求解Pareto Front多目标遗传算法的研究
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摘要
遗传算法是模拟达尔文的遗传选择和自然淘汰的生物进化
过程的一种新的迭代的全局优化搜索算法，已经广泛地应用到组合
优化问题求解、自适应控制、规划设计、机器学习和人工生命等领
域。由于现实世界中存在的问题往往呈现为多目标属性，而且需要
优化的多个目标之间又是相互冲突的，从而多目标遗传算法应运而
生，它使得进化群体并行搜寻多个目标，并逐渐找到问题的最优解。
本文对近十五年来多目标遗传算法的国内外研究现状进行了
较全面地阐述，其优化方法大致分为两大类：带参数的方法和不带
参数的方法。带参数的方法主要存在着参数难以选择及过于依赖参
数的选择等问题，不带参数的方法主要存在着速度比较慢的问题。
为此，本文的研究主要就是从提高寻找非支配集的速度，在保持群
体原有特性的前提下降低非支配集的大小，以及新群体的构造等方
面入手，通过基于分类和聚类的方法，有效提高多目标遗传算法总
体运行效率，降低其计算复杂性，使多目标遗传算法的收敛性能得
到进一步改善。该文以NSGA-Ⅱ为基准，对算法进行了改进，具
体提出了：用排除法构造非支配集、用聚集距离刻画个体间的内部
关系以及构造新群体，来提高运行速度和保持群体的多样性；用聚
类算法在保持原有特性的前提下，进一步改善收敛性能等。比较试
验结果表明，基于分类和聚类的多目标遗传算法，在运行效率与保
持群体多样性等方面取得了较好效果。
Genetic Algorithm (GA) is a set of new-global-optimistic search algorithm repeatedly which simulate the process of creature evolution that of Darwinian's genetic selection and natural elimination. It is widely applied to the domain of combinational evolutionary problem seeking, self-adapt controlling, planning devising, machine learning and artificial life etc. However, there are multi-objective attributes in real-world optimization problems that always conflict, so the multi-objective Genetic Algorithm (MOGA) is put forward. MOGA can deal simultaneously with many objections, and find gradually Pareto-optimal solutions.
This paper presents a critical review of MOGA' current researches mainly in the last 15 years. The multi-objective optimization techniques have two branches, one with parameters and another with no parameters. It's difficult for us to select parameters in the methods with parameters and its performance is highly dependent on an appropriate selection of the sharing factor. In addition, the work speed is very low in the methods with no parameters. Therefore, we focus on proceeding the algorithm's performance with increasing the speed of searching non-dominated solutions, reducing the number of non-dominated solutions in precondition of ensuring a better distribution of individuals, and constructing new populations. The multi-objective Genetic Algorithm based on sorting and clustering efficiently increase its run efficiency, debase its compute complexity and improve its convergence performance. In this paper, we take
NSGA-II as a benchmark. It has been improved, and specially proposed: Firstly, we has increased run speed and ensure the diversity of population is with constructing non-dominated set by throwing off the dominated solutions, expressing the interior relation of individuals each other by the crowding distance, and constructing new population. Secondly, we have further improved its convergence performance by clustering in precondition of ensuring a better distribution of individuals. Simulation results on six difficult optimization problems show that the multi-objective Genetic Algorithm based on sorting and clustering have ideal effects on the aspects of its speed and diversify.
