Evaluation of the Minimum Transportation Cost of Asymmetric/Symmetric Triangular Fuzzy Numbers with -Cut by the Row-Column Minima Method

Authors

DOI:

https://doi.org/10.21123/bsj.2024.10029

Keywords:

Asymmetric/symmetric triangular fuzzy transportation problem, Interval integer transportation problem, Row-column minima method, Transportation problem, α -cut method

Abstract

In this article, the main idea is to obtain the minimum transportation total fuzzy cost of the triangular transportation problem using the row-column minima (RCM) method. Here, the capacity of supply, the destination of demand, and transportation cost are all fully triangular fuzzy numbers with asymmetric or symmetric but not with the negative triangular fuzzy number (TFN). Vagueness plays an active role in many fields, such as science, engineering, medicine, management, etc. In this idea, the TFN problem is decomposed into two interval integer transportation problems (IITP) using the -cut method, by putting  and  to get the upper bound interval transportation problem and the lower bound interval transportation problem. These two interval problems are decomposed again into two problems: the right-bound transportation problem (RBTP) and the left-bound transportation problem (LBTP). First, compute an initial basic feasible solution for RBTP, then also obtain the optimum solution by the existing method; there is no need to solve LBTP directly because the solution of RBTP is the initial solution of LBTP. Apply the RCM method to LBTP, getting interval solutions for both interval transportation problems. Then the combined and computed the minimum fuzzy triangular transportation cost, in which an asymmetric or symmetric triangular fuzzy transportation problem (TFTP) is not changed into classical TP without using ranking methods, and the same result was obtained using the existing method. Some numerical examples are illustrated, and it is very suitable to clarify the idea of this concept. This idea is an easy way to understand the uncertainty that happens in a real-life situation.

References

Silmi Juman ZAM, Masoud M, Elhenawy M, Bhuiyan H, Komol MMR, Battaia O. A new algorithm for solving uncapacitated transportation problem with interval-defined demands and suppliers capacities. J. Intell. Fuzzy Syst. 2021; 41(1): 625-637. http://dx.doi.org/10.3233/JIFS-202436 .

Quddoos A, Habiba U. A New Method to Solve Interval Transportation Problems. Pak. J. Stat. Oper. Res. 2020; 16(4): 802-811. http://dx.doi.org/10.18187/pjsor.v16i4.3269.

Bisht DCS, Srivastava PK. Trisectional fuzzy trapezoidal approach to optimize interval data based transportation problem. J. King Saud Univ. Sci. 2020; 32(1): 195-199. https://doi.org/10.1016/j.jksus.2018.04.013.

Dalman H, Sivri M. A Fuzzy Logic Based Approach to Solve Interval Multiobjective Nonlinear Transportation Problem. Proc. Natl. Acad. Sci. India - Phys. Sci. 2019; 89(2): 279-289. http://dx.doi.org/10.1007/s40010-017-0469-z .

Das SK. An approach to optimize the cost of transportation problem based on triangular fuzzy programming problem. Complex Intell Syst. 2022; 8(1): 687-699. https://doi.org/10.1007/s40747-021-00535-2.

Fegade M, Muley AA. Solving Fuzzy transportation problem using zero suffix and robust ranking methodology. IOSR J. Eng. 2012; 2(7): 36-39. http://dx.doi.org/10.9790/3021-02723639

Holel MA, Hasan SQ. The Necessary and Sufficient Optimality Conditions for a System of FOCPs with Caputo–Katugampola Derivatives. Baghdad Sci. J. 2023; 20(5): 1713-1721. https://doi.org/10.21123/bsj.2023.7515.

Ebrahimnejad A. On solving transportation problems with triangular fuzzy numbers: Review with some extensions. 13th Iranian Conference on Fuzzy Systems (IFSC), Qazvin, Iran. IEEE. 2013; 1-4. http://dx.doi.org/10.1109/IFSC.2013.6675629 .

Singh S, Gupta G. A new approach for solving cost minimization balanced transportation problem under uncertainty. J. Transp. Secur. 2014; 7(4): 339-345. https://doi.org/10.1007/s12198-014-0147-1.

Kaur D, Mukherjee S, Basu K. A new fuzzy programming technique approach to solve fuzzy transportation problem. 2nd International Conference on Business and Information Management (ICBIM), Durgapur, India. IEEE. 2014; 144-150. http://dx.doi.org/10.1109/ICBIM.2014.6970977

Ezzati R, Khorram E, Enayati R. A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl. Math. Model. 2015; 39(12): 3183-3193. http://dx.doi.org/10.1016/j.apm.2013.03.014.

Gomathi SV, Jayalakshmi M. One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model. Symmetry. 2022; 14(10): 1-14. https://doi.org/10.3390/sym14102056.

Kumar PS. Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set. International Int. J. Syst. Assur. Eng. Manag. 2020; 11(1): 189-222. https://doi.org/10.1007/s13198-019-00941-3.

Alhindawee Z, Jafar R, Shahin H, Awad A. Solid Waste Treatment Using Multi-Criteria Decision Support Methods Case Study Lattakia City. Baghdad Sci. J. 2023; 20(5): 1575-1589. https://dx.doi.org/10.21123/bsj.2023.7472.

Hunwisai D, Kumam P. A method for solving a fuzzy transportation problem via Robust ranking technique and ATM. Cogent math. 2017; 4(1): 1283730. https://doi.org/10.1080/23311835.2017.1283730.

