灰色系统建模技术研究
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摘要
灰色系统理论是解决客观世界贫信息不确定性问题的方法,灰色系统建模是灰色系统理论的重要组成部分,已初步取得了一些成果,但是在理论研究方面还有许多亟待解决的问题。本文针对灰色系统建模的基础理论展开研究,系统地分析了灰色系统建模机理,重构了灰代数运算规则,提出了系列灰预测、灰关联、灰决策模型,并对这些灰色系统模型的性质进行了深入研究。本文的主要创新点有:
(1)从优化模型角度构建了灰数运算规则。在原有灰数运算规则的基础上,提出了简单灰数、合成灰数以及复合灰数的运算规则。实质上简单灰数和合成灰数运算都是复合灰数运算的特殊形式,而既有灰数运算规则的研究都属于简单灰数和合成灰数的范畴;本文从优化模型角度构建的复合灰数运算规则符合灰色系统的内涵,并能够有效解决其他运算规则难以解决的可逆性问题。
(2)提出了灰距离以及灰数大小比较的可能度规则。从灰数信息内涵分析入手,提出了离散型灰数距离和连续型灰数距离概念;基于灰数所蕴涵的实数信息,利用灰数取值的可能性构建了新的灰数大小比较的规则。
(3)构建了系列离散灰色模型。针对既有灰色模型存在的模型误差和初始点等问题,构建了离散灰色模型、优化离散灰色模型、近似非齐次指数增长序列离散灰色模型以及多变量离散灰色模型等新灰色预测模型,解决了既有灰色模型中存在的问题并拓展了灰色模型体系,研究了离散灰色模型和既有灰色模型的相互联系。
(4)研究了系列灰色模型的参数性质。针对灰色系统建模需要的数据处理问题,对系列离散灰色模型以及GM(n,h)模型的参数性质展开研究,并对GM(n,h)模型和多变量离散模型相互关系进行对比分析,将各类灰色模型的参数性质归入一个统一的分析体系。
(5)分析既有关联度模型的缺陷、提出灰关联模型所应满足的相关性质并构建了新的灰关联模型。从数据变换入手,找出了既有模型存在的理论缺陷,提出了平行性、一致性和保序性概念对灰关联公理加以补充,完善了关联度模型构建条件,并提出了几何关联度模型和灰数序列几何关联度模型。
(6)构建了多属性灰色决策模型、区间灰数互补判断矩阵排序模型和区间灰数互反判断矩阵排序模型。基于灰数大小比较的可能度公式以及灰数四则运算规则,构建了多属性灰色决策模型、区间灰数互补判断矩阵排序模型和区间灰数互反判断矩阵排序模型,并给出了相关算例验证这些模型的有效性。
Grey systems theory is a method to solve grey uncertainty problems in the world. Grey systems modeling is an important part of grey systems theory. In this field, some achievements have been made. But there are still many theoretical problems needed to be solved as soon as possible. This paper aims to study the basic theories of grey systems modeling. The main ideas are to analyze the mechanism of grey systems modeling, rebuild the algorithms of grey numbers and propose a series of grey forecasting models, grey relational models and grey decision-making models. In addition, some properties of these grey models are discussed. The main innovations of the paper are as follows.
The first innovation is to rebuild the algorithms of grey numbers with nonlinear optimized models. Based on the existing algorithms of grey numbers, the algorithms of simple grey numbers, complex grey numbers and multiple grey numbers are proposed. In fact, the algorithms of simple grey numbers and complex grey numbers are particular forms of that of multiple grey numbers. The existing algorithms belong to the algorithms of simple grey numbers and complex grey numbers. The algorithm of complex grey numbers proposed from the angel of optimized models are in accord with the connotations of grey numbers better and it can effectively solve the problem of reversibility which is difficult for other algorithms.
The second innovation is to propose the concept of grey distance and the probability rules on comparing grey numbers. The concepts of discrete grey distance and continuous grey distance are proposed with the comprehension on the information connotation of grey numbers. Based on the real numbers information contained in the grey number, the novel probability rules on comparing grey numbers are proposed.
The third innovation is to construct a series of discrete grey forecasting model. To overcome the problems of model error and initialized value of the existing grey model, several grey models are constructed, including discrete grey model, optimized discrete grey model, multiple-variables discrete grey model and discrete grey model based on non-homogeneous exponential data sequence and solve the problems of existing grey models and extend the grey model system. In addition, the relationship of discrete grey model and existent grey model are discussed.
The fourth innovation is to study the parameter properties of a series of grey forecasting models. The study has been made on the data processing of grey system modeling and the parameter properties of a series of discrete grey forecasting and GM (n, h) model are discussed. Then the relationship of GM (n, h) model and multiple-variable discrete grey model is discussed. The results indicate that the parameter properties of all kinds of grey models can be analyzed in a common system.
The fifth innovation is to analyze the limitations of existing grey relational models, put forward the corresponding properties the grey relational models should satisfy and form the new grey relational models. Beginning with data transformation, I find out the theoretical limitations of the existing models. Then the parallel, multiple and order-keeping properties are proposed to make the grey relational theory perfect. Grey geometry relational model and grey geometry relational model based on grey numbers sequence are also constructed.
