【作者】 陈勇明；

【导师】 向蓉美；

【作者基本信息】 西南财经大学 ， 统计学， 2008， 博士

【摘要】 从1982年我国学者邓聚龙教授提出灰色系统以来的20多年间,灰色系统理论的确取得了瞩目的成就,已经广泛应用于经济管理、投资决策、军事运筹等各个领域,科研成果也有的进入三大检索(SCI、EI、ISTP)。但是从世界范围的研究群体看,灰色系统的研究主要集中在中国学者和外籍华人学者;从研究内容看,应用研究较多,理论研究相对较少,应用的理论也多集中于灰色关联分析和GM(1,1)及其衍生模型。灰色系统理论在世界上还没有得到普遍认可,很多学者对灰色系统提出了种种质疑,其中一个重要原因在于一些灰色系统基本概念没有被严格定义,理论缺乏严密的数学基础,理论体系因而难以向纵深发展。为了完善灰色系统理论的理论基础,为了推广和更科学地应用灰色系统理论,本文重点研究了灰色系统的三个基本的理论问题,即包括灰靶变换计算公式与定义的不相容问题、灰靶理论与灰色关联分析的逆序问题、包络灰预测的人为随意性和局部信息丢失的问题。全文总共研究的5个方面内容如下:1、灰靶理论中灰靶变换的计算公式与灰靶变换定义的不相容问题。经过研究发现:邓聚龙教授给出的灰靶变换计算公式与灰靶变换定义是不相容的。将该问题分解成以下几个子问题逐个层层推进加以解决:首先通过构造特殊取值的适中值指标序列,以具体的数值实例证明了灰靶变换的计算公式与灰靶变换定义的不相容问题在理论上的存在性。接着研究了灰靶变换不相容问题的严重程度以说明不相容问题将影响灰靶理论的实际应用,为此采用统计模拟的方法,从几种常见的概率分布中抽取随机样本计算了不相容问题发生的频率,结果发现不相容问题发生的频率很大而且随着备择方案个数的增多而增加并迅速增加到1。然后利用几何概率的方法进一步分析了不相容问题发生的概率,与频率的变化情况一样,不相容问题发生的概率很大而且随着备择方案的增多而增加。再然后对不相容问题进行了实证分析,通过检验发表的应用灰靶理论的文献中用到的现实中的真实数据,证明了灰靶理论的不相容问题确实在应用中存在。在充分说明邓氏灰靶变换应当予以修正后,最后给出了灰靶变换计算公式的修正公式,证明了修正公式符合邓聚龙给出的灰靶变换的定义,并指出了修正计算公式不仅克服了原计算公式的缺陷,而且拓宽了指标序列的取值范围,将原公式只能取正数的范围拓宽到任意实数范围,从而拓宽了灰靶理论的应用范围。2、灰靶理论和灰色关联度的逆序问题逆序问题是多准则决策问题(MCDM, Multi-Criterion Decision Making problem)中一个重要问题,该问题最早在1982年由Valerie Belton和Tony Gear在研究层次分析法(AHP, Analytic Hierarchy Process)时提出,1991年在美国召开的ISAHP-2(the International Symposium on the Analytic Hierarchy Process)会议上学者们对这一问题进行了广泛的讨论。此后各种多准则决策算法的逆序问题受到广泛关注,但至今灰靶的逆序问题尚未有研究结果,本文首次研究了灰靶理论的逆序问题。通过构造具体的多准则决策矩阵,用数据证明了灰靶理论和灰色关联度逆序问题的存在性。并采用统计模拟的方法研究了灰靶理论逆序的严重程度。就其产生原因和逆序的消除方法等问题和另一个多准则决策方法TOPSIS作了比较分析。指出了灰靶理论逆序问题需要进一步研究的方向。3、包络灰预测的局部信息丢失问题现行的包络灰预测只反映了数据的整体趋势信息而忽略了局部趋势信息,与灰色系统主张的差异信息原理和最少信息原理是不协调的。而且在确定上、下包络线时人为地设定有很大的随意性。本文依照灰色系统的差异信息原理和最少信息原理,充分利用局部趋势信息提出了一种灰色预测射线投影算法。该算法不仅给出预测区间,而且给出区间里取值的信息,并且融入古典概率的思想给出了预测区间的可信度。指出与王福林、朱荣胜(2003)或宋中民(2002)给出的侧重提取总体趋势信息的预测模型配合使用均可以收到对总体趋势信息和局部趋势信息兼顾的效果。4、频繁集的自然分桶的Hash生成算法与灰色系统的现状研究在数据挖掘的关联规则挖掘中, Park(1995)提出的利用基于Hash技术的DHP算法生成频繁项目集时,存在两个主要问题:一是难以选择合适的Hash函数;二是容易产生Hash冲突.本文针对这两个问题,利用二维Hash算法的基本思想,提出了一种自然分桶的Hash算法,根据新的桶内容增加的需要自然地新增桶地址.该算法的突出优点在于不会产生Hash冲突和避免了寻找Hash函数的麻烦。利用所提出的算法,对灰色系统的发展状况进行了研究。以CNKI、万方博士论文数据库为数据源,以“灰色系统”为主题检索1982-2004期间与灰色系统理论有关的博士学位论文,利用频繁集自然分桶的Hash生成算法对检索结果进行分析,得到了反映这23年间灰色系统理论状况的变形Hash表,从该表可以清楚地发现灰色系统的应用领域较多,但是应用的内容主要局限于灰色关联分析和GM(1,1)及其衍生模型。该表也为从事灰色系统理论研究的科技工作者提供了一个方便的参考资料。5、灰色系统理论在经济管理领域的一个应用实例宜居城市是2005年由《北京城市总体规划(2004～2020年)》正式提出的概念,一经提出即引起强烈反响,相继有一系列文章对宜居城市的理解和衡量标准衡量指标的确定进行了研究,在此基础上,本文从定量的角度考虑宜居城市的问题,选取易量化的代表性指标体系,根据中国统计年鉴统计提供的数据,用修正灰靶变换公式对数据作预处理,利用灰靶理论对宜居城市进行了数量分析,分析结果表明西部地区与东部沿海地区有较大差距,要使各个地区协调发展,应当加大对西部地区投入的力度。本文的主要创新点有以下3个:1、发现了灰靶变换的计算公式与定义的不相容问题,用数值实例从逻辑上证明了不相容问题的存在性,并用统计模拟和概率计算研究了问题的程度性,并进一步进行了实证分析,最后给出了灰靶变换的修正的计算公式,证明了修正公式符合灰靶变换的定义而且新公式有明确的几何意义,拓宽了原公式的应用范围。2、发现了灰靶理论存在逆序问题,给出了由其中间计算环节灰色关联度引起逆序的具体数值例证,用统计模拟的方法研究了逆序的严重程度,就逆序的产生原因和消除方法与TOPSIS做了对比研究。3、指出了传统的包络灰预测在确定上下包络线时存在人为因素的随意性和丢失局部趋势信息的缺陷,与灰色系统本身主张的差异信息原理和最少信息原理是不协调的。提出了充分利用局部趋势信息的一种射线投影算法。指出了该算法与其它侧重总体趋势信息的算法配合使用可以起到兼顾局部和总体的互补效果。更多还原

