Solution of Fuzzy Maximal Flow Problems Using Fuzzy Linear Programming

In this paper, the fuzzy linear programming formulation of fuzzy maximal flow problems are proposed and on the basis of the proposed formulation a method is proposed to find the fuzzy optimal solution of fuzzy maximal flow problems. In the proposed method all the parameters are represented by triangular fuzzy numbers. By using the proposed method the fuzzy optimal solution of fuzzy maximal flow problems can be easily obtained. To illustrate the proposed method a numerical example is solved and the obtained results are discussed.

Authors:

• Amit Kumar
• Manjot Kaur

Keywords:

• Fuzzy linear programming
• Fuzzy maximal flow problem
• Ranking function
• Triangular fuzzy number

References:

[1] D. R. Fulkerson and G. B. Dantzig, Computation of Maximum Flow in
Network, Naval Research Logisics Quarterly, vol. 2, 1955, pp. 277-283.
[2] L. R. Ford, and D. R. Fulkerson, Maximal Flow Through a Network,
Canadian Journal of Mathematics, vol. 8, 1956, pp. 399-404.
[3] R. K. Ahuja, T. L. Magnanti and J. B. Orlin, Network Flows, Theory,
Algorithms and Applications, New Jersey, Prentice Hall, 1993.
[4] M. Bazaraa, J. Jarvis and H. F. Sherali, Linear Programming and Network
Flows, John Wiley, 1990.
[5] L. A. Zadeh, Fuzzy Sets, Information and Control, vol. 8, 1965, pp.
338-353.
[6] K. Kim and F. Roush, Fuzzy Flows on Network, Fuzzy Sets and Systems,
vol. 8, 1982, pp. 35-38.
[7] S. Chanas andW. Kolodziejczyk, Maximum Flow in a Network with Fuzzy
Arc Capacities, Fuzzy Sets and Systems, vol. 8, 1982, pp. 165-173.
[8] S. Chanas and W. Kolodziejczyk, Real-valued Flows in a Network with
Fuzzy Arc Capacities, Fuzzy Sets and Systems, vol. 13, 1984, pp. 139-
151.
[9] S. Chanas and W. Kolodziejczyk, Integer Flows in Network with Fuzzy
Capacity Constraints, Networks, vol. 16, 1986, pp. 17-31.
[10] S. Chanas, M. Delgado, J. L. Verdegay and M. Vila, Fuzzy Optimal Flow
on Imprecise Structures, European Journal of Operational Research, vol.
83, 1995, pp. 568-580.
[11] A. Diamond, A Fuzzy Max-flow Min-cut Theorem, Fuzzy Sets and
Systems, vol. 119, 2001, pp. 139-148.
[12] S. T. Liu and C. Kao, Network Flow Problems with Fuzzy Arc Lengths,
IEEE Transactions on Systems, Man and Cybernetics, vol. 34, 2004,
pp. 765-769.
[13] X. Ji, L. Yang and Z. Shao, Chance Constrained Maximum Flow Problem
with Arc Capacities, Lecture Notes in Computer Science, Springer-
Verlag, Berlin, Heidelberg, vol. 4114, 2006, pp. 11-19.
[14] F. Hernandes, M. T. Lamata, M. T. Takahashi, A. Yamakami, and J. L.
Verdegay, An Algorithm for the Fuzzy Maximum Flow Problem, IEEE
International Fuzzy Systems Conference, 2007, pp. 1-6.
[15] A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetics:
Theory and Applications, New York, Van Nostrand Reinhold, 1985.
[16] T. S. Liou and M. J. Wang, Ranking Fuzzy Numbers with Integral Value,
Fuzzy Sets and Systems, vol. 50, 1992, pp. 247-255.
[17] A. Kumar, J. Kaur and P. Singh, Fuzzy Optimal Solution of Fully Fuzzy
Linear Programming Problems with Inequality Constraints, International
Journal of Applied Mathematics and Computer Sciences, vol. 6, 2010,
pp. 37-41.