Skip to main content
SpringerLink
Log in

A Comprehensive Literature Review on Transportation Problems

  • Review Article
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

A systematic and organized overview of various existing transportation problems and their extensions developed by different researchers is offered in the review article. The article has gone through different research papers and books available in Google scholar, Sciencedirect, Z-library Asia, Springer.com, Research-gate, shodhganga, and many other E-learning platforms. The main purpose of the review paper is to recapitulate the existing form of various types of transportation problems and their systematic developments for the guidance of future researchers to help them classify the varieties of problems to be solved and select the criteria to be optimized.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

No data is used to prepare this article.

References

  1. Monge, G.: The founding fathers of optimal transport. Springer, Cham (1781)

    Google Scholar 

  2. Tolstoĭ, A.: On the history of the transportation and maximum flow problems. Math. Program. 91, 437–445 (1930)

    MathSciNet  Google Scholar 

  3. Kantorovich, L.V.: Mathematical methods of organizing and planning production. Manag. Sci. 6(4), 366–422 (1960). https://doi.org/10.1287/mnsc.6.4.366

    Article  MathSciNet  MATH  Google Scholar 

  4. Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20(1–4), 224–230 (1941). https://doi.org/10.1002/sapm1941201224

    Article  MathSciNet  MATH  Google Scholar 

  5. Koopmans, T.: A model of transportation. Act. Anal. Prod. Alloc. (1951). http://web.eecs.umich.edu/~pettie/matching/Koopmans-Reiter-mincost-flow-model-Cowlescommision-report.pdf. Accessed 12 Oct 2020

  6. Charnes, A., Cooper, W.W.: The stepping stone method of explaining linear programming calculations in transportation problems. Manag. Sci. 1(1), 49–69 (1954). https://doi.org/10.1287/mnsc.1.1.49

    Article  MathSciNet  MATH  Google Scholar 

  7. Dantzig, G.: Application of the simplex method to a transportation problem. Act. Anal. Prod. Alloc. (1951). https://ci.nii.ac.jp/naid/10021311930/. Accessed 12 Oct 2020

  8. Hitchcock, F.: The distribution of a product from several sources to numerous localities. Int. J. Pharm. Technol. 8(1), 3554–3570 (2016)

    Google Scholar 

  9. Sungeeta, S., Renu, T., Deepali, S.: A review on fuzzy and stochastic extensions of the Multi Index transportation problem. Yugoslav. J. Oper. Res. 27(1), 3–29 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gupta, R., Komal.: Literature Survey on Single and Multi-Objective Transportation Problems. Proceedings of International Conference on Sustainable Computing in Science, Technology and Management (SUSCOM), Amity University Rajasthan, Jaipur - India (2019)

  11. Klein, M.: A primal method for minimal cost flows with applications to the assignment and transportation problems. Manag. Sci. 14(3), 205–220 (1967). https://doi.org/10.1287/mnsc.14.3.205

    Article  MATH  Google Scholar 

  12. Lee, S.M., Moore, L.J.: Optimizing transportation problems with multiple objectives. AIIE Trans. 5(4), 333–338 (1973). https://doi.org/10.1080/05695557308974920

    Article  Google Scholar 

  13. Kwak, N., Schniederjans, M.J.: A goal programming model for improved transportation problem solutions. Omega 7, 367–370 (1979).https://www.sciencedirect.com/science/article/pii/0305048379900458. Accessed 12 Oct 2020

    Article  Google Scholar 

  14. Ahuja, R.K.: Algorithms for the minimax transportation problem. Nav. Res. Logist. Q. 33(4), 725–739 (1986). https://doi.org/10.1002/nav.3800330415

    Article  MathSciNet  MATH  Google Scholar 

  15. Currin, D.C.: Transportation problems with inadmissible routes. J. Oper. Res. Soc. 37(4), 387–396 (1986). https://doi.org/10.1057/jors.1986.66

    Article  MATH  Google Scholar 

  16. Shafaat, A., Goyal, S.K.: Resolution of degeneracy in transportation problems. J. Oper. Res. Soc. 39(4), 411–413 (1988). https://doi.org/10.1057/jors.1988.69

    Article  MATH  Google Scholar 

  17. Arsham, H., Kahn, A.B.: A simplex-type algorithm for general transportation problems: an alternative to stepping-stone. J. Oper. Res. Soc. 40(6), 581–590 (1989). https://doi.org/10.1057/jors.1989.95

    Article  MATH  Google Scholar 

  18. Kirca, Ö., Şatir, A.: A heuristic for obtaining an initial solution for the transportation problem. J. Oper. Res. Soc. 41(9), 865–871 (1990). https://doi.org/10.1057/jors.1990.124

    Article  MATH  Google Scholar 

  19. Goczyłla, K., Cielatkowski, J.: Optimal routing in a transportation network. Eur. J. Oper. Res. 87, 214–222 (1995)

    Article  MATH  Google Scholar 

  20. Adlakha, V., Kowalski, K.: An alternative solution algorithm for certain transportation problems. Int. J. Math. Educ. Sci. Technol. 30(5), 719–728 (2010). https://doi.org/10.1080/002073999287716

    Article  MathSciNet  MATH  Google Scholar 

  21. Minghe, S.: The transportation problem with exclusionary side constraints and two branch-and-bound algorithms. Eur. J. Oper. Res. 140, 629–647 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sharma, R., Gaur, A., Okunbor, D.: Management decision-making for transportation problems through goal programming. J. Acad. Bus. Econ. 4, 195 (2004)

    Google Scholar 

  23. Imam, T., Elsharawy, G., Gomah, M., Samy, I.: Solving transportation problem using object-oriented model. IJCSNS 9, 353 (2009)

