Discount replenishment opportunities with limited capacity: Optimal policy and pricing dynamics

We study an inventory model with occasional discounts on the fixed order cost, where a random capacity constraint limits the quantity ordered at the discount. We prove that the optimal policy resembles a can-order policy, characterized by a unique reorder, can-order, and order-up-to level. However, the random capacity constraints on the discount offers add complexity: the optimal policy features additional thresholds for inventory levels between the reorder and can-order point that determine when to use the discount. As these inventory-specific thresholds may render the optimal policy less straightforward to implement, we develop three heuristic replenishment policies inspired by the optimal policy structure. We compare their performance in a numerical study and derive an analytical bound on their worst-case optimality gaps in settings with an infinitely high backlog cost rate. With the buyer’s optimal response to capacitated discounts in mind, we investigate the supplier’s decision to set the discount price. A high discount relative to the regular order cost increases the probability that the buyer will accept the offer, but erodes the supplier’s margin on discount sales and may reduce regular sales. We show that a mutually beneficial outcome requires buyer-to-supplier compensation.