Optimal (s, S) policies with delivery time guarantees for manufacturing systems with early set-up

We consider optimal (s, S) policies with delivery time guarantees for production planning in one-machine manufacturing systems with early set-up. The machine produces one type of product and delivery time guarantee is offered to the customers for each unit of ordered product. The inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. In a (s, S) policy, the machine will shut down when an inventory level of S is attained and once the inventory level drops to s, the machine will re-start. A set-up time is required for the machine. We model the set-up by the exponential distribution. We obtained an analytical form of the steady state probability distribution for the inventory levels derived. The average profit of the system can be written in terms of this probability distribution. Hence the optimal (s, S) policy can be obtained by varying different possible values of s and S.