过程的一种新的迭代的全局优化搜索算法，已经广泛地应用到组合
优化问题求解、自适应控制、规划设计、机器学习和人工生命等领
域。由于现实世界中存在的问题往往呈现为多目标属性，而且需要
优化的多个目标之间又是相互冲突的，从而多目标遗传算法应运而
生，它使得进化群体并行搜寻多个目标，并逐渐找到问题的最优解。
本文对近十五年来多目标遗传算法的国内外研究现状进行了
较全面地阐述，其优化方法大致分为两大类：带参数的方法和不带
参数的方法。带参数的方法主要存在着参数难以选择及过于依赖参
数的选择等问题，不带参数的方法主要存在着速度比较慢的问题。
为此，本文的研究主要就是从提高寻找非支配集的速度，在保持群
体原有特性的前提下降低非支配集的大小，以及新群体的构造等方
面入手，通过基于分类和聚类的方法，有效提高多目标遗传算法总
体运行效率，降低其计算复杂性，使多目标遗传算法的收敛性能得
到进一步改善。该文以NSGA-Ⅱ为基准，对算法进行了改进，具
体提出了：用排除法构造非支配集、用聚集距离刻画个体间的内部
关系以及构造新群体，来提高运行速度和保持群体的多样性；用聚
类算法在保持原有特性的前提下，进一步改善收敛性能等。比较试
验结果表明，基于分类和聚类的多目标遗传算法，在运行效率与保
持群体多样性等方面取得了较好效果。
Genetic Algorithm (GA) is a set of new-global-optimistic search algorithm repeatedly which simulate the process of creature evolution that of Darwinian's genetic selection and natural elimination. It is widely applied to the domain of combinational evolutionary problem seeking, self-adapt controlling, planning devising, machine learning and artificial life etc. However, there are multi-objective attributes in real-world optimization problems that always conflict, so the multi-objective Genetic Algorithm (MOGA) is put forward. MOGA can deal simultaneously with many objections, and find gradually Pareto-optimal solutions.
This paper presents a critical review of MOGA' current researches mainly in the last 15 years. The multi-objective optimization techniques have two branches, one with parameters and another with no parameters. It's difficult for us to select parameters in the methods with parameters and its performance is highly dependent on an appropriate selection of the sharing factor. In addition, the work speed is very low in the methods with no parameters. Therefore, we focus on proceeding the algorithm's performance with increasing the speed of searching non-dominated solutions, reducing the number of non-dominated solutions in precondition of ensuring a better distribution of individuals, and constructing new populations. The multi-objective Genetic Algorithm based on sorting and clustering efficiently increase its run efficiency, debase its compute complexity and improve its convergence performance. In this paper, we take
NSGA-II as a benchmark. It has been improved, and specially proposed: Firstly, we has increased run speed and ensure the diversity of population is with constructing non-dominated set by throwing off the dominated solutions, expressing the interior relation of individuals each other by the crowding distance, and constructing new population. Secondly, we have further improved its convergence performance by clustering in precondition of ensuring a better distribution of individuals. Simulation results on six difficult optimization problems show that the multi-objective Genetic Algorithm based on sorting and clustering have ideal effects on the aspects of its speed and diversify.
引文
[1] 陈国良，王煦法，庄镇泉，王东生．遗传算法及其应用，人民邮电出版社，1999．
[2] David A. Van Veldhuizen and Gary B. Lamont. Multi-objective Evolutionary Algorithm Research: A History and Analysis. Technical and Computer Engineering, Graduate School of Engineering, Air Force Institute of Technology, Wright-Patterson AFB, 1998.
[3] Carlos M. Fonseca and Peter J. Fleming. An overview of evolutionary algorithms in multi-objective optimization. Technical report, Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, U. K.,1994.
[4] R. S. Rosenberg. Simulation of genetic populations with biochemical properties. PhD thesis, University of Michigan Ann Harbor, Michigan, 1967.
[5] H. W. Kuhn and A. W. Tucker. Nonlinear programming. In J. Neyman, editor, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pages 481-492, Berkeley, California, 1951. University of California Press.
[6] Gilbert Syswerda and Jeff Palmucci. The Application of Genetic Algorithms to Resource Scheduling. In Richard K. Booker, editors, Proceedings of the Fourth International Conference on Genetic Algorithms, pages 502-508, San Mateo, California, 1991. Morgan Kaufmann.
[7] W.Jakob, M. Gorges-Schleuter, and C.Blume. Application of genetic of genetic algorithms to task planning and learning. In R. Manner and B. Manderick, editors, Parallel Problem Solving form Nature, 2 nd Workshop, Lecture Notes in Computer Science, pages 291-300, Amsterdam, 1992. North-Holland Publishing Company.
