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.     

完备Brouwerian格上Fuzzy关系方程有极小解的条件

本文在有限论域上对完备Brouwerian格上Fuzzy关系方程极小的存在问题作了探讨，首先构造了Fuzzy关系方程有解但无极小解的一个例子，然后在解集非空时给出了对Fuzzy关系方程的每一个解都存在一个小于等于它的极小解的一个充分条件及一个充要条件，特别地，在充分条件下给出了一类Fuzzy关系方程所有极小解的个数的公式。     

The solution sets of infinite fuzzy relational equations with sup-conjunctor composition on complete distributive lattices

This paper deals with sup-conjunctor composition fuzzy relational equations in infinite domains and on complete distributive lattices. When its right-hand side is a continuous join-irreducible element or has an irredundant continuous join-decomposition, a necessary and sufficient condition describing an attainable solution (resp. an unattainable solution) is formulated and some properties of the attainable solution (resp. the unattainable solution) are shown. Further, the structure of solution sets is investigated.     

Solvability criteria for systems of fuzzy relation equations

By solving systems of fuzzy relation equations, qualitative process models can be obtained. To give more information on the solving procedure and to help constructing models, solvability criteria for-systems of fuzzy relation equations are necessary. In this article such criteria will be developed. Both methods are considered. In addition to some ideas on general , the is evaluated in detail. Criteria of practical use will be developed. These criteria will limit the variety of premise intersections to guarantee solvability. Nevertheless, they will still allow to model the significant behaviour of the processes.     

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.     

Fuzzy relation equations theory as a basis of fuzzy modelling: An overview

We give a wide overview on the applications of fuzzy relation equations theory to decision-making processes, to the construction of preference relations and to Knowledge Engineering, mainly fuzzy control and fuzzy pattern recognition. General theoretical results and methodological aspects are pointed out.     

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.     

Bivalent and other solutions of fuzzy relational equations via linguistic hedges

We show that the well-known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations. When the constraint is empty, constrained solutions are ordinary solutions. The generalization is obtained by employing intensifying and relaxing linguistic hedges, conceived in this paper as certain unary functions on the scale of truth degrees. One aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations. The other is to emphasize the role of linguistic hedges as constraints.     

On optimizing a linear objective function subjected to fuzzy relation inequalities

In this paper, we extend Guo and Xia’s necessary condition which has been presented by Guo and Xia (Fuzzy optimizat Decis Mak 5: 33–47, 2006) in order to study the finitely many constraints of fuzzy relation inequalities and optimize a linear objective function on this region which is defined by the fuzzy max–min operator. The new condition provides a means for removing the unnecessary paths resulting from Guo and Xia’s paths. Also, an algorithm and two numerical examples are offered to abbreviate and illustrate the steps of the resolution process of the problem.     

连续并既约元及其在刻画Fuzzy关系方程解集中的应用

本文首先引入连续并既约元(是并既约元但不是完全并既约元的元)的概念,并讨论了它的性质,然后应用连续并既约元的性质去刻画完备Brouwer格上无限Fuzzy关系方程A☉X=b的解集(其中A=(aj)j∈J和b已知,b为连续并既约元,X= (xj)j∈JT未知,“☉”表示“sup-inf”,J为无限集):给出了方程存在可达解与不可达解的充要条件及可达解与不可达解的一些性质,进一步刻画了方程的解集．     

集对Fuzzy格及其在格表示论中的应用

用幂集格构造了集对 Ｆｕｚｚｙ 格（这与用整数对构造有理数集有相似之处）,并用它证明了完整的软代数表示定理,即定义了到自身的映射且有最大元和最小元的格为软代数的充要条件是它与某个集对 Ｆｕｚｚｙ 格的子格同构．这样,与分配格在幂集 Ｂｏｏｌｅ 格中表示相对应,软代数在集对 Ｆｕｚｚｙ 格中有表示,在理论上是很完美的     

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.     

Resolution of composite fuzzy relation equations based on Archimedean triangular norms

Lately, the sup-t-norm composition of fuzzy relations has been used instead of the well-known max–min. Thus, there is a need for methods of studying and solving sup-t-norm fuzzy relation equations (t is any t-norm). In this paper, the solution existence problem is first studied and solvability criteria for composite fuzzy relation equations of any t-norm are given. Then, a methodology for solving fuzzy relation equations based on sup-t composition, where t is an Archimedean t-norm, is proposed. This resolution method is simpler and faster than those proposed for covering all the continuous t-norms. The result is important, since, as is shown in the paper, the only continuous t-norm that is not Archimedean is the “minimum”.     

Fuzzy matrix games via a fuzzy relation approach

A generalized model for a two person zero sum matrix game with fuzzy goals and fuzzy payoffs via fuzzy relation approach is introduced, and it is shown to be equivalent to two semi-infinite optimization problems. Further, in certain special cases, it is observed that the two semi-infinite optimization problems reduce to (finite) linear programming problems which are dual to each other either in the fuzzy sense or in the crisp sense.     

Bipolar fuzzy relation equations systems based on the product t‐norm

Bipolar fuzzy relation equations arise as a generalization of fuzzy relation equations considering unknown variables together with their logical connective negations. The occurrence of a variable and the occurrence of its negation simultaneously can give very useful information for certain frameworks where the human reasoning plays a key role. Hence, the resolution of bipolar fuzzy relation equations systems is a research topic of great interest. This paper focuses on the study of bipolar fuzzy relation equations systems based on the max‐product t‐norm composition. Specifically, the solvability and the algebraic structure of the set of solutions of these bipolar equations systems will be studied, including the case in which such systems are composed of equations whose independent term be equal to 0. As a consequence, this paper complements the contribution carried out by the authors on the solvability of bipolar max‐product fuzzy relation equations.     

A fuzzy solution to a majority voting game

In the Fiorina-Plott majority voting game experiment several puzzling phenomena were discovered: (1) In the experiment series where the payoffs were small in general, the predictive performance of the core was markedly worse than in the high payoff series. (2) The core outcome was found to be defeatable by another proposal. (3) Some of the majority winning social outcomes were Pareto-suboptimal. (4) Even when there was no theoretical equilibrium point, there was a clear clustering of outcomes. The paper outlines a fuzzy decision making procedure which seems to be capable of explaining all the puzzling features (1)–(4). In the absence of data on membership functions the explanation provided is of non-predictive nature; it renders the observed outcomes plausible without precisely predicting each one of them.     

Explicit solutions to a coupled integrable lattice equation

A coupled integrable lattice equation is derived from a 4 × 4 matrix spectral problem, then with the help of a special Darboux matrix, explicit solutions of the aforementioned equation are given by means of gauge transformation between the Lax pair. Finally, the density profiles of these exact solutions are presented to illustrate these solutions.     

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.     

The optimization problem over a distributive lattice

In this paper we give a necessary and sufficient condition for existence of minimal solution(s) of the linear system A * X ≥ b where A, b are fixed matrices and X is an unknown matrix over a lattice. Next, an algorithm which finds these minimal solutions over a distributive lattice is given. Finally, we find an optimal solution for the optimization problem min {Z = C * X | A * X ≥ b} where C is the given matrix of coefficients of objective function Z. This research was completed while the author was a visitor of the Center for Informatics and Applied Optimization, University of Ballarat, Ballarat, Australia.