American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimal decisions for a dual-channel supply chain under information asymmetry
  • JIMO Home
  • This Issue
  • Next Article
    On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints
July  2018, 14(3): 1007-1022. doi: 10.3934/jimo.2017087

Uniqueness of solutions to fuzzy relational equations regarding Max-av composition and strong regularity of the matrices in Max-av algebra

Jun Xie 1, , Jinlong Yuan 2,3,, , Dongxia Wang 1, , Weili Liu 1, and  Chongyang Liu 4,

1. 

Teaching and Research Office of Mathematics, Department of Basics, PLA Dalian Naval Academy, Dalian 116018, Liaoning, China
2. 

Department of Mathematics, Dalian Maritime University, Dalian 116026, Liaoning, China
3. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, PR China
4. 

School of Mathematics and Information Science, Shandong Institute of Business and Technology, , Yantai 264005, Shandong, China

* Corresponding author: Jinlong Yuan(yuanjinlong0613@163.com)

The reviewing process of this paper was handled by Changzhi Wu

Received  April 2016 Revised  December 2016 Published  September 2017

Fund Project: The second author is supported by the China Scholarship Council (Grant No. 201506060121) and Fundamental Research Funds for Central Universities in China. The fifth author is supported by the National Natural Science Foundation of China (Grant No. 11771008) and the Natural Science Foundation of Shandong Province in China (Grant Nos.: ZR2015FM014, ZR2015AL010 and ZR2017MA005).

The problem of solving a fuzzy relational equation plays an important role in fuzzy systems. In this paper, we investigate the uniqueness of solutions of fuzzy relational equations regarding Max-av composition through the relationship between minimal solutions and minimal coverage. A method for verifying the strong regularity of matrices in fuzzy Max-av algebra is proposed in the paper.

Keywords: Fuzzy relational equation, Max-av composition, unique solution, strong regularity.
Mathematics Subject Classification: Primary: 34A07; Secondary: 49N60.
Citation: Jun Xie, Jinlong Yuan, Dongxia Wang, Weili Liu, Chongyang Liu. Uniqueness of solutions to fuzzy relational equations regarding Max-av composition and strong regularity of the matrices in Max-av algebra. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1007-1022. doi: 10.3934/jimo.2017087
References:
[1]

U. Ahmed and M. Saqib, Optimal solution of fuzzy relation equation, Blekinge Institute of Technology, 2010. Google Scholar

[2]

K. Cechlarova, Unique solvability of max-min fuzzy equtaions and strong regularity of matrices over fuzzy algebra, Fuzzy Sets and Systems, 75 (1995), 165-177.  doi: 10.1016/0165-0114(95)00021-C.  Google Scholar

[3]

K. Cechlarova and K. Kolesar, An efficient algorithm to computing max-min inverse fuzzy relation for abductive reasoning, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 40 (2010), 158-169.   Google Scholar

[4]

T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, Third edition. MIT Press, Cambridge, MA, 2009.  Google Scholar

[5]

B. Davvaz, Strong regularity and fuzzy strong regularity in semihypergroups, Korean Society for Computational and Applied Mathematics and Korean SIGCAM, 2000. Google Scholar

[6]

S. C. Fang and J. Loetamonphong, An efficient solution procedure for fuzzy relation equations with max-product composition, IEEE Transactions on Fuzzy Systems, 7 (1999), 441-445.   Google Scholar

[7]

M. Gavalec, Solvability and unique solvability of max-min fuzzy equations, Fuzzy Sets and Systems, 124 (2001), 385-393.  doi: 10.1016/S0165-0114(01)00108-7.  Google Scholar

[8]

M. Gavalec and J. Plavka, Strong regularity of matrices in general max-min algebra, Linear Algebra and its Applications, 371 (2003), 241-254.  doi: 10.1016/S0024-3795(03)00462-2.  Google Scholar

[9]

S. M. Guu, Y. K. Wu and E. S. Lee, Multi-objective optimization with a max-t-norm fuzzy relational equation constraint, Computers and Mathematics with Applications, 61 (2011), 1559-1566.  doi: 10.1016/j.camwa.2011.01.023.  Google Scholar

