AI Chat Paper
Note: Please note that the following content is generated by AMiner AI. SciOpen does not take any responsibility related to this content.
{{lang === 'zh_CN' ? '文章概述' : 'Summary'}}
{{lang === 'en_US' ? '中' : 'Eng'}}
Chat more with AI
Powered by AMiner. Clicking this button will take you away from SciOpen.
Home Fuzzy Information and Engineering Article
PDF (885.9 KB)
Cite
EndNote(RIS) BibTeX
Collect
Submit Manuscript AI Chat Paper
Show Outline
Outline
Show full outline
Hide outline
Outline
Show full outline
Hide outline
Article | Open Access

Resolution of Fuzzy Relation Equations with Constraints

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Show Author Information

Abstract

This paper studies the resolution of the max-min compositional fuzzy relation equation with the constraint of ∑i=1nxi=1. The solvability and the unique solvability of this constrained fuzzy relation equation are characterized. Furthermore, this paper presents a resolution of it and designs a corresponding tabular method. Finally, a numerical example is provided to illustrate the resolution procedure.

Keywords

fuzzy relation equationfuzzy comprehensive evaluationconstraint condition

References

[1]

L. A. Zadeh, Fuzzy sets, Inf. Control., vol. 8, no. 3, pp. 338–353, 1965.
Crossref Google Scholar
[2]

E. Sanchez, Resolution of composite fuzzy relation equations, Inf. Control., vol. 30, no. 1, pp. 38–48, 1976.
Crossref Google Scholar
[3]
D. J. Dubois and H. M. Prade, Fuzzy Sets and Systems: Theory and Applications. New York, NY, USA: Academic Press, 1980.
[4]
H. J. Zimmermann, Fuzzy Set Theory and Its Applications. Beijing, China: World Publishing Corporation, 2011.
[5]

X. B. Qu and F. Sun, Solution sets of fuzzy relation equations with Addition-Min composition, J. Leshan Normal Univ., vol. 32, no. 4, pp. 6–10, 2017.
Google Scholar
[6]

X. Q. Zhu and Q. Q. Xiong, Solution sets of fuzzy relation equations with max-composition, J. Sichuan Normal Univ. (Nat. Sci.), vol. 41, no. 6, pp. 741–747, 2018.
Google Scholar
[7]

D. Zhang and X. P. Wang, Solutions of fuzzy relational equations with sup-T composition, Fuzzy Syst. Math., vol. 32, no. 1, pp. 125–129, 2018.
Google Scholar
[8]

F. Sun and X. B. Qu, Resolution of fuzzy relation equations with increasing operations over complete lattices, Inf. Sci., vol. 570, pp. 451–467, 2021.
Crossref Google Scholar
[9]

S. Stanimirović and I. Micić, On the solvability of weakly linear systems of fuzzy relation equations, Inf. Sci., vol. 607, pp. 670–687, 2022.
Crossref Google Scholar
[10]

V. L. Tiwari and A. Thapar, Solving max-Archimedean t-norm interval-valued fuzzy relation equations, Fuzzy Sets Syst., vol. 440, pp. 62–76, 2022.
Crossref Google Scholar
[11]

H. F. Wang and Y. C. Chang, Resolution of composite interval-valued fuzzy relation equations, Fuzzy Sets Syst., vol. 44, no. 2, pp. 227–240, 1991.
Crossref Google Scholar
[12]

S. Wang, S. C. Fang, and H. L. W. Nuttle, Solution sets of interval-valued fuzzy relational equations, Fuzzy Optim. Decis. Mak., vol. 2, no. 1, pp. 41–60, 2003.
Crossref Google Scholar
[13]

S. Wang, S. C. Fang, and H. L. W. Nuttle, Solution sets of interval-valued Min-S-norm fuzzy relational equations, Fuzzy Optim. Decis. Mak., vol. 4, no. 4, pp. 331–349, 2005.
Crossref Google Scholar
[14]

X. R. Wu and Z. M. Song, Simplified Tsukamoto method of solving max-min fuzzy relation equation, J. Luoyang Inst. Sci. Technol. (Nat. Sci.), vol. 27, no. 2, pp. 79–82, 2017.
Google Scholar
[15]
K. Y. Zhou, X. Y. Liang, W. Y. Zeng, and Q. Yin, Resolution of minimal solutions for max-lukasiewicz fuzzy relation equation by fuzzy neural network, J. Phys.: Conf. Ser.,doi:10.1088/1742-6596/2037/1/012041, 2021.
Crossref
[16]

S. Feng, J. Q. Li, and Y. C. Wen, A novel algorithm for obtaining minimal solutions of fuzzy relation equations based on neural networks, J. Beijing Norm. Univ. (Nat. Sci.), vol. 48, pp. 111–114, 2012.
Google Scholar
[17]

