基于新概率分布集的分布鲁棒Weber选址问题研究

发布时间：2018-03-07 07:21

本文选题：分布鲁棒优化　切入点：概率分布集　出处：《南京航空航天大学》2016年硕士论文　论文类型：学位论文

【摘要】：本文研究内容为用基于新的概率分布集的分布鲁棒优化方法研究不确定情形下的Weber选址问题,并通过数值实验表明我们的方法比常用的Min Max-Regret方法好,同时适当增大?时所需要的样本个数减少。论文的具体内容如下:第一章介绍了不确定设施选址的研究背景和研究现状,同时简单介绍了本文的研究工作。第二章针对选址问题中随机参数的协方差矩阵不一定是正定的这一事实,提出了新的概率分布集,这一改进的概率分布集及其性质通过构造新的虚拟随机向量而推导得到。第三章首先介绍了变分不等式和随机变分不等式的相关概念,然后构造了Weber选址问题的残量函数。当权重随机而顾客位置固定,同时目标函数使用1-范数时,通过分割平面,在各分割后形成的每个小矩形内,我们验证了残量函数满足分布鲁棒优化的假设条件,并可以将此问题通过对偶变换化为一个半定规划问题。通过分割平面法可以使得问题的约束个数的规模由指数阶降为多项式阶。第四章给出了具体的算法实现并得出了相应的结论。Min Max-Regret方法只考虑最坏情形下系统的表现,有时最坏情形发生的概率很小,用此方法做出的决策在很多情形下可能效果较差,同时此方法没有充分挖掘样本的概率统计信息;基于新的概率分布集的分布鲁棒优化方法克服了Min Max-Regret方法的不足之处,所作出的决策在大多数情形下比常用的Min Max-Regret方法要好,同时充分利用了样本里隐含的概率统计信息。通过数值实验验证了我们的方法在处理不确定Weber问题时比Min Max-Regret方法要好,同时表明当???I中的?增大时,所需要的样本个数减小,且对解的性态没有影响。第五章总结了全文并提出了展望。
[Abstract]:In this paper, we use the new distributed robust optimization method based on the new probability distribution set to study the Weber location problem in uncertain cases. The numerical experiments show that our method is better than the usual Min Max-Regret method, and at the same time, it is increased appropriately. The main contents of this paper are as follows: chapter one introduces the research background and research status of uncertain facility location. In chapter 2, a new probability distribution set is proposed for the fact that the covariance matrix of random parameters is not necessarily positive definite. This improved probability distribution set and its properties are derived by constructing a new virtual random vector. In chapter 3, the concepts of variational inequality and random variational inequality are introduced. Then, the residual function of Weber location problem is constructed. When the weight is random and the customer position is fixed, and the objective function uses 1-norm, by dividing the plane, each small rectangle is formed after each segmentation. We verify that the residual function satisfies the assumption of distributed robust optimization. The problem can be transformed into a semi-definite programming problem by dual transformation. The size of the constraint number of the problem can be reduced from exponential order to polynomial order by partitioning plane method. Chapter 4th gives the implementation of the algorithm. It is concluded that the Min Max-Regret method only considers the performance of the system in the worst-case scenario. Sometimes the probability of the worst case is very small, the decision made by this method may be poor in many cases, at the same time, the method does not fully mine the probability and statistics information of the sample. The new distributed robust optimization method based on the new probability distribution sets overcomes the shortcomings of the Min Max-Regret method and makes better decisions than the Min Max-Regret method in most cases. At the same time, we make full use of the probability and statistics information implied in the sample. The numerical experiments show that our method is better than the Min Max-Regret method in dealing with the uncertain Weber problem. ? ? In I? When the number of samples is increased, the number of samples is reduced, and the behavior of the solution is not affected. Chapter 5th summarizes the full text and puts forward the prospect.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O22

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