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Novel group decision making approach based on the rough soft approximations of graphs and hypergraphs

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Abstract

Rough soft knowledge is a key approach to understand and model uncertain, vague and not clearly defined situations in a parametric manner. Graphs, hypergraphs and other algebraic structures can be discussed more precisely when upper and lower approximate relations of objects are to be dealt with soft set theory. In this article, the notion of rough approximations is integrated with other algebraic structures under soft environment. Certain types of rough soft relations, rough soft graphs, rough soft (RS) hypergraphs are introduced with significant properties and results. The properties reflexivity, symmetry, transitivity and their negation for RS relations are discussed in detail. RS relations are described as the Cartesian product of single and two different approximation spaces with same and distinct parametric sets. Different types of RS graphs and RS digraphs are defined using RS relations. The algebraic connectivity of certain operations of RS graphs is computed with upper and lower bounds. The notion of RS relations on sets of more than two elements is illustrated to define the structure of soft hypergraphs and RS hypergraphs. The concepts of linearity, duality, connectedness, 2-section and their relations in RS hypergraphs is demonstrated with with isomorphism properties. The importance of RS information is described with a group decision making problem for matching a list of authors to different fields. The out-performance and advantages of the proposed approach over other existing concepts of uncertainty are highlighted in detail.

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References

  1. Akram, M., Arshad, M.: Fuzzy rough graph theory with applications. Int. J. Comput. Intell. Syst. 12(1), 90–107 (2018)

    Google Scholar 

  2. Akram, M., Arshad, M.: A new approach based on fuzzy rough digraphs for decision-making. J. Intell. Fuzzy Syst. 35(2), 2105–2121 (2018)

    Google Scholar 

  3. Akram, M., Luqman, A.: A new decision-making method based on bipolar neutrosophic directed hypergraphs. J. Appl. Math. Comput. 57(1), 547–575 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Akram, M., Nawaz, H.S.: Algorithms for the computation of regular single-valued neutrosophic soft hypergraphs applied to supranational Asian bodies. J. Appl. Math. Comput. 68(6), 4479–4506 (2022)

    MathSciNet  Google Scholar 

  5. Akram, M., Zafar, F.: Multi-criteria decision-making methods under soft rough fuzzy knowledge. J. Intell. Fuzzy Syst. 35(3), 3507–3528 (2018)

    Google Scholar 

  6. Akram, M., Zafar, F.: Hybrid Soft Computing Models Applied to Graph Theory. Studies in Fuzziness and Soft Computing, Springer, New York (2020)

    MATH  Google Scholar 

  7. Akram, M., Sarwar, M., Borzooei, R.A.: A novel decision-making approach based on hypergraphs in intuitionistic fuzzy environment. J. Intell. Fuzzy Syst. 35(2), 1905–1922 (2018)

    Google Scholar 

  8. Alcantud, J.C.R.: Softarisons: theory and practice. Neural Comput. Appl. 33(23), 16759–16771 (2021)

    Google Scholar 

  9. Alcantud, J.C.R., Feng, F., Yager, R.R.: An \(N\)-soft set approach to rough sets. IEEE Trans. Fuzzy Syst. 28(11), 2996–3007 (2019)

    Google Scholar 

  10. Almagrabi, A.O., Abdullah, S., Shams, M., Al-Otaibi, Y.D., Ashraf, S.: A new approach to q-linear Diophantine fuzzy emergency decision support system for COVID19. J. Ambient Intell. Hum. Comput. (2021). https://doi.org/10.1007/s12652-021-03130-y

    Article  Google Scholar 

  11. Atanassov, K.T.: Intuitionistic Fuzzy Sets, VII ITKR’s Session, Deposed in Central for Science-Technical Library of Bulgarian Academy of Sciences 1697/84. Sofia, Bulgaria (1983)

  12. Babitha, K.V., Sunil, J.: Soft set relations and functions. Comput. Math. Appl. 60(7), 1840–1849 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Barman, B., Patra, S.: Variable precision rough set based unsupervised band selection technique for hyperspectral image classification. Knowl.-Based Syst. 193, 105414 (2020)

    Google Scholar 

  14. Berge, C.: Graphs and Hypergraphs. North-Holland Publishing Company, Amsterdam, New York (1973)

    MATH  Google Scholar 

  15. Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. General Syst. 17(2–3), 191–209 (1990)

    MATH  Google Scholar 

  16. 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)

