分布估计算法及其在生产调度问题中的应用研究
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摘要
分布估计算法是进化计算领域新兴起的一类概率分析进化算法,它结合了智能计算和统计学习的知识,根据当前种群中若干较好个体的信息建立概率分布模型描述问题解空间的分布,并通过对概率模型随机采样产生新的种群,如此反复进行,实现种群的进化。分布估计算法通过建立概率模型描述变量之间的相关关系,能更有效地混合构造块并实现构造块重组,可以解决传统遗传算法难以解决的问题,尤其是解决非线性、高维复杂问题。分布估计算法求解问题的关键是建立一个能恰当描述问题解分布的概率模型,但是,概率模型的建立是一个非常复杂的问题,如果所建立的模型过于简单则不能正确反映问题的本质特征,影响求解效率,而模型过于复杂则会使算法学习复杂度增大。尤其对于复杂的组合优化问题,由于这类问题本身的复杂性,可行解中各变量间具有很强的相关性,建立一个能准确描述问题可行解的概率分布模型非常困难,在将分布估计算法用于求解复杂的组合优化问题时,如何建立一个能准确描述问题解分布的概率模型成为制约算法应用的瓶颈。
文中结合优良模式连接的思想、Bayesian统计推断理论与离散quasi-copula理论,对概率模型的建立方式进行改进,以提高算法求解复杂组合优化问题的能力,并将算法用于求解生产调度问题和旅行商问题,主要完成了以下工作:
1.根据遗传算法中的模式定理与积木块假设理论,通过概率把多个个体的相似特征结合起来考虑,发掘优势群体中个体结构的相似点,提出了基于优良模式连接的思想。在求解问题时,通过在优势群体中考虑个体相似点的信息,对以较高频率出现在后代中的多个相邻变量以概率为基础进行连接,组成一个连接块,令其为优良模式连接块,并在进化过程中以块为整体参与进化,增强那些适应度高于种群平均适应度的模式在下一代中出现的概率,相应地减少那些适应度低于种群平均适应度的模式在下一代中出现的概率。基于这种思想,使算法能有效避免构造块破坏问题,具有较好的连锁学习效果,同时为避免陷入局部最优,有条件的调整每个连接块内部各变量的排列顺序,从而有效提高分布估计算法的优化性能。
2.将基于优良模式连接的分布估计算法应用于求解Job Shop调度问题、柔性Job Shop调度问题和旅行商问题中。对于Job Shop调度问题和柔性Job Shop调度问题,在建立概率模型时充分利用了相邻工序在优势群体中的信息,通过概率值对以较高频率出现在优势群体中的相邻工序进行连接,从而使建立的概率模型能较好地反应调度问题中工序排序的特点。在旅行商问题中,在建立概率模型时充分考虑相邻城市出现在优势群体中的频率信息,并通过概率的大小对其进行连接,从而使建立的概率模型能较好地反应旅行商问题中个体的结构特征。仿真结果表明,所提出的算法在求解上述问题时表现出较好的性能。
3. Bayesian统计推断方法与贝叶斯网络优化方法不同,它不需要优化复杂的网络结构,而是直接利用样本提供的信息,通过建立一个后验分布对样本进行推断。因此,本文借鉴Bayesian统计推断理论中对样本的推断思想和方法,充分利用优势群体中个体的信息,建立了一种针对离散优化问题的基于Bayesian统计推理的分布估计算法。首先,针对个体结构的每一个位置,通过优势群体信息建立一种不断更新的先验分布概率模型,利用相邻变量出现在优势群体中的频率,通过计算每一个位置上的条件概率向量建立条件分布概率模型;然后,结合先验分布概率与条件分布概率,通过贝叶斯公式的转化,建立一种后验分布概率模型。这种后验概率分布模型综合了先验概率信息和样本信息,具有较好的统计推断效果,从后验概率模型中抽样产生新群体,通过对后验概率模型的不断更新,实现进化过程。
4.将基于Bayesian统计推理的分布估计算法应用于求解Job Shop调度问题、柔性Job Shop调度问题和旅行商问题中。根据Job Shop问题和柔性Job Shop调度问题的特点,针对工序排序的每一个位置建立先验分布概率模型,充分利用优势群体中各台机器上工序的排列信息,通过贝叶斯公式,建立一种能较好地反映Job Shop调度问题特点的后验概率模型,并从中抽样产生新群体;对于旅行商问题,充分利用各个城市间的排列位置信息建立先验分布概率模型和条件分布概率模型,通过贝叶斯公式获得一种后验概率模型并用以指导产生新群体。针对典型算例的仿真实验结果表明,该算法具有较好的寻优能力和鲁棒性,所建立的概率模型具有较好的稳定性。
5.在离散Quasi-copula理论的基础上,针对离散优化问题,提出了一种基于经验Copula的分布估计算法。对个体采用整数编码方式,采用经验分布函数求解各个变量的边缘分布函数,在估计经验Copula函数时,首先将以整数编码的个体映射到(0,1)区间上,然后对单位超立方体进行分割,等分成若干子超立方体,统计优势群体中的个体落入各个子超立方体中的个体数,构造多维经验Copula函数,得到一种针对离散变量的多变量相关的联合分布函数,并从中抽样产生新群体。由于考虑了多变量间的相关性,因而所建立的概率模型能较好地反映问题的特征。同时对算法的时间复杂性进行了分析。
6.将基于经验Copula的分布估计算法应用于求解旅行商问题和柔性Job Shop调度问题中。通过估计经验Copula,建立了上述问题的多变量相关的联合分布概率模型,并从中采样。仿真实验结果表明,该算法具有较好的求解能力。
Estimation of distribution algorithms is a class of novel evolutionary algorithm based on the probability models in the field of evolutionary computation which combines with intelligence computing theory and the knowledge of statistical studies. According to the information of some better individuals in the current population, the model of probability is built to describe the distribution of all the solutions. Then a new population can be obtained by sampling from this model of probability. This process improves the solution iteratively. The algorithm relies on the construction and maintenance of a probability model that characterizes the correlation between variables involved. So the building block is mixed and recombined effectively. It is shown a good performance on such problems that traditional GA is difficult to solve, especially on nonlinear and high-dimensional complex ones. The key of EDAs for solving problems is to construct a probability model which can characterize the promising solution accurately. However, it is very difficult to construct a proper model because the model can’t reflect the