Advertisement

A solid transportation problem in uncertain environment involving type-2 fuzzy variable

  • Amrit DasEmail author
  • Uttam Kumar Bera
  • Manoranjan Maiti
Original Article
  • 43 Downloads

Abstract

The main focus of this paper is to develop a new safety-based restricted fixed charge solid transportation problem with type-2 fuzzy parameter that minimizes both cost and time. Here we develop mainly two models, the first one has cost and time as type-2 fuzzy variables and the second one has cost, time and all the other parameters of the solid transportation problem as type-2 fuzzy variables. We also consider restrictions on the amount of transport goods. Both of these models are solved by two different techniques. First is using the usual credibility measure, and second is the generalized credibility measure. For the first technique, we use critical value (CV)-based reduction method to reduce a type-2 fuzzy set into a type-1 fuzzy set and then apply the centroid method to this reduced fuzzy set to find the corresponding crisp value. In the second case, a chance constrained programming model based on generalized credibility has been developed with the help of CV-based reduction method. The equivalent parametric programming problem in deterministic form is then solved under the weighted mean programming technique framework, the global criteria method and with the help of LINGO 13.0 software. Lastly, we have provided two real-life-based numerical examples to illustrate the models and also validate the results with the existing work. Some sensitivity analyses for the models are also presented with some logical comments. Finally the effects of total cost and time due to the changes of credibility levels of cost, time, demand, source, conveyance and safety are discussed.

Keywords

Solid transportation problem Type-2 fuzzy variable Credibility measure Critical value Weighted mean programming technique Safety constraint 

