Linear optimization problem subjected to fuzzy relational equations and fuzzy constraints

[1] O. A. Abdul-Rahman, M. Munetomo, An adaptive parameter binary-real coded genetic algorithm for constraint optimization
problems: Performance analysis and estimation of optimal control parameters, Information Sciences, 233 (2013), 54-86.

[2] M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear programming: Theory and algorithms, John Wiley and Sons, New
York, NY, 2006.

[3] C. W. Chang, B. S. Shieh, Linear optimization problem constrained by fuzzy max-min relation equations, Information Sciences,
234 (2013), 71-79.

[4] L. Chen, P. P. Wang, Fuzzy relation Equations (i): The general and specialized solving algorithms, Soft Computing, 6(5)
(2002), 428-435.

[5] M. Cornejo, D. Lobo, J. Medina, On the solvability of bipolar max-product fuzzy relation equations with the standard negation,
Fuzzy Sets and Systems, 410 (2021), 1-18.

[6] 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 (2012), 58-67.

[7] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy relational equations and their applications in knowledge engineering,
Dordrecht: Kluwer Academic Press, 1989.

[8] D. Dubey, S. Chandra, A. Mehra, Fuzzy linear programming under interval uncertainty based on IFS representation, Fuzzy
Sets and Systems, 188 (2012), 68-87.

[9] D. Dubois, H. Prade, Fundamentals of fuzzy sets, Kluwer, Boston, 2000.
[10] S. C. Fang, G. Li, Solving fuzzy relational equations with a linear objective function, Fuzzy Sets and Systems, 103 (1999),
107-113.

[11] S. Freson, B. De Baets, H. De Meyer, Linear optimization with bipolar max-min constraints, Information Sciences, 234
(2013), 3-15.

[12] Z. W. Geem, J. H. Kim, G. V. Loganathan, A new heuristic optimization algorithm: Harmony search, Simulation, 76(2)
(2001), 60-68.

[13] A. Ghodousian, An algorithm for solving linear optimization problems subjected to the intersection of two fuzzy relational
inequalities defined by Frank family of t-norms, International Journal in Foundations of Computer Science and Technology,
8(3) (2018), 1-20.

[14] 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.

[15] 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.

[16] A. Ghodousian, E. Khorram, Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with
max-min composition, Information Sciences, 178 (2008), 501-519.

[17] A. Ghodousian, E. Khorram, Linear optimization with an arbitrary fuzzy relational inequality, Fuzzy Sets and Systems, 206
(2012), 89-102.

[18] A. Ghodousian, M. Naeeimib, 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.

[19] 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.

[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.

[21] F. F. Guo, Z. Q. Xia, An algorithm for solving optimization problems with one linear objective function and finitely many
constraints of fuzzy relation inequalities, Fuzzy Optimization and Decision Making, 5 (2006), 33-47.

[22] 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.

[23] 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 (2010), 285-297.

[24] R. L. Haupt, E. Haupt, Practical genetic algorithms, Second ed., John Wiley and Sons, 2004.

[25] J. Kennedy, R. C. Eberhart, Particle swarm optimization, in: Proceedings of IEEE International Conference on Neural
Networks, 4 (1995), 1942-1948.

[26] 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.

[27] 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.

[28] P. K. Li, Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using Lukasiewicz triangular norm,
Soft Computing, 18 (2014), 1399-1404.

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

[30] J. L. Lin, On the relation between fuzzy max-Archimedean t-norm relational equations and the covering problem, Fuzzy Sets
and Systems, 160 (2009), 2328-2344.

[31] C. H. Lin, A rough penalty genetic algorithm for constrained optimization, Information Sciences, 241 (2013), 119-137.

[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.

[33] J. Loetamonphong, S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and
Systems, 118 (2001), 509-517.

[34] A. V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Sets and
Systems, 153 (2005), 261-273.

[35] H. D. Mazraeh, A. Molai, Resolution of nonlinear optimization problems subject to bipolar max-min fuzzy relation equation
constraints using genetic algorithm, Iranian Journal of Fuzzy Systems, (2018), 109-131.

[36] M. Mizumoto, H. J. Zimmermann, Comparison of fuzzy reasoning method, Fuzzy Sets and Systems, 8 (1982), 253-283.

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

[38] C. Pornsing, K. Saichareon, T. Karot, A modified particle swarm optimization for engineering constrained optimization
problems, International Journal of Computer Science and Electronics Engineering, 3(1) (2015), 2320-4028.

[39] 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.

[40] 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.), Fuzzy Automata and Decision Processes, North-Holland, New York, (1977),
221-234.

[41] B. S. Shieh, Minimizing a linear objective function under a fuzzy max-t-norm relation equation constraint, Information
Sciences, 181 (2011), 832-841.

[42] K. Socha, M. Dorigo, Ant colony optimization for continuous domain, European Journal of Operational Research, 185
(2008), 1155-1173.

[43] 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.

[44] 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.

[45] C. L. Sun, J. C. Zeng, J. S. Pan, An improved vector particle swarm optimization for constrained optimization problems,
Information Sciences, 181 (2011), 1153-1163.

[46] S. Wang, H. Li, Resolution of fuzzy relational inequalities with Boolean semi-tensor product composition, Mathematics, 9
(2021), 937.

[47] Y. K. Wu, Optimization of fuzzy relational equations with max-av composition, Information Sciences, 177 (2007), 4216-4229.

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

[49] 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.

[50] 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 (2008), 3347-3359.

[51] 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.

[52] 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.

[53] C. T. Yeh, On the minimal solutions of max-min fuzzy relation equations, Fuzzy Sets and Systems, 159 (2008), 23-39.