[8] Gareth Jones, Robert D. Brown, David E. Clark, Peter Willett, and Robert C. Glen. Searching Databases of Two-Dimensional and Three-Dimensional Chemical Structures using Genetic Algorithms. In Stephanie Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 597-602, San Mateo, California, 1993. Morgan Kaufmann.
[9] P.B. Wilson and M.D.Macleod. Low implementation cost ⅡR digital filter design using genetic algorithms. In IEE/IEEE Workshop on Natural Algorithms in Signal Processing, pages 4/1-4/8,Chelmsford, U.K., 1993.
[10] Xiaojian Liu, D.W. Begg, and R.J.Fishwick. Genetic approach to optimal
topology/controller design of adaptive structures. International Journal for Numerical Methods in Engineering,41:815-830, 1998.
[11] Xiaofeng Yang and Mitsuo Gen. Evolution program for bicriteria transportation problem. In M. Gen and T. Kobayashi, editors, Proceedings of the 16th International Conference on Computers and Industrial Engineering, pages 451-454, Ashikaga, Japan, 1994. Pergamon Press.
[12] M. Gen, K. Ida, and Y. Li. Solving bicriteria solid transportation problem with fuzzy numbers by genetic algorithm. International Journal of Computers and Industrial Engineering, 29:537-543, 1995.
[13] Mitsuo Gen and Runwei Cheng. Genetic Algorithms and Engineering Design. John Wiley and Sons,Inc., New York, 1997.
[14] Richardson,J.T.,Palmer, M.R.,Liepins,F.,&Hilliard,M.(1989).Some guidelines for genetic algorithms with penalty functions. Proceedings of the Third International Conference on Genetic A lgorithms.Morgan-Kauffman, 191-197.
[15] Schaffer, J.D.,(1984).Some experiments in machine learning using vector evaluated genetic algorithms. Unpublished doctoral dissertation, Vanderbilt University.
[16] J. David Schaffer. Multiple objective optimization with vector evaluated genetic algorithms. In Genetic Algorithms and their Applications: Proceedings of the First International Conference on Genetic Algorithms, pages 93-100. Lawrence Erlbaum, 1985.
[17] Patrick D. Surry, Nicholas J. Radcliffe, and Ian D. Boyd. A Multi-Objective Approach to Constrained Optimization of Gas Supply Networks: The COMOGA Method. In Terence C. Fogarty, editor, Evolutionary computing. AISB Workshop. Selected Papers, Lecture Notes in Computer Science, pages 166-180. Springer-Verlag, Sheffield, U. K., 1995.
[18] Dragan Cvetkovic, Ian Parmee, and Eric Webb, Multi-objective Optimization and Preliminary Airframe Design. In Ian Parmee, editor, The Integration of Evolutionary and Adaptive Computing Technologies with Product/System Design and Realization, pages 255-267, Plymouth, United Kingdom, April 1998. Plymouth Engineering Design Center, Springer-Verlag.
[19] Hisashi Tamaki, M. Mori, Araki, Y. Mishima, and H. Ogai. Multi-Criteria Optimization by Genetic Algorithms: A Case of Scheduling in Hot Rolling Process. In Proceedings of the 3
rd Conference of the Association of Asian-Pacific Operational Research Societies within IFORS(APORS'94), pages 374-381. World Scientific, 1995.
[20] Hisashi Tamaki, Hajime Kita, and Shigenobu Kobayashi. Multi-Objective Optimization by Genetic Algorithms: A Review. In Toshio Fukuda and Takeshi Furuhashi, editors, Proceedings of the 1996 International Conference on Evolutionary Computation (ICEC'96), pages 517-522, Nagoya, Japan, 1996. IEEE.
[21] Carlos M. Fonseca and Peter J. Fleming. Genetic Algorithms for Multi-objective Optimization: Formulation, Discussion and Generalization. In Stephanie Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 416-423, San Mateo, California, 1993. University of Illinois at Urbana-Champaign, Morgan Kauffman Publishers.