[10]

P. Ketty and K. Yordan, Algorithm for solving max-product fuzzy relational equations, Soft Computing, 11 (2007), 593-605.   Google Scholar

[11]

E. Khorram and A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Applied Mathematics and Computation, 173 (2006), 872-886.  doi: 10.1016/j.amc.2005.04.021.  Google Scholar

[12]

W. Y. Kuen, Optimization of fuzzy relational equations with max-av composition, Information Sciences, 177 (2007), 4216-4229.  doi: 10.1016/j.ins.2007.02.037.  Google Scholar

[13]

P. Li and Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using the Lukasiewicz triangular norm, Soft Computing, 18 (2014), 1399-1404.  doi: 10.1007/s00500-013-1152-1.  Google Scholar

[14]

P. K. Li and S. C. Fang, On the resolution and optimization of a system of fuzzy relational equations with sup-T composition, Fuzzy Optim Decis Making, 7 (2008), 169-214.  doi: 10.1007/s10700-008-9029-y.  Google Scholar

[15]

P. K. Li and S. C. Fang, On the unique solvability of fuzzy relational equations, Fuzzy Optim Decis Making, 10 (2011), 115-124.  doi: 10.1007/s10700-011-9100-y.  Google Scholar

[16]

J. L. Lin, W. Y. Kuen and S. M. Guu, On fuzzy relational equations and the covering problem, Information Sciences, 181 (2011), 2951-2963.  doi: 10.1016/j.ins.2011.03.004.  Google Scholar

[17]

J. Loetamonphong and S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems, 118 (2001), 509-517.  doi: 10.1016/S0165-0114(98)00417-5.  Google Scholar

[18]

A. V. Markovskii, Solution of fuzzy equations with max-product composition in inverse control and decision making problems, Automation and Remote Control, 65 (2004), 1486-1495.  doi: 10.1023/B:AURC.0000041426.51975.50.  Google Scholar

[19]

S. Martin and N. Lenka, Fuzzy relation equations-new solutions and solvability criteria, University of Ostrava, (2006). Google Scholar

[20]

K. Peeva, Resolution of fuzzy relational equations-method, algorithm and software with applications, Journal Information Sciences: an International Journal, 234 (2013), 44-63.  doi: 10.1016/j.ins.2011.04.011.  Google Scholar

[21]

S. M. Wang, S. C. Fang and H. L. M. Nuttle, Solution sets of interval-valued fuzzy relational equations, Fuzzy Optimization and Decision Making, 2 (2003), 41-60.  doi: 10.1023/A:1022800330844.  Google Scholar

[22]

Y. K. Wu and S. M. Guu, An efficient procedure for solving a fuzzy relational equation with max-Archimedean t-norm composition, IEEE Transactions on Fuzzy Systems, 16 (2008), 73-84.   Google Scholar

show all references

The reviewing process of this paper was handled by Changzhi Wu

References:
[1]

U. Ahmed and M. Saqib, Optimal solution of fuzzy relation equation, Blekinge Institute of Technology, 2010. Google Scholar

[2]

K. Cechlarova, Unique solvability of max-min fuzzy equtaions and strong regularity of matrices over fuzzy algebra, Fuzzy Sets and Systems, 75 (1995), 165-177.  doi: 10.1016/0165-0114(95)00021-C.  Google Scholar

[3]

K. Cechlarova and K. Kolesar, An efficient algorithm to computing max-min inverse fuzzy relation for abductive reasoning, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 40 (2010), 158-169.   Google Scholar

[4]

T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, Third edition. MIT Press, Cambridge, MA, 2009.  Google Scholar

[5]

B. Davvaz, Strong regularity and fuzzy strong regularity in semihypergroups, Korean Society for Computational and Applied Mathematics and Korean SIGCAM, 2000. Google Scholar

[6]

S. C. Fang and J. Loetamonphong, An efficient solution procedure for fuzzy relation equations with max-product composition, IEEE Transactions on Fuzzy Systems, 7 (1999), 441-445.   Google Scholar

[7]

M. Gavalec, Solvability and unique solvability of max-min fuzzy equations, Fuzzy Sets and Systems, 124 (2001), 385-393.  doi: 10.1016/S0165-0114(01)00108-7.  Google Scholar