【Abstract】 Since Chinese scholar Deng Julong put forward gray system theory in 1982, great success have been made in gray system theory during the past 25 years, it has been widely used in economic management, investment decision, military operations research and so on, some research results have been indexed by SCI, EI or ISTP. However, until today nearly all researchers of grey system theory are from Chinese speaking areas, and the theory is still hardly known nor accepted in the western world. Gray system’s theoretical research is much less than application research, and the content being studied is mainly limited to grey relational analysis, GM(1,1) model and its derived models, which are criticized by many scholars, the main reason is that some fundamental concepts are not well-defined and the theory is not built on solid mathematical foundations and not developed in a rigorous way.In order to make perfect the theoretical foundation of grey system theory, and to popularize and to use it scientifically, three fundamental problem of grey system are studied. The paper covers the following:1. The inconsistency problem on grey target transformation.It is discovered that Deng Julong’s grey target transformation formula is inconsistent with the definition of grey target transformation, which is also proposed by Deng. The problem is divided into several sub-problems and solved one by one.First of all, by constructing a special sequence of medium polarity, the existence of inconsistency problem is proved. Secondly, In order to investigate the seriousness of the inconsistency problem, the statistical simulation method is employed to calculate the frequency of the occurrence of the problem, the outcome of the experiment of the statistical simulation shows that the frequency of the problem is big and increases as the number of alternatives related to the considered decision making increases and increases to 1 in a high speed. And then, calculating geometric probability shows the same above outcome. Fourthly, by checking up on some papers published in journals, empirical analysis shows that the inconsistency problem exists in actual fact. After giving out sufficient reason for the grey target transformatin formula to be revised, finally, some revised formulae are put forward to solve the inconsistency problem. In a rigorous way, the new formulae are proved to be consistent with the definition of grey target transformation given by professor Deng Julong. Moreover, the new formulae not only overcame the inconsistency drawbacks of the classic one but also can be applied in a larger range, their taken value can be extended to negative numbers.2. The rank reversal problem of grey target theory and grey relational analysis.Rank reversal problem is an important problem of MCDM (Multi-Criterion Decision Making), the problem was first proposed by Valerie Belton and Tony Gear in 1982 when studying AHP (Analytic Hierarchy Process). When the ISAHP-2 (the Second International Symposium on the Analytic Hierarchy Process) was held in America in 1991, many participating experts discussed the problem widely. After that the rank reversal problem of those methods concerning MCDM have drawn great attention, however, up to now, the rank reversal problem of grey target theory is not reported. In this paper, the problem is studied the first time. By constructing a numeric decision matrix, the existence of rank reversal problem caused by grey relation incidence in grey target