Stochastic lead time with order crossover

Lead time plays an important role in many areas, including supply chain, economics, and marketing. A conventional assumption in most stochastic lead-time inventory models is that the lead times are independent and identically distributed (i.i.d.). However, it can be shown that applying such an assumption on practical lead time may not be valid in case of order crossover. An order crossover occurs when a later order received earlier. This becomes a common phenomenon in many business applications. Any inventory policy based on identically distributed practical lead times needs to be re-investigated. Although the crossover issue has been noticed in the literature, the exact solution for lead-time distribution under crossover remains primitive. Inventory model generally can be separated into two classes, periodic and continuous review. Our paper focuses on continuous-review models. When the lead-time sequence is stationary, some common techniques might be useful to estimate the statistics of the distribution, such as the autocorrelation function and partial autocorrelation function. However, the practical lead time in case of order crossover is not stationary. The proposed method reveals the joint distribution of the practical lead time instead. Any statistics, such as mean and correlation, can be estimated from the joint distribution directly. Subsequently, the optimal inventory police can be obtained in case of order crossover. An exponential distributed lead-time case study is used to demonstrate the use of the proposed method and the risk of mis-use i.i.d. practical lead times. Other phenomena can be studied by similar derivation.