Optimizing linear functions over novel fuzzy relation equations: Structure, feasibility, and global solutions

[1] C. W. Chang, B. S. Shieh, Linear optimization problem constrained by fuzzy max-min relation equations, Information Sciences, 234 (2013), 71-79. https://doi.org/10.1016/j.ins.2011.04.042

[2] L. Chen, P. P. Wang, Fuzzy relation equations (I): The general and specialized solving algorithms, Soft Computing, 6(5) (2002), 428-435. https://doi.org/10.1007/s00500-001-0157-3

[3] L. Chen, P. P. Wang, Fuzzy relation equations (II): The branch-point-solutions and the categorized minimal solutions, Soft Computing, 11(1) (2007), 33-40. https://doi.org/10.1007/s00500-006-0050-1

[4] M. Cornejo, D. Lobo, J. Medina, Bipolar fuzzy relation equations based on product t-norm, in: Proceedings of
2017 IEEE International Conference on Fuzzy Systems, 2017. https://doi.org/10.1109/FUZZ-IEEE.2017.
8015691

[5] S. Dempe, A. Ruziyeva, On the calculation of a membership function for the solution of a fuzzy linear optimization problem, Fuzzy Sets and Systems, 188(1) (2012), 58-67. https://doi.org/10.1016/j.fss.2011.07.014

[6] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy relational equations and their applications in knowledge
engineering, Dordrecht: Kluwer Academic Press, 1989. https://doi.org/10.1007/978-94-017-1650-5

[7] D. Dubey, S. Chandra, A. Mehra, Fuzzy linear programming under interval uncertainty based on IFS representation, Fuzzy Sets and Systems, 188(1) (2012), 68-87. https://doi.org/10.1016/j.fss.2011.09.008

[8] D. Dubois, H. Prade, An introduction to bipolar representations of information and preference, International
Journal of Intelligent Systems, 23(8) (2008), 866-877. https://doi.org/10.1002/int.20297

[9] D. Dubois, H. Prade, An overview of the asymmetric bipolar representation of positive and negative information
in possibility theory, Fuzzy Sets and Systems, 160 (2009), 1355-1366. https://doi.org/10.1016/j.fss.2008.
11.006

[10] Y. R. Fan, G. H. Huang, A. L. Yang, Generalized fuzzy linear programming for decision making under
uncertainty: Feasibility of fuzzy solutions and solving approach, Information Sciences, 241 (2013), 12-27.
https://doi.org/10.1016/j.ins.2013.04.004

[11] S. C. Fang, G. Li, Solving fuzzy relational equations with a linear objective function, Fuzzy Sets and Systems,
103(1) (1999), 107-113. https://doi.org/10.1016/S0165-0114(97)00184-X

[12] S. Freson, B. De Baets, H. De Meyer, Linear optimization with bipolar max-min constraints, Information
Sciences, 234 (2013), 3-15. https://doi.org/10.1016/j.ins.2011.06.009

[13] A. Ghodousian, Optimization of linear problems subjected to the intersection of two fuzzy relational inequalities
defined by Dubois-Prade family of t-norms, Information Sciences, 503 (2019), 291-306. https://doi.org/10.
1016/j.ins.2019.06.058

[14] A. Ghodousian, A. Babalhavaeji, An efficient genetic algorithm for solving nonlinear optimization problems
defined with fuzzy relational equations and max- Lukasiewicz composition, Applied Soft Computing, 69 (2018),
475-492. https://doi.org/10.1016/j.asoc.2018.04.029

[15] A. Ghodousian, E. Khorram, Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition, Information Sciences, 178(2) (2008), 501-519. https://doi.org/10.1016/
j.ins.2007.07.022

[16] A. Ghodousian, E. Khorram, Linear optimization with an arbitrary fuzzy relational inequality, Fuzzy Sets and
Systems, 206 (2012), 89-102. https://doi.org/10.1016/j.fss.2012.04.009

[17] A. Ghodousian, M. Naeeimi, A. Babalhavaeji, Nonlinear optimization problem subjected to fuzzy relational
equations defined by Dubois-Prade family of t-norms, Computers and Industrial Engineering, 119 (2018), 167-
180. https://doi.org/10.1016/j.cie.2018.03.038

[18] A. Ghodousian, M. Raeisian Parvari, A modified PSO algorithm for linear optimization problem subject to the
generalized fuzzy relational inequalities with fuzzy constraints (FRI-FC), Information Sciences, 418-419 (2017),
317-345. https://doi.org/10.1016/j.ins.2017.07.032

