Dynamic voltage scaling and adaptive body biasing have been shown to reduce dynamic and leakage power consumption effectively. In this paper, we optimally solve the combined supply voltage and body bias selection problem for multi-processor systems with imposed time constraints, explicitly taking into account the transition overheads implied by changing voltage levels. Both energy and time overheads are considered. We investigate the continuous voltage scaling as well as its discrete counterpart, and we prove NP-hardness in the discrete case. Furthermore, the continuous voltage scaling problem is formulated and solved using nonlinear programming with polynomial time complexity, while for the discrete problem we use mixed integer linear programming. Extensive experiments, conducted on several benchmarks and a real-life example, are used to validate the approaches.