Information Measures-Based Multi-criteria Decision-Making Problems for Interval-Valued Intuitionistic Fuzzy Environment

  • Pratibha RaniEmail author
  • Divya Jain
Research Article


In the present communication, new entropy and divergence measures are developed for interval-valued intuitionistic fuzzy sets and compared it with the existing measures. Numerical result reveals that the proposed entropy measure attains the accurate classification, which illustrates their efficiency. Further, to cope with the multi-criteria decision-making problems with non-commensurable and conflicting criteria, an extended VIKOR method is developed under interval-valued intuitionistic environment. On the basis of proposed divergence measure, the particular measure of closeness of each alternative is calculated to the interval-valued intuitionistic fuzzy positive ideal solution. To illustrate the applicability of the proposed method, a multi-criteria decision-making problem of supplier selection is discussed under incomplete and uncertain information situation, which employs its advantages and feasibility.


Divergence measure Entropy Interval-valued intuitionistic fuzzy set MCDM VIKOR 



  1. 1.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–356zbMATHGoogle Scholar
  2. 2.
    Hooda DS, Mishra AR, Jain D (2014) On generalized fuzzy mean code word lengths. Am J Appl Math 2(4):127–134Google Scholar
  3. 3.
    Hooda DS, Mishra AR (2015) On trigonometric fuzzy information measures. ARPN J Sci Technol 5(3):145–152Google Scholar
  4. 4.
    Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96MathSciNetzbMATHGoogle Scholar
  5. 5.
    Atanassov KT, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349MathSciNetzbMATHGoogle Scholar
  6. 6.
    Wei G, Wang X, (2007) Some geometric aggregation operators based on interval-valued intuitionistic fuzzy sets and their application to group decision making. In: Proceedings of the IEEE international conference on computational intelligence and security, pp. 495–499Google Scholar
  7. 7.
    Meng F, Chen X (2016) Entropy and similarity measure for Atanassov’s interval-valued intuitionistic fuzzy sets and their application. Fuzzy Opt Decision Making 15(1):75–101zbMATHGoogle Scholar
  8. 8.
    Nayagam VLG, Jeevaraj S, Dhanasekaran P (2017) An intuitionistic fuzzy multi-criteria decision-making method based on non-hesitance score for interval-valued intuitionistic fuzzy sets. Soft Comput 21(23):7077–7082zbMATHGoogle Scholar
  9. 9.
    Mishra AR, Rani P (2018) Interval-valued intuitionistic fuzzy WASPAS method: application in reservoir flood control management policy. Group Decision Negotiation 27(6):1047–1078Google Scholar
  10. 10.
    Mishra AR, Rani P (2018) Biparametric information measures based TODIM technique for interval-valued intuitionistic fuzzy environment. Arabian J Sci Eng 43(6):3291–3309zbMATHGoogle Scholar
  11. 11.
    Zadeh LA (1969) Biological applications of the theory of fuzzy sets and systems. In: Proceeding of an international symposium on biocybernetics of the central nervous system, pp. 199–206Google Scholar
  12. 12.
    De Luca A, Termini S (1972) A definition of a non-probabilistic entropy in the setting of fuzzy sets theory. Inf Control 20(4):301–312zbMATHGoogle Scholar
  13. 13.
    Pal NR, Pal SK (1989) Object background segmentation using new definitions of entropy. IEEE Proceedings 136(4):284–295Google Scholar
  14. 14.
    Mishra AR, Hooda DS, Jain D (2014) Weighted trigonometric and hyperbolic fuzzy information measures and their applications in optimization principles. Int J Comput Math Sci 3(7):62–68Google Scholar
  15. 15.
    Mishra AR, Hooda DS, Jain D (2015) On exponential fuzzy measures of information and discrimination. Int J Comput Appl 119(23):1–7Google Scholar
  16. 16.
