Yung-Yih Lur

On nilpotent fuzzy matrices

Abstract In this paper, we study the issue of nilpotent fuzzy matrices. We first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues. When a finite number of fuzzy matrices are simultaneously considered, we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices. On the other hand, it is well known that the max–min algebra is one kind of lattice. We shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice.

Convergence of max-arithmetic mean powers of a fuzzy matrix

Fuzzy matrices provide convenient representations for fuzzy relations on finite universes. In the literature, the powers of a fuzzy matrix with max-min/max-product/max-Archimedean t-norm compositions have been studied. It turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved. In this paper, the max-arithmetic mean composition is considered for the fuzzy relations. We show that the max-arithmetic mean powers of a fuzzy matrix always are convergent.

AN INEQUALITY FOR THE SPECTRAL RADIUS OF AN INTERVAL MATRIX

Abstract For an n × n interval matrix A = (Aij), we say that A is majorized by the point matrix A = (a ij ) if aij = |Aij| when the jth column of A has the property that there exists a power A m containing in the same jth column at least one interval not degenerated to a point interval, and aij = Aij otherwise. Denoting the generalized spectral radius (in the sense of Daubechies and Lagarias) of A by ϱ( A ), and the usual spectral radius of A by ϱ(A), it is proved that if A is majorized by A then ϱ(A) ⩽ ϱ( A ) . This inequality sheds light on the asymptotic stability theory of discrete-time linear interval systems.

On simultaneously nilpotent fuzzy matrices

Abstract Nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices. In this paper, we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices. The notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix. Properties of the simultaneous nilpotence will be established. In the study of consecutive powers of a fuzzy matrix, a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix. In this paper, we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices. Similar to the nilpotent characterization for a controllable fuzzy matrix, the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices.

A new proof of Mayer's theorem☆

This paper gives a new proof of Mayers theorem concerning the convergence of powers of an interval matrix.

On a recurrence algorithm for continuous-time linear fractional programming problems

In this paper, we develop a discrete approximation method for solving continuous-time linear fractional programming problems. Our method enables one to derive a recurrence structure which shall overcome the computational curse caused by the increasing numbers of decision variables in the approximate decision problems when the subintervals are getting smaller and smaller. Furthermore, our algorithm provides estimation for the error bounds of the approximate solutions. We also establish the convergence of our approximate solutions to the continuous-time linear fractional programming problems. Numerical examples are provided to illustrate the quality of the approximate solutions.

A recurrence method for a special class of continuous time linear programming problems

This article studies a numerical solution method for a special class of continuous time linear programming problems denoted by (SP). We will present an efficient method for finding numerical solutions of (SP). The presented method is a discrete approximation algorithm, however, the main work of computing a numerical solution in our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner, we provide a computational procedure which would yield an error bound introduced by the numerical approximation. We also demonstrate that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate the provided procedure.

Approximate solutions and error bounds for a class of continuous-time linear programming problems

In this article, we discuss a class of continuous-time linear programming (CLP) problems. We provide a discrete approximation procedure to find numerical solutions of CLP, establish the estimation for the error bound and prove that the searched sequence of approximate solution functions weakly star converges to an optimal solution of CLP. Finally, we provide some numerical examples to implement our proposed method and to show the quality of the proposed error bound.

Linear optimization of bipolar fuzzy relational equations with max-Łukasiewicz composition

According to the literature, a linear optimization problem subjected to a system of bipolar fuzzy relational equations with max-Łukasiewicz composition can be translated into a 0-1 integer linear programming problem and solved using integer optimization techniques. However, the technique of integer optimization may involve hight computation complexity. To improve computational efficiency for solving such an optimization problem, this paper proves that each component of an optimal solution obtained from such an optimization problem can either be the corresponding components lower bound or upper bound value. Because of this characteristic, a simple value matrix with some simplified rules can be proposed to reduce the problem size first. A simple solution procedure is then presented for determining optimal solutions without translating such an optimization problem into a 0-1 integer linear programming problem. Two examples are provided to illustrate the simplicity and efficiency of the proposed algorithm.

Pareto-optimal solution for multiple objective linear programming problems with fuzzy goals

Several methods have been addressed to attain fuzzy-efficient solution for the multiple objective linear programming problems with fuzzy goals (FMOLP) in the literature. Recently, Jimenez and Bilbao showed that a fuzzy-efficient solution may not guarantee to be a Pareto-optimal solution in the case that one of fuzzy goals is fully achieved. To show this point they employ Guu and Wu’s two-phase approach to obtain a fuzzy-efficient solution first, then a model like a conventional goal programming problem is proposed to find a Pareto-optimal solution. In this study, a new simplified two-phase approach is proposed to find a Pareto-optimal solution for FMOLP without relying on the results of Guu and Wu’s two-phase approach. This new simplified two-phase approach not only obtains the Pareto-optimal solution but also provides more potential information for decision makers. Precisely, decision makers can find out whether the fuzzy goals of objective function are overestimated or not and the amount of overestimation can easily be computed if it exists.