一类模糊数系数矩阵的模糊线性方程组的迭代算法

讨论模糊线性方程组X=A~X U解的存在条件及其迭代算法(其中,A~是以模糊数为元素的n×n矩阵,未知量X和常量U都是以模糊数为元素的n维向量,并且其加法和乘法均由Zadeh的扩张原理定义)。首先研究解的存在条件,尔后探讨求解的迭代算法及误差估计。     

一类模糊线性方程组的迭代算法及其误差估计

讨论模糊线性方程组X=AX+U解的存在条件及其迭代算法(其中,A是区间数为元素的n阶矩阵,未知量X和常量U都是以模糊数为元素的n维向量,并且其加法和乘法均由Zadeh的扩张原理定义).首先研究解的存在条件,尔后探讨求解的迭代算法及误差估计.     

模糊随机微分方程解的存在唯一性

将实数空间上的随机微分方程推广到模糊数空间,即为模糊随机微分方程.本文用Picard迭代的方法证明了其解的存在唯一性定理,推广了现有文献的结果,并且给出Picard迭代近似解误差的估计式.     

基于有界算子的三角模糊数运算

本文用sup-⊙(有界算子)合成代替通常模糊数运算中的sup-min合成,对三角模糊数讨论其加减乘除算术运算,证明了其和、差与数乘仍是三角模糊数,得到了积、商仍为三角模糊数的条件。并给出一个例子,说明以三角模糊数为系数的线性方程组有可能存在三角模糊数解。     

模糊映射的广义混合型强变分不等式

研究了新的一类模糊映射的广义混合型强变分不等式问题。证明了这类问题解的存在定理和收敛定理,给出解的带误差的Ishikawa型迭代算法。     

一类系数为梯形模糊数的两层多随从线性规划模型

针对一类系数为梯形模糊数的两层多随从线性规划问题,利用模糊结构元理论定义了模糊结构元加权序,证明了一类系数为梯形模糊数的两层多随从线性规划问题的最优解等价于两层多随从线性规划问题的最优解.根据线性规划的对偶定理和互补松弛性质,得到了两层多随从线性规划模型的最优化条件.最后,利用两层多随从线性规划模型的最优化条件,设计了求解一类系数为梯形模糊数的两层多随从线性规划问题的算法,并通过算例验证了该方法的可行性和合理性.     

C模糊系统及其摄动性与稳定性

还原性、摄动性与稳定性是模糊系统的重要属性,对其进行深入研究将有助于研制具有良好性能的模糊控制器.称能保持推理还原性的模糊系统为C模糊系统.设"if x is Ai then y is Bi,i=1,…,n"是一模糊规则组,模糊推理通过合成运算.来实现,则C模糊系统的输入输出关系可表示为A o R=B,其中A是模糊输入,B是模糊输出,R是模糊关系方程组Ai o X=Bi(i=1,…,n)的-个解.本文在模糊划分的条件下,分别就o为两种常用的合成算法∨-*和∧-→(*和→是BL代数中的伴随对)的情况讨论了上述方程组的解,从而给出C模糊系统可能存在的各种情形.在此基础上,给出了模糊系统的摄动性与稳定性的定义,并证明了基于前述两种合成运算的C模糊系统的若干特征定理,给出具有良好稳定性的C模糊系统的表示式.最后,研究了C模糊系统的解析形式,给出这类模糊系统的插值表示.     

一类系数为梯形模糊数的两层线性规划

讨论了一类系数为梯形模糊数的两层线性规划问题,首先是利用模糊结构元理论将梯形模糊数去模糊化,将其转化成常规的两层线性问题,并验证其去模糊化后的常规的两层线性规划的最优解与系数为梯形模糊数的两层线性规划问题的最优解一致,并给出具体的算法,数例进行验证.     

