Main Article Content
Abstract
Purpose of the study: The transportation problem has a huge application in logistics. Therefore, dealing a transportation problem with permissible constraints is always an interesting area. In the present paper, we have tried a version of transportation problem which is constrained in nature. We have explored strategies for dealing the subset constrained transportation problem.
Methodology: An updated version of matrix minima method is proposed in this paper. This version is used for subset constrained transportation problem only. We have also applied the mathematical model for solving the same subset constrained transportation problem.
Main Findings: The mathematical model gives an optimal solution of the constrained transportation problem with the help of software. It is found that the proposed updated matrix minima method does not guarantee the optimality. However, we use the matrix minima approach for having an idea of the closer feasible solution. It is found that the mathematical model is always a superior way to find an optimal solution.
Applications of this study: The study has a huge application in logistics.
Novelty/Originality of this study: An updated version of matrix minima method is proposed. The method is applied on constrained transportation problem and the results are compared with the optimal solution.
Keywords
Transportation Problem
Subset Constraint
Matrix Minima Method
Least Cost Method
Mathematical Modeling
Article Details
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How to Cite
Prajapati, R., Pal, J., & Dubey, O. P. (2023). Modified matrix minima method for subset constrained transportation problem and its performance evaluation with respect to the optimal solution by mathematical model. International Journal of Students’ Research in Technology & Management, 11(1), 23–30. https://doi.org/10.18510/ijsrtm.2023.1114
References
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1. Adlakha, V. and Kowalski, K. (2003). A simple heuristic for solving small fixed-charge transportation problems. Omega, 31(3), 205-211. https://doi.org/10.1016/S0305-0483(03)00025-2
2. Agadaga, G.O. and Akpan, N.P. (2017). Transshipment problem and its variants: A review. Mathematical Theory and Modeling, 7(2), 19-32.
3. Ardjmand, E., Young II, W.A., Weckman, G.R., Bajgiran, O.S., Aminipour, B. and Park, N. (2016). Applying genetic algorithm to a new bi-objective stochastic model for transportation, location, and allocation of hazardous materials. Expert systems with applications, 51, 49-58. https://doi.org/10.1016/j.eswa.2015.12.036
4. Babu, M.A., Helal, M.A., Hasan, M.S. and Das, U.K. (2013). Lowest Allocation Method (LAM): a new approach to obtain feasible solution of transportation model. International Journal of Scientific and Engineering Research, 4(11), 1344-1348.
5. Balakrishnan, N. (1990). Modified Vogel's approximation method for the unbalanced transportation problem. Applied Mathematics Letters, 3(2), 9-11. https://doi.org/10.1016/0893-9659(90)90003-T
6. Barr, R., Elam, J., Glover, F. and Klingman, D. (1980). A network augmenting path basis algorithm for transshipment problems. In Extremal methods and systems analysis, pp. 250-274. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46414-0_12
7. Brouer, B.D., Desaulniers, G., Karsten, C.V. and Pisinger, D. (2015). A matheuristic for the liner shipping network design problem with transit time restrictions. In International Conference on Computational Logistics, pp. 195-208. Springer, Cham. https://doi.org/10.1007/978-3-319-24264-4_14
8. Buson, E., Roberti, R. and Toth, P. (2014). A reduced-cost iterated local search heuristic for the fixed-charge transportation problem. Operations Research, 62(5), 1095-1106. https://doi.org/10.1287/opre.2014.1288
9. Deep, K., Dubey, O.P. and Nagar, A. (2012). Incorporating Genetic Algorithms in Transport Management. In Proceedings of the International Conference on Soft Computing for Problem Solving (SocProS 2011) December 20-22, 2011, pp. 177-192. Springer, India. https://doi.org/10.1007/978-81-322-0487-9_18
10. Dubey, O.P., Deep, K. and Nagar, A.K. (2014). Goal Programming Approach to Trans-shipment Problem. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, pp. 337-343. Springer, New Delhi.
11. Dwivedi, R.K., Mehta, N.N. and Dubey, O.P. (2009). Interactive Decision Making in Prioritized Unbalanced Transportation Problems. IUP Journal of Operations Management, 8(1), 67. https://doi.org/10.1007/978-81-322-1602-5_37
12. Gupta, K. and Arora, S.R. (2012). An algorithm to find optimum cost time trade off pairs in a fractional capacitated transportation problem with restricted flow. International Journal Of Research In Social Sciences, 2(2), 418.
13. Hakim, M.A. (2012). An alternative method to find initial basic feasible solution of a transportation problem. Annals of pure and applied mathematics, 1(2), 203-209.
