Abstract
In this paper, we have formulated a new model of multi-objective capacitated transportation problem (MOCTP) with mixed constraints. In this model, some objective functions are linear and some are fractional and are of conflicting in nature with each other. The main objective of this paper is to decide the optimum order of the product quantity which is to be shipped from source to the destination subject to the capacitated restriction on each route. Here the two situations have been discussed for the MOCTP model. In the first situation, we have considered that all the input information of the MOCTP model is exactly known and therefore a fuzzy goal programming approach have been directly used for obtaining the optimum order quantity of the product. While in the second situation the input information of the MOCTP model are uncertain in nature and this uncertainty have been studied and handled by the suitable approaches like trapezoidal fuzzy numbers, multi-choices, and probabilistic random variables respectively. Due to the presence of all these uncertainties and conflicting natures of objectives functions, we cannot solve this MOCTP directly. Therefore firstly we converted all these uncertainties into deterministic forms by using the appropriate transformation techniques. For this, the vagueness in MOCTP defined by trapezoidal fuzzy numbers has been converted into its crisp form by using the ranking function approach. Multichoices in input information parameters have been converted into its exact form by the binary variable transformation technique. Randomness in input information is defined by the Pareto probability distribution, and for conversion into deterministic form chance constrained programming has been used. After doing all these transformations, we have applied fuzzy goal programming approach for solving this resultant MOCTP model for obtaining the optimum order quantity. A case study has been done to illustrate the computational procedure.
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References
Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20, 224–230 (1941)
Koopmans, T.C., Reiter, S.: A model of transportation. In: Koopmans, T.J.C. (ed.) Activity Analysis of Production and Allocation—Proceedings of a Conference, pp. 222–259. Wiley, New York (1951)
Wagner, H.M.: On a class of capacitated transportation problems. Manag. Sci. 5(3), 304–318 (1959)
Lohgaonkar, M.H., Bajaj, V.H.: Fuzzy approach to solve multi-objective capacitated transportation problem. Int. J. Bioinform. Res. 2, 10–14 (2010)
Gupta, N., Bari, A.: Fuzzy multi-objective capacitated transportation problem with mixed constraints. J. Stat. Appl. Probab. 3(2), 1–9 (2014)
Gupta, N., Ali, I., Bari, A.: A compromise solution for multi-objective chance constraint capacitated transportation problem. Probstat Forum 6, 60–67 (2013)
Pramanik, S., Banerjee, D.: Multi-objective chance constrained capacitated transportation problem based on fuzzy goal programming. Int. J. Comput. Appl. 44(20), 42–46 (2012)
Sadia, S., Gupta, N., Ali, Q.M.: Multiobjective capacitated fractional transportation problem with mixed constraints. Math. Sci. Lett. 5(3), 235–242 (2016)
Bhargava, A.K., Singh, S.R., Bansal, D.: Multi-objective fuzzy chance constrained fuzzy goal programming for capacitated transportation problem. Int. J. Comput. Appl. 107(3), 18–23 (2014)
Ebrahimnejad, A.: A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 19, 171–176 (2014)
Ebrahimnejad, A.: An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers. J. Intell. Fuzzy Syst. 29(2), 963–974 (2015)
Giri, P.K., Maiti, M.K., Maiti, M.: Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Comput. Ind. Eng. 72, 160–168 (2014)
Pandian, P., Natarajan, G.: A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4(2), 79–90 (2010)
