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An algorithm for solving fuzzy relation equations with $$\max$$-$$T$$ composition operator. (English) Zbl 1136.03330
Summary: This paper studies the problem of solving a max-$$T$$ composite finite fuzzy relation equation, where $$T$$ is a special class of pseudo-t-norms. If the equation is solvable, then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions. Some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value. Under these conditions, some minimal solutions of the system can be obtained easily. Some procedures are also proposed in order to simplify the original system. The simplified system is then decomposed (if possible) into several subsystems with smaller dimensions, which are very easy to solve. Furthermore, a method is presented to solve each subsystem. By combining the method and those procedures, an efficient algorithm is proposed to obtain the set of feasible solutions of the original system. Two examples are also given to illustrate the algorithm.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
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##### References:
  Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy sets and systems, 7, 89-101, (1982) · Zbl 0483.04001  Dai, F.M.; Wang, Z.D., Fuzzy relations and fuzzy relation equations, Journal of yangzhou university, 5, 2, 8-10, (2002), (in Chinese) · Zbl 1012.03516  De Baets, B., An order-theoretic approach to solving sup-Tequations, (), 67-87 · Zbl 0874.04005  De Baets, B., Analytical solution methods for fuzzy relational equations, (), 291-340 · Zbl 0970.03044  Di Nola, A.; Pedrycz, W.; Sessa, S.; Wang, P.Z., Fuzzy relation equations under a class of triangular norms: a survey and new results, Stochastica, 2, 99-145, (1984) · Zbl 0581.04002  Di Nola, A.; Pedrycz, W.; Sessa, S., Fuzzy relation equations and algorithms of inference mechanism in expert systems, (), 355-367  Di Nola, A.; Pedrycz, W.; Sessa, S., Fuzzy relation equations under LSC and USC t-norms and their Boolean solutions, Stochastica, 11, 2-3, 151-183, (1987) · Zbl 0673.04003  Di Nola, A.; Pedrycz, W.; Sessa, S., Fuzzy relation equations with equality and difference composition operators, Fuzzy sets and systems, 25, 205-215, (1988) · Zbl 0645.04004  Di Nola, A.; Pedrycz, W.; Sessa, S., On some finite fuzzy relation equations, Information sciences, 50, 93-109, (1990) · Zbl 0688.90002  Di Nola, A.; Russo, C., Lukasiewicz transform and its application to compression and reconstruction of digital images, Information sciences, 177, 1481-1498, (2007) · Zbl 1114.06009  Di Nola, A.; Pedrycz, W.; Sessa, S., Some theoretical aspects of fuzzy-relation equations describing fuzzy systems, Information sciences, 34, 241-264, (1984) · Zbl 0557.93004  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  Fernandez, M.J.; Gil, P., Some specific types of fuzzy relation equations, Information sciences, 164, 189-195, (2004) · Zbl 1058.03058  Fernandez, M.J.; Suarez, F.; Gil, P., T-eigen fuzzy sets, Information sciences, 75, 63-80, (1993) · Zbl 0803.04006  Fodor, J.C., Strict preference relations based on weak t-norms, Fuzzy sets and systems, 43, 327-336, (1991) · Zbl 0756.90006  Fodor, J.C.; Keresztfalvi, T., Nonstandard conjunctions and implications in fuzzy logic, Internat. J. approx. reason., 12, 69-84, (1995) · Zbl 0815.03017  Gavalec, M., Solvability and unique solvability of max – min fuzzy equations, Fuzzy sets and systems, 124, 385-393, (2001) · Zbl 0994.03047  Gottwald, S., On the existence of solutions of systems of fuzzy equation, Fuzzy sets and systems, 12, 301-302, (1984) · Zbl 0556.04002  Gottwald, S., Generalized solvability criteria for fuzzy equations, Fuzzy sets and systems, 17, 258-296, (1985) · Zbl 0607.03015  Gottwald, S., Characterization of the solvability of fuzzy equations, Elektron. informationsverarb. kybernet., 22, 2/3, 67-91, (1986) · Zbl 0607.03016  Gottwald, S.; Pedrycz, W., Solvability of fuzzy relational