This article is concerned with the problem of computing the approximate preinverses of a fuzzy matrix with max-product composition. Employing the weighted$L_1$distance, a variety of evaluation functions are defined according to the preselected weights, and then, an analytical method is proposed to determine all approximate preinverses of equal quality by minimizing the preselected evaluation function. Since only the approximate preinverses with fewer or preferably no zeros are acceptable for many practical applications, a criterion for selecting weights is established to ensure that the determined approximate preinverses are desirable. Some examples are given to illustrate our results.

In the relevant literature, fuzzy relational inequalities with addition-min composition have been proposed to model the data transmission mechanism in a BitTorrent-like peer-to-peer file-sharing system. In this paper, we present a two-phase approach to find an optimal solution to data transmission that minimizes an associated function while its components are controlled to result in reduced network congestion. Numerical examples are given to illustrate the procedures of the two-phase approach.

In this paper, we study an optimization problem of minimizing a linear function subject to fuzzy relational inequalities with the addition-min composition. This optimization setting has recently been proposed to model the network cogestion issue when a BitTorrent-like peer-to-peer file-sharing system is used for data transmission. In a 2014 paper, a pseudominimal index (PMI)-based approach was proposed to search for an optimal solution. It turns out that the PMI-based approach may require to solve several to many linear programming problems in order to get an optimal solution. In this paper, we point out that the feasible domain is indeed convex. And we only need to solve a single linear programming problem to generate an optimal solution for the original optimization problem. Furthermore, our approach could be extended to the case with a nonlinear continuous objective function.

Fuzzy matrices provide convenient representations for fuzzy relations on finite universes. In the literature, the behavior of powers of a fuzzy matrix with maxmin/max-product/max-Archimedean t-norm/max-t-norm compositions have been studied. Conventionally, the algebraic operations involved in the study of powers of a fuzzy matrix usually belong to the max-t-norms. Recently the powers of a max-arithmetic mean fuzzy matrix have been studied. Typically, the max-arithmetic mean operation is not a max-t-norm. Since the max-arithmetic mean is a special example of the max-convex mean operations, we shall extend the study to powers of a max-convex mean fuzzy matrix in this paper.We show that its powers are always convergent.

In the literature, a necessary condition for minimal solutions of a fuzzy relational equation with max-product composition shows that each of its components is either zero or the corresponding component's value of the greatest solution. In this paper, we first extend this necessary condition to the situation with max-Archimedean triangular-norm (t-norm) composition. Based on this necessary condition, we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently. Furthermore, rather than work with the actual equations, we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions. Numerical examples with specific cases of the max-Archimedean t-norm composition are provided to illustrate the procedure.

In the literature, one of the minimal solutions is an optimal solution of solving a linear objective function subject to fuzzy relational equations with the max-Archimedean composition. Since the objective function is nonlinear so that this characteristic can't be employed again to the optimization problem with a linear fractional objective function. In this paper, according to the characteristics of feasible domain of fuzzy relational equations with max-Archimedean t-norm composition, some theoretical results are presented for exploring such an optimization problem. These results can be employed to cut down the feasible domain first. Hence, the work of computing an optimal solution can be simplified. Then the simplified problem can be converted into traditional linear fractional programming problems and a simple procedure is proposed for optimizing such a problem.

In this paper minimizing a linear objective function subject to a continuous max-i-norm fuzzy relational equation is considered. Our contributions are two folds. First, We show that this optimization problem can be divided into two subproblems by separating the decision variables associated with negative and nonnegative coefficients in the objective function. A 0-1 integer programming problem as an equivalent model can be derived for our current study. Our second contribution is to present an efficient procedure for solving a subclass of the max-t-norm-type optimization problems in which the max-product-type one is a special case, yet, the max-min-type one is not included. Numerical examples are provided to illustrate the procedure.

In literature, the optimization model with a linear objective function subject to fuzzy relation equations has been converted into a 0-1 integer programming problem by Fang and Li (1999). They proposed a jump-tracking branch-and-bound method to solve this 0-1 integer programming problem. In this paper, we propose an upper bound for the optimal objective value. Based on this upper bound and rearranging the structure of the problem, we present a backward jump-tracking branch-and-bound scheme for solving this optimization problem. A numerical example is provided to illustrate our scheme. Furthermore, testing examples show that the performance of our scheme is superior to the procedure in the paper by Fang and Li. Several testing examples show that our initial upper bound is sharp.