Chakraborty, A., Chakraborty, M.: Cost-time minimization in a transportation problem with fuzzy parameters: a case study. J. Transp. Syst. Eng. Inf. Technol. 10(6), 53–63 (2010)
Gupta, A., Kumar, A.: A new method for solving linear multi-objective transportation problems with fuzzy parameters. Appl. Math. Model. 36(4), 1421–1430 (2012)
Kaur, A., Kumar, A.: A new method for solving fuzzy transportation problems using ranking function. Appl. Math. Model. 35(12), 5652–5661 (2011)
Liu, S.T., Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153(3), 661–674 (2004)
Kundu, P., Kar, S., Maiti, M.: Multi-objective multi-item solid transportation problem in the fuzzy environment. Appl. Math. Model. 37(4), 2028–2038 (2013)
Chakraborty, D., Jana, D.K., Roy, T.K.: Arithmetic operations on the generalized intuitionistic fuzzy number and its applications to the transportation problem. Opsearch 52(3), 431–471 (2015)
Chang, C.T.: Multi-choice goal programming. Omega 35(4), 389–396 (2007)
Chang, C.T.: Revised multi-choice goal programming. Appl. Math. Model. 32(12), 2587–2595 (2008)
Acharya, S., Biswal, M.P.: Solving multi-choice multi-objective transportation problem. Int. J. Math. Oper. Res. 8(4), 509–527 (2016)
Biswal, M., Acharya, S.: Solving multi-choice linear programming problems by interpolating polynomials. Math. Comput. Model. 54(5), 1405–1412 (2011)
Roy, S.K.: Multi-choice stochastic transportation problem involving Weibull distribution. Int. J. Oper. Res. 21(1), 38–58 (2014)
Roy, S.K., Mahapatra, D.R., Biswal, M.P.: Multichoice stochastic transportation problem involving exponential distribution. J. Uncertain. Syst. 6(3), 200–213 (2012)
Mahapatra, D.R., Roy, S.K., Biswal, M.P.: Multichoice stochastic transportation problem involving extreme value distribution. Appl. Math. Model. 37(4), 2230–2240 (2013)
Maity, G., Roy, S.R.: Solving multi-choice multi-objective transportation problem: a utility function approach. J. Uncertain. Anal. Appl. 2, 11 (2014)
Maity, G., Kumar Roy, S.: Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand. Int. J. Manag. Sci. Eng. Manag. 11(1), 62–70 (2016)
Khalil, T.A., Raghav, Y.S., Badra, N.: Optimal solution of multi-choice mathematical programming problem using a new technique. Am. J. Oper. Res. 6, 167–172 (2016)
Biswal, M.P., Acharya, S.: Multi-choice multi-objective linear programming problem. J. Interdiscip. Math. 12(5), 606–637 (2009)
Biswas, A., Modak, N.: A Fuzzy Goal Programming Method for Solving Chance Constrained Programming with Fuzzy Parameters. In: Balasubramaniam, P. (ed.) Control, Computation and Information Systems. Communications in Computer and Information Science, vol. 140. Springer, Berlin, Heidelberg (2011)
Biswal, M.P., Samal, H.K.: Stochastic transportation problem with Cauchy random variables and multi-choice parameters. J. Phys. Sci. 17, 117–130 (2013)
Roy, S.K.J.: Transportation problem with multi-choice cost and demand and stochastic supply. J. Oper. Res. Soc. China (2016). https://doi.org/10.1007/s40305-016-0125-3
Barik, S.K.: Probabilistic fuzzy goal programming problems involving pareto distribution: some additive approaches. Fuzzy Inf. Eng. 7(2), 227–244 (2015)
Abbasbandy, S., Hajjari, T.: A new approach for ranking of trapezoidal fuzzy numbers. Comput. Math. Appl. 57(3), 413–419 (2009)
Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1, 45–55 (1978)
Lingo-User’s Guide: Lingo-user’s guide, Published by Lindo System Inc., 1415, North Dayton Street, Chicago, Illinois, 60622, USA (2016)
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Gupta, S., Ali, I. & Ahmed, A. Multi-objective capacitated transportation problem with mixed constraint: a case study of certain and uncertain environment. OPSEARCH 55, 447–477 (2018). https://doi.org/10.1007/s12597-018-0330-4
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DOI: https://doi.org/10.1007/s12597-018-0330-4