基于正态分布的连续多蚁群算法及其化工应用
|
摘要
进入21世纪以来，化学工业面临着经济、能源、环境以及社会等多方面的挑战，优化技术是迎接这些挑战的有效手段，能够应用于化工全价值链的各个环节。化工系统是一类典型的复杂系统，随着目标问题的规模越来越大，模型结构也越来越复杂，经典的优化方法已显乏力，对高效的智能化的优化技术的需求日益迫切。
蚁群算法是新近提出来的一种群智能优化方法。由于其优越的问题分布式求解模式，在离散优化问题的求解中取得了极大成功，引起了相关领域学者的广泛关注。但很多实际问题通常被表达成连续优化问题。如何有效地将全局优化性能优越但本质离散的蚁群算法用于优化连续空间的问题，此为亟待应对的挑战，这也是本文的主要研究内容。
蚁群算法在本质是一种基于解空间参数化的概率分布模型的搜索算法框架，这些参数就是信息素，而蚂蚁生成的解集合则可看作是用来更新概率分布参数的样本。因此信息素分布模型是影响蚁群算法最关键的因素，它决定了蚂蚁的行为与分布，设计一种好的信息素分布模型是构造高性能连续蚁群算法的关键。
基于此，本文通过对蚁群觅食的生物学模型中信息素分布的分析，用多元正态分布函数来模拟信息素的分布，提出了一种信息素呈多元正态分布的连续多蚁群算法(CMACO)。该算法通过对信息素分布函数的随机抽样来指导蚂蚁完成状态转移，信息素分布函数又随着蚂蚁的移动而被调整，实施信息素更新，进而引导蚂蚁在可行域中逐步向最优食物源聚集。为了提高算法的寻优性能，基于蚁群的成群募集机制，本文构建出多蚁群策略来有效地调配蚁群的行为以平衡其全局探索能力和局部挖掘能力。经多个经典函数的测试，表明CMACO适用于连续优化问题，具有良好的全局寻优性能。
对于终端时间给定、终端状态无约束的动态优化问题，本文通过控制变量参数化方法将其转换成静态优化问题，然后使用CMACO进行优化。按照该思路，将CMACO用于生产分泌蛋白的Park-Ramirez生物反应器以及生产外源蛋白的Lee-Ramirez生物反应器的补料流率优化问题。结果表明，CMACO在优化结果和计算代价上都有较好的性能。
复杂相平衡体系的Gibbs自由能函数存在多个局部解，应用局部优化算法易陷入局部解或者平凡解而难以得到全局解。本文采用CMACO直接最小化系统Gibbs自由能函数，无需考虑体系实际存在的相态，计算不依赖函数导数，能以较高概率收敛至全局解。
总之，论文对蚁群算法做了较为全面深入的分析和讨论，不仅提出了一种连续多蚁群算法，而且将其用于化工动态优化以及相平衡计算中。论文最后对所做工作进行了总结，并且对未来研究提出展望。
Since entering the 21st century, chemical industry is faced with pressures from economic, energy, environment and many other problems. Optimization technique to meet these challenges is an effective approach, which can be applied on any scales of the entire value chain in chemical industry. But chemical system is a typical complex system. As the scale of object problem becomes more and more large, of which the model structure becomes more and more complicated too, classical optimization method can't meet the demands of many practical problems. So the requirement for right efficient intelligent optimization methods has become increasingly urgent.
Ant colony optimization (ACO) is recently proposed as a class of intelligent optimization methods. Its predominant distributed pattern of problem solving achieves great success in combinational problems, and brings extensively attentions of related research area. But to many practical engineering problems, they are usually expressed as continuous optimization problems. So, it is an imperative challenge on how to apply the basic ant colony optimization strategy to the problems solving in continuous space, which is the major work of this thesis.
In nature, ant colony optimization is a unified searching algorithm framework based on the probability distribution model of solution space parameterization. The parameter is pheromone, and the solution set producing by ants can be considered as the samples for updating the probability distribution parameter. Hence, the model of pheromone distribution is the key factor in ant colony algorithm, which can thoroughly determine the behavior and distribution of ants. Therefore, the key issue for constructing high-performance continuous ant colony optimization is to design reasonable pheromone distribution model.
Analyzing the pheromone distribution in biological model of ant colony foraging, we use normal distribution to simulate pheromone distribution and
proposed a normal distribution of pheromone based continuous multi ant-colony algorithm, CMACO. Firstly, state transition of ant colony is implemented by stochastic sampling based on pheromone distribution function. Secondly, the distribution function is updated according to the quality of food source, by which the pheromone is updated. The iterative implementation can lead the ants to global optimal solution. Moreover, to improve optimization performance, multi-colonies strategy is introduced to balance the trade-off between global exploration and local exploitation ability based on group recruitment mechanism. Finally, CMACO was applied to several benchmark problems, and the results illustrate CMACO has well global optimization performance.
To solve dynamic optimization problems efficiently, control vector parameterization approach is introduced to transform the original dynamic optimization to static optimization problems, which is directly solved by CMACO. The efficiency of this method was illustrated with two challenging problems for optimizing feed-rate of fed-batch bioreactors, and the results show CMACO has well global optimization ability and fast convergence speed.
For complex phase equilibrium system, the Gibbs energy function has several local minima, so it's difficult to get the global minimum by the local optimization algorithms. CMACO was utilized to solve the phase equilibrium without reactions, and it need not considering the actual number and type of phases and needn't the derivative. The results show CMACO can find the global solution with high probability.
In short, ACO was adapted elaborately for continuous optimization in this work, and proposed a normal distribution based continuous multi ant-colony optimization, which was applied on dynamic optimization problems of chemical processes and complex phase equilibrium calculation.
