Ordering fuzzy sets over the real line: An approach based on decision making under uncertainty

The ordering of fuzzy sets over the real line is approached from the point of view of ordering intervals rather than ordering numbers. First, the maximax and maximin criteria which are commonly used for ordering intervals are expressed in terms of characteristic functions. These criteria and the Hurwicz criterion for decision making under complete ignorance are then reformulated in a manner that allows for their generalization to the fuzzy case. Transitivity is established for these ordering rules. A criterion based on the principle of diminishing marginal utility is also presented.     

Transitive fuzzy orderings of fuzzy numbers

In this paper we introduce a generalization of the Baas-Kwakernaak index by replacing the min operation in their definition by a t-norm. Some properties of the thus defined induced fuzzy ordering are established. In particular, it is shown that restrictions of the induced fuzzy ordering on some special classes of fuzzy numbers are reflexive fuzzy orders.     

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R³ with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies.     

Exact choice and fuzzy preferences

Agents often have to make exact choices on the basis of vague preferences. Therefore analysis of the way in which exact choices are induced by vague preferences is of considerable interest. In this paper we use the model of vague preferences as fuzzy orderings. One objective of this paper is conceptual in nature: we discuss several alternative notions of exact choice sets generated by a fuzzy preference ordering and corresponding notions of rationalizability of exact choices in terms of fuzzy preference orderings. The second objective of this paper is to explore conditions for rationalizability of exact choices in terms of a fuzzy preference ordering, under alternative definitions of such rationalizability.     

基于模糊数整体和局部特征的一种新排序方法

利用反映模糊数整体和局部特征的三个重要指标:模糊数的均值,截集的中点和扩展,本文提出一种新的模糊数排序方法.该方法将每个模糊数独立地映射到实数轴上,得到一个以数字大小为基准的自然顺序,不仅体现决策者对各排序指标的偏好,而且无需对模糊数进行两两比较,计算简便,易于理解,尤其是对三角和梯形模糊数而言,数值实验表明该方法在一定程度上克服了已有方法的缺陷.     

Questions of cardinality of finite fuzzy sets

In this paper we focus our attention on finite fuzzy sets. A complete, simple and easily applicable cardinality theory for them is presented. Questions of equipotency and non-classically understood cardinal numbers of finite fuzzy sets are discussed in detail. Also, problems of arithmetical operations (addition, subtraction, multiplication, division, and exponentiation) on as well as ordering relation for those cardinals are carefully investigated.     

Associative monotonic operations in fuzzy set theory

The properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets, on fuzzy numbers and on fuzzy probabilistic sets.     

Ramsey Numbers for Partially-Ordered Sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.     

Fuzzy Optimization Problems Based on Ordering Cones

The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients.     

On some chains of fuzzy sets

A partial order relation σ is defined in the set $\text{F}$(X) of the fuzzy sets in X. If this ordering is induced in the subset F(X) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on $\text{F}$(X) is defined and a lattice structure is induced in the set of infinite chains in L. Furthermore the set F′(X) of the fuzzy sets of F(X) which assume value in a finite subset of the real interval [0,1] is considered and the following properties are stated: any chain of elements of F′(X) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F′(X). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure.     

A fuzzy ordering on multi-dimensional fuzzy sets induced from convex cones

A fuzzy ordering for fuzzy sets on is presented by a fuzzy relation on which is induced by closed convex cones. The suitability of the fuzzy order is discussed using the axioms A₁–A₇ in (Fuzzy Sets and Systems 118 (2001) 375). For fuzzy sets on which are incomparable with respect to the fuzzy order, a method to evaluate the degree of satisfaction regarding the fuzzy order is presented by using a subsethood degree. Approximation by discrete cases is discussed for numerical calculation on the degree of the fuzzy order. Numerical examples are also given to illustrate our idea.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

RM approach for ranking of generalized trapezoidal fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization and forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples, it is proved that ranking method proposed by Chen and Chen (Expert Systems with Applications 36 (3): 6833) is incorrect. The main aim of this paper is to propose a new approach for the ranking of generalized trapezoidal fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as an RM approach. The main advantage of the proposed approach is that the proposed approach provides the correct ordering of generalized and normal trapezoidal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets and Systems 118 (3): 375).     

基于三角模糊数序列的灰色预测模型

以GM(1,1)模型为代表的灰色预测模型是以精确数序列为基础,难以满足实际需要.为了使灰色模型适应于模糊数序列,具体给出了一种基于三角模糊数序列的建模方法,这种方法也可以实现对二元区间模糊数和梯形模糊数序列的建模.首先由三角模糊数序列得出三个含有等量信息的精确数序列:重心序列、隶属函数的覆盖面积序列和中界点序列,对这三个序列分别建模后,再导出原始三角模糊数序列的三个界点的预测模型.这种建模方法既保持了模糊数的整体性又提高了建模序列的光滑度,提高了预测精度.最后进行了多组随机三角模糊数序列的数据模拟,验证了模型的有效性.     