Notes

Acknowledgements

The authors would like to thank to the editor and the anonymous reviewers for their suggestions which have led to an improvement in both the quality and clarity of the paper. Dr. Bera acknowledges the financial assistance from Department of Science and Technology, New Delhi, under the Research Project (F.No. SR/S4/MS:761/12).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20:224–230MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balinski ML (1961) Fixed-cost transportation problem. Naval Res Logist 8:41–54CrossRefzbMATHGoogle Scholar
  3. 3.
    Haley KB (1962) The solid transportation problem. Oper Res Int J 11:446–448zbMATHGoogle Scholar
  4. 4.
    Kundu P, Kar S, Maiti M (2013) Multi-objective multi-item solid transportation problem in fuzzy environment. Appl Math Model 37:2028–2038MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Liu ST (2006) Fuzzy total transportation cost measures for fuzzy solid transportation problem. Appl Math Comput 174:927–941MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ammar EE, Youness EA (2005) Study on multiobjective transportation problem with fuzzy numbers. Appl Math Comput 166:241–253MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bit AK, Biswal MP, Alam SS (1993) Fuzzy programming approach to multiobjective solid transportation problem. Fuzzy Sets Syst 57:183–194MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ojha A, Das B, Mondal S, Maiti M (2009) An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality. Math Comput Model 501(2):166–178MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kundu P, Kar S, Maiti M (2015) Multi-item solid transportation problems with type-2 fuzzy parameters. Appl Soft Comput 31:61–80CrossRefGoogle Scholar
  10. 10.
    Mahapatra DR, Roy SK, Biswal MP (2013) Multi-choice stochastic transportation problem involving extreme value distribution. Appl Math Model 37(4):2230–2240MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Baidya A, Bera UK, Maiti M (2013) Multi-item interval valued solid transportation problem with safety measure under fuzzy-stochastic environment. Int J Transp Secur 6(2):151–174CrossRefGoogle Scholar
  12. 12.
    Baidya A, Bera UK, Maiti M (2014) Solution of multi-item interval valued solid transportation problem with safety measure using different methods. Opsearch 51(1):1–22MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Adlakha V, Kowalski K (1999) On the fixed-charge transportation problem. OMEGA Int J Manag Scie 27:381–388CrossRefGoogle Scholar
  14. 14.
    Adlakha V, Kowalski K, Vemuganti RR, Lev B (2007) More-for-less algorithm for fixed-charge transportation problems. OMEGA Int J Manag Sci 35:116–127CrossRefGoogle Scholar
  15. 15.
    Xie F, Jia R (2012) Nonlinear fixed charge transportation problem by minimum cost flow-based genetic algorithm. Comput Ind Eng 63(4):763–778CrossRefGoogle Scholar
  16. 16.
    Raj K, Rajendran C (2012) A genetic algorithm for solving the fixed-charge transportation model: two-stage problem. Comput Oper Res 39:2016–2032CrossRefzbMATHGoogle Scholar
  17. 17.
    Yang L, Liu L (2007) Fuzzy fixed charge solid transportation problem and algorithm. Appl Soft Comput 7(3):879–889CrossRefGoogle Scholar
  18. 18.
    Yang L, Feng Y (2007) A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl Math Model 31:2668–2683CrossRefzbMATHGoogle Scholar
  19. 19.
    Ojha A, Das B, Mondal S, Maiti M (2010) A solid transportation problem for an item with fixed charge, vehicle cost and price discounted varying charge using genetic algorithm. Appl Soft Comput 10:100–110CrossRefGoogle Scholar
  20. 20.
    Kundu P, Kar S, Maiti M (2014) Fixed charge transportation problem with type-2 fuzzy variables. Inf Sci 255:170–186MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zadeh LA (1975) Concept of a linguistic variable and its application to approximate reasoning I. Inf Sci 8:199–249MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yager RR (1980) Fuzzy subsets of type-II in decisions. J Cybern 10(1–3):137–159MathSciNetCrossRefGoogle Scholar
  23. 23.
    Coupland S, John RI (2007) Geometric type-1 and type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst 15(1):3–15CrossRefzbMATHGoogle Scholar
  24. 24.
    Mendel JM (2001) Advances in type-2 fuzzy sets and systems. Inf Sci 177(1):84–110MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lv Z, Jin H, Yuan P (2009) The theory of triangle type-2 fuzzy sets. In: Proceedings of the 2009 IEEE international conference on computer and information technology, piscataway: IEEE Service Center, pp 57–62Google Scholar
  26. 26.
    Ling X, Zhang Y (2011) Operations on triangle type-2 fuzzy sets. Procedia Eng 15:3346–3350CrossRefGoogle Scholar
  27. 27.
    Karnik NN, Mendel JM (2001) Centroid of a type-2 fuzzy set. Inf Sci 132:195–220MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Greenfield S, John RI, Coupland S (2005) A novel sampling method for type-2 defuzzification. In: Proceedings of the UKCI 2005, LondonGoogle Scholar
  29. 29.
    Liu F (2008) An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Inf Sci 178:2224–2236MathSciNetCrossRefGoogle Scholar
  30. 30.
    Qin R, Liu YK, Liu ZQ (2011) Methods of critical value reduction for type-2 fuzzy variable and their applications. J Comput Appl Math 235:1454–1481MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Liu B, Iwamura K (1998) Chance constrained programming with fuzzy parameters. Fuzzy Sets Syst 94(2):227–237MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Das A, Bera UK, Maiti M (2016) Defuzzification of trapezoidal type-2 fuzzy variables and its application to solid transportation problem. J Intell Fuzzy Syst 30(4):2431–2445CrossRefzbMATHGoogle Scholar
  33. 33.
    Mendel JM, John RIB (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10(2):117–127CrossRefGoogle Scholar
  34. 34.
    Liu ZQ, Liu YK (2010) Type-2 fuzzy variables and their arithmetic. Soft Comput 14:729–747CrossRefzbMATHGoogle Scholar
  35. 35.
    Yager RR (1981) A procedure for ordering fuzzy subsets of the unit interval. Inf Sci 24:143–161MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zeng L (2006) Expected value method for fuzzy multiple attribute decision making. Tsinghua Sci Technol 11:102–106MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu B, Liu YK (2002) Expected value operator of fuzzy variable and fuzzy expected value models. IEEE Trans Fuzzy Syst 10(4):445–450CrossRefGoogle Scholar
  38. 38.
    Sugeno M (1985) An introductory survey of fuzzy control. Inf Sci 36:59–83MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Dalman H (2016) Uncertain programming model for multi-item solid transportation problem. Int J Mach Learn Cybern.  https://doi.org/10.1007/s13042-016-0538-7 Google Scholar
  40. 40.
    Chen L, Peng J, Zhang B (2017) Uncertain goal programming models for bicriteria solid transportation problem. Appl Soft Comput 51:49–59CrossRefGoogle Scholar
  41. 41.
    Das A, Bera UK, Maiti M (2017) A profit maximizing solid transportation model under a rough interval approach. IEEE Trans Fuzzy Syst 25(3):485–498CrossRefGoogle Scholar
  42. 42.
    Das A, Bera UK, Maiti M (2016) A breakable multi-item multi stage solid transportation problem under budget with Gaussian type-2 fuzzy parameters. Appl Intell 45(3):923–951CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyAgartala, Barjala, JiraniaIndia
  2. 2.Department of Applied MathematicsVidyasagar UniversityMidnaporeIndia
  3. 3.Department of Mathematics, School of Advanced SciencesVIT UniversityVelloreIndia

Personalised recommendations