Document Type : Research Paper


1 Department of Mathematics, Kosar University of Bojnord, Bojnord, Iran.

2 Department of Industrial Engineering, Birjand University of Technology, Birjand, Iran.


Neutrosophic set theory plays an important role in dealing with the impreciseness and inconsistency in data encountered in solving real-life problems. The current paper focuses on the neutrosophic fuzzy multiobjective linear programming problem (NFMOLPP), where the coefficients of the objective functions, constraints, and right-hand side parameters are single-valued trapezoidal neutrosophic numbers (NNs). From the viewpoint of complexity of the problem, a ranking function of NNs is proposed to convert the problem into equivalent MOLPPs with crisp parameters. Then suitable membership functions for each objective are formulated using their lowest and highest value. With the aim of linear programming techniques, a compromise optimal solution of NFMOLPP is obtained. The main advantage of the proposed approach is that it obtains a compromise solution by optimizing truth-membership, indeterminacy-membership, and falsity-membership functions, simultaneously. Finally, a transportation problem is introduced as an application to illustrate the utility and practicality of the approach.


Main Subjects

  • Ahmad, F., Ahmad, S., Zaindin, M., & Adhami, A. Y. (2021). A robust neutrosophic modeling and optimization approach for integrated energy-food-water security nexus management under uncertainty. Water13(2), 121.
  • Ahmad, S., Ahmad, F., & Sharaf, M. (2021). Supplier selection problem with type-2 fuzzy parameters: a neutrosophic optimization approach. International journal of fuzzy systems23, 755-775.
  • Ahmadini, A. A. H., & Ahmad, F. (2021). Solving intuitionistic fuzzy multiobjective linear programming problem under neutrosophic environment. AIMS mathematics, 6(5), 4556-4580.
  • Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems, 20(1), 87-96.
  • Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management science17(4), B-141.
  • Borovička, A. (2020). New complex fuzzy multiple objective programming procedure for a portfolio making under uncertainty. Applied soft computing96, 106607.
  • Das, S. K. (2022). An approach to optimize the cost of transportation problem based on triangular fuzzy programming problem. Complex & intelligent systems8(1), 687-699.
  • Kumar Das, S., Edalatpanah, S. A., & Dash, J. K. (2021). A novel lexicographical-based method for trapezoidal neutrosophic linear programming problem. Neutrosophic sets and systems46(1), 12.
  • Das, S. K. (2020). Application of transportation problem under pentagonal neutrosophic environment. Journal of fuzzy extension& applications, 1(1), 27- 41.
  • Deli, I., & Şubaş, Y. (2017). A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems. International journal of machine learning and cybernetics8(4), 1309-1322.
  • Khalifa, A. E. W. (2019). A signed distance method for solving multi-objective transportation problems in fuzzy environment. International journal of research in industrial engineering8(3), 274-282.
  • Khalifa, H. A. E. W., Kumar, P., & Mirjalili, S. (2021). A KKM approach for inverse capacitated transportation problem in neutrosophic environment. Sādhanā46(3), 1-8.
  • Khalifa, H. A. E. W., Kumar, P., & Alharbi, M. G. (2021). On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment. Journal of intelligent systems30(1), 620-635.
  • Khalifa, H., Elhenawy, M., Masoud, M., Bhuiyan, H., & Sabar, N. R. (2021). On multi-objective multi-item solid transportation problem in fuzzy environment. International journal of applied and computational mathematics7(1), 24.
  • Khalifa, H. A. E. W. (2020). Goal programming approach for solving heptagonal fuzzy transportation problem under budgetry constraint. Operations research and decisions30(1), 85-96.
  • Khalifa, H. A. E. W. (2019). Fuzzy compromise approach for solving interval-valued fractional multi-objective multi-product solid transportation problems. Journal of system management5(2), 1-20.
  • Liu, P., & Liu, X. (2018). The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making. International journal of machine learning and cybernetics9, 347-358.
  • Mahajan, S., & Gupta, S. K. (2021). On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions. Annals of operations research296, 211-241.
  • Maiti, I., Mandal, T., & Pramanik, S. (2020). Neutrosophic goal programming strategy for multi-level multi-objective linear programming problem. Journal of ambient intelligence and humanized computing11(8), 3175-3186.
  • Rizk-Allah, R. M., Abo-Sinna, M. A., & Hassanien, A. E. (2021). Intuitionistic fuzzy sets and dynamic programming for multi-objective non-linear programming problems. International journal of fuzzy systems23, 334-352.
  • Singh, S. K., & Yadav, S. P. (2015). Modeling and optimization of multi objective non-linear programming problem in intuitionistic fuzzy environment. Applied mathematical modelling39(16), 4617-4629.
  • Smarandache, F. (1999). A unifying field in logics: neutrosophic logic. In Philosophy(pp. 1-141). American Research Press.
  • Ye, J. (2017). Bidirectional projection method for multiple attribute group decision making with neutrosophic numbers. Neural computing and applications28, 1021-1029.
  • Ye, J. (2018). Neutrosophic number linear programming method and its application under neutrosophic number environments. Soft computing22, 4639-4646.
  • Ye, J. (2014). Prioritized aggregation operators of trapezoidal intuitionistic fuzzy sets and their application to multicriteria decision-making. Neural computing and applications25, 1447-1454.
  • Yu, G. F., Li, D. F., Liang, D. C., & Li, G. X. (2021). An intuitionistic fuzzy multi-objective goal programming approach to portfolio selection. International journal of information technology & decision making20(05), 1477-1497.
  • Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Infinite Study.
  • Wang, Q., Huang, Y., Kong, S., Ma, X., Liu, Y., Das, S. K., & Edalatpanah, S. A. (2021). A novel method for solving multiobjective linear programming problems with triangular neutrosophic numbers. Journal of mathematics2021, 1-8.
  • Zadeh, L. A. (1965) Fuzzy Sets. Information control, 8(3), 338-353.