D.C.隶属函数模糊集及其应用(II)——D.C.隶属函数模糊集的万能逼近性

本文是D.C.隶属函数模糊集及其应用系列研究的第二部分。指出在实际问题中普遍选用的三角形、半三角形、梯形、半梯形、高斯型、柯西型、S形、Z形、π形隶属函数模糊集等均为D.C.隶属函数模糊集,建立了D.C.隶属函数模糊集对模糊集的万有逼近性。探讨了D.C.隶属函数模糊集与模糊数之间的关系,给出了用D.C.隶属函数模糊集逼近模糊数的-εC e llina逼近形式,得到模糊数与D.C.函数之间的一个对应算子,指出了用模糊数表示D.C.函数的问题。     

D.C.隶属函数模糊集及其应用(Ⅱ)--D.C.隶属函数模糊集的万能逼近性

本文是D．C．隶属函数模糊集及其应用系列研究的第二部分。指出在实际问题中普遍选用的三角形、半三角形、梯形、半梯形、高斯型、柯西型、S形、Z形、π形隶属函数模糊集等均为D．C．隶属函数模糊集，建立了D．C．隶属函数模糊集对模糊集的万有逼近性。探讨了D．C．隶属函数模糊集与模糊数之间的关系，给出了用D．C．隶属函数模糊集逼近模糊数的e-Cellina逼近形式，得到模糊数与D．C．函数之间的一个对应算子，指出了用模糊数表示D．C．函数的问题。     

基于深度学习的直觉模糊集隶属度确定方法

直觉模糊集隶属度、非隶属度和犹豫度的确定方法是直觉模糊集理论与应用研究中一个十分重要的问题,其直接影响着相关方法的可扩展性及应用结果。然而,现有方法存在主观性强、标准难以统一等问题,并且大多基于模拟数据进行实验,难以应用至实际数据。针对上述问题以及大规模非结构化数据,提出一种基于深度学习的直觉模糊集隶属度、非隶属度和犹豫度确定方法。新方法克服了传统方法的技术和思维局限,拓展了直觉模糊集相关问题的研究思路,为其实际应用提供了更多可能。     

基于概率统计的模糊隶属函数计算研究

应用模糊数学方法解决实际问题的关键在于建立符合实际情况的隶属函数.常用计算模糊隶属函数的模糊统计法建立在大量调查数据和函数图形分析法的基础上,巨大的运算量和复杂的过程限制了模糊技术的实际应用.本文从统计学的角度提出了一种基于概率统计的模糊隶属函数计算方法,该方法相对简单并且能适应多种情况的隶属函数计算.最后通过实际算例对两种算法得到的隶属函数曲线进行了比对,验证了本文的方法具有较好的精度.     

基于隶属度与非隶属度交叉影响的直觉模糊集运算法则及其应用

考虑到不同直觉模糊集的隶属度与非隶属度之间可能存在着某些关联和相互影响,本文提出了直觉模糊集上的改进的加法运算、数乘运算、乘积运算和幂运算.在这些运算的基础上,我们重新给出了直觉模糊加权算术平均算子、直觉模糊有序加权算术平均算子、直觉模糊加权几何平均算子和直觉模糊有序加权几何平均算子的表达式,并研究了他们的一些性质.最后通过实例,说明了新的集成算子在决策应用中的有效性.     

模糊保并、保交映射的隶属函数

模糊关系与算子合成构成了模糊映射，即将模糊集映射为模糊集。本文证明了模糊保并、保交映射的隶属函数可以表示为模糊关系与算子的合成。     

连续模糊集的相关系数

相关系数在模糊决策领域发挥着重要的作用,但是其定义一直存在着问题。1985年,Murthy等~([1])提出了连续模糊集相关系数的两个计算公式,第一个公式与统计学中的相关系数具有相似的意义,但是其连续模糊隶属函数的均值定义是错误的。为此,借助于积分中值定理,定义了连续模糊隶属函数的均值以及方差和协方差,继而定义了连续模糊隶属函数的相关系数,从而彻底解决了Murthy等~([1])定义的第一个相关系数计算公式存在的问题。该相关系数与Chiang~([2])提出的离散隶属函数的相关系数一起,构成了完整的模糊集相关性理论。数值例子说明了,与Murthy~([1])第二个公式,Yu~([4])和Chaudhuri~([5])等提出的相关系数相比,我们提出的相关系数更合理有效。然后,将连续模糊隶属函数的相关系数概念推广到连续直觉模糊集,通过计算连续隶属函数以及连续非隶属函数的相关系数的平均值,定义了连续直觉模糊集的相关系数,该定义与Hung~([23])定义的离散直觉模糊集相关系数一起,构成了完整的直觉模糊集相关系数理论。最后,通过两个数值例子说明了连续直觉模糊集相关系数有效可行。     

