Abstract
In this paper, a new method is proposed to solve fully fuzzy transportation problems using the approach of the Hungarian and MODI algorithm. The objective of the proposed algorithm, namely, fuzzy Hungarian MODI algorithm, is to obtain the solution of fully fuzzy transportation problems involving triangular and trapezoidal fuzzy numbers. The introduced method together with Yager’s ranking technique gives the optimal solution of the problem. It also satisfies the conditions of optimality, feasibility, and positive allocation of cells using the elementwise subtraction of fuzzy numbers. A comparative study of the proposed method with existing procedure reveals that the solution of the proposed method satisfies the necessary conditions of a Transportation Problem (TP) to be an optimal solution in which the other methods do not guarantee. The proposed method is the extension of the Hungarian MODI method with fuzzy values. It is easy to understand and implement, as it follows the standard steps of the regular transportation problems. The method can be extended to other kinds of fuzzy transportation problems, such as unbalanced fuzzy TP, fuzzy degeneracy problem, fuzzy TP with prohibited routes, and many more.
Similar content being viewed by others
References
Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20, 224–230 (1941)
Dantzig, G.B., Thapa, M.N.: Springer: Linear Programming: 2: Theory and Extensions. Princeton University Press, New Jersey (1963)
Charnes, A., Cooper, W.W.: The stepping-stone method for explaining linear programming calculation in transportation problem. Manag. Sci. 1, 49–69 (1954)
Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, 141–164 (1970)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Chanas, S., Kolodziejczyk, W., Machaj, A.: A fuzzy approach to the transportation problem. Fuzzy Sets Syst. 13, 211–224 (1984)
Chanas, S., Kuchta, D.: A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst. 82, 299–305 (1996)
Liu, S.T., Chiang Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153, 661–674 (2004)
Verma, R., Biswal, M.P., Biswas, A.: Fuzzy programming technique to solve multi objective transportation problems with some non-linear membership functions. Fuzzy Sets Syst. 91, 37–43 (1997)
Liang, T.F., Chiu, C.S., Cheng, H.W.: Using possibilistic linear programming for fuzzy transportation planning decisions. Hsiuping J. 11, 93–112 (2005)
Liang, T.F.: Interactive multi objective transportation planning decisions using fuzzy linear programming. Asia-Pac. J. Op. Res. 25(01), 11–31 (2008)
Nagoor Gani, A., Abdul Razak, K.: Two stage fuzzy transportation problem. J. Phys. Sci. 10, 63–69 (2006)
Pandian, P., Natarajan, G.: A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4, 79–90 (2010)
Kaur, A., Kumar, A.: A new method for solving fuzzy transportation problems using ranking function. Appl. Math. Modell. 35, 5652–5661 (2011)
Yuan, Y.: Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst. 43(2), 139–157 (1991)
Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities-I. Fuzzy Sets Syst. 118(3), 375–385 (2001)
Chien, C.F., Chen, J.H., Wei, C.C.: Constructing a comprehensive modular fuzzy ranking framework and Illustration. J. Qual. 18(4), 333–349 (2011)
Yager, R.R.: A characterization of the extension principle. Fuzzy Sets Syst. 18, 205–217 (1986)
Kahraman, C., Cevik Onar, S., Oztaysi, B.: Fuzzy multicriteria decision-making: a literature review. Int. J. Comput. Intell. Syst. 8(4), 637–666 (2015)
Dhanasekar, S., Harikumar, K., Sattanathan, R.: A new approach for solving fuzzy assignment problems. J. Ultrascientist Phys. Sci. 24(A), 111–116 (2012)
Dhanasekar, S., Hariharan, S., Sekar, P.: Classical travelling salesman problem (TSP) based approach to solve fuzzy TSP using Yagers ranking. Int. J. Comput. Appl. 74, 1–4 (2013)
Dhanasekar, S., Hariharan, S., Sekar, P.: Classical replacement problem based approach to solve fuzzy replacement problem. Int. J. Appl. Eng. Res. 9(26), 9382–9385 (2014)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic-Theory and Applications. Prentice Hall, New York (1995)
Acknowledgments
The authors are thankful to the Editor-in-Chief Shun-Feng Su and the anonymous reviewers for their valuable comments and suggestions which have led to an improvement in both the quality and the clarity of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dhanasekar, S., Hariharan, S. & Sekar, P. Fuzzy Hungarian MODI Algorithm to Solve Fully Fuzzy Transportation Problems. Int. J. Fuzzy Syst. 19, 1479–1491 (2017). https://doi.org/10.1007/s40815-016-0251-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40815-016-0251-4