Optimal Control of Multi-Shop Systems. Part I: Parallel Shops. Part II: Series Shops.

This paper considers the optimal control structure for multi-shop Part I Parallel, Part II Series systems, where the input to the shop system is random and the shop output is determined by the number of workers in the shop. The number of workers available to the system is held constant, while control is exercised in discrete time by adjusting the allocation of workers to the various shops in the system. There is a cost for transferring workers. Additionally, there is a cost of holding backlog in the system. The control objective is to minimize the sum of these costs over an infinite horizon. It is shown that for some regions of the systems state space the optimal control policies are known exactly without resorting to computational methods. For other regions it is shown that the problem can be decomposed into subproblems of reduced complexity. Finally, an inertia hysteresis property is established which reduces the number of policy combinations which must be considered in some dases, and completely eliminates the necessity to determine policy in other cases. The net result is a substantial reduction in the computer storage and computational effort required to solve for the optimal control policy.