Minimizing the sum of earliness/tardiness in multi-machine scheduling: a mixed integer programming approach

We developed a mixed integer programming formulation for minimizing job earliness and tardiness (ET) in a multi-machine scheduling problem. Our formulation allows setup times to depend on the job-to-job sequence, processing times to depend on the job–machine combination, due dates and cost penalties to vary for each job, and machine characteristics to be non-uniform. The model can easily provide the optimal solution to problems involving about nine jobs and three machines. Thus, the formulation shown here can provide the optimal solutions necessary for the development and validation of industrial-scale, multi-machine ET heuristics, which to date, have only been tested against other heuristics.