We consider an inventory system with compound Poisson demands replenished by discrete production of units on a single-server facility. This facility may start a vacation at any production completion epoch; at the completion of a vacation the inventory level is inspected to decide whether or not to resume production. Unit production and vacation times are independent and identically distributed with general distributions. Under an (s, S) policy, production is terminated when the inventory level reaches a critical level S, and production is resumed when the inventory level, upon completion of a vacation, is at or below a prespecified value s. In this paper we prove that an (s, S) policy is optimal among all possible policies. We also derive a highly efficient algorithm for the determination and evaluation of an optimal (s, S) policy.
Federgruen, Awi, and Yu-Sheng Zheng. "Optimal control policies for stochastic inventory systems with endogenous supply." Probability in the Engineering and Informational Sciences 7 (1993): 257-272.
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