Maximum amplitude and its corresponding maximum-amplitude solution is defined in an addition-min fuzzy relation inequalities system. The maximum-amplitude solution could be viewed as a stable solution, since it is able to survive the biggest fluctuation or perturbation appearing in its components. For solving the maximum amplitude and maximum-amplitude solution, we construct a relevant optimization problem and then convert it into several subproblems. Some auxiliary functions and index sets are introduced to deal with these subproblems. Algorithm I is developed to find their optimal solutions. Based on the optimal solutions of the subproblems, we further propose the Algorithm II for obtaining the maximum amplitude and maximum-amplitude solution of an addition-min fuzzy relation inequalities system. Numerical examples illustrate the validness and efficiency of our proposed algorithms.

In order to minimize the intensity of electromagnetic radiation, as well as the operation cost of the base stations in a wireless communication station system, we construct a bilevel programming problem with max-product fuzzy relation inequalities constraint. We first investigate the first level programming problem based on the concept of minimal solution matrix. The complete optimal solution set of the first level problem, as a union of a finite number of closed intervals, is a non-convex infinite set in some cases. Hence the second level programming is a nonlinear optimization problem with non-convex feasible domain. For solving the second level programming, we convert it into a discrete optimization problem, in which the constraint is the set of all minimum max-norm matrix solutions. Furthermore, the discrete optimization problem is equivalently converted into a 0-1 integer programming problem, which could be solved by the famous branch-and-bound technique. Numerical examples are given to illustrate the feasibility and efficiency of our proposed algorithm. In addition, the bilevel programming problem is further generalized and discussed considering a wider range of managerial requirements.

In this paper, a class of large-scale systems with impulsive effect, input disturbance, and both variable and unbounded delays were investigated. On the assumption that all subsystems of the large-scale system can be exponentially stabilized, and the stabilizing feedbacks and corresponding Lyapunov functions (LFs) for the closed-loop systems are available, using the idea of vector Lyapunov method and M-matrix property, the intero-differential inequalities with variable and unbounded delays were constructed. By the stability analysis of the intero-differential inequalities, the sufficient conditions to ensure the robust exponential stability of the large-scale system were obtained. Finally, the correctness and validity of the methods was verified by two numerical examples.