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Markov chains and rough sets

  • Kavitha Koppula
  • Babushri Srinivas Kedukodi
  • Syam Prasad Kuncham
Methodologies and Application
  • 78 Downloads

Abstract

In this paper, we present a link between markov chains and rough sets. A rough approximation framework (RAF) gives a set of approximations for a subset of universe. Rough approximations using a collection of reference points gives rise to a RAF. We use the concept of markov chains and introduce the notion of a Markov rough approximation framework (MRAF), wherein a probability distribution function is obtained corresponding to a set of rough approximations. MRAF supplements well-known multi-attribute decision-making methods like TOPSIS and VIKOR in choosing initial weights for the decision criteria. Further, MRAF creates a natural route for deeper analysis of data which is very useful when the values of the ranked alternatives are close to each other. We give an extension to Pawlak’s decision algorithm and illustrate the idea of MRAF with explicit example from telecommunication networks.

Keywords

Rough set Markov chain Rough approximation framework Ring 

Notes

Acknowledgements

The authors acknowledge the anonymous reviewers and the editor for their valuable comments and suggestions. All authors acknowledge the support and encouragement of Manipal Institute of Technology, Manipal Academy of Higher Education (MAHE), India.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

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

References

  1. Akram M, Dudek WA (2008) Intuitionistic fuzzy left k-ideals of semirings. Soft Comput 12:881–890CrossRefzbMATHGoogle Scholar
  2. Abou-Zaid S (1991) On fuzzy subnearrings and ideals. Fuzzy Sets Syst 44(1):139–146MathSciNetCrossRefzbMATHGoogle Scholar
  3. Anderson FW, Fuller KR (1992) Rings and categories of modules. Springer, New YorkCrossRefzbMATHGoogle Scholar
  4. Bhavanari S, Kuncham SP (2013) Near rings. Fuzzy Ideals and graph theory. Chapman and Hall/CRC Press, LondonzbMATHGoogle Scholar
  5. Bhavanari S, Kuncham SP, Kedukodi BS (2010) Graph of a nearring with respect to an ideal. Commun Algebra 38:1957–1962MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bingzhen S, Weimin M, Haiyan Z (2016) An approach to emergency decision making based on decision-theoretic rough set over two universes. Soft Comput 20:3617–3628CrossRefGoogle Scholar
  7. Booth GL, Groenewald NJ, Veldsman S (1990) A Kurosh–Amitsur prime radical for near-rings. Commun Algebra 18(9):3111–3122MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen CT (2000) Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst 114:1–9CrossRefzbMATHGoogle Scholar
  9. Chen S, Hwang CL (1992) Fuzzy Multiple attribute decision making methods and applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  10. Ching W, Ng MK (2003) Advances in data mining and modeling. World Scientific, SingaporeCrossRefGoogle Scholar
  11. Ching W, Ng MK (2006) Markov chains: models, algorithms and applications. Springer, New YorkzbMATHGoogle Scholar
  12. Ching W, Yuen W, Loh A (2003) An inventory model with returns and lateral transshipments. J Oper Res Soc 54(6):636–641CrossRefzbMATHGoogle Scholar
  13. Chu ATW, Kalaba RE, Spingarn K (1979) A comparison of two methods for determining the weights of belonging to fuzzy sets. J Optim Theory Appl 27:531–538MathSciNetCrossRefzbMATHGoogle Scholar
  14. Ciucci D (2008) A unifying abstract approach for rough models. Lect Notes Artif Int 5009:371–378Google Scholar
  15. Davvaz B (2004) Roughness in rings. Inf Sci 164(1–4):147–163MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dudek WA (2008) Special types of intuitionistic fuzzy left h-ideals of hemirings. Soft Comput 12:359–364CrossRefzbMATHGoogle Scholar
  17. Davvaz B, Mahdavipour M (2006) Roughness in modules. Inf Sci 176(24):3658–3674MathSciNetCrossRefzbMATHGoogle Scholar
  18. Fan ZP (1996) Complicated multiple attribute decision making: theory and applications. Dissertation, Northeastern UniversityGoogle Scholar
  19. Haidong Z, Lan S, Shilong L (2017) Hesitant fuzzy rough set over two universes and its application in decision making. Soft Comput 21:1803–1816CrossRefzbMATHGoogle Scholar
  20. Harikrishnan PK, Lafuerza-Guillén B, Yeol JC, Ravindran KT (2017) New classes of generalized PN spaces and their normability. Acta Math Vietnam 42(4):727–746MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hwang CL, Yoon K (1981) Multiple attribute decision making methods and applications. Springer, BerlinCrossRefzbMATHGoogle Scholar
