Soft Computing

, Volume 22, Issue 7, pp 2275–2297 | Cite as

Defuzzification and application of trapezoidal type-2 fuzzy variables to green solid transportation problem

  • Amrit Das
  • Uttam Kumar Bera
  • Manoranjan Maiti
Methodologies and Application


The main objective of this investigation is to propose a defuzzification process of a trapezoidal type-2 fuzzy variable centred on critical value-based reduction method and nearest interval approximation, i.e. \(\alpha \)-cut of fuzzy number. In this context, this paper proposes some theorems with proof. Also as an application of the proposed defuzzification process, a new multi-objective green solid transportation model has been formulated with all of its parameters as trapezoidal type-2 fuzzy variables, where the objectives are profit maximization and minimization of carbon emission produced by the modes of transport depending upon their loads, fuel type used, type of engine, driving characteristics, etc. After defuzzification, to solve the equivalent crisp multi-objective solid transportation problem the intuitionistic fuzzy programming technique is used. Also we have proposed the MOGA and LINGO 13.0 iterative platform for the soft computation related to the problem. At the end, proposed methodologies are finally illustrated by providing numerical examples which incorporate some real-life data and demonstrate how a decision maker makes a balance between the maximum profit and minimum carbon emission. Also a comparative study with N–T method has been provided, and some managerial decisions are drawn.


Trapezoidal type-2 fuzzy variable Solid transportation problem Carbon emission Intuitionistic fuzzy programming technique 



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 Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology AgartalaJiraniaIndia
  2. 2.Department of Applied MathematicsVidyasagar UniversityMidnaporeIndia

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