Annals of Operations Research

, Volume 253, Issue 1, pp 599–620 | Cite as

Conic scalarization approach to solve multi-choice multi-objective transportation problem with interval goal

  • Sankar Kumar RoyEmail author
  • Gurupada Maity
  • Gerhard Wilhelm Weber
  • Sirma Zeynep Alparslan Gök


This paper explores the study of multi-choice multi-objective transportation problem (MCMTP) under the light of conic scalarizing function. MCMTP is a multi-objective transportation problem (MOTP) where the parameters such as cost, demand and supply are treated as multi-choice parameters. A general transformation procedure using binary variables is illustrated to reduce MCMTP into MOTP. Most of the MOTPs are solved by goal programming (GP) approach, but the solution of MOTP may not be satisfied all times by the decision maker when the objective functions of the proposed problem contains interval-valued aspiration levels. To overcome this difficulty, here we propose the approaches of revised multi-choice goal programming (RMCGP) and conic scalarizing function into the MOTP, and then we compare among the solutions. Two numerical examples are presented to show the feasibility and usefulness of our paper. The paper ends with a conclusion and an outlook on future studies.


Transportation problem Multi-objective decision making Multi-choice programming Goal programming Conic scalarization Interval uncertainty 



The second author is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [JRF(UGC)] scheme: Sanctioned letter number [F.17-130/1998(SA-I)] dated 26/06/2014. The authors are also very much thankful to the anonymous reviewers for their comments to improve the quality of the paper.


