A new result to generalize and extend a single machine scheduling problem solved by dynamic programming

The problem of scheduling jobs on an unreliable single machine is considered in this paper under a control-theoretical point of view. The scheduling problem is formalized as an optimal control problem and optimal control strategies (functions of the system state) are determined. In this way, optimal decisions can be taken also in presence of perturbations which affect the single machine (such as breakdowns and slowdowns). In this paper, a new result is proposed, which allows generalizing and extending the methodology proposed by the author in the recent past. With this result it is possible to remove the hypothesis that any unitary tardiness cost is greater than any unitary cost related to the deviation from the nominal processing time; moreover, it is possible to add the setup cost to the objective function which is minimized.