求解旅行商问题的微粒群算法研究
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摘要
旅行商(TSP)问题,是一个古老而又典型的NP-hard组合优化问题。该问题的有效求解,不仅有着重要的理论和学术价值,同时对于许多实际工程应用问题有着重要的指导意义。因此,寻求高效的TSP优化算法一直是优化领域的一个热点课题。微粒群算法是一种新型的群智能计算方法,算法模型简单、操作便捷,具有自组织、自适应、自学习等智能特性,已被证明是一种求解大规模复杂问题的有效工具,因此也是求解TSP问题的一种智能计算方法。然而,面对复杂的离散组合优化问题,现有的微粒群算法模型存在着效率低下,求解精度不高,易于早熟等缺陷。
为此,本论文基于TSP优化问题,着重围绕离散微粒群算法的模型设计和性能改进,进行以下内容的研究:
一、基于微粒群算法的优化机理,分析了离散微粒群算法设计的主要方法和基本原则,进而研究了离散微粒群算法的关键技术,包括离散问题的编码、个体微粒的评价及其主要进化操作,为后续的研究提供技术基础。
二、为提高离散微粒群算法的全局收敛性,提出了一种引入局部扰动策略的改进微粒群算法。首先基于整数编码,并结合遗传算法中的PMX算子,对离散微粒群算法中的速度、位置及其进化操作进行了重新定义,其次引入微粒的活力信息来实现控制参数的自适应调节,构造局部扰动策略以避免群体局部收敛。改进算法被用于典型TSP测试问题,其仿真结果表明该算法的有效性。
三、为提高离散微粒群算法对复杂问题的求解能力,提出一种基于边编码,采用表示边的两城市点之间所有点依次倒置插入的微粒更新操作的改进离散微粒群算法。并在对算法方程进行重新定义的基础之上,加入微粒适应值变优则更新,变差按一定概率更新的策略。通过在典型中型TSP问题上进行的实验,求解时间、求解精度上均令人满意。然后在这个改进算法的基础之上加入多级规约策略,为微粒群算法求解大规模问题提供了一种思路。
Traveling salesman problem (TSP) is an old and typical NP-hard combinatorial optimization problem, its valid and effective solutions not only has important theoretical and academic values, but also has important guiding significance for many practical engineering applications. So TSP is a hot topic in optimization field. Particle swarm optimization (PSO) is a new method based on swarm intelligence, which are characterized mainly by simple model, convenient operation, self-organizing, self-adaptive, self-learning, etc. Particle swarm optimization has been proved to be an effective technique to solve large-scale complex problems. Therefore, it is also an efficient method to solve TSP. However, the existing PSO also suffers from some shortcomings, such as low efficiency, low accuracy, premature convergence, etc., especially for the large scale and complex discrete combinatorial optimization problems.
The dissertation focuses on the model designing and performance improving of discrete particle swarm algorithm (DPSO) based on TSP, its main researches are listed as the following:
1. Based on the optimization mechanisms of swarm intelligence, the dissertation makes an analysis of the primary design methods and the basic principles for discrete particle swarm optimization firstly. After that, the key techniques of DPSO also are analyzed in details, including the code methods for discrete optimization problems, the evolutionary operations in discrete solution space, etc., which provide a valid technical foundation for the following study.
2. To improve the global convergence of DPSO, The dissertation develops an adaptive DPSO to solve TSP. Firstly, the improved DPSO adopts the integer code method and redesigns the particles’velocity, location, and their discrete evolution operations based on the PMX operator of genetic algorithm. Secondly, the improved DPSO introduces a local adjustment strategy for the parameters to avoid the local convergence of the swarm, which is controlled by the dynamic information of particles. The simulation results based on the typical TSP problems show that the proposed algorithm validly improves the global performance of PSO in discrete optimization problems.
3. To develop the performance of DPSO for complex discrete optimization problems, the dissertation proposed another improved DPSO, which takes advantage of side-code technique and a multilevel reduction strategy. The improved DPSO redesigns the particles’velocity, location, and their discrete evolution operations based on the side-code technique. After that, a multilevel reduction strategy is introduced to improve the global convergence of the algorithm. Finally, the improved algorithm is applied to solve medium or large scale TSP problems, the simulation results show that the proposed algorithm is a valid technique for complex TSP problems.
