Annals of Operations Research

, Volume 269, Issue 1–2, pp 667–692 | Cite as

Bi-criteria transportation problem with multiple parameters

  • Sonia Singh
  • Shalabh Singh


This paper introduces a bi-criteria transportation problem with multiple parameters which brings together the concept of price discrimination from the area of marketing management to the world of multi objective transportation problems. The problem deals with two objectives, the overall shipment cost and the bottleneck time of shipment. Both the objectives are of minimization type and have multi-choice coefficients pertaining to differential marketing strategies or different modes of transportation available. First, the problem with minimum availability and demand of goods is solved and then the problem is extended to the case of interval demand and supply. By iteratively solving multi-choice variants of a cost minimizing transportation problem/minimum cost flow problem, all Pareto optimal time–cost pairs are obtained. The proposed algorithm for both the variants is successfully implemented and solved using the CPLEX optimization package.


Multi-choice programming Transportation problem Time–cost trade-off 

Mathematics Subject Classification

90C29 90B50 90B06 



We are thankful to Mr. Ankit Khandelwal, Director, Analytics and Optimization Solutions, FICO, Singapore for his painstaking efforts to provide valuable comments and suggestions. We are also grateful to the anonymous reviewers for their invaluable suggestions and feedback that helped us to enhance our study.


  1. Ahuja, R. K. (1986). Algorithms for the minimax transportation problem. Naval Research Logistics Quarterly, 33(4), 725–739.CrossRefGoogle Scholar
  2. Aneja, Y. P., & Nair, K. P. K. (1979). Bicriteria transportation problem. Management Science, 25(1), 73–78.CrossRefGoogle Scholar
  3. Appa, G. M. (1973). The transportation problem and its variants. Operational Research Quarterly, 24, 79–99.CrossRefGoogle Scholar
  4. Bhatia, H. L., Swarup, K., & Puri, M. C. (1977). A procedure for time minimization transportation problem. Indian Journal of Pure and Applied Mathematics, 8(8), 920–929.Google Scholar
  5. Biswal, M. P., & Acharya, S. (2009). Transformation of a multi-choice linear programming problem. Applied Mathematics and Computation, 210(1), 182–188.CrossRefGoogle Scholar
  6. Biswal, M. P., & Acharya, S. (2011). Solving multi-choice linear programming problems by interpolating polynomials. Mathematical and Computer Modelling, 54(5), 1405–1412.CrossRefGoogle Scholar
  7. Chakraborty, D., Jana, D. K., & Roy, T. K. (2015). A new approach to solve multi-objective multi-choice multi-item Atanassov’s intuitionistic fuzzy transportation problem using chance operator. Journal of Intelligent and Fuzzy Systems, 28(2), 843–865.Google Scholar
  8. Chang, C. T. (2007). Multi-choice goal programming. Omega, 35(4), 389–396.CrossRefGoogle Scholar
  9. Chang, C. T. (2008). Revised multi-choice goal programming. Applied Mathematical Modelling, 32(12), 2587–2595.CrossRefGoogle Scholar
  10. Chang, C. T., Chen, H. M., & Zhuang, Z. Y. (2012). Multi-coefficients goal programming. Computers and Industrial Engineering, 62(2), 616–623.CrossRefGoogle Scholar
  11. Garfinkel, R. S., & Rao, M. R. (1971). The bottleneck transportation problem. Naval Research Logistics Quarterly, 18(4), 465–472.CrossRefGoogle Scholar
  12. Glickman, T. S., & Berger, P. D. (1977). Technical notecost/completion-date tradeoffs in the transportation problem. Operations Research, 25(1), 163–168.CrossRefGoogle Scholar
  13. Hammer, P. L. (1969). Time-minimizing transportation problems. Naval Research Logistics Quarterly, 16(3), 345–357.CrossRefGoogle Scholar
  14. Hammer, P. L. (1971). Communication on the bottleneck transportation problem and some remarks on the time transportation problem. Naval Research Logistics Quarterly, 18(4), 487–490.CrossRefGoogle Scholar
  15. Healy, W. C, Jr. (1964). Multiple choice programming (A procedure for linear programming with zero-one variables). Operations Research, 12(1), 122–138.CrossRefGoogle Scholar
  16. Liao, C. N. (2009). Formulating the multi-segment goal programming. Computers and Industrial Engineering, 56(1), 138–141.CrossRefGoogle Scholar
  17. Mahapatra, D. R., Roy, S. K., & Biswal, M. P. (2013). Multi-choice stochastic transportation problem involving extreme value distribution. Applied Mathematical Modelling, 37(4), 2230–2240.CrossRefGoogle Scholar
  18. Maity, G., & Roy, S. K. (2014). Solving multi-choice multi-objective transportation problem: a utility function approach. Journal of Uncertainty Analysis and Applications, 2(1), 11.CrossRefGoogle Scholar
  19. Orlin, J. B. (1993). A faster strongly polynomial minimum cost flow algorithm. Operations Research, 41(2), 338–350.CrossRefGoogle Scholar
  20. Prakash, S., Agarwal, A. K., & Shah, S. (1998). Nondominated solutions of cost–time trade-off transportation and assignment problems. Opsearch, 25(2), 126–131.Google Scholar
  21. Prakash, S., Kumar, P., Prasad, B., & Gupta, A. (2008). Pareto optimal solutions of a cost–time trade-off bulk transportation problem. European Journal of Operational Research, 188(1), 85–100.CrossRefGoogle Scholar
  22. Roy, S. K., Maity, G., Weber, G. W., & Gök, S. Z. A. (2017). Conic scalarization approach to solvemulti-choice multi-objective transportation problem with interval goal. Annals of Operations Research, 253(1), 599–620.CrossRefGoogle Scholar
  23. Sharma, J. K., & Swarup, K. (1977). Time minimizing transportation problems. Proceedings of the Indian Academy of Sciences-Section A, 86(6), 513–518.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Indian Institute of ManagementLucknowIndia

Personalised recommendations