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On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints
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 |
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.
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. |
[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. |
[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. |
[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. 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. |
[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). 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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). 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. |
[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. Google Scholar |
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