Document Type : Research Paper
Authors
Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran.
Abstract
This paper presents some novel theoretical results as well as practical algorithms and computational procedures on continuous fuzzy relational equations programming problems. The fuzzy relational programming problem is a minimization (maximization) problem with a linear objective function subject to fuzzy relational equalities or inequalities defined with certain algebraic operations. In the literature, the commonly seen frameworks for such optimization models are to assume that the operation takes minimum t-norm, strict continuous t-norms (e.g., product t-norm), nilpotent continuous t-norms (e.g., Lukasiewicz t-norm) or Archimedean continuous t-norms. Based on new concepts called partial solution sets, the current paper considers this problem\textbf{ }in the most general case where the fuzzy relational equality constraints are defined by an arbitrary continuous t-norm and capture some special characteristics of its feasible domain and the optimal solutions. It is shown that the current generalized results are automatically reduced to (apparently) different ones that hold for special operators when continuous t-norm is replaced by strict, nilpotent or Archimedean continuous t-norm. Also, the relationship between the results derived here and those of previous publications regarding this subject is also discussed. Finally, the proposed algorithm is outlined and illustrated by a numerical example where the continuous fuzzy relational equations is defined by Mayor-Torrens operator that is not an Archimedean t-norm (and then, neither strict nor nilpotent).
Keywords
- Fuzzy relational equations
- Strict t-norms
- Nilpotent t-norms
- Archimedean t-norms
- Continuous t-norms
- Global optimization
- Linear optimization
Main Subjects
[1] M. M. Bourke, D. G. Fisher, Solution algorithms for fuzzy relational equations with max-product composition, Fuzzy
Sets and Systems, 94 (1998), 61-69. https://doi.org/10.1016/S0165-0114(96)00246-1
Sets and Systems, 94 (1998), 61-69. https://doi.org/10.1016/S0165-0114(96)00246-1
[2] 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
Sciences, 234 (2013), 71-79. https://doi.org/10.1016/j.ins.2011.04.042
[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
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
IEEE International Conference on Fuzzy Systems, 2017. https://doi.org/10.1109/FUZZ-IEEE.2017.8015691
[5] E. Czogala, J. Drewniak, W. Pedrycz, Fuzzy relation equations on a finite set, Fuzzy Sets and Systems, 7 (1982),
89-101. https://doi.org/10.1016/0165-0114(82)90043-4
89-101. https://doi.org/10.1016/0165-0114(82)90043-4
[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. https://doi.org/10.1016/j.fss.2011.07.014
problem, Fuzzy Sets and Systems, 188 (2012), 58-67. https://doi.org/10.1016/j.fss.2011.07.014
[7] 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
engineering, Dordrecht: Kluwer Academic Press, 1989. https://doi.org/10.1007/978-94-017-1650-5
[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. https://doi.org/10.1016/j.fss.2011.09.008
Fuzzy Sets and Systems, 188 (2012), 68-87. https://doi.org/10.1016/j.fss.2011.09.008
[9] D. Dubois, H. Prade, Fundamentals of fuzzy sets, Kluwer, Boston, 2000. https://doi.org/10.1007/
978-1-4615-4429-6
978-1-4615-4429-6
[10] S. C. Fang, G. Li, Solving fuzzy relational equations with a linear objective function, Fuzzy Sets and Systems, 103
(1999), 107-113. https://doi.org/10.1016/S0165-0114(97)00184-X
(1999), 107-113. https://doi.org/10.1016/S0165-0114(97)00184-X
[11] 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
234 (2013), 3-15. https://doi.org/10.1016/j.ins.2011.06.009
[12] 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
defined by Dubois-Prade family of t-norms, Information Sciences, 503 (2019), 291-306. https://doi.org/10.1016/
j.ins.2019.06.058
[13] 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
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
[14] 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
Systems, 206 (2012), 89-102. https://doi.org/10.1016/j.fss.2012.04.009
[15] 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.
