An Investigation on the Fractional Transportation Problem via Weibull Distribution

Subject Areas : Transactions on Fuzzy Sets and Systems

Paraman Anukokila ^{1 *} , Aravasa Gounder Vel Murugan ² , Bheeman Radhakrishnan ³

1 - Department of Mathematics, PSG College of Arts and Science, Tamil Nadu, India.
2 - Department of Mathematics, PSG College of Arts and Science, Tamil Nadu, India.
3 - Department of Mathematics, PSG College of Technology, Tamil Nadu, India.

Received: 2024-10-10 Accepted : 2025-01-21 Published : 2025-06-13

Keywords: Chance constrained programming, Fractional transportation problem, Goal programming, Weibull distribution.,

Abstract :

In this work, multi-objective fractional stochastic solid transportation problem uncertainties are represented using the Weibull distribution. This transformation converts the multi-objective fractional stochastic solid transportation problem into a goal programming problem with chance constraints, incorporating probabilistic constraints into its formulation. Goal programming and hyperbolic membership functions also assist in solving the fractional transportation problem. The proposed models serve as the basis for numerical examples and approaches to solving the problem under validated uncertainty. Furthermore, we conduct a sensitivity analysis to assess the impact of parameter changes on the proposed method.

References:

[1] Arya R, Kumar V, Saxena A. Analyze the stochastic solid fuzzy transportation problem with mixed constraints through Weibull distribution. Transportation. 2024; 31(4s): 540-559. DOI: https://doi.org/10.52783/cana.v31.949
[2] Das A, Lee GM. A multi-objective stochastic solid transportation problem with the supply, demand, and conveyance capacity following the Weibull distribution. Mathematics. 2021; 9(15): 17-57. DOI: https://doi.org/10.3390/math9151757
[3] Mahapatra DR, Roy SK, Biswal MP. Multi-objective stochastic transportation problem involving lognormal. Journal of Physical Science. 2010; 14: 63-76.
[4] Schaible S, Ibaraki T. Fractional programming. European Journal of Operational Research. 1983; 12(4): 325-338. DOI: https://doi.org/10.1016/0377-2217(83)90153-4
[5] Ebrahimnejad A. A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Applied Soft Computing. 2014; 19: 171-176. DOI: https://doi.org/10.1016/j.asoc.2014.01.041
[6] Anukokila P, Radhakrishnan B. Goal programming approach to fully fuzzy fractional transportation problem. Journal of Taibah University for Science. 2019; 13(1): 864-874. DOI: https://doi.org/10.1080/16583655.2019.1651520
[7] Ojha A, Das B, Mondal S, Maiti M. A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process. Applied Mathematical Modelling. 2010; 34(8): 2256-2271. DOI: http://doi.org/10.1016/j.apm.2009.10.034
[8] Mahapatra DR. Multi-choice stochastic transportation problem involving Weibull distribution. International Journal of Optimization and Control: Theories & Applications (IJOCTA). 2014; 4(1): 45-55. DOI: http://doi.org/10.11121/ijocta.01.2014.00154
[9] Pr´ekopa A. Probabilistic programming. Handbooks in operations research and management science. 2003; 10: 267-351. DOI: https://doi.org/10.1016/S0927-0507(03)10005-9
[10] Williams AC. A stochastic transportation problem. Operations Research. 1963; 11(5): 759-770. DOI: https://doi.org/10.1287/opre.11.5.759
[11] Holmberg K, Tuy H. A production-transportation problem with stochastic demand and concave production costs. Mathematical Programming. 1999; 85: 157-179. DOI: https://doi.org/10.1007/s101070050050
[12] Charnes A, Cooper WW. Chance constraints and normal deviates. Journal of the American Statistical Association. 1962; 57 (297): 134-148. DOI: https://doi.org/10.1080/01621459.1962.10482155
[13] Sengupta D, Das A, Bera UK, Dutta A. A humanitarian green supply chain management considering minimum cost and time. International Journal of Business Analytics. 2021; 8(2): 63-82. DOI: https://doi.org/10.4018/IJBAN.2021040105
[14] Charnes A, Cooper WW. Chance-constrained programming. Management Science. 1959; 6(1): 73-79. DOI: https://doi.org/10.1287/mnsc.6.1.73
[15] Chalam GA. Fuzzy goal programming (FGP) approach to a stochastic transportation problem under budgetary constraint. Fuzzy Sets and Systems. 1994; 66(3): 293-299. DOI: https://doi.org/10.1016/0165- 0114(94)90096-5
[16] Kataoka S. A stochastic programming model. Econometrica. 1963; 31(1/2)181-196. DOI: https://doi.org/10.2307/1910956
[17] Bhattacharya UK. Fuzzy programming algorithm with linear and non-linear membership functions for a stochastic formulation of the multiple-objective transportation problem. Journal of Fuzzy Mathematics. 2007; 15(3): 5-41.
[18] Gupta A, Kumar A. A new method for solving linear multi-objective transportation problems with fuzzy parameters. Applied Mathematical Modeling. 2012; 36(4): 1421-1430. DOI: https://doi.org/10.1016/j.apm.2011.08.044
[19] Gessesse A, Mishra R. Genetic algorithm based fuzzy programming approach for multi-objective linear fractional stochastic transportation problem involving four-parameter Burr distribution. International Journal of System Assurance Engineering and Management. 2020; 11: 93-109. DOI:
https://doi.org/10.1007/s13198-019-00928-0
[20] Dambrosio C, Lodi A, Wiese S, Bragalli C. Mathematical programming techniques in water network optimization. European Journal of Operational Research. 2015; 243(3): 774-788. DOI: https://doi.org/10.1016/j.ejor.2014.12.039
[21] Guo H, Wang X, Zhou S. A transportation model with uncertain costs and supplies. International Journal of e-Navigation and Maritime Economy. 2015; 2: 1-11. DOI: https://doi.org/10.1016/j.enavi.2015.06.001
[22] Ruszczy´nski A, Shapiro A. Stochastic programming models. Handbooks in Operations Research and Management Science. 2003; 10: 1-64. DOI: https://doi.org/10.1016/S0927-0507(03)10001-1
[23] Gao X, Lee GM. A two-stage stochastic programming model for commodity redistribution under uncertainty in disaster response. Proceedings of International Conference on Computers and Industrial Engineering, CIE48, 2018. Dec. 2018, New Zealand, Auckland: University of Auckland; 2018.
[24] Agrawal P, Ganesh T. An artificial intelligence approach for solving stochastic transportation problems. Computers, Materials & Continua. 2022; 70(1): 817-829. DOI: https://doi.org/10.32604/cmc.2022.019685
[25] Gupta S, Garg H. Parameter estimation and optimization of multi-objective capacitated stochastic transportation problem for gamma distribution. Complex & Intelligent Systems. 2020; 6: 651-667. DOI: https://doi.org/10.1007/s40747-020-00156-1
[26] Weibull W. A statistical distribution function of wide applicability. Journal of Applied Mechanics. 1951; 18(3): 293-297. DOI: https://doi.org/10.1115/1.4010337
[27] Zimmermann HJ. Fuzzy set theory. Wiley interdisciplinary reviews: computational statistics. 2010; 2(3): 317-332. DOI: https://doi.org/10.1002/wics.82
[28] Schrage L. LINGO users guide. LINDO System Inc: Chicago, IL, USA: 2006.