problem’s essential properties if its structure is too simple while the complexity of algorithm will be increased rapidly if the structure is too complex. Especially, for combinatorial optimization problems, which have strong correlation among variables of the feasible solution because of the complexity of the problem itself. So it is a bottleneck problem to build an accurate probability distribution model to represent distribution of the feasible solutions of the problem when using the estimation of distribution algorithm to solve the complex combinatorial optimization problems.
In this paper, the ideas such as superior pattern connection,Bayesian Statistical Inference and Discrete Quasi-Copula theory, are introduced. The ability of solving the complex combinatorial optimization problem is improved by improving the way of building the probability model. Synchronously, the algorithm is used to solve the production scheduling problem and traveling salesman problem. The main achievement is as follows:
1. Based on the scheme theorem and the hypothesis of building blocks of genetic algorithm, the similar characteristics of some individuals are considered and explored together. The idea of superior pattern connection is proposed. Information of the similarities of individuals in superior population is considered when solving problems. Based on information of probability, multi-neighboring variables are connected into a whole block when their frequency appearing in superior population is higher. In addition, it is taken as the connection block of superior pattern and it appears as a whole part in iterative process. This enhances the probability of those patterns appearing in the next generation whose fitness values are higher than the average fitness value of population. At the same time, the probability of those schemes with lower fitness values than the average fitness of population in the next generation are decreased. This algorithm can effectively avoid the problem of building blocks destroyed and has better performances of chain learning. Besides, local adjustment to the variables within each sub-block is introduced under some conditions to avoid trapping in the local optimal, which can improve the the performance of algorithm effectively.
2. The estimation of distribution algorithm based on the superior pattern connection is used to solve job shop scheduling problem, flexible job shop scheduling problem and traveling salesman problem. For job shop scheduling problem and flexible job shop scheduling problem, the probability model is constructed using frequency information of pair-wise operations neighboring in superior population. These multi-neighboring operations are connected into a whole block based on information of probability. For the frequency appearing in superior population is higher, it makes the model of probability constructed well to reflect the characteristic of sequence of operations. For the traveling salesman problem, frequency information of pair-wise cities neighboring in superior population is used to build the probability model. These multi-neighboring cities are connected into a whole block according to the size of probability. So the model of probability constructed can well represent the structure of individual. The simulation results show that the proposed algorithm has a better performance for solving the problems mentioned above.
