International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

One Point Conventional Model to Optimize Trapezoidal Fuzzy Transportation Problem

Dinesh C. S. Bisht
Department of Mathematics, Jaypee Institute of Information Technology, Noida-201304, India.

Pankaj Kumar Srivastava
Department of Mathematics, Jaypee Institute of Information Technology, Noida-201304, India.

DOI https://dx.doi.org/10.33889/IJMEMS.2019.4.5-099

Received on July 20, 2018
  ;
Accepted on July 21, 2019

Abstract

This article puts forward a new one point approach to optimize trapezoidal fuzzy transportation problem. It proposes the method having point wise breakup of the trapezoidal number in such a way, that fuzzy transportation problem is converted into four crisp transportation problems. The method is equipped with minimum of supply and demand approach. In the end, the solutions are combined to construct the optimal solution. Modified distribution is applied on each crisp problem to develop optimal solution. The scheme presented is compared with competitive methods available in literature and it is found to be in good coordination with these. The scheme is equally good to be applied on unbalanced problems. Two numerical problems are considered to test the performance of the proposed approach.

Keywords- One point approach, Trapezoidal fuzzy number, Minimum demand supply, Modified distribution, Fuzzy transportation problem.

Citation

Bisht, D. C. S., & Srivastava, P. K. (2019). One Point Conventional Model to Optimize Trapezoidal Fuzzy Transportation Problem. International Journal of Mathematical, Engineering and Management Sciences, 4(5), 1251-1263. https://dx.doi.org/10.33889/IJMEMS.2019.4.5-099.

Conflict of Interest

Both authors have contributed equally in this work. The authors declare that there is no conflict of interest for this publication.

Acknowledgements

The authors extend their appreciation to the anonymous reviewers for their valuable suggestions.

References

Akyar, E., Akyar, H., & Düzce, S.A. (2012). A new method for ranking triangular fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 20(05), 729–740.

Allahviranloo, T., Lotfi, F.H., Kiasary, M.K., Kiani, N.A., & Alizadeh, L. (2008). Solving fully fuzzy linear programming problem by the ranking function. Applied Mathematical Sciences, 2(1), 19–32.

Baykasoglu, A., & Gocken, T. (2012). A direct solution approach to fuzzy mathematical programs with fuzzy decision variables. Expert Systems with Applications, 39(2), 1972–1978.

Bellman, R.E., & Zadeh, L.A. (1970). Decision-making in a fuzzy environment. Management Science, 17(4), B–141.

Bisht, D., & Srivastava, P.K. (2017, October). A unique conversion approach clubbed with a new ranking technique to optimize fuzzy transportation cost. In AIP Conference Proceedings, 1897(1), p. 020023. AIP Publishing.

Bisht, D.C.S., & Srivastava, P.K. (2018). Trisectional fuzzy trapezoidal approach to optimize interval data based transportation problem. Journal of King Saud University-Science (in press).

Bisht, D.C.S., & Srivastava, P.K. (2019). Fuzzy optimization and decision making. In Advanced Fuzzy Logic Approaches in Engineering Science, pp. 310-326. IGI Global.

Bisht, D.C.S, Srivastava, P.K., & Ram, M. (2018). Role of fuzzy logic in flexible manufacturing system. In Diagnostic Techniques in Industrial Engineering, (pp. 233-243). Springer, Cham.

Buckley, J.J., & Feuring, T. (2000). Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets and Systems, 109(1), 35–53.

Buckley, J.J., Feuring, T., & Hayashi, Y. (2001). Multi-objective fully fuzzified linear progamming. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9(05), 605–621.

Chanas, S., & Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems, 82(3), 299–305.

Chanas, S., & Kuchta, D. (1998). Fuzzy integer transportation problem. Fuzzy Sets and Systems, 98(3), 291–298.

Chanas, S., Kołodziejczyk, W., & Machaj, A. (1984). A fuzzy approach to the transportation problem. Fuzzy Sets and Systems, 13(3), 211–221.

Chhibber, D., Bisht, D.C.S., & Srivastava, P.K. (2019, January). Ranking approach based on incenter in triangle of centroids to solve type-1 and type-2 fuzzy transportation problem. In AIP Conference Proceedings, 2061(1), p. 020022. AIP Publishing.

Dehghan, M., Hashemi, B., & Ghatee, M. (2006). Computational methods for solving fully fuzzy linear systems. Applied Mathematics and Computation, 179(1), 328–343.

Dinagar, D.S., & Palanivel, K. (2009). The transportation problem in fuzzy environment. International Journal of Algorithms, Computing and Mathematics, 2(3), 65–71.

Dubois, D., & Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9(6), 613–626.

Ebrahimnejad, A. (2016). New method for solving fuzzy transportation problems with LR flat fuzzy numbers. Information Sciences, 357, 108–124.

Ganesh, A.H., & Jayakumar, S. (2014). Ranking of fuzzy numbers using radius of gyration of centroids. International Journal of Basic and Applied Sciences, 3(1), 17-22.

Ghazanfari, M., Yousefli, A., Ameli, M.S.J., & Bozorgi-Amiri, A. (2009). A new approach to solve time–cost trade-off problem with fuzzy decision variables. The International Journal of Advanced Manufacturing Technology, 42(3–4), 408–414.

