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Journal of Applied Mathematics
Volume 2012, Article ID 608681, 35 pages
http://dx.doi.org/10.1155/2012/608681
Research Article

Type-2 Fuzzy Soft Sets and Their Applications in Decision Making

College of Mathematics and Computer Sciences, Hebei University, Hebei, Baoding 071002, Hebei Province, China

Received 23 May 2012; Revised 26 September 2012; Accepted 10 October 2012

Academic Editor: Hector Pomares

Copyright © 2012 Zhiming Zhang and Shouhua Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Molodtsov, “Soft set theory-first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965. View at Google Scholar · View at Zentralblatt MATH
  3. Z. Pawlak, “Rough sets,” International Journal of Computer and Information Sciences, vol. 11, no. 5, pp. 341–356, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757–763, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. Z. Kong, L. Gao, L. Wang, and S. Li, “The normal parameter reduction of soft sets and its algorithm,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3029–3037, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. I. Ali, F. Feng, X. Liu, W. K. Min, and M. Shabir, “On some new operations in soft set theory,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1547–1553, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. Zou and Z. Xiao, “Data analysis approaches of soft sets under incomplete information,” Knowledge-Based Systems, vol. 21, no. 8, pp. 941–945, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. N. Çağman and S. Enginoğlu, “Soft matrix theory and its decision making,” Computers & Mathematics with Applications, vol. 59, no. 10, pp. 3308–3314, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. N. Çağman and S. Enginoğlu, “Soft set theory and uni-int decision making,” European Journal of Operational Research, vol. 207, no. 2, pp. 848–855, 2010. View at Publisher · View at Google Scholar
  12. T. Herawan and M. M. Deris, “A soft set approach for association rules mining,” Knowledge-Based Systems, vol. 24, no. 1, pp. 186–195, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. K. Gong, Z. Xiao, and X. Zhang, “The bijective soft set with its operations,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2270–2278, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. H. Aktaş and N. Çağman, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. B. Jun, “Soft BCK/BCI-algebras,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1408–1413, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Y. B. Jun and C. H. Park, “Applications of soft sets in ideal theory of BCK/BCI-algebras,” Information Sciences, vol. 178, no. 11, pp. 2466–2475, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. F. Feng, Y. B. Jun, and X. Zhao, “Soft semirings,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2621–2628, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. U. Acar, F. Koyuncu, and B. Tanay, “Soft sets and soft rings,” Computers &; Mathematics with Applications, vol. 59, no. 11, pp. 3458–3463, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. S. Yamak, O. Kazancı, and B. Davvaz, “Soft hyperstructure,” Computers & Mathematics with Applications, vol. 62, no. 2, pp. 797–803, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. M. Anvariyeh, S. Mirvakili, O. Kazancı, and B. Davvaz, “Algebraic hyperstructures of soft sets associated to semihypergroups,” Southeast Asian Bulletin of Mathematics, vol. 35, no. 6, pp. 911–925, 2011. View at Google Scholar
  21. P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001. View at Google Scholar · View at Zentralblatt MATH
  22. P. Majumdar and S. K. Samanta, “Generalised fuzzy soft sets,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1425–1432, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. W. Xu, J. Ma, S. Wang, and G. Hao, “Vague soft sets and their properties,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 787–794, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. X. Yang, T. Y. Lin, J. Yang, Y. Li, and D. Yu, “Combination of interval-valued fuzzy set and soft set,” Computers & Mathematics with Applications, vol. 58, no. 3, pp. 521–527, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. F. Feng, C. Li, B. Davvaz, and M. I. Ali, “Soft sets combined with fuzzy sets and rough sets: a tentative approach,” Soft Computing, vol. 14, no. 9, pp. 899–911, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. S. Bhattacharya and B. Davvaz, “Some more results on IF soft rough approximation space,” International Journal of Combinatorics, Article ID 893061, 10 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  27. P. K. Maji, R. Biswas, and A. R. Roy, “Intuitionistic fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 677–692, 2001. View at Google Scholar · View at Zentralblatt MATH
  28. P. K. Maji, A. R. Roy, and R. Biswas, “On intuitionistic fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 12, no. 3, pp. 669–683, 2004. View at Google Scholar · View at Zentralblatt MATH
  29. K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. Y. Jiang, Y. Tang, Q. Chen, H. Liu, and J. Tang, “Interval-valued intuitionistic fuzzy soft sets and their properties,” Computers & Mathematics with Applications, vol. 60, no. 3, pp. 906–918, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. A. Aygünoğlu and H. Aygün, “Introduction to fuzzy soft groups,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1279–1286, 2009. View at Publisher · View at Google Scholar
  32. J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, Upper-Saddle River, NJ, USA, 2001.
  33. L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning. I,” Information Sciences, vol. 8, pp. 199–249, 1975. View at Google Scholar · View at Zentralblatt MATH
  34. N. N. Karnik and J. M. Mendel, “Applications of type-2 fuzzy logic systems to forecasting of time-series,” Information Sciences, vol. 120, no. 1, pp. 89–111, 1999. View at Publisher · View at Google Scholar · View at Scopus
  35. N. N. Karnik and J. M. Mendel, “Operations on type-2 fuzzy sets,” Fuzzy Sets and Systems, vol. 122, no. 2, pp. 327–348, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. J. M. Mendel, R. I. John, and F. Liu, “Interval type-2 fuzzy logic systems made simple,” IEEE Transactions on Fuzzy Systems, vol. 122, pp. 327–348, 2001. View at Google Scholar
  37. H. B. Mitchell, “Pattern recognition using type-II fuzzy sets,” Information Sciences, vol. 170, no. 2–4, pp. 409–418, 2005. View at Publisher · View at Google Scholar · View at Scopus
  38. M. Mizumoto and K. Tanaka, “Fuzzy sets of type-2 under algebraic product and algebraic sum,” Fuzzy Sets and Systems, vol. 5, no. 3, pp. 277–290, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. H. Wu, Y. Wu, and J. Luo, “An interval type-2 fuzzy rough set model for attribute reduction,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 2, pp. 301–315, 2009. View at Publisher · View at Google Scholar · View at Scopus
  40. J. Zeng and Z. Q. Liu, “Type-2 fuzzy hidden Markov models and their application to speech recognition,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 3, pp. 454–467, 2006. View at Publisher · View at Google Scholar · View at Scopus
  41. F. Liu, “An efficient centroid type-reduction strategy for general type-2 fuzzy logic system,” Information Sciences, vol. 178, no. 9, pp. 2224–2236, 2008. View at Publisher · View at Google Scholar
  42. J. M. Mendel, F. Liu, and D. Zhai, “α-Plane representation for type-2 fuzzy sets: theory and applications,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 5, pp. 1189–1207, 2009. View at Publisher · View at Google Scholar · View at Scopus
  43. J. M. Mendel and D. R. Wu, Perceptual Computing: Aiding People in Making Subjective Judgments, Wiley and IEEE Press, 2010.
  44. A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007. View at Publisher · View at Google Scholar · View at Scopus
  45. Z. Kong, L. Gao, and L. Wang, “Comment on A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 540–542, 2009. View at Publisher · View at Google Scholar · View at Scopus
  46. F. Feng, Y. B. Jun, X. Liu, and L. Li, “An adjustable approach to fuzzy soft set based decision making,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 10–20, 2010. View at Publisher · View at Google Scholar
  47. Y. Jiang, Y. Tang, and Q. Chen, “An adjustable approach to intuitionistic fuzzy soft sets based decision making,” Applied Mathematical Modelling, vol. 35, no. 2, pp. 824–836, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH