In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article


Department of Mathematics‎, ‎Faculty of Science‎, ‎Gonbad Kavous University‎, ‎Gonbad Kavous‎, ‎Iran‎.


This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.


Main Subjects

[1] Abbasbandy, S., Asady, B. (2004). “Newton’s method for solving fuzzy nonlinear equations”, Applied Mathematics and Computation, 159, 349-356.
[2] Abuhijleh, E.A. (2023). “Complex hesitant fuzzy graph”, Fuzzy Information and Engineering, 15(2), 149-161.
[3] Alcantud, J.C.R., de Andrés, R., Torrecillas M.J.M. (2016). “Hesitant fuzzy worth: An innovative ranking methodology for hesitant fuzzy subsets”, Applied Soft Computing, 38, 232-243.
[4] Allahviranloo, T., Babakordi, F. (2017). “Algebraic solution of fuzzy linear system as: ÃX +B X =Y AX + BX = Y ”, Soft Computing, 21(24), 7463-7472.
[5] Amirfakhrian, M. (2012). “Analyzing the solution of a system of fuzzy linear equations by a fuzzy distance”, Soft Computing, 16(6), 1035-1041.
[6] Babakordi, F. (2020). “Hesitant fuzzy set and its types”, Journal of Decisions and Operations Research, 4(4), 353-361.
[7] Babakordi, F., Allahviranloo, T. (2022). “Introducing and solving the hesitant fuzzy system AX −B”, Systems Research Institute, Polish Academy of Sciences.
[8] Babakordi, F., Taghi-Nezhad, N.A. (2021). “Introducing hesitant fuzzy equations and determining market equilibrium price”, Control and Cybernetics, 50.
[9] Babakordi, F., Allahviranloo, T., Adabitabarfirozja, T. (2016). “An efficient method for solving LR fuzzy dual matrix system”, Journal of Intelligent & Fuzzy Systems, 30, 575-581.
[10] Baker kearfott, R. (1996). “Rigorous global search: Continues problems”, Kluwer Academic Publishers, The Netherlands.
[11] Buckley, J. (1991). “Solving fuzzy equations: A new solution concept”, Fuzzy Sets and Systems, 39(3), 291-301.
[12] Buckley, J.J., Feuring, T., Hayashi, Y. (2002). “Solving fuzzy equations using evolutionary algorithms and neural nets”, Soft Computing, 6, 116-123.
[13] Farahani, H., Nehi, H., Paripour, M. (2016). “Solving fuzzy complex system of linear equations using eigenvalue method”, Journal of Intelligent & Fuzzy Systems, 31(3), 1689-1699.
[14] Jeon, J., Krishnan, S., Manirathinam, T., Narayanamoorthy, S., Nazir Ahmad, M., Ferrara, M., Ahmadian, A. (2023). “An innovative probabilistic hesitant fuzzy set MCDM perspective for selecting flexible packaging bags after the prohibition on single-use plastics”, Scientific Reports, 13(1), 1-20.
[15] Khalili Goodarzi, F., Taghinezhad, N.A., Nasseri, S.H. (2014). “A new fuzzy approach to solve a novel model of open shop scheduling problem”, University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 76(3), 199-210.
[16] Kirişci, M. (2023). “Fermatean hesitant fuzzy sets for multiple criteria decision-making with applications”, Fuzzy Information and Engineering, 15(2), 100-127.
[17] Nasseri, S.H., Khalili, F., Taghi-Nezhad, N., Mortezania, S. (2014). “A novel approach for solving fully fuzzy linear programming problems using membership function concepts”, Annals of Fuzzy Mathematics and Informatics, 7(3), 355-368.
[18] Noor’ani, A., Kavikumar, J., Mustaf, M., Nor, S. (2011). “Solving dual fuzzy polynomial equation by ranking method”, Far East Journal of Mathematical Sciences, 51(2), 151-163.
[19] Taghi-Nezhad, N. (2019). “The p-median problem in fuzzy environment: Proving fuzzy vertex optimality theorem and its application”, Soft Computing.
[20] Taghi-Nezhad, N.A., Babakordi, F. (2023). “Fully hesitant parametric fuzzy equation”, Soft Computing, 1-12.
[21] Taleshian, F., Fathali, J., Taghi-Nezhad, N.A. (2018). “Fuzzy majority algorithms for the 1-median and 2-median problems on a fuzzy tree”, Fuzzy Information and Engineering, 1-24.
[22] Torra, V. (2010). “Hesitant Fuzzy Sets”, International Journal of Intelligent Systems, 25(6), 529-539.
[23] Torra, V., Narukawa, Y. (2009). “On hesitant fuzzy sets and decision”, In: 18-th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea.
[24] Wang, K., Lee, H. (1965). “First course on fuzzy theory and applications”, Advances in Soft Computing.
[25] Xia, M.M., Xu, Z.S. (2011). “Hesitant fuzzy information aggregation in decision making”, International Journal of Approximate Reasoning, 52(3), 395-407.
[26] Xia, M., Xu, Z., Chen, N. (2013).“Some hesitant fuzzy aggregation operators with their application in group decision making”, Group Decision and Negotiation, 22(2), 259-279.
[27] Xu, Y. (2023). “On uncertainty measures of the interval-valued hesitant fuzzy set”, Advance in Fuzzy System, Volume 2023, Article ID: 3228324, doi:10.1155/2023/3228324.
[28] Zhang, Z., Wang, C., Tian, D., Li, K. (2014). “Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making”, Computers & Industrial Engineering, 67, 116-138.
[29] Zhao, J., Xie, W. (2023). “Interval grey hesitant fuzzy set and its applications in decision-making”, Journal of Mathematics, Volume 2023, Article ID: 7694396, doi:10.1155/2023/7694396.
[30] Zhu, B., Xu, Z., Xia, M. (2011). “Hesitant fuzzy information aggregation in decision making”, International Journal of Approximate Reasoning, 52(3), 395-407.