Replenishing an inventory system permitting lost sales under Poisson demand and short lead times: Explicit performance and an efficient algorithm

In-store, on-shelf availability is a key tool that assists sellers to compete in a market whenever store loyalty is low. On the other hand, the increasing incentive to reduce stock places pressure on suppliers to shorten their lead times. In this work, a continuous-review inventory model is developed and investigated, under the assumptions of random demand with a Poisson distribution and an ordering cost. It is further assumed that unsatisfied demand becomes lost sales. Closed-form expressions are derived for the objective of maximizing the expected profit per unit time and for the expected cycle length, service level, holding cost, and lost-sales penalty cost (i.e., margin lost plus a goodwill loss). We assume the policy (Q,r), meaning that a replenishment of quantity Q (integer) is ordered once the (integer) level of the on-hand integer reaches r, known as the reorder point. We find that the searching domain is composed of a bounded two-dimensional domain and a line. An efficient algorithm with polynomial-time complexity that searches for the optimal integers is developed. The real-world applicability of the method and a sensitivity analysis of several key parameters are investigated via a numerical example. The superior performance of the proposed model with respect to an adjusted well-known newsvendor model is shown. Finally, we also compare these two models when the assumption of a short lead time is relaxed.