  In this article, we investigate a dynamic control problem of a
production-inventory system. Here, demands arrive at the production unit
according to a Poisson process and are processed in an FCFS manner. The
processing time of the customers' demand is the exponential distribution. The
production manufacturers produce the items on a make-to-order basis to meet
customer demands. The production is run until the inventory level becomes
sufficiently large. We assume that an item's production time follows
exponential distribution and the amount of time for the produced item to reach
the retail shop is negligible. Also, we assume that no new customer joins the
queue when there is a void inventory. This yields an explicit product-form
solution for the steady-state probability vector of the system. The optimal
policy that minimizes the discounted/average/pathwise average total cost per
production is derived using a Markov decision process approach. We find optimal
policy using value/policy iteration algorithms. Numerical examples are
discussed to verify the proposed algorithms.
