The multi-type component assignment problem (MCAP) aims to find the optimal assignment of multiple types of components to positions in a system that maximizes the system reliability. In MCAPs, a component of a particular type can only be assigned to a position of the same type in the system. The existing works on MCAPs assumed that a particular position can only be assigned a particular type of components. However, this assumption is not necessarily true in some real problems. This paper extends the existing works on MCAPs, and studies the two-type component assignment problem (TCAP) which is a special kind of the MCAP. In the studied TCAP, there are two types of components, and a particular position can be assigned at least one type of components. A Birnbaum importance-based two-stage approach is proposed to solve the TCAP. The proposed approach is illustrated by analytical examples, and numerical experiments are conducted to evaluate its performance. A real-world use case is given to show the feasibility and applicability of the proposed approach. The originality of this work lies in the proposition of a new mathematical model of the TCAP and a two-stage approach to solve such a problem.