[8]

M. Gavalec and J. Plavka, Strong regularity of matrices in general max-min algebra, Linear Algebra and its Applications, 371 (2003), 241-254.  doi: 10.1016/S0024-3795(03)00462-2.  Google Scholar

[9]

S. M. Guu, Y. K. Wu and E. S. Lee, Multi-objective optimization with a max-t-norm fuzzy relational equation constraint, Computers and Mathematics with Applications, 61 (2011), 1559-1566.  doi: 10.1016/j.camwa.2011.01.023.  Google Scholar

[10]

P. Ketty and K. Yordan, Algorithm for solving max-product fuzzy relational equations, Soft Computing, 11 (2007), 593-605.   Google Scholar

[11]

E. Khorram and A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Applied Mathematics and Computation, 173 (2006), 872-886.  doi: 10.1016/j.amc.2005.04.021.  Google Scholar

[12]

W. Y. Kuen, Optimization of fuzzy relational equations with max-av composition, Information Sciences, 177 (2007), 4216-4229.  doi: 10.1016/j.ins.2007.02.037.  Google Scholar

[13]

P. Li and Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using the Lukasiewicz triangular norm, Soft Computing, 18 (2014), 1399-1404.  doi: 10.1007/s00500-013-1152-1.  Google Scholar

[14]

P. K. Li and S. C. Fang, On the resolution and optimization of a system of fuzzy relational equations with sup-T composition, Fuzzy Optim Decis Making, 7 (2008), 169-214.  doi: 10.1007/s10700-008-9029-y.  Google Scholar

[15]

P. K. Li and S. C. Fang, On the unique solvability of fuzzy relational equations, Fuzzy Optim Decis Making, 10 (2011), 115-124.  doi: 10.1007/s10700-011-9100-y.  Google Scholar

[16]

J. L. Lin, W. Y. Kuen and S. M. Guu, On fuzzy relational equations and the covering problem, Information Sciences, 181 (2011), 2951-2963.  doi: 10.1016/j.ins.2011.03.004.  Google Scholar

[17]

J. Loetamonphong and S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems, 118 (2001), 509-517.  doi: 10.1016/S0165-0114(98)00417-5.  Google Scholar

[18]

A. V. Markovskii, Solution of fuzzy equations with max-product composition in inverse control and decision making problems, Automation and Remote Control, 65 (2004), 1486-1495.  doi: 10.1023/B:AURC.0000041426.51975.50.  Google Scholar

[19]

S. Martin and N. Lenka, Fuzzy relation equations-new solutions and solvability criteria, University of Ostrava, (2006). Google Scholar

[20]

K. Peeva, Resolution of fuzzy relational equations-method, algorithm and software with applications, Journal Information Sciences: an International Journal, 234 (2013), 44-63.  doi: 10.1016/j.ins.2011.04.011.  Google Scholar

[21]

S. M. Wang, S. C. Fang and H. L. M. Nuttle, Solution sets of interval-valued fuzzy relational equations, Fuzzy Optimization and Decision Making, 2 (2003), 41-60.  doi: 10.1023/A:1022800330844.  Google Scholar

[22]

Y. K. Wu and S. M. Guu, An efficient procedure for solving a fuzzy relational equation with max-Archimedean t-norm composition, IEEE Transactions on Fuzzy Systems, 16 (2008), 73-84.   Google Scholar

[1]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099
[2]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
[3]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024
[4]

İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021010
[5]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[6]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[7]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[8]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074
[9]

Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046
[10]

Haripriya Barman, Magfura Pervin, Sankar Kumar Roy, Gerhard-Wilhelm Weber. Back-ordered inventory model with inflation in a cloudy-fuzzy environment. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1913-1941. doi: 10.3934/jimo.2020052
[11]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
[12]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[13]

Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021018
[14]

Leon Mons. Partial regularity for parabolic systems with VMO-coefficients. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021041
[15]

Hirokazu Saito, Xin Zhang. Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021051
[16]

Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021070
[17]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[18]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036
[19]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[20]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (140)
  • HTML views (823)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences