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Generating Epsilon-Efficient Solutions in Multiobjective Optimization by Genetic Algorithm
Department of Mathematics, Faculty of Science, Taif University, Khurma, KSA
Faculty of Mathematics and Computer Science, University of Ƚódź, Ƚódź, Poland
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
- 1 Department of Mathematics, Faculty of Science, Taif University, Khurma, KSA
- 2 Faculty of Mathematics and Computer Science, University of Ƚódź, Ƚódź, Poland
- 3 Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
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Abstract
We develop a new evolutionary method of generating epsilon-efficient solutions of a continuous multiobjective programming problem. This is achieved by discretizing the problem and then using a genetic algorithm with some derived probabilistic stopping criteria to obtain all minimal solutions for the discretized problem. We prove that these minimal solutions are the epsilon-optimal solutions to the original problem. We also present some computational examples illustrating the efficiency of our method.
KeywordsVector OptimizationApproximate SolutionsGenetic AlgorithmStopping Criteria
- Ruzika, S. and Wiecek, M.M. (2005) Approximation Methods in Multiobjective Programming. Journal of Optimization Theory and Applications, 126, 473-501. https://doi.org/10.1007/s10957-005-5494-4
- Ghaznavi-Ghosoni, B.A., Khorram, E. and Soleimani-damaneh, M. (2013) Scalarization for Characterization of Approximate Strong/Weak/Proper Efficiency in Multi-Objective Optimization. Optimization, 62, 703-720. https://doi.org/10.1080/02331934.2012.668190
- Loridan, P. (1984) -Solutions in Vector Minimization Problems. Journal of Optimization Theory and Applications, 42, 265-276. https://doi.org/10.1007/BF00936165
- Engau, A. and Wiecek, M.M. (2007) Generating -Efficient Solutions in Multiobjective Programming. European Journal of Operational Research, 177, 1566-1579. https://doi.org/10.1016/j.ejor.2005.10.023
- Engau, A. and Wiecek, M.M. (2007) Exact Generation of Epsilon-Efficient Solutions in Multiple Objective Programming. OR Spectrum, 29, 335-350. https://doi.org/10.1007/s00291-006-0044-5
- Laumanns, M., Thiele, L., Deb, K. and Zitzler, E. (2002) Combining Convergence and Diversity in Evolutionary Multiobjective Optimization. Evolutionary Computation, 10, 263-282. https://doi.org/10.1162/106365602760234108
- Schutze, O., Laumanns, M., Tantar, E., Coello Coello, C.A. and Talbi, E.-G. (2007) Convergence of Stochastic Search Algorithms to Gap-Free Pareto Front Approximations. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2007), 892-899.
- Studniarski, M. (2011) Finding All Minimal Elements of a Finite Partially Ordered Set by Genetic Algorithm with a Prescribed Probability. Numerical Algebra, Control and Optimization, 1, 389-398. https://doi.org/10.3934/naco.2011.1.389
- Vose, M.D. (1999) The Simple Genetic Algorithm: Foundations and Theory. MIT Press, Cambridge.
- Reeves, C.R. and Rowe, J.E. (2003) Genetic Algorithms—Principles and Perspectives: A Guide to GA Theory. Kluwer, Boston.
- Rudolph, G. and Agapie, A. (2000) Convergence Properties of Some Multi-Objective Evolutionary Algorithms. In: Zalzala, A., et al., Eds., Proceedings of the 2000 Congress on Evolutionary Computation (CEC 2000), Vol. 2, IEEE Press, Piscataway, 1010-1016.
- Osman, M.S., Abo-Sinna, M.A. and Mousa, A.A. (2005) An Effective Genetic Algorithm Approach to Multiobjective Resource Allocation Problems (MORAPs). Applied Mathematics and Computation, 163, 755-768. https://doi.org/10.1016/j.amc.2003.10.057