Total completion time minimization in a computer system with a server and two parallel processors

The context of the problem tackled in this paper is a computer system composed by a single server and two identical parallel machines. The processing times on the server are assumed to be unary and the objective is to minimize the total completion time. The papers dealing with scheduling problems with a server generally consider that the setup activities require simultaneously the server and the machine. In this paper, this constraint is not considered and the studied problem is a two-stage hybrid flow shop with no-wait constraint between the two stages. An algorithm that can solve optimally this problem in O(n log(n)) time is proposed. Finally, it is shown that solving this problem optimally leads to an optimal solution to the problem without the no-wait constraint.