    Google Scholar 

  24. Adlakha, V.: Alternate solutions analysis for transportation problems (2009). https://clutejournals.com/index.php/JBER/article/view/2354. Accessed 12 Oct 2020

  25. Pandian, P., Natarajan, G.: A new method for finding an optimal solution for transportation problems. Int. J. Math. Sci. Eng. Appl. 4, 59–65 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  27. Sharma, G., Abbas, S., Gupta, V.: Solving transportation problem with the various method of a linear programming problem. Asian J. Curr. Eng. Maths 1, 81–83 (2012)

    Google Scholar 

  28. Sharma, G., Abbas, S., Gupta, V.K.: Solving transportation problem with the help of integer programming problem. IOSR J. Eng. 2, 1274–1277 (2012)

    Article  Google Scholar 

  29. Joshi, R.V.: Optimization techniques for transportation problems of three variables. IOSR J. Math. 9, 46–50 (2013)

    Article  Google Scholar 

  30. Rekha, S., Srividhya, B., Vidya, S.: Transportation cost minimization: max–min penalty approach. IOSR J. Math. 10, 6–8 (2014)

    Article  Google Scholar 

  31. Azad, S., Hossain, M., Rahman, M.: An algorithmic approach to solve transportation problems with the average total opportunity cost method. Int. J. Sci. Res. Publ. 7, 262–270 (2017)

    Google Scholar 

  32. Singh, S.: Note on transportation problem with a new method for the resolution of degeneracy. Univers. J. Ind. Bus. Manag. 3, 26–36 (2015)

    Article  Google Scholar 

  33. Palanievel, M., Suganya, M.: A new method to solve transportation problem-Harmonic Mean approach. Eng. Technol. Open Access J. 2, 1–3 (2018)

    Google Scholar 

  34. Charnes, A., Klingman, D.: The more-for-less paradox in the distribution model. Cahiers du Centre d’Etudes de Recherche Operationelle 13, 11–22 (1971)

    MathSciNet  MATH  Google Scholar 

  35. Klingman, D., Russell, R.: The transportation problem with mixed constraints. J. Oper. Res. Soc. 25(3), 447–455 (1974). https://doi.org/10.1057/jors.1974.78

    Article  MathSciNet  MATH  Google Scholar 

  36. Robb, D.J.: The ‘more for less’ paradox in distribution models: an intuitive explanation. IIE Trans. 22(4), 377–378 (2007). https://doi.org/10.1080/07408179008964192

    Article  MathSciNet  Google Scholar 

  37. Arora, S., Ahuja, A.: A paradox in a fixed charge transportation problem. Indian J. Pure Appl. Math. 31, 809–822 (2000)

    MathSciNet  MATH  Google Scholar 

  38. Adlakha, V., Kowalski, K.: A heuristic method for ‘more-for-less’ in distribution-related problems. Int. J. Math. Educ. Sci. Technol. 32(1), 61–71 (2001). https://doi.org/10.1080/00207390117225

    Article  Google Scholar 

  39. Adlakha, V., Kowalski, K., Lev, B.: Solving transportation problems with mixed constraints. Int. J. Manag. Sci. Eng. Manag. 1(1), 47–52 (2006). https://doi.org/10.1080/17509653.2006.10670996

    Article  Google Scholar 

  40. Storøy, S.: The transportation paradox revisited (2007). http://web.ist.utl.pt/mcasquilho/compute/_linpro/2007Storoy.pdf. Accessed 13 Oct 2020

  41. Pandian, P., Natarajan, G.: Fourier methods for solving transportation problems with mixed constraints. Int. J. Contemp. Math. Sci. 5, 1385–1395 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Joshi, V., Gupta, N.: Linear fractional transportation problem with varying demand and supply Vishwas Deep Joshi–Nilama Gupta. Matematiche (Catania) (2011). https://doi.org/10.4418/2011.66.2.1

    Article  Google Scholar 

  43. Joshi, V.D., Gupta, N.: Identifying more-for-less paradox in the linear fractional transportation problem using objective matrix (2012). https://matematika.utm.my/index.php/matematika/article/view/572. Accessed 13 Oct 2020

  44. Pandian, P., Anuradha, D.: Path method for finding a more-for-less optimal solution to transportation problems. In: International Conference on Mathematical Computer Engineering (2013)

  45. George, A.O., Jude, O., Anderson, C.N.: Paradox algorithm in application of a linear transportation problem. Am. J. Appl. Math. Stat. 2, 10–15 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

  47. Agarwal, S., Sharma, S.: A shootout method for time minimizing transportation problem with mixed constraints. Am. J. Math. Manag. Sci. 39(4), 299–314 (2020). https://doi.org/10.1080/01966324.2020.1730274

    Article  Google Scholar 

  48. Hammer, P.L.: Time-minimizing transportation problems. Nav. Res. Logist. Q. 16(3), 345–357 (1969). https://doi.org/10.1002/nav.3800160307

    Article  MathSciNet  MATH  Google Scholar 

  49. Garfinkel, R.S., Rao, M.R.: The bottleneck transportation problem. Nav. Res. Logist. Q. 18(4), 465–472 (1971). https://doi.org/10.1002/nav.3800180404

    Article  MathSciNet  MATH  Google Scholar 

  50. Szwarc, W.: Some remarks on the time transportation problem. Nav. Res. Logist. Q. 18(4), 473–485 (1971). https://doi.org/10.1002/nav.3800180405

    Article  MathSciNet  MATH  Google Scholar 

  51. Sharma, J., Swarup, K.: Time minimizing transportation problems. In: Proceedings of the Indian Academy of Sciences (1977)

  52. Varadarajan, R.: An optimal algorithm for 2× n bottleneck transportation problem. Oper. Res. Lett. 10, 525–529 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  53. Geetha, S., Nair, K.P.: A stochastic bottleneck transportation problem. J. Oper. Res. Soc. 45(5), 583–588 (1994). https://doi.org/10.1057/jors.1994.86