[19] A. Ghodousian, F. Samie Yousefi, Linear optimization problem subjected to fuzzy relational equations and fuzzy
constraints, Iranian Journal of Fuzzy Systems, 20(2) (2023), 1-20. https://doi.org/10.22111/ijfs.2023.
7552

[20] F. F. Guo, L. P. Pang, D. Meng, Z. Q. Xia, An algorithm for solving optimization problems with fuzzy relational
inequality constraints, Information Sciences, 252 (2013), 20-31. https://doi.org/10.1016/j.ins.2011.09.
030

[21] S. M. Guu, Y. K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy
Optimization and Decision Making, 12 (2002), 1568-4539. https://doi.org/10.1023/A:1020955112523

[22] S. M. Guu, Y. K. Wu, Minimizing a linear objective function under a max-t-norm fuzzy relational equation
constraint, Fuzzy Sets and Systems, 161(2) (2010), 285-297. https://doi.org/10.1016/j.fss.2009.03.007

[23] K. Hirota, W. Pedrycz, Solving fuzzy relational equations through logical filtering, Fuzzy Sets and Systems,
81(3) (1996), 355-363. https://doi.org/10.1016/0165-0114(95)00221-9

[24] F. Kouchakinejad, M. Mashinchi, R. Mesiar, Solution-set invariant matrices and vectors in fuzzy relation
inequalities based on max aggregation function composition, Iranian Journal of Fuzzy Systems, 13(7) (2016),
91-100. https://doi.org/10.22111/ijfs.2016.2945

[25] H. C. Lee, S. M. Guu, On the optimal three-tier multimedia streaming services, Fuzzy Optimization and Decision
Making, 2(1) (2003), 31-39. https://doi.org/10.1023/A:1022848114005

[26] P. K. Li, S. C. Fang, On the resolution and optimization of a system of fuzzy relational equations with
sup-t composition, Fuzzy Optimization and Decision Making, 7 (2008), 169-214. https://doi.org/10.1007/
s10700-008-9029-y

[27] P. Li, Q. Jin, On the resolution of bipolar max-min equations, Kybernetika, 52(4) (2016), 514-530. https:
//dml.cz/bitstream/handle/10338.dmlcz/145903/Kybernetika_52-2016-4_2.pdf

[28] P. Li, Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using Lukasiewicz triangular
norm, Soft Computing, 18 (2014), 1399-1404. https://doi.org/10.1007/s00500-013-1152-1

[29] J. X. Li, S. J. Yang, Fuzzy relation inequalities about the data transmission mechanism in bittorrent-like peer-topeer file sharing systems, in: Proceedings of the 9th International Conference on Fuzzy Systems and Knowledge
discovery (FSKD 2012), 452-456. https://doi.org/10.1109/FSKD.2012.6233956

[30] J. L. Lin, On the relation between fuzzy max-Archimedean t-norm relational equations and the covering problem,
Fuzzy Sets and Systems, 160(16) (2009), 2328-2344.https://doi.org/10.1016/j.fss.2009.01.012 .

[31] J. L. Lin, Y. K. Wu, S. M. Guu, On fuzzy relational equations and the covering problem, Information Sciences,
181(14) (2011), 2951-2963. https://doi.org/10.1016/j.ins.2011.03.004

[32] C. C. Liu, Y. Y. Lur, Y. K. Wu, Linear optimization of bipolar fuzzy relational equations with max- Lukasiewicz
composition, Information Sciences, 360 (2016), 149-162. https://doi.org/10.1016/j.ins.2016.04.041

[33] J. Loetamonphong, S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy
Sets and Systems, 118(3) (2001), 509-517. https://doi.org/10.1016/S0165-0114(98)00417-5

[34] J. Lu, S. C. Fang, Solving nonlinear optimization problems with fuzzy relation equations consraints, Fuzzy Sets
and Systems, 119(1) (2001), 1-20. https://doi.org/10.1016/S0165-0114(98)00471-0

[35] A. V. Markovskii, On the relation between equations with max-product composition and the covering problem,
Fuzzy Sets and Systems, 153(2) (2005), 261-273. https://doi.org/10.1016/j.fss.2005.02.010

[36] M. Mizumoto, H. J. Zimmermann, Comparison of fuzzy reasoning method, Fuzzy Sets and Systems, 8(3) (1982),
253-283. https://doi.org/10.1016/S0165-0114(82)80004-3

 [37] W. Pedrycz, Fuzzy relational equations with generalized connectives and their applications, Fuzzy Sets and
 Systems, 10(1-3) (1983), 185-201. https://doi.org/10.1016/S0165-0114(83)80114-6

 [38] W. Pedrycz, Granular computing: Analysis and design of intelligent systems, CRC Press, Boca Raton, 2013.

