Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions

We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonli...

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Published in:	Fuzzy optimization and decision making Vol. 11; no. 3; pp. 299 - 335
Main Authors:	Khorram, E., Ezzati, R., Valizadeh, Z.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2012 Springer Science + Business Media B.V Springer Nature B.V
Subjects:	Artificial Intelligence Calculus of Variations and Optimal Control; Optimization Decision making Fuzzy Fuzzy logic Fuzzy set theory Fuzzy sets Generators Genetic algorithms Inequalities Integer programming Linear programming Mathematical Logic and Foundations Mathematical models Mathematical programming Mathematics Mathematics and Statistics Methods Nonlinearity Norms Operations Research/Decision Theory Optimization Pareto optimum Probability Theory and Stochastic Processes Studies Efficient solutions Multi-objective optimization Archimedean t-norm Fuzzy relation inequality Genetic algorithms
ISSN:	1568-4539, 1573-2908
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Abstract	We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.
AbstractList	We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely "≤", "≥" and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.[PUBLICATION ABSTRACT] We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples. We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely " less than or equal to ", " greater than or equal to " and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.
Author	Valizadeh, Z. Khorram, E. Ezzati, R.
Author_xml	– sequence: 1 givenname: E. surname: Khorram fullname: Khorram, E. email: eskhor@aut.ac.ir organization: Department of Mathematics, Karaj Branch, Islamic Azad University, Faculty of Mathematics and Computer Science, Amirkabir University of Technology – sequence: 2 givenname: R. surname: Ezzati fullname: Ezzati, R. organization: Department of Mathematics, Karaj Branch, Islamic Azad University – sequence: 3 givenname: Z. surname: Valizadeh fullname: Valizadeh, Z. organization: Department of Mathematics, Karaj Branch, Islamic Azad University
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Cites_doi	10.1007/BF01584098 10.1016/j.fss.2004.09.010 10.1016/j.fss.2008.04.007 10.1016/S0165-0114(97)00184-X 10.1057/jors.1994.222 10.1016/S0165-0114(98)00417-5 10.1016/S0165-0114(03)00327-0 10.1016/S0165-0114(01)00052-5 10.1016/j.fss.2005.02.010 10.1016/S0165-0114(98)00471-0 10.1016/j.mcm.2008.10.018 10.1007/s10700-009-9059-0 10.1007/978-94-015-8449-4 10.1023/B:FODM.0000036862.45420.ea 10.1007/978-3-642-80808-1 10.1002/mcda.4020040103 10.1016/j.asoc.2009.12.014 10.1016/S0165-0114(03)00304-X 10.1016/S0019-9958(65)90241-X 10.1023/A:1020955112523 10.1016/j.fss.2003.06.007 10.1016/j.amc.2005.11.069 10.1007/s00500-001-0157-3 10.1016/j.fss.2009.03.007 10.1007/978-94-015-9540-7 10.1016/j.cie.2008.08.015 10.1016/S0019-9958(76)90446-0 10.1007/978-3-642-48320-2
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SubjectTerms	Artificial Intelligence Calculus of Variations and Optimal Control; Optimization Decision making Fuzzy Fuzzy logic Fuzzy set theory Fuzzy sets Generators Genetic algorithms Inequalities Integer programming Linear programming Mathematical Logic and Foundations Mathematical models Mathematical programming Mathematics Mathematics and Statistics Methods Nonlinearity Norms Operations Research/Decision Theory Optimization Pareto optimum Probability Theory and Stochastic Processes Studies
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Title	Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions
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