    Mishra AR, Jain D, Hooda DS (2016) On fuzzy distance and induced fuzzy information measures. J Inf Optim Sci 37(2):193–211MathSciNetGoogle Scholar
  17. 17.
    Mishra AR, Jain D, Hooda DS (2016) On logarithmic fuzzy measures of information and discrimination. J Optim Inf Sci 37(2):213–231MathSciNetGoogle Scholar
  18. 18.
    Burillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 78(3):305–316MathSciNetzbMATHGoogle Scholar
  19. 19.
    Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118(3):467–477MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hung WL, Yang MS (2006) Fuzzy entropy on intuitionistic fuzzy sets. Int J Intel Syst 21(4):443–451zbMATHGoogle Scholar
  21. 21.
    Mishra AR (2016) Intuitionistic fuzzy information with application in rating of township development. Ir J Fuzzy Syst 13(3):49–70MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mishra AR, Jain D, Hooda DS (2016) Intuitionistic fuzzy similarity and information measures with physical education teaching quality assessment. In: Proceeding of IC3T-2015, Springer—advances in intelligent systems and computing series-11156, 379, pp. 387–399Google Scholar
  23. 23.
    Ansari MD, Mishra AR, Ansari FT (2018) New divergence and entropy measures for intuitionistic fuzzy sets on edge detection. Int J Fuzzy Syst 20(2):474–487MathSciNetGoogle Scholar
  24. 24.
    Mishra AR, Jain D, Hooda DS (2017) Exponential intuitionistic fuzzy information measure with assessment of service quality. Int J Fuzzy Syst 19(3):788–798MathSciNetGoogle Scholar
  25. 25.
    Mishra AR, Rani P (2017) Shapley divergence measures with VIKOR method for multi-attribute decision-making problems. Neural Comput Appl. Google Scholar
  26. 26.
    Mishra AR, Rani P, Jain D (2017) Information measures based TOPSIS method for multicriteria decision making problem in intuitionistic fuzzy environment. Iran J Fuzzy Syst 14(6):41–63MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mishra AR, Kumari R, Sharma DK (2017) Intuitionistic fuzzy divergence measure-based multi-criteria decision-making method. Neural Comput Appl. Google Scholar
  28. 28.
    Mishra AR, Singh RK, Motwani D (2018) Multi-criteria assessment of cellular mobile telephone service providers using intuitionistic Fuzzy WASPAS method with similarity measures. Granul Comput. Google Scholar
  29. 29.
    Rani P, Jain D (2017) Intuitionistic fuzzy PROMETHEE technique for multi- criteria decision making problems based on entropy measure. Commun Comput Inf Sci (CCIS) 721:290–301Google Scholar
  30. 30.
    Rani P, Jain D, Hooda DS (2018) Extension of intuitionistic fuzzy TODIM technique for multi-criteria decision making method based on shapley weighted divergence measure Granul Comput
  31. 31.
    Liu XD, Zhang SH, Xiong FL, (2005), Entropy and Subsethood for General Interval-Valued Intuitionistic Fuzzy Sets, In Wang L, Jin Y (eds), Fuzzy Systems and Knowledge Discovery, Lecture Notes in Computer Science, 3613, 42-52Google Scholar
  32. 32.
    Chen Q, Xu ZS, Liu SS, Yu XH (2010) A method based on interval-valued intuitionistic fuzzy entropy for multiple attribute decision making. Int J Inf 13(1):67–77Google Scholar
  33. 33.
    Wei CP, Wang P, Zhang YZ (2011) Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications. Inf Sci 181(19):4273–4286MathSciNetzbMATHGoogle Scholar
  34. 34.
    Rani P, Jain D, Hooda DS (2018) Shapley function based interval-valued intuitionistic fuzzy VIKOR technique for correlative multi-criteria decision making problems. Iran J Fuzzy Syst 15(1):25–54MathSciNetzbMATHGoogle Scholar
  35. 35.
    Bhandari D, Pal NR (1993) Some new information measure for fuzzy sets. Inf Sci 67(3):209–228MathSciNetzbMATHGoogle Scholar
  36. 36.
    Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information-application to pattern recognition. Pattern Recogn Lett 28(2):197–206Google Scholar