 [39] K. Peeva, Y. Kyosev, Fuzzy relational calculus: Theory, applications and software, World Scientific Publishing
 Co Pte Ltd, 2005.

 [40] X. B. Qu, X. P. Wang, Minimization of linear objective functions under the constraints expressed by a system of
 fuzzy relation equations, Information Sciences, 178(17) (2008), 3482-3490. https://doi.org/10.1016/j.ins.
 2008.04.004

 [41] X. B. Qu, X. P. Wang, M. H. Lei, Conditions under which the solution sets of fuzzy relational equations over
 complete Brouwerian lattices form lattices, Fuzzy Sets and Systems, 234 (2014), 34-45. https://doi.org/10.
 1016/j.fss.2013.03.017

 [42] E. Sanchez, Resolution of Eigen fuzzy sets equations, Fuzzy Sets and Systems, 1(1) (1978), 69-75. https:
 //doi.org/10.1016/0165-0114(78)90033-7

 [43] E. Sanchez, Solution in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian
 logic, in: M.M. Gupta. G.N. Saridis, B.R. Games (Eds.), Readings in Fuzzy Sets for Intelligent Systems, (1993),
 159-165. https://doi.org/10.1016/B978-1-4832-1450-4.50017-1

 [44] B. S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Information Sciences, 178(8) (2008),
 1961-1967. https://doi.org/10.1016/j.ins.2007.12.006

 [45] B. S. Shieh, Minimizing a linear objective function under a fuzzy max-t-norm relation equation constraint,
 Information Sciences, 181(4) (2011), 832-841. https://doi.org/10.1016/j.ins.2010.10.024

 [46] G. B. Stamou, S. G. Tzafestas, Resolution of composite fuzzy relation equations based on Archimedean triangular
 norms, Fuzzy Sets and Systems, 120(3) (2001), 395-407. https://doi.org/10.1016/S0165-0114(99)00117-7

 [47] F. Sun, Conditions for the existence of the least solution and minimal solutions to fuzzy relation equations
 over complete Brouwerian lattices, Information Sciences, 205 (2012), 86-92. https://doi.org/10.1016/j.
 ins.2012.04.002

 [48] F. Sun, X. P. Wang, X. B. Qu, Minimal join decompositions and their applications to fuzzy relation equations
 over complete Brouwerian lattices, Information Sciences, 224 (2013), 143-151. https://doi.org/10.1016/j.
 ins.2012.10.038

 [49] Y. K. Wu, Optimization of fuzzy relational equations with max-av composition, Information Sciences, 177(19)
 (2007), 4216-4229. https://doi.org/10.1016/j.ins.2007.02.037

 [50] Y. K. Wu, S. M. Guu, Minimizing a linear function under a fuzzy max-min relational equation constraints,
 Fuzzy Sets and Systems, 150(1) (2005), 147-162.

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

 [52] Y. K. Wu, S. M. Guu, J. Y. Liu, Reducing the search space of a linear fractional programming problem under
 fuzzy relational equations with max-Archimedean t-norm composition, Fuzzy Sets and Systems, 159(24) (2008),
 3347-3359. https://doi.org/10.1016/j.fss.2008.04.007
Optimizing linear functions over novel fuzzy relation equations: Structure, feasibility, and global solutions
 179

  [53] Q. Q. Xiong, X. P. Wang, Fuzzy relational equations on complete Brouwerian lattices, Information Sciences,
 193 (2012), 141-152. https://doi.org/10.1016/j.ins.2011.12.030

 [54] S. J. Yang, An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities
 with addition-min composition, Fuzzy Sets and Systems, 255 (2014), 41-51. https://doi.org/10.1016/j.fss.
 2014.04.007

 [55] X. P. Yang, Resolution of bipolar fuzzy relation equations with max-Lukasiewicz composition, Fuzzy Sets and
 Systems, 397 (2020), 41-60. https://doi.org/10.1016/j.fss.2019.08.005

 [56] X. P. Yang, X. G. Zhou, B. Y. Cao, Latticized linear programming subject to max-product fuzzy relation
 inequalities with application in wireless communication, Information Sciences, 358-359 (2016), 44-55. https:
 //doi.org/10.1016/j.ins.2016.04.014

 [57] J. Zhou, Y. Yu, Y. Liu, Y. Zhang, Solving nonlinear optimization problems with bipolar fuzzy relational
 equations constraints, Journal of Inequalities and Applications, 126 (2016), 1-10. https://doi.org/10.1186/
 s13660-016-1056-6