Minimizing the sum of completion times with resource dependant processing times

We extend the classical problem of minimizing the sum of completion times to the case where the processing times are controllable by allocating a common, nonrenewable resource. The quality of a solution is measured by two criteria. The first criterion is the sum of completion times and the second is the total weighted resource consumption. We consider four problem variations for treating the two criteria. We prove that this problem is NP-hard for three of the four variations even if all resource consumption weights are equal. However, somewhat surprisingly, the variation of minimizing the integrated objective function is solvable in polynomial time. Although the sum of completion times is arguably the most important scheduling criterion, the complexity of this problem, until this paper, was an open question for three of the four variations. The results of this research have various implementations, including efficient battery use on mobile devices such as mobile computers, phones and GPS devices in order to prolong the life of their battery.