Y. P. Li and H. X. Li, Solution of fuzzy relation equation based on neural networks, J. Beijing Normal Univ. (Nat. Sci.), vol. 34, no. 2, pp. 168–173, 1998.
Google Scholar
[18]

P. He and X. P. Wang, Covering matrices and minimal solutions of fuzzy relational equations with max-min composition, Fuzzy Syst. Math., vol. 29, no. 2, pp. 109–117, 2015.
Google Scholar
[19]
Q. Zhang, Algorithms for calculating exactly one minimal solution of a finite fuzzy relation equation with max-min composition, Master Thesis, Sichuan Normal University, Chengdu, China, 2016.
[20]

Q. Zhang and X. P. Wang, An algorithm for calculating exactly one minimal solution of a fuzzy relation equation, Fuzzy Syst. Math., vol. 30, no. 4, pp. 18–22, 2016.
Google Scholar
[21]

E. Bartl, R. Belohlavek, and V. Vychodil, Bivalent and other solutions of fuzzy relational equations via linguistic hedges, Fuzzy Sets Syst., vol. 187, no. 1, pp. 103–112, 2012.
Crossref Google Scholar
[22]

E. Bartl and M. Trnecka, Covering of minimal solutions to fuzzy relational equations, Int. J. Gen. Syst., vol. 50, no. 2, pp. 117–138, 2021.
Crossref Google Scholar
[23]

T. Tang and X. P. Wang, The determinant method that a fuzzy relational equation with max-product composition has a unique solution, Fuzzy Syst. Math., vol. 29, no. 6, pp. 99–102, 2015.
Google Scholar
[24]

T. Tang and X. P. Wang, The conditions that a fuzzy relational equation with sup-conjunctor composition has a unique minimal solution, Fuzzy Syst. Math., vol. 30, no. 1, pp. 71–76, 2016.
Google Scholar
[25]
T. Tang, The conditions that fuzzy relational equations have a unique minimal solution, Master Thesis, Sichuan Normal University, Chengdu, China, 2016.
[26]
L. J. Liu and X. P. Wang, The set of solution-set-invariant coefficient matrices of fuzzy relational equations with max-min composition, Fuzzy Syst. Math., vol. 30, no. 1, pp. 65–70, 2016.
[27]
L. J. Liu, Sets of the solution-set-invariant coefficient matrices of fuzzy relation equations, Master Thesis, Sichuan Normal University, Chengdu, China, 2016.
[28]

X. P. Yang, Resolution of bipolar fuzzy relation equations with max-Łukasiewicz composition, Fuzzy Sets Syst., vol. 397, pp. 41–60, 2020.
Crossref Google Scholar
[29]

M. E. Cornejo, D. Lobo, and J. Medina, Bipolar fuzzy relation equations systems based on the product t-norm, Math. Meth. Appl. Sci., vol. 42, no. 1, pp. 5779–5793, 2019.
Crossref Google Scholar
[30]

Y. B. OuYang, Comment on Bipolar fuzzy relation equations systems based on the product t-norm, Math. Meth. Appl. Sci., vol. 45, no. 8, pp. 4661–4664, 2022.
Crossref Google Scholar
[31]

B. W. Fang, Minimizing a linear objective function under a max-overlap function fuzzy relational equation constraint, Fuzzy Sets Syst., vol. 447, pp. 1–21, 2022.
Crossref Google Scholar
[32]

H. C. Lee and S. M. Guu, On the optimal three-tier multimedia streaming services, Fuzzy Optim. Decis. Mak., vol. 2, no. 1, pp. 31–39, 2003.
Crossref Google Scholar
[33]

J. Loetamonphong and S. C. Fang, An efficient solution procedure for fuzzy relation equations with max-product composition, IEEE Trans. Fuzzy Syst., vol. 7, no. 4, pp. 441–445, 1999.
Crossref Google Scholar
[34]
C. Z. Luo, Introduction to Fuzzy Sets. Beijing, China: Beijing Normal University Press, 2005.
Fuzzy Information and Engineering
Volume 15 Issue 3,
September 2023
Pages 220-232
DOI: 10.26599/FIE.2023.9270017
Cite this article:
Xu X. Resolution of Fuzzy Relation Equations with Constraints. Fuzzy Information and Engineering, 2023, 15(3): 220-232. https://doi.org/10.26599/FIE.2023.9270017

611

Views

97

Downloads

0

Crossref

0

Web of Science

0

Scopus

Altmetrics
Received: 01 December 2022
Revised: 26 June 2023
Accepted: 29 July 2023
Published: 01 September 2023
© The Author(s) 2023. Published by Tsinghua University Press.

This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Return