Optimization of a Flow Shop System of Initially Controllable Machines

We consider an optimization problem for deterministic flow shop systems of traditional machines with service costs penalizing small service times. A regular completion-time cost is also included so as to complete jobs as early as possible. The service times are assumed to be initially controllable, i.e., they are set at the startup time. Assuming convexity of the cost functions, we formulate a convex optimization problem after linearization of the max constraints. The numeric solution of this problem demands a large memory limiting the solvable system sizes. In order to relieve the memory bottleneck, some waiting characteristics of jobs served in fixed-service-time flow shop systems are exploited to result in a simpler equivalent convex optimization problem. These characteristics and the benefit of CNC machines are demonstrated in a numerical example. We also show that the simplifications result in significant improvements in solvable system sizes and solution times.