In this work, we address the simultaneous design and control of a binary distillation column. The problem is first formulated as a mixed-integer dynamic optimization problem that is then transformed into a mixed-integer nonlinear programming problem using the simultaneous dynamic optimization approach (i.e., full discretization). This formulation is capable of designing the optimal feed tray location, tray sizing, optimal operating steady states, the optimal open-loop trajectory between them, and also the best controller paring and parameters that does the best tracking of the open-loop trajectory. For solving the complex mixed-integer dynamic optimization problem, an optimization decomposition strategy is proposed. The solution strategy is based on solving relaxed versions of the optimization problem and using the results to initialize complex problem versions. The full space problem was solved with the Bonmin solver. Two cases were analyzed, and in both the solver and the proposed decomposition strategy were capable of solving the problems successfully.