–5564 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akram, M., Ashraf, A., Sarwar, M.: Novel applications of intuitionistic fuzzy digraphs in decision support systems. Sci. World J. 2014, 1–11 (2014)Google Scholar
  3. 3.
    Akram, M., Alshehri, N., Davvaz, B., Ashraf, A.: Bipolar fuzzy digraphs in decision support systems. J. Mult. Value Log. Soft Comput. 27, 531–551 (2016)zbMATHGoogle Scholar
  4. 4.
    Akram, M., Ali, G., Alshehri, N.O.: A new multi-attribute decision-making method based on \(m\)-polar fuzzy soft rough sets. Symmetry 9(11), 271 (2017). CrossRefGoogle Scholar
  5. 5.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)CrossRefzbMATHGoogle Scholar
  6. 6.
    Banerjee, M., Pal, S.K.: Roughness of a fuzzy set. Inf. Sci. 93, 235–246 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Biswas, R.: On rough sets and fuzzy rough sets. Bull. Pol. Acad. Sci. Math. 42, 345–349 (1994)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Biswas, R.: On rough fuzzy sets. Bull. Pol. Acad. Sci. Math. 42, 352–355 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chakrabarty, K., Biswas, R., Nanda, S.: Fuzziness in rough sets. Fuzzy Sets Syst. 110, 247–251 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17, 191–209 (1990)CrossRefzbMATHGoogle Scholar
  11. 11.
    Feng, F.: Generalized rough fuzzy sets based on soft sets. In: Proceedings of the First International Workshop on Intelligent Systems and Applications, ISA2009, Wuhan, China, 23–24 May, pp. 825–828 (2009)Google Scholar
  12. 12.
    Feng, F., Li, C., Davvaz, B., Ali, M.I.: Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft. Comput. 14(9), 899–911 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Feng, F.: Soft rough sets applied to multicriteria group decision making. Ann. Fuzzy Math. Inform. 2(1), 69–80 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Feng, F., Liu, X., Leoreanu-Fotea, V., Jun, Y.B.: Soft sets and soft rough sets. Inf. Sci. 181(6), 1125–1137 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23, 610–614 (1993)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kauffman, A.: Introduction a la Theorie des Sous-emsembles Flous, Masson et Cie, Vol. 1 (1973)Google Scholar
  17. 17.
    Molodtsov, D.A.: Soft set theory-first results. Comput. Math. Appl. 37, 19–31 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mordeson, J.N., Peng, C.S.: Operations on fuzzy graphs. Inf. Sci. 79, 159–170 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nagoorgani, A.: Properties of \(\mu \)-complement of a fuzzy graph. Int. J. Algorithm Comput. Math. 2(3), 73–83 (2009)MathSciNetGoogle Scholar
  20. 20.
    Nagoorgani, A., Latha, S.R.: On irregular fuzzy graphs. Appl. Math. Sci. 6(11), 517–523 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inform. Sci. 11(5), 341–356 (1982)CrossRefzbMATHGoogle Scholar
  22. 22.
    Pawlak, Z.: Rough sets and fuzzy sets. Fuzzy Sets Syst. 17, 99–102 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pawlak, Z.: Rough sets, rough relations and rough functions. Fundam. Inform. 27(2), 103–108 (1996)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Turksen, I.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 20(2), 191–210 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wu, S.Y.: The compositions of fuzzy digraphs. J. Res. Educ. Sci. 31, 603–628 (1986)Google Scholar
  26. 26.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zafar, F., Akram, M.: A novel decision-making method based on rough fuzzy information. Int. J. Fuzzy Syst. (2017). Google Scholar
  28. 28.
    Zhan, J., Ali, M.I., Mehmood, N.: On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods. Appl. Soft Comput. 56, 446–457 (2017)CrossRefGoogle Scholar
  29. 29.
    Zhan, J., Liu, Q., Herawan, T.: A novel soft rough set: soft rough hemirings and its multicriteria group decision making. Appl. Soft Comput. 54, 393–402 (2017)CrossRefGoogle Scholar
  30. 30.
    Zhan, J., Zhu, K.: A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making. Soft. Comput. 21, 1923–1936 (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of the PunjabLahorePakistan

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