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

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


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.


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



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.


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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

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