为此,本论文基于TSP优化问题,着重围绕离散微粒群算法的模型设计和性能改进,进行以下内容的研究:
一、基于微粒群算法的优化机理,分析了离散微粒群算法设计的主要方法和基本原则,进而研究了离散微粒群算法的关键技术,包括离散问题的编码、个体微粒的评价及其主要进化操作,为后续的研究提供技术基础。
二、为提高离散微粒群算法的全局收敛性,提出了一种引入局部扰动策略的改进微粒群算法。首先基于整数编码,并结合遗传算法中的PMX算子,对离散微粒群算法中的速度、位置及其进化操作进行了重新定义,其次引入微粒的活力信息来实现控制参数的自适应调节,构造局部扰动策略以避免群体局部收敛。改进算法被用于典型TSP测试问题,其仿真结果表明该算法的有效性。
三、为提高离散微粒群算法对复杂问题的求解能力,提出一种基于边编码,采用表示边的两城市点之间所有点依次倒置插入的微粒更新操作的改进离散微粒群算法。并在对算法方程进行重新定义的基础之上,加入微粒适应值变优则更新,变差按一定概率更新的策略。通过在典型中型TSP问题上进行的实验,求解时间、求解精度上均令人满意。然后在这个改进算法的基础之上加入多级规约策略,为微粒群算法求解大规模问题提供了一种思路。
Traveling salesman problem (TSP) is an old and typical NP-hard combinatorial optimization problem, its valid and effective solutions not only has important theoretical and academic values, but also has important guiding significance for many practical engineering applications. So TSP is a hot topic in optimization field. Particle swarm optimization (PSO) is a new method based on swarm intelligence, which are characterized mainly by simple model, convenient operation, self-organizing, self-adaptive, self-learning, etc. Particle swarm optimization has been proved to be an effective technique to solve large-scale complex problems. Therefore, it is also an efficient method to solve TSP. However, the existing PSO also suffers from some shortcomings, such as low efficiency, low accuracy, premature convergence, etc., especially for the large scale and complex discrete combinatorial optimization problems.
The dissertation focuses on the model designing and performance improving of discrete particle swarm algorithm (DPSO) based on TSP, its main researches are listed as the following:
1. Based on the optimization mechanisms of swarm intelligence, the dissertation makes an analysis of the primary design methods and the basic principles for discrete particle swarm optimization firstly. After that, the key techniques of DPSO also are analyzed in details, including the code methods for discrete optimization problems, the evolutionary operations in discrete solution space, etc., which provide a valid technical foundation for the following study.
2. To improve the global convergence of DPSO, The dissertation develops an adaptive DPSO to solve TSP. Firstly, the improved DPSO adopts the integer code method and redesigns the particles’velocity, location, and their discrete evolution operations based on the PMX operator of genetic algorithm. Secondly, the improved DPSO introduces a local adjustment strategy for the parameters to avoid the local convergence of the swarm, which is controlled by the dynamic information of particles. The simulation results based on the typical TSP problems show that the proposed algorithm validly improves the global performance of PSO in discrete optimization problems.
3. To develop the performance of DPSO for complex discrete optimization problems, the dissertation proposed another improved DPSO, which takes advantage of side-code technique and a multilevel reduction strategy. The improved DPSO redesigns the particles’velocity, location, and their discrete evolution operations based on the side-code technique. After that, a multilevel reduction strategy is introduced to improve the global convergence of the algorithm. Finally, the improved algorithm is applied to solve medium or large scale TSP problems, the simulation results show that the proposed algorithm is a valid technique for complex TSP problems.
引文
[1]王晓陵,陆军.最优化方法和最优控制.哈尔滨,哈尔滨工程大学出版社,2006.
[2]邢文训,谢金星.现代化计算方法,北京,清华大学出版社,1999.
[3] Holland.J.Adaptation in natural and artificial systems:an introductory analysis wit -h application to biology, control, and artificial intelligence,2nd Edition.Cambridge,MA:MIT Press,1992.
[4] Kennedy.J,Eberhart.R.C,Shi.Y.Swarm intelligence,Morgan Kaufmann,2001.
[5] Bonabeau.E,Dorigo.M,Theraulaz.G..Swarm intelligence from natural to artificeal systems [M].Oxford Univergity Press Inc,1999,1-22.
[6]段海滨.蚁群算法原理及其应用.北京,科学出版社,2005.
