With the emergence of Role Based Access Control (RBAC) as the de facto access control model, organizations can now implement and manage many high level security policies. As a means of migration from traditional access control systems to RBAC, different role mining algorithms have been proposed in recent years for finding a minimal set of roles from existing user-permission assignments. While determining such roles, it is often required that certain security objectives are satisfied. A common goal is to enforce the role-usage cardinality constraint, which limits the maximum number of roles any user can have. In this paper, we propose two alternative approaches for role mining with an upper bound on the number of roles that can be assigned to each user, and validate their performance with benchmark data sets.