圈圈学论文（一）灰数概念及其运算

　　1.灰数的概念
　　灰色系统主要用灰数、灰色方程、灰色矩阵、灰色函数等来描述。其中，灰数是灰色系统的最基本单元。在系统研究过程中，由于人的认知能力有限，对反映系统行为的信息难以完全认知，所以人们只知道系统元素或参数的取值范围，通常把这种只知道取值范围而不知其确切的数成为灰数。在应用中，灰数实际上指在某个区间或某个一般数集内取值的不确定数。
　　灰数主要有以下几类：
　　（1）仅有下界的灰数。仅有下界而无上界的灰数记为：
　　（2）仅有上界的灰数。仅有上街而无下界的灰数记为：
　　（3）区间灰数。既有上确界又有下确界的灰数，记为：
　　（4）连续灰数与离散灰数。在某一期间内取有限个数值的灰数称为离散灰数。取值连续充满某一区间的灰数称为连续灰数。
　　（5）黑数与白数。黑数和白数是特殊的灰数。
　　（6）本征灰数和非本征灰数。本征灰数是指不能或暂时还不能找到一个白数作为其＂代表＂的灰数。非本征灰数是指凭借先验信息或是某种手段，可以找到一个白数作为其＂代表＂的灰数。称此白数为相应灰数的白化值，
　　2.相关定理
　　（1）预期概率度。
　　（2）区间灰数之间的大小比较：
　　（3）区间灰数之间的距离：
　　（4）设某灰数为闭区间或开区间的并集。扩展灰数（EGN）可表示为如下所示：
　　扩展灰数具有以下性质：
　　3.运算法则
　　灰数的四则运算法则和数乘运算法则，统称为灰数的基本运算法则。
　　英文学习：
　　1. The concept of gray number
　　The gray system is mainly described by gray numbers, gray equations, gray matrices, gray functions, etc. Among them, the gray number is the most basic unit of the gray system. In the process of system research, due to the limited cognitive ability of people, it is difficult to fully understand the information reflecting the system behavior. Therefore, people only know the value range of system elements or parameters. Usually, they only know the value range but do not know the value range. The exact number becomes the gray number. In application, gray numbers actually refer to uncertain numbers that take values in a certain interval or a certain general number set.
　　The gray numbers mainly fall into the following categories:
　　(1) Only the gray numbers of the lower bound. The gray number with only a lower bound but no upper bound is recorded as:
　　(2) There are only upper bound gray numbers. The gray number that only goes to the street without the lower bound is recorded as:
　　(3) Interval gray number. The grey number with both supremum and infimum is recorded as:
　　(4) Continuous gray number and discrete gray number. The gray number that takes a limited number of values in a certain period is called the discrete gray number. The ash number whose value is continuously filled in a certain interval is called continuous ash number.
　　(5) Black number and white number. The black number and white number are special gray numbers.
　　(6) Intrinsic gray number and extrinsic gray number. The intrinsic gray number refers to the gray number that cannot or temporarily cannot find a white number as its ＂representative＂. The extrinsic gray number refers to the gray number that can find a white number as its ＂representative＂ by virtue of prior information or some means. Call this white number the whitening value of the corresponding gray number,
　　2. Related theorems
　　(1) Expected probability.
　　(2) Comparison of the size of the interval gray numbers:
　　(3) The distance between the gray numbers in the interval:
　　(4) Let a certain gray number be the union of closed intervals or open intervals. The extended gray number (EGN) can be expressed as follows:
　　The extended gray number has the following properties:
　　3. Algorithm
　　英文：谷歌翻译
　　参考资料：
　　[1]张东兴. 三参数区间灰数信息下的多属性决策方法[D].河南农业大学,2019.
　　[1]  Zhou H , Wang J Q , Zhang H Y . Grey stochastic multi-criteria decision making based on regret theory and TOPSIS[J]. International Journal of Machine Learning and Cybernetics, 2015, 8(2):1-14.
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