Optimal randomized ordering policies for a capacitated two-echelon distribution inventory system

We propose a new formulation for controlling inventory in a two-echelon distribution system consisting of one warehouse and multiple non-identical retailers. In such a system, customer demand occurs based on a normal distribution at the retailers and propagates backward through the system. The warehouse and the retailers have a limited capacity for keeping inventory and if they are not able to fulfill the demand immediately, the demand will be lost. All the locations review their inventory periodically and replenish their inventory spontaneously based on a periodic Randomized Ordering (RO) policy. The RO policy determines order quantity of each location in each period by subtracting corresponding on-hand inventory at the beginning of that period from a deterministic decision variable. We propose a mathematical model to find the optimal RO policies such that an average systemwide cost consisting of ordering, holding, shortage, and surplus costs is minimized. We use the first and second moments of on-hand inventory as auxiliary variables. A remarkable advantage of this model is calculating the immediate fill rate of all locations without adding new variables and facing the curse of dimensionality. Using two numerical examples with stationary and non-stationary demand settings, we validate and evaluate the proposed model. For validation, we simulate the optimal RO policy and demonstrate that the optimal first and second moments of on-hand inventory from our model reasonably follow the corresponding moments obtained through simulation. Furthermore, we evaluate the RO policy by drawing a comparison between the optimal RO policy and the optimal well-known (R, S-n*, S-n*) policy. The results confirm that the RO policy could outperform (R, s, S) policy in terms of the average systemwide annual cost.