Three-Objective Programming with Continuous Variable Genetic Algorithm

Author(s)
Adugna Fita

ABSTRACT

The subject area of multiobjective optimization deals with the investigation of optimization problems that possess more than one objective function. Usually, there does not exist a single solution that optimizes all functions simultaneously; quite the contrary, we have solution set that is called nondominated set and elements of this set are usually infinite. It is from this set decision made by taking elements of nondominated set as alternatives, which is given by analysts. Since it is important for the decision maker to obtain as much information as possible about this set, our research objective is to determine a well-defined and meaningful approximation of the solution set for linear and nonlinear three objective optimization problems. In this paper a continuous variable genetic algorithm is used to find approximate near optimal solution set. Objective functions are considered as fitness function without modification. Initial solution was generated within box constraint and solutions will be kept in feasible region during mutation and recombination.

The subject area of multiobjective optimization deals with the investigation of optimization problems that possess more than one objective function. Usually, there does not exist a single solution that optimizes all functions simultaneously; quite the contrary, we have solution set that is called nondominated set and elements of this set are usually infinite. It is from this set decision made by taking elements of nondominated set as alternatives, which is given by analysts. Since it is important for the decision maker to obtain as much information as possible about this set, our research objective is to determine a well-defined and meaningful approximation of the solution set for linear and nonlinear three objective optimization problems. In this paper a continuous variable genetic algorithm is used to find approximate near optimal solution set. Objective functions are considered as fitness function without modification. Initial solution was generated within box constraint and solutions will be kept in feasible region during mutation and recombination.

KEYWORDS

Chromosome, Crossover, Heuristics, Mutation, Optimization, Population, Ranking, Genetic Algorithms, Multi-Objective, Pareto Optimal Solutions, Parent Selection

Chromosome, Crossover, Heuristics, Mutation, Optimization, Population, Ranking, Genetic Algorithms, Multi-Objective, Pareto Optimal Solutions, Parent Selection

Cite this paper

Fita, A. (2014) Three-Objective Programming with Continuous Variable Genetic Algorithm.*Applied Mathematics*, **5**, 3297-3310. doi: 10.4236/am.2014.521307.

Fita, A. (2014) Three-Objective Programming with Continuous Variable Genetic Algorithm.

References

[1] Sawaragi, Y., Nakayama, H. and Tanino, T. (1985) Theory of Multiobjective Optimization. Academic Press, Waltham.

[2] Steuer, R.E. (1986) Multiple Criteria: Theory, Computation and Application. John Wiley & Sons, New York.

[3] Ehrgott, M. (2005) Multicriteria Optimization. Springer, New York.

[4] Eichfelder, G. (2008) Vector Optimization. Springer, Berlin Heidelberg.

[5] Schaffer, J.D., Ed. (1989) Advances in Genetic Programming. Proceedings of the Third International Conference on Genetic Algorithms, Morgan Kaufmann, Burlington.

[6] Coley, D.A. (1999) An Introduction to Genetic Algorithms for Scientists and Engineers. World Scientific Publishing Co. Pte. Ltd., River Edge.

[7] Deb, K., Agrawal, S., Pratap, A. and Meyarivan, T. (2000) A Fast Elitist Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization: NSGA-II. Proceedings of the Parallel Problem Solving from Nature VI Conference, 849-858.

[8] Fonseca, C.M. and Fleming, P.J. (1995) An Overview of Evolutionary Algorithms in Multiobjective Optimization. Evolutionary Computation Journal, 3, 1-16.

[9] Knowles, J.D. and Corne, D.W. (1999) The Pareto Archived Evolution Strategy: A New Baseline Algorithm for Multiobjective Optimization. Proceedings of the 1999 Congress on Evolutionary Computation, Washington DC, 06 Jul-09 Jul 1999.

[10] Bot, R.I., Grad, S.-M. and Wanka, G. (2009) Duality in Vector Optimization. Springer-Verlag, Berlin Heidelberg.

http://dx.doi.org/10.1007/978-3-642-02886-1

[11] Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Boston.

[12] Deb, K. (2001) Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Hoboken.

[13] Sivanandam, S.N. and Deepa, S.N. (2008) Introduction to Genetic Algorithms. Springer-Verlag, Berlin Heidelberg.

[14] Semu, M. (2003) On Cone D.C Optimization and Conjugate Duality. Chinese Annals of Mathematics, 24, 521-528.

http://dx.doi.org/10.1142/S0252959903000529

[15] Goh, C.J. and Yang, X.Q. (2002) Duality in Optimization and Variational Inequalities. Taylor and Francis, New York.

http://dx.doi.org/10.1201/9781420018868

[16] Holland, J.H. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor. (2nd Edition, MIT Press, 1992.)