[7] Kennedy.J,Eberhart.R.A.Discrete binary version of the particle swarm algorit hm.Proceedings of the World Multiconferenceon Systemics,Cyberneticsan Inform atics [C],Piscataway,NJ:IEEE service Center,1997,4104- 4109.
[8] Eberhart.R.C,Kennedy.J.A New optimizer using particle swarm theory.Proc of Sixth Internationl Symposi numon MicroMachineand Human Science,Nagoya,Japan,1995,39-43.
[9] Keneny.J,Eberhart.R.C.A Discrete binary version of the particle swarm algorit hm.Proceedings of the World Multiconferen-ce on Systemics,Cybernetics and Informat ions [C],Piscataway,NJ:IEEE Service Center,1997,4104-4109.
[10] Heppner.F,Grenander.U.A.Stochastic nonlinear model for coordinated bird flocks,InKrasner S,Ed,The Ubiquity of Chaos,AAAS Publications,Washington, DC, 1990.
[11] Reynolds CW,Flocks.A distributed behavioral model,Computer Graphics,1987,21(4),25-34.
[12] Kennedy J,Mendes R.Population structure and particle swarm performance,Proc.IEEE Congress on Evolutionary Computation (CEC 2002),Hawaii USA, 2002,1671-1676.
[13] Shi.Y,Eberhart.R.C.A modified particle swarm optimizer,Proc. IEEE Congress on Evolutionary Computation, Piscataway,IEEE Press,1999,69-73.
[14] Shi.Y,Eberhart.R.C.Empirical study of particle swarm optimization,In:Proc. of Congress on Evolutionary Computation, Piscataway,NJ: IEEE Service Center,1999,1945–1950.
[15] Shi.Y,Eberhart.R.C.Fuzzy adaptive particl swarm optimization,New York: IEEE,Piscataway,NJ,USA,2001,101-106.
[16] Clerc.M,Kennedy.J.The particle swarm explosion, stability, and convergence in amulti dimensional complex space,IEEE Transactions on Evolutionary Computation,2002,6(1),58-73.
[17] Ozcan.E , Mohan.C.K . Analysis of a simple particle swarm optimization system.Intelligent Engineering Systems Through Artificial Neural Networks,1998,8, 252-258.
[18] Solis.F, Wets.R.Minimization by random search techniques.Mathematics of OperationsResearch, 1981, 6:19-30.
[19] Bergh.F.V.D.An analysis of particle swarm optimizers,Department of Computer Science, University of Pretoria,South Africa,2001.
[20]曾建潮,崔志华.一种保证全局收敛的PSO算法.计算机研究与发展,2004,41,8,1333-1338.
[21]介婧,曾建潮,韩崇昭.基于群体多样性反馈控制的自组织微粒群算法.计算机研究与发展,2008,45,3,464-471.
[22]曾建潮,王丽芳,一种广义微粒群算法模型.模式识别与人工智能,2005,18,6,685-688.
[23] Engelbrechtap,Berghf Van Den.Cooperative learning in neural netwo rks using particle swarm optimizers,South African Computer Journal,2000,26,84-90.
[24] Parsopoulos.K.E, Vrahatis.M.N.Recent approaches to global optimization proble -ms through particle swarm optimization,Natural Computing,2002,1,2-3,235-306.
[25] Tasgetiren.M..F,Sevkli.M.,Liang.Y.C,Gencyilmaz.G..Particle swarm optimization algorithm for permutation flowshop sequencing problem,Proceedings of the fourth international workshop on ant colony optimization and swarm intelligenc e,Lecture notes in computer science,Brussels, Belgium,vol,3172,382-390
[26] Kennedy.J,Eherhart.R.C.A discrete binary version of the particle swarm algori thmi CProc,the World Multiconferenee onSystemics,Cybernetics and Informatics,IEEE,1997,l,4104—4109.
[29] Ting.T,Rao.M.V.C,Loo.C.K.A novel approach for unit commitment problem via aneffective hybrid particle swarln optimization,IEEE Teansactions on Power Systems,2006,21,1,411-418.
[30] Correa..Elon.S,Freitas.Alex.A,Jonson..Coli.G.A new discrete particle swarm algorithm applied to attribute selection in abioin for matics data set.GEC06,2006.
