Fuzzy Solution of Z-Number-Based Multi-Objective Linear Programming Models with Different Membership Functions

  • Authors

    • Pandit Shinde Research Scholar, Bir Tikendrajit University, Imphal, Manipur, India, and Assistant Professor, Department of Engineering Sciences, AISSMS Institute of Information Technology, Pune, Maharashtra, India
    • Dr. Asit Sen Associate Professor, Bir Tikendrajit University, Imphal, Manipur, India
    • D. S. Shelar Assistant Professor, Department of Engineering Sciences, AISSMS Institute of Information Technology, Pune, Maharashtra, India https://orcid.org/0000-0002-3279-8964
    https://doi.org/10.14419/83myk890

    Received date: August 3, 2025

    Accepted date: September 22, 2025

    Published date: October 7, 2025

  • Z-Numbers; Multi-Objective LPP; Fuzzy Logic; Uncertainty; Decision-Making

  • Abstract

    In real-world decision-making, uncertainty plays a crucial role, especially when dealing with complex, multi-objective optimization problems. Traditional linear programming (LP) models often have the assumption that the data is precise and deterministic. However, this ‎assumption is often not realistic for many applications, as imprecision will always be present. This paper presents a fuzzy solution to ‎Z-Z-number-based multi-objective linear programming (ZMOLP) models. Z-numbers can combine fuzzy logic and Z-numbers to manage ‎uncertainty in (multi-objective) decision dilemmas. A Z-number consists of two parts: a fuzzy number for the uncertainty in the ‎data and a reliability score that indicates the degree of confidence in the data. The two aspects of Z-numbers make them ef‎fective for modeling imprecise data in multi-objective positioning. The effectiveness and overall computational efficiency of different ‎fuzzy membership functions (triangular, trapezoidal, Gaussian, etc.) were also explored to understand their impact on optimal solu‎tions. Z-numbers, through fuzzy numbers, provide a flexible and adaptable decision-making solution far superior to traditional methods ‎for managing imprecision that leads to a better representation of uncertainty related to objectives and constraints. To demonstrate the viability of Z-number-based approaches, practical applications were employed to illustrate their usage in healthcare decision-making and ‎engineering optimization‎.

