Research article

On the minimal solution for max-product fuzzy relation inequalities

  • Received: 13 July 2024 Revised: 17 October 2024 Accepted: 23 October 2024 Published: 28 October 2024
  • MSC : 90C70, 90C90

  • Minimal solutions play a crucial role in constructing the complete solution set of the max-product fuzzy relation inequalities, as well as in solving the corresponding fuzzy relation optimization problems. In this work, we propose a sufficient and necessary condition for checking whether a given solution is minimal in the max-product system. Our proposed approach is useful for eliminating non-minimal solutions from the set of all quasi-minimal solutions. Our proposed checking approach helps reduce computational complexity when solving the max-product system or related optimization problems.

    Citation: Guocheng Zhu, Zhining Wang, Xiaopeng Yang. On the minimal solution for max-product fuzzy relation inequalities[J]. AIMS Mathematics, 2024, 9(11): 30667-30685. doi: 10.3934/math.20241481

    Related Papers:

  • Minimal solutions play a crucial role in constructing the complete solution set of the max-product fuzzy relation inequalities, as well as in solving the corresponding fuzzy relation optimization problems. In this work, we propose a sufficient and necessary condition for checking whether a given solution is minimal in the max-product system. Our proposed approach is useful for eliminating non-minimal solutions from the set of all quasi-minimal solutions. Our proposed checking approach helps reduce computational complexity when solving the max-product system or related optimization problems.