    Article  MATH  Google Scholar 

  54. Nikolić, I.: Total time minimizing transportation problem. Yugosl. J. Oper. Res. 17, 125–133 (2007). https://doi.org/10.2298/YUJOR0701125N

    Article  MathSciNet  MATH  Google Scholar 

  55. Pandian, P., Natarajan, G.: A new method for solving bottleneck-cost transportation problems. In: International Mathematical Forum (2011)

  56. Jain, M., Saksena, P.K.: Time minimizing transportation problem with fractional bottleneck objective function. Yugosl. J. Oper. Res. 22, 115–129 (2012). https://doi.org/10.2298/YJOR100818004J

    Article  MathSciNet  MATH  Google Scholar 

  57. Kolman, P.: Time minimizing transportation problems with partial limitations of transported amount for transport participants. In: AIP Conference Proceedings, vol. 1648 (2015). https://doi.org/10.1063/1.4912945

  58. Waldherr, S., Poppenborg, J., Knust, S.: The bottleneck transportation problem with auxiliary resources. 4OR 13(3), 279–292 (2015). https://doi.org/10.1007/s10288-015-0284-9

    Article  MathSciNet  MATH  Google Scholar 

  59. Dhanapal, A., Sobana, V.E., Anuradha, D.: On solving bottleneck bi-criteria fuzzy transportation problems. Int. J. Eng. Technol. 7, 547–551 (2018)

    Article  Google Scholar 

  60. Vidhya, V., Ganesan, K.: A simple method for the solution of bottleneck-cost transportation problem under fuzzy environment. In: AIP Conference Proceedings, vol. 2277, no. 1, p. 090008 (2020). https://doi.org/10.1063/5.0026105

  61. Agarwal, S., Sharma, S.: A shootout method for time minimizing transportation problem with mixed constraints. Am. J. Math. Manag. Sci. (2020). https://doi.org/10.1080/01966324.2020.1730274

    Article  Google Scholar 

  62. Haley, K.B.: New methods in mathematical programming: the solid transportation problem. Oper. Res. 10(4), 448–463 (1962). https://doi.org/10.1287/opre.10.4.448

    Article  MATH  Google Scholar 

  63. Shell, E.: Distribution of product by several properties. In: Proceedings of the Second Symposium in Linear Programming (1955)

  64. Sharma, J.: Extensions and special cases of transportation problem: a survey (1978). Accessed 03 Dec 2020

  65. Haley, K.B.: The existence of a solution to the multi-index problem. J. Oper. Res. Soc. 16(4), 471–474 (1965). https://doi.org/10.1057/jors.1965.81

    Article  Google Scholar 

  66. Morávek, J., Vlach, M.: Letter to the Editor—On the necessary conditions for the existence of the solution of the multi-index transportation problem. Oper. Res. 15(3), 542–545 (1967). https://doi.org/10.1287/opre.15.3.542

    Article  MATH  Google Scholar 

  67. Smith, G.: A procedure for determining necessary and sufficient conditions for the existence of a solution to the multi-index problem. Aplikace matematiky 19(3), 177–183 (1974)

    MathSciNet  MATH  Google Scholar 

  68. Vlach, M.: Conditions for the existence of solutions of the three-dimensional planar transportation problem. Discrete Appl. Math. 13, 61–78 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  69. Junginger, W.: On representatives of multi-index transportation problems. Eur. J. Oper. Res. 66, 353–371 (1993)

    Article  MATH  Google Scholar 

  70. Kravtsov, M., Krachkovskii, A.: On some properties of three-index transportation polytopes (1999)

  71. Benterki, D., Zitouni, R., Keraghel, A., Benterki, D.: Elaboration and implantation of an algorithm solving a capacitated four-index transportation. Appl. Math. Sci. 1, 2643–2657 (2007). https://www.researchgate.net/publication/267118025. Accessed 14 Oct 2020

    MathSciNet  MATH  Google Scholar 

  72. Dhanapal, A., Pandian, P., Anuradha, D.: A new approach for solving solid transportation problems. Appl. Math. Sci. 4, 3603–3610 (2010)

    MathSciNet  MATH  Google Scholar 

  73. Pham, T., Dott, P.: Four indexes transportation problem with interval cost parameter for goods allocation planning. In: 2012 4th IEEE International Symposium on Logistics and Industrial Informatics (2012)

  74. Halder, S., Das, B., Panigrahi, G., Maiti, M.: Solving a solid transportation problem through fuzzy ranking. In: Lecture Notes Electrical Engineering, vol. 470, pp. 283–292 (2017). https://doi.org/10.1007/978-981-10-8585-7_27

  75. Bandopadhyaya, L., Puri, M.C.: Impaired flow multi-index transportation problem with axial constraints. J. Aust. Math. Soc. Ser. B 29, 296–309 (2018). https://doi.org/10.1017/S0334270000005828

    Article  MathSciNet  MATH  Google Scholar 

  76. Halder Jana, S., Giri, D., Das, B., Panigrahi, G., Jana, B., Maiti, M.: A solid transportation problem with additional constraints using Gaussian type-2 fuzzy environments. In: Springer Proceedings in Mathematics and Statistics, vol. 253, pp. 113–125. Springer, New York (2018)

  77. Das, A., Bera, U.K., Maiti, M.: A solid transportation problem in an uncertain environment involving a type-2 fuzzy variable. Neural Comput. Appl. 31(9), 4903–4927 (2019). https://doi.org/10.1007/s00521-018-03988-8

    Article  Google Scholar 

  78. Hirsch, W.M., Dantzig, G.B.: The fixed charge problem. Nav. Res. Logist. Q. 15(3), 413–424 (1968). https://doi.org/10.1002/nav.3800150306