Goyal, G., & Bisht, D.C.S. (2019, January). An aggregated higher order fuzzy logical relationships technique. In AIP Conference Proceedings, 2061(1), p. 020023. AIP Publishing.

Hitchcock, F.L. (1941). The distribution of a product from several sources to numerous localities. Studies in Applied Mathematics, 20(1–4), 224–230.

Jain, R. (1977). A procedure for multiple-aspect decision making using fuzzy sets. International Journal of Systems Science, 8(1), 1–7.

Jain, S., Bisht, D.C.S., & Mathpal, P.C. (2018). Particle swarm optimised fuzzy method for prediction of water table elevation fluctuation. International Journal of Data Analysis Techniques and Strategies, 10(2), 99-110.

Jain, S., Bisht, D.C.S., Singh, P., & Mathpal, P.C. (2017, October). Real coded genetic algorithm for fuzzy time series prediction. In AIP Conference Proceedings, 1897(1), p. 020021. AIP Publishing.

Jain, S., Mathpal, P.C., Bisht, D., & Singh, P. (2018). A unique computational method for constructing intervals in fuzzy time series forecasting. Cybernetics and Information Technologies, 18(1), 3-10.

Kaur, A., & Kumar, A. (2011). A new method for solving fuzzy transportation problems using ranking function. Applied Mathematical Modelling, 35(12), 5652–5661.

Kumar, A., & Kaur, A. (2011). Application of classical transportation methods for solving fuzzy transportation problems. Journal of Transportation Systems Engineering and Information Technology, 11(5), 68–80.

Kumar, A., & Kaur, A. (2012). Methods for solving unbalanced fuzzy transportation problems. Operational Research, 12(3), 287–316.

Liu, S.-T., & Kao, C. (2004). Solving fuzzy transportation problems based on extension principle. European Journal of Operational Research, 153(3), 661–674.

Lotfi, F.H., Allahviranloo, T., Jondabeh, M.A., & Alizadeh, L. (2009). Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Applied Mathematical Modelling, 33(7), 3151–3156.

Mathur, N., & Srivastava, P.K. (2019, January). A pioneer optimization approach for hexagonal fuzzy transportation problem. In AIP Conference Proceedings, 2061(1), p. 020030. AIP Publishing.

Mathur, N., Srivastava, P.K., & Paul, A. (2016). Trapezoidal fuzzy model to optimize transportation problem. International Journal of Modeling, Simulation, and Scientific Computing, 7(03), 1650028.

Mathur, N., Srivastava, P.K., & Paul, A. (2018). Algorithms for solving fuzzy transportation problem. International Journal of Mathematics in Operational Research, 12(2), 190–219.

Nagar, P., Srivastava, A., & Srivastava, P.K. (2019). Optimization of species transportation via an exclusive fuzzy trapezoidal centroid approach. Mathematics in Engineering, Science and Aerospace, 10(2), 271-280.

Nasrabadi, M.M., & Nasrabadi, E. (2004). A mathematical-programming approach to fuzzy linear regression analysis. Applied Mathematics and Computation, 155(3), 873–881.

Natarajan, P.P.G. (2010). A new method for finding an optimal solution of fully interval integer transportation problems. Applied Mathematical Sciences, 4(37), 1819–1830.

ÓhÉigeartaigh, M. (1982). A fuzzy transportation algorithm. Fuzzy Sets and Systems, 8(3), 235–243.

Pandian, P., & Natarajan, G. (2010). A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences, 4(2), 79–90.

Rezvani, S. (2013). Ranking generalized trapezoidal fuzzy numbers with Euclidean distance by the incentre of centroids. Mathematica Aeterna, 3(2), 103–114.

Saad, O.M., & Abass, S.A. (2002). A parametric study on transportation problem under fuzzy environment. Engineering Journal of University of Qatar, 15, 165–176.

Srivastava, P. K., Bisht, D., & Ram, M. (2018). Soft computing techniques and applications. In Advanced Mathematical Techniques in Engineering Sciences, pp. 57-69. Taylor & Francis Group.

Srivastava, P.K., & Bisht, D.C.S. (2018). Dichotomized incenter fuzzy triangular ranking approach to optimize interval data based transportation problem. Cybernetics and Information Technologies, 18(4), 111-119.

Srivastava, P.K., & Bisht, D.C.S. (2019). An efficient fuzzy minimum demand supply approach to solve fully fuzzy transportation problem. Mathematics in Engineering, Science and Aerospace, 10(2), 253-269.

Srivastava, P.K., & Bisht, D.C.S. (2019). Recent trends and applications of fuzzy logic. In Advanced Fuzzy Logic Approaches in Engineering Science (pp. 327-340). IGI Global.

Tanaka, H., Guo, P., & Zimmermann, H.-J. (2000). Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems. Fuzzy Sets and Systems, 113(2), 323–332.

Yager, R.R. (1980). On choosing between fuzzy subsets. Kybernetes, 9(2), 151–154.

Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.

Zimmermann, H.-J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1(1), 45–55.

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