Critical level policy for a production-inventory model with lost sales

We consider a storage process $W\lpar t\rpar$W(t) under the generalised order-up-to-level policy, based on a continuous-time Markov chain (CTMC). Specifically, the process starts at level S; whenever it drops to s, an order is sent, which is carried out after an exponential lead time. If during the lead time level S is reached, the order is cancelled, incurring some fee. This paper is written as an extension of Barron [2016. “An $(s,k,S)$(s,k,S) Fluid Inventory Model with Exponential Lead Times and Order Cancellations.” Stochastic Models 32 (2): 301–332]. While the latter paper considered a fluid inventory model with backlogging and focused on discounted analysis only, the case of lost sales was not solved. The present paper generalises the analysis to incorporate unsatisfied demand for the expected discounted costs and for the average costs per time unit. We consider four costs. There is a fixed nonzero ordering cost or a fee for each order cancellation, a purchase cost for each ordered item, a storage cost for the stock, and a penalty cost due to the unmet demand. Applying renewal theory, multi-dimensional martingales, and stopping time theory, we obtain explicit expressions of the cost components. Numerical study provides several guidelines on the optimal controls.