    Article  MathSciNet  MATH  Google Scholar 

  79. Balinski, M.L.: Fixed-cost transportation problems. Nav. Res. Logist. Q. 8(1), 41–54 (1961). https://doi.org/10.1002/nav.3800080104

    Article  MATH  Google Scholar 

  80. Kowalski, K., Lev, B.: On step fixed-charge transportation problem. Omega 36, 913–917 (2008)

    Article  Google Scholar 

  81. Kuhn, H.W., Baumol, W.J.: An approximative algorithm for the fixed-charges transportation problem. Nav. Res. Logist. Q. 9(1), 1–15 (1962). https://doi.org/10.1002/nav.3800090102

    Article  MATH  Google Scholar 

  82. Robers, P., Cooper, L.: A study of the fixed charge transportation problem. Comput. Math. Appl. 2, 125–135 (1976)

    Article  MATH  Google Scholar 

  83. Diaby, M.: Successive linear approximation procedure for generalized fixed-charge transportation problems. J. Oper. Res. Soc. 42(11), 991–1001 (1991). https://doi.org/10.1057/jors.1991.189

    Article  MathSciNet  MATH  Google Scholar 

  84. Kennington, J., Unger, E.: New branch-and-bound algorithm for the fixed-charge transportation problem. Manag. Sci. 22(10), 1116–1126 (1976). https://doi.org/10.1287/mnsc.22.10.1116

    Article  MathSciNet  MATH  Google Scholar 

  85. Gray, P.: Technical note-exact solution of the fixed-charge transportation problem. Oper. Res. 19(6), 1529 (1971). https://doi.org/10.1287/opre.19.6.1529

    Article  MATH  Google Scholar 

  86. Sandrock, K.: A simple algorithm for solving small, fixed-charge transportation problems. J. Oper. Res. Soc. 39(5), 467–475 (1988)

    Article  MATH  Google Scholar 

  87. Palekar, U.S., Karwan, M.H., Zionts, S.: A branch-and-bound method for the fixed charge transportation problem. Manag. Sci. 36(9), 1092–1105 (1990). https://doi.org/10.1287/mnsc.36.9.1092

    Article  MathSciNet  MATH  Google Scholar 

  88. Diaby, M.: Successive linear approximation procedure for generalized fixed-charge transportation problems. J. Oper. Res. Soc. 42, 991–1001 (1991)

    Article  MATH  Google Scholar 

  89. Hultberg, T., Cardoso, D.: The teacher assignment problem: a special case of the fixed charge transportation problem. Eur. J. Oper. Res. 101, 463–473 (1997)

    Article  MATH  Google Scholar 

  90. Adlakha, V., Kowalski, K.: On the fixed-charge transportation problem. Omega 27(3), 381–388 (1999). https://doi.org/10.1016/S0305-0483(98)00064-4

    Article  Google Scholar 

  91. Raj, K.A.A.D., Rajendran, C.: A hybrid genetic algorithm for solving single-stage fixed-charge transportation problems. Technol. Oper. Manag. 2(1), 1–15 (2011). https://doi.org/10.1007/s13727-012-0001-2

    Article  Google Scholar 

  92. Altassan, K.M., Moustafa El-Sherbiny, M., Sasidhar, B., El-Sherbiny, M.M.: Near-Optimal Solution For The Step Fixed Charge Transportation Problem. Appl. Math. Inf. Sci. 7(2), 661–669 (2013). https://doi.org/10.12785/amis/072L41

    Article  MathSciNet  Google Scholar 

  93. Molla-Alizadeh-Zavardehi, S., et al.: Step fixed charge transportation problems via the genetic algorithm. Indian J. Sci. Technol. 7, 949 (2014)

    Article  Google Scholar 

  94. Sagratella, S., Schmidt, M., Sudermann-Merx, N.: The noncooperative fixed charge transportation problem. Eur. J. Oper. Res. 284(1), 373–382 (2020). https://doi.org/10.1016/j.ejor.2019.12.024

    Article  MathSciNet  MATH  Google Scholar 

  95. Roy, S.K., Midya, S., Weber, G.W.: Multi-objective multi-item fixed-charge solid transportation problem under twofold uncertainty. Neural Comput. Appl. 31(12), 8593–8613 (2019). https://doi.org/10.1007/s00521-019-04431-2

    Article  Google Scholar 

  96. Biswas, A., Shaikh, A.A., Niaki, S.T.A.: Multi-objective non-linear fixed charge transportation problem with multiple modes of transportation in crisp and interval environments. Appl. Soft Comput. J. 80, 628–649 (2019). https://doi.org/10.1016/j.asoc.2019.04.011

    Article  Google Scholar 

  97. Midya, S., Roy, S.K.: Multi-objective fixed-charge transportation problem using rough programming. Int. J. Oper. Res. 37(3), 377–395 (2020). https://doi.org/10.1504/IJOR.2020.105444

    Article  MathSciNet  Google Scholar 

  98. Singh, G., Singh, A.: Solving multi-objective fixed charged transportation problem using a modified particle swarm optimization algorithm. In: Lecture Notes on Data Engineering and Communications Technologies, vol. 53, pp. 373–386. Springer (2021)

  99. Mahapatra, D.R.: Multi-choice stochastic transportation problem involving Weibull distribution. Int. Optim. Control Theor. Appl. 4(1), 45–55 (2013). https://doi.org/10.11121/ijocta.01.2014.00154

    Article  MathSciNet  MATH  Google Scholar 

  100. Maity, G., Roy, S.K.: Solving multi-choice multi-objective transportation problem: a utility function approach. J. Uncertain. Anal. Appl. (2014). https://doi.org/10.1186/2195-5468-2-11