蚁群算法是新近提出来的一种群智能优化方法。由于其优越的问题分布式求解模式，在离散优化问题的求解中取得了极大成功，引起了相关领域学者的广泛关注。但很多实际问题通常被表达成连续优化问题。如何有效地将全局优化性能优越但本质离散的蚁群算法用于优化连续空间的问题，此为亟待应对的挑战，这也是本文的主要研究内容。
蚁群算法在本质是一种基于解空间参数化的概率分布模型的搜索算法框架，这些参数就是信息素，而蚂蚁生成的解集合则可看作是用来更新概率分布参数的样本。因此信息素分布模型是影响蚁群算法最关键的因素，它决定了蚂蚁的行为与分布，设计一种好的信息素分布模型是构造高性能连续蚁群算法的关键。
基于此，本文通过对蚁群觅食的生物学模型中信息素分布的分析，用多元正态分布函数来模拟信息素的分布，提出了一种信息素呈多元正态分布的连续多蚁群算法(CMACO)。该算法通过对信息素分布函数的随机抽样来指导蚂蚁完成状态转移，信息素分布函数又随着蚂蚁的移动而被调整，实施信息素更新，进而引导蚂蚁在可行域中逐步向最优食物源聚集。为了提高算法的寻优性能，基于蚁群的成群募集机制，本文构建出多蚁群策略来有效地调配蚁群的行为以平衡其全局探索能力和局部挖掘能力。经多个经典函数的测试，表明CMACO适用于连续优化问题，具有良好的全局寻优性能。
对于终端时间给定、终端状态无约束的动态优化问题，本文通过控制变量参数化方法将其转换成静态优化问题，然后使用CMACO进行优化。按照该思路，将CMACO用于生产分泌蛋白的Park-Ramirez生物反应器以及生产外源蛋白的Lee-Ramirez生物反应器的补料流率优化问题。结果表明，CMACO在优化结果和计算代价上都有较好的性能。
复杂相平衡体系的Gibbs自由能函数存在多个局部解，应用局部优化算法易陷入局部解或者平凡解而难以得到全局解。本文采用CMACO直接最小化系统Gibbs自由能函数，无需考虑体系实际存在的相态，计算不依赖函数导数，能以较高概率收敛至全局解。
总之，论文对蚁群算法做了较为全面深入的分析和讨论，不仅提出了一种连续多蚁群算法，而且将其用于化工动态优化以及相平衡计算中。论文最后对所做工作进行了总结，并且对未来研究提出展望。
Since entering the 21st century, chemical industry is faced with pressures from economic, energy, environment and many other problems. Optimization technique to meet these challenges is an effective approach, which can be applied on any scales of the entire value chain in chemical industry. But chemical system is a typical complex system. As the scale of object problem becomes more and more large, of which the model structure becomes more and more complicated too, classical optimization method can't meet the demands of many practical problems. So the requirement for right efficient intelligent optimization methods has become increasingly urgent.
Ant colony optimization (ACO) is recently proposed as a class of intelligent optimization methods. Its predominant distributed pattern of problem solving achieves great success in combinational problems, and brings extensively attentions of related research area. But to many practical engineering problems, they are usually expressed as continuous optimization problems. So, it is an imperative challenge on how to apply the basic ant colony optimization strategy to the problems solving in continuous space, which is the major work of this thesis.
In nature, ant colony optimization is a unified searching algorithm framework based on the probability distribution model of solution space parameterization. The parameter is pheromone, and the solution set producing by ants can be considered as the samples for updating the probability distribution parameter. Hence, the model of pheromone distribution is the key factor in ant colony algorithm, which can thoroughly determine the behavior and distribution of ants. Therefore, the key issue for constructing high-performance continuous ant colony optimization is to design reasonable pheromone distribution model.
Analyzing the pheromone distribution in biological model of ant colony foraging, we use normal distribution to simulate pheromone distribution and
proposed a normal distribution of pheromone based continuous multi ant-colony algorithm, CMACO. Firstly, state transition of ant colony is implemented by stochastic sampling based on pheromone distribution function. Secondly, the distribution function is updated according to the quality of food source, by which the pheromone is updated. The iterative implementation can lead the ants to global optimal solution. Moreover, to improve optimization performance, multi-colonies strategy is introduced to balance the trade-off between global exploration and local exploitation ability based on group recruitment mechanism. Finally, CMACO was applied to several benchmark problems, and the results illustrate CMACO has well global optimization performance.
To solve dynamic optimization problems efficiently, control vector parameterization approach is introduced to transform the original dynamic optimization to static optimization problems, which is directly solved by CMACO. The efficiency of this method was illustrated with two challenging problems for optimizing feed-rate of fed-batch bioreactors, and the results show CMACO has well global optimization ability and fast convergence speed.
For complex phase equilibrium system, the Gibbs energy function has several local minima, so it's difficult to get the global minimum by the local optimization algorithms. CMACO was utilized to solve the phase equilibrium without reactions, and it need not considering the actual number and type of phases and needn't the derivative. The results show CMACO can find the global solution with high probability.
In short, ACO was adapted elaborately for continuous optimization in this work, and proposed a normal distribution based continuous multi ant-colony optimization, which was applied on dynamic optimization problems of chemical processes and complex phase equilibrium calculation.