[31] Hu.X.H,Eberhart.R.E,Shi.Y.H.Swarm intelligence for permutation optimization,a case study of n-queens problem.IEEE Swarm Intelligence Symposium,Indianapolis,2003,243—246.
[32] Bo Liu,Ling Wang,Yi hui Jin,De xian Huang.An Effective PSO Based Mem etic Algorithm for TSP.
[33]李宁,邹彤,孙德宝.带时间窗格路径问题的粒子群算法.系统工程理论与实践,2004,24,4,130-135.
[34] Clerc.M.Disecrete particle swarm optimization..On wuboluGC BabuBV,New Optimization Techniques in Engineening,Spring-Verlag,2004,219-240.
[35]钟一文,杨建刚,宁正远.求解TSP问题的离散粒子群优化算法.系统工程理论与实践,2006,6,88-94.
[36] Shi X H,Xing X L,Wang Q X.A discrete PSO method for generalized TSP problem Proc,The Third International Conferece on Machine Learning and Cyber nitics,2004,2378-2383.
[37]高海兵,周驰,高亮.广义粒子群优化模型.计算机学报,2006,12,1980-1987.
[38]高海昌,冯博琴,朱莉.智能优化算法求解TSP问题.控制与决策,2006,3,281-288.
[39]高尚,韩斌,吴小俊.求解旅行商问题的混合微粒群优化算法.控制与决策,2004,19,11,1286-128.
[40]高尚,蒋新姿,汤可宗等.蚁群算法和微粒群优化算法的混合算法.2006,7 Proceedings of the 25 Chinese Control Conference,7-11 August 2006,Harbin,Herlongjiang,1428-1432.
[41]郭文忠,李丽珊,林晓宇,求解TSP问题的模糊自适应粒子群算法.计算机科学,2006,33,161-185.
[42]蔡荣英,李丽珊,林晓宇.求解旅行商问题的自适应粒子群算法.计算机工程与设计,2007 2,28,261-263.
[43] Hopfield.J.J.Neural Networks and physical systems with emergent collective computational abilities.Proc Nat Acad Sci,1982,79,2554-2558.
[44]玄光男,程润伟.遗传算法与工程优化.北京,清华大学出版社,2004.
[45]曾建潮,介婧,崔志华.微粒群算法.北京,科学出版社,2004.
[46]郭涛.演化计算与优化武汉.武汉大学软件工程国家重点实验室,1999.
[47] Dofigo.M,Gambardella.L.M.Ant Colony System:a Cooperative Learning Approach to the Traveling Salesman Problem.IEEE Transactions on Evolutionary Computation,1997,1,53-66.
[48]王文峰,刘光远,温万慧.求解TSP问题的自逃逸混合离散粒子群算法研究.计算机科学,2006,8,34,143-145.
[49]李宁,刘飞,孙德宝.基于带变异算子粒子群优化算法的约束布局优化研究.计算机学报,2004,7,7,897-903.
[50]张军英,周斌.基于泛化竞争和局部渗透机制的自组织网TSP问题求解方法.计算机学报,2008,2,2,220-227.
[51]邹鹏,周智,陈国良.求解TSP问题的多级规约算法.软件学报,2003.1,1,35-43.
[52]戚玉涛,焦李成,刘芳.求解TSP问题的自适应规约免疫算法.软件学报,2008,6,6,1265-1273.
[53]焦李成,杜海峰,刘芳.免疫优化计算、学习与识别.北京,科学出版社
[54] LE.Louarn.F,Gendreau.M.Ceni Ants for Traveling Salesman Problem.Annals of Opereyions Reseach,2004,121,187-201.
[55] www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/tsp/.
[56] Jing Jie,Jianchao Zeng,.Chongzhao Han. Knowledge-based Cooperative Particle Swarm Optimization.Applied Mathematics and Computation. 2008, 205, 861-873
[57]Jing Jie,Jianchao Zeng,Chongzhao Han.Self-Organization Particle Swarm Optim ization Based on Information Feedback.Lecture Notes in Computer Science, 2006,4221,913 -922.
[2]邢文训,谢金星.现代化计算方法,北京,清华大学出版社,1999.
[3] Holland.J.Adaptation in natural and artificial systems:an introductory analysis wit -h application to biology, control, and artificial intelligence,2nd Edition.Cambridge,MA:MIT Press,1992.
