International Journal of Nonlinear Analysis and Applications

Solving linear fractional transportation problem with interval cost‎, ‎source and destination parameters

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Baghmalek Branch‎, ‎Islamic Azad University‎, Baghmalek‎, ‎Iran

2 Department of Mathematics, Yazd University, Yazd, Iran

10.22075/ijnaa.2023.32129.4773

Abstract

In this paper, we focus on the fractional transportation problem where the cost coefficient of the objective functions, and the source and destination parameters have been expressed as interval values. The variable transformation solves the linear fractional transportation problem with interval coefficients in the objective function. In this method, instead of intervals in the function, using a convex combination of the left limit and right limit of the interval, linear fractional transportation problems with Interval Coefficients are reduced to a nonlinear programming problem. Finally, the nonlinear problem is transformed into a linear programming problem with two more constraints and one more variable compared to the initial problem. The constraints with interval source and destination parameters have been converted into deterministic ones. Numerical examples are presented to clarify the idea of the proposed approach for three possible cases of the original problem.

Keywords

[1] G. Alefeld and J. Herzberger, Introduction to Interval computations, Academic Press, New York, 1983.
[2] B. Bajalinov, Linear-Fractional Programming: Theory, Methods, Applications and Software, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2013.
[3] M. Borza, A.S. Rambely, and M. Saraj, Solving linear fractional programming problems with interval coefficients in the objective function. A new approach, Appl. Math. Sci. 6 (2012), no. 69, 3443–3452.
[4] S. Chanas and D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets Syst. 82 (1996), no. 3, 299–305.
[5] A. Charnes and W.W. Cooper, Programming with linear fractional function, Naval Res. Logist. Quat. 9 (1962), 181–186.
[6] S.K. Das, A. Goswami, and S.S. Alam, Multiobjective transportation problem with interval cost, source and destination parameters, Eur. J. Oper. Res. 117 (1999), 100–112.
[7] H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, Eur. J. Oper. Res. 48 (1990), no. 2, 219–225.
[8] K.M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
[9] T. Shaocheng, Interval number and fuzzy number linear programming, Fuzzy Sets Syst. 66 (1994), no. 3 301–306.
[10] A. Sheikhi and M.J. Ebadi, On solving linear fractional programming transportation problems with fuzzy numbers, J. Fuzzy Exten. Appl. 4 (2023), no. 4, 327–339.
[11] A. Sheikhi and M.J. Ebadi, An efficient method for solving linear interval fractional transportation problems, J. Appl. Res. Ind. Eng. 12 (2025), no. 1, 133–143.
[12] A. Sheikhi, S.M. Karbassi, and N. Bidabadi, A novel algorithm for solving bi-objective fractional transportation problems with fuzzy numbers, J. Math. Exten. 14 (2019), 29–47.
[13] I.M. Stancu-Minasian, Fractional Programming: Theory, Methods and Applications, Kluwer Dordrecht, 1997.
[14] R.E. Steuer, Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res. 6 (1981), no. 3, 333–348.
[15] S.F. Tantawy, A new method for solving linear fractional programming problem, J. Basic Appl. Sci. 1 (2007), no. 2, 105–108.
[16] H.C. Wu, On interval-valued nonlinear programming problems, J. Math. Anal. Appl. 338 (2008), no. 1, 299–316.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 25 January 2025
  • Receive Date: 23 October 2023
  • Accept Date: 25 December 2023