[22] David E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Publishing Company, Reading, Massachusetts, 1989.
[23] David E. Goldberg and Kalyanmoy Deb. A comparison of selection schemes used in genetic algorithms. In G. J. E. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93. Morgan Kaufmann, San Mateo, California, 1991.
[24] N. Srinivas and Kalyanmoy Deb. Multi-objective Optimization Using Non-dominated Sorting in Genetic Algorithms. Evolutionary Computation, 2(3): 221-248, fall 1994.
[25] A. J. Chipperfield and P. J. Fleming. Gas Turbine Engine Controller Design using Multi-objective Genetic Algorithms. In A. M. S. Zalzala, editor, Proceedings of the First IEE/IEEE international Conference on Genetic Applications, GALESIA'95, pages 214-219, Halifax Hall, University of Sheffield, UK, September 1995. IEEE.
[26] Shigeru Obayashi. Pareto Genetic Algorithm for Aerodynamic Design using the Navier-Stokes Equations. In D. Quagliarella, J. Periaux, C. Poloni, and G. Winter, editors, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science. Recent Advances and Industrial Applications, chapter12,papgs245-266.John Wiley and sons,West Sussex,England, 1997.
[27] Karya Rodrfguez-vazquez, Carlos M. Fonseca, and Peter J. Fleming. Multi-objective Genetic Programming: A Nonlinear System Identification Application. in John R.Koza, editor, Late Breaking Papers at the Genetic Programming 1997 Conference, pages 207-212, Stanford University 1997 Conference, pages 207-212,Stanford University,
California, July 1997. Stanford Bookstore.
[28] John R. Koza. Genetic Programming. On the Programming of Computers by Means of Natural Selection. The MIT Press, 1992.
[29] Frank J. Aherne, Neil A. Thacker, and Peter I. Rockett. Optimal Pairwise Geometric Histograms. In Adrian F. Clark, editor, Electronic Proceedings of the Eighth British Machine Vision Conference,BMVC97, University of Essex, United Kingdom, September 1997.
[30] David S. Todd and Pratyush Sen. A Multiple Criteria Genetic Algorithm for Containership Loading. In Thomas Back, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 674-681, San Mateo, California, July 1997 Michigan State University, Morgan Kaufmann Publishers.
[31] Jeffrey Horn and Nicholas Nafpliotis. Multi-objective Optimization using the Niched Pareto Algorithm. Technical Report IlliGAl Report 93005, Univercity of Illinois at Urbana-Champaign, Urbana, Illinois,USA, 1993.
[32] Jeffrey Horn, Nicholas Nafpliotis and David E. Goldberg. A Niched Pareto Genetic Algorithm for Multi-objective Optimization. Accepted for publication in the Proceedings of the First IEEE Conference on Evolutionary Computation, IEEE World Congress on Computational Intelligence, Volume 1, 1994.
[33] A. D. Belegundu,D.V.Murthy, R.R.Salagame,andE.W.Constants.Multiobjective Optimization of Laminated Ceramic Composites Using Genetic Algorithms. In Fifth Analysis and Optimization, pages1015-1022, Panama City, Florida, 1994. ALAA. Paper84-4363-CP.
[34] Corlo Poloni and Valentino Pediroda. GA coupled with computationally expensive simulations: tools to improve efficiency. In D. Quagliarella,J. Periaux,C. Poloni, and G. Winter, editors, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science. Recent Advances and Industrial Applications, chapter 13,pages 167-288.John Wiley and Sons, West Sussex,England,1997.
[35] Carlos Artemio Coello Coello. An Empirical Study of Evolutionary Techniques for Multi-objective Optimization in Engineering Design. PhD thesis, Department of Computer Science, Tulane University, New Orleans, LA, apr 1996.