  1. Benson, H. P. (1979). An improved definition of proper efficiency for vector maximization with respect to cones. Journal of Mathematical Analysis and Applications, 71, 232–241.CrossRefGoogle Scholar
  2. Charnes, A., & Cooper, W. W. (1957). Management models and industrial applications of linear programming. Management Science, 4(1), 38–91.CrossRefGoogle Scholar
  3. Charnes, A., Cooper, W. W., & Ferguson, R. (1955). Optimal estimation of executive compensation by linear programming. Management Science, 1, 138–151.CrossRefGoogle Scholar
  4. Chang, C. T. (2007). Multi-choice goal programming. Omega, 35, 389–396.CrossRefGoogle Scholar
  5. Chang, C. T. (2008). Revised multi-choice goal programming. Applied Mathematical Modelling, 32, 2587–2595.CrossRefGoogle Scholar
  6. Ehrgott, M., Waters, C., Kasimbeyli, R., & Ustun, O. (2009). Multi-objective programming and multi-attribute utility functions in portfolio optimization. INFOR Information Systems and Operational Research, 47, 31–42.CrossRefGoogle Scholar
  7. Gasimov, R. N. (2001). Characterization of the Benson proper efficiency and scalarization in nonconvex vector optimization, Multiple Criteria Decision Making in the New Millennium. Book Series: Lecture Notes in Economics and Mathematical Systems, 507, 189–198.Google Scholar
  8. Gasimov, R. N. (2002). Augmented Lagrangian duality and non-differentiable optimization methods in non-convex programming. Journal of Global Optimization, 24, 187–203.CrossRefGoogle Scholar
  9. Gasimov, R. N., & Ozturk, G. (2006). Separation via polyhedral conic functions. Optimization Methods and Software, 21(4), 527–540.CrossRefGoogle Scholar
  10. Hannan, E. L. (1985). An assessment of some of the criticisms of goal programming. Journal Computers and Operations Research, 12(6), 525–541.CrossRefGoogle Scholar
  11. Ignizio, J. P. (1976). Goal programming and extensions. Lexington, MA: Lexington Books.Google Scholar
  12. Ignizio, J. P. (1978). A review of goal programming: A tool for multi-objective analysis. The Journal of the Operational Research Society, 29(11), 1109–1119.CrossRefGoogle Scholar
  13. Ignizio, J. P., & Romero, C. (2003). Goal programming. In H. Bidgoli (Ed.), Encyclopedia of information system (Vol. 2, pp. 489–500). San Diego, CA: Academic Press.CrossRefGoogle Scholar
  14. Lee, S.M., & Olson, D. (2000). Goal programming, In: Gal T., Stewart T. J., Hanne T. (Eds.), Multi-criteria decision making: Advances in MCDM models, algorithms, theory, and applications. Boston: Kluwer, Chapter 8, pp. 569–581.Google Scholar
  15. Lee, A. H. I., Kang, H.-Y., Yang, C.-Y., & Lin, C.-Y. (2010). An evaluation framework for product planning using FANP, QFD and multi-choice goal programming. International Journal of Production and Research, 48(13), 3977–3997.CrossRefGoogle Scholar
  16. Li, L., & Lai, K. K. (2000). A fuzzy approach to the multi-objective transportation problem. Computers and Operations Research, 27, 43–57.CrossRefGoogle Scholar
  17. Liao, C. N., & Kao, H. P. (2010). Supplier selection model using Taguchi loss function, analytical hierarchy process and multi-choice goal programming. Computers and Industrial Engineering, 58, 571–577.CrossRefGoogle Scholar
  18. 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
  19. Maity, G., & Roy, S. K. (2014). Solving multi-choice multi-objective transportation problem: A utility function approach. Journal of Uncertainty Analysis and Applications, 2, 11. doi: 10.1186/2195-5468-2-11.CrossRefGoogle Scholar
  20. Maity, G., & Roy, S. K. (2016). Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand. International Journal of Management Science and Engineering Management, 11(1), 62–70.CrossRefGoogle Scholar
  21. Midya, S., & Roy, S. K. (2014). Single-sink, fixed-charge, multi-objective, multi-index stochastic transportation problem. American Journal of Mathematical and Management Sciences, 33, 300–314.CrossRefGoogle Scholar
  22. Miettinen, K. (1999). Nonlinear multi-objective Optimization. Boston: Kluwer.Google Scholar
  23. Paksoy, T., & Chang, C. T. (2010). Revised multi-choice goal programming for multi-period, multi-stage inventory controlled supply chain model with popup stores in Guerilla marketing. Applied Mathematical Modelling, 34, 3586–3598.CrossRefGoogle Scholar
  24. Rahmati, S. H. A., Hajipour, V., & Niaki, S. T. A. (2013). A soft-computing Pareto-based meta-heuristic algorithm for a multi-objective multi-server facility location problem. Applied Soft Computing, 13, 1728–1740.CrossRefGoogle Scholar
  25. Romero, C. (2004). A general structure of achievement function for a goal programming model. European Journal of Operational Research, 153, 675–686.CrossRefGoogle Scholar
  26. Roy, S. K. (2014). Multi-choice stochastic transportation problem involving weibull distribution. International Journal of Operational Research, 21(1), 38–58.CrossRefGoogle Scholar
  27. Roy, S. K. (2015). Lagrange’s interpolating polynomial approach to solve multi-choice transportation problem. International Journal of Applied and Computational Mathematics, 1(4), 639–649.CrossRefGoogle Scholar
  28. Roy, S. K., & Mahapatra, D. R. (2011). Multi-objective interval-valued transportation probabilistic problem involving log-normal. International Journal of Computing Science and Mathematics, 1(2), 14–21.Google Scholar
  29. Roy, S. K., Mahaparta, D. R., & Biswal, M. P. (2012). Multi-choice stochastic transportation problem with exponential distribution. Journal of Uncertain Systems, 6(3), 200–213.Google Scholar
  30. Tamiz, M., Jones, D. F., & El-Darzi, E. (1995). A review of goal programming and its applications. Annals of Operations Research, 58, 39–53.CrossRefGoogle Scholar
  31. Tamiz M., Jones D.F., & Mirrazavi S.K. (1997). Intelligent solution and analysis of goal programmes: The GPSYS System, Technical Report, School of computer science and mathematics, University of Portsmouth, UK.Google Scholar
  32. Tamiz, M., Jones, D. F., & Romero, C. (1998). Goal programming for decision making: An overview of the current state-of-the-art. European Journal of Operational Research, 111(3), 569–581.CrossRefGoogle Scholar
  33. Ustun, O. (2012). Multi-choice goal programming formulation based on conic scalarization approach. Applied Mathematical Modelling, 34, 974–988.CrossRefGoogle Scholar
  34. Zeleny, M. (1982). The pros and cons of goal programming. Computers and Operations Research, 8, 357–359.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sankar Kumar Roy
    • 1
    Email author
  • Gurupada Maity
    • 1
  • Gerhard Wilhelm Weber
    • 2
  • Sirma Zeynep Alparslan Gök
    • 3
  1. 1.Department of Applied Mathematics with Oceanology and Computer ProgrammingVidyasagar UniversityMidnaporeIndia
  2. 2.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Department of Mathematics, Faculty of Arts and ScienceSuleyman Demirel UniversityIspartaTurkey

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