面向四棱直纹模糊信息处理的分析学原理和优化模型

There are numerous excellent studies on the fuzzy analysis and fuzzy optimization theory. However, a large number of results are limited to the case of one dimensional fuzzy numbers, which also restricts the development of fuzzy analysis and its application. Taking into account the fact that 2 dimensional fuzzy numbers can provide more flexibility to represent imprecise information, as well as apply in fuzzy optimization and so on. In this project, we study the function representation and algebraic operation of four-edges-fuzzy-numbers (a class of special 2 dimensional fuzzy numbers), and then the differentiability and integral of four-edges-fuzzy-number functions as well as fuzzy optimization are developed by functional analysis, real analysis and set-valued analysis. The major study context in this project includes those: (1) We established the differentiability of four-edges-fuzzy-number functions based on the function representation and algebraic operation of four-edges-fuzzy-numbers. Furthermore, a series of problems about the differential of four-edges-fuzzy- number functions are investigated in detail. (2) Based on the distance formula of four-edges-fuzzy-numbers which is set up in this project, Riemann integral of four-edges-fuzzy-number functions and its basic properties are researched. (3) The project makes a preliminary study of convexity for four-edges-fuzzy-number functions and fuzzy optimization, which are based on the ranking of four-edges- fuzzy-numbers. The research in this project cannot only encourage the development of fuzzy analysis, but also provide a new attempt for the application of four-edges-fuzzy -numbers in fuzzy optimization , fuzzy decision and so on.

关于模糊分析学和模糊优化理论已有很多优秀的研究成果，然而大量的结果仅局限于一维模糊数的情形，从而制约了模糊分析学及其应用的发展。考虑到2维模糊数在处理不确定信息时的灵活性和在模糊优化等领域的应用。本项目应用泛函分析、实分析、集值分析等手段，通过对四棱直纹模糊数（一类特殊2维模糊数）的函数表示与代数运算等问题的研究，发展四棱直纹模糊数值函数的分析学和优化理论。主要研究内容包括：（1）从研究四棱直纹模糊数的函数表示与代数运算出发，构建四棱直纹模糊数值函数的微分，进而探讨与之相关的一系列问题。（2）在建立四棱直纹模糊数空间距离的基础上，研究四棱直纹模糊数值函数的Riemann型积分及其性质。（3）基于四棱直纹模糊数的序结构，对四棱直纹模糊数值函数的凸性和模糊优化理论进行初步探究 。本项目的研究工作不仅可以促进模糊分析学的发展，而且为四棱直纹模糊数在模糊优化、模糊决策等方面的应用提供新的尝试。

模糊分析学和模糊优化理论的很多优秀研究成果仅局限于一维模糊数的情形，从而制约了模糊分析学及其应用的发展。考虑到二维模糊数在处理不确定信息时的灵活性和在模糊优化等领域的应用。本项目在国家自然科学基金的资助下，主要对四棱直纹模糊数（一类特殊二维模糊数）的函数表示与代数运算进行了研究，目的在于发展四棱直纹模糊数值函数的分析学和优化理论，为模糊分析学及其应用的进一步发展做基础性研究。三年来，基于n维方模糊数值函数在研究一类线性模糊数值函数微积分的基础上，我们对四棱直纹模糊数的函数表示与代数运算进行了初步研究，这对发展模糊分析学及其应用有重要的价值。在本项目研究过程中，发表SCI论文两篇，其中一篇被《Fuzzy Sets and Systems》杂志录用发表。本项目的研究成果为高维模糊数理论发展提供了重要的科学依据。