基于粗糙隶属函数的粗糙逻辑与推理

模糊推理是将论域上的模糊集看作是一个模糊命题,将对象的隶属度看作是命题的真值,进行的不确定推理;模糊集合的隶属度大都是通过经验或专家给出的,带有一定的主观性。为此,本文将粗糙隶属函数引入逻辑推理,它不需要任何先验信息,因此推理结果更具客观性。最后以实际例子说明了此方法的可行性和合理性。     

基于可变模糊集理论的城市用地适用性评估

依据可变模糊集理论,构造了城市用地适用性评估的相对差异函数模型和相对隶属函数模型,建立了求解综合相对隶属度的可变模糊评估模型,并应用该模型对2组待开发利用用地适用性进行了评估.研究表明:可变模糊评估是依据对立相对隶属函数来描述模糊概念,通过参数的可变性,自我验证可变模糊评估方法的可靠性,并通过级别特征值表达了评估对象属于某级别的程度,使评估结果更为精细,从而为城市用地适用性评估提供了一种有效的新方法.     

基于率模可靠性的客观确定非对称概型功能函数隶属函数的方法

摒弃目前以主观方法给出功能函数对结构安全模糊集隶属函数的做法，提出并从理论上证明了：当功能函数具有非对称概型时，将功能函数的线性函数假想为集值统计的随机集边界点，通过定积分运算获得隶属函数的方法。算例充分说明文中方法的科学性和客观性。     

Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems

The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.     

On fuzzy goals and maximizing decisions in stochastic optimal control

Fuzzy set theory has developed significantly in a mathematical direction during the past several years but few applications have emerged. This paper investigates the role of fuzzy set theory in certain optimal control formulations. In particular, it is shown that the well-known quadratic performance criterion in deterministic optimal control is equivalent to the exponential membership function of a certain fuzzy decision (set). In a stochastic setting, similar equivalences establish new definitions for “confluence of goals” and “maximizing decision” in fuzzy set theory. These and other definitions could lead to the development of a more applicable theory of fuzzy sets.     

Linear programming with multiple fuzzy goals

This paper describes the use of fuzzy set theory in goal programming (GP) problems. In particular, it is demonstrated how fuzzy or imprecise aspirations of the decision maker (DM) can be quantified through the use of piecewise linear and continuous functions. Models are then presented for the use of fuzzy goal programming with preemptive priorities, with Archimedean weights, and with the maximization of the membership function corresponding to the minimum goal. Examples are also provided.     

Correspondence between fuzzy measures and classical measures

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra $\text{σ}$ are discussed.     

Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process

Pythagorean fuzzy set, an extension of the intuitionistic fuzzy set which relax the condition of sum of their membership function to square sum of its membership functions is less than one. Under these environment and by incorporating the idea of the confidence levels of each Pythagorean fuzzy number, the present study investigated a new averaging and geometric operators namely confidence Pythagorean fuzzy weighted and ordered weighted operators along with their some desired properties. Based on its, a multi criteria decision-making method has been proposed and illustrated with an example for showing the validity and effectiveness of it. A computed results are compared with the aid of existing results.     

Optimal stopping in a fuzzy environment

The problem under consideration is that of optimally controlling and stopping either a deterministic or a stochastic system in a fuzzy environment. The optimal decision is the sequence of controls that maximizes the membership function of the intersection of the fuzzy constraints and a fuzzy goal. The fuzzy goal is a fuzzy set in the cartesian product of the state space with the set of possible stopping times. Dynamic programming is applied to yield a numerical solution. This approach yields an algorithm that corrects a result of Kacprzyk.     

Operation properties and -equalities of complex fuzzy sets

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define δ-equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be δ-equal if the distance between them is less than 1-δ. This paper shows how various operations between complex fuzzy sets affect given δ-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of δ-equalities of complex fuzzy sets in practice.     

On the calculation of a membership function for the solution of a fuzzy linear optimization problem

In the present paper the fuzzy linear optimization problem (with fuzzy coefficients in the objective function) is considered. Recent concepts of fuzzy solution to the fuzzy optimization problem based on the level-cut and the set of Pareto optimal solutions of a multiobjective optimization problem are applied. Chanas and Kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem, which is based on calculating the sum of lengths of certain intervals. The purpose of this paper is to determine a method for realizing this idea. We derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered.     

Multi-objectives fuzzy models for designing 3D trajectory in horizontal wells

In this paper, multi-objective models for designing 3D trajectory of horizontal wells are developed in a fuzzy environment. Here, the objectives of minimizing the length of the trajectory and the error of entry target point are fuzzy in nature. Some parameters, such as initial value, end value, lower bound and upper bound of the curvature radius, tool-face angle and the are length of each curve section, are also assumed to be vague and imprecise. The impreciseness in the above objectives have been expressed by fuzzy linear membership functions and that in the above parameters by triangular fuzzy numbers. Models have been solved by the fuzzy non-linear programming method based on Zimmermann [1] and Lee and Li [2]. Models are applied to practical design of the horizontal wells. Numerical results illustrate the accuracy and efficiency of the fuzzy models.