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Optimal control of a multi-state manufacturing system: control of production rate and temporary increase in capacity

In planning and scheduling of production systems manufacturers have two main strategies for responding to uncertainty: they build inventory to hedge against periods in which the production capacity is not sufficient to satisfy demand, or they temporarily increase the production capacity. We consider the problem of minimising the long-run average cost of holding inventory and/or purchasing extra capacity for a single facility producing a single part-type. The optimal production policy is of a hedging point policy type where two hedging levels are associated with each discrete capacity state of the system: a positive hedging level (inventory target) and a negative one (backlog level below which extra capacity should be purchased). We establish some ordering of the hedging levels, derive equations satisfied by the steady-state probability distribution of the inventory/backlog, and describe a program for specifying the optimal hedging levels.