In this paper, we address a problem for minimizing the number of items inspected in a back-light-unit (BLU) inspection system, which includes a K-stage inspection system, a source inspection shop, and a re-inspection shop. In order to formulate our objective, we make a steady-state flow analysis between nodes (or shops), and derive the steady-state amount of flows between nodes and defective rates by solving a nonlinear balance equation. We provide an enumeration method for determining an optimal value of K which minimizes the number of items inspected. Our methodology could be applied and extended to similar situations with slight modification.