    Article  Google Scholar 

  101. Quddoos, A., ull Hasan, M.G., Khalid, M.M.: Multi-choice stochastic transportation problem involving a general form of distributions. J. Korean Phys. Soc. 3(1), 1–9 (2014). https://doi.org/10.1186/2193-1801-3-565

    Article  Google Scholar 

  102. Roy, S.K.: Transportation problem with multi-choice cost and demand and stochastic supply. J. Oper. Res. Soc. China 4(2), 193–204 (2016). https://doi.org/10.1007/s40305-016-0125-3

    Article  MathSciNet  MATH  Google Scholar 

  103. Ranarahu, N., Dash, J.K., Acharya, S.: Computation of Multi-choice Multi-objective Fuzzy Probabilistic Transportation Problem, pp. 81–95. Springer, Singapore (2019)

    Google Scholar 

  104. Agrawal, P., Ganesh, T.: Multi-choice stochastic transportation problem involving logistic distribution. Adv. Appl. Math. Sci. 18, 45–58 (2018)

    Google Scholar 

  105. Al Qahtani, H., El-Hefnawy, A., El-Ashram, M.M., Fayomi, A.: A goal programming approach to multichoice multiobjective stochastic transportation problems with extreme value distribution. Adv. Oper. Res. (2019). https://doi.org/10.1155/2019/9714137

    Article  MathSciNet  Google Scholar 

  106. Nayak, J., et al.: Generalized binary variable approach to solving Multi-Choice transportation problem-Indian Journals. https://www.indianjournals.com/ijor.aspx?target=ijor:ijesm&volume=6&issue=5&article=012. Accessed 27 Jan 2021

  107. Agrawal, P., Ganesh, T.: Solution of stochastic transportation problem involving multi-choice random parameter using Newton’s divided difference interpolation. J. Inf. Optim. Sci. (2020). https://doi.org/10.1080/02522667.2019.1694741

    Article  Google Scholar 

  108. Chanas, S., Delgado, M., Verdegay, J.L., Vila, M.A.: Interval and fuzzy extensions of classical transportation problems. Transp. Plan. Technol. 17(2), 203–218 (1993). https://doi.org/10.1080/03081069308717511

    Article  Google Scholar 

  109. Baidya, A., Bera, U.K., Maiti, M.: Multi-item interval-valued solid transportation problem with safety measure under fuzzy-stochastic environment. J. Transp. Secur. 6(2), 151–174 (2013). https://doi.org/10.1007/s12198-013-0109-z

    Article  Google Scholar 

  110. Rani, D., Gulati, T.R.: Fuzzy optimal solution of interval-valued fuzzy transportation problems. Adv. Intell. Syst. Comput. 258, 881–888 (2014). https://doi.org/10.1007/978-81-322-1771-8_76

    Article  Google Scholar 

  111. Yu, V.F., Hu, K.J., Chang, A.Y.: An interactive approach for the multi-objective transportation problem with interval parameters. Int. J. Prod. Res. 53(4), 1051–1064 (2015). https://doi.org/10.1080/00207543.2014.939236

    Article  Google Scholar 

  112. Ebrahimnejad, A.: Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sadhana Acad. Proc. Eng. Sci. 41(3), 299–316 (2016). https://doi.org/10.1007/s12046-016-0464-0

    Article  MathSciNet  MATH  Google Scholar 

  113. Henriques, C.O., Coelho, D.: Multiobjective Interval Transportation Problems: A Short Review, pp. 99–116. Springer, Cham (2017)

    Google Scholar 

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

    Article  Google Scholar 

  115. Ramesh, G., Sudha, G., Ganesan, K.: A novel approach for the solution of multi-objective interval transportation problem. In: Journal of Physics: Conference Series, vol. 1000, no. 1 (2018). https://doi.org/10.1088/1742-6596/1000/1/012010

  116. Malik, M., Gupta, S.K.: Goal programming technique for solving fully interval-valued intuitionistic fuzzy multiple objective transportation problems. Soft Comput. 24(18), 13955–13977 (2020). https://doi.org/10.1007/s00500-020-04770-6

    Article  Google Scholar 

  117. Bharati, S.K.: Transportation problem with interval-valued intuitionistic fuzzy sets: impact of a new ranking. Prog. Artif. Intell. (2021). https://doi.org/10.1007/s13748-020-00228-w

    Article  Google Scholar 

  118. Chanas, S., Kołodziejczyk, W., Machaj, A.: A fuzzy approach to the transportation problem. Fuzzy Sects Syst. 13, 211–221 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  119. Chanas, S., Kuchta, D.: A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst. 82(3), 299–305 (1996). https://doi.org/10.1016/0165-0114(95)00278-2

    Article  MathSciNet  Google Scholar 

  120. Tada, M., Ishii, H.: An integer fuzzy transportation problem. Comput. Math. Appl. 31, 71–87 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  121. Liu, S., Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153, 661–674 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  122. Gani, A.N., Razak, K.A.: Two-stage fuzzy transportation problem (2006). Accessed 14 Oct 2020

  123. Gupta, P., Mehlawat, M.: An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit. TOP 15, 114–137 (2007). https://link.springer.com/content/pdf/10.1007/s11750-007-0006-3.pdf. Accessed 14 Oct 2020

    Article  MathSciNet  MATH  Google Scholar 

  124. Li, L., Huang, Z., Da, Q., Hu, J.: A new method based on goal programming for solving transportation problem with fuzzy cost. In: Proceedings—International Symposium on Information Processing, ISIP 2008 and International Pacific Workshop on Web Mining and Web-Based Application, WMWA 2008, pp. 3–8 (2008). https://doi.org/10.1109/ISIP.2008.9

