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A new measure of divergence with its application to multi-criteria decision making under fuzzy environment

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An Erratum to this article was published on 07 October 2016

Abstract

Divergence measure is an important tool for determining the amount of discrimination between two probability distributions. Since the introduction of fuzzy sets, divergence measures between two fuzzy sets have gained attention for their applications in various fields. Exponential entropy measure has some advantages over Shannon’s entropy. In this paper, we used the idea of Jensen–Shannon divergence to define a new divergence measure called ‘fuzzy Jensen-exponential divergence (FJSD)’ for measuring the discrimination/difference between two fuzzy sets. The measure is demonstrated to satisfy some very elegant properties, which shows its strength for applications in multi-criteria decision-making problems. Further, we develop a method to solve multi-criteria decision-making problems under fuzzy phenomenon by utilizing the proposed measure and demonstrate by a numerical example.

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References

  1. Burbea J, Rao CR (1982) On the convexity of some divergence measures based on entropy functions. IEEE Trans Inf Theory 28(3):489–495

    Article  MathSciNet  MATH  Google Scholar 

  2. Endres DM, Schindelin JE (2003) A new metric for probability distributions. IEEE Trans Inf Theory 49(7):1858–1860

    Article  MathSciNet  MATH  Google Scholar 

  3. Kullback S (1968) Information theory and statistics. Dover Publication, New York

    MATH  Google Scholar 

  4. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86

    Article  MathSciNet  MATH  Google Scholar 

  5. Rao C, Nayak T (1985) Cross entropy, dissimilarity measures and characterizations of quadratic entropy. IEEE Trans Inf Theory 31(5):589–593

    Article  MathSciNet  MATH  Google Scholar 

  6. Rènyi A (1961) On measures of entropy and information. Proc Forth Berkeley Symp Math Stat Probab 1:547–561

    MathSciNet  MATH  Google Scholar 

  7. Sharma BD, Mittal DP (1977) New non-additive measures of relative information. J Comb Inf Syst Sci 2:122–133

    MathSciNet  MATH  Google Scholar 

  8. Kailath T (1967) The divergence and Bhattacharyya distance measures in signal selection. IEEE Trans Commun 15(1):52–60

    Article  Google Scholar 

  9. Lin J (1991) Divergence measure based on Shannon entropy. IEEE Trans Inf Theory 37(1):145–151

    Article  MathSciNet  MATH  Google Scholar 

  10. Angulo JC, Antolin J, Rosa SL, Esquivel RO (2010) Jensen-Shannon divergence in conjugated spaces: entropy excess of atomic systems and sets with respect to their constituents. Phys A Stat Mech Appl 389(4):899–907

    Article  Google Scholar 

  11. Ghosh M, Das D, Chakraborty C, Roy AK (2010) Automated lecukocyte recognition using fuzzy divergence. Micron 41(7):840–846

    Article  Google Scholar 

  12. Grosse I, Bernaola-Galvan P, Carpena P et al (2002) Analysis of symbolic sequences using the Jensen-Shannon divergence. Phys Rev E 65(4):041905-1–041905-16

    Article  MathSciNet  MATH  Google Scholar 

  13. Gómez-Lopera JF, Martínez-Aroza J, Robles-Pérez AM et al (2000) An analysis of edge detection by using the Jensen-Shannon divergence. J Math Imag Vis 13(1):35–56

    Article  MathSciNet  MATH  Google Scholar 

  14. He Y, Hamza AB, Krim H (2003) A generalized divergence measure for robust image registration. IEEE Trans Signal Process 51(5):1211–1220

    Article  MathSciNet  Google Scholar 

  15. Lopera JFG, Aroza JM, Perez AMR et al (2000) An analysis of edge detection by using the Jensen-Shannon divergence. J Math Imag Vis 13(1):35–56

    Article  MathSciNet  MATH  Google Scholar 

  16. Majtey AP, Borras A, Casas M (2008) Jensen-Shannon divergence as a measure of the degree of entanglement. Int J Quant Inf 6(1):715–720

