Shortage decision policies for a fluid production model with MAP arrivals

We consider on a continuous production/inventory process where a single machine produces a certain product into a finite buffer. The demands arrive according to a Markov Additive Process governed by a continuous-time Markov chain, and their sizes are independent and have phase-type distributions depending on the type of arrival. Two shortage policies are considered: the backorder policy, in which any demand that cannot be satisfied immediately is backlogged, and the order policy, in which any demand that cannot be satisfied immediately is supplied (alternatively, the latter policy can be considered as lost sales). We assume that the total cost includes a production loss cost, a penalty cost, a fixed cost for an order and a variable cost for the ordered amount. By applying the regenerative theory, we use tools from the exit-time theorem for fluid processes to obtain the discounted cost functionals under both policies. In addition, the models are extended to include a non-zero safety stock. Numerical examples, sensitivity analysis and comparative study are included.