Advertisement

Multi-granulation interval-valued fuzzy probabilistic rough sets and their corresponding three-way decisions based on interval-valued fuzzy preference relations

  • Prasenjit Mandal
  • A. S. Ranadive
Original Paper

Abstract

In this paper, we study interval-valued fuzzy probabilistic rough sets (IVF-PRSs) based on multiple interval-valued fuzzy preference relations (IVFPRs) and consistency matrices, i.e., the multi-granulation interval-valued fuzzy preference relation probabilistic rough sets (MG-IVFPR-PRSs). First, in the proposed study, we convert IVFPRs into fuzzy preference relations (FPRs), and then construct the consistency matrix, the collective consistency matrix, the weighted collective preference relations, and the group collective preference relation (GCPR). Using this GCPR, four types of MG-IVFPR-PRSs are founded in terms of different constraints on parameter. Finally, to find a suitable way of explaining and determining these parameters in each model, three-way decisions are studied based on Bayesian minimum-risk procedure, i.e., the multi-granulation interval-valued fuzzy preference relation decision-theoretic rough set approach. An example is included to show the feasibility and potential of the theoretic results obtained.

Keywords

Interval-valued fuzzy probabilistic rough set Interval-valued fuzzy preference relation Multi-granulation Three-way decisions 

Notes

Acknowledgements

The authors would like to thank the Editor-in-Chief and reviewers for their thoughtful comments and valuable suggestions.

Compliance with ethical standards

Conflict of interest

Prasenjit Mandal and A. S. Ranadive declare that there is no conflict of interest.

Ethical approval

This article does not contain any study performed on humans or animals by the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

