On the nonlinear programming problems subject to a system of generalized bipolar fuzzy relational equalities defined with continuous t-norms

Document Type : Research Paper

Authors

1 Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran.

2 Department of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran

Abstract

As a starting point, this paper develops the system of bipolar fuzzy relational equations (FRE) to the most general case, where bipolar FREs are defined by an arbitrary continuous t-norm. Due to the fact that fuzzy relational equations are special cases of bipolar FREs, the proposed system can also be viewed as a generalization of traditional FREs, in which the fuzzy composition can be defined by a continuous t-norm. In order to determine the feasibility of the proposed system, some necessary and sufficient conditions are presented for studying continuous bipolar FREs. This is followed by a complete analysis of the set of feasible solutions to the problem. Contrary to FREs and bipolar FREs defined by continuous Archimedean t-norms, the feasible solutions set of generalized bipolar FREs consists of a finite number of compact sets that are not necessarily connected. Further, five techniques have been outlined in an attempt to simplify the current problem, and then an algorithm has been presented to find the feasible region of the problem. Next, we present a class of optimization models subject to continuous bipolar FRE constraints, in which the objective function incorporates a wide range of (non)linear functions, such as maximum functions, geometric mean functions, log-sum-exp functions, maximum eigenvalues of symmetric matrices, support functions for sets, etc. Considering that the problem has a finite number of local optimal solutions, the global optimal solution can always be obtained by choosing the point with the minimum objective value among these local optimal solutions. Lastly, as a means to illustrate the definitions, theorems, and algorithms presented in the paper, a step-by-step example is presented in several sections, in which the constraints are a system of bipolar FREs defined by the Dubois-Prade t-norm, which is a continuous non-Archimedean t-norm.

Keywords

Main Subjects

References

[1] M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear programming: Theory and algorithms, John wiley and Sons,
2006. https://doi.org/10.1002/0471787779
[2] S. Boyd, L. Vandenberghe, Convex optimization, Cambridge University Press, 2004. https://doi.org/10.1017/
cbo9780511804441
[3] 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
[4] L. Chen, P. P. Wang, Fuzzy relation equations (i): The general and specialized solving algorithms, Soft Computing,
6 (2002), 428-435. https://doi.org/10.1007/s00500-001-0157-3
[5] L. Chen, P. P.Wang, Fuzzy relation equations (ii): The branch-point-solutions and the categorized minimal solutions,
Soft Computing, 11 (2007), 33-40. https://doi.org/10.1007/s00500-006-0050-1
[6] M. E. Cornejo, D. Lobo, J. Medina, Bipolar fuzzy relation equations based on the product t-norm, In 2017 IEEE
International Conference on Fuzzy Systems (FUZZ-IEEE), (2017), 1-6. https://doi.org/10.1109/fuzz-ieee.
2017.8015691
[7] 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
[8] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy relation equations and their applications to knowledge engineering,
Springer Science and Business Media, 3 (2013). https://doi.org/10.1007/978-94-017-1650-5
[9] 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
[10] 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
[11] D. Dubois, H. Prade, An overview of the asymmetric bipolar representation of positive and negative information in
possibility theory, Fuzzy Sets and Systems, 160(10) (2009), 1355-1366. https://doi.org/i:10.1016/j.fss.2008.
11.006
[12] 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/i:
10.1016/j.ins.2013.04.004
[13] S. C. Fang, G. Li, Solving fuzzy relation 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
[14] 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
[15] 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
[16] 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
[17] 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
[18] 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
[19] 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
[20] 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
[21] A. Ghodousian, M. Sedigh Chopannavaz, Solving linear optimization problems subject to bipolar fuzzy relational
equalities defined with max-strict compositions, Information Sciences, 650 (2023), 119696. https://doi.org/10.
1016/j.ins.2023.119696
[22] 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
[23] S. M. Guu, Y. K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy
Optimization and Decision Making, 1 (2002), 347-360. https://doi.org/10.1016/j.fss.2009.03.007
[24] 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/10.1023/A:1020955112523
[25] 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
[26] H. C. Lee, S. M. Guu, On the optimal three-tier multimedia streaming services, Fuzzy Optimization and Decision
Making, 2 (2003), 31-39. https://doi.org/10.1023/A:1022848114005
[27] P. 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
[28] P. Li, Q. Jin, On the resolution of bipolar max-min equations, Kybernetika, 52(4) (2016), 514-530. https://doi.
org/10.14736/kyb-2016-4-0514
[29] P. Li, Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using the  Lukasiewicz triangular
norm, Soft Computing, 18(7) (2014), 1399-1404. https://doi.org/10.1007/s00500-013-1152-1
[30] J. X. Li, S. Yang, Fuzzy relation inequalities about the data transmission mechanism in bittorrentlike peer-to-peer
file sharing systems, In 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, (2012),
452-456. https://doi.org/10.1109/fskd.2012.6233956
[31] 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
[32] 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
[33] C. C. Liu, Y. Y. Lur, Y. K. Wu, Linear optimization of bipolar fuzzy relational equations with max- Lukasiewicz
co
[34] 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
[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 methods, 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, 2018. https://doi.
org/10.1201/9781315216737
[39] 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
[40] 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
[41] E. Sanchez, Resolution of eigen fuzzy sets equations, Fuzzy Sets and Systems, 1(1) (1987), 69-74. https://doi.
org/10.1016/0165-0114(78)90033-7
[42] E. Sanchez, Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic, In
Readings in Fuzzy Sets for Intelligent Systems, (1993), 159-165. https://doi.org/10.1016/b978-1-4832-1450-4.
50017-1
[43] 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
[44] 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
[45] 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
[46] 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
[47] 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
[48] Y. K. Wu, S. M. Guu, Minimizing a linear function under a fuzzy max-min relational equation constraint, Fuzzy
Sets and Systems, 150(1) (2005), 147-162. https://doi.org/10.1109/tfuzz.2007.902018
[49] Y. K. Wu, S. M. Guu, An efficient procedure for solving a fuzzy relational equation with max-Archimedean t-norm
composition, IEEE Transactions on Fuzzy Systems, 16(1) (2008), 73-84. https://doi.org/10.1016/j.fss.2004.
09.010
[50] Y. K. Wu, S. M. Guu, J. Y. C. 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
[51] 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
[52] 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
[53] 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
[54] 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 (2016), 44-55. https://doi.org/10.1016/
j.ins.2016.04.014
[55] J. Zhou, Y. Yu, Y. Liu, Y. Zhang, Solving nonlinear optimization problems with bipolar fuzzy relational
equation constraints, Journal of Inequalities and Applications, 2016 (2016), 1-10. https://doi.org/10.1186/
s13660-016-1056-6
Iranian Journal of Fuzzy Systems
Volume 22, Issue 3
May and June 2025
Pages 21-38

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