[17] Haupt, R.L. (2004) Sue Ellen Haupt: Practical Genetic Algorithms. 2nd Edition, John Wiley & Sons, Inc., Hoboken.

[18] Koza, J.R. (1992) Genetic Programming. Massachusetts Institute of Technology, Cambridge.

[19] Fita, A. (2014) Multiobjective Programming with Continuous Genetic Algorithm. International Journal of Scientific & Technology Research, 3, 135-149.

[20] Ikeda, K., Kita, H. and Kobayashi, S. (2001) Failure of Pareto-Based MOEAs: Does Nondominated Really Mean Near to Optimal. Proceedings of the Congress on Evolutionary Computation, 2, 957-962.

[21] Viennet, R., Fonteix, C. and Marc, I. (1996) Multicriteria Optimization Using a Genetic Algorithm for Determining a Pareto Set. International Journal of Systems Science, 27, 255-260.

[22] http://www.matworks.com

[1] Sawaragi, Y., Nakayama, H. and Tanino, T. (1985) Theory of Multiobjective Optimization. Academic Press, Waltham.

[2] Steuer, R.E. (1986) Multiple Criteria: Theory, Computation and Application. John Wiley & Sons, New York.

[3] Ehrgott, M. (2005) Multicriteria Optimization. Springer, New York.

[4] Eichfelder, G. (2008) Vector Optimization. Springer, Berlin Heidelberg.

[5] Schaffer, J.D., Ed. (1989) Advances in Genetic Programming. Proceedings of the Third International Conference on Genetic Algorithms, Morgan Kaufmann, Burlington.

[6] Coley, D.A. (1999) An Introduction to Genetic Algorithms for Scientists and Engineers. World Scientific Publishing Co. Pte. Ltd., River Edge.

[7] Deb, K., Agrawal, S., Pratap, A. and Meyarivan, T. (2000) A Fast Elitist Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization: NSGA-II. Proceedings of the Parallel Problem Solving from Nature VI Conference, 849-858.

[8] Fonseca, C.M. and Fleming, P.J. (1995) An Overview of Evolutionary Algorithms in Multiobjective Optimization. Evolutionary Computation Journal, 3, 1-16.

[9] Knowles, J.D. and Corne, D.W. (1999) The Pareto Archived Evolution Strategy: A New Baseline Algorithm for Multiobjective Optimization. Proceedings of the 1999 Congress on Evolutionary Computation, Washington DC, 06 Jul-09 Jul 1999.

[10] Bot, R.I., Grad, S.-M. and Wanka, G. (2009) Duality in Vector Optimization. Springer-Verlag, Berlin Heidelberg.

http://dx.doi.org/10.1007/978-3-642-02886-1

[11] Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Boston.

[12] Deb, K. (2001) Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Hoboken.

[13] Sivanandam, S.N. and Deepa, S.N. (2008) Introduction to Genetic Algorithms. Springer-Verlag, Berlin Heidelberg.

[14] Semu, M. (2003) On Cone D.C Optimization and Conjugate Duality. Chinese Annals of Mathematics, 24, 521-528.

http://dx.doi.org/10.1142/S0252959903000529

[15] Goh, C.J. and Yang, X.Q. (2002) Duality in Optimization and Variational Inequalities. Taylor and Francis, New York.

http://dx.doi.org/10.1201/9781420018868

[16] Holland, J.H. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor. (2nd Edition, MIT Press, 1992.)

[17] Haupt, R.L. (2004) Sue Ellen Haupt: Practical Genetic Algorithms. 2nd Edition, John Wiley & Sons, Inc., Hoboken.

[18] Koza, J.R. (1992) Genetic Programming. Massachusetts Institute of Technology, Cambridge.

[19] Fita, A. (2014) Multiobjective Programming with Continuous Genetic Algorithm. International Journal of Scientific & Technology Research, 3, 135-149.

[20] Ikeda, K., Kita, H. and Kobayashi, S. (2001) Failure of Pareto-Based MOEAs: Does Nondominated Really Mean Near to Optimal. Proceedings of the Congress on Evolutionary Computation, 2, 957-962.

[21] Viennet, R., Fonteix, C. and Marc, I. (1996) Multicriteria Optimization Using a Genetic Algorithm for Determining a Pareto Set. International Journal of Systems Science, 27, 255-260.

[22] http://www.matworks.com