equations and manipulation of fuzzy data, Fuzzy sets and systems, 18, 45-65, (1986) · Zbl 0607.94015  Gottwald, S.; Pedrycz, W., On the methodology of solving fuzzy relational equations and its impact on fuzzy modelling, (), 197-210  Gupta, M.M.; Qi, J., Design of fuzzy logic controllers based on generalized t-operators, Fuzzy sets and systems, 40, 473-486, (1991) · Zbl 0732.93050  Han, S.C.; Li, H.X., Notes on “pseudo-t-norms and implication operators on a complete Brouwerian lattice“ and “pseudo-t-norms and implication operators: direct products and direct product decompositions”, Fuzzy sets and systems, 153, 289-294, (2005) · Zbl 1086.03019  Han, S.-C.; Li, H.-X.; Wang, J.-Y., Resolution of finite fuzzy relation equations based on strong pseudo-t-norms, Applied mathematics letters, 19, 752-757, (2006) · Zbl 1121.03075  Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006  Hirota, K.; Pedrycz, W., Specificity shift in solving fuzzy relational equations, Fuzzy sets and systems, 106, 211-220, (1999)  Kagei, S., Fuzzy relational equation with defuzzification-algorithm for the largest solution, Fuzzy sets and systems, 123, 119-127, (2001) · Zbl 0991.65047  Kawaguchi, M.F.; Miyakoshi, M., Composite fuzzy relational equations with non-commutative conjunctions, Information sciences, 110, 113-125, (1998) · Zbl 0930.03074  Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall PTR, USA · Zbl 0915.03001  Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081  Lee, H.-C.; Guu, S.-M., On the optimal three-tier multimedia streaming services, Fuzzy optimization and decision making, 2, 31-39, (2003)  Li, X.; Ruan, D., Novel neural algorithms based on fuzzy δ rules for solving fuzzy relation equations: part II, Fuzzy sets and systems, 103, 473-486, (1999) · Zbl 0931.68100  Li, X.; Ruan, D., Novel neural algorithms based on fuzzy δ rules for solving fuzzy relation equations: part III, Fuzzy sets and systems, 109, 355-362, (2000) · Zbl 0956.68131  Liu, W.-J., On some systems of simultaneous equations in a completely distributive lattice, Information sciences, 50, 185-196, (1990) · Zbl 0701.06011  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  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  Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Information sciences, 171, 145-172, (2005) · Zbl 1078.68815  Lu, J.; Fang, S.-C., Solving nonlinear optimization problems with fuzzy relation equation constraints, Fuzzy sets and systems, 119, 1-20, (2001)  Luo, Y.; Li, Y., Decomposition and resolution of MIN-implication fuzzy relation equations based on S-implications, Fuzzy sets and systems, 148, 305-317, (2004) · Zbl 1060.03077  Luoh, L.; Wang, W.-J.; Liaw, Y.-K., New algorithms for solving fuzzy relation equations, Mathematics and computers in simulation, 59, 329-333, (2002) · Zbl 0999.03513  Miyakoshi, M.; Shimbo, M., Lower solutions of systems of fuzzy equations, Fuzzy sets and systems, 19, 37-46, (1986) · Zbl 0628.04004  Miyakoshi, M.; Shimbo, M., Solutions of composite fuzzy relational equations with triangular norms, Fuzzy sets and systems, 16, 53-63, (1985) · Zbl 0582.94031  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  Ohsato, A.; Sekiguchi, T., Convexly combined fuzzy relational equations and several aspects of their application to fuzzy information processing, Information sciences, 45, 275-313, (1988) · Zbl 0664.68112  Pedrycz, W., Fuzzy relational equations with triangular norms and their resolution, Busefal, 11, 24-32, (1982) · Zbl 0498.04004  Pedrycz, W., Numerical and applicational aspects of fuzzy relational equations, Fuzzy sets and systems, 11, 1-18, (1983) · Zbl 0517.93001  Pedrycz, W., On generalized fuzzy relational equations and their applications, J. math. anal. appl., 107, 520-536, (1985) · Zbl 0581.04003  Pedrycz, W., Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data, Fuzzy sets and systems, 16, 163-175, (1985) · Zbl 0601.68061  Pedrycz, W., Approximate solutions of fuzzy