    MATH  Google Scholar 

  17. Feng, F., Liu, X., Leoreanu-Fotea, V., Jun, Y.B.: Soft sets and soft rough sets. Inf. Sci. 181(6), 1125–1137 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Feng, F., Akram, M., Davvaz, B., Fotea, V.L.: Attribute analysis of information systems based on elementary soft implications. Knowl.-Based Syst. 70, 281–292 (2014)

    Google Scholar 

  19. Feng, F., Wan, Z., Alcantud, J.C.R., Garg, H.: Three-way decision based on canonical soft sets of hesitant fuzzy sets. AIMS Math. 7(2), 2061–2083 (2022)

    MathSciNet  Google Scholar 

  20. Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)

    MathSciNet  MATH  Google Scholar 

  21. Goetschel, R.H., Jr.: Introduction to fuzzy hypergraphs and Hebbian structures. Fuzzy Sets Syst. 76(1), 113–130 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Goetschel, R.H., Jr.: Fuzzy colorings of fuzzy hypergraphs. Fuzzy Sets Syst. 94(2), 185–204 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Hamed, A., Sobhy, A., Nassar, H.: Distributed approach for computing rough set approximations of big incomplete information systems. Inf. Sci. 547, 427–449 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Harary, F., Norman, R.Z.: Graph Theory as a Mathematical Model in Social Science (No. 2). University of Michigan, Institute for Social Research, Ann Arbor (1953)

    Google Scholar 

  25. He, T., Shi, K.: Rough graph and its structure. J. Shandong Univ. 41(6), 46–50 (2006)

    Google Scholar 

  26. Huang, B., Li, H., Feng, G., Zhou, X.: Dominance-based rough sets in multi-scale intuitionistic fuzzy decision tables. Appl. Math. Comput. 348, 487–512 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Kaufmann, A.: Introduction la Thorie des Sous-Ensembles Flous Lusage des Ingnieurs (Fuzzy Sets Theory). Masson, Paris (1973)

    Google Scholar 

  28. Kharal, A., Ahmad, B.: Mappings on soft classes. New Math. Nat. Comput. 7(3), 471–481 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Lee-Kwang, H., Lee, K.M.: Fuzzy hypergraph and fuzzy partition. IEEE Trans. Syst. Man Cybern. 25(1), 196–201 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Leoreanu-Fotea, V.: The lower and upper approximations in a hypergroup. Inf. Sci. 178(18), 3605–3615 (2008)

    MathSciNet  MATH  Google Scholar 

  31. Luo, J., Fujita, H., Yao, Y., Qin, K.: On modeling similarity and three-way decision under incomplete information in rough set theory. Knowl.-Based Syst. 191, 105251 (2020)

    Google Scholar 

  32. Ma, X., Akram, M., Zahid, K., Alcantud, J.C.R.: Group decision-making framework using complex Pythagorean fuzzy information. Neural Comput. Appl. (2020). https://doi.org/10.1007/s00521-020-05100-5

    Article  Google Scholar 

  33. Ma, X., Zhan, J., Sun, B., Alcantud, J.C.R.: Novel classes of coverings based multigranulation fuzzy rough sets and corresponding applications to multiple attribute group decision-making. Artif. Intell. Rev. (2020). https://doi.org/10.1007/s10462-020-09846-1

    Article  Google Scholar 

  34. Mahapatra, T., Pal, M.: An investigation on m-polar fuzzy threshold graph and its application on resource power controlling system. J. Ambient Intell. Hum. Comput. (2021). https://doi.org/10.1007/s12652-021-02914-6

    Article  Google Scholar 

  35. Maji, P.K., Biswas, R., Roy, A.R.: Fuzzy soft sets. J. Fuzzy Math. 9(3), 589–602 (2001)

    MathSciNet  MATH  Google Scholar 

  36. Meng, D., Zhang, X., Qin, K.: Soft rough fuzzy sets and soft fuzzy rough sets. Comput. Math. Appl. 62(12), 4635–4645 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Molodtsov, D.A.: Soft set theory-first results. Comput. Math. Appl. 37(4–5), 19–31 (1999)

    MathSciNet  MATH  Google Scholar 

  38. Mordeson, J.N., Nair, P.S.: Fuzzy Graphs and Fuzzy Hypergraphs. Studies in Fuzziness and Soft Computing, Springer-Verlag, Berlin, Heidelberg (2001)

    MATH  Google Scholar 

  39. Nicoletti, M.C., Uchôa, J.Q., Baptistini, M.T.: Rough relation properties. Int. J. Appl. Math. Comput. Sci. 11(3), 621–635 (2001)

