Research On Some Fundamental Problems Of Grey System By Statistical View

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.