  37. 37.
    Fan J, Xie W (1999) Distance measure and induced fuzzy entropy. Fuzzy Sets Syst 104(2):305–314MathSciNetzbMATHGoogle Scholar
  38. 38.
    Montes S, Couso I, Gil P, Bertoluzza C (2002) Divergence measure between fuzzy sets. Int J Approx Reason 30(2):91–105MathSciNetzbMATHGoogle Scholar
  39. 39.
    Montes I, Pal NR, Janis V, Montes S (2015) Divergence measures for intuitionistic fuzzy sets. IEEE Trans Fuzzy Syst 23(2):444–456Google Scholar
  40. 40.
    Kumari R, Mishra AR, Sharma DK, (2019), Intuitionistic Fuzzy Shapley-TOPSIS Method for Multi-Criteria Decision Making Problems based on Information Measures, Recent Patents on Computer Science,
  41. 41.
    Zhang QS, Jiang S, Jia B, Luo S (2010) Some information measures for interval-valued intuitionistic fuzzy sets. Inf Sci 180(24):5130–5145MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ye J (2011) Fuzzy cross entropy of interval-valued intuitionistic fuzzy sets and its optimal decision-making method based on the weights of alternatives. Expert Syst Appl 38(5):6179–6183Google Scholar
  43. 43.
    Meng F, Chen X (2015) Interval-valued intuitionistic fuzzy multi-criteria group decision making based on cross entropy and 2-additive measures. Soft Comput 19(7):2071–2082zbMATHGoogle Scholar
  44. 44.
    Qi X, Liang C, Zhang J (2015) Generalized cross-entropy based group decision making with unknown expert and attribute weights under interval-valued intuitionistic fuzzy environment. Comput Ind Eng 79:52–64Google Scholar
  45. 45.
    Mishra AR, Rani P, Pardasani KR (2018) Multiple-criteria decision-making for service quality selection based on Shapley COPRAS method under hesitant fuzzy sets. Granul Comput. Google Scholar
  46. 46.
    Liao H, Xu ZS, Zeng XJ (2015) Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making. IEEE Trans Fuzzy Syst 23(5):1343–1355Google Scholar
  47. 47.
    Opricovic S (1998) Multicriteria optimization of civil engineering systems. University of Belgrade, SerbiaGoogle Scholar
  48. 48.
    Opricovic S, Tzeng GH (2002) Multicriteria planning of post-earthquake sustainable reconstruction. Comput Aided Civil Infrastructure Eng 17(3):211–220Google Scholar
  49. 49.
    Opricovic S, Tzeng GH (2004) Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur J Oper Res 156(2):445–455zbMATHGoogle Scholar
  50. 50.
    Tzeng GH, Lin CW, Opricovic S (2005) Multi-criteria analysis of alternative-fuel buses for public transportation. Energy Policy 33(11):1373–1383Google Scholar
  51. 51.
    Opricovic S (2011) Fuzzy VIKOR with an application to water resources planning. Expert Syst Appl 38(10):12983–12990Google Scholar
  52. 52.
    Devi K (2011) Extension of VIKOR method in intuitionistic environment for robot selection. Expert Syst Appl 38(11):14163–14168Google Scholar
  53. 53.
    Liao H, Xu ZS (2013) A VIKOR-based method for hesitant multi-criteria decision making. Fuzzy Optim Decision Making 12(4):373–392MathSciNetzbMATHGoogle Scholar
  54. 54.
    Park JH, Cho HJ, Kwun YC (2011) Extension of the VIKOR method for group decision making with interval-valued intuitionistic fuzzy information. Fuzzy Optim Decision Making 10(3):233–253MathSciNetzbMATHGoogle Scholar
  55. 55.
    Dymova L, Sevastjanov P (2016) The operations on interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory. Inf Sci 360:256–272zbMATHGoogle Scholar
  56. 56.
    Izadikhah M (2012) Group decision making process for supplier selection with TOPSIS method under interval-valued intuitionistic fuzzy numbers. Adv Fuzzy Syst 2012 (Article ID 407942), 1–14Google Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of MathematicsMarwadi UniversityRajkotIndia
  2. 2.Department of MathematicsJaypee University of Engineering and TechnologyGunaIndia

Personalised recommendations