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    [1] W. Pedrycz, An identification algorithm in fuzzy relation systems, Fuzzy Set. Syst., 13 (1984), 153–167. https://doi.org/10.1016/0165-0114(84)90015-0 doi: 10.1016/0165-0114(84)90015-0
    [2] E. Sanchez, Resolution of composite fuzzy relation equations, Inform. Control, 30 (1976), 38–48. https://doi.org/10.1016/S0019-9958(76)90446-0 doi: 10.1016/S0019-9958(76)90446-0
    [3] J. Loetamonphong, S. C. Fang, An efficient solution procedure for fuzzy relational equations with max-product composition, IEEE T. Fuzzy Syst., 7 (1999), 441–445. https://doi.org/10.1109/91.784204 doi: 10.1109/91.784204
    [4] Y. K. Wu, S. M. Guu, Finding the complete set of minimal solutions for fuzzy relational equations with max-product composition, Int. J. Oper. Res., 1 (2004), 29–36. https://api.semanticscholar.org/CorpusID: 17494098
    [5] W. Pedrycz, On generalized fuzzy relational equations and their applications, J. Math. Anal. Appl., 107 (1985), 520–536. https://doi.org/10.1016/0022-247X(85)90329-4 doi: 10.1016/0022-247X(85)90329-4
    [6] A. A. Molai, Resolution of a system of the max-product fuzzy relation equations using L$\circ$U-factorization, Inform. Sciences, 234 (2013), 86–96. https://doi.org/10.1016/j.ins.2011.04.012 doi: 10.1016/j.ins.2011.04.012
    [7] X. P. Yang, X. G. Zhou, B. Y. Cao, Latticized linear programming subject to max-product fuzzy relation inequalities with application in wireless communication, Inform. Sciences, 358–359 (2016), 44–55. https://doi.org/10.1016/j.ins.2016.04.014 doi: 10.1016/j.ins.2016.04.014
    [8] B. S. Shieh, Deriving minimal solutions for fuzzy relation equations with max-product composition, Inform. Sciences, 178 (2008), 3766–3774. https://doi.org/10.1016/j.ins.2008.05.030 doi: 10.1016/j.ins.2008.05.030
    [9] A. V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Set. Syst., 153 (2005), 261–273. https://doi.org/10.1016/j.fss.2005.02.010 doi: 10.1016/j.fss.2005.02.010
    [10] X. P. Yang, D. H. Yuan, B. Y. Cao, Lexicographic optimal solution of the multi-objective programming problem subject to max-product fuzzy relation inequalities, Fuzzy Set. Syst., 341 (2018), 92–112. https://doi.org/10.1016/j.fss.2017.08.001 doi: 10.1016/j.fss.2017.08.001
    [11] M. Li, X. Wang, Remarks on minimal solutions of fuzzy relation inequalities with addition-min composition, Fuzzy Set. Syst., 410 (2021), 19–26. https://doi.org/10.1016/j.fss.2020.09.014 doi: 10.1016/j.fss.2020.09.014
    [12] S. Chen, K. Hayat, X. Yang, Upper bounded minimal solution of the max-min fuzzy relation inequality system, IEEE Access, 10 (2022), 84384–84397. https://doi.org/10.1109/ACCESS.2022.3197611 doi: 10.1109/ACCESS.2022.3197611
    [13] J. Loetamonphong, S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Set. Syst., 118 (2001), 509–517. https://doi.org/10.1016/S0165-0114(98)00417-5 doi: 10.1016/S0165-0114(98)00417-5
    [14] J. Lu, S. C. Fang, Solving nonlinear optimization problems with fuzzy relation equations constraints, Fuzzy Set. Syst., 119 (2001), 1–20. https://doi.org/10.1016/S0165-0114(98)00471-0 doi: 10.1016/S0165-0114(98)00471-0
    [15] A. Ghodousian, Optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by Dubois-Prade family of t-norms, Inform. Sciences, 503 (2019), 291–306. https://doi.org/10.1016/j.ins.2019.06.058 doi: 10.1016/j.ins.2019.06.058
    [16] E. Shivanian, E. Khorram, Optimization of linear objective function subject to fuzzy relation inequalities constraints with max-product composition, Iran. J. Fuzzy Syst., 7 (2010), 51–71. https://doi.org/10.22111/ijfs.2010.189 doi: 10.22111/ijfs.2010.189
    [17] E. Shivanian, E. Khorram, Monomial geometric programming with fuzzy relation inequality constraints with max-product composition, Comput. Ind. Eng., 56 (2009), 1386–1392. https://doi.org/10.1016/j.cie.2008.08.015 doi: 10.1016/j.cie.2008.08.015
    [18] A. A. Molai, A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints, Comput. Ind. Eng., 72 (2014), 306–314. https://doi.org/10.1016/j.cie.2014.03.024 doi: 10.1016/j.cie.2014.03.024
    [19] A. A. Molai, The quadratic programming problem with fuzzy relation inequality constraints, Comput. Ind. Eng., 62 (2012), 256–263. https://doi.org/10.1016/j.cie.2011.09.012 doi: 10.1016/j.cie.2011.09.012
    [20] C. A. Drossos, Generalized t-norm structures, Fuzzy Set. Syst., 104 (1999), 53–59. https://doi.org/10.1016/S0165-0114(98)00258-9 doi: 10.1016/S0165-0114(98)00258-9
    [21] D. Zhang, Triangular norms on partially ordered sets, Fuzzy Set. Syst., 153 (2005), 195–209. https://doi.org/10.1016/j.fss.2005.02.001 doi: 10.1016/j.fss.2005.02.001
    [22] G. D. Çaylı, Some methods to obtain t-norms and t-conorms on bounded lattices, Kybernetika, 55 (2019), 273–294. https://doi.org/10.14736/KYB-2019-2-0273 doi: 10.14736/KYB-2019-2-0273
    [23] B. D. Baets, Analytical solution methods for fuzzy relational equations, In D. Dubois and H. Prade Eds., Fundamentals of Fuzzy Sets, Boston: Kluwer Academic Publishers, 2000,291–340. https://doi.org/10.1007/978-1-4615-4429-6-7