  22. Jagadeesha B, Kedukodi BS, Kuncham SP (2016) Implications on a lattice. Fuzzy Inf Eng 8(4):411–425MathSciNetCrossRefGoogle Scholar
  23. Jagadeesha B, Kedukodi BS, Kuncham SP (2016) Interval valued L-fuzzy ideals based on t-norms and t-conorms. J Intell Fuzzy Syst 28(6):2631–2641MathSciNetCrossRefzbMATHGoogle Scholar
  24. Kedukodi BS, Kuncham SP, Bhavanari S (2007) C-prime fuzzy ideals of nearrings. Soochow J Math 33(4):891–901MathSciNetzbMATHGoogle Scholar
  25. Kedukodi BS, Kuncham SP, Bhavanari S (2009) Equiprime, 3-prime and c-prime fuzzy ideals of nearrings. Soft Comput 13(10):933–944CrossRefzbMATHGoogle Scholar
  26. Kedukodi BS, Kuncham SP, Bhavanari S (2010) Reference points and roughness. Inf Sci 180(17):3348–3361MathSciNetCrossRefzbMATHGoogle Scholar
  27. Kedukodi BS, Kuncham SP, Bhavanari S (2013) C-Prime fuzzy ideals of \(R^{n}[x]\). Int J Contemp Math Sci 8(3):133–137MathSciNetzbMATHGoogle Scholar
  28. Kedukodi BS, Kuncham SP, Jagadeesha B (2017) Interval valued L-fuzzy prime ideals, triangular norms and partially ordered groups. Soft Comput.  https://doi.org/10.1007/s00500-017-2798-x
  29. Kuncham SP, Jagadeesha B, Kedukodi BS (2016) Interval valued L-fuzzy cosets of nearrings and isomorphism theorems. Afr Math 27(3):393–408MathSciNetCrossRefzbMATHGoogle Scholar
  30. Kuroki N, Mordeson JN (1997) Structure of rough sets and rough groups. J Fuzzy Math 5(1):183–191MathSciNetzbMATHGoogle Scholar
  31. Lobillo FJ, Merino L, Navarro G, Santos E (2016) Rough ideals under relations associated to fuzzy ideals. Inf Sci 352–353:121–132CrossRefGoogle Scholar
  32. Ma X, Liu Q, Zhan J (2017) A survey of decision making methods based on certain hybrid soft set models. Artif Intell Rev 47:507–530CrossRefGoogle Scholar
  33. Ma X, Zhan J, Ali MI, Mehmood N (2018) A survey of decision making methods based on two classes of hybrid soft set models. Artif Intell Rev 49(4):511–529CrossRefGoogle Scholar
  34. Medhi J (2009) Stochastic processes. New age international Publishers Limited, New DelhizbMATHGoogle Scholar
  35. Nayak H, Kuncham SP, Kedukodi BS (2018) \(\theta \varGamma \) N-group. Mat Vesnik 70:64–78MathSciNetGoogle Scholar
  36. Opricovic S (1998) Multicriteria optimization of civil engineering systems. Fac Civ Eng Belgrade 2:5–21Google Scholar
  37. Opricovic S (2002) Multicriteria planning of post-earthquake sustainable reconstruction. Comput Aided Civ Infrastruct Eng 17:211–220CrossRefGoogle Scholar
  38. Page L, Brin S, Motwani R, Winograd T (1998) The PageRank citation ranking: Bringing order to the web. Technical report, Stanford digital library technologies project, USAGoogle Scholar
  39. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefzbMATHGoogle Scholar
  40. Pawlak Z (2002) Rough sets and intelligent data analysis. Inf Sci 147(1–4):1–12MathSciNetCrossRefzbMATHGoogle Scholar
  41. Sharma O (1995) Markovian queues. Ellis Horwood, ChichesterzbMATHGoogle Scholar
  42. Shenghai Z, Jingcheng Y, Yishan D, Xuanhua X (2017) Markov chain approximation to multi-stage interactive group decision-making method based on interval fuzzy number. Soft Comput 21:2701–2708CrossRefGoogle Scholar
  43. Zeleny M (1982) Multiple criteria decision making. Mc-Graw-Hill, New YorkzbMATHGoogle Scholar
  44. Zhan J, Alcantud JCR (2018) A survey of parameter reduction of soft sets and corresponding algorithms. Artif Intell Rev.  https://doi.org/10.1007/s10462-017-9592-0 Google Scholar
  45. Zhan J, Alcantud JCR (2018) A novel type of soft rough covering and its application to multicriteria group decision making. Artif Intell Rev.  https://doi.org/10.1007/s10462-018-9617-3 Google Scholar
  46. Zhan J, Ali MI, Mehmood N (2017) On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods. Appl Soft Comput 56:446–457CrossRefGoogle Scholar
  47. Zhan J, Davvaz B (2016) Notes on roughness in rings. Inf Sci 346–347:488–490MathSciNetCrossRefGoogle Scholar
  48. Zhan J, Liu Q, Herawan T (2017) A novel soft rough set: soft rough hemirings and its multicriteria group decision making. Appl Soft Comput 54:393–402CrossRefGoogle Scholar
  49. Zhan J, Liu Q, Zhu W (2017) Another approach to rough soft hemirings and corresponding decision making. Soft Comput 21:3769–3780CrossRefzbMATHGoogle Scholar
  50. Zhan J, Zhu K (2017) A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making. Soft Comput 21:1923–1936CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Manipal Institute of TechnologyManipal Academy of Higher Education (MAHE)ManipalIndia

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