    MathSciNet  MATH  Google Scholar 

  40. Parvathi, R., Thilagavathi, S., Karunambigai, M.G.: Intuitionistic fuzzy hypergraphs. Cybern. Inf. Technol. 9(2), 46–53 (2009)

    MathSciNet  Google Scholar 

  41. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982)

    MATH  Google Scholar 

  42. Pawlak, Z.: Rough sets, rough relations and rough functions. Fund. Inform. 27(2–3), 103–108 (1996)

    MathSciNet  MATH  Google Scholar 

  43. Raut, S., Pal, M.: Fuzzy intersection graph: a geometrical approach. J. Ambient Intell. Hum. Comput. (2021). https://doi.org/10.1007/s12652-021-03192-y

    Article  Google Scholar 

  44. Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and Their Applications, pp. 77–95. Academic Press, New York (1975)

    Google Scholar 

  45. Roy, A.R., Maji, P.K.: A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math. 203(2), 412–418 (2007)

    MATH  Google Scholar 

  46. Sarwar, M., Akram, M.: Certain hybrid rough models with type-2 soft information. J. Mult.-Valued Log. Soft Comput. 1–35 (In press)

  47. Sarwar, M., Ali, G., & Chaudhry, R.N.: Decision-making model for failure modes and effect analysis based on rough fuzzy integrated clouds. Appl. Soft Comput. 110148 (2023b)

  48. Sarwar, M.: A theoretical investigation based on the rough approximations of hypergraphs. J. Math. 2022 (2022)

  49. Sarwar, M.: Decision-making approaches based on color spectrum and \(D\)-TOPSIS method under rough environment. Comput. Appl. Math. 39(4), 1–32 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Sarwar, M., Akram, M., Liu, P.: An integrated rough ELECTRE II approach for risk evaluation and effects analysis in automatic manufacturing process. Artif. Intell. Rev. 54(6), 4449–4481 (2021)

    Google Scholar 

  51. Sarwar, M., Akram, M., Shahzadi, S.: Distance measures and \(\delta \)-approximations with rough complex fuzzy models. Granu. Comput. (2023). https://doi.org/10.1007/s41066-023-00371-4

    Article  Google Scholar 

  52. Shaheen, T., Mian, B., Shabir, M., Feng, F.: A novel approach to decision analysis using dominance-based soft rough sets. Int. J. Fuzzy Syst. 21(3), 954–962 (2019)

    MathSciNet  Google Scholar 

  53. Wang, C., Huang, Y., Shao, M., Fan, X.: Fuzzy rough set-based attribute reduction using distance measures. Knowl.-Based Syst. 164, 205–212 (2019)

    Google Scholar 

  54. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    MATH  Google Scholar 

  55. Zafar, F., Akram, M.: A novel decision-making method based on rough fuzzy information. Int. J. Fuzzy Syst. 20(3), 1000–1014 (2018)

    MathSciNet  Google Scholar 

  56. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994)

  57. Zhang, Z.: A rough set approach to intuitionistic fuzzy soft set based decision making. Appl. Math. Model. 36(10), 4605–4633 (2012)

    MathSciNet  MATH  Google Scholar 

  58. Zhang, K., Zhan, J., Wu, W.Z.: Novel fuzzy rough set models and corresponding applications to multi-criteria decision-making. Fuzzy Sets Syst. 383, 92–126 (2020)

    MathSciNet  MATH  Google Scholar 

  59. Zhu, G.N., Hu, J., Ren, H.: A fuzzy rough number-based AHP-TOPSIS for design concept evaluation under uncertain environments. Appl. Soft Comput. 91, 106228 (2020)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, GC Women University, Sialkot, 51310, Pakistan

    Musavarah Sarwar

  2. Department of Mathematics, University of Okara, Okara, 56300, Pakistan

    Fariha Zafar

  3. Department of Mathematics, University of the Punjab, New Campus, Lahore, 4590, Pakistan

    Muhammad Akram

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  1. Musavarah Sarwar
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  2. Fariha Zafar
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MS, FZ, MA conceptualized and designed the study, analysed the data, and wrote the manuscript.

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Correspondence to Muhammad Akram.

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Sarwar, M., Zafar, F. & Akram, M. Novel group decision making approach based on the rough soft approximations of graphs and hypergraphs. J. Appl. Math. Comput. 69, 2795–2830 (2023). https://doi.org/10.1007/s12190-023-01855-x

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