https://doi.org/10.1016/j.cie.2018.03.038
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
[16] 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
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
[17] A. Ghodousian, F. Samie Yousefi, Linear optimization problem subjected to fuzzy relational equations and fuzzy
constraints, Iranian Journal of Fuzzy Systems, 20 (2023), 1-20. https://doi.org/10.22111/IJFS.2023.7552
constraints, Iranian Journal of Fuzzy Systems, 20 (2023), 1-20. https://doi.org/10.22111/IJFS.2023.7552
[18] 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
Optimization and Decision Making, 12 (2002), 1568-4539. https://doi.org/10.1023/A:1020955112523
[19] 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. https://doi.org/10.1016/j.fss.2009.03.007
Fuzzy Sets and Systems, 161 (2010), 285-297. https://doi.org/10.1016/j.fss.2009.03.007
[20] M. Higashi, G. J. Klir, Resolution of finite fuzzy relation equations, Fuzzy Sets and Systems, 13 (1984), 65-82.
https://doi.org/10.1016/0165-0114(84)90026-5
https://doi.org/10.1016/0165-0114(84)90026-5
[21] E. P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper I: Basic analytical and algebraic properties,
Fuzzy Sets and Systems, 143 (2004), 5-26. https://doi.org/10.1016/j.fss.2003.06.007
Fuzzy Sets and Systems, 143 (2004), 5-26. https://doi.org/10.1016/j.fss.2003.06.007
[22] 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
norm, Soft Computing, 18 (2014), 1399-1404. https://doi.org/10.1007/s00500-013-1152-1
[23] J. L. Lin, Y. K. Wu, S. M. Guu, On fuzzy relational equations and the covering problem, Information Sciences, 181
(2011), 2951-2963. https://doi.org/10.1016/j.ins.2011.03.004
(2011), 2951-2963. https://doi.org/10.1016/j.ins.2011.03.004
[24] 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
composition, Information Sciences, 360 (2016), 149-162. https://doi.org/10.1016/j.ins.2016.04.041
[25] J. Loetamonphong, S. C. Fang, An efficient solution procedure for fuzzy relation equations with max-product composition, IEEE Transactions on Fuzzy Systems, 7 (1999), 441-445. https://doi.org/10.1109/91.784204
[26] J. Loetamonphong, S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets
and Systems, 118 (2001), 509-517. https://doi.org/10.1016/S0165-0114(98)00417-5
and Systems, 118 (2001), 509-517. https://doi.org/10.1016/S0165-0114(98)00417-5
[27] J. Lu, S. C. Fang, Solving nonlinear optimization problems with fuzzy relation equations constraints, Fuzzy Sets
and Systems, 119 (2001), 1-20. https://doi.org/10.1016/S0165-0114(98)00471-0
and Systems, 119 (2001), 1-20. https://doi.org/10.1016/S0165-0114(98)00471-0
[28] A. V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy
Sets and Systems, 153 (2005), 261-273. https://doi.org/10.1016/j.fss.2005.02.010
Sets and Systems, 153 (2005), 261-273. https://doi.org/10.1016/j.fss.2005.02.010
[29] M. Mizumoto, H. J. Zimmermann, Comparison of fuzzy reasoning method, Fuzzy Sets and Systems, 8 (1982),
253-283. https://doi.org/10.1016/S0165-0114(82)80004-3
253-283. https://doi.org/10.1016/S0165-0114(82)80004-3
[30] W. Pedrycz, Fuzzy relational equations with generalized connectives and their applications, Fuzzy Sets and Systems,
10 (1983), 185-201. https://doi.org/10.1016/S0165-0114(83)80114-6
10 (1983), 185-201. https://doi.org/10.1016/S0165-0114(83)80114-6
[31] W. Pedrycz, On generalized fuzzy relational equations and their applications, Journal of Mathematical Analysis
and Applications, 107 (1985), 520-536. https://doi.org/10.1016/0022-247X(85)90329-4
and Applications, 107 (1985), 520-536. https://doi.org/10.1016/0022-247X(85)90329-4
[32] W. Pedrycz, Proceeding in relational structures: Fuzzy relational equations, Fuzzy Sets and Systems, 40 (1991),
77-106. https://doi.org/10.1016/0165-0114(91)90047-T
77-106. https://doi.org/10.1016/0165-0114(91)90047-T
[33] W. Pedrycz, Granular computing: Analysis and design of intelligent systems, CRC Press, Boca Raton, 2013.
