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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. Index Terms--Convex optimization, discrete event dynamic systems, manufacturing.

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem. Index Terms--Constrained hybrid control, controllable processing times, flow shop, hierarchical decomposition, optimal control.