Empty container management has been a problem of containerized trade due to the imbalance between export and import. Recently, a platform, called container-xchange.com, was launched to share empty containers between suppliers and consumers. The suppliers, who have empty containers not used in a certain period of times, register their empty containers with the specified rental period and the number of empty containers. Similarly, the consumers, who want to use empty containers, also register the period and the quantities of empty containers. Instead of searching counterparts in the platform, we consider the problem of finding a matching of suppliers and consumers such that the number of the matched pairs is maximized. This problem can be modeled as an interval scheduling problem in which suppliers and consumers correspond to machines and jobs, respectively which have their own intervals of availability. In addition, hierarchies are also assigned to both machines and jobs and the assignment of jobs to machines is possible only if their hierarchies are compatible. Due to the computational intractability of the problem, we propose a heuristic and the performance of the heuristic is verified through numerical experiments. We found that the heuristic provides the feasible solutions faster than the off-shelf branch-and-bound method. The degradation of the quality of the solutions compared with the optimal solutions was less than 5% on average.