Baykasoğlu A, Subulan K. Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables. Expert Syst. Appl. 2017; 81: 193-222. https://doi.org/10.1016/j.eswa.2017.03.040.

Malik M, Gupta SK. Goal programming technique for solving fully interval-valued intuitionistic fuzzy multiple objective transportation problems. Soft Comput. 2020; 24: 13955-13977. https://doi.org/10.1007/s00500-020-04770-6.

Dhanasekar S, Hariharan S, Sekar P. Fuzzy Hungarian MODI Algorithm to solve fully fuzzy transportation problems Int. J. Fuzzy Syst. 2017; 19(5): 1479-1491. https://doi.org/10.1007/s40815-016-0251-4.

Akilbasha A, Pandian P, Natarajan G. An innovative exact method for solving fully interval integer transportation problems. Inform Med Unlocked. 2018; 11: 95-99. https://doi.org/10.1016/j.imu.2018.04.007.

Balasubramanian K, Subramanian S. Optimal solution of fuzzy transportation problems using ranking function. Int. J. Mech. Prod. Eng. Res. Dev. 2018; 8(4): 551-558. http://dx.doi.org/10.24247/ijmperdaug201856 .

Pandian P, Natarajan G, Akilbasha A. Fuzzy Interval Integer Transportation Problems. Int J Pure Appl Math. 2018, 119(9): 133-142.

Prabha SK, Vimala S. ATM for solving fuzzy transportation problem using method of magnitude. IAETSD J. Adv. Res. Appl. Sci. 2018; 5(3): 406-412.

Kumar RR, Gupta R, Karthiyayini O, Vatsala GA. An innovative approach to find the initial solution of a fuzzy transportation problem. AIP Conf. Proc. 2019; 2177(1): 020083. https://doi.org/10.1063/1.5135258.

Muthuperumal S, Titus P, Venkatachalapathy M. An algorithmic approach to solve unbalanced triangular fuzzy transportation problems. Soft Comput. 2020; 24(24):18689-18698. https://doi.org/10.1007/s00500-020-05103-3.

Srinivasan R, Karthikeyan N, Renganathan K, Vijayan DV. Method for solving fully fuzzy transportation problem to transform the materials. Mater. Today: Proc. 2020; 37(2): 431-433. https://doi.org/10.1016/j.matpr.2020.05.423.

Indira P, Jayalakshmi M. Fully interval integer transportation problem for finding optimal interval solution using row-column minima method. Int. J. Sci. Technol. Res. 2020; 9(4): 1777-1781.

Faizi S, Sałabun W, Ullah S, Rashid T, Więckowski J. A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique. Symmetry. 2020; 12(4): 516. https://doi.org/10.3390/sym12040516.

Vidhya V, Maheswari PU, Ganesan K. An alternate method for finding an optimal solution to Mixed Type Transportation Problem under a Fuzzy Environment. International Conference on Advances in Renewable and Sustainable Energy Systems (ICARSES 2020) 3rd -5th December, Chennai, India. IOP Conf. Ser.: Mater. Sci. Eng. 2021; 1130(1): 012064. http://dx.doi.org/10.1088/1757-899X/1130/1/012064 .

Vidhya V, Uma Maheswari P, Ganesan K. An alternate method for finding more for less solution to fuzzy transportation problem with mixed constraints. Soft Comput. 2021; 25(18): 11989-11996. https://doi.org/10.1007/s00500-021-05664-x.

Ammar ES, Emsimir A. A mathematical model for solving fuzzy integer linear programming problems with fully rough intervals. Granul. Comput. 2021: 6(3): 567-578. https://doi.org/10.1007/s41066-020-00216-4.

Sam'an M, Farikhin. A new fuzzy transportation algorithm for finding fuzzy optimal solution. Int. J. Math. Model. Numer. Optim. 2021; 11(1): 1-19. https://doi.org/10.1504/IJMMNO.2021.111715.

Deshmukh A. Fuzzy Transportation Problem By Using Triangular Fuzzy Numbers With Ranking Using Area Of Trapezium, Rectangle and Centroid At Different Level Of α-Cut. Turk. J. Comput. Math. Educ. 2021; 12(12): 3941-3951. https://doi.org/10.17762/turcomat.v12i12.8182.

Gupta S, Ali I, Ahmed A. An extended multi-objective capacitated transportation problem with mixed constraints in fuzzy environment. Int. J. Oper. Res. 2020; 37(3): 345-376. https://doi.org/10.1504/IJOR.2020.105443.

Gupta S, Ali I, Ahmed A. Multi-choice multi-objective capacitated transportation problem—a case study of uncertain demand and supply. Int. J. Stat. Manag. Syst. 2018; 21(3): 467-491. https://doi.org/10.1080/09720510.2018.1437943.

Gupta S, Ali I, Chaudhary S. Multi-objective capacitated transportation: a problem of parameters estimation, goodness of fit and optimization. Granul. Comput. 2020; 5: 119-134. https://doi.org/10.1007/s41066-018-0129-y.

Raina AA, Gupta S, Kour K. Fractional transportation problem with non-linear discount cost. Sri Lankan J. Appl. Stat. 2017; 18(3): 187-205. http://doi.org/10.4038/sljastats.v18i3.7935.

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Evaluation of the Minimum Transportation Cost of Asymmetric/Symmetric Triangular Fuzzy Numbers with -Cut by the Row-Column Minima Method. Baghdad Sci.J [Internet]. [cited 2024 May 18];21(11). Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/10029