Solving the two-machine open shop problem with a single server with respect to the makespan

We address in this paper the two-machine open shop problem with a single server to prepare jobs before going through the processing so as to minimize the makespan. The server is only needed during the preparation phase before becoming available again, leaving the prepared job to complete its processing. We present three lower bounds with respect to the makespan. In addition, we show the \(\mathcal{N}\mathcal{P}\) $\mathcal{N}\mathcal{P}$ -completeness of two restricted cases. Then, we present a well solvable case. Finally, we develop two mixed integer linear programming (MILP) models for the general problem along with an experimental study we conducted to analyze their performance.