We consider a (Q, r) perishable inventory system with state-dependent compound Poisson demands with a random batch size, general lead times, exponential shelf times, and lost sales. We assume \(r<Q\) and analyze the system using an embedded Markov process at the replenishment points. Using the queueing and Markov chain decomposition approach, we characterize the distribution of the residual lead time and derive the stationary distribution of the inventory level. We construct closed-form expressions for the expected total long-run average cost function. The closed form allows us to efficiently obtain, numerically, the optimal Q and r parameters. Numerical study provides several guidelines for the optimal control. For example, we show that approximating the lead time distribution by an exponential one only works when the optimal reorder point of the approximation is very small; in other cases the usage of the exact distribution can lead to substantial cost savings (up to 14%). We further provide intuition insight on the optimal controls and how they depend on different factors, e.g., the lead time variability, and the demand features (arrival rate, size and variability).

我们考虑具有状态依赖的复合Poisson需求的（Q，  r）易腐库存系统，该系统具有随机的批量大小，一般的交货时间，指数货架时间和销售损失。我们假设\（r <Q \）并在补给点使用嵌入式Markov过程分析系统。使用排队和马尔可夫链分解方法，我们可以表征剩余提前期的分布，并得出库存水平的平稳分布。我们为预期的总长期平均成本函数构造封闭形式的表达式。封闭形式使我们能够有效地从数值上获得最优的Q和r参数。数值研究为优化控制提供了一些指导。例如，我们表明，仅当近似值的最佳重排序点很小时，才可以用指数近似提前期分布。在其他情况下，使用精确的分配可以节省大量成本（最多14％）。我们进一步提供关于最佳控制以及它们如何取决于不同因素（例如提前期可变性和需求特征（到达率，大小和可变性））的直觉见解。