3. The theory of Bayesian statistical inference is different from the Bayesian network optimization method. It does not require the process of optimization for the complex network structure when solving the problems while the information of samples provided is directly used to construct the probability distribution. The samples are inferred through the establishment of a posterior distribution. Based on Bayesian statistical inference theory, a new estimation of distribution algorithm for discrete optimization problems is proposed by using the information of individuals in superior population. At first, according to each position of individual’s structure, the model of priori distribution probability is built by extracting the information of superior solutions updating. Then the model of conditional probability is also built by computing the vector of conditional probability on each position of individual and using the frequencies of neighboring variables appearing in superior population. After that, the model of posterior probability is given by combining the above two models with Bayesian formula. This model combines with information of priori distribution and conditional distribution. It has good results of statistical inference, and is used to guide new populations generation. The evolutionary process is achieved by updating iterately the model of posterior probability.
4. The estimation of distribution algorithm based on Bayesian statistical inference theory is used to solve job shop scheduling problem, flexible job shop scheduling problem and traveling salesman problem. According to the characteristics of job shop scheduling problem and flexible job shop scheduling problem, the model of priori distribution probability is built for each position of operations sequence. Making the best use of information of the sequence of operations on each machine, the posterior probability model that can well reflect the characteristics of problems is constructed by using Bayesian formula. The next population is generated by sampling from this model of posterior probability. For the traveling salesman problem, information of permutation between cities is used to build the models of priori probability and posterior probability. The model of posterior probability is constructed by Bayesian formula and the next population is generated by sampling from this model. The simulation results for typical instances show that the proposed algorithm has the preferable search ability and robustness and the model of probability constructed has better stability.
5. Based on discrete quasi-copula theory, an estimation of distribution algorithm based on the empirical copula function is proposed to solve discrete optimization problems.The individual is encoded with integer. The empirical distribution function is used to solve the marginal distribution function of each variable. When estimating the empirical copula function, firstly, the individuals of superior population are mapped into the interval (0,1). Then, the unit hypercube is equally partitioned into some sub-hypercubes and a multi-dimensional empirical copula function is constructed by counting the number of individuals in each sub-hypercube. A probability model of multivariate correlation for discrete variables is obtained and the next population is generated by sampling from the model. Because the proposed algorithm considers the multivariate correlation during constructing the model of probability, the probability models can well reflect the characteristics of problems’structure. Moreover, the time complexity of algorithm is analyzed.
