A heuristic for obtaining better initial feasible solution to the transportation problem


Vogel’s Approximation Method (VAM) is known as the best algorithm for generating an efficient initial feasible solution to the transportation problem. We demonstrate that VAM has some limitations and computational blunders. To overcome these limitations we develop an Improved Vogel’s Approximation Method (IVAM) by correcting these blunders. It is compared with VAM on obtained initial feasible solutions to a numerical example problem. Reduction in the total transportation cost over VAM by IVAM is found to be 2.27%. Besides, we have compared IVAM with each of twelve previously developed methods including VAM on solutions to numerical problems. IVAM leads to the minimal total cost solutions to seven, better solutions to four and the same better solution to the remaining one. Finally, a statistical analysis has been performed over the results of 1500 randomly generated transportation problems with fifteen distinct dimensions, where each of them has 100 problems instances. This analysis has demonstrated better performance of IVAM over VAM by reducing the total transportation cost in 71.8% of solved problems, especially for large size problems. Thus IVAM outperforms VAM by providing better initial feasible to the transportation problem.

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  1. 1.

    Hochbaum, D.S., Woeginger, G.J.: A linear-time algorithm for the bottleneck transportation problem with a fixed number of sources. Oper Res Lett 24(1), 25–28 (1999)

    Article  Google Scholar 

  2. 2.

    Baidya, A.: Stochastic supply chain, transportation models: implementations and benefits. OPSEARCH (2019). https://doi.org/10.1007/s12597-019-00370-7

    Article  Google Scholar 

  3. 3.

    Sharma, R.R.K., Sharma, K.D.: A new dual based procedure for the transportation problem. Eur. J. Oper. Res. 122(3), 611–624 (2000)

    Article  Google Scholar 

  4. 4.

    Sabbagh, M.S., Ghafari, H., Mousavi, S.R.: A new hybrid algorithm for the balanced transportation problem. Comput. Ind. Eng. 82, 115–126 (2015)

    Article  Google Scholar 

  5. 5.

    Dash, S., Mohanty, S.P.: Uncertain transportation model with rough unit cost, demand and supply. OPSEARCH (2018). https://doi.org/10.1007/s12597-017-0317-6

    Article  Google Scholar 

  6. 6.

    Liu, S.: The total cost bounds of the transportation problem with varying demand and supply. Omega 31(4), 247–251 (2003)

    Article  Google Scholar 

  7. 7.

    Juman, Z.A.M.S., Hoque, M.A.: A heuristic solution technique to attain the minimal total cost bounds of transporting a homogeneous product with varying demands and supplies. Eur. J. Oper. Res. 239(1), 146–156 (2014)

    Article  Google Scholar 

  8. 8.

    Ahmad, F., Adhami, A.Y.: Total cost measures with probabilistic cost function under varying supply and demand in transportation problem. OPSEARCH (2019). https://doi.org/10.1007/s12597-019-00364-5

    Article  Google Scholar 

  9. 9.

    Khurana, A., Adlakha, V.: On multi-index fixed charge bi-criterion transportation problem. OPSEARCH 52, 733 (2015). https://doi.org/10.1007/s12597-015-0212-y

    Article  Google Scholar 

  10. 10.

    Adlakha, V., Kowalski, K., Vemuganti, R.R.: Heuristic algorithms for the fixed-charge transportation problem. OPSEARCH (2006). https://doi.org/10.1007/BF03398770

    Article  Google Scholar 

  11. 11.

    Gupta, S., Ali, I., Ahmed, A.: Multi-objective capacitated transportation problem with mixed constraint: a case study of certain and uncertain environment. OPSEARCH (2018). https://doi.org/10.1007/s12597-018-0330-4

    Article  Google Scholar 

  12. 12.

    Gupta, K., Arora, R.: More for less method to minimize the unit transportation cost of a capacitated transportation problem with bounds on rim conditions. OPSEARCH (2017). https://doi.org/10.1007/s12597-016-0288-z

    Article  Google Scholar 

  13. 13.

    Charnes, A., Cooper, W.W., Henderson, A.: An Introduction to Linear programming. Wiley, New Work (1953)

    Google Scholar 

  14. 14.

    Reinfeld, N.V., Vogel, W.R.: Mathematical Programming. Prentice-Hall, Englewood Cliffs (1958)

    Google Scholar 

  15. 15.

    Babu, M.A., Helal, M.A., Hasan, M.S., Das, U.K.: Lowest allocation method (LAM): a new approach to obtain feasible solution of transportation model. Int J Sci Eng Res 4(11), 1344–1348 (2013)

    Google Scholar 

  16. 16.

    Babu, M.A., Helal, M.A., Hasan, M.S., Das, U.K.: Implied cost method (ICM): an alternative approach to find the feasible solution of transportation problem. Glob J Sci Front Res F Math Decis Sci 14(1), 5–13 (2014)

    Google Scholar 

  17. 17.

    Ali, M.A.M., Sik, Y.H.: Transportation problem: a special case for linear programing problems in mining engineering. Int J Min Sci Technol 22(3), 371–377 (2012)

    Article  Google Scholar 

  18. 18.

    Soomro, A.S., Junaid, M., Tularam, G.A.: Modified Vogel’s approximation method for solving transportation problems. Math Theory Model 5(4), 32–42 (2015)

    Google Scholar 

  19. 19.

