Fuzzy linear programming problems: models and solutions

  • Reza Ghanbari
  • Khatere Ghorbani-Moghadam
  • Nezam Mahdavi-AmiriEmail author
  • Bernard De Baets
Methodologies and Application


We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, \(\alpha \)-cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately.


Fuzzy linear programming Duality Ranking function Fuzzy number Fully fuzzy system 



The first author thanks the Research Council of Ferdowsi University of Mashhad; the second and third authors thank the Research Council of Sharif University of Technology; and the fourth author thanks the Research Council of Ghent University for supporting this work.

Compliance with ethical standards

Conflict of interest

Authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  2. 2.Faculty of Mathematical SciencesSharif University of TechnologyTehranIran
  3. 3.Department of Mathematical Modelling, Statistics and BioinformaticsGhent UniversityGhentBelgium

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