Optimal Time-Average Cost for Inventory Systems with Compound Poisson Demands and Lost-sales

Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a continuous replenishment rate. Consider a single-product inventory system in continuous review with constant replenishment and compound Poisson demands with lost-sales. The system is subject to carrying costs and lost-sale penalties, where the carrying cost is a linear function of on-hand inventory and the lost-sales penalty is incurred per lost-sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions, 2 constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform or Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes, we exhibit a closed form expression for the optimal replenishment rate and its corresponding cost. Finally, a numerical study is carried out to illustrate the results.