引文
[1] 21世纪化学科学的挑战委员会著，陈尔强等译．超越分子前沿．北京：科学出版社，2004
[2] Darton R C，Prince R G H，Wood D G，陈国华译．国际大师看化学工程．北京：化学工业出版社，2006
[3] 杨友麒，项曙光．化工过程模拟与优化．北京：化学工业出版社，2006
[4] 姚平经．化工过程系统工程．大连：大连理工大学出版社，1992
[5] 王弘轼．化工过程系统工程．北京：清华大学出版社，2006
[6] 沈静珠．过程系统优化．北京：机械工业出版社，1990
[7] 胡上序，陈海．化工过程的建模、仿真和优化．杭州：浙江大学出版社，1997
[8] 邓正龙．化工中的优化方法．北京：化学工业出版社，1992
[9] Edgar T F, Himmelblau D H, Lasdon L S. Optimization of Chemical Process (Second Edition). McGraw-Hill, 2001
[10] 程志刚．连续蚁群优化算法的研究及其化工应用．博士学位论文．杭州：浙江大学，2005．8
[11] 何小荣．化工过程优化．北京：清华大学出版社，2002
[12] 唐焕文，秦学志．实用最优化方法．大连：大连理工大学出版社，2004
[13] 张丽萍．粒子群优化算法的理论及实践．博士学位论文．杭州：浙江大学，2005．6
[14] 袁亚湘，孙文瑜．最优化理论与方法．北京：科学出版社，1997
[15] 刑文训，谢金星．现代优化计算方法．北京：清华大学出版社，1999
[16] Dorigo M, Stützle T. Ant Colony Optimization. Cambridge MA: The MIT Press, 2004
[17] Glover F.Tabu search: a tutorial. Interfaces. 1990, 20(4): 74-94
[18] 王正志，薄涛．进化计算．北京：国防工业出版社，2000
[19] 王小平，曹立明．遗传算法——理论、应用与软件实现．西安：西安交通大学出版社，2002
[20] 潘正君，康立山，陈毓屏．演化计算．北京：清华大学出版社，1998
[21] Bonabeau E, Dorigo M, Theraulaz G. Swarm Intelligence: From Natural to Artificial System. Oxford University Press, 1999
[22] 吴启迪，汪镭．智能蚁群算法及应用．上海：上海科技教育出版社，2004
[23] Eberhart R C, Shi Y. Guest Editorial Special Issue on Particle Swarm Optimization. IEEE Transaction on evolutionary computation,. 2004, 8(3): 201-203
[24] Dorigo M, Bonabeau E, Theraulaz G. Ant algorithms and stigmergy. Future Generation Computer Systems. 2000, 16(5): 851-871
[25] Colorni A, Dorigo M, Maniezzo Ⅴ. Distributed optimization by ant colonies, In:Proc. Of 1st European Confon Artificial Life. Pans, France. 1991:134-142
[26] 张兵．化工动态优化方法的研究与应用．博士学位沦为．杭州：浙江大学，2004
[27] 张兵，俞欢军，陈德钊．序贯优化化工动态问题的蚁群算法．高校化学工程学报．2006，20(1)：120-125
[28] Rajesh J, Gupta K, Kusumakar H S, Jayaraman V K, Kulkarni B D. Dynamic optimization of chemical processes using ant colony framework. Computers & Chemistry. 2001, 25(6):583-595
[29] 蒲黎明，俞欢军，陈德钊．连续多蚁群算法的构建及其在过程动态优化中的应用．高校化学工程学报．2007：已录用
[30] Jayaraman V K, Kulkami B D, Karale S, Shelokar P. Ant colony framework for optimal design and scheduling of batch plants. Computers & Chemical Engineering. 2000, 24(8):1901-1912
[31] Mathur M, Karale S, Priye S, Jayaraman V K, Kulkarni B D. Ant colony approach to continuous function optimization Ind. Eng. Chem. Res. 2000, 39 (10): 3814-3822
[32] Jayaraman V K, Kulkarni B D, Gupta K, Rajesh J, Kusumaker H S. Dynamic optimization of fed-batch bioreactors using the ant algorithm. Biotechnol Prog. 2001, 17(1): 81-88
[33] 贺益君，陈德钊，吴晓华．杂交蚁群系统的构建并用于反应动力学参数的估计．化工学报．2005，56(3)：487-491
[34] 贺益君，俞欢军，陈德钊．基于募集机制的连续蚁群系统及其应用．浙江大学学报(工学版)．2006，40(5)：748-752
[35] 贺益君，陈德钊．连续约束蚁群优化算法的构建及其在丁烯烷化过程中的应用．化工学报．2005，56(9)：1708-1713
[36] 贺益君，陈德钊．用于多目标优化的蚁群算法的构建及其应用．高技术通讯 2006，16(12)：1241-1245
[37] 程志刚，陈德钊，吴晓华．连续蚁群优化算法的研究．浙江大学学报(工学版)．2005，39(8)：1147-1151
[38] 程志刚，陈德钊，吴晓华，张兵．进化规划-蚁群优化算法的构建并用于化工过程操作优化．化工学报．2005，5 6(1 2)：2361-2366
[39] 程志刚，陈德钊，吴晓华．基于信息素正态分布的连续蚁群优化系统．系统工程与电子技术．2006，39(8)：458-462
[40] Shen Q, Jiang J-H, Tao J-C. Modified ant colony optimization algorithm for variable selection in QSAR Modeling: QSAR studies of cyclooxygenase inhibitors. Journal of Chemical Information and Modeling. 2005, 45(4): 1024-1029
[41] Xiao J, Zhou Z-K, Zhang G-X. Ant colony system algorithm for the optimization of beer fermentation control. Journal of Zhejiang University SCIENCE. 2004, 5(12): 1597-1603
[42] 刘业奎，万华，王黎，冯霄．催化剂计算机辅助设计的进展及蚁群算法．石油化工．2003，32(12)：1091-1094
[43] Papadopoulos a I, Linke P. On the synthesis and optimization of liquid-liquid extraction processes using stochastic search methods. Computers & Chemical Engineering. 2004, 28(11): 2394-2406
[44] Gambardella L M, Dorigo M. Ant-Q: A reinforcement learning approach to the traveling salesman problem. In: Proc of the 12th International Conf on Machine Learning, ML-95, Palo Alto. 1995:252-260
[45] Dorigo M, Gambardella L M. Ant colony system: A cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation. 1997, 1(1):53-66
[46] Dorigo M, Gambardella L M. Ant colonies for the traveling salesman problem. BioSystem. 1997, 43(2): 73-81
[47] Stützle T, Hoos H. MAX-MIN ant system and local search for the traveling salesman problem. In: Proceedings of the IEEE International Conference on Evolutionary Computation. Indianapolis:Indiana. 1997:309-314
[48] Stützle T, Hoos H. MAX-MIN ant system Future Generation Computer Systems. 2000, 16(8): 889-914
[49] Dorigo M, Di-Caro G. The Ant Colony Optimization Meta-heuristic. New Ideas in Optimization. UK: McGraw-Hill, 1999, 11-32
[50] Bonabeau E, Dorigo M, Theraulaz G. Inspiration for optimization from social insect behaviour. Nature. 2000, 406(6): 39-42
[51] Michael J B K, Jean-Bernard B, Laurent K. Ant-like task and recruitment in cooperative robots. Nature. 2000, 406(31): 992-995
[52] Dorigo M, G Di-Caro, Gambardella L. Ant algorithms for discrete optimization. Artificial Life. 1999, 5(3): 137-172
[53] 段海滨．蚁群算法原理及其应用．北京：科学出版社，2005
[54] Goss S, Aron S, Deneubourg J, Pasteels J. Self-organized shortcuts in the Argentine ant Naturwissensehaften. 1989, 76(12): 579-581
[55] 张纪会，徐心和．一种新的进化算法--蚁群算法．系统工程理论与实践．1999，19(3)：
[56] Colorni A, Dorigo M, Maniezzo Ⅴ. Ant system for Job-shop Scheduling. JORBEL-Belgian Journal of Operations Research, Statistics and Computer Science. 1994, 34(1): 39-53
[57] Dodgo M. Optimization, Learning and Natural Algorithms. Ph.D.Thesis. Italy: Politecnico di Milano, 1992
[58] Dorigo M, Maniezoo V, Colorni A. The Ant System: An autocatalytic optimization process. Technical Report 91-016, Dept. of Electronics, Politecnico di Milano, Italy. 1991:
[59] Dorigo M, Gambardella L. A study of some properties of Ant-Q In the Proceedings of PPSN Ⅳ-Fourth International Conference on Parallel Problem Solving From Nature. 1996:656-665
[60] 段海滨，王道波，于秀芬．蚁群算法的研究进展评述．自然杂志．2006，28(2)：102-105
[61] Dorigo M, Maniezzo V, Colorni A. Ant system: optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics-Part B. 1996, 26(1): 29-41
[62] Zlochin M, Dorigo M. Model-based search for combinatorial optimization: A comparative study In the Proceedings of PPSN Ⅳ-Six International Conference on Parallel Problem Solving From Nature. 2002:
[63] Zioehin M, Birattar M, Meuleau N, Dorigo M. Model-based search for combinatorial optimization. Technical Report TR/IRIDIA/2001-15, IRIDIA, Université Libre de Bruxelles. 2001:
[64] Gutjahr W J. A graph-based ant system andits convergence. Future Generation Computer Systems. 2000, 16(8): 873-888
[65] Stützle T, Dorigo M. A short convergence proof for a class of ant colony optimization Algorithms. IEEE Transactions on Evolutionary Computation. 2002, 6(4): 358-365
[66] Gutjahr W J. ACO algorithms with guaranteed convergence to optimal solution. Information Processing Letters. 2002, 82(3): 145-153
[67] 孙焘，王秀坤，刘业欣，张名举．一种简单蚂蚁算法及其收敛性分析．小型微型计算机系统．2003，24(8)：1524-1527
[68] 侯云鹤，熊信艮，吴耀武，鲁丽娟．基于广义蚁群算法的电力系统经济负荷分配．中国电机工程学报．2003，23(3)：59-64
[69] 段海滨，于道波，于秀芬．基本蚁群算法的AS收敛性研究．应用基础与工程科学学报． 2006, 14(2): 297-301
[70] Quinlan J. Combining instance-based and model-based learning. San Mateo: In Proceedings of the Tenth International Conference on Machine Learning. 1993:236-243
[71] Bertsekas D P. Nonlinear Programming. Belmont, MA: Athena Scientic, 1995
[72] Rubinstein R Y. The cross-entropy method for combinatorial and continuous optimization. Methodology and Computing in Applied probability. 1999, 1(2): 127-190
[73] 王笑蓉．蚁群优化的理论模型及在生产调度中的应用研究．博士学位论文．杭州：浙江大学，2003
[74] Muhlenbein H, Bendisch J, Voigt H M. From recombination of genes to the estimation of distributions of binary Parameters. In Proceedings of the 4th International Conference on Parallel Problem Solving from Nature. 1996:178-187
[75] Eberhart R C, Kennedy J F, Yuhui S. Swarm Intelligence. Morgan Kaufmann, 2001
[76] Bilchev G, Parmee I C. The ant colony metaphor for searching continuous design spaces. Lecture Notes in Computer Science. 1995, 993:25-39
[77] Wodrich M, Bilchev G. Cooperative distributed search: The ant's way. Control and Cybernetics. 1997, 26(3): 413-446
[78] 陈岐，沈洁，秦玲．蚁群算法求解连续空间优化问题的一种方法 软件学报．2002，13(12)：2317-2323
[79] 高尚，莫述军，钟娟．连续优化问题的蚁群算法研究．微机发展．2003，13(1)：21-22，69
[80] 段海滨，马冠军，王道波，于秀芬．一种求解连续空间优化问题的改进蚁群算法．系统仿真学报．2007，19(5)：974-977
[81] Dréo J, Siarry P. Continuous interacting ant colony algorithm based on dense heterarchy. Future Generation Computer Systems. 2004, 20(5): 841-856
[82] Dréo J, Siarry P. A new ant colony algorithm using the heterarchical concept aimed at optimization of multiminima continuous functions. Dorigoet al(Eds): ANTS 2002, LNCS 2463.216-221
[83] Monmarché N, Venturini G, Slimane M. On how the ants Pachycondyla apicalis suggesting a new search algorithm. Future Generation Computer Systems. 2000:937-946
[84] Fresneau D. Individual foraging and path fidelity in a ponerine ant. Paris 1985, 32(2):109-116
[85] Socha K. ACO for continuous and mixed-vaxiable optimization. Dorigoet al(Eds): ANTS2004, LNCS 3172.25-36
[86] Socha K, Dorigo M. Ant colony optimization for continuous domains. European Journal of Operational Research. 2006, doi: 10. 1016/j.ejor.2006.06.046
[87] 陈德钊．多元数据处理．北京：化学工业出版社，1998
[88] Luus R. Sensitivity of control policy on yield of a fed-batch reactor, IASTED International Conference on Modelling and Simulation, Pittsburgh, PA, 1995.
[89] Luus R. On the application of iterative dynamic programming to singular optimal control problems. IEEE Transactions on Automatic Control. 1992, 37(11): 1802-1806
[90] Luus R, Hennessy D. Optimization of fed-batch reactors by the Luus-Jaakola optimization procedure. Ind Eng Chem Res. 1998, 38:1945-1955
[91] Luus R. Iterative Dynamic Programming Chapman & Hall/CRC 2000
[92] Zhangbing, Chendezhao, Zhaoweixiang. Iterative ant-colony algorithm and its application to dynamic optimization of chemical process. Computers and Chemical Engineering. 2005, 29(2078-2086):
[93] Balsa-Canto E, Banga J R. Efficient optimal control of bioprocesses using second-order information. Ind Eng Chem Res. 2000, 39(11): 4287-4295
[94] Banga J R, Balsa-Canto E, Moles C G Dynamic optimisation of bioreactors: a review. Proceedings of the Indian Academy of Sciences. 2003, 69A (3-4): 1-21
[95] Park S, Ramirez W F. Optimal production of secreted protein in fed-batch reactors. AIChE Journal. 1988,34(9): 1550-1558
[96] Sarkar D, Modak J M. Optimisation of fed-batch bioreactors using genetics algorithmsChemical Engineering Science. 2003, 58: 2283-2296
[97] Irizarry R. A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm: Applications to paniculate processes and discrete dynamic systems. Chemical Engineering Science. 2005,60(21): 5663-5681
[98] Tholuder A, Ramirez W F. Obtaining smoother singular arc policies using a modified iterative dynamic programming algorithm. International Journal of Control. 1997, 68(5): 1115-1128
[99] Banga J R, Irrizarry R, Seider W D. Stochastic optimization for optimal and model-predictive control. Computers chem Engng. 1998,22(4/5): 603-612
[100]Balsa-Canto E, Banga J R, Alonso a A, Vassiliadis V S. Dynamic optimization of chemical and biochemical processes using restricted second-order information Computers & Chemical Engineering. 2001,25(539-546):
[101]Sarkar D, Modak J M. ANNSA: A hybrid artificial neural network/simulated annealing algorithm for optimal control problems. Chemical Engineering Science. 2003, 58(14): 3131-3142
[102]Lee J, Ramirez W F. Optimal fed-batch control of induced foreign protein produced by recombinant bacteria. AIChE Journal. 1994, 40(5): 899-907
[103]Carrsco E F, Banga J R. Dynamic optimisation of batch reactors using adaptive stochastic algorithms. Industrial and Engineering Chemistry Research. 1997, 36(6): 2252-2261
[104]Roubos J A, van Straten G, van Boxtel AJB. An evolutionary strategy for fed-batch bioreactor optimization; concepts and performance. Journal of Biotechnology. 1999, 67(2/3): 173-187
[105]Mekarapiruk W, Luus R. Optimal control by iterative dynamic programming with deterministic and random candidates for control Ind. Eng. Chem. Res. 2000, 39: 84-91
[106]Seader J D, Henley E J. Separation Process Principles. New York: Wiley & Sons, Inc, 1998
[107]Seider W D, Widagdo S. Multiphase equilibria of reactive systems. Fluid Phase Equilibria. 1996, 123:283-303
[108]Michelsen M L. The isothermal flash problem. Part 1. Stability. Fluid Phase Equilibria. 1982, 9: 1-19
[109]Michelsen M L. The isothermal flash problem: Part 2. Phase-Split Calculation. Fluid Phase Equilibria. 1982,9:21-35
[110]Michelsen M L. Phase Equilibrium calculation: what is easy and what is difficult? Comput. Chem. Eng.. 1993, 17(5/6): 431-439
[111]White W B, Johnson S M, Dantzig G B. Chemical equilibrium in complex mixtures. J. Chem.Phys. 1958,28(5): 751
[112]Seider W D, Gautam R. Computation of phase and chemical equilibrium- Part I: Local and constrained minima in Gibbs free energy. Journal of AICHE. 1979, 25(6): 911
[113]Seider W D, Gautam R. Computation of phase and chemical equilibrium-Part II: Phase splitting. Journal of AICHE. 1979, 25(6): 999
[114]Seider W D, Gautam R. Computation of phase and chemical equilibrium- Part III:Electrolytic solutions. Journal of AICHE. 1979, 25(6): 1006
[115]Castillo J. Grossmann I E. Computation of phase and chemical equilibria. Computers and Chemical Engineering. 1981, (5): 99-108
[116]Lantagne G, Marcos B, Cayrol B. Computation of complex equilibria by nonlinear optimization. Comput. Chem. Eng.. 1988,12(6): 589
[117]Flouda C A, Aggarwal A, Ciric A R. A global optimum search for nonconvex NLP and MINLP problems. Comput. Chem. Engng. 1989, 13(10): 1117
[118]Paules G E. Flouda C A. A new optimization approach for phase and chemical equilibrium problems. In the Annual AIChE Meeting. San Franciso, CA1989
[119]Mcdonald C M, Flouda C A. Global optimization and analysis for the Gibbs free energy function using the UNIFAC, Wilson, and ASOG equations. Industrial Engineering and Chemical Research. 1995,34: 1674-1687
[120]Mcdonald C M, Flouda C A. Global optimization for the phase and chemical equilibrium problem: application to the NRTL equation. Computers and Chemical Engineering. 1995, 19(11): 1111-1139
[121]Mcdonald C M, Flouda C A. Global optimization for the phase stability problem. Journal of AICHE. 1995,41(7): 1798-1814
[122]Rangaiah G P. Evaluation of genetic algorithms and simulated annealingfor phaseequilibrium and stability problems. Fluid Phase Equilibria. 2001, 187-188 & 83-109
[123]陈新志,蔡振云,胡望明。化工热力学。北京：化学工业出版社,2001
[124]Thompson M O, Ohanomah D W. Computation of multicomponent phase equilibria: Part I Vapour-liquid equilibria. Comput. Chem. Engng. 1984, 8(3): 147
[125]Thompson M O, Ohanomah D W. Computation of multicomponent phase equilibria -PartII: Liquid-liquid and solid-liquid equilibria. Comput. Chem. Engng. 1984, 8(3): 157
[126]Thompson M O, Ohanomah D W. Computation of multicomponent phase equilibria - PartIII: Multiphase equilibria. Comput. Chem. Engng. 1984, 8(3): 163
[127]Reneaume J M, Meyer M, Letourneau J J, Joulia X. A global MINLP approach for phase equilibrium calculations. Comput. Chem. Eng.. 1996, 20: 303-308
[128]Reid R C, Prausnitz J M. The properties of gases and liquids. New York: McGRAW-HILL, 1987
[2] Darton R C，Prince R G H，Wood D G，陈国华译．国际大师看化学工程．北京：化学工业出版社，2006
[3] 杨友麒，项曙光．化工过程模拟与优化．北京：化学工业出版社，2006
[4] 姚平经．化工过程系统工程．大连：大连理工大学出版社，1992
[5] 王弘轼．化工过程系统工程．北京：清华大学出版社，2006
[6] 沈静珠．过程系统优化．北京：机械工业出版社，1990
[7] 胡上序，陈海．化工过程的建模、仿真和优化．杭州：浙江大学出版社，1997
[8] 邓正龙．化工中的优化方法．北京：化学工业出版社，1992
[9] Edgar T F, Himmelblau D H, Lasdon L S. Optimization of Chemical Process (Second Edition). McGraw-Hill, 2001
[10] 程志刚．连续蚁群优化算法的研究及其化工应用．博士学位论文．杭州：浙江大学，2005．8
[11] 何小荣．化工过程优化．北京：清华大学出版社，2002
[12] 唐焕文，秦学志．实用最优化方法．大连：大连理工大学出版社，2004
[13] 张丽萍．粒子群优化算法的理论及实践．博士学位论文．杭州：浙江大学，2005．6
[14] 袁亚湘，孙文瑜．最优化理论与方法．北京：科学出版社，1997
[15] 刑文训，谢金星．现代优化计算方法．北京：清华大学出版社，1999
[16] Dorigo M, Stützle T. Ant Colony Optimization. Cambridge MA: The MIT Press, 2004
[17] Glover F.Tabu search: a tutorial. Interfaces. 1990, 20(4): 74-94
[18] 王正志，薄涛．进化计算．北京：国防工业出版社，2000
[19] 王小平，曹立明．遗传算法——理论、应用与软件实现．西安：西安交通大学出版社，2002
[20] 潘正君，康立山，陈毓屏．演化计算．北京：清华大学出版社，1998
[21] Bonabeau E, Dorigo M, Theraulaz G. Swarm Intelligence: From Natural to Artificial System. Oxford University Press, 1999
[22] 吴启迪，汪镭．智能蚁群算法及应用．上海：上海科技教育出版社，2004
[23] Eberhart R C, Shi Y. Guest Editorial Special Issue on Particle Swarm Optimization. IEEE Transaction on evolutionary computation,. 2004, 8(3): 201-203
[24] Dorigo M, Bonabeau E, Theraulaz G. Ant algorithms and stigmergy. Future Generation Computer Systems. 2000, 16(5): 851-871
[25] Colorni A, Dorigo M, Maniezzo Ⅴ. Distributed optimization by ant colonies, In:Proc. Of 1st European Confon Artificial Life. Pans, France. 1991:134-142
[26] 张兵．化工动态优化方法的研究与应用．博士学位沦为．杭州：浙江大学，2004
[27] 张兵，俞欢军，陈德钊．序贯优化化工动态问题的蚁群算法．高校化学工程学报．2006，20(1)：120-125
[28] Rajesh J, Gupta K, Kusumakar H S, Jayaraman V K, Kulkarni B D. Dynamic optimization of chemical processes using ant colony framework. Computers & Chemistry. 2001, 25(6):583-595
[29] 蒲黎明，俞欢军，陈德钊．连续多蚁群算法的构建及其在过程动态优化中的应用．高校化学工程学报．2007：已录用
[30] Jayaraman V K, Kulkami B D, Karale S, Shelokar P. Ant colony framework for optimal design and scheduling of batch plants. Computers & Chemical Engineering. 2000, 24(8):1901-1912
[31] Mathur M, Karale S, Priye S, Jayaraman V K, Kulkarni B D. Ant colony approach to continuous function optimization Ind. Eng. Chem. Res. 2000, 39 (10): 3814-3822
[32] Jayaraman V K, Kulkarni B D, Gupta K, Rajesh J, Kusumaker H S. Dynamic optimization of fed-batch bioreactors using the ant algorithm. Biotechnol Prog. 2001, 17(1): 81-88
[33] 贺益君，陈德钊，吴晓华．杂交蚁群系统的构建并用于反应动力学参数的估计．化工学报．2005，56(3)：487-491
[34] 贺益君，俞欢军，陈德钊．基于募集机制的连续蚁群系统及其应用．浙江大学学报(工学版)．2006，40(5)：748-752
[35] 贺益君，陈德钊．连续约束蚁群优化算法的构建及其在丁烯烷化过程中的应用．化工学报．2005，56(9)：1708-1713
[36] 贺益君，陈德钊．用于多目标优化的蚁群算法的构建及其应用．高技术通讯 2006，16(12)：1241-1245
[37] 程志刚，陈德钊，吴晓华．连续蚁群优化算法的研究．浙江大学学报(工学版)．2005，39(8)：1147-1151
[38] 程志刚，陈德钊，吴晓华，张兵．进化规划-蚁群优化算法的构建并用于化工过程操作优化．化工学报．2005，5 6(1 2)：2361-2366
[39] 程志刚，陈德钊，吴晓华．基于信息素正态分布的连续蚁群优化系统．系统工程与电子技术．2006，39(8)：458-462
[40] Shen Q, Jiang J-H, Tao J-C. Modified ant colony optimization algorithm for variable selection in QSAR Modeling: QSAR studies of cyclooxygenase inhibitors. Journal of Chemical Information and Modeling. 2005, 45(4): 1024-1029
[41] Xiao J, Zhou Z-K, Zhang G-X. Ant colony system algorithm for the optimization of beer fermentation control. Journal of Zhejiang University SCIENCE. 2004, 5(12): 1597-1603
[42] 刘业奎，万华，王黎，冯霄．催化剂计算机辅助设计的进展及蚁群算法．石油化工．2003，32(12)：1091-1094
[43] Papadopoulos a I, Linke P. On the synthesis and optimization of liquid-liquid extraction processes using stochastic search methods. Computers & Chemical Engineering. 2004, 28(11): 2394-2406
[44] Gambardella L M, Dorigo M. Ant-Q: A reinforcement learning approach to the traveling salesman problem. In: Proc of the 12th International Conf on Machine Learning, ML-95, Palo Alto. 1995:252-260
[45] Dorigo M, Gambardella L M. Ant colony system: A cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation. 1997, 1(1):53-66
[46] Dorigo M, Gambardella L M. Ant colonies for the traveling salesman problem. BioSystem. 1997, 43(2): 73-81
[47] Stützle T, Hoos H. MAX-MIN ant system and local search for the traveling salesman problem. In: Proceedings of the IEEE International Conference on Evolutionary Computation. Indianapolis:Indiana. 1997:309-314
[48] Stützle T, Hoos H. MAX-MIN ant system Future Generation Computer Systems. 2000, 16(8): 889-914
[49] Dorigo M, Di-Caro G. The Ant Colony Optimization Meta-heuristic. New Ideas in Optimization. UK: McGraw-Hill, 1999, 11-32
[50] Bonabeau E, Dorigo M, Theraulaz G. Inspiration for optimization from social insect behaviour. Nature. 2000, 406(6): 39-42
[51] Michael J B K, Jean-Bernard B, Laurent K. Ant-like task and recruitment in cooperative robots. Nature. 2000, 406(31): 992-995
[52] Dorigo M, G Di-Caro, Gambardella L. Ant algorithms for discrete optimization. Artificial Life. 1999, 5(3): 137-172
[53] 段海滨．蚁群算法原理及其应用．北京：科学出版社，2005
[54] Goss S, Aron S, Deneubourg J, Pasteels J. Self-organized shortcuts in the Argentine ant Naturwissensehaften. 1989, 76(12): 579-581
[55] 张纪会，徐心和．一种新的进化算法--蚁群算法．系统工程理论与实践．1999，19(3)：
[56] Colorni A, Dorigo M, Maniezzo Ⅴ. Ant system for Job-shop Scheduling. JORBEL-Belgian Journal of Operations Research, Statistics and Computer Science. 1994, 34(1): 39-53
[57] Dodgo M. Optimization, Learning and Natural Algorithms. Ph.D.Thesis. Italy: Politecnico di Milano, 1992
[58] Dorigo M, Maniezoo V, Colorni A. The Ant System: An autocatalytic optimization process. Technical Report 91-016, Dept. of Electronics, Politecnico di Milano, Italy. 1991:
[59] Dorigo M, Gambardella L. A study of some properties of Ant-Q In the Proceedings of PPSN Ⅳ-Fourth International Conference on Parallel Problem Solving From Nature. 1996:656-665
[60] 段海滨，王道波，于秀芬．蚁群算法的研究进展评述．自然杂志．2006，28(2)：102-105
[61] Dorigo M, Maniezzo V, Colorni A. Ant system: optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics-Part B. 1996, 26(1): 29-41
[62] Zlochin M, Dorigo M. Model-based search for combinatorial optimization: A comparative study In the Proceedings of PPSN Ⅳ-Six International Conference on Parallel Problem Solving From Nature. 2002:
[63] Zioehin M, Birattar M, Meuleau N, Dorigo M. Model-based search for combinatorial optimization. Technical Report TR/IRIDIA/2001-15, IRIDIA, Université Libre de Bruxelles. 2001:
[64] Gutjahr W J. A graph-based ant system andits convergence. Future Generation Computer Systems. 2000, 16(8): 873-888
[65] Stützle T, Dorigo M. A short convergence proof for a class of ant colony optimization Algorithms. IEEE Transactions on Evolutionary Computation. 2002, 6(4): 358-365
[66] Gutjahr W J. ACO algorithms with guaranteed convergence to optimal solution. Information Processing Letters. 2002, 82(3): 145-153
[67] 孙焘，王秀坤，刘业欣，张名举．一种简单蚂蚁算法及其收敛性分析．小型微型计算机系统．2003，24(8)：1524-1527
[68] 侯云鹤，熊信艮，吴耀武，鲁丽娟．基于广义蚁群算法的电力系统经济负荷分配．中国电机工程学报．2003，23(3)：59-64
[69] 段海滨，于道波，于秀芬．基本蚁群算法的AS收敛性研究．应用基础与工程科学学报． 2006, 14(2): 297-301
[70] Quinlan J. Combining instance-based and model-based learning. San Mateo: In Proceedings of the Tenth International Conference on Machine Learning. 1993:236-243
[71] Bertsekas D P. Nonlinear Programming. Belmont, MA: Athena Scientic, 1995
[72] Rubinstein R Y. The cross-entropy method for combinatorial and continuous optimization. Methodology and Computing in Applied probability. 1999, 1(2): 127-190
[73] 王笑蓉．蚁群优化的理论模型及在生产调度中的应用研究．博士学位论文．杭州：浙江大学，2003
[74] Muhlenbein H, Bendisch J, Voigt H M. From recombination of genes to the estimation of distributions of binary Parameters. In Proceedings of the 4th International Conference on Parallel Problem Solving from Nature. 1996:178-187
[75] Eberhart R C, Kennedy J F, Yuhui S. Swarm Intelligence. Morgan Kaufmann, 2001
[76] Bilchev G, Parmee I C. The ant colony metaphor for searching continuous design spaces. Lecture Notes in Computer Science. 1995, 993:25-39
[77] Wodrich M, Bilchev G. Cooperative distributed search: The ant's way. Control and Cybernetics. 1997, 26(3): 413-446
[78] 陈岐，沈洁，秦玲．蚁群算法求解连续空间优化问题的一种方法 软件学报．2002，13(12)：2317-2323
[79] 高尚，莫述军，钟娟．连续优化问题的蚁群算法研究．微机发展．2003，13(1)：21-22，69
[80] 段海滨，马冠军，王道波，于秀芬．一种求解连续空间优化问题的改进蚁群算法．系统仿真学报．2007，19(5)：974-977
[81] Dréo J, Siarry P. Continuous interacting ant colony algorithm based on dense heterarchy. Future Generation Computer Systems. 2004, 20(5): 841-856
[82] Dréo J, Siarry P. A new ant colony algorithm using the heterarchical concept aimed at optimization of multiminima continuous functions. Dorigoet al(Eds): ANTS 2002, LNCS 2463.216-221
[83] Monmarché N, Venturini G, Slimane M. On how the ants Pachycondyla apicalis suggesting a new search algorithm. Future Generation Computer Systems. 2000:937-946
[84] Fresneau D. Individual foraging and path fidelity in a ponerine ant. Paris 1985, 32(2):109-116
[85] Socha K. ACO for continuous and mixed-vaxiable optimization. Dorigoet al(Eds): ANTS2004, LNCS 3172.25-36
[86] Socha K, Dorigo M. Ant colony optimization for continuous domains. European Journal of Operational Research. 2006, doi: 10. 1016/j.ejor.2006.06.046
[87] 陈德钊．多元数据处理．北京：化学工业出版社，1998
[88] Luus R. Sensitivity of control policy on yield of a fed-batch reactor, IASTED International Conference on Modelling and Simulation, Pittsburgh, PA, 1995.
