Replenishment and fulfilment decisions for stores in an omni-channel retail network

Omni-channel retailing allows stores to be used to fulfil online orders. The allocation of online orders to stores can however be complicated, as one needs to take into account the inventory level of each store as well as potential future in-store demand. Current practice is therefore often to use myopic order allocation rules. However, such rules may cause inventory levels to become imbalanced across the retailers network, which might result in (expensive) excessive stocks for some stores while other stores face stock-outs. We therefore study the online fulfilment and replenishment decision of an omni-channel retailer with multiple stores to fulfil the online orders, considering future demand. In contrast to previous literature, we explicitly formulate it as a multi-period problem and formulate and solve it as a periodic Markov Decision Process (MDP). Each period (e.g., week) an ordering decision is made and replenishment happens after a lead time, while online orders are allocated at the end of each sub-period (e.g., day). As the problem easily becomes intractable with multiple stores, we find an approximation of the optimal policy by decomposing the MDP and applying a one-step policy improvement approach. In an extensive numerical study, we compare our policy with two well-known heuristics from the literature and practice. The results indicate that our approach outperforms the heuristics on both profit and service levels. Further analysis shows that our method is better at allocating the online orders to the stores, resulting in more balanced and less fluctuating inventory levels across the retailers network.