    [24] P. Li, S. C. Fang, On the resolution and optimization of a system of fuzzy relational equations with sup-$T$ composition, Fuzzy Optim. Decis. Ma., 7 (2008), 169–214. https://doi.org/10.1007/s10700-008-9029-y doi: 10.1007/s10700-008-9029-y
    [25] B. S. Shieh, Solutions of fuzzy relation equations based on continuous t-norms, Inform. Sciences, 177 (2007), 4208–4215. https://doi.org/10.1016/j.ins.2007.04.006 doi: 10.1016/j.ins.2007.04.006
    [26] P. Z. Wang, D. Z. Zhang, E. Sanchez, E. S. Lee, Latticized linear programming and fuzzy relation inequalities, J. Math. Anal. Appl., 159 (1991), 72–87. https://doi.org/10.1016/0022-247X(91)90222-L doi: 10.1016/0022-247X(91)90222-L
    [27] X. P. Yang, X. G. Zhou, B. Y. Cao, Y. H. Hong, Variable substitution method for solving single-variable term fuzzy relation geometric programming problem and its application, Int. J. Uncertain. Fuzz., 27 (2019), 537–557. https://doi.org/10.1142/S0218488519500247 doi: 10.1142/S0218488519500247
    [28] X. G. Zhou, X. P. Yang, B. Y. Cao, Posynomial geometric programming problem subject to max-min fuzzy relation equations, Inform. Sciences, 328 (2016), 15–25. https://doi.org/10.1016/j.ins.2015.07.058 doi: 10.1016/j.ins.2015.07.058
    [29] A. Ghodousiana, E. Khorram, Linear optimization with an arbitrary fuzzy relational inequality, Fuzzy Set. Syst., 206 (2012), 89–102. https://doi.org/10.1016/j.fss.2012.04.009 doi: 10.1016/j.fss.2012.04.009
    [30] A. Ghodousian, B. S. Rad, O. Ghodousian, A non-linear generalization of optimization problems subjected to continuous max-t-norm fuzzy relational inequalities, Soft Comput., 28 (2024), 4025–4036. https://doi.org/10.1007/s00500-023-09376-2 doi: 10.1007/s00500-023-09376-2
    [31] B. Hedayatfar, A. A. Molai, Geometric function optimization subject to mixed fuzzy relation inequality constraints, TWMS J. Appl. Eng. Math., 9 (2019), 434–445.
    [32] Z. Mashayekhi, E. Khorram, On optimizing a linear objective function subjected to fuzzy relation inequalities, Fuzzy Optim. Decis. Ma., 8 (2009), 103–114. https://doi.org/10.1007/s10700-009-9054-5 doi: 10.1007/s10700-009-9054-5
    [33] A. Ghodousiani, S. Falahatkar, A comparison between the resolution and linear optimization of FREs defined by product t-norm and geometric mean operator, J. Algorithms Comput., 54 (2022), 11–22. https://doi.org/10.22059/jac.2022.87918 doi: 10.22059/jac.2022.87918
    [34] E. Shivanian, F. Sohrabi, Monomial geometric programming with an arbitrary fuzzy relational inequality, Commun. Numer. Anal., 2015 (2015), 162–177. https://doi.org/10.5899/2015/cna-00243 doi: 10.5899/2015/cna-00243
    [35] X. Fu, C. Zhu, Z. Qin, Linear programming subject to max-product fuzzy relation inequalities with discrete variables, In: Proceeding of International Conference on Fuzzy Information & Engineering, Singapore: Springer, 2024, 37–48. https://doi.org/10.1007/978-981-97-2891-6-3
    [36] J. Yang, B. Cao, Posynomial fuzzy relation geometric programming, In: Proceeding of International Fuzzy Systems Association World Congress, Berlin/Heidelberg: Springer, 2007,563–572. https://doi.org/10.1007/978-3-540-72950-1-56
    [37] G. Singh, D. Pandey, A. Thapar, A posynomial geometric programming restricted to a system of fuzzy relation equations, Procedia Eng., 38 (2012), 3462–3476. https://doi.org/10.1016/j.proeng.2012.06.400 doi: 10.1016/j.proeng.2012.06.400
    [38] X. P. Yang, Linear programming method for solving semi-latticized fuzzy relation geometric programming with max-min composition, Int. J. Uncertain. Fuzz., 23 (2015), 781–804. https://doi.org/10.1142/S0218488515500348 doi: 10.1142/S0218488515500348
    [39] Z. Matusiewicz, J. Drewniak, Increasing continuous operations in fuzzy max-* equations and inequalities, Fuzzy Set. Syst., 232 (2013), 120–133. https://doi.org/10.1016/j.fss.2013.03.009 doi: 10.1016/j.fss.2013.03.009
    [40] J. Drewniak, Fuzzy relation equations and inequalities, Fuzzy Set. Syst., 14 (1984), 237–247. https://doi.org/10.1016/0165-0114(84)90084-8 doi: 10.1016/0165-0114(84)90084-8
    [41] A. Ghodousian, F. S. Yousefi, Linear optimization problem subjected to fuzzy relational equations and fuzzy constraints, Iran. J. Fuzzy Syst., 20 (2023), 1–20. https://doi.org/10.22111/IJFS.2023.7552 doi: 10.22111/IJFS.2023.7552
    [42] S. Wang, H. Li, Resolution of fuzzy relational inequalities with Boolean semi-tensor product composition, Mathematics, 9 (2021), 937. https://doi.org/10.3390/math9090937 doi: 10.3390/math9090937
    [43] Z. Matusiewicz, Minimizing and maximizing a linear objective function under a fuzzy max-* relational equation and an inequality constraint, Kybernetika, 58 (2022), 320–334. https://dml.cz/handle/10338.dmlcz/151033
    [44] A. A. Molai, Linear objective function optimization with the max-product fuzzy relation inequality constraints, Iran. J. Fuzzy Syst., 10 (2013), 47–61. https://doi.org/10.22111/IJFS.2013.1206 doi: 10.22111/IJFS.2013.1206
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