Reachable solution set of a fuzzy relation equation

The largest element of the solution set of a fuzzy relation equation has been found by E. Sanchez (Inform. and Control30 (1976), 38–48) but the smallest element does not exist. It is difficult to expose the solution of the fuzzy relation equation. In the case of the determinate relation equations, complete consequences have been found by Wang Peizhuang and Yuan Meng (“Relation Equation and Relation Inequalities,” Selected papers on fuzzy subsets, Beijing Normal University, March 1980). In the case of the fuzzy relation equations, Wang and Yuan have given a class of special solutions which probably possesses some minimality characterizations. In this paper, the reachable solution set of the fuzzy relation equation is given. For the fuzzy relation equation on the finite sets, a neat and efficient method for solving it is given.     

Decision-making with a fuzzy preference relation

In most decisio-making problems a preference relation in the set of alternatives is of a fuzzy nature, reflecting for instance on the fuzziness of experts estimates of the preferences. In this paper, the corresponding fuzzy equivalence and strict preference relations are defined for a given fuzzy non-strict preference relation in an unfuzzy set of alternatives which are used to introduce in a natural way the fuzzy set of nondominated alternatives. Two types of linearity of a fuzzy relation are introduced and the equivalence of the unfuzzy nondominated alternatives is studied. It is shown that unfuzzy nondominated solutions to the decision-making problem exist, provided the original fuzzy relation satisfies some topological requirements. A simple method of calculating these solutions is indicated.     

Monomial geometric programming with fuzzy relation equation constraints

Monomials are widely used. They are basic structural units of geometric programming. In the process of optimization, many objective functions can be denoted by monomials. We can often see them in resource allocation and structure optimization and technology management, etc. Fuzzy relation equations are important elements of fuzzy mathematics, and they have recently been widely applied in fuzzy comprehensive evaluation and cybernetics. In view of the importance of monomial functions and fuzzy relation equations, we present a fuzzy relation geometric programming model with a monomial objective function subject to the fuzzy relation equation constraints, and develop an algorithm to find an optimal solution based on the structure of the solution set of fuzzy relation equations. Two numerical examples are given to verify the developed algorithm. Our numerical results show that the algorithm is feasible and effective.     

Finding an inner estimation of the solution set of a fuzzy linear system

In this paper, we consider the problem of finding an inner estimation of the solution set of a fuzzy linear system with a real-valued coefficient matrix and a fuzzy-valued right-hand side vector. The proposed idea is based on the utilization of interval Gaussian elimination procedure to produce an inner estimation of the solutions set. To this end, firstly we apply interval Gaussian elimination procedure to obtain the solution set of a fuzzy linear system and secondly, by limiting it via solving a crisp linear system, we find an inner estimation of the solutions set, such that it satisfies the related fuzzy linear system. Finally, several numerical examples are given to show the efficiency and ability of our method.     

取大乘积型模糊关系几何规划

本文提出了一类目标函数为正项式,约束是取大乘积型模糊关系方程的优化 问题,我们在本文中阐述了取大乘积型模糊关系方程解的结构以及求解的方法,基于目标 函数中每个单项式的指数取值情况讨论了最优解,并且给出了解决此类优化问题的一个程 序,为了说明该方法的有效性给出了两个具体例子．     

模糊关系几何规划

提出一类目标函数为正项式,约束是模糊关系方程的优化问题。阐述模糊关系方程解集的结构以及求解的方法,基于目标函数中每个单项式的指数取值情况讨论最优解,并且给出解决此类优化问题的一个程序,为了说明该方法的有效性给出具体例子。     

On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies

In this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies. Necessary and sufficient conditions for the existence of a solution are given, and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique.     

Two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities

This paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities (FRI) with the max-Einstein composition operator. If its feasible domain is non-empty, then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions. Also, an efficient algorithm is proposed to solve the model based on the structure of FRI path, the concept of partial solution, and the branch-and-bound approach. The algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions. Some sufficient conditions are given that under them, some of the optimal components of the model are directly determined. Some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions. Then the reduced domain is decomposed (if possible) into several sub-domains with smaller dimensions that finding the components of the optimal solution in each sub-domain is very easy. In order to obtain an optimal solution of the original problem, we propose another more efficient algorithm which combines the first algorithm, these procedures, and the decomposition method. Furthermore, sufficient conditions are suggested that under them, the problem has a unique optimal solution. Also, a comparison between the recently proposed algorithm and the known ones will be made.     

