Authors:
Kumari Anupam,Tania Bose,Renu Bala,Gourav Gupta,Krishan Dutt Sharma,DOI NO:
https://doi.org/10.26782/jmcms.spl.11/2024.05.00012Keywords:
Transportation Problems,Physical Distribution Problem,Optimal Solution,Initial Basic Feasible Solution,Abstract
In today’s highly competitive world, the distribution of products plays a major role which makes it an important optimization problem related to determining the transportation route to transport a certain amount of products from supply points to demand points with minimum total transportation cost. This paper aims to introduce a new method to find the best and quick initial basic feasible solution for both balanced and unbalanced transportation problems. The proposed method always gives either optimal value or nearest to optimal value which is illustrated with two numerical illustrations i.e. one balanced and one unbalanced transportation problem. Also, the comparison of the results with some existing methods is also discussed.Refference:
I. A. Charnes, A. Henderson, and W. W. Cooper. : ‘An introduction to linear programming’. John Wiley and Sons. (1955).
II. A. Thamaraiselvi and R. Santhi. : ‘A new approach for optimization of real life transportation problem in neutrosophic environment’. Mathematical problems in Engineering. (2016), 5950747, https://doi.org/10.1155/2016/5950747
III. B. Prajwal, J. Manasa and R. Gupta. : ‘Determination of initial basic feasible solution for transportation problems by Supply-Demand Reparation Method and Continuous Al- location Method’. Logistics, Supply Chain and Financial Predictive Analytics. pp. 19–31, (2018). http://dx.doi.org/10.1007/978-981-13-0872-7_2
IV. C. S. Putcha and A. Shekaramiz. : ‘Development of a new optimal method for solution of transportation problems’. Proceedings of the World Congress on Engineering. 2010 III, WCE, (2010), https://www.iaeng.org/publication/WCE2010/WCE2010_pp1908-1912.pdf
V. F. L. Hitchcock. : ‘The distribution of a product from several sources to numerous localities’. Studies in Applied Mathematics. Vol. 20(1-4), pp. 224–230, (1941). http://dx.doi.org/10.1002/sapm1941201224
VI. G. B. Dantzig. : ‘Maximization of a linear function of variables subject to linear inequalities’. New York (1951).
VII. G. Gupta, and K. Anupam. : ‘An efficient method for solving intuitionistic fuzzy transportation problem of Type-2’. International Journal of Computational Mathematics. Vo.3(4), pp. 3795–3804, (2017). https://link.springer.com/article/10.1007/s40819-017-0326-4
VIII. G. Gupta, D. Rani, M. K. Kakkar. : ‘Optimal Solution of Fully Intuitionistic Fuzzy Transportation Problems: A Modified Approach, Manufacturing Technologies and Production Systems’. CRC Press. pp. 302-316, (2024), https://www.taylorfrancis.com/chapters/edit/10.1201/9781003367161-28/optimal-solution-fully-intuitionistic-fuzzy-transportation-problems-gourav-gupta-deepika-rani-mohit-kumar-kakkar
IX. G. Gupta, D. Rani and S. Singh. : ‘ An alternate for the initial basic feasible solution of category 1 uncertain transportation problems’. Proceedings of the National Academy of Sciences, India Section A, Vo. 90, pp. 157–167, (2019), http://dx.doi.org/10.1007/s40010-018-0557-8.
X. H. A. Hussein, A. K. S. Mushtak and S. M. Zabiba. : ‘A new revised efficient of VAM to find the initial solution for the transportation problem’. Journal of Physics: Conference Series. Vol. 1591, 012032, (2020). http://dx.doi.org/10.1088/1742-6596/1591/1/012032.
XI. H. S. Kasana and K. D. Kumar. : ‘Introductory operations research: theory and applications’. Springer-Verlag Berlin Heidelberg. (2004).
XII. J. Singla, G. Gupta, M. K. Kakkar and N. Garg. : ‘A novel approach to find initial basic feasible solution of transportation problems under uncertain environment’. AIP Conference Proceedings. 2357 (2022), https://doi.org/10.1063/5.0080755.
