Soft Computing

, Volume 21, Issue 9, pp 2297–2306 | Cite as

A solid transportation model with product blending and parameters as rough variables

  • Pradip Kundu
  • Mouhya B. Kar
  • Samarjit Kar
  • Tandra Pal
  • Manoranjan Maiti
Methodologies and Application


In this paper, we formulate a practical solid transportation problem with product blending which is a common issue in many operational and planning models in the chemical, petroleum, gasoline and process industries. In the problem formulation, we consider that raw materials from different sources with different quality (or purity) levels are to be transported to some destinations so that the materials received at each destination can be blended together into the final product to meet minimum quality requirement of that destination. The parameters such as transportation costs, availabilities, demands are considered as rough variables in designing the model. We construct a rough chance-constrained programming (RCCP) model for the problem with rough parameters based on trust measure. This RCCP model is then transformed into deterministic form to solve the problem. Numerical example is presented to illustrate the problem model and solution strategy. The results are obtained using the standard optimization solver LINGO.


Solid transportation problem Blending Rough variable Trust measure Chance-constrained programming 



Pradip Kundu sincerely acknowledges the support received from IISER Kolkata to carry out this research work.

Compliance with ethical standards

Conflict of interest



  1. Chanas S, Kuchta D (1996) A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst 82:299–305MathSciNetCrossRefGoogle Scholar
  2. DeWitt C, Lasdon L, Waren A, Brenner D, Melhem S (1989) OMEGA—an improved gasoline blending system for Texaco. Interfaces 19(1):85–101CrossRefGoogle Scholar
  3. Haley KB (1962) The sold transportation problem. Oper Res 10:448–463CrossRefMATHGoogle Scholar
  4. Hitchcoockk FL (1941) The distribution of product from several sources to numerous localities. J Math Phys 20:224–230MathSciNetCrossRefGoogle Scholar
  5. Jiménez F, Verdegay JL (1998) Uncertain solid transportation problems. Fuzzy Sets Syst 100:45–57MathSciNetCrossRefGoogle Scholar
  6. Jiménez F, Verdegay JL (1999) Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. Eur J Oper Res 117:485–510CrossRefMATHGoogle Scholar
  7. Kaur A, Kumar A (2012) A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Appl Soft Comput 12(3):1201–1213CrossRefGoogle Scholar
  8. Kelly JD, Mann JL (2003) Crude oil blend scheduling optimization: an application with multimillion dollar benefits. Hydrocarb Process 82(6):47–53Google Scholar
  9. Kundu P, Kar S, Maiti M (2013a) Multi-objective multi-item solid transportation problem in fuzzy environment. Appl Math Model 37:2028–2038MathSciNetCrossRefMATHGoogle Scholar
  10. Kundu P, Kar S, Maiti M (2013b) Some solid transportation models with crisp and rough costs. Int J Math Comput Phys Electr Comput Eng 73:185–192Google Scholar
  11. Kundu P, Kar S, Maiti M (2014) Multi-objective solid transportation problems with budget constraint in uncertain environment. Int J Syst Sci 45(8):1668–1682MathSciNetCrossRefMATHGoogle Scholar
  12. Liu B (2002) Theory and practice of uncertain programming. Physica-Verlag, HeidelbergCrossRefMATHGoogle Scholar
  13. Liu B (2003) Inequalities and convergence concepts of fuzzy and rough variables. Fuzzy Opt Decis Mak 2:87–100MathSciNetCrossRefGoogle Scholar
  14. Liu B (2004) Uncertainty theory: an introduction to its axiomatic foundations. Springer, BerlinCrossRefMATHGoogle Scholar
  15. Liu B, Iwamura K (1998) Chance constrained programming with fuzzy parameters. Fuzzy Sets Syst 94(2):227–237MathSciNetCrossRefMATHGoogle Scholar
