Advertisement

A novel possibility measure to interval-valued intuitionistic fuzzy set using connection number of set pair analysis and its applications

  • Harish GargEmail author
  • Kamal Kumar
Original Article

Abstract

The aim of this paper is to give a multi-attribute decision-making (MADM) method for the interval-valued intuitionistic fuzzy (IVIF) set using the set pair analysis (SPA) theory. IVIF set can express the uncertain information in a more valuable manner, while the connection number (CN) based on the “identity,” “discrepancy” and “contrary” degrees of the SPA theory handles the uncertainties and certainties systems. Based on these features, in this paper, we develop a new possibility degree measure based on the CNs to rank the different IVIF numbers. The properties and advantages of the measure are described in details. Later on, we present a novel MADM method and illustrate with several examples to validate it. A comparative as well as superiority analysis is performed to show the feasibility and validity of the approach.

Keywords

Interval-valued intuitionistic fuzzy set Set pair analysis Connection number Multi-attribute decision-making Possibility measures 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353zbMATHCrossRefGoogle Scholar
  2. 2.
    Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96zbMATHCrossRefGoogle Scholar
  3. 3.
    Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Garg H, Kumar K (2019) Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision-making problems. IEEE Trans Fuzzy Syst.  https://doi.org/10.1109/TFUZZ.2019.2897961 CrossRefGoogle Scholar
  5. 5.
    Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187CrossRefGoogle Scholar
  6. 6.
    Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Garg H (2016) Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making. Comput Ind Eng 101:53–69CrossRefGoogle Scholar
  8. 8.
    Garg H (2017) Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application. Eng Appl Arti Intel 60:164–174CrossRefGoogle Scholar
  9. 9.
    Xu Z, Chen J (2007) On geometric aggregation over interval-valued intuitionistic fuzzy information. In: Fourth international conference on fuzzy systems and knowledge discovery, FSKD 2007, vol 2, pp 466–471Google Scholar
  10. 10.
    Garg H (2018) Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision -making process. J Ind Manag Optim 14(1):283–308MathSciNetzbMATHGoogle Scholar
  11. 11.
    Arora R, Garg H (2018) Prioritized averaging/geometric aggregation operators under the intuitionistic fuzzy soft set environment. Sci Iran 25(1):466–482Google Scholar
  12. 12.
    Chen TY (2014) Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis. Inf Sci 261:149–169MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Garg H, Kumar K (2018) An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making. Soft Comput 22(15):4959–4970zbMATHCrossRefGoogle Scholar
  14. 14.
    Kaur G, Garg H (2018) Multi-attribute decision-making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment. Entropy 20(1):65.  https://doi.org/10.3390/e20010065 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kumar K, Garg H (2018) Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making. Appl Intel 48(8):2112–2119CrossRefGoogle Scholar
  16. 16.
    Park JH, Park IY, Kwun YC, Tan X (2011) Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy sets. Appl Math Model 35(5):2544–2556MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bai ZY (2013) An interval-valued intuitionistic fuzzy TOPSIS method based on an improved score function. Sci World J 2013: 879089 (6 pages)Google Scholar
  18. 18.
    Garg H (2016) A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Appl Soft Comput 38:988–999CrossRefGoogle Scholar
  19. 19.
    Sahin R (2016) Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets. Soft Comput 20(7):2557–2563zbMATHCrossRefGoogle Scholar
  20. 20.
    Xu ZS, Da QL (2003) Possibility degree method for ranking interval numbers and its application. J Syst Eng 18:67–70Google Scholar
  21. 21.
    Wei CP, Tang X (2010) Possibility degree method for ranking intuitionistic fuzzy numbers. In: 3rd IEEE/WIC/ACM international conference on web intelligence and intelligent agent technology (WI-IAT ’10), pp 142–145Google Scholar
  22. 22.
    Zhang X, Yue G, Teng Z (2009) Possibility degree of interval-valued intuitionistic fuzzy numbers and its application. In: Proceedings of the international symposium on information processing, pp 33 –36Google Scholar
  23. 23.
    Wan SP, Dong J (2014) A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J Comput Syst Sci 80(1):237–256MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Garg H, Kumar K (2019) Improved possibility degree method for ranking intuitionistic fuzzy numbers and their application in multiattribute decision-making. Granul Comput 4(2):237–247CrossRefGoogle Scholar
  25. 25.
    Dammak F, Baccour L, Alimi AM (2016) An exhaustive study of possibility measures of interval-valued intuitionistic fuzzy sets and application to multicriteria decision making. Adv Fuzzy Syst 2016, 9185706 (10 pages)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Gao F (2013) Possibility degree and comprehensive priority of interval numbers. Syst Eng Theory Pract 33(8):2033–2040Google Scholar
  27. 27.
    Garg H, Kumar K (2018) Group decision making approach based on possibility degree measures and the linguistic intuitionistic fuzzy aggregation operators using Einstein norm operations. J Mult Valued Log Soft Comput 31(1/2):175–209MathSciNetGoogle Scholar
