Applications and Research Progress of Max–Min Fuzzy Relation Inequalities

DOI: https://doi.org/10.62517/jnse.202617412

Author(s)

Shubin Chen, Xiaopeng Yang

Affiliation(s)

School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong, China

Abstract

Max–min fuzzy relation inequalities (max–min FRI) provide a mathematical framework for describing uncertain associations among demands, resources, and service capacities. This review traces the development of max–min FRI from basic solution structures to lexicographically minimal solutions, interval solutions, upper-bounded minimal solutions, and multi-objective optimization. Educational information resource allocation and P2P educational resource sharing are the main application contexts. The review highlights the distinction between Pareto-minimal solutions under upper-bound constraints and global resource minimization, and examines the transition from point solutions to solution-set descriptions. It also identifies current gaps in dynamic parameter updating, multi-objective trade-offs, and secondary selection of upper-bounded minimal solutions. The analysis indicates that max–min FRI has a solid foundation for educational resource allocation, but further work is needed to develop dynamic, robust, and practically implementable scheduling methods.

Keywords

Max–Min Fuzzy Relation Inequality; Minimal Solution; Upper-Bound Constraint; Pareto Optimality; Educational Information Resource; P2P Network; Resource Allocation

References

[1] ZADEH L A. Fuzzy sets. Information and Control, 1965, 8(3): 338–353. [2] SANCHEZ E. Resolution of composite fuzzy relation equations. Information and Control, 1976, 30: 38–48. [3] DI NOLA A, SESSA S, PEDRYCZ W, et al. Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Dordrecht: Kluwer Academic Publishers, 1989. [4] XIAO G, ZHU T, CHEN Y, et al. Linear searching method for solving approximate solution to system of max–min fuzzy relation equations with application in the instructional information resources allocation. IEEE Access, 2019, 7: 65019–65028. [5] MA Y B, YANG X B, CAO B Y. Fuzzy-relation-based lexicographic minimum solution to the P2P network system. IEEE Access, 2020, 8: 195447–195458. [6] CHEN Y, LIU X, ZHANG L. Interval solution to fuzzy relation inequality with application in P2P educational information resource sharing systems. IEEE Access, 2021, 9: 96166–96175. [7] CHEN S, HAYAT K, YANG X. Upper bounded minimal solution of the max–min fuzzy relation inequality system. IEEE Access, 2022, 10: 84384–84396. [8] MIYAKOSHI M, SHIMBO M. Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets and Systems, 1985, 16(1): 53–63. [9] MIYAKOSHI M, SHIMBO M. Sets of solution-set-invariant coefficient matrices of simple fuzzy relation equations with a monotone operation. Fuzzy Sets and Systems, 1986, 21(1): 59–83. [10] DREWNIAK J. Fuzzy relation equations and inequalities. Fuzzy Sets and Systems, 1984, 14(3): 237–242. [11] WANG P Z, ZHANG D Z, SANCHEZ E, et al. Latticized linear programming and fuzzy relation inequalities. Journal of Mathematical Analysis and Applications, 1991, 159(1): 72–87. [12] BOURKE M M, FISHER D G. Solution algorithms for fuzzy relational equations with max–product composition. Fuzzy Sets and Systems, 1998, 94(1): 61–69. [13] YANG X P, ZHOU X G, CAO B Y. Min–max programming problem subject to max–product fuzzy relation equations and its application. Journal of Intelligent & Fuzzy Systems, 2016, 31(1): 1–9. [14] SHU Q Y, YANG X P. Minimax programming problem subject to max–product fuzzy relational inequalities with max–plus composition. Fuzzy Information and Engineering, 2020, 12(3): 328–342. [15] WU Y K. Optimization of fuzzy relational equations with max–av composition. Fuzzy Sets and Systems, 2007, 158(8): 923–935. [16] GUO F F, PANG L P, MENG D, et al. An algorithm for solving optimization problems with fuzzy relational inequality constraints. Information