模糊网络最大流算法研究

将模糊数差值B~-A~视为模糊方程X~+A~=B~的解,进而探讨了模糊方程的求解问题,并基于目的规划理论,给出了模糊方程的广义解定义.运用目的规划的单纯型方法,得到了模糊方程广义解的计算公式及模糊方程广义解的若干性质.由模糊方程的广义解引申出了模糊数差值的定义.运用该定义将传统的网络最大流算法推广到模糊环境.结果表明,模糊数差值定义,克服了基于扩展原理意义下的模糊运算所产生的各种问题,解决了这些传统理论方法的拓展问题.     

一般梯形模糊逼近算子及其在模糊运输问题中的应用

研究运输成本信息为一般模糊数的模糊运输问题.首先,在保持一般模糊数的核不变的条件下,建立一般模糊数与一般梯形模糊数的距离最小优化模型,通过求解模型得到一般模糊数的一般梯形模糊逼近算子,并给出该逼近算子具有的性质如数乘不变性、平移不变性、连续性等.然后利用该逼近算子将一般模糊运输信息表转换成一般梯形模糊运输信息表,再根据已有GFLCM和GFMDM算法得到模糊运输问题的近似最优解,最后给出具体算例分析说明方法的有效性和合理性.     

硬聚类和模糊聚类的结合——双层FCM快速算法

模糊c均值(FCM)聚类算法在模式识别领域中得到了广泛的应用,但FCM算法在大数据集的情况下需要大量的CPU时间,令用户感到十分不便,提高算法的速度是一个急待解决的问题。本文提出的双层FCM聚类算法是一种快速算法,它体现了硬聚类和模糊聚类的结合,以硬聚类的结果对模糊聚类的初始值进行指导,从而明显地缩短了迭代过程。双层FCM算法所用的CPU时间仅为FCM算法的十三分之一,因而具有很强的实用价值。     

Solution and Application of the Matrix Equation AX~=b~

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Evaluation of fully fuzzy matrix equations by fuzzy neural network

In this paper, a new hybrid method based on fuzzy neural network for approximate solution of fully fuzzy matrix equations of the form AX=D$\mathit{AX}=D$, where A and D are two fuzzy number matrices and the unknown matrix X is a fuzzy number matrix, is presented. Then, we propose some definitions which are fuzzy zero number, fuzzy one number and fuzzy identity matrix. Based on these definitions, direct computation of fuzzy inverse matrix is done using fuzzy matrix equations and fuzzy neural network. It is noted that the uniqueness of the calculated fuzzy inverse matrix is not guaranteed. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution of fuzzy matrix equations that supposedly has a unique fuzzy solution, a simple algorithm from the cost function of the fuzzy neural network is proposed. To illustrate the easy application of the proposed method, numerical examples are given and the obtained results are discussed.     

Shortest Path Problem on a Network with Imprecise Edge Weight

A network with its arc lengths as imprecise number, instead of a real number, namely, interval number and triangular fuzzy number is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. To obtain a fuzzy shortest path from a source vertex to all other vertices, a common algorithm is developed which works well on both types of imprecise numbers under consideration. In the proposed algorithm, a decision-maker is to negotiate with the obtained fuzzy shortest paths according to his/her view only when the means are same but the widths are different of the obtained paths. Otherwise, a fuzzy optimal path is obtained to which the decision-maker always satisfies with different grades of satisfaction. All pairs fuzzy shortest paths can be found by repeated use of the proposed algorithm.     

The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties

Many multiple attribute decision analysis (MADA) problems are characterised by both quantitative and qualitative attributes with various types of uncertainties. Incompleteness (or ignorance) and vagueness (or fuzziness) are among the most common uncertainties in decision analysis. The evidential reasoning (ER) approach has been developed in the 1990s and in the recent years to support the solution of MADA problems with ignorance, a kind of probabilistic uncertainty. In this paper, the ER approach is further developed to deal with MADA problems with both probabilistic and fuzzy uncertainties.In this newly developed ER approach, precise data, ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure, leading to a fuzzy belief decision matrix. A utility-based grade match method is proposed to transform both numerical data and qualitative (fuzzy) assessment information of various formats into the fuzzy belief structure. A new fuzzy ER algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix, resulting in an aggregated fuzzy distributed assessment for each alternative. Different from the existing ER algorithm that is of a recursive nature, the new fuzzy ER algorithm provides an analytical means for combining all attributes without iteration, thus providing scope and flexibility for sensitivity analysis and optimisation. A numerical example is provided to illustrate the detailed implementation process of the new ER approach and its validity and wide applicability.     

