Skip to main content
SpringerLink
Log in

Multi-objective capacitated transportation problem with mixed constraint: a case study of certain and uncertain environment

  • Application Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

In this paper, we have formulated a new model of multi-objective capacitated transportation problem (MOCTP) with mixed constraints. In this model, some objective functions are linear and some are fractional and are of conflicting in nature with each other. The main objective of this paper is to decide the optimum order of the product quantity which is to be shipped from source to the destination subject to the capacitated restriction on each route. Here the two situations have been discussed for the MOCTP model. In the first situation, we have considered that all the input information of the MOCTP model is exactly known and therefore a fuzzy goal programming approach have been directly used for obtaining the optimum order quantity of the product. While in the second situation the input information of the MOCTP model are uncertain in nature and this uncertainty have been studied and handled by the suitable approaches like trapezoidal fuzzy numbers, multi-choices, and probabilistic random variables respectively. Due to the presence of all these uncertainties and conflicting natures of objectives functions, we cannot solve this MOCTP directly. Therefore firstly we converted all these uncertainties into deterministic forms by using the appropriate transformation techniques. For this, the vagueness in MOCTP defined by trapezoidal fuzzy numbers has been converted into its crisp form by using the ranking function approach. Multichoices in input information parameters have been converted into its exact form by the binary variable transformation technique. Randomness in input information is defined by the Pareto probability distribution, and for conversion into deterministic form chance constrained programming has been used. After doing all these transformations, we have applied fuzzy goal programming approach for solving this resultant MOCTP model for obtaining the optimum order quantity. A case study has been done to illustrate the computational procedure.

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

References

  1. Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20, 224–230 (1941)

    Article  Google Scholar 

  2. Koopmans, T.C., Reiter, S.: A model of transportation. In: Koopmans, T.J.C. (ed.) Activity Analysis of Production and Allocation—Proceedings of a Conference, pp. 222–259. Wiley, New York (1951)

    Google Scholar 

  3. Wagner, H.M.: On a class of capacitated transportation problems. Manag. Sci. 5(3), 304–318 (1959)

    Article  Google Scholar 

  4. Lohgaonkar, M.H., Bajaj, V.H.: Fuzzy approach to solve multi-objective capacitated transportation problem. Int. J. Bioinform. Res. 2, 10–14 (2010)

    Article  Google Scholar 

  5. Gupta, N., Bari, A.: Fuzzy multi-objective capacitated transportation problem with mixed constraints. J. Stat. Appl. Probab. 3(2), 1–9 (2014)

    Article  Google Scholar 

  6. Gupta, N., Ali, I., Bari, A.: A compromise solution for multi-objective chance constraint capacitated transportation problem. Probstat Forum 6, 60–67 (2013)

    Google Scholar 

  7. Pramanik, S., Banerjee, D.: Multi-objective chance constrained capacitated transportation problem based on fuzzy goal programming. Int. J. Comput. Appl. 44(20), 42–46 (2012)

    Google Scholar 

  8. Sadia, S., Gupta, N., Ali, Q.M.: Multiobjective capacitated fractional transportation problem with mixed constraints. Math. Sci. Lett. 5(3), 235–242 (2016)

    Article  Google Scholar 

  9. Bhargava, A.K., Singh, S.R., Bansal, D.: Multi-objective fuzzy chance constrained fuzzy goal programming for capacitated transportation problem. Int. J. Comput. Appl. 107(3), 18–23 (2014)

    Google Scholar 

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

    Article  Google Scholar 

  11. Ebrahimnejad, A.: An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers. J. Intell. Fuzzy Syst. 29(2), 963–974 (2015)

    Article  Google Scholar 

  12. Giri, P.K., Maiti, M.K., Maiti, M.: Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Comput. Ind. Eng. 72, 160–168 (2014)

    Article  Google Scholar 

  13. 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)

    Google Scholar 

  14. Chakraborty, A., Chakraborty, M.: Cost-time minimization in a transportation problem with fuzzy parameters: a case study. J. Transp. Syst. Eng. Inf. Technol. 10(6), 53–63 (2010)

    Google Scholar 

  15. Gupta, A., Kumar, A.: A new method for solving linear multi-objective transportation problems with fuzzy parameters. Appl. Math. Model. 36(4), 1421–1430 (2012)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Kundu, P., Kar, S., Maiti, M.: Multi-objective multi-item solid transportation problem in the fuzzy environment. Appl. Math. Model. 37(4), 2028–2038 (2013)