Lexicographically ordered trees

We characterize trees whose lexicographic ordering produces an order isomorphic copy of some sets of real numbers, or an order isomorphic copy of some set of ordinal numbers. We characterize trees whose lexicographic ordering is order complete, and we investigate lexicographically ordered ω-splitting trees that, under the open-interval topology of their lexicographic orders, are of the first Baire category. Finally we collect together some folklore results about the relation between Aronszajn trees and Aronszajn lines, and use earlier results of the paper to deduce some topological properties of Aronszajn lines.     

Preference relations on a set of fuzzy utilities as a basis for decision making

In fuzzy decision problems, we often encounter situations of choosing among alternatives which are assigned fuzzy utilities. These problems have been approached using fuzzy implications or direct comparisons among fuzzy utilities. In the literature, however, there are few attempts to investigate the issues addressing reasonable choice or reasonable ordering using fuzzy sets theory. This paper first introduces some fundamental properties of fuzzy binary relations and certain conditions of reasonable orderings of fuzzy utilities. Then a method for constructing a fuzzy preference relation on a given set of fuzzy utilities is proposed for the sake of rational decision making. This procedure employs the concepts of the extended minimum and the Hamming distance between the greatest upper sets or the greatest lower sets of fuzzy utilities. Finally it is shown that the proposed fuzzy preference relations have reasonable properties as fuzzy orderings for decision making.     

两类判断矩阵转换研究

针对多属性决策中的常用三种类型(实数型、区间型、三角模糊型)判断矩阵,提出了具有一致性的两类(互反、互补)判断矩阵之间的转换新公式,并从保持方案序关系的意义上说明了两类判断矩阵的等价性.     

Elementary fuzzy calculus

In [2] Dubois and Prade introduced the notion of fuzzy numbers and defined the basic operations of addition, subtraction, multiplication, and division. A slightly modified definition of fuzzy numbers was presented in [4], and in that paper a metric was defined for this family of fuzzy sets. Another less restrictive definition of fuzzy numbers can be found in [1].In the present paper we consider fuzzy numbers from a somewhat different perspective. Basically, we shall view fuzzy numbers in a topological vector space setting. Using the customary vector space operations together with the metric given in [4] we will define differentiation and integration of fuzzy-valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration.     

Comparison of topological and uniform structures for fuzzy numbers and the fixed point problem

The goal of this paper is to compare certain topological and uniform structures on sets of fuzzy numbers. Since fuzzy sets can be viewed as a kind of subsets they can be endowed with uniform structures in ways similar to those used for classic subsets. The Hausdorff construction is one such approach. The structures we study are the myope structure, the uniform convergence of the cuts in the sense of Hausdorff, and a structure induced by a metric defined by Goetschel and Voxman. We show how these structures are related to fixed point problems for some classes of fuzzy numbers.     

Ranking fuzzy quantities based on the angle of the reference functions

Ordering fuzzy quantities and their comparison play a key tool in many applied models in the world and in particular decision-making procedures. However a huge number of researches is attracted to this filed but until now there is any unique accepted method to rank the fuzzy quantities. In fact, each proposed method may has some shortcoming. So we are going to present a novel method based on the angle of the reference functions to cover a wide range of fuzzy quantities by over coming the draw backs of some existing methods. In the mentioned firstly, the angle between the left and right membership functions (the reference functions) of every fuzzy set is called Angle of Fuzzy Set (AFS), and then in order to extend ranking of two fuzzy sets the angle of fuzzy sets and α-cuts is used. The method is illustrated by some numerical examples and in particular the results of ranking by the proposed method and some common and existing methods for ranking fuzzy sets is compared to verify the advantage of the new approach. In particular, based on the results of comparison of our method with well known methods which are exist in the literature, we will see that against of most existing ranking approaches, our proposed approach can rank fuzzy numbers that have the same mode and symmetric spreads. In fact, the proposed method in this paper can effectively rank symmetric fuzzy numbers as well as the effective methods which are appeared in the literature. Moreover, unlike of most existing ranking approaches, our proposed approach can rank non-normal fuzzy sets. Finally, we emphasize that the concept of fuzzy ordering is one of key role in establishing the numerical algorithms in operations research such as fuzzy primal simplex algorithms, fuzzy dual simplex algorithms and as well as discussed in the works of Ebrahimnejad and Nasseri and coworkers , , , , ,  and .