14. Joo, C.M. and Kim, B.S. (2014). Block transportation scheduling under delivery restriction in shipyard using meta-heuristic algorithms. Expert systems with applications, 41(6), 2851-2858. https://doi.org/10.1016/j.esw a.2013.10.020
15. Kaur, P., Verma, V. and Dahiya, K. (2017). Capacitated two-stage time minimization transportation problem with restricted flow. RAIRO-Operations Research, 51(2), 447-467. https://doi.org/10.1051/ro/2016033
16. Kavita, G. and Shri Ram, A. (2013). Linear plus linear fractional capacitated transportation problem with restricted flow. American Journal of Operations Research, 2013.
17. Khurana, A. (2015). Variants of transshipment problem. European Transport Research Review, 7(2), 1-19. https://doi.org/10.1007/s12544-015-0154-8
18. Kousalya, P. and Malarvizhi, P. (2016). A new technique for finding initial basic feasible solution to transportation problem. International Journal of Engineering and Management Research (IJEMR), 6(3), 26-32.
19. Lee, S.M. and Moore, L.J. (1973). Optimizing transportation problems with multiple objectives. AIIE transactions, 5(4), 333-338. https://doi.org/10.1080/05695557308974920
20. Loch, G.V. and da Silva, A.C.L., (2014). A computational study on the number of iterations to solve the transportation problem. Applied Mathematical Sciences, 8(92), 4579-4583. https://doi.org/10.12988/ams.2 014.46435
21. Misra, S. and Das, C. (1981). Three-dimensional transportation problem with capacity restriction. NZOR, 9(1), 47-58.
22. Prajapati, R., Pal, J., & Dubey, O. P. (2022). Transportation problem with restriction on arcs and their Solution by mathematical modeling. International Journal of Students’ Research in Technology & Management, 10(1), 67-74. https://doi.org/10.18510/ijsrtm.2022.1015
23. Quddoos, A., Javaid, S. and Khalid, M.M. (2012). A new method for finding an optimal solution for transportation problems. International Journal on Computer Science and Engineering, 4(7), 1271.
24. Quddoos, A., Javaid, S. and Khalid, M.M. (2016). A revised version of asm-method for solving transportation problem. International Journal Agriculture, Statistics, Science, 12(1), 267-272.
25. Rachev, S.T. and Rüschendorf, L. (1998). Mass Transportation Problems: Volume I: Theory (Vol. 1). Springer Science & Business Media.
26. Rachev, S.T. and Rüschendorf, L. (2006). Mass transportation problems: Applications. Springer Science & Business Media.
27. Samuel, A.E. and Venkatachalapathy, M. (2011). Modified Vogel’s approximation method for fuzzy transportation problems. Applied Mathematical Sciences, 5(28), 1367-1372.
28. Sharma, J.K. (2009). Operations research theory and application. Macmilan Publishers.
29. Sharma, V., Dahiya, K. and Verma, V. (2010). Capacitated two-stage time minimization transportation problem. Asia-Pacific journal of operational research, 27(04), 457-476. https://doi.org/10.1142 /S021759591000279X
30. Singh, P. and Saxena, P.K. (1998). Total shipping cost/completion-date trade-off in transportation problem with additional restriction. Journal of Interdisciplinary Mathematics, 1(2-3), 161-174. https://doi.org/10.1080/097 20502.1998.10700251
31. Staniec, C.J. (1987). Solving the multicommodity transshipment problem. Naval Postgraduate School Monterey CA. https://doi.org/10.21236/ADA199164
32. Uddin, M.S., Khan, A.R., Kibria, C.G. and Raeva, I. (2016). Improved least cost method to obtain a better IBFS to the transportation problem. Journal of Applied Mathematics and Bioinformatics, 6(2), 1.
33. Vancroonenburg, W., Della Croce, F., Goossens, D. and Spieksma, F.C. (2014). The red–blue transportation problem. European Journal of Operational Research, 237(3), 814-823. https://doi.org/10.1016/j.ejo r.2014.02.055
References
1. Adlakha, V. and Kowalski, K. (2003). A simple heuristic for solving small fixed-charge transportation problems. Omega, 31(3), 205-211. https://doi.org/10.1016/S0305-0483(03)00025-2
2. Agadaga, G.O. and Akpan, N.P. (2017). Transshipment problem and its variants: A review. Mathematical Theory and Modeling, 7(2), 19-32.