6. The estimation of distribution algorithm based on empirical copula is used to solve traveling salesman problem and flexible job shop scheduling problem. A model of multivariate correlation for these two problems is constructed by estimating the empirical copula and sampling from them. The simulation results show that the proposed algorithm has a better search ability.
文中结合优良模式连接的思想、Bayesian统计推断理论与离散quasi-copula理论,对概率模型的建立方式进行改进,以提高算法求解复杂组合优化问题的能力,并将算法用于求解生产调度问题和旅行商问题,主要完成了以下工作:
1.根据遗传算法中的模式定理与积木块假设理论,通过概率把多个个体的相似特征结合起来考虑,发掘优势群体中个体结构的相似点,提出了基于优良模式连接的思想。在求解问题时,通过在优势群体中考虑个体相似点的信息,对以较高频率出现在后代中的多个相邻变量以概率为基础进行连接,组成一个连接块,令其为优良模式连接块,并在进化过程中以块为整体参与进化,增强那些适应度高于种群平均适应度的模式在下一代中出现的概率,相应地减少那些适应度低于种群平均适应度的模式在下一代中出现的概率。基于这种思想,使算法能有效避免构造块破坏问题,具有较好的连锁学习效果,同时为避免陷入局部最优,有条件的调整每个连接块内部各变量的排列顺序,从而有效提高分布估计算法的优化性能。
2.将基于优良模式连接的分布估计算法应用于求解Job Shop调度问题、柔性Job Shop调度问题和旅行商问题中。对于Job Shop调度问题和柔性Job Shop调度问题,在建立概率模型时充分利用了相邻工序在优势群体中的信息,通过概率值对以较高频率出现在优势群体中的相邻工序进行连接,从而使建立的概率模型能较好地反应调度问题中工序排序的特点。在旅行商问题中,在建立概率模型时充分考虑相邻城市出现在优势群体中的频率信息,并通过概率的大小对其进行连接,从而使建立的概率模型能较好地反应旅行商问题中个体的结构特征。仿真结果表明,所提出的算法在求解上述问题时表现出较好的性能。
3. Bayesian统计推断方法与贝叶斯网络优化方法不同,它不需要优化复杂的网络结构,而是直接利用样本提供的信息,通过建立一个后验分布对样本进行推断。因此,本文借鉴Bayesian统计推断理论中对样本的推断思想和方法,充分利用优势群体中个体的信息,建立了一种针对离散优化问题的基于Bayesian统计推理的分布估计算法。首先,针对个体结构的每一个位置,通过优势群体信息建立一种不断更新的先验分布概率模型,利用相邻变量出现在优势群体中的频率,通过计算每一个位置上的条件概率向量建立条件分布概率模型;然后,结合先验分布概率与条件分布概率,通过贝叶斯公式的转化,建立一种后验分布概率模型。这种后验概率分布模型综合了先验概率信息和样本信息,具有较好的统计推断效果,从后验概率模型中抽样产生新群体,通过对后验概率模型的不断更新,实现进化过程。
4.将基于Bayesian统计推理的分布估计算法应用于求解Job Shop调度问题、柔性Job Shop调度问题和旅行商问题中。根据Job Shop问题和柔性Job Shop调度问题的特点,针对工序排序的每一个位置建立先验分布概率模型,充分利用优势群体中各台机器上工序的排列信息,通过贝叶斯公式,建立一种能较好地反映Job Shop调度问题特点的后验概率模型,并从中抽样产生新群体;对于旅行商问题,充分利用各个城市间的排列位置信息建立先验分布概率模型和条件分布概率模型,通过贝叶斯公式获得一种后验概率模型并用以指导产生新群体。针对典型算例的仿真实验结果表明,该算法具有较好的寻优能力和鲁棒性,所建立的概率模型具有较好的稳定性。
5.在离散Quasi-copula理论的基础上,针对离散优化问题,提出了一种基于经验Copula的分布估计算法。对个体采用整数编码方式,采用经验分布函数求解各个变量的边缘分布函数,在估计经验Copula函数时,首先将以整数编码的个体映射到(0,1)区间上,然后对单位超立方体进行分割,等分成若干子超立方体,统计优势群体中的个体落入各个子超立方体中的个体数,构造多维经验Copula函数,得到一种针对离散变量的多变量相关的联合分布函数,并从中抽样产生新群体。由于考虑了多变量间的相关性,因而所建立的概率模型能较好地反映问题的特征。同时对算法的时间复杂性进行了分析。
6.将基于经验Copula的分布估计算法应用于求解旅行商问题和柔性Job Shop调度问题中。通过估计经验Copula,建立了上述问题的多变量相关的联合分布概率模型,并从中采样。仿真实验结果表明,该算法具有较好的求解能力。
Estimation of distribution algorithms is a class of novel evolutionary algorithm based on the probability models in the field of evolutionary computation which combines with intelligence computing theory and the knowledge of statistical studies. According to the information of some better individuals in the current population, the model of probability is built to describe the distribution of all the solutions. Then a new population can be obtained by sampling from this model of probability. This process improves the solution iteratively. The algorithm relies on the construction and maintenance of a probability model that characterizes the correlation between variables involved. So the building block is mixed and recombined effectively. It is shown a good performance on such problems that traditional GA is difficult to solve, especially on nonlinear and high-dimensional complex ones. The key of EDAs for solving problems is to construct a probability model which can characterize the promising solution accurately. However, it is very difficult to construct a proper model because the model can’t reflect the problem’s essential properties if its structure is too simple while the complexity of algorithm will be increased rapidly if the structure is too complex. Especially, for combinatorial optimization problems, which have strong correlation among variables of the feasible solution because of the complexity of the problem itself. So it is a bottleneck problem to build an accurate probability distribution model to represent distribution of the feasible solutions of the problem when using the estimation of distribution algorithm to solve the complex combinatorial optimization problems.