    Alkubaisi, M.: Modified VOGEL method to find initial basic feasible solution (IBFS)—introducing a new methodology to find best IBFS. Bus Manag Res 4(2), 22–36 (2015)

    Article  Google Scholar 

  20. 20.

    Akpan, S., Usen, J., Ajah, O.: A modified Vogel approximation method for solving balanced transportation problems. Am Sci Res J Eng Technol Sci (ASRJETS) 14(3), 289–302 (2015)

    Google Scholar 

  21. 21.

    Kasana, H.S., Kumar, K.D.: Introductory Operations Research: Theory and Applications. Springer, New Delhi (2005)

    Google Scholar 

  22. 22.

    Kirca, O., Satir, A.: A heuristic for obtaining an initial solution for the transportation problem. J. Oper. Res. Soc. 41(9), 865–871 (1990)

    Article  Google Scholar 

  23. 23.

    Korukoğlu, S., Balli, S.: An improved Vogel’s approximation method for the transportation problem. Math Comput Appl 16(2), 370–381 (2011)

    Google Scholar 

  24. 24.

    Ullah, M.W., Uddin, M.A., Kawser, R.: A modified Vogel’s approximation method for obtaining a good primal solution of transportation problems. Ann Pure Appl Math 11(1), 63–71 (2016)

    Google Scholar 

  25. 25.

    Mathirajan, M., Meenakshi, B.: Experimental analysis of some variants of Vogel’s approximation method. Asia Pac J Oper Res 21, 447–462 (2004)

    Article  Google Scholar 

  26. 26.

    Shimshak, D.G., Kaslik, J.A., Barclay, T.D.: A modification of Vogel’s approximation method through the use of heuristic. INEOR 19, 259–263 (1981)

    Google Scholar 

  27. 27.

    Goyal, S.K.: Improving VAM for unbalanced transportation problems. J. Oper. Res. Soc. 35(12), 1113–1114 (1984)

    Article  Google Scholar 

  28. 28.

    Balakrishnan, N.: Modified Vogel’s approximation method for the unbalanced transportation problem. Appl Math Lett 3(2), 9–11 (1990)

    Article  Google Scholar 

  29. 29.

    Ramakrishnan, C.S.: An improvement to Goyal’s modified VAM for the unbalanced transportation problem. J. Oper. Res. Soc. 39(6), 609–610 (1988)

    Article  Google Scholar 

  30. 30.

    Brenner, U.: A faster polynomial algorithm for the unbalanced Hitchcock transportation problem. Oper Res Lett 36(4), 408–413 (2008)

    Article  Google Scholar 

  31. 31.

    Juman, Z.A.M.S., Hoque, M.A.: An efficient heuristic to obtain a better initial feasible solution to the transportation problem. Appl. Soft Comput. 34, 813–826 (2015)

    Article  Google Scholar 

  32. 32.

    Juman, Z.A.M.S., Hoque, M.A., Buhari, M.I.: A sensitivity analysis and an implementation of the well-known Vogel’s approximation method for solving an unbalanced transportation problem, Malays. J Sci 32(1), 66–72 (2013)

    Google Scholar 

  33. 33.

    Vasko, F.J., Storozhyshina, N.: Balancing a transportation problem: is it really that simple? OR Insight 24(3), 205–214 (2011)

    Article  Google Scholar 

  34. 34.

    Das, U.K., Babu, M.A., Khan, A.R., Uddin, M.S.: Advanced Vogel’s approximation method (AVAM): a new approach to determine penalty cost for better feasible solution of transportation problem. Int J Eng Res Technol (IJERT) 3(1), 182–187 (2014)

    Article  Google Scholar 

  35. 35.

    Das, U.K., Babu, M.A., Khan, A.R., Uddin, M.S.: Logical development of Vogel’s approximation method (LD-VAM): an approach to find basic feasible solution of transportation problem. Int J Sci Technol Res 3(2), 42–48 (2014)

    Google Scholar 

  36. 36.

    Babu, M.A., Das, U.K., Khan, A.R., Uddin, M.S.: A simple experimental analysis on transportation problem: a new approach to allocate zero supply or demand for all transportation algorithm. Int J Eng Res Appl (IJERA) 4(1), 418–422 (2014)

    Google Scholar 

  37. 37.

    Das, U.K., Babu, M.A., Uddin, M.S.: OTPA-Optimized Transportation Problem Algorithm, A Web Based Software Tool. http://www.otpa.info. (2014c)

  38. 38.

    Taha, H.A.: TORA-Temporary Ordered Routing Algorithm

  39. 39.

    DasGupta, A.: Normal approximations and the central limit theorem. In: Fundamentals of Probability: A First Course. Springer texts in statistics, Springer, New York (2010). https://doi.org/10.1007/978-1-4419-5780-1_10

    Google Scholar 

  40. 40.

    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I. Wiley, New York (1968)

    Google Scholar 

  41. 41.

    Feller, W.: Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1971)

    Google Scholar 

  42. 42.

    Pitman, J.: Probability. Springer, New York (1992)

    Google Scholar 

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Babu, M.A., Hoque, M.A. & Uddin, M.S. A heuristic for obtaining better initial feasible solution to the transportation problem. OPSEARCH 57, 221–245 (2020). https://doi.org/10.1007/s12597-019-00429-5

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  • Transportation problem
  • VAM
  • IVAM
  • Initial feasible solution
  • Minimal cost solution