[4] Kennedy.J,Eberhart.R.C,Shi.Y.Swarm intelligence,Morgan Kaufmann,2001.
[5] Bonabeau.E,Dorigo.M,Theraulaz.G..Swarm intelligence from natural to artificeal systems [M].Oxford Univergity Press Inc,1999,1-22.
[6]段海滨.蚁群算法原理及其应用.北京,科学出版社,2005.
[7] Kennedy.J,Eberhart.R.A.Discrete binary version of the particle swarm algorit hm.Proceedings of the World Multiconferenceon Systemics,Cyberneticsan Inform atics [C],Piscataway,NJ:IEEE service Center,1997,4104- 4109.
[8] Eberhart.R.C,Kennedy.J.A New optimizer using particle swarm theory.Proc of Sixth Internationl Symposi numon MicroMachineand Human Science,Nagoya,Japan,1995,39-43.
[9] Keneny.J,Eberhart.R.C.A Discrete binary version of the particle swarm algorit hm.Proceedings of the World Multiconferen-ce on Systemics,Cybernetics and Informat ions [C],Piscataway,NJ:IEEE Service Center,1997,4104-4109.
[10] Heppner.F,Grenander.U.A.Stochastic nonlinear model for coordinated bird flocks,InKrasner S,Ed,The Ubiquity of Chaos,AAAS Publications,Washington, DC, 1990.
[11] Reynolds CW,Flocks.A distributed behavioral model,Computer Graphics,1987,21(4),25-34.
[12] Kennedy J,Mendes R.Population structure and particle swarm performance,Proc.IEEE Congress on Evolutionary Computation (CEC 2002),Hawaii USA, 2002,1671-1676.
[13] Shi.Y,Eberhart.R.C.A modified particle swarm optimizer,Proc. IEEE Congress on Evolutionary Computation, Piscataway,IEEE Press,1999,69-73.
[14] Shi.Y,Eberhart.R.C.Empirical study of particle swarm optimization,In:Proc. of Congress on Evolutionary Computation, Piscataway,NJ: IEEE Service Center,1999,1945–1950.
[15] Shi.Y,Eberhart.R.C.Fuzzy adaptive particl swarm optimization,New York: IEEE,Piscataway,NJ,USA,2001,101-106.
[16] Clerc.M,Kennedy.J.The particle swarm explosion, stability, and convergence in amulti dimensional complex space,IEEE Transactions on Evolutionary Computation,2002,6(1),58-73.
[17] Ozcan.E , Mohan.C.K . Analysis of a simple particle swarm optimization system.Intelligent Engineering Systems Through Artificial Neural Networks,1998,8, 252-258.
[18] Solis.F, Wets.R.Minimization by random search techniques.Mathematics of OperationsResearch, 1981, 6:19-30.
[19] Bergh.F.V.D.An analysis of particle swarm optimizers,Department of Computer Science, University of Pretoria,South Africa,2001.
[20]曾建潮,崔志华.一种保证全局收敛的PSO算法.计算机研究与发展,2004,41,8,1333-1338.
[21]介婧,曾建潮,韩崇昭.基于群体多样性反馈控制的自组织微粒群算法.计算机研究与发展,2008,45,3,464-471.
[22]曾建潮,王丽芳,一种广义微粒群算法模型.模式识别与人工智能,2005,18,6,685-688.
[23] Engelbrechtap,Berghf Van Den.Cooperative learning in neural netwo rks using particle swarm optimizers,South African Computer Journal,2000,26,84-90.
[24] Parsopoulos.K.E, Vrahatis.M.N.Recent approaches to global optimization proble -ms through particle swarm optimization,Natural Computing,2002,1,2-3,235-306.
[25] Tasgetiren.M..F,Sevkli.M.,Liang.Y.C,Gencyilmaz.G..Particle swarm optimization algorithm for permutation flowshop sequencing problem,Proceedings of the fourth international workshop on ant colony optimization and swarm intelligenc e,Lecture notes in computer science,Brussels, Belgium,vol,3172,382-390
[26] Kennedy.J,Eherhart.R.C.A discrete binary version of the particle swarm algori thmi CProc,the World Multiconferenee onSystemics,Cybernetics and Informatics,IEEE,1997,l,4104—4109.
[29] Ting.T,Rao.M.V.C,Loo.C.K.A novel approach for unit commitment problem via aneffective hybrid particle swarln optimization,IEEE Teansactions on Power Systems,2006,21,1,411-418.
