**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32586

##### A Method for Solving a Bi-Objective Transportation Problem under Fuzzy Environment

**Authors:**
Sukhveer Singh,
Sandeep Singh

**Abstract:**

A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost and fuzzy time of transportation without according priorities to them is considered. To the best of our knowledge, there is no method in the literature to find efficient solutions of the bi-objective transportation problem under uncertainty. In this paper, a bi-objective transportation problem in an uncertain environment has been formulated. An algorithm has been proposed to find efficient solutions of the bi-objective transportation problem under uncertainty. The proposed algorithm avoids the degeneracy and gives the optimal solution faster than other existing algorithms for the given uncertain transportation problem.

**Keywords:**
Transportation problem,
efficient solution,
ranking function,
fuzzy transportation problem.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1131343

**References:**

[1] E. E. Ammar and E. A. Youness, “Study on multiobjective transportation problem with fuzzy numbers.” Appl. Math. Comput., Vol. 166, pp. 241–253, 2005.

[2] R. E. Bellmannand and L. A. Zadeh, “Decision making in fuzzy environment,” Management sciences, Vol. 17, pp. 141-164, 1970.

[3] A. K. Bit, “Fuzzy programming with Hyperbolic membership functions for Multi-objective capacitated transportation problem, OPSEARCH, Vol. 41, pp. 106-120, 2004.

[4] A. K. Bit, M.P. Biswal and S.S. Alam, “Fuzzy programming approach to multicriteria decision making transportation problem,” Fuzzy sets and systems, Vol. 50, pp. 35-41, 1992.

[5] S. Chanas, W. Kolodziejckzy and Machaj, “A fuzzy approach to the transportation problem,” Fuzzy Sets and Systems, Vol. 13, pp. 211-221, 1984.

[6] S. K. Das, A. Goswami and S. S Alam, “Multiobjective transportation problem with interval cost, source and destination parameters.” Eur. J. Oper. Res., Vol. 117, pp. 100–112, 1999.

[7] S. C. Fang, C. F. Hu., H. F. Wang and S. Y. Wu., “Linear programming with fuzzy coefficients in constraints”, Computers and mathematics with applications, Vol. 37, pp. 63-76, 1999.

[8] P. Gupta and M. K. Mehlawat, “An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit,” Top., Vol. 15, pp. 114–137, 2007.

[9] A. Kaufmann and M. M. Gupta, “Introduction to fuzzy arithmetics, theory and applications,” Van Nostrand Reinhold, New York, 1991.

[10] Y. J. Lai and C. L. Hawng, Fuzzy Mathematical Programming, Lecture notes in Economics and Mathematical systems, Springer-Verlag, 1992.

[11] T. S. Liou and M. J. Wang, “Ranking fuzzy numbers with integral values,” Fuzzy sets and systems, Vol. 50, pp. 247-255, 1992.

[12] H. M. Nehi, H. R. Maleki and M. Mashinchi, “Solving fuzzy number linear programming problem by lexicographic ranking function,” Italian journal of pure and applied mathematics, Vol. 16, pp. 9-20, 2004.

[13] A. A. Noora and P. Karami, “Ranking functions and its applications to fuzzy DEA,” International mathematical forum, Vol. 3, pp. 1469-1480, 2008.

[14] S. Pramanik, T.K. Roy, “Multiobjective transportation model with fuzzy parameters: priority based fuzzy goal programming approach.” J. Transp. Syst. Eng. Inform. Technol., Vol. 8, pp. 40–48, 2008.

[15] S. Prakash, “Transportation problem with objectives to minimizes the total cost and duration of transportation”, OPSEARCH, Vol. 18, pp. 235-238, 1981.

[16] S. Prakash, A. K. Agarwal and S. Shah, “Non-dominated solutions of cost–time trade-off transportation and assignment problems”, OPSEARCH, Vol. 25, pp. 126–131, 1988.

[17] S. Purushotam, S. Prakash and P. Dhyani, “A transportation problem with minimization of duration and total cost of transportation as high and low priority objectives respectively”, Bulletin of the technical university of Istanbul, Vol. 37, pp. 1-11, 1984.

[18] H. Rommelfanger, J. Wolf and R. Hanuscheck, “Linear programming with fuzzy coefficients, Fuzzy sets and systems”, Vol. 29, pp. 195-206, 1989.

[19] C. R. Seshan and V. G. Tikekar, “On Sharma-Sawrup algorithm for time minimizing transportation problems”, Proceeding of the Indian Academy of Sciences, Mathematical Sciences, Vol. 89, pp. 101-102, 1980.

[20] J. K. Sharma and K. Sawrup, “Bi-level time minimizing transportation problem”, Discrete optimization, Vol. 5, pp. 714–723, 1977.

[21] Sonia and M. C. Puri, “Two level hierarchical time minimizing transportation problem”, TOP., Vol. 12, pp. 301-330, 2004.

[22] Sonia, A. Khandelwal and M. C. Puri, “Bi-level time minimizing transportation problem”, Discrete optimization, Vol. 5, pp. 714-723, 2008.

[23] H. Tanaka, K. Asai, “Fuzzy linear programming problems with fuzzy numbers”, Fuzzy Sets and Systems, Vol. 13, pp. 1-10, 1984.

[24] H. Tanaka, Ichihashi and K. Asai, “A formulation of fuzzy linear programming based on comparison of fuzzy numbers”, Control and cybernetics, Vol. 13, pp. 185-194, 1984.

[25] L. A. Zadeh, “Fuzzy sets”, Information and Control, Vol. 8, pp. 338-353, 1965.

[26] H. J. Zimmermann, “Fuzzy programming and linear programming with several objective functions”, Fuzzy sets and System, Vol. 1, pp. 45- 55, 1978.