Advertisement

Sādhanā

, 44:105 | Cite as

Multi-objective solid transportation problem under stochastic environment

  • Shubham Singh
  • Avik PradhanEmail author
  • M P Biswal
Article
  • 40 Downloads

Abstract

In real life, three-dimensional (solid) transportation problem is an uncertain multi-objective decision-making (MODM) problem. In particular, it involves searching for the best transportation set-up that meets the decision maker’s preferences by considering the conflicting objectives/criteria such as transportation cost, transportation time, environmental and social issues. To tackle such complex situations, this paper proposes a general formulation of the multi-objective solid transportation problem (STP) with some random parameters. The paper makes the following contributions: (i) proposes a solution methodology based on chance-constraint programming technique to solve an STP with the uncertainty characterized by gamma distribution, (ii) proposes the initial feasibility conditions for the problem and (iii) extends fuzzy programming approach for solving the multi-objective stochastic problems. A numerical example is presented to illustrate the model and methodology.

Keywords

Solid transportation problem multi-objective decision making stochastic programming chance-constraint programming gamma distribution 

Notes

Acknowledgements

The authors are thankful to the reviewers for their thoughtful comments and suggestions, which improved the quality and presentation of the article.

References

  1. 1.
    Hitchcock F L 1941 The distribution of a product from several sources to numerous localities. Journal of Mathematics and Physics 20: 224–230MathSciNetCrossRefGoogle Scholar
  2. 2.
    Schell E D 1955 Distribution of a product by several properties. In: Proceedings of the Second Symposium in Linear Programming, DCS/Comptroller HQ, US Air Force, Washington, DC, vol. 2, pp. 615–642Google Scholar
  3. 3.
    Haley K B 1962 New methods in mathematical programming—the solid transportation problem. Operations Research 10(4): 448–463CrossRefGoogle Scholar
  4. 4.
    Chen Lin, Jin P and Zhang B 2017 Uncertain goal programming models for bicriteria solid transportation problem. Applied Soft Computing 51: 49–59CrossRefGoogle Scholar
  5. 5.
    Bit A, Biswal M P and Alam S 1993 Fuzzy programming approach to multiobjective solid transportation problem. Fuzzy Sets and Systems 57: 183–194MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gen M, Ida K, Li Y and Kubota E 1995 Solving bicriteria solid transportation problem with fuzzy numbers by a genetic algorithm. Computers and Industrial Engineering 29: 537–541CrossRefGoogle Scholar
  7. 7.
    Jiménez F and Verdegay J 1998 Uncertain solid transportation problems. Fuzzy Sets and Systems 100: 45–57MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yang L and Liu L 2007 Fuzzy fixed charge solid transportation problem and algorithm. Applied Soft Computing 7: 879–889CrossRefGoogle Scholar
  9. 9.
    Liu S T 2006 Fuzzy total transportation cost measures for fuzzy solid transportation problem. Applied Mathematics and Computation 174: 927–941MathSciNetCrossRefGoogle Scholar
  10. 10.
    Rani D and Gulati T R 2016 Uncertain multi-objective multi-product solid transportation problems. Sadhana 41(5): 531–539MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ebrahimnejad A 2016 Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sadhana 41(3): 299–316MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yang L and Feng Y 2007 A bicriteria solid transportation problem with fixed charge under stochastic environment. Applied Mathematical Modelling 31: 2668–2683CrossRefGoogle Scholar
  13. 13.
    Roy S K 2014 Multi-choice stochastic transportation problem involving Weibull distribution. International Journal of Operational Research 21: 38–58MathSciNetCrossRefGoogle Scholar
  14. 14.
    Roy S, Mahapatra D and Biswal M P 2012 Multi-choice stochastic transportation problem with exponential distribution. Journal of Uncertain Systems 6: 200–213Google Scholar
  15. 15.
    Cui Q and Sheng Y 2012 Uncertain programming model for solid transportation problem. Information 15: 342–348Google Scholar
  16. 16.
    Das A and Bera U K 2015 A bi-objective solid transportation model under uncertain environment. In: Facets of uncertainties and applications. Springer, New Delhi, pp. 261–275 CrossRefGoogle Scholar
  17. 17.
    Zhang B, Peng J, Li S and Chen L 2016 Fixed charge solid transportation problem in uncertain environment and its algorithm. Computers and Industrial Engineering 102: 186–197CrossRefGoogle Scholar
  18. 18.
    Chen B, Liu Y and Zhou T 2017 An entropy based solid transportation problem in uncertain environment. Journal of Ambient Intelligence and Humanized Computing 10(1): 357–363CrossRefGoogle Scholar
  19. 19.
    Dalman H and Sivri M 2017 Multi-objective solid transportation problem in uncertain environment. Iranian Journal of Science and Technology, Transactions A: Science 41: 505–514CrossRefGoogle Scholar
  20. 20.
    Rani D, Gulati T R and Kumar A 2014 A method for unbalanced transportation problems in fuzzy environment. Sadhana 39(3): 573–581MathSciNetCrossRefGoogle Scholar
  21. 21.
    Charnes A and Cooper W W 1959 Chance-constrained programming. Management Science 6: 73–79MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ulungu E L and Teghem J 1994 Multi-objective combinatorial optimization problems: a survey. Journal of Multi-Criteria Decision Analysis 3: 83–104CrossRefGoogle Scholar
  23. 23.
    Deb K 1999 Solving goal programming problems using multi-objective genetic algorithms. In: Proceedings of the 1999 Congress on Evolutionary Computation, IEEE, Washington, DC, pp. 77–84Google Scholar
  24. 24.
    Zadeh L 1963 Optimality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control 8: 59–60CrossRefGoogle Scholar
  25. 25.
    Koski J 1988 Multicriteria truss optimization. In: Multicriteria optimization in engineering and in the sciences. Springer, Boston, MA, pp. 263–307CrossRefGoogle Scholar
  26. 26.
    Lin J 1976 Multiple-objective problems: Pareto-optimal solutions by method of proper equality constraints. IEEE Transactions on Automatic Control 21: 641–650MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mavrotas G 2009 Effective implementation of the \(\epsilon \)-constraint method in multi-objective mathematical programming problems. Applied Mathematics and Computation 213: 455–465MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zimmermann H J 1978. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1: 45–55MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hulsurkar S, Biswal M P and Sinha S B 1997 Fuzzy programming approach to multi-objective stochastic linear programming problems. Fuzzy Sets and Systems 88: 173–181MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kumar M, Vrat P and Shankar R 2004 A fuzzy goal programming approach for vendor selection problem in a supply chain. Computers and Industrial Engineering 46: 69–85CrossRefGoogle Scholar
  31. 31.
    El-Wahed W F A and Lee S M 2006 Interactive fuzzy goal programming for multi-objective transportation problems. Omega 34(2): 158–166CrossRefGoogle Scholar
  32. 32.
    Hu C F, Teng C J and Li S Y 2007 A fuzzy goal programming approach to multi-objective optimization problem with priorities. European Journal of Operational Research 176(3): 1319–1333MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lai Y J, Liu T Y and Hwang C L 1994 Topsis for MODM. European Journal of Operational Research 76(3): 486–500CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.School of SciencesIndrashil UniversityMehsanaIndia

Personalised recommendations