3. Ardjmand, E., Young II, W.A., Weckman, G.R., Bajgiran, O.S., Aminipour, B. and Park, N. (2016). Applying genetic algorithm to a new bi-objective stochastic model for transportation, location, and allocation of hazardous materials. Expert systems with applications, 51, 49-58. https://doi.org/10.1016/j.eswa.2015.12.036
4. Babu, M.A., Helal, M.A., Hasan, M.S. and Das, U.K. (2013). Lowest Allocation Method (LAM): a new approach to obtain feasible solution of transportation model. International Journal of Scientific and Engineering Research, 4(11), 1344-1348.
5. Balakrishnan, N. (1990). Modified Vogel's approximation method for the unbalanced transportation problem. Applied Mathematics Letters, 3(2), 9-11. https://doi.org/10.1016/0893-9659(90)90003-T
6. Barr, R., Elam, J., Glover, F. and Klingman, D. (1980). A network augmenting path basis algorithm for transshipment problems. In Extremal methods and systems analysis, pp. 250-274. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46414-0_12
7. Brouer, B.D., Desaulniers, G., Karsten, C.V. and Pisinger, D. (2015). A matheuristic for the liner shipping network design problem with transit time restrictions. In International Conference on Computational Logistics, pp. 195-208. Springer, Cham. https://doi.org/10.1007/978-3-319-24264-4_14
8. Buson, E., Roberti, R. and Toth, P. (2014). A reduced-cost iterated local search heuristic for the fixed-charge transportation problem. Operations Research, 62(5), 1095-1106. https://doi.org/10.1287/opre.2014.1288
9. Deep, K., Dubey, O.P. and Nagar, A. (2012). Incorporating Genetic Algorithms in Transport Management. In Proceedings of the International Conference on Soft Computing for Problem Solving (SocProS 2011) December 20-22, 2011, pp. 177-192. Springer, India. https://doi.org/10.1007/978-81-322-0487-9_18
10. Dubey, O.P., Deep, K. and Nagar, A.K. (2014). Goal Programming Approach to Trans-shipment Problem. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, pp. 337-343. Springer, New Delhi.
11. Dwivedi, R.K., Mehta, N.N. and Dubey, O.P. (2009). Interactive Decision Making in Prioritized Unbalanced Transportation Problems. IUP Journal of Operations Management, 8(1), 67. https://doi.org/10.1007/978-81-322-1602-5_37
12. Gupta, K. and Arora, S.R. (2012). An algorithm to find optimum cost time trade off pairs in a fractional capacitated transportation problem with restricted flow. International Journal Of Research In Social Sciences, 2(2), 418.
13. Hakim, M.A. (2012). An alternative method to find initial basic feasible solution of a transportation problem. Annals of pure and applied mathematics, 1(2), 203-209.
14. Joo, C.M. and Kim, B.S. (2014). Block transportation scheduling under delivery restriction in shipyard using meta-heuristic algorithms. Expert systems with applications, 41(6), 2851-2858. https://doi.org/10.1016/j.esw a.2013.10.020
15. Kaur, P., Verma, V. and Dahiya, K. (2017). Capacitated two-stage time minimization transportation problem with restricted flow. RAIRO-Operations Research, 51(2), 447-467. https://doi.org/10.1051/ro/2016033
16. Kavita, G. and Shri Ram, A. (2013). Linear plus linear fractional capacitated transportation problem with restricted flow. American Journal of Operations Research, 2013.
17. Khurana, A. (2015). Variants of transshipment problem. European Transport Research Review, 7(2), 1-19. https://doi.org/10.1007/s12544-015-0154-8
18. Kousalya, P. and Malarvizhi, P. (2016). A new technique for finding initial basic feasible solution to transportation problem. International Journal of Engineering and Management Research (IJEMR), 6(3), 26-32.
19. Lee, S.M. and Moore, L.J. (1973). Optimizing transportation problems with multiple objectives. AIIE transactions, 5(4), 333-338. https://doi.org/10.1080/05695557308974920
20. Loch, G.V. and da Silva, A.C.L., (2014). A computational study on the number of iterations to solve the transportation problem. Applied Mathematical Sciences, 8(92), 4579-4583. https://doi.org/10.12988/ams.2 014.46435
21. Misra, S. and Das, C. (1981). Three-dimensional transportation problem with capacity restriction. NZOR, 9(1), 47-58.
22. Prajapati, R., Pal, J., & Dubey, O. P. (2022). Transportation problem with restriction on arcs and their Solution by mathematical modeling. International Journal of Students’ Research in Technology & Management, 10(1), 67-74. https://doi.org/10.18510/ijsrtm.2022.1015
23. Quddoos, A., Javaid, S. and Khalid, M.M. (2012). A new method for finding an optimal solution for transportation problems. International Journal on Computer Science and Engineering, 4(7), 1271.
