Skip to main content
SpringerLink
Log in

Arithmetic Operations on Generalized Parabolic Fuzzy Numbers and Its Application

  • Research Article
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we have studied the basic arithmetic operations for two generalized positive parabolic fuzzy numbers by using the concept of the distribution and complementary distribution functions. The major advantage of these operations is that they do not need the computation of \(\alpha \)-cut of the fuzzy number and hence it becomes more powerful where the standard method i.e., \(\alpha \)-cuts method fails. Based on these operations, some elementary applications on mensuration have been illustrated and compared their results with generalized triangular fuzzy numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Kaufmann A, Gupta MM (1991) Introduction to fuzzy arithmetic. Van Nostrand, New York

    MATH  Google Scholar 

  3. Dubois D, Prade H (1988) Possibility theory. Plenum Press, Berlin

    Book  MATH  Google Scholar 

  4. Klir GJ (1997) Fuzzy arithmetic with requistic constraints. Fuzzy Sets Syst 91:165–175

    Article  MATH  Google Scholar 

  5. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(28):3–28

    Article  MathSciNet  MATH  Google Scholar 

  6. Piegat A (2005) A new definion of the fuzzy set. Int J Appl Math Comput Sci 15(1):125–140

    MathSciNet  MATH  Google Scholar 

  7. Stefanini L, Guerra ML (2006) On fuzzy arithmetic operations: some properties and distributive approximations. Int J Appl Math 19:171–199

    MathSciNet  MATH  Google Scholar 

  8. Gao S, Zhang Z, Cao C (2009) Multiplication operation on fuzzy numbers. J Softw 4(4):331–338

    Article  Google Scholar 

  9. Akther T, Ahmad SU (2009) A computational method for fuzzy arithmetic operations. Daffodil Int Univ J Sci Technol 4(1):18–22

    Google Scholar 

  10. Mahanta S, Chutia R, Baruah HK (2010) Fuzzy arithmetic witwith using the method of \(\alpha \)-cuts. Int J Latest Trends Comput 1(2):73–80

    Google Scholar 

  11. Taleshian A, Rezvani S (2011) Multiplication operation on trapezoidal fuzzy numbers. J Phys Sci 11:17–26

    MathSciNet  MATH  Google Scholar 

  12. Chutia R, Mahanta S, Datta D (2011) Arithmetic of triangular fuzzy variable from credibility theory. Int J Energy Inf Commun 2(3):9–20

    Google Scholar 

  13. Bansal A (2011) Trapezoidal fuzzy numbers (a,b,c,d): arithmetic behavior. Int J Phys Math Sci 2:39–44

    Google Scholar 

  14. Oussalah M (2002) On the compatibility between defuzzification and fuzzy arithmetic operations. Fuzzy Sets Syst 128(2):247–260

    Article  MathSciNet  MATH  Google Scholar 

  15. Kechagias P, Papadopoulos BK (2007) Computational method to evaluate fuzzy arithmetic operations. Appl Math Comput 185:169–177

    MathSciNet  MATH  Google Scholar 

  16. Deschrijver G (2007) Arithmetic operators in interval-valued fuzzy set theory. Inf Sci 177:2906–2924

    Article  MathSciNet  MATH  Google Scholar 

  17. Xue F, Tang W, Zhao R (2008) The expected value of a function of a fuzzy variable with a continuous membership function. Comput Math Appl 55:1215–1224

    Article  MathSciNet  MATH  Google Scholar 

  18. Banerjee S, Roy TK (2012) Arithmetic operations on generalized trapzoidal fuzzy number and its applications. Turk J Fuzzy Syst 3(1):16–44

    Google Scholar 

  19. Garg H (2014) A novel approach for analyzing the behavior of industrial systems using weakest t-norm and intuitionistic fuzzy set theory. ISA Trans 53:1199–1208

    Article  Google Scholar 

  20. Vahidi J, Rezvani S (2013) Arithmetic operations on trapezoidal fuzzy numbers. J Nonlinear Anal Appl, Article ID jnaa–00,111, 8

  21. Garg H (2016) A novel approach for analyzing the reliability of series-parallel system using credibility theory and different types of intuitionistic fuzzy numbers. J Braz Soc Mech Sci Eng 38(3):1021–1035

    Article  Google Scholar 

  22. Ross TJ (2004) Fuzzy logic with engineering applications, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  23. Liu B, Liu YK (2002) Expected value of fuzzy variable and fuzzy expected value model. IEEE Trans Fuzzy Syst 10(4):445–450

    Article  Google Scholar 

  24. Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1:97–110

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Thapar University Patiala, Patiala, 147004, India

    Harish Garg &  Ansha

Authors
  1. Harish Garg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ansha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harish Garg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garg, H., Ansha Arithmetic Operations on Generalized Parabolic Fuzzy Numbers and Its Application. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 88, 15–26 (2018). https://doi.org/10.1007/s40010-016-0278-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-016-0278-9

Keywords

Search

Navigation