Computational and Applied Mathematics

, Volume 37, Issue 3, pp 2943–2964 | Cite as

An (RS)-norm fuzzy information measure with its applications in multiple-attribute decision-making

  • Rajesh JoshiEmail author
  • Satish Kumar


In this paper, we introduce a quantity measure which is called (RS)-norm entropy and discuss some of its major properties with Shannon’s and other entropies in the literature. Based on this (RS)-norm entropy, we have proposed a new (RS)-norm fuzzy information measure and discussed its validity and properties. Further, we have given its comparison with other fuzzy information measures to prove its effectiveness. Attribute weights play an important role in multiple-attribute decision-making problems. In the present communication, two methods of determining the attribute weights are introduced. First is the case when the information regarding attribute weights is incompletely known or completely unknown and second is when we have partial information about attribute weights. For the first case, the extension of ordinary entropy weight method is used to calculate attribute weights and minimum entropy principle method based on solving a linear programming model is used in the second case. Finally, two methods are explained through numerical examples.


R-norm entropy Shannon’s entropy Convex and concave function (RS)-norm information measure (RS)-norm fuzzy information measure MADM 

Mathematics Subject Classification

94A15 94A24 26D15 



The authors are thankful to anonymous referees for their valuable comments and suggestions to improve this manuscript.


