Efficient arc-flow formulations for makespan minimisation on parallel machines with a common server

We consider the problem of scheduling non preemptively a set of jobs on parallel identical machines with prior setup operations on a single shared server, where the objective is to minimise the makespan. We develop an arc-flow formulation to the problem with two multigraphs, one for the machines and one for the server, with a same set of nodes representing points in time, and arcs associated with job execution, and with machines or server idleness. The resulting formulation, called Flow–Flow formulation (FFF), and its tuned version (FFT) are compared with the best existing model in the literature, a time-indexed variable formulation ($F2$), on benchmark instances with up to 200 jobs and 10 machines. Computational results showed that our Flow–Flow models outperformed $F2$ especially for instances with more than 50 jobs and optimally solved a majority of problems with 150 and 200 jobs for which $F2$ found only very few optimal solutions.