  125. Lin, F.: Solving the transportation problem with fuzzy coefficients using genetic algorithms. In: 2009 IEEE International Conference on Fuzzy Systems (2009)

  126. Pandian, P., Natarajan, G.: A new algorithm for finding an optimal fuzzy solution for fuzzy transportation problems. Appl. Math. Sci. 4, 79–90 (2010)

    MATH  Google Scholar 

  127. Güzel, N.: Fuzzy transportation problem with the fuzzy amounts and the fuzzy costs. World Appl. Sci. J. 8(5), 543–549 (2010)

    Google Scholar 

  128. Kumar, A., Kaur, A.: Application of classical transportation methods to find the fuzzy optimal solution of fuzzy transportation problems. Fuzzy Inf. Eng 1(1), 81–99 (2011). https://doi.org/10.1007/s12543-011-0068-7

    Article  MathSciNet  MATH  Google Scholar 

  129. Gani, A.N., Samuel, A.E., Anuradha, D.: Simplex type algorithm for solving fuzzy transportation problem. Tamsui Oxf. J. Inf. Math. Sci. 27(1), 89–98 (2011). https://doi.org/10.13140/2.1.1865.7929

    Article  Google Scholar 

  130. Kumar, B., Murugesan, S.: On fuzzy transportation problem using triangular fuzzy numbers with the modified, revised simplex method. Int. J. Eng. Sci. Technol. 4(2012), 285–294 (2012)

    Google Scholar 

  131. Ebrahimnejad, A.: A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl. Soft. Comput. 19, 171–176 (2014)

    Article  Google Scholar 

  132. Das, U.K., Ashraful-Babu, R., Khan, A., Helal, U.: 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 

  133. Elmaghraby, S.E.: Allocation under uncertainty when the demand has continuous D.F. Manag. Sci. 6(3), 270–294 (1960). https://doi.org/10.1287/mnsc.6.3.270

    Article  MathSciNet  MATH  Google Scholar 

  134. Williams, A.C.: A stochastic transportation problem. Oper. Res. 11(5), 759–770 (1963). https://doi.org/10.1287/opre.11.5.759

    Article  MathSciNet  MATH  Google Scholar 

  135. Szwarc, W.: The transportation problem with stochastic demand. Manag. Sci. 11(1), 33–50 (1964). https://doi.org/10.1287/mnsc.11.1.33

    Article  MathSciNet  Google Scholar 

  136. Wilson, D.: An a priori bounded model for transportation problems with stochastic demand and integer solutions. AIIE Trans. 4(3), 186–193 (1972). https://doi.org/10.1080/05695557208974848

    Article  Google Scholar 

  137. Cooper, L., Leblanc, L.J.: Stochastic transportation problems and other network-related convex problems. Nav. Res. Logist. Q. 24(2), 327–337 (1977). https://doi.org/10.1002/nav.3800240211

    Article  MathSciNet  MATH  Google Scholar 

  138. LeBlanc, L.J.: A heuristic approach for large scale discrete stochastic transportation-location problems. Comput. Math. Appl. 3, 87–94 (1977)

    Article  Google Scholar 

  139. Holmberg, K., Joernsten, K.: Cross decomposition applied to the stochastic transportation problem. Eur. J. Oper. Res. 17(1984), 361–368 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  140. Qi, L.: Forest iteration method for stochastic transportation problem. Math. Program. Study 25, 142–163 (1985). https://doi.org/10.1007/bfb0121081

    Article  MathSciNet  MATH  Google Scholar 

  141. Freling, R., Romeijn, H.E., Morales, D.R., Wagelmans, A.P.M.: A branch-and-price algorithm for the multiperiod single-sourcing problem. Oper. Res. 51(6), 922–939 (2003). https://doi.org/10.1287/opre.51.6.922.24914

    Article  MathSciNet  MATH  Google Scholar 

  142. Larsson, T., Patriksson, M., Rydergren, C., Daneva, M.: A comparison of feasible direction methods for the stochastic transportation problem. Comput. Optim. Appl. 46(3), 451–466 (2008). https://doi.org/10.1007/s10589-008-9199-0

    Article  MathSciNet  MATH  Google Scholar 

  143. Mahapatra, D.R., Roy, S.K., Biswal, M.P.: Stochastic based on multi-objective transportation problems involving normal randomness. Adv. Model. Optim. 12(2), 205–223 (2010)

    MathSciNet  MATH  Google Scholar 

  144. Ge, Y., Ishii, H.: Stochastic bottleneck transportation problem with flexible supply and demand quantity. Kybernetika 47, 560–571 (2011)

    MathSciNet  MATH  Google Scholar 

  145. Akdemir, H.G., Tiryaki, F., Günay Akdemir, H.: Bilevel stochastic transportation problem with exponentially distributed demand. Bitlis Eren Univ. J. Sci. Technol. (2012). https://doi.org/10.17678/beuscitech.47150

    Article  Google Scholar 

  146. Biswal, M.P., Samal, H.K.: Stochastic transportation problem with cauchy random variables and multi choice parameters (2013). Accessed 15 Oct 2020

  147. Hinojosa, Y., Puerto, J., Saldanha-da-Gama, F.: A two-stage stochastic transportation problem with fixed handling costs and a priori selection of the distribution channels. TOP 22, 1123–1147 (2014). https://link.springer.com/content/pdf/10.1007/s11750-014-0321-4.pdf. Accessed 15 Oct 2020

    Article  MathSciNet  MATH  Google Scholar 

  148. Stewart, T.J., Ittmann, H.W.: Two-stage optimization in a transportation problem. J. Oper. Res. Soc. 30(10), 897–904 (1979). https://doi.org/10.1057/jors.1979.210