[89] Luus R. On the application of iterative dynamic programming to singular optimal control problems. IEEE Transactions on Automatic Control. 1992, 37(11): 1802-1806
[90] Luus R, Hennessy D. Optimization of fed-batch reactors by the Luus-Jaakola optimization procedure. Ind Eng Chem Res. 1998, 38:1945-1955
[91] Luus R. Iterative Dynamic Programming Chapman & Hall/CRC 2000
[92] Zhangbing, Chendezhao, Zhaoweixiang. Iterative ant-colony algorithm and its application to dynamic optimization of chemical process. Computers and Chemical Engineering. 2005, 29(2078-2086):
[93] Balsa-Canto E, Banga J R. Efficient optimal control of bioprocesses using second-order information. Ind Eng Chem Res. 2000, 39(11): 4287-4295
[94] Banga J R, Balsa-Canto E, Moles C G Dynamic optimisation of bioreactors: a review. Proceedings of the Indian Academy of Sciences. 2003, 69A (3-4): 1-21
[95] Park S, Ramirez W F. Optimal production of secreted protein in fed-batch reactors. AIChE Journal. 1988,34(9): 1550-1558
[96] Sarkar D, Modak J M. Optimisation of fed-batch bioreactors using genetics algorithmsChemical Engineering Science. 2003, 58: 2283-2296
[97] Irizarry R. A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm: Applications to paniculate processes and discrete dynamic systems. Chemical Engineering Science. 2005,60(21): 5663-5681
[98] Tholuder A, Ramirez W F. Obtaining smoother singular arc policies using a modified iterative dynamic programming algorithm. International Journal of Control. 1997, 68(5): 1115-1128
[99] Banga J R, Irrizarry R, Seider W D. Stochastic optimization for optimal and model-predictive control. Computers chem Engng. 1998,22(4/5): 603-612
[100]Balsa-Canto E, Banga J R, Alonso a A, Vassiliadis V S. Dynamic optimization of chemical and biochemical processes using restricted second-order information Computers & Chemical Engineering. 2001,25(539-546):
[101]Sarkar D, Modak J M. ANNSA: A hybrid artificial neural network/simulated annealing algorithm for optimal control problems. Chemical Engineering Science. 2003, 58(14): 3131-3142
[102]Lee J, Ramirez W F. Optimal fed-batch control of induced foreign protein produced by recombinant bacteria. AIChE Journal. 1994, 40(5): 899-907
[103]Carrsco E F, Banga J R. Dynamic optimisation of batch reactors using adaptive stochastic algorithms. Industrial and Engineering Chemistry Research. 1997, 36(6): 2252-2261
[104]Roubos J A, van Straten G, van Boxtel AJB. An evolutionary strategy for fed-batch bioreactor optimization; concepts and performance. Journal of Biotechnology. 1999, 67(2/3): 173-187
[105]Mekarapiruk W, Luus R. Optimal control by iterative dynamic programming with deterministic and random candidates for control Ind. Eng. Chem. Res. 2000, 39: 84-91
[106]Seader J D, Henley E J. Separation Process Principles. New York: Wiley & Sons, Inc, 1998
[107]Seider W D, Widagdo S. Multiphase equilibria of reactive systems. Fluid Phase Equilibria. 1996, 123:283-303
[108]Michelsen M L. The isothermal flash problem. Part 1. Stability. Fluid Phase Equilibria. 1982, 9: 1-19
[109]Michelsen M L. The isothermal flash problem: Part 2. Phase-Split Calculation. Fluid Phase Equilibria. 1982,9:21-35
[110]Michelsen M L. Phase Equilibrium calculation: what is easy and what is difficult? Comput. Chem. Eng.. 1993, 17(5/6): 431-439
[111]White W B, Johnson S M, Dantzig G B. Chemical equilibrium in complex mixtures. J. Chem.Phys. 1958,28(5): 751
[112]Seider W D, Gautam R. Computation of phase and chemical equilibrium- Part I: Local and constrained minima in Gibbs free energy. Journal of AICHE. 1979, 25(6): 911
[113]Seider W D, Gautam R. Computation of phase and chemical equilibrium-Part II: Phase splitting. Journal of AICHE. 1979, 25(6): 999
[114]Seider W D, Gautam R. Computation of phase and chemical equilibrium- Part III:Electrolytic solutions. Journal of AICHE. 1979, 25(6): 1006
[115]Castillo J. Grossmann I E. Computation of phase and chemical equilibria. Computers and Chemical Engineering. 1981, (5): 99-108
[116]Lantagne G, Marcos B, Cayrol B. Computation of complex equilibria by nonlinear optimization. Comput. Chem. Eng.. 1988,12(6): 589
[117]Flouda C A, Aggarwal A, Ciric A R. A global optimum search for nonconvex NLP and MINLP problems. Comput. Chem. Engng. 1989, 13(10): 1117
[118]Paules G E. Flouda C A. A new optimization approach for phase and chemical equilibrium problems. In the Annual AIChE Meeting. San Franciso, CA1989
[119]Mcdonald C M, Flouda C A. Global optimization and analysis for the Gibbs free energy function using the UNIFAC, Wilson, and ASOG equations. Industrial Engineering and Chemical Research. 1995,34: 1674-1687
[120]Mcdonald C M, Flouda C A. Global optimization for the phase and chemical equilibrium problem: application to the NRTL equation. Computers and Chemical Engineering. 1995, 19(11): 1111-1139
[121]Mcdonald C M, Flouda C A. Global optimization for the phase stability problem. Journal of AICHE. 1995,41(7): 1798-1814
[122]Rangaiah G P. Evaluation of genetic algorithms and simulated annealingfor phaseequilibrium and stability problems. Fluid Phase Equilibria. 2001, 187-188 & 83-109
[123]陈新志,蔡振云,胡望明。化工热力学。北京：化学工业出版社,2001
[124]Thompson M O, Ohanomah D W. Computation of multicomponent phase equilibria: Part I Vapour-liquid equilibria. Comput. Chem. Engng. 1984, 8(3): 147
[125]Thompson M O, Ohanomah D W. Computation of multicomponent phase equilibria -PartII: Liquid-liquid and solid-liquid equilibria. Comput. Chem. Engng. 1984, 8(3): 157
[126]Thompson M O, Ohanomah D W. Computation of multicomponent phase equilibria - PartIII: Multiphase equilibria. Comput. Chem. Engng. 1984, 8(3): 163
[127]Reneaume J M, Meyer M, Letourneau J J, Joulia X. A global MINLP approach for phase equilibrium calculations. Comput. Chem. Eng.. 1996, 20: 303-308
[128]Reid R C, Prausnitz J M. The properties of gases and liquids. New York: McGRAW-HILL, 1987