Understanding the lead-time effects in stochastic inventory systems with discounted costs

This paper explores qualitative effects of lead-time uncertainty in a basic continuous-time single-item inventory model with Poisson demand and stochastic lead times. The objective is to minimize the infinite-horizon expected total discounted cost. Order costs are linear, so a base-stock policy is optimal. Traditionally and intuitively, people believe that a (stochastically) longer lead time results in a larger lead-time demand and therefore that the system should have a higher optimal base-stock level. In fact, Song shows that this is true for models with the average cost criterion. Here, it is shown that this intuition is not always validated for models with the discounted cost criterion. Certain systems are identified in which the lead times do have a monotonic impact on the optimal base-stock levels. Examples include systems with constant, geometric, gamma, or uniform lead-time distributions. In addition it is found that a shorter lead time does not necessarily result in a smaller optimal system cost.