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Lexicographic Methods for Fuzzy Linear Programming

Author

Listed:
  • Boris Pérez-Cañedo

    (Department of Mathematics, University of Cienfuegos, Cienfuegos 55100, Cuba)

  • José Luis Verdegay

    (Department of Computer Science and Artificial Intelligence, University of Granada, 18071 Granada, Spain)

  • Eduardo René Concepción-Morales

    (Department of Informatics, University of Cienfuegos, Cienfuegos 55100, Cuba)

  • Alejandro Rosete

    (Facultad de Ingeniería Informática, Universidad Tecnológica de La Habana José Antonio Echeverría (Cujae), Marianao La Habana 19390, Cuba)

Abstract

Fuzzy Linear Programming (FLP) has addressed the increasing complexity of real-world decision-making problems that arise in uncertain and ever-changing environments since its introduction in the 1970s. Built upon the Fuzzy Sets theory and classical Linear Programming (LP) theory, FLP encompasses an extensive area of theoretical research and algorithmic development. Unlike classical LP, there is not a unique model for the FLP problem, since fuzziness can appear in the model components in different ways. Hence, despite fifty years of research, new formulations of FLP problems and solution methods are still being proposed. Among the existing formulations, those using fuzzy numbers (FNs) as parameters and/or decision variables for handling inexactness and vagueness in data have experienced a remarkable development in recent years. Here, a long-standing issue has been how to deal with FN-valued objective functions and with constraints whose left- and right-hand sides are FNs. The main objective of this paper is to present an updated review of advances in this particular area. Consequently, the paper briefly examines well-known models and methods for FLP, and expands on methods for fuzzy single- and multi-objective LP that use lexicographic criteria for ranking FNs. A lexicographic approach to the fuzzy linear assignment (FLA) problem is discussed in detail due to the theoretical and practical relevance. For this case, computer codes are provided that can be used to reproduce results presented in the paper and for practical applications. The paper demonstrates that FLP that is focused on lexicographic methods is an active area with promising research lines and practical implications.

Suggested Citation

  • Boris Pérez-Cañedo & José Luis Verdegay & Eduardo René Concepción-Morales & Alejandro Rosete, 2020. "Lexicographic Methods for Fuzzy Linear Programming," Mathematics, MDPI, vol. 8(9), pages 1-21, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1540-:d:410912
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    References listed on IDEAS

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    1. R. E. Bellman & L. A. Zadeh, 1970. "Decision-Making in a Fuzzy Environment," Management Science, INFORMS, vol. 17(4), pages 141-164, December.
    2. Amit Kumar & Anila Gupta, 2011. "Methods for solving fuzzy assignment problems and fuzzy travelling salesman problems with different membership functions," Fuzzy Information and Engineering, Springer, vol. 3(1), pages 3-21, March.
    3. P. L. Yu, 1973. "A Class of Solutions for Group Decision Problems," Management Science, INFORMS, vol. 19(8), pages 936-946, April.
    4. Jagdeep Kaur & Amit Kumar, 2013. "A New Method to Find the Unique Fuzzy Optimal Value of Fuzzy Linear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 529-534, February.
    5. H. W. Kuhn, 1955. "The Hungarian method for the assignment problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 2(1‐2), pages 83-97, March.
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    1. Figueroa–García, Juan Carlos & Hernández, Germán & Franco, Carlos, 2022. "A review on history, trends and perspectives of fuzzy linear programming," Operations Research Perspectives, Elsevier, vol. 9(C).
    2. Sandra González-Gallardo & Ana B. Ruiz & Mariano Luque, 2021. "Analysis of the Well-Being Levels of Students in Spain and Finland through Interval Multiobjective Linear Programming," Mathematics, MDPI, vol. 9(14), pages 1-27, July.

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