Scheduling Multiple Variable-Speed Machines

We examine scheduling problems where we control not only the assignment of jobs to machines, but also the time used by the job on the machine. For instance, many tooling machines allow control of the speed at which a job is run. Increasing the speed incurs costs due to machine wear but also increases throughput. We discuss some fundamental scheduling problems in this environment and give algorithms for some interesting cases. Some cases are inherently diicult so for these we give heuristics. Our approach illustrates the exploitation of underlying network structure in combinatorial optimization problems. We create heuristics that optimally schedule a large portion of the jobs and then attempt to t in the remainder. This also gives a method for quickly nding valid inequalities violated by the linear relaxation solution. For the problem of minimizing the sum of makespan and production costs, a rounding heuristic is within a constant factor of optimal. Our heuristics are compared to other, classical, heuristics. Traditional sequencing and scheduling models assume that the time a job requires on a machine is not under control: either the time is xed, or it is a random variable determined by outside forces. Some problems require not only the assignment of job