Abstract
Transportation problem (TP) is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of TP involving interval-valued trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a fuzzy linear programming approach for solving interval-valued trapezoidal fuzzy numbers transportation problem based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To illustrate the proposed approach an application example is solved. It is demonstrated that study of interval-valued trapezoidal fuzzy numbers transportation problem gives rise to the same expected results as those obtained for TP with trapezoidal fuzzy numbers.
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References
Zimmermann H I 1978 Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1(1): 45–55
Oheigeartaigh M 1982 A fuzzy transportation algorithm. Fuzzy Sets Syst. 8(3): 235–243
Chanas S, Kolodziejczyk W and Machaj A 1984 A fuzzy approach to the transportation problem. Fuzzy Sets Syst. 13(3): 211–221
Chanas S, Delgado M, Verdegay J L and Vila M A 1993 Interval and fuzzy extensions of classical transportation problems. Transp. Plann. Technol. 17(2): 203–218
Chanas S and Kuchta D 1996 A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst. 82(2): 299–305
Jimenez F and Verdegay J L 1998 Uncertain solid transportation problem. Fuzzy Sets Syst. 100(1–3): 45–57
Jimenez F and Verdegay J L 1999 Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. Eur. J. Operat. Res. 117(3): 485–510
Liu S T and Kao C 2004 Solving fuzzy transportation problems based on extension principle. Eur. J. Operat. Res. 153(3): 661–674
Gani A and Razak K A 2006 Two stage fuzzy transportation problem. J. Phys. Sci. 10: 63–69
Li L, Huang Z, Da Q and Hu J 2008 A new method based on goal programming for solving transportation problem with fuzzy cost. International Symposiums on Information Processing 3–8
Lin F T 2009 Solving the transportation problem with fuzzy coefficients using genetic algorithms. IEEE International Conference on Fuzzy Systems 1468–1473
Dinagar D S and Palanivel K 2009 The transportation problem in fuzzy environment. Int. J. Algorithms Comput. Math. 2(3): 65–71
Pandian P and Natarajan G 2010 A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4(2): 79–90
Kumar A and Kaur A 2010 Application of linear programming for solving fuzzy transportation problems. J. Appl. Math. Informatics 29(3–4): 831–846
Gupta A, Kumar A and Kaur A 2012 Mehar’s method to find exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems. Optimiz. Lett. 6: 1737–1751
Ebrahimnejad A 2015 A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems. Int. J. Syst. Sci. 46(11): 2048–2060
Shanmugasundari M and Ganesan K 2013 A novel approach for the fuzzy optimal solution of fuzzy transportation problem. Int. J. Eng. Res. Appl. 3(1): 1416–1421
Sudhagar S and Ganesan K 2012 A fuzzy approach to transport optimization problem. Optimiz. Eng. 10.1007/s11081-012-9202-6
Ebrahimnejad A 2015 Note on a fuzzy approach to transport optimization problem. Optimiz. Eng. 10.1007/s11081-015-9277-y
Kumar A and Kaur A 2014 Optimal way of selecting cities and conveyances for supplying coal in uncertain environment. Sadhana. 10.1007/s12046-013-0207-4
Ebrahimnejad A 2015 An improved approach for solving transportation problem with triangular fuzzy numbers. J. Intell. Fuzzy Syst. 29(2): 963–974
Kumar A and Kaur A 2011 A new method for solving fuzzy transportation problems using ranking function. Appl. Math. Modell. 35(12): 5652–5661
Kaur A and Kumar A 2012 A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 12(3): 1201–1213
Ebrahimnejad A 2014 A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 19: 171–176
Chiang J 2005 The optimal solution of the transportation problem with fuzzy demand and fuzzy product. J. Inf. Sci. Eng. 21: 439–451
Gupta A and Kumar A 2012 A new method for solving linear multi-objective transportation problems with fuzzy parameters. Appl. Math. Modell. 36: 1421–1430
Chiang J 2001 Fuzzy linear programming based on statistical confidence interval and interval-valued fuzzy set. Eur. J. Operat. Res. 129: 65–86
Yao J S and Lin F T 2002 Constructing a fuzzy flow-shop sequencing model based on statistical data. Int. J. Approximation Reason. 29: 215–234
Wei S H and Chen S M 2009 Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Syst. Appl. 36: 2285–2299
Farhadinia B 2014 Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems . Appl. Math. Modell. 38(1): 50–62
Kumar A and Kaur A 2011 Application of classical transportation methods to find the fuzzy optimal solution of fuzzy transportation problems. Fuzzy Inf. Eng. 3(1): 81–99
Kumar A and Kaur A 2012 Methods for solving unbalanced fuzzy transportation problems. Operat. Res. 12(3): 287–316
Bazaraa M S, Jarvis J J and Sherali H D 2010 Linear programming and network flows. New York: John Wiley and Sons
Yager R R 1981 A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24: 143–161
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The author thanks to the editor and the anonymous reviewers for their suggestions that improved this paper significantly. The author greatly appreciates the office of vice chancellor for research and technology of Islamic Azad University, Qaemshahr Branch for financial support too.
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EBRAHIMNEJAD, A. Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sādhanā 41, 299–316 (2016). https://doi.org/10.1007/s12046-016-0464-0
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DOI: https://doi.org/10.1007/s12046-016-0464-0