Deriving minimal solutions for fuzzy relation equations with max-product composition.

*(English)*Zbl 1151.03345Summary: This work considers fuzzy relation equations with max-product composition. The critical problem in solving such equations is to determine the minimal solutions when an equation is solvable. However, this problem is NP-hard and difficult to solve [A. V. Markovskii, “On the relation between equations with max-product composition and the covering problem”, Fuzzy Sets Syst. 153, 261–273 (2005; Zbl 1073.03538)]. This work first examines the attributes of a solvable equation and characteristics of minimal solutions, then reduces the equation to an irreducible form, and converts the problem into a covering problem, for which minimal solutions are correspondingly determined. Furthermore, for theoretical and practical applications, this work presents a novel method for obtaining minimal solutions. The proposed method easily derives a minimal solution, and obtains other minimal solutions from this predecessor using a back-tracking step. The proposed method is compared with an existing algorithm, and some applications are described in detail.

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

##### Keywords:

fuzzy relation equation; solvability; max-product composition; minimal solutions; continuous t-norm; NP-hard problem
Full Text:
DOI

##### References:

[1] | Baets, B.D., Analytical solution methods for fuzzy relational equations, fundamentals of fuzzy sets, (), 291-340 · Zbl 0970.03044 |

[2] | Bourke, M.; Fisher, D., Solution algorithms for fuzzy relation equations with MAX-product composition, Fuzzy sets and systems, 94, 61-69, (1998) · Zbl 0923.04003 |

[3] | Di Nola, A.; Pedrycz, W.; Sessa, S., On measures of fuzziness of solutions of fuzzy relation equations with generalized connectives, Journal of mathematical analysis and applications, 106, 443-453, (1985) · Zbl 0593.03011 |

[4] | Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Dordrecht · Zbl 0694.94025 |

[5] | Dubois, D.; Prade, H., New results about properties and semantics of fuzzy set-theoretic operators, (), 59-75 |

[6] | Fernandez, M.J.; Gil, P., Some specific types of fuzzy relation equations, Information sciences, 164, 189-195, (2004) · Zbl 1058.03058 |

[7] | Gupta, M.M.; Qi, J., Design of fuzzy logic controllers based on generalized T-operators, Fuzzy sets and systems, 40, 473-489, (1991) · Zbl 0732.93050 |

[8] | Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006 |

[9] | Hirota, K.; Pedrycz, W., Fuzzy relational compression, IEEE transactions on systems man and cybernetics, part B, 29, 407-415, (1999) |

[10] | Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic theory and applications, (1995), Prentice Hall PTR, Pearson Education, Inc. · Zbl 0915.03001 |

[11] | Loetamonphong, J.; Fang, S.C., Optimization of fuzzy relation equations with MAX-product composition, Fuzzy sets and systems, 118, 509-517, (2001) · Zbl 1044.90533 |

[12] | 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 |

[13] | 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 |

[14] | Nobuhara, H.; Pedrycz, W.; Hirota, K., Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction, IEEE transactions on fuzzy systems, 8, 325-334, (2000) |

[15] | Pedrycz, W., Fuzzy relational equations with generalized connectives and their applications, Fuzzy sets and systems, 10, 185-201, (1983) · Zbl 0525.04004 |

[16] | 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 |

[17] | Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048 |

[18] | Shieh, B.S., Solutions of fuzzy relation equations based on continuous t-norms, Information sciences, 177, 4208-4215, (2007) · Zbl 1122.03054 |

[19] | Shieh, B.S., Infinite fuzzy relation equations with continuous t-norms, Information sciences, 178, 1961-1967, (2008) · Zbl 1135.03346 |

[20] | 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 |

[21] | Thole, U.; Zimmermann, H.J.; Zysno, P., On the suitability of minimum and product operators for intersection of fuzzy sets, Fuzzy sets and systems, 2, 167-180, (1979) · Zbl 0408.94030 |

[22] | Yager, R.R., On a general class of fuzzy connectives, Fuzzy sets and systems, 4, 235-242, (1980) · Zbl 0443.04008 |

[23] | Yager, R.R., Some procedures for selecting fuzzy set-theoretic operators, International journal of general systems, 8, 235-242, (1982) · Zbl 0488.04005 |

[24] | Zimmermann, H.J.; Zysno, P., Latent connectives in human decision-making, Fuzzy sets and systems, 4, 37-51, (1980) · Zbl 0435.90009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.