In this paper, the ideas such as superior pattern connection,Bayesian Statistical Inference and Discrete Quasi-Copula theory, are introduced. The ability of solving the complex combinatorial optimization problem is improved by improving the way of building the probability model. Synchronously, the algorithm is used to solve the production scheduling problem and traveling salesman problem. The main achievement is as follows:
1. Based on the scheme theorem and the hypothesis of building blocks of genetic algorithm, the similar characteristics of some individuals are considered and explored together. The idea of superior pattern connection is proposed. Information of the similarities of individuals in superior population is considered when solving problems. Based on information of probability, multi-neighboring variables are connected into a whole block when their frequency appearing in superior population is higher. In addition, it is taken as the connection block of superior pattern and it appears as a whole part in iterative process. This enhances the probability of those patterns appearing in the next generation whose fitness values are higher than the average fitness value of population. At the same time, the probability of those schemes with lower fitness values than the average fitness of population in the next generation are decreased. This algorithm can effectively avoid the problem of building blocks destroyed and has better performances of chain learning. Besides, local adjustment to the variables within each sub-block is introduced under some conditions to avoid trapping in the local optimal, which can improve the the performance of algorithm effectively.
2. The estimation of distribution algorithm based on the superior pattern connection is used to solve job shop scheduling problem, flexible job shop scheduling problem and traveling salesman problem. For job shop scheduling problem and flexible job shop scheduling problem, the probability model is constructed using frequency information of pair-wise operations neighboring in superior population. These multi-neighboring operations are connected into a whole block based on information of probability. For the frequency appearing in superior population is higher, it makes the model of probability constructed well to reflect the characteristic of sequence of operations. For the traveling salesman problem, frequency information of pair-wise cities neighboring in superior population is used to build the probability model. These multi-neighboring cities are connected into a whole block according to the size of probability. So the model of probability constructed can well represent the structure of individual. The simulation results show that the proposed algorithm has a better performance for solving the problems mentioned above.
3. The theory of Bayesian statistical inference is different from the Bayesian network optimization method. It does not require the process of optimization for the complex network structure when solving the problems while the information of samples provided is directly used to construct the probability distribution. The samples are inferred through the establishment of a posterior distribution. Based on Bayesian statistical inference theory, a new estimation of distribution algorithm for discrete optimization problems is proposed by using the information of individuals in superior population. At first, according to each position of individual’s structure, the model of priori distribution probability is built by extracting the information of superior solutions updating. Then the model of conditional probability is also built by computing the vector of conditional probability on each position of individual and using the frequencies of neighboring variables appearing in superior population. After that, the model of posterior probability is given by combining the above two models with Bayesian formula. This model combines with information of priori distribution and conditional distribution. It has good results of statistical inference, and is used to guide new populations generation. The evolutionary process is achieved by updating iterately the model of posterior probability.
4. The estimation of distribution algorithm based on Bayesian statistical inference theory is used to solve job shop scheduling problem, flexible job shop scheduling problem and traveling salesman problem. According to the characteristics of job shop scheduling problem and flexible job shop scheduling problem, the model of priori distribution probability is built for each position of operations sequence. Making the best use of information of the sequence of operations on each machine, the posterior probability model that can well reflect the characteristics of problems is constructed by using Bayesian formula. The next population is generated by sampling from this model of posterior probability. For the traveling salesman problem, information of permutation between cities is used to build the models of priori probability and posterior probability. The model of posterior probability is constructed by Bayesian formula and the next population is generated by sampling from this model. The simulation results for typical instances show that the proposed algorithm has the preferable search ability and robustness and the model of probability constructed has better stability.
5. Based on discrete quasi-copula theory, an estimation of distribution algorithm based on the empirical copula function is proposed to solve discrete optimization problems.The individual is encoded with integer. The empirical distribution function is used to solve the marginal distribution function of each variable. When estimating the empirical copula function, firstly, the individuals of superior population are mapped into the interval (0,1). Then, the unit hypercube is equally partitioned into some sub-hypercubes and a multi-dimensional empirical copula function is constructed by counting the number of individuals in each sub-hypercube. A probability model of multivariate correlation for discrete variables is obtained and the next population is generated by sampling from the model. Because the proposed algorithm considers the multivariate correlation during constructing the model of probability, the probability models can well reflect the characteristics of problems’structure. Moreover, the time complexity of algorithm is analyzed.
6. The estimation of distribution algorithm based on empirical copula is used to solve traveling salesman problem and flexible job shop scheduling problem. A model of multivariate correlation for these two problems is constructed by estimating the empirical copula and sampling from them. The simulation results show that the proposed algorithm has a better search ability.
引文
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