模糊分析中的结构元方法(Ⅰ)

Method of structuring element in fuzzy analysis(Ⅰ)

模糊数和模糊值函数是模糊分析中的最基本概念,在模糊分析中模糊数与模糊值函数的运算通常都是基于扩张原理的形式给出的,而模糊值函数的微分和积分也都是基于区间值函数的相应结果利用表现定理形式给出的。它们的共同特点都是对元素遍历某个条件所对应的全体结果进行运算,或取l遍历[0,1]所对应的全体结果的并运算。这种运算中的遍历过程给模糊分析理论的应用带来了极大的不便,使得操作无法进行。因此,需要寻找模糊数和模糊值函数运算的其它有效的表达方式。本文提出了模糊结构元的概念, 并研究了模糊结构元的性质,给出了模糊数的结构元表现定理。利用模糊数的结构元表现形式可以使模糊数的运算变成普通实数与模糊结构元之间的运算,使得过去必须依赖扩张原理和表现定理来刻画的模糊数运算变得更加简单与直观。模糊结构元理论与技术不仅仅为模糊分析计算的简化提供了工具,同时也为模糊分析理论与应用的研究开创了一条新的途径。

Fuzzy number and fuzzy-valued functions are the basic concepts in fuzzy analysis. Generally, the operations of fuzzy numbers and fuzzy-valued functions are provided with the form based on the extension principle. In paper [1], presented the concept of fuzzy structuring element, and discussed some of its useful properties, proved the structuring element representation theorem of fuzzy number and fuzzy-valued function. In this paper, gave the differential and integral calculate method of fuzzy-valued functions using fuzzy structuring element method. The theory of fuzzy structuring element not only provides a simple and effective tool for the fuzzy analysis and calculations, but also makes a new path to study the theory and application of fuzzy analysis.