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Optimization of fuzzy relational equations with max-av composition. (English) Zbl 1140.90523
Summary: Max-min and max-product compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations. Both are members in the class of max-\(t\)-norm composition. In this study, the max-av composition is considered for the same optimization model, which does not belong to the max-\(t\)-norm composition. However, max-av composition generates some properties of the solution set that are similar to the max-product composition. Thanks to these properties, a simple value matrix with rules can be applied to reduce problem size. Thus, this study proposes an efficient procedure for obtaining optimal solutions without decomposing the problem into two sub-problems or finding all the potential minimal solutions.

MSC:
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
03E72 Theory of fuzzy sets, etc.
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[1] Chen, L.; Wang, P.P., Fuzzy relation equations (I): the general and specialized solving algorithms, Soft computing, 6, 428-435, (2002) · Zbl 1024.03520
[2] Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy sets and systems, 7, 89-101, (1982) · Zbl 0483.04001
[3] Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Academic Press Dordrecht · Zbl 0694.94025
[4] Di Nola, A.; Pedrycz, W.; Sessa, S.; Sanchez, E., Fuzzy relation equations theory as a basis of fuzzy modelling: an overview, Fuzzy sets and systems, 40, 415-429, (1991) · Zbl 0727.04005
[5] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press San Diego · Zbl 0444.94049
[6] Fang, S.-C.; Li, G., Solving fuzzy relation equations with a linear objective function, Fuzzy sets and systems, 103, 107-113, (1999) · Zbl 0933.90069
[7] Fernández, M.J.; Gil, P., Some specific types of fuzzy relation equations, Information sciences, 164, 189-195, (2004) · Zbl 1058.03058
[8] Guu, S.-M.; Wu, Y.-K., Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy optimization and decision making, 1, 4, 347-360, (2002) · Zbl 1055.90094
[9] Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006
[10] Hirota, K.; Pedrycz, W., Fuzzy relational compression, IEEE transactions on systems, man, and cybernetics-part B, 29, 3, 407-415, (1999)
[11] Khorram, E.; Ghodousian, A., Linear objective function optimization with fuzzy relation equation constraints regarding MAX-av composition, Applied mathematics and computation, 173, 872-886, (2006) · Zbl 1091.65057
[12] Lee, H.-C.; Guu, S.-M., On the optimal three-tier multimedia streaming services, Fuzzy optimization and decision making, 2, 3, 31-39, (2002)
[13] Loetamonphong, J.; Fang, S.-C., Optimization of fuzzy relational equations with MAX-product composition, Fuzzy sets and systems, 118, 509-517, (2001) · Zbl 1044.90533
[14] Loetamonphong, J.; Fang, S.-C.; Young, R.E., Multi-objective optimization problems with fuzzy relation equation constraints, Fuzzy sets and systems, 127, 141-164, (2002) · Zbl 0994.90130
[15] Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Information sciences, 171, 145-172, (2005) · Zbl 1078.68815
[16] Lu, J.; Fang, S.-C., Solving nonlinear optimization problems with fuzzy relation equations constraints, Fuzzy sets and systems, 119, 1-20, (2001)
[17] Markovskii, A.V., On the relation between equations with MAX-product composition and the covering problem, Fuzzy sets and systems, 153, 261-273, (2005) · Zbl 1073.03538
[18] Nobuhara, H.; Bede, B.; Hirota, K., On various eigen fuzzy sets and their application to image reconstruction, Information sciences, 176, 2988-3010, (2006) · Zbl 1102.68697
[19] H. Nobuhara, W. Pedrycz, K. Hirota, A digital watermarking algorithm using image compression method based on fuzzy relational equation, in: Proceeding of IEEE International Conference on Fuzzy Systems (12-17/5/2002, Honolulu, Hawaii, USA), vol. 2, 2002, pp. 1568-1573.
[20] Nobuhara, H.; Pedrycz, W.; Hirota, K., Fast solving method of fuzzy relational equation and its application to lossy image compression/recompression, IEEE transactions on fuzzy systems, 8, 3, 325-334, (2000)
[21] Pedrycz, W., On generalized fuzzy relational equations and their applications, Journal of mathematical analysis and applications, 107, 520-536, (1985) · Zbl 0581.04003
[22] Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048
[23] Wang, H.-F., A multi-objective mathematical programming problem with fuzzy relation constraints, Journal of multi-criteria decision analysis, 4, 23-35, (1995) · Zbl 0843.90131
[24] Wang, P.Z.; Zhang, D.Z.; Sanchez, E.; Lee, E.S., Latticized linear programming and fuzzy relation inequalities, Journal of mathematical analysis and applications, 159, 72-87, (1991) · Zbl 0746.90081
[25] Wang, S.; Fang, S.-C.; Nuttle, H.L.W., Solution sets of interval-valued fuzzy relational equations, Fuzzy optimization and decision making, 2, 1, 41-60, (2003) · Zbl 1178.03071
[26] Wu, Y.-K.; Guu, S.-M., A note on fuzzy relation programming problems with MAX-strict-t-norm composition, Fuzzy optimization and decision making, 3, 3, 271-278, (2004) · Zbl 1091.90087
[27] Wu, Y.-K.; Guu, S.-M.; Liu, J.Y.-C., An accelerated approach for solving fuzzy relation equations with a linear objective function, IEEE transactions on fuzzy systems, 10, 4, 552-558, (2002)
[28] Wu, Y.-K.; Guu, S.-M., Minimizing a linear function under a fuzzy max – min relational equation constraint, Fuzzy sets and systems, 150, 147-162, (2005) · Zbl 1074.90057
[29] Yang, J.-H.; Cao, B.-Y., Geometric programming with fuzzy relation equation constraints, Proceedings of the IEEE international conference on fuzzy systems, 557-560, (2005)
[30] Zadeh, L.A., Toward a generalized theory of uncertainty (GTU)-an outline, Information sciences, 172, 1-40, (2005) · Zbl 1074.94021
[31] Zimmermann, H.-J., Fuzzy set theory and its application, (1991), Kluwer Academic Publishers Boston/Dordrecht/London, pp. 75-82
[32] Zimmermann, H.-J., Results of empirical studies in fuzzy set theory, (), 303-312
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