  16. Liu L, Zhu Y (2007) Rough variables with values in measurable spaces. Inf Sci 177:4678–4685MathSciNetCrossRefMATHGoogle Scholar
  17. Liu S (2006) Fuzzy total transportation cost measures for fuzzy solid transportation problem. Appl Math Comput 174:927–941MathSciNetMATHGoogle Scholar
  18. Méndez CA, Grossmann IE, Harjunkoski I, Kaboré P (2006) A simultaneous optimization approach for off-line blending and scheduling of oil-refinery operations. Comput Chem Eng 30:614–634CrossRefGoogle Scholar
  19. Misener R, Floudas CA (2009) Advances for the pooling problem: modeling, global optimization, and computational studies survey. Appl Comput Math 8(1):3–22MathSciNetMATHGoogle Scholar
  20. Mondal M, Maity AK, Maiti MK, Maiti M (2013) A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money. Appl Math Model 37:3200–3215MathSciNetCrossRefMATHGoogle Scholar
  21. Nagarjan A, Jeyaraman K (2010) Solution of chance constrained programming problem for multi-objective interval solid transportation problem under stochastic environment using fuzzy approach. Int J Comput Appl 10(9):19–29Google Scholar
  22. Ojha A, Das B, Mondal S, Maity M (2009) An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality. Math Comput Model 50(1–2):166–178MathSciNetCrossRefMATHGoogle Scholar
  23. Papageorgiou DJ, Toriello A, Nemhauser GL, Savelsbergh MWP (2012) Fixed-charge transportation with product blending. Transp Sci 46(2):281–295CrossRefGoogle Scholar
  24. Pawlak Z (1982) Rough sets. Int J Inf Comput Sci 11(5):341–356CrossRefMATHGoogle Scholar
  25. Pawlak Z (1991) Rough sets—theoretical aspects of reasoning about data. Kluwer Academatic Publishers, BostonMATHGoogle Scholar
  26. Pawlak Z, Skowron A (2007) Rough sets: some extensions. Inf Sci 177:28–40MathSciNetCrossRefMATHGoogle Scholar
  27. Polkowski L (2002) Rough sets, mathematical foundations. Physica-Verlag, HeidelbergGoogle Scholar
  28. Rigby B, Lasdon L, Waren A (1995) The evolution of Texaco’s blending systems: from OMEGA to StarBlend. Interfaces 25(5):64–83CrossRefGoogle Scholar
  29. Romo F, Tomasgar A, Hellemo L, Fodstad M, Eidesen BH, Pedersen B (2009) Optimizing the Norwegian natural gas production and transport. Interfaces 39(1):46–56CrossRefGoogle Scholar
  30. Tao Z, Xu J (2012) A class of rough multiple objective programming and its application to solid transportation problem. Inf Sci 188:215–235MathSciNetCrossRefMATHGoogle Scholar
  31. Xu J, Yao L (2010) A class of two-person zero-sum matrix games with rough payoffs. Int J Math Math Sci. doi: 10.1155/2010/404792 MathSciNetMATHGoogle Scholar
  32. Xu J, Li B, Wu D (2009) Rough data envelopment analysis and its application to supply chain performance evaluation. Int J Prod Econ 122:628–638CrossRefGoogle Scholar
  33. Yang L, Feng Y (2007) A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl Math Model 31:2668–2683CrossRefMATHGoogle Scholar
  34. Yang L, Liu L (2007) Fuzzy fixed charge solid transportation problem and algorithm. Appl Soft Comput 7:879–889CrossRefGoogle Scholar
  35. Youness E (2006) Characterizing solutions of rough programming problems. Eur J Oper Res 168:1019–1029MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Pradip Kundu
    • 1
  • Mouhya B. Kar
    • 2
  • Samarjit Kar
    • 3
  • Tandra Pal
    • 4
  • Manoranjan Maiti
    • 5
  1. 1.Department of Mathematics and StatisticsIISER KolkataMohanpurIndia
  2. 2.Department of Computer ScienceHeritage Institute of TechnologyKolkataIndia
  3. 3.Department of MathematicsNational Institute of Technology (NIT)DurgapurIndia
  4. 4.Department of Computer Science and EngineeringNational Institute of Technology (NIT)DurgapurIndia
  5. 5.Department of Applied MathematicsVidyasagar UniversityMidnapurIndia

Personalised recommendations