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Dynamic voltage scaling and adaptive body biasing have been shown to reduce dynamic and leakage power consumption effectively. The authors report an optimal solution to the combined supply voltage and body bias selection problem for multiprocessor systems with imposed time constraints, explicitly taking into account the transition overheads implied by changing voltage levels, and considering both energy and time overheads. They investigate continuous voltage scaling as well as its discrete counterpart, and strongly 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 they use mixed integer linear programming. Extensive experiments, conducted on several benchmarks and a real-life example, are used to validate the approaches.