    Article  Google Scholar 

  17. Molladavoudi S, Zainuddin H, Tim CK (2012) Jensen-Shannon divergence and non-linear quantum dynamics. Phys Lett A 376(26–27):1955–1961

    Article  MathSciNet  MATH  Google Scholar 

  18. Neagoe M, Popescu D, Niculescu V (2014) Applications of entropic divergence measures for DNA segmentation into high variable regions of cryptosporidium spp. gp60 gene. Rom Rep Phys 66(4):1078–1087

    Google Scholar 

  19. Naghshvar M, Javidi T, Wigger M (2015) Extrinsic Jensen-Shannon divergence: applications to variable-length coding. IEEE Trans Inf Theory 61(4):2148–2164

    Article  MathSciNet  MATH  Google Scholar 

  20. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  21. Bhandari D, Pal NR (1993) Some new information measure for fuzzy sets. Inf Sci 67(3):209–228

    Article  MathSciNet  MATH  Google Scholar 

  22. Hooda DS (2004) On generalized measures of fuzzy entropy. Math Slovaca 54(3):315–325

    MathSciNet  MATH  Google Scholar 

  23. Bajaj RK, Hooda DS (2010) On some new generalized measures of fuzzy information. World Acad Sci Eng Technol 4(2):657–663

    Google Scholar 

  24. Shang X, Jiang G (1997) A note on fuzzy information measures. Pattern Recogn Lett 18(5):425–432

    Article  Google Scholar 

  25. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423

    Article  MathSciNet  MATH  Google Scholar 

  26. Pal NR, Pal SK (1989) Object-background segmentation using new definitions of entropy. IEE Proc 136(4):284–295

    Google Scholar 

  27. Pal NR, Pal SK (1991) Entropy: a new definition and its applications. IEEE Trans Syst Man Cybern 5(21):1260–1270

    Article  MathSciNet  Google Scholar 

  28. Kvalseth TO (2000) On exponential entropies. IEEE Int Conf Syst Man Cybernet 4:2822–2826

    Google Scholar 

  29. Verma R, Sharma BD (2011) On generalized exponential fuzzy entropy. World Acad Sci Eng Technol 60:1402–1405

    Google Scholar 

  30. Fan J, Xie W (1999) Distance measures and induced fuzzy entropy. Fuzzy Sets Syst 104(2):305–314

    Article  MathSciNet  MATH  Google Scholar 

  31. Merigó JM, Gil-Lafuente AM, Yager RR (2015) An overview of fuzzy research with bibliometric indicators. Appl Soft Comput 27:420–433

    Article  Google Scholar 

  32. Hardy GH, Littlewood JE, Polya G (1951) Inequalities, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  33. Luca AD, Termini S (1972) A definition of non-probabilistic entropy in the setting of fuzzy sets theory. Inf Control 20(4):301–312

    Article  MATH  Google Scholar 

  34. Kaharaman C (2008) Fuzzy multi-criteria decision-making theory and applications with recent developments. Springer, New York

    Book  Google Scholar 

  35. Verma R, Sharma BD (2011) A measure of inaccuracy between two fuzzy sets. Cybern Inf Technol 11(2):13–23

    MathSciNet  Google Scholar 

  36. Verma R, Sharma BD (2014) A new inaccuracy measure for fuzzy sets and its application in multi-criteria decision making. Int J Intell Syst Appl 6(5):62–69

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Sciences, HMR Institute of Technology and Management, Guru Gobind Singh Indraprastha University, Hamidpur, Delhi, 110036, India

    Rajkumar Verma

  2. Department of Mathematics, Jaypee Institute of Information Technology, Noida, Uttar Pradesh, 201307, India

    Shikha Maheshwari

Authors
  1. Rajkumar Verma
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  2. Shikha Maheshwari
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Correspondence to Rajkumar Verma.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00521-016-2613-0.

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Verma, R., Maheshwari, S. A new measure of divergence with its application to multi-criteria decision making under fuzzy environment. Neural Comput & Applic 28, 2335–2350 (2017). https://doi.org/10.1007/s00521-016-2311-y

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