Solution sets of interval-valued fuzzy relational equations.

*(English)*Zbl 1178.03071
Fuzzy Optim. Decis. Mak. 2, No. 1, 41-60 (2003); erratum ibid. 6, No. 3, 297 (2007).

Summary: This paper introduces the concepts of tolerable solution set, united solution set, and controllable solution set of interval-valued fuzzy relational equations. Given a continuous t-norm, it is proved that each of the three types of the solution sets of interval-valued fuzzy relational equations with a max-t-norm composition, if nonempty, is composed of one maximum solution and a finite number of minimal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. Examples are given to illustrate the procedures.

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

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\textit{S. Wang} et al., Fuzzy Optim. Decis. Mak. 2, No. 1, 41--60 (2003; Zbl 1178.03071)

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##### References:

[1] | Bourke, M. M. and D. G. Fisher. (1998). “Solution Algorithms for Fuzzy Relational Equations with Max-Product Composition,” Fuzzy Sets and Systems 94, 61-69. · Zbl 0923.04003 |

[2] | De Baets, B. (2000). “Analytical Solution Methods for Fuzzy Relational Equations,” Fundamentals of Fuzzy Sets. Boston: Kluwer Academic Publishers. · Zbl 0970.03044 |

[3] | Di Nola, A., S. Sessa, W. Pedrycz, and E. Sanchez. (1989). Fuzzy Relation Equations and Their Applications in Knowledge Engineering. Dordrecht: Kluwer Academic Publishers. · Zbl 0694.94025 |

[4] | Fang, S.-C. and G. Li, (1999). “Solving Fuzzy Relation Equations with a Linear Objective Function,” Fuzzy Sets and Systems 103, 107-113. · Zbl 0933.90069 |

[5] | Kulpa, Z. and K. Roslaniec. (1999). “Solution Sets for Systems of Linear Interval Equations,” Proceedings of the XIVth Polish Conference on Computer Methods in Mechanics (PCCMM’99), Rzeszow, Poland, May 26-29. · Zbl 0973.65026 |

[6] | Li, G. and S.-C. Fang. (1998). “Solving Interval-Valued Fuzzy Relation Equations,” IEEE Transactions on Fuzzy Systems 6, 321-324. |

[7] | Loetamonphong, J. and S.-C. Fang. (1999). “An Efficient Solution Procedure for Fuzzy Relation Equations with Max-Product Composition,” IEEE Transactions on Fuzzy Systems 7, 441-445. |

[8] | Loetamonphong, J. and S.-C. Fang. (2001). “Optimization of Fuzzy Relation Equations with Max-Product Composition,” Fuzzy Sets and Systems 118, 509-517. · Zbl 1044.90533 |

[9] | Loetamonphong, J., S.-C. Fang, and R. E. Young. (2002). “Multi-Objective Optimization Problems with Fuzzy Relation Equation Constraints,” Fuzzy Sets and Systems 127, 141-164. · Zbl 0994.90130 |

[10] | Lu, J. and S.-C. Fang. (2001). “Solving Nonlinear Optimization Problems with Fuzzy Relation Equation Constraints,” Fuzzy Sets and Systems 119, 1-20. |

[11] | Neumaier, A. (1990). Interval Methods for Systems of Equations. Cambridge: Cambridge University Press. · Zbl 0715.65030 |

[12] | Pappis, C. P. and M. Sugeno. (1985). “Fuzzy Relational Equations and Inverse Problem,” Fuzzy Sets and Systems 15, 79-90. · Zbl 0561.04003 |

[13] | Pedrycz, W. (1983). “Fuzzy Relational Equations with Generalized Connectives and their Applications,” Fuzzy Sets and Systems 10, 185-201. · Zbl 0525.04004 |

[14] | Pedrycz, W. (1985). “On Generalized Fuzzy Relational Equations and their Applications,” Journal of Mathematical Analysis and Applications 107, 520-536. · Zbl 0581.04003 |

[15] | Sanchez, E. (1976). “Resolution of Composite Fuzzy Relation Equations,” Information and Control 30, 38-48. · Zbl 0326.02048 |

[16] | Schweizer, B. and A. Sklar. (1961). “Associative Functions and Statistical Triangle Inequalities,” Publications of Mathematical Debrecen 8, 169-186. · Zbl 0107.12203 |

[17] | Wang H.-F. and Y.-C. Chang. (1991). “Resolution of Composite Interval-Valued Fuzzy Relation Equations,” Fuzzy Sets and Systems 44, 227-240. · Zbl 0738.04003 |

[18] | Zadeh, L. A. (1965). “Fuzzy Sets,” Information and Control 8, 338-353. · Zbl 0139.24606 |

[19] | Zimmermann, H. |

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