A continuous production-inventory problem with regeneration cycles

Relevant costs are a linear inventory holding cost, a linear backorder cost and, a fixed set up cost for initiating a production run. As a service measure, we consider the average amount of lost sales per unit time. The corresponding optimization problem is discussed. The optimal control policy is determined with the help of theory of regenerative and renewal processes which enables us to find the precise optimal policy instead of using approximations. As a special case we obtain the solution for completely backlogged excess demand.