  • References

    1. H. Kawamura, “Fuzzy multi-objective and multi-stage optimization—an application of fuzzy theory to artificial life,” in Proc. 1995 IEEE Int. Conf. Fuzzy Systems, Yokohama, Japan: IEEE, 1995, pp. 701–708. https://doi.org/10.1109/FUZZY.1995.409760.
    2. R. Klimek, “Ambient-aware continuous aid for mountain rescue activities,” Information Sciences, vol. 653, no. 119772, pp. 1-31, Jan. 2024. https://doi.org/10.1016/j.ins.2023.119772.
    3. P. Gulia et al., “A systematic review on fuzzy-based multi-objective linear programming methodologies: Concepts, challenges and applications,” Arch. Computat. Methods Eng., vol. 30, no. 8, pp. 4983–5022, Nov. 2023. https://doi.org/10.1007/s11831-023-09966-1.
    4. S. K. Singh and S. P. Yadav, “Intuitionistic fuzzy multi-objective linear programming problem with various membership functions,” Ann. Oper. Res., vol. 269, no. 12, pp. 693–707, Oct. 2018. https://doi.org/10.1007/s10479-017-2551-y.
    5. F. Hasankhani, B. Daneshian, T. Allahviranloo, and F. M. Khiyabani, “A new method for solving linear programming problems using Z-numbers ranking,” Math. Sci., vol. 17, no. 2, pp. 121–131, Jun. 2023. https://doi.org/10.1007/s40096-021-00446-w.
    6. I. A. Baky, “Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach,” Appl. Math. Modelling, vol. 34, no. 9, pp. 2377–2387, Sep. 2010. https://doi.org/10.1016/j.apm.2009.11.004.
    7. Y. L. P. Thorani and N. R. Shankar, “Fuzzy multi-objective assignment linear programming problem based on L-R fuzzy numbers,” Int. J. Comput. Appl., vol. 63, no. 5, pp. 38–49, Feb. 2013. https://doi.org/10.5120/10466-5185.
    8. R. R. Aliyev, “Multi-attribute decision making based on Z-valuation,” Procedia Comput. Sci., vol. 102, pp. 218–222, 2016. https://doi.org/10.1016/j.procs.2016.09.393.
    9. M. Akram, I. Ullah, and T. Allahviranloo, “An interactive method for the solution of fully Z-number linear programming models,” Granul. Com-put., vol. 8, no. 6, pp. 1205–1227, Nov. 2023. https://doi.org/10.1007/s41066-023-00402-0.
    10. A. Rehmat, H. Saeed, and M. S. Cheema, “Fuzzy multi-objective linear programming approach,” Pak. J. Stat. Oper. Res., vol. 3, no. 2, p. 87-98, Jul. 2007. https://doi.org/10.18187/pjsor.v3i2.62.
    11. O. Guenounou, A. Belmehdi, and B. Dahhou, “Multi-objective optimization of TSK fuzzy models,” Expert Syst. Appl., vol. 36, no. 4, pp. 7416– 7423, May 2009. https://doi.org/10.1016/j.eswa.2008.09.044.
    12. A. Akarc¸ay and N. Y. Pehlivan, “Fuzzy multi-objective nonlinear programming problems under various membership functions: A comparative anal-ysis,” J. Eng. Sci. Des., vol. 11, no. 3, pp. 857–872, Sep. 2023. https://doi.org/10.21923/jesd.1062118.
    13. Sˇ. Sˇtevic´, E. K. Zavadskas, F. M. O. Tawfiq, F. Tchier, and T. Davidov, “Fuzzy multicriteria decision-making model based on Z-numbers for the evaluation of information technology for order picking in warehouses,” Appl. Sci., vol. 12, no. 24, p. 12533, Dec. 2022. https://doi.org/10.3390/app122412533.
    14. N. M. F. H. N. B. Alam, K. M. N. Ku Khalif, N. I. Jaini, and A. Gegov, “The application of Z-numbers in fuzzy decision making: The state of the art,” Information, vol. 14, no. 7, p. 400, Jul. 2023. https://doi.org/10.3390/info14070400.
    15. E. Osgooei and S. Jafarzadeh Ghoushchi, “An extended mathematical model for solving Z-number linear programming problems using additive weighting method based on best-worst method,” J. Fuzzy Ext. Appl., Online First, Jan. 2025.
    16. H. Liao, F. Liu, Y. Xiao, Z. Wu, and E. K. Zavadskas, “A survey on Z- number-based decision analysis methods and applications: What’s going on and how to go further?,” Information Sciences, vol. 663, p. 120234, Mar. 2024. https://doi.org/10.1016/j.ins.2024.120234.
    17. M. Yuan, B. Zeng, J. Chen, and C. Wang, “Z-based maximum expected linear programming model with applications,” Mathematics, vol. 11, no. 17, p. 3750, Aug. 2023. https://doi.org/10.3390/math11173750.
    18. V. Nourani and H. Najafi, “A Z-number based multi-attribute decision- making algorithm for hydro-environmental system management,” Neural Comput. Appl., vol. 35, no. 9, pp. 6405–6421, Mar. 2023. https://doi.org/10.1007/s00521-022-08025-3.
    19. D. Qiu, R. Dong, S. Chen, and A. Li, “On an optimization method based on Z-numbers and the multi-objective evolutionary algorithm,” AUTO-SOFT, vol. 24, no. 1, pp. 147–150, Jan. 2018. https://doi.org/10.1080/10798587.2017.1327153.
    20. H. Cheng, W. Huang, Q. Zhou, and J. Cai, “Solving fuzzy multi- objective linear programming problems using deviation degree measures and weighted max-min method,” Appl. Math. Modelling, vol. 37, no. 10– 11, pp. 6855–6869, Jun. 2013. https://doi.org/10.1016/j.apm.2013.01.048.

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  • How to Cite

    Shinde, P., Sen, D. A., & Shelar, D. S. . (2025). Fuzzy Solution of Z-Number-Based Multi-Objective Linear Programming Models with Different Membership Functions. International Journal of Basic and Applied Sciences, 14(6), 88-98. https://doi.org/10.14419/83myk890