    Article  Google Scholar 

  19. Chakraborty, D., Jana, D.K., Roy, T.K.: Arithmetic operations on the generalized intuitionistic fuzzy number and its applications to the transportation problem. Opsearch 52(3), 431–471 (2015)

    Article  Google Scholar 

  20. Chang, C.T.: Multi-choice goal programming. Omega 35(4), 389–396 (2007)

    Article  Google Scholar 

  21. Chang, C.T.: Revised multi-choice goal programming. Appl. Math. Model. 32(12), 2587–2595 (2008)

    Article  Google Scholar 

  22. Acharya, S., Biswal, M.P.: Solving multi-choice multi-objective transportation problem. Int. J. Math. Oper. Res. 8(4), 509–527 (2016)

    Google Scholar 

  23. Biswal, M., Acharya, S.: Solving multi-choice linear programming problems by interpolating polynomials. Math. Comput. Model. 54(5), 1405–1412 (2011)

    Article  Google Scholar 

  24. Roy, S.K.: Multi-choice stochastic transportation problem involving Weibull distribution. Int. J. Oper. Res. 21(1), 38–58 (2014)

    Article  Google Scholar 

  25. Roy, S.K., Mahapatra, D.R., Biswal, M.P.: Multichoice stochastic transportation problem involving exponential distribution. J. Uncertain. Syst. 6(3), 200–213 (2012)

    Google Scholar 

  26. Mahapatra, D.R., Roy, S.K., Biswal, M.P.: Multichoice stochastic transportation problem involving extreme value distribution. Appl. Math. Model. 37(4), 2230–2240 (2013)

    Article  Google Scholar 

  27. Maity, G., Roy, S.R.: Solving multi-choice multi-objective transportation problem: a utility function approach. J. Uncertain. Anal. Appl. 2, 11 (2014)

    Article  Google Scholar 

  28. Maity, G., Kumar Roy, S.: Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand. Int. J. Manag. Sci. Eng. Manag. 11(1), 62–70 (2016)

    Google Scholar 

  29. Khalil, T.A., Raghav, Y.S., Badra, N.: Optimal solution of multi-choice mathematical programming problem using a new technique. Am. J. Oper. Res. 6, 167–172 (2016)

    Article  Google Scholar 

  30. Biswal, M.P., Acharya, S.: Multi-choice multi-objective linear programming problem. J. Interdiscip. Math. 12(5), 606–637 (2009)

    Article  Google Scholar 

  31. Biswas, A., Modak, N.: A Fuzzy Goal Programming Method for Solving Chance Constrained Programming with Fuzzy Parameters. In: Balasubramaniam, P. (ed.) Control, Computation and Information Systems. Communications in Computer and Information Science, vol. 140. Springer, Berlin, Heidelberg (2011)

    Google Scholar 

  32. Biswal, M.P., Samal, H.K.: Stochastic transportation problem with Cauchy random variables and multi-choice parameters. J. Phys. Sci. 17, 117–130 (2013)

    Google Scholar 

  33. Roy, S.K.J.: Transportation problem with multi-choice cost and demand and stochastic supply. J. Oper. Res. Soc. China (2016). https://doi.org/10.1007/s40305-016-0125-3

    Google Scholar 

  34. Barik, S.K.: Probabilistic fuzzy goal programming problems involving pareto distribution: some additive approaches. Fuzzy Inf. Eng. 7(2), 227–244 (2015)

    Article  Google Scholar 

  35. Abbasbandy, S., Hajjari, T.: A new approach for ranking of trapezoidal fuzzy numbers. Comput. Math. Appl. 57(3), 413–419 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  37. Lingo-User’s Guide: Lingo-user’s guide, Published by Lindo System Inc., 1415, North Dayton Street, Chicago, Illinois, 60622, USA (2016)

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, UP, India

    Srikant Gupta, Irfan Ali & Aquil Ahmed

Authors
  1. Srikant Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irfan Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aquil Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irfan Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gupta, S., Ali, I. & Ahmed, A. Multi-objective capacitated transportation problem with mixed constraint: a case study of certain and uncertain environment. OPSEARCH 55, 447–477 (2018). https://doi.org/10.1007/s12597-018-0330-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-018-0330-4

Keywords

Search

Navigation