[36] Jacques Periaux, Mourad Sefrioui, and Bertrand Mantel. GA Multiple Objective
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[37] Ganesh Vedarajan, Louis Chi Chan, and David E. Goldberg. Investment Portfolio Optimization using Genetic Algorithms. In John R. Koza, editor, Late Breaking Papers at the Genetic Programming 1997 Conference, pages 255-263, Stanford University, California, July 1997. Stanford Bookstore.
[38] N.Srinivas and Kalyanmoy Deb. Multi-objective optimization using non-dominated sorting in genetic algorithms. Technical report, Department of Mechanical Engineering, Indian Institute of Technology, Kanput, India, 1993.
[39] E. Michielssen and D. S. Weile. Electromagnetic System Design using Genetic Algorithms. in Genetic Algorithms and Evolution Strategies in Engineering and Computer Science, pages 267-288. John Wiley and Sons, England, 1995.
[40] Eckart Zitzler and Lothar Thiele. Multi-objective Optimization Using Evolutionary Algorithms—A Comparative Study. In A. E. Eiben, editor, Problem Solving from Nature V, pages 292-301, Amsterdam, September 1998. Springer-Verlag.
[41] Eckart Zitzler and Lothar Thiele. An Evolutionary Algorithm for Multi-objective Optimization: The Strength Pareto Approach. Technical Report 43, Computer Engineering and Communication Networks Lab (TIK), Swiss Federal Institute of Technology (EYH), Zurich Switzerland, May 1998.
[42] Zilzler, E. and Thiele,L.(1998) Multi-objective optimization using evolutionary algorithms—A comparative case study. In Eiben,A.E. and Schwefel, H., editors, Parallel Problem Solving from Nature, V, pages 292-301,Springer, Berlin,Germany.
[43] Knowles,J.and Corne,D.(1999)The Pareto archived evolution strategy: A new baseline algorithm for multi-objective optimisation. Proceedings of the 1999 Congress on Evolutionary Computation, Piscatway: New Jersey: IEEE Service Center, 98-105.
[44] Rudolph,G.(1999)Evolutionary search under partially ordered sets. Technical Report No.CI-67/99,Dortmund: Department of Computer Science/LS11,Unicersity of Dortmund, Germany.
[45] Deb, K., Agrawal, S., Pratap, A. and Meyarivan, T.(2000).A Fast Elitist Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization: NSGA-Ⅱ.Technical Report No.2000001. Kanpur: Indian Institute of Technology Kanpur, India.
[46] 朱学军，陈彤等．多个体参与交叉的Pareto多目标遗传算法．电子学报，2001，29(1)：106-109．
[47] 崔逊学，李淼，方廷建．基于免疫原理的多目标进化算法群体多样性研究．模式识别与人工智能，2001，14(3)：291-295．
[48] 张铭钧，韩金华，郎毅翔．水下机器人运动规划中多目标遗传算法的操作方法．哈尔滨工程大学学报，2000，21，(2)：23-27．
[49] E. Zitzler. Evolutionary Algorithms for Multiobjective Optimization. EUROGEN 2001-Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems, September 2001, to appear.
[50] Kalyanmoy Deb, Amrit Pratap, Sameer Agrawal and T. Meyrivan. A Fast and Elitist Multi-objective Genetic Algorithm: NSGA-11. 1EEE Transactions on Evolutionary Computation, 6(2):182-197, April 2002.
[51] Deb, K, Mohan, M. and Mishra, S. (February, 2003). A Fast Multi-objective Evolutionary Algorithm for Finding Well-Spread Pareto-Optimal Solutions. KanGAL Report No. 2003002.
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[54] 曹元大，向尕．基于多目标规划的QoS路由选择的数学模型及优化算法．计算机工程，2003，29(2)：122-124．
[55] 杨子晨等．基于多目标遗传算法求解多边谈判问题的Pareto解．计算机工程与应用，2002，38(1)：39-41．
[56] Carlos A. Coello Coello. An Updated Survey of GA-Based Multiobjective Optimization Techniques. ACM Computing Surveys, 32(2): 109-143, June 2000.
