Multi-objective optimization problems with fuzzy relation equation constraints.

*(English)*Zbl 0994.90130Summary: This paper studies a new class of optimization problems which have multiple objective functions subject to a set of fuzzy relation equations. Since the feasible domain of such a problem is in general non-convex and the objective functions are not necessarily linear, traditional optimization methods may become ineffective and inefficient. Taking advantage of the special structure of the solution set, a reduction procedure is developed to simplify a given problem. Moreover, a genetic-based algorithm is proposed to find the “Pareto optimal solutions”. The major components of the proposed algorithm together with some encouraging test results are reported.

##### MSC:

90C29 | Multi-objective and goal programming |

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90C59 | Approximation methods and heuristics in mathematical programming |

##### Software:

Genocop
PDF
BibTeX
XML
Cite

\textit{J. Loetamonphong} et al., Fuzzy Sets Syst. 127, No. 2, 141--164 (2002; Zbl 0994.90130)

Full Text:
DOI

**OpenURL**

##### References:

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

[2] | Bezdek, J.C., Pattern recognition with fuzzy objective function algorithms, (1981), Plenum Press New York · Zbl 0503.68069 |

[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.; Li, G., Solving fuzzy relation equations with a linear objective function, Fuzzy sets and systems, 103, 107-113, (1999) · Zbl 0933.90069 |

[6] | M.P. Fourman, Compaction of symbolic layout using genetic algorithms, Proc. 1st ICGA, 1985, pp. 141-153. |

[7] | C.M. Fonseca, P.J. Fleming, Genetic algorithms for multiobjective optimization: formulation, discussion and generalization, Proc. 5th ICGA, 1993, pp. 416-423. |

[8] | Fonseca, C.M.; Fleming, P.J., An overview of evolutionary algorithms in multiobjective optimization, Evol. comput., 3, 1, 1-16, (1995) |

[9] | Goldberg, D.E., Genetic algorithms in search, optimization, and machine learning, (1989), Addison-Wesley Reading, MA · Zbl 0721.68056 |

[10] | 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 |

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

[12] | J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan, 1975. |

[13] | J. Horn, N. Nafpliotis, D.E. Goldberg, A niched Pareto genetic algorithm for multiobjective optimization, Proc. 1st ICEC, 1994, pp. 82-87. |

[14] | Jakob, W.; Gorges-schleuter, M.; Blume, C., Application of genetic algorithms to task planning and learning, (), 291-300 |

[15] | G. Jones, R.D. Brown, D.E. Clark, P. Willet, R.C. Glen, Searching databases of two-dimensional and three-dimensional chemical structures using genetic algorithms, Genetic Algorithms: Proc. 5th Internat. Conf., 1993, pp. 597-602. |

[16] | Klir, G.J.; Folger, T.A., Fuzzy sets, uncertainty and information, (1988), Prentice-Hall NJ · Zbl 0675.94025 |

[17] | F. Kursawe, A variant of evolution strategies for vector optimization, Parallel Problem Solving from Nature, 1991, pp. 193-197. |

[18] | Lettieri, A.; Liguori, F., Characterization of some fuzzy relation equations provided with one solution on a finite set, Fuzzy sets and systems, 13, 83-94, (1984) · Zbl 0553.04004 |

[19] | G. Li, S.-C. Fang, On the resolution of finite fuzzy relation equations, OR Report No. 322, North Carolina State University, Raleigh, North Carolina, May 1996, to appear in Fuzzy Sets and Systems. |

[20] | Li, G.; Fang, S.-C., Solving interval-valued fuzzy relation equations, IEEE trans. fuzzy systems, 6, 321-324, (1998) |

[21] | Loetamonphong, J.; Fang, S.-C., An efficient solution procedure for fuzzy relation equations with MAX-product composition, IEEE trans. fuzzy systems, 7, 441-445, (1999) |

[22] | 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 |

[23] | Lu, J.; Fang, S.-C., Solving nonlinear optimization problems with fuzzy relation equations constraints, Fuzzy sets and systems, 119, 1-20, (2001) |

[24] | Michalewicz, Z., Genetic algorithms + data structures = evolution program, (1996), Springer Berlin/Heidelberg/New York · Zbl 0841.68047 |

[25] | Prevot, M., Algorithm for the solution of fuzzy relations equations, Fuzzy sets and systems, 5, 319-322, (1981) · Zbl 0451.04004 |

[26] | Reeves, C.R., Modern heuristic techniques for combinatorial problems, (1993), Wiley New York · Zbl 0942.90500 |

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

[28] | J.D. Schaffer, Multiple objective optimization with vector evaluated genetic algorithms, Proc. 1st ICGA, 1985, pp. 141-153. |

[29] | Srinivas, N.; Deb, K., Multiobjective optimization using nondominated sorting in genetic algorithms, Evol. comput., 2, 221-248, (1994) |

[30] | Steuer, R.E., Multiple criteria optimization: theory, computation, and application, (1986), Wiley New York · Zbl 0663.90085 |

[31] | G. Syswerda, J. Palmucci, The application of genetic algorithms to resource scheduling, Genetic Algorithms: Proc. 4th Internat. Conf., 1991, pp. 502-508. |

[32] | H. Tamaki, H. Kita, S. Kobayashi, Multi-objective optimization by genetic algorithms: a review, Proc. 1996 IEEE ICEC, 1996, pp. 517-522. |

[33] | Wang, W.F., A multi-objective mathematical programming problem with fuzzy relation constraints, J. multi-criteria dec. anal., 4, 23-35, (1995) · Zbl 0843.90131 |

[34] | Yager, R.R., Fuzzy decision making including unequal objectives, Fuzzy sets and systems, 1, 87-95, (1978) · Zbl 0378.90011 |

[35] | Yager, R.R., Some procedures for selecting fuzzy set-theoretic operators, Int. J. general systems, 8, 235-242, (1982) · Zbl 0488.04005 |

[36] | 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.