Sciences, 2013, 252: 20–31. [17] SHU Q Y, YANG X P. Maximin optimization problem subject to min–product fuzzy relation inequalities with application in supply and demand scheme. Complexity, 2019, 2019: 4960638. [18] FRESON S, JANSSENS G, DE BAETS B, et al. Linear optimization with bipolar max–min constraints. Information Sciences, 2013, 234: 3–15. [19] LI Y Y, WANG X P, YANG X P. Max–product bipolar fuzzy relation equations with applications. Fuzzy Sets and Systems, 2022, 436: 140–156. [20] YANG X P. Linear programming method for solving semi-latticized fuzzy relation geometric programming with max–min composition. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2015, 23(5): 781–804. [21] LI Y Y, WANG X P, YANG X P. Max–Łukasiewicz bipolar fuzzy relation equations. Fuzzy Sets and Systems, 2021, 407: 142–160. [22] MOLAI A. Fuzzy relation max–min programming with a monotone function objective. Fuzzy Information and Engineering, 2010, 2(3): 249–267. [23] QIU J J, XUE H B, LI G R, et al. Fuzzy relation bilevel optimization model in the wireless communication station system. IEEE Access, 2020, 8: 60811–60823. [24] YANG X P, LI J Z. Fuzzy relation inequality-based consistency of the wireless communication basic-station system considering the non-working state stations. Soft Computing, 2022, 26(11): 5131–5142. [25] LI G R, QIU J J, YANG X P. Weighted minimax programming subject to the two-sides fuzzy relation inequalities with max–product composition. Journal of Intelligent & Fuzzy Systems, 2020, 39(1): 593–605. [26] CHEN Y, LIU X, ZHANG L. Maximum interval solution of max–min fuzzy relation equations. Journal of Intelligent & Fuzzy Systems, 2021, 40(3): 1–12. [27] CHEN Y, LIU X, ZHANG L. The widest maximum interval solution of max–min fuzzy relation equations. Fuzzy Sets and Systems, 2022, 436: 102–118. [28] CHEN M, ZHU G, ISLAM S, et al. Centralized solution in max–min fuzzy relation inequalities. AIMS Mathematics, 2025, 10(4): 7864–7890. [29] FANG S C, LI G Z. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems, 1999, 103(1): 107–113. [30] GUO F F, XIA Z Q. An algorithm for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities. Fuzzy Optimization and Decision Making, 2006, 5(1): 33–47. [31] WU Y K, GUU S M. Minimizing a linear function under a fuzzy max–min relational equation constraint. Fuzzy Sets and Systems, 2005, 150(1): 147–162. [32] GUU S M, WU Y K. Minimizing a linear objective function under a max–t-norm fuzzy relational equation constraint. Fuzzy Sets and Systems, 2010, 161(2): 285–297. [33] ZHOU X G, ZHONG X Y, LIN H T, et al. Lexicographic maximum solution of min–product fuzzy relation inequalities for modeling the optimal pricing with fixed priority grade in supply chain. IEEE Access, 2018, 6: 71306–71316. [34] ZHONG Y B, XIAO G, YANG X P. Fuzzy relation lexicographic programming for modelling P2P file sharing system. Soft Computing, 2019, 23(11): 3605–3614. [35] PEDRYCZ W. Fuzzy relational equations with generalized connectives and their applications. Fuzzy Sets and Systems, 1983, 10(2): 185–201. [36] BAAJ I. The Chebyshev approximate solution to the system of max–t fuzzy relational equations. Fuzzy Sets and Systems, 2022, 435: 94–110. [37] LIN H T, YANG X P. Approximate solutions of fuzzy relation inequalities with max–min composition. Journal of Intelligent & Fuzzy Systems, 2020, 38(3): 2613–2622. [38] YANG X P. Approximate solutions and approximate reasoning of fuzzy relation inequalities. Fuzzy Information and Engineering, 2021, 13(4): 433–446. [39] CHEN M, SHIBGHATULLAH A S, SUBRAMANIAM K, et al. Weighted mini-max programming subject to the maxmin fuzzy relation inequalities. AIMS Mathematics, 2024, 9(6): 13624–13641. [40] LEI H, FENG Y, YANG X, et al. Genetic algorithm for bi-objective optimization under min–product fuzzy relation inequality constraints in supply chains. Engineering Applications of Artificial Intelligence, 2026, 181: 115358.