Model predictive control (MPC) is a computationally demanding control technique that allows dealing with multiple-input and multiple-output systems while handling constraints in a systematic way. The necessity of solving an optimization problem at every sampling instant often 1) limits the application scope to slow dynamical systems and/or 2) results in expensive computational hardware implementations. Traditional MPC design is based on the manual tuning of software and computational hardware design parameters, which leads to suboptimal implementations. This brief proposes a framework for automating the MPC software and computational hardware codesign while achieving an optimal tradeoff between computational resource usage and controller performance. The proposed approach is based on using a biobjective optimization algorithm, namely BiMADS. Two test studies are considered: a central processing unit and field-programmable gate array implementations of fast gradient-based MPC. Numerical experiments show that the optimization-based design outperforms Latin hypercube sampling, a statistical sampling-based design exploration technique.