Fuzzy relation equations on a finite set

The family of all the solutions of a fuzzy relation equation on a finite set is considered. It is characterized by the set of all the lower solutions, which can be obtained by a combinatorial algorithm.     

Solution Sets of Interval-Valued Min-S-Norm Fuzzy Relational Equations

This paper introduces the concepts of tolerable solution set, united solution set, and controllable solution set for interval-valued fuzzy relational equations. Given a continuous s-norm, it is shown that each of the three types of the solution sets of interval-valued fuzzy relational equations with a min-s-norm composition, if nonempty, is composed of one minimum solution and a finite number of maximal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. An example is given to illustrate the proposed procedures. An erratum to this article is available at .     

Solution Sets of Interval-Valued Fuzzy Relational Equations

This paper introduces the concepts of tolerable solution set, united solution set, and controllable solution set of interval-valued fuzzy relational equations. Given a continuous t-norm, it is proved that each of the three types of the solution sets of interval-valued fuzzy relational equations with a max-t-norm composition, if nonempty, is composed of one maximum solution and a finite number of minimal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. Examples are given to illustrate the procedures. An erratum to this article is available at .     

Fuzzy相似矩阵方程X2=X与最优模糊等价矩阵的存在性

在文［１］基础上,对Ｆｕｚｚｙ 相似矩阵方程Ｘ２＝ Ｘ 的解的结构进行了进一步研究。首先提出了Ｆｕｚｚｙ 等价标准型的概念,为解的表达提供了工具; 第二,指出了相应标准分解过程的参数系的唯一性; 第三,在群作用观点下和平移等价类的意义下,讨论了解的类数计算公式; 第四,给出了解的分类表达式; 最后,证明了“失真”最小的模糊等价阵,即Ｆｕｚｚｙ 最优等价阵的存在性,为Ｆｕｚｚｙ 聚类提供了理论依据     

不确定离散模糊随机系统的鲁棒方差约束输出反馈控制

对一类具有范数有界不确定性的离散T-S模糊随机系统。研究不仅使整个闭环模糊系统全局渐近稳定。而且每个模糊子系统的稳态状态方差满足给定上界性能指标约束的输出反馈鲁棒方差控制律的设计问题。利用线性矩阵不等式(LMI)技术，导出输出反馈鲁棒方差控制律的存在条件，并基于矩阵相似变换给出其可解性条件，同时用一组线性矩阵不等式的可行解。给出输出反馈鲁棒方差控制律的一个参数化表达形式。     

Fuzzy modelling with tolerances

We consider the problem of the resolution of fuzzy equation systems: A certain number of fuzzy inputs and outputs is given. How can we determine a fuzzy relation which describes the behaviour of the object under consideration? When attributing this problem to fuzzy equations in the classical sense it may happen that a solution does not exist. Therefore we introduce fuzzy tolerances as an extension of the solvability conception. For this problem we get the whole solution set by means of posets. Moreover, we consider the inverse problem: How to determine tolerances to get consistency (i.e. at least one solution) for the arising equation system.     

Some properties of minimal solutions for a fuzzy relation equation

Some properties of the solution set and minimal solutions of a fuzzy relation equation are considered. In this paper, we show the necessary and sufficient condition for existence of a minimal solution of a finite fuzzy relation equation defined on infinite index sets.     

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.     

求解具有max-t-norm合成算子的模糊关系不等式

研究一类具有max-t-norm合成算子的模糊关系不等式,该问题是模糊关系方程以及区间值模糊关系方程的推广.通过分析其解集的特点,提出一个基于筛选原则求解该类问题的算法,并给出数值算例说明该算法的有效性.     

模糊关系方程在预测深部开采地面下沉中的应用

根据深部矿体开采岩体移动变形的特点,建立了地面下沉值与影响因素间的模糊关系方程.确定出各指标间的模糊关系并对模糊关系方程进行了求解;然后通过模糊变换得出采区下沉的预测值.通过工程实例对比分析结果表明,理论分析结果与实测资料是一致的.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.