https://doi.org/10.1201/9781315216737
https://doi.org/10.1201/9781315216737
[34] I. Perfilieva, Fuzzy function as an approximate solution to a system of fuzzy relation equations, Fuzzy Sets and
Systems, 147 (2004), 363-383. https://doi.org/10.1016/j.fss.2003.12.007
Systems, 147 (2004), 363-383. https://doi.org/10.1016/j.fss.2003.12.007
[35] M. Prevot, Algorithm for the solution of fuzzy relations, Fuzzy Sets and Systems, 5 (1981), 319-322. https:
//doi.org/10.1016/0165-0114(81)90059-2
//doi.org/10.1016/0165-0114(81)90059-2
[36] 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 (2008), 3482-3490. https://doi.org/10.1016/j.ins.2008.04.004
relation equations, Information Sciences, 178 (2008), 3482-3490. https://doi.org/10.1016/j.ins.2008.04.004
[37] 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. https://doi.org/10.1007/978-94-017-1650-5
in: M. M. Gupta. G. N. Saridis, B. R. Games (Eds.), Fuzzy Automata and Decision Processes, North-Holland, New
York, (1977), 221-234. https://doi.org/10.1007/978-94-017-1650-5
[38] B. S. Shieh, Minimizing a linear objective function under a fuzzy max-t-norm relation equation constraint, Information
Sciences, 181 (2011), 832-841. https://doi.org/10.1016/j.ins.2010.10.024
Sciences, 181 (2011), 832-841. https://doi.org/10.1016/j.ins.2010.10.024
[39] G. B. Stamou, S. G. Tzafestas, Resolution of composite fuzzy relation equations based on Archimedean triangular
norms, Fuzzy Sets and Systems, 120 (2001), 395-407.
norms, Fuzzy Sets and Systems, 120 (2001), 395-407.
[40] 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
complete Brouwerian lattices, Information Sciences, 224 (2013), 143-151. https://doi.org/10.1016/j.ins.2012.
10.038
[41] Y. K. Wu, Optimization of fuzzy relational equations with max-av composition, Information Sciences, 177 (2007),
4216-4229. https://doi.org/10.1016/j.ins.2007.02.037
4216-4229. https://doi.org/10.1016/j.ins.2007.02.037
[42] 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. https://doi.org/10.1016/j.fss.2004.09.010
Sets and Systems, 150 (2005), 147-162. https://doi.org/10.1016/j.fss.2004.09.010
[43] Y. K. Wu, S. M. Guu, Y. C. Liu, An accelerated approach for solving fuzzy relation fuzzy relation equations with a
linear objective function, IEEE Transactions on Fuzzy Systems, 10(4) (2002), 552-558. https://doi.org/10.1109/
TFUZZ.2002.800657
linear objective function, IEEE Transactions on Fuzzy Systems, 10(4) (2002), 552-558. https://doi.org/10.1109/
TFUZZ.2002.800657
[44] Y. K.Wu, S. M. Guu, J. Y. Liu, A note on fuzzy relation programming problems with max-strict-t-norm composition,
Fuzzy Optimization and Decision Making, 3 (2004), 271-278. https://doi.org/10.1023/B:FODM.0000036862.
45420.ea
Fuzzy Optimization and Decision Making, 3 (2004), 271-278. https://doi.org/10.1023/B:FODM.0000036862.
45420.ea
[45] 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.
https://doi.org/10.1016/j.fss.2008.04.007
relational equations with max-Archimedean t-norm composition, Fuzzy Sets and Systems, 159 (2008), 3347-3359.
https://doi.org/10.1016/j.fss.2008.04.007
[46] 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
addition-min composition, Fuzzy Sets and Systems, 255 (2014), 41-51. https://doi.org/10.1016/j.fss.2014.
04.007
[47] X. P. Yang, Resolution of bipolar fuzzy relation equations with max- Lukasiewicz composition, Fuzzy Sets and
Systems, (2020). https://doi.org/10.1016/j.fss.2019.08.005
Systems, (2020). https://doi.org/10.1016/j.fss.2019.08.005
[48] 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
with application in wireless communication, Information Sciences, 358-359 (2016), 44-55. https://doi.org/10.
1016/j.ins.2016.04.014
)