References

  1. Chen SM (1996) A fuzzy reasoning approach for rule-based systems based on fuzzy logics. IEEE Trans Systems Man Cybern Part B Cybern 26(5):769–778CrossRefGoogle Scholar
  2. Chen SM (1997) Interval-valued fuzzy hypergraph and fuzzy partition. IEEE Trans Systems Man Cybern Part B Cybern 27(4):725–733CrossRefGoogle Scholar
  3. Chen SJ, Chen SM (2008) Fuzzy risk analysis based on measures of similarity between interval-valued fuzzy numbers. Comput Math Appl 55:1670–1685MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen SM, Chen JH (2009) Fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators. Expert Syst Appl 36(3):6309–6317CrossRefGoogle Scholar
  5. Chen SM, Chen CD (2011) Handling forecasting problems based on high-order fuzzy logical relationships. Expert Syst Appl 38(4):3857–3864CrossRefGoogle Scholar
  6. Chen SM, Chien CY (2011) Parallelized genetic ant colony systems for solving the traveling salesman problem. Expert Syst Appl 38(4):3873–3883CrossRefGoogle Scholar
  7. Chen SM, Chung NY (2006) Forecasting enrollments of students using fuzzy time series and genetic algorithms. Int J Inf Manag Sci 17(3):1–17zbMATHGoogle Scholar
  8. Chen SM, Hong JA (2014) Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation of fuzzy sets. Inf Sci 286:63–74CrossRefGoogle Scholar
  9. Chen SM, Horng YJ (1996) Finding inheritance hierarchies in interval-valued fuzzy concept-networks. Fuzzy Sets Syst 84:75–83MathSciNetCrossRefzbMATHGoogle Scholar
  10. Chen SM, Hsiao WH (2000) Bidirectional approximate reasoning for rule-based systems using interval-valued fuzzy sets. Fuzzy Sets Syst 113:185–203MathSciNetCrossRefzbMATHGoogle Scholar
  11. Chen SM, Huang CM (2003) Generating weighted fuzzy rules from relational database systems for estimating values using genetic algorithms. IEEE Trans Fuzzy Syst 11(4):495–506CrossRefGoogle Scholar
  12. Chen SM, Kao PY (2013) Taiex forecasting based on fuzzy time series, particle swarm optimization techniques and support vector machines. Inf Sci 247:62–71MathSciNetCrossRefGoogle Scholar
  13. Chen SM, Lee LW (2010) Fuzzy multiple criteria hierarchical group decision-making based on interval type-2 fuzzy sets. IEEE Trans Syst Man Cybern Part A Syst Hum 40(5):1120–1128MathSciNetCrossRefGoogle Scholar
  14. Chen SM, Niou SJ (2011) Fuzzy multiple attributes group decision-making based on fuzzy preference relations. Expert Syst Appl 38(4):3865–3872CrossRefGoogle Scholar
  15. Chen SM, Sanguansat K (2011) Analyzing fuzzy risk based on similarity measures between interval-valued fuzzy numbers. Expert Syst Appl 38(7):8612–8621CrossRefGoogle Scholar
  16. Chen SM, Wang HY (2009) Evaluating students’ answerscripts based on interval-valued fuzzy grade sheets. Expert Syst Appl 36(6):9839–9846CrossRefGoogle Scholar
  17. Chen SM, Hsiao WH, Jong WT (1997) Bidirectional approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst 91:339–353MathSciNetCrossRefzbMATHGoogle Scholar
  18. Chen SM, Wang NY, Pan JS (2009) Forecasting enrollments using automatic clustering techniques and fuzzy logical relationships. Expert Syst Appl 36(8):11,070–11,076CrossRefGoogle Scholar
  19. Chen SM, Lin TE, Lee LW (2014) Group decision making using incomplete fuzzy preference relations based on the additive consistency and the order consistency. Inf Sci 259:1–15MathSciNetCrossRefzbMATHGoogle Scholar
  20. Chen SM, Cheng SH, Lin TE (2015) Group decision making systems using group recommendations based on interval fuzzy preference relations and consistency matrices. Inf Sci 298:555–567MathSciNetCrossRefzbMATHGoogle Scholar
  21. Du WS, Hu BQ (2016) Dominance-based rough set approach to incomplete ordered information systems. Inf Sci 346:106–129MathSciNetCrossRefGoogle Scholar
  22. Feng T, Mi JS (2016) Variable precision multigranulation decision-theoretic fuzzy rough sets. Knowl Based Syst 91:93–101CrossRefGoogle Scholar
  23. Horng YJ, Chen SM, Chang YC, Lee CH (2005) A new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques. IEEE Trans Fuzzy Syst 13(2):216–228CrossRefGoogle Scholar
  24. Lee LW, Chen SM (2008) Fuzzy multiple attributes group decision-making based on the extension of TOPSIS method and interval type-2 fuzzy sets. In: Proceedings of the 2008 international conference on machine learning and cybernetics, Kunming, China, vol 6, pp 3260–3265Google Scholar
  25. Lee LW (2012) Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency. Expert Syst Appl 39(14):11,666–11,676CrossRefGoogle Scholar
  26. Liang D, Liu D (2014) Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets. Inf Sci 276:186–203CrossRefGoogle Scholar
  27. Liang DC, Liu D (2015) Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets. Inf Sci 300:28–48MathSciNetCrossRefzbMATHGoogle Scholar
  28. Liang JY, Li R, Qian YH (2012) Distance: a more comprehensible perspective for measures in rough set theory. Knowl Based Syst 27:126–136CrossRefGoogle Scholar
  29. Lin G, Qian Y, Liang J (2012) Nmgrs: neighborhood-based multigranulation rough sets. Int J Approx Reason 53:1404–1418MathSciNetCrossRefzbMATHGoogle Scholar
  30. Liu C, Miao D, Qian J (2014) On multi-granulation covering rough sets. Int J Approx Reason 55:1404–1418MathSciNetCrossRefzbMATHGoogle Scholar
  31. Liu D, Liang DC, Wang CC (2016) A novel three-way decision model based on incomplete information system. Knowl Based Syst 91:32–45CrossRefGoogle Scholar
  32. Liu F, Peng YN, Yu Q, Zhao H (2018) A decision-making model based on interval additive reciprocal matrices with additive approximation-consistency. Inf Sci 422:161–176MathSciNetCrossRefGoogle Scholar
  33. Lv Y, Chen Q, Wu L (2013) Multi-granulation probabilistic rough set model. Int Conf Fuzzy Syst Knowl Discov 10:146–151Google Scholar
  34. Ma W, Sun B (2012) Probabilistic rough set over two universes and rough entropy. Int J Approx Reason 53:608–619MathSciNetCrossRefzbMATHGoogle Scholar
