A triple (S,s,ℓ)-thresholds base-stock policy subject to uncertainty environment, returns and order cancellations

This paper studies a continuous-review (S,s,ℓ) inventory model that extends the familiar (S,s)-policy by including returns and order cancellations. The demand and return processes follow independent compound Poisson processes. We assume a random shelf life, a random lead time and early losses. Our objective is to find a replenishment level S, a reorder level s, and a threshold cancellation level ℓ that minimizes the total expected cost per time unit under lost sales. The cost structure includes variable and fixed costs for an order, a holding cost, a penalty cost, a handling cost for returns, a relocation cost, and costs for end of life and early loss. Via Markovian framework, we derive the steady-state probabilities for the inventory level, and construct closed-form expressions for the average cost functions. Using a numerical study, we investigate how the best S,s and ℓ, are impacted by the system's parameters, such as returns, demands (rate, mean and variability), and order cancellations. We illustrate the impact of the return flow on the best policy and the cost, and show that ignoring returns may yield a substantial loss. Consequently, we compare our model to a relatively simple one with no returns but with less demand, and specify when the latter model can be used as a good approximation. Finally, we identify situations where an earlier-cancellation policy is more cost effective. Our framework is completed by providing two efficient search algorithms for numerically finding the optimal parameters. We show that both algorithms significantly reduce the computational effort compared to the basic algorithm.