基于区间与模糊Shapley值的合作收益分配策略

将经典Shapley值三条公理进行拓广,提出具有模糊支付合作对策的Shapley值公理体系。研究一种特殊的模糊支付合作对策,即具有区间支付的合作对策,并且给出了该区间Shapley值形式。根据模糊数和区间数的对应关系,提出模糊支付合作对策的Shapley值,指出该模糊Shapley值是区间支付模糊合作对策的自然模糊延拓。结果表明:对于任意给定置信水平α,若α=1,则模糊Shapley值对应经典合作对策的Shapley值,否则对应具有区间支付合作对策的区间Shapley值。通过模糊数的排序,给出了最优的分配策略。由于对具有模糊支付的合作对策进行比较系统的研究,从而为如何求解局中人参与联盟程度模糊化、支付函数模糊化的合作对策,奠定了一定的基础。     

基于模糊测度与累积前景理论的区间二型模糊多准则决策方法

针对准则值为区间二型模糊数且准则间存在关联关系的风险型多准则决策问题, 本文提出一种基于模糊测度理论与累积前景理论的区间二型模糊多准则决策方法。首先, 为全面反映准则间的关联关系, 本文提出Shapley区间二型模糊Choquet积分算子, 并证明该算子的一些性质。其次, 为反映专家行为偏好, 本文定义区间二型模糊前景效应与前景价值函数, 并提出累积前景Shapley区间二型模糊Choquet积分算子。然后, 为确定准则集的模糊测度, 本文建立基于区间二型模糊双向投影与Shapley函数的权重优化模型。在此基础上, 本文给出一种用于解决准则值为区间二型模糊数, 准则间存在关联关系, 专家存在风险偏好以及准则权重部分未知的多准则决策方法。最后, 通过风险投资实例佐证所提出的方法的适用性与科学性。     

Filtering-based robust fault detection of fuzzy jump systems

This paper studies the robust fault detection filter (RFDF) design problems for uncertain nonlinear Markov jump systems with state delays and parameter uncertainties. By means of Takagi-Sugeno fuzzy models, the dynamics of filtering error generator and the fuzzy RFDF system are constructed. With the aid of the selected weighting matrix function, the design objective is to find an optimal RFDF which results in a minimal difference between the reference model (ideal solution) and the RFDF (real solution) to be designed. A sufficient condition is firstly established on the stochastic stability by using stochastic Lyapunov-Krasovskii functional approach. Then in terms of linear matrix inequalities techniques, sufficient conditions on the existence of fuzzy RFDF are presented and proved. Finally, the design problem is formulated as an optimization algorithm. Simulation results illustrate that the proposed RFDF can detect the faults shortly after the occurrences.     

基于粗糙集与模糊集的连续值域决策表的离散化算法

将粗糙集理论与模糊集理论结合起来,给出一种连续值域决策表的离散化算法。该算法从已知数据的初始决策系统出发,首先构造对像的相似矩阵,然后根据相似矩阵的传递闭包及粗糙集正域的思想得出决策表的条件类,再根据条件类将连续值决策表化为区间值决策表,最后根据各区间值将连续值域决策表化为离散决策表。     

Inner and outer bounds for the solution set of parametric linear systems

Consider linear systems involving affine-linear dependencies on interval parameters. Presented is a free C-XSC software implementing a generalized parametric fixed-point iteration method for verified enclosure of the parametric solution set. Some specific features of the corresponding algorithm concerning sharp enclosure of the contracting matrix and inner approximation of the solution enclosure are discussed.