XIII. J. Singla, G. Gupta, M. K. Kakkar and N. Garg. : ‘Revised Algorithm of Vogel’s Approximation Method (RA-VAM): An Approach to Find Basic Initial Feasible Solution of Transportation Problem’. ECS Transactions. Vol. 107(1), pp. 8757, (2022). http://dx.doi.org/10.1149/10701.8757ecst.
XIV. K. Karagul and Y. Sahin. : ‘A novel approximation method to obtain initial basic feasible solution of transportation problem’. Journal of King Saud University-Engineering Sciences. Vol. 32, pp. 211–218, (2020). http://dx.doi.org/10.1016/j.jksues.2019.03.003 .
XV. K. M. S. Reza, A. R. M. J. U. Jamali and B. Biswas. : ‘A modified algorithm for solving unbalanced transportation problems’. Journal of Engineering Science. Vol. 10(1), pp. 93– 101, (2019).
XVI. K. Swarup, P. K. Gupta and M. Mohan. : ‘Operations research’. Sultan Chand & sons educational publishers. New Delhi, (2003).
XVII. M. M. Ahmed, M. A. Islam, M. Katun, S. Yesmin and M.S. Uddin. : ‘New procedure of finding an initial basic feasible solution of the time minimizing transportation problems’. Open Journal of Applied Sciences. Vol. 5, pp. 634–640, (2015). 10.4236/ojapps.2015.510062.
XVIII. M. M. Ahmed, A. R. Khan, F. Ahmed and M. S. Uddin. : ‘Incessant allocation method for solving transportation problems’. American Journal of Operations Research. Vol. 6, pp. 236–244, (2016) 10.4236/ajor.2016.63024.
XIX. M. M. Gothi, R. G. Patel and B. S. Patel. : ‘A concept of an optimal solution of the transportation problem using the weighted arithmetic mean’. Advances in Mathematics: Scientific Journal. Vol. 10(3), pp. 1707–1720, (2021). https://doi.org/10.37418/amsj.10.3.52.
XX. M. Sathyavathy and M. Shalini. : ‘Solving transportation problem with four different pro- posed mean method and comparison with existing methods for optimum solution’. Journal of Physics: Conference Series. Vol. 1362, 012083, (2019). http://dx.doi.org/10.1088/1742-6596/1362/1/012088.
XXI. Neha Garg, Mohit Kumar Kakkar, Gourav Gupta, Jajji Singla. : ‘Meta heuristic algorithm for vehicle routing problem with uncertainty in customer demand’. ECS Transactions. Vol. 107(1), 6407, (2022). https://iopscience.iop.org/article/10.1149/10701.6407ecst.
XXII. P. R. Murthy. : ‘Operations research (linear programming)’. New age international (p) limited publishers. Section 4.4.3, Example 4.1, 145, (2007).
XXIII. R. G. Patel and P. H. Bhathawala. : ‘An innovative approach to optimal solution of a transportation problem’. International Journal of Innovative Research in Science, Engineering and Technology. Vol. 5(4), pp. 5695–5700, (2016). 10.15680/IJIRSET.2016.0504082.
XXIV. R. H. Ballou. : ‘Business Logistics Management’. Prentice-Hall International. New Jersey, (1999).
XXV. S. Korukoglu, and S. Balli. : ‘An improved Vogel’s approximation method for the transportation problem’. Association for Scientific Research, Mathematical and Computational Application. Vol. 16, pp. 370–381, (2011). http://dx.doi.org/10.3390/mca16020370.
XXVI. S. K. Mohanta. : ‘Sequential approach towards the optimal solution of transportation problem’. International Journal of Mathematics and Its Applications. Vol. 9(4), pp. 13–24, (2021). https://www.researchgate.net/publication/333559366_ON_OPTIMAL_SOLUTION_OF_TRANSPORTATION_PROBLEM_LOGICAL_APPROACH#fullTextFileContent .
XXVII. Z. A. M. S. Juman and N. G. S. A. Nawarathne. : ‘An efficient alternative approach to solve a transportation problem’. Ceylon Journal of Science. Vol. 48(1), pp. 19–29, (2019), http://dx.doi.org/10.4038/cjs.v48i1.7584.