Fuzzy Relation Bilevel Optimization Model in the Wireless Communication Station System

Abstract

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.

Keywords

• Fuzzy relation inequality
• bilevel programming problem
• max-product composition
• nonlinear optimization
• minimum max-norm matrix solution