The last innovation is study on grey decision-making model. Based on the algorithms of grey numbers and the probability rules on comparing grey numbers, several grey decision-making models are constructed, including multi-attribute grey number decision-making model, sorting model on grey reciprocal judgment matrix and sorting model on grey complementary judgment matrix. And at the same time, the corresponding examples have been given to validate the effectiveness of these models.
(1)从优化模型角度构建了灰数运算规则。在原有灰数运算规则的基础上,提出了简单灰数、合成灰数以及复合灰数的运算规则。实质上简单灰数和合成灰数运算都是复合灰数运算的特殊形式,而既有灰数运算规则的研究都属于简单灰数和合成灰数的范畴;本文从优化模型角度构建的复合灰数运算规则符合灰色系统的内涵,并能够有效解决其他运算规则难以解决的可逆性问题。
(2)提出了灰距离以及灰数大小比较的可能度规则。从灰数信息内涵分析入手,提出了离散型灰数距离和连续型灰数距离概念;基于灰数所蕴涵的实数信息,利用灰数取值的可能性构建了新的灰数大小比较的规则。
(3)构建了系列离散灰色模型。针对既有灰色模型存在的模型误差和初始点等问题,构建了离散灰色模型、优化离散灰色模型、近似非齐次指数增长序列离散灰色模型以及多变量离散灰色模型等新灰色预测模型,解决了既有灰色模型中存在的问题并拓展了灰色模型体系,研究了离散灰色模型和既有灰色模型的相互联系。
(4)研究了系列灰色模型的参数性质。针对灰色系统建模需要的数据处理问题,对系列离散灰色模型以及GM(n,h)模型的参数性质展开研究,并对GM(n,h)模型和多变量离散模型相互关系进行对比分析,将各类灰色模型的参数性质归入一个统一的分析体系。
(5)分析既有关联度模型的缺陷、提出灰关联模型所应满足的相关性质并构建了新的灰关联模型。从数据变换入手,找出了既有模型存在的理论缺陷,提出了平行性、一致性和保序性概念对灰关联公理加以补充,完善了关联度模型构建条件,并提出了几何关联度模型和灰数序列几何关联度模型。
(6)构建了多属性灰色决策模型、区间灰数互补判断矩阵排序模型和区间灰数互反判断矩阵排序模型。基于灰数大小比较的可能度公式以及灰数四则运算规则,构建了多属性灰色决策模型、区间灰数互补判断矩阵排序模型和区间灰数互反判断矩阵排序模型,并给出了相关算例验证这些模型的有效性。
Grey systems theory is a method to solve grey uncertainty problems in the world. Grey systems modeling is an important part of grey systems theory. In this field, some achievements have been made. But there are still many theoretical problems needed to be solved as soon as possible. This paper aims to study the basic theories of grey systems modeling. The main ideas are to analyze the mechanism of grey systems modeling, rebuild the algorithms of grey numbers and propose a series of grey forecasting models, grey relational models and grey decision-making models. In addition, some properties of these grey models are discussed. The main innovations of the paper are as follows.
The first innovation is to rebuild the algorithms of grey numbers with nonlinear optimized models. Based on the existing algorithms of grey numbers, the algorithms of simple grey numbers, complex grey numbers and multiple grey numbers are proposed. In fact, the algorithms of simple grey numbers and complex grey numbers are particular forms of that of multiple grey numbers. The existing algorithms belong to the algorithms of simple grey numbers and complex grey numbers. The algorithm of complex grey numbers proposed from the angel of optimized models are in accord with the connotations of grey numbers better and it can effectively solve the problem of reversibility which is difficult for other algorithms.
The second innovation is to propose the concept of grey distance and the probability rules on comparing grey numbers. The concepts of discrete grey distance and continuous grey distance are proposed with the comprehension on the information connotation of grey numbers. Based on the real numbers information contained in the grey number, the novel probability rules on comparing grey numbers are proposed.
The third innovation is to construct a series of discrete grey forecasting model. To overcome the problems of model error and initialized value of the existing grey model, several grey models are constructed, including discrete grey model, optimized discrete grey model, multiple-variables discrete grey model and discrete grey model based on non-homogeneous exponential data sequence and solve the problems of existing grey models and extend the grey model system. In addition, the relationship of discrete grey model and existent grey model are discussed.
The fourth innovation is to study the parameter properties of a series of grey forecasting models. The study has been made on the data processing of grey system modeling and the parameter properties of a series of discrete grey forecasting and GM (n, h) model are discussed. Then the relationship of GM (n, h) model and multiple-variable discrete grey model is discussed. The results indicate that the parameter properties of all kinds of grey models can be analyzed in a common system.
The fifth innovation is to analyze the limitations of existing grey relational models, put forward the corresponding properties the grey relational models should satisfy and form the new grey relational models. Beginning with data transformation, I find out the theoretical limitations of the existing models. Then the parallel, multiple and order-keeping properties are proposed to make the grey relational theory perfect. Grey geometry relational model and grey geometry relational model based on grey numbers sequence are also constructed.
The last innovation is study on grey decision-making model. Based on the algorithms of grey numbers and the probability rules on comparing grey numbers, several grey decision-making models are constructed, including multi-attribute grey number decision-making model, sorting model on grey reciprocal judgment matrix and sorting model on grey complementary judgment matrix. And at the same time, the corresponding examples have been given to validate the effectiveness of these models.
引文
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