Fuzzy function approximation for multi-choice goal programming in transportation problems

Document Type : Research Paper

Authors

Department of mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

Transportation problems are widely used decision-making models in logistics, production, and supply-chain management. In real-world applications, the input parameters such as costs, supplies, and demands are often uncertain or imprecise, making classical crisp formulations inadequate. To address this challenge, this study proposes a fuzzy multi-choice goal programming (FMCGP) model enhanced with fuzzy function-approximation techniques. Unlike previous works, where fuzzy transportation problems are treated using direct defuzzification or ranking approaches, our method integrates fuzzy least-squares linear regression and a fuzzy binary polynomial approximation to represent and approximate multi-choice fuzzy goals flexibly. This dual approach allows the decision-maker to simultaneously handle multiple fuzzy objectives and constraints within a unified framework. A key feature of the proposed methodology is that all comparisons between fuzzy and crisp values are evaluated using the necessity measure with a degree of 0.8, ensuring mathematically consistent and practically interpretable inequality relations. To demonstrate the model's applicability, we present a case study of a transportation planning problem under uncertainty. The numerical experiments illustrate how the proposed approach outperforms existing fuzzy transportation methods in terms of solution feasibility, interpretability, and computational efficiency. The results confirm that the FMCGP model with fuzzy function approximation provides a powerful and flexible tool for decision-making under uncertainty, offering improved accuracy and robustness compared with classical fuzzy transportation approaches. In addition, the framework is general enough to be extended to other types of fuzzy optimization problems beyond transportation.

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References

[1] Allahdadi, M., & Rivaz, S. (2024). Some new results on rough interval linear programming problems and their application to scheduling and  xed-charge transportation problems. RAIRO-Operations Research, 58(5), 3697-3714.
[2] Arami, Z., Arabameri, M., & Mishmast Nehi, H. (2023). Fuzzy approximating functions and its application in solving fuzzy multi-choice linear programming models. Journal of Mathematical Modeling, 11(4), 649-663. doi: 10.22124/JMM.2023.24269.2178
[3] Biswal, M. P., & Acharya, S. (2009). Multi-choice multi-objective linear programming problem. Journal of Interdisciplinary Mathematics, 12(5), 606-637. doi.org/10.1080/09720502.2009.10700650
[4] Biswal, M. P., & Acharya, S. (2009). Transformation of a multi-choice linear programming problem. Applied Mathematics and Computation, 210(1), 182-188. doi.org/10.1016/j.amc.2008.12.080
[5] Chang, C. T. (2007). Multi-choice goal programming. Omega, 35(4), 389-396. doi.org/10.1016/j.omega.2005.07.009
[6] Chang, C. T. (2008). Revised multi-choice goal programming. Applied Mathematical Modelling, 32(12), 2587-2595. doi.org/10.1016/j.apm.2007.09.008
[7] Charnes, A., Cooper, W. W., & Ferguson, R. O. (1955). Optimal estimation of executive compensation by linear programming. Management Science, 1(2), 138-151. doi.org/10.1287/mnsc.1.2.138.
[8] Ebrahimnejad, A. (2015). An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers. Journal of Intelligent & Fuzzy Systems, 29(2), 963-974. doi: 10.3233/IFS-151625.
[9] Ebrahimnejad, A. (2014). A simpli ed new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Applied Soft Computing, 19, 171-176.
[10] Ebrahimnejad, A., & Verdegay, J. L. (2016). An ecient computational approach for solving type-2 intuitionistic fuzzy numbers based transportation problems. International Journal of Computational Intelligence Systems, 9(6), 1154-1173.
[11] Hitchcock, F. L. (1941). The distribution of a product from several sources to numerous localities. Journal of Mathematics and Physics, 20(1-4), 224-230.doi.org/10.1002/sapm1941201224
[12] Ignizio, J. P. (1976). Goal programming and extensions. (Lexington Books: Lexington, MA, 1976).
[13] Kaur, A., & Kumar, A. (2012). A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Applied Soft Computing, 12(3), 1201-1213. doi.org/10.1016/j.asoc.2011.10.014
[14] Kaur, A., & Kumar, A. (2011). A new method for solving fuzzy transportation problems using ranking function. Applied Mathematical Modelling, 35(12), 5652-5661. doi.org/10.1016/j.apm.2011.05.012
[15] Koopmans, T. C. (1949). Optimum utilization of the transportation system. Econometrica: Journal of the Econometric Society, 136-146. doi.org/10.2307/1907301
[16] Lee, A. H., Kang, H. Y., Yang, C. Y., & Lin, C. Y. (2010). An evaluation framework for product planning using FANP, QFD and multi-choice goal programming. International Journal of Production Research, 48(13), 3977-3997. doi.org/10.1080/00207540902950845
[17] Mahapatra, D. R., Roy, S. K., & Biswal, M. P. (2013). Multi-choice stochastic transportation problem involving extreme value distribution. Applied Mathematical Modelling, 37(4), 2230-2240. doi.org/10.1016/j.apm.2012.04.024
[18] Maity, G., & Roy, S. K. (2017). Solving fuzzy transportation problem using multi-choice goal programming. Discrete Mathematics, Algorithms and Applications, 9(06), 1750076. doi.org/10.1142/S1793830917500768
[19] Maity, G., & Roy, S. K. (2014). Solving multi-choice multi-objective transportation problem: a utility function approach. Journal of uncertainty Analysis and Applications, 2, 1-20. doi. www.juaa-journal.com/content/2/1/11
[20] Maity, G., & Roy, S. K. (2016). Solving multi-objective transportation problem with interval goal using utility function approach. International Journal of Operational Research, 27(4), 513-529. doi.org/10.1504/IJOR.2016.080143
[21] Midya, S., & Roy, S. K. (2017). Analysis of interval programming in di erent environments and its application to  xed-charge transportation problem. Discrete Mathematics, Algorithms and Applications, 9(03), 1750040. doi:10.1142/S1793830917500409
[22] Midya, S., & Roy, S. K. (2014). Solving single-sink,  xed-charge, multi-objective, multiindex stochastic transportation problem. American Journal of Mathematical and Management Sciences, 33(4), 300-314. doi.org/10.1080/01966324.2014.942474
[23] Narasimhan, R. (1980). Goal programming in a fuzzy environment. Decision Sciences, 11(2), 325-336. doi.org/10.1111/j.1540-5915.1980.tb01142.x
[24] Rivaz, S., Nasseri, S. H., & Ziaseraji, M. (2020). A fuzzy goal programming approach to multiobjective transportation problems. Fuzzy information and engineering, 12(2), 139-149.
[25] Roy, S. K., & Maity, G. (2017). Minimizing cost and time through single objective function in multi-choice interval valued transportation problem. Journal of Intelligent & Fuzzy Systems, 32(3), 1697-1709. doi:10.3233/JIFS-151656
[26] Roy, S. K., Maity, G., &Weber, G. W. (2017). Multi-objective two-stage grey transportation problem using utility function with goals. Central European Journal of Operations Research, 25(2), 417-439.
[27] Tamiz, M., Jones, D., & Romero, C. (1998). Goal programming for decision making: An overview of the current state-of-the-art. European Journal of Operational Research, 111(3), 569-581. doi.org/10.1016/S0377-2217(97)00317-2 

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History

  • Receive Date: 21 June 2025
  • Revise Date: 10 January 2026
  • Accept Date: 17 May 2026

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