    Article  MATH  Google Scholar 

  149. Fulya, M.G., Lin, A.L., Gen, M., Lin, L., Altiparmak, F.: A genetic algorithm for two-stage transportation problem using priority-based encoding. OR Spectr. (2006). https://doi.org/10.1007/s00291-005-0029-9

    Article  MathSciNet  MATH  Google Scholar 

  150. Tang, L., Gong, H.: A hybrid two-stage transportation and batch scheduling problem. Appl. Math. Model. 32(12), 2467–2479 (2008). https://doi.org/10.1016/j.apm.2007.09.028

    Article  MathSciNet  MATH  Google Scholar 

  151. Sudhakar, V.J., Kumar, V.N.: Solving the multiobjective two-stage fuzzy transportation problem by zero suffix method (2010). [Online]. Available: www.ccsenet.org/jmr. Accessed 22 Feb 2021

  152. Pandian, P., Natarajan, G.: Solving two-stage transportation problems. In: Communications in Computer and Information Science, 2011, vol. 140, CCIS, pp. 159–165 (2011). https://doi.org/10.1007/978-3-642-19263-0_20

  153. Raj, K.A.A.D., Rajendran, C.: A genetic algorithm for solving the fixed-charge transportation model: two-stage problem. Comput. Oper. Res. 39(9), 2016–2032 (2012). https://doi.org/10.1016/j.cor.2011.09.020

    Article  MATH  Google Scholar 

  154. Calvete, H.I., Galé, C., Iranzo, J.A.: An improved evolutionary algorithm for the two-stage transportation problem with the fixed charge at depots. OR Spectr. 38(1), 189–206 (2016). https://doi.org/10.1007/s00291-015-0416-9

    Article  MathSciNet  MATH  Google Scholar 

  155. Roy, S.K., Maity, G., Weber, G.W.: Multi-objective two-stage grey transportation problem using utility function with goals. Cent. Eur. J. Oper. Res. 25(2), 417–439 (2017). https://doi.org/10.1007/s10100-016-0464-5

    Article  MathSciNet  MATH  Google Scholar 

  156. Malhotra, R.: A polynomial algorithm for a two-stage time minimizing transportation problem. Opsearch 39(5–6), 251–266 (2002). https://doi.org/10.1007/bf03399188

    Article  MathSciNet  MATH  Google Scholar 

  157. Cosma, O., Pop, P.C., Sabo, C.: A novel hybrid genetic algorithm for the two-stage transportation problem with fixed charges associated to the routes. In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), Jan 2020, vol. 12011, LNCS, pp. 417–428 (2020). https://doi.org/10.1007/978-3-030-38919-2_34

  158. Khanna, S., Puri, M.C.: A paradox in linear fractional transportation problems with mixed constraints. Optimization 27(4), 375–387 (1993). https://doi.org/10.1080/02331939308843896

    Article  MathSciNet  MATH  Google Scholar 

  159. Stancu-Minasian, I.M.: Fractional Transportation Problem, pp. 336–364. Springer, Dordrecht (1997)

    Google Scholar 

  160. Joshi, V.D., Gupta, N.: Linear fractional transportation problem with varying demand and supply. Matematiche (Catania) LXVI, 3–12 (2011). https://doi.org/10.4418/2011.66.2.1

    Article  MathSciNet  MATH  Google Scholar 

  161. Saxena, A., Singh, P., Saxena, P.K.: Quadratic fractional transportation problem with additional impurity restrictions. J. Stat. Manag. Syst. 10(3), 319–338 (2007). https://doi.org/10.1080/09720510.2007.10701257

    Article  MATH  Google Scholar 

  162. Khurana, A., Arora, S.R.: The sum of a linear and a linear fractional transportation problem with the restricted and enhanced flow. J. Interdiscip. Math. 9(2), 373–383 (2006). https://doi.org/10.1080/09720502.2006.10700450

    Article  MathSciNet  MATH  Google Scholar 

  163. Liu, S.: Fractional transportation problem with fuzzy parameters. Soft Comput. (2015). https://doi.org/10.1007/s00500-015-1722-5

    Article  Google Scholar 

  164. Mohanaselvi, S., Ganesan, K.: A new approach for solving linear fuzzy fractional transportation problem. Int. J. Civ. Eng. Technol. 8(8), 1123–1129 (2017)

    Google Scholar 

  165. Anukokila, P., Anju, A., Radhakrishnan, B.: Optimality of intuitionistic fuzzy fractional transportation problem of type-2. Arab J. Basic Appl. Sci. 26(1), 519–530 (2019). https://doi.org/10.1080/25765299.2019.1691895

    Article  MATH  Google Scholar 

  166. Anukokila, P., Radhakrishnan, B.: Goal programming approach to the fully fuzzy fractional transportation problem. J. Taibah Univ. Sci. 13(1), 864–874 (2019). https://doi.org/10.1080/16583655.2019.1651520

    Article  Google Scholar 

  167. El Sayed, M.A., Abo-Sinna, M.A.: A novel approach for fully intuitionistic fuzzy multi-objective fractional transportation problem. Alex. Eng. J. 60(1), 1447–1463 (2021). https://doi.org/10.1016/j.aej.2020.10.063

    Article  Google Scholar 

  168. Rubin, P., Narasimhan, R.: Fuzzy goal programming with nested priorities. Fuzzy Sets Syst. 14, 115–129 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  169. Charnes, A., Cooper, W.W.: Management models and industrial applications of linear programming. Manag. Sci. 4(1), 38–91 (1957). https://doi.org/10.1287/mnsc.4.1.38

    Article  MathSciNet  MATH  Google Scholar 

  170. Zimmermann, H.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1, 45–55 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  171. Diaz, J., Ja, D.: Finding a complete description of all efficient solutions to a multiobjective transportation problem (1979). Accessed 22 Nov 2020

