Analysis of batch ordering inventory models with setup cost and capacity constraint

Stochastic periodic-review batch ordering inventory problems appear in many industrial settings. However, few literature deals with the optimal ordering polices for such problems, no mention to the inclusion of the fixed ordering cost and the production capacity. In this paper, we consider a single-item periodic-review batch ordering inventory system with the consideration of the setup cost and the capacity constraint for each order over a finite planning horizon. By proposing several new convex notions, we show that a batch-based (s,S) policy is optimal for the unlimited ordering capacity case, while for the limited ordering capacity case, a modified (r,Q) policy is optimal for the setting with zero ordering setup cost, and a batch-based X–Y band policy for the setting with positive ordering setup cost. Moreover, we analytically study the sensitivity of the policy parameters with respect to the capacity and batch order size, and derive the bounds on the optimal policy parameters. We further extend our analysis to the infinite horizon setting and show that the structure of the optimal policy remains similar. Finally, the numerical experiments provide some insights into the impact of model parameters on the benefit of reducing the batch size and increasing the ordering capacity, and indicate that ignoring batch requirement may lead to a significant cost increment.