  28. 28.
    Shui XZ, Li DQ (2003) A possibility based method for priorities of interval judgment matrix. Chin J Manag Sci 11(1):63–65Google Scholar
  29. 29.
    Chen SM, Cheng SH, Lan TC (2016) Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf Sci 367–368:279–295CrossRefGoogle Scholar
  30. 30.
    Sivaraman G, Nayagam VLG, Ponalagusamy R (2013) Multi-criteria interval valued intuitionistic fuzzy decision making using a new score function. In: KIM 2013 knowledge and information management conference, pp 122–131Google Scholar
  31. 31.
    Zhao K (1989) Set pair and set pair analysis-a new concept and systematic analysis method. In: Proceedings of the national conference on system theory and regional planning, pp 87–91Google Scholar
  32. 32.
    Jiang YL, Xu CF, Yao Y, Zhao KQ (2004) Systems information in set pair analysis and its applications. In: Proceedings of 2004 international conference on machine learning and cybernetics, vol 3, pp 1717–1722Google Scholar
  33. 33.
    ChangJian W (2007) Application of the set pair analysis theory in multiple attribute decision-making. J Mech Strength 6(029):1009–1012Google Scholar
  34. 34.
    Garg H, Kumar K (2019) An advanced study on operations of connection number based on set pair analysis. Nat Acad Sci Lett.  https://doi.org/10.1007/s40009-018-0748-5 MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lü WS, Zhang B (2012) Set pair analysis method of containing target constraint mixed interval multi-attribute decision-making. Appl Mech Mater 226:2222–2226CrossRefGoogle Scholar
  36. 36.
    Xie Z, Zhang F, Cheng J, Li L (2013) Fuzzy multi-attribute decision making methods based on improved set pair analysis. Sixth Int Symp Comput Intel Des 2:386–389Google Scholar
  37. 37.
    Cao YX, Zhou H, Wang JQ (2018) An approach to interval-valued intuitionistic stochastic multi-criteria decision-making using set pair analysis. Int J Mach Learn Cybern 9(4):629–640CrossRefGoogle Scholar
  38. 38.
    Kumar K, Garg H (2018) TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. Comput Appl Math 37(2):1319–1329MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Fu S, Zhou H (2016) Triangular fuzzy number multi-attribute decision-making method based on set-pair analysis. J Softw Eng 6(4):52–58MathSciNetGoogle Scholar
  40. 40.
    Wang JQ, Gong L (2009) Interval probability stochastic multi-criteria decision-making approach based on set pair analysis. Control Decis 24:1877–1880Google Scholar
  41. 41.
    Garg H, Kumar K (2018) Distance measures for connection number sets based on set pair analysis and its applications to decision making process. Appl Intel 48(10):3346–3359CrossRefGoogle Scholar
  42. 42.
    Garg H, Kumar K (2018) A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory. Artif Intel Rev.  https://doi.org/10.1007/s10462-018-9668-5 CrossRefGoogle Scholar
  43. 43.
    Hu J, Yang L (2011) Dynamic stochastic multi-criteria decision making method based on cumulative prospect theory and set pair analysis. Syst Eng Proc 1:432–439CrossRefGoogle Scholar
  44. 44.
    Garg H, Kumar K (2018) A novel correlation coefficient of intuitionistic fuzzy sets based on the connection number of set pair analysis and its application. Sci Iran E 25(4):2373–2388Google Scholar
  45. 45.
    Garg H, Kumar K (2018) Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis. Arab J Sci Eng 43(6):3213–3227zbMATHCrossRefGoogle Scholar
  46. 46.
    Chaokai H, Meng W (2015) A new reputation model for p2p network based on set pair analysis. Open Cybern Syst J 9:1393–1398CrossRefGoogle Scholar
  47. 47.
    Garg H (2016) Some series of intuitionistic fuzzy interactive averaging aggregation operators. Springer Plus 5(1):999.  https://doi.org/10.1186/s40064-016-2591-9 CrossRefGoogle Scholar
  48. 48.
    Shen F, Ma X, Li Z, Xu ZS, Cai D (2018) An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation. Inf Sci 428:105–119MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wang CY, Chen SM (2017) Multiple attribute decision making based on interval-valued intuitionistic fuzzy sets, linear programming methodology, and the extended TOPSIS method. Inf Sci 397:155–167CrossRefGoogle Scholar
  50. 50.
    Arora R, Garg H (2019) Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties. Comput Appl Math 38(2):1–36.  https://doi.org/10.1007/s40314-019-0764-1 MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Nancy, Garg H (2019) A novel divergence measure and its based TOPSIS method for multi criteria decision-making under single-valued neutrosophic environment. J Intel Fuzzy Syst 36(1):101–115CrossRefGoogle Scholar
  52. 52.
    Rani D, Garg H (2019) Some modified results of the subtraction and division operations on interval neutrosophic sets. J Exp Theor Artif Intel.  https://doi.org/10.1080/0952813X.2019.1592236 CrossRefGoogle Scholar
  53. 53.
    Singh S, Garg H (2018) Symmetric triangular interval type-2 intuitionistic fuzzy sets with their applications in multi criteria decision making. Symmetry 10(9):401.  https://doi.org/10.3390/sym10090401 CrossRefzbMATHGoogle Scholar
  54. 54.
    Garg H, Rani D (2019) Some results on information measures for complex intuitionistic fuzzy sets. Int J Intel Syst.  https://doi.org/10.1002/int.22127 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsThapar Institute of Engineering and Technology, Deemed UniversityPatialaIndia

Personalised recommendations