Solving fuzzy relation equations with a linear objective function.

*(English)*Zbl 0933.90069Summary: An optimization model with a linear objective function subject to a system of fuzzy relation equations is presented. Due to the non-convexity of its feasible domain defined by fuzzy relation equations, designing an efficient solution procedure for solving such problems is not a trivial job. In this paper, we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0-1 integer programming with a branch-and-bound solution technique. After presenting our solution procedure, a concrete example is included for illustration purpose.

##### MSC:

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90C08 | Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) |

90C09 | Boolean programming |

##### Keywords:

optimization; linear objective function; fuzzy relation equations; 0-1 integer programming; branch-and-bound solution
PDF
BibTeX
XML
Cite

\textit{S.-C. Fang} and \textit{G. Li}, Fuzzy Sets Syst. 103, No. 1, 107--113 (1999; Zbl 0933.90069)

Full Text:
DOI

##### References:

[1] | Adamopoulos, G.I.; Pappis, C.P., Some results on the resolution of fuzzy relation equations, Fuzzy sets and systems, 60, 83-88, (1993) · Zbl 0794.04005 |

[2] | Adlassnig, K.P., Fuzzy set theory in medical diagnosis, IEEE trans. systems man cybernet., 16, 260-265, (1986) |

[3] | Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy sets and systems, 7, 89-101, (1982) · Zbl 0483.04001 |

[4] | Di Nola, A., Relational equations in totally ordered lattices and their complete resolution, J. math. appl., 107, 148-155, (1985) · Zbl 0588.04006 |

[5] | Fang, S.-C.; Puthenpura, S., Linear optimization and extensions: theory and algorithm, (1993), Prentice-Hall Englewood Cliffs, NJ |

[6] | Guo, S.Z.; Wang, P.Z.; Di Nola, A.; Sessa, S., Further contributions to the study of finite fuzzy relation equations, Fuzzy sets and systems, 26, 93-104, (1988) · Zbl 0645.04003 |

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

[8] | Li, G.; Fang, S.-C., On the resolution of finite fuzzy relation equations, () |

[9] | Prevot, M., Algorithm for the solution of fuzzy relations equations, Fuzzy sets and systems, 5, 319-322, (1985) · Zbl 0451.04004 |

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

[11] | Wang, H.-F., An algorithm for solving iterated composite relation equations, (), 242-249 |

[12] | Wang, P.Z.; Sessa, S.; Di Nola, A.; Pedrycz, W., How many lower solutions does a fuzzy relation equation have?, Bull. pour. sous. ens. flous. appl. (BUSEFAL), 18, 67-74, (1984) · Zbl 0581.04001 |

[13] | Winston, W.L., Introduction to mathematical programming: application and algorithms, (1995), Duxbury Press Belmont, CA · Zbl 0837.90086 |

[14] | Zimmermann, H.-J., Fuzzy set theory and its applications, (1991), Kluwer Academic Publishers Boston/Dordrecht/London · Zbl 0719.04002 |

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.