  1. Aczel J, Daroczy Z (1975) On measures of information and their characterization. Academic Press, New YorkzbMATHGoogle Scholar
  2. Boekee DE, Vander Lubbe JCA (1980) The R-norm information measure. Inf Control 45:136–155MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bhandari D, Pal NR (1993) Some new information measures for fuzzy sets. Inf Sci 67(3):204–228MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bloch I (2015) Fuzzy sets for image processing and understanding. Fuzzy Sets Syst 281:280–291MathSciNetCrossRefGoogle Scholar
  5. De Luca A, Termini S (1972) A definition of a non-probabilistic entropy in the setting of fuzzy sets theory. Inf Control 20:301–312CrossRefzbMATHGoogle Scholar
  6. De SK, Biswas R, Roy AR (2000) Some operations on intuitionistic fuzzy sets. Fuzzy Sets Syst 114:477–484MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ebanks BR (1983) On measure of fuzziness and their representation. J Math Anal Appl 94:301–312MathSciNetCrossRefGoogle Scholar
  8. Ekel PY (2002) Fuzzy sets and methods of decision making. Comput Math Appl 44(7):863–875MathSciNetCrossRefzbMATHGoogle Scholar
  9. Güneralpa B, Gertnera G, Mendozaa G, Anderson A (2007) Evaluating probabilistic data with a possibilistic criterian in land-restoration decision-making: effects on the precision of results. Fuzzy Sets Syst 158:1546–1560CrossRefGoogle Scholar
  10. Havdra JH, Charvat F (1967) Quantification method classification process: concept of structral \(\alpha \)-entropy. Kybernetika 3:30–35MathSciNetGoogle Scholar
  11. Hooda DS (2004) On generalized measures of fuzzy entropy. Math Slov 54:315–325MathSciNetzbMATHGoogle Scholar
  12. Hwang CH, Yang MS (2008) On entropy of fuzzy sets. Int J Uncertainty Fuzziness Knowl-Based Syst 16:519–527MathSciNetCrossRefzbMATHGoogle Scholar
  13. Higashi M, Klir GJ (1982) On measures of fuzziness and fuzzy complements. Int J Gen Syst 8:169–180MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hung WL, Yang MS (2006) Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 21:443–451CrossRefzbMATHGoogle Scholar
  15. Joshi R, Kumar S (2016a) \((R, S)\)-norm information measure and a relation between coding and questionnaire theory. Open Syst Inf Dyn 23(3):1–12MathSciNetCrossRefzbMATHGoogle Scholar
  16. Joshi R, Kumar S (2016b) A new approach in multiple attribute decision making using \(R\)-norm entropy and Hamming distance measure. Int J Inf Manag Sci 27(3):253–268Google Scholar
  17. Joshi R, Kumar S (2017a) A new intuitionistic fuzzy entropy of order-\(\alpha \) with applications in multiple attribute decision making. Adv Intell Syst Comput 546:212–219Google Scholar
  18. Joshi R, Kumar S (2017b) A new exponential fuzzy entropy of order \((\alpha, \beta )\) and its application in multiple attribute decision-making problems. Commun Math Stat 5:213–229MathSciNetCrossRefzbMATHGoogle Scholar
  19. Joshi R, Kumar S (2017c) Parametric \((R, S)\)-norm entropy on intuitionistic fuzzy sets with a new approach in multiple attribute decision making. Fuzzy Inf Eng 9:181–203MathSciNetGoogle Scholar
  20. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22:79–86MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kapur JN (1997) Measures of fuzzy information. Mathematical Science Trust Society, New DelhiGoogle Scholar
  22. Kaufman A (1980) Fuzzy subsets: fundamental theoretical elements, vol 3. Academic Press, New YorkGoogle Scholar
  23. Kosko B (1986) Fuzzy entropy and conditioning. Inf Sci 40(2):165–174MathSciNetCrossRefzbMATHGoogle Scholar
  24. Li P, Liu B (2008) Entropy of credibility distributions for fuzzy variables. IEEE Trans Fuzzy Syst 16:123–129CrossRefGoogle Scholar
  25. Liu M, Ren H (2014) A new intuitionistic fuzzy entropy and application in multi-attribute decision-making. Information 5:587–601CrossRefGoogle Scholar
  26. Pal NR, Pal SK (1989) Object background segmentation using new definition of entropy. Proc Inst Electron Eng 136:284–295Google Scholar
  27. Pal NR, Pal SR (1992) Higher order fuzzy entropy and hybrid entropy of a set. Inf Sci 61(3):211–231MathSciNetCrossRefzbMATHGoogle Scholar
  28. Pedrycz W (1997) Fuzzy sets in pattern recognition: accomplishments and challenges. Fuzzy Sets Syst 90(2):171–176MathSciNetCrossRefGoogle Scholar
  29. Renyi A (1961) On measures of entropy and information. In: Proceedings of the 4th Barkley symposium on Mathemtaical statistics and probability, vol 1. University of California Press, p 547Google Scholar
  30. Ramze R, Goedhart B, Lelieveldt BPF, Reiber JHC (1999) Fuzzy feature selection. Pattern Recognit 32(12):2011–2019CrossRefGoogle Scholar
  31. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(378–423):623–656MathSciNetCrossRefzbMATHGoogle Scholar
  32. Tsalli C (1988) Possible generalization of Boltzman–Gibbs statistics. J Stat Phys 52:480–487Google Scholar
  33. Torres A, Nieto JJ (2006) Fuzzy logic in medicines and bioinformatics. J Biomed Bio-Technol 2006:1–7CrossRefGoogle Scholar
  34. Taneja IJ (1975) A study of generalized measures of information theory. Ph.D. thesis, University of DelhiGoogle Scholar
  35. Verma R, Sharma BD (2011) On generalized exponential fuzzy entropy. World Acad Sci Eng Technol 60:1402–1405Google Scholar
  36. Verma R, Sharma BD (2014) On intuitionistic fuzzy entropy of order-\(\alpha \). Adv Fuzzy Syst 2014:1–8 (article ID 789890)MathSciNetCrossRefGoogle Scholar
  37. Wang J, Wang P (2012) Intutionistic linguistic fuzzy multi-criteria decision-making method based on intutionistic fuzzy entropy. Control Decis 27:1694–1698MathSciNetGoogle Scholar
  38. Xia M, Xu Z (2012) Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Inf Fusion 13:31–47CrossRefGoogle Scholar
  39. Yager RR (1979) On the measure of fuzziness and negation. Part 1: membership in the unit interval. Int J Gen Syst 5:221–229CrossRefzbMATHGoogle Scholar
  40. Zadeh LA (1965) Fuzzy sets. Inf Control 8:221–229CrossRefGoogle Scholar
  41. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Department of MathematicsMaharishi Markandeshwar UniversityMullana-AmbalaIndia

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