基于信息密度的广义不确定直觉模糊集成算子及其应用

Intuitionistic fuzzy set is the generalization of traditional fuzzy set, which includes the degree of membership, the degree of non-membership and degree of indeterminancy. Thus it can describe the essence of fuzzy things in details. For complex system, it usually uses the interval intuitionistic fuzzy number, triangular intuitionistic fuzzy number or trapezoidal intuitionistic fuzzy number to characterize the fuzzy information. In view of the interactions between membership degree and non-membership degree, which is caused by indeterminancy degree of uncertain intuitionistic fuzzy number, this project proposes novel operational rules of uncertain intuitionistic fuzzy number. Aim at the density structure of uncertain intuitionistic fuzzy numbers and the fusion of various kinds of intuitionistic fuzzy numbers information, we construct an information density weighting model and propose the definition of density-based generalized unceratain intuitionistic fuzzy information aggregation operators. The operational rules of intuitionistic fuzzy numbers and the mathematic properties of aggregation operator, which includes monotonicity, idempotency, homogeneity, boundedness and commutativity, are studied. Furthermore, we define the orness measure and dispersion measure of the proposed operators, by which we can characterize the attitude of the valuators. The developed operators can be applied to the clustering and comprehensive assessment of ecological environment. We believe that this project can improve the operational rules of intuitionistic fuzzy sets and enrich the theory of information aggregation operators. This project has many practical applications in real life.

直觉模糊集是传统模糊集的推广，它含有隶属度、非隶属度和犹豫度三个因素的信息，因而能更加细腻地描述模糊事物的本质。对于复杂的系统，其信息表征形式往往是区间直觉模糊数、三角直觉模糊数、梯形直觉模糊数等不确定的直觉模糊信息。本项目拟考虑不确定直觉模糊数的犹豫度对隶属度和非隶属度的交叉影响，提出新的不确定直觉模糊数运算规则，针对不确定直觉模糊数的密度结构特征和多种不确定直觉模糊信息的融合，构建信息密度赋权模型，并提出基于信息密度的广义不确定直觉模糊集成算子的概念。研究直觉模糊数的运算规则和集成算子的若干数学性质，包括单调性、幂等性、齐次性、介值性、置换不变性等。同时定义该类算子的orness测度和离散测度，用于刻画评价者的态度特征，并将其应用于生态环境的聚类和综合评价之中。本项目的研究不仅可以丰富和完善直觉模糊集的运算规则和信息集成算子相关理论，而且在实际中具有较强的应用价值。

本研究在探讨语言数据、区间数据和直觉模糊数据运算法则基础上，结合模块化信息集成理论，提出一系列新的信息集成算子，如有序Quai模块化平均（OQMA)）算子、广义有序模块化平均（GOMA）算子、广义语言加权对数平均（GLWLA）算子、广义语言有序加权对数平均（GLOWLA）算子、基于信息密度的广义不确定直觉模糊集成(DGUIFA)算子。同时通过数理推导，证明这些算子具有单调性、幂等性、有界性性、置换不变性等数理性质。进而，构造相应的orness测度，用以反映算子的乐观（or-like）程度。另外，基于新的区间乘性偏好关系的连续对数相容性定义和组合优化模型实现区间乘性偏好信息的赋权，同时引入合作博弈理论实现语言偏好信息的赋权。最后建立基于这些算子的群决策方法，并通过实证分析验证方法的适用性和有效性。本项目的研究丰富和完善信息集成算子相关理论，为决策者选择最优的多属性决策方案提供支持。