24. Quddoos, A., Javaid, S. and Khalid, M.M. (2016). A revised version of asm-method for solving transportation problem. International Journal Agriculture, Statistics, Science, 12(1), 267-272.
25. Rachev, S.T. and Rüschendorf, L. (1998). Mass Transportation Problems: Volume I: Theory (Vol. 1). Springer Science & Business Media.
26. Rachev, S.T. and Rüschendorf, L. (2006). Mass transportation problems: Applications. Springer Science & Business Media.
27. Samuel, A.E. and Venkatachalapathy, M. (2011). Modified Vogel’s approximation method for fuzzy transportation problems. Applied Mathematical Sciences, 5(28), 1367-1372.
28. Sharma, J.K. (2009). Operations research theory and application. Macmilan Publishers.
29. Sharma, V., Dahiya, K. and Verma, V. (2010). Capacitated two-stage time minimization transportation problem. Asia-Pacific journal of operational research, 27(04), 457-476. https://doi.org/10.1142 /S021759591000279X
30. Singh, P. and Saxena, P.K. (1998). Total shipping cost/completion-date trade-off in transportation problem with additional restriction. Journal of Interdisciplinary Mathematics, 1(2-3), 161-174. https://doi.org/10.1080/097 20502.1998.10700251
31. Staniec, C.J. (1987). Solving the multicommodity transshipment problem. Naval Postgraduate School Monterey CA. https://doi.org/10.21236/ADA199164
32. Uddin, M.S., Khan, A.R., Kibria, C.G. and Raeva, I. (2016). Improved least cost method to obtain a better IBFS to the transportation problem. Journal of Applied Mathematics and Bioinformatics, 6(2), 1.
33. Vancroonenburg, W., Della Croce, F., Goossens, D. and Spieksma, F.C. (2014). The red–blue transportation problem. European Journal of Operational Research, 237(3), 814-823. https://doi.org/10.1016/j.ejo r.2014.02.055
2. Agadaga, G.O. and Akpan, N.P. (2017). Transshipment problem and its variants: A review. Mathematical Theory and Modeling, 7(2), 19-32.
3. Ardjmand, E., Young II, W.A., Weckman, G.R., Bajgiran, O.S., Aminipour, B. and Park, N. (2016). Applying genetic algorithm to a new bi-objective stochastic model for transportation, location, and allocation of hazardous materials. Expert systems with applications, 51, 49-58. https://doi.org/10.1016/j.eswa.2015.12.036
4. Babu, M.A., Helal, M.A., Hasan, M.S. and Das, U.K. (2013). Lowest Allocation Method (LAM): a new approach to obtain feasible solution of transportation model. International Journal of Scientific and Engineering Research, 4(11), 1344-1348.
5. Balakrishnan, N. (1990). Modified Vogel's approximation method for the unbalanced transportation problem. Applied Mathematics Letters, 3(2), 9-11. https://doi.org/10.1016/0893-9659(90)90003-T
6. Barr, R., Elam, J., Glover, F. and Klingman, D. (1980). A network augmenting path basis algorithm for transshipment problems. In Extremal methods and systems analysis, pp. 250-274. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46414-0_12
7. Brouer, B.D., Desaulniers, G., Karsten, C.V. and Pisinger, D. (2015). A matheuristic for the liner shipping network design problem with transit time restrictions. In International Conference on Computational Logistics, pp. 195-208. Springer, Cham. https://doi.org/10.1007/978-3-319-24264-4_14
8. Buson, E., Roberti, R. and Toth, P. (2014). A reduced-cost iterated local search heuristic for the fixed-charge transportation problem. Operations Research, 62(5), 1095-1106. https://doi.org/10.1287/opre.2014.1288
9. Deep, K., Dubey, O.P. and Nagar, A. (2012). Incorporating Genetic Algorithms in Transport Management. In Proceedings of the International Conference on Soft Computing for Problem Solving (SocProS 2011) December 20-22, 2011, pp. 177-192. Springer, India. https://doi.org/10.1007/978-81-322-0487-9_18
10. Dubey, O.P., Deep, K. and Nagar, A.K. (2014). Goal Programming Approach to Trans-shipment Problem. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, pp. 337-343. Springer, New Delhi.
