Economic lot-sizing and dynamic quantity competition

We study a problem of dynamic quantity competition in continuous time with two competing retailers facing different replenishment cost structures. Retailer 1 faces fixed ordering costs and variable procurement costs and all inventory kept in stock is subject to holding costs. Retailer 2 only faces variable procurement costs. Both retailers are allowed to change their sales quantities dynamically over time. Following the structure of the economic order quantity (EOQ) model, retailer 1 places replenishment orders in batches and retailer 2 follows a just-in-time (JIT) policy. The objective of both retailers is to maximize their individual average profit anticipating the competitor's replenishment and output decisions. The problem is solved by a two-stage hierarchical optimization approach using backwards induction. The second-stage model is a differential game in output quantities between the two retailers for a given cycle length. At the first stage, the replenishment policy is determined. We prove the existence of a unique optimal solution and derive an open-loop Nash equilibrium. We show that both retailers follow contrary output strategies over the order cycle. The EOQ retailer, driven by inventory holding costs, decreases his market share whereas the output of the JIT retailer increases. Moreover, depending on the cost structure, the EOQ retailer might partially be a monopolist. At the first stage, the EOQ retailer determines the cycle length, anticipating the optimal output trajectories at the second stage.