  172. Isermann, H.: The enumeration of all efficient solutions for a linear multiple-objective transportation problem. Nav. Res. Logist. Q. 26(1), 123–139 (1979). https://doi.org/10.1002/nav.3800260112

    Article  MathSciNet  MATH  Google Scholar 

  173. Leberling, H.: On finding compromise solutions in multicriteria problems using the fuzzy min-operator. Fuzzy Sets Syst. 6, 105–118 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  174. Majumdar, M., Mitra, T.: Dynamic optimization with a non-convex technology: the case of a linear objective function. Rev. Econ. Stud. 50, 143–151 (1983)

    Article  MATH  Google Scholar 

  175. Słowiński, R.: A multicriteria fuzzy linear programming method for water supply system development planning. Fuzzy Sets Syst. 19(3), 217–237 (1986). https://doi.org/10.1016/0165-0114(86)90052-7

    Article  MathSciNet  MATH  Google Scholar 

  176. Ringuest, J., Rinks, D.: Interactive solutions for the linear multiobjective transportation problem. Eur. J. Oper. Res. 32, 96–106 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  177. Bit, A., Biswal, M., Alam, S.: Fuzzy programming approach to multicriteria decision making transportation problem. Fuzzy Sets Syst. 50, 135–141 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  178. Verma, R., Biswal, M., Biswas, A.: Fuzzy programming technique to solve multi-objective transportation problems with some non-linear membership functions. Fuzzy Sets Syst. 91, 37–43 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  179. Gen, M., Li, Y., Gen, M., Ida, K.: Solving multi-objective transportation problem by spanning tree-based genetic algorithm. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 82, 2802–2810 (2000)

    Google Scholar 

  180. Das, S., Goswami, A., Alam, S.: Multiobjective transportation problem with interval cost, source and destination parameters. Eur. J. oper. Res. 117, 100–112 (1999)

    Article  MATH  Google Scholar 

  181. Li, L., Lai, K.: A fuzzy approach to the multiobjective transportation problem. Comput. Oper. Res. 27, 43–57 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  182. Abd El-Wahed, W.: A multi-objective transportation problem under fuzziness Fuzzy Approach. Fuzzy Sets Syst. 117, 27–33 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  183. Ammar, E., Youness, E.: Study on multiobjective transportation problem with fuzzy numbers. Appl. Math. Comput. 166, 241–253 (2005)

    MathSciNet  MATH  Google Scholar 

  184. Abd El-Wahed, W.F., Lee, S.M.: Interactive fuzzy goal programming for multi-objective transportation problems. Omega 34, 158–166 (2006). https://doi.org/10.1016/j.omega.2004.08.006

    Article  Google Scholar 

  185. Zangiabadi, M., Maleki, H.R.: Fuzzy goal programming for multiobjective transportation problems. J. Appl. Math. Comput. 24(1–2), 449–460 (2007). https://doi.org/10.1007/BF02832333

    Article  MathSciNet  MATH  Google Scholar 

  186. Lau, H.C.W., Chan, T.M., Tsui, W.T., Chan, F.T.S., Ho, G.T.S., Choy, K.L.: A fuzzy guided multi-objective evolutionary algorithm model for solving transportation problem. Expert Syst. Appl. 36(4), 8255–8268 (2009). https://doi.org/10.1016/j.eswa.2008.10.031

    Article  Google Scholar 

  187. Lohgaonkar, M., Bajaj, V.: Fuzzy approach to solve the multi-objective capacitated transportation problem. Int. J. Bioinform. 2, 10–14 (2010)

    Google Scholar 

  188. Pal, B.B., Kumar, M., Sen, S.: Priority based fuzzy goal programming approach for fractional multilevel programming problems. Int. Rev. Fuzzy Math. 6(2), 1–14 (2011)

    Google Scholar 

  189. Zaki, S., Allah, A.A., Geneedi, H., Elmekawy, A.; Efficient multiobjective genetic algorithm for solving transportation, assignment, and transshipment problems (2012). Accessed 22 Nov 2020

  190. Maity, G., Roy, S.K.: Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand. Int. J. Manag. Sci. Eng. Manag. 11(1), 62–70 (2016). https://doi.org/10.1080/17509653.2014.988768

    Article  Google Scholar 

  191. Roy, S.K., Maity, G., Weber, G.-W.: Multi-objective two-stage grey transportation problem using utility function with goals. Artic. Cent. Eur. J. Oper. Res. (2017). https://doi.org/10.1007/s10100-016-0464-5

    Article  MATH  Google Scholar 

  192. Biswas, A., Shaikh, A., Niaki, S.: Multi-objective non-linear fixed charge transportation problem with multiple modes of transportation in crisp and interval environments. Appl. Soft. Comput. 80, 628–649 (2019)

    Article  Google Scholar 

  193. Bera, R.K., Mondal, S.K.: Analyzing a two-staged multi-objective transportation problem under quantity dependent credit period policy using q-fuzzy number. Int. J. Appl. Comput. Math. (2020). https://doi.org/10.1007/s40819-020-00901-7

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

First author (Yadvendra Kacher) acknowledges the financial support as Junior research fellowship (JRF) received from CSIR (Govt. of India) through HRDG(CSIR) senction Letter No./File No.: 09/1032(0019)/2019-EMR-I.

Author information

Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, 211004, India

    Yadvendra Kacher & Pitam Singh

Authors
  1. Yadvendra Kacher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pitam Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both the authors contributed equally in developing the whole article.

Corresponding author

Correspondence to Pitam Singh.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kacher, Y., Singh, P. A Comprehensive Literature Review on Transportation Problems. Int. J. Appl. Comput. Math 7, 206 (2021). https://doi.org/10.1007/s40819-021-01134-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-01134-y

Keywords

Search

Navigation