[2] David A. Van Veldhuizen and Gary B. Lamont. Multi-objective Evolutionary Algorithm Research: A History and Analysis. Technical and Computer Engineering, Graduate School of Engineering, Air Force Institute of Technology, Wright-Patterson AFB, 1998.
[3] Carlos M. Fonseca and Peter J. Fleming. An overview of evolutionary algorithms in multi-objective optimization. Technical report, Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, U. K.,1994.
[4] R. S. Rosenberg. Simulation of genetic populations with biochemical properties. PhD thesis, University of Michigan Ann Harbor, Michigan, 1967.
[5] H. W. Kuhn and A. W. Tucker. Nonlinear programming. In J. Neyman, editor, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pages 481-492, Berkeley, California, 1951. University of California Press.
[6] Gilbert Syswerda and Jeff Palmucci. The Application of Genetic Algorithms to Resource Scheduling. In Richard K. Booker, editors, Proceedings of the Fourth International Conference on Genetic Algorithms, pages 502-508, San Mateo, California, 1991. Morgan Kaufmann.
[7] W.Jakob, M. Gorges-Schleuter, and C.Blume. Application of genetic of genetic algorithms to task planning and learning. In R. Manner and B. Manderick, editors, Parallel Problem Solving form Nature, 2 nd Workshop, Lecture Notes in Computer Science, pages 291-300, Amsterdam, 1992. North-Holland Publishing Company.
[8] Gareth Jones, Robert D. Brown, David E. Clark, Peter Willett, and Robert C. Glen. Searching Databases of Two-Dimensional and Three-Dimensional Chemical Structures using Genetic Algorithms. In Stephanie Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 597-602, San Mateo, California, 1993. Morgan Kaufmann.
[9] P.B. Wilson and M.D.Macleod. Low implementation cost ⅡR digital filter design using genetic algorithms. In IEE/IEEE Workshop on Natural Algorithms in Signal Processing, pages 4/1-4/8,Chelmsford, U.K., 1993.
[10] Xiaojian Liu, D.W. Begg, and R.J.Fishwick. Genetic approach to optimal
topology/controller design of adaptive structures. International Journal for Numerical Methods in Engineering,41:815-830, 1998.
[11] Xiaofeng Yang and Mitsuo Gen. Evolution program for bicriteria transportation problem. In M. Gen and T. Kobayashi, editors, Proceedings of the 16th International Conference on Computers and Industrial Engineering, pages 451-454, Ashikaga, Japan, 1994. Pergamon Press.
[12] M. Gen, K. Ida, and Y. Li. Solving bicriteria solid transportation problem with fuzzy numbers by genetic algorithm. International Journal of Computers and Industrial Engineering, 29:537-543, 1995.
[13] Mitsuo Gen and Runwei Cheng. Genetic Algorithms and Engineering Design. John Wiley and Sons,Inc., New York, 1997.
[14] Richardson,J.T.,Palmer, M.R.,Liepins,F.,&Hilliard,M.(1989).Some guidelines for genetic algorithms with penalty functions. Proceedings of the Third International Conference on Genetic A lgorithms.Morgan-Kauffman, 191-197.
[15] Schaffer, J.D.,(1984).Some experiments in machine learning using vector evaluated genetic algorithms. Unpublished doctoral dissertation, Vanderbilt University.
[16] J. David Schaffer. Multiple objective optimization with vector evaluated genetic algorithms. In Genetic Algorithms and their Applications: Proceedings of the First International Conference on Genetic Algorithms, pages 93-100. Lawrence Erlbaum, 1985.
[17] Patrick D. Surry, Nicholas J. Radcliffe, and Ian D. Boyd. A Multi-Objective Approach to Constrained Optimization of Gas Supply Networks: The COMOGA Method. In Terence C. Fogarty, editor, Evolutionary computing. AISB Workshop. Selected Papers, Lecture Notes in Computer Science, pages 166-180. Springer-Verlag, Sheffield, U. K., 1995.