11. Dwivedi, R.K., Mehta, N.N. and Dubey, O.P. (2009). Interactive Decision Making in Prioritized Unbalanced Transportation Problems. IUP Journal of Operations Management, 8(1), 67. https://doi.org/10.1007/978-81-322-1602-5_37
12. Gupta, K. and Arora, S.R. (2012). An algorithm to find optimum cost time trade off pairs in a fractional capacitated transportation problem with restricted flow. International Journal Of Research In Social Sciences, 2(2), 418.
13. Hakim, M.A. (2012). An alternative method to find initial basic feasible solution of a transportation problem. Annals of pure and applied mathematics, 1(2), 203-209.
14. Joo, C.M. and Kim, B.S. (2014). Block transportation scheduling under delivery restriction in shipyard using meta-heuristic algorithms. Expert systems with applications, 41(6), 2851-2858. https://doi.org/10.1016/j.esw a.2013.10.020
15. Kaur, P., Verma, V. and Dahiya, K. (2017). Capacitated two-stage time minimization transportation problem with restricted flow. RAIRO-Operations Research, 51(2), 447-467. https://doi.org/10.1051/ro/2016033
16. Kavita, G. and Shri Ram, A. (2013). Linear plus linear fractional capacitated transportation problem with restricted flow. American Journal of Operations Research, 2013.
17. Khurana, A. (2015). Variants of transshipment problem. European Transport Research Review, 7(2), 1-19. https://doi.org/10.1007/s12544-015-0154-8
18. Kousalya, P. and Malarvizhi, P. (2016). A new technique for finding initial basic feasible solution to transportation problem. International Journal of Engineering and Management Research (IJEMR), 6(3), 26-32.
19. Lee, S.M. and Moore, L.J. (1973). Optimizing transportation problems with multiple objectives. AIIE transactions, 5(4), 333-338. https://doi.org/10.1080/05695557308974920
20. Loch, G.V. and da Silva, A.C.L., (2014). A computational study on the number of iterations to solve the transportation problem. Applied Mathematical Sciences, 8(92), 4579-4583. https://doi.org/10.12988/ams.2 014.46435
21. Misra, S. and Das, C. (1981). Three-dimensional transportation problem with capacity restriction. NZOR, 9(1), 47-58.
22. Prajapati, R., Pal, J., & Dubey, O. P. (2022). Transportation problem with restriction on arcs and their Solution by mathematical modeling. International Journal of Students’ Research in Technology & Management, 10(1), 67-74. https://doi.org/10.18510/ijsrtm.2022.1015
23. Quddoos, A., Javaid, S. and Khalid, M.M. (2012). A new method for finding an optimal solution for transportation problems. International Journal on Computer Science and Engineering, 4(7), 1271.
24. Quddoos, A., Javaid, S. and Khalid, M.M. (2016). A revised version of asm-method for solving transportation problem. International Journal Agriculture, Statistics, Science, 12(1), 267-272.
25. Rachev, S.T. and Rüschendorf, L. (1998). Mass Transportation Problems: Volume I: Theory (Vol. 1). Springer Science & Business Media.
26. Rachev, S.T. and Rüschendorf, L. (2006). Mass transportation problems: Applications. Springer Science & Business Media.
27. Samuel, A.E. and Venkatachalapathy, M. (2011). Modified Vogel’s approximation method for fuzzy transportation problems. Applied Mathematical Sciences, 5(28), 1367-1372.
28. Sharma, J.K. (2009). Operations research theory and application. Macmilan Publishers.
29. Sharma, V., Dahiya, K. and Verma, V. (2010). Capacitated two-stage time minimization transportation problem. Asia-Pacific journal of operational research, 27(04), 457-476. https://doi.org/10.1142 /S021759591000279X
30. Singh, P. and Saxena, P.K. (1998). Total shipping cost/completion-date trade-off in transportation problem with additional restriction. Journal of Interdisciplinary Mathematics, 1(2-3), 161-174. https://doi.org/10.1080/097 20502.1998.10700251
31. Staniec, C.J. (1987). Solving the multicommodity transshipment problem. Naval Postgraduate School Monterey CA. https://doi.org/10.21236/ADA199164
32. Uddin, M.S., Khan, A.R., Kibria, C.G. and Raeva, I. (2016). Improved least cost method to obtain a better IBFS to the transportation problem. Journal of Applied Mathematics and Bioinformatics, 6(2), 1.
33. Vancroonenburg, W., Della Croce, F., Goossens, D. and Spieksma, F.C. (2014). The red–blue transportation problem. European Journal of Operational Research, 237(3), 814-823. https://doi.org/10.1016/j.ejo r.2014.02.055