relational equations, Fuzzy sets and systems, 28, 183-202, (1988) · Zbl 0669.04002  Pedrycz, W., Algorithms for solving fuzzy relational equations in a probabilistic setting, Fuzzy sets and systems, 38, 313-327, (1990) · Zbl 0727.04006  Perfilieva, I., Fuzzy function as an approximate solution to a system of fuzzy relation equations, Fuzzy sets and systems, 147, 363-383, (2004) · Zbl 1050.03036  Perfilieva, I.; Novak, V., System of fuzzy relation equations as a continuous model of IF-THEN rules, Information sciences, 177, 3218-3227, (2007) · Zbl 1124.03029  Pavlica, V.; Petrovacki, D., About simple fuzzy control and fuzzy control based on fuzzy relational equations, Fuzzy sets and systems, 101, 41-47, (1999)  Sanchez, E., Solutions of fuzzy equations with extended operations, Fuzzy sets and systems, 12, 237-248, (1984) · Zbl 0556.04001  Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048  Shieh, B.-S., Solutions of fuzzy relation equations based on continuous t-norms, Information sciences, 177, 4208-4215, (2007) · Zbl 1122.03054  Stamou, G.B.; Tzafestas, S.G., Resolution of composite fuzzy relation equations based on Archimedean triangular norms, Fuzzy sets and systems, 120, 395-407, (2001) · Zbl 0979.03042  Stamou, G.B.; Tzafestas, S.G., Neural fuzzy relational systems with a new learning algorithm, Mathematics and computers in simulation, 51, 301-314, (2000)  Turunen, E., On generalized fuzzy relation equations: necessary and sufficient conditions for the existence of solutions, Acta univ. carolinae math. phys., 28, 33-37, (1987) · Zbl 0661.04004  Umeyama, S., The complementary process of fuzzy medical diagnosis and its properties, Information sciences, 38, 229-242, (1986) · Zbl 0603.92001  W.B. Vasantha Kandasamy, F. Smarandache, Fuzzy relational maps and neutrosophic relational maps, hexis church rock (2004) (see chapters one and two) at: http://mat.iitm.ac.in/ wbv/book13.htm.  Wagenknecht, M.; Hartmann, K., On direct and inverse problems for fuzzy equation systems with tolerances, Fuzzy sets and systems, 24, 93-102, (1987) · Zbl 0633.04001  H.-F. Wang, An algorithm for solving iterated composite relation equations, in: Proc. NAFIPS, 1988, pp. 242-249.  Wang, P.Z.; Sessa, S.; Di Nola, A.; Pedrycz, W., How many lower solutions does a fuzzy relation equation have?, Bull. pour. sons. ens. flous. appl.(BUSEFAL), 18, 67-74, (1984) · Zbl 0581.04001  Wang, X.P., Method of solution to fuzzy relation equations in a complete Brouwerian lattice, Fuzzy sets and systems, 120, 409-414, (2001) · Zbl 0981.03055  Wang, Z.D.; Dai, F.M., The sets of solutions of L-relation equations T(a,x)=b and I(a,x)=b, Journal of yangzhou university, 6, 3, 8-10, (2003), (in Chinese)  Wang, Z.D.; Yu, Y.D., Pseudo-t-norms and implication operators on a complete Brouwerian lattice, Fuzzy sets and systems, 132, 113-124, (2002) · Zbl 1013.03020  Wang, Z.D.; Yu, Y.D., Pseudo-t-norms and implication operators: direct products and direct product decompositions, Fuzzy sets and systems, 139, 673-683, (2003) · Zbl 1032.03023  Wang, X.-P., Infinite fuzzy relational equations on a complete Brouwerian lattice, Fuzzy sets and systems, 138, 657-666, (2003) · Zbl 1075.03026  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, 271-278, (2004) · Zbl 1091.90087  Wu, Y.-K., Optimization of fuzzy relational equations with MAX-av composition, Information sciences, 177, 4216-4229, (2007) · Zbl 1140.90523  Xiong, Q.-Q.; Wang, X.-P., Some properties of sup – min fuzzy relational equations on infinite domains, Fuzzy sets and systems, 151, 393-402, (2005) · Zbl 1062.03053  Yager, R., An approach to inference in approximate reasoning, International journal of man – machine studies, 13, 323-338, (1980)  Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606  Zhao, C.K., On matrix equations in a class of complete and completely distributive lattices, Fuzzy sets and systems, 22, 303-320, (1987) · Zbl 0621.06006
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