[18] Dragan Cvetkovic, Ian Parmee, and Eric Webb, Multi-objective Optimization and Preliminary Airframe Design. In Ian Parmee, editor, The Integration of Evolutionary and Adaptive Computing Technologies with Product/System Design and Realization, pages 255-267, Plymouth, United Kingdom, April 1998. Plymouth Engineering Design Center, Springer-Verlag.
[19] Hisashi Tamaki, M. Mori, Araki, Y. Mishima, and H. Ogai. Multi-Criteria Optimization by Genetic Algorithms: A Case of Scheduling in Hot Rolling Process. In Proceedings of the 3
rd Conference of the Association of Asian-Pacific Operational Research Societies within IFORS(APORS'94), pages 374-381. World Scientific, 1995.
[20] Hisashi Tamaki, Hajime Kita, and Shigenobu Kobayashi. Multi-Objective Optimization by Genetic Algorithms: A Review. In Toshio Fukuda and Takeshi Furuhashi, editors, Proceedings of the 1996 International Conference on Evolutionary Computation (ICEC'96), pages 517-522, Nagoya, Japan, 1996. IEEE.
[21] Carlos M. Fonseca and Peter J. Fleming. Genetic Algorithms for Multi-objective Optimization: Formulation, Discussion and Generalization. In Stephanie Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 416-423, San Mateo, California, 1993. University of Illinois at Urbana-Champaign, Morgan Kauffman Publishers.
[22] David E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Publishing Company, Reading, Massachusetts, 1989.
[23] David E. Goldberg and Kalyanmoy Deb. A comparison of selection schemes used in genetic algorithms. In G. J. E. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93. Morgan Kaufmann, San Mateo, California, 1991.
[24] N. Srinivas and Kalyanmoy Deb. Multi-objective Optimization Using Non-dominated Sorting in Genetic Algorithms. Evolutionary Computation, 2(3): 221-248, fall 1994.
[25] A. J. Chipperfield and P. J. Fleming. Gas Turbine Engine Controller Design using Multi-objective Genetic Algorithms. In A. M. S. Zalzala, editor, Proceedings of the First IEE/IEEE international Conference on Genetic Applications, GALESIA'95, pages 214-219, Halifax Hall, University of Sheffield, UK, September 1995. IEEE.
[26] Shigeru Obayashi. Pareto Genetic Algorithm for Aerodynamic Design using the Navier-Stokes Equations. In D. Quagliarella, J. Periaux, C. Poloni, and G. Winter, editors, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science. Recent Advances and Industrial Applications, chapter12,papgs245-266.John Wiley and sons,West Sussex,England, 1997.
[27] Karya Rodrfguez-vazquez, Carlos M. Fonseca, and Peter J. Fleming. Multi-objective Genetic Programming: A Nonlinear System Identification Application. in John R.Koza, editor, Late Breaking Papers at the Genetic Programming 1997 Conference, pages 207-212, Stanford University 1997 Conference, pages 207-212,Stanford University,
California, July 1997. Stanford Bookstore.
[28] John R. Koza. Genetic Programming. On the Programming of Computers by Means of Natural Selection. The MIT Press, 1992.
[29] Frank J. Aherne, Neil A. Thacker, and Peter I. Rockett. Optimal Pairwise Geometric Histograms. In Adrian F. Clark, editor, Electronic Proceedings of the Eighth British Machine Vision Conference,BMVC97, University of Essex, United Kingdom, September 1997.
[30] David S. Todd and Pratyush Sen. A Multiple Criteria Genetic Algorithm for Containership Loading. In Thomas Back, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 674-681, San Mateo, California, July 1997 Michigan State University, Morgan Kaufmann Publishers.
[31] Jeffrey Horn and Nicholas Nafpliotis. Multi-objective Optimization using the Niched Pareto Algorithm. Technical Report IlliGAl Report 93005, Univercity of Illinois at Urbana-Champaign, Urbana, Illinois,USA, 1993.
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