  35. Mandal P, Ranadive AS (2017) Multi-granulation bipolar-valued fuzzy probabilistic rough sets and their corresponding three-way decisions over two universes. Soft Comput.  https://doi.org/10.1007/s00500-017-2765-6 Google Scholar
  36. Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821CrossRefGoogle Scholar
  37. Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11:341–356CrossRefzbMATHGoogle Scholar
  38. Pawlak Z, Wong SKW, Ziarko W (1988) Rough sets: probabilistic versus deterministic approach. Int J Man Mach Stud 29:81–95CrossRefzbMATHGoogle Scholar
  39. Pedrycz W, Chen SM (2011) Granular computing and intelligent systems: design with information granules of high order and high type. Springer, HeidelbergCrossRefGoogle Scholar
  40. Pedrycz W, Chen SM (2015a) Granular computing and decision-making: interactive and iterative approaches. Springer, HeidelbergCrossRefGoogle Scholar
  41. Pedrycz W, Chen SM (2015b) Information granularity, big data, and computational intelligence. Springer, HeidelbergCrossRefGoogle Scholar
  42. Polkowski L (1996) Rough mereology: a new paradigm for approximate reasoning. Int J Approx Reason 15:333–365MathSciNetCrossRefzbMATHGoogle Scholar
  43. Qian Y, Liang J, Dang C (2010a) Incomplete multigranulation rough set. IEEE Trans Syst Man Cybern 20:420–431CrossRefGoogle Scholar
  44. Qian YH, Liang JY, Yao YY, Dang CY (2010b) MGRS: a multigranulation rough set. Inf Sci 180:949–970CrossRefzbMATHGoogle Scholar
  45. Qian YH, Zhang H, Sang YL, Liang JY (2014) Multigranulation decision-theoretic rough sets. Int J Approx Reason 55:225–237MathSciNetCrossRefzbMATHGoogle Scholar
  46. Sun B, Ma W, Zhao H (2014) Decision-theoretic rough fuzzy set model and application. Inf Sci 283:180–196MathSciNetCrossRefzbMATHGoogle Scholar
  47. Tanio T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets Syst 12:117–131MathSciNetCrossRefGoogle Scholar
  48. Tsai PW, Pan JS, Chen SM, Liao BY, Hao SP (2008) Parallel cat swarm optimization. In: Proceedings of the 2008 international conference on machine learning and cybernetics, Kunming, China, vol 6, pp 3328–3333Google Scholar
  49. Tsai PW, Pan JS, Chen SM, Liao BY (2012) Enhanced parallel cat swarm optimization based on the taguchi method. Expert Syst Appl 39(7):6309–6319CrossRefGoogle Scholar
  50. Wang JY, Xu LM (2002) Probabilistic rough set model. Comput Sci 29(8):76–78Google Scholar
  51. Wang ZH, Wang H, Feng QR, Shu L (2015) The approximation number function and the characterization of covering approximation space. Inf Sci 305:196–207MathSciNetCrossRefzbMATHGoogle Scholar
  52. Wei SH, Chen SM (2009) Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Syst Appl 16(2):2285–2299CrossRefGoogle Scholar
  53. Wei L, Zhang WX (2004) Probabilistic rough sets characterized by fuzzy sets. Int J Uncertain Fuzziness Knowl Based Syst 12:47–60MathSciNetCrossRefzbMATHGoogle Scholar
  54. Xu Z (2011) Consistency of interval fuzzy preference relations in group decision making. Appl Soft Comput 11(5):3898–3909CrossRefGoogle Scholar
  55. Xu W, Wang Q, Zhang X (2011) Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int J Gen Syst 13:246–259MathSciNetGoogle Scholar
  56. Xu W, Wang Q, Zhang X (2012) A generalized multi-granulation rough set approach. Lect Notes Bioinf 1(6840):681–689Google Scholar
  57. Xu W, Wang Q, Zhang X (2013) Multi-granulation rough sets based on tolerance relations. Soft Comput 17:1241–1252CrossRefzbMATHGoogle Scholar
  58. Xu W, Wang Q, Luo S (2014) Multi-granulation fuzzy rough sets. J Intell Fuzzy Syst 26:1323–1340MathSciNetzbMATHGoogle Scholar
  59. Yang HL, Liao X, Wang S, Wang J (2013) Fuzzy probabilistic rough set model on two universes and its applications. Int J Approx Reason 54:1410–1420MathSciNetCrossRefzbMATHGoogle Scholar
  60. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation. Inf Sci 111:239–259MathSciNetCrossRefzbMATHGoogle Scholar
  61. Yao YY (2003) Probabilistic approaches to rough sets. Expert Syst 20:287–297CrossRefGoogle Scholar
  62. Yao YY (2008) Probabilistic rough set approximations. Int J Approx Reason 49:255–271CrossRefzbMATHGoogle Scholar
  63. Yao YY (2009) Three-way decision: an interpretation of rules in rough set theory. Lect Notes Comput Sci 5589:642–649CrossRefGoogle Scholar
  64. Yao YY (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180:341–353MathSciNetCrossRefGoogle Scholar
  65. Yao YY, Wong SKM (1992) A decision theoretic framework for approximating concepts. Int J Man Mach Stud 37:793–809CrossRefGoogle Scholar
  66. Yao JT, Yao YY, Ziarko W (2008) Probabilistic rough sets: approximations, decision-makings, and applications. Int J Approx Reason 49:253–254CrossRefGoogle Scholar
  67. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–358CrossRefzbMATHGoogle Scholar
  68. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427MathSciNetCrossRefzbMATHGoogle Scholar
  69. Zhang HY, Yang SY, Ma JM (2016) Ranking interval sets based on inclusion measures and applications to three-way decisions. Knowl Based Syst 91:62–70CrossRefGoogle Scholar
  70. Zhang XX, Chen DG, Tsang ECC (2017) Generalized dominance rough set models for the dominance intuitionistic fuzzy information systems. Inf Sci 378:1–25MathSciNetCrossRefGoogle Scholar
  71. Zhao XR, Hu BQ (2015) Fuzzy and interval-valued decision-theoretic rough set approaches based on the fuzzy probability measure. Inf Sci 298:534–554MathSciNetCrossRefzbMATHGoogle Scholar
  72. Zhao XR, Hu BQ (2016) Fuzzy probabilistic rough sets and their corresponding three-way decisions. Knowl Based Syst 91:126–142CrossRefGoogle Scholar
  73. Ziarko W (2005) Probabilistic rough sets. Lect Notes Comput Sci 3641:283–293CrossRefzbMATHGoogle Scholar
  74. Ziarko W (2008) Probabilistic approach to rough sets. Int J Approx Reason 49:272–284MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bhalukdungri Jr. High SchoolPuruliaIndia
  2. 2.Department of Pure and Applied MathematicsGuru Ghasidas UniversityBilaspurIndia

Personalised recommendations