# American Institute of Mathematical Sciences

• Previous Article
On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints
• JIMO Home
• This Issue
• Next Article
Optimal decisions for a dual-channel supply chain under information asymmetry
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

 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.

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. [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. [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. [4] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, Third edition. MIT Press, Cambridge, MA, 2009. [5] B. Davvaz, Strong regularity and fuzzy strong regularity in semihypergroups, Korean Society for Computational and Applied Mathematics and Korean SIGCAM, 2000. [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. [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. [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. [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. [10] P. Ketty and K. Yordan, Algorithm for solving max-product fuzzy relational equations, Soft Computing, 11 (2007), 593-605. [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. [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. [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. [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. [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. [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. [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. [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. [19] S. Martin and N. Lenka, Fuzzy relation equations-new solutions and solvability criteria, University of Ostrava, (2006). [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. [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. [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.

show all references

##### References:
 [1] U. Ahmed and M. Saqib, Optimal solution of fuzzy relation equation, Blekinge Institute of Technology, 2010. [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. [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. [4] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, Third edition. MIT Press, Cambridge, MA, 2009. [5] B. Davvaz, Strong regularity and fuzzy strong regularity in semihypergroups, Korean Society for Computational and Applied Mathematics and Korean SIGCAM, 2000. [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. [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. [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. [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. [10] P. Ketty and K. Yordan, Algorithm for solving max-product fuzzy relational equations, Soft Computing, 11 (2007), 593-605. [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. [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. [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. [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. [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. [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. [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. [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. [19] S. Martin and N. Lenka, Fuzzy relation equations-new solutions and solvability criteria, University of Ostrava, (2006). [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. [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. [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.
 [1] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [2] Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157 [3] Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631 [4] Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895 [5] Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193 [6] Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-11. doi: 10.3934/jimo.2018099 [7] Xinlong Feng, Yinnian He. On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5387-5400. doi: 10.3934/dcds.2016037 [8] Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053 [9] Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012 [10] Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047 [11] Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163 [12] Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435 [13] Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709 [14] Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873 [15] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [16] Saeid Abbasi-Parizi, Majid Aminnayeri, Mahdi Bashiri. Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1271-1295. doi: 10.3934/jimo.2018083 [17] Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015 [18] Joseph W. Jerome. Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2465-2482. doi: 10.3934/dcdsb.2012.17.2465 [19] Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211 [20] Kenji Kimura, Jen-Chih Yao. Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 167-181. doi: 10.3934/jimo.2008.4.167

2017 Impact Factor: 0.994