Integer multiple production schedules for multi-stage systems with finite production rates and startup delays

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1977

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The problem considered is that of production scheduling for a multi-stage system in which semi-finished units are passed from stage to stage before emerging as finished product. Each stage is assumed to have a constant finite rate of production greater than the constant rate of demand for the final product. In addition, a continuous review re-order policy and an infinite time horizon are assumed. An optimal production schedule is defined as that set of fixed production cycle times (or, equivalently, fixed production lot sizes) which minimizes the time-averaged total of the setup (and shutdown) costs incurred at each stage, and the costs of holding the inventories which accumulate throughout the system. The basic difficulty encountered is that of accounting for the interdependence of successive stages. In all but the most restricted cases, the initial startup at stage 1 must be delayed relative to that at stage i+1 (the immediate predecessor stage) to avoid stockouts in the inventory held between the two stages. The average size of this inventory is directly effected by the length of such a delay. Since the required delay at stage i is, in general, a non-analytical function of the cycle times chosen for stages i and i+1, the analytical derivation of optimal cycle times is not possible. The approach taken herein is that of first deriving an analytical expression for the lower bound on the total cost associated with a given set of cycle times (i.e., a given production schedule). A theorem is then stated and proved establishing both a necessary and sufficient condition for identifying production schedules with associated total costs achieving their respective lower bounds. Alternative computerized algorithms are presented for finding the least cost solution from among this set of solutions. This approach is extended to allow for finished goods backorders, constraints on average inventory levels, converging branches, and stochastic lead times for raw materials orders.

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