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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.

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##### 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.
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