Skip to main content
SpringerLink
Log in

A Novel Kind of Arithmetic Operations on Trapezoidal Fuzzy Numbers and Its Applications to Optimize the Transportation Cost

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

The main objective of this paper is to define novel kind of operations on trapezoidal fuzzy numbers that is totally different from the existing operations. Significance of the proposed operations is that the transportation cost is even reduced while the same problem solved by existing operations. To validate this, two real-life examples are solved.

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.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data Availability

We do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

References

  1. Chanas, S., Kuchta, D.: A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst. 82(299), 299–305 (1996)

    Article  MathSciNet  Google Scholar 

  2. Chanas, S., Kołodziejczyk, W., Machaj, A.: A fuzzy approach to the transportation problem. Fuzzy Sets Syst. 13(3), 211–221 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, S.H.: Operations on fuzzy numbers with function principal. Tamkang J. Manag. Sci. 6, 13–25 (1985)

  4. Chen, S.J., Chen, S.M.: Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers. Appl. Intell. 26(1), 1–11 (2007)

    Article  Google Scholar 

  5. Chen, S.J., Hwang, C.L.: Fuzzy multiple attribute decision making methods. In: Fuzzy Multiple Attribute Decision Making, pp. 289–486. Springer, Berlin (1992).

  6. Dinagar, D.S., Palanivel, K.: The transportation problem in fuzzy environment. Int. J. Algorithms Comput. Math. 2(3), 65–71 (2009)

    MATH  Google Scholar 

  7. Dubois, D., Prade, H.: Operations on fuzzy numbers. International Journal of systems science 9(6), 613–626 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ebrahimnejad, A.: A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Applied Soft Computing 19, 171–176 (2014)

    Article  Google Scholar 

  9. Ganesan, K., Veeramani, P.: On arithmetic operations of interval numbers. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 13(06), 619–631 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ganesan, K., Veeramani, P.: Fuzzy linear programs with trapezoidal fuzzy numbers. Ann. Oper. Res. 143(1), 305–315 (2006)

    Article  MATH  Google Scholar 

  11. Gani, A.N., Razak, K.A.: Two stage fuzzy transportation problem. J. Phys. Sci. 10, 63–69 (2006)

  12. George, G., Maheswari, P.U., Ganesan, K.: A modified method to solve fuzzy transportation problem involving trapezoidal fuzzy numbers. In: AIP Conference Proceedings, vol. 2277, No. 1, p. 090005. AIP Publishing LLC, Melville (2020)

  13. Gupta, P., Mehlawat, M.K.: An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit. TOP 15(1), 114–137 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Heilpern, S.: The expected value of a fuzzy number. Fuzzy Sets Syst. 47(1), 81–86 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. Journal of mathematics and physics 20(1–4), 224–230 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaufmann, A., Gupta, M.M.: Introduction to Fuzzy Arithmetic: Theory and Applications. Van Nostrand Reinhold Co., New York (1985)

  17. Kaur, A., Kumar, A.: A new method for solving fuzzy transportation problems using ranking function. Appl. Math. Model. 35(12), 5652–5661 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liou, T.S., Wang, M.J.J.: Ranking fuzzy numbers with integral value. Fuzzy Sets Syst. 50(3), 247–255 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, S.T., Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153(3), 661–674 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. ÓhÉigeartaigh, M.: A fuzzy transportation algorithm. Fuzzy Sets Syst. 8(3), 235–243 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pandian, P., Natarajan, G.: A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4(2), 79–90 (2010)

    MATH  Google Scholar 

  22. Rezvani, S., Molani, M.: Representation of trapezoidal fuzzy numbers with shape function. Ann. Fuzzy Math. Inform 8(1), 89–112 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Saad, O.M., Abass, S.A.: A parametric study on transportation problem under fuzzy environment. J. Fuzzy Math. 11, 115–124 (2002)

  24. Zadeh, L.A.: Information and control. Fuzzy Sets 8(3), 338–353 (1965)

    Google Scholar 

  25. Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1(1), 45–55 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor of the paper and the referees for their precise remarks to improve the presentation of the paper.

Funding

I do not have sources of funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, K. Ramakrishnan College of Engineering, Tiruchirappalli, Tamilnadu, India

    M. Edwin Antony Raj

  2. Department of Mathematics, St. Joseph’s College (Autonomous), Tiruchirappalli, Tamilnadu, India

    Geetha Sivaraman & P. Vishnukumar

Authors
  1. M. Edwin Antony Raj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geetha Sivaraman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Vishnukumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to the writing of this manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to M. Edwin Antony Raj.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interest.

Human Participants and/or Animals

The research does not involve human participants and/or animals.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raj, M.E.A., Sivaraman, G. & Vishnukumar, P. A Novel Kind of Arithmetic Operations on Trapezoidal Fuzzy Numbers and Its Applications to Optimize the Transportation Cost. Int. J. Fuzzy Syst. 25, 1069–1076 (2023). https://doi.org/10.1007/s40815-022-01425-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-022-01425-6

Keywords

Search

Navigation