theory is studied. The seriousness of rank reversal problem is studied by statistical simulation. A comparion to TOPSIS is also considered. The prolem needing further study is pointed out.3. The problem of local information losing in grey envelope forecast.The present grey envelope forecast only care about the data’s global information but ignored its local information, which is inconsistent with the grey system’s axiom of difference information and the axiom of least information. In addition, To decide the lower and upper envelope curve is in an arbitrary way. In this paper, according to the grey system’s axiom of difference information and the axiom of least information, a grey forecast method called Project Half-line Algorithm is put forward to make full use of local trend information. The algorithm gives out not only the forecast interval but also its confidence by classical probability. Wang Fulin(2003) and Song Zhongmin(2002)’s models emphasize global information, if Project Half-line Algorithm and Wang or Song’s model were used at the same time, it can give consideration to local and global information.4. Hash Algorithm Based on Naturally Assigning Bucket for Searching FrequentItem-set and the mining the advance information in grey system theory. To mine association rules in data mining, when hash-based DHP algorithm is employed to search frequent item sets, two problems will be encountered: one is that it is difficult to select appropriate Hash function, the other is there exists conflict of Hash bucket address. In order to solve the two problems, the fundamental idea of Two Dimension Hash Algorithm is accepted to propose a Hash Algorithm Based on Naturally Assigning Bucket (HABONAB), which naturally add bucket address when necessary for new content. The outstanding merits of HABONAB are no Hash conflicts and avoiding Hash function. In addition, HABONAB has quality of high efficiency with respect to certain database such as sails record of supermarket. Based on the CNKI and Wan-fang Doctoral Dissertation database, those doctoral dissertations during 1982 to 2004 are retrieved by subject title“grey system”, then the HABONAB is applied to obtain the deformed Hash table which reflect the development of grey system during the 23 years. From the Hash table, it is clearly shown that grey system is widely used, however, the applied contents of grey system are mainly limited to grey relational analysis, GM(1,1) and its derived models. In addition, the hash table may serve as useful reference material for people who engage in grey system’s research.5. An application of grey target theory.Fitting-residence city, a concept formally raised in Overall Urban Planning of Beijing (2004~2020) in 2005, caused strong repercussions once raised. A series of articles have focused on the research of understanding and the determination of measure criteria of fitting-residence city. From the qualitative aspect, This paper, combined with the statistic data offered by China Statistical Yearbook and the Grey Target Theory, chose representative index series and made an analysis on fitting-residence city from the quantitative aspect.更多还原