A Variant of Lb-Convexity and its Application to Inventory Models with Batch Ordering

Previous studies show that the concept of L-b-convexity is helpful in characterizing the optimal policy for some inventory models with positive leadtimes. Such examples include the lost-sales inventory model by Zipkin (2008). On the structure of lost-sales inventory models. Operations Research, 56(4) 937-944. and the inventory-pricing model by Pang et al. (2012). A note on the structure of joint inventory-pricing control with leadtimes. Operations Research, 60(3), 581-587. However, when taking batch ordering into account, L-b-convexity does not work anymore. In this paper, we extend L-b-convexity to a more general concept termed as Q-jump-L-b-convexity and apply it to batch ordering inventory models including a lost-sales inventory model and an inventory-pricing model with batch ordering and positive leadtimes. By utilizing this new concept, we can partially characterize the structure of the optimal policies for both the models. Moreover, we are able to evaluate the sensitivity of the optimal decisions with respect to system states. Our results can also be applied to the serial and the assembly inventory systems with lost-sales and batch ordering.