[30] Correa..Elon.S,Freitas.Alex.A,Jonson..Coli.G.A new discrete particle swarm algorithm applied to attribute selection in abioin for matics data set.GEC06,2006.
[31] Hu.X.H,Eberhart.R.E,Shi.Y.H.Swarm intelligence for permutation optimization,a case study of n-queens problem.IEEE Swarm Intelligence Symposium,Indianapolis,2003,243—246.
[32] Bo Liu,Ling Wang,Yi hui Jin,De xian Huang.An Effective PSO Based Mem etic Algorithm for TSP.
[33]李宁,邹彤,孙德宝.带时间窗格路径问题的粒子群算法.系统工程理论与实践,2004,24,4,130-135.
[34] Clerc.M.Disecrete particle swarm optimization..On wuboluGC BabuBV,New Optimization Techniques in Engineening,Spring-Verlag,2004,219-240.
[35]钟一文,杨建刚,宁正远.求解TSP问题的离散粒子群优化算法.系统工程理论与实践,2006,6,88-94.
[36] Shi X H,Xing X L,Wang Q X.A discrete PSO method for generalized TSP problem Proc,The Third International Conferece on Machine Learning and Cyber nitics,2004,2378-2383.
[37]高海兵,周驰,高亮.广义粒子群优化模型.计算机学报,2006,12,1980-1987.
[38]高海昌,冯博琴,朱莉.智能优化算法求解TSP问题.控制与决策,2006,3,281-288.
[39]高尚,韩斌,吴小俊.求解旅行商问题的混合微粒群优化算法.控制与决策,2004,19,11,1286-128.
[40]高尚,蒋新姿,汤可宗等.蚁群算法和微粒群优化算法的混合算法.2006,7 Proceedings of the 25 Chinese Control Conference,7-11 August 2006,Harbin,Herlongjiang,1428-1432.
[41]郭文忠,李丽珊,林晓宇,求解TSP问题的模糊自适应粒子群算法.计算机科学,2006,33,161-185.
[42]蔡荣英,李丽珊,林晓宇.求解旅行商问题的自适应粒子群算法.计算机工程与设计,2007 2,28,261-263.
[43] Hopfield.J.J.Neural Networks and physical systems with emergent collective computational abilities.Proc Nat Acad Sci,1982,79,2554-2558.
[44]玄光男,程润伟.遗传算法与工程优化.北京,清华大学出版社,2004.
[45]曾建潮,介婧,崔志华.微粒群算法.北京,科学出版社,2004.
[46]郭涛.演化计算与优化武汉.武汉大学软件工程国家重点实验室,1999.
[47] Dofigo.M,Gambardella.L.M.Ant Colony System:a Cooperative Learning Approach to the Traveling Salesman Problem.IEEE Transactions on Evolutionary Computation,1997,1,53-66.
[48]王文峰,刘光远,温万慧.求解TSP问题的自逃逸混合离散粒子群算法研究.计算机科学,2006,8,34,143-145.
[49]李宁,刘飞,孙德宝.基于带变异算子粒子群优化算法的约束布局优化研究.计算机学报,2004,7,7,897-903.
[50]张军英,周斌.基于泛化竞争和局部渗透机制的自组织网TSP问题求解方法.计算机学报,2008,2,2,220-227.
[51]邹鹏,周智,陈国良.求解TSP问题的多级规约算法.软件学报,2003.1,1,35-43.
[52]戚玉涛,焦李成,刘芳.求解TSP问题的自适应规约免疫算法.软件学报,2008,6,6,1265-1273.
[53]焦李成,杜海峰,刘芳.免疫优化计算、学习与识别.北京,科学出版社
[54] LE.Louarn.F,Gendreau.M.Ceni Ants for Traveling Salesman Problem.Annals of Opereyions Reseach,2004,121,187-201.
[55] www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/tsp/.
[56] Jing Jie,Jianchao Zeng,.Chongzhao Han. Knowledge-based Cooperative Particle Swarm Optimization.Applied Mathematics and Computation. 2008, 205, 861-873
[57]Jing Jie,Jianchao Zeng,Chongzhao Han.Self-Organization Particle Swarm Optim ization Based on Information Feedback.Lecture Notes in Computer Science, 2006,4221,913 -922.